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1,800 | 2,635 | Heuristics for Ordering Cue Search in
Decision Making
Peter M. Todd
Anja Dieckmann
Center for Adaptive Behavior and Cognition
MPI for Human Development
Lentzeallee 94, 14195 Berlin, Germany
[email protected]
[email protected]
Abstract
Simple lexicographic decision heuristics that consider cues one at a
time in a particular order and stop searching for cues as soon as a
decision can be made have been shown to be both accurate and
frugal in their use of information. But much of the simplicity and
success of these heuristics comes from using an appropriate cue
order. For instance, the Take The Best heuristic uses validity order
for cues, which requires considerable computation, potentially
undermining the computational advantages of the simple decision
mechanism. But many cue orders can achieve good decision
performance, and studies of sequential search for data records have
proposed a number of simple ordering rules that may be of use in
constructing appropriate decision cue orders as well. Here we
consider a range of simple cue ordering mechanisms, including
tallying, swapping, and move-to-front rules, and show that they can
find cue orders that lead to reasonable accuracy and considerable
frugality when used with lexicographic decision heuristics.
1
O ne -Re ason De c i si on M aki ng and O r de r e d Se ar c h
How do we know what information to consider when making a decision? Imagine
the problem of deciding which of two objects or options is greater along some
criterion, such as which of two cities is larger. We may know various facts about
each city, such as whether they have a major sports team or a university or airport.
To decide between them, we could weight and sum all the cues we know, or we
could use a simpler lexicographic rule to look at one cue at a time in a particular
order until we find a cue that discriminates between the options and indicates a
choice [1]. Such lexicographic rules are used by people in a variety of decision
tasks [2]-[4], and have been shown to be both accurate in their inferences and frugal
in the amount of information they consider before making a decision. For instance,
Gigerenzer and colleagues [5] demonstrated the surprising performance of several
decision heuristics that stop information search as soon as one discriminating cue is
found; because only that cue is used to make the decision, and no integration of
information is involved, they called these heuristics ?one-reason? decision
mechanisms. Given some set of cues that can be looked up to make the decision,
these heuristics differ mainly in the search rule that determines the order in which
the information is searched. But then the question of what information to consider
becomes, how are these search orders determined?
Particular cue orders make a difference, as has been shown in research on the Take
The Best heuristic (TTB) [6], [7]. TTB consists of three building blocks. (1) Search
rule: Search through cues in the order of their validity, a measure of accuracy equal
to the proportion of correct decisions made by a cue out of all the times that cue
discriminates between pairs of options. (2) Stopping rule: Stop search as soon as
one cue is found that discriminates between the two options. (3) Decision rule:
Select the option to which the discriminating cue points, that is, the option that has
the cue value associated with higher criterion values.
The performance of TTB has been tested on several real-world data sets, ranging
from professors? salaries to fish fertility [8], in cross-validation comparisons with
other more complex strategies. Across 20 data sets, TTB used on average only a
third of the available cues (2.4 out of 7.7), yet still outperformed multiple linear
regression in generalization accuracy (71% vs. 68%). The even simpler Minimalist
heuristic, which searches through available cues in a random order, was more frugal
(using 2.2 cues on average), yet still achieved 65% accuracy. But the fact that the
accuracy of Minimalist lagged behind TTB by 6 percentage points indicates that
part of the secret of TTB?s success lies in its ordered search. Moreover, in
laboratory experiments [3], [4], [9], people using lexicographic decision strategies
have been shown to employ cue orders based on the cues? validities or a
combination of validity and discrimination rate (proportion of decision pairs on
which a cue discriminates between the two options).
Thus, the cue order used by a lexicographic decision mechanism can make a
considerable difference in accuracy; the same holds true for frugality, as we will
see. But constructing an exact validity order, as used by Take The Best, takes
considerable information and computation [10]. If there are N known objects to
make decisions over, and C cues known for each object, then each of the C cues
must be evaluated for whether it discriminates correctly (counting up R right
decisions), incorrectly (W wrong decisions), or does not discriminate between each
of the N?(N-1)/2 possible object pairs, yielding C?N?(N-1)/2 checks to perform to
gather the information needed to compute cue validities (v = R/(R+W)) in this
domain. But a decision maker typically does not know all of the objects to be
decided upon, nor even all the cue values for those objects, ahead of time?is there
any simpler way to find an accurate and frugal cue order?
In this paper, we address this question through simulation-based comparison of a
variety of simple cue-order-learning rules. Hope comes from two directions: first,
there are many cue orders besides the exact validity ordering that can yield good
performance; and second, research in computer science has demonstrated the
efficacy of a range of simple ordering rules for a closely related search problem.
Consequently, we find that simple mechanisms at the cue-order-learning stage can
enable simple mechanisms at the decision stage, such as lexicographic one-reason
decision heuristics, to perform well.
2
Si mpl e appr oac he s to c onstr uc ti ng c ue s e ar c h or de r s
To compare different cue ordering rules, we evaluate the performance of different cue
orders when used by a one-reason decision heuristic within a particular well-studied
sample domain: large German cities, compared on the criterion of population size using 9
cues ranging from having a university to the presence of an intercity train line [6], [7].
Examining this domain makes it clear that there are many good possible cue orders.
When used with one-reason stopping and decision building blocks, the mean accuracy of
the 362,880 (9!) cue orders is 70%, equivalent to the performance expected from
Minimalist. The accuracy of the validity order, 74.2%, falls toward the upper end of the
accuracy range (62-75.8%), but there are still 7421 cue orders that do better than the
validity order. The frugality of the search orders ranges from 2.53 cues per decision to
4.67, with a mean of 3.34 corresponding to using Minimalist; TTB has a frugality of 4.23,
implying that most orders are more frugal. Thus, there are many accurate and frugal cue
orders that could be found?a satisficing decision maker not requiring optimal
performance need only land on one.
An ordering problem of this kind has been studied in computer science for nearly four
decades, and can provide us with a set of potential heuristics to test. Consider the case of
a set of data records arranged in a list, each of which will be required during a set of
retrievals with a particular probability pi. On each retrieval, a key is given (e.g. a record?s
title) and the list is searched from the front to the end until the desired record, matching
that key, is found. The goal is to minimize the mean search time for accessing the
records in this list, for which the optimal ordering is in decreasing order of pi. But if
these retrieval probabilities are not known ahead of time, how can the list be ordered after
each successive retrieval to achieve fast access? This is the problem of self-organizing
sequential search [11], [12].
A variety of simple sequential search heuristics have been proposed for this problem,
centering on three main approaches: (1) transpose, in which a retrieved record is moved
one position closer to the front of the list (i.e., swapping with the record in front of it); (2)
move-to-front (MTF), in which a retrieved record is put at the front of the list, and all
other records remain in the same relative order; and (3) count, in which a tally is kept of
the number of times each record is retrieved, and the list is reordered in decreasing order
of this tally after each retrieval. Because count rules require storing additional
information, more attention has focused on the memory-free transposition and MTF
rules. Analytic and simulation results (reviewed in [12]) have shown that while
transposition rules can come closer to the optimal order asymptotically, in the short run
MTF rules converge more quickly (as can count rules). This may make MTF (and count)
rules more appealing as models of cue order learning by humans facing small numbers of
decision trials. Furthermore, MTF rules are more responsive to local structure in the
environment (e.g., clumped retrievals over time of a few records), and transposition can
result in very poor performance under some circumstances (e.g., when neighboring pairs
of ?popular? records get trapped at the end of the list by repeatedly swapping places).
It is important to note that there are important differences between the selforganizing sequential search problem and the cue-ordering problem we address
here. In particular, when a record is sought that matches a particular key, search
proceeds until the correct record is found. In contrast, when a decision is made
lexicographically and the list of cues is searched through, there is no one ?correct?
cue to find?each cue may or may not discriminate (allow a decision to be made).
Furthermore, once a discriminating cue is found, it may not even make the right
decision. Thus, given feedback about whether a decision was right or wrong, a
discriminating cue could potentially be moved up or down in the ordered list. This
dissociation between making a decision or not (based on the cue discrimination
rates), and making a right or wrong decision (based on the cue validities), means
that there are two ordering criteria in this problem?frugality and accuracy?as
opposed to the single order?search time?for records based on their retrieval
probability pi . Because record search time corresponds to cue frugality, the
heuristics that work well for the self-organizing sequential search task are likely to
produce orders that emphasize frugality (reflecting cue discrimination rates) over
accuracy in the cue-ordering task. Nonetheless, these heuristics offer a useful
starting point for exploring cue-ordering rules.
2.1 The cue-ordering rules
We focus on search order construction processes that are psychologically plausible by
being frugal both in terms of information storage and in terms of computation. The
decision situation we explore is different from the one assumed by Juslin and Persson
[10] who strongly differentiate learning about objects from later making decisions about
them. Instead we assume a learning-while-doing situation, consisting of tasks that have
to be done repeatedly with feedback after each trial about the adequacy of one?s decision.
For instance, we can observe on multiple occasions which of two supermarket checkout
lines, the one we have chosen or (more likely) another one, is faster, and associate this
outcome with cues including the lines? lengths and the ages of their respective cashiers.
In such situations, decision makers can learn about the differential usefulness of cues for
solving the task via the feedback received over time.
We compare several explicitly defined ordering rules that construct cue orders for
use by lexicographic decision mechanisms applied to a particular probabilistic
inference task: forced choice paired comparison, in which a decision maker has to
infer which of two objects, each described by a set of binary cues, is ?bigger? on a
criterion?just the task for which TTB was formulated. After an inference has been
made, feedback is given about whether a decision was right or wrong. Therefore,
the order-learning algorithm has information about which cues were looked up,
whether a cue discriminated, and whether a discriminating cue led to the right or
wrong decision. The rules we propose differ in which pieces of information they
use and how they use them. We classify the learning rules based on their memory
requirement?high versus low?and their computational requirements in terms of
full or partial reordering (see Table 1).
Table 1: Learning rules classified by memory and computational requirements
High memory load,
complete reordering
High memory load,
local reordering
Low memory load,
local reordering
Validity: reorders cues
based on their
current validity
Tally swap: moves
cue up (down) one
position if it has
made a correct
(incorrect) decision
if its tally of correct
minus incorrect
decisions is ? (?)
than that of next
higher (lower) cue
Simple swap: moves
cue up one position
after correct decision,
and down after an
incorrect decision
Tally: reorders cues
by number of
correct minus
incorrect decisions
made so far
Associative/delta rule:
reorders cues by
learned association
strength
Move-to-front (2 forms):
Take The Last (TTL):
moves discriminating
cue to front
TTL-correct: moves
cue to front only if it
correctly discriminates
The validity rule, a type of count rule, is the most demanding of the rules we
consider in terms of both memory requirements and computational complexity. It
keeps a count of all discriminations made by a cue so far (in all the times that the
cue was looked up) and a separate count of all the correct discriminations.
Therefore, memory load is comparatively high. The validity of each cue is
determined by dividing its current correct discrimination count by its total
discrimination count. Based on these values computed after each decision, the rule
reorders the whole set of cues from highest to lowest validity.
The tally rule only keeps one count per cue, storing the number of correct decisions
made by that cue so far minus the number of incorrect decisions. If a cue
discriminates correctly on a given trial, one point is added to its tally, if it leads to
an incorrect decision, one point is subtracted. The tally rule is less demanding in
terms of memory and computation: Only one count is kept, no division is required.
The simple swap rule uses the transposition rather than count approach. This rule
has no memory of cue performance other than an ordered list of all cues, and just
moves a cue up one position in this list whenever it leads to a correct decision, and
down if it leads to an incorrect decision. In other words, a correctly deciding cue
swaps positions with its nearest neighbor upwards in the cue order, and an
incorrectly deciding cue swaps positions with its nearest neighbor downwards.
The tally swap rule is a hybrid of the simple swap rule and the tally rule. It keeps a
tally of correct minus incorrect discriminations per cue so far (so memory load is
high) but only locally swaps cues: When a cue makes a correct decision and its tally
is greater than or equal to that of its upward neighbor, the two cues swap positions.
When a cue makes an incorrect decision and its tally is smaller than or equal to that
of its downward neighbor, the two cues also swap positions.
We also evaluate two types of move-to-front rules. First, the Take The Last (TTL)
rule moves the last discriminating cue (that is, whichever cue was found to
discriminate for the current decision) to the front of the order. This is equivalent to
the Take The Last heuristic [6], [7], which uses a memory of cues that discriminated
in the past to determine cue search order for subsequent decisions. Second, TTLcorrect moves the last discriminating cue to the front of the order only if it correctly
discriminated; otherwise, the cue order remains unchanged. This rule thus takes
accuracy as well as frugality into account.
Finally, we include an associative learning rule that uses the delta rule to update cue
weights according to whether they make correct or incorrect discriminations, and
then reorders all cues in decreasing order of this weight after each decision. This
corresponds to a simple network with nine input units encoding the difference in cue
value between the two objects (A and B) being decided on (i.e., ini = -1 if
cuei(A)<cuei(B), 1 if cuei(A)>cuei(B), and 0 if cuei(A)=cuei(B) or cuei was not
checked) and with one output unit whose target value encodes the correct decision (t
= 1 if criterion(A)>criterion(B), otherwise -1), and with the weights between inputs
and output updated according to ?wi = lr ? (t - ini?wi) ? ini with learning rate lr = 0.1.
We expect this rule to behave similarly to Oliver?s rule initially (moving a cue to
the front of the list by giving it the largest weight when weights are small) and to
swap later on (moving cues only a short distance once weights are larger).
3
Si mul ati on Study of Si mpl e O r de r i ng Rul e s
To test the performance of these order learning rules, we use the German cities data set
[6], [7], consisting of the 83 largest-population German cities (those with more than
100,000 inhabitants), described on 9 cues that give some information about population
size. Discrimination rate and validity of the cues are negatively correlated (r = -.47). We
present results averaged over 10,000 learning trials for each rule, starting from random
initial cue orders. Each trial consisted of 100 decisions between randomly selected
decision pairs. For each decision, the current cue order was used to look up cues until a
discriminating cue was found, which was used to make the decision (employing a onereason or lexicographic decision strategy). After each decision, the cue order was
updated using the particular order-learning rule. We start by considering the cumulative
accuracies (i.e., online or amortized performance?[12]) of the rules, defined as the total
percentage of correct decisions made so far at any point in the learning process. The
contrasting measure of offline accuracy?how well the current learned cue order would
do if it were applied to the entire test set?will be subsequently reported (see Figure 1).
For all but the move-to-front rules, cumulative accuracies soon rise above that of the
Minimalist heuristic (proportion correct = .70) which looks up cues in random order and
thus serves as a lower benchmark. However, at least throughout the first 100 decisions,
cumulative accuracies stay well below the (offline) accuracy that would be achieved by
using TTB for all decisions (proportion correct = .74), looking up cues in the true order of
their ecological validities. Except for the move-to-front rules, whose cumulative
accuracies are very close to Minimalist (mean proportion correct in 100 decisions: TTL:
.701; TTL-correct: .704), all learning rules perform on a surprisingly similar level, with
less than one percentage point difference in favor of the most demanding rule (i.e., delta
rule: .719) compared to the least (i.e., simple swap: .711; for comparison: tally swap:
.715; tally: .716; validity learning rule: .719). Offline accuracies are slightly higher, again
with the exception of the move to front rules (TTL: .699; TTL-correct: .702; simple
swap: .714; tally swap: .719; tally: .721; validity learning rule: .724; delta rule: .725; see
Figure 1). In longer runs (10,000 decisions) the validity learning rule is able to converge
on TTB?s accuracy, but the tally rule?s performance changes little (to .73).
Figure 1: Mean offline accuracy of
order learning rules
Figure 2: Mean offline frugality of
order learning rules
All learning rules are, however, more frugal than TTB, and even more frugal than
Minimalist, both in terms of online as well as offline frugality. Let us focus on their
offline frugality (see Figure 2): On average, the rules look up fewer cues than Minimalist
before reaching a decision. There is little difference between the associative rule, the
tallying rules and the swapping rules (mean number of cues looked up in 100 decisions:
delta rule: 3.20; validity learning rule: 3.21; tally: 3.01; tally swap: 3.04; simple swap:
3.13). Most frugal are the two move-to front rules (TTL-correct: 2.87; TTL: 2.83).
Consistent with this finding, all of the learning rules lead to cue orders that show positive
correlations with the discrimination rate cue order (reaching the following values after
100 decisions: validity learning rule: r = .18; tally: r = .29; tally swap: r = .24; simple
swap: r = .18; TTL-correct: r = .48; TTL: r = .56). This means that cues that often lead to
discriminations are more likely to end up in the first positions of the order. This is
especially true for the move-to-front rules. In contrast, the cue orders resulting from all
learning rules but the validity learning rule do not correlate or correlate negatively with
the validity cue order, and even the correlations of the cue orders resulting from the
validity learning rule after 100 decisions only reach an average r = .12.
But why would the discrimination rates of cues exert more of a pull on cue order than
validity, even when the validity learning rule is applied? As mentioned earlier, this is
what we would expect for the move-to-front rules, but it was unexpected for the other
rules. Part of the explanation comes from the fact that in the city data set we used for the
simulations, validity and discrimination rate of cues are negatively correlated. Having a
low discrimination rate means that a cue has little chance to be used and hence to
demonstrate its high validity. Whatever learning rule is used, if such a cue is displaced
downward to the lower end of the order by other cues, it may have few chances to escape
to the higher ranks where it belongs. The problem is that when a decision pair is finally
encountered for which that cue would lead to a correct decision, it is unlikely to be
checked because other, more discriminating although less valid, cues are looked up
before and already bring about a decision. Thus, because one-reason decision making is
intertwined with the learning mechanism and so influences which cues can be learned
about, what mainly makes a cue come early in the order is producing a high number of
correct decisions and not so much a high ratio of correct discriminations to total
discriminations regardless of base rates.
This argument indicates that performance may differ in environments where cue
validities and discrimination rates correlate positively. We tested the learning rules on
one such data set (r=.52) of mammal species life expectancies, predicted from 9 cues. It
also differs from the cities environment with a greater difference between TTB?s and
Minimalist?s performance (6.5 vs. 4 percentage points). In terms of offline accuracy, the
validity learning rule now indeed more closely approaches TTB?s accuracy after 100
decisions (.773 vs. .782)., The tally rule, in contrast, behaves very much as in the cities
environment, reaching an accuracy of .752, halfway between TTB and Minimalist
(accuracy =.716). Thus only some learning rules can profit from the positive correlation.
4
D i s c u s s i on
Most of the simpler cue order learning rules we have proposed do not fall far behind a
validity learning rule in accuracy, and although the move-to-front rules cannot beat the
accuracy achieved if cues were selected randomly, they compensate for this failure by
being highly frugal. Interestingly, the rules that do achieve higher accuracy than
Minimalist also beat random cue selection in terms of frugality.
On the other hand, all rules, even the delta rule and the validity learning rule, stay below
TTB?s accuracy across a relatively high number of decisions. But often it is necessary to
make good decisions without much experience. Therefore, learning rules should be
preferred that quickly lead to orders with good performance. The relatively complex rules
with relatively high memory requirement, i.e., the delta and the validity learning rule, but
also the tally learning rule, more quickly rise in accuracy compared the rules with lower
requirements. Especially the tally rule thus represents a good compromise between cost,
correctness and psychological plausibility considerations.
Remember that the rules based on tallies assume full memory of all correct minus
incorrect decisions made by a cue so far. But this does not make the rule implausible, at
least from a psychological perspective, even though computer scientists were reluctant to
adopt such counting approaches because of their extra memory requirements. There is
considerable evidence that people are actually very good at remembering the frequencies
of events. Hasher and Zacks [13] conclude from a wide range of studies that frequencies
are encoded in an automatic way, implying that people are sensitive to this information
without intention or special effort. Estes [14] pointed out the role frequencies play in
decision making as a shortcut for probabilities. Further, the tally rule and the tally swap
rule are comparatively simple, not having to keep track of base rates or perform divisions
as does the validity rule. From the other side, the simple swap and move to front rules
may not be much simpler, because storing a cue order may be about as demanding as
storing a set of tallies. We have run experiments (reported elsewhere) in which indeed the
tally swap rule best accounts for people?s actual processes of ordering cues.
Our goal in this paper was to explore how well simple cue-ordering rules could work in
conjunction with lexicographic decision strategies. This is important because it is
necessary to take into account the set-up costs of a heuristic in addition to its application
costs when considering the mechanism?s overall simplicity. As the example of the
validity search order of TTB shows, what is easy to apply may not necessarily be so easy
to set up. But simple rules can also be at work in the construction of a heuristic?s building
blocks. We have proposed such rules for the construction of one building block, the
search order. Simple learning rules inspired by research in computer science can enable a
one-reason decision heuristic to perform only slightly worse than if it had full knowledge
of cue validities from the very beginning. Giving up the assumption of full a priori
knowledge for the slight decrease in accuracy seems like a reasonable bargain: Through
the addition of learning rules, one-reason decision heuristics might lose some of their
appeal to decision theorists who were surprised by the performance of such simple
mechanisms compared to more complex algorithms, but they gain psychological
plausibility and so become more attractive as explanations for human decision behavior.
References
[1] Fishburn, P.C. (1974). Lexicographic orders, utilities and decision rules: A survey.
Management Science, 20, 1442-1471.
[2] Payne, J.W., Bettman, J.R., & Johnson, E.J. (1993). The adaptive decision maker. New York:
Cambridge University Press.
[3] Br?der, A. (2000). Assessing the empirical validity of the ?Take-The-Best? heuristic as a model
of human probabilistic inference. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 26 (5), 1332-1346.
[4] Br?der, A. (2003). Decision making with the ?adaptive toolbox?: Influence of environmental
structure, intelligence, and working memory load. Journal of Experimental Psychology: Learning,
Memory, & Cognition, 29, 611-625.
[5] Gigerenzer, G., Todd, P.M., & The ABC Research Group (1999). Simple heuristics that make
us smart. New York: Oxford University Press.
[6] Gigerenzer, G., & Goldstein, D.G. (1996). Reasoning the fast and frugal way: Models of
bounded rationality. Psychological Review, 103 (4), 650-669.
[7] Gigerenzer, G., & Goldstein, D.G. (1999). Betting on one good reason: The Take The Best
Heuristic. In G. Gigerenzer, P.M. Todd & The ABC Research Group, Simple heuristics that make
us smart. New York: Oxford University Press.
[8] Czerlinski, J., Gigerenzer, G., & Goldstein, D.G. (1999). How good are simple heuristics? In G.
Gigerenzer, P.M. Todd & The ABC Research Group, Simple heuristics that make us smart. New
York: Oxford University Press.
[9] Newell, B.R., & Shanks, D.R. (2003). Take the best or look at the rest? Factors influencing
?one-reason? decision making. Journal of Experimental Psychology: Learning, Memory, and
Cognition, 29, 53-65.
[10] Juslin, P., & Persson, M. (2002). PROBabilities from EXemplars (PROBEX): a ?lazy?
algorithm for probabilistic inference from generic knowledge. Cognitive Science, 26, 563-607.
[11] Rivest, R. (1976). On self-organizing sequential search heuristics. Communications of the
ACM, 19(2), 63-67.
[12] Bentley, J.L. & McGeoch, C.C. (1985). Amortized analyses of self-organizing sequential
search heuristics. Communications of the ACM, 28(4), 404-411.
[13] Hasher, L., & Zacks, R.T. (1984). Automatic Processing of fundamental information: The case
of frequency of occurrence. American Psychologist, 39, 1372-1388.
[14] Estes, W.K. (1976). The cognitive side of probability learning. Psychological Review, 83, 3764.
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1,801 | 2,636 | Analysis of a greedy active learning strategy
Sanjoy Dasgupta?
University of California, San Diego
[email protected]
Abstract
We abstract out the core search problem of active learning schemes, to
better understand the extent to which adaptive labeling can improve sample complexity. We give various upper and lower bounds on the number
of labels which need to be queried, and we prove that a popular greedy
active learning rule is approximately as good as any other strategy for
minimizing this number of labels.
1
Introduction
An increasingly common phenomenon in classification tasks is that unlabeled data is abundant, whereas labels are considerably harder to come by. Genome sequencing projects, for
instance, are producing vast numbers of peptide sequences, but reliably labeling even one
of these with structural information requires time and close attention.
This distinction between labeled and unlabeled data is not captured in standard models like
the PAC framework, and has motivated the field of active learning, in which the learner
is able to ask for the labels of specific points, but is charged for each label. These query
points are typically chosen from an unlabeled data set, a practice called pool-based learning
[10]. There has also been some work on creating query points synthetically, including a
rich body of theoretical results [1, 2], but this approach suffers from two problems: first,
from a practical viewpoint, the queries thus produced can be quite unnatural and therefore
bewildering for a human to classify [3]; second, since these queries are not picked from the
underlying data distribution, they might have limited value in terms of generalization. In
this paper, we focus on pool-based learning.
We are interested in active learning with generalization guarantees. Suppose the hypothesis
class has VC dimension d and we want a classifier whose error rate on distribution P over
the joint (input, label) space, is less than > 0. The theory tells us that in a supervised
setting, we need some m = m(, d) labeled points drawn from P (for a fixed level of
confidence, which we will henceforth ignore). Can we get away with substantially fewer
than m labels if we are given unlabeled points from P and are able to adaptively choose
which points to label? How much fewer, and what querying strategies should we follow?
Here is a toy example illustrating the potential of active learning. Suppose the data lie on
the real line, and the classifiers are simple thresholding functions, H = {hw : w ? R}:
hw (x) = 1(x ? w).
VC theory tells us that if the underlying distribution P can be classified perfectly by some
hypothesis in H (called the realizable case), then it is enough to draw m = O(1/) random
labeled examples from P , and to return any classifier consistent with them. But suppose
we instead draw m unlabeled samples from P . If we lay these points down on the line,
their hidden labels are a sequence of 0?s followed by a sequence of 1?s, and the goal is to
discover the point w at which the transition occurs. This can be accomplished with a simple
binary search which asks for just log m labels. Thus active learning gives us an exponential
improvement in the number of labels needed: by adaptively querying log m labels, we can
automatically infer the rest of them.
Generalized binary search?
So far we have only looked at an extremely simple learning problem. For more complicated
hypothesis classes H, is a sort of a generalized binary search possible? What would the
search space look like? For supervised learning, in the realizable case, the usual bounds
specify a sample complexity of (very roughly) m ? d/ labeled points if the target error
rate is . So let?s pick this many unlabeled points, and then try to find a hypothesis consistent with all the hidden labels by adaptively querying just a few of them. We know via
Sauer?s lemma that H can classify these m points (considered jointly) in at most O(m d )
different ways ? in effect, the size of H is reduced to O(md ). This finite set is the effective
b (In the 1-d example, H
b has size m + 1, corresponding to the intervals
hypothesis class H.
into which the points xi split the real line.) The most we can possibly learn about the target
hypothesis, even if all labels are revealed, is to narrow it down to one of these regions. Is it
possible to pick among these O(md ) possibilities using o(m) labels? If binary search were
possible, just O(d log m) labels would be needed.
Unfortunately, we cannot hope for a generic positive result of this kind. The toy example
above is a 1-d linear separator. We show that for d ? 2, the situation is very different:
Pick any collection of m (unlabeled) points on the unit sphere in Rd ,
for d ? 2, and assume their hidden labels correspond perfectly to some
b which cannot be
linear separator. Then there are target hypotheses in H
identified without querying all the labels.
(What if the active learner is not required to identify exactly the right hypothesis, but something close to it? This and other little variations don?t help much.) Therefore, even in the
most benign situations, we cannot expect that every target hypothesis will be identifiable
using o(m) labels. To put it differently, in the worst case over target hypotheses, active
learning gives no improvement in sample complexity.
But hopefully, on average (with respect to some distribution over target hypotheses), the
number of labels needed is small. For instance, when d = 2 in the bad case above, a
b can be identified by querying just
target hypothesis chosen uniformly at random from H
O(log m) labels in expectation. This motivates the main model of this paper.
An average-case model
We will count the expected number of labels queried when the target hypothesis is chosen
b This can be interpreted as a Bayesian setting, but it is
from some distribution ? over H.
more accurate to think of ? merely as a device for averaging query counts, which has no
b
bearing on the final generalization bound. A natural choice is to make ? uniform over H.
b but H
b reflects the
Most existing active learning schemes work with H rather than H;
underlying combinatorial structure of the problem, and it can?t hurt to deal with it directly.
b for instance, if H is the set of linear
Often ? can chosen to mask the structure of H;
b is a set of convex regions of H, and ? can be made proportional to the
separators, then H
volume of each region. This makes the problem continuous rather than combinatorial.
What is the expected number of labels needed to identify a target hypothesis chosen from ??
In this average-case setting, is it always possible to get away with o(m) labels, where m is
the sample complexity of the supervised learning problem as defined above? We show that
the answer, once again, is sadly no. Thus the benefit of active learning is really a function
of the specific hypothesis class and the particular pool of unlabeled data. Depending on
these, the expected number of labels needed lies in the following range (within constants):
ideal case:
worst case:
d log m
m
perfect binary search
all labels, or randomly chosen queries
Notice the exponential gap between the top and bottom of this range. Is there some simple querying strategy which always achieves close to the minimum (expected) number of
labels, whatever this minimum number might be?
Our main result is that this property holds for a variant of a popular greedy scheme: always
ask for the label which most evenly divides the current effective version space weighted by
?. This doesn?t necessarily minimize the number of queries, just as a greedy decision tree
algorithm need not produce trees of minimum size. However:
b the expected number of labels needed by this
When ? is uniform over H,
b times that of any other strategy.
greedy strategy is at most O(ln |H|)
We also give a bound for arbitrary ?, and show corresponding lower bounds in both the
uniform and non-uniform cases.
Variants of this greedy scheme underlie many active learning heuristics, and are often described as optimal in the literature. This is the first rigorous validation of the scheme in
b = O(d log m),
a general setting. The performance guarantee is significant: recall log |H|
the minimum number of queries possible.
2
Preliminaries
Let X be the input space, Y = {0, 1} the space of labels, and P an unknown underlying
distribution over X ? Y. We want to select a hypothesis (a function X ? Y) from some
class H of VC dimension d < ?, which will accurately predict labels of points in X . We
will assume that the problem is realizable, that is, there is some hypothesis in H which
gives a correct prediction on every point. Suppose that points (x1 , y1 ) . . . , (xm , ym ) are
drawn randomly from P . Standard bounds give us a function m(, d) such that if we want
a hypothesis of error ? (on P , modulo some fixed confidence level), and if m ? m(, d),
then we need only pick a hypothesis h ? H consistent with these labeled points [9].
Now suppose just the pool of unlabeled data x1 , . . . , xm is available. The possible labelings
of these points form a subset of {0, 1}m , the effective hypothesis class
b ?
H
= {(h(x1 ), . . . , h(xm )) : h ? H}.
b = O(md ). We want to pick the unique h ? H
b which is
Sauer?s lemma [9] tells us |H|
consistent with all the hidden labels, by querying just a few of them.
Any deterministic search strategy can be represented as a binary tree whose internal nodes
b We can also
are queries (?what is the xi ?s label??), and whose leaves are elements of H.
accommodate randomization ? for instance, to allow a random choice of query point ? by
letting internal nodes of the tree be random coin flips. Our main result, Theorem 3, is
unaffected by this generalization.
xm
x1
x2
L3
x3
x4
Figure 1: To identify target hypotheses like L3 , we need to see all the labels.
3
Some bad news
Claim 1 Let H be the hypothesis class of linear separators in R2 . For any set of m distinct
data points on the perimeter of the unit circle, there are always some target hypotheses in
b which cannot be identified without querying all m labels.
H
Proof. To see this, consider the following realizable labelings (Figure 1):
? Labeling L0 : all points are negative.
? Labeling Li (1 ? i ? m): all points are negative except xi .
It is impossible to distinguish these cases without seeing all the labels.1
Remark. To rephrase this example in terms of learning a linear separator with error ? ,
suppose the input distribution P (X ) is a density over the perimeter of the unit circle. No
matter what this density is, there are always target hypotheses in H which force us to ask
for ?(1/) labels: no improvement over the sample complexity of supervised learning.
In this example, the bad target hypotheses have a large imbalance in probability mass between their positive and negative regions. By adding an extra dimension and an extra point,
exactly the same example can be modified to make the bad hypotheses balanced.
Let?s return to the original 2-d case. Some hypotheses must lie at depth m in any query tree;
but what about the rest? Well, suppose for convenience that x1 , . . . , xm are in clockwise
b = {hij : 1 ? i 6= j ? m} ? {h0 , h1 }, where hij
order around the unit circle. Then H
labels xi ? ? ? xj?1 positive (if j < i it wraps around) and the remaining points negative, and
h0 , h1 are everywhere negative/positive. It is possible to construct a query tree in which
each hij lies at depth ? 2(m/|j ? i| + log |j ? i|). Thus, if the target hypothesis is chosen
b the expected number of labels queried is at most
uniformly from H,
P
1
2m
+
2(m/|j
?
i|
+
log
|j
?
i|)
= O(log m).
i6=j
m(m?1)+2
This is why we place our hopes in an average-case analysis.
1
What if the final hypothesis ? considered as a point in {0, 1}m ? doesn?t have to be exactly right,
but within Hamming distance k of the correct one? Then a similar example forces ?(m/k) queries.
4
Main result
b we will analyze search strategies according to the number
Let ? be any distribution over H;
of labels they require, averaged over target hypotheses drawn from ?. In terms of query
trees, this is the average depth of a leaf chosen according to ?. Specifically, let T be any
tree whose leaves include the support of ?. The quality of this tree is
P
P
Q(T, ?) =
b ?(h) ? leaf-depth(h).
b ?(h) ? (# labels needed for h) =
h?H
h?H
Is there always a tree of average depth o(m)? The answer, once again, is sadly no.
Claim 2 Pick any d ? 2 and any m ? 2d. There is an input space X of size m and a
hypothesis class H of VC dimension d, defined on domain X , with the following property:
b then any query tree T has Q(T, ?) ? m/8.
if ? is chosen to be uniform over H = H,
Proof. Let X consist of any m points x1 , . . . , xm , and let H consist of all hypotheses
h : X ? {0, 1} which are positive on exactly d inputs. In order to identify a particular
element h ? H, any querying method must discover exactly the d points xi on which h is
nonzero. By construction, the order in which queries are asked is irrelevant ? it might as
well be x1 , x2 , . . .. The rest is a simple probability calculation.
In our average-case model, we have seen one example in which intelligent querying results
in an exponential improvement in the number of labels required, and one in which it is
no help at all. Is there some generic scheme which always comes close to minimizing the
number of queries, whatever the minimum number might be? Here?s a natural candidate:
b be the current version space. For each
Greedy strategy. Let S ? H
unlabeled xi , let Si+ be the hypotheses which label xi positive and Si?
the ones which label it negative. Pick the xi for which these sets are most
nearly equal in ?-mass, that is, for which min{?(Si+ ), ?(Si? )} is largest.
We show this is almost as good at minimizing queries as any other strategy.
b Suppose that the optimal query tree requires
Theorem 3 Let ? be any distribution over H.
?
Q labels in expectation, for target hypotheses chosen according to ?. Then the expected
number of labels needed by the greedy strategy is at most 4Q? ln 1/(minh ?(h)).
b We also show
For the case of uniform ?, the approximation ratio is thus at most 4 ln |H|.
almost-matching lower bounds in both the uniform and non-uniform cases.
5
5.1
Analysis of the greedy active learner
Lower bounds on the greedy scheme
The greedy approach is not optimal because it doesn?t take into account the way in which a
query reshapes the search space ? specifically, the effect of a query on the quality of other
b might consist of several dense clusters, each of which permits
queries. For instance, H
rapid binary search. However, the version space must first be whittled down to one of
these subregions, and this process, though ultimately optimal, might initially be slower at
shrinking the hypothesis space than more shortsighted alternatives. A concrete example of
this type gives rise to the following lower bound.
b n of size n
Claim 4 For any n ? 16 which is a power of two, there is a concept class H
such that: under uniform ?, the optimal tree has average height at most qn = ?(log n),
but the greedy active learning strategy produces a tree of average height ?(q n ? logloglogn n ).
For non-uniform ?, the greedy scheme can deviate more substantially from optimality.
b with 2n + 1 elements and a distriClaim 5 For any n ? 2, there is a hypothesis class H
b
bution ? over H, such that: (a) ? ranges in value from 1/2 to 1/2n+1 ; (b) the optimal tree
has average depth less than 3; (c) the greedy tree has average depth at least n/2.
Proofs of these lower bounds appear in the full paper, available at the author?s website.
5.2
Upper bound
Overview. The lower bounds on the quality of a greedy learner are sobering, but things
cannot get too much worse than this. Here?s the basic argument for uniform ?: we show
that if the optimal tree T ? requires Q? queries in expectation, then some query must (again
b of ?-mass ?(1/Q? ). Therefore, the root query of
in expectation) ?cut off? a chunk of H
the greedy tree TG is at least this good (cf. Johnson?s set cover analysis [8]). Things get
trickier when we try to show that the rest of TG is also good, because although T ? uses just
Q? queries on average, it may need many more queries for certain hypotheses. Subtrees
of TG could correspond to version spaces for which more than Q? queries are needed, and
the roots of these subtrees might not cut down the version space much...
For a worst-case model, a proof of approximate optimality is known in a related context
[6]; as we saw in Claim 1, that model is trivial in our situation. The average-case model,
and especially the use of arbitrary weights ?, require more care.
Details. For want of space, we only discuss some issues that arise in proving the main
theorem, and leave the actual proof to the full paper.
The key concept we have to define is the quality of a query, and it turns out that we need this
to be monotonically decreasing, that is, it should only go down as active learning proceeds
and the version space shrinks. This rules out some natural entropy-based notions.
b and a possible next query is xj . If
Suppose we are down to some version space S ? H,
S + is the subset of S which labels xj positive, and S ? are the ones that label it negative,
then on average the probability mass (measured by ?) eliminated by xj is
?(S + )
?
?(S) ?(S )
+
?(S ? )
+
?(S) ?(S )
=
2?(S + )?(S ? )
.
?(S)
We say xj shrinks (S, ?) by this much, with the understanding that this is in expectation.
Shrinkage is easily seen to have the monotonicity property we need.
b ?) by ?, then it shrinks (S, ?) by at most ? for any S ? H.
b
Lemma 6 If xj shrinks (H,
We would expect that if the optimal tree is short, there must be at least one query which
b ?) considerably. More concretely, the definition of shrinkage seems to suggest
shrinks (H,
that if all queries provide shrinkage at most ?, and the current version space has mass ?(S),
then at least about ?(S)/? more queries are needed. This isn?t entirely true, because of a
second effect: if |S| = 2, then we need just one query, regardless of ?(S).
Roughly speaking, when there are lots of hypotheses with significant mass left in S, the first
effect dominates; thereafter the second takes over. To smoothly incorporate both effects,
we use the
Pnotion of collision probability. For a distribution ? over support Z, this is
CP(?) = z?Z ?(z)2 , the chance that two random draws from ? are identical.
b ?) by at most ? > 0. Pick any S ? H,
b
Lemma 7 Suppose every query shrinks (H,
and any query tree T whose leaves include S. If ?S is the restriction of ? to S (that is,
?S (h) = ?(h)/?(S) for h ? S), then Q(T, ?S ) ? (1 ? CP(?S )) ? ?(S)/?.
b and any tree T whose leaves include all of S. Then there
Corollary 8 Pick any S ? H
must exist a query which shrinks (S, ?S ) by at least (1 ? CP(?S ))/Q(T, ?S ).
b is such that ?S has small collision probability, some
So if the current version space S ? H
query must split off a sizeable chunk of S. This can form the basis of a proof by induction.
But what if CP(?S ) is large, say greater than 1/2? In this case, the mass of some particular
hypothesis h0 ? S exceeds that of all the others combined, and S could shrink by just
an insignificant amount during the subsequent greedy query, or even during the next few
iterations of greedy queries. It turns out, however, that within roughly the number of iterations that the optimal tree needs for target h0 , the greedy procedure will either reject h0 or
identify it as the target. If it is rejected, then by that time S will have shrunk considerably.
By combining the two cases for CP(?S ), we get the following lemma, which is proved in
the full paper and yields our main theorem as an immediate consequence.
Lemma 9 Let T ? denote any particular query tree for ?, and let T be the greedilyb which corresponds to a subtree TS of T ,
constructed query tree. For any S ? H
?(S)
Q(TS , ?S ) ? 4Q(T ? , ?S ) ln
.
minh?S ?(h)
6
Related work and promising directions
Rather than attempting to summarize the wide range of proposed active learning methods,
for instance [5, 7, 10, 13, 14], we will discuss three basic techniques upon which they rely.
Greedy search. This is the technique we have abstracted and rigorously validated in this
paper. It is the foundation of most of the schemes cited above. Algorithmically, the main
problem is that the query selection rule is not immediately tractable, so approximations are
b consists of convex sets, and if ? is chosen to be pronecessary. For linear separators, H
portional to volume, query selection involves estimating volumes of convex regions, which
is tractable but (using present techniques) inconvenient. Tong and Koller [13] investigate
margin-based approximations which are efficiently computable using SVM technology.
Opportunistic priors. This is a trick in which the learner takes a look at the unlabeled
b leads to standard
data and then places bets on hypotheses. A uniform bet over all of H
generalization bounds. But if the algorithm places more weight on certain hypotheses (for
instance, those with large margin), then its final error bound is excellent if it guessed right,
and worse-than-usual if it guessed wrong. This technique is not specific to active learning,
and has been analyzed elsewhere (eg. [12]). One interesting line of work investigates a
flexible family of priors specified by pairwise similarities between data points, eg. [14].
Bayesian assumptions. In our analysis, although ? can be seen as some sort of prior belief,
there is no assumption that nature shares this belief; in particular, the generalization bound
does not depend on it. A Bayesian assumption has an immediate benefit for active learning:
if at any stage the remaining version space (weighted by prior ?) is largely in agreement
on the unlabeled data, it is legitimate to stop and output one of these remaining hypotheses
[7]. In a non-Bayesian setting this is not legitimate.
When the hypothesis class consists of probabilistic classifiers, the Bayesian assumption
has also been used in another way: to approximate the greedy selection rule using the
MAP estimate instead of an expensive summation over the posterior (eg. [11]).
In terms of theoretical results, another work which considers the tradeoff between labels
and generalization error is [7], in which a greedy scheme, realized using sampling, is analyzed in a Bayesian setting. The authors show that it is possible to achieve an exponential
improvement in the number of labels needed to learn linear separators, when both data and
target hypothesis are chosen uniformly from the unit sphere. It is an intriguing question
whether this holds for more general data distributions.
Other directions. We have looked at the case where the acceptable error rate is fixed and
the goal is to minimize the number of queries. What about fixing the number of queries and
asking for the best (average) error rate possible? In other words, the query tree has a fixed
b Treating each
depth, and each leaf is annotated with its remaining version space S ? H.
element of S as a point in {0, 1}m (its predictions on the pool of data), the error at this leaf
depends on the Hamming diameter of S. What is a good querying strategy for producing
low-diameter leaves?
The most widely-used classifiers are perhaps linear separators. Existing active learning
b an arrangement of hyperplanes [4].
schemes ignore the rich algebraic structure of H,
Acknowledgements. I am very grateful to the anonymous NIPS reviewers for their careful
and detailed feedback.
References
[1] D. Angluin. Queries and concept learning. Machine Learning, 2:319?342, 1988.
[2] D. Angluin. Queries revisited. Proceedings of the Twelfth International Conference on Algorithmic Learning Theory, pages 12?31, 2001.
[3] E.B. Baum and K. Lang. Query learning can work poorly when a human oracle is used. International Joint Conference on Neural Networks, 1992.
[4] A. Bjorner, M. Las Vergnas, B. Sturmfels, N. White, and G. Ziegler. Oriented matroids. Cambridge University Press, 1999.
[5] D. Cohn, Z. Ghahramani, and M. Jordan. Active learning with statistical models. Journal of
Artificial Intelligence Research, 4:129?145, 1996.
[6] S. Dasgupta, P.M. Long, and W.S. Lee. A theoretical analysis of query selection for collaborative filtering. Machine Learning, 51:283?298, 2003.
[7] Y. Freund, S. Seung, E. Shamir, and N. Tishby. Selective sampling using the query by committee
algorithm. Machine Learning, 28:133?168, 1997.
[8] D.S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer
and System Sciences, 9:256?278, 1974.
[9] M.J. Kearns and U.V. Vazirani. An introduction to computational learning theory. MIT Press,
1993.
[10] A. McCallum and K. Nigam. Employing em and pool-based active learning for text classification. Fifteenth International Conference on Machine Learning, 1998.
[11] N. Roy and A. McCallum. Toward optimal active learning through sampling of error reduction.
Twentieth International Conference on Machine Learning, 2003.
[12] J. Shawe-Taylor, P. Bartlett, R. Williamson, and M. Anthony. Structural risk minimization over
data-dependent hierarchies. IEEE Transactions on Information Theory, 1998.
[13] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. Journal of Machine Learning Research, 2001.
[14] X. Zhu, J. Lafferty, and Z. Ghahramani. Combining active learning and semi-supervised learning using gaussian fields and harmonic functions. ICML workshop, 2003.
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1,802 | 2,637 | Online Bounds for Bayesian Algorithms
Sham M. Kakade
Computer and Information Science Department
University of Pennsylvania
Andrew Y. Ng
Computer Science Department
Stanford University
Abstract
We present a competitive analysis of Bayesian learning algorithms in the
online learning setting and show that many simple Bayesian algorithms
(such as Gaussian linear regression and Bayesian logistic regression) perform favorably when compared, in retrospect, to the single best model in
the model class. The analysis does not assume that the Bayesian algorithms? modeling assumptions are ?correct,? and our bounds hold even
if the data is adversarially chosen. For Gaussian linear regression (using logloss), our error bounds are comparable to the best bounds in the
online learning literature, and we also provide a lower bound showing
that Gaussian linear regression is optimal in a certain worst case sense.
We also give bounds for some widely used maximum a posteriori (MAP)
estimation algorithms, including regularized logistic regression.
1
Introduction
The last decade has seen significant progress in online learning algorithms that perform well
even in adversarial settings (e.g. the ?expert? algorithms of Cesa-Bianchi et al. (1997)). In
the online learning framework, one makes minimal assumptions on the data presented to
the learner, and the goal is to obtain good (relative) performance on arbitrary sequences. In
statistics, this philosophy has been espoused by Dawid (1984) in the prequential approach.
We study the performance of Bayesian algorithms in this adversarial setting, in which the
process generating the data is not restricted to come from the prior?data sequences may
be arbitrary. Our motivation is similar to that given in the online learning literature and the
MDL literature (see Grunwald, 2005) ?namely, that models are often chosen to balance
realism with computational tractability, and often assumptions made by the Bayesian are
not truly believed to hold (e.g. i.i.d. assumptions). Our goal is to study the performance of
Bayesian algorithms in the worst-case, where all modeling assumptions may be violated.
We consider the widely used class of generalized linear models?focusing on Gaussian
linear regression and logistic regression?and provide relative performance bounds (comparing to the best model in our model class) when the cost function is the logloss. Though
the regression problem has been studied in a competitive framework and, indeed, many
ingenious algorithms have been devised for it (e.g., Foster, 1991; Vovk, 2001; Azoury and
Warmuth, 2001) , our goal here is to study how the more widely used, and often simpler,
Bayesian algorithms fare. Our bounds for linear regression are comparable to the best
bounds in the literature (though we use the logloss as opposed to the square loss).
The competitive approach to regression started with Foster (1991), who provided competitive bounds for a variant of the ridge regression algorithm (under the square loss).
Vovk (2001) presents many competitive algorithms and provides bounds for linear regression (under the square loss) with an algorithm that differs slightly from the Bayesian one.
Azoury and Warmuth (2001) rederive Vovk?s bound with a different analysis based on
Bregman distances. Our work differs from these in that we consider Bayesian Gaussian
linear regression, while previous work typically used more complex, cleverly devised algorithms which are either variants of a MAP procedure (as in Vovk, 2001) , or that involve
other steps such as ?clipping? predictions (as in Azoury and Warmuth, 2001) . These distinctions are discussed in more detail in Section 3.1.
We should also note that when the loss function is the logloss, multiplicative weights algorithms are sometimes identical to Bayes rule with particular choices of the parameters (see
Freund and Schapire, 1999) . Furthermore, Bayesian algorithms have been used in some
online learning settings, such as the sleeping experts setting of Freund et al. (1997) and the
online boolean prediction setting of Cesa-Bianchi et al. (1998). Ng and Jordan (2001) also
analyzed an online Bayesian algorithm but assumed that the data generation process was
not too different from the model prior. To our knowledge, there have been no studies of
Bayesian generalized linear models in an adversarial online learning setting (though many
variants have been considered as discussed above).
We also examine maximum a posteriori (MAP) algorithms for both Gaussian linear regression (i.e., ridge regression) and for (regularized) logistic regression. These algorithms are
often used in practice, particularly in logistic regression where Bayesian model averaging is computationally expensive, but the MAP algorithm requires only solving a convex
problem. As expected, MAP algorithms are somewhat less competitive than full Bayesian
model averaging, though not unreasonably so.
2
Bayesian Model Averaging
We now consider the Bayesian model averaging (BMA) algorithm and give a bound on its
worst-case online loss. We start with some preliminaries. Let x ? Rn denote the inputs of
a learning problem and y ? R the outputs. Consider a model from the generalized linear
model family (see McCullagh and Nelder, 1989) , that can be written p(y|x, ?) = p(y|?T x),
where ? ? Rn are the parameters of our model (?T denotes the transpose of ?). Note that
the predicted distribution of y depends only on ?T x, which is linear in ?. For example, in
the case of Gaussian linear regression, we have
?(?T x ? y)2
1
exp
p(y|x, ?) = ?
,
(1)
2? 2
2?? 2
where ? 2 is a fixed, known constant that is not a parameter of our model. In logistic
regression, we would have
1
1
log p(y|x, ?) = y log
+
(1
?
y)
log
1
?
,
(2)
1 + exp(??T x)
1 + exp(??T x)
where we assume y ? {0, 1}.
Let S = {(x(1) , y (1) ), (x(2) , y (2) ), . . . , (x(T ) , y (T ) )} be an arbitrary sequence of examples,
possibly chosen by an adversary. We also use St to denote the subsequence consisting of
only the first t examples. We assume throughout this paper that ||x(t) || ? 1 (where || ? ||
denotes the L2 norm).
Assume that we are going to use a Bayesian algorithm to make our online predictions.
Specifically, assume that we have a Gaussian prior on the parameters:
p(?) = N (?; ~0, ? 2 In ),
where In is the n-by-n identity matrix, N (?; ?, ?) is the Gaussian density with mean ? and
covariance ?, and ? 2 > 0 is some fixed constant governing the prior variance. Also, let
Q
t
(i) (i)
i=1 p(y |x , ?) p(?)
pt (?) = p(?|St ) = R Q
t
(i) |x(i) , ?) p(?)d?
p(y
i=1
?
be the posterior distribution over ? given the first t training examples. We have that p0 (?) =
p(?) is just the prior distribution.
On iteration t, we are given the input x(t) , and our algorithm makes a prediction using the
posterior distribution over the outputs:
Z
p(y|x(t) , St?1 ) = p(y|x(t) , ?)p(?|St?1 )d?.
?
We are then given the true label y (t) , and we suffer logloss ? log p(y (t) |x(t) , St?1 ). We
define the cumulative loss of the BMA algorithm after T rounds to be
T
X
LBMA (S) =
? log p(y (t) |x(t) , St?1 ).
t=1
Importantly, note that even though the algorithm we consider is a Bayesian one, our theoretical results do not assume that the data comes from any particular probabilistic model.
In particular, the data may be chosen by an adversary.
We are interested in comparing against the loss of any ?expert? that uses some fixed parameters ? ? Rn . Define `? (t) = ? log p(y (t) |x(t) , ?), and let
T
T
X
X
L? (S) =
`? (t) =
? log p(y (t) |x(t) , ?).
t=1
t=1
Sometimes, we also wish to
R compare against distributions over experts. Given a distribution
Q over ?, define `Q (t) = ? ?Q(?) log p(y (t) |x(t) , ?)d?, and
Z
T
X
LQ (S) =
`Q (t) = Q(?)L? (S)d?.
t=1
?
This is the expected logloss incurred by a procedure that first samples some ? ? Q and
then uses this ? for all its predictions. Here, the expectation is over the random ?, not over
the sequence of examples. Note that the expectation is of the logloss, which is a different
type of averaging than in BMA, which had the expectation and the log in the reverse order.
2.1 A Useful Variational Bound
The following lemma provides a worst case bound of the loss incurred by Bayesian algorithms and will be useful for deriving our main result in the next section. A result very
similar to this (for finite model classes) is given by Freund et al. (1997). For completeness,
we prove the result here in its full generality, though our proof is similar to theirs. As usual,
R
q(?)
.
define KL(q||p) = ? q(?) log p(?)
Lemma 2.1: Let Q be any distribution over ?. Then for all sequences S
LBMA (S) ? LQ (S) + KL(Q||p0 ).
Proof: Let Y = {y (1) , . . . , y (T ) } and X = {x(1) , . . . , x(T ) }. The chain rule of conditional
probabilities implies that LBMA (S) = ? log p(Y |X) and L? (S) = ? log p(Y |X, ?). So
Z
LBMA (S) ? LQ (S) = ? log p(Y |X) + Q(?) log p(Y |X, ?)d?
?
Z
p(Y |X, ?)
=
Q(?) log
d?
p(Y |X)
?
|X,?)p0 (?)
By Bayes rule, we have that pT (?) = p(Yp(Y
. Continuing,
|X)
Z
pT (?)
=
Q(?) log
d?
p0 (?)
Z?
Z
Q(?)
Q(?)
=
Q(?) log
d? ? Q(?) log
d?
p
(?)
p
0
T (?)
?
?
= KL(Q||p0 ) ? KL(Q||pT ).
Together with the fact that KL(Q||pT ) ? 0, this proves the lemma.
2.2 An Upper Bound for Generalized Linear Models
For the theorem that we shortly present, we need one new definition. Let fy (z) =
? log p(y|?T x = z). Thus, fy(t) (?T x(t) ) = `? (t). Note that for linear regression (as
defined in Equation 1), we have that for all y
1
(3)
|fy00 (z)| = 2
?
and for logistic regression (as defined in Equation 2), we have that for y ? {0, 1}
|fy00 (z)| ? 1 .
Theorem 2.2: Suppose fy (z) is continuously differentiable. Let S be a sequence such that
||x(t) || ? 1 and such that for some constant c, |fy00(t) (z)| ? c (for all z). Then for all ?? ,
1
n
T c? 2
? 2
LBMA (S) ? L?? (S) + 2 ||? || + log 1 +
(4)
2?
2
n
The ||?? ||2 /2? 2 term can be interpreted as a penalty term from our prior. The log term is
how fast our loss could grow in comparison to the best ?? . Importantly, this extra loss is
only logarithmic in T in this adversarial setting.
This bound almost identical to those provided by Vovk (2001); Azoury and Warmuth (2001)
and Foster (1991) for the linear regression case (under the square loss); the only difference
is that in their bounds, the last term is multiplied by an upper bound on y (t) . In contrast,
we require no bound on y (t) in the Gaussian linear regression case due to the fact that we
deal with the logloss (also recall |fy00 (z)| = ?12 for all y).
Proof: We use Lemma 2.1 with Q(?) = N (?; ?? , 2 In ) being a normal distribution with
mean ?? and covariance 2 In . Here, 2 is a variational parameter that we later tune to get
the tightest possible bound. Letting H(Q) = n2 log 2?e2 be the entropy of Q, we have
?1
Z
1
1 T
KL(Q||p0 ) =
Q(?) log
exp
?
?
?
d? ? H(Q)
2? 2
(2?)n/2 |? 2 In |1/2
?
Z
1
n
= n log ? + 2
Q(?)?T ?d? ? ? n log
2? ?
2
1
n
= n log ? + 2 ||?? ||2 + n2 ? ? n log .
(5)
2?
2
To prove the result, we also need to relate the error of LQ to that of L?? . By taking a Taylor
expansion of fy (assume y ? S), we have that
(z ? z ? )2
fy (z) = fy (z ? ) + fy0 (z ? )(z ? z ? ) + fy00 (?(z))
,
2
for some appropriate function ?. Thus, if z is a random variable with mean z ? , we have
(z ? z ? )2
Ez [fy (z)] = fy (z ? ) + fy0 (z ? ) ? 0 + Ez fy00 (?(z))
2
? 2
(z ? z )
? fy (z ? ) + cEz
2
c
= fy (z ? ) + Var(z)
2
Consider a single example (x, y). We can apply the argument above with z ? = ?? T x, and
z = ?T x, where ? ? Q. Note that E[z] = z ? since Q has mean ?? . Also, Var(?T x) =
xT (2 In )x = ||x||2 2 ? 2 (because we previously assumed that ||x|| ? 1). Thus, we have
c2
E??Q [fy (?T x)] ? fy (?? T x) +
2
Since `Q (t) = E??Q [fy(t) (?T x(t) )] and `?? (t) = fy(t) (?? T x(t) ), we can sum both sides
from t = 1 to T to obtain
Tc 2
LQ (S) ? L?? (S) +
2
Putting this together with Lemma 2.1 and Equation 5, we find that
n
Tc 2
1
LBMA (S) ? L?? (S) +
+ n log ? + 2 ||?? ||2 + n2 ? ? n log .
2
2?
2
n? 2
and
simplifying,
Theorem
2.2
follows.
Finally, by choosing 2 = n+T
c? 2
2.3 A Lower Bound for Gaussian Linear Regression
The following lower bound shows that, for linear regression, no other prediction scheme is
better than Bayes in the worst case (when our penalty term is ||?? ||2 ). Here, we compare to
an arbitrary predictive distribution q(y|x(t) , St?1 ) for prediction at time t, which suffers an
instant loss `q (t) = ? log q(y (t) |x(t) , St?1 ). In the theorem, b?c denotes the floor function.
Theorem 2.3: Let L?? (S) be the loss under the Gaussian linear regression model using the
parameter ?? , and let ? 2 = ? 2 = 1. For any set of predictive distributions q(y|x(t) , St?1 ),
there exists an S with ||x(t) || ? 1 such that
T
X
1 ? 2
n
T
? (S) +
`q (t) ? inf
(L
||?
||
)
+
log
1
+
?
?
?
2
2
n
t=1
Proof: (sketch) If n = 1 and if S is such that x(t) = 1, one can show the equality:
1
1
LBMA (S) = inf
(L?? (S) + ||?? ||2 ) + log(1 + T )
??
2
2
(1)
(T )
Let Y = {y , . . . , y } and X = {1, . . . , 1}. By the chain rule of conditional probabilities, LBMA (S) = ? log p(Y |X) (where p is the Gaussian linear regression model), and
PT
q?s loss is t=1 `q (t) = ? log q(Y |X). For any predictive distribution q that differs from
p, there must exist some sequence S such that ? log q(Y |X) is greater than ? log p(Y |X)
(since probabilities are normalized). Such a sequence proves the result for n = 1.
The modification for n dimensions follows: S is broken into bT /nc subsequences where
(t)
in every subsequence only one dimension has xk = 1 (and the other dimensions are set to
0). The result follows due to the additivity of the losses on these subsequences.
3
MAP Estimation
We now present bounds for MAP algorithms for both Gaussian linear regression (i.e., ridge
regression) and logistic regression. These algorithms use the maximum ??t?1 of pt?1 (?) to
form their predictive distribution p(y|x(t) , ??t?1 ) at time t, as opposed to BMA?s predictive
distribution of p(y|x(t) , St?1 ). As expected these bounds are weaker than BMA, though
perhaps not unreasonably so.
3.1 The Square Loss and Ridge Regression
Before we provide the MAP bound, let us first present the form of the posteriors and the
Pt
T
predictions for Gaussian linear regression. Define At = ?12 In + ?12 i=1 x(i) x(i) , and
Pt
bt = i=1 x(i) y (i) . We now have that
?t ,
pt (?) = p(?|St ) = N ?; ??t , ?
(6)
?1
?
where ??t = A?1
t bt , and ?t = At . Also, the predictions at time t + 1 are given by
p(y (t+1) |x(t+1) , St ) = N y (t+1) ; y?t+1 , s2t+1
(7)
T
? t x(t+1) + ? 2 . In contrast, the prediction of a
where y?t+1 = ??tT x(t+1) , s2t+1 = x(t+1) ?
?
fixed expert using parameter ? would be
p(y (t) |x(t) , ?? ) = N y (t) ; yt? , ? 2 ,
(8)
where yt? = ?? T x(t) .
Now the BMA loss is:
LBMA (S) =
T
q
X
1 (t) ?T (t) 2
2?s2t
(y
?
?
x
)
+
log
t?1
2
2s
t
t=1
(9)
Importantly, note how Bayes is adaptively weighting the squared term with the inverse
variances 1/st (which depend on the current observation x(t) ). The logloss of using a fixed
expert ?? is just:
L?? (S) =
T
X
?
1
(y (t) ? ?? T x(t) )2 + log 2?? 2
2
2?
t=1
(10)
The MAP procedure (referred to as ridge regression) uses p(y|x(t) , ??t?1 ) which has a fixed
variance. Hence, the MAP loss is essentially the square loss and we define it as such:
T
T
X
X
T
e ?? (S) = 1
e MAP (S) = 1
(y (t) ? ??t?1
x(t) )2 , L
(y (t) ? ?? T x(t) )2 ,
L
2 t=1
2 t=1
(11)
where ??t is the MAP estimate (see Equation 6).
Corollary 3.1: Let ? 2 = ? 2 + ? 2 . For all S such that ||x(t) || ? 1 and for all ?? , we have
2
2
2
2
e MAP (S) ? ? L
e ?? (S) + ? ||?? ||2 + ? n log 1 + T ?
L
?2
2? 2
2
?2 n
Proof: Using Equations (9,10) and Theorem 2.2, we have
T
X
1 (t) ?T (t) 2
? ?t?1 x )
2 (y
2s
t
t=1
?
T
X
1
1
(y (t) ? ?? T x(t) )2 + 2 ||?? ||2
2
2?
2?
t=1
?
X
T
T c? 2
2?? 2
n
+ log 1 +
+
log p
2
n
2?s2t
t=1
Equations (6, 7) imply that ? 2 ? s2t ? ? 2 + ? 2 . Using this, the result follows by noting
that the last term is negative and by multiplying both sides of the equation by ? 2 + ? 2 .
We might have hoped that MAP were more competitive in that the leading coefficient,
e ?? (S) term in the bound, be 1 (similar to Theorem 2.2) rather than ? 22 .
in front of the L
?
Crudely, the reason that MAP is not as effective as BMA is that MAP does not take into
account the uncertainty in its predictions?thus the squared terms cannot be reweighted to
take variance into account (compare Equations 9 and 11).
Some previous (non-Bayesian) algorithms did in fact have bounds with this coefficient
being unity. Vovk (2001) provides such an algorithm, though this algorithm differs from
MAP in that its predictions at time t are a nonlinear function of x(t) (it uses At instead
of At?1 at time t). Foster (1991) provides a bound with this coefficient being 1 with
more restrictive assumptions. Azoury and Warmuth (2001) also provide a bound with a
coefficient of 1 by using a MAP procedure with ?clipping.? (Their algorithm thresholds the
T
prediction y?t = ??t?1
x(t) if it is larger than some upper bound. Note that we do not assume
any upper bound on y (t) .)
As the following lower bound shows, it is not possible for the MAP linear regression algoe ?? (S), with a reasonable regret bound. A similar lower
rithm to have a coefficient of 1 for L
bound is in Vovk (2001), which doesn?t apply to our setting where we have the additional
constraint ||x(t) || ? 1.
Theorem 3.2: Let ? 2 = ? 2 + ? 2 . There exists a sequence S with ||x(t) || ? 1 such that
e MAP (S) ? inf (L
e ?? (S) + 1 ||?? ||2 ) + ?(T )
L
??
2
Proof: (sketch)
Let S be a length T + 1 sequence, with n = 1, where for the first T steps,
?
x(t) = 1/ T and y (t) = 1, and at T + 1, x(T +1) = 1 and y (T +1) = 0. Here, one can show
e ?? (S) + 1 ||?? ||2 ) = T /4 and L
e MAP (S) ? 3T /8, and the result follows.
that inf ?? (L
2
3.2 Logistic Regression
MAP estimation is often used for regularized logistic regression, since it requires only
solving a convex program (while BMA has to deal with a high dimensional integral over
? that is intractable to compute exactly). Letting ??t?1 be the maximum of the posterior
PT
pt?1 (?), define LMAP (S) = t=1 ? log p(y (t) |x(t) , ??t?1 ). As with the square loss case,
the bound we present is multiplicatively worse (by a factor of 4).
Theorem 3.3: In the logistic regression model with ? ? 0.5, we have that for all sequences
S such that ||x(t) || ? 1 and y (t) ? {0, 1} and for all ??
T ?2
2
LMAP (S) ? 4L?? (S) + 2 ||?? ||2 + 2n log 1 +
?
n
Proof: (sketch) Assume n = 1 (the general case is analogous). The proof consists of
showing that `??t?1 (t) = ? log p(y (t) |x(t) , ??t?1 ) ? 4`BMA (t). Without loss of generality,
assume y (t) = 1 and xR(t) ? 0, and for convenience, we just write x instead of x(t) . Now
the BMA prediction is ? p(1|?, x)pt?1 (?)d?, and `BMA (t) is the negative log of this. Note
? = ? gives probability 1 for y (t) = 1 (and this setting of ? minimizes the loss at time t).
Since we do not have a closed form solution of the posterior pt?1 , let us work with another
distribution q(?) in lieu of pt?1 (?) that satisfies certain properties. Define pq =
R
p(1|?,
x)q(?)d?, which can be viewed as the prediction using q rather than the posterior.
?
We choose q to be the rectification of the Gaussian N (?; ??t?1 , ? 2 In ), such that there is
positive probability only for ? ? ??t?1 (and the distribution is renormalized). With this
choice, we first show that the loss of q, ? log pq , is less than or equal to `BMA (t). Then we
complete the proof by showing that `??t?1 (t) ? ?4 log pq , since ? log pq ? `BMA (t).
Consider the q which maximizes pq subject to the following constraints: let q(?) have its
maximum at ??t?1 ; let q(?) = 0 if ? < ??t?1 (intuitively, mass to the left of ??t?1 is just
making the pq smaller); and impose the constraint that ?(log q(?))00 ? 1/? 2 . We now
argue that for such a q, ? log pq ? `BMA (t). First note that due to the Gaussian prior p0 , it
is straightforward to show that ?(log pt?1 )00 (?) ? ?12 (the prior imposes some minimum
curvature). Now if this posterior pt?1 were rectified (with support only for ? ? ??t?1 )
and renormalized, then such a modified distribution clearly satisfies the aforementioned
constraints, and it has loss less than the loss of pt?1 itself (since the rectification only
increases the prediction). Hence, the maximizer, q, of pq subject to the constraints has loss
less than that of pt?1 , i.e. ? log pq ? `BMA (t).
We now show that such a maximal q is the (renormalized) rectification of the Gaussian
N (?; ??t?1 , ? 2 In ), such that there is positive probability only for ? > ??t?1 . Assume some
other q2 satisfied these constraints and maximized pq . It cannot be that q2 (??t?1 ) < q(??t?1 ),
else one can show q2 would not be normalized (since with q2 (??t?1 ) < q(??t?1 ), the curvature constraint imposes that this q2 cannot cross q). It also cannot be that q2 (??t?1 ) >
q(??t?1 ). To see this, note that normalization and curvature imply that q2 must cross pt only
once. Now a sufficiently slight perturbation of this crossing point to the left, by shifting
more mass from the left to the right side of the crossing point, would not violate the curvature constraint and would result in a new distribution with larger pq , contradicting the
maximality of q2 . Hence, we have that q2 (??t?1 ) = q(??t?1 ). This, along with the curvature
constraint and normalization, imply that the rectified Gaussian, q, is the unique solution.
To complete the proof, we show ` ? (t) = ? log p(1|x, ??t?1 ) ? ?4 log pq . We consider
?t?1
two cases, ??t?1 < 0 and ??t?1 ? 0. We start with the case ??t?1 < 0. Using the boundedness
of the derivative |? log p(1|x, ?)/??| < 1 and that q only has support for ? ? ??t?1 , we have
Z
pq =
exp(log p(1|x, ?))q(?)d?
Z?
?
exp log(p(1|x, ??t?1 ) + ? ? ??t?1 q(?)d? ? 1.6p(1|x, ??t?1 )
?
R
where we have used that ? exp(? ? ??t?1 )q(?)d? < 1.6 (which can be verified numerically
using the definition of q with ? ? 0.5). Now observe that for ??t?1 ? 0, we have the lower
bound ? log p(1|x, ??t?1 ) ? log 2. Hence, ? log pq ? ? log p(1|x, ??t?1 ) ? log 1.6 ?
(? log p(1|x, ??t?1 ))(1 ? log 1.6/ log 2) ? 0.3`??t?1 (t), which shows `??t?1 (t) ? ?4 log pq .
?
Now for
R the case ?t?1 ? 0. Let ? be the sigmoid function, so p(1|x, ?) = ?(?x) and
pq = ? ?(x?)q(?)d?. Since the sigmoid is concave
R for ? >0 and, for this case, q only has
support from positive ?, we have that pq ? ? x ? ?q(?)d? . Using the definition of q, we
then have that pq ? ?(x(??t?1 + ?)) ? ?(??t?1 + ?), where the last inequality follows from
??t?1 + ? > 0 and x ? 1. Using properties of ?, one can show |(log ?)0 (z)| < ? log ?(z)
(for all z). Hence, for all ? ? ??t?1 , |(log ?)0 (?)| < ? log ?(?) ? ? log ?(??t?1 ). Using
this derivative condition along with the previous bound on pq , we have that ? log pq ?
? log ?(??t?1 + ?) ? (? log ?(??t?1 ))(1 ? ?) = `??t?1 (t)(1 ? ?), which shows that
` ? (t) ? ?4 log pq (since ? ? 0.5). This proves the claim when ??t?1 ? 0.
?t?1
Acknowledgments. We thank Dean Foster for numerous helpful discussions. This work
was supported by the Department of the Interior/DARPA under contract NBCHD030010.
References
Azoury, K. S. and Warmuth, M. (2001). Relative loss bounds for on-line density estimation with the
exponential family of distributions. Machine Learning, 43(3).
Cesa-Bianchi, N., Freund, Y., Haussler, D., Helmbold, D., Schapire, R., and Warmuth, M. (1997).
How to use expert advice. J. ACM, 44.
Cesa-Bianchi, N., Helmbold, D., and Panizza, S. (1998). On Bayes methods for on-line boolean
prediction. Algorithmica, 22.
Dawid, A. (1984). Statistical theory: The prequential approach. J. Royal Statistical Society.
Foster, D. P. (1991). Prediction in the worst case. Annals of Statistics, 19.
Freund, Y. and Schapire, R. (1999). Adaptive game playing using multiplicative weights. Games and
Economic Behavior, 29:79?103.
Freund, Y., Schapire, R., Singer, Y., and Warmuth, M. (1997). Using and combining predictors that
specialize. In STOC.
Grunwald, P. (2005). A tutorial introduction to the minimum description length principle.
McCullagh, P. and Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall.
Ng, A. Y. and Jordan, M. (2001). Convergence rates of the voting Gibbs classifier, with application
to Bayesian feature selection. In Proceedings of the 18th Int?l Conference on Machine Learning.
Vovk, V. (2001). Competitive on-line statistics. International Statistical Review, 69.
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1,803 | 2,638 | Synergistic Face Detection and Pose Estimation
with Energy-Based Models
Margarita Osadchy
NEC Labs America
Princeton NJ 08540
[email protected]
Matthew L. Miller
NEC Labs America
Princeton NJ 08540
[email protected]
Yann Le Cun
The Courant Institute
New York University
[email protected]
Abstract
We describe a novel method for real-time, simultaneous multi-view face
detection and facial pose estimation. The method employs a convolutional network to map face images to points on a manifold, parametrized
by pose, and non-face images to points far from that manifold. This
network is trained by optimizing a loss function of three variables: image, pose, and face/non-face label. We test the resulting system, in a
single configuration, on three standard data sets ? one for frontal pose,
one for rotated faces, and one for profiles ? and find that its performance
on each set is comparable to previous multi-view face detectors that can
only handle one form of pose variation. We also show experimentally
that the system?s accuracy on both face detection and pose estimation is
improved by training for the two tasks together.
1 Introduction
The detection of human faces in natural images and videos is a key component in a wide
variety of applications of human-computer interaction, search and indexing, security, and
surveillance. Many real-world applications would profit from multi-view detectors that can
detect faces under a wide range of poses: looking left or right (yaw axis), up or down (pitch
axis), or tilting left or right (roll axis).
In this paper we describe a novel method that not only detects faces independently of their
poses, but simultaneously estimates those poses. The system is highly-reliable, runs at near
real time (5 frames per second on standard hardware), and is robust against variations in
yaw (?90? ), roll (?45? ), and pitch (?60? ).
The method is motivated by the idea that multi-view face detection and pose estimation are
so closely related that they should not be performed separately. The tasks are related in the
sense that they must be robust against the same sorts of variation: skin color, glasses, facial
hair, lighting, scale, expressions, etc. We suspect that, when trained together, each task can
serve as an inductive bias for the other, yielding better generalization or requiring fewer
training examples [2].
To exploit the synergy between these two tasks, we train a convolutional network to map
face images to points on a face manifold, and non-face images to points far away from
that manifold. The manifold is parameterized by facial pose. Conceptually, we can view
the pose parameter as a latent variable that can be inferred through an energy-minimization
process [4]. To train the machine we derive a new type of discriminative loss function that
is tailored to such detection tasks.
Previous Work: Learning-based approaches to face detection abound, including real-time
methods [16], and approaches based on convolutional networks [15, 3]. Most multi-view
systems take a view-based approach, which involves building separate detectors for different views and either applying them in parallel [10, 14, 13, 7] or using a pose estimator to
select a detector [5]. Another approach is to estimate and correct in-plane rotations before
applying a single pose-specific detector [12]. Closer to our approach is that of [8], in which
a number of Support Vector Regressors are trained to approximate smooth functions, each
of which has a maximum for a face at a particular pose. Another machine is trained to convert the resulting values to estimates of poses, and a third is trained to convert the values
into a face/non-face score. The resulting system is very slow.
2 Integrating face detection and pose estimation
To exploit the posited synergy between face detection and pose estimation, we must design
a system that integrates the solutions to the two problems. We hope to obtain better results
on both tasks, so this should not be a mere cascaded system in which the answer to one
problem is used to assist in solving the other. Both answers must be derived from one
underlying analysis of the input, and both tasks must be trained together.
Our approach is to build a trainable system that can map raw images X to points in a
low-dimensional space. In that space, we pre-define a face manifold F (Z) that we parameterize by the pose Z. We train the system to map face images with known poses to the
corresponding points on the manifold. We also train it to map non-face images to points far
away from the manifold. Proximity to the manifold then tells us whether or not an image
is a face, and projection to the manifold yields an estimate of the pose.
Parameterizing the Face Manifold: We will now describe the details of the parameterizations of the face manifold. Let?s start with the simplest case of one pose parameter
Z = ?, representing, say, yaw. If we want to preserve the natural topology and geometry
of the problem, the face manifold under yaw variations in the interval [?90? , 90? ] should
be a half circle (with constant curvature). We embed this half-circle in a three-dimensional
space using three equally-spaced shifted cosines.
? ?
(1)
Fi (?) = cos(? ? ?i ); i = 1, 2, 3; ? = [? , ]
2 2
When we run the network on an image X, it outputs a vector G(X) with three components
that can be decoded analytically into corresponding pose angle:
P3
Gi (X) cos(?i )
? = arctan Pi=1
(2)
3
i=1 Gi (X) sin(?i )
The point on the manifold closest to G(X) is just F (?).
The same idea can be applied to any number of pose parameters. Let us consider the set
of all faces with yaw in [?90, 90] and roll in [?45, 45]. In an abstract way, this set is
isomorphic to a portion of the surface of a sphere. Consequently, we encode the pose with
the product of the cosines of the two angles:
Fij (?, ?) = cos(? ? ?i ) cos(? ? ?j ); i, j = 1, 2, 3;
(3)
For convenience we rescale the roll angles to the range of [?90, 90]. With these parameterizations, the manifold has constant curvature, which ensures that the effect of errors will
be the same regardless of pose. Given nine components of the network?s output Gij (X),
we compute the corresponding pose angles as follows:
P
P
cc = Pij Gij (X) cos(?i ) cos(?j ); cs = Pij Gij (X) cos(?i ) sin(?j )
sc =
ss =
ij Gij (X) sin(?i ) cos(?j );
ij Gij (X) sin(?i ) sin(?j )
(4)
? = 0.5(atan2(cs + sc, cc ? ss) + atan2(sc ? cs, cc + ss))
? = 0.5(atan2(cs + sc, cc ? ss) ? atan2(sc ? cs, cc + ss))
Note that the dimension of the face manifold is much lower than that of the embedding
space. This gives ample space to represent non-faces away from the manifold.
3 Learning Machine
To build a learning machine for the proposed approach we refer to the Minimum Energy
Machine framework described in [4].
Energy Minimization Framework: We can view our system as a scalar-value function
EW (Y, Z, X), where X and Z are as defined above, Y is a binary label (Y = 1 for face,
Y = 0 for a non-face), and W is a parameter vector subject to learning. EW (Y, Z, X)
can be interpreted as an energy function that measures the degree of compatibility between
X, Z, Y . If X is a face with pose Z, then we want: EW (1, Z, X) ? EW (0, Z 0 , X) for
any pose Z 0 , and EW (1, Z 0 , X) ? EW (1, Z, X) for any pose Z 0 6= Z.
Operating the machine consists in clamping X to the observed value (the image), and
finding the values of Z and Y that minimize EW (Y, Z, X):
(Y , Z) = argminY ?{Y }, Z?{Z} EW (Y, Z, X)
(5)
where {Y } = {0, 1} and {Z} = [?90, 90]?[?45, 45] for yaw and roll variables. Although
this inference process can be viewed probabilistically as finding the most likely configuration of Y and Z according to a model that attributes high probabilities to low-energy
configurations (e.g. a Gibbs distribution), we view it as a non probabilistic decision making process. In other words, we make no assumption as to the finiteness of integrals over
{Y }and {Z}that would be necessary for a properly normalized probabilistic model. This
gives us considerable flexibility in the choice of the internal architecture of EW (Y, Z, X).
Our energy function for a face EW (1, Z, X) is defined as the distance between the point
produced by the network GW (X) and the point with pose Z on the manifold F (Z):
EW (1, Z, X) = kGW (X) ? F (Z)k
(6)
The energy function for a non-face EW (0, Z, X) is equal to a constant T that we can
interpret as a threshold (it is independent of Z and X). The complete energy function is:
EW (Y, Z, X) = Y kGW (X) ? F (Z)k + (1 ? Y )T
(7)
The architecture of the machine is depicted in Figure 1. Operating this machine (finding the output label and pose with the smallest energy) comes down to first finding:
Z = argminZ?{Z} ||GW (X) ? F (Z)||, and then comparing this minimum distance,
kGW (X) ? F (Z)k, to the threshold T . If it smaller than T , then X is classified as a face,
otherwise X is classified as a non-face. This decision is implemented in the architecture as
a switch, that depends upon the binary variable Y .
Convolutional Network: We employ a Convolutional Network as the basic architecture for
the GW (X) image-to-face-space mapping function. Convolutional networks [6] are ?endto-end? trainable system that can operate on raw pixel images and learn low-level features
and high-level representation in an integrated fashion. Convolutional nets are advantageous
because they easily learn the types of shift-invariant local features that are relevant to image
recognition; and more importantly, they can be replicated over large images (swept over
every location) at a fraction of the cost of replicating more traditional classifiers [6]. This
is a considerable advantage for building real-time systems.
We employ a network architecture similar to LeNet5 [6]. The difference is in the number
of maps. In our architecture we have 8 feature maps in the bottom convolutional and
subsampling layers and 20 maps in the next two layers. The last layer has 9 outputs to
encode two pose parameters.
Training with a Discriminative Loss Function for Detection: We define the loss function
as follows:
1 X
1 X
L1 (W, Z i , X i ) +
L0 (W, X i )
(8)
L(W ) =
|S1 |
|S0 |
i?S1
i?S0
Figure 1: Architecture of the Minimum Energy Machine.
where S1 is the set of training faces, S0 the set of non-faces, L1 (W, Z i , X i ) and L0 (W, X i )
are loss functions for a face sample (with a known pose) and non-face, respectively1 .
The loss L(W ) should be designed so that its minimization for a particular positive training
sample (X i , Z i , 1), will make EW (1, Z i , X i ) < EW (Y, Z, X i ) for Y 6= Y i or Z 6= Z i .
To satisfy this, it is sufficient to make EW (1, Z i , X i ) < EW (0, Z, X i ). For a particular negative training sample (X i , 0), minimizing the loss should make EW (1, Z, X i ) >
EW (0, Z, X i ) = T for any Z. To satisfy this, it is sufficient to make EW (1, Z, X i ) > T .
Let W be the current parameter value, and W 0 be the parameter value after an update
caused by a single sample. To cause the machine to achieve the desired behavior, we need
the parameter update to decrease the difference between the energy of the desired label and
the energy of the undesired label. In our case, since EW (0, Z, X) = T is constant, the
following condition on the update is sufficient to ensure the desired behavior:
Condition 1. for a face example (X, Z, 1), we must have: EW 0 (1, Z, X) < EW (1, Z, X)
For a non-face example (X, 1), we must have: EW 0 (1, Z, X) > EW (1, Z, X)
We choose the following forms for L1 and L0 :
L1 (W, 1, Z, X) = EW (1, Z, X)2 ;
L0 (W, 0, X) = K exp[?E(1, Z, X)]
(9)
where K is a positive constant.
Next we show that minimizing (9) with an incremental gradient-based algorithm will satisfy
condition 1. With gradient-based optimization algorithms, the parameter update formula
?L
. where A is a judiciously chosen symmetric
is of the form: ?W = W 0 ? W = ??A ?W
positive semi-definite matrix, and ? is a small positive constant.
2
(1,Z,X)
For Y = 1 (face): An update step will change the parameter by ?W = ??A ?EW ?W
=
(1,Z,X)
?2?EW (1, Z, X)A ?EW?W
. To first order (for small values of ?), the resulting change
in EW (1, Z, X) is given by:
?EW (1, Z, X)
?W
T
?W = ?2?EW (1, Z, X)
?EW (1, Z, X)
?W
T
A
?EW (1, Z, X)
<0
?W
because EW (1, Z, X) > 0 (it?s a distance), and the quadratic form is positive. Therefore
EW 0 (1, Z, X) < EW (1, Z, X).
1
Although face samples whose pose is unknown can easily be accommodated, we will not discuss
this possibility here.
95
Percentage of yaws correctly estimated
100
95
Percentage of faces detected
100
90
85
80
75
70
65
60
Pose + detection
Detection only
55
50
0
2
4
6
8
10
12
14
16
18
20
90
85
80
75
70
65
60
Pose + detection
Pose only
55
50
False positive rate
0
5
10
15
20
25
30
Yaw-error tolerance (degrees)
Figure 2: Synergy test. Left: ROC curves for the pose-plus-detection and detection-only networks.
Right: frequency with which the pose-plus-detection and pose-only networks correctly estimated the
yaws within various error tolerances.
For Y = 0 (non-face): An update step will change the parameter by ?W =
(1,Z,X)
??A ?K exp[?E(1,Z,X)]
= ?K exp[?EW (1, Z, X)] ?EW?W
. To first order (for small
?W
values of ?), the resulting change in EW (1, Z, X) is given by:
?EW (1, Z, X)
?W
T
?W = ?K exp[?EW (1, Z, X)]
?EW (1, Z, X)
?W
T
A
?EW (1, Z, X)
>0
?W
Therefore EW 0 (1, Z, X) > EW (1, Z, X).
Running the Machine: Our detection system works on grayscale images
? and it applies
the network to each image at a range of scales, stepping by a factor of 2. The network
is replicated over the image at each scale, stepping by 4 pixels in x and y (this step size
is a consequence of having two, 2x2 subsampling layers). At each scale and location,
the network outputs are compared to the closest point on the manifold, and the system
collects a list of all instances closer than our detection threshold. Finally, after examining
all scales, the system identifies groups of overlapping detections in the list and discards all
but the strongest (closest to the manifold) from each group. No attempt is made to combine
detections or apply any voting scheme. We have implemented the system in C. The system
can detect, locate, and estimate the pose of faces that are between 40 and 250 pixels high
in a 640 ? 480 image at roughly 5 frames per second on a 2.4GHz Pentium 4.
4 Experiments and results
Using the above architecture, we built a detector to locate faces and estimate two pose
parameters: yaw from left to right profile, and in-plane rotation from ?45 to 45 degrees.
The machine was trained to be robust against pitch variation.
In this section, we first describe the training regimen for this network, and then give the
results of two sets of experiments. The first set of experiments tests whether training for
the two tasks together improves performance on both. The second set allows comparisons
between our system and other published multi-view detectors.
Training: Our training set consisted of 52, 850, 32x32-pixel faces from natural images
collected at NEC Labs and hand annotated with appropriate facial poses (see [9] for a
description of how the annotation was done). These faces were selected from a much
larger annotated set to yield a roughly uniform distribution of poses from left profile to right
profile, with as much variation in pitch as we could obtain. Our initial negative training data
consisted of 52, 850 image patches chosen randomly from non-face areas of a variety of
images. For our second set of tests, we replaced half of these with image patches obtained
by running the initial version of the detector on our training images and collecting false
detections. Each training image was used 5 times during training, with random variations
95
Percentage of poses correctly estimated
100
95
Percentage of faces detected
100
90
85
80
75
70
65
Frontal
Rotated in plane
Profile
60
55
50
0
0.5
1
1.5
2
2.5
3
3.5
False positives per image
4
4.5
5
90
85
80
75
70
65
60
In-plane rotation
Yaw
55
50
0
5
10
15
20
25
30
Pose-error tolerance (degrees)
Figure 3: Results on standard data sets. Left: ROC curves for our detector on the three data
sets. The x axis is the average number of false positives per image over all three sets, so each point
corresponds to a single detection threshold. Right: frequency with which yaw and roll are estimated
within various error tolerances.
?
?
in scale (from x 2 to x(1 + 2)), in-plane rotation (?45? ), brightness (?20), contrast
(from 0.8 to 1.3).
To train the network, we made 9 passes through this data, though it mostly converged after
about the first 6 passes. Training was performed using LUSH [1], and the total training
time was about 26 hours on a 2Ghz Pentium 4. At the end of training, the network had
converged to an equal error rate of 5% on the training data and 6% on a separate test set of
90,000 images.
Synergy tests: The goal of the synergy test was to verify that both face detection and pose
estimation benefit from learning and running in parallel. To test this claim we built three
networks with almost identical architectures, but trained to perform different tasks. The
first one was trained for simultaneous face detection and pose estimation (combined), the
second was trained for detection only and the third for pose estimation only. The ?detection
only? network had only one output for indicating whether or not its input was a face. The
?pose only? network was identical to the combined network, but trained on faces only (no
negative examples). Figure 2 shows the results of running these networks on our 10,000
test images. In both these graphs, we see that the pose-plus-detection network had better
performance, confirming that training for each task benefits the other.
Standard data sets: There is no standard data set that tests all the poses our system is
designed to detect. There are, however, data sets that have been used to test more restricted
face detectors, each set focusing on a particular variation in pose. By testing a single
detector with all of these sets, we can compare our performance against published systems.
As far as we know, we are the first to publish results for a single detector on all these data
sets. The details of these sets are described below:
? MIT+CMU [14, 11] ? 130 images for testing frontal face detectors. We count 517 faces
in this set, but the standard tests only use a subset of 507 faces, because 10 faces are in
the wrong pose or otherwise not suitable for the test. (Note: about 2% of the faces in the
standard subset are badly-drawn cartoons, which we do not intend our system to detect.
Nevertheless, we include them in the results we report.)
? TILTED [12] ? 50 images of frontal faces with in-plane rotations. 223 faces out of 225
are in the standard subset. (Note: about 20% of the faces in the standard subset are outside
of the ?45? rotation range for which our system is designed. Again, we still include these
in our results.)
? PROFILE [13] ? 208 images of faces in profile. There seems to be some disagreement
about the number of faces in the standard set of annotations: [13] reports using 347 faces
of the 462 that we found, [5] reports using 355, and we found 353 annotations. However,
these discrepencies should not significantly effect the reported results.
We counted a face as being detected if 1) at least one detection lay within a circle centered
on the midpoint between the eyes, with a radius equal to 1.25 times the distance from that
point to the midpoint of the mouth, and 2) that detection came at a scale within a factor of
Figure 4: Some example face detections. Each white box shows the location of a detected face. The
angle of each box indicates the estimated in-plane rotation. The black crosshairs within each box
indicate the estimated yaw.
Data set ?
False positives per image ?
Our detector
Jones & Viola [5] (tilted)
Jones & Viola [5] (profile)
Rowley et al [11]
Schneiderman & Kanade [13]
TILTED
4.42
90%
90%
26.90
97%
95%
x
89%
PROFILE
.47
67%
x
70%
83%
x
96%
x
3.36
83%
86%
93%
MIT+CMU
.50
83%
1.28
88%
x
x
x
x
Table 1: Comparisons of our results with other multi-view detectors. Each column shows the detection rates for a given average number of false positives per image (these rates correspond to those for
which other authors have reported results). Results for real-time detectors are shown in bold. Note
that ours is the only single detector that can be tested on all data sets simultaneously.
two of the correct scale for the face?s size. We counted a detection as a false positive if it
did not lie within this range for any of the faces in the image, including those faces not in
the standard subset.
The left graph in Figure 3 shows ROC curves for our detector on the three data sets. Figure 4
shows a few results on various poses. Table 1 shows our detection rates compared against
other systems for which results were given on these data sets. The table shows that our
results on the TILTED and PROFILE sets are similar to those of the two Jones & Viola
detectors, and even approach those of the Rowley et al and Schneiderman & Kanade nonreal-time detectors. Those detectors, however, are not designed to handle all variations in
pose, and do not yield pose estimates.
The right side of Figure 3 shows our performance at pose estimation. To make this graph,
we fixed the detection threshold at a value that resulted in about 0.5 false positives per
image over all three data sets. We then compared the pose estimates for all detected faces
(including those not in the standard subsets) against our manual pose annotations. Note that
this test is more difficult than typical tests of pose estimation systems, where faces are first
localized by hand. When we hand-localize these faces, 89% of yaws and 100% of in-plane
rotations are correctly estimated to within 15? .
5 Conclusion
The system we have presented here integrates detection and pose estimation by training
a convolutional network to map faces to points on a manifold, parameterized by pose,
and non-faces to points far from the manifold. The network is trained by optimizing a loss
function of three variables ? image, pose, and face/non-face label. When the three variables
match, the energy function is trained to have a small value, when they do not match, it is
trained to have a large value.
This system has several desirable properties:
? The use of a convolutional network makes it fast. At typical webcam resolutions, it can
process 5 frames per second on a 2.4Ghz Pentium 4.
? It is robust to a wide range of poses, including variations in yaw up to ?90? , in-plane
rotation up to ?45? , and pitch up to ?60? . This has been verified with tests on three
standard data sets, each designed to test robustness against a single dimension of pose
variation.
? At the same time that it detects faces, it produces estimates of their pose. On the standard
data sets, the estimates of yaw and in-plane rotation are within 15? of manual estimates
over 80% and 95% of the time, respectively.
We have shown experimentally that our system?s accuracy at both pose estimation and face
detection is increased by training for the two tasks together.
References
[1] L. Bottou and Y. LeCun. The Lush Manual. http://lush.sf.net, 2002.
[2] R. Caruana. Multitask learning. Machine Learning, 28:41?75, 1997.
[3] C. Garcia and M. Delakis. A neural architecture for fast and robust face detection. IEEE-IAPR
Int. Conference on Pattern Recognition, pages 40?43, 2002.
[4] F. J. Huang and Y. LeCun. Loss functions for discriminative training of energy-based graphical
models. Technical report, Courant Institute of Mathematical Science, NYU, June 2004.
[5] M. Jones and P. Viola. Fast multi-view face detection. Technical Report TR2003-96, Mitsubishi
Electric Research Laboratories, 2003.
[6] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to document
recognition. Proceedings of the IEEE, 86(11):2278?2324, November 1998.
[7] S. Z. Li, L. Zhu, Z. Zhang, A. Blake, H. Zhang, and H. Shum. Statistical learning of multi-view
face detection. In Proceedings of the 7th European Conference on Computer Vision-Part IV,
2002.
[8] Y. Li, S. Gong, and H. Liddell. Support vector regression and classification based multi-view
face detection and recognition. In Face and Gesture, 2000.
[9] H. Moon and M. L. Miller. Estimating facial pose from sparse representation. In International
Conference on Image Processing, Singapore, 2004.
[10] A. Pentland, B. Moghaddam, and T. Starner. View-based and modular eigenspaces for face
recognition. In CVPR, 1994.
[11] H. A. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. PAMI, 20:22?38,
1998.
[12] H. A. Rowley, S. Baluja, and T. Kanade. Rotation invariant neural network-based face detection.
In Computer Vision and Pattern Recognition, 1998.
[13] H. Schneidermn and T. Kanade. A statistical method for 3d object detection applied to faces
and cars. In Computer Vision and Pattern Recognition, 2000.
[14] K. Sung and T. Poggio. Example-based learning of view-based human face detection. PAMI,
20:39?51, 1998.
[15] R. Vaillant, C. Monrocq, and Y. LeCun. Original approach for the localisation of objects in
images. IEE Proc on Vision, Image, and Signal Processing, 141(4):245?250, August 1994.
[16] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In
Proceedings IEEE Conf. on Computer Vision and Pattern Recognition, pages 511?518, 2001.
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1,804 | 2,639 | A Probabilistic Model for Online Document
Clustering with Application to Novelty Detection
Jian Zhang?
?School of Computer Science
Cargenie Mellon University
Pittsburgh, PA 15213
[email protected]
Zoubin Ghahramani??
? Gatsby Computational Neuroscience Unit
University College London
London WC1N 3AR, UK
[email protected]
Yiming Yang?
?School of Computer Science
Cargenie Mellon University
Pittsburgh, PA 15213
[email protected]
Abstract
In this paper we propose a probabilistic model for online document clustering. We use non-parametric Dirichlet process prior to model the growing number of clusters, and use a prior of general English language
model as the base distribution to handle the generation of novel clusters.
Furthermore, cluster uncertainty is modeled with a Bayesian Dirichletmultinomial distribution. We use empirical Bayes method to estimate
hyperparameters based on a historical dataset. Our probabilistic model
is applied to the novelty detection task in Topic Detection and Tracking
(TDT) and compared with existing approaches in the literature.
1
Introduction
The task of online document clustering is to group documents into clusters as long as
they arrive in a temporal sequence. Generally speaking, it is difficult for several reasons:
First, it is unsupervised learning and the learning has to be done in an online fashion,
which imposes constraints on both strategy and efficiency. Second, similar to other learning
problems in text, we have to deal with a high-dimensional space with tens of thousands of
features. And finally, the number of clusters can be as large as thousands in newswire data.
The objective of novelty detection is to identify the novel objects from a sequence of data,
where ?novel? is usually defined as dissimilar to previous seen instances. Here we are interested in novelty detection in the text domain, where we want to identify the earliest report
of every new event in a sequence of news stories. Applying online document clustering
to the novelty detection task is straightforward by assigning the first seed of every cluster
as novel and all its remaining ones as non-novel. The most obvious application of novelty
detection is that, by detecting novel events, systems can automatically alert people when
new events happen, for example.
In this paper we apply Dirichlet process prior to model the growing number of clusters, and
propose to use a General English language model as a basis of newly generated clusters.
In particular, the new clusters will be generated according to the prior and a background
General English model, and each document cluster is modeled using a Bayesian Dirichletmultinomial language model. The Bayesian inference can be easily carried out due to
conjugacy, and model hyperparameters are estimated using a historical dataset by the empirical Bayes method. We evaluate our online clustering algorithm (as well as its variants)
on the novelty detection task in TDT, which has been regarded as the hardest task in that
literature [2].
The rest of this paper is organized as follows. We first introduce our probabilistic model
in Section 2, and in Section 3 we give detailed information on how to estimate model
hyperparameters. We describe the experiments in Section 4, and related work in Section 5.
We conclude and discuss future work in Section 6.
2
A Probabilistic Model for Online Document Clustering
In this section we will describe the generative probabilistic model for online document
(x)
(x)
(x)
clustering. We use x = (n1 , n2 , . . . , nV ) to represent a document vector where each
(x)
element nv denotes the term frequency of the v th corresponding word in the document
x, and V is the total size of the vocabulary.
2.1
Dirichlet-Multinomial Model
The multinomial distribution has been one of the most frequently used language models for
modeling documents in information retrieval. It assumes that given the set of parameters
? = (?1 , ?2 , . . . , ?V ), a document x is generated with the following probability:
PV
V
(x)
nv )! Y n(x)
(
p(x|?) = QVv=1 (x)
?v v .
v=1 nv ! v=1
From the formula we can see the so-called naive assumption: words are assumed to be independent of each other. Given a collection of documents generated from the same model,
the parameter ? can be estimated with Maximum Likelihood Estimation (MLE).
In a Bayesian approach we would like to put a Dirichlet prior over the parameter (? ?
Dir(?)) such that the probability
R of generating a document is obtained by integrating over
the parameter space: p(x) = p(?|?)p(x|?)d?. This integration can be easily written
down due to the conjugacy between Dirichlet and multinomial distributions. The key difference between the Bayesian approach and the MLE is that the former uses a distribution
to model the uncertainty of the parameter ?, while the latter gives only a point estimation.
2.2
Online Document Clustering with Dirichlet Process Mixture Model
In our system documents are grouped into clusters in an online fashion. Each cluster is
modeled with a multinomial distribution whose parameter ? follows a Dirichlet prior. First,
a cluster is chosen based on a Dirichlet process prior (can be either a new or existing
cluster), and then a document is drawn from that cluster.
We use Dirichlet Process (DP) to model the prior distribution of ??s, and our hierarchical
model is as follows:
xi |ci ? M ul(.|? (ci ) )
?i
G
iid.
?
?
G
DP (?, G0 )
(1)
where ci is the cluster indicator variable, ? i is the multinomial parameter 1 for each document, and ? (ci ) is the unique ? for the cluster ci . G is a random distribution generated from
the Dirichlet process DP (?, G0 ) [4], which has a precision parameter ? and a base distribution G0 . Here our base distribution G0 is a Dirichlet distribution Dir(??1 , ??2 , . . . , ??V )
PV
with t=1 ?t = 1, which reflects our expected knowledge about G. Intuitively, our G0
distribution can be treated as the prior over general English word frequencies, which has
been used in information retrieval literature [6] to model general English documents.
The exact cluster-document generation process can be described as follows:
1. Let xi be the current document under processing (the ith document in the input
sequence), and C1 , C2 , . . . , Cm are already generated clusters.
2. Draw a cluster ci based on the following Dirichlet process prior [4]:
|C |
Pmj
p(ci = Cj ) =
(j = 1, 2, . . . , m)
? + j=1 |Cj |
p(ci = Cm+1 )
=
?+
?
Pm
j=1
(2)
|Cj |
Pm
where |Cj | stands for the cardinality of cluster j with j=1 |Cj | = i ? 1, and
with certain probability a new cluster Cm+1 will be generated.
3. Draw the document xi from the cluster ci .
2.3
Model Updating
Our models for each cluster need to be updated based on incoming documents. We can
write down the probability that the current document xi is generated by any cluster as
Z
p(xi |Cj ) = p(?(Cj ) |Cj )p(xi |?(Cj ) )d?(Cj ) (j = 1, 2, . . . , m, m + 1)
where p(? (Cj ) |Cj ) is the posterior distribution of parameters of the j th cluster (j =
1, 2, . . . , m) and we use p(? (Cm+1 ) |Cm+1 ) = p(? (Cm+1 ) ) to represent the prior distribution of the parameters of the new cluster for convenience. Although the dimensionality of
? is high (V ? 105 in our case), closed-form solution can be obtained under our Dirichletmultinomial assumption. Once the conditional probabilities p(xi |Cj ) are computed, the
probabilities p(Cj |xi ) can be easily calculated using the Bayes rule:
p(Cj )p(xi |Cj )
p(Cj |xi ) = Pm+1
0
0
j 0 =1 p(Cj )p(xi |Cj )
where the prior probability of each cluster is calculated using equation (2).
Now there are several choices we can consider on how to update the cluster models. The
first choice, which is correct but obviously intractable, is to fork m + 1 children of the
current system where the j th child is updated with document xi assigned to cluster j, while
the final system is a probabilistic combination of those children with the corresponding
probabilities p(Cj |xi ). The second choice is to make a hard decision by assigning the
current document xi to the cluster with the maximum probability:
p(Cj )p(xi |Cj )
ci = arg max p(Cj |xi ) = Pm+1
.
Cj
0
0
j 0 =1 p(Cj )p(xi |Cj )
For ? we use ?v to denote the v th element in the vector, ? i to denote the parameter vector that
generates the ith document, and ? (j) to denote the parameter vector for the j th cluster.
1
The third choice is to use a soft probabilistic updating, which is similar in spirit to the
Assumed Density Filtering (ADF) [7] in the literature. That is, each cluster is updated by
exponentiating the likelihood function with probabilities:
p(Cj |xi )
p(?(Cj ) |xi , Cj ) ?
p(xi |?(Cj ) )
p(?(Cj ) |Cj )
However, we have to specially deal with the new cluster since we cannot afford both timewise and space-wise to generate a new cluster for each incoming document. Instead, we
will update all existing clusters as above, and new cluster will be generated only if c i =
Cm+1 . We will use HD and PD (hard decision and probabilistic decision) to denote the
last two candidates in our experiments.
3
Learning Model Parameters
In the above probabilistic model there are still several hyperparameters not specified,
namely the ? and ? in the base distribution G0 = Dir(??1 , ??2 , . . . , ??V ), and the precision parameter ? in the DP (?, G0 ). Since we can obtain a partially labeled historical
dataset 2 , we now discuss how to estimate those parameters respectively.
We will mainly use the empirical Bayes method [5] to estimate those parameters instead
of taking a full Bayesian approach, since it is easier to compute and generally reliable
when the number of data points is relatively large compared to the number of parameters.
Because the ? i ?s are iid. from the random distribution G, by integrating out the G we get
X
?
1
?i |?1 , ?2 , . . . , ? i?1 ?
G0 +
? j
?+i?1
? + i ? 1 j<i ?
where the distribution is a mixture of continuous and discrete distributions, and the ? ?
denotes the probability measure giving point mass to ?.
Now suppose we have a historical dataset H which contains K labeled clusters H j (j =
1, 2, . . . , K), with the k th cluster Hk = {xk,1 , xk,2 , . . . , xk,mk } having mk documents.
The joint probability of ??s of all documents can be obtained as
p(?1 , ?2 , . . . , ? |H| ) =
|H|
Y
(
i=1
X
?
1
G0 +
? j)
?+i?1
? + i ? 1 j<i ?
where |H| is the total number of documents. By integrating over the unknown parameter
??s we can get
?
?
Z Y
|H|
? p(xi |?i )? p(?1 , ?2 , . . . , ? |H| )d?1 d?2 . . . d? |H|
p(H) =
i=1
=
?
Z
|H|
Y
? p(xi |?i )(
i=1
1
?
G0 +
?+i?1
?+i?1
X
j<i
?
??j )d?i ?
(3)
Empirical Bayes method can be applied to equation (3) to estimate the model parameters
by maximization3 . In the following we discuss how to estimate parameters individually in
detail.
2
Although documents are grouped into clusters in the historical dataset, we cannot make directly
use of those labels due to the fact that clusters in the test dataset are different from those in the
historical dataset.
3
Since only a subset of documents are labeled in the historical dataset H, the maximization is
only taken over the union of the labeled clusters.
Estimating ?t ?s
3.1
Our hyperparameter ? vector contains V number of parameters for the base distribution G 0 ,
which can be treated as the expected distribution of G ? the prior of the cluster parameter
??s.
Although ? contains V ? 105 number of actual parameters in our case, we can still use
the empirical Bayes to do a reliable point estimation since the amount of data we have to
represent general English is large (in our historical dataset there are around 10 6 documents,
around 1.8 ? 108 English words in total) and highly informative about ?. We use the
P
(H)
(H)
(H)
(H)
(x)
smoothed estimation ? ? (1 + n1 , 1 + n2 , . . . , 1 + nV ) where nt = x?H nt
PV
is the total number of times that term t happened in the collection H, and t=1 ?t should
be normalized to 1. Furthermore, the pseudo-count one is added to alleviate the out-ofvocabulary problem.
Estimating ?
3.2
Though ? is just a scalar parameter, it has the effect to control the uncertainty of the prior
knowledge about how clusters are related to the general English model with the parameter
?. We can see that ? controls how far each new cluster can deviate from the general English
model 4 . It can be estimated as follows:
K
K Z
Y
Y
?? = arg max
p(Hk |?) = arg max
p(Hk |?(k) )p(?(k) |?)d? (k)
(4)
?
?
k=1
k=1
?? can be numerically computed by solving the following equation:
K?(?) ? K
V
X
?(??v )?v +
v=1
K X
V
X
k)
?(??v + n(H
)?v ?
v
k=1 v=1
where the digamma function ?(x) is defined as ?(x) ?
K
X
?(? +
k=1
d
dx
V
X
k)
n(H
)=0
v
v=1
ln ?(x).
Alternatively we can choose ? by evaluating over the historical dataset. This is applicable
(though computationally expensive) since it is only a scalar parameter and we can precompute its possible range based on equation (4).
Estimating ?
3.3
The precision parameter ? of the DP is also very important for the model, which controls
how far the random distribution G can deviate from the baseline model G0 . In our case, it is
also the prior belief about how quickly new clusters will be generated in the sequence. Similarly we can use equation (3) to estimate ?, since items related to ? can be factored out as
Q|H| ?yi
L
i=1 ?+i?1 . Suppose we have a labeled subset H = {(x1 , y1 ), (x2 , y2 ), . . . , (xM , yM )}
of training data, where yi is 1 if xi is a novel document or 0 otherwise. Here we describe
two possible choices:
1. The simplest way is to assume that ? is a fixed constant during the process, and it
?y t
L
? = arg max? Q
can be computed as ?
i?H L ?+i?1 , here H denotes the subset of
indices of labeled documents in the whole sequence.
2. The assumption that ? is fixed maybe restrictive in reality, especially considering
the fact that it reflects the generation rate of new clusters. More generally, we
4
The mean and variance of a Dirichlet distribution (?1 , ?2 , . . . , ?V ) ? Dir(??1 , ??2 , . . . , ??V )
(1??v )
are: E[?v ] = ?v and Var[?v ] = ?v(?+1)
.
can assume that ? is some function of variable i. In particular, we assume ? =
a/i + b + ci where a, b and c are non-negative numbers. This formulation is a
generalization of the above case, where the i?1 term allows a much faster decrease
at the beginning, and c is the asymptotic rate of events happening as i ? ?.
Again the parameters a, b and c are estimated by MLE over the training dataset:
yi
Q
a
?, ?b, c? = arg maxa,b,c>0 i?H L (a/i+b+ci)
a/i+b+ci+i .
4
Experiments
We apply the above online clustering model to the novelty detection task in Topic Detection
and Tracking (TDT). TDT has been a research community since its 1997 pilot study, which
is a research initiative that aims at techniques to automatically process news documents in
terms of events. There are several tasks defined in TDT, and among them Novelty Detection
(a.k.a. First Story Detection or New Event Detection) has been regarded as the hardest task
in this area [2]. The objective of the novelty detection task is to detect the earliest report
for each event as soon as that report arrives in the temporal sequence of news stories.
4.1
Dataset
We use the TDT2 corpus as our historical dataset for estimating parameters, and use the
TDT3 corpus to evaluate our model 5 . Notice that we have a subset of documents in the
historical dataset (TDT2) for which events labels are given. The TDT2 corpus used for
novelty detection task consists of 62,962 documents, among them 8,401 documents are
labeled in 96 clusters. Stopwords are removed and words are stemmed, and after that there
are on average 180 words per document. The total number of features (unique words) is
around 100,000.
4.2
Evaluation Measure
In our experiments we use the standard TDT evaluation measure [1] to evaluate our results.
The performance is characterized in terms of the probability of two types of errors: miss
and false alarm (PM iss and PF A ). These two error probabilities are then combined into a
single detection cost, Cdet , by assigning costs to miss and false alarm errors:
Cdet = CM iss ? PM iss ? Ptarget + CF A ? PF A ? Pnon?target
where
1. CM iss and CF A are the costs of a miss and a false alarm, respectively,
2. PM iss and PF A are the conditional probabilities of a miss and a false alarm, respectively, and
3. Ptarget and Pnon?target is the priori target probabilities (Ptarget = 1 ?
Pnon?target ).
It is the following normalized cost that is actually used in evaluating various TDT systems:
(Cdet )norm =
Cdet
min(CM iss ? Ptarget , CF A ? Pnon?target )
where the denominator is the minimum of two trivial systems. Besides, two types of evaluations are used in TDT, namely macro-averaged (topic-weighted) and micro-averaged
5
Strictly speaking we only used the subsets of TDT2 and TDT3 that is designated for the novelty
detection task.
(story-weighted) evaluations. In macro-averaged evaluation, the cost is computed for every
event, and then the average is taken. In micro-averaged evaluation the cost is averaged over
all documents? decisions generated by the system, thus large event will have bigger impact
on the overall performance. Note that macro-averaged evaluation is used as the primary
evaluation measure in TDT. In addition to the binary decision ?novel? or ?non-novel?, each
system is required to generated a confidence score for each test document. The higher the
score is, the more likely the document is novel. Here we mainly use the minimum cost to
evaluate systems by varying the threshold, which is independent of the threshold setting.
4.3
Methods
One simple but effective method is the ?GAC-INCR? clustering method [9] with cosine
similarity metric and TFIDF term weighting, which has remained to be the top performing
system in TDT 2002 & 2003 official evaluations. For this method the novelty confidence
score we used is one minus the similarity score between the current cluster xi and its nearest
neighbor cluster: s(xi ) = 1.0 ? maxj<i sim(ci , cj ), where ci and cj are the clusters that
xi and xj are assigned to, respectively, and the similarity is taken to be the cosine similarity
between two cluster vectors, where the ltc TFIDF term weighting scheme is used to scale
each dimension of the vector. Our second method is to train a logistic regression model
which combines multiple features generated by the GAC-INCR method. Those features
not only include the similarity score used by the first method, but also include the size of
its nearest cluster, the time difference between the current cluster and the nearest cluster,
etc. We call this method ?Logistic Regression?, where we use the posterior probability
p(novelty|xi ) as the confidence score. Finally, for our online clustering algorithm we
choose the quantity s(xi ) = log p(C0 |xi ) as the output confidence score.
4.4
Experimental Results
Our results for three methods are listed in Table 1, where both macro-averaged and microaveraged minimum normalized costs are reported 6 . The GAC-INCR method performs
very well, so does the logistic regression method. For our DP results, we observed that
using the optimized ?? will get results (not listed in the table) that are around 10% worse
than using the ? obtained through validation, which might be due to the flatness of the
optimal function value as well as the sample bias of the clusters in the historical dataset 7 .
Another observation is that the probabilistic decision does not actually improve the hard
decision performance, especially for the ?var option. Generally speaking, our DP methods
are comparable to the other two methods, especially in terms of topic-weighted measure.
Table 1: Results for novelty detection on TDT3 corpus
Method
GAC-INCR
Logistic Regression
DP with ?f ix , HD
DP with ?var , HD
DP with ?f ix , PD
DP with ?var , PD
Topic-weighted Cost
COST (Miss, FA)
0.6945 (0.5614, 0.0272)
0.7027 (0.5732, 0.0264)
0.7054 (0.4737, 0.0473)
0.6901 (0.5789, 0.0227)
0.7054 (0.4737, 0.0473)
0.9025 (0.8772, 0.0052)
Story-weighted Cost
COST (Miss, FA)
0.7090 (0.5614, 0.0301)
0.6911 (0.5732, 0.0241)
0.7744 (0.5965, 0.0363)
0.7541 (0.5789, 0.0358)
0.7744 (0.5965, 0.0363)
0.9034 (0.8772, 0.0053)
6
In TDT official evaluation there is also the DET curve, which is similar in spirit to the ROC curve
that can reflects how the performance changes as the threshold varies. We will report those results in
a longer version of this paper.
7
It is known that the cluster labeling process of LDC is biased toward topics that will be covered
in multiple languages instead of one single language.
5
Related Work
Zaragoza et al. [11] applied a Bayesian Dirichlet-multinomial model to the ad hoc information retrieval task and showed that it is comparable to other smoothed language models.
Blei et al. [3] used Chinese Restaurant Processes to model topic hierachies for a collection of documents. West et al. [8] discussed the sampling techniques for base distribution
parameters in the Dirichlet process mixture model.
6
Conclusions and Future Work
In this paper we used a hierarchical probabilistic model for online document clustering.
We modeled the generation of new clusters with a Dirichlet process mixture model, where
the base distribution can be treated as the prior of general English model and the precision
parameter is closely related to the generation rate of new clusters. Model parameters are
estimated with empirical Bayes and validation over the historical dataset. Our model is
evaluated on the TDT novelty detection task, and results show that our method is promising.
In future work we would like to investigate other ways of estimating parameters and use
sampling methods to revisit previous cluster assignments. We would also like to apply our
model to the retrospective detection task in TDT where systems do not need to make decisions online. Though its simplicity, the unigram multinomial model has its well-known
limitation, which is the naive assumption about word independence. We also plan to explore richer but still tractable language models in this framework. Meanwhile, we would
like to combine this model with the topic-conditioned framework [10] as well as incorporate hierarchical mixture model so that novelty detection will be conditioned on some topic,
which will be modeled by either supervised or semi-supervised learning techniques.
References
[1] The 2002 topic detection & tracking task definition
http://www.nist.gov/speech/tests/tdt/tdt2002/evalplan.htm, 2002.
and
evaluation
plan.
[2] Allan, J., Lavrenko, V. & Jin, H. First story detection in tdt is hard. In Proc. of CIKM 2000.
[3] Blei, D., Griffiths, T., Jordan, M. & Tenenbaum, J. Hierarchical topic models and the nested
chinese restaurant process. Advances in Neural Information Processing Systems, 15, 2003.
[4] Ferguson, T. A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1:209?
230, 1973.
[5] Gelman A., Carlin, J., Stern, H. & Rubin, D. Bayesian Data Analysis (2nd ed.). CHAPMAN &
HALL/CRC, 2003.
[6] Miller, D., Leek, T. & Schwartz, R. Bbn at trec 7: Using hidden markov models for information
retrieval. In TREC-7, 1999.
[7] Minka, T. A family of algorithms for approximate Bayesian inference. Ph.D. thesis, MIT, 2001.
[8] West, M., Mueller, P. & Escobar, M.D. Hierarchical priors and mixture models, with application
in regression and density estimation. In Aspects of Uncertainty: A tribute to D. V. Lindley, A.F.M.
Smith and P. Freeman, (eds.), Wiley, New York.
[9] Yang, Y., Pierce, T. & Carbonell, J. A Study on Retrospective and On-line Event Detection. In
Proc. of SIGIR 1998.
[10] Yang, Y., Zhang, J., Carnobell, J. & Jin, C. Topic-conditioned novelty detection. In Proc. of 8th
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2002.
[11] Zaragoza, H., Hiemstra, D., Tipping, D. & Robertson, S. Bayesian extension to the language
model for ad hoc information retrieval. In Proc. SIGIR 2003.
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1,805 | 264 | 298
Okamoto, Kawato, Ioui aod Miyake
Model Based Image Compression and
Adaptive Data Representation by
Interacting Filter Banks
SeiMiyake
Toshiaki Okamoto, Mitsuo Kawato, Toshio Ioui
ATR Auditory and
Visual Perception Research Laboratories
Sanpeidani, Inuidani. Seika-cho. Soraku-gun
Kyoto 619-02. Japan
NHK Science and
Technical Research Laboratories
1-10-11. Kinuta. Setagaya
Tokyo 157 ? Japan
Abstract
To achieve high-rate image data compression
while maintainig a high quality reconstructed
image, a good image model and an efficient
way to represent the specific data of each
image must be introduced. Based on the
physiological knowledge of multi - channel
characteristics and inhibitory interactions
between them in the human visual system,
a mathematically coherent parallel architecture
for image data compression which utilizes the
Markov random field Image model and
interactions between a vast number of filter
banks, is proposed.
1. Introduction
Data compression has been one of the most important
and active areas in information theory and computer science.
The goal of image coding is reducing the number of bits
in data representation as much as possible, and reconstructing
a faithful duplicate of the original image. In order to achieve
a high compression ratio while maintaining the high quality
Model Based Image Compression
of the reconstructed image, a good image model and an
efficient way to represent image data must be found. Based
on physiological knowledge of the human visual system, we
propose a mathematically coherent parallel architecture for
the image data compression, which utilizes a stochastic iInage
model and interactions between a vast number of filter banks.
2. Model based image compression and
dynamic spatial filtering
The process of reconstructing an original image from
compressed data is an ill-posed problem, since an infinite
number of original images lead to the same compressed data
and solutions to the inverse problem can not uniquely be
determined. The coupled Markov random field (MRF) image
model proposed by Geman and Geman is introduced to resolve
this ill-posedness. The mean field approximation of the MRF
is equivalent to a recurrent type neural network with the
Ljapunov function (see Koch. Marroquin and Yuille as a
special case where the form of the Ljapunov function is
predetermined). Correspondingly, a similar deterministic
framework of image compression in which the MRF is
replaced by the recurrent network, can be developed.
Further, even if a good MRF model is introduced for
a family of images, the data for each image must be known
in order to reconstruct it. In previous studies of image data
compression, representation of image data is fixed in each
schema. On the other hand, in this paper, an adaptive data
representation is proposed, tuned to each image by competion
and cooperation of a vast number of filter banks.
Fig. 1 shows a block diagram of the proposed
communication system. Procedures at the encoder side are
(1) partial partition and segmentation of the image by the
299
300
Okamoto, Kawato, Inui and Miyake
line process of the MRF which represents the image
discontinuity, (2) learning of energy parameters which uses
the line process to define the MRF model in each segmented
area of the image, (3) adaptive data representation of images
by cooperation and competition of a vast number of filter
banks. (4) Information about energy value parameters, the
types of selected filter and their outputs, and the line
processes is transmitted, through communication channel.
(5) Image reconstruction is carried out at the decorder site
by stochastic relaxation based on the aquired MRF model.
output from the selected filters, and the line process. These
procedures are explained in detail below.
1. The set of line processes represents discontinuities in
the 3-dimensional world such as occluding contours or
boundaries between different objects. It is not necessarily
closed, but it can posess a strong tendency to do so if the
MRF model is appropriately chosen. Based on this property,
the image can be partially segmented into several regions.
2. If we adopt the MRF image model, the occurrence
probability
n(w) of each configuration w is Gibbsian:
n(w)= exp{-U(w)/T}
Z
Furthermore, the energy U(w) can be expressed as a summation
of local potential Vc(w) , which depends on the configuration
only in the clique C.
U(w)= L Vc(w)
CeSc
Determination of the local energy Vc is equivalent to defining
a specific MRF model of the image. Determination of the
local energy is equivalent to assigning a real value VEo to
Model Based Image Compression
every possible configuration within the clique C. These energy
parameters are estimated so that the Kullback divergence
G between the real image distribution P and the model image
distribution P' is minimized:
P(w)
G(V)=~P(w)log{p'(w. Vd}
The following learning equation can be derived In
approximately the same way as the learning rule of the
Boltzmann machine (Ackley, Hinton, Sejnowski. 1985).
Here L(C)
IS the characteristic function of the specific
configuration 'i of the clique C, that is. MC)=l if{Ys;sEC}='i
otherwise, I;(C)=O.
The first term on the right side is the
average number of configurations in the real image. The
second term on the right side is the average number of each
configuration
generated in the MRF with the energy Vc
when part of the image configuration is fixed to the given
'I
image.
3. This
procedure
is
based
on
the
multi - channel
characteristics of the human visual system. inhibitory
interaction between X-cell and Y -cell systems. and interactions
between columns with different orientation selectivity. etc.
We prepare a vast number of filters centered at each site
s in a variety of sizes. shapes and orientations. In particular)
we use two-dimensional Gaussian filters Gs(w) to represent
the DC components (i.e. average luminance) of the gray
level, and use the first-order derivative of the Gaussian filters
VGs(w) to represent the gradient of the gray levels. The filters
whose receptive fields significantly intersect with the line
process are inhibited. Inhibitory interactions between filters
of similar, shape and orientation at nearby sites are introduced
301
302
Okamoto, Kawato, Inui and Miyake
as well as self excitation to find the N-maximum outputs
of 'YGs, and to find the N-minimum outputs of the Laplacian
Gaussian ~Gs. Of course, 2N must be less than the number
of sites to attain data compression.
4. We transmit the local potential energy, the site of the
line process, and the outputs from the N - maximum, and
the outputs from the N Gaussain filters which correspond
to the N - minimum Laplacian Gaussain filters.
5. Image reconstruction is carried out by the usual
stochastic relaxation, that is, energy minimization with
simulated annealing. However, because we have data
constraints as output from the 2N selected filters, we need
to minimize the sum of the MRF model energy and the data
constraint energy:
If we do not further
compress the filter outputs, the
regularization parameter is increased to infinity during
constrained stochastic relaxation.
3. Experimental results
First, we ascertained that the proposed energy learning
rule works well for various images. Here, we report only
one example from the data compression experiments. We used
the shown in Fig. 2a to examine the potential of our scheme.
The image data consists of 256 pixels, each of which has
8 bit gray levels. We used the dynamic spatial sampling
of filter banks. Fig 2a also shows selected sample points
in the image as black dots, as well as a few examples of
selected filter shapes. Note that not only the density of
the sampling points, but also the selected filter shapes are
\
Model Based Image Compression
appropriate local characteristics of the image. Fig. 2b shows
the reconstructed image after 20 iterations of the relaxation
computation. The signal to noise ratio of the reconstructed
images was a bou t 38dB.
References
D. H., Ackley, G. E. Hinton, and T. J. Sejnowski,: "A Learning
algorithm for Boltzmann Machines.", Cognitive Science, vol.
9, pp.147 - 169, (1985).
S. Geman and D. Geman,: "Stochastic relaxation, Gibbs
distribution, and the Basian restoration of images", IEEE Trans.
vol. PAMI - 6, pp.721 - 741, (1984).
S. Hongo, M. Kawato, T. Inui, and S. Miyake,; "Contour
extraction of images on parallel computer", Proc. of 1th IJCNN,
(1989).
T. Inui, M. Kawato and R. Suzuki: "The mechanism of mental
scannIng In foveal vision", BioI. Cybern. vol. 30, pp.147 - 155,
(1978).
C. Koch, J. Marroquin, and A. Yuille: "Analog 'neural'
networks in early vision", Proc. Natl. Acad. Sci. USA, vol. 83,
pp.4263 - 4267, (1986).
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305
PART IV:
OPTIMIZATION AND CONTROL
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1,806 | 2,640 | A Three Tiered Approach for Articulated Object
Action Modeling and Recognition
Le Lu
Gregory D. Hager
Department of Computer Science
Johns Hopkins University
Baltimore, MD 21218
lelu/[email protected]
Laurent Younes
Center of Imaging Science
Johns Hopkins University
Baltimore, MD 21218
[email protected]
Abstract
Visual action recognition is an important problem in computer vision.
In this paper, we propose a new method to probabilistically model and
recognize actions of articulated objects, such as hand or body gestures,
in image sequences. Our method consists of three levels of representation. At the low level, we first extract a feature vector invariant to scale
and in-plane rotation by using the Fourier transform of a circular spatial
histogram. Then, spectral partitioning [20] is utilized to obtain an initial
clustering; this clustering is then refined using a temporal smoothness
constraint. Gaussian mixture model (GMM) based clustering and density
estimation in the subspace of linear discriminant analysis (LDA) are then
applied to thousands of image feature vectors to obtain an intermediate
level representation. Finally, at the high level we build a temporal multiresolution histogram model for each action by aggregating the clustering
weights of sampled images belonging to that action. We discuss how this
high level representation can be extended to achieve temporal scaling invariance and to include Bi-gram or Multi-gram transition information.
Both image clustering and action recognition/segmentation results are
given to show the validity of our three tiered representation.
1 Introduction
Articulated object action modeling, tracking and recognition has been an important research issue in computer vision community for decades. Past approaches [3, 13, 4, 6, 23, 2]
have used many different kinds of direct image observations, including color, edges, contour or moments [14], to fit a hand or body?s shape model and motion parameters.
In this paper, we propose to learn a small set of object appearance descriptors, and then
to build an aggregated temporal representation of clustered object descriptors over time.
There are several obvious reasons to base gesture or motion recognition on a time sequence
of observations. First, most hand or body postures are ambiguous. For example, in American Sign Language, ?D? and ?G?, ?H? and ?U? have indistinguishable appearance from some
viewpoints. Furthermore, these gestures are difficult to track from frame to frame due to
motion blur, lack of features, and complex self-occlusions. By modeling hand/body gesture
as a sequential learning problem, appropriate discriminative information can be retrieved
and more action categories can be handled.
In related work, Darrell and Pentland [7] describe dynamic time warping (DTW) to align
and recognize a space-time gesture against a stored library. To build the library, key views
are selected from incoming an video by choosing views that have low correlation with all
current views. This approach is empirical and does not guarantee any sort of global consistency of the chosen views. As a result, recognition may be unstable. In comparision, our
method describes image appearances uniformly and clusters them globally from a training
set containing different gestures.
For static hand posture recognition, Tomasi et al. [24] apply vector quantization methods
to cluster images of different postures and different viewpoints. This is a feature-based
approach, with thousands of features extracted for each image. However, clustering in a
high dimensional space is very difficult and can be unstable. We argue that fewer, more
global features are adequate for the purposes of gesture recognition. Furthermore, the
circular histogram representation has adjustable spatial resolution to accomodate differing
appearance complexities, and it is translation, rotation, and scale invariant.
In other work, [27, 9] recognize human actions at a distance by computing motion information between images and relying on temporal correlation on motion vectors across
sequences. Our work also makes use of motion information, but does not rely exclusively
on it. Rather, we combine appearance and motion cues to increase sensitivity beyond what
either can provide alone. Since our method is based on the temporal aggregation of image
clusters as a histogram to recognize an action, it can also be considered to be a temporal texton-like method [17, 16]. One advantage of the aggregated histogram model in a
time-series is that it is straightforward to accommodate temporal scaling by using a sliding
window. In addition, higher order models corresponding to bigrams or trigrams of simpler
?gestemes? can also be naturally employed to extend the descriptive power of the method.
In summary, there are four principal contributions in this paper. First, we propose a new
scale/rotation-invariant hand image descriptor which is stable, compact and representative.
Second, we introduce a methods for sequential smoothing of clustering results. Third,
we show LDA/GMM with spectral partitioning initialization is an effective way to learn
well-formed probability densities for clusters. Finally, we recognize image sequences as
actions efficiently based on a flexible histogram model. We also discuss improvement to
the method by incorporating motion information.
2 A Three Tiered Approach
We propose a three tiered approach for dynamic action modeling comprising low level
feature extraction, intermediate level feature vector clustering and high level histogram
recognition as shown in Figure 1.
Low Level: Rotation Invariant Feature
Extraction
Probabilistic Foreground Map
(GMM Color Segmentation,
Probabilistic Appearance Modeling,
Dynamic Texture Segmentation by
GPCA)
Intermediate Level: Clustering Presentation
for Image Frames
High Level: Aggregated Histogram Model
for Action Recognition
GMM/LDA Density Modeling for
Clusters
Temporal Aggregated
Multiresolution Histogram for
Action
Feature Extraction
via Circular/Fourier Representation
Temporally Constrained
Clustering (Refinement)
From Unigram to Bigram,
Multigram Histogram Model
Feature Dimension Reduction
via Variance Analysis
Framewise Clustering via
Spectral Segmentation
(Initialization)
Temporal Pyramids and Scaling
Figure 1: Diagram of a three tier approach for dynamic articulated object action modeling.
(a)
(b)
(c)
Figure 2: (a) Image after background subtraction (b) GMM based color segmentation
(c) Circular histogram for feature extraction.
2.1
Low Level: Rotation Invariant Feature Extraction
In the low level image processing, our goals are to locate the region of interest in an image and to extract a scale and in-plane rotation invariant feature vector as its descriptor.
In order to accomplish this, a reliable and stable foreground model of the target in question is expected. Depending on the circumstances, a Gaussian mixture model (GMM) for
segmentation [15], probabilistic appearance modeling [5], or dynamic object segmentation
by Generalized Principal Component Analysis (GPCA) [25] are possible solutions. In this
paper, we apply a GMM for hand skin color segmentation.
We fit a GMM by first performing a simple background subtraction to obtain a noisy foreground containing a hand object (shown in Figure 2 (a)). From this, more than 1 million RGB pixels are used to train skin and non-skin color density models with 10 Gaussian kernels for each class. Having done this, for new images a probability density ratio
Pskin /Pnonskin of these two classes is computed. If Pskin /Pnonskin is larger than 1, the
pixel is considered as skin (foreground) and is otherwise background. A morphological
operator is then used to clean up this initial segmentaion and create a binary mask for the
hand object. We then compute the centroid and second central moments of this 2D mask.
A circle is defined about the target by setting its center as the centroid and its radius as
2.8 times largest eigenvalues of the second central moment matrix (covering over 99% skin
pixels in Figure 2 (c)). This circle is then divided to have 6 concentric annuli which contain 1, 2, 4, 8, 16, 32 bins from inner to outer, respectively. Since the position and size of
this circular histogram is determined by the color segmentation, it is translation and scale
invariant.
We then normalize the density value Pskin + Pnonskin = 1 for every pixel within the
foreground mask (Figure 2) over the hand region. For each bin of the circular histogram,
we calculate the mean of Pskin ( ?log(Pskin ), or ?log(Pskin /Pnonskin ) are also possible
choices) of pixels in that bin as its value. The values of all bins along each circle form a
vector, and 1D Fourier transform is applied to this vector. The power spectra of all annuli
are ordered into a linear list producing a feature vector f~(t) of 63 dimensions representing
the appearance of a hand image.1 Note that the use of the Fourier power spectrum of the
annuli makes the representation rotation invariant.
2.2
Intermediate Level: Clustering Presentation for Image Frames
After the low level processing, we obtain a scale and rotation invariant feature vector as
an appearance representation for each image frame. The temporal evolution of feature
vectors represent actions. However, not all the images are actually unique in appearance.
1
An optional dimension reduction of feature vectors can be achieved by eliminating dimensions
which have low variance. It means that feature values of those dimensions do not change much in the
data, therefore are non-informative.
At the intermediate level, we cluster images from a set of feature vectors. This frame-wise
clustering is critical for dimension reduction and the stability of high level recognition.
Initializing Clusters by Spectral Segmentation There are two critical problems with
clustering algorithms: determining the true number of clusters and initializing each cluster. Here we use a spectral clustering method [20, 22, 26, 18] to solve both problems.
We first build the affinity matrix of pairwise distances between feature fectors2 . We then
perform a singular value decomposition on the affinity matrix with proper normalization
[20]. The number of clusters is determined by choosing the n dominant eigenvalues. The
corresponding eigenvectors are taken as an orthogonal subspace for all the data.
To get n cluster centers, we take the approach of [20] and choose vectors that minimize the
absolute value of cosine between any two cluster centers:
rand(0, N ) :
k=1
Pk?1
ID(k) =
(1)
n
n
~
~
arg mint=1..N c=1 | cos(f (ID(c)), f (t))| : n ? k > 1
where f~n (t) is the feature vector of image frame t after numerical normalization in [20]
and ID(k) is the image frame number chosen for the center of cluster k. N is the number of
images used for spectral clustering. For better clustering results, multiple restarts are used
for initialization.
Unlike [18], we find this simple clustering procedure is sufficient to obtain a good set of
clusters from only a few restarts. After initialization, the Kmeans [8] is used to smooth the
centers. Let C1 (t) denote the class label for image t, and ~g (c) = f~(ID(c)); c = 1 . . . n
denote cluster centers.
Refinement: Temporally Constrained Clustering Spectral clustering methods are designed for an unordered ?bag? of feature vectors, but, in our case, the temporal ordering
of image is an important source of information. In particular, the stablity of appearance is
easily computed by computing the motion energy3 between two frames. Let M (t) denote
the motion energy
between frames t and t?1. Define Tk,j = {t|C1 (t) = k, C1 (t?1) = j}
? (k, j) = P
and M
t?Tk,j M (t)/|Tk,j |. We now create a regularized clustering cost function
as
?
?
kg(c)?g(C2 (t?1))k
?
? e?kf (t)?g(c)k
M (t)
e?
+
?
(2)
C2 (t) = arg maxc=1..n Pn
kg(c)?g(C2 (t?1))k
Pn
?
? M
? c=1 e?kf (t)?g(c)k
? (c,C (t?1))
2
e
c=1
where ? is the weighting parameter. Here motion energy M (t) plays a role as the temperature T in simulated annealing. When it is high (strong motion between frames), the
motion continuity condition is violated and the labels of successive frames can change
freely; when it is low, the smoothness term constrains the possible transitions of classes
? (k, j).
with low M
With this in place, we now scan through the sequence searching for C2 (t) of maximum
value given C2 (t ? 1) is already fixed. 4 This temporal smoothing is most relevant with
images with motions, and static frames are already stably clustered and therefore their
cluster labels to not change.
2
The exponent of either Euclidean distance or Cosine distance between two feature vectors can
be used in this case.
3
A simple method is to compute motion energy as the Sum of Squared Differences (SSD) by
subtracting two Pskin density masses from successive images.
4
? (k, j) changes after scanning the labels of the image sequence once, thus more
Note that M
iterations could be used to achieve more accurate temporal smoothness of C3 (t), t = 1..N . From
our experiments, more iterations does not change the result much.
GMM for Density Modeling and Smoothing Given clusters, we build a probability density model for each. A Gaussian Mixture Model [11, 8] is used to gain good local relaxation
based on the initial clustering result provided by the above method and good generalization
for new data. Due to the curse of dimensionality, it is difficult to obtain a good estimate
of a high dimensional density function with limited and largely varied training data. We
introduce an iterative method incorporating Linear Discriminative Analysis (LDA) [8] and
a GMM in an EM-like fashion to perform dimensional reduction. The initial clustering
labels help to build the scatter matrices for LDA. The optimal projection matrix of LDA is
then obtained from the decomposition of clusters? scatter matrices [8]. The original feature
vectors can be further projected into a low dimensional space, which improves the estimation of multi-variate Gaussian density function. With the new clustering result from GMM,
LDA?s scatter matrices and projection matrix can be re-estimated, and GMM can also be
re-modeled in the new LDA subspace. This loop converges within 3 ? 5 iterations from
our experiments. Intuitively, LDA projects the data into a low dimensional subspace where
the image clusters are well separated, which helps to have a good parameter estimation for
GMM with limited data. Given more accurate GMM, more accurate clustering results are
obtained, which also causes better estimate of LDA. The theoretical proof of convergence
is undertaken. After this process, we have a Gaussian density model for each cluster.
2.3
High Level: Aggregated Histogram Model for Action Recognition
Given a set of n clusters, define w(t) = [pc1 (f (t)), pc2 (f (t)), ..., pcn (f (t))]T where px (y)
denotes the density value of the vector y with respect to the GMM for cluster x. An action
is then a trajectory of [w(t1 ), w(t1 + 1), ..., w(t2 )]T in <n . For recognition purposes, we
want to calculate some discriminative statistics from each trajectory. One natural way is to
Pt2
w(t)/(t2 ? t1 + 1) over time which is a temporal weighted
use its mean Ht1 ,t2 = t=t
1
histogram. Note that the histogram Ht1 ,t2 bins are precisely corresponding to the trained
clusters.
From the training set, we aggregate the cluster weights of images within a given hand action
to form a histogram model. In this way, a temporal image sequence corresponding to one
action is represented by a single vector. The matching of different actions is equivalent to
compute the similarity of two histograms which has variants. Here we use Bhattacharyya
similarity metric [1] which has has several useful properties including: it is an approximation of ?2 test statistics with fixed bias; it is self-consistent; it does not have the singularity
problem while matching empty histogram bins; and its value is properly bounded within
?
[0, 1]. Assume we have a library of action histograms H1? , H2? , ..., HM
, the class label of a
? t1 ,t2 is determined by the following equation.
new action H
?
# 12 ?
"
n q
?
?
X
? t1 ,t2 ) = arg min
? t1 ,t2 (c)
? t1 ,t2 ) = 1 ?
(3)
L(H
D(Hl? , H
Hl? (c) ? H
l=1..M ?
?
c=1
This method is low cost because only one exemplar per action category is needed.
One problem with this method is that all sequence information has been compressed, e.g.,
we cannot distinguish an opening hand gesture from a closing hand using only one histogram. This problem can be easily solved by subdividing the sequence and histogram
model into m parts: Htm1 ,t2 = [Ht1 ,(t1 +t2 )/m , ..., H(t1 +t2 )?(m?1)/m,t2 ]T . For an extreme
case when one frame is a subsequence, the histogram model simply becomes exactly the
vector form of the representative surface.
We intend to classify hand actions with speed differences into the same category. To
achieve this, the image frames within a hand action can be sub-sampled to build a set
of temporal pyramids. In order to segment hand gestures from a long video sequence, we
create several sliding windows with different frame sampling rates. The proper time scaling
magnitude is found by searching for the best fit over temporal pyramids.
Taken together, the histogram representation achieves an adjustable multi-resolution measurement to describe actions. A Hidden Markov Model (HMM) with discrete observations
could be also employed to train models for different hand actions, but more template samples per gesture class are required. The histogram recognition method has the additional
advantage that it does not depend on extremely accurate frame-wise clustering. A small
proportion of incorrect labels does not effect the matching value much. In comparison,
in an HMM with few training samples, outliers seriously impact the accuracy of learning.
From the viewpoint of considering hand actions as a language process, our model is an integration of individual observations (by labelling each frame with a set of learned clusters)
from different time slots. The labels? transitions between successive frames are not used
to describe the temporal sequence. By subdividing the histogram, we are extending the
representation to contain bigram, trigram, etc. information.
3 Results
We have tested our three tiered method on the problem of recognizing sequences of hand
spelling gestures.
Framewise clustering. We first evaluate the low level representation of single images and
intermediate clustering algorithms. A training set of 3015 images are used. The frame-toframe motion energy is used to label images as static or dynamic. For spectral clustering,
3 ? 4 restarts from both the dynamic and static set are sufficient to cover all the modes in
the training set. Then, temporal smoothing is employed and a Gaussian density is calculated
for each cluster in a 10 dimensional subspace of the LDA projection. As a result, 24 clusters
are obtained which contain 16 static and 8 dynamic modes. Figure 3 shows 5 frames
closest to the mean of the probability density of cluster 1, 3, 19, 5, 13, 8, 21, 15, 6, 12. It
can be seen that clustering results are insensitive to artifacts of skin segmentation. From
Figure 3, it is also clear that dynamic modes have significantly larger determinants than
static ones. The study of the eigenvalues of covariance matrices shows that their superellipsoid shapes are expanded within 2 ? 3 dimensions or 6 ? 8 dimensions for static
or dynamic clusters. Taken together, this means that static clusters are quite tight, while
dynamic clusters contain much more in-class variation. From Figure 4 (c), dynamic clusters
gain more weight during the smoothing process incorporating the temporal constraint and
subsequent GMM refinement.
Figure 3: Image clustering results after low and intermediate level processing.
Action recognition and segmentation. For testing images, we first project their feature
A
A
YA
YA
W
A
XA
Y
A
Y
A
W
A
X
A
(a)
(b)
(c)
(d)
Figure 4: (a) Affinity matrix of 3015 images. (b) Affinity matrices of cluster centoids (from upper
left to lower right) after spectral clustering, temporal smoothing and GMM. (c) Labelling results of
3015 images (red squares are frames whose labels changed with smoothing process after spectral
clustering). (d) The similarity matrix of segmented hand gestures. The letters are labels of gestures.
vectors into the LDA subspace. Then, the GMM is used to compute their weights with
respect to each cluster. We manually choose 100 sequences for testing purposes, and compute their similarities with respect to a library of 25 gestures. The length of the action
sequences was 9 ? 38 frames. The temporal scale of actions in the same category ranged
from 1 to 2.4. The results were recognition rates of 90% and 93% without/with temporal
smoothing (Equation 2). Including the top three candidates, the recognition rates increase
to 94% and 96%, respectively. We also used the learned model and a sliding window with
temporal scaling to segment actions from a 6034 frame video sequence containing dynamic
gestures and static hand postures. The similarity matrices among 123 actions found in the
video is shown in Figure 4 (d). 106 out of 123 actions (86.2%) are correctly segmented and
recognized.
Integrating motion information. As noted previously, our method cannot distinguish
opening/closing hand gestures without temporally subdividing histograms. An alternative
solution is to integrate motion information5 between frames. Motion feature vectors are
also clustered, which results a joint (appearance and motion) histogram model for actions.
We assume independence of the data and therefore simple contatenate these two histograms
into a single action representation. From our preliminary experiments, both motion integration and histogram subdivision are comparably effective to recognize gestures with
opposite direction.
4 Conclusion and Discussion
We have presented a method for classifying the motion of articulated gestures using
LDA/GMM-based clustering methods and a histogram-based model of temporal evolution.
Using this model, we have obtained extremely good recognition results using a relatively
coarse representation of appearance and motion in images.
There are mainly three methods to improve the performance of histogram-based classification, i.e., adaptive binning, adaptive subregion, and adaptive weighting [21]. In our
approach, adaptive binning of the histogram is automatically learned by our clustering
algorithms; adaptive subregion is realized by subdividing action sequences to enrich the
histogram?s descriptive capacity in the temporal domain; adaptive weighting is achieved
from the trained weights of Gaussian kernels in GMM.
Our future work will focus on building a larger hand action database containing 50 ? 100
5
Motion information can be extracted by first aligning two hand blobs, subtracting two skin-color
density masses, then using the same circular histogram in section 2.1 to extract a feature vector for
positive and negative density residues respectively. Another simple way is to subtract two frames?
feature vectors directly.
categories for more extensive testing, and on extending the representation to include other
types of image information (e.g. contour information). Also, by finding an effective foreground segmentation module, we intend to apply the same methods to other applications
such as recognizing stylized human body motion.
References
[1] F. Aherne, N. Thacker, and P. Rockett, The Bhattacharyya Metric as an Absolute Similarity
Measure for Frequency Coded Data, Kybernetika, 34:4, pp. 363-68, 1998.
[2] V. Athitsos and S. Sclaroff, Estimating 3D Hand Pose From a Cluttered Image, CVPR, 2003.
[3] M. Brand, Shadow Puppetry, ICCV, 1999.
[4] R. Bowden and M. Sarhadi, A Non-linear of Shape and Motion for Tracking Finger Spelt
American Sign Language, Image and Vision Computing, 20:597-607, 2002.
[5] T. Cootes, G. Edwards and C. Taylor, Active Appearance Models, IEEE Trans. PAMI, 23:6, pp.
681-685, 2001.
[6] D. Cremers, T. Kohlberger and C. Schnrr, Shape statistics in Kernel Space for Variational Image
Segmentation, Pattern Recognition, 36:1929-1943, 2003.
[7] T. J. Darrell and A. P. Pentland, Recognition of Space-Time Gestures using a Distributed Representation, MIT Media Laboratory Vision and Modeling TR-197.
[8] R. O. Duda, P. E. Hart and D. G. Stork, Pattern Classification, Wiley Interscience, 2002.
[9] A. Efros, A. Berg, G. Mori and J. Malik, Recognizing Action at a Distance. ICCV, pp. 726?733,
2003.
[10] W. T. Freeman and E. H. Adelson, The Design and Use of Steerable Filters, IEEE Trans. PAMI,
13:9, pp. 891-906, 1991.
[11] T. Hastie and R. Tibshirani, Discriminant Analysis by Gaussian Mixtures. Journal of Royal
Statistical Society Series B, 58(1):155-176.
[12] W. Hawkins, P. Leichner and N. Yang, The Circular Harmonic Transform for SPECT Reconstruction and Boundary Conditions on the Fourier Transform of the Sinogram, IEEE Trans. on
Medical Imaging, 7:2, 1988.
[13] A. Heap and D. Hogg, Wormholes in Shape Space: Tracking through Discontinuous Changes
in Shape, ICCV, 1998.
[14] M. K. Hu, Visual pattern recognition by moment invariants, IEEE Trans. Inform. Theory,
8:179-187, 1962.
[15] M. J. Jones and J. M. Rehg, Statistical Color Models with Application to Skin Detection Int. J.
of Computer Vision, 46:1 pp: 81-96, 2002.
[16] B. Julesz, Textons, the elements of texture perception, and their interactions. Nature, 290:91-97,
1981.
[17] T. Leung and J. Malik, Representing and Recognizing the Visual Appearance of Materials using
Three-dimensional Textons, Int. Journal of Computer Vision, 41:1, pp. 29-44, 2001.
[18] M. Maila and J. Shi, Learning Segmentation with Random Walk, NIPS 2001.
[19] B. Moghaddam and A. Pentland, Probabilistic Visual Learning for Object Representation, IEEE
Trans. PAMI 19:7, 1997.
[20] A. Ng, M. Jordan and Y. Weiss, On Spectral Clustering: Analysis and an algorithm, NIPS,
2001.
[21] S. Satoh, Generalized Histogram: Empirical Optimization of Low Dimensional Features for
Image Matching, ECCV, 2004.
[22] J. Shi and J. Malik, Normalized Cuts and Image Segmentation, IEEE Trans. on PAMI, 2000.
[23] B. Stenger, A. Thayananthan, P. H. S. Torr, and R. Cipolla, Filtering Using a Tree-Based
Estimator, ICCV, II:1063-1070, 2003.
[24] C. Tomasi, S. Petrov and A. Sastry, 3D tracking = classification + interpolation, ICCV, 2003.
[25] R. Vidal and R. Hartley, Motion Segmentation with Missing Data using PowerFactorization
and GPCA, CVPR, 2004.
[26] Y. Weiss, Segmentation using eigenvectors: A Unifying view. ICCV, 1999.
[27] Lihi Zelnik-Manor and Michal Irani, Event-based video analysis, CVPR, 2001.
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1,807 | 2,641 | Incremental Learning for Visual Tracking
?
Jongwoo Lim? David Ross? Ruei-Sung Lin? Ming-Hsuan Yang?
University of Illinois ? University of Toronto ? Honda Research Institute
[email protected] [email protected] [email protected] [email protected]
Abstract
Most existing tracking algorithms construct a representation of a target
object prior to the tracking task starts, and utilize invariant features to
handle appearance variation of the target caused by lighting, pose, and
view angle change. In this paper, we present an efficient and effective online algorithm that incrementally learns and adapts a low dimensional eigenspace representation to reflect appearance changes of the target, thereby facilitating the tracking task. Furthermore, our incremental
method correctly updates the sample mean and the eigenbasis, whereas
existing incremental subspace update methods ignore the fact the sample
mean varies over time. The tracking problem is formulated as a state
inference problem within a Markov Chain Monte Carlo framework and
a particle filter is incorporated for propagating sample distributions over
time. Numerous experiments demonstrate the effectiveness of the proposed tracking algorithm in indoor and outdoor environments where the
target objects undergo large pose and lighting changes.
1
Introduction
The main challenges of visual tracking can be attributed to the difficulty in handling appearance variability of a target object. Intrinsic appearance variabilities include pose variation
and shape deformation of a target object, whereas extrinsic illumination change, camera
motion, camera viewpoint, and occlusions inevitably cause large appearance variation. Due
to the nature of the tracking problem, it is imperative for a tracking algorithm to model such
appearance variation.
Here we developed a method that, during visual tracking, constantly and efficiently updates a low dimensional eigenspace representation of the appearance of the target object.
The advantages of this adaptive subspace representation are several folds. The eigenspace
representation provides a compact notion of the ?thing? being tracked rather than treating
the target as a set of independent pixels, i.e., ?stuff? [1]. The use of an incremental method
continually updates the eigenspace to reflect the appearance change caused by intrinsic
and extrinsic factors, thereby facilitating the tracking process. To estimate the locations
of the target objects in consecutive frames, we used a sampling algorithm with likelihood
estimates, which is in direct contrast to other tracking methods that usually solve complex
optimization problems using gradient-descent approach.
The proposed method differs from our prior work [14] in several aspects. First, the proposed algorithm does not require any training images of the target object before the tracking
task starts. That is, our tracker learns a low dimensional eigenspace representation on-line
and incrementally updates it as time progresses (We assume, like most tracking algorithms,
that the target region has been initialized in the first frame). Second, we extend our sampling method to incorporate a particle filter so that the sample distributions are propagated
over time. Based on the eigenspace model with updates, an effective likelihood estimation
function is developed. Third, we extend the R-SVD algorithm [6] so that both the sample
mean and eigenbasis are correctly updated as new data arrive. Though there are numerous
subspace update algorithms in the literature, only the method by Hall et al. [8] is also able
to update the sample mean. However, their method is based on the addition of a single column (single observation) rather than blocks (a number of observations in our case) and thus
is less efficient than ours. While our formulation provides an exact solution, their algorithm
gives only approximate updates and thus it may suffer from numerical instability. Finally,
the proposed tracker is extended to use a robust error norm for likelihood estimation in the
presence of noisy data or partial occlusions, thereby rendering more accurate and robust
tracking results.
2
Previous Work and Motivation
Black et al. [4] proposed a tracking algorithm using a pre-trained view-based eigenbasis
representation and a robust error norm. Instead of relying on the popular brightness constancy working principal, they advocated the use of subspace constancy assumption for
visual tracking. Although their algorithm demonstrated excellent empirical results, it requires to build a set of view-based eigenbases before the tracking task starts. Furthermore,
their method assumes that certain factors, such as illumination conditions, do not change
significantly as the eigenbasis, once constructed, is not updated.
Hager and Belhumeur [7] presented a tracking algorithm to handle the geometry and illumination variations of target objects. Their method extends a gradient-based optical flow
algorithm to incorporate research findings in [2] for object tracking under varying illumination conditions. Prior to the tracking task starts, a set of illumination basis needs to be
constructed at a fixed pose in order to account for appearance variation of the target due to
lighting changes. Consequently, it is not clear whether this method is effective if a target
object undergoes changes in illumination with arbitrary pose.
In [9] Isard and Blake developed the Condensation algorithm for contour tracking in which
multiple plausible interpretations are propagated over time. Though their probabilistic approach has demonstrated success in tracking contours in clutter, the representation scheme
is rather primitive, i.e., curves or splines, and is not updated as the appearance of a target
varies due to pose or illumination change.
Mixture models have been used to describe appearance change for motion estimation [3]
[10]. In Black et al. [3] four possible causes are identified in a mixture model for estimating
appearance change in consecutive frames, and thereby more reliable image motion can be
obtained. A more elaborate mixture model with an online EM algorithm was recently
proposed by Jepson et al. [10] in which they use three components and wavelet filters to
account for appearance changes during tracking. Their method is able to handle variations
in pose, illumination and expression. However, their WSL appearance model treats pixels
within the target region independently, and therefore does not have notion of the ?thing?
being tracked. This may result in modeling background rather than the foreground, and fail
to track the target.
In contrast to the eigentracking algorithm [4], our algorithm does not require a training
phase but learns the eigenbases on-line during the object tracking process, and constantly
updates this representation as the appearance changes due to pose, view angle, and illumination variation. Further, our method uses a particle filter for motion parameter estimation
rather than the Gauss-Newton method which often gets stuck in local minima or is distracted by outliers [4]. Our appearance-based model provides a richer description than
simple curves or splines as used in [9], and has notion of the ?thing? being tracked. In
addition, the learned representation can be utilized for other tasks such as object recognition. In this work, an eigenspace representation is learned directly from pixel values
within a target object in the image space. Experiments show that good tracking results
can be obtained with this representation without resorting to wavelets as used in [10], and
better performance can potentially be achieved using wavelet filters. Note also that the
view-based eigenspace representation has demonstrated its ability to model appearance of
objects at different pose [13], and under different lighting conditions [2].
3
Incremental Learning for Tracking
We present the details of the proposed incremental learning algorithm for object tracking
in this section.
3.1 Incremental Update of Eigenbasis and Mean
The appearance of a target object may change drastically due to intrinsic and extrinsic
factors as discussed earlier. Therefore it is important to develop an efficient algorithm to
update the eigenspace as the tracking task progresses. Numerous algorithms have been
developed to update eigenbasis from a time-varying covariance matrix as more data arrive
[6] [8] [11] [5]. However, most methods assume zero mean in updating the eigenbasis
except the method by Hall et al. [8] in which they consider the change of the mean when
updating eigenbasis as each new datum arrives. Their update algorithm only handles one
datum per update and gives approximate results, while our formulation handles multiple
data at the same time and renders exact solutions.
We extend the work of the classic R-SVD method [6] in which we update the eigenbasis
while taking the shift of the sample mean into account. To the best of our knowledge, this
formulation with mean update is new in the literature.
Given a d ? n data matrix A = {I1 , . . . , In } where each column Ii is an observation (a ddimensional image vector in this paper), we can compute the singular value decomposition
(SVD) of A, i.e., A = U ?V >. When a d?m matrix E of new observations is available, the
0
0
0
0
R-SVD algorithm efficiently computes the SVD of the matrix A = (A|E) = U ? V >
based on the SVD of A as follows:
? of E, and U 0 = (U |E).
?
1. Apply QR decomposition to and get orthonormal basis E
0
2. Let V = V0 I0m where Im is an m ? m identity matrix. It follows then,
>
U >AV U >E ? U >E
0
0
0
0
V 0
U
? = U >A V = E
= 0 E? >E .
? > (A|E) 0 Im = E
? >AV E
? >E
0
??
? V? > and the SVD of A0 is
3. Compute the SVD of ? = U
0
0
??
? V? >)V 0 > = (U 0 U
? )?(
? V? >V 0 > ).
A = U (U
Exploiting the properties of orthonormal bases and block structures, the R-SVD algorithm
computes the new eigenbasis efficiently. The computational complexity analysis and more
details are described in [6].
One problem with the R-SVD algorithm is that the eigenbasis U is computed from AA >
with the zero mean assumption. We modify the R-SVD algorithm and compute the eigenbasis with mean update. The following derivation is based on scatter matrix, which is same
as covariance matrix except a scalar factor.
Proposition 1 Let Ip = {I1 , I2 , . . . , In }, Iq = {In+1 , In+2 , . . . , In+m }, and Ir = (Ip |Iq ).
Denote the means and scatter matrices of Ip , Iq , Ir as ?Ip , ?Iq , ?Ir , and Sp , Sq , Sr respecnm ?
tively, then Sr = Sp + Sq + n+m
(Iq ? ?Ip )(?Iq ? ?Ip )> .
n ?
m ?
m ?
Ip + n+m
Iq , ?Ip ? ?Ir = n+m
Proof: By definition, ?Ir = n+m
(Ip ? ?Iq ); ?Iq ? ?Ir =
n
?
?
n+m (Iq ? Ip ) and,
Pn
Pn+m
Sr = Pi=1 (Ii ? ?Ir )(Ii ? ?Ir )> + i=n+1 (Ii ? ?Ir )(Ii ? ?Ir )>
n
=
(Ii ? ?Ip + ?Ip ? ?Ir )(Ii ? ?Ip + ?Ip ? ?Ir )> +
Pi=1
n+m
?
?
?
?
?
? >
i=m+1 (Ii ? Iq + Iq ? Ir )(Ii ? Iq + Iq ? Ir )
= Sp + n(?Ip ? ?Ir )(?Ip ? ?Ir )> + Sq + m(?Iq ? ?Ir )(?Iq ? ?Ir )>
nm2
?
? ?
? > + Sq + n2 m 2 (?Ip ? ?Iq )(?Ip ? ?Iq )>
= Sp + (n+m)
2 (Ip ? Iq )(Ip ? Iq )
(n+m)
nm ?
= Sp + Sq + n+m
(Ip ? ?Iq )(?Ip ? ?Iq )>
? and I?r = {I1 ?
Let I?p = {I1 ? ?Ip , . . . , In ? ?Ip }, I?q = {In+1 ? ?Iq , . . . , In+m ?
Iq },
q
?Ir , . . . , In+m ? ?Ir }, and the SVD of I?r = Ur ?r V > . Let E
? = I?q | nm (?Ip ? ?Iq ) ,
r
n+m
? I?p |E)
? >. Therefore, we compute SVD on (I?p |E)
? to
and use Proposition 1, Sr = (I?p |E)(
get Ur . This can be done efficiently by the R-SVD algorithm as described above.
In summary, given the mean ?Ip and the SVD of existing data Ip , i.e., Up ?p Vp> and new
data Iq , we can compute the the mean ?Ir and the SVD of Ir , i.e., Ur ?r Vr> easily:
q
n ?
m ?
? = Iq ? ?Ir 1(1?m) | nm (?Ip ? ?Iq ) .
1. Compute ?Ir = n+m
Ip + n+m
Iq , and E
n+m
? to obtain (Ur ?r Vr>).
2. Compute R-SVD with (Up ?p Vp>) and E
In numerous vision problems, we can further exploit the low dimensional approximation of
image data and put larger weights on the recent observations, or equivalently downweight
the contributions of previous observations. For example as the appearance of a target object
gradually changes, we may want to put more weights on recent observations in updating
the eigenbasis since they are more likely to be similar to the current appearance of the
target. The forgetting factor f can be used under this premise as suggested in [11] , i.e.,
0
0
A = (f A |E) = (U (f ?)V >|E) where A and A are original and weighted data matrices,
respectively.
3.2 Sequential Inference Model
The visual tracking problem is cast as an inference problem with a Markov model and
hidden state variable, where a state variable Xt describes the affine motion parameters
(and thereby the location) of the target at time t. Given a set of observed images I t =
{I1 , . . . , It }. we aim to estimate the value of the hidden state variable Xt . Using Bayes?
theorem, we have
Z
p(Xt | It ) ? p(It |Xt )
p(Xt |Xt?1 ) p(Xt?1 | It?1 ) dXt?1
The tracking process is governed by the observation model p(It |Xt ) where we estimate the
likelihood of Xt observing It , and the dynamical model between two states p(Xt |Xt?1 ).
The Condensation algorithm [9], based on factored sampling, approximates an arbitrary
distribution of observations with a stochastically generated set of weighted samples. We
use a variant of the Condensation algorithm to model the distribution over the object?s
location, as it evolves over time.
3.3 Dynamical and Observation Models
The motion of a target object between two consecutive frames can be approximated by
an affine image warping. In this work, we use the six parameters of affine transform
to model the state transition from Xt?1 to Xt of a target object being tracked. Let
Xt = (xt , yt , ?t , st , ?t , ?t ) where xt , yt , ?t , st , ?t , ?t , denote x, y translation, rotation
angle, scale, aspect ratio, and skew direction at time t. Each parameter in Xt is modeled
independently by a Gaussian distribution around its counterpart in Xt?1 . That is,
p(Xt |Xt?1 ) = N (Xt ; Xt?1 , ?)
where ? is a diagonal covariance matrix whose elements are the corresponding variances
of affine parameters, i.e., ?x2 , ?y2 , ??2 , ?s2 , ??2 , ??2 .
Since our goal is to use a representation to model the ?thing? that we are tracking, we
model the image observations using a probabilistic interpretation of principal component
analysis [16]. Given an image patch predicated by Xt , we assume the observed image It
was generated from a subspace spanned by U centered at ?. The probability that a sample
being generated from the subspace is inversely proportional to the distance d from the
sample to the reference point (i.e., center) of the subspace, which can be decomposed into
the distance-to-subspace, dt , and the distance-within-subspace from the projected sample
to the subspace center, dw . This distance formulation, based on a orthonormal subspace
and its complement space, is similar to [12] in spirit.
The probability of a sample generated from a subspace, pdt (It |Xt ), is governed by a Gaussian distribution:
pdt (It | Xt ) = N (It ; ?, U U >+ ?I)
where I is an identity matrix, ? is the mean, and ?I term corresponds to the additive Gaussian noise in the observation process. It can be shown [15] that the negative exponential
distance from It to the subspace spanned by U , i.e., exp(?||(It ? ?) ? U U >(It ? ?)||2 ),
is proportional to N (It ; ?, U U >+ ?I) as ? ? 0.
Within a subspace, the likelihood of the projected sample can be modeled by the Mahalanobis distance from the mean as follows:
pdw (It | Xt ) = N (It ; ?, U ??2 U >)
where ? is the center of the subspace and ? is the matrix of singular values corresponding
to the columns of U . Put together, the likelihood of a sample being generated from the
subspace is governed by
p(It |Xt ) = pdt (It |Xt ) pdw (It |Xt ) = N (It ; ?, U U > + ?I) N (It ; ?, U ??2 U > ) (1)
Given a drawn sample Xt and the corresponding image region It , we aim to compute
p(It |Xt ) using (1). To minimize the effects of noisy pixels, we utilize a robust error norm
2
[4], ?(x, ?) = ?2x+x2 instead of the Euclidean norm d(x) = ||x||2 , to ignore the ?outlier?
pixels (i.e., the pixels that are not likely to appear inside the target region given the current
eigenspace). We use a method similar to that used in [4] in order to compute dt and dw .
This robust error norm is helpful especially when we use a rectangular region to enclose
the target (which inevitably contains some noisy background pixels).
4
Experiments
To test the performance of our proposed tracker, we collected a number of videos recorded
in indoor and outdoor environments where the targets change pose in different lighting conditions. Each video consists of 320 ? 240 gray scale images and are recorded at 15 frames
per second unless specified otherwise. For the eigenspace representation, each target image
region is resized to 32 ? 32 patch, and the number of eigenvectors used in all experiments
is set to 16 though fewer eigenvectors may also work well. Implemented in MATLAB
with MEX, our algorithm runs at 4 frames per second on a standard computer with 200
particles. We present some tracking results in this section and more tracking results as well
as videos can be found at http://vision.ucsd.edu/?jwlim/ilt/.
4.1 Experimental Results
Figure 1 shows the tracking results using a challenging sequence recorded with a moving digital camera in which a person moves from a dark room toward a bright area while
changing his pose, moving underneath spot lights, changing facial expressions and taking
off glasses. All the eigenbases are constructed automatically from scratch and constantly
updated to model the appearance of the target object while undergoing appearance changes.
Even with the significant camera motion and low frame rate (which makes the motions between frames more significant, or equivalently to tracking fast moving objects), our tracker
stays stably on the target throughout the sequence.
The second sequence contains an animal doll moving in different pose, scale, and lighting
conditions as shown in Figure 2. Experimental results demonstrate that our tracker is able
to follow the target as it undergoes large pose change, cluttered background, and lighting
variation. Notice that the non-convex target object is localized with an enclosing rectangular window, and thus it inevitably contains some background pixels in its appearance
representation. The robust error norm enables the tracker to ignore background pixels and
estimate the target location correctly. The results also show that our algorithm faithfully
Figure 1: A person moves from dark toward bright area with large lighting and pose changes. The
images in the second row shows the current sample mean, tracked region, reconstructed image, and
the reconstruction error respectively. The third and forth rows shows 10 largest eigenbases.
Figure 2: An animal doll moving with large pose, lighting variation in a cluttered background.
models the appearance of the target, as shown in eigenbases and reconstructed images, in
the presence of noisy background pixels.
We recorded a sequence to demonstrate that our tracker performs well in outdoor environment where lighting conditions change drastically. The video was acquired when a person
walking underneath a trellis covered by vines. As shown in Figure 3, the cast shadow
changes the appearance of the target face drastically. Furthermore, the combined pose and
lighting variation with low frame rate makes the tracking task extremely difficult. Nevertheless, the results show that our tracker successfully follows the target accurately and
robustly. Due to heavy shadows and drastic lighting change, other tracking methods based
on gradient, contour, or color information are unlikely to perform well in this case.
4.2 Discussion
The success of our tracker can be attributed to several factors. It is well known that
the appearance of an object undergoing pose change can be modeled well by view-based
Figure 3: A person moves underneath a trellis with large illumination change and cast shadows
while changing his pose. More results can be found in the project web page.
representation [13]. Meanwhile at fixed pose, the appearance of an object in different
illumination conditions can be approximated well by a low dimensional subspace [2]. Our
empirical results show that these variations can be learned on-line without any prior training
phase, and also the changes caused by cast and attached shadows can still be approximated
by a linear subspace to some extent.
We show a few failure cases at our the web site
mentioned earlier. Typically, the failure happens when there is a combination of fast pose
change and drastic illumination change.
In this paper, we do not directly address the partial occlusion problems. Empirical results
show that temporary and partial occlusions can be handled by our method through the
robust error norm and the constant update of the eigenspace. Nevertheless situations arise
where we may have prior knowledge of the objects being tracked, and can exploit such
information for better occlusion handling.
To demonstrate the potency of our modified R-SVD algorithm in faithfully modeling the
object appearance, we compare the reconstructed images using our method and a conventional SVD algorithm. In Figure 4 first row contains a set of images tracked by our
tracker, and the second and fourth rows show the reconstructed images using 16 eigenvectors obtained after 121 incremental updates of 605 frame (block size is set to 5), and the
top 16 eigenvectors obtained by conventional SVD algorithm using all 605 tracked images.
Note that we only maintained 16 eigenvectors during tracking, and discarded the remaining
eigenvectors at each update. The residue images are presented in the third and fifth rows,
and the average L2 reconstruction error per pixel is 5.73?10?2 and 5.65?10?2 for our
modified R-SVD method and the conventional SVD algorithm respectively. The figure and
average reconstruction error shows that our modified R-SVD method is able to effectively
model the object appearance without losing detailed information.
5
Conclusions and Future Work
We have presented an appearance-based tracker that incrementally learns a low dimensional
eigenspace representation for object tracking while the target undergoes pose, illumination
and appearance changes. Whereas most tracking algorithms operate on the premise that
the object appearance or ambient environment lighting condition does not change as time
progresses, our method adapts the model representation to reflect appearance variation of
the target, thereby facilitating the tracking task. In contrast to the existing incremental
subspace methods, our R-SVD method updates the mean and eigenbasis accurately and
efficiently, and thereby learns a good eigenspace representation to faithfully model the
appearance of the target being tracked. Our experiments demonstrate the effectiveness of
the proposed tracker in indoor and outdoor environments where the target objects undergo
large pose and lighting changes.
The current dynamical model in our sampling method is based on a Gaussian distribution,
but the dynamics could be learned from exemplars for more efficient parameter estimation.
Our algorithm can be extended to construct a set of eigenbases for modeling nonlinear
aspects of appearance variation more precisely and automatically. We aim to address these
issues in our future work.
Figure 4: Reconstructed images and errors using our and the conventional SVD algorithms.
References
[1] E. H. Adelson and J. R. Bergen. The plenoptic function and the elements of early vision. In M. Landy and
J. A. Movshon, editors, Computational Models of Visual Processing, pp. 1?20. MIT Press, 1991.
[2] P. Belhumeur and D. Kreigman. What is the set of images of an object under all possible lighting conditions.
In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp. 270?277, 1997.
[3] M. J. Black, D. J. Fleet, and Y. Yacoob. A framework for modeling appearance change in image sequence.
In Proceedings of the Sixth IEEE International Conference on Computer Vision, pp. 660?667, 1998.
[4] M. J. Black and A. D. Jepson. Eigentracking: Robust matching and tracking of articulated objects using
view-based representation. In Proceedings of European Conference on Computer Vision, pp. 329?342,
1996.
[5] M. Brand. Incremental singular value decomposition of uncertain data with missing values. In Proceedings
of the Seventh European Conference on Computer Vision, volume 4, pp. 707?720, 2002.
[6] G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 1996.
[7] G. Hager and P. Belhumeur. Real-time tracking of image regions with changes in geometry and illumination. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, pp. 403?410,
1996.
[8] P. Hall, D. Marshall, and R. Martin. Incremental eigenanalysis for classification. In Proceedings of British
Machine Vision Conference, pp. 286?295, 1998.
[9] M. Isard and A. Blake. Contour tracking by stochastic propagation of conditional density. In Proceedings
of the Fourth European Conference on Computer Vision, volume 2, pp. 343?356, 1996.
[10] A. D. Jepson, D. J. Fleet, and T. F. El-Maraghi. Robust online appearance models for visual tracking. In
Proceedings of IEEE Conference on Computer Vision and Pattern Recognition, volume 1, pp. 415?422,
2001.
[11] A. Levy and M. Lindenbaum. Sequential Karhunen-Loeve basis extraction and its application to images.
IEEE Transactions on Image Processing, 9(8):1371?1374, 2000.
[12] B. Moghaddam and A. Pentland. Probabilistic visual learning for object recognition. IEEE Transactions
on Pattern Analysis and Machine Intelligence, 19(7):696?710, 1997.
[13] H. Murase and S. Nayar. Visual learning and recognition of 3d objects from appearance. International
Journal of Computer Vision, 14(1):5?24, 1995.
[14] D. Ross, J. Lim, and M.-H. Yang. Adaptive probabilistic visual tracking with incremental subspace update.
In Proceedings of the Eighth European Conference on Computer Vision, volume 2, pp. 470?482, 2004.
[15] S. Roweis. EM algorithms for PCA and SPCA. In Advances in Neural Information Processing Systems
10, pp. 626?632, 1997.
[16] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 61(3):611?622, 1999.
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1,808 | 2,642 | Adaptive Discriminative Generative Model
and Its Applications
?
Ruei-Sung Lin? David Ross? Jongwoo Lim? Ming-Hsuan Yang?
University of Illinois ? University of Toronto ? Honda Research Institute
[email protected] [email protected] [email protected] [email protected]
Abstract
This paper presents an adaptive discriminative generative model that generalizes the conventional Fisher Linear Discriminant algorithm and renders a proper probabilistic interpretation. Within the context of object
tracking, we aim to find a discriminative generative model that best separates the target from the background. We present a computationally
efficient algorithm to constantly update this discriminative model as time
progresses. While most tracking algorithms operate on the premise that
the object appearance or ambient lighting condition does not significantly
change as time progresses, our method adapts a discriminative generative model to reflect appearance variation of the target and background,
thereby facilitating the tracking task in ever-changing environments. Numerous experiments show that our method is able to learn a discriminative generative model for tracking target objects undergoing large pose
and lighting changes.
1 Introduction
Tracking moving objects is an important and essential component of visual perception,
and has been an active research topic in computer vision community for decades. Object
tracking can be formulated as a continuous state estimation problem where the unobservable states encode the locations or motion parameters of the target objects, and the task is
to infer the unobservable states from the observed images over time. At each time step,
a tracker first predicts a few possible locations (i.e., hypotheses) of the target in the next
frame based on its prior and current knowledge. The prior knowledge includes its previous
observations and estimated state transitions. Among these possible locations, the tracker
then determines the most likely location of the target object based on the new observation. An attractive and effective prediction mechanism is based on Monte Carlo sampling
in which the state dynamics (i.e., transition) can be learned with a Kalman filter or simply
modeled as a Gaussian distribution. Such a formulation indicates that the performance of
a tracker is largely based on a good observation model for validating all hypotheses. Indeed, learning a robust observation model has been the focus of most recent object tracking
research within this framework, and is also the focus of this paper.
Most of the existing approaches utilize static observation models and construct them before
a tracking task starts. To account for all possible variation in a static observation model,
it is imperative to collect a large set of training examples with the hope that it covers all
possible variations of the object?s appearance. However, it is well known that the appearance of an object varies significantly under different illumination, viewing angle, and shape
deformation. It is a daunting, if not impossible, task to collect a training set that enumerates
all possible cases. An alternative approach is to develop an adaptive method that contains
a number of trackers that track different features or parts of a target object [3]. Therefore,
even though each tracker may fail under certain circumstances, it is unlikely all of them fail
at the same time. The tracking method then adaptively selects the trackers that are robust at
current situation to predict object locations. Although this approach improves the flexibility
and robustness of a tracking method, each tracker has a static observation model which has
to be trained beforehand and consequently restricts its application domains severely. There
are numerous cases, e.g., robotics applications, where the tracker is expected to track a previously unseen target once it is detected. To the best of our knowledge, considerably less
attention is paid to developing adaptive observation models to account for appearance variation of a target object (e.g., pose, deformation) or environmental changes (e.g., lighting
conditions and viewing angles) as tracking task progresses.
Our approach is to learn a model for determining the probability of a predicted image location being generated from the class of the target or the background. That is, we formulate
a binary classification problem and develop a discriminative model to distinguish observations from the target class and the background class. While conventional discriminative
classifiers simply predict the class of each test sample, a good model within the abovementioned tracking framework needs to select the most likely sample that belongs to target
object class from a set of samples (or hypotheses). In other words, an observation model
needs a classifier with proper probabilistic interpretation.
In this paper, we present an adaptive discriminative generative model and apply it to object
tracking. The proposed model aims to best separate the target and the background in the
ever-changing environment. The problem is formulated as a density estimation problem,
where the goal is, given a set of positive (i.e., belonging to the target object class) and negative examples (i.e., belonging to the background class), to learn a distribution that assigns
high probability to the positive examples and low probability to the negative examples. This
is done in a two-stage process. First, in the generative stage, we use a probabilistic principal
component analysis to model the density of the positive examples. The result of this state is
a Gaussian, which assigns high probability to examples lying in the linear subspace which
captures the most variance of the positive examples. Second, in the discriminative stage,
we use negative examples (specifically, negative examples that are assigned high probability by our generative model) to produce a new distribution which reduces the probability
of the negative examples. This is done by learning a linear projection that, when applied
to the data and the generative model, increases the distance between the negative examples
and the mean. Toward that end, it is formulated as an optimization problem and we show
that this is a direct generalization of the conventional Fisher Linear Discriminant algorithm
with proper probabilistic interpretation. Our experimental results show that our algorithm
can reliably track moving objects whose appearance changes under different poses, illumination, and self deformation.
2 Probabilistic Tracking Algorithm
We formulate the object tracking problem as a state estimation problem in a way similar
to [5] [9]. Denote ot as an image region observed at time t and Ot = {o1 , . . . , ot } is a set
of image regions observed from the beginning to time t. An object tracking problem is a
process to infer state st from observation Ot , where state st contains a set of parameters
referring to the tracked object?s 2-D position, orientation, and scale in image ot . Assuming
a Markovian state transition, this inference
Z problem is formulated with a recursive equation:
p(st |Ot ) = kp(ot |st ) p(st |st?1 )p(st?1 |Ot?1 )dst?1
(1)
where k is a constant, and p(ot |st ) and p(st |st?1 ) correspond to the observation model and
dynamic model, respectively.
In (1), p(st?1 |Ot?1 ) is the state estimation given all the prior observations up to time t ? 1,
and p(ot |st ) is the likelihood that observing image ot at state st . Put together, the posterior
estimation p(st |Ot ) can be computed efficiently. For object tracking, an ideal distribution
of p(st |Ot ) should peak at ot , i.e., st matching the observed object?s location ot . While
the integral in (1) predicts the regions where object is likely to appear given all the prior
observations, the observation model p(ot |st ) determines the most likely state that matches
the observation at time t.
In our formulation, p(ot |st ) measures the probability of observing ot as a sample being
generated by the target object class. Note that Ot is an image sequence and if the images
are acquired at high frame rate, it is expected that the difference between ot and ot?1
is small though object?s appearance might vary according to different of viewing angles,
illuminations, and possible self-deformation. Instead of adopting a complex static model
to learn p(ot |st ) for all possible ot , a simpler model suffices by adapting this model to
account for the appearance changes. In addition, since ot and ot?1 are most likely similar
and computing p(ot |st ) depends on p(ot?1 |st?1 ), the prior information p(ot?1 |st?1 ) can
be used to enhance the distinctiveness between the object and its background in p(ot |st ).
The idea of using an adaptive observation model for object tracking and then applying
discriminative analysis to better predict object location is the focus of the rest the paper. The
observation model we use is based on probabilistic principle component analysis (PPCA)
[10]. Object Tracking using PCA models have been well exploited in the computer vision
community [2]. Nevertheless, most existing tracking methods do not update the observation
models as time progresses. In this paper, we follow the work by Tipping and Bishop [10]
and propose an adaptive observation model based on PCA within a formal probabilistic
framework. Our result is a generalization of the conventional Fisher Linear Discriminant
with proper probabilistic interpretation.
3 A Discriminative Generative Observation Model
In this work, we track a target object based on its observations in the videos, i.e., ot . Since
the size of image region ot might change according to different st , we first convert ot to
a standard size and use it for tracking. In the following, we denote yt as the standardized
appearance vector of ot .
The dimensionality of the appearance vector yt is usually high. In our experiments, the
standard image size is a 19 ? 19 patch and thus yt is a 361-dimensional vector. We thus
model the appearance vector with a graphical model of low-dimensional latent variables.
3.1 A Generative Model with Latent Variables
A latent model relates a n-dimensional appearance vector y to a m-dimensional vector of
latent variables x:
y = Wx + ? +
(2)
where W is a n ? m projection matrix associating y and x, ? is the mean of y, and is
additive noise. As commonly assumed in factor analysis [1] and other graphical models [6],
the latent variables x are independent with unit variance, x ? N (0, Im ), where Im is the
m-dimensional identity matrix, and is zero mean Gaussian noise, ? N (0, ? 2 In ). Since
x and are both Gaussians, it follows that y is also a Gaussian distribution, y ? N (?, C),
where C = W W T + ? 2 I and In is an n-dimensional identity matrix. Together with (2),
we have a generative observation model:
p(ot |st ) = p(yt |W, ?, ) ? N (yt |?, W W T + ? 2 In )
(3)
This latent variable model follows the form of probabilistic principle component analysis,
and its parameters can be estimated from a set of examples [10]. Given a set of appearance
P
samples Y = {y1 , . . . , yN }, the covariance matrix of Y is denoted as S = N1 (y ?
?)(y ? ?)T . Let {?i |i = 1, . . . , N } be the eigenvalues of S arranged in descending order,
i.e., ?i ? ?j if i < j. Also, define the diagonal matrix ?m = diag(?1 , . . . , ?m ), and let
Um be the eigenvectors that corresponds to the eigenvalues in ?m . Tipping and Bishop
[10] show that the the maximum likelihood estimate of ?, W and can be obtained by
N
n
X
1 X
1
yi , W = Um (?m ? ? 2 Im )1/2 R, ? 2 =
?i
(4)
?=
N i=1
n ? m i=m+1
where R is an arbitrary m ? m orthogonal rotation matrix.
To model all possible appearance variations of a target object (due to pose, illumination
and view angle change), one could resort to a mixture of PPCA models. However, it not
only requires significant computation for estimating the model parameters but also leads
to other serious questions such as the number of components as well as under-fitting or
over-fitting. On the other hand, at any given time a linear PPCA model suffices to model
gradual appearance variation if the model is constantly updated. In this paper, we use a
single PPCA, and dynamically adapt the model parameters W , ?, and ? 2 to account for
appearance change.
3.1.1 Probability computation with Probabilistic PCA
Once the model parameters are known, we can compute the probability that a vector y is a
sample of this generative appearance model. From (4), the log-probability is computed by
1
(5)
L(W, ?, ? 2 ) = ? N log 2? + log |C| + y T C ?1 y
2
where y = y ? ?. Neglecting the constant terms, the log-probability is determined by
y T C ?1 y. Together with C = W W T + ? 2 In and (4), it follows that
1 T
T
T
yT C ?1 y = y T Um ??1
(6)
m Um y + 2 y (In ? Um Um )y
?
T
Here yT Um ??1
m Um y is the Mahalanobis distance of y in the subspace spanned by Um , and
T
T
y (In ? Um Um )y is the shortest distance from y to this subspace spanned by Um . Usually
? is set to a small value, and consequently the probability will be determined solely by the
distance to the subspace. However, the choice of ? is not trivial. From (6), if the ? is set to
a value much smaller than the actual one, the distance to the subspace will be favored and
ignore the contribution of Mahalanobis distance, thereby rendering an inaccurate estimate.
The choice of ? is even more critical in situations where the appearance changes dynamically and requires ? to be adjusted accordingly. This topic will be further examined in the
following section.
3.1.2 Online Learning of Probabilistic PCA
Unlike the analysis in the previous section where model parameters are estimated based on
a fixed set of training examples, our generative model has to learn and update its parameters
on line. Starting with a single example (the appearance of the tracked object in the first
video frame), our generative model constantly updates its parameters as new observations
arrive.
The equations for updating parameters are derived from (4). The update procedure of Um
and ?m is complicated since it involves the computations of eigenvalues and eigenvectors.
Here we use a forgetting factor ? to put more weights on the most recent data. Denote the
newly arrived samples at time t as Y = {y 1 , . . . , y M }, and assume the mean ? is fixed,
t
Um
and ?tm can be obtained by performing singular value decomposition (SVD) on
p
?
(7)
[ ?Um,t?1 (?m,t?1 )1/2 | (1 ? ?)Y ]
1/2
where Y = [y 1 ? ?, . . . , y M ? ?]. ?m,t and Um,t will contain the m-largest singular values
and the corresponding singular vectors respectively at time t. This update procedure can
be efficiently implemented using the R-SVD algorithm, e.g., [4] [7].
If the mean ? constantly changes, the above update procedure can not be applied. We
1/2
recently proposed a method [8] to compute SVD with correct updated mean in which ?m,t
and Um,t can be obtained by computing SVD on
p
p
h?
i
?Um,t?1 (?m,t?1 )1/2 (1 ? ?)Y (1 ? ?)?(?t?1 ? ?Y )
(8)
P
M
1
i
where Y = [y 1 ? ?Y , . . . , y M ? ?Y ] and ?Y = M
i=1 y . This formulation is similar to
the SVD computation with the fixed mean case, and the same incremental SVD algorithm
1/2
can be used to compute ?m,t and Um,t with an extra term shown in (8).
Computing and updating ? is more difficult than the form in (8). In the previous section,
we show that an inaccurate value of ? will severely affect probability estimates. In order
to have an accurate estimate of ? using (4), a large set of training examples is usually
required. Our generative model starts with a single example and gradually adapts the model
parameters. If we update ? based on (4), we will start with a very small value of ? since
there are only a few samples at our disposal at the outset, and the algorithm could quickly
lose track of the target because of an inaccurate probability estimate. Since the training
examples are not permanently stored in memory, ?i in (4) and consequently ? may not be
accurately updated if the number of drawn samples is insufficient. These constraints lead
us to develop a method that adaptively adjusts ? according to the newly arrived samples,
which will be explained in the next section.
3.2 Discriminative Generative Model
As is observed in Section 2, the object?s appearance at ot?1 and ot do not change much.
Therefore, we can use the observation at ot?1 to boost the likelihood measurement in ot .
That is, we draw a set samples (i.e., image patches) parameterized by {sit?1 |i = 1, ..., k}
in ot?1 that have large p(ot?1 |sit?1 ), but the low posterior p(sit?1 |Ot?1 ). These are treated
as the negative samples (i.e., samples that are not generated from the class of the target
object) that the generative model is likely to confuse at Ot .
Given a set samples Y 0 = {y 1 , . . . , y k } where y i is the appearance vector collected in ot?1
based on state parameter sit?1 , we want to find a linear projection V ? that projects Y 0 onto
a subspace such that the likelihood of Y 0 in the subspace is minimized. Let V be a p ? n
matrix and since p(y|W, ?, ?) is a Gaussian, p(V y|V, W, ?, ?) ? N (V ?, V CV T ) is a also
Gaussian. The log likelihood is computed by
k
(9)
L(V, W, ?, ?) = ? p log(2?) + log |V CV T | + tr((V CV T )?1 V S 0 V T )
2
P
where S 0 = k1 ki=1 (y i ? ?)(y i ? ?)T .
To facilitate the following analysis we first assume V projects Y 0 to a 1-D space, i.e., p = 1
and V = v T , and thus
k
vT S 0 v
T
L(V, W, ?, ?) = ?
log(2?) + log |v Cv| + T
(10)
2
v Cv
Note that v T Cv is the variance of the object samples in the projected space, and we need
to impose a constraint, e.g., v t Cv = 1, to ensure that the minimum likelihood solution of
v does not increase the variance in the projected space. Let v T Cv = 1, the optimization
problem becomes
vT S 0 v
v ? = arg max v T S 0 v = arg max T
(11)
v
v Cv
{v|v T Cv=1}
Thus, we obtain an equation exactly like the Fisher discriminant analysis for a binary classification problem. In (11), v is a projection that keeps the object?s samples in the projected
space close to the ? (with variance v T Cv = 1), while keeping negative samples in Y 0 away
from ?. The optimal value of v is the generalized eigenvector of S 0 and C that corresponds
to largest eigenvalue. In a general case, it follows that
|V S 0 V T |
(12)
V ? = arg max |V S 0 V T | = arg max
V
|V CV T |
{V CV T =I}
where V ? can be obtained by solving a generalized eigenvalue problem of S 0 and C. By
projecting observation samples onto a low dimensional subspace, we enhance the discriminative power of the generative model. In the meanwhile, we reduce the time required to
compute probabilities, which is also a critical improvement for real time applications like
object tracking.
3.2.1 Online Update of Discriminative Analysis
The computation of the projection matrix V depends on matrices C and S 0 . In section
3.1.2, we have shown the procedures to update C. The same procedures can be used to
P
P
update S 0 . Let ?Y 0 = k1 ki=1 y i and SY 0 = k1 ki=1 (y i ? ?Y 0 )(y i ? ?Y 0 )T ,
S0 =
k
1X i
(y ? ?)(y i ? ?)T = Sy + (? ? ?Y 0 )(? ? ?Y 0 )T
k i=1
(13)
Given S 0 and C, V is computed by solving a generalized eigenvalue problem. If we decompose S 0 = AT A and C = B T B, then we can find V more efficiently using generalized
singular value decomposition. Denote UY 0 and ?Y 0 as the SVD of SY 0 , it follows that by
1/2
1/2
letting A = [UY 0 ?Y 0 | (? ? ?Y 0 )]T and B = [Um ?m |? 2 I]T , we obtain S 0 = AT A and
T
C = B B.
As is detailed in [4] , V can be computed by first performing a QR factorization:
A
QA
=
R
(14)
B
QB
and computing the singular value decomposition of QA
(15)
QA = UA DA VAT
?1
, we then obtain V = R VA . The rank of A is usually small in vision applications, and V
can be computed efficiently, thereby facilitating tracking the process.
4 Proposed Tracking Algorithm
In this section, we summarize the proposed tracking algorithm and demonstrate how the
abovementioned learning and inference algorithms are incorporated for object tracking.
Our algorithm localizes the tracked object in each video frame using a rectangular window.
A state s is a length-5 vector, s = (x, y, ?, w, h), that parameterizes the windows position
(x, y), orientation (?) and width and height (w, h). The proposed algorithm is based on
maximum likelihood estimate (i.e., the most probable location of the object) given all the
observations up to that time instance, s?t = arg maxst p(st |Ot ).
We assume that state transition is a Gaussian distribution, i.e.,
p(st |st?1 ) ? N (st?1 , ?s )
(16)
where ?s is a diagonal matrix. According to this distribution, the tracker then draws N
samples St = {c1 , . . . , cN } which represent the possible locations of the target. Denote yti
as the appearance vector of ot , and Yt = {yt1 , . . . , ytN } as a set of appearance vectors that
corresponds to the set of state vectors St . The posterior probability that the tracked object
is at ci in video frame ot is then defined as
p(st = ci |Ot ) = ?p(yti |V, W, ?, ?)p(st = ci |s?t?1 )
(17)
where ? is a constant. Therefore, s?t = arg maxci ?St p(st = ci |Ot ).
Once s?t is determined, the corresponding observation yt? will be a new example to update
W and ?. Appearance vectors yti with large p(yti |V, W, ?, ?) but whose corresponding state
parameters ci are away from s?t will be used as new examples to update V .
Our tracking assumes o1 and s?1 are given (through object detection) and thus obtains the
first appearance vector y1 which in turn is used an the initial value of ?, but V and W are
unknown at the outset. When V and W are not available, our tracking algorithm is based
on template matching (with ? being the template). The matrix W is computed after a small
number of appearance vectors are observed. When W is available, we can then start to
compute and update V accordingly.
As mentioned in the Section 3.1.1, it is difficult to obtain an accurate estimate of ?. In our
tracking the system, we adaptively adjust ? according to ?m in W . We set ? be a fixed
fraction of the smallest eigenvalues in ?m . This will ensure the distance measurement
in (6) will not be biased to favor either the Mahalanobis distance in the subspace or the
distance to the subspace.
5 Experimental Results
We tested the proposed algorithm with numerous object tracking experiments. To examine whether our model is able to adapt and track objects in the dynamically changing environment, we recorded videos containing appearance deformation, large illumination change, and large pose variations. All the image sequences consist of 320 ? 240
pixel grayscale videos, recorded at 30 frames/second and 256 gray-levels per pixel. The
forgetting term is empirically selected as 0.85, and the batch size for update is set to 5
as a trade-off of computational efficiency as well as effectiveness of modeling appearance change due to fast motion. More experimental results and videos can be found at
http://www.ifp.uiuc.edu/?rlin1/adgm.html.
Figure 1: A target undergoes pose and lighting variation.
Figures 1 and 2 show snapshots of some tracking results enclosed with rectangular windows. There are two rows of images below each video frame. The first row shows the
sampled images in the current frame that have the largest likelihoods of being the target locations according our discriminative generative model. The second row shows the sample
images in the current video frame that are selected online for updating the discriminative
generative model.
The results in Figure 1 show the our method is able to track targets undergoing pose and
lighting change. Figure 2 shows tracking results where the object appearances change
significantly due to variation in pose and lighting as well as cast shadows. These experiments demonstrate that our tracking algorithm is able to follow objects even when there
is a large appearance change due to pose or lighting variation. We have also tested these
two sequences with conventional view-based eigentracker [2] or template-based method.
Empirical results show that such methods do not perform well as they do not update the
object representation to account for appearance change.
Figure 2: A target undergoes large lighting and pose variation with cast shadows.
6 Conclusion
We have presented a discriminative generative framework that generalizes the conventional
Fisher Linear Discriminant algorithm with a proper probabilistic interpretation. For object
tracking, we aim to find a discriminative generative model that best separates the target
class from the background. With a computationally efficient algorithm that constantly update this discriminative model as time progresses, our method adapts the discriminative
generative model to account for appearance variation of the target and background, thereby
facilitating the tracking task in different situations. Our experiments show that the proposed model is able to learn a discriminative generative model for tracking target objects
undergoing large pose and lighting changes. We also plan to apply the proposed method to
other problems that deal with non-stationary data stream in our future work.
References
[1] T. W. Anderson. An Introduction to Multivariate Statistical Analysis. Wiley, New York, 1984.
[2] M. J. Black and A. D. Jepson. Eigentracking: Robust matching and tracking of articulated objects using
view-based representation. In B. Buxton and R. Cipolla, editors, Proceedings of the Fourth European
Conference on Computer Vision, LNCS 1064, pp. 329?342. Springer Verlag, 1996.
[3] R. T. Collins and Y. Liu. On-line selection of discriminative tracking features. In Proceedings of the Ninth
IEEE International Conference on Computer Vision, volume 1, pp. 346?352, 2003.
[4] G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press, 1996.
[5] M. Isard and A. Blake. Contour tracking by stochastic propagation of conditional density. In B. Buxton
and R. Cipolla, editors, Proceedings of the Fourth European Conference on Computer Vision, LNCS 1064,
pp. 343?356. Springer Verlag, 1996.
[6] M. I. Jordan, editor. Learning in Graphical Models. MIT Press, 1999.
[7] A. Levy and M. Lindenbaum. Sequential Karhunen-Loeve basis extraction and its application to images.
IEEE Transactions on Image Processing, 9(8):1371?1374, 2000.
[8] R.-S. Lin, D. Ross, J. Lim, and M.-H. Yang. Incremental subspace update with running mean.
Technical report, Beckman Institute, University of Illinois at Urbana-Champaign, 2004. available at
http://www.ifp.uiuc.edu/?rlin1/isuwrm.pdf.
[9] D. Ross, J. Lim, and M.-H. Yang. Adaptive probabilistic visual tracking with incremental subspace update.
In T. Pajdla and J. Matas, editors, Proceedings of the Eighth European Conference on Computer Vision,
LNCS 3022, pp. 470?482. Springer Verlag, 2004.
[10] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Journal of the Royal Statistical Society, Series B, 61(3):611?622, 1999.
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1,809 | 2,643 | Hierarchical Bayesian Inference in
Networks of Spiking Neurons
Rajesh P. N. Rao
Department of Computer Science and Engineering
University of Washington, Seattle, WA 98195
[email protected]
Abstract
There is growing evidence from psychophysical and neurophysiological
studies that the brain utilizes Bayesian principles for inference and decision making. An important open question is how Bayesian inference
for arbitrary graphical models can be implemented in networks of spiking neurons. In this paper, we show that recurrent networks of noisy
integrate-and-fire neurons can perform approximate Bayesian inference
for dynamic and hierarchical graphical models. The membrane potential
dynamics of neurons is used to implement belief propagation in the log
domain. The spiking probability of a neuron is shown to approximate the
posterior probability of the preferred state encoded by the neuron, given
past inputs. We illustrate the model using two examples: (1) a motion detection network in which the spiking probability of a direction-selective
neuron becomes proportional to the posterior probability of motion in
a preferred direction, and (2) a two-level hierarchical network that produces attentional effects similar to those observed in visual cortical areas
V2 and V4. The hierarchical model offers a new Bayesian interpretation
of attentional modulation in V2 and V4.
1
Introduction
A wide range of psychophysical results have recently been successfully explained using
Bayesian models [7, 8, 16, 19]. These models have been able to account for human responses in tasks ranging from 3D shape perception to visuomotor control. Simultaneously,
there is accumulating evidence from human and monkey experiments that Bayesian mechanisms are at work during visual decision making [2, 5]. The versatility of Bayesian models
stems from their ability to combine prior knowledge with sensory evidence in a rigorous
manner: Bayes rule prescribes how prior probabilities and stimulus likelihoods should be
combined, allowing the responses of subjects or neural responses to be interpreted in terms
of the resulting posterior distributions.
An important question that has only recently received attention is how networks of cortical neurons can implement algorithms for Bayesian inference. One powerful approach has
been to build on the known properties of population coding models that represent information using a set of neural tuning curves or kernel functions [1, 20]. Several proposals have
been made regarding how a probability distribution could be encoded using population
codes ([3, 18]; see [14] for an excellent review). However, the problem of implementing
general inference algorithms for arbitrary graphical models using population codes remains
unresolved (some encouraging initial results are reported in Zemel et al., this volume). An
alternate approach advocates performing Bayesian inference in the log domain such that
multiplication of probabilities is turned into addition and division to subtraction, the latter
operations being easier to implement in standard neuron models [2, 5, 15] (see also the
papers by Deneve and by Yu and Dayan in this volume). For example, a neural implementation of approximate Bayesian inference for a hidden Markov model was investigated in
[15]. The question of how such an approach could be generalized to spiking neurons and
arbitrary graphical models remained open.
In this paper, we propose a method for implementing Bayesian belief propagation in networks of spiking neurons. We show that recurrent networks of noisy integrate-and-fire
neurons can perform approximate Bayesian inference for dynamic and hierarchical graphical models. In the model, the dynamics of the membrane potential is used to implement
on-line belief propagation in the log domain [15]. A neuron?s spiking probability is shown
to approximate the posterior probability of the preferred state encoded by the neuron, given
past inputs. We first show that for a visual motion detection task, the spiking probability
of a direction-selective neuron becomes proportional to the posterior probability of motion
in the neuron?s preferred direction. We then show that in a two-level network, hierarchical
Bayesian inference [9] produces responses that mimic the attentional effects seen in visual
cortical areas V2 and V4.
2
Modeling Networks of Noisy Integrate-and-Fire Neurons
2.1 Integrate-and-Fire Model of Spiking Neurons
We begin with a recurrently-connected network of integrate-and-fire (IF) neurons receiving
feedforward inputs denoted by the vector I. The membrane potential of neuron i changes
according to:
X
X
dvi
?
= ?vi +
wij Ij +
uij vj0
(1)
dt
j
j
where ? is the membrane time constant, Ij denotes the synaptic current due to input neuron
j, wij represents the strength of the synapse from input j to recurrent neuron i, vj0 denotes the synaptic current due to recurrent neuron j, and uij represents the corresponding
synaptic strength. If vi crosses a threshold T , the neuron fires a spike and vi is reset to the
potential vreset . Equation 1 can be rewritten in discrete form as:
X
X
vi (t + 1) = vi (t) + (?vi (t) +
wij Ij (t)) +
uij vj0 (t))
(2)
j
i.e.
vi (t + 1)
=
X
wij Ij (t) +
X
j
u0ij vj0 (t)
(3)
j
j
where is the integration rate, u0ii = 1 + (uii ? 1) and for i 6= j, u0ij = uij .
A more general integrate-and-fire model that takes into account some of the effects of nonlinear filtering in dendrites can be obtained by generalizing Equation 3 as follows:
X
X
vi (t + 1) = f
wij Ij (t) + g
u0ij vj0 (t)
(4)
j
j
where f and g model potentially different dendritic filtering functions for feedforward and
recurrent inputs.
2.2 Stochastic Spiking in Noisy IF Neurons
To model the effects of background inputs and the random openings of membrane channels,
one can add a Gaussian white noise term to the right hand side of Equations 3 and 4. This
makes the spiking of neurons in the recurrent network stochastic. Plesser and Gerstner [13]
and Gerstner [4] have shown that under reasonable assumptions, the probability of spiking
in such noisy neurons can be approximated by an ?escape function? (or hazard function)
that depends only on the distance between the (noise-free) membrane potential v i and the
threshold T . Several different escape functions were studied. Of particular interest to the
present paper is the following exponential function for spiking probability suggested in [4]
for noisy integrate-and-fire networks:
P (neuron i spikes at time t) = ke(vi (t)?T )/c
(5)
where k and c are arbitrary constants. We used a model that combines Equations 4 and 5
to generate spikes, with an absolute refractory period of 1 time step.
3
Bayesian Inference using Spiking Neurons
3.1 Inference in a Single-Level Model
We first consider on-line belief propagation in a single-level dynamic graphical model and
show how it can be implemented in spiking networks. The graphical model is shown in
Figure 1A and corresponds to a classical hidden Markov model. Let ?(t) represent the
hidden state of a Markov model at time t with transition probabilities given by P (?(t) =
?i |?(t ? 1) = ?j ) = P (?it |?jt?1 ) for i, j = 1 . . . N . Let I(t) be the observable output
governed by the probabilities P (I(t)|?(t)). Then, the forward component of the belief
propagation algorithm [12] prescribes the following ?message? for state i from time step t
to t + 1:
X
mt,t+1
= P (I(t)|?it )
P (?it |?jt?1 )mt?1,t
(6)
i
j
j
m0,1
i
If
= P (?i ) (the prior distribution over states), then it is easy to show using Bayes
rule that mt,t+1
= P (?it , I(t), . . . , I(1)). If the probabilities are normalized at each update
i
step:
X
P (?it |?jt?1 )mt?1,t
/nt?1,t
(7)
mt,t+1
= P (I(t)|?it )
j
i
j
P
where nt?1,t = j mt?1,t
, then the message becomes equal to the posterior probability
j
of the state and current input, given all past inputs:
mt,t+1
= P (?it , I(t)|I(t ? 1), . . . , I(1))
i
(8)
3.2 Neural Implementation of the Inference Algorithm
By comparing the membrane potential equation (Eq. 4) with the on-line belief propagation equation (Eq. 7), it is clear that the first equation can implement the second if belief
propagation is performed in the log domain [15], i.e., if:
vi (t + 1) ? log mt,t+1
i
X
f
wij Ij (t) = log P (I(t)|?it )
(9)
(10)
j
g
X
j
u0ij vj0 (t)
=
log(
X
P (?it |?jt?1 )mt?1,t
/nt?1,t )
j
(11)
j
In this model, the dendritic filtering functions f and g approximate the logarithm function1 , the synaptic currents Ij (t) and vj0 (t) are approximated by the corresponding instantaneous firing rates, and the recurrent synaptic weights u0ij encode the transition probabilities P (?it |?jt?1 ). Normalization by nt?1,t is implemented by subtracting log nt?1,t using
inhibition.
1
An alternative approach, which was also found to yield satisfactory results, is to approximate the
log-sum with a linear weighted sum [15], the weights being chosen to minimize the approximation
error.
t
t+1
?t
t+1
? t+1
? t+1
1
?t
1
I(t+1)
I(t)
? t+1
2
?t
2
t
I(t)
B
A
I(t)
I(t+1)
I(t)
D
C
Figure 1: Graphical Models and their Neural Implementation. (A) Single-level dynamic graph-
ical model. Each circle represents a node denoting the state variable ? t which can take on values
?1 , . . . , ?N . (B) Recurrent network for implementing on-line belief propagation for the graphical
model in (A). Each circle represents a neuron encoding a state ? i . Arrows represent synaptic connections. The probability distribution over state values at each time step is represented by the entire
population. (C) Two-level dynamic graphical model. (D) Two-level network for implementing online belief propagation for the graphical model in (C). Arrows represent synaptic connections in the
direction pointed by the arrow heads. Lines without arrow heads represent bidirectional connections.
Finally, since the membrane potential vi (t + 1) is assumed to be proportional to log mt,t+1
i
(Equation 9), we have:
vi (t + 1) = c log mt,t+1
+T
(12)
i
for some constants c and T . For noisy integrate-and-fire neurons, we can use Equation 5 to
calculate the probability of spiking for each neuron i as:
P (neuron i spikes at time t + 1)
? e(vi (t+1)?T )/c
= e
log mt,t+1
i
=
mt,t+1
i
(13)
(14)
Thus, the probability of spiking (or equivalently, the instantaneous firing rate) for neuron i
in the recurrent network is directly proportional to the posterior probability of the neuron?s
preferred state and the current input, given all past inputs. Figure 1B illustrates the singlelevel recurrent network model that implements the on-line belief propagation equation 7.
3.3 Hierarchical Inference
The model described above can be extended to perform on-line belief propagation and
inference for arbitrary graphical models. As an example, we describe the implementation
for the two-level hierarchical graphical model in Figure 1C.
As in the case of the 1-level dynamic model, we define the following ?messages? within a
(message from state i to other states at level 1
particular level and between levels: mt,t+1
1,i
from time step t to t + 1), mt1?2,i (?feedforward? message from state i at level 1 sent to
(message from state i to other states at level 2 from time step t
level 2 at time t), mt,t+1
2,i
to t + 1), and mt2?1,i (?feedback? message from state i at level 2 sent to level 1 at time
t). Each of these messages can be calculated based on an on-line version of loopy belief
propagation [11] for the multiply connected two-level graphical model in Figure 1C:
XX
t?1
t
t
t
mt1?2,i =
P (?1,k
|?2,i
, ?1,j
)mt?1,t
(15)
1,j P (I(t)|?1,k )
j
mt2?1,i
=
X
j
k
t?1
t
P (?2,i
|?2,j
)mt?1,t
2,j
(16)
mt,t+1
1,i
t
)
= P (I(t)|?1,i
XX
j
mt,t+1
2,i
X
= mt1?2,i
t?1
t
t
|?2,j
, ?1,k
P (?1,i
)mt2?1,j mt?1,t
1,k
(17)
k
t?1
t
P (?2,i
|?2,j
)mt?1,t
2,j
(18)
j
Note the similarity between the last equation and the equation for the single-level model
(Equation 6). The equations above can be implemented in a 2-level hierarchical recurrent
network of integrate-and-fire neurons in a manner similar to the 1-level case. We assume
that neuron i in level 1 encodes ?1,i as its preferred state while neuron i in level 2 encodes ?2,i . We also assume specific feedforward and feedback neurons for computing and
conveying mt1?2,i and mt2?1,i respectively.
Taking the logarithm of both sides of Equations 17 and 18, we obtain equations that can
be computed using the membrane potential dynamics of integrate-and-fire neurons (Equation 4). Figure 1D illustrates the corresponding two-level hierarchical network. A modification needed to accommodate Equation 17 is to allow bilinear interactions between synaptic
inputs, which changes Equation 4 to:
X
XX
wij Ij (t) + g
(19)
vi (t + 1) = f
u0ijk vj0 (t)xk (t)
j
j
k
Multiplicative interactions between synaptic inputs have previously been suggested by several authors (e.g., [10]) and potential implementations based on active dendritic interactions
have been explored. The model suggested here utilizes these multiplicative interactions
within dendritic branches, in addition to a possible logarithmic transform of the signal before it sums with other signals at the soma. Such a model is comparable to recent models
of dendritic computation (see [6] for more details).
4
Results
4.1 Single-Level Network: Probabilistic Motion Detection and Direction Selectivity
We first tested the model in a 1D visual motion detection task [15]. A single-level recurrent
network of 30 neurons was used (see Figure 1B). Figure 2A shows the feedforward weights
for neurons 1, . . . , 15: these were recurrently connected to encode transition probabilities biased for rightward motion as shown in Figure 2B. Feedforward weights for neurons
16, . . . , 30 were identical to Figure 2A but their recurrent connections encoded transition
probabilities for leftward motion (see Figure 2B). As seen in Figure 2C, neurons in the
network exhibited direction selectivity. Furthermore, the spiking probability of neurons
reflected the posterior probabilities over time of motion direction at a given location (Figure 2D), suggesting a probabilistic interpretation of direction selective spiking responses in
visual cortical areas such as V1 and MT.
4.2
Two-Level Network: Spatial Attention as Hierarchical Bayesian Inference
We tested the two-level network implementation (Figure 1D) of hierarchical Bayesian inference using a simple attention task previously used in primate studies [17]. In an input
image, a vertical or horizontal bar could occur either on the left side, right side, or both
sides (see Figure 3). The corresponding 2-level generative model consisted of two states
at level 2 (left or right side) and four states at level 1: vertical left, horizontal left, vertical
right, horizontal right. Each of these states was encoded by a neuron at the respective level.
The feedforward connections at level 1 were chosen to be vertically or horizontally oriented
Gabor filters localized to the left or right side of the image. Since the experiment used static
images, the recurrent connections at each level implemented transition probabilities close
to 1 for the same state and small random values for other states. The transition probabilities
t?1
t
t
P (?1,k
|?2,i
, ?1,j
) were chosen such that for ?2t = left side, the transition probabilities for
?t
w15
w1
1
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Figure 2: Responses from the Single-Level Motion Detection Network. (A) Feedforward weights
for neurons 1, . . . , 15 (rightward motion selective neurons). Feedforward weights for neurons
16, . . . , 30 (leftward motion selective) are identical. (B) Recurrent weights encoding the transition
probabilities P (?it+1 |?jt ) for i, j = 1, . . . , 30. Probability values are proportional to pixel brightness.
(C) Spiking responses of three of the first 15 neurons in the recurrent network (neurons 8, 10, and
12). As is evident, these neurons have become selective for rightward motion as a consequence of
the recurrent connections specified in (B). (D) Posterior probabilities over time of motion direction
(at a given location) encoded by the three neurons for rightward and leftward motion.
states ?1t coding for the right side were set to values close to zero (and vice versa, for ? 2t =
right side). As shown in Figure 3, the response of a neuron at level 1 that, for example,
prefers a vertical edge on the right mimics the response of a V4 neuron with and without
attention (see figure caption for more details). The initial setting of the priors at level 2 is
the crucial determinant of attentional modulation in level 1 neurons, suggesting that feedback from higher cortical areas may convey task-specific priors that are integrated into V4
responses.
5
Discussion and Conclusions
We have shown that recurrent networks of noisy integrate-and-fire neurons can perform
approximate Bayesian inference for single- and multi-level dynamic graphical models. The
model suggests a new interpretation of the spiking probability of a neuron in terms of the
posterior probability of the preferred state encoded by the neuron, given past inputs. We
illustrated the model using two problems: inference of motion direction in a single-level
network and hierarchical inference of object identity at an attended visual location in a twolevel network. In the first case, neurons generated direction-selective spikes encoding the
probability of motion in a particular direction. In the second case, attentional effects similar
to those observed in primate cortical areas V2 and V4 emerged as a result of imposing
appropriate priors at the highest level.
The results obtained thus far are encouraging but several important questions remain. How
does the approach scale to more realistic graphical models? The two-level model explored
in this paper assumed stationary objects, resulting in simplified dynamics for the two levels
in our recurrent network. Experiments are currently underway to test the robustness of the
proposed model when richer classes of dynamics are introduced at the different levels. An-
A
B
30
Spikes/second
Spikes/second
30
Ref Att Away
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20
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C
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D
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30
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20
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0
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Figure 3: Responses from the Two-Level Hierarchical Network. (A) Top panel: Input image
(lasting the first 15 time steps) containing a vertical bar (?Reference?) on the right side. Each input was convolved with a retinal spatiotemporal filter. Middle: Three sample spike trains from the
1st level neuron whose preferred stimulus was a vertical bar on the right side. Bottom: Posterior
probability of a vertical bar (= spiking probability or instantaneous firing rate of the neuron) plotted
over time. (B) Top panel: An input containing two stimuli (?Pair?). Below: Sample spike trains and
posterior probability for the same neuron as in (A). (C) When ?attention? is focused on the right side
(depicted by the white oval) by initializing the prior probability encoded by the 2nd level right-coding
neuron at a higher value than the left-coding neuron, the firing rate for the 1st level neuron in (A) increases to a level comparable to that in (A). (D) Responses from a neuron in primate area V4 without
attention (top panel, Ref Att Away and Pair Att Away; compare with (A) and (B)) and with attention
(bottom panel, Pair Att Ref; compare with (C)) (from [17]). Similar responses are seen in V2 [17].
other open question is how active dendritic processes could support probabilistic integration
of messages from local, lower-level, and higher-level neurons, as suggested in Section 3.
We intend to investigate this question using biophysical (compartmental) models of cortical
neurons. Finally, how can the feedforward, feedback, and recurrent synaptic weights in the
networks be learned directly from input data? We hope to investigate this question using
biologically-plausible approximations to the expectation-maximization (EM) algorithm.
Acknowledgments. This research was supported by grants from ONR, NSF, and the Packard Foundation. I am grateful to Wolfram Gerstner, Michael Shadlen, Aaron Shon, Eero Simoncelli, and Yair
Weiss for discussions on topics related to this paper.
References
[1] C. H. Anderson and D. C. Van Essen. Neurobiological computational systems. In Computational Intelligence: Imitating Life, pages 213?222. New York, NY: IEEE Press, 1994.
[2] R. H. S. Carpenter and M. L. L. Williams. Neural computation of log likelihood in control of
saccadic eye movements. Nature, 377:59?62, 1995.
[3] S. Deneve and A. Pouget. Bayesian estimation by interconnected neural networks (abstract no.
237.11). Society for Neuroscience Abstracts, 27, 2001.
[4] W. Gerstner. Population dynamics of spiking neurons: Fast transients, asynchronous states, and
locking. Neural Computation, 12(1):43?89, 2000.
[5] J. I. Gold and M. N. Shadlen. Neural computations that underlie decisions about sensory stimuli.
Trends in Cognitive Sciences, 5(1):10?16, 2001.
[6] M. Hausser and B. Mel. Dendrites: bug or feature? Current Opinion in Neurobiology, 13:372?
383, 2003.
[7] D. C. Knill and W. Richards. Perception as Bayesian Inference. Cambridge, UK: Cambridge
University Press, 1996.
[8] K. P. K?ording and D. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427:244?
247, 2004.
[9] T. S. Lee and D. Mumford. Hierarchical Bayesian inference in the visual cortex. Journal of the
Optical Society of America A, 20(7):1434?1448, 2003.
[10] B. W. Mel. NMDA-based pattern discrimination in a modeled cortical neuron. Neural Computation, 4(4):502?517, 1992.
[11] K. Murphy, Y. Weiss, and M. Jordan. Loopy belief propagation for approximate inference: An
empirical study. In Proceedings of UAI (Uncertainty in AI), pages 467?475. 1999.
[12] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA, 1988.
[13] H. E. Plesser and W. Gerstner. Noise in integrate-and-fire neurons: From stochastic input to
escape rates. Neural Computation, 12(2):367?384, 2000.
[14] A. Pouget, P. Dayan, and R. S. Zemel. Inference and computation with population codes.
Annual Review of Neuroscience, 26:381?410, 2003.
[15] R. P. N. Rao. Bayesian computation in recurrent neural circuits. Neural Computation, 16(1):1?
38, 2004.
[16] R. P. N. Rao, B. A. Olshausen, and M. S. Lewicki. Probabilistic Models of the Brain: Perception
and Neural Function. Cambridge, MA: MIT Press, 2002.
[17] J. H. Reynolds, L. Chelazzi, and R. Desimone. Competitive mechanisms subserve attention in
macaque areas V2 and V4. Journal of Neuroscience, 19:1736?1753, 1999.
[18] M. Sahani and P. Dayan. Doubly distributional population codes: Simultaneous representation
of uncertainty and multiplicity. Neural Computation, 15:2255?2279, 2003.
[19] Y. Weiss, E. P. Simoncelli, and E. H. Adelson. Motion illusions as optimal percepts. Nature
Neuroscience, 5(6):598?604, 2002.
[20] R. S. Zemel, P. Dayan, and A. Pouget. Probabilistic interpretation of population codes. Neural
Computation, 10(2):403?430, 1998.
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1,810 | 2,644 | A Temporal Kernel-Based Model for Tracking
Hand-Movements from Neural Activities
Lavi Shpigelman12 Koby Crammer1 Rony Paz23 Eilon Vaadia23 Yoram Singer1
1
School of computer Science and Engineering
2
Interdisciplinary Center for Neural Computation
3
Dept. of Physiology, Hadassah Medical School
The Hebrew University Jerusalem, 91904, Israel
Email for correspondance: [email protected]
Abstract
We devise and experiment with a dynamical kernel-based system for
tracking hand movements from neural activity. The state of the system
corresponds to the hand location, velocity, and acceleration, while the
system?s input are the instantaneous spike rates. The system?s state dynamics is defined as a combination of a linear mapping from the previous
estimated state and a kernel-based mapping tailored for modeling neural
activities. In contrast to generative models, the activity-to-state mapping
is learned using discriminative methods by minimizing a noise-robust
loss function. We use this approach to predict hand trajectories on the
basis of neural activity in motor cortex of behaving monkeys and find
that the proposed approach is more accurate than both a static approach
based on support vector regression and the Kalman filter.
1
Introduction
The paper focuses on the problem of tracking hand movements, which constitute smooth
spatial trajectories, from spike trains of a neural population. We do so by devising a dynamical system which employs a tailored kernel for spike trains along with a linear mapping
corresponding to the states? dynamics. Consider a situation where a subject performs free
hand movements during a task that requires accurate space and time precision. In the lab,
it may be a constrained reaching task while in real life it may be an every day task such as
eating. We wish to track the hand position given only spike trains from a recorded neural
population. The rationale of such an undertaking is two fold. First, this task can be viewed
as a stem towards the development of a Brain Machine Interface (BMI) which gradually and
rapidly become a possible future solution for the motor disabled patients. Recent studies of
BMIs [13, 3, 10] (being on-line and feedback enabled) show that a relatively small number
of cortical units can be used to move a cursor or a robot effectively, even without generation of hand movements and that training of the subjects improves the overall success of
the BMIs. Second, an open loop (off-line) movement decoding (see e.g. [7, 1, 15, 11, 8]),
while inappropriate for BMIs, is computationally less expensive, easier to implement and
allows repeated analysis thus providing a handle to understandings of neural computations
in the brain.
Early studies [6] showed that the direction of arm movement is reflected by the population
vector of preferred directions weighted by current firing rates, suggesting that intended
movement is encoded in the firing rate which, in turn, is modulated by the angle between a
unit?s preferred direction (PD) and the intended direction. This linear regression approach
is still prevalent and is applied, with some variation of the learning methods, in closed and
open loop settings. There is relatively little work on the development of dedicated nonlinear
methods.
Both movement and neural activity are dynamic and can therefore be naturally modeled by
dynamical systems. Filtering methods often employ generative probabilistic models such
as the well known Kalman filter [16] or more neurally specialized models [1] in which a
cortical unit?s spike count is generated by a probability function of its underlying firing
rate which is tuned to movement parameters. The movement, being a smooth trajectory,
is modeled as a linear transition with (typically additive Gaussian) noise. These methods
have the advantage of being aware of the smooth nature of movement and provide models
of what neurons are tuned to. However, the requirement of describing a neural population?s
firing probability as a function of movement state is hard to satisfy without making costly
assumptions. The most prominent is the assumption of statistical independence of cells
given the movement.
Kernel based methods have been shown to achieve state of the art results in many application domains. Discriminative kernel methods, such as Support Vector Regression (SVR)
forgo the task of modeling neuronal tuning functions. Furthermore, the construction of
kernel induced feature spaces, lends itself to efficient implementation of distance measures
over spike trains that are better suited to comparing two neural population trajectories than
the Euclidean distance in the original space of spike counts per bins [11, 5]. However,
SVR is a ?static? method that does not take into account the smooth dynamics of the predicted movement trajectory which imposes a statistical dependency between consecutive
examples.
This paper introduces a kernel based regression method that incorporates linear dynamics
of the predicted trajectories. In Sec. 2 we formally describe the problem setting. We introduce the movement tracking model and the associated learning framework in Sec. 3. The
resulting learning problem yields a new kernel for linear dynamical systems. We provide
an efficient calculation of this kernel and describe our dual space optimization method for
solving the learning problem. The experimental method is presented in Sec. 4. Results,
underscoring the merits of our algorithm are provided in Sec. 5 and conclusions are given
in Sec. 6.
2
Problem Setting
Our training set contains m trials. Each trial (typically indexed by i or j) consists of a pair
tiend
of movement and neural recordings, designated by Yi , Oi . Yi = yti t=1
is a time
d
i
series of movement state values and yt ? R is the movement state vector at time t in
trial i. We are interested in reconstructing position, however, for better modeling, yti may
be a vector of position, velocity and acceleration (as is the case in Sec. 4). This trajectory is
tiend
observed during model learning and is the inference target. Oi = {ot }t=1
is a time series
of neural spike counts and oit ? Rq is a vector
of spike counts from q cortical units at time
t. We wish to learn a function zit = f Oi1:t that is a good estimate (in a sense formalized
in the sequel) of the movement yti . Thus, f is a causal filtering method.
We confine ourselves to a causal setting since we plan to apply the proposed method in a
closed loop scenario where real-time output is required. The partition into separate trajectories is a natural one in a setting where a session is divided into many trials, each consisting
of one attempt at accomplishing the basic task (such as reaching movements to displayed
targets). In tasks that involve no hitting of objects, hand movements are typically smooth.
End point movement in small time steps is loosely approximated as having constant acceleration. On the other hand, neural spike counts (which are typically measured in bins
of 50 ? 100ms) vary greatly from one time step to the next. In summary, our goal is to
devise a dynamic mapping from sequences of neural activities ending at a given time to the
instantaneous hand movement characterization (location, velocity, and acceleration).
3
Movement Tracking Algorithm
Our regression method is defined as follows: given a series O ? Rq?tend of observations
and, possibly, an initial state y0 , the predicted trajectory Z ? Rd?tend is,
zt = Azt?1 + W? (ot ) , tend ? t > 0 ,
(1)
where z0 = y0 , A ? Rd?d is a matrix describing linear movement dynamics and
W ? Rd?q is a weight matrix. ? (ot ) is a feature vector of the observed spike trains
at time t and is later replaced by a kernel operator (in the dual formulation to follow).
Thus, the state transition is a linear transformation of the previous state with the addition
of a non-linear effect of the observation.
Pt
Note that unfolding the recursion in Eq. (1) yields zt = At y0 + k=1 At?k W? (ok ) .
Assuming that A describes stable dynamics (the real parts of the eigenvalues of A are les
than 1), then the current prediction depends, in an exponentially decaying manner, on the
previous observations. We further assume that A is fixed and wish to learn W (we describe
our choice of A in Sec. 4). In addition, ot may also encompass a series of previous spike
counts in a window ending at time t (as is the case in Sec. 4). Also, note that this model (in
its non-kernelized version) has an algebraic form which is similar to the Kalman filter (to
which we compare our results later).
Primal Learning Problem: The optimization problem presented here is identical to the
standard SVR learning problem (see, for example [12]) with the exception that zit is defined
as in Eq. (1) while in standard SVR, zt = W? (ot
) (i.e. without the linear dynamics).
m
Given a training set of fully observed trials Yi , Oi i=1 we define the learning problem
to be
i
min
W
m tX
d
end X
X
i
1
2
zt ? yti .
kWk + c
s
s ?
2
i=1 t=1 s=1
(2)
P
2
2
Where kWk = a,b (W)ab (is the Forbenius norm). The second term is a sum of training
errors (in all trials, times and movement dimensions). | ? |? is the ? insensitive loss and is
defined as |v|? = max {0, |v| ? ?}. The first term is a regularization term that promotes
small weights and c is a fixed constant providing a tradeoff between the regularization
term and the training error. Note that to compensate for different units and scales of the
movement dimensions one could either define a different ?s and cs for each dimension of
the movement or, conversely, scale the sth movement dimension. The tracking method,
combined with the optimization specified here, defines the complete algorithm. We name
this method the Discriminative Dynamic Tracker or DDT in short.
A Dual Solution: The derivation of the dual of the learning problem defined in Eq. (2)
is rather mundane (e.g. [12]) and is thus omitted. Briefly, we replace the ?-loss with pairs
of slack variables. We then write a Lagrangian of the primal problem and replace zit with
its (less-standard) definition. We then differentiate the Lagrangian with respect to the slack
variables and W and obtain a dual optimization problem. We present the dual dual problem
in a top-down manner, starting with the general form and finishing with a kernel definition.
The form of the dual is
T
T
T
max? ? 12 (?? ? ?) G (?? ? ?) + (?? ? ?) y ? (?? + ?)
?,?
s.t. ?, ?? ? [0, c]
.
(3)
Note that the above expression conforms to the dual form of SVR. Let ` equal the size of the
movement space (d), multiplied by the total number of time steps in all the training trajectories. ?, ?? ? R` are vectors of Lagrange multipliers, y ? R` is a column concatenation of
T T
1 T
m
all the training set movement trajectories y1 ? ? ? ytm
, = [?, . . . , ?]T ? R`
end
and G ? R`?` is a Gram matrix (vT denotes transposition). One obvious difference between our setting and the standard SVR lies within the size of the vectors and Gram matrix.
In addition, a major difference is the definition of G. We define G here in a hierarchical
manner. Let i, j ? {1, . . . , m} be trajectory (trial) indexes. G is built from blocks indexed
by Gij , which are in turn made from basic blocks, indexed by Kij
tq as follows
?
?
?
?
ij
K11
???
Kij
G11 ? ? ? G1m
1tj
?
?
..
..
?
?
..
..
ij
..
?,
G = ? ...
? , G =?
.
.
.
.
.
?
?
ij
ij
Kti 1 ? ? ? Kti tj
Gm1 ? ? ? Gmm
end
end end
where block Gij refers to a pair of trials (i and j). Finally Each basic block, Kij
tq refers to a
j
i
pair of time steps t and q in trajectories i and j respectively. tend , tend are the time lengths
of trials i and j. Basic blocks are defined as
q
t X
X
ij
T
=
Kij
At?r krs
Aq?s ,
(4)
tq
r=1 s=1
ij
= k oir , ojs is a (freely chosen) basic kernel between the two neural observawhere krs
tions oir and ojs at times r and s in trials i and j respectively. For an explanation of kernel
operators we refer the
readerto [14] and mention that the kernel operator can be viewed
as computing ? oir ? ? ojs where ? is a fixed mapping to some inner product space.
The choice of kernel (being the choice of feature space) reflects a modeling decision that
specifies how similarities
P between neural patterns are measured. The resulting dual form
of the tracker is zt = k ?k Gtk where Gt is the Gram matrix row of the new example.
It is therefore clear from Eq. (4) that the linear dynamic characteristics of DDT results in
a Gram matrix whose entries depend on previous observations. This dependency is exponentially decaying as the time difference between events in the trajectories grow. Note
that solution of the dual optimization problem in Eq. (3) can be calculated by any standard quadratic programming optimization tool. Also, note that direct calculation of G is
inefficient. We describe an efficient method in the sequel.
Efficient Calculation of the Gram Matrix Simple, straight-forward calculation of the
Gram matrix is time consuming. To illustrate this, suppose each trial is of length tiend = n,
then calculation of each basic block would take ?(n2 ) summation steps. We now describe
a procedure based on dynamic-programming method for calculating the Gram matrix in a
constant number of operations for each basic block.
Omitting the indexing over trials to ease notation, we are interested in calculating the basic
Pt
block Ktq . First, define Btq = k=1 kkq At?k . the basic block Ktq can be recursively
calculated in three different ways:
Ktq = Kt(q?1) AT + Btq
(5)
Ktq
= AK(t?1)q + (Bqt )
T
T
(6)
T
Ktq = AK(t?1)(q?1) A + (Bqt ) + Btq ? ktq .
(7)
Thus, by adding Eq. (5) to Eq. (6) and subtracting Eq. (7) we get
Ktq = AK(t?1)q + Kt(q?1) AT ? AK(t?1)(q?1) AT + ktq I .
Btq (and the entailed summation) is eliminated in exchange for a 2D dynamic program with
initial conditions: K11 = k11 I , K1q = K1(q?1) AT + k1q I , Kt1 = AK(t?1)1 + kt1 I.
Table 1: Mean R2 , MAE? & MSE (across datasets, folds, hands and directions) for each algorithm.
Algorithm
Kalman filter
DDT-linear
SVR-Spikernel
DDT-Spikernal
pos.
0.64
0.59
0.61
0.73
R2
vel.
0.58
0.49
0.64
0.67
accl.
0.30
0.17
0.37
0.40
pos.
0.40
0.63
0.44
0.37
MAE?
vel.
0.15
0.41
0.14
0.14
accl.
0.37
0.58
0.34
0.34
pos.
0.78
0.97
0.76
0.50
MSE
vel. accl.
0.27 1.16
0.50 1.23
0.20 0.98
0.16 0.91
DDT?Spikernel, R2 Scores
1
0.8
0.6
0.4
left hand, X dir.
left hand, Y dir.
0.2
right hand, X dir.
right hand, Y dir.
0
0
0.2
0.4
0.6
0.8
Kalman filter, R2 Scores
1
0
0.2
0.4
0.6
0.8
DDT?linear, R2 Scores
1
0
0.2
0.4
0.6
0.8
SVR?Spikernel, R2 Scores
Figure 1: Correlation coefficients (R2 , of predicted and observed hand positions) comparisons of
the DDT-Spikernel versus the Kalman filter (left), DDT-linear (center) and SVR-Spikernel (right).
Each data point is the R2 values obtained by the DDT-Spikernel and by another method in one fold
of one of the datasets for one of the two axes of movement (circle / square) and one of the hands
(filled/non-filled). Results above the diagonals are cases were the DDT-Spikernel outperformes.
Suggested Optimization Method. One possible way to solve the optimization problem
(essentially, a modification of the method described in [4] for classification) is to sequentially solve a reduced problem with respect to a single constraint at a time. Define:
X ?
X ?
?i =
?j ? ?j Gij ? yi ? min
?j ? ?j Gij ? yi .
?
?i ,?i ?[0,c]
j
j
?
?
Then ?i is the amount of ?-insensitive error that can be corrected for example i by keeping
(?)
(?)
all ?j6=i constant and changing ?i . Optimality is reached by iteratively choosing the
(?)
example with the largest ?i and changing its ?i
error for this example.
4
within the [0, c] limits to minimize the
Experimental Setting
The data used in this work was recorded from the primary motor cortex of a Rhesus
(Macaca Mulatta) monkey (?4.5 kg). The monkey sat in a dark chamber, and up to 8
electrodes were introduced into MI area of each hemisphere. The electrode signals were
amplified, filtered and sorted. The data used in this report was recorded on 8 different days
and includes hand positions, sampled at 500Hz, spike times of single units (isolated by signal fit to a series of windows) and of multi units (detection by threshold crossing) sampled
at 1ms precision. The monkey used two planar-movement manipulanda to control 2 cursors on the screen to perform a center-out reaching task. Each trial began when the monkey
centered both cursors on a central circle. Either cursor could turn green, indicating the hand
to be used in the trial. Then, one of eight targets appeared (?go signal?), the center circle
disappeared and the monkey had to move and reach the target to receive liquid reward. The
number of multi-unit channels ranged from 5 to 15, the number of single units was 20-27
and the average total was 34 units per dataset. The average spike rate per channel was 8.2
spikes/sec. More information on the recordings can be found in [9].
1
DDT (Spikernel)
DDT (Spikernel)
75%
88.1%
100%
100%
Kalman Filter
78.12%
80.0%
96.3%
Kalman Filter
DDT (Linear)
91.88%
SVR (Spikernel)
98.7%
86.3%
95.6%
86.8%
62.5%
DDT (Linear)
78.7%
87.5%
SVR (Spikernel)
99.4%
63.75%
SVR (Spikernel)
DDT (Spikernel)
Kalman Filter
84.4%
DDT (Linear)
Figure 2: Comparison of R2 -performance between algorithms. Each algorithm is represented by a
vertex. The weight of an edge between two algorithms is the fraction of tests in which the algorithm
on top achieves higher R2 score than the other. A bold edge indicates a fraction higher than 95%.
Graphs from left to right are for position, velocity, and acceleration respectively.
The results that we present here refer to prediction of instantaneous hand movements during
the period from ?Go Signal? to ?Target Reach? times of both hands in successful trials.
Note that some of the trials required movement of the left hand while keeping the right
hand steady and vise versa. Therefore, although we considered only movement periods
of the trials, we had to predict both movement and non-movement for each hand. The
cumulative time length of all the datasets was about 67 minutes. Since the correlation
between the movements of the two hands tend to zero - we predicted movement for each
T
hand separately, choosing the movement space to be [x, y, vx , vy , ax , ay ] for each of the
T
hands (preliminary results using only [x, y, vx , vy ] were less accurate).
We preprocessed the spike trains into spike counts in a running windows of 100ms (choice
of window size is based on previous experience [11]). Hand position, velocity and acceleration were calculated using the 500Hz recordings. Both spike counts and hand movement
were then sampled at steps of 100ms (preliminary results
with
step size 50ms were negligibly different for all algorithms). A labeled example yti , oit for time t in trial i consisted
of the previous 10 bins of population spike counts and the state, as a 6D vector for the left
or right hand. Two such consecutive examples would than have 9 time bins of spike count
overlap. For example, the number of cortical units q in the first dataset was 43 (27 single
and 16 multiple) and the total length of all the trials that were used in that dataset is 529
seconds. Hence in that session there are 5290 consecutive examples where each is a 43?10
matrix of spike counts along with two 6D vectors of end point movement.
In order to run our algorithm we had to choose base kernels, their parameters, A and c (and
?, to be introduced below). We used the Spikernel [11], a kernel designed to be used with
spike rate patterns, and the simple dot product (i.e. linear regression). Kernel parmeters and
c were chosen (and subsequently held fixed) by 5 fold cross validation over half of the first
dataset only. We compared DDT with the Spikernel and with the linear kernel to standard
SVR using the Spikernel and the Kalman filter. We also obtained tracking results using
both DDT and SVR with the standard exponential kernel. These results were slightly less
accurate on average than with the Spikernel and are therefore omitted here. The Kalman
filter was learned assuming the standard state space model (yt = Ayt?1 + ? , ot =
Hyt +?, where ?, ? are white Gaussian noise with appropriate correlation matrices) such as
in [16]. y belonged to the same 6D state space as described earlier. To ease the comparison
- the same matrix A that was learned for the Kalman filter was used in our algorithm
(though we show that it is not optimal for DDT), multiplied by a scaling parameter ?. This
parameter was selected to produce best position results on the training set. The selected ?
value is 0.8.
The figures that we show in Sec. 5 are of test results in 5 fold cross validation on the rest
of the data. Each of the 8 remaining datasets was divided into 5 folds. 4/5 were used for
MSE
X
# Support
Position
MAE
14K
Y
position
2
R
Acceleration
8K
6K
?
?
?
?
Figure 3: Effect of ? on R2 , MAE? ,MSE and
number of support vectors.
velocity
10K
Actual
DDT?Spikernel
SVR?Spikernel
acceleration
Velocity
12K
Figure 4: Sample of tracking with the DDTSpikernel and the SVR-Spikernel.
training (with the parameters obtained previously and the remaining 1/5 as test set). This
process was repeated 5 times for each hand. Altogether we had 8sets ? 5folds ? 2hands = 80
folds.
5
Results
We begin by showing average results across all datasets, folds, hands and X/Y directions for
the four algorithms that are compared. Table. 1 shows mean Correlation Coefficients (R2 ,
between recorded and predicted movement values), Mean ? insensitive Absolute Errors
(MAE? ) and Mean Square Errors (MSE). R2 is a standard performance measure, MAE ? is
the error minimized by DDT (subject to the regularization term) and MSE is minimized by
the Kalman filter. Under all the above measures the DDT-Spikernel outperforms the rest
with the SVR-Spikernel and the Kalman Filter alternating in second place.
To understand whether the performance differences are significant we look at the distribution of position (X and Y) R2 values at each of the separate tests (160 altogether). Figure 1
shows scatter plots of R2 results for position predictions. Each plot compares the DDTSpikernel (on the Y axis) with one of the other three algorithms (on the X axes). It is
clear that in spite large differences in accuracy across datasets, the algorithm pairs achieve
similar success with the DDT-Spikernel achieving a better R2 score in almost all cases.
To summarize the significance of R2 differences we computed the number of tests in which
one algorithm achieved a higher R2 value than another algorithm (for all pairs, in each of
the position, velocity and acceleration categories). The results of this tournament between
the algorithms are presented in Figure 2 as winning percentages. The graphs produce a
ranking of the algorithms and the percentages are the significances of the ranking between
pairs. The DDT-Spikernel is significantly better then the rest in tracking position.
The matrix A in use is not optimal for our algorithm. The choice of ? scales its effect. When
? = 0 we get the standard SVR algorithm (without state dynamics). To illustrate the effect
of ? we present in Figure 3 the mean (over 5 folds, X/Y direction and hand) R2 results on
the first dataset as a function of ?. It is clear that the value chosen to minimize position error
is not optimal for minimizing velocity and acceleration errors. Another important effect of
? is the number of the support patterns in the learned model, which drops considerably
(by about one third) when the effect of the dynamics is increased. This means that more
training points fall strictly within the ?-tube in training, suggesting that the kernel which
tacitly results from the dynamical model is better suited for the problem. Lastly, we show a
sample of test tracking results for the DDT-Spikernel and SVR-Spikernel in Figure 4. Note
that the acceleration values are not smooth and are, therefore, least aided by the dynamics of
the model. However, adding acceleration to the model improves the prediction of position.
6
Conclusion
We described and reported experiments with a dynamical system that combines a linear
state mapping with a nonlinear observation-to-state mapping. The estimation of the system?s parameters is transformed to a dual representation and yields a novel kernel for temporal modelling. When a linear kernel is used, the DDT system has a similar form to the
Kalman filter as t ? ?. However, the system?s parameters are set so as to minimize the
regularized ?-insensitive `1 loss between state trajectories. DDT also bares similarity to
SVR, which employs the same loss yet without the state dynamics. Our experiments indicate that by combining a kernel-induced feature space, linear state dynamics, and using a
robust loss we are able to leverage the trajectory prediction accuracy and outperform common approaches. Our next step toward an accurate brain-machine interface for predicting
hand movements is the development of a learning procedure for the state dynamic mapping
A and further developments of neurally motivated and compact representations.
Acknowledgments This study was partly supported by a center of excellence grant (8006/00)
administered by the ISF, BMBF-DIP, by the U.S. Israel BSF and by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. L.S. is supported
by a Horowitz fellowship.
References
[1] A. E. Brockwell, A. L. Rojas, and R. E. Kass. Recursive bayesian decoding of motor cortical
signals by particle filtering. Journal of Neurophysiology, 91:1899?1907, 2004.
[2] E. N. Brown, L. M. Frank, D. Tang, M. C. Quirk, and M. A. Wilson. A statistical paradigm for
neural spike train decoding applied to position prediction from ensemble firing patterns of rat
hippocampal place cells. Journal of Neuroscience, 18(7411?7425), 1998.
[3] J. M. Carmena, M. A. Lebedev, R. E. Crist, J. E. O?Doherty, D. M. Santucci, D. F. Dimitrov,
P. G. Patil, C. S. Henriques, and M. A. L. Nicolelis. Learning to control a brain-machine
interface for reaching and grasping by primates. PLOS Biology, 1(2):001?016, 2003.
[4] K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based
vector machines. Jornal of Machine Learning Research, 2:265?292, 2001.
[5] J. Eichhorn, A. Tolias, A. Zien, M. Kuss, C. E. Rasmussen, J. Weston, N. Logothetis, and
B. Sch?olkopf. Prediction on spike data using kernel algorithms. In NIPS 16. MIT Press, 2004.
[6] A. P. Georgopoulos, J. Kalaska, and J. Massey. Spatial coding of movements: A hypothesis
concerning the coding of movement direction by motor cortical populations. Experimental
Brain Research (Supp), 7:327?336, 1983.
[7] R. E. Isaacs, D. J. Weber, and A. B. Schwartz. Work toward real-time control of a cortical
neural prothesis. IEEE Trans Rehabil Eng, 8(196?198), 2000.
[8] C. Mehring, J. Rickert, E. Vaadia, S. C. de Oliveira, A. Aertsen, and S. Rotter. Inference of
hand movements from local field potentials in monkey motor cortex. Nature Neur., 6(12), 2003.
[9] R. Paz, T. Boraud, C. Natan, H. Bergman, and E. Vaadia. Preparatory activity in motor cortex
reflects learning of local visuomotor skills. Nature Neur., 6(8):882?890, August 2003.
[10] M. D. Serruya, N. G. Hatsopoulos, L. Paninski, M. R. Fellows, and J. P. Donoghue. Instant
neural control of a movement signal. Nature, 416:141?142, March 2002.
[11] L. Shpigelman, Y. Singer, R. Paz, and E. Vaadia. Spikernels: Embedding spiking neurons in
inner product spaces. In NIPS 15, Cambridge, MA, 2003. MIT Press.
[12] A. Smola and B. Scholkop. A tutorial on support vector regressio. In NeuroCOLT2 Technical
Report, 1998.
[13] S. I. H. Tillery, D. M. Taylor, and A. B. Schwartz. Training in cortical control of neuroprosthetic devices improves signal extraction from small neuronal ensembles. Reviews in the Neurosciences, 14:107?119, 2003.
[14] V. Vapnik. The Nature of Statistical Learning Theory. Springer, N.Y., 1995.
[15] J. Wessberg, C. R. Stambaugh, J. D. Kralik, P. D. Beck, M. Laubach, J. K. Chapin, J. Kim,
J. Biggs, M. A. Srinivasan, and M. A. Nicolelis. Real-time prediction of hand trajectory by
ensembles of cortical neurons in primates. Nature, 408(16), November 2000.
[16] W. Wu, M. J. Black, Y. Gao, E. Bienenstock, M. Serruya, and J. P. Donoghue. Inferring hand
motion from multi-cell recordings in motor cortex using a kalman filter. In SAB02, pages 66?73,
Edinburgh, Scotland (UK), 2002.
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1,811 | 2,645 | Confidence Intervals for the Area under the
ROC Curve
Corinna Cortes
Google Research
1440 Broadway
New York, NY 10018
[email protected]
Mehryar Mohri
Courant Institute, NYU
719 Broadway
New York, NY 10003
[email protected]
Abstract
In many applications, good ranking is a highly desirable performance for
a classifier. The criterion commonly used to measure the ranking quality
of a classification algorithm is the area under the ROC curve (AUC). To
report it properly, it is crucial to determine an interval of confidence for
its value. This paper provides confidence intervals for the AUC based
on a statistical and combinatorial analysis using only simple parameters
such as the error rate and the number of positive and negative examples.
The analysis is distribution-independent, it makes no assumption about
the distribution of the scores of negative or positive examples. The results
are of practical use and can be viewed as the equivalent for AUC of the
standard confidence intervals given in the case of the error rate. They
are compared with previous approaches in several standard classification
tasks demonstrating the benefits of our analysis.
1
Motivation
In many machine learning applications, the ranking quality of a classifier is critical. For
example, the ordering of the list of relevant documents returned by a search engine or
a document classification system is essential. The criterion widely used to measure the
ranking quality of a classification algorithm is the area under an ROC curve (AUC). But, to
measure and report the AUC properly, it is crucial to determine an interval of confidence
for its value as it is customary for the error rate and other measures. It is also important to
make the computation of the confidence interval practical by relying only on a small and
simple number of parameters. In the case of the error rate, such intervals are often derived
from just the sample size N .
We present an extensive theoretical analysis of the AUC and show that a similar confidence
interval can be derived for its value using only simple parameters such as the error rate k/N ,
the number of positive examples m, and the number of negative examples n = N ? m.
Thus, our results extend to AUC the computation of confidence intervals from a small
number of readily available parameters.
Our analysis is distribution-independent in the sense that it makes no assumption about the
distribution of the scores of negative or positive examples. The use of the error rate helps
determine tight confidence intervals. This contrasts with existing approaches presented in
the statistical literature [11, 5, 2] which are based either on weak distribution-independent
assumptions resulting in too loose confidence intervals, or strong distribution-dependent
assumptions leading to tight but unsafe confidence intervals.
We show that our results are of practical use. We also compare them with previous approaches in several standard classification tasks demonstrating the benefits of our analysis.
Our results are also useful for testing the statistical significance of the difference of the
AUC values of two classifiers.
The paper is organized as follows. We first introduce the definition of the AUC, its connection with the Wilcoxon-Mann-Whitney statistic (Section 2), and briefly review some
essential aspects of the existing literature related to the computation of confidence intervals
for the AUC. Our computation of the expected value and variance of the AUC for a fixed
error rate requires establishing several combinatorial identities. Section 4 presents some
existing identities and gives the proof of novel ones useful for the computation of the variance. Section 5 gives the reduced expressions for the expected value and variance of the
AUC for a fixed error rate. These can be efficiently computed and used to determine our
confidence intervals for the AUC (Section 6). Section 7 reports the result of the comparison
of our method with previous approaches, including empirical results for several standard
tasks.
2
Definition and Properties of the AUC
The Receiver Operating Characteristics (ROC) curves were originally introduced in signal
detection theory [6] in connection with the study of radio signals, and have been used
since then in many other applications, in particular for medical decision-making. Over the
last few years, they have found increased interest in the machine learning and data mining
communities for model evaluation and selection [14, 13, 7, 12, 16, 3]. The ROC curve for
a binary classification problem plots the true positive rate as a function of the false positive
rate. The points of the curve are obtained by sweeping the classification threshold from the
most positive classification value to the most negative. For a fully random classification,
the ROC curve is a straight line connecting the origin to (1, 1). Any improvement over
random classification results in an ROC curve at least partially above this straight line. The
AUC is defined as the area under the ROC curve.
Consider a binary classification task with m positive examples and n negative examples.
Let C be a fixed classifier that outputs a strictly ordered list for these examples. Let
x1 , . . . , xm be the output of C on the positive examples and y1 , . . . , yn its output on the
negative examples and denote by 1X the indicator function of a set X. Then, the AUC, A,
associated to C is given by:
Pm Pn
i=1
j=1 1xi >yj
(1)
A=
mn
which is the value of the Wilcoxon-Mann-Whitney statistic [10]. Thus, the AUC is closely
related to the ranking quality of the classification. It can be viewed as a measure based on
pairwise comparisons between classifications of the two classes. It is an estimate of the
probability Pxy that the classifier ranks a randomly chosen positive example higher than
a negative example. With a perfect ranking, all positive examples are ranked higher than
the negative ones and A = 1. Any deviation from this ranking decreases the AUC, and the
expected AUC value for a random ranking is 0.5.
3
Overview of Related Work
This section briefly describes some previous distribution-dependent approaches presented
in the statistical literature to derive confidence intervals for the AUC and compares them
to our method. The starting point for these analyses is a formula giving the variance of the
AUC, A, for a fixed distribution of the scores Px of the positive examples and Py of the
negative examples [10, 1]:
2
?A
=
A(1 ? A) + (m ? 1)(Pxxy ? A2 ) + (n ? 1)(Pxyy ? A2 )
mn
(2)
where Pxxy is the probability that the classifier ranks two randomly chosen positive examples higher than a negative one, and Pxyy the probability that it ranks two randomly chosen
negative examples lower than a positive one. To compute the variance exactly using Equation 2, the distributions Px and Py must be known.
Hanley and McNeil [10] argue in favor of exponential distributions, loosely claiming that
this upper-bounds the variance of normal distributions with various means and ratios of
A
2A2
variances. They show that for exponential distributions Pxxy = 2?A
and Pxyy = 1+A
.
The resulting confidence intervals are of course relatively tight, but their validity is questionable since they are based on a strong assumption about the distributions of the positive
and negative scores that may not hold in many cases.
An alternative considered by several authors to the exact computation of the variance is to
determine instead the maximum of the variance over all possible continuous distributions
with the same expected value of the AUC. For all such distributions, one can fix m and
n and compute the expected AUC and its variance. The maximum variance is denoted by
2
?max
and is given by [5, 2]:
2
?max
=
A(1 ? A)
1
?
min {m, n}
4 min {m, n}
(3)
Unfortunately, this often yields loose confidence intervals of limited practical use.
Our approach for computing the mean and variance of the AUC is distribution-independent
and inspired by the machine learning literature where analyses typically center on the error
rate. We require only that the error rate be measured and compute the mean and variance of
the AUC over all distributions Px and Py that maintain the same error rate. Our approach
is in line with that of [5, 2] but it crucially avoids considering the maximum of the variance. We show that it is possible to compute directly the mean and variance of the AUC
assigning equal weight to all the possible distributions. Of course, one could argue that not
all distributions Px and Py are equally probable, but since these distributions are highly
problem-dependent, we find it risky to make any general assumption on the distributions
and thereby limit the validity of our results. Our approach is further justified empirically
by the experiments reported in the last section.
4
Combinatorial Analysis
The analysis of the statistical properties of the AUC given a fixed error rate requires various
combinatorial calculations. This section describes several of the combinatorial identities
that are used in our computation of the confidence intervals. For all q ? 0, let Xq (k, m, n)
be defined by:
0
k
X
M
M
Xq (k, m, n) =
xq
(4)
x
x0
x=0
where M = m ? (k ? x) + x, M 0 = n + (k ? x) ? x, and x0 = k ? x. In previous work,
we derived the following two identities which we used to compute the expected value of
the AUC [4]:
k
k
X
X
n+m+1
(k ? x)(m ? n) + k n + m + 1
X1 (k, m, n) =
X0 (k, m, n) =
x
2
x
x=0
x=0
To simplify the expression of the variance of the AUC, we need to compute X2 (k, m, n).
Proposition 1 Let k, m, n be non-negative integers such that k ? min{m, n}, then:
k
X
m+n+1
X2 (k, m, n) =
P2 (k, m, n, x)
x
x=0
(5)
where P2 is the following 4th-degree polynomial: P2 (k, m, n, x) = (k ? x)/12(?2x3 +
2x2 (2m ? n + 2k ? 4) + x(?3m2 + 3nm + 3m ? 5km ? 2k 2 + 2 + k + nk + 6n) +
(3(k ? 1)m2 ? 3nm(k ? 1) + 6km + 5m + k 2 m + 8n + 8 ? 9nk + 3k + k 2 + k 2 n)).
Proof.
5
The proof of the proposition is left to a longer version of this paper.
Expectation and Variance of the AUC
This section presents the expression of the expectation and variance of the AUC for a fixed
error rate k/N assuming that all classifications or rankings with k errors are equiprobable.
For a given classification, there may be x, 0 ? x ? k, false positive examples. Since the
number of errors is fixed, there are x0 = k ? x false negative examples. The expression Xq
discussed in the previous section represents the q-th moment of x over all classifications
with exactly k errors. In previous work, we gave the exact expression of the expectation of
the AUC for a fixed number of errors k:
Proposition 2 ([4]) Assume that a binary classification task with m positive examples
and n negative examples is given. Then, the expected value of the AUC, A, over all
classifications with k errors is given by:
Pk?1 m+n !
k
k
(n ? m)2 (m + n + 1)
x
E[A] = 1 ?
?
? Pk x=0 m+n+1
.
m+n
4mn
m+n
x=0
x
Note that the two sums in this expression cannot be further simplified since they are known
not to admit a closed form [9]. We also gave the expression of the variance of the AUC in
terms of the function F defined for all Y by:
Pk
M M0
x=0 x
x0 Y
F (Y ) = Pk
(6)
.
M M0
x=0
x
x0
The following proposition reproduces that result:
Proposition 3 ([4]) Assume that a binary classification task with m positive examples
and n negative examples is given. Then, the variance of the AUC A over all classifications with k errors is given by: ? 2 (A) = F ((1 ?
k?x
x
n+ m
)2 )
2
2
2
F ( mx +n(k?x) +(m(m+1)x+n(n+1)(k?x))?2x(k?x)(m+n+1)
).
12m2 n2
? F ((1 ?
k?x
x
n+ m
2
))2 +
Because of the products of binomial terms, the computation of the variance using this
expression is inefficient even for relatively small values of m and n. This expression can
however be reduced using the identities presented in the previous section which leads to
significantly more efficient computations that we have been using in all our experiments.
Corollary 1 ([4]) Assume that a binary classification task with m positive examples and n
negative examples is given. Then, the variance of the AUC A over all classifications with
((m+n?2)Z4 ?(2m?n+3k?10)Z3 )
k errors is given by: ? 2 (A) = (m+n+1)(m+n)(m+n?1)T72m
+
2 n2
(m+n+1)(m+n)T (m2 ?nm+3km?5m+2k2 ?nk+12?9k)Z2
48m2 n2
(m+n+1)Q1 Z1
kQ0
+
2
2
2
72m
n
144m n2 with:
Pk?i
m+n+1?i
Zi =
x=0
Pk
(
x=0
x
(m+n+1
)
x
)
?
(m+n+1)2 (m?n)4 Z12
16m2 n2
?
, T = 3((m ? n)2 + m + n) + 2, and:
Q0 = (m + n + 1)T k2 + ((?3n2 + 3mn + 3m + 1)T ? 12(3mn + m + n) ? 8)k + (?3m2 +
7m + 10n + 3nm + 10)T ? 4(3mn + m + n + 1)
Q1 = T k3 + 3(m ? 1)T k2 + ((?3n2 + 3mn ? 3m + 8)T ? 6(6mn + m + n))k + (?3m2 +
7(m + n) + 3mn)T ? 2(6mn + m + n)
Proof. The expression of the variance given in Proposition 3 requires the computation
of Xq (k, m, n), q = 0, 1, 2. Using the identities giving the expressions of X0 and X1 and
Proposition 1, which provides the expression of X2 , ? 2 (A) can be reduced to the expression
given by the corollary.
6
Theory and Analysis
Our estimate of the confidence interval for the AUC is based on a simple and natural assumption. The main idea for its computation is the following. Assume that a confidence
interval E = [e1 , e2 ] is given for the error rate of a classifier C over a sample S, with the
confidence level 1 ? . This interval may have have been derived from a binomial model of
C, which is a standard assumption for determining a confidence interval for the error rate,
or from any other model used to compute that interval. For a given error rate e ? E, or
equivalently for a given number of misclassifications, we can use the expectation and variance computed in the previous section and Chebyshev?s inequality to predict a confidence
interval Ae for the AUC at the confidence level 1 ? 0 . Since our equiprobable model for
the classifications is independent of the model used to compute the interval of confidence
for the error rate, we can use E and Ae , e ? E, to compute a confidence interval of the
AUC at the level (1 ? )(1 ? 0 ).
Theorem 1 Let C be a binary classifier and let S be a data sample of size N with m
positive examples and n negative examples, N = m + n. Let E = [e1 , e2 ] be a confidence
interval for the error rate of C over S at the confidence level 1 ? . Then, for any 0 ,
0 ? 0 ? 1, we can compute a confidence interval for the AUC value of the classifier C at
the confidence level (1 ? )(1 ? 0 ) that depends only on , 0 , m, n, and the interval E.
Proof. Let k1 = N e1 and k2 = N e2 be the number of errors associated to the error rates
e1 and e2 , and let IK be the interval IK = [k1 , k2 ]. For a fixed k ? IK , by Propositions
2 and Corollary 1, we can compute the exact value of the expectation E[Ak ] and variance
? 2 (Ak ) of the AUC Ak . Using Chebyshev?s inequality, for any k ? IK and any k > 0,
?(Ak )
P |Ak ? E[Ak ]| ? ?
? k
(7)
k
where E[Ak ] and ?(Ak ) are the expressions given in Propositions 2 and Corollary 1, which
depend only on k, m, and n. Let ?1 and ?2 be defined by:
?(Ak )
?(Ak )
?2 = max E[Ak ] + ?
(8)
?1 = min E[Ak ] ? ?
k?IK
k?IK
k
k
?1 and ?2 only depend on IK (i.e., on e1 and e2 ), and on k, m, and n. Let IA be the
confidence interval defined by IA = [?1 , ?2 ] and let k = 0 for all k ? IK . Using the
fact that the confidence interval E is independent of our equiprobability model for fixed-k
AUC values and the Bayes? rule:
X
P(A ? IA ) =
P (A ? IA | K = k)P (K = k)
(9)
k?R+
?
X
k?IK
P (A ? IA | K = k)P (K = k)
? (1 ? 0 )
X
k?IK
P (K = k) ? (1 ? 0 )(1 ? )
(10)
(11)
where we used the property of Eq. 7 and the definitions of the intervals IK and IA . Thus,
IA constitutes a confidence interval for the AUC value of C at the confidence level (1 ?
)(1 ? 0 ).
In practice, the confidence interval E is often determined as a result of the assumption that
C follows a binomial law. This leads to the following theorem.
.020
.035
Standard deviation
Standard deviation
.030
.015
.010
Max
Distribution?dependent
Distribution?independent
.005
.025
.020
.015
Max
Distribution?dependent
Distribution?independent
.010
.005
0.75
0.80
0.85
0.90
0.95
1.00
0.6
0.7
0.8
AUC
AUC
(a)
(b)
0.9
1.0
Figure 1: Comparison of the standard deviations for three different methods with: (a) m = n = 500;
(b) m = 400 and n = 200. The curves are obtained by computing the expected AUC and its standard
deviations for different values of the error rate using the maximum-variance approach (Eq. 3), our
distribution-independent method, and the distribution-dependent approach of Hanley [10].
Theorem 2 Let C be a binary classifier, let S be a data sample of size N with m positive
examples and n negative examples, N = m + n, and let k0 be the number of misclassifications of C on S. Assume that C follows a binomial law, then, for any , 0 ? ? 1, we can
compute a confidence interval of the AUC value of the classifier C at the confidence level
1 ? that depends only on , k0 , m, and n.
Proof.
Assume that C follows a binomial law with coefficient p. Then, Chebyshev?s
inequality yields:
1
p(1 ? p)
?
(12)
P(|C ? E[C]| ? ?) ?
2
N?
4N ? 2
1
1
Thus, E = [ kN0 ? ? ?
, k0 + ? ?
] forms a confidence interval for the
2 (1? 1?)N N
2 (1? 1?)N
?
error rate of C at the confidence level 1 ? . By Theorem
1, we can?compute for the
?
AUC value a confidence interval at the level (1 ? (1 ? 1 ? ))(1 ? (1 ? 1 ? )) = 1 ?
depending only on , m, n, and the interval E, i.e., k0 , N = m + n, and .
For large N , we can use the normal approximation of the binomial law to determine a finer
interval E. Indeed, for large N ,
?
(13)
P(|C ? E[C]| ? ?) ? 2?(2 N ?)
with ?(u) =
R?
u
2
/2
e?x
?
dx.
2?
Thus, E = [ kN0 ?
?
??1 ( 1? 21? ) k0
?
,N
2 N ?
confidence interval for the error rate at the confidence level
?
+
??1 ( 1? 21? )
?
]
2 N
is the
1 ? .
For simplicity, in the proof of Theorem 2, k was chosen to be a constant (k = 0 ) but, in
general, it can be another function of k leading to tighter confidence intervals. The results
presented in the next section were obtained with k = a0 exp((k ? k0 )2 /2a21 ), where a0
and a1 are constants selected so that the inequality 11 be verified.
7
Experiments and Comparisons
The analysis in the previous section provides a principled method for computing a confidence interval of the AUC value of a classier C at the confidence level 1 ? that depends
only on k, n and m. As already discussed, other expressions found in the statistical literature lead to either too loose or unsafely narrow confidence intervals based on questionable assumptions on the probability functions Px and Py [10, 15]. Figure 1 shows a
comparison of the standard deviations obtained using the maximum-approach (Eq. 3), the
distribution-dependent expression from [10], and our distribution-independent method for
NAME
m+n
n
m+n
AUC
k
m+n
?indep
?A
?dep
?max
368
700
303
1159
2473
201
0.63
0.67
0.54
0.17
0.10
0.37
0.70
0.63
0.87
0.85
0.84
0.85
0.24
0.26
0.13
0.05
0.03
0.13
0.0297
0.0277
0.0176
0.0177
0.0164
0.0271
0.0440
0.0330
0.0309
0.0161
0.0088
0.0463
0.0269
0.0215
0.0202
0.0176
0.0161
0.0306
0.0392
0.0317
0.0281
0.0253
0.0234
0.0417
pima
yeast
credit
internet-ads
page-blocks
ionosphere
Table 1: Accuracy and AUC values for AdaBoost [8] and estimated standard deviations for several
datasets from the UC Irvine repository. ?indep is a distribution-independent standard deviation obtained using our method (Theorem 2). ?A is given by Eq. (2) with the values of A, Pxxy , and Pxyy
derived from data. ?dep is the distribution-dependent standard deviation of Hanley [10], which is
based on assumptions that may not always hold. ?max is defined by Eq. (3). All results were obtained
on a randomly selected test set of size m + n.
various error rates. For m = n = 500, our distribution-independent method consistently
leads to tighter confidence intervals (Fig. 1 (a)). It also leads to tighter confidence intervals for AUC values more than .75 for the uneven distribution m = 400 and n = 200
(Fig. 1 (b)). For lower AUC values, the distribution-dependent approach produces tighter
intervals, but its underlying assumptions may not hold.
A different comparison was made using several datasets available from the UC Irvine repository (Table 1). The table shows that our estimates of the standard deviations (?indep ) are in
general close to or tighter than the distribution-dependent standard deviation ?dep of Hanley
[10]. This is despite we do not make any assumption about the distributions of positive
and negative examples. In contrast, Hanley?s method is based on specific assumptions
about these distributions. Plots of the actual ranking distribution demonstrate that these
assumptions are often violated however. Thus, the relatively good performance of Hanley?s approach on several data sets can be viewed as fortuitous and is not general. Our
distribution-independent method provides tight confidence intervals, in some cases tighter
than those derived from ?A , in particular because it exploits the information provided by
the error rate. Our analysis can also be used to determine if the AUC values produced by
two classifiers are statistically significant by checking if the AUC value of one falls within
the confidence interval of the other.
8
Conclusion
We presented principled techniques for computing useful confidence intervals for the AUC
from simple parameters: the error rate, and the negative and positive sample sizes. We
demonstrated the practicality of these confidence intervals by comparing them to previous
approaches in several tasks. We also derived the exact expression of the variance of the
AUC for a fixed k, which can be of interest in other analyses related to the AUC.
The Wilcoxon-Mann-Whitney statistic is a general measure of the quality of a ranking that
is an estimate of the probability that the classifier ranks a randomly chosen positive example higher than a negative example. One could argue that accuracy at the top or the
bottom of the ranking is of higher importance. This, however, contrarily to some belief,
is already captured to a certain degree by the definition of the Wilcoxon-Mann-Whitney
statistic which penalizes more errors at the top or the bottom of the ranking. It is however an interesting research problem to determine how to incorporate this bias in a stricter
way in the form of a score-specific weight in the ranking measure, a weighted WilcoxonMann-Whitney statistic, or how to compute the corresponding expected value and standard
deviation in a general way and design machine learning algorithms to optimize such a mea-
sure. A preliminary analysis suggests, however, that the calculation of the expectation and
the variance are likely to be extremely complex in that case. Finally, it could also be interesting but difficult to adapt our results to the distribution-dependent case and compare
them to those of [10].
Acknowledgments
We thank Rob Schapire for pointing out to us the problem of the statistical significance
of the AUC, Daryl Pregibon for the reference to [11], and Saharon Rosset for various
discussions about the topic of this paper.
References
[1] D. Bamber. The Area above the Ordinal Dominance Graph and the Area below the
Receiver Operating Characteristic Graph. Journal of Math. Psychology, 12, 1975.
[2] Z. W. Birnbaum and O. M. Klose. Bounds for the Variance of the Mann-Whitney
Statistic. Annals of Mathematical Statistics, 38, 1957.
[3] J-H. Chauchat, R. Rakotomalala, M. Carloz, and C. Pelletier. Targeting Customer
Groups using Gain and Cost Matrix; a Marketing Application. Technical report, ERIC
Laboratory - University of Lyon 2, 2001.
[4] Corinna Cortes and Mehryar Mohri. AUC Optimization vs. Error Rate Minimization.
In Advances in Neural Information Processing Systems (NIPS 2003), volume 16, Vancouver, Canada, 2004. MIT Press.
[5] D. Van Dantzig. On the Consistency and Power of Wilcoxon?s Two Sample Test. In
Koninklijke Nederlandse Akademie van Weterschappen, Series A, volume 54, 1915.
[6] J. P. Egan. Signal Detection Theory and ROC Analysis. Academic Press, 1975.
[7] C. Ferri, P. Flach, and J. Hern?andez-Orallo. Learning Decision Trees Using the Area
Under the ROC Curve. In Proceedings of the 19th International Conference on Machine Learning. Morgan Kaufmann, 2002.
[8] Yoav Freund and Robert E. Schapire. A Decision Theoretical Generalization of OnLine Learning and an Application to Boosting. In Proceedings of the Second European Conference on Computational Learning Theory, volume 2, 1995.
[9] Ronald L. Graham, Donald E. Knuth, and Oren Patashnik. Concrete Mathematics.
Addison-Wesley, Reading, Massachusetts, 1994.
[10] J. A. Hanley and B. J. McNeil. The Meaning and Use of the Area under a Receiver
Operating Characteristic (ROC) Curve. Radiology, 1982.
[11] E. L. Lehmann. Nonparametrics: Statistical Methods Based on Ranks. Holden-Day,
San Francisco, California, 1975.
[12] M. C. Mozer, R. Dodier, M. D. Colagrosso, C. Guerra-Salcedo, and R. Wolniewicz.
Prodding the ROC Curve: Constrained Optimization of Classifier Performance. In
Neural Information Processing Systems (NIPS 2002). MIT Press, 2002.
[13] C. Perlich, F. Provost, and J. Simonoff. Tree Induction vs. Logistic Regression: A
Learning Curve Analysis. Journal of Machine Learning Research, 2003.
[14] F. Provost and T. Fawcett. Analysis and Visualization of Classifier Performance:
Comparison under Imprecise Class and Cost Distribution. In Proceedings of the Third
International Conference on Knowledge Discovery and Data Mining. AAAI, 1997.
[15] Saharon Rosset. Ranking-Methods for Flexible Evaluation and Efficient Comparison
of 2-Class Models. Master?s thesis, Tel-Aviv University, 1999.
[16] L. Yan, R. Dodier, M. C. Mozer, and R. Wolniewicz. Optimizing Classifier Performance via the Wilcoxon-Mann-Whitney Statistics. In Proceedings of the International Conference on Machine Learning, 2003.
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1,812 | 2,646 | Face Detection ? Efficient and Rank Deficient
Wolf Kienzle, G?okhan Bak?r, Matthias Franz and Bernhard Scho? lkopf
Max-Planck-Institute for Biological Cybernetics
Spemannstr. 38, D-72076 T?ubingen, Germany
{kienzle, gb, mof, bs}@tuebingen.mpg.de
Abstract
This paper proposes a method for computing fast approximations to support vector decision functions in the field of object detection. In the
present approach we are building on an existing algorithm where the set
of support vectors is replaced by a smaller, so-called reduced set of synthesized input space points. In contrast to the existing method that finds
the reduced set via unconstrained optimization, we impose a structural
constraint on the synthetic points such that the resulting approximations
can be evaluated via separable filters. For applications that require scanning large images, this decreases the computational complexity by a significant amount. Experimental results show that in face detection, rank
deficient approximations are 4 to 6 times faster than unconstrained reduced set systems.
1
Introduction
It has been shown that support vector machines (SVMs) provide state-of-the-art accuracies
in object detection. In time-critical applications, however, they are of limited use due to
their computationally expensive decision functions. In particular, the time complexity of
an SVM classification operation is characterized by two parameters. First, it is linear in the
number of support vectors (SVs). Second, it scales with the number of operations needed
for computing the similarity between an SV and the input, i.e. the complexity of the kernel
function. When classifying image patches of size h ? w using plain gray value features, the
decision function requires an h ? w dimensional dot product for each SV. As the patch size
increases, these computations become extremely expensive. As an example, the evaluation
of a single 20 ? 20 patch on a 320 ? 240 image at 25 frames per second already requires
660 million operations per second.
In the past, research towards speeding up kernel expansions has focused exclusively on the
first issue, i.e. on how to reduce the number of expansion points (SVs) [1, 2]. In [2], Burges
introduced a method that, for a given SVM, creates a set of so-called reduced set vectors
(RSVs) that approximate the decision function. This approach has been successfully applied in the image classification domain ? speedups on the order of 10 to 30 have been reported [2, 3, 4] while the full accuracy was retained. Additionally, for strongly unbalanced
classification problems such as face detection, the average number of RSV evaluations can
be further reduced using cascaded classifiers [5, 6, 7]. Unfortunately, the above example
illustrates that even with as few as three RSVs on average (as in [5]), such systems are not
competitive for time-critical applications.
The present work focuses on the second issue, i.e. the high computational cost of the kernel
evaluations. While this could be remedied by switching to a sparser image representation
(e.g. a wavelet basis), one could argue that in connection with SVMs, not only are plain
gray values straightforward to use, but they have shown to outperform Haar wavelets and
gradients in the face detection domain [8]. Alternatively, in [9], the authors suggest to compute the costly correlations in the frequency domain. In this paper, we develop a method
that combines the simplicity of gray value correlations with the speed advantage of more
sophisticated image representations. To this end, we borrow an idea from image processing:
by constraining the RSVs to have a special structure, they can be evaluated via separable
convolutions. This works for most standard kernels (e.g. linear, polynomial, Gaussian and
sigmoid) and decreases the average computational complexity of the RSV evaluations from
O(h ? w) to O(r ? (h + w)), where r is a small number that allows the user to balance between speed and accuracy. To evaluate our approach, we examine the performance of these
approximations on the MIT+CMU face detection database (used in [10, 8, 5, 6]).
2
Burges? method for reduced set approximations
The present section briefly describes Burges? reduced set method [2] on which our work is
based. For reasons that will become clear below, h ? w image patches are written as h ? w
matrices (denoted by bold capital letters) whose entries are the respective pixel intensities.
In this paper, we refer to this as the image-matrix notation.
Assume that an SVM has been successfully trained on the problem at hand. Let
{X1 , . . . Xm } denote the set of SVs, {?1 , . . . ?m } the corresponding coefficients, k(?, ?)
the kernel function and b the bias of the SVM solution. The decision rule for a test pattern
X reads
!
m
X
f (X) = sgn
yi ?i k(Xi , X) + b .
(1)
i=1
In SVMs, the decision surface induced by f corresponds to a hyperplane in the reproducing
kernel Hilbert space (RKHS) associated with k. The corresponding normal
?=
m
X
yi ?i k(Xi , ?)
(2)
i=1
can be approximated using a smaller, so-called reduced set (RS) {Z1 , . . . Zm0 } of size
m0 < m, i.e. an approximation to ? of the form
0
0
? =
m
X
?i k(Zi , ?).
(3)
i=1
This speeds up the decision process by a factor of m/m0 . To find such ?0 , we fix a desired
set size m0 and solve
min k? ? ?0 k2RKHS
(4)
for ?i and Zi . Here, k ? kRKHS denotes the Euclidian norm in the RKHS. The resulting RS
decision function f 0 is then given by
? 0
?
m
X
f 0 (X) = sgn ?
?i k(Zi , X) + b? .
(5)
i=1
In practice, ?i , Zi are found using a gradient based optimization technique. Details can be
found in [2].
3
From separable filters to rank deficient reduced sets
We now describe the concept of separable filters in image processing and show how this
idea extends to a broader class of linear filters and to a special class of nonlinear filters,
namely those used by SVM decision functions. Using the image-matrix notation, it will
become clear that the separability property boils down to a matrix rank constraint.
3.1
Linear separable filters
Applying a linear filter to an image amounts to a two-dimensional convolution of the image
with the impulse response of the filter. In particular, if I is the input image, H the impulse
response, i.e. the filter mask, and J the output image, then
J = I ? H.
(6)
If H has size h ? w, the convolution requires O(h ? w) operations for each output pixel.
However, in special cases where H can be decomposed into two column vectors a and b,
such that
H = ab>
(7)
holds, we can rewrite (6) as
J = [I ? a] ? b> ,
(8)
>
>
since the convolution is associative and in this case, ab = a ? b . This splits the original
problem (6) into two convolution operations with masks of size h?1 and 1?w, respectively.
As a result, if a linear filter is separable in the sense of equation (7), the computational
complexity of the filtering operation can be reduced from O(h ? w) to O(h + w) per pixel
by computing (8) instead of (6).
3.2
Linear rank deficient filters
In view of (7) being equivalent to rank(H) ? 1, we now generalize the above concept to
linear filters with low rank impulse responses. Consider the singular value decomposition
(SVD) of the h ? w matrix H,
H = USV> ,
(9)
and recall that U and V are orthogonal matrices of size h ? h and w ? w, respectively,
whereas S is diagonal (the diagonal entries are the singular values) and has size h ? w.
Now let r = rank(H). Due to rank(S) = rank(H), we may write H as a sum of r rank
one matrices
r
X
s i u i vi >
(10)
H=
i=1
where si denotes the ith singular value of H and ui , vi are the iths columns of U and V
(i.e. the ith singular vectors of the matrix H), respectively. As a result, the corresponding linear filter can be evaluated (analogously to (8)) as the weighted sum of r separable
convolutions
r
X
J=
si [I ? ui ] ? vi >
(11)
i=1
and the computational complexity drops from O(h ? w) to O(r ? (h + w)) per output
pixel. Not surprisingly, the speed benefit depends on r, which can be seen to measure the
structural complexity1 of H. For square matrices (w = h) for instance, (11) does not give
any speedup compared to (6) if r > w/2.
1
In other words, the flatter the spectrum of HH> , the less benefit can be expected from (11).
3.3
Nonlinear rank deficient filters and reduced sets
Due to the fact that in 2D, correlation is identical with convolution if the filter mask is
rotated by 180 degrees (and vice versa), we can apply the above idea to any image filter
f (X) = g(c(H, X)) where g is an arbitrary nonlinear function and c(H, X) denotes the
correlation between images patches X and H (both of size h ? w). In SVMs this amounts
to using a kernel of the form
k(H, X) = g(c(H, X)).
(12)
If H has rank r, we may split the kernel evaluation into r separable correlations plus a
scalar nonlinearity. As a result, if the RSVs in a kernel expansion such as (5) satisfy
this constraint, the average computational complexity decreases from O(m0 ? h ? w) to
O(m0 ? r ? (h + w)) per output pixel. This concept works for many off-the-shelf kernels
used in SVMs. While linear, polynomial and sigmoid kernels are defined as functions of
input space dot products and therefore immediately satisfy equation (12), the idea applies
to kernels based on the Euclidian distance as well. For instance, the Gaussian kernel reads
k(H, X) = exp(?(c(X, X) ? 2c(H, X) + c(H, H))).
(13)
Here, the middle term is the correlation which we are going to evaluate via separable filters.
The first term is independent of the SVs ? it can be efficiently pre-computed and stored in
a separate image. The last term is merely a constant scalar independent of the image data.
Finally, note that these kernels are usually defined on vectors. Nevertheless, we can use
our image-matrix notation due to the fact that the squared Euclidian distance between two
vectors of gray values x and z may be written as
kx ? zk2 = kX ? Zk2F ,
(14)
whereas the dot product amounts to
1
x> z =
kXk2F + kZk2F ? kX ? Zk2F ,
(15)
2
where X and Z are the corresponding image patches and k ? kF is the Frobenius norm for
matrices.
4
Finding rank deficient reduced sets
In our approach we consider a special class of the approximations given by (3), namely
those where the RSVs can be evaluated efficiently via separable correlations. In order to
obtain such approximations, we use a constrained version of Burges? method. In particular,
we restrict the RSV search space to the manifold spanned by all image patches that ?
viewed as matrices ? have a fixed, small rank r (which is to be chosen a priori by the user).
To this end, the Zi in equation (3) are replaced by their singular value decompositions
Z i ? U i S i Vi > .
(16)
The rank constraint can then be imposed by allowing only the first r diagonal elements of
Si to be non-zero. Note that this boils down to using an approximation of the form
0
?0r =
m
X
?i k(Ui,r Si,r Vi,r > , ?)
(17)
i=1
with Si,r being r ? r (diagonal) and Ui,r , Vi,r being h ? r, w ? r (orthogonal2 ) matrices,
respectively. Analogously to (4) we fix m0 and r and find Si,r , Ui,r , Vi,r and ?i that minimize the approximation error ? = k? ? ?0r k2RKHS . The minimization problem is solved via
2
In this paper we call a non-square matrix orthogonal if its columns are pairwise orthogonal and
have unit length.
gradient decent. Note that when computing gradients, the image-matrix notation (together
with (14) or (15), and the equality kXk2F = tr(XX> )) allows a straightforward computation of the kernel derivatives w.r.t. the components of the decomposed RSV image patches,
i.e. the row, column and scale information in Vi,r , Ui,r and Si,r , respectively. However,
while the update rules for ?i and Si,r follow immediately from the respective derivatives,
care must be taken in order to keep Ui,r and Vi,r orthogonal during optimization. This
can be achieved through re-orthogonalization of these matrices after each gradient step.
In our current implementation, however, we perform those updates subject to the so-called
Stiefel constraints [11]. Intuitively, this amounts to rotating (rather than translating) the
columns of Ui,r and Vi,r , which ensures that the resulting matrices are still orthogonal,
i.e. lie on the Stiefel manifold. Let S(h, r) be the manifold of orthogonal h ? r matrices,
the (h, r)-Stiefel manifold. Further, let U?
i,r denote an orthogonal basis for the orthogonal complement of the subspace spanned by the columns of Ui,r . Now, given the ?free?
gradient G = ??/?Ui,r we compute the ?constrained? gradient
? = G ? Ui,r G> Ui,r ,
G
(18)
which is the projection of G onto the tangent space of S(h, r) at Ui,r . The desired rotation
is then given [11] by the (matrix) exponential of the h ? h skew-symmetric matrix
!
? > Ui,r ?(G
? > U? )>
G
i,r
A=t?
(19)
? > U?
G
0
i,r
where t is a user-defined step size parameter. For details, see [11]. A Matlab library is
available at [12].
5
Experiments
This section shows the results of two experiments. The first part illustrates the behavior of
rank deficient approximations for a face detection SVM in terms of the convergence rate
and classification accuracy for different values of r. In the second part, we show how an
actual face detection system, similar to that presented in [5], can be sped up using rank
deficient RSVs. In both experiments we used the same training and validation set. It
consisted of 19 ? 19 gray level image patches containing 16081 manually collected faces
(3194 of them kindly provided by Sami Romdhani) and 42972 non-faces automatically
collected from a set of 206 background scenes. Each patch was normalized to zero mean
and unit variance. The set was split into a training set (13331 faces and 35827 non-faces)
and a validation set (2687 faces and 7145 non-faces). We trained a 1-norm soft margin
SVM on the training set using a Gaussian kernel with ? = 10. The regularization constant
C was set to 1. The resulting decision function (1) achieved a hit rate of 97.3% at 1.0%
false positives on the validation set using m = 6910 SVs. This solution served as the
approximation target ? (see equation (2)) during the experiments described below.
5.1
Rank deficient faces
In order to see how m0 and r affect the accuracy of our approximations, we compute rank
deficient reduced sets for m0 = 1 . . . 32 and r = 1 . . . 3 (the left array in Figure 1 illustrates
the actual appearance of rank deficient RSVs for the m0 = 6 case). Accuracy of the resulting decision functions is measured in ROC score (the area under the ROC curve) on the
validation set. For the full SVM, this amounts to 0.99. The results for our approximations
are depicted in Figure 2. As expected, we need a larger number of rank deficient RSVs
than unconstrained RSVs to obtain similar classification accuracies, especially for small
r. Nevertheless, the experiment points out two advantages of our method. First, a rank as
m' = 1
r=ful
=
+
+
r=3
=
+
+
r=2
=
+
r=1
r=1
r=2
r=3
r=full
m' = 6
=
+ ...
Figure 1: Rank deficient faces. The left array shows the RSVs (Zi ) of the unconstrained
(top row) and constrained (r decreases from 3 to 1 down the remaining rows) approximations for m0 = 6. Interestingly, the r = 3 RSVs are already able to capture face-like
structures. This supports the fact that the classification accuracy for r = 3 is similar to
that of the unconstrained approximations (cf. Figure 2, left plot). The right array shows the
m0 = 1 RSVs (r = full, 3, 2, 1, top to bottom row) and their decomposition into rank one
matrices according to (10). For the unconstrained RSV (first row) it shows an approximate
(truncated) expansion based on the three leading singular vectors. While for r = 3 the
decomposition is indeed similar to the truncated SVD, note how this similarity decreases
for r = 2, 1. This illustrates that the approach is clearly different from simply finding unconstrained RSVs and then imposing the rank constraint via SVD (in fact, the norm (4) is
smaller for the r = 1 RSV than for the leading singular vector of the r = full RSV).
low as three seems already sufficient for our face detection SVM in the sense that for equal
sizes m0 there is no significant loss in accuracy compared to the unconstrained approximation (at least for m0 > 2). The associated speed benefit over unconstrained RSVs is shown
in the right plot of Figure 2: the rank three approximations achieve accuracies similar to
the unconstrained functions, while the number of operations reduces to less than a third.
Second, while for unconstrained RSVs there is no solution with a number of operations
smaller than h ? w = 361 (in the right plot, this is the region beyond the left end of the solid
line), there exist rank deficient functions which are not only much faster than this, but yield
considerably higher accuracies. This property will be exploited in the next experiment.
5.2
A cascade-based face detection system
In this experiment we built a cascade-based face detection system similar to [5, 6], i.e.
a cascade of RSV approximations of increasing size m0 . As the benefit of a cascaded
classifier heavily depends on the speed of the first classifier which has to be evaluated on
the whole image [5, 6], our system uses a rank deficient approximation as the first stage.
Based on the previous experiment, we chose the m0 = 3, r = 1 classifier. Note that this
function yields an ROC score of 0.9 using 114 multiply-adds, whereas the simplest possible
unconstrained approximation m0 = 1, r = full needs 361 multiply-adds to achieve a ROC
score of only 0.83 (cf. Figure 2). In particular, if the threshold of the first stage is set to
yield a hit rate of 95% on the validation set, scanning the MIT+CMU set (130 images, 507
faces) with m0 = 3, r = 1 discards 91.5% of the false positives, whereas the m0 = 1,
r = full can only reject 70.2%. At the same time, when scanning a 320 ? 240 image 3 , the
three separable convolutions plus nonlinearity require 55ms, whereas the single, full kernel
evaluation takes 208ms on a Pentium 4 with 2.8 GHz. Moreover, for the unconstrained
3
For multi-scale processing the detectors are evaluated on an image pyramid with 12 different
scales using a scale decay of 0.75. This amounts to scanning 140158 patches for a 320 ? 240 image.
1
0.9
0.9
0.8
r=1
r=2
r=3
r=full
0.7
0.6
0
10
1
10
#RSVs (m')
ROC score
ROC score
1
0.8
r=1
r=2
r=3
r=full
0.7
0.6
2
10
3
10
4
10
#operations (m'?r?(h+w))
Figure 2: Effect of the rank parameter r on classification accuracies. The left plots shows
the ROC score of the rank deficient RSV approximations (cf. Section 4) for varying set sizes
(m0 = 1 . . . 32, on a logarithmic scale) and ranks (r = 1 . . . 3). Additionally, the solid line
shows the accuracy of the RSVs without rank constraint (cf. Section 2), here denoted by
r = full. The right plot shows the same four curves, but plotted against the number of
operations needed for the evaluation of the corresponding decision function when scanning
large images (i.e. m0 ? r ? (h + w) with h = w = 19), also on a logarithmic scale.
Figure 3: A sample output from our demonstration system (running at 14 frames per second).
In this implementation, we reduced the number
of false positives by adjusting the threshold of
the final classifier. Although this reduces the
number of detections as well, the results are still
satisfactory. This is probably due to the fact that
the MIT+CMU set contains several images of
very low quality that are not likely to occur in
our setting, using a good USB camera.
cascade to catch up in terms of accuracy, the (at least) m0 = 2, r = full classifier (also
with an ROC score of roughly 0.9) should be applied afterwards, requiring another 0.3 ?
2 ? 208 ms ? 125ms.
The subsequent stages of our system consist of unconstrained RSV approximations of size
m0 = 4, 8, 16, 32, respectively. These sizes were chosen such that the number of false
positives roughly halves after each stage, while the number of correct detections remains
close to 95% on the validation set (with the decision thresholds adjusted accordingly). To
eliminate redundant detections, we combine overlapping detections via averaging of position and size if they are closer than 0.15 times the estimated patch size. This system yields
93.1% correct detections and 0.034% false positives on the MIT+CMU set. The current
system was incorporated into a demo application (Figure 3). For optimal performance, we
re-compiled our system using the Intel compiler (ICC). The application now classifies a
320x240 image within 54ms (vs. 238ms with full rank RSVs only) on a 2.8 GHz PC. To
further reduce the number of false positives, additional bootstrapped (as in [5]) stages need
to be added to the cascade. Note that this will not significantly affect the speed of our system (currently 14 frames per second) since 0.034% false positives amounts to merely 47
patches to be processed by subsequent classifiers.
6
Discussion
We have presented a new reduced set method for SVMs in image processing, which creates sparse kernel expansions that can be evaluated via separable filters. To this end, the
user-defined rank (the number of separable filters into which the RSVs are decomposed)
provides a mechanism to control the tradeoff between accuracy and speed of the resulting
approximation. Our experiments show that for face detection, the use of rank deficient
RSVs leads to a significant speedup without losing accuracy. Especially when rough approximations are required, our method gives superior results compared to the existing reduced set methods since it allows for a finer granularity which is vital in cascade-based
detection systems. Another property of our approach is simplicity. At run-time, rank deficient RSVs can be used together with unconstrained RSVs or SVs using the same canonical
image representation. As a result, the required changes in existing code, such as in [5], are
small. In addition, our approach allows the use of off-the-shelf image processing libraries
for separable convolutions. Since such operations are essential in image processing, there
exist many (often highly optimized) implementations. Finally, the method can well be used
to train a neural network, i.e. to go directly from the training data to a sparse, separable
function as opposed to taking the SVM ?detour?. A comparison of that approach to the
present one, however, remains to be done.
References
[1] E. Osuna and F. Girosi. Reducing the run-time complexity in support vector machines. In
B. Sch?olkopf, C. J. C. Burges, and A. J. Smola, editors, Advances in Kernel Methods ? Support
Vector Learning, pages 271?284, Cambridge, MA, 1999. MIT Press.
[2] C. J. C. Burges. Simplified support vector decision rules. In International Conference on
Machine Learning, pages 71?77, 1996.
[3] C. J. C. Burges and B. Scho? lkopf. Improving the accuracy and speed of support vector machines. In M. C. Mozer, M. I. Jordan, and T. Petsche, editors, Advances in Neural Information
Processing Systems, volume 9, page 375. MIT Press, 1997.
[4] E. Osuna, R. Freund, and F. Girosi. Training support vector machines: an application to face
detection. In Proceedings IEEE Conference on Computer Vision and Pattern Recognition, 1997.
[5] S. Romdhani, P. Torr, B.Sch?olkopf, and A. Blake. Computationally efficient face detection. In
Proceedings of the International Conference on Computer Vision, pages 695?700, 2001.
[6] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In
Proceedings IEEE Conference on Computer Vision and Pattern Recognition, 2001.
[7] G. Blanchard and D. Geman. Hierarchical testing designs for pattern recognition. Technical
Report 2003-07, Universit Paris-Sud, 2003.
[8] B. Heisele, T. Poggio, and M. Pontil. Face detection in still gray images. AI Memo 1687, MIT,
May 2000. CBCL Memo 187.
[9] S. Ben-Yacoub, B. Fasel, and J. Luettin. Fast face detection using MLP and FFT. In Proceedings
International Conference on Audio and Video-based Biometric Person Authentication, 1999.
[10] H. A. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(1):23?38, January 1998.
[11] A. Edelman, T. Arias, and S. Smith. The geometry of algorithms with orthogonality constraints.
SIAM Journal on Matrix Analysis and Applications, 20:303?353, 1998.
[12] RDRSLIB ? a matlab library for rank deficient reduced sets in object detection,
http://www.kyb.mpg.de/bs/people/kienzle/rdrs/rdrs.htm.
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1,813 | 2,647 | Non-Local Manifold Tangent Learning
Yoshua Bengio and Martin Monperrus
Dept. IRO, Universit?e de Montr?eal
P.O. Box 6128, Downtown Branch, Montreal, H3C 3J7, Qc, Canada
{bengioy,monperrm}@iro.umontreal.ca
Abstract
We claim and present arguments to the effect that a large class of manifold learning algorithms that are essentially local and can be framed as
kernel learning algorithms will suffer from the curse of dimensionality,
at the dimension of the true underlying manifold. This observation suggests to explore non-local manifold learning algorithms which attempt to
discover shared structure in the tangent planes at different positions. A
criterion for such an algorithm is proposed and experiments estimating a
tangent plane prediction function are presented, showing its advantages
with respect to local manifold learning algorithms: it is able to generalize very far from training data (on learning handwritten character image
rotations), where a local non-parametric method fails.
1
Introduction
A central issue of generalization is how information from the training examples can be
used to make predictions about new examples, and without strong prior assumptions, i.e.
in non-parametric models, this may be fundamentally difficult as illustrated by the curse
of dimensionality. There has been in recent years a lot of work on unsupervised learning based on characterizing a possibly non-linear manifold near which the data would lie,
such as Locally Linear Embedding (LLE) (Roweis and Saul, 2000), Isomap (Tenenbaum,
de Silva and Langford, 2000), kernel Principal Components Analysis (PCA) (Sch o? lkopf,
Smola and M?uller, 1998), Laplacian Eigenmaps (Belkin and Niyogi, 2003), and Manifold Charting (Brand, 2003). These are all essentially non-parametric methods that can be
shown to be kernel methods with an adaptive kernel (Bengio et al., 2004), and which represent the manifold on the basis of local neighborhood relations, very often constructed using the nearest neighbors graph (the graph with one vertex per observed example, and arcs
between near neighbors). The above methods characterize the manifold through an embedding which associates each training example (an input object) with a low-dimensional
coordinate vector (the coordinates on the manifold). Other closely related methods characterize the manifold as well as ?noise? around it. Most of these methods consider the
density as a mixture of flattened Gaussians, e.g. mixtures of factor analyzers (Ghahramani
and Hinton, 1996), Manifold Parzen windows (Vincent and Bengio, 2003), and other local
PCA models such as mixtures of probabilistic PCA (Tipping and Bishop, 1999). This is
not an exhaustive list, and recent work also combines modeling through a mixture density
and dimensionality reduction (Teh and Roweis, 2003; Brand, 2003).
In this paper we claim that there is a fundamental weakness with such kernel methods, due
to the locality of learning: we show that the local tangent plane of the manifold at a point
x is defined based mostly on the near neighbors of x according to some possibly data-
dependent kernel KD . As a consequence, it is difficult with such methods to generalize to
new combinations of values x that are ?far? from the training examples xi , where being
?far? is a notion that should be understood in the context of several factors: the amount of
noise around the manifold (the examples do not lie exactly on the manifold), the curvature
of the manifold, and the dimensionality of the manifold. For example, if the manifold
curves quickly around x, neighbors need to be closer for a locally linear approximation to
be meaningful, which means that more data are needed. Dimensionality of the manifold
compounds that problem because the amount of data thus needed will grow exponentially
with it. Saying that y is ?far? from x means that y is not well represented by its projection
on the tangent plane at x.
In this paper we explore one way to address that problem, based on estimating the tangent
planes of the manifolds as a function of x, with parameters that can be estimated not only
from the data around x but from the whole dataset. Note that there can be more than one
manifold (e.g. in vision, one may imagine a different manifold for each ?class? of object),
but the structure of these manifolds may be related, something that many previous manifold
learning methods did not take advantage of. We present experiments on a variety of tasks
illustrating the weaknesses of the local manifold learning algorithms enumerated above.
The most striking result is that the model is able to generalize a notion of rotation learned
on one kind of image (digits) to a very different kind (alphabet characters), i.e. very far
from the training data.
2
Local Manifold Learning
By ?local manifold learning?, we mean a method that derives information about the local
structure of the manifold (i.e. implicitly its tangent directions) at x based mostly on the
training examples ?around? x. There is a large group of manifold learning methods (as
well as the spectral clustering methods) that share several characteristics, and can be seen
as data-dependent kernel PCA (Bengio et al., 2004). These include LLE (Roweis and Saul,
2000), Isomap (Tenenbaum, de Silva and Langford, 2000), kernel PCA (Sch o? lkopf, Smola
and M?uller, 1998) and Laplacian Eigenmaps (Belkin and Niyogi, 2003). They first build a
data-dependent Gram matrix M with n ? n entries KD (xi , xj ) where D = {x1 , . . . , xn }
is the training set and KD is a data-dependent kernel, and compute the eigenvectoreigenvalue pairs {(vk , ?k )} of M . The embedding of the training set is obtained directly
from the principal eigenvectors vk of M (the
q i-th element of vk gives the k-th coordinate of xi ?s embedding, possibly scaled by ?nk , i.e. ek (xi ) = vik ) and the embedding
for a new example
can be estimated using the Nystro? m formula (Bengio et al., 2004):
Pn
ek (x) = ?1k i=1 vki KD (x, xi ) for the k-th coordinate of x, where ?k is the k-th eigenq
value of M (the optional scaling by ?nk would also apply). The above equation says
that the embedding for a new example x is a local interpolation of the manifold coori)
dinates of its neighbors xi , with interpolating weights given by KD?(x,x
. To see more
k
clearly how the tangent plane may depend only on the neighbors of x, consider the relation between the tangent plane and the embedding function: the tangent plane at x is
k (x)
simply the subspace spanned by the vectors ?e?x
. In the case of very ?local? kernels
like that of LLE, spectral clustering with Gaussian kernel, Laplacian Eigenmaps or kernel
PCA with Gaussian kernel, that derivative only depends significantly on the near neighk (x)
bors of x. Consider for example kernel PCA with a Gaussian kernel: then ?e?x
can be
closely approximated by a linear combination of the difference vectors (x ? x j ) for xj
near x. The weights of that combination may depend on the whole data set, but if the
ambiant space has many more dimensions then the number of such ?near? neighbors of
x, this is a very strong locally determined constraint on the shape of the manifold. The
case of Isomap is less obvious but we show below that it is also local. Let D(a, b) denote
the graph geodesic distance going only through a, b and points from the training set. As
shown in (Bengio et al., 2004), thePcorresponding data-dependent kernel canPbe defined as
2
? i + D)
? where D
?i = 1
KD (x, xi ) = ? 21 (D(x, xi )2 ? n1 j D(x, xj )2 ? D
j D(xi , xj )
n
P
1
?=
?
and D
j Dj . Let N (x, xi ) denote the index j of the training set example xj that is
n
a neighbor of x that minimizes ||x ? xj || + D(xj , xi ). Then
?
?
X
X
(x ? xN (x,xj ) )
(x ? xN (x,xi ) )
?ek (x)
1
1
?
=
vki ?
D(x, xj )
? D(x, xi )
?x
?k i
n j
||x ? xN (x,xj ) ||
||x ? xN (x,xi ) ||
(1)
which is a linear combination of vectors (x ? xk ), where xk is a neighbor of x. This clearly
shows that the tangent plane at x associated with Isomap is also included in the subspace
spanned by the vectors (x ? xk ) where xk is a neighbor of x.
There are also a variety of local manifold learning algorithms which can be classified as
?mixtures of pancakes? (Ghahramani and Hinton, 1996; Tipping and Bishop, 1999; Vincent and Bengio, 2003; Teh and Roweis, 2003; Brand, 2003). These are generally mixtures
of Gaussians with a particular covariance structure. When the covariance matrix is approximated using its principal eigenvectors, this leads to ?local PCA? types of methods. For
these methods the local tangent directions directly correspond to the principal eigenvectors
of the local covariance matrices. Learning is also local since it is mostly the examples
around the Gaussian center that determine its covariance structure. The problem is not so
much due to the form of the density as a mixture of Gaussians. The problem is that the local
parameters (e.g. local principal directions) are estimated mostly based on local data. There
is usually a non-local interaction between the different Gaussians, but its role is mainly of
global coordination, e.g. where to set the Gaussian centers to allocate them properly where
there is data, and optionally how to orient the principal directions so as to obtain a globally
coherent coordinate system for embedding the data.
2.1
Where Local Manifold Learning Would Fail
It is easy to imagine at least four failure causes for local manifold learning methods, and
combining them will create even greater problems:
? Noise around the manifold: data are not exactly lying on the manifold. In the case
of non-linear manifolds, the presence of noise means that more data around each pancake
region will be needed to properly estimate the tangent directions of the manifold in that
region.
? High curvature of the manifold. Local manifold learning methods basically approximate the manifold by the union of many locally linear patches. For this to work, there must
be at least d close enough examples in each patch (more with noise). With a high curvature
manifold, more ? smaller ? patches will be needed, and the number of required patches
will grow exponentially with the dimensionality of the manifold. Consider for example the
manifold of translations of a high-contrast image.The tangent direction corresponds to the
change in image due a small translation, i.e. it is non-zero only at edges in the image. After
a one-pixel translation, the edges have moved by one pixel, and may not overlap much with
the edges of the original image if it had high contrast. This is indeed a very high curvature
manifold.
? High intrinsic dimension of the manifold. We have already seen that high manifold
dimensionality d is hurtful because O(d) examples are required in each patch and O(r d )
patches (for some r depending on curvature and noise) are necessary to span the manifold.
In the translation example, if the image resolution is increased, then many more training
images will be needed to capture the curvature around the translation manifold with locally
linear patches. Yet the physical phenomenon responsible for translation is expressed by a
simple equation, which does not get more complicated with increasing resolution.
? Presence of many manifolds with little data per manifold. In many real-world contexts there is not just one global manifold but a large number of manifolds which however
share something about their structure. A simple example is the manifold of transformations (view-point, position, lighting,...) of 3D objects in 2D images. There is one manifold
per object instance (corresponding to the successive application of small amounts of all
of these transformations). If there are only a few examples for each such class then it
is almost impossible to learn the manifold structures using only local manifold learning.
However, if the manifold structures are generated by a common underlying phenomenon
then a non-local manifold learning method could potentially learn all of these manifolds
and even generalize to manifolds for which a single instance is observed, as demonstrated
in the experiments below.
3
Non-Local Manifold Tangent Learning
Here we choose to characterize the manifolds in the data distribution through a matrixvalued function F (x) that predicts at x ? Rn a basis for the tangent plane of the manifold
near x, hence F (x) ? Rd?n for a d-dimensional manifold. Basically, F (x) specifies
?where? (in which directions) one expects to find near neighbors of x.
We are going to consider a simple supervised learning setting to train this function. As
with Isomap, we consider that the vectors (x ? xi ) with xi a near neighbor of x span a
noisy estimate of the manifold tangent space. We propose to use them to define a ?target?
for training F (x). In our experiments we simply collected the k nearest neighbors of each
example x, but better selection criteria might be devised. Points on the predicted tangent
subspace can be written F 0 (x)w with w ? Rd being local coordinates in the basis specified
by F (x). Several criteria are possible to match the neighbors differences with the subspace
defined by F (x). One that yields to simple analytic calculations is simply to minimize
the distance between the x ? xj vectors and their projection on the subspace defined by
F (x). The low-dimensional local coordinate vector wtj ? Rd that matches neighbor xj
of example xt is thus an extra free parameter that has to be optimized, but is obtained
analytically. The overall training criterion involves a double optimization over function F
and local coordinates wtj of what we call the relative projection error:
min
F,{wtj }
X X
t
j?N (xt )
||F 0 (xt )wtj ? (xt ? xj )||2
||xt ? xj ||2
(2)
where N (x) denotes the selected set of near neighbors of x. The normalization by ||x t ?
xj ||2 is to avoid giving more weight to the neighbors that are further away. The above
ratio amounts to minimizing the square of the sinus of the projection angle. To perform
the above minimization, we can do coordinate descent (which guarantees convergence to
a minimum), i.e. alternate changes in F and changes in w?s which at each step go down
the total criterion. Since the minimization over the w?s can be done separately for each
example xt and neighbor xj , it is equivalent to minimize
||F 0 (xt )wtj ? (xt ? xj )||2
||xt ? xj ||2
(3)
over vector wtj for each such pair (done analytically) and compute the gradient of the
above over F (or its parameters) to move F slightly (we used stochastic gradient on the
parameters of F ). The solution for wtj is obtained by solving the linear system
F (xt )F 0 (xt )wtj = F (xt )
(xt ? xj )
.
||xt ? xj ||2
(4)
In our implementation this is done robustly through a singular value decomposition
F 0 (xt ) = U SV 0 and wtj = B(xt ? xj ) where B can be precomputed for all the neighbors
Pd
of xt : B = ( k=1 1Sk > V.k V.k0 /Sk2 )F (xt ). The gradient of the criterion with respect to
the i-th row of F (xt ), holding wtj fixed, is simply
X
wtji
(F 0 (xt )w ? (xt ? xj ))
2
||x
?
x
||
t
j
j
(5)
where wtji is the i-th element of wtj . In practice, it is not necessary to store more than one
wtj vector at a time. In the experiments, F (?) is parameterized as a an ordinary one hidden
layer neural network with n inputs and d ? n outputs. It is trained by stochastic gradient
descent, one example xt at a time.
Although the above algorithm provides a characterization of the manifold, it does not directly provide an embedding nor a density function. However, once the tangent plane
function is trained, there are ways to use it to obtain all of the above. The simplest method
is to apply existing algorithms that provide both an embedding and a density function based
on a Gaussian mixture with pancake-like covariances. For example one could use (Teh and
Roweis, 2003) or (Brand, 2003), the local covariance matrix around x being constructed
from F 0 (x)diag(? 2 (x))F (x), where ?i2 (x) should estimate V ar(wi ) around x.
3.1
Previous Work on Non-Local Manifold Learning
The non-local manifold learning algorithm presented here (find F (?) which minimizes the
criterion in eq. 2) is similar to the one proposed in (Rao and Ruderman, 1999) to estimate the generator matrix of a Lie group. That group defines a one-dimensional manifold
generated by following the orbit x(t) = eGt x(0), where G is an n ? n matrix and t is a
scalar manifold coordinate. A multi-dimensional manifold can be obtained by replacing
Gt above by a linear combination of multiple generating matrices. In (Rao and Ruderman,
1999) the matrix exponential is approximated to first order by (I + Gt), and the authors
estimatePG for a simple signal undergoing translations, using as a criterion the minimiza?||2 , where x
? is a neighbor of x in the data. Note that in
tion of x,?x mint ||(I + Gt)x ? x
this model the tangent plane is a linear function of x, i.e. F1 (x) = Gx. By minimizing
the above across many pairs of examples, a good estimate of G for the artificial data was
recovered by (Rao and Ruderman, 1999). Our proposal extends this approach to multiple
dimensions and non-linear relations between x and the tangent planes. Note also the earlier
work on Tangent Distance (Simard, LeCun and Denker, 1993), in which the tangent planes
are not learned but used to build a nearest neighbor classifier that is based on the distance
between the tangent subspaces around two examples to be compared. The main advantage
of the approach proposed here over local manifold learning is that the parameters of the
tangent plane predictor can be estimated using data from very different regions of space,
thus in principle allowing to be less sensitive to all four of the problems described in 2.1,
thanks to sharing of information across these different regions.
4
Experimental Results
The objective of the experiments is to validate the proposed algorithm: does it estimate
well the true tangent planes? does it learn better than a local manifold learning algorithm?
Error Measurement In addition to visualizing the results for the low-dimensional data,
we measure performance by considering how well the algorithm learns the local tangent
distance, as measured by the normalized projection error of nearest neighbors (eq. 3). We
compare the errors of four algorithms, always on test data not used to estimate the tangent plane: (a) true analytic (using the true manifold?s tangent plane at x computed analytically), (b) tangent learning (using the neural-network tangent plane predictor F (x),
trained using the k ? d nearest neighbors in the training set of each training set example), (c) Isomap (using the tangent plane defined on Eq. 1), (d) Local PCA (using the d
principal components of the k nearest neighbors of x in the training set).
0.16
Generalization of Tangent Learning
Analytic
Local PCA
Isomap
Tangent Learning
4
0.15
0.14
3
0.13
0.12
2
0.11
1
0.1
0.09
0
0.08
0.07
?1
0.06
?2
0
1
2
3
4
5
6
7
8
9
Figure 1: Task 1 2-D data with 1-D sinusoidal manifolds: the method indeed captures the tangent planes. The small segments are the estimated tangent planes.
Red points are training examples.
1
2
3
4
5
6
7
8
9
10
10
Figure 2: Task 2 relative projection error for k-th nearest neighbor, w.r.t. k, for
compared methods (from lowest to highest at k=1: analytic, tangent learning, local PCA, Isomap). Note U-shape due to
opposing effects of curvature and noise.
Task 1 We first consider a low-dimensional but multi-manifold problem. The data {x i }
are in 2 dimensions and coming from a set of 40 1-dimensional manifolds. Each manifold is composed of 4 near points obtained from a randomly based sinus, i.e ?i ? 1..4,
xi = (a + ti , sin(a + ti ) + b, where a, b, and ti are randomly chosen. Four neighbors
were used for training both the Tangent Learning algorithm and the benchmark local nonparametric estimator (local PCA of the 4 neighbors). Figure 1 shows the training set and
the tangent planes recovered, both on the training examples and generalizing away from
the data. The neural network has 10 hidden units (chosen arbitrarily). This problem is
particularly difficult for local manifold learning, which does very poorly here: the out-ofsample relative prediction error are respectively 0.09 for the true analytic plane, 0.25 for
non-local tangent learning, and 0.81 for local PCA.
Task 2 This is a higher dimensional manifold learning problem, with 41 dimensions.
The data are generated by sampling Gaussian curves. Each curve is of the form x(i) =
2
et1 ?(?2+i/10) /t2 with i ? {0, 1, . . . , 40}. Note that the tangent vectors are not linear in x.
The manifold coordinates are t1 and t2 , sampled uniformly, respectively from (?1, 1) and
(.1, 3.1). Normal noise (standard deviation = 0.001) is added to each point. 100 example
curves were generated for training and 200 for testing. The neural network has 100 hidden
units. Figure 2 shows the relative projection error for the four methods on this task, for the
k-th nearest neighbor, for increasing values of k. First, the error decreases because of the
effect of noise (near noisy neighbors may form a high angle with the tangent plane). Then,
it increases because of the curvature of manifold (further away neighbors form a larger
angle).
Task 3 This is a high-dimensional multi-manifold task, involving digit images to which
we have applied slight rotations, in such a way as to have the knowledge of the analytic
formulation of the manifolds. There is one rotation manifold for each instance of digit from
the database, but only two examples for each manifold: one real image from the MNIST
dataset and one slightly rotated image. 1000?2 examples are used for training and 1000?2
for testing. In this context we use k = 1 nearest neighbor only and manifold dimension
is 1. The average relative projection error for the nearest neighbor is 0.27 for the analytic
tangent plane, 0.43 with tangent learning (100 hidden units), and 1.5 with Local PCA.
Here the neural network would probably overfit if trained too much (here only 100 epochs).
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Figure 3: Left column: original image. Middle: applying a small amount of the predicted
rotation. Right: applying a larger amount of the predicted rotation. Top: using the estimated tangent plane predictor. Bottom: using local PCA, which is clearly much worse.
An even more interesting experiment consists in applying the above trained predictor on
novel images that come from a very different distribution but one that shares the same
manifold structure: it was applied to images of other characters that are not digits. We
have used the predicted tangent planes to follow the manifold by small steps (this is very
easy to do in the case of a one-dimensional manifold). Figure 3 shows for example on a
letter ?M? image the effect of a few such steps and a larger number of steps, both for the
neural network predictor and for the local PCA predictor.
This example illustrates the crucial point that non-local tangent plane learning is able to
generalize to truly novel cases, where local manifold learning fails.
In all the experiments we found that all the randomly initialized neural networks converged
to similarly good solutions. The number of hidden units was not optimized, although preliminary experimentation showed phenomena of over-fitting and under-fitting due to too
small or too large number hidden units was possible.
5
Conclusion
The central claim of this paper is that there are fundamental problems with non-parametric
local approaches to manifold learning, essentially due to the curse of dimensionality (at the
dimensionality of the manifold), but worsened by manifold curvature, noise, and the presence of several disjoint manifolds. To address these problems, we propose that learning
algorithms should be designed in such a way that they can share information, coming from
different regions of space, about the structure of the manifold. In this spirit we have proposed a simple learning algorithm based on predicting the tangent plane at x with a function
F (x) whose parameters are estimated based on the whole data set. Note that the same fundamental problems are present with non-parametric approaches to semi-supervised learning (e.g. as in (Szummer and Jaakkola, 2002; Chapelle, Weston and Scholkopf, 2003;
Belkin and Niyogi, 2003; Zhu, Ghahramani and Lafferty, 2003)), which rely on proper
estimation of the manifold in order to propagate label information.
Future work should investigate how to better handle the curvature problem, e.g. by follow-
ing the manifold (using the local tangent estimates), to estimate a manifold-following path
between pairs of neighboring examples. The algorithm can also be extended in a straightforward way to obtain a Gaussian mixture or a mixture of factor analyzers (with the factors
or the principal eigenvectors of the Gaussian centered at x obtained from F (x)). This view
can also provide an alternative criterion to optimize F (x) (the local log-likelihood of such
a Gaussian). This criterion also tells us how to estimate the missing information (the variances along the eigenvector directions). Since we can estimate F (x) everywhere, a more
ambitious view would consider the density as a ?continuous? mixture of Gaussians (with
an infinitesimal component located everywhere in space).
Acknowledgments
The authors would like to thank the following funding organizations for support: NSERC,
MITACS, IRIS, and the Canada Research Chairs.
References
Belkin, M. and Niyogi, P. (2003). Using manifold structure for partially labeled classification. In
Becker, S., Thrun, S., and Obermayer, K., editors, Advances in Neural Information Processing
Systems 15, Cambridge, MA. MIT Press.
Bengio, Y., Delalleau, O., Le Roux, N., Paiement, J.-F., Vincent, P., and Ouimet, M. (2004). Learning
eigenfunctions links spectral embedding and kernel PCA. Neural Computation, 16(10):2197?
2219.
Brand, M. (2003). Charting a manifold. In Becker, S., Thrun, S., and Obermayer, K., editors,
Advances in Neural Information Processing Systems 15. MIT Press.
Chapelle, O., Weston, J., and Scholkopf, B. (2003). Cluster kernels for semi-supervised learning. In
Becker, S., Thrun, S., and Obermayer, K., editors, Advances in Neural Information Processing
Systems 15, Cambridge, MA. MIT Press.
Ghahramani, Z. and Hinton, G. (1996). The EM algorithm for mixtures of factor analyzers. Technical
Report CRG-TR-96-1, Dpt. of Comp. Sci., Univ. of Toronto.
Rao, R. and Ruderman, D. (1999). Learning lie groups for invariant visual perception. In Kearns,
M., Solla, S., and Cohn, D., editors, Advances in Neural Information Processing Systems 11,
pages 810?816. MIT Press, Cambridge, MA.
Roweis, S. and Saul, L. (2000). Nonlinear dimensionality reduction by locally linear embedding.
Science, 290(5500):2323?2326.
Sch?
olkopf, B., Smola, A., and M?
uller, K.-R. (1998). Nonlinear component analysis as a kernel
eigenvalue problem. Neural Computation, 10:1299?1319.
Simard, P., LeCun, Y., and Denker, J. (1993). Efficient pattern recognition using a new transformation
distance. In Giles, C., Hanson, S., and Cowan, J., editors, Advances in Neural Information
Processing Systems 5, pages 50?58, Denver, CO. Morgan Kaufmann, San Mateo.
Szummer, M. and Jaakkola, T. (2002). Partially labeled classification with markov random walks.
In Dietterich, T., Becker, S., and Ghahramani, Z., editors, Advances in Neural Information
Processing Systems 14, Cambridge, MA. MIT Press.
Teh, Y. W. and Roweis, S. (2003). Automatic alignment of local representations. In Becker, S.,
Thrun, S., and Obermayer, K., editors, Advances in Neural Information Processing Systems 15.
MIT Press.
Tenenbaum, J., de Silva, V., and Langford, J. (2000). A global geometric framework for nonlinear
dimensionality reduction. Science, 290(5500):2319?2323.
Tipping, M. and Bishop, C. (1999). Mixtures of probabilistic principal component analysers. Neural
Computation, 11(2):443?482.
Vincent, P. and Bengio, Y. (2003). Manifold parzen windows. In Becker, S., Thrun, S., and Obermayer, K., editors, Advances in Neural Information Processing Systems 15, Cambridge, MA.
MIT Press.
Zhu, X., Ghahramani, Z., and Lafferty, J. (2003). Semi-supervised learning using gaussian fields and
harmonic functions. In ICML?2003.
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1,814 | 2,648 | Semi-Markov Conditional Random Fields for
Information Extraction
Sunita Sarawagi
Indian Institute of Technology
Bombay, India
[email protected]
William W. Cohen
Center for Automated Learning & Discovery
Carnegie Mellon University
[email protected]
Abstract
We describe semi-Markov conditional random fields (semi-CRFs), a conditionally trained version of semi-Markov chains. Intuitively, a semiCRF on an input sequence x outputs a ?segmentation? of x, in which
labels are assigned to segments (i.e., subsequences) of x rather than to
individual elements xi of x. Importantly, features for semi-CRFs can
measure properties of segments, and transitions within a segment can
be non-Markovian. In spite of this additional power, exact learning and
inference algorithms for semi-CRFs are polynomial-time?often only a
small constant factor slower than conventional CRFs. In experiments
on five named entity recognition problems, semi-CRFs generally outperform conventional CRFs.
1
Introduction
Conditional random fields (CRFs) are a recently-introduced formalism [12] for representing a conditional model Pr(y|x), where both x and y have non-trivial structure (often
sequential). Here we introduce a generalization of sequential CRFs called semi-Markov
conditional random fields (or semi-CRFs). Recall that semi-Markov chain models extend
hidden Markov models (HMMs) by allowing each state si to persist for a non-unit length of
time di . After this time has elapsed, the system will transition to a new state s0 , which depends only on si ; however, during the ?segment? of time between i and i + di , the behavior
of the system may be non-Markovian. Generative semi-Markov models are fairly common
in certain applications of statistics [8, 9], and are also used in reinforcement learning to
model hierarchical Markov decision processes [19].
Semi-CRFs are a conditionally trained version of semi-Markov chains. In this paper, we
present inference and learning methods for semi-CRFs. We also argue that segments often
have a clear intuitive meaning, and hence semi-CRFs are more natural than conventional
CRFs. We focus here on named entity recognition (NER), in which a segment corresponds
to an extracted entity; however, similar arguments might be made for several other tasks,
such as gene-finding [11] or NP-chunking [16].
In NER, a semi-Markov formulation allows one to easily construct entity-level features
(such as ?entity length? and ?similarity to other known entities?) which cannot be easily
encoded in CRFs. Experiments on five different NER problems show that semi-CRFs often
outperform conventional CRFs.
2
2.1
CRFs and Semi-CRFs
Definitions
A CRF models Pr(y|x) using a Markov random field, with nodes corresponding to elements of the structured object y, and potential functions that are conditional on (features
of) x. Learning is performed by setting parameters to maximize the likelihood of a set of
(x, y) pairs given as training data. One common use of CRFs is for sequential learning
problems like NP chunking [16], POS tagging [12], and NER [15]. For these problems
the Markov field is a chain, and y is a linear sequence of labels from a fixed set Y. For instance, in the NER application, x might be a sequence of words, and y might be a sequence
in {I, O}|x| , where yi = I indicates ?word xi is inside a name? and yi = O indicates the
opposite.
Assume a vector f of local feature functions f = hf 1 , . . . , f K i, each of which maps a pair
(x, y) and an index i to a measurement f k (i, x, y) ? R. Let f (i, x, y) be the vector of these
P|x|
measurements, and let F(x, y) = i f (i, x, y). For example, in NER, the components
of f might include the measurement f 13 (i, x, y) = [[xi is capitalized]] ? [[yi = I]], where
the indicator function [[c]] = 1 if c if true and zero otherwise; this implies that F 13 (x, y)
would be the number of capitalized words xi paired with the label I. Following previous
work [12, 16] we will define a conditional random field (CRF) to be an estimator of the
form
1 W?F(x,y)
e
(1)
Pr(y|x, W) =
Z(x)
P W?F(x,y0 )
where W is a weight vector over the components of F, and Z(x) = y0 e
.
To extend this to the semi-Markov case, let s = hs1 , . . . , sp i denote a segmentation of x,
where segment sj = htj , uj , yj i consists of a start position tj , an end position uj , and a
label yj ? Y . Conceptually, a segment means that the tag yj is given to all xi ?s between
i = tj and i = uj , inclusive. We assume segments have positive length, and completely
cover the sequence 1 . . . |x| without overlapping: that is, that tj and uj always satisfy
t1 = 1, up = |x|, 1 ? tj ? uj ? |x|, and tj+1 = uj + 1. For NER, a valid segmentation
of the sentence ?I went skiing with Fernando Pereira in British Columbia? might be s =
h(1, 1, O), (2, 2, O), (3, 3, O), (4, 4, O), (5, 6, I), (7, 7, O), (8, 9, I)i, corresponding to the
label sequence y = hO, O, O, O, I, I, O, I, Ii.
We now assume a vector g of segment feature functions g = hg 1 , . . . , g K i, each of
which maps a triple (j, x, s) to a measurement g k (j, x, s) ? R, and define G(x, s) =
P|s|
j g(j, x, s). We also make a restriction on the features, analogous to the usual Markovian assumption made in CRFs, and assume that every component g k of g is a function
only of x, sj , and the label yj?1 associated with the preceding segment sj?1 . In other
words, we assume that every g k (j, x, s) can be rewritten as
g k (j, x, s) = g 0k (yj , yj?1 , x, tj , uj )
(2)
for an appropriately defined g 0k . In the rest of the paper, we will drop the g 0 notation and
use g for both versions of the segment-level feature functions. A semi-CRF is then an
estimator of the form
1 W?G(x,s)
Pr(s|x, W) =
e
(3)
Z(x)
P W?G(x,s0 )
where again W is a weight vector for G and Z(x) = s0 e
.
2.2
An efficient inference algorithm
The inference problem for a semi-CRF is defined as follows: given W and x, find the
best segmentation, argmax s Pr(s|x, W), where Pr(s|x, W) is defined by Equation 3. An
efficient inference algorithm is suggested by Equation 2, which implies that
X
argmax s Pr(s|x, W) = argmax s W ? G(x, s) = argmax s W ?
g(yj , yj?1 , x, tj , uj )
j
Let L be an upper bound on segment length. Let si:y denote the set of all partial segmentations starting from 1 (the first index of the sequence) to i, such that the last segment has
the label y and ending position i. Let Vx,g,W (i, y) denote the largest value of W ? G(x, s0 )
for any s0 ? si:y . Omitting the subscripts, the following recursive calculation implements
a semi-Markov analog of the usual Viterbi algorithm:
(
maxy0 ,d=1...L V (i ? d, y 0 ) + W ? g(y, y 0 , x, i ? d + 1, i) if i > 0
0
if i = 0
V (i, y) =
(4)
??
if i < 0
The best segmentation then corresponds to the path traced by maxy V (|x|, y).
2.3
Semi-Markov CRFs vs order-L CRFs
Since conventional CRFs need not maximize over possible segment lengths d, inference
for semi-CRFs is more expensive. However, Equation 4 shows that the additional cost is
only linear in L. For NER, a reasonable value of L might be four or five.1 Since in the
worst case L ? |x|, the semi-Markov Viterbi algorithm is always polynomial, even when
L is unbounded.
For fixed L, it can be shown that semi-CRFs are no more expressive than order-L CRFs.
For order-L CRFs, however the additional computational cost is exponential in L. The
difference is that semi-CRFs only consider sequences in which the same label is assigned
to all L positions, rather than all |Y|L length-L sequences. This is a useful restriction, as it
leads to faster inference.
Semi-CRFs are also a natural restriction, as it is often convenient to express features in
terms of segments. As an example, let dj denote the length of a segment, and let ?
be the average length of all segments with label I. Now consider the segment feature
g k1 (j, x, s) = (dj ? ?)2 ? [[yj = I]]. After training, the contribution of this feature toward
2
Pr(s|x) associated with a length-d entity will be proportional to ewk ?(d??) ?i.e., it allows
the learner to model a Gaussian distribution of entity lengths.
An exponential model for lengths could be implemented with the feature g k2 (j, x, y) =
dj ? [[yj = I]]. In contrast to the Gaussian-length feature above, g k2 is ?equivalent to? a
local feature function f (i,P
x, y) = [[yi = I]], in
Pthe following sense: for every triple x, y, s,
where y is the tags for s, j g k2 (j, x, s) = i f (i, s, y). Thus a semi-CRF model based
on the single feature g k2 could also be represented by a conventional CRF.
In general, a semi-CRF model can be factorized in terms of an equivalent order-1 CRF
model if and only if the sum of the segment features can be rewritten as a sum of local
features. Thus the degree to which semi-CRFs are non-Markovian depends on the feature
set.
2.4
Learning algorithm
During training the goal is to maximize log-likelihood over a given training set T =
{(x` , s` )}N
`=1 . Following the notation of Sha and Pereira [16], we express the loglikelihood over the training sequences as
X
X
L(W) =
log Pr(s` |x` , W) =
(W ? G(x` , s` ) ? log ZW (x` ))
(5)
`
1
`
Assuming that non-entity words are placed in unit-length segments, as we do below.
We wish to find a W that maximizes L(W). Equation 5 is convex, and can thus be maximized by gradient ascent, or one of many related methods. (In our implementation we use
a limited-memory quasi-Newton method [13, 14].) The gradient of L(W) is the following:
P
0
W?G(x` ,s0 )
X
0 G(s , x` )e
?L(W) =
G(x` , s` ) ? s
(6)
ZW (x` )
`
X
=
G(x` , s` ) ? EPr(s0 |W) G(x` , s0 )
(7)
`
The first set of terms are easy to compute. However, to compute the the normalizer, ZW (x` ), and the expected value of the features under the current weight vector,
EPr(s0 |W) G(x` , s0 ), we must use the Markov property of G to construct a dynamic programming algorithm, similar for forward-backward. We thus define ?(i, y) as the value of
P
W?G(s0 ,x)
where again si:y denotes all segmentations from 1 to i ending at i and
s0 ?si:y e
labeled y. For i > 0, this can be expressed recursively as
?(i, y) =
L X
X
0
?(i ? d, y 0 )eW?g(y,y ,x,i?d+1,i)
d=1 y 0 ?Y
with the base cases defined as ?(0,
Py) = 1 and ?(i, y) = 0 for i < 0. The value of ZW (x)
can then be written as ZW (x) = y ?(|x|, y).
P
0
A similar approach can be used to compute the expectation s0 G(x` , s0 )eW?G(x` ,s ) .
For the k-th component of G, let ? k (i, y) be the value of the sum
P
k 0
W?G(x` ,s0 )
, restricted to the part of the segmentation ending at
s0 ?si:y G (s , x` )e
position i. The following recursion2 can then be used to compute ? k (i, y):
? k (i, y) =
PL P
0
(i?d, y 0 ) + ?(i?d, y 0 )g k (y, y 0 , x, i?d + 1, i))eW?g(y,y ,x,i?d+1,i)
P
Finally we let EPr(s0 |W) Gk (s0 , x) = ZW1(x) y ? k (|x|, y).
d=1
3
3.1
y 0 ?Y (?
k
Experiments with NER data
Baseline algorithms and datasets
In our experiments, we trained semi-CRFs to mark entity segments with the label I, and
put non-entity words into unit-length segments with label O. We compared this with two
versions of CRFs. The first version, which we call CRF/1, labels words inside and outside
entities with I and O, respectively. The second version, called CRF/4, replaces the I tag
with four tags B, E, C, and U , which depend on where the word appears in an entity [2].
We compared the algorithms on five NER problems, associated with three different corpora.
The Address corpus contains 4,226 words, and consists of 395 home addresses of students
in a major university in India [1]. We considered extraction of city names and state names
from this corpus. The Jobs corpus contains 73,330 words, and consists of 300 computerrelated job postings [4]. We considered extraction of company names and job titles. The
18,121-word Email corpus contains 216 email messages taken from the CSPACE email
corpus [10], which is mail associated with a 14-week, 277-person management game. Here
we considered extraction of person names.
2
As in the forward-backward algorithm for chain CRFs [16], space requirements here can be
reduced from M L|Y| to M |Y|, where M is the length of the sequence, by pre-computing an appropriate set of ? values.
3.2
Features
As features for CRF, we used indicators for specific words at location i, or locations within
three words of i. Following previous NER work [7]), we also used indicators for capitalization/letter patterns (such as ?Aa+? for a capitalized word, or ?D? for a single-digit
number).
As features for semi-CRFs, we used the same set of word-level features, as well their
logical extensions to segments. Specifically, we used indicators for the phrase inside a
segment and the capitalization pattern inside a segment, as well as indicators for words
and capitalization patterns in 3-word windows before and after the segment. We also used
indicators for each segment length (d = 1, . . . , L), and combined all word-level features
with indicators for the beginning and end of a segment.
To exploit more of the power of semi-CRFs, we also implemented a number of dictionaryderived features, each of which was based on different dictionary D and similarity function
sim. Letting xsj denote the subsequence hxtj . . . xuj i, a dictionary feature is defined as
g D,sim (j, x, s) = argmax u?D sim(xsj , u)?i.e., the distance from the word sequence xsj
to the closest element in D.
For each of the extraction problems, we assembled one external dictionary of strings, which
were similar (but not identical) to the entity names in the documents. For instance, for
city names in the Address data, we used a web page listing cities in India. Due to variations in the way entity names are written, rote matching these dictionaries to the data
gives relatively low F1 values, ranging from 22% (for the job-title extraction task) to 57%
(for the person-name task). We used three different similarity metrics (Jaccard, TFIDF,
and JaroWinkler) which are known to work well for name-matching in data integration
tasks [5]. All of the distance metrics are non-Markovian?i.e., the distance-based segment
features cannot be decomposed into sums of local features. More detail on the distance
metrics, feature sets, and datasets above can be found elsewhere [6].
We also extended the semi-CRF algorithm to construct, on the fly, an internal segment
dictionary of segments labeled as entities in the training data. To make measurements on
training data similar to those on test data, when finding the closest neighbor of xsj in the
internal dictionary, we excluded all strings formed from x, thus excluding matches of xsj to
itself (or subsequences of itself). This feature could be viewed as a sort of nearest-neighbor
classifier; in this interpretation the semi-CRF is performing a sort of bi-level stacking [21].
For completeness in the experiments, we also evaluated local versions of the dictionary
features. Specifically, we constructed dictionary features of the form f D,sim (i, x, y) =
argmax u?D sim(xi , u), where D is either the external dictionary used above, or an internal
word dictionary formed from all words contained in entities. As before, words in x were
excluded in finding near neighbors to xi .
3.3
Results and Discussion
We evaluated F1-measure performance3 of CRF/1, CRF/4, and semi-CRFs, with and without internal and external dictionaries. A detailed tabulation of the results are shown in Table 1, and Figure 1 shows F1 values plotted against training set size for a subset of three of
the tasks, and four of the learning methods. In each experiment performance was averaged
over seven runs, and evaluation was performed on a hold-out set of 30% of the documents.
In the table the learners are trained with 10% of the available data?as the curves show,
performance differences are often smaller with more training data. Gaussian priors were
used for all algorithms, and for semi-CRFs, a fixed value of L was chosen for each dataset
based on observed entity lengths. This ranged between 4 and 6 for the different datasets.
In the baseline configuration in which no dictionary features are used, semi-CRFs perform
3
F1 is defined as 2*precision*recall/(precision+recall.)
Address_City
Email_Person
90
90
85
85
F1 span accuracy
F1 span accuracy
F1 span accuracy
Address_State
100
90
80
70
60
50
40
CRF/4
30
SemiCRF+int
20
CRF/4+dict
10
SemiCRF+int+dict
0
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Fraction of available training data
80
75
CRF/4
SemiCRF+int
CRF/4+dict
SemiCRF+int+dict
70
65
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Fraction of available training data
80
75
70
CRF/4
SemiCRF+int
CRF/4+dict
SemiCRF+int+dict
65
0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Fraction of available training data
Figure 1: F1 as a function of training set size. Algorithms marked with ?+dict? include external
dictionary features, and algorithms marked with ?+int? include internal dictionary features. We do
not use internal dictionary features for CRF/4 since they lead to reduced accuracy.
baseline
F1
CRF/1
state
title
person
city
company
CRF/4
state
title
person
city
company
semi-CRF
state
title
person
city
company
+internal dict
F1 ?base
+external dict
F1 ?base
+both dictionaries
F1 ?base ?extern
20.8
28.5
67.6
70.3
51.4
44.5
3.8
48.0
60.0
16.5
113.9
-86.7
-29.0
-14.7
-67.9
69.2
38.6
81.4
80.4
55.3
232.7
35.4
20.4
14.4
7.6
55.2
19.9
64.7
69.8
15.6
165.4
-30.2
-4.3
-0.7
-69.6
-67.3
-65.6
-24.7
-15.1
-77.2
15.0
23.7
70.9
73.2
54.8
25.4
7.9
64.5
70.6
20.6
69.3
-66.7
-9.0
-3.6
-62.4
46.8
36.4
82.5
80.8
61.2
212.0
53.6
16.4
10.4
11.7
43.1
14.6
74.8
76.3
25.1
187.3
-38.4
5.5
4.2
-54.2
-24.7
-92.0
-10.9
-6.1
-65.9
25.6
33.8
72.2
75.9
60.2
35.5
37.5
74.8
75.3
59.7
38.7
10.9
3.6
-0.8
-0.8
62.7
41.1
82.8
84.0
60.9
144.9
21.5
14.7
10.7
1.2
65.2
40.2
83.7
83.6
60.9
154.7
18.9
15.9
10.1
1.2
9.8
-2.5
1.2
-0.5
0.0
Table 1: Comparing various methods on five IE tasks, with and without dictionary features. The
column ?base is percentage change in F1 values relative to the baseline. The column ?extern is is
change relative to using only external-dictionary features.
best on all five of the tasks. When internal dictionary features are used, the performance
of semi-CRFs is often improved, and never degraded by more than 2.5%. However, the
less-natural local version of these features often leads to substantial performance losses for
CRF/1 and CRF/4. Semi-CRFs perform best on nine of the ten task variants for which
internal dictionaries were used. The external-dictionary features are helpful to all the algorithms. Semi-CRFs performs best on three of five tasks in which only external dictionaries
were used.
Overall, semi-CRF performs quite well. If we consider the tasks with and without external
dictionary features as separate ?conditions?, then semi-CRFs using all available information4 outperform both CRF variants on eight of ten ?conditions?.
We also compared semi-CRF to order-L CRFs, with various values of L.5 In Table 2 we
show the result for L = 1, L = 2, and L = 3, compared to semi-CRF. For these tasks, the
performance of CRF/4 and CRF/1 does not seem to improve much by simply increasing
4
I.e., the both-dictionary version when external dictionaries are available, and the internaldictionary only version otherwise.
5
Order-L CRFs were implemented by replacing the label set Y with Y L . We limited experiments
to L ? 3 for computational reasons.
Address State
Address City
Email persons
L=1
20.8
70.3
67.6
CRF/1
L=2
20.1
71.0
63.7
L=3
19.2
71.2
66.7
L=1
15.0
73.2
70.9
CRF/4
L=2
16.4
73.9
70.7
semi-CRF
L=3
16.4
73.7
70.4
25.6
75.9
72.2
Table 2: F1 values for different order CRFs
order.
4
Related work
Semi-CRFs are similar to nested HMMs [1], which can also be trained discriminitively [17]. The primary difference is that the ?inner model? for semi-CRFs is of short,
uniformly-labeled segments with non-Markovian properties, while nested HMMs allow
longer, diversely-labeled, Markovian ?segments?.
Discriminative learning methods can be used for conditional random fields with architectures more complex than chains (e.g., [20, 18]), and one of these methods has also been
applied to NER [3]. Further, by creating a random variable for each possible segment, one
can learn models strictly more expressive than the semi-Markov models described here.
However, for these methods, inference is not tractable, and hence approximations must be
made in training and classification. An interesting question for future research is whether
the tractible extension to CRF inference considered here can can be used to improve inference methods for more expressive models.
In recent prior work [6], we investigated semi-Markov learning methods for NER based
on a voted perceptron training algorithm [7]. The voted perceptron has some advantages
in ease of implementation, and efficiency. (In particular, the voted perceptron algorithm
is more stable numerically, as it does not need to compute a partition function. ) However, semi-CRFs perform somewhat better, on average, than our perceptron-based learning
algorithm. Probabilistically-grounded approaches like CRFs also are preferable to marginbased approaches like the voted perceptron in certain settings, e.g., when it is necessary to
estimate confidences in a classification.
5
Concluding Remarks
Semi-CRFs are a tractible extension of CRFs that offer much of the power of higher-order
models without the associated computational cost. A major advantage of semi-CRFs is that
they allow features which measure properties of segments, rather than individual elements.
For applications like NER and gene-finding [11], these features can be quite natural.
Appendix
An implementation of semi-CRFs is available at http://crf.sourceforge.net, and a NER
package using this package is available on http://minorthird.sourceforge.net.
References
[1] V. R. Borkar, K. Deshmukh, and S. Sarawagi. Automatic text segmentation for extracting
structured records. In Proc. ACM SIGMOD International Conf. on Management of Data, Santa
Barabara,USA, 2001.
[2] A. Borthwick, J. Sterling, E. Agichtein, and R. Grishman. Exploiting diverse knowledge sources
via maximum entropy in named entity recognition. In Sixth Workshop on Very Large Corpora
New Brunswick, New Jersey. Association for Computational Linguistics., 1998.
[3] R. Bunescu and R. J. Mooney. Relational markov networks for collective information extraction. In Proceedings of the ICML-2004 Workshop on Statistical Relational Learning (SRL2004), Banff, Canada, July 2004.
[4] M. E. Califf and R. J. Mooney. Bottom-up relational learning of pattern matching rules for
information extraction. Journal of Machine Learning Research, 4:177?210, 2003.
[5] W. W. Cohen, P. Ravikumar, and S. E. Fienberg. A comparison of string distance metrics for
name-matching tasks. In Proceedings of the IJCAI-2003 Workshop on Information Integration
on the Web (IIWeb-03), 2003.
[6] W. W. Cohen and S. Sarawagi. Exploiting dictionaries in named entity extraction: Combining
semi-markov extraction processes and data integration methods. In Proceedings of the Tenth
ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2004.
[7] M. Collins. Discriminative training methods for hidden Markov models: Theory and experiments with perceptron algorithms. In Empirical Methods in Natural Language Processing
(EMNLP), 2002.
[8] X. Ge. Segmental Semi-Markov Models and Applications to Sequence Analysis. PhD thesis,
University of California, Irvine, December 2002.
[9] J. Janssen and N. Limnios. Semi-Markov Models and Applications. Kluwer Academic, 1999.
[10] R. E. Kraut, S. R. Fussell, F. J. Lerch, and J. A. Espinosa. Coordination in teams: evidence from
a simulated management game. To appear in the Journal of Organizational Behavior, 2005.
[11] A. Krogh. Gene finding: putting the parts together. In M. J. Bishop, editor, Guide to Human
Genome Computing, pages 261?274. Academic Press, 2nd edition, 1998.
[12] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for
segmenting and labeling sequence data. In Proceedings of the International Conference on
Machine Learning (ICML-2001), Williams, MA, 2001.
[13] D. C. Liu and J. Nocedal. On the limited memory BFGS method for large-scale optimization.
Mathematic Programming, 45:503?528, 1989.
[14] R. Malouf. A comparison of algorithms for maximum entropy parameter estimation. In Proceedings of The Sixth Conference on Natural Language Learning (CoNLL-2002), pages 49?55,
2002.
[15] A. McCallum and W. Li. Early results for named entity recognition with conditional random
fields, feature induction and web-enhanced lexicons. In Proceedings of The Seventh Conference
on Natural Language Learning (CoNLL-2003), Edmonton, Canada, 2003.
[16] F. Sha and F. Pereira. Shallow parsing with conditional random fields. In Proceedings of HLTNAACL, 2003.
[17] M. Skounakis, M. Craven, and S. Ray. Hierarchical hidden Markov models for information
extraction. In Proceedings of the 18th International Joint Conference on Artificial Intelligence,
Acapulco, Mexico. Morgan Kaufmann., 2003.
[18] C. Sutton, K. Rohanimanesh, and A. McCallum. Dynamic conditional random fields: Factorized probabilistic models for labeling and segmenting sequence data. In ICML, 2004.
[19] R. Sutton, D. Precup, and S. Singh. Between MDPs and semi-MDPs: A framework for temporal
abstraction in reinforcement learning. Artificial Intelligence, 112:181?211, 1999.
[20] B. Taskar, P. Abbeel, and D. Koller. Discriminative probabilistic models for relational data. In
Proceedings of Eighteenth Conference on Uncertainty in Artificial Intelligence (UAI02), Edmonton, Canada, 2002.
[21] D. H. Wolpert. Stacked generalization. Neural Networks, 5:241?259, 1992.
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1,815 | 2,649 | Optimal information decoding from neuronal
populations with specific stimulus selectivity
Marcelo A. Montemurro
The University of Manchester
Faculty of Life Sciences
Moffat Building
PO Box 88, Manchester M60 1QD, UK
[email protected]
Stefano Panzeri ?
The University of Manchester
Faculty of Life Sciences
Moffat Building
PO Box 88, Manchester M60 1QD, UK
[email protected]
Abstract
A typical neuron in visual cortex receives most inputs from other cortical
neurons with a roughly similar stimulus preference. Does this arrangement of inputs allow efficient readout of sensory information by the target cortical neuron? We address this issue by using simple modelling of
neuronal population activity and information theoretic tools. We find that
efficient synaptic information transmission requires that the tuning curve
of the afferent neurons is approximately as wide as the spread of stimulus preferences of the afferent neurons reaching the target neuron. By
meta analysis of neurophysiological data we found that this is the case
for cortico-cortical inputs to neurons in visual cortex. We suggest that
the organization of V1 cortico-cortical synaptic inputs allows optimal information transmission.
1
Introduction
A typical neuron in visual cortex receives most of its inputs from other visual cortical neurons. The majority of cortico-cortical inputs arise from afferent cortical neurons with a
preference to stimuli which is similar to that of the target neuron [1, 2, 3]. For example, orientation selective neurons in superficial layers in ferret visual cortex receive more
than 50% of their cortico-cortical excitatory inputs from neurons with orientation preference which is less than 30o apart. However, this input structure is rather broad in terms
of stimulus-specificity: cortico-cortical connections between neurons tuned to dissimilar
stimulus orientation also exist [4]. The structure and spread of the stimulus specificity of
cortico-cortical connections has received a lot of attention because of its importance for
understanding the mechanisms of generation of orientation tuning (see [4] for a review).
However, little is still known on whether this structure of inputs allows efficient transmission of sensory information across cortico-cortical synapses.
It is likely that efficiency of information transmission across cortico-cortical synapses also
depends on the width of tuning curves of the afferent cortical neurons to stimuli. In fact,
theoretical work on population coding has shown that the width of the tuning curves has
?
Corresponding author
an important influence on the quality and the nature of the information encoding in cortical
populations [5, 6, 7, 8]. Another factor that may influence the efficiency of cortico-cortical
synaptic information transmission is the biophysical capability of the target neuron. To
conserve all information during synaptic transmission, the target neuron must conserve the
?label? of the spikes arriving from multiple input neurons at different places on its dendritic
tree [9]. Because of biophysical limitations, a target neuron that e.g. can only sum inputs
at the soma may lose a large part of the information present in the afferent activity. The
optimal arrangement of cortico-cortical synapses may also depend on the capability of
postsynaptic neurons in processing separately spikes from different neurons.
In this paper, we address the problem of whether cortico-cortical synaptic systems encode
information efficiently. We introduce a simple model of neuronal information processing
that takes into account both the selective distribution of stimulus preferences typical of
cortico-cortical connections and the potential biophysical limitations of cortical neurons.
We use this model and information theoretic tools to investigate whether there is an optimal trade-off between the spread of distribution of stimulus preference across the afferent
neurons and the tuning width of the afferent neurons itself. We find that efficient synaptic
information transmission requires that the tuning curve of the afferent neurons is approximately as wide as the spread of stimulus preferences of the afferent fibres reaching the
target neuron. By reviewing anatomical and physiological data, we argue that this optimal
trade-off is approximately reached in visual cortex. These results suggest that neurons in
visual cortex are wired to decode optimally information from a stimulus-specific distribution of synaptic inputs.
2
Model of the activity of the afferent neuronal population
We consider a simple model for the activity of the afferent neuronal population based on
the known tuning properties and spatial and synaptic organisation of sensory areas.
2.1
Stimulus tuning of individual afferent neurons
We assume that the the population is made of a large number N of neurons (for a real
cortical neuron, the number N of afferents is in the order of few thousands [10]). The
response of each neuron rk (k = 1, ? ? ? , N ) is quantified as the number of spikes fired in
a salient post-stimulus time window of a length ? . Thus, the overall neuronal population
response is represented as a spike count vector r = (r1 , ? ? ? , rN ).
We assume that the neurons are tuned to a small number D of relevant stimulus parameters
[11, 12], such as e.g. orientation, speed or direction of motion of a visual object. The
stimulus variable will thus be described as a vector s = (s1 , . . . , sD ) of dimension D. The
number of stimulus features that are encoded by the neuron will be left as a free parameter
to be varied within the range 1-5 for two reasons. First, although there is evidence that the
number of stimulus features encoded by a single neuron is limited [11, 12], more research
is still needed to determine exactly how many stimulus parameters are encoded in different
areas. Second, a previous related study [8] has shown that, when considering large neuronal
populations with a uniform distribution of stimulus preferences (such as an hypercolumn
in V1 containing all stimulus orientations) the tuning width of individual neurons which is
optimal for population coding depends crucially on the number of stimulus features being
encoded. Thus, it is interesting to investigate how the optimal arrangement of corticocortical synaptic systems depends on the number of stimulus features being encoded.
The neuronal tuning function of the k ? th neuron (k = 1, ? ? ? , N ), which quantifies the
mean spike count of the k ? th neuron to the presented stimulus, is modelled as a Gaussian
distribution, characterised by the following parameters: preferred stimulus s(k) , tuning
width ?f , and response modulation m:
?
f (k) (s) = me
(s?s(k) )2
2?f 2
(1)
The Gaussian tuning curve is a good description of the tuning properties of e.g. V1 or
MT neurons to variables such as stimulus orientation motion direction [13, 14, 15], and is
hence widely used in models of sensory coding [16, 17]. Large values of ?f indicate coarse
coding, whereas small values of ?f indicate sharp tuning.
Spike count responses of each neuron on each trial are assumed to follow a Poisson distribution whose mean is given by the above neuronal tuning function (Eq. 1). The Poisson
model is widely used because it is the simplest model of neuronal firing that captures the
salient property of neuronal firing that the variance of spike counts is proportional to its
mean. The Poisson model neglects all correlations between spikes. This assumption is
certainly a simplification but it is sufficient to account for the majority of the information
transmitted by real cortical neurons [18, 19, 20] and, as we shall see later, it is mathematically convenient because it makes our model tractable.
2.2
Distribution of stimulus preferences among the afferent population
Neurons in sensory cortex receive a large number of inputs from other neurons with a variety of stimulus preferences. However, the majority of their inputs come from neurons with
roughly similar stimulus preference [1, 2, 3]. To characterise correctly this type of spread
of stimulus preference among the afferent population, we assume (unlike in previous studies [8]), that the probability distribution of the preferred stimulus among afferent neurons
follows a Gaussian distribution:
P (?s) =
1
(2?)D/2 ?pD
?
e
(?
s??
s0 )2
2
2?p
(2)
In Eq. (2) the parameter ?s0 represents the the center of the distribution, thus being the
most represented preferred stimulus in the population. (we set, without loss of generality, ?s0 = 0.) The parameter ?p controls the spread of stimulus preferences of the afferent
neuronal population: a small value of ?p indicates that a large fraction of the population
have similar stimulus preferences, and a large value of ?p indicates that all stimuli are
represented similarly. A Gaussian distribution of stimulus preferences of the afferent population fits well empirical data on distribution of preferred orientations of synaptic inputs
of neurons in both deep and superficial layers of ferret primary visual cortex [3].
3
Width of tuning and spread of stimulus preferences in visual cortex
To estimate the width of tuning ?f and the spread of stimulus preferences ?p of corticocortical afferent populations in visual cortex, we reviewed critically published anatomical
and physiological data. We concentrated on excitatory synaptic inputs, which form the
majority of inputs to a cortical pyramidal neuron [10]. We computed ?p by fitting (by a least
square method) the published histograms of synaptic connections as function of stimulus
preference of the input neuron to Gaussian distributions. Similarly, we determined ?f by
fitting spike count histograms to Gaussian tuning curves.
When considering a target neuron in ferret primary visual cortex and using orientation as
the stimulus parameters, the spread of stimulus preferences ?p of its inputs is ? 20o for
layer 5/6 neurons [3], and 16o [3] to 23o [21] for layer 2/3 neurons. The orientation tuning
width ?f of the cortical inputs to the V1 target neuron is that of other V1 neurons that
project to it. This ?f is 17o for Layer 4 neurons [22], and it is similar for neurons in deep
and superficial layers [3]. When considering a target neuron in Layer 4 of cat visual cortex
and orientation tuning, the spread of stimulus preference ?p is 20o [2] and ?f is ? 17o .
When considering a target neuron in ferret visual cortex and motion direction tuning, the
spread of tuning of its inputs ?p is ? 30 o [1]. Motion direction tuning widths of macaque
neurons is ? 28o , and this width is similar across species (see [13]).
The most notable finding of our meta-analysis of published data is that ?p and ?f appear
to be approximately of the same size and their ratio ?f /?p is distributed around 1, in the
range 0.7 to 1.1 for the above data. We will use our model to understand whether this range
of ?f /?p corresponds to an optimal way to transmit information across a synaptic system.
4
Information theoretic quantification of population decoding
To characterise how a target neuronal system can decode the information about sensory
stimuli contained in the activity of its afferent neuronal population, we use mutual information [23]. The mutual information between a set of stimuli and the neuronal responses
quantifies how well any decoder can discriminate among stimuli by observing the neuronal
responses. This measure has the advantage of being independent of the decoding mechanism used, and thus puts precise constraints on the information that can be decoded by any
biological system operating on the afferent activity.
Previous studies on the information content of an afferent neuronal population [7, 8] have
assumed that the target neuronal decoding system can extract all the information during
synaptic transmission. To do so, the target neuron must conserve the ?label? of the spikes
arriving from multiple neurons at different sites on its dendritic tree [9]. Given the potential biophysical difficulty in processing each spike separately, a simple alternative to spike
labelling has been proposed, - spike pooling [10, 24]. In this scheme, the target neuron
simply sums up the afferent activity. To characterize how the decoding of afferent information would work in both cases, we compute both the information that can be decoded by
either a system that processes separately spikes from different neurons (the ?labeled-line
information?) and the information available to a decoder that sums all incoming spikes (the
?pooled information?) [9, 24]. In the next two subsections we define these quantities and
we explain how we compute it in our model.
4.1
The information available to the the labeled-line decoder
The mutual information between the set of the stimuli and the labeled-line neuronal population activity is defined as follows [9, 24]:
P (r|s)
LL
I (S, R) = dsP (s)
(3)
P (r|s) log
P (r)
r
where P (s) is the probability of stimulus occurrence (here taken for simplicity as a uniform distribution over the hypersphere of D dimensions and ?radius? s? ). P (r|s) is the
probability of observing a neuronal population response r conditional to the occurrence of
stimulus s, and P (r) = dsP (s)P (r|s). Since the response vector r keeps separate the
spike counts of each neuron, the amount of information in Eq. (3) is only accessible to a
decoder than can keep the label of which neuron fired which spike [9, 24]. The probability
P (r|s) is computed according to the Poisson distribution, which is entirely determined by
the knowledge of the tuning curves [5]. The labeled-line mutual information is difficult to
compute for large populations, because it requires the knowledge of the probability of the
large-dimensional response vector r. However, since in our model we assume that we have
a very large number of independent neurons in the population and that the total activity of
the system is of the order of its size, then we can use the following simpler (but still exact)
expression[16, 25]:
1
D
ln (2?e) +
ds P (s) ln (|J (s)|)
(4)
2
2
where H(S) is the entropy of the prior stimulus presentation distribution P (S), J (s) is
the Fisher information matrix and | . . . | stands for the determinant. The Fisher information
matrix is a D ? D matrix who?s elements i, j are defined as follows:
2
?
Ji,j (s) = ?
P (r|s)
log P (r|s) ,
(5)
?si sj
r
I LL (S, R) = H(S) ?
Fisher information is a useful measure of the accuracy with which a particular stimulus can
be reconstructed from a single trial observation of neuronal population activity. However,
in this paper it is used only as a step to obtain a computationally tractable expression for the
labeled-line mutual information. The Fisher information matrix can be computed by taking
into account that for a population of Poisson neurons is just the sum of the Fisher information for individual neurons, and the latter has a simple expression in terms of tuning curves
[16]. Since the neuronal population size N is is large, the sum over Fisher information
of individual neurons can be replaced by an integral over the stimulus preferences of the
neurons in the population, weighted by their probability density P (?s). After performing
the integral over the distribution of preferred stimuli, we arrived at the following result for
the elements of the Fisher information matrix:
?
? 2 1+?
N?m
? D?2
2)
2
(
?
+
?
(?
+
?
?
)
e
i,j
i,j
i
j
D
?p2 (1 + ? 2 ) 2 +2
2
Ji,j (s) =
(6)
where we have introduced the following short-hand notation ?f /?p ? ? and s/?p ? ?;
?i,j stands for the Kroneker Delta. From Eq. (6) it is possible to compute explicitly the
determinant |J (s)|, which has the following form:
|J (s)| =
D
?i = ?(?)D (1 + ? 2 )D?1 1 + ? 2 (1 + ? 2 )
(7)
i=1
where ?(?) is given by:
?
?
N?m
? D?2
e 2(1+?2 )
D
2
?p (1 + ? 2 ) 2 +1
2
?(?) =
(8)
Inserting Eq. (7) into Eq. (4), one obtains a tractable but still exact expression for the
mutual information , which has the advantage over Eq. (3) of requiring only an integral
over a D-dimensional stimulus rather than a sum over an infinite population.
We have studied numerically the dependence of the labeled-line information on the parameters ?f and ?p as a function of the number of encoded stimulus features D 1 . We
investigated this by fixing ?p and then varying the ration ?f /?p over a wide range. Results
(obtained for ?p = 1 but representative of a wide ?f range) are reported in Fig. 1. We
found that, unlike the case of a uniform distribution of stimulus preferences [8], there is a
finite value of the width of tuning ?f that maximizes the information for all D ? 2. Interestingly, for D ? 2 the range 0.7 ? ?f /?p ? 1.1 found in visual cortex either contains the
maximum or corresponds to near optimal values of information transmission. For D = 1,
information is maximal for very narrow tuning curves. However, also in this case the information values are still efficient in the cortical range ?f /?p ? 1, in that the tail of the
D = 1 information curve is avoided in that region. Thus, the range of values of ?f and ?p
found in visual cortex allows efficient synaptic information transmission over a wide range
of number of stimulus features encoded by the neuron.
We found (data not shown) that other parameters such as m and ? , had a weak or null effect on
the optimal configuration; see [17] for a D = 1 example in a different context.
1
ILL(S,R)
D=1
D=5
0
2
? /? 4
f
p
6
8
Figure 1: Mutual labeled-line information as a function of the ratio of tuning curve width
and stimulus preference spread ?f /?p . The curves for each stimulus dimensionality D
were shifted by a constant factor to separate them for visual inspection (lower curves correspond to higher values of D). The y-axis is thus in arbitrary units. The position of the
maximal information for each stimulus dimension falls either inside the range of values
of ?f /?p found in visual cortex, or very close to it (see text) . Parameters are as follows:
s? = 2, rmax = 50Hz, ? = 10ms.
4.2
The information available to the the pooling decoder
We now consider the case in which the target neuron cannot process separately spikes
from different neurons (for example, a neuron that just sums up post-synaptic potentials
of approximately equal weight at the soma). In this case the label of the neuron that fired
each spike is lost by the target neuron, and it can only operate on the pooled neuronal
signal, in which the identity of each spike is lost. Pooling mechanisms have been proposed
as simple information processing strategies for the nervous system. We now study how
pooling changes the requirements for efficient decoding by the target neuron.
Mathematically speaking,
pooling maps the vector r onto a scalar ? equal to the sum of the
individual activities: ? = rk . Thus, the mutual information that can be extracted by any
decoder that only pools it inputs is given by the following expression:
P (?|s)
(9)
I P (S, R) = dsP (s)
P (?|s) log
P (?)
?
where P (?|s) and P (?) are the the stimulus-conditional and stimulus-unconditional probability of observing a pooled population response ? on a single trial. The probability P (?|s)
can be computed by noting that a sum of Poisson-distributed responses is still a Poissondistributed response whose tuning curve to stimuli is just the sum of the individual tuning
curves. The pooled mutual information is thus a function of a single Poisson-distributed
response variables and can be computed easily also for large populations.
The dependence of the pooled information on the parameters ?f and ?p as a function of
the number of encoded stimulus features D is reported in Fig. 2. There is one important
difference with respect to the labeled-line information transmission case. The difference is
that, for the pooled information, there is a finite optimal value for information transmission
also when the neurons are tuned to one-dimensional stimulus feature. For all cases of stimulus dimensionality considered, the efficient information transmission though the pooled
IP(S,R)
D=1
D=3
0
1
2
?f/?p
3
4
Figure 2: Pooled mutual information as a function of the ratio of tuning curve width and
stimulus preference spread ?f /?p . The maxima are inside the range of experimental values
of ?f /?p found in the visual cortex, or very close to it (see text). As for Fig. 1, the curves
for each stimulus dimensionality D were shifted by a constant factor to separate them for
visual inspection (lower curves correspond to higher values of D). The y-axis is thus in
arbitrary units. Parameters are as follows: s? = 2, rmax = 50 Hz,? = 10ms.
neuronal decoder can still be reached in the visual cortical range 0.7 ? ?f ?p ? 1.1. This
finding shows that the range of values of ?f and ?p found in visual cortex allows efficient synaptic information transmission even if the target neuron does not rely on complex
dendritic processing to conserve the label of the neuron that fired the spike.
5
Conclusions
The stimulus specificity of cortico-cortical connections is important for understanding the
mechanisms of generation of orientation tuning (see [4]) for a review). Here, we have
shown that the stimulus-specific structure of cortico-cortical connections may have also implications for understanding cortico-cortical information transmission. Our results suggest
that, whatever the exact role of cortico-cortical synapses in generating orientation tuning,
their wiring allows efficient transmission of sensory information.
Acknowledgments
We thanks A. Nevado and R. Petersen for many interesting discussions. Research supported
by ICTP, HFSP, Royal Society and Wellcome Trust 066372/Z/01/Z.
References
[1] B. Roerig and J. P. Y. Kao. Organization of intracortical circuits in relation to direction
preference maps in ferret visual cortex. J. Neurosci., 19:RC44(105), 1999.
? T?oth, and Z. F. Kisv?arday. Orien[2] T. Yousef, T. Bonhoeffer, D-S. Kim, U. T. Eysel, E.
tation topography of layer 4 lateral networks revealed by optical imaging in cat visual
cortex (area 18). European J. Neurosci., 11:4291?4308, 1999.
[3] B. Roerig and B. Chen. Relations of local inhibitory and excitatory circuits to orientation preference maps in ferret visual cortex. Cerebral Cortex, 12:187?198, 2002.
[4] K. A. C. Martin. Microcircuits in visual cortex. Current Opinion in Neurobiology,
12:418?425, 2002.
[5] P. Dayan and L. F. Abbott. Theoretical Neuroscience. MIT Press, 2001.
[6] D. C. Fitzpatrick, R. Batra, T. R. Stanford, and S. Kuwada. A neuronal population
code for sound localization. Nature, 388:871?874, 1997.
[7] A. Pouget, S. Deneve, J-C. Ducom, and P.E. Latham. Narrow versus wide tuning
curves: what?s best for a population code? Neural Computation, 11:85?90, 1999.
[8] K. Zhang and T.J. Sejnowski. Neuronal tuning: to sharpen or to broaden? Neural
Computation, 11:75?84, 1999.
[9] D. S. Reich, F. Mechler, and J. D. Victor. Independent and redundant information in
nearby cortical neurons. Science, 294:2566?2568, 2001.
[10] M. N. Shadlen and W. T. Newsome. The variable discharge of cortical neurons:
implications for connectivity, computation and coding. J. Neurosci., 18(10):3870?
3896, 1998.
[11] N. Brenner, W. Bialek, and R. de Ruyter van Steveninck. Adaptive rescaling maximizes information transmission. Neuron, 26:695?702, 2000.
[12] J. Touryan, B. Lau, and Y. Dan. Isolation of relevant visual features from random
stimuli for cortical complex cells. J. Neurosci, 22:10811?10818, 2002.
[13] T. D. Albright. Direction and orientation selectivity of neurons in visual area MT of
the macaque. J. Neurophysiol., 52:1106?1130, 1984.
[14] K.H. Britten, M. N. Shadlen, W. T. Newsome, and J. A. Movshon. The analysis
of visual-motion - a comparison of neuronal and psychophysical performance. J.
Neurosci., 12:4745?4765, 1992.
[15] K Kang, RM Shapley, and H Sompolinsky. Information tuning of population of neurons in primary visual cortex. J. Neurosci., 24:3726?3735, 2004.
[16] N. Brunel and J. P. Nadal. Mutual information, fisher information and population
coding. Neural Computation, 10:1731?1757, 1998.
[17] A. Nevado, M.P. Young, and S. Panzeri. Functional imaging and neural information
coding. Neuroimage, 21:1095?1095, 2004.
[18] S. Nirenberg, S. M. Carcieri, A.L. Jacobs, and P. E. Latham. Retinal ganglion cells
act largely as independent encoders. Nature, 411:698?701, 2001.
[19] R. S. Petersen, S. Panzeri, and M.E. Diamond. Population coding of stimulus location
in rat somatosensory cortex. Neuron, 32:503?514, 2001.
[20] M. W. Oram, N.G. Hatsopoulos, B.J. Richmond, and J.P. Donoghue. Excess synchrony in motor cortical neurons provides redundant direction information with that
from coarse temporal measures. J. Neurophysiol., 86:1700?1716, 2001.
[21] M. B. Dalva, M. Weliky, and L. Katz. Relations between local synaptic connections
and orientation domains in primary visual cortex. Neuron, 19:871?880, 1997.
[22] W. M. Usrey, M. P. Sceniak, and B. Chapman. Receptive fields and response properties of neurons in layer 4 of ferret visual cortex. J. Neurophysiol., 89:1003?1015,
2003.
[23] T.M. Cover and J.A. Thomas. Elements of information theory. John Wiley, 1991.
[24] S. Panzeri, F. Petroni, R.S. Petersen, and M.E. Diamond. Decoding neuronal population activity in rat somatosensory cortex: role of columnar organization. Cerebral
Cortex, 13:45?52, 2003.
[25] K Kang and H Sompolinsky. Mutual information of population codes and distance
measures in probability space. Phys. Rev. Lett., 86:4958?4961, 2001.
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1,816 | 265 | 542
Kassebaum, Thnorio and Schaefers
The Cocktail Party Problem:
Speech/Data Signal Separation Comparison
between Backpropagation and SONN
John Kassebaum
[email protected]
Manoel Fernando Tenorio
[email protected]
Christoph Schaefers
Parallel Distributed Structures Laboratory
School of Electrical Engineering
Purdue University
W. Lafayette, IN. 47907
ABSTRACT
This work introduces a new method called Self Organizing Neural
Network (SONN) algorithm and compares its performance with Back
Propagation in a signal separation application. The problem is to
separate two signals; a modem data signal and a male speech signal,
added and transmitted through a 4 khz channel. The signals are sampled at 8 khz, and using supervised learning, an attempt is made to
reconstruct them. The SONN is an algorithm that constructs its own
network topology during training, which is shown to be much smaller
than the BP network, faster to trained, and free from the trial-anderror network design that characterize BP.
1. INTRODUCTION
The research in Neural Networks has witnessed major changes in algorithm design
focus, motivated by the limitations perceived in the algorithms available at the
time. With the extensive work performed in that last few years using multilayered
networks, it was soon discovered that these networks present limitations in tasks
The Cocktail Party Problem:
that: (a) are difficult to determine problem complexity a priori, and thus design
network of the correct size, (b) training not only takes prohibitively long times,
but requires a large number of samples as well as fine parameter adjustment,
without guarantee of convergence, (c) such networks do not handle the system
identification task efficiently for systems whose time varying structure changes
radically, and, (d) the trained network is little more than a black box of weights
and connections, revealing little about the problem structure; being hard to find
the justification for the algorithm weight choice, or an explanation for the output
decisions based on an input vector. We believe that this need is sparking the
emergence of a third generation of algorithms to address such questions.
2. THE SELF
ALGORITHM
ORGANIZING
NEURAL
NETWORK
2.1 SELF ORGANIZING NETWORK FAMILY
A family of Self Organizing Structure (SOS) Algorithms can be readily designed
with our present knowledge, and can be used as a tool to research the motivating
questions. Each individual algorithm in this family might have different characteristics, which are summarized in the following list:
- A search strategy for the structure of the final model
- A rule of connectivity
- A performance criteria
- A transfer function set with appropriate training rule
As we will show here, by varying each one of these components, a different
behavior of the algorithm can be imposed.
Self organizing structure algorithms are not new. These algorithms have been
present in the statistical literature since the mid 70's in a very different context.
As far as we know, the first one to propose such an algorithm was Ivahnenko
[1971] which was followed by a host of variations on that original proposal
[Duffy&Franklin, 1975; Ikeda, et al., 1976; Tomura&Kondo, 1980; Farlow,1989].
Ivahnenko's subfamily of algorithms (GMDH - Group Method of Data Handling)
can be characterized in our classification by the same four-tuple criterion: (1) gradient descent local search, (2) creation of regular feedforward layers with elements
pairwisely connected, (3) least-mean-squares estimation, and (4) a single element
set comprised of a 2 order bivariate function.
Here we want to present our subfamily (SON - Self Organizing Networks) of the
SOS algorithm family, characterized differently by: (1) global optimization search,
(2) arbitrary connectivity based on an arbitrary number of neuron inputs, (3)
Structure Estimation Criteria (SEC) (a variation of Rissanen's [1983]. Minimum
Description Length Criteria, extended to the hierarchical case), and, (4) for training speed, activation functions are restricted to be linear on the parameters and
the output functions need to be invertible, no other restriction is imposed in kind
or number. The particular algorithm presented here is called the Self Organizing
543
544
Kassebaum, Tenorio and Schaefers
Neural Network (SONN) [Tenorio&Lee, 1988,1989; Tenorio 1990 a,b]. It was composed of: (1) a graph synthesis procedure based on Simulated Annealing [Kirkpatrick et a.1., 1983]; (2) two input neurons that a.re arbitrarily connected; (3) the
Structure Estimation Criteria; a.nd, (4) a set of a.ll polynomials that a.re special
ca.ses of 2nd order bivariates a.nd inclusive, followed or not by sigmoid functions.
The SONN a.lgorithm performs a. search in the model space by the construction
of hypersurfa.ces. A network of nodes, each node representing a. hypersurface, is
organized to be a.n a.pproximate model of the real system. Below, the components
of SONN a.re discussed.
2.2 THE ALGORITHM STRUCTURE
The mechanisms behind the a.lgorithm works as follows. First, create a. set of terminals which a.re the output of the nodes a.vailable for connection to other nodes.
This set is initialized with the output of the input nodes; in other words, the input
variables themselves. From this set, with uniform probability, select a subset (2 in
our case) of terminals, a.nd used them as inputs to the new node. To construct the
new node, select a.ll the function of the set of prototype functions (activation followed by output function), a.nd evaluate the SEC using the terminals as inputs.
Selecting the best function, test for the acceptance of that node according to the
Simulated Annealing move a.cceptance criterion. If the new node is a.ccepted, place
its output in the set of terminals and iterate until the optimum model is found.
The details or the a.lgorithm can be found in [Tenorio&Lee, 1989].
2.2.1 The Prototype Functions
Consider the Mahalanobis distance:
Yj
=sig{(x-/-LPC- 1 (x-/-L)t}
(1 )
This distance ca.n be rewritten as a second order function, whose parameters are
the indirect representation of the covariance matrix X and the mean vector /-L.
This function is linear in the parameters, which makes it easy to perform training,
a.nd it is the function with the smallest degree of non linearity; only simpler is the
linear case. Interestingly enough, this is the same prototype function used in the
GMDH a.lgorithm to form the Ivahnenko polynomial for apparently completely
different reasons. In the SONN, this function is taken to be 2-input and all its possible variations (32) by setting parameters to zero are included in the set of
a.ctivation functions. This set combined with the output function (the identify or
sigmoid), for the set of prototype functions, used by the a.lgorithm in the node
construction.
2.2.2 Evaluation of the Model Based on the MDL Criterion
The selection rule of the neuron transfer function was based on a modification of
the Minimal Description Length (MOL) information criterion. In [Rissanen, 1978],
the principle of minimal description for statistical estimation was developed. The
reason for the choice of such a criterion is that, in general the accuracy of the
model can increase at the expense of simplicity in the number oC parameters. The
The Cocktail Party Problem:
increase of complexity might also be accompanied by the overfitting of the model.
To overcome this problem, the MDL provides a trade-oft' between the accuracy
and the complexity of the model by including the structure estimation term of the
final model. The final model (with the minimal MDL) is optimum in the sense of
being a consistent estimate of the number of parameters while achieving the
minImUm error [Rissanen, 1980]. Given a sequence of observations
Xl ,X2 , ??? ,XN from the random variable X, the dominant term of the MDL in
[Rissanen, 1978] is:
MDL = -log f(x
Ie) +0.5 k log N
(2)
Ie)
where f(x
is the estimated probability density function of the model, k is the
number of parameters, and N is the number of observations. The first term is
actually the negative of the maximum likelihood (ML) with respect to the
estimated parameter. The second term describes the structure of the models and it
is used as a penalty for the complexity of the model.
3. EXAMPLE - THE COCKTAIL PARTY PROBLEM
The Cocktail Party Problem is the name given to the phenomenon that people
can understand and track speech in a noisy environment, even when the noise is
being made by other speakers. A simpler version of this problem is presented here:
a 4 khz channel is excited with male speech and modem data additively at the
same time. The task presented to the network is to separate both signals.
To compare the accuracy of the signal separation between the SONN and the
Back Propagation algorithms a normalized RMSE is used as a performance
index:
normalized RMSE _ _ _ _R_M_S_E
_ __
StandardDevision
(3)
3.1. EXPERIMENTS WITH BACK PROPAGATION
In order to design a filter using Back Propagation for this task, several architectures were considered. Since the input and output to the problem are time series,
and such architectures are static, modifications to the original paradigm is
required to deal with the time dimension. Several proposals have been made in
this respect: tapped delay filters, recurrent architectures, low pass filter transfer
functions, modified discriminant functions, and self excitatory connections (see
[Wah, Tenorio, Merha, and Fortes, 90] ). The best result for this task was
achieved by two tapped delay lines in the input layer, one for the input signal, the
other for the output signal. The network was trained to recognize the speech signal from the mixed signal. The mixed signal had a speech to modem data energy
ratio of 4:1, or 2.5 dB.
The network was designed to be a feedforward with 42 inputs (21 delayed versions
of the input signal, and similarly for the output signal), 15 hidden units, and a
single output unit. The network was trained with a single phoneme, taking about
545
546
Kassebaum, Tenorio and Schaefers
10 cpu-hours on a Sequent machine. The network when presented with the trained
phoneme added to the modem data, produced a speech reconstruct ability error
equal to a nRMSE of 0.910. Previously several different configurations of the network were tried as well as different network parameters, and signal ratios of 1:1;
all with poor results. Few networks actually converged to a final solution. A
major problem with the BP architecture is that it can perfectly filter the signal in
the first few samples, just to later demonstrate increasing amounts of cumulative
errors; this instability may be fruit of the recurring nature of the architecture,
and suboptimal weight training (Figure 2). The difficulty in finding and fine tuning the architecture, the training convergence, and time requirements led us to
later stop pursuing the design of these filters with Back Propagation strategies.
3.2. EXPERIMENTS WITH SONN
At that time, the SONN algorithm had been successfully used for identification
and prediction tasks [Tenorio&Lee; 88,89,90]. To make the task more realistic
with possible practical utilization of this filter (Data-Over-Voice Circuits), the
energy ratio between the voice and the modem data was reduced to 1:1, or 0 dB.
A tapped delay line containing 21 delayed versions of the mixed signal was
presented to the algorithm. Two sets of prototype functions were used, and both
contained the full set of 32 variations of 2nd order bivariates. The first set had
the identity (SONN-I experiments) and the second had a sigmoid (SONN-SIG
experiments) as the output function for each node.
SONN-I created 370 nodes, designing a final model with 5 nodes. The final symbolic transfer function which represents the closed form function of the network
was extracted. Using a Gould Powernode 9080, this search took 98.6 sec, with an
average of 3.75 nodes/sec. The final model had an nRMSE of 0.762 (Figure 3) for
reconstructed speech with the same BP data; with 19 weights. Training with the
modem signalled to nRMSE of 0.762 (Figure 4) for the BP data. A search using
the SONN-SIG model was allowed to generate 1000 nodes, designing a final model
with 5 nodes. With the same computer, the second search took 283.42 sec, with an
average 3.5 nodes/sec. The final model had an nRMSE comparable to the SONN-I
(better by 5-10%); with 20 weights. The main characteristics of both signals were
captured, specially if one looks at the plots and notices the same order of nonlinearity between the real and estimated signals (no over or under estimation).
Because of the forgiving nature of the human speech perception, the voice after
reconstruction, although sightly muffled, remains of good quality; and the reconstructed modem signal can be used to reconstruct the original digital message,
without much further post processing. The SONN does not present cumulative
errors during the reconstruction, and when test with different (unseen, from the
same speaker) speech data, performed as well as with the test data. We have yet
to fully explore the implication of that to different speakers and with speaker of
different gender or language. These results will be reported elsewhere.
4. COMPARISON BETWEEN THE TWO ALGORITHMS
Below we outline the comparison between the two algorithms drawn from our
experience with this signal separation problem.
The Cocktail Party Problem:
4.1. ADVANTAGES
The following were advantages of the SONN approach over the BP paradigm.
The most striking difference was found in the training times, and in the amount
of data required for training. The BP required 42 inputs (memories), where as the
SONN functioned with 21 inputs, actually using as few as 4 in the final model
(input variable selection). The SONN removed the problem of model estimation
and architecture design. The number of connections with the SONN models is as
low as 8 for 20 weights (relevant connections), as compared with 645 connections
and weights for the BP model. The accuracy and complexity of the model can be
trade for learning time as in BP, but the models that were more accurate also
required less parameters than BP. The networks are not required to be homogeneous, thus contributing to smaller models as well. Above all, the SONN can produce both the C code for the network as well as the sequence of individual node
symbolic functions; the SONN-I can also produce the symbolic representation of
the closed form function of he entire network.
4.2. DISADVANTAGES
Certain disadvantages of using self-organizing topology networks with stochastic
optimization algorithms were also apparent. The learning time of the SONN is
non deterministic, and depends on the model complexity and starting point.
Those are characteristic of the Simulated Annealing (SA) algorithm. These disadvantages are also present in the BP approach for different reasons. The connectivity of the model is not known a priori, which does not permit hardware implementation algorithms with direct connectivity emulation. Because the SONN
selects nodes from a growing set with uniform probability, the probability of
choosing a pair of nodes decreases with the inverse of the square of the number of
nodes. Thus algorithm effectiveness decreases with processing time. Careful plotting of the SEC, nRMSE, and complexity trajectories during training reveal that
the first 10% of the processing time achieves 90% of the final steady state values.
Biasing the node selection procedure might be an alternative to modify this
behavior. Simulated Annealing also required parametric tuning of the algorithm
by setting" the initial and final temperature, the duration of the search at each
temperature and the temperature decay. Alternative algorithms such as A * might
produce a better alternative to stochastic search algorithms.
6. CONCLUSION AND FUTURE WORK
In this study, we proposed a new approach for the signal separation filter design
neural network (SONN) algorithm. The varibased on a flexible, self-organizi
able structure provides the oppo; llity to search and construct the optimal model
based on input-output observations. The hierarchical v' ton of the MDL, ' lIed
the Structure Estimation Criteria, was used to guide
.; trade-off betwel the
model complexity and the accuracy of the estimation. The SONN approach
demonstrates potential usefulness as a tool for non linear signal processing function design.
We would like to explore the use of high level knowledge for function selection
547
548
Kassebaum, Thnorio and Schaefers
and connectivity. Also, the issues involving estimator and deterministic searches
are still open. Currently we are exploring the use of SONN for digital circuit synthesis, and studying how close the architecture generated here can approach the
design of natural structures when performing similar functions. More classification
problems, and problems involving dynamical systems (adaptive control and signal
processing) need to be explored to give us the experience needed to tackle the
problems for which it was designed.
6.
NOTE
The results reported here were originally intended for two papers accepted for
presentation at the NIPS'89. The organizing committee asked us to fuse the into a
single presentation for organizational purposes. In the limited time and the small
space allocated for the presentation of these results, we sought a compromise
between the reporting of the results and the description and comments on our
experience with the algorithm. The interested reader should look at the other
references about the SONN listed here and forthcoming papers.
REFERENCE
A. G. Ivakhnenko, (1971) "Polynomial Theory of Complex Systems," IEEE Trans.
S.M.C, Vol. SMC-1, no.4, pp. 364-378, Oct.
J. J. Duffy and M. A. Franklin, (1975) "A Learning Identification Algorithm and
its Application to an Environmental System," IEEE Trans. S. M. C., Vol. SMC-5,
no. 2, pp. 226-240.
S. Ikeda, M. Ochiai and Y. Sawarogi, (1976) "Sequential GMDH Algorithm and its
Application to River Flow Prediction," IEEE Trans S.M.C., Vol. SMC-6, no.7, pp.
473-479, July.
H. Tamura, T. Kondo, (1980) "Heuristics Free Group Method of Data Handling
Algorithm of Generating Optimal Partial Polynomials with Application to Air
Pollution Predication," Int. J. Systems Sci., 11,no.9, pp. 1095-1111.
J. Rissanen (1978) "Modeling by Shortest Data Description," Automatica, Vol.14,
pp. 465-471.
J. Rissanen, (1980) "Consistent Order Estimation of Autoregression Processes by
Shortest Description of Data," Analysis and Optimation of Stochastic System,
Jacobs et al eds. NY Academic.
J. Rissanen, (1983) "A Universal Prior for Integers and Estimation by Minimum
Description Length," Annuals of Statistics, Vol. 11 , no. 2, pp.416-431.
S.Kirkpatrick, C.D. Gelatt, M.P. Vecchi, (1983) "Optimization by Simulated
Annealing," Science, vol.220, pp. 671-680, May.
M. F. M. Tenorio and W.-T. Lee, (1988) "Self-Organizing Neural Network for the
Identification Problem," Advances in Neural Information Processing Systems I,
David S. Touretzky ed., pp. 57-64.
M. F. M. Tenorio and W.-T. Lee, (1989) "Self-Organizing Neural Network for the
The Cocktail Party Problem:
549
Identification Problem," School of Electrical Engineering, Purdue University,
Tech Report TR-EE 89-20, June.
M. F. M. Tenorio and W. -T. Lee, (1990) "Self-Organizing Network for the
Identification Problem," (expanded) IEEE Trans. on the Neural Networks, to
appear.
M. F. M. Tenorio, (1990) "The Self-Organizing Neural Network Algorithm:
Adapting Topology for Optimum Supervised Learning," IEEE Hawaii Conference
in Systems Science, 22, January.
M. F. Tenorio, (1990) "Self-Organizing Neural Network for the Signal Separation
Problem," to be submitted.
B. Wah, M. Tenorio, P. Mehra, J. Fortes, (1990) "Artificial Neural Networks:
Theory, Algorithms, Application and Implementations," IEEE press.
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1,817 | 2,650 | Newscast EM
Wojtek Kowalczyk
Department of Computer Science
Vrije Universiteit Amsterdam
The Netherlands
[email protected]
Nikos Vlassis
Informatics Institute
University of Amsterdam
The Netherlands
[email protected]
Abstract
We propose a gossip-based distributed algorithm for Gaussian mixture
learning, Newscast EM. The algorithm operates on network topologies
where each node observes a local quantity and can communicate with
other nodes in an arbitrary point-to-point fashion. The main difference
between Newscast EM and the standard EM algorithm is that the M-step
in our case is implemented in a decentralized manner: (random) pairs
of nodes repeatedly exchange their local parameter estimates and combine them by (weighted) averaging. We provide theoretical evidence and
demonstrate experimentally that, under this protocol, nodes converge exponentially fast to the correct estimates in each M-step of the EM algorithm.
1
Introduction
Advances in network technology, like peer-to-peer networks on the Internet or sensor networks, have highlighted the need for efficient ways to deal with large amounts of data that
are distributed over a set of nodes. Examples are financial data reported on the Internet,
weather data observed by a set of sensors, etc. In particular, in many data mining applications we are interested in learning a global model from such data, like a probability
distribution or a clustering of the data, without first transferring all the data to a central
repository. Ideally, we would like to have a fully decentralized algorithm that computes
and disseminates aggregates of the data, with minimal processing and communication requirements and good fault-tolerant behavior.
A recent development in distributed systems technology is the use of gossip-based models
of computation [1, 2, 3]. Roughly, in a gossip-based protocol each node repeatedly contacts some other node at random and the two nodes exchange information. Gossip-based
protocols are very simple to implement, while they enjoy strong performance guarantees
as a result of randomization. Their use in data mining and machine learning applications is
currently finding inroads [4, 5].
In this paper we propose a gossip-based, fully decentralized implementation of the
Expectation-Maximization (EM) algorithm for Gaussian mixture learning [6]. Our algorithm, which we call ?Newscast EM?, assumes a set of data {xi } that are drawn independently from a common Gaussian mixture and are distributed over the nodes of a network
(one data point per node). Newscast EM utilizes a gossip-based protocol in its M-step to
learn a global Gaussian mixture model p(x) from the data. The main idea is to perform the
M-step in a number of cycles. Each node starts with a local estimate of the model parameters. Then, in every cycle, each node contacts some other node that is chosen at random
from a list of known nodes, and the two nodes replace their local model estimates by their
(weighted) averages. As we show below, under such a protocol the (erroneous) local models of the individual nodes converge exponentially fast to the (correct) global model in each
M-step of the algorithm.
Our approach is fundamentally different from other distributed exact implementations of
the EM algorithm that resort on global broadcasting [7] or routing trees [8]. In the latter,
for instance, data sufficient statistics are propagated through a spanning tree in the network,
combined with an incremental learning scheme as in [9]. A disadvantage of that approach
is that only one node is carrying out computations at any time step, whereas in Newscast
EM all nodes are running the same protocol in parallel. This results in a batch M-step that
has average runtime at most logarithmic in the number of nodes, as we will see next.
2
Gaussian mixtures and the EM algorithm
A k-component Gaussian mixture for a random vector x ? IRd is defined as the convex
combination
k
X
p(x) =
?s p(x|s)
(1)
s=1
?d/2
of k Gaussian densities p(x|s) = (2?)
|Cs |?1/2 exp[?(x ? ms )> Cs?1 (x ? ms )/2],
each parameterized by its mean ms and covariance matrix Cs . The components of the
mixture are indexed by the random variable s that takes values from 1 to k, and ? s = p(s)
defines a discrete prior distribution over the components. Given a set {x1 , . . . , xn } of
independent and identically distributed samples from p(x), the learning task is to estimate
k
the parameter vector
Pn ? = {?s , ms , Cs }s=1 of the k components that maximizes the loglikelihood L = i=1 log p(xi ; ?). Throughout we assume that the likelihood function is
bounded from above (e.g., by placing appropriate bounds on the components covariance
matrices).
Maximization of the data log-likelihood L can be carried out by the EM algorithm [6]
which can be seen as iteratively maximizing a lower bound of L [9]. This bound F is
a function of the current mixture parameters ? and a set of ?responsibility? distributions
{qi (s)}, i = 1, . . . , n, where each qi (s) corresponds to a data point xi and defines an
arbitrary discrete distribution over s. This lower bound is given by:
F=
n X
k
X
i=1 s=1
qi (s) log p(xi , s; ?) ? log qi (s) .
(2)
In the E-step of the EM algorithm, the responsibility qi (s) for each point xi is set to the
Bayes posterior p(s|xi ) given the parameters found in the previous step. In the M-step we
solve for the unknown parameters of the mixture by maximizing F for fixed qi (s). This
yields the following updates:
Pn
Pn
Pn
qi (s)xi x>
i
i=1 qi (s)
i=1 qi (s)xi
?s =
,
ms =
,
Cs = i=1
?ms m>
s . (3)
n
n?s
n?s
Note that the main operation of the M-step is averaging: ?s is the average of qi (s), ms
is the average of products qi (s)xi (divided by ?s ), and the covariance matrix Cs is the
>
average of matrices qi (s)xi x>
i (divided by ?s and decreased by ms ms ). This observation
is essential for the proposed algorithm, as we will shortly see.
3
Newscast computing and averaging
The proposed distributed EM algorithm for Gaussian mixture learning relies on the use of
the Newscast protocol for distributed computing [3]. Newscast is a gossip-based protocol
that applies to networks where arbitrary point-to-point communication between nodes is
possible, and it involves repeated data exchange between nodes using randomization: with
constant frequency each node contacts some other node at random, and the two nodes
exchange application-specific data as well as caches with addresses of other nodes. The
protocol is very robust, scalable, and simple to implement?its Java implementation is
only a few kBytes of code and can run on small network-enabled computing devices such
as mobile phones, PDAs, or sensors.
As with other gossip-based protocols [2], Newscast can be used for computing the mean
of a set of values that are distributed over a network. Suppose that values v1 , . . . , vn are
stored in the nodes of a network, one value per node. Moreover
Pn suppose that each node
knows the addresses of all other nodes. To compute ? = n1 i=1 vi , each node i initially
sets ?i = vi as its local estimate of ?, and then runs the following protocol for a number of
cycles:
Uniform Newscast (for node i)
1. Contact a node j = f (i) that is chosen uniformly at random from 1, . . . , n.
2. Nodes i and j update their estimates as follows: ?0i = ?0j = (?i + ?j )/2.
For the purpose of analysis we will assume that in each cycle every node initiates a single
contact (but in practice the algorithm can be fully asynchronous). Note that the mean of
the local estimates {?i } is always the correct mean ?, while for their variance holds:
Lemma 1. In each cycle of uniform Newscast the variance of the local estimates drops on
1
the average by factor ?, with ? ? 2?
.
e
Pn
Proof. 1 Let ?t = i=1 (?i ? ?)2 be the unnormalized variance of the local estimates ?i
at cycle t. Suppose, without loss of generality, that within cycle t nodes initiate contacts in
the order 1, 2, . . . , n. The new variance after node?s 1 contact is:
? + ?
2
1
f (1)
?1 = ?t ? (?1 ? ?)2 ? (?f (1) ? ?)2 + 2
??
(4)
2
1
1
(5)
= ?t ? (?1 ? ?)2 ? (?f (1) ? ?)2 + (?1 ? ?)(?f (1) ? ?).
2
2
Taking expectation over f , and using the fact that P [f (i) = j] = n1 for all i and j, gives:
n
1 X
1
1
1
(?j ? ?)2 = 1 ?
? ? (?1 ? ?)2 . (6)
E[?1 |?t = ?] = ? ? (?1 ? ?)2 ?
2
2n j=1
2n
2
After n such exchanges, the variance ?t+1 is on the average:
n
1 n
1 X
1 n?i
E[?t+1 |?t = ?] = 1 ?
??
1?
(?i ? ?)2 .
2n
2 i=1
2n
Bounding the term (1 ?
1 n?i
2n )
by (1 ?
1 n
2n )
finally gives:
1 n
?
E[?t+1 |?t = ?] ?
1?
?? ? .
2
2n
2 e
1
(7)
1
See [3] for an alternative proof of the same bound.
(8)
Thus after t cycles of uniform Newscast,
? the original variance ?0 of the local estimates is
reduced on the average to ?t ? ?0 /(2 e)t . The fact that the variance drops at an exponential rate means that the nodes learn the correct average very fast. Indeed, using Chebyshev?s
inequality Pt [|?i ? ?| ? ?] ? ?t /(n?2 ) we see that for any ? > 0, the probability that
some node makes an estimation error larger than ? is dropping exponentially fast with the
number of cycles t. In particular, we can derive a bound on the number of cycles that are
needed in order to guarantee with high probability that all nodes know the correct answer
with some specific accuracy:
Theorem 1. With probability 1 ? ? , after d0.581(log n + 2 log ? + 2 log 1? + log 1? )e cycles
of uniform Newscast holds maxi |?i ? ?| ? ?, for any ? > 0 and data variance ? 2 .
?
Proof. Using Lemma 1 and the fact that ?0 = n? 2 , we obtain E[?t ] ? n? 2 /(2 e)t .
?
2
2
Setting ? = log( n?
?2 ? )/ log(2 e) we obtain E[?? ] ? ? ?. Using Markov inequality, with
2
probability at least 1 ? ? holds ?? ? ? . Therefore, since ?? is the sum of local terms, for
each of them must hold |?i ? ?| ? ?. It is straightforward to show by induction over ? that
the same inequality will hold for any time ? 0 > ? .
For example, for unit-variance data and a network with n = 104 nodes we need 49 cycles
to guarantee with probability 95% that each node is within 10?10 from the correct answer.
Note that in uniform Newscast, each node in the network is assumed to know the addresses
of all other nodes, and therefore can choose in each cycle one node uniformly at random
to exchange data with. In practice, however, each node can only have a limited cache of
addresses of other nodes. In this case, the averaging algorithm is modified as follows:
Non-uniform Newscast (for node i)
1. Contact a node j = f (i) that is appropriately chosen from i?s local cache.
2. Nodes i and j update their estimates as follows: ?0i = ?0j = (?i + ?j )/2.
3. Nodes i and j update their caches appropriately.
Step 3 implements a ?membership management? schedule which effectively defines a dynamically changing random graph topology over the network. In our experiments we
adopted the protocol of [10] which roughly operates as follows. Each entry k in node?s i
cache contains an ?age? attribute that indicates the number of cycles that have been elapsed
since node k created that entry. In step 1 above, node i contacts the node j with the largest
age from i?s cache, and increases by one the age of all other entries in i?s cache. Then
node i exchanges estimates with node j as in step 2. In step 3, both nodes i and j select a
random subset of their cache entries and mutually exchange them, filling empty slots and
discarding self-pointers and duplicates. Finally node i creates an entry with i?s address in
it and age zero, which is added in j?s cache. The resulting protocol is particularly effective
and, as we show in the experiments below, in some cases it even outperforms the uniform
Newscast. We refer to [10] for more details.
4
The Newscast EM algorithm
Newscast EM (NEM) is a gossip-based distributed implementation of the EM algorithm for
Gaussian mixture learning, that applies to the following setting. We are given a set of data
{xi } that are distributed over the nodes of a network (one data point per node). The data
are assumed independent samples from a common k-component Gaussian mixture p(x)
with (unknown) parameters ? = {?s , ms , Cs }ks=1 . The task is to learn the parameters of
the mixture with maximum likelihood in a decentralized manner: that is, all learning steps
should be performed locally at the nodes, and they should involve as little communication
as possible.
The NEM algorithm is a direct application of the averaging protocol of Section 3 for estimating the parameters ? of p(x) using the EM updates (3). The E-step of NEM is identical
to the E-step of the standard EM algorithm, and it can be performed by all nodes in parallel. The novel characteristic of NEM is the M-step which is implemented as a sequence
of gossip-based cycles: At the beginning of each M-step, each node i starts with a local
estimate ?i of the ?correct? parameter vector ? (correct according to EM and for the current
EM iteration). Then, in every cycle, each node contacts some other node at random, and
the two nodes replace their local estimates ?i by their (weighted) averages. At the end of
the M-step each node has converged (within machine precision) to the correct parameter ?.
To simplify notation, we will denote by ?i = {?si , msi , C?si } the local estimates of node i
for the parameters of component s at any point of the algorithm. The parameter C?si is
defined such that Csi = C?si ? msi m>
si . The complete algorithm, which runs identically
and in parallel for each node, is as follows:
Newscast EM (for node i)
1. Initialization. Set qi (s) to some random positive value and then normalize all qi (s) to
sum to 1 over all s.
2. M-step. Initialize i?s local estimates for each component s as follows: ?si = qi (s),
msi = xi , C?si = xi x>
i . Then repeat for ? cycles:
a. Contact a node j = f (i) from i?s local cache.
b. Nodes i and j update their local estimates for each component s as follows:
?si + ?sj
,
2
?si msi + ?sj msj
m0si = m0sj =
,
?si + ?sj
?si C?si + ?sj C?sj
0
0
.
C?si
= C?sj
=
?si + ?sj
0
0
?si
= ?sj
=
(9)
(10)
(11)
c. Nodes i and j update their caches appropriately.
3. E-step. Compute new responsibilities qi (s) = p(s|xi ) for each component s using the
M-step estimates ?si , msi , and Csi = C?si ? msi m>
si .
4. Loop. Go to step 2, unless a stopping criterion is satisfied that involves the parameter
estimates themselves or the energy F.2
A few observations about the algorithm are in order. First note that both the initialization
of the algorithm (step 1) as well as the E-step are completely local to each node. Similarly,
a stopping criterion involving the parameter estimates can be implemented locally if each
node caches its estimates from the previous EM-iteration. The M-step involves a total of
k[1 + d + d(d + 1)/2] averages, for each one of the k components and for dimensionality d, which are computed with the Newscast protocol. Given that all nodes agree on the
number ? of Newscast cycles in the M-step, and assuming that ? is large enough to guarantee convergence to the correct parameter estimates, the complete NEM algorithm can be
performed identically and in parallel by all nodes in the network.
It is easy to see that at any cycle of an M-step, and for any component s, the weighted
2
Note that F is a sum of local terms, and thus it can also be computed using the same protocol.
averages over all nodes of the local estimates are always the EM-correct estimates, i.e.,
Pn
i=1 ?si msi
P
= ms
(12)
n
i=1 ?si
and similarly for the C?si . Moreover, note that the weighted averages of the msi in (10)
and the C?si in (11), with weights given by (9), can be written as unweighted averages of
the corresponding products ?si msi and ?si C?si . In other words, each local estimate can be
written as the ratio of two local estimates that converge to the correct values at the same
exponential rate (as shown in the previous section). The above observations establish the
following:
Theorem 2. In every M-step of Newscast EM, each node converges exponentially fast to
the correct parameter estimates for each component of the mixture.
Similarly, the number of cycles ? for each M-step can be chosen according to Theorem 1.
However, note that in every M-step each node has to wait ? cycles before its local estimates
have converged, and only then can it use these estimates in a new next E-step. We describe
here a modification of NEM that allows a node to run a local E-step before its M-step has
converged. This ?partial? NEM algorithm is based on the following ?self-correction? idea:
instead of waiting until the M-step converges, after a small number of cycles each node runs
a local E-step, adjusts its responsibilities, and propagates appropriate corrections through
the network.
Such a scheme additionally requires that each node caches its responsibilities from the
previous E-step, denoted by q?i (s). The only modification is in the initialization of the Mstep: instead of fully resetting the local estimates as in step 2 above, a node makes the
following corrections to its current estimates ?si , msi , C?si for each component s:
0
?si
= ?si + qi (s) ? q?i (s),
m0si
0
C?si
(13)
0
q?i (s)]}/?si
,
= {msi ?si + xi [qi (s) ?
0
= {C?si ?si + xi x>
?i (s)]}/?si
.
i [qi (s) ? q
(14)
(15)
After these corrections, the Newscast averaging protocol is executed for a number of cycles (smaller than the number ? of the ?full? NEM). These corrections may increase the
variance of the local estimates, but in most cases the corresponding increase of variance is
relatively small. This results in speed-ups (often as large as 10-fold), however guaranteed
convergence is hard to establish.3
5
Experiments
To get an insight into the behavior of the presented algorithms we ran several experiments
using a Newscast simulator.4 In Fig. 1 we demonstrate the the performance of uniform
and non-uniform Newscast in typical averaging tasks involving zero-mean unit-variance
data. In Fig. 1(left) we plot the variance reduction rate ? (mean and one standard deviation
for 50 runs) as a function of the number of cycles, for averaging problems involving n =
105 data. Note
? that in uniform Newscast the observed rate is always below the derived
bound 1/(2 e) ? 0.303 from Lemma 1. Moreover note that in non-uniform Newscast the
variance drops faster than in uniform Newscast. This is due to the fact that the dynamic
cache exchange scheme of [10] results in in-degree network distributions that are very
peaked around the cache size. In practice this means that on the average each node is
3
This would require, for instance, that individual nodes have estimates of the total variance over
the network, which is not obvious how it can be done.
4
Available from http://www.cs.vu.nl/?steen/globesoul/sim.tgz
42.5
Uniform Newscast
Non?uniform (cache 20)
Uniform Newscast
Non?uniform (cache 20)
42
0.3
Cycles for convergence
Variance reduction rate
0.305
0.295
0.29
41
40.5
40
39.5
0.285
0
41.5
5
10
15
20
25
Number of cycles
30
35
40
39 3
10
4
5
10
10
6
10
Number of nodes
Figure 1: (Left) Variance reduction rate of uniform and non-uniform Newscast, in averaging tasks involving n = 105 nodes. (Right) Number of cycles to achieve convergence
within ? = 10?10 for unit-variance datasets of various sizes.
equally often contacted to by other nodes in each cycle of the protocol. We also observed
that the variance reduction rate is on the average unaffected by the network size, while
larger networks result in smaller deviations. For n = 8 ? 105 , for instance, the standard
deviation is half the one plotted above.
In Fig. 1(right) we plot the number of cycles that are required to achieve model accuracy at
all nodes within ? = 10?10 as a function of the network size. Note that all observed quantities are below the derived bound of Theorem 1, while non-uniform Newscast performs
slightly better than uniform Newscast.
We also ran experiments involving synthetic data drawn from Gaussian mixtures of different number of data points, where we observed results essentially identical to those obtained
by the standard (centralized) EM. We also performed some experiments with the ?partial?
NEM, where it turned out that in most cases we could obtain the same model accuracy with
a much smaller number of cycles (5?10 times than the ?full? NEM), but in some cases the
algorithm did not converge.
6
Summary and extensions
We presented Newscast EM, a distributed gossip-based implementation of the EM algorithm for learning Gaussian mixture models. Newscast EM applies on networks where
each one of a (large) number of nodes observes a local quantity, and can communicate with
other nodes in a point-to-point fashion. The algorithm utilizes a gossip-based protocol in
its M-step to learn a global Gaussian mixture model from the data: each node starts with
a local estimate of the parameters of the mixture and then, for a number of cycles till convergence, pairs of nodes repeatedly exchange their local parameter estimates and combine
them by (weighted) averaging. Newscast EM implements a batch M-step that has average runtime logarithmic in the network size. We believe that gossip-based protocols like
Newscast can be used in several other algorithms that learn models from distributed data.
Several extensions of the algorithm are possible. Here we have assumed that each node
in the network observes one data point. We can easily generalize this to situations where
each node observes (and stores) a collection of points, like in [8]. On the other hand, if the
locally observed data are too many, one may consider storing only some sufficient statistics of these data locally, and appropriately bound the energy F in each iteration to get a
convergent EM algorithm [11]. Another interesting extension is to replace the averaging
step 2 of uniform and non-uniform Newscast with weighted averaging (for some choice of
weights), and study the variance reduction rate in this case. Another interesting problem is
when the E-step cannot be performed locally at a node but it requires distributing some information over the network. This could be the case, for instance, when each node observes
only a few elements of a vector-valued quantity while, for instance, all nodes together observe the complete sample. We note that if the component models factorize, several useful
quantities can be computed by averaging in the log domain. Finally, it would be interesting
to investigate the applicability of the Newscast protocol in problems involving distributed
inference/learning in more general graphical models [12].
Acknowledgments
We want to thank Y. Sfakianakis for helping in the experiments, T. Pylak for making his
Newscast simulator publicly available, and D. Barber, Z. Ghahramani, and J.J. Verbeek for
their comments. N. Vlassis is supported by PROGRESS, the embedded systems research
program of the Dutch organization for Scientific Research NWO, the Dutch Ministry of
Economic Affairs and the Technology Foundation STW, project AES 5414.
References
[1] R. Karp, C. Schindelhauer, S. Shenker, and B. Vo? cking. Randomized rumour spreading. In
Proc. 41th IEEE Symp. on Foundations of Computer Science, Redondo Beach, CA, November
2000.
[2] D. Kempe, A. Dobra, and J. Gehrke. Gossip-based computation of aggregate information. In
Proc. 44th IEEE Symp. on Foundations of Computer Science, Cambridge, MA, October 2003.
[3] M. Jelasity, W. Kowalczyk, and M. van Steen. Newscast computing. Technical report, Dept. of
Computer Science, Vrije Universiteit Amsterdam, 2003. IR-CS-006.
[4] C. C. Moallemi and B. Van Roy. Distributed optimization in adaptive networks. In S. Thrun,
L. Saul, and B. Sch?olkopf, editors, Advances in Neural Information Processing Systems 16.
MIT Press, Cambridge, MA, 2004.
[5] D. Kempe and F. McSherry. A decentralized algorithm for spectral analysis. In Proc. 36th ACM
Symp. on Theory of Computing, Chicago, IL, June 2004.
[6] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via
the EM algorithm. J. Roy. Statist. Soc. B, 39:1?38, 1977.
[7] G. Forman and B. Zhang. Distributed data clustering can be efficient and exact. ACM SIGKDD
Explorations, 2(2):34?38, 2000.
[8] R. D. Nowak. Distributed EM algorithms for density estimation and clustering in sensor networks. IEEE Trans. on Signal Processing, 51(8):2245?2253, August 2003.
[9] R. M. Neal and G. E. Hinton. A view of the EM algorithm that justifies incremental, sparse, and
other variants. In M. I. Jordan, editor, Learning in graphical models, pages 355?368. Kluwer
Academic Publishers, 1998.
[10] S. Voulgaris, D. Gavidia, and M. van Steen. Inexpensive membership management for unstructured P2P overlays. Journal of Network and Systems Management, 2005. To appear.
[11] J. R. J. Nunnink, J. J. Verbeek, and N. Vlassis. Accelerated greedy mixture learning. In Proc.
Belgian-Dutch Conference on Machine Learning, Brussels, Belgium, January 2004.
[12] M. A. Paskin and C. E. Guestrin. Robust probabilistic inference in distributed systems. In Proc.
20th Int. Conf. on Uncertainty in Artificial Intelligence, Banff, Canada, July 2004.
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1,818 | 2,651 | Multiple Relational Embedding
Roland Memisevic
Department of Computer Science
University of Toronto
[email protected]
Geoffrey Hinton
Department of Computer Science
University of Toronto
[email protected]
Abstract
We describe a way of using multiple different types of similarity relationship to learn a low-dimensional embedding of a dataset. Our method
chooses different, possibly overlapping representations of similarity by
individually reweighting the dimensions of a common underlying latent
space. When applied to a single similarity relation that is based on Euclidean distances between the input data points, the method reduces to
simple dimensionality reduction. If additional information is available
about the dataset or about subsets of it, we can use this information to
clean up or otherwise improve the embedding. We demonstrate the potential usefulness of this form of semi-supervised dimensionality reduction on some simple examples.
1
Introduction
Finding a representation for data in a low-dimensional Euclidean space is useful both for
visualization and as prelude to other kinds of data analysis. The common goal underlying the many different methods that accomplish this task (such as ISOMAP [1], LLE [2],
stochastic neighbor embedding [3] and others) is to extract the usually small number of
factors that are responsible for the variability in the data. In making the underlying factors
explicit, these methods help to focus on the kind of variability that is important and provide
representations that make it easier to interpret and manipulate the data in reasonable ways.
Most dimensionality reduction methods are unsupervised, so there is no way of guiding
the method towards modes of variability that are of particular interest to the user. There
is also no way of providing hints when the true underlying factors are too subtle to be
discovered by optimizing generic criteria such as maximization of modeled variance in
PCA, or preservation of local geometry in LLE. Both these difficulties can be alleviated by
allowing the user to provide more information than just the raw data points or a single set
of pairwise similarities between data points.
As an example consider images of faces. Nonlinear methods have been shown to find
embeddings that nicely reflect the variability in the data caused by variation in face identity,
pose, position, or lighting effects. However, it is not possible to tell these methods to
extract a particular single factor for the purpose of, say intelligent image manipulation or
pose identification, because the extracted factors are intermingled and may be represented
simultaneously across all latent space dimensions.
Here, we consider the problem of learning a latent representation for data based on knowl-
edge that is provided by a user in the form of several different similarity relations. Our
method, multiple relational embedding (MRE), finds an embedding that uses a single latent data representation, but weights the available latent space dimensions differently to
allow the latent space to model the multiple different similarity relations. By labeling a
subset of the data according to the kind of variability one is interested in, one can encourage the model to reserve a subset of the latent dimensions for this kind of variability.
The model, in turn, returns a ?handle? to that latent space in the form of a corresponding
learned latent space metric. Like stochastic neighbor embedding, MRE can also be derived
as a simplification of Linear Relational Embedding[4].
1.1
Related work
The problem of supplementing methods for unsupervised learning with ?side-information?
in order to influence their solutions is not new and many different approaches have been
suggested. [5], for example, describes a way to inform a PCA model by encouraging it to
preserve a user-defined grouping structure; [6] consider the problem of extracting exactly
two different kinds of factors, which they denote ?style? and ?content?, by using bilinear
models; more recently, [7] and [8] took a quite different approach to informing a model.
They suggest pre-processing the input data by learning a metric in input space that makes
the data respect user defined grouping constraints.
Our approach differs from these and other methods in two basic ways. First, in all the
methods mentioned above, the side-information has to be defined in terms of equivalence
constraints. That is, a user needs to define a grouping structure for the input data by informing the model which data-points belong together. Here, we consider a rather different
approach, where the side-information can be encoded in the form of similarity relations.
This allows arbitrary continuous degrees of freedom to constrain the low-dimensional embeddings. Second, our model can deal with several, possibly conflicting, kinds of sideinformation. MRE dynamically ?allocates? latent space dimensions to model different
user-provided similarity relations. So inconsistent relations are modeled in disjoint subspaces, and consistent relations can share dimensions. This scheme of sharing the dimensions of a common latent space is reminiscent of the INDSCAL method [9] that has been
popular in the psychometric literature.
A quite different way to extend unsupervised models has recently been introduced by [10]
and [11], where the authors propose ways to extract common factors that underlie two
or more different datasets, with possibly different dimensionalities. While these methods rely on a supervision signal containing information about correspondences between
data-points in different datasets, MRE can be used to discover correspondences between
different datasets using almost no pre-defined grouping constraints.
2
Multiple Relational Embedding
In the following we derive MRE as an extension to stochastic neighbor embedding (SNE).
Let X denote the matrix of latent space elements arranged column-wise, and let ? 2 be some
real-valued neighborhood variance or ?kernel bandwidth?. SNE finds a low-dimensional
representation for a set of input data points y i (i = 1, . . . , N ) by first constructing a similarity matrix P with entries
exp(? ?12 ky i ? y j k2 )
Pij := P
1
i
k 2
k exp(? ? 2 ky ? y k )
(1)
and then minimizing (w.r.t. the set of latent space elements xi (i = 1, . . . , N )) the mismatch
between P and the corresponding latent similarity matrix Q(X) defined by
exp(?kxi ? xj k2 )
.
Qij (X) := P
i
k 2
k exp(?kx ? x k )
(2)
The (row-) normalization of both matrices arises from SNE?s probabilistic formulation in
which the (i, j)th entry of P and Q is interpreted as the probability that the ith data-point
will pick the j th point as its neighbor (in observable and latent space, respectively). The
mismatch is defined as the sum of Kullback-Leibler-divergences between the respective
rows [3].
Our goal is to extend SNE so that it learns latent data representations that not only approximate the input space distances well, but also reflect additional characteristics of the input
data that one may be interested in. In order to accommodate these additional characteristics, instead of defining a single similarity-matrix that is based on Euclidean distances in
data space, we define several matrices P c , (c = 1, . . . , C), each of which encodes some
known type of similarity of the data. Proximity in the Euclidean data-space is typically one
of the types of similarity that we use, though it can easily be omitted. The additional types
of similarity may reflect any information that the user has access to about any subsets of
the data provided the information can be expressed as a similarity matrix that is normalized
over the relevant subset of the data.
At first sight, a single latent data representation seems to be unsuitable to accommodate the
different, and possibly incompatible, properties encoded in a set of P c -matrices. Since our
goal, however, is to capture possibly overlapping relations, we do use a single latent space
and in addition we define a linear transformation Rc of the latent space for each of the C
different similarity-types that we provide as input. Note that this is equivalent to measuring
distances in latent space using a different Mahalanobis metric for each c corresponding to
the matrix Rc T Rc .
In order to learn the transformations Rc from the data along with the set of latent representations X we consider the loss function
X
E(X) =
E c (X),
(3)
c
where we define
1 X c
E c (X) :=
P log
N i,j ij
Pijc
Qcij
!
and
Qcij := Qij (Rc X).
(4)
Note that in the case of C = 1, R1 = I (and fixed) and P 1 defined as in Eq. (1) this
function simplifies to the standard SNE objective function. One might consider weighting
the contribution of each similarity-type using some weighting factor ?c . We found that
the solutions are rather robust with regard to different sets of ?c and weighted all error
contributions equally in our experiments.
As indicated above, here we consider diagonal R-matrices only, which simply amounts to
using a rescaling factor for each latent space dimension. By allowing each type of similarity
to put a different scaling factor on each dimension the model allows similarity relations
that ?overlap? to share dimensions. Completely unrelated or ?orthogonal? relations can be
encoded by using disjoint sets of non-zero scaling factors.
The gradient of E(X) w.r.t. a single latent space element xl takes a similar form to the
gradient of the standard SNE objective function and is given by
?E(X)
2 XX c
=
(Pil + Plic ? Qcli ? Qcil ) Rc T Rc (xl ? xi ),
(5)
l
?x
N c i
2
0
?2
?4
?6 REucl
0.5
0.5
0
0
?0.5
1
?0.5
0.5
1
0.5
0
0
0.5
0
?0.5
?0.5
?1
2
3
0
?2
?4
0
?1
3
2
0.5
1
?0.5
?0.5
2
?1
?6 RClass
?1
1
Figure 1: Embedding of images of rotated objects. Left: SNE, right: MRE. Latent representatives are colored on a gray-scale corresponding to angle of rotation in the original
images. The rightmost plots show entries on the diagonals of latent space transformations
REucl and RClass .
the gradient w.r.t. to a single entry of the diagonal of Rc reads
?
?
?E(X)
2 c ?X X c
j
R
Pij ? Qcij (xil ? xl )2 ? ,
=
c
?Rll
N ll i j
(6)
where xil denotes the lth component of the ith latent representative.
As an illustrative example we ran MRE on a set of images from the Columbia object images
library (COIL) [12]. The dataset contains (128 ? 128)-dimensional gray-scale images of
different objects that vary only by rotation, i.e. by a single degree of freedom. We took
three subsets of images depicting toy-cars, where each subset corresponds to one of three
different kinds of toy-cars, and embedded the first 30 images of each of these subsets in a
three-dimensional space. We used two similarity relations: The first, P Eucl , corresponds
to the standard SNE objective; the second, P Class , is defined as a block diagonal matrix
1
that contains homogeneous blocks of size 30 ? 30 with entries ( 30
) and models class
membership, i.e. we informed the model using the information that images depicting the
same object class belong together.
We also ran standard SNE on the same dataset1 . The results are depicted in figure 1. While
SNE?s unsupervised objective to preserve Euclidean distances leads to a representation
where class-membership is intermingled with variability caused by object rotation (leftmost plot), in the MRE approximation the contribution of class-membership is factored out
and represented in a separate dimension (next plot). This is also reflected in the entries
on the diagonal of the corresponding R-matrices, depicted in the two right-most plots.
RClass is responsible for representing class membership and can do so using just a single
dimension. REucl on the other hand makes use of all dimensions to some degree, reflecting
the fact that the overall variability in ?pixel-space? depends on class-membership, as well
as on other factors (here mainly rotation). Note that with the variability according to class1
For training we set ? 2 manually to 5 ? 107 for both SNE and MRE and initialized all entries in X
and the diagonals of all Rc with small normally distributed values. In all experiments we minimized
the loss function defined in Eq. (3) using Carl Rasmussens? matlab function ?minimize? for 200
iterations (simple gradient descent worked equally well, but was much slower).
membership factored out, the remaining two dimensions capture the rotational degree of
freedom very cleanly.
2.1
Partial information
In many real world situations there might be side-information available only for a subset
of the data-points, because labelling a complete dataset could be too expensive or for other
reasons impossible. A partially labelled dataset can in that case still be used to provide
a hint about the kind of variability that one is interested in. In general, since the corresponding transformation Rc provides a way to access the latent space that represents the
desired similarity-type, a partially labelled dataset can be used to perform a form of supervised feature extraction in which the labelled data is used to specify a kind of feature ?by
example?. It is straightforward to modify the model to deal with partially labelled data.
For each type of similarity c that is known to hold for a subset containing N c examples,
the corresponding P c -matrix references only this subset of the complete dataset and is thus
an N c ? N c -matrix. To keep the latent space elements not corresponding to this subset
unaffected by this error contribution, we can define for each c an index set I c containing
just the examples referenced by P c and rewrite the loss for that type of similarity as
!
c
X
P
1
ij
E c (X) := c
.
(7)
Pijc log
N
Qcij
c
i,j?I
3
Experiments
3.1
Learning correspondences between image sets
In extending the experiment described in section 2 we trained MRE to discover correspondences between sets of images, in this case with different dimensionalities. We picked 20
successive images from one object of the COIL dataset described above and 28 images
(112 ? 92 pixels) depicting a person under different viewing angles taken from the UMIST
dataset[13]. We chose this data in order to obtain two sets of images that vary in a ?similar?
or related way. Note that, because the datasets have different dimensionalities, here it is not
possible to define a single relation describing Euclidean distance between all data-points.
Instead we constructed two relations P Coil and P Umist (for both we used Eq. (1) with ? 2
set as in the previous experiment), with corresponding index-sets I Coil and I Umist containing the indices of the points in each of the two datasets. In addition we constructed one
class-membership relation in the same way as before and two identical relations P 1 and
P 2 that take the form of a 2 ? 2-matrix filled with entries 12 . Each of the corresponding
index sets I 1 and I 2 points to two images (one from each dataset) that represent the end
points of the rotational degree of freedom, i.e. to the first and the last points if we sort the
data according to rotation (see figure 2, left plot). These similarity types are used to make
sure that the model properly aligns the representations of the two different datasets. Note
that the end points constitute the only supervision signal; we did not use any additional
information about the alignment of the two datasets.
After training a two-dimensional embedding2 , we randomly picked latent representatives
of the COIL images and computed reconstructions of corresponding face images using a
kernel smoother (i.e. as a linear combination of the face images with coefficients based on
latent space distances). In order to factor out variability corresponding to class membership
we first multiplied all latent representatives by the inverse of R class . (Note that such a
strategy will in general blow up the latent space dimensions that do not represent class
membership, as the corresponding entries in Rclass may contain very small values. The
2
Training was done using 500 iterations with a setup as in the previous experiment.
Figure 2: Face reconstructions by alignment. Left: Side-information in form of two image
pairs in correspondence. Right: Reconstructions of face images from randomly chosen cat
images.
kernel smoother consequently requires a very large kernel bandwidth, with the net effect
that the latent representation effectively collapses in the dimensions that correspond to class
membership ? which is exactly what we want.) The reconstructions, depicted in the right
plot of figure 2, show that the model has captured the common mode of variability.
3.2
Supervised feature extraction
To investigate the ability of MRE to perform a form of ?supervised feature extraction? we
used a dataset of synthetic face images that originally appeared in [1]. The face images
vary according to pose (two degrees of freedom) and according to the position of a lighting
source (one degree of freedom). The corresponding low-dimensional parameters are available for each data-point. We computed an embedding with the goal of obtaining features
that explicitly correspond to these different kinds of variability in the data.
We labelled a subset of 100 out of the total of 698 data-points with the three mentioned
degrees of freedom in the following way: After standardizing the pose and lighting parameters so that they were centered and had unit variance, we constructed three corresponding
similarity matrices (P Pose1 , P Pose2 , P Lighting ) for a randomly chosen subset of 100 points
using Eq. (1) and the three low-dimensional parameter sets as input data. In addition we
used a fourth similarity relation P Ink , corresponding to overall brightness or ?amount of
ink?, by constructing for each image a corresponding feature equal to the sum of its pixel
intensities and then defining the similarity matrix as above. We set the bandwidth parameter ? 2 to 1.0 for all of these similarity-types3 . In addition we constructed the standard SNE
relation P Eucl (defined for all data-points) using Eq. (1) with ? 2 set4 to 100.
We initialized the model as before and trained for 1000 iterations of ?minimize? to find
an embedding in a four-dimensional space. Figure 3 (right plot) shows the learned latent
space metrics corresponding to the five similarity-types. Obviously, MRE devotes one
dimension to each of the four similarity-types, reflecting the fact that each of them describes
a single one-dimensional degree of freedom that is barely correlated with the others. Dataspace similarities in contrast are represented using all dimensions. The plots on the left
of figure 3 show the embedding of the 598 unlabelled data-points. The top plot shows the
embedding in the two dimensions in which the two ?pose?-metrics take on their maximal
values, the bottom plot shows the dimensions in which the ?lighting?- and ?ink?-metric take
on their maximal values. The plots show that MRE generalizes over unlabeled data: In each
dimension the unlabeled data is clearly arranged according to the corresponding similarity
type, and is arranged rather randomly with respect to other similarity types. There are
a few correlations, in particular between the first pose- and the ?ink?-parameter, that are
inherent in the dataset, i.e. the data does not vary entirely independently with respect to
these parameters. These correlations are also reflected in the slightly overlapping latent
3
This is certainly not an optimal choice, but we found the solutions to be rather robust against
changes in the bandwidth, and this value worked fine.
4
See previous footnote.
0.3
5
REucl
0.2
0
x1
0.1
?5
1
0
5
?0.1
2
3
4
3
4
3
4
2
3
4
2
3
4
RLights
0
?0.2
?5
1
?0.3
?0.2
?0.1
0
0.1
0.2
0.3
x4
5
2
RPose2
0
0.3
?5
1
0.2
5
x3
0.1
2
RPose1
0
0
?5
1
?0.1
Ink
5
?0.2
R
0
?0.3
?5
?0.2
?0.1
0
0.1
x2
0.2
0.3
1
Figure 3: Left: Embedding of faces images that were not informed about their lowdimensional parameters. For a randomly chosen subset of these (marked with a circle),
the original images are shown next to their latent representatives. Right: Entries on the
diagonals of five latent space transformations.
space weight sets. MRE gets the pose-embedding wrong for a few very dark images that
are apparently too far away in the data space to be associated with the correct labeled datapoints.
4
Conclusions
We introduced a way to embed data in a low-dimensional space using a set of similarity
relations. Our experiments indicate that the informed feature extraction that this method facilitates will be most useful in cases where conventional dimensionality reduction methods
fail because of their completely unsupervised nature. Although we derived our approach
as an extension to SNE, it should be straightforward to apply the same idea to other dimensionality reduction methods.
Acknowledgements: Roland Memisevic is supported by a Government of Canada Award.
Geoffrey Hinton is a fellow of CIAR and holds a CRC chair. This research was also supported by grants from NSERC and CFI.
References
[1] Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A global geometric framework for
nonlinear dimensionality reduction. Science, pages 2319?2323, 2000.
[2] S.T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding.
Science, 290, 2000.
[3] Geoffrey Hinton and Sam Roweis. Stochastic neighbor embedding. In Advances in Neural
Information Processing Systems 15, pages 833?840. MIT Press, 2003.
[4] A. Paccanaro and G. E. Hinton. Learning hierarchical structures with linear relational embedding. In Advances in Neural Information Processing Systems 14, Cambridge, MA, 2002. MIT
Press.
[5] David Cohn. Informed projections. In Advances in Neural Information Processing Systems 15,
pages 849?856. MIT Press, 2003.
[6] Joshua B. Tenenbaum and William T. Freeman. Separating style and content with bilinear
models. Neural Computation, 12(6):1247?1283, 2000.
[7] Eric P. Xing, Andrew Y. Ng, Michael I. Jordan, and Stuart Russell. Distance metric learning with application to clustering with side-information. In Advances in Neural Information
Processing Systems 15, pages 505?512. MIT Press, Cambridge, MA, 2003.
[8] Michinari Momma Tijl De Bie and Nello Cristianini. Efficiently learning the metric using sideinformation. In Proc. of the 14th International Conference on Algorithmic Learning Theory,
2003.
[9] J. Douglas Carroll and Jih-Jie Chang. Analysis of individual differences in multidimensional
scaling via an n-way generalization of ?eckart-young? decomposition. Psychometrika, 35(3),
1970.
[10] J. H. Ham, D. D. Lee, and L. K. Saul. Learning high dimensional correspondences from low
dimensional manifolds. In In Proceedings of the ICML 2003 Workshop on The Continuum from
Labeled to Unlabeled Data in Machine Learning and Data Mining, pages 34?41, Washington,
D.C., 2003.
[11] Jakob J. Verbeek, Sam T. Roweis, and Nikos Vlassis. Non-linear cca and pca by alignment of local models. In Advances in Neural Information Processing Systems 16. MIT Press, Cambridge,
MA, 2004.
[12] S. A. Nene, S. K. Nayar, and H. Murase. Columbia object image library (coil-20). Technical
report, 1996.
[13] Daniel B Graham and Nigel M Allinson. Characterizing virtual eigensignatures for general
purpose face recognition. 163, 1998.
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1,819 | 2,652 | Conditional Random Fields for Object
Recognition
Ariadna Quattoni Michael Collins Trevor Darrell
MIT Computer Science and Artificial Intelligence Laboratory
Cambridge, MA 02139
{ariadna, mcollins, trevor}@csail.mit.edu
Abstract
We present a discriminative part-based approach for the recognition of
object classes from unsegmented cluttered scenes. Objects are modeled
as flexible constellations of parts conditioned on local observations found
by an interest operator. For each object class the probability of a given
assignment of parts to local features is modeled by a Conditional Random Field (CRF). We propose an extension of the CRF framework that
incorporates hidden variables and combines class conditional CRFs into
a unified framework for part-based object recognition. The parameters of
the CRF are estimated in a maximum likelihood framework and recognition proceeds by finding the most likely class under our model. The main
advantage of the proposed CRF framework is that it allows us to relax the
assumption of conditional independence of the observed data (i.e. local
features) often used in generative approaches, an assumption that might
be too restrictive for a considerable number of object classes.
1 Introduction
The problem that we address in this paper is that of learning object categories from supervised data. Given a training set of n pairs (xi , yi ), where xi is the ith image and yi is the
category of the object present in xi , we would like to learn a model that maps images to
object categories. In particular, we are interested in learning to recognize rigid objects such
as cars, motorbikes, and faces from one or more fixed view-points.
The part-based models we consider represent images as sets of patches, or local features,
which are detected by an interest operator such as that described in [4]. Thus an image
xi can be considered to be a vector {xi,1 , . . . , xi,m } of m patches. Each patch xi,j has
a feature-vector representation ?(xi,j ) ? Rd ; the feature vector might capture various
features of the appearance of a patch, as well as features of its relative location and scale.
This scenario presents an interesting challenge to conventional classification approaches in
machine learning, as the input space xi is naturally represented as a set of feature-vectors
{?(xi,1 ), . . . , ?(xi,m )} rather than as a single feature vector. Moreover, the local patches
underlying the local feature vectors may have complex interdependencies: for example,
they may correspond to different parts of an object, whose spatial arrangement is important
to the classification task.
The most widely used approach for part-based object recognition is the generative model
proposed in [1]. This classification system models the appearance, spatial relations and
co-occurrence of local parts. One limitation of this framework is that to make the model
computationally tractable one has to assume the independence of the observed data (i.e.,
local features) given their assignment to parts in the model. This assumption might be too
restrictive for a considerable number of object classes made of structured patterns.
A second limitation of generative approaches is that they require a model P (xi,j |hi,j ) of
patches xi,j given underlying variables hi,j (e.g., hi,j may be a hidden variable in the
model, or may simply be yi ). Accurately specifying such a generative model may be challenging ? in particular in cases where patches overlap one another, or where we wish to
allow a hidden variable hi,j to depend on several surrounding patches. A more direct approach may be to use a feature vector representation of patches, and to use a discriminative
learning approach. We follow an approach of this type in this paper.
Similar observations concerning the limitations of generative models have been made in
the context of natural language processing, in particular in sequence-labeling tasks such as
part-of-speech tagging [7, 5, 3] and in previous work on conditional random fields (CRFs)
for vision [2]. In sequence-labeling problems for NLP each observation xi,j is typically
the j?th word for some input sentence, and hi,j is a hidden state, for example representing
the part-of-speech of that word. Hidden Markov models (HMMs), a generative approach,
require a model of P (xi,j |hi,j ), and this can be a challenging task when features such as
word prefixes or suffixes are included in the model, or where hi,j is required to depend
directly on words other than xi,j . This has led to research on discriminative models for sequence labeling such as MEMM?s [7, 5] and conditional random fields (CRFs)[3]. A strong
argument for these models as opposed to HMMs concerns their flexibility in terms of representation, in that they can incorporate essentially arbitrary feature-vector representations
?(xi,j ) of the observed data points.
We propose a new model for object recognition based on Conditional Random Fields. We
model the conditional distribution p(y|x) directly. A key difference of our approach from
previous work on CRFs is that we make use of hidden variables in the model. In previous
work on CRFs (e.g., [2, 3]) each ?label? yi is a sequence hi = {hi,1 , hi,2 , . . . , hi,m } of
labels hi,j for each observation xi,j . The label sequences are typically taken to be fully
observed on training examples. In our case the labels yi are unstructured labels from some
fixed set of object categories, and the relationship between yi and each observation xi,j is
not clearly defined. Instead, we model intermediate part-labels hi,j as hidden variables in
the model. The
P model defines conditional probabilities P (y, h | x), and hence indirectly
P (y | x) = h P (y, h | x), using a CRF. Dependencies between the hidden variables
h are modeled by an undirected graph over these variables. The result is a model where
inference and parameter estimation can be carried out using standard graphical model algorithms such as belief propagation [6].
2 The Model
2.1 Conditional Random Fields with Hidden Variables
Our task is to learn a mapping from images x to labels y. Each y is a member of a set Y
of possible image labels, for example, Y = {background, car}. We take each image x
to be a vector of m ?patches? x = {x1 , x2 , . . . , xm }.1 Each patch xj is represented by a
feature vector ?(xj ) ? Rd . For example, in our experiments each xj corresponds to a patch
that is detected by the feature detector in [4]; section [3] gives details of the feature-vector
representation ?(xj ) for each patch. Our training set consists of labeled images (xi , yi ) for
i = 1 . . . n, where each yi ? Y, and each xi = {xi,1 , xi,2 , . . . , xi,m }. For any image x
we also assume a vector of ?parts? variables h = {h1 , h2 , . . . , hm }. These variables are
not observed on training examples, and will therefore form a set of hidden variables in the
1
Note that the number of patches m can vary across images, and did vary in our experiments. For
convenience we use notation where m is fixed across different images; in reality it will vary across
images but this leads to minor changes to the model.
model. Each hj is a member of H where H is a finite set of possible parts in the model.
Intuitively, each hj corresponds to a labeling of xj with some member of H. Given these
definitions of image-labels y, images x, and part-labels h, we will define a conditional
probabilistic model:
e?(y,h,x;?)
.
(1)
P (y, h | x, ?) = P
?(y 0 ,h,x;?)
y 0 ,h e
Here ? are the parameters of the model, and ?(y, h, x; ?) ? R is a potential function
parameterized by ?. We will discuss the choice of ? shortly. It follows that
P ?(y,h,x;?)
X
e
(2)
P (y | x, ?) =
P (y, h | x, ?) = P h ?(y0 ,h,x;?) .
y 0 ,h e
h
Given a new test image x, and parameter values ?? induced from a training example, we
will take the label for the image to be arg maxy?Y P (y | x, ?? ). Following previous work
on CRFs [2, 3], we use the following objective function in training the parameters:
X
1
(3)
L(?) =
log P (yi | xi , ?) ? 2 ||?||2
2?
i
The first term in Eq. 3 is the log-likelihood of the? data. The
? second term is the log of a
Gaussian prior with variance ? 2 , i.e., P (?) ? exp 2?1 2 ||?||2 . We will use gradient ascent
to search for the optimal parameter values, ?? = arg max? L(?), under this criterion.
We now turn to the definition of the potential function ?(y, h, x; ?). We assume an undirected graph structure, with the hidden variables {h1 , . . . , hm } corresponding to vertices
in the graph. We use E to denote the set of edges in the graph, and we will write (j, k) ? E
to signify that there is an edge in the graph between variables hj and hk . In this paper we
assume that E is a tree.2 We define ? to take the following form:
?(y, h, x; ?) =
m X
X
j=1
fl1 (j, y, hj , x)?l1 +
l
X X
(j,k)?E
fl2 (j, k, y, hj , hk , x)?l2
(4)
l
where fl1 , fl2 are functions defining the features in the model, and ?l1 , ?l2 are the components
of ?. The f 1 features depend on single hidden variable values in the model, the f 2 features
can depend on pairs of values. Note that ? is linear in the parameters ?, and the model in
Eq. 1 is a log-linear model. Moreover the features respect the structure of the graph, in that
no feature depends on more than two hidden variables hj , hk , and if a feature does depend
on variables hj and hk there must be an edge (j, k) in the graph E.
Assuming that the edges in E form a tree, and that ? takes the form in Eq. 4, then exact
methods exist for inference and parameter estimation in the model. This follows because
belief propagation [6] can be used to calculate the following quantities in O(|E||Y|) time:
X
?y ? Y, Z(y | x, ?) =
exp{?(y, h, x; ?)}
h
?y ? Y, j ? 1 . . . m, a ? H,
X
P (hj = a | y, x, ?) =
P (h | y, x, ?)
h:hj =a
?y ? Y, (j, k) ? E, a, b ? H,
P (hj = a, hk = b | y, x, ?) =
X
P (h | y, x, ?)
h:hj =a,hk =b
2
This will allow exact methods for inference and parameter estimation in the model, for example
using belief propagation. If E contains cycles then approximate methods, such as loopy beliefpropagation, may be necessary for inference and parameter estimation.
The first term Z(y | x, ?) is a partition function defined by a summation over
the h variables.
Terms of this form can be used to calculate P (y | x, ?) =
P
Z(y | x, ?)/ y0 Z(y 0 | x, ?). Hence inference?calculation of arg max P (y | x, ?)?
can be performed efficiently in the model. The second and third terms are marginal distributions over individual variables hj or pairs of variables hj , hk corresponding to edges in
the graph. The next section shows that the gradient of L(?) can be defined in terms of these
marginals, and hence can be calculated efficiently.
2.2 Parameter Estimation Using Belief Propagation
This section considers estimation of the parameters ?? = arg max L(?) from a training
sample, where L(?) is defined in Eq. 3. In our work we used a conjugate-gradient method
to optimize L(?) (note that due to the use of hidden variables, L(?) has multiple local
minima, and our method is therefore not guaranteed to reach the globally optimal point).
In this section we describe how the gradient of L(?) can be calculated efficiently. Consider
the likelihood term that is contributed by the i?th training example, defined as:
!
? P
?(yi ,h,xi ;?)
e
(5)
Li (?) = log P (yi | xi , ?) = log P h ?(y0 ,h,x ;?) .
i
y 0 ,h e
We first consider derivatives with respect to the parameters ?l1 corresponding to features
fl1 (j, y, hj , x) that depend on single hidden variables. Taking derivatives gives
?Li (?) X
??(yi , h, xi ; ?) X
??(y 0 , h, xi ; ?)
=
P (h | yi , xi , ?)
?
P (y 0 , h | xi , ?)
1
1
??l
??l
??l1
0
h
=
X
y ,h
P (h | yi , xi , ?)
=
fl1 (j, yi , hj , xi ) ?
P (hj = a |
X
P (y 0 , h | xi , ?)
y 0 ,h
j=1
h
X
m
X
yi , xi , ?)fl1 (j, yi , a, xi )
?
X
m
X
fl1 (j, y 0 , hj , xi )
j=1
0
P (hj = a, y | xi , ?)fl1 (j, y 0 , a, xi )
y 0 ,j,a
j,a
can be expressed in terms of components P (hj = a | xi , ?) and
It follows that ?L??i (?)
1
l
P (y | xi , ?), which can be calculated using belief propagation, provided that the graph E
forms a tree structure. A similar calculation gives
?Li (?)
??l2
X
=
P (hj = a, hk = b | yi , xi , ?)fl2 (j, k, yi , a, b, xi )
(j,k)?E,a,b
X
?
P (hj = a, hk = b, y 0 | xi , ?)fl2 (j, k, y 0 , a, b, xi )
y 0 ,(j,k)?E,a,b
hence ?Li (?)/??l2 can also be expressed in terms of expressions that can be calculated
using belief propagation.
2.3 The Specific Form of our Model
We now turn to the specific form for the model in this paper. We define
X
X
X
?(y, h, x; ?) =
?(xj ) ? ?(hj ) +
?(y, hj ) +
?(y, hj , hk )
j
j
(6)
(j,k)?E
Here ?(k) ? Rd for k ? H is a parameter vector corresponding to the k?th part label. The
inner-product ?(xj ) ? ?(hj ) can be interpreted as a measure of the compatibility between
patch xj and part-label hj . Each parameter ?(y, k) ? R for k ? H, y ? Y can be
interpreted as a measure of the compatibility between part k and label y. Finally, each
parameter ?(y, k, l) ? R for y ? Y, and k, l ? H measures the compatibility between an
edge with labels k and l and the label y. It is straightforward to verify that the definition in
Eq. 6 can be written in the same form as Eq. 4. Hence belief propagation can be used for
inference and parameter estimation in the model.
The patches xi,j in each image are obtained using the SIFT detector [4]. Each patch xi,j
is then represented by a feature vector ?(xi,j ) that incorporates a combination of SIFT and
relative location and scale features.
The tree E is formed by running a minimum spanning tree algorithm over the parts hi,j ,
where the cost of an edge in the graph between hi,j and hi,k is taken to be the distance
between xi,j and xi,k in the image. Note that the structure of E will vary across different
images. Our choice of E encodes our assumption that parts conditioned on features that are
spatially close are more likely to be dependent. In the future we plan to experiment with
the minimum spanning tree approach under other definitions of edge-cost. We also plan to
investigate more complex graph structures that involve cycles, which may require approximate methods such as loopy belief propagation for parameter estimation and inference.
3 Experiments
We carried out three sets of experiments on a number of different data sets.3 The first
two experiments consisted of training a two class model (object vs. background) to distinguish between a category from a single viewpoint and background. The third experiment
consisted of training a multi-class model to distinguish between n classes.
The only parameter that was adjusted in the experiments was the scale of the images upon
which the interest point detector was run. In particular, we adjusted the scale on the car
side data set: in this data set the images were too small and without this adjustment the
detector would fail to find a significant amount of features.
For the experiments we randomly split each data set into three separate data sets: training,
validation and testing. We use the validation data set to set the variance parameters ? 2 of
the gaussian prior.
3.1 Results
In figure 2.a we show how the number of parts in the model affects performance. In the case
of the car side data set, the ten-part model shows a significant improvement compared to
the five parts model while for the car rear data set the performance improvement obtained
by increasing the number of parts is not as significant. Figure 2.b shows a performance
comparison with previous approaches [1] tested on the same data set (though on a different
partition). We observe an improvement between 2 % and 5 % for all data sets.
Figures 3 and 4 show results for the multi-class experiments. Notice that random performance for the animal data set would be 25 % across the diagonal. The model exhibits
best performance for the Leopard data set, for which the presence of part 1 alone is a clear
predictor of the class. This shows again that our model can learn discriminative part distributions for each class. Figure 3 shows results for a multi-view experiment where the task
is two distinguish between two different views of a car and background.
3
The images were obtained from http://www.vision.caltech.edu/html-files/archive.html and the
car side images from http://l2r.cs.uiuc.edu/ cogcomp/Data/Car/. Notice, that since our algorithm
does not currently allow for the recognition of multiple instances of an object we test it on a partition
of the the training set in http://l2r.cs.uiuc.edu/ cogcomp/Data/Car/ and not on the testing set in that
site. The animals data set is a subset of Caltech?s 101 categories data set.
Figure 1: Examples of the most likely assignment of parts to features for the two class
experiments (car data set).
Data set
(a) Car Side
Car Rear
5 parts
94 %
91 %
10 parts
99 %
91.7 %
Data set
Car Side
Car Rear
(b) Face
Plane
Motorbike
Our Model
99 %
94.6 %
99 %
96 %
95 %
Others [1]
90.3 %
96.4 %
90.2 %
92.5 %
Figure 2: (a) Equal Error Rates for the car side and car rear experiments with different
number of parts. (b) Comparative Equal Error Rates.
Figure 1 displays the Viterbi labeling4 for a set of example images showing the most likely
assignment of local features to parts in the model. Figure 6 shows the mean and variance
of each part?s location for car side images and background images. The mean and variance
of each part?s location for the car side images were calculated in the following manner:
First we find for every image classified as class a the most likely part assignment under our
model. Second, we calculate the mean and variance of positions of all local features that
were assigned to the same part. Similarly Figure 5 shows part counts among the Viterbi
paths assigned to examples of a given class.
As can be seen in Figure 6 , while the mean location of a given part in the background
images and the mean location of the same part in the car images are very similar, the parts
in the car have a much tighter distribution which seems to suggest that the model is learning
the shape of the object.
As shown in Figure 5 the model has also learnt discriminative part distributions for each
class, for example the presence of part 1 seems to be a clear predictor for the car class. In
general part assignments seem to rely on a combination of appearance and relative location.
Part 1, for example, is assigned to wheel like patterns located on the left of the object.
4
This is the labeling h? = arg maxh P (h | y, x, ?) where x is an image and y is the label for
the image under the model.
Data set
Car Side
Car Rear
Precision
87.5 %
87.4 %
Recall
98 %
86.5 %
Figure 3: Precision and recall results for 3 class experiment.
Data set
Leopards
Llamas
Rhinos
Pigeons
Leopards
91 %
0%
0%
0%
Llamas
2%
50 %
40 %
30 %
Rhinos
0%
27 %
46 %
20 %
Pigeons
7%
23 %
14 %
50 %
Figure 4: Confusion table for 4 class experiment.
However, the parts might not carry semantic meaning. It appears that the model has learnt
a vocabulary of very general parts with significant variability in appearance and learns to
discriminate between classes by capturing the most likely arrangement of these parts for
each class.
In some cases the model relies more heavily on relative location than appearance because
the appearance information might not be very useful for discriminating between the two
classes. One of the reasons for this is that the detector produces a large number of false detections, making the appearance data too noisy for discrimination. The fact that the model
is able to cope with this lack of discriminating appearance information illustrates its flexible
data-driven nature. This can be a desirable model property of a general object recognition
system, because for some object classes appearance is the important discriminant (i.e., in
textured classes) while for others shape may be important (i.e., in geometrically constrained
classes).
One noticeable difference between our model and similar part-based models is that our
model learns large parts composed of small local features. This is not surprising given how
the part dependencies were built (i.e., through their position in minimum spanning tree):
the potential functions defined on pairs of hidden variables tend to smooth the allocation of
parts to patches.
1
3
3
8
4
5
8
76
5
4
Figure 5: Graph showing part counts for the background (left) and car side images (right)
4 Conclusions and Further Work
In this work we have presented a novel approach that extends the CRF framework by incorporating hidden variables and combining class conditional CRFs into an unified framework for object recognition. Similarly to CRFs and other maximum entropy models our
approach allows us to combine arbitrary observation features for training discriminative
classifiers with hidden variables. Furthermore, by making some assumptions about the
joint distribution of hidden variables one can derive efficient training algorithms based on
dynamic programming.
Background Shape
Car Shape
200
200
150
150
100
100
3
50
50
4
5
0
1
8
?50
?50
?100
?100
?150
?150
?200
?200
?150
?100
?50
0
(a)
3
5
0
50
100
150
200
?200
?200
?150
?100
?50
9
7
6
4
8
0
50
100
150
200
(b)
Figure 6: (a) Graph showing mean and variance of locations for the different parts for the
car side images; (b) Mean and variance of part locations for the background images.
The main limitation of our model is that it is dependent on the feature detector picking up
discriminative features of the object. Furthermore, our model might learn to discriminate
between classes based on the statistics of the feature detector and not the true underlying
data, to which it has no access. This is not a desirable property since it assumes the feature
detector to be consistent. As future work we would like to incorporate the feature detection
process into the model.
References
[1] R. Fergus, P. Perona,and A. Zisserman. Object class recognition by unsupervised scale-invariant
learning. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,volume 2, pages 264-271, June 2003.
[2] S. Kumar and M. Hebert. Discriminative random fields: A framework for contextual interaction
in classification. In IEEE Int Conference on Computer Vision,volume 2, pages 1150-1157, June
2003.
[3] J. Lafferty,A. McCallum and F. Pereira. Conditional random fields: Probabilistic models for
segmenting and labeling sequence data. In Proc. Int Conf. on Machine Learning, 2001.
[4] D. Lowe. Object Recognition from local scale-invariant features. In IEEE Int Conference on
Computer Vision, 1999.
[5] A. McCallum, D. Freitag, and F. Pereira. Maximum entropy markov models for information
extraction and segmentation. In ICML-2000, 2000.
[6] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann,1988.
[7] A. Ratnaparkhi. A maximum entropy part-of-speech tagger. In EMNLP, 1996.
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1,820 | 2,653 | A Topographic Support Vector Machine:
Classification Using Local Label Configurations
Johannes Mohr
Clinic for Psychiatry and Psychotherapy
Charit?e Medical School
and
Bernstein Center for Computational Neuroscience Berlin
10117 Berlin, Germany
Klaus Obermayer
Department of Electrical Engineering and Computer Science
Berlin University of Technology
and
Bernstein Center for Computational Neuroscience Berlin
10587 Berlin, Germany
[email protected], [email protected]
Abstract
The standard approach to the classification of objects is to consider the
examples as independent and identically distributed (iid). In many real
world settings, however, this assumption is not valid, because a topographical relationship exists between the objects. In this contribution we
consider the special case of image segmentation, where the objects are
pixels and where the underlying topography is a 2D regular rectangular
grid. We introduce a classification method which not only uses measured
vectorial feature information but also the label configuration within a topographic neighborhood. Due to the resulting dependence between the
labels of neighboring pixels, a collective classification of a set of pixels
becomes necessary. We propose a new method called ?Topographic Support Vector Machine? (TSVM), which is based on a topographic kernel
and a self-consistent solution to the label assignment shown to be equivalent to a recurrent neural network. The performance of the algorithm is
compared to a conventional SVM on a cell image segmentation task.
1
Introduction
The segmentation of natural images into semantically meaningful subdivisions can be considered as one or more binary pixel classification problems, where two classes of pixels are
characterized by some measurement data (features). For each binary problem the task is
to assign a set of new pixels to one of the two classes using a classifier trained on a set of
labeled pixels (training data).
In conventional classification approaches usually the assumption of iid examples is made,
so the classification result is determined solely by the measurement data. Natural images,
however, possess a topographic structure, in which there are dependencies between the
labels of topographic neighbors, making the data non-iid. Therefore, not only the measurement data, but also the labels of the topographic neighbors can be used in the classification
of a pixel. It has been shown for a number of problems that dependencies between instances can improve model accuracy. A Conditional Random Field approach approach has
been used for labeling text sequences by [1]. Combining this idea with local discriminative
models, in [2] a discriminative random field was used to model the dependencies between
the labels of image blocks in a probabilistic framework. A collective classification relational dependency network was used in [3] for movie box-office receipts prediction and
paper topic classification. The maximization of the per label margin of pairwise Markov
networks was applied in [4] to handwritten character recognition and collective hypertext
classification. There, the number of variables and constraints of the quadratic programming
problem was polynomial in the number of labels.
In this work, we propose a method which is also based on margin maximization and allows
the collective assignments of a large number of binary labels which have a regular grid
topography. In contrast to [4] the number of constraints and variables does not depend on
the number of labels. The method called topographic support vector machine (TSVM) is
based on the assumption that knowledge about the local label configuration can improve the
classification of a single data point. Consider as example the segmentation of a collection
of images depicting physical objects of similar shape, but high variability in gray level and
texture. In this case, the measurements are dissimilar, while the local label configurations
show high similarity.
Here, we apply the TSVM to the supervised bottom-up segmentation of microscopic images of Papanicolaou stained cervical cell nuclei from the CSSIP pap smear dataset 1 . Segmentation of these images is important for the detection of cervical cancer or precancerous
cells. The final goal is to use so-called malignancy associated changes (MACs), e.g. a
slight shift of the distribution of nuclear size not yet visual to the human observer, in order
to detect cancer at an early stage [5]. A previously used bottom-up segmentation approach
for this data using morphological watersheds was reported to have difficulties with weak
gradients and the presence of other large gradients adjacent to the target [5]. Top-down
methods like active contour models have successfully been used [6], but require heuristic
initialization and error correction procedures.
2
Classification using a Topographic Support Vector Machine
Let O = {o1 , ..., on } be a set of n sites on a 2D pixel-grid and G = {Go , o ? O} be
a neighborhood system for O, where Go is the set of neighbors of o and neighborhood
is defined by o 6? Go and o ? Gp ? p ? Go . For each pixel site oi from the set O,
a binary label yi ? {?1, +1} giving the class assignment is assumed to be known. To
simplify the notation, in the following we are going to make use of multi-indices written in
the form of vectors, referring to pairs of indices on a two-dimensional grid. We define the
neighborhood of order c as G c = {Gi , i ? O}; Gi = {k ? O : 0 < (k?i)2 ? c}. This way,
G 1 describes the first order neighborhood system (4 neighbors), G 2 the second order system
(8 neighbors), and so on. Each pixel site is characterized by some measurement vector. This
could for example be the vector of gray value intensities at a pixel site, the gray value patch
around a central pixel location, or the responses to a bank of linear or nonlinear filters (e.g.
Gabor coefficients). Using a training set composed of (possibly several) sets of pixel sites,
each accompanied by a set of measurement vectors X = {xi , ?i ? [1..n]} and a set of
1
Centre for Sensor Signal and Information Processing, University of Queensland
labels Y = {yi , ?i ? [1..n]} (e.g. a manually labeled image), the task of classification is
to assign class labels to a set of ? pixels sites U = {u1 , ..., u? } of an unlabeled image, for
? = {?
which a set of measurements X
xi , ?i ? [1..?]} is available. For the classification we
will use a support vector machine.
2.1
Support Vector Classification
In Support Vector Classification (SVC) methods ([7]), a kernel is used to solve a complex
classification task in a usually high-dimensional feature space via a separating hyperplane.
Results from statistical learning theory ([8]) show that maximizing the margin (the distance
of the closest data point to the hyperplane) leads to improved generalization abilities. In
practice, the optimal margin hyperplane can be obtained solving a quadratic programming
problem. Several schemes have been introduced to deal with noisy measurements via the
introduction of slack variables. In the following we will shortly review one such scheme,
the C-SVM, which is also later used in the experiments. For a canonical separating hyperplane (w, b) in a higher dimensional feature space H, to which the n variables x i are
mapped by ?(x), and n slack variables ?i the primal objective function of a C-SVM can
be formulated as
n
1
X
w
2 + C
min n
?i ,
(1)
w?H,??R
2
n i=1
subject to yi (wT ?(xi ) + b) ? 1 ? ?i ,
?i ? 0,
C > 0,
i = 1, ..., n.
In order to classify a new object h with unknown label, the following decision rule is
evaluated:
m
X
f (xh ) = sgn
?i yi K(xh , xi ) + b ,
(2)
i=1
where the sum runs over all m support vectors.
2.2
Topographic Kernel
We now assume that the label of each pixel is determined by both the measurement and
the set of labels of its topographic neighbors. We define a vector yGh where the labels of
the q topographic neighbors of the pixel h are concatenated in an arbitrary, but fixed order.
We propose a support vector classifier using an extended kernel, which in addition to the
measurement vector xh , also includes the vector yGh :
K(xh , xj , yGh , yGj ) = K1 (xh , xj ) + ? ? K2 (yGh , yGj ),
(3)
where ? is a hyper-parameter. Kernel K1 can be an arbitrary kernel working on the measurements. For kernel K2 an arbitrary dot-product kernel might be used. In the following
we restrict ourselves to a linear kernel (corresponding to the normalized Hamming distance
between the local label configurations)
K2 (yGh , yGj ) =
1
hyGh |yGj i,
q
(4)
where h...|...i denotes a scalar product. The kernel K2 defined in eq. (4) thus consists of a
dot-product between these vectors divided by their length. For a neighborhood G hc of order
c we obtain
X
1
K2 (yGh , yGj ) =
yh+s ? yj+s
(5)
q
?
|s|< c,s6=0
The linear kernel K2 in (4) takes on its maximum value, if the label configurations are
identical, and its lowest value if the label configuration is inverted.
2.3
Learning phase
If a SVM is trained using the topographic kernel (3), the topographic label configuration is
included in the learning process. The resulting support vectors will still contain the relevant
information about the measurements, but additionally the label neighborhood information
relevant for a good distinction of the classes.
2.4
Classification phase
In order to collectively classify a set of ? new pixel sites with unknown topographic label
configuration, we propose the following iterative approach to achieve a self-consistent solution to the classification problem. We denote the labels at step ? as yh (? ), ?h. At each
step ? new labels are assigned according to
m
X
yh (? ) = sgn
j=1
?j ? yv(j) ? K(xh , xv(j) , yGh (? ? 1), yGv(j) ) + b , ?h.
(6)
The sum runs over all m support vectors, whose indices on the 2D grid are denoted by
the vector v(j). Since initially the labels are unknown, we use at step ? = 0 the results
from a conventional support-vector machine (? = 0) as initialization for the labels. For the
following steps some estimates of the neighboring labels are available from the previous
iteration. Using this new topographic label information in addition to the measurement
information, using (6) a new assignment decision for the labels is made. This leads to an
iterative assignment of new labels.
If we write the contributions from kernel K1 , which depend only on x and do not change
with ? , as ch (j) = ?j ? yv(j) ? K1 (xh , xv(j) ) equation (6) becomes
yh (? ) = sgn
m
X
j=1
??j yv(j) K2 (yGh (? ? 1), yGv(j) ) + ch (j) + b , ?h.
(7)
Putting in the linear kernel from equation (5), we get
yh (? ) = sgn
m
X
j=1
?j yv(j)
?
q
X
?
|s|< c,s6=0
[yh+s (? ? 1) ? yv(j)+s ] + ch (j) + b , ?h. (8)
Interchanging the sums, using the definitions
? Pm
j=1 ?j yv(j) yv(j)+(k?h)
q
wh,k =
0
and
?h = ?
m
X
j=1
we obtain
yh (? ) = sgn
X
k
: k ? Gh
: k?
/ Gh
ch (j) + b ,
yk (? ? 1) ? wh,k ? ?h , ?h.
(9)
(10)
(11)
This corresponds to the equations describing the dynamics of a recurrent neural network
composed of McCulloch-Pitts neurons [9]. The condition for symmetric weights w h,k =
wk,h is equivalent to an inversion symmetry of the label configurations of the support vectors in the neighborhood topology, therefore the weights in equation (9) are not necessarily
symmetric. A suitable stopping criterion for the iteration is that the net reaches either a
fixed point yh (? ) = yh (? ? 1), ?h, or an attractor cycle yh (? ) = yh (?), ? < ? ? 1, ?h.
The network described by eq. (11) corresponds to a diluted network of ? binary neurons
with no self-interaction and asymmetric weights. One can see from eq.(9) that the network has only local connections, corresponding to the topographic neighborhood G h . The
measurement xh only influences the individual unit threshold ?h of the network, via the
weighted sum over all support vectors of the contributions from kernel K1 (eq. (10)). The
label configurations of the support vectors, on the other hand, are contained in the network
weights via eq.(9). The weights are multiplied by the hyper-parameter ?, which determines
how much the label configuration influences the class decision in comparison to the measurements. It has to be adjusted to yield optimal results for a class of data sets. For ? = 0
the TSVM becomes a conventional SVM.
2.5
Symmetrization of the weights
In order to ensure convergence, we suggest to use an inversion symmetric version K 2sym of
kernel K2 . For the pixel grid we can define the inversion operation as l + t ? l ? t, t ?
? . Taking the inverse of the vector yGl ,
N2 , ?l + t ? Gl , and denote the inverse of a by a
in which the set yGl is concatenated in an arbitrary but fixed order, leads to a reordering of
the components of the vector. The benefit from the chosen inversion symmetric kernel is
that the self consistency equations for the labels will turn out to be equivalent to a Hopfield
net, which has proven convergence properties. We define the new kernel as
K2sym (yGh , yGj ) =
1
(hyGh |yGj i + hyGh |?
yGj i).
q
(12)
Although only the second argument is inverted within the kernel, the value of this kernel
does not depend on the order of the arguments.
Proof It follows from the definition of the inversion operator and the dot product that
hyGh |yGj i = h?
yGh |?
yGj i = hyGj |yGh i = h?
yGj |?
yGh i and h?
yGh |yGj i = hyGh |?
yG j i =
hyGj |?
yGh i = h?
yGj |yGh i. Therefore,
K2sym (yGh , yGj )
=
=
1
1
(hyGh |yGj i + hyGh |?
yGj i) = (hyGj |yGh i + h?
yGj |yGh i)
q
q
1
(hyGj |yGh i + hyGj |?
yGh i) = K2sym (yGj , yGh )
q
Putting kernel (12) into eq.(7) and defining
? Pm
sym
j=1 ?j yv(j) (yv(j)+(k?h) + yv(j)?(k?h) )
q
wh,k =
0
: k ? Gh
: k?
/ Gh
.
(13)
we get
yh (? ) = sgn
X
k
sym
yk (? ? 1) ? wh,k
? ?h , ?h.
(14)
sym
Since the network weights wh,j
defined in eq.(13) are symmetric this corresponds to the
equation describing the dynamics during the retrieval phase of a Hopfield network [10].
Instead of taking the sum over all patterns, the sum is taken over all support vectors. The
weight between two neurons in the original Hopfield net corresponds to the correlation
between two components (over all fundamental patterns). In (13) the weight only depends
on the difference vector k-h between the two neurons on the 2D grid and is proportional to
the correlation (over all support vectors) between the label of a support vector and the label
in the distance k-h.
Table 1: Average misclassification rate R and the standard deviation of the mean ? at
optimal hyper-parameters C, S and ?.
algorithm
SVM
STSVM
TSVM
3
log2 C
4
4
2
log2 S
0.5
0.5
0.5
?
0
1.2
1.4
R[%]
2.23
1.96
1.86
?[%]
0.05
0.06
0.05
Experiments
We applied the above algorithms to the binary classification of pixel sites of cell images
from the CSSIP pap smear dataset. The goal was to assign the label +1 to the pixels belonging to the nucleus, and -1 to all others. The dataset contains three manual segmentations
of the nucleus? boundaries, from which we generated a ?ground truth? label for the area
of the nucleus using a majority voting. Only the first 300 images were used in the experiments. As a measurement vector we took a column-ordering of a 3x3 gray value patch
centered on a pixel site. In order to measure the classification performance for a noniid data set, we estimated the test error based on the collective classification of all pixels
in several randomly chosen test images. We compared three algorithms: A conventional
SVM, the ?TSVM? with the topographic kernel K2 from eq.(4) and the ?STSVM? with the
inversion symmetric topographic kernel K2sym from eq.(12). In the experiments we used
a label neighborhood of order 32, which corresponds to q = 100 neighbors. For kernel
K1 we used an RBF kernel K1 (x1 , x2 ) = exp(?kx1 ? x2 k2 /S 2 ) with hyper-parameter
S. Since the data set was very large, no cross-validation or resampling techniques were
required, and only a small subset of the available training data could be used for training.
We randomly sampled several disjoint training sets in order to improve the accuracy of
the error estimation. First, the hyper-parameters S and C (for TSVM and STSVM also ?)
were optimized via a grid search in parameter space. This was done by measuring the average test error over 20 test images and 5 training sets. Then, the test of the classifiers was
conducted at the in each case optimal hyper-parameters for 20 yet unused test images and
50 randomly sampled disjoint training sets. In all experiments using synchronous update
either a fixed point or an attractor cycle of length two was reached. The average number of
iterations needed was 12 (TSVM) and 13 (STSVM). Although the convergence properties
have only been formally proven for the symmetric weight STSVM, experimental evidence
suggests the same convergence properties for the TSVM. The results for synchronous update are shown in table 1 (results using asynchronous update differed only by 0.01%). The
performance of both topographic algorithms is superior to the conventional SVM, while
the TSVM performed slightly better than the STSVM. For the top-down method in [6] the
results were only qualitatively assessed by a human expert, not quantitatively compared to
a manual segmentation, therefore a direct comparison to our results was not possible. To
illustrate the role of the hyper-parameter ?, fig.1 shows 10 typical test images and their
segmentations achieved by an STSVM at different values of ? for fixed S and C. For
increasing ? the label images become less noisy, and at ? = 0.4 most artifacts have disappeared. This is caused by the increasing weight put on the label configuration via kernel
K2sym . Increasing ? even further will eventually lead to the appearance of spurious artifacts, as the influence of the label configuration will dominate the classification decision.
4
Conclusions
We have presented a classification method for a special case of non-iid data in which the
objects are linked by a regular grid structure. The proposed algorithm is composed of two
Figure 1: Final labels assigned by the STSVM at fixed hyper-parameters C = 2 6 , S = 22 .
(a) ? = 0, (b) ? = 0.1, (c) ? = 0.2, (d) ? = 0.3, (e) ? = 0.4.
components: The first part is a topographic kernel which integrates conventional feature
information and the information of the label configurations within a topographic neighborhood. The second part consists of a collective classification with recurrent neural network
dynamics which lets local label configurations converge to attractors determined by the
label configurations of the support vectors. For the asymmetric weight TSVM, the dimensionality of the problem is increased by the neighborhood size as compared to a conventional SVM (twice the neighborhood size for the symmetric weight STSVM). However,
the number of variables and constraints does not increase with the number of data points to
be labeled. Therefore, the TSVM and the STSVM can be applied to image segmentation
problems, where a large number of pixel labels have to be assigned simultaneously.
The algorithms were applied to the bottom-up cell nucleus segmentation in pap smear images needed for the detection of cervical cancer. The classification performance of the
TSVM and STSVM were compared to a conventional SVM, and it was shown that the inclusion of the topographic label configuration lead to a substantial decrease in the average
misclassification rate. The two topographic algorithms were much more resistant to noise
and smaller artifacts. A removal of artifacts which have similar size and the same measurement features as some of the nuclei cannot be achieved by a pure bottom-up method, as
this requires a priori model knowledge. In practice, the lower dimensional TSVM is to be
preferred over the STSVM, since it is faster and performed slightly better.
Acknowledgments
This work was funded by the BMBF (grant 01GQ0411). We thank Sepp Hochreiter for
useful discussions.
References
[1] J. Lafferty; A. McCallum; F. Pereira. Conditional random fields: Probabilistic models for
segmenting and labeling sequence data. In Proc. Int. Conf. on Machine Learning, 2001.
[2] S. Kumar; M. Hebert. Discriminative fields for modeling spatial dependencies in natural images. In Sebastian Thrun, Lawrence Saul, and Bernhard Scho? lkopf, editors, Advances in Neural
Information Processing Systems 16. MIT Press, Cambridge, MA, 2004.
[3] J. Neville and D. Jensen. Collective classification with relational dependency networks. In
Proc. 2nd Multi-Relational Data Mining Workshop, 9th ACM SIGKDD Intern. Conf. Knowledge
Discovery and Data Mining, 2003.
[4] B. Taskar, C. Guestrin, and D. Koller. Max-margin markov networks. In Sebastian Thrun,
Lawrence Saul, and Bernhard Scho? lkopf, editors, Advances in Neural Information Processing
Systems 16. MIT Press, Cambridge, MA, 2004.
[5] P. Bamford. Segmentation of Cell Images with Application to Cervical Cancer Screening. PhD
thesis, University of Queensland, 1999.
[6] P. Bamford and B. Lovell. Unsupervised cell nucleus segmentation with active contours. Signal
Processing Special Issue: Deformable Models and Techniques for Image and Signal Processing, 71(2):203?213, 1998.
[7] B. Sch?olkopf and A. Smola. Learning with Kernels. The MIT Press, 2002.
[8] V. Vapnik. Statistical Learning Theory. Springer, New York, 1998.
[9] W. McCulloch and W. Pitts. A logical calculus of the ideas immanent in nervous activity.
Bulletin of mathematical physics, 5:115?133, 1943.
[10] S. Haykin. Neural Networks. Macmillan College Publishing Company Inc., 1994.
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1,821 | 2,654 | Surface Reconstruction using
Learned Shape Models
Jan Erik Solem
School of Technology and Society
Malm?o University, Sweden
[email protected]
Fredrik Kahl
RSISE, Australian National University
ACT 0200, Australia
[email protected]
Abstract
We consider the problem of geometrical surface reconstruction from one
or several images using learned shape models. While humans can effortlessly retrieve 3D shape information, this inverse problem has turned out
to be difficult to perform automatically. We introduce a framework based
on level set surface reconstruction and shape models for achieving this
goal. Through this merging, we obtain an efficient and robust method for
reconstructing surfaces of an object category of interest.
The shape model includes surface cues such as point, curve and silhouette features. Based on ideas from Active Shape Models, we show how
both the geometry and the appearance of these features can be modelled
consistently in a multi-view context. The complete surface is obtained by
evolving a level set driven by a PDE, which tries to fit the surface to the
inferred 3D features. In addition, an a priori 3D surface model is used to
regularize the solution, in particular, where surface features are sparse.
Experiments are demonstrated on a database of real face images.
1 Introduction
Humans have a remarkable ability of perceiving 3D shape information - even from a single
photograph. Exactly how the human visual system works or how shape is represented is to
a large extent unknown. It is clear that the capabilities rely on a strong prior model and the
efficient use of different surface cues. The corresponding task of automatically recovering
a 3D surface model for a computer has turned out to be a challenging problem, even with
the addition of multiple images. In the present paper, we try to solve the problem in the
case of a single object category, more specifically, faces. At the heart of our approach lies
the combination of sophisticated surface reconstruction techniques and a strong statistical
model of surface features.
The first part concerns the statistical model of surface features for inferring 3D shape. The
features are primarily geometric ones, such as point, curve and silhouette features. Both the
geometric relations and their appearances (in terms of image intensities) are modelled. The
distributions are learned from real data. Also, a 3D model of the complete surface is used as
a weak regularizer where surface features are sparse. The motivations for introducing such
a model are several. We are interested in automatically recovering a surface model given
new image data of the object of interest. It is a hard problem to robustly extract curves and
apparent contours (i.e. silhouettes) without any a priori model. Moreover, many objects are
hard to reliably reconstruct due to specularities and illumination effects. By using distinct
geometric features and strong priors, we will still be able to obtain reliable results. Another
problem is textureless areas, and more generally, lack of information in the input. Our a
priori model will work as a domain-specific regularizer.
The second part of this work deals with fitting surfaces to points and curves in 3D space,
and at the same time, fitting the projections of surface contours to apparent contours in the
images. The approach taken here is a variational one - we define a functional which we will
try to minimize with respect to some parameters describing the geometry of the surfaces.
This variational problem leads to a surface evolution, driven by a PDE. The surface is
represented implicitly as the level set of a real-valued function [1].
1.1 Related Work
In the area of statistical shape models, our work is related to and inspired by Active Shape
Models (ASM) [2]. One distinction is that we will model both 2D and 3D data, while
ASM have mainly have been applied to 2D objects. In contrast to standard ASM, our
observations are from multiple views. In [3] a multi-view model is utilized, but no explicit
(or even consistent) 3D data is maintained within the model. In order to do inference on
our model, we have adopted the ideas of Probabilistic PCA (PPCA) [4].
In the seminal work [5], a complete 3D model is built from a database of 200 laser scans
of faces. The so-called morphable model can be fitted to an image with very impressive
results. The model itself is quite complex (approximately 70000 vertices), resulting in long
computation times. Although recent advances, their method still requires manual intervention [6]. A generic face model has also been used in [7] for computing regularized structure
and motion as well as in [8] based on silhouettes obtained from background subtraction.
In the area of using level set surface representations for fitting surfaces to 3D data, this paper
is related to the work in [9] where surfaces are fitted to points using a distance potential. In
[10] apparent contours were incorporated for level set based surface fitting. Surfaces can
also be estimated using 2D dense data, as in [11] based on photo-consistency. Shape priors
for level sets have previously been applied to segmentation, e.g. [12].
1.2 Contribution of the Paper
The main contribution of this paper is the approach itself - the combination of a state-of-theart surface reconstruction technique and the learned statistical model of surface features.
This merging results in a robust and efficient method for surface reconstruction without
any need for manual intervention. There is no need for an abundant number of images, in
fact, a single image is sufficient.
Another key contribution is the introduction of a multi-view feature model which is capable of representing both 2D and 3D data in a consistent manner. As the model is fully
probabilistic, the missing data problem can be handled in a natural way. By only incorporating distinct surface features, compared to a full morphable model, we not only gain
computational efficiency, but also robustness to specularities and other illumination effects.
This point is a also valid when compared to other surface reconstruction methods based on
image-correlation [11]. In the case of face modelling, this is known to cause problems due
to the complexity of the BRDF for human skin. The main contribution within the field of
level sets is in the incorporation of an a priori 3D model used for surface regularization.
2 Part I: A Learned Shape Model
In this section, we develop a statistical model which is needed in order to be able to automatically extract and compute 3D surface features from the input images. The output
from the model will serve as input to the level set reconstruction algorithm in order to get a
complete surface model. In addition, we will use an a priori 3D surface model as a (weak)
regularizer.
2.1 The Feature Model
Suppose we have a number of elements in a d-dimensional vector t, for example, a collection of 3D point coordinates. Suppose t can be related to some latent vector u of dimension
q where the relationship is linear:
t = W u + ?,
(1)
where W is a matrix of size d ? q and ? is d-vector allowing for non-zero mean. However, our measurements take place in the images, which is a non-linear function of the 3D
features according to the perspective camera model. Denote the projection function with
f : Rd ? Re , projecting all 3D features to 2D image features, for one or several images 1 .
Also, we need to change coordinate system of the 3D features to suit the actual projection function. Denote this mapping by T : Rd ? Rd . Thus, f (T (t)) will project all
normalized 3D data to all images.
Finally, a noise model needs to be specified. We assume that the image measurements
are independent and normally distributed, likewise, the latent variables are assumed to be
Gaussian with unit variance u ? N (0, I). Thus, in summary:
t2D = f (T (t)) + = f (T (W u + ?)) + ,
(2)
where ? N (0, ? 2 I) for some scalar ?.
Before the model can be used, its parameters need to be estimated from training data.
Given that it is a probabilistic model, this is best done with maximum likelihood (ML).
Suppose we are given n examples {t2D,i }ni=1 , the ML estimate for W and ? is obtained
by minimizing:
n
X
1
2
2
||(t2D,i ? f (Ti (W ui + ?))|| + ||ui || ,
(3)
?2
i=1
over all unknowns. The standard deviation ? is estimated a priori from the data. Once the
model parameters W and ? have been learned from examples, they are kept fix. In practice,
to minimize (3) we alternatively optimize over (W, ?) and the latent variables {u i }ni using
gradient descent. Initial estimates can be obtained by intersecting 3D structure from each
set of images and then applying standard PPCA algorithms for the linear part [4]. The
normalization Ti (?) is chosen such that each normalized 3D sample has zero mean and unit
variance.
A 3D point which is visible in m > 1 images will be represented in the vector t with its 3D
coordinates (X, Y, Z). For points visible in only one image, m = 1, no depth information
is available, and such points are represented similarly to apparent contour points. A curve
will be represented in the model by a number of points along the curve. In the training
of the model, it is important to parameterize each 3D curve such that each point on the
curve approximately corresponds to the same point on the corresponding curve in the other
examples. As for curves, we sample the apparent contours (in the images) using arc-length
parametrization. However, there is no 3D information available for the apparent contours
as they are view-dependent. A simple way is to treat contours points as 3D points with a
constant, approximate (but crude) depth estimate.
1
In the experiments, f (?) will model the projection of three calibrated perspective cameras.
2.2 The Grey-Level Model
The missing component in the model is the relationship between 2D image features and
the underlying grey-level (or color) values at these pixels. Again, we adopt a linear model
(PPCA). Using the same notation as in (1), but now with the subscript gl for grey-level, the
model can be written
tgl = Wgl ugl + ?gl + gl ,
where tgl is a vector containing the grey-level values of all the 2D image features and
gl is Gaussian noise in the measurements. In the training phase, each data sample of
grey-levels is normalized by subtracting the mean and scaling to unit variance. The MLestimate of Wgl and ?gl is computed with the EM-algorithm [4]. The complete statistical
two-layer model with one feature model and one grey-level model is very similar to the
concept of ASM [2]. In principle, the same techniques as used for ASM can be applied to
automatically compute the latent variables of the system, i.e. u and u gl .
2.3 The 3D Model
The two-layer feature model produces only a sparse set of features in 3D space. Even if
these cues are characteristic for a particular sample (or individual), it is often not enough
in order to infer a complete surface model, in particular, in regions where the features
are sparse. Therefore, we introduce a 3D surface model consisting of the complete mean
surface serving as domain-specific regularizer. The mean surface is obtained from laser
scans with the technique described in [13].
3 Part II: Surface Reconstruction
3.1 Level Set Formulation
Let x be a point in the open set ? ? R3 . The time dependent surface ?(t) is represented
implicitly as the zero level set of a function ?(x, t) : ? ? R+ ? R as
?(t) = {x ; ?(x, t) = 0} ,
(4)
where ? is defined such that ?(x, t) < 0 inside ? and ?(x, t) > 0 outside. Using the
definition above gives the outward unit normal n and the mean curvature ? as
n=
??
|??|
and ? = ? ?
??
.
|??|
(5)
One important, frequently used example is the signed distance function, where the requirement |??(x)| = 1 is imposed.
The zero set of ?(x, t) represents ?(t) at all times t. This means that ?(x(t), t) ? 0 for a
point on the curve x(t) ? ?(t). Differentiating with respect to t gives
?t + v ? ?? = 0 ? ?t + vn |??| = 0 ,
(6)
where v = dx(t)/dt and vn is the velocity normal to the surface. This PDE is solved
in order to move the surface according to some derived velocity v. For a more thorough
treatment of level set surfaces, see [1].
3.2 Surface Fitting to Points
In [9] surfaces are fitted to a point set S using the level set method. An initial surface is
deformed in the gradient direction of an energy functional which involves elastic energy
and potential energy. The energy is expressed using a distance potential as the surface
integral
Z
EP (?) =
d(x) d? ,
(7)
?
where ? is the surface, d? surface area and d(x) = dist(x, S) is the distance from the point
x to S. The gradient descent of (7) is, cf. [9],
?t = (?d(x) ? n + d(x)?) |??| ,
(8)
where n is the surface normal and ? the mean curvature.
This motion is known to interfere with surface regularization, since all surface points are
attracted to the 3D features. Therefore we cut the influence of the potential by setting
d(x) = min(d(x), dmax ).
3.3 Surface Fitting to Apparent Contours
Let ? be an apparent contour in an image parameterized as ?(s) : I ? R ? R 2 . The
back-projected cone, written in homogeneous coordinates,
C(s) = c + ?P + ?(s) ,
(9)
+
(where c is the camera center and P the pseudo-inverse of the 3 ? 4 camera matrix of
a perspective projection) should graze the surface at the location of the contour generator.
It is undesirable to attract the surface to the entire back-projected cone of the apparent
contour. The cone should only touch the surface along a curve - the so called contour
generator.
We propose to solve this in the following manner. For each point on the curve m = ?(s),
let x? denote the point closest to ? (If there are several, then choose the one with smallest
depth). The function ? is kept to be a signed distance function, i.e. |??| = 1. This means
that the point x? is easily found by checking the values of ? along the line of sight given
by (9). The set of these points, S 0 , will be a (possibly discontinuous) space curve ? ? (?, s).
This set is then added to the distance potential as
d(x) = min(d(x), d0 (x)) ,
(10)
0
0
where d (x) = dist(x, S ) is updated at appropriate intervals as the surface evolves.
3.4 Adding a 3D Shape Prior
Since the data, i.e. the points, curves and contours, are sparse it is customary to use a prior
for the regions where there is no information. Instead of the common choice of minimal
surface type models, we propose to use a learned shape model, as described in Section 2.3.
By first aligning the mean shape to the data, the deviation can be expressed similar to (7)
as
Z
EP rior (?) =
dP rior (x) d? ,
(11)
?
where dP rior (x) is the distance potential of the aligned mean shape.
3.5 The Combined Motion PDE
Adding all the components above gives a functional ET ot = EP +?EP rior , where ? ? R+
determines the weight of the prior shape. Combining the components (8), (10) and (11)
above leads to a PDE for the motion of the surface as
?t = [(?d(x) + ??dP rior (x)) ? n + (d(x) + ?dP rior (x))?]|??| .
(12)
This PDE is solved iteratively until a steady state is reached, which yields a (local) minimum of ET ot .
Figure 1: Extracted image features for one image triple and the reconstructed 3D surface.
4 Experiments
4.1 The Shape Model
All images were taken by a stereo setup with three (synchronized) cameras. The setup
was pre-calibrated for both extrinsic and intrinsic camera parameters. Example images are
given in Figures 1 and 2. In total a database of 28 image triplets were collected of faces
from different persons (23 males and 5 females of ages ranging from 7 to 65). 25 of these
were used for training and 3 for testing the two-layer feature model. The 3D mean surface
was computed from a database of 24 persons (different from the 28 persons above) using a
laser scanner as described in [13].
The 28 triplets in the training and test set were manually labelled - 36 points, 8 curves and
5 apparent contours were extracted for each person, see Figure 1. The two-layer model has
q = 12 elements in the latent variable u for the geometrical part and q gl = 15 elements
in ugl for the grey-level model. These numbers were found empirically in order to capture
most of the variations in the data. The model is able to adopt quite well to the test set. In
fact, even for one (frontal) input image of the test set, the model predicts the two profile
views remarkably well. As the pose over all faces is (essentially) constant for both test and
training images, the normalizing coordinate mapping T is restricted to scale and translation.
4.2 Surface Reconstruction
Once 3D data has been obtained by fitting the two-layer shape model to the input images,
surfaces are fitted to this data by solving the PDE (12). In the standard level set formulation,
the surface must be closed in the computational domain in order to make ? continuous. We
use the technique in [14] for initializing and evolving open implicit surfaces.
Surfaces were reconstructed for a number of persons, a selection is shown in Figures 1 and 2, where the zero set is visualized and triangulated using the marching cubes
algorithm . The value of ? was 0.3 and dmax was set from the maximum distance of the
feature points to the initial surface (4-5 voxels). For the reconstructions in Figure 2, the
mean and median distances (measured in voxel width) of the feature points to the reconstructed surface have been computed. See Figure 3 for a typical histogram and a table of
the results. Most points have sub-voxel distance to the surface, i.e. the deviation is of the
same order as the surface resolution. This shows that we are not fitting a surface to the
mean shape but that it really fits the feature data. The reader is encouraged to zoom in on
the reconstructions in Figure 2 to verify the quality.
5 Conclusions and Future Work
In this paper, a framework for reconstructing surface geometry based on learned shape
models and level sets has been presented. The approach relies solely on images as input
and the output of the system is a geometric 3D surface consistent with the input images. A
new regularization approach is introduced in the level set framework, where the distance to
an a priori 3D model is used instead of the common mean curvature regularization.
Figure 2: Input images and the reconstructed surface for three persons, two in the training
data and one (bottom) in the test data. For each person: One of the input images, triangulated surfaces and surfaces with texture. Note that the profiles above are different from the
ones in the input.
Histogram for distance to the surface for all 3D points
70
60
50
Person 1
Person 2
Person 3
40
30
20
10
0
0
mean
0.978
0.805
0.861
median
0.788
0.601
0.843
1
2
3
Distance in voxel width
Figure 3: Histogram of the deviations of the feature points from the surface for the second
person in Figure 2. The table displays mean and median deviations for all three persons.
Our current work focuses on incorporating a robust photo-consistency measure in the energy potential (7) to enable more detailed geometry. Also, the total number of faces is
quite small in the database and we will collect and label more images. Currently, only
images taken with the tri-stereo setup have been used with heads facing the middle camera (cf. Figure 1). Once the statistical model has been learned, it can be utilized for other
(nearby) poses as well, but to what extent is yet to be explored.
References
[1] J.A. Sethian. Level Set Methods and Fast Marching Methods Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge
University Press, 1999.
[2] T. F. Cootes and Taylor C. J. Active shape model search using local grey-level models: A
quantatitative evaluation. In British Machine Vision Conf., pages 639?648, 1993.
[3] T.F. Cootes, G.V. Wheeler, K.N. Walker, and C.J. Taylor. View-based active appearance models.
Image and Vision Computing, 20(9-10):657?664, 2002.
[4] M. E. Tipping and C. M. Bishop. Probabilistic principal component analysis. Phil. Trans. Royal
Soc. London B, 61(3):611?622, 1999.
[5] V. Blanz and T. Vetter. A morphable model for the synthesis of 3d faces. In SIGGRAPH, pages
187?194, 1999.
[6] S. Romdhani and T. Vetter. Efficient, robust and accurate fitting of a 3d morphable model. In
Int. Conf. Computer Vision, pages 59?66, Nice, France, 2003.
[7] P. Fua. Regularized bundle-adjustment to model heads from image sequences without calibration data. Int. J. Comput. Vision, 38(2):153?171, 2000.
[8] B. Moghaddam, J. Lee, H. Pfister, and R. Machiraju. Model-based 3d face capture with shapefrom-silhouettes. In IEEE International Workshop on Analysis and Modeling of Faces and
Gestures (AMFG), pages 20?27, 2003.
[9] H.K. Zhao, S. Osher, B. Merriman, and M. Kang. Implicit and non-parametric shape reconstruction from unorganized points using a variational level set method. In Computer Vision and
Image Understanding, pages 295?319, 2000.
[10] J.E. Solem and F. Kahl. Surface reconstruction from the projection of points, curves and contours. In 2nd Int. Symposium on 3D Data Processing, Visualization and Transmission, Thessaloniki, Greece, 2004.
[11] O. Faugeras and R. Keriven. Variational principles, surface evolution, PDEs, level set methods,
and the stereo problem. IEEE Transactions on Image Processing, 7(3):336?344, 1998.
[12] M. Rousson and N. Paragios. Shape priors for level set representations. In Proc. European
Conf. on Computer Vision, volume 2351 of Lecture Notes in Computer Science. Springer, 2002.
[13] K. Skoglund. Three-dimensional face modelling and analysis. Master?s thesis, Informatics and
Mathematical Modelling, Technical University of Denmark, DTU, 2003.
[14] J.E. Solem and A. Heyden. Reconstructing open surfaces from unorganized data points. In
International Conference on Computer Vision and Pattern Recognition, Washington DC, 2004.
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1,822 | 2,655 | Maximum-Margin Matrix Factorization
Nathan Srebro
Dept. of Computer Science
University of Toronto
Toronto, ON, CANADA
[email protected]
Jason D. M. Rennie
Tommi S. Jaakkola
Computer Science and Artificial Intelligence Lab
Massachusetts Institute of Technology
Cambridge, MA, USA
jrennie,[email protected]
Abstract
We present a novel approach to collaborative prediction, using low-norm
instead of low-rank factorizations. The approach is inspired by, and has
strong connections to, large-margin linear discrimination. We show how
to learn low-norm factorizations by solving a semi-definite program, and
discuss generalization error bounds for them.
1
Introduction
Fitting a target matrix Y with a low-rank matrix X by minimizing the sum-squared error is
a common approach to modeling tabulated data, and can be done explicitly in terms of the
singular value decomposition of Y . It is often desirable, though, to minimize a different
loss function: loss corresponding to a specific probabilistic model (where X are the mean
parameters, as in pLSA [1], or the natural parameters [2]); or loss functions such as hinge
loss appropriate for binary or discrete ordinal data. Loss functions other than squared-error
yield non-convex optimization problems with multiple local minima. Even with a squarederror loss, when only some of the entries in Y are observed, as is the case for collaborative
filtering, local minima arise and SVD techniques are no longer applicable [3].
Low-rank approximations constrain the dimensionality of the factorization X = U V 0 .
Other constraints, such as sparsity and non-negativity [4], have also been suggested for
better capturing the structure in Y , and also lead to non-convex optimization problems.
In this paper we suggest regularizing the factorization by constraining the norm of U and
V ?constraints that arise naturally when matrix factorizations are viewed as feature learning for large-margin linear prediction (Section 2). Unlike low-rank factorizations, such
constraints lead to convex optimization problems that can be formulated as semi-definite
programs (Section 4). Throughout the paper, we focus on using low-norm factorizations
for ?collaborative prediction?: predicting unobserved entries of a target matrix Y , based on
a subset S of observed entries YS . In Section 5, we present generalization error bounds for
collaborative prediction using low-norm factorizations.
2
Matrix Factorization as Feature Learning
Using a low-rank model for collaborative prediction [5, 6, 3] is straightforward: A lowrank matrix X is sought that minimizes a loss versus the observed entries YS . Unobserved
entries in Y are predicted according to X. Matrices of rank at most k are those that can
be factored into X = U V 0 , U ? Rn?k , V ? Rm?k , and so seeking a low-rank matrix is
equivalent to seeking a low-dimensional factorization.
If one of the matrices, say U , is fixed, and only the other matrix V 0 needs to be learned, then
fitting each column of the target matrix Y is a separate linear prediction problem. Each row
of U functions as a ?feature vector?, and each column of V 0 is a linear predictor, predicting
the entries in the corresponding column of Y based on the ?features? in U .
In collaborative prediction, both U and V are unknown and need to be estimated. This can
be thought of as learning feature vectors (rows in U ) for each of the rows of Y , enabling
good linear prediction across all of the prediction problems (columns of Y ) concurrently,
each with a different linear predictor (columns of V 0 ). The features are learned without
any external information or constraints which is impossible for a single prediction task (we
would use the labels as features). The underlying assumption that enables us to do this in
a collaborative filtering situation is that the prediction tasks (columns of Y ) are related, in
that the same features can be used for all of them, though possibly in different ways.
Low-rank collaborative prediction corresponds to regularizing by limiting the dimensionality of the feature space?each column is a linear prediction problem in a low-dimensional
space. Instead, we suggest allowing an unbounded dimensionality for the feature space, and
regularizing by requiring a low-norm factorization, while predicting with large-margin.
Consider adding to the loss a penalty term which is the sum of squares of entries in U and
2
2
V , i.e. kU kFro + kV kFro (kkFro denotes the Frobenius norm). Each ?conditional? problem
(fitting U given V and vice versa) again decomposes into a collection of standard, this time
regularized, linear prediction problems. With an appropriate loss function, or constraints
on the observed entries, these correspond to large-margin linear discrimination problems.
For example, if we learn a binary observation matrix by minimizing a hinge loss plus such
a regularization term, each conditional problem decomposes into a collection of SVMs.
3
Maximum-Margin Matrix Factorizations
Matrices with a factorization X = U V 0 , where U and V have low Frobenius norm (recall
that the dimensionality of U and V is no longer bounded!), can be characterized in several
equivalent ways, and are known as low trace norm matrices:
Definition 1. The trace norm1 kXk? is the sum of the singular values of X.
2
2
Lemma 1. kXk? = minX=U V 0 kU kFro kV kFro = minX=U V 0 12 (kU kFro + kV kFro )
The characterization in terms of the singular value decomposition allows us to characterize
low trace norm matrices as the convex hull of bounded-norm rank-one matrices:
n
o
2
2
Lemma 2. {X | kXk? ? B} = conv uv 0 |u ? Rn , v ? Rm , |u| = |v| = B
In particular, the trace norm is a convex function, and the set of bounded trace norm matrices is a convex set. For convex loss functions, seeking a bounded trace norm matrix
minimizing the loss versus some target matrix is a convex optimization problem.
This contrasts sharply with minimizing loss over low-rank matrices?a non-convex problem. Although the sum-squared error versus a fully observed target matrix can be minimized efficiently using the SVD (despite the optimization problem being non-convex!),
minimizing other loss functions, or even minimizing a squared loss versus a partially observed matrix, is a difficult optimization problem with multiple local minima [3].
1
Also known as the nuclear norm and the Ky-Fan n-norm.
In fact, the trace norm has been suggested as a convex surrogate to the rank for various
rank-minimization problems [7]. Here, we justify the trace norm directly, both as a natural
extension of large-margin methods and by providing generalization error bounds.
To simplify presentation, we focus on binary labels, Y ? {?1}n?m . We consider hardmargin matrix factorization, where we seek a minimum trace norm matrix X that matches
the observed labels with a margin of one: Yia Xia ? 1 for all ia ? S. We also consider
soft-margin learning, where we minimize a trade-off between the trace norm of X and its
hinge-loss relative to YS :
X
minimize kXk? + c
max(0, 1 ? Yia Xia ).
(1)
ia?S
As in maximum-margin linear discrimination, there is an inverse dependence between the
norm and the margin. Fixing the margin and minimizing the trace norm is equivalent
to fixing the trace norm and maximizing the margin. As in large-margin discrimination
with certain infinite dimensional (e.g. radial)
pkernels, the data is always separable with
sufficiently high trace norm (a trace norm of n|S| is sufficient to attain a margin of one).
The max-norm variant Instead of constraining the norms of rows in U and V on average, we can constrain all rows of U and V to have small L2 norm, replacing the trace norm
with kXkmax = minX=U V 0 (maxi |Ui |)(maxa |Va |) where Ui , Va are rows of U, V . Lowmax-norm discrimination has a clean geometric interpretation. First, note that predicting
the target matrix with the signs of a rank-k matrix corresponds to mapping the ?items?
(columns) to points in Rk , and the ?users? (rows) to homogeneous hyperplanes, such that
each user?s hyperplane separates his positive items from his negative items. Hard-margin
low-max-norm prediction corresponds to mapping the users and items to points and hyperplanes in a high-dimensional unit sphere such that each user?s hyperplane separates his
positive and negative items with a large-margin (the margin being the inverse of the maxnorm).
4
Learning Maximum-Margin Matrix Factorizations
In this section we investigate the optimization problem of learning a MMMF, i.e. a low
norm factorization U V 0 , given a binary target matrix. Bounding the trace norm of U V 0 by
2
2
1
2 (kU kFro + kV kFro ), we can characterize the trace norm in terms of the trace of a positive
semi-definite matrix:
Lemma 3 ([7, Lemma 1]). For any X ? Rn?m
and t ? R: kXk? ? t iff there exists
A X
n?n
m?m
2
A?R
and B ? R
such that X 0 B < 0 and tr A + tr B ? 2t.
Proof. Note that for any matrix W , kW kFro = tr W W 0 . If XA0 X
< 0, we can write it as
B
2
2
1
0
U
0
0
a product [ V ] [ U V ]. We have X = U V and 2 (kU kFro + kV kFro ) = 21 (tr A + tr B) ? t,
0
establishing kXk? ? t. Conversely, if kXk? ? t we can write
it as X = U V with
0
0
0
U
U
X
tr U U + tr V V ? 2t and consider the p.s.d. matrix X 0 V V 0 .
Lemma 3 can be used in order to formulate minimizing the trace norm as a semi-definite
optimization problem (SDP). Soft-margin matrix factorization (1), can be written as:
X
yia Xia ? 1 ? ?ia
1
A X
min (tr A + tr B) + c
?ia s.t.
< 0,
?ia ? S (2)
0
X B
?ia ? 0
2
ia?S
2
A < 0 denotes A is positive semi-definite
Associating a dual variable Qia with each constraint on Xia , the dual of (2) is [8, Section
5.4.2]:
X
I
(?Q ? Y )
max
Qia s.t.
< 0, 0 ? Qia ? c
(3)
(?Q ? Y )0
I
ia?S
where Q ? Y denotes the sparse matrix (Q ? Y )ia = Qia Yia for ia ? S and zeros
elsewhere. The problem is strictly feasible, and there is no duality gap. The p.s.d. constraint
in the dual (3) is equivalent to bounding the spectral norm of Q ? Y , and the dual can also
be written as an optimization problem subject to a bound on the spectral norm, i.e. a bound
on the singular values of Q ? Y :
X
kQ ? Y k2 ? 1
max
Qia s.t.
(4)
0 ? Qia ? c ?ia ? S
ia?S
In typical collaborative prediction problems, we observe only a small fraction of the entries
in a large target matrix. Such a situation translates to a sparse dual semi-definite program,
with the number of variables equal to the number of observed entries. Large-scale SDP
solvers can take advantage of such sparsity.
The prediction matrix X ? minimizing (1) is part of the primal optimal solution of (2), and
can be extracted from it directly. Nevertheless, it is interesting to study how the optimal
prediction matrix X ? can be directly recovered from a dual optimal solution Q? alone.
Although unnecessary when relying on interior point methods used by most SDP solvers (as
these return a primal/dual optimal pair), this can enable us to use specialized optimization
methods, taking advantage of the simple structure of the dual.
Recovering X ? from Q? As for linear programming, recovering a primal optimal solution directly from a dual optimal solution is not always possible for SDPs. However, at
least for the hard-margin problem (no slack) this is possible, and we describe below how
an optimal prediction matrix X ? can be recovered from a dual optimal solution Q? by
calculating a singular value decomposition and solving linear equations.
Given a dual optimal Q? , consider its singular value decomposition Q? ? Y = U ?V 0 .
Recall that all singular values of Q? ?Y are bounded by one, and consider only the columns
? ? Rn?p of U and V? ? Rm?p of V with singular value one. It is possible to show [8,
U
Section 5.4.3], using complimentary slackness, that for some matrix R ? Rp?p , X ? =
? RR0 V? 0 is an optimal solution to the maximum margin matrix factorization problem (1).
U
Furthermore, p(p+1)
is bounded above by the number of non-zero Q?ia . When Q?ia > 0,
2
and assuming hard-margin constraints, i.e. no box constraints in the dual, complimentary
?
?i RR0 V? 0 = Yia , providing us with a linear equation on
slackness dictates that Xia
= U
a
p(p+1)
0
the
entries
in
the
symmetric
RR
. For hard-margin matrix factorization, we can
2
0
therefore recover the entries of RR by solving a system of linear equations, with a number
of variables bounded by the number of observed entries.
Recovering specific entries The approach described above requires solving a large system of linear equations (with as many variables as observations). Furthermore, especially
when the observations are very sparse (only a small fraction of the entries in the target
matrix are observed), the dual solution is much more compact then the prediction matrix:
the dual involves a single number for each observed entry. It might be desirable to avoid
storing the prediction matrix X ? explicitly, and calculate a desired entry Xi?0 a0 , or at least
its sign, directly from the dual optimal solution Q? .
Consider adding the constraint Xi0 a0 > 0 to the primal SDP (2). If there exists an optimal
solution X ? to the original SDP with Xi?0 a0 > 0, then this is also an optimal solution to
the modified SDP, with the same objective value. Otherwise, the optimal solution of the
modified SDP is not optimal for the original SDP, and the optimal value of the modified
SDP is higher (worse) than the optimal value of the original SDP.
Introducing the constraint Xi0 a0 > 0 to the primal SDP (2) corresponds to introducing a
new variable Qi0 a0 to the dual SDP (3), appearing in Q ? Y (with Yi0 a0 = 1) but not in the
objective. In this modified dual, the optimal solution Q? of the original dual would always
be feasible. But, if Xi?0 a0 < 0 in all primal optimal solutions, then the modified primal
SDP has a higher value, and so does the dual, and Q? is no longer optimal for the new dual.
By checking the optimality of Q? for the modified dual, e.g. by attempting to re-optimize
it, we can recover the sign of Xi?0 a0 .
We can repeat this test once with Yi0 a0 = 1 and once with Yi0 a0 = ?1, corresponding
to Xi0 a0 < 0. If Yi0 a0 Xi?0 a0 < 0 (in all optimal solutions), then the dual solution can be
improved by introducing Qi0 a0 with a sign of Yi0 a0 .
Predictions for new users So far, we assumed that learning is done on the known entries
in all rows. It is commonly desirable to predict entries in a new partially observed row of
Y (a new user), not included in the original training set. This essentially requires solving
a ?conditional? problem, where V is already known, and a new row of U is learned (the
predictor for the new user) based on a new partially observed row of X. Using maximummargin matrix factorization, this is a standard SVM problem.
Max-norm MMMF as a SDP The max-norm variant can also be written as a SDP, with
the primal and dual taking the forms:
min t + c
X
?ia s.t.
ia?S
max
X
ia?S
5
Qia s.t.
A
X0
?
(?Q ? Y )0
X
<0
B
Aii , Baa ? t ?i, a
yia Xia ? 1 ? ?ia
?ia ? S
?ia ? 0
(?Q ? Y )
<0
?
?, ? are diagonal
tr ? + tr ? = 1
0 ? Qia ? c ?ia ? S
(5)
(6)
Generalization Error Bounds for Low Norm Matrix Factorizations
Similarly to standard feature-based prediction approaches, collaborative prediction methods can also be analyzed in terms of their generalization ability: How confidently can we
predict entries of Y based on our error on the observed entries YS ? We present here generalization error bounds that holds for any target matrix Y , and for a random subset of
observations S, and bound the average error across all entries in terms of the observed
margin error3 . The central assumption, paralleling the i.i.d. source assumption for standard
feature-based prediction, is that the observed subset S is picked uniformly at random.
Theorem 4. For all target matrices Y ? {?1}n?m and sample sizes |S| > n log n, and
for a uniformly selected sample S of |S| entries in Y , with probability at least 1 ? ? over
3
The bounds presented here are special cases of bounds for general loss functions that we present
and prove elsewhere [8, Section 6.2]. To prove the bounds we bound the Rademacher complexity of
bounded trace norm and bounded max-norm matrices (i.e. balls w.r.t. these norms). The unit trace
norm ball is the convex hull of outer products of unit norm vectors. It is therefore enough to bound
the Rademacher complexity of such outer products, which boils down to analyzing the spectral norm
of random matrices. As a consequence of Grothendiek?s inequality, the unit max-norm ball is within
a factor of two of the convex hull of outer products of sign vectors. The Rademacher complexity of
such outer products can be bounded by considering their cardinality.
the sample selection, the following holds for all matrices X ? Rn?m and all ? > 0:
1
1
|{ia|Xia Yia ? 0}| <
|{ia ? S|Xia Yia ? ?}|+
nm
|S|
s
s
s
ln(1 + | log kXk? /?|)
kXk? ?
(n + m) ln n
ln(4/?)
4
ln m
K ?
+
+
|S|
|S|
2|S|
? nm
(7)
and
1
1
|{ia|Xia Yia ? 0}| <
|{ia ? S|Xia Yia ? ?}|+
nm
|S|
s
s
r
kXkmax n + m
ln(1 + | log kXk? /?|)
ln(4/?)
12
+
+
?
|S|
|S|
2|S|
(8)
Where K is a universal constant that does not depend on Y ,n,m,? or any other quantity.
To understand the scaling of these bounds, consider n ? m matrices X = U V 0 where
the norms of rows of U and V are bounded by?
r, i.e. matrices with kXkmax ? r2 . The
2
trace norm of such matrices is bounded by r / nm, and so the two bounds agree up to
log-factors?the cost of allowing the norm to be low on-average but not uniformly. Recall
that the conditional problem, where V is fixed and only U is learned, is a collection of
low-norm (large-margin) linear prediction problems. When the norms of rows in U and V
are bounded by r, aqsimilar generalization error bound on the conditional problem would
2
n
include the term r? |S|
, matching the bounds of Theorem 4 up to log-factors?learning
both U and V does not introduce significantly more error than learning just one of them.
Also
? of interest is the comparison with bounds for low-rank matrices, for which kXk? ?
?rank X kXkFro . In particular, for n ? m rank-k X with entries bounded by B, kXk? ?
knmB, and the second term in the right-hand side of (7) becomes:
s
B?
k(n + m) ln n
4
K
ln m
(9)
?
|S|
Although this is the best (up to log factors) that can be expected from scale-sensitive
bounds4 , taking a combinatorial approach, the dependence on the magnitude of the entries
in X (and the margin) can be avoided [9].
6
Implementation and Experiments
Ratings In many collaborative prediction tasks, the labels are not binary, but rather are
discrete ?ratings? in several ordered levels (e.g. one star through five stars). Separating R
levels by thresholds ?? = ?0 < ?1 < ? ? ? < ?R = ?, and generalizing hard-margin
constraints for binary labels, one can require ?Yia + 1 ? Xia ? ?Yia +1 ? 1. A soft-margin
version of these constraints, with slack variables for the two constraints on each observed
rating, corresponds to a generalization of the hinge loss which is a convex bound on the
zero/one level-agreement error (ZOE) [10]. To obtain a loss which is a convex bound on
the mean-absolute-error (MAE?the difference, in levels, between the predicted level and
the true level), we introduce R ? 1 slack variables for each observed rating?one for each
4
For general loss functions, bounds as in Theorem 4 depend only on the Lipschitz constant of
the loss, and (9) is the best (up to log factors) that can be achieved without explicitly bounding the
magnitude of the loss function.
of the R ? 1 constraints Xia ? ?r for r < Yia and Xia ? ?r for r ? Yia . Both of
these soft-margin problems (?immediate-threshold? and ?all-threshold?) can be formulated
as SDPs similar to (2)-(3). Furthermore, it is straightforward to learn also the thresholds
(they appear as variables in the primal, and correspond to constraints in the dual)?either
a single set of thresholds for the entire matrix, or a separate threshold vector for each row
of the matrix (each ?user?). Doing the latter allows users to ?use ratings differently? and
alleviates the need to normalize the data.
Experiments We conducted preliminary experiments on a subset of the 100K MovieLens
Dataset5 , consisting of the 100 users and 100 movies with the most ratings. We used CSDP
[11] to solve the resulting SDPs6 . The ratings are on a discrete scale of one through five,
and we experimented with both generalizations of the hinge loss above, allowing per-user
thresholds. We compared against WLRA and K-Medians (described in [12]) as ?Baseline?
learners. We randomly split the data into four sets. For each of the four possible test sets,
we used the remaining sets to calculate a 3-fold cross-validation (CV) error for each method
(WLRA, K-medians, trace norm and max-norm MMMF with immediate-threshold and allthreshold hinge loss) using a range of parameters (rank for WLRA, number of centers for
K-medians, slack cost for MMMF). For each of the four splits, we selected the two MMMF
learners with lowest CV ZOE and MAE and the two Baseline learners with lowest CV ZOE
and MAE, and measured their error on the held-out test data. Table 1 lists these CV and
test errors, and the average test error across all four test sets. On average and on three of
the four test sets, MMMF achieves lower MAE than the Baseline learners; on all four of
the test sets, MMMF achieves lower ZOE than the Baseline learners.
Test
Set
1
2
3
4
Method
WLRA rank 2
WLRA rank 2
WLRA rank 1
WLRA rank 2
Avg.
1
2
3
4
max-norm C=0.0012
trace norm C=0.24
max-norm C=0.0012
max-norm C=0.0012
Avg.
ZOE
CV
Test
0.547 0.575
0.550 0.562
0.562 0.543
0.557 0.553
0.558
0.543 0.562
0.550 0.552
0.551 0.527
0.544 0.550
0.548
Method
K-Medians K=2
K-Medians K=2
K-Medians K=2
K-Medians K=2
max-norm C=0.0012
max-norm C=0.0011
max-norm C=0.0012
max-norm C=0.0012
MAE
CV
Test
0.678 0.691
0.686 0.681
0.700 0.681
0.685 0.696
0.687
0.669 0.677
0.675 0.683
0.668 0.646
0.667 0.686
0.673
Table 1: Baseline (top) and MMMF (bottom) methods and parameters that achieved the
lowest cross validation error (on the training data) for each train/test split, and the error for
this predictor on the test data. All listed MMMF learners use the ?all-threshold? objective.
7
Discussion
Learning maximum-margin matrix factorizations requires solving a sparse semi-definite
program. We experimented with generic SDP solvers, and were able to learn with up to
tens of thousands of labels. We propose that just as generic QP solvers do not perform
well on SVM problems, special purpose techniques, taking advantage of the very simple
structure of the dual (3), are necessary in order to solve large-scale MMMF problems.
SDPs were recently suggested for a related, but different, problem: learning the features
5
http://www.cs.umn.edu/Research/GroupLens/
Solving with immediate-threshold loss took about 30 minutes on a 3.06GHz Intel Xeon.
Solving with all-threshold loss took eight to nine hours. The MATLAB code is available at
www.ai.mit.edu/?nati/mmmf
6
(or equivalently, kernel) that are best for a single prediction task [13]. This task is hopeless
if the features are completely unconstrained, as they are in our formulation. Lanckriet et al
suggest constraining the allowed features, e.g. to a linear combination of a few ?base feature spaces? (or base kernels), which represent the external information necessary to solve
a single prediction problem. It is possible to combine the two approaches, seeking constrained features for multiple related prediction problems, as a way of combining external
information (e.g. details of users and of items) and collaborative information.
An alternate method for introducing external information into our formulation is by adding
to U and/or V additional fixed (non-learned) columns representing the external features.
This method degenerates to standard SVM learning when Y is a vector rather than a matrix.
An important limitation of the approach we have described, is that observed entries are
assumed to be uniformly sampled. This is made explicit in the generalization error bounds.
Such an assumption is typically unrealistic, as, e.g., users tend to rate items they like. At
an extreme, it is often desirable to make predictions based only on positive samples. Even
in such situations, it is still possible to learn a low-norm factorization, by using appropriate
loss functions, e.g. derived from probabilistic models incorporating the observation process. However, obtaining generalization error bounds in this case is much harder. Simply
allowing an arbitrary sampling distribution and calculating the expected loss based on this
distribution (which is not possible with the trace norm, but is possible with the max-norm
[8]) is not satisfying, as this would guarantee low error on items the user is likely to want
anyway, but not on items we predict he would like.
Acknowledgments We would like to thank Sam Roweis for pointing out [7].
References
[1] T. Hofmann. Unsupervised learning by probabilistic latent semantic analysis. Machine Learning Journal, 42(1):177?196, 2001.
[2] M. Collins, S. Dasgupta, and R. Schapire. A generalization of principal component analysis to
the exponential family. In Advances in Neural Information Processing Systems 14, 2002.
[3] Nathan Srebro and Tommi Jaakkola. Weighted low rank approximation. In 20th International
Conference on Machine Learning, 2003.
[4] D.D. Lee and H.S. Seung. Learning the parts of objects by non-negative matrix factorization.
Nature, 401:788?791, 1999.
[5] T. Hofmann. Latent semantic models for collaborative filtering. ACM Trans. Inf. Syst.,
22(1):89?115, 2004.
[6] Benjamin Marlin. Modeling user rating profiles for collaborative filtering. In Advances in
Neural Information Processing Systems, volume 16, 2004.
[7] Maryam Fazel, Haitham Hindi, and Stephen P. Boyd. A rank minimization heuristic with application to minimum order system approximation. In Proceedings American Control Conference,
volume 6, 2001.
[8] Nathan Srebro. Learning with Matrix Factorization. PhD thesis, Massachusetts Institute of
Technology, 2004.
[9] N. Srebro, N. Alon, and T. Jaakkola. Generalization error bounds for collaborative prediction
with low-rank matrices. In Advances In Neural Information Processing Systems 17, 2005.
[10] Amnon Shashua and Anat Levin. Ranking with large margin principle: Two approaches. In
Advances in Neural Information Proceedings Systems, volume 15, 2003.
[11] B. Borchers. CSDP, a C library for semidefinite programming. Optimization Methods and
Software, 11(1):613?623, 1999.
[12] B. Marlin. Collaborative filtering: A machine learning perspective. Master?s thesis, University
of Toronto, 2004.
[13] G. Lanckriet, N. Cristianini, P. Bartlett, L. El Ghaoui, and M. Jordan. Learning the kernel matrix
with semidefinite programming. Journal of Machine Learning Research, 5:27?72, 2004.
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1,823 | 2,656 | Outlier Detection with One-class Kernel Fisher
Discriminants
Volker Roth
ETH Zurich, Institute of Computational Science
Hirschengraben 84, CH-8092 Zurich
[email protected]
Abstract
The problem of detecting ?atypical objects? or ?outliers? is one of the
classical topics in (robust) statistics. Recently, it has been proposed to
address this problem by means of one-class SVM classifiers. The main
conceptual shortcoming of most one-class approaches, however, is that in
a strict sense they are unable to detect outliers, since the expected fraction
of outliers has to be specified in advance. The method presented in this
paper overcomes this problem by relating kernelized one-class classification to Gaussian density estimation in the induced feature space. Having
established this relation, it is possible to identify ?atypical objects? by
quantifying their deviations from the Gaussian model. For RBF kernels
it is shown that the Gaussian model is ?rich enough? in the sense that it
asymptotically provides an unbiased estimator for the true density. In order to overcome the inherent model selection problem, a cross-validated
likelihood criterion for selecting all free model parameters is applied.
1
Introduction
A one-class-classifier attempts to find a separating boundary between a data set and the
rest of the feature space. A natural application of such a classifier is estimating a contour
line of the underlying data density for a certain quantile value. Such contour lines may
be used to separate ?typical? objects from ?atypical? ones. Objects that look ?sufficiently
atypical? are often considered to be outliers, for which one rejects the hypothesis that they
come from the same distribution as the majority of the objects. Thus, a useful application
scenario would be to find a boundary which separates the jointly distributed objects from
the outliers. Finding such a boundary defines a classification problem in which, however,
usually only sufficiently many labeled samples from one class are available. Usually no
labeled samples from the outlier class are available at all, and it is even unknown if there
are any outliers present.
It is interesting to notice that the approach of directly estimating a boundary, as opposed to
first estimating the whole density, follows one of the main ideas in learning theory which
states that one should avoid solving a too hard intermediate problem. While this line of reasoning seems to be appealing from a theoretical point of view, it leads to a severe problem
in practical applications: when it comes to detecting outliers, the restriction to estimating
only a boundary makes it impossible to derive a formal characterization of outliers without prior assumptions on the expected fraction of outliers or even on their distribution. In
practice, however, any such prior assumptions can hardly be justified. The fundamental
problem of the one-class approach lies in the fact that outlier detection is a (partially) unsupervised task which has been ?squeezed? into a classification framework. The missing part
of information has been shifted to prior assumptions which can probably only be justified,
if the solution of the original problem was known in advance.
This paper aims at overcoming this problem by linking kernel-based one-class classifiers
to Gaussian density estimation in the induced feature space. Objects which have an ?unexpected? high Mahalanobis distance to the sample mean are considered as ?atypical objects?
or outliers. A particular Mahalanobis distance is considered to be unexpected, if it is very
unlikely to observe an object that far away from the mean vector in a random sample of
a certain size. We will formalize this concept in section 3 by way of fitting linear models
in quantile-quantile plots. The main technical ingredient of our method is the one-class
kernel Fisher discriminant classifier (OC-KFD), for which the relation to Gaussian density
estimation is shown. From the classification side, the OC-KFD-based model inherits the
simple complexity control mechanism by using regularization techniques. The explicit relation to Gaussian density estimation, on the other hand, makes it possible to formalize the
notion of atypical objects by observing deviations from the Gaussian model. It is clear that
these deviations will heavily depend on the chosen model parameters. In order to derive
an objective characterization of atypical objects it is, thus, necessary to select a suitable
model in advance. This model-selection problem is overcome by using a likelihood-based
cross-validation framework for inferring the free parameters.
2
Gaussian density estimation and one-class LDA
Let X denote the n ? d data matrix which contains the n input vectors xi ? Rd as rows. It
has been proposed to estimate a one-class decision boundary by separating the dataset from
the origin [12], which effectively coincides with replicating all xi with opposite sign and
separating X and ?X. Typically, a ?-SVM classifier with RBF kernel function is used.
The parameter ? bounds the expected number of outliers and must be selected a priori. The
method proposed here follows the same idea of separating the data from their negatively
replicated counterparts. Instead of a SVM, however, a Kernel Fisher Discriminant (KFD)
classifier is used [7, 10]. The latter has the advantage that is is closely related to Gaussian
density estimation in the induced feature space. By making this relation explicit, outliers
can be identified without specifying the expected fraction of outliers in advance. We start
with a linear discriminant analysis (LDA) model, and then kernels will be introduced.
Let X 0 = (X, ?X)> denote the augmented (2n ? d) data matrix which also contains
the negative
P samples ?xi . Without loss of generality we assume that the sample mean
?+ ?
i xi > 0, so that the sample means of the positive data and the negative data
differ: ?+ 6= ?? . Let us now assume that our data are realizations of a normally distributed
random variable in d dimensions: X ? Nd (?, ?). Denoting by X c the centered data
? ? W = (1/n)X c> X c .
matrix, the estimator for ? takes the form ?
>
The LDA solution ? ? maximizes the between-class scatter ? >
? B? ? with B = ?+ ?+ +
>
>
?? ?? under the constraint on the within-class scatter ? ? W ? ? = 1. Note that in our
special case with X 0 = (X, ?X)> the usual pooled within-class matrix W simply reduces
to the above defined W = (1/n)X c> X c . Denoting by y 0 = (2, . . . , 2, ?2, . . . , ?2)> a
2n-indicator vector for class membership in class ?+? or ???, it is well-known (see e.g. [1])
that the LDA solution (up to a scaling factor) can be found by minimizing a least-squares
? = arg min? ky 0 ?X 0 ?k2 . In [3] a slightly more general form of the problem
functional: ?
is described where the above functional is minimized under a constrained on ?, which in
the simplest case amounts to adding a term ?? > ? to the functional. Such a ridge regression
model assumes a penalized total covariance of the form T = (1/(2n)) ? X 0> X 0 + ?I =
?
(1/n) ? X > X + ?I. Defining an n-vector of ones y = (1, . . . , 1)> , the solution vector ?
reads
? = (X 0> X 0 + ?I)?1 X 0> y 0 = (X > X + ?I)?1 X > y.
?
(1)
2
According to [3], an appropriate scaling factor is defied in terms of the quantity s =
? which leads us to the correctly scaled LDA vector ? ? =
(1/n) ? y > y? = (1/n) ? y > X ?,
?1
2 ?1/2 ?
s (1 ? s )
? that fulfills the normalization condition ? >
? W ? ? = 1.
One further derives from [3] that the mean vector of X, projected onto the 1-dimensional
LDA-subspace has the coordinate value m+ = s(1 ? s2 )?1/2 , and that the Mahalanobis
distance from a vector x to the sample mean ?+ is the sum of the squared Euclidean
distance in the projected space and an orthogonal distance term:
>
2
2
2
> ?1
D(x, ?+ ) = (? >
x.
? x ? m+ ) + D? with D? = ?(1 ? s )(? ? x) + x T
(2)
Note that it is the term D? which makes the density estimation model essentially different
from OC-classification: while the latter considers only distances in the direction of the
projection vector ?, the true density model additionally takes into account the distances in
the orthogonal subspace.
Since the assumption X ? Nd (?, ?) is very restrictive, we propose to relax it by assuming
that we have found a suitable transformation of our input data ? : Rd 7? Rp , x 7? ?(x),
such that the transformed data are Gaussian in p dimensions. If the transformation is carried
out implicitly by introducing a Mercer kernel k(xi , xj ), we arrive at an equivalent problem
in terms of the kernel matrix K = ??> and the expansion coefficients ?:
?
? = (K + ?I)?1 y.
(3)
From [11] it follows that the mapped vectors can be represented in Rn as ?(x) =
K ?1/2 k(x), where k(x) denotes the kernel vector k(x) = (k(x, x1 ), . . . , k(x, xn ))> .
Finally we derive the following form of the Mahalanobis distances which again consist of
the Euclidean distance in the classification subspace plus an orthogonal term:
2
2
>
2
D(x, ?+ ) = (?>
? k(x) ? m+ ) ? (1 ? s )(?? k(x)) + n?(x),
(4)
where ?(x) = ?> (x)(?> ? + ?I)?1 ?(x) = k> (x)(K + ?I)?1 K ?1 k(x),
m+ = s(1 ? s2 )?1/2 , s2 = (1/n) ? y > y? = (1/n) ? y > K ?,
? and ?? = s?1 (1 ? s2 )?1/2 ?.
?
Equation (4) establishes the desired link between OC-KFD and Gaussian density estimation, since for our outlier detection mechanism only Mahalanobis distances are needed.
While it seems to be rather complicated to estimate a density by the above procedure, the
main benefit over directly estimating the mean and the covariance lies in the inherent complexity regulation properties of ridge regression. Such a complexity control mechanism is
of particular importance in highly nonlinear kernel models. Moreover, for ridge-regression
models it is possible to analytically calculate the effective degrees of freedom, a quantity
that will be of particular interest when it comes to detecting outliers.
3
Detecting outliers
Let us assume that the model is completely specified, i.e. both the kernel function k(?, ?)
and the regularization parameter ? are fixed. The central lemma that helps us to detect
outliers can be found in most statistical textbooks:
Lemma 1. Let X be a Gaussian random variable X ? Nd (?, ?). Then ? ? (X ?
?)> ??1 (X ? ?) follows a chi-square (?2 ) distribution on d degrees of freedom.
For the penalized regression models, it might be more appropriate to use the effective degrees of freedom df instead of d in the above lemma. In the case of one-class LDA with
ridge penalties we can easily estimate it as df = trace(X(X > X + ?I)?1 X > ), [8], which
for a kernel model translates into df = trace(K(K + ?I)?1 ). The intuitive interpretation
of the quantity df is the following: denoting by V the matrix of eigenvectors of K and by
{?i }ni=1 the corresponding eigenvalues, the fitted values y? read
y? = V diag {?i = ?i /(?i + ?)} V > y.
(5)
It follows that compared to the unpenalized case, where all eigenvectors v i are constantly
weighted by 1, the contribution of the i-th eigenvector v i is down-weighted by a factor
?i /1 = ?i . If the ordered eigenvalues decrease rapidly, however, the values ?i are either
close to zero or close to one, and df determines the number of terms that are ?essentially
different? from zero. The same is true for the orthogonal distance term in eq. (4): note that
?(x) = k> (x)(K + ?I)?1 K ?1 k(x) = k> V diag ?i0 = ((?i + ?)?i )?1 V > k(x). (6)
Compared to the unpenalized case (the contribution of v i is weighted by ??2
i ), the contribution of v i is down-weighted by the same factor ?i0 /??2
= ?i .
i
From lemma 1 we conclude that if the data are well described by a Gaussian model in
the kernel feature space, the observed Mahalanobis distances should look like a sample
from a ?2 -distribution with df degrees of freedom. A graphical way to test this hypothesis
is to plot the observed quantiles against the theoretical ?2 quantiles, which in the ideal
case gives a straight line. Such a quantile-quantile plot is constructed as follows: Let
?(i) denote the observed Mahalanobis distances ordered from lowest to highest, and p i the
cumulative proportion before each ?(i) given by pi = (i ? 1/2)/n. Let further zi = F ?1 pi
denote the theoretical quantile at position pi , where F is the cumulative ?2 -distribution
function. The quantile-quantile plot is then obtained by plotting ?(i) against zi . Deviations
from linearity can be formalized by fitting a linear model on the observed quantiles and
calculating confidence intervals around the fit. Observations falling outside the confidence
interval are then treated as outliers. A potential problem of this approach is that the outliers
themselves heavily influence the quantile-quantile fit. In order to overcome this problem,
the use of robust fitting procedures has been proposed in the literature, see e.g. [4]. In
the experiments below we use an M-estimator with Huber loss function. For estimating
confidence intervals around the fit we use the standard formula (see [2, 5])
p
?(?(i) ) = b ? (?2 (zi ))?1 (pi (1 ? pi ))/n,
(7)
which can be intuitively understood as the product of the slope b and the standard error of
the quantiles. A 100(1 ? ?)% envelope around the fit is then defined as ?(i) ? z?/2 ?(?(i) )
where z?/2 is the 1 ? (1 ? ?)/2 quantile of the standard normal distribution.
The choice of the confidence level ? is somewhat arbitrary, and from a conceptual viewpoint
one might even argue that the problem of specifying one free parameter (i.e. the expected
fraction of outliers) has simply been transferred into the problem of specifying another
one. In practice, however, selecting ? is a much more intuitive procedure than guessing the
fraction of outliers. Whereas the latter requires problem-specific prior knowledge which is
hardly available in practice, the former depends only on the variance of a linear model fit.
Thus, ? can be specified in a problem independent way.
4
Model selection
In our model the data are first mapped into some feature space, in which then a Gaussian
model is fitted. Mahalanobis distances to the mean of this Gaussian are computed by
evaluating (4). The feature space mapping is implicitly defined by the kernel function, for
which we assume that it is parametrized by a kernel parameter ?. For selecting all free
parameters in (4), we are, thus, left with the problem of selecting ? = (?, ?)> .
The idea is now to select ? by maximizing the cross-validated likelihood. From a theoretical viewpoint, the cross-validated (CV) likelihood framework is appealing, since in [13]
the CV likelihood selector has been shown to asymptotically perform as well as the optimal benchmark selector which characterizes the best possible model (in terms of KullbackLeibler divergence to the true distribution) contained in the parametric family.
For kernels that map into a space with dimension p > n, however, two problems arise: (i)
the subspace spanned by the mapped samples varies with different sample sizes; (ii) not
the whole feature space is accessible for vectors in the input space. As a consequence, it
is difficult to find a ?proper? normalization of the Gaussian density in the induced feature
space. We propose to avoid this problem by considering the likelihood in the input space
rather than in the feature space, i.e. we are looking for a properly normalized density model
p(x|?) in Rd such that p(x|?) has the same contour lines as the Gaussian model in the feature
space: p(xi |?) = p(xi |?) ? p(?(xi )|?) = p(?(xj )|?). Denoting by Xn = {xi }ni=1 a
sample from p(x) from which the kernel matrix K is built, a natural input space model is
R
pn (x|Xn , ?) = Z ?1 exp{? 21 D(x; Xn , ?)}, with Z = Rd pn (x|Xn , ?) dx, (8)
where D(x; Xn , ?) denotes the (parametrized) Mahalanobis distances (4) of a Gaussian
model in the feature space. Note that this density model in the input space has the same
form as our Gaussian model in the feature space, except for the different normalization
constant Z. Computing this constant Z requires us to solve a normalization integral over the
whole d-dimensional input space. Since in general this integral is not analytically tractable
for nonlinear kernel models, we propose to approximate Z by a Monte Carlo sampling
method. In our experiments, for instance, the VEGAS algorithm [6], which implements a
mixed importance-stratified sampling approach, showed to be a reasonable method for up
to 10 input dimensions.
By using the CV likelihood framework we are guaranteed to (asymptotically) perform as
well as the best model in the parametrized family. Thus, the question arises whether the
family of densities defined by a Gaussian model in a kernel-induced feature space is ?rich
enough? such that no systematic errors occur. For RBF kernels, the following lemma provides a positive answer to this question.
Lemma 2. Let k(xi , xj ) = exp(?kxi ? xj k2 /?). As ? ? 0 , pnP
(x|Xn , ?) converges to
n
a Parzen window with vanishing kernel width: pn (x|Xn , ?) ? n1 i=1 ?(x ? xi ).
A formal proof is omitted due to space limitations. The basic ingredients of the proof
are: (i) In the limit ? ? 0 the expansion coefficients approach ?
? ? 1/(1 + ?)1. Thus,
y? = K ?
? ? 1/(1 + ?)1 and s2 ? 1/(1 + ?). (ii) D(x; ?, ?) ?
PnC(x) < ?, if x ?
{xi }ni=1 , and D(x; ?, ?) ? ?, else. Finally pn (x|Xn , ?, ?) ? n1 i=1 ?(x ? xi ).
Note that in the limit ? ? 0 a Parzen window becomes an unbiased estimator for any
continuous density, which provides an asymptotic justification for our approach: the crossvalidated likelihood framework guarantees us to convergence to a model that performs as
well as the best model in our model class as n ? ?. The latter, however, is ?rich enough?
in the sense that it contains models which in the limit ? ? 0 converge to an unbiased
estimator for every continuous p(x). Since contour lines of pn (x) are contour lines of a
Gaussian model in the feature space, the Mahalanobis distances are expected to follow a
?2 distribution, and atypical objects can be detected by observing the distribution of the
empirical Mahalanobis distances as described in the last section.
It remains to show that describing the data as a Gaussian in a kernel-induced feature space
is a statistically sound model. This is actually the case, since there exist decay rates for the
kernel width ? such that n grows at a higher rate as the effective degrees of freedom df :
Lemma 3. Let k(xi , xj ) = exp(?kxi ? xj k2 /?) and pn (x|Xn , ?, ?) defined by (8). If
? ? 1 decays like O(n?1/2 ), and for fixed ? ? 1, the ratio df /n ? 0 as n ? ?.
A formal proof is omitted due to space limitations. The basic ingredients of the proof are:
? i as n ? ?, (ii) the eigenvalue spectrum of
(i) the eigenvalues ?0i of (1/n)K converge to ?
? i ? exp(??i2 ), (iii) for
a Gaussian RBF kernel decays atPan exponential-quadratic rate: ?
n
?1/2 2
?i )] ? n1/2 ? ?1 log(n/?)
n sufficiently large, it holds that i=1 1/[1 + (?/n) exp(n
(proof by induction, using the fact that ln(n + 1) ? ln(n) ? 1/(n2 + n) which follows
from a Taylor expansion of the logarithm) ? df (n)/n ? 0.
5
Experiments
The performance of the proposed method is demonstrated for an outlier detection task in the field of face recognition.
The Olivetti face database (see
http://www.uk.research.att.com/facedatabase.html) contains ten different images of each of
40 distinct subjects, taken under different lighting conditions and at different facial expressions and facial details (glasses / no glasses). None of the subjects, however, wears
sunglasses. All the images are taken against a homogeneous background with the subjects
in an upright, frontal position. In this experiment we additionally corrupted the dataset by
including two images in which we have artificially changed normal glasses to ?sunglasses?
as can be seen in figure 1. The goal is to demonstrate that the proposed method is able to
identify these two atypical images without any problem-dependent prior assumptions.
Figure 1: Original and corrupted images with in-painted ?sunglasses?.
Each of the 402 images is characterized by a 10-dimensional vector which contains the
projections onto the leading 10 eigenfaces (eigenfaces are simply the eigenvectors of the
images treated as pixel-wise vectorial objects). These vectors are feed into a RBF kernel
of the form k(xi , xj ) = exp(?kxi ? xj k2 /?). In a first step, the free model parameters
(?, ?) are selected by maximizing the cross-validated likelihood. A simple 2-fold cross
validation scheme is used: the dataset is randomly split into a training set and a test set of
equal size, the model is build from the training set (including the numerical solution of the
normalization integral), and finally the likelihood is evaluated on the test set. This procedure is repeated for different values of (?, ?). In order to simplify the selection process
we kept ? = 10?4 fixed and varied only ?. Both the test likelihood and the corresponding
model complexity measured in terms of the effective degrees of freedom (df ) are plotted
in figure 2. One can clearly identify both an overfitting and an underfitting regime, separated by a broad plateau of models with similarly high likelihood. The df -curve, however,
shows a similar plateau, indicating that all these models have comparable complexity. This
observation suggests that the results should be rather insensitive to variations of ? over values contained in this plateau. This suggestion is indeed confirmed by the results in figure
2, where we compared the quantile-quantile plot for the maximum likelihood parameter
value with that of a slightly suboptimal model. Both quantile plots look very similar, and
in both cases two objects clearly fall outside a 99% envelope around the linear fit. Outside the plateau (no figure due to space limitations) the number of objects considered as
outlies drastically increases in overfitting regime (? too small), or decreases to zero in the
underfitting regime (? too large).
In figure 3 again the quantile plot for the most likely model is depicted. This time, however,
both objects identified as outliers are related to the corresponding original images, which
in fact are the artificially corrupted ones. In addition, the uncorrupted images are localized
in the plot, indicating that they look rather typical.
Some implementation details. Presumably the easiest way of implementing the model is
to carry
Pout an eigenvalue decomposition of K. Both the the effective degrees of freedom
df = i ?i /(?i + ?) and the Mahalanobis distances in eq. (4) can then by derived easily
10
?400
20
?450
5
10
?500
30
? (i)
test log?likelihood
? (i)
30
?350
20
15
?300
10
40
?250
effective degrees of freedom
40
50
?200
?550
?600
5
10
15
20
25
? 2 quantiles
30
6
7
8
9
log(?)
10
11
12
13
5
10
15
20
25
?2 quantiles
30
50
Figure 2: Middle panel: Selecting the kernel width ? by cross-validated likelihood (solid
line). The dotted line shows the corresponding effective degrees of freedom (df ). Left +
right panels: quantile plot for optimal model (left) and slightly suboptimal model (right).
40
99.99%
20
30
? (i)
99%
99%
10
99.99%
5
10
15
? 2 quantiles
20
25
30
Figure 3: Quantile plot with linear fit (solid) and envelopes (99% and 99.99 %, dashed).
from this decomposition (see (5) and (6)). Efficient on-line variants can be implemented by
using standard update formulas for matrix inversion by partitioning. For an implementation
of the VEGAS algorithm see [9]. The R package ?car? provides a comfortable implementation of quantile-quantile plots and robust line fitting (see also http://www.R-project.org).
6
Conclusion
Detecting outliers by way of one-class classifiers aims at finding a boundary that separates
?typical? objects in a data sample from the ?atypical? ones. Standard approaches of this
kind suffer from the problem that they require prior knowledge about the expected fraction
of outliers. For the purpose of outlier detection, however, the availability of such prior
information seems to be an unrealistic (or even contradictory) assumption. The method
proposed in this paper overcomes this shortcoming by using a one-class KFD classifier
which is directly related to Gaussian density estimation in the induced feature space. The
model benefits from both the built-in classification method and the explicit parametric density model: from the former it inherits the simple complexity regulation mechanism based
on only two tuning parameters. Moreover, within the classification framework it is possible to quantify the model complexity in terms of the effective degrees of freedom df . The
Gaussian density model, on the other hand, makes it possible to derive a formal description of atypical objects by way of hypothesis testing: Mahalanobis distances are expected
to follow a ?2 -distribution in df dimensions, and deviations from this distribution can be
quantified by confidence intervals around a fitted line in a quantile-quantile plot. Since
the density model is parametrized by both the kernel function and the regularization constant, it is necessary to select these free parameters before the outlier detection phase. This
parameter selection is achieved by observing the cross-validated likelihood for different
parameter values, and choosing those parameters which maximize this quantity. The theoretical motivation for this selection process follows from [13] where it has been shown
that the cross-validation selector asymptotically performs as well as the so called benchmark selector which selects the best model contained in the parametrized family of models.
Moreover, for RBF kernels it is shown in lemma 2 that the corresponding model family is
?rich enough? in the sense that it contains an unbiased estimator for the true density (as
long as it is continuous) in the limit of vanishing kernel width. Lemma 3 shows that there
exist decay rates for the kernel width such that the ratio of effective degrees of freedom and
sample size approaches zero.
The experiment on detecting persons wearing sunglasses within a collection of rather heterogeneous face images effectively demonstrates that the proposed method is able to detect
atypical objects without prior assumptions on the expected number of outliers. In particular, it demonstrates that the whole processing pipeline consisting of model selection by
cross-validated likelihood, fitting linear quantile-quantile models and detecting outliers by
considering confidence intervals around the fit works very well in practical applications
with reasonably small input dimensions. For input dimensions 10 the numerical solution of the normalization integral becomes rather time consuming when using the VEGAS
algorithm. Evaluating the usefulness of more sophisticated sampling models like MarkovChain Monte-Carlo methods for this particular task will be subject of future work.
Acknowledgments. The author would like to thank Tilman Lange, Mikio Braun and
Joachim M. Buhmann for helpful discussions and suggestions.
References
[1] R. Duda, P. Hart, and D. Stork. Pattern Classification. Wiley & Sons, 2001.
[2] J. Fox. Applied Regression, Linear Models, and Related Methods. Sage, 1997.
[3] T. Hastie, A. Buja, and R. Tibshirani. Penalized discriminant analysis. Annals of Statistics,
23:73?102, 1995.
[4] P.J. Huber. Robust Statistics. Wiley, 1981.
[5] M. Kendall and A. Stuart. The Advanced Theory of Statistics, volume 1. McMillan, 1977.
[6] G.P. Lepage. Vegas: An adaptive multidimensional integration program. Technical Report
CLNS-80/447, Cornell University, 1980.
[7] S. Mika, G. R?atsch, J. Weston, B. Sch?olkopf, and K.-R. M?uller. Fisher discriminant analysis
with kernels. In Y.-H. Hu, J. Larsen, E. Wilson, and S. Douglas, editors, Neural Networks for
Signal Processing IX, pages 41?48. IEEE, 1999.
[8] J. Moody. The effective number of parameters: An analysis of generalisation and regularisation
in nonlinear learning systems. In J. Moody, S. Hanson, and R. Lippmann, editors, NIPS 4, 1992.
[9] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery. Numerical Recipies in C.
Cambridge University Press, 1992.
[10] V. Roth and V. Steinhage. Nonlinear discriminant analysis using kernel functions. In S.A. Solla,
T.K. Leen, and K.-R. M?uller, editors, NIPS 12, pages 568?574. MIT Press, 2000.
[11] B. Sch?olkopf, S. Mika, C. Burges, P. Knirsch, K.-R. Mu? ller, G. R?atsch, and A. Smola. Input
space vs. feature space in kernel-based methods. IEEE Trans. Neural Networks, 10(5), 1999.
[12] B. Sch?olkopf, R.C. Williamson, A. Smola, and J. Shawe-Taylor. SV estimation of a distribution?s support. In S. Solla, T. Leen, and K.-R. Mu? ller, editors, NIPS 12, pages 582?588. 2000.
[13] M.J. van der Laan, S. Dudoit, and S. Keles. Asymptotic optimality of likelihood-based crossvalidation. Statistical Applications in Genetics and Molecular Biology, 3(1), 2004.
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1,824 | 2,657 | Class-size Independent Generalization Analsysis
of Some Discriminative Multi-Category
Classification Methods
Tong Zhang
IBM T.J. Watson Research Center
Yorktown Heights, NY 10598
[email protected]
Abstract
We consider the problem of deriving class-size independent generalization bounds for some regularized discriminative multi-category classification methods. In particular, we obtain an expected generalization
bound for a standard formulation of multi-category support vector machines. Based on the theoretical result, we argue that the formulation over-penalizes misclassification error, which in theory may lead to
poor generalization performance. A remedy, based on a generalization
of multi-category logistic regression (conditional maximum entropy), is
then proposed, and its theoretical properties are examined.
1
Introduction
We consider the multi-category classification problem, where we want to find a predictor
p : X ? Y, where X is a set of possible inputs and Y is a discrete set of possible outputs.
In many applications, the output space Y can be extremely large, and may be regarded as
infinity for practical purposes. For example, in natural language processing and sequence
analysis, the input can be an English sentence, and the output can be a parse or a translation
of the sentence. For such applications, the number of potential outputs can be exponential
of the length of the input sentence. As another example, in machine learning based webpage search and ranking, the input is the keywords and the output space consists of all
web-pages.
In order to handle such application tasks, from the theoretical point of view, we do not
need to assume that the output space Y is finite, so that it is crucial to obtain generalization
bounds that are independent of the size of Y. For such large scale applications, one often
has a routine that maps each x ? X to a subset of candidates GEN(x) ? Y, so that the
desired output associated with x belongs to GEN(x). For example, for web-page search,
GEN(x) consists of all pages that contain one or more keywords in x. For sequence
annotation, GEN(x) may include all annotation sequences that are consistent. Although
the set GEN(x) may significantly reduce the size of potential outputs Y, it can still be
large. Therefore it is important that our learning bounds are independent of the size of
GEN(x).
We consider the general setting of learning in Hilbert spaces since it includes the popular
kernel methods. Let our feature space H be a reproducing kernel Hilbert space with dot
product ?. For a weight vector w ? H, we use notation kwk2H = w ? w. We associate each
possible input/output pair (x, y) ? X ? Y with a feature vector fx,y ? H. Our classifier is
characterized by a weight vector w ? H, with the following classification rule:
pw (x) = arg
max
c?GEN(x)
w ? fx,c .
(1)
Note that computational issues are ignored in this paper. In particular, we assume that the
above decision can be computed efficiently (either approximately or exactly) even when
GEN(x) is large. In practice, this is often possible either by heuristic search or dynamic
programming (when GEN(X) has certain local-dependency structures). In this paper, we
are only interested in the learning performance, so that we will not discuss the computational aspect.
We assume that the input/output pair (x, y) ? X ?Y is drawn from an unknown underlying
distribution D. The quality of the predictor w is measured by some loss function. In this
paper, we focus on the expected classification error with respect to D:
`D (w) = E(X,Y ) I(pw (X), Y ),
(2)
where (X, Y ) is drawn from D, and I is the standard 0-1 classification error: I(Y 0 , Y ) = 0
when Y 0 = Y and I(Y 0 , Y ) = 1 when Y 0 6= Y .
The general set up we described above is useful for many application problems, and has
been investigated, for example, in [2, 6]. The important issue of class-size independent (or
weakly dependent) generalization analysis has also been discussed there.
Consider a set of training data S = {(Xi , Yi ), i = 1, . . . , n}, where we assume that for
? S ? H such that the classification error
each i, Yi ? GEN(Xi ). We would like to find w
? S ) is as small as possible. This paper studies regularized discriminative learning
`D (w
? S ? H by solving the following optimization
methods that estimate a weight vector w
problem:
#
" n
?
1X
2
? S = arg min
L(w, Xi , Yi ) + kwkH ,
(3)
w
w?H n
2
i=1
where ? ? 0 is an appropriately chosen regularization parameter, and L(w, X, Y ) is a
loss function which is convex of w. In this paper, we focus on some loss functions of the
following form:
?
?
X
L(w, X, Y ) = ? ?
?(w ? (fX,Y ? fX,c ))? ,
c?GEN(X)\Y
where ? and ? are appropriately chosen real-valued functions.
Typically ? is chosen as an increasing function and ? as a decreasing function, selected so that (3) is a convex optimization problem. The intuition behind this method
is that the resulting optimization formulation favors large values w ? (fXi ,Yi ? fXi ,c )
for all c ? GEN(Xi )\Yi . Therefore, it favors a weight vector w ? H such that
w ? fXi ,Yi = arg maxc?GEN(Xi ) w ? fXi ,c , which encourages the correct classification
rule in (1). The regularization term ?2 kwk2H is included for capacity control, which has
become the standard practice in machine learning nowadays.
Two of the most important methods used in practice, multi-category support vector machines [7] and penalized multi-category logistic regression (conditional maximum entropy
with Gaussian smoothing [1]), can be regarded as special cases of (3). The purpose of this
paper is to study their generalization behaviors. In particular, we are interested in generalization bounds that are independent of the size of GEN(Xi ).
2
Multi-category Support Vector Machines
We consider the multi-category support vector machine method proposed in [7]. It is a
? S computed based on the following formula:
special case of (3) with w
?
?
n
X
X
?
1
? S = arg min ?
h(w ? (fXi ,Yi ? ?fXi ,c )) + kwk2H ? ,
(4)
w
w?H n
2
i=1
c?GEN(Xi )\Yi
where h(z) = max(1 ? z, 0) is the hinge loss used in the standard SVM formulation.
From the asymptotic statistical point of view, this formulation has some drawbacks in that
there are cases such that the method does not lead to a classifier that achieves the Bayes
error [9] (inconsistency). A Bayes consistent remedy has been proposed in [4]. However,
method based on (4) has some attractive properties, and has been successfully used for
some practical problems.
We are interested in the generalization performance of (4). As we shall see, this formulation
performs very well in the linearly separable (or near separable) case. Our analysis also
reveals a problem of this method for non-separable problems. Specifically, the formulation
over-penalizes classification error. Possible remedies will be suggested at the end of the
section.
We start with the following theorem, which specifies a generalization bound in a form often
referred to as the oracle inequality. That is, it bounds the generalization performance of
the SVM method (4) in terms of the best possible true multi-category SVM loss. Proof is
left to Appendix B.
Theorem 2.1 Let M = supX supY,Y 0 ?GEN(X) kfX,Y ? fX,Y 0 kH . The expected generalization error of (4) can be bounded as:
? S ) ? ES E(X,Y )
ES `D (w
? S ? (fX,Y ? fX,c ))
h(w
sup
c?GEN(X)\Y
2
?
max(?n, M ) + M
?n
?
2
inf ?E(X,Y )
w?H
X
h(w ? (fX,Y ? fX,c )) +
c?GEN(X)\Y
?nkwk2H
2(n + 1)
?
?,
where ES is the expectation with respect to the training data.
Note that the generalization bound does not depend on the size of GEN(X), which is
what we want to achieve. The left-hand side of the theorem bounds the classification error
? S ? (fX,Y ? fX,c )),
of the multi-category SVM classifier in terms of supc?GEN(X)\Y h(w
P
while the right hand side in terms of c?GEN(X)\Y h(w ? (fX,Y ? fX,c )). There is a
mismatch here. The latter is a very loose bound since it over-counts classification errors
in the summation when multiple errors are made at the same point. In fact, although the
class-size dependency does
Pnot come into our generalization analysis, it may well come
into the summation term c?GEN(X)\Y h(w ? (fX,Y ? fX,c )) when multiple errors are
made at the same point. We believe that this is a serious flaw of the method, which we
will try
Pto remedy later. However, the bound can be quite tight in the near separable case,
? S ? (fX,Y ? fX,c )) is small. The following Corollary gives such
when c?GEN(X)\Y h(w
a result:
Corollary 2.1 Assume that there is a large margin separator w? ? H such that for each
data point (X, Y ), the following margin condition holds:
?c ? GEN(X)\Y : w? ? fX,Y ? w? ? fX,c + 1.
Then in the limit of ? ? 0, the expected generalization error of (4) can be bounded as:
? S) ?
ES `D (w
kw? k2H
sup
sup
kfX,Y ? fX,Y 0 k2H ,
n + 1 X Y,Y 0 ?GEN(X)
where ES is the expectation with respect to the training data.
Proof. Just choose w? on the right hand side of Theorem 2.1. 2
The above result for (4) gives a class-size independent bound for large margin separable
problems. The bound generalizes a similar result for two-class hard-margin SVM. It also
matches a bound for multi-class perceptron in [2]. To our knowledge, this is the first result
showing that the generalization performance of a batch large margin algorithm such as (4)
can be class-size independent (at least in the separable case). Previous results in [2, 6],
relying on the covering number analysis, lead to bounds that depend on the size of Y
(although the result in [6] is of a different style).
Our analysis also implies that the multi-category classification method (4) has good generalization behavior for separable problems. However, as pointed out earlier, for nonseparable problems, the formulation over-penalize classification error since in the summation, it may count classification error at a point multiple times when multiple
mistakes
P
are made at the point. A remedy is to replace the summation symbol c?GEN(Xi )\Yi
in (4) by the sup operator supc?GEN(Xi )\Yi , as we have used for bounding the classification error on the left hand side of Theorem 2.1. This is done in [3]. However, like
(4), the resulting formulation is also inconsistent. Instead of using a hard-sup operator,
we may also use a soft-sup operator, which can possibly
lead to consistency. For examP
ple, consider the equality supc |hc | = limp?? ( c |hc |p )1/p , we may approximate the
right hand side limit with a P
large p. Another more interesting formulation is to consider
supc hc = limp?? p?1 ln( c exp(phc )), which leads to a generalization of the conditional maximum entropy method.
3
Large Margin Discriminative Maximum Entropy Method
Based on the motivation given at the end of the last section, we propose the following generalization of maximum entropy (multi-category logistic regression) with Gaussian prior
(see [1]). It introduces a margin parameter into the standard maximum entropy formulation, and can be regarded as a special case of (3):
?
n
X
1
1
? S = arg min ?
ln ?1 +
w
w?H
n i=1 p
?
?
X
c?GEN(Xi )\Yi
p(??w?(fXi ,Yi ?fXi ,c )) ?
e
?
?
2 ?
+ kwkH ,
2
(5)
where ? is a margin condition, and p > 0 is a scaling factor (which in theory can also be
removed by a redefinition of w and ?).
If we choose ? = 0, then this formulation is equivalent to the standard maximum entropy
method. If we pick the margin parameter ? = 1, and let p ? ?, then
?
?
X
1 ?
ln 1 +
ep(??w?(fXi ,Yi ?fXi ,c )) ? ?
sup
h(w?(fXi ,Yi ?fXi ,c )),
p
c?GEN(Xi )\Yi
c?GEN(Xi )\Yi
where h(z)
P = max(0, 1 ? z) is used in (4). In this case, the formulation reduces to (4)
but with c?GEN(Xi )\Yi replaced by supc?GEN(Xi )\Yi . As discussed at the end of last
section, this solves the problem of over-counting the classification error.
In general, even with a finite scaling factor p, the log-transform in (4) guarantees that
one penalizes misclassification error at most p1 ln |GEN(Xi )| times at a point, where
|GEN(Xi )| is the size of GEN(Xi ), while in (4), one may potentially over-penalize
|GEN(Xi )| times. Clearly this is a desirable effect for non-separable problems. Methods in (5) have many attractive properties. In particular, we are able to derive class-size
independent generalization bounds for this method. The proof of the following theorem is
given in Appendix C.
Theorem 3.1 Let M = supX supY,Y 0 ?GEN(X) kfX,Y ? fX,Y 0 kH . Define loss L(w, x, y)
as:
?
?
X
1
ep(??w?(fx,y ?fx,c )) ? ,
L(w, x, y) = ln ?1 +
p
c?GEN(x)\y
and let
E(X,Y ) L(w, X, Y ) +
?n
kwk2H .
w?H
2(n + 1)
The expected generalization error of (5) can be bounded as:
Q? = inf
? S , X, Y ) ? Q? +
ES E(X,Y ) L(w
M2
(1 ? e?pQ? ).
?n
where ES is the expectation with respect to the training data.
Theorem 3.1 gives a class-size independent generalization bound for (5). Note that the left
? S from (5), and the right hand size is specified in terms
hand side is the true loss of the w
of the best possible regularized true loss Q? , plus a penalty term that is no larger than
M 2 /(?n). It is clear that this generalization bound is class-size independent. Moreover,
unlike Theorem 2.1, the loss function on the left hand side matches the loss function on
the right hand side in Theorem 3.1. These are not trivial properties. In fact, most learning
methods do not have these desirable properties. We believe this is a great advantage for the
maximum entropy-type discriminative learning method in (5). It implies that this class of
algorithms are suitable for problems with large number of classes. Moreover, we can see
that the generalization performance is well-behaved no matter what values of p and ? we
choose.
If we take ? = 0 and p = 1, then we obtain a generalization bound for the popular maximum entropy method with Gaussian prior, which has been widely used in natural language
processing applications. To our knowledge, this is the first generalization bound derived for
this method. Our result not only shows the importance of Gaussian prior regularization, but
also implies that the regularized conditional maximum entropy method has very desirable
generalization behavior.
Another interesting special case of (5) is to let ? = 1 and p ? ?. For simplicity we only
consider the case that |GEN(X)| is finite (but can be arbitrarily large). In this case, we
note that 0 ? L(w, X, Y ) ? supc?GEN(X)\Y h(w ? (fX,Y ? fX,c )) ? ln |GEN(X)|
. We
p
thus obtain from Theorem 3.1 a bound
ES E(X,Y )
c?GEN(X)\Y
"
+ inf
w?H
E(X,Y )
EX ln |GEN(X)| M 2
+
p
?n
#
?kwk2H
? fX,c )) +
.
2
? S ? (fX,Y ? fX,c )) ?
h(w
sup
sup
c?GEN(X)\Y
h(w ? (fX,Y
Now we can take a sufficiently large p such that the term EX ln |GEN(X)|/p becomes
negligible. Let p ? ?, the result implies a bound for the SVM method in [3]. For
non-separable problems, this bound is clearly superior
Pto the SVM bound in Theorem 2.1
since the right hand side replaces the summation c?GEN(X)\Y by the sup operator
supc?GEN(X)\Y . In theory, this satisfactorily solves the problem of over-penalizing misclassification error. Moreover, an advantage over [3] is that for some p, consistency can be
achieved. Our analysis also establishes a bridge between the Gaussian smoothed maximum
entropy method [1] and the SVM method in [3].
4
Conclusion
We studied the generalization performance of some regularized multi-category classification methods. In particular, we derived a class-size independent generalization bound for
a standard formulation of multi-category support vector machines. Based on the theoretical investigation, we showed that this method works well for linearly separable problems.
However, it over-penalizes mis-classification error, leading to loose generalization bounds
in the non-separable case. A remedy, based on a generalization of the maximum entropy
method, is proposed. Moreover, we are able to derive class-size independent bounds for the
newly proposed formulation, which implies that this class of methods (including the standard maximum entropy) are suitable for classification problems with very large number of
classes. We showed that in theory, the new formulation provides a satisfactory solution to
the problem of over-penalizing mis-classification error.
A
A general stability bound
The following lemma is essentially a variant of similar stability results for regularized learning systems used in [8, 10]. We include the proof Sketch for completeness.
Lemma A.1 Consider a sequence of convex functions Li (w) for i = 1, 2, . . . Define for
k = 1, 2, . . .
#
" k
X
?n
2
kwkH .
Li (w) +
wk = arg min
w
2
i=1
Then for all k ? 1, there exists subgradient (cf. [5]) ?Lk+1 (wk+1 ) of Li at wk+1 such
that
k+1
1 X
1
wk+1 = ?
?Li (wk+1 ), kwk ? wk+1 kH ?
k?Lk+1 (wk+1 )kH .
?n i=1
?n
Proof Sketch. The first equality is the first-order condition for the optimization problem [5]
where wk+1 is the solution. Now, subtracting this equality at wk and wk+1 , we have:
??n(wk+1 ? wk ) = ?Lk+1 (wk+1 ) +
k
X
(?Li (wk+1 ) ? ?Li (wk )).
i=1
Multiply the two sides by wk+1 ? wk , we obtain
??nkwk+1 ?wk k2H = ?Lk+1 (wk+1 )?(wk+1 ?wk )+
k
X
(?Li (wk+1 )??Li (wk ))?(wk+1 ?wk ).
i=1
Note that ?Li (wk+1 ) ? ?Li (wk )) ? (wk+1 ? wk ) = dLi (wk , wk+1 ) + dLi (wk+1 , wk ),
where dL (w, w0 ) = L(w0 ) ? L(w) ? ?L(w) ? (w0 ? w) is often called the Bregman
divergence of L, which is well-known to be non-negative for any convex function L (this
claim is also easy to verify by definition). We thus have (?Li (wk+1 )??Li (wk ))?(wk+1 ?
wk ) ? 0. It follows that
??nkwk+1 ? wk k2H ? ?Lk+1 (wk+1 ) ? (wk+1 ? wk ) ? ?k?Lk+1 (wk+1 )kH kwk+1 ? wk kH .
By canceling the factor kwk+1 ? wk kH , we obtain the second inequality. 2
B
Proof Sketch of Theorem 2.1
? k be the solution of (4) with
Consider training samples (Xi , Yi ) for i = 1, . . . , n + 1. Let w
Pn+1
the training sample (Xk , Yk ) removed from the set (that is, the summation is i=1,i6=k ),
P
P
? be the solution of (4) but with the summation ni=1 replaced by n+1
and let w
i=1 . Now for
? ? (fXk ,Yk ? fXk ,c ) for c ? GEN(X). It follows from
notation simplicity, we let zk,c = w
Lemma A.1 that
? 2H = ?
kwk
n+1
1 X
?n
X
h0 (zk,c )zk,c ,
? k ? wk
? H ??
kw
k=1 c?GEN(X)
M
?n
X
h0 (zk,c ),
c?GEN(X)\Y
0
where h (?) denotes a subgradient of h(?). Therefore using the inequality ?h0 (z) ? h(z) ?
h0 (z)z, we have
? k ? (fXk ,Yk ? fXk ,c )) ? h(zk,c )] ? kw
? k ? wk
? HM
[h(w
sup
c?GEN(Xk )\Yk
??
M2
?n
X
h0 (zk,c ) ?
c?GEN(Xk )\Yk
M2
?n
X
[h(zk,c ) ? h0 (zk,c )zk,c ].
c?GEN(Xk )\Yk
Summing over k = 1, . . . , n + 1, we obtain
n+1
X
? k ? (fXk ,Yk ? fXk ,c )) ? h(zk,c )]
[h(w
sup
k=1 c?GEN(Xk )\Yk
M2
?
?n
X
M
?n
[h(zk,c ) ? h0 (zk,c )zk,c ]
c?GEN(Xk )\Yk k=1
2
=
n+1
X
X
n+1
X
? 2H M 2 .
h(zk,c ) + kwk
c?GEN(Xk )\Yk k=1
Therefore given an arbitrary w ? H, we have
n+1
X
? k ? (fXk ,Yk ? fXk ,c ))
h(w
sup
k=1 c?GEN(Xk )\Yk
?(1 +
M2
)
?n
X
n+1
X
? 2H M 2
h(zk,c ) + kwk
c?GEN(Xk )\Yk k=1
?
?
n+1
X
X
M 2 2M 2 ?
?n
? 2H ?
)
h(zk,c ) +
kwk
? max(1 +
,
?n ?n
2
c?GEN(Xk )\Yk k=1
?
?
n+1
X
X
M 2 2M 2 ?
?n
? max(1 +
,
)
h(w ? (fXk ,Yk ? fXk ,c )) +
kwk2H ? .
?n ?n
2
c?GEN(Xk )\Yk k=1
Now, taking expectation with respect to the training data, we obtain the bound.
C
Proof Sketch of Theorem 3.1
Similar to the proof of Theorem 2.1, we consider training samples (Xi , Yi ) for i =
? k be the solution of (5) with the training sample (Xk , Yk ) removed
1, . . . , n + 1. Let w
Pn+1
? be the solution of (5) but with
from the set (that is, the summation is i=1,i6=k ), and let w
Pn
Pn+1
the summation i=1 replaced by i=1 . It follows from Lemma A.1 that
1
M
? k ,Yk )
? k ? wk
? H?
? Xk , Yk )kH ?
kw
k?L(w,
(1 ? e?pL(w,X
).
?n
?n
Therefore
M2
? k ,Yk )
? Xk , Yk ) ?
? k , Xk , Yk ) ? L(w,
(1 ? e?pL(w,X
).
L(w
?n
Now summing over k, we obtain
n+1
n+1
1 X
1 X
M2
? k , Xk , Yk ) ?
? Xk , Yk ) +
L(w
L(w,
n+1
n+1
?n
k=1
k=1
n+1
1 X ?pL(w,X
?
k ,Yk )
1?
e
n+1
!
.
k=1
Taking expectation with respect to the training data, and using the following Jensen?s
inequality:
n+1
Pn+1
1
1 X ?pL(w,X
? k ,Yk )
? k ,Yk )
,
?ES
e
? ?e?pES n+1 k=1 L(w,X
n+1
k=1
we obtain
k
? , Xk , Yk ) ? ES
ES E(Xk ,Yk ) L(w
n+1
X
k=1
Pn+1
? Xk , Yk ) M 2
L(w,
1 ? e?pES k=1
+
n+1
?n
?
L(w,X
k ,Yk )
n+1
Pn+1
? Xk , Yk ) ? (n+1)Q? (which follows from the optimal
Now, using the fact ES k=1 L(w,
? we obtain the theorem.
property of w),
References
[1] Stanley Chen and Ronald Rosenfeld. A survey of smoothing techniques for ME models. IEEE Trans. Speech and Audio Processing, 8:37?50, 2000.
[2] Michael Collins.
Parameter estimation for statistical parsing models: Theory and practice of distribution-free methods. In IWPT, 2001. available at
http://www.ai.mit.edu/people/mcollins/publications.html.
[3] Koby Crammer and Yoram Singer. On the algorithmic implementation of multiclass
kernel-based vector machines. Journal of Machine Learning Research, 2:265?292,
2001.
[4] Y. Lee, Y. Lin, and G. Wahba. Multicategory support vector machines, theory, and
application to the classification of microarray data and satellite radiance data. Journal
of American Statistical Association, 99:67?81, 2004.
[5] R. Tyrrell Rockafellar. Convex analysis. Princeton University Press, Princeton, NJ,
1970.
[6] Ben Taskar, Carlos Guestrin, and Daphne Koller. Max-margin markov networks.
In Sebastian Thrun, Lawrence Saul, and Bernhard Sch?olkopf, editors, Advances in
Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004.
[7] J. Weston and C. Watkins. Multi-class support vector machines. Technical Report
CSD-TR-98-04, Royal Holloway, 1998.
[8] Tong Zhang. Leave-one-out bounds for kernel methods. Neural Computation,
15:1397?1437, 2003.
[9] Tong Zhang. Statistical analysis of some multi-category large margin classification
methods. Journal of Machine Learning Research, 5:1225?1251, 2004.
[10] Tong Zhang. Statistical behavior and consistency of classification methods based on
convex risk minimization. The Annals of Statitics, 32:56?85, 2004. with discussion.
.
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1,825 | 2,658 | Generative Affine Localisation and Tracking
John Winn
Andrew Blake
Microsoft Research Cambridge
Roger Needham Building
7 J. J. Thomson Avenue
Cambridge CB3 0FB, U.K
http://research.microsoft.com/mlp
Abstract
We present an extension to the Jojic and Frey (2001) layered sprite model
which allows for layers to undergo affine transformations. This extension
allows for affine object pose to be inferred whilst simultaneously learning the object shape and appearance. Learning is carried out by applying
an augmented variational inference algorithm which includes a global
search over a discretised transform space followed by a local optimisation. To aid correct convergence, we use bottom-up cues to restrict the
space of possible affine transformations. We present results on a number
of video sequences and show how the model can be extended to track an
object whose appearance changes throughout the sequence.
1
Introduction
Generative models provide a powerful and intuitive way to analyse images or video sequences. Because such models directly represent the process of image generation, it is
straightforward to incorporate prior knowledge about the imaging process and to interpret
results. Since the entire data set is modelled, generative models can give improved accuracy and reliability over feature-based approaches and they also allow for selection between
models using Bayesian model comparison. Finally, it is possible to sample from generative
models, for example, for the purposes of image or video editing.
One popular type of generative model represents images as a composition of layers [1]
where each layer corresponds to the appearance and shape of an individual object or the
background. If the generative model is expressed probabilistically, Bayesian learning and
inference techniques can then be applied to reverse the imaging process and infer the shape
and appearance of individual objects in an unsupervised fashion [2].
The difficulty with generative models is how to apply Bayesian inference efficiently. In a
layered model, inference involves localising the pose of the layers in each image, which
is hard because of the large space of possible object transformations that needs to be explored. Previously, this has been dealt with by imposing restrictions on the space of object
transformations, such as allowing only similarity transformations [3]. Alternatively, if the
images are known to belong to a video sequence, tracking constraints can be used to focus the search on a small area of transformation space consistent with a dynamic model of
object motion [4]. However, even in a video sequence, this technique relies on the object
remaining in frame and moving relatively slowly.
In this paper, we extend the work of [3] and present an approach to object localisation which
allows objects to undergo planar affine transformations and works well both in the frames of
a video sequence and in unordered sets of images. A two-layer generative model is defined
and inference performed using a factorised variational approximation, including a global
search over a discretised transform space followed by a local optimisation using conjugate
gradients. Additionally, we exploit bottom-up cues to constrain the space of transforms
being explored. Finally, we extend our generative model to allow the object appearance in
one image to depend on its appearance in the previous one. Tracking appearance in this
way gives improved performance for objects whose appearance changes slowly over time
(e.g. objects undergoing non-planar rotation).
If the images are not frames of a video, or the object is out-of-frame or occluded in the
previous image, then the system automatically reverts to using a learned foreground appearance model.
2
The generative image model
This section describes the generative image model, which is illustrated in the Bayesian
network of Figure 1. This model consists of two layers, a foreground layer containing a
single object and a background layer.
We denote our image set as {x1 , . . . , xN }, where xi is a vector of the pixel intensities in the
ith image. The background layer is assumed to be stationary and so its appearance vector b
is set to be the same size as the image. A mask mi has binary elements that indicate which
b
f
T
m
x
N
Figure 1: The Bayesian network for the generative image model. The rounded rectangle is a
plate, indicating that there are N copies of each contained node (one for each image). Common
to all images are the background b, foreground object appearance f and mask prior ?. An affine
transform T gives the position and pose of the object in each image. The binary mask m defines
the area of support of the foreground object and has a prior given by a transformed ?. The observed
image x is generated by adding noise ? separately to the transformed foreground appearance and the
background and composing them together using the mask. For illustration, the images underneath
each node of the graph represent the inferred value of that node given a data set of hand images. A
priori, the appearance and mask of the object are not known.
pixels of the ith image are foreground. The mask is set to be slightly larger than the image
to allow the foreground object to overlap the edge of the image.
The foreground layer is represented by an appearance image vector f and a prior over its
mask ?, both of which are to be inferred from the image set. The elements of ? are real
numbers in the range [0, 1] which indicate the probability that the corresponding mask
pixels are on, as suggested in [5]. The object appearance and mask prior are defined in a
canonical, normalised pose; the actual position and pose of the object in the ith image is
given by an affine transformation Ti . With our images in vector form, we can consider a
transformation T to be a sparse matrix where the jth row defines the linear interpolation
of pixels that gives the jth pixel in the transformed image. For example, a translation of an
integer number of pixels is represented as a matrix whose entries Tjk are 1 if the translation
of location k in the source image is location j in the destination image, and 0 otherwise.
Hence, the transformed foreground appearance is given by Tf and the transformed mask
prior by T?. Given the transformed mask prior, the conditional distribution for the kth
mask pixel is
P (mk = 1 | ?, T) = (T?)k .
(1)
The observed image x is generated by a composition of the transformed foreground appearance and the background plus some noise. The conditional distribution for the kth image
pixel is given by
P (xk | b, f , m, T, ?) = N (xk | (Tf )k , ?f?1 )mk N (xk | bk , ?b?1 )1?mk
(2)
where ? = (?f , ?b ) are the noise precisions for the foreground layer and the background
layer respectively. The elements of both b and f are given broad Gaussian priors and the
prior on each ? is a broad Gamma distribution. The prior on each element of ? is a Beta
distribution.
3
Factorised variational inference
Given the above model and a set of images {x1 , . . . , xN }, the inference task is to learn
a posterior distribution over all other variables including the background, the foreground
appearance and mask prior, the transformation and mask for each image and the noise
precisions. Direct application of Bayes?s theorem is intractable because this would require
integrating over all unobserved variables. Instead, we turn to the approximate inference
technique of variational inference [6].
Variational inference involves defining a factorised variational distribution Q and then optimising to minimise the Kullback-Leibler divergence between Q and the true posterior
distribution. The motivation behind this methodology is that we expect the posterior to be
unimodal and tightly peaked and so it can be well approximated by a separable distribution.
In this paper, we choose our variational distribution to be factorised with respect to each element of b, f , m and ? and also with respect to ?f and ?b . The factor of Q corresponding
to each of these variables has the same form as the prior over that variable. For example,
the factor for the kth element of ? is a Beta distribution Q(?k ) = ?(?k | a0 , b0 ). The choice
of approximation to the posterior over the affine transform Q(T) is a more complex one,
and will be discussed below.
The optimisation of the Q distribution is achieved by firstly initialising the parameters of all
factors and then iteratively updating each factor in turn so as to minimise the KL divergence
whilst keeping all other factors fixed. If we define H to be the set of all hidden variables,
then the factor for the ith member Hi is updated using
log Q(Hi ) = hlog P ({x1 , . . . , xN }, H)i?Q(Hi ) + const.
(3)
f
b
b
b2
f
f2
? f T (m * x)
?
1
2
? fT m
T
log ?
x
? 12 (1 ? m) * ( x ? b) 2
?
1
2
m * ( x ? Tf )
T log ?
T log(1 ? ? )
? f m* x
? 12 ? f m
? b (1 ? m) * x
? 12 ? b (1 ? m)
1
2
1
2
2
T ?1m
1 ? T ?1m
?1
Tf
Tf 2
?
log ?
log(1 ? ? )
?1
m
1? m
m
1? m
m
log ? b ? ? b ( x ? b) 2
log ? f ? ? f ( x ? Tf ) 2
N
Figure 2: The messages passed when VMP is applied to the generative model. The messages to
or from T are not shown (see text). Where a message is shown as leaving the N plate, the destination
node receives a set of N messages, one from each copy of the nodes within the plate. Where a
message is shown entering the N plate, the message is sent to all copies of the destination node. All
expectations are with respect to the variational distribution Q.
where h.i?Q(Hi ) means the expectation under the distribution given by the product of all
factors of Q except Q(Hi ).
When the model is a Bayesian network, this optimisation procedure can be carried out
in a modular fashion by applying Variational Message Passing (VMP) [7, 8]. Using VMP
makes it very much simpler and quicker to extend, modify, combine or compare probabilistic models; it gives the same results as applying factorised variational inference by hand and
places no additional constraints on the form of the model. In VMP, messages consisting
of vectors of real numbers are sent to each node from its parent and children in the graph.
In our model, the messages to and from all nodes (except T) are shown in Figure 2. By
expressing each variational factor as an exponential family distribution, the ?natural parameter vector? [8] of that distribution can be optimised using (3) by adding messages received
at the corresponding node. For example, if the prior over b is N (b | ?, ? ?1 ), the parameter
vector of the factor Q(b) = N (b | ?0 , ? 0?1 ) is updated from the messages received at b
using
prior
natural param. vector
z
}| {
?0 ? 0
? 21 ? 0
=
z
received messages
z
}| { N
}|
{
X h? (1 ? m ) ? x i
??
bi
i
i
+
.
? 12 ?
? 12 h?bi (1 ? mi )i
(4)
i=1
The form of the natural parameter vector varies for different exponential family distributions (Gaussian, Gamma, Beta, discrete . . . ) but the update equation remains the same.
Following this update, the message being sent from b is recomputed to reflect the new parameters of Q(b). For details of the derivation this update equation and how to determine
VMP messages for a given model, see [8].
Where a set of similar messages are sent corresponding to the pixels of an image, it is
convenient to think instead of a single message where each element is itself an image. It
is efficient to structure the implementation in this way because message computation and
parameter updates can then be carried out using block operations on entire images.
4
Learning the object transformation
Following [3], we decompose the layer transformation into a product of transformations
and define a variational distribution that is separable over each. To allow for affine transformations, we choose to decompose T into three transformations applied sequentially,
T = Txy Trs Ta .
(5)
In this expression, Txy is a two-dimensional translation belonging to a finite set of translations Txy . Similarly, Trs is a rotation and uniform scaling and so the space of transforms
is also two-dimensional and is discretised to form a finite set Trs . The third transformation
Ta is a freeform (non-discretised) affine transform. The variational distribution over the
combined transform T is given by
Q(T) = Q(Txy )Q(Trs )Q(Ta ).
(6)
Because Txy and Trs are discretised, Q(Txy ) and Q(Trs ) are defined to be discrete distributions. We can apply (3) to determine the update equations for these distributions,
= hmi . (Txy hTrs Ta log ?i) + h1 ? mi . (Txy hTrs Ta log(1 ? ?)i)
(7)
+ ?f hmi . x ? Txy hTrs Ta f i ? 12 Txy Trs Ta f 2 + z xy
?1
?1
log Q(Trs ) = Txy m . (Trs hTa log ?i) + Txy (1 ? m) . (Trs hTa log(1 ? ?)i)
?1
1
+ ?f T?1
Trs Ta f 2 + z rs (8)
xy (m ? x) . (Trs hTa f i) ? 2 ?f Txy m
log Q(Txy )
where z xy and z rs are constants which can be found by normalisation.
As described in [3], the evaluation of (7) and (8) for all Txy ? Txy and all Trs ? Trs
can be carried out efficiently using Fast Fourier Transforms in either Cartesian or log-polar
co-ordinate systems. The use of FFTs allows us to make both Txy and Trs large: we set Txy
to contain all translations of a whole number of pixels and Trs to contain 360 rotations (at
1? intervals) and 50 scalings (where each scaling represents a ? 1.5% increase in length
scale). FFTs can be used within the VMP framework as both (7) and (8) involve quantities
that are contained in messages to T (see Figure 2).
Finally, we define the variational distribution over Ta to be a delta function,
Q(Ta ) = ?(Ta ? T?a ).
(9)
Unlike all the other variational factors, this cannot be optimised analytically. To minimise
the KL divergence, we need to find the value of T?a that maximises
?1 ?1
?1
?
?
Fa = T?1
rs Txy m . (Ta hlog ?i) + Trs Txy (1 ? m) . (Ta hlog(1 ? ?)i)
?1 ?1
?1
T?a f 2 .
(10)
+ ?f Trs Txy (m ? x) . (T?a hf i) ? 12 ?f T?1
rs Txy m
This local maximisation is achieved efficiently by using a trust-region Newton method. The
assumption is that the search through Txy and Trs has located the correct posterior mode in
transform space and that it is only necessary to use gradient methods to find the peak of that
mode. This assumption appeared valid for the image sequences used in our experiments,
even when the transformation of the foreground layer was not well approximated by a
similarity transform alone.
Inference in this model is made harder due to an inherent non-identifiability problem. The
pose of the learned appearance and mask prior is undefined and so applying a transform to f
and ? and the inverse of the transform to each Ti results in an unchanged joint distribution.
When applying a variational technique, such non-identifiability leads to many more local
minima in the KL divergence. We partially resolve this issue by adding a constraint to this
model that the expected mask h?i is centred, so that its centre of gravity is in the middle
of the latent image. This constraint is applied by shifting the parameters of Q(?) directly
following each update (and also shifting Q(f ) and each Q(T) appropriately).
Background
Example frame #1
Foreground #1
Normalised frame #1
Object appearance & mask
Example frame #2
Foreground #2
Normalised frame #2
Figure 3: Tracking a hand undergoing extreme affine transformation. The first column shows
the learned background and masked object appearance. The second and third columns contain two
frames from the sequence along with the foreground segmentation for each. The final column shows
each frame transformed by the inverse of the inferred object transform. In each image the red outline
surrounds the area where the transformed mask prior ? is greater than 0.5.
4.1
Using bottom-up information to improve inference
Given that ? is centred, we can significantly improve convergence by using bottom-up information about the translation of the object. For example, the inferred mask m i for each
frame is very informative about the location of the object in that frame. Using sufficient
data, we could learn a conditional model P (Txy | hmi i) and bound Txy by only considering translations with non-negligible posterior mass under this conditional model. Instead,
we use a conservative, hand-constructed bound on Txy based on the assumption that, during inference, the most probable mask under Q(mi ) consists of a (noisy) subset of the true
mask pixels. Suppose the true mask contains M non-zero pixels with second moment of
area IM and the current most probable mask contains V non-zero pixels (V ? M ) with
second moment of area IV . A bound on c, the position of the centre of the inferred mask
relative to the centre of true mask, is given by
diag(ccT ) ? (M ? V ) diag(IM /V ? IV /M ).
(11)
We can gain a conservative estimate of M and IM by using the maximum values of V
and IV across all frames, multiplied by a constant ? ? 1.2. The bound is deliberately
constructed to be conservative; its purpose is to discard settings of Txy that have negligible
probability under the model and so avoid local minima in the variational optimisation. The
bound is updated at each iteration and applied by setting Q(Txy ) = 0 for values of Txy
outside the bound. Q(Txy ) is then re-normalised.
The use of this bound on Txy is intended as a very simple example of incorporating bottomup information to improve inference within a generative model. In future work, we intend
to investigate using more informative bottom-up cues, such as optical flow or tracked interest points, to propose probable transformations within this model. Incorporating such
proposals or bounds into a variational inference framework both speeds convergence and
helps avoid local minima.
5
Experimental results
We present results on two video sequences. The first is of a hand rotating both parallel
to the image plane and around its own axis, whilst also translating in three dimensions.
The sequence consists of 59 greyscale frames, each of size 160 ? 120 pixels (excluding
the border). Our Matlab implementation took about a minute per frame to analyse the se-
Appearance & mask
Example frame #1
Foreground #1
Example frame #2
Foreground #2
Figure 4: Affine tracking of a semi-transparent object.
Appearance & mask
First frame
Foreground
Last frame
Foreground
Figure 5: Tracking an object with changing appearance. A person is tracked throughout a sequence despite their appearance changing dramatically from between first and last frames. The blue
outline shows the inferred mask m which differs slightly from ? due to the object changing shape.
quence, over half of which was spent on the conjugate gradient optimisation step. Figure 3
shows the expected values of the background and foreground layers under the optimised
variational distribution, along with foreground segmentation results for two frames of the
sequence. The right hand column gives another indication of the accuracy of the inferred
transformations by applying the inverse transformation to the entire frame and showing that
the hand then has a consistent normalised position and size. In a video of the hand showing
the tracked outline,1 the outline appears to move smoothly and follow the hand with a high
degree of accuracy, despite the system not using any temporal constraints.
Results for a second sequence showing a cyclist are given in Figure 4. Although the cyclist
and her shadow are tracked correctly, the learned appearance is slightly inaccurate as the
system is unable to capture the perspective foreshortening of the bicycle. This could be
corrected by allowing Ta to include projective transformations.
6
Tracking objects with changing appearance
The model described so far makes the assumption that the appearance of the object does
not change significantly from frame to frame. If the set of images are actually frames from
a video, we can model objects whose appearance changes slowly by allowing the model
to use the object appearance in the previous frame as the basis for its appearance in the
current frame. However, we may not know if the images are video frames and, even if we
do, the object may be occluded or out-of-frame in the previous image. We can cope with
this uncertainty by inferring automatically whether to use the previous frame or the learned
appearance f . Switching between two methods in this way is similar to [9].
The model is extended by introducing a binary variable si for each frame and define a new
appearance variable gi = si f + (1 ? si )T?1
i?1 xi?1 . Hence gi either equals the foreground
appearance f (if si = 1) or the transform-normalised previous frame (if si = 0). For the
first frame, we fix s1 = 1. We then replace f with gi in (2) and then apply VMP within the
resulting Bayesian network.
The extended model is able to track an object even when its appearance changes significantly throughout the image sequence (see Figure 5). The binary variable s i is found to
have an expected value ? 0 for all frames (except the first). Using the tracked appearance
1
Videos of results are available from http://johnwinn.org/Research/affine.
allows the foreground segmentation of each frame to be accurate even though the object is
poorly modelled by the inferred appearance image. If we introduce an abrupt change into
the sequence, for example by reversing the second half of the sequence, hs i i is found to be
? 1 for the frame following the change. In other words, the system has detected not to use
the previous frame at this point, but to revert to using the latent appearance image f .
7
Discussion
We have proposed a method for localising an object undergoing affine transform whilst
simultaneously learning its shape and appearance. This power of this method has been
demonstrated by tracking moving objects in several real videos, including where the appearance of the object changes significantly from start to end. The system makes no assumptions about the speed of motion of the object, requires no special initialisation and is
robust to the object being temporarily occluded or moving out of frame.
A natural extension to this work is to allow multiple layers, with each layer having its own
latent shape and appearance and set of affine transformations. Unfortunately, as the number of latent variables increases, the inference problem becomes correspondingly harder
and an exhaustive search becomes less practical. Instead, we are investigating performing inference in a simpler model where a subset of the variables have been approximately
marginalised out. The results of using this simpler model can then be used to guide inference in the full model. A further interesting addition to the model would be to allow layers
to be grouped into rigid or articulated three-dimensional objects.
Acknowledgments
The authors would like to thank Nebojsa Jojic for suggesting the use of a binary switch
variable for tracking and Tom Minka for helpful discussions.
References
[1] J. Y. A. Wang and E. H. Adelson. Representing moving images with layers. In IEEE Transactions
on Image Processing, volume 3, pages 625?638, 1994.
[2] N. Jojic and B. Frey. Learning flexible sprites in video layers. In Proc. of IEEE Conf. on
Computer Vision and Pattern Recognition, 2001.
[3] B. Frey and N. Jojic. Fast, large-scale transformation-invariant clustering. In Advances in Neural
Information Processing Systems 14, 2001.
[4] M. K. Titsias and C. K. I. Williams. Fast unsupervised greedy learning of multiple objects
and parts from video. 2004. To appear in Proc. Generative-Model Based Vision Workshop,
Washington DC, USA.
[5] C.K.I. Williams and M. K. Titsias. Greedy learning of multiple objects in images using robust
statistics and factorial learning. Neural Computation, 16(5):1039?1062, 2004.
[6] 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.
[7] C. M. Bishop, J. M. Winn, and D. Spiegelhalter. VIBES: A variational inference engine for
Bayesian networks. In Advances in Neural Information Processing Systems, volume 15, 2002.
[8] J. M. Winn and C. M. Bishop. Variational Message Passing. 2004. To appear in Journal of
Machine Learning Research. Available from http://johnwinn.org.
[9] A. Jepson, D. Fleet, and T. El-Maraghi. Robust online appearance models for visual tracking. In
Proc. IEEE Conf. Computer Vision and Pattern Recognition, volume I, pages 415?422, 2001.
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1,826 | 2,659 | Learning syntactic patterns for automatic
hypernym discovery
Rion Snow
Daniel Jurafsky
Andrew Y. Ng
Computer Science Department
Stanford University
Stanford, CA 94305
Linguistics Department
Stanford University
Stanford, CA 94305
Computer Science Department
Stanford University
Stanford, CA 94305
[email protected]
[email protected]
[email protected]
Abstract
Semantic taxonomies such as WordNet provide a rich source of knowledge for natural language processing applications, but are expensive to
build, maintain, and extend. Motivated by the problem of automatically
constructing and extending such taxonomies, in this paper we present a
new algorithm for automatically learning hypernym (is-a) relations from
text. Our method generalizes earlier work that had relied on using small
numbers of hand-crafted regular expression patterns to identify hypernym pairs. Using ?dependency path? features extracted from parse trees,
we introduce a general-purpose formalization and generalization of these
patterns. Given a training set of text containing known hypernym pairs,
our algorithm automatically extracts useful dependency paths and applies
them to new corpora to identify novel pairs. On our evaluation task (determining whether two nouns in a news article participate in a hypernym
relationship), our automatically extracted database of hypernyms attains
both higher precision and higher recall than WordNet.
1
Introduction
Semantic taxonomies and thesauri such as WordNet [5] are a key source of knowledge for
natural language processing applications, and provide structured information about semantic relations between words. Building such taxonomies, however, is an extremely slow
and labor-intensive process. Further, semantic taxonomies are invariably limited in scope
and domain, and the high cost of extending or customizing them for an application has
often limited their usefulness. Consequently, there has been significant recent interest in
finding methods for automatically learning taxonomic relations and constructing semantic
hierarchies. [1, 2, 3, 4, 6, 8, 9, 13, 15, 17, 18, 19, 20, 21]
In this paper, we build an automatic classifier for the hypernym/hyponym relation. A noun
X is a hyponym of a noun Y if X is a subtype or instance of Y. Thus ?Shakespeare? is a
hyponym of ?author? (and conversely ?author? is a hypernym of ?Shakespeare?), ?dog? is
a hyponym of ?canine?, ?desk? is a hyponym of ?furniture?, and so on.
Much of the previous work on automatic semantic classification of words has been based
on a key insight first articulated by Hearst [8], that the presence of certain ?lexico-syntactic
patterns? can indicate a particular semantic relationship between two nouns. Hearst noticed
that, for example, linking two noun phrases (NPs) via the constructions ?Such NP Y as
NP X ?, or ?NP X and other NP Y ?, often implies that NP X is a hyponym of NP Y , i.e., that
NP X is a kind of NP Y . Since then, several researchers have used a small number (typically
less than ten) of hand-crafted patterns like these to try to automatically label such semantic
relations [1, 2, 6, 13, 17, 18]. While these patterns have been successful at identifying some
examples of relationships like hypernymy, this method of lexicon construction is tedious
and severely limited by the small number of patterns typically employed.
Our goal is to use machine learning to automatically replace this hand-built knowledge.
We first use examples of known hypernym pairs to automatically identify large numbers of
useful lexico-syntactic patterns, and then combine these patterns using a supervised learning algorithm to obtain a high accuracy hypernym classifier. More precisely, our approach
is as follows:
1. Training:
(a) Collect noun pairs from corpora, identifying examples of hypernym pairs
(pairs of nouns in a hypernym/hyponym relation) using WordNet.
(b) For each noun pair, collect sentences in which both nouns occur.
(c) Parse the sentences, and automatically extract patterns from the parse tree.
(d) Train a hypernym classifier based on these features.
2. Test:
(a) Given a pair of nouns in the test set, extract features and use the classifier to
determine if the noun pair is in the hypernym/hyponym relation or not.
The rest of the paper is structured as follows. Section 2 introduces our method for automatically discovering patterns indicative of hypernymy. Section 3 then describes the setup of
our experiments. In Section 4 we analyze our feature space, and in Section 5 we describe
a classifier using these features that achieves high accuracy on the task of hypernym identification. Section 6 shows how this classifier can be improved by adding a new source of
knowledge, coordinate terms.
2
Representing lexico-syntactic patterns with dependency paths
The first goal of our work is to automatically identify lexico-syntactic patterns indicative
of hypernymy. In order to do this, we need a representation space for expressing these
patterns. We propose the use of dependency paths as a general-purpose formalization of
the space of lexico-syntactic patterns. Dependency paths have been used successfully in
the past to represent lexico-syntactic relations suitable for semantic processing [11].
A dependency parser produces a dependency tree that represents the syntactic relations
between words by a list of edge tuples of the form:
(word1 ,CATEGORY 1 :RELATION:CATEGORY 2 , word2 ). In this formulation each word is
the stemmed form of the word or multi-word phrase (so that ?authors? becomes ?author?),
and corresponds to a specific node in the dependency tree; each category is the part of
speech label of the corresponding word (e.g., N for noun or P REP for preposition); and
the relation is the directed syntactic relationship exhibited between word1 and word2
(e.g., OBJ for object, MOD for modifier, or CONJ for conjunction), and corresponds to a
specific link in the tree. We may then define our space of lexico-syntactic patterns to be all
shortest paths of four links or less between any two nouns in a dependency tree. Figure 1
shows the partial dependency tree for the sentence fragment ?...such authors as Herrick and
Shakespeare? generated by the broad-coverage dependency parser MINIPAR [10].
...
-N:pre:PreDet
authors
-N:mod:Prep
-N:punc:U
such
-Prep:pcomp-n:N
as
Herrick
and
-N:conj:N
-Prep:pcomp-n:N
Shakespeare
Figure 1: MINIPAR dependency tree example with transform
NP X and other NP Y :
NP X or other NP Y :
NP Y such as NP X :
Such NP Y as NP X :
NP Y including NP X :
NP Y , especially NP X :
(and,U: PUNC :N),-N: CONJ :N, (other,A: MOD :N)
(or,U: PUNC :N),-N: CONJ :N, (other,A: MOD :N)
N: PCOMP - N :P REP,such as,such as,P REP : MOD :N
N: PCOMP - N :P REP,as,as,P REP : MOD :N,(such,P RE D ET: PRE :N)
N: OBJ :V,include,include,V: I :C,dummy node,dummy node,C: REL :N
-N: APPO :N,(especially,A: APPO - MOD :N)
Table 1: Dependency path representations of Hearst?s patterns
We then remove the original nouns in the noun pair to create a more general pattern. Each
dependency path may then be presented as an ordered list of dependency tuples. We extend
this basic MINIPAR representation in two ways: first, we wish to capture the fact that certain function words like ?such? (in ?such NP as NP?) or ?other? (in ?NP and other NP?) are
important parts of lexico-syntactic patterns. We implement this by adding optional ?satellite links? to each shortest path, i.e., single links not already contained in the dependency
path added on either side of each noun. Second, we capitalize on the distributive nature of
the syntactic conjunction relation (nouns linked by ?and? or ?or?, or in comma-separated
lists) by distributing dependency links across such conjunctions. For example, in the simple 2-member conjunction chain of ?Herrick? and ?Shakespeare? in Figure 1, we add the
entrance link ?as, -P REP : PCOMP - N :N? to the single element ?Shakespeare? (as a dotted
line in the figure). Our extended dependency notation is able to capture the power of the
hand-engineered patterns described in the literature. Table 1 shows the six patterns used in
[1, 2, 8] and their corresponding dependency path formalizations.
3
Experimental paradigm
Our goal is to build a classifier which, when given an ordered pair of nouns, makes the
binary decision of whether the nouns are related by hypernymy.
All of our experiments are based on a corpus of over 6 million newswire sentences.1 We
first parsed each of the sentences in the corpus using MINIPAR. We extract every pair of
nouns from each sentence.
752,311 of the resulting unique noun pairs were labeled as Known Hypernym or Known
Non-Hypernym using WordNet.2 A noun pair (ni , nj ) is labeled Known Hypernym if nj
is an ancestor of the first sense of ni in the WordNet hypernym taxonomy, and if the only
?frequently-used?3 sense of each noun is the first noun sense listed in WordNet. Note that
nj is considered a hypernym of ni regardless of how much higher in the hierarchy it is with
respect to ni . A noun pair may be assigned to the second set of Known Non-Hypernym
pairs if both nouns are contained within WordNet, but neither noun is an ancestor of the
other in the WordNet hypernym taxonomy for any senses of either noun. Of our collected
noun pairs, 14,387 were Known Hypernym pairs, and we assign the 737,924 most frequently occurring Known Non-Hypernym pairs to the second set; this number is selected
to preserve the roughly 1:50 ratio of hypernym-to-non-hypernym pairs observed in our
hand-labeled test set (discussed below).
We evaluated our binary classifiers in two ways. For both sets of evaluations, our classifier
was given a pair of nouns from an unseen sentence and had to make a hypernym vs. nonhypernym decision. In the first style of evaluation, we compared the performance of our
classifiers against the Known Hypernym versus Known Non-Hypernym labels assigned by
1
The corpus contains articles from the Associated Press, Wall Street Journal, and Los Angeles
Times, drawn from the T IPSTER 1, 2, 3, and T REC 5 corpora [7]. Our most recent experiments
(presented in Section 6) include articles from Wikipedia (a popular web encyclopedia), extracted
with the help of Tero Karvinen?s Tero-dump software.
2
We access WordNet 2.0 via Jason Rennie?s WordNet::QueryData interface.
3
A noun sense is determined to be ?frequently-used? if it occurs at least once in the sense-tagged
Brown Corpus Semantic Concordance files (as reported in the cntlist file distributed as part of
WordNet 2.0). This determination is made so as to reduce the number of false hypernym/hyponym
classifications due to highly polysemous nouns (nouns which have multiple meanings).
Hypernym Classifiers on WordNet-labeled dev set
1
Logistic Regression (Buckets)
Logistic Regression (Binary)
Hearst Patterns
And/Or Other Pattern
0.9
0.8
Precision
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 2: Hypernym pre/re for all features
0
0.1
0.2
0.3
0.4
0.5
Recall
0.6
0.7
0.8
0.9
1
Figure 3: Hypernym classifiers
WordNet. This provides a metric for how well our classifiers do at ?recreating? WordNet?s
judgments.
For the second set of evaluations we hand-labeled a test set of 5,387 noun pairs from
randomly-selected paragraphs within our corpus (with part-of-speech labels assigned by
MINIPAR). The annotators were instructed to label each ordered noun pair as one of
?hyponym-to-hypernym?, ?hypernym-to-hyponym?, ?coordinate?, or ?unrelated? (the coordinate relation will be defined in Section 6). As expected, the vast majority of pairs
(5,122) were found to be unrelated by these measures; the rest were split evenly between
hypernym and coordinate pairs (134 and 131, resp.).
Interannotator agreement was obtained between four labelers (all native speakers of English) on a set of 511 noun pairs, and determined for each task according to the averaged
F-Score across all pairs of the four labelers. Agreement was 83% and 64% for the hypernym and coordinate term classification tasks, respectively.
4
Features: pattern discovery
Our first study focused on discovering which dependency paths might prove useful features
for our classifiers. We created a feature lexicon of 69,592 dependency paths, consisting of
every dependency path that occurred between at least five unique noun pairs in our corpus.
To evaluate these features, we constructed a binary classifier for each pattern, which simply
classifies a noun pair as hypernym/hyponym if and only if the specific pattern occurs at least
once for that noun pair. Figure 2 depicts the precision and recall of all such classifiers (with
recall at least .0015) on the WordNet-labeled data set.4 Using this formalism we have been
able to capture a wide variety of repeatable patterns between hypernym/hyponym noun
pairs; in particular, we have been able to rediscover the hand-designed patterns originally
proposed in [8] (the first five features, marked in red)5 , in addition to a number of new
patterns not previously discussed (of which four are marked as blue triangles in Figure
2 and listed in Table 2. This analysis gives a quantitative justification to Hearst?s initial
intuition as to the power of hand-selected patterns; nearly all of Hearst?s patterns are at the
high-performance boundary of precision and recall for individual features.
NP Y like NP X :
NP Y called NP X :
NP X is a NP Y :
NP X , a NP Y (appositive):
N: PCOMP - N :P REP,like,like,P REP : MOD :N
N: DESC :V,call,call,V: VREL :N
N: S :VBE,be,be,-VBE: PRED :N
N: APPO :N
Table 2: Dependency path representations of other high-scoring patterns
4
Redundant features consisting of an identical base path to an identified pattern but differing only
by an additional ?satellite link? are marked in Figure 2 by smaller versions of the same symbol.
5
We mark the single generalized ?conjunction other? pattern -N: CONJ :N, (other,A: MOD :N) to
represent both of Hearst?s original ?and other? and ?or other? patterns.
Best Logistic Regression (Buckets):
Best Logistic Regression (Binary):
Best Multinomial Naive Bayes:
Best Complement Naive Bayes:
Hearst Patterns:
?And/Or Other? Pattern:
0.3480
0.3200
0.3175
0.3024
0.1500
0.1170
Table 3: Average maximum F-scores for cross validation on WordNet-labeled training set
5
A hypernym-only classifier
Our first hypernym classifier is based on the intuition that unseen noun pairs are more
likely to be a hypernym pair if they occur in the test set with one or more lexico-syntactic
patterns found to be indicative of hypernymy. We record in our noun pair lexicon each
noun pair that occurs with at least five unique paths from our feature lexicon discussed in
the previous section. We then create a feature count vector for each such noun pair. Each
entry of the 69,592-dimension vector represents a particular dependency path, and contains
the total number of times that that path was the shortest path connecting that noun pair in
some dependency tree in our corpus. We thus define as our task the binary classification of
a noun pair as a hypernym pair based on its feature vector of dependency paths.
We use the WordNet-labeled Known Hypernym / Known Non-Hypernym training set defined in Section 3. We train a number of classifiers on this data set, including multinomial
Naive Bayes, complement Naive Bayes [16], and logistic regression. We perform model
selection using 10-fold cross validation on this training set, evaluating each model based
on its maximum F-Score averaged across all folds. The summary of average maximum
F-scores is presented in Table 3, and the precision/recall plot of our best models is presented in Figure 3. For comparison, we evaluate two simple classifiers based on past work
using only a handful of hand-engineered features; the first simply detects the presence of
at least one of Hearst?s patterns, arguably the previous best classifier consisting only of
lexico-syntactic patterns, and as implemented for hypernym discovery in [2]. The second
classifier consists of only the ?NP and/or other NP? subset of Hearst?s patterns, as used
in the automatic construction of a noun-labeled hypernym taxonomy in [1]. In our tests
we found greatest performance from a binary logistic regression model with 14 redundant
threshold buckets spaced at the exponentially increasing intervals {1, 2, 4, ...4096, 8192};
our resulting feature space consists of 974,288 distinct binary features. These buckets are
defined such that a feature corresponding to pattern p at threshold t will be activated by
a noun pair n if and only if p has been observed to occur as a shortest dependency path
between n at least t times.
Our classifier shows a dramatic improvement over previous classifiers; in particular, using
our best logistic regression classifier trained on newswire corpora, we observe a 132%
relative improvement of average maximum F-score over the classifier based on Hearst?s
patterns.
6
Using coordinate terms to improve hypernym classification
While our hypernym-only classifier performed better than previous classifiers based on
hand-built patterns, there is still much room for improvement. As [2] points out, one problem with pattern-based hypernym classifiers in general is that within-sentence hypernym
pattern information is quite sparse. Patterns are useful only for classifying noun pairs which
happen to occur in the same sentence; many hypernym/hyponym pairs may simply not occur in the same sentence in the corpus. For this reason [2], following [1] suggests relying
on a second source of knowledge: ?coordinate? relations between nouns. The WordNet
glossary defines coordinate terms as ?nouns or verbs that have the same hypernym?. Here
we treat the coordinate relation as a symmetric relation that exists between two nouns that
share at least one common ancestor in the hypernym taxonomy, and are therefore ?the
same kind of thing? at some level. Many methods exist for inferring that two nouns are
coordinate terms (a common subtask in automatic thesaurus induction). We expect that
Interannotator Agreement:
Distributional Similarity Vector Space Model:
Thresholded Conjunction Pattern Classifier:
Best WordNet Classifier:
0.6405
0.3327
0.2857
0.2630
Table 4: Summary of maximum F-scores on hand-labeled coordinate pairs
Coordinate term classifiers on hand-labeled test set
1
0.9
0.8
0.8
0.7
0.6
0.5
0.4
0.6
0.5
0.4
0.3
0.3
0.2
0.2
0.1
0
Interannotator Agreement
TREC+Wikipedia
TREC, Hybrid
TREC, Hypernym-only
WordNet Classifiers
Hearst Patterns
And/Or Other Pattern
0.9
Precision
Precision
0.7
Hypernym Classifiers on hand-labeled test set
1
Interannotator Agreement
Distributional Similarity
Conjunct Pattern
WordNet
0.1
0
0.1
0.2
0.3
0.4
0.5
Recall
0.6
0.7
Figure 4: Coordinate classifiers on
hand-labeled test set
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
Recall
0.6
0.7
0.8
0.9
Figure 5: Hypernym classifiers on
hand-labeled test set
using coordinate information will increase the recall of our hypernym classifier: if we are
confident that two nouns ni , nj are coordinate terms, and that nj is a hyponym of nk , we
may then infer with higher probability that ni is similarly a hyponym of nk ?despite never
having encountered the pair (ni , nk ) within a single sentence.
6.1 Coordinate Term Classification
Prior work for identifying coordinate terms includes automatic word sense clustering methods based on distributional similarity (e.g., [12, 14]) or on pattern-based techniques, specifically using the coordination pattern ?X, Y, and Z? (e.g., [2]). We construct both types of
classifier. First we construct a vector-space model similar to [12] using single MINIPAR
dependency links as our distributional features.6 We use the normalized similarity score
from this model for coordinate term classification. We evaluate this classifier on our handlabeled test set, where of 5,387 total pairs, 131 are labeled as ?coordinate?. For purposes
of comparison we construct a series of classifiers from WordNet, which make the binary
decision of determining whether two nouns are coordinate according to whether they share
a common ancestor within k nouns higher up in the hypernym taxonomy, for all k from 1
to 6. Also, we compare a simple pattern-based classifier based on the conjunction pattern,
which thresholds simply on the number of conjunction patterns found between a given pair.
Results of this experiment are shown in Table 4 and Figure 4.
The strong performance of the simple conjunction pattern model suggests that it may be
worth pursuing an extended pattern-based coordinate classifier along the lines of our hypernym classifier; for now, we proceed with our distributional similarity vector space model
(with a 16% relative F-score improvement over the conjunction model) in the construction
of a combined hypernym-coordinate hybrid classifier.
6.2 Hybrid hypernym-coordinate classification
We now combine our hypernym and coordinate models in order to improve hypernym classification. We define two probabilities of pair relationships between nouns: P (n i < nj ),
H
6
We use the same 6 million MINIPAR-parsed sentences used in our hypernym training set. Our
feature lexicon consists of the 30,000 most frequent noun-connected dependency edges. We construct
feature count vectors for each of the most frequently occurring 163,198 individual nouns. As in [12]
we normalize these feature counts with pointwise mutual information, and compute as our measure
of similarity the cosine coefficient between these normalized vectors.
Interannotator Agreement:
TREC+Wikipedia Hypernym-only Classifier (Logistic Regression):
TREC Hybrid Linear Interpolation Hypernym/Coordinate Model:
TREC Hypernym-only Classifier (Logistic Regression):
Best WordNet Classifier:
Hearst Patterns Classifier:
?And/Or Other? Pattern Classifier:
0.8318
0.3592
0.3268
0.2714
0.2339
0.1417
0.1386
Table 5: Maximum F-Score of hypernym classifiers on hand-labeled test set
representing the probability that noun ni has nj as an ancestor in its hypernym hierarchy,
and P (ni ? nj ), the probability that nouns ni and nj are coordinate terms, i.e., that they
C
share a common hypernym ancestor at some level. Defining the probability produced by
our best hypernym-only classifier as Pold (ni < nj ), and a probability obtained by normalH
izing the similarity score from our coordinate classifier as P (ni ? nj ), we apply a simple
C
linear interpolation scheme to compute a new hypernymy probability. Specifically, for each
pair of nouns (ni , nk ), we recompute the probability that nk is a hypernym of ni as:7
P
Pnew (ni < nk ) ? ?1 Pold (ni < nk ) + ?2 j P (ni ? nj )Pold (nj < nk )
H
7
H
C
H
Results
Our hand-labeled dataset allows us to compare our classifiers with WordNet and the previous feature-based methods, now using the human labels as ground truth. Figure 5 shows
the performance of each method in a precision/recall plot. We evaluated several classifiers
based on the WordNet hypernym taxonomy.8 The best WordNet-based results are plotted
in Figure 5. Our logistic regression hypernym-only model trained on the newswire corpora has a 16% relative F-score improvement over the best WordNet classifier, while the
combined hypernym/coordinate model has a 40% relative F-score improvement. Our bestperforming classifier is a hypernym-only model additionally trained on the Wikipedia corpus, with an expanded feature lexicon of 200,000 dependency paths; this classifier shows a
54% improvement over WordNet. In Table 5 we list the maximum F-scores of each method.
In Table 6 we analyze the disagreements between the highest F-score WordNet classifier
and our combined hypernym/coordinate classifier.9
8
Conclusions
Our experiments demonstrate that automatic methods can be competitive with WordNet
for the identification of hypernym pairs in newswire corpora. In future work we will use
the presented method to automatically generate flexible, statistically-grounded hypernym
taxonomies directly from corpora. These taxonomies will be made publicly available to
complement existing semantic resources.
7
We constrain our parameters ?1 , ?2 such that ?1 +?2 = 1; we set these parameters using 10-fold
cross-validation on our hand-labeled test set. For our final evaluation we use ? 1 = 0.7.
8
We tried all combinations of the following parameters: the maximum number of senses of a hyponym for which to find hypernyms, the maximum distance between the hyponym and its hypernym
in the WordNet taxonomy, and whether or not to allow synonyms. The WordNet model achieving the
maximum F-score uses only the first sense of a hyponym and allows a maximum distance of 4 links
between a hyponym and hypernym.
9
There are 31 such disagreements, with WordNet agreeing with the human labels on 5 and our hybrid model agreeing on the other 26. We additionally inspect the types of noun pairs where our model
improves upon WordNet, and find that at least 30% of our model?s improvements are not restricted
to Named Entities; given that the distribution of Named Entities among the labeled hypernyms in our
test set is over 60%, this gives us hope that our classifier will perform well at the task of hypernym
induction even in more general, non-newswire domains.
Type of Noun Pair
NE: Person
NE: Place
NE: Company
NE: Other
Not Named Entity:
Count
7
7
2
1
9
Example Pair
?John F. Kennedy / president?, ?Marlin Fitzwater / spokesman?
?Diamond Bar / city?, ?France / place?
?American Can / company?, ?Simmons / company?
?Is Elvis Alive / book?
?earthquake / disaster?, ?soybean / crop?
Table 6: Analysis of improvements over WordNet
Acknowledgments
We thank Kayur Patel, Mona Diab, Allison Buckley, and Todd Huffman for useful discussions and assistance annotating data. R. Snow is supported by an NDSEG Fellowship sponsored by the DOD and AFOSR. This work is also supported by the ARDA
AQUAINT program, and by the Department of the Interior/DARPA under contract number
NBCHD030010.
References
[1] Caraballo, S.A. (2001) Automatic Acquisition of a Hypernym-Labeled Noun Hierarchy from
Text. Brown University Ph.D. Thesis.
[2] Cederberg, S. & Widdows, D. (2003) Using LSA and Noun Coordination Information to Improve
the Precision and Recall of Automatic Hyponymy Extraction. Proc. of CoNLL-2003, pp. 111?118.
[3] Ciaramita, M. & Johnson, M. (2003) Supersense Tagging of Unknown Nouns in WordNet. Proc.
of EMNLP-2003.
[4] Ciaramita, M., Hofmann, T., & Johnson, M. (2003) Hierarchical Semantic Classification: Word
Sense Disambiguation with World Knowledge. Proc. of IJCAI-2003.
[5] Fellbaum, C. (1998) WordNet: An Electronic Lexical Database. Cambridge, MA: MIT Press.
[6] Girju, R., Badulescu A., & Moldovan D. (2003) Learning Semantic Constraints for the Automatic
Discovery of Part-Whole Relations. Proc. of HLT-2003.
[7] Harman, D. (1992) The DARPA TIPSTER project. ACM SIGIR Forum 26(2), Fall, pp. 26?28.
[8] Hearst, M. (1992) Automatic Acquisition of Hyponyms from Large Text Corpora. Proc. of the
Fourteenth International Conference on Computational Linguistics, Nantes, France.
[9] Hearst, M. & Sch?utze, H. (1993) Customizing a lexicon to better suit a computational task. In
Proc. of the ACL SIGLEX Workshop on Acquisition of Lexical Knowledge from Text.
[10] Lin, D. (1998) Dependency-based Evaluation of MINIPAR. Workshop on the Evaluation of
Parsing Systems, Granada, Spain
[11] Lin, D. & Pantel P. (2001) Discovery of Inference Rules for Question Answering. Natural
Language Engineering, 7(4), pp. 343?360.
[12] Pantel, P. (2003) Clustering by Committee. Ph.D. Dissertation. Department of Computing Science, University of Alberta.
[13] Pantel, P. & Ravichandran, D. (2004) Automatically Labeling Semantic Classes. Proc. of
NAACL-2004.
[14] Pereira, F., Tishby, N., & Lee, L. (1993) Distributional Clustering of English Words. Proc. of
ACL-1993, pp. 183?190.
[15] Ravichandran, D. & Hovy, E. (2002) Learning Surface Text Patterns for a Question Answering
system. Proc. of ACL-2002.
[16] Rennie J., Shih, L., Teevan, J., & Karger, D. (2003) Tackling the Poor Assumptions of Naive
Bayes Text Classifiers. Proc. of ICLM-2003.
[17] Riloff, E. & Shepherd, J. (1997) A Corpus-Based Approach for Building Semantic Lexicons.
Proc of EMNLP-1997.
[18] Roark, B. & Charniak, E. (1998) Noun-phrase co-occurerence statistics for semi-automaticsemantic lexicon construction. Proc. of ACL-1998, 1110?1116.
[19] Tseng, H. (2003) Semantic classification of unknown words in Chinese. Proc. of ACL-2003.
[20] Turney, P.D., Littman, M.L., Bigham, J. & Shanyder, V. (2003) Combining independent modules to solve multiple-choice synonym and analogy problems. Proc. of RANLP-2003, pp. 482?489.
[21] Widdows, D. (2003) Unsupervised methods for developing taxonomies by combining syntactic
and statistical information. Proc. of HLT/NAACL 2003, pp. 276?283.
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Viola
Neurally Inspired Plasticity in Oculomotor
Processes
Paul A. Viola
Artificial Intelligence Laboratory
M"assachusetts Institute of Technology
Cambridge, MA 02139
ABSTRACT
We have constructed a two axis camera positioning system which
is roughly analogous to a single human eye. This Artificial-Eye (Aeye) combines the signals generated by two rate gyroscopes with
motion information extracted from visual analysis to stabilize its
camera. This stabilization process is similar to the vestibulo-ocular
response (VOR); like the VOR, A-eye learns a system model that
can be incrementally modified to adapt to changes in its structure,
performance and environment. A-eye is an example of a robust sensory system that performs computations that can be of significant
use to the designers of mobile robots.
1
Introduction
We have constructed an "artificial eye" (A-eye), an autonomous robot that incorporates a two axis camera positioning system (figure 1). Like a the human oculomotor
system, A-eye can estimate the rotation rate of its body with a gyroscope and estimate the rotation rate of its "eye" by measuring image slip acr~ its "retina". Using
the gyroscope to sense rotation, A-eye attempts to stabilize its camera by driving
the camera motors to counteract body motion. The conversion of gyro output to
motor command is dependent on the characteristics of the gyroscope, the structure
of camera lensing system and the response of the motors. A correctly functioning stabilization system must model the characteristics of each of these external
variables.
Neurally Inspired Plasticity in Oculomotor Processes
Figure 1: The construction of A-eye can be viewed in rough analogy to the human
oculomotor system. In place of an eye, A-eye has a camera on a two axis positioning platform. In place of the circular canals of the inner ear, A-eye has two rate
gyroscopes that measure rotation in perpendicular axes.
Since camera motion implies stabilization error, A-eye uses a visual estimate of
camera motion to incrementally update its system model. \Vhen the camera is correctly stabilized there is no statistically significant slip. \Vhenever a particular gyro
measurement is associated with a result camera motion, A-eye makes an incremental change to its response to that particular measurement to reduce that error in
the future.
A-eye was built for two reasons: to facilitate the operation of complex visually
guided mobile robots and to explore the applicability of simple learning techniques
to the construction of a robust robot.
2
Autonomous Robots
An autonomous robot must function correctly for long periods of time without human intervention. It is certainly difficult to create an autonomous robot or process
that will function accurately, both initially and perpetually. To achieve such a goal,
autonomous processes must be able to adapt both to unforeseen aspects of the environment and inaccuracies in construction. One approach to attaining successful
autonomous performance would entail the full characterization of the robot's structure, its performance requirements, and its relationship with the environment. Since
clearly both the robot and its environment are susceptible to change any characterization could not be static. In contrast, our approach only partially categorizes
the robot's structure, environment, and task. Without more detailed information
initial performance is inaccurate. However, by using a measure of error in performance initially partial categorization can incrementally improved. In addition, a
change to system performance can be compensated continually. In this way the
extensive analysis and engineering that would be required to characterize, foresee
291
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Viola
and circumvent variability can be greatly reduced.
3
The VOR
The oculomotor processes found in vertebrates are well studied. examples of adaptive, visually guided processing [Gou85]. The three oculomotor processes found
almost universally in vertebrates (the vestibulo-ocular response, the optokinetic
system, and the saccadic system), accurately perform ocular positioning tasks with
little or no conscious direction. The response times of these systems demonstrate
that little high level, "conscious", processing could take place. In a limited sense
these processes are autonomous, and it should come as no surprise that they are
quite plastic. Such plasticity is necessary to counteract the foreseeable changes in
the eye due to growth and aging and the unforeseeable changes due to illness and
injury.
The VOR works to counteract the motion of a creature in its environment. A correctly functioning VOR ensures that a creature "sees" as little unintended motion
as possible. Miles [FAM81] and others have demonstrated that the VOR is an
adaptive motor response, capable of significant recalibration in a matter of days.
Adaptation can be demonstrated by the use of inverting or magnifying spectacles.
While wearing these glasses the correct orbital motion of the eye, given a particular head motion, is significantly different from the normal response. Initially, the
response to head motion is an incorrect eye motion. With time eye motion begins
to approach the correct counteracting motion. This kind of adaptation allows an
animal to continue functioning in spite of injury or illness.
4
The Device
A-eye is a small autonomous robot that incorporates a CCD camera, a three wheel
base, a two axis pitch/yaw camera positioning platform, and two rate gyroscopes.
On board processing includes a Motorola microcontroller and 68020 based video
processing board. Including batteries, A-eye is a foot high cylinder that is 12 inches
wide. In its present configuration A-eye can run autonomously for up to three hours
(figure 2).
A-eye's goal is to learn how to keep its camera stable as its base trundles down
corridors. There are two sources of information regarding the motion of A-eye's
base: gyro rotation measurements and optical flow. Rate gyroscopes measure base
rotation rate directly. Visual analysis can be used to estimate motion by a number
of methods of varying complexity (see [Hil83] for a good overview). By attempting
to measure only camera rotation from slip complexity can be avoided. The simple
method we have chosen measures the slip of images across the retina.
4.1
Visual Rotation Estimation
Our approach to camera rotation estimation uses a pre-processing subunit commonly known as a "Reichard detector" which for clarity we will call a shift and
correlate .nit [PR73]. A shift and cOJTelate unit has as its inputs a set of samples
Neurally Inspired Plasticity in Oculomotor Processes
Figure 2: A photo of the current state of A-eye.
from a blurred area of the retina. It shifts these inputs spatially and correlates them
with a previous, unshifted set of inputs. \"hen two succeeding images are identical except for a spatial shift, the units which perform that shift respond strongly.
Clearly the activity of a shift and correlate anit contains information about retinal
motion. Due to the size and direction of shift, some detectors will be sensitive
to small motions, others large motions, and each will be sensitive to a particular
direction of motion.
The input from the shift and correlate units is used to build value-unit encoded
retinal velocity map, in which each unit is sensitive to a different direction and
range of velocities. The map has 9 units in a 3 by 3 grid (fig 3). To create such
a map, each of the shift and correlate uniu is connected to every map unit. By
moving the camera, displaced images that are examples of motion, are generated.
The motor command that generated this motion example corresponds to a unit
in the visual velocity map. Connection weights are updated by a standard least
squares learning rule. In operation, the most ac.tive unit represents the estimate of
visual motion.
4.2
Gyroscope Rotation Estimation
Contrary to first intuition, vertebrates do not rely on visual information to stabilize their eyes. Instead head rotation information measured by the inner ear, or
the vestibula, is used keep the eyes stable. Animals do not measure ocular motion
directly from visual information for two reasons: a) the response rates of photoreceptors prevent useful visual processing during rapid eye movements [Gou85] b) the
293
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Viola
(s)CD0
808
0CDG)
Figure 3: The 9 unit velocity map has 1 unit for each of the 8 "chess moves" .
Base
Rotation
Rate
--.
Gyroscope
Transfa
Function
Inverting
Transfer
Function
Motor
Transfer
Function
......
Eye
Rotation
Rate
Figure 4: Open-loop control of ocular position based on gyroscope output.
required visual analysis takes approximately lOOms l . These difficulties combine to
prevent rapid response to unexpected head and body motions. A-eye is beset with
similar limitations and we have chosen a similar solution.
The output of the gyroscope is some function of head rotation rate. Stabilization
is achieved by driving the ocular motors directly in opposition to the measured
velocity (fig 4). This counteract rotation of the base in one direction by moving
the camera in the opposite direction. Such an open-loop system is very simple and
can perform well; they are unfortunately very reliant on proper calibration and
recalibration to maintain performance [Oga70).
A-eye maintains calibration information in the form of a function from gyroscope
output to motor velocity command. This function is an 8 unit gaussian radial basis
approximation network (TP89). Basis function approximation has excellent computational properties while representing wide variety of smooth functions. Weights
are modified with a simple least squares update rule, based errors in camera motion
detected visually.
5
Training A-eye
A-eye learns to perform the VOR in a two phase process. First, the measurement of
visual motion is calibrated to the generation of camera motion commands. Second,
lOcular following, the tendency to follow the motion of a !Cene in the &beence of head motion,
has a typical latency of lOOms [FM87).
Neurally Inspired Plasticity in Oculomotor Processes
the stabilizing motor responses to gyroscope measurements are approximated. This
approximation is modified based on a visual estimate of camera motion .
By observing motor commands and comparing them to the resulting visual motion,
a map from visual motion to appropriate motor command can be learned. To train
the visual motion map, A-eye performs a set of characteristic motions and observes
the results. Each motor command is categorized as one of the 9 distinct motions
encoded by the visual motion map. With each motion, the connections from shift
and correlate units to the visual motion map are updated so that issuing a motor
command results in activity in the correct visual motion unit. Because no reference
is made to external variables, this measure of visual motion is completely relative
to the function of the camera motors. The visual motion map plays the role of ertor
signal for later learning.
By observing both the gyroscope output and the visual response from head motion,
A-eye learns the appropriate compensating eye motion for all head motions. Eye
compensation motions are the result of motor commands generated by the approximation network applied to the gyroscope output. Incorrect responses will cause
visual mot.ion. This motion, as measured by the visual motion detector, is the error
signal that drives the modification of the approximation network. This is the heart
of the adaptation in the VOR.
5.1
Results
While training the motion detector and approximation network there are 5 training
events per second (the visual analysis takes about 200 msec). Training the visual
motion detector can take up to 10 minutes (in a few environments the weights refuse
to settle on the correct values). While it is possible to hand wire a detector that
is 95% accurate, most learned detectors worked well, attaining 85% accuracy. In
both cases, the detectors have the desirable capability of rejecting object motion
whenever there is actual camera motion (this is due to the global nature of the
analysis).
The approximation network converges to a function that performs well in minutes
(figure 5) . Analysis of the images generated by the camera leads us to bound the
cumulative error in rotation over a 1 minute trial at 5 degrees (we believe this
approaches the accuracy limitations inherent in the gyroscope).
An approach to reducing this gyroscope error involves yet another oculomotor pr~
cess: optokinetic nystagmus (OKN). This is the tendency for an otherwise undirected eye to follow visual motion in the absence of vestibular cues. A-eye's visual
motion map is in motor coordinates. By directing the camera in the opposite direction from observed motion, residual errors in VOR can be reduced.
6
Application
We claim that the stabilization that results from a correctly calibrated VOR is
useful both for navigation and scene analysis. A stable inertial reference can act
to assist tactical navigation when traversing rough terrain. Large body attitude
295
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Viola
4000.
2000.
-60.
-40.
20.
40.
60.
-2000.
-4000.
Figure 5: A correct transfer function (rough) and the learned (smoother) approximation.
changes, that can result from such travel, make it difficult to maintain a navigational
bearing. However, when there exists a relatively stable inertial reference fr :lme less
analysis need be performed to predict or sense changes in bearing by other means.
The VOR is especially applicable to legged vehicles, where the terrain and the form
of locomotion can cause constant rapid changes in attitude [Rai89] [Ang89]. The
task of adapting conventional vision systems to such vehicles is formidable. As
the rate of pitching increases. the quality of video images degrade, while the task
of finding a correspondence between successive images will increase in complexity.
With the addition of the visual stabilization that A-eye can provide, an otherwise
complex visual analysis task can be much simplified.
7
Conclusions
A-eye is in part a response to the observation that static calibration is a disastrous
weakness. Static calibration not only forces an engineer to expend additional effort
at design time. it requires constant performance monitoring and recalibration. By
creating a device that monitors its own performance and adapts to changes. significant work can be saved in design and at numerous times during the lifetime of the
device.
A-eye is also in part a confirmation that simple. tractable and reliable learning
. mechanisms are sufficient to perform useful motor learning.
Finally. A-eye is in part a demonstration that useful visual processing can be performed in real-time with an reasonable amount of computation. This processing
yields the additional side-benefit of simplifying the complex task of visual recognition.
Neurally Inspired Plasticity in Oculomotor Processes
Acknowledgements
This report describes research done at the Artificial Intelligence Laboratory of the
Massachusetts Institute of Technology. Support for this research was provided by
Hughes Artificial Intelligence Center contract #SI-804475-D, the Office of Naval
Research contract NOOOI4-86-K-0685, and the Defense Advanced Research Projects
Agency under Office of Naval Research contract ~OOOI4-85-K-0124.
References
[Ang89] Colin Angle. Genghis, a six legged autonomous walking robot. Masterts
thesis, MIT, 1989.
[FAM81] S. G. Lisberger F . A. Miles. Plasticity in the vestibulo-ocular reflex: A
new hypothesis. Ann. Rev. Neurosci., 4:273-299,1981.
[FM87]
K. Kawano F.A. Miles. Visual stabilization of the eyes. TINS, 4(10):153158, 1987. Reference on Opto-kinetic nystagmus latency.
[Gou85] Peter Gouras. Oculomotor system. In James Schwartz Eric Kandel, editor, Principles of Neuroscience, chapter 34. Elsevier Science Publishing,
1985.
[Hil83]
Ellen C. Hildreth. The Measurement of Visual Motion. The MIT Press,
1983. Good book on the extraction of motion from edges.
[Oga70]
Katsuhiko Ogata. Modem Control Engineering. Prentice-Hall, Englewood Cliffs, N.J., 1970. Steady State Frequency Response (page 372).
[PR73]
T. Poggio and W. Reichard. Considerations on models of movement detection. Kybernetic, 13:223-227, 1973.
[Rai89]
Marc H. Raibert. Trotting, pacing, and bounding by a quadruped robot.
Journal of Biomechanics, 1989.
[TP89]
Federico Girosi Tomaso Poggio. A theory of networks for approximation
and learning. AI Memo 1140, MIT, 1989.
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1,828 | 2,660 | An Information Maximization Model of
Eye Movements
Laura Walker Renninger, James Coughlan, Preeti Verghese
Smith-Kettlewell Eye Research Institute
{laura, coughlan, preeti}@ski.org
Jitendra Malik
University of California, Berkeley
[email protected]
Abstract
We propose a sequential information maximization model as a
general strategy for programming eye movements. The model
reconstructs high-resolution visual information from a sequence of
fixations, taking into account the fall-off in resolution from the
fovea to the periphery. From this framework we get a simple rule
for predicting fixation sequences: after each fixation, fixate next at
the location that minimizes uncertainty (maximizes information)
about the stimulus. By comparing our model performance to human
eye movement data and to predictions from a saliency and random
model, we demonstrate that our model is best at predicting fixation
locations. Modeling additional biological constraints will improve
the prediction of fixation sequences. Our results suggest that
information maximization is a useful principle for programming
eye movements.
1
In trod u ction
Since the earliest recordings [1, 2], vision researchers have sought to understand the
non-random yet idiosyncratic behavior of volitional eye movements. To do so, we
must not only unravel the bottom-up visual processing involved in selecting a
fixation location, but we must also disentangle the effects of top-down cognitive
factors such as task and prior knowledge. Our ability to predict volitional eye
movements provides a clear measure of our understanding of biological vision.
One approach to predicting fixation locations is to propose that the eyes move to
points that are ?salient?. Salient regions can be found by looking for centersurround contrast in visual channels such as color, contrast and orientation, among
others [3, 4]. Saliency has been shown to correlate with human fixation locations
when observers ?look around? an image [5, 6] but it is not clear if saliency alone
can explain why some locations are chosen over others and in what order. Task as
well as scene or object knowledge will play a role in constraining the fixation
locations chosen [7]. Observations such as this led to the scanpath theory, which
proposed that eye movement sequences are tightly linked to both the encoding and
retrieval of specific object memories [8].
1.1
Our Approach
We propose that during natural, active vision, we center our fixation on the most
informative points in an image in order to reduce our overall uncertainty about what
we are looking at. This approach is intuitive and may be biologically plausible, as
outlined by Lee & Yu [9]. The most informative point will depend on both the
observer?s current knowledge of the stimulus and the task. The quality of the
information gathered with each fixation will depend greatly on human visual
resolution limits. This is the reason we must move our eyes in the first place, yet it
is often ignored. A sequence of eye movements may then be understood within a
framework of sequential information maximization.
2
Human eye movements
We investigated how observers examine a novel shape when they must rely heavily
on bottom-up stimulus information. Because eye movements will be affected by the
task of the observer, we constructed a learn-discriminate paradigm. Observers are
asked to carefully study a shape and then discriminate it from a highly similar one.
2.1
Stimuli and Design
We use novel silhouettes to reduce the influence of object familiarity on the pattern
of eye movements and to facilitate our computations of information in the model.
Each silhouette subtends 12.5? to ensure that its entire shape cannot be characterized
with a single fixation.
During the learning phase, subjects first fixated a marker and then pressed a button
to cue the appearance of the shape which appeared 10? to the left or right of fixation.
Subjects maintained fixation for 300ms, allowing for a peripheral preview of the
object. When the fixation marker disappeared, subjects were allowed to study the
object for 1.2 seconds while their eye movements were recorded. During the
discrimination phase, subjects were asked to select the shape they had just studied
from a highly similar shape pair (Figure 1). Performance was near 75% correct,
indicating that the task was challenging yet feasible. Subjects saw 140 shapes and
given auditory feedback.
release fixation,
view object freely
(1200ms)
maintain fixation
(300ms)
Which shape is a match?
fixate, initiate trial
Figure 1. Temporal layout of a trial during the learning phase (left). Discrimination of
learned shape from a highly similar one (right).
2.2
Apparatus
Right eye position was measured with an SRI Dual Purkinje Image eye tracker while
subjects viewed the stimulus binocularly. Head position was fixed with a bitebar. A
25 dot grid that covered the extent of the presentation field was used for calibration.
The points were measured one at a time with each dot being displayed for 500ms.
The stimuli were presented using the Psychtoolbox software [10].
3
Model
We wish to create a model that builds a representation of a shape silhouette given
imperfect visual information, and which updates its representation as new visual
information is acquired. The model will be defined statistically so as to explicitly
encode uncertainty about the current knowledge of the shape silhouette. We will use
this model to generate a simple rule for predicting fixation sequences: after each
fixation, fixate next at the location that will decrease the model?s uncertainty as
much as possible. Similar approaches have been described in an ideal observer
model for reading [11], an information maximization algorithm for tracking
contours in cluttered images [12] and predicting fixation locations during object
learning [13].
3.1
Representing information
The information in silhouettes clearly resides at its contour, which we represent
with a collection of points and associated tangent orientations. These points and
their associated orientations are called edgelets, denoted e1, e2, ... eN, where N is the
total number of edgelets along the boundary. Each edgelet ei is defined as a triple
ei=(xi, yi, zi) where (xi, yi) is the 2D location of the edgelet and zi is the orientation
of the tangent to the boundary contour at that point. zi can assume any of Q possible
values 1, 2, ?, Q, representing a discretization of Q possible orientations ranging
from 0 to ? , and we have chosen Q=8 in our experiments. The goal of the model is
to infer the most likely orientation values given the visual information provided by
one or more fixations.
3.2
Updating knowledge
The visual information is based on indirect measurements of the true edgelet values
e1, e2, ... eN. Although our model assumes complete knowledge of the number N and
locations (xi, yi) of the edgelets, it does not have direct access to the orientations zi.1
Orientation information is instead derived from measurements that summarize the
local frequency of occurrence of edgelet orientations, averaged locally over a coarse
scale (corresponding to the spatial scale at which resolution is limited by the human
visual system). These coarse measurements provide indirect information about
individual edgelet orientations, which may not uniquely determine the orientations.
We will use a simple statistical model to estimate the distribution of individual
orientation values conditioned on this information.
Our measurements are defined to model the resolution limitations of the human
visual system, with highest resolution at the fovea and lower resolution in the
1
Although the visual system does not have precise knowledge of location coordinates, the
model is greatly simplified by assuming this knowledge. It is reasonable to expect that
location uncertainty will be highly correlated with orientation uncertainty, so that the
inclusion of location should not greatly affect the model's decisions of where to fixate next.
periphery. Distance to the fovea is rmeasured as eccentricity E, the visual angle
between any point and the fovea. If x = ( x, y ) is the location of a point in an image
r
and f = ( f x , f y ) is the fixation (i.e. foveal) location in the image then the
r
eccentricity is E = xr ? f , measured in units of visual degrees. The effective
resolution of orientation discrimination falls with increasing eccentricity as
r (E ) = FPH ( E + E 2 ) where r(E) is an effective radius over which the visual system
spatially pools information and FPH =0.1 and E2=0.8 [14].
Our model represents pooled information as a histogram of edge orientations within
the effective radius. For each edgelet ei we define the histogram of all edgelet
orientations ej within radius ri = r(E) of ei , where E is the eccentricity of xri = ( xi , yi )
r
r
relative to the current fixation f , i.e. E = xri ? f . To define the histogram more
precisely we will introduce the neighborhood set Ni of all indices j corresponding to
r r
edgelets within radius ri of ei : N i = all j s.t. xi ? x j ? ri , with number of
{
}
neighborhood edgelets |Ni|. The (normalized) histogram centered at edgelet ei is then
defined as
hiz =
1
Ni
??
j?N i
z,z j
,
which is the proportion of edgelet orientations that assume value z in the
(eccentricity-dependent) neighborhood of edgelet ei.2
Figure 2. Relation between eccentricity E and radius r(E) of the neighborhood (disk)
which defines the local orientation histogram (hiz ). Left and right panels show two
fixations for the same object.
Up to this point we have restricted ourselves to the case of a single fixation. To
designate a sequence of multiple fixations we will index them byrk=1, 2, ?, K (for
K total fixations). The k th fixation location is denoted by f ( k ) = ( f xk , f yk ) . The
quantities ri , Ni and hiz depend on fixation location and so to make this dependence
explicit we will augment them with superscripts as ri(k ) , N i(k ) , and hiz(k ) .
2
? x, y is the Kronecker delta function, defined to equal 1 if x = y and 0 if x ? y .
Now we describe the statistical model of edgelet orientations given information
obtained from multiple fixations. Ideally we would like to model the exact
distribution
of
orientations
conditioned
on
the
histogram
data:
(1)
( 2)
(K )
(k )
represents
all
histogram
,
where
P(zi , z 2 , ... z N | {hiz }, {hiz },K, {hiz })
{hiz }
r
components z at every edgelet ei for fixation f (k ) . This exact distribution is
intractable, so we will use a simple approximation. We assume the distribution
factors over individual edgelets:
N
P(zi , z 2 , ... z N | {hiz(1) }, {hiz( 2 ) },K, {hiz( K ) }) = ? g i(zi )
i =1
where gi(zi) is the marginal distribution of orientation zi. Determining these
marginal distributions is still difficult even with the factorization assumption, so we
K
will make an additional approximation: g (z ) = 1
hiz( k ) , where Zi is a suitable
?
i i
Z i k =1
(k )
normalization factor. This approximation corresponds to treating hiz as a likelihood
function over z, with independent likelihoods for each fixation k. While the
approximation has some undesirable properties (such as making the marginal
distribution gi(zi) more peaked if the same fixation is made repeatedly), it provides a
simple mechanism for combining histogram evidence from multiple, distinct
fixations.
3.3
Selecting the next fixation
r ( K +1)
Given the past K fixations, the next fixation f
is chosen to minimize the model
r ( K +1)
entropy of the edgelet orientations. In other words, f
is chosen to minimize
r ( K +1)
H( f
) = entropy[ P(zi , z2 , ... z N | {hiz(1) }, {hiz( 2 ) },K, {hiz( K +1) })] , where the entropy of a
distribution P(x) is defined as ? ? P( x) log P ( x) . In practice, we minimize the
x
r
entropy by evaluating it across a set of candidate locations f ( K +1) which forms a
regularly sampled grid across the image.3 We note that this selection rule makes
decisions that depend, in general, on the full history of previous K fixations.
4
Results
Figure 3 shows an example of one observer?s eye movements superimposed over the
shape (top row), the prediction from a saliency model (middle row) [3] and the
prediction from the information maximization model (bottom row). The information
maximization model updates its prediction after each fixation.
An ideal sequence of fixations can be generated by both models. The saliency model
selects fixations in order of decreasing salience. The information maximization
model selects the maximally informative point after incorporating information from
the previous fixations. To provide an additional benchmark, we also implemented a
3
This rule evaluates the entropy resulting from every possible next fixation before making a
decision. Although this rule is suitable for our modeling purposes, it would be inefficient to
implement in a biological or machine vision system. A practical decision rule would use
current knowledge to estimate the expected (rather than actual) entropy.
Figure 3. Example eye movement pattern, superimposed over the stimulus (top row),
saliency map (middle row) and information maximization map (bottom row).
model that selects fixations at random. One way to quantify the performance is to
map a subject?s fixations onto the closest model predicted fixation locations,
ignoring the sequence in which they were made. In this analysis, both the saliency
and information maximization models are significantly better than random at
predicting candidate locations (p < 0.05; t-test) for three observers (Figure 4, left).
The information maximization model performs slightly but significantly better than
the saliency model for two observers (lm, kr). If we match fixation locations while
retaining the sequence, errors become quite large, indicating that the models cannot
account for the observed behavior (Figure 4, right).
Sequence Error
Visual Angle (deg)
Location Error
R S I
R S I
R S I
R S I
R S I
R S I
Figure 4. Prediction error of three models: random (R), saliency (S) and information
maximization (I) for three observers (pv, lm, kr). The left panel shows the error in
predicting fixation locations, ignoring sequence. The right panel shows the error when
sequence is retained before mapping. Error bars are 95% confidence intervals.
The information maximization model incorporates resolution limitations, but there
are further biological constraints that must be considered if we are to build a model
that can fully explain human eye movement patterns. First, saccade amplitudes are
typically around 2-4? and rarely exceed 15? [15]. When we move our eyes, the
image of the visual world is smeared across the retina and our perception of it is
actively suppressed [16]. Shorter saccade lengths may be a mechanism to reduce
this cost. This biological constraint would cause a fixation to fall short of the
prediction if it is distant from the current fixation (Figure 5).
Figure 5. Cost of moving the eyes. Successive fixations may fall short of the maximally
salient or informative point if it is very distant from the current fixation.
Second, the biological system may increase its sampling efficiency by planning a
series of saccades concurrently [17, 18]. Several fixations may therefore be made
before sampled information begins to influence target selection. The information
maximization model currently updates after each fixation. This would create a
discrepancy in the prediction of the eye movement sequence (Figure 6).
Figure 6. Three fixations are made to a location that is initially highly informative
according to the information maximization model. By the fourth fixation, the subject
finally moves to the next most informative point.
5
D i s c u s s i on
Our model and the saliency model are using the same image information to
determine fixation locations, thus it is not surprising that they are roughly similar in
their performance of predicting human fixation locations. The main difference is
how we decide to ?shift attention? or program the sequence of eye movements to
these locations. The saliency model uses a winner-take-all and inhibition-of-return
mechanism to shift among the salient regions. We take a completely different
approach by saying that observers adopt a strategy of sequential information
maximization. In effect, the history of where we have been matters because our
model is continually collecting information from the stimulus. We have an implicit
?inhibition-of-return? because there is little to be gained by revisiting a point.
Second, we attempt to take biological resolution limits into account when
determining the quality of information gained with each fixation. By including
additional biological constraints such as the cost of making large saccades and the
natural time course of information update, we may be able to improve our prediction
of eye movement sequences.
We have shown that the programming of eye movements can be understood within a
framework of sequential information maximization. This framework is portable to
any image or task. A remaining challenge is to understand how different tasks
constrain the representation of information and to what degree observers are able to
utilize the information.
Acknowledgments
Smith-Kettlewell Eye Research Institute, NIH Ruth L. Kirschstein NRSA, ONR #N0001401-1-0890, NSF #IIS0415310, NIDRR #H133G030080, NASA #NAG 9-1461.
References
[1] Buswell (1935). How people look at pictures. Chicago: The University of Chicago Press.
[2] Yarbus (1967). Eye movements and vision. New York: Plenum Press.
[3] Itti & Koch (2000). A saliency-based search mechanism for overt and covert shifts of
visual attention. Vision Research, 40, 1489-1506.
[4] Kadir & Brady (2001). Scale, saliency and image description. International Journal of
Computer Vision, 45(2), 83-105.
[5] Parkhurst, Law, and Niebur (2002). Modeling the role of salience in the allocation of
overt visual attention. Vision Research, 42(1), 107-123.
[6] Nothdurft (2002). Attention shifts to salient targets. Vision Research, 42, 1287-1306.
[7] Oliva, Torralba, Castelhano & Henderson (2003). Top-down control of visual attention in
object detection. Proceedings of the IEEE International Conference on Image Processing,
Barcelona, Spain.
[8] Noton & Stark (1971). Scanpaths in eye movements during pattern perception. Science,
171, 308-311.
[9] Lee & Yu (2000). An information-theoretic framework for understanding saccadic
behaviors. Advanced in Neural Processing Systems, 12, 834-840.
[10] Brainard (1997). The psychophysics toolbox. Spatial Vision, 10 (4), 433-436.
[11] Legge, Hooven, Klitz, Mansfield & Tjan (2002). Mr.Chips 2002: new insights from an
ideal-observer model of reading. Vision Research, 42, 2219-2234.
[12] Geman & Jedynak (1996). An active testing model for tracking roads in satellite images. IEEE
Trans. Pattern Analysis and Machine Intel, 18(1), 1-14.
[13] Renninger & Malik (2004). Sequential information maximization can explain eye
movements in an object learning task. Journal of Vision, 4(8), 744a.
[14] Levi, Klein & Aitesbaomo (1985). Vernier acuity, crowding and cortical magnification.
Vision Research, 25(7), 963-977.
[15] Bahill, Adler & Stark (1975). Most naturally occurring human saccades have
magnitudes of 15 degrees or less. Investigative Ophthalmology, 14, 468-469.
[16] Burr, Morrone & Ross (1994). Selective suppression of the magnocellular visual
pathway during saccadic eye movements. Nature, 371, 511-513.
[17] Caspi, Beutter & Eckstein (2004). The time course of visual information accrual guiding
eye movement decisions. Proceedings of the Nat?l Academy of Science, 101(35), 13086-90.
[18] McPeek, Skavenski & Nakayama (2000). Concurrent processing of saccades in visual
search. Vision Research, 40, 2499-2516.
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correlated:1 |
1,829 | 2,661 | Expectation Consistent Free Energies for
Approximate Inference
Manfred Opper
ISIS
School of Electronics and
Computer Science
University of Southampton
SO17 1BJ, United Kingdom
[email protected]
Ole Winther
Informatics and
Mathematical Modelling
Technical University of Denmark
DK-2800 Lyngby, Denmark
[email protected]
Abstract
We propose a novel a framework for deriving approximations for intractable probabilistic models. This framework is based on a free energy
(negative log marginal likelihood) and can be seen as a generalization
of adaptive TAP [1, 2, 3] and expectation propagation (EP) [4, 5]. The
free energy is constructed from two approximating distributions which
encode different aspects of the intractable model such a single node constraints and couplings and are by construction consistent on a chosen set
of moments. We test the framework on a difficult benchmark problem
with binary variables on fully connected graphs and 2D grid graphs. We
find good performance using sets of moments which either specify factorized nodes or a spanning tree on the nodes (structured approximation).
Surprisingly, the Bethe approximation gives very inferior results even on
grids.
1
Introduction
The development of tractable approximations for the statistical inference with probabilistic data models is of central importance in order to develop their full potential. The most
prominent and widely developed [6] approximation technique is the so called Variational
Approximation (VA) in which the true intractable probability distribution is approximated
by the closest one in a tractable family. The most important tractable families of distributions are multivariate Gaussians and distributions which factorize in all or in certain groups
of variables [7]. Both choices have their drawbacks. While factorizing distributions neglect
correlations, multivariate Gaussians allow to retain a significant amount of dependencies
but are restricted to continuous random variables which have the entire real space as their
natural domain (otherwise KL divergences becomes infinite).
More recently a variety of non variational approximations have been developed which can
be understood from the idea of global consistency between local approximations. E.g.,
in the Bethe?Kikuchi approach [8] the local neighborhood of each variable in a graphical
model is implicitly approximated by a tree-like structure. Consistency is achieved by the
matching of marginal distributions at the connecting edges of the graph. Thomas Minka?s
Expectation Propagation (EP) framework seems to provide a general framework for developing and unifying such consistency approximations [4, 5]. Although the new frameworks
have led to a variety of promising applications, often outperforming VA schemes, the unsatisfactory division between the treatment of constrained and unconstrained, continuous
random variables seems to persist.
In this paper we propose an alternative approach which we call the expectation consistent
(EC) approximation which is not plagued by this problem. We require consistency between
two complimentary global approximations (say, a factorizing & a Gaussian one) to the same
probabilistic model which may have different support. Our method is a generalization of
the adaptive TAP approach (ADATAP) [2, 3] developed for inference on densely connected
graphical models which has been applied successfully to a variety of problems ranging from
probabilistic ICA over Gaussian process models to bootstrap methods for kernel machines.
2
Approximative inference
We consider the problem of computing expectations, i.e. certain sums or integrals involving
a probability distribution with density
p(x) =
1
f (x) ,
Z
(1)
for
R a vector of random variables x = (x1 , x2 , . . . , xN ) with the partition function Z =
dxf (x). We assume that the necessary exact operations are intractable, where the intractability arises either because the necessary sums are over a too large number of variables or because multivariate integrals cannot be evaluated exactly. In a typical scenario,
f (x) is expressed as a product of two functions
f (x) = f1 (x)f2 (x)
(2)
with f1,2 (x) ? 0, where f1 is ?simple? enough to allow for tractable computations.
The idea of many approximate inference methods is to approximate the ?complicated?
part f2 (x) by replacing
it with a ?simpler?
function, say of some exponential form
P
K
T
exp ? g(x) ? exp
j=1 ?j gj (x) . The vector of functions g is chosen in such a
way that the desired sums or integrals can be calculated in an efficient way and the parameters ? are adjusted to optimize certain criteria. Hence, the word tractability should always
be understood as relative to some approximating set of functions g.
Our novel framework of approximation will be restricted to problems, where both parts f 1
and f2 can be considered as tractable relative to some suitable g, and the intractability of
the density p arises from forming their product. Take, as an example, the density (with
respect to the Lebesgue measure in RN ) given by
?
?
Y
X
p(x) =
?? (x? ) exp ?
xi Jij xj ? ,
(3)
?
i<j
where the x? denote tractable potentials defined on disjoint subsets of variables x? .
In order to have a non-trivial problem, the ?? should be a non-Gaussian function.
One may approximate
p(x) by a factorizing distribution, thereby replacing f 2 (x) ?
P
exp
i<j xi Jij xj by some function which factorizes in the components x i . AlternaQ
tively, one can consider replacing f1 (x) = i ?i (xi ) by a Gaussian function to make
the whole distribution Gaussian. Both approximations are not ideal. The first completely
neglects correlations of the variables but leads to marginal distributions of the x i , which
might qualitatively resemble the non Gaussian shape of the true marginal. The second
one neglects the non Gaussian effects but incorporates correlations which might be used
in order to approximate the two variable covariance functions. While within the VA both
approximations would appear independent from each others, we will, in the following develop an approach for combining two approximations which ?communicate? by matching
the corresponding expectations of the functions g(x). We do not have to assume that either
choice is a reasonably good approximation for the global joint density p(x) as done in the
VA. In fact, we apply the approach to a case where the KL divergence between one of them
and p is even infinite!
3
Gibbs free energy
Free energies (FE) provide a convenient formalism for dealing with probabilistic approximation problems. In this framework, the true, intractable distribution p(x) = f (x)
Z is
implicitly characterized as the solution of an optimization problem defined through the the
relative entropy or KL divergence
Z
q(x)
KL(q, p) = dx q(x) ln
(4)
p(x)
between p and other trial distributions q. In contrast to the usual formulation of the VA,
where one minimizes the KL divergence directly within a tractable family, it is more convenient to consider the following two stage optimization process. In the first step, one
constrains the trial distributions q by fixing the values of a set of generalized moments
hg(x)iq . This will be helpful later to enable the communication between approximations.
?) as
We define the Gibbs Free Energy G(?
?) = min {KL(q, p) | hg(x)iq = ? } ? ln Z .
G(?
q
(5)
We have subtracted the term ln Z to make the expression independent of the intractable
partition function Z. In a second step, the moments of the distribution and also the partition
function Z are found within the same approach by relaxing the constraints and further
?) with respect to the ?.
minimizing G(?
?) = ? ln Z
min G(?
?
and
?) .
hgi = argmin G(?
(6)
?
We will next give a short summary of properties of the Gibbs free energy (GFE). The
optimizing density in (5) is given by
f (x)
q(x) =
exp ?T g(x) ,
(7)
Z(?)
?) is
with a normalizing partition function Z(?). The set of Lagrange parameters ? = ?(?
chosen such that the conditions hg(x)iq = ? are fulfilled, i.e. ? satisfies
? ln Z(?)
=?.
??
(8)
Inserting the optimizing distribution eq. (7) into the definition of the Gibbs free energy eq.
(5), we get the explicit expression:
n
o
?) = ? ln Z(?(?
?)) + ?T (?
?)?
? = max ? ln Z(?) + ?T ? ,
G(?
(9)
?
i.e., G is the Legendre transform or dual of ? ln Z(?). Hence, G is a convex function of
?)
its arguments and ?G(?
? = ?.
??
?) can be used to generate moments, e.g.
G(?
?)
? 2 G(?
???
?T
??
=
??
=
?T
??
?T
??
??
?1
?1
= hg(x)gT (x)i ? hg(x)ihg(x)iT
, (10)
where the expectations are over the density eq. (7).
The derivative with respect to a parameter t contained in the probability density p(x|t) =
f (x,t)
Zt can be calculated using (9) and (8) as
?, t)
dG(?
? ln Z(?, t)
? ln Z(?, t) d?T
? ln Z(?, t)
=?
+ ??
=?
.
dt
?t
??
dt
?t
(11)
The important message is that we only need to take the explicit t dependence into account,
i.e. we can keep ? fixed upon differentiation.
3.1
Free energy examples
Here we give the free energies for three tractable models and choices of moments that will
be used subsequently in the free energy framework.
Q
Completely factorized, i.e. p(x) = i ?i (xi ). For simplicity we will consider biased
binary variables: ?i (xi ) = [?(xi + 1) + ?(xi ? 1)]e?i xi and fix the first moments m = hxi.
Denoting the conjugate Lagrange parameters by ?:
X
G(m) =
Gi (mi )
with Gi (mi ) = max {? ln Zi (?i ) + mi ?i }
(12)
?i
i
with Zi (?i ) =
R
dxi ?i (?)e?i xi = 2 cosh(?i + ?i ).
Tree-connected graph. For the case where either the couplings and the moments together define a tree-connected graph, we can write the free energy in term of single- and
two-node free energies. Considering again completely factorized binary variables, all nontrivial moments on the graph (ij) ? G are the means m and correlations of linked nodes
Mij = hxi xj i:
X
X
(1 ? ni )Gi (mi ) ,
(13)
G(m, {Mij }(ij)?G ) =
Gij (mi , mj , Mij ) +
(ij)?G
i
where Gij (mi , mj , Mij ) is the two-node free energy defined in a similar fashion as the
one-node free energy, ni the number of links to node i and Gi (mi ) is the one-node free
energy.
Gaussian distribution. We set ? = (m, M) with all first moments m and an arbitrary
subset of second moments M for a Gaussian model ?i (xi ) ? exp[ai xi ? b2i x2i ] and p(x)
given by eq. (3). We introduce conjugate variables ? and ??/2. ? can be eliminated
analytically, whereas we get a log-determinant maximization problem for ?:
G(m, M)
1
1X
= ? mT Jm ? mT a +
Mii bi
2
2 i
1
1
T
ln det(? ? J) ? Tr ?(M ? mm ) .
+ max
?
2
2
(14)
4
Exact interpolation representation
If the density p factors into a tractable f1 and an intractable part f2 , according to eq. (2),
we can construct a representation of the Gibbs free energy which also separates into two
corresponding parts. We treat f2 (x) as a perturbation which is smoothly turned on using a
parameter 0 ? t ? 1. We define f2 (x, t) to be a smooth interpolation between the trivial
f2 (x, t = 0) = 1 and the ?full? intractable f2 (x, t = 1) = f2 (x). Hence, we define
parametric densities and the corresponding free energy by p(x|t) = Z1t f1 (x)f2 (x, t) and
1
f1 (x)f2 (x, t) exp ?T g(x)
(15)
q(x|t) =
Zq (?, t)
n
o
?, t) = max ? ln Zq (?, t) + ?T ? .
Gq (?
(16)
?
D
E
we derive the following
Using eq. (11), and the fact that ? ln Z(?,t) = d ln f2 (x,t)
?t
dt
q(x|t)
?, t)
exact identity for the free energy G(?
?, 1) ? Gq (?
?, 0) = ?
Gq (?
Z
1
dt
0
d ln f2 (x, t)
dt
.
(17)
q(x|t)
?) = G(?
?, t = 1) and tractable
to relate the Gibbs free energy of the intractable model Gq (?
?, t = 0). An simple approximation is obtained for the case f2 (x, t) = [f2 (x)]t ,
model G(?
when the expectation over q(x|t) is replaced by the expectation over the tractable q(x|0),
ie
Z 1
d ln f2 (x, t)
?) ? G(?
?, 0) ?
?, 0) ? hln f2 (x)iq(x|0) . (18)
G(?
dt
= G(?
dt
0
q(x|0)
This result coincides with the variational approximation when we restrict the family of
optimizing functions to be of the form q(x|0).
4.1
Expectation Consistent Approximation
Our goal is to go beyond the variational approximation and capture more of the t dependence of the intractable part f2 in the term eq. (17). We will now use our assumption that
besides the family of distributions eq. (15), there is a second family which can be used as
an approximation to the distribution p(x|t). It is given by
1
r(x|t) =
f2 (x, t) exp ?T g(x) ,
(19)
Zr (?, t)
where the parameters ? will be chosen in such a way as to guarantee consistency for the
expectations of g, i.e. hg(x)ir(x|t) = ? . Using eq. (19) in place of q(x|t) in eq. (17), we
get the approximation
Z 1
d ln f2 (x, t)
?, 1) ? Gq (?
?, 0) ?
?, 1) ? Gr (?
?, 0) ,
Gq (?
dt
= Gr (?
(20)
dt
0
r(x|t)
where the last equality is derived from the fact that both types of densities eqs. (15) and
(19) contain the same exponential family. This allows us to carry out the integral over the
interaction strength t in eq. (20) in closed form without specifying the interpolating term
f2 (x, t) explicitly. Hence, the expectation consistent (EC) approximation is
?, 1) ? Gq (?
?, 0) + Gr (?
?, 1) ? Gr (?
?, 0) ? GEC (?
?) .
Gq (?
(21)
?, 0), Gr ? Gr (?
?, 1) and Gs ? Gr (?
?, 0) in
To simplify notation, we will write Gq ? Gq (?
the following.
5
Models with pair-wise interactions
Our framework works very naturally to the class of models eq. (3). The EC approximation eq. (21) will be based on approximating neglected correlations in the factorizing ap?, 1) and
proximation using a Gaussian distributions. The corresponding free energies G(?
?, 0) appearing in eq. (21) are thus found using the Gaussian free energy eq. (14) with J
r(?
and J = 0
1
(22)
GEC (m, M) = Gq (m, M, 0) ? mT Jm
2
1
1
+ max
ln det(? ? J) ? Tr ?(M ? mmT )
?
2
2
1
1
? max
ln det ? ? Tr ?(M ? mmT ) ,
?
2
2
where the free energy Gq (m, M, 0) will depend explicitly upon the potentials ?? (x? ).
6
Free energy minimization algorithms
In our approach, inference is based on the minimization of the free energy with respect to
its arguments ? . While the exact free energy is by construction a convex function in ? ,
our free energy approximation GEC = Gq + Gr ? Gs contains the concave contribution
?Gs and may not be convex. Hence, we may have potentially many local minima and
other stationary points, like saddlepoints. Moreover, the expression derived for G EC is
not a simple explicit functions of ? but contains additional optimizations over Lagrange
parameters. Consequently, we cannot expect that message-passing algorithms similar to
loopy belief propagation or expectation propagation (EP) [4] which sequentially update
moments and Lagrange parameters are guaranteed to converge.
6.1
Guaranteed convergence ? variational bounding
An iterative algorithms which is guaranteed to find at least stationary points of the Free
Energy is similar in spirit to the so-called double-loop approaches [9, 10]. The basic idea
is to minimize a decreasing sequence of convex upper bounds to GEC . Each convex upper
?) at the present iteration ? ? , i.e. usbound is derived by linearizing the concave term ?Gs (?
?
lbound
T
? ) ? Gs
?) = ?C? +?
? ?s , with C? ? ln Zq (??s ) and ??s = ?s (?
?? ). Since
ing Gs (?
(?
it is usually much easier to deal with the Lagrange parameters we convert the resulting
convex minimization problem into a concave maximization problem for the ??s
?) + Gr (?
?) ? ? T ??s + C?
? Gq (?
= min max ? ln Zq (?q ) ? ln Zr (?r ) + ? T (?q + ?r ? ??s ) + C?
? ? ,?
q
r
= max {? ln Zq (?q ) ? ln Zr (?r )|?q + ?r = ??s } + C?
?q ,?r
= max {? ln Zq (??s ? ?r ) ? ln Zr (?r ) + C? } .
(23)
?r
This can be summarized in the following double loop algorithm which is guaranteed not to
increase our free energy approximation.
?)
GEC (?
?) by
1. Outer loop: For fixed old value ? ? , bound the concave term ?Gs (?
?) go get the convex upper bound to GEC (?
?).
?Glbound
(?
s
2. Inner loop: Solve the concave maximization problem
max L
with L = ? ln Zq (??s ? ?r ) ? ln Zr (?r ) .
(24)
?r
Inserting the solution into ? (?r ) = hg(x)ir gives the new value ? ? for ? .
Currently, we either solve the non-linear inner-loop optimization by a sequential approach
that are computationally efficient when Gr is the free energy of a multivariate Gaussian or
by interior point methods [11, 12].
7
Simulations
We have tested the EC framework in a benchmark set-up proposed by Wainwright and
Jordan [12]. The stochastic variables are binary xi = ?1 with pair-wise couplings are
used. The N = 16 nodes are either fully connected or connected to nearest neighbors
in a 4-by-4 grid. The external field (observation) strengths ?i are drawn from a uniform
distribution ?i ? U[?dobs , dobs ] with dobs = 0.25. Three types of coupling strength
statistics are considered: repulsive (anti-ferromagnetic) Jij ? U[?2dcoup , 0], mixed Jij ?
U[?dcoup , +dcoup ] and attractive (ferromagnetic) Jij ? U[0,P
+2dcoup ] with dcoup > 0.
We compute the average one-norm error on the marginals:
i |p(xi = 1) ? p(xi =
1|Method)|/N , p(xi = 1) = (1 + mi )/2 over 100 trials testing the following Methods:
SP = sum-product (aka loopy belief propagation (BP) or Bethe approximation) and LD =
log-determinant maximization [12], EC factorized and EC structured. Results for SP and
LD are taken from Ref. [12]. For EC, we are minimizing the EC free energy eq. (22) where
Gq (m, M, 0) depend upon the approximation we are using. For the factorized model we
use the free energy eq. (12) and for the structured model we assume a single tractable
potential ?(x) in eq. (3) which contains all couplings on a spanning tree. For G q , we use
the free energy eq. (13). The spanning tree is defined by the following simple heuristic:
choose as next pair of nodes to link, the (so far unlinked) pair with strongest absolute
coupling |Jij | that will not cause a loop in the graph.
The results are summarized in table 1. The Bethe approximation always give inferior results compared to EC (note that only loopy BP convergent problem instances were used to
calculate the error [12]). This might be a bit surprising for the sparsely connected grids.
This indicates that loopy BP and too a lesser degree extensions building upon BP [5] are
only to be applied to really sparse graphs and/or weakly coupled nodes, where the error
induced by not using a properly normalized distribution can be expected to be small. We
also speculate that a structured variational approximation, using the same heuristics as described above to construct the spanning tree, in many cases will be superior to the Bethe
approximation as also observed by Ref. [5]. LD is a robust method which seems to be
limited in it?s achievable precision. EC structured is uniformly superior to all other approaches. Additional simulations (not included in the paper) also indicate that EC give
much improved estimates of free energies and two-node marginals when compared to the
Bethe- and Kikuchi-approximation.
8
Conclusion and outlook
We have introduced a novel method for approximate inference which tries to overcome
certain limitations of single approximating distributions by achieving consistency for two
of these on the same problem. While we have demonstrated its accuracy in this paper
only for a model with binary elements, it can also be applied to models with continuous
random variables or hybrid models with both discrete and continuous variables. We expect
that our method becomes most powerful when certain tractable substructures of variables
with strong dependencies can be identified in a model. Our approach would then allow
to deal well with the weaker dependencies between the groups. A generalization of our
method to treat graphical models beyond pair-wise interaction is obtained by iterating the
approximation. This is useful in cases, where an initial three term approximation G EC =
Table 1: The average one-norm error on marginals for the Wainwright-Jordan set-up.
Problem type
Graph
Full
Grid
Coupling
Repulsive
Repulsive
Mixed
Mixed
Attractive
Attractive
Repulsive
Repulsive
Mixed
Mixed
Attractive
Attractive
dcoup
0.25
0.50
0.25
0.50
0.06
0.12
1.0
2.0
1.0
2.0
1.0
2.0
SP
Mean
0.037
0.071
0.004
0.055
0.024
0.435
0.294
0.342
0.014
0.095
0.440
0.520
LD
Mean
0.020
0.018
0.020
0.021
0.027
0.033
0.047
0.041
0.016
0.038
0.047
0.042
Method
EC fac
Mean
0.003
0.031
0.002
0.022
0.004
0.117
0.153
0.198
0.011
0.082
0.125
0.177
EC struct
Mean
0.0017
0.0143
0.0013
0.0151
0.0031
0.0211
0.0031
0.0021
0.0018
0.0068
0.0028
0.0024
Gq + Gr ? Gs still contains non-tractable component free energies G.
References
[1] M. Opper and O. Winther, ?Gaussian processes for classification: Mean field algorithms,?
Neural Computation, vol. 12, pp. 2655?2684, 2000.
[2] M. Opper and O. Winther, ?Tractable approximations for probabilistic models: The adaptive
Thouless-Anderson-Palmer mean field approach,? Phys. Rev. Lett., vol. 86, pp. 3695, 2001.
[3] M. Opper and O. Winther, ?Adaptive and self-averaging Thouless-Anderson-Palmer mean field
theory for probabilistic modeling,? Phys. Rev. E, vol. 64, pp. 056131, 2001.
[4] T. P. Minka, ?Expectation propagation for approximate Bayesian inference,? in UAI 2001,
2001, pp. 362?369.
[5] T. Minka and Y. Qi, ?Tree-structured approximations by expectation propagation,? in NIPS 16,
S. Thrun, L. Saul, and B. Sch?olkopf, Eds. MIT Press, Cambridge, MA, 2004.
[6] Christopher M. Bishop, David Spiegelhalter, and John Winn, ?Vibes: A variational inference
engine for bayesian networks,? in Advances in Neural Information Processing Systems 15,
S. Thrun S. Becker and K. Obermayer, Eds., pp. 777?784. MIT Press, Cambridge, MA, 2003.
[7] H. Attias, ?A variational Bayesian framework for graphical models,? in Advances in Neural
Information Processing Systems 12, T. Leen et al., Ed. 2000, MIT Press, Cambridge.
[8] J. S. Yedidia, W. T. Freeman, and Y. Weiss, ?Generalized belief propagation,? in Advances in
Neural Information Processing Systems 13, T. K. Leen, T. G. Dietterich, and V. Tresp, Eds.,
2001, pp. 689?695.
[9] A. L. Yuille, ?CCCP algorithms to minimize the Bethe and Kikuchi free energies: convergent
alternatives to belief propagation,? Neural Comput., vol. 14, no. 7, pp. 1691?1722, 2002.
[10] T. Heskes, K. Albers, and H. Kappen, ?Approximate inference and constrained optimization,?
in UAI-03, San Francisco, CA, 2003, pp. 313?320, Morgan Kaufmann Publishers.
[11] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, 2004.
[12] M. J. Wainwright and M. I. Jordan, ?Semidefinite methods for approximate inference on graphs
with cycles,? Tech. Rep. UCB/CSD-03-1226, UC Berkeley CS Division, 2003.
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1,830 | 2,662 | Methods Towards Invasive Human
Brain Computer Interfaces
Thomas Navin Lal1 , Thilo Hinterberger2 , Guido Widman3 ,
Michael Schr?oder4 , Jeremy Hill1 , Wolfgang Rosenstiel4 ,
Christian E. Elger3 , Bernhard Sch?olkopf1 and Niels Birbaumer2,5
Max-Planck-Institute for Biological Cybernetics, Tu? bingen, Germany
{navin,jez,bs}@tuebingen.mpg.de
2
Eberhard Karls University, Dept. of Medical Psychology and
Behavioral Neurobiology, T?ubingen, Germany
{thilo.hinterberger,niels.birbaumer}@uni-tuebingen.de
3
University of Bonn, Department of Epileptology, Bonn, Germany
{guido.widman,christian.elger}@ukb.uni-bonn.de
4
Eberhard Karls University, Dept. of Computer Engineering, Tu? bingen,
Germany {schroedm,rosenstiel}@informatik.uni-tuebingen.de
5
Center for Cognitive Neuroscience, University of Trento, Italy
1
Abstract
During the last ten years there has been growing interest in the development of Brain Computer Interfaces (BCIs). The field has mainly been
driven by the needs of completely paralyzed patients to communicate.
With a few exceptions, most human BCIs are based on extracranial electroencephalography (EEG). However, reported bit rates are still low. One
reason for this is the low signal-to-noise ratio of the EEG [16]. We are
currently investigating if BCIs based on electrocorticography (ECoG) are
a viable alternative. In this paper we present the method and examples
of intracranial EEG recordings of three epilepsy patients with electrode
grids placed on the motor cortex. The patients were asked to repeatedly imagine movements of two kinds, e.g., tongue or finger movements.
We analyze the classifiability of the data using Support Vector Machines
(SVMs) [18, 21] and Recursive Channel Elimination (RCE) [11].
1
Introduction
Completely paralyzed patients cannot communicate despite intact cognitive functions. The
disease Amyotrophic Lateral Sclerosis (ALS) for example, leads to complete paralysis of
the voluntary muscular system caused by the degeneration of the motor neurons. Birbaumer
et al. [1, 9] developed a Brain Computer Interface (BCI), called the Thought Translation
Device (TTD), which is used by several paralyzed patients. In order to use the interface,
patients have to learn to voluntary regulate their Slow Cortical Potentials (SCP). The system
then allows its users to write text on the screen of a computer or to surf the web. Although it
presents a major breakthrough, the system has two disadvantages. Not all patients manage
Figure 1: The left picture schematically shows the position of the 8x8 electrode grid of patient II. It was placed on the right hemisphere. As shown in the right picture the electrodes
are connected to the amplifier via cables that are passed through the skull.
to control their SCP. Furthermore the bit rate is quite low. A well-trained user requires
about 30 seconds to write one character.
Recently there has been increasing interest on EEG-based BCIs in the machine learning
community. In contrast to the TTD, in many BCI-systems the computer learns rather than
the system?s user [2, 5, 11]. Most such BCIs require a data collection phase during which
the subject repeatedly produces brain states of clearly separable locations. Machine learning techniques like Support Vector Machines or Fisher Discriminant are applied to the data
to derive a classifying function. This function can be used in online applications to identify
the different brain states produced by the subject.
The majority of BCIs is based on extracranial EEG-recordings during imagined limb
movements. We restrict ourselves to mentioning just a few publications [14, 15, 17, 22].
Movement-related cortical potentials in humans on the basis of electrocorticographical data
have also been studied, e.g. by [20]. Very recently the first work describing BCIs based on
electrocorticographic recordings was published [6, 13]. Successful approaches have been
developed using BCIs based on single unit, multiunit or field potentials recordings of primates. Serruya et al. taught monkeys to control a cursor on the basis of potentials from 7-30
motor cortex neurons [19]. The BCI developed by [3] enables monkeys to reach and grasp
using a robot arm. Their system is based on recordings from frontoparietal cell ensembles.
Driven by the success of BCIs for primates based on single unit or multiunit recordings,
we are currently developing a BCI-system that is based on ECoG recordings, as described
in the present paper.
2
Electrocorticography and Epilepsy
All patients presented suffer from a focal epilepsy. The epileptic focus - the part of the
brain which is responsible for the seizures - is removed by resection. Prior to surgery, the
epileptic focus has to be localized. In some complicated cases, this must be done by placing
electrodes onto the surface of the cortex as well as into deeper regions of the brain. The
skull over the region of interest is removed, the electrodes are positioned and the incision is
sutured. The electrodes are connected to a recording device via cables (cf. Figure 1). Over
a period of a 5 to 14 days ECoG is continuously recorded until the patient has had enough
seizures to precisely localize the focus [10]. Prior to surgery the parts of the cortex that are
covered by the electrodes are identified by the electric stimulation of electrodes.
In the current setup, the patients keep the electrode implants for one to two weeks. After
the implantation surgery, several days of recovery and follow-up examinations are needed.
Due to the tight time constraints, it is therefore not possible to run long experiments. Furthermore most of the patients cannot concentrate for a long period of time. Therefore only
a small amount of data could be collected.
Table 1: Positions of implanted electrodes. All three patients had an electrode grid implanted that partly covered the right or the left motor cortex.
patient
implanted electrodes
I
64-grid right hemisphere,
two 4-strip interhemisphere
64-grid right hemisphere
20-grid central,
four 16-strips frontal
II
III
3
task
trials
left vs. right hand
200
little left finger vs. tongue
little right finger vs. tongue
150
100
Experimental Situation and Data Acquisition
The experiments were performed in the department of epileptology of the University of
Bonn. We recorded ECoG data from three epileptic patients with a sampling rate of
1000Hz.
The electrode grids were placed on the cortex under the dura mater and covered the primary motor and premotor area as well as the fronto-temporal region either of the right or
left hemisphere. The grid-sizes ranged from 20 to 64 electrodes. Furthermore two of the
patients had additional electrodes implanted on other parts of the cortex (cf. Table 1). The
imagery tasks were chosen such that the involved parts of the brain
? were covered by the electrode grid
? were represented spatially separate in the primary motor cortex.
The expected well-localized signal in motor-related tasks suggested discrimination tasks
using imagination of hand, little finger, or tongue movements.
The patients were seated in a bed facing a monitor and were asked to repeatedly imagine
two different movements. At the beginning of each trial, a small fixation cross was displayed in the center of the screen. The 4 second imagination phase started with a cue that
was presented in the form of a picture showing either a tongue or a little finger for patients
II and III. The cue for patient I was an arrow pointing left or right. There was a short break
between the trials. The images which were used as a cue are shown in Figure 5.
4
Preprocessing
Starting half a second after the visualization of the task-cue, we extracted a window of
length 1.5 seconds from the data of each electrode. For every trial and every electrode we
thus obtained an EEG sequence that consisted of 1500 samples. The linear trend from every
sequence was removed. Following [8, 11, 15] we fitted a forward-backward autoregressive
model of order three to each sequence. The concatenated model parameters of the channels
together with the descriptor of the imagined task (i.e. +1, -1) form one training point. For
a given number n of EEG channels, a training point (x, y) is therefor a point in R 3n ?
{?1, 1}.
5
Channel Selection
The number of available training points is relatively small compared to the dimensionality
of the data. The data of patient III for example, consists of only 100 training points of
Figure 2: The patients were asked to repeatedly imagine two different movements that are
represented separately at the primary cortex, e.g. tongue and little finger movements. This
figure shows two stimuli that were used as a cue for imagery. The trial structure is shown
on the right. The imagination phase lasted four seconds. We extracted segments of 1.5
seconds from the ECoG recordings for the analysis.
dimension 252. This is a typical setting in which features selection methods can improve
classification accuracy.
Lal et al. [11] recently introduced a feature selection method for the special case of EEG
data. Their method is based on Recursive Feature Elimination (RFE) [7]. RFE is a backward feature selection method. Starting with the full data set, features are iteratively removed from the data until a stopping criteria is met. In each iteration a Support Vector
Machine (SVM) is trained and its weight vector is analyzed. The feature that corresponds
to the smallest weight vector entry is removed.
Recursive Channel Elimination (RCE) [11] treats features that belong to the data of a channel in a consistent way. As in RFE, in every iteration one SVM is trained. The evaluation
criteria that determines which of the remaining channels will be removed is the mean of the
weight vector entries that correspond to a channel?s features. All features of the channel
with the smallest mean value are removed from the data. The output of RCE is a list of
ranked channels.
6
Data Analysis
To begin with, we are interested in how well SVMs can learn from small ECoG data sets.
Furthermore we would like to understand how localized the classification-relevant information is, i.e. how many recording positions are necessary to obtain high classification
accuracy. We compare how well SVMs can generalize given the data of different subsets
of ECoG-channels:
(i) the complete data, i.e. all channels
(ii) the subset of channels suggested by RCE. In this setting we use the list of ranked
channels from RCE in the following way: For every l in the range of one to the
total number of channels, we calculate a 10-fold cross-validation error on the data
of the l best-ranked channels. We use the subset of channels which leads to the
lowest error estimate.
(iii) the two best-ranked channels by RCE. The underlying assumption used here is
that the classification-relevant information is extremely localized and that two correctly chosen channels contain sufficient information for classification purposes.
(iv) two channels drawn at random.
Throughout the paper we use linear SVMs. For regularization purposes we use a ridge on
the kernel matrix which corresponds to a 2-norm penalty on the slack variables [4].
C4
muV
C3
C2
C1
0
500
1000
1500
time [ms]
2000
2500
3000
3500
Figure 3: This plot shows ECoG recordings from 4 channels while the patient was imagining movements. The distance of two horizontal lines decodes 100?V . The amplitude of
the recordings ranges roughly from -100 ?V to +100 ?V which is on the order of five to
ten times the amplitude measured with extracranial EEG.
To evaluate the classification performance of an SVM that is trained on a specific subset
of channels we calculate its prediction error on a separate test set. We use a double-crossvalidation scheme - the following procedure is repeated 50 times:
We randomly split the data into a training set (80%) and a test set (20%). Via 10-fold
cross-validation on the training set we estimate all parameters for the different considered
subsets (i)-(iv):
(i) The ridge is estimated.
(ii) On the basis of the training set RCE suggests a subset of channels. We restrict the
training set as well as the test set to these channels. A ridge-value is then estimated
from the restricted training set.
(iii) We restrict the training set and the test set to the 2 best ranked channels by RCE.
The ridge is then estimated on the restricted training set.
(iv) The ridge is estimated.
We then train an SVM on the (restricted) training set using the estimated ridge. The trained
model is tested on the (restricted) test set. For (i)-(iv) we obtain 50 test error estimates from
the 50 repetitions for each patient. Table 2 summarizes the results.
7
Results
The results in Table 2 show that the generalization ability can significantly be increased by
RCE. For patient I the error decreases from 38% to 24% when using the channel subsets
suggested by RCE. In average RCE selects channel subsets of size 5.8. For patient II the
number of channels is reduced to one third but the channel selection process does not yield
an increased accuracy. The error of 40% can be reduced to 23% for patient III using in
average 5 channels selected by RCE.
For patients I and III the choice of the best 2 ranked channels leads to a much lower error
as well. The direct comparison of the results using the two best ranked channels to two
randomly chosen channels shows how well the RCE ranking method works: For patient
three the error drops from chance level for two random channels to 18 % using the two
best-ranked channels.
The reason why there is such a big difference in performance for patient III when comparing (i) and (iii) might be, that out of the 84 electrodes, only 20 are located over or close to
the motor cortex. RCE successfully identifies the important electrodes.
In contrast to patient III, the electrodes of patient II are all more or less located close to
Table 2: Classification Results. We compare the classification accuracy of SVMs trained
on the data of different channel subsets: (i) all ECoG-channels, (ii) the subset determined
by Recursive Channel Elimination (RCE), (iii) the subset consisting of the two best ranked
channels by RCE and (iv) two randomly drawn channels. The mean errors of 50 repetitions
are given along with the standard deviations. The test error can significantly be reduced by
RCE for two of the three patients. Using the two best ranked channels by RCE also yields
good results for two patients. SVMs trained on two random channels show performance
better than chance only for patient II.
pat
I
II
III
all channels (i)
#channels error
74
64
84
0.382 ? 0.071
0.257 ?0.076
0.4 ?0.1
RCE cross-val. (ii)
#channels error
5.8
21.5
5.0
0.243 ? 0.063
0.268 ? 0.080
0.233 ?0.13
RCE top 2 (iii)
error
random 2 (iv)
error
0.244 ? 0.078
0.309 ? 0.086
0.175 ? 0.078
chance level
0.419 ? 0.123
chance level
the motor cortex. This explains why data from two randomly drawn channels can yield
a classification rate better than chance. Furthermore patient II had the fewest electrodes
implanted and thus the chance of randomly choosing an electrode close to an important
location is higher than for the other two patients.
8
Discussion
We recorded ECoG-data from three epilepsy patients during a motor imagery experiment.
Although only few data were collected, the following conclusions can be drawn:
? The data of all three patients is reasonably well classifiable. The error rates range
from 17.5% to 23.3%. This is still high compared to the best error rates from BCI
based on extracranial EEG which are as low as 10% (e.g. [12]). Please note that
we used 1.5 seconds data from each trial only and that very few training points
(100-200) were available. Furthermore, extracranial EEG has been studied and
developed for a number of years.
? Recursive Channel Elimination (RCE) shows very good performance. RCE successfully identifies subsets of ECoG-channels that lead to good classification performance. On average, RCE leads to a significantly improved classification rate
compared to a classifier that is based on the data of all available channels.
? Poor classification rates using two randomly drawn channels and high classification rates using the two best-ranked channels by RCE suggest that classification
relevant information is focused on small parts of the cortex and depends on the
location of the physiological function.
? The best ranked RCE-channels correspond well with the results from the electric
stimulation (cf. Figure 8).
9
Ongoing Work and Further Research
Although our preliminary results indicate that invasive Brain Computer Interfaces may
be feasible, a number of questions need to be investigated in further experiments. For
instance, it is still an open question whether the patients are able to adjust to a trained
classifier and whether the classifying function can be transferred from session to session.
Moreover, experiments that are based on tasks different from motor imaginary need to
X X X
X X
X
X X X
Figure 4: Electric stimulation of the implanted electrodes helps to identify the parts of the
cortex that are covered by the electrode grid. This information is necessary for the surgery.
The red (solid) dots on the left picture mark the motor cortex of patient II as identified
by the electric stimulation method. The positions marked with yellow crosses correspond
to the epileptic focus. The red points on the right image are the best ranked channels by
Recursive Channel Elimination (RCE). The RCE-channels correspond well to the results
from the electro stimulation diagnosis.
be implemented and tested. It is quite conceivable that the tasks that have been found to
work well for extracranial EEG are not ideal for ECoG. Likewise, it is unclear whether our
preprocessing and machine learning methods, originally developed for extracranial EEG
data, are well adapted to the different type of data that ECoG delivers.
Acknowledgements
This work was supported in part by the Deutsche Forschungsgemeinschaft (SFB 550, B5
and grant RO 1030/12), by the National Institute of Health (D.31.03765.2), and by the
IST Programme of the European Community, under the PASCAL Network of Excellence,
IST-2002-506778. T.N.L. was supported by a grant from the Studienstiftung des deutschen
Volkes. Special thanks go to Theresa Cooke.
References
[1] N. Birbaumer, N. Ghanayim, T. Hinterberger, I. Iversen, B. Kotchoubey, A. K u? bler,
J. Perelmouter, E. Taub, and H. Flor. A spelling device for the paralysed. Nature,
398:297?298, 1999.
[2] B. Blankertz, G. Curio, and K. M?uller. Classifying single trial EEG: Towards brain
computer interfacing. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances
in Neural Information Processing Systems, volume 14, Cambridge, MA, USA, 2001.
MIT Press.
[3] J.M. Carmena, M.A Lebedev, R.E Crist, J.E O?Doherty, D.M. Santucci, D. Dimitrov,
P.G. Patil, C.S Henriquez, and M.A. Nicolelis. Learning to control a brain-machine
interface for reaching and grasping by primates. PLoS Biology, 1(2), 2003.
[4] C. Cortes and V. Vapnik. Support-vector networks. Machine Learning, 20:273?297,
1995.
[5] J. del R. Millan, F. Renkens, J. Mourino, and W. Gerstner. Noninvasive brain-actuated
control of a mobile robot by human eeg. IEEE Transactions on Biomedical Engineering. Special Issue on Brain-Computer Interfaces, 51(6):1026?1033, June 2004.
[6] B. Graimann, J. E. Huggins, S. P. Levine, and G. Pfurtscheller. Towards a direct brain
interface based on human subdural recordings and wavelet packet analysis. IEEE
Trans. IEEE Transactions on Biomedical Engineering, 51(6):954?962, 2004.
[7] I. Guyon, J. Weston, S. Barnhill, and V. Vapnik. Gene selection for cancer classification using support vector machines. Journal of Machine Learning Research,
3:1439?1461, March 2003.
[8] S. Haykin. Adaptive Filter Theory. Prentice-Hall International, Inc., Upper Saddle
River, NJ, USA, 1996.
[9] T. Hinterberger, J. Kaiser, A. Kbler, N. Neumann, and N. Birbaumer. The Thought
Translation Device and its Applications to the Completely Paralyzed. In Diebner,
Druckrey, and Weibel, editors, Sciences of the Interfaces. Genista-Verlag, T u? bingen,
2001.
[10] J. Engel Jr. Presurgical evaluation protocols. In Surgical Treatment of the Epilepsies,
pages 740?742. Raven Press Ltd., New York, 2nd edition, 1993.
[11] T.N. Lal, M. Schr?oder, T. Hinterberger, J. Weston, M. Bogdan, N. Birbaumer, and
B. Sch?olkopf. Support Vector Channel Selection in BCI. IEEE Transactions on
Biomedical Engineering. Special Issue on Brain-Computer Interfaces, 51(6):1003?
1010, June 2004.
[12] S. Lemm, C. Sch?afer, and G. Curio. BCI Competition 2003 - Data Set III: Probabilistic Modeling of Sensorimotor mu-Rhythms for Classification of Imaginary Hand
Movements. IEEE Transactions on Biomedical Engineering. Special Issue on BrainComputer Interfaces, 51(6):1077?1080, June 2004.
[13] E. C. Leuthardt, G. Schalk, J. R. Wolpaw, J. G. Ojemann, and D. W. Moran. A
braincomputer interface using electrocorticographic signals in humans. Journal of
Neural Engineering, 1:63?71, 2004.
[14] D.J. McFarland, L.M. McCane, S.V. David, and J.R. Wolpaw. Spatial filter selection
for EEG-based communication. Electroencephalography and Clinical Neurophysiology, 103:386?394, 1997.
[15] G. Pfurtscheller., C. Neuper amd A. Schlo? gl, and K. Lugger. Separability of EEG
signals recorded during right and left motor imagery using adaptive autoregressive
parameters. IEEE Transactions on Rehabilitation Engineering, 6(3):316?325, 1998.
[16] J. Raethjen, M. Lindemann, M. D?umpelmann, R. Wenzelburger, H. Stolze, G. Pfister,
C. E. Elger, J. Timmer, and G. Deuschl. Corticomuscular coherence in the 6-15 hz
band: is the cortex involved in the generation of physiologic tremor? Experimental
Brain Research, 142:32?40, 2002.
[17] H. Ramoser, J. M?uller-Gerking, and G. Pfurtscheller. Optimal spatial filtering of single trial EEG during imagined hand movement. IEEE Transactions on Rehabilitation
Engineering, 8(4):441?446, 2000.
[18] B. Sch?olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, USA,
2002.
[19] M.D. Serruya, N.G Hatsopoulos, L. Paninski, M.R. Fellows, and Donoghue J.P. Instant neural control of a movement signal. Nature, 416:141?142, 2002.
[20] C. Toro, G. Deuschl, R. Thatcher, S. Sato, C. Kufta, and M. Hallett. Event-related
desynchronization and movement-related cortical potentials on the ECoG and EEG.
Electroencephalography Clinical Neurophysiology, 5:380?389, 1994.
[21] V. N. Vapnik. Statistical Learning Theory. John Wiley and Sons, New York, USA,
1998.
[22] R. Wolpaw and D.J McFarland. Multichannel EEG-based brain-computer communication. Electroencephalography and Clinical Neurophysiology, 90:444?449, 1994.
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1,831 | 2,663 | Contextual models for object detection using
boosted random fields
Antonio Torralba
MIT, CSAIL
Cambridge, MA 02139
[email protected]
Kevin P. Murphy
UBC, CS
Vancouver, BC V6T 1Z4
[email protected]
William T. Freeman
MIT, CSAIL
Cambridge, MA 02139
[email protected]
Abstract
We seek to both detect and segment objects in images. To exploit both local image data as well as contextual information, we introduce Boosted
Random Fields (BRFs), which uses Boosting to learn the graph structure and local evidence of a conditional random field (CRF). The graph
structure is learned by assembling graph fragments in an additive model.
The connections between individual pixels are not very informative, but
by using dense graphs, we can pool information from large regions of
the image; dense models also support efficient inference. We show how
contextual information from other objects can improve detection performance, both in terms of accuracy and speed, by using a computational
cascade. We apply our system to detect stuff and things in office and
street scenes.
1
Introduction
Our long-term goal is to build a vision system that can examine an image and describe what
objects are in it, and where. In many images, such as Fig. 5(a), objects of interest, such as
the keyboard or mouse, are so small that they are impossible to detect just by using local
features. Seeing a blob next to a keyboard, humans can infer it is likely to be a mouse; we
want to give a computer the same abilities.
There are several pieces of related work. Murphy et al [9] used global scene context to
help object recognition, but did not model relationships between objects. Fink and Perona
[4] exploited local dependencies in a boosting framework, but did not allow for multiple
rounds of communication between correlated objects. He et al [6] do not model connections
between objects directly, but rather they induce such correlations indirectly, via a bank of
hidden variables, using a ?restricted Boltzmann machine? architecture.
In this paper, we exploit contextual correlations between the object classes by introducing
Boosted Random Fields (BRFs). Boosted random fields build on both boosting [5, 10]
and conditional random fields (CRFs) [8, 7, 6]. Boosting is a simple way of sequentially
constructing ?strong? classifiers from ?weak? components, and has been used for singleclass object detection with great success [12]. Dietterich et al [3] combine boosting and
1D CRFs, but they only consider the problem of learning the local evidence potentials; we
consider the much harder problem of learning the structure of a 2D CRF.
Standard applications of MRFs/ CRFs to images [7] assume a 4-nearest neighbor grid
structure. While successful in low-level vision, this structure will fail in capturing important long distance dependencies between whole regions and across classes. We propose
a method for learning densely connected random fields with long range connections. The
topology of these connections is chosen by a weak learner which has access to a library
of graph fragments, derived from patches of labeled training images, which reflect typical
spatial arrangments of objects (similar to the segmentation fragments in [2]). At each round
of the learning algorithm, we add more connections from other locations in the image and
from other classes (detectors). The connections are assumed to be spatially invariant, which
means this update can be performed using convolution followed by a sigmoid nonlinearity.
The resulting architecture is similar to a convolutional neural network, although we used a
stagewise training procedure, which is much faster than back propagation.
In addition to recognizing things, such as cars and people, we are also interested in recognizing spatially extended ?stuff? [1], such as roads and buildings. The traditional sliding
window approach to object detection does not work well for detecting ?stuff?. Instead, we
combine object detection and image segmentation (c.f., [2]) by labeling every pixel in the
image. We do not rely on a bottom-up image segmentation algorithm, which can be fragile
without top-down guidance.
2
Learning potentials and graph structure
A conditional random field (CRF) is a distribution of the form
Y
1 Y
?i (Si )
?i,j (Si , Sj )
P (S|x) =
Z i
j?Ni
where x is the input (e.g., image), Ni are the neighbors of node i, and Si are labels. We
have assumed pairwise potentials for notational simplicity. Our goal is to learn the local
evidence potentials, ?i , the compatibility potentials ?, and the set of neighbors Ni .
We propose the following simple approximation: use belief propagation (BP) to estimate
the marginals, P (Si |x), and then use boosting to maximize the likelihood of each node?s
training data with respect to ?i and ?.
In more detail, the algorithm is as follows. At iteration t, the goal is to minimize the
negative log-likelihood of the training data. As in [11], we consider the per-label loss (i.e.,
we use marginal probabilities), as opposed to requiring that the joint labeling be correct (as
in Viterbi decoding). Hence the cost function to be minimized is
Y
YY
YY
?
?
Jt =
Jit = ?
bti,m (Si,m ) = ?
bti,m (+1)Si,m bti,m (?1)1?Si,m (1)
m
i
m
i
i
?
= (Si,m +
where Si,m ? {?1, +1} is the true label for pixel i in training case m, Si,m
t
1)/2 ? {0, 1} is just a relabeling, and bi,m = [P (Si = ?1|xm , t), P (Si = 1|xm , t)] is the
belief state at node i given input image xm after t iterations of the algorithm.
The belief at node i is given by the following (dropping the dependence on case m)
bti (?1) ? ?ti (?1) Mit (?1) where Mit is the product of all the messages coming into i
from all its neighbors at time t and where the message that k sends to i is given by
Y
X
btk (sk )
t+1
(2)
Mit+1 (?1) =
?t+1
(?1)
?
(?1)
=
?
(s
,
?1)
k,i
k
k?i
k?i
?ti?k (sk )
k?Ni
sk ?{?1,+1}
where ?k,i is the compatility between nodes k and i. If we assume that the local potentials
t
t
have the form ?ti (si ) = [eFi /2 ; e?Fi /2 ], where Fit is some function of the input data, then:
bti (+1) = ?(Fit + Gti ),
Gti = log Mit (+1) ? log Mit (?1)
(3)
?u
where ?(u) = 1/(1 + e ) is the sigmoid function. Hence each term in Eq. 1 simplifies to
a cost function similar to that used in boosting:
X
t
t
log Jit =
log 1 + e?Si,m (Fi,m +Gi,m ) .
(4)
m
1. Input: a set of labeled pairs {xi,m ; Si,m }, bound T
Output: Local evidence functions fit (x) and message update functions git (bNi ).
t=0
t=0
2. Initialize: bt=0
i,m = 0; Fi,m = 0; Gi,m = 0
3. For t=1..T.
(a) Fit local potential fi (xi,m ) by weighted LS to
t
t
t
Yi,m
= Si,m (1 + e?Si,m (Fi +Gi,m ) )
t
. compatibilities git (bt?1
(b) Fit
Ni ,m ) to Yi,m by weighted LS.
t?1
t
(c) Compute local potential Fi,m
= Fi,m
+ fit (xi,m )
(e) Update
(f) Update
Pt
g n (bt?1
Ni ,m )
n=1 i
t
t
t
the beliefs bi,m = ?(Fi,m + Gi,m )
t+1
weights wi,m
= bti,m (?1) bti,m (+1)
(d) Compute compatibilities Gti,m =
Figure 1: BRF training algorithm.
We assume that the graph is very densely connected so that the information that
one single node sends to another is so small that we can make the approximation
t+1
?t+1
k?i (+1)/?k?i (?1) ? 1. (This is a reasonable approximation in the case of images,
where each node represents a single pixel; only when the influence of many pixels is taken
into account will the messages become informative.) Hence
P
btk,m (sk )
sk ?[?1,+1] ?k,i (sk , +1) ?ti?k (sk )
Mit+1 (+1) X
t+1
Gi = log t+1
=
log P
(5)
btk,m (sk )
Mi (?1)
?
(s
,
?1)
k
t
sk ?[?1,+1] k,i k
?i?k (sk )
P
t
X
s ?[?1,+1] ?k,i (sk , +1) bk,m (sk )
?
log P k
(6)
t
sk ?[?1,+1] ?k,i (sk , ?1) bk,m (sk )
k
With this simplification,
is now a non-linear function of the beliefs Gt+1
(~btm ) at
i
iteration t. Therefore, We can write the beliefs at iteration t as a function of the local
evidences and the beliefs at time t ? 1: bti (+1) = ?(Fit (xi,m ) + Gti (~bt?1
m )). The key idea
behind BRFs is to use boosting to learn the G functions, which approximately implement
message passing in densely connected graphs. We explain this in more detail below.
Gt+1
i
2.1
Learning local evidence potentials
Defining Fit (xi,m ) = Fit?1 (xi,m ) + fit (xi,m ) as an additive model, where xi,m are the
features of training sample m at node i, we can learn this function in a stagewise fashion
by optimizing the second order Taylor expansion of Eq. 4 wrt fit , as in logitBoost [5]:
X
t
t
t
arg min
log
J
?
arg
min
wi,m
(Yi,m
? fit (xi,m ))2
(7)
i
t
t
fi
fi
m
?Si,m (Fit +Gti,m )
t
where Yi,m
= Si,m (1+e
). In the case that the weak learner is a ?regression
stump?, fi (x) = ah(x)+b, we can find the optimal a, b by solving a weighted least squares
t
problem, with weights wi,m
= bti (?1) bti (+1); we can find the best basis function h(x) by
searching over all elements of a dictionary.
2.2
Learning compatibility potentials and graph structure
In this section, we discuss how to learn the compatibility functions ?ij , and hence the
structure of the graph. Instead of learning the compatibility functions ?ij , we propose to
1. Input: a set of inputs {xi,m } and functions fit , git
Output: Set of beliefs bi,m and MAP estimates Si,m .
t=0
t=0
2. Initialize: bt=0
i,m = 0; Fi,m = 0; Gi,m = 0
3. From t = 1 to T , repeat
t?1
t
(a) Update local evidences Fi,m
= Fi,m
+ fit (xi,m )
(b) Update compatibilities Gti,m =
(c) Compute current beliefs
bti,m
=
Pt
g n (bt?1
Ni ,m )
n=1 i
t
?(Fi,m + Gti,m )
4. Output classification is Si,m = ? bti,m > 0.5
Figure 2: BRF run-time inference algorithm.
t+1
learn directly the function Gt+1
as we
iP . We propose to use an additive model for Gi
t
t+1
n ~t
t
did for learning F : Gi,m = n=1 gi (bm ), where ~bm is a vector with the beliefs of all
nodes in the graph at iteration t for the training sample m. The weak learners gin (~btm ) can
be regression stumps with the form gin (~btm ) = a?(w
~ ? ~btm > ?) + b, where a, b, ? are the
parameters of the regression stump, and w
~ i is a set of weights selected from a dictionary.
In the case of a graph with weak and almost symmetrical connections (which holds if
?(s1 , s2 ) ? 1, for all (s1 , s2 ), which implies the messages are not very informative) we can
further simplify the function Gt+1
by approximating it as a linear function of the beliefs:
i
X
Gt+1
?k,i btk,m (+1) + ?k,i
(8)
i,m =
k?Ni
This step reduces the computational cost. The weak learners gin (~btm ) will also be linear
t
functions. Hence the belief update simplifies to bt+1
?i ?~btm + ?i + Fi,m
), which
i,m (+1) = ?(~
is similar to the mean-field update equations. The neighborhood Ni over which we sum
incoming messages is determined by the graph structure, which is encoded in the non-zero
values of ?i . Each weak learner gin will compute a weighted combination of the beliefs of
the some subset of the nodes; this subset may change from iteration to iteration, and can be
quite large. At iteration t, we choose the weak learner git so as to minimize
Pt?1 n t?1
X
t
t t?1
log Jit (bt?1 ) = ?
log 1 + e?Si,m (Fi,m +gi (bm )+ n=1 gi (bm ))
m
which reduces to a weighted least squares problem similar to Eq. 7. See Fig. 1 for the
pseudo-code for the complete learning algorithm, and Fig. 2 for the pseudo-code for runtime inference.
3
BRFs for multiclass object detection and segmentation
With the BRF training algorithm in hand, we describe our approach for multiclass object
detection and region-labeling using densely connected BRFs.
3.1
Weak learners for detecting stuff and things
The square sliding window approach does not provide a natural way of working with irregular objects. Using region labeling as an image representation allows dealing with irregular
and extended objects (buildings, bookshelf, road, ...). Extended stuff [1] may be a very
important source of contextual information for other objects.
(a) Examples from the dictionary of about 2000 patches and masks, Ux,y , Vx,y .
(b) Examples from the dictionary of 30 graphs, Wx,y,c .
t=1
f
+
t=2
F
+
S
... =
Truth
f
Output
f
t=0
(c) Example feedforward segmentation for screens.
Figure 3: Examples of patches from the dictionary and an example of the segmentation obtained
using boosting trained with patches from (a).
The weak learners we use for the local evidence potentials are based on the segmentation
fragments proposed in [2]. Specifically, we create a dictionary of about 2000 image patches
U , chosen at random (but overlapping each object), plus a corresponding set of binary (inclass/ out-of-class) image masks, V : see Fig. 3(a). At each round t, for each class c, and
for each dictionary entry, we construct the following weak learner, whose output is a binary
matrix of the same size as the image I:
v(I) = ((I ? U ) > ?) ? V > 0
(9)
where ? represents normalized cross-correlation and ? represents convolution. The intuition behind this is that I ? U will produce peaks at image locations that contain this
patch/template, and then convolving with V will superimpose the segmentation mask on
top of the peaks. As a function of the threshold ?, the feature will behave more as a template
detector (? ? 1) or as a texture descriptor (? << 1).
To be able to detect objects at multiple scales, we first downsample the image to scale ?,
compute v(I ? ?), and then upsample the result. The final weak learner does this for
multiple scales, ORs all the results together, and then takes a linear transformation.
f (I) = ? (?? [v(I ? ?) ? ?]) + ?
(10)
Fig. 3(c) shows an example of segmentation obtained by using boosting without context.
The weak learners we use for the compatibility functions have a similar form:
!
C
X
gc (b) = ?
bc? ? Wc? + ?
(11)
c? =1
where bc? is the image formed by the beliefs at all pixels for class c? . This convolution
corresponds to eq. 8 in which the node i is one pixel x, y of class c. The binary kernels
(graph fragments) W define, for each node x, y of object class c, all the nodes from which it
will receive messages. These kernels are chosen by sampling patches of various sizes from
the labeling of images from the training set. This allows generating complicated patterns
of connectivity that reflect the statistics of object co-occurrences in the training set. The
overall incoming message is given by adding the kernels obtained at each boosting round.
(This is the key difference from mutual boosting [4], where the incoming message is just
the output of a single weak learner; thus, in mutual boosting, previously learned inter-class
connections are only used once.) Although it would seem to take O(t) time to compute Gt ,
we can precompute a single equivalent kernel W ? , so at runtime the overall complexity is
still linear in the number of boosting rounds, O(T ).
!
C
t
C
X
X
X def X
t
n
n
b c? ?
Gx,y,c =
bc? ? Wc?? + ? ?
? Wc? +
?n =
c? =1
n=1
n
c? =1
car car
building car
road car
F
Road
b=?(F+G)
Car
x
car building building building road building
Building
car road
building road
a) Incoming messages
to a car node.
b) Compatibilities (W?).
t=1
t=2
t=4
G
road road
y
t=20
c) A car out of context
(outside 3rd floor windows)
is less of a car.
t=40
Final labeling
b(car)
S(all)
d) Evolution of the beliefs for the car nodes (b) and labeling (S) for road, building, car.
Figure 4: Street scene. The BRF is trained to detect cars, buildings and the road.
In Fig. 4(a-b), we show the structures of the graph and the weights W ? defined by GT for
a BRF trained to detect cars, buildings and roads in street scenes.
3.2
Learning and inference
For training we used a labeled dataset of office and street scenes with about 100 images in
each set. During the training, in the first 5 rounds we only update the local potentials, to
allow local evidence to accrue. After the 5th iteration we start updating also the compatibility functions. At each round, we update only the local potential and compatibility function
associated with a single object class that reduces the most the multiclass cost. This allows
objects that need many features to have more complicated local potentials.
The algorithm learns to first detect easy (and large) objects, since these reduce the error of
all classes the fastest. The easy-to-detect objects can then pass information to the harder
ones. For instance, in office scenes, the system first detects screens, then keyboards, and
finally computer mice. Fig. 5 illustrates this behavior on the test set. A similar behavior is
obtained for the car detector (Fig. 4(d)). The detection of building and road provides strong
constraints for the locations of the car.
3.3
Cascade of classifiers with BRFs
The BRF can be turned into a cascade [12] by thresholding the beliefs. Computations
can then be reduced by doing the convolutions (required for computing f and g) only in
pixels that are still candidates for the presence of the target. At each round we update a
t
binary rejection mask for each object class, Rx,y,c
, by thresholding the beliefs at round t:
t
t?1
t
t
Rx,y,c = Rx,y,c ?(bx,y,c > ?c ). A pixel in the rejection mask is set to zero when we can
decide that the object is not present (when btx,y,c is below the threshold ?ct ? 0), and it is set
to 1 when more processing is required. The threshold ?ct is chosen so that the percentage
of missed detections is below a predefined level (we use 1%). Similarity we can define a
detection mask that will indicate pixels in which we decide the object is present. The mask
is then used for computing the features v(I) and messages G by applying the convolutions
only on the pixels not yet classified. We can denote those operators as ?R and ?R . This
Input image
screen
mouse
Ground truth
Output labeling
keyboard
t=5
b (screen)
t=10
t=15
b (screen)
b (screen)
t=25
b (screen)
t=50
b (screen)
F
G
b (keyboard)
b (keyboard)
b (keyboard)
b (keyboard)
b (keyboard)
b (mouse)
b (mouse)
b (mouse)
b (mouse)
b (mouse)
F
G
F
Area under ROC
G
1
Screen
BRF
0.5
t=0
Mouse
Keyboard
t=20
Boosting
Iteration (t)
t=50
Figure 5: Top. In this desk scene, it is easy to identify objects like the screen, keyboard and mouse,
even though the local information is sometimes insufficient. Middle: the evolution of the beliefs
(b and F and G) during detection for a test image. Bottom. The graph bellow shows the average
evolution of the area under the ROC for the three objects on 120 test images.
results in a more efficient classifier with only a slight decrease of performance. In Fig. 6 we
compare the reduction of the search space when implementing a cascade using independent
boosting (which reduces to Viola and Jones [12]), and when using BRF?s. We see that for
objects for which context is the main source of information, like the mouse, the reduction
in search space is much more dramatic using BRFs than using boosting alone.
4
Conclusion
The proposed BRF algorithm combines boosting and CRF?s, providing an algorithm that
is easy for both training and inference. We have demonstrated object detection in cluttered
scenes by exploiting contextual relationships between objects. The BRF algorithm is computationally efficient and provides a natural extension of the cascade of classifiers by integrating evidence from other objects in order to quickly reject certain image regions. The
BRF?s densely connected graphs, which efficiently collect information over large image
regions, provide an alternative framework to nearest-neighbor grids for vision problems.
Acknowledgments
This work was sponsored in part by the Nippon Telegraph and Telephone Corporation as
part of the NTT/MIT Collaboration Agreement, by BAE systems, and by DARPA contract
DABT63-99-1-0012.
Size of search space
Detection rate
Screen
Keyboard
Mouse
100%
Boosting
Boosting
Boosting
50%
BRF
BRF
BRF
Round
0%
0
1
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30
40
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0
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0.4
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1 0
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1
Figure 6: Contextual information reduces the search space in the framework of a cascade and improves performances. The search space is defined as the percentage of pixels that require further
processing before a decision can be reached at each round. BRF?s provide better performance and
requires fewer computations. The graphs (search space and ROCs) correspond to the average results
on a test set of 120 images.
References
[1] E. H. Adelson. On seeing stuff: the perception of materials by humans and machines. In Proc.
SPIE, volume 4299, pages 1?12, 2001.
[2] E. Borenstein and S. Ullman. Class-specific, top-down segmentation. In Proc. European Conf.
on Computer Vision, 2002.
[3] T. Dietterich, A. Ashenfelter, and Y. Bulatov. Training conditional random fields via gradient
tree boosting. In Intl. Conf. on Machine Learning, 2004.
[4] M. Fink and P. Perona. Mutual boosting for contextual influence. In Advances in Neural Info.
Proc. Systems, 2003.
[5] J. Friedman, T. Hastie, and R. Tibshirani. Additive logistic regression: a statistical view of
boosting. Annals of statistics, 28(2):337?374, 2000.
[6] Xuming He, Richard Zemel, and Miguel Carreira-Perpinan. Multiscale conditional random
fields for image labelling. In Proc. IEEE Conf. Computer Vision and Pattern Recognition,
2004.
[7] S. Kumar and M. Hebert. Discriminative random fields: A discriminative framework for contextual interaction in classification. In IEEE Conf. on Computer Vision and Pattern Recognition,
2003.
[8] J. Lafferty, A. McCallum, and F. Pereira. Conditional random fields: Probabilistic models for
segmenting and labeling sequence data. In Intl. Conf. on Machine Learning, 2001.
[9] K. Murphy, A. Torralba, and W. Freeman. Using the forest to see the trees: a graphical model
relating features, objects and scenes. In Advances in Neural Info. Proc. Systems, 2003.
[10] R. Schapire. The boosting approach to machine learning: An overview. In MSRI Workshop on
Nonlinear Estimation and Classification, 2001.
[11] B. Taskar, C. Guestrin, and D. Koller. Max-margin markov networks. In Advances in Neural
Info. Proc. Systems, 2003.
[12] P. Viola and M. Jones. Robust real-time object detection. Intl. J. Computer Vision, 57(2):137?
154, 2004.
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1,832 | 2,664 | Dynamic Bayesian Networks for
Brain-Computer Interfaces
Pradeep Shenoy
Department of Computer Science
University of Washington
Seattle, WA 98195
[email protected]
Rajesh P. N. Rao
Department of Computer Science
University of Washington
Seattle, WA 98195
[email protected]
Abstract
We describe an approach to building brain-computer interfaces (BCI)
based on graphical models for probabilistic inference and learning. We
show how a dynamic Bayesian network (DBN) can be used to infer
probability distributions over brain- and body-states during planning and
execution of actions. The DBN is learned directly from observed data
and allows measured signals such as EEG and EMG to be interpreted in
terms of internal states such as intent to move, preparatory activity, and
movement execution. Unlike traditional classification-based approaches
to BCI, the proposed approach (1) allows continuous tracking and prediction of internal states over time, and (2) generates control signals based
on an entire probability distribution over states rather than binary yes/no
decisions. We present preliminary results of brain- and body-state estimation using simultaneous EEG and EMG signals recorded during a
self-paced left/right hand movement task.
1
Introduction
The problem of building a brain-computer interface (BCI) has received considerable attention in recent years. Several researchers have demonstrated the feasibility of using EEG
signals as a non-invasive medium for building human BCIs [1, 2, 3, 4, 5] (see also [6] and
articles in the same issue). A central theme in much of this research is the postulation of
a discrete brain state that the user maintains while performing one of a set of physical or
imagined actions. The goal is to decode the hidden brain state from the observable EEG
signal, and to use the decoded state to control a robot or a cursor on a computer screen.
Most previous approaches to BCI (e.g., [1, 2, 4]) have utilized classification methods applied to time slices of EEG data to discriminate between a small set of brain states (e.g., left
versus right hand movement). These methods typically involve various forms of preprocessing (such as band-pass filtering or temporal smoothing) as well as feature extraction on
time slices known to contain one of the chosen set of brain states. The output of the classifier is typically a yes/no decision regarding class membership. A significant drawback of
such an approach is the need to have a ?point of reference? for the EEG data, i.e., a synchronization point in time where the behavior of interest was performed. Also, classifier-based
approaches typically do not model the uncertainty in their class estimates. As a result, it
is difficult to have a continuous estimate of the brain state and to associate an uncertainty
with the current estimate.
In this paper, we propose a new framework for BCI based on probabilistic graphical models [7] that overcomes some of the limitations of classification-based approaches to BCI.
We model the dynamics of hidden brain- and body-states using a Dynamic Bayesian Network (DBN) that is learned directly from EEG and EMG data. We show how a DBN can be
used to infer probability distributions over hidden state variables, where the state variables
correspond to brain states useful for BCI (such as ?Intention to move left hand?, ?Left hand
in motion?, etc). Using a DBN gives us several advantages in addition to providing a continuous probabilistic estimate of brain state. First, it allows us to explicitly model the hidden
causal structure and dependencies between different brain states. Second, it facilitates the
integration of information from multiple modalities such as EEG and EMG signals, allowing, for example, EEG-derived estimates to be bootstrapped from EMG-derived estimates.
In addition, learning a dynamic graphical model for time-varying data such as EEG allows
other useful operations such as prediction, filling in of missing data, and smoothing of state
estimates using information from future data points. These capabilities are difficult to obtain while working exclusively in the frequency domain or using whole slices of the data
(or its features) for training classifiers. We illustrate our approach in a simple Left versus
Right hand movement task and present preliminary results showing supervised learning and
Bayesian inference of hidden state for a dataset containing simultaneous EEG and EMG
recordings.
2
The DBN Framework
We study the problem of modeling spontaneous movement of the left/right arm using EEG
and EMG signals. It is well known that EEG signals show a slow potential drift prior to
spontaneous motor activity. This potential drift, known as the Bereitschaftspotential (BP,
see [8] for an excellent survey), shows variation in distribution over scalp with respect to
the body part being moved. In particular, the BP related to movement of left versus right
arm shows a strong lateral asymmetry. This allows one to not only estimate the intent to
move prior to actual movement, but also distinguish between left and right movements.
Previous approaches [1, 2] have utilized BP signals in classification-based BCI protocols
based on synchronization cues that identify points of movement onset. In our case, the
challenge was to model the structure of BPs and related movement signals using the states
of the DBN, and to recognize actions without explicit synchronization cues.
Figure 1 shows the complete DBN (referred to as Nf ull in this paper) used to model the leftright hand movement task. The hidden state Bt in Figure 1(a) tracks the higher-level brain
state over time and generates the hidden EEG and EMG states Et and Mt respectively.
These hidden states in turn generate the observed EEG and EMG signals. The dashed
arrows indicate that the hidden states make transitions over time. As shown in Figure 1(b),
the state Bt is intended to model the high-level intention of the subject. The figure shows
both the values Bt can take as well the constraints on the transition between values. The
actual probabilities of the allowed transitions are learned from data.
The hidden states Et and Mt are intended to model the temporal structure of the EEG and
EMG signals, which are generated using a mixture of Gaussians conditioned on E t and
Mt respectively. In the same way as the values of Bt are customized for our particular
experiment, we would like the state transitions of Et and Mt to also reflect their respective
constraints. This is important since it allows us to independently learn the simpler DBN
Nemg consisting of only the node Mt and the observed EMG signal. Similarly, we can
also independently learn the model Neeg consisting of the node Et and the observed EEG
signal. We use the models shown in Figure 2 for allowed transitions of the states M t and Et
respectively. In particular, Figure 2(a) indicates that the EMG state can transition along one
transition
Bt
Bt+1
Et
transition
Et+1
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Movt
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(b) Allowed Brain States Bt
(a) The Complete Network
Figure 1: Dynamic graphical model for modeling brain and body processes in a selfpaced movement task: (a) At each time instant t, the brain state Bt generates the EEG
and EMG internal states Et and Mt respectively, which in turn generate the observed EEG
and EMG. The dotted arrows represent transitions to a state at the next time step. (b) The
transition graph for the brain state Bt . The probability of each allowed transition is learned
from input data.
of three chains of states (labeled (1), (2), and (3)), representing the rest state, a left-hand
action and a right-hand action respectively. In each chain, the state Mt in each time step
either retains its old value with a given probability (self-pointing arrow) or transitions to the
next state value in that particular chain. The transition graph of Figure 2(b) shows similar
constraints on the EEG, except that the left and right action chains are further partitioned
into intent, action, and post-action subgroups of states, since each of these components are
discernible from the BP in EEG (but not from EMG) signals.
(chain of states)
e1
(chain of states)
mp+1
mq
(2)
(1)
m0
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(a) Transition Constraints on Mt
LM
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(3)
mp
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(chain of states)
m1
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ek
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(b) Transition Constraints onEt
Figure 2: Constrained transition graphs for the hidden EMG and EEG states E t and
Mt respectively. (a) The EMG state transitions between its values mi are constrained to
be in one of three chains: the chains model (1) rest, (2) left arm movement, and (3) right
arm movement. (b) In the EEG state transition graph, the left and right movement chains
are further divided into state values encoding intent (LI/RI), movement (LM/RM), and post
movement (LPM/RPM).
3
3.1
Experiments and Results
Data Collection and Processing
The task: The subject pressed two distinct keys on a keyboard with the left hand or right
hand at random at a self-initiated pace. We recorded 8 EEG channels around the motor area
of cortex (C3, Cz, C4, FC1, FC2, CP1, CP2, Pz) using averaged ear electrodes as reference,
and 2 differential pairs of EMG (one on each arm). Data was recorded at 2048Hz for a
period of 20 minutes, with the movements being separated by approximately 3-4s.
Average for Left Hand
Average for Right Hand
4
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Figure 3: Movement-related potential drift recorded during the hand-movement task:
The two plots show the EEG signals averaged over all trials from the motor-related channels
C3 and C4 for left (left panel) and right hand movement (right panel). The averages indicate
the onset and laterality of upcoming movements.
Processing: The EEG channels were bandpass-filtered 0.5Hz-5Hz, before being downsampled and smoothed at 128Hz. The EMG channels were converted to RMS values computed
over windows for an effective sampling rate of 128Hz.
Data Analysis: The recorded data were first analyzed in the traditional manner by averaging across all trials. Figure 3 shows the average of EEG channels C3 and C4 for left
and right hand movement actions respectively. As can be seen, the averages for both channels are different for the two classes. Furthermore, there is a slow potential drift preceding
the action and a return to the baseline potential after the action is performed. Previous
researchers [1] have classified EEG data over a window leading up to the instant of action
with high accuracy (over 90%) into left or right movement classes. Thus, there appears to
be a reliable amount of information in the EEG signal for at least discriminating between
left versus right movements.
Data Evaluation using SVMs: To obtain a baseline and to evaluate the quality of our
recorded data, we tested the performance of linear support vector machines (SVMs) on
classifying our EEG data into left and right movement classes. The choice of linear SVMs
was motivated by their successful use on similar problems by other researchers [1]. Time
slices of 0.5 seconds before each movement were concatenated from all EEG channels and
used for classification. We performed hyper-parameter selection using leave-one-out crossvalidation on 15 minutes of data and obtained an error of 15% on the remaining 5 minutes
of data. Such an error rate is comparable to those obtained in previous studies on similar
tasks, suggesting that the recorded data contains sufficient movement-related information
to be tested in experiments involving DBNs.
Learning the parameters of the DBN: We used the Graphical Models Toolkit
(GMTK) [9] for learning the parameters of our DBN. GMTK provides support for expressing constraints on state transitions (as described in Section 2). It learns the constrained
conditional probability tables and the parameters for the mixture of Gaussians using the
expectation-maximization (EM) algorithm.
We constructed a supervisory signal from the recorded key-presses as follows: A period of
100ms around each keystroke was labeled ?motor action? for the appropriate hand. This
signal was used to train the network Nemg in a supervised manner. To generate a supervisory signal for the network Neeg , or the full combined network Nf ull (Figure 1), we
added prefixes and postfixes of 150ms each to each action in this signal, and labeled them
?preparatory? and ?post-movement? activity respectively. These time-periods were chosen
by examining the average EEG and EMG activity over all actions. Thus, we can use partial
(EEG only) or full evidence in the inference step to obtain probability distributions over
brain state. The following sections describe our learning procedure and inference results in
greater detail.
3.2
Learning and Inference with EMG
Our first step is to learn the simpler model Nemg that has only the hidden Mt state and the
observed EMG signal. This is to test inference using the EMG signal alone. The parameters
of this DBN were learned in a supervised manner.
We used 15 minutes of EMG data to train our simplified model, and then tested it on the
remaining 5 minutes of data. The model was tested using Viterbi decoding (a single pass
of max-product inference over the network). In other words, the maximum a posteriori
(MAP) sequence of values for hidden states was computed. Figure 4 shows a 100s slice of
data containing 2 channels of EMG, and the predicted hidden EMG state Mt . The states 0,
1 and 2 correspond to ?no action?, left, and right actions respectively. In the shown figure,
the state Mt successfully captures not only all the obvious arm movements but also the
actions that are obscured by noise.
3.3
Learning the EEG Model
We used the supervisory signal described earlier to learn the corresponding EEG model
Neeg . Note that the brain-state can be inferred from the hidden EEG state Et directly, since
the state space is appropriately partitioned as shown in Figure 2(b).
Figure 5 shows the result of inference on the learned model Neeg using only the EEG
signals as evidence. The figure shows a subset of the EEG channels (C3,Cz,C4), the supervisory signal, and the predicted brain state Bt (the MAP estimate). The figure shows that
many of the instances of action (but not all) are correctly identified by the model.
Our model gives us at each time instant a MAP-estimated state sequence that best describes
the past, and the probability associated with that state sequence. This gives us, at each time
instant, a measure of how likely each brain state Bt is, with reference to the others. For
convenience, we can use the probability associated with the REST state (see Figure 1) as
reference. Figure 6 shows a graphical illustration of this instantaneous time estimate. The
plotted graphs are, in order, the supervisory signal (i.e., the ?ground truth value?) and the
instantaneous measures of likelihood of intention/movement/post-movement states for the
left and right hand respectively. For convenience, we represent the likelihood ratio of each
state?s MAP probability estimate to that of the rest state, and use a logarithmic scale. We
see that the true hand movements are correctly inferred in a surprisingly large number of
cases (log likelihood ratio crosses 0). Furthermore, the actual likelihood values convey a
measure of the uncertainty in the inference, a property that would be of great value for
critical BCI applications such as controlling a robotic wheelchair.
In summary, our graphical models Nemg and Neeg have shown promising results in correctly identifying movement onset from EMG and EEG signals respectively. Ongoing work
is focused on improving accuracy by using features extracted from EEG, and inference using both EEG and EMG in Nf ull (the full model).
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Figure 4: Bayesian Inference of Movement using EMG: The figure shows 100 seconds of
EMG data from two channels along with the MAP state sequence predicted by our trained
EMG model. The states 0,1,2 correspond to ?no action?, left, and right actions respectively.
Our model correctly identifies the obscured spikes in the noisy right EMG channel
4
Discussion and Conclusion
We have shown that dynamic Bayesian networks (DBNs) can be used to model the transitions between brain- and muscle-states as a subject performs a motor task. In particular,
a two-level hierarchical network was proposed for simultaneously estimating higher-level
brain state and lower-level EEG and EMG states in a left/right hand movement task. The
results demonstrate that for a self-paced movement task, hidden brain states useful for BCI
such as intention to move the left or right hand can be decoded from a DBN learned directly
from EEG and EMG data.
Previous work on BCIs can be grouped into two broad classes: self-regulatory BCIs and
BCIs based on detecting brain state. Self-regulatory BCIs rely on training the user to regulate certain features of the EEG, such as cortical positivity [10], or oscillatory activity (the
? rhythm, see [5]), in order to control, for example, a cursor on a display. The approach
presented in this paper falls in the second class of BCIs, those based on detecting brain
states [1, 2, 3, 4]. However, rather than employing classification methods, we use probabilistic graphical models for inferring brain state and learning the transition probabilities
between brain states.
Successfully learning a dynamic graphical model as suggested in this paper offers several
advantages over traditional classification-based schemes for BCI. It allows one to explicitly model the hidden causal structure and dependencies between different brain states. It
provides a probabilistic framework for integrating information from multiple modalities
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Figure 5: Bayesian Inference of Brain State using EEG: The figure shows 1 minute of
EEG data (at 128Hz) for the channels C3, Cz, C4, along with the ?true? brain state and
the brain state inferred using our DBN model with only EEG evidence. State 0 is the rest
state, states 1 through 3 represent left hand movement, and 4 through 6 represent right hand
movement (see Figure 1(b)).
such as EEG and EMG signals, allowing, for example, EEG-derived estimates to be bootstrapped from EMG-derived estimates. A dynamic graphical model for time-varying data
such as EEG also allows prediction, filling in of missing data, and smoothing of state estimates using information from future data points, properties not easily achieved in methods
that work exclusively in the frequency domain or use data slices for training classifiers. Our
current efforts are focused on investigating methods for learning dynamic graphical models
for motor tasks of varying complexity and using these models to build robust, probabilistic
BCI systems.
References
[1] B. Blankertz, G. Curio, and K.R. Mueller. Classifying single trial EEG: Towards
brain computer interfacing. In Advances in Neural Information Processing Systems
12, 2001.
[2] G. Dornhege, B. Blankertz, G. Curio, and K.-R. Mueller. Combining features for
BCI. In Advances in Neural Information Processing Systems 15, 2003.
[3] J. D. Bayliss and D. H. Ballard. Recognizing evoked potentials in a virtual environment. In Advances in Neural Information Processing Systems 12, 2000.
[4] P. Meinicke, M. Kaper, F. Hoppe, M. Heumann, and H. Ritter. Improving transfer
rates in brain computer interfacing: a case study. In Advances in Neural Information
Processing Systems 15, 2003.
[5] J.R. Wolpaw, D.J. McFarland, and T.M. Vaughan. Brain-computer interfaces for communication and control. IEEE Trans Rehab Engg, pages 222?226, 2000.
6
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Left action Log(P(state)/P(rest))
zero
pre
movt
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movt
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Right action Log(P(state)/P(rest))
Figure 6: Probabilistic Estimation of Brain State: The figure shows the supervisory
signal, along with a probabilistic measure of the current state for left and right actions
respectively. The measure shown is the log ratio of the instantaneous MAP estimate for the
relevant state and the estimate for the rest state.
[6] J. R. Wolpaw et al. Brain-computer interface technology: a review of the first international meeting. IEEE Trans Rehab Engg, 8:164?173, 2000.
[7] R. E. Neapolitan. Learning Bayesian Networks. Prentice Hall, NJ, 2004.
[8] M. Jahanshahi and M. Hallet. The Bereitschaftspotential: movement related cortical
potentials. Kluwer Academic, New York, 2002.
[9] J. Bilmes and G. Zweig. The graphical models toolkit: An open source software system for speech and time-series processing. In IEEE Intl. Conf. on Acoustics, Speech,
and Signal Processing, Orlando FL, 2002.
[10] N. Birbaumer, N. Ghanayim, T. Hinterberger, I. Iverson, B. Kotchubey, A. Kiibler,
J. Perelmouter, E. Taub, and H. Flor. A spelling device for the paralyzed. In Nature,
398: 297-298, 1999.
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1,833 | 2,665 | Hierarchical Eigensolver for Transition Matrices
in Spectral Methods
?
Chakra Chennubhotla? and Allan D. Jepson?
Department of Computational Biology, University of Pittsburgh
?
Department of Computer Science, University of Toronto
Abstract
We show how to build hierarchical, reduced-rank representation for large
stochastic matrices and use this representation to design an efficient algorithm for computing the largest eigenvalues, and the corresponding
eigenvectors. In particular, the eigen problem is first solved at the coarsest level of the representation. The approximate eigen solution is then
interpolated over successive levels of the hierarchy. A small number of
power iterations are employed at each stage to correct the eigen solution.
The typical speedups obtained by a Matlab implementation of our fast
eigensolver over a standard sparse matrix eigensolver [13] are at least a
factor of ten for large image sizes. The hierarchical representation has
proven to be effective in a min-cut based segmentation algorithm that we
proposed recently [8].
1 Spectral Methods
Graph-theoretic spectral methods have gained popularity in a variety of application domains: segmenting images [22]; embedding in low-dimensional spaces [4, 5, 8]; and clustering parallel scientific computation tasks [19]. Spectral methods enable the study of properties global to a dataset, using only local (pairwise) similarity or affinity measurements between the data points. The global properties that emerge are best understood in terms of a
random walk formulation on the graph. For example, the graph can be partitioned into clusters by analyzing the perturbations to the stationary distribution of a Markovian relaxation
process defined in terms of the affinity weights [17, 18, 24, 7]. The Markovian relaxation
process need never be explicitly carried out; instead, it can be analytically expressed using
the leading order eigenvectors, and eigenvalues, of the Markov transition matrix.
In this paper we consider the practical application of spectral methods to large datasets. In
particular, the eigen decomposition can be very expensive, on the order of O(n 3 ), where n
is the number of nodes in the graph. While it is possible to compute analytically the first
eigenvector (see ?3 below), the remaining subspace of vectors (necessary for say clustering)
has to be explicitly computed. A typical approach to dealing with this difficulty is to first
sparsify the links in the graph [22] and then apply an efficient eigensolver [13, 23, 3].
In comparison, we propose in this paper a specialized eigensolver suitable for large stochastic matrices with known stationary distributions. In particular, we exploit the spectral properties of the Markov transition matrix to generate hierarchical, successively lower-ranked
approximations to the full transition matrix. The eigen problem is solved directly at the
coarsest level of representation. The approximate eigen solution is then interpolated over
successive levels of the hierarchy, using a small number of power iterations to correct the
solution at each stage.
2 Previous Work
One approach to speeding up the eigen decomposition is to use the fact that the columns
of the affinity matrix are typically correlated. The idea then is to pick a small number
of representative columns to perform eigen decomposition via SVD. For example, in the
Nystrom approximation procedure, originally proposed for integral eigenvalue problems,
the idea is to randomly pick a small set of m columns; generate the corresponding affinity
matrix; solve the eigenproblem and finally extend the solution to the complete graph [9, 10].
The Nystrom method has also been recently applied in the kernel learning methods for
fast Gaussian process classification and regression [25]. Other sampling-based approaches
include the work reported in [1, 2, 11].
Our starting point is the transition matrix generated from affinity weights and we show
how building a representational hierarchy follows naturally from considering the stochastic matrix. A closely related work is the paper by Lin on reduced rank approximations of
transition matrices [14]. We differ in how we approximate the transition matrices, in particular our objective function is computationally less expensive to solve. In particular, one
of our goals in reducing transition matrices is to develop a fast, specialized eigen solver for
spectral clustering. Fast eigensolving is also the goal in ACE [12], where successive levels
in the hierarchy can potentially have negative affinities. A graph coarsening process for
clustering was also pursued in [21, 3].
3 Markov Chain Terminology
We first provide a brief overview of the Markov chain terminology here (for more details
see [17, 15, 6]). We consider an undirected graph G = (V, E) with vertices vi , for i =
{1, . . . , n}, and edges ei,j with non-negative weights ai,j . Here the weight ai,j represents
the affinity between vertices vi and vj . The affinities are represented by a non-negative,
symmetric n ? n matrix
ai,j as elements. The degree of a node j is
PnA having weights
Pn
defined to be: dj =
a
=
a
, where we define D = diag(d1 , . . . , dn ).
i,j
j,i
i=1
j=1
A Markov chain is defined using these affinities by setting a transition probability matrix
M = AD ?1 , where the columns of M each sum to 1. The transition probability matrix
defines the random walk of a particle on the graph G.
The random walk need never be explicitly carried out; instead, it can be analytically expressed using the leading order eigenvectors, and eigenvalues, of the Markov transition
matrix. Because the stochastic matrices need not be symmetric in general, a direct eigen
decomposition step is not preferred for reasons of instability. This problem is easily circumvented by considering a normalized affinity matrix: L = D ?1/2 AD?1/2 , which is related
to the stochastic matrix by a similarity transformation: L = D ?1/2 M D1/2 . Because L
is symmetric, it can be diagonalized: L = U ?U T , where U = [~u1 , ~u2 , ? ? ? , ~un ] is an
orthogonal set of eigenvectors and ? is a diagonal matrix of eigenvalues [?1 , ?2 , ? ? ? , ?n ]
sorted in decreasing order. The eigenvectors have unit length k~uk k = 1 and from the
form of A and D it can be shown that the eigenvalues ?i ? (?1, 1], with at least one
eigenvalue equal to one. Without loss of generality, we take ?1 = 1. Because L and M
are similar we can perform an eigen decomposition of the Markov transition matrix as:
M = D1/2 LD?1/2 = D1/2 U ? U T D?1/2 . Thus an eigenvector ~u of L corresponds to
an eigenvector D 1/2 ~u of M with the same eigenvalue ?.
The Markovian relaxation process after ? iterations, namely M ? , can be represented as:
M ? = D1/2 U ?? U T D?1/2 . Therefore, a particle undertaking a random walk with an
initial distribution p~ 0 acquires after ? steps a distribution p~ ? given by: p~ ? = M ? p~ 0 .
Assuming the graph is connected, as ? ? ?,
Pnthe Markov chain approaches a unique
stationary distribution given by ~? = diag(D)/ i=1 di , and thus, M ? = ~? 1T , where 1 is
a n-dim column vector of all ones. Observe that ~? is an eigenvector of M as it is easy to
show that M~? = ~? and the corresponding eigenvalue is 1. Next, we show how to generate
hierarchical, successively low-ranked approximations for the transition matrix M .
4 Building a Hierarchy of Transition Matrices
The goal is to generate a very fast approximation, while simultaneously achieving sufficient
f as some
accuracy. For notational ease, we think of M as a fine-scale representation and M
f further, we can generate
coarse-scale approximation to be derived here. By coarsening M
successive levels of the representation hierarchy. We use the stationary distribution ~? to
construct a corresponding coarse-scale stationary distribution ~?. As we just discussed a
critical property of the fine scale Markov matrix M is that it is similar to the symmetric
matrix L and we wish to preserve this property at every level of the representation hierarchy.
4.1 Deriving Coarse-Scale Stationary Distribution
We begin by expressing the stationary distribution ~? as a probabilistic mixture of latent
distributions. In matrix notation, we have
~? = K ~?,
(1)
where ~? is an unknown mixture coefficient vector of length m, K is an n ? P
m non-negative
n
kernel matrix whose columns are latent distributions that each sum to 1: i=1 Ki,j = 1
~
and m n. It is easy to derive a maximum likelihood approximation
of ? using an EM
~
type algorithm [16]. The main step is to find a stationary point ?, ? for the Lagrangian:
E??
n
X
i=1
?i ln
m
X
Ki,j ?j + ?
j=1
X
m
j=1
?j ? 1 .
(2)
An implicit step in this EM procedure is to compute the the ownership probability r i,j of
the j th kernel (or node) at the coarse scale for the ith node on the fine scale and is given by
?j Ki,j
.
ri,j = Pm
k=1 ?k Ki,k
(3)
The EM procedure allows for an update of both ~? and the latent distributions in the kernel
matrix K (see ?8.3.1 in [6]).
For initialization, ~? is taken to be uniform over the coarse-scale states. But in choosing
kernels K, we provide a good initialization for the EM procedure. Specifically, the Markov
matrix M is diffused using a small number of iterations to get M ? . The diffusion causes
random walks from neighboring nodes to be less distinguishable. This in turn helps us
select a small number of columns of M ? in a fast and greedy way to be the kernel matrix
K. We defer the exact details on kernel selection to a later section (?4.3).
4.2 Deriving the Coarse-Scale Transition Matrix
f, the coarse-scale transition matrix, we break it down into three steps.
In order to define M
First, the Markov chain propagation at the coarse scale can be defined as:
f~q k ,
~q k+1 = M
(4)
where ~q is the coarse scale probability distribution after k steps of the random walk.
Second, we expand ~q k into the fine scale using the kernels K resulting in a fine scale
probability distribution p~ k :
p~ k = K~q k .
(5)
k
Finally, we lift p~ k back into the coarse scale by using the ownership probability of the j th
kernel for the ith node on the fine grid:
qjk+1 =
n
X
i=1
ri,j pik
(6)
Substituting for Eqs.(3) and (5) in Eq. 6 gives
qjk+1 =
n
X
i=1
ri,j
m
X
Ki,t qtk =
t=1
n
X
i=1
? K
Pm j i,j
k=1 ?k Ki,k
X
m
Ki,t qtk .
(7)
t=1
We can write the preceding equation in a matrix form:
?1
K~q k .
~q k+1 = diag( ~? ) K T diag K ~?
f as:
Comparing this with Eq. 4, we can derive the transition matrix M
?1
f = diag( ~? ) K T diag K ~?
M
K.
(8)
(9)
f~?, so ~? is the stationary distribution for M
f. Following the
It is easy to see that ~? = M
~
f
definition of M , and its stationary distribution ?, we can generate a symmetric coarse scale
e given by
affinity matrix A
?1
e=M
fdiag(~?) = diag( ~? ) K T
diag K ~?
A
Kdiag(~?) ,
(10)
f from Eq. 9. The coarse-scale affinity matrix A
e is
where we substitute for the expression M
then normalized to get:
e=D
e ?1/2 A
eD
e ?1/2 ;
L
e = diag(de1 , de2 , ? ? ? , dem ),
D
(11)
e (see
where dej is the degree of node j in the coarse-scale graph represented by the matrix A
f is precisely similar to a
?3 for degree definition). Thus, the coarse scale Markov matrix M
e
symmetric matrix L.
4.3 Selecting Kernels
For demonstration purpose, we present the kernel selection details on the image of an eye
shown below. To begin with, a random walk is defined where each pixel in the test image
is associated with a vertex of the graph G. The edges in G are defined by the standard
8-neighbourhood of each pixel. For the demonstrations in this paper, the edge weight
ai,j between neighbouring pixels xi and xj is given by a function of the difference in the
corresponding intensities I(xi ) and I(xj ): ai,j = exp(?(I(xi ) ? I(xj ))2 /2?a2 ), where
?a is set according to the median absolute difference |I(xi ) ? I(xj )| between neighbours
measured over the entire image. The affinity matrix A with the edge weights is then used
to generate a Markov transition matrix M .
The kernel selection process we use is fast and greedy. First, the fine scale Markov matrix
M is diffused to M ? using ? = 4. The Markov matrix M is sparse as we make the
affinity matrix A sparse. Every column in the diffused matrix M ? is a potential kernel. To
facilitate the selection process, the second step is to rank order the columns of M ? based
on a probability value in the stationary distribution ~? . Third, the kernels (i.e. columns of
M ? ) are picked in such a way that for a kernel Ki all of the neighbours of pixel i which are
within the half-height of the the maximum value in the kernel Ki are suppressed from the
selection process. Finally, the kernel selection is continued until every pixel in the image is
within a half-height of the peak value of at least one kernel. If M is a full matrix, to avoid
the expense of computing M ? explicitly, random kernel centers can be selected, and only
the corresponding columns of M ? need be computed.
We show results from a three-scale hierarchy on the eye image (below). The image has
25 ? 20 pixels but is shown here enlarged for clarity. At the first coarse scale 83 kernels
are picked. The kernels each correspond to a different column in the fine scale transition
matrix and the pixels giving rise to these kernels are shown numbered on the image.
Using these kernels as an initialization, the EM procedure derives a coarse-scale stationary distribution ~?
21
14
26
4
(Eq. 2), while simultaneously updating the kernel ma12
27
2
19
trix. Using the newly updated kernel matrix K and the
5
8 13
23 30
f
18
6 9
derived stationary distribution ~? a transition matrix M
28
20
15
32
10
22
is
generated
(Eq.
9).
The
coarse
scale
Markov
matrix
24
17
7
f? , again using ? = 4. The kernel
is then diffused to M
Coarse Scale 1 Coarse Scale 2
selection algorithm is reapplied, this time picking 32
kernels for the second coarse scale. Larger values of ? cause the coarser level to have fewer
elements. But the exact number of elements depends on the form of the kernels themselves.
For the random experiments that we describe later in ?6 we found ? = 2 in the first iteration
and 4 thereafter causes the number of kernels to be reduced by a factor of roughly 1/3 to
1/4 at each level.
72
28 35 44 51
64 82
4 12 31
56
19
77
36 45
52 65
13
57
23 37
5
40
53 63 73
14 29
6
66
38
74
47
24
7
30 41 54 71 78
58
15
8 20
25 39 48 59 67
68 79
21
16 26 42 49
2 11
55
60 75
32
83
43
9
76
50
17 27
61
33
69 80
3
46
18
70 81
34
10
62
22
1
25
11
1
3
16
31
29
At coarser levels of the hierarchy, we expect the kernels to get less sparse and so will the
affinity and the transition matrices. In order to promote sparsity at successive levels of
e by zeroing out elements associated with ?small? transition
the hierarchy we sparsify A
f
probabilities in M . However, in the experiments described later in ?6, we observe this
sparsification step to be not critical. To summarize, we use the stationary distribution ~?
f, and its stationary distribution ~?, at the
at the fine-scale to derive a transition matrix M
e and
coarse-scale. The coarse scale transition in turn helps to derive an affinity matrix A
e It is obvious that this procedure can be repeated recursively. We
its normalized version L.
describe next how to use this representation hierarchy for building a fast eigensolver.
5 Fast EigenSolver
Our goal in generating a hierarchical representation of a transition matrix is to develop a
fast, specialized eigen solver for spectral clustering. To this end, we perform a full eigen
decomposition of the normalized affinity matrix only at the coarsest level. As discussed
in the previous section, the affinity matrix at the coarsest level is not likely to be sparse,
hence it will need a full (as opposed to a sparse) version of an eigen solver. However it is
typically the case that e ? m n (even in the case of the three-scale hierarchy that we
just considered) and hence we expect this step to be the least expensive computationally.
The resulting eigenvectors are interpolated to the next lower level of the hierarchy by a
process which will be described next. Because the eigen interpolation process between
every adjacent pair of scales in the hierarchy is similar, we will assume we have access to
e (size: m ? e) for the normalized affinity matrix L
e (size: m ? m)
the leading eigenvectors U
and describe how to generate the leading eigenvectors U (size: n ? e), and the leading
eigenvalues S (size: e ? 1), for the fine-scale normalized affinity matrix L (size: n ? n).
There are several steps to the eigen interpolation process and in the discussion that follows
we refer to the lines in the pseudo-code presented below.
e can be interpolated using the kernel matrix K to
First, the coarse-scale eigenvectors U
e
generate U = K U , an approximation for the fine-scale eigenvectors (line 9). Second,
interpolation alone is unlikely to set the directions of U exactly aligned with U L , the
vectors one would obtain by a direct eigen decomposition of the fine-scale normalized
affinity matrix L. We therefore update the directions in U by applying a small number of
power iterations with L, as given in lines 13-15.
e , S)
e
function (U, S) = CoarseToFine(L, K, U
1: INPUT
2:
L, K ? {L is n ? n and K is n ? m where m n}
e /Se ? {leading coarse-scale eigenvectors/eigenvalues of L.
e U
e is of size m ? e, e ? m}
3:
U
4: OUTPUT
5:
U, S ? {leading fine-scale eigenvectors/eigenvalues of L. U is n ? e and S is e ? 1.}
x 10
0.4
3
0.96
0.94
0.92
0.9
0.35
2.5
Relative Error
Absolute Relative Error
0.98
Eigen Value
|??|??1
?3
Eigen Spectrum
1
2
1.5
1
5
10
15
20
Eigen Index
(a)
25
30
0.2
0.15
0.1
0.5
0.88
0.3
0.25
0.05
5
10
15
20
Eigen Index
(b)
25
30
5
10
15
20
Eigen Index
25
30
(c)
Figure 1: Hierarchical eigensolver results. (a) comparing ground truth eigenvalues S L
(red circles) with multi-scale eigensolver spectrum S (blue line) (b) Relative absolute error
L|
between eigenvalues: |S?S
(c) Eigenvector mismatch: 1 ? diag |U T UL | , between
SL
eigenvectors U derived by the multi-scale eigensolver and the ground truth U L . Observe
the slight mismatch in the last few eigenvectors, but excellent agreement in the leading
eigenvectors (see text).
6: CONSTANTS: TOL = 1e-4; POWER ITERS = 50
7:
?
?
e
8: TPI = min POWER ITERS, log(e ? eps/TOL)/ log(min(S))
{eps: machine accuracy}
e
9: U = K U {interpolation from coarse to fine}
10: while not converged do
11: Uold = U {n ? e matrix, e n}
12: for i = 1 to TPI do
13:
U ? LU
14: end for
15: U ? Gram-Schmidt(U ) {orthogonalize U }
16: Le = U T LU {L may be sparse, but Le need not be.}
17: Ue Se UeT = svd(Le ) {eigenanalysis of Le , which is of size e ? e.}
18: U ? U Ue {update the leading eigenvectors of L}
19: S = diag(Se ) {grab the leading eigenvalues of L}
T
U ) {1 is a e ? 1 vector of all ones}
20: innerProd = 1 ? diag( Uold
21: converged = max[abs(innerProd)] < TOL
22: end while
The number of power iterations TPI can be bounded as discussed next. Suppose ~v =
U c where U is a matrix of true eigenvectors and c is a coefficient vector for an arbitrary
vector ~v . After TPI power iterations ~v becomes ~v = U diag(S TPI )c, where S has the exact
eigenvalues. In order for the component of a vector ~v in the direction Ue (the eth column of
U ) not to be swamped by other components, we can limit it?s decay after TPI iterations as
TPI
follows: (S(e)/S(1)) >= e?eps/TOL, where S(e) is the exact eth eigenvalue, S(1) =
1, eps is the machine precision, TOL is requested accuracy. Because we do not have access
to the exact value S(e) at the beginning of the interpolation procedure, we estimate it from
e This leads to a bound on the power iterations TPI, as derived on
the coarse eigenvalues S.
the line 9 above. Third, the interpolation process and the power iterations need not preserve
orthogonality in the eigenvectors in U . We fix this by Gram-Schmidt orthogonalization
procedure (line 16). Finally, there is a still a problem with power iterations that needs
to be resolved, in that it is very hard to separate nearby eigenvalues. In particular, for
the convergence of the power iterations the ratio that matters is between the (e + 1) st and
eth eigenvalues. So the idea we pursue is to use the power iterations only to separate the
reduced space of eigenvectors (of dimension e) from the orthogonal subspace (of dimension
n ? e). We then use a full SVD on the reduced space to update the leading eigenvectors
U , and eigenvalues S, for the fine-scale (lines 17-20). This idea is similar to computing the
Ritz values and Ritz vectors in a Rayleigh-Ritz method.
6 Interpolation Results
Our multi-scale decomposition code is in Matlab. For the direct eigen decomposition, we
have used the Matlab program svds.m which invokes the compiled ARPACKC routine
[13], with a default convergence tolerance of 1e-10.
In Fig. 1a we compare the spectrum S obtained from a three-scale decomposition on the
eye image (blue line) with the ground truth, which is the spectrum SL resulting from direct
eigen decomposition of the fine-scale normalized affinity matrices L (red circles). There
is an excellent agreement in the leading eigenvalues. To illustrate this, we show absolute
L|
in Fig. 1b. The spectra agree mostly, except for
relative error between the spectra: |S?S
SL
the last few eigenvalues. For a quantitative comparison between the eigenvectors, we plot
in Fig. 1c the following measure: 1 ? diag(|U T UL |), where U is the matrix of eigenvectors
obtained by the multi-scale approximation, UL is the ground-truth resulting from a direct
eigen decomposition of the fine-scale affinity matrix L and 1 is a vector of all ones. The
relative error plot demonstrates a close match, within the tolerance threshold of 1e-4 that
we chose for the multi-scale method, in the leading eigenvector directions between the
two methods. The relative error is high with the last few eigen vectors, which suggests
that the power iterations have not clearly separated them from other directions. So, the
strategy we suggest is to pad the required number of leading eigen basis by about 20%
before invoking the multi-scale procedure. Obviously, the number of hierarchical stages
for the multi-scale procedure must be chosen such that the transition matrix at the coarsest
scale can accommodate the slight increase in the subspace dimensions. For lack of space
we are omitting extra results (see Ch.8 in [6]).
Next we measure the time the hierarchical eigensolver takes to compute the leading eigenbasis for various input sizes, in comparison with the svds.m procedure [13]. We form
images of different input sizes by Gaussian smoothing of i.i.d noise. The Gaussian function has a standard deviation of 3 pixels. The edges in graph G are defined by the standard
8-neighbourhood of each pixel. The edge weights between neighbouring pixels are simply given by a function of the difference in the corresponding intensities (see ?4.3). The
affinity matrix A with the edge weights is then used to generate a Markov transition matrix
M . The fast eigensolver is run on ten different instances of the input image of a given size
and the average of these times is reported here. For a fair comparison between the two
procedures, we set the convergence tolerance value for the svds.m procedure to be 1e-4,
the same as the one used for the fast eigensolver. We found the hierarchical representation
derived from this tolerance threshold to be sufficiently accurate for a novel min-cut based
segmentation results that we reported in [8]. Also, the subspace dimensionality is fixed to
be 51 where we expect (and indeed observe) the leading 40 eigenpairs derived from the
multi-scale procedure to be accurate. Hence, while invoking svds.m we compute only the
leading 41 eigenpairs.
In the table shown below, the first column corresponds to the number of nodes in the graph,
while the second and third columns report the time taken in seconds by the svds.m procedure and the Matlab implementation of the multi-scale eigensolver respectively. The
fourth column reports the speedups of the multi-scale eigensolver over svds.m procedure
on a standard desktop (Intel P4, 2.5GHz, 1GB RAM). Lowering the tolerance threshold for
svds.m made it faster by about 20 ? 30%. Despite this, the multi-scale algorithm clearly
outperforms the svds.m procedure. The most expensive step in the multi-scale algorithm
is the power iteration required in the last stage, that is interpolating eigenvectors from the
first coarse scale to the required fine scale. The complexity is of the order of n ? e where
e is the subspace dimensionality and n is the size of the graph. Indeed, from the table we
can see that the multi-scale procedure is taking time roughly proportional to n. Deviations
from the linear trend are observed at specific values of n, which we believe are due to the
n
322
632
642
652
1002
1272
1282
1292
1602
2552
2562
2572
5112
5122
5132
6002
7002
8002
svds.m
1.6
10.8
20.5
12.6
44.2
91.1
230.9
96.9
179.3
819.2
2170.8
871.7
7977.2
20269
7887.2
10841.4
15048.8
Multi-Scale
1.5
4.9
5.5
5.1
13.1
20.4
35.2
20.9
34.4
90.3
188.7
93.3
458.8
739.3
461.9
644.2
1162.4
1936.6
Speedup
1.1
2.2
3.7
2.5
3.4
4.5
6.6
4.6
5.2
9.1
11.5
9.3
17.4
27.4
17.1
16.8
12.9
variations in the difficulty of the specific eigenvalue problem (eg. nearly multiple eigenvalues).
The hierarchical representation has proven to be
effective in a min-cut based segmentation algorithm that we proposed recently [8]. Here we explored the use of random walks and associated
spectral embedding techniques for the automatic
generation of suitable proposal (source and sink)
regions for a min-cut based algorithm. The multiscale algorithm was used to generate the 40 leading eigenvectors of large transition matrices (eg.
size 20K ? 20K). In terms of future work, it will
be useful to compare our work with other approximate methods for SVD such as [23].
Ack: We thank S. Roweis, F. Estrada and M. Sakr for valuable comments.
References
[1] D. Achlioptas and F. McSherry. Fast Computation of Low-Rank Approximations. STOC, 2001.
[2] D. Achlioptas et al Sampling Techniques for Kernel Methods. NIPS, 2001.
[3] S. Barnard and H. Simon Fast Multilevel Implementation of Recursive Spectral Bisection for
Partitioning Unstructured Problems. PPSC, 627-632.
[4] M. Belkin et al Laplacian Eigenmaps and Spectral Techniques for Embedding. NIPS, 2001.
[5] M. Brand et al A unifying theorem for spectral embedding and clustering. AI & STATS, 2002.
[6] C. Chennubhotla. Spectral Methods for Multi-scale Feature Extraction and Spectral Clustering.
http://www.cs.toronto.edu/?chakra/thesis.pdf Ph.D Thesis, Department of
Computer Science, University of Toronto, Canada, 2004.
[7] C. Chennubhotla and A. Jepson. Half-Lives of EigenFlows for Spectral Clustering. NIPS, 2002.
[8] F. Estrada, A. Jepson and C. Chennubhotla. Spectral Embedding and Min-Cut for Image Segmentation. Manuscript Under Review, 2004.
[9] C. Fowlkes et al Efficient spatiotemporal grouping using the Nystrom method. CVPR, 2001.
[10] S. Belongie et al Spectral Partitioning with Indefinite Kernels using Nystrom app. ECCV,
2002.
[11] A. Frieze et al Fast Monte-Carlo Algorithms for finding low-rank approximations. FOCS,
1998.
[12] Y. Koren et al ACE: A Fast Multiscale Eigenvectors Computation for Drawing Huge Graphs
IEEE Symp. on InfoVis 2002, pp. 137-144
[13] R. B. Lehoucq, D. C. Sorensen and C. Yang. ARPACK User Guide: Solution of Large Scale
Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM 1998.
[14] J. J. Lin. Reduced Rank Approximations of Transition Matrices. AI & STATS, 2002.
[15] L. Lova?sz. Random Walks on Graphs: A Survey Combinatorics, 1996, 353?398.
[16] G. J. McLachlan et al Mixture Models: Inference and Applications to Clustering. 1988
[17] M. Meila and J. Shi. A random walks view of spectral segmentation. AI & STATS, 2001.
[18] A. Ng, M. Jordan and Y. Weiss. On Spectral Clustering: analysis and an algorithm NIPS, 2001.
[19] A. Pothen Graph partitioning algorithms with applications to scientific computing. Parallel
Numerical Algorithms, D. E. Keyes et al (eds.), Kluwer Academic Press, 1996.
[20] G. L. Scott et al Feature grouping by relocalization of eigenvectors of the proximity matrix.
BMVC, pg. 103-108, 1990.
[21] E. Sharon et al Fast Multiscale Image Segmentation CVPR, I:70-77, 2000.
[22] J. Shi and J. Malik. Normalized cuts and image segmentation. PAMI, August, 2000.
[23] H. Simon et al Low-Rank Matrix Approximation Using the Lanczos Bidiagonalization Process
with Applications SIAM J. of Sci. Comp. 21(6):2257-2274, 2000.
[24] N. Tishby et al Data clustering by Markovian Relaxation NIPS, 2001.
[25] C. Williams et al Using the Nystrom method to speed up the kernel machines. NIPS, 2001.
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1,834 | 2,666 | An Investigation of Practical Approximate
Nearest Neighbor Algorithms
Ting Liu, Andrew W. Moore, Alexander Gray and Ke Yang
School of Computer Science
Carnegie-Mellon University
Pittsburgh, PA 15213 USA
{tingliu, awm, agray, yangke}@cs.cmu.edu
Abstract
This paper concerns approximate nearest neighbor searching algorithms,
which have become increasingly important, especially in high dimensional perception areas such as computer vision, with dozens of publications in recent years. Much of this enthusiasm is due to a successful new
approximate nearest neighbor approach called Locality Sensitive Hashing (LSH). In this paper we ask the question: can earlier spatial data
structure approaches to exact nearest neighbor, such as metric trees, be
altered to provide approximate answers to proximity queries and if so,
how? We introduce a new kind of metric tree that allows overlap: certain
datapoints may appear in both the children of a parent. We also introduce new approximate k-NN search algorithms on this structure. We
show why these structures should be able to exploit the same randomprojection-based approximations that LSH enjoys, but with a simpler algorithm and perhaps with greater efficiency. We then provide a detailed
empirical evaluation on five large, high dimensional datasets which show
up to 31-fold accelerations over LSH. This result holds true throughout
the spectrum of approximation levels.
1
Introduction
The k-nearest-neighbor searching problem is to find the k nearest points in a dataset X ?
RD containing n points to a query point q ? RD , usually under the Euclidean distance. It
has applications in a wide range of real-world settings, in particular pattern recognition,
machine learning [7] and database querying [11]. Several effective methods exist for this
problem when the dimension D is small (e.g. 1 or 2), such as Voronoi diagrams [26],
or when the dimension is moderate (i.g. up to the 10?s), such as kd-trees [8] and metric
trees. Metric trees [29], or ball-trees [24], so far represent the practical state of the art for
achieving efficiency in the largest dimensionalities possible [22, 6]. However, many realworld problems are posed with very large dimensionalities which are beyond the capability
of such search structures to achieve sub-linear efficiency, for example in computer vision,
in which each pixel of an image represents a dimension. Thus, the high-dimensional case
is the long-standing frontier of the nearest-neighbor problem.
Approximate searching. One approach to dealing with this apparent intractability has
been to define a different problem, the (1 + ?) approximate k-nearest-neighbor searching
problem, which returns points whose distance from the query is no more than (1 + ?) times
the distance of the true kth nearest-neighbor. Further, the problem is often relaxed to only
do this with high probability, and without a certificate property telling the user when it
has failed to do so, nor any guarantee on the actual rank of the distance of the points
returned, which may be arbitrarily far from k [4]. Another commonly used modification to
the problem is to perform the search under the L1 norm rather than L2 .
Locality Sensitive Hashing. Several methods of this general nature have been proposed
[17, 18, 12], and locality-sensitive hashing (LSH) [12] has received considerable recent
attention because it was shown that its runtime is independent of the dimension D and
has been put forth as a practical tool [9]. Roughly speaking, a locality sensitive hashing
function has the property that if two points are ?close,? then they hash to same bucket
with ?high? probability; if they are ?far apart,? then they hash to same bucket with ?low?
probability. Formally, a function family H = {h : S ? U} is (r1 , r2 , p1 , p2 )-sensitive, where
r1 < r2 , p1 > p2, for distance function D if for any two points p, q ? S, the following
properties hold:
1. if p ? B(q, r1 ), then Prh?H [h(q) = h(p)] ? p1 , and
2. if p 6? B(q, r2 ), then Prh?H [h(q) = h(p)] ? p2 ,
where B(q, r) denotes a hypersphere of radius r centered at q. By defining a LSH scheme,
namely a (r, r(1 + ?), p1 , p2 )-sensitive hash family, the (1 + ?)-NN problem can be solved
by performing a series of hashing and searching within the buckets. See [12, 13] for details.
Applications such as computer vision, e.g. [23, 28] have found (1 + ?) approximation to
be useful, for example when the k-nearest-neighbor search is just one component in a large
system with many parts, each of which can be highly inaccurate. In this paper we explore
the extent to which the most successful exact search structures can be adapted to perform
(1 + ?) approximate high-dimensional searches. A notable previous approach along this
line is a simple modification of kd-trees [3] ? ours takes the more powerful metric trees as
a starting point. We next review metric trees, then introduce a variant, known as spill trees.
2
Metric Trees and Spill Trees
2.1 Metric Trees
The metric tree [29, 25, 5] is a data structure that supports efficient nearest neighbor search.
We briefly A metric tree organizes a set of points in a spatial hierarchical manner. It is a
binary tree whose nodes represent a set of points. The root node represents all points, and
the points represented by an internal node v is partitioned into two subsets, represented by
its two children. Formally, if we use N(v) to denote the set of points represented by node
v, and use v.lc and v.rc to denote the left child and the right child of node v, then we have
N(v) = N(v.lc) ? N(v.rc)
(1)
0/ = N(v.lc) ? N(v.rc)
(2)
for all the non-leaf nodes. At the lowest level, each leaf node contains very few points.
Partitioning. The key to building a metric-tree is how to partition a node v. A typical
way is as follows. We first choose two pivot points from N(v), denoted as v.lpv and v.rpv.
Ideally, v.lpv and v.rpv are chosen so that the distance between them is the largest of allpair distances within N(v). More specifically, ||v.lpv ? v.rpv|| = max p1,p2?N(v) ||p1 ? p2||.
However, it takes O(n2 ) time to find the optimal v.lpv and v.rpv. In practice, we resort to a
linear-time heuristic that is still able to find reasonable pivot points.1 After v.lpv and v.rpv
are found, we can go ahead to partition node v.
Here is one possible strategy for partitioning. We first project all the points down to the
~
~ ? v.lpv,
vector ~u = v.rpv
and then find the median point A along ~u. Next, we assign all
the points projected to the left of A to v.lc, and all the points projected to the right of A
to v.rc. We use L to denote the d-1 dimensional plane orthogonal to ~u and goes through
A. It is known as the decision boundary since all points to the left of L belong to v.lc
1 Basically, we first randomly pick a point p from v. Then we search for the point that is the
farthest to p and set it to be v.lpv. Next we find a third point that is farthest to v.lpv and set it as v.rpv.
Figure 1: partitioning in a metric tree.
Figure 2: partitioning in a spill tree.
and all points to the right of L belong to v.rc (see Figure 1). By using a median point to
split the datapoints, we can ensure that the depth of a metric-tree is log n. However, in our
~ + v.rpv))
~
implementation, we use a mid point (i.e. the point at 12 (v.lpv
instead, since it is
more efficient to compute, and in practice, we can still have a metric tree of depth O(log n).
Each node v also has a hypersphere B, such that all points represented by v fall in the ball
centered at v.center with radius v.r, i.e. we have N(v) ? B(v.center, v.r). Notice that the
balls of the two children nodes are not necessarily disjoint.
Searching. A search on a metric-tree is simply a guided DFS (for simplicity, we assume
that k = 1). The decision boundary L is used to decide which child node to search first. If
the query q is on the left of L, then v.lc is searched first, otherwise, v.rc is searched first. At
all times, the algorithm maintains a ?candidate NN?, which is the nearest neighbor it finds
so far while traversing the tree. We call this point x, and denote the distance between q and
x by r. If DFS is about to exploit a node v, but discovers that no member of v can be within
distance r of q, then it prunes this node (i.e., skip searching on this node, along with all its
descendants). This happens whenever kv.center ? qk ? v.r ? r. We call this DFS search
algorithm MT-DFS thereafter.
In practice, the MT-DFS algorithm is very efficient for NN search, and particularly when
the dimension of a dataset is low (say, less than 30). Typically for MT-DFS, we observe
an order of magnitude speed-up over na??ve linear scan and other popular data structures
such as SR-trees. However, MT-DFS starts to slow down as the dimension of the datasets
increases. We have found that in practice, metric tree search typically finds a very good NN
candidate quickly, and then spends up to 95% of the time verifying that it is in fact the true
NN. This motivated our new proposed structure, the spill-tree, which is designed to avoid
the cost of exact NN verification.
2.2 Spill-Trees
A spill-tree (sp-tree) is a variant of metric-trees in which the children of a node can ?spill
over? onto each other, and contain shared datapoints. The partition procedure of a metrictree implies that point-sets of v.lc and v.rc are disjoint: these two sets are separated by the
decision boundary L. In a sp-tree, we change the splitting criteria to allow overlaps between
two children. In other words, some datapoints may belong to both v.lc and v.rc.
We first explain how to split an internal node v. See Figure 2 as an example. Like a metrictree, we first choose two pivots v.lpv and v.rpv, and find the decision boundary L that goes
through the mid point A, Next, we define two new separating planes, LL and LR, both of
which are parallel to L and at distance ? from L. Then, all the points to the right of plane
LL belong to the child v.rc, and all the points to the left of plane LR belong to the child v.lc.
Mathematically, we have
N(v.lc) = {x | x ? N(v), d(x, LR) + 2? > d(x, LL)}
N(v.rc) = {x | x ? N(v), d(x, LL) + 2? > d(x, LR)}
(3)
(4)
Notice that points fall in the region between LL and LR are shared by v.lc and v.rc. We call
this region the overlapping buffer, and we call ? the overlapping size. For v.lc and v.rc, we
can repeat the splitting procedure, until the number of points within a node is less than a
specific threshold, at which point we stop.
3
Approximate Spill-tree-based Nearest Neighbor Search
It may seem strange that we allow overlapping in sp-trees. The overlapping obviously
makes both the construction and the MT-DFS less efficient than regular metric-trees, since
the points in the overlapping buffer may be searched twice. Nonetheless, the advantage of
sp-trees over metric-trees becomes clear when we perform the defeatist search, an (1 + ?)NN search algorithm based on sp-trees.
3.1 Defeatist Search
As we have stated, the MT-DFS algorithm typically spends a large fraction of time backtracking to prove a candidate point is the true NN. Based on this observation, a quick
revision would be to descends the metric tree using the decision boundaries at each level
without backtracking, and then output the point x in the first leaf node it visits as the NN of
query q. We call this the defeatist search on a metric-tree. Since the depth of a metric-tree
is O(log n), the complexity of defeatist search is O(log n) per query.
The problem with this approach is very low accuracy. Consider the case where q is very
close to a decision boundary L, then it is almost equally likely that the NN of q is on the
same side of L as on the opposite side of L, and the defeatist search can make a mistake
with probability close to 1/2. In practice, we observe that there exists a non-negligible
fraction of the query points that are close to one of the decision boundaries. Thus the
average accuracy of the defeatist search algorithm is typically unacceptably low, even for
approximate NN search.
This is precisely the place where sp-trees can help: the defeatist search on sp-trees has much
higher accuracy and remains very fast. We first describe the algorithm. For simplicity, we
continue to use the example shown in Figure 2. As before, the decision boundary at node
v is plane L. If a query q is to the left of L, we decide that its nearest neighbor is in v.lc. In
this case, we only search points within N(v.lc), i.e., the points to the left of LR. Conversely,
if q is to the right of L, we only search node v.rc, i.e. points to the right of LL. Notice that
in either case, points in the overlapping buffer are always searched. By introducing this
buffer of size ?, we can greatly reduce the probability of making a wrong decision. To see
this, suppose that q is to the left of L, then the only points eliminated are the one to the right
of plane LR, all of which are at least distance ? away from q.
3.2 Hybrid Sp-Tree Search
One problem with spill-trees is that their depth varies considerably depending on the overlapping size ?. If ? = 0, a sp-tree turns back to a metric tree with depth O(log n). On the
other hand, if ? ? ||v.rpv ? v.lpv||/2, then N(v.lc) = N(v.rc) = N(v). In other words, both
children of node v contain all points of v. In this case, the construction of a sp-tree does
not even terminate and the depth of the sp-tree is ?.
To solve this problem, we introduce hybrid sp-trees and actually use them in practice. First
we define a balance threshold ? < 1, which is usually set to 70%. The constructions of
a hybrid sp-tree is similar to that of a sp-tree except the following. For each node v, we
first split the points using the overlapping buffer. However, if either of its children contains
more than ? fraction of the total points in v, we undo the overlapping splitting. Instead, a
conventional metric-tree partition (without overlapping) is used, and we mark v as a nonoverlapping node. In contrast, all other nodes are marked as overlapping nodes. In this
way, we can ensure that each split reduces the number of points of a node by at least a
constant factor and thus we can maintain the logarithmic depth of the tree.
The NN search on a hybrid sp-tree also becomes a hybrid of the MT-DFS search and the
defeatist search. We only do defeatist search on overlapping nodes, for non-overlapping
nodes, we still do backtracking as MT-DFS search. Notice that we can control the hybrid
by varying ?. If ? = 0, we have a pure sp-tree with defeatist search ? very efficient but
not accurate enough; if ? ? ||v.rpv ? v.lpv||/2, then every node is a non-overlapping node
(due to the balance threshold mechanism) ? in this way we get back to the traditional
metric-tree with MT-DFS, which is perfectly accurate but inefficient. By setting ? to be
somewhere in between, we can achieve a balance of efficiency and accuracy. As a general
rule, the greater ? is, the more accurate and the slower the search algorithm becomes.
3.3 Further Efficiency Improvement Using Random Projection
The hybrid sp-tree search algorithm is much more efficient than the traditional MT-DFS
algorithm. However, this speed-up becomes less pronounced when the dimension of a
dataset becomes high (say, over 30). In some sense, the hybrid sp-tree search algorithm
also suffer from the curse of dimensionality, only much less severely than MT-DFS.
However, a well-known technique, namely, random projection is readily available to deal
with the high-dimensional datasets. In particular, the Johnson-Lindenstrauss Lemma [15]
states that one can embed a dataset of n points in a subspace of dimension O(log n) with little distortion on the pair-wise distances. Furthermore, the embedding is extremely simple:
one simply picks a random subspace S and project all points to S.
In our (1 + ?)-NN search algorithm, we use random projection as a pre-processing step:
project the datapoints to a subspace of lower dimension, and then do the hybrid sptree search. Both the construction of sp-tree and the search are conducted in the lowdimensional subspace. Naturally, by doing random projection, we will lose some accuracy.
But we can easily fix this problem by doing multiple rounds of random projections and doing one hybrid sp-tree search for each round. Assume the failure probability of each round
is ?, then by doing L rounds, we drive down this probability to ?L .
The core idea of the hash function used in [9] can be viewed as a variant of random projection.2 Random projection can be used as a pre-processing step in conjunction with other
? conducted a series of experiments which
techniques such as conventional MT-DFSWe
show that a modest speed-up is obtained by using random projection with MT-DFS (about
4-fold), but greater (up to 700-fold) speed-up when used with sp-tree search. Due to limited
space these results will appear in the full version of this paper [19].
4
Experimental Results
We report our experimental results based on hybrid sp-tree search on a variety of real-world
datasets, with the number of datapoints ranging from 20,000 to 275,465, and dimensions
from 60 to 3,838. The first two datasets are same as the ones used in [9], where it is
demonstrated that LSH can have a significant speedup over SR-trees.
Aerial Texture feature data contain 275,465 feature vectors of 60 dimensions representing texture
information of large aerial photographs [21, 20].
Corel hist 20,000 histograms (64-dimensional) of color thumbnail-sized images taken from the
COREL STOCK PHOTO library. However, of the 64 dimensions, only 44 of them contain non-zero entries. See [27] for more discussions. We are unable to obtain the original
dataset used in [9] from the authors, and so we reproduced our own version, following their
description. We expect that the two datasets to be almost identical.
Corel uci 68,040 histograms (64-dimensional) of color images from the COREL library. This
dataset differs significantly from Corel hist and is available from the UCI repository [1].
Disk trace 40,000 content traces of disk-write operations, each being a 1 Kilo-Byte block (therefore
having dimension 1,024). The traces are generated from a desktop computer running SuSe
Linux during daily operation.
Galaxy Spectra of 40,000 galaxies from the Sloan Digital Sky Survey, with 4000 dimensions.
Besides the sp-tree search algorithm, we also run a number of other algorithms:
LSH The original LSH implementation used in [9] is not public and we were unable to
obtain it from the authors. So we used our own efficient implementation. Experiments (described later) show that ours is comparable to the one in [9].
Na??ve The na??ve linear-scan algorithm.
2 The Johnson-Lindenstrauss Lemma only works for L norm. The ?random sampling? done in
2
the LSH in [9] roughly corresponds to the L1 version of the Johnson-Lindenstrauss Lemma.
SR-tree We use the implementation of SR-trees by Katayama and Satoh [16].
Metric-Tree This is highly optimized k-NN search based on metric trees [29, 22], and
code is publicly available [2].
The experiments are run on a dual-processor AMD Opteron machine of 1.60 GHz with
8 GB RAM. We perform 10-fold cross-validation on all the datasets. We measure the
CPU time and accuracy of each algorithm. Since all the experiments are memory-based
(all the data fit into memory completely), there is no disk access during our experiments.
To measure accuracy, we use the effective distance error [3, 9], which is defined as
d
E = Q1 ?q?Q dalg
? ? 1 , where dalg is the distance from a query q to the NN found by the
algorithm, and d ? is the distance from q to the true NN. The sum is taken over all queries.
For the k-NN case where (k > 1), we measure separately the distance ratios between the
closest points found to the nearest neighbor, the 2nd closest one to the 2nd nearest neighbor
and so on, and then take the average. Obviously, for all exact k-NN algorithms, E = 0, for
all approximate algorithms, E ? 0.
4.1 The Experiments
First, as a benchmark, we run the Na??ve, SR-tree, and the Metric-tree algorithms. All of
them find exact NN. The results are summarized in Table 1.
Table 1: the CPU time of exact SR-tree, Metric-tree, and Na??ve search
Algorithm
(%)
Naive
SR-tree
Metric-tree
Aerial
43620
23450
3650
Corel hist
(k = 1) (k = 10)
462
465
184
330
58.4
91.2
Corel uci
Disk trace
Galaxy
5460
3230
791
27050
n/a
19860
46760
n/a
6600
All the datasets are rather large, and the metric-tree is consistently the fastest. On the other
hand, the SR-tree implement only has limited speedup over the Na??ve algorithm, and it fails
to run on Disk trace and Galaxy, both of which have very high dimensions.
Then, for approximate NN search, we compare sp-tree with three other algorithms: LSH,
traditional Metric-tree and SR-tree. For each algorithm, we measure the CPU time needed
for the error E to be 1%, 2%, 5%, 10% and 20%, respectively. Since Metric-tree and SRtree are both designed for exact NN search, we also run them on randomly chosen subsets
of the whole dataset to produce approximate answers. We show the comparison results of
all algorithms for the Aerial and the Corel hist datasets, both for k = 1, in Figure 3. We
also examine the speed-up of sp-tree over other algorithms. In particular, the CPU time and
the speedup of sp-tree search over LSH is summarized in Table 2.
Aerial (D=60, n=275,476, k=1)
25000
15000
CPU time(s)
Sp-tree
LSH
Metric-tree
SR-tree
20000
CPU time(s)
Corel_hist (D=64, n=20,000, k=1)
10000
5000
0
12
5
10
Error (%)
20
400
350
300
250
200
150
100
50
0
Sp-tree
LSH
Metric-tree
SR-tree
1 2
5
10
Error (%)
20
Figure 3: CPU time (s) vs. Error (%) for selected datasets.
Since we used our own implementation of LSH, we first need to verify that it has comparable performance as the one used in [9]. We do so by examining the speedup of both
implementations over SR-tree on the Aerial and Corel hist datasets, with both k = 1 and
Table 2: the CPU time (s) of Sp-tree and its speedup (in parentheses) over LSH
Error
(%)
20
10
5
2
1
Aerial
33.5 (31)
73.2 (27)
138 (31)
286 (26)
426 (23)
Corel
(k = 1)
1.67 (8.3)
2.79 (9.8)
4.5 (11)
8 (9.5)
13.5 (6.4)
hist
(k = 10)
3.27 (6.3)
5.83 (7.0)
9.58 (6.8)
20.6 (4.2)
27.9 (4.1)
Corel uci
Disk trace
Galaxy
8.7 (8.0)
19.1 (4.9)
33.1 (4.8)
61.9 (4.4)
105 (4.1)
13.2 (5.3)
43.1 (2.9)
123 (3.7)
502 (2.5)
1590 (3.2)
24.9 (5.5)
44.2 (7.8)
76.9 (11)
110 (14)
170 (12)
k = 10 for the latter.3 For the Aerial dataset, in the case where E varies from 10% to 20%,
the speedup of LSH in [9] over SR-tree varies from 4 to 6, and as for our implementation,
the speedup varies from 4.5 to 5.4. For Corel hist, when E ranges from 2% to 20%, in
the case k = 1, the speedups of LSH in [9] ranges from 2 to 7, ours from 2 to 13. In the
case k = 10, the speedup in [9] is from 3 to 12, and ours from 4 to 16. So overall, our
implementation is comparable to, and often outperforms the one in [9].
Perhaps a little surprisingly, the Metric-tree search algorithm (MT-DFS) performs very
well on Aerial and Corel hist datasets. In both cases, when the E is small (1%), MTDFS outperforms LSH by a factor of up to 2.7, even though it aims at finding the exact
NN, while LSH only finds an approximate NN. Furthermore, the approximate MT-DFS
algorithm (conventional metric-tree based search using a random subset of the training
data) consistently outperforms LSH across the entire error spectrum on Aerial. We believe
that it is because that in both datasets, the intrinsic dimensions are quite low and thus the
Metric-tree does not suffer from the curse of dimensionality.
For the rest of the datasets, namely, Corel uci, Disk trace, and Galaxy, metric-tree becomes
rather inefficient because of the curse of dimensionality, and LSH becomes competitive.
But in all cases, sp-tree search remains the fastest among all algorithms, frequently achieving 2 or 3 orders of magnitude in speed-up. Space does not permit a lengthy conclusion, but
the summary of this paper is that there is empirical evidence that with appropriate redesign
of the data structures and search algorithms, spatial data structures remain a useful tool in
the realm of approximate k-NN search.
5
Related Work
The idea of defeastist search, i.e., non-backtracking search, has been explored by various
researchers in different contexts. See, for example, Goldstein and Ramakrishnan [10],
Yianilos [30], and Indyk [14]. The latter also proposed a data structure similar to the spilltree, where the decision boundary needs to be aligned with a coordinate and there is no
hybrid version. Indyk proved how this data structure can be used to solve approximate NN
in the L? norm.
References
[1] http://kdd.ics.uci.edu/databases/CorelFeatures/CorelFeatures.data.html.
[2] http://www.autonlab.org/autonweb/showsoftware/154/.
[3] S. Arya, D. Mount, N. Netanyahu, R. Silverman, and A. Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. Journal of the ACM, 45(6):891?923,
1998.
[4] Kevin Beyer, Jonathan Goldstein, Raghu Ramakrishnan, and Uri Shaft. When is ?nearest neighbor? meaningful? Lecture Notes in Computer Science, 1540:217?235, 1999.
[5] P. Ciaccia, M. Patella, and P. Zezula. M-tree: An efficient access method for similarity search
in metric spaces. In Proceedings of the 23rd VLDB International Conference, September 1997.
3 The comparison in [9] is on disk access while we compare CPU time. So strictly speaking, these
results are not comparable. Nonetheless we expect them to be more or less consistent.
[6] K. Clarkson. Nearest Neighbor Searching in Metric Spaces: Experimental Results for sb(S). ,
2002.
[7] R. O. Duda and P. E. Hart. Pattern Classification and Scene Analysis. John Wiley & Sons,
1973.
[8] J. H. Friedman, J. L. Bentley, and R. A. Finkel. An algorithm for finding best matches in logarithmic expected time. ACM Transactions on Mathematical Software, 3(3):209?226, September
1977.
[9] A. Gionis, P. Indyk, and R. Motwani. Similarity Search in High Dimensions via Hashing. In
Proc 25th VLDB Conference, 1999.
[10] J. Goldstein and R. Ramakrishnan. Constrast Polots and P-Sphere Trees: Speace vs. Time in
Nearest Neighbor Searches. In Proc. 26th VLDB conference, 2000.
[11] A. Guttman. R-trees: A dynamic index structure for spatial searching. In Proceedings of
the Third ACM SIGACT-SIGMOD Symposium on Principles of Database Systems. Assn for
Computing Machinery, April 1984.
[12] P. Indyk and R. Motwani. Approximate nearest neighbors: towards removing the curse of
dimensionality. In STOC, pages 604?613, 1998.
[13] Piotr Indyk. High Dimensional Computational Geometry. PhD. Thesis, 2000.
[14] Piotr Indyk. On approximate nearest neighbors under l? norm. J. Comput. Syst. sci., 63(4),
2001.
[15] W. Johnson and J. Lindenstrauss. Extensions of lipschitz maps into a hilbert space. Contemp.
Math., 26:189?206, 1984.
[16] Norio Katayama and Shin?ichi Satoh. The SR-tree: an index structure for high-dimensional
nearest neighbor queries. pages 369?380, 1997.
[17] J. Kleinberg. Two Algorithms for Nearest Neighbor Search in High Dimension. In Proceedings
of the Twenty-ninth Annual ACM Symposium on the Theory of Computing, pages 599?608,
1997.
[18] E. Kushilevitz, R. Ostrovsky, and Y. Rabani. Efficient Search for Approximate Nearest Neighbors in High Dimensional Spaces. In Proceedings of the Thirtieth Annual ACM Symposium on
the Theory of Computing, 1998.
[19] T. Liu, A. W. Moore, A. Gray, and Ke. Yang. An investigation of practical approximate nearest
neighbor algorithms (full version). Manuscript in preparation.
[20] B. S. Manjunath. Airphoto dataset, http://vivaldi.ece.ucsb.edu/Manjunath/research.htm.
[21] B. S. Manjunath and W. Y. Ma. Texture features for browsing and retrieval of large image data.
IEEE Transactions on Pattern Analysis and Machine Intelligence, 18(8):837?842, 1996.
[22] A. W. Moore. The Anchors Hierarchy: Using the Triangle Inequality to Survive HighDimensional Data. In Twelfth Conference on Uncertainty in Artificial Intelligence. AAAI Press,
2000.
[23] G. Mori, S. Belongie, and J. Malik. Shape contexts enable efficient retrieval of similar shapes.
In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, 2001.
[24] S. M. Omohundro. Efficient Algorithms with Neural Network Behaviour. Journal of Complex
Systems, 1(2):273?347, 1987.
[25] S. M. Omohundro. Bumptrees for Efficient Function, Constraint, and Classification Learning.
In R. P. Lippmann, J. E. Moody, and D. S. Touretzky, editors, Advances in Neural Information
Processing Systems 3. Morgan Kaufmann, 1991.
[26] F. P. Preparata and M. Shamos. Computational Geometry. Springer-Verlag, 1985.
[27] Y. Rubnet, C. Tomasi, and L. J. Guibas. The earth mover?s distance as a metric for image
retrieval. International Journal of Computer Vision, 40(2):99?121, 2000.
[28] Gregory Shakhnarovich, Paul Viola, and Trevor Darrell. Fast pose estimation with parameter
sensitive hashing. In Proceedings of the International Conference on Computer Vision, 2003.
[29] J. K. Uhlmann. Satisfying general proximity/similarity queries with metric trees. Information
Processing Letters, 40:175?179, 1991.
[30] P. Yianilos. Excluded middle vantage point forests for nearest neighbor search. In DIMACS
Implementation Challenge, 1999.
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1,835 | 2,667 | Using Random Forests in the Structured
Language Model
Peng Xu and Frederick Jelinek
Center for Language and Speech Processing
Department of Electrical and Computer Engineering
The Johns Hopkins University
{xp,jelinek}@jhu.edu
Abstract
In this paper, we explore the use of Random Forests (RFs) in the structured language model (SLM), which uses rich syntactic information in
predicting the next word based on words already seen. The goal in this
work is to construct RFs by randomly growing Decision Trees (DTs) using syntactic information and investigate the performance of the SLM
modeled by the RFs in automatic speech recognition.
RFs, which were originally developed as classifiers, are a combination
of decision tree classifiers. Each tree is grown based on random training
data sampled independently and with the same distribution for all trees in
the forest, and a random selection of possible questions at each node of
the decision tree. Our approach extends the original idea of RFs to deal
with the data sparseness problem encountered in language modeling.
RFs have been studied in the context of n-gram language modeling and
have been shown to generalize well to unseen data. We show in this paper
that RFs using syntactic information can also achieve better performance
in both perplexity (PPL) and word error rate (WER) in a large vocabulary
speech recognition system, compared to a baseline that uses Kneser-Ney
smoothing.
1
Introduction
In many systems dealing with speech or natural language, such as Automatic Speech
Recognition and Statistical Machine Translation, a language model is a crucial component
for searching in the often prohibitively large hypothesis space. Most state-of-the-art systems use n-gram language models, which are simple and effective most of the time. Many
smoothing techniques that improve language model probability estimation have been proposed and studied in the n-gram literature [1]. There has so far been work in exploring
Decision Tree (DT) language models [2, 3], which attempt to cluster similar histories together to achieve better probability estimation. However, the results were negative [3]:
decision tree language models failed to improve upon the baseline n-gram models with the
same order n.
Random Forest (RF) language models, which are generalizations of DT language models,
have been recently applied to word n-grams [4]. DT growing is randomized to construct
RFs efficiently. Once constructed, the RFs function as a randomized history clustering,
which helps in dealing with the data sparseness problem. In general, the weakness of some
trees can be compensated for by other trees. The collective contribution of all DTs in an
RF n-gram model results in significant improvements in both perplexity (PPL) and word
error rate (WER) in a large vocabulary speech recognition system.
Language models can also be improved with better representations of the history. Recent
efforts have studied various ways of using information from a longer history span than that
usually captured by normal n-gram language models, as well as ways of using syntactic
information that is not available to the word-based n-gram models [5, 6, 7]. All these language models are based on stochastic parsing techniques that build up parse trees for the
input word sequence and condition the generation of words on syntactic and lexical information available in the parse trees. Since these language models capture useful hierarchical
characteristics of language, they can improve PPL and WER significantly for various tasks.
However, due to the n-gram nature of the components of the syntactic language models,
the data sparseness problem can be severe.
In order to reduce the data sparseness problem for using rich syntactic information in the
context, we study the use of RFs in the structured language model (SLM) [5]. Our results
show that although the components of the SLM have high order n-grams, our RF approach
can still achieve better performance, reducing both the perplexity (PPL) and word error
rate (WER) in a large vocabulary speech recognition system compared to a Kneser-Ney
smoothing baseline.
2
Basic Language Modeling
The purpose of a language model is to estimate the probability of a word string. Let W
denote a string of N words, that is, W = w1 , w2 , . . . , wN . Then, by the chain rule of
probability, we have
Q
P (W )=P (w1 )? N
(1)
i=2 P (wi |w1 ,...,wi?1 ).
In order to estimate the probabilities P (wi |w1 , . . . , wi?1 ), we need a training corpus consisting of a large number of words. However, in any practical natural language system of
even moderate vocabulary size, it is clear that the number of probabilities to be estimated
and stored is prohibitively large. Therefore, histories w1 , . . . , wi?1 for a word wi are usually grouped into equivalence classes. The most widely used language models, n-gram
language models, use the identities of the last n ? 1 words as equivalence classes. In an
n-gram model, we then have
P (W )=P (w1 )?
QN
i=2
i?1
P (wi |wi?n+1
),
(2)
i?1
where we have used wi?n+1
to denote the word sequence wi?n+1 , . . . , wi?1 .
If we could handle unlimited amounts of training data, the maximum likelihood (ML)
i?1
estimate of P (wi |wi?n+1
) would be the best:
i?1
P (wi |wi?n+1
)=
i
C(wi?n+1
)
,
i?1
C(w
)
i?n+1
(3)
i
where C(wi?n+1
) is the number of times the string wi?n+1 , . . . , wi is seen in the training
data.
2.1
Language Model Smoothing
An n-gram model when n = 3 is called a trigram model. For a vocabulary of size |V | =
104 , there are |V |3 = 1012 trigram probabilities to be estimated. For any training data of a
manageable size, many of the probabilities will be zero if the ML estimate is used.
In order to solve this problem, many smoothing techniques have been studied (see [1]
and the references therein). Smoothing adjusts the ML estimates to produce more accurate
probabilities and to assign nonzero probabilities to any word string. Details about various
smoothing techniques will not be presented in this paper, but we will outline a particular
way of smoothing, namely interpolated Kneser-Ney smoothing [8], for later reference.
Interpolated Kneser-Ney smoothing assumes the following form:
i?1
PKN (wi |wi?n+1
)
=
i
max(C(wi?n+1
)?D,0)
i?1
C(w
)
i?n+1
(4)
i?1
i?1
+?(wi?n+1
)PKN (wi |wi?n+2
),
i?1
where D is a discounting constant and ?(wi?n+1
) is the interpolation weight for the lower
order probabilities ((n ? 1)-gram). The discount constant is often estimated using the
n1
leave-one-out method, leading to the approximation D = n1 +2n
, where n1 is the number
2
of n-grams with count one and n2 is the number of n-grams with count two. To ensure that
the probabilities sum to one, we have
D
P
i?1
?(wi?n+1
)=
wi :C(wi
)>0
i?n+1
i?1
C(w
)
i?n+1
1
.
The lower order probabilities in interpolated Kneser-Ney smoothing can be estimated as
(assuming ML estimation):
P
i?1
PKN (wi |wi?n+2
)= P
1
wi?n+1 :C(wi
)>0
i?n+1
.
1
wi?n+1 ,wi :C(wi
)>0
i?n+1
(5)
Note that the lower order probabilities are usually recursively smoothed using Equation 4.
2.2
Language Model Evalution
A commonly used task-independent quality measure for a given language model is related
to the cross-entropy of the underlying model and is referred to as perplexity (PPL):
P P L=exp(?1/N
PN
i=1
log [P (wi |w1i?1 )]),
(6)
where w1 , . . . , wN is the test text that consists of N words.
For different tasks, there are different task-dependent quality measures of language models. For example, in an automatic speech recognition system, the performance is usually
measured by word error rate (WER).
3
The Structured Language Model (SLM)
The SLM uses rich syntactic information beyond regular word n-grams to improve language model quality. An extensive presentation of the SLM can be found in Chelba and
Jelinek, 2000 [5]. The model assigns a probability P (W, T ) to every sentence W and every possible binary parse T . The terminals of T are the words of W with POS tags, and
the nodes of T are annotated with phrase headwords and non-terminal labels. Let W be
h_{-m} = (<s>, SB)
h_{-1}
h_0 = (h_0.word, h_0.tag)
(<s>, SB) ....... (w_p, t_p) (w_{p+1}, t_{p+1}) ........ (w_k, t_k) w_{k+1}.... </s>
Figure 1: A word-parse k-prefix
a sentence of length n words to which we have prepended the sentence beginning marker
<s> and appended the sentence end marker </s> so that w0 =<s> and wn+1 =</s>.
Let Wk = w0 . . . wk be the word k-prefix of the sentence ? the words from the beginning
of the sentence up to the current position k ? and Wk Tk the word-parse k-prefix. Figure 1
shows a word-parse k-prefix; h_0, .., h_{-m} are the exposed heads, each head being a pair (headword, non-terminal label), or (word, POS tag) in the case of a root-only
tree. The exposed heads at a given position k in the input sentence are a function of the
word-parse k-prefix [5].
The joint probability P (W, T ) of a word sequence W and a complete parse T comes from
contributions of three components: WORD-PREDICTOR, TAGGER and CONSTRUCTOR. The SLM works in the following way: first, the WORD-PREDICTOR predicts a
word based on the word-parse prefix; the TAGGER then assigns a POS tag to the predicted
word based on the word itself and the word-parse prefix; the CONSTRUCTOR takes a
series of actions each of which turns a parse prefix into a new parse prefix (the series of
actions ends with a NULL action which tells the WORD-PREDICTOR to predict the next
word). Details about the three components can be found in [5]. Each of the three components can be seen as an n-gram model and can be estimated independently because of the
product form of the joint probability. They are parameterized (approximated) as follows:
P (wk |Wk?1 Tk?1 )
=
P (wk |h0 .tag,h0 .word,h?1 .tag,h?1 .word),
P (tk |wk ,Wk?1 Tk?1 )
=
P (tk |wk ,h0 .tag,h?1 .tag),
k
k
P (pk
i |Wk?1 Tk?1 ,wk ,tk ,p1 ...pi?1 )
=
P (pk
i |h0 .tag,h?1 .tag,h?2 .tag,h0 .word,h?1 .word),
(7)
(8)
(9)
where pki is the ith CONSTRUCTOR action after the k th word and POS tag have been
predicted. Since the number of parses for a given word prefix Wk grows exponentially
with k, |{Tk }| ? O(2k ), the state space of our model is huge even for relatively short
sentences. Thus we must use a search strategy that prunes the allowable parse set. One
choice is a synchronous multi-stack search algorithm [5] which is very similar to a beam
search.
The language model probability assignment for the word at position k + 1 in the input
sentence is made using:
PSLM (wk+1 |Wk )
=
P
?(Wk ,Tk )
=
P (Wk Tk )/
Tk ?Sk
P (wk+1 |Wk Tk )??(Wk ,Tk ),
P
T ?S P (Wk Tk ),
k
k
(10)
which ensures a proper probability normalization over strings of words, where S k is the set
of all parses present in the stacks at the current stage k and P (Wk Tk ) is the joint probability
of word-parse prefix Wk Tk .
Each model component ?WORD-PREDICTOR, TAGGER, CONSTRUCTOR? is estimated independently from a set of parsed sentences after undergoing headword percolation
and binarization (see details in [5]).
4
4.1
Using Random Forests in the Structured Language Model
Random Forest n-gram Modeling
A Random Forest (RF) n-gram model is a collection of randomly constructed decision tree
(DT) n-gram models. Unlike RFs in classification and regression tasks [9, 10, 11], RFs
are used in language modeling to deal with the data sparsenes problem [4]. Therefore, the
training data is not randomly sampled for each DT. Figure 2 shows the algorithm DT-Grow
and Node-Split used for generating random DT language models.
We define a position in the history as the distance between a word in the history and the
predicted word. The randomization is carried out in two places: a random selection of
Algorithm DT-Grow
Input: counts for training and heldout data
Algorithm Node-Split(p)
Input: node p and training data associated
Create a root node containing all
Initialize: histories in the training data and
put it in set ?
Initialize:
While ? is not empty
1. Get a node p from ?
2. If Node-Split(p) is successful, eliminate p from ? and put the two children of p in ?
Foreach internal node p in the tree
Randomly select a subset of positions I in the history
Foreach position i in I
1. Group all histories into basic elements
?(v)
2. Randomly split the elements ?(v)
into sets L and R
3. While there are elements moved, Do
1. LH
p ? normalized likelihood of heldout data associated with p, using
training data statistics in p
(a) Move each element from L to
R if the move results in positive
gain in training data likelihood
2. Get the set of leaves P rooted in p
3. LH
P ? normalized likelihood of heldout data associated with all leaves in
P, using training data statistics in the
corresponding leaves
H
4. if LH
P ? Lp < 0, prune the subtree
rooted in p
Output: a Decistion Tree language model
(b) Move each element from R to
L if the move results in positive
gain in training data likelihood
Select the position from I that results in the
largest gain
Output: a split L and R, or failure if the largest
gain is not positive
Figure 2: The algorithm DT-Grow and Node-Split
positions in the history and an initial random split of basic elements. Since our splitting
criterion is to maximize the log-likelihood of the training data, each split uses only statistics
(from training data) associated with the node under consideration. Smoothing is not needed
in the splitting and we can use a fast exchange algorithm [12] in Node-Split. Given a
position i in the history, ?(v) is defined to be the set of histories belonging to the node p,
such that they all have word v at position i. It is clear that for every position i in the history,
the union ?v ?(v) is all histories in the node p.
In DT-Grow, after a DT is fully grown, we use some heldout data to prune it. Pruning is
done in such a way that we maximize the likelihood of the heldout data, where smoothing
is applied according to Equation 4:
i?1
PDT (wi |?DT (wi?n+1
))
=
i?1
))?D,0)
i?n+1
i?1
C(?DT (w
))
i?n+1
max(C(wi ,?DT (w
(11)
i?1
i?1
+?(?DT (wi?n+1
))PKN (wi |wi?n+2
)
i?1
where ?DT (?) is one of the DT nodes the history can be mapped to and PKN (wi |wi?n+2
)
is as defined in Equation 5. This pruning is similar to the pruning strategy used in
CART [13].
Once we get the DTs, we only use the leaf nodes as equivalence classes of histories. If
a new history is encountered, it is very likely that we will not be able to place it at a leaf
i?1
node in the DT. In this case, ?(?DT (wi?n+1
)) = 1 in Equation 11 and we simply use
i?1
PKN (wi |wi?n+2 ) to get the probabilities.
The randomized version of the DT growing algorithm can be run many times and finally
we will get a collection of randomly grown DTs: a Random Forest (RF). Since each DT is
a smoothed language model, we simply aggregate all DTs in our RF to get the RF language
model. Suppose we have M randomly grown DTs, DT1 , . . . , DTM . In the n-gram case,
the RF language model probabilities can be computed as:
i?1
1
PRF (wi |wi?n+1
)= M
i?1
?DTj (wi?n+1
)
PM
j=1
(12)
i?1
PDTj (wi |?DTj (wi?n+1
))
i?1
wi?n+1
i?1
wi?n+1
where
maps the history
to a leaf node in DTj . If
can not
be mapped to a leaf node in some DT, we back-off to the lower order KN probability as
mentioned earlier.
It can be shown by the Law of Large Numbers that the probability in Equation
12 converges
i?1
as the number of DTs grows. It converges to ET PT (wi |?T (wi?n+1
) where T is a
random variable representing the random DTs. The advantage of the RF approach over the
KN smoothing lies in the fact that different DTs have different weaknesses and strengths
for word prediction. As the number of trees grows, the weakness of some trees can be
compensated for by some other trees. This advantage and the convergence have been shown
experimentally in [4].
4.2
Using RFs in the SLM
Since the three model components in the SLM as in Equation 7-9 can be estimated independently, we can construct an RF for each component using the algorithm DT-Grow in
the previous section. The only difference is that we will have different n-gram orders and
different items in the history for each model.
Ideally, we would like to use RFs for each component in the SLM. However, due to the
nature of the SLM, there are difficulties. The SLM uses a synchronous multi-stack search
algorithm to dynamically construct stacks and compute the language model probabilities
as in Equation 10. If we use RFs for all components, we need to load all DTs in the RFs
into memory at runtime. This is impractical for RFs of any reasonable size.
There is a different approach that can take advantage of the randomness in the RFs. Supa
pose we have M randomly grown DTs, DT1a , . . . , DTM
for each component a of the SLM,
where a ? {P, T, C} for WORD-PREDICTOR, TAGGER and CONSTRUCTOR, respectively. The DTs are grouped into M triples {DTjP , DTjT , DTjC } j = 1, . . . , M . We
calculate the joint probability P (W, T ) for the j th DT triple according to:
Pj (W,T )=
Qn+1
k=1 [PDT P
j
(wk |Wk?1 Tk?1 )?PDT T (tk |Wk?1 Tk?1 ,wk )?
j
Q Nk
k
k
k
i=1 PDT C (pi |Wk?1 Tk?1 ,wk ,tk ,p1 ...pi?1 )].
j
(13)
Then, the language model probability assignment for the j th DT triple is made using:
Pj (wk+1 |Wk )
=
?j (Wk ,Tkj )
=
j
j
j PDT P (wk+1 |Wk Tk )??j (Wk ,Tk ),
j
T ?Sk
j
k
P
j
Pj (Wk Tkj )/
j
j Pj (Wk Tk ),
T ?S
k
k
P
(14)
which is achieved by running the synchronous multi-stack algorithm using the j th DT triple
as a model. Finally, after the SLM is run M times, the RF language model probability is
an average of the probabilities above:
1
PRF (wk+1 |Wk )= M
PM
j=1
Pj (wk+1 |Wk ).
(15)
The triple {DTjP , DTjT , DTjC } can be considered as a single DT in which the root node
has three children corresponding to the three root nodes of DTjP , DTjT and DTjC . The
root node of this DT asks the question: Which model component does the history belong
to? According to the answer, we can proceed to one of the three children nodes (one of the
three components, in fact). Since the multi-stack search algorithm is deterministic given
the DT, the probability in Equation 15 can be shown to converge.
5
5.1
Experiments
Perplexity (PPL)
We have used the UPenn Treebank portion of the WSJ corpus to carry out our experiments.
The UPenn Treebank contains 24 sections of hand-parsed sentences, for a total of about
one million words. We used section 00-20 for training our models, section 21-22 as heldout
data for pruning the DTs, and section 23-24 to test our models. Before carrying out our
experiments, we normalized the text in the following ways: numbers in arabic form were
replaced by a single token ?N?, punctuation was removed, all words were mapped to lower
case, extra information in the parse trees was ignored, and, finally, traces were ignored. The
word vocabulary contains 10k words including a special token for unknown words. There
are 40 items in the part-of-speech set and 54 items in the non-terminal set, respectively.
The three components in the SLM were treated independently during training. We trained
an RF for each component and each RF contained 100 randomly grown DTs. The baseline
SLM used KN smoothing (KN-SLM). The 100 probability sequences from the 100 triples
were aggregated to get the final PPL. The results are shown in Table 1. We also interpolated
the SLM with the KN-trigram to get further improvements. The interpolation weight ? in
Table 1 is on KN-trigram. The RF-SLM achieved a 10.9% and a 7.5% improvement over
the KN-SLM, before and after interpolation with KN-trigram, respectively. Compared to
the improvements reported in [4] (10.5% from RF-trigram to KN-trigram), the RF-SLM
achieved greater improvement by using syntactic information. Figure 3 shows the convergence of the PPL as the number of DTs grows from 1 to 100.
160
155
Model
KN-SLM
RF-SLM
Gain
150
PPL
145
140
135
?=0.0
137.9
122.8
10.9%
?=0.4
127.2
117.6
7.5%
?=1.0
145.0
145.0
-
130
Table 1: PPL comparison between KNSLM and RF-SLM, interpolated with
KN-trigram
125
120
20
40
60
Number of DTs
80
100
Figure 3: PPL convergence
5.2
Word Error Rate by N -best Re-scoring
To test our RF modeling approach in the context of speech recognition, we evaluated the
models in the WSJ DARPA?93 HUB1 test setup. The size of the test set is 213 utterances,
3446 words. The 20k word open vocabulary and baseline 3-gram model are the standard
ones provided by NIST and LDC ? see [5] for details. The N -best lists were generated
using the standard 3-gram model trained on 40M words of WSJ. The N -best size was at
most 50 for each utterance, and the average size was about 23. For the KN-SLM and RFSLM, we used 20M words automatically parsed, binarized and enriched with headwords
and NT/POS tag information. As the size of RF-SLM becomes very large, we only used
RF for the WORD-PREDICTOR component (RF-SLM-P). The other two components used
KN smoothing. The results are reported in Table 2.
Model
KN-SLM
RF-SLM-P
?=0.0
12.8
11.9
?=0.2
12.5
12.2
?=0.4
12.6
12.3
?=0.6
12.7
12.3
?=0.8
12.7
12.6
Table 2: N-best rescoring WER results
For purpose of comparison, we interpolated all models with the KN-trigram built from
40M words at different level of interpolation weights ? (on KN-trigram). However, it is the
? = 0.0 column that is the most interesting. We can see that the RF approach improved
over the regular KN approach with an absolute WER reduction of 0.9%.
6
Conclusions
Based on the idea of Random Forests in classification and regression, we developed algorithms for constructing and using Random Forests in language modeling. In particular, we
applied this new probability estimation technique to the Structured Language Model, in
which there are three model components that can be estimated independently. The independently constructed Random Forests can be considered as a more general single Random
Forest, which ensures the convergence of the probabilities as the number of Decision Trees
grows. The results on a large vocabulary speech recognition system show that we can
achieve significant reduction in both perplexity and word error rate, compared to a baseline
using Kneser-Ney smoothing.
References
[1] Stanley F. Chen and Joshua Goodman, ?An empirical study of smoothing techniques for language modeling,? Tech. Rep. TR-10-98, Computer Science Group, Harvard University, Cambridge, Massachusetts, 1998.
[2] L. Bahl, P. Brown, P. de Souza, and R. Mercer, ?A tree-based statistical language model for
natural language speech recognition,? in IEEE Transactions on Acoustics, Speech and Signal
Processing, July 1989, vol. 37, pp. 1001?1008.
[3] Gerasimos Potamianos and Frederick Jelinek, ?A study of n-gram and decision tree letter
language modeling methods,? Speech Communication, vol. 24(3), pp. 171?192, 1998.
[4] Peng Xu and Frederick Jelinek, ?Random forests in language modeling,? in Proceedings of the
2004 Conference on Empirical Methods in Natural Language Processing, Barcelona, Spain,
July 2004.
[5] Ciprian Chelba and Frederick Jelinek, ?Structured language modeling,? Computer Speech and
Language, vol. 14, no. 4, pp. 283?332, October 2000.
[6] Eugene Charniak, ?Immediate-head parsing for language models,? in Proceedings of the 39th
Annual Meeting and 10th Conference of the European Chapter of ACL, Toulouse, France, July
2001, pp. 116?123.
[7] Brian Roark, Robust Probabilistic Predictive Syntactic Processing: Motivations, Models and
Applications, Ph.D. thesis, Brown University, Providence, RI, 2001.
[8] Reinhard Kneser and Hermann Ney, ?Improved backing-off for m-gram language modeling,?
in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing, 1995, vol. 1, pp. 181?184.
[9] Y. Amit and D. Geman, ?Shape quantization and recognition with randomized trees,? Neural
Computation, , no. 9, pp. 1545?1588, 1997.
[10] Leo Breiman, ?Random forests,? Tech. Rep., Statistics Department, University of California,
Berkeley, Berkeley, CA, 2001.
[11] T.K. Ho, ?The random subspace method for constructing decision forests,? IEEE Trans. on
Pattern Analysis and Machine Intelligence, vol. 20, no. 8, pp. 832?844, 1998.
[12] S. Martin, J. Liermann, and H. Ney, ?Algorithms for bigram and trigram word clustering,?
Speech Communication, vol. 24(3), pp. 171?192, 1998.
[13] L. Breiman, J.H. Friedman, R.A. Olshen, and C.J. Stone, Classification and Regression Trees,
Chapman and Hall, New York, 1984.
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1,836 | 2,668 | A Hidden Markov Model for de Novo Peptide
Sequencing
Bernd Fischer, Volker Roth, Joachim M. Buhmann
Institute of Computational Science
ETH Zurich
CH-8092 Zurich, Switzerland
[email protected]
Jonas Grossmann, Sacha Baginsky,
Wilhelm Gruissem
Institute of Plant Sciences
ETH Zurich
CH-8092 Zurich, Switzerland
Franz Roos,
Peter Widmayer
Inst. of Theoretical Computer Science
ETH Zurich
CH-8092 Zurich, Switzerland
Abstract
De novo Sequencing of peptides is a challenging task in proteome research. While there exist reliable DNA-sequencing methods, the highthroughput de novo sequencing of proteins by mass spectrometry is still
an open problem. Current approaches suffer from a lack in precision
to detect mass peaks in the spectrograms. In this paper we present a
novel method for de novo peptide sequencing based on a hidden Markov
model. Experiments effectively demonstrate that this new method significantly outperforms standard approaches in matching quality.
1
Introduction
The goal of de novo peptide sequencing is to reconstruct an amino acid sequence from a
given mass spectrum. De novo sequencing by means of mass spectrometry is a very challenging task, since many practical problems like measurement errors or peak suppression
have to be overcome. It is, thus, not surprising that current approaches to reconstruct the
sequence from mass spectra are usually limited to those species for which genome information is available. This case is a simplified problem of the de novo sequencing problem,
since the hypothesis space of possible sequences is restricted to the known ones contained
in a sequence database.
In this paper we present a Hidden Markov Model (HMM) for de novo sequencing. The
main difference to standard methods which are all based on dynamic programming [2, 1]
lies in the fully probabilistic model. Our trained HMM defines a generative model for
mass spectra which, for instance, is used for scoring observed spectra according to their
likelihood given a peptide sequence. Besides predicting the most likely sequence, however,
the HMM framework is far more general in the sense that it additionally allows us to specify
the confidence in the predictions.
2
Tandem Mass Spectrometry
In a typical sequencing experiment by mass spectrometry a protein is digested with the
help of an enzyme. This digestion reaction breaks the protein into several peptides, each
of which consists of a short sequence of typically 10 to 20 amino acid residues, with an
additional H-atom at the N-terminus and an OH-group at the C-terminus.
DSRSCK
AYSARDGFSHEK
DGDGYASDZROPGFSHEK
1. MS
m/z
AYSARDGFSHEK
AYSAR
A
K
DGFSHEK
AY
EK
HEK
AYS
2. MS
m/z
Figure 1: In the first mass measurement the parent mass is selected. In the second measurement the peptide is dissociated and the mass of the ion fragments is measured.
There are two measurement steps in a tandem mass spectrometer. The first step is responsible for filtering peptides of a certain total mass (also called the parent mass). The difficulty
in measuring the parent mass arises from different 12 C/13 C isotope proportions of the
approximately 30-80 C-atoms contained in a peptide. Fluctuations of the 13 C fraction result in a binomial distribution of parent masses in the measurement. Given such an ?ion
count distribution? one can roughly estimate the mono-isotopic parent mass of the peptide,
where the term mono-isotopic here refers to a peptide that contains exclusively 12 C atoms.
In practice, all isotope configurations of a peptide with parent masses that do not exceed
the estimated mono-isotopic mass by more than a predefined offset are separated from the
other peptides and passed to the second spectrometer.
Figure 2: Top: The ideal peaks of a peptide sequence are drawn. Bottom: The spectrum of
the corresponding peptide.
In the second mass measurement, a peptide is split into two fragments by means of collision
induced dissociation with a noble gas. In almost all cases the peptide is broken between
two amino acids. Thus, an ideal spectrum is composed of the masses of all prefix and suffix
sequences of the peptide. Deviations from this ideal case are e.g. caused by problems in
determining the exact mono-isotopic mass of the fragments due to isotope shifts. Further
complications are caused by an accidental loss of water (H2 O), ammonia (NH3 ) or other
molecules in the collision step. Moreover, the ion counts are not uniformly distributed over
the spectrum. And last but not least, the measurements are noisy.
3
The Hidden Markov Model for de Novo Peptide Sequencing
A peptide can formally be described as a sequence of symbols from a fixed alphabet A of
20 amino acids. We will denote amino acids with ? ? A and the mass of an amino acid
with M (?). The input data is a spectrum of ion counts over all mass units. The ion count
for mass m is denoted by x(m). The spectra are discretized to approximately one Dalton
mass units and normalized such that the mean ion count per Dalton is constant.
The mono-isotopic parent mass m0p of the peptide P = (?1 , . . . , ?n ) with ?i ? A is
the sum of all amino acid masses plus a constant mass for the N- and C-termini. m 0p =
Pn
constN + i=1 M (?i ) + constC. For the sake of simplicity it is assumed that the N- and
C-termini are not present and thus the parent mass considered in the sequel is
Pn
(1)
mp = i=1 M (?i ) .
In the HMM framework a spectrum is regarded as a realization of a random process. The
physical process that generates spectra is based on the fact that a peptide is randomly broken
into two parts by interaction with a noble gas. Each of these parts is detected in the mass
spectrometer and increases the ion-count in the corresponding mass interval. Finally, a
histogram over many such events is measured. In order to derive a model of the generation
process, we make the simplifying assumptions that (i) breaks occur only at amino acid
boundaries, and (ii) the probability of observing a break after a certain amino acid depends
only on the amino acid itself. These assumptions allow us to model the generative process
by way of a Markov process on a finite state automation. In such a model, the process of
generating a spectrum for a peptide of known parent mass is formalized as a path through
the automaton in 1 Dalton steps until the constraint on the parent mass is satisfied.
3.1
Finite State Automaton
s0
sA
1
sA
2
sA
3
sR
1
sR
2
sR
3
sV
1
sV
2
sV
3
sA
71
s?
sR
156
s+
sV
99
Figure 3: The finite state machine of the Hidden Markov Model. For each amino acid ?
there is a list of M (?) states.
The finite state automaton (fig. 3) has one initial state s0 . For each amino acid ? ? A
?
there exists a list of M (?) states s?
1 , . . . , sM (?) . Together with the end states s+ and s?
the complete set of states is
S = {s0 } ? s?
(2)
j | ? ? A, 1 ? j ? M (?) ? {s+ , s? } .
The bold edges in the graph correspond to state transition probabilities a(s, t) from state s
to state t. Once the automation is in the first state s?
1 of a state list of one amino acid ?, it
has to pass through all other states within the specific list. Thus for the next M (?) steps
the list corresponding to amino acid ? is linearly traversed. If the automaton is in the last
?0
0
state s?
M (?) of a list, it can reach the start states s1 of any other amino acid ? . The random
variable for the state sequence is denoted by Y1 , . . . , Ymp . The transition probabilities are
a(s, t) = P {Yi+1
?
? 1
r?
= t|Yi = s} =
?
0
?
? ? ? A, 1 ? i < M (?) : s = s?
i ? t = si+1
?
?
? ? ? A, ? ? A : s = sm(?) ? t = s1
else .
(3)
The first row (a(s, t) = 1) describes the case where the automaton is in a non-terminating
state of a list of amino acid ? (1 ? i < M (?) : s = s?
i ), where the following state is
accepted with probability 1. The second row, on the contrary, refers to a terminating state
of a list. In such a case, the starting state of any other amino acid is selected with probability
r? . The probabilities r? are the probabilities of occurrence of amino acid ?.
The transition probabilities a(s0 , t) from the start state s0 are the occurrence probabilities
of the amino acids.
r? ? ? ? A : t = s ?
1
(4)
a(s0 , t) =
0
else
Finally one has to ensure that the parent mass constraint is fulfilled. In order to satisfy
the constraint we device a time dependent hidden Markov model in which the transition
probability changes with a heavy side function at time mp from a(s, t) to a0 (s, t). The
dotted arrows in figure 3 show the transition probabilities a0 (s, t) into the end states s+ and
s? .
?
? 1 ? ? ? A : s = s?
M (?) , t = s+
0
a (s, t) =
(5)
1 ? ? ? A, 1 ? i < M (?) : s = s?
i , t = s?
?
0 else
If the automaton is in the last state s?
M (?) of an amino acid state list, it changes to the
positive end state s+ with probability 1 since the parent mass constraint is satisfied. If the
automaton is in one of the other states, it changes to the negative end state s? since the
parent mass constraint is violated. It is important to realize that all amino acid sequences
that fulfill the parent mass constraint can be transformed into state sequences that end in
the positive state s+ and vice versa.
3.2
Emission Probabilities
0.1
0.1
0.35
0.09
0.09
0.3
0.08
0.08
0.07
0.25
0.07
0.06
0.06
0.2
0.05
H2O
0.05
NH 3
0.15
0.04
0.04
0.03
CO+NH 3
0.02
H2O+NH 3
0.03
0.1
CO
?40
NH 3
0.05
0.01
0.01
0
?50
H2O
0.02
2H2O
?30
?20
(a)
?10
0
10
0
0
?50
?50
?40
?40
?30
?30
?20
?20
?10
?10
0
0
10
10
(b)
Figure 4: Mean height of ion counts for different shifts with respect to the ideal prefix
fragments (a) and suffix fragments (b).
At each state of the finite state automaton an ion count value is emitted. Figure 4 shows
the mean ion count for different positions relative to the amino acid bound averaged over
all amino acids. The histograms are taken over the training examples described in the
experimental section. It happens quite frequently that an amino acid looses water (H 2 O)
or ammonia (NH3 ). The ion count patterns for the prefix fragments (fig. 4 a) and the
suffix fragments (fig. 4 b) are quite different due to chemical reasons. For instance, carbon
monoxide loss in the suffix fragments is an unlikely event. Suffix fragments are more stable
than prefix fragments: the central peak at position 0 (amino acid boundary) is three times
higher for the suffix fragments than for the prefix fragments. Note that in figure 4 b) we
used two different scales.
s+
m=m p
m=0
s0
Figure 5: Folding the spectrum in the middle makes the intern mirror symmetry of the
problem visible. The Markov chain models a sequence with three amino acids. The filled
circles correspond to the amino acid boundaries. Each amino acid bound generates an ion
count pattern for the prefix fragment and one for the suffix fragment.
Breaking a peptide in the second mass spectrometer produces both a prefix and a suffix
fragment. To simultaneously process peaks of both types of fragments, we use one forward
and one backward Markov chain which are independent of each other. Due to the inherent
mirror symmetry of the problem (fig. 5) it is sufficient to limit the length of both models to
mp /2. For the recognition process we assume that we simultaneously observe two peaks
xm,1 = x(m) and xm,2 = x(mp ? m) in step m. The joint observation of the prefix and
the suffix peaks is an essential modeling step in our method.
The forward and the backward Markov chains are extended to hidden Markov models to
describe the ion counts in the mass spectra. The emission probabilities depend on the two
states of the prefix and suffix sequence, since these states give rise to ion counts in the
measurements. We define
? m = xm = (x(m), x(mp ? m)) | Y?m = (s, s0 )
bs,s0 (xm ) = P X
(6)
as the emission probabilities of ion counts.
? m are the (coupled) random variables of the ion counts. The hidden variables for the state
X
? m describes the transition
sequence are denoted by Y?m . This notion of coupled variables X
from two independent Markov chains to one coupled hidden Markov model with a squared
number of states (2-tuple states).
The joint probability of observable and hidden variables given the parent mass m p is
P {X = x, Y = y | s+ , mp } = a (s0 , y1 ) a0 ymp , s+ ?
(7)
? m ?1
?
p
2
?
? Y
??
bym ,ymp ?m (xm )a (ym , ym+1 ) a ymp ?m , ymp ?m+1 ? a y mp ?1 , y mp ?1 +1
m=1
2
2
This formula holds for parent masses with an odd Dalton value, an equivalent formula can
be derived for the even case. The first term in eq. (7) is the joint probability from s 0 to
y1 in the prefix model and the transition ymp to s+ in the suffix model. In each term of
the product, two peaks are observed on both sides of the spectrum: one at position m and
the other at the mirror position mp ? m. The joint probability of emissions is defined by
bym ,ymp ?m (xm , xmp ?m ). Furthermore, the transition probabilities of the prefix and suffix
sequences are multiplied which reflects the independence assumption of the Markov model.
The two chains are connected by the transition probability a(y(mp ?1)/2 , y(mp ?1)/2+1 ) of
traversing from the last state of the forward Markov chain to the first state of the backward
chain.
3.3
Most Probable Sequence
The input spectrum usually comes with an estimate of the parent mass with a tolerance of
about 1 Dalton. Using a maximum likelihood approach the parent mass estimate is
X
P {X = x, Y = y | s+ , mp } . (8)
m
? p = argmax P {X = x | s+ , mp } = argmax
mp
mp
y
The sum over all sequences can be computed efficiently by dynamic programming using
the forward algorithm.
One result of de novo peptide sequencing is the computation of the best sequence generating a given spectrum. Given the estimated parent mass m
? p the maximum posterior estimate
of the sequence is
y ? = argmax P {Y = y | X = x, s+ , m
? p } = argmax P {X = x, Y = y | s+ , m
? p} .
y
y
(9)
The best sequence can efficiently be found by the Viterbi algorithm. To compute the posterior probability one has to normalize the joint probability P {X = x, Y = y | s+ , m
? p } by
the evidence P {X = x | s+ , m
? p } using the forward-backward algorithm.
In the mass spectra ions with very low mass or almost parent mass are less frequently
observed than ions with a medium mass. Therefore it becomes quite difficult to estimate the
whole sequence with a high score. It is also possible to give a score for each subsequence
of the peptide, especially a score for each amino acid. An amino acid is a subsequence
yp , . . . , yq of the state sequence y1 , . . . , ymp .
=
P {yp , . . . , yq | s+ , x, mp }
P
P
y1 ,...yp?1
yq+1 ,...,ym P y1 , . . . , ymp , x | s+ , mp
p
P {x | s+ , mp }
(10)
(11)
This can again be computed by some variation of the forward and backward algorithm.
3.4
Simplification of the Model
The coupled hidden Markov model has 2 3752 = 5 640 625 states that leads to a runtime
of 20 minutes per peptide which for practical applications is problematic. A significant
simplification is achieved by assuming that there are two spectra observed, where the second one is the mirror version of the first one. The emission probabilities in this simplified
model only depend on the states of the prefix Markov chain (fig. 6). Thus the emission of
mirror peaks x(mp ?m) is deterministically coupled to the emission of the peak xm . Since
this model has only 2 375 states, the computation time reduces to 1-2 seconds per peptide.
4
Experiments
In our experiments a protein probe of plant cell vacuoles (Arabidopsis thaliana) was digested with trypsin. The mass spectrometer gave an output of 7056 different candidate spectra. From a database search with SEQUEST [3] and further validation with
PeptideProphet [4], 522 spectra with a confidence larger than 90% were extracted.
It was shown that the PeptideProphet score is a very reliable scoring method for peptide identification by database search. The database output was used as training data. The
m=m p
m=0
m=m p
m=0
s0
s+
Figure 6: In the simplified model two mirrored spectra are observed. The emission of
symbols is coupled with the amino acid bounds of the prefix sequence.
quality of the HMM inference is measured by the ratio of common amino acid boundaries
and the number of amino acids in the database sequence. The performance of the HMM
was tested by leave-one-out cross validation: in each training step the emission probabilities and the amino acid occurrence probabilities are re-estimated, with one sequence
excluded from the training set. To estimate the emission probabilities, the ion count is discretized to a fixed number of bins, in such a way that all bins contain an equal number of
counts. The leave-one-out scheme is repeated for different numbers of discretization levels.
1
0.9
0.8
0.7
recall
0.6
0.5
0.4
0.3
0.2
0.1
0
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
number of bins
Figure 7: Cross validation of recall rates for different number of bins in the discretization
process. Depicted are the lower quartile, the median and the upper quartile.
The resulting performance recall rate are depicted in figure 7. Choosing 5 bins yields the
highest recall value.
We have chosen the prominent de novo sequencing programs LUTEFISK [6] and PEAKS
[5] as competitors for the simplified HMM. We compared the sequence from the HMM
with the highest scoring sequences from the other programs. In figure 8 a) the estimated
parent masses compared to the database parent mass is drawn. The plot demonstrates that
all de novo sequencing methods tend to overestimate the parent mass. The best one is the
HMM with 89.1% correct estimations, whereas only 59.3% of the LUTEFISK estimates
and 58.1% of the PEAKS estimates are correct. In figure 8 b) boxplots of the recognition
rate of peak positions is drawn. The three lines in the box correspond to the lower quartile,
the median and the upper quartile of the distribution. The median recall of the HMM is
75.0%, for Lutefisk 53.9% and for Peaks 56.7%. Note that the lower quartile of the HMM
results is above 50%, whereas it is below 10% for the other programs.
5
Conclusion and Further Work
A novel method for the analysis of mass spectra in de novo peptide sequencing is presented
in this paper. The proposed hidden Markov model is a fully probabilistic model for the
generation process of mass spectra. The model was tested on mass spectra from vacuola
1
1
1
0.9
0.9
0.9
1
0.9
0.8
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
0.5
0.5
0.5
0.4
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.2
0.1
0.1
0.1
0.1
0.8
0.7
Recall
0.6
0.5
0.4
0
<?2 ?2 ?1 0 1 2 3
Lutefisk
0
?3 ?2 ?1 0 1 2 3
Peaks
(a)
0
0.3
0
?3 ?2 ?1 0 1 2 3
HMM
Lutefisk
Peaks
HMM
(b)
Figure 8: a) Histogram on difference of estimated parent mass and database output. b)
Recall of peak positions.
proteins. The HMM clearly outperforms its competitors in recognition of the parent mass
and peak localization. In further work additional model parameters will be introduced to
represent and to detect amino acids with post-translational modifications. Reliable subsequences can further be used for a tagged database search to identify peptides with posttranslational modifications. Our method shows a large potential for high throughput de
novo sequencing of proteins which is unmatched by competing techniques.
Acknowledgment This work has been partially supported by DFG grant # Buh 914/5.
References
[1] Sacha Baginsky, Mark Cieliebak, Wilhelm Gruissem, Torsten Kleffmann, Zsuzsanna
Lipt?ak, Matthias M?uller, and Paolo Penna. Audens: A tool for automatic de novo
peptide sequencing. Technical Report 383, ETH Zurich, Dept. of Computer Science,
2002.
[2] Ting Chen, Ming-Yang Kao, Matthew Tepel, John Rush, and George M. Church. A
dynamic programming approach to de novo peptide sequencing via tandem mass spectrometry. Journal of Computational Biology, 8(3):325?337, 2001.
[3] Jimmy K. Eng, Ashley L. McCormack, and John R. Yates. An approach to correlate
tandem mass spectral data of peptides with amino acid sequences in a protein database.
American Society for Mass Spectrometry, 5(11):976?989, 1994.
[4] Andrew Keller, Alexey I. Nesvizhskii, Eugene Kolker, and Ruedi Aebersold. Empirical
statistical model to estimate the accuracy of peptide identifications made by MS/MS
and database search. Analytical Chemistry, 2002.
[5] Bin Ma, Kaizhong Zhang, Christopher Hendrie, Chengzhi Liang, Ming Li, Amanda
Doherty-Kirby, and Gilles Lajoie. Peaks: Powerful software for peptide de novo sequencing by tandem mass spectrometry. Rapid Communication in Mass Spectrometry,
17(20):2337?2342, 2003.
[6] J. Alex Taylor and Richard S. Johnson. Implementation and uses of automated de novo
peptide sequencing by tandem mass spectrometry. Analytical Chemistry, 73:2594?
2604, 2001.
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1,837 | 2,669 | Maximal Margin Labeling for Multi-Topic Text
Categorization
Hideto Kazawa, Tomonori Izumitani, Hirotoshi Taira and Eisaku Maeda
NTT Communication Science Laboratories
Nippon Telegraph and Telephone Corporation
2-4 Hikaridai, Seikacho, Sorakugun, Kyoto 619-0237 Japan
{kazawa,izumi,taira,maeda}@cslab.kecl.ntt.co.jp
Abstract
In this paper, we address the problem of statistical learning for multitopic text categorization (MTC), whose goal is to choose all relevant topics (a label) from a given set of topics. The proposed algorithm, Maximal Margin Labeling (MML), treats all possible labels as independent
classes and learns a multi-class classifier on the induced multi-class categorization problem. To cope with the data sparseness caused by the huge
number of possible labels, MML combines some prior knowledge about
label prototypes and a maximal margin criterion in a novel way. Experiments with multi-topic Web pages show that MML outperforms existing
learning algorithms including Support Vector Machines.
1
Multi-topic Text Categorization (MTC)
This paper addresses the problem of learning for multi-topic text categorization (MTC),
whose goal is to select all topics relevant to a text from a given set of topics. In MTC,
multiple topics may be relevant to a single text. We thus call a set of topics label, and say
that a text is assigned a label, not a topic.
In almost all previous text categorization studies (e.g. [1, 2]), the label is predicted by
judging each topic?s relevance to the text. In this decomposition approach, the features
specific to a topic, not a label, are regarded as important features. However, the approach
may result in inefficient learning as we will explain in the following example.
Imagine an MTC problem of scientific papers where quantum computing papers are assigned multi-topic label ?quantum physics (QP) & computer science (CS)?. (QP and CS
are topics in this example.) Since there are some words specific to quantum computing
such as ?qbit1 ?, one can say that efficient MTC learners should use such words to assign
label QP & CS. However, the decomposition approach is likely to ignore these words since
they are only specific to a small portion of the whole QP or CS papers (there are many more
QP and CS papers than quantum computing papers), and therefore are not discriminative
features for either topic QP or CS.
1
Qbit is a unit of quantum information, and frequently appears in real quantum computing literatures, but rarely seen in other literatures.
Symbol
x(? Rd )
t1 , t2 , . . . , tl
T
L, ?(? T )
L[j]
?(= 2T )
{(xi , Li )}m
i=1
Meaning
A document vector
Topics
The set of all topics
A label
The binary representation of L. 1 if tj ? L and 0 otherwise.
The set of all possible labels
Training samples
Table 1: Notation
Parametric Mixture Model (PMM) [3] adopts another approach to MTC. It is assumed in
PMM that multi-topic texts are generated from a mixture of topic-specific word distributions. Its decision on labeling is done at once, not separately for each topic. However, PMM
also has a problem with multi-topic specific features such as ?qbit? since it is impossible
for texts to have such features given PMM?s mixture process.
These problems with multi-topic specific features are caused by dependency assumptions
between labels, which are explicitly or implicitly made in existing methods. To solve these
problems, we propose Maximal Margin Labeling, which treats labels as independent
classes and learns a multi-class classifier on the induced multi-class problem.
In this paper, we first discuss why multi-class classifiers cannot be directly applied to MTC
in Section 2. We then propose MML in Section 3, and address implementation issues in
Section 4. In Section 5, MML is experimentally compared with existing methods using a
collection of multi-topic Web pages. We summarize this paper in Section 6.
2
Solving MTC as a Multi-Class Categorization
To discuss why existing multi-class classifiers do not work in MTC, we start from the
multi-class classifier proposed in [4]. Hereafter we use the notation given in Table 1. The
multi-class classifier in [4] categorizes an object into the class whose prototype vector is
the closest to the object?s feature vector. By substituting label for class, the classifier can
be written as follows.
f (x) = arg max hx, m? iX
(1)
???
where h, iX is the the inner product of Rd , and m? ? Rd is the prototype vector of label ?.
Following the similar argument as in [4], the prototype vectors are learned by solving the
following maximal margin problem2 .
X
X
1
min kM k2 + C
?i?
M 2
1?i?m ???,?6=Li
s.t.
hxi , mLi iX ? hxi , m? iX ? 1 ? ?i?
for 1 ? i ? m, ?? 6= Li ,
(2)
where M is the prototype matrix whose columns are the prototype vectors, and kM k is the
Frobenius matrix norm of M .
Note that Eq. (1) and Eq. (2) cover only training samples? labels, but also all possible labels.
This is because the labels unseen in training samples may be relevant to test samples. In
2
In Eq.(2), we penalize all violation of the margin constraints. On the other hand, Crammer
and Singer penalize only the largest violation of the margin constraint for each training sample [4].
We chose the ?penalize-all? approach since it leads to an optimization problem without equality
constraints (see Eq.(7)), which is much easier to solve than the one in [4].
usual multi-class problems, such unseen labels seldom exist. In MTC, however, the number
of labels is generally very large (e.g. one of our datasets has 1,054 labels (Table 2)), and
unseen labels often exist. Thus it is necessary to consider all possible labels in Eq. (1) and
Eq. (2) since it is impossible to know which unseen labels are present in the test samples.
There are two problems with Eq. (1) and Eq. (2). The first problem is that they involve the
prototype vectors of seldom or never seen labels. Without the help of prior knowledge about
where the prototype vectors should be, it is impossible to obtain appropriate prototype
vectors for such labels. The second problem is that these equations are computationally too
demanding since they involve combinatorial maximization and summation over all possible
labels, whose number can be quite large. (For example, the number is around 230 in the
datasets used in our experiments.) We will address the first problem in Section 3 and the
second problem in Section 4.
3
Maximal Margin Labeling
In this section, we incorporate some prior knowledge about the location of prototype vectors into Eq. (1) and Eq. (2), and propose a novel MTC learning algorithm, Maximal
Margin Labeling (MML).
As prior knowledge, we simply assume that the prototype vectors of similar labels should
be placed close to each other. Based on this assumption, we first rewrite Eq. (1) to yield
f (x) = arg max hM T x, e? iL ,
(3)
???
where h, iL is the inner product of R|?| and {e? }??? is the orthonormal basis of R|?| . The
classifier of Eq. (3) can be interpreted as a two-step process: the first step is to map the
vector x into R|?| by M T , and the second step is to find the closest e? to image M T x.
Then we replace {e? }??? with (generally) non-orthogonal vectors {?(?)}??? whose geometrical configuration reflects label similarity. More formally speaking, we use vectors
{?(?)}??? that satisfy the condition
h?(?1 ), ?(?2 )iS = S(?1 , ?2 ) for ??1 , ?2 ? ?,
(4)
where h, iS is an inner product of the vector space spanned by {?(?)}??? , and S is a
Mercer kernel [5] on ? ? ? and is a similarity measure between labels. We call the vector
space spanned by {?(?)} VS .
With this replacement, MML?s classifier is written as follows.
f (x) = arg max hW x, ?(?)iS ,
(5)
???
where W is a linear map from Rd to VS . W is the solution of the following problem.
m
X
X
1
min
kW k2 + C
?i?
W
2
i=1 ???,?6=Li
?(Li )??(?)
s.t.
W xi ,
? 1??i? , ?i? ? 0 for 1 ? i ? m, ?? 6= Li . (6)
k?(Li )??(?)k
Note that if ?(?) is replaced by e? , Eq. (6) becomes identical to Eq. (2) except for a scale
factor. Thus Eq. (5) and Eq. (6) are natural extensions of the multi-class classifier in [4].
We call the MTC classifier of Eq. (5) and Eq. (6) ?Maximal Margin Labeling (MML)?.
Figure 1 explains the margin (the inner product in Eq. (6)) in MML. The margin represents
the distance from the image of the training sample xi to the boundary between the correct
label Li and wrong label ?. MML optimizes the linear map W so that the smallest margin
between all training samples and all possible labels becomes maximal, along with a penalty
C for the case that samples penetrate into the margin.
Figure 1: Maximal Margin Labeling
Dual Form For numerical computation, the following Wolfe dual form of Eq. (6) is more
convenient. (We omit its derivation due to space limits.)
X
S(Li , Li? )?S(Li , ?? )?S(?, Li? )+S(?, ?? )
1 X X ? ??
p
?i ?i? (xi ?xi? )
max
?i? ?
2
??
2 (1?S(Li , ?))(1?S(Li? , ?? ))
i
?
?
i,? i ,?
i,?
s.t. 0 ? ?i? ? C for 1 ? i ? m, ?? 6= Li ,
(7)
Pm P
P
?
where we denote i=1 ???,??6=Li by i,? , and ?i are the dual variables corresponding
to the first inequality constraints in Eq. (6). Note that Eq. (7) does not contain ?(?): all the
computations involving ? can be done through the label similarity S. Additionally xi only
appears in the inner products, and therefore can be replaced by any kernel of x.
Using the solution ?i? of Eq. (7), the MML?s classifier in Eq. (5) can be written as follows.
X
S(Li , L)?S(?, L)
.
(8)
f (x) = arg max
?i? (x?xi ) p
2(1?S(Li , ?))
L??
i,?
Label Similarity3 As examples of label similarity, we use two similarity measures: Dice
measure and cosine measure.
4
Dice measure
Cosine measure
4
4.1
Pl
2 j=1 ?1 [j]?2 [j]
2|?1 ??2 |
SD (?1 , ?2 ) =
= Pl
.
(9)
Pl
|?1 |+|?2 |
j=1 ?1 [j] +
j=1 ?2 [j]
Pl
|?1 ? ?2 |
j=1 ?1 [j]?2 [j]
p
qP
= qP
SC (?1 , ?2 ) = p
.(10)
l
l
|?1 | |?2 |
j=1 ?1 [j]
j=1 ?2 [j]
Efficient Implementation
Approximation in Learning
Eq. (7) contains the sum over all possible labels. As the number of topics (l) increases, this
summation rapidly becomes intractable since |?| grows exponentially as 2l . To circumvent
3
The following discussion is easily extended to include the case that both ?1 and ?2 are empty
although we do not discuss the case due to space limits.
this problem, we approximate the sum over all possible labels in Eq. (7) by the partial sum
over ?i? of |(A ? B c ) ? (Ac ? B)| = 1 and set all the other ?i? to zero. This approximation
reduces the burden of the summation quite a lot: the number of summands is reduced from
2l to l, which is a huge reduction especially when many topics exist.
To understand the rationale behind the approximation, first note that ?i? is the dual variable
corresponding to the first inequality constraint (the margin constraint) in Eq. (7). Thus ?i?
is non-zero if and only if W xi falls in the margin between ?(Li ) and ?(?). We assume
that this margin violation mainly occurs when ?(?) is ?close? to ?(Li ), i.e. |(A ? B c ) ?
(Ac ? B)| = 1. If this assumption holds well, the proposed approximation of the sum will
lead to a good approximation of the exact solution.
4.2
Polynomial Time Algorithms for Classification
The classification of MML (Eq. (8)) involves the combinatorial maximization over all possible labels, so it can be a computationally demanding process. However, efficient classification algorithms are available when either the cosine measure or dice measure is used as
label similarity.
Eq. (8) can be divided into the subproblems by the number of topics in a label.
f (x) =
arg max g(x, L),
?n
L
(11)
? 1 ,L
? 2 ,...,L
?l}
L?{L
=
arg max g(x, L).
(12)
L??,|L|=n
where g(x) is
g(x, L)
=
l
X
cn [j]L[j],
j=1
cn [j]
=
? P
??
2Li [j]
2?[j]
i (x?xi )
?
?
.
?
?
i,?
|?|+n
2(1?SD (Li ,?)) |Li |+n
?
P
?
(x?x
)
?[j]
L
[j]
i
i
?
i
?
?
?
?
?
? ?
i,?
2(1?SC (Li ,?))
|Li | n
|?| n
if SD is used.
(13)
if SC is used.
Here n = |L|. The computational cost of Eq. (13) for all j is O(n? l) (n? is the number of
non-zero ?), and that of Eq. (12) is O(l log l). Thus the total cost of the classification by
Eq. (11) is O(n? l2 + l2 log l). On the other hand, n? is O(ml) under the approximation
described above. Therefore, the classification can be done within O(ml3 ) computational
steps, which is a significant reduction from the case that the brute force search is used in
Eq. (8).
5
Experiments
In this section, we report experiments that compared MML to PMM [3], SVM5 [6], and
BoosTexter [2] using a collection of Web pages. We used a normalized linear kernel
k(x, x? ) = x ? x? /kxkkx? k in MML and SVM. As for BoosTexter, ?real abstaining AdaBoost.MH? was used as the weak learner.
5.1
Experimental Setup
The datasets used in our experiment represent the Web page collection used in [3] (Table 2). The Web pages were collected through the hyperlinks from Yahoo!?s top directory
5
For each topic, an SVM classifier is trained to predict whether the topic is relevant (positive) or
irrelevant (negative) to input doucments.
Dataset Name (Abbrev.)
Arts & Humanities (Ar)
Business & Economy (Bu)
Computers & Internet (Co)
Education (Ed)
Entertainment (En)
Health (He)
Recreation (Rc)
Reference (Rf)
Science (Si)
Social Science (SS)
Society & Culture (SC)
#Text
#Voc
7,484
11,214
12,444
12,030
12,730
9,205
12,828
8,027
6,428
12,111
14,512
23,146
21,924
34,096
27,534
32,001
30,605
30,324
39,679
37,187
52,350
31,802
#Tpc #Lbl
26
30
33
33
21
32
22
33
40
39
27
Label Size Frequency (%)
1
2
3
4 ?5
599 55.6 30.5 9.7 2.8 1.4
233 57.6 28.8 11.0 1.7 0.8
428 69.8 18.2 7.8 3.0 1.1
511 66.9 23.4 7.3 1.9 0.6
337 72.3 21.1 4.5 1.0 1.1
335 53.2 34.0 9.5 2.4 0.9
530 69.2 23.1 5.6 1.4 0.6
275 85.5 12.6 1.5 0.3 0.1
457 68.0 22.3 7.3 1.9 0.5
361 78.4 17.0 3.7 0.7 0.3
1054 59.6 26.1 9.2 2.9 2.2
Table 2: A summary of the web page datasets. ?#Text? is the number of texts in the dataset,
?#Voc? the number of vocabularies (i.e. features), ?#Tpc? the number of topics, ?#Lbl? the
number of labels, and ?Label Size Frequency? is the relative frequency of each label size.
(Label size is the number of topics in a label.)
Method
MML
PMM
SVM
Boost
Feature Type
TF, TF?IDF
TF
TF, TF?IDF
Binary
Parameter
C = 0.1, 1, 10
Model1, Model2
C = 0.1, 1, 10
R = {2, 4, 6, 8, 10}?103
Table 3: Candidate feature types and learning parameters. (R is the number of weak hypotheses.) The underlined fetures and parameters were selected for the evaluation with the
test data.
(www.yahoo.com), and then divided into 11 datasets by Yahoo?s top category. Each
page is labeled with the Yahoo?s second level sub-categories from which the page is hyperlinked. (Thus, the sub-categories are topics in our term.) See [3] for more details about
the collection. Then the Web pages were converted into three types of feature vectors: (a)
Binary vectors, where each feature indicates the presence (absence) of a term by 1 (0); (b)
TF vectors, where each feature is the number of appearances of a term (term frequency);
and (c) TF?IDF vectors, where each feature is the product of term frequency and inverse
document frequency [7].
To select the best combinations of feature types and learning parameters such as the penalty
C for MML, the learners were trained on 2,000 Web pages with all combinations of feature and parameter listed in Table 3, and then were evaluated by labeling F-measure on
independently drawn development data. The combinations which achieve the best labeling
F-measures (underlined in Table 3) were used in the following experiments.
5.2
Evaluation Measures
We used three measures to evaluate labeling performance: labeling F-measure, exact match
ratio, and retrieval F-measure. In the following definitions, {Lpred
}ni=1 and {Ltrue
}ni=1
i
i
mean the predicted labels and the true labels, respectively.
Labeling F-measure Labeling F-measure FL evaluates the average labeling performance
while taking partial match into account.
Pl
pred
n
n
true
? Ltrue
|
1 X 2 j=1 Li [j]Li [j]
1 X 2|Lpred
i
i
FL =
=
.
(14)
P
n i=1 |Lpred
n i=1 lj=1 (Lpred
| + |Ltrue
|
[j] + Ltrue
[j])
i
i
i
i
Dataset
Ar
Bu
Co
Ed
En
He
Rc
Rf
Si
SS
SC
Avg
MD
0.55
0.80
0.62
0.56
0.64
0.74
0.63
0.67
0.61
0.73
0.60
0.65
Labeling F-measure
MC PM SV BO
0.44 0.50 0.46 0.38
0.81 0.75 0.76 0.75
0.59 0.61 0.55 0.47
0.43 0.51 0.48 0.37
0.52 0.61 0.54 0.49
0.74 0.66 0.67 0.60
0.46 0.55 0.49 0.44
0.58 0.63 0.56 0.50
0.54 0.52 0.47 0.39
0.71 0.66 0.64 0.59
0.55 0.54 0.49 0.44
0.58 0.59 0.56 0.49
MD
0.44
0.63
0.51
0.45
0.55
0.58
0.54
0.60
0.52
0.65
0.44
0.54
Exact Match Ratio
MC PM SV
0.32 0.21 0.29
0.62 0.48 0.57
0.46 0.35 0.41
0.34 0.19 0.30
0.44 0.31 0.42
0.53 0.34 0.47
0.38 0.25 0.37
0.51 0.39 0.49
0.43 0.22 0.36
0.60 0.45 0.55
0.40 0.21 0.32
0.46 0.31 0.41
BO
0.22
0.53
0.34
0.23
0.36
0.39
0.33
0.41
0.28
0.49
0.27
0.35
MD
0.30
0.25
0.27
0.25
0.37
0.35
0.47
0.29
0.37
0.36
0.29
0.32
Retrieval F-measure
MC PM SV BO
0.26 0.24 0.29 0.22
0.27 0.20 0.29 0.20
0.25 0.19 0.30 0.17
0.23 0.21 0.25 0.16
0.33 0.30 0.35 0.29
0.35 0.23 0.35 0.26
0.39 0.36 0.40 0.33
0.25 0.24 0.25 0.16
0.35 0.28 0.31 0.19
0.35 0.18 0.31 0.15
0.28 0.25 0.26 0.20
0.30 0.24 0.31 0.21
Table 4: The performance comparison by labeling F-measure (left), exact match ratio (middle) and retrieval F-measure (right). The bold figures are the best ones among the five
methods, and the underlined figures the second best ones. MD, MC, PM, SV, and BO
represent MML with SD , MML with SC , PMM, SVM and BoosTexter, respectively.
Exact Match Ratio Exact match ratio EX counts only exact matches between the predicted label and the true label.
n
EX =
1X
I[Lpred
= Ltrue
],
i
i
n i=1
(15)
where I[S] is 1 if the statement S is true and 0 otherwise.
Retrieval F-measure6 For real tasks, it is also important to evaluate retrieval performance, i.e. how accurately classifiers can find relevant texts for a given topic. Retrieval
F-measure FR measures the average retrieval performance over all topics.
Pn
l
[j]Ltrue
[j]
1 X 2 i=1 Lpred
i
i
FR =
.
(16)
Pn
true
l j=1 i=1 (Lpred
[j] + Li [j])
i
5.3
Results
First we trained the classifiers with randomly chosen 2,000 samples. We then calculated
the three evaluation measures for 3,000 other randomly chosen samples. This process was
repeated five times, and the resulting averaged values are shown in Table 4. Table 4 shows
that the MMLs with Dice measure outperform other methods in labeling F-measure and
exact match ratio. The MMLs also show the best performance with regard to retrieval Fmeasure although the margins to the other methods are not as large as observed in labeling
F-measure and exact match ratio. Note that no classifier except MML with Dice measure
achieves good labeling on all the three measures. For example, PMM shows high labeling
F-measures, but its performance is rather poor when evaluated in retrieval F-measure.
As the second experiment, we evaluated the classifiers trained with 250?2000 training samples on the same test samples. Figure 2 shows each measure averaged over all datasets. It is
observed that the MMLs show high generalization even when training data is small. An interesting point is that MML with cosine measure achieves rather high labeling F-measures
and retrieval F-measure with training data of smaller size. Such high-performace, however,
does not continue when trained on larger data.
6
FR is called ?the macro average of F-measures? in the text categorization community.
Figure 2: The learning curve of labeling F-measure (left), exact match ratio (middle) and
retrieval F-measure (right). MD, MC, PM, SV, BO mean the same as in Table 4.
6
Conclusion
In this paper, we proposed a novel learning algorithm for multi-topic text categorization.
The algorithm, Maximal Margin Labeling, embeds labels (sets of topics) into a similarityinduced vector space, and learns a large margin classifier in the space. To overcome the
demanding computational cost of MML, we provide an approximation method in learning
and efficient classification algorithms. In experiments on a collection of Web pages, MML
outperformed other methods including SVM and showed better generalization.
Acknowledgement
The authors would like to thank Naonori Ueda, Kazumi Saito and Yuji Kaneda of Nippon
Telegraph and Telephone Corporation for providing PMM?s codes and the datasets.
References
[1] Thorsten Joachims. Text categorization with support vector machines: learning with
many relevant features. In Claire N?edellec and C?eline Rouveirol, editors, Proc. of the
10th European Conference on Machine Learning, number 1398, pages 137?142, 1998.
[2] Robert E. Schapire and Yoram Singer. BoosTexter: A boosting-based system for text
categorization. Machine Learning, 39(2/3):135?168, 2000.
[3] Naonori Ueda and Kazumi Saito. Parametoric mixture models for multi-topic text. In
Advances in Neural Information Processing Systems 15, pages 1261?1268, 2003.
[4] Koby Crammer and Yoram Singer. On the algorithmic implementation of multiclass
kernel-based vector machines. Journal of Machine Learning Research, 2:265?292,
2001.
[5] Klaus-Robert M?uller, Sebastian Mika, Gunnar R?atsch, Koji Tsuda, and Bernhard
Sch?olkopf. An introduction to kernel-based learning algorithms. IEEE Transactions
on Neural Networks, 12(2):181?201, 2001.
[6] Vladimir N. Vapnik. Statistical Learning Theory. John Wiley & Sons, Inc., 1998.
[7] Ricardo Baeza-Yates and Berthier Ribeiro-Neto. Modern Information Retrieval.
Addison-Wealy, 1999.
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1,838 | 267 | Recognizing Hand-Printed Letters and Digits
Recognizing Hand-Printed Letters and Digits
Gale L. Martin James A. Pittman
MCC, Austin, Texas 78759
ABSTRACT
We are developing a hand-printed character recognition system using a multilayered neural net trained through backpropagation. We report on results of
training nets with samples of hand-printed digits scanned off of bank checks
and hand-printed letters interactively entered into a computer through a stylus digitizer. Given a large training set, and a net with sufficient capacity to
achieve high performance on the training set, nets typically achieved error
rates of 4-5% at a 0% reject rate and 1-2% at a 10% reject rate. The topology
and capacity of the system, as measured by the number of connections in the
net, have surprisingly little effect on generalization. For those developing
practical pattern recognition systems, these results suggest that a large and
representative training sample may be the single, most important factor in
achieving high recognition accuracy. From a scientific standpoint, these results raise doubts about the relevance to backpropagation of learning models
that estimate the likelihood of high generalization from estimates of capacity.
Reducing capacity does have other benefits however, especially when the reduction is accomplished by using local receptive fields with shared weights.
In this latter case, we find the net evolves feature detectors resembling those
in visual cortex and Linsker's orientation-selective nodes.
Practical interest in hand-printed character recognition is fueled by two current technology trends: one toward systems that interpret hand-printing on hard-copy documents and one toward notebook-like computers that replace the keyboard with a stylus
digitizer. The stylus enables users to write and draw directly on a flat panel display.
In this paper, we report on results applying multi-layered neural nets trained through
backpropagation (Rumelhart, Hinton, & Williams, 1986) to both cases.
Developing pattern recognition systems is typically a two-stage process. First, intuition
and experimentation are used to select a set of features to represent the raw input pattern. Then a variety of well-developed techniques are used to optimize the classifier
system that assumes this featural representation. Most applications of backpropagation learning to character recognition use the learning capabilities only for this latter
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Martin and Pittman
stage--developing the classifier system (Burr, 1986; Denker, Gardner, Graf, Henderson, Howard, Hubbard, Jackel, Baird, & Guyon, 1989; Mori & Yokosawa, 1989; Weideman, Manry, & Yau, 1989). However, backpropagation learning affords the opportunity to optimize feature selection and pattern classification simultaneously. We avoid
using pre-determined features as input to the net in favor of using a pre- segmented,
size-normalized grayscale array for each character. This is a first step toward the goal
of approximating the raw input projected onto the human retina, in that no pre-processing of the input is required.
We report on results for both hand-printed digits and letters. The hand-printed digits
come from a set of 40,000 hand-printed digits scanned from the numeric amount region
of "real-world" bank checks. They were pre-segmented and size-normalized to a
15x24 grayscale array. The test set consists of 4,000 samples and training sets varied
from 100 to 35,200 samples. Although it is always difficult to compare recognition rates
arising from different pattern sets, some appreciation for the difficulty of categorization can be gained using human performance data as a benchmark. An independent
person categorizing the test set of pre-segmented, size-normalized digits achieved an
error rate of 3.4%. This figure is considerably below the near-perfect performance of
operators keying in numbers directly from bank checks, because the segmentation algorithm is flawed .
Working with letters, as well as digits, enables tests of the generality of results on a
different pattern set having more than double the number of output categories. The
hand-printed letters come from a set of 8,600 upper-case letters collected from over
110 people writing with a stylus input device on a flat panel display. The stylUS collects
a sequence of x-y coordinates at 200 points per second at a spatial resolution of 1000
points per inch. The temporal sequence for each character is first converted to a sizenormalized bitmap array, keeping aspect ratio constant. We have found that recognition accuracy is significantly improved if these bitmaps are blurred through convolution
with a gaussian distnbution. Each pattern is represented as a 15x24 grayscale image.
A test set of 2,368 samples was extracted by selecting samples from 18 people, so that
training sets were generated by people different from those generating the test set.
Training set sizes ranged from 500 to roughly 6,300 samples.
1 HIGH RECOGNITION ACCURACY
We find relatively high recognition accuracy for both pattern sets. Thble 11 reports
the minimal error rates achieved on the test samples for both pattern sets, at various
reject rates. In the case of the hand-printed digits, the 4% error rate (0% rejects) ap1. Eff~cts of the number.of training samples and network capacity and topology are reported in the
next sectIon. Nets were tramed to error rates of 2-3%. 1i"aining began with a learning rate of .05 and
a mome~tum value of .9. The learning rate was decreased when training accuracy began to oscillate or
had stabtlized for a large number of training epochs. We evaluate the output vector on a winner-takeall basis, as this consistently improves accuracy and results in network parameters having a smaller
effect on perfonnance.
Recognizing Hand-Printed Letters and Digits
proaches the 3.4% errors made by the human judge. This suggests that further improvements to generalization will require improving segmentation accuracy. The fact
that an error rate of 5% was achieved for letters is promising. Accuracy is fairly high,
Table 1:
Error rates of best nets trained on largest sample sets and tested
on new samples
REJECT RATE
0%
5%
10%
35%
DIGITS
LETTERS
4%
5%
3%
2%
.003%
3%
1%
.001%
even though there are a large number of categories (26). This error rate may be adequate for applications where contextual constraints can be used to significantly boost
accuracy at the word-level.
2 MINIMAL NETWORK CAPACITY AND TOPOLOGY EFFECTS
The effects of network parameters on generalization have both practical and scientific
significance. The practical developer of pattern recognition systems is interested in
such effects to determine whether limited resources should be spent on trying to optimize network parameters or on collecting a larger, more representative training set.
For the scientist, effects of capacity bear on the relevance of learning models to backpropagation.
A central premise of most general models of learning-by-example is that the size of
the initial search space-the capacity of the system-determines the number of training samples needed to achieve high generalization performance. Learning is conceptualized as a search for a function that maps all possible inputs to their correct outputs.
Learning occurs by comparing successive samples of input-output pairs to functions
in a search space. Functions inconsistent with training samples are rejected. Very large
training sets narrow the search down to a function that closely approximates the desired function and yields high generalization. The capacity of a learning system-the
number of functions it can represent--determines generalization, since a larger initial
search space requires more training samples to narrow the search sufficiently . This
suggests that to improve generalization, capacity should be minimized. Unfortunately,
it is typically unclear how to minimize capacity without eliminating the desired function
from the search space. A heuristic, which is often suggested, is that simple is usually
better. It receives support from experience in curve fitting. Low-order polynomials typically extrapolate and interpolate better than high-order polynomials (Duda & Hart,
1973).
Extensions of the heuristic to neural net learning propose reducing capacity by reducing the number of connections or the number of bits used to represent each connection
407
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Martin and Pittman
weight (Baum & Haussler, 1989; Denker, Schwartz, Wittner, Solla, Howard, Jackel,
& Hopfield,1987). We manipulated the capacity of nets in a number of ways: 1) varying
the number of hidden nodes, 2) limiting connectivity between layers so that nodes received input from only local areas, and 3) sharing connection weights between hidden
nodes. We found only negligible effects on generalization.
2.1 NUMBER OF HIDDEN NODES
Figure 1 presents generalization results as a function of training set size for nets having
one hidden layer and varying numbers of hidden nodes. The number of free parameters
(i.e., number of connections and biases) in each case is presented in parentheses. Despite considerable variation in the number of free parameters, using nets with fewer
hidden nodes did not improve generalization.
Baum & Haussler (1989) estimate the number of training samples required to achieve
an error rate e (where 0 < e ~ 1/8) on the generalization test, when an error rate
of el2 has been achieved on the training set. They assume a feed-forward net with
one hidden layer and W connections. The estimates are distribution-free in the sense
that calculations assume an arbitrary to-be-learned function. If the number of training
samples is of order : log ~ ,where N refers to the number of nodes, then it is a near
certainty that the net will achieve generalization rates of (1 - e). This estimate is the
upper-bound on the number of training samples needed. They also provide a lower
Digits
Letters
100
100
,
./
.;/
,';/
I
~
-
I
,
~
t:::
0
u
~
Number of Hidden Nodes
50
100
I
75
75
1000
I
/
,
Number of Hidden Nodes
50
(18,560)
170
(63,080)
170
(65,816)
383 (142,103)
365
(141,281)
10000 100000
Training Set Size
50
100
1000
10000 100000
'll'aining Set Size
Figure 1. Effect of number of hidden nodes and training set size on generalization.
Recognizing Hand-Printed Letters and Digits
bound estimate, on the order of W/e. Using fewer than this number of samples will,
for some functions, fail to achieve generalization rates of (1- e). The fact that we find
no advantage to reducing the number of connections conflicts with Baum & Haussler's
estimates and the underlying assumption that capacity plays a strong role in determining generalization.
Baum & Haussler also suggest using a constant of proportionality of 1 in their estimates. This implies that achieving error rates of 10% or less on new samples requires
about 10 times as many training examples as there are connection weights in the net.
For our largest nets, this implies a requirement of roughly a million training samples,
which most developers would regard as prohibitively large. We found that about 5,000
samples were sufficient. Thus, a sufficiently large training sample does not imply aprohibitively large sample, at least for character recognition. We find that sample sizes of
the order of thousands to tens of thousands yield performance very close to human levels. One reason for the discrepancy is that Baum & Haussler'S estimates are distribution-free in the sense that they reflect worst-case scenarios across all possible functions the net might learn. Presumably, the functions underlying most natural pattern
recognition tasks are not representative of the set of all possible functions. These results raise doubts about the relevance to natural pattern recognition of learning models
based on worst-case analyses, because content may greatly impact generalization.
2.2 LOCAL CONNECTMTY AND SHARED WEIGHTS
A more biologically plausible way to reduce capacity is to limit connectivity between
layers to local areas and to use shared weights. For example, visual cortex contains
neurons, each of which is responsive to a feature such as an oriented line appearing
in a small, local region on the retina (Hubel & Wiesel, 1979). A given oriented line-detector is essentially replicated across the visual field, so that the same feature can be
detected wherever it appears. In this sense, the connections feeding into an orientedline detector are shared across all similar line-detectors for different areas of the visual
field. In an artificial neural net, local structure is achieved by limiting connectivity.
A given hidden node receives input from only local areas in the input or hidden layer
preceding it. Weight sharing is achieved by linking the incoming weights across a set
of hidden nodes. Corresponding weights leading into these nodes are randomly initialized to the same values and forced to have equivalent updates during learning. In this
way the net evolves the same local feature set that is invariant across the input array.
Several demonstrations exist indicating that local connectivity and shared weights improve generalization performance in tasks where position invariance is required (Ie
Cun, 1989; Rumelhart, Hinton, & Williams, 1986).
We examined the benefits of using local receptive fields with shared weights for handprinted character recognition, where position invariance was not required. This does
not minimize the importance of position invariance. However, it is only one of many
necessary invariants underlying reliable pattern recognition. Unfortunately, most of
these invariants have not been explicitly specified. So we don't know how to bias a net
toward discovering them. Testing the role of local receptive fields with shared weights
409
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Martin and Pittman
in situations where position invariance is not required is relevant to discovering whether these constraints have a role other than in promoting position invariance.
As indicated in Figure 2, we find only slightly improved generalization in moving from
nets with global connectivity between layers to nets with local receptive fields or to nets
with local receptive fields and shared weights. This is true despite the fact that the
number of free parameters is substantially reduced. The positive effects that do occur
are at relatively small training set sizes. This may explain why others have reported
a greater degree of improved generalization by using local receptive fields (Honavar
& Uhr, 1988). The data reported are for networks with two hidden layers. Global nets
had 150 nodes in the first layer and 50 nodes in the second. In the Local nets, first hidden layer nodes (540) received input from 5x8local and overlapping regions (offset by
2 pixels) on the input array. Second hidden layer nodes (100) and output layer nodes
had global receptive fields. The Local. Shared nets had 540 nodes in the first hidden
layer with shared weights and, at the second hidden layer, either 102 (digits) or 180 (letters) nodes with local, overlapping, and shared receptive fields of size 4x6 on the 1st
100
Digits
100
h
Letters
,~
, ,#
. ' .,f
.....
u
Q)
t:::
0
u
75
75
~
50
100
- - Global
Local
(62.210)
(77.250)
Global (63,026)
Local,
Shared
( 4,857)
Local,
1 6
Shared (11, 4 )
1000
10000 100000
Training Set Size
Figure 2.
Local
(78,866)
50 +-~:;:::==::::;::::==:::::;::::::.
100
1000 10000 100000
Training Set Size
Effects of net capacity and topology on generalization.
hidden layer. We have experimented with a large variety of different net architectures
of this sort, varying the number of hidden nodes, the sizes and overlap of local receptive
fields, and the use of local receptive fields with and without shared weights in one or
both hidden layers. The fact that we've found little difference in generalization for two
different pattern sets across such variations in network architectures argues for the
generality of the results.
Recognizing Hand-Printed Letters and Digits
2.3
DISCUSSION
Given an architecture that enables relatively high training performance, we find only
small effects of network capacity and topology on generalization performance. A large
training set yields relatively high recognition accuracy in a robust way across most net
architectures with which we've worked. These results suggest some practical advice to
those developing hand-printed character recognition systems. IT optimizing generalization performance is the goal, it is probably better to devote limited resources to collecting a very large, representative training set than to extensive experimentation with
different net architectures. The variations in net capacity and topolOgy we've examined
do not substantially affect generalization performance for sufficiently large training
sets. Sufficiently large should be interpreted as on the order of a thousand to tens of
thousands of samples for hand-printed character recognition.
From a theoretical standpoint, the negligible effects of network capacity on generalization performance contradicts the central premise of machine learning that the size of
the initial hypothesis space determines learning performance. This challenges the relevance, to backpropagation learning, of statistical models that estimate likelihood of
high generalization performance from estimates of capacity. Due to the gradient descent nature of backpropagation learning, not all functions that can be represented will
be visited during learning. The negligible effects of capacity suggest that the number
of functions visited during learning constitutes only a very small percentage of the total
possible functions that can be represented.
There are a number of reasons for believing that capacity might impact generalization
performance in other circumstances. We regularly train only to 2-3% error rates. This
helps to avoid the possibility of overfitting the data, although we have seen no indication of this when we have trained to higher levels, as long as we use large training sets.
It is also possible that the number of connections is not a good measure of capacity.
For example, the amount of information that can be passed on by a given connection
may be a better measure than the number of connections. At this conference, Ie Cun,
Denker, Solla, Howard, & Jackel have also presented evidence that removing unimportant weights from a network may be a better way to reduce capacity. However, the
fact that generalization rates come very close to human accuracy levels, even for nets
with extremely large numbers of free parameters, suggests that general effects of net
capacity and topology are, at best, small in comparison to effects of training set size.
We don't deny that there are likely to be net topologies that push performance up to
human accuracy levels, presumably by biasing the net toward discovering the range of
invariants that underlie human pattern recognition. The problem is that only a few
of these invariants have been explicitly specified (e.g., position, size, rotation), and so
it is not possible to bias a net toward discovering the full range.
411
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Martin and Pittman
3 ADVANTAGES OF REDUCING CAPACITY
Although reducing gross indicators of capacity may not significantly improve generalization, there are good practical and scientific reasons for doing it. A good reason to
reduce the number of connections is to speed processing. Also, using local receptive
fields with shared weights biases a net toward position invariance, and toward producing a simpler, more modular internal representation which can be replicated across a
large retina. This has important implications for developing nets that combine character segmentation with recognition.
Using local receptive fields with shared weights also offers promise for increasing our
understanding of how the net correctly classifies patterns because the number of distinct receptive fields is greatly reduced. Figure 3 depicts Hinton diagrams of local reDigits
Letters
Figure 3. Receptive fields that evolved in 1st hidden layer nodes in nets with
local receptive fields having shared weights.
ceptive fields from 1st hidden layer nodes in nets with shared weights trained on digits
or letters. Each of the eight large, gray rectangles corresponds to the receptive field
for a hidden node. The four on the left came from a net trained on digits; those on
the right from a net trained on letters. Within each ofthese eight, the black rectangles
correspond to negative weights and the white to positive weights. The size of the black
and white rectangles reflects the magnitude of the weights.
The local feature detectors that develop for both pattern sets appear to be oriented
line and edge detectors. These are similar to oriented line and edge detectors found
in visual cortex (Hubel & Wiesel, 1979) and to Linsker's (1986,1988) orientation-selective nodes, which emerge from a self-adaptive net exposed to random patterns. In
Linsker's case, the feature detectors develop as an emergent property of the principle
that the signal transformation occurring from one layer to the next should maximize
the information that output signals convey about input signals. The fact that similar
Recognizing Hand-Printed Letters and Digits
feature detectors emerge in backpropagation nets trained on "natural" patterns is interesting because there were no explicit constraints to maximize information flow between layers in the backpropagation nets and because categorization is typically viewed
as an abstraction process involving considerable loss of category-irrelevant information.
References
Baum, E. and Haussler, D. (1989) What size net gives valid generalization? in D. S.
Touretzky (Ed.) Advances in neural information processing systems I, Morgan
Kaufman.
Burr, D. J. (1986) A neural network digit recognizer. Proceedings of the 1986
International conference on systems, man and cybernetics, Atlanta, Georgia.
pp. 1621-1625.
Denker, J. S., Gardner, W. R., Graf, H. P., Henderson, D., Howard, R. E., Hubbard,
W., Jackel, L. D., Baird, H. S., and Guyon, I. (1989) Neural network recognizer
for hand-written zip code digits in D. S. Touretzky (Ed.) Advances in neural
information processing systems I, Morgan Kaufman.
Denker, J. S., Schwartz, D., Wittner, B., SolIa, S., Howard, R. E., Jackel, L. D., &
Hopfield, J. (1987) Large automatic learning, rule extraction and generalization.
Complex Systems, 1, pp. 877-933.
Duda, R. 0., and Hart, P. E. (1973) Pattern classification and scene analysis. NY:
John Wiley & Sons.
Honavar, V. and Uhr, L. (1988) Experimental results indicate that generation, local
receptive fields and global convergence improve perceptual learning in
connectionist networks. CS-TR 805. Computer Science Department, University
of Wisconsin-Madison.
Hubel, D. H. and Wiesel, T. N. (1979) Brain mechanisms of vision. Scientific
American, 241, pp. 150-162.
Ie Cun, Y. (1989) Generalization and network design strategies. Thchnical Report
CRG-TR-89-4, Department of Computer Science, University of Thronto.
Linsker, R. (1986) From basic network principles to neural architecture; Emergence
of orientation-selective cells. Proceedings of the National Academy of Sciences,
USA, 83, pp. 8390-8394.
Linsker, R. (1988) Thwards an organizing principle for a layered perceptual network
in D. Anderson (Ed.) Neural information processing systems. American Institute
of Physics.
413
414
Martin and Pittman
Mori, Y. and Yokosawa, K. (1989) Neural networks that learn to discriminate similar
kanji characters. in. D. S. Touretzky (Ed.) Advances in neural information
processing systems I, Morgan Kaufman.
Rumelhart, D. E., Hinton, O. E., & Williams, R. J. Learning internal representations
by error propagation in D. E. Rumelhart & J. L. McClelland (Editors) Parallel
distributed processing: V. 1. Cambridge, Mass.: MIT Press, 1986
Weideman, W. E., Manry, M T. & Yau, H. C. (1989) A comparison of a nearest
neighbor classifier and a neural network for numeric handprint character
recognition. IEEE International Conference on Neural Networks, Washington,
D. c., 1989.
Acknowledgements
We would like to thank the NCR corporation for loaning us the set of hand-printed
digits and Joyce Conner, Janet Kilgore, and Kay Bauer for their invaluable help in collecting the set of hand-printed letters.
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1,839 | 2,670 | A Second order Cone Programming
Formulation for Classifying Missing Data
Chiranjib Bhattacharyya
Department of Computer Science and Automation
Indian Institute of Science
Bangalore, 560 012, India
[email protected]
Pannagadatta K. S.
Department of Electrical Engineering
Indian Institute of Science
Bangalore, 560 012, India
[email protected]
Alexander J. Smola
Machine Learning Program
National ICT Australia and ANU
Canberra, ACT 0200, Australia
[email protected]
Abstract
We propose a convex optimization based strategy to deal with uncertainty
in the observations of a classification problem. We assume that instead
of a sample (xi , yi ) a distribution over (xi , yi ) is specified. In particular, we derive a robust formulation when the distribution is given by a
normal distribution. It leads to Second Order Cone Programming formulation. Our method is applied to the problem of missing data, where it
outperforms direct imputation.
1
Introduction
Denote by (x, y) ? X ?Y patterns with corresponding labels. The typical machine learning
formulation only deals with the case where (x, y) are given exactly. Quite often, however,
this is not the case ? for instance in the case of missing values we may be able (using a
secondary estimation procedure) to estimate the values of the missing variables, albeit with
a certain degree of uncertainty. It is therefore only natural to take the decreased reliability of
such data into account and design estimators accordingly. What we propose in the present
paper goes beyond the traditional imputation strategy where missing values are estimated
and then used as if they had actually been observed. The key difference in what follows is
? i is drawn from a
that we will require that with high probability any (?
xi , yi ) pair, where x
distribution of possible xi , will be estimated correctly. For the sake of simplicity we limit
ourselves to the case of binary classification.
The paper is organized as follows: Section 2 introduces the problem of classification with
uncertain data. We solve the equations arising in the context of normal random variables
in Section 3 which leads to a Second Order Cone Program (SOCP). As an application
the problem of classification with missing variables is described in Section 4. We report
experimental results in Section 5.
2
Linear Classification using Convex Optimization
Assume we have m observations (xi , yi ) drawn iid (independently and identically distributed) from a distribution over X ? Y, where X is the set of patterns and Y = {?1} are
the labels (e.g. the absence/presence of a particular object). It is our goal to find a function
f : X ? Y which classifies observations x into classes +1 and ?1.
2.1
Classification with Certainty
Assume that X is a dot product space and f is a linear function
f (x) = sgn(hw, xi + b).
(1)
In the case of linearly separable datasets we can find (w, b) which separates the two classes.
Unfortunately, such separation is not always possible and we need to allow for slack in the
separation of the two sets. Consider the formulation
minimize
w,b,?
m
X
?i
(2a)
i=1
subject to yi (hw, xi i + b) ? 1 ? ?i , ?i ? 0, kwk ? W for all 1 ? i ? m
(2b)
It is well known that this problem minimizes an upper bound on the number of errors. The
latter occur
p whenever ?i ? 1, where ?i are the slack variables. The Euclidean norm of
kwk = hw, wi, is upper bounded by a user defined constant W . This is equivalent
to lower bounding the margin, or the separation between the two classes. The resulting
discriminant surface is called the generalized optimal hyperplane [9]. The statement of (2)
is slightly nonstandard. Typically one states the SVM optimization problem as follows [3]:
m
X
1
?i
minimize kwk2 + C
w,b,?
2
i=1
(3a)
subject to yi (hw, xi i + b) ? 1 ? ?i , ?i ? 0 for all 1 ? i ? m
(3b)
Instead of the user defined parameter W , the formulation (3) uses another parameter C.
For a proper choice of C and W the two formulations are equivalent. For the purpose of
the present paper, however, (2) will be much more easily amenable to modifications and to
cast the resulting problem as a second order cone program (SOCP).
2.2
Classification with Uncertainty
So far we assumed that the (xi , yi ) pairs are known with certainty. We now relax this to the
assumption that we only have a distribution over the xi , that is (Pi , yi ) at our disposition
(due to a sampling procedure, missing variables, etc.). Formally xi ? Pi . In this case it
makes sense to replace the constraints (2b) of the optimization problem (2) by
subject to Pr {yi (hw, xi i + b) ? 1 ? ?i } ? ?i , ?i ? 0, kwk ? W ? 1 ? i ? m
(4)
Here we replaced the linear classification constraint by a probabilistic one, which is required to hold with probability ?i ? (0, 1]. This means that by choosing a value of ?i close
to 1 we can find a conservative classifier which will classify even very infrequent (x i , yi )
pairs correctly. Hence ?i provides robustness of the estimate with respect to deviating xi .
It is clear that
Pmunless we impose further restrictions on Pi , it will be difficult to minimize the
objective i=1 ?i with the constraints (4) efficiently. In the following we will consider the
special cases of gaussian uncertainty for which a mathematical programming formulation
can be found.
3
Normal Distributions
For the purpose of this section we assume that Pi = N (?
xi , ?i ), i.e., xi is drawn from a
Gaussian distribution with mean x
?i and covariance ?i . We will not require that ?i has full
rank. This means that the uncertainty about xi may be limited to individual coordinates or
to a subspace of X . As we shall see, this problem can be posed as SOCP.
3.1
Robust Classification
Under the above assumptions, the probabilistic constraint (4) becomes
subject to Pr {yi (hw, xi i + b) ? 1 ? ?i } ? ?i where xi ? N (?
x i , ?i )
?i ? 0, kwk ? W for all 1 ? i ? m
The stochastic constraint can be restated as a deterministic optimization problem
zi ? z i
yi b + ? i ? 1 ? z i
? ?i
Pr
?
? zi
? zi
(5a)
(5b)
(6)
where zi := ?yi w> xi is a normal random variable with mean z?i and variance ?z2i :=
w> ?i w. Consequently (zi ? z?i )/?zi is a random variable with zero mean and unit variance
and we can compute the lhs of (6) by evaluating the cumulative distribution function for
normal distributions
Z u
s2
1
?(u) := ?
e? 2 ds.
2? ??
In summary, (6) is equivalent to the condition
yi b + ? i ? 1 ? z i
?
? ?i .
? zi
which can be solved (since ?(u) is monotonic and invertible), for the argument of ? and
obtain a condition on its argument
p
(7)
yi (w> x
?i + b) ? 1 ? ?i + ?i wT ?i w , ?i = ??1 (?i )
We now proceed to deriving a mathematical programming formulation.
3.2
Second Order Cone Programming Formulation
Depending on ?i we can distinguish between three different cases. First consider the case
where ?i = 0 or ?i = 0.5. This means that the second order cone part of the constraint (7)
reduces to the linear inequality of (2b). In other words, we recover the linear constraint of
a standard SVM.
Secondly consider the case ?i < 0 or ?i < 0.5. This means that the constraint (7) describes
a concave set, which turns the linear classification task into a hard optimization problem.
However, it is not very likely that anyone would like to impose such constraints which hold
only with low probability. After all, uncertain data requires the constraint to become more
restrictive in holding not only for a guaranteed point xi but rather for an entire set.
Lastly consider the case ?i > 0 or ?i > 0.5 second order cone constraint. In this case (7)
describes a convex set in in w, b, ?i . We obtain the following optimization problem:
m
X
minimize
?i
(8a)
w,b,?
i=1
1
subject to yi (w> xi + b) ? 1 ? ?i + ?i k?i2 wk and ?i ? 0 ? 1 ? i ? m
kwk ? W
(8b)
(8c)
These problems can be solved efficiently by publicly available codes: recent advances in
Interior point methods for convex nonlinear optimization [8] have made such problems
feasible. As a special case of convex nonlinear optimization SOCPs have gained much
attention in recent times. For a further discussion of efficient algorithms and applications
of SOCP see [6].
3.3
Worst Case Prediction
Note that if at optimality ?i > 0, the hyperplane intersects with the constraint set
B(xi , ?i , ?i ). Moreover, at a later stage we will need to predict the class label to asses
on which side of the hyperplane B lies. If the hyperplane intersects B we will end up with
different predictions for points in the different half spaces. In such a scenario a worst case
prediction, y can be
hw, xi i + b
y = sgn(z) sgn(h ? ?) where ? = ??1 (?), z = ?
and h = |z|.
(9)
w> ?w
Here sgn(z) gives us the sign of the point in the center of the ellipsoid and (h ? ?) is
the distance of z from the center. If the hyperplane intersects the ellipsoid, the worst case
prediction is then the prediction for all points which are in the opposite half space of the
center (xi ). Plugging ? = 0.5, i.e., ? = 0 into (9) yields the standard prediction (1).
In such a case h can serve as a measure of confidence as to how well the discriminating
hyperplane classifies the mean(xi ) correctly.
3.4
Set Constraints
The same problem as (8) can also be obtained by considering that the uncertainty in each
datapoint is characterized by an ellipsoid
2
B(xi , ?i , ?i ) = {x : (x ? xi )> ??1
(10)
i (x ? xi ) ? ?i }
in conjunction with the constraint
yi (hw, xi + b) ? 1 ? ?i for all x ? Si
(11)
where Si = B(xi , ?i , ?i ) As before ?i = ??1 (?i ) for ?i ? 0. In other words, we have
?i = 0 only when the hyperplane w> x + b = 0 does not intersect the ball B(xi , ?i , ?i ).
Note that this puts our optimization setting into the same category as the knowledge-based
SVM, and SDP for invariances as all three deal with the above type of constraint (11).
More to the point, in [5] Si = S(xi , ?) is a polynomial in ? which describes the set
of invariance transforms of xi (such as distortion or translation). [4] define Si to be a
polyhedral ?knowledge? set, specified by the intersection of linear constraints.
Such considerations suggest yet another optimization setting: instead of specifying a polyhedral set Si by constraints we can also specify it by its vertices. In particular, we may set
Si to be the convex hull of a set as in Si = co{xij for 1 ? j ? mi }. By the convexity of
the constraint set itself it follows that a necessary and sufficient condition for (11) to hold
is that the inequality holds for all x ? {xij for 1 ? j ? mi }. Consequently we can replace
(11) by yi (hw, xij i + b) ? 1 ? ?i Note that the index ranges over j rather than i. Such a
setting allows us to deal with uncertainties, e.g. regarding the range of variables, which are
just given by interval boundaries, etc. The table below summarizes the five cases:
Name
Plain SVM[3]
Knowledge Based SVM[4]
Invariances [5]
Normal Distribution
Convex Hull
Set Si
{xi }
Polyhedral set
trajectory of polynomial
B(xi , ?i , ?i )
co{xij ? 1 ? j ? mi }
Optimization Problem
Quadratic Program
Quadratic Program
Semidefinite Program
Second Order Cone Program
Quadratic Program
Clearly all the above constraints can be mixed and matched and it is likely that there will be
more additions to this table in the future. More central is the notion of stating the problems
via (11) as a starting point.
4
Missing Variables
In this section we discuss how to address the missing value problem. Key is how to obtain
estimates of the uncertainty in the missing variables. Since our optimization setting allows
for uncertainty in terms of a normal distribution we attempt to estimate the latter directly.
In other words, we assume that x|y is jointly normal with mean ?y and covariance ?y .
Hence we have the following two-stage procedure to deal with missing variables:
? Estimate ?y , ?y from incomplete data, e.g. by means of the EM algorithm.
? Use the conditionally normal estimates of xmissing |(xobserved , y) in the optimization problem. This can then be cast in terms of a SOCP as described in the previous
section.
Note that there is nothing to prevent us from using other estimates of uncertainty and use
e.g. the polyhedral constraints subsequently. However, for the sake of simplicity we focus
on normal distributions in this paper.
4.1
Estimation of the model parameters
We now detail the computation of the mean and covariance matrices for the datapoints
which have missing values. We just sketch the results, for a detailed derivation see e.g. [7].
Let x ? Rd , where xa ? Rda be the vector whose values are known, while xm ? Rd?da
be the vector consisting of missing variables. Assuming a jointly normal distribution in x
with mean ? and covariance ? it follows that
>
?1
xm |xa ? N (?m + ?am ??1
aa (xa ? ?a ), ?mm ? ?am ?aa ?am ).
Here we decomposed ?, ? according to (xa , xm ) into
?aa
? = (?a , ?m ) and ? =
?>
am
?am
?mm
.
(12)
(13)
Hence, knowing ?, ? we can estimate the missing variables and determine their degree of
uncertainty. One can show that [7] to obtain ?, ? the EM algorithm reads as follows:
1. Initialize ?, ?.
2. Estimate xm |xa for all observations using (12).
3. Recompute ?, ? using the completed data set and go to step 2.
4.2
Robust formulation for missing values
As stated above, we model the missing variables as Gaussian random variables, with its
mean and covariance given by the model described in the previous section. The standard
practice for imputation is to discard the covariance and treat the problem as a deterministic
problem, using the mean as surrogate. But using the robust formulation (8) one can as well
account for the covariance.
Let ma be number of datapoints for which all the values are available, while mm be the
number of datapoints containing missing values. Then the final optimization problem reads
as follows:
minimize
w,b,?
m
X
?i
(14)
i=1
subject to yi (hw, xi i + b) ? 1 ? ?i
1
2
yj (w> xj + b) ? 1 ? ?j + ??1 (?j )k?j wk
?i ? 0
kwk ? W
?1 ? i ? ma
?ma + 1 ? j ? ma + mm
?1 ? i ? ma + mm
The mean xj has two components; xaj has values available, while the imputed vector is
given by x
?mj , via (12). The matrix ?j has all entries zero except those involving the
missing values, given by Cj , computed via (12).
The formulation (14) is an optimization problem which involves minimizing a linear objective over linear and second order cone constraints. At optimality the values of w, b,
can be used to define a classifier (1). The resulting discriminator can be used to predict
the the class label of a test datapoint having missing variables by a process of conditional
imputation as follows.
Perform the imputation process assuming that the datapoint comes from class 1(class with
label y = 1). Specifically compute the mean and covariance, as outlined in section 4.1,
and denote them by ?1 and ?1 (see (13)) respectively. The training dataset of class 1 is to
be used in the computation of ?1 and ?1 . Using the estimated ?1 and ?1 compute h as
defined in (9), and denote it by h1 . Compute the label of ?1 with the rule (1), call it y1 .
Assuming that the test data comes from class 2 (with label y = ?1) redo the entire process
and denote the resulting mean, covariance, and h by ?2 , ?2 , h2 respectively. Denote by y2
the label of ?2 as predicted by (1). We decide that the observation belongs to class with
label y? as
y? = y2 if h1 < h2 and y? = y1 otherwise
(15)
The above rule chooses the prediction with higher h value or in other words the classifier
chooses the prediction about which it is more confident. Using y? , h1 , h2 as in (15), the
worst case prediction rule (9) can be modified as follows
y = y? sgn(h ? ?) where ? = ??1 (?) and h = max(h1 , h2 )
(16)
It is our hypothesis that the formulation (14) along with this decision rule is robust to
uncertainty in the data.
5
Experiments with the Robust formulation for missing values
Experiments were conducted to evaluate the proposed formulation (14), against the standard imputation strategy. The experiment methodology consisted of creating a dataset of
missing values from a completely specified dataset. The robust formulation (14) was used
to learn a classifier on the dataset having missing values. The resulting classifier was used
to give a worst case prediction (16), on the test data. Average number of disagreements was
taken as the error measure. In the following we describe the methodology in more detail.
Consider a fully specified dataset, D = {(xi , yi )|xi ? Rd , yi ? {?1}1 ? i ? N } having
N observations, each observation is a d dimensional vector (xi ) and labels yi . A certain
fraction(f ) of the observations were randomly chosen. For each of the chosen datapoints
dm (= 0.5d) entries were randomly deleted. This then creates a dataset having N datapoints
out of which Nm (= f N, 0 ? f ? 1) of them have missing values. This data is then
randomly partitioned into test set and training set in the ratio 1 : 9 respectively. We do this
exercise to generate 10 different datasets and all our results are averaged over them.
Assuming that the conditional probability distribution of the missing variables given the
? j ) can be estimated by the
other variables is a gaussian, the mean(xj ) and the covariance (C
methods described in (4.1). The robust optimization problem was then solved for different
values of ?. The parameter ?j (= ?) is set to the same value for all the Nm datapoints. For
each value of ? the worst case error is recorded.
Experimental results are reported for three public domain datasets downloaded from uci
repository ([2]). Pima(N = 768, d = 8), Heart ( N = 270, d = 13), and Ionosphere(N =
351, d = 34), were used for experiments.
Setting ? = 0.5, yields the generalized optimal hyperplane formulation, (2). The generalized optimal hyperplane will be referred to as the nominal classifier. The nominal classifier
considers the missing values are well approximated by the mean (xj ), and there is no uncertainty.
0.5
0.6
robust
nomwc
robustwc
robust
nomwc
robustwc
robust
nomwc
robustwc
0.55
0.5
0.4
0.45
0.4
0.4
0.35
0.3
0.3
0.25
0.5
0.6
0.7
0.8
0.9
1
0.2
0.5
0.41
0.6
0.7
0.8
0.9
1
0.4
0.42
0.2
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.45
robust
nomwc
robustwc
robust
nomwc
robustwc
robust
nomwc
robustwc
0.4
0.4
0.39
0.3
0.38
0.35
0.37
0.36
0.3
0.35
0.2
0.34
0.25
0.33
0.5
0.32
0.5
0.6
0.7
0.8
0.9
1
0.6
0.7
0.8
0.9
1
0.2
0.5
Figure 1: Performance of the robust programming solution for various datasets of the UCI
database. From left to right: Pima, Ionosphere, and Heart dataset. Top: small fraction
of data with missing variables (50%), Bottom: large number of observations with missing
variables (90%)
The experimental results are summarized by the graphs(1). The robust classifier almost
always outperforms the nominal classifier in the worst case sense (compare nomwc and
robustwc). Results are presented for low(f = 0.5), and high (f = 0.9) number of missing
values. The results show that for low number of missing values(f = 0.5) the robust classifier is marginally better than the nominal classifier the gain but for large f = 0.9 the gain
is significant. This confirms that the imputation strategy fails for high noise.
The standard misclassification error for the robust classifier, using the standard prediction
(1), is also shown in the graph with the legend robust. As expected the robust classifier
performance does not deteriorate in the standard misclassification sense as ? is increased.
In summary the results seems to suggest that for low noise level the nominal classifier
trained on imputed data performs as good as the robust formulation. But for high noise
level the robust formulation yields dividends in the worst case sense.
6
Conclusions
An SOCP formulation was proposed for classifying noisy observations and the resulting
formulation was applied to the missing data case. In the worst case sense the classifier
shows a better performance over the standard imputation strategy. Closely related to this
work is the Total Support Vector Classification(TSVC) formulation, presented in [1]. The
TSVC formulation tries to reconstruct the original maximal margin classifier in the presence of noisy data. Both TSVC formulation and the approach in this paper address the issue
of uncertainty in input data and it would be an important research direction to compare the
two approaches.
Acknowledgements CB was partly funded by ISRO-IISc Space technology cell (Grant
number IST/ECA/CB/152). National ICT Australia is funded through the Australian Government?s Backing Australia?s Ability initiative, in part through the Australian Research
Council. AS was supported by grants of the ARC. We thank Laurent ElGhaoui, Michael
Jordan, Gunnar R?atsch, and Frederik Schaffalitzky for helpful discussions and comments.
References
[1] J. Bi and T. Zhang. Support vector classification with input data uncertainty. In Advances in Neural Information Processing Systems. MIT Press, 2004.
[2] C. L. Blake and C. J. Merz. UCI repository of machine learning databases, 1998.
[3] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:273?297,
1995.
[4] G. Fung, O. L. Mangasarian, and Jude Shavlik. Knowledge-based support vector machine classifiers. In Advances in Neural Information Processing Systems. MIT Press,
2002.
[5] Thore Graepel and Ralf Herbrich. Invariant pattern recognition by semidefinite programming machines. In Advances in Neural Information Processing Systems 16, Cambridge, MA, 2003. MIT Press.
[6] M.S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret. Applications of second-order
cone programming. Linear Algebra and its Applications, 284(1?3):193?228, 1998.
[7] K. V. Mardia, J. T. Kent, and J. M. Bibby. Multivariate Analysis. Academic Press,
1979.
[8] Y. Nesterov and A. Nemirovskii. Interior Point Algorithms in Convex Programming.
Number 13 in Studies in Applied Mathematics. SIAM, Philadelphia, 1993.
[9] V. Vapnik. The Nature of Statistical Learning Theory. Springer, New York, 1995.
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1,840 | 2,671 | At the Edge of Chaos: Real-time Computations and
Self-Organized Criticality in Recurrent Neural Networks
Thomas Natschl?ager
Software Competence
Center Hagenberg
A-4232 Hagenberg, Austria
[email protected]
Nils Bertschinger
Max Planck Institute for
Mathematics in the Sciences
D-04103 Leipzig, Germany
[email protected]
Robert Legenstein
Institute for Theoretical
Computer Science, TU Graz
A-8010 Graz, Austria
[email protected]
Abstract
In this paper we analyze the relationship between the computational capabilities of randomly connected networks of threshold gates in the timeseries domain and their dynamical properties. In particular we propose
a complexity measure which we find to assume its highest values near
the edge of chaos, i.e. the transition from ordered to chaotic dynamics.
Furthermore we show that the proposed complexity measure predicts the
computational capabilities very well: only near the edge of chaos are
such networks able to perform complex computations on time series. Additionally a simple synaptic scaling rule for self-organized criticality is
presented and analyzed.
1
Introduction
It has been proposed that extensive computational capabilities are achieved by systems
whose dynamics is neither chaotic nor ordered but somewhere in between order and chaos.
This has led to the idea of ?computation at the edge of chaos?. Early evidence for this
hypothesis has been reported e.g. in [1]. The results of numerous computer simulations
carried out in these studies suggested that there is a sharp transition between ordered and
chaotic dynamics. Later on this was confirmed by Derrida and others [2]. They used
ideas from statistical physics to develop an accurate mean-field theory which allowed to
determine the critical parameters analytically. Because of the physical background, this
theory focused on the autonomous dynamics of the system, i.e. its relaxation from an
initial state (the input) to some terminal state (the output) without any external influences.
In contrast to such ?off-line? computations, we will focus in this article on time-series
computations, i.e. mappings, also called filters, from a time-varying input signal to a timevarying output signal. Such ?online? or real-time computations describe more adequately
the input to output relation of systems like animals or autonomous robots which must react
in real-time to a continuously changing stream of sensory input.
The purpose of this paper is to analyze how the computational capabilities of randomly
connected recurrent neural networks in the domain of real-time processing and the type
of dynamics induced by the underlying distribution of synaptic weights are related to each
other. In particular, we will show that for the types of neural networks considered in this paper (defined in Sec. 2) there also exists a transition from ordered to chaotic dynamics. This
phase transition is determined using an extension of the mean-field approach described in
[3] and [4] (Sec. 3). As the next step we propose a novel complexity measure (Sec. 4) which
input
neuron #
network activity
10
20
30
0
20
40
0
timesteps
20
40
0
20
timesteps
40
timesteps
0.2
0.8
?
0
0.6
?0.4
?0.6
ordered
0.1
ical
crit
?0.2
1
chaotic
0.4
mean activity
0.4
10
?2
Figure 1: Networks of randomly connected threshold gates can exhibit ordered, critical and
chaotic dynamics. In the upper row examples of the temporal evolution of the network state
xt are shown (black: xi,t = 1, white: xi,t = 0, input as indicated above) for three different
networks with parameters taken from the ordered, critical and chaotic regime, respectively.
Parameters: K = 5, N = 500, u
? = ?0.5, r = 0.3 and ? and ? 2 as indicated in the phase
plot below. The background of the phase plot shows the mean activity a? (see Sec. 3) of
the networks depending on the parameters ? and ? 2 .
can be calculated using the mean-field theory developed in Sec. 3 and serves as a predictor for the computational capability of a network in the time-series domain. Employing a
recently developed framework for analyzing real-time computations [5, 6] we investigate
in Sec. 5 the relationship between network dynamics and the computational capabilities in
the time-series domain. In Sec. 6 of this paper we propose and analyze a synaptic scaling
rule for self-organized criticality (SOC) for the types of networks considered here. In contrast to previous work [7], we do not only check that the proposed rule shows adaptation
towards critical dynamics, but also show that the computational capabilities of the network
are actually increased if the rule is applied.
Relation to previous work: In [5], the so-called liquid state machine (LSM) approach was
proposed and used do analyze the computational capabilities in the time-series domain of
randomly connected networks of biologically inspired network models (composed of leaky
integrate-and-fire neurons). We will use that approach to demonstrate that only near the
edge of chaos, complex computations can be performed (see Sec. 5). A similar analysis for
a restricted case (zero mean of synaptic weights) of the network model considered in this
paper can be found in [4].
2
The Network Model and its Dynamics
We consider input driven recurrent networks consisting of N threshold gates with states
xi ? {0, 1}. Each node i receives nonzero incoming weights wij from exactly K randomly
chosen nodes j. Each nonzero connection weight wij is randomly drawn from a Gaussian
distribution with mean ? and variance ? 2 . Furthermore, the network is driven by an external input signal u(?) which is injected into each node. Hence, in summary, the update of
PN
the network state xt = (x1,t , . . . , xN,t ) is given by xi,t = ?( j=1 wij ? xj,t?1 + ut?1 )
which is applied to all neurons in parallel and where ?(h) = 1 if h ? 0 and ?(h) = 0
otherwise. In the following we consider a randomly drawn binary input signal u (?) : at each
time step ut assumes the value u
? + 1 with probability r and the value u
? with probability
1 ? r. This network model is similar to the one we have considered in [4]. However it
differs in two important aspects: a) By using states xi ? {0, 1} we emphasis the asymmetric information encoding by spikes prevalent in biological neural systems and b) it is more
general in the sense that the Gaussian distribution from which the non-zero weights are
drawn is allowed to have an arbitrary mean ? ? R. This implies that the network activity
PN
at = N1 i=1 xi,t can vary considerably for different parameters (compare Fig. 1) and
enters all the calculations discussed in the rest of the paper.
The top row of Fig. 1 shows typical examples of ordered, critical and chaotic dynamics (see
the next section for a definition of order and chaos). The system parameters corresponding
to each type of dynamics are indicated in the lower panel (phase plot). We refer to the
(phase) transition from the ordered to the chaotic regime as the critical line (shown as the
solid line in the phase plot). Note that increasing the variance ? 2 of the weights consistently
leads to chaotic behavior.
3
The Critical Line: Order and Fading Memory versus Chaos
To define the chaotic and ordered phase of an input driven network we use an approach
which is similar to that proposed by Derrida and Pomeau [2] for autonomous systems:
consider two (initial) network states with a certain (normalized) Hamming distance. These
states are mapped to their corresponding successor states (using the same weight matrix)
with the same input in each case and the change in the Hamming distance is observed. If
small distances tend to grow this is a sign of chaos whereas if the distance tends to decrease
this is a signature of order.
Following closely the arguments in [4, 3] we developed a mean-field theory (see [8] for
all details) which allows to calculate the update dt+1 = f (dt , at , ut ) of the normalized
? t as well as the
Hamming distance dt = |{i : xi,t 6= x
?i,t }|/N between two states xt and x
update at+1 = A(at , ut ) of the network activity in one time step. Note that dt+1 depends
on the input ut (in contrast to [3]) and also on the activity at (in contrast to [4]). Hence the
two-dimensional map Fu (dt , at ) := (dt+1 , at+1 ) = (f (dt , at , ut ), A(at , ut )) describes
the time evolution of dt and at given the input times series u(?) .
Let us consider the steady state of the averaged Hamming distance f ? as well as the steady
state of the averaged network activity a? , i.e. (f ? , a? ) = limt?? hFut i.1 If f ? = 0 we
know that any state differences will eventually die out and the network is in the ordered
phase. If on the other hand a small difference is amplified and never dies out we have
f ? 6= 0 and the network is in the chaotic phase. Whether f ? = 0 or f ? 6= 0 can be decided
by looking at the slope of the function f (?, ?, ?) at its fixed point f ? = 0. Since at does not
depend on dt we calculate the averaged steady state activity a? and determine the slope ??
of the map rf (d, a, u
? + 1) + (1 ? r)f (d, a, u
?) at the point (d, a) = (0, a? ). Accordingly
we say that the network is in the ordered, critical or chaotic regime if ? ? < 1, ?? = 1 or
?? > 1 respectively. In [8] it is shown that the so called critical line ?? = 1 where the
phase transition from ordered to chaotic behavior occurs is given by
Pbf =
K?1
X
n=0
K ? 1 ?n
1
a (1 ? a? )K?1?n (rQ(1, n, u
? + 1) + (1 ? r)Q(1, n, u
?)) =
(1)
n
K
where Pbf denotes the probability (averaged over the inputs and the network activity) that
a node will change its output if a single out of its K input bits is flipped.2 Examples of
Fut denotes t-fold composition of the map Fu (?, ?) where in the k-th iteration the input uk is
applied and h?i denotes the average over all possible initial conditions and all input signals with a
given statistics determined by u
? and r.
2
The actual single bit-flip probability Q depends on the number n of inputs which are 1 and the
1
K=5
K = 10
0.4
0.1
0.3
0.2
?
0
0.06
?0.1
?0.2
0.04
?0.3
0.3
0.07
0.2
0.06
0.1
0.05
0
?
0.1
NM?Separation
0.08
?0.1
0.04
?0.2
0.03
NM?Separation
0.4
?0.3
0.02
?0.4
0.02
?0.4
?0.5
?0.5
?0.6
?0.6
0
0.1
1
?2
10
0.01
0
0.1
1
?2
10
Figure 2: N M -separation assumes high values on the critical line. The gray coded image
shows the N M -separation in dependence on ? and ? 2 for K denoted in the panels, r = 0.3,
u
? = ?0.5 and b = 0.1. The solid line marks the critical values for ? and ? 2 .
critical lines that were calculated from this formula (marked by the solid lines) can be seen
in Fig. 2 for K = 5 and K = 10.3
The ordered phase can also be described by using the notion of fading memory (see [5] and
the references therein). Intuitively speaking in a network with fading memory a state x t is
fully determined by a finite history ut?T , ut?T +1 , . . . , ut?1 , ut of the input u(?) . A slight
reformulation of this property (see [6] and the references therein) shows that it is equivalent
to the requirement that all state differences vanish, i.e. being in the ordered phase. Fading
memory plays an important role in the ?liquid state machine? framework [5] since together
with the separation property (see below) it would in principle allow an appropriate readout
function to deduce the recent input, or any function of it, from the network state. If on
the other hand the network does not have fading memory (i.e. is in the chaotic regime)
a given network state xt also contains ?spurious? information about the initial conditions
and hence it is hard or even impossible to deduce any features of the recent input.
4
NM-Separation as a Predictor for Computational Power
The already mentioned separation property [5] is especially important if a network is to
be useful for computations on input time-series: only if different input signals separate
the network state, i.e. different inputs result in different states, it is possible for a readout
function to respond differently. Hence it is necessary that any two different input time
series for which the readout function should produce different outputs drive the recurrent
network into two sufficiently different states.
The mean field theory we have developed (see [8]) can be extended to describe the
update dt+1 = s(dt , ...) of the Hamming distance that result from applying different inputs u(?) and u
?(?) with a mean distance of b := Pr {ut 6= u
?t }, i.e. the separation. In summary the three-dimensional map Su,?u (dt , at , a
?t ) := (dt+1 , at+1 , a
?t+1 ) =
(s(dt , at , a
? t , ut , u
?t ), A(at , ut ), A(?
at , u
?t )) fully describes the time evolution of the Hamming distance and the network activities. Again we consider the steady state of the averaged
t
Hamming distance s? and the network activities a? , a
?? , i.e. (s? , a? , a
?? ) = limt?? Su,?
u .
The overall separation for a given input statistics (determined by u
?, r, and b) is then given
by s? . However, this overall separation measure can not be directly related to the computa-
R ?u
external input u and is given by Q(1, n, u) = ?? ?(?, n?, n? 2 ) 1 ? ?(?u ? ?, ?, ? 2 ) d? +
R?
?(?, n?, n? 2 )?(?u ? ?, ?, ? 2 )d? where ?, ? denote the Gaussian density and cumulative den?u
sity respectively (see [8] for a detailed explanation).
3
For each value of ? = ?0.6 + k ? 0.01, k = 0 . . . 100 a search was conducted to find the value
for ? 2 such that ?? = 1. Numerical iterations of the function A were used to determine a ? .
A
B
3bit parity (K = 5)
C5bit random boolean functions
3bit parity (K = 10)
0.4
5
0.8
4
0.2
?0.2
2
?0.4
?0.6
0.01
3
2
1
2
?
10
100
0
0.4
0.2
1
1
0.1
0.6
MC (MI)
?
3
MC (MI)
0
0.01
0.1
1
2
?
10
100
0
mean MI
4
0.01
0.1
1
2
?
10
100
0
Figure 3: Real-time computation at the edge of chaos. A The gray coded image (an interpolation between the data points marked with open diamonds) shows the performance
of trained networks in dependence of the parameters ? and ? 2 for the
P delayed 3-bit parity task. Performance is measured as the memory capacity M C = ? I(v, y (? ) ) where
I(v, y (? ) ) is the mutual information between the classifier output v(?) and the target func(? )
tion yt = PARITY(ut?? , ut?? ?1 , ut?? ?2 ) measured on a test set. B Same as panel A
but for K = 10. C Same as panel A but for an average over 50 randomly drawn Boolean
functions f of the last 5 time steps, i.e. yt = f (ut , ut?1 , ..., ut?4 ).
tional power since chaotic networks separate even minor differences in the input to a very
high degree. The part of this separation that is caused by the input distance b and not by the
distance of some initial state is therefore given by s? ? f ? because f ? measures the state
distance that is caused by differences in the initial states and remains even after long runs
with the same inputs (see Sec. 3). Note that f ? is always zero in the ordered phase and
non-zero in the chaotic phase.
Since we want the complexity measure, which we will call N M -separation, to be a
predictor for computational power we correct s? ? f ? by a term which accounts for
the separation due to an all-dominant input drive. A suitable measure for this ?immediate separation? i? is the average increase in the Hamming distance if the system is
run for a long time (t ? ?) with equal inputs u(?) = u
?(?) and then a single step
with an input pair (v, v?) with an average difference
of b = Pr {v, 6= v? } is applied:
P1
t
i? = limt?? v,?v=0 rv (1?r)1?v b|v??v| (1?b)1?|v??v| s(?, ?, ?, v, v?) ? Su,u
?f ? . Hence
a measure of the network mediated separation N Msep due to input differences is given by
N Msep = s? ? f ? ? i?
(2)
In Fig. 2 the N M -separation resulting from an input difference of b = 0.1 is shown in
dependence of the network parameters ? and ? 2 .4 Note that the N M -separation peaks
very close to the critical line. Because of the computational importance of the separation
property this also suggests that the computational capabilities of the networks will peak at
the onset of chaos, which is confirmed in the next section.
5
Real-Time Computations at the Edge of Chaos
To access the computational power of a network we make use of the so called ?liquid state
machine? framework which was proposed by Maass et.al. [5] and independently by Jaeger
[6]. They put forward the idea that any complex time-series computation can be implemented by composing a system which consists of two conceptually different parts: a) a
4
For each value of ? = ?0.6 + k ? 0.05, k = 0 . . . 20, 10 values for ? 2 where chosen near
the critical line and 10 other values where equally spaced (on a logarithmic scale) over the interval
[0.02,50]. For each such pair (?, ? 2 ) extensive numerical iterations of the map S where performed
to obtain accurate estimates of s? , f ? and i? . Hopefully these numerical estimates can be replaced
by analytic results in the future.
properly chosen general-purpose recurrent network with ?rich? dynamics and b) a readout function that is trained to map the network state to the desired outputs (see [5, 6, 4]
for more details). This approach is potentially successful if the general-purpose network
encodes the relevant features of the input signal in the network state in such a way that
the readout function can easily extract it. We will show that near the critical line the networks considered in this paper encode the input in such a way that a simple linear classifier
C(xt ) = ?(w ? xt + w0 ) suffices to implement a broad range of complex nonlinear filters. Note that in order to train the network for a given task only the parameters w ? R N ,
w0 ? R of the linear classifier are adjusted such that the actual network output vt = C(xt )
is as close as possible to the target values yt .
To access the computational power in a principled way networks with different parameters
were tested on a delayed 3-bit parity task for increasing delays and on randomly drawn
Boolean functions of the last 5 input bits. Note that these tasks are quite complex for
the networks considered here since most of them are not linear separable (i.e. the parity
function) and require memory. Hence to achieve good performance it is necessary that a
state xt contains information about several input bits ut0 , t0 < t in a nonlinear transformed
form such that a linear classifier C is sufficient to perform the nonlinear computations.
The results are summarized in Fig. 3 where the performance (measured in terms of mutual
information) on a test set between the network output and the target signal is shown for
various parameter settings (for details see [4]). The highest performance is clearly achieved
for parameter values close to the critical line where the phase transition occurs. This has
been noted before [1]. In contrast to these previous results the networks used here are
not optimized for any specific task but their computational capabilities are assessed by
evaluating them for many different tasks. Therefore a network that is specifically designed
for a single task will not show a good performance in this setup. These considerations
suggest the following hypotheses regarding the computational function of generic recurrent
neural circuits: to serve as a general-purpose temporal integrator, and simultaneously as a
kernel (i.e., nonlinear projection into a higher dimensional space) to facilitate subsequent
(linear) readout of information whenever it is needed.
6
Self-Organized Criticality via Synaptic Scaling
Since the computational capabilities of a network depend crucially on having almost critical
dynamics an adaptive system should be able to adjust its dynamics accordingly.
Equ. (1) states that critical dynamics are achieved if the probability Pbf that a single bitflip in the input shows up in the output should on average (over the external and internal
1
input statistics given by u
?, r and a? respectively) be equal to K
. To allow for a rule that
can adjust the weights of each node a local estimate of Pbf must be available. This can
be accomplished by estimating Pbf from the margin of each node, i.e. the distance of the
internal activation from the firing threshold. Intuitively a node with an activation that is
much higher or lower than its firing threshold is rather unlikely to change its output if a
i
single bit in its input is flipped. Formally Pbf
of node i is given by the average (over the
internal and external input statistics) of the following quantity:
1
K
N
X
? (wij (1 ? 2xj,t?1 )(1 ? 2xi,t ) ? mi,t )
(3)
j=1,wij 6=0
P
N
where mi,t = j=1 wij xj,t?1 + ut?1 denotes the margin of node i (see [8] for details).
Each node now applies synaptic scaling to adjust itself towards the critical line. Accordingly we arrive at the following SOC-rule:
1
esti
1
if Pbf
(t) > K
1+? ? wij
wij (t + 1) =
(4)
esti
1
(1 + ?) ? wij (t) if Pbf (t) < K
A
neuron #
50
100
150
200
100
200
300
400
500
600
700
timesteps
B
C
1
1
K*P
K*P
bf
1.5
bf
1.5
0.5
0
0
0.5
100
200
300
400
timesteps
500
600
700
0
0
100
200
300
400
500
600
700
timesteps
Figure 4: Self-organized criticality. A Time evolution of the network state x t starting
in a chaotic regime while the SOC-rule (4) is active (black: xi,t = 1, white: xi,t = 0).
Parameters: N = 500, K = 5, u
? = ?0.5, r = 0.3, ? = 0 and initial ? 2 = 100. B
Estimated Pbf . The dotted line shows how the node averaged estimate of Pbf evolves over
time for the network shown in A. The running average of this estimate (thick black line) as
used by the SOC-rule clearly shows that Pbf approaches its critical value (dashed line). C
Same as B but for K = 10 and initial ? 2 = 0.01 in the ordered regime.
esti
where 0 < ? 1 is the learning rate and Pbf
(t) is a running average of the formula
i
in Equ. (3) to estimate Pbf . Applying this rule in parallel to all nodes of the network is
then able to adjust the network dynamics towards criticality as shown in Fig. 4 5 . The upper
row shows the time evolution of the network states xt while the SOC-rule (4) is running.
It is clearly visible how the network dynamics changes from chaotic (the initial network
had the parameters K = 5, ? = 0 and ? 2 = 100) to critical dynamics that respect the
input signal. The lower row of Fig. 4 shows how the averaged estimated bit-flip probability
PN
esti
1
i=1 Pbf (t) approaches its critical value for the case of the above network and one
N
that started in the ordered regime (K = 10, ? = 0, ? 2 = 0.01).
Since critical dynamics are better suited for information processing (see Fig. 3) it is expected that the performance on the 3-bit parity task improves due to SOC. This is confirmed in Fig. 5 which shows how the memory capacity M C (defined in Fig. 3) grows for
networks that were initialized in the chaotic and ordered regime respectively. Note that the
performance reached by these networks using the SOC-rule (4) is as high as for networks
where the critical value for ? 2 is chosen apriori and stays at this level. This shows that
rule (4) is stable in the sense that it keeps the dynamics critical and does not destroy the
computational capabilities.
7
Discussion
We developed a mean-field theory for input-driven networks which allows to determine the
position of the transition line between ordered and chaotic dynamics with respect to the
5
Here a learning rate of ? = 0.01 and an exponentially weighted running average with a time
constant of 15 time steps were used.
A
B
K = 5, start ?2 = 100 (chaotic)
K = 10, start ?2 = 0.01 (ordered)
5
4
MC [bits]
MC [bits]
4
3
2
3
2
1
1
0
0
0
500
1000
SOC steps
1500
2000
0
500
1000
1500
2000
SOC steps
Figure 5: Time evolution of the performance with activated SOC-rule. A The plot shows
the memory capacity M C (see Fig. 3) on the 3-bit parity task averaged over 25 networks (?
standard deviation as error-bars) evaluated at the indicated time steps. At each evaluation
time step the network weights were fixed and the M C was measured as in Fig. 3 by training
the corresponding readouts from scratch. The networks were initialized in the chaotic
regime. B Same as in A but for K = 10 and networks initialized in the ordered regime.
parameters controlling the network connectivity and input statistics. Based on this theory
we proposed a complexity measure (called N M -separation) which assumes its highest
values at the critical line and shows a clear correlation with the computational power for
real-time time-series processing. These results provide further evidence for the idea of
?computation at the edge of chaos? [1] and support the hypothesis that dynamics near
the critical line are expected to be a general property of input driven dynamical systems
which support complex real-time computations. Therefore our analysis and the proposed
complexity measure provide a new approach towards discovering dynamical principles that
enable biological systems to do sophisticated information processing.
Furthermore we have shown that a local rule for synaptic scaling is able to adjust the
weights of a network towards critical dynamics. Additionally networks adjusting themselves by this rule have been found to exhibit enhanced computational capabilities. Thereby
systems can combine task-specific optimization provided by (supervised) learning rules
with self-organization of its dynamics towards criticality. This provides an explanation
how specific information can be processed while still being able to react to incoming signals in a flexible way.
Acknowledgement This work was supported in part by the PASCAL project #IST-2002-506778
of the European Community.
References
[1] C. G. Langton. Computation at the edge of chaos. Physica D, 42, 1990.
[2] B. Derrida and Y. Pomeau. Random networks of automata: A simple annealed approximation.
Europhys. Lett., 1:45?52, 1986.
[3] B. Derrida. Dynamical phase transition in non-symmetric spin glasses. J. Phys. A: Math. Gen.,
20:721?725, 1987.
[4] N. Bertschinger and T. Natschl?ager. Real-time computation at the edge of chaos in recurrent
neural networks. Neural Computation, 16(7):1413?1436, 2004.
[5] W. Maass, T. Natschl?ager, and H. Markram. Real-time computing without stable states: A new
framework for neural computation based on perturbations. Neural Computation, 14(11), 2002.
[6] H. Jaeger and H. Haas. Harnessing nonlinearity: Predicting chaotic systems and saving energy
in wireless communication. Science, 304(5667):78?80, 2004.
[7] S. Bornholdt and T. R?ohl. Self-organized critical neural networks. Physical Review E, 67:066118,
2003.
[8] N. Bertschinger and T. Natschl?ager.
Supplementary information to the meanfield theory for randomly connected recurrent networks of threshold gates, 2004.
http://www.igi.tugraz.at/tnatschl/edge-of-chaos/mean-field-supplement.pdf.
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1,841 | 2,672 | Exponential Family Harmoniums
with an Application to Information Retrieval
Max Welling & Michal Rosen-Zvi
Information and Computer Science
University of California
Irvine CA 92697-3425 USA
[email protected]
Geoffrey Hinton
Department of Computer Science
University of Toronto
Toronto, 290G M5S 3G4, Canada
[email protected]
Abstract
Directed graphical models with one layer of observed random variables
and one or more layers of hidden random variables have been the dominant modelling paradigm in many research fields. Although this approach has met with considerable success, the causal semantics of these
models can make it difficult to infer the posterior distribution over the
hidden variables. In this paper we propose an alternative two-layer model
based on exponential family distributions and the semantics of undirected models. Inference in these ?exponential family harmoniums? is
fast while learning is performed by minimizing contrastive divergence.
A member of this family is then studied as an alternative probabilistic
model for latent semantic indexing. In experiments it is shown that they
perform well on document retrieval tasks and provide an elegant solution
to searching with keywords.
1 Introduction
Graphical models have become the basic framework for generative approaches to probabilistic modelling. In particular models with latent variables have proven to be a powerful
way to capture hidden structure in the data. In this paper we study the important subclass
of models with one layer of observed units and one layer of hidden units.
Two-layer models can be subdivided into various categories depending on a number of
characteristics. An important property in that respect is given by the semantics of the
graphical model: either directed (Bayes net) or undirected (Markov random field). Most
two-layer models fall in the first category or are approximations derived from it: mixtures
of Gaussians (MoG), probabilistic PCA (pPCA), factor analysis (FA), independent components analysis (ICA), sigmoid belief networks (SBN), latent trait models, latent Dirichlet
allocation (LDA, otherwise known as multinomial PCA, or mPCA) [1], exponential family PCA (ePCA), probabilistic latent semantic indexing (pLSI) [6], non-negative matrix
factorization (NMF), and more recently the multiple multiplicative factor model (MMF)
[8].
Directed models enjoy important advantages such as easy (ancestral) sampling and easy
handling of unobserved attributes under certain conditions. Moreover, the semantics of
directed models dictates marginal independence of the latent variables, which is a suitable
modelling assumption for many datasets. However, it should also be noted that directed
models come with an important disadvantage: inference of the posterior distribution of the
latent variables given the observations (which is, for instance, needed within the context of
the EM algorithm) is typically intractable resulting in approximate or slow iterative procedures. For important applications, such as latent semantic indexing (LSI), this drawback
may have serious consequences since we would like to swiftly search for documents that
are similar in the latent topic space.
A type of two-layer model that has not enjoyed much attention is the undirected analogue
of the above described family of models. It was first introduced in [10] where it was named
?harmonium?. Later papers have studied the harmonium under various names (the ?combination machine? in [4] and the ?restricted Boltzmann machine? in [5]) and turned it into
a practical method by introducing efficient learning algorithms. Harmoniums have only
been considered in the context of discrete binary variables (in both hidden and observed
layers), and more recently in the Gaussian case [7]. The first contribution of this paper is to
extend harmoniums into the exponential family which will make them much more widely
applicable.
Harmoniums also enjoy a number of important advantages which are rather orthogonal to
the properties of directed models. Firstly, their product structure has the ability to produce
distributions with very sharp boundaries. Unlike mixture models, adding a new expert may
decrease or increase the variance of the distribution, which may be a major advantage in
high dimensions. Secondly, unlike directed models, inference in these models is very fast,
due to the fact that the latent variables are conditionally independent given the observations.
Thirdly, the latent variables of harmoniums produce distributed representations of the input.
This is much more efficient than the ?grandmother-cell? representation associated with
mixture models where each observation is generated by a single latent variable. Their most
important disadvantage is the presence of a global normalization factor which complicates
both the evaluation of probabilities of input vectors1 and learning free parameters from
examples. The second objective of this paper is to show that the introduction of contrastive
divergence has greatly improved the efficiency of learning and paved the way for large scale
applications.
Whether a directed two-layer model or a harmonium is more appropriate for a particular application is an interesting question that will depend on many factors such as prior
(conditional) independence assumptions and/or computational issues such as efficiency of
inference. To expose the fact that harmoniums can be viable alternatives to directed models we introduce an entirely new probabilistic extension of latent semantic analysis (LSI)
[3] and show its usefulness in various applications. We do not want to claim superiority
of harmoniums over their directed cousins, but rather that harmoniums enjoy rather different advantages that deserve more attention and that may one day be combined with the
advantages of directed models.
2 Extending Harmoniums into the Exponential Family
Let xi , i = 1...Mx be the set of observed random variables and hj , j = 1...Mh be the set
of hidden (latent) variables. Both x and h can take values in either the continuous or the
discrete domain. In the latter case, each variable has states a = 1...D.
To construct an exponential family harmonium (EFH) we first choose Mx independent
distributions pi (xi ) for the observed variables and Mh independent distributions pj (hj )
1
However, it is easy to compute these probabilities up to a constant so it is possible to compare
probabilities of data-points.
for the hidden variables from the exponential family and combine them multiplicatively,
Mx
Y
p({xi }) =
ri (xi ) exp [
X
Mh
Y
p({hj }) =
?ia fia (xi ) ? Ai ({?ia })]
(1)
a
i=1
sj (hj ) exp [
X
j=1
?jb gjb (hj ) ? Bj ({?jb })]
(2)
b
where {fia (xi ), gjb (hj )} are the sufficient statistics for the models (otherwise known as
features), {?ia , ?jb } the canonical parameters of the models and {Ai , Bj } the log-partition
functions (or log-normalization factors). In the following we will consider log(ri (xi )) and
log(sj (hj )) as additional features multiplied by a constant.
Next, we couple the random variables in the log-domain by the introduction of a quadratic
interaction term,
X
X
X jb
p({xi , hj }) ? exp [
?ia fia (xi ) +
?jb gjb (hj ) +
Wia fia (xi )gjb (hj )]
(3)
ia
jb
ijab
Note that we did not write the log-partition function for this joint model in order to indicate our inability to compute it in general. For some combinations of exponential family
jb
distributions it may be necessary to restrict the domain of Wia
in order to maintain norjb
jb
malizability of the joint probability distribution (e.g. Wia ? 0 or Wia
? 0). Although
we could also have mutually coupled the observed variables (and/or the hidden variables)
using similar interaction terms we refrain from doing so in order to keep the learning and
inference procedures efficient. Consequently, by this construction the conditional probability distributions are a product of independent distributions in the exponential family with
shifted parameters,
p({xi }|{hj }) =
p({hj }|{xi }) =
Mx
Y
exp [
X
??ia fia (xi ) ? Ai ({??ia })]
i=1
a
Mh
Y
X
exp [
j=1
??ia = ?ia +
X
jb
Wia
gjb (hj ) (4)
jb
? jb gjb (hj ) ? Bj ({?
? jb })] ?
? jb = ?jb +
?
X
jb
Wia
fia (xi )(5)
ia
b
P
P
Finally, using the following identity, y exp a ?a fa (y) = exp A({?a }) we can also
compute the marginal distributions of the observed and latent variables,
X
X
X jb
p({xi }) ? exp [
?ia fia (xi ) +
Bj ({?jb +
Wia fia (xi )})]
(6)
ia
p({hj }) ? exp [
X
jb
j
?jb gjb (hj ) +
X
i
ia
Ai ({?ia +
X
jb
Wia
gjb (hj )})]
(7)
jb
Note that 1) we can only compute the marginal distributions up to the normalization constant and 2) in accordance with the semantics of undirected models, there is no marginal
independence between the variables (but rather conditional independence).
2.1 Training EF-Harmoniums using Contrastive Divergence
Let p?({xi }) denote the data distribution (or the empirical distribution in case we observe a
finite dataset), and p the model distribution. Under the maximum likelihood objective the
learning rules for the EFH are conceptually simple2 ,
? jb )ip? ? hB 0 (?
? jb )ip
??ia ? hfia (xi )ip? ? hfia (xi )ip
??jb ? hB 0 (?
(8)
jb
2
jb
These learning rules are derived by taking derivatives of the log-likelihood objective using Eqn.6.
0 ?
0 ?
?Wijab ? hfia (xi )Bjb
(?jb )ip? ? hfia (xi )Bjb
(?jb )ip
(9)
0
? jb )/? ?
? jb with ?
? jb defined in Eqn.5. One should note
where we have defined Bjb
= ?Bj (?
that these learning rules are changing the parameters in an attempt to match the expected
sufficient statistics of the data distribution and the model distribution (while maximizing
entropy). Their simplicity is somewhat deceptive, however, since the averages h?ip are
intractable to compute analytically and Markov chain sampling or mean field calculations
are typically wheeled out to approximate them. Both have difficulties: mean field can only
represent one mode of the distribution and MCMC schemes are slow and suffer from high
variance in their estimates.
In the case of binary harmoniums (restricted BMs) it was shown in [5] that contrastive divergence has the potential to greatly improve on the efficiency and reduce the variance of
the estimates needed in the learning rules. The idea is that instead of running the Gibbs
sampler to its equilibrium distribution we initialize Gibbs samplers on each data-vector
and run them for only one (or a few) steps in parallel. Averages h?ip in the learning rules
Eqns.8,9 are now replaced by averages h?ipCD where pCD is the distribution of samples
that resulted from the truncated Gibbs chains. This idea is readily generalized to EFHs.
Due to space limitations we refer to [5] for more details on contrastive divergence learning3 . Deterministic learning rules can also be derived straightforwardly by generalizing the
results described in [12] to the exponential family.
3 A Harmonium Model for Latent Semantic Indexing
To illustrate the new possibilities that have opened up by extending harmoniums to the
exponential family we will next describe a novel model for latent semantic indexing (LSI).
This will represent the undirected counterpart of pLSI [6] and LDA [1].
One of the major drawbacks of LSI is that inherently discrete data (word counts) are being
modelled with variables in the continuous domain. The power of LSI on the other hand is
that it provides an efficient mapping of the input data into a lower dimensional (continuous)
latent space that has the effect of de-noising the input data and inferring semantic relationships among words. To stay faithful to this idea and to construct a probabilistic model on
the correct (discrete) domain we propose the following EFH with continuous latent topic
variables, hj , and discrete word-count variables, xia ,
p({hj }|{xia }) =
Mh
Y
j=1
p({xia }|{hj }) =
Mx
Y
i=1
X j
Nhj [
Wia xia , 1]
(10)
ia
S{xia } [?ia +
X
j
hj Wia
]
(11)
j
P
Note that {xia } represent indicator variables satisfying a xia = 1 ?i, where xia = 1
means that word ?i? in the vocabulary was observed ?a? times. Nh [?, ?] denotes a normal
PD
distribution with mean ? and std.? and S{xa } [?a ] ? exp ( a=1 ?a xa ) is the softmax
function defining a probability distribution over x. Using Eqn.6 we can easily deduce the
marginal distribution of the input variables,
X
1X X j
(
Wia xia )2 ]
(12)
p({xia }) ? exp [
?ia xia +
2
j
ia
ia
3
Non-believers in contrastive divergence are invited to simply run the the Gibbs sampler to equilibrium before they do an update of the parameters. They will find that due to the special bipartite
structure of EFHs learning is still more efficient than for general Boltzmann machines.
j
We observe that the role of the components Wia
is that of templates or prototypes: input
P
j
vectors xia with large inner products ia Wia xia ?j will have high probability under this
model. Just like pLSI and LDA can be considered as natural generalizations of factor
analysis (which underlies LSI) into the class of directed models on the discrete domain, the
above model can be considered as the natural generalization of factor analysis into class
of undirected models on the discrete domain. This idea is supported by the result that the
same model with Gaussian units in both hidden and observed layers is in fact equivalent to
factor analysis [7].
3.1 Identifiability
From the form of the marginal distribution Eqn.12 we can derive a number of transformations of the parameters that will leave the distribution invariant. First we note that the comP
j
j
k
ponents Wia
can be rotated and mirrored arbitrarily in latent space4 : Wia
? k U jk Wia
T
with U U = I. Secondly, we note that observed variables xia satisfy a constraint,
P
j
a xia = 1 ?i. This results in a combined shift invariance for the components Wia and
the offsets ?ia . Taken together, this results in the following set of transformations,
X
XX j
j
j
k
Wia
?
U jk (Wia
+ Vik )
?ia ? (?ia + ?i ) ?
(
Vl )(Wia
)
(13)
j
k
l
where U T U = I. Although these transformations leave the marginal distribution over the
observable variables invariant, they do change the latent representation and as such may
have an impact on retrieval performance (if we use a fixed similarity measure between
topic representations of documents). To fix the spurious degrees of freedom we have choP
j n
xia . First,
sen to impose conditions on the representations in latent space: hnj = ia Wia
we center the latent representations which has the effect of minimizing the ?activity? of
the latent variables and moving as much log-probability as possible to the constant component ?ia . Next we align the axes in latent space with the eigen-directions of the latent
covariance matrix. This has the effect of approximately decorrelating the marginal latent
activities. This follows because the marginal distribution in latent space can be approxP
P Q
j n
xia , 1]/N where we have used Eqn.10 and
imated by: p({hj }) ? n j Nhj [ ia Wia
replaced p({xia }) by its empirical distribution. Denoting by ? and ? = U T ?U the sample
mean and sample covariance of {hnj }, it is not hard to show that the following transformation will have the desired effect5 :
?
?
X
X
1 k
j
j
jk
k
?
?ia ? ?ia +
?j Wia
(14)
Wia ?
U
Wia ?
Mx
j
k
One could go one step further than the de-correlation process described above by introducing covariances ? in the conditional Gaussian distribution of the latent variables Eqn.10.
This would not result in a more general model because the effect of this on the marginal
P
j
k
KK T = ?.
distribution over the observed variables is given by: Wia
? k K jk Wia
However, the extra freedom can be used to define axes in latent space for which the projected data become approximately independent and have the same scale in all directions.
4
Technically we call this the Euclidean group of transformations.
P
Some spurious degrees of freedom remain since shifts ?i and shifts Vij that satisfy i Vij = 0
will not affect the projection into latent space. One could decide to fix the remaining degrees of
freedom by for
(subject
to
P in Lj2 norm
Pexample requiring that components are asjsmall asj possible
P
j
1
1
W
+
the constraint i Vij = 0), leading to the further shifts, Wia
? Wia ? D
W
ia
ia
a
ia
DMx
P
1
and ?ia ? ?ia ? D a ?ia .
5
Precision vs. Recall for Newsgroups Dataset
0.8
Precision vs. Recall for Newsgroups Dataset
Precision vs. Recall for Newgroups Dataset
0.8
0.65
0.6
EFH
0.7
EFH+MF
LSA
EFH
0.7
EFH+MF
0.55
LSA
0.6
0.6
EFH
0.4
TFIDF
Precision
Precision
Precision
0.5
0.45
0.5
0.35
TFIDF
0.2
0.1 ?5
10
?4
10
0.3
0.25
RANDOM
0.2
?3
10
?2
10
Recall
(a)
?1
10
0
10
0.15 ?5
10
TFIDF
0.4
0.3
0.3
LSA
0.5
0.4
0.2
RANDOM
?4
10
?3
10
Recall
(b)
?2
10
?1
10
0
10
0.1 ?5
10
RANDOM
?4
10
?3
10
?2
10
?1
10
0
10
Recall
(c)
Figure 1: Precision-recall curves when the query was (a) entire documents, (b) 1 keyword, (c) 2
keywords for the EFH with and without 10 MF iterations, LSI , TFIDF weighted words and random
guessing. PR curves with more keywords looked very similar to (c). A marker at position k (counted
from the left along a curve) indicates that 2k?1 documents were retrieved.
4 Experiments
Newsgroups:
We have used the reduced version of the ?20newsgroups? dataset prepared for MATLAB
by Roweis6 . Documents are presented as 100 dimensional binary occurrence vectors and
tagged as a member of 1 out of 4 domains. Documents contain approximately 4% of the
words, averaged across the 16242 postings.
An EFH model with 10 latent variables was trained on 12000 training cases using stochastic gradient descent on mini-batches of 1000 randomly chosen documents (training
time approximately 1 hour on a 2GHz PC). A momentum term was added to speed up
convergence. To test the quality of the trained model we mapped the remaining 4242
P
j
j
xia and where {Wia
, ?ia } were
query documents into latent space using hj = ia Wia
?gauged? as in Eqns.14. Precision-recall curves were computed by comparing training
and query documents using the usual ?cosine coefficient? (cosine of the angle between
documents) and reporting success when the retrieved document was in the same domain
as the query (results averaged over all queries). In figure 1a we compare the results
with LSI (also 10 dimensions) [3] where we preprocessed the data in the standard way
(x ? log(1 + x) and entropy weighting of the words) and to similarity in word space
using TF-IDF weighting of the words. In figure 1b,c we show PR curves when only 1 or
2 keywords were provided corresponding to randomly observed words in the query document. The EFH model allows a principled way to deal with unobserved entries by inferring
them using the model (in all other methods we insert 0 for the unobserved entries which
corresponds to ignoring
them). We have used a few iterations of mean field to achieve
P P
k
k
Wjb
+ ?jb )?
xjb ]/?i where ?i is a normalization constant
that: x
?ia ? exp [ jb ( k Wia
PD
?ia = 1 ?i. We note that this is
and where x
?ia represent probabilities: x
?ia ? [0, 1],
a=1 x
still highly efficient and achieves a significant improvement in performance. In all cases we
find that without any preprocessing or weighting EFH still outperforms the other methods
except when large numbers of documents were retrieved.
In the next experiment we compared performance of EFH, LSI and LDA by training models on a random subset of 15430 documents with 5 and 10 latent dimensions (this was
found to be close to optimal for LDA). The EFH and LSI models were trained as in the
previous experiment while the training and testing details7 for LDA can be found in [9].
For the remaining test documents we clamped a varying number of observed words and
6
7
http://www.cs.toronto.edu/?roweis/data.html
The approximate inference procedure was implemented using Gibbs sampling.
Latent Representations of Newsgroups Data
8
6
0.25
EFH+MF 10D
EFH+MF 5D
4
?4
0.2
LSI 5D
0.15
LDA 5D
LDA 10D
?2
0
2
4
0.1
LSI 10D
6
8
0.05
10
0
0
2
4
6
number of keywords
(a)
8
10
12
6
4
2
0
?2
?4
?6
(b)
Retrieval Performance for NIPS Dataset
fraction documents also retrieved by TF?IDF
fraction observed words correctly predicted
Document Reconstruction for Newsgroup Dataset
0.3
0.52
0.5
0.48
0.46
0.44
0.42
0.4
0.38
0
20
40
60
80
number of retrieved documents
100
(c)
Figure 2: (a) Fraction of observed words that was correctly observed by EFH, LSI and LDA using 5
and 10 latent variables when we vary the number of keywords (observed words that were ?clamped?),
(b) latent 3-D representations of newsgroups data, (c) Fraction of documents retrieved by EFH on the
NIPS dataset which was also retrieved by the TF-IDF method.
asked the models to predict the remaining observed words in the documents by computing the probabilities for all words in the vocabulary to be present and ranking them (see
previous paragraph for details). By comparing the list of the R remaining observed words
in the document with the top-R ranked inferred words we computed the fraction of correctly predicted words. The results are shown in figure 2a as a function of the number of
clamped words. To provide anecdotal evidence that EFH can infer semantic relationships
we clamped the words ?drive? ?driver? and ?car? which resulted in: ?car? ?drive? ?engine?
?dealer? ?honda? ?bmw? ?driver? ?oil? as the most probable words in the documents. Also,
clamping ?pc? ?driver? and ?program? resulted in: ?windows? ?card? ?dos? ?graphics? ?software? ?pc? ?program? ?files?.
NIPS Conference Papers:
Next we trained a model with 5 latent dimensions on the NIPS dataset8 which has a large
vocabulary size (13649 words) and contains 1740 documents of which 1557 were used
for training and 183 for testing. Count values were redistributed in 12 bins. The array
W contains therefore 5 ? 13649 ? 12 = 818940 parameters. Training was completed in the
order of a few days. Due to the lack of document labels it is hard to assess the quality of the
trained model. We choose to compare performance on document retrieval with the ?golden
standard?: cosine similarity in TF-IDF weighted word space. In figure 2c we depict the
fraction of documents retrieved by EFH that was also retrieved by TF-IDF as we vary the
number of retrieved documents. This correlation is indeed very high but note that EFH
computes similarity in a 5-D space while TF-IDF computes similarity in a 13649-D space.
5 Discussion
The main point of this paper was to show that there is a flexible family of 2-layer probabilistic models that represents a viable alternative to 2-layer causal (directed) models. These
models enjoy very different properties and can be trained efficiently using contrastive divergence. As an example we have studied an EFH alternative for latent semantic indexing
where we have found that the EFH has a number of favorable properties: fast inference allowing fast document retrieval and a principled approach to retrieval with keywords. These
were preliminary investigations and it is likely that domain specific adjustments such as a
more intelligent choice of features or parameterization could further improve performance.
Previous examples of EFH include the original harmonium [10], Gaussian variants thereof
[7], and the PoT model [13] which couples a gamma distribution with the covariance of a
8
Obtained from http://www.cs.toronto.edu/?roweis/data.html.
normal distribution. Some exponential family extensions of general Boltzmann machines
were proposed in [2], [14], but they do not have the bipartite structure that we study here.
While the components of the Gaussian-multinomial EFH act as prototypes or templates
for highly probable input vectors, the components of the PoT act as constraints (i.e. input
vectors with large inner product have low probability). This can be traced back to the
shape of the non-linearity B in Eqn.6. Although by construction B must be convex (it
is the log-partition function), for large input values it can both be positive (prototypes,
e.g. B(x) = x2 ) or negative (constraints, e.g. B(x) = ?log(1 + x)). It has proven
difficult to jointly model both prototypes and constraints in the this formalism except for
the fully Gaussian case [11]. A future challenge is therefore to start the modelling process
with the desired non-linearity and to subsequently introduce auxiliary variables to facilitate
inference and learning.
References
[1] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning
Research, 3:993?1022, 2003.
[2] C.K.I.Williams. Continuous valued Boltzmann machines. Technical report, 1993.
[3] S.C. Deerwester, S.T. Dumais, T.K. Landauer, G.W. Furnas, and R.A. Harshman. Indexing by
latent semantic analysis. Journal of the American Society of Information Science, 41(6):391?
407, 1990.
[4] Y. Freund and D. Haussler. Unsupervised learning of distributions of binary vectors using
2-layer networks. In Advances in Neural Information Processing Systems, volume 4, pages
912?919, 1992.
[5] G.E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:1771?1800, 2002.
[6] Thomas Hofmann. Probabilistic latent semantic analysis. In Proc. of Uncertainty in Artificial
Intelligence, UAI?99, Stockholm, 1999.
[7] T. K. Marks and J. R. Movellan. Diffusion networks, products of experts, and factor analysis.
Technical Report UCSD MPLab TR 2001.02, University of California San Diego, 2001.
[8] B. Marlin and R. Zemel. The multiple multiplicative factor model for collaborative filtering. In
Proceedings of the 21st International Conference on Machine Learning, volume 21, 2004.
[9] M. Rosen-Zvi, T. Griffiths, M. Steyvers, and P. Smyth. The author-topic model for authors
and documents. In Proceedings of the Conference on Uncertainty in Artificial Intelligence,
volume 20, 2004.
[10] P. Smolensky. Information processing in dynamical systems: foundations of harmony theory.
In D.E. Rumehart and J.L. McClelland, editors, Parallel Distributed Processing: Explorations
in the Microstructure of Cognition. Volume 1: Foundations. McGraw-Hill, New York, 1986.
[11] M. Welling, F. Agakov, and C.K.I. Williams. Extreme components analysis. In Advances in
Neural Information Processing Systems, volume 16, Vancouver, Canada, 2003.
[12] M. Welling and G.E. Hinton. A new learning algorithm for mean field Boltzmann machines.
In Proceedings of the International Conference on Artificial Neural Networks, Madrid, Spain,
2001.
[13] M. Welling, G.E. Hinton, and S. Osindero. Learning sparse topographic representations with
products of student-t distributions. In Advances in Neural Information Processing Systems,
volume 15, Vancouver, Canada, 2002.
[14] R. Zemel, C. Williams, and M. Mozer. Lending direction to neural networks. Neural Networks,
8(4):503?512, 1995.
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1,842 | 2,673 | Convergence and No-Regret in Multiagent
Learning
Michael Bowling
Department of Computing Science
University of Alberta
Edmonton, Alberta
Canada T6G 2E8
[email protected]
Abstract
Learning in a multiagent system is a challenging problem due to two key
factors. First, if other agents are simultaneously learning then the environment is no longer stationary, thus undermining convergence guarantees. Second, learning is often susceptible to deception, where the other
agents may be able to exploit a learner?s particular dynamics. In the
worst case, this could result in poorer performance than if the agent was
not learning at all. These challenges are identifiable in the two most common evaluation criteria for multiagent learning algorithms: convergence
and regret. Algorithms focusing on convergence or regret in isolation
are numerous. In this paper, we seek to address both criteria in a single
algorithm by introducing GIGA-WoLF, a learning algorithm for normalform games. We prove the algorithm guarantees at most zero average
regret, while demonstrating the algorithm converges in many situations
of self-play. We prove convergence in a limited setting and give empirical results in a wider variety of situations. These results also suggest
a third new learning criterion combining convergence and regret, which
we call negative non-convergence regret (NNR).
1
Introduction
Learning to select actions to achieve goals in a multiagent setting requires overcoming a
number of key challenges. One of these challenges is the loss of the stationarity assumption
when multiple agents are learning simultaneously. Another challenge is guaranteeing that
the learner cannot be deceptively exploited by another agent. Both of these challenges distinguish the multiagent learning problem from traditional single-agent learning, and have
been gaining recent attention as multiagent applications continue to proliferate.
In single-agent learning tasks, it is reasonable to assume that the same action from the
same state will result in the same distribution over outcomes, both rewards and next states.
In other words, the environment is stationary. In a multiagent task with other learning
agents, the outcomes of an agent?s action will vary with the changing policies of the other
agents. Since most of the convergence results in reinforcement learning depend upon the
environment being stationary, convergence is often difficult to obtain in multiagent settings.
The desirability of convergence has been recently contested. We offer some brief insight
into this debate in the introduction of the extended version of this paper [1].
Equilibrium learners [2, 3, 4] are one method of handling the loss of stationarity. These algorithms learn joint-action values, which are stationary, and in certain circumstances guarantee these values converge to Nash (or correlated) equilibrium values. Using these values,
the player?s strategy corresponds to the player?s component of some Nash (or correlated)
equilibrium. This convergence of strategies is guaranteed nearly independently of the actions selected by the other agents, including when other agents play suboptimal responses.
Equilibrium learners, therefore, can fail to learn best-response policies even against simple
non-learning opponents.1 Best-response learners [5, 6, 7] are an alternative approach that
has sought to learn best-responses, but still considering whether the resulting algorithm
converges in some form. These approaches usually examine convergence in self-play, and
have included both theoretical and experimental results.
The second challenge is the avoidance of exploitation. Since learning strategies dynamically change their action selection over time, it is important to know that the change cannot
be exploited by a clever opponent. A deceptive strategy may ?lure? a dynamic strategy
away from a safe choice in order to switch to a strategy where the learner receives much
lower reward. For example, Chang and Kaelbling [8] demonstrated that the best-response
learner PHC [7] could be exploited by a particular dynamic strategy. One method of measuring whether an algorithm can be exploited is the notion of regret. Regret has been
explored both in game theory [9] and computer science [10, 11]. Regret measures how
much worse an algorithm performs compared to the best static strategy, with the goal to
guarantee at least zero average regret, no-regret, in the limit.
These two challenges result in two completely different criteria for evaluation: convergence and no-regret. In addition, they have almost exclusively been explored in isolation.
For example, equilibrium learners can have arbitrarily large average regret. On the other
hand, no-regret learners? strategies rarely converge in self-play [12] in even the simplest
of games.2 In this paper, we seek to explore these two criteria in a single algorithm for
learning in normal-form games. In Section 2 we present a more formal description of the
problem and the two criteria. We also examine key related work in applying gradient ascent
algorithms to this learning problem. In Section 3 we introduce GIGA-WoLF, an algorithm
with both regret and convergence properties. The algorithm is followed by theoretical and
experimental analyses in Sections 4 and 5, respectively, before concluding.
2
Online Learning in Games
A game in normal form is a tuple, (n, A1...n , R1...n ), where n is the number of players in
the game, Ai is a set of actions available to player i (A = A1 ?. . . ?An ), and Ri : A ? <
is a mapping from joint actions to player i?s reward. The problem of learning in a normalform game is one of repeatedly selecting an action and receiving a reward, with a goal of
maximizing average reward against an unknown opponent. If there are two players then it
is convenient to write a player?s reward function as a |A1 | ? |A2 | matrix. Three example
normal-form games are shown in Table 1.
Unless stated otherwise we will assume the learning algorithm is player one. In the context
of a particular learning algorithm and a particular opponent, let rt ? <|A1 | be the vector
of actual rewards that player one would receive at time t for each of its actions. Let x t ?
1
This work is not restricted to zero-sum games and our use of the word ?opponent? refers simply
to other players in the game.
2
A notable exception is Hart and Mas-Colell?s algorithm that guarantees the empirical distribution of play converges to that of a correlated equilibrium. Neither strategies nor expected values
necessarily converge, though.
Table 1: Examples of games in normal-form.
H
H
R1 =
T
?
AB
T
1 ?1
?1 1
?
R2 = ?R1
(a) Matching Pennies
R1 =
A
B
?
R2 =
A
B
?
R
0 3
1 2
?
3 2
0 1
?
(b) Tricky Game
P
S
0
1
R
0 ?1 1
R1 = P @ 1 0 ?1 A
S
?1 1 0
R2 = ?R1
(c) Rock?Paper?Scissors
P D(A1 ) be the algorithm?s strategy at time t, selected from probability distributions over
actions. So, player one?s expected payoff at time t is (rt ? xt ). Let 1a be the probability
distribution that assigns probability 1 to action a ? A1 . Lastly, we will assume the reward
2
for any action is bounded by rmax and therefore ||rt ||2 ? |A1 |rmax
.
2.1
Evaluation Criteria
One common evaluation criterion for learning in normal-form games is convergence. There
are a number of different forms of convergence that have been examined
P in the literature.
These include, roughly increasingPin strength: average reward (i.e., (rt ? xt )/T ), empirical distribution of actions (i.e.,
xt /T ), expected reward (i.e., (rt ? xt )), and strategies
(i.e., xt ). We focus in this paper on convergence of strategies as this implies the other three
forms of convergence as well. In particular, we will say an algorithm converges against a
particular opponent if and only if limt?? xt = x? .
The second common evaluation criterion is regret. Total regret3 is the difference between
the maximum total reward of any static strategy given the past history of play and the
algorithm?s total reward.
RT ? max
a?A1
T
X
t=1
((rt ? 1a ) ? (rt ? xt ))
Average regret is just the asymptotic average of total regret, limT ?? RT /T . An algorithm
has no-regret if and only if the average regret is less than or equal to zero against all opponent strategies. The no-regret property makes a strong claim about the performance of
the algorithm: the algorithm?s expected average reward is at least as large as the expected
average award any static strategy could have achieved. In other words, the algorithm is
performing at least as well as any static strategy.
2.2
Gradient Ascent Learning
Gradient ascent is a simple and common technique for finding parameters that optimize
a target function. In the case of learning in games, the parameters represent the player?s
strategy, and the target function is expected reward. We will examine three recent results
evaluating gradient ascent learning algorithms in normal-form games.
Singh, Kearns, and Mansour [6] analyzed gradient ascent (IGA) in two-player, two-action
games, e.g., Table 1(a) and (b). They examined the resulting strategy trajectories and payoffs in self-play, demonstrating that strategies do not always converge to a Nash equilibrium, depending on the game. They proved, instead, that average payoffs converge (a
3
Our analysis focuses on expectations of regret (total and average), similar to [10, 11]. Although
note that for any self-oblivious behavior, including GIGA-WoLF, average regret of at most zero on
expectation implies universal consistency, i.e., regret of at most with high probability [11].
weaker form of convergence) to the payoffs of the equilibrium. WoLF-IGA [7] extended
this work to the stronger form of convergence, namely convergence of strategies, through
the use of a variable learning rate. Using the WoLF (?Win or Learn Fast?) principle, the algorithm would choose a larger step size when the current strategy had less expected payoff
than the equilibrium strategy. This results in strategies converging to the Nash equilibrium
in a variety of games including all two-player, two-action games.4 Zinkevich [11] looked
at gradient ascent using the evaluation criterion of regret. He first extended IGA beyond
two-player, two-action games. His algorithm, GIGA (Generalized Infinitesimal Gradient
Ascent), updates strategies using an unconstrained gradient, and then projects the resulting
strategy vector back into the simplex of legal probability distributions,
xt+1 = P (xt + ?t rt )
where
P (x) = argmin ||x ? x0 ||,
(1)
x0 ?P D(A1 )
?t is the stepsize at time t, and || ? || is the standard L2 norm. He proved GIGA?s total regret
is bounded by,
?
?
2
RT ?
T + |A|rmax
( T ? 1/2).
(2)
Since GIGA is identical to IGA in two-player, two-action games, we also have that GIGA
achieves the weak form of convergence in this subclass of games. It is also true, though,
that GIGA?s strategies do not converge in self-play even in simple games like matching
pennies.
In the next section, we present an algorithm that simultaneously achieves GIGA?s no-regret
result and part of WoLF-IGA?s convergence result. We first present the algorithm and then
analyze these criteria both theoretically and experimentally.
3
GIGA-WoLF
GIGA-WoLF is a gradient based learning algorithm that internally keeps track of two different gradient-updated strategies, xt and zt . The algorithm chooses actions according to
the distribution xt , but updates both xt and zt after each iteration. The update rules consist
of three steps.
(1) x
?t+1
(2) zt+1
?t+1
(3) xt+1
= P (xt + ?t rt )
= P (zt + ?t rt /3)
||zt+1 ? zt ||
= min 1,
||zt+1 ? x
?t+1 ||
= x
?t+1 + ?t+1 (zt+1 ? x
?t+1 )
Step (1) updates xt according to GIGA?s standard gradient update and stores the result as
x
?t+1 . Step (2) updates zt in the same manner, but with a smaller step-size. Step (3) makes
a final adjustment on xt+1 by moving it toward zt+1 . The magnitude of this adjustment is
limited by the change in zt that occurred in step (2).
A key factor in understanding this algorithm is the observance that a strategy a receives
higher reward than a strategy b if and only if the gradient at a is in the direction of b (i.e.,
rt ? (b ? a) > 0). Therefore, the step (3) adjustment is in the direction of the gradient
if and only if zt received higher reward than xt . Notice also that, as long as xt is not
near the boundary, the change due to step (3) is of lower magnitude than the change due
4
WoLF-IGA may, in fact, be a limited variant of the extragradient method [13] for variational
inequality problems. The extragradient algorithm is guaranteed to converge to a Nash equilibrium
in self-play for all zero-sum games. Like WoLF-IGA, though, it does not have any known regret
guarantees, but more importantly requires the other players? payoffs to be known.
to step (1). Hence, the combination of steps (1) and (3) result in a change with two key
properties. First, the change is in the direction of positive gradient. Second, the magnitude
of the change is larger when zt received higher reward than xt . So, we can interpret the
update rule as a variation on the WoLF principle of ?win or learn fast?, i.e., the algorithm is
learning faster if and only if its strategy x is losing to strategy z. GIGA-WoLF is a major
improvement on the original presentation of WoLF-IGA, where winning was determined
by comparison with an equilibrium strategy that was assumed to be given. Not only is less
knowledge required, but the use of a GIGA-updated strategy to determine winning will
allow us to prove guarantees on the algorithm?s regret.
In the next section we present a theoretical examination of GIGA-WoLF?s regret in nplayer, n-action games, along with a limited guarantee of convergence in two-player, twoaction games. In Section 5, we give experimental results of learning using GIGA-WoLF,
demonstrating that convergence extends beyond the theoretical analysis presented.
4
Theoretical Analysis
We begin by examining GIGA-WoLF?s regret against an unknown opponent strategy. We
will prove the following bound on average regret.
?
Theorem 1 If ?t = 1/ t, the regret of GIGA-WoLF is,
?
?
2
(2 T ? 1).
RT ? 2 T + |A|rmax
Therefore, limT ?? RT /T ? 0, hence GIGA-WoLF has no-regret.
Proof. We begin with a brief overview of the proof. We will find a bound on the regret of
the strategy xt with respect to the dynamic strategy zt . Since zt is unmodified GIGA, we
already have a bound on the regret of zt with respect to any static strategy. Hence, we can
bound the regret of xt with respect to any static strategy.
We start by examining the regret of xt with respect to zt using a similar analysis as used
by Zinkevich [11]. Let ?x?z
refer to the difference in expected payoff between zt and xt
t
at time t, and Rx?z
be the sum of these differences, i.e., the total regret of xt with respect
T
to zt ,
T
X
x?z
x?z
?t
= rt ? (zt ? xt )
RT
?
?x?z
.
t
t=1
We will use the following potential function, ?t ? (xt ? zt )2 /2?t . We can examine how
this potential changes with each step of the update. ??1t , ??2t , and ??3t refers to the
change in potential caused by steps (1), (2), and (3), respectively. ??4t refers to the change
in potential caused by the learning rate change from ?t?1 to ?t . This gives us the following
equations for the potential change.
??1t+1
=
??2t+1
=
??3t+1
=
??4t+1
=
??t+1
=
1/2?t ((?
xt+1 ? zt )2 ? (xt ? zt )2 )
1/2?t ((?
xt+1 ? zt+1 )2 ? (?
xt+1 ? zt )2 )
1/2?t ((xt+1 ? zt+1 )2 ? (?
xt+1 ? zt+1 )2 )
(1/2?t+1 ? 1/2?t ) (xt+1 ? zt+1 )2
??1t+1 + ??2t+1 + ??3t+1 + ??4t+1
Notice that if ?t+1 = 1 then xt+1 = zt+1 . Hence ?t+1 = 0, and ??2t+1 + ??3t+1 ? 0.
If ?t+1 < 1, then ||xt+1 ? x
?t+1 || = ||zt+1 ? zt ||. Notice also that in this case xt+1 is
co-linear and between x
?t+1 and zt+1 . So,
||?
xt+1 ? zt+1 ||
= ||?
xt+1 ? xt+1 || + ||xt+1 ? zt+1 ||
= ||zt+1 ? zt || + ||xt+1 ? zt+1 ||
We can bound the left with the triangle inequality,
||?
xt+1 ? zt+1 ||
||xt+1 ? zt+1 ||
? ||?
xt+1 ? zt || + ||zt ? zt+1 ||
? ||?
xt+1 ? zt ||.
So regardless of ?t+1 , ??2t+1 + ??3t+1 < 0. Hence, ??t+1 ? ??1t+1 + ??4t+1 .
We will now use this bound on the change in the potential to bound the regret of x t with
respect to zt . We know from Zinkevich that,
(?
xt+1 ? zt )2 ? (xt ? zt )2 ? 2?t rt ? (zt ? xt ) + ?t2 rt2 .
Therefore,
?x?z
t
1
(?
xt+1 ? zt )2 ? (xt ? zt )2 ? rt2
2?t
? ???1t+1 + 1/2?t rt2 ? ???t+1 + ??4t+1 + 1/2?t rt2 .
? ?
2
Since we assume rewards are bounded?by rmax we can bound rt2 by |A|rmax
. Summing up
regret and using the fact that ?t = 1/ t, we get the following bound.
Rx?z
T
T
X
?t
2
|A|rmax
2
t=1
T
2
X
1
|A|rmax
? (?1 ? ?T ) +
?1 +
?t
?T
2
t=1
?
?
2
?
T + |A|rmax
( T ? 1/2)
?
???t + ??4t +
We know that GIGA?s regret with respect to any strategy is bounded by the same value (see
Inequality 2). Hence,
?
?
2
(2 T ? 1),
RT ? 2 T + |A|rmax
as claimed.
The second criterion we want to consider is convergence. As with IGA, WoLF-IGA, and
other algorithms, our theoretical analysis will be limited to two-player, two-action generalsum games. We further limit ourselves to the situation of GIGA-WoLF playing ?against?
GIGA. These restrictions are a limitation of the proof method, which uses a case-by-case
analysis that is combinatorially impractical for the case of self-play. This is not necessarily
a limitation on GIGA-WoLF?s convergence. This theorem along with the empirical results
we present later in Section 5 give a strong sense of GIGA-WoLF?s convergence properties.
The full proof can be found in [1].
Theorem 2 In a two-player, two-action repeated game, if one player follows the GIGAWoLF algorithm and the other follows the GIGA algorithm, then their strategies will converge to a Nash equilibrium.
5
Experimental Analysis
We have presented here two theoretical properties of GIGA-WoLF relating to guarantees
on both regret and convergence. There have also been extensive experimental results performed on GIGA-WoLF in a variety of normal-form games [1]. We summarize the results
here. The purpose of these experiments was to demonstrate the theoretical results from the
previous section as well as explore the extent to which the results (convergence, in particular) can be generalized. In that vein, we examined the same suite of normal-form games
used in experiments with WoLF-PHC, the practical variant of WoLF-IGA [7].
One of the requirements of GIGA-WoLF (and GIGA) is knowledge of the entire reward
vector (rt ), which requires knowledge of the game and observation of the opponent?s action. In practical situations, one or both of these are unlikely to be available. Instead, only
the reward of the selected action is likely to be observable. We have relaxed this requirement in these experiments by providing GIGA-WoLF (and GIGA) with only estimates of
the gradient from stochastic approximation. In particular, after selecting action a and receiving reward r?a , we update the current estimate of action a?s component of the reward
vector, rt+1 = rt + ?t (?
ra ? 1a ? rt )1a , where ?t is the learning rate. This is a standard
method of estimation commonly used in reinforcement learning (e.g., Q-learning).
For almost all of the games explored, including two-player, two-action games as well as
n-action zero-sum games, GIGA-WoLF strategies converged in self-play to equilibrium
strategies of the game. GIGA?s strategies, on the other hand, failed to converge in self-play
over the same suite of games. These results are nearly identical to the PHC and WoLF-PHC
experiments over the same games. A prototypical example of these results is provided in
Figure 1(a) and (b), showing strategy trajectories while learning in Rock-Paper-Scissors.
GIGA?s strategies do not converge, while GIGA-WoLF?s strategies do converge. GIGAWoLF also played directly against GIGA in this game resulting in convergence, but with
a curious twist. The resulting expected and average payoffs are shown in Figure 1(c).
Since both are no-regret learners, average payoffs are guaranteed to go to zero, but the
short-term payoff is highly favoring GIGA-WoLF. This result raises an interesting question
about the relative short-term performance of no-regret learning algorithms, which needs to
be explored further.
1
0.4
0.2
Reward
0.6
0.6
0.4
0
0
0.2 0.4 0.6 0.8
Pr(Rock)
(a) GIGA
1
0.04
0
-0.04
0.2
0
Average Reward
Expected Reward
0.08
0.8
Pr(Paper)
0.8
Pr(Paper)
0.12
1
-0.08
0
0.2 0.4 0.6 0.8
Pr(Rock)
(b) GIGA-WoLF
1
0
500000
Iterations
1e+06
(c) GIGA v. GIGA-WoLF
Figure 1: Trajectories of joint strategies in Rock-Paper-Scissors when both players use
GIGA (a) or GIGA-WoLF (b). Also shown (c) are the expected and average payoffs of the
players when GIGA and GIGA-WoLF play against each other.
GIGA-WoLF did not lead to convergence in all of the explored games. The ?problematic?
Shapley?s game, for which many similarly convergent algorithms fail in, also resulted in
non-convergence for GIGA-WoLF. On the other hand, this game has the interesting property that both players? when using GIGA-WoLF (or GIGA) actually achieve negative regret.
In other words, the algorithms are outperforming any static strategy to which they could
converge upon. This suggests a new desirable property for future multiagent (or online)
learning algorithms, negative non-convergence regret (NNR). An algorithm has NNR, if it
satisfies the no-regret property and either (i) achieves negative regret or (ii) its strategy converges. This property combines the criteria of regret and convergence, and GIGA-WoLF is
a natural candidate for achieving this compelling result.
6
Conclusion
We introduced GIGA-WoLF, a new gradient-based algorithm, that we believe is the first
to simultaneously address two criteria: no-regret and convergence. We proved GIGAWoLF has no-regret. We also proved that in a small class of normal-form games, GIGAWoLF?s strategy when played against GIGA will converge to a Nash equilibrium. We
summarized experimental results of GIGA-WoLF playing in a variety of zero-sum and
general-sum games. These experiments verified our theoretical results and exposed two
interesting phenomenon that deserve further study: short-term performance of no-regret
learners and the new desiderata of negative non-convergence regret. We expect GIGAWoLF and these results to be the foundation for further understanding of the connections
between the regret and convergence criteria.
References
[1] Michael Bowling. Convergence and no-regret in multiagent learning. Technical Report TR04-11, Department of Computing Science, University of Alberta, 2004.
[2] Michael L. Littman. Markov games as a framework for multi-agent reinforcement
learning. In Proceedings of the Eleventh International Conference on Machine Learning, pages 157?163, 1994.
[3] Junling Hu and Michael P. Wellman. Multiagent reinforcement learning: Theoretical
framework and an algorithm. In Proceedings of the Fifteenth International Conference on Machine Learning, pages 242?250, 1998.
[4] Amy Greenwald and Keith Hall. Correlated Q-learning. In Proceedings of the AAAI
Spring Symposium Workshop on Collaborative Learning Agents, 2002.
[5] Caroline Claus and Craig Boutilier. The dynamics of reinforcement learning in cooperative multiagent systems. In Proceedings of the Fifteenth National Conference on
Artificial Intelligence, pages 746?752, 1998.
[6] Satinder Singh, Michael Kearns, and Yishay Mansour. Nash convergence of gradient dynamics in general-sum games. In Proceedings of the Sixteenth Conference on
Uncertainty in Artificial Intelligence, pages 541?548, 2000.
[7] Michael Bowling and Manuela Veloso. Multiagent learning using a variable learning
rate. Artificial Intelligence, 136:215?250, 2002.
[8] Yu-Han Chang and Leslie Pack Kaelbling. Playing is believing: the role of beliefs
in multi-agent learning. In Advances in Neural Information Processing Systems 14,
2001.
[9] Sergiu Hart and Andreu Mas-Colell. A simple adaptive procedure leading to correlated equilibrium. Econometrica, 68:1127?1150, 2000.
[10] Peter Auer, Nicol`o Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. Gambling in
a rigged casino: The adversarial multi-arm bandit problem. In 36th Annual Symposium on Foundations of Computer Science, pages 322?331, 1995.
[11] Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the Twentieth International Conference on Machine
Learning, pages 928?925, 2003.
[12] Amir Jafari, Amy Greenwald, David Gondek, and Gunes Ercal. On no-regret learning,
fictitious play, and nash equilibrium. In Proceedings of the Eighteenth International
Conference on Machine Learning, pages 226?223, 2001.
[13] G. M. Korpelevich. The extragradient method for finding saddle points and other
problems. Matecon, 12:747?756, 1976.
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1,843 | 2,674 | Maximising Sensitivity in a Spiking Network
Anthony J. Bell,
Redwood Neuroscience Institute
1010 El Camino Real, Suite 380
Menlo Park, CA 94025
[email protected]
Lucas C. Parra
Biomedical Engineering Department
City College of New York
New York, NY 10033
[email protected]
Abstract
We use unsupervised probabilistic machine learning ideas to try to explain the kinds of learning observed in real neurons, the goal being
to connect abstract principles of self-organisation to known biophysical processes. For example, we would like to explain Spike TimingDependent Plasticity (see [5,6] and Figure 3A), in terms of information
theory. Starting out, we explore the optimisation of a network sensitivity measure related to maximising the mutual information between input
spike timings and output spike timings. Our derivations are analogous to
those in ICA, except that the sensitivity of output timings to input timings is maximised, rather than the sensitivity of output ?firing rates? to
inputs. ICA and related approaches have been successful in explaining
the learning of many properties of early visual receptive fields in rate coding models, and we are hoping for similar gains in understanding of spike
coding in networks, and how this is supported, in principled probabilistic
ways, by cellular biophysical processes. For now, in our initial simulations, we show that our derived rule can learn synaptic weights which can
unmix, or demultiplex, mixed spike trains. That is, it can recover independent point processes embedded in distributed correlated input spike
trains, using an adaptive single-layer feedforward spiking network.
1
Maximising Sensitivity.
In this section, we will follow the structure of the ICA derivation [4] in developing the
spiking theory. We cannot claim, as before, that this gives us an information maximisation
algorithm, for reasons that we will delay addressing until Section 3. But for now, to first
develop our approach, we will explore an interim objective function called sensitivity which
we define as the log Jacobian of how input spike timings affect output spike timings.
1.1
How to maximise the effect of one spike timing on another.
Consider a spike in neuron j at time tl that has an effect on the timing of another spike in
neuron i at time tk . The neurons are connected by a weight wij . We use i and j to index
neurons, and k and l to index spikes, but sometimes for convenience we will use spike
indices in place of neuron indices. For example, wkl , the weight between an input spike l
and an output spike k, is naturally understood to be just the corresponding wij .
dtl
threshold potential
dtk
du
u(t)
R(t)
resting potential
tk
output spikes
tl
input spikes
Figure 1: Firing time tk is determined by the time of threshold crossing. A change of an
input spike time dtl affects, via a change of the membrane potential du the time of the
output spike by dtk .
In the simplest version of the Spike Response Model [7], spike l has an effect on spike k that
depends on the time-course of the evoked EPSP or IPSP, which we write as R kl (tk ? tl ).
In general, this Rkl models both synaptic and dendritic linear responses to an input spike,
and thus models synapse type and location. For learning, we need only consider the value
of this function when an output spike, k, occurs.
In this model, depicted in Figure 1, a neuron adds up its spiking inputs until its membrane potential, ui (t), reaches threshold at time tk . This threshold we will often, again for
convenience, write as uk ? ui (tk , {tl }), and it is given by a sum over spikes l:
X
uk =
wkl Rkl (tk ? tl ) .
(1)
l
To maximise timing sensitivity, we need to determine the effect of a small change in the
input firing time tl on the output firing time tk . (A related problem is tackled in [2].) When
tl is changed by a small amount dtl the membrane potential will change as a result. This
change in the membrane potential leads to a change in the time of threshold crossing dt k .
The contribution to the membrane potential, du, due to dtl is (?uk /?tl )dtl , and the change
in du corresponding to a change dtk is (?uk /?tk )dtk . We can relate these two effects
by noting that the total change of the membrane potential du has to vanish because u k is
defined as the potential at threshold. ie:
du =
?uk
?uk
dtk +
dtl = 0 .
?tk
?tl
(2)
This is the total differential of the function uk = u(tk , {tl }), and is a special case of the
implicit function theorem. Rearranging this:
dtk
?uk
?uk
=?
= ?wkl R? kl /u? k .
(3)
dtl
?tl
?tk
Now, to connect with the standard ICA
P derivation [4], recall the ?rate? (or sigmoidal) neuron, for which yi = gi (ui ) and ui = j wij xj . For this neuron, the output dependence on
input is ?yi /?xj = wij gi0 while the learning gradient is:
?yi
?
= 1 ? fi (ui )xj
log
?wij
?xj wij
(4)
The analogous learning gradient for the spiking case, from (3), is:
P
?
dtk
?
1
a j(a)Rka
log
?
.
=
?wij
dtl wij
u? k
(5)
where the ?score functions?, fi , are defined in terms of a density estimate on the summed
?
?
inputs: fi (ui ) = ?u
log gi0 = ?u
log p?(ui ).
i
i
where j(a) = 1 if spike a came from neuron j, and 0 otherwise.
Comparing the two cases in (4) and (5), we see that the input variable xj has become
the temporal derivative of the sum of the EPSPs coming from synapse j, and the output
variable (or score function) fi (ui ) has become u? ?1
k , the inverse of the temporal derivative
of the membrane potential at threshold. It is intriguing (A) to see this quantity appear as
analogous to the score function in the ICA likelihood model, and, (B) to speculate that
experiments could show that this? voltage slope at threshold? is a hidden factor in STDP
data, explaining some of the scatter in Figure 3A. In other words, an STDP datapoint should
lie on a 2-surface in a 3D space of {?w, ?t, u? k }. Incidentally, u? k shows up in any
learning rule optimising an objective function involving output spike timings.
1.2
How to maximise the effect of N spike timings on N other ones.
Now we deal with the case of a ?square? single-layer feedforward mapping between spike
timings. There can be several input and output neurons, but here we ignore which neurons
are spiking, and just look at how the input timings affect the output timings. This is captured
in a Jacobian matrix of all timing dependencies we call T. The entries of this matrix are
Tkl ? ?tk /?tl . A multivariate version of the sensitivity measure introduced in the previous
section is the log of the absolute determinant of the timing matrix, ie: log |T|. The full
derivation for the gradient ?W log |T| is in the Appendix. Here, we again draw out the
analogy between Square ICA [4] and this gradient, as follows. Square ICA with a network
y = g(Wx) is:
?W ? ?W log |J| = W?1 ? f (u)xT
(6)
where the Jacobian J has entries ?yi /?xj and the score functions are now, fi (u) =
Q
?
? ?u
log p?(u) for the general likelihood case, with p?(u) = i gi0 being the special case of
i
ICA. We will now split the gradient in (6) according to the chain rule:
?W log |J|
=
[?J log |J|] ? [?W J]
?T
j(l)
= J
? Jkl i(k)
? fk (u)xj
.
wkl
(7)
[?T log |T|] ? [?W T]
!#
"
P
?
?T
j(l)
a j(a)Rka
= T
? Tkl i(k)
?
wkl
u? k
(9)
(8)
In this equation, i(k) = ?ik and j(l) = ?jl . P
The righthand term is a 4-tensor with entries
?Jkl /?wij , and ? is defined as A ? Bij = kl Akl Bklij . We write the gradient this way
to preserve, in the second term, the independent structure of the 1 ? 1 gradient term in
(4), and to separate a difficult derivation into two easy parts. The structure of (8) holds up
when we move to the spiking case, giving:
?W log |T| =
(10)
where i(k) is now defined as being 1 if spike k occured in neuron i, and 0 otherwise. j(l)
and j(a) are analogously defined.
Because the T matrix is much bigger than the J matrix, and because it?s entries are more
complex, here the similarity ends. When (10) is evaluated for a single weight influencing a
single spike coupling (see the Appendix for the full derivation), it yields:
?wkl ?
? log |T|
Tkl ?1
=
T lk ? 1 ,
?wkl
wkl
(11)
This is a non-local update involving a matrix inverse at each step. In the ICA case of (6),
such an inverse was removed by the Natural Gradient transform (see [1]), but in the spike
timing case, this has turned out not to be possible, because of the additional asymmetry
introduced into the T matrix (as opposed to the J matrix) by the R? kl term in (3).
2
Results.
Nonetheless, this learning rule can be simulated. It requires running the network for a
while to generate spikes (and a corresponding T matrix), and then for each input/output
spike coupling, the corresponding synapse is updated according to (11). When this is done,
and the weights learn, it is clear that something has been sacrificed by ignoring the issue
of which neurons are producing the spikes. Specifically, the network will often put all the
output spikes on one output neuron, with the rates of the others falling to zero. It is happy
to do this, if a large log |T| can thereby be achieved, because we have not included this
?which neuron? information in the objective. We will address these and other problems in
Section 3, but now we report on our simulation results on demultiplexing.
2.1
Demultiplexing spike trains.
An interesting possibility in the brain is that ?patterns? are embedded in spatially distributed
spike timings that are input to neurons. Several patterns could be embedded in single input
trains. This is called multiplexing. To extract and propagate these patterns, the neurons
must demultiplex these inputs using its threshold nonlinearity. Demultiplexing is the ?point
process? analog of the unmixing of independent inputs in ICA. We have been able to robustly achieve demultiplexing, as we now report.
We simulated a feed-forward network with 3 integrate-and-fire neurons and inputs from 3
presynaptic neurons. Learning followed (11) where we replace the inverse by the pseudoinverse computed on the spikes generated during 0.5 s. The pseudo-inverse is necessary
because even though on average, the learning matches number of output spikes to number
of input spikes, the matrix T is still not usually square and so its actual inverse cannot be
taken.
In addition, in these simulations, an additional term is introduced in the learning to make
sure all the output neurons fire with equal probability. This partially counters the ignoral of
the ?which neuron? information, which we explained above. Assuming Poisson spike count
ni for the ith output neuron with equalP
firing rate n
? i it is easy to derive in an approximate
term that will control the spike count, i (?
ni ? ni ). The target firing rates n
? i were set to
match the ?source? spike train in this example.
The network learns to demultiplex mixed spike trains, as shown in Figure 2. This demultiplexing is a robust property of learning using (11) with this new spike-controlling term.
Finally, what about the spike-timing dependendence of the observed learning? Does it
match experimental results? The comparison is made in Figure 3, and the answer is no.
There is a timing-dependent transition between depression and potentiation in our result
Spike Trains
mixing
mixed input trains
1
1
0.8
2
0.6
3
0
50
100
150
200
250
300
350
400
450
0.4
500
0.2
output
1
0
2
3
synaptic weights
0
50
100
150
200
250
300
350
400
450
500
original spike train
1
1
0.5
2
0
3
0
50
100
150
200
250
time in ms
300
350
400
450
500
?0.5
Figure 2: Unmixed spike trains. The input (top lef) are 3 spike trains which are a mixture
of three independent Poison processes (bottom left). The network unmixes the spike train
to approximately recover the original (center left). In this example 19 spikes correspond to
the original with 4 deletion and 2 insertions. The two panels at the right show the mixing
(top) and synaptic weight matrix after training (bottom).
in Figure 3B, but it is not a sharp transition like the experimental result in Figure 3A. In
addition, it does not transition at zero (ie: when tk ? tl = 0), but at a time offset by the
rise time of the EPSPs. In earlier experiments, in which we tranformed the gradient in
(11) by an approximate inverse Hessian, to get an approximate Natural Gradient method,
a sharp transition did emerge in simulations. However, the approximate inverse Hessian
was singular, and we had to de-emphasise this result. It does suggest, however, that if the
Natural Gradient transform can be usefully done on some variant of this learning rule, it
may well be what accounts for the sharp transition effect of STDP.
3
Discussion
Although these derivations started out smoothly, the reader possibly shares the authors?
frustration at the approximations involved here. Why isn?t this simple, like ICA? Why
don?t we just have a nice maximum spikelihood model, ie: a density estimation algorithm
for multivariate point processes, as ICA was a model in continuous space? We are going
to be explicit about the problems now, and will propose a direction where the solution may
lie. The over-riding problem is: we are unable to claim that in maximising log |T|, we are
maximising the mutual information between inputs and outputs because:
1. The Invertability Problem. Algorithms such as ICA which maximise log Jacobians can
only be called Infomax algorithms if the network transformation is both deterministic and
invertable. The Spike Response Model is deterministic, but it is not invertable in general.
When not invertable, the key formula (considering here vectors of input and output timings,
tin and tout )is transformed from simple to complex. ie:
Z
p(tin )
p(tin )
p(tout ) =
becomes p(tout ) =
d tin
(12)
|T|
solns tin |T|
Thus when not invertable, we need to know the Jacobians of all the inputs that could have
caused an output (called here ?solns?), something we simply don?t know.
2. The ?Which Neuron? Problem. Instead of maximising the mutual information
I(tout , tin ), we should be maximising I(tiout , tiin ), where the vector ti is the timing
(B) Gradient
150
100
100
? w (a.u.)
? w / w (%)
(A) STDP
150
50
0
?50
?100
?100
50
0
?50
?50
0
? t (ms)
50
100
?100
?20
0
20
40
60
? t (ms)
80
100
Figure 3: Dependence of synaptic modification on pre/post inter-spike interval. Left (A):
From Froemke & Dan, Nature (2002)]. Dependence of synaptic modification on pre/post
inter-spike interval in cat L2/3 visual cortical pyramidal cells in slice. Naturalistic spike
trains. Each point represents one experiment. Right (B): According to Equation (11).
Each point corresponds to an spike pair between approximately 100 input and 100 output
spikes.
vector, t, with the vector, i, of corresponding neuron indices, concatenated. Thus, ?who
spiked?? should be included in the analysis as it is part of the information.
3. The Predictive Information Problem. In ICA, since there was no time involved, we did
not have to worry about mutual informations over time between inputs and outputs. But in
the spiking model, output spikes may well have (predictive) mutual information with future
input spikes, as well as the usual (causal) mutual information with past input spikes. The
former has been entirely missing from our analysis so far.
These temporal and spatial infomation dependencies missing in our analysis so far, are
thrown into a different light by a single empirical observation, which is that Spike TimingDependent Plasticity is not just a feedforward computation like the Spike Response Model.
Specifically, there must be at least a statistical, if not a causal, relation between a real
synapse?s plasticity and its neuron?s output spike timings, for Figure 3B to look like it
does.
It seems we have to confront the need for both a ?memory? (or reconstruction) model, such
as the T we have thus far dealt with, in which output spikes talk about past inputs, and a
?prediction? model, in which they talk about future inputs. This is most easily understood
from the point of view of Barber & Agakov?s variational Infomax algorithm [3]. They
argue for optimising a lower bound on mutual information, which, for our neurons?, would
be expressed using an inverse model p?, as follows:
e in , tiout ) = H(tiin ) ? hlog p?(tiin |tiout )ip(ti ,ti ) ? I(tiin , tiout )
I(ti
in
out
(13)
In a feedforward model, H(tiin ) may be disregarded in taking gradients, leading us to the
optimisation of a ?memory-prediction? model p?(tiin |tiout ) related to something supposedly happening in dendrites, somas and at synapses. In trying to guess what this might be,
it would be nice if the math worked out. We need a square Jacobian matrix, T, so that
|T| = p?(tiin |tiout ) can be our memory/prediction model. Now let?s rename our feedfor?
?
ward timing Jacobian T (?up the dendritic trees?), as T, and let?s fantasise that there is
?
?
some, as yet unspecified, feedback Jacobian T (?down the dendritic trees?), which covers
?
?
electrotonic influences as they spread from soma to synapse, and which T can be combined with by some operation ??? to make things square. Imagine further, that doing this
?
? ?
?
yields a memory/prediction model on the inputs. Then the T we are looking for is T ? T,
? ?
?
?
and the memory-prediction model is: p?(tiin |tiout ) = T ? T
?
?
?
?
Ideally, the entries of T should be as before, ie: T kl = ?tk /?tl . What should the entries
?
?
?
?
?
?
of T be? Becoming just one step more concrete, suppose T had entries T lk = ?cl /?tk ,
where cl is some, as yet unspecified, value, or process, occuring at an input synapse when
spike l comes in. What seems clear is that ? should combine the correctly tensorised forms
?
?
?
?
?
? ?
?
of T and T (giving them each 4 indices ijkl), so that T = T ? T sums over the spikes
k and l to give a I ? J matrix, where I is the number of output neurons, and J the number
of input neurons. Then our quantity, T, would represent all dependencies of input neuronal
activity on output activity, summed over spikes.
?
?
Further, we imagine that T contains reverse (feedback) electrotonic transforms from soma
?
?
to synapse R lk that are somehow symmetrically related to the feedforward Spike Re?
?
sponses from synapse to soma, which we now rename R kl . Thinking for a moment in terms
of somatic k and synaptic l, voltages V , currents I and linear cable theory, the synapse to
?
?
?
?
soma transform, R kl would be related to an impedance in Vk = Il Z kl , while the soma
?
?
?
?
to synapse transform, R lk would be related to an admittance in Il = Vk Y lk [8]. The
?
?
?
?
symmetry in these equations is that Z kl is just the inverse conjugate of Y lk .
Finally, then, what is cl ? And what is its relation to the calcium concentration, [Ca2+ ]l ,
at a synapse, when spike l comes in? These questions naturally follow from considering
the experimental data, since it is known that the calcium level at synapses is the critical
integrating factor in determining whether potentiation or depression occurs [5].
4
Appendix: Gradient of log |T| for the full Spike Response Model.
Here we give full details of the gradient for Gerstner?s Spike Response Model [7]. This is
a general model for which Integrate-and-Fire is a special case. In this model the effect of
a presynaptic spike at time tl on the membrane potential at time t is described by a post
synaptic potential or spike response, which may also depend on the time that has passed
since the last output spike tk?1 , hence the spike response is written as R(t ? tk?1 , t ? tl ).
This response is weighted by the synaptic strength wl . Excitatory or inhibitory synapses are
determined by the sign of wl . Refractoriness is incorporated by adding a hyper-polarizing
contribution (spike-afterpotential) to the membrane potential in response to the last preceding spike ?(t ? tk?1 ). The membrane potential as a function of time is therefore given
by
X
u(t) = ?(t ? tk?1 ) +
wl R(t ? tk?1 , t ? tl ) .
(14)
l
We have ignored here potential contributions from external currents which can easily be included without modifying the following derivations. The output firing times t k are defined
as the times for which u(t) reaches firing threshold from below. We consider a dynamic
threshold, ?(t ? tk?1 ), which may depend on the time since that last spike tk?1 , together
then output spike times are defined implicitly by:
t = tk : u(t) = ?(t ? tk?1 ) and
du(t)
> 0.
dt
For this more general model Tkl is given by
?1
? k ? tk?1 , tk ? tl , )
dtk
?u
??
?u
wkl R(t
Tkl =
=?
?
=
,
? k ? tk?1 )
dtl
?tk
?tk
?tl
u(t
? k ) ? ?(t
(15)
(16)
?
? t), u(t),
where R(s,
?
and ?(t)
are derivatives with respect to t. The dependence of Tkl on
tk?1 should be implicitly assumed. It has been omitted to simplify the notation.
Now we compute the derivative of log |T| with respect to wkl . For any matrix T we have
? log |T|/?Tab = [T?1 ]ba . Therefore:
? log |T| X ? log |T| ?Tab X ?1 ?Tab
=
[T ]ba
.
(17)
?wkl
?Tab ?wkl
?wkl
ab
ab
Utilising the Kronecker delta ?ab = (1 if a = b, else 0), the derivative of (16) with respect
to wkl gives:
#
"
? a ? ta?1 , ta ? tb )
?Tab
?
wab R(t
=
P
? a ? ta?1 )
? a ? ta?1 , ta ? tc ) ? ?(t
?wkl
?wkl ?(ta ? ta?1 ) + c wac R(t
? a ? ta?1 , ta ? tb )
R(t
= ?ak ?bl
? a ? ta?1 )
u(t
? a ) ? ?(t
? a ? ta?1 , ta ? tb )?ak R(t
? a ? ta?1 , ta ? tl )
wab R(t
2
? a ? ta?1 )
u(t
? a ) ? ?(t
Tal
?bl
?
.
= ?ak Tab
wab
wal
?
Therefore:
? log |T|
?wkl
=
X
[T?1 ]ba ?ak Tab
ab
=
Tkl
wkl
[T?1 ]lk ?
?bl
Tal
?
wab
wal
X
b
[T?1 ]bk Tkl
!
(18)
(19)
=
Tkl
[T?1 ]lk ? 1 .
wkl
(20)
Acknowledgments
We are grateful for inspirational discussions with Nihat Ay, Michael Eisele, Hong Hui Yu,
Jim Crutchfield, Jeff Beck, Surya Ganguli, Sophi`e Deneve, David Barber, Fabian Theis,
Tony Zador and Arunava Banerjee. AJB thanks all RNI colleagues for many such discussions.
References
[1] Amari S-I. 1997. Natural gradient works efficiently in learning, Neural Computation,
10, 251-276
[2] Banerjee A. 2001. On the Phase-Space Dynamics of Systems of Spiking Neurons.
Neural Computation, 13, 161-225
[3] Barber D. & Agakov F. 2003. The IM Algorithm: A Variational Approach to Information Maximization. Advances in Neural Information Processing Systems 16, MIT Press.
[4] Bell A.J. & Sejnowski T.J. 1995. An information maximization approach to blind
separation and blind deconvolution, Neural Computation, 7, 1129-1159
[5] Dan Y. & Poo M-m. 2004. Spike timing-dependent plasticity of neural circuits, Neuron,
44, 23-30
[6] Froemke R.C. & Dan Y. 2002. Spike-timing-dependent synaptic modification induced
by natural spike trains. Nature, 28, 416: 433-8
[7] Gerstner W. & Kistner W.M. 2002. Spiking neuron models, Camb. Univ. Press
[8] Zador A.M., Agmon-Snir H. & Segev I. 1995. The morphoelectrotonic transform: a
graphical approach to dendritic function, J. Neurosci., 15(3): 1669-1682
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1,844 | 2,675 | Probabilistic computation in spiking populations
Richard S. Zemel
Dept. of Comp. Sci.
Univ. of Toronto
Quentin J. M. Huys
Gatsby CNU
UCL
Rama Natarajan
Dept. of Comp. Sci.
Univ. of Toronto
Peter Dayan
Gatsby CNU
UCL
Abstract
As animals interact with their environments, they must constantly update
estimates about their states. Bayesian models combine prior probabilities, a dynamical model and sensory evidence to update estimates optimally. These models are consistent with the results of many diverse
psychophysical studies. However, little is known about the neural representation and manipulation of such Bayesian information, particularly
in populations of spiking neurons. We consider this issue, suggesting a
model based on standard neural architecture and activations. We illustrate the approach on a simple random walk example, and apply it to
a sensorimotor integration task that provides a particularly compelling
example of dynamic probabilistic computation.
Bayesian models have been used to explain a gamut of experimental results in tasks which
require estimates to be derived from multiple sensory cues. These include a wide range
of psychophysical studies of perception;13 motor action;7 and decision-making.3, 5 Central
to Bayesian inference is that computations are sensitive to uncertainties about afferent and
efferent quantities, arising from ignorance, noise, or inherent ambiguity (e.g., the aperture
problem), and that these uncertainties change over time as information accumulates and
dissipates. Understanding how neurons represent and manipulate uncertain quantities is
therefore key to understanding the neural instantiation of these Bayesian inferences.
Most previous work on representing probabilistic inference in neural populations has focused on the representation of static information.1, 12, 15 These encompass various strategies
for encoding and decoding uncertain quantities, but do not readily generalize to real-world
dynamic information processing tasks, particularly the most interesting cases with stimuli changing over the same timescale as spiking itself.11 Notable exceptions are the recent, seminal, but, as we argue, representationally restricted, models proposed by Gold and
Shadlen,5 Rao,10 and Deneve.4
In this paper, we first show how probabilistic information varying over time can be represented in a spiking population code. Second, we present a method for producing spiking
codes that facilitate further processing of the probabilistic information. Finally, we show
the utility of this method by applying it to a temporal sensorimotor integration task.
1
TRAJECTORY ENCODING AND DECODING
We assume that population spikes R(t) arise stochastically in relation to the trajectory X(t)
of an underlying (but hidden) variable. We use RT and XT for the whole trajectory and
spike trains respectively from time 0 to T . The spikes RT constitute the observations and
are assumed to be probabilistically related to the signal by a tuning function f (X, ? i ):
P (R(i, T )|X(T )) ? f (X, ?i )
(1)
for the spike train of the ith neuron, with parameters ?i . Therefore, via standard Bayesian
inference, RT determines a distribution over the hidden variable at time T , P (X(T )|RT ).
We first consider a version of the dynamics and input coding that permits an analytical
examination of the impact of spikes. Let X(t) follow a stationary Gaussian process such
that the joint distribution P (X(t1 ), X(t2 ), . . . , X(tm )) is Gaussian for any finite collection
of times, with a covariance matrix which depends on time differences: Ctt0 = c(|t ? t0 |).
Function c(|?t|) controls the smoothness of the resulting random walks. Then,
R
P (X(T )|RT ) ? p(X(T )) X(T ) dX(T )P (RT |X(T ))P (X(T )|X(T ))
(2)
where P (X(T )|X(T )) is the distribution over the whole trajectory X(T ) conditional on
the value of X(T ) at its end point. If RT are a set of conditionally independent inhomogeneous Poisson processes, we have
Q
P R
P (RT |X(T )) ? i? f (X(ti? ), ?i ) exp ? i ? d? f (X(? ), ?i ) ,
(3)
where ti? ?? are the spike times ? of neuron i in RT . Let ? = [X(ti? )] be the vector of
stimulus positions at the times at which we observed a spike and ? = [?(ti? )] be the vector
2
2
of spike positions. If the tuning
P Rfunctions are Gaussian f (X, ?i ) ? exp(?(X ? ?i ) /2? )
and sufficiently dense that i ? d? f (X, ?i ) is independent of X (a standard assumption
in population coding), then P (RT |X(T )) ? exp(?k? ? ?k2 /2? 2 ) and in Equation 2, we
can marginalize out X(T ) except at the spike times ti? :
R
2
?1
P (X(T )|RT ) ? p(X(T )) ? d? exp ?[?, X(T )]T C 2 [?, X(T )] ? k???k
(4)
2
2?
and C is the block covariance matrix between X(ti? ), x(T ) at the spike times [tt? ] and the
final time T . This Gaussian integral has P (X(T )|RT ) ? N (?(T ), ?(T )), with
?(T ) = CT t (Ctt + I? 2 )?1 ? = k?
?(T ) = CT T ? kCtT
(5)
CT T is the T, T th element of the covariance matrix and CT t is similarly a row vector. The
dependence in ? on past spike times is specified chiefly by the inverse covariance matrix,
and acts as an effective kernel (k). This kernel is not stationary, since it depends on factors
such as the local density of spiking in the spike train RT .
For example, consider where X(t) evolves according to a diffusion process with drift:
dX = ??Xdt + ? dN (t)
(6)
where ? prevents it from wandering too far, N (t) is white Gaussian noise with mean zero
and ?2 variance. Figure 1A shows sample kernels for this process.
Inspection of Figure 1A reveals some important traits. First, the monotonically decreasing
kernel magnitude as the time span between the spike and the current time T grows matches
the intuition that recent spikes play a more significant role in determining the posterior over
X(T ). Second, the kernel is nearly exponential, with a time constant that depends on the
time constant of the covariance function and the density of the spikes; two settings of these
parameters produced the two groupings of kernels in the figure. Finally, the fully adaptive
kernel k can be locally well approximated by a metronomic kernel k<R> (shown in red in
Figure 1A) that assumes regular spiking. This takes advantage of the general fact, indicated
by the grouping of kernels, that the kernel depends weakly on the actual spike pattern, but
strongly on the average rate. The merits of the metronomic kernel are that it is stationary
and only depends on a single mean rate rather than the full spike train RT . It also justifies
s
Kernels k and k
?0.5
C
5
0
0.03 0.06 0.09
0.04 0.06 0.08
t?t
Time
spike
True stimulus and means
0.1
D
Full kernel
E
Regular, stationary kernel
?0.5
0
?0.5
0.03
0.04
0.05
0.06
0.07
Time
0.08
0.09
0
0.5
Space
0
Space
?4
10
Space
Variance ratio
10
?2
10
0.5
B
?2 / ?2
Kernel size (weight)
A
0.1
0
0.5
0.03
0.04
0.05
0.06
0.07
Time
0.08
0.09
0.1
Figure 1: Exact and approximate spike decoding with the Gaussian process prior. Spikes
are shown in yellow, the true stimulus in green, and P (X(T )|RT ) in gray. Blue: exact
inference with nonstationary and red: approximate inference with regular spiking. A Kernel samples for a diffusion process as defined by equations 5, 6. B, C: Mean and variance
of the inference. D: Exact inference with full kernel k and E: approximation based on
metronomic kernel k<R> . (Equation 7).
the form of decoder used for the network model in the next section.6 Figure 1D shows an
example of how well Equation 5 specifies a distribution over X(t) through very few spikes.
Finally, 1E shows a factorized approximation with the stationary kernel similar to that used
by Hinton and Brown6 and in our recurrent network:
Pt
Q
s
P? (X(t)|R(t)) ? i f (X, ?i ) j=0 kj tij = exp(?E(X(t), R(t), t)),
(7)
By design, the mean is captured very well, but not the variance, which in this example
grows too rapidly for long interspike intervals (Figure 1B, C). Using a slower kernel improves performance on the variance, but at the expense of the mean. We thus turn to the network model with recurrent connections that are available to reinstate the spike-conditional
characteristics of the full kernel.
2
NETWORK MODEL FORMULATION
Above we considered how population spikes RT specify a distribution over X(T ). We now
extend this to consider how interconnected populations of neurons can specify distributions
over time-varying variables. We frame the problem and our approach in terms of a two-level
network, connecting one population of neurons to another; this construction is intended to
apply to any level of processing. The network maps input population spikes R(t) to output
population spikes S(t), where input and output evolve over time. As with the input spikes,
ST indicates the output spike trains from time 0 to T , and these output spikes are assumed
to determine a distribution over a related hidden variable.
For the recurrent and feedforward computation in the network, we start with the deceptively simple goal9 of producing output spikes in such a way that the distribution
Q(X(T )|ST ) they imply over the same hidden variable X(T ) as the input, faithfully
matches P (X(T )|RT ). This might seem a strange goal, since one could surely just listen to the input spikes. However, in order for the output spikes to track the hidden variable,
the dynamics of the interactions between the neurons must explicitly capture the dynamics
of the process X(T ). Once this ?identity mapping? problem has been solved, more general,
complex computations can be performed with ease. We illustrate this on a multisensory
integration task, tracking a hidden variable that depends on multiple sensory cues.
The aim of the recurrent network is to take the spikes R(t) as inputs, and produce output
spikes that capture the probabilistic dynamics. We proceed in two steps. We first consider
the probabilistic decoding process which turns ST into Q(X(t)|ST ). Then we discuss the
recurrent and feedforward processing that produce appropriate ST given this decoder. Note
that this decoding process is not required for the network processing; it instead provides a
computational objective for the spiking dynamics in the system.
We use a simple log-linear decoder based on a spatiotemporal kernel:6
Q(X(T )|ST ) ? exp(?E(X(T ), ST , T )) , where
P PT
E(X, ST , T ) = j ? =0 S(j, T ? ? )?j (X, ? )
(8)
(9)
is an energy function, and the spatiotemporal kernels are assumed separable: ? j (X, ? ) =
gj (X)?(? ). The spatial kernel gj (X) is related to the receptive field f (X, ?j ) of neuron j
and the temporal kernel ?j (X, ? ) to k<RT > .
The dynamics of processing in the network follows a standard recurrent neural architecture
for modeling cortical responses, in the case that network inputs R(t) and outputs S(t) are
spikes. The effect of a spike on other neurons in the network is assumed to have some
simple temporal dynamics, described here again by the temporal kernel ?(? ):
PT
PT
ri (t) = ? =0 R(i, T ? ? )?(? ) sj (t) = ? =0 S(j, T ? ? )?(? )
where T is the extent of the kernel. The response of an output neuron is governed by a
stochastic spiking rule, where the probability that neuron j spikes at time t is given by:
P
P
P (S(j, t) = 1) = ?(uj (t)) = ? ( i wij ri (t) + k vkj sk (t ? 1))
(10)
where ?() is the logistic function, and W and V are the feedforward and recurrent weights.
If ?(? ) = exp(??? ), then uj (T ) = ?(0)(Wj ? R(T ) + Vj ? S(T )) + ?(1)uj (T ? 1); this
corresponds to a discretization of the standard dynamics for the membrane potential of a
du
leaky integrate-and-fire neuron: ? dtj = ??uj +WR+VS, where the leak ? is determined
by the temporal kernel.
The task of the network is to make Q(X(T )|ST ) of Equation 8 match P (X(T )|RT ) coming from one of the two models above (exact dynamic or approximate stationary kernel).
We measure the discrepancy using the Kullback-Leibler (KL) divergence:
P
J = t KL [P (X(T )|RT )||Q(X(T )|ST )]
(11)
and, as a proof of principle in the experiments below, find optimal W and V by
minimizing the KL divergence J using back-propagation through time (BPTT). In order to implement this in the most straightforward way, we convert the stochastic spikingP
rule (EquationP10) to a deterministic rule via the mean-field assumption: S j (t) =
? ( i wij ri (t) + k vkj sk (t ? 1)). The gradients are tedious, but can be neatly expressed in a temporally recursive form. Note that our current focus in the system is on
the representational capability of the system, rather than its learning. Our results establish
that the system can faithfully represent the posterior distribution. We return to the issue of
more plausible learning rules below.
The resulting network can be seen as a dynamic spiking analogue of the recurrent network
scheme of Pouget et al.:9 both methods formulate feedforward and recurrent connections so
that a simple decoding of the output can match optimal but complex decoding applied to the
inputs. A further advantage of the scheme proposed here is that it facilitates downstream
processing of the probabilistic information, as the objective encourages the formation of
distributions at the output that factorize across the units.
3
RELATED MODELS
Ideas about the representation of probabilistic information in spiking neurons are in vogue.
One treatment considers Poisson spiking in populations with regular tuning functions, assuming that stimuli change slowly compared with the inter-spike intervals.8 This leads
to a Kalman filter account with much formal similarity to the models of P (X(T )|R T ).
However, because of the slow timescale, recurrent dynamics can be allowed to settle to an
underlying attractor. In another approach, the spiking activity of either a single neuron 4 or
a pair of neurons5 is considered as reporting (logarithmic) probabilistic information about
an underlying binary hypothesis. A third treatment proposes that a population of neurons
directly represents the (logarithmic) probability over the state of a hidden Markov model. 10
Our method is closely related to the latter two models. Like Deneve?s4 we consider the
transformation of input spikes to output spikes with a fixed assumed decoding scheme so
that the dynamics of an underlying process is captured. Our decoding mechanism produces
something like the predictive coding apparent in Deneve?s scheme, except that here, a neuron may not need to spike not only if it itself has recently spiked and thereby conveyed
the appropriate information; but also if one of its population neighbors has recently spiked.
This is explicitly captured by the recurrent interactions among the population. Our scheme
also resembles Rao?s10 approach in that it involves population codes. Our representational
scheme is more general, however, in that the spatiotemporal decoder defines the relationship between output spikes and Q(X(T )|ST ), whereas his method assumes a direct encoding, with each output neuron?s activity proportional to log Q(X(T )|ST ). Our decoder
can produce such a direct encoding if the spatial and temporal kernels are delta functions,
but other kernels permit coordination amongst the population to take into account temporal
effects, and to produce higher fidelity in the output distribution.
4
EXPERIMENTS
1. Random walk. We describe two experiments. For ease of presentation and comparison, these simulations treat X(t) as a discrete variable, so that the encoding model is a
hidden Markov model (HMM) rather than the Gaussian process defined above. The first
is a random walk, as in Equation 6 and Figure 1, which allows us to make comparisons
with the exact statistics. In a discrete setting, the walk parameters ? and ? determine the
entries in the transition matrix of the corresponding HMM; in a continuous one, the covariance function C of the Gaussian process. Input spikes are generated according to Gaussian
tuning functions (Equation 1). In the recurrent network model, the spatiotemporal kernels are fixed: the spatial kernels are based on the regular locations of the output units j,
gj (X) = |X ? Xj |2 /(1 + |X ? Xj |2 ) and the temporal kernel is ?(? ) = exp(??? ),
where ? = 2 is set to match the walk dynamics. In the following simulations, the network
contained 20 inputs, 60 states, and 20 outputs.
Results on two walk trajectories with different dynamics are shown in Figure 2. The network is trained on walks with parameters (? = 0.2, ? = 2) that force the state to move to
and remain near the center. Figures 2A & B show that in intervals without input spikes, the
inferred mean quickly shifts towards the center and remains there until evidence is received
in the form of input spikes. The feedforward weights (Fig. 2F) show strong connections
between an input unit and its corresponding output, while the learned recurrent weights
(Fig. 2E) reflect the transition probabilities: units coding for extreme values have strong
connections to those nearer the center, and units with preferred values near the center have
strong self-connections. Fig. 2C&D) shows the results of testing this trained network on
walks with different dynamics (? = 0.8, ? = 7). The resulting mismatch between the
mean approximated trajectory (yellow line) and true stimulus (dashed line) (Fig. 2D), and
the variance (Fig. 2H), shows that the learned weights capture the input dynamics.
A
B
E
F
2
3
10
4
2
6
20
8
0
0.5
Variance of Encoded and Decoded Distributions
1
12
0
14
16
?1
18
8
10
12
14
16
18
20
Full Inference
Approximation
120
12
100
?2
0
10
30
12
?0.5
?4
40
14
10
8
6
?1
?6
16
50
18
80
60
40
4
?1.5
?8
?2
20
140
Full Inference
Approximation
14
Variance
10
Output
Output Neurons
Neurons
Input Neurons
2
8
Variance of Encoded and Decoded Distributions
16
1
6
Output Neurons
H
1.5
Variance
4
6
G
4
4
4
D
Recurrent
Weights
Recurrent Weights
Feed?forward Weights
2
2
C
20
60
?2
2
104
6 20 8
10
30
12
40
14
Output Neurons
Output
Neurons
16 50 18
20
60
20
2
0
0
2
4
6
8
10
Time
12
14
16
18
20
0
0
2
4
6
8
10
12
14
16
18
20
Time
Figure 2: Comparison between full inference using hidden Markov model and approximation using network model. Top Row: Full Inference (A,C) and approximation (B,D) results
from two walks. Input spikes (RT ) are shown as green circles; output spikes (ST > .9) as
magenta stars; true stimulus as dashed line; mean inferred trajectory as red line; mean approximated trajectory as yellow line; distributions P (X(t)|RT ) and Q(X(t)|ST ) at each
timestep in gray. Bottom Row: Feedfoward (E); recurrent weights (F); variance of exact
and approximate inference from walks 1 (G) and 2 (H).
2. Sensorimotor task. We next applied our framework to a recent experiment on probabilistic computation during sensorimotor processing.7 Here, human subjects tried to move
a cursor on a display to a target by moving a (hidden) finger. The cursor was shown before
the start of the movement, it was then hidden, except for one point of blurry visual feedback
in the middle of the movement (with variances 0 = ?0 < ?L < ?M < ?? = ?). Unbeknownst to them, on the onset of movement, the cursor was displaced by ?X, drawn from
a prior distribution P (?X). The subjects must estimate ?X in order to compensate for the
displacement and land the cursor on the target. The key result is that subjects learned and
used the prior information P (?X), and indeed integrated it with the visual information
in a way that was appropriately sensitive to the amount of blur (figure 3A). The authors
showed that a simple Bayesian model could account for these data.
We model a population representation of the 2D cursor position X(t) on the screen.
Two spiking input codes?from vision (RT v ) and proprioception (RT p ), present also
in the absence of visual feedback?are mapped into an output code ST representing
P (X(t)|RT v , RT p ). This is a neural instantiation of Bayesian cue combination, and also
an extension of the previous model to the dynamic case.
The first experiment involved a Gaussian prior distribution: P (?X) ? N (1, .5). During
initial experience subjects learn about this prior from trajectories; we determine the parameters of the HMM. We use BPTT to learn feedforward weights for the input spikes from
the different modalities, and recurrent weights between the output units. The input population had 64 units per modality, while the state space and output population each had 100
units. Input spikes were Poisson based on tuning functions centered on a grid within the
2D space; spatiotemporal kernels were based on the (gridded) output units j. The model
was tested in conditions directly matching the experiment, with the cursor and finger moving along a straight line trajectory from the current model?s estimate of the cursor position,
< X(t) >Q(X(t)|ST ) , to the target location. The model captures the main effects of the
experiment (see Figure 3) with respect to visual blur. The prior was ignored when the sensory evidence was precise (?0 ), it dominated on trials without feedback (?? ), and the two
Final Deviation
0
1
2
1
1
1
0
0
0
?1
0
1
2
?1
0
1
?1
0
2
1
2
Imposed displacement (?X)
Final Deviation
Figure 3: (a) Results for a typical subject from the first Ko? rding-Wolpert experiment,7 for
different degrees of blur in the visual feedback (?{0,M,L,?} ). The average lateral displacement of the cursor from the target location at the trial end, as a function of the imposed
displacement of the cursor from the finger location (?X), which is drawn from N (1, .5).
(b) Model results under the same testing conditions. See text for model details.
(a)
(b)
(c)
?2
0
2
Figure 4: (a) Bimodal prior P (?X) ? N (?2, 0.5) for cursor displacement in second
K?ording-Wolpert experiment.7 (b) Results from human subjects. (c) Model results.
factors combined on intermediate degrees of blurriness.
In the second experiment the prior was bimodal (Fig. 4A) and feedback was blurred (? L ).
For this prior, the final cursor location should be based on the more prevalent displacements, so responses based on optimal inference should be non-linear. may modify the
posterior estimate of cursor location. Finally, its 2DIndeed, this is the case (Fig. 4B;C). Intuitively, the blurry visual feedback inadequately defines the true finger position, and thus
the posterior P (X(t)|RT ) is influenced by the learned bimodal prior; the network model
accurately reconstructs this optimal posterior.
Our model generalizes the simple Bayesian account, and suggests new avenues for predictions. The dynamic nature of the system permits modeling the integration of several visual
cues during the trial, as well as differential effects of the timing of visual feedback. The integration of cues in our model also allows it to capture interactions between them. Finally,
its 2D nature allows our system to model other aspects of combining visual and proprioceptive cues, such as their varying and contrasting degrees of certainty across space. 14
5
DISCUSSION
We proposed a spiking population coding scheme for representing and propagating uncertainty which operates at the fine-grained timescale of individual inter-spike intervals. We
motivated the key spatio-temporal spike kernels in the model from analytical results in a
Gaussian process, and suggested two approximations to the exact decoding provided by
these adaptive spatiotemporal kernels. The first is a regular stationary kernel while the
second is a recurrent network model. We showed how gradient descent can set model parameters to match the requirements on the output distribution and capture the dynamics
underlying a hidden variable. This is a dynamic and spiking extension of DPC,15 and a
population extension of Deneve.4 We showed its proficiency by comparison with exact
inference in a random walk, and a neural model that does not use a population code.
The most important direction concerns biologically plausible learning in the full spiking
form of the model. One possibility is to view spikes as a primitive action chosen by a
neuron. In this case, we can use the analog of direct policy methods in partially observable
Markov decision processes,2 with faithful tracking of X(t) leading to reward. It is also
possible that simpler, Hebbian rules will suffice. A second future direction concerns inference of one variable from another using our spiking population code model. This problem
involves marginalizing over intermediate variables, which is difficult in direct representations of distributions over these variables, due to approximating logs of sums with sums of
logs;10 we are investigating how well our scheme can approximate this computation.
We applied the model to a challenging sensorimotor integration task which has been used to
demonstrate Bayesian inference. Since it offers a dynamic account, we can make a number
of predictions about the consequences of variations to the experiment. Most interesting
would be the case in which a bimodal likelihood is combined with a unimodal (or bimodal)
prior (rather than vice-versa), or indeed two instances of visual feedback during the task.
Acknowledgements
We thank Sophie Deneve and Jon Pillow for helpful discussions. RZ & RN funded by NSERC, CIHR
NET program; PD & QH by Gatsby Charitable Fdtn., BIBA consortium, UCL MB/PhD program.
References
[1] Anderson C.H. & Van Essen D.C. (1994). Neurobiological computational systems. In: Computational Intelligence: Imitating Life, Zurada, Marks, Robinson (ed.), IEEE Press, 213-222.
[2] Baxter, J. & Bartlett, P. (2001). Infinite-horizon policy-gradient estimation. JAIR, 319 - 350.
[3] Carpenter, R. H. S. & Williams, M. L. L. (1995). Neural computation of log likelihood in the
control of saccadic eye movements. Nature, 377: 59-62.
[4] Deneve, S. (2004). Bayesian inference in spiking neurons. NIPS-17.
[5] Gold, JI & Shadlen, MN (2001). Neural computations that underlie decisions about sensory
stimuli. Trends in Cognitive Sciences 5:10-16.
[6] Hinton, GE & Brown, AD (2000) Spiking Boltzmann machines. NIPS-12: 122-129.
[7] K?ording, KP & Wolpert, D (2004) Bayesian integration in sensorimotor learning. Nature
427:244-247.
[8] Latham, P., Deneve, S., & Pouget, A., (2004). Optimal computation with attractor networks. J
Physiology, Paris.
[9] Pouget, A., Zhang, K, Deneve, S, & Latham, P. (1998) Statistically efficient estimation using
population codes. NeuralComputation, 10: 373-401.
[10] Rao, R. (2004). Bayesian computation in recurrent neural circuits. Neural Computation, 16(1).
[11] Rieke, F, Warland, D, de Ruyter v. Steveninck, & Bialek, W. (1999). Spikes. MIT Press.
[12] Sahani, M & Dayan, P (2003) Doubly distributional population codes: Simultaneous representation of uncertainty and multiplicity. Neural Computation 15.
[13] Saunders, J.A. & Knill, D.C. (2001). Perception of 3D surface orientation from skew symmetry.
Vision Research, 41 (24) 3163 - 3183.
[14] Van Beers, R. J., Sittig, A.C., & Denier, J.J. (1999). Integration of propriocetive and visual
position-information J Neurophysiol, 81: 1355-1364.
[15] Zemel, R.S., Dayan, P. & Pouget A. (1998). Probabilistic interpretation of population codes.
Neural Computation, 10, 403-430.
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1,845 | 2,676 | Using the Equivalent Kernel to Understand
Gaussian Process Regression
Peter Sollich
Dept of Mathematics
King?s College London
Strand, London WC2R 2LS, UK
[email protected]
Christopher K. I. Williams
School of Informatics
University of Edinburgh
5 Forrest Hill, Edinburgh EH1 2QL, UK
[email protected]
Abstract
The equivalent kernel [1] is a way of understanding how Gaussian process regression works for large sample sizes based on a continuum limit.
In this paper we show (1) how to approximate the equivalent kernel of the
widely-used squared exponential (or Gaussian) kernel and related kernels, and (2) how analysis using the equivalent kernel helps to understand
the learning curves for Gaussian processes.
Consider the supervised regression problem for a dataset D with entries (xi , yi ) for i =
1, . . . , n. Under Gaussian Process (GP) assumptions the predictive mean at a test point x ?
is given by
f?(x? ) = k> (x? )(K + ? 2 I)?1 y,
(1)
where K denotes the n ? n matrix of covariances between the training points with entries
k(xi , xj ), k(x? ) is the vector of covariances k(xi , x? ), ? 2 is the noise variance on the
observations and y is a n ? 1 vector holding the training targets. See e.g. [2] for further
details.
We can define a vector of functions h(x? ) = (K + ? 2 I)?1 k(x? ) . Thus we have f?(x? ) =
h> (x? )y, making it clear that the mean prediction at a point x? is a linear combination of
the target values y. Gaussian process regression is thus a linear smoother, see [3, section
2.8] for further details. For a fixed test point x? , h(x? ) gives the vector of weights applied
to targets y. Silverman [1] called h> (x? ) the weight function.
Understanding the form of the weight function is made complicated by the matrix inversion
of K + ? 2 I and the fact that K depends on the specific locations of the n datapoints.
Idealizing the situation one can consider the observations to be ?smeared out? in x-space
at some constant density of observations. In this case analytic tools can be brought to bear
on the problem, as shown below. By analogy to kernel smoothing Silverman [1] called the
idealized weight function the equivalent kernel (EK).
The structure of the remainder of the paper is as follows: In section 1 we describe how
to derive the equivalent kernel in Fourier space. Section 2 derives approximations for the
EK for the squared exponential and other kernels. In section 3 we show how use the EK
approach to estimate learning curves for GP regression, and compare GP regression to
kernel regression using the EK.
1
Gaussian Process Regression and the Equivalent Kernel
It is well known (see e.g. [4]) that the posterior mean for GP regression can be obtained as
the function which minimizes the functional
n
1 X
1
(yi ? f (xi ))2 ,
J[f ] = kf k2H + 2
(2)
2
2?n i=1
where kf kH is the RKHS norm corresponding to kernel k. (However, note that the GP
framework gives much more than just this mean prediction, for example the predictive
variance and the marginal likelihood p(y) of the data under the model.)
Let ?(x) = E[y|x] be the target function for our regression problem and write E[(y ?
f (x))2 ] = E[(y ? ?(x))2 ] + (?(x) ? f (x))2 . Using the fact that the first term on the RHS
is independent of f motivates considering a smoothed version of equation 2,
Z
?
1
J? [f ] =
(?(x) ? f (x))2 dx + kf k2H ,
2? 2
2
where ? has dimensions of the number of observations per unit of x-space
(length/area/volume etc. as appropriate). If we consider kernels that are stationary,
k(x, x0 ) = k(x ? x0 ), the natural basis in which to analyse
equation 1 is the Fourier
R
basis of complex sinusoids so that f (x) is represented as f?(s)e2?is?x ds and similarly for
?(x). Thus we obtain
!
Z
1
? ?
|f?(s)|2
2
J? [f ] =
ds,
|f (s) ? ??(s)| +
2
?2
S(s)
R
as kf k2HR =
|f?(s)|2 /S(s)ds where S(s) is the power spectrum of the kernel k,
?2?is?x
S(s) = k(x)e
dx. J? [f ] can be minimized using calculus of variations to ob?(s) = S(s)?(s)/(? 2 /? + S(s)) which is recognized as the convolution f (x? ) =
tain
f
R
h(x? ? x)?(x)dx. Here the Fourier transform of the equivalent kernel h(x) is
?
h(s)
=
1
S(s)
=
.
S(s) + ? 2 /?
1 + ? 2 /(?S(s))
(3)
?
The term ? 2 /? in the first expression for h(s)
corresponds to the power spectrum of a
white noise process, whose delta-function covariance function becomes a constant in the
Fourier domain. This analysis is known as Wiener filtering; see, e.g. [5, ?14-1]. Notice
that as ? ? ?, h(x) tends to the delta function. If the input density is non-uniform the
analysis above should be interpreted as computing the equivalent kernel for np(x) = ?.
This approximation will be valid if the scale of variation of p(x) is larger than the width of
the equivalent kernel.
2
The EK for the Squared Exponential and Related Kernels
For certain kernels/covariance functions the EK h(x) can be computed exactly by Fourier
inversion. Examples include the Ornstein-Uhlenbeck process in D = 1 with covariance
k(x) = e??|x|
(see [5, p. 326]), splines in D = 1 corresponding
to the regularizer
R
R
kP f k2 = (f (m) )2 dx [1, 6], and the regularizer kP f k2 = (?2 f )2 dx in two dimensions, where the EK is given in terms of the Kelvin function kei [7].
We now consider the commonly used squared exponential (SE) kernel k(r) =
exp(?r 2 /2`2 ), where r 2 = ||x?x0 ||2 . (This is sometimes called the Gaussian or radial basis function kernel.) Its Fourier transform is given by S(s) = (2?`2 )D/2 exp(?2? 2 `2 |s|2 ),
where D denotes the dimensionality of x (and s) space.
From equation 3 we obtain
? SE (s) =
h
1
,
1 + b exp(2? 2 `2 |s|2 )
where b = ? 2 /?(2?`2 )D/2 . We are unaware of an exact result in this case, but the following
initial approximation is simple but effective. For large ?, b will be small. Thus for small
? SE ' 1, but for large s it is approximately 0. The change takes
s = |s| we have that h
place around the point sc where b exp(2? 2 `2 s2c ) = 1, i.e. s2c = log(1/b)/2? 2 `2 . As
? SE between 1 and 0 can be expected
exp(2? 2 `2 s2 ) grows quickly with s, the transition of h
to be rapid, and thus be well-approximated by a step function.
Proposition 1 The approximate form of the equivalent kernel for the squared-exponential
kernel in D-dimensions is given by
s D/2
c
hSE (r) =
JD/2 (2?sc r).
r
Proof: hSE (s) is a function of s = |s| only, and for D > 1 the Fourier integral can
be simplified by changing to spherical polar coordinates and integrating out the angular
variables to give
Z ? ?+1
s
? SE (s) ds
hSE (r) = 2?r
J? (2?rs)h
(4)
r
0
Z sc ?+1
s D/2
s
c
' 2?r
J? (2?rs) ds =
JD/2 (2?sc r).
r
r
0
where ? = D/2 ? 1, J? (z) is a Bessel function of the first kind and we have used the
identity z ?+1 J? (z) = (d/dz)[z ?+1 J?+1 (z)].
Note that in D = 1 by computing the Fourier transform of the boxcar function we obtain
hSE (x) = 2sc sinc(2?sc x) where sinc(z) = sin(z)/z. This is consistent with Proposition
1 and J1/2 (z) = (2/?z)1/2 sin(z). The asymptotic form of the EK in D = 2 is shown in
Figure 2(left) below.
Notice that sc scales as (log(?))1/2 so that the width of the EK (which is proportional to
1/sc ) will decay very slowly as ? increases. In contrast for a spline of order m (with power
spectrum ? |s|?2m ) the width of the EK scales as ??1/2m [1].
If instead of RD we consider the input set to P
be the unit circle, a stationary kernel can
be periodized by the construction kp (x, x0 ) = n?Z k(x ? x0 + 2n?). This kernel will
be represented as a Fourier series (rather than with a Fourier transform) because of the
periodicity. In this case the step function in Fourier space approximation would give rise
to a Dirichlet kernel as the EK (see [8, section 4.4.3] for further details on the Dirichlet
kernel).
We now show that the result of Proposition 1 is asymptotically exact for ? ? ?, and calculate the leading corrections for finite ?. The scaling of the width of the EK as 1/s c suggests
writing hSE (r) = (2?sc )D g(2?sc r). Then from equation 4 and using the definition of sc
?+1
Z ?
z
2?sc s
J? (zs/sc )
g(z) =
ds
sc (2?sc )D 0
z
1 + exp[2? 2 `2 (s2 ? s2c )]
Z ?
u ?+1
J? (zu)
=z
du
(5)
2 `2 s2 (u2 ? 1)]
2?z
1
+
exp[2?
0
c
where we have rescaled s = sc u in the second step. The value of sc , and hence ?, now
enters only in the exponential via a = 2? 2 `2 s2c . For a ? ?, the exponential tends to zero
for u < 1 and to infinity for u > 1. The factor 1/[1 + exp(. . .)] is therefore a step function
?(1 ? u) in the limit and Proposition 1 becomes exact, with g? (z) ? lima?? g(z) =
(2?z)?D/2 JD/2 (z). To calculate corrections to this, one uses that for large but finite a the
difference ?(u) = {1 + exp[a(u2 ? 1)]}?1 ? ?(1 ? u) is non-negligible only in a range
of order 1/a around u = 1. The other factors in the integrand of equation 5 can thus be
Taylor-expanded around that point to give
Z ?
?
X
Ik dk u ?+1
J? (zu) , Ik =
g(z) = g? (z) + z
?(u)(u ? 1)k du
k! duk
2?z
0
u=1
k=0
The problem is thus reduced to calculating the integrals Ik . Setting u = 1 + v/a one has
Z ?
Z 0
vk
1
k
?
1
v
dv
+
dv
ak+1 Ik =
2
1 + exp(v 2 /a + 2v)
0
?a 1 + exp(v /a + 2v)
Z a
Z ?
(?1)k+1 v k
vk
=
dv
+
dv
2
1 + exp(v 2 /a + 2v)
0 1 + exp(?v /a + 2v)
0
In the first integral, extending the upper limit to ? gives an error that is exponentially
small in a. Expanding the remaining 1/a-dependence of the integrand one then gets, to
leading order in 1/a, I0 = c0 /a2 , I1 = c1 /a2 while all Ik with k ? 2 are smaller by
at least 1/a2 . The numerical constants are ?c0 = c1 = ? 2 /24. This gives, using that
(d/dz)[z ?+1 J? (z)] = z ? J? (z) + z ?+1 J??1 (z) = (2? + 1)z ? J? (z) ? z ?+1 J?+1 (z):
Proposition 2 The equivalent kernel for the squared-exponential kernel is given for large
? by hSE (r) = (2?sc )D g(2?sc r) with
n
o
1
z
1
g(z) =
JD/2 (z) + 2 (c0 + c1 (D ? 1))JD/2?1 (z) ? c1 zJD/2 (z) +O( 4 )
D
a
a
(2?z) 2
For e.g. D = 1 this becomes g(z) = ? ?1 {sin(z)/z ? ? 2 /(24a2 )[cos(z) + z sin(z)]}. Here
and in general, by comparing the second part of the 1/a2 correction with the leading order
term, one estimates that the correction is of relative size z 2 /a2 . It will therefore provide
a useful improvement as long as z = 2?sc r < a; for larger z the expansion in powers of
1/a becomes a poor approximation because the correction terms (of all orders in 1/a) are
comparable to the leading order.
2.1
Accuracy of the approximation
To evaluate the accuracy of the approximation we can compute the EK numerically as
follows: Consider a dense grid of points in RD with a sampling density ?grid . For making
2
predictions at the grid points we obtain the smoother matrix K(K + ?grid
I)?1 , where1
2
2
?grid = ? ?grid /?, as per equation 1. Each row of this matrix is an approximation to the
EK at the appropriate location, as this is the response to a y vector which is zero at all points
except one. Note that in theory one should use a grid over the whole of RD but in practice
one can obtain an excellent approximation to the EK by only considering a grid around the
point of interest as the EK typically decays with distance. Also, by only considering a finite
grid one can understand how the EK is affected by edge effects.
2
To understand this scaling of ?grid
consider the case where ?grid > ? which means that the
effective variance at each of the ?grid points per unit x-space is larger, but as there are correspondingly
more points this effect cancels out. This can be understood by imagining the situation where there
2
are ?grid /? independent Gaussian observations with variance ?grid
at a single x-point; this would
2
be equivalent to one Gaussian observation with variance ? . In effect the ? observations per unit
x-space have been smoothed out uniformly.
1
0.16
0.35
0.35
Numerical
Proposition 1
Proposition 2
0.3
0.25
0.14
0.12
0.1
0.08
0.2
0.2
0.15
0.15
0.1
0.1
0.05
0.05
0
0
?0.05
?0.05
?0.1
0
5
10
Numerical
Proposition 1
Proposition 2
0.3
0.25
?0.1
0
15
5
10
15
0.06
0.04
0.02
0
?0.02
Numerical
Proposition 1
Sample
?0.04
?0.5
?0.4
?0.3
?0.2
?0.1
0
0.1
0.2
0.3
0.4
0.5
Figure 1: Main figure: plot of the weight function corresponding to ? = 100 training
points/unit length, plus the numerically computed equivalent kernel at x = 0.0 and the sinc
approximation from Proposition 1. Insets: numerically evaluated g(z) together with sinc
and Proposition 2 approximations for ? = 100 (left) and ? = 104 (right).
Figure 1 shows plots of the weight function for ? = 100, the EK computed on the grid as
described above and the analytical sinc approximation. These are computed for parameter
values of `2 = 0.004 and ? 2 = 0.1, with ?grid /? = 5/3. To reduce edge effects, the
interval [?3/2, 3/2] was used for computations, although only the centre of this is shown
in the figure. There is quite good agreement between the numerical computation and the
analytical approximation, although the sidelobes decay more rapidly for the numerically
computed EK. This is not surprising because the absence of a truly hard cutoff in Fourier
space means one should expect less ?ringing? than the analytical approximation predicts.
The figure also shows good agreement between the weight function (based on the finite
sample) and the numerically computed EK. The insets show the approximation of Proposition 2 to g(z) for ? = 100 (a = 5.67, left) and ? = 104 (a = 9.67, right). As expected, the
addition of the 1/a2 -correction gives better agreement with the numerical result for z < a.
Numerical experiments also show that the mean squared error between the numerically
computed EK and the sinc approximation decreases like 1/ log(?). The is larger than the
na??ve estimate (1/a2 )2 ? 1/(log(?))4 based on the first correction term from Proposition
2, because the dominant part of the error comes from the region z > a where the 1/a
expansion breaks down.
2.2
Other kernels
Our analysis is not in fact restricted to the SE kernel. Consider an isotropic kernel, for
which the power spectrum S(s) depends on s = |s| only. Then we can again define from
equation 3 an effective cutoff sc on the range of s in the EK via ? 2 /? = S(sc ), so that
?
h(s)
= [1 + S(sc )/S(s)]?1 . The EK will then have the limiting form given in Proposi?
approaches a step function ?(sc ? s), i.e. if it becomes infinitely ?steep?
tion 1 if h(s)
around the point s = sc for sc ? ?. A quantitative criterion for this is that the slope
? 0 (sc )| should become much larger than 1/sc , the inverse of the range of the step func|h
? 0 (s) = S 0 (s)S(sc )S ?2 (s)[1 + S(sc )/S(s)]?2 , this is equivalent to requiring
tion. Since h
0
that ?sc S (sc )/4S(sc ) ? ?d log S(sc )/d log sc must diverge for sc ? ?. The result
of Proposition 1 therefore applies to any kernel whose power spectrum S(s) decays more
rapidly than any positive power of 1/s.
A trivial example of a kernel obeying this condition would be a superposition of finitely
many SE kernels with different lengthscales `2 ; the asymptotic behaviour of sc is then
governed by the smallest `. A less obvious case is the ?rational quadratic? k(r) =
[1 + (r/l)2 ]?(D+1)/2 which has an exponentially decaying power spectrum S(s) ?
exp(?2?`s). (This relationship is often used in the reverse direction, to obtain the power
spectrum of the Ornstein-Uhlenbeck (OU) kernel exp(?r/`).) Proposition 1 then applies,
with the width of the EK now scaling as 1/sc ? 1/ log(?).
The previous example is a special case of kernels which can be written
as superpositions
R
of SE kernels with a distribution p(`) of lengthscales `, k(r) = exp(?r 2 /2`2 )p(`) d`.
This is in fact the most general representation for an isotropic kernel which defines a valid
covariance function in any dimension D, see [9, ?2.10]. Such a kernel has power spectrum
Z ?
D/2
`D exp(?2? 2 `2 s2 )p(`) d`
S(s) = (2?)
(6)
0
and one easily verifies that the rational quadratic kernel, which has S(s) ? exp(?2?` 0 s),
is obtained for p(`) ? `?D?2 exp(?`20 /2`2 ). More generally, because the exponential
R 1/s
factor in equation 6 acts like a cutoff for ` > 1/s, one estimates S(s) ? 0 `D p(`) d`
for large s. This will decay more strongly than any power of 1/s for s ? ? if p(`) itself
decreases more strongly than any power of ` for ` ? 0. Any such choice of p(`) will
therefore yield a kernel to which Proposition 1 applies.
3
Understanding GP Learning Using the Equivalent Kernel
We now turn to using EK analysis to get a handle on average case learning curves for Gaussian processes. Here the setup is that a function ? is drawn from a Gaussian process, and we
obtain ? noisy observations of ? per unit x-space at random x locations. We are concerned
with the mean squared error (MSE) between the GP prediction f and ?. Averaging over
the noise process, the x-locations of the training data and the prior over ? we obtain the
average MSE as a function of ?. See e.g. [10] and [11] for an overview of earlier work on
GP learning curves.
To understand the asymptotic behaviour of for large ?, we now approximate Rthe true
GP predictions with the EK predictions from noisy data, given by fEK (x) = h(x ?
x0 )y(x0 )dx0 in the continuum limit of ?smoothed out? input locations. We assume as before
that y = target + noise, i.e. y(x) = ?(x) + ?(x) where E[?(x)?(x0 )] = (??2 /?)?(x ? x0 ).
Here ??2 denotes the true noise variance, as opposed to the noise variance assumed in the
EK; the Rscaling ofR ??2 with ? is explained in footnote 1. For a fixed target ?, the MSE
is = ( dx)?1 [?(x) ? fEK (x)]2 dx. Averaging over the noise process ? and target
function ? gives in Fourier space
Z n
o
2 Z
(? 2 /?)S? (s)/S 2 (s) + ??2 /? 2
2
?
? 2 (s) ds = ?
=
S? (s)[1 ? h(s)]
+ (??2 /?)h
ds
?
[1 + ? 2 /(?S(s))]2
(7)
where S? (s) is the power spectrum of the prior over target functions. In the case S(s) =
S? (s) and ? 2 = ??2 where the kernel is exactly matched to theRstructure of the target,
equation 7 gives the Bayes error B and simplifies to B = (? 2 /?) [1 + ? 2 /(?S(s))]?1 ds
(see also [5, eq. 14-16]). Interestingly, this is just the analogue (for a continuous power
spectrum of the kernel rather than a discrete set of eigenvalues) of the lower bound of [10]
?=2
0.5
0.03
0.025
?
0.02
0.015
0.01
?=4
0.1
0.005
0
?0.005
1
0.05
1
0.5
0.5
0
0
?0.5
?0.5
?1
25
?1
50
100
?
250
500
Figure 2: Left: plot of the asymptotic form of the EK (sc /r)J1 (2?sc r) for D = 2 and
? = 1225. Right: log-log plot of against log(?) for the OU and Matern-class processes
(? = 2, 4 respectively). The dashed lines have gradients of ?1/2 and ?3/2 which are the
predicted rates.
on the MSE of standard GP prediction from finite datasets. In experiments this bound
provides a good approximation to the actual average MSE for large dataset size n [11].
This supports our approach of using the EK to understand the learning behaviour of GP
regression.
Treating the denominator in the expression for B again as a hard cutoff at s = sc , which is
justified for large ?, one obtains for an SE target and learner ? ? 2 sc /? ? (log(?))D/2 /?.
To get analogous predictions for the mismatched case, one can write equation 7 as
Z
Z
[1 + ? 2 /(?S(s))] ? ? 2 /(?S(s))
??2
S? (s)
=
ds +
ds.
?
[1 + ? 2 /(?S(s))]2
[S(s)?/? 2 + 1]2
The first integral is smaller than (??2 /? 2 )B and can be neglected as long as B . In the
second integral we can again make the cutoff
R ? approximation?though now with s having
to be above sc ? to get the scaling ? sc sD?1 S? (s) ds. For target functions with a
?
power-law decay S? (s) ? s?? of the power spectrum at large s this predicts ? sD??
c
(log(?))(D??)/2 . So we generically get slow logarithmic learning, consistent with the
observations in [12]. For D = 1 and an OU target (? = 2) we obtain ? (log(?)) ?1/2 , and
for the Matern-class covariance function k(r) = (1 + r/`) exp(?r/`) (which has power
spectrum ? (3/`2 + 4? 2 s2 )?2 , so ? = 4) we get ? (log(?))?3/2 . These predictions
were tested experimentally using a GP learner with SE covariance function (` = 0.1 and
assumed noise level ? 2 = 0.1) against targets from the OU and Matern-class priors (with
` = 0.05) and with noise level ??2 = 0.01, averaging over 100 replications for each value
of ?. To demonstrate the predicted power-law dependence of on log(?), in Figure 2(right)
we make a log-log plot of against log(?). The dashed lines show the gradients of ?1/2
and ?3/2 and we observe good agreement between experimental and theoretical results
for large ?.
3.1
Using the Equivalent Kernel in Kernel Regression
Above we have used the EK to understand how standard GP regression works. One could
alternatively envisage using P
the EK to perform kernel regression, on given finite data sets,
producing a prediction ??1 i h(x? ? xi )yi at x? . Intuitively this seems appealing as a
cheap alternative to full GP regression, particularly for kernels such as the SE where the
EK can be calculated analytically, at least to a good approximation. We now analyze briefly
how such an EK predictor would perform compared to standard GP prediction.
Letting h?iRdenote averaging over noise, training input points and the test point and setting
f? (x? ) = h(x, x? )?(x)dx, the average MSE of the EK predictor is
P
pred = h[?(x) ? (1/?) i h(x, xi )yi ]2 i
R
?2 R
= h[?(x) ? f? (x)]2 + ?? h2 (x, x0 )dx0 i + ?1 h h2 (x, x0 )? 2 (x0 )dx0 i ? ?1 hf?2 (x)i
Z
Z
?2
(? 2 /?)S? (s)/S 2 (s) + ??2 /? 2
h? 2 i
ds
=
ds
+
2
2
2
?
[1 + ? /(?S(s))]
?
[1 + ? /(?S(s))]2
R
R
R
Here we have set h? 2 i = ( dx)?1 ? 2 (x) dx = S? (s) ds for the spatial average of the
squared target amplitude. Taking the matched case, (S? (s) = S(s) and ??2 = ? 2 ) as an
example, the first term (which is the one we get for the prediction from ?smoothed out?
2 D
training inputs, see eq. 7) is of order ? 2 sD
c /?, while the second one is ? h? isc /?. Thus
both terms scale in the same way, but the ratio of the second term to the first is the signalto-noise ratio h? 2 i/? 2 , which in practice is often large. The EK predictor will then perform
significantly worse than standard GP prediction, by a roughly constant factor, and we have
confirmed this prediction numerically. This result is somewhat surprising given the good
agreement between the weight function h(x? ) and the EK that we saw in figure 1, leading
to the conclusion that the detailed structure of the weight function is important for optimal
prediction from finite data sets.
In summary, we have derived accurate approximations for the equivalent kernel (EK) of
GP regression with the widely used squared exponential kernel, and have shown that the
same analysis in fact extends to a whole class of kernels. We have also demonstrated that
EKs provide a simple means of understanding the learning behaviour of GP regression,
even in cases where the learner?s covariance function is not well matched to the structure
of the target function. In future work, it will be interesting to explore in more detail the use
of the EK in kernel smoothing. This is suboptimal compared to standard GP regression as
we saw. However, it does remain feasible even for very large datasets, and may then be
competitive with sparse methods for approximating GP regression. From the theoretical
point of view, the average error of the EK predictor which we calculated may also provide
the basis for useful upper bounds on GP learning curves.
Acknowledgments: This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. This
publication only reflects the authors? views.
References
[1] B. W. Silverman. Annals of Statistics, 12:898?916, 1984.
[2] C. K. I. Williams. In M. I. Jordan, editor, Learning in Graphical Models, pages 599?621.
Kluwer Academic, 1998.
[3] T. J. Hastie and R. J. Tibshirani. Generalized Additive Models. Chapman and Hall, 1990.
[4] F. Girosi, M. Jones, and T. Poggio. Neural Computation, 7(2):219?269, 1995.
[5] A. Papoulis. Probability, Random Variables, and Stochastic Processes. McGraw-Hill, New
York, 1991. Third Edition.
[6] C. Thomas-Agnan. Numerical Algorithms, 13:21?32, 1996.
[7] T. Poggio, H. Voorhees, and A. Yuille. Tech. Report AI Memo 833, MIT AI Laboratory, 1985.
[8] B. Sch?olkopf and A. Smola. Learning with Kernels. MIT Press, 2002.
[9] M. L. Stein. Interpolation of Spatial Data. Springer-Verlag, New York, 1999.
[10] M. Opper and F. Vivarelli. In NIPS 11, pages 302?308, 1999.
[11] P. Sollich and A. Halees. Neural Computation, 14:1393?1428, 2002.
[12] P. Sollich. In NIPS 14, pages 519?526, 2002.
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1,846 | 2,677 | Exponentiated Gradient Algorithms for
Large-margin Structured Classification
Peter L. Bartlett
U.C.Berkeley
Michael Collins
MIT CSAIL
[email protected]
[email protected]
Ben Taskar
Stanford University
David McAllester
TTI at Chicago
[email protected]
[email protected]
Abstract
We consider the problem of structured classification, where the task is
to predict a label y from an input x, and y has meaningful internal structure. Our framework includes supervised training of Markov random
fields and weighted context-free grammars as special cases. We describe
an algorithm that solves the large-margin optimization problem defined
in [12], using an exponential-family (Gibbs distribution) representation
of structured objects. The algorithm is efficient?even in cases where the
number of labels y is exponential in size?provided that certain expectations under Gibbs distributions can be calculated efficiently. The method
for structured labels relies on a more general result, specifically the application of exponentiated gradient updates [7, 8] to quadratic programs.
1
Introduction
Structured classification is the problem of predicting y from x in the case where y has
meaningful internal structure. For example x might be a word string and y a sequence of
part of speech labels, or x might be a Markov random field and y a labeling of x, or x
might be a word string and y a parse of x. In these examples the number of possible labels
y is exponential in the size of x. This paper presents a training algorithm for a general
definition of structured classification covering both Markov random fields and parsing.
We restrict our attention to linear discriminative classification. We assume that pairs hx, yi
can be embedded in a linear feature space ?(x, y), and that a predictive rule is determined
by a direction (weight vector) w in that feature space. In linear discriminative prediction we
select the y that has the greatest value for the inner product h?(x, y), wi. Linear discrimination has been widely studied in the binary and multiclass setting [6, 4]. However, the case
of structured labels has only recently been considered [2, 12, 3, 13]. The structured-label
case takes into account the internal structure of y in the assignment of feature vectors, the
computation of loss, and the definition and use of margins.
We focus on a formulation where each label y is represented as a set of ?parts?, or equivalently, as a bit-vector. Moreover, we assume that the feature vector for y and the loss
for y are both linear in the individual bits of y. This formulation has the advantage that it
naturally covers both simple labeling problems, such as part-of-speech tagging, as well as
more complex problems such as parsing.
We consider the large-margin optimization problem defined in [12] for selecting the classification direction w given a training sample. The starting-point for these methods is a
primal problem that has one constraint for each possible labeling y; or equivalently a dual
problem where each y has an associated dual variable. We give a new training algorithm
that relies on an exponential-family (Gibbs distribution) representation of structured objects. The algorithm is efficient?even in cases where the number of labels y is exponential
in size?provided that certain expectations under Gibbs distributions can be calculated efficiently. The computation of these expectations appears to be a natural computational
problem for structured problems, and has specific polynomial-time dynamic programming
algorithms for some important examples: for example, the clique-tree belief propagation
algorithm can be used in Markov random fields, and the inside-outside algorithm can be
used in the case of weighted context-free grammars.
The optimization method for structured labels relies on a more general result, specifically
the application of exponentiated gradient (EG) updates [7, 8] to quadratic programs (QPs).
We describe a method for solving QPs based on EG updates, and give bounds on its rate of
convergence. The algorithm uses multiplicative updates on dual parameters in the problem.
In addition to their application to the structured-labels task, the EG updates lead to simple
algorithms for optimizing ?conventional? binary or multiclass SVM problems.
Related work [2, 12, 3, 13] consider large-margin methods for Markov random fields and
(weighted) context-free grammars. We consider the optimization problem defined in [12].
[12] use a row-generation approach based on Viterbi decoding combined with an SMO
optimization method. [5] describe exponentiated gradient algorithms for SVMs, but for
binary classification in the ?hard-margin? case, without slack variables. We show that the
EG-QP algorithm converges significantly faster than the rates shown in [5]. Multiplicative
updates for SVMs are also described in [11], but unlike our method, the updates in [11] do
not appear to factor in a way that allows algorithms for MRFs and WCFGs based on Gibbsdistribution representations. Our algorithms are related to those for conditional random
fields (CRFs) [9]. CRFs define a linear model for structured problems, in a similar way
to the models in our work, and also rely on the efficient computation of marginals in the
training phase. Finally, see [1] for a longer version of the current paper, which includes
more complete derivations and proofs.
2
The General Setting
We consider the problem of learning a function f : X ? Y, where X is a set and Y is a
countable set. We assume a loss function L : X ? Y ? Y ? R+ . The function L(x, y, y?)
measures the loss when y is the true label for x, and y? is a predicted label; typically, y? is
the label proposed by some function f (x). In general we will assume that L(x, y, y?) = 0
for y = y?. Given some distribution over examples (X, Y ) in X ? Y, our aim is to find a
function with low expected loss, or risk, EL(X, Y, f (X)).
We consider functions f which take a linear form. First, we assume a fixed function G
which maps an input x to a set of candidates G(x). For all x, we assume that G(x) ? Y,
and that G(x) is finite. A second component to the model is a feature-vector representation
? : X ? Y ? Rd . Given a parameter vector w ? Rd , we consider functions of the form
fw (x) = arg max h?(x, y), wi.
y?G(x)
Given n independent training examples (xi , yi ) with the same distribution as (X, Y ), we
will formalize a large-margin optimization problem that is a generalization of support vector methods for binary classifiers, and is essentially the same as the formulation in [12]. The
optimal parameters are taken to minimize the following regularized empirical risk function:
X
1
2
kwk + C
max (L(xi , yi , y) ? mi,y (w))
y
2
+
i
where mi,y (w) = hw, ?(xi , yi )i ? hw, ?(xi , y)i is the ?margin? on (i, y) and (z)+ =
max{z, 0}. This optimization can be expressed as the primal problem in Figure 1. Following [12], the dual of this problem is also shown in Figure 1. The dual is a quadratic
Primal
problem:
P
2
minw,? 21 kwk + C i i
Subject to the constraints:
?i, ?y ? G(xi ), hw, ?i,y i ? Li,y ? i
?i, i ? 0
Dual problem:
?), where
P max?? F (?
F (?
?) = C i,y ?i,y Li,y ?
P P
1 2
i,y
j,z ?i,y ?j,z h?i,y , ?j,z i
2C
Subject to the constraints:
X
?i,
?i,y = 1 ; ?i, y, ?i,y ? 0
y
P
?
?i,y where w? is the
Relationship between optimal values: w ? = C i,y ?i,y
?
arg min of the primal problem, and ?
? is the arg max of the dual problem.
Figure 1: The primal and dual problems. We use the definitions Li,y = L(xi , yi , y), and ?i,y =
?(xi , yi ) ? ?(xi , y). We assume that for all i, Li,y = 0 for y = yi . The constant C dictates the
relative penalty for values of the slack variables i which are greater than 0.
program F (?
?) in the dual variables ?i,y for all i = 1 . . . n, y ? G(xi ). The dual variables
for each example are constrained to form a probability distribution over Y.
2.1 Models for structured classification
The problems we are interested in concern structured labels, which have a natural decomposition into ?parts?. Formally, we assume some countable set of parts, R. We also assume
a function R which maps each object (x, y) ? X ? Y to a finite subset of R. Thus R(x, y)
is the set of parts belonging to a particular object. In addition we assume a feature-vector
representation ? of parts: this is a function ? : X ? R ? Rd . The feature vector for an
object (x, y) is then a sum of the feature vectors for its parts, and we also assume that the
loss function L(x, y, y?) decomposes into a sum over parts:
X
X
?(x, y) =
?(x, r)
L(x, y, y?) =
l(x, y, r)
r?R(x,y)
r?R(x,?
y)
Here ?(x, r) is a ?local? feature vector for part r paired with input x, and l(x, y, r) is
a ?local? loss for part r when proposed for the pair (x, y). For convenience we define
indicator variables I(x, y, r) which are 1 if r ? R(x, y), 0 otherwise. We also define sets
R(xi ) = ?y?G(xi ) R(xi , y) for all i = 1 . . . n.
Example 1: Markov Random Fields (MRFs) In an MRF the space of labels G(x),
and their underlying structure, can be represented by a graph. The graph G = (V, E) is
a collection of vertices V = {v1 , v2 , . . . vl } and edges E. Each vertex vi ? V has a set
of possible labels, Yi . The set G(x) is then defined as Y1 ? Y2 . . . ? Yl . Each clique in
the graph has a set of possible configurations: for example, if a particular clique contains
vertices {v3 , v5 , v6 }, the set of possible configurations of this clique is Y3 ? Y5 ? Y6 . We
define C to be the set of cliques in the graph, and for any c ? C we define Y(c) to be the set
of possible configurations for that clique. We decompose each y ? G(x) into a set of parts,
by defining R(x, y) = {(c, a) ? R : c ? C, a ? Y(c), (c, a) is consistent with y}. The
feature vector representation ?(x, c, a) for each part can essentially track any characteristics of the assignment a for clique c, together with any features of the input x. A number
of choices for the loss function l(x, y, (c, a)) arePpossible. For example, consider the Hamming loss used in [12], defined as L(x, y, y?) = i Iyi 6=y?i . To achieve this, first assign each
vertex vi to a single one of the cliques in which it appears. Second, define l(x, y, (c, a)) to
be the number of labels in the assignment (c, a) which are both incorrect and correspond
to vertices which have been assigned to the clique c (note that assigning each vertex to a
single clique avoids ?double counting? of label errors).
Example 2: Weighted Context-Free Grammars (WCFGs). In this example x is an
input string, and y is a ?parse tree? for that string, i.e., a left-most derivation for x under
some context-free grammar. The set G(x) is the set of all left-most derivations for x
Inputs: A learning rate ?.
Data structures: A vector ?? of variables, ?i,r , ?i, ?r ? R(xi ).
? = exp(P
?i,r )/Zi where Zi is a normalization term.
Definitions: ?i,y (?)
r?R(xi ,y)
Algorithm:
? Choose initial values ??1 for the ?i,r variables (these values can be arbitrary).
? For t = 1 . . . T + 1:
P
? For i = 1 . . . n, r ? R(xi ), calculate ?ti,r = y ?i,y (??t )I(xi , y, r).
P
P
t
? Set wt = C
i,r?R(xi ,yi ) ?i,r ?
i,r?R(xi ) ?i,r ?i,r
? For i = 1 . . . n, r ? R(xi ),
t+1
t
calculate updates ?i,r
= ?i,r
+ ?C (li,r + hwt , ?i,r i)
Output: Parameter values wT +1
Figure 2: The EG algorithm for structured problems. We use ?i,r = ?(xi , r) and li,r = l(xi , yi , r).
under the grammar. For convenience, we restrict the grammar to be in Chomsky-normal
form, where all rules in the grammar are of the form hA ? B Ci or hA ? ai, where
A, B, C are non-terminal symbols, and a is some terminal symbol. We take a part r to
be a CF-rule-tuple hA ? B C, s, m, ei. Under this representation A spans words s . . . e
inclusive in x; B spans words s . . . m; and C spans words (m + 1) . . . e. The function
R(x, y) maps a derivation y to the set of parts which it includes. In WCFGs ?(x, r) can
be any function mapping a rule production and its position in the sentence x, to a feature
vector. One example of a loss function would be to define l(x, y, r) to be 1 only if r?s
non-terminal A is not seen spanning words s . . . e in the derivation y. This would lead
to L(x, y, y?) tracking the number of ?constituent errors? in y?, where a constituent is a
(non-terminal, start-point, end-point) tuple such as (A, s, e).
3
EG updates for structured objects
We now consider an algorithm for computing ?
? ? = arg max???
F (?
?), where F (?
?) is the
?
dual form of the maximum margin problem, as in Figure 1. In particular, we are interested
in P
the optimal values of the primal form parameters, which are related to ?
? ? by w? =
?
C i,y ?i,y ?i,y . A key problem is that in many of our examples, the number of dual
variables ?i,y precludes dealing with these variables directly. For example, in the MRF
case or the WCFG cases, the set G(x) is exponential in size, and the number of dual
variables ?i,y is therefore also exponential.
We describe an algorithm that is efficient for certain examples of structured objects such
as MRFs or WCFGs. Instead of representing the ?i,y variables explicitly, we will instead
manipulate a vector ?? of variables ?i,r for i = 1 . . . n, r ? R(xi ). Thus we have one of
these ?mini-dual? variables for each part seen in the training data. Each of the variables
?i,r can take any value in the reals. We now define the dual variables ?i,y as a function of
? which takes the form of a Gibbs distribution:
the vector ?,
P
exp( r?R(xi ,y) ?i,r )
?
P
?i,y (?) = P
.
y 0 exp(
r?R(xi ,y 0 ) ?i,r )
Figure 2 shows an algorithm for maximizing F (?
?). The algorithm defines a sequence of
values ??1 , ??2 , . . .. In the next section we prove that the sequence F (?
?(??1 )), F (?
?(??2 )), . . .
converges to max? F (?
?). The algorithm can be implemented efficiently, independently
of the dimensionality of
? , provided that there is an efficient algorithm for computing
P?
?
?
marginal terms ?i,r = i,y ?i,y (?)I(x
i , y, r) for all i = 1 . . . n, r ? R(xi ), and all ?. A
P
P
?
key property is that the primal parameters w = C i,y ?i,y (?)?i,y = C i ?(xi , yi ) ?
C
P
i,y
?
?i,y (?)?(x
i , y) can be expressed in terms of the marginal terms, because:
X
i,y
?
?i,y (?)?(x
i , y) =
X
i,y
?
?i,y (?)
X
?(xi , r) =
r?R(xi ,y)
X
?i,r ?(xi , r)
i,r?R(xi )
P
P
and hence w = C i ?(xi , yi ) ? C i,r?R(xi ) ?i,r ?(xi , r). The ?i,r values can be calculated for MRFs and WCFGs in many cases, using standard algorithms. For example, in
the WCFG case, the inside-outside algorithm can be used, provided that each part r is a
context-free rule production, as described in Example 2 above. In the MRF case, the ? i,r
values can be calculated efficiently if the tree-width of the underlying graph is small.
?
Note that the main storage requirements of the algorithm in Figure 2 concern the vector ?.
This is a vector which has as many components as there are parts in the training set. In
practice, the number of parts in the training data can become extremely large. Fortunately,
an alternative, ?primal form? algorithm is possible. Rather than explicitly storing the ? i,r
variables, we can store a vector zt of the same dimensionality as wt . The ?i,r values can
be computed from zt . More explicitly, the main body of the algorithm in Figure 2 can be
replaced with the following:
? Set z1 to some initial value. For t = 1 . . . T + 1:
? Set wt = 0
t
t
? For i = 1 . . . n: Compute ?ti,r for r? R(xi ), using ?i,r
= ?C((t ? 1)li,r +
hz , ?i,r i);
P
P
t
Set wt = wt + C
r?R(xi ,yi ) ?i,r ?
r?R(xi ) ?i,r ?i,r
? Set zt+1 = zt + wt
1
It can be verified that if ?i, r, ?i,r
= ?Ch?i,r , z1 i, then this alternative algorithm defines
t
the same sequence of (implicit) ?i,r values, and (explicit) wt values, as the original algorithm. In the next section we show that the original algorithm converges for any choice of
1
initial values ??1 , so this restriction on ?i,r
should not be significant.
4
Exponentiated gradient (EG) updates for quadratic programs
We now prove convergence properties of the algorithm in Figure 2. We show that it is
an instantiation of a general algorithm for optimizing quadratic programs (QPs), which
relies on Exponentiated Gradient (EG) updates [7, 8]. In the general problem we assume a
positive semi-definite matrix A ? Rm?m , and a vector b ? Rm , specifying a loss function
Q(?
?) = b0 ?
? + 12 ?
? 0 A?
?. Here ?
? is an m-dimensional vector of reals. We assume
? is
P that ?
formed by the concatenation of n vectors ?
? i ? Rmi for i = 1 . . . n, where i mi = m.
We assume that each ?
? i lies in a simplex of dimension
mi , so that the feasible set is
mi
X
m
?i,j = 1; for all i, j, ?i,j ? 0}.
(1)
? = {?
?:?
? ? R ; for i = 1 . . . n,
j=1
Our aim is to find arg min???
Q(?
?). Figure 3 gives an algorithm?the ?EG-QP?
?
algorithm?for finding the minimum. In the next section we give a proof of its convergence properties.
The EG-QP algorithm can be used to find the minimum of ?F (?
?), and hence the maximum
of the dual objective F (?
?). We justify the algorithm in Figure 2 by showing that it is
equivalent to minimization of ?F (?
?) using the EG-QP algorithm. We give the following
theorem:
P
P P
Theorem 1 Define F (?
?) = C i,y ?i,y Li,y ? 21 C 2 i,y j,z ?i,y ?j,z h?i,y , ?j,z i,
P
and assume as in section 2 that Li,y =
r?R(xi ,y) l(xi , y, r) and ?(xi , y) =
P
1
?
? (? ) . . . ?
? (??T +1 ) defined by the algorithm
r?R(xi ,y) ?(xi , r). Consider the sequence ?
in Figure 2, and the sequence ?
?1 . . . ?
? T +1 defined by the EG-QP algorithm when applied
to Q(?
?) = ?F (?
?). Then under the assumption that ?
? (??1 ) = ?
? 1 , it follows that ?
? (??t ) = ?
?t
for t = 1 . . . (T + 1).
Inputs: A positive semi-definite matrix A, and a vector b, specifying a loss function
Q(?
?) = b ? ?
? + 12 ?
? 0 A?
?. Each vector ?
? is in ?, where ? is defined in Eq. 1.
Algorithm:
? Initialize ?
? 1 to a point in the interior of ?. Choose a learning rate ? > 0.
? For t = 1 . . . T
? Calculate s?t = ?Q(?
?t ) = b + A?
?t .
P t
t+1
t+1
t
? Calculate ?
?
as: ?i, j, ?i,j = ?i,j
exp{??sti,k }
exp{??sti,j }/ k ?i,k
Output: Return ?
? T +1 .
Figure 3: The EG-QP algorithm for quadratic programs.
Proof. We can write F (?)
? = C i,y ?i,y Li,y ? 21 C 2 k i ?(xi , yi ) ? i,y ?i,y ?(xi , y)k2 . It
t
P
(?
? )
follows that ?F
= CLi,y + Ch?(xi , y), wt i = C r?R(xi ,y) li,r + h?i,r , wt i where as
??i,y
P
P
t
?(xi , y)). The rest of the proof proceeds by inbefore wt = C( i ?(xi , yi ) ? i,y ?i,y
duction; due to space constraints we give a sketch of the proof here. The idea is to show that
?
? (??t+1 ) = ?
? t+1 under the inductive hypothesis that ?(
? ??t ) = ?
? t . This follows immediately
t
t+1
?
?
from the definitions of the mappings ?
? (? ) ? ?(
? ? ) and ?
?t ? ?
? t+1 in the two algot
P
?F
(
?
?
)
rithms, together with the identities sti,y = ? ??i,y = ?C r?R(xi ,y) (li,r + h?i,r , wt i)
P
P
P
t+1
t
and ?i,r
? ?i,r
= ?C (li,r + h?i,r , wt i).
4.1
Convergence of the exponentiated gradient QP algorithm
The following theorem shows how the optimization algorithm converges to an optimal solution. The theorem compares the value of the objective function for the algorithm?s vector
?
? t to the value for a comparison vector u ? ?. (For example, consider u as the solution
of the QP.) The convergence result is in terms of several properties of the algorithm and
the comparison vector u. The distance between u and ?
? 1 is measured using the KullbackLiebler (KL) divergence. Recall
that
the
KL-divergence
between two probability vectors
P
u
?, v? is defined as D(?
u, v?) = i ui log uvii . For sequences of probability vectors, u
? ? ?
with u
? = (?
u1 , . . . , u
?n ) and u
?i = (ui,1 , . . . , ui,m
),
we
can
define
a
divergence
as
the
sum
i
? u, v?) = Pn D(?
u
,
v
?
).
Two
other
key
parameters
of KL-divergences: for u
?, v? ? ?, D(?
i
i
i=1
are ?, the largest eigenvalue of the positive semidefinite symmetric matrix A, and
?))i ? min (?Q(?
?))i ? 2 n max |Aij | + max |bi | .
B = max max (?Q(?
???
?
i
i
ij
i
Theorem 2 For all u
? ? ?,
T
X
? u, ?
1
D(?
?1)
e?B ? 1 ? ?B
Q(?
?1 ) ? Q(?
?T +1 )
Q(?
?t ) ? Q(?
u) +
+ 2 2
.
?B
T t=1
?T
? B (1 ? ?(B + ?)e )
T
Choosing ? = 0.4/(B + ?) ensures that
T
? u, ?
1X
D(?
?1)
Q(?
?1 ) ? Q(?
?T +1 )
Q ?
? T +1 ?
Q(?
?t ) ? Q(?
u) + 2.5(B + ?)
+ 1.5
.
T t=1
T
T
The first lemma we require is due to Kivinen and Warmuth [8].
Lemma 1 For any u
? ? ?,
? u, ?
? u, ?
? ?t , ?
?Q(?
?t ) ? ?Q(?
u) ? D(?
? t ) ? D(?
? t+1 ) + D(?
? t+1 )
We focus on the third term. Define ?(i) Q(?
?) as the segment of the gradient vector corresponding
to
the
component
?
?
of
?,
?
and
define the random variable Xi,t , satisfying
i
t
Pr Xi,t = ? ?(i) Q(?
? ) j = ?i,j .
? ?t , ?
Lemma 2 D(?
? t+1 ) =
n
X
h
log E e
i=1
? ?t , ?
Proof. D(?
? t+1 ) =
n X
X
i=1
=
=
j
n X
X
i=1
n
X
t
?ij
log
t
?ij
?(Xi,t ?EXi,t )
log
?
e?B ? 1 ? ?B
B2
t
?ij
log
X
t
?ik
exp(???i,k )
k
X
X
n
var(Xi,t ).
i=1
t+1
?ij
j
i=1
n
X
i
t
?ik
exp ???i,k +
??
? it
!
? ?i
k
+ ??i,j
!
!
n
h
i e?B ? 1 ? ?B X
?(Xi,t ?EXi,t )
=
log E e
?
var(Xi,t ).
B2
i=1
i=1
This last inequality is at the heart of the proof of Bernstein?s inequality; e.g., see [10].
The second part of the proof of the theorem involves bounding this variance in terms of
the loss. The following lemma relies on the fact that this variance is, to first order, the
decrease in the quadratic loss, and that the second order term in the Taylor series expansion
of the loss is small compared to the variance, provided the steps are not too large. The
lemma and its proof require several definitions. For any d, let ? : Rd ? (0, 1)d be the
? i = exp(?i )/ Pd exp(?j ), for ?? ? Rd . We shall work in the
softmax function, ?(?)
j=1
exponential parameter space: let ??t be the exponential parameters at step t, so that the
updates are ??t+1 = ??t ? ??Q(?
?t ), and
?
? t = ?(??it ). Define the
the QP variables satisfy
i
t
random variables Xi,t,??, satisfying Pr Xi,t,?? = ? ?(i) Q(?
? ) j = ?(??i ) j . This takes
the same values as Xi,t , but
is given by a different exponential
parameter
its distribution
(??i instead of ??it ). Define ??t , ??t+1 = a??t + (1 ? a)??t+1 : a ? [0, 1] .
Lemma 3 For some ?? ? [??t , ??t+1 ],
?
n
X
var(Xi,t ) ? ? 2 (B + ?)
n
X
var(Xi,t,??) ? Q(?
?t ) ? Q(?
?t+1 ),
i=1
i=1
but for all ?? ? [??t , ??t+1 ], var(Xi,t,??) ? e?B var(Xi,t ). Hence,
n
X
var(Xi,t ) ?
i=1
1
Q(?
?t ) ? Q(?
?t+1 ) .
?B
? (1 ? ?(B + ?)e )
Thus, for ? < 0.567/(B + ?), Q(?
?t ) is non-increasing in t.
The proof is in [1]. Theorem 2 follows from an easy calculation.
5
Experiments
We compared an online1 version of the Exponentiated Gradient algorithm with the factored
Sequential Minimal Optimization (SMO) algorithm in [12] on a sequence segmentation
task. We selected the first 1000 sentences (12K words) from the CoNLL-2003 named
entity recognition challenge data set for our experiment. The goal is to extract (multiword) entity names of people, organizations, locations and miscellaneous entities. Each
word is labelled by 9 possible tags (beginning of one of the four entity types, continuation
of one of the types, or not-an-entity). We trained a first-order Markov chain over the tags,
In the online algorithm we calculate marginal terms, and updates to the w t parameters, one
training example at a time. As yet we do not have convergence bounds for this method, but we have
found that it works well in practice.
1
14
14
12
12
10
10
8
8
6
SMO
6
SMO
EG(th-2.7)
EG (eta .5)
EG(th-3)
4
EG (eta 1)
(a)
99
92
78
71
64
57
50
43
36
29
22
8
15
99
92
85
78
71
64
57
50
43
36
29
22
8
0
15
0
1
2
1
EG(th-4.5)
2
85
4
(b)
Figure 4: Number of iterations over training set vs. dual objective for the SMO and EG algorithms.
(a) Comparison with different ? values; (b) Comparison with ? = 1 and different initial ? values.
where our cliques are just the nodes for the tag of each word and edges between tags of
consecutive words. The feature vector for each node assignment consists of the word itself,
its capitalization and morphological features, etc., as well as the previous and consecutive
words and their features. Likewise, the feature vector for each edge assignment consists of
the two words and their features as well as surrounding words.
Figure 4 shows the growth of the dual objective function after each pass through the data
for SMO and EG, for several settings of the learning rate ? and the initial setting of the ?
parameters. Note that SMO starts up very quickly but slows down in a suboptimal region,
while EG lags at the start, but overtakes SMO and achieves a larger than 10% increase in
the value of the objective. These preliminary results suggest that a hybrid algorithm could
get the benefits of both, by starting out with several SMO updates and then switching to EG.
The key issue is to switch from the marginal ? representation SMO maintains to the Gibbs ?
representation that EG uses. We can find ? that produces ? by first computing conditional
?probabilities? that correspond to our marginals (e.g. dividing edge marginals by node
marginals in this case) and then letting ??s be the logs of the conditional probabilities.
References
[1] Long version of this paper. Available at http://www.ai.mit.edu/people/mcollins.
[2] Y. Altun, I. Tsochantaridis, and T. Hofmann. Hidden markov support vector machines. In
ICML, 2003.
[3] Michael Collins. Parameter estimation for statistical parsing models: Theory and practice of
distribution-free methods. In Harry Bunt, John Carroll, and Giorgio Satta, editors, New Developments in Parsing Technology. Kluwer, 2004.
[4] K. Crammer and Y. Singer. On the algorithmic implementation of multiclass kernel-based
vector machines. Journal of Machine Learning Research, 2(5):265?292, 2001.
[5] N. Cristianini, C. Campbell, and J. Shawe-Taylor. Multiplicative updatings for support-vector
learning. Technical report, NeuroCOLT2, 1998.
[6] N. Cristianini and J. Shawe-Taylor. An Introduction to Support Vector Machines and Other
Kernel-Based Learning Methods. Cambridge University Press, 2000.
[7] J. Kivinen and M. Warmuth. Exponentiated gradient versus gradient descent for linear predictors. Information and Computation, 132(1):1?63, 1997.
[8] J. Kivinen and M. Warmuth. Relative loss bounds for multidimensional regression problems.
Journal of Machine Learning Research, 45(3):301?329, 2001.
[9] John Lafferty, Andrew McCallum, and Fernando Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Proceedings of ICML-01, 2001.
[10] D. Pollard. Convergence of Stochastic Processes. Springer-Verlag, 1984.
[11] F. Sha, L. Saul, and D. Lee. Multiplicative updates for large margin classifiers. In COLT, 2003.
[12] B. Taskar, C. Guestrin, and D. Koller. Max margin Markov networks. In NIPS, 2003.
[13] I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun. Support vector machine learning for
interdependent and structured output spaces. ICML, 2004 (To appear).
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1,847 | 2,678 | Probabilistic Inference of Alternative Splicing
Events in Microarray Data
Ofer Shai, Brendan J. Frey, and Quaid D. Morris
Dept. of Electrical & Computer Engineering
University of Toronto, Toronto, ON
Qun Pan, Christine Misquitta, and Benjamin J. Blencowe
Banting & Best Dept. of Medical Research
University of Toronto, Toronto, ON
Abstract
Alternative splicing (AS) is an important and frequent step in mammalian
gene expression that allows a single gene to specify multiple products,
and is crucial for the regulation of fundamental biological processes. The
extent of AS regulation, and the mechanisms involved, are not well understood. We have developed a custom DNA microarray platform for
surveying AS levels on a large scale. We present here a generative model
for the AS Array Platform (GenASAP) and demonstrate its utility for
quantifying AS levels in different mouse tissues. Learning is performed
using a variational expectation maximization algorithm, and the parameters are shown to correctly capture expected AS trends. A comparison of
the results obtained with a well-established but low through-put experimental method demonstrate that AS levels obtained from GenASAP are
highly predictive of AS levels in mammalian tissues.
1 Biological diversity through alternative splicing
Current estimates place the number of genes in the human genome at approximately 30,000,
which is a surprisingly small number when one considers that the genome of yeast, a singlecelled organism, has 6,000 genes. The number of genes alone cannot account for the complexity and cell specialization exhibited by higher eukaryotes (i.e. mammals, plants, etc.).
Some of that added complexity can be achieved through the use of alternative splicing,
whereby a single gene can be used to code for a multitude of products.
Genes are segments of the double stranded DNA that contain the information required by
the cell for protein synthesis. That information is coded using an alphabet of 4 (A, C, G,
and T), corresponding to the four nucleotides that make up the DNA. In what is known
as the central dogma of molecular biology, DNA is transcribed to RNA, which in turn is
translated into proteins. Messenger RNA (mRNA) is synthesized in the nucleus of the cell
and carries the genomic information to the ribosome. In eukaryotes, genes are generally
comprised of both exons, which contain the information needed by the cell to synthesize
proteins, and introns, sometimes referred to as spacer DNA, which are spliced out of the
pre-mRNA to create mature mRNA. An estimated 35%-75% of human genes [1] can be
C1
(a)
C1
A
C1
C2
C1 A 3? C2
(b)
(c)
C2
C1 A 3?
A1
A2
C1
C2
C1
A1
C1
A2
C2
C2
C1
(d)
C1 A 5? C2
A 5? C2
C1
C1
C1
C2
C2
A
C2
C2
C2
C1
C2
Figure 1: Four types of AS. Boxes represent exons and lines represent introns, with the possible
splicing alternatives indicated by the connectors. (a) Single cassette exon inclusion/exclusion. C1
and C2 are constitutive exons (exons that are included in all isoforms) and flank a single alternative
exon (A). The alternative exon is included in one isoform and excluded in the other. (b) Alternative
3? (or donor) and alternative 5? (acceptor) splicing sites. Both exons are constitutive, but may contain alternative donor and/or acceptor splicing sites. (c) Mutually exclusive exons. One of the two
alternative exons (A1 and A2 ) may be included in the isoform, but not both. (d) Intron inclusion. An
intron may be included in the mature mRNA strand.
spliced to yield different combinations of exons (called isoforms), a phenomenon referred
to as alternative splicing (AS). There are four major types of AS as shown in Figure 1.
Many multi-exon genes may undergo more than one alternative splicing event, resulting in
many possible isoforms from a single gene. [2]
In addition to adding to the genetic repertoire of an organism by enabling a single gene to
code for more than one protein, AS has been shown to be critical for gene regulation, contributing to tissue specificity, and facilitating evolutionary processes. Despite the evident
importance of AS, its regulation and impact on specific genes remains poorly understood.
The work presented here is concerned with the inference of single cassette exon AS levels
(Figure 1a) based on data obtained from RNA expression arrays, also known as microarrays.
1.1 An exon microarray data set that probes alternative splicing events
Although it is possible to directly analyze the proteins synthesized by a cell, it is easier, and
often more informative, to instead measure the abundance of mRNA present. Traditionally,
gene expression (abundance of mRNA) has been studied using low throughput techniques
(such as RT-PCR or Northern blots), limited to studying a few sequences at a time and
making large scale analysis nearly impossible.
In the early 1990s, microarray technology emerged as a method capable of measuring the
expression of thousands of DNA sequences simultaneously. Sequences of interest are deposited on a substrate the size of a small microscope slide, to form probes. The mRNA
is extracted from the cell and reverse-transcribed back into DNA, which is labelled with
red and green fluorescent dye molecules (cy3 and cy5 respectively). When the sample of
tagged DNA is washed over the slide, complementary strands of DNA from the sample hybridize to the probes on the array forming A-T and C-G pairings. The slide is then scanned
and the fluorescent intensity is measured at each probe. It is generally assumed that the
intensity measure at the probe is linearly related to the abundance of mRNA in the cell over
a wide dynamic range.
Despite significant improvements in microarray technologies in recent years, microarray
data still presents some difficulties in analysis. Low measurements tend to have extremely
low signal to noise ratio (SNR) [7] and probes often bind to sequences that are very similar,
but not identical, to the one for which they were designed (a process referred to as cross-
C1
A
C1
A
C1:A
C1
C2
C2
3 Body probes
A:C2
A
C2
2 Inclusion junction probes
C1:C2
C1
C2
1 Exclusion junction probe
Figure 2: Each alternative splicing event is studied using six probes. Probes were chosen to measure
the expression levels of each of the three exons involved in the event. Additionally, 3 probes are used
that target the junctions that are formed by each of the two isoforms. The inclusion isoform would
express the junctions formed by C1 and A, and A and C2 , while the exclusion isoform would express
the junction formed by C1 and C2
hybridization). Additionally, probes exhibit somewhat varying hybridization efficiency,
and sequences exhibit varying labelling efficiency.
To design our data sets, we mined public sequence databases and identified exons that were
strong candidates for exhibiting AS (the details of that analysis are provided elsewhere
[4, 3]). Of the candidates, 3,126 potential AS events in 2,647 unique mouse genes were
selected for the design of Agilent Custom Oligonucleotide microarray. The arrays were
hybridized with unamplified mRNA samples extracted from 10 wild-type mouse tissues
(brain, heart, intestine, kidney, liver, lung, salivary gland, skeletal muscle, spleen, and
testis). Each AS event has six target probes on the arrays, chosen from regions of the
C1 exon, C2 exon, A exon, C1 :A splice junction, A:C2 splice junction, and C1 :C2 splice
junction, as shown in Figure 2.
2 Unsupervised discovery of alternative splicing
With the exception of the probe measuring the alternative exon, A (Figure 2), all probes
measure sequences that occur in both isoforms. For example, while the sequence of the
probe measuring the junction A:C1 is designed to measure the inclusion isoform, half of it
corresponds to a sequence that is found in the exclusion isoform. We can therefore safely
assume that the measured intensity at each probe is a result of a certain amount of both
isoforms binding to the probe. Due to the generally assumed linear relationship between
the abundance of mRNA hybridized at a probe and the fluorescent intensity measured,
we model the measured intensity as a weighted sum of the overall abundance of the two
isoforms.
A stronger assumption is that of a single, consistent hybridization profile for both isoforms
across all probes and all slides. Ideally, one would prefer to estimate an individual hybridization profile for each AS event studied across all slides. However, in our current
setup, the number of tissues is small (10), resulting in two difficulties. First, the number of
parameters is very large when compared to the number of data point using this model, and
second, a portion of the events do not exhibit tissue specific alternative splicing within our
small set of tissues. While the first hurdle could be accounted for using Baysian parameter
estimation, the second cannot.
2.1 GenASAP - a generative model for alternative splicing array platform
Using the setup described above, the expression vector x, containing the six microarray
measurements as real numbers, can be decomposed as a linear combination of the abundance of the two splice isoforms, represented by the real vector s, with some added noise:
x = ?s + noise, where ? is a 6 ? 2 weight matrix containing the hybridization profiles for
s1
x^ C
x^ C
1
2
s2
x^ A
x^ C :A
x^ A:C
2
x^ C1:C2
2
xC1:C2
1
r
xC
xC
2
xA
xC :A
xA:C
oC1
oC2
oA
oC :A
oA:C
1
1
1
2
oC :C
1
2
?n 2
Figure 3: Graphical model for alternative splicing. Each measurement in the observed expression
profile, x, is generated by either using a scale factor, r, on a linear combination of the isoforms, s, or
drawing randomly from an outlier model. For a detailed description of the model, see text.
the two isoforms across the six probes. Note that we may not have a negative amount of
a given isoform, nor can the presence of an isoform deduct from the measured expression,
and so both s and ? are constrained to be positive.
Expression levels measured by microarrays have previously been modelled as having
expression-dependent noise [7]. To address this, we rewrite the above formulation as
x = r(?s + ?),
(1)
where r is a scale factor and ? is a zero-mean normally distributed random variable with
a diagonal covariance matrix, ?, denoted as p(?) = N (?; 0, ?). The prior distribution for
the abundance of the splice isoforms is given by a truncated normal distribution, denoted
as p(s) ? N (s, 0, I)[s ? 0], where [?] is an indicator function such that [s ? 0] = 1 if
?i, si ? 0, and [s ? 0] = 0 otherwise.
Lastly, there is a need to account for aberrant observations (e.g. due to faulty probes, flakes
of dust, etc.) with an outlier model. The complete GenASAP model (shown in Figure 3)
accounts for the observations as the outcome of either applying equation (1) or an outlier
model. To avoid degenerate cases and ensure meaningful and interpretable results, the
number of faulty probes considered for each AS event may not exceed two, as indicated by
the filled-in square constraint node in Figure 3.
The distribution of x conditional on the latent variables, s, r, and o, is:
p(x|s, r, o) =
Y
N (xi ; r?i s, r2 ?i )[oi =0] N (xi ; Ei , Vi )[oi =1] ,
(2)
i
where oi ? {0, 1} is a bernoulli random variable indicating if the measurement at probe xi
is the result of the AS model or the outlier model parameterized by p(oi = 1) = ?i . The
parameters of the outlier model, E and V, are not optimized and are set to the mean and
variance of the data.
2.2 Variational learning in the GenASAP model
To infer the posterior distribution over the splice isoform abundances while at the same time
learning the model parameters we use a variational expectation-maximization algorithm
(EM). EM maximizes the log likelihood of the data by iteratively estimating the posterior
distribution of the model given the data in the expectation (E) step, and maximizing the
log likelihood with respect to the parameters, while keeping the posterior fixed, in the
maximization (M) step. Variational EM is used when, as in the case of GenASAP, the exact
posterior is intractable. Variational EM minimizes the free energy of the model, defined as
the KL-divergence between the joint distribution of the latent and observed variables and
the approximation to the posterior under the model parameters [5, 6].
We approximate the true posterior using the Q distribution given by
Q({s(t) }, {o(t) }, {r(t) }) =
T
Y
Q(r(t) )Q(o(t) |r(t) )
t=1
Y
(t)
(t)
Q(si |oi , r(t) )
i
T
?1 Y
=Z (t)
(3)
d
d
(t)
(t)
?(t) ? (t) N (s(t) ; ?(t)
? 0],
ro , ?ro )[s
t=1
where Z is a normalization constant, the superscript d indicates that ? is constrained to be
diagonal, and there are T iid AS events. For computational efficiency, r is selected from
a finite set, r ? {r1 , r2 , . . . , rC } with uniform probability. The variational free energy is
given by
XXZ
Q({s(t) }, {o(t) }, {r(t) })
.
F(Q, P ) =
Q({s(t) }, {o(t) }, {r(t) }) log
P ({s(t) }, {o(t) }, {r(t) }, {x(t) })
s
r
o
(4)
Variational EM minimizes the free energy by iteratively updating the Q distribution?s vari(t)d
(t)d
ational parameters (?(t) , ? (t) , ?ro , and ?ro ) in the E-step, and the model parameters (?,
?, {r1 , r2 , . . . , rC }, and ?) in the M-step. The resulting updates are too long to be shown
in the context of this paper and are discussed in detail elsewhere [3]. A few particular points
regarding the E-step are worth covering in detail here.
If the prior on s was a full normal distribution, there would be no need for a variational
approach, and exact EM is possible. For a truncated normal distribution, however, the mixing proportions, Q(r)Q(o|r) cannot be calculated analytically except for the case where s
is scalar, necessitating the diagonality constraint. Note that if ? was allowed to be a full
covariance matrix, equation (3) would be the true posterior, and we could find the sufficient
statistics of Q(s(t) |o(t) , r(t) ):
?1
T
(t) T ?1
?(t)
(I ? O(t) )?)?1 ?T (I ? O(t) )T ??1 x(t) r(t)
ro = (I + ? (I ? O ) ?
?1
?(t)
ro
= (I + ?T (I ? O(t) )T ??1 (I ? O(t) )?)
(5)
(6)
where O is a diagonal matrix with elements Oi,i = oi . Furthermore, it can be easily
shown that the optimal settings for ?d and ?d approximating a normal distribution with
full covariance ? and mean ? is
?doptimal = ?
?1
?doptimal
= diag(?
(7)
?1
)
(8)
In the truncated case, equation (8) is still true. Equation (7) does not hold, though, and
?doptimal cannot be found analytically. In our experiments, we found that using equation
(7) still decreases the free energy every E-step, and it is significantly more efficient than
using, for example, a gradient decent method to compute the optimal ?d .
Intuitive Weigh Matrix
Optimal Weight Matrix
50
50
40
40
30
30
20
20
10
0
10
Inclusion Isoform
0
Exclusion Isoform
Inclusion Isoform
(a)
Exclusion Isoform
(b)
Figure 4: (a) An intuitive set of weights. Based on the biological background, one would expect to
see the inclusion isoform hybridize to the probes measuring C1 , C2 , A, C1 :A, and A:C2 , while the
exclusion isoform hybridizes to C1 , C2 , and C1 :C2 . (b) The learned set of weights closely agrees
with the intuition, and captures cross hybridization between the probes
Contribution of
exclusion isoform
Contribution of
inclusion isoform
AS model
Original Data
(a)
(b)
RT?PCR
AS model
measurement
prediction
(% exclusion) (% exclusion)
14%
27%
72%
70%
8%
22%
outliers
(c)
Figure 5: Three examples of data cases and their predictions. (a) The data does not follow our notion
of single cassette exon AS, but the AS level is predicted accurately by the model.(b) The probe C1 :A
is marked as outlier, allowing the model to predict the other probes accurately. (c) Two probes are
marked as outliers, and the model is still successful in predicting the AS levels.
3 Making biological predictions about alternative splicing
The results presented in this paper were obtained using two stages of learning. In the first
step, the weight matrix, ?, is learned on a subset of the data that is selected for quality.
Two selection criteria were used: (a) sequencing data was used to select those cases for
which, with high confidence, no other AS event is present (Figure 1) and (b) probe sets
were selected for high expression, as determined by a set of negative controls. The second
selection criterion is motivated by the common assumption that low intensity measurements
are of lesser quality (see Section 1.1). In the second step, ? is kept fixed, and we introduce
the additional constraint that the noise is isotropic (? = ?I) and learn on the entire data
set. The constraint on the noise is introduced to prevent the model from using only a subset
of the six probes for making the final set of predictions.
We show a typical learned set of weights in Figure 4. The weights fit well with our intuition
of what they should be to capture the presence of the two isoforms. Moreover, the learned
weights account for the specific trends in the data. Examples of model prediction based on
the microarray data are shown in Figure 5.
Due to the nature of the microarray data, we do not expect all the inferred abundances to be
equally good, and we devised a scoring criterion that ranks each AS event based on its fit to
the model. Intuitively, given two input vectors that are equivalent up to a scale factor, with
inferred MAP estimations that are equal up to the same scale factor,
P we would like their
scores to be identical. The scoring criterion used, therefore is k (xk ? r?k s)2 /(xk +
Rank
500
1000
2000
5000
10000
15000
20000
30000
Pearson?s correlation
coefficient
0.94
0.95
0.95
0.79
0.79
0.78
0.75
0.65
False positive
rate
0.11
0.08
0.05
0.2
0.25
0.29
0.32
0.42
Table 1: Model performance evaluated at various ranks. Using 180 RT-PCR measurements, we are
able to predict the model?s performance at various ranks. Two evaluation criteria are used: Pearson?s
correlation coefficient between the model?s predictions and the RT-PCR measurements and false
positive rate, where a prediction is considered to be false positive if it is more than 15% away from
the RT-PCR measurement.
r?k s)2 , where the MAP estimations for r and s are used. This scoring criterion can be
viewed as proportional to the sum of noise to signal ratios, as estimated using the two
values given by the observation and the model?s best prediction of that observation.
Since it is the relative amount of the isoforms that is of most interest, we need to use the
inferred distribution of the isoform abundances to obtain an estimate for the relative levels
of AS. It is not immediately clear how this should be done. We do, however, have RTPCR measurements for 180 AS events to guide us (see figure 6 for details). Using the
top 50 ranked RT-PCR measurement, we fit three parameters, {a1 , a2 , a3 }, such that the
s2
+ a3 , where s1 is the
proportion of excluded isoform present, p, is given by p = a1 s1 +a
2 s2
MAP estimation of the abundance of the inclusion isoform, s2 is the MAP estimation of the
abundance of the exclusion isoform, and the RT-PCR measurement are used for target p.
The parameters are fitted using gradient descent on a least squared error (LSE) evaluation
criterion.
We used two criteria to evaluate the quality of the AS model predictions. Pearson?s correlation coefficient (PCC) is used to evaluate the overall ability of the model to correctly
estimate trends in the data. PCC is invariant to affine transformation and so is independent
of the transformation parameters a1 and a3 discussed above, while the parameter a2 was
found to effect PCC very little. The PCC stays above 0.75 for the top two thirds ranked predictions. The second evaluation criterion used is the false positive rate, where a prediction
is considered to be false positive if it is more than 15% away from the RT-PCR measurement. This allows us to say, for example, that if a prediction is within the top 10000, we
are 75% confident that it is within 15% of the actual levels of AS.
4 Summary
We designed a novel AS model for the inference of the relative abundance of two alternatively spliced isoforms from six measurements. Unsupervised learning in the model is
performed using a structured variational EM algorithm, which correctly captures the underlying structure of the data, as suggested by its biological nature. The AS model, though
presented here for a cassette exon AS events, can be used to learn any type of AS, and with
a simple adjustment, multiple types.
The predictions obtained from the AS model are currently being used to verify various
claims about the role of AS in evolution and functional genomics, and to help identify
sequences that affect the regulation of AS.
% Exclusion isoform
RT?PCR measurement Vs. AS model predictions
RT?PCR measurements:
90
80
AS model prediction
Int
es
Te tine
sti
Kid s
n
Sa ey
liva
Br ry
ain
Sp
le
Liv en
er
Mu
sc
Lu le
ng
100
70
60
50
40
30
14 22 27 32 47 46 66 78 63
20
AS model prediction:
10
27 24 26 26 51 75 60 85 100
(a)
0
0
20
40
60
80
RT?PCR measurement
100
(b)
Figure 6: (a) Sample RT-PCR. RNA extracted from the cell is reverse-transcribed to DNA, amplified
and labelled with radioactive or fluorescent molecules. The sample is pulled through a viscous gel in
an electric field (DNA, being an acid, is positively charged). Shorter strands travel further through
the gel than longer ones, resulting in two distinct bands, corresponding to the two isoforms, when
exposed to a photosensitive or x-ray film. (b) A scatter plot showing the RT-PCR measurements as
compared to the AS model predictions. The plot shows all available RT-PCR measurements with a
rank of 8000 or better.
The AS model presented assumes a single weight matrix for all data cases. This is an
oversimplified view of the data, and current work is being carried out in identifying probe
specific expression profiles. However, due to the low dimensionality of the problem (10 tissues, six probes per event), care must be taken to avoid overfitting and to ensure meaningful
interpretations.
Acknowledgments
We would like to thank Wen Zhang, Naveed Mohammad, and Timothy Hughes for their
contributions in generating the data set. This work was funded in part by an operating and
infrastructure grants from the CIHR and CFI, and a operating grants from NSERC and a
Premier?s Research Excellence Award.
References
[1] J. M. Johnson et al. Genome-wide survey of human alternative pre-mrna splicing with exon
junction microarrays. Science, 302:2141?44, 2003.
[2] L. Cartegni et al. Listening to silence and understanding nonsense: exonic mutations that affect
splicing. Nature Gen. Rev., 3:285?98, 2002.
[3] Q. Pan et al. Revealing global regulatory features of mammalian alternative splicing using a
quantitative microarray platform. Molecular Cell, 16(6):929?41, 2004.
[4] Q. Pan et al. Alternative splicing of conserved exons is frequently species specific in human and
mouse. Trends Gen., In Press, 2005.
[5] M. I. Jordan, Z. Ghahramani, T. Jaakkola, and Lawrence K. Saul. An introduction to variational
methods for graphical models. Machine Learning, 37(2):183? 233, 1999.
[6] R. M. Neal and G. E. Hinton. A view of the em algorithm that justifies incremental, sparse, and
other variants. In Learning in Graphical Models. Cambridge, MIT Press, 1998.
[7] D. M. Rocke and B. Durbin. A model for measurement error for gene expression arrays. Journal
of Computational Biology, 8(6):557?69, 2001.
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1,848 | 2,679 | Message Errors in Belief Propagation
Alexander T. Ihler, John W. Fisher III, and Alan S. Willsky
Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
[email protected], [email protected], [email protected]
Abstract
Belief propagation (BP) is an increasingly popular method of performing approximate inference on arbitrary graphical models. At times,
even further approximations are required, whether from quantization or
other simplified message representations or from stochastic approximation methods. Introducing such errors into the BP message computations
has the potential to adversely affect the solution obtained. We analyze
this effect with respect to a particular measure of message error, and show
bounds on the accumulation of errors in the system. This leads both to
convergence conditions and error bounds in traditional and approximate
BP message passing.
1
Introduction
Graphical models and message-passing algorithms defined on graphs are a growing field of
research. In particular, the belief propagation (BP, or sum-product) algorithm has become
a popular means of solving inference problems exactly or approximately. One part of
its appeal is its optimality for tree-structured graphical models (models which contain no
loops). However, its is also widely applied to graphical models with cycles. In these cases
it may not converge, and if it does its solution is approximate; however in practice these
approximations are often good. Recently, further justifications for loopy belief propagation
have been developed, including a few convergence results for graphs with cycles [1?3].
The approximate nature of loopy BP is often a more than acceptable price for efficient inference; in fact, it is sometimes desirable to make additional approximations. There may be
a number of reasons for this?for example, when exact message representation is computationally intractable, the messages may be approximated stochastically [4] or deterministically by discarding low-likelihood states [5]. For BP involving continuous, non-Gaussian
potentials, some form of approximation is required to obtain a finite parametrization for
the messages [6?8]. Additionally, graph simplification by edge removal may be regarded
as a coarse form of message approximation. Finally, one may wish to approximate the
messages and reduce their representation size for another reason?to decrease the communications required for distributed inference applications. In a distributed environment, one
may approximate the transmitted message to reduce its representational cost [9], or discard
it entirely if it is deemed ?sufficiently similar? to the previously sent version [10]. This may
significantly reduce the amount of communication required.
Given that message approximation may be desirable, we would like to know what effect
the introduced errors have on our overall solution. To characterize the effect in graphs
with cycles, we analyze the deviation from a solution given by ?exact? loopy BP (not, as is
typically considered, the deviation of loopy BP from the true marginal distributions). In the
process, we also develop some results on the convergence of loopy BP. Section 3 describes
the major themes of the paper; but first we provide a brief summary of belief propagation.
2
Graphical Models and Belief Propagation
Graphical models provide a convenient means of representing conditional independence
relations among large numbers of random variables. Specifically, each node s in a graph
is associated with a random variable xs , while the set of edges E is used to describe the
conditional dependency structure of the variables. A distribution satisfies the conditional
independence relations specified by an undirected graph if it factors into a product of potential functions ? defined on the cliques (fully-connected subsets) of the graph; the converse
is also true if p(x) is strictly positive [11]. Here we consider graphs with at most pairwise
interactions (a typical assumption in BP), where the distribution factors according to
Y
Y
p(x) =
?st (xs , xt )
?s (xs )
(1)
s
(s,t)?E
The goal of belief propagation [12], or BP, is to compute the marginal distribution p(xt ) at
each node t. BP takes the form of a message-passing algorithm between nodes, expressed
in terms of an update to the outgoing message from each node t to each neighbor s in terms
of the (previous iteration?s) incoming messages from t?s neighbors ?t ,
Z
Y
mts (xs ) ? ?ts (xt , xs )?t (xt )
mut (xt )dxt
(2)
u??t \s
Typically each message is normalized so as to integrate to unity (and we assume that such
normalization is possible). At any iteration, one may calculate the belief at node t by
Y
Mti (xt ) ? ?t (xt )
miut (xt )
(3)
u??t
For tree-structured graphical models, belief propagation can be used to efficiently perform
exact marginalization. Specifically, the iteration (2) converges in a finite number of iterations (at most the length of the longest path in the graph), after which the belief (3) equals
the correct marginal p(xt ). However, as observed by [12], one may also apply belief propagation to arbitrary graphical models by following the same local message passing rules
at each node and ignoring the presence of cycles in the graph; this procedure is typically
referred to as ?loopy? BP.
For loopy BP, the sequence of messages defined
by (2) is not guaranteed to converge to a fixed
point after any number of iterations. Under relatively mild conditions, one may guarantee the existence of fixed points [13]. However, they may
not be unique, nor are the results exact (the belief Mti does not converge to the true marginal).
In practice however the procedure often arrives at
a reasonable set of approximations to the correct
marginal distributions.
1
1
3
2
4
2
3
4
4
1
3
2
3
1
1
4
1
2
Figure 1: For a graph with cycles, one
may show an equivalence between n iterations of loopy BP and the n-level
computation tree (shown here for n = 3
and rooted at node 1; example from [2]).
It is sometimes convenient to think of loopy BP
in terms of its computation tree [2]. The n-level
computation tree rooted at some node t is a treestructured ?unrolling? of the graph, so that n iterations of loopy BP on the original graph
is equivalent at the node t to exact inference on the computation tree. An example of this
structure is shown in Figure 1.
log m/m
?
m(x)
m(x)
?
(a)
} log d (e)
(b)
Figure 2: (a) A message m(x), solid, and its approximation m(x),
?
dashed. (b) Their log-ratio
log m(x)/m(x);
?
log d (e) characterizes their similarity by measuring the error?s dynamic range.
3
Overview of Results
To orient the reader, we lay out the order and general results which are obtained in this paper. We begin by considering multiplicative error functions which describe the difference
between a ?true? message m(x) (typically meaning consistent with some BP fixed-point)
and some approximation m(x)
?
= m(x) ? e(x). We apply a particular functional measure
d (e) (defined below) and show how this measure behaves with respect to the BP equations (2) and (3). When applied to traditional BP, this results in a novel sufficient condition
for its convergence to a unique solution, specifically
X d (?ut )2 ? 1
max
< 1,
(4)
2
(s,t)?E
d (?ut ) + 1
u??t \s
and may be further improved in most cases. The condition (4) is shown to be slightly
stronger than the sufficient condition given in [2]. More importantly, however, the method
in which it is derived allows us to generalize to many other situations:
? The condition (4) is easily improved for graphs with irregular geometry or potential
strengths
? The method also provides a bound on the distance between any two BP fixed points.
? The same methodology may be applied to the case of quantized or otherwise approximated messages, yielding bounds on the ensuing error (our original motivation).
? By regarding message errors as a stochastic process and applying a few additional
assumptions, a similar analysis obtains alternate, tighter estimates (though not
necessarily bounds) of performance.
4
Message Approximations
In order to discuss the effects and propagation of errors introduced to the BP messages,
we first require a measure of the difference between two messages. Although there are
certainly other possibilities, it is very natural to consider the message deviations (which we
denote ets ) to be multiplicative, or additive in the log-domain, and examine a measure of
the error?s dynamic range:
p
m
? ts (xs ) = mts (xs )ets (xs )
d (ets ) = max ets (a)/ets (b)
(5)
a,b
Then, we have that mts (x) = m
? ts (x)?x if and only if log d (ets ) = 0. This measure may
also be related to more traditional error measures, including an absolute error on log m(x),
a floating-point precision on m(x), and the Kullback-Leibler divergence D(m(x)km(x));
?
for details, see [14]. In this light our analysis of message approximation (Section 5.3)
may be equivalently regarded as a statement about the required precision for an accurate
implementation of loopy BP. Figure 2 shows an example message m(x) and approximation
m(x)
?
along with their associated error e(x).
To facilitate our analysis, we split the message update operation (2) into two parts. In the
first, we focus on the message products
Y
Y
Mts (xt ) ? ?t (xt )
mut (xt )
Mt (xt ) ? ?t (xt )
mut (xt )
(6)
u??t \s
u??t
where as usual, the proportionality constant is chosen to normalize M . We show the message error metric is (sub-)additive, i.e. that the errors in each incoming message (at most)
add in their impact on M . The secondZoperation is the message convolution
mts (xs ) ?
?ts (xt , xs )Mts (xt )dxt
(7)
where M is a normalized message or product of messages. We demonstrate a level of
contraction, that is, the approximation of mts is measurably better than the approximation
of Mts used to construct it.
We use the convention that lowercase quantities (mts , ets , . . .) refer to messages and message errors, while uppercase ones (Mts , Ets , Mt , . . .) refer to products of messages or
errors?all incoming messages to node t (Mt and Et ), or all except the one from s (Mts
and Ets ). Due to space constraints, many omitted details and proofs can be found in [14].
4.1
Additivity and Error Contraction
The log of (5) is sub-additive, since
for several
incoming
? ut (x)} we have
?
?
?Y messages
? X{m
?
log d (Ets ) = log d Mts /Mts = log d
eut ?
log d (eut )
(8)
We may also derive a minimum rate of contraction on the errors. We consider the message
from t to s; since all quantities in this section relate to mts and Mts we suppress the
subscripts. The error measure Rd (e) is given by
R
?(xt , a)M (xt )E(xt )dxt
?(xt , b)M (xt )dxt
2
2
R
R
d (e) = d (m/m)
?
= max
?
(9)
a,b
?(xt , a)M (xt )dxt
?(xt , b)M (xt )E(xt )dxt
subject to certain constraints, such
R as positivity
R of the messages and potentials. Since
?f, g > 0,
f (x) dx / g(x) dx ? max f (x)/g(x)
(10)
x
log d (e) ?
we can directly obtain the two bounds:
2
2
2
4
(11)
d (e) ? d (E) and d (e) ? d (?)
where we have extended the measure d (?) to
functions of two variables (describing a minimum rate of mixing across the potential) by
log d (E)
2
log d (?)
?(a, b)
2
.
(12)
d (?) = max
d(E)+1
a,b,c,d ?(c, d)
log d(?)
d(?) +d(E)
However, with some work one may show [14] the
stronger measure of contraction,
log d (E) ?
2
d (?) d (E) + 1
.
(13)
d (e) ?
2
Figure 3: Bounds on the error output
d (?) + d (E)
2
2
d (e) as a function of the error in the
Sketch of proof: While the full proof is rather involved,
product of incoming messages d (E).
we outline the procedure here. First, use (10) to show
that the maximum of (9) given d (?) is attained by potentials of the form ?(x, a) ? 1 + K?A (x)
and ?(x, b) ? 1 + K?B (x), where K = d (?)2 ? 1 and ?A and ?B take
R on only values {0, 1},
along
with
a
similar
form
for
E(x).
Then
define
the
variables
M
=
M (x)?A (x), MAE =
A
R
M (x)?A (x)?E (x), etc., and optimize given the constraints 0 ? MA , MB , ME ? 1, MAE ?
min[MA , ME ], and MBE ? max[0, ME ? (1 ? MB )] (where the last constraint arises from the
fact that ME + MB ? MBE ? 1). Simplifying and taking the square root yields (13).
The bound (13) is shown in Figure 3; note that it improves both error bounds (11), shown
as straight lines. In the next section, we use (8)-(13) to analyze the behavior of loopy BP.
5
Implications in Graphs with Cycles
We begin by examining loopy BP with exact message passing, using the previous results
to derive a new sufficient condition for convergence to a unique fixed point. When this
condition is not satisfied, we instead obtain a bound on the relative distances between any
two fixed points of the loopy BP equations. We then consider the effect of introducing
additional errors into the messages passed at each iteration, showing sufficient conditions
for this operation to converge, and a bound on the resulting error from exact loopy BP.
5.1
Convergence of Loopy BP & Fixed Point Distance
Tatikonda and Jordan [2] showed that the convergence and fixed points of loopy BP may
be considered in terms of a Gibbs measure on the graph?s computation tree, implying that
loopy BP is guaranteed to converge if the graph satisfies Dobrushin?s condition [15]. Dobrushin?s condition is a global measure and difficult to verify; given in [2] is a sufficient
condition (often called Simon?s condition):
X
max
log d (?ut ) < 1
(14)
t
u??t
where d (?) is defined as in (12). Using the previous section?s analysis, we may argue
something slightly stronger. Let us take the ?true? messages mts to be any fixed point of
BP, and ?approximate? them at each iteration by performing loopy BP from some arbitrary
initial conditions. Now suppose that the largest message-product error log d (Eut ) in any
node u with parent t at level i of the computation tree (corresponding to iteration n ? i out
of n total iterations of loopy BP) is bounded above by some constant log ?i . Note that this
2
is trivially true (at any i) for the constant log ?i = max(u,t)?E |?t | log d (?ut ) . Now, we
may bound d (Ets ) at any replicate of node t with parent s on level i ? 1 of the tree by
2
X
d (?ut ) ?i + 1
log d (Ets ) ? gts (log ?i ) =
log
.
(15)
2
d (?ut ) + ?i
u??t \s
and we may define log ?i?1 = maxt,s gts (log ?i ) to bound the error at level i?1. Loopy BP
will converge if the sequence ?i , ?i?1 , . . . is strictly decreasing for all ? > 1, i.e. gts (z) < z
0
00
for all z > 0. This is guaranteed by the conditions gts (0) = 0, gts
(0) < 1 and gts
(z) < 0.
0
The first is easy to show, the third can be verified by algebra, and the condition gts
(0) < 1
can be rewritten to give the convergence criterion
X d (?ut )2 ? 1
<1
(16)
max
2
(s,t)?E
d (?ut ) + 1
u??t \s
We may relate (16) to Simon?s condition (14) by expanding the set ?t \ s to the larger ?t
2
and noting that log x ? xx2 ?1
+1 for all x ? 1 with equality as x ? 1. Doing so, we see
that Simon?s condition is sufficient to guarantee (16), but that (16) may be true (implying
convergence) when Simon?s condition is not satisfied. The improvement over Simon?s condition becomes negligible as connectivity increases (assuming the graph has approximately
equal-strength potentials), but can be significant for low connectivity. For example, if the
graph consists of a single loop then each node t has at most two neighbors. In this case,
the contraction (16) tells us that the outgoing message in either direction is always closer
to the BP fixed point than the incoming message. Thus we obtain the result of [1], that (for
finite-strength potentials) BP always converges to a unique fixed point on graphs containing
a single loop. Simon?s condition, on the other hand, is too loose to demonstrate this fact.
If the condition (16) is not satisfied, then the sequence {?i } is not always decreasing and
there may be multiple fixed points. In this case, the sequence {?i } as defined will decrease
until it reaches the largest value ? such that maxts gts (log ?) = log ?. Since the choice of
initialization was arbitrary, we may opt to initialize to any other fixed point, and observe
that the difference Et between these two fixed point beliefs is bounded by
2
X
d (?ut ) ? + 1
log d (Et ) ?
log
(17)
2
d (?ut ) + ?
u??t
log d (Et )
10
8
6
Simple bound, grids (a) and (b)
Nonuniform bound, grid (a)
Nonuniform bound, grid (b)
Simons condition
4
2
0
0
(a)
(b)
??
0.5
1
1.5
2
2.5
(c)
Figure 4: Two small (5 ? 5) grids, with (a) all equal-strength potentials log d (?)2 = ? and (b)
several weaker ones (log d (?)2 = .5?, thin lines). The methods described provide bounds (c) on the
distance between any two fixed points as a function of potential strength ?, all of which improve on
Simon?s condition. See text for details.
Thus, the fixed points of BP lie in some potentially small set. If log ? is small (the condition (16) is nearly satisfied) then although we cannot guarantee convergence to a unique
fixed point, we can guarantee that every fixed point and our estimate are all mutually close
(in a log-ratio sense).
5.2
Improving the Bounds by Path-counting
If we are willing to put a bit more effort into our bound-computation, we may be able to
improve it.In particular the proofs of (16)-(17) assume that, as a message error propagates
through the graph, repeated convolution with only the strongest set of potentials is possible.
But often even if the worst potentials are quite strong, every cycle which contains them
also contains several weaker potentials. Using an iterative algorithm much like BP itself,
we may obtain a more globally aware estimate of error propagation.
Let us consider a message-passing procedure (potentially
offline) in which node
? performed
?
i
on the message error d eits at iteration i to its neighbor s.
t passes a (scalar) bound ?ts
2
1
The bound may be initialized to ?ts
= d (?ts ) , and the next iteration?s (updated) outgoing
bound is given by the pair of equations
2
X
d (?ts ) ?its + 1
i+1
i
i
log ?ts
= log
log
?
=
log ?ut
(18)
ts
2
d (?ts ) + ?its
u??t \s
Here, as in Section 5.1, ?its bounds the error d (Ets ) in the product of incoming messages.
i
If (18) converges to log ?ts
? 0 for all t, s P
we may guarantee a unique fixed point for
i
loopy BP; if not, we may compute log ?it = ?t log ?ut
to obtain a bound on the belief
error at each node t. If every node is identical (same number of neighbors, same potential
strengths) this yields the same bound as (17); however, if the graph or potential strengths are
inhomogeneous it provides a strictly stronger bound on loopy BP convergence and errors.
This situation is illustrated in Figure 4?we specify two 5?5 grids in terms of their potential
strengths and compute bounds on the log-range of their fixed point beliefs. (While potential
strength does not completely specify the graphical model, it is sufficient for all the bounds
2
considered here.) One grid (a) has equal-strength potentials log d (?) = ?, while the other
has many weaker potentials (?/2). The worst-case bounds are the same (since both have
a node with four strong neighbors), shown as the solid curve in (c). However, the dashed
curves show the estimate of (18), which improves only slightly for the strongly coupled
graph (a) but considerably for the weaker graph (b). All three bounds give considerably
more information than Simon?s condition (dotted vertical line).
5.3
Introducing additional errors
As discussed in the introduction, we may wish to introduce or allow additional errors in our
messages at each stage, in order to improve the computational or communication efficiency
of the algorithm. This may be the result of an actual distortion imposed on the message
(perhaps to decrease its complexity, for example quantization), from censoring the message
update (reusing the message from the previous iteration) when the two are sufficiently
similar, or from approximating or quantizing the model parameters (potential functions).
Any of these additional errors can be easily incorporated into our framework.
If at each iteration, we introduce an additional (perhaps stochastic) error to each message
which has a dynamic range bounded by some constant ?, the relationships of (18) become
2
X
d (?ts ) ?its + 1
i+1
i
i
log ?ts
= log
+
log
?
log
?
=
log ?ut
(19)
ts
2
d (?ts ) + ?its
u??t \s
and gives a bound on the steady-state error (distance from a fixed point) in the system.
5.4
Stochastic Analysis
Unfortunately, the above bounds are often pessimistic compared to actual performance. By
treating the perturbations as stochastic we may obtain a more realistic estimate (though no
longer a strict bound) on the resulting error. Specifically, let us describe the error func2
tions log ets (xs ) for each xs as a random variable with mean zero and variance ?ts
. By
assuming that the errors inP
each incoming message are uncorrelated, we obtain additivity
2
of their variances: ?2ts = u??t \s ?ut
. The assumption of uncorrelated errors is clearly
questionable since propagation around loops may couple the incoming message errors, but
is common in quantization analysis, and we shall see that it appears reasonable in practice.
We would also like to estimate the contraction of variance incurred in the convolution step.
We may do so by applying a simple sigma-point quadrature (?unscented?) approximation [16], in which the standard deviation of the convolved function mts (xs ) is estimated
by applying the same nonlinearity (13) to the standard deviation of the error on the incoming product Mts . Thus, similarly to (18) and (19), we have
!2
?
2
X
d (?ts ) ?ts + 1
2
2
+ (log ?)2
(log ?ts )2 =
?ut
(20)
?ts = log
2
?ts + d (?ts )
u??t \s
The steady-state
solution of (20) yields an estimate of the variances of the log-belief log pt
P
2
by ?t2 = u??t ?ut
; this estimate is typically much smaller than the bound (18) due to the
strict sub-additive relationship between the standard deviations. Although it is not a bound,
using a Chebyshev-like argument we may conclude that, for example, the 2?t distance will
be greater than the typical errors observed in practice.
6
Experiments
We demonstrate the error bounds for perturbed messages with a set of Monte Carlo trials.
In particular, for each trial we construct a binary-valued 5 ? 5 grid with uniform potential strengths, which are either (1) all positively correlated, or (2) randomly chosen to be
positively or negatively correlated (equally likely); we also assign random single-node potentials to each xs . We then run a quantized version of BP, rounding each log-message
to discrete values separated by 2 log ? (ensuring that the newly introduced error satisifies
d (e) ? ?). Figure 5 shows the maximum belief error in each of 100 trials of this procedure
for various values of ?.
Also shown are the bound on belief error developed in Section 5.3 and the 2? estimate
computed assuming uncorrelated message errors. As can be seen, the stochastic estimate
is often a much tighter, more accurate assessment of error, but it does not possess the
same strong theoretical guarantees. Since, as observed in analysis of quantization and
stability in digital filtering [17], the errors introduced by quantization are typically close to
independent, the assumptions of the stochastic estimate are reasonable and empirically we
observe that the estimate and actual errors behave similarly.
max log d (Et )
max log d (Et )
Strict bound
Stochastic estimate
Positive corr. potentials
Mixed corr. potentials
1
10
0
10
10
10
10
1
2
log ? ?
3
3
10
10
2
10
1
2
0
10
(a) log d (?) = .25
10 1
100
10 -1
10 -2
log ? ?
10 -3 -3
10
10-2
10-1
100
2
(b) log d (?) = 1
Figure 5: Maximum belief errors incurred as a function of the quantization error. The scatterplot
indicates the maximum error measured in the graph for each of 200 Monte Carlo runs; this is strictly
bounded above by the solution of (18), solid, and bounded with high probability (assuming uncorrelated errors) by (20), dashed.
7
Conclusions
We have described a particular measure of distortion on BP messages and shown that it is
sub-additive and measurably contractive, leading to sufficient conditions for loopy BP to
converge to a unique fixed point. Furthermore, this enables analysis of quantized, stochastic, or other approximate forms of BP, yielding sufficient conditions for convergence and
bounds on the deviation from exact message passing. Assuming the perturbations are uncorrelated can often give tighter estimates of the resulting error. For additional details as
well as some further consequences and extensions, see [14].
The authors would like to thank Erik Sudderth, Martin Wainwright, Tom Heskes, and Lei Chen for
many helpful discussions. This research was supported in part by MIT Lincoln Laboratory under
Lincoln Program 2209-3023 and by ODDR&E MURI through ARO grant DAAD19-00-0466.
References
[1] Y. Weiss. Correctness of local probability propagation in graphical models with loops. Neural
Computation, 12(1), 2000.
[2] S. Tatikonda and M. Jordan. Loopy belief propagation and gibbs measures. In UAI, 2002.
[3] T. Heskes. On the uniqueness of loopy belief propagation fixed points. To appear in Neural
Computation, 2004.
[4] D. Koller, U. Lerner, and D. Angelov. A general algorithm for approximate inference and its
application to hybrid Bayes nets. In UAI 15, pages 324?333, 1999.
[5] J. M. Coughlan and S. J. Ferreira. Finding deformable shapes using loopy belief propagation.
In ECCV 7, May 2002.
[6] E. B. Sudderth, A. T. Ihler, W. T. Freeman, and A. S. Willsky. Nonparametric belief propagation.
In CVPR, 2003.
[7] M. Isard. PAMPAS: Real?valued graphical models for computer vision. In CVPR, 2003.
[8] T. Minka. Expecatation propagation for approximate bayesian inference. In UAI, 2001.
[9] A. T. Ihler, J. W. Fisher III, and A. S. Willsky. Communication-constrained inference. Technical
Report TR-2601, Laboratory for Information and Decision Systems, 2004.
[10] L. Chen, M. Wainwright, M. Cetin, and A. Willsky. Data association based on optimization in
graphical models with application to sensor networks. Submitted to Mathematical and Computer Modeling, 2004.
[11] P. Clifford. Markov random fields in statistics. In G. R. Grimmett and D. J. A. Welsh, editors,
Disorder in Physical Systems, pages 19?32. Oxford University Press, Oxford, 1990.
[12] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufman, San Mateo, 1988.
[13] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free energy approximations and
generalized belief propagation algorithms. Technical Report 2004-040, MERL, May 2004.
[14] A. T. Ihler, J. W. Fisher III, and A. S. Willsky. Message errors in belief propagation. Technical
Report TR-2602, Laboratory for Information and Decision Systems, 2004.
[15] Hans-Otto Georgii. Gibbs measures and phase transitions. Studies in Mathematics. de Gruyter,
Berlin / New York, 1988.
[16] S. Julier and J. Uhlmann. A general method for approximating nonlinear transformations of
probability distributions. Technical report, RRG, Dept. of Eng. Science, Univ. of Oxford, 1996.
[17] A. Willsky. Relationships between digital signal processing and control and estimation theory.
Proc. IEEE, 66(9):996?1017, September 1978.
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1,849 | 268 | 372
Touretzky and Wheeler
A Computational Basis for Phonology
David S. Touretzky
School of Computer Science
Carnegie Mellon University
Pittsburgh, PA 15213
Deirdre W. Wheeler
Department of Linguistics
University of Pittsburgh
Pittsburgh, PA 15260
ABSTRACT
The phonological structure of human languages is intricate, yet highly
constrained. Through a combination of connectionist modeling and
linguistic analysis, we are attempting to develop a computational basis
for the nature of phonology. We present a connectionist architecture
that performs multiple simultaneous insertion, deletion, and mutation
operations on sequences of phonemes, and introduce a novel additional
primitive, clustering. Clustering provides an interesting alternative to
both iterative and relaxation accounts of assimilation processes such as
vowel harmony. Our resulting model is efficient because it processes
utterances entirely in parallel using only feed-forward circuitry.
1 INTRODUCTION
Phonological phenomena can be quite complex, but human phonological behavior is
also highly constrained. Many operations that are easily learned by a perceptron-like
sequence mapping network are excluded from real languages. For example, as Pinker
and Prince (1988) point out in their critique of the Rumelhart and McClelland (1986)
verb learning model, human languages never reverse the sequence of segments in a
word, but this is an easy mapping for a network to learn. On the other hand, we note that
some phonological processes that are relatively common in human languages, such as
vowel harmony, appear difficult for a sequence-mapping architecture to learn. Why are
only certain types of sequence operations found in human languages, and not others? We
suggest that this is a reflection of the limitations of an underlying, genetically-determined,
specialized computing architecture. We are searching for this architecture.
A Computational Basis for Phonology
Our work was initially inspired by George Lakoff's theory of cognitive phonology
(Lakoff, 1988, 1989), which is in tum a development of the ideas of John Goldsmith
(to appear). Lakoff proposes a three-level representation scheme. The M (morphophonemic) level represents the underlying form of an utterance, the P (phonemic) level
is an intermediate form, and the F (phonetic) level is the derived surface form.
Lakoff uses a combination of inter-level mapping rules and intra-level well-formedness
conditions to specify the relationships between P- and F-Ievel representations and the
M-Ievel input. In a connectionist implementation, the computations performed by the
mapping rules are straightforward, but we find the well-formedness conditions troubling.
Goldsmith's proposal was that phonology is a goal-directed constraint satisfaction system that operates via parallel relaxation. He cites Smolensky's hannony theoryl Lakoff
has adopted this appeal to hannony theory in his description of how well-formedness
conditions could work.
In our model, we further develop the Goldmsith and Lakoff mapping scheme, but we reject
harmony-based well-formedness conditions for several reasons. First, harmony theory
involves simulated annealing search. The timing constraints of real nervous systems rule
out simulated annealing. Second, it is not clear how to construct an energy function for
a connectionist network that performs complex discrete phonological operations. Finally
there is our desire to explain why certain types of processes occur in human languages
and others do not. Harmony theory alone is too unconstrained for this purpose.
We have implemented a model called M3 p (for "Many Maps" Model of Phonology) that
allows us to account for virtually all of the phenomena in (Lakoff, 1989) using a tighUyconstrained, purely-feedforward computing scheme. In the next section we describe the
mapping matrix architecture that is the heart of M3 p . Next we give an example of an
iterative process, Yawelmani vowel harmony,2, which Lakoff models with a P-Ievel wellformedness condition. Such a condition would have to be implemented by relaxation
search for a "minimum energy state" in the P-Ievel representation, which we wish to
avoid. Finally we present our alternative approach to vowel harmony, using a novel
clustering mechanism that eliminates the need for relaxation.
2 THE MAPPING MATRIX ARCHITECTURE
Figure 1 is an overview of our "many maps" model. M-P constructions compute how
to go from the M-Ievel representation of an utterance to the P-Ievel representation. The
derivation is described as a set of explicit changes to the M-Ievel string. M-P constructions read the segments in the M-Ievel buffer and write the changes, phrased as mutation,
deletion, and insertion requests, into slots of a buffer called P-deriv. The M-Ievel and
P-deriv buffers are then read by the M-P mapping matrix, which produces the P-Ievel
representation as its output. The process is repeated at the next level, with P-F constructions writing changes into an F-deriv buffer, and a P-F map deriving an F-Ievel
1Srnolensky' s "hannony theory" should not be confused with the linguistic phenomenon of "vowel hannony."
2Yawelmani is a dialect of Yokuts, an American Indian language from California. Our Yawelmani data is
drawn from Kenstowicz and Kisseberth (1979), as is Lakoff's.
373
374
Touretzky and Wheeler
M-Level Buffer
I
M-P Constructions
P-Level Buffer
I
P-F Constructions
-----4~~ F-Level Buffer
I
Canonicalization
I
Surface Phonetic ...
Representation ....----'
Figure 1: Overview of the "many maps" model.
representation. A final step called "canonicalization" cleans up the representations of the
individual segments.
Figure 2 shows the effect of an M-P construction that breaks up CCC consonant clusters
by inserting a vowel after the first consonant, producing CiCCo The input in this case
is the Yawelmani word /?ugnhin/ "drinks", and the desired insertion is indicated in Pderiv. The mapping matrix derives the P-Ievel representation right-justified in the buffer,
with no segment gaps or collisions. It can do this even when mutliple simultaneous
insertions and deletions are being performed. But it cannot perform arbitrary sequence
manipulations, such as reversing all the segments of an utterance. Further details of the
matrix architecture are given in (Touretzlcy, 1989) and (Wheeler and Touretzky, 1989).
3 ITERATIVE PHENOMENA
Several types of phonological processes operate on groups of adjacent segments, often by
making them more similar to an immediately preceding (or following) trigger segment.
Vowel harmony and voicing assimilation are two examples. In Yawelmani, vowel harmony takes the following form: an [ahigh] vowel that is preceded by an [ahigh] round
vowel becomes round and back. In the form Ido:s+aV "might report", the non-round,
back vowel Ia! is [-high], as is the preceding round vowel/o/. Therefore the Ia! becomes
round, yielding the surface form [do:soIJ. Similarly, in Idub+hin/ "leads by the hand", the
[+highJ vowel Ii! is preceded by the [+high] round vowel lui, so the (II becomes round and
back, giving [dubhun]. In /bok'+hinl "finds", the Ii! does not undergo harmony because
it differs in height from the preceding vowel.
A Computational Basis for Phonology
P-Deriv:
mut
del
ins
-
-
-
-
-
-
i
h
-
-
-
-
1
-
-
n
g
u
?
M-Level:
-
-
+
n
-
-
n
1
h
h
n
n
P-Level:
i
i
g
u
M-PMapping
Matrix
?
Figure 2: Perfonning an insertion via the M-P mapping matrix.
Hannony is described as an iterative process because it can apply to entire sequences of
vowels, as in the following derivation:
It'ul+sit+hin/
It'ul+sut+hinl
It'ul+sut+hunl
"burns for"
harmony on second vowel
harmony on third vowel
In Yawelmani we saw an epenthesis process that inserts a high vowel Ii! to break up
lengthy consonant clusters. Epenthetic vowels may either undergo or block hannony.
With the word /logw+xa! "let's pulverize", epenthesis inserts an Ii! to break up the Igwx!
cluster, producing /logiw+xa!. Now the Ia! is preceded by a [+high, -round] vowel, so
hannony does not apply, whereas in Ido:s+al/, which has the same sequence of underlying
vowels, it did. This is an instance of epenthesis blocking hannony. In other environments
the epenthetic vowel may itself undergo hannony. For example:
/?ugn+hinl
/?uginhinl
/?ugunhin/
I?ugunhun/
"drinks"
epenthesis
harmony on epenthetic vowel
harmony on third vowel
The standard generative phonology analysis of hannony utilizes the following rule, applying after epenthesis, that is supposed to iterate through the utterance from left to right,
changing one vowel at a time:
375
376
Touretzky and Wheeler
+SYll ]
+round
]
[ +syll
[
[
1
high -- +back
/
:~;~d Co _
a
Lakoff offers an alternative account of epenthesis and harmony that eliminates iteration.
He states epenthesis as an M-P construction:
M:
C
P:
[]
I
C
i
{C.#}
I
[]
The harmony rule is stated as a P-Ievel well-formedness condition that applies simultaneously throughout the buffer:
P:
If [+syll. +round. ahigh] Co X.
then if X = [+syll. ahigh]. then X = [+round. +back].
Starting with /?ugn+hin/ at M-Ievel. Lakoff?s model would settle into a representation
of nugunhun/ at P-Ievel. We repeat again the crucial point that this representation is
not derived by sequential application of rules; it is merely licensed by one application
of epenthesis and two of harmony. The actual computation of the P-Ievel representation
would be performed by a parallel relaxation process. perhaps using simulated annealing.
that somehow determines the sequence that best satisfies all applicable constraints at
P-Ievel.
4 THE CLUSTERING MECHANISM
Our account of vowel harmony must differ from LakofCs because we do not wish to
rely on relaxation in our model. Instead. we introduce special clustering circuitry to
recognize sequences of segments that share certain properties. The clustering idea is
meant to be analogous to perceptual grouping in vision. Sequences of adjacent visuallysimilar objects are naturally perceived as a whole. A similar mechanism operating on
phonological sequences. although unprecedented in linguistic theory. does not appear
implausible. Crucial to our model is the principle that perceived sequences may be
operated on as a unit. This allows us to avoid iteration and give a fully-parallel account
of vowel harmony.
The clustering mechanism is controlled by a small number of language-specific parameters. The rule shown below is the P-F clustering rule for Yawelmani. Cluster type
[+syllabic] indicates that the rule looks only at vowels. (This is implemented by an
additional mapping matrix that extracts the vowel projection of the P-Ievel buffer. The
clustering mechanism actually looks at the output of this matrix rather than at the P-Ievel
buffer directly.) The trigger of a cluster is a round vowel of a given height. and the
elements are the subsequent adjacent vowels of matching height. Application of the rule
causes elements (but not triggers) to undergo a change; in this case. they become round
and back.
A Computational Basis for Phonology
Yawelmani vowel harmony - P-F mapping:
Cluster type: [+syllabic]
[+round, ahigh]
Trigger:
Element:
[ahigh]
Change:
[+round, +back]
The following hypothetical vowel sequence illustrates the application of this clustering
rule. Consonants are omitted for clarity:
I
234
5
6
7
8
9
1
U
i
e
0
0
a
i
+
+
trigger:
element:
i
+
+
+
+
The second vowel is round, so it's a trigger. Since the third and fourth vowels match it
in height, they become elements. The fifth vowel is [-highl, so it is not included in the
cluster. The sixth vowel triggers a new cluster because it's round; it is also [-high]. The
seventh and eighth vowels are also [-highl, so they can be elements, but the ninth vowel
is excluded from the cluster because is [+highl. Note that vowel 7 is an element, but
it also meets the specification for a trigger. Given a choice, our model prefers to mark
segments as elements rather than triggers because only elements undergo the specified
change. The distinction is moot in Yawlemani, where triggers are already round and
back, but it matters in other languages; see (Wheeler and Touretzky, 1989) for details.
Figures 2 and 3 together show the derivation of the Yawelmani word [?ugunhunl from
the underlying form /?ugn+hin/. In figure 2 an M-P construction inserted a high vowel.
In figure 3 the P-F clustering circuitry has examined the P-Ievel buffer and marked the
triggers and elements. Segments that were marked as elements then have the change
[+round, +backl written into their corresponding mutation slots in F-deriv. Finally, the
P-F mapping matrix produces the sequence /?ugunhun/ as the F-Ievel representation of
the utterance.
5 DISCUSSION
We could not justify the extra circuitry required for clustering if it were suitable only
for Yawelmani vowel harmony. The same mechanism handles a variety of other iterative
phenomena, including Slovak and Gidabal vowel shortening, Icelandic umlaut, and Russian voicing assimilation. The full mechanism has some additional parameters beyond
those covered in the discussion of Yawelmani. For example, clustering may proceed from
right-to-Ieft (as is the case in Russian) instead of from left-to-right Also, clusters may
be of either bounded or unbounded length. Bounded clusters are required for alternation
processes, such as Gidabal shortening. They cover exactly two segments: a trigger and
one element We are making a deliberate analogy here with metrical phonology (stress
systems), where unbounded feet may be of arbitrary length, but bounded feet always
contain exactly two syllables. No language has strictly trisyllabic feet We predict a similar constraint will hold for iterative phenomena when they are reformulated in parallel
clustering terms, i.e., no language requires bounded-length clusters with more than one
element
377
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Touretzky and Wheeler
?
P-Ievel:
u
.
g
h
n
1
n
1
trigger
element
Clustering:
F-deriv:
.....-
n
u
n
I--
11
I-F-Ievel:
n
U
-~
u
h
..
n
u
g
u
u
T
"""--
J
.
P-FMapplng
Matrix
?
Figure 3: Clustering applied to Yawelmani vowel hannony.
A Computational Basis for Phonology
Our model makes many other predictions of constraints on human phonology, based
on limitations of the highly-structured "many maps" architecture. We are attempting to
verify these predictions, and also to extend the model to additional aspects of phonological
behavior, such as syllabification and stress.
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 George Lakoff for encouragement
and support, John Goldsmith for helpful correspondence, and Gillette Elvgren III for
implementing the simulations.
References
Goldsmith, J. (to appear) Phonology as an intelligent system. To appear in a festschrift
for Leila Gleitman, edited by D. Napoli and J. Kegl.
Kenstowicz, M., and Kisseberth, C. (1979) Generative Phonology: Description and
Theory. San Diego, CA: Academic Press.
Lakoff, G. (1988) A suggestion for a linguistics with connectionist foundations. In D. S.
Touretzky, G. E. Hinton, and T. J. Sejnowski (eds.), Proceedings of the 1988 Connectionist
Models Summer School, pp. 301-314. San Mateo, CA: Morgan Kaufmann.
Lakoff, G. (1989) Cognitive phonology. Draft of paper presented at the UC-Berkeley
Workshop on Constraints vs Rules, May 1989.
Pinker, S., and Prince, A. (1988) On language and connectionism: analysis of a parallel
distributed processing model of language acquisition. In S. Pinker & J. Mehler (eds.),
Connections and Symbols. Cambridge, Massachusetts: MIT Press.
Rumelhart, D. E., and McClelland, J. L. (1986) On learning the past tenses of English
verbs. In J. L. McClelland and D. E. Rumelhart (eds.), Parallel Distributed Processing:
Explorations in the MicroStructJ&re of Cognition, volume 2. Cambridge, Massachusetts:
MIT Press.
Smolensky, P. (1986) Information processing in dynamical systems: foundations of harmony theory. In D. E. Rumelhart and J. L. McClelland (eds.), Parallel Distributed
Processing: Explorations in the MicroStructure of Cognition, volume 1. Cambridge,
Massachusetts: MIT Press.
Touretzky, D. S. (1989) Toward 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: Erlbaum.
Wheeler, D. W., and Touretzky, D. S. (1989) A connectionist implementation of cognitive
phonology. Technical report CMU-CS-89-144, Carnegie Mellon University, School of
Computer Science. To appear in G. Lakoff and L. Hyman (eds.), Proceedings of the UCBerkeley Phonology Workshop on Constraints vs. Rules. University of Chicago Press.
379
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1,850 | 2,680 | New Criteria and a New Algorithm for Learning
in Multi-Agent Systems
Rob Powers
Computer Science Department
Stanford University
Stanford, CA 94305
[email protected]
Yoav Shoham
Computer Science Department
Stanford University
Stanford, CA 94305
[email protected]
Abstract
We propose a new set of criteria for learning algorithms in multi-agent
systems, one that is more stringent and (we argue) better justified than
previous proposed criteria. Our criteria, which apply most straightforwardly in repeated games with average rewards, consist of three requirements: (a) against a specified class of opponents (this class is a parameter
of the criterion) the algorithm yield a payoff that approaches the payoff
of the best response, (b) against other opponents the algorithm?s payoff
at least approach (and possibly exceed) the security level payoff (or maximin value), and (c) subject to these requirements, the algorithm achieve
a close to optimal payoff in self-play. We furthermore require that these
average payoffs be achieved quickly. We then present a novel algorithm,
and show that it meets these new criteria for a particular parameter class,
the class of stationary opponents. Finally, we show that the algorithm
is effective not only in theory, but also empirically. Using a recently
introduced comprehensive game theoretic test suite, we show that the
algorithm almost universally outperforms previous learning algorithms.
1
Introduction
There is rapidly growing interest in multi-agent systems, and in particular in learning algorithms for such systems. There is a growing body of algorithms proposed, and some arguments about their relative merits and domains of applicability (for example, [14] and [17]).
In [15] we survey much of this literature, and argue that it suffers from not having a clear
objective criteria with which to evaluate each algorithm (this shortcoming is not unique to
the relatively small computer science literature on multi-agent learning, and is shared by
the much vaster literature on learning in game theory). In [15] we also define five different
coherent agendas one could adopt, and identify one of them ? the agent-centric one ? as
particularly relevant from the computer science point of view.
In the agent-centric agenda one asks how an agent can learn optimally in the presence of
other independent agents, who may also be learning. To make the discussion precise we
will concentrate on algorithms for learning in known, fully observable two-player repeated
games, with average rewards. We start with the standard definition of a finite stage game
(aka normal form game):
Definition 1 A two-player stage game is a tuple G = (A1 , A2 , u1 , u2 ), where
? Ai is a finite set of actions available to player i
? ui : A1 ? A2 ? ? is a utility function for player i
Figure 1 shows two well-known games from the literature, to which we?ll refer again later.
In a repeated game the stage game is repeated, finitely or infinitely. The agent accumulates
rewards at each round; in the finite case the agent?s aggregate reward is the average of the
stage-game rewards, and in the infinite case it is the limit average (we ignore the subtlety
that arises when the limit does not exist, but this case does not present an essential problem).
While the vast majority of the literature on multi-agent learning (surprisingly) does not
start with a precise statement of objectives, there are some exceptions, and we discuss
them in the next section, including their shortcomings. In the following section we propose a stronger set of criteria that, we believe, does not suffer from these limitations. We
then present an algorithm that provably meets these stronger requirements. However, we
believe that all formal requirements ? including our own ? are merely baseline guarantees,
and any proposed algorithm must be subjected to empirical tests. While many previous
proposals provide empirical results, we think it is fair to say that our level of empirical
validation is unprecedented in the literature. We show the results of tests for all pairwise comparisons of major existing algorithms, using a recently-developed game theoretic
testbed called GAMUT [13] to systematically sample a very large space of games.
2
Previous criteria for multi-agent learning
To our knowledge, Bowling and Veloso [1] were the first in the AI community to explicitly
put forth formal requirements. Specifically they proposed two criteria:
Rationality: If the other players? policies converge to stationary policies then the learning
algorithm will converge to a stationary policy that is a best-response (in the stage game) to
the other players? policies.
Convergence: The learner will necessarily converge to a stationary policy.
Throughout this paper, we define a stationary policy as one that selects an action at each
point during the game by drawing from the same distribution, regardless of past history.
Bowling and Veloso considered known repeated games and proposed an algorithm that
provably meets their criteria in 2x2 games (games with two players and two actions per
player). Later, Conitzer and Sandholm [5] adopted the same criteria, and demonstrated an
algorithm meeting the criteria for all repeated games.
At first glance these criteria are reasonable, but a deeper look is less satisfying. First, note
that the property of convergence cannot be applied unconditionally, since one cannot ensure that a learning procedure converges against all possible opponents without sacrificing
rationality. So implicit in that requirement is some limitation on the class of opponents.
And indeed both [1] and [5] acknowledge this and choose to concentrate on the case of
self-play, that is, on opponents that are identical to the agent in question.
Dare
Y ield
Dare
0, 0
4, 1
Y ield
1, 4
2, 2
(a) Chicken
Cooperate
Def ect
Cooperate
3, 3
0, 4
Def ect
4, 0
1, 1
(b) Prisoner?s Dilemma
Figure 1: Example stage games. The payoff for the row player is given first in each cell,
with the payoff for the column player following.
We will have more to say about self-play later, but there are other aspects of these criteria
that bear discussion. While it is fine to consider opponents playing stationary policies,
there are other classes of opponents that might be as relevant or even more relevant; this
should be a degree of freedom in the definition of the problem. For instance, one might be
interested in the classes of opponents that can be modeled by finite automata with at most
k states; these include both stationary and non-stationary strategies.
We find the property of requiring convergence to a stationary strategy particularly hard to
justify. Consider the Prisoner?s Dilemma game in Figure 1. The Tit-for-Tat algorithm1
achieves an average payoff of 3 in self-play, while the unique Nash equilibrium of the
stage game has a payoff of only 1. Similarly, in the game of Chicken, also shown in
Figure 1, a strategy that alternates daring while its opponent yields and yielding while its
opponent dares achieves a higher expected payoff in self-play than any stationary policy
could guarantee. This problem is directly addressed in [2] and a counter-proposal made for
how to consider equilibria in repeated games. But there is also a fundamental issue with
these two criteria; they can both be thought of as a requirement on the play of the agent,
rather than the reward the agent receives.
Our final point regarding these two criteria is that they express properties that hold in the
limit, with no requirements whatsoever on the algorithm?s performance in any finite period.
But this question is not new to the AI community and has been addressed numerous times
in game theory, under the names of universal consistency, no-regret learning, and the Bayes
envelope, dating back to [9] (see [6] for an overview of this history). There is a fundamental
similarity in approach throughout, and we will take the two criteria proposed in [7] as being
representative.
Safety: The learning rule must guarantee at least the minimax payoff of the game. (The
minimax payoff is the maximum expected value a player can guarantee against any possible
opponent.)
Consistency: The learning rule must guarantee that it does at least as well as the best
response to the empirical distribution of play when playing against an opponent whose
play is governed by independent draws from any fixed distribution.
They then define universal consistency as the requirement that a learning rule do at least as
well as the best response to the empirical distribution regardless of the actual strategy the
opponent is employing (this implies both safety and consistency) and show that a modification of the fictitious play algorithm [3] achieves this requirement. A limitation common
to these game theory approaches is that they were designed for large-population games and
therefore ignore the effect of the agent?s play on the future play of the opponent. But this
can pose problems in smaller games. Consider the game of Prisoner?s Dilemma once again.
Even if the opponent is playing Tit-for-Tat, the only universally consistent strategy would
be to defect at every time step, ruling out the higher payoff achievable by cooperating.
3
A new set of criteria for learning
We will try to take the best of each proposal and create a joint set of criteria with the
potential to address some of the limitations mentioned above.
We wish to keep the notion of optimality against a specific set of opponents. But instead of
restricting this set in advance, we?ll make this a parameter of the properties. Acknowledging that we may encounter opponents outside our target set, we will also incorporate the
requirement of safety, which guarantees we achieve at least the security value, also known
1
The Tit-for-Tat algorithm cooperates in the first round and then for each successive round plays
the action its opponent played in the previous round.
as the maximin payoff, for the stage game. As a possible motivation for our approach,
consider the game of Rock-Paper-Scissors, which despite its simplicity has motivated several international tournaments. While the unique Nash equilibrium policy is to randomize,
the winners of the tournaments are those players who can most effectively exploit their
opponents who deviate without being exploited in turn.
The question remains of how best to handle self-play. One method would be to require that
a proposed algorithm be added to the set of opponents it is required to play a best response
to. While this may seem appealing at first glance, it can be a very weak requirement on the
actual payoff the agent receives. Since our opponent is no longer independent of our choice
of strategy, we can do better than settling for just any mutual best response, and try to maximize the value we achieve as well. We therefore propose requiring the algorithm achieve
at least the value of some Nash equilibrium that is Pareto efficient over the set of Nash
equilibria.2 Similarly, algorithms exist that satisfy ?universal consistency? and if played
by all agents will converge to a correlated equilibria[10], but this result provides an even
weaker constraint on the actual payoff received than convergence to a Nash equilibrium.
Let k be the number of outcomes for the game and b the maximum possible difference in
payoffs across the outcomes. We require that for any choice of ? > 0 and ? > 0 there exist
a T0 , polynomial in 1? , 1? , k, and b, such that for any number of rounds t > T0 the algorithm
achieves the following payoff guarantees with probability at least 1 ? ?.
Targeted Optimality: When the opponent is a member of the selected set of opponents,
the average payoff is at least VBR ??, where VBR is the expected value of the best response
in terms of average payoff against the actual opponent.
Compatibility: During self-play, the average payoff is at least Vself P lay ? ?, where
Vself P lay is defined as the minimum value achieved by the player in any Nash equilibrium
that is not Pareto dominated by another Nash equilibrium.
Safety: Against any opponent, the average payoff is at least Vsecurity ? ?, with Vsecurity
defined as max?1 ??1 min?2 ??2 EV (?1 , ?2 ).3
4
An algorithm
While we feel designing algorithms for use against more complex classes of opponent is
critical, as a minimal requirement we first show an algorithm that meets the above criteria
for the class of stationary opponents that has been the focus of much of the existing work.
Our method incorporates modifications of three simple strategies: Fictitious Play [3], Bully
[12], and the maximin strategy in order to create a more powerful hybrid algorithm.
Fictitious Play has been shown to converge in the limit to the best response against a stationary opponent. Each round it plays its best response to the most likely stationary opponent given the history of play. Our implementation uses a somewhat more generous
best-response calculation so as to achieve our performance requirements during self-play.
BR? (?) ? arg max(EOV (x, ?)), 4
x?X(?,?)
where X(?, ?) = {y ? ?1 : EV (y, ?) ? max (EV (z, ?)) ? ?}
z??1
2
An outcome is Pareto efficient over a set if there is no other outcome in that set with a payoff at
least as high for every agent and strictly higher for at least one agent.
3
Throughout the paper, we use EV (?1 , ?2 ) to indicate the expected payoff to a player for playing
strategy ?1 against an opponent playing ?2 and EOV (?1 , ?2 ) as the expected payoff the opponent
achieves. ?1 and ?2 are the sets of mixed strategies for the agent and its opponent respectively.
4
Note that BR0 (?) is a member of the standard set of best response strategies to ?.
We extend the Bully algorithm to consider the full set of mixed strategies and again maximize our opponent?s value when multiple strategies yield equal payoff for our agent.
BullyM ixed ? arg max(EOV (x, BR(x))),
x?X
where X = {y ? ?1 : EV (y, BR0 (y)) = max (EV (z, BR0 (z)))}
z??1
The maximin strategy is defined as
M aximin ? arg max min EV (?1 , ?2 )
?1 ??1
?2 ??2
We will now show how to combine these strategies into a single method satisfying all
three criteria. In the code shown below, t is the current round, AvgV aluem is the average value achieved by the agent during the last m rounds, VBully is shorthand for
EV (BullyM ixed, BR0 (BullyM ixed)), and dtt21 represents the distribution of opponent
actions for the period from round t1 to round t2 .
Set strategy = BullyMixed
for ?1 time steps
Play strategy
for ?2 time steps
if (strategy == BullyMixed AND AvgV alueH < VBully ? ?1 )
With probability, p, set strategy = BR?2 (dt0 )
Play strategy
if ||d?01 ? dtt??1 || < ?3
Set bestStrategy = BR?2 (dt0 )
else if (strategy == BullyMixed AND AvgV alueH > VBully ? ?1 )
Set bestStrategy = BullyMixed
else
Set bestStrategy = BestResponse
while not end of game
if avgV aluet??0 < Vsecurity ? ?0
Play maximin strategy for ?3 time steps
else
Play bestStrategy for ?3 time steps
The algorithm starts out with a coordination/exploration period in which it attempts to determine what class its opponent is in. At the end of this period it chooses one of three
strategies for the rest of the game. If it determines its opponent may be stationary it settles
on a best response to the history up until that point. Otherwise, if the BullyMixed strategy
has been performing well it maintains it. If neither of these conditions holds, it adopts a
default strategy, which we have set to be the BestResponse strategy. This strategy changes
each round, playing the best response to the maximum likelihood opponent strategy based
on the last H rounds of play. Once one of these strategies has been selected, the algorithm plays according to it whenever the average value meets or exceeds the security level,
reverting to the maximin strategy if the value drops too low.
Theorem 1 Our algorithm satisfies the three properties stated in section 3 for the class of
stationary opponents, with a T0 proportional to ( b? )3 1? .
This theorem can be proven for all three properties using a combination of basic probability
theory and repeated applications of the Hoeffding inequality [11], but the proof itself is
prohibitively long for inclusion in this publication.
5
Empirical results
Although satisfying the criteria we put forth is comforting, we feel this is but a first step
in making a compelling argument that an approach might be useful in practice. Traditionally, researchers suggesting a new algorithm also include an empirical comparison of the
algorithm to previous work. While we think this is a critical component of evaluating an
algorithm, most prior work has used tests against just one or two other algorithms on a
very narrow set of test environments, which often vary from researcher to researcher. This
practice has made it hard to consistently compare the performance of different algorithms.
In order to address this situation, we?ve started to code a collection of existing algorithms.
Combining this set of algorithms with a wide variety of repeated games from GAMUT
[13], a game theoretic test suite, we have the beginnings of a comprehensive testbed for
multi-agent learning algorithms. In the rest of this section, we?ll concentrate on the results
for our algorithm, but we hope that this testbed can form the foundation for a broad, consistent framework of empirical testing in multi-agent learning going forward. For all of
our environments we conducted our tests using a tournament format, where each algorithm
plays all other algorithms including itself.
0.500
MetaStrategy
StochFP
StochIGA
WoLF-PHC
BullyMixed
MiniMax
0.400
0.300
0.200
0.100
-M
ax
Jo
in
tQ
Hy
pe
r-Q
Lo
ca
lQ
St
oc
hF
P
St
oc
hI
GA
W
oL
FPH
C
m
et
aS
tra
te
gy
M
Ra
nd
o
Bu
lly
M
ini
M
ax
Bu
lly
M
i xe
d
-
Figure 2: Average value for last 20K rounds (of 200K) across all games in GAMUT.
Let us first consider the results of a tournament over a full set of games in GAMUT. Figure
2 portrays the average value achieved by each agent (y-axis) averaged over all games, when
playing different opponents (x-axis). The set of agents includes our strategy (MetaStrategy), six different adaptive learning approaches (Stochastic Fictitious Play [3,8], Stochastic
IGA[16], WoLF-PHC[1], Hyper-Q learning[18], Local Q-learning[19], and JointQ-Max[4]
(which learns Q-values over the joint action space but assumes its opponent will cooperate
to maximize its payoff)), and four fixed strategies (BullyMixed, Bully[12], the maximin
strategy, and Random (which selects a stationary mixed strategy at random)). We have
chosen a subset of the most successful algorithms to display on the graph. Against the four
stationary opponents, all of the adaptive learners fared equally well, while fixed strategy
players achieved poor rewards. In contrast, BullyMixed fared well against the adaptive
algorithms. As desired, our new algorithm combined the best of these characteristics to
achieve the highest value against all opponents except itself. It fares worse than BullyMixed
since it will always yield to BullyMixed, giving away the more advantageous outcome in
games like Chicken. However, when comparing how each agent performs in self-play,
our algorithm scores quite well, finishing a close second to Hyper-Q learning while the
two Bully algorithms finish near last. Hyper-Q is able to gain in self-play by occasionally
converging to outcomes with high social welfare that our strategy does not consider.
MetaStrategy
WoLF-PHC
StochFP
BullyMixed
StochIGA
MiniMax
100%
95%
90%
85%
80%
TwoByTwoGame
ShapleysGame
TravelersDilemma
Rochambeau
RandomZeroSum
RandomGame
RandomCompound
PrisonersDilemma
MinimumEffort
MajorityVoting
MatchingPennies
HawkAndDove
GrabTheDollar
GameWActionPar
DispersionGame
CovariantGame
CoordinationGame
Chicken
BertrandOligopoly
BattleOfTheSexes
70%
CollaborationGame
75%
Figure 3: Percent of maximum value for last 20K rounds (of 200K) averaged across all
opponents for selected games in GAMUT. The rewards were divided by the maximum
reward achieved by any agent to make visual comparisons easier.
So far we?ve seen that our new algorithm performs well when playing against a variety of
opponents. In Figure 3 we show the reward for each agent, averaged across the set of possible opponents for a selection of games in GAMUT. Once again our algorithm outperforms
the existing algorithms in nearly all games. When it fails to achieve the highest reward it
often appears to be due to its policy of ?generosity?; in games where it has multiple actions
yielding equal value, it chooses a best response that maximizes its opponent?s value.
The ability to study how individual strategies fare in each class of environment reflects an
advantage of our more comprehensive testing approach. In future work, this data can be
used both to aid in the selection of an appropriate algorithm for a new environment and to
pinpoint areas where an algorithm might be improved. Note that we use environment here
to indicate a combination of both the game and the distribution over opponents.
6
Conclusions and Future Work
Our objective in this work was to put forth a new set of criteria for evaluating the performance of multi-agent learning algorithms as well as propose a more comprehensive method
for empirical testing. In order to motivate this new approach for vetting algorithms, we have
presented a novel algorithm that meets our criteria and outperforms existing algorithms in a
wide variety of environments. We are continuing to work actively to extend our approach.
In particular, we wish to demonstrate the generality of our approach by providing algorithms that calculate best response to different sets of opponents (conditional strategies,
finite automata, etc.) Additionally, the criteria need to be generalized for n-player games
and we hope to combine our method for known games with methods for learning the structure of the game, ultimately devising new algorithms for unknown stochastic games.
Acknowledgements
This work was supported in part by a Benchmark Stanford Graduate Fellowship, DARPA
grant F30602-00-2-0598, and NSF grant IIS-0205633.
References
[1] Bowling, M. & Veloso, M. (2002). Multiagent learning using a variable learning rate. In Artificial
Intelligence, 136, pp. 215-250.
[2] Brafman, R. & Tennenholtz, M. (2002). Efficient Learning Equilibrium. In Advances in Neural
Information Processing Systems 15.
[3] Brown, G. (1951). Iterative Solution of Games by Fictitious Play. In Activity Analysis of Production and Allocation. New York: John Wiley and Sons.
[4] Claus, C. & Boutilier, C. (1998). The dynamics of reinforcement learning in cooperative multiagent systems. In Proceedings of the National Conference on Artificial Intelligence , pp. 746-752.
[5] Conitzer, V. & Sandholm, T. (2003). AWESOME: A General Multiagent Learning Algorithm that
Converges in Self-Play and Learns a Best Response Against Stationary Opponents. In Proceedings
of the 20th International Conference on Machine Learning, pp. 83-90, Washington, DC.
[6] Foster, D. & Vohra, R. (1999). Regret in the on-line decision problem. ?Games and Economic
Behavior? 29:7-36.
[7] Fudenberg, D. & Levine, D. (1995) Universal consistency and cautious fictitious play. Journal of
Economics Dynamics and Control 19:1065-1089.
[8] Fudenberg, D. & Levine, D. (1998). The theory of learning in games. MIT Press.
[9] Hannan, J. (1957) Approximation to Bayes risk in repeated plays. Contributions to the Theory of
Games 3:97-139.
[10] Hart, S. & Mas-Colell, A. (2000). A simple adaptive procedure leading to correlated equilibrium.
In Econometrica, Vol. 68, No. 5, pages 1127-1150.
[11] Hoeffding, W. (1956). On the distribution of the number of successes in independent trials.
Annals of Mathematical Statistics 27:713-721.
[12] Littman, M. & Stone, P. (2001). Implicit Negotiation in Repeated Games. In Proceedings of the
Eighth International Workshop on Agent Theories, Architectures, and Languages, pp. 393-404.
[13] Nudelman, E., Wortman, J., Leyton-Brown, K., & Shoham, Y. (2004). Run the GAMUT: A
Comprehensive Approach to Evaluating Game-Theoretic Algorithms. AAMAS-2004. To Appear.
[14] Sen, S. & Weiss, G. (1998). Learning in multiagent systems. In Multiagent systems: A modern
introduction to distributed artificial intelligence, chapter 6, pp. 259-298, MIT Press.
[15] Shoham, Y., Powers, R., & Grenager, T. (2003). Multi-Agent Reinforcement Learning: a critical
survey. Technical Report.
[16] Singh, S., Kearns, M., & Mansour, Y. (2000). Nash convergence of gradient dynamics in generalsum games. In Proceedings of UAI-2000, pp. 541-548, Morgan Kaufman.
[17] Stone, P. & Veloso, M. (2000). Multiagent systems: A survey from a machine learning perspective. Autonomous Robots, 8(3).
[18] Tesauro, G. (2004). Extending Q-Learning to General Adaptive Multi-Agent Systems. In Advances in Neural Information Processing Systems 16.
[19] Watkins, C. & Dayan, P. (1992). Technical note: Q-learning. Machine Learning, 8(3):279-292.
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1,851 | 2,681 | Proximity graphs for clustering and manifold
learning
? Carreira-Perpin?
? an
Miguel A.
Richard S. Zemel
Dept. of Computer Science, University of Toronto
6 King?s College Road. Toronto, ON M5S 3H5, Canada
Email: {miguel,zemel}@cs.toronto.edu
Abstract
Many machine learning algorithms for clustering or dimensionality reduction take as input a cloud of points in Euclidean space, and construct
a graph with the input data points as vertices. This graph is then partitioned (clustering) or used to redefine metric information (dimensionality reduction). There has been much recent work on new methods for
graph-based clustering and dimensionality reduction, but not much on
constructing the graph itself. Graphs typically used include the fullyconnected graph, a local fixed-grid graph (for image segmentation) or a
nearest-neighbor graph. We suggest that the graph should adapt locally
to the structure of the data. This can be achieved by a graph ensemble
that combines multiple minimum spanning trees, each fit to a perturbed
version of the data set. We show that such a graph ensemble usually produces a better representation of the data manifold than standard methods;
and that it provides robustness to a subsequent clustering or dimensionality reduction algorithm based on the graph.
1
Introduction
Graph-based algorithms have long been popular, and have received even more attention
recently, for two of the fundamental problems in machine learning: clustering [1?4] and
manifold learning [5?8]. Relatively little attention has been paid to the properties and
construction methods for the graphs that these algorithms depend on.
A starting point for this study is the question of what constitutes a good graph. In the
applications considered here, the graphs are an intermediate form of representation, and
therefore their utility to some extent depends on the algorithms that they will ultimately be
used for. However, in the case of both clustering and manifold learning, the data points are
assumed to lie on some small number of manifolds. Intuitively, the graph should represent
these underlying manifolds well: it should avoid shortcuts that travel outside a manifold,
avoid gaps that erroneously disconnect regions of a manifold, and be dense within the
manifold and clusters. Also, while the algorithms differ with respect to connectedness, in
that clustering wants the graph to be disconnected, while for manifold learning the graph
should be connected, they both want at least the inside of the clusters, or dense areas of the
manifold, to be enhanced relative to the between-cluster, or sparse manifold connections.
Dataset
PSfrag replacements
MST
k-NNG
-ball graph
Delaunay
triangulation
Perturbed
dataset
Figure 1: Sensitivity to noise of proximity graphs. Top row: several proximity graphs
constructed on a noisy sample of points lying on a circle. Bottom row: the same graphs
constructed on a different sample; specifically, we added to each point Gaussian noise of
standard deviation equal to the length of the small segment shown in the centre of the
dataset (top row), built the graph, and drew it on the original dataset. This small perturbation can result in large changes in the graphs, such as disconnections, shortcuts or changes
in connection density.
Many methods employ simple graph constructions. A fully-connected graph is used for
example in spectral clustering and multidimensional scaling, while a fixed grid, with each
point connecting to some small fixed number of neighbors in a pre-defined grid of locations, is generally used in image segmentation. An -ball, in which each point connects
to points within some distance , and k-nearest neighbors (k-NNG) are generalization of
these approaches, as they take into account distance in some features associated with each
point instead of simply the grid locations. The -ball or k-NNG provide an improvement
over the fully-connected graph or fixed grid (clustering: [3, 9]; manifold learning: [5, 7]).
These traditional methods contain parameters (, k) that strongly depend on the data; they
generally require careful, costly tuning, as typically graphs must be constructed for a range
of parameter values, the clustering or dimensionality-reduction algorithm run on each, and
then performance curves compared to determine the best settings. Figure 1 shows that these
methods are quite sensitive to sparsity and noise in the data points, and that the parameters
should ideally vary within the data set. It also shows that other traditional graphs (e.g. the
Delaunay triangulation) are not good for manifolds, since they connect points nonlocally.
In this paper we propose a different method of graph construction, one based on minimum spanning trees (MSTs). Our method involves an ensemble of trees, each built on a
perturbed version of the data. We first discuss the motivation for this new type of graph,
and then examine its robustness properties, and its utility to both subsequent clustering or
dimensionality reduction methods.
2
Two new types of proximity graphs
A minimum spanning tree is a tree subgraph that contains all the vertices and has a minimum sum of edge weights. As a skeleton of a data set, the MST has some good properties:
it tends to avoid shortcuts between branches (typically caused by long edges, which are
contrary to the shortest-length criterion) and it gives a connected graph (usually a problem
for other methods with often-occurring random small groupings of points). In fact, the
MST was an early approach to clustering [10]. However, the MST is too sparse (having
only N ? 1 edges for an N -point data set, and no cycles) and is sensitive to noise. One
way to flesh it out and attain robustness to noise is to form an MST ensemble that combines
multiple MSTs; we give two different algorithms for this.
2.1
Perturbed MSTs (PMSTs)
Perturbed MSTs combine a number of MSTs, each fit to a perturbed version of the data
set. The perturbation is done through a local noise model that we estimate separately for
each data point based on its environment: point xi is perturbed by adding to it zero-mean
uniform noise of standard deviation si = rdi where di is the average distance to the k
nearest neighbors of xi , and r ? [0, 1]. In this paper we use k = 5 throughout and study the
effect of r. The locality of the noise model allows points to move more or less depending
on the local data structure around them and to connect to different numbers of neighbors at
different distances?in effect we achieve a variable k and .
To build the PMST ensemble, we generate T > 1 perturbed copies of the entire data set
according to the local noise model and fit an MST to each. The PMST ensemble assigns
a weight eij ? [0, 1] to the edge between points xi and xj equal to the average number
of times that edge appears on the trees. For T = 1 this gives the MST of the unperturbed
data set; for T ? ? it gives a stochastic graph where eij is the probability (in the Laplace
sense) of that edge under the noise model. The PMST ensemble contains at most T (N ? 1)
edges (usually much less). Although the algorithm is randomized, the PMST ensemble for
large T is essentially deterministic, and insensitive to noise by construction. In practice a
small T is enough; we use T = 20 in the experiments.
2.2
Disjoint MSTs (DMSTs)
Here we build a graph that is a deterministic collection of t MSTs that satisfies the property
that the nth tree (for n = 1, . . . , t) is the MST of the data subject to not using any edge
already in the previous 1, . . . , t ? 1 trees. One possible construction algorithm is an extension of Kruskal?s algorithm for the MST where we pick edges without replacement and
restart for every new tree. Specifically, we sort the list of N (N2?1) edges eij by increasing
distance dij and visit each available edge in turn, removing it if it merges two clusters (or
equivalently does not create a cycle); whenever we have removed N ? 1 edges, we go back
to the start of the list. We repeat the procedure t times in total.
The DMST ensemble consists of the union of all removed edges and contains t(N ? 1)
edges each of weight 1. The t parameter controls the overall density of the graph, which is
always connected; unlike or k (for the -ball or k-NNG), t is not a parameter that depends
locally on the data, and again points may connect to different numbers of neighbors at
different distances. We obtain the original MST for t = 1; values t = 2?4 (and usually
quite larger) work very well in practice. t need not be integer, i.e., we can fix the total
number of edges instead. In any case we should use t N2 .
2.3
Computational complexity
For a data set with N points, the computational complexity is approximately
O(T N 2 log N ) (PMSTs) or O(N 2 (log N + t)) (DMSTs). In both cases the resulting
graphs are sparse (number of edges is linear on number of points N ). If imposing an a
priori sparse structure (e.g. an 8-connected grid in image segmentation) the edge list is
much shorter, so the graph construction is faster. For the perturbed MST ensemble, the
perturbation of the data set results in a partially disordered edge list, which one should be
able to sort efficiently. The bottleneck in the graph construction itself is the computation of
pairwise distances, or equivalently of nearest neighbors, of a set of N points (which affects
the -ball and k-NNG graphs too): in 2D this is O(N log N ) thanks to properties of planar
geometry, but in higher dimensions the complexity quickly approaches O(N 2 ).
Overall, the real computational bottleneck is the graph postprocessing, typically O(N 3 ) in
spectral methods (for clustering or manifold learning). This can be sped up to O(cN 2 ) by
using sparsity (limiting a priori the edges allowed, thus approximating the true solution)
but then the graph construction is likewise sped up. Thus, even if our graphs are slightly
more costly to construct than the -ball or k-NNG, the computational savings are very large
if we avoid having to run the spectral technique multiple times in search for a good or k.
3
Experiments
We present two sets of experiments on the application of the graphs to clustering and manifold learning, respectively.
3.1
Clustering
In affinity-based clustering, our data is an N ? N affinity matrix W that defines a graph
(where nonzeros define edge weights) and we seek a partition of the graph that optimizes
a cost function, such as mincut [1] or normalized cut [2]. Typically the affinities are
wij = exp(? 12 (dij /?)2 ) (where dij is the problem-dependent distance between points
xi and xj ) and depend on a scale parameter ? ? (0, ?). This graph partitioning problem
is generally NP-complete, so approximations are necessary, such as spectral clustering algorithms [2]. In spectral clustering we seek to cluster in the leading eigenvectors of the
P
1
1
normalized affinity matrix N = D? 2 WD? 2 (where D = diag ( i wij ), and discarding
a constant eigenvector associated with eigenvalue 1). Spectral clustering succeeds only for
a range of values of ? where N displays the natural cluster structure of the data; if ? is
too small W is approximately diagonal and if ? is too large W is approximately a matrix
of ones. It is thus crucial to determine a good ?, which requires computing clusterings
over a range of ? values?an expensive computation since each eigenvector computation
is O(N 3 ) (or O(cN 2 ) under sparsity conditions).
Fig. 2 shows segmentation results for a grayscale image from [11] where the objective is
to segment the occluder from the underlying background, a hard task given the intensity
gradients. We use a standard weighted Euclidean distance on the data points (pixels) x =
(pixel location, intensity). One method uses the 8-connected grid (where each pixel is connected to its 8 neighboring pixels). The other method uses the PMST or DMST ensemble
(constrained to contain only edges in the 8-connected grid) under different values of the r, t
parameters; the graph has between 44% and 98% the number of edges in the 8-grid, depending on the parameter value. We define the affinity matrix as wij = eij exp(? 21 (dij /?)2 )
(where eij ? [0, 1] are the edge values). In both methods we apply the spectral clustering algorithm of [2]. The plot shows the clustering error (mismatched occluder area) for a
range of scales. The 8-connected grid succeeds in segmenting the occluder for ? ? [0.2, 1]
approximately, while the MST ensembles (for all parameter values tested) succeed for a
wider range?up to ? = ? in many cases. The reason for this success even for such high
? is that the graph lacks many edges around the occluder, so those affinities are zero no
matter how high the scale is. In other words, for clustering, our graphs enhance the inside
of clusters with respect to the bridges between clusters, and so ease the graph partitioning.
3.2
Manifold learning
For dimensionality reduction, we concentrate on applying Isomap [5], a popular and powerful algorithm. We first estimate the geodesic distances (i.e., along the manifold) g?ij be-
???? First 5 eigenvectors (except the constant one) ????
8-grid
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8-grid
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Segmentation
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Figure 2: Using a proximity graph increases the scale range over which good segmentations
are possible. We consider segmenting the greyscale image at the bottom (an occluder over
a background) with spectral clustering, asking for K = 5 clusters. The color diagrams
represent the segmentation (column 1) and first 5 eigenvectors of the affinity matrix (except
the constant eigenvector, columns 2?4) obtained with spectral clustering, using a PMST
ensemble with r = 0.4 (upper row) or an 8-connectivity graph (lower row), for 3 different
scales: ?1 = 0.5, ?2 = 1.6 and ? = ?. The PMST ensemble succeeds at all scales
(note how several eigenvectors are constant over the occluder), while the 8-connectivity
graph progressively deteriorates as ? increases to give a partition of equal-sized clusters
for large scale. In the bottom part of the figure we show: the PMST ensemble graph in
3D space; the clustering error vs ? (where the right end is ? = ?) for the 8-connectivity
graph (thick blue line) and for various other PMST and DMST ensembles under various
parameters (thin lines). The PMST and DMST ensembles robustly (for many settings of
their parameters) give an almost perfect segmentation over a large range of scales.
tween pairs of points in the data set as the shortest-path lengths in a graph learned from the
data. Then we apply multidimensional scaling to these distances to obtain a collection of
low-dimensional points {yi }N
i=1 that optimally preserves the estimated geodesic distances.
In fig. 3 we show the results of applying Isomap using different graphs to two data sets
(ellipse and Swiss roll) for which we know the true geodesic distances gij . In a real application, since the true geodesic distances are unknown, error and variance cannot be
computed; an estimated residual variance has been proposed [5] to determine the optimal
graph parameter. For the perturbed MST ensemble, we binarize the edge values by making
1 any eij > 0. (It is often possible to refine the graph by zeroing edges with small eij ,
since this removes shortcuts that may have arisen by chance, particularly if T is large; but
it is difficult to estimate the right threshold reliably.) The plots show 3 curves as a function
of the graph parameter: the average error E in the geodesic distances; Isomap?s estimated
residual variance V? ; and the true residual variance V . From the plots we can see that V?
correlates well with V (though it underestimates it) and also with E for the Swiss roll, but
not for the ellipse; this can make the optimal graph parameter difficult to determine in a
real application. Given this, the fact that our graphs work well over a larger region of their
parameter space than the -ball or k-NNG graphs makes them particularly attractive.
The plots for the Swiss roll show that, while for the low noise case the -ball or k-NNG
graphs work well over a reasonable region of their parameter space, for the high noise case
this region decreases a lot, almost vanishing for the -ball. This is because for low values
of the parameter the graph is disconnected, while for high values it has multiple shortcuts;
the difficulty of the task is compounded by the small number of points used, N = 500 (an
unavoidable fact in high dimensions). However, for the PMSTs the region remains quite
wide and for the DMSTs the approximate region t ? [2, 8] gives good results. For very low
r or t = 1 the graph is the single MST, thus the large errors.
It is also important to realize that the range of the r parameter of the PMST ensemble does
not depend on the data, while the range for and k does. The range of the t parameter of
the DMST ensemble does depend on the data, but we have found empirically that t = 2?4
gives very good results with all data sets we have tried.
4
Discussion
One main contribution of this paper is to highlight the relatively understudied problem of
converting a data set into a graph, which forms an intermediate representation for many
clustering and manifold learning algorithms. A second contribution is novel construction
algorithms, which are: easy to implement, not expensive to compute, robust across many
noise levels and parameter settings, and useful for clustering and manifold learning. In
general, a careful selection of the graph construction algorithm makes the results of these
machine learning methods robust, and avoids or limits the required parameter search. Finally, the combination of many graphs, formed from perturbed versions of the data, into an
ensemble of graphs, is a novel approach to the construction problem.
Our idea of MST ensembles is an extension to graphs of the well-known technique of combining predictors by averaging (regression) or voting (classification), as is the regularizing
effect of training with noise [12]. An ensemble of predictors improves the generalization to
unseen data if the individual predictors are independent from each other and disagree with
each other, and can be explained by the bias-variance tradeoff. Unlike regression or classification, unsupervised graph learning lacks at present an error function, so it seems difficult
to apply the bias-variance framework here. However, we have conducted a wide range of
empirical tests to understand the properties of the ensemble MSTs, and to compare them
to the other graph construction methods, in terms of the error in the geodesic distances (if
known a priori). In summary, we have found that the variance of the error for the geodesic
Ellipse, high noise
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Figure 3: Performance of Isomap with different graphs in 3 data sets: ellipse with N = 100
points, high noise; Swiss roll with N = 500 points, low and high noise (where high noise
means Gaussian with standard deviation equal to 9% of the separation between branches).
All plots show on the X axis the graph parameter (, k, r or t); on the left Y axis the
PN
average error in the geodesic distances (red curve, E = N12 i,j=1 |?
gij ? gij |); and on the
? DY ))
right Y axis Isomap?s estimated residual variance (solid blue curve, V? = 1 ? R2 (G,
2
?
and true residual variance (dashed blue curve, V = 1 ? R (G, DY )), where G and G
are the matrices of estimated and true geodesic distances, respectively, DY is the matrix of
Euclidean distances in the low-dimensional embedding, and R(A, B) is the standard linear
correlation coefficient, taken over all entries of matrices A and B. Where the curves for
-ball and k-NNG are missing, the graph was disconnected.
distances decreases for the ensemble when the individual graphs are sparse (e.g. MSTs as
used here, or -ball and k-NNG with low or k); but not necessarily when the graphs are
not sparse.
The typical cut [9, 13] is a clustering criterion that is based on the probability pij that
points xi and xj are in the same cluster over all possible partitions (under the Boltzmann
distribution for the mincut cost function). The pij need to be estimated: [9] use SwendsenWang sampling, while [13] use randomized trees sampling. However, these trees are not
used to define a proximity graph, unlike in our work.
An important direction for future work concerns the noise model for PMSTs. The model
we propose is isotropic, in that every direction of perturbation is equally likely. A better
way is to perturb points more strongly in directions likely to lie within the manifold and less
strongly in directions away from the manifold, using a method such as k nearest neighbors
to estimate appropriate directions. Preliminary experiments with such a manifold-aligned
model are very promising, particularly when the data is very noisy or its distribution on the
manifold is not uniform. The noise model can also be extended to deal with non-Euclidean
data by directly perturbing the similarities.
Acknowledgements
Funding provided by a CIHR New Emerging Teams grant.
References
[1] Zhenyu Wu and Richard Leahy. An optimal graph theoretic approach to data clustering: Theory
and its application to image segmentation. IEEE Trans. on Pattern Anal. and Machine Intel.,
15(11):1101?1113, November 1993.
[2] Jianbo Shi and Jitendra Malik. Normalized cuts and image segmentation. IEEE Trans. on
Pattern Anal. and Machine Intel., 22(8):888?905, August 2000.
[3] Pedro F. Felzenszwalb and Daniel P. Huttenlocher. Efficient graph-based image segmentation.
Int. J. Computer Vision, 59(2):167?181, September 2004.
[4] Romer Rosales, Kannan Achan, and Brendan Frey. Learning to cluster using local neighborhood structure. In ICML, 2004.
[5] Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A global geometric framework for
nonlinear dimensionality reduction. Science, 290(5500):2319?2323, December 22 2000.
[6] Sam T. Roweis and Lawrence K. Saul. Nonlinear dimensionality reduction by locally linear
embedding. Science, 290(5500):2323?2326, December 22 2000.
[7] Mikhail Belkin and Partha Niyogi. Laplacian eigenmaps for dimensionality reduction and data
representation. Neural Computation, 15(6):1373?1396, June 2003.
[8] Kilian Q. Weinberger and Lawrence K. Saul. Unsupervised learning of image manifolds by
semidefinite programming. In CVPR, 2004.
[9] Marcelo Blatt, Shai Wiseman, and Eytan Domany. Data clustering using a model granular
magnet. Neural Computation, 9(8):1805?1842, November 1997.
[10] C. T. Zahn. Graph-theoretical methods for detecting and describing gestalt clusters. IEEE
Trans. Computers, C?20(1):68?86, April 1971.
[11] Chakra Chennubhotla and Allan Jepson. EigenCuts: Half-lives of EigenFlows for spectral
clustering. In NIPS, 2003.
[12] Christopher M. Bishop. Neural Networks for Pattern Recognition. Oxford University Press,
New York, Oxford, 1995.
[13] Yoram Gdalyahu, Daphna Weinshall, and Michael Werman. Self organization in vision: Stochastic clustering for image segmentation, perceptual grouping, and image database organization. IEEE Trans. on Pattern Anal. and Machine Intel., 23(10):1053?1074, October 2001.
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1,852 | 2,682 | Optimal sub-graphical models
Mukund Narasimhan? and Jeff Bilmes?
Dept. of Electrical Engineering
University of Washington
Seattle, WA 98195
{mukundn,bilmes}@ee.washington.edu
Abstract
We investigate the problem of reducing the complexity of a graphical
model (G, PG ) by finding a subgraph H of G, chosen from a class of
subgraphs H, such that H is optimal with respect to KL-divergence. We
do this by first defining a decomposition tree representation for G, which
is closely related to the junction-tree representation for G. We then give
an algorithm which uses this representation to compute the optimal H ?
H. Gavril [2] and Tarjan [3] have used graph separation properties to
solve several combinatorial optimization problems when the size of the
minimal separators in the graph is bounded. We present an extension of
this technique which applies to some important choices of H even when
the size of the minimal separators of G are arbitrarily large. In particular,
this applies to problems such as finding an optimal subgraphical model
over a (k ? 1)-tree of a graphical model over a k-tree (for arbitrary k)
and selecting an optimal subgraphical model with (a constant) d fewer
edges with respect to KL-divergence can be solved in time polynomial in
|V (G)| using this formulation.
1
Introduction and Preliminaries
The complexity of inference in graphical models is typically exponential in some parameter of the graph, such as the size of the largest clique. Therefore, it is often required to find
a subgraphical model that has lower complexity (smaller clique size) without introducing
a large error in inference results. The KL-divergence between the original probability distribution and the probability distribution on the simplified graphical model is often used to
measure the impact on inference. Existing techniques for reducing the complexity of graphical models including annihilation and edge-removal [4] are greedy in nature and cannot
make any guarantees regarding the optimality of the solution. This problem is NP-complete
[9] and so, in general, one cannot expect a polynomial time algorithm to find the optimal
solution. However, we show that when we restrict the problem to some sets of subgraphs,
the optimal solution can be found quite quickly using a dynamic programming algorithm
in time polynomial in the tree-width of the graph.
1.1
Notation and Terminology
A graph G = (V, E) is said to be triangulated if every cycle of length greater than 3
has a chord. A clique of G is a non-empty set S ? V such that {a, b} ? E for all
?
This work was supported by NSF grant IIS-0093430 and an Intel Corporation Grant.
{b, c, d}
d
{c, f, g}
{b, c}
{b, e, c}
b
c
{f, c}
{c, e}
{e, c, f }
g
{b, e}
a
e
f
{a, b, e}
Figure 1: A triangulated graph G and a junction-tree for G
a, b ? S. A clique S is maximal if S is not properly contained in another clique. If
? and ? are non-adjacent vertices of G then a set of vertices S ? V \ {?, ?} is called
an (?, ?)-separator if ? and ? are in distinct components of G[V \ S]. S is a minimal
(?, ?)-separator if no proper subset of S is an (?, ?)-separator. S is said to be a minimal
separator if S is a minimal (?, ?)-separator for some non adjacent a, b ? V . If T =
(K, S) is a junction-tree for G (see [7]), then the nodes K of T correspond to the maximalcliques of G, while the links S correspond to minimal separators of G (We reserve the
terms vertices/edges for elements of G, and nodes/links for the elements of T ). If G is
triangulated, then the number of maximal cliques is at most |V |. For example, in the graph
G shown in Figure 1, K = {{b, c, d} , {a, b, e} , {b, e, c} , {e, c, f } , {c, f, g}}. The links
S of T correspond to minimal-separators of G in the following way. If Vi Vj ? S (where
Vi , Vj ? K and hence are cliques of G), then Vi ? Vj 6= ?. We label each edge Vi Vj ? S
with the set Vij = Vi ? Vj , which is a non-empty complete separator in G. The removal of
any link Vi Vj ? S disconnects T into two subtrees which we denote T (i) and T (j) (chosen
so that T (i) contains Vi ). We will let K(i) be the nodes of T (i) , and V (i) = ?V ?K (i) V be
the set of vertices corresponding to the subtree T (i) . The junction tree property ensures that
V (i) ? V (j) = Vi ? Vj = Vij . We will let G(i) be the subgraph induced by V (i) .
A graphical model is a pair (G, P ) where P is the joint probability distribution for random
variables X1 , X2 , . . . , Xn , and G is a graph with vertex set V (G) = {X1 , X2 , . . . , Xn }
such that the separators in G imply conditional independencies in P (so P factors according
to G). If G is triangulated, then the junction-tree algorithm can be used for exact inference
in the probability distribution P . The complexity of this algorithm grows with the treewidth
of G (which is one less than the size of the largest clique in G when G is triangulated). The
growth is exponential when P is a discrete probability distribution, thus rendering exact
inference for graphs with large treewidth impractical. Therefore, we seek another graphical
model (H, PH ) which allows tractable inference (so H should have lower treewidth than
G has). The general problem of finding a graphical model of tree-width at most k so as
to minimize the KL-divergence from a specified probability distribution is NP complete
for general k ([9]) However, it is known that this problem is solvable in polynomial time
(in |V (G)|) for some special cases cases (such as when G has bounded treewidth or when
k = 1 [1]). If (G, PG ) and (H, PH ) are graphical models, then we say that (H, PH ) is a
subgraphical model of (G, PG ) if H is a spanning subgraph of G. Note in particular that
separators in G are separators in H, and hence (G, PH ) is also a graphical model.
2
Graph Decompositions and Divide-and-Conquer Algorithms
For the remainder of the paper, we will be assuming that G = (V, E) is some triangulated
graph, with junction tree T = (K, S). As observed above, if Vi Vj ? S, then the removal
{b, c, d}
d
{b, c}
{b, e, c}
b
c
c
{c, f, g}
{f, c}
{e, c, f }
g
{b, e}
a
e
e
f
{a, b, e}
Figure 2: The graphs G(i) , G(j) and junction-trees T (i) and T (j) resulting from the removal
of the link Vij = {c, e}
of Vij = Vi ? Vj disconnects G into two (vertex-induced) subgraphs G(i) and G(j) which
are both triangulated, with junction-trees T (i) and T (j) respectively. We can recursively
decompose each of G(i) and G(j) into smaller and smaller subgraphs till the resulting subgraphs are cliques. When the size of all the minimal separators are bounded, we may use
these decompositions to easily solve problems that are hard in general. For example, in [5]
it is shown that NP-complete problems like vertex coloring, and finding maximum independent sets can be solved in polynomial time on graphs with bounded tree-width (which
are equivalent to spanning graphs with bounded size separators). We will be interested in
finding (triangulated) subgraphs of G that satisfy some conditions, such as a bound on the
number of edges, or a bound on the tree-width and which optimize separable objective
functions (described in Section 2)
One reason why problems such as this can often be solved easily when the tree-width of
G is bounded by some constant is this : If Vij is a separator decomposing G into G(i)
and G(j) , then a divide-and-conquer approach would suggest that we try and find optimal
subgraphs of G(i) and G(j) and then splice the two together to get an optimal subgraph of
G. There are two issues with this approach. First, the optimal subgraphs of G (i) and G(j)
need not necessarily match up on Vij , the set of common vertices. Second, even if the two
subgraphs agree on the set of common vertices, the graph resulting from splicing the two
subgraphs together need not be triangulated (which could happen even if the two subgraphs
individually are triangulated). To rectify the situation, we can do the following. We partition the set of subgraphs of G(i) and G(j) into classes, so that any subgraph of G(i) and any
subgraph G(j) corresponding to the same class are compatible in the sense that they match
up on their intersection namely Vij , and so that by splicing the two subgraphs together, we
get a subgraph of G which is acceptable (and in particular is triangulated). Then given optimal subgraphs of both G(i) and G(j) corresponding to each class, we can enumerate over
all the classes and pick the best one. Of course, to ensure that we do not repeatedly solve
the same problem, we need to work bottom-up (a.k.a dynamic programming) or memoize
our solutions. This procedure can be carried out in polynomial (in |V |) time as long as
we have only a polynomial number of classes. Now, if we have a polynomial number of
classes, these classes need not actually be a partition of all the acceptable subgraphs, though
the union of the classes must cover all acceptable subgraphs (so the same subgraph can be
contained in more than one class). For our application, every class can be thought of to be
the set of subgraphs that satisfy some constraint, and we need to pick a polynomial number
of constraints that cover all possibilities. The bound on the tree-width helps us here. If
k
|Vij | = k, then in any subgraph H of G, H[Vij ] must be one of the 2(2) possible subgraphs
k
of G[Vij ]. So, if k is sufficiently small (so 2(2) is bounded by some polynomial in |V |),
then this procedure results in a polynomial time algorithm. In this paper, we show that in
some cases we can characterize the space H so that we still have a polynomial number of
constraints even when the tree-width of G is not bounded by a small constant.
2.1 Separable objective functions
For cases where exact inference in the graphical model (G, PG ) is intractable, it is natural to
try to find a subgraphical model (H, PH ) such that D(PG kPH ) is minimized, and inference
using H is tractable. We will denote by H the set of subgraphs of G that are tractable for
inference. For example, this set could be the set of subgraphs of G with treewidth one less
than the treewidth of G, or perhaps the set of subgraphs of G with at d fewer edges. For a
specified subgraph H of G, there is a unique probability distribution PH factoring over H
that minimizes D(PG kPH ). Hence, finding a optimal subgraphical model is equivalent to
finding a subgraph H for which D(PG kPH ) is minimized. If Vij is a separator of G, we
will attempt to find optimal subgraphs of G by finding optimal subgraphs of G (i) and G(j)
and splicing them together. However, to do this, we need to ensure that the objective criteria
also decomposes along the separator Vij . Suppose that H is any triangulated subgraph of G.
Let PG(i) and PG(j) be the (marginalized) distributions of PG on V (i) and V (j) respectively,
and PH (i) and PH (j) be the (marginalized) distributions of the distribution PH on V (i) and
V (j) where H (i) = H[V (i) ] and H (j) = H[V (j) ], The following result assures us that the
KL-divergence also factors according to the separator Vij .
Lemma 1. Suppose that (G, PG ) is a graphical model, H is a triangulated subgraph of
G, and PH factors over H. Then D(PG kPH ) = D(PG(i) kPH (i) ) + D(PG(j) kPH (j) ) ?
D(PG[Vij ] kPH[Vij ] ).
Proof. Since H is a subgraph of G, and Vij is a separator of G, Vij must also be a sepa
P (i) ({Xv }v?V (i) )?PH (j) ({Xv }v?V (j) )
rator of H. Therefore, PH {Xv }v?V = H
. The result
)
PH[V ] ({Xv }
ij
v?Vij
follows immediately.
Therefore, there is hope that we can reduce our our original problem of finding an optimal
subgraph H ? H as one of finding subgraphs of H (i) ? G(i) and H (j) ? G(j) that are
compatible, in the sense that they match up on the overlap Vij , and for which D(PG kPH )
is minimized. Throughout this paper, for the sake of concreteness, we will assume that
the objective criterion is to minimize the KL-divergence. However, all the results can
be extended to other objective functions, as long as they ?separate? in the sense that for
any separator, the objective function is the sum of the objective functions of the two parts,
possibly modulo some correction factor which is purely a function of the separator. Another
example might be the complexity r(H) of representing the graphical model H. A very
natural representation satisfies r(G) = r(G(i) ) + r(G(j) ) if G has a separator G(i) ? G(j) .
Therefore, the representation cost reduction would satisfy r(G) ? r(H) = (r(G (i) ) ?
r(H (i) )) + (r(G(j) ) ? r(H (j) )), and so also factors according to the separators. Finally
note that any linear combinations of such separable functions is also separable, and so this
technique could also be used to determine tradeoffs (representation cost vs. KL-divergence
loss for example). In Section 4 we discuss some issues regarding computing this function.
2.2
Decompositions and decomposition trees
For the algorithms considered in this paper, we will be mostly interested in the decompositions that are specified by the junction tree, and we will represent these decompositions
by a rooted tree called a decomposition tree. This representation was introduced in [2, 3],
and is similar in spirit to Darwiche?s dtrees [6] which specify decompositions of directed
acyclic graphs. In this section and the next, we show how a decomposition tree for a graph
may be constructed, and show how it is used to solve a number of optimization problems.
abd; ce; gf
a; be; cd
d; bc; e
abe
dbc
ebc
e; cf ; g
cef
cf g
Figure 3: The separator tree corresponding to Figure 1
A decomposition tree for G is a rooted tree whose vertices correspond to separators and
cliques of G. We describe the construction of the decomposition tree in terms of a junctiontree T = (K, S) for G. The interior nodes of the decomposition tree R(T ) correspond to
S (the links of T and hence the minimal separators of G). The leaf or terminal nodes
represent the elements of K (the nodes of T and hence the maximal cliques of G). R(T )
can be recursively constructed from T as follows : If T consists of just one node K, (and
hence no edges), then R consists of just one node, which is given the label K as well. If
however, T has more than one node, then T must contain at least one link. To begin, let
Vi Vj ? S be any link in T . Then removal of the link Vi Vj results in two disjoint junctiontrees T (i) and T (j) . We label the root of R by the decomposition (V (i) ; Vij ; V (j) ). The rest
of R is recursively built by successively picking links of T (i) and T (j) (decompositions of
G(i) and G(j) ) to form the interior nodes of R. The effect of this procedure on the junction
tree of Figure 1 is shown in Figure 3, where the decomposition associated with the interior
nodes is shown inside the nodes. Let M be the set of all nodes of R(T ). For any interior
node M induced by the the link Vi Vj ? S of T , then we will let M (i) and M (j) represent
the left and right children of M , and R(i) and R(j) be the left and right trees below M .
3
3.1
Finding optimal subgraphical models
Optimal sub (k ? 1)-trees of k-trees
Suppose that G is a k-tree. A sub (k ? 1)-tree of G is a subgraph H of G that is (k ? 1)tree. Now, if Vij is any minimal separator of G, then both G(i) and G(j) are k-trees on
vertex sets V (i) and V (j) respectively. It is clear that the induced subgraphs H[V (i) ] and
H[V (j) ] are subgraphs of G(i) and G(j) and are partial (k ? 1)-trees. We will be interested
in finding sub (k ? 1)-trees of k trees and this problem is trivial by the result of [1] when
k = 2. Therefore, we assume that k ? 3. The following result characterizes the various
possibilities for H[Vij ] in this case.
Lemma 2. Suppose that G is a k-tree, and S = Vij is a minimal separator of G corresponding to the link ij of the junction-tree T . In any (k ? 1)-tree H ? G either
1. There is a u ? S such that u is not connected to vertices in both V (i) \ S and
V (j) \ S. Then S \ {u} is a minimal separator in H and hence is complete.
2. Every vertex in S is connected to vertices in both V (i) \S and V (j) \S. Then there
are vertices {x, y} ? S such that the edge H[S] is missing only the edge {x, y}.
Further either H[V (i) ] or H[V (j) ] does not contain a unchorded x-y path.
Proof. We consider two possibilities. In the first, there is some vertex u ? S such that u is
not connected to vertices in both V (i) \S and V (j) \. Since the removal of S disconnects G,
the removal of S must also disconnect H. Therefore, S must contain a minimal separator
of H. Since H is a (k ? 1)-tree, all minimal separators of H must contain k ? 1 vertices
which must therefore be S \{u}. This corresponds to case (1) above. Clearly this possiblity
can occur.
If there is no such u ? S, then every vertex in S is connected to vertices in both V (i) \ S
and V (j) \ S. If x ? S is connected to some yi ? V (i) \ S and yj ? V (j) \ S, then x is
contained in every minimal yi /yj separator (see [5]). Therefore, every vertex in S is part of
a minimal separator. Since each minimal separator contains k ? 1 vertices, there must be at
least two distinct minimum separators contained in S. Let Sx = S \ {x} and Sy = S \ {y}
be two distinct minimal separators. We claim that H[S] contains all edges except the edge
{x, y}. To see this, note that if z, w ? S, with z 6= w and {z, w} 6= {x, y} (as sets),
then either {z, w} ? Sy or {z, w} ? Sx . Since both Sx and Sy are complete in H, this
edge must be present in H. The edge {x, y} is not present in H[S] because all minimal
separators in H must be of size k ? 1. Further, if both V (i) and V (j) contain an unchorded
path between x and y, then by joining the two paths at x and y, we get a unchorded cycle
in H which contradicts the fact that H is triangulated.
Therefore, we may associate k2 ? 2 + 2 ? k constraints with each separator Vij of G as
follows. There are k possible constraints corresponding to case (1) above (one for each
choice of x), and k2 ? 2 choices corresponding to case (2) above. This is because for each
pair {x, y} corresponding to the missing edge, we have either V (i) contains no unchorded
xy paths or V (j) contains no unchorded xy paths. More explicitly, we can encode the set
of constraints CM associated with each separator S corresponding to an interior node M of
the decomposition tree as follows: CM = { (x, y, s) : x ? S, y ? S, s ? {i, j}}. If y = x,
then this corresponds to case (1) of the above lemma. If s = i, then x is connected only to
H (i) and if s = j, then x is connected only to H (j) . If y 6= x, then this corresponds to case
(2) in the above lemma. If s = i, then H (i) does not contain any unchorded path between
x and y, and there is no constraint on H (j) . Similarly if s = j, then H (j) does not contain
any unchorded path between x and y, and there is no constraint on H (i) .
Now suppose that H (i) and H (j) are triangulated subgraphs of G(i) and G(j) respectively,
then it is clear that if H (i) and H (j) both satisfy the same constraint they must match up
on the common vertices Vij . Therefore to splice together two solutions corresponding to
the same constraint, we only need to check that the graph obtained by splicing the graphs
is triangulated.
Lemma 3. Suppose that H (i) and H (j) are triangulated subgraphs of G(i) and G(j) respectively such that both of them satisfy the same constraint as described above. Then the
graph H obtained by splicing H (i) and H (j) together is triangulated.
Proof. Suppose that both H (i) and H (j) are both triangulated and both satisfy the same
constraint. If both H (i) and H (j) satisfy the same constraint corresponding to case (1)
in Lemma 2 and H has an unchorded cycle, then this cycle must involve elements of both
H (i) and H (j) . Therefore, there must be two vertices of S \{u} on the cycle, and hence this
cycle has a chord as S \ {u} is complete. This contradiction shows that H is triangulated.
So assume that both of them satisfy the constraint corresponding to case (2) of Lemma 2.
Then if H is not triangulated, there must be a t-cycle (for t ? 4) with no chord. Now, since
{x, y} is the only missing edge of S in H, and because H (i) and H (j) are individually
triangulated, the cycle must contain x, y and vertices of both V (i) \ S and V (j) \ S. We
may split this unchorded cycle into two unchorded paths, one contained in V (i) and one in
V (j) thus violating our assumption that both H (i) and H (j) satisfy the same constraint.
If |S| = k, then there are 2k + 2 ? k2 ? O(k 2 ) ? O(n2 ). We can use a divide and conquer
strategy to find the optimal sub (k ? 1) tree once we have taken care of the base case, where
G is just a single clique (of k + 1) elements. However, for this case, it is easily checked that
any subgraph of G obtained by deleting exactly one edge results in a (k ? 1) tree, and every
sub (k?1)-tree results from this operation. Therefore, the optimal (k?1)-tree can be found
using Algorithm 1, and in this case, the complexity of Algorithm 1 is O(n(k + 1) 2 ). This
procedure can be generalized to find the optimal sub (k ? d)- tree for any fixed d. However,
the number of constraints grows exponentially with d (though it is still polynomial in n).
Therefore for small, fixed values of d, we can compute the optimal sub (k ? d)-tree of G.
While we can compute (k ? d)-trees of G by first going from a k tree to a (k ? 1) tree, then
from a (k ? 1)-tree to a (k ? 2)-tree, and so on in a greedy fashion, this will not be optimal
in general. However, this might be a good algorithm to try when d is large.
3.2
Optimal triangulated subgraphs with |E(G)| ? d edges
Suppose that we are interested in a (triangulated) subgraph of G that contains d fewer edges
that G does. That is, we want to find an optimal subgraph H ? G such that |E(H)| =
|E(G)| ? d. Note that by the result of [4] there is always a triangulated subgraph with d
fewer edges (if d < |E(G)|). Two possibilities for finding such an optimal subgraph are
1. Use the procedure described in [4]. This is a greedy procedure which works in
d steps by deleting an edge at each step. At each state, the edge is picked from
the set of edges whose deletion leaves a triangulated graph. Then the edge which
causes the least increase in KL-divergence is picked at each stage.
2. For each possible subset A of E(G) of size d, whose deletion leaves a triangulated
graph, compute the KL divergence using
the formula above, and then pick the
|E(G)|
such sets, this can be done in polynomial
optimal one. Since there are
d
time (in |V (G)|) when d is a constant.
The first greedy algorithm is not guaranteed to yield the optimal solution. The second takes
time that is O(n2d ). Now, let us solve this problem using the framework we?ve described.
Let H be the set of subgraphs of G which may be obtained by deletion of d
edges.
For each M = ij ? M corresponding to the o
separator Vij , let CM =
n
E(G[Vij ])
(l, r, c, s, A) : l + r ? c = d, s a d bit string, A ?
. The constraint reprec
sented by (l, r, c, A) is this : A is a set of d edges of G[Vij ] that are missing in H, l edges
are missing from the left subgraph, and r edges are missing from the right subgraph. c represents the double count, and so is subtracted from the total. If k is the size of the largest
k
clique, then the total number of such constraints is bounded by 2d ? 2d ? (2) ? O(k 2d )
d
which could be better than O(n2d ) and is polynomial in |V | when d is constant. See [10]
for additional details.
4
Conclusions
Algorithm 1 will compute the optimal H ? H for the two examples discussed above and
is polynomial (for fixed constant d) even if k is O(n). In [10] a generalization is presented
which will allow finding the optimal solution for other classes of subgraphical models.
Now, we assume an oracle model for computing KL-divergences of probability distributions on vertex sets of cliques. It is clear that these KL-divergences can be computed
R ? separator-tree for G;
for each vertex M of R in order of increasing height (bottom up) do
for each constraint cM of M do
if M is an interior vertex of R corresponding to edge ij of the junction tree then
Let Ml and Mr be the left and right children of M ;
Pick constraint cl ? CMl compatible with cM to minimize table[Ml , cl ];
Pick constraint cr ? CMr compatible with cM to minimize table[Mr , cr ];
loss ? D(PG [M ]kPH [M ]);
table[M, cM ] ? table[Ml , cl ] + table[Mr , cr ] ? loss;
else
table[M, cM ] ? D(PG [M ]kPH [M ]);
end
end
end
Algorithm 1: Finding optimal set of constraints
efficiently for distributions like Gaussians, but for discrete distributions this may not be
possible when k is large. However even in this case this algorithm will result in only
polynomial calls to the oracle. The standard algorithm [3] which is exponential in the
treewidth will make O(2k ) calls to this oracle. Therefore, when the cost of computing the
KL-divergence is large, this algorithm becomes even more attractive as it results in exponential speedup over the standard algorithm. Alternatively, if we can compute approximate
KL-divergences, or approximately optimal solutions, then we can compute an approximate
solution by using the same algorithm.
References
[1] C. Chow and C. Liu, ?Approximating discrete probability distributions with dependence trees?, IEEE Transactions on Information Theory, v. 14, 1968, Pages 462?467.
[2] F. Gavril, ?Algorithms on clique separable graphs?, Discrete Mathematics v. 9 (1977),
pp. 159?165.
[3] R. E. Tarjan. ?Decomposition by Clique Separators?, Discrete Mathematics, v. 55
(1985), pp. 221?232.
[4] U. Kjaerulff. ?Reduction of computational complexity in Bayesian networks through
removal of weak dependencies?, Proceedings of the Tenth Annual Conference on
Uncertainty in Artificial Intelligence, pp. 374?382, 1994.
[5] T. Kloks, ?Treewidth: Computations and Approximations?, Springer-Verlag, 1994.
[6] A. Darwiche and M. Hopkins. ?Using recursive decomposition to construct elimination orders, jointrees and dtrees?, Technical Report D-122, Computer Science Dept.,
UCLA.
[7] S. Lauritzen. ?Graphical Models?, Oxford University Press, Oxford, 1996.
[8] T. A. McKee and F. R. McMorris. ?Topics in Intersection Graph Theory?, SIAM
Monographs on Discrete Mathematics and Applications, 1999.
[9] D. Karger and N. Srebro. ?Learning Markov networks: Maximum bounded tree-width
graphs.? In Symposium on Discrete Algorithms, 2001, Pages 391-401.
[10] M. Narasimhan and J. Bilmes. ?Optimization on separator-clique trees.?, Technical
report UWEETR 2004-10, June 2004.
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1,853 | 2,683 | Rate- and Phase-coded Autoassociative Memory
M?t? Lengyel
Peter Dayan
Gatsby Computational Neuroscience Unit, University College London
17 Queen Square, London WC1N 3AR, United Kingdom
{lmate,dayan}@gatsby.ucl.ac.uk
Abstract
Areas of the brain involved in various forms of memory exhibit patterns
of neural activity quite unlike those in canonical computational models.
We show how to use well-founded Bayesian probabilistic autoassociative
recall to derive biologically reasonable neuronal dynamics in recurrently
coupled models, together with appropriate values for parameters such as
the membrane time constant and inhibition. We explicitly treat two cases.
One arises from a standard Hebbian learning rule, and involves activity
patterns that are coded by graded firing rates. The other arises from a
spike timing dependent learning rule, and involves patterns coded by the
phase of spike times relative to a coherent local field potential oscillation.
Our model offers a new and more complete understanding of how neural
dynamics may support autoassociation.
1
Introduction
Autoassociative memory in recurrently coupled networks seems fondly regarded as having been long since solved, at least from a computational perspective. Its neurobiological
importance, as a model of episodic (event) memory storage and retrieval (from noisy and
partial inputs) in structures such as area CA3 in the hippocampus, is of course clear [1].
This perhaps suggests that it is only the exact mapping of the models to the neural substrate
that holds any remaining theoretical interest.
However, the characteristic patterns of activity in areas such as CA3 that are involved in
memory are quite unlike those specified in the bulk of models. In particular neurons (for
instance hippocampal place cells) show graded activity during recall [2], prominent theta
frequency oscillations [3] and an apparent variety of rules governing synaptic plasticity
[4, 5]. The wealth of studies of memory capacity of attractor networks of binary units does
not give many clues to the specification, analysis or optimization of networks acting in
these biologically relevant regimes. In fact, even theoretical approaches to autoassociative
memories with graded activities are computationally brittle.
Here, we generalize previous analyses [6, 7] to address these issues. Formally, these models interpret recall as Bayesian inference based on information given by the noisy input,
the synaptic weight matrix, and prior knowledge about the distribution of possible activity patterns coding for memories. More concretely (see section 2), the assumed activity
patterns and synaptic plasticity rules determine the term in neuronal update dynamics that
describes interactions between interconnected cells. Different aspects of biologically reasonable autoassociative memories arise from different assumptions. We show (section 3)
We thank Boris Gutkin for helpful discussions on the phase resetting characteristics of different
neuron types. This work was supported by the Gatsby Charitable Foundation.
that for neurons are characterized by their graded firing rates, the regular rate-based characterization of neurons effectively approximates optimal Bayesian inference. Optimal values
for parameters of the update dynamics, such as level of inhibition or leakage conductance,
are inherently provided by our formalism. We then extend the model (section 4) to a setting involving spiking neurons in the context of a coherent local field potential oscillation
(LFPO). Memories are coded by the the phase of the LFPO at which each neuron fires, and
are stored by spike timing dependent plasticity. In this case, the biophysically plausible
neuronal interaction function takes the form of a phase reset curve: presynaptic firing accelerates or decelerates the postsynaptic cell, depending on the relative timing of the two
spikes, to a degree that is proportional to the synaptic weight between the two cells.
2
MAP autoassociative recall
The first requirement is to specify the task for autoassociative recall in a probabilistically
sound manner. This specification leads to a natural account of the dynamics of the neurons
during recall, whose form is largely determined by the learning rule. Unfortunately, the
full dynamics includes terms that are not purely local to the information a post-synaptic
neuron has about pre-synaptic activity, and we therefore consider approximations that restore essential characteristics necessary to satisfy the most basic biological constraints. We
validate the quality of the approximations in later sections.
The construction of the objective function: Consider an autoassociative network which
has stored information about M memories x1 . . . xM in a synaptic weight matrix, W between a set of N neurons. We specify these quantities rather generally at first to allow for
different ways of construing the memories later. The most complete probabilistic description of its task is to report the conditional distribution P [x|?
x, W] over the activities x given
noisy inputs x
? and the weights. The uncertainty in this posterior distribution has two roots.
First, the activity pattern referred to by the input is unclear unless there is no input noise.
Second, biological synaptic plasticity rules are data-lossy ?compression algorithms?, and
so W specifies only imprecise information about the stored memories.
In an ideal case, P [x|?
x, W] would have support only on the M stored patterns x1 . . . xM .
However, biological storage methods lead to weights W that permit a much greater range
of possibilities. We therefore consider methods that work in the full space of activities x.
In order to optimize the probability of extracting just the correct memory, decision theory
encourages us to maximize the posterior probability [8]:
x
? := argmaxx P [x|?
x, W] , P [x|?
x, W] ? P [x] P [?
x|x] P [W|x]
(1)
The first term in Eq.1 imports prior knowledge
Q of the statistical characteristics of the memories, and is assumed to factorize: P [x] := i Px [xi ]. The second term describes the noise
process corrupting the inputs. For unbiased noise it will be a term in x that is effectively
centered on x
?. We assume that the noise corrupting each element
Q of the patterns
Q is independent, and independent of the original pattern, so P [?
x|x] := i P [?
xi |x] := i P [?
xi |xi ].
The third term assesses the likelihood that the weight matrix came from a training set of
size M including pattern x.1 Biological constraints encourage consideration of learning
updates for the synapse from neuron j to neuron i that are local to the pre-synaptic (xm
j )
m
and post-synaptic (xm
i ) activities of connected neurons when pattern x is stored:
m
m
?wi,j
:= ? xm
(2)
i , xj
We assume the contributions of individual training patterns are additive, Wi,j :=
P
m
m ?wi,j , and that there are no autapses in the network, Wi,i := 0.
1
Uncertainty about M could also be incorporated into the model, but is neglected here.
Storing a single random pattern drawn from the prior distribution will result in a synaptic
weight change with a distribution determined by the prior
and the learning rule, having
2
??w = h? (x1 , x2 )iPx [x1 ]?Px [x2 ] mean, and ??w
= ?2 (x1 , x2 ) P [x ]?P [x ] ? ?2?w
x 1
x 2
variance. Storing M ? 1 random patterns means adding M ? 1 iid. random variables and
thus, for moderately large M , results in a synaptic weight with an approximately Gaussian
distribution P [Wi,j ] ' G (Wi,j ; ?W , ?W ), with mean ?W = (M ? 1) ??w and variance
2
2
?W
= (M ? 1) ??w
. Adding a further particular pattern x is equivalent to adding a
random variable with a mean determined by the learning rule, and zero variance, thus:
P [Wi,j |xi , xj ] ' G (Wi,j ; ?W + ? (xi , xj ) , ?W )
(3)
We also make the approximation that
Q elements of the synaptic weight matrix are independent, and thus write: P [W|x] := i,j6=i P [Wi,j |xi , xj ].
Having restricted our horizons to maximum a posteriori (MAP) inference, we can consider
as an objective function the log of the posterior distribution. In the light of our factorizability assumptions, this is
O (x) = log
[?
x|x] + log P [W|x]
P P [x] + log PP
P
= i log P [xi ] + i log P [?
xi |xi ] + i,j6=i log P [Wi,j |xi , xj ]
(4)
Neuronal update dynamics: Finding the global maximum of the objective function, as
stated in equation 1, is computationally extravagant, and biologically questionable. We
therefore specify neuronal dynamics arising from gradient ascent on the objective function:
?x x? ? ?x O (x) .
(5)
Combining equations 4 and 5 we get
i
?x dx
dt =
?
?xi
log P [W|x] =
?
?xi
log P [x] +
?
j6=i ?xi
P
?
?xi
?
?xi log P [W|x] , where
?
?xi log P [Wj,i |xj , xi ] .
log P [?
x|x] +
log P [Wi,j |xi , xj ] +
(6)
(7)
The first two terms in equation 6 only depend on the activity of the neuron itself and its
input. For example, for a Gaussian prior Px [xi ] = G (Wi,j ; ?x , ?x ) and unbiased Gaussian
noise on the input P [?
xi |xi ] = G (?
xi ; xi , ?x? ), these would be:
d
d
i
i
= ??x2 ? ?12 + ?12 xi + ?x?2i (8)
xi |xi ] = ?x??x
+ x?i??x
2
2
dxi log P [xi ] + dxi log P [?
x
x
?
x
x
x
?
x
?
The first term on the right-hand side of the last equality expresses a constant bias; the
second involves self-decay; and the third describes the effect of the input.
The terms in equation 7 indicate how a neuron should take into account the activity of other
neurons based on the synaptic weights. From equation 3, the terms are
?
?xi
?
?xi
log P [Wi,j |xi , xj ]=
log P [Wj,i |xj , xi ]=
1
2
?W
h
(Wi,j ? ?W )
i
(9)
1
2
?W
h
i
?
?
(Wj,i ? ?W ) ?x
?(xj , xi ) ? ?(xj , xi ) ?x
?(xj , xi )
i
i
(10)
?
?(xi , xj )
?xi
? ?(xi , xj )
?
?(xi , xj )
?xi
Two aspects of the above formul? are biologically troubling. The last terms in each express the effects of other cells, but without there being corresponding synaptic weights. We
approximate these terms using their mean values over the prior distribution. In this case
?
?
?i+ = h? (xi , xj ) ?x
? (xi , xj )iPx [xj ] and ?i? = h? (xj , xi ) ?x
? (xj , xi )iPx [xj ] coni
i
tribute terms that only depend on the activity of the updated cell, and so can be lumped
with the prior- and input-dependent terms of Eq.8.
Further, equation 10 includes synaptic weights, Wj,i , that are postsynaptic with respect
to the updated neuron. This would require the neuron to change its activity depending
on the weights of its postsynaptic synapses. One simple work-around is to approximate
a postsynaptic weight by the mean of its conditional distribution given the corresponding
presynaptic weight: Wj,i ' hP [Wj,i |Wi,j ]i. In the simplest case of perfectly symmetric
or anti-symmetric learning, with ? (xi , xj ) = ?? (xj , xi ), we have Wj,i = ?Wj,i and
?i+ = ?i? = ?i . In the anti-symmetric case ?w = 0.
Making these assumptions, the neuronal interaction function simplifies to
?
? (xi , xj )
(11)
H (xi , xj ) = (Wi,j ? ?W ) ?x
i
i
and ?22
j6=i H (xi , xj ) ? (N ? 1) ?i is the weight-dependent term of equation 7.
W
Equation 11 shows that there is a simple relationship between the synaptic plasticity rule,
? (xi , xj ), and the neuronal interaction function, H (xi , xj ), that is approximately optimal
for reading out the information that is encoded in the synaptic weight matrix by that synaptic plasticity rule. It also shows that the magnitude of this interaction should be proportional
to the synaptic weight connecting the two cells, Wi,j .
hP
We specialize this analysis to two important cases with (a) graded, rate-based, or (b) spiking, oscillatory phase-based, activities. We derive appropriate dynamics from learning
rules, and show that, despite the approximations, the networks have good recall performance.
3
Rate-based memories
The most natural assumption about pattern encoding is that the activity of each unit is
interpreted directly as its firing rate. Note, however, that most approaches to autoassociative
memory assume binary patterns [9], sitting ill with the lack of saturation in cortical or
hippocampal neurons in the appropriate regime. Experiments [10] suggest that regulating
activity levels in such networks is very tricky, requiring exquisitely carefully tuned neuronal
dynamics. There has been work on graded activities in the special case of line or surface
attractor networks [11, 12], but these also pose dynamical complexitiese. By contrast,
graded activities are straightforward in our framework.
Consider Hebbian covariance learning: ?cov (xi , xj ) := Acov (xi ? ?x ) (xj ? ?x ), where
Acov > 0 is a normalizing constant and ?x is the mean of the prior distribution of the
patterns to be stored. The learning rule is symmetric, and so, based on Eq.11, the optimal
neuronal interaction function is Hcov (xi , xj ) = Acov (Wi,j ? ?W ) (xj ? ?x ). This leads
to a term in the dynamics which is the conventional weighted sum of pre-synaptic firing
rates. The other key term in the dynamics is ?i = ?A2cov ?x2 (xi ? ?x ), where ?x2 is the
variance of the prior distribution, expressing self-decay to a baseline activity level determined by ?x . The prior- and input-dependent terms also contribute to self-decay as shown
in Eq.8. Integration of the weighted sum of inputs plus decay to baseline constitute the
widely used leaky integrator reduction of a single neuron [10].
Thus, canonical models of synaptic plasticity (the Hebbian covariance rule) and single neuron firing rate dynamics are exactly matched for autoassociative recall. Optimal values for
all parameters of single neuron dynamics (except the membrane time constant determining the speed of gradient ascent) are directly implied. This is important, since it indicates
how to solve the problem for graded autoassociative memories (as opposed to saturing ones
[14, 15]), that neuronal dynamics have to be finely tuned. As examples, the leak conductance is given by the sum of the coefficients of all terms linear in xi , the optimal bias current
is the sum of all terms independent of xi , and the level of inhibition can be determined from
the negative terms in the interaction function, ??W and ??x .
Since our derivation embodies a number of approximations, we performed numerical simulations. To gauge the performance of the Bayes-optimal network we compared it to
networks of increasing complexity (Fig. 1A,B). A trivial lower bound of performance is
A
B
prior
input
ideal observer
Bayesian: prior + input
Bayesian: prior + input + synapses
3
2
0.7
prior
input
Bayesian: prior + input
Bayesian: prior + input + synapses
1
0.5
Frequency
Recalled activity
0.6
0
?1
0.4
0.3
0.2
?2
0.1
?3
?3
?2
?1
0
1
Stored activity
2
?4
?3
?2
?1
0
Error
1
2
3
4
5
D
C
1.2
Average normalized error
1.2
prior
input
ideal observer
Bayesian
Treves
1.4
prior
input
ideal observer
Bayesian
Treves
1.4
1
Average error
0
?5
3
0.8
0.6
0.4
1
0.8
0.6
0.4
0.2
0.2
0
0
1
10
Number of stored patterns
100
1
10
Number of stored patterns
100
Figure 1: Performance of the rate-coded Bayesian inference network (), compared to a Bayesian
network that only takes into account evidence from the prior and the input but not from the synaptic
weight matrix (?), a network that randomly generates patterns from the prior distribution (?), a
network that transmits its input to its output (+), and the ?ideal observer? having access to the list
of stored patterns (?). A. Firing rates of single units at the end of the recall process (y-axis) against
firing rates in the original pattern (x-axis). B. Frequency histograms of errors (difference between
recalled and stored firing rates). The ideal observer is not plotted because its error distribution was a
Dirac-delta at 0. C, D. Benchmarking the Bayesian network against the network of Treves [13] (?)
on patterns of non-negative firing rates. Average error is the square root of the mean squared error
(C), average normalized error measures only the angle difference between true and recalled activities
(D). (These measures are not exactly the same as that used to derive the dynamics (equation 1), but
are reasonably appropriate.) The prior distribution was Gaussian with ?x = 0 mean and ?x2 = 1
variance (A,B), or a Gaussian with ?x = 0.5 mean and ?x2 = 0.25 variance truncated below 0 (C)
(yielding approximately a = 0.5 density), or ternary with a = 0.5 mean and density (D). The input
was corrupted by unbiased Gaussian noise of ?x2? = 1 variance (A,B), or ?x2? = 1.5 variance (C,D) and
cut at 0 (C,D). The learning rule was the covariance rule with Acov = 1 (A,B), or with Acov = 1/N a2
(C,D). The number of cells in the network was N = 50 (A,B) and N = 100 (C,D), and the number of
memories stored was M = 2 (A,B) or varied between M = 2 . . . 100 (C,D, note logarithmic scale).
For each data point, 10 different networks were simulated with a different set of stored patterns, and
for each network, 10 attempts at recall were made, with a noisy version of a randomly chosen pattern
as the input and with activities initialized at this input.
given by a network that generates random patterns from the same prior distribution from
which the patterns to be stored were drawn (P [x]). Another simple alternative is a network that simply transmits its input (?
x) to its output. (Note that the ?input only? network
is not necessarily superior to the ?prior only? network: their relative effectiveness depends
on the relative variances of the prior and noise distributions, a narrow prior with a wide
noise distribution would make the latter perform better, as in Fig. 1D). The Bayesian inference network performs considerably better than any of these simple networks. Crucially,
this improvement depends on the information encoded in synaptic weights: the network
practically falls back to the level of the ?input only? network (or the ?prior only? network,
whichever is the better, data not shown) if this information is ignored at the construction of
the recall dynamics (by taking the third term in Eq. 6 to be 0).
An upper bound on the performance of any network using some biological form of synaptic
plasticity comes from an ?ideal observer? which knows the complete list of stored patterns
(rather than its distant reflection in the synaptic weight matrix) and computes and compares
the probability that each was corrupted to form the input x
? to find the best match (rather
than using neural dynamics). Such an ideal observer only makes errors when both the
number of patterns stored and the noise in the input is sufficiently large, so that corrupting
a stored pattern is likely to make it more similar to another stored pattern. In the case
shown in Fig. 1A,B, this is not the case, since only two patterns were stored, and the ideal
observer performs perfectly as expected. Nevertheless, there may be situations in which
perfect performance is out of reach even for an ideal observer (Fig. 1C,D), which makes it
a meaningful touchstone. In summary, the performance of any network can be assessed by
measuring where it lies between the better one of the ?prior only? and ?input only? networks
and the ideal observer.
As a further challenge, we also benchmarked our model against the model of Treves [13]
(Fig. 1C,D), which we chose because it is a rare example of a network that was designed to have near optimal recall performance in the face of non-binary patterns. In
this work,
drawn from the distribution P [xi ] :=
Treves considered ternary patterns,
1 ? 43 a ? (xi ) + a? xi ? 21 + a3 ? xi ? 32 , where ? (x) is the Dirac-delta function.
Here, a = ?x quantifies the density of the patterns (i.e. how non-sparse they are). The
patterns are stored using the covariance rule as stated above (with Acov := N1a2 ). Neuronal
update in the model is discrete, asynchronous, and involves two steps. First the ?local field?
P
P
3
is calculated as hi := j6=i Wi,j xj ? k ( i xi ? N ) + Input, then the output of the
neuron is calculated as a threshold linear function of the local field: xi := g (hi ? hThr )
if hi > hThr and xi := 0 otherwise, where g := 0.53 a/ (1 ? a) is the gain parameter,
and hThr := 0 is the threshold, and the value of k is set by iterative search to optimize
performance.
The comparison between Treves? network as we implemented it and our network is imperfect, since the former is optimized for recalling ternary patterns while, in the absence of
neural evidence for ternary patterns, we used the simpler and more reasonable neural dynamics for our network that emerge from an assumption that the distribution over the stored
patterns is Gaussian. Further, we corrupted the inputs by unbiased additive Gaussian noise
(with variance ?x2? = 1.5), but truncated the activities at 0, though did not adjust the dynamics of our network in the light of the truncation. Of course, these can only render our
network less effective. Still, the Bayesian network clearly outperformed the Treves network
when the patterns were drawn from a truncated Gaussian (Fig. 1C). The performance of the
Bayesian network stayed close to that of an ideal observer assuming non-truncated Gaussian input, showing that most of the errors were caused by this assumption and not from
suboptimality of neural interactions decoding the information in synaptic weights. Despite
extensive efforts to find the optimal parameters for the Treves network, its performance did
not even reach that of the ?input only? network.
Finally, again for ternary patterns, we also considered only penalizing errors about the direction of the vectors of recalled activities ignoring errors about their magnitudes (Fig. 1D).
The Treves network did better in this case, but still not as well as the Bayesian network.
Importantly, in both cases, in the regime where synaptic weights were saturated in the
M ? N limit and thus it was no longer possible to extract any useful information from the
synaptic weights, the Bayesian network still only fell back to the level of the ?prior only?
network, but the Treves network did not seem to have any such upper bound on its errors.
4
Phase-based memories
Brain areas known to be involved in memory processing demonstrate prominent oscillations (LFPOs) under a variety of conditions, including both wake and sleep states [16].
Under these conditions, the phases of the spikes of a neuron relative to the LFPO have been
shown to be carefully controlled [17], and even to convey meaningful stimulus information,
e.g. about the position of an animal in its environment [3] or retrieved odor identity [18].
The discovery of spike timing dependent plasticity (STDP) in which the relative timing
of pre- and postsynaptic firings determines the sign and extent of synaptic weight change,
offered new insights into how the information represented by spike times may be stored
in neural networks [19]. However, bar some interesting suggestions about neuronal resonance [20], it is less clear how one might correctly recall information thereby stored in the
synaptic weights.
The theory laid out in Section 2 allows us to treat this problem systematically. First,
neuronal activities, xi , will be interpreted as firing times relative to a reference phase
of the ongoing LFPO, such as the peak of theta oscillation in the hippocampus, and
will thus be circular variables drawn from a circular Gaussian. Next, our learning rule
is an exponentially decaying Gabor-function of the phase difference between pre- and
postsynaptic firing: ?STDP (xi , xj ) := ASTDP exp[?STDP cos(??i,j )] sin(??i,j ? ?STDP )
with ??i,j = 2? (xi ? xj ) /TSTDP . STDP characteristics in different brain regions are
well captured by this general formula, but the parameters determining their exact shapes
greatly differ among regions. We constrain our analysis to the antisymmetric case, so
that ?STDP = 0, and set other parameters to match experimental data on hippocampal
STDP [5]. The neuronal interaction function
that satisfies Eq.11 is HSTDP (x
i , xj ) =
2?ASTDP /TSTDP Wi,j exp[?STDP cos(??i,j )] cos(??i,j ) ? ?STDP sin2 (??i,j ) . This interaction function decreases firing phase, and thus accelerates the postsynaptic cell if the
presynaptic spike precedes postsynaptic firing, and delays the postsynaptic cell if the presynaptic spike arrives just after the postsynaptic cell fired. This characteristic is the essence
of the biphasic phase reset curve of type II cells [21], and has been observed in various
types of neurons, including neocortical cells [22]. Thus again, our derivation directly couples STDP, a canonical model of synaptic plasticity, and phase reset curves in a canonical
model of neural dynamics.
Numerical simulations tested again the various approximations. Performance of the network is shown in Fig.2 as is comparable to that of the rate coded network (Fig.2). Further
simulations will be necessary to map out the performance of our network over a wider
range of parameters, such as the signal-to-noise ratio.
5
Discussion
We have described a Bayesian approach to recall in autoassociative memories. This permits
the derivation of neuronal dynamics appropriate to a synaptic plasticity rule, and we used
this to show a coupling between canonical Hebbian and STDP plasticity rules and canonical
rate-based and phase-based neuronal dynamics respectively. This provides an unexpectedly
close link between optimal computations and actual implementations. Our method also
leads to networks that are highly competent at recall.
There are a number of important direction for future work. First, even in phase-based
networks, not all neurons fire in each period of the oscillation. This suggests that neurons
may employ a dual code ? the more rate-based probability of being active in a cycle, and
the phase-based timing of the spike relative to the cycle [24]. The advantages of such a
scheme have yet to be fully characterized.
Second, in the present framework the choice of the learning rule is arbitrary, as long as the
A
0.35
0.3
B
prior
input
Bayesian
6
0.25
5
Average error
Frequency
7
0.2
0.15
4
3
0.1
2
0.05
1
0
?60
?40
?20
0
Error
20
40
60
input
ideal observer
Bayesian
0
1
10
Number of stored patterns
100
Figure 2: Performance of the phase-coded network. Error distribution for the ideal observer was
a Dirac-delta at 0 (B) and was thus omitted from A. Average error of the ?prior only? network was
too large (A) to be plotted in B. The prior was a von Mises distribution with ?x = 0 mean, ?x =
0.5 concentration on a T? = 125 ms long cycle matching data on theta frequency modulation
of pyramidal cell population activity in the hippocampus [23]. Input was corrupted by unbiased
circular Gaussian (von Mises) noise with ?x = 10 concentration. Learning rule was circular STDP
rule with ASTDP = 0.03, ?STDP = 4 and TSTDP = T? parameters matching experimental data on
hippocampal STDP [5] and theta periodicity. The network consisted of N = 100 cells, and the
number of memories stored was M = 10 (A) or varied between M = 2 . . . 100 (B, note logarithmic
scale). For further explanation of symbols and axes, see Figure 1.
recall dynamics is optimally matched to it. Our formalism also suggests that there may be
a way to optimally choose the learning rule itself in the first place, by matching it to the
prior distribution of patterns. This approach would thus be fundamentally different from
those seeking ?globally? optimal learning rules [25], and may be more similar to those used
to find optimal tuning curves appropriately matching stimulus statistics [26].
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
Marr D. Philos Trans R Soc Lond B Biol Sci 262:23, 1971.
O?Keefe J. Exp Neurol 51:78, 1976.
O?Keefe J, Recce ML. Hippocampus 3:317, 1993.
Bliss TVP, L?mo T. J Physiol (Lond) 232:331, 1973.
Bi GQ, Poo MM. J Neurosci 18:10464, 1998.
MacKay DJC. In Maximum entropy and Bayesian methods, 237, 1990.
Sommer FT, Dayan P. IEEE Trans Neural Netw 9:705, 1998.
Jaynes ET. Probability theory: the logic of science. Cambridge University Press, 2003.
Amit DJ. Modeling brain function. Cambridge University Press, 1989.
Dayan P, Abbott LF. Theoretical neuroscience. MIT Press, 2001.
Zhang K. J Neurosci 16:2112, 1996.
Seung HS. Proc Natl Acad Sci USA 93:13339, 1996.
Treves A. Phys Rev A 42:2418, 1990.
Hopfield JJ. Proc Natl Acad Sci USA 76:2554, 1982.
Hopfield JJ. Proc Natl Acad Sci USA 81:3088, 1984.
Buzs?ki Gy. Neuron 33:325, 2002.
Harris KD, et al. Nature 424:552, 2003.
Li Z, Hopfield JJ. Biol Cybern 61:379, 1989.
Abbott LF, Nelson SB. Nat Neurosci 3:1178, 2000.
Scarpetta S, et al. Neural Comput 14:2371, 2002.
Ermentrout B, et al. Neural Comput 13:1285, 2001.
Reyes AD, Fetz FE. J Neurophysiol 69:1673, 1993.
Klausberger T, et al. Nature 421:844, 2003.
Mueller R, et al. In BioNet?96 , 70, 1976.
Gardner E, Derrida B. J Phys A 21:271, 1988.
Laughlin S. Z Naturforsch 36:901, 1981.
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1,854 | 2,684 | Joint MRI Bias Removal Using Entropy
Minimization Across Images
Erik G. Learned-Miller
Department of Computer Science
University of Massachusetts, Amherst
Amherst, MA 01003
Parvez Ahammad
Division of Electrical Engineering
University of California, Berkeley
Berkeley, CA 94720
Abstract
The correction of bias in magnetic resonance images is an important
problem in medical image processing. Most previous approaches have
used a maximum likelihood method to increase the likelihood of the pixels in a single image by adaptively estimating a correction to the unknown
image bias field. The pixel likelihoods are defined either in terms of a
pre-existing tissue model, or non-parametrically in terms of the image?s
own pixel values. In both cases, the specific location of a pixel in the image is not used to calculate the likelihoods. We suggest a new approach
in which we simultaneously eliminate the bias from a set of images of
the same anatomy, but from different patients. We use the statistics from
the same location across different images, rather than within an image, to
eliminate bias fields from all of the images simultaneously. The method
builds a ?multi-resolution? non-parametric tissue model conditioned on
image location while eliminating the bias fields associated with the original image set. We present experiments on both synthetic and real MR
data sets, and present comparisons with other methods.
1
Introduction
The problem of bias fields in magnetic resonance (MR) images is an important problem
in medical imaging. This problem is illustrated in Figure 1. When a patient is imaged in
the MR scanner, the goal is to obtain an image which is a function solely of the underlying
tissue (left of Figure 1). However, typically the desired anatomical image is corrupted by a
multiplicative bias field (2nd image of Figure 1) that is caused by engineering issues such
as imperfections in the radio frequency coils used to record the MR signal. The result is a
corrupted image (3rd image of Figure 1). (See [1] for background information.) The goal
of MR bias correction is to estimate the uncorrupted image from the corrupted image.
A variety of statistical methods have been proposed to address this problem. Wells et
al. [7] developed a statistical model using a discrete set of tissues, with the brightness
distribution for each tissue type (in a bias-free image) represented by a one-dimensional
Guassian distribution. An expectation-maximization (EM) procedure was then used to
simultaneouly estimate the bias field, the tissue type, and the residual noise. While this
method works well in many cases, it has several drawbacks: (1) Models must be developed
a priori for each type of acquistion (for each different setting of the MR scanner), for each
Figure 1: On the left is an idealized mid-axial MR image of the human brain with little
or no bias field. The second image is a simulated low-frequency bias field. It has been
exaggerated for ease of viewing. The third image is the result of pixelwise multiplication
of the image by the bias field. The goal of MR bias correction is to recover the low-bias
image on the left from the biased image on the right. On the right is the sine/cosine basis,
used to construct band-limited bias fiels (see text).
new area of the body, and for different patient populations (like infants and adults). (2)
Models must be developed from ?bias-free? images, which may be difficult or impossible
to obtain in many cases. (3) The model assumes a fixed number of tissues, which may
be inaccurate. For example, during development of the human brain, there is continuous
variability between gray matter and white matter. In addition, a discrete tissue model does
not handle so-called partial volume effects in which a pixel represents a combination of
several tissue types. This occurs frequently since many pixels occur at tissue boundaries.
Non-parametric approaches have also been suggested, as for example by Viola [10]. In
that work, a non-parametric model of the tissue was developed from a single image. Using
the observation that the entropy of the pixel brightness distribution for a single image is
likely to increase when a bias field is added, Viola?s method postulates a bias-correction
field by minimizing the entropy of the resulting pixel brightness distribution. This approach
addresses several of the problems of fixed-tissue parametric models, but has its own drawbacks: (1) The statistical model may be weak, since it is based on data from only a single
image. (2) There is no mechanism for distinguishing between certain low-frequency image
components and a bias field. That is, the method may mistake signal for noise in certain
cases when removal of the true signal reduces the entropy of the brightness distriibution.
We shall show that this is a problem in real medical images.
The method we present overcomes or improves upon problems associated with both of
these methods and their many variations (see, e.g., [1] for recent techniques). It models tissue brightness non-parametrically, but uses data from multiple images to provide improved
distribution estimates and alleviate the need for bias-free images for making a model. It
also conditions on spatial location, taking advantage of a rich information source ignored
in other methods. Experimental results demonstrate the effectiveness of our method.
2
The Image Model and Problem Formulation
We assume we are given a set I of observed images Ii with 1 ? i ? N, as shown on the
left side of Figure 2. Each of these images is assumed to be the product of some bias-free
image Li and a smooth bias field Bi ? B . We shall refer to the bias-free images as latent
images (also called intrinsic images by some authors). The set of all latent images shall be
denoted L and the set of unknown bias fields B. Then each observed image can be written
as the product Ii (x, y) = Li (x, y) ? Bi (x, y), where (x, y) gives the pixel coordinates of each
point, with P pixels per image.
Consider again Figure 2. A pixel-stack through each image set is shown as the set of pixels
corresponding to a particular location in each image (not necessarily the same tissue type).
Our method relies on the principle that the pixel-stack values will have lower entropy when
the bias fields have been removed. Figure 3 shows the simulated effect, on the distribution
of values in a pixel-stack, of adding different bias fields to each image.
The latent image generation model assumes that each pixel is drawn from a fixed distribution px,y (?) which gives the probability of each gray value at the the location (x, y) in the
image. Furthermore, we assume that all pixels in the latent image are independent, given
the distributions from which they are drawn. It is also assumed that the bias fields for each
image are chosen independently from some fixed distribution over bias fields. Unlike most
models for this problem which rely on statistical regularities within an image, we take a
completely orthogonal approach by assuming that pixel values are independent given their
image locations, but that pixel-stacks in general have low entropy when bias fields are
removed.
We formulate the problem as a maximum a posteriori (MAP) problem, searching for the
most probable bias fields given the set of observed images. Letting B represent the 25dimensional product space of smooth bias fields (corresponding to the 25 basis images of
Figure 1), we wish to find
arg maxP(B|I)
(a)
=
B?B
arg maxP(I|B)P(B)
(1)
B?B
(b)
=
arg maxP(I|B)
(2)
B?B
(c)
=
arg maxP(L(I, B))
(3)
B?B
N
=
arg max ? ? px,y (Li (x, y))
B?B
(4)
x,y i=1
N
=
arg max ? ? log px,y (Li (x, y))
(5)
arg min ? H(px,y )
(6)
arg min ? H? Vasicek (L1 (x, y), ..., LN (x, y))
(7)
B?B
(d)
?
B?B
(e)
?
B?B
=
x,y i=1
x,y
x,y
arg min ? H? Vasicek (
B?B
x,y
I1 (x, y)
IN (x, y)
, ...,
).
B1 (x, y)
BN (x, y)
(8)
Here H is the Shannon entropy (?E(log P(x))) and H? Vasicek is a sample-based entropy
estimator.1 (a) is just an application of Bayes rule. (b) assumes a uniform prior over the
allowed bias fields. The method can easily be altered to incorporate a non-uniform prior.
1 The
entropy estimator used is similar to Vasicek?s estimator [6], given (up to minor details) by
N (i+m)
1 N?m
(i)
H? Vasicek (Z 1 , ..., Z N ) =
log
(Z
?
Z
)
,
(9)
?
N ? m i=1
m
where Z i ?s represent the values in a pixel-stack, Z (i) ?s represent those same values in rank order, N is
the number of values in the pixel-stack and m is a function of N (like N 0.5 ) such that m/N goes to 0
as m and N go to infinity. These entropy estimators are discussed at length elsewhere [3].
Figure 2: On the left are a set of mid-coronal brain images from eight different infants,
showing clear signs of bias fields. A pixel-stack, a collection of pixels at the same point
in each image, is represented by the small square near the top of each image. Although
there are probably no more than two or three tissue types represented by the pixel-stack,
the brightness distribution through the pixel-stack has high empirical entropy due to the
presence of different bias fields in each image. On the right are a set of images that have
been corrected using our bias field removal algorithm. While the images are still far from
identical, the pixel-stack entropies have been reduced by mapping similar tissues to similar
values in an ?unsupervised? fashion, i.e. without knowing or estimating the tissue types.
(c) expresses the fact that the probability of the observed image given a particular bias field
is the same as the probability of the latent image associated with that observed image and
bias field. The approximation (d) replaces the empirical mean of the log probability at each
pixel with the negative entropy of the underlying distribution at that pixel. This entropy is
in turn estimated (e) using the entropy estimator of Vasicek [6] directly from the samples in
the pixel-stack, without ever estimating the distributions px,y explicitly. The inequality (d)
becomes an equality as N grows large by the law of large numbers, while the consistency
of Vasicek?s entropy estimator [2] implies that (e) also goes to equality with large N. (See
[2] for a review of entropy estimators.)
3
The Algorithm
Using these ideas, it is straightforward to construct algorithms for joint bias field removal.
As mentioned above, we chose to optimize Equation (8) over the set of band-limited bias
fields. To do this, we parameterize the set of bias fields using the sine/cosine basis images
shown on the right of Figure 1:
25
Bi =
? ? j ? j (x, y).
j=1
We optimize Equation (8) by simultaneously updating the bias field estimates (taking a step
along the numerical gradient) for each image to reduce the overall entropy. That is, at time
step t, the coefficients ? j for each bias field are updated using the latent image estimates
and entropy estimates from time step t ? 1. After all ??s have been updated, a new set of
latent images and pixel-stack entropies are calculated, and another gradient step is taken.
Though it is possible to do a full gradient descent to convergence by optimizing one image
at a time, the optimization landscape tends to have more local minima for the last few
images in the process. The appeal of our joint gradient descent method, on the other hand,
is that the ensemble of images provides a natural smoothing of the optimization landscape
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
0
1
0
50
100
150
200
250
0
0
50
100
150
200
250
Figure 3: On the left is a simulated distribution from a pixel-stack taken through a particular set of bias-free mid-axial MR images. The two sharp peaks in the brightness distribution
represent two tissues which are commonly found at that particular pixel location. On the
right is the result of adding an independent bias field to each image. In particular, the
spread, or entropy, of the pixel distribution increases. In this work, we seek to remove bias
fields by seeking to reduce the entropy of the pixel-stack distribution to its original state.
in the joint process. It is in this sense that our method is ?multi-resolution?, proceeding
from a smooth optimization in the beginning to a sharper one near the end of the process.
We now summarize the algorithm:
1. Initialize the bias field coefficients for each image to 0, with the exception of
the coefficient for the DC-offset (the constant bias field component), which is
initialized to 1. Initialize the gradient descent step size ? to some value.
2. Compute the summed pixelwise entropies for the set of images with initial ?neutral? bias field corrections. (See below for method of computation.)
3. Iterate the following loop until no further changes occur in the images.
(a) For each image:
i. Calculate the numerical gradient ?? HVasicek of (8) with respect to the bias
field coefficients (? j ?s) for the current image.
ii. Set ? = ? + ??? H? Vasicek .
(b) Update ? (reduce its value according to some schedule).
Upon convergence, it is assumed that the entropy has been reduced as much as possible by
changing the bias fields, unless one or more of the gradient descents is stuck in a local minimum. Empirically, the likelihood of sticking in local minima is dramatically reduced by
increasing the number of images (N) in the optimization. In our experiments described below with only 21 real infant brains, the algorithm appears to have found a global minimum
of all bias fields, at least to the extent that this can be discerned visually.
Note that for a set of identical images, the pixel-stack entropies are not increased by multiplying each image by the same bias field (since all images will still be the same). More
generally, when images are approximately equivalent, their pixel-stack entropies are not
signficantly affected by a ?common? bias field, i.e. one that occurs in all of the images.2
This means that the algorithm cannot, in general, eliminate all bias fields from a set of images, but can only set all of the bias fields to be equivalent. We refer to any constant bias
field remaining in all of the images after convergence as the residual bias field.
2 Actually, multiplying each image by a bias field of small magnitude can artificially reduce the
entropy of a pixel-stack, but this is only the result of the brightness values shrinking towards zero.
Such artificial reductions in entropy can be avoided by normalizing a distribution to unit variance
between iterations of computing its entropy, as is done in this work.
Fortunately, there is an effect that tends to minimize the impact of the residual bias field
in many test cases. In particular, the residual bias field tends to consist of components for
each ? j that approximate the mean of that component across images. For example, if half
of the observed images have a positive value for a particular component?s coefficient, and
half have a negative coefficient for that component, the residual bias field will tend to have
a coefficient near zero for that component. Hence, the algorithm naturally eliminates bias
field effects that are non-systematic, i.e. that are not shared across images.
If the same type of bias field component occurs in a majority of the images, then the algorithm will not remove it, as the component is indistinguishable, under our model, from the
underlying anatomy. In such a case, one could resort to within-image methods to further
reduce the entropy. However, there is a risk that such methods will remove components
that actually represent smooth gradations in the anatomy. This can be seen in the bottom
third of Figure 4, and will be discussed in more detail below.
4
Experiments
To test our algorithm, we ran two sets of experiments, the first on synthetic images for
validation, and the second on real brain images. We obtained synthetic brain images from
the BrainWeb project [8, 9] such as the one shown on the left of Figure 1. These images can
be considered ?idealized? MR images in the sense that the brightness values for each tissue
are constant (up to a small amount of manually added isotropic noise). That is, they contain
no bias fields. The initial goal was to ensure that our algorithm could remove synthetically
added bias fields, in which the bias field coefficients were known. Using K copies of a
single ?latent? image, we added known but different bias fields to each one. For as few as
five images, we could reliably recover the known bias field coefficients, up to a fixed offset
for each image, to within 1% of the power of the original bias coefficients.
More interesting are the results on real images, in which the latent images come from
different patients. We obtained 21 pre-registered3 infant brain images (top of Figure 4)
from Brigham and Women?s Hospital in Boston, Massachusetts. Large bias fields can be
seen in many of the images. Probably the most striking is a ?ramp-like? bias field in the
sixth image of the second row. (The top of the brain is too bright, while the bottom is too
dark.) Because the brain?s white matter is not fully developed in these infant scans, it is
difficult to categorize tissues into a fixed number of classes as is typically done for adult
brain images; hence, these images are not amenable to methods based on specific tissue
models developed for adults (e.g. [7]).
The middle third of Figure 4 shows the results of our algorithm on the infant brain images.
(These results must be viewed in color on a good monitor to fully appreciate the results.)
While a trained technician can see small imperfections in these images, the results are
remarkably good. All major bias artifacts have been removed.
It is interesting to compare these results to a method that reduces the entropy of each image
individually, without using constraints between images. Using the results of our algorithm
as a starting point, we continued to reduce the entropy of the pixels within each image
(using a method akin to Viola?s [10]), rather than across images. These results are shown
in the bottom third of Figure 4. Carefully comparing the central brain regions in the middle
section of the figure and the bottom section of the figure, one can see that the butterfly
shaped region in the middle of the brain, which represents developing white matter, has
3 It
is interesting to note that registration is not strictly necessary for this algorithm to work. The
proposed MAP method works under very broad conditions, the main condition being that the bias
fields do not span the same space as parts of the actual medical images. It is true, however, that as the
latent images become less registered or differ in other ways, that a much larger number of images is
needed to get good estimates of the pixel-stack distributions.
been suppressed in the lower images. This is most likely because the entropy of the pixels
within a particular image can be reduced by increasing the bias field ?correction? in the
central part of the image. In other words, the algorithm strives to make the image more
uniform by removing the bright part in the middle of the image. However, our algorithm,
which compares pixels across images, does not suppress these real structures, since they
occur across images. Hence coupling across images can produce superior results.
5
Discussion
The idea of minimizing pixelwise entropies to remove nuisance variables from a set of images is not new. In particular, Miller et al. [4, 5] presented an approach they call congealing
in which the sum of pixelwise entropies is minimized by separate affine transforms applied
to each image. Our method can thus be considered an extension of the congealing process
to non-spatial transformations. Combining such approaches to do registration and bias removal simulataneously, or registration and lighting rectification of faces, for example, is an
obvious direction for future work.
This work uses information unused in other methods, i.e. information across images. This
suggests an iterative scheme in which both types of information, both within and across
images, are used. Local models could be based on weighted neighborhoods of pixels, pixel
cylinders, rather than single pixel-stacks, in sparse data scenarios. For ?easy? bias correction problems, such an approach may be overkill, but for difficult problems in bias correction, where the bias field is difficult to separate from the underlying tissue, as discussed in
[1], such an approach could produce critical extra leverage.
We would like to thank Dr. Terrie Inder and Dr. Simon Warfield for graciously providing
the infant brain images for this work. The images were obtained under NIH grant P41
RR13218. Also, we thank Neil Weisenfeld and Sandy Wells for helpful discussions. This
work was partially supported by Army Research Office grant DAAD 19-02-1-0383.
References
[1] Fan, A., Wells, W., Fisher, J., Cetin, M., Haker, S., Mulkern, C., Tempany, C., Willsky, A. A
unified variational approach to denoising and bias correction in MR. Proceedings of IPMI, 2003.
[2] Beirlant, J., Dudewicz, E., Gyorfi, L. and van der Meulen, E. Nonparametric entropy estimation:
An overview. International Journal of Mathematical and Statistical Sciences, 6. pp.17-39. 1997.
[3] Learned-Miller, E. G. and Fisher, J. ICA using spacings estimates of entropy. Journal of Machine
Learning Research, Volume 4, pp. 1271-1295, 2003.
[4] Miller, E. G., Matsakis, N., Viola, P. A. Learning from one example through shared densities on
transforms. IEEE Conference on Computer Vision and Pattern Recognition. 2000.
[5] Miller, E. G. Learning from one example in machine vision by sharing probability densities.
Ph.D. thesis. Massachusetts Institute of Technology. 2002.
[6] Vasicek, O. A test for normality based on sample entropy. Journal of the Royal Statistical Society
Series B, 31. pp. 632-636, 1976.
[7] Wells, W. M., Grimson, W. E. L., Kikinis, R., Jolesz, F. Adaptive segmentation of MRI data.
IEEE Transactions on Medical Imaging, 15. pp. 429-442, 1996.
[8] Collins, D.L., Zijdenbos, A.P., Kollokian, J.G., Sled, N.J., Kabani, C.J., Holmes, C.J., Evans,
A.C. Design and Construction of a realistic digital brain phantom. IEEE Transactions on Medical
Imaging, 17. pp. 463-468, 1998.
[9] http://www.bic.mni.mcgill.ca/brainweb/
[10] Viola, P.A. Alignment by maximization of mutual information. Ph.D. Thesis. Massachusetts
Institute of Technology. 1995.
Figure 4: NOTE: This image must be viewed in color (preferably on a bright display) for
full effect. Top. Original infant brain images. Middle. The same images after bias removal
with our algorithm. Note that developing white matter (butterfly-like structures in middle
brain) is well-preserved. Bottom. Bias removal using a single image based algorithm.
Notice that white matter structures are repressed.
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1,855 | 2,685 | The Laplacian PDF Distance: A Cost
Function for Clustering in a Kernel
Feature Space
Robert Jenssen1?, Deniz Erdogmus2 , Jose Principe2 , Torbj?rn Eltoft1
1
2
Department of Physics, University of Troms?, Norway
Computational NeuroEngineering Laboratory, University of Florida, USA
Abstract
A new distance measure between probability density functions
(pdfs) is introduced, which we refer to as the Laplacian pdf distance. The Laplacian pdf distance exhibits a remarkable connection to Mercer kernel based learning theory via the Parzen window
technique for density estimation. In a kernel feature space defined
by the eigenspectrum of the Laplacian data matrix, this pdf distance is shown to measure the cosine of the angle between cluster
mean vectors. The Laplacian data matrix, and hence its eigenspectrum, can be obtained automatically based on the data at hand,
by optimal Parzen window selection. We show that the Laplacian
pdf distance has an interesting interpretation as a risk function
connected to the probability of error.
1
Introduction
In recent years, spectral clustering methods, i.e. data partitioning based on the
eigenspectrum of kernel matrices, have received a lot of attention [1, 2]. Some
unresolved questions associated with these methods are for example that it is not
always clear which cost function that is being optimized and that is not clear how
to construct a proper kernel matrix.
In this paper, we introduce a well-defined cost function for spectral clustering. This
cost function is derived from a new information theoretic distance measure between
cluster pdfs, named the Laplacian pdf distance. The information theoretic/spectral
duality is established via the Parzen window methodology for density estimation.
The resulting spectral clustering cost function measures the cosine of the angle
between cluster mean vectors in a Mercer kernel feature space, where the feature
space is determined by the eigenspectrum of the Laplacian matrix. A principled
approach to spectral clustering would be to optimize this cost function in the feature
space by assigning cluster memberships. Because of space limitations, we leave it
to a future paper to present an actual clustering algorithm optimizing this cost
function, and focus in this paper on the theoretical properties of the new measure.
?
Corresponding author. Phone: (+47) 776 46493. Email: [email protected]
An important by-product of the theory presented is that a method for learning the
Mercer kernel matrix via optimal Parzen windowing is provided. This means that
the Laplacian matrix, its eigenspectrum and hence the feature space mapping can
be determined automatically. We illustrate this property by an example.
We also show that the Laplacian pdf distance has an interesting relationship to the
probability of error.
In section 2, we briefly review kernel feature space theory. In section 3, we utilize
the Parzen window technique for function approximation, in order to introduce the
new Laplacian pdf distance and discuss some properties in sections 4 and 5. Section
6 concludes the paper.
2
Kernel Feature Spaces
Mercer kernel-based learning algorithms [3] make use of the following idea: via a
nonlinear mapping
? : Rd ? F, x ? ?(x)
(1)
the data x1 , . . . , xN ? Rd is mapped into a potentially much higher dimensional
feature space F. For a given learning problem one now considers the same algorithm
in F instead of in Rd , that is, one works with ?(x1 ), . . . , ?(xN ) ? F.
Consider a symmetric kernel function k(x, y). If k : C ? C ? R is a continuous
kernel of a positive integral operator in a Hilbert space L2 (C) on a compact set
C ? Rd , i.e.
Z
?? ? L2 (C) :
C
k(x, y)?(x)?(y)dxdy ? 0,
(2)
then there exists a space F and a mapping ? : Rd ? F, such that by Mercer?s
theorem [4]
NF
X
k(x, y) = h?(x), ?(y)i =
?i ?i (x)?i (y),
(3)
i=1
where h?, ?i denotes an inner product, the ?i ?s are the orthonormal eigenfunctions
of the kernel and NF ? ? [3]. In this case
p
p
?(x) = [ ?1 ?1 (x), ?2 ?2 (x), . . . ]T ,
(4)
can potentially be realized.
In some cases, it may be desirable to realize this mapping. This issue has been
addressed in [5]. Define the (N ? N ) Gram matrix, K, also called the affinity, or
kernel matrix, with elements Kij = k(xi , xj ), i, j = 1, . . . , N . This matrix can be
diagonalized as ET KE = ?, where the columns of E contains the eigenvectors of K
?1 , . . . , ?
?N , ?
?1 ?
and ? is a diagonal matrix containing the non-negative eigenvalues ?
?
? ? ? ? ?N . In [5], it was shown that the eigenfunctions and eigenvalues of (4) can
?
??
be approximated as ?j (xi ) ? N eji , ?j ? Nj , where eji denotes the ith element
of the jth eigenvector. Hence, the mapping (4), can be approximated as
q
q
?
? N e N i ]T .
?(xi ) ? [ ?1 e1i , . . . , ?
(5)
Thus, the mapping is based on the eigenspectrum of K. The feature space data set
may be represented in matrix form as ?N ?N = [?(x1 ), . . . , ?(xN )]. Hence, ? =
1
? 2 ET . It may be desirable to truncate the mapping (5) to C-dimensions. Thus,
? It is well-known that K
? =?
?T?
? is
only the C first rows of ? are kept, yielding ?.
the best rank-C approximation to K wrt. the Frobenius norm [6].
The most widely used Mercer kernel is the radial-basis-function (RBF)
||x ? y||2
k(x, y) = exp ?
.
2? 2
3
(6)
Function Approximation using Parzen Windowing
Parzen windowing is a kernel-based density estimation method, where the resulting
density estimate is continuous and differentiable provided that the selected kernel
is continuous and differentiable [7]. Given a set of iid samples {x1 , . . . , xN } drawn
from the true density f (x), the Parzen window estimate for this distribution is [7]
N
1 X
?
f(x)
=
W?2 (x, xi ),
(7)
N i=1
where W?2 is the Parzen window, or kernel, and ? 2 controls the width of the kernel.
The Parzen window must integrate to one, and is typically chosen to be a pdf itself
with mean xi , such as the Gaussian kernel
||x ? xi ||2
1
?
W?2 (x, xi ) =
,
(8)
d exp
2? 2
(2?? 2 ) 2
which we will assume in the rest of this paper. In the conclusion, we briefly discuss
the use of other kernels.
Consider a function h(x) = v(x)f (x), for some function v(x). We propose to
estimate h(x) by the following generalized Parzen estimator
N
1 X
?
v(xi )W?2 (x, xi ).
(9)
h(x)
=
N i=1
This estimator is asymptotically unbiased, which can be shown as follows
(
) Z
N
1 X
Ef
v(xi )W?2 (x, xi ) = v(z)f (z)W?2 (x, z)dz = [v(x)f (x)] ? W?2 (x),
N i=1
(10)
where Ef (?) denotes expectation with respect to the density f (x). In the limit as
N ? ? and ?(N ) ? 0, we have
lim [v(x)f (x)] ? W?2 (x) = v(x)f (x).
(11)
N ??
?(N )?0
Of course, if v(x) = 1 ?x, then (9) is nothing but the traditional Parzen estimator
of h(x) = f (x). The estimator (9) is also asymptotically consistent provided that
the kernel width ?(N ) is annealed at a sufficiently slow rate. The proof will be
presented in another paper.
Many approaches have been proposed in order to optimally determine the size of
the Parzen window, given a finite sample data set. A simple selection rule was
proposed by Silverman [8], using the mean integrated square error (MISE) between
the estimated and the actual pdf as the optimality metric:
1
?opt = ?X 4N ?1 (2d + 1)?1 d+4 ,
(12)
P
2
?1
where d is the dimensionality of the data and ?X = d
i ?Xii , where ?Xii are the
diagonal elements of the sample covariance matrix. More advanced approximations
to the MISE solution also exist.
4
The Laplacian PDF Distance
Cost functions for clustering are often based on distance measures between pdfs.
The goal is to assign memberships to the data patterns with respect to a set of
clusters, such that the cost function is optimized.
Assume that a data set consists of two clusters. Associate the probability density
function p(x) with one of the clusters, and the density q(x) with the other cluster.
Let f (x) be the overall probability density function of the dataR set. Now define the
f ?1 weighted inner product between p(x) and q(x) as hp, qif ? p(x)q(x)f ?1 (x)dx.
In such an inner product space, the Cauchy-Schwarz inequality holds, that is,
2
hp, qif ? hp, pif hq, qif . Based on this discussion, an information theoretic distance
measure between the two pdfs can be expressed as
hp, qif
DL = ? log q
? 0.
hp, pif hq, qif
(13)
We refer to this measure as the Laplacian pdf distance, for reasons that we discuss
next. It can be seen that the distance DL is zero if and only if the two densities
are equal. It is non-negative, and increases as the overlap between the two pdfs
decreases. However, it does not obey the triangle inequality, and is thus not a
distance measure in the strict mathematical sense.
We will now show that the Laplacian pdf distance is also a cost function for clustering in a kernel feature space, using the generalized Parzen estimators discussed
in the previous section. Since the logarithm is a monotonic function, we will derive
the expression for the argument of the log in (13). This quantity will for simplicity
be denoted by the letter ?L? in equations.
Assume that we have available the iid data points {xi }, i = 1, . . . , N1 , drawn from
p(x), which is the density of cluster C1 , and the iid {xj }, j = 1, . . . , N2 , drawn from
1
1
q(x), the density of C2 . Let h(x) = f ? 2 (x)p(x) and g(x) = f ? 2 (x)q(x). Hence, we
may write
R
h(x)g(x)dx
q
.
(14)
L= R
R
h2 (x)dx g 2 (x)dx
We estimate h(x) and g(x) by the generalized Parzen kernel estimators, as follows
N1
1 X
1
?
h(x) =
f ? 2 (xi )W?2 (x, xi ),
N1 i=1
N2
1 X
1
g?(x) =
f ? 2 (xj )W?2 (x, xj ).
N2 j=1
(15)
The approach taken, is to substitute these estimators into (14), to obtain
Z
h(x)g(x)dx
N1
N2
1
1
1 X
1 X
f ? 2 (xi )W?2 (x, xi )
f ? 2 (xj )W?2 (x, xj )
N1 i=1
N2 j=1
?
Z
=
Z
NX
1 ,N2
1
? 12
? 12
f (xi )f (xj ) W?2 (x, xi )W?2 (x, xj )dx
N1 N2 i,j=1
=
NX
1 ,N2
1
1
1
f ? 2 (xi )f ? 2 (xj )W2?2 (xi , xj ),
N1 N2 i,j=1
(16)
where in the last step, the convolution theorem for Gaussians has been employed.
Similarly, we have
Z
h2 (x)dx ?
N1 ,N1
1
1 X ?1
f 2 (xi )f ? 2 (xi0 )W2?2 (xi , xi0 ),
2
N1 0
Z
g 2 (x)dx ?
N2 ,N2
1
1 X ?1
f 2 (xj )f ? 2 (xj 0 )W2?2 (xj , xj 0 ).
N22 0
(17)
i,i =1
(18)
j,j =1
Now we define the matrix Kf , such that
1
1
Kfij = Kf (xi , xj ) = f ? 2 (xi )f ? 2 (xj )K(xi , xj ),
(19)
where K(xi , xj ) = W2?2 (xi , xj ) for i, j = 1, . . . , N and N = N1 + N2 . As a
consequence, (14) can be re-written as follows
PN1 ,N2
i,j=1 Kf (xi , xj )
L = qP
(20)
PN2 ,N2
N1 ,N1
0)
0)
K
(x
,
x
K
(x
,
x
0
0
f
i
i
f
j
j
i,i =1
j,j =1
The key point of this paper, is to note that the matrix K = Kij = K(xi , xj ), i, j =
1, . . . , N , is the data affinity matrix, and that K(xi , xj ) is a Gaussian RBF kernel
function. Hence, it is also a kernel function that satisfies Mercer?s theorem. Since
K(xi , xj ) satisfies Mercer?s theorem, the following by definition holds [4]. For any
set of examples {x1 , . . . , xN } and any set of real numbers ?1 , . . . , ?N
N X
N
X
i=1 j=1
?i ?j K(xi , xj ) ? 0,
(21)
in analogy to (3). Moreover, this means that
N X
N
X
1
1
?i ?j f ? 2 (xi )f ? 2 (xj )K(xi , xj ) =
i=1 j=1
N X
N
X
i=1 j=1
?i ?j Kf (xi , xj ) ? 0,
(22)
hence Kf (xi , xj ) is also a Mercer kernel.
Now, it is readily observed that the Laplacian pdf distance can be analyzed in terms
of inner products in a Mercer kernel-based Hilbert feature space, since Kf (xi , xj ) =
h?f (xi ), ?f (xj )i. Consequently, (20) can be written as follows
PN1 ,N2
i,j=1 h?f (xi ), ?f (xj )i
L = qP
PN2 ,N2
N1 ,N1
i,i0 =1 h?f (xi ), ?f (xi0 )i
j,j 0 =1 h?f (xj ), ?f (xj 0 )i
D P
E
PN2
N1
1
1
i=1 ?f (xi ), N2
j=1 ?f (xj )
N1
= rD
ED P
E
N2
1 P N1
1 P N1
1
1 P N2
i=1 ?f (xi ), N1
i0 =1 ?f (xi0 )
j=1 ?f (xj ), N2
j 0 =1 ?f (xj 0 )
N1
N2
m1f , m2f
=
= cos 6 (m1f , m2f ),
(23)
||m1f ||||m2f ||
P Ni
where mif = N1i l=1
?f (xl ), i = 1, 2, that is, the sample mean of the ith cluster
in feature space.
This is a very interesting result. We started out with a distance measure between
densities in the input space. By utilizing the Parzen window method, this distance
measure turned out to have an equivalent expression as a measure of the distance
between two clusters of data points in a Mercer kernel feature space. In the feature
space, the distance that is measured is the cosine of the angle between the cluster
mean vectors.
The actual mapping of a data point to the kernel feature space is given by the
eigendecomposition of Kf , via (5). Let us examine this mapping in more detail.
1
Note that f 2 (xi ) can be estimated from the data by the traditional Parzen pdf
estimator as follows
v
u
N
u1 X
p
1
f 2 (xi ) = t
W?f2 (xi , xl ) = di .
(24)
N
l=1
Define the matrix D = diag(d1 , . . . , dN ). Then Kf can be expressed as
1
1
Kf = D? 2 KD? 2 .
(25)
Quite interestingly, for ?f2 = 2? 2 , this is in fact the Laplacian data matrix.
1
The above discussion explicitly connects the Parzen kernel and the Mercer kernel.
Moreover, automatic procedures exist in the density estimation literature to optimally determine the Parzen kernel given a data set. Thus, the Mercer kernel is
also determined by the same procedure. Therefore, the mapping by the Laplacian
matrix to the kernel feature space can also be determined automatically. We regard
this as a significant result in the kernel based learning theory.
As an example, consider Fig. 1 (a) which shows a data set consisting of a ring
with a dense cluster in the middle. The MISE kernel size is ?opt = 0.16, and
the Parzen pdf estimate is shown in Fig. 1 (b). The data mapping given by the
corresponding Laplacian matrix is shown in Fig. 1 (c) (truncated to two dimensions
for visualization purposes). It can be seen that the data is distributed along two lines
radially from the origin, indicating that clustering based on the angular measure
we have derived makes sense.
The above analysis can easily be extended to any number of pdfs/clusters. In the
C-cluster case, we define the Laplacian pdf distance as
L=
C?1
XX
i=1 j6=i
C
q
hpi , pj if
.
(26)
hpi , pi if hpj , pj if
In the kernel feature space, (26), corresponds to all cluster mean vectors being
pairwise as orthogonal to each other as possible, for all possible unique pairs.
4.1
Connection to the Ng et al. [2] algorithm
Recently, Ng et al. [2] proposed to map the input data to a feature space determined
by the eigenvectors corresponding to the C largest eigenvalues of the Laplacian matrix. In that space, the data was normalized to unit norm and clustered by the
C-means algorithm. We have shown that the Laplacian pdf distance provides a
1
It is a bit imprecise to refer to Kf as the Laplacian matrix, as readers familiar with
spectral graph theory may recognize, since the definition of the Laplacian matrix is L =
I ? Kf . However, replacing Kf by L does not change the eigenvectors, it only changes the
eigenvalues from ?i to 1 ? ?i .
0
0
(a) Data set
(b) Parzen pdf estimate
(c) Feature space data
Figure 1: The kernel size is automatically determined (MISE), yielding the Parzen
estimate (b) with the corresponding feature space mapping (c).
clustering cost function, measuring the cosine of the angle between cluster means,
in a related kernel feature space, which in our case can be determined automatically. A more principled approach to clustering than that taken by Ng et al. is to
optimize (23) in the feature space, instead of using C-means. However, because of
the normalization of the data in the feature space, C-means can be interpreted as
clustering the data based on an angular measure. This may explain some of the
success of the Ng et al. algorithm; it achieves more or less the same goal as clustering based on the Laplacian distance would be expected to do. We will investigate
this claim in our future work. Note that we in our framework may choose to use
only the C largest eigenvalues/eigenvectors in the mapping, as discussed in section
2. Since we incorporate the eigenvalues in the mapping, in contrast to Ng et al.,
the actual mapping will in general be different in the two cases.
5
The Laplacian PDF distance as a risk function
We now give an analysis of the Laplacian pdf distance that may further motivate its
use as a clustering cost function. Consider again the two cluster case. The overall
data distribution can be expressed as f (x) = P1 p(x) + P2 q(x), were Pi , i = 1, 2, are
the priors. Assume that the two clusters are well separated, such that for xi ? C1 ,
f (xi ) ? P1 p(xi ), while for xi ? C2 , f (xi ) ? P2 q(xi ). Let us examine the numerator
R
of (14) in this case. It can be approximated as p(x)q(x)
f (x) dx
Z
Z
Z
Z
p(x)q(x)
1
1
p(x)q(x)
?
dx +
dx ?
q(x)dx +
p(x)dx.
(27)
f (x)
f (x)
P1 C 1
P2 C 2
C1
C2
By performing a similar calculation for the denominator of (14), it can be shown to
be approximately equal to ?P1 P . Hence, the Laplacian pdf distance can be written
1 1
as a risk function, given by
Z
Z
p
1
1
L ? P 1 P2
q(x)dx +
p(x)dx .
(28)
P1 C 1
P2 C 2
Note that if P1 = P2 = 12 , then L = 2Pe , where Pe is the probability of error when
assigning data points to the two clusters, that is
Z
Z
Pe = P 1
q(x)dx + P2
p(x)dx.
(29)
C1
C2
Thus, in this case, minimizing L is equivalent to minimizing Pe . However, in the case
that P1 6= P2 , (28) has an even more interesting interpretation. In that situation,
it can be seen that the two integrals in the expressions (28)Rand (29) are weighted
exactly oppositely. For example, if P1 is close to one, L ? C2 p(x)dx, while Pe ?
R
q(x)dx. Thus, the Laplacian pdf distance emphasizes to cluster the most unC1
likely data points correctly. In many real world applications, this property may be
crucial. For example, in medical applications, the most important points to classify
correctly are often the least probable, such as detecting some rare disease in a group
of patients.
6
Conclusions
We have introduced a new pdf distance measure that we refer to as the Laplacian
pdf distance, and we have shown that it is in fact a clustering cost function in a
kernel feature space determined by the eigenspectrum of the Laplacian data matrix.
In our exposition, the Mercer kernel and the Parzen kernel is equivalent, making
it possible to determine the Mercer kernel based on automatic selection procedures
for the Parzen kernel. Hence, the Laplacian data matrix and its eigenspectrum can
be determined automatically too. We have shown that the new pdf distance has an
interesting property as a risk function.
The results we have derived can only be obtained analytically using Gaussian kernels. The same results may be obtained using other Mercer kernels, but it requires
an additional approximation wrt. the expectation operator. This discussion is left
for future work.
Acknowledgments.
0300340.
This work was partially supported by NSF grant ECS-
References
[1] Y. Weiss, ?Segmentation Using Eigenvectors: A Unifying View,? in International Conference on Computer Vision, 1999, pp. 975?982.
[2] A. Y. Ng, M. Jordan, and Y. Weiss, ?On Spectral Clustering: Analysis and an
Algorithm,? in Advances in Neural Information Processing Systems, 14, 2001,
vol. 2, pp. 849?856.
[3] K. R. M?
uller, S. Mika, G. R?
atsch, K. Tsuda, and B. Sch?
olkopf, ?An Introduction
to Kernel-Based Learning Algorithms,? IEEE Transactions on Neural Networks,
vol. 12, no. 2, pp. 181?201, 2001.
[4] J. Mercer, ?Functions of Positive and Negative Type and their Connection with
the Theory of Integral Equations,? Philos. Trans. Roy. Soc. London, vol. A, pp.
415?446, 1909.
[5] C. Williams and M. Seeger, ?Using the Nystr?
om Method to Speed Up Kernel
Machines,? in Advances in Neural Information Processing Systems 13, Vancouver, Canada, USA, 2001, pp. 682?688.
[6] M. Brand and K. Huang, ?A Unifying Theorem for Spectral Embedding and
Clustering,? in Ninth Int?l Workshop on Artificial Intelligence and Statistics,
Key West, Florida, USA, 2003.
[7] E. Parzen, ?On the Estimation of a Probability Density Function and the
Mode,? Ann. Math. Stat., vol. 32, pp. 1065?1076, 1962.
[8] B. W. Silverman, Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, 1986.
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1,856 | 2,686 | Efficient Kernel Discriminant Analysis via QR
Decomposition
Tao Xiong
Department of ECE
University of Minnesota
[email protected]
Jieping Ye
Department of CSE
University of Minnesota
[email protected]
Vladimir Cherkassky
Department of ECE
University of Minnesota
[email protected]
Qi Li
Department of CIS
University of Delaware
[email protected]
Ravi Janardan
Department of CSE
University of Minnesota
[email protected]
Abstract
Linear Discriminant Analysis (LDA) is a well-known method for feature extraction and dimension reduction. It has been used widely in
many applications such as face recognition. Recently, a novel LDA algorithm based on QR Decomposition, namely LDA/QR, has been proposed,
which is competitive in terms of classification accuracy with other LDA
algorithms, but it has much lower costs in time and space. However,
LDA/QR is based on linear projection, which may not be suitable for data
with nonlinear structure. This paper first proposes an algorithm called
KDA/QR, which extends the LDA/QR algorithm to deal with nonlinear data by using the kernel operator. Then an efficient approximation of
KDA/QR called AKDA/QR is proposed. Experiments on face image data
show that the classification accuracy of both KDA/QR and AKDA/QR
are competitive with Generalized Discriminant Analysis (GDA), a general kernel discriminant analysis algorithm, while AKDA/QR has much
lower time and space costs.
1
Introduction
Linear Discriminant Analysis [3] is a well?known method for dimension reduction. It has
been used widely in many applications such as face recognition [2]. Classical LDA aims
to find optimal transformation by minimizing the within-class distance and maximizing
the between-class distance simultaneously, thus achieving maximum discrimination. The
optimal transformation can be readily computed by computing the eigen-decomposition on
the scatter matrices.
Although LDA works well for linear problems, it may be less effective when severe nonlinearity is involved. To deal with such a limitation, nonlinear extensions through kernel
functions have been proposed. The main idea of kernel-based methods is to map the input
data to a feature space through a nonlinear mapping, where the inner products in the feature
space can be computed by a kernel function without knowing the nonlinear mapping explicitly [9]. Kernel Principal Component Analysis (KPCA) [10], Kernel Fisher Discriminant
Analysis (KFDA) [7] and Generalized Discriminant Analysis (GDA) [1] are, respectively,
kernel-based nonlinear extensions of the well known PCA, FDA and LDA methods.
To our knowledge, there are few efficient algorithms for general kernel based discriminant
algorithms ? most known algorithms effectively scale as O(n3 ) where n is the sample
size. In [6, 8], S. Mika et al. made a first attempt to speed up KFDA through a greedy
approximation technique. However the algorithm was developed to handle the binary classification problem. For multi-class problem, the authors suggested the one against the rest
scheme by considering all two-class problems.
Recently, an efficient variant of LDA, namely LDA/QR, was proposed in [11, 12]. The
essence of LDA/QR is the utilization of QR-decomposition on a small size matrix. The
time complexity of LDA/QR is linear in the size of the training data, as well as the number
of dimensions of the data. Moreover, experiments in [11, 12] show that the classification
accuracy of LDA/QR is competitive with other LDA algorithms.
In this paper, we first propose an algorithm, namely KDA/QR1 , which is a nonlinear extension of LDA/QR. Since KDA/QR involves the whole kernel matrix, which is not scalable
for large datasets, we also propose an approximation of KDA/QR, namely AKDA/QR. A
distinct property of AKDA/QR is that it scales as O(ndc), where n is the size of the data,
d is the dimension of the data, and c is the number of classes.
We apply the proposed algorithms on face image datasets and compare them with LDA/QR,
and Generalized Discriminant Analysis (GDA) [1], a general method for kernel discriminant analysis. Experiments show that: (1) AKDA/QR is competitive with KDA/QR and
GDA in classification; (2) both KDA/QR and AKDA/QR outperform LDA/QR in classification; and (3) AKDA/QR has much lower costs in time and space than GDA.
2
LDA/QR
In this section, we give a brief review of the LDA/QR algorithm [11, 12]. This algorithm
has two stages. The first stage maximizes the separation between different classes via QR
Decomposition [4]. The second stage addresses the issue of minimizing the within-class
distance, while maintaining low time/space complexity.
Let A ? IRd?n be the data matrix, where each column ai is a vector in d-dimensional space.
Assume A is partitioned into c classes {?i }ci=1 , and the size of the ith class |?i | = ni .
Define between-class, within-class, and total scatter matrices Sb , Sw , and St respectively,
?
as follows?[3]: Sb = Hb Hbt , Sw = Hw Hwt , and St = Ht Htt , where Hb = [ N1 (m1 ?
m), ? ? ? , Nc (mc ? m)] ? Rd?c , Hw = A ? [m1 et1 , ? ? ? , mc etc ] ? Rd?n , and Ht =
A ? met ? Rd?n , ei = (1, ? ? ? , 1)t ? Rni ?1 , e = (1, ? ? ? , 1)t ? Rn?1 , mi is the mean of
the ith class, and m is the global mean. It is easy to check that St = Sb + Sw .
The first stage of LDA/QR aims to solve the following optimization problem,
G = arg max
trace(Gt Sb G).
t
G G=I
(1)
Note that this optimization only addresses the issue of maximizing the between-class distance. The solution can be obtained by solving the eigenvalue problem on Sb . The solution
can also be obtained through QR Decomposition on the centroid matrix C [12], where
C = [m1 , m2 , ? ? ? , mc ] consists of the c centroids. More specifically, let C = QR be
the QR Decomposition of C, where Q ? IRn?c has orthonormal columns and R ? IRc?c
1
KDA/QR stands for Kernel Discriminant Analysis via QR-decomposition
Algorithm 1: LDA/QR
/* Stage I: */
1. Construct centroid matrix C;
2. Compute QR Decomposition of C as C = QR, where Q ? IRd?c , R ? IRc?c ;
/* Stage II: */
3. Y ? Hbt Q;
4. Z ? Htt Q;
5. B ? Y t Y ; /*Reduced between-class scatter matrix*/
6. T ? Z t Z; /*Reduced total scatter matrix*/
7. Compute the c eigenvectors ?i of (T + ?Ic )?1 B with decreasing eigenvalues;
8. G ? QV , where V = [?1 , ? ? ? , ?c ].
is upper triangular. Then G = QV , for any orthogonal matrix V , solves the optimization problem in Eq. (1). Note that the choice of orthogonal matrix V is arbitrary, since
trace(Gt Sb G) = trace(V t Gt Sb GV ), for any orthogonal matrix V .
The second stage of LDA/QR refines the first stage by addressing the issue of minimizing
the within-class distance. It incorporates the within-class scatter information by applying a
relaxation scheme on V (relaxing V from an orthogonal matrix to an arbitrary matrix). In
the second stage of LDA/QR, we look for a transformation matrix G such that G = QV , for
some V . Note that V is not required to be orthogonal. The original problem of computing G
is equivalent to computing V . Since Gt Sb G = V t (Qt Sb Q)V , Gt Sw G = V t (Qt Sw Q)V ,
and Gt St G = V t (Qt St Q)V , the original problem of finding optimal G is equivalent to
finding V , with B = Qt Sb Q, W = Qt Sw Q, and T = Qt St Q as the ?reduced? betweenclass, within-class and total scatter matrices, respectively. Note that B has much smaller
size than the original scatter matrix Sb (similarly for W and T ).
The optimal V can be computed efficiently using many existing LDA-based methods, since
we are dealing with matrices B, W , and T of size c by c. We can compute the optimal V
by simply applying regularized LDA; that is, we compute V , by solving a small eigenvalue
problem on (W + ?Ic )?1 B or (T + ?Ic )?1 B (note T = B + W ), for some positive
constant ? [3]. The pseudo-code for this algorithm is given in Algorithm 1. We use the
total scatter instead of the within-class scatter in Lines 4, 6, and 7, mainly for convenience
of presentation of the kernel methods in Section 3 and Section 4.
3
Kernel discriminant analysis via QR-decomposition (KDA/QR)
In this section, the KDA/QR algorithm, a nonlinear extension of LDA/QR through kernel
functions, is presented. Let ? be a mapping to the feature space and ?(A) be the data
matrix in the feature space. Then, the centroid matrix C ? in the feature space is
?
1
1
?
?
?(ai ), ? ? ? ,
?(ai ) .
(2)
C = m1 , ? ? ? , mc =
n1
nc
i??1
i??c
The global centroid in the feature space can be computed as m? = n1 i ni m?
i . To maximize between-class distance in the feature space, as discussed in Section 2, we perform QR
decomposition on C ? , i.e., C ? = Q? R? . A key observation is that R? can be computed
as (C ? )t C ? = (R? )t R? by applying the Cholesky decomposition on (C ? )t C ? [4].
Note that C ? = A? M , where A? = ?(A) = [?(a1 ) . . . ?(an )], and the ith column
of M is (0, ? ? ? , 0, n1i , ? ? ? , n1i , 0, ? ? ? , 0)t . Let K be the kernel matrix with K(i, j) =
?(ai ), ?(aj ). Then
(C ? )t C ? = M t KM.
(3)
Algorithm 2: KDA/QR
/* Stage I: */
1. Construct kernel matrix K;
2. Compute (C ? )t C ? = M t (KM ) as in Eq. (3);
3. Compute R? from the Cholesky Decomposition of (C ? )t C ? ;
/* Stage II: */
4. Y ? ? N t M t KM (R? )?1 ;
5. Z ? ? E t KM (R? )?1 ;
6. B ? ? (Y ? )t Y ? ;
7. T ? ? (Z ? )t Z ? ;
?
?1 ?
B , with decreasing eigenvalues;
8. Compute the c eigenvectors ??
i of (T + ?Ic )
?
?
,
?
,
?
?
?
,
?
];
9. V ? ? [??
c
1
2
10. G? ? C ? (R? )?1 V ? ;
With the computed R? , Q? = C ? (R? )?1 . The matrices Y ? , Z ? , B ? , and W ? in the
feature space (corresponding to the second stage in LDA/QR) can be computed as follows.
In the feature space, we
have Hb? = C ? N , where the ith column of N is
?
?
n
((0, ? ? ? , ni , ? ? ? 0)t ? n i (n1 , ? ? ? , nc )t . It follows that Y ? = (Hb? )t Q? =
N t (C ? )t C ? (R? )?1 = N t M t KM (R? )?1 . Similarly, Ht? = A? E and Z ? =
(Ht? )t Q? = E t (A? )t C ? (R? )?1 = E t (A? )t A? M (R? )?1 = E t KM (R? )?1 , where
E = I ? n1 eet .
Since Sb? = Hb? (Hb? )t and St? = Ht? (Ht? )t , we have
B?
T?
= (Q? )t Sb? Q? = (Q? )t Hb? (Hb? )t Q? = (Y ? )t Y ? ,
= (Q? )t St? Q? = (Q? )t Ht? (Ht? )t Q? = (Z ? )t Z ? .
c
?
?1 ?
We proceed by computing the c eigenvectors {??
B . Define V ? =
i }i=1 of (T + ?Ic )
?
?
?
[?1 , ?2 , ? ? ? , ?c ]. The final transformation matrix can be computed as
G? = Q? V ? = C ? (R? )?1 V ? .
(4)
=
For a given data point z, its projection by G? is (G? )t ?(z)
(V ? )t ((R? )?1 )t (C ? )t ?(z) = (V ? )t ((R? )?1 )t M t Ktz , where Ktz ? IRn and
Ktz (i) = ?(ai ), ?(z).
The pseudo-code for the KDA/QR algorithm is given in Algorithm 2.
3.1
Complexity analysis of KDA/QR
The cost to formulate the kernel matrix in Line 1 is O(n2 d). The computation of (C ? )t C ?
in Line 2 takes O(n2 ), taking advantage of the sparse structure of M . The Cholesky decomposition in Line 3 takes O(c3 ) [4]. Lines 4 takes O(c3 ), as M t KM is already computed in
Line 2. In Line 5, the
computation of Z ?
= E t KM (R? )?1 = (I ? n1 eet )KM (R? )?1 =
KM (R? )?1 ? n1 e (et KM )(R? )?1 in the given order takes O(nc2 ), assuming KM
is kept in Line 2. Lines 6, 7, and 8 take O(c3 ), O(nc2 ) and O(c3 ), respectively. Hence,
the total complexity of the kernel LDA/QR algorithm is O(n2 d). Omitting the cost for
evaluating the kernel matrix K, which is required in all kernel-based algorithms, the total
cost is O(n2 ). Note that all other general discriminant analysis algorithms scale as O(n3 ).
4
Approximate KDA/QR (AKDA/QR)
In this section, we present the AKDA/QR algorithm, which is an efficient approximation
of the KDA/QR algorithm from the last section. Note that the bottleneck of KDA/QR is the
explicit formation of the large kernel matrix K for the computation of (C ? )t C ? in Line
2 of Algorithm 2. The AKDA/QR algorithm presented in this section avoids the explicit
construction of K, thus reducing the computational cost significantly.
? t ?
?
The key to AKDA/QR is the
=
efficient computation of (C ) C , ?where C
1
?
?
?
[m1 , ? ? ? , mc ] and mj = nj i??j ?(ai ). AKDA/QR aims to find xj in the original
?
space such that ?(x?j ) approximates m?
j . Mathematically, the optimal xj can be computed
by solving the following optimization problem:
1
min ?(xj ) ?
?(ai )2 for j = 1, ? ? ? , c.
(5)
d
nj
xj ?R
i??j
To proceed, we only consider Gaussian kernels for AKDA/QR, as they are the most widely
used ones in the literature [9]. Furthermore, the optimization problem in (5) can be simplified by focusing on the Gaussian kernels, as shown in the following lemma.
Lemma 4.1. Consider Gaussian kernel function exp(?x ? y2 /?), where ? is the bandwidth parameter. The optimization problem in (5) is convex if
2
(xj ? ai ) ? 1
for each j = 1, ? ? ? , c and for all i ? ?j ,
(6)
?
Proof. It is easy to check that, for the Gaussian kernel, the optimization problem in (5)
reduces to:
min f (xj )
xj ?Rd
for j = 1, ? ? ? , c,
(7)
where f (x) = i??j fi (x) and fi (x) = ?exp(?x ? ai 2 /?). The Hessian matrix of
fi (x) is H(fi ) = ?2 exp(?x ? ai 2 /?)(I ? ?2 (x ? ai )(x ? ai )t ). It is easy to show that
if ?2 (x ? ai ) ? 1, for all i ? ?j , then H(fi ) is positive semi-definite, that is, fi (x) is
convex. Thus, f (x), the sum of convex functions is also convex.
For applications involving high-dimensional data, such as face recognition, ? is usually
large (typically ranging from thousands to hundreds of thousands [13]), and the condition
in Lemma 4.1 holds if we restrict our search space to the convex hull of each class in the
original space. Therefore, the global minimum of the optimization problem in (7) can be
found very efficiently using Newton?s or gradient decent methods. A key observation is
that for relatively large ?, the centroid of each class in the original space will map very
close to the centroid in the feature space [9], which can serve as the approximatesolution
of the optimization problem in (7). Experiments show that choosing x?j = n1j i??j ai
produces results close to the one by solving the optimization problem in (7). We thus use
it in all the following experiments.
With the computed x?j , for j = 1, . . . , c, the centroid matrix C ? can be approximated by
C ? ? [?(x?1 ) . . . ?(x?c )] (? C? ? )
(8)
?
(C? ? )t C? ? = K,
(9)
and
Algorithm 3: AKDA/QR
/* Stage I: */
1. Compute x?j = n1j i??j ai , for j = 1, ? ? ? , c;
? as in Eq. (9);
2. Construct kernel matrix K
? ? from the Cholesky Decomposition of K;
?
3. Compute R
/* Stage II: */
? R
? ? )?1 ;
4. Y? ? ? N t K(
? ? )?1 ;
? tc (R
5. Z? ? ? E t K
?
? t ??
?
?
6. B ? (Y ) Y ;
7. T?? ? (Z? ? )t Z? ? ;
?1 ? ?
??
8. Compute the c eigenvectors ???
B , with decreasing eigenvalues;
i of (T + ?Ic )
?
?
?
?
9. V? ? [??1 , ??2 , ? ? ? , ??c ];
? ? ? C? ? (R
? ? )?1 V? ? ;
10. G
time
space
PCA
O(n2 d)
O(nd)
LDA/QR
O(ndc)
O(nc)
GDA
O(n2 d + n3 )
O(n2 )
KDA/QR
O(n2 d)
O(n2 )
AKDA/QR
O(ndc)
O(nc)
Table 1: Comparison of time & space complexities of several dimension reduction algorithms: n is the size of the data, d is the dimension, and c is the number of classes.
? j) = ?(x? ), ?(x? ) and K
? ? Rc?c . The Cholesky decomposition of K
? will
where K(i,
i
j
?
?
t
?
? .
? = (R
? )R
? by K
give us R
? ? )?1 ,
? ? = C? ? N , and Y? ? = N t K(
? R
? ? )?1 . Similarly, Z? ? = E t K
? tc (R
It follows that H
b
?
? tc (i, j) = ?(ai ), ?(x ).
where N and E are defined as in Section 3, and K
j
The following steps will be the same as the KDA/QR algorithm. The pseudo-code for
AKDA/QR is given in Algorithm 3.
4.1
Complexity analysis of AKDA/QR
? in Line 2 takes O(c2 d). The
It takes O(dn) in Line 1. The construction of the matrix K
3
Cholesky Decomposition in Line 3 takes O(c ) [4]. Lines 4 and 5 take O(c3 ) and O(ndc)
respectively. It then takes O(c3 ) and O(nc2 ) for matrix multiplications in Lines 6 and 7,
respectively. Line 8 computes the eigen-decomposition of a c by c matrix, hence takes
O(c3 ) [4]. Thus, the most expensive step in Algorithm 3 is Line 5, which takes O(ndc).
Table 1 lists the time and space complexities of several dimension reduction algorithms. It
is clear from the table that AKDA/QR is more efficient than other kernel based methods.
5
Experimental results
In this section, we evaluate both the KDA/QR and AKDA/QR algorithms. The performance is measured by classification accuracy. Note that both KDA/QR and AKDA/QR
have two parameters: ? for the kernel function and ? for the regularization. Experiments show that choosing ? = 100000 and ? = 0.15 for KDA/QR, and ? = 100000
and ? = 0.10 for AKDA/QR produce good overall results. We thus use these values
in all the experiments. 1-Nearest Neighbor (1-NN) method is used as the classifier. We
randomly select p samples of each person from the dataset for training and the rest for
0.9
0.99
0.8
0.98
0.7
Accuracy
Accuracy
1
0.97
0.96
PCA
LDA/QR
KDA/QR
AKDA/QR
0.95
0.6
PCA
LDA/QR
KDA/QR
AKDA/QR
0.5
0.4
0.94
0.3
0.93
0.2
3
4
5
6
7
Number of training samples per class
8
3
4
5
6
7
Number of training samples per class
8
Figure 1: Comparison of classification accuracy on PIX (left) and AR (right).
testing. We repeat the experiments 20 times and report the average recognition accuracy
of each method. The MATLAB codes for the KDA/QR and AKDA/QR algorithms may be
accessed at http://www.cs.umn.edu/?jieping/Kernel.
Datasets: We use the following three datasets in our study, which are publicly available:
PIX contains 300 face images of 30 persons. The image size of PIX image is 512 ? 512.
We subsample the images down to a size of 100 ? 100 = 10000; ORL is a well-known
dataset for face recognition. It contains ten different face images of 40 persons, for a total
of 400 images. The image size is 92 ? 112 = 10304; AR is a large face image datasets. We
use a subset of AR. This subset contains 1638 face images of 126 persons. Its image size
is 768 ? 576. We subsample the images down to a size of 60 ? 40 = 2400. Each dataset is
normalized to have zero mean and unit variance.
KDA/QR and AKQA/QR vs. LDA/QR: In this experiment, we compare the performance of AKDA/QR and KDA/QR with that of several other linear dimension reduction
algorithms including PCA, LDA/QR on two face datasets. We use 100 principal components for PCA as it produces good overall results. The results are summarized in Fig. 1,
where the x-axis denotes the number of samples per class in the training set and the y-axis
denotes the classification accuracy. Fig. 1 shows that KDA/QR and AKQA/QR consistently outperform LDA/QR and PCA. The most interesting result lies in the AR dataset,
where AKDA/QR and KDA/QR outperform LDA/QR by a large margin. It is known that
the images in the AR dataset contain pretty large area of occlusion due to sun glasses and
scarves, which makes linear algorithms such as LDA/QR less effective. Another interesting observation is that the approximate AKQA/QR algorithm is competitive with its exact
version KDA/QR in all cases.
KDA/QR and AKQA/QR vs. GDA: In this experiment, we compare the performance of
AKDA/QR and KDA/QR with Generalized Discriminant Analysis (GDA) [1]. The comparison is made on the ORL face dataset, as the result of GDA on ORL is available in
[5]. We also include the results on PCA and LDA/QR. The results are summarized in Table 2. The main observation from this experiment is that both KDA/QR and AKDA/QR are
competitive with GDA, while AKDA/QR is much more efficient than GDA (see Table 1).
Similar to the first experiment, Table 2 shows that KDA/QR and AKDA/QR consistently
outperform the PCA and LDA/QR algorithms in terms of recognition accuracy.
6
Conclusions
In this paper, we first present a general kernel discriminant analysis algorithm, called
KDA/QR. Using Gaussian kernels, we then proposed an approximate algorithm to
p
3
4
5
6
7
8
PCA
0.8611
0.8938
0.9320
0.9512
0.9633
0.9713
LDA/QR
0.8561
0.9083
0.9385
0.9444
0.9692
0.9713
GDA
0.8782
0.9270
0.9535
0.9668
0.9750
0.9938
KDA/QR
0.9132
0.9321
0.9625
0.9737
0.9825
0.9875
AKDA/QR
0.9118
0.9300
0.9615
0.9744
0.9815
0.9875
Table 2: Comparison of classification accuracy on ORL face image dataset. p is the number
of training samples per class. The results on GDA are taken from [5].
KDA/QR, which we call AKDA/QR. Our experimental results show that the accuracy
achieved by the two algorithms is very competitive with GDA, a general kernel discriminant algorithms, while AKDA/QR is much more efficient. In particular, the computational
complexity of AKDA/QR is linear in the number of the data points in the training set as
well as the number of dimensions and the number of classes.
Acknowledgment Research of J. Ye and R. Janardan is sponsored, in part, by the Army High Performance Computing Research Center under the auspices of the Department of the Army, Army
Research Laboratory cooperative agreement number DAAD19-01-2-0014, the content of which does
not necessarily reflect the position or the policy of the government, and no official endorsement
should be inferred.
References
[1] G. Baudat and F. Anouar. Generalized discriminant analysis using a kernel approach. Neural
Computation, 12(10):2385?2404, 2000.
[2] P.N. Belhumeour, J.P. Hespanha, and D.J. Kriegman. Eigenfaces vs. fisherfaces: Recognition
using class specific linear projection. IEEE TPAMI, 19(7):711?720, 1997.
[3] K. Fukunaga. Introduction to Statistical Pattern Classification. Academic Press, San Diego,
California, USA, 1990.
[4] G. H. Golub and C. F. Van Loan. Matrix Computations. The Johns Hopkins University Press,
Baltimore, MD, USA, third edition, 1996.
[5] Q. Liu, R. Huang, H. Lu, and S. Ma. Kernel-based optimized feature vectors selection and
discriminant analysis for face recognition. In ICPR Proceedings, pages 362 ? 365, 2002.
[6] S. Mika, G. R?atsch, and K.-R. M?uller. A mathematical programming approach to the kernel
fisher algorithm. In NIPS Proceedings, pages 591 ? 597, 2001.
[7] S. Mika, G. Ratsch, J. Weston, B. Sch?okopf, and K.-R. M?uller. Fisher discriminant analysis
with kernels. In IEEE Neural Networks for Signal Processing Workshop, pages 41 ? 48, 1999.
[8] S. Mika, A.J. Smola, and B. Sch?olkopf. An improved training algorithm for kernel fisher discriminants. In AISTATS Proceedings, pages 98?104, 2001.
[9] B. Sch?okopf and A. Smola. Learning with Kernels: Support Vector Machines, Regularization,
Optimization and Beyond. MIT Press, 2002.
[10] B. Sch?okopf, A. Smola, and K. M?uller. Nonlinear component analysis as a kernel eigenvalue
problem. Neural Computation, 10(5):1299?1319, 1998.
[11] J. Ye and Q. Li. LDA/QR: An efficient and effective dimension reduction algorithm and its
theoretical foundation. Pattern recognition, pages 851?854, 2004.
[12] J. Ye, Q. Li, H. Xiong, H. Park, R. Janardan, and V. Kumar. IDR/QR: An incremental dimension
reduction algorithm via QR decomposition. In ACM SIGKDD Proceedings, pages 364?373,
2004.
[13] W. Zheng, L. Zhao, and C. Zou. A modified algorithm for generalized discriminant analysis.
Neural Computation, 16(6):1283?1297, 2004.
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1,857 | 2,687 | Multi-agent Cooperation
in Diverse Population Games
K. Y. Michael Wong, S. W. Lim and Z. Gao
Hong Kong University of Science and Technology, Hong Kong, China.
{phkywong, swlim, zhuogao}@ust.hk
Abstract
We consider multi-agent systems whose agents compete for resources by
striving to be in the minority group. The agents adapt to the environment
by reinforcement learning of the preferences of the policies they hold.
Diversity of preferences of policies is introduced by adding random biases to the initial cumulative payoffs of their policies. We explain and
provide evidence that agent cooperation becomes increasingly important
when diversity increases. Analyses of these mechanisms yield excellent
agreement with simulations over nine decades of data.
1
Introduction
In the intelligent control of large systems, the multi-agent approach has the advantages of
parallelism, robustness, scalability, and light communication overhead [1]. Since it involves
many interacting adaptive agents, the behavior becomes highly complex. While a standard
analytical approach is to study their steady state behavior described by the Nash equilibria
[2], it is interesting to consider the dynamics of how the steady state is approached. Of
particular interest is the case of heterogeneous agents, which have diversi?ed preferences
in decision making. In such cases, the cooperation of agents becomes very important.
Speci?cally, we consider the dynamics of a version of large population games which models the collective behavior of agents simultaneously and adaptively competing for limited
resources. The game is a variant of the Minority Game, in which the agents strive to
make the minority decision, thereby balancing the load distributed between the majority
and minority choices [3]. Previous work showed that the system behavior depends on the
input dimension of the agents? policies. When the policy dimension is too low, many agents
share identical policies, and the system suffers from the maladaptive behavior of the agents,
meaning that they prematurely rush to adapt to system changes in bursts [4].
Recently, we have demonstrated that a better system ef?ciency can be attained by introducing diversity [5]. This is done by randomly assigning biases to the initial preference of
policies of the agents, so that agents sharing common policies may not adopt them at the
same time, and maladaptation is reduced. As a result, the population difference between the
majority and minority groups decreases. For typical control tasks such as the distribution of
shared resources, this corresponds to a high system ef?ciency. In contrast to the maladaptive regime, in which agents blindly respond to environmental signals, agent cooperation
becomes increasingly important in the diverse regime. Namely, there are fewer agents ad-
justing their policy perferences at each step of the steady state, but there emerges a more
coordinated pattern of policy adjustment among them. Hence, it is interesting to study the
mechanisms by which they adapt mutually, and their effects on the system ef?ciency.
In this paper, we explain the cooperative mechanisms which appear successively when
the diversity of the agents? preference of policies increases, as recently proposed in [6].
We will provide experimental evidence of these effects, and sketch their analyses which
yield excellent agreement with simulations. While we focus on the population dynamics
of the Minority Game, we expect that the observed cooperative mechanisms are relevant to
reinforcement learning in multi-agent systems more generally.
2
The Minority Game
The Minority Game consists of a population of N agents competing sel?shly to maximize
their individual utility in an environment of limited resources, N being odd [3]. Each agent
makes a decision + or ? at each time step, and the minority group wins. For typical control
tasks such as the resource allocation, the decisions + and ? may represent two alternative
resources, so that less agents utilizing a resource implies more abundance. The decisions
of each agent are prescribed by policies, which are binary functions mapping the history of
the winning bits of the game in the most recent m steps to decisions + or ?. Hence, m is
the memory size. Before the game starts, each agent randomly picks s policies out of the
set of 2D policies with replacement, where D ? 2m is the number of input states.
The long-term goal of an agent is to maximize her cumulative payoff, which is the sum
of the undiscounted payoffs received during the game history. For the decision
P ? i (t) of
agent i at time t (?i (t) = ?1), the payoff is ??i (t)G(A(t)), where A(t) ? i ?i (t)/N ,
and G(A) satis?es the property signG(A) = signA. She tries to achieve her goal by
choosing at each step, out of her s policies, the most successful one so far, and outputing
her decision accordingly. The success of a policy is measured by its cumulative payoff,
updated every step irrespective of whether it is adopted or not. This reinforcement learning
provides an agent with adaptivity. Though we only consider random policies instead of
organized ones, we expect that the model is suf?cient to capture the collective behavior of
large population games. In this paper, we consider a step payoff function, G(A) = signA.
The cumulative payoffs then take integer values. Note that an agent gains in payoff when
she makes a decision opposite to A(t), and loses otherwise, re?ecting the winning of the
minority group.
It is natural to consider systems with diverse preferences of policies [5]. This means that
the initial cumulative payoffs of policies ? (? = 1, . . . , s ? 1) of agent i with respect to
her sth policy have random biases ?i? . Diversity is important in reducing the maladaptive
behavior of the agents, since otherwise the same policy of all agents accumulates the same
payoffs, and would be adopted at the same time. In this paper, we consider the case s = 2,
and the biases are the sums of ?1 randomly drawn R times. In particular, when R is not too
small, the bias distribution approaches a Gaussian distribution with mean 0 and variance R.
The ratio ? ? R/N is referred to as the diversity. For odd R, no two policies have the same
cumulative payoffs throughout the process, and the dynamics is deterministic, resulting in
highly precise simulation results useful for re?ned comparison with theories.
The population averages of the decisions oscillate around 0 at the steady state. Since a
large difference between the majority and minority populations implies inef?cient resource
allocation, the inef?ciency of the system is often measured by the variance ? 2 /N of the
population making decision +, and is given by
?
?
?2
N
? h[A? (t) (t) ? hA? (t) (t)it ]2 it ,
(1)
N
4
where h it denotes time average at the steady state. Its dependence on the diversity is
10
0
2
4
? /N
2
? /N
10
10
?2
10
?3
N = 127
N = 511
N = 2047
N = 8191
N = 32767
10
N = 127
N = 511
N = 2047
?4
10
?6
10
?2
10
2
10
6
?
10
?4
10
2
10
4
10
6
?
Figure 1: (a) The dependence of the variance of the population making decision + on the
diversity at m = 1 and s = 2. Symbols: simulation results averaged over 1,024 samples of
initial conditions. Lines: theory. Dashed-dotted line: scaling prediction. (b) Comparison
between simulation results (symbols), theory with kinetic sampling only (dashed lines),
one-wait approximation (dash-dotted lines), and many-wait approximation (lines).
shown in Fig. 1. Several modes of agent cooperation can be identi?ed, and explained in the
following sections.
3
Statistical Cooperation
For each curve with a given N in Fig. 1(a), and besides the ?rst few data points where
? ? N ?1 and ? 2 /N ? N , the behavior of the variance is dominated by the scaling relation
? 2 /N ? ??1 for ? ? 1. To interpret this result, we describe the macroscopic dynamics of
the system by de?ning the D-dimensional vector A ? (t), which is the sum of the decisions
of all agents responding to history ? of their policies, normalized by N . While only one of
the D components corresponds to the historical state ?? (t) of the system, the augmentation
to D components is necessary to describe the attractor structure and the transient behavior
of the system dynamics.
The key to analysing the system dynamics is the observation that the cumulative payoffs
of all policies displace by exactly the same amount when the game proceeds. Hence for
a given pair of policies, the pro?le of the relative cumulative payoff distribution remains
unchanged, but the peak position shifts with the game dynamics. Let us consider the change
in A? (t) when ? is the historical state ?? (t). We let S?? (?) be the number of agents
holding policies ? and ? (with ? < ?), and the bias of ? with respect to ? is ?. If the
cumulative payoff of policy ? at time t is ?? (t), then the agents holding policies ? and ?
make decisions according to policy ? if ? + ?? (t) ? ?? (t) > 0, and policy ? otherwise.
Hence ? + ?? (t) ? ?? (t) is referred to as the preference of ? with respect to ?. At time
t, the cumulative payoff of policy ? changes from ?? (t) to ?? (t) ? ??? signA? (t), where
??? is the decision of policy ? at state ?. Only the ?ckle agents, that is, those agents with
preferences on the verge of switching signs, contribute to the change in A? (t), namely,
? + ?? (t) ? ?? (t) = ?1 and ??? ? ??? = ?2signA? (t). Hence we have
A? (t + 1) ? A? (t) =
?signA? (t)
2 X X
S?? (r ? ?? (t) + ?? (t))
N
r=?1
?<?
??(??? ? ??? ? 2rsignA? (t))
(2)
where ?(n) = 1 if n = 0, and 0 otherwise. In the region where D ? ln N , we have
Prob =
1
8
1
8
A
?
Q
0.6
P
Prob =
3
8
A
R
Prob =
3
8
+
average number of agents
Prob =
0.4
0.2
0.0
?3001
?1
3001
preference
Figure 2: (a) The attractor in the Minority Game with m = 1, following the period-4
sequence of P-Q-R-Q in the phase space of A+ and A? . There are 4 approaches to the
attractor indicated by the arrows, and the respective probabilities are obtained by considering the detailed dynamics from the different initial positions and states. (b) Experimental
evidence of the kinetic sampling effect: steady-state preference dependence of the average number of agents holding the identity policy and its complement, immediately before
state Q enters state R, at ? = N = 1, 023 and averaged over 100,000 samples of initial
conditions.
S?? (?) ? 1, and Eq. (2) is self-averaging. Following the derivation in [5], we arrive at
r
2
?
?
?
A (t + 1) = A (t) ? signA (t)
?(? ? ?? (t)).
(3)
?R
Equation (3) shows that the dynamics proceeds in the direction
which reduces the magnip
tude of the population vector, each time by a step of size 2/?R. At the steady state, each
component oscillates between positive and negative, as shown in the example of m = 1
in Fig. 2(a). Due to the maladaptive nature of the dynamics, it never reaches p
the zero
value. As a result, each state is con?ned in a D-dimensional hypercube of size 2/?R,
irrespective of the initial position of the population vector. This con?nement enables us
to compute the variance of the decisions, given by ? 2 /N = f (?)/2??, where f (?) is a
smooth function of ?, which approaches (1 ? 1/4D)/3 for ? ? 1. The physical picture of
this scaling relation comes from the broadening of the preference distribution due to bias
diversity. The fraction of ?ckle agents at every time step consists of those who have ?1
preferences, which scales as the height of the bias?distribution near its center. Since
the
?
distribution is a Gaussian with standard deviation R, the step sizes scale as 1/ R, and
variances ? 2 /N as ??1 . The scaling relation shows that agent cooperation in this regime is
described at the level of statistical distributions of policy preferences, since
? the number of
agents making an adaptive move at each step is suf?ciently numerous (? N ).
4
Kinetic Sampling
As shown in Fig. 1(a), ? 2 /N deviates above the scaling with ??1 when ? ? N . To consider
the origin of this deviation, we focus in Fig. 2(b) on how the average number of agents, who
hold the identity policy with ??? = ? and its complementary policy ??? = ??, depends on
the preference ?+?? ??? , when the system reaches the steady state in games with m = 1.
Since the preferences are time dependent, we sample their frequencies at a ?xed time, say,
immediately before the state changes from Q to R in Fig. 2(a). One would expect that the
bias distribution is reproduced. However, we ?nd that a peak exists at ? + ? ? ? ?? = ?1.
This value of the preference corresponds to that of the attractor step from Q to R when at
state ?, decision + loses and decision ? wins, and ? + ?? ? ?? changes from ?1 to +1.
The peak at the attractor step shows that its average size is self-organized to be larger than
those of the transient steps described by the background distribution.
This effect that favors the cooperation of larger clusters of agents is referred to as the
kinetic sampling effect. When ? ? N , A? (t + 1) ? A? (t) scales as N ?1 and is no longer
self-averaging. Rather, Eq. (2) shows that it is equal to 2/N times
? the number of ?ckle
agents atptime t, which is Poisson distributed with a mean of N/ 2?R = ?/2, where
? ? N 2/?R is the average step size. However, since the attractor is formed by steps
which reverse the sign of A? , the average step size in the attractor is larger than that in
the transient state, because a long jump in the vicinity of the attractor is more likely to get
trapped.
To describe this effect, we consider the probability Patt (?A) of step sizes ?A in the
attractor (with ?A? > 0 for all ?). Assuming
Pthat all states of the phase space are equally
likely to be accessed, we have Patt (?A) = A Patt (?A, A), where Patt (?A, A) is the
probability of ?nding the position A with displacement ?A in the attractor. Consider the
example of m = 1, where there is only one step Q
along each axis A? . The sign reversal
condition implies that Patt (?A, A) ? PPoi (?A) ? ?[?A? (A? + ?A? )], where ?(x)
is the step function of x, Q
and PPoi (?A) is the Poisson distribution of step sizes, yielding
Patt (?A) ? PPoi (?A) ? ?A? . We note that the extra factors of ?A? favor larger step
sizes. Thus, the attractor averages h(?A? )2 iatt are given by
h(?A? )2 iatt =
h(?A? )2 ?A+ ?A? iPoi
.
h?A+ ?A? iPoi
(4)
There are agents who contribute to both ?A+ and ?A? , giving rise to their correlations.
In Eq. (2), the strategies of the agents contributing to ?A+ and ?A? satisfy ??+ ? ??+ =
?2r and ??? ? ??? = 2r respectively. Among the agents contributing to ?A+ , the extra
requirement of ??? ? ??? = 2r implies that an average of 1/4 of them also contribute to
?A? . Hence, the number of agents contributing to both steps is a Poisson variable with
mean ?/8, and those exclusive to the individual steps are Poisson variables with mean
3?/8. This yields, for example,
?a0 e? 3?
? 3? ?a+ e? 3?
? 3? ?a?
P
?? ?
8
8
h?A+ ?A? iPoi = N42 a0 ,a+ ,a? ea08! ?
8
a+ !
8
a? !
8
(a0 + a+ )(a0 + a? ).
(5)
Together with similar expressions of the numerator in Eq. (4), we obtain
h(?A? )2 iatt =
2?3 + 15?2 + 20? + 4
.
N 2 (2? + 1)
(6)
The attractor states are given by A? = m? /N and m? /N ? ?A? , where m? =
1, 3, . . . , N ?A? ? 1. This yields a variance of
?2
7h(N ?A+ )2 iatt + 7h(N ?A? )2 iatt ? 8
=
,
N
192N
which gives, on combining with Eq. (6),
?2
14?3 + 105?2 + 132? + 24
=
.
N
96N (2? + 1)
(7)
(8)
When the diversity is low, ? ? 1, and Eq. (8) reduces to ? 2 /N = 7/48??, agreeing with
the scaling result of the previous section. When ? ? N , Eq. (8) has excellent agreement
with simulation results, which signi?cantly deviate above the scaling relation.
5
Waiting Effect
As shown in Fig. 1(b), ? 2 /N further deviates above the predictions of kinetic sampling
when ? ? N . To study the origin of this effect, we consider the example of m = 1. As
shown in Fig. 2(a), the attractor consists of both hops along the A? axes. Analysis shows
that only those agents holding the identity policy and its complement can complete both
hops after they have adjusted their preferences to ? + ?? ? ?? = ?1. Since there are
fewer and fewer ?ckle agents in the limit ? ? N , one would expect that a single agent of
this type would dominate the game dynamics, and ? 2 /N would approach 0.25/N , as also
predicted by Eq. (8).
However, attractors having 2 ?ckle agents are about 10 times more common in the extremely diverse limit. As illustrated in Fig. 3(a) for a typical case, one of the two agents
?rst arrives at the status of ?1 preference of her policies and stay there waiting. Meanwhile, the preference of the second agent is steadily reduced. Once she has arrived at the
status of ?1 preference of her policies, both agents can then cooperate to complete the
dynamics of the attractor. In this example, both agents do not belong to the correct type
that can complete the dynamics alone, but waiting is crucial for them to complete the hops
in the attractor, even though one would expect that the probability of ?nding more than one
?ckle agents at a time step is drastically less than that for one. Thus, the composition of
the group of ?ckle agents is self-organized through this waiting effect, and consequently
the step sizes and variance increase above those predicted by kinetic sampling.
The analysis of the waiting effect is lengthy. Here the agents are so diverse that the average
step size is approaching 0. At each state in the phase space, the system remains stationary
for many time steps, waiting for some agent to reduce the magnitude of her preference until
policy switching can take place. For illustration, we sketch the approximation of including
up to one wait. As shown in Fig. 2(a), the attractor may be approached from the arm (P
or R) or from the corner (Q). Consider the case of the state approaching from P, waiting
up to k times at Q to move to R, and ending the transient dynamics thereafter. Then the
cumulative payoffs of a policy ? can be written as ?? + ??+ at P, ?? , . . . , ?? ? k??? at Q
and, in the attractor of period 4, repeating the sequence of ?? ? k??? ? ??? at R, ?? ? k???
at Q, ?? ? k??? + ??+ at P, and ?? ? k??? at Q. The
P movement of the cumulative payoffs
can be conveniently represented by writing ?? = ? k ? ??? , where k ? denotes the number
of wins minus losses of decision 1 at state ? in the game history. For m = 1, these steps
are plotted in the space of k + and k ? in Fig. 3(b).
The size of each step is 2/N times the number of ?ckle agents at that step, which is Poisson
distributed with average ?/2. The average numbers of agents appearing simultaneously in
different steps positioned along the directions k + ?k ? = constant and k ? = constant are,
respectively, ?/8 and ?/4, and 0 for other directions. Thus, the average number of agents
common in the pairs of steps {PQ, QQ1 }, {QQk , QP}, {QP, QR}, {PQ, QP} are ?/8,
?/8, ?/8 and ?/4 respectively. The rest of the combinations of steps are uncorrelated.
The number of agents involved in the steps are described in Table 1.
The variance of the step sizes is given by
X
1
h [(?A+ )2 +(?A? )2 ]iatt =
Pj
2
j
?P
1
+ 2
i=0,1 h 2 [(?A )
P
+ (?A? )2 ]?A+ ?A? ii,j
i=0,1 h?A
+ ?A? i
i,j
!
,
(9)
where j = arm or corner. The variance of decisions can then be obtained from Eq. (7). For
illustration, we consider the derivation of the Poisson average h?A+ ?A? i for one-wait
2000
k
2nd agent
?
QQ 1
PQ
Preference
1000
0
QQ r
1st agent
?1000
QQ k
?2000
QP
QR
0
400
800
1200
Time
k
+
Figure 3: (a) Experimental evidence of the waiting effect: a typical example of the evolution of the preference of the 2 agents switching policies at the attractor in a game with
m = 1, N = 127, and R = 224 ?1. The system converges to the attractor at t = 1, 086. (b)
The space of k + and k ? describing the movement of the cumulative payoffs in the game
with m = 1. Thick arrows: non-vanishing steps. Thin arrows: waiting steps. Thick double
arrows: attractor steps. The dashed lines link those steps that share common agents.
arm approach. Noting that the only non-vanishing steps are PQ, QR and QP, we obtain
P?
h?A+ ?A? i1,arm = N42 k=1 h[1 ? ?(ai )?(aturn,1 )?(acum )]?(aturn,1 )
Qk
r=1 ?(await,r )?(aturn,2 )(a? + a0 )(acum + aturn,2 + a0 )iPoi
n ? h ? ?2
i
io
h ? ?2
?
?
?
? 7?
8
?
e
.
(10)
= N42 1? ? e? 2 12 ?
+
4
+
8
8
8
8
1?e
2
We note that the number a0 accounts for the agents who contribute to both steps in the
attractor, and thus can complete the attractor dynamics alone in the extremely diverse limit.
On the other hand, the number acum arises from the ?rst step PQ arriving at Q. Once
present, it will appear in the attractor step QP, irrespective of the duration k of the wait at
Q. These acum agents can wait to complete the attractor dynamics together with the a?
agents who contribute independently to the step from Q to R, as well as the a0 agents who
contribute to both attractor steps. As a result, the average step size increases due to this
waiting effect. In the former case, cooperation between individual types of agents becomes
indispensable in reaching the steady state behavior.
Other Poisson averages in Eq. (9) can be derived similarly. As shown in Fig. 1(b), the waiting effect causes the variance to increase beyond the kinetic sampling prediction, agreeing
with the trend of the simulation results. In particular, the variance approaches 0.34/N in
the extremely diverse limit, signi?cantly greater than the limit of 0.25/N in the absence of
waiting effects. Further approximation including multiple waiting steps results in the theoretical curves with excellent agreement with the simulation results, as shown in Fig. 1(b).
In the extremely diverse limit, the theoretical predictions approach 0.42/N , very close to
the simulation result of 0.43/N .
6
Conclusion
We have studied the dynamical mechanisms of cooperation, which emerges automatically
in a multi-agent system with adaptive agents competing sel?shly for ?nite resources. At
low diversity, agent cooperation proceeds at the statistical level, resulting in the scaling
relation of the variance with diversity. At high diversity, when kinetic sampling becomes
Label
PQ
QQ1
QQr
QQk
QR
QP
Table 1: The number of ?ckle agents in the steps of one wait.
Steps
No. of agents
Poisson averages
?? + ??+ ? ?? ai + aturn,1 + acum hai i = ?/8, haturn,1 i = ?/8,
hacum i = ?/4.
?? ? ?? ? ??? await,1 + aturn,1
hawait,1 i = 3?/8.
?? ? (r ? 1)???
? ?? ? r???
?? ? (k ? 1)???
? ?? ? k???
?? ? k??? ?
?? ? (k + 1)???
?? ? k??? ?
?? ? k??? + ??+
await,r
await,k + aturn,2
a? + a 0
acum + aturn,2 + a0
hawait,r i = ?/2,
(2 ? r ? k ? 1).
hawait,k i = 3?/8,
haturn,2 i = ?/8.
ha? i = 3?/8,
ha0 i = ?/8.
signi?cant, we ?nd that the attractor dynamics favors the cooperation of larger clusters
of agents. In extremely diverse systems, we further discover a waiting mechanism, when
agents who are unable to complete the attractor dynamics alone wait for other agents to
collaborate with them. When waiting is present, cooperation between individual types
of agents becomes indispensable in reaching the steady state behavior. Together, these
mechanisms yield theoretical predictions of the population variance in excellent agreement
with simulations over nine decades of data.
We expect that the observed mechanisms of agent cooperation can be found in reinforcement learning of multi-agent systems in general, due to their generic nature. The mechanisms of statistical cooperation, kinetic sampling and waiting illustrate the importance of
dynamical considerations in describing the system behavior, and the capability of multiagent systems to self-organize in their collective dynamics. In particular, it is interesting to
note that given enough waiting time, agents with limited abilities can cooperate to achieve
dynamics unachievable by individuals. This is relevant to evolutionary approaches to multiagent control, since it allows limited changes to accumulate into bigger improvements.
Acknowledgments
We thank C. H. Yeung, Y. S. Ting and B. H. Wang for fruitful discussions. This
work is supported by the Research Grant Council of Hong Kong (HKUST6153/01P,
HKUST6062/02P) and DAG04/05.SC25.
References
[1] G. Wei? and S. Sen, Adaption and Learning in Multi-agent Systems, Lecture Notes in Computer
Science 1042 (Springer, Berlin, 1995).
[2] E. Rasmusen, Games and Information (Basil Blackwell, Oxford, 2001).
[3] D. Challet and Y. C. Zhang, Emergence of Cooperation and Organization in an Evolutionary
Game, Physica A 246, pp. 407-418 (1997).
[4] R. Savit, R. Manuca, and R. Riolo, Adaptive Competition, Market Ef?ciency, and Phase Transitions, Phys. Rev. Lett. 82, pp. 2203-2206 (1999).
[5] K. Y. M. Wong, S. W. Lim, and P. Luo, Diversity and Adaptation in Large Population Games,
Int. J. Mod. Phys. B 18, 2422-2431 (2004).
[6] K. Y. M. Wong, S. W. Lim, and Z. Gao, Dynamical Mehanisms of Adaptation in Multi-agent
Systems, Phys. Rev. E 70, 025103(R) (2004).
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1,858 | 2,688 | Modelling Uncertainty in the Game of Go
David H. Stern
Department of Physics
Cambridge University
[email protected]
Thore Graepel
Microsoft Research
Cambridge, U.K.
[email protected]
David J. C. MacKay
Department of Physics
Cambridge University
[email protected]
Abstract
Go is an ancient oriental game whose complexity has defeated attempts to automate it. We suggest using probability in a Bayesian
sense to model the uncertainty arising from the vast complexity
of the game tree. We present a simple conditional Markov random field model for predicting the pointwise territory outcome of
a game. The topology of the model reflects the spatial structure of
the Go board. We describe a version of the Swendsen-Wang process for sampling from the model during learning and apply loopy
belief propagation for rapid inference and prediction. The model
is trained on several hundred records of professional games. Our
experimental results indicate that the model successfully learns to
predict territory despite its simplicity.
1
Introduction
The game of Go originated in China over 4000 years ago. Its rules are simple (See
www.gobase.org for an introduction). Two players, Black and White, take turns
to place stones on the intersections of an N ? N grid (usually N = 19 but smaller
boards are in use as well). All the stones of each player are identical. Players place
their stones in order to create territory by occupying or surrounding areas of the
board. The player with the most territory at the end of the game is the winner. A
stone is captured if it has been completely surrounded (in the horizontal and vertical
directions) by stones of the opponent?s colour. Stones in a contiguous ?chain? have
the common fate property: they are captured all together or not at all [1].
The game that emerges from these simple rules has a complexity that defeats attempts to apply minimax search. The best Go programs play only at the level of
weak amateur Go players and Go is therefore considered to be a serious AI challenge
not unlike Chess in the 1960s. There are two main reasons for this state of affairs:
firstly, the high branching factor of Go (typically 200 to 300 potential moves per
position) prevents the expansion of a game tree to any useful depth. Secondly, it
is difficult to produce an evaluation function for Go positions. A Go stone has no
intrinsic value; its value is determined by its relationships with other stones. Go
players evaluate positions using visual pattern recognition and qualitative intuitions
which are difficult to formalise.
Most Go programs rely on a large amount of hand-tailored rules and expert knowl-
edge [2]. Some machine learning techniques have been applied to Go with limited
success. Schraudolph, Dayan and Sejnowski [3] trained a multi-layer perceptron to
evaluate board positions by temporal difference learning. Enzenberger [4] improved
on this by structuring the topologies of his neural networks according to the relationships between stones on the board. Graepel et al. [1] made use of the common
fate property of chains to construct an efficient graph-based representation of the
board. They trained a Support Vector Machine to use this representation to solve
Go problems.
Our starting point is the uncertainty about the future course of the game that
arises from the vast complexity of the game tree. We propose to explicitly model
this uncertainty using probability in a Bayesian sense. The Japanese have a word,
aji, much used by Go players. Taken literally it means ?taste?. Taste lingers, and
likewise the influence of a Go stone lingers (even if it appears weak or dead) because
of the uncertainty of the effect it may have in the future. We use a probabilistic
model that takes the current board position and predicts for every intersection of
the board if it will be Black or White territory. Given such a model the score of
the game can be predicted and hence an evaluation function produced. The model
is a conditional Markov random field [5] which incorporates the spatial structure of
the Go board.
2
Models for Predicting Territory
Consider the Go board as an undirected Graph G = (N , E) with N = Nx ?Ny nodes
n ? N representing vertices on the board and edges e ? E connecting vertically
and horizontally neighbouring points. We can denote a position as the vector c ?
{Black, White, Empty}N for cn = c(n) and similarly the final territory outcome
of the game as s ? {+1, ?1}N for sn = s(n). For convenience we score from
the point of view of Black so elements of s representing Black territory are valued
+1 and elements representing white territory are valued ?1. Go players will note
that we are adopting the Chinese method of scoring empty as well as occupied
intersections. The distribution we wish to model is P (s|c), that is, the distribution
over final territory outcomes given the current position. Such a model would be
useful for several reasons.
? Most importantly, the detailed outcomes provide us with a simple
P evaluation function for Go positions by the expected score, u(c) := h i si iP (s|c) .
An alternative (and probably better) evaluation function is givenPby the
probability of winning which takes the form P (Black wins) = P ( i si >
komi), where komi refers to the winning threshold for Black.
? Connectivity of stones is vital because stones can draw strength from other
stones. Connectivity could be measured by the correlation between nodes
under the distribution P (s|c). This would allow us to segment the board
into ?groups? of stones to reduce complexity.
? It would also be useful to observe cases where we have an anti-correlation
between nodes in the territory prediction. Japanese refer to such cases as
miai in which only one of two desired results can be achieved at the expense
of the other - a consequence of moving in turns.
? The fate of a group of Go stones could be estimated from the distribution
P (s|c) by marginalising out the nodes not involved.
The way stones exert long range influence can be considered recursive. A stone
influences its neighbours, who influence their neighbours and so on. A simple model
which exploits this idea is to consider the Go board itself as an undirected graphical
model in the form of a Conditional Random Field (CRF) [5]. We factorize the
distribution as
?
?
Y
X
1
1
P (s|c) =
?f (sf , cf , ? f ) =
exp ?
log(?f (sf , cf , ? f ))? .
Z(c, ?)
Z(c, ?)
f ?F
f ?F
(1)
The simplest form of this model has one factor for each pair of neighbouring nodes
i, j so ?f (sf , cf , ? f ) = ?f (si , sj , ci , cj , ? f ).
Boltzmann5 For our first model we decompose the factors into ?coupling? terms
and ?external field? terms as follows:
?
?
X
1
P (s|c) =
exp ?
{w(ci , cj )si sj + h(ci )si + h(cj )sj }?
(2)
Z(c, ?)
(i,j)?F
This gives a Boltzmann machine whose connections have the grid topology of the
board. The couplings between territory-outcome nodes depend on the current board
position local to those nodes and the external field at each node is determined
by the state of the board at that location. We assume that Go positions with
their associated territory positions are symmetric with respect to colour reversal so
?f (si , sj , ci , cj , ? f ) = ?f (?si , ?sj , ?ci , ?cj , ? f ). Pairwise connections are also invariant to direction reversal so ?f (si , sj , ci , cj , ? f ) = ?f (sj , si , cj , ci , ? f ). It follows
that the model described in 2 can be specified by just five parameters:
?
?
?
?
?
wchains = w(Black, Black) = w(White, White),
winter?chain = w(Black, White) = w(White, Black),
wchain?empty = w(Empty, White) = w(Empty, Black),
wempty = w(Empty, Empty),
hstones = h(Black) = ?h(White),
and h(empty) is set to zero by symmetry. We will refer to this model as Boltzmann5.
This simple model is interesting because all these parameters are readily interpreted.
For example we would expect wchains to take on a large positive value since chains
have common fate.
BoltzmannLiberties A feature that has particular utility for evaluating Go positions is the number of liberties associated with a chain of stones. A liberty of a
chain is an empty vertex adjacent to it. The number of liberties indicates a chain?s
safety because the opponent would have to occupy all the liberties to capture the
chain. Our second model takes this information into account:
?
?
X
1
P (s|c) =
exp ?
w(ci , cj , si , sj , li , lj )? ,
(3)
Z(c, ?)
(i,j)?F
where li is element i of a vector l ? {+1, +2, +3, 4 or more}N the liberty count of
each vertex on the Go board. A group with four or more liberties is considered
relatively safe. Again we can apply symmetry arguments and end up with 78
parameters. We will refer to this model as BoltzmannLiberties.
We trained the two models using board positions from a database of 22,000 games
between expert Go players1 . The territory outcomes of a subset of these games
1
The GoGoD database, April 2003. URL:http://www.gogod.demon.co.uk
(a) Gibbs Sampling
(b) Swendsen Wang
Figure 1: Comparing ordinary Gibbs with Swendsen Wang sampling for Boltzmann5. Shown are the differences between the running averages and the exact
marginals for each of the 361 nodes plotted as a function of the number of wholeboard samples.
were determined using the Go program GnuGo2 to analyse their final positions.
Each training example comprised a board position c, with its associated territory
outcome s. Training was performed by maximising the likelihood ln P (s 0 |c) using
gradient descent. In order to calculate the likelihood it is necessary to perform
inference to obtain the marginal expectations of the potentials.
3
Inference Methods
It is possible to perform exact inference on the model by variable elimination [6].
Eliminating nodes one diagonal at a time gave an efficient computation. The cost
of exact inference was still too high for general use but it was used to compare other
inference methods.
Sampling The standard method for sampling from a Boltzmann machine is to use
Gibbs sampling where each node is updated one at a time, conditional on the others.
However, Gibbs sampling mixes slowly for spin systems with strong correlations.
A generalisation of the Swendsen-Wang process [7] alleviates this problem. The
original Swendsen-Wang algorithm samples from a ferromagnetic Ising model with
no external field by adding an additional set of ?bond? nodes d, one attached to
each factor (edge) in the original graph. Each of these nodes can either be in the
state ?bond? or ?no bond?. The new factor potentials ?f (sf , cf , df , ? f ) are chosen
such that if a bond exists between a pair of spins then they are forced to be in
the same state. Conditional on the bonds, each cluster has an equal probability
of having all its spins in the ?up? state or all in the ?down? state. The algorithm
samples from P (s|d, c, ?) and P (d|s, c, ?) in turn (flipping clusters and forming
bonds respectively). It can be generalised to models with arbitrary couplings and
biases [7, 8]. The new factor potentials ?f (sf , cf , df , ? f ) have the following effect:
if the coupling is positive then when the d node is in the ?bond? state it forces the
two spins to be in the same state; if the coupling is negative the ?bond? state forces
the two spins to be opposite. The probability of each cluster being in each state
depends on the sum of the biases involved. Figure 1 shows that the mixing rate
of the sampling process is improved by using Swendsen-Wang allowing us to find
accurate marginals for a single position in a couple of seconds.
2
URL:http://www.gnu.org/software/gnugo/gnugo.html
Loopy Belief Propagation In order to perform very rapid (approximate) inference we used the loopy belief propagation (BP) algorithm [9] and the results are
examined in Section 4. This algorithm is similar to an influence function [10], as
often used by Go programmers to segment the board into Black and White territory
and for this reason is laid out below.
For each board vertex j ? N , create a data structure called a node containing:
1. A(j), the set of nodes corresponding to the neighbours of vertex j,
2. a set of new messages mnew
ij (sj ) ? Mnew , one for each i ? A(j),
3. a set of old messages mold
ij (sj ) ? Mold , one for each i ? A(j),
4. a belief bj (sj ).
repeat
for all j ? N do
for all i ? A(j) do
for all sj ? {Black, White} do
let variable SUM := 0,
for all si ? {Black, White} do
SUM := SUM + ?(i,j) (si , sj )
Q
mold
qi (si ),
q?A(i)\j
end for
mnew
ij (sj ) := SUM,
end for
end for
end for
for all messages, mnew
xy (sy ) ? Mnew do
new
old
mxy (sy ) := ?mxy (sy ) + (1 ? ?)mnew
xy (sy ),
end for
until completed I iterations (typically I=50)
Belief Update:
for all j ? N do
for all sj ? {Black,
White} do
Q
bj (sj ) :=
mnew
qj (sj )
q?A(j)
end for
end for
Here, ? (typically 0.5), damps any oscillations. ?(i,j) (si , sj ) is the factor potential (see (1)) and in the case of Boltzmann5 takes on the form ?(i,j) (si , sj ) =
exp (w(ci , cj )si sj + h(ci )si + h(cj )sj ). Now the probability of each vertex being
Black or White territory is found by normalising the beliefs at each node. For
example P (sj = Black) = bj (Black)/Z where Z = bj (Black) + bj (White). The
accuracy of the loopy BP approximation appears to be improved by using it during
the parameter learning stage in cases where it is to be used in evaluation.
4
Results for Territory Prediction
Some Learnt Parameters Here are some parameters learnt for the Boltzmann5
model (2). This model was trained on 290 positions from expert Go games at move
80. Training was performed by maximum likelihood as described in Section 2.
(a) Boltzmann5 (Exact)
(b) Boltzmann5 (Loopy BP)
Figure 2: Comparing territory predictions for a Go position from a professional
game at move 90. The circles represent stones. The small black and white squares
at each vertex represent the average territory prediction at that vertex, from ?1
(maximum white square) to +1 (maximum black square).
? hstones = 0.265
? wempty = 0.427
? wchain?empty = 0.442
? wchains = 2.74
? winter?chain = 0.521
The values of these parameters can be interpreted. For example wchains corresponds
to the correlation between the likely territory
outcome of two adjacent vertices in a chain of
connected stones. The high value of this parameter derives from the ?common fate? property of chains as described in Section 1.
Interestingly, the value of the parameter wempty (corresponding to the coupling
between territory predictions of neighbouring vertices in empty space) is 0.427 which
is close to the critical coupling for an Ising model, 0.441.
Territory Predictions Figure 2 gives examples of territory predictions generated
by Boltzmann5. In comparison, Figure 3 shows the prediction of BoltzmannLiberties
and a territory prediction from The Many Faces of Go [2]. Go players confirm that
the territory predictions produced by the models are reasonable, even around loose
groups of Black and White stones. Compare Figures 2 (a) and 3 (a); when liberty
counts are included as features, the model can more confidently identify which of the
two small chains competing in the bottom right of the board is dead. Comparing
Figure 2 (a) and (b) Loopy BP appears to give over-confident predictions in the top
right of the board where few stones are present. However, it is a good approximation
where many stones are present (bottom left).
Comparing Models and Inference Methods Figure 4 shows cross-entropies
between model territory predictions and true final territory outcomes for a dataset
of expert games. As we progress through a game, predictions become more accurate
(not surprising) but the spread of the accuracy increases, possibly due to incorrect
assessment of the life-and-death status of groups. Swendsen-Wang performs better
than Loopy BP, which may suffer from its over-confidence. BoltzmannLiberties
performs better than Boltzmann5 (when using Swendsen-Wang) the difference in
(a) BoltzmannLiberties (Exact)
(b) Many Faces of Go
Figure 3: Diagram (a) is produced by exact inference (training was also by Loopy
BP). Diagram (b) shows the territory predicted by The Many Faces of Go (MFG)
[2]. MFG uses of a rule-based expert system and its prediction for each vertex has
three possible values: ?White?, ?Black? or ?unknown/neutral?.
performance increasing later in the game when liberty counts become more useful.
5
Modelling Move Selection
In order to produce a Go playing program we are interested in modelling the selection of moves. A measure of performance of such a model is the likelihood it assigns
to professional moves as measured by
X X
log P (move|model).
(4)
games moves
We can obtain a probability over moves by choosing a Gibbs distribution with the
negative energy replaced by the evaluation function,
0
P (move|model, w) =
e?u(c ,w)
Z(w)
(5)
where u(c0 , w) is an evaluation function evaluated at the board position c0 resulting
from a given move. The inverse temperature parameter ? determines the degree to
which the move made depends on its evaluation. The territory predictions from the
models Boltzmann5 and BoltzmannLiberties can be combined with the evaluation
function of Section 2 to produce position evaluators.
6
Conclusions
We have presented a probabilistic framework for modelling uncertainty in the game
of Go. A simple model which incorporates the spatial structure of a board position
can perform well at predicting the territory outcomes of Go games. The models
described here could be improved by extracting more features from board positions
and increasing the size of the factors (see (1)).
1.0
Cross Entropy
0.0
0.5
1.0
0.5
0.0
Cross Entropy
1.5
Loopy BP
1.5
Swendsen?Wang
B5
BLib
Move 20
B5
BLib
Move 80
B5
BLib
Move 150
B5
BLib
Move 20
B5
BLib
Move 80
B5
BLib
Move 150
PN
Figure 4: Cross entropies N1 n [s0n log sn + (1 ? s0n ) log(1 ? sn )] between actual and
predicted territory outcomes, s0n and n for 327 Go positions. Sampling is compared
with Loopy BP (training and testing). 3 board positions were analysed for each
game (moves 20, 80 and 150). The Boltzmann5 (B5) and the BoltzmannLiberties
(BLib) models are compared.
Acknowledgements We thank I. Murray for helpful discussions on sampling and
T. Minka for general advice about probabilistic inference. This work was supported
by a grant from Microsoft Research UK.
References
[1] Thore Graepel, Mike Goutrie, Marco Kruger, and Ralf Herbrich. Learning on graphs
in the game of Go. In Proceedings of the International Conference on Artificial Neural
Networks, ICANN 2001, 2001.
[2] David Fotland.
Knowledge representation in the many faces of go.
ftp://www.joy.ne.jp/welcome/igs/Go/computer/mfg.tex.Z, 1993.
URL:
[3] Nicol N. Schrauldolph, Peter Dayan, and Terrance J. Sejnowski. Temporal difference
learning of position evaluation in the game of go. In Advances in Neural Information
Processing Systems 6, pages 817?824, San Fransisco, 1994. Morgan Kaufmann.
[4] Markus Enzenberger. The integration of a priori knowledge into a Go playing neural
network. URL: http://www.markus-enzenberger.de/neurogo.html, 1996.
[5] John Lafferty, Andrew McCallum, and Fernando Pereira. Conditional random fields:
Probabilistic models for segmenting and labeling sequence data. In Proc. Int. Conf.
on Machine Learning, 2001.
[6] Fabio Gagliardi Cozman. Generalizing variable elimination in Bayesian networks. In
Proceedings of the IBERAMIA/SBIA 2000 Workshops, pages 27?32, 2000.
[7] R. H. Swendsen and J-S Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Physical Review Letters, 58:86?88, 1987.
[8] Robert G. Edwards and Alan D. Sokal. Generalisation of the Fortuin-KasteleynSwendsen-Wang representation and Monte Carlo algorithm. Physical Review Letters,
38(6), 1988.
[9] Yair Weiss. Belief propagation and revision in networks with loops. Technical report,
AI Lab Memo, MIT, Cambridge, 1998.
[10] A. L. Zobrist. Feature Extractions and Representations for Pattern Recognition and
the Game of Go. PhD thesis, Graduate School of the University of Wisconsin, 1970.
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1,859 | 2,689 | Modeling Nonlinear Dependencies in
Natural Images using Mixture of
Laplacian Distribution
Hyun Jin Park and Te Won Lee
Institute for Neural Computation, UCSD
9500 Gilman Drive, La Jolla, CA 92093-0523
{hjinpark, tewon}@ucsd.edu
Abstract
Capturing dependencies in images in an unsupervised manner is
important for many image processing applications. We propose a
new method for capturing nonlinear dependencies in images of
natural scenes. This method is an extension of the linear Independent
Component Analysis (ICA) method by building a hierarchical model
based on ICA and mixture of Laplacian distribution. The model
parameters are learned via an EM algorithm and it can accurately
capture variance correlation and other high order structures in a
simple manner. We visualize the learned variance structure and
demonstrate applications to image segmentation and denoising.
1
In trod u ction
Unsupervised learning has become an important tool for understanding biological
information processing and building intelligent signal processing methods. Real
biological systems however are much more robust and flexible than current artificial
intelligence mostly due to a much more efficient representations used in biological
systems. Therefore, unsupervised learning algorithms that capture more sophisticated
representations can provide a better understanding of neural information processing
and also provide improved algorithm for signal processing applications. For example,
independent component analysis (ICA) can learn representations similar to simple cell
receptive fields in visual cortex [1] and is also applied for feature extraction, image
segmentation and denoising [2,3]. ICA can approximate statistics of natural image
patches by Eq.(1,2), where X is the data and u is a source signal whose distribution is
a product of sparse distributions like a generalized Laplacian distribution.
X = Au
(1)
P (u ) = ? P (u i )
(2)
But the representation learned by the ICA algorithm is relatively low-level. In
biological systems there are more high-level representations such as contours,
textures and objects, which are not well represented by the linear ICA model. ICA
learns only linear dependency between pixels by finding strongly correlated linear
axis. Therefore, the modeling capability of ICA is quite limited. Previous approaches
showed that one can learn more sophisticated high-level representations by capturing
nonlinear dependencies in a post-processing step after the ICA step [4,5,6,7,8].
The focus of these efforts has centered on variance correlation in natural images. After
ICA, a source signal is not linearly predictable from others. However, given variance
dependencies, a source signal is still ?predictable? in a nonlinear manner. It is not
possible to de-correlate this variance dependency using a linear transformation.
Several researchers have proposed extensions to capture the nonlinear dependencies.
Portilla et al. used Gaussian Scale Mixture (GSM) to model variance dependency in
wavelet domain. This model can learn variance correlation in source prior and showed
improvement in image denoising [4]. But in this model, dependency is defined only
between a subset of wavelet coefficients. Hyvarinen and Hoyer suggested using a
special variance related distribution to model the variance correlated source prior.
This model can learn grouping of dependent sources (Subspace ICA) or topographic
arrangements of correlated sources (Topographic ICA) [5,6]. Similarly, Welling et al.
suggested a product of expert model where each expert represents a variance
correlated group [7]. The product form of the model enables applications to image
denoising. But these models don?t reveal higher-order structures explicitly.
Our model is motivated by Lewicki and Karklin who proposed a 2-stage model where
the 1st stage is an ICA model (Eq. (3)) and the 2 nd-stage is a linear generative model
where another source v generates logarithmic variance for the 1st stage (Eq. (4)) [8].
This model captures variance dependency structure explicitly, but treating variance as
an additional random variable introduces another level of complexity and requires
several approximations. Thus, it is difficult to obtain a simple analytic PDF of source
signal u and to apply the model for image processing problems.
(
P (u | ? ) = c exp ? u / ?
q
)
(3)
log[? ] = Bv
(4)
We propose a hierarchical model based on ICA and a mixture of Laplacian
distribution. Our model can be considered as a simplification of model in [8] by
constraining v to be 0/1 random vector where only one element can be 1. Our model is
computationally simpler but still can capture variance dependency. Experiments show
that our model can reveal higher order structures similar to [8]. In addition, our model
provides a simple parametric PDF of variance correlated priors, which is an important
advantage for adaptive signal processing. Utilizing this, we demonstrate simple
applications on image segmentation and image denoising. Our model provides an
improved statistic model for natural images and can be used for other applications
including feature extraction, image coding, or learning even higher order structures.
2
Modeling nonlinear dependencies
We propose a hierarchical or 2-stage model where the 1 st stage is an ICA source signal
model and the 2nd stage is modeled by a mixture model with different variances (figure
1). In natural images, the correlation of variance reflects different types of regularities
in the real world. Such specialized regularities can be summarized as ?context?
information. To model the context dependent variance correlation, we use mixture
models where Laplacian distributions with different variance represent different
contexts. For each image patch, a context variable Z ?selects? which Laplacian
distribution will represent ICA source signal u. Laplacian distributions have 0-mean
but different variances. The advantage of Laplacian distribution for modeling context
is that we can model a sparse distribution using only one Laplacian distribution. But
we need more than two Gaussian distributions to do the same thing. Also conventional
ICA is a special case of our model with one Laplacian. We define the mixture model
and its learning algorithm in the next sections.
Figure 1: Proposed hierarchical model (1st stage is ICA generative model. 2nd stage is
mixture of ?context dependent? Laplacian distributions which model U. Z is a random
variable that selects a Laplacian distribution that generates the given image patch)
2.1
Mixture of Laplacian Distribution
We define a PDF for mixture of M-dimensional Laplacian Distribution as Eq.(5),
where N is the number of data samples, and K is the number of mixtures.
N
N K
M
N K
r
r r
P(U | ?, ?) = ? P(u n | ?, ?) = ?? ? k P(u n | ?k ) = ?? ? k ?
n
n
k
n
k
m
1
(2? )
k ,m
? u n,m
exp? ?
? ?k , m
?
?
?
?
?
(5)
r
r r
r r
un = (un,1 , un , 2 , , , un,M ) : n-th data sample, U = (u1 , u 2 , , , ui , , , u N )
r
r r
r
r
?k = (?k ,1 , ?k , 2 ,..., ?k ,M ) : Variance of k-th Laplacian distribution, ? = (?1 , ?2 , , , ?k , , , ?K )
?k
: probability of Laplacian distribution k, ? = (? 1 , , , ? K ) and
?
k
?k =1
It is not easy to maximize Eq.(5) directly, and we use EM (expectation maximization)
algorithm for parameter estimation. Here we introduce a new hidden context variable
Z that represents which Laplacian k, is responsible for a given data point. Assuming
we know the hidden variable Z, we can write the likelihood of data and Z as Eq.(6),
zkn
?K ?
??
N
?
u n ,m
r
? ? 1 ??
z kn
?
?
P(U , Z | ?, ? ) = ? P(u n , Z | ?, ? ) = ? ? (? k ) ? ? ?
? exp? ? z kn
?
?
?
?
?k , m
n
n
k
m ? ? 2?k ,m ?
?
?? ??
?
N
? ??? ??
? ?
? ??? ?
? ??? ??
(6)
z kn : Hidden binary random variable, 1 if n-th data sample is generated from k-th
Laplacian, 0 other wise. ( Z = (z kn ) and ? z kn = 1 for all n = 1?N)
k
2.2
EM algorithm for learning the mixture model
The EM algorithm maximizes the log likelihood of data averaged over hidden variable
Z. The log likelihood and its expectation can be computed as in Eq.(7,8).
?
?
u
1
log P(U , Z | ?, ? ) = ? ? z kn log(? k ) + ? z kn ? log(
) ? n ,m
?
2
?
?
n ,k ?
m
k ,m
k ,m
?
?
??
??
??
??
(7)
?
?
u
1
E {log P (U , Z | ?, ? )} = ? E z kn ?log(? k ) + ? ? log(
) ? n ,m
?
2? k , m
?k , m
n ,k
m ?
??
{ }
??
??
??
??
(8)
The expectation in Eq.(8) can be evaluated, if we are given the data U and estimated
parameters ? and ?. For ? and ?, EM algorithm uses current estimation ?? and ??.
{ } {
} ? z P( z
E z kn ? E z kn | U , ? ' , ? ' =
1
zkn =0
n
k
n
k
| u n , ?' , ? ' ) = P ( z kn = 1 | u n , ? ' , ? ' )
(9)
=
P (u n | z kn = 1, ?' , ? ' ) P( z kn = 1 | ? ' , ? ' )
P(u n | ?' , ? ' )
=
M
u n ,m
1
1
1
exp(?
) ?? k ' =
?
?k , m '
P (u n | ? ' , ? ' ) m 2?k ,m '
cn
M
?k '
? 2?
m
k ,m
'
exp(?
u n ,m
?k , m '
)
Where the normalization constant can be computed as
K
K
M
k
k =1
m =1
cn = P (u n | ? ' , ? ' ) = ? P (u n | z kn , ? ' , ? ' ) P ( z kn | ? ' , ? ' ) = ? ? k ?
1
(2? )
exp( ?
k ,m
u n ,m
?k ,m
)
(10)
The EM algorithm works by maximizing Eq.(8), given the expectation computed from
Eq.(9,10). Eq.(9,10) can be computed using ?? and ?? estimated in the previous
iteration of EM algorithm. This is E-step of EM algorithm. Then in M-step of EM
algorithm, we need to maximize Eq.(8) over parameter ? and ?.
First, we can maximize Eq.(8) with respect to ?, by setting the derivative as 0.
?? 1
u n,m ??
?E{log P (U , Z | ?, ? )}
?? = 0
= ? E z kn ?? ?
+
?
?? k ,m
?
(? k ,m ) 2 ???
???
n
k ,m
?
{ }
? ? k ,m
? E{z }? u
=
? E{z }
n
k
n ,m
n
(11)
n
k
n
Second, for maximization of Eq.(8) with respect to ?, we can rewrite Eq.(8) as below.
(12)
E {log P (U , Z | ? , ? )} = C + ? E {z kn' }log(? k ' )
n ,k '
As we see, the derivative of Eq.(12) with respect to ? cannot be 0. Instead, we need to
use Lagrange multiplier method for maximization. A Lagrange function can be
defined as Eq.(14) where ? is a Lagrange multiplier.
{ }
(13)
L (? , ? ) = ? ? E z kn' log(? k ' ) + ? (? ? k ' ? 1)
n,k '
k'
By setting the derivative of Eq.(13) to be 0 with respect to ? and ?, we can simply get
the maximization solution with respect to ?. We just show the solution in Eq.(14).
?L(?, ? )
?L(?, ? )
=0
= 0,
??
??
?
? ?
?
? ? k = ? ? E z kn ? / ? ?? E z kn ?
?
? ? k n
? n
{ }
{ }
(14)
Then the EM algorithm can be summarized as figure 2. For the convergence criteria,
we can use the expectation of log likelihood, which can be calculated from Eq. (8).
?k =
{ }
, ?k , m = E um + e (e is small random noise)
2. Calculate the Expectation by
1. Initialize
1
K
u n ,m
1 M ?k '
exp( ?
)
?
c n m 2? k , m '
? k ,m '
3. Maximize the log likelihood given the Expectation
{ } {
}
E z kn ? E z kn | U , ? ' , ? ' =
?
? ?
?
?k ,m ? ? ? E {z kn }? u n,m ? / ? ? E {z kn }? ,
?
? ?
?
? k ? ? ? E {z kn }? / ? ?? E {z kn }?
? ? n
?
? n
? ? k n
?
4. If (converged) stop, otherwise repeat from step 2.
?
n
Figure 2: Outline of EM algorithm for Learning the Mixture Model
3
Experimental Results
Here we provide examples of image data and show how the learning procedure is
performed for the mixture model. We also provide visualization of learned variances
that reveal the structure of variance correlation and an application to image denoising.
3.1
Learning Nonlinear Dependencies in Natural images
As shown in figure 1, the 1 st stage of the proposed model is simply the linear ICA. The
ICA matrix A and W(=A-1) are learned by the FastICA algorithm [9]. We sampled
105(=N) data from 16x16 patches (256 dim.) of natural images and use them for both
first and second stage learning. ICA input dimension is 256, and source dimension is
set to be 160(=M). The learned ICA basis is partially shown in figure 1. The 2nd stage
mixture model is learned given the ICA source signals. In the 2 nd stage the number of
mixtures is set to 16, 64, or 256(=K). Training by the EM algorithm is fast and several
hundred iterations are sufficient for convergence (0.5 hour on a 1.7GHz Pentium PC).
For the visualization of learned variance, we adapted the visualization method from
[8]. Each dimension of ICA source signal corresponds to an ICA basis (columns of A)
and each ICA basis is localized in both image and frequency space. Then for each
Laplacian distribution, we can display its variance vector as a set of points in image
and frequency space. Each point can be color coded by variance value as figure 3.
(a1)
(a2)
(b1)
(b2)
Figure 3: Visualization of learned variances (a1 and a2 visualize variance of
Laplacian #4 and b1 and 2 show that of Laplacian #5. High variance value is mapped
to red color and low variance is mapped to blue. In Laplacian #4, variances for
diagonally oriented edges are high. But in Laplacian #5, variances for edges at
spatially right position are high. Variance structures are related to ?contexts? in the
image. For example, Laplacian #4 explains image patches that have oriented textures
or edges. Laplacian #5 captures patches where left side of the patch is clean but right
side is filled with randomly oriented edges.)
A key idea of our model is that we can mix up independent distributions to get nonlinearly dependent distribution. This modeling power can be shown by figure 4.
Figure 4: Joint distribution of nonlinearly dependent sources. ((a) is a joint histogram
of 2 ICA sources, (b) is computed from learned mixture model, and (c) is from learned
Laplacian model. In (a), variance of u2 is smaller than u1 at center area (arrow A), but
almost equal to u1 at outside (arrow B). So the variance of u2 is dependent on u1. This
nonlinear dependency is closely approximated by mixture model in (b), but not in (c).)
3.2
Unsupervised Image Segmentation
The idea behind our model is that the image can be modeled as mixture of different
variance correlated ?contexts?. We show how the learned model can be used to
classify different context by an unsupervised image segmentation task. Given learned
model and data, we can compute the expectation of a hidden variable Z from Eq. (9).
Then for an image patch, we can select a Laplacian distribution with highest
probability, which is the most explaining Laplacian or ?context?. For segmentation,
we use the model with 16 Laplacians. This enables abstract partitioning of images and
we can visualize organization of images more clearly (figure 5).
Figure 5: Unsupervised image segmentation (left is original image, middle is color
labeled image, right image shows color coded Laplacians with variance structure.
Each color corresponds to a Laplacian distribution, which represents surface or
textural organization of underlying contexts. Laplacian #14 captures smooth surface
and Laplacian #9 captures contrast between clear sky and textured ground scenes.)
3.3
Application to Image Restoration
The proposed mixture model provides a better parametric model of the ICA source
distribution and hence an improved model of the image structure. An advantage is in
the MAP (maximum a posterior) estimation of a noisy image. If we assume Gaussian
noise n, the image generation model can be written as Eq.(15). Then, we can compute
MAP estimation of ICA source signal u by Eq.(16) and reconstruct the original image.
(15)
X = Au + n
(16)
u? = argmax log P (u | X , A) = argmax (log P ( X | u , A) + log P (u ) )
u
u
Since we assumed Gaussian noise, P(X|u,A) in Eq. (16) is Gaussian. P(u) in Eq. (16)
can be modeled as a Laplacian or a mixture of Laplacian distribution. The mixture
distribution can be approximated by a maximum explaining Laplacian. We evaluated
3 different methods for image restoration including ICA MAP estimation with simple
Laplacian prior, same with Laplacian mixture prior, and the Wiener filter. Figure 6
shows an example and figure 7 summarizes the results obtained with different noise
levels. As shown MAP estimation with the mixture prior performs better than the
others in terms of SNR and SSIM (Structural Similarity Measure) [10].
Figure 6: Image restoration results (signal variance 1.0, noise variance 0.81)
16
ICA MAP (Mixture prior)
ICA MAP (Laplacian prior)
W iener
14
0.8
SSIM Index
SNR
12
10
8
6
0.6
0.4
0.2
4
2
ICA MAP(Mixture prior)
ICA MAP(Laplacian prior)
W iener
Noisy Image
1
0
0.5
1
1.5
Noise variance
2
2.5
0
0
0.5
1
1.5
Noise variance
2
2.5
Figure 7: SNR and SSIM for 3 different algorithms (signal variance = 1.0)
4
D i s c u s s i on
We proposed a mixture model to learn nonlinear dependencies of ICA source signals
for natural images. The proposed mixture of Laplacian distribution model is a
generalization of the conventional independent source priors and can model variance
dependency given natural image signals. Experiments show that the proposed model
can learn the variance correlated signals grouped as different mixtures and learn highlevel structures, which are highly correlated with the underlying physical properties
captured in the image. Our model provides an analytic prior of nearly independent and
variance-correlated signals, which was not viable in previous models [4,5,6,7,8].
The learned variances of the mixture model show structured localization in image and
frequency space, which are similar to the result in [8]. Since the model is given no
information about the spatial location or frequency of the source signals, we can
assume that the dependency captured by the mixture model reveals regularity in the
natural images. As shown in image labeling experiments, such regularities correspond
to specific surface types (textures) or boundaries between surfaces.
The learned mixture model can be used to discover hidden contexts that generated
such regularity or correlated signal groups. Experiments also show that the labeling of
image patches is highly correlated with the object surface types shown in the image.
The segmentation results show regularity across image space and strong correlation
with high-level concepts.
Finally, we showed applications of the model for image restoration. We compare the
performance with the conventional ICA MAP estimation and Wiener filter. Our
results suggest that the proposed model outperforms other traditional methods. It is
due to the estimation of the correlated variance structure, which provides an improved
prior that has not been considered in other methods.
In our future work, we plan to exploit the regularity of the image segmentation result
to lean more high-level structures by building additional hierarchies on the current
model. Furthermore, the application to image coding seems promising.
References
[1] A. J. Bell and T. J. Sejnowski, The ?Independent Components? of Natural Scenes are Edge
Filters, Vision Research, 37(23):3327?3338, 1997.
[2] A. Hyvarinen, Sparse Code Shrinkage: Denoising of Nongaussian Data by Maximum
Likelihood Estimation,Neural Computation, 11(7):1739-1768, 1999.
[3] T. Lee, M. Lewicki, and T. Sejnowski., ICA Mixture Models for unsupervised
Classification of non-gaussian classes and automatic context switching in blind separation.
PAMI, 22(10), October 2000.
[4] J. Portilla, V. Strela, M. J. Wainwright and E. P Simoncelli, Image Denoising using Scale
Mixtures of Gaussians in the Wavelet Domain, IEEE Trans. On Image Processing, Vol.12, No.
11, 1338-1351, 2003.
[5] A. Hyvarinen, P. O. Hoyer. Emergence of phase and shift invariant features by
decomposition of natural images into independent feature subspaces. Neurocomputing, 1999.
[6] A. Hyvarinen, P.O. Hoyer, Topographic Independent component analysis as a model of V1
Receptive Fields, Neurocomputing, Vol. 38-40, June 2001.
[7] M. Welling and G. E. Hinton, S. Osindero, Learning Sparse Topographic Representations
with Products of Student-t Distributions, NIPS, 2002.
[8] M. S. Lewicki and Y. Karklin, Learning higher-order structures in natural images, Network:
Comput. Neural Syst. 14 (August 2003) 483-499.
[9] A.Hyvarinen, P.O. Hoyer, Fast ICA matlab code.,
http://www.cis.hut.fi/projects/compneuro/extensions.html/
[10] Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, The SSIM Index for Image
Quality Assessment, IEEE Transactions on Image Processing, vol. 13, no. 4, Apr. 2004.
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1,860 | 269 | Predicting Weather Using a Genetic Memory
Predicting Weather Using a Genetic Memory: a Combination of Kanerva's Sparse Distributed Memory with
Holland's Genetic Algorithms
David Rogers
Research Institute for Advanced Computer Science
MS 230-5, NASA Ames Research Center
Moffett Field, CA 94035
ABSTRACT
Kanerva's sparse distributed memory (SDM) is an associative-memory model based on the mathematical properties of high-dimensional
binary address spaces. Holland's genetic algorithms are a search technique for high-dimensional spaces inspired by evolutionary processes
of DNA. "Genetic Memory" is a hybrid of the above two systems,
in which the memory uses a genetic algorithm to dynamically reconfigure its physical storage locations to reflect correlations between
the stored addresses and data. For example, when presented with
raw weather station data, the Genetic Memory discovers specific features in the weather data which correlate well with upcoming rain,
and reconfigures the memory to utilize this information effectively.
This architecture is designed to maximize the ability of the system
to scale-up to handle real-world problems.
INTRODUCTION
The future success of neural networks depends on an ability to "scale-up" from
small networks and low-dimensional toy problems to networks of thousands or millions of nodes and high-dimensional real-world problems. (The dimensionality of a
problem refers to the number of variables needed to describe the problem domain.)
Unless neural networks are shown to be scalable to real-world problems, they will
likely remain restricted to a few specialized applications.
Scaling-up adds two types of computational demands to a system. First, there is a
linear increase in computational demand proportional to the increased number of variables. Second, there is a greater, nonlinear increase in computational demand due to
455
456
Rogers
the number of interactions that can occur between the variables. This latter effect is
primarily responsible for the difficulties encountered in scaling-up many systems.
In general, it is difficult to scale-up a system unless it is specifically designed to
function well in high-dimensional domains.
Two systems designed to function well in high-dimensional domains are Kanerva' s
sparse distributed memory (Kanerva, 1988) and Holland's genetic algorithms
(Holland, 1986). I hypothesized that a hybrid of these two systems would preserve
this ability to operate well in high-dimensional environments, and offer grater functionality than either individually. I call this hybrid Genetic Memory. To test its
capabilities, I applied it to the problem of forecasting rain from local weather data.
Kanerva's sparse distributed memory (SDM) is an associative-memory model based
on the mathematical properties of high-dimensional binary address spaces. It can be
represented as a three-layer neural-network with an extremely large nwnber of
nodes (I ,000,000+) in the middle layer. In its standard formulation, the connections between the input layer and the hidden layer (the input representation used by
the system) are flXed, and learning is done by changing the values of the connections
between the hidden layer and the output layer.
Holland's genetic algorithms are a search technique for high-dimensional spaces inspired by evolutionary processes of DNA. Members of a set of binary strings competes for the opportunity to recombine. Recombination is done by selecting two
"successful" members of the population to be the parents. A new string is created
by splicing together pieces of each parent. Finally, the new string is placed into the
set, and some "unsuccessful" older string removed.
"Genetic Memory" is a hybrid of the above two systems. In this hybrid, a genetic algorithm is used to reconfigure the connections between the input layer and the hidden layer. The connections between the hidden layer and the output layer are
changed using the standard method for a sparse distributed memory. The "success"
of an input representation is determined by how well it reflects correlations between addresses and data, using my previously presented work on statistical prediction (Rogers, 1988). Thus, we have two separate learning algorithms in the two levels. The memory uses the genetic algorithm to dynamically reconfigure its input
representation to better reflect correlations between collections of input variables
and the stored data.
I applied this Genetic Memory architecture to the problem of predicting rain given
only local weather features such as the air pressure, the cloud cover, the month, the
temperature, etc. The weather data contained 15 features, sampled every 4-hours
over a 2O-year period on the Australian coast. I coded each state into a 256-bit address, and stored at that address a single bit which denoted whether it rained in the 4
hours following that weather state. I allowed the genetic algorithm to reconfigure
the memory while it scanned the file of weather states.
The success of this procedure was measured in two ways. First, once the training
was completed, the Genetic Memory was better at predicting rain than was the standard sparse distributed memory. Second, I had access to the input representations
discovered by the Genetic Memory and could view the specific combinations of features that predicted rain. Thus, unlike many neural networks, the Genetic Memory
allows the user to inspect the internal representations it discovers during training.
Predicting Weather Using a Genetic Memory
Reference Address
01010101101
~
~
Input Data
Radius
o
I 0 I 01
Dist
Select
1010101010
1011101100
0010101111
1101101101
0100000110
0110101001
1011010110
1100010111
""
? _;1
1101100111
0000011110
0011011001
I tI 0 11 I
+ + + ++ ,
0111 11 11 0
?
?
-1 -1
-1
1
1
II
1 -1
1 -1 1
?
..._--
?
O?
?
0 ?
?
?
0 ?
1?
-1 -1
-1
1
1
1 -1
1 -1
o
-2 0
2
2
0
0 0
-1
-1 -1
1 1
1
-1
1
0 0
1 -1 1
1 1 1 1 1 100 1 1
Location
Addresses
Data
Counters
, , , , + ++ ++
'L
I -31 31 -31 3 I
Threshold at 0 + + + + + + + + + +
Output Data I 0 I 0 I 0 I tI tI tI 0 I 1 I 0 11 I
Sums 1-31
-s 1-31 sis 13
Figure 1: Structure of a Sparse Distributed Memory
SPARSE DISTRmUTED MEMORY
Sparse distributed memory can be best illustrated as a variant of random-access memory (RAM). The structure of a twelve-location SDM with ten-bit addresses and
ten-bit data is shown in figure 1.
A memory location is a row in this figure. The location addresses are set to random
addresses. The data counters are initialized to zero. All operations begin with addressing the memory; this entails fmding the Hamming distance between the reference address and each of the location addresses. If this distance is less than or equal
to the Hamming radius. the select-vector entry is set to 1. and that location is
termed selected. The ensemble of such selected locations is called the selected set.
Selection is noted in the figure as non-gray rows. A radius is chosen so that only a
small percentage of the memory locations are selected for a given reference address.
When writing to the memory. all selected counters beneath elements of the input data equal to 1 are incremented. and all selected counters beneath elements of the input
data equal to 0 are decremented. This completes a write operation. When reading
from the memory. the selected data counters are summed columnwise into the register swns. If the value of a sum is greater than or equal to zero. we set the corresponding bit in the output data to 1; otherwise. we set the bit in the output data to
O. (When reading. the contents of the input data are ignored.)
457
458
Rogers
This example makes clear that a datum is distributed over the data counters of the selected locations when writing. and that the datum is reconstructed during reading by
averaging the sums of these counters. However, depending on what additional data
were written into some of the selected locations, and depending on how these data
correlate with the original data. the reconstruction may contain noise.
The SDM model can also be described as a fully-connected three-layer feed-forward
neural network. In this model. the location addresses are the weights between the
input layer and the hidden units. and the data counters are the weights between the
hidden units and the output layer. Note that the number of hidden-layer nodes (at
least 1,000 and possibly up to 1,000,(00) is much larger than is commonly used for
artificial neural networks. It is unclear how well standard algorithms. such as backpropagation, would perform with such a large number of units in the hidden layer.
HOLLAND'S GENETIC ALGORITHMS
Genetic Algorithms are a search technique for high-dimensional spaces inspired by
the evolutionary processes of DNA. The domain of a genetic algorithm is a population ofrued-length binary strings and a fitness function, which is a method for evaluating the fitness of each of the members. We use this fitness function to select two
highly-ranked members for recombination. and one lowly-ranked member for replacement (The selection may be done either absolutely. with the best and worst
members always being selected. or probabilisticly. with the members being chosen
proportional to their fitness scores.)
The member selected as bad is removed from the population. The two members selected as good are then recombined to create a new member to take its place in the
population. In effect, the genetic algorithm is a search over a high-dimensional
space for strings which are highly-rated by the fitness function.
The process used to create new members of the population is called crossover. In a
crossover. we align the two good candidates end-to-end and segment them at one or
more crossover-points. We then create a new string by starting the transcription of
bits at one of the parent strings, and switching the transcription to the other parent
at the crossover-points. This new string is placed into the population. taking the
place of the poorly-rated member.
11??I'~IOllllll ????
111:1"1.1 10 ...
Second parent
1101101101:
~
~
First parent
~
New member
Figure 2: Crossover of two binary strings
By running the genetic algorithm over the population many times, the population
"evolves" towards members which are rated more fit by our fitness function.
Predicting Weather Using a Genetic Memory
Output Layer
Weights changed using
perceptron rule
Hidden Unit Layer
-..
Weights changed using
Genetic Algorithm
Input Layer
Figure 3: Structure of a Genetic Memory
Holland has a lIIaUIt~lIIaUl:at pruuI Uta&. g~l1~Ul: atgunuul1s uastAI UI1 Ule crossover
procedure are an extremely efficient method for searching a high-dimensional space.
GENETIC MEMORY
Genetic Memory is a hybrid of Kanerva's Sparse Distributed Memory and Holland's
Genetic Algorithms. In this hybrid, the location addresses of the SDM are not held
constant; rather, a Genetic Algorithm is used to move them to more advantageous
positions in the address space. If we view SDM as a neural net, this hybrid uses a genetic algorithm to change the weights in the connections between the input layer and
the hidden unit layer, while the connections between the hidden unit layer and the
output layer at changed using the standard method for a SDM.
Most other work which combined neural networks and genetic algorithms kept multiple networks; the Genetic Algorithm was used to recombine the more successful
of these networks to create new entire networks.
In a Genetic Memory there is a single network with different algorithms changing
the weights in different layers. Thus, a Genetic Memory incorporates the Genetic
Algorithm directly into the operation of a single network.
AUSTRALIAN WEATHER DATA
The weather data was collected at a single site on the Australian coast. A sample
was taken every 4 hours for 25 years; the me contains over 58,000 weather samples
The file contained 15 distinct features, including year, month, day of the month,
time of day, pressure, dry bulb temperature, wet bulb temperature, dew point, wind
speed, wind direction, cloud cover, and whether it rained in the past four hours.
For this work, I coded each weather sample into a 256-bit word. Each weather sample was coded into a 256-bit binary address, giving each feature a 16-bit field in that
address. The feature values were coarse-coded into a simple thennometer-style
code. For example, figure 4 shows the code used for month.
PROCEDURE FOR WEATHER PREDICTION
In the standard SDM model, the locations addresses are held constant. In a Genetic
Memory, the location addresses are reconfigured using a Genetic Algorithm.
459
460
Rogers
JAN:
FEB:
MAR:
APR:
MAY:
JUN:
1111111100000000
0111111111000000
0011111111100000
0000111111110000
0000011111111000
0000001111111110
JUL: 1000000001111111
AUG:
SEP:
OCT:
NOV:
DEC:
1100000000111111
1111000000011111
1111100000001111
1111110000000011
1111111000000001
Figure 4: 16-bit code used for month
The fitness function used is based on my work on statistical prediction and presented
at NIPS-88 [Rogers 1988]. This work assigns a number to each physical storage location (a row in the figure) which is a measure of the predictiveness of that location.
Highly-predictive locations are recombined using crossover; the newly-created location address is given to a location which is relatively unpredictive. The data
counter is a measure of the co"elation between the selection of a location and the
occurrence of a given bit value. Thus, we can use the data counters to judge the fitness, i.e., the predictiveness, of each memory location.
To train the memory, we present the memory with each weather state in turn. The
memory is not shown the data a multiple number of times. For each state, the memory is addressed with the 256-bit address which represents it. non is written to the
memory if it does not rain in the next four hours, and "1" if it does. After the memory has seen a given number of weather samples, the Genetic Algorithm is performed to replace a poorly-predictive location with a new address created from two
predictive addresses.
The procedure is continued until the memory has seen 50,000 weather samples, and
has performed -5,000 genetic recombinations.
ANAL YSIS OF RESULTS
The initial results from the Genetic Memory procedure was conducted on a memory
with 1,000 storage locations. The weather sample set consisted of a sequence of
weather samples taken every 4 hours over a period of 20 years. In the sample set, it
rained in the next 4 hours for -10% of the samples, and was dry in the next four
hours in -90% of the samples.
The Genetic Memory was testing by storing -50,000 weather samples. The samples
were given to the memory in chronological order. During the course of storage, the
memory reconfigured itself with -5,000 genetic recombinations. A Genetic Memory and a standard Sparse Distributed Memory were tested against 1,000 previously
unseen weather samples. In initial experiments, the Genetic Memory had 50% fewer
errors than the Sparse Distributed Memory.
However, the Genetic Memory does not only show an improvement in performance,
it allows the user to analyze the genetically-determined memory locations to discover how the memory improved its performance.
By studying highly-rated memory locations in the Genetic Memory, we can open the
black box: that is, access the parameters the memory has decided are the most effective in associating the sample addresses with the sample data. This ability to access
the parameters the system found effective has two important implications. First,
Predicting Weather Using a Genetic Memory
the parameters may offer insights into the underlying physical processes in the system under study. Second. knowledge of how the system predicts may be vital for determining the robustness and the envelope of applicability of the memory prior to
embedding into a real-world system.
Simply scoring the performance of a system is not enough. We must be able to
"open the black box" to study why the system performs as it does.
OPENING THE BLACK BOX
When the training is completed. we can analyze the structure of memory locations
which performed well to discover which features they found most discriminatory
and which values of those features were preferred. For example. here is a memory
location which was rated highly-fit for predicting rain after training:
1101001100000011 1111011110101011 0111111100010000 1100000011011010
0100110011111011 1111110000000011 0111111011()()()()()() 001110110110()l10
000000101111011001100000010000100001001110110100 0100000111111111
0000000111111110 0000000011111111 0011011111111111 0100110000001000
By measuring the distance between a given 16-bit field and all possible values for
that field. we can discover which values of the feature are most desired. (Closer in
hamming distance is better.) The absolute range of values is the sensitivity of the location to changes along that feature dimension. Figure 5 shows an analysis of the 16bit field for month in the given memory location:
Location's 16-bit field
for month:0111111100010000
t
values for months
Distance
Less desirable
JAN: 1111111100000000
2
Value desirability
FEB: 0111111110000000
2
More desirable
MAR: 0011111111000000
4
APR: 0000111111110000
6
. .. etc ...
~
Feature (sensitivity
Month (12)
IOVf\ IS f-
S
f-
?
I
I
-
\-
11
JFMAMJJASOND
Values
Figure 5: Analyzing a location field
In this case. the location finds January and February the most desirable months for
rain. and July and August the least desirable months.
The relative sensitivity towards different features measures which features are most
important in making the prediction of rain. In this case. we have a change of distance of 12 bits. which makes this location very sensitive to the value of the month.
We can estimate which features are the most important in predicting rain by looking
at the relative sensitivity of the different fields in the location to changes in their
feature. The following graphs show the most sensitive features of the previously
shown memory location towards predicting rain; that is. the location is very sensitive to the combination of all these fields with the proper values.
461
462
Rogers
Dry bulb temp (12)
Cloud cover (13)
Iii
Iii
10
ou.........&.J...u...o....o.J....&............"Ju.........~
None
Low
oL........L.~...L..L..J..........L...............-"--'-J
210
High
Pressure (12)
270
240
Month (12)
IIi
Iii
10
10100
JFMAMJJASOND
Figure 6: The four most sensitive features
The "most preferred values" of these fields are the minima of these graphs. For example, this location greatly prefers January and February over June and JUly. The preferences of this location are for the month to be January or February, for low pressure. high cloud cover, and low temperature. Surprisingly. whether it rained in the
last four hours is not one of the most important features for this location.
We can also look some of the least sensitive features. The following graphs show
the least sensitive features of the memory location towards predicting rain; that is,
the location is relatively insensitive to the values of these features.
Wet bulb temp (5)
Year (5)
Wind direction (4)
Iii
10
61
73
80
210
240
270
N
E
S
W
Figure 7: The three least sensitive features
This set contains some fields that one would expect to be relatively unimportant,
such as year. Fields such as wind direction is unimportant to this location, but interestingly other highly-rated locations fmd it to be very useful in other regions of the
weather space.
Predicting Weather Using a Genetic Memory
COMPARISON WITH DAVIS' METHOD
Davis' Algorithm has been shown to be a powerful new method for augmenting the
power of a backpropagation-based system. The following is an attempt to contrast
our approaches, without denigrating the importance his groundbreaking work. The
reader is referred to his book for detailed information about his approach.
It is difficult to directly compare the performance of these techniques given the preliminary nature of the experiments done with Genetic Memory. However, it is possible to compare architectural features of the systems, and estimate the relative
strengths a weaknesses.
? 8ackpropagation vs. Associative Memories: Davis' approach relies on the performance of the backpropagation algorithm for the central learning cycle of the system. Associative memories have a far quicker learning cycle than backpropagation
networks, and have been shown to have preferential characteristics after training in
some domains. A system based on an associative memory may share these advantages
over a system based on backpropagation.
? Scalability: Many issues concerning the scalability of backpropagation networks
remain unresolved. It is not simple to build backpropagation networks of thousands
or hundreds of thousands of units. In contrast, Kanerva's Sparse Distributed Memory is specifically designed for such massive construction; one implementation on the
Connection Machine can contain l,O(XM)()() hidden units. The Genetic Memory shares
this property.
? Unity: Davis' algorithm has two levels of processing. The first level consists of
standard backpropagation networks, and the second is a meta-level which manipulates these networks. The Genetic Memory has incorporated both algorithms into a
single network; both algorithms are operating simultaneously.
My intuition is that different algorithms may be best suited for the different layers
of a neural network. Layers with a large fan-out (such as the input layer to the layer of hidden units) may be best driven by an algorithm suited to high-dimensional
searching, such as Genetic Algorithms or a Kohonen-style self-organizing system.
Layers with a large fan-in (such as the hidden-unit layer to the output layer) may be
best driven by a hill-climbing algorithms such a backpropagation.
CONCLUSIONS
? Real-world problems are often "high-dimensional", that is, are described by large
numbers of dependent variables. Algorithms must be specifically designed to function well in such high-dimensional spaces. Genetic Memory is such an algorithm .
? Genetic Memory, while sharing some features with Davis' approach, has fundamental differences that may make it more appropriate to some problems and easier to
scale to extremely-large (> 100,000 node) systems.
? The incorporation of the Genetic Algorithm improves the recall performance of a
standard associative memory.
? The structure of the Genetic Memory allows the user to access the parameters discovered by the Genetic Algorithm and used to assist in making the associations
stored in the memory.
463
464
Rogers
Acknowledgments
This work was supported in part by Cooperative Agreements NCC 2-408 and NCC
2-387 from the National Aeronautics and Space Administration (NASA) to the Universities Space Research Association (USRA). Funding related to the Connection
Machine was jointly provided by NASA and the Defense Advanced Research Projects
Agency (DARPA). All agencies involved were very helpful in promoting this
work, for which I am grateful.
The entire RIACS staff and the SDM group has been supportive of my work. Bruno
Olshausen was a vital sounding-board. Pentti Kanerva trusted my intuitions even
when the payoff wasn't yet clear. And finally, thanks to Doug Brockman, who decided to wait for me.
References
Davis, L., Genetic algoritluns and simulated annealing. London, England: Pitman
Publishing (1987).
Holland, J. H., Adaptation in natural and artificial systems. Ann Arbor: University of Michigan Press (1975).
Holland, J. H., "Escaping brittleness: the possibilities of general-purpose learning
algorithms applied to parallel rule-based systems," in Machine learning. an
artificial intelligence approach. Volume II, R. J. Michalski, J. G. Carbonell,
and T. M. Mitchell, eds. Los Altos, California: Morgan Kaufmann (1986).
Kanerva, Pentti., "Self-propagating Search: A Unified Theory of Memory," Center
for the Study of Language and Infonnation Report No. CSLI-84-7 (1984).
Kanerva, Pentti., Sparse distributed memory. Cambridge. Mass: MIT Press, 1988.
Rogers. David, "Using data-tagging to improve the perfonnance of Kanerva's sparse
distributed memory," Research Institute for Advanced Computer Science
Technical Report 88.1, NASA Ames Research Center (l988a).
Rogers, David, "Kanerva's Sparse Distributed Memory: an Associative Memory Algorithm Well-Suited to the Connection Machine," Int. J. High-Speed Comput., 2, pp. 349-365 (1989).
Rogers, David, "Statistical Prediction with Kanerva's Sparse Distributed Memory,"
Advances in Neural Information Processing Systems I, San Mateo: MorganKaufman (1989).
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1,861 | 2,690 | On-Chip Compensation of Device-Mismatch
Effects in Analog VLSI Neural Networks
Miguel Figueroa
Department of Electrical Engineering, Universidad de Concepci?on
Casilla 160-C, Correo 3, Concepci?on, Chile
[email protected]
Seth Bridges and Chris Diorio
Computer Science & Engineering, University of Washington
Box 352350, Seattle, WA 98195-2350, USA
{seth, diorio}@cs.washington.edu
Abstract
Device mismatch in VLSI degrades the accuracy of analog arithmetic
circuits and lowers the learning performance of large-scale neural networks implemented in this technology. We show compact, low-power
on-chip calibration techniques that compensate for device mismatch. Our
techniques enable large-scale analog VLSI neural networks with learning performance on the order of 10 bits. We demonstrate our techniques
on a 64-synapse linear perceptron learning with the Least-Mean-Squares
(LMS) algorithm, and fabricated in a 0.35?m CMOS process.
1
Introduction
Modern embedded and portable electronic systems operate in unknown and mutating environments, and use adaptive filtering and machine learning techniques to discover the statistics of the data and continuously optimize their performance. Artificial neural networks
are an attractive substrate for implementing these techniques, because their regular computation and communication structures makes them a good match for custom VLSI implementations. Portable systems operate under severe power dissipation and space constraints,
and VLSI implementations provide a good tradeoff between computational throughput and
power/area cost. More specifically, analog VLSI neural networks perform their computation using the physical properties of transistors with orders of magnitude less power and
die area than their digital counterparts. Therefore, they could enable large-scale real-time
adaptive signal processing systems on a single die with minimal power dissipation.
Despite the promises delivered by analog VLSI, an important factor has prevented the success of large-scale neural networks using this technology: device mismatch. Gradients in
the parameters of the fabrication process create variations in the physical properties of silicon devices across a single chip. These variations translate into gain and offset mismatches
in the arithmetic blocks, which severely limit the overall performance of the system. As
a result, the accuracy of analog implementations rarely exceeds 5-6 bits, even for smallscale networks. This limitation renders these implementations useless for many important
applications. Although it is possible to combat some of these effects using careful design
techniques, they come at the cost of increased power and area, making an analog solution
less attractive.
(a) Single-layer LMS perceptron.
(b) Block diagram for the synapse.
Figure 1: A single-layer perceptron and synapse. (a) The output z of the perceptron is the inner
product between the input and weight vectors. The LMS algorithm updates the weights based on the
inputs and an error signal e. (b) The synapse stores the weight in an analog memory cell. A Gilbert
multiplier computes the product between the input and the weight and outputs a differential current.
The LMS block updates the weight.
We have built a 64-synapse analog neural network with an learning performance of 10 bits,
representing an improvement of more than one order of magnitude over that of traditional
analog designs, with a modest increase in power and die area. We fabricated our network
using a double-poly, 4-metal 0.35?m CMOS process available from MOSIS. We achieve
this performance by locally calibrating the critical analog blocks after circuit fabrication
using a combination of one-time (or periodic) and continuous calibration using the same
feedback as the network?s learning algorithm. We chose the Least Mean Squares (LMS) algorithm because of its simplicity and wide applicability in supervised learning techniques
such as adaptive filtering, adaptive inverse control, and noise canceling. Moreover, several useful unsupervised-learning techniques, such as adaptive orthogonalization, principal components analysis (PCA), independent components analysis (ICA) and decisionfeedback learning, use simple generalizations of LMS.
2
A linear LMS perceptron
Fig. 1(a) shows our system architecture, a linear perceptron with scalar output that performs
the function:
N
X
z(i) = bw0 (i) +
xj (i) wj (i)
(1)
j=1
where i represents time, z(i) is the output, xj (i) are the inputs, wj (i) are the synaptic
weights, and b is a constant bias input. We clarify the role of b in Section 3.1. After each
presentation of the input, the LMS algorithm updates the weights using the learning rule:
wj (i + 1) = wj (i) + ? xj (i) e(i)
i = 0 . . . N, x0 (i) = b
(2)
where ? is a constant learning rate, and e(i) is the error between the output and a reference
signal r(i) such that e(i) = r(i) ? z(i).
3
The synapse
Fig. 1(b) shows a block diagram of our synapse. We store the synaptic weights in a memory
cell that implements nonvolatile analog storage with linear updates. A circuit transforms
the single-ended voltage output of the memory cell (Vw ) into a differential voltage signal
(Vw+ , Vw? ), with a constant common mode. A Gilbert multiplier computes the 4-quadrant
product between this signal and the input (also represented as a differential voltage Vx+ ,
Vx? ). The output is a differential analog current pair (Io+ , Io? ), which we sum across all
synapses by connecting them to common wires.
(a) Measured output vs. input value.
(b) Measured output vs. weight value.
Figure 2: Gilbert multiplier response for 8 synapses.(a) Our multiplier maximizes the linearity of
xi , achieving a linear range of 600mV differential. Gain mismatch is 2:1 and offset mismatch is up to
200mV. (b) Our multiplier maximizes weight range at the cost of weight linearity (1V single-ended,
2V differential). The gain variation is lower, but the offset mismatch exceeds 60% of the range.
Because we represent the perceptron?s output and the reference with differential currents,
we can easily compute the error using simple current addition. We then transform (off-chip
in our current implementation) the resulting analog error signal using a pulse-density modulation (PDM) representation [1]. In this scheme, the value of the error is represented as
the difference between the density (frequency) of two fixed-width, fixed-amplitude digital
pulse trains (Pe+ and Pe? in Fig. 1(b)). These properties make the PDM representation
largely immune to amplitude and jitter noise. The performance of the perceptron is highly
sensitive to the resolution of the error signal; therefore the PDM representation is a good
match for it. The LMS block in the synapse takes the error and input values and computes
update pulses (also using PDM) according to Eqn. 2.
In the rest of this section, we analyze the effects of device mismatch in the performance
of the major blocks, discuss their impact in overall system performance, and present the
techniques that we developed to deal with them. We illustrate with experimental results
taken from silicon implementation of the perceptron in a 0.35?m CMOS process. All data
presented in this paper, unless otherwise stated, comes from this silicon implementation.
3.1 Multiplier
A Gilbert multiplier implements a nonlinear function of the product between two differential voltages. Device mismatch in the multiplier has two main effects: First, it creates
offsets in the inputs. Second, mismatch across the entire perceptron creates variations in
the offsets, gain, and linearity of the product. Thus, Eqn. 1 becomes:
N
X
z(i) =
aj fjx xj (i) ? dxj fjw wj (i) ? dw
x0 (i) = b
(3)
j
j=0
where aj represents the gain mismatch between multipliers, fjx and fjw are the nonlinearities applied to the inputs and weights (also mismatched across the perceptron), and dxj and
dw
j are the mismatched offsets of the inputs and weights.
Our analysis and simulations of the LMS algorithm [2] determine that the performance of
the algorithm is much more sensitive to the linearity of fjx than to the linearity of fjw , because the inputs vary over their dynamic range with a large bandwidth, while the bandwidth
of the weights is much lower than the adaptation time-constant. Therefore, the adaptation
compensates for mild nonlinearities in the weights as long as fjw remains a monotonic odd
function [2]. Consequently, we sized the transistors in the Gilbert multiplier to maximize
the linearity of fjx , but paid less attention (in order to minimize size and power) to fjw .
Fig. 2(a) shows the output of 8 synapses in the system as a function of the input value. The
(a) Memory cell circuit.
(b) Measured weight updates
Figure 3: A simple PDM analog memory cell. (a) We store each weight as nonvolatile analog charge
on the floating gate FG. The weight increments and decrements are proportional to the density of the
pulses on Pinc and Pdec . (b) Memory updates as a function of the increment and decrement pulse
densities for 8 synapses. The updates show excellent linearity (10 bits), but also poor matching both
within a synapse and between synapses.
response is highly linear. The gain mismatch is about 2:1, but the LMS algorithm naturally
absorbs it into the learned weight value. Fig. 2(b) shows the multiplier output as a function
of the single-ended weight value Vw . The linearity is visibly worse in this case, but the
LMS algorithm compensates for it.
The graphs in the Fig. 2 also show the input and weight offsets. Because of the added
mismatch in the single-ended to differential converter, the weights present an offset of up
to ?300mV, or 30% of the weight range. The LMS algorithm will also compensate for this
offset by absorbing it into the weight, as shown in the analysis of [3] for backprogagation
neural networks. However, this will only occur if the weight range is large enough accomodate for the offset mismatch. Consequently, we sacrifice weight linearity to increase
the weight range. Input offsets pose a harder problem, though. The offsets are small (up
100mV), but because of the restricted input range (to maximize linearity), they are large
enough to dramatically affect the learning performance of the perceptron. Our solution was
to use the bias synapse w0 to compensate for the accumulated input offset. Assuming that
the multiplier is linear, offsets translate into nonzero-mean inputs, which a bias synapse
trained with LMS can remove as demonstrated in [4]. To guarantee sufficient gain, we
provide a stronger bias current to the multiplier in the bias synapse.
3.2 Memory cell
A synapse transistor [5] is a silicon device that provides compact, accurate, nonvolatile
analog storage as charge on its floating gate. Fowler-Nordheim tunneling adds charge to
the floating gate and hot-electron injection removes charge. Both mechanisms can be used
to accurately update the stored value during normal device operation. Because of these
properties, synapse transistors have been a popular choice for weight storage in recent
silicon learning systems [6, 7].
Despite the advantages listed above, it is hard to implement linear learning rules such as
LMS using tunneling and injection. This is because their dynamics are exponential with respect to their control variables (floating-gate voltage, tunneling voltage and injection drain
current), which naturally lead to weight-dependent nonlinear update rules. This is an important problem because the learning performance of the perceptron is strongly dependent
on the accuracy of the weight updates; therefore distortions in the learning rule will degrade
performance. The initial design of our memory cell, shown in Fig. 3(a) and based on the
work presented in [8], solves this problem: We store the analog weight as charge on the
floating gate FG of synapse transistor M1 . Pulses on Pdec and Pinc activate tunneling and
injection and add or remove charge from the floating gate, respectively. The operational
(a) Calibrated memory cell circuit.
(b) Measured calibrated weight updates.
Figure 4: PDM memory cell with local calibration. (a) We first match the tunneling rate across all
synapses by locally changing the voltage at the floating gate FGdec . Then, we modify the injection
rate to match the local tunneling rate using the floating gate FGinc . (b) The calibrated updates are
symmetric and uniform within 9-10 bits.
amplifier sets the floating-gate voltage at the global voltage Vbias . Capacitor Cw integrates
the charge updates, changing the output Vout by ?Vout = ?Q/C. Because the floatinggate voltage is constant and so are the pulse widths and amplitudes, the magnitude of the
updates depends on the density of the pulses Pinc and Pdec . Fig. 3(b) shows the magnitude
of the weight updates as a function of the density of pulses in Pinc (positive slopes) and
Pdec (negative slopes) for 8 synapses. The linearity of the updates, measured as the integral
nonlinearity (INL) of the transfer functions depicted in Fig. 3(b), exceeds 10 bits.
Fig. 3(b) highlights an important problem caused by device mismatch: the strengths of
tunneling and injection are poorly balanced within a synapse (the slopes show up to a 4:1
mismatch). Moreover, they show a variation of more than 3:1 across different synapses
in the perceptron. This translates into asymmetric update rules that are also nonuniform
across synapses. The local asymmetry of the learning rate translates into offsets between
the learned and target weights, degrading the learning performance of the perceptron. The
nonuniformity between learning rates across the perceptron changes Eqn. 2 into:
wj (i + 1) = wj (i) + ?j xj (i) e(i)
i = 0 . . . N, x0 (i) = b
(4)
where ?j are the different learning rates for each synapse. Generalizing the conventional
stability analysis of LMS [9], we can show that the condition for the stability of the weight
vector is: 0 < ?max < 1/?max , where ?max is the maximal eigenvalue of the input?s
correlation matrix and ?max = maxj (?j ). Therefore, learning rate mismatch does not
affect the accuracy of the learned weights, but it does slow down convergence because we
need to scale all learning rates globally to limit the value of the maximal rate.
To maintain good learning performance and convergence speed, we need to make learning
rates symmetric and uniform across the perceptron. We modified the design of the memory
cell to incorporate local calibration mechanisms that achieve this goal. Fig. 4(a) shows our
new design. The first step is to equalize tunneling rates: The voltage at the new floating gate
FGdec sets the voltage at the floating-gate FG and controls the ratio between the strength
of tunneling and injection onto FG: Raising the voltage at FGdec increases the drain-tochannel voltage and reduces the gate-to-tunneling-junction voltage at M1 , thus increasing
injection efficiency and reducing tunneling strength [5]. We set the voltage at FGdec by first
tunneling using the global line erase dec, and then injecting on transistor M3 by lowering
the local line set dec to equalize the tunneling rates across all synapses. To compare the
tunneling rates, we issue a fixed number of pulses at Pdec and compare the memory cell
outputs using a double-sampling comparator (off-chip in the current implementation). To
control the injection rate, we add transistor M2 , which limits the current through M1 and
(a) LMS block.
(b) Measured RMS error.
Figure 5: LMS block at each synapse. (a) The difference between the densities of Pinc and Pdec
is proportional to the product between the input and the error, and thus constitutes an LMS update
rule. (b) RMS error for a single-synapse with a constant input and reference, including a calibrated
memory cell with symmetric updates, a simple synapse with asymmetric updates, and a simulated
ideal synapse.
thus the injection strength of the pulse at Pinc . We control the current limit with the voltage
at the new floating gate FGinc : we first remove electrons from the floating gate using the
global line erase inc. Then we inject on transistor M4 by lowering the local line set inc to
match the injection rates across all synapses. The entire process is controlled by a simple
state machine (also currently off-chip). Fig. 4(b) shows the tunneling and injection rates
after calibration as a function of the density of pulses Pinc and Pdec . Comparing the graph
to Fig. 4(b), it is clear that the update rates are now symmetric and uniform across all
synapses (they match within 9-10 bits). Note that we could also choose to calibrate just for
learning rate symmetry and not uniformity across synapses, thus eliminating the floating
gate FGinc and its associated circuitry. This optimization would result in approximately a
25% reduction in memory cell area (6% reduction in total synapse area), but would also
cause an increase of more than 200% in convergence time, as illustrated in Section 4.
3.3 The LMS block
Fig. 5(a) shows a block diagram of the LMS-update circuit at each synapse. A pulsedensity modulator [10] transforms the synaptic input into a pair of digital pulse-trains of
?
fixed width (P+
x , Px ). The value of the input is represented as the difference between the
density (frequency) of the pulse trains. We implement the memory updates of Eqn. 2 by
?
digitally combining the input and error pulses (P+
e , Pe ) such that:
Pinc
Pdec
= (Px+ AN D Pe+ ) OR (Px? AN D Pe? )
= (Px+ AN D Pe? ) OR (Px? AN D Pe+ )
(5)
(6)
This technique was used previously in a synapse-transistor based circuit that learns correlations between signals [11], and to multiply and add signals [1]. If the pulse trains are
asynchronous and sparse, then using Eqn. 5 and Eqn. 6 to increment and decrement the
synaptic weight implements the LMS learning rule of Eqn. 2.
To validate our design, we first trained a single synapse with a DC input to learn a constant
reference. Because the input is constant, the linearity and offsets in the input signal do not
affect the learning performance; therefore this experiment tests the resolution of the feedback path (LMS circuit and memory cell) isolated from the analog multipliers. Fig. 5(b)
shows the evolution of the RMS value of the error for a synapse using the original and
calibrated memory cells. The resolution of the pulse-density modulators is about 8 bits,
which limits the resolution of the error signal. We also show the RMS error for a simulated (ideal) synapse learning from the same error. We plot the results in a logarithmic
scale to highlight the differences between the three curves. The RMS error of the calibrated synapse converges to about 0.1nA. Computing the equivalent resolution in bits as
(a) Measured RMS error.
(b) Measured weight evolution.
Figure 6: Results for 64-synapse experiment. (a) Asymmetric learning rates and multiplier offsets
limit the output resolution to around 3 bits. Symmetric learning rates and a bias synapse brings
the resolution up to more 10 bits, and uniform updates reduce convergence time. (b) Synapse 4
shows a larger mismatch than synapse 1 and therefore it deviates from its theoretical target value
to compensate. The bias synapse in the VLSI perceptron converges to a value that compensates for
offsets in the inputs xi to the multipliers.
RM S error
rb = ?log2 0.5 output
, we find that for a 2?A output range, this error represents an
range
output resolution of about 13 bits. The difference with the simulated synapse is due to the
discrete weight updates in the PDM memory cell. Without calibration, the RMS error converges to 0.4nA (or about 11 bits), due to the offset in the learned weights introduced by the
asymmetry in the learning rate. As discussed in Section 4, the degradation of the learning
performance in a larger-scale system due to asymmetric learning rates is drastically larger.
4
A 64-synapse perceptron
To test our techniques in a larger-scale system, we fabricated a 64-synapse linear perceptron in a 0.35?m CMOS process. The circuit uses 0.25mm2 of die area and dissipates
200?W. Fig. 6(a) shows the RMS error of the output in a logarithmic scale as we introduce
different compensation techniques. We used random zero-mean inputs selected from a uniform distribution over the entire input range, and trained the network using the response
from a simulated perceptron with ideal multipliers and fixed weights as a reference. In
our first experiments, we trained the network without using any compensation. The error
settles to 10?A RMS, which corresponds to an output resolution of about 3 bits for a full
range of 128?A differential. Calibrating the synapses for symmetric learning rates only
improves the RMS error to 5?A (4 bits), but the error introduced by the multiplier offsets
still dominates the residual error. Introducing the bias synapse and keeping the learning
rates symmetric (but nonuniform across the perceptron) compensates for the offsets and
brings the error down to 60nA RMS, corresponding to an output resolution better than 10
bits. Further calibrating the synapses to achieve uniform, symmetric learning rates maintains the same learning performance, but reduces convergence time to less than one half, as
predicted by the analysis in Section 3.2. A simulated software perceptron with ideal multipliers and LMS updates that uses an error signal of the same resolution as our experiments
gives an upper bound of just under 12 bits for the learning performance.
Fig. 6(b) depicts the evolution of selected weights in the silicon perceptron with on-chip
compensation and the software version. The graph shows that synapse 1 in our VLSI implementation suffers from little mismatch, and therefore its weight virtually converges to
the theoretical value given by the software implementation. Because the PDM updates are
discrete, the weight shows a larger oscillation around its target value than the software version. Synapse 4 shows a larger mismatch; therefore it converges to a visibly different value
from the theoretical in order to compensate for it. The bias weight in the software percep-
tron converges to zero because the inputs have zero mean. In the VLSI perceptron, input
offsets in the multipliers create nonzero-mean inputs; therefore the bias synapse converges
to a value that compensates for the aggregated effect of the offsets. The normalized value
of -1.2 reflects the gain boost given to this multiplier to increase its dynamic range.
5
Conclusions
Device mismatch prevents analog VLSI neural networks from delivering good learning
performance for large-scale applications. We identified the key effects of mismatch and
presented on-chip compensation techniques. Our techniques rely both on one-time (or
periodic) calibration, and on the adaptive operation of the system to achieve continuous
calibration. Combining these techniques with careful circuit design enables an improvement of more than one order of magnitude in accuracy compared to traditional analog
designs, at the cost of an off-line calibration phase and a modest increase in die area and
power. We illustrated our techniques with a 64-synapse analog-VLSI linear perceptron that
adapts using the LMS algorithm. Future work includes extending these techniques to unsupervised learning algorithms such as adaptive orthogonalization, principal components
analysis (PCA) and independent components analysis (ICA).
Acknowledgements
This work was financed in part by the Chilean government through FONDECYT grant
#1040617. We fabricated our chips through MOSIS.
References
[1] Y. Hirai and K. Nishizawa, ?Hardware implementation of a PCA learning network by an asynchronous PDM digital circuit,? in IEEE-INNS-ENNS International Joint Conference on Neural
Networks (IJCNN), vol. 2, pp. 65?70, 2000.
[2] M. Figueroa, Adaptive Signal Processing and Correlational Learning in Mixed-Signal VLSI.
Ph.D. Thesis, University of Washington, 2005.
[3] B. K. Dolenko and H. C. Card, ?Tolerance to analog hardware of on-chip learning in backpropagation networks,? IEEE Transactions on Neural Networks, vol. 6, no. 5, pp. 1045?1052,
1995.
[4] F. Palmieri, J. Zhu, and C. Chang, ?Anti-Hebbian learning in topologically constrained linear
networks: A tutorial,? IEEE Transactions on Neural Networks, vol. 4, no. 5, pp. 748?761, 1993.
[5] C. Diorio, P. Hasler, B. Minch, and C. Mead, ?A complementary pair of four-terminal silicon
synapses,? Analog Integrated Circuits and Signal Processing, vol. 13, no. 1/2, pp. 153?166,
1997.
[6] C. Diorio, D. Hsu, and M. Figueroa, ?Adaptive CMOS: from biological inspiration to systemson-a-chip,? Proceedings of the IEEE, vol. 90, no. 3, pp. 345?357, 2002.
[7] J. Dugger and P. Hasler, ?Improved correlation learning rule in continuously adapting floatinggate arrays using logarithmic pre-distortion of input and learning signals,? in IEEE Intl. Symposium on Circuits and Systems (ISCAS), vol. 2, pp. 536?539, 2002.
[8] C. Diorio, S. Mahajan, P. Hasler, B. A. Minch, and C. Mead, ?A high-resolution nonvolatile
analog memory cell,? in IEEE Intl. Symp. on Circuits and Systems, vol. 3, pp. 2233?2236,
1995.
[9] B. Widrow and E. Walach, Adaptive Inverse Control. Upper Saddle River, NJ: Prentice-Hall,
1996.
[10] C. Mead, Analog VLSI and Neural Systems. Reading, MA: Addison-Wesley, 1989.
[11] A. Shon, D. Hsu, and C. Diorio, ?Learning spike-based correlations and conditional probabilities in silicon,? in Neural Information Processing Systems (NIPS), (Vancouver, BC), 2001.
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1,862 | 2,691 | Hierarchical Distributed Representations for
Statistical Language Modeling
John Blitzer, Kilian Q. Weinberger, Lawrence K. Saul, and Fernando C. N. Pereira
Department of Computer and Information Science, University of Pennsylvania
Levine Hall, 3330 Walnut Street, Philadelphia, PA 19104
{blitzer,kilianw,lsaul,pereira}@cis.upenn.edu
Abstract
Statistical language models estimate the probability of a word occurring
in a given context. The most common language models rely on a discrete
enumeration of predictive contexts (e.g., n-grams) and consequently fail
to capture and exploit statistical regularities across these contexts. In this
paper, we show how to learn hierarchical, distributed representations of
word contexts that maximize the predictive value of a statistical language
model. The representations are initialized by unsupervised algorithms for
linear and nonlinear dimensionality reduction [14], then fed as input into
a hierarchical mixture of experts, where each expert is a multinomial distribution over predicted words [12]. While the distributed representations
in our model are inspired by the neural probabilistic language model of
Bengio et al. [2, 3], our particular architecture enables us to work with
significantly larger vocabularies and training corpora. For example, on a
large-scale bigram modeling task involving a sixty thousand word vocabulary and a training corpus of three million sentences, we demonstrate
consistent improvement over class-based bigram models [10, 13]. We
also discuss extensions of our approach to longer multiword contexts.
1
Introduction
Statistical language models are essential components of natural language systems for
human-computer interaction. They play a central role in automatic speech recognition [11],
machine translation [5], statistical parsing [8], and information retrieval [15]. These models estimate the probability that a word will occur in a given context, where in general
a context specifies a relationship to one or more words that have already been observed.
The simplest, most studied case is that of n-gram language modeling, where each word is
predicted from the preceding n?1 words. The main problem in building these models is
that the vast majority of word combinations occur very infrequently, making it difficult to
estimate accurate probabilities of words in most contexts.
Researchers in statistical language modeling have developed a variety of smoothing techniques to alleviate this problem of data sparseness. Most smoothing methods are based on
simple back-off formulas or interpolation schemes that discount the probability of observed
events and assign the ?leftover? probability mass to events unseen in training [7]. Unfortunately, these methods do not typically represent or take advantage of statistical regularities
among contexts. One expects the probabilities of rare or unseen events in one context to be
related to their probabilities in statistically similar contexts. Thus, it should be possible to
estimate more accurate probabilities by exploiting these regularities.
Several approaches have been suggested for sharing statistical information across contexts.
The aggregate Markov model (AMM) of Saul and Pereira [13] (also discussed by Hofmann
and Puzicha [10] as a special case of the aspect model) factors the conditional probability
table of a word given its context by a latent variable representing context ?classes?. However, this latent variable approach is difficult to generalize to multiword contexts, as the
size of the conditional probability table for class given context grows exponentially with
the context length.
The neural probabilistic language model (NPLM) of Bengio et al. [2, 3] achieved significant improvements over state-of-the-art smoothed n-gram models [6]. The NPLM encodes
contexts as low-dimensional continuous vectors. These are fed to a multilayer neural network that outputs a probability distribution over words. The low-dimensional vectors and
the parameters of the network are trained simultaneously to minimize the perplexity of the
language model. This model has no difficulty encoding multiword contexts, but its training
and application are very costly because of the need to compute a separate normalization for
the conditional probabilities associated to each context.
In this paper, we introduce and evaluate a statistical language model that combines the
advantages of the AMM and NPLM. Like the NPLM, it can be used for multiword contexts, and like the AMM it avoids per-context normalization. In our model, contexts are
represented as low-dimensional real vectors initialized by unsupervised algorithms for dimensionality reduction [14]. The probabilities of words given contexts are represented by
a hierarchical mixture of experts (HME) [12], where each expert is a multinomial distribution over predicted words. This tree-structured mixture model allows a rich dependency
on context without expensive per-context normalization. Proper initialization of the distributed representations is crucial; in particular, we find that initializations from the results
of linear and nonlinear dimensionality reduction algorithms lead to better models (with
significantly lower test perplexities) than random initialization.
In practice our model is several orders of magnitude faster to train and apply than the
NPLM, enabling us to work with larger vocabularies and training corpora. We present results on a large-scale bigram modeling task, showing that our model also leads to significant
improvements over comparable AMMs.
2
Distributed representations of words
Natural language has complex, multidimensional semantics. As a trivial example, consider
the following four sentences:
The vase broke.
The window broke.
The vase contains water.
The window contains water.
The bottom right sentence is syntactically valid but semantically meaningless. As shown by
the table, a two-bit distributed representation of the words ?vase? and ?window? suffices to
express that a vase is both a container and breakable, while a window is breakable but cannot be a container. More generally, we expect low dimensional continuous representations
of words to be even more effective at capturing semantic regularities.
Distributed representations of words can be derived in several ways. In a given corpus of
text, for example, consider the matrix of bigram counts whose element C
Pij records the
number of times that word wj follows word wi . Further, let pij = Cij / k Cik denote
the conditional frequencies derived from these counts, and let p~i denote the V -dimensional
frequency vector with elements pij , where V is the vocabulary size. Note that the vectors p~i
themselves provide a distributed representation of the words wi in the corpus. For large
vocabularies and training corpora, however, this is an extremely unwieldy representation,
tantamount to storing the full matrix of bigram counts. Thus, it is natural to seek a lower
dimensional representation that captures the same information. To this end, we need to map
each vector p~i to some d-dimensional vector ~xi , with d V . We consider two methods in
dimensionality reduction for this problem. The results from these methods are then used to
initialize the HME architecture in the next section.
2.1
Linear dimensionality reduction
The simplest form of dimensionality reduction is principal component analysis (PCA).
PCA computes a linear projection of the frequency vectors p~i into the low dimensional
subspace that maximizes their variance. The variance-maximizing subspace of dimensionality d is spanned by the top d eigenvectors of the frequency vector covariance matrix. The
eigenvalues of the covariance matrix measure the variance captured by each axis of the
subspace. The effect of PCA can also be understood as a translation and rotation of the
frequency vectors p~i , followed by a truncation that preserves only their first d elements.
2.2
Nonlinear dimensionality reduction
Intuitively, we would like to map the vectors p~i into a low dimensional space where semantically similar words remain close together and semantically dissimilar words are far
apart. Can we find a nonlinear mapping that does this better than PCA? Weinberger et al.
recently proposed a new solution to this problem based on semidefinite programming [14].
Let ~xi denote the image of p~i under this mapping. The mapping is discovered by first
2
learning the V ?V matrix of Euclidean squared distances [1] given by Dij = |~xi ? ~xj | .
This is done by balancing two competing goals: (i) to co-locate semantically similar words,
and (ii) to separate semantically dissimilar words. The first goal is achieved by fixing the
distances between words with similar frequency vectors to their original values. In particular, if p~j and p~k lie within some small neighborhood of each other, then the corresponding
2
element Djk in the distance matrix is fixed to the value |~
pj ? p~k | . The second goal is
achieved by maximizing the sum of pairwise squared distances ?ij Dij . Thus, we push the
words in the vocabulary as far apart as possible subject to the constraint that the distances
between semantically similar words do not change.
The only freedom in this optimization is the criterion for judging that two words are semantically similar. In practice, we adopt a simple criterion such as k-nearest neighbors in
the space of frequency vectors p~i and choose k as small as possible so that the resulting
neighborhood graph is connected [14].
The optimization is performed over the space of Euclidean squared distance matrices [1].
Necessary and sufficient conditions for the matrix D to be interpretable as a Euclidean
squared distance matrix are that D is symmetric and that the Gram matrix1 derived from
G = ? 12 HDH T is semipositive definite, where H = I ? V1 11T . The optimization can
thus be formulated as the semidefinite programming problem:
Maximize ?ij Dij subject to: (i) DT = D, (ii) ? 12 HDH 0, and
2
(iii) Dij = |~
pi ? p~j | for all neighboring vectors p~i and p~j .
1
Assuming without loss of generality that the vectors ~
xi are centered on the origin, the dot products Gij = ~
xi ? ~
xj are related to the pairwise squared distances Dij = |~
xi ? ~
xj |2 as stated above.
PCA
SDE
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1: Eigenvalues from principal component analysis (PCA) and semide?nite embedding (SDE), applied to bigram distributions of the 2000 most frequently occuring words in
the corpus. The eigenvalues, shown normalized by their sum, measure the relative variance
captured by individual dimensions.
The optimization is convex, and its global maximum can be computed in polynomial
time [4]. The optimization here differs slightly from the one used by Weinberger et al. [14]
in that here we only preserve local distances, as opposed to local distances and angles.
After computing the matrix Dij by semide?nite programming, a low dimensional embedding ~xi is obtained by metric multidimensional scaling [1, 9, 14]. The top eigenvalues of
the Gram matrix measure the variance captured by the leading dimensions of this embedding. Thus, one can compare the eigenvalue spectra from this method and PCA to ascertain
if the variance of the nonlinear embedding is concentrated in fewer dimensions. We refer
to this method of nonlinear dimensionality reduction as semide?nite embedding (SDE).
Fig. 1 compares the eigenvalue spectra of PCA and SDE applied to the 2000 most frequent
words2 in the corpus described in section 4. The ?gure shows that the nonlinear embedding
by SDE concentrates its variance in many fewer dimensions than the linear embedding by
PCA. Indeed, Fig. 2 shows that even the ?rst two dimensions of the nonlinear embedding
preserve the neighboring relationships of many words that are semantically similar. By
contrast, the analogous plot generated by PCA (not shown) reveals no such structure.
MONDAY
TUESDAY
WEDNESDAY
THURSDAY
FRIDAY
SATURDAY
SUNDAY
MAY, WOULD, COULD, SHOULD,
MIGHT, MUST, CAN, CANNOT,
COULDN'T, WON'T, WILL
ONE, TWO, THREE,
FOUR, FIVE, SIX,
SEVEN, EIGHT, NINE,
TEN, ELEVEN,
TWELVE, THIRTEEN,
FOURTEEN, FIFTEEN,
SIXTEEN,
SEVENTEEN,
EIGHTEEN
MILLION
BILLION
ZERO
JANUARY
FEBRUARY
MARCH
APRIL
JUNE
JULY
AUGUST
SEPTEMBER
OCTOBER
NOVEMBER
DECEMBER
Figure 2: Projection of the normalized bigram counts of the 2000 most frequent words
onto the ?rst two dimensions of the nonlinear embedding obtained by semide?nite programming. Note that semantically meaningful neighborhoods are preserved, despite the
massive dimensionality reduction from V = 60000 to d = 2.
2
Though convex, the optimization over distance matrices for SDE is prohibitively expensive for
large matrices. For the results in this paper?on the corpus described in section 4?we solved the
semide?nite program in this section to embed the 2000 most frequent words in the corpus, then used
a greedy incremental solver to embed the remaining 58000 words in the vocabulary. Details of this
incremental solver will be given elsewhere. Though not the main point of this paper, the nonlinear
embedding of V = 60000 words is to our knowledge one of the largest applications of recently
developed spectral methods for nonlinear dimensionality reduction [9, 14].
3
Hierarchical mixture of experts
The model we use to compute the probability that word w0 follows word w is known as a
hierarchical mixture of experts (HME) [12]. HMEs are fully probabilistic models, making
them ideally suited to the task of statistical language modeling. Furthermore, like multilayer neural networks they can parameterize complex, nonlinear functions of their input.
Figure 3 depicts a simple, two-layer HME. HMEs are tree-structured mixture models in
which the mixture components are ?experts? that lie at the leaves of the tree. The interior
nodes of the tree perform binary logistic regressions on the input vector to the HME, and
the mixing weight for a leaf is computed by multiplying the probabilities of each branch
(left or right) along the path to that leaf. In our model, the input vector ~x is a function of
the context word w, and the expert at each leaf specifies a multinomial distribution over
the predicted word w0 . Letting ? denote a path through the tree from root to leaf, the HME
computes the probability of a word w0 conditioned on a context word w as
X
Pr(w0 |w) =
Pr(?|~x(w)) ? Pr(w0 |?).
(1)
?
We can compute the maximum likelihood parameters for the HME using an ExpectationMaximization (EM) algorithm [12]. The E-step involves computing the posterior probability over paths Pr(?|w, w0 ) for each observed bigram in the training corpus. This can be
done by a recursive pass through the tree. In the M-step, we must maximize the EM auxiliary function with respect to the parameters of the logistic regressions and multinomial
leaves as well as the input vectors ~x(w). The logistic regressions in the tree decouple and
can be optimized separately by Newton?s method, while the multinomial leaves have a simple closed-form update. Though the input vectors are shared across all logistic regressions
in the tree, we can compute their gradients and hessians in one recursive pass and update
them by Newton?s method as well.
The EM algorithm for HMEs converges to a local maximum of the log-likelihood, or equivalently, a local minimum of the training perplexity
?
?? C1
?Y
?
Ptrain =
Pr(wj |wi )Cij
,
(2)
?
?
ij
P
where C = ij Cij is the total number of observed bigrams in the training corpus. The
algorithm is sensitive to the choice of initialization; in particular, as we show in the next
word
initialized by
PCA or SDE
input
vector
logistic
regression
logistic
regression
logistic
regression
multinomial
distribution
multinomial
distribution
multinomial
distribution
multinomial
distribution
Figure 3: Two-layer HME for bigram modeling. Words are mapped to input vectors; probabilities of next words are computed by summing over paths through the tree. The mapping
from words to input vectors is initialized by dimensionality reduction of bigram counts.
Ptest
init
random
PCA
SDE
Ptest
m
8
16
32
64
d
4
468
406
385
8
407
364
361
12
378
362
360
16
373
351
355
Table 1: Test perplexities of HMEs
with different input dimensionalities
and initializations.
4
435
385
350
336
d
8
429
361
328
308
12
426
360
320
298
16
428
355
317
294
Table 2: Test perplexities of HMEs
with different input dimensionalities
and numbers of leaves.
section, initialization of the input vectors by PCA or SDE leads to significantly better models than random initialization. We initialized the logistic regressions in the HME to split
the input vectors recursively along their dimensions of greatest variance. The multinomial
distributions at leaf nodes were initialized by uniform distributions.
For an HME with m multinomial leaves and d-dimensional input vectors, the number of
parameters scales as O(V d + V m + dm). The resulting model can be therefore be much
more compact than a full bigram model over V words.
4
Results
We evaluated our models on the ARPA North American Business News (NAB) corpus.
Our training set contained 78 million words from a 60,000 word vocabulary. In the interest
of speed, we truncated the lowest-count bigrams from our training set. This left us with
a training set consisting of 1.7 million unique bigrams. The test set, untruncated, had 13
million words resulting in 2.1 million unique bigrams.
4.1
Empirical evaluation
Table 1 reports the test perplexities of several HMEs whose input vectors were initialized
in different ways. The number of mixture components (i.e., leaves of the HME) was fixed
at m = 16. In all cases, the inputs initialized by PCA and SDE significantly outperformed
random initialization. PCA and SDE initialization performed equally well for all but the
lowest-dimensional inputs. Here SDE outperformed PCA, most likely because the first few
eigenvectors of SDE capture more variance in the bigram counts than those of PCA (see
Figure 1).
Table 2 reports the test perplexities of several HMEs initialized by SDE, but with varying
input dimensionality (d) and numbers of leaves (m). Perplexity decreases with increasing
tree depth and input dimensionality, but increasing the dimensionality beyond d = 8 does
not appear to give much gain.
4.2
Comparison to a class-based bigram model
w
z
w'
Figure 4: Belief network for AMM.
We obtained baseline results from an AMM [13]
trained on the same corpus. The model (Figure 4)
has the form
X
Pr(w0 |w) =
Pr(z|w) ? Pr(w0 |z). (3)
z
The number of estimated parameters in AMMs
scales as 2?|Z|?V , where |Z| is the size of the latent variable (i.e., number of classes)
and V is the number of words in the vocabulary.
parameters (*1000)
960
1440
2400
4320
Ptest (AMM)
456
414
353
310
Ptest (HME)
429
361
328
308
improvement
6%
13%
7%
1%
Table 3: Test perplexities of HMEs and AMMs with roughly equal parameter counts.
Table 3 compares the test perplexities of several HMEs and AMMs with similar numbers
of parameters. All these HMEs had d = 8 inputs initialized by SDE. In all cases, the HMEs
match or outperform the AMMs. The performance is nearly equal for the larger models,
which may be explained by the fact that most of the parameters of the larger HMEs come
from the multinomial leaves, not from the distributed inputs.
4.3
Comparison to NPLM
The most successful large-scale application of distributed representations to language modeling is the NPLM of Bengio et al. [2, 3], which in part inspired our work. We now compare
the main aspects of the two models.
?
m
8
16
32
64
d
4
1
2
4
9
8
1
2
4
10
12
1
2
4
10
16
1
2
4
10
Table 4: Training times ? in hours
for HMEs with m leaves.
The NPLM uses softmax to compute the probability of a word w0 given its context, thus requiring a
separate normalization for each context. Estimating the parameters of this softmax requires O(V )
computation per observed context and accounts
for almost all of the computational resources required by the model. Because of this, the NPLM
vocabulary size was restricted to 18000 words,
and even then it required more than 3 weeks using 40 CPUs to finish 5 epochs of training [2].
By contrast, our HMEs require O(md) computation per observed bigram. As Table 4 shows, actual training times are rather insensitive to input dimensionality. This allowed us to use
a 3.5? larger vocabulary and a larger training corpus than were used for the NPLM, and
still complete training our largest models in a matter of hours. Note that the numbers in
Table 4 do not include the time to compute the initial distributed representations by PCA
(30 minutes) or SDE (3 days), but these computations do not need to be repeated for each
trained model.
The second difference between our model and the NPLM is the choice of initialization.
Bengio et al. [3] report negligible improvement from initializing the NPLM input vectors
by singular value decomposition. By contrast, we found that initialization by PCA or SDE
was essential for optimal performance of our models (Table 1).
Finally, the NPLM was applied to multiword contexts. We have not done these experiments yet, but our model extends naturally to multiword contexts, as we explain in the next
section.
5
Discussion
In this paper, we have presented a statistical language model that exploits hierarchical
distributed representations of word contexts. The model shares the advantages of the
NPLM [2], but differs in its use of dimensionality reduction for effective parameter ini-
tialization and in the significant speedup provided by the HME architecture. We can consequently scale our models to larger training corpora and vocabularies. We have also demonstrated that our models consistently match or outperform a baseline class-based bigram
model.
The class-based bigram model is nearly as effective as the HME, but it has the major drawback that there is no straightforward way to extend it to multiword contexts without exploding its parameter count. Like the NPLM, however, the HME can be easily extended.
We can form an input vector for a multiword history (w1 , w2 ) simply by concatenating the
vectors ~x(w1 ) and ~x(w2 ). The parameters of the corresponding HME can be learned by
an EM algorithm similar to the one in this paper. Initialization from dimensionality reduction is also straightforward: we can compute the low dimensional representation for each
word separately. We are actively pursuing these ideas to train models with hierarchical
distributed representations of multiword contexts.
References
[1] A. Y. Alfakih, A. Khandani, and H. Wolkowicz. Solving Euclidean distance matrix completion problems via semidefinite programming. Computational Optimization Applications, 12(13):13?30, 1999.
[2] Y. Bengio, R. Ducharme, P. Vincent, and C. Janvin. A neural probabilistic language model.
Journal of Machine Learning Research, 3:1137?1155, 2003.
[3] Y. Bengio, R. Ducharme, P. Vincent, and C. Jauvin. A neural probabilistic language model. In
T. K. Leen, T. G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing
Systems, volume 13, Cambridge, MA, 2001. MIT Press.
[4] D. B. Borchers. CSDP, a C library for semidefinite programming. Optimization Methods and
Software, 11(1):613?623, 1999.
[5] P. Brown, S. D. Pietra, V. D. Pietra, and R. Mercer. The mathematics of statistical machine
translation: parameter estimation. Computational Linguistics, 19(2):263?311, 1991.
[6] P. F. Brown, V. J. D. Pietra, P. V. deSouza, J. C. Lai, and R. L. Mercer. Class-based n-gram
models of natural language. Computational Linguistics, 18(4):467?479, 1992.
[7] S. Chen and J. Goodman. An empirical study of smoothing techniques for language modeling.
In Proceedings of the 34th Annual Meeting of the ACL, pages 310?318, 1996.
[8] M. Collins. Three generative, lexicalised models for statistical parsing. In Proceedings of the
35th Annual Meeting of the Association for Computational Linguistics, 1997.
[9] J. Ham, D. D. Lee, S. Mika, and B. Sch?olkopf. A kernel view of the dimensionality reduction of
manifolds. In Proceedings of the Twenty First International Conference on Machine Learning
(ICML-04), Banff, Canada, 2004.
[10] T. Hofmann and J. Puzicha. Statistical models for co-occurrence and histogram data. In Proceedings of the International Conference Pattern Recognition, pages 192?194, 1998.
[11] F. Jelinek. Statistical Methods for Speech Recognition. MIT Press, 1997.
[12] M. I. Jordan and R. A. Jacobs. Hierarchical mixtures of experts and the EM algorithm. Neural
Computation, 6:181?214, 1994.
[13] L. K. Saul and F. C. N. Pereira. Aggregate and mixed-order Markov models for statistical
language processing. In C. Cardie and R. Weischedel, editors, Proceedings of the Second Conference on Empirical Methods in Natural Language Processing (EMNLP-97), pages 81?89,
New Providence, RI, 1997.
[14] K. Q. Weinberger, F. Sha, and L. K. Saul. Learning a kernel matrix for nonlinear dimensionality
reduction. In Proceedings of the Twenty First International Confernence on Machine Learning
(ICML-04), Banff, Canada, 2004.
[15] C. Zhai and J. Lafferty. A study of smoothing methods for language models applied to information retrieval. ACM Transactions on Information Systems, 22(2):179?214, 2004.
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1,863 | 2,692 | Limits of Spectral Clustering
Ulrike von Luxburg and Olivier Bousquet
Max Planck Institute for Biological Cybernetics
Spemannstr. 38, 72076 T?ubingen, Germany
{ulrike.luxburg,olivier.bousquet}@tuebingen.mpg.de
Mikhail Belkin
The University of Chicago, Department of Computer Science
1100 E 58th st., Chicago, USA
[email protected]
Abstract
An important aspect of clustering algorithms is whether the partitions
constructed on finite samples converge to a useful clustering of the whole
data space as the sample size increases. This paper investigates this
question for normalized and unnormalized versions of the popular spectral clustering algorithm. Surprisingly, the convergence of unnormalized
spectral clustering is more difficult to handle than the normalized case.
Even though recently some first results on the convergence of normalized spectral clustering have been obtained, for the unnormalized case
we have to develop a completely new approach combining tools from
numerical integration, spectral and perturbation theory, and probability.
It turns out that while in the normalized case, spectral clustering usually
converges to a nice partition of the data space, in the unnormalized case
the same only holds under strong additional assumptions which are not
always satisfied. We conclude that our analysis gives strong evidence for
the superiority of normalized spectral clustering. It also provides a basis
for future exploration of other Laplacian-based methods.
1
Introduction
Clustering algorithms partition a given data set into several groups based on some notion
of similarity between objects. The problem of clustering is inherently difficult and often
lacks clear criteria of ?goodness?. Despite the difficulties in determining the quality of a
given partition, it is still possible to study desirable properties of clustering algorithms from
a theoretical point of view. In this paper we study the consistency of spectral clustering,
which is an important property in the general framework of statistical pattern recognition.
A clustering algorithm is consistent if it produces a well-defined (and, hopefully, sensible)
partition, given sufficiently many data points. The consistency is a basic sanity check, as an
algorithm which is not consistent would change the partition indefinitely as we add points to
the dataset, and, consequently, no reasonable small-sample performance could be expected
at all. Surprisingly, relatively little research into consistency of clustering algorithms has
been done so far, exceptions being only k-centers (Pollard, 1981) and linkage algorithms
(Hartigan, 1985).
While finite-sample properties of spectral clustering have been studied from a theoretical
point of view (Spielman and Teng, 1996; Guattery and Miller, 1998; Kannan et al., 2000;
Ng et al., 2001; Meila and Shi, 2001) we focus on the limit behavior for sample size tending
to infinity. In this paper we develop a new strategy to prove convergence results for spectral
clustering algorithms. Unlike our previous attempts this strategy allows to obtain results
for both normalized and unnormalized spectral clustering. As a first result we can recover
the main theorem of von Luxburg et al. (2004), which had been proved with different and
more restrictive methods, and, in brief, states that usually normalized spectral clustering
converges. We also extend that result to the case of multiple eigenvectors. Our second result
concerns the case of unnormalized spectral clustering, for which no convergence properties
had been known so far. This case is much more difficult to treat than the normalized
case, as the limit operators have a more complicated form. We show that unnormalized
spectral clustering also converges, but only under strong additional assumptions. Contrary
to the normalized case, those assumptions are not always satisfied, as we can show by
constructing an example, and in this case there is no hope for convergence. Even worse,
on a finite sample it is impossible to verify whether the assumptions hold or not. As a third
result we prove statements about the form of the limit clustering. It turns out that in case
of convergence, the structure of the clustering constructed on finite samples is conserved in
the limit process. From this we can conclude that if convergence takes place, then the limit
clustering presents an intuitively appealing partition of the data space.
It is also interesting to note that several recent methods for semi-supervised and transductive learning are based on eigenvectors of similarity graphs (cf. Belkin and Niyogi, 2003;
Chapelle et al., 2003; Zhu et al., 2003). Our theoretical framework can also be applied to
investigate the consistency of those algorithms with respect to the unlabeled data.
There is an ongoing debate on the advantages of the normalized versus unnormalized graph
Laplacians for spectral clustering. It has been found empirically that the normalized version
performs as well or better than the unnormalized version (e.g., Van Driessche and Roose,
1995; Weiss, 1999; in the context of semi-supervised learning see also Zhou et al., 2004).
We are now able to provide additional evidence to this effect from a theoretical point of
view. Normalized spectral clustering is a well-behaved algorithm which always converges
to a sensible limit clustering. Unnormalized spectral clustering on the other hand should be
treated with care as consistency can only be asserted under strong assumptions which are
not always satisfied and, moreover, are difficult to check in practice.
2
Graph Laplacians and spectral clustering on finite samples
In the following we denote by ?(T ) the spectrum of a linear operator, by C(X ) the space
of continuous functions on X with infinity norm, and by rg(d) the range of a function
d ? C(X ). For given sample points X1 , ..., Xn drawn iid according to an (unknown) distribution P on some data space X we denote the empirical distribution by Pn . For a nonnegative, symmetric similarity function s :PX ?X ? IR we define the similarity matrix
n
as Kn := (s(Xi , Xj ))i,j=1,...,n , set di := j=1 s(Xi , Xj ), and define the degree matrix
Dn as the diagonal matrix with entries di . The unnormalized Laplacian matrix is defined
?1/2
?1/2
as Ln := Dn ? Kn , and two common ways of normalizing it are L?n := Dn Ln Dn
??
?1
or Ln := Dn Ln . In the following we always arrange the eigenvalues of the Laplacian
matrices in non-decreasing order 0 = ?1 ? ?2 ... ? ?n respecting their multiplicities. In its
simplest form, unnormalized (resp. normalized) spectral clustering partitions the sample
points Xi into two groups according to whether the i-th coordinate of the second eigenvector is larger or smaller than a certain threshold b ? IR. Often, instead of considering
only the second eigenvector, one uses the first r eigenvectors (for some small number r)
simultaneously to obtain a partition into several sets. For an overview of different spectral
clustering algorithms see for example Weiss (1999).
3
Limit results
In this section we want to state and discuss our main results. The general assumptions
in the following three theorems are that the data space X is a compact metric space from
which the sample points (Xi )i?IN are drawn independently according to an unknown probability distribution P . Moreover we require the similarity function s : X ?X ? IR to be
continuous, and in the normalized case to be bounded away from 0, that is s(x, y) > l > 0
for all x, y ? X and some l ? IR. By d ? C(X ) we will denote the ?degree function?, and
U ? and U will denote the ?limit operators? of L?n and Ln for n ? ?. The exact definitions
of these functions and operators, as well as all further mathematical details, definitions, and
proofs will be postponed to Section 4.
Let us start with the first question raised in the introduction: does the spectral clustering
constructed on a finite sample converge to a partition of the whole data space if the sample
size increases? In the normalized case, convergence results have recently been obtained
in von Luxburg et al. (2004). However, those methods were specifically designed for the
normalized Laplacian and cannot be used in the unnormalized case. Here we state a convergence result for the normalized case in the form how it can be obtained with our new
methods. The theorem is formulated for the symmetric normalization L?n , but it holds
similarly for the normalization L??n .
Theorem 1 (Convergence of normalized spectral clustering) Under the general assumptions, if the first r eigenvalues of the limit operator U ? have multiplicity 1, then the
same holds for the first r eigenvalues of L?n for sufficiently large n. In this case, the first r
eigenvalues of L?n converge to the first r eigenvalues of U ? , and the corresponding eigenvectors converge almost surely. The partitions constructed by normalized spectral clustering
from the first r eigenvectors on finite samples converge almost surely to a limit partition of
the whole data space.
Our new result about the convergence in the unnormalized case is the following:
Theorem 2 (Convergence of unnormalized spectral clustering) Under the general assumptions, if the first r eigenvalues of the limit operator U have multiplicity 1 and are not
element of rg(d), then the same holds for the first r eigenvalues of n1 Ln for sufficiently
large n. In this case, the first r eigenvalues of n1 Ln converge to the first r eigenvalues of U ,
and the the corresponding eigenvectors converge almost surely. The partitions constructed
by unnormalized spectral clustering from the first r eigenvectors on finite samples converge
almost surely to a limit partition of the whole data space.
On the first glance, this theorem looks very similar to Theorem 1: if the general assumptions are satisfied and the first eigenvalues are ?nice?, then unnormalized spectral clustering converges. However, the difference between Theorems 1 and 2 is what it means for
an eigenvalue to be ?nice?. In Theorem 1 we only require the eigenvalues to have multiplicity 1 (and in fact, if the multiplicity is larger than 1 we can still prove convergence
of eigenspaces instead of eigenvectors). In Theorem 2, however, the condition ? 6? rg(d)
has to be satisfied. In the proof this is needed to ensure that the eigenvalue ? is isolated
in the spectrum of U , which is a fundamental requirement for applying perturbation theory
to the convergence of eigenvectors. If this condition is not satisfied, perturbation theory
is in principle unsuitable to obtain convergence results for eigenvectors. The reason why
this condition appears in the unnormalized case but not in the normalized case lies in the
structure of the respective limit operators, which, surprisingly, is more complicated in the
unnormalized case than in the normalized one. In the next section we will construct an example where the second eigenvalue indeed lies within rg(d). This means that there actually
exist situations in which Theorem 2 cannot be applied, and hence unnormalized spectral
clustering might not converge.
Now we want to turn to the second question raised in the introduction: In case of convergence, is the limit clustering a reasonable clustering of the whole data space? To answer
this question we analyze the structure of the limit operators (for simplicity we state this
for the unnormalized case only). Assume that we are given a partition X = ?ki=1 Xi
of the data space into k disjoint sets. If we order the sample points according to their
memberships in the sets Xi , then we can write the Laplacian in form of a block matrix
Ln ? (Lij,n )i,j=1,...,k where each sub-matrix Lij,n contains the rows of Ln corresponding
to points in set Xi and the columns corresponding to points in Xj . In a similar way, the limit
operator U can be decomposed into a matrix of operators Uij : C(Xj ) ? C(Xi ). Now we
can show that for all i, j = 1, ..., k the sub-matrices n1 Lij,n converge to the corresponding
sub-operators Uij such that their spectra converge in the same way as in Theorems 1 and 2.
This is a very strong result as it means that for every given partition of X , the structure of
the operators is preserved in the limit process.
Theorem 3 (Structure of the limit operators) Let X = ?ki=1 Xi be a partition of the data
space. Let Lij,n be the sub-matrices of Ln introduced above, Uij : C(Xj ) ? C(Xi ) the
?
?
restrictions of U corresponding to the sets Xi and Xj , and Uij,n
and Uij
the analogous
1
quantities for the normalized case. Then under the general assumptions, n Lij,n converges
?
compactly to Uij a.s. and L?ij,n converges compactly to Uij
a.s.
With this result it is then possible to give a first answer on the question how the limit
partitions look like. In Meila and Shi (2001) it has been established that normalized spectral
clustering tries to find a partition such that a random walk on the sample points tends to
stay within each of the partition sets Xi instead of jumping between them. With the help
of Theorem 3, the same can now be said for the normalized limit partition, and this can
also be extended to the unnormalized case. The operators U ? and U can be interpreted as
diffusion operators on the data space. The limit clusterings try to find a partition such that
the diffusion tends to stay within the sets Xi instead of jumping between them. In particular,
the limit partition segments the data space into sets such that the similarity within the sets
is high and the similarity between the sets is low, which intuitively is what clustering is
supposed to do.
4
Mathematical details
In this section we want to explain the general constructions and steps that need to be
taken to prove Theorems 1, 2, and 3. However, as the proofs are rather technical we only
present proof sketches that convey the overall strategy. Detailed proofs can be found in von
Luxburg (2004) where all proofs are spelled out in full length. Moreover, we will focus on
the proof of Theorem 2 as the other results can be proved similarly.
To be able to define convergence of linear operators, all operators have to act on the same
space. As this is not the case for the matrices Ln for different n, for each Ln we will
construct a related operator Un on the space C(X ) which will be used instead of Ln . In
Step 2 we show that the interesting eigenvalues and eigenvectors of n1 Ln and Un are in a
one-to-one relationship. Then we will prove that the Un converge in a strong sense to some
limit operator U on C(X ) in Step 3. As we can show in Step 4, this convergence implies
the convergence of eigenvalues and eigenvectors of Un . Finally, assembling the parts will
finish the proof of Theorem 2.
Step 1: Construction of the operators Un on C(X ).
We first define the empirical and true degree functions in C(X ) as
Z
Z
dn (x) := s(x, y)dPn (y)
and
d(x) := s(x, y)dP (y).
Corresponding to the matrices Dn and Kn we introduce the following multiplication and
integral operators on C(X ):
Mdn f (x) := dn (x)f (x)
Z
Sn f (x) := s(x, y)f (y)dPn (y)
and
and
Md f (x) := d(x)f (x)
Z
Sf (x) := s(x, y)f (y)dP (y).
Note that dn (Xi ) = n1 di , and for f ? C(X ) and v := (f (X1 ), ..., f (Xn ))? it holds
that n1 (Dn v)i = Mdn f (Xi ) and n1 (Kn v)i = Sn f (Xi ). Hence the function dn and the
operators Mdn and Sn are the counterparts of the discrete degrees n1 di and the matrices
1
1
n Dn and n Kn . The scaling factor 1/n comes from the hidden 1/n-factor in the empirical
distribution Pn . The natural pointwise limits of dn , Mdn , and Sn for n ? ? are given
by d, Md , and S. The operators corresponding to the unnormalized Laplacians n1 Ln =
1
n (Dn ? Kn ) and its limit operator are
Un f (x) := Mdn f (x) ? Sn f (x)
and
U f (x) := Md f (x) ? Sf (x).
1
Step 2: Relations between ?( n
Ln ) and ?(Un ).
Proposition 4 (Spectral properties) 1. The spectrum of Un consists of rg(dn ), plus some
isolated eigenvalues with finite multiplicity. The same holds for U and rg(d).
2. If f ? C(X ) is an eigenfunction of Un with arbitrary eigenvalue ?, then the vector
v ? IRn with vi = f (Xi ) is an eigenvector of the matrix n1 Ln with eigenvalue ?.
3. If v is an eigenvector of the matrix n1 Ln with eigenvalue ? 6? rg(dn ), then the
P
function f (x) = n1 ( j s(x, Xj )vj )/(dn (x) ? ?) is the unique eigenfunction of Un with
eigenvalue ? satisfying f (Xi ) = vi .
Proof. It is well-known that the (essential) spectrum of a multiplication operator coincides
with the range of the multiplier function. Moreover, the spectrum of a sum of a bounded
operator with a compact operator contains the essential spectrum of the bounded operator.
Additionally, it can only contain some isolated eigenvalues with finite multiplicity (e.g.,
Theorem IV.5.35 in Kato, 1966). The proofs of the other parts of this proposition can be
obtained by elementary shuffling of eigenvalue equations and will be skipped.
,
Step 3: Convergence of Un to U .
Dealing with the randomness. Recall that the operators Un are random operators as
they depend on the given sample points X1 , ..., Xn via the empirical distribution Pn . One
important tool to cope with this randomness will be the following proposition:
Proposition 5 (Glivenko-Cantelli class) Let (X , d) be a compact metric space and s :
X ?X ? RIR continuous. Then
R F := {s(x, ?); x ? X } is a Glivenko-Cantelli class, that is
supx?X | s(x, y)dPn (y) ? s(x, y)dP (y)| ? 0 almost surely.
Proof. This proposition follows from Theorem 2.4.1. of v. d. Vaart and Wellner (1996). ,
Note that one direct consequence of this proposition is that kdn ? dk? ? 0 a.s.
Types of convergence. Let E be an arbitrary Banach space and B its unit
S ball. A sequence
(Sn )n of linear operators on E is called collectively compact if the set n Sn B is relatively
compact in E (with respect to the norm topology). A sequence of operators converges
collectively compactly if it converges pointwise and if there exists some N ? IN such that
the operators (Sn ? S)n>N are collectively compact. A sequence of operators converges
compactly if it converges pointwise and if for every sequence (xn )n in B, the sequence
(S?Sn )xn is relatively compact. See Anselone (1971) and Chatelin (1983) for background
reading. A sequence (xn )n in E converges up to a change of sign to x ? E if there exists
a sequence (an )n of signs an ? {?1, +1} such that the sequence (an xn )n converges to x.
Proposition 6 (Un converges compactly to U a.s.) Let X be a compact metric space and
s : X ?X ? IR continuous. Then Un converges to U compactly a.s.
Proof. (a) Sn converges to S collectively compactly a.s. With the help of the GlivenkoCantelli property in Proposition 5 it is easy to see that Sn converges to S pointwise, that
is kSn f ? Sf k? ? 0 a.s. for all f ? C(X ). As the limit operator S is compact, to
prove that (Sn ? S)n are collectively compact a.s. it is enough to prove that (Sn )n are
collectively compact a.s. This can be done by the Arzela-Ascoli theorem.
(b) Mdn converges to Md in operator norm a.s. This is a direct consequence of the
Glivenko-Cantelli properties of Proposition 5.
(c) Un = Sn ? Mdn converges to U = S ? Md compactly a.s. Both operator
norm convergence and collectively compact convergence imply compact convergence
(cf. Proposition 3.18 of Chatelin, 1983). Moreover, it is easy to see that the sum of two
compactly converging operators converges compactly.
,
Step 4: Convergence of the eigenfunctions of Un to those of U .
It is a result of perturbation theory (see the comprehensive treatment in Chatelin, 1983,
especially Section 5.1) that compact convergence of operators implies the convergence of
eigenvalues and spectral projections in the following way. If ? is an isolated eigenvalue in
?(U ) with finite multiplicity, then there exists a sequence ?n ? ?(Un ) of isolated eigenvalues with finite multiplicity such that ?n ? ?. If the first r eigenvalues of T have
multiplicity 1, then the same holds for the first r eigenvalues of Tn for sufficiently large
n, and the i-th eigenvalues of Tn converge to the i-th eigenvalue of T . The corresponding
eigenvectors converge up to a change of sign. If the multiplicity of the eigenvalues is larger
than 1 but finite, then the corresponding eigenspaces converge. Note that for eigenvalues
which are not isolated in the spectrum, convergence cannot be asserted, and the same holds
for the corresponding eigenvectors (e.g., Section IV.3 of Kato, 1966).
In our case, by Proposition 4 we know that the spectrum of U consists of the whole interval rg(d), plus eventually some isolated eigenvalues. Hence an eigenvalue ? ? ?(U )
is isolated in the spectrum iff ? 6? rg(d) holds, in which case convergence holds as stated
above.
Step 5: Convergence of unnormalized spectral clustering.
Now we can to put together the different parts. In the first two steps we transferred the
problem of the convergence of the eigenvectors of n1 Ln to the convergence of eigenfunctions of Un . In Step 3 we showed that Un converges compactly to the limit operator U ,
which according to Step 4 implies the convergence of the eigenfunctions of Un . In terms
of the eigenvectors of n1 Ln this means the following: if ? denotes the j-th eigenvalue
of U with eigenfunction f ? C(X ) and ?n the j-th eigenvalue of n1 Ln with eigenvector vn = (vn,1 , ..., vn,n )? , then there exists a sequence of signs ai ? {?1, +1} such that
supi=1,...,n |ai vn,i ? f (Xi )| ? 0 a.s. As spectral clustering is constructed from the coordinates of the eigenvectors, this leads to the convergence of spectral clustering in the
unnormalized case. This completes the proof of Theorem 2.
,
The proof for Theorem 1 can be obtained in a very similar way. Here the limit operator is
Z
p
U ? f (x) := (I ? S ? )f (x) := f (x) ? (s(x, y)/ d(x)d(y) )f (y)dP (y).
The main difference to the unnormalized case is that the operator Md in U gets replaced by
the identity operator I in U ? . This simplifies matters as one can easily express the spectrum
of (I ? S ? ) via the spectrum of the compact operator S ? . From a different point of view,
consider the identity operator as the operator of multiplication by the constant one function
1. Its range is the single point rg(1) = {1}, and hence the critical interval rg(d) ? ?(U )
shrinks to the point 1 ? ?(U ? ), which in general is a non-isolated eigenvalue with infinite
multiplicity.
Finally, note that it is also possible to prove more general versions of Theorems 1 and 2
where the eigenvalues have finite multiplicity larger than 1. Instead of the convergence of
the eigenvectors we then obtain the convergence of the projections on the eigenspaces.
The proof of Theorem 3 works as the ones of the other two theorems. The exact definitions
of the operators considered in this case are
Z
q
?
Uij : C(Xj ) ? C(Xi ), ?ij fi (x) ? (sij (x, y)/ di (x)dj (y) )fj (y)dPj (y)
Z
Uij : C(Xj ) ? C(Xi ), Uij f (x) = ?ij di (x)fi (x) ? sij (x, y)fj (y)dPj (y)
where di , fi , Pi , and sij denote the restrictions of the functions to Xi and Xi ?Xj , respectively, and ?ij is 1 if i = j and 0 otherwise. For the diffusion interpretation, note that if
there exists an ideal partition of the data space (that is, s(xi , xj ) = 0 for xi , xj in different
?
sets Xi and Xj ), then the off-diagonal operators Uij
and Uij with i 6= j vanish, and the
?
first k eigenvectors of U and U can be reconstructed by the piecewise constant eigenvectors of the diagonal operators Uii? and Uii . In this situation, spectral clustering recovers
the ideal clustering. If there exists no ideal clustering, but there exists a partition such that
the off-diagonal operators are ?small? and the diagonal operators are ?large?, then it can
be seen by perturbation theory arguments that spectral clustering will find such a partition.
The off-diagonal operators can be interpreted as diffusion operators between different clusters (note that even in the unnormalized case, the multiplication operator only appears in
the diagonal operators). Hence, constructing a clustering with small off-diagonal operators
corresponds to a partition such that few diffusion between the clusters takes place.
Finally, we want to construct an example where the second eigenvalue of U satisfies ? ?
rg(d). Let X = [1, 2] ? IR, s(x, y) := xy, and p a piecewise constant probability density
on X with p(x) = c if 4/3 ? x < 5/3 and p(x) = (3 ? c)/2 otherwise, for some fixed
constant c ? [0, 3] (e.g., for small c this density has two clearly separated high density
regions). The degree function in this case is d(x) = 1.5x (independently of c) and has
range [1.5, 3] on X . We can see that an eigenfunction of U Rfor eigenvalue ? 6? rg(d) has
the form f (x) = ?x/(3x ? ?), where the equation ? = x2 /(3x ? ?)p(x)dx has to
be satisfied. This means that ? 6? rg(d) is an eigenvalue of U iff the equation g(?) :=
R2 2
!
x /(3x ? ?)p(x)dx = 1 is satisfied. For our simple density function p, this integral
1
can be solved analytically. It can then been seen that g(?) = 1 is only satisfied for ? = 0,
hence the only eigenvalue outside of rg(d) is the trivial eigenvalue 0.
Note that in applications of spectral clustering, we do not know the limit operator U and
hence cannot test whether its relevant eigenvalues are in its essential spectrum rg(d) or
not. If, for some special reason, one really wants to use unnormalized spectral clustering,
one should at least estimate the critical region rg(d) by [mini di /n, maxi di /n] and check
whether the relevant eigenvalues of n1 Ln are inside or close to this interval or not. This observation then gives an indication whether the results obtained can considered to be reliable
or not. However, this observation is not a valid statistical test.
5
Conclusions
We have shown that under standard assumptions, normalized spectral clustering always
converges to a limit partition of the whole data space which depends only on the probability
distribution P and the similarity function s. For unnormalized spectral clustering, this can
only be guaranteed under the strong additional assumption that the first eigenvalues of the
Laplacian do not fall inside the range of the degree function. As shown by our example,
this condition has to be taken seriously.
Consistency results are a basic sanity check for behavior of statistical learning algorithms.
Algorithms which do not converge cannot be expected to exhibit reliable results on finite
samples. Therefore, in the light of our theoretical analysis we assert that the normalized
version of spectral clustering should be preferred in practice. This suggestion also extends
to other applications of graph Laplacians including semi-supervised learning.
References
P. Anselone. Collectively compact operator approximation theory. Prentice-Hall, 1971.
M. Belkin and P. Niyogi. Using manifold structure for partially labeled classification. In
Advances in Neural Information Processing Systems 15, 2003.
O. Chapelle, J. Weston, and B. Sch?olkopf. Cluster kernels for semi-supervised learning. In
Advances in Neural Information Processing Systems 15, 2003.
F. Chatelin. Spectral Approximation of Linear Operators. Academic Press, 1983.
S. Guattery and G. L. Miller. On the quality of spectral separators. SIAM Journal of Matrix
Anal. Appl., 19(3), 1998.
J. Hartigan. Statistical theory in clustering. Journal of classification, 2:63?76, 1985.
R. Kannan, S. Vempala, and A. Vetta. On clusterings - good, bad and spectral. Technical
report, Computer Science Department, Yale University, 2000.
T. Kato. Perturbation theory for linear operators. Springer, Berlin, 1966.
M. Meila and J. Shi. A random walks view of spectral segmentation. In 8th International
Workshop on Artificial Intelligence and Statistics, 2001.
A. Ng, M. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In
Advances in Neural Information Processing Systems 14, 2001.
D. Pollard. Strong consistency of k-means clustering. Ann. of Stat., 9(1):135?140, 1981.
D. Spielman and S. Teng. Spectral partitioning works: planar graphs and finite element
meshes. In 37th Annual Symposium on Foundations of Computer Science, 1996.
A. v. d. Vaart and J. Wellner. Weak Convergence and Empirical Processes. Springer, 1996.
R. Van Driessche and D. Roose. An improved spectral bisection algorithm and its application to dynamic load balancing. Parallel Comput., 21(1), 1995.
U. von Luxburg. Statistical Learning with Similarity and Dissimilarity Functions. PhD
thesis, draft, available at http://www.kyb.tuebingen.mpg.de/?ule, 2004.
U. von Luxburg, O. Bousquet, and M. Belkin. On the convergence of spectral clustering
on random samples: the normalized case. In COLT, 2004.
Y. Weiss. Segmentation using eigenvectors: A unifying view. In Proceedings of the International Conference on Computer Vision, pages 975?982, 1999.
D. Zhou, O. Bousquet, T. Lal, J. Weston, and B. Sch?olkopf. Learning with local and global
consistency. In Advances in Neural Information Processing Systems 16, 2004.
X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields
and harmonic functions. In ICML, 2003.
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1,864 | 2,693 | Semi-parametric exponential family PCA
Sajama
Alon Orlitsky
Department of Electrical and Computer Engineering
University of California at San Diego, La Jolla, CA 92093
[email protected] and [email protected]
Abstract
We present a semi-parametric latent variable model based technique for
density modelling, dimensionality reduction and visualization. Unlike
previous methods, we estimate the latent distribution non-parametrically
which enables us to model data generated by an underlying low dimensional, multimodal distribution. In addition, we allow the components
of latent variable models to be drawn from the exponential family which
makes the method suitable for special data types, for example binary or
count data. Simulations on real valued, binary and count data show favorable comparison to other related schemes both in terms of separating
different populations and generalization to unseen samples.
1
Introduction
Principal component analysis (PCA) is widely used for dimensionality reduction with applications ranging from pattern recognition and time series prediction to visualization. One
important limitation of PCA is that it is not based on a probability model. A probabilistic formulation of PCA can offer several advantages like allowing statistical testing,
application of Bayesian inference methods and naturally accommodating missing values
[1]. Latent variable models are commonly used in statistics to summarize observations
[2]. A latent variable model assumes that the distribution of data is determined by a latent or mixing
distribution P (?) and a conditional or component distribution P (x|?), i.e.,
R
P (x) = P (?)P (x|?)d?.
Probabilistic PCA (PPCA) [1] borrows from one such popular model called factor analysis to propose a probabilistic alternative PCA. A key feature of this probabilistic model
is that the latent distribution P (?) is also assumed to be Gaussian since it leads to simple
and fast model estimation, i.e., the density of x is approximated by a Gaussian distribution whose covariance matrix is aligned along a lower dimensional subspace. This may
be a good approximation when data is drawn from a single population and the goal is to
explain the data in terms of a few variables. However, in machine learning we often deal
with data drawn from several populations and PCA is used to reduce dimensions to control
computational complexity of learning. A mixture model with Gaussian latent distribution
would not be able to capture this information. The projection obtained using a Gaussian
latent distribution tends to be skewed toward the center [1] and hence the distinction between nearby sub-populations may be lost in the visualization space. For these reasons,
it is important not to make restrictive assumptions about the latent distribution. Several
recently proposed dimension reduction methods can, like PPCA, be thought of as special
cases of latent variable modelling which differ in the speci?c assumptions they make about
the latent and conditional distributions.
We present an alternative probabilistic formulation, called semi-parametric PCA (SPPCA), where no assumptions are made about the distribution of the latent random variable ?. Non-parametric latent distribution estimation allows us to approximate data density
better than previous schemes and hence gives better low dimensional representations. In
particular, multi-modality of the high dimensional density is better preserved in the projected space. When the observed data is composed of several clusters, this technique can
be viewed as performing simultaneous clustering and dimensionality reduction. To make
our method suitable for special data types, we allow the conditional distribution P (x|?) to
be any member of the exponential family of distributions. Use of exponential family distributions for P (x|?) is common in statistics where it is known as latent trait analysis and they
have also been used in several recently proposed dimensionality reduction schemes [3, 4].
We use Lindsay?s non-parametric maximum likelihood estimation theorem to reduce the
estimation problem to one with a large enough discrete prior. It turns out that this choice
gives us a prior which is ?conjugate? to all exponential family distributions, allowing us to
give a uni?ed algorithm for all data types. This choice also makes it possible to ef?ciently
estimate the model even in the case when different components of the data vector are of
different types.
2
The constrained mixture model
We assume that the d-dimensional observation vectors x
R 1 , . . . , xn are outcomes of iid
draws of a random variable whose distribution P (x) = P (?)P (x|?)d? is determined
by the latent distribution P (?) and the conditional distribution P (x|?). This can also be
viewed as a mixture density with P (?) being the mixing distribution, the mixture components labelled by ? and P (x|?) being the component distribution corresponding to ?. The
latent distribution is used to model the interdependencies among the components of x and
the conditional distribution to model ?noise?. For example in the case of a collection of
documents we can think of the ?content? of the document as a latent variable since it cannot
be measured. For any given content, the words used in the document and their frequency
may depend on random factors - for example what the author has been reading recently,
and this can be modelled by P (x|?).
Conditional distribution P (x|?): We assume that P (?) adequately models the dependencies among the components of x and hence that the components of x are independent
when conditioned upon ?, i.e., P (x|?) = ?j P (xj |?j ), where xj and ?j are the j?th components of x and ?. As noted in the introduction, using Gaussian means and constraining
them to a lower dimensional subspace of the data space is equivalent to using Euclidean
distance as a measure of similarity. This Gaussian model may not be appropriate for other
data types, for instance the Bernoulli distribution may be better for binary data and Poisson
for integer data. These three distributions, along with several others, belong to a family
of distributions known as the exponential family [5]. Any member of this family can be
written in the form
log P (x|?) = log P0 (x) + x? ? G(?)
where ? is called the natural parameter and G(?) is a function that ensures that the probabilities sum to one. An important property of this family is that the mean ? of a distribution
and its natural parameter ? are related through a monotone invertible, nonlinear function
? = G0 (?) = g(?). It can be shown that the negative log-likelihoods of exponential family
distributions can be written as Bregman distances (ignoring constants) which are a family
of generalized metrics associated with convex functions [4]. Note that by using different
distributions for the various components of x, we can model mixed data types.
Latent distribution P (?): Like previous latent variable methods, including PCA, we
constrain the latent variable ? to an `-dimensional Euclidean subspace of R d to model the
belief that the intrinsic dimensionality of the data is smaller than d. One way to represent
the (unknown) linear constraint on values that ? can take is to write it as an invertible linear
transformation of another random variable which takes values a ? R` ,
? = aV + b
where V is an ` ? d rotation matrix and b is a d-dimensional displacement vector. Hence
any distribution P? (?) satisfying the low dimensional constraints can be represented using a triple (P (a), V, b), where P (a) is a distribution over R` . Lindsay?s mixture nonparametric maximum likelihood estimation (NPMLE) theorem states that for ?xed (V ,b),
the maximum likelihood (ML) estimate of P (a) exists and is a discrete distribution with no
more than n distinct points of support [6]. Hence if ML is the chosen parameter estimation
technique, the SP-PCA model can be assumed (without loss of generality) to be a constrained ?nite mixture model with at most n mixture components. The number of mixture
components in the model, n, grows with the amount of data and we propose to use pruning
to reduce the number of components during model estimation to help both in computational
speed and model generalization. Finally, we note that instead of the natural parameter, any
of its invertible transformations could have been constrained to a lower dimensional space.
Choosing to linearly constrain the natural parameter affords us computational advantages
similar to those available when we use the canonical link in generalized linear regression.
Low dimensional representation:
There are several ways in which low dimensional
representations can be obtained using the constrained mixture model. We would ideally
like to represent a given observation x by the unknown ? (or the corresponding a related
to ? by ? = aV + b) that generated it, since the conditional distribution P (x|?) is used
to model random effects. However, the actual value of a is not known to us and all of our
knowledge of a is contained in the posterior distribution P (a|x) = P (a)P (x|a)/P (x).
Since a belongs to an `-dimensional space, any of its estimators like the posterior mean
or mode (MAP estimate) can be used to represent x in ` dimensions. For presenting the
simulation results in this paper, we use the posterior mean as the representation point.
This representation has been used in other latent variable methods to get meaningful low
dimensional views [1, 3]. Another method is to represent x by that point ? on (V, b) that
is closest according to the appropriate Bregman distance (it can be shown that there is a
unique such ? opt on the plane). This representation is a generalization of the standard
Euclidean projection and was used in [4].
The Gaussian case: When the exponential family distribution chosen is Gaussian, the
model is a mixture of n spherical Gaussians all of whose means lie on a hyperplane in
the data space. This can be thought of as a ?soft? version of PCA, i.e., Gaussian case of
SP-PCA is related to PCA in the same manner as Gaussian mixture model is related to
K-means. The use of arbitrary mixing distribution over the plane allows us to approximate
arbitrary spread of data along the hyperplane. Use of ?xed variance spherical Gaussians
ensures that like PCA, the direction perpendicular to the plane (V, b) is irrelevant in any
metric involving relative values of likelihoods P (x|? k ), including the posterior mean.
Consider the case when data density P (x) belongs to our model space, i.e., it is speci?ed by
{A, V, b, ?, ?} and let D be any direction parallel to the plane (V, b) along which the latent
distribution P (?) has non-zero variance. Since Gaussian noise with variance ? is added to
this latent distribution to obtain P (x), variance of P (x) along D will be greater than ?.
The variance of P (x) along any direction perpendicular to (V, b) will be exactly ?. Hence,
PCA of P (x) yields the subspace (V, b) which is the same as that obtained using SP-PCA
(this may not be true when P (x) does not belong to our model space). We found that
SP-PCA differs signi?cantly from PPCA in the predictive power of the low-dimensional
density model (see Section 5).
3
Model estimation
Algorithm for ML estimation: We present an EM algorithm for estimating parameters
of a ?nite mixture model with the components constrained to an `-dimensional Euclidean
subspace. We propose an iterative re-weighted least squares (IRLS) method for the maximization step along the lines of generalized linear model estimation. Use of weighted least
squares does not guarantee monotone increase in data likelihood. To guarantee convergence
of the algorithm, we can check the likelihood of data at the IRLS update and decrease step
size if necessary. Let x1 , . . . , xn be iid samples drawn from a d-dimensional density P (x),
c be the number of mixture components and let the mixing density be ? = (?1 , . . . , ?c ).
Associated with each mixture component (indexed by k) are parameter vectors ? k and ak
which are related by ? k = ak V + b. In this section we will work with the assumption
that all components of x correspond to the same exponential family for ease of notation.
For each observed xi there is an unobserved ?missing? variable zi which is a c-dimensional
binary vector whose k?th component is one if the k?th mixture component was the outcome
in the i?th random draw and zero otherwise. If yl is a vector, we use ylm to denote its m?th
component. (Derivation of the algorithm is omitted for lack of space, for details please see
[7]).
The E-step is identical to unconstrained ?nite mixture case,
?k P (xi /? k )
m=1 ?m P (xi /? m )
z?ik = E(zik ) = Pc
;
x
?kj =
In the M-step we update ?, V , b, and ak in the following manner
?k = P n
Pn
?ik
i=1 z
P
c
i=1
m=1 zim
=
Pn
;
[GRi ]l1 =
Pn
?ik xij
i=1 z
P
n
?ik
i=1 z
?ik
i=1 z
n
ai is updated by adding ?ai calculated using
(V ?i V 0 )?ai = GRi
;
[?i ]qq =
?g(?iq )
??iq
d
X
(?
xij ? g(?ij ))Vlj
j=1
Here the function g(?) is as de?ned in Section 2 and depends on the member of the exponential family that is being used. Each column of the matrix V , vs , is updated by adding
?vs calculated using
(A0 ?s A)?vs = GRs
;
[?s ]kk =
?g(?ks )
??ks
;
[GRs ]l1 =
c
X
(?
xk0 s ? g(?k0 s ))Ak0 l
k0 =1
Each component of vector b, bs , is updated by adding ?bs calculated using
Hs ?bs = GRs
;
Hs =
c
X
?g(?k0 s )
??k0 s
0
k =1
;
GRs =
c
X
(?
xk0 s ? g(?k0 s ))
k0 =1
Pruning the mixture components: Redundant mixture components can be pruned between the EM iterations in order to improve speed of the algorithm and generalization
properties while retaining the full capability to approximate P (x). We propose the following criteria for pruning
? Starved components : If ?k < C1 , then drop the k?th component
? Nearby components : If max
i |P (xi |? k1 ) ? P (x| ? k2 )| < C2 , then drop either k1?th
or k2?th component
The value of C1 should be ?(1/n) since we want to measure how starved a component
is based on what percentage of the data it is ?responsible? for. To measure the nearness of
components we use the ?-norm of the difference between probabilities the components
assign to observations since we do not want to lose mixture components that are distinguished with respect to a small number of observation vectors. In the case of clustering
this means that we do not ignore under-represented clusters. C2 should be chosen to be a
small constant, depending on how much pruning is desired.
Convergence of the EM iterations and computational complexity: It is easy to verify
that the SP-PCA model satis?es the continuity assumptions of Theorem 2, [8], and hence
we can conclude that any limit point of the EM iterations is a stationary point of the log
likelihood function. The computational complexity of the E-step is O(cdn) and of the Mstep is O(cd`2 ). For the Gaussian case, the E-step only takes O(c`n) since we only need
to take into account the variation of data along the subspace given by current value of V
(see Section 2). The most expensive step is computation of P (xi |? j ). The k-d tree data
structure is often used to identify relevant mixture components to speed up this step.
Model selection: While any of the standard model selection methods based on penalizing
complexity could be used to choose `, an alternative method is to pick ` which minimizes
a validation or bootstrap based estimate of the prediction error (negative log likelihood per
sample). For the Gaussian case, a fast method to pick ` would be to plot the variance of data
along the principal directions (found using PCA) and look for the dimension at which there
is a ?knee? or a sudden drop in variance or where the total residual variance falls below a
chosen threshold.
Consistency of the Maximum Likelihood estimator: We propose to use the ML estimator to ?nd the latent space (V, b) and the latent distribution P (a). Usually a parametric
form is assumed for P (a) and the consistency of the ML estimate is well known for this
task where the parameter space is a subset of a ?nite dimensional Euclidean space. In
the SP-PCA model, one of the parameters (P (a)) ranges over the space of all distribution
functions on R` and hence we need to do more to verify the validity of our estimator. Exponential family mixtures are not identi?able in general. This, however, is not a problem
for us since we are only interested in approximating P (x) well and not in the actual parameters corresponding to the distribution. Hence we use the de?nition of consistency of
an estimator given by Redner. Let ?0 be the ?true? parameter from which observed samples
are drawn. Let C0 be the set of all parameters ? corresponding to the ?true? distribution
F (x/?0 ) (i.e., C0 = {? : F (x/?) = F (x/?0 ) ? x}). Let ??n be an estimator of ? based
? be the quotient topological space obtained from ?
on n observed samples of X and let ?
obtained by identifying the set C0 to a point ??0 .
De?nition The sequence of estimators {?
? n , n = 1, . . . , ?} is said to be strongly
consistent in the sense of Redner if limm?? ??n = ??0 almost surely.
Theorem If P (a) is assumed to be zero outside a bounded subset of R` , the ML estimator of parameter (V, b, P (a)) is strongly consistent for Gaussian, Binary and Poisson
conditional distributions.
The theorem follows by verifying that the assumptions of Kiefer et. al. [9] are satis?ed
by the SP-PCA model. The assumption that P (a) is zero outside a bounded region is not
restrictive in practice since we expect the observations xi belong to a bounded region of
Rd . (Proof omitted for lack of space, please see [7]).
Table 1: Bootstrap estimates of prediction error for PPCA and SP-PCA.
D ENSITY
ERROR
4
I SOTROPIC
GAUSSIAN
`=1
PPCA
`=2
`=3
`=1
SP-PCA
`=2
`=3
F ULL
GAUSSIAN
50.39
38.03
34.71
34.76
36.85
30.99
28.54
343.83
Relationship to past work
SP-PCA is a factor model that makes fewer assumptions about latent distribution than
PPCA [1]. Mixtures of probabilistic principal component analyzers (also known as mixtures of factor analyzers) is a generalization of PPCA which overcomes the limitation of
global linearity of PCA via local dimensionality reduction. Mixtures of SP-PCA?s can be
similarly de?ned and used for local dimensionality reduction. Collins et. al. [4] proposed a
generalization of PCA using exponential family distributions. Note that this generalization
is not associated with a probability density model for the data. SP-PCA can be thought of
as a ?soft? version of this generalization of PCA, in the same manner as Gaussian mixtures
are a soft version of K-means. Generative topographic mapping (GTM) is a probabilistic
alternative to Self organizing map which aims at ?nding a nonlinear lower dimensional
manifold passing close to data points. An extension of GTM using exponential family distributions to deal with binary and count data is described in [3]. Apart from the fact that
GTM is a non-linear dimensionality reduction technique while SP-PCA is globally linear
like PCA, one main feature that distinguishes the two is the choice of latent distribution.
GTM assumes that the latent distribution is uniform over a ?nite and discrete grid of points.
Both the location of the grid and the nonlinear mapping are to be given as an input to the
algorithm. Tibshirani [10] used a semi-parametric latent variable model for estimation of
principle curves. Discussion of these and other dimensionality reduction schemes based on
latent trait and latent class models can be found in [7].
5
Experiments
In this section we present simulations on synthetic and real data to demonstrate the properties of SP-PCA. In factor analysis literature, it is commonly believed that choice of prior
distribution is unimportant for the low dimensional data summarization (see [2], Sections
2.3, 2.10 and 2.16). Through the examples below we argue that estimating the prior instead
of assuming it arbitrarily can make a difference when latent variable models are used for
density approximation, data analysis and visualization.
Use of SP-PCA as a low dimensional density model:
The Tobamovirus data which
consists of 38 18-dimensional examples was used in [1] to illustrate properties of PPCA.
PPCA and SP-PCA can be thought of as providing a range of low-dimensional density
models for the data. The complexity of these densities increases with and is controlled by
the value of ` (the projected space dimension) starting with the zero dimensional model of
an isotropic Gaussian. For a ?xed lower dimension `, SP-PCA has greater approximation
capability than PPCA. In Table 1, we present bootstrap estimates of the predictive power of
PPCA and SP-PCA for various values of L. SP-PCA has lower prediction error than PPCA
for ` = 1, 2 and 3. This indicates that SP-PCA combines ?exible density estimation and
excellent generalization even when trained on a small amount of data.
Simulation results on discrete datasets:
We present experiments on 20 Newsgroups
dataset comparing SP-PCA to PCA, exponential family GTM [3] and Exponential family
PCA [4]. Data for the ?rst set of simulations was drawn from comp.sys.ibm.pc.hardware,
comp.sys.mac.hardware and sci.med newsgroups. A dictionary size of 150 words was
chosen and the words in the dictionary were picked to be those which have maximum
mutual information with class labels. 200 documents were drawn from each of the three
newsgroups to form the training data. Two-dimensional representations obtained using
various methods are shown in Fig. 1. In the projection obtained using PCA, Exponential
family PCA and Bernoulli GTM, the classes comp.sys.ibm.pc.hardware and comp.sys.mac.hardware were not well separated in the 2D space. This result (Fig. 1(c)) was presented
in [3] and the the overlap between the two groups was attributed to the fact that they are
very similar and hence share many words in common. However, SP-PCA was able to
separate the three sets reasonably well (Fig. 1(d)). One way to quantify the separation of
dissimilar groups in the two-dimensional projections is to use the training set classi?cation
error of projected data using SVM. The accuracy of the best SVM classi?er (we tried a
range of SVM parameter values and picked the best for each projected data set) was 75% for
bernoulli GTM projection and 82.3% for SP-PCA projection (the difference corresponds to
44 data points while the total number of data points is 600). We conjecture that the reason
comp.sys.ibm.pc.hardware and comp.sys.mac.hardware have overlap in projection using
Bernoulli GTM is that the prior is assumed to be over a pre-speci?ed grid in latent space
and the spacing between grid points happened to be large in the parameter space close to the
two news groups. In contrast to this, in SP-PCA there is no grid and the latent distribution
is allowed to adapt to the given data set. Note that a standard clustering algorithm could be
used on the data projected using SP-PCA to conclude that data consisted of three kinds of
documents.
20
1
1
100
0.8
10
0.5
0.6
0
80
0.4
0
60
0.2
?0.5
0
?10
?1
40
?0.2
?20
20
?0.4
?1.5
?0.6
0
?2
?30
?0.8
?2.5
?20
?1
?1.5
?1
?0.5
0
0.5
1
1.5
(a) PCA
2
2.5
3
?40
?50
?40
?30
?20
?10
0
10
(b) Expontl. PCA
20
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
(c) GTM
0.6
0.8
1
?60
?40
?20
0
20
40
60
80
100
(d) SP-PCA
Figure 1: Projection by various methods of binary data from 200 documents each from
comp.sys.ibm.pc.hardware (?), comp.sys.mac.hardware (?) and sci.med (.)
Data for the second set of simulations was drawn from sci.crypt, sci.med, sci.space and
soc.culture.religion.christianity newsgroups. A dictionary size of 100 words was chosen
and again the words in the dictionary were picked to be those which have maximum mutual information with class labels. 100 documents were drawn from each of the newsgroups
to form the training data and 100 more to form the test data. Fig. 2 shows two-dimensional
representations of binary data obtained using various methods. Note that while the four
newsgroups are bunched together in the projection obtained using Exponential family PCA
[4] (Fig. 2(b)), we can still detect the presence four groups from this projection and in
this sense this projection is better than the PCA projection. This result is pleasing since it
con?rms our intuition that using negative log-likelihood of Bernoulli distribution as a measure of similarity is more appropriate than squared Euclidean distance for binary data. We
conjecture that the reason the four groups are not well separated in this projection is that
a conjugate prior has to be used in its estimation for computational purposes [4] and the
form and parameters of this prior are considered ?xed and given inputs to the algorithm.
Both SP-PCA (Fig. 2(c)) and Bernoulli GTM (Fig. 2(e)) were able to clearly separate the
clusters in the training data. Figures 2(d) and 2(f) show representation of test data using the
models estimated by SP-PCA and Bernoulli GTM respectively. To measure generalization
of these methods, we use a K-nearest neighbors based non-parametric estimate of the density of the projected training data. The percentage difference between the log-likelihoods
of training and test data with respect to this density was 9.1% for SP-PCA and 17.6% for
GTM for K=40 (SP-PCA had smaller percentage change in log-likelihood for most values
of K that we tried between 10 and 40). This indicates that SP-PCA generalizes better than
GTM. This can be seen visually by comparing Figures 2(e) and 2(f) where the projections
of training and test data of sci.space (?) differ signi?cantly.
1
14
0.5
12
0
50
10
0
?0.5
8
?1
?50
6
?1.5
4
?100
?2
2
?2.5
?150
?3
?4
?3
?2
?1
0
1
2
0
?20
?15
(a) PCA
?10
?5
0
5
10
15
20
25
30
?200
?150
(b) Exponential PCA
50
?100
?50
0
(c) SP-PCA
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0
0.2
0.2
0
0
?0.2
?0.2
?0.4
?0.4
?0.6
?0.6
?50
?100
?0.8
?0.8
?150
?1
?200
?150
?100
?50
(d) Test data - SP-PCA
0
?1
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
(e) Bernoulli GTM
0.8
1
?1
?0.8
?0.6
?0.4
?0.2
0
0.2
0.4
0.6
0.8
1
(f) Test data - GTM
Figure 2: Projection by various methods of binary data from 100 documents each from
sci.crypt (?), sci.med (?), sci.space (?) and soc.culture.religion.christianity (+)
Acknowledgments
We thank Sanjoy Dasgupta and Thomas John for helpful conversations.
References
[1] M. Tipping and C. Bishop. Probabilistic principal component analysis. Journal of the Royal
Statistical Society, Series B, 61(3):611?622, 1999.
[2] David J. Bartholomew and Martin Knott. Latent variable models and Factor analysis, volume 7
of Kendall?s Library of Statistics. Oxford University Press, 2nd edition, 1999.
[3] A. Kaban and M. Girolami. A combined latent class and trait model for the analysis and visualization of discrete data. IEEE Transaction on Pattern Analysis and Machine Intelligence,
23(8):859?872, August 2001.
[4] M. Collins, S. Dasgupta, and R. E. Schapire. A generalization of principal components analysis
to the exponential family. In Advances in Neural Information Processing Systems 14, 2002.
[5] P. McCullagh and J. A. Nelder. Generalized Linear Models. Monographs on Statistics and
Applied Probability. Chapman and Hall, 1983.
[6] B. G. Lindsay. The geometry of mixture likelihoods : A general theory. The Annals of Statistics,
11(1):86?04, 1983.
[7] Sajama and A. Orlitsky. Semi-parametric exponential family PCA : Reducing dimensions via
non-parametric latent distribution estimation. Technical Report CS2004-0790, University of
California at San Diego, http://cwc.ucsd.edu/? sajama, 2004.
[8] C. F. J. Wu. On the convergence properties of the EM algorithm. Annals of Statistics, 11(1):95?
103, 1983.
[9] J. Kiefer and J. Wolfowitz. Consistency of the maximum likelihood estimator in the presence
of in?nitely many incidental parameters. The Annals of Mathematical Statistics, 27:887?906,
1956.
[10] R. Tibshirani. Principal curves revisited. Statistics and Computation, 2:183?190, 1992.
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1,865 | 2,694 | Saliency-Driven Image Acuity Modulation on a
Reconfigurable Silicon Array of Spiking Neurons
R. Jacob Vogelstein1 , Udayan Mallik2 , Eugenio Culurciello3 ,
Gert Cauwenberghs2 and Ralph Etienne-Cummings2
1
Dept. of Biomedical Engineering, Johns Hopkins University, Baltimore, MD
2
Dept. of Electrical & Computer Engineering, Johns Hopkins University, Baltimore, MD
3
Dept. of Electrical Engineering, Yale University, New Haven, CT
{jvogelst,umallik1,gert,retienne}@jhu.edu, [email protected]
Abstract
We have constructed a system that uses an array of 9,600 spiking silicon neurons, a fast microcontroller, and digital memory, to implement
a reconfigurable network of integrate-and-fire neurons. The system is
designed for rapid prototyping of spiking neural networks that require
high-throughput communication with external address-event hardware.
Arbitrary network topologies can be implemented by selectively routing address-events to specific internal or external targets according to a
memory-based projective field mapping. The utility and versatility of the
system is demonstrated by configuring it as a three-stage network that
accepts input from an address-event imager, detects salient regions of the
image, and performs spatial acuity modulation around a high-resolution
fovea that is centered on the location of highest salience.
1
Introduction
The goal of neuromorphic engineering is to design large-scale sensory information processing systems that emulate the brain. In many biological neural systems, the information
received by a sensory organ passes through multiple stages of neural computations before
a judgment is made. A convenient way to study this functionality is to design separate
chips for each stage of processing and connect them with a fast data bus. However, it is not
always advisable to fabricate a new chip to test a hypothesis regarding a particular neural
computation, and software models of spiking neural networks cannot typically execute or
communicate with external devices in real-time. Therefore, we have designed specialized
hardware that implements a reconfigurable array of spiking neurons for rapid prototyping
of large-scale neural networks.
Neuromorphic sensors can generate up to millions of spikes per second (see, e.g., [1]),
so a proper communication protocol is required for multi-chip systems. ?Address-Event
Representation? (AER) was developed for this purpose over a decade ago and has since
become the common ?language? of neuromorphic chips [2?7]. The central idea of AER
is to use time-multiplexing to emulate extensive connectivity between neurons. Although
it was originally proposed to implement a one-to-one connection topology, AER has been
extended to allow convergent and divergent connectivity [5, 8, 9], and has even been used
DAC
driving potential
(analog)
DIO
MCU
driving
potential
(digital)
postsynaptic
neuron addr.
presynaptic
neuron addr.
RAM
synapse
parameters
postsynaptic
neuron addr.
MCU
I&F
synaptic
weight
outgoing
spike addr.
incoming
spike addr.
I&F
recurrent
mode
DIO
(a)
feed-forward
mode
RAM
DAC
(b)
Figure 1: (a) Block diagram of IFAT system. Incoming and outgoing address-events are
communicated through the digital I/O port (DIO), with handshaking executed by the microcontroller (MCU). The digital-to-analog converter (DAC) is controlled by the MCU and
provides the synaptic driving potential (?E? in Figure 2) to the integrate-and-fire neurons
(I&F), according to the synapse parameters stored in memory (RAM). Modified from [18].
(b) Printed circuit board integrating all components of the IFAT system.
for functions in addition to inter-chip communication [10?12]. Within our hardware array, all inter-neuron communication is performed using AER; the absence of hardwired
connections is the feature that allows for reconfigurability.
A few examples of AER-based reconfigurable neural array transceivers can be found in the
literature [8, 9], but our Integrate-and-Fire Array Transceiver system (IFAT) differs in its
size and flexibility. With four custom aVLSI chips [13] operating in parallel and 128 MB
of digital RAM, the system contains 9,600 neurons and up to 4,194,304 synapses. Because
it was designed from the start for generality and biological realism, every silicon neuron
implements a discrete-time version of the classical biological ?membrane equation? [14],
a simple conductance-like model of neural function that allows for emulating an unlimited
number of synapse types by dynamically varying two parameters [13]. By using a memorybased projective field mapping to route incoming address-events to different target neurons,
the system can implement arbitrarily complex network topologies, limited only by the capacity of the RAM.
To demonstrate the functionality of the IFAT, we designed a three-stage feed-forward model
of salience-based attention and implemented it entirely on the reconfigurable array. The
model is based on a biologically-plausible architecture that has been used to explain human
visual search strategies [15, 16]. Unlike previous hardware implementations (e.g. [17]), we
use a multi-chip system and perform all computations with spiking neurons. The network
accepts spikes from an address-event imager as inputs, computes spatial derivatives of light
intensity as a measure of local information content, identifies regions of high salience, and
foveates a location of interest by reducing the resolution in the surrounding areas. These
capabilities are useful for smart, low-bandwidth, wide-angle surveillance networks.
2
Hardware
From the perspective of an external device, the IFAT system (Figure 1) operates as an
AER transceiver, both receiving and transmitting spikes over a bidirectional address-event
(AE) bus. Internally, incoming events are routed to any number of integrate-and-fire (I&F)
Vdd
?2
?1
Vbp
M5
Vdd
Vcomp
Rscan
M11
Vm
E
X1
Creq
Vdd
M6
M15
M7 M8
M4
C1
C2
M12
?1
?2
W0
M1
W1
M2
W2
M3
Rreq
M14
X2
C0
M13
Cack
Cm
M9
Rack
M10
Vthresh
Vbn
Vreset
Figure 2: Silicon neuron. The ?general-purpose? synapse is shown inside the dashed
box [13], with event generation circuitry shown right [9].
neurons according to a look-up table stored in RAM. When the inputs are sufficient to
cause a neuron to spike, the output is either directed to other internal neurons (for recurrent
networks) or to an external device via the AE bus. The following two sections will describe
the system and silicon neurons in more detail.
2.1
The IFAT system
A block diagram of the IFAT system and its physical implementation are shown in Figure 1.
The primary components are a 100 MHz FPGA (Xilinx XC2S100PQ208), 128 MB of nonvolatile SRAM (TI bq4017), a high-speed DAC (TI TLC7524CN), a 68-pin digital I/O
interface (DIO), and 4 custom aVLSI chips that implement a total of 9,600 I&F neurons.
The FPGA controls access to both an internal and external AE bus, and communicates
address-events between both the I&F neurons and external devices in bit-parallel using a
four-phase asynchronous handshaking scheme.
The 128 MB of RAM is arranged as a 4 MB ? 32 array. Each 32-bit entry contains a
complete description of a single synapse, specifying the postsynaptic target, the synaptic
equilibrium potential, and the synaptic weight. The weight field can be further subdivided
into three parts, corresponding to three ways in which biological neurons can control the
synaptic weight (w) [14, p. 91]:
w = npq
(1)
where n is the number of quantal neurotransmitter sites, p is the probability of neurotransmitter release per site, and q is a measure of the postsynaptic effect of the neurotransmitter.
In the IFAT system, the FPGA can implement p with a simple pseudo-random number algorithm, it can control n by sending multiple outgoing spikes for each incoming spike, and
it sends the value of q to the I&F neuron chips (see Section 2.2).
Instead of hardwired connections between neurons, the IFAT implements ?virtual connections? by serially routing incoming events to their appropriate targets at a rate of up to
1,000,000 events per second. When the IFAT receives an AE from an external device, the
FPGA observes the address, appends some ?chip identifier? (CID) bits, and stores the resulting binary number as a base address. It then adds additional offset bits to form
a complete 22-bit RAM address, which it uses to look up a set of synaptic parameters.
After configuring q and instructing the DAC to produce the analog synaptic equilibrium
potential, the FPGA activates a target neuron by placing its address on the internal AE
bus and initiating asynchronous handshaking with the appropriate I&F chip. It then increments the offset by one and repeats the process for the next synapse, stopping when it
sees a reserved code word in the data field. Recurrent connections can be implemented
simply by appending a different CID to events generated by the on-board I&F neurons,
while connections to external devices are achieved by specifying the appropriate CID for
Row Decoder
E (V) W
Column Decoder
Spike Threshold
V
m
1.5
1
60 x 40 Array of Neurons
0.5
Column Arbitration
111
000
5
0
0
0.1
0.2
0.3
Time (Arbitrary Units)
0.4
(a)
Row Arbitration
LFSR
0
(b)
Figure 3: (a) Data collected from one neuron during operation of the chip. The lower trace
illustrates the membrane potential (Vm ) of a single neuron in the array as a series of events
are sent at times marked at the bottom of the figure. The synaptic equilibrium potential (E)
and synaptic weight (W ) are drawn in the top two traces. Figure from [13]. (b) Integrateand-fire chip micrograph. The linear-feedback shift register (LFSR) implements a pseudorandom element for resolving arbitration conflicts. Modified from [13].
the postsynaptic target. With this infrastructure, arbitrary patterns of connectivity can be
implemented, limited only by the memory?s capacity.
2.2
Integrate-and-Fire Neurons
As described above, the IFAT system includes four custom aVLSI chips [13] that contain a
total of 9,600 integrate-and-fire neurons. All the neurons are identical and each implements
a simple conductance-like model of a single, ?general purpose? synapse using a switchedcapacitor architecture (Figure 2). The synapses have two internal parameters that can be
dynamically modulated for each incoming event: the synaptic equilibrium potential (E)
and the synaptic weight (W0-W2). Values for both parameters are stored in RAM; the
3-bit q is used by the FPGA to selectively enable binary-sized capacitors C0-C2, while E
is converted to an analog value by the DAC. By varying these parameters, it is possible
to emulate a large number of different kinds of synapses impinging on the same cell. An
example of one neuron in operation is shown in Figure 3a.
A micrograph of the integrate-and-fire chip is shown in Figure 3b. Incoming address-events
are decoded and sent to the appropriate neuron in the 60 ? 40 array. When a neuron?s
membrane potential exceeds an externally-provided threshold voltage, it requests service
from the peripheral arbitration circuitry. After request is acknowledged, the neuron is reset and its address is placed on the IFAT system?s internal AER bus. Conflicts between
simultaneously active neurons are resolved by a novel arbitration scheme that includes a
pseudo-random element on-chip [19].
3
Experimental Design and Results
To demonstrate the functionality of the IFAT system, we designed and implemented a threestage network for salience-based foveation [16] of an address-event imager. This work
is motivated by the fact that wide-angle image sensors in a monitoring sensor network
(a)
(b)
Figure 4: (a) Test image. (b) Output from Octopus Retina.
extract a large quantity of data from the environment, most of which is irrelevant. Because
bandwidth is limited and data transmission is energy-intensive, it is desirable to reduce the
amount of information sent over the communication channel. Therefore, if a particular
region of the visual field can be identified as having high salience, that part of the image
can be selectively transmitted with high resolution and the surrounding scene can be be
compressed.
The input to the first stage of the network is a stream of address-events generated by an
asynchronous imager called the ?Octopus Retina? (OR) [1,20]. The OR contains a 60 ? 80
array of light-sensitive ?neurons? that each represent local light intensity as a spike rate. In
other words, pixels that receive a lot of light spike more frequently than those that receive
a little light. For these experiments, we collected 100,000 events from the OR over the
course of about one second while it was viewing a grayscale picture mounted on a white
background. The test image and OR output are shown in Figure 4.
To identify candidate regions of salience, the first stage of the network is configured to
compute local changes in contrast. Every 2 ? 2 block of pixels in the OR corresponds
to four neurons on the IFAT that respond to light-to-dark or dark-to-light transitions in
the rightward or downward direction (Figure 5a). Each IFAT cell computes local changes
in contrast due to a receptive field (RF) that spans four OR pixels in either the horizontal or vertical dimension, with two of its inputs being excitatory and the other two being
inhibitory. When a given IFAT cell?s RF falls on a region of visual space with uniform
brightness, all of the OR pixels projecting to that cell will have the same mean firing rate,
so the excitatory and inhibitory inputs will cancel. However, if a cell?s excitatory inputs
are exposed to high light intensity and its inhibitory inputs are exposed to low light intensity, the cell will receive more excitatory inputs than inhibitory inputs and will generate an
output spike train with spike frequency proportional to the contrast. The output from the
4,800 IFAT neurons in the first stage of the network in response to the OR input is shown
in Figure 5b.
The second stage of processing is designed to pool inputs from neighboring contrastsensitive cells to identify locations of high salience. Our underlying assumption is that
regions of interest will contain more detail than their surroundings, producing a large output from the first stage. Blocks of 8 ? 8 IFAT cells from the first stage project to single
cells in the second stage, and each 8 ? 8 region overlaps the next by 4 neurons (Figure 6a).
Therefore, every IFAT cell in the second stage has an 8 ? 8 RF. Although it is not necessary to normalize the firing rates of the first and second stages, because every second stage
IFAT cell receives 64 inputs, we reduce the strength of the synaptic connections between
the two stages to conserve bandwidth. The output from the 300 IFAT neurons in the second
1
4
2
3
(a)
(b)
Figure 5: (a) Stage 1 network for computing local changes in contrast. Squares in the
center represent OR pixels. Circles represent IFAT neurons. Excitatory synapses are represented by triangles, and inhibitory synapses as circles. Only four IFAT neurons with nonoverlapping receptive fields are shown for clarity. (b) Output of stage 1, as implemented on
the IFAT, with Figure 4b from the OR as input.
stage of the network in response to the output from the first stage IFAT neurons is shown
in Figure 6b.
The final stage of processing modulates the spatial acuity of the original image to reduce the
resolution outside the region of highest salience. This is achieved by a foveation network
that pools inputs from neighboring pixels using overlapping Gaussian kernels (Figure 7a)
[18]. The shape of the kernel functions is implemented by varying the synaptic weight
and synaptic equilibrium potential between OR neurons and IFAT cells in the third stage:
within every pooled block, the strongest connections originate from the center pixels and
the weakest connections come from the outermost pixels. Instead of physically moving the
OR imager to center the fovea on the region of interest, we relocate the fovea by performing
simple manipulations in the address domain. First, the address space of incoming events
is enlarged beyond the range provided by the OR and the fovea is centered within this
virtual visual field (Figure 7a). Then, the row and column address of the second stage IFAT
neuron with the largest output is subtracted from the address of the center of the fovea,
and the result is stored as a constant offset. This offset is then added to the addresses of
all incoming events from the OR, resulting in a shift of the OR image in the virtual visual
field so that the fovea will be positioned over the region of highest salience. The output
from the 1,650 IFAT neurons in the third stage network is shown in Figure 7b. With a
32 ? 32 pixel high-resolution fovea, the network allows for a 66% reduction in the number
of address-events required to reconstruct the image.
4
Conclusion
We have demonstrated a multi-chip neuromorphic system for performing saliency-based
spatial acuity modulation. An asynchronous imager provides the input and communicates
with a reconfigurable array of spiking silicon neurons using address-events. The resulting
output is useful for efficient spatial and temporal bandwidth allocation in low-power vision
sensors for wide-angle video surveillance. Future work will concentrate on extending the
functionality of the multi-chip system to perform stereo processing on address-event data
from two imagers.
(a)
(b)
Figure 6: (a) Stage 2 network for computing local changes in contrast. Blocks of 8 ? 8
IFAT neurons from stage 1 (shown as regions alternately shaded white and gray) project
to single IFAT neurons in stage 2 (not shown). Blocks are shown as non-overlapping for
clarity. (b) Output of stage 2, as implemented on the IFAT, with Figure 5b from stage 1 as
input.
Acknowledgments
This work was partially funded by NSF Awards #0120369, #9896362, and IIS-0209289;
ONR Award #N00014-99-1-0612; and a DARPA/ONR MURI #N00014-95-1-0409. Additionally, RJV is supported by an NSF Graduate Research Fellowship.
References
[1] E. Culurciello, R. Etienne-Cummings, and K. A. Boahen, ?A biomorphic digital image sensor,?
IEEE J. Solid-State Circuits, vol. 38, no. 2, 2003.
[2] M. Sivilotti, Wiring considerations in analog VLSI systems, with application to fieldprogrammable networks. PhD thesis, California Institute of Technology, Pasadena, CA, 1991.
[3] M. Mahowald, An analog VLSI system for stereoscopic vision. Boston, MA: Kluwer Academic
Publishers, 1994.
[4] J. Lazzaro, J. Wawrzynek, M. Mahowald, M. Sivilotti, and D. Gillespie, ?Silicon auditory processors as computer peripherals,? IEEE Trans. Neural Networks, vol. 4, no. 3, pp. 523?528,
1993.
[5] K. A. Boahen, ?Point-to-point connectivity between neuromorphic chips using address events,?
IEEE Trans. Circuits & Systems II, vol. 47, no. 5, pp. 416?434, 2000.
[6] C. M. Higgins and C. Koch, ?Multi-chip neuromorphic motion processing,? in Proc. 20th Anniversary Conference on Advanced Research in VLSI (D. S. Wills and S. P. DeWeerth, eds.),
(Los Alamitos, CA), pp. 309?323, IEEE Computer Society, 1999.
[7] S.-C. Liu, J. Kramer, G. Indiveri, T. Delbru? ck, and R. Douglas, ?Orientation-selective aVLSI
spiking neurons,? in Advances in Neural Information Processing Systems 14 (T. G. Dietterich,
S. Becker, and Z. Ghahramani, eds.), Cambridge, MA: MIT Press, 2002.
[8] G. Indiveri, A. M. Whatley, and J. Kramer, ?A reconfigurable neuromorphic VLSI multi-chip
system applied to visual motion computation,? in Proc. MicroNeuro?99, Apr. 1999.
[9] D. H. Goldberg, G. Cauwenberghs, and A. G. Andreou, ?Probabilistic synaptic weighting in a
reconfigurable network of VLSI integrate-and-fire neurons,? Neural Networks, vol. 14, no. 6-7,
pp. 781?793, 2001.
(a)
(b)
Figure 7: (a) Stage 3 foveation network. The 32 ? 32 pixel high-resolution fovea (center)
is surrounded by lower-resolution areas where 2 ? 2, 4 ? 4, and 8 ? 8 blocks of OR
neurons (shown as non-overlapping for clarity) project to single IFAT cells. The address
space for inputs to the foveation network is 128 ? 128. [18]. (b) Output of stage 3, as
implemented on the IFAT, with the fovea centered on the location with the maximum firing
rate in Figure 6b, from stage 2. Peripheral pixels that receive no input are not shown.
[10] S. R. Deiss, R. J. Douglas, and A. M. Whatley, ?A pulse-coded communications infrastructure
for neuromorphic systems,? in Pulsed Neural Networks (W. Maass and C. M. Bishop, eds.),
pp. 157?178, Cambridge, MA: MIT Press, 1999.
[11] M. Mahowald and R. Douglas, ?A silicon neuron,? Nature, vol. 354, pp. 515?518, 1991.
[12] R. J. Vogelstein, F. Tenore, R. Philipp, M. S. Adlerstein, D. H. Goldberg, and G. Cauwenberghs,
?Spike timing-dependent plasticity in the address domain,? in Advances in Neural Information
Processing Systems 15 (S. Becker, S. Thrun, and K. Obermayer, eds.), Cambridge, MA: MIT
Press, 2003.
[13] R. J. Vogelstein, U. Mallik, and G. Cauwenberghs, ?Silicon spike-based synaptic array and
address-event transceiver,? in Proc. ISCAS?04, vol. 5, (Vancouver, BC), pp. 385?388, 2004.
[14] C. Koch, Biophysics of Computation: Information Processing in Single Neurons. New York,
NY: Oxford University Press, 1999.
[15] C. Koch and S. Ullman, ?Shifts in selective visual attention: towards the underlying neural
circuitry,? Human Neurobiology, vol. 4, pp. 219?227, 1985.
[16] L. Itti, E. Niebur, and C. Koch, ?A model of saliency-based fast visual attention for rapid scene
analysis,? IEEE Trans. Pattern Analysis & Machine Intelligence, vol. 20, no. 11, pp. 1254?1259,
1998.
[17] T. Horiuchi, T. Morris, C. Koch, and S. P. DeWeerth, ?Analog VLSI circuits for attentionbased, visual tracking,? in Advances in Neural Information Processing Systems 9, pp. 706?712,
Cambridge, MA: MIT Press, 1997.
[18] R. J. Vogelstein, U. Mallik, E. Culurciello, G. Cauwenberghs, and R. Etienne-Cummings, ?Spatial acuity modulation of an address-event imager,? in ICECS?04, 2004.
[19] R. J. Vogelstein, U. Mallik, and G. Cauwenberghs, ?Reconfigurable silicon array of spiking
neurons,? IEEE Trans. Neural Networks, 2005. (Submitted).
[20] E. Culurciello, R. Etienne-Cummings, and K. Boahen, ?Second generation of high dynamic
range, arbitrated digital imager,? in Proc. ISCAS?04, vol. 4, (Vancouver, BC), pp. 828?831,
2004.
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1,866 | 2,695 | Breaking SVM Complexity
with Cross-Training
G?okhan H. Bak?r
Max Planck Institute
for Biological Cybernetics,
T?ubingen, Germany
[email protected]
L?eon Bottou
NEC Labs America
Princeton NJ, USA
[email protected]
Jason Weston
NEC Labs America
Princeton NJ, USA
[email protected]
Abstract
We propose to selectively remove examples from the training set using
probabilistic estimates related to editing algorithms (Devijver and Kittler,
1982). This heuristic procedure aims at creating a separable distribution
of training examples with minimal impact on the position of the decision
boundary. It breaks the linear dependency between the number of SVs
and the number of training examples, and sharply reduces the complexity
of SVMs during both the training and prediction stages.
1
Introduction
The number of Support Vectors (SVs) has a dramatic impact on the efficiency of Support
Vector Machines (Vapnik, 1995) during both the learning and prediction stages. Recent
results (Steinwart, 2004) indicate that the number k of SVs increases linearly with the
number n of training examples. More specifically,
k/n ?? 2BK
(1)
where n is the number of training examples and BK is the smallest classification error
achievable with the SVM kernel K. When using a universal kernel such as the Radial Basis
Function kernel, BK is the Bayes risk B, i.e. the smallest classification error achievable with
any decision function.
The computational requirements of modern SVM training algorithms (Joachims, 1999;
Chang and Lin, 2001) are very largely determined by the amount of memory required to
store the active segment of the kernel matrix. When this amount exceeds the available
memory, the training time increases quickly because some kernel matrix coefficients must
be recomputed multiple times. During the final phase of the training process, the active
segment always contains all the k(k + 1)/2 dot products between SVs. Steinwart?s result
(1) then suggests that the critical amount of memory scales at least like B 2 n2 . This can be
practically prohibitive for problems with either big training sets or large Bayes risk (noisy
problems). Large numbers of SVs also penalize SVMs during the prediction stage as the
computation of the decision function requires a time proportional to the number of SVs.
When the problem is separable, i.e. B = 0, equation (1) suggests1 that the number k of
SVs increases less than linearly with the number n of examples. This improves the scaling
laws for the SVM computational requirements.
1
See also (Steinwart, 2004, remark 3.8)
In this paper, we propose to selectively remove examples from the training set using probabilistic estimates inspired by training set editing algorithms (Devijver and Kittler, 1982).
The removal procedure aims at creating a separable set of training examples without modifying the location of the decision boundary. Making the problem separable breaks the
linear dependency between the number of SVs and the number of training examples.
2
2.1
Related work
Salient facts about SVMs
We focus now on the C-SVM applied to the two-class pattern recognition problem. See
(Burges, 1998) for a concise reference. Given n training patterns xi and their associated
classes yi = ?1, the SVM decision function is:
f (x) =
n
X
(2)
?i? yi K(xi , x) + b?
i=1
The coefficient ?i? in (2) are obtained by solving a quadratic programing problem:
??
=
1X
?i ?j yi yj K(xi , xj )
2 i,j
?
i
X
subject to ?i, 0 ? ?i ? C and
?i yi = 0
arg max
X
?i ?
(3)
i
This optimization yields three categories of training examples depending on ? i? . Within
each category, the possible values of the margins yi f (xi ) are prescribed by the KarushKuhn-Tucker optimality conditions.
- Examples such that ?i? = C are called bouncing SVs or margin errors and satisfy
yi f (xi ) < 1. The set of bouncing SVs includes all training examples misclassified
by the SVM, i.e. those which have a negative margin yi f (xi ) < 0.
- Examples such that 0 < ?i? < C are called ordinary SVs and satisfy yi f (xi ) = 1.
- Examples such that ?i? = 0 satisfy relation yi f (xi ) > 1. These examples play no
role in the SVM decision function (2). Retraining after discarding these examples
would still yield the same SVM decision function (2).
These facts provide some insight into Steinwart?s result (1). The SVM decision function,
like any other decision rule, must asymptotically misclassify at least Bn examples, where
B is the Bayes risk. All these examples must therefore become bouncing SVs.
To illustrate dependence on the Bayes risk, we perform a linear classification task in two
dimensions under varying amount of class overlap. The class distributions were uniform
on a unit square with centers c1 and c2 . Varying the distance between c1 and c2 allows us
to control the Bayes risk. The results are shown in figure 1.
2.2
A posteriori reduction of the number of SVs.
Several techniques aim to reduce the prediction complexity of SVMs by expressing the
SVM solution (2) with a smaller kernel expansion. Since one must compute the SVM solution before applying these post-processing techniques, they are not suitable for reducing
the complexity of the training stage.
Reduced Set Construction. Burges (Burges, 1996) proposes to construct new patterns
zj in order to define a compact approximation of the decision function (2). Reduced set
construction usually involves solving a non convex optimization problem and is not applicable on arbitrary inputs such as graphs or strings.
3
10
# ?=C for SVM
# ?<C for SVM
rank K
sv
1.5
y.f(x)
1
2
10
log #SV
0.5
?
0
?0.5
1
10
?1
?1.5
1000 2000 3000 4000 5000 6000 7000
0
10
0
0.05
0.1
0.15
0.2
Bayes risk
0.25
0.3
0.35
Figure 1: Effect of noise on the number of sup-
Figure 2: Histogram of SVs selected by the
port vectors. The number of ordinary SVs stays
almost constant whereas the number of bouncing SVs grows. Additional support vectors do
not give extra information as indicated by the
rank of the kernel matrix. See section 2.1.
`1 penalization method on the MNIST 3-8 discrimination task. The initial SVs have been ordered on the x-axis by increasing margin yf (x)
and decreasing ?. See last paragraph in section
2.2.
Reduced Set Selection. The set of basis2 functions K(xi , ?) associated with the SVs xi
do not necessarily constitute a linearly independent family. The same decision function f (?)
can then be expressed by multiple linear combination of the functions K(xi , ?). Reduced
set selection methods attempt to select a subset of the SVs that is sufficient to express
the SVM decision function. For instance, (Downs, Gates and Masters, 2001) propose to
compute the row echelon form of the kernel matrix and discard SVs that lead to zero rows.
This approach maintains the original SVM decision function.
In contrast, the `1 penalization method suggested in (Sch?olkopf and Smola, 2002,
sect. 18.4.2) simply attempts to construct a sufficiently good approximation of the original SVM decision function by solving
2
X
X
X
?
arg min
?i yi K(xi , ?) ?
?i yi K(xi , ?)
+ ?
|?i |
(4)
?
i
i
K
i
where parameter ? trades accuracy versus sparsity, and k?kK denotes the Reproducing Kernel Hilbert Space norm (Sch?olkopf and Smola, 2002, definition 2.9). Simplifying expression (4) yields a numerically tractable quadratic programming problem.
Which examples are selected? We have investigated the `1 penalization method (4) as
follows. We train a first SVM to discriminate digits 3 and 8 on the MNIST dataset (see
section 4.2) after randomly swapping 10% of the class labels in the training set. We then
select a subset of the resulting support vectors using the `1 penalization method.
Choosing ? is quite difficult in practice. To evaluate the accuracy of the procedure, we train
a second SVM on the selected vectors, compare its recognition accuracy with that of the
first SVM. This was best achieved by enforcing the constraint ?i ? 0 in (4) because the
second SVM cannot return an expansion with negative coefficients.
Figure 2 shows the histogram of selected SVs. The initial support vectors have been ordered on the x-axis by increasing values of yi f (xi ), and, in the case of margin SVs, by
decreasing values of ?i . The selected SVs includes virtually no misclassified SVs, but
instead concentrates on SVs with large ?i .
This result suggests that simple pre-processing methods might indicate which training examples are really critical for SVM classification.
2
We use the customary name basis functions despite linear dependence. . .
2.3
Training set editing techniques
We now consider techniques for reducing the set of training examples before running a
training algorithm. Reducing the amount of training data is indeed an obvious way to
reduce the complexity of training. Quantization and clustering methods might be used to
achieve this goal. These methods however reduce the training data without considering the
loss function of interest, and therefore sacrifice classification accuracy. We focus instead
on editing techniques, i.e. techniques for discarding selected training examples with the
aim of achieving similar or better classification accuracy.
Two prototypical editing techniques, MULTIEDIT and CONDENSE, have been thoroughly
studied (Devijver and Kittler, 1982, chapter 3) in the context of the nearest neighbor (1NN) classification rule.
Removing interior examples. The CONDENSE algorithm was first described by (Hart,
1968). This algorithm selects a subset of the training examples whose 1-NN decision
boundary still classifies correctly all of the initial training examples:
Algorithm 1 (C ONDENSE).
1 Select a random training example and put it in set R.
2 For each training example i = 1, . . . , n : classify example i using the 1-NN rule with set
R as the training set, and insert it into R if it is misclassified.
3 Return to step 2 if R has been modified during the last pass.
4 The final contents of R constitute the condensed training set.
This is best understood when both classes form homogeneous clusters in the feature space.
Algorithm 1 discards training examples located in the interior of each cluster.
This strategy works poorly when there is a large overlap between the pattern distributions
of both classes, that is to say when the Bayes risk B is large. Consider for instance a feature
space region where P (y = +1 | x) > P (y = ?1 | x) > 0. A small number of training
examples of class y = ?1 can still appear in such a region. We say that they are located on
the wrong side of the Bayes decision boundary. Asymptotically, all such training examples
belong to the condensed training set in order to ensure that they are properly recognized as
members of class y = ?1.
Removing noise examples. The Edited Nearest Neighbor rule (Wilson, 1972) suggests
to first discard all training examples that are misclassified when applying the 1-NN rule
using all n ? 1 remaining examples as the training set. It was shown that removing these
examples improves the asymptotic performance of the nearest neighbor rule. Whereas
the 1-NN risk is asymptotically bounded by 2B, the Edited 1-NN risk is asymptotically
bounded by 1.2 B, where B is the Bayes risk.
The MULTIEDIT algorithm (Devijver and Kittler, 1982, section 3.11) asymptotically discards all the training examples located on the wrong side of the Bayes decision boundary.
The asymptotic risk of the multi-edited nearest neighbor rule is the Bayes risk B.
Algorithm 2 (M ULTI E DIT).
1 Divide randomly the training data into s splits S1 , . . . , Ss . Let us call fi the 1-NN classifier
that uses Si as the training set.
2 Classify all examples in Si using the classifier f(i+1) mod s and discard all misclassified
examples.
3 Gather all the remaining examples and return to step 1 if any example has been discarded
during the last T iterations.
4 The remaining examples constitute the multiedited training set.
By discarding examples located on the wrong side of the Bayes decision boundary, algorithm MULTIEDIT constructs a new training set whose apparent distribution has the same
Bayes decision boundary as the original problem, but with Bayes risk equal to 0. Devijver
and Kittler claim that MULTIEDIT produces an ideal training set for CONDENSE.
Algorithm MULTIEDIT also discards some proportion of training examples located on the
correct side of Bayes decision boundary. Asymptotically this does not matter. However
this is often a problem in practice. . .
2.4
Editing algorithms and SVMs
Training examples recognized with high confidence usually do not appear in the SVM
solution (2) because they do not become support vectors. On the other hand, outliers always
become support vectors. Intuitively, SVMs display the properties of the CONDENSE but
lack the properties of the MULTIEDIT algorithm.
The mathematical proofs for the asymptotic properties of MULTIEDIT depend on the specific nature of the 1-NN classification rule. The MULTIEDIT algorithm itself could be identically defined for any classifier. This suggests (but does not prove) that these properties
might remain valid for SVM classifiers3 .
This contribution is an empirical attempt to endow Support Vector Machines with the properties of the MULTIEDIT algorithm.
Editing SVM training sets implicitly modifies the SVM loss function in a way that relates
to robust statistics. Editing alters the apparent distribution of training examples such that
the class distributions P (x | y = 1) and P (x | y = ?1) no longer overlap. If the class
distributions were known, this could be done by trimming the tails of the class distributions.
A similar effect could be obtained by altering the SVM loss function (the hinge loss) into a
non convex loss function that gives less weight to outliers.
3
Cross-Training
Cross-Training is a representative algorithm of such combinations of SVMs and editing
algorithms. It begins with creating s subsets of the training set with r examples each. Independent SVMs are then trained on each subset. The decision functions of these SVMs
are then used to discard two types of training examples: those which are confidently recognized, as in CONDENSE, and those which are misclassified, as in MULTIEDIT. A final SVM
is then trained using the remaining examples.
Algorithm 3 (C ROSS T RAINING).
1 Create s subsets of size r by randomly drawing r/2 examples of each class.
2 Train s independent SVMs f1 , . . . , fs using each of the subsets as the training set.
3 For each training example (xi , yi ) estimate the margin average mi and variance vi :
P
P
mi = 1s sr=1 yi fr (xi )
vi = 1s sr=1 (mi ? yi fr (xi ))2
4 Discard all training examples for which mi + vi < 0.
5 Discard all training examples for which mi ? vi > 1.
6 Train a final SVM on the remaining training examples.
The apparent simplicity of this algorithm hides a lot of hyperparameters. The value of the
C parameters for the SVMs at steps [2] and [6] has a considerable effect on the overall
performance of the algorithm.
For the first stage SVMs, we choose the C parameter which yields the best performance
on training sets of size r. For the second stage SVMs, we choose the C parameter which
yields the best overall performance measured on a separate validation set.
Furthermore, we discovered that the discarding steps tend to produce a final set of training
examples with very different numbers of examples for each class. Specific measures to
alleviate this problem are discussed in section 4.3.
3
Further comfort comes from the knowledge that a SVM with the RBF kernel and without threshold term b implements the 1-NN rule when the RBF radius tends to zero.
0.2
LIBSVM
X?Train LIBSVM
0.19
2500
500
LIBSVM
X?Train LIBSVM
450
400
0.18
2000
350
300
SVs
Test Error
0.17
1500
250
0.16
1000
200
150
0.15
100
500
0
0
LIBSVM
X?Train LIBSVM
Time (secs)
3000
0.14
1000
2000
3000 4000 5000
Training set size
6000
7000
8000
0.13
0
50
1000
2000
3000 4000 5000
Training set size
6000
7000
8000
0
0
1000
2000
3000 4000 5000
Training set size
6000
7000
8000
Figure 3: Comparing LIBSVM and Cross-Training on a toy problem of two Gaussian clouds for
increasing number of training points. Cross-Training gives an almost constant number of support
vectors (left figure) for increasing training set size, whereas in LIBSVM the number of support vectors increases linearly. The error rates behave similarly (middle figure), and Cross-Training gives an
improved training time (right figure). See section 4.1.
4
Experiments
4.1
Artificial Data
We first constructed artificial data, by generating two classes from two Gaussian clouds in
10 dimensions with means (1, 1, 1, 1, 1, 0, 0, 0, 0, 0) and (?1, ?1, ?1, ?1, ?1, 0, 0, 0, 0, 0)
and standard deviation 4. We trained a linear SVM for differing amounts of training points,
selecting C via cross validation. We compare the performance of LIBSVM4 with CrossTraining using LIBSVM with s = 5, averaging over 10 splits. The results given in figure
3 show a reduction in SVs and computation time using Cross-Training, with no loss in
accuracy.
4.2
Artificial Noise
Our second experiment involves the discrimination of digits 3 and 8 in the MNIST5
database. Artificial noise was introduced by swapping the labels of 0%, 5%, 10% and
15% of the examples. There are 11982 training examples and 1984 testing examples. All
experiments were carried out using LIBSVM?s ?-SVM (Chang and Lin, 2001) with the
RBF kernel (? = 0.005). Cross-Training was carried out by splitting the 11982 training
examples into 5 subsets. Figure 4 reports our results for various amounts of label noise.
The number of SVs (left figure) increases linearly for the standard SVM and stays constant
for the Cross-Training SVM. The test errors (middle figure) seem similar. Since our label
noise is artificial, we can also measure the misclassification rate on the unmodified testing set (right figure). This measurement shows a slight loss of accuracy without statistical
significance.
4.3
Benchmark Data
Finally the cross-training algorithm was applied to real data sets from both the ANU repository6 and from the the UCI repository7 .
Experimental results were quite disappointing until we realized that the discarding steps
tends to produce training sets with very different numbers of examples for each class. To
alleviate this problem, after training each SVM, we choose the value of the threshold b ?
4
5
6
7
http://www.csie.ntu.edu.tw/?cjlin/libsvm/
http://yann.lecun.com/exdb/mnist
http://mlg.anu.edu.au/?raetsch/data/index.html
ftp://ftp.ics.uci.edu/pub/machine-learning-databases
8000
2.5%
15.0%
2.0%
6000
1.5%
10.0%
4000
1.0%
5.0%
2000
0.5%
0
0.0%
0%
5%
10%
15%
0.0%
0%
5%
10%
15%
0%
5%
10%
15%
Figure 4: Number of SVs (left figure) and test error (middle figure) for varying amounts of label
noise on the MNIST 3-8 discrimination task. The x-axis in all graphs shows the amount of label noise;
white squares correspond to LIBSVM; black circles to Cross-Training; dashed lines to bagging the
first stage Cross-Training SVMs. The last graph (right figure) shows the test error measured without
label noise. See section 4.2
in (2) which achieves the best validation performance. We also attempt to balance the
final training set by re-inserting examples discarded during step [5] of the cross-training
algorithm.
Experiments were carried out using RBF kernels with the kernel width reported in the
literature. In the SVM experiments, the value of parameter C was determined by crossvalidation and then used for training a SVM on the full dataset. In the cross-training experiments, we make a validation set by taking r/3 examples from the training set. These
examples are only used for choosing the values of C and for adjusting the SVM thresholds.
Details and source code are available8 .
Dataset
Banana
Waveform
Splice
Adult
Adult
Forest
Forest
Forest
Train
Size
400
400
1000
3185
32560
50000
90000
200000
Test
Size
4900
4600
2175
16280
16280
58100
58100
58100
SVM
Perf.[%]
89.0
90.2
90.0
84.2
85.1
90.3
91.6
?
SVM
#SV
111
172
601
1207
11325
12476
18983
?
XTrain
Subsets
5?200
5?200
5?300
5?700
5?6000
5?10000
5?18000
8?30000
XTrain
Perf.[%]
88.2
88.7
89.9
84.2
84.8
89.2
90.7
92.1
XTrain
#SV
51
87
522
606
1194
7967
13023
19526
Table 1: Comparison of SVM and Cross-Training results on standard benchmark data sets.
The columns in table 1 contain the dataset name, the size of the training set used for the
experiment, the size of the test set, the SVM accuracy and number of SVs, the CrossTraining subset configuration, accuracy, and final number of SVs. Bold typeface indicates
which differences were statistically significant according to a paired test. These numbers
should be considered carefully because they are impacted by the discrete nature of the grid
search for parameter C. The general trend still indicates that Cross-Training causes a slight
loss of accuracy but requires much less SVs.
Our largest training set contains 200000 examples. Training a standard SVM on such a
set takes about one week of computation. We do not report this result because it was not
practical to determine a good value of C for this experiment. Cross-Training with specified
hyperparameters runs overnight. Cross-Training with hyperparameter grid searches runs in
two days.
We do not report detailled timing results because much of the actual time can be attributed
8
http://www.kyb.tuebingen.mpg.de/bs/people/gb/xtraining
to the search for the proper hyperparameters. Timing results would then depend on loosely
controlled details of the hyperparameter grid search algorithms.
5
Discussion
We have suggested to combine SVMs and training set editing techniques to break the linear relationship between number of support vectors and number of examples. Such combinations raise interesting theoretical questions regarding the relative value of each of the
training examples.
Experiments with a representative algorithm, namely Cross-Training, confirm that both the
training and the recognition time are sharply reduced. On the other hand, Cross-Training
causes a minor loss of accuracy, comparable to that of reduced set methods (Burges, 1996),
and seems to be more sensitive than SVMs in terms of parameter tuning.
Despite these drawbacks, Cross-Training provides a practical means to construct kernel
classifiers with significantly larger training sets.
6
Acknowledgement
We thank Hans Peter Graf, Eric Cosatto and Vladimir Vapnik for their advice and support.
Part of this work was funded by NSF grant CCR-0325463.
References
Burges, C. J. C. (1996). Simplified Support Vector Decision Rules. In Saitta, L., editor, Proceedings of the 13th International Conference on Machine Learning, pages 71?77, San Mateo, CA.
Morgan Kaufmann.
Burges, C. J. C. (1998). A Tutorial on Support Vector Machines for Pattern Recognition. Data
Mining and Knowledge Discovery, 2(2):121?167.
Chang, C.-C. and Lin, C.-J. (2001). Training ?-Support Vector Classifiers: Theory and Algorithms.
Neural Computation, 13(9):2119?2147.
Devijver, P. and Kittler, J. (1982). Pattern Recogniton, A statistical approach. Prentice Hall, Englewood Cliffs.
Downs, T., Gates, K. E., and Masters, A. (2001). Exact Simplification of Support Vector Solutions.
Journal of Machine Learning Research, 2:293?297.
Hart, P. (1968). The condensed nearest neighbor rule. IEEE Transasctions on Information Theory,
14:515?516.
Joachims, T. (1999). Making Large?Scale SVM Learning Practical. In Sch o? lkopf, B., Burges, C.
J. C., and Smola, A. J., editors, Advances in Kernel Methods ? Support Vector Learning, pages
169?184, Cambridge, MA. MIT Press.
Sch?olkopf, B. and Smola, A. J. (2002). Learning with Kernels. MIT Press, Cambridge, MA.
Steinwart, I. (2004). Sparseness of Support Vector Machines?Some Asymptotically Sharp Bounds.
In Thrun, S., Saul, L., and Scho? lkopf, B., editors, Advances in Neural Information Processing
Systems 16. MIT Press, Cambridge, MA.
Vapnik, V. N. (1995). The Nature of Statistical Learning Theory. Springer Verlag, New York.
Wilson, D. L. (1972). Asymptotic properties of the nearest neighbor rules using edited data. IEEE
Transactions on Systems, Man, and Cybernetics, 2:408?420.
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1,867 | 2,696 | Identifying protein-protein interaction sites on a
genome-wide scale
Haidong Wang? Eran Segalo Asa Ben-Hur? Daphne Koller? Douglas L. Brutlag?
?
Computer Science Department, Stanford University, CA 94305
{haidong, koller}@cs.stanford.edu
o
Center for Studies in Physics and Biology, Rockefeller University, NY 10021
[email protected]
?
Department of Genome Sciences, University of Washington, WA 98195
[email protected]
?
Department of Biochemistry, Stanford University, CA 94305
[email protected]
Abstract
Protein interactions typically arise from a physical interaction of one or
more small sites on the surface of the two proteins. Identifying these sites
is very important for drug and protein design. In this paper, we propose
a computational method based on probabilistic relational model that attempts to address this task using high-throughput protein interaction data
and a set of short sequence motifs. We learn the model using the EM
algorithm, with a branch-and-bound algorithm as an approximate inference for the E-step. Our method searches for motifs whose presence in a
pair of interacting proteins can explain their observed interaction. It also
tries to determine which motif pairs have high affinity, and can therefore
lead to an interaction. We show that our method is more accurate than
others at predicting new protein-protein interactions. More importantly,
by examining solved structures of protein complexes, we find that 2/3 of
the predicted active motifs correspond to actual interaction sites.
1
Introduction
Many cellular functions are carried out through physical interactions between proteins.
Discovering the protein interaction map can therefore help to better understand the workings of the cell. Indeed, there has been much work recently on developing high-throughput
methods to produce a more complete map of protein-protein interactions [1, 2, 3].
Interactions between two proteins arise from physical interactions between small regions on the surface of the proteins [4] (see Fig. 2(b)). Finding interaction sites is an
important task, which is of particular relevance to drug design. There is currently no highthroughput experimental method to achieve this goal, so computational methods are required. Existing methods either require solving a protein?s 3D structure (e.g., [5]), and
therefore are computationally very costly and not applicable on a genome-wide scale, or
use known interaction sites as training data (e.g., [6]), which are relatively scarce and hence
have poor coverage. Other work focuses on refining the highly noisy high-throughput interaction maps [7, 8, 9], or on assessing the confidence levels of the observed interactions [10].
In this paper, we propose a computational method for predicting protein interactions
P1
d
d
a
Aa
P5
P2
Ad
Ab
P5
Ad
Ad
P1
Aab
Bab S Bdb
b
d
c
b
P2
b
P3
(a)
Adb
P4
T12 O
Aad
Add
Bab S Bab
T15 O
Abd
Bab S
T25 O
(b)
Figure 1: (a) Simple illustration of our assumptions for protein-protein interactions. The small
elements denote motif occurrences on proteins, with red denoting active and gray denoting inactive
motifs. (b) A fragment of our probabilistic model, for the proteins P 1 , P2 , P5 . We use yellow to
denote an assignment of the value true, and black to denote the value false; full circles denote an
assignment observed in the data, and patterned circles an assignment hypothesized by our algorithm.
The dependencies involving inactive motif pairs were removed from the graph because they do not
affect the rest of the model.
and the sites at which the interactions take place, which uses as input only high-throughput
protein-protein interaction data and the protein sequences. In particular, our method assumes no knowledge of the 3D protein structure, or of the sites at which binding occurs.
Our approach is based on the assumption that interaction sites can be described using a
limited repertoire of conserved sequence motifs [11]. This is a reasonable assumption since
interaction sites are significantly more conserved than the rest of the protein surface [12].
Given a protein interaction map, our method tries to explain the observed interactions by
identifying a set of sites of motif occurrence on every pair of interacting proteins through
which the interaction is mediated. To understand the intuition behind our approach, consider the example of Fig. 1(a). Here, the interaction pattern of the protein P1 can best be
explained using the motif pair a, b, where a appears in P1 and b in the proteins P2 , P3 , P4
but not in P5 . By contrast, the motif pair d, b is not as good an explanation, because d
also appears in P5 , which has a different interaction pattern. In general, our method aims
to identify motif pairs that have high affinity, potentially leading to interaction between
protein pairs that contain them.
However, a sequence motif might be used for a different purpose, and not give rise to an
active binding site; it might also be buried inside the protein, and thus be inaccessible for
interaction. Thus, the appearance of an appropriate motif does not always imply interaction.
A key feature of our approach is that we allow each motif occurrence in a protein to be
either active or inactive. Interactions are then induced only by the interactions of highaffinity active motifs in the two proteins. Thus, in our example, the motif d in p2 is inactive,
and hence does not lead to an interaction between p2 and p4 , despite the affinity between
the motif pair c, d. We note that Deng et al. [8] proposed a somewhat related method for
genome-wide analysis of protein interaction data, based on protein domains. However,
their method is focused on predicting protein-protein interactions and not on revealing the
site of interaction, and they do not allow for the possibility that some domains are inactive.
Our goal is thus to identify two components: the affinities between pairs of motifs, and
the activity of the occurrences of motifs in different proteins. Our algorithm addresses this
problem by using the framework of Bayesian networks [13] and probabilistic relational
models [14], which allows us to handle the inherent noise in the protein interaction data
and the uncertain relationship between interactions and motif pairs. We construct a model
encoding our assumption that protein interactions are induced by the interactions of active
motif pairs. We then use the EM algorithm [15], to fill in the details of the model, learning
both the motif affinities and activities from the observed data of protein-protein interactions
and protein motif occurrences. We address the computational complexity of the E-step in
these large, densely connected models by using an approximate inference procedure based
on branch-and-bound.
We evaluated our model on protein-protein interactions in yeast and Prosite motifs [11].
As a basic performance measure, we evaluated the ability of our method to predict new
protein-protein interactions, showing that it achieves better performance than several other
models. In particular, our results validate our assumption that we can explain interactions
via the interactions of active sequence motifs. More importantly, we analyze the ability
of our method to discover the mechanism by which the interaction occurs. Finally, we
examined co-crystallized protein pairs where the 3D structure of the interaction is known,
so that we can determine the sites at which the interaction took place. We show that our
active motifs are more likely to participate in interactions.
2
The Probabilistic Model
The basic entities in our probabilistic model are the proteins and the set of sequence motifs
that can mediate protein interactions. Our model therefore contains a set of protein entities
P = {P1 , . . . , Pn }, with the motifs that occur in them. Each protein P is associated with
the set of motifs that occur in it, denoted by P.M . As we discussed, a key premise of
our approach is that a specific occurrence of a sequence motif may or may not be active.
Thus, each motif occurrence a ? P.M is associated with a binary-value variable P.A a ,
which takes the value true if Aa is active in protein P and false otherwise. We structure the
|
prior probability P (P.Aa = true) = min{0.8, 3+0.1?|P.M
}, to capture our intuition that
|P.M |
the number of active motifs in a protein is roughly a constant fraction of the total number
of motifs in the protein, but that even proteins with few motifs tend to have at least some
number of active motifs.
A pair of active motifs in two proteins can potentially bind and induce an interaction
between the corresponding proteins. Thus, in our model, a pair of proteins interact if each
contains an active motif, and this pair of motifs bind to each other. The probability with
which two motifs bind to each other is called their affinity. We encode this assumption by
including in our model entities Tij corresponding to a pair of proteins Pi , Pj . For each
pair of motifs a ? Pi .M and b ? Pj .M , we introduce a variable Tij .Aab , which is a
deterministic AND of the activity of these two motifs. Intuitively, this variable represents
whether the pair of motifs can potentially interact. The probability with which two active
motif occurrences bind is their affinity. We model the binding event between two motif
occurrences using a variable Tij .Bab , and define: P (Tij .Bab = true | Tij .Aab = true) =
?ab and P (Tij .Bab = true | Tij .Aab = false) = 0, where ?ab is the affinity between motifs
a and b. This model reflects our assumption that two motif occurrences can bind only if
they are both active, but their actual binding probability depends on their affinity. Note that
this affinity is a feature of the motif pair and does not depend on the proteins in which they
appear.
We must also account for interactions that are not explained by our set of motifs,
whether because of false positives in the data, or because of inadequacies of our model
or of our motif set. Thus, we add a spurious binding variable Tij .S, for cases where an
interaction between Pi and Pj exists, but cannot be explained well by our set of active
motifs. The probability that a spurious binding occurs is given by P (Tij .S = true) = ?S .
Finally, we observe an interaction between two proteins if and only if some form of
binding occurs, whether by a motif pair or a spurious binding. Thus, we define a variable
Tij .O, which represents whether protein i was observed to interact with protein j, to be
a deterministic OR of all the binding variables Tij .S and Tij .Bab . Overall, Tij .O is a
noisy-OR [13] of all motif pair variables Tij .Aab .
Note that our model accounts for both types of errors in the protein interaction data.
False negatives (missing interactions) in the data are addressed through the fact that the
presence of an active motif pair only implies that binding takes place with some probability.
False positives (wrong interactions) in the data are addressed through the introduction of
the spurious interaction variables.
The full model defines a joint probability distribution over the entire set of attributes:
Q Q
P (P.A,T.A,
Q T.B, T.S, T.O) = i a?Pi .M P (Pi .Aa )
Q
a?Pi .M,b?Pj .M P (Tij .Aab | Pi .Aa , Pj .Ab )P (Tij .Bab | Tij .Aab )
ij
P (Tij .S)P (Tij .O | Tij .B, Tij .S)
where each of these conditional probability distributions is as specified above. We use
? to denote the entire set of model parameters {?a,b }a,b ? {?S }. An instantiation of our
probabilistic model is illustrated in Fig. 1(b).
3
Learning the Model
We now turn to the task of learning the model from the data. In a typical setting, we are
given as input a protein interaction data set, specifying a set of proteins P and a set of
observed interacting pairs T.O. We are also given a set of potentially relevant motifs, and
the occurrences of these motifs in the different proteins in P. Thus, all the variables except
for the O variables are hidden. Our learning task is thus twofold: we need to infer the values
of the hidden variables, both the activity variables P.A, T.A, and the binding variables
T.B, T.S; we also need to find a setting of the model parameters ?, which specify the
motif affinities. We use a variant of the EM algorithm [15] to find both an assignment to
the parameters ?, and an assignment to the motif variables P.A, which is a local maximum
of the likelihood function P (T.O, P.A | ?). Note that, to maximize this objective, we
search for a MAP assignment to the motif activity variables, but sum out over the other
hidden variables. This design decision is reasonable in our setting, where determining motif
activities is an important goal; it is a key assumption for our computational procedure.
As in most applications of EM, our main difficulty arises in the E-step, where we need
to compute the distribution over the hidden variables given the settings of the observed
variables and the current parameter settings. In our model, any two motif variables (both
within the same protein and across different proteins) are correlated, as there exists a path
of influence between them in the underlying Bayesian network (see Fig. 1(c)). These correlations make the task of computing the posterior distribution over the hidden variables
intractable, and we must resort to an approximate computation. While we could apply a
general purpose approximate inference algorithm such as loopy belief propagation [16],
such methods may not converge in densely connected model such as this one, and there
are few guarantees on the quality of the results even if they do converge. Fortunately, our
model turns out to have additional structure that we can exploit. We now describe an approximate inference algorithm that is tailored to our model, and is guaranteed to converge
to a (strong) local maximum.
Our first observation is that the only variables that correlate the different protein pairs
Tij are the motif variables P.A. Given an assignment to these activity variables, the network decomposes into a set of independent subnetworks, one for each protein pair. Based
on this observation, we divide our computation of the E-step into two parts. In the first,
we find an assignment to the motif variables in each protein, P.A; in the second, we compute the posterior probability over the binding motif pair variables T.B, T.S, given the
assignment to the motif variables.
We begin by describing the second phase. We observe that, as all the motif pair variables, T.A, are fully determined by the motif variables, the only variables left to reason
about are the binding variables T.B and T.S. The variables for any pair Tij are independent of the rest of the model given the instantiation to T.A and the interaction evidence. That fact, combined with the noisy-OR form of the interaction, allows us to compute the posterior probability required in the E-step exactly and efficiently. Specifically,
the computation for the variables associated with a particular protein pair Tij is as follows, where we omit the common prefix Tij to simplify notation. If Aab = false, then
P (Bab = true | Aab = false, O, ?) = 0. Otherwise, if Aab = true, then
P (Bab = true | A, O, ?)
=
P (Bab | A, ?)P (O | Bab = true, A, ?)
.
P (O | A, ?)
The first term in the numerator is simply the motif affinity ?ab ; the second term is 1 if
O = true andQ0 otherwise. The numerator can easily be computed as P (O | A, ?) =
1 ? (1 ? ?S ) Aa,b =true (1 ? ?ab ). The computation for P (S) is very similar.
We now turn to the first phase, of finding a setting to all of the motif variables. Unfortunately, as we discussed, the model is highly interconnected, and a finding an optimal
joint setting to all of these variables P.A is intractable. We thus approximate finding this
joint assignment using a method that exploits our specific structure. Our method iterates
over proteins, finding in each iteration the optimal assignment to the motif variables of each
protein given the current assignment to the motif activities in the remaining proteins. The
process repeats, iterating over proteins, until convergence.
As we discussed, the likelihood of each assignment to Pi .A can be easily computed
using the method described above. However, the computation for each protein is still exponential in the number of motifs it contains, which can be large (e.g., 15). However, in
our specific model, we can apply the following branch-and-bound algorithm (similar to an
approach proposed by Henrion [17] for BN2O networks) to find the globally optimal assignment to the motif variables of each protein. The idea is that we search over the space
of possible assignments Pi .A for one that maximizes the objective we wish to maximize.
We can show that if making a motif active relative to one assignment does not improve the
objective, it will also not improve the objective relative to a large set of other assignments.
More precisely, let f (Pi .A) = P (Pi .A, P?i .A|O, ?) denote the objective we wish
to maximize, where P?i .A is the fixed assignment to motif variables in all proteins
except Pi . Let Pi .A?a denote the assignment to all the motif variables in Pi except
for Aa . We
Q compute the ratio of f after we switch Pi .Aa from false to true. Let
ha (Pj ) = Pj .Ab =true (1 ? ?ab ) denote the probability that motif a does not bind with
any active motif in Pj . We can now compute:
?a (Pi .A?a ) =
?
Y
1?j?n
Tij .O=false
f (Pi .Aa = true, Pi .A?a )
g
=
f (Pi .Aa = false, Pi .A?a )
1?g
Q
Y
1 ? (1 ? ?S )ha (Pj ) a6=b,Pi .Ab =true hb (Pj )
Q
ha (Pj ) ?
1 ? (1 ? ?S ) a6=b,Pi .Ab =true hb (Pj )
1?j?n
(1)
Tij .O=true
where g is the prior probability for a motif in protein Pi to be active.
Now, consider a different point in the search, where our current motif activity assignment is Pi .A0?a , which has all the active motifs in Pi .A?a and some additional ones. The
first two terms in the product of Eq. (1) are the same for ?a (Pi .A?a ) and ?a (Pi .A0?a ).
For the final term (the large fraction), one can show using some algebraic manipulation
that this term in ?a (Pi .A0?a ) is lower than that for ?a (Pi .A?a ). We conclude that
?a (Pi .A?a ) ? ?a (Pi .A0?a ), and hence that:
f (Pi .Aa = true, Pi .A0?a )
f (Pi .Aa = true, Pi .A?a )
?1 ?
? 1.
f (Pi .Aa = false, Pi .A?a )
f (Pi .Aa = false, Pi .A0?a )
It follows that, if switching motif a from inactive to active relative to Pi .A decreases f , it
will also decrease f if we have some additional active motifs.
We can exploit this property in a branch-and-bound algorithm in order to find the globally optimal assignment Pi .A. Our algorithm keeps a set V of viable candidates for motif
assignments. For presentation, we encode assignments via the set of active motifs they
contain. Initially, V contains only the empty assignment {}. We start out by considering
motif assignments with a single active motif. We put such an assignment {a} in V if its
f -score is higher than f ({}). Now, we consider assignments {a, b} that have two active
motifs. We consider {a, b} only if both {a} and {b} are in V . If so, we evaluate its f -score,
and add it to V if this score is greater than that of {a} and {b}. Otherwise, we throw it
away. We continue this process for all assignments of size k: For each assignment with
active motif set S, we test whether S ? {a} ? V for all a ? S; if we compare f (S) to each
f (S ? {a}), and add it if it dominates all of them. The algorithm terminates when, from
some k, no assignment of size k is saved.
To understand the intuition behind this pruning procedure, consider a candidate assignment {a, b, c, d}, and assume that {a, b, c} ? V , but {b, c, d} 6? V . In this case, we must
have that {b, c} ? V , but adding d to that assignment reduces the f -score. In this case, as
shown by our analysis, adding d to the superset {a, b, c} would also reduce the f -score.
This algorithm is still exponential in worst case. However, in our setting, a protein with
many motifs has a low prior probability that each of them is active. Hence, adding new
motifs is less likely to increase the f -score, and the algorithm tends to terminate quickly.
As we show in Section 4, this algorithm significantly reduces the cost of our procedure.
Our E-step finds an assignment to P.A which is a strong local optimum of the objective function max P (P.A | T.O, ?): The assignment has higher probability than any
assignment that changes any of the motif variables for any single protein. For that assignment, our algorithm also computes the distribution over all of the binding variables, as
described above. Using this completion, we can now easily compute the (expected) sufficient statistics for the different parameters in the model. As each of these parameters is a
simple binomial distribution, the maximum likelihood estimation in the M-step is entirely
standard; we omit details.
4
Results
We evaluated our model on reliable S. cerevisiae protein interactions data from MIPS [2]
and DIP [3] databases. As for non-interaction data, we randomly picked pairs of proteins
that have no common function and cellular location. This results in a dataset of 2275
proteins, 4838 interactions (Tij .O = true), and 9037 non-interactions (Tij .O = false). We
used sequence motifs from the Prosite database [11] resulting in a dataset of 516 different
motifs with an average of 7.1 motif occurrences per protein. If a motif pair doesn?t appear
between any pair of interacting proteins, we initialize its affinity to be 0 to maximize the
joint likelihood. Its affinity will stay at 0 during the EM iterations and thus simplify our
model structure. We set the initial affinity for the remaining 8475 motif pairs to 0.03.
We train our model with motifs initialized to be either all active (P.A = true) or all
inactive (P.A = false). We get similar results with these two different initializations,
indicating the robustness of our algorithm. Below we only report the results based on all
motifs initialized to be active. Our branch-and-bound algorithm is able to significantly
reduce the number of motif activity assignments that need to be evaluated. For a protein
with 15 motifs, the number of assignments evaluated is reduced from 215 = 32768 in
exhaustive search to 802 using our algorithm. Since majority of the computation is spent
on finding the activity assignments, this resulted in a 40 fold reduction in running time.
Predicting protein-protein interactions. We test our model by evaluating its performance
in predicting interactions. We test this performance using 5-fold cross validation on the set
of interacting and non-interacting protein pairs. In each fold, we train a model and predict
P (Tij .O) = true for pairs Pi , Pj in the held-out interactions.
Many motif pairs are over-represented in interacting proteins. We thus compare our
method to a baseline method that ranks pairs of proteins on the basis of the maximum
enrichment of over-represented motif pairs (see [18] for details). We also compare it to
a model where all motifs are set to be active; this is analogous to the method of Deng
et al. [8]. For completeness, we compare the two variants of the model using data on the
domain (Pfam and ProDom [19]) content of the proteins as well as the Prosite motif content.
Proportion of all interactions
predicted to interact
1
0.8
0.6
Association (Sprinzak & Margalit)
0.4
Prosite motif (Allow inactive motif. P.A = {0, 1})
Prosite motif (All motifs active. P.A = 1)
0.2
Pfam&ProDom (All domains active. P.A = 1.
Deng et al)
0
0
0.2
0.4
0.6
0.8
1
Proportion of all non-interactions predicted to interact
(a)
(b)
Figure 2: (a) ROC curve for different methods. The X-axis is the proportion of all non-interacting
protein pairs in the training data predicted to interact. Y-axis is the proportion of all interacting protein
pairs in the training data predicted to interact. Points are generated using different cutoff probabilties.
A larger area under the curve indicates better prediction. Our method (square marker) outperforms
all other methods. (b) Two protein chains that form a part of the 1ryp complex in PDB, interacting at
the site of two short sequence motifs.
The ROC curves in Fig. 2(a) show that our method outperforms the other methods, and that
the additional degree of freedom of allowing motifs to be inactive is essential. These results
validate our modeling assumptions; they also show that our method can be used to suggest
new interactions and to assign confidence levels on observed interactions, which is much
needed in view of the inaccuracies and large fraction of missing interactions in current
interaction databases.
Evaluating predicted active motifs. A key feature of our approach is its ability to detect
pairs of interacting motifs. We evaluate these predictions against the data from Protein Data
Bank (PDB) [20], which contains some solved structures of interacting proteins Fig. 2(b).
While the PDB data is scarce, it provides the ultimate evaluation of our predicted active
motifs. We extracted all structures from PDB that have at least two co-crystallized chains,
and whose chains are nearly identical to yeast proteins. From the residues that are in contact
between two chains (distance < 5 Angstr?oms), we infer which protein motifs participate in
interactions. Among our training data, 105 proteins have co-crystallized structure in PDB.
On these proteins, our data contained a total of 620 motif occurrences, of which 386 are
predicted to be active. Among those motifs predicted to be active, 257 of them (66.6%)
are interacting in PDB. Among the 234 motifs predicted to be inactive, only 120 of them
(51.3%) are interacting. The chi-square p-value is 10?4 . On the residue level, our predicted
active motifs consist of 3736 amino acids, and 1388 of them (37.2%) are interacting. In
comparison, our predicted inactive motifs consist of 3506 amino acids, and only 588 of
them (16.0%) are interacting. This significant enrichment provides support for the ability
of our method to detect motifs that participate in interactions. In fact, the set of interactions
in PDB is only a subset of the interactions those proteins participate in. Therefore, the
actual rate of false positive active motifs is likely to be lower than we report here.
5
Discussion and Conclusions
In this paper, we presented a probabilistic model which explicitly encodes elements in the
protein sequence that mediate protein-protein interactions. By using a variant of the EM al-
gorithm and a branch-and-bound algorithm for the E-step, we make the learning procedure
tractable. Our result shows that our method successfully uncovers motif activities and binding affinities, and uses them to predict both protein interactions and specific binding sites.
The ability of our model to predict structural elements, without a full structure analysis,
provides support for the viability of our approach.
Our use of a probabilistic model provides us with a general framework to incorporate
different types of data into our model, allowing it to be extended in varies ways. First,
we can incorporate additional signals for protein interactions, such as gene expression data
(as in [9]), cellular location, or even annotations from the literature (as in [7]). We can
also integrate protein interaction data across multiple species; for example, we might try to
use the yeast interaction data to provide more accurate predictions for the protein-protein
interactions in fly [10].
References
[1] P. Uetz, et al. A comprehensive analysis of protein-protein interactions in saccharomyces cerevisiae. Nature, 403(6770):623?7, 2000. 0028-0836 Journal Article.
[2] H. W. Mewes, et al. Mips: a database for genomes and protein sequences. Nucleic Acids Res,
2002.
[3] I. Xenarios, et al.Dip ; the database of interacting proteins: a research tool for studying cellular
networks of protein interactions. Nucleic Acids Research, 30(1):303?305, 2002. (c) 2002 Inst.
For Sci. Info.
[4] P. Chakrabarti and J. Janin. Dissecting protein protein recognition sites. PROTEINS: Structure,
Function, and Genetics, 47:334?343, 2002.
[5] J. J. Gray, et al.Protein protein docking with simultaneous optimization of rigid-body displacement and side-chain conformations. Journal of Molecular Biology, 331:281?299, 2003.
[6] Y. Ofran and B. Rost. Predicted protein-protein interaction sites from local sequence information. FEBS Lett., 544(1-3):236?239, 2003.
[7] R. Jansen, et al. A bayesian networks approach for predicting protein-protein interactions from
genomic data. Science, 302:449?53, 2003.
[8] M. Deng, S. Mehta, F. Sun, and T. Chen. Inferring domain-domain interactions from proteinprotein interactions. Genome Res, 12(10):1540?8, 2002. 22253763 1088-9051 Journal Article.
[9] E. Segal, H. Wang, and D. Koller. Discovering molecular pathways from protein interaction and
gene expression data. Bioinformatics, 19 Suppl 1:I264?I272, 2003. 1367-4803 Journal Article.
[10] L. Giot, et al. A protein interaction map of drosophila melanogaster. Science, 302(5651):1727?
36, 2003.
[11] L. Falquet, et al. The PROSITE database, its status in 2002. Nucliec Acids Research, 30:235?
238, 2002.
[12] D. R. Caffrey, et al. Are protein protein interfaces more conserved in sequence than the rest of
the protein surface? Protein Science, 13:190?202, 2003.
[13] J. Pearl. Probabilistic Reasoning in Intelligent Systems. Morgan Kaufmann, 1988.
[14] D. Koller and A. Pfeffer. Probabilistic frame-based systems. In Proc. AAAI, pages 580?587,
1998.
[15] A.P. Dempster, N.M. Laird, and D.B. Rubin. Maximum likelihood from incomplete data via
the em algorithm. J. Roy. Stat. Soc., B(39):1?39, 1977.
[16] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Generalized belief propagation. In NIPS, pages
689?695, 2000.
[17] M. Henrion. Search-based methods to bound diagnostic probabilities in very large belief nets.
In Uncertainty in Artificial Intelligence, pages 142?150, 1991.
[18] E. Sprinzak and H. Margalit. Correlated sequence-signatures as markers of protein-protein
interaction. Journal of Molecular Biology, 311:681?692, 2001.
[19] R. Apweiler, et al. The interpro database, an integrated documentation resource for protein
families, domains and functional sites. Nucleic Acids Res, 29(1):37?40, 2001. 1362-4962
Journal Article.
[20] H.M. Berman, et al. The protein data bank. Nucleic Acids Research, 28:235?242, 2000.
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1,868 | 2,697 | Theory of Localized Synfire Chain:
Characteristic Propagation Speed of
Stable Spike Patterns
Kosuke Hamaguchi
RIKEN Brain Science Institute
Wako, Saitama 351-0198, JAPAN
[email protected]
Masato Okada
Dept. of Complexity Science and
Engineering, University of Tokyo,
Kashiwa, Chiba, 277-8561, JAPAN
[email protected]
Kazuyuki Aihara
Institute of Industrial Science, University of Tokyo &
ERATO Aihara Complexity Modeling Project JST
Meguro, Tokyo 153-8505, JAPAN
[email protected]
Abstract
Repeated spike patterns have often been taken as evidence for the synfire
chain, a phenomenon that a stable spike synchrony propagates through
a feedforward network. Inter-spike intervals which represent a repeated
spike pattern are influenced by the propagation speed of a spike packet.
However, the relation between the propagation speed and network structure is not well understood. While it is apparent that the propagation
speed depends on the excitatory synapse strength, it might also be related
to spike patterns. We analyze a feedforward network with Mexican-Hattype connectivity (FMH) using the Fokker-Planck equation. We show
that both a uniform and a localized spike packet are stable in the FMH
in a certain parameter region. We also demonstrate that the propagation
speed depends on the distinct firing patterns in the same network.
1 Introduction
Neurons transmit information through spikes, but how the information is encoded remains a
matter of debate. The classical view is that the firing rates of neurons encode information,
while a recent view is that spatio-temporal spike patterns encode the information. For
example, the synchrony of spikes observed in the cortex is thought to play functional roles
in cognitive functions [1]. The mechanism of synchrony has been studied theoretically
for several neural network models. Especially, the model of spike synchrony propagation
through a feedforward network is called the synfire chain [2].
The mechanism of generating synchrony in a feedforward network can be described as follows. When feedforward connections are homogeneous with excitatory efficacy as a whole,
?
firing
rate
synaptic
input
r(?? , t)
?in (?, t)
?
....
??
position
?
inhibitory excitatory
- +
W(? - ??)
Mexican-Hat-type Connectivity
Figure 1: Network architecture. Each layer consists of N neuron arranged in a circle. Each
neuron projects its axon to a post-synaptic layer with Mexican-Hat-type connectivity.
post-synaptic neurons accept similar synaptic inputs. If neurons receive similar temporally
modulated inputs, the resultant spike timings will be also similar, or roughly synchronized
even though the membrane potentials fluctuate because of noise [3]. The question in a feedforward network is, whether the timing of spikes within a layer becomes more synchronized
or not as the spike packet propagates through a sequence of neural layers. Detailed analyses
of the activity propagation in feedforward networks have shown that homogeneous feedforward networks with excitatory synapses have a stable spike synchrony propagation mode
[4, 5, 6]. Neurons, however, are embedded in more structured networks with excitatory and
inhibitory synapses. Thus, such network structure would generate inhomogeneous inputs
to neurons and whether spike synchrony is stable is not a trivial issue.
One simple way to detect the synfire chain phenomena would be to record from several
neurons and find significant repeated patterns. If a spike packet propagates through a feedforward network, a statistically significant number of spike pairs would be found that have
fixed relative time lags (or inter-spike intervals (ISIs).) Such correlated activity has been
experimentally observed in the anterior forebrain pathway of songbirds [7], in the prefrontal cortex of primates [8], both in vivo and in vitro [9], and in an artificially constructed
network in vitro [10]. To generate fixed ISIs by spike packet propagation, the propagation speed of a spike packet must be constant over several trials. The speed depends on
spike patterns as well as the structure of the network. Conventional homogeneous feedforward networks have only one stable spike pattern, namely a spatially uniform synchronized
activity, but structured networks can generally produce spatially inhomogeneous spike patterns. In those networks, the relation between propagation speed and differences in the
spike pattern is not well understood.
It is therefore an important problem to study a biologically realistic, structured network in
the context of the synfire chain. Among suggested network structures, Mexican-Hat-type
(MH) connectivity is one of the most widely accepted as being representative of connectivity in the cortex [11]. Studies of a feedforward network with the MH connectivity (FMH)
have been reported [12]. In a FMH, a localized activity propagates through the network, and
this network is preferable to a homogeneous feedforward network because it can transmit
analog information regarding position [12], and both position and intensity [13]. However,
no detailed analytical work on the structured synfire chain has been reported. In this paper,
we use the Fokker-Planck equation to analyze the FMH. The method of the Fokker-Planck
equation enables us to analyze the collective behavior of the membrane potentials in an
identical neural population [14, 15]. When it is applied to the synfire chain [5], the detailed
analysis of the flow diagram, the effect of the membrane potential distribution on the spike
packet evolution, and the interaction of spike packets is possible [5].
This paper thus examines the feedforward neural network model with Mexican-Hat-type
connectivity. Our strategy is, first, to describe the evolution of firing states through order
parameters, which allows us to measure the macroscopic quantity of the network. Second,
we relate the input order parameters to the output ones through the Fokker-Planck equation.
Finally, we analyze the evolution of spike packets with various shapes, and investigate
stable firing patterns and their propagation speeds.
2 Model
We analyze the dynamics of a structured feedforward network composed of identical single compartment, Leaky Integrate-and-Fire (LIF) neurons. Each neuron is aligned in a ring
neural layer, and projects its axon to the next neural layer with the Mexican-Hat-type connectivity (Fig.1). The input to one neuron generally includes both outputs from pre-synaptic
neurons and a random noisy synaptic current from ongoing activity. If we assume that the
thousands of synaptic background inputs that connect to one neuron are independent Poissonian, instantaneous synapse, and have small amplitude of post synaptic potential (PSP),
we can approximate the sum of noisy background inputs as a Gaussian white noise fluctuating around the mean of the total noisy inputs. The membrane potential of a neuron at
position ? at time t, which receives many random Poisson excitatory and inhibitory inputs,
can be approximately described through a stochastic differential equation as follows:
dv(?, t)
v(?, t)
?
C
= Iin
+?
? + D0 ?(t),
(1)
(?, t) ?
dt
R
Z ?
d?0
?
Iin
W (? ? ?0 )r? (?0 , t),
(2)
(?, t) =
?? 2?
Z 0
r? (?, t) =
dt0 r(?, t ? t0 )?(t0 ),
(3)
??
W (?) = W0 + W1 cos(?),
(4)
?
where C is membrane capacitance, R is membrane resistance, I in (?, t) is the synaptic
current to a neuron at position ?, ?
? is proportional to the mean of the total noisy input, and
?(t) is a Gaussian random variable with h?(t)i = 0 and h?(t)?(t0 )i = ?(t ? t0 ). Here, D0 is
?
the amplitude of Gaussian noise. The input current Iin
(?, t) is obtained from the weighted
?
sum of output currents r (?, t) generated by pre-synaptic neurons. The synaptic current is
derived from the convolution of its firing rate r(?, t) with the PSP time course ?(t). Here,
?(t) = ??2 t exp(??t) where ? is chosen such that a single EPSP generates a 0.0014
mV elevation from the resting potential. The Mexican-Hat-type connectivity consists of
a uniform term W0 and a spatial modulation term W1 cos(?). We set the reset potential
and threshold potential as V0 and Vth , respectively. We start simulations from stationary
distribution of membrane potentials. The input to the initial layer is formulated in terms of
the firing rate on the virtual pre-synaptic layer,
(t ? t?)2
r0 + r1 cos(?)
?
exp(?
),
(5)
r(?, t) =
2? 2
2?? 2
where r0 and r1 are input parameters, and the temporal profile of activity is assumed to
be the Gaussian with ? = 1 and t? = 10. Throughout this paper, parameter values are
given as follows: C = 100 pF, R = 100 M?, Vth = 15 mV, D0 = 100, ?
? = 0.075 pA,
V0 = 0 mV, Vth = 15 mV, ? = 2, and ? = 0.00017. Space is divided into 50 regions
for the Fokker-Planck equation approach, and 10000 LIF neurons per layer are used in
simulations.
C
A
0.2 averaged membrane
potential distribution
mV
10
0
?10
P? = 0 (v,t)
10
0
?10
5
LIF
10
time [ms]
15
20
[P(v,t)]?
B
0
0.1
t=10.5 [ms]
mV
PDE
P?(v,t=10.5)
??
?
0
?
0
?15 ?10
V0
Vth
0
mV
10
15
Figure 2: A: Dynamics of membrane potential distribution at ? = 0. B: A snapshot of
the membrane potential distributions for a range of [?? ?] at t = 10.5. C: A snapshot
of the membrane potential distribution averaged with position ?. Results by a numerical
simulation (LIF) and the Fokker-Planck equation (PDE) are shown.
3 Theory
The prerequisites for a full description of the network activity are time series of order
parameters at an arbitrary time t defined as follows:
Z
p
1
r0 (t) =
(6)
d? r(?, t),
r1 (t) = (rc (t))2 + (rs (t))2 ,
2?
Z
Z
1
1
rc (t) =
d? cos(?)r(?, t),
rs (t) =
d? sin(?)r(?, t),
(7)
2?
2?
where r0 (t) is the population firing rate of the neuron population, and r c (t) and rs (t) are
coefficients of the Fourier transformation of the spatial firing pattern which represent the
spatial eccentricity of activity at time t. rc (t) and rs (t) depends on the position of the
localized activity, but r1 (t) does not. These order parameters play an important role in two
ways. First, we can describe the input to the next layer neuron at ? in a closed form of order
parameters. Second, their time integrals, which will be introduced later, become indices of
the spike packet shape. Input currents are described with the order parameters as follows:
?
Iin
(?, t) = W0 r0? (t) + W1 (rc? (t) cos(?) + rs? (t) sin(?)) ,
Z 0
?
r{0,c,s}
(t) =
dt0 ?(t0 ) r{0,c,s} (t ? t0 ).
(8)
(9)
??
Given the time sequence of order parameters in the pre-synaptic layer, the order parameters
in the post-synaptic layer are obtained through the following calculations. The analytical
method we use here is the Fokker-Planck equation which describes the distribution of the
membrane potential of a pool of identical neurons as the probability density P ? (v, t) of
voltage v at time t. The suffix ? denotes that this neuron population is located at position ?.
We assume that there are a large number of neurons at position ?. Equation (1) is equivalent
to the Fokker-Planck equation [16] within the limit of a large neuron number N ? ?,
?
?
P? (v, t) =
J? (v, t) + ?(v ? V0 )J? (Vth , t),
?t
?v
v
I ? (?, t) + ?
?
?
J? (v, t) =
? in
+
D P? (v, t),
?
C
?v
(10)
(11)
where J? (v, t) is a probability flux and D =
1
2
D0
C
2
. Boundary conditions are
P? (Vth , t) = 0,
r(?, t) =
J? (V0+ , t)
(12)
?
J? (V0? , t)
= J? (Vth , t).
(13)
Equation (12) is the absorbing boundary condition at the threshold potential, and Eq. (13) is
the current source at the reset potential. From Eq. (13), we obtain the firing rate of a postsynaptic neuron r(?, t). The Fokker-Planck equations are solved based on the modified
Chang-Cooper algorithm [17].
Figure 2 shows the actual distribution of the initial layer?s membrane potentials and their
dynamics which accepts virtual layer activity with parameter r0 = 500 and r1 = 350.
Figure 2A shows the evolution of the probability density P?=0 (v, t). From white to black,
the probability becomes higher. Figure 2B is a snapshot of the probability density at time
t = 10.5 over the region from ?? to ?. As a result of a localized input injection, part
of neuronal membrane potentials is strongly depolarized. The membrane potential distribution averaged over the neural layer is illustrated in Fig. 2C. It shows the consistency
between the numerical simulations and the Fokker-Planck equations. The probability flow
dropping out from the threshold potential Vth is a firing rate. Combined with these firing
rates at each position ? and definitions of order parameters in Eqs. (6)-(7), the order parameters on the post-synaptic neural layer are again calculated. The closed forms of order
parameters have been obtained.
Spatio-temporal patterns of firing rates and dynamics of order parameters in response to
a localized input (r0 = 600, r1 = 300) and a uniform input (r0 = 900, r1 = 0) are
shown in Fig. 3. When an input is spatially localized, the spatio-temporal firing pattern
is localized with a slightly distorted shape (Fig. 3A). On the other hand, when a uniform
input is applied, the spatio-temporal firing pattern is uniform as illustrated in Fig. 3B. We
show an example of the time course of r0 (t) and r1 (t) in Fig. 3C and 3D for both the
numerical simulation of 10, 000 LIF neurons and the Fokker-Planck equation. Elevation of
time course of r1 (t) in Fig. 3C indicates the localized firing. In contrast, the uniform input
generates no response in r1 (t) parameter as illustrated in Fig. 3D.
To quantitatively evaluate the spike packet shape and propagation speed, we define indices
r0 , r1 , and ?. r0 and r1 can be directly defined as
Z
Z
r0 = dt r0 (t) ? spontaneous firing rate, r1 = dt r1 (t).
(14)
r0 corresponds to the total population activity, and r1 corresponds to spatial eccentricity of
the activity. r0 and r1 are a natural extension of an index used in the study of the synfire
chain [4] in the sense that an index corresponds to the area of a time varying parameters
of the system, such as the population firing rate (r0 (t)) above the spontaneous firing rate,
or spatial eccentricity (r1 (t)). The basic idea of characterizing the spike packet was to
approximate the firing rate curve through a Gaussian function [4] as in Eq. (5). Here, the
approximated Gaussian curve is obtained by minimizing the mean squared error with r 0 (t)
and the Gaussian. We also use the index ? obtained from the variance of the Gaussian, and
t? as an index for the arrival time of the spike synchrony taken from the peak time of the
Gaussian (Fig. 3C).
4 Results
Our observation of the activities of the FMH with various parameter sets reveals two types
of stable spike packets. Figure 4 shows the activity of the FMH with four characteristic
parameter sets of W0 and W1 . Here we use r0 = 500 and r1 = 350 for the upper figures
A
0
??
5
10
time [ms]
15
0
??
20
C
5
10
time [ms]
15
20
D
?
500
t
0
5
1000
FP
LIF
r0(t)
[spk/s]
1000 Gaussian
Approximation
[spk/s]
firing rate
?
?
?
B
firing rate
?
r1(t)
10
time [ms]
15
20
r0(t)
FP
LIF
500
r1(t)
0
5
10
time [ms]
15
20
Figure 3: Activity profiles in response to a localized input (A,C) and a uniform input (B,D).
A,B: Spatio-temporal pattern of the firing rates from neurons at position ?? to ?. C,D:
Time courses of order parameters (r0 (t), r1 (t)) calculated from numerical simulations of
a population of LIF neurons (squares and crosses) and the Fokker-Planck equation (solid
lines). The time course of order parameters in response to a single stimulus is approximated
by using a Gaussian function. In C, Gaussian approximation of r 0 (t) is also shown. The
variance of the Gaussian ? and the mean value t? are used as the indices of a spike packet.
as a localized input and r0 = 900 and r1 = 0 for the lower ones as a uniform input. The
common parameter is ? = 1 and t? = 2. When both W0 and W1 are small, no spike packet
can propagate (Non-firing). When the uniform activation term W 0 is sufficiently strong,
a uniform spike packet is stable (Uniform Activity). Note that even though the localized
input elicits localized spike packets with several layers, it finally decays to the uniform
spike packet. When the Mexican-Hat term W1 is strong enough, only the localized spike
packet is stable (Localized Activity). When W0 and W1 are balanced within a certain ratio,
there exists a novel firing mode where both the uniform and the localized spike packet are
stable depending on the initial layer input (Multi-stable). The results show that there are
four types of phase and two types of spike packet in the FMH. The stability of a spike packet
depends on W0 and W1 . In addition, the difference of the arrival times of propagating spike
packets in the 8th layer shown in the Multi-stable phase indicates that the propagation speed
of spike packets might differ.
It is apparent that the propagation speed depends on the strength of the excitatory synapse
efficacy, however, our results in the Multi-stable phase in Fig. 4 suggest that a spike pattern
also determines the propagation speed. To investigate this effect, we plotted propagation
time ?t?, the difference between propagation time t?post and t?pre for various W1 (Fig. 5B).
The speed is analyzed after the spike packet indices r0 , r1 and ? have converged. The
convergences of spike packets are shown in the flow diagram in Fig. 5A for (W 0 , W1 ) =
(1, 1.5) case. In Fig. 5B, each triangle indicates the speed of the localized activity, and each
circle corresponds to that of the uniform activity. Within the plotted region (W 1 = 1.4 ?
2), both the uniform and localized activities are stable, and no bursting activity is observed.
This indicates that as W1 rises the propagation speed of localized activity becomes higher.
In contrast, the propagation speed of the uniform activity does not depends on W 1 because
Layer
Uniform Input Localized Input
Non-firing
(W0,W1) = (0.7, 1)
1
2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
0
5
10
15
Uniform Activity
(W0,W1) = (1, 0.6)
20 0
5
10
15
Localized Activity Multi-stable phase
(W0,W1) = (0.7, 2.5) (W0,W1) = (1, 1.4)
20 0
5
10
15
20 0
5
10
15
20
time [ms]
Figure 4: Four characteristic FMH with different stable states. The evolutions of firing rate
propagation are shown. The upper row panels show the response to an identical localized
pulse input, and the lower row panels show the response to a uniform pulse input.
uniform activity generates rc (t) = rs (t) = 0.
5 Summary
We have found that there are four phases in the W0 ? W1 parameter space; Non-firing,
Localized activity, Uniform activity, and Multi-stable phase. Multi-stable phase is the most
intriguing in that an identical network has completely different firing modes in response to
different initial inputs. In this phase, the effect of spike pattern on the propagation speed
of the spike packet can be directly studied. By the analysis of the Fokker-Planck equation,
we found that the propagation speed depends on the distinct firing patterns in the same
network. It implies that observation of repeated spike patterns requires an appropriately
controlled input if the network structure produces a multi-stable state. The characteristic
speed of the spike packet also suggests that the speed of information processing in the brain
depends on the spiking pattern, or the representation of the information.
Acknowledgment
This study is partially supported by the Advanced and Innovational Research Program in
Life Sciences, a Grant-in-Aid No. 15016023 for Scientific Research on Priority Areas
(C) Advanced Brain Science Project, a Grand-in-Aid No. 14084212, from the Ministry of
Education, Culture, Sports, Science, and Technology, the Japanese Government.
References
[1] C. M. Gray, P. K?onig, A. K. Engel, and W. Singer, ?Oscillatory responses in cat visual cortex exhibit inter-columnar synchronization which reflects global stimulus properties,? Nature,
vol. 338, pp. 334?337, 1989.
[2] M. Abeles, Corticonics: neural circuits of the cerebral cortex. Cambridge UP, 1991.
[3] Z. Mainen and T. Sejnowski, ?Reliability of spike timing in neocortical neurons,? Science,
vol. 268, pp. 1503?1506, 1995.
[4] M. Diesmann, M.-O. Gewaltig, and A. Aertsen, ?Stable propagation of synchronous spiking in
cortical neural networks,? Nature, vol. 402, pp. 529?533, 1999.
A
w0 = 1
B
1.2
local
uniform
-
? t [ms]
500
300
r1
Localized
100
Uniform
0
300
200
100
?/?
0
200
600
r0
1000
1
0.8
1.4
1.6
1.8
w1
2
Figure 5: A: Flow diagram with parameter (W0 , W1 ) = (1, 1.5) where both localized and
uniform spike packets are stable. Two attractors in high r1 region (Localized) and high r0
with r1 = 0 region (Uniform) are shown. A sequence of arrows indicates the evolution of
a spike packet in the (r0 , ?/?, r1 ) space. B: Plot of propagation time ?t? that is necessary
time for a spike packet to propagate one neural layer. These results indicate that localized
spike packets propagate slower than the uniform ones.
[5] H. C?ateau and T. Fukai, ?Fokker-planck approach to the pulse packet propagation in synfire
chain,? Neural Networks, vol. 14, pp. 675?685, 2001.
[6] W. M. Kistler and W. Gerstner, ?Stable propagation of activity pulses in populations of spiking
neurons,? Neural Comp., vol. 14, pp. 987?997, 2002.
[7] R. Kimpo, F. Theunissen, and A. Doupe, ?Propagation of correlated activity through multiple
stages of a neural circuit,? J. Neurosci., vol. 23, no. 13, pp. 5750?5761, 2003.
[8] M. Abeles, H. Bergman, E. Margalit, and E. Vaadia, ?Spatiotemporal firing patterns in the
frontal cortex of behaving monkeys,? J. Neurophysiol., vol. 70, pp. 1629?1638, 1993.
[9] Y. Ikegaya, G. Aaron, R. Cossart, D. Aronov, I. Lampl, D. Ferster, and R. Yuste, ?Synfire
chains and cortical songs: Temporal modules of cortical activity,? Science, vol. 304, pp. 559?
564, 2004.
[10] A. Reyes, ?Synchrony-dependent propagation of firing rate in iteratively constructed networks
in vitro,? Nature Neuroscience, vol. 6, pp. 593 ? 599, 2003.
[11] D. H. Hubel and T. N. Wiesel, ?Receptive fields, binocular interaction and functional architecture in the cat?s visual cortex,? J. Physiol., vol. 160, pp. 106?154, 1962.
[12] M. C. W. van Rossum, G. G. Turrigiano, and S. B. Nelson, ?Fast propagation of firing rates
through layered networks of noisy neurons,? J. Neurosci., vol. 22, pp. 1956?1966, 2002.
[13] K. Hamaguchi and K. Aihara, ?Quantitative information transfer through layers of spiking neurons connected by mexican-hat type connectivity,? Neurocomputing, vol. 58-60, pp. 85?90,
2004.
[14] D. J. Amit and N. Brunel, ?Dynamics of a recurrent network of spiking neurons before and
following learning,? Network: Computation in Neural Systems, vol. 8, no. 4, pp. 373?404,
1997.
[15] N. Brunel and V. Hakim, ?Fast Global Oscillations in Networks of Integrate-and-Fire Neurons
with Low Firing Rates,? Neural Comp., vol. 11, no. 7, pp. 1621?1671, 1999.
[16] H. Risken, The Fokker-Planck Equation. Springer-Verlag, 1996.
[17] J. S. Chang and G. Cooper, ?A practical difference scheme for fokker-planck equations,? J.
Comp. Phys., vol. 6, pp. 1?16, 1970.
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1,869 | 2,698 | Sharing Clusters Among Related Groups:
Hierarchical Dirichlet Processes
Yee Whye Teh(1) , Michael I. Jordan(1,2), Matthew J. Beal(3) and David M. Blei(1)
(1)
(3)
Computer Science Div., (2) Dept. of Statistics
Dept. of Computer Science
University of California at Berkeley
University of Toronto
Berkeley CA 94720, USA
Toronto M5S 3G4, Canada
{ywteh,jordan,blei}@cs.berkeley.edu
[email protected]
Abstract
We propose the hierarchical Dirichlet process (HDP), a nonparametric
Bayesian model for clustering problems involving multiple groups of
data. Each group of data is modeled with a mixture, with the number of
components being open-ended and inferred automatically by the model.
Further, components can be shared across groups, allowing dependencies
across groups to be modeled effectively as well as conferring generalization to new groups. Such grouped clustering problems occur often in
practice, e.g. in the problem of topic discovery in document corpora. We
report experimental results on three text corpora showing the effective
and superior performance of the HDP over previous models.
1 Introduction
One of the most significant conceptual and practical tools in the Bayesian paradigm is
the notion of a hierarchical model. Building on the notion that a parameter is a random
variable, hierarchical models have applications to a variety of forms of grouped or relational
data and to general problems involving ?multi-task learning? or ?learning to learn.? A
simple and classical example is the Gaussian means problem, in which a grand mean ?0
is drawn from some distribution, a set of K means are then drawn independently from a
Gaussian with mean ?0 , and data are subsequently drawn independently from K Gaussian
distributions with these means. The posterior distribution based on these data couples the
means, such that posterior estimates of the means are shrunk towards each other. The
estimates ?share statistical strength,? a notion that can be made precise within both the
Bayesian and the frequentist paradigms.
Here we consider the application of hierarchical Bayesian ideas to a problem in ?multi-task
learning? in which the ?tasks? are clustering problems, and our goal is to share clusters
among multiple, related clustering problems. We are motivated by the task of discovering
topics in document corpora [1]. A topic (i.e., a cluster) is a distribution across words while
documents are viewed as distributions across topics. We want to discover topics that are
common across multiple documents in the same corpus, as well as across multiple corpora.
Our work is based on a tool from nonparametric Bayesian analysis known as the Dirichlet
process (DP) mixture model [2, 3]. Skirting technical definitions for now, ?nonparametric?
can be understood simply as implying that the number of clusters is open-ended. Indeed,
at each step of generating data points, a DP mixture model can either assign the data point
to a previously-generated cluster or can start a new cluster. The number of clusters is a
random variable whose mean grows at rate logarithmic in the number of data points.
Extending the DP mixture model framework to the setting of multiple related clustering
problems, we will be able to make the (realistic) assumption that we do not know the
number of clusters a priori in any of the problems, nor do we know how clusters should be
shared among the problems.
When generating a new cluster, a DP mixture model selects the parameters for the cluster
(e.g., in the case of Gaussian mixtures, the mean and covariance matrix) from a distribution
G0 ?the base distribution. So as to allow any possible parameter value, the distribution
G0 is often assumed to be a smooth distribution (i.e., non-atomic). Unfortunately, if we
now wish to extend DP mixtures to groups of clustering problems, the assumption that G0
is smooth conflicts with the goal of sharing clusters among groups. That is, even if each
group shares the same underlying base distribution G0 , the smoothness of G0 implies that
they will generate distinct cluster parameters (with probability one). We will show that this
problem can be resolved by taking a hierarchical Bayesian approach. We present a notion
of a hierarchical Dirichlet process (HDP) in which the base distribution G0 for a set of DPs
is itself a draw from a DP. This turns out to provide an elegant and simple solution to the
problem of sharing clusters among multiple clustering problems.
The paper is organized as follows. In Section 2, we provide the basic technical definition
of DPs and discuss related representations involving stick-breaking processes and Chinese
restaurant processes. Section 3 then introduces the HDP, motivated by the requirement
of a more powerful formalism for the grouped data setting. As for the DP, we present
analogous stick-breaking and Chinese restaurant representations for the HDP. We present
empirical results on a number of text corpora in Section 5, demonstrating various aspects of
the HDP including its nonparametric nature, hierarchical nature, and the ease with which
the framework can be applied to other realms such as hidden Markov models.
2 Dirichlet Processes
The Dirichlet process (DP) and the DP mixture model are mainstays of nonparametric
Bayesian statistics (see, e.g., [3]). They have also begun to be seen in applications in machine learning (e.g., [7, 8, 9]). In this section we give a brief overview with an eye towards
generalization to HDPs. We begin with the definition of DPs [4]. Let (?, B) be a measurable space, with G0 a probability measure on the space, and let ?0 be a positive real number.
A Dirichlet process is the distribution of a random probability measure G over (?, B) such
that, for any finite partition (A1 , . . . , Ar ) of ?, the random vector (G(A1 ), . . . , G(Ar )) is
distributed as a finite-dimensional Dirichlet distribution:
(G(A1 ), . . . , G(Ar )) ? Dir ?0 G0 (A1 ), . . . , ?0 G0 (Ar ) .
(1)
We write G ? DP(?0 , G0 ) if G is a random probability measure distributed according to
a DP. We call G0 the base measure of G, and ?0 the concentration parameter.
The DP can be used in the mixture model setting in the following way. Consider a set
of data, x = (x1 , . . . , xn ), assumed exchangeable. Given a draw G ? DP(?0 , G0 ),
independently draw n latent factors from G: ?i ? G. Then, for each i = 1, . . . , n,
draw xi ? F (?i ), for a distribution F . This setup is referred to as a DP mixture model.
If the factors ?i were all distinct, then this setup would yield an (uninteresting) mixture
model with n components. In fact, the DP exhibits an important clustering property, such
that the draws ?i are generally not distinct. Rather, the number of distinct values grows as
O(log n), and it is this that defines the random number of mixture components.
There are several perspectives on the DP that help to understand this clustering property.
In this paper we will refer to two: the Chinese restaurant process (CRP), and the stickbreaking process. The CRP is a distribution on partitions that directly captures the clustering of draws from a DP via a metaphor in which customers share tables in a Chinese restaurant [5]. As we will see in Section 4, the CRP refers to properties of the joint distribution
of the factors {?i }. The stick-breaking process, on the other hand, refers to properties of
G, and directly reveals its discrete nature [6]. For k = 1, 2 . . ., let:
Qk?1
?k ? G0
?k0 ? Beta(1, ?0 )
?k = ?k0 l=1 (1 ? ?k0 ).
(2)
P?
Then with probability one the random measure defined by G = k=1 ?k ??k is a sample
from DP(?0 , G0 ). The construction for ?1 , ?2 , . . . in (2) can be understood as taking a
stick of unit length, and repeatedly breaking off segments of length ?k . The stick-breaking
construction shows that DP mixture models can be viewed as mixture models with a countably infinite number of components. To see this, identify each ?k as the parameter of the
k th mixture component, with mixing proportion given by ?k .
3 Hierarchical Dirichlet Processes
We will introduce the hierarchical Dirichlet process (HDP) in this section. First we describe the general setting in which the HDP is most useful?that of grouped data. We
assume that we have J groups of data, each consisting of nj data points (xj1 , . . . , xjnj ).
We assume that the data points in each group are exchangeable, and are to be modeled with
a mixture model. While each mixture model has mixing proportions specific to the group,
we require that the different groups share the same set of mixture components. The idea is
that while different groups have different characteristics given by a different combination
of mixing proportions, using the same set of mixture components allows statistical strength
to be shared across groups, and allows generalization to new groups.
The HDP is a nonparametric prior which allows the mixture models to share components.
It is a distribution over a set of random probability measures over (?, B): one probability
measure Gj for each group j, and a global probability measure G0 . The global measure G0
is distributed as DP(?, H), with H the base measure and ? the concentration parameter,
while each Gj is conditionally independent given G0 , with distribution Gj ? DP(?0 , G0 ).
To complete the description of the HDP mixture model, we associate each xji with a factor
?ji , with distributions given by F (?ji ) and Gj respectively. The overall model is given in
Figure 1 left, with conditional distributions:
G0 | ?, H ? DP(?, H)
?ji | Gj ? Gj
Gj | ?, G0 ? DP(?0 , G0 )
xji | ?ji ? F (?ji ) .
(3)
(4)
The stick-breaking construction
(2) shows that a draw of G0 can be expressed as a weighted
P?
sum of point masses: G0 = k=1 ?k ??k . This fact that G0 is atomic plays an important
role in ensuring that mixture components are shared across different groups. Since G0 is
the base distribution for the individual Gj ?s, (2) again shows that the atoms of the individual
Gj are samples from G0 . In particular, since G0 places non-zero mass only on the atoms
? = (?k )?
k=1 , the atoms of Gj must also come from ?, hence we may write:
P
P
G0 = ?
Gj = ?
(5)
k=1 ?k ??k
k=1 ?jk ??k .
Identifying ?k as the parameters of the k th mixture component, we see that each submodel
corresponding to distinct groups share the same set of mixture components, but have differing mixing proportions, ? j = (?jk )?
k=1 .
Finally, it is useful to explicitly describe the relationships between the mixing proportions
? and (? j )Jj=1 . Details are provided in [10]. Note that the weights ?j are conditionally independent given ? since each Gj is independent given G0 . Applying (1) to finite partitions
H
?
?0
?0
G1
?1i
global
?11
?13
G0
?0
G2
x1i
x2i
n1
?23 ?32
G3
?18 ?11
?16
? ?15
?11 ?13 ?12 ?12
14
?17
n3
?22
?21 ?21
?25
?23
?24
?23 ?22 ?26
?24
?13
x3i
n2
?31
?22
?12 ?3
?24 ?21
?3i
?2i
?2
?1
?36
?31 ?35
?31
?32
?34
?33
?32
?28 ?27
group j=2
group j=1
group j=3
Figure 1: Left: graphical model of an example HDP mixture model with 3 groups. Corresponding
to each DP node we also plot a sample draw from the DP using the stick-breaking construction.
Right: an instantiation of the CRF representation for the 3 group HDP. Each of the 3 restaurants has
customers sitting around tables, and each table is served a dish (which corresponds to customers in
the Chinese restaurant for the global DP).
of ?, we get ? j ? DP(?0 , ?), where we interpret ? and ? j as probability measures over
the positive integers. Hence ? is simply the putative mixing proportion over the groups.
We may in fact obtain an explicit stick-breaking construction for the ? j ?s as well. Applying
(1) to partitions ({1, . . . , k ? 1}, {k}, {k + 1, . . .}) of positive integers, we have:
Pk
0
0 Qk?1
0
?jk
? Beta ?0 ?k , ?0 1 ? l=1 ?l
?jk = ?jk
(6)
l=1 (1 ? ?jl ) .
4 The Chinese Restaurant Franchise
We describe an alternative view of the HDP based directly upon the distribution a HDP induces on the samples ?ji , where we marginalize out G0 and Gj ?s. This view directly leads
to an efficient Gibbs sampler for HDP mixture models, which is detailed in the appendix.
Consider, for one group j, the distribution of ?j1 , . . . , ?jnj as we marginalize out Gj . Recall that since Gj ? DP(?0 , G0 ) we can describe this distribution by describing how to
generate ?j1 , . . . , ?jnj using the CRP. Imagine nj customers (each corresponds to a ?ji )
at a Chinese restaurant with an unbounded number of tables. The first customer sits at the
first table. A subsequent customer sits at an occupied table with probability proportional
to the number of customers already there, or at the next unoccupied table with probability
proportional to ?0 . Suppose customer i sat at table tji . The conditional distributions are:
X
P njt
tji | tj1 , . . . , tji?1 , ?0 ?
?t + P 0 n?00 +?0 ?tnew ,
(7)
0 n 0 +?0
t
t
jt
t
jt
where njt is the number of customers currently at table t. Once all customers have sat down
the seating plan corresponds to a partition of ?j1 , . . . , ?jnj . This is an exchangeable process in that the probability of a partition does not depend on the order in which customers
sit down. Now we associate with table t a draw ?jt from G0 , and assign ?ji = ?jtji .
Performing this process independently for each group j, we have now integrated out all the
Gj ?s, and have an assignment of each ?ji to a sample ?jtji from G0 , with the partition
structures given by CRPs. Notice now that all ?jt ?s are simply i.i.d. draws from G0 , which
is again distributed according to DP(?, H), so we may apply the same CRP partitioning
process to the ?jt ?s. Let the customer associated with ?jt sit at table kjt . We have:
X
P mk
kjt | k11 , . . . , k1n1 , k21 , . . . , kjt?1 , ? ?
?k + P 0 m? 0 +?0 ?knew . (8)
0 m 0 +?
k
k
jk
k
k
Perplexity on test abstacts of LDA and HDP mixture
Posterior over number of topics in HDP mixture
1050
15
LDA
HDP Mixture
Number of samples
Perplexity
1000
950
900
850
10
5
800
750
10
20
30
40
50 60 70 80 90 100 110 120
Number of LDA topics
0
61 62 63 64 65 66 67 68 69 70 71 72 73
Number of topics
Figure 2: Left: comparison of LDA and HDP mixture. Results are averaged over 10 runs, with error
bars being 1 standard error. Right: histogram of the number of topics the HDP mixture used over 100
posterior samples.
Finally we associate with table k a draw ?k from H and assign ?jt = ?kjt . This completes
the generative process for the ?ji ?s, where we marginalize out G0 and Gj ?s. We call this
generative process the Chinese restaurant franchise (CRF). The metaphor is as follows: we
have J restaurants, each with nj customers (?ji ?s), who sit at tables (?jt ?s). Now each table
is served a dish (?k ?s) from a menu common to all restaurants. The customers are sociable,
prefering large tables with many customers present, and also prefer popular dishes.
5 Experiments
We describe 3 experiments in this section to highlight the various aspects of the HDP: its
nonparametric nature; its hierarchical nature; and the ease with which we can apply the
framework to other models, specifically the HMM.
Nematode biology abstracts. To demonstrate the strength of the nonparametric approach
as exemplified by the HDP mixture, we compared it against latent Dirichlet allocation
(LDA), which is a parametric model similar in structure to the HDP [1]. In particular,
we applied both models to a corpus of nematode biology abstracts1 , evaluating the perplexity of both models on held out abstracts. Here abstracts correspond to groups, words
correspond to observations, and topics correspond to mixture components, and exchangeability correspond to the typical bag-of-words assumption. In order to study specifically the
nonparametric nature of the HDP, we used the same experimental setup for both models2 ,
except that in LDA we had to vary the number of topics used between 10 and 120, while
the HDP obtained posterior samples over this automatically.
The results are shown in Figure 2. LDA performs best using between 50 and 80 topics,
while the HDP performed just as well as these. Further, the posterior over the number of
topics used by HDP is consistent with this range. Notice however that the HDP infers the
number of topics automatically, while LDA requires some method of model selection.
NIPS sections. We applied HDP mixture models to a dataset of NIPS 1-12 papers organized into sections3 . To highlight the transfer of learning achievable with the HDP, we
1
Available at http://elegans.swmed.edu/wli/cgcbib. There are 5838 abstracts in total. After removing
standard stop words and words appearing less than 10 times, we are left with 476441 words in total
and a vocabulary size of 5699.
2
In both models, we used a symmetric Dirichlet distribution with weights of 0.5 for the prior H
over topic distributions, while the concentration parameters are integrated out using a vague gamma
prior. Gibbs sampling using the CRF is used, while the concentration parameters are sampled using
a method described in [10]. This also applies to the NIPS sections experiment on next page.
3
To ensure we are dealing with informative words in the documents, we culled stop words as well
Average perplexity over NIPS sections of 3 models
Generalization from LT, AA, AP to VS
6000
5000
M1: additional sction ignored
M2: flat, additional section
M3: hierarchical, additional section
5500
Perplexity
Perplexity
5000
LT
AA
AP
4500
4500
4000
4000
3500
3500
3000
3000
2500
0
10
20
30
40
50
60
Number of VS training documents
70
80
2500
0
10
20
30
40
50
60
Number of VS training documents
70
80
Figure 3: Left: perplexity of test VS documents given training documents from VS and another
section for 3 different models. Curves shown are averaged over the other sections and 5 runs. Right:
perplexity of test VS documents given LT, AA and AP documents respectively, using M3, averaged
over 5 runs. In both, the error bars are 1 standard error.
show improvements to the modeling of a section when the model is also given documents
from another section. Our test section is always the VS (vision sciences) section, while
the additional section is varied across the other eight. The training set always consist of
80 documents from the other section (so that larger sections like AA (algorithms and architecures) do not get an unfair advantage), plus between 0 and 80 documents from VS.
There are 47 test documents, which are held fixed as we vary over the other section and the
number N of training VS documents. We compared 3 different models for this task. The
first model (M1) simply ignores documents from the additional section, and uses a HDP to
model the VS documents. It serves as a baseline. The second model (M2) uses a HDP mixture model, with one group per document, but lumping together training documents from
both sections. The third model (M3) takes a hierarchical approach and models each section
separately using a HDP mixture model, and places another DP prior over the common base
distributions for both submodels4 .
As we see in Figure 3 left, the more hierarchical approach of M3 performs best, with perplexity decreasing drastically with modest values of N , while M1 does worst for small N .
However with increasing N , M1 improves until it is competitive with M3 but M2 does
worst. This is because M2 lumps all the documents together, so is not able to differentiate
between the sections, as a result the influence of documents from the other section is unduly strong. This result confirms that the hierarchical approach to the transfer-of-learning
problem is a useful one, as it allows useful information to be transfered to a new task (here
the modeling of a new section), without the data from the previous tasks overwhelming
those in the new task.
We also looked at the performance of the M3 model on VS documents given specific other
sections. This is shown in Figure 3 right. As expected, the performance is worst given LT
(learning theory), and improves as we move to AA and AP (applications). In Table 1 we
show the topics pertinent to VS discovered by the M3 model. First we trained the model
on all documents from the other section. Then, keeping the assignments of words to topics
fixed in the other section, we introduced VS documents and the model decides to reuse
some topics from the other section, as well as create new ones. The topics reused by VS
documents confirm to our expectations of the overlap between VS and other sections.
as words occurring more than 4000 or less than 50 times in the documents. As sections differ over
the years, we assigned by hand the various sections to one of 9 prototypical sections: CS, NS, LT,
AA, IM, SP, VS, AP and CN.
4
Though we have only described the 2 layer HDP the 3 layer extension is straightforward. In
fact on our website http://www.cs.berkeley.edu/?ywteh/research/npbayes we have an implementation of the
general case where DPs are coupled hierarchically in a tree-structured model.
LT
AA
task
representation
pattern
processing
trained
representations
three process
unit patterns
CS
cells cell
activity
response
neuron visual
patterns
pattern single
fig
NS
signal layer
gaussian cells
fig nonlinearity
nonlinear rate
eq cell
algorithms test
approach
methods based
point problems
form large
paper
processing
pattern
approach
architecture
single shows
simple based
large control
IM
visual images
video language
image pixel
acoustic delta
lowpass flow
SP
approach
based trained
test layer
features table
classification
rate paper
AP
ii tree pomdp
observable
strategy class
stochastic
history
strategies
density
CN
examples
concept
similarity
bayesian
hypotheses
generalization
numbers
positive classes
hypothesis
visual cells
cortical
orientation
receptive
contrast spatial
cortex stimulus
tuning
large examples
form point see
parameter
consider
random small
optimal
distance
tangent image
images
transformation
transformations
pattern vectors
convolution
simard
motion visual
velocity flow
target chip eye
smooth
direction optical
signals
separation
signal sources
source matrix
blind mixing
gradient eq
image images
face similarity
pixel visual
database
matching facial
examples
policy optimal
reinforcement
control action
states actions
step problems
goal
Table 1: Topics shared between VS and the other sections. Shown are the two topics with most
numbers of VS words, but also with significant numbers of words from the other section.
Alice in Wonderland. The infinite hidden Markov model (iHMM) is a nonparametric
model for sequential data where the number of hidden states is open-ended and inferred
from data [11]. In [10] we show that the HDP framework can be applied to obtain a cleaner
formulation of the iHMM, providing effective new inference algorithms and potentially hierarchical extensions. In fact the original iHMM paper [11] served as inspiration for this
work and first coined the term ?hierarchical Dirichlet processes??though their model is
not hierarchical in the Bayesian sense, involving priors upon priors, but is rather a set of
coupled urn models similar to the CRF. Here we report experimental comparisons of the
iHMM against other approaches on sentences taken from Lewis Carroll?s Alice?s Adventures in Wonderland.
Perplexity on test sentences of Alice
50
ML
MAP
VB
Perplexity
40
30
20
10
0
0
5
10
15
20
Number of hidden states
25
30
Figure 4: Comparing iHMM (horizontal line)
versus ML, MAP and VB trained HMMs. Error bars are 1 standard error (those for iHMM too
small to see).
ML, MAP, and variational Bayesian (VB)
[12] models with numbers of states ranging from 1 to 30 were trained multiple
times on 20 sentences of average length
51 symbols (27 distinct symbols, consisting of 26 letters and ? ?), and tested on
40 sequences of average length 100. Figure 4 shows the perplexity of test sentences. For VB, the predictive probability
is intractable to compute, so the modal setting of parameters was used. Both MAP
and VB models were given optimal settings of the hyperparameters found in the
iHMM. We see that the iHMM has a lower
perlexity than every model size for ML,
MAP, and VB, and obtains this with one
countably infinite model.
6 Discussion
We have described the hierarchical Dirichlet process, a hierarchical, nonparametric model
for clustering problems involving multiple groups of data. HDP mixture models are able
to automatically determine the appropriate number of mixture components needed, and
exhibit sharing of statistical strength across groups by having components shared across
groups. We have described the HDP as a distribution over distributions, using both the
stick-breaking construction and the Chinese restaurant franchise. In [10] we also describe
a fourth perspective based on the infinite limit of finite mixture models, and give detail for
how the HDP can be applied to the iHMM. Direct extensions of the model include use of
nonparametric priors other than the DP, building higher level hierarchies as in our NIPS
experiment, as well as hierarchical extensions to the iHMM.
Appendix: Gibbs Sampling in the CRF
The CRF is defined by the variables t = (tji ), k = (kjt ), and ? = (?k ). We describe an
inference procedure for the HDP mixture model based on Gibbs sampling t, k and ? given
data items x. For the full derivation see [10]. Let f (?|?) and h be the density functions for
?jt
F (?) and H respectively, n?i
be
jt be the number of tji0 ?s equal to t except tji , and mk
the number of kj 0 t0 ?s equal to k except kjt . The conditional probability for tji given the
other variables is proportional to the product of a prior and likelihood term. The prior term
is given by (7) where, by exchangeability, we can take tji to be the last one assigned. The
likelihood is given by f (xji |?kjt ) where for t = tnew we may sample kjtnew using (8), and
?knew ? H. The distribution is then:
?0 f (xji |?kjt ) if t = tnew
p(tji = t | t\tji , k, ?, x) ?
(9)
n?i
jt f (xji |?kjt ) if t currently used.
Similarly the conditional distributions for kjt and ?k are:
( Q
? i:tji =t f (xji |?k )
if k = k new
p(kjt = k | t, k\kjt , ?, x) ?
?t Q
mk
i:tji =t f (xji |?k ) if k currently used.
Y
p(?k | t, k, ?\?k , x) ? h(?k )
f (xji |?k )
(10)
(11)
ji:kjtji =k
where ?knew ? H. If H is conjugate to F (?) we have the option of integrating out ?.
References
[1] D.M. Blei, A.Y. Ng, and M.I. Jordan. Latent Dirichlet allocation. JMLR, 3:993?1022, 2003.
[2] M.D. Escobar and M. West. Bayesian density estimation and inference using mixtures. Journal
of the American Statistical Association, 90:577?588, 1995.
[3] S.N. MacEachern and P. M?uller. Estimating mixture of Dirichlet process models. Journal of
Computational and Graphical Statistics, 7:223?238, 1998.
[4] T.S. Ferguson. A Bayesian analysis of some nonparametric problems. Annals of Statistics,
1(2):209?230, 1973.
?
[5] D. Aldous. Exchangeability and related topics. In Ecole
d??et?e de probabilit?es de Saint-Flour
XIII?1983, pages 1?198. Springer, Berlin, 1985.
[6] J. Sethuraman. A constructive definition of Dirichlet priors. Statistica Sinica, 4:639?650, 1994.
[7] R.M. Neal. Markov chain sampling methods for Dirichlet process mixture models. Journal of
Computational and Graphical Statistics, 9:249?265, 2000.
[8] C.E. Rasmussen. The infinite Gaussian mixture model. In NIPS, volume 12, 2000.
[9] D.M. Blei, T.L. Griffiths, M.I. Jordan, and J.B. Tenenbaum. Hierarchical topic models and the
nested Chinese restaurant process. NIPS, 2004.
[10] Y.W. Teh, M.I. Jordan, M.J. Beal, and D.M. Blei. Hierarchical dirichlet processes. Technical
Report 653, Department of Statistics, University of California at Berkeley, 2004.
[11] M.J. Beal, Z. Ghahramani, and C.E. Rasmussen. The infinite hidden Markov model. In NIPS,
volume 14, 2002.
[12] M.J. Beal. Variational Algorithms for Approximate Bayesian Inference. PhD thesis, Gatsby
Unit, University College London, 2004.
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1,870 | 2,699 | Optimal Aggregation of Classifiers and Boosting
Maps in Functional Magnetic Resonance
Imaging
Vladimir Koltchinskii
Department of Mathematics and Statistics
University of New Mexico
Albuquerque, NM, 87131
Manel Mart??nez-Ram?on
Department of Electrical and Computer Engineering
University of New Mexico
Albuquerque, NM, 87131
Stefan Posse
Department of Psychiatry and The Mind Institute
University of New Mexico
Albuquerque, NM, 87131
Abstract
We study a method of optimal data-driven aggregation of classifiers in a
convex combination and establish tight upper bounds on its excess risk
with respect to a convex loss function under the assumption that the solution of optimal aggregation problem is sparse. We use a boosting type
algorithm of optimal aggregation to develop aggregate classifiers of activation patterns in fMRI based on locally trained SVM classifiers. The
aggregation coefficients are then used to design a ?boosting map? of the
brain needed to identify the regions with most significant impact on classification.
1
Introduction
We consider a problem of optimal aggregation (see [1]) of a finite set of base classifiers in
a complex aggregate classifier. The aggregate classifiers we study are convex combinations
of base classifiers and we are using boosting type algorithms as aggregation tools. Building
upon recent developments in learning theory, we show that such boosting type aggregation
yields a classifier with a small value of excess risk in the case when optimal aggregate
classifiers are sparse and that, moreover, the procedure provides reasonably good estimates
of aggregation coefficients. Our primary goal is to use this approach in the problem of
classification of activation patterns in functional Magnetic Resonance Imaging (fMRI) (see,
e.g., [2]).
In these problems it is of interest not only to classify the patterns, but also to determine
areas of the brain that are relevant for a particular classification task. Our approach is based
on splitting the image into a number of functional areas, training base classifiers locally in
each area and then combining them into a complex aggregate classifier. The aggregation
coefficients are used to create a special representation of the image we call the boosting
map of the brain. It is needed to identify the functional areas with the most significant
impact on classification.
Previous work has focused on classifying patterns within subject [2] and these patterns were located in the occipital lobe. Here we are considering a different problem, that
is widely distributed patterns in multiple brain regions across groups of subjects. We use
prior knowledge from functional neuroanatomical brain atlases to subdivide the brain into
Regions of Interest, which makes this problem amenable to boosting. Classification across
subjects requires spatial normalization to account for inter-subject differences in brain size
and shape, but also needs to be robust with respect to inter-subject differences in activation
patterns ?shape and amplitude.
Since fMRI patterns are very high dimensional and the amount of training data is
typically limited, some form of ?bet on sparsity? principle (?use a procedure that does well
in sparse problems, since no procedure does well in dense problems? see [3]) becomes
almost unavoidable and our theoretical analysis shows that boosting maps might have a
good chance of success in sparse problems (when only few functional areas are relevant for
classification).
2
Optimal aggregation of classifiers
Although we developed a multiclass extension of the method, for simplicity, we are dealing
here with a standard binary classification. Let (X, Y ) be a random couple with distribution
P, X being an instance in some space S (e.g., it might be an fMRI pattern) and Y ?
{?1, 1} being a binary label. Here and in what follows all the random variables are defined
on a probability space (?, ?, P), E denotes the expectation. Functions f : S 7? R will
be used as classifiers, sign(f (x)) being a predictor of the label for an instance x ? S (no
decision is being made if f (x) = 0). The quantity P {(x, y) : yf (x) ? 0} (the probability
of misclassification or abstaining) is called the generalization error or the risk of f. Suppose
that H := {h1 , . . . , hN } is a given family of classifiers taking values in [?1, 1]. Let
X
N
N
X
|?j | ? 1
conv(H) :=
? j hj :
j=1
j=1
be the symmetric convex hull of H. One of the versions of optimal aggregation problem
would be to find a convex combination f ? conv(H) that minimizes the generalization
error of f in conv(H). For a given f ? conv(H) its quality is measured by
E(f ) := P {(x, y) : yf (x) ? 0} ?
inf
g?conv(H)
P {(x, y) : yg(x) ? 0},
which is often called the excess risk of f. Since the true distribution P of (X, Y ) is unknown, the solution of the optimal aggregation problem is to be found based on the training
data (X1 , Y1 ), . . . , (Xn , Yn ) consisting of n independent copies of (X, Y ).
Let Pn denote the empirical measure based on the training data, i.e., Pn (A) represents
the frequency of training examples in a set A ? S?{?1, 1}. In what follows, we denote P h
or Pn h the integrals of a function h on S ? {?1, 1} with respect to P or Pn , respectively.
We use the same notation for functions on S with an obvious meaning.
Since the generalization error is not known, it is tempting to try to estimate the optimal
convex aggregate classifier by minimizing the training error Pn {(x, y) : yf (x) ? 0}
over the convex hull conv(H). However, this minimization problem is not computationally
feasible and, moreover, the accuracy of empirical approximation (approximation of P by
Pn ) over the class of sets {{(x, y) : yf (x) ? 0} : f ? conv(H)} is not good enough
when H is a large class. An approach that allows one to overcome both difficulties and that
proved to be very successful in the recent years is to replace the minimization of the training
error by the minimization of the empirical risk with respect to a convex loss function. To
be specific, let ? be a nonnegative decreasing convex function on R such that ?(u) ? 1 for
u ? 0. We will denote (? ? f )(x, y) := ?(yf (x)). The quantity
Z
P (? ? f ) = (? ? f )dP = E?(Y f (X))
is called the risk of f with respect to the loss ?, or the ?-risk of f. We will call a function
f0 :=
N
X
?0j hj ? conv(H)
j=1
an ?-otimal aggregate classifier if it minimizes the ?-risk over conv(H). Similarly to the
excess risk, one can define the excess ?-risk of f as
E? (f ) := P (? ? f ) ?
inf
g?conv(H)
P (? ? g).
Despite the fact that we concentrate in what follows on optimizing the excess ?-risk
(?-optimal aggregation) it often provides also a reasonably good solution of the problem
of minimizing the generalization error (optimal aggregation), as it follows from simple
inequalities relating the two risks and proved in [4].
As before, since P is unknown, the minimization of ?-risk has to be replaced by the
corresponding empirical risk minimization problem
Pn (? ? f ) =
n
1X
? Yj f (Xj ) ?? min, f ? conv(H),
n i=1
PN ?
whose solution f? := j=1 ?
j hj is called an empirical ?-optimal aggregate classifier.
? j are small for most of the values
We will show that if f0 , f? are ?sparse? (i.e., ?0j , ?
of j), then the excess ?-risk of the empirical ?-optimal aggregate classifier is small and,
moreover, the coefficients of f? are close to the coefficients of f0 in ?1 -distance.
The sparsity assumption is almost unavoidable in many problems because of the ?bet
on sparsity? principle (see the Introduction).
At a more formal level, if there exists a small subset J ? {1, 2, . . . , N } such that the
sets of random variables {Y, hj (X), j ? J} and {hj (X), j 6? J} are independent and, in
addition, Ehj (X) = 0, j 6? J, then, using Jensen?s inequality, it is easy to check that in an
?-optimal aggregate classifier f0 one can take ?0j = 0, j 6? J.
PN
We will define a measure of sparsity of a function f := j=1 ?j hj ? conv(H) that
is somewhat akin to sparsity charactersitics considered in [5, 6]. For 0 ? d ? N, let
X
?(f ; d) := min
|?j | : J ? {1, . . . , N }, card(J) = d
j6?J
and let ?n (d) :=
Define
d log(N n/d)
.
n
p
dn (f ) := min d : 1 ? d ? N, ?n (d) ? ?(d) .
Of course, if there exists J ? {1, . . . , N } such that ?j = 0 for all j 6? J and
card(J) = d, then dn (f ) ? d.
We will also need the following measure of linear independence of functions in H :
?1
X
?(d) := ?(H; d) =
inf
?j hj
.
P inf
J?{1,...,N },card(J)=d
j?J
|?j |=1
j?J
L2 (P )
Finally, we need some standard conditions on the loss function ? (as, for instance, in
[4]). Assume that ? is Lipschitz on [?1, 1] with some constant L, |?(u) ? ?(v)| ? L|u ?
v|, u, v ? [?1, 1], and the following condition on the convexity modulus of ? holds with
??L:
?(u) + ?(v)
u+v
??
? ?|u ? v|2 , u, v ? [?1, 1].
2
2
In fact, ?(u) is often replaced by a function ?(uM ) with a large enough M (in other
words, the ?-risk is minimized over M conv(H)). This is the case, for instance, for so called
regularized boosting [7]. The theorem below applies to this case as well, only a simple
rescaling of the constants is needed.
Theorem 1 There exist constants K1 , K2 > 0 such that for all t > 0
r
^ log N
t
L2
?
?
?n (dn (f ))
+
? e?t
P E? (f ) ? K1
?
n
n
and
r
X
N
L
0
? j ? ? | ? K2 ?(dn (f?) + dn (f0 )) ?n (dn (f?) + dn (f0 )) + t ? e?t .
P
|?
j
?
n
j=1
Our proof requires some background material on localized Rademacher complexities
and their role in bounding of excess risk (see [8]). We defer it to the full version of the
paper. Note that the first bound depends only on dn (f?) and the second on dn (f?), dn (f0 ).
Both quantities can be much smaller than N despite the fact that empirical risk minimization occurs over the whole N -dimensional convex hull. However, the approach to convex
aggregation based on minimization of the empirical ?-risk over the convex hull does not
guarantee that f? is sparse even if f0 is. To address this problem, we also studied another
approach based on minimization of the penalized empirical ?-risk with the penalty based
on the number of nonzero coefficients of the classifier, but the size of the paper does not
allow us to discuss it.
3
Classification of fMRI patterns and boosting maps
We are using optimal aggregation methods described above in the problem of classification
of activation patterns in fMRI. Our approach is based on dividing the training data into
two parts: for local training and for aggregation. Then, we split the image into N functional areas and train N local classifiers h1 , . . . , hN based on the portions of fMRI data
corresponding to the areas. The data reserved for aggregation is then used to construct an
aggregate classifier. In applications, we are often replacing direct minimization of empirical
risk with convex loss by the standard AdaBoost algorithm (see, e.g., [9]), which essentially
means choosing the loss function as ?(u) = e?u . A weak (base) learner for AdaBoost simply chooses in this case a local classifier among h1 , . . . , hN with the smallest weighted
training error [in more sophisticated versions, we choose a local classifier at random with
probability depending on the size of its weighted training error] and after a number of
rounds AdaBoost returns a convex combination of local classifiers. The coefficients of this
aggregate classifier are then used to create a new visual representation of the brain (the
boosting map) that highlights the functional areas with significant impact on classification.
In principle, it is also possible to use the same data for training of local classifiers and for
aggregation (retraining the local classifiers at each round of boosting), but this approach is
time consuming.
We use statistical parametric model (SPM) t-maps of MRI scans [10]. Statistical parametric maps (SPMs) are image processes with voxel1 values that are, under the null hypothesis, distributed according to a known probability density function, usually the Student?s
1
A voxel is the amplitude of a position in the 3-D MRI image matrix.
Figure 1: Masks used to split the image into functional areas in multi-slice and 3 orthogonal
slice display representations.
T or F distributions. These are known colloquially as t- or f-maps. Namely, one analyzes
each and every voxel using any standard (univariate) statistical test. The resulting statistical
parameters are assembled into an image - the SPM.
The classification system essentially transforms the t-map of the image into the boosting map and at the same time it returns the aggregate classifier. The system consists of the
data preprocessing block that splits the image into functional areas based on specified
masks, and also splits the data into portions corresponding to the areas. In one of our examples, we use the main functional areas brainstem, cerebellum, occipital, temporal, parietal,
subcortical and frontal. We split these masks in left and right, having in total 14 of them.
The classifier block then trains local classifiers based on local data (in the current version
we are using SVM classifiers). Finally, the aggregation or boosting block computes and
outputs the aggregate classifier and the boosting map of the image. We developed a version of the system that deals with multi-class problems in spirit of [11], but the details go
beyond the scope of this paper. The architecture of the network allows us also to train it
sequentially. Let f be a classifier produced by the network in the previous round of work,
let (X1 , Y1 ), . . . , (Xn , Yn ) be either the same or a new training data set and let h1 , . . . , hN
be local classifiers (based either on the same, or on a new set of masks). Then one can
?Yj f (Xj )
, where Z is a standard
assign to the training examples the initial weights wj = e Z
normalizing constant, instead of usually chosen uniform weights. After this, the AdaBoost
can proceed in a normal fashion creating at the end an aggregate of f and of new local classifiers. The process can be repeated recursively updating both the classifier and the boosting
map.
5
5
5
10
10
10
15
15
15
20
20
20
25
25
25
30
30
30
35
35
5
10
15
20
25
30
35
35
5
10
15
20
25
30
35
5
10
15
20
25
30
35
Figure 2: Left and center: Patterns corresponding to two classes of data. Right: Locations
of the learners chosen by the boosting procedure (white spots). The background image
corresponds to the two patterns of left and center figures superimposed.
5
5
10
10
15
15
20
20
25
25
30
30
35
35
5
10
15
20
25
30
35
5
10
15
20
25
30
35
Figure 3: Patterns corrupted with noise in the gaussian parameters, artifacts, and additive
noise used in the synthetic data experiment.
Figure 4: Two t-maps corresponding to visual (left) and motor activations in the same
subject used in the real data experiment.
As a synthetic data example, we generate 40 ? 40 pixels images of two classes. Each
class of images consists of three gaussian clusters placed in different positions. We generate
the set of images by adding gaussian noise of standard deviation 0.1 to the standard deviation and position of the clusters. Then, we add 10 more clusters with random parameters,
and finally, additive noise of standard deviation 0.1. Figure 2 (left and center) shows the
averages of class 1 and class 2 images respectively. Two samples of the images can be seen
in Figure 3
We apply a base learner to each one of the 1600 pixels of the images. Learners have
been trained with 200 data, 100 of each class, and the aggregation has been trained with
200 more data. The classifier has been tested with 200 previously unknown data. The error
averaged over 100 trials is of 9.5%. The same experiment has been made with a single linear
SVM, producing an error which exceeds 20%, although this rate can be slightly improved
by selecting C by cross validation.
The resulting boosting map can be seen in Fig. 2 (right). As a proof of concept, we
remark that the map is able to focus in the areas in which the clusters corresponding to each
class are, discarding those areas in which only randomly placed clusters are present.
In order to test the algorithm in a real fMRI experiment, we use 20 images taken from
10 healthy subjects on a 1.5 Tesla Siemens Sonata scanner. Stimuli were presented via MR
compatible LCD goggles and headphones. The paradigm consists of four interleaved tasks:
visual (8 Hz checkerboard stimulation), motor (2 Hz right index finger tapping), auditory
Figure 5: Boosting map of the brain corresponding to the classification problem with visual
and motor activations. Darker regions correspond to higher values.
left brainstem:
left cerebellum:
left parietal:
left temporal:
left occipital:
left subcortical:
left frontal:
0
0.15
0.02
0.03
0.29
0
0
right brainstem:
right cerebellum:
right parietal:
right temporal:
right occipital:
right subcortical:
right frontal:
0
0.16
0.06
0.15
0.15
0
0
Table 1: Values of the convex aggregation.
(syllable discrimination) and cognitive (mental calculation). These tasks are arranged in
randomized blocks (8 s per block). Finger tapping in the motor task was regulated with
an auditory tone, subjects were asked to tap onto a button-response pad. During the auditory task, subjects were asked to respond on a button-response pad for each ?Ta? (25% of
sounds), but not to similar syllables. Mental calculation stimuli consisted of three singledigit numbers heard via headphone. Participants had to sum them and divide by three,
responding by button press when there was no remainder (50% of trials).
Functional MRI data were acquired using single-shot echo-planar imaging with TR:
2 s, TE: 50 ms, flip angle: 90 degrees, matrix size: 64 ? 64 pixels, FOV: 192 mm. Slices
were 6 mm thick, with 25% gap, 66 volumes were collected for a total measurement time
of 132 sec per run. Statistical parametric mapping was performed to generate t-maps that
represent brain activation changes.
The t-maps are lowpass filtered and undersampled to obtain 32 ? 32 ? 24 t-maps (Fig.
4). The resulting t-maps are masked to obtain 14 subimages, then the data is normalized
in amplitude. We proceed as mentioned to train a set of 14 Support Vector Machines. The
used kernel is a gaussian one with ? = 2 and C = 10. These parameters have been chosen
to provide an acceptable generalization. A convex aggregation of the classifier outputs is
then trained.
We tested the algorithm in binary classification of visual against auditory activations.
We train the base learners with 10 images, and the boosting with 9. Then, we train the
base learners again with 19, leaving one for testing. We repeat the experiment leaving
out a different image each trial. None of the images was misclassified. The values for the
aggregation are in Table 1. The corresponding boosting map is shown in Fig 5. It highlights
the right temporal and both occipital areas, where the motor and visual activations are
present (see Fig. 4). Also, there is activation in the cerebellum area in some of the motor
t-maps, which is highlighted by the boosting map.
In experiments for the six binary combination of activation stimuli, the average error
was less than 10%. This is an acceptable result if we take into account that the data included
ten different subjects, whose brain activation patterns present noticeable differences.
4
Future goals
Boosting maps we introduced in this paper might become a useful tool in solving classification problems for fMRI data, but there is a number of questions to be answered before it
is the case. The most difficult problem is the choice of functional areas and local classifiers
so that the ?true? boosting map is identifiable based on the data. As our theoretical analysis
shows, this is related to the degree of linear independence of local classifiers quantified by
the function ?(d). If ?(d) is too large for d = dn (f0 ) ? dn (f?), the empirical boosting map
can become very unstable and misleading. In such cases, there is a challenging model selection problem (how to choose a ?good? subset of H or how to split H into ?almost linearly
independent clusters? of functions) that has to be addressed to develop this methodology
further.
Acknowledgments
We want to acknowledge to Jeremy Bockholt (MIND Institute) for providing the brain
masks, generated with BRAINS2. Partially supported by NSF Grant DMS-0304861 and
NIH Grant NIBIB 1 RO1 EB002618-01, Dept. of Mathematics and Statistics, Dept. of
Electrical and Computing Engineering, Dept. of Psychiatry and The MIND Institute.
References
[1] Tsybakov, A. (2003) Optimal rates of aggregation. In: COLT2003, Lecture Notes in Artificial
Intelligence, Eds.: M. Warmuth and B. Schoelkopf, Springer.
[2] Cox, D.D., Savoy, R.L. (2003) Functional magnetic resonance imaging (fMRI) ?brain reading?: detecting and classifying distributed patterns of fMRI activity in human visual cortex,
Neuroimage19, 2, 261?70.
[3] Friedman, J., Hastie, T., Rosset, S., Tibshirani, R. and Zhu, J. (2004) Discussion on Boosting,
Annals of Statistics, 32, 1, 102?107.
[4] Bartlett, P. L., Jordan, M.I., McAuliffe, J. D. (2003) Convexity, classification, and risk
bounds. Technical Report 638, Department of Statistics, U.C. Berkeley, 2003. Journal of
the American Statistical Association.To appear.
[5] Koltchinskii, V., Panchenko, D. and Lozano, F. (2003) Bounding the generalization error of
combined classifiers: balancing the dimensionality and the margins. A. Appl. Prob. , 13, 1.
[6] Koltchinskii, V., Panchenko, D. and Andonova, S. (2003) Generalization bounds for voting classifiers based on sparsity and clustering. In: COLT2003, Lecture Notes in Artificial
Intelligence, Eds.: M. Warmuth and B. Schoelkopf, Springer.
[7] Blanchard, G., Lugosi, G. and Vayatis, N. (2003) On the rates of convergence of regularized
boosting classifiers. Journal of Machine Learning Research 4, 861-894.
[8] Koltchinskii, V. (2003) Local Rademacher Complexities and Oracle Inequalities in Risk
Minimization. Preprint.
[9] Schapire, R. E. (1999) A brief Introduction to Boosting. In: Proc. of the 6th Intl. Conf. on
Artificial Inteligence.
[10] Friston, K., Frith, C., Liddle, P. and Frackowiak, R. (1991) Comparing functional (PET)
images: the assessment of significant change. J. Cereb. Blood Flow Met.11, 690-699
[11] Allwein, E. L., Schapire, R. E., and Singer, Y. (2000) Reducing multiclass to binary: A
unifying approach for margin classifiers. J. Machine Learning Research, 1, 113-141.
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1,871 | 27 | 573
BIT - SERIAL NEURAL NETWORKS
Alan F. Murray, Anthony V . W. Smith and Zoe F. Butler.
Department of Electrical Engineering, University of Edinburgh,
The King's Buildings, Mayfield Road, Edinburgh,
Scotland, EH93JL.
ABSTRACT
A bit - serial VLSI neural network is described from an initial architecture for a
synapse array through to silicon layout and board design. The issues surrounding bit
- serial computation, and analog/digital arithmetic are discussed and the parallel
development of a hybrid analog/digital neural network is outlined. Learning and
recall capabilities are reported for the bit - serial network along with a projected
specification for a 64 - neuron, bit - serial board operating at 20 MHz. This technique is extended to a 256 (2562 synapses) network with an update time of 3ms,
using a "paging" technique to time - multiplex calculations through the synapse
array.
1. INTRODUCTION
The functions a synthetic neural network may aspire to mimic are the ability to consider many solutions simultaneously, an ability to work with corrupted data and a
natural fault tolerance. This arises from the parallelism and distributed knowledge
representation which gives rise to gentle degradation as faults appear. These functions are attractive to implementation in VLSI and WSI. For example, the natural
fault - tolerance could be useful in silicon wafers with imperfect yield, where the
network degradation is approximately proportional to the non-functioning silicon
area.
To cast neural networks in engineering language, a neuron is a state machine that is
either "on" or "off', which in general assumes intermediate states as it switches
smoothly between these extrema. The synapses weighting the signals from a
transmitting neuron such that it is more or less excitatory or inhibitory to the receiving neuron. The set of synaptic weights determines the stable states and represents
the learned information in a system.
The neural state, VI' is related to the total neural activity stimulated by inputs to
the neuron through an activation junction, F. Neural activity is the level of excitation of the neuron and the activation is the way it reacts in a response to a change
in activation. The neural output state at time t, V[, is related to x[ by
V[
= F (xf)
(1)
The activation function is a "squashing" function ensuring that (say) Vi is 1 when
is large and -1 when Xi is small. The neural update function is therefore straightforward:
Xi
.
,
XI,+1 - XI
? ????
+ 0~
i-n-l
~
~
T ii V'J
(2)
J-O
where 8 represents the rate of change of neural activity, Tij is the synaptic weight
and n is the number of terms giving an n - neuron array [1].
Although the neural function is simple enough, in a totally interconnected n - neuron network there are n 2 synapses requiring n 2 multiplications and summations and
? American Institute of Physics 1988
574
a large number of interconnects. The challenge in VLSI is therefore to design a simple, compact synapse that can be repeated to build a VLSI neural network with
manageable interconnect. In a network with fixed functionality, this is relatively
straightforward. H the network is to be able to learn, however, the synaptic weights
must be programmable, and therefore more complicated.
2. DESIGNING A NEURAL NETWORK IN VLSI
There are fundamentally two approaches to implementing any function in silicon digital and analog. Each technique has its advantages and disadvantages, and these
are listed below, along with the merits and demerits of bit - serial architectures in
digital (synchronous) systems.
Digital vs. analog: The primary advantage of digital design for a synapse array is
that digital memory is well understood, and can be incorporated easily. Learning
networks are therefore possible without recourse to unusual techniques or technologies. Other strengths of a digital approach are that design techniques are advanced,
automated and well understood and noise immunity and computational speed can
be high. Unattractive features are that digital circuits of this complexity need to be
synchronous and all states and activities are quantised, while real neural networks
are asynchronous and unquantised. Furthermore, digital multipliers occupy a large
silicon area, giving a low synapse count on a single chip.
The advantages of analog circuitry are that asynchronous behaviour and smooth
neural activation are automatic. Circuit elements can be small, but noise immunity
is relatively low and arbitrarily high precision is not possible. Most importantly, no
reliable analog, non - volatile memory technology is as yet readily available. For
this reason, learning networks lend themselves more naturally to digital design and
implementation.
Several groups are developing neural chips and boards, and the following listing
does not pretend to be exhaustive. It is included, rather, to indicate the spread of
activity in this field. Analog techniques have been used to build resistor I operational amplifier networks [2,3] similar to those proposed by Hopfield and Tank [4].
A large group at Caltech is developing networks implementing early vision and
auditory processing functions using the intrinsic nonlinearities of MaS transistors in
the subthreshold regime [5,6]. The problem of implementing analog networks with
electrically programmable synapses has been addressed using CCDIMNOS technology [7]. Finally, Garth [8] is developing a digital neural accelerator board ("Netsim") that is effectively a fast SIMD processor with supporting memory and communications chips.
Bit - serial vs. bit - parallel: Bit - serial arithmetic and communication is efficient
for computational processes, allowing good communication within and between
VLSI chips and tightly pipelined arithmetic structures. It is ideal for neural networks as it minimises the interconnect requirement by eliminating multi - wire
busses. Although a bit - parallel design would be free from computational latency
(delay between input and output), pipelining makes optimal use of the high bit rates possible in serial systems, and makes for efficient circuit usage.
2.1 An asynchronous pulse stream VLSI neural network:
In addition to the digital system that forms the substance of this paper, we are
developing a hybrid analOg/digital network family. This work is outlined here, and
has been reported in greater detail elsewhere [9, 10, 11]. The generic (logical and
layout) architecture of a single network of n totally interconnected neurons is shown
575
schematically in figure 1. Neurons are represented by circles, which signal their
states, Vi upward into a matrix of synaptic operators. The state signals are connected to a n - bit horizontal bus running through the synaptic array, with a connection to each synaptic operator in every column. All columns have n operators
(denoted by squares) and each operator adds its synaptic contribution, Tij V j , to the
running total of activity for the neuron i at the foot of the column. The synaptic
function is therefore to multiply the signalling neuron state, Vj , by the synaptic
weight, T ij , and to add this product to the running total. This architecture is common to both the bit - serial and pulse - stream networks.
Synapse
States { Vj }
Neurons
Figure 1. Generic architecture for a network of n totally interconnected neurons.
This type of architecture has many attractions for implementation in 2 - dimensional
j=II -1
silicon as the summation
2
Tij
Vj is distributed in space.
The interconnect
j=O
requirement (n inputs to each neuron) is therefore distributed through a column,
reducing the need for long - range wiring. The architecture is modular, regular and
can be easily expanded.
In the hybrid analog/digital system, the circuitry uses a "pulse stream" signalling
method similar to that in a natural neural system. Neurons indicate their state by
the presence or absence of pulses on their outputs, and synaptic weighting is
achieved by time - chopping the presynaptic pulse stream prior to adding it to the
postsynaptic activity summation. It is therefore asynchronous and imposes no fundamental limitations on the activation or neural state. Figure 2 shows the pulse
stream mechanism in more detail. The synaptic weight is stored in digital memory
local to the operator. Each synaptic operator has an excitatory and inhibitory pulse
stream input and output. The resultant product of a synaptic operation, Tij V j , is
added to the running total propagating down either the excitatory or inhibitory
channel. One binary bit (the MSBit) of the stored Tij determines whether the contribution is excitatory or inhibitory.
The incoming excitatory and inhibitory pulse stream inputs to a neuron are
integrated to give a neural activation potential that varies smoothly from 0 to 5 V.
This potential controls a feedback loop with an odd number of logic inversions and
576
.
??
XT ??
V,
.u.u,
?
Figure 2. Pulse stream arithmetic. Neurons are denoted by 0 and synaptic operators
by D.
thus forms a switched "ring - oscillator". H the inhibitory input dominates, the feedback loop is broken. H excitatory spikes subsequently dominate at the input, the
neural activity rises to 5V and the feedback loop oscillates with a period determined
by a delay around the loop. The resultant periodic waveform is then converted to a
series of voltage spikes, whose pulse rate represents the neural state, Vi' Interestingly, a not dissimilar technique is reported elsewhere in this volume, although the
synapse function is executed differently [12].
3. A 5 - STATE BIT - SERIAL NEURAL NETWORK
The overall architecture of the 5 - state bit - serial neural network is identical to
that of the pulse stream network. It is an array of n 2 interconnected synchronous
synaptic operators, and whereas the pulse stream method allowed Vj to assume all
values between "off' and "on", the 5 - state network VJ is constrained to 0, ?0.5 Qr
? 1. The resultant activation function is shown in Figure 3. Full digital multiplication is costly in silicon area, but multiplication of Tij by Vj = 0.5 merely requires
the synaptic weight to be right - shifted by 1 bit. Similarly, multiplication by 0.25
involves a further right - shift of Til' and multiplication by 0.0 is trivially easy. VJ
< 0 is not problematic, as a switchable adder/subtractor is not much more complex
than an adder. Five neural states are therefore feasible with circuitry that is only
slightly more complex than a simple serial adder. The neural state expands from a 1
bit to a 3 bit (5 - state) representation, where the bits represent "add/subtract?",
"shift?" and "multiply by O?".
Figure 4 shows part of the synaptic array. Each synaptic operator includes an 8 bit
shift register memory block holding the synaptic weight, Til' A 3 bit bus for the 5
neural states runs horizontally above each synaptic row. Single phase dynamic
CMOS has been used with a clock frequency in excess of 20 MHz [13). Details of
a synaptic operator are shown in figure 5. The synaptic weight Til cycles around the
shift register and the neural state Vj is present on the state bus. During the first
clock CYCle, the synaptic weight is multiplied by the neural state and during the
second, the most significant bit (MSBit) of the resultant Tij Vj is sign - extended for
577
State VJ
lHRESHOLD
.....-------=-------..
s?
"5
Activity sJ
STATE"
"Sharper"
"Smoother"
~.....::~-"'--x.&..t------
Activity "J
Figure 3. "Hard - threshold", 5 - state and sigmoid activation functions.
J-a-1 T v
~
J-li
.. J
v,
v,
Figure 4. Section of the synaptic array of the 5 - state activation function neural network.
8 bits to allow for word growth in the running summation. A least significant bit
(LSBit) signal running down the synaptic columns indicates the arrival of the LSBit
of the Xj running total. If the neural state is ?O.5 the synaptic weight is right
shifted by 1 bit and then added to or subtracted from the running total. A multiplication of ? 1 adds or subtracts the weight from the total and multiplication by 0
578
.0.5
.0.0
Add/Subtract
Add!
Subtract
Carry
Figure S. The synaptic operator with a 5 - state activation function.
does not alter the running summation.
The final summation at the foot of the column is thresholded externally according
to the 5 - state activation function in figure 3. As the neuron activity Xj' increases
through a threshold value x" ideal sigmoidal activation represents a smooth switch
of neural state from -1 to 1. The 5 - state "staircase" function gives a superficially
much better approximation to the sigmoid form than a (much simpler to implement) threshold function. The sharpness of the transition can be controlled to
"tune" the neural dynamics for learning and computation. The control parameter is
referred to as temperature by analogy with statistical functions with this sigmoidal
form. High "temperature" gives a smoother staircase and sigmoid, while a temperature of 0 reduces both to the ''Hopfield'' - like threshold function. The effects of
temperature on both learning and recall for the threshold and 5 - state activation
options are discussed in section 4.
4. LEARNING AND RECALL WITH VLSI CONSTRAINTS
Before implementing the reduced - arithmetic network in VLSI, simulation experiments were conducted to verify that the 5 - state model represented a worthwhile
enhancement over simple threshold activation. The "benchmark" problem was
chosen for its ubiquitousness, rather than for its intrinsic value. The implications
for learning and recall of the 5 - state model, the threshold (2 - state) model and
smooth sigmoidal activation ( 00 - state) were compared at varying temperatures
with a restricted dynamic range for the weights T ij ? In each simulation a totally
interconnected 64 node network attempted to learn 32 random patterns using the
delta rule learning algorithm (see for example [14]). Each pattern was then corrupted with 25% noise and recall attempted to probe the content addressable
memory properties under the three different activation options.
During learning, individual weights can become large (positive or negative). When
weights are "driven" beyond the maximum value in a hardware implementation,
579
which is determined by the size of the synaptic weight blocks, some limiting
mechanism must be introduced. For example, with eight bit weight registers, the
limitation is -128 S Tij S 127. With integer weights, this can be seen to be a problem of dynamic range, where it is the relationship between the smallest possible
weight (? 1) and the largest (+ 127/-128) that is the issue.
Results: Fig. 6 shows examples of the results obtained, studying learning using 5 state activation at different temperatures, and recall using both 5 - state and threshold activation. At temperature T=O, the 5 - state and threshold models are
degenerate, and the results identical. Increasing smoothness of activation (temperature) during learning improves the quality of learning regardless of the activation
function used in recall, as more patterns are recognised successfully. Using 5 - state
activation in recall is more effective than simple threshold activation. The effect of
dynamic range restrictions can be assessed from the horizontal axis, where T/j:6. is
shown. The results from these and many other experiments may be summarised as
follows:5 - State activation vs. threshold:
1) Learning with 5 - state activation was protracted over the threshold activation,
as binary patterns were being learnt, and the inclusion of intermediate values
added extra degrees of freedom.
2) Weight sets learnt using the 5 - state activation function were "better" than
those learnt via threshold activation, as the recall properties of both 5 - state
and threshold networks using such a weight set were more robust against
noise.
3) Full sigmoidal activation was better than 5 - state, but the enhancement was
less significant than that incurred by moving from threshold - 5 - state. This
suggests that the law of diminishing returns applies to addition of levels to the
neural state Vi' This issue has been studied mathematically [15], with results
that agree qualitatively with ours.
Weight Saturation:
Three methods were tried to deal with weight saturation. Firstly, inclusion of a
decay, or "forgetting" term was included in the learning cycle [1]. It is our view
that this technique can produce the desired weight limiting property, but in the time
available for experiments, we were unable to "tune" the rate of decay sufficiently
well to confirm it. Renormalisation of the weights (division to bring large weights
back into the dynamic range) was very unsuccessful, suggesting that information
distributed throughout the numerically small weights was being destroyed. Finally,
the weights were allowed to "clip" (ie any weight outside the dynamic range was set
to the maximum allowed value). This method proved very successful, as the learning algorithm adjusted the weights over which it still had control to compensate for
the saturation effect. It is interesting to note that other experiments have indicated
that Hopfield nets can "forget" in a different way, under different learning control,
giving preference to recently acquired memories [16]. The results from the saturation experiments were:1) For the 32 pattemJ64 node problem, integer weights with a dynamic range
greater than ?30 were necessary to give enough storage capability.
2) For weights with maximum values TiJ = 50-70, "clipping" occurs, but network performance is not seriously degraded over that with an unrestricted
weight set.
580
15
c 10
"0
~
e
,
=
.2
-
~
T=20
T=10
T=O
....--- -- - -
- - ,- ., ...
;A .......;.. f:'-:' :::::7.:::.::-:::-: f'-.
I
en
e
u
T=30 _._.-.-
15
,...?.-.....-.?. _.?. -.-._.-..
,. ".'
i
,..
i
j''''-,,'i
~-------------
!
!
,
5
-..,..
i
j
'" ?????? ? ???????????????? ??????
j
I
I
,
I
0
O~~~~--~~
0
20
30
40 50 60 70
Limit
5 . state activation function recal1
o
__~~__
40 50 60 70
Limit
tlHopficld" activation function recall
20
30
Figure 6. Recall of patterns learned with the 5 . state activation function and subsequently restored using the 5-state and the hard - threshold activation functions.
T is the "temperature", or smoothness of the activation function, and "limit" the value
ofTI; ?
These results showed that the 5 - state model was worthy of implementation as a
VLSI neural board, and suggested that 8 - bit weights were sufficient.
S. PROJECTED SPECIFICATION OF A HARDWARE NEURAL BOARD
The specification of a 64 neuron board is given here, using a 5 - state bit - serial 64
x 64 synapse array with a derated clock speed of 20 MHz. The synaptic weights are
8 bit words and the word length of the running summation XI is 16 bits to allow for
growth. A 64 synapse column has a computational latency of 80 clock cycles or
bits, giving an update time of 4 .... s for the network. The time to load the weights
into the array is limited to 6O .... s by the supporting RAM, with an access time of
12Ons. These load and update times mean that the network is executing 1 x 10'
operations/second, where one operation is ? Tlj Vj ? This is much faster than a
natural neural network, and much faster than is necessary in a hardware accelerator. We have therefore developed a "paging" architecture, that effectively "trades off" some of this excessive speed against increased network size.
A "moving - patch" neural board: An array of the 5 - state synapses is currently
being fabricated as a VLSI integrated circuit. The shift registers and the
adderlsubtractor for each synapse occupy a disappointingly large silicon area, allowing only a 3 x 9 synaptic array. To achieve a suitable size neural network from this
array, several chips need to be included on a board with memory and control circuitry. The "moving patch" concept is shown in figure 7, where a small array of
synapses is passed over a much larger n x n synaptic array.
Each time the array is "moved" to represent another set of synapses, new weights
must be loaded into it. For example, the first set of weights will be T 11 ?. , T;J ... T 21
... T 2j to Tjj , the second set T j + 1 ,l to Tu etc.. The final weight to be loaded will be
581
Smaller "Patch"
n neurons .. om synaptic array
moves over array
rr~
~'-
_____)__-..
>
Figure 7. The "moving patch" concept, passing a small synaptic "patch" over a larger
run synapse array.
TNt? Static, off - the - shelf RAM is used to store the weights and the whole opera-
tion is pipelined for maximum efficiency. Figure 8 shows the board level design for
the network.
Control
Synaptic Accelerator Chips
HOST
Figure 8. A "moving patch" neural network board.
The small "patch" that moves around the array to give n neurons comprises 4 VLSI
synaptic accelerator chips to give a 6 x 18 synaptic array. The number of neurons to
be simulated is 256 and the weights for these are stored in 0.5 Mb of RAM with a
load time of 8ms. For each "patch" movement, the partial runnin~ summatinn ;.
582
calculated for each column, is stored in a separate RAM until it is required to be
added into the next appropriate summation. The update time for the board is 3ms
giving 2 x 107 operations/second. This is slower than the 64 neuron specification,
but the network is 16 times larger, as the arithmetic elements are being used more
efficiently. To achieve a network of greater than 256 neurons, more RAM is
required to store the weights. The network is then slower unless a larger number of
accelerator chips is used to give a larger moving "patch".
6. CONCLUSIONS
A strategy and design method has been given for the construction of bit - serial
VLSI neural network chips and circuit boards. Bit - serial arithmetic, coupled to a
reduced arithmetic style, enhances the level of integration possible beyond more
conventional digital, bit - parallel schemes. The restrictions imposed on both synaptic weight size and arithmetic precision by VLSI constraints have been examined
and shown to be tolerable, using the associative memory problem as a test.
While we believe our digital approach to represent a good compromise between
arithmetic accuracy and circuit complexity, we acknowledge that the level of
integration is disappointingly low. It is our belief that, while digital approaches
may be interesting and useful in the medium term, essentially as hardware accelerators for neural simulations, analog techniques represent the best ultimate option in 2
- dimensional silicon. To this end, we are currently pursuing techniques for analog
pseudo - static memory, using standard CMOS technology. In any event, the full
development of a nonvolatile analog memory technology, such as the MNOS technique [7], is key to the long - term future of VLSI neural nets that can learn.
7. ACKNOWLEDGEMENTS
The authors acknowledge the support of the Science and Engineering Research
Council (UK) in the execution of this work.
References
1.
2.
3.
4.
5.
6.
S. Grossberg, "Some Physiological and Biochemical Consequences of Psychological Postulates," Proc. Natl. Acad. Sci. USA, vol. 60, pp. 758 - 765, 1968.
H. P. Graf, L. D. Jackel, R. E. Howard, B. Straughn, J. S. Denker, W.
Hubbard, D. M. Tennant, and D. Schwartz, "VLSI Implementation of a
Neural Network Memory with Several Hundreds of Neurons," Proc. AlP
Conference on Neural Networks for Computing. Snowbird, pp. 182 - 187, 1986.
W. S. Mackie, H. P. Graf, and J. S. Denker, "Microelectronic Implementation of Connectionist Neural Network Models," IEEE Conference on Neural
Information Processing Systems. Denver, 1987.
J . J. Hopfield and D. W. Tank, "Neural" Computation of Decisions in Optimisation Problems," BioI. Cybern., vol. 52, pp. 141 - 152, 1985.
M. A. Sivilotti, M. A. Mahowald, and C. A. Mead, Real - Time Visual Computations Using Analog CMOS Processing Arrays, 1987. To be published
C. A. Mead, "Networks for Real - Time Sensory Processing," IEEE Conference on Neural Information Processing Systems, Denver, 1987.
583
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
J. P. Sage, K. Thompson. and R. S. Withers, "An Artificial Neural Network
Integrated Circuit Based on MNOSlCCD Principles," Proc. AlP Conference on
Neural Networlcs for Computing, Snowbird, pp. 381 - 385, 1986.
S. C. J. Garth, "A Chipset for High Speed Simulation of Neural Network Systems," IEEE Conference on Neural Networlc.s, San Diego, 1987.
A. F. Murray and A. V. W. Smith, "A Novel Computational and Signalling
Method for VLSI Neural Networks," European Solid State Circuits Conference
, 1987.
A. F. Murray and A. J. W. Smith, "Asynchronous Arithmetic for VLSI
Neural Systems," Electronics Letters, vol. 23, no. 12, p. 642, June, 1987.
A. F. Murray and A. V. W. Smith, "Asynchronous VLSI Neural Networks
using Pulse Stream Arithmetic," IEEE Journal of Solid-State Circuits and Systems, 1988. To be published
M. E. Gaspar, "Pulsed Neural Networks: Hardware, Software and the Hopfield AID Converter Example," IEEE Conference on Neural Information Processing Systems. Denver, 1987.
M. S. McGregor, P. B. Denyer, and A. F. Murray, "A Single - Phase Clocking Scheme for CMOS VLSI," Advanced Research in VLSI " Proceedings of the
1987 Stanford Conference, 1987.
D. E. Rumelhart, G. E. Hinton, and R. J. Williams, "Learning Internal
Representations by Error Propagation," Parallel Distributed Processing "
Explorations in the Microstructure of Cognition, vol. 1, pp. 318 - 362, 1986.
M. Fleisher and E. Levin, "The Hopfiled Model with Multilevel Neurons
Models," IEEE Conference on Neural Information Processing Systems. Denver,
1987.
G. Parisi, "A Memory that Forgets," J. Phys. A .' Math. Gen., vol. 19, pp.
L617 - L620, 1986.
| 27 |@word manageable:1 eliminating:1 inversion:1 chopping:1 pulse:13 simulation:4 tried:1 solid:2 disappointingly:2 carry:1 electronics:1 initial:1 series:1 seriously:1 ours:1 interestingly:1 activation:34 yet:1 must:3 readily:1 update:5 v:3 signalling:3 scotland:1 smith:4 math:1 node:2 preference:1 sigmoidal:4 simpler:1 firstly:1 five:1 along:2 become:1 mayfield:1 acquired:1 forgetting:1 themselves:1 multi:1 totally:4 increasing:1 circuit:9 medium:1 sivilotti:1 developed:1 extremum:1 fabricated:1 pseudo:1 every:1 expands:1 growth:2 oscillates:1 uk:1 control:6 schwartz:1 appear:1 before:1 positive:1 engineering:3 understood:2 multiplex:1 local:1 limit:3 consequence:1 acad:1 mead:2 approximately:1 studied:1 examined:1 suggests:1 limited:1 range:7 grossberg:1 block:2 implement:1 addressable:1 area:4 word:3 road:1 regular:1 pipelined:2 operator:11 storage:1 cybern:1 restriction:2 conventional:1 imposed:1 layout:2 straightforward:2 regardless:1 williams:1 thompson:1 sharpness:1 rule:1 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expanded:1 relatively:2 department:1 developing:4 according:1 electrically:1 smaller:1 slightly:1 postsynaptic:1 restricted:1 recourse:1 agree:1 bus:4 count:1 mechanism:2 merit:1 end:1 unusual:1 studying:1 junction:1 available:2 operation:4 multiplied:1 probe:1 eight:1 worthwhile:1 denker:2 generic:2 appropriate:1 tolerable:1 tlj:1 subtracted:1 slower:2 assumes:1 running:10 pretend:1 giving:5 murray:5 build:2 move:2 added:4 zoe:1 spike:2 occurs:1 primary:1 costly:1 restored:1 strategy:1 enhances:1 unable:1 separate:1 simulated:1 sci:1 presynaptic:1 reason:1 length:1 relationship:1 executed:1 demerit:1 sharper:1 holding:1 negative:1 rise:2 sage:1 design:8 implementation:7 allowing:2 neuron:27 wire:1 benchmark:1 acknowledge:2 howard:1 supporting:2 extended:2 incorporated:1 communication:3 hinton:1 worthy:1 tnt:1 introduced:1 cast:1 required:2 immunity:2 connection:1 learned:2 chipset:1 able:1 beyond:2 suggested:1 parallelism:1 below:1 pattern:5 regime:1 challenge:1 saturation:4 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1,872 | 270 | 660
Geiger and Girosi
Coupled Markov Random Fields and
Mean Field Theory
Davi Geigerl
Artificial Intelligence
Laboratory, MIT
545 Tech. Sq. # 792
Cambridge, MA 02139
and
Federico Girosi
Artificial Intelligence
Laboratory, MIT
545 Tech. Sq. # 788
Cambridge, MA 02139
ABSTRACT
In recent years many researchers have investigated the use of Markov
Random Fields (MRFs) for computer vision. They can be applied
for example to reconstruct surfaces from sparse and noisy depth
data coming from the output of a visual process, or to integrate
early vision processes to label physical discontinuities. In this paper we show that by applying mean field theory to those MRFs
models a class of neural networks is obtained. Those networks can
speed up the solution for the MRFs models. The method is not
restricted to computer vision.
1
Introduction
In recent years many researchers (Geman and Geman, 1984) (Marroquin et. al.
1987) (Gamble et. al. 1989) have investigated the use of Markov Random Fields
(MRFs) for early vision. Coupled MRFs models can be used for the reconstruction
of a function starting from a set of noisy sparse data, such as intensity, stereo, or
motion data. They have also been used to integrate early vision processes to label
physical discontinuities. Two fields are usually required in the MRFs formulation
of a problem: one represents the function that has to be reconstructed, and the
other is associated to its discontinuities. The reconstructed function, say I, has
1 New
address is Siemens Corporate Research, 755 College Road East, Princeton NJ 08540
Coupled Markov Random Fields and Mean Field Theory
Figure 1: The 6quare lattice with the line proceu I and the field
pizel6.
J defined at 60me
a continuous range and the discontinuity field, say I, is a binary field (1 if there
is a discontinuity and 0 otherwise, see figure 1). The essence of the MRFs model
is that the probability distribution of the configuration of the fields, for a given
a set of data, has a Gibbs distribution for some cost functional dependent upon
a small neighborhood. Since the fields have a discrete range, to find the solution
becomes a combinatorial optimization problem, that can be solved by means of
methods like the Monte Carlo one (simulated annealing (Kirkpatrick and all, 1983),
for example). However it has a main drawback: the amount of computer time
needed for the implementation.
We propose to approximate the solution of the problem formulated in the MRFs
frame with its "average solution." The mean field theory (MFT) allows us to find
deterministic equations for MRFs whose solution approximates the solution of the
statistical problem. A class of neural networks can naturally solve these equations
(Hopfield, 1984) (Koch et. al., 1985) (Geiger and Yuille, 1989). An advantage of
such an approach is that the solution of the networks is faster than the Monte Carlo
techniques, commonly used to deal with MRFs.
A main novelty in this work, and a quite general one, is to show that the binary
field representing the discontinuities can be averaged out to yield an effective theory independent of the binary field. The possibility of writing a set of equations
describing the network is also useful for a better understanding of the nature of the
solution and of the parameters of the model. We show the network performance in
an example of image reconstruction from sparse data.
661
662
Geiger and Girosi
2
MRFs and Bayes approach
One of the main attractions of MRFs models in vision is that they can deal directly
with discontinuities. We consider coupled MRFs depending upon two fields, J
and I. For the problem of image reconstruction the field J represents the field to
be smoothed and I represents the discontinuities. In this case I is a binary field,
assuming the values 1 if there is a discontinuity and 0 otherwise. The Markov
property asserts that the probability of a certain value of the field at any given
site in the lattice depends only upon neighboring sites. According to the CliffordHammersley theorem, the prior probability of a state of the fields J and I has the
Gibbs form:
P(j, I)
1
= _e-fjU(J,I)
Z
(2.1)
where J and I are the fields, e.g. the surface-field and its discontinuities, Z is the
normalization constant also known as the partition function, U(j, I) = Ei Ui(J, I)
is an energy function that can be computed as the sum of local contributions from
each lattice site i, and f3 is a parameter that is called the inverse of the natural
temperature of the field. If a sparse observation 9 for any given surface-field / is
given and a model of the noise is available then one knows the conditional probability
P(gIJ, I). Bayes theorem then allows us to write the posterior distribution:
.!.
-fjv(JI,)
P(J II ) = P(gIJ, I)P(j, I) =
, 9
P(g)
- Ze
.
(2.2)
For the case of a sparse image corrupted by white gaussian noise
V(j,llg) = L~i(ji _gi)2
+ Ui(j,l)
(2.3)
i
=
where ~i;
1 or 0 depending on whether data are available or not. V(J,llg) is
sometimes called the visual cost !unction. The solution for the problem is the given
by some estimate of the fields. The maximum of the posterior distribution or other
related estimates of the "true" data-field value can not be computed analytically,
but sample distributions of the field with the probability distribution of (2.2) can
be obtained using Monte Carlo techniques such as the Metropolis algorithm. These
algorithms sample the space of possible values of the fields accwding to the proba.bility distribution P(j,llg).
A drawback of coupled MRFs has been the amount of computer time used in the
Metropolis algorithm or in simulated annea.ling (Kirkpatrick et. al., 1983).
A justification for using the mean field (MF) as a measure of the fields, J for example, resides in the fact that it represents the minimum variance Bayes estimator.
More precisely, the average variance of the field J is given by
Coupled Markov Random Fields and Mean Field Theory
Va".!
= LU -
/)2 PU, llg)
I,l
where / is a given estimate of the field, the EJ,l represents the sum over all the
possible configurations of / and " and Va".! is the variance. Minimizing Va".! with
respect to all possible values of / we obtain
This equation for / defines the deterministic MF equations.
2.1
MFT and Neural Networks
To connect MRFs to neural networks, we use Mean field theory (MFT) to obtain
deterministic equations from MRFs that represent a class of neural networks.
The mean field for the values f and I at site i are given by
" = L "PU, llg)
and
r. = L
1.1
The sum over the binary process, Ii
approximation,
Ii =
L
e-tn~i(J'-"i)2+Ui(J.f#i.I,=1)]
----Z-.- - -
(2.5)
?
where the partition function Z where factorized as
=L
(2.4)
0,1 gives for (2.3), using the mean field
I
Zi
'i P(/, Ilg)
e- fJ >'i(Ji-"i)2 (e-fJUi(/,f#,.li=O)
TIi Zi'
In this case
+ e- fJUi (J,T#i,I,=l?).
I
Another way to write the equation for / is
_fJV.-llecti ..
"
" = L.J"
I
where
e
?
Z.
?
(2.6)
663
664
Geiger and Girosi
The important result obtained here is that the effective potential does not dependend on the binary field Ii. The line process field has been eliminated to yield a
temperature dependent effective potential (also called visual cost function). The
interaction of the field f with itself has changed after the line process has been
averaged out. We interpret this result as the effect of the interaction of the line
processes with the field f to yield a new temperature dependent potential.
The computation of the sum over all the configurations of the field f is hard and
we use the saddle point approximation. In this case is equivalent to minimize
veJJeeti""(f). A dynamical equation to find the minimum of veJJeeti'Oe is given by
introducing a damping force
that brings the system to equilibrium. Therefore the
mean field equation under the mean field and saddle point approximation becomes
*
.!!.... ~eJJeeti'Oe(1 r = 8h
8h ?
,'J
(2.8)
8t
Equation (2.8) represents a class of unsupervised neural networks coupled to (2.5).
The mean field solution is given by the fixed point of (2.8) and (2.5) it is attained
after running (2.8) and (2.5) as t ........ 00. This network is better understood with an
example of image reconstruction.
3
Example: Image reconstruction
To reconstruct images from sparse data and to detect discontinuities we use the
weak membrane model where Ui(J, I) in two dimensions is given by
u..'" ?(f, h , v)
=
Q
~[(-I. . -
L...J J."
i,;
J',,. . 1)2(1-h.'" ')+(/''" '- J.-"
-I. 1 .)2(1_v . .)]+"V(J.. ?+V? .)
'"
'''i"
'"
I
(3.1)
and
Q
and l' are positive parameters.
The first term, contains the interaction between the field and the line processes: if
the horizon tal or vertical gradient is very high at site (i, j) the corresponding line
1), to make the visual cost
process will be very likely to be active (~,; 1 or Vi,;
function decrease and signal a discontinuity. The second term takes into account
the price we pay each time we create a discontinuity and is necessary to prevent
the creation of discontinuities everywhere. The effective cost function (2.7) then
becomes
=
=
Coupled Markov Random Fields and Mean Field Theory
665
Figure 2: The network i& repre,ented for the one dimen&ional ca,e. The line, are
the connection,
Vai" = ~ ["\ii(Ji,i-9i,i )2+a(a~i)2+(ai,i)2- ~ln[(I+e-"('Y-a4t/?)(1+e-"('Y-a4i./?]]
',J
(3.2)
where a~i
-
= Ii.; -
fi-1,i, ar,i
= Ii,i -
1
----..",.....~-',J - 1 + e"('Y-a(f'.j-la-l,j)2)
h? . -
Ji';-l
and
and (2.5) is then given by
1
, = 1 + e"('Y- a ( J,'.j- f,'.j-d 2 )
Vi i
(3.3).
we point out here that while the line process field is a binary field, its mean value
is a continuous (analog) function in the range between 0 and 1.
Discretizing (2.8) in time and applying for (3.2), we obtain
1.';+1 =
I.j - w [..\ii(h~i -
-a(l.~i -1."_1,;)(1 where h.,i
algorithm.
by (3.4) at
J again at
a(l.~i -1.~i-1)(1 - v~i) + a(l.~i+1 -1.~i)(1 - v~i+d
hf.i) + a(l."+1'i -1.~i)(l- hf+1,i)]
9i,i) -
and vi,i are given by the network (3.3) and n is the time step on the
We notice that (3.4) is coupled with (3.3) such that the field fis updated
step n and then (3.3) updates the field h and v before (3.4) updates field
step n + 1.
This is a simple unsupervised neural network where the imput are the fields J and
the output is the line process field h or v. This network is coupled to the network
(2.8) to solve for the field J and then constitute the global network for this problem
(see figure 2). It has been shown by many authors and (Geiger and Yuille, 1989)
that these class of networks is equivalent to Hcpfield networks (Hopfield, 1984)
(Koch et. al., 1985).
(3.4)
666
Geiger and Girosi
Figure 3: a. The .dill life image 128 x 128 pizel6. The image 6moothed with
I = 1400 and Q = 4 for 9 iteration6. The line proceS6 field (needs thinning). b. A
face image of 128 x 128 pizel6. Randomly chosen 50 % of the original image (for
di6play the other 50% are filled with white dot6). c. The network described above i6
applied to 6mooth and fill in using the same parameters and for 10 iterations. For
comparison we show the results of simply bluring the 6par6e data (no line process
field).
An important connection we make is to show (Geiger and Girosi, 1989) (Geiger,
1989) that the work of Blake and Zisserman (Blake and Zisserman, 1987) can be
seen as an approximation of these results.
Coupled Markov Random Fields and Mean Field Theory
In the zero temperature limit (f3 -+ 00) (3.3) becomes the Heaviside function (1
or 0) and the interpretation is simple: when the horizontal or vertical gradient are
larger than a threshold (JI) a vertical or horizontal discontinuity is created.
4
Results
We applied the network to a real still life image and the result was an enhancement
of specular edges, shadow edges and some other contours while smoothing out the
noise (see Figure 3a). This result is consistent with all the images we have used.
From one face image we produced sparse data by randomly suppressing 50% of
the data. (see Figure 3b). We then applied the neural network to reconstruct the
lmage.
AcknowledgeIllents
We are grateful to Tomaso Poggio for his guidance and support.
References
A. Blake and A.
MIT Press.
Zisserman. (1987) Vi,mal Recondruction. Cambridge, Mass:
E. Gamble and D. Geiger and T. Poggio and D. Weinshall. (1989) Integration of
vision modulea and labeling of aurface diacontinuitiea. Invited paper to IEEE Trans.
Sustems, Man & Cybernetics, December.
D. Geiger and F. Girosi. (1989) Parallel and deterministic algorithma for MRFs:
surface reconstruction and integration. A.!, Memo No.1114. Artificial Intelligence
Laboratory of MIT.
D. Geiger. (1989) Viaual modela with datiatical field theory. Ph.D. thesis. MIT,
Physics department and Artificial Intelligence Laboratory.
D. Geiger and A. Yuille. (1989) A common framework for image segmentation
and surface recon..truction. Harvard Robotics Laboratory Technical Report, 89-7,
Harvard, August.
S. Geman and D. Geman. (1984) Stochadic Relazation, Gibba Dutributiona, and
the Bayeaian Redoration of Imagea. Pattern Analysis and Machine Intelligence,
PAMI-6:721-741.
J. J. Hopfield. (1984) Neurona with graded reaponse have collective computational
properties like those of two-state neurona. Proc. Natl. Acad. ScL, 81:3088-3092,
S. Kirkpatrick and C. D. Gelatt and M. P.
Simulated Annealina. Science. 220:219-227.
Vecchio
(1983) Optimization by
C. Koch and J. Marroquin and A. Yuille. (1985) Analog 'Neuronal' Networka in
Early Vision. Proc. Natl. Acad. SeL, 83:4263-4267.
J. L. Marroquin and S. Mitter and T. Poggio. (1987) Probabilutic Solution of
nl-Poled Problema in Computational Viaion. J. Amer. Stat. Assoc., 82:76-89.
667
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1,873 | 2,700 | Generalization Error Bounds for Collaborative
Prediction with Low-Rank Matrices
Nathan Srebro
Department of Computer Science
University of Toronto
Toronto, ON, Canada
[email protected]
Noga Alon
School of Mathematical Sciences
Tel Aviv University
Ramat Aviv, Israel
[email protected]
Tommi S. Jaakkola
Computer Science and Artificial Intelligence Laboratory
Massachusetts Institute of Technology
Cambridge, MA, USA
[email protected]
Abstract
We prove generalization error bounds for predicting entries in a partially
observed matrix by fitting the observed entries with a low-rank matrix. In
justifying the analysis approach we take to obtain the bounds, we present
an example of a class of functions of finite pseudodimension such that
the sums of functions from this class have unbounded pseudodimension.
1
Introduction
?Collaborative filtering? refers to the general task of providing users with information on
what items they might like, or dislike, based on their preferences so far and how they relate
to the preferences of other users. This approach contrasts with a more traditional featurebased approach where predictions are made based on features of the items.
For feature-based approaches, we are accustomed to studying prediction methods in terms
of probabilistic post-hoc generalization error bounds. Such results provide us a (probabilistic) bound on the performance of our predictor on future examples, in terms of its
performance on the training data. These bounds hold without any assumptions on the true
?model?, that is the true dependence of the labels on the features, other than the central
assumptions that the training examples are drawn i.i.d. from the distribution of interest.
In this paper we suggest studying the generalization ability of collaborative prediction
methods. By ?collaborative prediction? we indicate that the objective is to be able to predict user preferences for items, that is, entries in some unknown target matrix Y of useritem ?ratings?, based on observing a subset YS of the entries in this matrix1 . We present
1
In other collaborative filtering tasks, the objective is to be able to provide each user with a few
items that overlap his top-rated items, while it is not important to be able to correctly predict the users
ratings for other items. Note that it is possible to derive generalization error bounds for this objective
based on bounds for the ?prediction? objective.
arbitrary source distribution
random training set
hypothesis
training error
generalization error
?
target matrix Y
? random set S of observed entries
?
predicted matrix X
? observed discrepancy DS (X; Y )
?
true discrepancy D(X; Y )
Figure 1: Correspondence with post-hoc bounds on the generalization error for standard
feature-based prediction tasks
Pn Pm
1
bounds on the true average overall error D(X; Y ) = nm
i=1
a=1 loss(Xia ; Yia ) of
the P
predictions X in terms of the average error over the observed entries DS (X; Y ) =
1
ia?S loss(Xia ; Yia ), without making any assumptions on the true nature of the pref|S|
erences Y . What we do assume is that the subset S of entries that we observe is chosen
uniformly at random. This strong assumption parallels the i.i.d. source assumption for
feature-based prediction.
In particular, we present generalization error bounds on prediction using low-rank models.
Collaborative prediction using low-rank models is fairly straight forward. A low-rank matrix X is sought that minimizes the average observed error DS (X; Y ). Unobserved entries
in Y are then predicted according to X. The premise behind such a model is that there
are only a small number of factors influencing the preferences, and that a user?s preference
vector is determined by how each factor applies to that user. Different methods differ in
how they relate real-valued entries in X to preferences in Y , and in the associated measure
of discrepancy. For example, entries in X can be seen as parameters for a probabilistic
models of the entries in Y , either mean parameters [1] or natural parameters [2], and a
maximum likelihood criterion used. Or, other loss functions, such as squared error [3, 2],
or zero-one loss versus the signs of entries in X, can be minimized.
Prior Work Previous results bounding the error of collaborative prediction using a lowrank matrix all assume the true target matrix Y is well-approximated by a low-rank matrix.
This corresponds to a large eigengap between the top few singular values of Y and the
remaining singular values. Azar et al [3] give asymptotic results on the convergence of
the predictions to the true preferences, assuming they have an eigengap. Drineas et al [4]
analyze the sample complexity needed to be able to predict a matrix with an eigengap, and
suggests strategies for actively querying entries in the target matrix. To our knowledge, this
is the first analysis of the generalization error of low-rank methods that do not make any
assumptions on the true target matrix.
Generalization error bounds (and related online learning bounds) were previously discussed
for collaborative prediction applications, but only when prediction was done for each user
separately, using a feature-based method, with the other user?s preferences as features [5,
6]. Although these address a collaborative prediction application, the learning setting is a
standard feature-based setting. These methods are also limited, in that learning must be
performed separately for each user.
Shaw-Taylor et al [7] discuss assumption-free post-hoc bounds on the residual errors of
low-rank approximation. These results apply to a different setting, where a subset of the
rows are fully observed, and bound a different quantity?the distance between rows and
the learned subspace, rather then the distance to predicted entries.
Organization In Section 2 we present a generalization error bound for zero-one loss,
based on a combinatorial result which we prove in Section 3. In Section 4 we generalize
the bound to arbitrary loss functions. Finally, in Section 5 we justify the combinatorial
approach taken, by considering an alternate approach (viewing rank-k matrices as combination of k rank-1 matrices) and showing why it does not work.
2
Generalization Error Bound for Zero-One Error
We begin by considering binary labels Yia ? ? and a zero-one sign agreement loss:
loss? (Xia ; Yia ) = 1Yia Xia ?0
(1)
Theorem 1. For any matrix Y ? {?1}n?m , n, m > 2, ? > 0 and integer k, with probability at least 1 ? ? over choosing a subset S of entries in Y uniformly among all subsets
of |S| entries, the discrepancy with respect to the zero-one sign agreement loss satisfies2 :
s
k(n + m) log 16em
? log ?
k
?X,rank X<k D? (X; Y ) < D?
(X;
Y
)
+
S
2|S|
To prove the theorem we employ standard arguments about the generalization error for
finite hypothesis classes with bounded cardinality.
First fix Y as well as X ? Rn?m . When an index pair (i, a) is chosen uniformly at random,
loss(Xia ; Yia ) is a Bernoulli random variable with probability D? (X; Y ) of being one.
If the entries of S are chosen independently and uniformly, |S|D?
S (X; Y ) is Binomially
distributed with mean |S|D? (X; Y ) and using Chernoff?s inequality:
?2|S|2
Pr D? (X; Y ) ? D?
(2)
S (X; Y ) + ? e
S
The distribution of S in Theorem 1 is slightly different, as S is chosen without repetitions.
The mean of D?
S (X; Y ) is the same, but it is more concentrated, and (2) still holds.
Now consider all rank-k matrices. Noting that loss(Xia ; Yia ) depends only on the sign
of Xia , it is enough to consider the equivalence classes of matrices with the same sign
patterns. Let f (n, m, k) be the number of such equivalence classes, i.e. the number of
possible sign configurations of n ? m matrices of rank at most k:
F (n, m, k) = {sign X ? {?, 0, +}n?m |X ? Rn?m , rank X ? k}
f (n, m, k) = ]F (n, m, k)
(
where sign X denotes the element-wise sign matrix (sign X)ia =
1
0
?1
If Xia > 0
If Xia = 0 .
If Xia < 1
For all matrices in an equivalence class, the random variable D?
S (X; Y ) is the same, and
taking a union bound of the events D? (X; Y ) ? D?
(X;
Y
)+
for
each of these f (n, m, k)
S
random variables we have:
s
!
log f (n, m, k) ? log ?
?
?
? ? (3)
Pr ?X,rank X?k D (X; Y ) ? DS (X; Y ) +
S
2|S|
q
?
by using (2) and setting = log f (n,m,k)?log
. The proof of Theorem 1 rests on bounding
2|S|
f (n, m, k), which we will do in the next section.
Note that since the equivalence classes we defined do not depend on the sample set, no
symmetrization argument is necessary.
2
All logarithms are base two
3
Sign Configurations of a Low-Rank Matrix
In this section, we bound the number f (n, m, k) of sign configurations of n ? m rankk matrices over the reals. Such a bound was previously considered in the context of
unbounded error communication complexity. Alon, Frankl and R?odl [8] showed that
(n+m)k+h+m
f (n, m, k) ? minh (8dnm/he)
, and used counting arguments to establish
that some (in fact, most) binary matrices can only be realized by high-rank matrices, and
therefore correspond to functions with high unbounded error communication complexity.
Here, we follow a general course outlined by Alon [9] to obtain a simpler, and slightly
tighter, bound based on the following result due to Warren:
Let P1 , . . . , Pr be real polynomials in q variables, and let C be the complement of the
variety defined by ?i Pi , i.e. the set of points in which all the m polynomials are non-zero:
C = {x ? Rq |?i Pi (x) 6= 0}
Theorem 2 (Warren [10]). If all r polynomials are of degree at most d, then the number
of connected components of C is at most:
q
q
X
r
4edr
c(C) ? 2(2d)q
2i
?
i
q
i=0
where the second inequality holds when r > q > 2.
The signs of the polynomials P1 , . . . , Pr are fixed inside each connected component of C.
And so, c(C) bounds the number of sign configurations of P1 , . . . , Pr that do not contain
zeros. To bound the overall number of sign configurations the polynomials are modified
slightly (see Appendix), yielding:
Corollary 3 ([9, Proposition 5.5]). The number of -/0/+ sign configurations of r polynomials, each of degree at most d, over q variables, is at most (8edr/q)q (for r > q > 2).
In order to apply these bounds to low-rank matrices, recall that any matrix X of rank at
most k can be written as a product X = U V 0 where U ? Rn?k and V ? Rk?m . Consider
the k(n+m) entries of U, V as variables, and the nm entries of X as polynomials of degree
two over these variables:
k
X
Ui? Va?
Xia =
?=1
Applying Corollary 3 we obtain:
k(n+m)
8e?2?nm
? (16em/k)k(n+m)
Lemma 4. f (n, m, k) ? k(n+m)
Substituting this bound in (3) establishes Theorem 1. The upper bound on f (n, m, k) is
tight up to a multiplicative factor in the exponent:
1
Lemma 5. For m > k 2 , f (n, m, k) ? m 2 (k?1)n
Proof. Fix any matrix V ? Rm?k with rows in general position, and consider the number
f (n, V, k) of sign configurations of matrices U V 0 , where U varies over all n ? k matrices.
Focusing only on +/? sign configurations (no zeros in U V 0 ), each row of sign U V 0 is a
homogeneous linear classification
of the
rows of V , i.e. of m vectors in general position
P
k?1
in Rk . There are exactly 2 i=0 mi
possible homogeneous linear classifications of m
vectors in general position in Rk , and so these many options for each row of sign U V 0 . We
can therefore bound:
!n
k?1
n(k?1)
X
1
m
m n
m
f (n, m, k) ? f (n, V, k) ? 2
?
?
= m 2 (k?1)n
i
k?1
k?1
i=0
4
Generalization Error Bounds for Other Loss Functions
In Section 2 we considered generalization error bounds for a zero-one loss function. More
commonly, though, other loss functions are used, and it is desirable to obtain generalization
error bounds for general loss functions.
When dealing with other loss functions, the magnitude of the entries in the matrix are
important, and not only their signs. It is therefore no longer enough to bound the number
of sign configurations. Instead, we will bound not only the number of ways low rank
matrices behave with regards to a threshold of zero, but the number of possible ways lowrank matrices can behave relative to any set of thresholds. That is, for any threshold matrix
T ? Rn?m , we will show that the number of possible sign configurations of (X ? T ),
where X is low-rank, is small. Intuitively, this captures the complexity of the class of
low-rank matrices not only around zero, but throughout all possible values.
We then use standard results from statistical machine learning to obtain generalization error
bounds from the bound on the number of relative sign configurations. The number of relative sign configurations serves as a bound on the pseudodimension?the maximum number
of entries for which there exists a set of thresholds such that all relative sign configurations
(limited to these entries) is possible. The pseudodimension can in turn be used to show the
existence of a small -net, which is used to obtain generalization error bounds.
Recall the definition of the pseudodimension of a class of real-valued functions:
Definition 1. A class F of real-valued functions pseudo-shatters the points x1 , . . . , xn
with thresholds t1 , . . . , tn if for every binary labeling of the points (s1 , . . . , sn ) ? {+, ?}n
there exists f ? F s.t. f (xi ) ? ti iff si = ?. The pseudodimension of a class F is the
supremum over n for which there exist n points and thresholds that can be shattered.
In order to apply known results linking the pseudodimension to covering numbers, we
consider matrices X ? Rn?m as real-valued functions X : [n] ? [m] ? R over index
pairs to entries in the matrix. The class Xk of rank-k matrices can now be seen as a class
of real-valued functions over the domain [n] ? [m]. We bound the pseudodimension of
this class by bounding, for any threshold matrix T ? Rn?m the number of relative sign
matrices:
FT (n, m, k) = {sign (X ? T ) ? {?, 0, +}n?m |X ? Rn?m , rank X ? k}
fT (n, m, k) = ]FT (n, m, k)
Lemma 6. For any T ? Rn?m , we have fT (n, m, k) ?
16em k(n+m)
.
k
Proof. We take a similar approach to that of Lemma 4, writing rank-k matrices as a product
X = U V 0 where U ? Rn?k and V ? Rk?m . Consider the k(n + m) entries of U, V as
variables, and the nm entries of X ? T as polynomials of degree two over these variables:
(X ? T )ia =
k
X
Ui? Va? ? Tia
?=1
Applying Corollary 10 yields the desired bound.
Corollary 7. The pseudodimension of the class Xk of n ? m matrices over the reals of
rank at most k, is at most k(n + m) log 16em
k .
We can now invoke standard generalization error bounds in terms of the pseudodimension
(Theorem 11 in the Appendix) to obtain:
Theorem 8. For any monotone loss function with |loss| ? M , any matrix Y ? {?1}n?m ,
n, m > 2, ? > 0 and integer k, with probability at least 1 ? ? over choosing a subset S of
entries in Y uniformly among all subsets of |S| entries:
v
u
u k(n + m) log 16em log M |S| ? log ?
t
k
k(n+m)
?X,rank X<k D(X; Y ) < DS (X; Y ) + 6
|S|
5
Low-Rank Matrices as Combined Classifiers
Rank-k matrices are those matrices which are a sum of k rank-1 matrices. If we view
matrices as functions from pairs of indices to the reals, we can think of rank-k matrices as
?combined? classifiers, and attempt to bound their complexity as such, based on the low
complexity of the ?basis? functions, i.e. rank-1 matrices.
A similar approach is taken in related work on learning with low-norm (maximum margin)
matrix factorization [11, 12], where the hypothesis class can be viewed as a convex combination of rank-1 unit-norm matrices. Scale-sensitive (i.e. dependent on the margin, or the
slope of the loss function) generalization error bounds for this class are developed based on
the graceful behavior of scale-sensitive complexity measures (e.g. log covering numbers
and the Rademacher complexity) with respect to convex combinations. Taking a similar
view, it is possible to obtain scale-sensitive generalization error bounds for low-rank matrices. In this Section we question whether it is possible to obtain scale-insensitive bounds,
similar to Theorems 1 and 8, by viewing low-rank matrices as combined classifiers.
It cannot be expected that scale-insensitive complexity would be preserved when taking
convex combinations of an unbounded number of base functions. However, the VCdimension, a scale-insensitive measure of complexity, does scale gracefully when taking
linear combinations of a bounded number of functions from a low VC-dimension class of
indicator function. Using this, we can obtain generalization error bounds for linear combinations of signs of rank-one matrices, but not signs of linear combinations of rank-one
matrices. An alternate candidate scale-insensitive complexity measure is the pseudodimension of a class of real-valued functions. If we could bound the pseudodimension of the
class of sums of k functions from a bounded-pseudodimension base class of real valued
functions, we could avoid the sign-configuration counting and obtain generalization error
bounds for rank-k matrices. Unfortunately, the following counterexample shows that this
is not possible.
Theorem 9. There exists a family F closed under scalar multiplication whose pseudodimension is at most five, and such that {f1 + f2 |f1 , f2 ? F} does not have a finite pseudodimension.
Proof. We describe a class F of real-valued functions over the positive integers N. To do
so, consider a one-to-one mapping of finite sets of positive integers to the positive integers.
For each A ? N define two functions3 , fA (x) = 2xA + 1x?A and gA (x) = 2xA . Let F be
the set of all scalar multiplications of these functions.
For every A ? N , fA ? gA is the indicator function of A, implying that every finite subset
can be shattered, and the pseudodimension of {f1 + f2 : f1 , f2 ? F} is unbounded.
It remains to show that the pseudodimension of F is less than six. To do so, we note that
there are no positive integers A < B and x < y and positive reals ?, ? > 0 such that
?(2xB + 1) > ?2xA and ?2yB < ?(2yA + 1). It follows that for any A < B and any
?, ? > 0, on an initial segment (possibly empty) of N we have ?gB ? ?fB ? ?gA ? ?fA
while on the rest of N we have ?gA ? ?fA < ?gB ? ?fB . In particular, any pair of
3
We use A to refer both to a positive integer and the finite set it maps to.
functions (?fA , ?fB ) or (?fA , ?gB ) or (?gA , ?gB ) in F that are not associated with the
same subset (i.e. A 6= B), cross each other at most once. This holds also when ? or ? are
negative, as the functions never change signs.
For any six naturals x1 < x2 < ? ? ? < x6 and six thresholds, consider the three labellings
(+, ?, +, ?, +, ?), (?, +, ?, +, ?, +), (+, +, ?, ?, +, +). The three functions realizing
these labellings must cross each other at least twice, but by the above arguments, there are
no three functions in F such that every pair crosses each other at least twice.4
6
Discussion
Alon, Frankl and R?odl [8] use a result of Milnor similar to Warren?s Theorem 2. Milnor?s
and Warren?s theorems were previously used for bounding the VC-dimension of certain
geometric classes [13], and of general concept classes parametrized by real numbers, in
terms of the complexity of the boolean formulas over polynomials used to represent them
[14]. This last general result can be used to bound the VC-dimension of signs of n?m rankk matrices by 2k(n + m) log(48enm), yielding a bound similar to Theorem 1 with an extra
log |S| term. In this paper, we take a simpler path, applying Warren?s theorem directly, and
thus avoiding the log |S| term and reducing the other logarithmic term. Applying Warren?s
theorem directly also enables us to bound the pseudodimension and obtain the bound of
Theorem 8 for general loss functions.
Another notable application of Milnor?s result, which likely inspired these later uses, is for
bounding the number of configurations of n points in Rd with different possible linear classifications [15, 16]. Viewing signs of rank-k n ? m matrices as n linear classification of m
points in Rk , this bound can be used to bound f (n, m, k) < 2km log 2n+k(k+1)n log n without using Warren?s Theorem directly [8, 12]. The bound of Lemma 4 avoids the quadratic
dependence on k in the exponent.
Acknowledgments We would like to thank Peter Bartlett for pointing out [13, 14]. N.S. and
T.J. would like to thank Erik Demaine for introducing them to oriented matroids.
A
Proof of Corollary 3
Consider a set R ? Rq containing one variable configuration for each possible sign pattern. Set
.
= 12 min1?i?q,x?RPi (x)6=0 |Pi (x)| > 0. Now consider the 2q polynomials Pi+ (x) = Pi (x) +
and Pi? (x) = Pi (x) ? and C 0 = x ? Rq |?i Pi+ (x) 6= 0, Pi? (x) 6= 0 . Different points in R
(representing all sign configurations) lie in different connected components of C 0 . Invoking Theorem
2 on C 0 establishes Corollary 3.
The count in Corollary 3 differentiates between positive, negative and zero signs. However, we are
only concerned with the positivity of Yia Xia (in the proof of Theorem 1) or of Xia ? Tia (in the
proof of Theorem 8), and do not need to differentiate
between zero and negative values. Invoking
Theorem 2 on C + = x ? Rq |?i Pi+ (x) 6= 0 , yields:
Corollary 10. The number of -/+ sign configurations (where zero is considered negative) of r polynomials, each of degree at most d, over q variables, is at most (4edr/q)q (for r > q > 2).
P
Applying Corollary 10 on the nm degree-two polynomials Yia k?=1 Ui? Va? establishes that for
any Y , the number of configurations of sign agreements of rank-k matrices with Y is bounded by
(8em/k)k(n+m) and yields a constant of 8 instead of 16 inside the logarithm in Theorem 1. Applying
Corollary 10 instead of Corollary 3 allows us to similarly tighten in the bounds in Corollary 7 and in
Theorem 8.
4
A more careful analysis shows that F has pseudodimension three.
B
Generalization Error Bound in terms of the Pseudodimension
Theorem 11. Let F be a class of real-valued functions f : X ? R with pseudodimension d, and
loss : R ? Y ? R be a bounded monotone loss function (i.e. for all y, loss(x, y) is monotone in x), with loss < M . For any joint distribution over (X, Y ), consider an i.i.d. sample
S = (X1 , Y1 ), . . . , (Xn , Yn ). Then for any > 0:
!
d
n
2 n
1X
32eM
Pr ?f ?F EX,Y [loss(f (X), Y )] >
loss(f (Xi ), Yi ) + < 4e(d + 1)
e? 32
S
n i=1
The bound is a composition of a generalization error bound in terms of the L1 covering number [17,
Theorem 17.1], a bound on the L1 covering number in terms of the pseudodimension [18] and the
observation that composition with a monotone function does not increase the pseudodimension [17,
Theorem 12.3].
References
[1] T. Hofmann. Latent semantic models for collaborative filtering. ACM Trans. Inf. Syst.,
22(1):89?115, 2004.
[2] Nathan Srebro and Tommi Jaakkola. Weighted low rank approximation. In 20th International
Conference on Machine Learning, 2003.
[3] Yossi Azar, Amos Fiat, Anna R. Karlin, Frank McSherry, and Jared Saia. Spectral analysis of
data. In ACM Symposium on Theory of Computing, pages 619?626, 2001.
[4] Petros Drineas, Iordanis Kerenidis, and Prabhakar Raghavan. Competitive recommendation
systems. In ACM Symposium on Theory of Computing, 2002.
[5] K. Crammer and Y. Singer. Pranking with ranking. In Advances in Neural Information Processing Systems, volume 14, 2002.
[6] Sanjoy Dasgupta, Wee Sun Lee, and Philip M. Long. A theoretical analysis of query selection
for collaborative filtering. Machine Learning, 51(3):283?298, 2003.
[7] John Shawe-Taylor, Nello Cristianini, and Jaz Kandola. On the concentration of spectral properties. In Advances in Neural Information Processing Systems, volume 14, 2002.
[8] N. Alon, P Frankl, and V. R?odel. Geometrical realization of set systems and probabilistic
communication complexity. In Foundations of Computer Science (FOCS), 1985.
[9] Noga Alon. Tools from higher algebra. In M. Gr?otschel R.L. Graham and L. Lov?asz, editors,
Handbook of Combinatorics, chapter 32, pages 1749?1783. North Holland, 1995.
[10] H. E. Warren. Lower bounds for approximation by nonlinear manifolds. Transactions of the
American Mathematical Society, 133:167?178, 1968.
[11] Nathan Srebro, Jason Rennie, and Tommi Jaakkola. Maximum margin matrix factorization. In
Advances in Neural Information Processing Systems, volume 17, 2005.
[12] Nathan Srebro. Learning with Matrix Factorization. PhD thesis, Massachusetts Institute of
Technology, 2004.
[13] Shai Ben-David and Michael Lindenbaum. Localization vs. identification of semi-algebraic
sets. Machine Learning, 32(3):207?224, 1998.
[14] Paul Goldberg and Mark Jerrum. Bounding the vapnik-chervonenkis dimension of concept
classes parameterized by real numbers. Machine Learning, 18(2-3):131?148, 1995.
[15] Jacob Goodman and Richard Pollack. Upper bounds for configurations and polytopes in Rd .
Discrete and Computational Geometry, 1:219?227, 1986.
[16] Noga Alon. The number of polytopes, configurations and real matroids. Mathematika, 33:62?
71, 1986.
[17] Martin Anthony and Peter L. Bartlett. Neural Networks Learning: Theoretical Foundations.
Cambridge University Press, 1999.
[18] David Haussler. Sphere packing numbers for subsets of the boolean n-cube with bounded
Vapnick-Chernovenkis dimension. J. Comb. Thoery, Ser. A, 69(2):217?232, 1995.
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srebro:4 versus:1 querying:1 foundation:2 degree:6 editor:1 pi:10 row:6 course:1 last:1 free:1 warren:8 institute:2 taking:4 matroids:2 distributed:1 regard:1 xia:13 dimension:5 xn:2 avoids:1 fb:3 forward:1 made:1 commonly:1 saia:1 far:1 tighten:1 transaction:1 supremum:1 dealing:1 handbook:1 xi:2 latent:1 why:1 nature:1 tel:1 anthony:1 domain:1 anna:1 accustomed:1 bounding:6 azar:2 paul:1 x1:3 position:3 candidate:1 lie:1 theorem:26 rk:5 formula:1 showing:1 exists:3 vapnik:1 phd:1 magnitude:1 margin:3 logarithmic:1 likely:1 partially:1 scalar:2 recommendation:1 holland:1 applies:1 corresponds:1 acm:3 ma:1 viewed:1 careful:1 change:1 determined:1 uniformly:5 edr:3 justify:1 useritem:1 reducing:1 lemma:5 sanjoy:1 ya:1 milnor:3 mark:1 crammer:1 avoiding:1 ex:1 |
1,874 | 2,701 | Algebraic Set Kernels with Application to
Inference Over Local Image Representations
Amnon Shashua and Tamir Hazan ?
Abstract
This paper presents a general family of algebraic positive definite similarity functions over spaces of matrices with varying column rank. The
columns can represent local regions in an image (whereby images have
varying number of local parts), images of an image sequence, motion trajectories in a multibody motion, and so forth. The family of set kernels
we derive is based on a group invariant tensor product lifting with parameters that can be naturally tuned to provide a cook-book of sorts covering
the possible ?wish lists? from similarity measures over sets of varying
cardinality. We highlight the strengths of our approach by demonstrating the set kernels for visual recognition of pedestrians using local parts
representations.
1
Introduction
In the area of learning from observations there are two main paths that are often mutually
exclusive: (i) the design of learning algorithms, and (ii) the design of data representations.
The algorithm designers take pride in the fact that their algorithm can generalize well given
straightforward data representations (most notable example is SVM [11]), whereas those
who work on data representations demonstrate often remarkable results with sophisticated
data representations using only straightforward learning algorithms (e.g. [5, 10, 6]). This
dichotomy is probably most emphasized in the area of computer vision, where image understanding from observations involve data instances of images or image sequences containing
huge amounts of data. A straightforward representation treating all the measurements as
a single vector, such as the raw pixel data, or a transformed raw-pixel data, places unreasonable demands on the learning algorithm. The ?holistic? representations suffer also
from sensitivity to occlusions, invariance to local and global transformations, non-rigidity
of local parts of the object, and so forth.
Practitioners in the area of data representations have long noticed that a collection of local
representations (part-based representations) can be most effective to ameliorate changes of
appearance [5, 10, 6]. The local data representations vary in their sophistication, but share
the same principle where an image corresponds to a collection of points each in a relatively
small dimensional space ? instead of a single point in high-dimensional space induced
by holistic representations. In general, the number of points (local parts) per image may
vary and the dimension of each point may vary as well. The local representations tend
?
School of Engineering and Computer Science, Hebrew University of Jerusalem, Jerusalem
91904, Israel
to be robust against occlusions, local and global transformations and preserve the original
resolution of the image (the higher the resolution the more parts are generated per image).
The key for unifying local and holistic representations for inference engines is to design
positive definite similarity functions (a.k.a. kernels) over sets (of vectors) of varying cardinalities. A Support Vector Machine (SVM) [11] can then handle sets of vectors as a single
instance via application of those ?set kernels?. A set kernel would be useful also to other
types of inference engines such as kernel versions of PCA, LDA, CCA, ridge regression and
any algorithm which can be mapped onto inner-products between pairs of data instances
(see [8] for details on kernel methods).
Formally, we consider an instance being represented by a collection of vectors, which for
the sake of convenience, form the columns of a matrix. We would like to find an algebraic
family of similarity functions sim(A, B) over matrices A, B which satisfy the following
requirements: (i) sim(A, B) is an inner product, i.e., sim(A, B) = ?(A)> ?(B) for some
mapping ?() from matrices to vectors, (ii) sim(A, B) is built over local kernel functions
k(ai , bj ) over columns ai and bj of A, B respectively, (iii) The column cardinality (rank
of column space) of A and B need not be the same (number of local parts may differ from
image to image), and (iv) the parameters of sim(A, B) should induce the properties of invariance to order (alignement) of parts, part occlusions, and degree of interactions between
local parts. In a nutshell, our work provides a cook-book of sorts which fundamentally
covers the possible algebraic kernels over collections of local representations built on top
of local kernels by combining (linearly and non-linearly) local kernels to form a family of
global kernels over local representations.
The design of a kernel over sets of vectors has been recently attracting much attention in the
computer vision and machine learning literature. A possible approach is to fit a distribution
to the set of vectors and define the kernel as a distribution matching measure [9, 12, 4].
This has the advantage that the number of local parts can vary but at the expense of fitting
a distribution to the variation over parts. The variation could be quite complex at times,
unlikely to fit into a known family of distributions in many situations of interest, and in
practice the sample size (number of columns of A) is not sufficiently large to reliably fit
a distribution. The alternative, which is the approach taken in this paper, is to create a
kernel over sets of vectors in a direct manner. When the column cardinality is equal it is
possible to model the similarity measure as a function over the principal angles between the
two column spaces ([14] and references therein) while for varying column cardinality only
heuristic similarity measures (which are not positive definite) have so far been introduced
[13].
It is important to note that although we chose SVM over local representations as the application to demonstrate the use of set kernels, the need for adequately working with instances
made out of sets of various cardinalities spans many other application domains. For example, an image sequence may be represented by a set (ordered or unordered) of vectors,
where each vector stands for an image, the pixels in an image can be represented as a tuple
consisting of position, intensity and other attributes, motion trajectories of multiply moving bodies can be represented as a collection of vectors, and so on. Therefore, the problem
addressed in this paper is fundamental both theoretically and from a practical perspective
as well.
2
The General Family of Inner-Products over Matrices
We wish to derive the general family of positive definite similarity measures sim(A, B)
over matrices A, B which have the same number of rows but possibly different column
rank (in particular, different number of columns). Let A be of dimensions n ? k and
B of dimension n ? q where n is fixed and k, q can vary at will over the application of
sim(?, ?) on pairs of matrices. Let m = max{n, k, q} be the upper bound over all values
of k, q encountered by the data. Let ai , bj be the column vectors of matrices A, B and
let k(ai , bj ) be the local kernel function. For example, in the context where the column
vectors represent local parts of an image, then the matching function k(?, ?) between pairs
of local parts provides the building blocks of the overall similarity function. The local
kernel is some positive definite function k(x, y) = ?(x)> ?(y) which is the inner-product
between the ?feature?-mapped vectors x, y for some feature map ?(?). For example, if ?(?)
is the polynomial map of degree up to d, then k(x, y) = (1 + x> y)d .
The local kernels can be combined in a linear or non-linear manner. When the combination
is linear the similarity becomes the analogue of the inner-product between vectors extended
to matrices. We will refer to the linear family as sim(A, B) =< A, B > and that will be
the focus of this section. In the next section we will derive the general (algebraic) nonlinear family which is based on ?lifting? the input matrices A, B onto higher dimensional
spaces and feeding the result onto the < ?, ? > machinery developed in this section, i.e.,
sim(A, B) =< ?(A), ?(B) >.
We will start by embedding A, B onto m ? m matrices by zero padding as follows. Let
ei denote the i?th standard basis vector (0, .., 0, 1, 0, .., 0) of Rm . The the embedding is
represented by linear combinations of tensor products:
A?
n X
k
X
aij ei ? ej ,
B?
q
n X
X
blt el ? et .
l=1 t=1
i=1 j=1
Note that A, B are the upper-left blocks of the zero-padded
matrices. Let S be a positive
Pp
semi definite m2 ? m2 matrix represented by S = r=1 Gr ? Fr where Gr , Fr are m ? m
? r be the n ? n
matrices1 . Let F?r be the q ? k upper-left sub-matrix of Fr> , and let G
upper-left sub-matrix of Gr . We will be using the following three identities:
Gx1 ? F x2 = (G ? F )(x1 ? x2 ),
(G ? F )(G0 ? F 0 ) = GG0 ? F F 0 ,
>
< x1 ? x2 , y1 ? y2 >= (x>
1 y1 )(x2 y2 ).
The inner-product < A, B > over all p.s.d. matrices S has the form:
X
X
X
< A, B > = <
aij ei ? ej , (
Gr ? Fr )
blt el ? et >
r
i,j
=
XX
r
XX
=
XX
r
=
>
aij blt (e>
i Gr el )(ej Fr et )
i,j,l,t
aij blt (Gr )il (Fr )jt
i,j,l,t
XX
r
aij blt < ei ? ej , Gr el ? Fr et >
i,j,l,t
=
r
l,t
? r B)jt (Fr )jt
(A> G
lt
!
= trace
X
? r B)F?r
(A G
>
r
We have represented the inner product < A, B > using the choice of m ? m matrices
Gr , Fr instead of the choice of a single m2 ? m2 p.s.d. matrix S. The matrices Gr , Fr
1
Any S can be represented as a sum over tensor products: given column-wise ordering, the matrix
G ? F is composed of n ? n blocks of the form fij G. Therefore, take Gr to be the n ? n blocks of
S and Fr to be the elemental matrices which have ?1? in coordinate r = (i, j) and zero everywhere
else.
Pp
must be selected such that r=1 Gr ? Fr is positive semi definite. The problem of deciding on the the necessary conditions on Fr and Gr such that the sum over tensor products is
p.s.d is difficult. Even deciding whether a given S has a separable decomposition is known
to be NP-hard [3]. The sufficient
conditions are easy ? choosing Gr , Fr to be positive
Pp
semi definite would make r=1 Gr ? Fr positive semi definite as well. In this context
(of separable S) we need one more constraint in order to work with non-linear local ker?r = M
? r> M
? r must ?distribute with the kernel?,
nels k(x, y) = ?(x)> ?(y): the matrices G
namely there exist Mr such that
? >M
? r ?(y) = ?(x)> G
? r ?(y).
k(Mr x, Mr y) = ?(Mr x)> ?(Mr y) = ?(x)> M
r
To summarize the results so far, the most general, but seperable, analogue of the innerproduct over vectors to the inner-product of matrices of varying column cardinality has the
form:
X
< A, B >=
trace(Hr F?r )
(1)
r
Where the entries of Hr consists of k(Mr ai , Mr bj ) over the columns of A, B after possibly
? r ), and F?r are the q ? k upperundergoing global coordinate changes by Mr (the role of G
>
left sub-matrix of positive definite m ? m matrices Fr .
? r is to perform global coordinate changes of Rn before applicaThe role of the matrices G
tion of the kernel k() on the columns of A, B. These global transformations include projections (say onto prototypical ?parts?) that may be given or ?learned? from a training set.
The matrices F?r determine the range of interaction between columns of A and columns of
? r = I then < A, B >= trace(A> B F? ) where F? is the upper-left
B. For example, when G
P
submatrix with the appropriate dimension of some fixed m ? m p.s.d matrix F = r Fr .
Note that entries of A> B are k(ai , bj ). In other words, when
P Gr = I, < A, B > boils
down to a simple linear super-position of the local kernels, ij k(ai , bj )fij where the entries fij are part of the upper-left block of a fixed positive definite matrix F where the
block dimensions are commensurate with the number of columns of A and those of B. The
various choices of F determine the type of invariances one could obtain from the similarity measure. For example, when F = I the similarity is simply the sum (average) of
the local kernels k(ai , bi ) thereby assuming we have a strict alignment between the local
parts represented by A and the local parts represented by B. On the other end of the invariance spectrum, when F = 11> (all entries are ?1?) the similarity measure averages
over all interactions of local parts k(ai , bj ) thereby achieving an invariance to the order of
the parts. A decaying weighted interaction such as fij = ? ?|i?j| would provide a middle
ground between the assumption of strict alignment and the assumption of complete lack of
alignment. In the section below we will derive the non-linear version of sim(A, B) based
on the basic machinery of < A, B > of eqn. (1) and lifting operations on A, B.
3
Lifting Matrices onto Higher Dimensions
The family of sim(A, B) =< A, B > forms a weighted linear superposition of the
local kernel k(ai , bj ). Non-linear combinations of local kernels emerge using mappings ?(A) from the input matrices onto other higher-dimensional matrices, thus forming
sim(A, B) =< ?(A), ?(B) >. Additional invariance properties and parameters controlling the perfromance of sim(A, B) emerge with the introduction of non-linear combinations of local kernels, and those will be discussed later on in this section.
Consider the general d-fold lifting ?(A) = A?d which can be viewed as a nd ? k d matrix.
Let Fr be a p.s.d. matrix of dimension md ? md and F?r be the upper-left q d ? k d block of
? r )?d be a p.s.d matrix of dimension nd ? nd where G
? r is p.s.d. n ? n
Fr . Let Gr = (G
matrix. Using the identity (A?d )> B ?d = (A> B)?d we obtain the inner-product in the
lifted space:
< A?d , B ?d >=
X
? r B)?d F?r .
trace (A> G
r
?l
By taking linear combinations of < A , B ?l >, l = 1, ..., d, we get the general nonhomogenous d-fold inner-product simd (A, B). A this point the formulation is general but
somewhat unwieldy computational-wise. The key for computational simplification lay in
the fact that choices of Fr determine not only local interactions (as in the linear case) but
also group invariances. The group invariances are a result of applying symmetric operators
on the tensor product space ? we will consider two of those operators here, known as the
the d-fold alternating tensor A?d = A ? .... ? A and the d-fold symmetric tensor A?d =
A ? ... ? A. These lifting operations introduce the determinant and permanent operations on
? r B, as described below.
submatrices of A> G
The alternating tensor is a multilinear map of Rn , (A ? .... ? A)(x1 ? ... ? xd ) = Ax1 ?
... ? Axd , where
1 X
x1 ? ... ? xd =
sign(?)x?(1) ? .... ? x?(d) ,
d!
??Sd
where Sd is the symmetric group over d letters and ? ? Sd are the permutations of the
group. If x1 , ..., xn form a basis of Rn , then the nd elements xi1 ? ... ? xid , where 1 ?
i1 < ... < id ? n form a basis of the alternating d ? f old tensor product of Rn , denoted
as ?d Rn . If A ? Rn?k is a linear map on Rn sending points to Rk , then A?d is a
linear map on ?d Rn sending x1 ? ... ? xd to Ax1 ? ... ? Axd , i.e., sending points in
?d Rn to points in ?d Rk . The matrix representation of A?d is called the ?d?th compound
matrix? Cd (A) whose (i1 , ..., id |j1 , ..., jd ) entry has the value det(A[i1 , ..., id : j1 , ..., jd ])
where the determinant is of the d ? d block constructed by choosing
the rows
i1 , ..., id
and the columns j1 , ..., jd of A. In other words, Cd (A) has nd rows and kd columns
(instead of nd ? k d necessary for A?d ) whose entries are equal to the d ? d minors of A.
When k = d, Ck (A) is a vector known as the Grasmanian of A, and when n = k = d
then Cd (A) = det(A). Finally, the identity (A?d )> B ?d = (A> B)?d specializes to
(A?d )> B ?d = (A> B)?d which translates to the identity Cd (A)> Cd (B) = Cd (A> B)
known as the Binet-Cauchy theorem [1]. Taken together, the ?d-fold alternating kernel?
?d (A, B) is defined by:
X
? r B)F?r , (2)
?d (A, B) =< A?d , B ?d >=< Cd (A), Cd (B) >=
trace Cd (A> G
r
m
where F?r is the
?
upper-left submatrix of the p.s.d m
d ? d matrix Fr . Note
? r B)ij = k(Mr ai , Mr bj ) where G
?r =
that the local kernel plugs in as the entries of (A> G
>
Mr Mr .
q
d
k
d
Another symmetric operator on the tensor product space is via the d-fold symmetric tensor
space Symd Rn whose points are:
1 X
x?(1) ? .... ? x?(d) .
x1 ? ? ? xd =
d!
??Sd
The analogue of Cd (A) is the ?d?th power matrix? Rd (A) whose (i1 , ..., id |j1 , ..., jd ) entry
has the value perm(A[i1 , ..., id : j1 , ..., jd ]) and which stands for the map A?d
(A ? ? ? A)(x1 ? ? ? xd ) = Ax1 ? ? ? Axd .
In other words, Rd (A) has n+d?1
rows and k+d?1
columns whose entries are equal to
d
d
the d?d permanents of A. The analogue of the Binet-Cauchy theorem is Rd (A)> Rd (B) =
Rd (A> B). The ensuing kernel similarity function, referred to as the ?d-fold symmetric
kernel? is:
X
? r B)F?r (3)
Symd (A, B) =< A?d , B ?d >=< Rd (A), Rd (B) >=
trace Rd (A> G
r
k+d?1
where F?r is the q+d?1
?
upper-left submatrix of the positive definite m+d?1
?
d
d
d
n+d?1
matrix
F
.
Due
to
lack
of
space
we
will
stop
here
and
spend
the
remainder
of
this
r
d
section in describing in laymen terms what are the properties of these similarity measures,
how they can be constructed in practice and in a computationally efficient manner (despite
the combinatorial element in their definition).
3.1 Practical Considerations
To recap, the family of similarity functions sim(A, B) comprise of the linear version
< A, B > (eqn. 1) and non-linear versions ?l (A, B), Syml (A, B) (eqns. 2,3) which are
group projections of the general kernel < A?d , B ?d >. These different similarity functions are controlled by the choice of three items: Gr , Fr and the parameter d representing
the degree of the tensor product operator. Specifically, we will focus on the case Gr = I
? r is fairly inand on ?d (A, B) as a representative of the non-linear family. The role of G
teresting as it can be viewed as a projection operator from ?parts? to prototypical parts that
can be learned from a training set but we leave this to the full length article that will appear
later.
Practically, to compute ?d (A, B) one needs to run over all d ? d blocks of the k ? q matrix A> B (whose entries are k(ai , bj )) and for each block compute the determinant. The
similarity function is a weighted sum of all those determinants weighted by fij . By appropriate selection of F one can control both the complexity (avoid running over all possible
d ? d blocks) of the computation and the degree of interaction between the determinants.
These determinants have an interesting geometric interpretation if those are computed over
unitary matrices ? as described next.
Let A = QA RA and B = QB RB be the QR factorization of the matrices, i.e., QA has
orthonormal columns which span the column space of A, then it has been recently shown
?1
[14] that RA
can be computed from A using only operations over k(ai , aj ). Therefore,
?T >
?1
the product Q>
A QB , which is equal to RA A BRB , can be computed using only local
?1
kernel applications. In other words, for each A compute RA
(can be done using only
inner-products over columns of A), then when it comes to compute A> B compute in?T >
?1
stead RA
A BRB
which is equivalent to computing Q>
A QB . Thus effectively we have
replaced every A with QA (unitary matrix).
Now, ?d (QA , QB ) for unitary matrices is the sum over the product of the cosine principal
angles between d-dim subspaces spanned by columns of A and B. The value of each
determinant of the d ? d blocks of Q>
A QB is equal to the product of the cosine principal
angles between the respective d-dim subspaces determined by corresponding selection of
d columns from A and d columns from B. For example, the case k = q = d produces
>
?d (QA , QB ) = det(Q>
A QB ) which is the product of the eigenvalues of the matrix QA QB .
Those eigenvalues are the cosine of the principal angles between the column space of A
and the column space of B [2]. Therefore, det(Q>
A QB ) measures the ?angle? between the
two subspaces spanned by the respective columns of the input matrices ? in particular is
invariant to the order of the columns. For smaller values of d we obtain the sum over such
products between subspaces spanned by subsets of d columns between A and B.
The advantage of smaller values of d is two fold: first it enables to compute the similarity
when k 6= q and second breaks down the similarity between subspaces into smaller pieces.
The entries of the matrix F determine which subspaces are being considered and the interaction between subspaces in A and B. A diagonal F compares corresponding subspaces
(a)
(b)
Figure 1: (a) The configuration of the nine sub-regions is displayed over the gradient image. (b)
some of the positive examples ? note the large variation in appearance, pose and articulation.
between A and B whereas off-diagonal entries would enable comparisons between different choices of subspaces in A and in B. For example, we may want to consider choices
of d columns arranged in a ?sliding? fashion, i.e., column sets {1, .., d}, {2, ..., d + 1}, ...
and so forth, instead of the combinatorial number of all possible choices. This selection
is associated with a sparse diagonal F where the non-vanishing entries along the diagonal
have the value of ?1? and correspond to the sliding window selections.
To conclude, in the linear version < A, B > the role of F is to determine the range of
interaction between columns of A and columns of B, whereas with the non-linear version
it is the interaction between d-dim subspaces rather than individual columns. We could
select all possible interactions (exponential number) or any reduced interaction set such as
the sliding window rule (linear number of choices) as described above.
4
Experiments
We examined the performance of sim(A, B) on part-based representations for pedestrian
detection using SVM for the inference engine. The dataset we used (courtesy of Mobileye
Ltd.) covers a challenging variability of appearance, viewing position and body articulation
(see Fig. 1). We ran a suit of comparative experiments using sim(A, B) =< A, B >
with three versions of F = {I, 11> , decay} with local kernels covering linear, d?th degree
polynomial (d = 2, 6) and RBF kernel, and likewise with sim(A, B) = ?d (A, B) with d =
2, sparse diagonal F (covering a sliding window configuration) and with linear, polynomial
and RBF local kernels. We compared our results to the conventional down-sampled holistic
representation where the raw images were down-sampled to size 20 ? 20 and 32 ? 32.
Our tests also included simulation of occlusions (in the test images) in order to examine
the sensitivity of our sim(A, B) family to occlusions. For the local part representation,
the input image was divided into 9 fixed regions where for each image local orientation
statistics were were generated following [5, 7] with a total of 22 numbers per region (see
Fig 1a), thereby making a 22 ? 9 matrix representation to be fed into sim(A, B). The
size of the training set was 4000 split evenly between positive and negative examples and
a test set of 4000 examples was used to evaluate the performance of each trial. The table
below summarizes the accuracy results for the raw-pixel (holistic) representation over three
trials: (i) images down-sampled to 20 ? 20, (ii) images down-sampled to 32 ? 32, and
(iii) test images were partially occluded (32 ? 32 version). The accuracy figures are the
ratio between the sum of the true positives and true negatives and the total number of test
examples.
20 ? 20
32 ? 32
occlusion
raw linear
78%
78%
73.5%
poly d = 2
83%
84%
72%
poly d = 6
84%
85%
77%
RBF
86%
82%
76.5%
The table below displays sim(A, B) with linear and RBF local kernels.
local kernel
linear
RBF
< A, B >, F = I
< A, B >, F = 11>
< A, B >, fij = 2?|i?j|
?2 (A, B)
90.8%
91.2%
85%
85%
90.6%
90.4%
88%
90%
One can see that the local part representation provides a sharp increase in accuracy compared to the raw pixel holistic representation. The added power of invariance to order of
parts induced by < A, B >, F = 11> is not required since the parts are aligned and therefore the accuracy is the highest for the linear combination of local RBF < A, B >, F = I.
The same applies for the non-linear version ?d (A, B) ? the additional invariances that
come with a non-linear combination of local parts are apparently not required. The power
of non-linearity associated with the combination of local parts comes to bear when the
test images have occluded parts, i.e., at random one of the columns of the input matrix is
removed (or replaced with a random vector), as shown in the table below:
local kernel
linear
RBF
< A, B >, F = I
?2 (A, B)
62%
83%
87%
88%
One can notice that a linear combination of local parts suffers from reduced accuracy
whereas the non-linear combination maintains a stable accuracy (compare the right-most
columns of the two tables above). Although the experiments above are still preliminary
they show the power and potential of the sim(A, B) family of kernels defined over local
kernels. With the principles laid down in Section 3 one can construct a large number (we
touched only a few) of algebraic kernels which combine the local kernels in non-linear
ways thus creating invariances to order and increased performance against occlusion. Further research is required for sifting through the various possibilities with this new family
of kernels and extracting their properties, their invariances and behavior under changing
parameters (Fr , Gr , d).
References
[1] A.C. Aitken. Determinants and Matrices. Interscience Publishers Inc., 4th edition, 1946.
[2] G.H. Golub and C.F. Van Loan. Matrix computations. John Hopkins University Press, 1989.
[3] L. Gurvits. Classical deterministic complexity of edmonds? problem and quantum entanglement. In ACM Symp. on Theory of Computing, 2003.
[4] R. Kondor and T. Jebara. A kernel between sets of vectors. In International Conference on
Machine Learning, ICML, 2003.
[5] D.G. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of
Computer Vision, 2004.
[6] C. Schmidt and R. Mohr. Local grey-value invariants for image retrieval. IEEE Transactions
on Pattern Analysis and Machine Intelligence, 19(5):530?535, 1997.
[7] A. Shashua, Y. Gdalyahu, G. Hayun and L. Mann. ?Pedestrian Detection for Driving Assistance
Systems?. IEEE Intelligent Vehicles Symposium (IV2004), June. 2004, Parma Italy.
[8] B. Sch?olkopf and A.J. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002.
[9] G. Shakhnarovich, J.W. Fisher, and T. Darrell. Face recognition from long-term observations.
In Proceedings of the European Conference on Computer Vision, 2002.
[10] S. Ullman, M. Vidal-Naquet, and E. Sali. Visual features of intermediate complexity and their
use in classification. Nature Neuroscience, 5(7):1?6, 2002.
[11] V.N. Vapnik. The nature of statistical learning. Springer, 2nd edition, 1998.
[12] N. Vasconcelos, P. Ho, and P. Moreno. The kullback-leibler kernel as a framework for discriminant and localized representations for visual recognition. In Proceedings of the European
Conference on Computer Vision, pages 430?441, Prague, Czech Republic, May 2004.
[13] C. Wallraven, B. Caputo, and A. Graf. Recognition with local features: the kernel recipe. In
Proceedings of the International Conference on Computer Vision, 2003.
[14] L. Wolf and A. Shashua. Learning over sets using kernel principal angles. Journal of Machine
Learning Research (JMLR), 4(10):913?931, 2003.
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1,875 | 2,702 | Nonparametric Transforms of Graph Kernels
for Semi-Supervised Learning
Xiaojin Zhu?
?
Jaz Kandola?
School of Computer Science
Carnegie Mellon University
5000 Forbes Avenue
Pittsburgh, PA 15213 USA
Zoubin Ghahramani??
?
John Lafferty?
Gatsby Computational Neuroscience Unit
University College London
17 Queen Square
London WC1N 3AR UK
Abstract
We present an algorithm based on convex optimization for constructing
kernels for semi-supervised learning. The kernel matrices are derived
from the spectral decomposition of graph Laplacians, and combine labeled and unlabeled data in a systematic fashion. Unlike previous work
using diffusion kernels and Gaussian random field kernels, a nonparametric kernel approach is presented that incorporates order constraints
during optimization. This results in flexible kernels and avoids the need
to choose among different parametric forms. Our approach relies on
a quadratically constrained quadratic program (QCQP), and is computationally feasible for large datasets. We evaluate the kernels on real
datasets using support vector machines, with encouraging results.
1
Introduction
Semi-supervised learning has been the focus of considerable recent research. In this learning problem the data consist of a set of points, with some of the points labeled and the
remaining points unlabeled. The task is to use the unlabeled data to improve classification
performance. Semi-supervised methods have the potential to improve many real-world
problems, since unlabeled data are often far easier to obtain than labeled data.
Kernel-based methods are increasingly being used for data modeling and prediction because of their conceptual simplicity and good performance on many tasks. A promising
family of semi-supervised learning methods can be viewed as constructing kernels by transforming the spectrum of a ?local similarity? graph over labeled and unlabeled data. These
kernels, or regularizers, penalize functions that are not smooth over the graph [7]. Informally, a smooth eigenvector has the property that two elements of the vector have similar
values if there are many large weight paths between them on the graph. This results in the
desirable behavior of the labels varying smoothly over the graph, as sought by, e.g., spectral
clustering approaches [2], diffusion kernels [5], and the Gaussian random field approach
[9]. However, the modification to the spectrum, called a spectral transformation, is often
a function chosen from some parameterized family. As examples, for the diffusion kernel
the spectral transformation is an exponential function, and for the Gaussian field kernel the
transformation is a smoothed inverse function.
In using a parametric approach one faces the difficult problem of choosing an appropriate
family of spectral transformations. For many familes the number of degrees of freedom
in the parameterization may be insufficient to accurately model the data. In this paper
we propose an effective nonparametric method to find an optimal spectral transformation
using kernel alignment. The main advantage of using kernel alignment is that it gives
us a convex optimization problem, and does not suffer from poor convergence to local
minima. A key assumption of a spectral transformation is monotonicity, so that unsmooth
functions over the data graph are penalized more severly. We realize this property by
imposing order constraints. The optimization problem in general is solved using semidefinite programming (SDP) [1]; however, in our approach the problem can be formulated
in terms of quadratically constrained quadratic programming (QCQP), which can be solved
more efficiently than a general SDP.
This paper is structured as follows. In Section 2 we review some graph theoretic concepts
and relate them to the construction of kernels for semi-supervised learning. In Section 3
we introduce convex optimization via QCQP and relate it to the more familiar linear and
quadratic programming commonly used in machine learning. Section 4 poses the problem
of kernel based semi-supervised learning as a QCQP problem with order constraints. Experimental results using the proposed optimization framework are presented in Section 5.
The results indicate that the semi-supervised kernels constructed from the learned spectral
transformations perform well in practice.
2
Semi-supervised Kernels from Graph Spectra
We are given a labeled dataset consisting of input-output pairs {(x1 , y1 ), . . . , (xl , yl )} and
a (typically much larger) unlabeled dataset {xl+1 , . . . , xn } where x is in some general
input space and y is potentially from multiple classes. Our objective is to construct a kernel
that is appropriate for the classification task. Since our methods use both the labeled and
unlabeled data, we will refer to the resulting kernels as semi-supervised kernels. More
specifically, we restrict ourselves to the transductive setting where the unlabeled data also
serve as the test data. As such, we only need to find a good Gram matrix on the points
{x1 , . . . , xn }. For this approach to be effective such kernel matrices must also take into
account the distribution of unlabeled data, in order that the unlabeled data can aid in the
classification task. Once these kernel matrices have been constructed, they can be deployed
in standard kernel methods, for example support vector machines.
In this paper we motivate the construction of semi-supervised kernel matrices from a
graph theoretic perspective. A graph is constructed where the nodes are the data instances
{1, . . . , n} and an edge connects nodes i and j if a ?local similarity? measure between xi
and xj suggests they may have the same label. For example, the local similarity measure
can be the Euclidean distance between feature vectors if x ? Rm , and each node can connect to its k nearest neighbors with weight value equal to 1. The intuition underlying the
graph is that even if two nodes are not directly connected, they should be considered similar
as long as there are many paths between them. Several semi-supervised learning algorithms
have been proposed under the general graph theoretic theme, based on techniques such as
random walks [8], diffusion kernels [5], and Gaussian fields [9]. Many of these methods
can be unified into the regularization framework proposed by [7], which forms the basis of
this paper.
The graph can be represented by an n ? n weight matrix W = [wij ] where wij is the edge
weight between nodes i and j, with wij = 0 if there is no edge. We require the entries of W
to be non-negative, and assume that it forms a symmetric matrix; it is not necessary for W
itself to be positive semi-definite. In semi-supervised learning W is an essential quantity;
we assume it is provided by domain experts,
P and hence do not study its construction. Let
D be a diagonal matrix where dii =
j wij is the degree of node i. This allows us
to define the combinatorial graph Laplacian as L = D ? W (the normalized Laplacian
? = D?1/2 LD?1/2 can be used as well). We denote L?s eigensystem by {?i , ?i }, so
L
Pn
>
that L =
i=1 ?i ?i ?i where we assume the eigenvalues are sorted in non-decreasing
order. The matrix L has many interesting properties [3]; for instance, it is always positive
semi-definite, even if W is not. Perhaps the most important property of the Laplacian
related to semi-supervised learning is the following: a smaller P
eigenvalue ? corresponds
to a smoother eigenvector ? over the graph; that is, the value ij wij (?(i) ? ?(j))2 is
small. In a physical system the smoother eigenvectors correspond to the major vibration
modes. Assuming the graph structure is correct, from a regularization perspective we want
to encourage smooth functions, to reflect our belief that labels should vary slowly over the
graph. Specifically, [2] and [7] suggest a general principle for creating a semi-supervised
kernel K from the graph Laplacian L: transform the eigenvalues ? into r(?), where the
spectral transformation r is a non-negative and decreasing function1
K=
n
X
r(?i ) ?i ?>
i
(1)
i=1
Note that it may be that r reverses the order of the eigenvalues,
so that smooth ?i ?s have
P
larger eigenvalues in K. A ?soft labeling? function f =
cP
?
i i in a kernel machine has
a penalty term in the RKHS norm given by ?(||f ||2K ) = ?( c2i /r(?i )). Since r is decreasing, a greater penality is incurred for those terms of f corresponding to eigenfunctions
that are less smooth. In previous work r has often been chosen from a parametric family.
2
For example, the diffusion kernel [5] corresponds to r(?) = exp(? ?2 ?) and the Gaussian
1
field kernel [10] corresponds to r(?) = ?+
. Cross validation has been used to find the
hyperparameters ? or for these spectral transformations. Although the general principle
of equation (1) is appealing, it does not address question of which parametric family to use
for r. Moreover, the number of degrees of freedom (or the number of hyperparameters)
may not suit the task at hand, resulting in overly constrained kernels. The contribution of
the current paper is to address these limitations using a convex optimization approach by
imposing an ordering constraint on r but otherwise not assuming any parametric form for
the kernels.
3
Convex Optimization using QCQP
Let Ki = ?i ?>
matrices of the eigenvectors. The semii , i = 1 ? ? ? n be the outer productP
n
supervised kernel K is a linear combination K = i=1 ?i Ki , where ?i ? 0. We formulate
the problem of finding the spectral transformation as one that finds the interpolation coefficients {r(?i ) = ?i } by optimizing some convex objective function on K. To maintain the
positive semi-definiteness constraint on K, one in general needs to invoke SDPs [1]. Semidefinite optimization can be described as the problem of optimizing a linear function of a
symmetric matrix subject to linear equality constraints and the condition that the matrix be
positive semi-definite. The well-known linear programming problem can be generalized
to a semi-definite optimization by replacing the vector of variables with a symmetric matrix, and replacing the non-negativity constraints with a positive semi-definite constraints.
This generalization inherits several properties: it is convex, has a rich duality theory and
allows theoretically efficient solution algorithms based on iterating interior point methods
to either follow a central path or decrease a potential function. However, a limitation of
SDPs is their computational complexity [1], which has restricted their application to small
scale problems [6]. However, an important special case of SDPs are quadratically constrained quadratic programs (QCQP) which are computationally more efficient. Here both
the objective function and the constraints are quadratic as illustrated below,
minimize
subject to
1
1 >
x P0 x + q0> x + r0
2
1 >
x Pi x + qi> x + ri ? 0
2
Ax = b
(2)
i = 1???m
We use a slightly different notation where r is the inverse of that in [7].
(3)
(4)
n
n
where Pi ? S+
, i = 1, . . . , m, where S+
defines the set of square symmetric positive
semi-definite matrices. In a QCQP, we minimize a convex quadratic function over a feasible
region that is the intersection of ellipsoids. The number of iterations required to reach the
solution is comparable to the number required for linear programs, making the approach
feasible for large datasets. However, as observed in [1], not all SDPs can be relaxed to
QCQPs. For the semi-supervised kernel learning task presented here solving an SDP would
be computationally infeasible.
Recent work [4, 6] has proposed kernel target alignment that can be used not only to assess
the relationship between the feature spaces generated by two different kernels, but also to
assess the similarity between spaces induced by a kernel and that induced by the labels
themselves. Desirable properties of the alignment measure can be found in [4]. The crucial aspect of alignnement for our purposes is that its optimization can be formulated as a
QCQP. The objective function is the empirical kernel alignment score:
hKtr , T iF
? tr , T ) = p
A(K
hKtr , Ktr iF hT, T iF
(5)
where Ktr is the kernel matrix restricted to the training points,
P hM, N iF denotes the Frobenius product between two square matrices hM, N iF = ij mij nij = T r(M N > ), and T
is the target matrix on training data, with entry Tij set to +1 if yi = yj and ?1 otherwise.
Note for binary {+1, ?1} training labels y this is simply the rank one matrix T = yy> . K
is guaranteed to be positive semi-definite by constraining ?i ? 0. Previous work using kernel alignment did not take into account that the Ki ?s were derived from the graph Laplacian
with the goal of semi-supervised learning. As such, the ?i ?s can take arbitrary values and
there is no preference to penalize components that do not vary smoothly over the graph.
This can be rectified by requiring smoother eigenvectors to receive larger coefficients, as
shown in the next section.
4
Semi-Supervised Kernels with Order Constraints
As stated above, we would like to maintain a decreasing order on the spectral transformation ?i = r(?i ) to encourage smooth functions over the graph. This motivates the set of
order constraints
?i ? ?i+1 ,
i = 1???n ? 1
(6)
And we can specify the desired semi-supervised kernel as follows.
Definition 1 An order constrained semi-supervised kernel K is the solution to the following convex optimization problem:
maxK
subject to
? tr , T )
A(K
Pn
K = i=1 ?i Ki
?i ? 0
trace(K) = 1
?i ? ?i+1 ,
i = 1???n ? 1
(7)
(8)
(9)
(10)
(11)
where T is the training target matrix, Ki = ?i ?>
i and ?i ?s are the eigenvectors of the graph
Laplacian.
The formulation is an extension to [6] with order constraints, and with special components
Ki ?s from the graph Laplacian. Since ?i ? 0 and Ki ?s are outer products, K will automatically be positive semi-definite and hence a valid kernel matrix. The trace constraint is
needed to fix the scale invariance of kernel alignment. It is important to notice the order
constraints are convex, and as such the whole problem is convex. Let vec(A) be the column
vectorization of a matrix A. Defining M = vec(K1,tr ) ? ? ? vec(Km,tr ) , it is not hard to
show that the problem can then be expressed as
max?
subject to
vec(T )> M ?
||M ?|| ? 1
?i ? 0
?i ? ?i+1 ,
i = 1???n ? 1
(12)
(13)
(14)
(15)
The objective function is linear in ?, and there is a simple cone constraint, making it a
quadratically constrained quadratic program (QCQP).
An improvement of the above order constrained semi-supervised kernel can be obtained
by studying the Laplacian eigenvectors with zero eigenvalues. For a graph Laplacian there
will be k zero eigenvalues if the graph has k connected subgraphs. The k eigenvectors are
piecewise constant over individual subgraphs, and zero elsewhere. This is desirable when
k > 1, with the hope that subgraphs correspond to different classes. However if k = 1, the
graph is connected. The first eigenvector ?1 is a constant vector. The corresponding K1 is
a constant matrix, and acts as a bias term. In this situation we do not want to impose the
order constraint ?1 ? ?2 on the constant bias term. Instead we let ?1 vary freely during
optimization.
Definition 2 An improved order constrained semi-supervised kernel K is the solution to
the same problem in Definition 1, but the order constraints (11) apply only to non-constant
eigenvectors:
?i ? ?i+1 ,
i = 1 ? ? ? n ? 1, and ?i not constant
(16)
In practice we do not need all n eigenvectors of the graph Laplacian, or equivalently all n
Ki ?s. The first m < n eigenvectors with the smallest eigenvalues work well empirically.
Also note we could have used the fact that Ki ?s are from orthogonal eigenvectors ?i to
further simplify the expression. However we neglect this observation, making it easier to
incorporate other kernel components if necessary.
It is illustrative to compare and contrast the order constrained semi-supervised kernels to
other semi-supervised kernels with different spectral transformation. We call the original
kernel alignment solution in [6] a maximal-alignment kernel. It is the solution to Definition 1 without the order constraints (11). Because it does not have the additional constraints,
it maximizes kernel alignment among all spectral transformation. The hyperparameters ?
and of the Diffusion kernel and Gaussian fields kernel (described earlier) can be learned
by maximizing the alignment score also, although the optimization problem is not necessarily convex. These kernels use different information from the original Laplacian eigenvalues ?i . The maximal-alignment kernels ignore ?i altogether. The order constrained
semi-supervised kernels only use the order of ?i and ignore their actual values. The diffusion and Gaussian field kernels use the actual values. In terms of the degree of freedom in
choosing the spectral transformation ?i ?s, the maximal-alignment kernels are completely
free. The diffusion and Gaussian field kernels are restrictive since they have an implicit
parametric form and only one free parameter. The order constrained semi-supervised kernels incorporates desirable features from both approaches.
5
Experimental Results
We evaluate the order constrained kernels on seven datasets. baseball-hockey (1993 instances / 2 classes), pc-mac (1943/2) and religion-atheism (1427/2) are document categorization tasks taken from the 20-newsgroups dataset. The distance measure is the standard
cosine similarity between tf.idf vectors. one-two (2200/2), odd-even (4000/2) and ten
digits (4000/10) are handwritten digits recognition tasks. one-two is digits ?1? vs. ?2?;
odd-even is the artificial task of classifying odd ?1, 3, 5, 7, 9? vs. even ?0, 2, 4, 6, 8? digits,
such that each class has several well defined internal clusters; ten digits is 10-way classification. isolet (7797/26) is isolated spoken English alphabet recognition from the UCI
repository. For these datasets we use Euclidean distance on raw features. We use 10NN
unweighted graphs on all datasets except isolet which is 100NN. For all datasets, we use
the smallest m = 200 eigenvalue and eigenvector pairs from the graph Laplacian. These
values are set arbitrarily without optimizing and do not create a unfair advantage to the
proposed kernels. For each dataset we test on five different labeled set sizes. For a given
labeled set size, we perform 30 random trials in which a labeled set is randomly sampled
from the whole dataset. All classes must be present in the labeled set. The rest is used as
unlabeled (test) set in that trial. We compare 5 semi-supervised kernels (improved order
constrained kernel, order constrained kernel, Gaussian field kernel, diffusion kernel2 and
maximal-alignment kernel), and 3 standard supervised kernels (RBF (bandwidth learned
using 5-fold cross validation),linear and quadratic). We compute the spectral transformation for order constrained kernels and maximal-alignment kernels by solving the QCQP
using standard solvers (SeDuMi/YALMIP). To compute accuracy we use a standard SVM.
We choose the the bound on slack variables C with cross validation for all tasks and kernels. For multiclass classification we perform one-against-all and pick the class with the
largest margin.
The results3 are shown in Table 1, which has two rows for each cell: The upper row is
the average test set accuracy with one standard deviation; The lower row is the average
training set kernel alignment, and in parenthesis the average run time in seconds for SeDuMi/YALMIP on a 3GHz Linux computer. Each number is averaged over 30 random
trials. To assess the statistical significance of the results, we perform paired t-test on test
accuracy. We highlight the best accuracy in each row, and those that can not be determined
as different from the best, with paired t-test at significance level 0.05. The semi-supervised
kernels tend to outperform standard supervised kernels. The improved order constrained
kernels are consistently among the best. Figure 1 shows the spectral transformation ?i of
the semi-supervised kernels for different tasks. These are for the 30 trials with the largest
labeled set size in each task. The x-axis is in increasing order of ?i (the original eigenvalues
of the Laplacian). The mean (thick lines) and ?1 standard deviation (dotted lines) of only
the top 50 ?i ?s are plotted for clarity. The ?i values are scaled vertically for easy comparison among kernels. As expected the maximal-alignment kernels? spectral transformation is
zigzagged, diffusion and Gaussian field?s are very smooth, while order constrained kernels?
are in between. The order constrained kernels (green) have large ?1 because of the order
constraint. This seems to be disadvantageous ? the spectral transformation tries to balance
it out by increasing the value of other ?i ?s so that the constant K1 ?s relative influence is
smaller. On the other hand the improved order constrained kernels (black) allow ?1 to be
small. As a result the rest ?i ?s decay fast, which is desirable.
6
Conclusions
We have proposed and evaluated a novel approach for semi-supervised kernel construction
using convex optimization. The method incorporates order constraints, and the resulting
convex optimization problem can be solved efficiently using a QCQP. In this work the base
kernels were derived from the graph Laplacian, and no parametric form for the spectral
transformation was imposed, making the approach more general than previous approaches.
Experiments show that the method is both computationally feasible and results in improvements to classification performance when used with support vector machines.
2
The hyperparameters ? 2 and are learned with the fminbnd() function in Matlab to maximize
kernel alignment.
3
Results on baseball-hockey and odd-even are similar and omitted for space. Full results can be
found at http://www.cs.cmu.edu/?zhuxj/pub/ocssk.pdf
PC vs. MAC
Religion vs. Atheism
1
1
Improved order
Order
Max?align
Gaussian field
Diffusion
0.9
0.8
0.7
0.7
0.6
0.6
? scaled
? scaled
0.8
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0
Improved order
Order
Max?align
Gaussian field
Diffusion
0.9
0.1
0
5
10
15
20
25
30
35
40
45
0
50
0
5
10
15
20
rank
Ten Digits (10 classes)
35
40
45
50
1
Improved order
Order
Max?align
Gaussian field
Diffusion
0.9
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
5
10
15
20
25
rank
30
35
40
45
Improved order
Order
Max?align
Gaussian field
Diffusion
0.9
? scaled
? scaled
30
ISOLET (26 classes)
1
0
25
rank
50
0
0
5
10
15
20
25
30
35
40
45
50
rank
Figure 1: Comparison of spectral transformation for the 5 semi-supervised kernels.
References
[1] S. Boyd and L. Vandenberge. Convex Optimization. Cambridge University Press, Cambridge
UK, 2004.
[2] O. Chapelle, J. Weston, and B. Sch?olkopf. Cluster kernels for semi-supervised learning. In
Advances in Neural Information Processing Systems, 15, volume 15, 2002.
[3] F. R. K. Chung. Spectral graph theory, Regional Conference Series in Mathematics, No. 92.
American Mathematical Society, 1997.
[4] N. Cristianini, J. Shawe-Taylor, A. Elisseeff, and J. Kandola. On kernel-target alignment. In
Advances in NIPS, 2001.
[5] R. I. Kondor and J. Lafferty. Diffusion kernels on graphs and other discrete input spaces. In
Proc. 19th International Conf. on Machine Learning, 2002.
[6] G. Lanckriet, N. Cristianini, P. Bartlett, L. E. Ghaoui, and M. Jordan. Learning the kernel matrix
with semidefinite programming. Journal of Machine Learning Research, 5:27?72, 2004.
[7] A. Smola and R. Kondor. Kernels and regularization on graphs. In Conference on Learning
Theory, COLT/KW, 2003.
[8] M. Szummer and T. Jaakkola. Partially labeled classification with Markov random walks. In
Advances in Neural Information Processing Systems, 14, volume 14, 2001.
[9] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and
harmonic functions. In ICML-03, 20th International Conference on Machine Learning, 2003.
[10] X. Zhu, J. Lafferty, and Z. Ghahramani. Semi-supervised learning: From Gaussian fields to
Gaussian processes. Technical Report CMU-CS-03-175, Carnegie Mellon University, 2003.
Training
set size
Improved
Order
pc-mac
10
87.0 ? 5.0
0.71 ( 1)
30
90.3 ? 1.3
0.68 ( 8)
50
91.3 ? 0.9
0.64 (31)
70
91.5 ? 0.6
0.63 (70)
90
91.5 ? 0.6
0.63 (108)
religion-atheism
10
72.8 ?11.2
0.50 ( 1)
30
84.2 ? 2.4
0.38 ( 8)
50
84.5 ? 2.3
0.31 (28)
70
85.7 ? 1.4
0.29 (55)
90
86.6 ? 1.3
0.27 (86)
one-two
10
96.2 ? 2.7
0.87 ( 2)
20
96.4 ? 2.8
0.87 ( 3)
30
98.2 ? 2.1
0.84 ( 8)
40
98.3 ? 1.9
0.84 (13)
50
98.4 ? 1.9
0.83 (31)
Ten digits (10 classes)
50
76.6 ? 4.3
0.47 (26)
100
84.8 ? 2.6
0.47 (124)
150
86.5 ? 1.7
0.48 (310)
200
88.1 ? 1.3
0.47 (708)
250
89.1 ? 1.1
0.47 (942)
isolet (26 classes)
50
56.0 ? 3.5
0.27 (26)
100
64.6 ? 2.1
0.26 (105)
150
67.6 ? 2.6
0.26 (249)
200
71.0 ? 1.8
0.26 (441)
250
71.8 ? 2.3
0.26 (709)
Order
semi-supervised kernels
Gaussian
Diffusion
Field
Max-align
RBF
standard kernels
Linear
Quadratic
84.9 ? 7.2
0.57 ( 1)
89.6 ? 2.3
0.49 ( 8)
90.5 ? 1.7
0.46 (31)
90.8 ? 1.3
0.46 (56)
91.3 ? 1.3
0.45 (98)
56.4 ? 6.2
0.32
76.4 ? 6.1
0.19
81.1 ? 4.6
0.16
84.6 ? 2.1
0.14
86.3 ? 2.3
0.13
57.8 ?11.5
0.35
79.6 ?11.2
0.23
87.5 ? 2.8
0.20
90.5 ? 1.2
0.19
91.3 ? 1.1
0.18
71.1 ? 9.7
0.90 ( 1)
85.4 ? 3.9
0.74 ( 6)
88.4 ? 2.1
0.68 (25)
89.6 ? 1.6
0.66 (59)
90.3 ? 1.0
0.65 (84)
51.6 ? 3.4
0.11
62.6 ? 9.6
0.03
67.8 ? 9.0
0.02
74.7 ? 7.4
0.01
79.0 ? 6.4
0.01
63.0 ? 5.1
0.30
71.8 ? 5.5
0.18
77.6 ? 4.8
0.14
80.2 ? 4.6
0.12
82.5 ? 4.2
0.11
62.3 ? 4.2
0.25
71.2 ? 5.3
0.13
75.7 ? 5.4
0.10
74.3 ? 8.7
0.08
79.1 ? 7.3
0.08
70.9 ?10.9
0.42 ( 1)
83.0 ? 2.9
0.31 ( 6)
83.5 ? 2.5
0.26 (23)
85.3 ? 1.6
0.25 (42)
86.4 ? 1.5
0.24 (92)
55.2 ? 5.8
0.31
71.2 ? 6.3
0.20
80.4 ? 4.1
0.17
83.0 ? 2.9
0.16
84.5 ? 2.1
0.15
60.9 ?10.7
0.31
80.3 ? 5.1
0.22
83.5 ? 2.7
0.20
85.4 ? 1.8
0.19
86.2 ? 1.6
0.18
60.7 ? 7.5
0.85 ( 1)
74.4 ? 5.4
0.60 ( 7)
77.4 ? 6.1
0.48 (27)
82.3 ? 3.0
0.43 (51)
82.8 ? 2.6
0.40 (85)
55.8 ? 5.8
0.13
63.4 ? 6.5
0.05
69.3 ? 6.5
0.04
73.1 ? 5.8
0.03
77.7 ? 5.1
0.02
60.1 ? 7.0
0.30
63.7 ? 8.3
0.18
69.4 ? 7.0
0.15
75.7 ? 6.0
0.13
74.6 ? 7.6
0.12
61.2 ? 4.8
0.26
70.1 ? 6.3
0.15
70.7 ? 8.5
0.11
71.0 ?10.0
0.10
70.0 ?11.5
0.09
90.6 ?14.0
0.66 ( 1)
93.9 ? 8.7
0.64 ( 4)
97.2 ? 2.5
0.61 ( 7)
96.5 ? 2.4
0.61 (15)
95.6 ? 9.0
0.60 (37)
58.2 ?17.6
0.43
87.0 ?16.0
0.38
98.1 ? 2.2
0.35
98.9 ? 1.8
0.36
99.4 ? 0.5
0.35
59.4 ?18.9
0.53
83.2 ?19.8
0.50
98.1 ? 2.7
0.47
99.1 ? 1.4
0.48
99.6 ? 0.3
0.46
85.4 ?11.5
0.95 ( 1)
94.5 ? 1.6
0.90 ( 3)
96.4 ? 2.1
0.86 ( 6)
96.3 ? 2.3
0.86 (11)
96.6 ? 2.3
0.84 (25)
78.7 ?14.3
0.38
90.4 ? 4.6
0.33
93.6 ? 3.1
0.30
94.0 ? 2.7
0.29
96.1 ? 2.4
0.28
85.1 ? 5.7
0.26
86.0 ? 9.4
0.22
89.6 ? 5.9
0.17
91.6 ? 6.3
0.18
93.0 ? 3.6
0.17
85.7 ? 4.8
0.30
90.9 ? 3.7
0.25
92.9 ? 2.8
0.24
94.9 ? 2.0
0.21
95.8 ? 2.3
0.20
71.5 ? 5.0
0.21 (26)
83.4 ? 2.6
0.17 (98)
86.4 ? 1.3
0.18 (255)
88.0 ? 1.3
0.16 (477)
89.3 ? 1.0
0.16 (873)
41.4 ? 6.8
0.15
63.7 ? 3.5
0.12
75.1 ? 3.0
0.11
80.4 ? 2.5
0.10
84.6 ? 1.4
0.10
49.8 ? 6.3
0.16
72.5 ? 3.3
0.13
80.4 ? 2.1
0.13
84.4 ? 1.6
0.11
87.2 ? 1.3
0.11
70.3 ? 5.2
0.51 (25)
80.7 ? 2.6
0.49 (100)
84.5 ? 1.9
0.50 (244)
86.0 ? 1.5
0.49 (523)
87.2 ? 1.3
0.49 (706)
57.0 ? 4.0
-0.62
69.4 ? 1.9
-0.64
75.2 ? 1.4
-0.66
78.3 ? 1.3
-0.65
80.4 ? 1.4
-0.65
50.2 ? 9.0
-0.50
56.0 ? 7.8
-0.52
56.2 ? 7.2
-0.53
60.8 ? 7.3
-0.54
61.3 ? 7.6
-0.54
66.3 ? 3.7
-0.25
77.2 ? 2.3
-0.29
81.4 ? 2.2
-0.31
84.3 ? 1.7
-0.33
85.7 ? 1.3
-0.33
42.0 ? 5.2
0.13 (25)
59.0 ? 3.6
0.10 (127)
65.2 ? 3.0
0.09 (280)
70.9 ? 2.3
0.08 (570)
73.6 ? 1.5
0.08 (836)
41.2 ? 2.9
0.03
58.5 ? 2.9
-0.02
65.4 ? 2.6
-0.05
70.6 ? 1.9
-0.07
73.7 ? 1.2
-0.07
29.0 ? 2.7
0.11
47.4 ? 2.7
0.08
57.2 ? 2.7
0.07
64.8 ? 2.1
0.06
69.8 ? 1.5
0.06
50.1 ? 3.7
0.31 (24)
63.2 ? 1.9
0.29 (102)
67.9 ? 2.5
0.27 (221)
72.3 ? 1.7
0.27 (423)
74.2 ? 1.5
0.27 (665)
28.7 ? 2.0
-0.89
46.3 ? 2.4
-0.90
57.6 ? 1.5
-0.90
63.9 ? 1.6
-0.91
68.8 ? 1.5
-0.91
30.0 ? 2.7
-0.80
46.6 ? 2.7
-0.82
57.3 ? 1.8
-0.83
64.2 ? 2.0
-0.83
69.5 ? 1.7
-0.84
23.7 ? 2.4
-0.65
42.0 ? 2.9
-0.69
53.8 ? 2.2
-0.70
60.5 ? 1.6
-0.72
66.2 ? 1.4
-0.72
Table 1: Accuracy, alignment scores, and run times on the datasets. The table compares 8
kernels. Each cell has two rows: The upper row is test set accuracy with standard error;
the lower row is training set alignment (SeDuMi/YALMIP run time in seconds is given in
parentheses). All numbers are averaged over 30 random trials. Accuracies in boldface are
the best as determined by a paired t-test at the 0.05 significance level.
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1,876 | 2,703 | The cerebellum chip:
an analog VLSI implementation of a
cerebellar model of classical conditioning
Constanze Hofst?tter, Manuel Gil, Kynan Eng,
Giacomo Indiveri, Matti Mintz, J?rg Kramer* and Paul F. M. J. Verschure
Institute of Neuroinformatics
University/ETH Zurich
CH-8057 Zurich, Switzerland
[email protected]
Abstract
We present a biophysically constrained cerebellar model of
classical conditioning, implemented using a neuromorphic analog
VLSI (aVLSI) chip. Like its biological counterpart, our cerebellar
model is able to control adaptive behavior by predicting the
precise timing of events. Here we describe the functionality of the
chip and present its learning performance, as evaluated in
simulated conditioning experiments at the circuit level and in
behavioral experiments using a mobile robot. We show that this
aVLSI model supports the acquisition and extinction of adaptively
timed conditioned responses under real-world conditions with
ultra-low power consumption.
I n tro d u cti o n
1
The association of two correlated stimuli, an initially neutral conditioned stimulus
(CS) which predicts a meaningful unconditioned stimulus (US), leading to the
acquisition of an adaptive conditioned response (CR), is one of the most essential
forms of learning. Pavlov introduced the classical conditioning paradigm in the
early 20th century to study associative learning (Pavlov 1927). In classical
conditioning training an animal is repeatedly exposed to a CS followed by a US
after a certain inter-stimulus interval (ISI). The animal learns to elicit a CR
matched to the ISI, reflecting its knowledge about an association between the CS,
US, and their temporal relationship. Our earlier software implementation of a
*
J?rg Kramer designed the cerebellum chip that was first tested at the 2002 Telluride
Neuromorphic Engineering Workshop. Tragically, he died soon afterwards while hiking
on Telescope Peak on 24 July, 2002.
biophysically constrained model of the cerebellar circuit underlying classical
conditioning (Verschure and Mintz 2001; Hofst?tter et al. 2002) provided an
explanation of this phenomenon by assuming a negative feedback loop between the
cerebellar cortex, deep nucleus and inferior olive. It could acquire and extinguish
correctly timed CRs over a range of ISIs in simulated classical conditioning
experiments, as well as in associative obstacle avoidance tasks using a mobile
robot. In this paper we present the analog VLSI (aVLSI) implementation of this
cerebellum model ? the cerebellum chip ? and the results of chip-level and
behavioral robot experiments.
2
T h e mo d el ci r cu i t a n d a VL S I i mp l eme n ta ti o n
Figure 1: Anatomy of the cerebellar model circuit (left) and the block diagram of
the corresponding chip (right).
The model (Figure 1) is based on the identified cerebellar pathways of CS, US and
CR (Kim and Thompson 1997) and includes four key hypotheses which were
implemented in the earlier software model (Hofst?tter et al. 2002):
1.
CS related parallel fiber (pf) and US related climbing fiber (cf) signals
converge at Purkinje cells (PU) in the cerebellum (Steinmetz et al. 1989). The
direction of the synaptic changes at the pf-PU-synapse depends on the temporal
coincidence of pf and cf activity. Long-term depression (LTD) is induced by pf
activity followed by cf activity within a certain time interval, while pf activity
alone induces long-term potentiation (LTP) (Hansel et al. 2001).
2.
A prolonged second messenger response to pf stimulation in the dendrites of
PU constitutes an eligibility trace from the CS pathway (Sutton and Barto
1990) that bridges the ISI (Fiala et al. 1996).
3.
A microcircuit (Ito 1984) comprising PU, deep nucleus (DN) and inferior olive
(IO) forms a negative feedback loop. Shunting inhibition of IO by DN blocks
the reinforcement pathway (Thompson et al. 1998), thus controlling the
induction of LTD and LTP at the pf-PU-synapse.
4.
DN activity triggers behavioral CRs (McCormick and Thompson 1984). The
inhibitory PU controls DN activity by a mechanism called rebound excitation
(Hesslow 1994): When DN cells are disinhibited from PU input, their
membrane potential slowly repolarises and spikes are emitted if a certain
threshold is reached. Thereby, the correct timing of CRs results from the
adaptation of a pause in PU spiking following the CS.
In summary, in the model the expression of a CR is triggered by DN rebound
excitation upon release from PU inhibition. The precise timing of a CR is
dependent on the duration of an acquired pause in PU spiking following a CS. The
PU response is regulated by LTD and LTP at the pf-PU-synapse under the control
of a negative feedback loop comprising DN, PU and IO.
We implemented an analog VLSI version of the cerebellar model using a standard
1.6?m CMOS technology, and occupying an area of approximately 0.25 mm2. A
block diagram of the hardware model is shown in Figure 1. The CS block receives
the conditioned stimulus and generates two signals: an analog long-lasting, slowly
decaying trace (cs_out) and an equally long binary pulse (cs_wind). Similarly, the
US block receives an unconditioned stimulus and generates a fast pulse (us_out).
The two pulses cs_wind and us_out are sent to the LT-ISI block that is responsible
for perfoming LTP and LTD, upregulating or downregulating the synaptic weight
signal w. This signal determines the gain by which the cs_out trace is multiplied in
the MU block. The output of the multiplier MU is sent on to the PU block, together
with the us_out signal. It is a linear integrate-and-fire neuron (the axon-hillock
circuit) connected to a constant current source that produces regular spontaneous
activity. The current source is gated by the digital cf_wind signal, such that the
spontaneous activity is shut off for the duration of the cs_out trace.
The chip allowed one of three learning rules to be connected. Experiments showed
that an ISI-dependent learning rule with short ISIs resulting in the strongest LTD
was the most useful (Kramer and Hofst?tter 2002). Two elements were added to
adapt the model circuit for real-world robot experiments. Firstly, to prevent the
expression of a CR after a US had already been triggered, an inhibitory connection
from IO to CRpathway was added. Secondly, the transduction delay (TD) from the
aVLSI circuit to any effectors (e.g. motor controls of a robot) had to be taken into
account, which was done by adding a delay from DN to IO of 500ms.
The chip?s power consumption is conservatively estimated at around 100 W
(excluding off-chip interfacing), based on measurements from similar integrateand-fire neuron circuits (Indiveri 2003). This figure is an order of magnitude lower
than what could be achieved using conventional microcontrollers (typically 1-10
mW), and could be improved further by optimising the circuit design.
3
S i mu l a ted co n d i ti o n i n g ex p eri men ts
The aim of the ?in vitro? simulated conditioning experiments was to understand the
learning performance of the chip. To obtain a meaningful evaluation of the
performance of the learning system for both the simulated conditioning
experiments and the robot experiments, the measure of effective CRs was used. In
acquisition experiments CS-US pairs are presented with a fixed ISI. Whenever a
CR occurs that precedes the US, the US signal is not propagated to PU due to the
inhibitory connection from DN to IO. Thus in the context of acquisition
experiments a CR is defined as effective if it prevents the occurrence of a US spike
at PU. In contrast, in robot experiments an effective CR is defined at the
behavioral level, including only CRs that prevent the US from occurring.
Figure 2: Learning related response changes in the cerebellar aVLSI chip. The most
relevant neural responses to a CS-US pair (ISI of 3s, ITI of 12s) are presented for a
trial before (naive) significant learning occurred and when a correctly timed CR is
expressed (trained). US-related pf and CS/CR-related cf signals are indicated by
vertical lines passing through the subplots. A CS-related pf-signal evokes a
prolonged response in the pf-PU-synapse, the CS-trace (Trace subplot). While an
active CS-trace is present, an inhibitory element (I) is active which inactivates an
element representing the spontaneous activity of PU (Hofst?tter et al. 2002). (A)
The US-related cf input occurs while there is an active CS-trace (Trace subplot), in
this case following the CS with an ISI of 3s. LTD predominates over LTP under
these conditions (Weight subplot). Because the PU membrane potential (PU)
remains above spiking threshold, PU is active and supplies constant inhibition to
DN (DN) while in the CS-mode. Thus, DN cannot repolarize and remains inactive
so that no CR is triggered. (B) Later in the experiment, the synaptic weight of the
pf-PU-synapse (Weight) has been reduced due to previous LTD. As a result,
following a CS-related pf input, the PU potential (PU subplot) falls below the
spiking threshold, which leads to a pause in PU spiking. The DN membrane
potential repolarises, so that rebound spikes are emitted (DN subplot). This
rebound excitation triggers a CR. DN inhibition of IO prevents US related cfactivity. Thus, although a US signal is still presented to the circuit, the reinforcing
US pathway is blocked. These conditions induce only LTP, raising the synaptic
weight of the pf-PU-synapse (Weight subplot).
The results we obtained were broadly consistent with those reported in the
biological literature (Ito 1984; Kim and Thompson 1997). The correct operation of
the circuit can be seen in the cell traces illustrating the properties of the aVLSI
circuit components before significant learning (Figure 2 A), and after a CR is
expressed (Figure 2B). Long-term acquisition experiments (25 blocks of 10 trials
each over 50 minutes) showed that chip functions remained stable over a long time
period. In each trial the CS was followed by a US with a fixed ISI of 3s; the inter
trial interval (ITI) was 12s. The number of effective CRs shows an initial fast
learning phase followed by a stable phase with higher percentages of effective CRs
(Figure 3B). In the stable phase the percentage of effective CRs per block
fluctuates around 80-90%. There are fluctuations of up to 500ms in the CR latency
caused by the interaction of LTD and LTP in the stable phase, but the average CR
latency remains fairly constant.
Figure 4 shows the average of five acquisition experiments (5 blocks of 10 trials
per experiment) for ISIs of 2.5s, 3s and 3.5s. The curves are similar in shape to the
ones in the long-term experiment. The CR latency quickly adjusts to match the ISI
and remains stable thereafter (Figure 4A). The effect of the ISI-dependent learning
rule can be seen in two ways: firstly, the shorter the ISI, the faster the stable phase
is reached, denoting faster learning. Secondly, the shorter the ISI, the better the
performance in terms of percentage of effective CRs (Figure 4B). The parameters
of the chip were tuned to optimally encode short ISIs in the range of 1.75s to 4.5s.
Separate experiments showed that the chip could also adapt rapidly to changes in
the ISI within this range after initial learning.
(Error bar = 1 std. dev.)
Figure 3: Long-term changes in CR latency (A) and % effective CRs (B) per block
of 10 CSs during acquisition. Experiment length = 50min., ISI = 3s, ITI = 12s.
(Error bar = 1 std. dev.)
Figure 4: Average of five acquisition experiments per block of 10 CSs for ISIs of
2.5s ( ), 3s (*) and 3.5s ( ). (A) Avg. CR latency. (B) Avg. % effective CRs.
4
Ro b o t a s s o ci a ti v e l ea rn i n g ex p eri men t s
The ?in vivo? learning capability of the chip was evaluated by interfacing it to a
robot and observing its behavior in an unsupervised obstacle avoidance task.
Experiments were performed using a Khepera microrobot (K-team, Lausanne,
Switzerland, Figure 5A) in a circular arena with striped walls (Figure 5C). The
robot was equipped with 6 proximal infra-red (IR) sensors (Figure 5B). Activation
of these sensors (US) due to a collision triggered a turn of ~110? in the opposite
direction (UR). A line camera (64 pixels x 256 gray-levels) constituted the distal
sensor, with detection of a certain spatial frequency (~0.14 periods/degree)
signalling the CS. Visual CSs and collision USs were conveyed to CSpathway and
USpathway on the chip. The activation of CRpathway triggered a motor CR: a 1s
long regression followed by a turn of ~180?. Communication between the chip and
the robot was performed using Matlab on a PC. The control program could be
downloaded to the robot's processor, allowing the robot to act fully autonomously.
In each experiment, the robot was placed in the circular arena exploring its
environment with a constant speed of ~4 cm/s. A spatial frequency CS was
detected at some distance when the robot approached the wall, followed by a
collision with the wall, stimulating the IR sensors and thus triggering a US.
Consequently the CS was correlated with the US, predicting it. The ISIs of these
stimuli were variable, due to noise in sensor sampling, and variations in the angle
at which the robot approached the wall.
Figure 5: (A) Khepera microrobot with aVLSI chip mounted on top. (B) Only the
forward sensors were used during the experiments. (C) The environment: a 60cm
diameter circular arena surrounded by a 15cm high wall. A pattern of vertical,
equally sized black and white bars was placed on the wall.
Associative learning mediated by the cerebellum chip significantly altered the
robot's behavior in the obstacle avoidance task (Figure 6) over the course of each
experiment. In the initial learning phase, the behavior was UR driven: the robot
drove forwards until it collided with the wall, only then performing a turn (Figure
6A1). In the trained phase, the robot usually turned just before it collided with the
wall (Figure 6A2), reducing the number of collisions. The positions of the robot
when a CS, US or CR event occurred in these two phases are shown in Figure 6B1
and B2. The CRs were not expressed immediately after the CSs, but rather with a
CR latency adjusted to just prevent collisions (USs). Not all USs were avoided in
the trained phase due to some excessively short ISIs (Figure 7) and normal
extinction processes over many unreinforced trials. After the learning phase the
percentage of effective CRs fluctuated between 70% and 100% (Figure 7).
Figure 6: Learning performance of the robot. (Top row) Trajectories of the robot.
The white circle with the black dot in the center indicates the beginning of
trajectories. (Bottom row) The same periods of the experiment examined at the
circuit level: = CS, * = US, = CR. (A1, B1) Beginning of the experiment (CS
3-15). (A2, B2) Later in the experiment (CS 32-44).
Figure 7: Trends in learning behavior (average of 5 experiments, 25 min. each). 90
CSs were presented in each experiment. Error bars indicate one standard deviation.
(A) Average percentage of effective CRs over 9 blocks of 10 CSs. (B) Number of
CS occurrences ( ), US occurrences (*) and CR occurrences ( ).
5
Di s cu s s i o n
We have presented one of the first examples of a biologically constrained model of
learning implemented in hardware. Our aVLSI cerebellum chip supports the
acquisition and extinction of adaptively timed responses under noisy, real world
conditions. These results provide further evidence for the role of the cerebellar
circuit embedded in a synaptic feedback loop in the learning of adaptive behavior,
and pave the way for the creation of artefacts with embedded ultra low-power
learning capabilities.
6
Ref eren ces
Fiala, J. C., Grossberg, S. and Bullock, D. (1996). Metabotropic glutamate receptor
activation in cerebellar Purkinje cells as substrate for adaptive timing of the classical
conditioned eye-blink response. Journal of Neuroscience 16: 3760-3774.
Hansel, C., Linden, D. J. and D'Angelo, E. (2001). Beyond parallel fiber LTD, the diversity
of synaptic and nonsynaptic plasticity in the cerebellum. Nature Neuroscience 4: 467-475.
Hesslow, G. (1994). Inhibition of classical conditioned eyeblink response by stimulation of
the cerebellar cortex in decerebrate cat. Journal of Physiology 476: 245-256.
Hofst?tter, C., Mintz, M. and Verschure, P. F. M. J. (2002). The cerebellum in action: a
simulation and robotics study. European Journal of Neuroscience 16: 1361-1376.
Indiveri, G. (2003). A low-power adaptive integrate-and-fire neuron circuit. IEEE
International Symposium on Circuits and Systems, Bangkok, Thailand, 4: 820-823.
Ito, M. (1984). The modifiable neuronal network of the cerebellum. Japanese Journal of
Physiology 5: 781-792.
Kim, J. J. and Thompson, R. F. (1997). Cerebellar circuits and synaptic mechanisms
involved in classical eyeblink conditioning. Trends in the Neurosciences 20(4): 177-181.
Kim, J. J. and Thompson, R. F. (1997). Cerebellar circuits and synaptic mechanisms
involved in classical eyeblink conditioning. Trend. Neurosci. 20: 177-181.
Kramer, J. and Hofst?tter, C. (2002). An aVLSI model of cerebellar mediated associative
learning. Telluride Workshop, CO, USA.
McCormick, D. A. and Thompson, R. F. (1984). Neuronal response of the rabbit cerebellum
during acquisition and performance of a classical conditioned nictitating membrane-eyelid
response. J. Neurosci. 4: 2811-2822.
Pavlov, I. P. (1927). Conditioned Reflexes, Oxford University Press.
Steinmetz, J. E., Lavond, D. G. and Thompson, R. F. (1989). Classical conditioning in
rabbits using pontine nucleus stimulation as a conditioned stimulus and inferior olive
stimulation as an unconditioned stimulus. Synapse 3: 225-233.
Sutton, R. S. and Barto, A. G. (1990). Time derivate models of Pavlovian Reinforcement
Learning and Computational Neuroscience: Foundations of Adaptive Networks., MIT press:
chapter 12, 497-537.
Thompson, R. F., Thompson, J. K., Kim, J. J. and Shinkman, P. G. (1998). The nature of
reinforcement in cerebellar learning. Neurobiology of Learning and Memory 70: 150-176.
Verschure, P. F. M. J. and Mintz, M. (2001). A real-time model of the cerebellar circuitry
underlying classical conditioning: A combined simulation and robotics study.
Neurocomputing 38-40: 1019-1024.
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1,877 | 2,704 | VDCBPI: an Approximate Scalable Algorithm
for Large POMDPs
Pascal Poupart
Department of Computer Science
University of Toronto
Toronto, ON M5S 3H5
[email protected]
Craig Boutilier
Department of Computer Science
University of Toronto
Toronto, ON M5S 3H5
[email protected]
Abstract
Existing algorithms for discrete partially observable Markov decision
processes can at best solve problems of a few thousand states due to
two important sources of intractability: the curse of dimensionality and
the policy space complexity. This paper describes a new algorithm
(VDCBPI) that mitigates both sources of intractability by combining the
Value Directed Compression (VDC) technique [13] with Bounded Policy Iteration (BPI) [14]. The scalability of VDCBPI is demonstrated on
synthetic network management problems with up to 33 million states.
1 Introduction
Partially observable Markov decision processes (POMDPs) provide a natural and expressive framework for decision making, but their use in practice has been limited by the lack
of scalable solution algorithms. Two important sources of intractability plague discrete
model-based POMDPs: high dimensionality of belief space, and the complexity of policy
or value function (VF) space. Classic solution algorithms [4, 10, 7], for example, compute
value functions represented by exponentially many value vectors, each of exponential size.
As a result, they can only solve POMDPs with on the order of 100 states. Consequently,
much research has been devoted to mitigating these two sources of intractability.
The complexity of policy/VF space has been addressed by observing that there are often
very good policies whose value functions are representable by a small number of vectors.
Various algorithms such as approximate vector pruning [9], point-based value iteration
(PBVI) [12, 16], bounded policy iteration (BPI) [14], gradient ascent (GA) [11, 1] and
stochastic local search (SLS) [3] exploit this fact to produce (often near-optimal) policies
of low complexity (i.e., few vectors) allowing larger POMDPs to be solved. Still these
scale to problems of only roughly 1000 states, since each value vector may still have exponential dimensionality. Conversely, it has been observed that belief states often carry
more information than necessary. Hence, one can often reduce vector dimensionality by
using compact representations such as decision trees (DTs) [2], algebraic decision diagrams (ADDs) [8, 9], or linear combinations of small basis functions (LCBFs) [6], or by
indirectly compressing the belief space into a small subspace by a value-directed compression (VDC) [14] or exponential PCA [15]. Once compressed, classic solution methods can
be used. However, since none of these approaches address the exponential complexity of
policy/VF space, they can only solve slightly larger POMDPs (up to 8250 states [15]).
Scalable POMDP algorithms can only be realized when both sources of intractability are
tackled simultaneously. While Hansen and Feng [9] implemented such an algorithm by
combining approximate state abstraction with approximate vector pruning, they didn?t
demonstrate the scalability of the approach on large problems. In this paper, we describe
how to combine value directed compression (VDC) with bounded policy iteration (BPI)
and demonstrate the scalability of the resulting algorithm (VDCBPI) on synthetic network
management problems of up to 33 million states. Among the techniques that deal with the
curse of dimensionality, VDC offers the advantage that the compressed POMDP can be directly fed into existing POMDP algorithms with no (or only slight) adjustments. This is not
the case for exponential-PCA, nor compact representations (DTs, ADDs, LCBFs). Among
algorithms that mitigate policy space complexity, BPI distinguishes itself by its ability to
avoid local optima (cf. GA), its efficiency (cf. SLS) and the fact that belief state monitoring
is not required (cf. PBVI, approximate vector pruning). Beyond the combination of VDC
with BPI, we offer two other contributions. We propose a new simple heuristic to compute
good lossy value directed compressions. We also augment BPI with the ability to bias its
policy search to reachable belief states. As a result, BPI can often find a much smaller
policy of similar quality for a given initial belief state.
2 POMDP Background
A POMDP is defined by: states S; actions A; observations Z; transition function T , where
T (s, a, s0 ) denotes Pr(s0 |s, a); observation function Z, where Z(s, z) is the probability
Pr(z|s, a) of observation z in state s after executing a; and reward function R, where
R(s, a) is the immediate reward associated with s when executing a. We assume discrete
state, action and observation sets and focus on discounted, infinite horizon POMDPs with
discount factor 0 ? ? < 1.
Policies and value functions for POMDPs are typically defined over belief space B, where
a belief state b is a distribution over S capturing an agent?s knowledge about the current
state of the world. Belief state b can be updated in response to a specific action-observation
pair ha, zi using Bayes rule. We denote the (unnormalized) belief update mapping by T a,z ,
where Tija,z = Pr(sj |a, si ) Pr(z|sj ). A factored POMDP, with exponentially many states,
thus gives rise to a belief space of exponential dimensionality.
Policies represented by finite state controllers (FSCs) are defined by a (possibly cyclic) directed graph ? = hN , Ei, where nodes n ? N correspond to stochastic action choices and
edges e ? E to stochastic transitions. An FSC can be viewed as a policy ? = h?, ?i, where
action strategy ? associates each node n with a distribution over actions ?(n) = Pr(a|n),
and observation strategy ? associates each node n and observation z with a distribution
over successor nodes ?(n, z) = Pr(n0 |n, z) (corresponding to the edge from n labeled
with z). The value function V ? of FSC ? is given by:
X
X
X
?
0
0
0
?
0
0
V (n, s) =
Pr(a|n)R(s, a) + ?
a
Pr(s |s, a) Pr(z|s , a)
Pr(n |n, z)V (n , s ) (1)
n0
z
The value V (n, b) of each node n is thus linear w.r.t the belief state; hence the value
function of the controller is piecewise-linear and convex. The optimal value function V ?
often has a large (if not infinite) number of vectors, each corresponding to a different node.
The optimal value function V ? satisfies Bellman?s equation:
X
?
a
V (b) = max R(b, a) + ?
Pr(z|b, a)V (bz )
a
z
(2)
max
s.t.
V (n,
P
Ps) + ?
0
|s, a) Pr(z|s0 , a) Pr(a, n0 |n, z)V (n0 , s0 )], ?s
P a [Pr(a|n)R(s, a)P+ ? s0 ,z Pr(s
0
Pr(a|n) = 1;
Pr(a, n |n, z) = Pr(a|n), ?a
a
n0
Pr(a|n) ? 0, ?a;
Pr(a, n0 |n, z) ? 0, ?a, z
Table 1: LP to uniformly improve the value function of a node.
max
s.t.
P
o(s, n)s,n
s,n
V (n,
s)
P + s,n ?
P
0
|s, a) Pr(z|s0 , a) Pr(a, n0 |n, z)V (n0 , s0 )], ?s
P a [Pr(a|n)R(s, a)P+ ? s0 ,z Pr(s
0
Pr(a|n) = 1;
Pr(a, n |n, z) = Pr(a|n), ?a
a
n0
Pr(a|n) ? 0, ?a;
Pr(a, n0 |n, z) ? 0, ?a, z
Table 2: LP to improve the value function of a node in a non-uniform way according to the
steady state occupancy o(s, n).
3 Bounded Policy Iteration
We briefly review the bounded policy iteration (BPI) algorithm (see [14] for details) and
describe a simple extension to bias its search toward reachable belief states. BPI incrementally constructs an FSC by alternating policy improvement and policy evaluation. Unlike policy iteration [7], this is done by slowly increasing the number of nodes (and value
vectors). The policy improvement step greedily improves each node n by optimizing its
action and observation strategies by solving the linear program (LP) in Table 1. This LP
uniformly maximizes the improvement in the value function by optimizing n?s distributions Pr(a, n0 |n, z). The policy evaluation step computes the value function of the current
controller by solving Eq. 1. The algorithm monotonically improves the policy until convergence to a local optimum, at which point new nodes are introduced to escape the local
optimum. BPI is guaranteed to converge to a policy that is optimal at the ?tangent? belief
states while slowly growing the size of the controller [14].
In practice, we often wish to find a policy suitable for a given initial belief state. Since
only a small subset of belief space is often reachable, it is generally possible to construct
much smaller policies tailored to the reachable region. We now describe a simple way to
bias BPI?s efforts toward the reachable region. Recall that the LP in Table 1 optimizes the
parameters of a node to uniformly improve its value at all belief states. We propose a new
LP (Table 2) that weighs the improvement by the (unnormalized) discounted occupancy
distribution induced by the current policy. This accounts for belief states reachable for the
node by aggregating them together. The (unnormalized) discounted occupancy distribution
is given by:
o(s0 , n0 ) = b0 (s0 , n0 ) + ?
X
o(s, n) Pr(a|n) Pr(z|a, s) Pr(n0 |n, z)
?s0 , n0
s,a,z,n
The LP in Table
P 2 is obtained by introducing variables s,n for each s, replacing the objective by s,n o(s, n)s,n and replacing in each constraint by the corresponding s,n .
When using the modified LP, BPI naturally tries to improve the policy at the reachable belief states before the others. Since the modification ensures that the value function doesn?t
decrease at any belief state, focusing the efforts on reachable belief states won?t decrease
policy value at other belief states. Furthermore, though the policy is initially biased toward
reachable states, BPI will eventually improve the policy for all belief states.
f
b
T?
~?
T
~
b
b?
~?
T
~
b?
~
R
R
T?
~
R
R
r
r?
Figure 1: Functional flow of a POMDP (dotted arrows) and a compressed POMDP (solid
arrows).
4 Value-Directed Compression
We briefly review the sufficient conditions for a lossless compression of POMDPs [13] and
describe a simple new algorithm to obtain good lossy compressions. Belief states constitute
a sufficient statistic summarizing all information available to the decision maker (i.e., past
actions and observations). However, as long as enough information is available to evaluate
the value of each policy, one can still choose the best policy. Since belief states often
contain information irrelevant to the estimation of future rewards, one can often compress
belief states into some lower-dimensional representation. Let f be a compression function
that maps each belief state b into some lower dimensional compressed belief state ?b (see
Figure 1). Here ?b can be viewed as a bottleneck that filters the information contained in
b before it is used to estimate future rewards. We desire a compression f such that ?b
corresponds to the smallest statistic sufficient for accurately predicting the current reward
r as well as the next compressed belief state ?b0 (since it captures all the information in b0
necessary to accurately predict subsequent rewards). Such a compression f exists if we can
? such
also find compressed transition dynamics T? a,z and a compressed reward function R
that:
? ? f and f ? T a,z = T? a,z ? f ?a ? A, z ? Z
R=R
(3)
a,z
?
?
Given an f , R and T satisfying Eq. 3, we can evaluate any policy ? using the compressed
POMDP dynamics to obtain V? ? . Since V ? = V? ? ?f , the compressed POMDP is equivalent
to the original.
When restricting f to be linear (represented by matrix F ), we can rewrite Eq. 3
? and T a,z F = F T? a,z ?a ? A, z ? Z
R = FR
(4)
a,z
That is, the column space of F spans R and is invariant w.r.t. each T . Hence, the
columns of the best linear lossless compression mapping F form a basis for the smallest
invariant subspace (w.r.t. each T a,z ) that spans R, i.e., the Krylov subspace. We can find the
columns of F by Krylov iteration: multiplying R by each T a,z until the newly generated
vectors are linear combinations of previous ones.1 The dimensionality of the compressed
space is equal to the number of columns of F , which is necessarily smaller than or equal
? and
to the dimensionality of the original belief space. Once F is found, we can compute R
a,z
?
each T
by solving the system in Eq. 4.
Since linear lossless compressions are not always possible, we can extend the technique of
[13] to find good lossy compressions with early stopping of the Krylov iteration. We retain
only the vectors that are ?far? from being linear combinations of prior vectors. For instance,
if v is a linearP
combination of v1 , v2 , . . . , vn , then there are coefficients c1 , c2 , . . . , cn s.t.
the error ||v ? i ci vi ||2 is zero. Given a threshold or some upper bound k on the desired
number of columns in F , we run Krylov iteration, retaining only the vectors with an error
greater than , or the k vectors with largest error. When F is computed by approximate
1
For numerical stability, one must orthogonalize each vector before multiplying by T a,z .
? and T? a,z by solving the linear system in Eq. 4?
Krylov iteration, we cannot compute R
due to the lossy nature of the compression, the system is overconstrained. But we can find
? and T? a,z by computing a least square approximation, solving:
suitable R
? and F > T a,z F = F > F T? a,z ?a ? A, z ? Z
F >R = F >F R
While compression is required when the dimensionality of belief space is too large, unfortunately, the columns of F have the same dimensionality. Factored POMDPs of exponential
dimension can, however, admit practical Krylov iteration if carried out using a compact
? and each T? a,z .
representation (e.g., DTs or ADDs) to efficiently compute F , R
5 Bounded Policy Iteration with Value-Directed Compression
In principle, any POMDP algorithm can be used to solve the compressed POMDPs produced by VDC. If the compression is lossless and the POMDP algorithm exact, the computed policy will be optimal for the original POMDP. In practice, POMDP algorithms are
usually approximate and lossless compressions are not always possible, so care must be
taken to ensure numerical stability and a policy of high quality for the original POMDP.
We now discuss some of the integration issues that arise when combining VDC with BPI.
Since V = F V? , maximizing the compressed value vector V? of some node n automatically
maximizes the value V of n w.r.t. the original POMDP when F is nonnegative; hence it is
essential that F be nonnegative. Otherwise, the optimal policy of the compressed POMDP
may not be optimal for the original POMDP. Fortunately, when R is nonnegative then F
is guaranteed to be nonnegative by the nature of Krylov iteration. If some rewards are
negative, we can add a sufficiently large constant to R to make it nonnegative without
changing the decision problem.
Since most algorithms, including BPI, compute approximately optimal policies it is also
critical to normalize the columns of F . Suppose F has two columns f 1 and f2 with L1 lengths 1 and 100, respectively. Since V = F V? = v?1 f1 + v?2 f2 , changes in v?2 have a
much greater impact on V than changes in v?1 . Such a difference in sensitivity may bias the
search for a good policy to an undesirable region of the belief space, or may even cause
the algorithm to return a policy that is far from optimal for the original POMDP despite the
fact that it is -optimal for the compressed POMDP.
We note that it is ?safer? to evaluate policies iteratively by successive approximation rather
than solving the system in Eq. 1. By definition, the transition matrices T a,z have eigenvalues with magnitude ? 1. In contrast, lossy compressed transition matrices T? a,z are not
guaranteed to have this property. Hence, solving the system in Eq. 1 may not correspond
to policy evaluation. It is thus safer to evaluate policies by successive approximation for
lossy compressions.
Finally several algorithms including BPI compute witness belief states to verify the dominance of a value vector. Since the compressed belief space B? is different from the original
belief space B, this must be approached with care. B is a simplex corresponding to the
convex hull of the state points. In contrast, since each row vector of F is the compressed
version of some state point, B? corresponds to the convex hull of the row vectors of F .
When F is non-negative, it is often possible to ignore this difference. For instance, when
verifying the dominance of a value vector, if there is a compressed witness ?b, there is always an uncompressed witness b, but not vice-versa. This means that we can properly
identify all dominating value vectors, but we may erroneously classify a dominated vector
as dominating. In practice, this doesn?t impact the correctness of algorithms such as policy
iteration, bounded policy iteration, incremental pruning, witness algorithm, etc. but it will
slow them down since they won?t be able to prune as many value vectors as possible.
cycle16
cycle19
100
95
90
250
Expected Rewards
120
Expected Rewards
Expected Rewards
105
cycle22
115
110
105
250
200
150
100
50
# of basis fns
20
40
60
100
80
150
100
50
# of basis fns
# of nodes
cycle25
20
40
60
80
100
200
120
150
100
140
135
110
105
100
150
50
40
60
100
80
150
100
50
# of basis fns
# of nodes
100
120
# of nodes
130
125
120
115
20
40
60
80
100
200
120
150
100
50
# of basis fns
# of nodes
3legs25
3legs22
20
40
60
80
100
120
# of nodes
cycle25
150
145
140
135
130
Time (1000 seconds)
12
Expected Rewards
Expected Rewards
80
250
200
120
60
110
250
200
40
135
115
95
250
20
3legs19
Expected Rewards
Expected Rewards
Expected Rewards
145
# of basis fns
50
# of basis fns
# of nodes
120
20
120
3legs16
150
100
125
250
200
120
130
160
155
150
145
140
250
125
250
200
150
100
# of basis fns
50
20
40
60
80
100
# of nodes
120
10
8
6
4
2
250
200
150
100
# of basis fns
50
20
40
60
80
100
120
200
150
100
# of basis fns
# of nodes
50
20
40
60
80
100
120
# of nodes
Figure 2: Experimental results for cycle and 3legs network configurations of 16, 19,
22 and 25 machines. The bottom right graph shows the running time of BPI on compressed
versions of a cycle network of 25 machines.
VDCBPI
heuristic
doNothing
16
120.9
100.6
98.4
3legs
19
22
137.0 151.0
118.3 138.3
112.9 133.5
25
164.8
152.3
147.1
16
103.9
102.5
91.6
cycle
19
22
121.3 134.3
117.9 130.2
105.4 122.0
25
151.4
152.3
140.1
Table 3: Comparison of the best policies achieved by VDCBPI to the doNothing and
heuristic policies.
The above tips work well when VDC is integrated with BPI. We believe they are sufficient
to ensure proper integration of VDC with other POMDP algorithms, though we haven?t
verified this empirically.
6 Experiments
We report on experiments with VDCBPI on some synthetic network management problems
similar to those introduced in [5]. A system administrator (SA) maintains a network of
machines. Each machine has a 0.1 probability of failing at any stage; but this increases to
0.333 when a neighboring machine is down. The SA receives a reward of 1 per working
machine and 2 per working server. At each stage, she can either reboot a machine, ping a
machine or do nothing. She only observes the status of a machine (with 0.95 accuracy) if
she reboots or pings it. Costs are 2.5 (rebooting), 0.1 (pinging), and 0 (doing nothing). An
n-machine network induces to a POMDP with 2n states, 2n + 1 actions and 2 observations.
We experimented with networks of 16, 19, 22 and 25 machines organized in two configurations: cycle (a ring) and 3legs (a tree of 3 branches joined at the root). Figure 2 shows
the average expected reward earned by policies computed by BPI after the POMDP has
been compressed by VDC. Results are averaged over 500 runs of 60 steps, starting with a
belief state where all machines are working.2 As expected, decision quality increases as we
increase the number of nodes used in BPI and basis functions used in VDC. Also interesting
are some of the jumps in the reward surface of some graphs, suggesting phase transitions
w.r.t. the dimensionality of the compression. The bottom right graph in Fig. 2 shows the
time taken by BPI on a cycle network of 25 machines (other problems exhibit similar
behavior). VDC takes from 4902s. to 12408s. (depending on size and configuration) to
compress POMDPs to 250 dimensions.3
In Table 3 we compare the value of the best policy with less than 120 nodes found by
VDCBPI to two other simple policies. The doNothing policy lets the network evolve
without any rebooting or pinging. The heuristic policy estimates at each stage the
probability of failure4 of each machine and reboots the machine most likely to be down
if its failure probability is greater than threshold p1 or pings it if greater than threshold
p2 . Settings of p1 = 0.8 and p2 = 0.15 were used.5 This heuristic policy performs very
well and therefore offers a strong competitor to VDCBPI. But it is possible to do better
than the heuristic policy by optimizing the choice of the machine that the SA may reboot
or ping. Since a machine is more likely to fail when neighboring machines are down, it
is sometimes better to choose (for reboot) a machine surrounded by working machines.
However, since the SA doesn?t exactly know which machines are up or down due to partial
observability, such a tradeoff is difficult to evaluate and sometimes not worthwhile. With a
sufficient number of nodes and basis functions, VDCBPI outperforms the heuristic policy
on the 3legs networks and matches it on the cycle networks. This is quite remarkable
given the fact that belief states were compressed to 250 dimensions or less compared to the
original dimensionality ranging from 65,536 to 33,554,432.
7 Conclusion
We have described a new POMDP algorithm that mitigates both high belief space dimensionality and policy/VF complexity. By integrating value-directed compression with
2
The ruggedness of the graphs is mainly due to the variance in the reward samples.
Reported running times are the cputime measured on 3GHz linux machines.
4
Due to the large state space, approximate monitoring was performed by factoring the joint.
5
These values were determined through enumeration of all threshold combinations in increments
of 0.05, choosing the best for 25-machine problems.
3
bounded policy iteration, we can solve synthetic network management POMDPs of 33 million states (3 orders of magnitude larger than previously solved discrete POMDPs). Note
that the scalability of VDCBPI is problem dependent, however we hope that new, scalable, approximate POMDP algorithms such as VDCBPI will allow POMDPs to be used to
model real-world problems, with the expectation that they can be solved effectively. We
also described several improvements to the existing VDC and BPI algorithms.
Although VDC offers the advantage that any existing solution algorithm can be used to
solve compressed POMDPs, it would be interesting to combine BPI or PBVI with a factored representation such as DTs or ADDs, allowing one to directly solve large scale
POMDPs without recourse to an initial compression. Beyond policy space complexity
and high dimensional belief spaces, further research will be necessary to deal with exponentially large action and observation spaces.
References
[1] D. Aberdeen and J. Baxter. Scaling internal-state policy-gradient methods for POMDPs. Proc.
of the Nineteenth Intl. Conf. on Machine Learning, pp.3?10, Sydney, Australia, 2002.
[2] C. Boutilier and D. Poole. Computing optimal policies for partially observable decision processes using compact representations. Proc. AAAI-96, pp.1168?1175, Portland, OR, 1996.
[3] D. Braziunas and C. Boutilier. Stochastic local search for POMDP controllers. Proc. AAAI-04,
to appear, San Jose, CA, 2004.
[4] A. R. Cassandra, M. L. Littman, and N. L. Zhang. Incremental pruning: A simple, fast, exact
method for POMDPs. Proc. UAI-97, pp.54?61, Providence, RI, 1997.
[5] C. Guestrin, D. Koller, and R. Parr. Max-norm projections for factored MDPs. Proc. IJCAI-01,
pp.673?680, Seattle, WA, 2001.
[6] C. Guestrin, D. Koller, and R. Parr. Solving factored POMDPs with linear value functions.
IJCAI-01 Wkshp. on Planning under Uncertainty and Incomplete Information, Seattle, 2001.
[7] E. A. Hansen. Solving POMDPs by searching in policy space. Proc. UAI-98, pp.211?219,
Madison, Wisconsin, 1998.
[8] E. A. Hansen and Z. Feng. Dynamic programming for POMDPs using a factored state representation. Proc. AIPS-2000, pp.130?139, Breckenridge, CO, 2000.
[9] E. A. Hansen and Z. Feng. Approximate planning for factored POMDPs. Proc. ECP-2001,
Toledo, Spain, 2000.
[10] L. P. Kaelbling, M. Littman, and A. R. Cassandra. Planning and acting in partially observable
stochastic domains. Artif. Intel., 101:99?134, 1998.
[11] N. Meuleau, L. Peshkin, K. Kim, and L. P. Kaelbling. Learning finite-state controllers for
partially observable environments. Proc. UAI-99, pp.427?436, Stockholm, 1999.
[12] J. Pineau, G. Gordon, and S. Thrun. Point-based value iteration: an anytime algorithm for
POMDPs. IJCAI-03, Acapulco, Mexico, 2003.
[13] P. Poupart and C. Boutilier. Value-directed compressions of POMDPs. Advances in Neural
Information Processing Systems, pp.1547?1554, Vancouver, Canada, 2002.
[14] P. Poupart and C. Boutilier. Bounded finite state controllers. Advances in Neural Information
Processing Systems, Vancouver, Canada, 2003.
[15] N. Roy and G. Gordon. Exponential family PCA for belief compression in pomdps. Advances
in Neural Information Processing Systems, pp.1635?1642, Vancouver, BC, 2002.
[16] M. T. J. Spaan and N. Vlassis. A point-based pomdp algorithm for robot planning. IEEE Intl.
Conf. on Robotics and Automation, to appear, New Orleans, 2004.
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1,878 | 2,705 | A Generalized Bradley-Terry Model: From
Group Competition to Individual Skill
Tzu-Kuo Huang
Chih-Jen Lin
Department of Computer Science
National Taiwan University
Taipei 106, Taiwan
Ruby C. Weng
Department of Statistics
National Chenechi University
Taipei 116, Taiwan
Abstract
The Bradley-Terry model for paired comparison has been popular in
many areas. We propose a generalized version in which paired individual
comparisons are extended to paired team comparisons. We introduce a
simple algorithm with convergence proofs to solve the model and obtain
individual skill. A useful application to multi-class probability estimates
using error-correcting codes is demonstrated.
1
Introduction
The Bradley-Terry model [2] for paired comparisons has been broadly applied in many
areas such as statistics, sports, and machine learning. It considers the model
?i
P (individual i beats individual j) =
,
(1)
?i + ? j
where ?i is the overall skill of the ith individual. Given k individuals and rij as the number
of times that i beats j, an approximate skill pi can be found by minimizing the negative log
likelihood of the model (1):
?
X?
pi
pj
min
l(p) = ?
rij log
+ rji log
p
pi + p j
pi + p j
i<j
subject to
0 ? pi , i = 1, . . . , k,
k
X
pi = 1.
(2)
i=1
Thus, from paired comparisons, we can obtain individual performance. This model dates
back to [14] and has been extended to more general settings. Some reviews are, for example, [5, 6]. Problem (2) can be solved by a simple iterative procedure:
Algorithm 1
1. Start with any initial p0j > 0, j = 1, . . . , k.
2. Repeat (t = 0, 1, . . .)
a. Let s = (t mod k) + 1. For j = 1, . . . , k, define
? P
si
? P i:i6=rs r+r
if j = s,
si
is
t,n
i:i6=s pt +pt
pj ?
s
i
? t
pj
if j 6= s.
(3)
b. Normalize pt,n to be pt+1 .
until ?l(pt )/?pj = 0, j = 1, . . . , k are satisfied.
This algorithm is so simple that there is no need to use sophisticated optimization techniques. If rij > 0, ?i, j, Algorithm 1 globally converges to the unique minimum of (2). A
systematic study of the convergence is in [9].
Several machine learning work have used the Bradley-Terry model and one is to obtain
multi-class probability estimates from pairwise coupling [8]. For any data instance x, if
nij is the number of training data in the ith or jth class, and
rij ? nij P (x in class i | x in class i or j)
is available, solving (2) obtains the estimate of P (x in class i), i = 1, . . . , k. [13] tried to
extend this algorithm to other multi-class settings such as ?one-against-the rest? or ?errorcorrecting coding,? but did not provide a convergence proof. In Section 5.2 we show that
the algorithm proposed in [13] indeed has some convergence problems.
In this paper, we propose a generalized Bradley-Terry model where each comparison is
between two disjoint subsets of subjects. Then from the results of team competitions, we
can approximate the skill of each individual. This model has many potential applications.
For example, from records of tennis or badminton doubles (or singles and doubles combined), we may obtain the rank of all individuals. A useful application in machine learning
is multi-class probability estimates using error-correcting codes. We then introduce a simple iterative method to solve the generalized model with a convergence proof. Experiments
on multi-class probability estimates demonstrate the viability of the proposed model and
algorithm. Due to space limitation, we omit all proofs in this paper.
2
Generalized Bradley-Terry Model
We propose a generalized Bradley-Terry model where, using team competition results, we
can approximate individual skill levels. Consider a group of k individuals: {1, . . . , k}.
Two disjoint subsets Ii+ and Ii? form teams for games and ri ? 0 (ri? ? 0) is the number
of times that Ii+ beats Ii? (Ii? beats Ii+ ). Thus, we have Ii ? {1, . . . , k}, i = 1, . . . , m so
that
Ii = Ii+ ? Ii? ,
Ii+ 6= ?, Ii? 6= ?, and Ii+ ? Ii? = ?.
Under the model that
P
P
j?Ii+ ?j
j?I + ?j
+
?
P
P (Ii beats Ii ) = P
= P i
,
j?Ii ?j
j?I + ?j +
j?I ? ?j
i
we can define
qi ?
X
pj ,
j?Ii
qi+ ?
i
X
j?Ii+
pj ,
qi? ?
X
pj ,
j?Ii?
and minimize the negative log likelihood
min
p
l(p) = ?
m
X
?
?
ri log(qi+ /qi ) + ri? log(qi? /qi ) ,
i=1
under the same constraints of (2). If Ii , i = 1, . . . , k(k ? 1)/2 are as the following:
Ii+
Ii?
ri
ri?
{1}
{2} r12
r21
..
..
..
..
.
.
.
.
{k ? 1} {k} rk?1,k rk,k?1
(4)
then (4) goes back to (2). The difficulty of solving (4) over solving (2) is that now l(p)
is expressed in terms of qi+ , qi? , qi but the real variable is p. The original Bradley-Terry
model is a special case of other statistical models such as log-linear or generalized linear
model, so methods other than Algorithm 1 (e.g., iterative scaling and iterative weighted
least squares) can also be used. However, (4) is not in a form of such models and hence
these methods cannot be applied. We propose the following algorithm to solve (4).
Algorithm 2
1. Start with p0j > 0, j = 1, . . . , k and corresponding qi0,+ , qi0,? , qi0 , i = 1, . . . , m.
2. Repeat (t = 0, 1, . . .)
a. Let s = (t mod k) + 1. For j = 1, . . . , k, define
?P
P
ri?
ri
?
? i:s?Ii+ qit,+ + i:s?Ii? qit,? t
pj if j = s,
P
ri +r ?
i
(5)
pt,n
j ?
i:s?Ii
qt
?
i
? t
pj
if j 6= s.
b. Normalize pt,n to pt+1 .
c. Update qit,+ , qit,? , qit to qit+1,+ , qit+1,? , qit+1 , i = 1, . . . , m.
until ?l(pt )/?pj = 0, j = 1, . . . , k are satisfied.
For the multiplicative factor in (5) to be well defined (i.e., non-zero denominator), we
need Assumption 1, which will be discussed in Section 3. Eq. (5) is a simple fixed-point
type update; in each iteration, only one component (i.e., pts ) is modified while the others
remain the same. It is motivated from using a descent direction to strictly decrease l(p): If
?l(pt )/?ps 6= 0, then
? ?
!
?2 ! . ? X
?l(pt )
?l(pt )
ri + ri?
t,n
t
t
< 0,
(6)
? (ps ? ps ) = ?
ps
?ps
?ps
qit
i:s?Ii
where
?l(p)
?ps
pt,n
s
= ?
X r?
X ri + r?
X ri
i
i
.
+ ?
? +
q
q
q
i
i
i
?
+
i:s?I
i:s?Ii
i:s?Ii
i
pts
? is a descent direction in optimization since a sufficiently small step along
Thus,
this direction guarantees the strict decrease of the function value. Since now we take the
whole direction without searching for the step size, more efforts are needed to prove the
strict decrease in Lemma 1. However, (6) does hint that (5) is a reasonable update.
Lemma 1 If pts > 0 is the index to be updated and ?l(pt )/?ps 6= 0, then l(pt+1 ) < l(pt ).
If we apply the update rule (5) on the pairwise model,
P
P
rsi
i:i6=s pts
i:i6=s rsi
t
P
P
rsi
ris ps = P
rsi +ris and (5) goes back to (3).
i:i6=s pt +pt +
i:i6=s pt +pt
i:i6=s pt +pt
s
3
i
s
i
s
i
Convergence of Algorithm 2
For any point satisfying ?l(p)/?pj = 0, j = 1, . . . , k and constraints of (4), it is a stationary point of (4)1 . We will prove that Algorithm 2 converges to such a point. If
1
A stationary point means a Karash-Kunh-Tucker (KKT) point for constrained optimization problems like (2) and (4). Note that here ?l(p)/?pj = 0 implies (and is more restricted than) the KKT
condition.
it stops in a finite number of iterations, then ?l(p)/?pj = 0, j = 1, . . . , k, which
means a stationary point of (4) is already obtained. Thus, we only need to handle the
case where {pt } is an infinite sequence. As {pt }?
t=0 is in a compact (i.e., closed and
Pk
bounded) set {p | 0 ? pj ? 1, j=1 pj = 1}, it has at least one convergent subsequence. Assume p? is one such convergent point. In the following we will prove that
?l(p? )/?pj = 0, j = 1, . . . , k.
To prove the convergence of a fixed-point type algorithm, we need that if p?s > 0 and
?l(p? )/?ps 6= 0, then from p?s we can use (5) to update it to p?,n
6= p?s . We thus make the
s
following assumption to ensure that p?s > 0 (see also Theorem 1).
Assumption 1 For each j ? {1, . . . , k},
?i:i?A Ii = {1, . . . , k}, where A = {i | (Ii+ = {j}, ri > 0) or (Ii? = {j}, ri? > 0)}.
That is, each individual forms a winning (losing) team in some competitions which together
involve all subjects.
An issue left in Section 2 is whether the multiplicative factor in (5) is well defined. With
Assumption 1 and initial p0j > 0, j = 1, . . . , k, one can show by induction that ptj > 0, ?t
and hence the denominator of (5) is never zero: If ptj > 0, Assumption 1 implies that
P
P
t,+
or i:j?I ? ri? /qit,? is positive. Thus, both numerator and denominator in
i:j?I + ri /qi
i
i
the multiplicative factor are positive, and so is pt+1
j .
If rij > 0, the original Bradley-Terry model satisfies Assumption 1. No matter the model
satisfies the assumption or not, an easy way to fulfill it is to add an additional term
!
?
k
X
ps
(7)
??
log Pk
s=1
j=1 pj
to l(p), where ? is a small positive number. That is, for each s, we make an Ii = {1, . . . , k}
Pk
with Ii+ = {s}, ri = ?, and ri? = 0. As j=1 pj = 1 is one of the constraints, (7) reduces
Pk
to ?? s=1 log ps , which is a barrier term in optimization to ensure that ps does not go to
zero. The property p?s > 0 and the convergence of Algorithm 2 are in Theorem 1:
Theorem 1 Under Assumption 1, any convergent point p? of Algorithm 2 satisfies p?s >
0, s = 1, . . . , k and is a stationary point of (4).
4
Asymptotic Distribution of the Maximum Likelihood Estimator
For the standard Bradley-Terry model, asymptotic distribution of the MLE (i.e., p) has been
discussed in [5]. In this section, we discuss the asymptotic distribution for the proposed
estimator. To work on
?, we define
P the real probability
P
P
q?i ? j?Ii ?j ,
q?i+ ? j?I + ?j ,
q?i? ? j?I ? ?j ,
i
i
and consider ni ? ri + ri? as a constant. Note that ri ? BIN(ni , q?i+ /?
qi ) is a random
variable representing the number of times that Ii+ beats Ii? in ni competitions. By defining
for s, t = 1, . . . , k,h
i P
P
n q??
n q?+
= i:s?I + q?+i q?i2 + i:s?I ? q??i q?i2 ,
?ss ? var ?l(?)
?ps
i
i
i i
h
i Pi i
q?? n
?l(?)
?st ? cov ?l(?)
= i:s,t?I + q?i+ q?2i ?
?ps , ?pt
i
i i
P
P
P
q?i+ ni
ni
ni
+
?
?
i:(s,t)?Ii ?Ii q?i2
i:(s,t)?Ii? ?Ii+ q?i2 +
i:s,t?Ii? q?? q?2 , s 6= t,
i i
we have the following theorem:
Theorem
? 2 Let n be the?total number of comparisons. If ri is independent of rj , ?i 6= j,
then n(p1 ? ?1 ), . . . , n(pk?1 ? ?k?1 ) have for large samples the multivariate normal
distribution with zero means and dispersion matrix [??st ]?1 , where
??st = ?st ? ?sk ? ?tk + ?kk , s, t = 1, . . . , k ? 1.
5
Application to Multi-class Probability Estimates
Many classification methods are two-class based approaches and there are different ways
to extend them for multi-class cases. Most existing studies focus on predicting class labels
but not probability estimates. In this section, we discuss how the generalized Bradley-Terry
model can be applied to multi-class probability estimates.
Error-correction coding [7, 1] is a general method to construct binary classifiers and combine them for multi-class prediction. It suggests some ways to construct Ii+ and Ii? ; both
are subsets of {1, . . . , k}. Then one trains a binary model using data from classes in Ii+
(Ii? ) as positive (negative). Simple and commonly used methods such as ?one-against-one?
and ?one-against-the rest? are its special cases. Given ni the number of training data with
classes in Ii = Ii+ ? Ii? , we assume here that for any data x,
ri ? ni P (x in classes of Ii+ | x in classes of Ii+ or Ii? )
(8)
is available, and the task is to approximate P (x in class s), s = 1, . . . , k. In the rest of this
section we discuss the special case ?one-against-the rest? and the earlier results in [13].
5.1
Properties of the ?One-against-the rest? Approach
For this approach, Ii , i = 1, . . . , m are
Ii+
Ii?
{1} {2, . . . , k}
{2} {1, 3, . . . , k}
..
..
.
.
ri
r1
r2
..
.
ri?
1 ? r1
1 ? r2
..
.
Now n1 = ? ? ? = nm = the total number of training data, so the solution to (4) is not
affected by ni . Thus, we remove it from (8), so ri + ri? = 1. As ?l(p)/?ps = 0 becomes
k
X
rj?
r1 1 ? r1
rk 1 ? rk
rs X rj?
+
= k, we have ?
= ??? =
?
= k?
= ?,
ps
1 ? pj
p1 1 ? p 1
pk 1 ? p k
1 ? pj
j=1
j:j6=s
wherep? is a constant. These equalities provide another
p way to solve p, and ps = ((1 +
?) ? (1 + ?)2 ? 4rs ?)/2?. Note that ((1 + ?) + (1 + ?)2 ? 4rs ?)/2? also satisfies
the equalities, but it is negative when ? < 0, and greater than 1 when ? > 0. By solving
P
m
s=1 ps = 1, we obtain ? and the optimal p.
From the formula of ps , if ? > 0, larger ps implies smaller (1 + ?)2 ? 4rs ? and hence larger
rs . It is similar for ? < 0. Thus, the order of p1 , . . . , pk is the same as that of r1 , . . . , rk :
Theorem 3 If rs ? rt , then ps ? pt .
5.2
The Approach in [13] for Error-Correcting Codes
[13] was the first attempt to address the probability estimates using general error-correcting
codes. By considering the same optimization problem (4), it proposes a heuristic update
rule
P
P
?
i:s?Ii+ ri +
i:s?Ii? ri
t,n
ps ? P
pt ,
(9)
P
ni qit,+
ni qit,? s
+
?
+
t
t
i:s?I
i:s?I
q
q
i
i
i
i
but does not provide a convergence proof. For a fixed-point update, we expect that at the
optimum, the multiplicative factor in (9) is one. However, unlike (5), when the factor is one,
(9) does not relate to ?l(p)/?ps = 0. In fact, a simple example shows that this algorithm
Pk
may never converge. Taking the ?one-against-the rest? approach, if we keep i=1 pti = 1
and assume ni = 1, then ri + ri?P= 1 and the factor in the update rule (9) is
rs + i:i6=s ri?
P
pts + i:i6=s (1?pti )
=
P
k?1+2rs ? k
i=1 ri
.
k?2+2pts
Pk
If the algorithm converges and the factor approaches one, then ps = (1 + 2rs ? i=1 ri )/2
Pk
Pk
but they may not satisfy s=1 ps = 1. Therefore, if in the algorithm we keep i=1 pti = 1
as [13] did, the factor may not approach one and the algorithm does not converge. More
generally, if Ii = {1, . . . , k}, ?i, the algorithm may not converge. As qit = 1, the condition
Pk
that the factor equals one can be written as a linear equation of p. Together with i=1 pi =
1, there is an over-determined linear system (i.e., k + 1 equations and k variables).
6
Experiments on Multi-class Probability Estimates
6.1
Simulated Examples
We consider the same settings in [8, 12] by defining three possible class probabilities:
(a) p1 = 1.5/k, pj = (1 ? p1 )/(k ? 1), j = 2, . . . , k.
(b) k1 = k/2 if k is even, and (k + 1)/2 if k is odd; then p1 = 0.95 ? 1.5/k1 , pi =
(0.95 ? p1 )/(k1 ? 1) for i = 2, . . . , k1 , and pi = 0.05/(k ? k1 ) for i = k1 + 1, . . . , k.
(c) p1 = 0.95 ? 1.5/2, p2 = 0.95 ? p1 , and pi = 0.05/(k ? 2), i = 3, . . . , k.
Classes are competitive in case (a), but only two dominate in case (c). We then generate ri
by adding some noise to qi+ /qi :
q+
ri = min(max(?, qii (1 + 0.1N (0, 1))), 1 ? ?).
Then ri? = 1 ? ri . Here ? = 10?7 is used so that all ri , ri? are positive. We consider the
four encodings used in [1] to generate Ii :
1. ?1vs1?: the pairwise approach (eq. (2)).
2. ?1vsrest?: the ?one-against-the rest? approach in Section 5.1.
3. ?dense?: Ii = {1, . . . , k} for all i. Ii is randomly split to two equally-sized sets Ii+ and
Ii? . [10 log2 k] such splits are generated2 . Following [1], we repeat this procedure 100
times and select the one whose [10 log2 k] splits have the smallest distance.
4. ?sparse?: Ii+ , Ii? are randomly drawn from {1, . . . , k} with E(|Ii+ |) = E(|Ii? |) = k/4.
Then [15 log2 k] such splits are generated. Similar to ?dense,? we repeat the procedure
100 times to find a good coding.
Figure 1 shows averaged accuracy rates over 500 replicates for each of the four methods
when k = 22 , 23 , . . . , 26 . ?1vs1? is good for (a) and (b), but suffers some losses in (c),
where the class probabilities are highly unbalanced. [12] has observed this and proposed
some remedies. ?1vsrest? is quite competitive in all three scenarios. Furthermore, ?dense?
and ?sparse? are less competitive in cases (a) and (b) when k is large. Due to the large
|Ii+ | and |Ii? |, the model is unable to single out a clear winner when probabilities are more
balanced. We also analyze the (relative) mean square error (MSE) in Figure 2:
? k
!
500
k
X
1 X X j
MSE =
(?
p ? pi )2 /
p2i ,
(10)
500 j=1 i=1 i
i=1
? j is the probability estimate obtained in the jth of the 500 replicates. Results of
where p
Figures 2(b) and 2(c) are consistent with those of the accuracy. Note that in Figure 2(a), as
P
? j ) are balanced, ki=1 (?
p (and p
pji ? pi )2 is small. Hence, all approaches have small MSE
though some have poor accuracy.
2
We use [x] to denote the nearest integer value of x.
1
0.8
0.8
0.8
0.6
0.4
0.2
Test Accuracy
1
Test Accuracy
Test Accuracy
1
0.6
0.4
0.2
0
2
3
4
log k
5
0
2
6
0.6
0.4
0.2
3
2
4
log k
5
0
2
6
3
2
(a)
4
log k
5
6
2
(b)
(c)
0.4
0.014
0.35
0.012
0.3
0.01
0.25
0.008
0.4
0.35
0.3
0.2
0.15
0.006
0.004
0.1
0.002
0.05
0
2
0.45
MSE
0.016
MSE
MSE
Figure 1: Accuracy by the four encodings: ?1vs1? (dashed line, square), ?1vsrest? (solid
line, cross), ?dense? (dotted line, circle), ?sparse? (dashdot line, asterisk)
3
4
log2 k
5
6
(a)
0
2
0.25
0.2
0.15
0.1
0.05
3
4
log2 k
(b)
5
6
0
2
3
4
log2 k
5
6
(c)
Figure 2: MSE by the four encodings: legend the same as Figure 1
6.2
Experiments on Real Data
In this section we present experimental results on some real-world multi-class problems.
They have been used in [12], which provides more information about data preparation. Two
problem sizes, 300/500 and 800/1,000 for training/testing, are used. 20 training/testing
splits are generated and the testing error rates are averaged. All data used are available
at http://www.csie.ntu.edu.tw/?cjlin/papers/svmprob/data. We use
the same four ways in Section 6.1 to generate Ii . All of them have |I1 | ? ? ? ? ? |Im |. With
the property that these multi-class problems are reasonably balanced, we set ni = 1 in (8).
Since there are no probability values available for these problems, we compare the accuracy by predicting the label with the largest probability estimate. The purpose here is to
compare the four probability estimates but not to check the difference from existing multiclass classification techniques. We consider support vector machines (SVM) [4] with the
RBF kernel as the binary classifier. An improved version [10] of [11] obtains ri . Full
SVM parameter selection is conducted before testing, although due to space limitation, we
omit details here. The code is modified from LIBSVM [3], a library for support vector
machines. The resulting accuracy is in Table 1 for smaller and larger training/testing sets.
Except ?1vs1,? the other three approaches are quite competitive. These results indicate that
practical problems are more similar to the case of (c) in Section 6.1, where few classes
dominate. This observation is consistent with the findings in [12]. Moreover, ?1vs1? suffers some losses when k is larger (e.g., letter), the same as in Figure 1(c); so for ?1vs1,?
[12] proposed using a quadratic model instead of the Bradley-Terry model.
In terms of the computational time, because the number of binary problems for ?dense? and
?sparse? ([10 log2 k] and [15 log2 k], respectively) is larger than k, and each binary problem
involves many classes of data (all and one half), their training time is longer than ?1vs1?
and ?1vsrest.? ?Dense? is particularly time consuming. Note that though ?1vs1? solves
k(k ? 1)/2 binaries, it is efficient as each binary problem involves only two classes of data.
Table 1: Average of 20 test errors (in percentage) by four encodings (lowest boldfaced)
300 training and 500 testing
800 training and 1,000 testing
Problem k
1vs1 1vsrest dense sparse
1vs1 1vsrest dense sparse
dna
3 10.47
10.33 10.45 10.19
6.21
6.45
6.415
6.345
waveform 3 15.01
15.35 15.66 15.12 13.525 13.635
13.76
13.99
satimage 6 14.22
15.08 14.72
14.8
11.54
11.74 11.865 11.575
segment 7
6.24
6.69
6.62
6.19
3.295
3.605
3.52
3.25
10.89 10.81 11.14
7.78
7.49
7.31 7.575
USPS 10 11.37
MNIST 10 13.84
12.56
13.0 12.29
8.11
7.37
7.59
7.535
35.17 33.86 33.88
21.11 19.685
20.14 19.49
letter
26 39.73
In summary, we propose a generalized Bradley-Terry model which gives individual skill
from group competition results. A useful application to general multi-class probability
estimate is demonstrated.
References
[1] E. L. Allwein, R. E. Schapire, and Y. Singer. Reducing multiclass to binary: a unifying approach
for margin classifiers. Journal of Machine Learning Research, 1:113?141, 2001.
[2] R. A. Bradley and M. Terry. The rank analysis of incomplete block designs: I. the method of
paired comparisons. Biometrika, 39:324?345, 1952.
[3] C.-C. Chang and C.-J. Lin. LIBSVM: a library for support vector machines, 2001. Software
available at http://www.csie.ntu.edu.tw/?cjlin/libsvm.
[4] C. Cortes and V. Vapnik. Support-vector network. Machine Learning, 20:273?297, 1995.
[5] H. A. David. The method of paired comparisons. Oxford University Press, New York, second
edition, 1988.
[6] R. R. Davidson and P. H. Farquhar. A bibliography on the method of paired comparisons.
Biometrics, 32:241?252, 1976.
[7] T. G. Dietterich and G. Bakiri. Solving multiclass learning problems via error-correcting output
codes. Journal of Artificial Intelligence Research, 2:263?286, 1995.
[8] T. Hastie and R. Tibshirani. Classification by pairwise coupling. In M. I. Jordan, M. J. Kearns,
and S. A. Solla, editors, Advances in Neural Information Processing Systems 10. MIT Press,
Cambridge, MA, 1998.
[9] D. R. Hunter. MM algorithms for generalized Bradley-Terry models. The Annals of Statistics,
32:386?408, 2004.
[10] H.-T. Lin, C.-J. Lin, and R. C. Weng. A note on Platt?s probabilistic outputs for support vector
machines. Technical report, Department of Computer Science, National Taiwan University,
2003.
[11] J. Platt. Probabilistic outputs for support vector machines and comparison to regularized likelihood methods. In A. Smola, P. Bartlett, B. Sch?olkopf, and D. Schuurmans, editors, Advances
in Large Margin Classifiers, Cambridge, MA, 2000. MIT Press.
[12] T.-F. Wu, C.-J. Lin, and R. C. Weng. Probability estimates for multi-class classification by pairwise coupling. In S. Thrun, L. Saul, and B. Sch?olkopf, editors, Advances in Neural Information
Processing Systems 16. MIT Press, Cambridge, MA, 2004.
[13] B. Zadrozny. Reducing multiclass to binary by coupling probability estimates. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing
Systems 14, pages 1041?1048. MIT Press, Cambridge, MA, 2002.
[14] E. Zermelo. Die berechnung der turnier-ergebnisse als ein maximumproblem der wahrscheinlichkeitsrechnung. Mathematische Zeitschrift, 29:436?460, 1929.
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1,879 | 2,706 | Making Latin Manuscripts Searchable using
gHMM?s
Jaety Edwards Yee Whye Teh David Forsyth Roger Bock Michael Maire
{jaety,ywteh,daf,bock,mmaire}@cs.berkeley.edu
Grace Vesom
Department of Computer Science
UC Berkeley
Berkeley, CA 94720
Abstract
We describe a method that can make a scanned, handwritten mediaeval
latin manuscript accessible to full text search. A generalized HMM is
fitted, using transcribed latin to obtain a transition model and one example each of 22 letters to obtain an emission model. We show results for
unigram, bigram and trigram models. Our method transcribes 25 pages
of a manuscript of Terence with fair accuracy (75% of letters correctly
transcribed). Search results are very strong; we use examples of variant spellings to demonstrate that the search respects the ink of the document. Furthermore, our model produces fair searches on a document
from which we obtained no training data.
1. Intoduction
There are many large corpora of handwritten scanned documents, and their number is growing rapidly. Collections range from the complete works of Mark Twain to thousands of
pages of zoological notes spanning two centuries. Large scale analyses of such corpora
is currently very difficult, because handwriting recognition works poorly. Recently, Rath
and Manmatha have demonstrated that one can use small bodies of aligned material as
supervised data to train a word spotting mechanism [7]. The result can make scanned
handwritten documents searchable.
Current techniques assume a closed vocabulary ? one can search only for words in the
training set ? and search for instances of whole words. This approach is particularly
unattractive for an inflected language, because individual words can take so many forms
that one is unlikely to see all in the training set. Furthermore, one would like the method
used to require very little aligned training data, so that it is possible to process documents
written by different scribes with little overhead. Mediaeval Latin manuscripts are a natural
first corpus for studying this problem, because there are many scanned manuscripts and
because the handwriting is relatively regular. We expect the primary user need to be search
over a large body of documents ? to allow comparisons between documents ? rather
than transcription of a particular document (which is usually relatively easy to do by hand).
Desirable features for a system are: First, that it use little or no aligned training data (an
ideal, which we believe may be attainable, is an unsupervised learning system). Second,
that one can search the document for an arbitrary string (rather than, say, only complete
words that appear in the training data). This would allow a user to determine whether a
document contains curious or distinctive spellings, for example (figure 7).
We show that, using a statistical model based on a generalized HMM, we can search a medieval manuscript with considerable accuracy, using only one instance each of each letter
in the manuscript to train the method (22 instances in total; Latin has no j, k, w, or z). Furthermore, our method allows fairly accurate transcription of the manuscript. We train our
system on 22 glyphs taken from a a 12th century latin manuscript of Terence?s Comedies
(obtained from a repository of over 80 scanned medieval works maintained by Oxford University [1]). We evaluate searches using a considerable portion of this manuscript aligned
by hand; we then show that fair search results are available on a different manuscript (MS.
Auct. D. 2. 16, Latin Gospels with beast-headed evangelist portraits made at Landvennec,
Brittany, late 9th or early 10th century, from [1]) without change of letter templates.
1.1. Previous Work
Handwriting recognition is a traditional problem, too well studied to review in detail here
(see [6]). Typically, online handwriting recognition (where strokes can be recorded) works
better than offline handwriting recognition. Handwritten digits can now be recognized
with high accuracy [2, 5]. Handwritten amounts can be read with fair accuracy, which
is significantly improved if one segments the amount into digits at the same time as one
recognizes it [4, 5]. Recently several authors have proposed new techniques for search
and translation in this unrestricted setting. Manmatha et al [7] introduce the technique of
?word spotting,? which segments text into word images, rectifies the word images, and
then uses an aligned training set to learn correspondences between rectified word images
and strings. The method is not suitable for a heavily inflected language, because words
take so many forms. In an inflected language, the natural unit to match to is a subset of a
word, rather than a whole word, implying that one should segment the text into blocks ?
which may be smaller than words ? while recognizing. Vinciarelli et al [8] introduce a
method for line by line recognition based around an HMM and quite similar to techniques
used in the speech recognition community. Their method uses a window that slides along
the text to obtain features; this has the difficulty that the same window is in some places too
small (and so uninformative) and in others too big (and so spans more than one letter, and is
confusing). Their method requires a substantial body of aligned training data, which makes
it impractical for our applications. Close in spirit to our work is the approach to machine
translation of Koehn and Knight [3]. They demonstrate that the statistics of unaligned
corpora may provide as powerful constraints for training models as aligned bitexts.
2. The Model
Our models for both search and transcription are based on the generalized HMM and differ
only in their choice of transition model. In an HMM, each hidden node ct emits a single
evidence node xt . In a generalized HMM, we allow each ct to emit a series of x?s whose
length is itself a random variable. In our model, the hidden nodes correspond to letters
and each xt is a single column of pixels. Allowing letters to emit sets of columns lets us
accomodate letter templates of variable width. In particular, this means that we can unify
segmenting ink into letters and recognizing blocks of ink; figure 3 shows an example of
how useful this is.
2.1. Generating a line of text
Our hidden state consists of a character label c, width w and vertical position y. The
statespace of c contains the characters ?a?-?z?, a space ? ?, and a special end state ?. Let T c
be the template associated with character c, Tch , Tcw be respectively the height and width
of that template, and m be the height of the image.
Figure 1: Left, a full page of our manuscript, a 12?th century manuscript of Terence?s
Comedies obtained from [1]. Top right, a set of lines from a page from that document and
bottom right, some words in higher resolution. Note: (a) the richness of page layout; (b)
the clear spacing of the lines; (c) the relatively regular handwriting.
Figure 2: Left, the 22 instances, one per letter, used to train our emission model. These
templates are extracted by hand from the Terence document. Right, the five image channels
for a single letter.
Beginning at image column 1 (and assuming a dummy space before the first character),
?
?
?
?
choose character c ? p(c|c?1...?n ) (an n-gram letter model)
choose length w ? Uniform(Tcw ? k, Tcw + k) (for some small k)
choose vertical position y ? Uniform(1, m ? Tch )
z,y and Tch now define a bounding box b of pixels. Let i and j be indexed from
the top left of that bounding box.
? draw pixel (i, j) ? N (Tcij , ?cij ) for each pixel in b
? draw all pixels above and below b from background gaussian N (?0 , ?0 )
(See 2.2 for greater detail on pixel emission model)
? move to column w + 1 and repeat until we enter the end state ?.
Inference on a gHMM is a relatively straighforward business of dynamic programming. We have used unigram, bigram and trigram models, with each transition
model fitted using an electronic version of Caesar?s Gallic Wars, obtained from
http://www.thelatinlibrary.com. We do not believe that the choice of author
should significantly affect the fitted transition model ? which is at the level of characters
? but have not experimented with this point. The important matter is the emission model.
2.2. The Emission Model
Our emission model is as follows: Given the character c and width w, we generate a template of the required length. Each pixel in this template becomes the mean of a gaussian
which generates the corresponding pixel in the image. This template has a separate mean
image for each pixel channel. The channels are assumed independent given the means.
We train the model by cutting out by hand a single instance of each letter from our corpus
(figure 2). This forms the central portion of the template. Pixels above and below this
Model
Perfect transcription
unigram
bigram
trigram
matching chars
21019
14603
15572
15788
substitutions
0
5487
4597
4410
insertions
0
534
541
507
deletions
0
773
718
695
Table 1: Edit distance between our transcribed Terence and the editor?s version. Note the
trigram model produces significantly fewer letter errors than the unigram model, but that
the error rate is still a substantial 25%.
central box are generated from a single gaussian used to model background pixels (basically
white pixels). We add a third variable yt to our hidden state indicating the vertical position
of the central box. However, since we are uninterested in actually recovering this variable,
during inference we sum it out of the model. The width of a character is constrained to be
close to the width (tw ) of our hand cut example by setting p(w|c) = 0 for w < tw ? k and
w > tw + k. Here k is a small, user defined integer. Within this range, p(w|c) is distributed
uniformly, larger templates are created by appending pixels from the background model to
the template and smaller ones by simply removing the right k-most columns of the hand
cut example.
For features, we generate five image representations, shown in figure 2. The first is a grayscale version of the original color image. The second and third are generated by convolving
the grayscale image with a vertical derivative of gaussian filter, separating the positive
and negative components of this response, and smoothing each of these gradient images
separately. The fourth and fifth are generated similarly but with a horizontal derivative of
gaussian filter. We have experimented with different weightings of these 5 channels. In
practice we use the gray scale channel and the horizontal gradient channels. We emphasize
the horizontal pieces since these seem the more discriminative.
2.3. Transcription
For transcription, we model letters as coming from an n-gram language model, with no
dependencies between words. Thus, the probability of a letter depends on the k letters
before it, where k = n unless this would cross a word boundary in which case the history
terminates at this boundary. We chose not to model word to word transition probabilities
since, unlike in English, word order in Latin is highly arbitrary. This transition model is
fit from a corpus of ascii encoded latin. We have experimented with unigram (i.e. uniform
transition probabilities), bigram and trigram letter models. We can perform transcription by
fitting the maximum likelihood path through any given line. Some results of this technique
are shown in figure 3.
2.4. Search
For search, we rank lines by the probability that they contain our search word. We set up
a finite state machine like that in figure 4. In this figure, ?bg? represents our background
model for that portion of the line not generated by our search word. We can use any of
the n-gram letter models described for transcription as the transition model for ?bg?. The
probability that the line contains the search word is the probability that this FSM takes
path 1. We use this FSM as the transition model for our gHMM, and output the posterior
probability of the two arrows leading into the end state. 1 and 2 are user defined weights,
but in practice the algorithm does not appear to be particular sensitive to the choice of these
parameters. The results presented here use the unigram model.
Editorial translation Orator ad vos venio ornatu prologi:
unigram
b u rt o r
a d u o s
u em
o
o r n a t u
p r o l o g r
b u rt o r
a d v o s
v em
o
o r u a t u
p r o l o g r
fo r a t o r
a d v o s
v en i o
o r n a t u
p r o l o g i
bigram
trigram
Figure 3: We transcribe the text by finding the maximum likelihood path through the
gHMM. The top line shows the standard version of the line (obtained by consensus among
editors who have consulted various manuscripts; we obtained this information in electronic form from http://www.thelatinlibrary.com). Below, we show the line
as segmented and transcribed by unigram, bigram and trigram models; the unigram and
bigram models transcribe one word as ?vemo?, but the stronger trigram model forces the
two letters to be segmented and correctly transcribes the word as ?venio?, illustrating the
considerable benefit to be obtained by segmenting only at recognition time.
1 ? ?1
Path 1
1 ? ?2
a
b
bg
?1
?
bg
Path 2
?2
Figure 4: The finite state machine to search for the word ?ab.? ?bg? is a place holder for the
larger finite state machine defined by our language model?s transition matrix.
3. Results
Figure 1 shows a page from our collection. This is a scanned 12th century manuscript
of Terence?s Comedies, obtained from the collection at [1]. In preprocessing, we extract
individual lines of text by rotating the image to various degrees and projecting the sum of
the pixel values onto the y-axis. We choose the orientation whose projection vector has the
lowest entropy, and then segment lines by cutting at minima of this projection.
Transcription is not our primary task, but methods that produce good transcriptions are
going to support good searches. The gHMM can produce a surprisingly good transcription,
given how little training data is used to train the emission model. We aligned an editors
version of Terence with 25 pages from the manuscript by hand, and computed the edit distance between the transcribed text and the aligned text; as table 1 indicates, approximately
75% of letters are read correctly.
Search results are strong. We show results for two documents. The first set of results
refers to the edition of Terence?s Comedies, from which we took the 22 letter instances.
In particular, for any given search term, our process ranks the complete set of lines. We
used a hand alignment of the manuscript to determine which lines contained each term;
figure 5 shows an overview of searches performed using every word that appears in the
50
100
150
200
250
300
350
400
450
500
550
Figure 5: Our search ranks 587 manuscript lines, with higher ranking lines more likely
to contain the relevant term. This figure shows complete search results for each term that
appears more than three times in the 587 lines. Each row represents the ranked search
results for a term, and a black mark appears if the search term is actually in the line; a
successful search will therefore appear as a row which is wholly dark to the left, and then
wholly light. All 587 lines are represented. More common terms are represented by lower
rows. More detailed results appear in figure 5 and figure 6; this summary figure suggests
almost all searches are highly successful.
document more than three times, in particular, showing which of the ranked set of lines
actually contained the search term. For almost every search, the term appears mainly in
the lines with higher rank. Figure 6 contains more detailed information for a smaller set of
words. We do not score the position of a word in a line (for practical reasons).
Figure 7 demonstrates (a) that our search respects the ink of the document and (b) that for
the Terence document, word positions are accurately estimated. The spelling of mediaeval
documents is typically cleaned up by editors; in our manuscript, the scribe reliably spells
?michi? for the standard ?mihi?. A search on ?michi? produces many instances; a search
on ?mihi? produces none, because the ink doesn?t have any. Notice this phenomenon also
in the bottom right line of figure 7, the scribe writes ?habet, ut consumat nunc cum nichil
obsint doli? and the editor gives ?habet, ut consumat nunc quom nil obsint doli.? Figure 8
shows that searches on short strings produce many words containing that string as one
would wish.
4. Discussion
We have shown that it is possible to make at least some handwritten mediaeval manuscripts
accessible to full text search, without requiring an aligned text or much supervisory data.
Our documents have very regular letters, and letter frequencies ? which can be obtained
from transcribed Latin ? appear to provide so powerful a cue that relatively little detailed
information about letter shapes is required. Linking letter segmentation and recognition
has thoroughly beneficial effects. This suggests that the pool of manuscripts that can be
made accessible in this way is large. In particular, we have used our method, trained on
22 instances of letters from one document, to search another document. Figure 9 shows
the results from two searches of our second document (MS. Auct. D. 2. 16, Latin Gospels
with beast-headed evangelist portraits made at Landvennec, Brittany, late 9th or early 10th
century, from [1]). No information from this document was used in training at all; but letter
1tu
arbitror
pater
etiam
nisi
factum
primum
siet
vero
illi
inter
hic
michi
ibi
qui
tu
ibi
michi
0.9
0.8
0.7 qui
hic
0.6
inter
0.5
illi
0.4
siet
0.3
vero
0.2
nisi
0.1
50
100
150
200
250
300
350
400
450
500
550
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Figure 6: On the left, search results for selected words (indicated on the leftmost column).
Each row represents the ranked search results for a term, and a black mark appears if
the search term is actually in the line; a successful search will therefore appear as a row
which is wholly dark to the left, and then wholly light. Note only the top 300 results are
represented, and that lines containing the search term are almost always at or close to the
top of the search results (black marks to the left). On the right, we plot precision against
recall for a set of different words by taking the top 10, 20, ... lines returned from the search,
and checking them against the aligned manuscript. Note that, once all cases have been
found, if the size of the pool is increased the precision will fall with 100% recall; many
words work well, with most of the first 20 or so lines returned containing the search term.
shapes are sufficiently well shared that the search is still useful.
All this suggests that one might be able to use EM to link three processes: one that clusters
to determine letter shapes; one that segments letters; and one that imposes a language
model. Such a system might be able to make handwritten Latin searchable with no training
data.
References
[1] Early Manuscripts at Oxford University.
Bodleian library ms. auct. f. 2.13.
http://image.ox.ac.uk/.
[2] Serge Belongie, Jitendra Malik, and Jan Puzicha. Shape matching and object recognition using shape contexts. IEEE T. Pattern Analysis and Machine Intelligence,
24(4):509?522, 2002.
[3] Philipp Koehn and Kevin Knight. Estimating word translation probabilities from unrelated monolingual corpora. In Proc. of the 17th National Conf. on AI, pages 711?715.
AAAI Press / The MIT Press, 2000.
[4] Y. LeCun, L. Bottou, and Y. Bengio. Reading checks with graph transformer networks.
In International Conference on Acoustics, Speech, and Signal Processing, volume 1,
pages 151?154, Munich, 1997. IEEE.
[5] Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner. Gradient-based learning applied to
document recognition. Proceedings of the IEEE, 86(11):2278?2324, 1998.
[6] R. Plamondon and S.N. Srihari. Online and off-line handwriting recognition: a comprehensive survey. IEEE Transactions on Pattern Analysis and Machine Intelligence,
22(1):63?84, 2000.
[7] T. M. Rath and R. Manmatha. Word image matching using dynamic time warping.
In Proc. of the Conf. on Computer Vision and Pattern Recognition (CVPR), volume 2,
pages 521?527, 2003.
[8] Alessandro Vinciarelli, Samy Bengio, and Horst Bunke. Offline recognition of unconstrained handwritten texts using hmms and statistical language models. IEEE Trans.
Pattern Anal. Mach. Intell., 26(6):709?720, 2004.
michi: Spe incerta certum mihi laborem sustuli,
mihi: Faciuntne intellegendo ut nil intellegant?
michi: Nonnumquam conlacrumabat. placuit tum id mihi.
mihi: Placuit: despondi. hic nuptiis dictust dies.
michi: Sto exspectans siquid mi imperent. venit una, "heus tu" inquit "Dore,
mihi: Meam ne tangam? CH. Prohibebo inquam. GN. Audin tu? hic furti se adligat:
mihi: Habet, ut consumat nunc quom nil obsint doli;
michi: Quando nec gnatu? neque hic mi quicquam obtemperant,
Figure 7: The handwritten text does not fully correspond to the transcribed version; for
example, scribes commonly write ?michi? for the standard ?mihi?. Our search process
reflects the ink fairly faithfully, however. Left the first four lines returned for a search on
the string ?michi?; right the first four lines returned for a search on the string ?mihi?,
which does not appear in the document. Note that our search process can offer scholars
access to the ink in a particular document, useful for studying variations in transcription,
etc.
tu: Quid te futurum censes quem adsidue exedent?
tu: Quae ibi aderant forte unam aspicio adulescentulam
Figure 8: Searches on short strings produce substrings of words as well as words (we show
the first two lines returned from a search for ?tu?).
sunt
interrogaverunt
Figure 9: The first six lines returned from the second manuscript, (MS. Auct. D. 2. 16, Latin
Gospels with beast-headed evangelist portraits made at Landvennec, Brittany, late 9th or
early 10th century, from [1]), in response to the queries ?interrogeraverunt? (left; lines
three and six contain the word, which is localized largely correctly) and ?sunt? (right;
lines one and four contain the word). We do not have aligned text, so cannot measure the
recall and precision for searches on this document. The recall and precision are clearly
not as good as those for the Terence document, the search is reasonably satisfactory, given
that no training information from this document was available.
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1,880 | 2,707 | Neural network computation by
in vitro transcriptional circuits
Jongmin Kim1 , John J. Hopfield3 , Erik Winfree2
Biology , CNS and Computer Science2 , California Institute of Technology.
Molecular Biology3 , Princeton University.
{jongmin,winfree}@dna.caltech.edu, [email protected]
1
Abstract
The structural similarity of neural networks and genetic regulatory networks to digital circuits, and hence to each other, was noted from the
very beginning of their study [1, 2]. In this work, we propose a simple
biochemical system whose architecture mimics that of genetic regulation and whose components allow for in vitro implementation of arbitrary circuits. We use only two enzymes in addition to DNA and RNA
molecules: RNA polymerase (RNAP) and ribonuclease (RNase). We
develop a rate equation for in vitro transcriptional networks, and derive a correspondence with general neural network rate equations [3].
As proof-of-principle demonstrations, an associative memory task and a
feedforward network computation are shown by simulation. A difference
between the neural network and biochemical models is also highlighted:
global coupling of rate equations through enzyme saturation can lead
to global feedback regulation, thus allowing a simple network without
explicit mutual inhibition to perform the winner-take-all computation.
Thus, the full complexity of the cell is not necessary for biochemical
computation: a wide range of functional behaviors can be achieved with
a small set of biochemical components.
1
Introduction
Biological organisms possess an enormous repertoire of genetic responses to everchanging combinations of cellular and environmental signals. Characterizing and decoding the
connectivity of the genetic regulatory networks that govern these responses is a major challenge of the post-genome era [4]. Understanding the operation of biological networks is intricately intertwined with the ability to create sophisticated biochemical networks de novo.
Recent work developing synthetic genetic regulatory networks has focused on engineered
circuits in bacteria wherein protein signals are produced and degraded [5, 6]. Although
remarkable, such network implementations in bacteria have many unknown and uncontrollable parameters.
We propose a biochemical model system ? a simplified analog of genetic regulatory circuits
? that provides well-defined connectivity and uses nucleic acid species as fuel and signals
that control the network. Our goal is to establish an explicit model to guide the laboratory
construction of synthetic biomolecular systems in which every component is known and
RNAP
transcript
RNase
inhibitor
(A)
activator
DNA switch
(B)
Figure 1: (A) The components of an in vitro circuit. The switch template (blue) is shown
with the activator (red) attached. The dotted box indicates the promoter sequence and the
downstream direction. (B) The correspondence between a neural network and an in vitro
biochemical network. Neuron activity corresponds to RNA transcript concentration, while
synaptic connections correspond to DNA switches with specified input and output.
where quantitative predictions can be tested. Only two enzymes are used in addition to synthetic DNA templates: RNA polymerase, which recognizes a specific promoter sequence
in double-stranded DNA and transcribes the downstream DNA to produce an RNA transcript, and ribonuclease, which degrades RNA but not DNA. In this system, RNA transcript
concentrations are taken as signals. Synthetic DNA templates may assume two different
conformations with different transcription efficiency: ON or OFF. Upon interaction with
a RNA transcript of the appropriate sequence, the DNA template switches between different conformations like a gene regulated by transcription factors. The connectivity ? which
RNA transcripts regulate which DNA templates ? is dictated by Watson?Crick base-pairing
rules and is easy to program. The network computation is powered by rNTP that drives the
synthesis of RNA signals by RNAP, while RNase forces transient signals to decay. With
a few assumptions, we find that this stripped-down analog of genetic regulatory networks
is mathematically equivalent to recurrent neural networks, confirming that a wide range of
programmable dynamical behaviors is attainable.
2
Construction of the transcriptional network
The DNA transcriptional switch. The elementary unit of our networks will be a DNA
switch, which serves the role of a gene in a genetic regulatory circuit. The basic requirements for a DNA switch are to have separate input and output domains, to transcribe poorly
by itself [7], and to transcribe efficiently when an activator is bound to it. A possible mechanism of activation is the complementation of an incomplete promoter region, allowing
more favorable binding of RNAP to the DNA template. Figure 1A illustrates our proposed
design for DNA transcriptional switches and circuits. We model a single DNA switch with
the following binding reactions:
A + I
AI
OFF
D +A
DA
ON
ON
DA + I
D + AI
OFF
where D (blue) is a DNA template with an incomplete promoter region, A (red) is an
activator that complements the incomplete promoter region, and I (green) is an inhibitor
complementary to A. Thus, I can bind free A. Furthermore, activator A contains a ?toehold? region [8] that overhangs past the end of D, allowing inhibitor I to strip off A from
the DA complex. D is considered OFF and DA is considered ON, based on their efficiency
as templates for transcription. This set of binding reactions provides a means to choose the
threshold of the sigmoidal activation function, as will be explained later.
RNAP and RNase drive changes in RNA transcript concentration; their activity is modeled
using a first-order approximation for enzyme kinetics. For the moment, we assume that the
input species (activator and inhibitor) are held at constant levels by external control.
By RNA polymerase
By RNase
kp
k
R ?d ?
DA ? DA + R
?kp
D ? D+R
where 0 < ? < 1 due to lack of activation and ? represents the complete degradation of
RNA products by RNase. kd and kp are set by the concentration of enzymes.
In general, a set of chemical reactions obeying mass action have dynamics described by
?
d[Xi ] X Y
=
k?
[Xj ]rj (p?i ? ri? )
dt
j
?
ri?
where k? is the rate constant,
is the stoichiometry of species Xi as a reactant (typically
0 or 1), and p?i is the stoichiometry of Xi as a product in reaction ?. Analysis of our
system is greatly simplified by the assumption that the binding reactions are fast and go
to completion. We define D tot as the sum of free and bound species:D tot = [D] + [DA].
Similarly, I tot = [I]+[AI] and Atot = [A]+[DA]+[AI]. Then, [DA] depends on D tot and
?, where ? = Atot ? I tot . Because I can scavenge A whether the latter is free or bound
to D, A can activate D only when ? > 0. The amount of [DA] is proportional to ? when
0 < ? < D tot , as shown in Figure 2A. It is convenient to represent this nonlinearity using a
piecewise-linear approximation of a sigmoidal function, specifically, ?(x) = |x+1|?|x?1|
.
2
? where
Thus, we can represent [DA] using ? and a rescaled ?: [DA] = 12 Dtot (1 + ?(?)),
? = 2?
?
D tot ? 1 is called the signal activity. At steady-state, kd [R] = kp [DA] + ?kp [D];
thus,
1 kp tot
? + 1 + ?) .
[R] =
D ((1 ? ?)?(?)
2 kd
If we consider the activator concentration as an input and the steady-state transcript concentration as an output, then the (presumed constant) inhibitor concentration, I tot , sets the
threshold, and the function assumes a sigmoidal shape (Fig. 2D). Adjusting the amount of
template, D tot , sets the magnitude of the output signal and the width of the transition region (Fig. 2C). We can adjust the width of the transition region independent of the threshold
such that a step function would be achieved in the limit. Thus, we have a sigmoidal function with an adjustable threshold, without reliance on cooperative binding of transcription
factors as is common in biological systems [9].
Networks of transcriptional switches. The input domain of a DNA switch is upstream
of the promoter region; the output domain is downstream of the promoter region. This
separation of domains allows us to design DNA switches that have any desired connectivity.
[ DA ]
D
? (x)
tot
1
D
tot
?1
?
[R]
1
[R]
x
?1
(A)
(B)
(C)
A
tot
(D)
A
tot
Figure 2: (A) [DA] as a function of ?. (B) The sigmoid ?(x). (C,D) [R] as a function of
Atot for three values of D tot and I tot , respectively.
We assume that distinct signals in the network are represented as distinct RNA sequences
that have negligible crosstalk (undesired binding of two molecules representing different
signals). The set of legitimate binding reactions is as follows:
OFF
A j+ I j
A jI j
D ij+ A j
D ij Aj
ON
D ij + A j I j
OFF
D ij Aj + I j
ON
where Dij is the DNA template that has the jth input domain and ith output domain, the
activator Aj complements the incomplete promoter region of Dij , and the inhibitor Ij is
complementary to Aj . Note that Ij can strip off Aj from the Dij Aj complex, thus imposing
a sharp threshold as before. Again, we assume fast and complete binding reactions.
The set of enzyme reactions for the transcriptional network is as follows:
By RNA polymerase
kp
Dij Aj ? Dij Aj + Ai
By RNase
if sij = 1
?kp
Dij ? Dij + Ai
kp
Dij Aj ? Dij Aj + Ii
?kp
Dij ? Dij + Ii
k
Ij ?d ?
k
if sij = ?1
Aj ?d ?
k
Aj Ij ?d ?
k
Dij Aj ?d Dij
where sij ? {+1, ?1} indicates whether switch ij will produce an activator or an inhibitor.
This notation reflects that the production of Ii is equivalent to the consumption of Ai . The
tot
change of RNA concentrations over time is easy to express with ?i = Atot
i ? Ii :
X
d?i
= ?kd ? ?i + kp
sij ([Dij Aj ] + ?[Dij ]) .
(1)
dt
j
Network equivalence. We show next that the time evolution of this biochemical network
model is equivalent to that of a general Hopfield neural network model [3]:
X
dxi
wij ?(xj ) + ?i .
(2)
= ?xi +
?
dt
j
Equation 1 can be rewritten to use the same nonlinear activation function ? defined earlier.
i
? i = 2?
Let ?
tot ? 1 be a rescaled difference between activator and inhibitor concentrations,
D?i
tot
where DP
?i is the load on Ai , i.e., the total concentration of all switches that bind to A i :
tot
tot
tot
D?i = j Dji
and Dij
= [Dij Aj ] + [Dij ]. Then, we can derive the following rate
? i plays the role of unit i?s activity xi :
equation, where ?
X
tot
tot
X kp
?i
Dij
Dij
kp
1 d?
?
?
= ? ?i +
(1 ? ?)sij tot ?(?j )+
(1 + ?)sij tot ? 1 .
kd dt
kd
D?i
kd
D?i
j
j
(3)
Given the set of constants describing an arbitrary transcriptional network, the constants for
an equivalent neural network can be obtained immediately by comparing Equations 2 and
3. The time constant ? is the inverse of the RNase degradation rate: fast turnover of RNA
molecules leads to fast response of the network. The synaptic weight wij is proportional to
the concentration of switch template ij, attenuated by the load on Ai . However, the threshold ?i is dependent on the weights, perhaps implying a lack of generality. To implement
an arbitrary neural network, we must introduce two new types of switches to the transcriptional network. To achieve arbitrary thresholds, we introduce bias switches D iB which
have no input domain and thus produce outputs constitutively; this adds an adjustable constant to the right hand side of Equation 3. To balance the load on Ai , we add null switches
tot
D0i which bind to Ai but have no output domain; this allows us to ensure that all D?i
are
equal. Consequently, given any neural network with weights wij and thresholds ?i , we can
tot
specify concentrations Dij
such that the biochemical network has identical dynamics, for
some ? .
Michaelis?Menten enzyme reactions. Next, we explore the validity of our assumption
that enzyme kinetics are first-order reactions. A basic but more realistic model is the
Michaelis?Menten mechanism [10], in which the enzyme and substrate bind to form an
enzyme-substrate complex. For example, if E is RNAP,
k+
kcat
E + Dij Aj *
) EDij Aj ? E + Dij Aj + Ii /Ai .
k?
An important ramification of Michaelis?Menten reactions is that there is competition for
the enzyme by the substrates, because the concentration of available enzymes is reduced
as they bind to substrates, leading to saturation when the enzyme concentration is limiting. Using the steady-state assumption for Michaelis?Menten reactions, we establish the
following relations to the rate constants of first-order reactions:
kp =
E tot kcat
?
1 + L KM
? ? kp =
0
E tot kcat
? 0
1 + L KM
kd =
Edtot
kd,cat
?
1 + Ld Kd,M
(4)
where kcat and KM = (k? + kcat )/k+ are the catalytic constant (enzyme?s speed) and
0
Michaelis constant (enzyme?s affinity to target) of RNAP for the ON state switch, k cat
and
0
KM are for the OFF state switch, and kd,cat and Kd,M are the constants of RNase. E tot
P [D A ]
and Edtot are the concentrations of RNAP and RNase, respectively. L = i,j KijM j +
P [Aj ]+[Ij ]+[Aj Ij ]+[Dij Aj ]
P [Dij ]
is the load on RNAP and Ld =
is the load on
0
i,j
i,j KM
Kd,M
RNase (i.e., the total concentration of binding targets divided by the Michaelis constants of
the enzymes), both of which may be time varying. To make the first-order approximation
valid, we must keep L and Ld constant. Introduction of a new type of switch with different
Michaelis constants can make L constant by balancing the load on the enzyme. A scheme
to keep Ld constant is not obvious, so we set reaction conditions such that Ld 1.
3
Example computations by transcriptional networks
Feed-forward networks.
We first consider a feed-forward network to compute
f (x, y, z) = x
?yz + y?z + x. From the Boolean circuit shown in Figure 3A, we can construct
an equivalent neural network. We label units 1 through 6: units 1, 2, 3 correspond to inputs
x, y, z whereas units 4, 5, 6 are computation units. Using the conversion rule discussed
in the network equivalence section, we can calculate the parameters of the transcriptional
network. Under the first-order approximation of Equation 3, the simulation result is exact
(Fig. 3C). For comparison, we also explicitly simulated mass action dynamics for the full
set of chemical equations with the Michaelis?Menten enzyme reactions, using biologically
plausible rate constants and with E tot and Edtot calculated from Equation 4 using estimated
values of L and Ld . The full model performs the correct calculation of f for all eight 3-bit
inputs, although the magnitude of signals is exaggerated due to an underestimate of RNase
load (Fig. 3C).
Associative memories. Figure 4A shows three 4-by-4 patterns to be memorized in a continuous neural network [3]. We chose orthogonal patterns because a 16 neuron network has
limited capacity. Our training algorithm is gradient descent combined with the perceptron
learning rule. After training, the parameters of the neural network are converted to the
parameters of the transcriptional network as previously described. Starting from a random
6
f
1
?1
y
y
z
(B)
z
4
?2 1
1
2
2
1
f
1
0
?2
?1
1
(A)
1
?1
^
?
i
x
x
?4
(C)
0
200
400
600
time(sec)
800
1000
Figure 3: (A,B) A Boolean circuit and a neural network to compute f (x, y, z) = x
?yz+?
y z+
x. (C) The activity of computation units (first-order approximation: solid lines; MichaelisMenten reaction: dotted lines) for x=True=1, y=False=?1, z=True=1.
3
2
^
?
i
1
0
?1
?2
(A)
(B)
?3
0
200
400
time(sec)
600
Figure 4: (A) The three patterns to be memorized. (B) Time-course for the transcriptional
network recovery of the third pattern. (odd columns: blue lines, even columns: red lines)
initial state, a typical response of the transcriptional network (with the first-order approximation of Equation 3) is shown in Figure 4B. Thus, our in vitro transcriptional networks
can support complex sets of stable steady-states.
A winner-take-all network. Instead of trying to compensate for the saturation phenomena
of Michaelis?Menten reactions, we can make use of it for computation. As an example,
consider the winner-take-all computation [11], which is commonly implemented as a neural network with O(N 2 ) mutually inhibitory connections (Fig. 5A), but which can also be
implemented as an electrical circuit with O(N ) interconnections by using a single global
inhibitory feedback gate [12]. In a biochemical system, a limited global resource, such as
RNAP, can act to regulate all the DNA switches and thus similarly produce global inhibition. This effect is exploited by the simple transcriptional network shown in Figure 5B, in
which the output from each DNA switch activates the same DNA switch itself, and mutual
inhibition is achieved by competition for RNAP. Specifically, we have switch templates D ii
with fixed thresholds set by Ii , and Dii produces Ai as its output RNA. With the instant
binding assumption, we then derive the following equation:
dAtot
E tot
kcat
kd,cat tot
E tot
k0
i
=? d
?
Ai +
[Dii Ai ] + cat
[D
]
.
(5)
ii
0
dt
1 + Ld Kd,M
1 + L KM
KM
The production rate of Ai depends on Atot
and on L, while the degradation rate of Ai
i
depends on Atot
and
on
L
,
as
shown
in
Figure
6A. For a winner-take-all network, an ON
d
i
state switch draws more RNAP than an OFF state switch (because of the smaller Michaelis
constant for the ON state). Thus, if the other switches are turned OFF, the load on RNAP
(L) becomes small, leading to faster production of the remaining ON switches. When the
production rate curve and the degradation rate curve have three intersections, bistability is
achieved such that the switches remain ON or OFF, depending on their current state.
Consider n equivalent switches starting with initial activator concentrations above the
threshold, and with the highest concentration at least ? above the rest (as a percentage).
Analysis indicates that a less leaky system (small ?) and sufficient differences in initial
activator concentrations (large ?) can guarantee the existence of a unique winner. Simulations of a 10-switch winner-take-all network confirm this analysis, although we do not see
perfect behavior (Fig. 6B). Figure 6C shows a time-course of a unique winner situation.
Switches get turned OFF one by one whenever the activator level approaches the threshold,
until only one switch remains ON.
?1
?1
0.5
?1
0.5
?1
(A)
1
?1
0.5
0.5
0.5
0.5
1
1
1
?1
1
(B)
1
Figure 5: (A) A 3-unit WTA network with explicit mutual inhibition. (B) An equivalent
biochemical network.
1
25
0.8
20
1/?
0.6
15
0.4
L : high
10
tot
Ii
tot
tot
Ii +D ii
tot
Ai
(B)
2
1.5
1
0.2
5
(A)
3
2.5
i
L : low
dt
30
[A ] / ?M
tot
dAi
0.5
5
10
?(%)
15
0
(C)
0
0
5000
10000
time(sec)
15000
Figure 6: For WTA networks: (A) Production rates (solid lines) for two different L?s,
compared to a linear degradation rate (dotted line). (B) Empirical probability of correct
output as a function of ? and ?. (C) Time-course with ? = 0.33% and ? = 0.04.
Similarly, we can consider a k-WTA network where k winners persist. If we set the parameters appropriately such that k winners are stable but k + 1 winners are unstable, the
simulation result recovered k winners most of the time. Even a single k-WTA gate can
provide impressive computational power [13].
4
Discussion
We have shown that, if we treat transcriptionally controlled DNA switches as synapses
and the concentrations of RNA species as the states of neurons, then the in vitro transcriptional circuit is equivalent to the neural network model and therefore can be programmed
to carry out a wide variety of tasks. The structure of our biochemical networks differs
from that of previous formal models of genetic regulatory circuits [14, 15, 16]. For example, consider the work of [16], which established a connection to the class of Boltzmann
machines. There, the occupancy of regulatory binding sites corresponds to the state of neurons, the weights are set by the cooperative interaction among transcription factors, and the
thresholds are the effective dissociation constants at a binding site. Thus, implementing a
general N -unit neural network requires only O(N ) biochemical species, but up to O(N 2 )
significant binding interactions must be encoded in the molecular sequences. Changing
or tuning a network is therefore non-trivial. In contrast, in our transcriptional networks,
each weight and threshold is represented by the continuously adjustable concentration of a
distinct species, and the introduction or deletion of any node is straightforward.
Each synapse is represented by a DNA switch with a single input?output specification, so
the number of DNA switches grows as O(N 2 ) for a fully recurrent neural network with
N neurons (unlike the circuits of [16]). This constraint may be relieved because, in many
networks of interest, most nodes have a small number of connections [17, 18]. The time
for computation will increase as O(N ) due to finite hybridization rates because, if the total
concentration of all RNA signals is capped, the concentration of any given species will
decrease as 1/N . The weights are capped by the maximum gain of the system, which is the
production rate divided by the degradation rate. Since the time constant of the network is
the inverse of the degradation rate, if we wish to implement a network with large weights,
we must increase the time constant.
We can analyze the cost of computing by considering basic physical chemistry. The energy
consumption is about 20kT (= 10?19 J) per nucleotide incorporated, and 1 bit of informa-
tion is encoded by a sequence containing tens of nucleotides. The encoding energy is large,
since the molecule for each bit must contain specific instructions for connectivity, unlike
spatially arranged digital circuits where a uniform physical signal carrier can be used. Furthermore, many copies (e.g., 1013 for a 1?M signal in 20?l) of a given species must be
produced to change the concentration in a bulk sample. Worse yet, because degradation is
not modulated in the transcriptional network, switching relies on selective change of production rates, thus continually using energy to maintain an ON state. Devising a scheme to
minimize maintenance energy costs, such as in CMOS technology for electrical circuits, is
an important problem.
The theory presented here is meant to serve as a guide for the construction of real biochemical computing networks. Naturally, real systems will deviate considerably from the
idealized model (although perhaps less so than do neural network models from real neurons). For example, hybridization is neither instantaneous nor irreversible, strands can have
undesired conformations and crosstalk, and enzyme reactions depend on the sequence and
are subject to side reactions that generate incomplete products. Some problems, such as hybridization speed and crosstalk, can be reduced by slowing the enzyme reactions and using
proper sequence design [19]. Ultimately, some form of fault tolerance will be necessary at
the circuit level. Restoration of outputs to digital values, achieved by any sufficiently highgain sigmoidal activation function, provides some level of immunity to noise at the gate
level, and attractor dynamics can provide restoration at the network level. A full understanding of fault tolerance in biochemical computing remains an important open question.
Future directions include utilizing the versatility of active RNA molecules (such as aptamers, ribozymes, and riboswitches [20, 21]) for more general chemical input and output,
devising a biochemical learning scheme analogous to neural network training algorithms
[22], and studying the stochastic behavior of the transcriptional network when a very small
number of molecules are involved in small volumes [5].
Acknowledgements. We thank Michael Elowitz, Paul Rothemund, Casimir Wierzynski,
Dan Stick and David Zhang for valuable discussions, and ONR and NSF for funding.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
McCulloch WS, Pitts W, Bull. Math. Biophys. 5 (1943), 115.
Monod J, Jacob F, Cold Spring Harb. Symp. Quant. Biol. 26 (1961), 389-401.
Hopfield JJ, Proc. Nat. Acad. Sci. USA 81 (1984), 3088-3092.
Hasty J, McMillen D, Issacs F, Collins JJ, Nat. Rev. Genet. 2 (2001), 268-279.
Elowitz MB, Leibler S, Nature 403 (2000), 335-338.
Gardner TS, Cantor CR, Collins JJ, Nature 403 (2000), 339-342.
Martin CT, Coleman JE, Biochemistry 26 (1987), 2690-2696.
Yurke B, Mills AP Jr., Genetic Programming and Evolvable Machines 4 (2003), 111-122.
Shea MA, Ackers GK, J. Mol. Biol. 181 (1985), 211-230.
Hammes GG, Thermodynamics and kinetics for the biological sciences, Wiley (2000).
Yuille AL, Gieger D, in The Handbook of Brain Theory and Neural Networks, Arbib MA, ed.,
MIT Press (1995), 1056-1060.
Tank DW, Hopfield JJ, IEEE Trans. on Circuits and Systems 33 (1986), 533-541.
Maass W, Neural Computation 12 (2000), 2519-2535.
Glass L, Kauffman SA, J. Theo. Biol. 39 (1973), 103-129.
Mjolsness E, Sharp DH, Reinitz J, J. Theo. Biol. 152 (1991), 429-453.
Buchler NE, Gerland U, Hwa T, Proc. Nat. Acad. Sci. USA 100 (2003), 5136-5141.
Bray D, Science 301 (2003), 1864-1865.
Reed RD, IEEE Trans. on Neural Networks, 4 (1993), 740-744.
Dirks R, Lin M, Winfree E, Pierce NA, Nucleic Acids Research 32 (2004), 1392-1403.
Lilley DM, Trends Biochem. Sci. 28 (2003), 495-501.
Nudler E, Mironov AS, Trends Biochem. Sci. 29 (2004), 11-17.
Mills AP Jr., Yurke B, Platzman PM, Biosystems 52 (1999), 175-180.
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1,881 | 2,708 | Who?s in the Picture?
Tamara L. Berg, Alexander C. Berg, Jaety Edwards and D.A. Forsyth
Computer Science Division
U.C. Berkeley
Berkeley, CA 94720
[email protected]
Abstract
The context in which a name appears in a caption provides powerful cues
as to who is depicted in the associated image. We obtain 44,773 face images, using a face detector, from approximately half a million captioned
news images and automatically link names, obtained using a named entity recognizer, with these faces. A simple clustering method can produce fair results. We improve these results significantly by combining
the clustering process with a model of the probability that an individual
is depicted given its context. Once the labeling procedure is over, we
have an accurately labeled set of faces, an appearance model for each
individual depicted, and a natural language model that can produce accurate results on captions in isolation.
1
Introduction
It is a remarkable fact that pictures and their associated annotations are complementary.
This observation has been used to browse museum collections ([1]) and organize large
image collections ([2, 7, 12, 13]). All of these papers use fairly crude ?bag of words?
models, treating words as floating tags and looking at the co-occurrence of image regions
and annotated words. In this paper, we show that significant gains are available by treating
language more carefully.
Our domain is a large dataset of news photographs with associated captions. A face detector
is used to identify potential faces and a named entity recognizer to identify potential names.
Multiple faces and names from one image-caption pair are quite common. The problem is
to find a correspondence between some of the faces and names. As part of the solution
we learn an appearance model for each pictured name and the likelihood of a particular
instance of a name being pictured based on the surrounding words and punctuation.
Face recognition cannot be surveyed reasonably in the space available. Reviews appear
in [6, 10, 11]. Although face recognition is well studied, it does not work very well in
practice [15]. One motivation for our work is to take the large collection of news images
and captions as semi-supervised input and and produce a fully supervised dataset of faces
labeled with names. The resulting dataset exhibits many of the confounding factors that
make real-world face recognition difficult, in particular modes of variation that are not
found in face recognition datasets collected in laboratories. It is important to note that this
task is easier than general face recognition because each face has only a few associated
names.
N1
F1
N1
F1
N2
F2
N2
F2
Null
F1 =
F2 =
N1 = John Furlong,
Null
Null
N1
F1
N1
F1
N2
F2
N2
F2
Null
Null
Null
Null
N1
F1
N1
F1
N2
F2
N2
F2
N2 = Winter Olympics,
Null
Null
President and Chief Operating Officer of the Vancouver,
British Columbia 2010 Bid Corporation John Furlong
(rear) smiles while celebrating with compatriots their
victory in obtaining the 2010 Winter Olympics bid on late
July 2, 2003 in Prague. Vancouver won with 56 votes
against 53 votes for Pyeonchang in the second round of
balloting at an IOC gathering in Prague. REUTERS/Petr
Josek
Null
Null
N1
F1
N2
F2
Null
Null
Null
All possible name/face assignments
Figure 1: Left: A typical data item consisting of a picture and its associated caption.
Center: Detected faces and names for this data item. Right: The set of possible correspondences for this data item. Our model allows each face to be assigned to at most one
name, each name to be assigned to at most one face, and any face or name to be assigned
to Null. Our named entity recognizer occasionally identifies strings that do not refer to
actual people (e.g. ?Winter Olympics?) . These names are assigned low probability under
our model and therefore their assignment to a face is unlikely. EM iterates between computing the expectation of the possible face-name correspondences and updating the face
clusters and language model. Unusually, we can afford to compute all possible face-name
correspondences for a data item since the number of possibilities is small. For this item,
we correctly choose the best matching ?F1 to Null?, ?N2 to Null?, and ?F2 to N1?.
Language: Quite simple common phenomena in captions suggest using a language model.
First, our named entity recognizer occasionally marks incorrect names like ?United Nations?. The context in which these incorrect detections occur suggest that they do not refer
to actual people. Second, name-context pairs can be weighted according to their probability. In a caption such as ?Michael Jackson responds to questioning Thursday, Nov. 14, 2002
in Santa Maria Superior Court in San ta Maria, Calif., during a $21 million lawsuit brought
against him by Marcel Avram for failing to appear at two millennium concerts...?, Michael
Jackson appears in a more favorable context (at the beginning of the caption, followed by
a verb) than Marcel Avram (near the middle of the caption, followed by a preposition).
Our approach combines a simple appearance model using kPCA and LDA, with a language model, based on context. We evaluate both an EM and maximum likelihood clustering and show that incorporating language with appearance produces better results than
using appearance alone. We also show the results of the learned natural language classifier
applied to a set of captions in isolation.
2
Linking a face and language model with EM
A natural way of thinking about name assignment is as a hidden variable problem where
the hidden variables are the correct name-face correspondences for each picture. This suggests using an expectation maximization (EM) procedure. EM iterates between computing
before ? CEO Summit
after ? Martha Stewart
before ? U.S. Joint
after ? Null
before ? U.S. House
before ? Julia Vakulenko
after ? Andrew Fastow after ? Jennifer Capriati
before ? Angelina Jolie
after ? Jon Voight
before ? Ric Pipino
after ? Heidi Klum
before ? Vice President before ? Marcel Avram
Dick Cheney
after ? Michael Jackson
after ? President George W.
before ? U.S. Open
after ? David Nalbandian
before ? al Qaeda
after ? Null
before ? James Bond
after ? Pierce Brosnan
before ? James Ivory
after ? Naomi Watts
Figure 2: Names assigned using our raw clustering procedure (before) and incorporating a
language model (after). Our named entity recognizer occasionally detects incorrect names
(e.g. ?CEO Summit?), but based on context the language model assigns low probabilities
to these names, making their assignment unlikely. When multiple names are detected like
?Julia Vakulenko? and ?Jennifer Capriati?, the probability for each name depends on its
context. The caption for this picture reads ?American Jennifer Capriati returns the ball to
her Ukrainian opponent Julia Vakulenko in Paris during...? The language model prefers to
assign the name ?Jennifer Capriati? because its context (beginning of the caption followed
by a present tense verb) indicates it is more likely to be pictured than ?Julia Vakulenko?
(middle of the caption followed by a preposition). For pictures such the man labeled ?al
Qaeda? to ?Null? where the individual is not named in the caption, the language model
correctly assigns ?Null? to the face. As table 1 shows, incorporating a language model
improves our face clusters significantly.
the expected values of the set of face-name correspondences (given a face clustering and
language model) and updating the face clusters and language model given the correspondences. Unusually, it is affordable to compute the expected value of all possible face-name
correspondences for a data item since the number of possibilities is small.
To use EM we need a model of how pictures are generated. Generative model:
To generate a data item:
1. Choose N, the number of names, and F, the
number of faces.
2. Generate N name, context pairs.
3. For each of these name, context pairs,
generate a binary variable pictured conditioned on the context alone (from
P (pictured|context)).
4. For each pictured = 1, generate
a face
P
from P (f ace|name) (F n =
pictured).
name, context
pictured
N
face_u
face_n
Fu
Fn
D
5. Generate F u = F ? F n other faces from
P (f ace).
The parameters necessary for the EM process are P (f ace|name) (sec 2.2), the probability
that a face is generated by a given name, P (pictured|context) (sec 2.3), the probability
that a name is pictured given its context, and P (f ace) the probability that a face is generated without a name.
Elizabeth Dole
Angelina Jolie
Elizabeth Dole
Angelina Jolie
Donald Fehr
Donald Fehr
U.S. Open
Abraham Lincoln
Abraham Lincoln
U.S. Open
empty
empty
Anastasia Myskina
Without language model
Anastasia Myskina
With language model
Figure 3: Left: Example clusters using only appearance to cluster. Right: The same
clusters, but using appearance + language to cluster. Some clusters get larger (Elizabeth
Dole, Angelina Jolie) because of the inclusion of more correct faces. Some clusters get
smaller (Anastasia Myskina) because of the exclusion of incorrect faces. All clusters get
more accurate because the language model is breaking ambiguities and giving the clustering a push in the right direction. Some clusters that do not refer to actual people, ?U.S.
Open? completely disappear using the language model. Other clusters like ?Abraham Lincoln? (who is a person, but whose associated pictures most often portray people other than
?Abraham Lincoln?) become empty when using the language model, presumably because
these faces are assigned to the correct names.
2.1
Name Assignment
For each image-caption pair, we calculate the costs of all possible assignments of names to
faces (dependent upon the associated faces and names) and use the best such assignment.
An example of the extracted names, faces and all possible assignments can be seen in figure
1.
The likelihood of picture xi under assignment aj , of names to faces under our generative
model is:
L(xi , aj ) =P (N )P (F )P (n1 , c1 )...P (nn , cn )?
Y
Y
Y
P (pictured? |c? )P (f?(?) |n? )
(1 ? P (pictured? |c? ))
P (f? )
?
?
?
Where P(N) is the probability of generating N names, P(F) is the probability of generating F
faces, and P (ni , ci ) are the probabilities of generating namei and context ci . In assignment
aj , ? indexes into the names that are pictured, ?(?) indexes into the faces assigned to the
pictured names, ? indexes into the names that are not pictured and ? indexes into the faces
without assigned names. The terms P (N )P (F )P (n1 , c1 )...P (nn , cn ) are not dependent
on the assignment so we can ignore them when calculating the probability of an assignment
and focus on the remaining terms.
IN Pete Sampras IN of the U.S. celebrates his victory over Denmark?s OUT Kristian Pless
OUT at the OUT U.S. Open OUT at Flushing Meadows August 30, 2002. Sampras won the
match 6-3 7- 5 6-4. REUTERS/Kevin Lamarque
Germany?s IN Chancellor Gerhard Schroeder IN, left, in discussion with France?s IN President Jacques Chirac IN on the second day of the EU summit at the European Council headquarters in Brussels, Friday Oct. 25, 2002. EU leaders are to close a deal Friday on finalizing entry
talks with 10 candidate countries after a surprise breakthrough agreement on Thursday between
France and Germany regarding farm spending.(AP Photo/European Commission/HO)
?The Right Stuff? cast members IN Pamela Reed IN, (L) poses with fellow cast member IN
Veronica Cartwright IN at the 20th anniversary of the film in Hollywood, June 9, 2003. The
women played wives of astronauts in the film about early United States test pilots and the space
program. The film directed by OUT Philip Kaufman OUT, is celebrating its 20th anniversary
and is being released on DVD. REUTERS/Fred Prouser
Kraft Foods Inc., the largest U.S. food company, on July 1, 2003 said it would take steps, like
capping portion sizes and providing more nutrition information, as it and other companies face
growing concern and even lawsuits due to rising obesity rates. In May of this year, San Francisco attorney OUT Stephen Joseph OUT, shown above, sought to ban Oreo cookies in California ? a suit that was withdrawn less than two weeks later. Photo by Tim Wimborne/Reuters
REUTERS/Tim Wimborne
Figure 4: Our new procedure gives us not only better clustering results, but also a natural
language classifier which can be tested on captions in isolation. Above: a few captions
labeled with IN (pictured) and OUT (not pictured) using our learned language model. Our
language model has learned which contexts have high probability of referring to pictured
individuals and which contexts have low probabilities. We observe an 85% accuracy of
labeling who is portrayed in a picture using only our language model. The top 3 labelings
are all correct. The last incorrectly labels ?Stephen Joseph? as not pictured when in fact
he is the subject of the picture. Some contexts that are often incorrectly labeled are those
where the name appears near the end of the caption (usually a cue that the individual
named is not pictured). Some cues we could add that should improve the accuracy of our
language model are the nearness of words like ?shown?, ?pictured?, or ?photographed?.
The complete data log likelihood is:
?
?
X X
?
(Pij log(L(xi , aj ))?
i?pics
j?ass
Where Pij is an indicator variable telling which correspondence occurred in this data item.
The Pij are missing data whose expectations are computed in the E step.
This gives a straightforward EM procedure:
? E ? update the Pij according to the normalized probability of picture i with assignment j.
? M ? maximize the parameters P (f ace|name) and P (pictured|context) using
soft counts.
2.2
Modeling the appearance of faces ? P (f ace|name)
We model appearance using a mixture model with one mixture element per name in our
lexicon. We need a representation for faces in a feature space where comparisons are
helpful. Our representation is obtained by rectification of the faces followed by kernel
principal components analysis (kPCA) and linear discriminant analysis (LDA) (details in
[5]). We model the distributions P (f ace|name) using gaussians with fixed covariance.
To obtain features we first automatically rectify all faces to a canonical pose. Five support
vector machines are trained as feature detectors (corners of the left and right eyes, corners
Model
Appearance Model, No Lang Model
Appearance Model + Lang Model
EM
56%
72%
MM
67%
77%
Table 1: Above: We randomly selected a set of 1000 faces from our dataset and hand
labeled them with their correct names. Here we show what percentage of those faces are
correctly labeled by each of our methods (EM and maximal correspondence clustering,
MM). For both methods, incorporating a language model improves their respective clusterings greatly. Standard statistical knowledge says that using the expected values should
perform better than simply choosing the maximal assignment at each step (MM). However,
we have found that using the maximal assignment works better than taking an expectation.
One reason this could be true is that EM averages incorrect faces into the appearance
model, making the mean unstable.
of the mouth and the tip of the nose) using features consisting of the geometric blur of [4]
applied to grayscale patches. We then use kPCA ([16]) to reduce the dimensionality of our
data and compute linear discriminants ([3]) on the single name, single face pictures.
Because of the large size of our dataset, we cannot compute the kernel matrix, K for kPCA,
directly. Instead we use an approximation to calculate the eigenvectors of K, the Nystro? m
approximation (cf [17, 9]). The Nystr?om approximation computes two exact subsets of K
and uses these to efficiently approximate the rest of K and its eigenvectors (details in [5]).
2.3
Language Model ? P (pictured|context)
Our language model assigns a probability to each name based on its context within the
caption. These distributions, P (pictured|context), are learned using counts of how often
each context appears describing an assigned name, versus how often that context appears
describing an unassigned name. We have one distribution for each possible context cue,
and assume that context cues are modeled independently (because we lack enough data to
model them jointly).
For context, we use a variety of cues: the part of speech tags of the word immediately
prior to the name and immediately after the name within the caption (modeled jointly), the
location of the name in the caption, and the distances to the nearest ?,?, ?.?, ?(?, ?)?, ?(L)?,
?(R)? and ?(C)?. We tried adding a variety of other language model cues, but found that
they did not increase the assignment accuracy.
The probability of being pictured given multiple context cues (where Ci are the different
independent context cues) can be formed using Bayes rule:
P (pictured|C1 , C2 , ...Cn ) =
P (pictured|C1 )...P (pictured|Cn )
P (pictured)n?1
We compute maximum likelihood estimates of each of P (pictured|Ci ) and P (pictured)
using soft counts.
2.4
Best correspondence and mean correspondence
Given our hidden variable problem of determining correct name-face assignments, from a
statistics point of view EM seems like the most favorable choice. However, many computer
vision problems have observed better results by choosing maximum over expected values.
We have tried both methods and found that using the maximal assignment produced better
results (table 1). One reason this might be true is that for cases where there is a clear best
assignment the max and the average are basically equivalent. For cases where there is no
clear best, EM averages over assignments, producing a mean that has no real meaning since
it is an average of different people?s faces.
Classifier
Baseline
EM Labeling with Language Model
MM Labeling with Language Model
labels correct
67%
76%
84%
IN correct
100%
95%
87%
OUT correct
0%
56%
76%
Table 2: Above: Results of applying our learned language model to a test set of 430
captions (text alone). In our test set, we hand labeled each detected name with IN/OUT
based on whether the referred name was pictured within the corresponding picture. We
then tested how well our language model could predict those labels (?labels correct? refers
to the total percentage of names that were correctly labeled, ?IN correct? the percentage
of pictured names correctly labeled, and ?OUT correct? the percentage of not pictured
names correctly labeled). The baseline figure gives the accuracy of labeling all names
as pictured. Using EM to learn a language model gives an accuracy of 76% while using
a maximum likelihood clustering gives 84%. Again the maximum likelihood clustering
outperforms EM. Names that are most often mislabeled are those that appear near the end
of the caption or in other contexts that usually denote a name being not pictured.
The Maximal Assignment process is nearly the same as the EM process except instead of
calculating the expected value of each assignment only the maximal assignment is nonzero.
The Maximal Assignment procedure:
? M1 ? set the maximal Pij to 1 and all others to 0.
? M2 ? maximize the parameters P (f ace|name) and P (pictured|context) using
counts.
3
Results
We have collected a dataset consisting of approximately half a million news pictures and
captions from Yahoo News over a period of roughly two years.
Faces: Using the face detector of [14], we extract 44,773 large well detected face images.
Since these pictures were taken ?in the wild? rather than under fixed laboratory conditions,
they represent a broad range of individuals, pose, expression, illumination conditions and
time frames. Our face recognition dataset is more varied than any other to date.
Names: We use an open source named entity recognizer ([8]) to detect proper names in
each of the associated captions. This gives us a set of names associated with each picture.
Scale: We obtain 44,773 large and reliable face detector responses. We reject face images
that cannot be rectified satisfactorily, leaving 34,623. Finally, we concentrate on images
within whose captions we detect proper names, leaving 30,281, the final set we cluster on.
3.1
Quantitative Results
Incorporating a natural language model into face clustering produces much better results
than clustering on appearance alone. As can be seen in table 1, using, only appearance
produces an accuracy of 67% while appearance + language gives 77%. For face labeling,
using the maximum likelihood assignment (MM) rather than the average (EM) produces
better results (77% vs 72%).
One neat by-product of our clustering is a natural language classifier. We can evaluate that
classifier on text without associated pictures. In table 2, we show results for labeling names
with pictured and not pictured using our language model. Using the language model we
correctly label 84% of the names while the baseline (labeling everyone as pictured) only
gives 67%. The maximum likelihood assignment also produces a better language model
than EM (76% vs 84%). A few things that our language model learns as indicative of being
pictured are being near the beginning of the caption, being followed by a present tense verb,
and being near ?(L)?, ?(R)?, or ?(C)?.
4
Discussion
We have shown previously ([5]) that a good clustering can be created using names and
faces. In this work, we show that by analyzing language more carefully we can produce
a much better clustering (table 1). Not only do we produce better face clusters, but we
also learn a natural language classifier that can be used to determine who is pictured from
text alone (table 2). We have coupled language and images, using language to learn about
images and images to learn about language.
The next step will be to try to learn a language model for free text on a webpage. One area
we would like to apply this to is improving google image search results. Using a simple
image representation and a modified context model perhaps we could link google images
with the words on the surrounding webpages to improve search results.
References
[1] K. Barnard, D.A. Forsyth, ?Clustering Art,? Computer Vision and Pattern Recognition,
2001
[2] K. Barnard, P. Duygulu, N. de Freitas, D.A. Forsyth, D. Blei, and M.I. Jordan, ?Matching Words and Pictures,? Journal of Machine Learning Research, Vol 3, pp 1107-1135,
2003.
[3] P. Belhumeur, J. Hespanha, D. Kriegman ?Eigenfaces vs. Fisherfaces: Recognition
Using Class Specific Linear Projection? Transactions on Pattern Analysis and Machine
Intelligence, Special issue on face recognition, pp. 711-720, July 1997.
[4] A.C. Berg, J. Malik, ?Geometric Blur for Template Matching,? Computer Vision and
Pattern Recognition,Vol I, pp. 607-614, 2001.
[5] T.L. Berg, A.C. Berg, J. Edwards, M. Maire, R. White, E. Learned-Miller, D.A. Forsyth
?Names and Faces in the News? Computer Vision and Pattern Recognition, 2004.
[6] V. Blanz, T. Vetter, ?Face Recognition Based on Fitting a 3D Morphable Model,?
Transactions on Pattern Analysis and Machine Intelligence Vol. 25 no.9, 2003.
[7] C. Carson, S. Belongie, H. Greenspan, J. Malik, ?Blobworld ? Image segmentation using expectationmaximization and its application to image querying,? IEEE Transactions
on Pattern Analysis and Machine Intelligence, 24(8), pp. 1026?1038, 2002.
[8] H. Cunningham, D. Maynard, K. Bontcheva, V. Tablan, ?GATE: A Framework and
Graphical Development Environment for Robust NLP Tools and Applications,? 40th
Anniversary Meeting of the Association for Computational Linguistics?, Philadelphia,
July 2002.
[9] C. Fowlkes, S. Belongie, F. Chung, J. Malik, ?Spectral Grouping Using The Nystr o? m
Method,? TPAMI, Vol. 26, No. 2, February 2004.
[10] R. Gross, J. Shi and J. Cohn, ?Quo Vadis Face Recognition?,? Third Workshop on
Empirical Evaluation Methods in Computer Vision, December, 2001.
[11] R. Gross, I. Matthews, and S. Baker, ?Appearance-Based Face Recognition and LightFields,? Transactions on Pattern Analysis and Machine Intelligence, 2004.
[12] V. Lavrenko, R. Manmatha., J. Jeon, ?A Model for Learning the Semantics of Pictures,? Neural Information Processing Systems, 2003
[13] J. Li and J. Z. Wang, ?Automatic Linguistic Indexing of Pictures by a Statistical Modeling Approach,? Transactions on Pattern Analysis and Machine Intelligence, vol. 25,
no. 9, pp. 1075-1088, 2003
[14] K. Mikolajczyk ?Face detector,? Ph.D report, INRIA Rhone-Alpes
[15] J. Scheeres,
?Airport face scanner failed?,
Wired News,
2002.
http://www.wired.com/news/privacy/0,1848,52563,00.html.
[16] B. Scholkopf, A. Smola, K.-R. Muller ?Nonlinear Component Analysis as a Kernel
Eigenvalue Problem? Neural Computation, Vol. 10, pp. 1299-1319, 1998.
[17] C. Williams, M. Seeger ?Using the Nystro? m Method to Speed up Kernel Machines,?
Advances in Neural Information Processing Systems, Vol 13, pp. 682-688, 2001.
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1,882 | 2,709 | Semigroup Kernels on Finite Sets
Marco Cuturi
Computational Biology Group
Ecole des Mines de Paris
35 rue Saint Honor?e
77300 Fontainebleau
[email protected]
Jean-Philippe Vert
Computational Biology Group
Ecole des Mines de Paris
35 rue Saint Honor?e
77300 Fontainebleau
[email protected]
Abstract
Complex objects can often be conveniently represented by finite sets of
simpler components, such as images by sets of patches or texts by bags
of words. We study the class of positive definite (p.d.) kernels for two
such objects that can be expressed as a function of the merger of their
respective sets of components. We prove a general integral representation of such kernels and present two particular examples. One of them
leads to a kernel for sets of points living in a space endowed itself with a
positive definite kernel. We provide experimental results on a benchmark
experiment of handwritten digits image classification which illustrate the
validity of the approach.
1
Introduction
Suppose we are to deal with complex (e.g non-vectorial) objects from a set Z on which
we wish to apply existing kernel methods [1] to perform tasks such as classification or
regression. Assume furthermore that the latter objects can be meaningfully described by
small components contained in a set X . Namely, we suppose that we can define an a
priori mapping ? which maps any z ? Z into a finite unordered list of elements of X ,
? (z) = [x1 , x2 , ..., xn ], through a sampling process which may be exhaustive, heuristic or
random both in the quantity of sampled components n and in the way those components
are extracted. Comparing two such complex objects through the direct comparison of their
respective lists of components has attracted much attention recently, namely through the
definition of p.d. kernel on such ? -lists. Most recent approaches to compare two ? -lists
involve the estimation of two distributions pz and pz? on X within a parametric class of
models that fit (e.g. in maximum likelihood (ML) sense) respectively ? (z) and ? (z ? ) seen
as a samples from laws on X , where each resulting law could be identified with z and z ?
respectively. Such a kernel is then defined between pz and pz? , as seen for example in [2]
with the Information Diffusion kernel, in [3] with the family of Mutual Information Kernels
or in [4] with the use of the Battacharyya affinity between pz and pz? . An alternative
and non-parametric approach to ? -lists comparison that studies the subspaces generated by
points of ? (z) and ? (z ? ) in a feature space was also proposed in [5], recalling elements
presented in Kernel-Canonical Correlation Analysis [6].
We explore in this contribution a different direction to kernel design for lists by studying the
class of kernels whose value computed on two lists is only defined through its value on their
concatenation. This approach was already used in [7], where a particular kernel for strings
that only compares two strings through their concatenation is presented. In this paper, the
approach is extended to a more general and abstract setting of ? -lists, but the motivation
remains the same as in [7]: if two ? -lists are similar, e.g. in terms of the distribution of
the components they describe, then their concatenation will be more ?concentrated? than if
they are very different, in which case it might look more like a reunion of two disjoint sets
of points. As a result, one can expect to get a relevant measure of similarity, and hence a
kernel, by studying properties of the concatenation of two lists such as its concentration.
After an example of a valid kernel for lists seen as measures on the space of components
(Section 2), we provide a complete characterization for this class of kernels (Section 3)
by casting them in the context of semigroup kernels. This leads to the definition of a
second kernel based on exponential densities on X , which boils down after a numerical
approximation to the computation of the entropy of the maximum likelihood density (taken
in the considered exponential family) of the points numbered by lists of components. This
kernel is extended in Section 4 to points taken in a reproducing kernel Hilbert space defined
by a kernel ? on X , and is then tested on a problem of image classification, where images
are seen as bags of pixels and a non-linear kernel between pixels is used (Section 5).
2
The entropy kernel
As a warm-up, let us assume that the set X is measurable, e.g. X = Rd , and that to
any point x ? X we can associate a probability measure on X with density ?x with
respect to a common measure (e.g. the Borel uniform measure), with finite entropy
R
def
h(?) = ? X ? ln ?. Consider for example a Gaussian distribution with mean x and
n
fixed variance. A natural way
Pnto represent an unordered list ? (z) = [x1 , x2 , ..., xn ] ? X
is by the density ?? = 1/n i=1 ?xi . In that case, a p.d. kernel k between unordered lists
? and ? ? that only depends on their concatenation ? (z) ? ? (z ? ) is equivalent to a p.d. kernel
between densities ? and ?? that only depends on ?+?? . Hence we are looking for a p.d. kernel on the set P of probability densities of finite entropy of the form ?(?, ?? ) = ?(? + ?? ).
An example of such a kernel is provided in the following proposition. Recall that a negative
g : X 2 ? R that satisfies
Pn definite (n.d.) kernel on a set X is a symmetric function
n
c
c
g(x
,
x
)
?
0
for
any
n
?
N,
(x
,
.
.
.
,
x
)
?
X
,
and
(c1 . . . , cn ) ? Rn with
i j
i
j
1
n
Pi,j=1
n
i=1 ci = 0. A useful link between p.d. and n.d. kernels is that g is n.d. if and only if
exp(?tg) is p.d. for all t > 0 [8, Theorem 3.2.2.].
?
Proposition 1. The function g : ?, ?? 7? h( ?+?
2 ) is negative definite on P, making
def
?+??
kh (?, ?? ) = e?th( 2
between two measures.
)
a p.d. kernel on P for any t > 0. We call kh the entropy kernel
The entropy kernel is already a satisfactory answer to our initial motivation to look at
merger of points. Observe that if ?x is a probability density around x, then ?? can often
be thought of as an estimate of the distribution of the points in ? , and (?? + ?? ? )/2 is an
estimate of the distribution of the points enumerated in ? ? ? ? . If the latter estimate has a
small entropy we can guess that the points in ? and ? ? are likely to have similar distributions
which is exactly the similarity that is quantified by the entropy kernel.
Proof of Proposition 1. It is known that the real-valued function r : y 7? ?y ln y is n.d.
on R+ as a semigroup endowed with addition [8, Example 6.5.16]. As a consequence the
function f 7? r ? f is n.d. on P as a pointwise application of r, and so is its summation on
X . For any real-valued n.d. kernel k and any real-valued function g, we have trivially that
?
(y, y ? ) 7? k(y, y ? ) + g(y) + g(y ? ) remains negative definite, hence h( f +f
2 ) is n.d. through
?
1
ln 2
?
?
h( f +f
2 ) = 2 h(f + f ) + 2 (|f | + |f |), yielding positive definiteness of kh . ?
3
Semigroups and integral representations of p.d. kernels on finite
Radon measures
In order to generalize the example presented in the previous section, let us briefly recall
the concept of p.d. kernels on semigroups [8]. A nonempty set S is called an Abelian
(autoinvolutive) semigroup if it is equipped with an Abelian associative composition ?
admitting a neutral element in S. A function ? : S 7? R is called a positive definite
(resp. negative definite) function on the semigroup (S, ?) if (s, t) 7? ?(s ? t) is a p.d. (resp.
n. d.) kernel on S ? S.
The entropy kernel defined in Proposition 1 is therefore a p.d. kernel on the semigroup
of measures with finite entropy endowed with usual addition. This can be generalized by
assuming that X is a Hausdorff space, which suffices to consider the set of finite Radon
b
b
measures M+
(X ) [8]. For ? ? M+
(X ), we Rnote |?| = ?(X ) < +?. For a Borel
X
measurable function f ? R , we note ?[f ] = X f d?. Endowed with the usual Abelian
b
addition between measures, (M+
(X ), +) is an Abelian semigroup. The reason to consider
this semigroup is that there is a natural semigroup homomorphism between
Pn finite lists of
b
points and elements of M+
(X ) given by ? = [x1 , ..., xn ] 7? ?? =
i=1 ?xi , where
b
?x ? M+
(X ) is an arbitrary finite measure associated with each x ? X . We discussed in
section 2 the case where ?x has a density, but more general measures are allowed, such as
?x = ?x , the Dirac measure. Observe that when we talk about lists, it should be understood
that some objects might appear with some multiplicity which should be taken
account
Pinto
n
(specially when X is finite), making us consider weighted measures ? = i=1 ci ?xi in
the general case. We now state the main result of this section which characterizes bounded
b
p.d. functions on the semigroup M+
(X ),
b
Theorem 1. A bounded real-valued function ? on M+
(X ) such that ?(0) = 1 is p.d. if
and only if it has an integral representation:
Z
?(?) =
e??[f ] d?(f ),
C + (X )
where ? is a uniquely determined positive radon measure on C + (X ), the space of nonnegative-valued continuous functions of RX endowed with the topology of pointwise convergence.
b
Proof. (sketch) Endowed with the topology of weak convergence, M+
(X ) is a Hausdorff
space [8, Proposition 2.3.2]. The general result of integral representation of bounded p.d.
function [8, Theorem 4.2.8] therefore applies. It can be shown that bounded semicharacters
b
on M+
(X ) are exactly the functions of the form ? 7? exp(??[f ]) where f ? C + (X ) by
using the characterization of semicharacters on (R+ , +) [8, Theorem 6.5.8] and the fact
b
that atomic measures is a dense subset of M+
(X ) [8, Theorem 2.3.5].
As a constructive application to this general representation theorem, let us consider the
case ?x = ?x and consider, as a subspace of C + (X ), the linear span of N non-constant,
continuous, real-valued and linearly independent functions f1 , ..., fN on X . As we will see
below, this is equivalent to considering a set of densities defined by an exponential model,
PN
namely of the form p? (x) = exp( j=1 ?j fj (x)??(?)) where ? = (? j )j=1..N ? ? ? RN
is variable and ? is a real-valued function defined on ? to ensure normalization of the
densities p? . Considering a prior ? on the parameter space ? is equivalent to defining
a Radon measure taking positive values on the subset of C + (X ) spanned by functions
f1 , ..., fN . We now have ( see [9] for a geometric point of view) that:
Theorem 2. ??? ? ? being the ML parameter associated with ? and noting p? = p??? ,
Z
?|?|h(p? )
?? (?) = e
e?|?|d(p? ||p? ) ?(d?),
?
is a p.d. kernel on the semigroup of measures, where d(p||q) =
Kullback-Leibler divergence between p and q.
R
supp(q)
p ln pq is the
Although an exact calculation of the latter equation is feasible in certain cases (see [10, 7]),
an approximation can be computed using Laplace?s approximation. If for example the prior
on the densities is taken to be Jeffrey?s prior [9, p.44] then the following approximation
holds:
? ? N2
2?
.
(1)
?(?) ? ?(?)
?
:= e?|?|h(p? )
|?|
|?|??
2
The ML estimator being unaffected by the total weight |?|, we have ?(2?)
?
= ?(?)
? 2 ( |?|
4? )
which we use to renormalize our kernel on its diagonal:
? p
! N2
?(|?+?? |)h(p?+?? )
?|
2
|?||?
e
k(?, ?? ) = ?|?|h(p )?|?? |h(p ? )
?
?
|?| + |?? |
e
N
Two problems call now for a proper renormalization: First, if |?? | ? |?| (which would
be the case if ? describes far more elements than ? ? ), the entropy h(p?+?? ) will not take
into account the elements enumerated in ?? . Second, the value taken by our p.d function ??
decreases exponentially with |?| as can be seen in equation (1). This inconvenient scaling
behavior leads in practice to bad SVM classification results due to diagonal dominance of
the Gram matrices produced by such kernels (see [11] for instance). Recall however that
the Laplace approximation can be accurate only when |?| ? 0. To take into account this
tradeoff on the ideal range of |?|, we rewrite the previous expression using a common width
parameter ? after having applied a renormalization on ? and ?? :
?2?
?
?
k? (?, ? ) = k( ?, ? ?? ) = e
|?| |? |
?
?
h(p??? )?
h(p? )+h(p ? )
?
2
?
,
(2)
?
?
+ |??? | . ? should hence be big enough in practical applications to ensure the
where ??? = |?|
consistency of Laplace?s approximation and thus positive definiteness, while small enough
to avoid diagonal dominance. We will nowP
always suppose that our atomic measures are
n
normalized, meaning that their total weight i=1 ci always sums up to 1.
Let us now review a practical case when X is Rk , and that some kind of gaussianity among
points makes sense. We can use k-dimensional normal distributions pm,? ? N (m, ?)
(where ? is a k ? k p.d. matrix)
Pn to define our densities.
Pn The ML parameters?of a measure
? are in that case : ?
? = i=1 ci xi and ?? = i=1 ci (xi ? ?
?)(xi ? ?
?) . Supposing
that the span of the n vectors xi covers Rk yields non-degenerated covariance matrices.
This ensures the existence of the entropy of the ML estimates through the formula [12]:
h(pm,? ) = 12 ln ((2?e)n |?|). The value of the normalized kernel in (2) is then:
?p
!2?
|?? ||??? |
?
k? (?, ? ) =
.
|???? |
This framework is however limited to vectorial data for which the use of Gaussian laws
makes any sense. An approach designed to bypass this double restriction is presented in
the next section, taking advantage of a prior knowledge on the components space through
the use of a kernel ?.
4
A kernel defined through regularized covariance operators
Endowing X (now also considered 2-separable) with a p.d. kernel ? bounded on the diagonal, we make use in this section of its corresponding reproducing kernel Hilbert space
(RKHS, see [13] for a complete survey). This RKHS is denoted by ?, and its feature map
by ? : x 7? ?(x, ?). ? is infinite dimensional in the general case, preventing any systematical use of exponential densities on that feature space. We bypass this issue through
a generalization of the previous section by still assuming some ?gaussianity? among the
elements numbered by atomic measures ?, ?? and ??? which, once mapped in the feature
space, are now functions. More precisely, our aim when dealing with Euclidean spaces
was to estimate finite dimensional covariance matrices ?? , ??? , ???? and compare them in
terms of their spectrum or more precisely through their determinant. In this section we
use such finite samples to estimate, diagonalize and regularize three covariance operators
S? , S?? , S??? associated with each measure on ?, and compare them by measuring their respective dispersion in a similar way. We note for ? ? ? its dual ? ? (namely the linear form
? ? R s.t. ? 7? ? ? ? = h?, ?i? ) and ||?||2 = ? ? ?. Let (ei )i?N be a complete orthonormal
base of ? (i.e. such that span(ei )i?N = ? and e?i ej = ?ij ). Given a family of positive real
?
numbers (ti )i?N
P, we note?St,e the bilinear symmetric operator which maps ?, ? 7? ? St,e ?
where St,e = i?N ti ei ei .
def
For an atomic measure ? and notingP
??i = (?i ? ?[?]) its n centered points in ?, the
n
empirical covariance operator S? = i=1 ci ?i ?i? on ? can be described through such a
diagonal representation by finding its principal eigenfunctions, namely orthogonal functions in ? which maximize the expected (w.r.t to ?) variance of the normalized dot-product
def v ? ?
hv (?) = ||v||
here defined for any v of ?. Such functions can be obtained through the
following recursive maximizations:
n
1 X ??
ci vj ?i .
vj =
argmax
var? (hv (?)) =
argmax
2
v??,v?{v1 ,...,vj?1 }
v??,v?{v1 ,...,vj?1 } ||vj || i=1
As in the framework of Kernel PCA [1] (from which this calculus only differs by considering weighted points in the feature space) we have by the representer theorem [1] that all the
solutions of these successive maximizations lie in span({??i }i=1..n ). Thus for each vj there
Pd
? ? ?j where K
?? =
exists a vector ?j of Rn such that vj = i=1 ?j,i ??i with ||vj ||2 = ?j? K
(In ? 1n,n ?c )K? (In ? ?c 1n,n ) is the centered Gram matrix K? = [?(xi , xj )]1?i,j?n of
the points taken in the support of ?, with 1n,n being the n ? n matrix composed of ones
and ?c the n ? n diagonal matrix of ci coefficients. Our latter formulation is however
? ? . We
ill-defined, since any ?j is determined up to the addition of any element of ker K
def
? ? ? Rn to consider functions of positive
thus restrict our parameters ? to lie in E = ker K
?
squared norm, having now:
?j =
? ? ?c K
? ?? ?
?
?? K
= var? (hvj (?))
?
?
? K? ?
? k =0
??E:?k<j,?? K?
argmax
? ? ?c K
? ? and K
? ? being symmetric positive definite on E (one can
Both endomorphism K
?
?
? ? ), the right-hand argument of the previous equaeasily prove that ker K? = ker K? ?c K
? ? ?c K
? ? by K
? ? , can be maximized through a
tion, known as the Rayleigh quotient of K
Hermitian generalized eigenvalue decomposition. This computation yields a basis ?j of E
? ? ?i = 0 for i < j ? dim(E), and with corresponding positive eigenvalues
such that ?j? K
Pd
in decreasing order ?1 , ..., ?dim(E) . Through vj = i=1 ?j,i ??i and writing r = dim(E),
this also yields an orthogonal basis (vj )i?r of span{(??i )i?n }, which can be completed
to span ? through a Gram-Schmidt orthonormalization process using the original basis
(ei )i?N . The orthonormal base corresponding to S? is thus (vi )i?N , where the r first vectors are the original eigenvectors obtained through the previous maximization. Such a
diagonal representation of S? takes the form S? = S?,v where ? = (?1 , ..., ?r , 0, ...). This
bilinear form is however degenerated on ? and facing the same problem encountered in
[4, 6] we also propose to solve this issue through a regularization by adding a component
? on every vector of the base, i.e. defining ?? = (?1 + ?, ..., ?r + ?, ?, ...) with ? > 0, to
propose a regularization of S? as:
S?? ,v =
r
X
i=1
(?i + ?)vi vi? +
X
? vi vi? .
i>r
The entropy of a covariance operator St,e not being defined, we bypass this issue by considering the entropy of its marginal distribution on its first d eigenfunctions, namely introducPd
ing the quantity |St,e |d = d2 ln(2?e) + 12 i=1 ln ti . Let us sum up ideas now and consider
?
three normalized measures ?, ?? and ??? = ?+?
2 , which yield three different orthonormal
bases vi , vi? and vi?? of ? and three different families of weights ?? = (?i?r +?, ?, ...), ??? =
(??i?r? + ?, ?, ...) and ???? = (???i?r?? + ?, ?, ...). Though working on different bases, those
respective d first directions allow us to express an approached form of kernel (2) limited to
different subspaces of ? of arbitrary size d ? r?? ? max(r, r? ):
?
?
!!
|S?? ,v |d + |S??? ,v? |d
?
kd,? (?, ? ) = exp ?2? |S???? ,v?? |d ?
2
? qQ
?
2?
(3)
??i
r
?i Qr ?
1
+
1
+
i=1
i=1
?
?
? ,
=?
Qr??
???
i
i=1 1 + ?
The latter expression is independent of d, while letting d go to infinity lets every base on
which are computed our entropies span the entire space ?. Though the latter hint does not
establish a valid theoretical proof of the positive definiteness of this kernel, we use this final
formula for the following classification experiments.
5
Experiments
Following the previous work of [4], we have conducted experiments on an extraction of
500 images (28 ? 28 pixels) taken in the MNIST database of handwritten digits, with 50
images for each digit. To each image z we randomly associate a set ? (z) of 25 to 30
pixels among black points (intensity superior to 191 on a 0 to 255 scale ) in the image,
where X is {1, .., 28} ? {1, .., 28} in this case. In all our experiments we set ? to be 21
which always yielded positive definite Gram matrices in practice. To define our RKHS
? we used both the linear kernel, ?a ((x1 , y1 ), (x2 , y2 )) = (x1 x2 + y1 y2 )/272 and the
2
2
1 ?y2 )
Gaussian kernel of width ?, namely ?b ((x1 , y1 ), (x2 , y2 )) = e? (x1 ?x227) 2+(y
. The
?2? 2
linear case boils down to the simple application presented in the end of section 3 where we
fit Gaussian bivariate-laws on our three measures and define similarity through variance
analysis. The resulting diagonal variances (?1,1 , ?2,2 ),(??1,1 , ??2,2 ) and (???1,1 , ???2,2 ) measure
of our data for each of the three measures, yielding a kernel value of
? the dispersion
?1,1 ?2,2 ??1,1 ??2,2
equal to 0.382 in the case shown in figure 1. The linear kernel man??
???
1,1 ?2,2
ages a good discrimination between clearly defined digits such as 1 and 0 but fails at
doing so when considering numbers whose pixels? distribution cannot be properly characterized by ellipsoid-like shapes. Using instead the Gaussian kernel brings forward a
non-linear perspective to the previous approach since it maps now all pixels into Gaussian bells, providing thus a much richer function class for ?. In this case two parameters
(a)
?1,1 = 0.0552
?2,2 = 0.0013
(b)
?1 = 0.276
??1,1 = 0.0441
??2,2 = 0.0237
???1,1 = 0.0497
???2,2 = 0.0139
??1 = 0.168
???1 = 0.184
0
Figure 1: First Eigenfunction of three empirical measures ?1 , ?0 and ?1 +?
using the linear
2
(a) and the Gaussian (b, with ? = 0.01, ? = 0.1) kernel. Below each image are the corresponding eigenvalues which correspond to the variance captured by each eigenfunction,
the second eigenvalue being also displayed in the linear case (a).
require explicit tuning: ? (the width of ?) controls the range of the typical eigenvalues
found in the spectrum of our regularized operators whereas ? acts as a scaling parameter for the latter values as can be seen in equation (3). An efficient choice can thus only
be defined on pairs of parameter, which made us use two ranges of values for ? and ?
based on preliminary attempts: ? ? 10?2 ? {0.1, 0.3, 0.5, 0.8, 1, 1.5, 2, 3, 5, 8, 10, 20} and
? ? 10?1 ?{0.5, 1, 1.2, 1.5, 1.8, 2, 2.5, 3}. For each kernel computed on the base of a (?, ?)
couple, we used a balanced training fold of our dataset to train 10 binary SVM classifiers,
namely one for each digit versus all other 9 digits. The class of the remaining images of the
test fold was then predicted to be the one with highest SVM score among the the 10 previously trained binary SVMs. Splitting our data into test and training sets was led through
a 3-fold cross validation (roughly 332 training images and 168 for testing), averaging the
test error on 5 random fold splits of the original data. Those results were obtained using
the spider toolbox1 and graphically displayed in figure (2). Note that the best testing errors
were reached using a ? value of 0.12 with an ? parameter within 0.008 and 0.02, this error
being roughly 19.5% with a standard deviation inferior to 1% in all the region corresponding to an error lower than 22%. To illustrate the sensibility of our method to the number of
sampled points in ? we show in the same figure the decrease of this error when the number
of sampled points ranges from 10 to 30 with independently chosen random points for each
computation. As in [4], we also compared our results to the standard RBF kernel on images
seen as vectors of {0, 1}27?27 , using a fixed number of 30 sampled points and the formula
||z?z ? ||
k(z, z ? ) = e? 30?2?2 . We obtained similar results with an optimal error rate of roughly
44.5% for ? ? {0.12, 0.15, 0.18}. Our results didn?t improve by choosing different soft
margin C parameters, which we hence just set to be C = ? as is chosen by default by the
spider toolbox.
1
see http://www.kyb.tuebingen.mpg.de/bs/people/spider/
50%
2
10 ?
0.1
0.3
0.5
0.8
1
1.5
2
3
5
8
10
20
45%
0.05
e < 22 %
e < 19.5 %
0.12
0.15
?
e < 22 %
0.18
40%
Averaged error rate
0.1
35%
30%
25%
0.2
20%
0.25
15%
10
0.3
(a)
15
20
# points
25
30
(b)
Figure 2: (a) Average test error (displayed as a grey level) of different SVM handwritten
character recognition experiments using 500 images from the MNIST database (each seen
as a set of 25 to 30 randomly selected black pixels), carried out with 3-fold (2 for training, 1
for test) cross validations with 5 repeats, where parameters ? (regularization) and ? (width
of the Gaussian kernel) have been tuned to different values. (b) Curve of the same error
(with ? = 0.01, ? = 0.12 fixed) depending now on the size of the sets of randomly selected
black pixels for each image, this size varying between 10 and 30.
Acknowledgments
The authors would like to thank Francis Bach, Kenji Fukumizu and J?er?emie Jakubowicz
for fruitful discussions and Xavier Dupr?e for his help on the MNIST database.
References
[1] B. Sch?olkopf and A.J. Smola. Learning with Kernels: Support Vector Machines, Regularization,
Optimization, and Beyond. MIT Press, Cambridge, MA, 2002.
[2] J. Lafferty and G. Lebanon. Information diffusion kernels. In Advances in Neural Information
Processing Systems 14, Cambridge, MA, 2002. MIT Press.
[3] M. Seeger. Covariance kernels from bayesian generative models. In Advances in Neural Information Processing Systems 14, pages 905?912, Cambridge, MA, 2002. MIT Press.
[4] R. Kondor and T. Jebara. A kernel between sets of vectors. In Machine Learning, Proceedings
of the Twentieth International Conference (ICML 2003), pages 361?368. AAAI Press, 2003.
[5] L. Wolf and A. Shashua. Learning over sets using kernel principal angles. Journal of Machine
Learning Research, 4:913?931, 2003.
[6] F. Bach and M. Jordan. Kernel independent component analysis. Journal of Machine Learning
Research, 3:1?48, 2002.
[7] M. Cuturi and J.-P. Vert. A mutual information kernel for sequences. In IEEE International
Joint Conference on Neural Networks, 2004.
[8] C. Berg, J.P.R. Christensen, and P. Ressel. Harmonic Analysis on Semigroups. Springer, 1984.
[9] S. Amari and H. Nagaoka. Methods of information geometry. AMS vol. 191, 2001.
[10] F. M. J. Willems, Y. M. Shtarkov, and Tj. J. Tjalkens. The context-tree weighting method: basic
properties. IEEE Transancations on Information Theory, pages 653?664, 1995.
[11] J.-P. Vert, H. Saigo, and T. Akutsu. Local alignment kernels for protein sequences. In
B. Schoelkopf, K. Tsuda, and J.-P. Vert, editors, Kernel Methods in Computational Biology.
MIT Press, 2004.
[12] T. Cover and J. Thomas. Elements of Information Theory. Wiley & Sons, New-York, 1991.
[13] N. Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical
Society, 68:337 ? 404, 1950.
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1,883 | 2,710 | Bayesian Regularization and Nonnegative
Deconvolution for Time Delay Estimation
Yuanqing Lin, Daniel D. Lee
GRASP Laboratory, Department of Electrical and System Engineering
University of Pennsylvania, Philadelphia, PA 19104
linyuanq, [email protected]
Abstract
Bayesian Regularization and Nonnegative Deconvolution (BRAND) is
proposed for estimating time delays of acoustic signals in reverberant
environments. Sparsity of the nonnegative filter coefficients is enforced
using an L1 -norm regularization. A probabilistic generative model is
used to simultaneously estimate the regularization parameters and filter
coefficients from the signal data. Iterative update rules are derived under
a Bayesian framework using the Expectation-Maximization procedure.
The resulting time delay estimation algorithm is demonstrated on noisy
acoustic data.
1 Introduction
Estimating the time difference of arrival is crucial for binaural acoustic sound source
localization[1]. A typical scenario is depicted in Fig. 1 where the azimuthal angle ? to
the sound source is determined by the difference in direct propagation times of the sound
to the two microphones. The standard signal processing algorithm for determining the
time delay between twoRsignals s(t) and x(t) relies upon computing the cross-correlation
function[2]: C(?t) = dt x(t)s(t ? ?t) and determining the time delay ?t that maximizes the cross-correlation. In the presence of uncorrelated white noise, this procedure is
equivalent to the optimal matched filter for detection of the time delayed signal.
However, a typical room environment is reverberant and the measured signal is contaminated with echoes from multiple paths as shown in Fig. 1. In this case, the cross-correlation
and related algorithms may not be optimal for estimating the time delays. An alternative
approach would be to estimate the multiple time delays as a linear deconvolution problem:
X
min kx(t) ?
?i s(t ? ?ti )k2
(1)
?
i
Unfortunately, this deconvolution can be ill-conditioned resulting in very noisy solutions
for the coefficients ?. Recently, we proposed incorporating nonnegativity constraints ? ?
0 in the deconvolution to overcome the ill-conditioned linear solutions [3]. The use of
these constraints is justified by acoustic models that describe the theoretical room impulse
response with nonnegative filter coeffients [4]. The resulting optimization problem can be
written as the nonnegative quadratic programming problem:
min kx ? S?k2
(2)
??0
s
?t?
?t1
?t2
?
x1
x2
d
Figure 1: The typical scenario of reverberant signal. x2 (t) comes from the direct path (?t2 ) and
echo paths(?tE ).
2.2
11
x 10
2
Phase Alignment Transform
Cross-correlation
800
a
600
1.8
400
1.6
200
1.4
0
1.2
-10
-5
0
5
time delay(T s )
10
Linear Deconvolution
200
-200
-10
1
c
100
0.8
0
0.6
-100
0.4
-200
0.2
-300
-10
-5
0
5
time delay(Ts )
10
b
-5
0
5
time delay(T s )
10
Nonnegative Deconvolution
d
0
-10
-5
0
5
time delay(Ts )
10
Figure 2: Time delay estimation of a speech signal with a) cross-correlation, b) phase alignment
transform, c) linear deconvolution, d) nonnegative deconvolution. The observed signal x(t) = s(t ?
Ts ) + 0.5s(t ? 8.75Ts ) contains an additional time-delayed echo. Ts is the sampling interval.
where x = {x(t1 ) x(t2 ) . . . x(tN )}T is a N ? 1 data vector, S = {s(t ? ?t1 ) s(t ?
?t2 ) . . . s(t ? ?tM )} is an N ? M matrix, and ? is a M ? 1 vector of nonnegative
coefficients.
Figure 2 compares the performance of cross-correlation, phase alignment transform(a generalized cross-correlation algorithm), linear deconvolution, and nonnegative deconvolution
for estimating the time delays in a clean speech signal containing an echo. From the structure of the estimated coefficients, it is clear that nonnegative deconvolution can successfully
discover the structure of the time delays present in the signal. However, in the presence of
large background noise, it may be necessary to regularize the nonnegative quadratic optimization to prevent overfitting. In this case, we propose using an L1 -norm regularization
to favor sparse solutions [5]:
X
?
min kx ? S?k2 + ?
?i
(3)
??0
i
? (?
? ? 0) describes the trade-off between fitting the observed
In this formula, the parameter ?
data and enforcing sparse solutions. The proper choice of this parameter may be crucial in
obtaining the optimal time delay estimates. In the rest of this manuscript, we introduce a
proper generative model for these regularization parameters and filter coefficients within a
probabilistic Bayesian framework. We show how these parameters can be efficiently determined using appropriate iterative estimates. We conclude by demonstrating and discussing
the performance of our algorithm on noisy acoustic signals in reverberant environments.
2 Bayesian regularization
Instead of arbitrarily setting values for the regularization parameters, we show how a
Bayesian framework can be used to automatically estimate the correct values from the
data. Bayesian regularization has previously been successfully applied to neural network
learning [6], model selection, and relevance vector machine (RVM) [7]. In these works, the
fitting coefficients are assumed to have Gaussian priors, which lead to an L2 -norm regularization. In our model, we use L1 -norm sparsity regularization, and Bayesian framework
will be used to optimally determine the appropriate regularization parameters.
Our probabilistic model assumes the observed data signal is generated by convolving the
source signal with a nonnegative filter describing the room impulse response. This signal
is then contaminated by additive Gaussian white noise with zero-mean and covariance ? 2 :
?
?
1
1
2
kx
?
S?k
.
(4)
P (x|S, ?, ? 2 ) =
exp
?
2? 2
(2?? 2 )N/2
To enforce sparseness in the filter coefficients ?, an exponential prior distribution is used.
This prior only has support in the nonnegative orthant and the sharpness of the distribution
is given by the regularization parameter ?:
P (?|?) = ?M exp{??
M
X
?i }, ? ? 0 .
(5)
i=1
In order to infer the optimal settings of the regularization parameters ? 2 and ?, Bayes rule
is used to maximize the posterior distribution:
P (?, ? 2 |x, S) =
P (x|?, ? 2 , S)P (?, ? 2 )
.
P (x|S)
(6)
Assuming that P (?, ? 2 ) is relatively flat [8], estimating ? 2 and ? is then equivalent to
maximizing the likelihood:
Z
?M
2
P (x|?, ? , S) =
d? exp[?F (?)]
(7)
(2?? 2 )N/2 ??0
where
1
(x ? S?)T (x ? S?) + ?eT ?
(8)
F (?) =
2? 2
T
and e = [1 1 . . . 1] .
Unfortunately, the integral in Eq. 7 cannot be directly maximized. Previous approaches
to Bayesian regularization have used iterative updates heuristically derived from selfconsistent fixed point equations. In our model, the following iterative update rules for
? and ? 2 can be derived using Expectation-Maximization:
Z
1
1
??
d? eT ?Q(?)
(9)
?
M ??0
Z
1
d? (x ? S?)T (x ? S?)Q(?)
(10)
? 2 ??
N ??0
where the expectations are taken over the distribution
Q(?) =
exp[?F (?)]
,
Z?
(11)
R
with normalization Z? = ??0 d? exp[?F (?)]. These updates have guaranteed convergence properties and can be intuitively understood as iteratively reestimating ? and ? 2
based upon appropriate expectations over the current estimate for Q(?).
2.1 Estimation of ?M L
The integrals in Eqs. 9?10 are dominated by ? ? ?M L where the most likely ?M L is
given by:
1
?M L = arg min 2 (x ? S?)T (x ? S?) + ?T ?.
(12)
??0 2?
This optimization is equivalent to the nonnegative quadratic programming problem in Eq. 3
? = ?? 2 . To efficiently compute ?M L , we have recently developed two distinct
with ?
methods for optimizing Eq. 12.
The first method is based upon a multiplicative update rule for nonnegative quadratic programming [9]. We first write the problem in the following form:
1
min ?T A? + bT ?,
??0 2
where A =
1 T
? 2 S S,
(13)
and b = ? ?12 ST x.
First, we decompose the matrix A = A+ ? A? into its positive and negative components
such that:
?
?
Aij if Aij > 0
0
if Aij ? 0
?
A+
A
(14)
=
=
ij
ij
0
if Aij ? 0
?Aij if Aij < 0
Then the following is an auxiliary function that upper bounds Eq. 13 [9]:
? = bT ? +
G(?, ?)
? i 2 1X ?
1 X (A+ ?)
?i ?j
?i ?
A ?
?i?
? j (1 + ln
).
2 i
?
?i
2 i,j ij
?
?i?
?j
(15)
Minimizing Eq. 15 yields the following iterative multiplicative rule with guaranteed convergence to ?M L :
p
?bi + b2i + 4(A+ ?)i (A? ?)i
.
(16)
?i ?? ?i
2(A+ ?)i
The iterative formula in Eq. 16 is used to efficiently compute a reasonable estimate for
?M L from an arbitrary initialization. However, its convergence is similar to other interior
point methods in that small components of ?M L will continually decrease but never equal
zero. In order to truly sparsify the solution, we employ an alternative method based upon
the simplex algorithm for linear programming.
Our other optimization method is based upon finding a solution ?M L that satistifies the
Karush-Kuhn-Tucker (KKT) conditions for Eq. 13:
A? + b = ?, ? ? 0, ? ? 0, ?i ?i = 0, i = 1, 2, . . . , M.
(17)
By introducing additional artificial
variables a, the KKT conditions can be transformed into
P
the linear optimization min i ai subject to the constraints:
a ? 0
(18)
? ? 0
(19)
? ? 0
(20)
A? ? ? + sign(?b)a = ?b
(21)
?i ?i = 0, i = 1, 2, . . . , M.
(22)
The only nonlinear constraint is the product ?i ?i = 0. However, this can be effectively
implemented in the simplex procedure by modifying the selection of the pivot element
to ensure that ?i and ?i are never both in the set of basic variables. With this simple
modification of the simplex algorithm, the optimal ?M L can be efficiently computed.
2.2 Approximation of Q(?)
Once the most likely ?M L has been determined, the simplest approach for estimating the
new ? and ? 2 in Eqs. 9?10 is to replace Q(?) ? ?(? ? ?M L ) in the integrals. Unfortunately, this simple approximation will cause ? and ? to diverge from bad initial estimates.
To overcome these difficulties, we use a slightly more sophisticated method of estimating
the expectations to properly consider variability in the distribution Q(?).
We first note that the solution ?M L of the nonnegative quadratic optimization in Eq. 12
naturally partitions the elements of the vector ? into two distinct subsets ?I and ?J , consisting of components i ? I such that (?M L )i = 0, and components j ? J such that
(?M L )j > 0, respectively. It will then be useful to approximate the distribution Q(?) as
the factored form:
Q(?) ? QI (?I )QJ (?J )
(23)
Consider the components ?J near the maximum likelihood solution ?M L . Among these
components, none of nonnegativity constraints are active, so it is reasonable to approximate
the distribution QJ (?J ) by the unconstrained Gaussian:
QJ (?J ) ? exp[?F (?J |?I = 0)]
(24)
ML
This Gaussian distribution has mean ?J and inverse covariance given by the submatrix
AJJ of A = ?12 ST S.
For the other components ?I , it is important to consider the nonnegativity constraints, since
L
?M
= 0 is on the boundary of the distribution. We can represent QI (?I ) with the first
I
two order Tyler expansion:
?F
1
)|?M L ]TI ?I ? ?TI AII ?I )},
QI (?I ) ? exp{?[(
??
2
1
ML
T
? exp[?(A?
+ b)I ?I ? ?TI AII ?I ]
2
?I ? 0.
(25)
? I (?I ) so that the
QI (?I ) is then approximated with factorial exponential distribution Q
integrals in Eqs. 9?10 can be easily evaluated.
Y 1
? I (?I ) =
e??i /?i , ?I ? 0
(26)
Q
?i
i?I
which has support only for nonnegative ?I ? 0. The mean-field parameters ? are optimally obtained by minimizing the KL-divergence:
Z
?
? I (?I ) ln QI (?I ) .
(27)
min
d?I Q
??0 ?I ?0
QI (?I )
This integral can easily be computed in terms of the parameters ? and yields the minimization:
X
? T ? + 1 ?T A?,
?
(28)
min ?
ln ?i + b
I
??0
2
i?I
? I = (A?M L + b)I , A
? = AII + diag(AII ). To solve this minimization problem,
where b
we use an auxiliary function for Eq. 28 similar to the auxiliary function for nonnegative
quadratic programming:
? =?
G(?, ?)
X
i?I
?T ? + 1
ln ?i + b
I
2
X (A
? + ?)
? i
i?I
?
?i
?2i ?
1 X ??
?i ?j
Aij ?
?i ?
?j (1 + ln
), (29)
2
?
?i ?
?j
i,j?I
? =A
? + ?A
? ? is the decomposition of A
? into its positive and negative components.
where A
Minimization of this auxiliary function yields the following multiplicative update rules for
?i :
q
? + ?)i [(A
? ? ?)i + 1 ]
??bi + ?b2i + 4(A
?i
.
(30)
?i ?? ?i
+
?
2(A ?)i
These iterations are then guaranteed to converge to the optimal mean-field parameters for
the distribution QI (?I ).
? I (?I )QJ (?J ), the expectations in Eqs. 9?10 can be
Given the factorized approximation Q
analytically calculated. The mean value of ? under this distribution is given by:
?
?iM L if i ? J
(31)
?
?i =
?i
if i ? I
and its covariance C is:
?
Cij =
(AJJ ?1 )ij
?2i ?ij
if i, j ? J
otherwise
(32)
The update rules for ? and ? 2 are then given by:
?
??
?2
??
M
P
?i
i?
1
? T (x ? S?)
? + Tr(ST SC)]
[(x ? S?)
N
(33)
(34)
To summarize, the complete algorithm consists of the following steps:
1. Initialize ? and ? 2 .
2. Determine ?M L by solving the nonnegative quadratic programming in Eq. 12.
? I (?I )QJ (?J ) by solving the mean field
3. Approximate the distribution Q(?) ? Q
?I .
equations for ? in Q
? and covariance C for this distribution.
4. Calculate the mean ?
5. Reestimate regularization parameters ? and ? 2 using Eqs. 33?34.
6. Go back to Step 2 until convergence.
3
1.8
10
1.6
-40dB
40dB
-20dB
20dB
- 5dB
5dB
2
10
1.4
0
10
1
22
??
?/?
?/?
1
10
-40dB
40dB
-20dB
20dB
- 5dB
5dB
1.2
0.8
1
-1
10
0.6
2
-2
10
0.4
3
-3
10
0.2
0
4
-4
0
5
10
Iteration Number
15
20
10
0
5
10
Iteration Number
15
20
Figure 3: Iterative estimation of ? (M/? in the figure, indicating the reverberation level) and ? 2
when x(t) is contaminated by background white noise at -5 dB, -20 dB, and -40 dB levels. The
horizontal dotted lines indicate the true levels.
3 Results
We illustrate the performance of our algorithm in estimating the regularization parameters
as well as the nonnegative filter coefficients of a speech source signal s(t). The observed
signal x(t) is simulated by a time-delayed version of the source signal mixed with an echo
along with additive Gaussian white noise ?(t):
x(t) = s(t ? Ts ) + 0.5s(t ? 16.5Ts ) + ?(t).
(35)
We compare the results of the algorithm as the noise level is changed. Fig. 3 shows the
convergence of the estimates for ? and ? 2 as the noise level is varied between -5 dB and
-40 dB. There is rapid convergence of both parameters even with bad initial estimates. The
resulting value of the ? 2 parameter is very close to the true noise level. Additionally, the
estimated ? parameter is inversely related to the reverberation level of the environment,
given by the sum of the true filter coefficients.
Fig. 4 demonstrates the importance of correctly determining the regularization parameters
in estimating the time delay structure in the presence of noise. Using the Bayesian regularization procedure, the resulting estimate for ?M L correctly models the direct path time
delay as well as the secondary echo. However, if the regularization parameters are manually set incorrectly to over-sparsify the solution, the resulting estimates for the time delays
may be quite inaccurate.
4 Discussion
In summary, we propose using a Bayesian framework to automatically regularize nonnegative deconvolutions for estimating time delays in acoustic signals. We present two methods
for efficiently solving the resulting nonnegative quadratic programming problem. We also
derive an iterative algorithm from Expectation-Maximization to estimate the regularization
parameters. We show how these iterative updates can simulataneously estimate the timedelay structure in the signal, as well as the background noise level and reverberation level
of the room. Our results indicate that the algorithm is able to quickly converge to an optimal
solution, even with bad initial estimates. Preliminary tests with an acoustic robotic platform
indicate that these algorithms can successfully be implemented on a real-time system.
?=50, ?2 = 0.12
1
a
0.8
?
0.6
0.4
0.2
0
-20
-15
-10
-5
0
5
10
15
20
5
10
15
20
Time delay (Ts )
2
?? =200
1
b
0.8
?
0.6
0.4
0.2
0
-20
-15
-10
-5
0
Time delay (Ts )
Figure 4: Estimated time delay structure from ?M L with different regularizations: a) Bayesian
regularization, b) manually set regularization. Dotted lines indicate the true positions of the time
delays.
We are currently working to extend the algorithm to the situation where the source signal
needs to also be estimated. In this case, priors for the source signal are used to regularize
the source estimates. These priors are similar to those used for blind source separation. We
are investigating algorithms that can simultaneously estimate the hyperparameters for these
priors in addition to the other parameters within a consistent Bayesian framework.
References
[1] E. Ben-Reuven and Y. Singer, ?Discriminative binaural sound localization,? in Advances in Neural Information Processing Systems, S. T. Suzanna Becker and K. Obermayer, Eds., vol. 15. The
MIT Press, 2002.
[2] C. H. Knapp and G. C. Carter., ?The generalized correlation method for estimation of time delay,?
IEEE Transactions on ASSP, vol. 24, no. 4, pp. 320?327, 1976.
[3] Y. Lin, D. D. Lee, and L. K. Saul, ?Nonnegative deconvolution for time of arrival estimation,? in
ICASSP, 2004.
[4] J. B. Allen and D. A. Berkley, ?Image method for efficient simulating small-room acoustics,? J.
Acoust. Soc. Am., vol. 65, pp. 943?950, 1979.
[5] B. Olshausen and D. Field, ?Emergence of simple-cell receptive field properties by learning a
sparse code for nature images,? Nature, vol. 381, pp. 607?609, 1996.
[6] D. Foresee and M. Hagan, ?Gauss- Newton approximation to Bayesian regularization,? in Proceedings of the 1997 International Joint Conference on Neural Networks, 1997, pp. 1930?1935.
[7] M. E. Tipping, ?Sparse Bayesian learning and the relevance vector machine,? Journal of Machine
Learning Research, vol. 1, pp. 211?244, 2001.
[8] D. MacKay, ?Bayesian interpolation,? Neural Computation, vol. 4, pp. 415?447, 1992.
[9] F. Sha, L. K. Saul, and D. Lee, ?Multiplicative updates for nonnegative quadratic programming in support vector machines,? in Advances in Neural Information Processing Systems, S. T.
Suzanna Becker and K. Obermayer, Eds., vol. 15. The MIT Press, 2002.
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1,884 | 2,711 | Similarity and discrimination in classical
conditioning: A latent variable account
Aaron C. Courville*1,3 , Nathaniel D. Daw4 and David S. Touretzky2,3
1
Robotics Institute, 2 Computer Science Department,
3
Center for the Neural Basis of Cognition,
Carnegie Mellon University, Pittsburgh, PA 15213
4
Gatsby Computational Neuroscience Unit, University College London
{aaronc,dst}@cs.cmu.edu; [email protected]
Abstract
We propose a probabilistic, generative account of configural learning
phenomena in classical conditioning. Configural learning experiments
probe how animals discriminate and generalize between patterns of simultaneously presented stimuli (such as tones and lights) that are differentially predictive of reinforcement. Previous models of these issues
have been successful more on a phenomenological than an explanatory
level: they reproduce experimental findings but, lacking formal foundations, provide scant basis for understanding why animals behave as they
do. We present a theory that clarifies seemingly arbitrary aspects of previous models while also capturing a broader set of data. Key patterns
of data, e.g. concerning animals? readiness to distinguish patterns with
varying degrees of overlap, are shown to follow from statistical inference.
1 Introduction
Classical conditioning experiments probe how organisms learn to predict significant events
such as the receipt of food or shock. While there is a history of detailed quantitative theories about these experiments, only recently has there been a sustained attempt to understand
them in terms of sound statistical prediction [1]. A statistical foundation helps to identify
key theoretical issues (such as uncertainty) underlying these experiments, to explain otherwise puzzling results, and to connect these behavioral theories with theories of neural
computation, which are also increasingly framed in statistical terms.
A cluster of issues that has received great experimental and theoretical attention in conditioning ? but not yet from a statistically grounded perspective ? concerns discrimination
and generalization between patterns of sensory input. Historically, these issues arose in the
context of nonlinear discriminations, such as the XOR problem (in which, e.g., a light and
a tone each predict shock when presented alone, but not together). While animals can learn
such a discrimination, the seminal model of Rescorla and Wagner [2] cannot, since it assumes that the prediction is linear in the stimuli. Traditionally, this problem was solved by
introducing extra discriminative features to the model?s input (known as ?configural units,?
since they detect conjunctions of stimuli such as tone plus light), rendering the augmented
problem linearly solvable [3]. On this foundation rests a wealth of work probing how ani-
mals learn and predict given compounds of stimuli. Here, we reinterpret these issues from
a Bayesian perspective.
Previous work posits an informal division (or perhaps a spectrum) between ?elemental?
and ?configural? approaches to stimulus patterns, distinguished by whether a compound?s
association with reinforcement is derived from its individual stimuli (lights, tones), or rests
collectively in the full compound (light and tone together). The prototypical elemental
model is the original Rescorla?Wagner model, without configural units, in which the aggregate prediction is linear in the elements. The standard configural model is that of Pearce
[4], in which responding to a compound is determined by previous experience with that
and other similar compounds, through a process of generalization and weighted averaging.
Both theories match an impressive range of experimental data, but each is refuted by some
experiments that the other captures. It is not clear how to move beyond this stalemate.
Because the theories lack formal foundations, their details ? particularly those on which
they differ ? are ad-hoc and poorly understood. For instance, what circumstances justify
the introduction of a new configural unit, and what should be the form of generalization
between compounds?
Here we leverage our Bayesian theory of conditioning [5] to shed new light on these issues.
Our model differs from traditional ones in a number of ways. Notably, analogizing conditioning to classification, we take a generative rather than a discriminative approach. That
is, we assume animals are modeling their complete sensory experience (lights, tones, and
shocks) rather than only the chance of shock conditioned on lights and tones. We assume
that stimuli are correlated with each other, and with reinforcement, through shared latent
variables. Because a latent variable can trigger multiple events, these causes play a role
akin to configural units in previous theories, but offer stronger normative guidance. Questions about generalization (what is the probability that a latent variable is active given a
particular constellation of inputs) are seen as standard statistical inference; questions about
model structure (how many ?configural units? should there be and with what constellations
of stimuli are they associated) are answerable using Bayesian model averaging, which we
have suggested animals can approximate [5]. Such inferences also determine whether an
animal?s experience on a trial is best explained by multiple causes interacting additively, in
the style of Rescorla?Wagner, or by a single cause triggering multiple events like one of
Pearce?s configural units. This allows our theory to capture patterns of data that seem to
favor each of its predecessors.
Our theory is meant to shed light on the normative reasons why animals behave as they do,
rather than on how they might carry out computations like those we describe. In practice,
the inferences we discuss can be computed only approximately, and we intend no claim that
animals are using the same approximations to them as we are. More mechanistic models,
such as Pearce?s, can broadly be viewed as plausible implementations for approximating
some aspects of our more general framework.
2 Theories of Learning with Compound Stimuli
Classical conditioning experiments probe animals? anticipation of a reinforcer R such as
food or footshock, given the presentation of initially neutral stimuli such as lights and tones.
Expectation is assessed via reflexive conditioned responses such as salivation or freezing,
which are thought to reveal animals? predictions of reinforcement. By studying responding
as a function of the pattern of previous reinforcer / stimulus pairings, the experiments assess
learning. To describe a conditioning task abstractly, we use capital letters for the stimuli
and + and ? to indicate whether they are reinforced. For instance, the XOR task can be
written as A+, B+, AB?, where AB? denotes simultaneous presentation of both stimuli
unreinforced. Typically, each type of trial is delivered repeatedly, and the development of
responding is assessed.
We now describe the treatment of compound stimuli in the models of Rescorla and Wagner
[2] and Pearce [4]. In both models, the set of stimuli present on a trial is converted into
an input vector x. The strength of the conditioned response is modeled as proportional to
a prediction of reinforcement v = x ? w, the dot product between the input and a weight
vector. Finally, one or more weights are updated proportionally to the mismatch r ? v
between observed and predicted reinforcement.
For both theories, x includes an element (or ?unit?) corresponding to each individual stimulus.1 In Pearce?s model, and in augmented ?added elements? versions of the Rescorla?
Wagner model [3], additional ?configural? units are also included, corresponding to conjunctions of stimuli. In particular, it is assumed that a unique configural unit is added for
each stimulus compound observed, such as ABC. Note that this assumption is both arbitrary (e.g. we might very well include elements for subcompounds such as AB) and
unrealistic (given the profusion of uncontrolled stimuli simultaneously present in a real
experiment).
The theories differ as to how they apportion activation over x and learning over w. In the
Rescorla?Wagner model, the input vector is binary: xi = 1 if the ith stimulus (or an exactly
matching compound) is present, 0 otherwise. For learning, the weight corresponding to
each active input is updated. The Pearce model instead spreads graded activation over x,
based on a measure of similarity between the observed stimulus compound (or element)
and the compounds represented by the model?s configural units. In particular, if we denote
the number of stimulus elements present in an observed stimulus pattern a as size(a), and
in the pattern represented by the ith configural unit as size(i), then the activation of unit
i by pattern a is given by xi = size(overlap(a, i))2 /(size(a) ? size(i)). The learning
phase updates only the weight corresponding to the configural unit that exactly matches
the observed stimulus configuration.
As neither scheme has much formal basis, there seems to be no theoretical reason to prefer
one over the other, nor over any other ad-hoc recipe for apportioning representation and
learning. Empirical considerations also provide ambivalent guidance, as we discuss next.
3 Data on Learning with Compound Stimuli
Both the elemental and configural models reproduce a number of well known experimental
phenomena. Here we review several basic patterns of results. Notably, each theory has
a set of experiments that seems to support it over the other. Later, we will show that our
normative theory accounts for all of these results.
Overshadowing When a pair of stimuli AB+ is reinforced together, then tested separately, responding to either individual stimulus is often attenuated compared to a control in
which the stimulus is trained alone (A+). Both models reproduce overshadowing, though
Rescorla?Wagner incorrectly predicts that it takes at least two AB+ pairings to materialize.
Summation The converse of overshadowing is summation: when two stimuli are individually reinforced, then tested together, there is often a greater response to the pair than to
either element alone. In a recent variation by Rescorla [6], animals were trained on a pair
of compounds AB+ and CD+, then responses were measured to the trained compounds,
the individual elements A, B, etc., and the novel transfer compounds AD and BC. The
strongest response was elicited by the trained compounds. The transfer compounds elicited
a moderate response, and the individual stimuli produced the weakest responding.
1
In Pearce?s presentation of his model, these units are added only after elements are observed
alone. We include them initially, which does not affect the model?s behavior, to stress similarity with
the Rescorla-Wagner model.
The added elements Rescorla?Wagner model predicts this result due to the linear summation of the influences of all the units (A through D, AB, and CD ? note that the added
configural units are crucial). However, because of the normalization term in the generalization rule, Pearce?s model often predicts no summation. Here it predicts equal responding
to the individual stimuli and to the transfer compounds. There is controversy as to whether
the model can realistically be reconciled with summation effects [4, 7], but on the whole,
these phenomena seem more parsimoniously explained with an elemental account.
Overlap A large number of experiments (see [4] for a review) demonstrate that the more
elements shared by two compounds, the longer it takes animals to learn to discriminate
between them. Though this may seem intuitive, elemental theories predict the opposite.
In one example, Redhead and Pearce [8] presented subjects with the patterns A+, BC+
reinforced and ABC? unreinforced. Differential responding between A and ABC was
achieved in fewer trials than that between BC and ABC.
Pearce?s configural theory predicts this result because the extra overlap between BC and
ABC (compared to A vs. ABC) causes each compound to activate the other?s configural
unit more strongly. Thus, larger weights are required to produce a differentiated prediction.
Rescorla?Wagner predicts the opposite result, because compounds with more elements, e.g.
BC, accumulate more learning on each trial.
4 A latent variable model of stimulus generalization
In this section we present a generative model of how stimuli and reinforcers are jointly
delivered. We will show how the model may be used to estimate the conditional probability of reinforcement (the quantity we assume drives animals? responding) given some
pattern of observed stimuli. The theory is based on the one we presented in [5], and casts
conditioning as inference over a set of sigmoid belief networks. Our goal here is to use this
formalism to explain configural learning phenomena.
4.1 A Sigmoid Belief Network Model of Conditioning
Consider a vector of random variables S representing stimuli on a trial, with the jth stimulus present when Sj = 1 and absent when Sj = 0. One element of S is distinguished as the
reinforcer R; the remainder (lights and tones) is denoted as Stim. We encode the correlations between all stimuli (including the reinforcer) through common connections to a vector
of latent variables, or causes, x where xi ? {0, 1}. According to the generative process,
on each trial the state of the latent variables is determined by independent Bernoulli draws
(each latent variable has a weight determining its chance of activation [5]). The probability
of stimulus j being present is then determined by its relationship to the latent variables:
(j) T
P (Sj | m, wm , x) = (1 + exp(?(wm
) x ? wbias ))?1 ,
(1)
(j)
where the weight vector wm encodes the connection strengths between x and Sj for the
model structure m. The bias weight wbias is fixed at ?6, ensuring that spontaneous events
are rare. Some examples of the type of network structure under consideration are shown as
graphical models in Figure 1(c)?(d) and Figure 2(c)?(e).
We assume animals learn about the model structure itself, analogous to the experiencedependent introduction of configural units in previous theories. In our theory, animals
use experience to infer which network structures (from a set of candidates) and weights
likely produced the observed stimuli and reinforcers. These in turn determine predictions
of future reinforcement. Details of this inference are laid out below.
4.2 Generalization: inference over latent variables
Generalization between observed stimulus patterns is a key aspect of previous models. We
now describe how generalization arises in our theory.
Given a particular belief net structure m, weights wm , and previous conditioning experience D, the probability of reinforcement R given observed stimuli Stim can be computed
by integrating over the possible settings x of the latent variables:
X
P (R | Stim, m, wm , D) =
P (R | m, wm , x)P (x | Stim, m, wm , D)
(2)
x
The first term is given by Equation 1. By Bayes? rule, the second term weighs particular
settings of the hidden causes proportionally to the likelihood that they would give rise to the
observed stimuli. This process is a counterpart to Pearce?s generalization rule for configural
units. Unlike Pearce?s rule, inference over x considers settings of the individual causes xi
jointly (allowing for explaining away effects) and incorporates prior probabilities over each
cause?s activation. Nevertheless, the new rule broadly resembles its predecessor in that a
cause is judged likely to be active (and contributes to predicting R) if the constellation of
stimuli it predicts is similar to what is observed.
4.3 Learning to discriminate: inference over models
We treat the model weights wm and the model structure m as uncertain quantities subject
to standard Bayesian inference. We assume that, given a model structure, the weights are
mutually independent a priori and each distributed according to a Laplace distribution.2
Conditioning on the data D produces a posterior distribution over the weights, over which
we integrate to predict R:
Z
P (R | Stim, m, D) = P (R | Stim, m, wm , D)P (wm | m, D)dwm
(3)
Uncertainty over model structure is handled analogously. Integrating over posterior model
uncertainty we arrive at the prediction of reinforcement:
X
P (R | Stim, D) =
P (R | Stim, m, D)P (m | D),
(4)
m
where P (m | D) ? P (D | m)P (m) and the marginal likelihood P (D | m) is computed
similarly to equation 3, by integration over the weights. The prior over models, P (m) is
expressed as a distribution over nx , the number of latent variables, and over
Qnlxi , the number
P (li ). We
of links between the stimuli and each latent variable: P (m) = P (nx ) i=1
assume that P (nx ) and each P (li ) are given by geometric distributions (param. = 0.1),
renormalized to sum to unity over the maximum of 5 latents and 5 stimuli. This prior
reflects a bias against complex models. The marginal likelihood term also favors simplicity,
due to the automatic Occam?s razor (see [5]). For our simulations, we approximately
evaluated Equation 4 using reversible-jump Markov Chain Monte Carlo (see [5] for details).
Progressively conditioning on experience to resolve prior uncertainty in the weights and
model structure produces a gradual change in predictions akin to the incremental learning
rules of previous models. The extent to which a particular model structure m participates
in predicting R in Equation 4 is, by Bayes? rule, proportional to its prior probability, P (m),
and to the extent that it explains the data, P (D | m). Thus a prior preference for simpler
models competes against better data fidelity for more complex models. As data accumulate,
1 ?|y??|/b
The Laplace distribution is given by f (y) = 2b
e
. In our simulations ? = 0 and b = 2.
As a prior, it encodes a bias for sparsity consistent with a preference for simpler model structures.
2
1
1
0.8
0.8
P(R | Probe,D )
P(R | Probe,D )
x1
0.6
0.4
0.2
0
0.6
A
0.4
AB
Control
Probe Stimulus
(a) Overshadowing
0
R
(c) Overshadowing
x
0.2
A
B
x
1
Trained
Transfer
Probe Stimulus
(b) Summation
2
Element
A
B
C
D
R
(d) Summation
Figure 1: Results of MCMC simulation. (a) Overshadowing (AB+): the predicted probability of
reinforcement in response to presentations of the element A, the compound AB, and an individually
trained control element (A+). (b) Summation experiment (AB+, CD+): the predicted probability
of reinforcement in response to separate presentations of the trained compounds (AB, CD), the
transfer compounds (AD, BC) and the elements (A, B, etc.). (c) Depiction of the MAP model
structure after overshadowing training. (d) The MAP model structure after AB+ CD+ training.
the balance shifts toward the latter, and predictions become more accurate. Analogously,
weights are small a priori but can grow with experience.
Together with the generalization effects discussed above, these inference effects explain
why animals can learn more readily to discriminate stimulus compounds that have less
overlap. Key to the discrimination is inferring that different compounds are produced by
separate latent variables; the more the compounds overlap, the more accurately will the
data be approximated by a model with a single latent variable (preferred a priori), which
biases the complexity-fidelity tradeoff toward simplicity and retards acquisition.
5 Results
Overshadowing Overshadowing exemplifies our account of between-compound generalization; our model?s performance is illustrated in Figure 1(a). After 5 AB+ pairings,
the network with highest posterior probability, depicted in (c), contains one latent variable correlated with both stimuli and the reinforcer. Consistent with experimental results,
testing on A produces attenuated responding. This is because predicting whether A is reinforced requires balancing the relative probabilities of two unlikely events: that the stimulus
occurred spontaneously (with x1 inactive), versus that it was caused by x1 being active,
but that B uncharacteristically failed to occur (this probability measures generalization between the patterns A and AB). Overall, this tradeoff decreases the chance that x1 is active,
suppressing the prediction of reinforcement relative to the control treatment, where A is
reinforced in isolation (A+). Unlike the Rescorla?Wagner model, ours correctly predicts
that overshadowing can occur after even a single AB+ presentation.
Summation Figure 1(b) shows our model?s performance on Rescorla?s AB+ CD+ summation and transfer experiment [6], which is one of several summation experiments our
model explains. Compounds were reinforced 10 times. Consistent with experimental findings, the model predicts greatest responding to the trained compounds (AB, CD), moderate responding to transfer compounds (AD, BC), and least responding to the elements (A,
B, etc.). The maximum a posteriori (MAP) model structure (Figure 1(d)) mimics the training compounds, with one latent variable connected to A, B, and R and another connected
to C, D, and R. The results follow from a combination of generalization and additivity.
The training compounds activate one latent variable strongly; the transfer compounds acti-
3.5
Average number of latent variables
1
0.8
0.6
P(R | A, D )
P(R | B,C, D )
P(R | A,B,C, D )
0.4
0.2
0
0
5
10
15
20
Trial Blocks
25
1
A
B
R
(c) 4 Trial Blocks
A
1
0.5
5
10
15
20
Trial Blocks
25
30
(b) Model growth with training
x
x
B
C
1
C
2
1.5
0
0
30
(a) Overlap: learning curves
x
3
2.5
x
2
x
1
R
(d) 10 Trial Blocks
A
x
2
B
3
C
R
(e) 20 Trial Blocks
Figure 2: Summary of MCMC simulation results on the A+, BC+, ABC? experiment. The
estimated error due to MCMC sampling is small and not shown. (a) Learning curves showing the
predicted probability of reinforcement in response to separate presentations of A, BC, and ABC
as a function of number of trial blocks. (b) The average number of latent variables over the 10000
MCMC sample models. (c) - (e) Representations of MAP model structures after training with 4, 10,
and 20 trial blocks (edge widths represent mean weight strength).
vate both latents weakly (together additively influencing the probability of reinforcement);
the elements weakly activate only a single latent variable.
Overlap Figure 2(a) shows the model?s learning curves from the overlapping compound
experiment, A+, BC+, ABC?. Each trial block contains one trial of each type. The
model correctly predicts faster discrimination between A and ABC than between BC and
ABC. This pattern results from progressive increase in the number of inferred latent variables (b). Early in training, probability density concentrates on small models with a single
latent variable correlating all stimuli and the reinforcer (c). After more trials, models with
two latent variables become more probable, one correlating A and R and the other correlating B and C with both A and R, attempting to capture both BC+ and ABC? trial
types. (d). With further training, the most likely models are those with three latents, each
encoding one trial type (e). Our theory captures many similar experiments demonstrating
the difficulty of discriminating overlapping compounds.
6 Discussion
The configural unit is an ad-hoc device that nonetheless plays a key role in previous experimental and theoretical work in conditioning. Its inclusion in models like that of Rescorla?
Wagner invites a number of questions. Which configurations should be represented? How
should activation and learning be apportioned between them? These issues are contentious,
admitting no clear answer, precisely because of the arbitrary nature of the device. We
have shown how a latent variable correlated with a constellation of stimuli provides a well
founded counterpart to the configural unit, and how a range of experimental phenomena
concerning similarity and discrimination can be accounted for with the assumption that
animals are carrying out inference about these variables. While data exist that tend to fa-
vor each of the two major previous models of configural learning over the other, the new
model accounts for the full pattern, balancing the strengths of both theories. Our theory
also improves on its predecessors in other ways; for instance, because it includes learning
about stimulus interrelationships it can explain second-order conditioning [5], which is not
addressed by either the Pearce or the Rescorla?Wagner accounts.
Of course, many issues remain. A full account of summation phenomena, in particular, is
beyond the scope of the present model. We treat reinforcer delivery as binary and model
a limited, saturating, summation in probabilities. However, realistic summation almost
certainly concerns reinforcement magnitudes as well (see, for example, [9]), and our model
would need to be augmented to address them. Because we have assumed that trials are IID,
the model cannot yet account for effects of trial ordering (e.g. the difference between partial
reinforcement and extinction). These could be addressed by incorporating dynamics into
the generative model, so that inference requires tracking the changing model parameters.
Also for future work is exploring how different priors might give rise to different behavior.
An advantage of Bayesian modeling is that because the free parameters are formulated as
priors, they represent concrete assertions about the world (e.g. how often particular kinds
of events occur), and can thus be constrained and even experimentally manipulated.
We have focused only on two previous models and only on animal behavioral experiments.
Issues of similarity and discrimination are also studied in the rather different setting of human category judgments, where Bayesian generative approaches have also proved useful
[10]. There is also a tradition of more neurophysiological models of the hippocampal substrates of configural learning [11, 12]. Given the large body of theory and experiment on
these issues, this seems a promising direction for future work connecting our behavioral
theory with neurophysiological ones. In one of the hippocampal theories, Gluck and Myers [12] augment the Rescorla?Wagner model with an input representation learned by an
autoencoder. Since autoencoders perform probabilistic density modeling, this is probably
the most statistically minded of prior approaches to configural representation and has clear
parallels with our work.
Acknowledgments
This work was supported by National Science Foundation grants IIS-9978403 and DGE9987588. ND is funded by a Royal Society USA Research Fellowship and the Gatsby
Foundation. We thank Peter Dayan, Yael Niv and Geoff Gordon for helpful discussions.
References
[1] P. Dayan, T. Long, Advances in Neural Information Processing Systems 10 (1998), pp. 117?123.
[2] R. A. Rescorla, A. R. Wagner, Classical Conditioning II, A. H. Black, W. F. Prokasy, eds.
(Appleton-Century-Crofts, 1972), pp. 64?99.
[3] R. A. Rescorla, Journal of Comparative and Physiological Psychology 79, 307 (1972).
[4] J. M. Pearce, Psychological Review 101, 587 (1994).
[5] A. C. Courville, N. D. Daw, G. J. Gordon, D. S. Touretzky, Advances in Neural Information
Processing Systems 16 (2004).
[6] R. A. Rescorla, Quarterly Journal of Experimental Psychology 56B, 161 (2003).
[7] R. A. Rescorla, Animal Learning and Behavior 25, 200 (1997).
[8] E. S. Redhead, J. M. Pearce, Quarterly Journal of Experimental Psychology 48B, 46 (1995).
[9] E. F. Kremer, Journal of Experimental Psychology: Animal Behavior Processes 4, 22 (1978).
[10] J. B. Tenenbaum, T. L. Griffiths, Behavioral and Brain Sciences 24, 629 (2001).
[11] R. C. O?Reilly, J. W. Rudy, Psychological Review 108, 311 (2001).
[12] M. A. Gluck, C. Myers, Hippocampus 3, 491 (1993).
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1,885 | 2,712 | A harmonic excitation state-space approach to
blind separation of speech
Rasmus Kongsgaard Olsson and Lars Kai Hansen
Informatics and Mathematical Modelling
Technical University of Denmark, 2800 Lyngby, Denmark
rko,[email protected]
Abstract
We discuss an identification framework for noisy speech mixtures. A
block-based generative model is formulated that explicitly incorporates
the time-varying harmonic plus noise (H+N) model for a number of latent
sources observed through noisy convolutive mixtures. All parameters
including the pitches of the source signals, the amplitudes and phases of
the sources, the mixing filters and the noise statistics are estimated by
maximum likelihood, using an EM-algorithm. Exact averaging over the
hidden sources is obtained using the Kalman smoother. We show that
pitch estimation and source separation can be performed simultaneously.
The pitch estimates are compared to laryngograph (EGG) measurements.
Artificial and real room mixtures are used to demonstrate the viability
of the approach. Intelligible speech signals are re-synthesized from the
estimated H+N models.
1 Introduction
Our aim is to understand the properties of mixtures of speech signals within a generative
statistical framework. We consider convolutive mixtures, i.e.,
xt =
L?1
X
Ak st?k + nt ,
(1)
k=0
where the elements of the source signal vector, st , i.e., the ds statistically independent
source signals, are convolved with the corresponding elements of the filter matrix, Ak .
The multichannel sensor signal, xt , is furthermore degraded by additive Gaussian white
noise.
It is well-known that separation of the source signals based on second order statistics is
infeasible in general. Consider the second order statistic
hxt x>
t0 i =
L?1
X
>
Ak hst?k s>
t0 ?k0 iAk0 + R,
(2)
k,k0 =0
where R is the (diagonal) noise covariance matrix. If the sources can be assumed stationary
white noise, the source covariance matrix can be assumed proportional to the unit matrix
without loss of generality, and we see that the statistic is symmetric to a common rotation
of all mixing matrices Ak ? Ak U. This rotational invariance means that the acquired
statistic is not informative enough to identify the mixing matrix, hence, the source time
series.
However, if we consider stationary sources with known, non-trivial, autocorrelations
0
hst s>
t0 i = G(t ? t ), and we are given access to measurements involving multiple val0
ues of G(t ? t ), the rotational degrees of freedom are constrained and we will be able to
recover the mixing matrices up to a choice of sign and scale of each source time series.
Extending this argument by the observation that the mixing model (1) is invariant to filtering of a given column of the convolutive filter provided that the inverse filter is applied to
corresponding source signal, we see that it is infeasible to identify the mixing matrices if
these arbitrary inverse filters can be chosen to that they are allowed to ?whiten? the sources,
see also [1].
For non-stationary sources, on the other hand, the autocorrelation functions vary through
time and it is not possible to choose a single common whitening filter for each source. This
means that the mixing matrices may be identifiable from multiple estimates of the second
order correlation statistic (2) for non-stationary sources. Analysis in terms of the number
of free parameters vs. the number of linear conditions is provided in [1] and [2].
Also in [2], the constraining effect of source non-stationarity was exploited by the simultaneous diagonalization of multiple estimates of the source power spectrum. In [3] we
formulated a generative probabilistic model of this process and proved that it could estimate sources and mixing matrices in noisy mixtures. Blind source separation based on
state-space models has been studied, e.g., in [4] and [5]. The approach is especially useful
for including prior knowledge about the source signals and for handling noisy mixtures.
One example of considerable practical importance is the case of speech mixtures.
For speech mixtures the generative model based on white noise excitation may be improved
using more realistic priors. Speech models based on sinusoidal excitation have been quite
popular in speech modelling since [6]. This approach assumes that the speech signal is
a time-varying mixture of a harmonic signal and a noise signal (H+N model). A recent
application of this model for pitch estimation can be found in [7]. Also [8] and [9] exploit
the harmonic structure of certain classes of signals for enhancement purposes. A related
application is the BSS algorithm of [10], which uses the cross-correlation of the amplitude
in different frequency. The state-space model naturally leads to maximum-likelihood estimation using the EM-algorithm, e.g. [11], [12]. The EM algorithm has been used in related
models: [13] and [14].
In this work we generalize our previous work on state space models for blind source separation to include harmonic excitation and demonstrate that it is possible to perform simultaneous un-mixing and pitch tracking.
2 The model
The assumption of time variant source statistics help identify parameters that would otherwise not be unique within the model. In the following, the measured signals are segmented
into frames, in which they are assumed stationary. The mixing filters and observation noise
covariance matrix are assumed stationary across all frames.
The colored noise (AR) process that was used in [3] to model the sources is augmented to
include a periodic excitation signal that is also time-varying. The specific choice of periodic
basis function, i.e. the sinusoid, is motivated by the fact that the phase is linearizable,
facilitating one-step optimization. In frame n, source i is represented by:
sni,t
=
p
X
n n
fi,t
0 si,t?t0 +
t0 =1
=
p
X
K
X
n
n
n
?i,k
sin(?0,i
kt + ?in ) + vi,t
k=1
n n
fi,t
0 si,t?t0 +
t0 =1
K
X
n
n
n
cni,2k?1 sin(?0,i
kt) + cni,2k cos(?0,i
kt) + vi,t
(3)
k=1
n
where n ? {1, 2, .., N } and i ? {1, 2, .., ds }. The innovation noise, vi,t
, is i.i.d Gausn
sian. Clearly, (3) represents a H+N model. The fundamental frequency, ?0,i
, enters the
estimation problem in an inherent non-linear manner.
In order to benefit from well-established estimation theory, the above recursion is fitted into the framework of Gaussian linear models, see [15]. The Kalman filter model
is an instance of this model. The augmented state space is constructed by including a history of past samples for each source. Source vector i in frame n is defined:
?
?>
sni,t = sni,t sni,t?1 . . . sni,t?p+1
. All sni,t ?s are stacked in the total source vec?
?>
tor: ?snt = (sn1,t )> (sn2,t )> . . . (snds ,t )>
. The resulting state-space model is:
?snt
xnt
? tn
= Fn?snt?1 + Cn unt + v
n
n
= A?st + nt
? t ? N (0, Q), nt ? N (0, R) and ?sn1 ? N (?n , ?n ). The combined harmonics inwhere v
?>
?
, where the harmonics
put vector is defined: unt = (un1,t )> (un2,t )> . . . (unds ,t )>
corresponding to source i in frame n are:
?
?>
n
n
n
n
t) cos(?0,i
t) . . . sin(K?0,i
t) cos(K?0,i
t)
uni,t = sin(?0,i
It is apparent that the matrix multiplication by A
sources, where the dx ? ds channel filters are:
? >
..
a>
a11
12
>
? a>
..
a
22
? 21
A=? .
..
..
.
? .
.
.
a>
dx 1
In order to implement the
follows:
? n
???
F1 0
0
Fn2 ? ? ?
?
Fn = ?
..
..
? ...
.
.
0
0
???
? n
Q1 0
???
0
Qn2 ? ? ?
?
Qn = ?
..
..
? ...
.
.
0
0
???
? n
C1 0
???
0
Cn2 ? ? ?
?
Cn = ?
..
..
? ...
.
.
0
0
???
a>
dx 2
constitutes a convolutive mixing of the
?
a>
1ds
a>
2ds
..
.
?
?
?
?
a>
dx ds
..
H+N source model, the parameter matrices are constrained as
?
0
0
?
? ,
..
?
.
n
Fds
?
0
0
?
? ,
..
?
.
n
Qds
?
0
0
?
? ,
..
?
.
Cnds
?
?
?
Fni = ?
?
?
n
fi,1
1
0
..
.
n
fi,2
0
1
..
.
???
???
???
..
.
n
fi,p?1
0
0
..
.
n
fi,p
0
0
..
.
0
0
???
1
0
(Qni )jj 0 =
?
Cni
?
?
=?
?
?
n
qin
0
j = j 0W= 1
j 6= 1 j 0 6= 1
cni,1
0
0
..
.
cni,2
0
0
..
.
???
???
???
..
.
cni,2K
0
0
..
.
0
0
???
0
?
?
?
?
?
?
?
?
?
?
?
?
3 Learning
Having described the convolutive mixing problem in the general framework of linear Gaussian models, more specifically the Kalman filter model, optimal inference of the sources is
obtained by the Kalman smoother. However, since the problem at hand is effectively blind,
we also need to estimate the parameters. Along the lines of, e.g. [15], we will invoke an EM
approach. The log-likelihood is bounded
from below: L(?) ? F (?, p?) ?
p),
R
R J (?, p?) ? R(?
with the definitions J (?, p?) ? dS?
p(S) log p(X, S|?) and R(?
p) ? dS?
p(S) log p?(S).
In accordance with standard EM theory, J (?, p?) is optimized wrt. ? in the M-step. The
E-step infers the relevant moments of the marginal posterior, p? = p(S|X, ?). For the Gaussian model the means are also source MAP estimates. The combined E and M steps are
guaranteed not to decrease L(?).
3.1 E-step
The forward-backward recursions which comprise the Kalman smoother are employed in
the E-step to infer moments of the source posterior, p(S|X, ?), i.e. the joint posterior of
the sources conditioned on all observations. The relevant second-order statistic of this
n
distribution in segment n is the marginal posterior mean, ??st ? h?snt i, and autocorrelation,
n
n
n
>
n
n
n >
Mi,t ? hsi,t (si,t ) i ? [ mi,1,t mi,2,t .. mi,L,t ] , along with the marginal lag-one
1,n
>
>
n
n
covariance, M1,n
m1,n
.. m1,n
i,2,t
i,t ? hsi,t (si,t?1 ) i ? [ mi,1,t
i,L,t ] . In particular,
n
n
mi,t is the first element of mi,1,t . All averages are performed over p(S|X, ?). The forward
recursion also yields the log-likelihood, L(?).
3.2 M-step
The M-step utility function, J (?, p?), is defined:
J (?, p?) = ?
ds
ds
N
X
1XX
[
log det ?ni + (? ? 1)
log qin
2 n=1 i=1
i=1
+? log det R +
ds
X
h(sni,1 ? ?ni )T (?ni )?1 (sni,1 ? ?ni )i
i=1
+
ds
? X
X
t=2 i=1
h
?
X
1 n
n T n 2
(s
?
(d
)
z
)
i
+
h(xnt ? A?snt )T R?1 (xnt ? A?snt )i]
i
i,t
qin i,t
t=1
where h?i signifies averaging over the source posterior from the previous E-step, p(S|X, ?)
and ? is the frame length. The linear source parameters are grouped as
?>
?
?>
?
dni ? (fin )> (cni )>
, zni ? (sni,t?1 )> (uni,t )>
where
fin ? [ fi,1
fi,2
..
fi,p ]
>
,
cni ? [ ci,1
ci,2
>
.. ci,p ]
n
Optimization of J (?, p?) wrt. ? is straightforward (except for the ?0,i
?s). Relatively minor
changes are introduced to the estimators of e.g. [12] in order to respect the special constrained format of the parameter matrices and to allow for an external input to the model.
More details on the estimators for the correlated source model are given in [3].
n
It is in general difficult to maximize J (?, p?) wrt. to ?i,0
, since several local maxima exist,
n
e.g. at multiples of ?i,0
, see e.g. [6]. This problem is addressed by narrowing the search
range based on prior knowledge of the domain, e.g. that the pitch of speech lies in the range
n
50-400Hz. A candidate estimate for ?i,0
is obtained by computing the autocorrelation
n
n > n
function of si,t ? (fi ) si,t?1 . Grid search is performed in the vicinity of the candidate.
For each point in the grid we optimize dni :
" ? ?
?#?1 X
?
? ?
n
n >
X (Mn )
?
s
(u
)
mni,t,t?1
n
i,t?1
i,t?1
i,t
(4)
di,new =
s?ni,t uni,t
uni,t (?sni,t?1 )> uni,t (uni,t )>
t=2
t=2
F [Hz] (source 1)
At each step of the EM-algorithm, the parameters are normalized by enforcing ||Ai || = 1,
4000
3000
2000
1000
F [Hz] (source 2)
0
4000
3000
2000
1000
0
0
0.5
1
t [sec]
0
0.5
1
t [sec]
0
0.5
1
t [sec]
Figure 1: Amplitude spectrograms of the frequency range 0-4000Hz, from left to right: the
true sources, the estimated sources and the re-synthesized source.
that is enforcing a unity norm on the filter coefficients related to source i.
4 Experiment I: BSS and pitch tracking in a noisy artificial mixture
The performance of a pitch detector can be evaluated using electro-laryngograph (EGG)
recordings, which are obtained from electrodes placed on the neck, see [7]. In the following
experiment, speech signals from the TIMIT [16] corpus is used for which the EGG signals
were measured, kindly provided by the ?festvox? project (http://festvox.org).
Two male speech signals (Fs = 16kHz) were mixed through known mixing filters and
degraded by additive white noise (SNR ?20dB), constructing two observation signals. The
pitches of the speech signals were overlapping. The filter coefficients (of 2 ? 2 = 4 FIR
filter impulse responses) were:
?
?
1.00 0.35 ?0.20
0.00 0.00, 0.00 0.00 ?0.50 ?0.30 0.20
A =
0.00 0.00
0.70 ?0.20 0.15, 1.30 0.60
0.30
0.00 0.00
The signals were segmented into frames, ? = 320 ? 20ms, and the order of the ARprocess was set to p = 1. The number of harmonics was limited to K = 40. The pitch
grid search involved 30 re-estimations of dni . In figure 1 is shown the spectrograms of
F [Hz] (source 1)
160
140
0
120
100
160
140
120
100
0
F [Hz] (source 2)
80
80
0.4
0.6
0.8
t [sec]
1
Figure 2: The estimated (dashed) and EGG-provided (solid) pitches as a function of time.
The speech mixtures were artificially mixed from TIMIT utterances and white noise was
added.
approximately 1 second of 1) the original sources, 2) the MAP source estimates and 3) the
resynthesized sources (from the estimated model parameters). It is seen that the sources
were well separated. Also, the re-synthesizations are almost indistinguishable from the
source estimates. In figure 2, the estimated pitch of both speech signals are shown along
with the pitch of the EGG measurements.1 The voiced sections of the speech were manually
preselected, this step is easily automated. The estimated pitches do follow the ?true? pitches
as provided by the EGG. The smoothness of the estimates is further indicating the viability
of the approach, as the pitch estimates are frame-local.
5 Experiment II: BSS and pitch tracking in a real mixture
The algorithm was further evaluated on real room recordings that were also used in [17].2
Two male speakers synchronously count in English and Spanish (Fs = 16kHz). The mixtures were degraded with noise (SNR ?20dB). The filter length, the frame length, the order
of the AR-process and the number of harmonics were set to L = 25, ? = 320, p = 1 and
K = 40, respectively. Figure 3 shows the MAP source estimates and the re-synthesized
sources. Features of speech such as amplitude modulation are clearly evident in estimates
and re-synthesizations.3 A listening test confirms: 1) the separation of the sources and
2) the good quality of the synthesized sources, reconfirming the applicability of the H+N
model. Figure 4 displays the estimated pitches of the sources, where the voiced sections
were manually preselected. Although, the ?true? pitch is unavailable in this experiment, the
smoothness of the frame-local pitch-estimates is further support for the approach.
1
The EGG data are themselves noisy measurements of the hypothesized ?truth?. Bandpass filtering
was used for preprocessing.
2
The mixtures were obtained from http://inc2.ucsd.edu/?tewon/ica_cnl.html.
3
Note that the ?English? counter lowers the pitch throughout the sentence.
F [Hz] (source 1)
3000
2000
1000
F [Hz] (source 2)
0
3000
2000
1000
0
0
1
2
30
t [sec]
1
2
3
t [sec]
Figure 3: Spectrograms of the estimated (left) and re-synthesized sources (right) extracted
from the ?one two . . . ? and ?uno dos . . . ? mixtures, source 1 and 2, respectively
6 Conclusion
It was shown that prior knowledge on speech signals and quasi-periodic signals in general
can be integrated into a linear non-stationary state-space model. As a result, the simultaneous separation of the speech sources and estimation of their pitches could be achieved. It
was demonstrated that the method could cope with noisy artificially mixed signals and real
room mixtures. Future research concerns more realistic mixtures in terms of reverberation
time and inclusion of further domain knowledge. It should be noted that the approach is
computationally intensive, we are also investigating means for approximate inference and
parameter estimation that would allow real time implementation.
Acknowledgement
This work is supported by the Danish ?Oticon Fonden?.
References
[1] E. Weinstein, M. Feder and A.V. Oppenheim, Multi-channel signal separation by decorrelation,
IEEE Trans. on speech and audio processing, vol. 1, no. 4, pp. 405-413,1993.
[2] Parra, L., Spence C., Convolutive blind separation of non-stationary sources. IEEE Trans. on
speech and audio processing, vol. 5, pp. 320-327, 2000.
[3] Olsson, R. K., Hansen L. K., Probabilistic blind deconvolution of non-stationary source. Proc.
EUSIPCO, 2004, accepted. Olsson R. K., Hansen L. K., Estimating the number of sources in a
noisy convolutive mixture using BIC. International conference on independent component analysis 2004, accepted. Preprints may be obtained from http://www.imm.dtu.dk/?rko/
research.htm.
[4] Gharbi, A.B.A., Salam, F., Blind separtion of independent sources in linear dynamical media. NOLTA, Hawaii, 1993. http://www.egr.msu.edu/bsr/papers/blind_
separation/nolta93.pdf
120
100
0
F [Hz] (source 2)
0
F [Hz] (source 1)
140
80
180
160
140
120
100
80
0
2
4
t [sec]
6
8
Figure 4: Pitch tracking in ?one two . . . ?/?uno dos . . . ? mixtures.
[5] Zhang, L., Cichocki, A., Blind Deconvolution of dynamical systems: a state space appraoch,
Journal of signal processing, vol. 4, no. 2, pp. 111-130, 2000.
[6] McAulay, R.J., Quateri. T.F., Speech analysis/synthesis based on a sinusoidal representation,
IEEE Trans. on acoustics, speech and signal processing, vol. 34, no. 4, pp. 744-754, 1986.
[7] Parra, L., Jain U., Approximate Kalman filtering for the harmonic plus noise model. IEEE Workshop on applications of signal processing to audio and acoustics, pp. 75-78, 2001.
[8] Nakatani, T., Miyoshi, M., and Kinoshita, K., One microphone blind dereverberation based on
quasi-periodicity of speech signals, Advances in Neural Information Processing Systems 16 (to
appear), MIT Press, 2004.
[9] Hu, G. Wang, D., Monaural speech segregation based on pitch tracking and amplitude modulation, IEEE Trans. neural networks, in press, 2004.
[10] Anem?uller, J., Kollmeier, B., Convolutive blind source separation of speech signals based on
amplitude modulation decorrelation, Journal of the Acoustical Society of America, vol. 108, pp.
2630, 2000.
[11] A. P. Dempster, N. M. Laird, and Rubin D. B., Maximum liklihood from incomplete data via
the EM algorithm, Journal of the Royal Statistical Society, vol. 39, pp. 1?38, 1977.
[12] Shumway, R.H., Stoffer, D.S., An approach to time series smoothing and forecasting using the
EM algorithm. Journal of time series analysis, vol. 3, pp. 253-264. 1982.
[13] Moulines E., Cardoso J. F., Gassiat E., Maximum likelihood for blind separation and deconvolution of noisy signals using mixture models, ICASSP, vol. 5, pp. 3617-20, 1997.
[14] Cardoso, J.F., Snoussi, H. , Delabrouille, J., Patanchon, G., Blind separation of noisy Gaussian
stationary sources. Application to cosmic microwave background imaging, Proc. EUSIPCO, pp
561-564, 2002.
[15] Roweis, S., Ghahramani, Z., A unifying review of linear Gaussian models. Neural Computation,
vol. 11, pp. 305-345, 1999.
[16] Center for Speech Technology Research, University of Edinburgh,http://www.cstr.ed.
ac.uk/
[17] Lee, T.-W., Bell, A.J., Orglmeister, R., Blind source separation of real world signals, Proc. IEEE
international conference neural networks, pp 2129-2135, 1997.
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1,886 | 2,713 | The Entire Regularization Path for the Support
Vector Machine
Trevor Hastie
Department of Statistics
Stanford University
Stanford, CA 94305, USA
[email protected]
Saharon Rosset
IBM Watson Research Center
P.O. Box 218
Yorktown Heights, N.Y. 10598
[email protected]
Robert Tibshirani
Department of Statistics
Stanford University
Stanford, CA 94305, USA
[email protected]
Ji Zhu
Department of Statistics
University of Michigan
Ann Arbor, MI 48109-1092
[email protected]
Abstract
In this paper we argue that the choice of the SVM cost parameter can be
critical. We then derive an algorithm that can fit the entire path of SVM
solutions for every value of the cost parameter, with essentially the same
computational cost as fitting one SVM model.
1
Introduction
We have a set of n training pairs xi , yi , where xi ? Rp is a p-vector of real valued predictors
(attributes) for the ith observation, yi ? {?1, +1} codes its binary response. The standard
criterion for fitting the linear SVM )[1, 2, 3] is
1
min ||?||2 + C
?i ,
?0 ,? 2
i=1
n
subject to, for each i: yi (?0 + xTi ?)
(1)
?
1 ? ?i .
Here the ?i are non-negative slack variables that allow points to be on the wrong side of their
?soft margin? (f (x) = ?1), as well as the decision boundary, and C is a cost parameter that
controls the amount of overlap. If the data are separable, then for sufficiently large C the
solution achieves the maximal margin separator; if not, the solution achieves the minimum
overlap solution with largest margin.
Alternatively, we can formulate the problem using a (hinge) Loss + Penalty criterion [4, 5]:
min
?0 ,?
n
i=1
[1 ? yi (?0 + ? T xi )]+ +
?
||?||2 .
2
The regularization parameter ? in (2) corresponds to 1/C, with C in (1).
(2)
This latter formulation emphasizes the role of regularization. In many situations we have
sufficient variables (e.g. gene expression arrays) to guarantee separation. We may nevertheless avoid the maximum margin separator (? ? 0), which is governed by observations
on the boundary, in favor of a more regularized solution involving more observations.
The nonlinear
kernel SVMs can be represented in this form as well. With kernel K and
n
f (x) = ?0 + i=1 ?i K(x, xi ), we solve [5]
min
?0 ,?
n
[1 ? yi (?0 +
i=1
n
?
?j ?j K(xj , xj ).
2 j=1
n
?i K(xi , xj ))] +
j=1
n
(3)
j =1
Often the regularization parameter C (or ?) is regarded as a genuine ?nuisance?. Software
packages, such as the widely used SVMlight [6], provide default settings for C.
To illustrate the effect of regularization, we generated data from a pair of mixture densities,
described in detail in [5, Chapter 2]. We used an SVM with a radial kernel K(x, x ) =
exp(??||x ? x ||2 ). Figure 1 shows the test error as a function of C for these data, using
four different values for ?. Here we see a dramatic range in the correct choice for C (or
? = 1/C). When ? = 5, the most regularized model is called for; when ? = 0.1, the least
regularized.
Test Error Curves ? SVM with Radial Kernel
?=1
? = 0.5
? = 0.1
0.30
0.25
0.20
Test Error
0.35
?=5
1e?01
1e+01
1e+03
1e?01
1e+01
1e+03
1e?01
1e+01
1e+03
1e?01
1e+01
1e+03
C = 1/?
Figure 1: Test error curves for the mixture example, using four different values for the radial kernel
parameter ?.
One of the reasons that investigators avoid extensive exploration of C is the computational
cost involved. In this paper we develop an algorithm which fits the entire path of SVM
solutions [?0 (C), ?(C)], for all possible values of C, with essentially the computational
cost of fitting a single model for a particular value of C. Our algorithm exploits the fact
that the Lagrange multipliers implicit in (1) are piecewise-linear in C. This also means that
?
the coefficients ?(C)
are also piecewise-linear in C. This is true for all SVM models, both
linear and nonlinear kernel-based SVMs.
2
Problem Setup
We use a criterion equivalent to (1), implementing the formulation in (2):
n
?
min
?i + ? T ? subject to 1 ? yi f (xi ) ? ?i ; ?i ? 0; f (x) = ?0 + ? T x.
?,?0
2
i=1
(4)
Initially we consider only linear SVMs to get the intuitive flavor of our procedure; we then
generalize to kernel SVMs.
We construct the Lagrange primal function
LP :
n
n
n
? T
?i + ? ? +
?i (1 ? yi f (xi ) ? ?i ) ?
?i ?i
2
i=1
i=1
i=1
(5)
and set the derivatives to zero. This gives
?
:
??
?
:
??0
1
?i yi xi
? i=1
n
?=
n
yi ?i = 0,
(6)
(7)
i=1
along with the KKT conditions
?i (1 ? yi f (xi ) ? ?i ) = 0
?i ?i = 0
1 ? ?i ? ?i = 0
(8)
(9)
(10)
We see that 0 ? ?i ? 1, with ?i = 1 when ?i > 0 (which is when yi f (xi ) < 1). Also
when yi f (xi ) > 1, ?i = 0 since no cost is incurred, and ?i = 0. When yi f (xi ) = 1, ?i
can lie between 0 and 1.
The usual Lagrange multipliers associated with the solution to (1) are ?i = ?i /? = C?i .
We prefer our formulation here since our ?i ? [0, 1], and this simplifies the definition of
the paths we define.
We wish to find the entire solution path for all values of ? ? 0. Our basic idea is as follows.
We start with ? large and decrease it toward zero, keeping track of all the events that occur
along the way. As ? decreases, ||?|| increases, and hence the width of the margin decreases.
As this width decreases, points move from being inside to outside their margins. Their
corresponding ?i change from ?i = 1 when they are inside their margin (yi f (xi ) < 1) to
?i = 0 when they are outside their margin (yi f (xi ) > 1). By continuity, points must linger
on the margin (yi f (xi ) = 1) while their ?i decrease from 1 to 0. We will see that the ?i (?)
trajectories are piecewise-linear in ?, which affords a great computational savings: as long
as we can establish the break points, all values in between can be found by simple linear
interpolation. Note that points can return to the margin, after having passed through it.
It is easy to show that if the ?i (?) are piecewise linear in ?, then both ?i (C) = C?i (C)
and ?(C) are piecewise linear in C. It turns out that ?0 (C) is also piecewise linear in C.
Our algorithm keeps track of the following sets:
? M = {i : yi f (xi ) = 1, 0 ? ?i ? 1}, M for Margin
? I = {i : yi f (xi ) < 1, ?i = 1}, I for Inside the margin
? O = {i : yi f (xi ) > 1, ?i = 0}, O for Outside the margin
3
The Algorithm
Due to space restrictions, we show some details here; the rest can be found in [7].
Initialization
The initial conditions depend on whether the classes are balanced or not (n+ = n? ). The
balanced case is easier. For very large ?, ||?|| is small, and the the margin is very wide,
all points are in O, and hence ?i = 1?i. From (6) this means the orientation of ? is fixed
until the ?i change. The margin narrows as ? decreases, but the orientation remains fixed.
Because of (7), the narrowing margin must connect with an outermost member of each
class simultaneously. These points are easily identified, and this establishes the first event,
the first tenants of M, and ?0 .
When n? = n+ , the setup is more complex. In order to satisfy the constraint (7), a
quadratic programming algorithm is needed to obtain the initial configuration. See [7] for
details.
Kernels
The development so far has been in the original feature space. It is easy to see that the
entire development carries through with ?kernels? as well. In this case f (x) = ?0 + g(x),
and the only change that occurs is that (6) is changed to
1
?j yj K(xi , xj ), i = 1, . . . , n,
? j=1
n
g(xi ) =
(11)
or ?j (?) = ?j yj /? using the notation in (3). Hereafter we will develop our algorithm for
this more general kernel case.
The Path
The algorithm hinges on the set of points M sitting on the margin. We consider M at the
point that an event has occurred:
1. The initial event, which means 2 or more points start in M, with their initial values
of ? ? [0, 1].
2. A point from I has just entered M, with its value of ?i initially 1.
3. A point from O has reentered M, with its value of ?i initially 0.
4. One or more points in M has left the set, to join either O or I.
Whichever the case, for continuity reasons this set will stay stable until the next event
occurs, since to pass through M, a point?s ?i must change from 0 to 1 or vice versa. Since
all points in M have yi f (xi ) = 1, we can establish a path for their ?i .
We use the subscript to index the sets above immediately after the th event has occurred.
Suppose |M | = m, and let ?i , ?0 and ? be the values of these parameters at the point of
entry. Likewise f is the function at this point. For convenience we define ?0 = ??0 , and
hence ?0 = ? ?0 .
Since
?
?
n
1 ?
yj ?j K(x, xj ) + ?0 ? ,
f (x) =
? j=1
for ? > ? > ?+1 we can write
?
?
f (x) =
f (x) ? f (x) + f (x)
?
?
?
?
1?
=
(?j ? ?j )yj K(x, xj ) + (?0 ? ?0 ) + ? f (x)? .
?
j?M
(12)
(13)
The second line follows because all the observations in I have their ?i = 1, and those in
O have their ?i = 0, for this range of ?. Since each of the m points xi ? M are to stay
on the margin, we have that
?
?
1?
(?j ? ?j )yi yj K(xi , xj ) + yi (?0 ? ?0 ) + ? ? = 1, ?i ? M .
(14)
?
j?M
Writing ?j = ?j ? ?j , from (14) we have
?j yi yj K(xi , xj ) + yi ?0 = ? ? ?, ?i ? M .
(15)
j?M
Furthermore, since at all times
n
i=1
yi ?i = 0, we have that
yj ?j = 0.
(16)
j?M
Equations (15) and (16) constitute m + 1 linear equations in m + 1 unknowns ?j , and can
be solved. The ?j and hence ?j will change linearly in ?, until the next event occurs:
?j = ?j ? (? ? ?)bj , j ? {0} ? M .
(17)
See [7] for more precise details on solving these equations.
From (13) we have
f (x) =
?
f (x) ? h (x) + h (x),
?
where
h (x) =
yj bj K(x, xj ) + b0
(18)
(19)
j?M
Thus the function itself changes in a piecewise-inverse manner in ?.
Finding ?+1
The paths continue until one of the following events occur:
1. One of the ?i for i ? M reaches a boundary (0 or 1). For each i the value of ?
for which this occurs is easily established.
2. One of the points in I or O attains yi f (xi ) = 1.
By examining these conditions, we can establish the largest ? < ? for which an event
occurs, and hence establish ?+1 and update the sets.
Termination
In the separable case, we terminate when I becomes empty. At this point, all the ?i in (4)
are zero, and further movement increases the norm of ? unnecessarily.
In the non-separable case, ? runs all the way down to zero. For this to happen without f
?blowing up? in (18), we musthave f ? h = 0, and hence the boundary and margins
remain fixed at a point where i ?i is as small as possible, and the margin is as wide as
possible subject to this constraint.
3.1
Computational Complexity
?=1
? = 0.5
? = 0.1
1e?04
1e?02
1e+00
?
1e?03
Eigenvalue
?=5
1e?07
1 1111111 11111
11 111
1
111
11111
11
11
1e?11
80
60
40
Margin Size
20
0
4
3
23
43
2
2
33
422
1111
222
33322
41111
332211111
4 1111
2 111111111111111111
33222
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44
11111111111111
1111111111111111
33 2222222
4
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3333
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4
33
22222
11111111111
44
3333
22
1111111111
22222
333
4
111111111
222
4
11111111
333
2222
3333
11111111
22222
444
111111
333
2222
111111
4
333
2222
1111111
44
3333
222222
11111
3
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44
33
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44
33
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33
22222
4
33333
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44
33
11
22
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333
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222
44
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22
33
22
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4
333
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44444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444
2
22
3
2
4
333333333333333333333333333333333333333333
222222222
4
3
2
?=5
1 1 1
11
11 11
1111
11 11 11111
111
1 111
111111 1 1
1111 11
11 1
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1 11 1 111 1
11 1 1111
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1
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1
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11
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2
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2
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1
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1
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2
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At any update event along the path of our algorithm, the main computational burden is
solving the system of equations of size m = |M |. While this normally involves O(m3 )
computations, since M+1 differs from M by typically one observation, inverse updating can reduce the computations to O(m2 ). The computation of h (xi ) in (19) requires
O(nm ) computations. Beyond that, several checks of cost O(n) are needed to evaluate
the next move.
?=1
? = 0.5
? = 0.1
0
50
100
150
200
Sequence Number
Figure 2: [Left] The margin sizes |M | as a function of ?, for different values of the radial-kernel
parameter ?. The vertical lines show the positions used to compare the times with libsvm. [Right]
The eigenvalues (on the log scale) for the kernel matrices K? corresponding to the four values
of ?.The larger eigenvalues correspond in this case to smoother eigenfunctions, the small ones to
rougher. The rougher eigenfunctions get penalized exponentially more than the smoother ones. For
smaller values of ?, the effective dimension of the space is truncated.
Although we have no hard results, our experience so far suggests that the total number ?
of moves is O(k min(n+ , n? )), for k around 4 ? 6; hence typically some small multiple
c of n. If the average size of M is m, this suggests the total computational burden is
O(cn2 m + nm2 ), which is similar to that of a single SVM fit.
Our R function SvmPath computes all 632 steps in the mixture example (n+ = n? =
100, radial kernel, ? = 1) in 1.44(0.02) secs on a Pentium 4, 2Ghz Linux machine; the svm
function (using the optimized code libsvm, from the R library e1071) takes 9.28(0.06)
seconds to compute the solution at 10 points along the path. Hence it takes our procedure
about 50% more time to compute the entire path, than it costs libsvm to compute a typical
single solution.
4
Mixture simulation continued
The ? in Figure 1 are the entire collection of change points as described in Section 3. We
were at first surprised to discover that not all these sequences achieved zero training errors
on the 200 training data points, at their least regularized fit. In fact the minimal training errors, and the corresponding values for ? are summarized in Table 1. It is sometimes argued
that the implicit feature space is ?infinite dimensional? for this kernel, which suggests that
perfect separation is always possible. The last row of the table shows the effective rank of
the 200?200 kernel Gram matrix K (which we defined to be the number of singular values
greater than 10?12 ). In general a full rank K is required to achieve perfect separation. This
rank-deficiency of the Gram matrix has been noted by a number of other authors.
This emphasizes the fact that not all features in the feature map implied by K are of equal
?
Training Errors
Effective Rank
5
0
200
1
12
177
0.5
21
143
0.1
33
76
Table 1: The number of minimal training errors for different values of the radial kernel scale parameter ?, for the mixture simulation example. Also shown is the effective rank of the 200 ? 200 Gram
matrix K? .
stature; many of them are shrunk way down to zero. Rephrasing, the regularization in (3)
penalizes unit-norm features by the inverse of their eigenvalues, which effectively annihilates some, depending on ?. Small ? implies wide, flat kernels, and a suppression of
wiggly, ?rough? functions.
Writing (3) in matrix form,
min L[y, K?] +
?0 ,?
? T
? K?,
2
(20)
we reparametrize using the eigen-decomposition of K = UDUT . Let K? = U? ? where
? ? = DUT ?. Then (20) becomes
min? L[y, U? ? ] +
?0 ,?
? ? T ?1 ?
? D ? .
2
(21)
Now the columns of U are unit-norm basis functions (in R2 ) spanning the column space
of K; from (21) we see that those members corresponding to near-zero eigenvalues (the
elements of the diagonal matrix D) get heavily penalized and hence ignored. Figure 2
shows the elements of D for the four values of ?.
5
Discussion
Our work on the SVM path algorithm was inspired by early work on exact path algorithms
in other settings. ?Least Angle Regression? [8] show that the coefficient path for the sequence of ?lasso? coefficients is piecewise linear. The lasso uses a quadratic criterion,
with an L1 constraint. In fact, any model with an L1 constraint and a quadratic, piecewise
quadratic, piecewise linear, or mixed quadratic and linear loss function, will have piecewise
linear coefficient paths, which can be calculated exactly and efficiently for all values of ?
[9]. This includes the L1 SVM [10].
The SVM model has a quadratic constraint and a piecewise linear (?hinge?) loss function.
This leads to a piecewise linear path in the dual space, hence the Lagrange coefficients ?i
are piecewise linear.
Of course, quadratic criterion + quadratic constraints also lead to exact path solutions, as
in the classic ridge regression case, since a closed form solution is obtained via the SVD.
The general techniques employed in this paper are known as parametric programming in
convex optimization. After completing this work, it was brought to our attention that [11]
reported on the picewise-linear nature of the lagrange multipliers, although they did not
develop the path algorithm. [12, 13] employ techniques similar to ours in incremental
learning for SVMs. These authors do not construct exact paths as we do, but rather focus
on updating and downdating the solutions as more (or less) data arises. [14] allow for
updating the parameters as well, but again do not construct entire solution paths.
The SvmPath has been implemented in the R computing environment, and is available
from the R website.
Acknowledgements
The authors thank Jerome Friedman for helpful discussions, and Mee-Young Park for assisting with some of the computations. Trevor Hastie was partially supported by grant
DMS-0204162 from the National Science Foundation, and grant RO1-EB0011988-08 from
the National Institutes of Health.
References
[1] B. Boser, I. Guyon, and V. Vapnik. A training algorithm for optimal margin classifiers.
In Proceedings of COLT II, Philadelphia, PA, 1992.
[2] C. Cortes and V. Vapnik. Support vector networks. Machine Learning, 20:1?25, 1995.
[3] Bernard Sch?olkopf and Alex Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond (Adaptive Computation and Machine Learning). MIT Press, 2001.
[4] G. Wahba, Y. Lin, and H. Zhang. Gacv for support vector machines. In A.J. Smola,
P.L. Bartlett, B. Sch?olkopf, and D. Schuurmans, editors, Advances in Large Margin
Classifiers, pages 297?311, Cambridge, MA, 2000. MIT Press.
[5] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning; Data
mining, Inference and Prediction. Springer Verlag, New York, 2001.
[6] Thorsten Joachims. Practical Advances in Kernel Methods ? Support Vector Learning, chapter Making large scale SVM learning practical. MIT Press, 1999. see
http://svmlight.joachims.org.
[7] Trevor Hastie, Saharon Rosset, Robert Tibshirani, and Ji Zhu. The entire regularization path for the support vector machine. Journal of Machine Learning Research,
(5):1391?1415, 2004.
[8] B. Efron, T. Hastie, I. Johnstone, and R.. Tibshirani. Least angle regression. Technical
report, Stanford University, 2002.
[9] Saharon Rosset and Ji Zhu.
Piecewise linear regularized solution
paths.
Technical
report,
Stanford
University,
2003.
http://www-stat.stanford.edu/?saharon/papers/piecewise.ps.
[10] Ji Zhu, Saharon Rosset, Trevor Hastie, and Robert Tibshirani. L1 norm support vector
machines. Technical report, Stanford University, 2003.
[11] Massimiliano Pontil and Alessandro Verri. Properties of support vector machines.
Neural Comput., 10(4):955?974, 1998.
[12] Shai Fine and Katya Scheinberg. Incas: An incremental active set method for svm.
Technical report, IBM Research Labs, Haifa, 2002.
[13] G. Cauwenberghs and T. Poggio. Incremental and decremental support vector machine learning. In Advances in Neural Information Processing Systems (NIPS*2000),
volume 13. MIT Press, Cambridge, MA, 2001.
[14] Christopher Diehl and Gert Cauwenberghs. Svm incremental learning, adaptation and
optimization. In Proceedings of the 2003 International Joint Conference on Neural
Networks, pages 2685?2690, 2003. Special series on Incremental Learning.
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arises:1 investigator:1 evaluate:1 |
1,887 | 2,714 | Harmonising Chorales by Probabilistic Inference
Moray Allan and Christopher K. I. Williams
School of Informatics, University of Edinburgh
Edinburgh EH1 2QL
[email protected], [email protected]
Abstract
We describe how we used a data set of chorale harmonisations composed
by Johann Sebastian Bach to train Hidden Markov Models. Using a probabilistic framework allows us to create a harmonisation system which
learns from examples, and which can compose new harmonisations. We
make a quantitative comparison of our system?s harmonisation performance against simpler models, and provide example harmonisations.
1
Introduction
Chorale harmonisation is a traditional part of the theoretical education of Western classical
musicians. Given a melody, the task is to create three further lines of music which will
sound pleasant when played simultaneously with the original melody. A good chorale
harmonisation will show an understanding of the basic ?rules? of harmonisation, which
codify the aesthetic preferences of the style. Here we approach chorale harmonisation as
a machine learning task, in a probabilistic framework. We use example harmonisations
to build a model of harmonic processes. This model can then be used to compose novel
harmonisations.
Section 2 below gives an overview of the musical background to chorale harmonisation.
Section 3 explains how we can create a harmonisation system using Hidden Markov Models. Section 4 examines the system?s performance quantitatively and provides example
harmonisations generated by the system. In section 5 we compare our system to related
work, and in section 6 we suggest some possible enhancements.
2
Musical Background
Since the sixteenth century, the music of the Lutheran church had been centred on the
?chorale?. Chorales were hymns, poetic words set to music: a famous early example is
Martin Luther?s ?Ein? feste Burg ist unser Gott?. At first chorales had only relatively simple melodic lines, but soon composers began to arrange more complex music to accompany
the original tunes. In the pieces by Bach which we use here, the chorale tune is taken generally unchanged in the highest voice, and three other musical parts are created alongside
it, supporting it and each other. By the eighteenth century, a complex system of rules had
developed, dictating what combinations of notes should be played at the same time or following previous notes. The added lines of music should not fit too easily with the melody,
but should not clash with it too much either. Dissonance can improve the music, if it is
resolved into a pleasant consonance.
Figure 1: Hidden state representations (a) for harmonisation, (b) for ornamentation.
The training and test chorales used here are divided into two sets: one for chorales in
?major? keys, and one for chorales in ?minor? keys. Major and minor keys are based around
different sets of notes, and musical lines in major and minor keys behave differently.
The representation we use to model harmonisations divides up chorales into discrete timesteps according to the regular beat underlying their musical rhythm. At each time-step we
represent the notes in the various musical parts by counting how far apart they are in terms
of all the possible ?semitone? notes.
3
3.1
Harmonisation Model
HMM for Harmonisation
We construct a Hidden Markov model in which the visible states are melody notes and the
hidden states are chords. A sequence of observed events makes up a melody line, and a
sequence of hidden events makes up a possible harmonisation for a melody line. We denote
the sequence of melody notes as Y and the harmonic motion as C, with yt representing the
melody at time t, and ct the harmonic state.
Hidden Markov Models are generative models: here we model how a visible melody line is
emitted by a hidden sequence of harmonies. This makes sense in musical terms, since we
can view a chorale as having an underlying harmonic structure, and the individual notes of
the melody line as chosen to be compatible with this harmonic state at each time step. We
will create separate models for chorales in major and minor keys, since these groups have
different harmonic structures.
For our model we divide each chorale into time steps of a single beat, making the assumption that the harmonic state does not change during a beat. (Typically there are three or
four beats in a bar.) We want to create a model which we can use to predict three further
notes at each of these time steps, one for each of the three additional musical lines in the
harmonisation.
There are many possible hidden state representations from which to choose. Here we represent a choice of notes by a list of pitch intervals. By using intervals in this way we
represent the relationship between the added notes and the melody at a given time step,
without reference to the absolute pitch of the melody note. These interval sets alone would
be harmonically ambiguous, so we disambiguate them using harmonic labels, which are
included in the training data set. Adding harmonic labels means that our hidden symbols
not only identify a particular chord, but also the harmonic function that the chord is serving. Figure 1(a) shows the representation used for some example notes. Here (an A major
chord) the alto, tenor and bass notes are respectively 4, 9, and 16 semitones below the
soprano melody. The harmonic label is ?T?, labelling this as functionally a ?tonic? chord.
Our representation of both melody and harmony distinguishes between a note which is
continued from the previous beat and a repeated note.
We make a first-order Markov assumption concerning the transition probabilities between
the hidden states, which represent choices of chord on an individual beat:
P (ct |ct?1 , ct?2 , . . . , c0 ) = P (ct |ct?1 ).
We make a similar assumption concerning emission probabilities to model how the observed event, a melody note, results from the hidden state, a chord:
P (yt |ct , . . . , c0 , yt?1 , . . . , y0 ) = P (yt |ct ).
In the Hidden Markov Models used here, the ?hidden? states of chords and harmonic symbols are in fact visible in the data during training. This means that we can learn transition
and emission probabilities directly from observations in our training data set of harmonisations. We use additive smoothing (adding 0.01 to each bin) to deal with zero counts in the
training data.
Using a Hidden Markov Model framework allows us to conduct efficient inference over
our harmonisation choices. In this way our harmonisation system will ?plan? over an entire
harmonisation rather than simply making immediate choices based on the local context.
This means, for example, that we can hope to compose appropriate ?cadences? to bring our
harmonisations to pleasant closes rather than finishing abruptly.
Given a new melody line, we can use the Viterbi algorithm to find the most likely state
sequence, and thus harmonisation, given our model. We can also provide alternative harmonisations by sampling from the posterior [see 1, p. 156], as explained below.
3.2
Sampling Alternative Harmonisations
Using ?t?1 (j), the probability of seeing the observed events of a sequence up to time t ? 1
and finishing in state j, we can calculate the probability of seeing the first t ? 1 events,
finishing in state j, and then transitioning to state k at the next step:
P (y0 , y1 , . . . , yt?1 , ct?1 = j, ct = k) = ?t?1 (j)P (ct = k|ct?1 = j).
We can use this to calculate ?t (j|k), the probability that we are in state j at time t ? 1 given
the observed events up to time t ? 1, and given that we will be in state k at time t:
?t?1 (j)P (ct = k|ct?1 = j)
?t (j|k) = P (ct?1 = j|y0 , y1 , . . . , yt?1 , ct = k) = P
.
l ?t?1 (l)P (ct = k|ct?1 = l)
To sample from P (C|Y ) we first choose the final state by sampling from its probability
distribution according to the model:
?T (j)
P (cT = j|y0 , y1 , . . . , yT ) = P
.
l ?T (l)
Once we have chosen a value for the final state cT , we can use the variables ?t (j|k) to
sample backwards through the sequence:
P (ct = j|y0 , y1 , . . . , yT , ct+1 ) = ?t+1 (j|ct+1 ).
3.3
HMM for Ornamentation
The chorale harmonisations produced by the Hidden Markov Model described above harmonise the original melody according to beat-long time steps. Chorale harmonisations are
Table 1: Comparison of predictive power achieved by different models of harmonic sequences on training and test data sets (nats).
? T1
? T1
? T1
? T1
ln
PP (C|Y )
P ln P (ct |yt )
P ln P (ct |ct?1 )
ln P (ct )
Training (maj)
Test (maj)
Training (min)
Training (min)
2.56
3.00
5.41
6.43
4.90
3.22
7.08
7.61
2.66
3.52
5.50
6.57
5.02
4.33
7.21
7.84
not limited to this rhythmic form, so here we add a secondary ornamentation stage which
can add passing notes to decorate these harmonisations. Generating a harmonisation and
adding the ornamentation as a second stage greatly reduces the number of hidden states
in the initial harmonisation model: if we went straight to fully-ornamented hidden states
then the data available to us concerning each state would be extremely limited. Moreover,
since the passing notes do not change the harmonic structure of a piece but only ornament
it, adding these passing notes after first determining the harmonic structure for a chorale is
a plausible compositional process.
We conduct ornamentation by means of a second Hidden Markov Model. The notes added
in this ornamentation stage generally smooth out the movement between notes in a line of
music, so we set up the visible states in terms of how much the three harmonising musical
lines rise or fall from one time-step to the next. The hidden states describe ornamentation
of this motion in terms of the movement made by each part during the time step, relative
to its starting pitch. This relative motion is described at a time resolution four times as fine
as the harmonic movement. On the first of the four quarter-beats we always leave notes
as they were, so we have to make predictions only for the final three quarter-beats. Figure
1(b) shows an example of the representation used. In this example, the alto and tenor lines
remain at the same pitch for the second quarter-beat as they were for the first, and rise by
two semitones for the third and fourth quarter-beats, so are both represented as ?0,0,2,2?,
while the bass line does not change pitch at all, so is represented as ?0,0,0,0?.
4
Results
Our training and test data are derived from chorale harmonisations by Johann Sebastian
Bach.1 These provide a relatively large set of harmonisations by a single composer, and are
long established as a standard reference among music theorists. There are 202 chorales in
major keys of which 121 were used for training and 81 used for testing; and 180 chorales
in minor keys (split 108/72).
Using a probabilistic framework allows us to give quantitative answers to questions about
the performance of the harmonisation system. There are many quantities we could compute, but here we will look at how high a probability the model assigns to Bach?s own
harmonisations given the respective melody lines. We calculate average negative log probabilities per symbol, which describe how predictable the symbols are under the model.
These quantities provide sample estimates of cross-entropy. Whereas verbal descriptions
of harmonisation performance are unavoidably vague and hard to compare, these figures
allow our model?s performance to be directly compared with that of any future probabilistic
harmonisation system.
Table 1 shows the average negative log probability per symbol of Bach?s chord symbol
1
We used a computer-readable edition of Bach?s chorales downloaded from ftp://i11ftp.
ira.uka.de/pub/neuro/dominik/midifiles/bach.zip
Figure 2: Most likely harmonisation under our model of chorale K4, BWV 48
Figure 3: Most likely harmonisation under our model of chorale K389, BWV 438
sequences given their respective melodic symbol sequences, ? T1 ln P (C|Y ), on training
and test data sets of chorales in major and minor keys. As a comparison we give analogous negative log probabilities
for a model predicting chord states from their respective
P
ln P (ct |yt ), for a simple Markov chain between the chord states,
melody notes, ? T1
P
? T1
ln P (ct |ct?1 ), and for a model which assumes that the chord states are indepenP
dently drawn, ? T1
ln P (ct ). The Hidden Markov Model here has 5046 hidden chord
states and 58 visible melody states.
The Hidden Markov Model finds a better fit to the training data than the simpler models:
to choose a good chord for a particular beat we need to take into account both the melody
note on that beat and the surrounding chords. Even the simplest model of the data, which
assumes that each chord is drawn independently, performs worse on the test data than the
training data, showing that we are suffering from sparse data. There are many chords, chord
to melody note emissions, and especially chord to chord transitions, that are seen in the test
data but never occur in the training data. The models? performance with unseen data could
be improved by using a more sophisticated smoothing method, for example taking into
account the overall relative frequencies of harmonic symbols when assigning probabilities
to unseen chord transitions. However, this lower performance with unseen test data is not
a problem for the task we approach here, of generating new harmonisations, as long as we
can learn a large enough vocabulary of events from the training data to be able to find good
harmonisations for new chorale melodies.
Figures 2 and 3 show the most likely harmonisations under our model for two short
chorales. The system has generated reasonable harmonisations. We can see, for example,
passages of parallel and contrary motion between the different parts. There is an appropriate harmonic movement through the harmonisations, and they come to plausible cadences.
The generated harmonisations suffer somewhat from not taking into account the flow of
the individual musical lines which we add. There are large jumps, especially in the bass
line, more often than is desirable ? the bass line suffers most since has the greatest variance
with respect to the soprano melody. This excessive jumping also feeds through to reduce
the performance of the ornamentation stage, creating visible states which are unseen in the
training data. The model structure means that the most likely harmonisation leaves these
states unornamented. Nevertheless, where ornamentation has been added it fits with its
context and enhances the harmonisations.
The authors will publish further example harmonisations, including MIDI files, online at
http://www.tardis.ed.ac.uk/?moray/harmony/.
5
Relationship to previous work
Even while Bach was still composing chorales, music theorists were catching up with musical practice by writing treatises to explain and to teach harmonisation. Two famous examples, Rameau?s Treatise on Harmony [2] and the Gradus ad Parnassum by Fux [3],
show how musical style was systematised and formalised into sets of rules. The traditional
formulation of harmonisation technique in terms of rules suggests that we might create an
automatic harmonisation system by finding as many rules as we can and encoding them as
a consistent set of constraints. Pachet and Roy [4] provide a good overview of constraintbased harmonisation systems. For example, one early system [5] takes rules from Fux and
assigns penalties according to the seriousness of each rule being broken. This system then
conducts a modified best-first search to produce harmonisations. Using standard constraintsatisfaction techniques for harmonisation is problematic, since the space and time needs of
the solver tend to rise extremely quickly with the length of the piece.
Several systems have applied genetic programming techniques to harmonisation, for example McIntyre [6]. These are similar to the constraint-based systems described above,
but instead of using hard constraints they encode their rules as a fitness function, and try
to optimise that function by evolutionary techniques. Phon-Amnuaisuk and Wiggins [7]
are reserved in their assessment of genetic programming for harmonisation. They make a
direct comparison with an ordinary constraint-based system, and conclude that the performance of each system is related to the amount of knowledge encoded in it rather than the
particular technique it uses. In their comparison the ordinary constraint-based system actually performs much better, and they argue that this is because it possesses implicit control
knowledge which the system based on the genetic algorithm lacks.
Even if they can be made more efficient, these rule-based systems do not perform the full
task of our harmonisation system. They take a large set of rules written by a human and
attempt to find a valid solution, whereas our system learns its rules from examples.
Hild et al. [8] use neural networks to harmonise chorales. Like the Hidden Markov Models
in our system, these neural networks are trained using example harmonisations. However,
while two of their three subtasks use only neural networks trained on example harmonisations, their second subtask, where chords are chosen to instantiate more general harmonies,
includes constraint satisfaction. Rules written by a human penalise undesirable combinations of notes, so that they will be filtered out when the best chord is chosen from all
those compatible with the harmony already decided. In contrast, our model learns all its
harmonic ?rules? from its training data.
Ponsford et al. [9] use n-gram Markov models to generate harmonic structures. Unlike in
chorale harmonisation, there is no predetermined tune with which the harmonies need to fit.
The data set they use is a selection of 84 saraband dances, by 15 different seventeen-century
French composers. An automatically annotated corpus is used to train Markov models
using contexts of different lengths, and the weighted sum of the probabilities assigned by
these models used to predict harmonic movement. Ponsford et al. create new pieces first by
random generation from their models, and secondly by selecting those randomly-generated
pieces which match a given template. Using templates gives better results, but the great
majority of randomly-generated pieces will not match the template and so will have to be
discarded. Using a Hidden Markov Model rather than simple n-grams allows this kind of
template to be included in the model as the visible state of the system: the chorale tunes
in our system can be thought of as complex templates for harmonisations. Ponsford et
al. note that even with their longest context length, the cadences are poor. In our system
the ?planning? ability of Hidden Markov Models, using the combination of chords and
harmonic labels encoded in the hidden states, produces cadences which bring the chorale
tunes to harmonic closure.
This paper stems from work described in the first author?s MSc thesis [10] carried out in
2002. We have recently become aware that similar work has been carried out independently
in Japan by a team led by Prof S. Sagayama [11, 12]. To our knowledge this work has
been published only in Japanese2 . The basic frameworks are similar, but there are several
differences. First, their system only describes the harmonisation in terms of the harmonic
label (e.g. T for tonic) and does not fully specify the voicing of the three harmony lines or
ornamentation. Secondly, they do not give a quantitative evaluation of the harmonisations
produced as in our Table 1. Thirdly, in [12] a Markov model on blocks of chord sequences
rather than on individual chords is explored.
6
Discussion
Using the framework of probabilistic influence allows us to perform efficient inference to
generate new chorale harmonisations, avoiding the computational scaling problems suffered by constraint-based harmonisation systems. We described above neural network
and genetic algorithm techniques which were less compute-intensive than straightforward
constraint satisfaction, but the harmonisation systems using these techniques retain a preprogrammed knowledge base, whereas our model is able to learn its harmonisation constraints from training data.
Different forms of graphical model would allow us to take into account more of the dependencies in harmonisation. For example, we could use a higher-order Markov structure,
although this by itself would be likely to greatly increase the problems already seen here
with sparse data. An alternative might be to use an Autoregressive Hidden Markov Model
[13], which models the transitions between visible states as well as the hidden state transitions modelled by an ordinary Hidden Markov Model.
Not all of Bach?s chorale harmonisations are in the same style. Some of his harmonisations
are intentionally complex, and others intentionally simple. We could improve our harmonisations by modelling this stylistic variation, either manually annotating training chorales
according to their style or by training a mixture of HMMs.
As we only wish to model the hidden harmonic state given the melody, rather than construct
a full generative model of the data, Conditional Random Fields (CRFs) [14] provide a
related but alternative framework. However, note that training such models (e.g. using
iterative scaling methods) is more difficult than the simple counting methods that can be
applied to the HMM case. On the other hand the use of the CRF framework would have
2
We thank Yoshinori Shiga for explaining this work to us.
some advantages, in that additional features could be incorporated. For example, we might
be able to make better predictions by taking into account the current time step?s position
within its musical bar. Music theory recognises a hierarchy of stressed beats within a bar,
and harmonic movement should correlated with these stresses. The ornamentation process
especially might benefit from a feature-based approach.
Our system described above only considers chords as sets of intervals, and thus does not
have a notion of the key of a piece (other than major or minor). However, voices have a
preferred range and thus the notes that should be used do depend on the key, so the key
signature could also be used as a feature in a CRF. Taking into account the natural range of
each voice would prevent the bass line from descending too low and keep the three parts
closer together. In general more interesting harmonies result when musical lines are closer
together and their movements are more constrained. Another dimension that could be
explored with CRFs would be to take into account the words of the chorales, since Bach?s
own harmonisations are affected by the properties of the texts as well as of the melodies.
Acknowledgments
MA gratefully acknowledges support through a research studentship from Microsoft Research Ltd.
References
[1] R. Durbin, S. R. Eddy, A. Krogh, and G. Mitchison. Biological sequence analysis. Cambridge
University Press, 1998.
[2] J.-P. Rameau. Trait?e de l?Harmonie reduite a` ses principes naturels. Paris, 1722.
[3] J. J. Fux. Gradus ad Parnassum. Vienna, 1725.
[4] F. Pachet and P. Roy. Musical harmonization with constraints: A survey. Constraints, 6(1):
7?19, 2001.
[5] B. Schottstaedt.
CCRMA, 1989.
Automatic species counterpoint.
Technical report, Stanford University
[6] R. A. McIntyre. Bach in a box: The evolution of four-part baroque harmony using the genetic
algorithm. In Proceedings of the IEEE Conference on Evolutionary Computation, 1994.
[7] S. Phon-Amnuaisuk and G. A. Wiggins. The four-part harmonisation problem: a comparison
between genetic algorithms and a rule-based system. In Proceedings of the AISB?99 Symposium
on Musical Creativity, 1999.
[8] H. Hild, J. Feulner, and W. Menzel. HARMONET: A neural net for harmonizing chorales in
the style of J.S. Bach. In R.P. Lippman, J.E. Moody, and D.S. Touretzky, editors, Advances in
Neural Information Processing 4, pages 267?274. Morgan Kaufmann, 1992.
[9] D. Ponsford, G. Wiggins, and C. Mellish. Statistical learning of harmonic movement. Journal
of New Music Research, 1999.
[10] M. M. Allan. Harmonising Chorales in the Style of Johann Sebastian Bach. Master?s thesis,
School of Informatics, University of Edinburgh, 2002.
[11] T. Kawakami. Hidden Markov Model for Automatic Harmonization of Given Melodies. Master?s thesis, School of Information Science, JAIST, 2000. In Japanese.
[12] K. Sugawara, T. Nishimoto, and S. Sagayama. Automatic harmonization for melodies based on
HMMs including note-chain probability. Technical Report 2003-MUS-53, Acoustic Society of
Japan, December 2003. In Japanese.
[13] P. C. Woodland. Hidden Markov Models using vector linear prediction and discriminative
output distributions. In Proc ICASSP, volume I, pages 509?512, 1992.
[14] J. D. Lafferty, A. McCallum, and F. C. N. Pereira. Conditional Random Fields: probabilistic
models for segmenting and labeling sequence data. In Proc ICML, pages 282?289, 2001.
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1,888 | 2,715 | Theories Of Access Consciousness
Michael D. Colagrosso
Department of Computer Science
Colorado School of Mines
Golden, CO 80401 USA
[email protected]
Michael C. Mozer
Institute of Cognitive Science
University of Colorado
Boulder, CO 80309 USA
[email protected]
Abstract
Theories of access consciousness address how it is that some mental states but
not others are available for evaluation, choice behavior, and verbal report. Farah,
O?Reilly, and Vecera (1994) argue that quality of representation is critical; Dehaene, Sergent, and Changeux (2003) argue that the ability to communicate representations is critical. We present a probabilistic information transmission or
PIT model that suggests both of these conditions are essential for access consciousness. Having successfully modeled data from the repetition priming literature in the past, we use the PIT model to account for data from two experiments
on subliminal priming, showing that the model produces priming even in the absence of accessibility and reportability of internal states. The model provides a
mechanistic basis for understanding the dissociation of priming and awareness.
Philosophy has made many attempts to identify distinct aspects of consciousness. Perhaps
the most famous effort is Block?s (1995) delineation of phenomenal and access consciousness. Phenomenal consciousness has to do with ?what it is like? to experience chocolate
or a pin prick. Access consciousness refers to internal states whose content is ?(1) inferentially promiscuous, i.e., poised to be used as a premise in reasoning, (2) poised for control
of action, and (3) poised for rational control of speech.? (p. 230) The scientific study of consciousness has exploded in the past six years, and an important catalyst for this explosion
has been the decision to focus on the problem of access consciousness: how is it that some
mental states but not others become available for evaluation, choice behavior, verbal report,
and storage in working memory. Another reason for the recent explosion of consciousness
research is the availability of functional imaging techniques to explore differences in brain
activation between conscious and unconscious states, as well as the development of clever
psychological experiments that show that a stimulus that is not consciously perceived can
nonetheless influence cognition, which we describe shortly.
1
Subliminal Priming
The phenomena we address utilize an experimental paradigm known as repetition priming.
Priming refers to an improvement in efficiency in processing a stimulus item as a result
of previous exposure to the item. Efficiency is defined in terms of shorter response times,
lower error rates, or both. A typical long-term perceptual priming experiment consists of
a study phase during which participants are asked to read aloud a list of words, and a test
phase during which participants must name or categorize a series of words, presented one
at a time. Reaction time is lower and/or accuracy is higher for test words that were also on
the study list. Repetition priming occurs without strategic effort on the part of participants,
and therefore appears to be a low level mechanism of learning, which likely serves as the
mechanism underlying the refinement of cognitive skills with practice.
In traditional studies, priming is supraliminal?the prime is consciously perceived. In the
studies we model here, primes are subliminal. Subliminal priming addresses fundamental
issues concerning conscious access: How is it that a word or image that cannot be identified,
detected, or even discriminated in forced choice can nonetheless influence the processing
of a subsequent stimulus word? Answering this question in a computational framework
would be a significant advance toward understanding the nature of access consciousness.
2
Models of Conscious and Unconscious Processing
In contrast to the wealth of experimental data, and the large number of speculative and
philosophical papers on consciousness, concrete computational models are rare. The domain of consciousness is particularly ripe for theoretical perspectives, because it is a significant contribution to simply provide an existence proof of a mechanism that can explain
specific experimental data. Ordinarily, a theorist faces skepticism when presenting a model;
it often seems that hundreds of alternative, equally plausible accounts must exist. However,
when addressing data deemed central to issues of consciousness, simply providing a concrete handle on the phenomena serves to demystify consciousness and bring it into the
realm of scientific understanding.
We are familiar with only three computational models that address specific experimental
data in the domain of consciousness. We summarize these models, and then present a
novel model and describe its relationship to the previous efforts. Farah, O?Reilly, and Vecera (1994) were the first to model specific phenomena pertaining to consciousness in a
computational framework. The phenomena involve prosopagnosia, a deficit of overt face
recognition following brain damage. Nonetheless, prosopagnosia patients exhibit residual
covert recognition by a variety of tests. For example, when patients are asked to categorize
names as famous or nonfamous, their response times are faster to a famous name when the
name is primed by a picture of a semantically related face (e.g., the name ?Bill Clinton?
when preceded by a photograph of Hillary), despite the fact that they could not identify the
related face. Farah et al. model face recognition in a neural network, and show that when
the network is damaged, it loses the ability to perform tasks requiring high fidelity representations (e.g., identification) but not tasks requiring only coarse information (e.g., semantic
priming). They argue that conscious perception is associated with a certain minimal quality
of representation.
Dehaene and Naccache (2001) outline a framework based on Baars? (1989) notion of conscious states as residing in a global workspace. They describe the workspace as a ?distributed neural system...with long-distance connectivity that can potentially interconnect
multiple specialized brain areas in a coordinated, though variable manner.? (p. 13) Dehaene, Sergent, and Changeaux (2003) implement this framework in a complicated architecture of integrate-and-fire neurons and show that the model can qualitatively account for
the attentional blink phenomenon. The attentional blink is observed in experiments where
participants are shown a rapid series of stimuli, which includes two targets (T1 and T2).
If T2 appears shortly after T1, the ability to report T2 drops, as if attention is distracted.
Dehane et al. explain this phenomenon as follows. When T1 is presented, its activation
propagates to frontal cortical areas (the global workspace). Feedback connections lead to
a resonance between frontal and posterior areas, which strengthen T1 but block T2 from
entering the workspace. If the T1-T2 lag is sufficiently great, habituation of T1 sufficiently
weakens the representation such that T2 can enter the workspace and suppress T1. In this
account, conscious access is achieved via resonance between posterior and frontal areas.
Although the Farah et al. and Dehaene et al. models might not seem to have much in
common, they both make claims concerning what is required to achieve functional connectivity between perceptual and response systems. Farah et al. focus on aspects of the
representation; Dehaene et al. focus on a pathway through which representations can be
communicated. These two aspects are not incompatible, and in fact, a third model incorporates both. Mathis and Mozer (1996) describe an architecture with processing modules for
perceptual and response processes, implemented as attractor neural nets. They argue that in
order for a representation in some perceptual module to be assured of influencing a response
module, (a) it must have certain characteristics?temporal persistence and well-formedness?
which is quite similar to Farah et al.?s notion of quality, and (b) the two modules must be
interconnected?which is the purpose of Dehaene et al.?s global workspace. The model has
two limitations that restrict its value as a contemporary account of conscious access. First,
it addressed classical subliminal priming data, but more reliable data has recently been
reported. Second, like the other two models, Mathis and Mozer used a complex neural
network architecture with arbitrary assumptions built in, and the sensitivity of the model?s
behavior to these assumptions is far from clear. In this paper, we present a model that embodies the same assumptions as Mathis and Mozer, but overcomes its two limitations, and
explains subliminal-priming data that has yet to be interpreted via a computational model.
3
The Probabilistic Information Transmission (PIT) Framework
Our model is based on the probabilistic information transmission or PIT framework of
Mozer, Colagrosso, and Huber (2002, 2003). The framework characterizes the transmission
of information from perceptual to response systems, and how the time course of information transmission changes with experience (i.e., priming). Mozer et al. used this framework
to account for a variety of facilitation effects from supraliminal repetition priming.
The framework describes cognition in terms of a collection of information-processing pathways, and supposes that any act of cognition involves coordination among multiple pathways. For example, to model a letter-naming task where a letter printed in upper or lower
case is presented visually and the letter must be named, the framework would assume a
perceptual pathway that maps the visual input to an identity representation, and a response
pathway that maps a identity representation to a naming response. The framework is formalized as a probabilistic model: the pathway input and output are random variables and
microinference in a pathway is carried out by Bayesian belief revision.
The framework captures the time course of information processing for a single experimental trial. To elaborate, consider a pathway whose input at time t is a discrete random
variable, denoted X(t), which can assume values x1 , x2 , x3 , . . . , xnx corresponding to alternative input states. Similarly, the output of the pathway at time t is a discrete random
variable, denoted Y (t), which can assume values y1 , y2 , y3 , . . . , yny . For example, in the
letter-naming task, the input to the perceptual pathway would be one of nx = 52 visual
patterns corresponding to the upper- and lower-case letters of the alphabet, and the output
is one of ny = 26 letter identities. To present a particular input alternative, say xi , to the
model for T time steps, we specify X(t) = xi for t = 1 . . . T , and allow the model to
compute P(Y (t) | X(1) . . . X(t)).
A pathway is modeled as a dynamic Bayes network; the minimal version of the model
used in the present simulations is simply a hidden Markov model, where the X(t) are
observations and the Y (t) are inferred state (see Figure 1a). In typical usage, an HMM
is presented with a sequence of distinct inputs, whereas we maintain the same input for
many successive time steps; and an HMM transitions through a sequence of distinct hidden
states, whereas we attempt to converge with increasing confidence on a single state.
Figure 1b illustrates the time course of inference in a single pathway with 52 input and
26 output alternatives and two-to-one associations. The solid line in the Figure shows, as
a function of time t, P(Y (t) = yi | X(1) = x2i . . . X(t) = x2i ), i.e., the probability
that input i (say, the visual pattern of an upper case O) will produce its target output (the
letter identity). Evidence for the target output accumulates gradually over time, yielding a
speed-accuracy curve that relates the number of iterations to the accuracy of identification.
Y0
Y1
X1
Y2
X2
P(Output)
1
Y3
X3
(a)
(b)
0.8
0.6
O
0.4
0.2
0
Q
Time
Figure 1: (a) basic pathway architecture?a hidden Markov model; (b) time course of inference in a
pathway when the letter O is presented, causing activation of both O and the visually similar Q.
The exact shape of the speed-accuracy curve?the pathway dynamics?are determined by
three probability distributions, which embody the knowledge and past experience of the
model. First, P(Y (0)) is the prior distribution over outputs in the absence of any information about the input. Second, P(Y (t) | Y (t ? 1)) characterizes how the pathway output
evolves over time. We assume the transition probability matrix serves as a memory with
diffusion, i.e., P(Y (t) = yi |Y (t ? 1) = yj ) = (1 ? ?)?ij + ?P(Y (0) = yi ), where ? is the
diffusion constant and ?ij is the Kronecker delta. Third, P(X(t) | Y (t)) characterizes the
strength of association between inputs and outputs. The greater the association strength,
the more rapidly that information about X will be communicated
to Y . We parameterize
P
this distribution as P(X(t) = xi |Y (t) = yj ) ? 1 + k ?ik ?kj , where ?ij indicates the
frequency of experience with the association between states xi and yj , and ?ik specifies the
similarity between states xi and xk . (Although the representation of states is localist, the ?
terms allow us to design in the similarity structure inherent in a distributed representation.)
These association strengths are highly constrained by the task structure and the similarity
structure and familiarity of the inputs.
Fundamental to the framework is the assumption that with each experience, a pathway becomes more efficient at processing an input. Efficiency is reflected by a shift in the speedaccuracy curve to the left. In Mozer, Colagrosso, and Huber (2002, 2003), we propose two
distinct mechanisms to model phenomena of supraliminal priming. First, the association
frequencies, ?ij , are increased following a trial in which xi leads to activation of yj , resulting in more efficient transmission of information, corresponding to an increased slope
of the solid line in Figure 1b. The increase is Hebbian, based on the maximum activation
achieved by xi and yj : ??ij = ? maxt P(X(t) = xi )P(Y (t) = yj ), where ? is a step size.
Second, the priors, which serve as a model of the environment, are increased to indicate a
greater likelihood of the same output occurring again in the future. In modeling data from
supraliminal priming, we found that the increases to association frequencies are long lasting, but the increases to the priors decay over the course of a few minutes or a few trials.
As a result, the prior updating does not play into the simulation we report here; we refer
the reader to Mozer, Colagrosso, and Huber (2003) for details.
4
Access Consciousness and PIT
We have described the operation of a single pathway, but to model any cognitive task,
we require a series of pathways in cascade. For a simple choice task, we use a percpetual pathway cascaded to a response pathway. The interconnection between the pathways
is achieved by copying the output of the perceptual pathway, Y p (t), to the input of the
response pathway, X r (t), at each time t.
This multiple-pathway architecture allows us to characterize the notion of access consciousness. Considering the output of the perceptual pathway, access is achieved when:
(1) the output representation is sufficient to trigger the correct behavior in the response
pathway, and (2) the perceptual and response pathways are functionally interconnected. In
more general terms, access for a perceptual pathway output requires that these two condi-
tions be met not just for a specific response pathway, but for arbitrary response pathways
(e.g., pathways for naming, choice, evaluation, working memory, etc.). In Mozer and Colagrosso (in preparation) we characterize the sufficiency requirements of condition 1; they
involve a representation of low entropy that stays active for long enough that the representation can propagate to the next pathway.
As we will show, a briefly presented stimulus fails to achieve a representation that supports
choice and naming responses. Nonetheless, the stimulus evokes activity in the perceptual
pathway. Because perceptual priming depends on the magnitude of the activation in the
perceptual pathway, not on the activation being communicated to response pathways, the
framework is consistent with the notion of priming occurring in the absence of awareness.
4.1
Simulation of Bar and Biederman (1998)
Bar and Biederman (1998) presented a sequence of masked line drawings of objects and
asked participants to name the objects, even if they had to guess. If the guess was incorrect,
participants were required to choose the object name from a set of four alternatives. Unbeknownst to the participant, some of the drawings in the series were repeated, and Bar and
Biederman were interested in whether participants would benefit from the first presentation even if it could not be identified. The repeated objects could be the same or a different
exemplar of the object, and it could appear in either the same or a different display position.
Participants were able to name 13.5% of drawings on presentation 1, but accuracy jumped
to 34.5% on presentation 2. Accuracy did improve, though not as much, if the same shape
was presented in a different position, but not if a different drawing of the same object
was presented, suggesting a locus of priming early in the visual stream. The improvement
in accuracy is not due to practice in general, because accuracy rose only 4.0% for novel
control objects over the course of the experiment. The priming is firmly subliminal, because
participants were not only unable to name objects on the first presentation, but their fouralternative forced choice (4AFC) performance was not much above chance (28.5%).
To model these phenomena, we created a response pathway with fifty states representing
names of objects that are used in the experiment, e.g., chair and lamp. We also created a
perceptual pathway with states representing visual patterns that correspond to the names
in the response pathway. Following the experimental design, every object identity was
instantiated in two distinct shapes, and every shape could be in one of nine different visualfield positions, leading to 900 distinct states in the perceptual pathway to model the possible
visual stimuli. The following parameters were fit to the data. If two perceptual states, xi
and xk are the same shape in different positions, they are assigned a similarity coefficient
?ik = 0.95; all other similarity coefficients are zero. The association frequency, ?, for
valid associations in the perceptual pathway was 22, and the response pathway 18. Other
parameters were ? p = .05, ? r = .01, and ? = 1.0.
The PIT model achieves a good fit to the human experimental data (Figure 2). Specifically, priming is greatest for the same shape in the same position, some priming occurs for
the same shape in a different position, and no substantial priming occurs for the different
shape. Figure 3a shows the time course of activation of a stimulus representation in the
perceptual pathway when the stimulus is presented for 50 iterations, on both the first and
third presentations. The third presentation was chosen instead of the second to make the
effect of priming clearer.
Even though a shape cannot be named on the first presentation, partial information about
the shape may nonetheless be available for report. The 4AFC test of Bar and Biederman provides a more sensitive measure of residual stimulus information. In past work,
we modeled forced-choice tasks using a response pathway with only the alternatives under
consideration. However, in this experiment, forced-choice performance must be estimated
conditional on incorrect naming. In PIT framework, we achieve this using naming and
40
40
First Block
Second Block
30
25
20
15
10
5
First Block
35
Percent Correct Naming
Percent Correct Naming
35
Second Block
30
25
20
15
10
5
0
0
Control
Objects
Prime
SHAPE: Same
Objects POSITION: Same
Same
Different
Different Different Second
Same
Different Control
Control
Objects
Prime
SHAPE: Same
Objects POSITION: Same
Same
Different
Different Different Second
Same
Different Control
Figure 2: (left panel) Data from Bar and Biederman (1998) (right panel) Simulation of PIT. White
bar: accuracy on first presentation of a prime object. Black bars: the accuracy when the object is
repeated, either with the same or different shape, and in the same or different position. Grey bars:
accuracy for control objects at the beginning and the end of the experiment.
forced-choice output pathways having output distributions N (t) and F (t), which are linked
via the perceptual state, Y p (t). F (t) must be reestimated with the evidence that N (t) is
not the target state. This inference problem is intractable. We therefore used a shortcut in
which a single response pathway is used, augmented with a simple three-node belief net
(Figure 3b) to capture the dependence between naming and forced choice. The belief net
has a response pathway node Y r (t) connected to F (t) and N (t), with conditional distribution P (N (t) = ni |Y r (t) = yj ) = ??ij + (1 ? ?)/|Y r |, and an analogous distribution for
P (F (t) = fi |Y r (t) = yj ). The free parameter ? determines how veridically naming and
forced-choice actions reflect response-pathway output. Over a range of ?, ? < 1, the model
obtains forced-choice performance near chance on the first presentation when the naming
response is incorrect. For example, with ? = 0.72, the model produces a forced-choice
accuracy on presentation 1 of 26.1%. (Interestingly, the model also produces below chance
performance on presentation 2 if the object is not named correctly?23.5%?which is also
found in the human data?20.0%.) Thus, by the stringent criterion of 4AFC, the model
shows no access consciousness, and therefore illustrates a dissociation between priming
and access consciousness. In our simulation, we followed the procedure of Bar and Biederman by including distractor alternatives with visual and semantic similarity to the target.
These distractors are critical: with unrelated distractors, the model?s 4AFC performance is
significantly above chance, illustrating that a perceptual representation can be adequate to
support some responses but not others, as Farah et al. (1994) also argued.
4.2
Simulation of Abrams and Greenwald (2000)
During an initial phase of the experiment, participants categorized 24 clearly visible target
words as pleasant (e.g., HUMOR) or unpleasant (e.g., SMUT). They became quite familiar
with the task by categorizing each word a total of eight times. In a second phase, participants were asked to classify the same targets and were given a response deadline to induce
errors. The targets were preceded by masked primes that could not be identified. Of interest
is the effective valence (or EV) of the target for different prime types, defined as the error
rate difference between unpleasant and pleasant targets. A positive (negative) EV indicates
that responses are biased toward a pleasant (unpleasant) interpretation by the prime. As one
would expect, pleasant primes resulted in a positive EV, unpleasant primes in a negative EV.
Of critical interest is the finding that a nonword prime formed by recombining two pleasant targets (e.g., HULIP from HUMOR and TULIP) or unpleasant targets (e.g., BIUT from
BILE and SMUT ) also served to bias the targets. More surprising, a positive EV resulted
from unpleasant prime words formed by recombining two pleasant targets (TUMOR from
TULIP and HUMOR ), indicating that subliminal priming arises from word fragments, not
words as unitary entities, and providing further evidence for an early locus of subliminal
priming. Note that the results depend critically on the first phase of the experiment, which
gave participants extensive practice on a relatively small set of words that were then used
as and recombined to form primes. Words not studied in the first phase (orphans) provided
Probability
0.6
0.5
0.4
0.3
0.2
0.1
0
object, first presentation
object, third presentation
different object
N(t)
F(t)
Yr(t)
1
50
1000
(a)
(b)
Time (msec)
Figure 3: (a) Activation of the perceptual representation in PIT as a function of processing iterations
Effective Valence
on the ?rst (thin solid line) and third (thick solid line) presentations of target. (b) Bayes net for
performing 4AFC conditional on incorrect naming response.
0.4
Experiment
Model
0.3
0.2
0.1
0
targets
hulip-type
tumor-type
orphans
Figure 4: Effective valence of primes in the Abrams and Greenwald (2000) experiment for human
subjects (black bars) and PIT model (grey bars). HULIP-type primes are almost as strong as target
repetitions, and TUMOR-type primes have a positive valence, contrary to the meaning of the word.
no signi?cant EV effect when used as primes.
In this simulation, we used a three pathway model: a perceptual pathway that maps visual patterns to orthography with 200 input states corresponding both to words, nonwords,
and nonword recombinations of words; a semantic pathway that maps to 100 distinct lexical/semantic states; and a judgement pathway that maps to two responses, pleasant and
unpleasant. In the perceptual pathway, similarity structure was based on letter overlap,
so that HULIP was similar to both TULIP and HUMOR, with ? = 0.837. No similarity
was assumed in the semantic state representation; consistent with the previous simulation,
? p = .05, ? s = .01, ? j = .01, and ? = .01. At the outset of the simulation, ? frequencies
for correct associations were 15, 19, and 25 in the perceptual, semantic, and judgement
pathways. The initial phase of the experiment was simulated by repeated supraliminal
presentation of words, which increased the association frequencies in all three pathways
through the ??ij learning rule.
Long-term supraliminal priming is essential in establishing the association strengths, as
we?ll explain. Short-term subliminal priming also plays a key role in the experiment. During the second phase of the experiment, residual activity from the prime?primarily in the
judgement pathway?biases the response to the target. Residual activation of the prime
is present even if the representation of the prime does not reach suf?cient strength that it
could be named or otherwise reported.
The outcome of the simulation is consistent with the human data (Figure 4). When a
HULIP -type prime is presented, HUMOR and TULIP become active in the semantic pathway because of their visual similarity to HULIP. Partial activation of these two practiced
words pushes the judgement pathway toward a pleasant response, resulting in a positive
EV. When a TUMOR-type prime is presented, three different words become active in the
semantic pathway: HUMOR, TULIP, and TUMOR itself. Although TUMOR is more active, it
was not one of the words studied during the initial phase of the experiment, and as a result,
it has a relatively weak association to the unpleasant judgement, in contrast to the other
two words which have strong associations to the pleasant judgement. Orphan primes have
little effect because they were not studied during the initial phase of the experiment, and
consequently their association to pleasant and unpleasant judgements is also weak. In summary, activation of the prime along a critical, well-practiced pathway may not be suf?cient
to support an overt naming response, yet it may be suf?cient to bias the processing of the
immediately following target.
5
Discussion
An important contribution of this work has been to demonstrate that specific experimental
results relating to access consciousness and subliminal priming can be interpreted in a concrete computational framework. By necessity, the PIT framework, which we previously
used to model supraliminal priming data, predicts the existence of subliminal priming,
because the mechanisms giving rise to priming depend on degree of activation of a representation, whereas the processes giving rise to access consciousness also depend on the
temporal persistence of a representation.
Another contribution of this work has been to argue that two previous computational models each tell only part of the story. Farah et al. argue that quality of representation is
critical; Dehaene et al. argue that pathways to communicate representations is critical. The
PIT framework argues that both of these features are necessary for access consciousness.
Although the PIT framework is not completely developed, it nonetheless makes a clear
prediction: that subliminal priming is can never be stronger than supraliminal priming,
because the maximal activation of subliminal primes is never greater than that of supraliminal primes. One might argue that many theoretical frameworks might predict the same,
but no other computational model is sufficiently well developed?in terms of addressing
both priming and access consciousness?to make this prediction.
In its current stage of development, a weakness of the PIT framework is that it is silent
as to how perceptual and response pathways become flexibly interconnected based on task
demands. However, the PIT framework is not alone in failing to address this critical issue:
The Dehaene et al. model suggests that once a representation enters the global workspace,
all response modules can access it, but the model does not specify how the appropriate perceptual module wins the competition to enter the global workspace, or how the appropriate
response module is activated. Clearly, flexible cognitive control structures that perform
these functions are intricately related to mechanisms of consciousness.
Acknowledgments
This research was supported by NIH/IFOPAL R01 MH61549?01A1.
References
Abrams, R. L., & Greenwald, A. G. (2000). Parts outweigh the whole (word) in unconscious analysis
of meaning. Psychological Science, 11(2), 118?124.
Baars, B. (1989). A cognitive theory of consciousness. Cambridge: Cambridge University Press.
Bar, M., & Biederman, I. (1998). Subliminal visual priming. Psychological Science, 9(6), 464?468.
Block, N. (1995). On a confusion about a function of consciousness. Brain and Behavioral Sciences,
18(2), 227?247.
Dehaene, S., & Naccache, L. (2001). Towards a cognitive neuroscience of consciousness: basic
evidence and a workspace framework. Cognition, 79, 1?37.
Dehaene, S., Sergent, C., & Changeux, J.-P. (2003). A neuronal network model linking subjective reports and objective physiological data during conscious perception. Proceedings of the
National Academy of Sciences, 100, 8520?8525.
Farah, M. J., O?Reilly, R. C., & Vecera, S. P. (1994). Dissociated overt and covert recognition as an
emergent property of a lesioned neural network. Psychological Review, 100, 571?588.
Mathis, D. W., & Mozer, M. C. (1996). Conscious and unconscious perception: a computational
theory. In G. Cottrell (Ed.), Proceedings of the Eighteenth Annual Conference of the Cognitive
Science Society (pp. 324?328). Hillsdale, NJ: Erlbaum & Associates.
Mozer, M. C., Colagrosso, M. D., & Huber, D. E. (2002). A rational analysis of cognitive control
in a speeded discrimination task. In T. G. Dietterich, S. Becker, & Z. Ghahramani (Eds.),
Advances in Neural Information Processing Systems 14. Cambridge, MA: MIT Press.
Mozer, M. C., Colagrosso, M. D., & Huber, D. E. (2003). Mechanisms of long-term repetition
priming and skill refinement: A probabilistic pathway model. In Proceedings of the TwentyFifth Annual Conference of the Cognitive Science Society. Hillsdale, NJ: Erlbaum Associates.
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1,889 | 2,716 | Bayesian inference in spiking neurons
Sophie Deneve?
Gatsby Computational Neuroscience Unit
University College London
London, UK WC1N 3AR
[email protected]
Abstract
We propose a new interpretation of spiking neurons as Bayesian integrators accumulating evidence over time about events in the external world
or the body, and communicating to other neurons their certainties about
these events. In this model, spikes signal the occurrence of new information, i.e. what cannot be predicted from the past activity. As a result,
firing statistics are close to Poisson, albeit providing a deterministic representation of probabilities. We proceed to develop a theory of Bayesian
inference in spiking neural networks, recurrent interactions implementing a variant of belief propagation.
Many perceptual and motor tasks performed by the central nervous system are probabilistic, and can be described in a Bayesian framework [4, 3]. A few important but hidden
properties, such as direction of motion, or appropriate motor commands, are inferred from
many noisy, local and ambiguous sensory cues. These evidences are combined with priors
about the sensory world and body. Importantly, because most of these inferences should
lead to quick and irreversible decisions in a perpetually changing world, noisy cues have to
be integrated on-line, but in a way that takes into account unpredictable events, such as a
sudden change in motion direction or the appearance of a new stimulus.
This raises the question of how this temporal integration can be performed at the neural
level. It has been proposed that single neurons in sensory cortices represent and compute
the log probability that a sensory variable takes on a certain value (eg Is visual motion in
the neuron?s preferred direction?) [9, 7]. Alternatively, to avoid normalization issues and
provide an appropriate signal for decision making, neurons could represent the log probability ratio of a particular hypothesis (eg is motion more likely to be towards the right than
towards the left) [7, 6]. Log probabilities are convenient here, since under some assumptions, independent noisy cues simply combine linearly. Moreover, there are physiological
evidence for the neural representation of log probabilities and log probability ratios [9, 6, 7].
However, these models assume that neurons represent probabilities in their firing rates. We
argue that it is important to study how probabilistic information are encoded in spikes.
Indeed, it seems spurious to marry the idea of an exquisite on-line integration of noisy cues
with an underlying rate code that requires averaging on large populations of noisy neurons
and long periods of time. In particular, most natural tasks require this integration to take
place on the time scale of inter-spike intervals. Spikes are more efficiently signaling events
?
Institute of Cognitive Science, 69645 Bron, France
than analog quantities. In addition, a neural theory of inference with spikes will bring us
closer to the physiological level and generate more easily testable predictions.
Thus, we propose a new theory of neural processing in which spike trains provide a deterministic, online representation of a log-probability ratio. Spikes signals events, eg that
the log-probability ratio has exceeded what could be predicted from previous spikes. This
form of coding was loosely inspired by the idea of ?energy landscape? coding proposed
by Hinton and Brown [2]. However, contrary to [2] and other theories using rate-based
representation of probabilities, this model is self-consistent and does not require different
models for encoding and decoding: As output spikes provide new, unpredictable, temporally independent evidence, they can be used directly as an input to other Bayesian neurons.
Finally, we show that these neurons can be used as building blocks in a theory of approximate Bayesian inference in recurrent spiking networks. Connections between neurons
implement an underlying Bayesian network, consisting of coupled hidden Markov models.
Propagation of spikes is a form of belief propagation in this underlying graphical model.
Our theory provides computational explanations of some general physiological properties
of cortical neurons, such as spike frequency adaptation, Poisson statistics of spike trains,
the existence of strong local inhibition in cortical columns, and the maintenance of a tight
balance between excitation and inhibition. Finally, we discuss the implications of this
model for the debate about temporal versus rate-based neural coding.
1 Spikes and log posterior odds
1.1 Synaptic integration seen as inference in a hidden Markov chain
We propose that each neuron codes for an underlying ?hidden? binary variable, xt , whose
state evolves over time. We assume that xt depends only on the state at the previous time
step, xt?dt , and is conditionally independent of other past states. The state xt can switch
1
limdt?0 P (xt = 1|xt?dt = 0), and from 1 to
from 0 to 1 with a constant rate ron = dt
0 with a constant rate roff . For example, these transition rates could represent how often
motion in a preferred direction appears the receptive field and how long it is likely to stay
there.
The neuron infers the state of its hidden variable from N noisy synaptic inputs, considered
to be observations of the hidden state. In this initial version of the model, we assume
that these inputs are conditionally independent homogeneous Poisson processes, synapse
i
i emitting a spike between time t and t + dt (sit = 1) with constant probability qon
dt if
i
xt = 1, and another constant probability qoff dt if xt = 0. The synaptic spikes are assumed
to be otherwise independent of previous synaptic spikes, previous states and spikes at other
synapses. The resulting generative model is a hidden Markov chain (figure 1-A).
However, rather than estimating the state of its hidden variable and communicating this
estimate to other neurons (for example by emitting a spike when sensory evidence for
xt = 1 goes above a threshold) the neuron reports and communicates its certainty that the
current state is 1. This certainty takes the form of the log of the ratio of the probability
that the hidden state is 1, and the probability that the state is 0, given all the synaptic inputs
? P (xt =1|s0?t ) ?
received so far: Lt = log P
(xt =0|s0?t ) . We use s0?t as a short hand notation for the N
synaptic inputs received at present and in the past. We will refer to it as the log odds ratio.
Thanks to the conditional independencies assumed in the generative model, we can compute this Log odds ratio iteratively. Taking the limit as dt goes to zero, we get the following
differential equation:
?
?
?
? P
L? = ron 1 + e?L ? roff 1 + eL + i wi ?(sit ? 1) ? ?
B.
A.
xt dt
ron .roff
qon , qoff
st dt
xt
ron .roff
xt dt
qon , qoff
st
i
t
s
qon , qoff
st
Ot
j
It
?
Ot
Lt
st dt
?
Gt
t
t
C.
E.
2
0
-2
-4
D.
500
1000
1500
2000
2500
2
3000
Count
Log odds
4
20
Lt 0
-2
0
500
1000
1500
2000
2500
Time
Ot
3000
0
200
400
600
ISI
Figure 1: A. Generative model for the synaptic input. B. Schematic representation of log
odds ratio encoding and decoding. The dashed circle represents both eventual downstream
elements and the self-prediction taking place inside the model neuron. A spike is fired only
when Lt exceeds Gt . C. One example trial, where the state switches from 0 to 1 (shaded
area) and back to 0. plain: Lt , dotted: Gt . Black stripes at the top: corresponding spikes
train. D. Mean Log odds ratio (dark line) and mean output firing rate (clear line). E. Output
spike raster plot (1 line per trial) and ISI distribution for the neuron shown is C. and D.
Clear line: ISI distribution for a poisson neuron with the same rate.
wi , the synaptic weight, describe how informative synapse i is about the state of the hidden
? qi ?
variable, e.g. wi = log qion . Each synaptic spike (sit = 1) gives an impulse to the log
off
odds ratio, which is positive if this synapse is more active when the hidden state if 1 (i.e it
increases the neuron?s confidence that the state is 1), and negative if this synapse is more
active when xt = 0 (i.e it decreases the neuron?s confidence that the state is 1).
The bias, ?, is determined by how informative it is not to receive any spike, e.g. ? =
P
i
i
i qon ? qoff . By convention, we will consider that the ?bias? is positive or zero (if not,
we need simply to invert the status of the state x).
1.2 Generation of output spikes
The spike train should convey a sparse representation of Lt , so that each spike reports new
information about the state xt that is not redundant with that reported by other, preceding,
spikes. This proposition is based on three arguments: First, spikes, being metabolically
expensive, should be kept to a minimum. Second, spikes conveying redundant information
would require a decoding of the entire spike train, whereas independent spike can be taken
into account individually. And finally, we seek a self consistent model, with the spiking
output having a similar semantics to its spiking input.
To maximize the independence of the spikes (conditioned on xt ), we propose that the neuron fires only when the difference between its log odds ratio Lt and a prediction Gt of this
log odds ratio based on the output spikes emitted so far reaches a certain threshold. Indeed,
supposing that downstream elements predicts Lt as best as they can, the neuron only needs
to fire when it expects that prediction to be too inaccurate (figure 1-B). In practice, this
will happen when the neuron receives new evidence for xt = 1. Gt should thereby follow
the same dynamics as Lt when spikes are not received. The equation for Gt and the output
Ot (Ot = 1 when an output spike is fired) are given by:
G? =
Ot
=
?
?
?
?
ron 1 + e?L ? roff 1 + eL + go ?(Ot ? 1)
go
1. when Lt > Gt + , 0 otherwise,
2
(1)
(2)
Here go , a positive constant, is the only free parameter, the other parameters being constrained by the statistics of the synaptic input.
1.3 Results
Figure 1-C plots a typical trial, showing the behavior of L, G and O before, during and after
presentation of the stimulus. As random synaptic inputs are integrated, L fluctuates and
eventually exceeds G + 0.5, leading to an output spike. Immediately after a spike, G jumps
to G + go , which prevents (except in very rare cases) a second spike from immediately
following the first. Thus, this ?jump? implements a relative refractory period.
? on ? However,
. Thus L
G decays as it tends to converge back to its stable level gstable = log rroff
eventually exceeds G again, leading to a new spike. This threshold crossing happens more
often during stimulation (xt = 1) as the net synaptic input alters to create a higher overall
level of certainty, Lt .
Mean Log odds ratio and output firing rate
? t of the Bayesian neuron during presentation of its preferred stimulus
The mean firing rate O
(i.e. when xt switches from 0 to 1 and back to 0) is plotted in figure 1-D, together with the
? t , both averaged over trials. Not surprisingly, the log-posterior
mean log posterior ratio L
ratio reflects the leaky integration of synaptic evidence, with an effective time constant that
depends on the transition probabilities ron , roff . If the state is very stable (ron = roff ? 0),
synaptic evidence is integrated over almost infinite time periods, the mean log posterior
ratio tending to either increase or decrease linearly with time. In the example in figure 1D, the state is less stable, so ?old? synaptic evidence are discounted and Lt saturates.
? t tracks the state of xt almost perfectly. This
In contrast, the mean output firing rate O
is because, as a P
form of predictive coding, the output spikes reflect the new synaptic
i
evidence, It =
i ?(st ? 1) ? ?, rather than the log posterior ratio itself. In particular, the mean output firing rate is a rectified linear function of the mean input, e. g.
h
i+
? = 1 I? = P wi q i
?? .
O
go
i
on(off)
Analogy with a leaky integrate and fire neuron
We can get an interesting insight into the computation performed by this neuron by linearizing L and G around their mean levels over trials. Here we reduce the analysis to prolonged,
statistically stable periods when the state is constant (either ON or OFF). In this case, the
? and its output prediction G
? are also constant over time. We make
mean level of certainty L
the rough approximation that the post spike jump, go , and the input fluctuations are small
?
compared to the mean level of certainty L.
Rewriting Vt = Lt ? Gt +
go
2
as the ?membrane potential? of the Bayesian neuron:
V? = ?kL? V + It ? ?go ? go Ot
?
?
where kL? = ron e?L + roff eL , the ?leak? of the membrane potential, depends on the overall
level of certainty. ?go is positive and a monotonic increasing function of go .
A.
s t1 dt
s t1
s t1 dt
B.
C.
x t1
x t3 dt
x t3
x t3 dt
x t1 dt
x t1
x t1 dt
x t2
x t3
x t1
?
x tn
x t3
x t2
?
x tn
?
Lx2
D.
x t2 dt
s t2 dt
x t2
s t2
x t2 dt
s t2 dt
Log odds
10
No inh
-0.5
-1
-1
-2-1.5
5
Feedback
500 1000 1500 2000
Tiger
Stripes
0
-5
-10
500
1000
1500
2000
2500
Time
Figure 2: A. Bayesian causal network for yt (tiger), x1t (stripes) and x2t (paws). B. A network feedforward computing the log posterior for x1t . C. A recurrent network computing
the log posterior odds for all variables. D. Log odds ratio in a simulated trial with the net2
1
1
work in C (see text). Thick line: Lxt , thin line: Lxt , dash-dotted: Lxt without inhibition.
2
Insert: Lxt averaged over trials, showing the effect of feedback.
The linearized Bayesian neuron thus acts in its stable regime as a leaky integrate and fire
(LIF) neuron. The membrane potential Vt integrates its input, Jt = It ? ?go , with a leak
kL? . The neuron fires when its membrane potential reaches a constant threshold go . After
each spikes, Vt is reset to 0.
Interestingly, for appropriately chosen compression factor go , the mean input to the linearized neuron J? = I? ? ?go ? 0 1 . This means that the membrane potential is purely
driven to its threshold by input fluctuations, or a random walk in membrane potential. As
a consequence, the neuron?s firing will be memoryless, and close to a Poisson process. In
particular, we found Fano factor close to 1 and quasi-exponential ISI distribution (figure 1E) on the entire range of parameters tested. Indeed, LIF neurons with balanced inputs have
been proposed as a model to reproduce the statistics of real cortical neurons [8]. This balance is implemented in our model by the neuron?s effective self-inhibition, even when the
synaptic input itself is not balanced.
Decoding
As we previously said, downstream elements could predict the log odds ratio Lt by computing Gt from the output spikes (Eq 1, fig 1-B). Of course, this requires an estimate of
the transition probabilities ron , roff , that could be learned from the observed spike trains.
However, we show next that explicit decoding is not necessary to perform bayesian inference in spiking networks. Intuitively, this is because the quantity that our model neurons
receive and transmit, eg new information, is exactly what probabilistic inference algorithm
propagate between connected statistical elements.
1
Even if go is not chosen optimally, the influence of the drift J? is usually negligible compared to
the large fluctuations in membrane potential.
2 Bayesian inference in cortical networks
The model neurons, having the same input and output semantics, can be used as building blocks to implement more complex generative models consisting of coupled Markov
chains. Consider, for example, the example in figure 2-A. Here, a ?parent? variable x1t
(the presence of a tiger) can cause the state of n other ?children? variables ([xkt ]k=2...n ),
of whom two are represented (the presence of stripes,x2t , and motion, x3t ). The ?children? variables are Bayesian neurons identical to those described previously. The resulting
bayesian network consist of n + 1 coupled hidden Markov chains. Inference in this architecture corresponds to computing the log posterior odds ratio for the tiger, x1t , and the log
posterior of observing stripes or motion, ([xkt ]k=2...n ), given the synaptic inputs received
by the entire network so far, i.e. s20?t , . . . , sk0?t .
Unfortunately, inference and learning in this network (and in general in coupled Markov
chains) requires very expensive computations, and cannot be performed by simply propagating messages over time and among the variable nodes. In particular, the state of a child
variable xtk depends on xkt?dt , skt , x1t and the state of all other children at the previous
time step, [xjt?dt ]2<j<n,j6=i . In contrast, our network can only implement pairwise interactions, a connection between two spiking neurons implementing the conditional probability
linking the two corresponding binary variables.
Thus, we need to assume additional conditional independencies between the nodes in the
generative model, so that their joint probability can be pairwise factorized: p(xt , xt?1 ) =
Q
j Q
1
i
i
i
ij ?(xt , xt )
i ?(xt , xt?dt ). In words, it means that variables bias each other?s probZ
abilities, but do not influence each other?s dynamics, i.e they do not affect each other?s
transition probabilities. For example, a tiger does not affect the probability that stripes
appear or disappear, but increases their probability of being present.
Naive implementation
In this restricted case, marginal posterior probabilities can be computed iteratively by propagating beliefs in time and between the variables, or, in our model, by propagating spikes
in a neural network. This is because the probability of a variable xkt can be directly updated by the conditional probability
of observing the synaptic input to another connected
P
neuron, slt , eg p(slt |xkt ) = xl p(slt |xlt )p(xlt |xkt ), marginalizing out the hidden state xl . Of
t
course, rather than using slt , we use Otl , the output of the Bayesian neuron coding for xlt .
As we said previously, this output directly represents the new synaptic evidence received
by neuron l. The resulting equation is identical to the one derived previously for poisson
input,
P
k
k
L? x = fk (Lx ) + l wlk ?(Otl ? 1) ? ?k
k
k
(1 + e?x ) ? roff
(1 + ex ). As previously, wlk , the synaptic weight,
where fk (x) = ron
describes how informative it is for neuron k to receive a spike from neuron (or synapse)
P (O k =1|y =1)
l, wlk = log( P (Otk =1|ytt =0) ), while ?k is how informative it is not to receive a spike,
t
P
?k = dt l P (Otl = 1|xkt = 1) ? P (Otl = 1|xkt = 0).
This shows that our model is self-consistent. Except at the first stage of processing (eg in
the retina), all inputs are proposed to come from other Bayesian neurons.
Results
We implemented these update rules in a spiking neural network (figure 2-B) representing
the generative model in figure 2-A, with 100 possible children for x1t . We first consider
the case where there is no feedback connections, meaning that w1k = 0 for all k. In this
case the network computes the probability of a tiger at time t, integrating multiple sensory
cues such as the presence of stripes or motion in the visual scene.
In the example trials plotted in figure 2-D, we fixed the state of x1t and x2t : the tiger and
the stripes are present in the shaded temporal window, and absent outside of it. We then
sample the states of the other children (i.e is there motion or not?) and the corresponding
?observed? synaptic inputs, sk0?tmax , from the generative model. Once this synaptic input
has been generated, it is used as an input to the network in figure 2-B. What is plotted in
the Log odds ratio for the tiger, Lxt 1 and the stripes, Lxt 1 , as a function of time. As we can
see, the stripes receive very noisy synaptic input and can only provide weak evidence that
they are present. However, the tiger neuron is able to combine inputs from its 100 children
and get a much higher certainty.
Unfortunately, bayesian inference in this feedforward network is incomplete: The presence
of a tiger affects the probability of stripes, not only the other way round. To implement
this, we also need feedback connections, w1k . The network with feedforward and feedback
processing fails miserably, given that its activity explode, as illustrated in figure 2-D.
Balanced excitation/inhibition
This failure is due to the presence of loops, whereby a spike from neuron k increases
the certainty of neuron l (and its probability of firing) by wkl , and a spike from
neuron l increases in turn the certainty of neuron k by wlk . These loops result in
spikes reverberating through the network, ad infinitum, without reporting new information, a phenomena akin to loopy belief propagation [10]. To avoid overcounting of
evidence, we thus have to discount the reverberated ?old evidence? from the synaptic input:
P
P
k
k
k
L? x = fk (Lx ) + l wkl ?(Otl ? 1) ? l wkl wlk ?(Ot?dt
? 1) ? ?k .
We implemented this discounting using inhibitory neurons recurrently connected to each
excitatory neuron (figure 2-C). The inhibitory neurons are used to predict the redundant
feedback a bayesian neuron will receive and substract this prediction so that, once again,
only new information are taken into account and communicated to other neurons. Each
excitatory loop is compensated by an inhibitory loop, resulting in a balance between
excitation and inhibition at the level of each neuron within the network.
The result on one trial is plotted in figure 2-D. The ?tiger? log odds ratio is almost indistinguishable from the feedforward case, and is not plotted. The ?stripes? Log odds ratio
increases during presentation of the tiger due to the feedback. In other words, the stripes
neuron can take into account no only its own its own synaptic input, but also the synaptic
input to the other children neurons (such as evidence for motion), thanks to the presence of
a common source (the tiger).
Over many trials, we found that the statistics of the bayesian neuron were still poisson,
and their output firing is stillP
a rectified linear function of the input firing rate in a stable
?l +
?k = [
statistical regime, i.e. O
l wkl O ] .
Discussion
We started from an interpretation of synaptic integration in single neurons as a form of
inference in a hidden Markov chain. We derived a model of spiking neurons and their
interactions able to compute the marginal posterior probabilities of sensory and motor variables given evidence received in the entire network. In this view, the brain implements
an underlying bayesian network in an interconnected neural architecture, with conditional
probabilities represented by synaptic weights. The model makes a rich set of predictions
for the general properties of neuron and synaptic dynamics, such as a time constant that
depends on the overall level of inputs, specific forms of frequency dependant spike and
synaptic adaptation (not shown here) and micro-balanced excitation and inhibition. However, it is still restricted to probabilistic computations involving binary variables. In a
related work similar ideas are applied population encoding of log probability distribution
for analog variables (Zemel, Huys and Dayan, submitted to NIPS 2004).
Despite non-linear processing at the single neural level, the emerging picture is relatively
simple: The neuron acts as a leaky integrate and fire neuron driven by noise. The output
firing rate is a rectified weighted sum of the input firing rates, while the firing statistics are
Poisson. However, these output spike trains are a deterministic function of the input spike
trains. Spikes report fluctuations in the level of certainty that could not be predicted either
from the stability of its stimulus (contribution from Gt ) or the loops in the network (contribution from the inhibitory neuron). Thus firing will be, by definition, unpredictable. This
last observation leads us to suggest that the irregular firing and Poisson statistics observed
in cortical neurons [1] arises as a direct consequence of the random fluctuations in the sensory inputs and the instability of the real word, but are not due to unreliable or ?chaotic?
neural processing.
Finally, it is crucial for the biological realism of the model to find adaptive neural dynamics
and synaptic plasticity able to learn the parameters of the internal model and conditional
probabilities, and we are currently exploring these issues. Fortunately, the required learning
rules are local and unsupervised. According to our preliminary work, the synaptic weights
and bias depend on the joint probability of presynaptic/postsynaptic spikes and can be
learned with the spike time dependent plasticity observed in hippocampus and cortex [5].
Meanwhile, the transition probabilities simply correspond to how often the neuron switches
between an active and an inactive state.
Acknowledgments
We thank Peter Dayan, Peter Latham, Zoubin Ghahramani, Jean Laurens and Jacques
Droulez for helpful discussions and comments. This work was supported by a Dorothy
Hodgkin fellowship for the Royal Society and the BIBA European Project.
References
[1] K. H. Britten, M. N. Shadlen, W. T. Newsome, and J. A. Movshon. The analysis of visual
motion: A comparison of neuronal and psychophysical performance. Journal of Neuroscience,
12:4745?4765, 1992.
[2] G. Hinton and A. Brown. Spiking boltzmann machines. In S. Solla, T. Leen, and K. Muller, editors, Neural Information Processing System, volume 12, pages 122?8. MIT Press, Cambridge,
MA, 2000.
[3] D. Knill and W. Richards. Perception as Bayesian inference. Cambridge University Press,
Cambridge, MA, 1996.
[4] K. Kording and D. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427:244?7,
2004.
[5] H. Markram and M. Tsodyks. Redistribution of synaptic efficacy between neocortical pyramidal
neurons. Nature, 382:807?19, 1996.
[6] M. Mazurek, J. Roitman, J. Ditterich, and M. Shadlen. A role for neural integrators in perceptual
decision making. Cerebral cortex, 13(11):1257?69, 2003.
[7] R. Rao. Bayesian computation in recurrent neural circuits. Neural Computation, 16(1):1?38,
2003.
[8] M. Shadlen and W. Newsome. Noise, neural codes and cortical organization. Current Opinion
in Neurobiology, 4:569?579, 1994.
[9] Y. Weiss and D. Fleet. Velocity likelihood in biological and machine vision. In R. Rao, B. Olshausen, and M. Lewicki, editors, Probabilistic Models of the Brain: Perception and Neural
Function, pages 77?96. MIT Press, Cambridge, MA, 2002.
[10] Y. Weiss and W. Freeman. Correctness of belief propagation in gaussian graphical models of
arbitrary topology. Neural Computation, 13:2173?200, 2001.
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1,890 | 2,717 | Binet-Cauchy Kernels
S.V.N. Vishwanathan, Alexander J. Smola
National ICT Australia, Machine Learning Program, Canberra, ACT 0200, Australia
{SVN.Vishwanathan, Alex.Smola}@nicta.com.au
Abstract
We propose a family of kernels based on the Binet-Cauchy theorem and its extension to Fredholm operators. This includes as special cases all currently known
kernels derived from the behavioral framework, diffusion processes, marginalized
kernels, kernels on graphs, and the kernels on sets arising from the subspace angle
approach. Many of these kernels can be seen as the extrema of a new continuum
of kernel functions, which leads to numerous new special cases. As an application,
we apply the new class of kernels to the problem of clustering of video sequences
with encouraging results.
1
Introduction
Recent years have see a combinatorial explosion of results on kernels for structured and
semi-structured data, including trees, strings, graphs, transducers and dynamical systems
[6, 8, 15, 13]. The fact that these kernels are very specific to the type of discrete data under
consideration is a major cause of confusion to the practitioner. What is required is a) an
unified view of the field and b) a recipe to design new kernels easily.
The present paper takes a step in this direction by unifying these diverse kernels by means
of the Binet-Cauchy theorem. Our point of departure is the work of Wolf and Shashua [17],
or more specifically, their proof that det A? B is a kernel on matrices A, B ? Rm?n . We
extend the results of [17] in the following three ways:
1. There exists an operator-valued equivalent of the Binet-Cauchy theorem.
2. Wolf and Shashua only exploit the Binet-Cauchy theorem for one particular choice of
parameters. It turns out that the continuum of these values corresponds to a large class
of kernels some of which are well known and others which are novel.
3. The Binet-Cauchy theorem can be extended to semirings. This points to a close connection with rational kernels [3].
Outline of the paper: Section 2 contains the main result of the present paper: the definition of Binet-Cauchy kernels and their efficient computation. Subsequently, section 3
discusses a number of special cases, which allows us to recover well known kernel functions. Section 4 applies our derivations to the analysis of video sequences, and we conclude
with a discussion of our results.
2
Binet-Cauchy Kernels
In this section we deal with linear mappings from X = Rn to Y = Rm (typically denoted
by matrices), their coordinate free extensions to Fredholm operators (here Rn and Rm are
replaced by measurable sets), and their extensions to semirings (here addition and multiplication are replaced by an abstract class of symbols (?, ?) with the same distributive
properties).
2.1
The General Composition Formula
We begin by defining compound matrices. They arise by picking subsets of entries of a
matrix and computing their determinants.
Definition 1 (Compound Matrix) Let A ? Rm?n , let q ? min(m, n) and let Iqn = {i =
(i1 , i2 , . . . , iq ) : 1 ? i1 < . . . < iq ? n, ii ? N} and likewise Iqm . Then the compound
matrix of order q is defined as
[Cq (A)]i,j := det(A(ik , jl ))qk,l=1 where i ? Iqn and j ? Iqm .
(1)
Here i, j are assumed to be arranged in lexicographical order.
Theorem 2 (Binet-Cauchy) Let A ? Rl?m and, B ? Rl?n . For q ? min(m, n, l) we
have Cq (A? B) = Cq (A)? Cq (B).
When q = m = n = l we have Cq (A) = det(A) and the Binet-Cauchy theorem becomes
the well known identity det(A? B) = det(A) det(B). On the other hand when q = 1 we
have C1 (A) = A, so Theorem 2 reduces to a tautology.
Theorem 3 (Binet-Cauchy for Semirings) When the common semiring (R, +, ?, 0, 1) is
replaced by an abstract semiring (K, ?, ?, ?0, ?1) the equality Cq (A? B) = Cq (A)? Cq (B)
still holds. Here all operations occur on the monoid K, addition and multiplication are
replaced by ?, ?, and (?
0, ?
1) take the role of (0, 1).
A second extension of Theorem 2 is to replace matrices by Fredholm operators, as they
can be expressed as integral operators with corresponding kernels. In this case, Theorem 2
becomes a statement about convolutions of integral kernels.
Definition 4 (Fredholm Operator) A Fredholm operator is a bounded linear operator between two Hilbert spaces with closed range and whose kernel and co-kernel are finitedimensional.
Theorem 5 (Kernel Representation of Fredholm Operators) Let A : L2 (Y) ? L2 (X)
and, B : L2 (Y) ? L2 (Z) be Fredholm operators. Then there exists some kA : X ? Y ? R
such that for all f ? L2 (X) we have
Z
[Af ](x) =
kA (x, y)f (y)dy.
(2)
Y
?
Moreover, for the composition A B we have kA? B (x, z) =
R
Y
kA? (x, y)kB (y, z)dy.
Here the convolution of kernels kA and kB plays the same role as the matrix multiplication. To extend the Binet-Cauchy theorem we need to introduce the analog of compound
matrices:
Definition 6 (Compound Kernel and Operator) Denote by X, Y ordered sets and let k :
X ? Y ? R. Define IqX = {x ? Xq : x1 ? . . . ? xq } and likewise IqY . Then the compound
kernel of order q is defined as
k [q] (x, y) := det(k(xk , yl ))qk,l=1 where x ? IqX and y ? IqY .
(3)
If k is the integral kernel of an operator A we define Cq (A) to be the integral operator
corresponding to k [q] .
Theorem 7 (General Composition Formula [11]) Let X, Y, Z be ordered sets and let A :
L2 (Y) ? L2 (X), B : L2 (Y) ? L2 (Z) be Fredholm operators. Then for q ? N we have
Cq (A? B) = Cq (A)? Cq (B).
To recover Theorem 2 from Theorem 7 set X = [1..m], Y = [1..n] and Z = [1..l].
(4)
2.2
Kernels
The key idea in turning the Binet-Cauchy theorem and its various incarnations into a kernel
is to exploit the fact that tr A? B and det A? B are kernels on operators A, B. We extend
this by replacing A? B with some functions ?(A)? ?(B) involving compound operators.
Theorem 8 (Trace and Determinant Kernel) Let A, B : L2 (X) ? L2 (Y) be Fredholm
operators and let S : L2 (Y) ? L2 (Y), T : L2 (X) ? L2 (X) be positive trace-class operators. Then the following two kernels are well defined and they satisfy Mercer?s condition:
k(A, B) = tr SA? T B
(5)
?
k(A, B) = det SA T B .
(6)
Note that determinants are not defined in general for infinite dimensional operators, hence
our restriction to Fredholm operators A, B in (6).
Proof Observe that S and T are positive and compact. Hence they admit a decomposition
?
into S = VS VS? and
virtue of the commutativity of the trace we have
T = VT VT . By
?
that k(A, B) = tr [VT AVS ] [VT BVS ] . Analogously, using the Binet-Cauchy theorem,
we can decompose the determinant. The remaining terms VT AVS and VT BVS are again
Fredholm operators for which determinants are well defined.
Next we use special choices of A, B, S, T involving compound operators directly to state
the main theorem of our paper.
Theorem 9 (Binet-Cauchy Kernel) Under the assumptions
of Theorem 8 it follows
that
for all q ? N the kernels k(A, B) = tr Cq SA? T B and k(A, B) = det Cq SA? T B
satisfy Mercer?s condition.
Proof We exploit the factorization S = VS VS? , T = VT? VT and apply Theorem 7. This
yields Cq (SA? T B) = Cq (VT AVS )? Cq (VT BVS ), which proves the theorem.
Finally, we define a kernel based on the Fredholm determinant itself. It is essentially a
weighted combination of Binet-Cauchy kernels. Fredholm determinants are defined as
follows [11]:
?
X
?q
tr Cq (A).
(7)
D(A, ?) :=
q!
q=1
This series converges for all ? ? C and it is an entire function of ?. It suggests a kernel
involving weighted combinations of the kernels of Theorem 9. We have the following:
Corollary 10 (Fredholm Kernel) Let A, B, S, T as in Theorem 9 and let ? > 0. Then the
following kernel satisfies Mercer?s condition:
k(A, B) := D(A? B, ?) where ? > 0.
(8)
?
D(A B, ?) is a weighted combination of the kernels discussed above. The exponential
1
down-weighting via q!
ensures that the series converges even in the case of exponential
growth of the values of the compound kernel.
2.3
Efficient Computation
At first glance, computing the kernels of Theorem 9 and Corollary 10 presents a formidable
m?n
computational task
, the matrix
even in the finite dimensional case. If A, B ? R
n
?
Cq (A B) has q rows and columns and each of the entries requires the computation of
a determinant of a q-dimensional matrix. A brute-force approach would involve O(q 3 nq )
operations (assuming 2q ? n). Clearly we need more efficient techniques.
When computing determinants, we can take recourse to Franke?s Theorem [7] which states
that
n?1
det Cq (A) = (det A)( q?1 ) .
(9)
n?1
and consequently k(A, B) = det Cq [SA? T B] = (det[SA? T B])( q?1 ) .1 This indicates
that the determinant kernel may be of limited use, due to the typically quite high power in
the exponent. Kernels building on tr Cq are not plagued by this problem and we give an
efficient recursion below. It follows from the ANOVA kernel recursion of [1]:
Lemma 11 Denote by A ? Cm?m a square matrix and let ?1 , . . . , ?m be its eigenvalues.
Then tr Cq (A) can be computed by the following recursion:
q
tr Cq (A) =
n
X q
1X
(?1)j+1 C?q?j (A)C?j (A) where C?q (A) =
?j .
q j=1
j=1
(10)
Proof We begin by writing A in its Jordan normal form as A = P DP ?1 where D is a
block diagonal, upper triangular matrix. Furthermore, the diagonal elements of D consist
of the eigenvalues of A. Repeated application of the Binet-Cauchy Theorem yields
tr Cq (A) = tr Cq (P )Cq (D)Cq (P ?1 ) = tr Cq (D)Cq (P ?1 )Cq (P ) = tr Cq (D)
(11)
For a triangular matrix the determinant is the product of its diagonal entries. Since all
the square submatrices of D are also upper triangular, to construct tr(Cq (D)) we need to
sum over all products of exactly q eigenvalues. This is analog to the requirement of the
ANOVA kernel of [1]. In its simplified version it can be written as (10), which completes
the proof.
We can now compute the Jordan normal form of SA? T B in O(n3 ) time and apply
Lemma 11 directly to it to compute the kernel value.
Finally, in the case of Fredholm determinants, we can use the recursion directly, because
for n-dimensional matrices the sum terminates after n terms. This is no more expensive
than computing tr Cq directly. Note that in the general nonsymmetric case (i.e. A 6= A? )
no such efficient recursions are known.
3
Special Cases
We now focus our attention on various special cases to show how they fit into the general
framework which we developed in the previous section. For this to succeed, we will map
various systems such as graphs, dynamical systems, or video sequences into Fredholm
operators. A suitable choice of this mapping and of the operators S, T of Theorem 9 will
allow us to recover many well-known kernels as special cases.
3.1
Dynamical Systems
We begin by describing a partially observable discrete time LTI (Linear Time Invariant)
model commonly used in control theory. Its time-evolution equations are given by
yt = P xt + wt
xt = Qxt?1 + vt
where wt ? N(0, R)
where vt ? N(0, S).
(12a)
(12b)
Here yt ? Rm is observed, xt ? Rn is the hidden or latent variable, and P ? Rm?n , Q ?
Rn?n , R ? Rm?m and, S ? Rn?n , moreover R, S 0. Typically m ? n. similar model
exists for continuous LTI. Further details on it can be found in [14].
Following the behavioral framework of [16] we associate dynamical systems, X :=
(P, Q, R, S, x0 ), with their trajectories, that is, the set of yt with t ? N for discrete time
systems (and t ? [0, ?) for the continuous-time case). These trajectories can be interpreted
1
Eq. (9) can be seen as follows: the compound matrix of an orthogonal matrix is orthogonal and
consequently its determinant is unity. Subsequently use an SVD factorization of the argument of the
compound operator to compute the determinant of the compound matrix of a diagonal matrix.
as linear operators mapping from Rm (the space of observations y) into the time domain
(N or [0, ?)) and vice versa. The diagram below depicts this mapping:
X
/ Traj(X)
/ Cq (Traj(X))
Finally, Cq (Traj(X)) is weighted in a suitable fashion by operators S and T and the trace
is evaluated. This yields an element from the family of Binet-Cauchy kernels.
In the following we discuss several kernels and we show that they differ essentially
in how the mapping into a dynamical system occurs (discrete-time or continuous time,
fully observed or partial observations), whether any other preprocessing is carried out on
Cq (Traj(X)) (such as QR decomposition in the case of the kernel proposed by [10] and
rediscovered by [17]), or which weighting S, T is chosen.
3.2
Dynamical Systems Kernels
We begin with kernels on dynamical systems (12) as proposed in [14]. Set S = 1, q = 1
and T to be the diagonal operator with entries e??t . In this case the Binet-Cauchy kernel
between systems X and X ? becomes
?
X
tr Cq (S Traj(X) T Traj(X ? )? ) =
e??t yt? yt? .
(13)
i=1
Since yt , yt? are random variables, we also need to take expectations over wt , vt , wt? , vt? .
Some tedious yet straightforward algebra [14] allows us to compute (13) as follows:
1
?
tr [SM2 + R] ,
(14)
k(X, X ? ) = x?
0 M1 x0 + ?
e ?1
where M1 , M2 satisfy the Sylvester equations:
M1 = e?? Q? P ? P ? Q? + e?? Q? M1 Q? and M2 = P ? P ? + e?? Q? M2 Q? .
(15)
3
Such kernels can be computed in O(n ) time [5]. Analogous expressions for continuoustime systems exist [14]. In Section 4 we will use this kernel to compute similarities between
video sequences, after having encoded the latter as a dynamical system. This will allow us
to compare sequences of different length, as they are all mapped to dynamical systems in
the first place.
3.3
Martin Kernel
A characteristic property of (14) is that it takes initial conditions of the dynamical system
into account. If this is not desired, one may choose to pick only the subspace spanned by
the trajectory yt . This is what was proposed in [10].2
More specifically, set S = T = 1, consider the trajectory upto only a finite number of time
steps, say up to n, and let q = n. Furthermore let Traj(X) = QX RX denote the QRdecomposition of Traj(X), where QX is an orthogonal matrix and RX is upper triangular.
Then it can be easily verified the kernel proposed by [10] can be written as
?
k(X, X ? ) = tr Cq (SQX T Q?
X ? ) = det(QX QX ? ).
(16)
This similarity measure has been used by Soatto, Doretto, and coworkers [4] for the analysis and computation of similarity in video sequences. Subsequently Wolf and Shashua
[17] modified (16) to allow for kernels: to deal with determinants on a possibly infinitedimensional feature space they simply project the trajectories on a reduced set of points in
feature space.3 This is what [17] refer to as a kernel on sets.
2
Martin [10] suggested the use of Cepstrum coefficients of a dynamical system to define a Euclidean metric. Later De Cock and Moor [2] showed that this distance is, indeed, given by the
computation of subspace angles, which can be achieved by computing the QR-decomposition.
3
To be precise, [17] are unaware of the work of [10] or of [2] and they rediscover the notion of
subspace angles for the purpose of similarity measures.
3.4
Graph Kernels
Yet another class of kernels can be seen to fall into this category: the graph kernels proposed
in [6, 13, 9, 8]. Denote by G(V, E) a graph with vertices V and edges E. In some cases,
such as in the analysis of molecules, the vertices will be equipped with labels L. For
recovering these kernels from our framework we set q = 1 and systematically map graph
kernels to dynamical systems.
We denote by xt a probability distribution over the set of vertices at time t. The timeevolution xt ? xt+1 occurs by performing a random walk on the graph G(V, E). This
yields xt+1 = W D?1 xt , where W is the connectivity matrix of the graph and D is a
diagonal matrix where Dii denotes the outdegree of vertex i. For continuous-time systems
?
? is the normalized graph Laplacian [9].
one uses x(t) = exp(?Lt)x(0),
where L
In the graph kernels of [9, 13] one assumes that the variables xt are directly observed and
no special mapping is required in order to obtain yt . Various choices of S and T yield the
following kernels:
? [9] consider a snapshot of the diffusion process at t = ? . This amounts to choosing
T = 1 and a S which is zero except for a diagonal entry at ? .
? The inverse Graph-Laplacian kernel proposed in [13] uses a weighted combination of
diffusion process and corresponds to S = W a diagonal weight matrix.
? The model proposed in [6] can be seen as using a partially observable model: rather
than observing the states directly, we only observe the labels emitted at the states. If we
associate this mapping from states to labels with the matrix P of (12), set T = 1 and
let S be the projector on the first n time instances, we recover the kernels from [6].
So far, we deliberately made no distinction between kernels on graphs and kernels between
graphs. This is for good reason: the trajectories depend on both initial conditions and the
dynamical system itself. Consequently, whenever we want to consider kernels between initial conditions, we choose the same dynamical system in both cases. Conversely, whenever
we want to consider kernels between dynamical systems, we average over initial conditions.
This is what allows us to cover all the aforementioned kernels in one framework.
3.5
Extensions
Obviously the aforementioned kernels are just specific instances of what is possible by
using kernels of Theorem 9. While it is pretty much impossible to enumerate all combinations, we give a list of suggestions for possible kernels below:
? Use the continuous-time diffusion process and a partially observable model. This would
extend the diffusion kernels of [9] to comparisons between vertices of a labeled graph
(e.g. atoms in a molecule).
? Use diffusion processes to define similarity measures between graphs.
? Compute the determinant of the trajectory associated with an n-step random walk on a
graph, that is use Cq with q = n instead of C1 . This gives a kernel analogous to the
one proposed by Wolf and Shashua [17], however without the whitening incurred by
the QR factorization.
? Take Fredholm determinants of the above mentioned trajectories.
? Use a nonlinear version of the dynamical system as described in [14].
4
Experiments
To test the utility of our kernels we applied it to the task of clustering short video clips.
We randomly sampled 480 short clips from the movie Kill Bill and model them as linear
ARMA models (see Section 3.1).
The sub-optimal procedure outlined in [4] was used
for estimating the model parameters P , Q, R and,
S and the kernels described in Section 3.2 were applied to these models. Locally Linear Embedding
(LLE) [12] was used to cluster and embed the clips
in two dimensions. The two dimensional embedding obtained by LLE is depicted in Figure 1. We
randomly selected a few data points from Figure 1
and depict the first frame of the corresponding clips
in Figure 2.
Observe the linear cluster (with a projecting arm) in
Figure 1: LLE embeddings of 480 ran- Figure 1. This corresponds to clips which are temdom clips from Kill Bill
porally close to each other and hence have similar
dynamics.
For
instance
clips
in
the
far
right
depict
a
person rolling in the snow while those in the far left corner depict a sword fight while clips
in the center involve conversations between two characters. A naiv?e comparison of the intensity values or a dot product of the actual clips would not be able to extract such semantic
information. Even though the camera angle varies with time our kernel is able to successfully pick out the underlying dynamics of the scene. These experiments are encouraging
and future work will concentrate on applying this to video sequence querying.
2
1
0
?1
?2
?3
?4
?5
?4
?3
?2
?1
0
1
2
3
4
Figure 2: LLE embeddings of a subset of our dataset. A larger version is available from
http://mlg.anu.edu.au/?vishy/papers/KillBill.png
5
Discussion
In this paper, we introduced a unifying framework for defining kernels on discrete objects
using the Binet-Cauchy theorem on compounds of the Fredholm operators. We demonstrated that many of the previously known kernels can be explained neatly by our framework. In particular many graph kernels and dynamical system related kernels fall out as
natural special cases. The main advantage of our unifying framework is that it allows kernel engineers to use domain knowledge in a principled way to design kernels for solving
real life problems.
Acknowledgement We thank Stephane Canu and Ren?e Vidal for useful discussions. National ICT Australia is supported by the Australian Government?s Program Backing Australia?s Ability. This work was partly supported by grants of the Australian Research Council. This work was supported by the IST Programme of the European Community, under
the Pascal Network of Excellence, IST-2002-506778.
References
[1] C. J. C. Burges and V. Vapnik. A new method for constructing artificial neural networks. Interim technical report, ONR contract N00014 - 94-c-0186, AT&T Bell Laboratories, 1995.
[2] K. De Cock and B. De Moor. Subspace angles between ARMA models. Systems and
Control Letter, 46:265 ? 270, 2002.
[3] C. Cortes, P. Haffner, and M. Mohri. Rational kernels. In Proceedings of Neural
Information Processing Systems 2002, 2002. in press.
[4] G. Doretto, A. Chiuso, Y.N. Wu, and S. Soatto. Dynamic textures. International
Journal of Computer Vision, 51(2):91 ? 109, 2003.
[5] J. D. Gardiner, A. L. Laub, J. J. Amato, and C. B. Moler. Solution of the Sylvester matrix equation AXB ? + CXD? = E. ACM Transactions on Mathematical Software,
18(2):223 ? 231, 1992.
[6] T. G?artner, P.A. Flach, and S. Wrobel. On graph kernels: Hardness results and efficient alternatives. In B. Sch?olkopf and M. Warmuth, editors, Sixteenth Annual Conference on Computational Learning Theory and Seventh Kernel Workshop, COLT.
Springer, 2003.
[7] W. Gr?obner. Matrizenrechnung. BI Hochschultaschenb?ucher, 1965.
[8] H. Kashima, K. Tsuda, and A. Inokuchi. Marginalized kernels between labeled
graphs. In Proceedings of the 20th International Conference on Machine Learning
(ICML), Washington, DC, United States, 2003.
[9] I.R. Kondor and J. D. Lafferty. Diffusion kernels on graphs and other discrete structures. In Proceedings of the ICML, 2002.
[10] R.J. Martin. A metric for ARMA processes. IEEE Transactions on Signal Processing,
48(4):1164 ? 1170, 2000.
[11] A. Pinkus. Spectral properties of totally positive kernels and matrices. In M. Gasca
and C. A. Miccheli, editors, Total Positivity and its Applications, volume 359 of Mathematics and its Applications, pages 1?35. Kluwer, March 1996.
[12] S. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323 ? 2326, December 2000.
[13] A.J. Smola and I.R. Kondor. Kernels and regularization on graphs. In B. Sch?olkopf
and M. K. Warmuth, editors, Proceedings of the Annual Conference on Computational Learning Theory, Lecture Notes in Computer Science. Springer, 2003.
[14] A.J. Smola, R. Vidal, and S.V.N. Vishwanathan. Kernels and dynamical systems.
Automatica, 2004. submitted.
[15] S.V.N. Vishwanathan and A.J. Smola. Fast kernels on strings and trees. In Proceedings of Neural Information Processing Systems 2002, 2002.
[16] J. C. Willems. From time series to linear system. I. Finite-dimensional linear time
invariant systems. Automatica J. IFAC, 22(5):561 ? 580, 1986.
[17] L. Wolf and A. Shashua. Learning over sets using kernel principal angles. Jounal of
Machine Learning Research, 4:913 ? 931, 2003.
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expressed:1 ordered:2 partially:3 applies:1 springer:2 wolf:5 corresponds:3 satisfies:1 acm:1 succeed:1 identity:1 consequently:3 replace:1 axb:1 specifically:2 infinite:1 except:1 wt:4 lemma:2 engineer:1 total:1 principal:1 partly:1 svd:1 latter:1 alexander:1 |
1,891 | 2,718 | Semi-supervised Learning on Directed Graphs
Dengyong Zhou? , Bernhard Sch?olkopf? , and Thomas Hofmann??
?
Max Planck Institute for Biological Cybernetics
72076 Tuebingen, Germany
{dengyong.zhou, bernhard.schoelkopf}@tuebingen.mpg.de
?
Department of Computer Science, Brown University
Providence, RI 02912 USA
[email protected]
Abstract
Given a directed graph in which some of the nodes are labeled, we investigate the question of how to exploit the link structure of the graph to infer
the labels of the remaining unlabeled nodes. To that extent we propose a
regularization framework for functions defined over nodes of a directed
graph that forces the classification function to change slowly on densely
linked subgraphs. A powerful, yet computationally simple classification
algorithm is derived within the proposed framework. The experimental
evaluation on real-world Web classification problems demonstrates encouraging results that validate our approach.
1
Introduction
We consider semi-supervised classification problems on weighted directed graphs, in which
some nodes in the graph are labeled as positive or negative, and where the task consists in
classifying unlabeled nodes. Typical examples of this kind are Web page categorization
based on hyperlink structure [4, 11] and document classification or recommendation based
on citation graphs [10], yet similar problems exist in other domains such as computational
biology. For the sake of concreteness, we will mainly focus on the Web graph in the sequel,
i.e. the considered graph represents a subgraph of the Web, where nodes correspond to Web
pages and directed edges represent hyperlinks between them (cf. [3]).
We refrain from utilizing attributes or features associated with each node, which may or
may not be available in applications, but rather focus on the analysis of the connectivity of
the graph as a means for classifying unlabeled nodes. Such an approach inevitably needs
to make some a priori premise about how connectivity and categorization of individual
nodes may be related in real-world graphs. The fundamental assumption of our framework
is the category similarity of co-linked nodes in a directed graph. This is a slightly more
complex concept than in the case of undirected (weighted) graphs [1, 18, 12, 15, 17], where
a typical assumption is that an edge connecting two nodes will more or less increase the
likelihood of the nodes belonging to the same category. Co-linkage on the other hand
seems a more suitable and promising concept in directed graphs, as is witnessed by its
successful use in Web page categorization [4] as well as co-citation analysis for information
retrieval [10]. Notice that co-linkage comes in two flavors: sibling structures, i.e. nodes
with common parents, and co-parent structures, i.e. nodes with common children. In most
Web and citation graph related application, the first assumption, namely that nodes with
highly overlapping parent sets are likely to belong to the same category, seems to be more
relevant (cf. [4]), but in general this will depend on the specific application.
One possible way of designing classifiers based on graph connectivity is to construct a
kernel matrix based on pairwise links [11] and then to adopt a standard kernel method,
e.g. Support Vector Machines (SVMs) [16] as a learning algorithm. However, a kernel
matrix as the one proposed in [11] only represents local relationships among nodes, but
completely ignores the global structure of the graph. The idea of exploiting global rather
than local graph structure is widely used in other Web-related techniques, including Web
page ranking [2, 13], finding similar Web pages [7], detecting Web communities [13, 9]
and so on. The major innovation of this paper is a general regularization framework on
directed graphs, in which the directionality and global relationships are considered, and
a computationally attractive classification algorithm, which is derived from the proposed
regularization framework.
2
2.1
Regularization Framework
Preliminaries
A directed graph ? = (V, E) consists of a set of vertices, denoted by V and a set of edges,
denoted by E ? V ? V . Each edge is an ordered pair of nodes [u, v] representing a
directed connection from u to v. We do not allow self loops, i.e. [v, v] 6? E for all v ? V .
In a weighted directed graph, a weight function w : V ? V ? R+ is associated with ?,
satisfying w([u, v]) = 0 if and only if [u, v] 6? E. Typically, we can equip a directed graph
with a canonical weight function by defining w([u, v]) ? 1 if and only if [u, v] ? E. The
in-degree p(v) and out-degree q(v) of a vertex v ? V , respectively, are defined as
X
X
p(v) ?
w([u, v]), and q(v) ?
w([v, u]) .
(1)
{u|[u,v]?E}
{u|[v,u]?E}
Let H(V ) denote the space of functions f : V ? R, which assigns a real value f (v) to
each vertex v. The function f can be represented as a column vector in R|V | , where |V |
denotes the number of the vertices in V . The function space H(V ) can be endowed with
the usual inner product:
X
hf, gi =
f (v)g(v).
(2)
v
Accordingly, the norm of the function induced from the inner product is kf k =
2.2
p
hf, f i.
Bipartite Graphs
A bipartite graph G = (H, A, L) is a special type of directed graph that consists of two
sets of vertices, denoted by H and A respectively, and a set of edges (or links), denoted by
L ? H ? A. In a bipartite graph, each edge connects a vertex in H to a vertex in A. Any
directed graph ? = (V, E) can be regarded as a bipartite graph using the following simple
construction [14]: H ? {h|h ? V, q(h) > 0}, A ? {a|a ? V, p(a) > 0}, and L ? E.
Figure 1 depicts the construction of the bipartite graph. Notice that vertices of the original
graph ? may appear in both vertex sets H and A of the constructed bipartite graph.
The intuition behind the construction of the bipartite graph is provided by the so-called hub
and authority web model introduced by Kleinberg [13]. The model distinguishes between
two types of Web pages: authoritative pages, which are pages relevant to some topic,
and hub pages, which are pages pointing to relevant pages. Note that some Web pages can
Figure 1: Constructing a bipartite graph from a directed one. Left: directed graph. Right:
bipartite graph. The hub set H = {1, 3, 5, 6}, and the authority set A = {2, 3, 4}. Notice
that the vertex indexed by 3 is simultaneously in the hub and authority set.
simultaneously be both hub and authority pages (see Figure 1). Hubs and authorities exhibit
a mutually reinforcement relationship: a good hub node points to many good authorities
and a good authority node is pointed to by many good hubs. It is interesting to note that in
general there is no direct link from one authority to another. It is the hub pages that glue
together authorities on a common topic.
According to Kleinberg?s model, we suggestively call the vertex set H in the bipartite graph
the hub set, and the vertex set A the authority set.
2.3
Smoothness Functionals
If two distinct vertices u and v in the authority set A are co-linked by vertex h in the hub
set H as shown in the left panel of Figure 2, then we think that u and v are likely to be
related, and the co-linkage strength induced by h between u and v can be measured by
ch ([u, v]) =
w([h, u])w([h, v])
.
q(h)
(3)
In addition, we define ch (v, v) = 0 for all v in the authority set A and for all h in the hub
set H. Such a relevance measure can be naturally understood in the situation of citation
networks. If two articles are simultaneously cited by some other article, then this should
make it more likely that both articles deal with a similar topics. Moreover, the more articles
cite both articles together, the more significant the connection. A natural question arising
in this context is why the relevance measure is further normalized by out-degree. Let us
consider the following two web sites: Yahoo! and kernel machines. General interest portals
like Yahoo! consists of pages having a large number of diverse hyperlinks. The fact that
two web pages are co-linked by Yahoo! does not establish a significant connection between
them. In contrast, the pages on the kernel machine Web site have much fewer hyperlinks,
but the Web pages pointed to are closely related in topic.
Let f denote a function defined on the authority set A. The smoothness of function f can
be measured by the following functional:
2
f (u)
f (v)
1 XX
ch ([u, v]) p
?p
.
(4)
?A (f ) =
2 u,v
p(u)
p(v)
h
The smoothness functional penalizes large differences in function values for vertices in the
authority set A that are strongly related. Notice that the function values are normalized by
Figure 2: Link and relevance. Left panel: vertices u and v in the authority set A are colinked by vertex h in the hub set H. Right panel: vertices u and v in the hub set H co-link
vertex a in the authority set A.
in-degree. For the Web graph, the explanation is similar to the one given before. Many
web pages contain links to popular sites like the Google search engine. This does not mean
though that all these Web pages share a common topic. However, if two web pages point to
web page like the one of the Learning with Kernels book, it is likely to express a common
interest for kernel methods.
Now define a linear operator T : H(A) ? H(H) by
X w([h, a])
p
f (a).
(T f )(h) =
q(h)p(a)
a
Then its adjoint T ? : H(H) ? H(A) is given by
X w([h, a])
p
(T ? f )(a) =
f (h).
q(h)p(a)
h
(5)
(6)
These two operators T and T ? were also implicitly suggested by [8] for developing a new
Web page ranking algorithm. Further define the operator SA : H(A) ? H(A) by composing T and T ? , i.e.
SA = T ? T,
(7)
and the operator ?A : H(A) ? H(A) by
?A = I ? S A ,
(8)
where I denotes the identity operator. Then we can show the following (See Appendix A
for the proof):
Proposition 1. ?A (f ) = hf, ?A f i.
Comparing with the combinatorial Laplace operator defined on undirected graphs [5], we
can think of the operator ?A as a Laplacian but defined on the authority set of directed
graphs. Note that Proposition 1 also shows that the Laplacian ?A is positive semi-definite.
In fact, we can further show that the eigenvalues of the operator SA are scattered in [0, 1],
and accordingly the eigenvalues of the Laplacian ?A fall into [0, 1].
Similarly, if two distinct vertices u and v co-link vertex a in the authority set A as shown in
right panel of Figure 2, then u and v are also thought to be related. The co-linkage strength
between u and v induced by a can be measured by
ca ([u, v]) =
w([u, a])w([v, a])
.
p(a)
(9)
and the smoothness of function f on the hub set H can be measured by:
2
1 XX
f (u)
f (v)
?H (f ) =
ca ([u, v]) p
?p
.
2 u,v a
q(u)
q(v)
(10)
As before, one can define the operators SH = T T ? and ?H = I ? SH leading to the
corresponding statement:
Proposition 2. ?H (f ) = hf, ?H f i.
Convexly combining together the two smoothness functionals (4) and (10), we obtain a
smoothness measure of function f defined on the whole vertex set V :
?? (f ) = ??A (f ) + (1 ? ?)?H (f ), 0 ? ? ? 1,
(11)
where the parameter ? weighs the relative importance between ?A (f ) and ?H (f ). Extend
the operator T to H(V ) by defining (T f )(v) = 0 if v is only in the authority set A and
not in the hub set H. Similarly extend T ? by defining (T ? f )(v) = 0 if v is only in the
hub set H and not in the authority set A. Then, if the remaining operators are extended
correspondingly, one can define the operator S? : H(V ) ? H(V ) by
S? = ?SA + (1 ? ?)SH ,
(12)
and the Laplacian on directed graphs ?? : H(V ) ? H(V ) by
?? = I ? S ? .
(13)
Clearly, ?? = ??A + (1 ? ?)?H . By Proposition 1 and 2, it is easy to see that:
Proposition 3. ?? (f ) = hf, ?? f i.
2.4
Regularization
Define a function y in H(V ) in which y(v) = 1 or ?1 if vertex v is labeled as positive
or negative, and 0, if it is not labeled. The classification problem can be regarded as the
problem of finding a function f , which reproduces the target function y to a sufficient
degree of accuracy while being smooth in a sense quantified by the above smoothness
functional. A formalization of this idea leads to the following optimization problem:
o
n
?
(14)
f ? = argmin ?? (f ) + kf ? yk2 .
2
f ?H(V )
The final classification of vertex v is obtained as sign f ? (v). The first term in the bracket
is called the smoothness term or regularizer, which measures the smoothness of function
f, and the second term is called the fitting term, which measures its closeness to the given
function y. The trade-off between these two competitive terms is captured by a positive
parameter ?. Successively smoother solutions f ? can be obtained by decreasing ? ? 0.
Theorem 4. The solution f ? of the optimization problem (14) satisfies
?? f ? + ?(f ? ? y) = 0.
Proof. By Proposition 3, we have
??? (f )
.
?f v
Differentiating the cost function in the bracket of (14) with respect to function f completes
the proof.
(?? f )(v) =
Corollary 5. The solution f ? of the optimization problem (14) is
f ? = (1 ? ?)(I ? ?S? )?1 y, where ? = 1/(1 + ?).
It is worth noting that the closed form solution presented by Corollary 5 shares the same
appearance as the algorithm proposed by [17], which operates on undirected graphs.
3
Experiments
We considered the Web page categorization task on the WebKB dataset [6]. We only
addressed a subset which contains the pages from the four universities: Cornell, Texas,
Washington, Wisconsin. We removed pages without incoming or outgoing links, resulting in 858, 825, 1195 and 1238 pages respectively, for a total of 4116. These pages were
manually classified into the following seven categories: student, faculty, staff, department,
course, project and other. We investigated two different classification tasks. The first is
used to illustrate the significance of connectivity information in classification, whereas the
second one stresses the importance of preserving the directionality of edges. We may assign a weight to each hyperlink according to the textual content of web pages or the anchor
text contained in hyperlinks. However, here we are only interested in how much we can
obtain from link structure only and hence adopt the canonical weight function defined in
Section 2.1.
We first study an extreme classification problem: predicting which university the pages
belong to from very few labeled training examples. Since pages within a university are
well-linked, and cross links between different universities are rare, we can imagine that
few training labels are enough to exactly classify pages based on link information only. For
each of the universities, we in turn viewed corresponding pages as positive examples and
the pages from the remaining universities as negative examples. We have randomly draw
two pages as the training examples under the constraint that there is at least one labeled
instance for each class. Parameters were set to ? = 0.50, and ? = 0.95. (In fact, in
this experiment the tuning parameters have almost no influence on the result.) Since the
Web graph is not connected, some small isolated subgraphs possibly do not contain labeled
instances. The values of our classifying function on the pages contained in these subgraphs
will be zeros and we simply think of these pages as negative examples. This is consistent
with the search engine ranking techniques [2, 13]. We compare our method with SVMs
using a kernel matrix K constructed as K = W T W [11], where W denotes the adjacency
matrix of the web graph and W T denotes the transpose of W . The test errors averaged over
100 training sample sets for both our method and SVMs are summarized into the following
table:
our method
SVMs
Cornel
0.03 (? 0.00)
0.42 (? 0.03)
Texas
0.02 (? 0.01)
0.39 (? 0.03)
Washington
0.01 (? 0.00)
0.40 (? 0.02)
Wisconsin
0.02 (? 0.00)
0.43 (? 0.02)
However, to be fair, we should state that the kernel matrix that we used in the SVM may not
be the best possible kernel matrix for this task ? this is an ongoing research issue which is
not the topic of the present paper.
The other investigated task is to discriminate the student pages in a university from the
non-student pages in the same university. As a baseline we have applied our regularization
method on the undirected graph [17] obtained by treating links as undirected or bidirectional, i.e., the affinity matrix is defined to be W T + W . We use the AUC scores to
measure the performances of the algorithms. The experimental results in Figure 3(a)-3(d)
clearly demonstrate that taking the directionality of edges into account can yield substantial accuracy gains. In addition, we also studied the influence of different choices for the
parameters ? and ?; we used the Cornell Web for that purpose and sampled 10 labeled
training pages. Figure 3(e) show that relatively small values of ? are more suitable. We
think that is because the subgraphs in each university are quite small, limiting the information conveyed in the graph structure. The influence of ? is shown in Figure 3(f). The
performance curve shows that large values for ? are preferable. This confirms the conjecture that co-link structure among authority pages is much more important than within the
hub set.
0.8
0.8
0.7
0.7
0.7
0.6
0.6
0.6
directed (?=1, ?=0.10)
undirected (?=0.10)
2
4
6
8
10
12
14
# labeled points
16
18
directed (?=1, ?=0.10)
undirected (?=0.10)
0.4
20
2
4
6
(a) Cornell
8
10
12
14
16
# labeled points
18
2
0.8
0.84
0.75
0.82
0.8
0.5
0.78
0.4
0.76
directed (?=1, ?=0.10)
undirected (?=0.10)
4
6
8
10
12
14
# labeled points
(d) Wisconsin
16
18
20
8
10
12
14
# labeled points
16
18
20
0.7
0.65
0.6
0.55
0.5
0.74
0.1
6
0.85
0.86
AUC
0.6
AUC
AUC
0.7
4
(c) Washington
0.88
2
directed (?=1, ?=0.10)
undirected (?=0.10)
0.3
20
(b) Texas
0.8
0.3
0.5
0.4
0.5
0.5
0.4
AUC
0.9
0.8
AUC
AUC
0.9
0.2
0.3
0.4
0.5
0.6
0.7
parameter values
(e) ? (Cornell)
0.8
0.9
0.45
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
parameter values
1
(f) ? (Cornell)
Figure 3: Classification on the WebKB dataset. Figure (a)-(d) depict the AUC scores of
the directed and undirected regularization methods on the classification problem student
vs. non-student in each university. Figure (e)-(f) illustrate the influences of the different
choices of the parameters ? and ?.
4
Conclusions
We proposed a general regularization framework on directed graphs, which has been validated on a real-word Web data set. The remaining problem is how to choose the suitable
parameters contained in this approach. In addition, it is worth noticing that this framework
can be applied without any essential changes to bipartite graphs, e.g. to graphs describing
customers? purchase behavior in market basket analysis. Moreover, in the absence of labeled instances, this framework can be utilized in an unsupervised setting as a (spectral)
clustering method for directed or bipartite graphs. Due to lack of space, we have not been
able to give a thorough discussion of these topics.
Acknowledgments We would like to thank David Gondek for his help on this work.
References
[1] M. Belkin, I. Matveeva, and P. Niyogi. Regularization and regression on large graphs. In COLT,
2004.
[2] S. Brin and L. Page. The anatomy of a large scale hypertextual web search engine. In Proc. 7th
Intl. WWW Conf., 1998.
[3] A. Broder, R. Kumar, F. Maghoul, P. Raghavan, S. Rajagopalan, R. Stata, A. Tomkins, and
J. Wiener. Graph structure in the Web. In Proc. 9th Intl. WWW Conf., 2000.
[4] S. Chakrabarti, B. Dom, and P. Indyk. Enhanced hypertext categorization using hyperlinks. In
Proc. ACM SIGMOD Conf., 1998.
[5] F. Chung. Spectral Graph Theory. Number 92 in Regional Conference Series in Mathematics.
American Mathematical Society, 1997.
[6] M. Craven, D. DiPasquo, D. Freitag, A. McCallum, T. Mitchell, K. Nigam, and S. Slattery.
Learning to extract symbolic knowledge from the World Wide Web. In Proc. 15th National
Conf. on Artificial Intelligence, 1998.
[7] J. Dean and M. Henzinger. Finding related Web pages in the World Wide Web. In Proc. 8th
Intl. WWW Conf., 1999.
[8] C. Ding, X. He, P. Husbands, H. Zha, and H. D. Simon. PageRank, HITS and a unified framework for link analysis. In Proc. 25th ACM SIGIR Conf., 2001.
[9] G. Flake, S. Lawrence, C. L. Giles, and F. Coetzee. Self-organization and identification of Web
communities. IEEE Computer, 35(3):66?71, 2002.
[10] C. Lee Giles, K. Bollacker, and S. Lawrence. CiteSeer: An automatic citation indexing system.
In Proc. 3rd ACM Conf. on Digital Libraries, 1998.
[11] T. Joachims, N. Cristianini, and J. Shawe-Taylor. Composite kernels for hypertext categorisation. In ICML, 2001.
[12] T. Joachims. Transductive learning via spectral graph partitioning. In ICML, 2003.
[13] J. Kleinberg. Authoritative sources in a hyperlinked environment.
46(5):604?632, 1999.
Journal of the ACM,
[14] R. Lempel and S. Moran. SALSA: the stochastic approach for link-structure analysis. ACM
Transactions on Information Systems, 19(2):131?160, 2001.
[15] A. Smola and R. Kondor. Kernels and regularization on graphs. In Learning Theory and Kernel
Machines. Springer-Verlag, Berlin-Heidelberg, 2003.
[16] V. N. Vapnik. Statistical learning theory. Wiley, NY, 1998.
[17] D. Zhou, O. Bousquet, T. N. Lal, J. Weston, and B. Sch o? lkopf. Learning with local and global
consistency. In NIPS, 2003.
[18] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using Gaussian fields and
harmonic functions. In ICML, 2003.
A
Proof of Proposition 1
Expand the right site of Equ. (4):
?A (f )
=
=
f (u)f (v)
f 2 (u)
?p
p(u)
p(u)p(v)
u,v h
2
X XX
f (u)f (v)
f (u) X X
?
ch ([u, v]) p
.
ch ([u, v])
p(u)
p(u)p(v)
u,v
u
v
XX
ch ([u, v])
(15)
h
h
By substituting Equ. (3), the first term in the above equality can be rewritten as
X X X w([h, u])w([h, v]) f 2 (u)
q(h)
p(u)
u
v
h
X X w([h, u]) 2
X 2
=
f (u) =
f (u).
p(u)
u
u
(16)
h
In addition, the second term in Equ. (15) can be transformed into
X X w([h, u])w([h, v]) f (u)f (v)
p
q(h)
p(u)p(v)
u,v h
XX
w([h, u]) w([h, v])
p
=
f (u) p
f (v).
q(h)p(u) q(h)p(v)
u,v h
Substituting Equ. (16) and (17) into (15), we have
X 2
XX
w([h, u]) w([h, v])
p
?A (f ) =
f (u) ?
f (u) p
f (v).
q(h)p(u) q(h)p(v)
u
u,v h
This completes the proof.
(17)
(18)
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|
1,892 | 2,719 | On Semi-Supervised Classification
Balaji Krishnapuram, David Williams, Ya Xue, Alex Hartemink, Lawrence Carin
Duke University, USA
M?ario A. T. Figueiredo
Instituto de Telecomunicac?o? es, Instituto Superior T?ecnico, Portugal
Abstract
A graph-based prior is proposed for parametric semi-supervised classification. The prior utilizes both labelled and unlabelled data; it also integrates features from multiple views of a given sample (e.g., multiple
sensors), thus implementing a Bayesian form of co-training. An EM
algorithm for training the classifier automatically adjusts the tradeoff between the contributions of: (a) the labelled data; (b) the unlabelled data;
and (c) the co-training information. Active label query selection is performed using a mutual information based criterion that explicitly uses the
unlabelled data and the co-training information. Encouraging results are
presented on public benchmarks and on measured data from single and
multiple sensors.
1
Introduction
In many pattern classification problems, the acquisition of labelled training data is costly
and/or time consuming, whereas unlabelled samples can be obtained easily. Semisupervised algorithms that learn from both labelled and unlabelled samples have been the
focus of much research in the last few years; a comprehensive review up to 2001 can be
found in [13], while more recent references include [1, 2, 6, 7, 16?18].
Most recent semi-supervised learning algorithms work by formulating the assumption that
?nearby? points, and points in the same structure (e.g., cluster), should have similar labels
[6,7,16]. This can be seen as a form of regularization, pushing the class boundaries toward
regions of low data density. This regularization is often implemented by associating the
vertices of a graph to all the (labelled and unlabelled) samples, and then formulating the
problem on the vertices of the graph [6, 16?18].
While current graph-based algorithms are inherently transductive ? i.e., they cannot be
used directly to classify samples not present when training ? our classifier is parametric and the learned classifier can be used directly on new samples. Furthermore, our algorithm is trained discriminatively by maximizing a concave objective function; thus we
avoid thorny local maxima issues that plague many earlier methods.
Unlike existing methods, our algorithm automatically learns the relative importance of the
labelled and unlabelled data. When multiple views of the same sample are provided (e.g.
features from different sensors), we develop a new Bayesian form of co-training [4]. In
addition, we also show how to exploit the unlabelled data and the redundant views of the
sample (from co-training) in order to improve active label query selection [15].
The paper is organized as follows. Sec. 2 briefly reviews multinomial logistic regression.
Sec. 3 describes the priors for semi-supervised learning and co-training. The EM algorithm
derived to learn the classifiers is presented in Sec. 4. Active label selection is discussed in
Sec. 5. Experimental results are shown in Sec. 6, followed by conclusions in Sec. 7.
2
Multinomial Logistic Regression
In an m-class supervised learning problem, one is given a labelled training set DL =
{(x1 , y 1 ), . . . , (xL , y L )}, where xi ? Rd is a feature vector and y i the corresponding
(1)
(m)
class label. In ?1-of-m? encoding, y i = [yi , . . . , yi ] is a binary vector, such that
(c)
(j)
yi = 1 and yi = 0, for j 6= c, indicates that sample i belongs to class c. In multinomial
logistic regression [5], the posterior class probabilities are modelled as
Pm
log P (y (c) = 1|x) = xT w(c) ? log k=1 exp(xT w(k) ), for c = 1, . . . , m, (1)
Pm
where w(c) ? Rd is the class-c weight vector. Notice that since c=1 P (y (c) = 1|x) = 1,
one of the weight vectors is redundant; we arbitrarily choose to set w(m) = 0, and consider
the (d (m?1))-dimensional vector w = [(w(1) )T , ..., (w(m?1) )T ]T . Estimation of w may
be achieved by maximizing the log-likelihood (with Y ? {y 1 , ..., y L }) [5]
Pm
PL Pm (c) T (c)
T (j)
y
x
w
?
log
`(w) ? log P (Y|w) = i=1
exp(x
w
)
. (2)
i
i
c=1 i
j=1
In the presence of a prior p(w), we seek a maximum a posteriori (MAP) estimate,
b = arg maxw {`(w) + log p(w)}. Actually, if the training data is separable, `(w) is
w
unbounded, and a prior is crucial.
Although we focus on linear classifiers, we may see the d-dimensional feature vectors x as
having resulted from some deterministic, maybe nonlinear, transformation of an input raw
feature vector r; e.g., in a kernel classifier, xi = [1, K(r i , r 1 ), ..., K(r i , r L )] (d = L + 1).
3
3.1
Graph-Based Data-Dependent Priors
Graph Laplacians and Regularization for Semi-Supervised Learning
Consider a scalar function f = [f1 , ..., f|V | ]T , defined on the set V = {1, 2, ..., |V |} of
vertices of an undirected graph (V, E). Each edge of the graph, joining vertices i and j, is
given a weight kij = kji ? 0, and we collect all the weights in a |V | ? |V | matrix K. A
natural way to measure how much f varies across the graph is by the quantity
XX
kij (fi ? fj )2 = 2 f T ? f ,
(3)
i
j
P
P
where ? = diag{ j k1j , ..., j k|V |j }?K is the so-called graph Laplacian [2]. Notice
that kij ? 0 (for all i, j) guarantees that ? is positive semi-definite and also that ? has (at
least) one null eigenvalue (1T?1 = 0, where 1 has all elements equal to one).
In semi-supervised learning, in addition to DL , we are given U unlabelled samples DU =
{xL+1 , . . . , xL+U }. To use (3) for semi-supervised learning, the usual choice is to assign
one vertex of the graph to each sample in X = [x1 , . . . , xL+U ]T (thus |V | = L + U ),
and to let kij represent some (non-negative) measure of ?similarity? between xi and xj . A
Gaussian random field (GRF) is defined on the vertices of V (with inverse variance ?)
p(f ) ? exp{?? f T? f /2},
in which configurations that vary more (according to (3)) are less probable. Most graphbased approaches estimate the values of f , given the labels, using p(f ) (or some modification thereof) as a prior. Accordingly, they work in a strictly transductive manner.
3.2
Non-Transductive Semi-Supervised Learning
We first consider two-class problems (m = 2, thus w ? Rd ). In contrast to previous uses of
graph-based priors, we define f as the real function f (defined over the entire observation
space) evaluated at the graph nodes. Specifically, f is defined as a linear function of x,
and at the graph node i, fi ? f (xi ) = wT xi . Then, f = [f1 , ..., f|V | ]T = Xw, and p(f )
induces a Gaussian prior on w, with precision matrix A = XT ?X,
p(w) ? exp{?(?/2) wT XT? Xw} = exp{?(?/2) wT Aw}.
(4)
Notice that since ? is singular, A may also be singular, and the corresponding prior may
therefore be improper. This is no problem for MAP estimation of w because (as is well
known) the normalization factor of the prior plays no role in this estimate. If we include
extra regularization, by adding a non-negative diagonal matrix to A, the prior becomes
p(w) ? exp ?(1/2) wT (?0 A + ?) w ,
(5)
where we may choose ? = diag{?1 , ..., ?d }, ? = ?1 I, or even ? = 0.
For m > 2, we define (m?1) identical independent priors, one for each w(c) , c = 1, ..., m.
The joint prior on w = [(w(1) )T , ..., (w(m?1) )T ]T is then
m?1
Y
1
1
(c)
p(w|?) ?
exp{? (w(c) )T ?0 A + ?(c) w(c) } = exp{? wT ?(?)w}, (6)
2
2
c=1
(c)
(c)
(c)
where ? is a vector containing all the ?i parameters, ?(c) = diag{?1 , ..., ?d }, and
(1)
(m?1)
?(?) = diag{?0 , ..., ?0
} ? A + block-diag{?(1) , ..., ?(m?1) }.
(7)
Finally, since all the ??s are inverses of variances, the conjugate priors are Gamma [3]:
(c)
(c)
(c)
(c)
p(?0 |?0 , ?0 ) = Ga(?0 |?0 , ?0 ), and p(?i |?1 , ?1 ) = Ga(?i |?1 , ?1 ), for c =
1, ..., m ? 1 and i = 1, ..., d. Usually, ?0 , ?0 , ?1 , and ?1 are given small values indicating
diffuse priors. In the zero limit, we obtain scale-invariant (improper) Jeffreys hyper-priors.
Summarizing, our model for semi-supervised learning includes the log-likelihood (2), a
prior (6), and Gamma hyper-priors. In Section 4, we present a simple and computationally
efficient expectation-maximization (EM) algorithm for obtaining the MAP estimate of w.
3.3
Exploiting Features from Multiple Sensors: The Co-Training Prior
In some applications several sensors are available, each providing a different set of features.
For simplicity, we assume two sensors s ? {1, 2}, but everything discussed here is easily
(s)
extended to any number of sensors. Denote the features from sensor s, for sample i, as xi ,
and Ss as the set of sample indices for which we have features from sensor s (S1 ?S2 =
{1, ..., L + U }). Let O = S1 ? S2 be the indices for which both sensors are available, and
OU = O ? {L + 1, ..., L + U } the unlabelled subset of O.
By using the samples in S1 and S2 as two independent training sets, we may obtain two sepb 1 and w
b 2 ). However, we can coordinate the information from
arate classifiers (denoted w
b1
both sensors by using an idea known as co-training [4]: on the OU samples, classifiers w
2
b should agree as much as possible. Notice that, in a logistic regression framework,
and w
the disagreement between the two classifiers on the OU samples can be measured by
P
1 T (1)
2 T (2) 2
T
(8)
i?OU [(w ) xi ? (w ) xi ] = ? C ?,
P
where ? = [(w1 )T (w2 )T ]T and C = i?OU [(x1i )T (?x2i )T ]T [(x1i )T (?x2i )T ]. This
suggests the ?co-training prior? (where ?co is an inverse variance):
p(w1 , w2 ) = p(?) ? exp ?(?co /2) ? TC ? .
(9)
This Gaussian prior can be combined with two smoothness Gaussian priors on w1 and w2
(obtained as described in Section 3.2); this leads to a prior which is still Gaussian,
p(w1 , w2 ) = p(?) ? exp ?(1/2) ? T ?co C + block-diag{?1 , ?2 } ? ,
(10)
where ?1 and ?2 are the two graph-based precision matrices (see (7)) for w1 and w2 .
We can again adopt a Gamma hyper-prior for ?co . Under this prior, and with a logistic
regression likelihood as above, estimates of w1 and w2 can easily be found using minor
modifications to the EM algorithm described in Section 4. Computationally, this is only
slightly more expensive than separately training the two classifiers.
4
Learning Via EM
b we use the EM algorithm, with ? as missing data, which
To find the MAP estimate w,
is equivalent to integrating out ? from the full posterior before maximization [8]. For
simplicity, we will only describe the single sensor case (no co-training).
E-step: We compute the expected value of the complete log-posterior, given Y and the
b Q(w|w)
b ? E[log p(w, ?|Y)|w].
b Since
current parameter estimate w:
log p(w, ?|Y) = log p(Y|w) ? (1/2)wT ?(?)w + K,
(11)
(where K collects all terms independent of w) is linear w.r.t. all the ? parameters (see (6)
and (7)), we just have to plug their conditional expectations into (11):
b = log p(Y|w) ? (1/2)wT E[?(?)|w]
b w = `(w) ? (1/2)wT ?(w)
b w. (12)
Q(w|w)
We consider several different choices for the structure of the ? matrix. The necessary
expectations have well-known closed forms, due to the use of conjugate Gamma hyper(c)
priors [3]. For example, if the ?0 are m ? 1 free non-negative parameters, we have
(c)
(c)
b = (2 ?0 + d) [2 ?0 + (w
b (c) )T Aw
b (c) ]?1 .
?0 ? E[?0 |w]
(c)
for c = 1, ..., m ? 1. For ?0 = ?0 , we still have a simple closed-form expres(c)
b and the same is true for the ?i parameters, for i > 0. Finally,
sion for E[?0 |w],
b ? E[?(?)|w]
b results from replacing the ??s in (7) by the corresponding conditional
?(w)
expectations.
b the M-step reduces to a logistic regression problem with a
M-step: Given matrix ?(w),
quadratic regularizer, i.e., maximizing (12). To this end, we adopt the bound optimization
approach (see details in [5, 11]). Let B be a positive definite matrix such that ?B bounds
below (in the matrix sense) the Hessian of `(w), which is negative definite, and g(w) is
b
the gradient of `(w). Then, we have the following lower bound on Q(w|w):
T
T
T
b ? l(w)
b + (w ? w)
b g(w)
b ? [(w ? w)
b B(w ? w)
b + w ?(w)w]/2.
b
Q(w|w)
b new = (B + ?(w))
b ?1 (Bw
b + g(w)),
b is guaranteed
The maximizer of this lower bound, w
new
b |w)
b ? Q(w|
b w),
b and we thus obtain a monotonic gento increase the Q-function, Q(w
eralized EM algorithm [5, 11]. This (maybe costly) matrix inversion can be avoided by a
sequential approach where we only maximize w.r.t. one element of w at a time, preserving
the monotonicity of the procedure. The sequential algorithm visits one particular element
of w, say wu , and updates its estimate by maximizing the bound derived above, while
keeping all other variables fixed at their previous values. This leads to
b ? (?(w)
b w)
b u ] [(B + ?(w))
b uu ]?1 ,
w
bunew = w
bu + [gu (w)
(13)
and
=w
bv , for v 6= u. The total time required by a full sweep for all u = 1, ..., d is
O(md(L + d)); this may be much better than the O((dm)3 ) of the matrix inversion.
w
bvnew
5
Active Label Selection
If we are allowed to obtain the label for one of the unlabelled samples, the following question arises: which sample, if labelled, would provide the most information?
b provided by EM. Our approach uses a Laplace approximaConsider the MAP estimate w
b H?1 ), where H is the posterior precision matrix,
tion of the posterior p(w|Y) ' N (w|w,
i.e., the Hessian of minus the log-posterior H = ?2 (? log p(w|Y)). This approximation is
b
known to be accurate for logistic regression under a Gaussian prior [14]. By treating ?(w)
(the expectation of ?(?)) as deterministic, we obtain an evidence-type approximation [14]
P
T
T
b
b + L
H = ?2 [? log(p(Y|w)p(w|?(w)))]
= ?(w)
i=1 (diag{pi } ? pi pi ) ? xi xi ,
where pi is the (m ? 1)-dimensional vector computed from (1), the c-th element of which
indicates the probability that sample xi belongs to class c.
Now let x? ? DU be an unlabelled sample and y ? its label. Assume that the MAP estib remains unchanged after including y ? . In Sec. 7 we will discuss the merits and
mate w
shortcomings of this assumption, which is only strictly valid when L ? ?. Accepting it
implies that after labeling x? , and regardless of y ? , the posterior precision changes to
(14)
H0 = H + (diag{p? } ? p? pT? ) ? x? xT? .
Since the entropy of a Gaussian with precision H is (?1/2) log |H| (up to an additive
constant), the mutual information (MI) between y ? and w (i.e., the expected decrease in
entropy of w when y ? is observed) is I(w; y ? ) = (1/2) log {|H0 |/|H|}. Our criterion
is then: the best sample to label is the one that maximizes I(w; y ? ). Further insight into
I(w; y ? ) can be obtained in the binary case (where p is a scalar); here, the matrix identity
|H + p? (1 ? p? )x? xT? | = |H|(1 + p? (1 ? p? )xT? H?1 x? ) yields
I(w; y ? ) = (1/2) log(1 + p? (1 ? p? )xT? H?1 x? ).
(15)
This MI is larger when p? ? 0.5, i.e., for samples with uncertain classifications. On the
other hand, with p? fixed, I(w; y ? ) grows with xT? H?1 x? , i.e., it is large for samples with
high variance of the corresponding class probability estimate. Summarizing, (15) favors
samples with uncertain class labels and high uncertainty in the class probability estimate.
6
Experimental Results
We begin by presenting two-dimensional synthetic examples to visually illustrate our semisupervised classifier. Fig. 1 shows the utility of using unlabelled data to improve the deci-
Figure 1: Synthetic two-dimensional examples. (a) Comparison of the supervised logistic
linear classifier (boundary shown as dashed line) learned only from the labelled data (shown
in color) with the proposed semi-supervised classifier (boundary shown as solid line) which
also uses the unlabelled samples (shown as dots). (b) A RBF kernel classifier obtained by
our algorithm, using two labelled samples (shaded circles) and many unlabelled samples.
Figure 2: (a)-(c) Accuracy (on UCI datasets) of the proposed method, the supervised SVM,
and the other semi-supervised classifiers mentioned in the text; a subset of samples is labelled and the others are treated as unlabelled samples. In (d), a separate holdout set is used
to evaluate the accuracy of our method versus the amount of labelled and unlabelled data.
sion boundary in linear and non-linear (kernel) classifiers (see figure caption for details).
Next we show results with linear classifiers on three UCI benchmark datasets. Results
with nonlinear kernels are similar, and therefore omitted to save space. We compare our
method against state-of-the-art semi-supervised classifiers: the GRF method of [18], the
SGT method of [10], and the transductive SVM (TSVM) of [9]. For reference, we also
present results for a standard SVM. To avoid unduly helping our method, we always use
a k=5 nearest neighbors graph, though our algorithm is not very sensitive to k. To avoid
disadvantaging other methods that do depend on such parameters, we use their best settings.
Since these adjustments cannot be made in practice, the difference between our algorithm
and the others is under-represented. Each point on the plots in Fig. 2(a)-(c) is an average
of 20 trials: we randomly select 20 labelled sets which are used by every method. All
remaining samples are used as unlabelled by the semi-supervised algorithms.
Figs. 2(a)-(c) are transductive, in the sense that the unlabelled and test data are the same.
Our logistic GRF is non-transductive: after being trained, it may be applied to classify new
data without re-training. In Fig. 2(d) we present non-transductive results for the Ionosphere
data. Training took place using labelled and unlabelled data, and testing was performed on
200 new unseen samples. The results suggest that semi-supervised classifiers are most
relevant when the labelled set is small relative to the unlabelled set (as is often the case).
Our final set of results address co-training (Sec. 3.3) and active learning (Sec. 5), applied to
airborne sensing data for the detection of surface and subsurface land mines. Two sensors
were used: (1) a 70-band hyper-spectral electro-optic (EOIR) sensor; (2) an X-band synthetic aperture radar (SAR). A simple (energy) ?prescreener? detected potential targets; for
each of these, two feature vectors were extracted, of sizes 420 and 9, for the EOIR and SAR
sensors, respectively. 123 samples have features from the EOIR sensor alone, 398 from the
Figure 3: (a) Land mine detection ROC curves of classifiers designed using only hyperspectral (EOIR) features, only SAR features, and both. (b) Number of landmines detected
during the active querying process (dotted lines), for active training and random selection
(for the latter the bars reflect one standard deviation about the mean). ROC curves (solid)
are for the learned classifier as applied to the remaining samples.
SAR sensor alone, and 316 from both. This data will be made available upon request.
We first consider supervised and semi-supervised classification. For the purely supervised
case, a sparseness prior is used (as in [14]). In both cases a linear classifier is employed. For
the data for which only one sensor is available, 20% of it is labelled (selected randomly).
For the data for which both sensors are available, 80% is labelled (again selected randomly).
The results presented in Fig. 3(a) show that, in general, the semi-supervised classifiers
outperform the corresponding supervised ones, and the classifier learned from both sensors
is markedly superior to classifiers learned from either sensor alone.
In a second illustration, we use the active-learning algorithm (Sec. 5) to only acquire the
100 most informative labels. For comparison, we also show average results over 100 independent realizations for random label query selection (error bars indicate one standard
deviation). The results in Fig. 3(b) are plotted in two stages: first, mines and clutter are selected during the labeling process (dashed curves); then, the 100 labelled examples are used
to build the final semi-supervised classifier, for which the ROC curve is obtained using the
remaining unlabelled data (solid curves). Interestingly, the active-learning algorithm finds
almost half of the mines while querying for labels. Due to physical limitations of the sensors, the rate at which mines are detected drops precipitously after approximately 90 mines
are detected ? i.e., the remaining mines are poorly matched to the sensor physics.
7
7.1
Discussion
Principal Contributions
Semi-supervised vs. Transductive: Unlike most earlier methods, after the training stage our
algorithm can directly classify new samples without computationally expensive re-training.
Tradeoff between labelled and unlabelled data: Automatically addressing the inherent
tradeoff between their relative contributions, we have ensured that even a small amount
of labelled data does not get overlooked because of an abundance of unlabelled samples.
Bayesian co-training: Using the proposed prior, classifiers for all sensors are improved
using: (a) the label information provided on the other types of data, and (b) samples drawn
from the joint distribution of features from multiple sensors.
Active label acquisition: We explicitly account for the knowledge of the unlabelled data and
the co-training information while computing the well known mutual information criterion.
7.2
Quality of Assumptions and Empirically Observed Shortcomings
The assumption that the mode of the posterior distribution of the classifier remains unchanged after seeing an additional label is clearly not true at the beginning of the active
learning procedure. However, we have empirically found it a very good approximation after the active learning procedure has yielded as few as 15 labels. This assumption allows a
tremendous saving in the computational cost, since it helps us avoid repeated re-training of
classifiers in the active label acquisition process while evaluating candidate queries.
A disturbing fact that has been reported in the literature (e.g., in [12]) and that we have
confirmed (in unreported experiments) is that the error rate of the active query selection
increases slightly when the number of labelled samples grows beyond an optimal number.
We conjecture that this may be caused by keeping the hyper-prior parameters ?0 , ?0 , ?1 , ?1
fixed at the same value; in all of our experiments we have set them to 10?4 , corresponding
to an almost uninformative hyper-prior.
References
[1] M. Belkin, I. Matveeva, and P. Niyogi. Regularization and regression on large graphs. In Proc.
Computational Learning Theory ? COLT?04, Banff, Canada, 2004.
[2] M. Belkin and P. Niyogi. Using manifold structure for partially labelled classification. In NIPS
15, MIT Press, Cambridge, MA, 2003.
[3] J. Bernardo and A. Smith. Bayesian Theory. J. Wiley & Sons, Chichester, UK, 1994.
[4] A. Blum and T. Mitchell. Combining labelled and unlabelled data with co-training. In Proc.
Computational Learning Theory ? COLT?98, Madison, WI, 1998.
[5] D. B?ohning. Multinomial logistic regression algorithm. Annals Inst. Stat. Math., vol. 44,
pp. 197?200, 1992.
[6] O. Chapelle, J. Weston, and B. Sch?olkopf. Cluster kernels for semi-supervised learning. In
NIPS 15, MIT Press, Cambridge, MA, 2003.
[7] A. Corduneanu and T. Jaakkola. On Information regularization. In Proc. Uncertainty in Artificial Intelligence ? UAI?03, Acapulco, Mexico, 2003.
[8] M. Figueiredo. Adaptive sparseness using Jeffreys? prior. In NIPS 14, MIT Press, 2002.
[9] T. Joachims. Transductive inference for text classification using support vector machines. In
Int. Conf. Machine Learning ? ICML?99, 1999.
[10] T. Joachims. Transductive learning via spectral graph partitioning. In ICML?03, 2003.
[11] K. Lange, D. Hunter, and I. Yang. Optimization transfer using surrogate objective functions. J.
Computational and Graphical Statistics, vol. 9, pp. 1?59, 2000.
[12] G. Schohn and D. Cohn. Less is more: Active learning with support vector machines. Intern.
Conf. on Mach. Learn. ? ICML?00.
[13] M. Seeger. Learning with labelled and unlabelled data. Tech. Rep., Institute for Adaptive and
Neural Computation, University of Edinburgh, UK, 2001.
[14] M. Tipping. Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn.
Research, vol. 1, pp. 211?244, 2001.
[15] S. Tong and D. Koller. Support vector machine active learning with applications to text classification. In J. Mach. Learn. Research, vol. 2, pp. 45?66, 2001.
[16] D. Zhou, O. Bousquet, T. Lal, J. Weston, and B. Sch?olkopf. Semi-supervised learning by
maximizing smoothness. J. of Mach. Learn. Research, 2004 (submitted).
[17] X. Zhu, J. Lafferty and Z. Ghahramani. Combining active learning and semi-supervised learning
using Gaussian fields and harmonic functions. In ICML?03 Workshop on The Continuum from
Labelled to Unlabelled Data in Mach. Learning, 2003.
[18] X. Zhu, J. Lafferty and Z. Ghahramani. Semi-supervised learning: From Gaussian fields to
Gaussian processes. Tech. Rep. CMU-CS-03-175, School of CS, CMU, 2003.
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1,893 | 272 | 36
Bialek, Rieke, van Steveninck and Warland
Reading a Neural Code
William Bialek, Fred Rieke, R. R. de Ruyter van Steveninck 1 and
David Warland
Department of Physics, and
Department of Molecular and Cell Biology
University of California at Berkeley
Berkeley, California 94720
ABSTRACT
Traditional methods of studying neural coding characterize the encoding of known stimuli in average neural responses. Organisms
face nearly the opposite task - decoding short segments of a spike
train to extract information about an unknown, time-varying stimulus. Here we present strategies for characterizing the neural code
from the point of view of the organism, culminating in algorithms
for real-time stimulus reconstruction based on a single sample of
the spike train. These methods are applied to the design and analysis of experiments on an identified movement-sensitive neuron in
the fly visual system. As far as we know this is the first instance in
which a direct "reading" of the neural code has been accomplished.
1
Introduction
Sensory systems receive information at extremely high rates, and much of this information must be processed in real time. To understand real-time signal processing
in biological systems we must understand the representation of this information in
neural spike trains. \Ve ask several questions in particular:
? Does a single neuron signal only the occurrence of particular stimulus '"features," or can the spike train represent a continuous time-varying input?
1 Rijksuniversiteit
Groningen, Postbus 30.001,9700 RB Groningen The Netherlands
Reading a Neural Code
? How much information is carried by the spike train of a single neuron?
? Is the reliability of the encoded signal limited by noise at the sensory input
or by noise and inefficiencies in the subsequent layers of neural processing?
? Is the neural code robust to errors in spike timing, or do realistic levels of
synaptic noise place significant limits on information transmission?
? Do simple analog computations on the encoded signals correspond to simple
manipulations of the spike trains?
Although neural coding has been studied for more than fifty years, clear experimental answers to these questions have been elusive (Perkel & Bullock, 1968; de Ruyter
van Steveninck & Bialek, 1988). Here we present a new approach to the characterization of the neural code which provides explicit and sometimes surprising answers
to these questions when applied to an identified movement-sensitive neuron in the
fly visual system.
We approach the study of spiking neurons from the point of view of the organism,
which, based only on the spike train, must estimate properties of an unknown timevarying stimulus. Specifically we try to solve the problem of decoding the spike train
to recover the stimulus in real time. As far as we know our work is the first instance
in which it has been possible to "read" the neural code in this literal sense. Once
we can read the code, we can address the questions posed above. In this paper we
focus on the code reading algorithm, briefly summarizing the results which follow.
2
Theoretical background
The traditional approach to the study of neural coding characterizes the encoding
process: For an arbitrary stimulus waveform s( r), what can we predict about the
spike train? This process is completely specified by the conditional probability
distribution P[{tdls(r)] of the spike arrival times {til conditional on the stimulus
s( r). In practice one cannot characterize this distribution in its entirety; most
experiments result in only the lowest moment - the firing rate as function of time
given the stimulus.
The classic experiments of Adrian and others established that, for static stimuli, the
resulting constant firing rate provides a measure of stimulus strength. This concept
is easily extended to any stimulus waveform which is characterized by constant
parameters, such as a single frequency or fixed amplitude sine wave. l\'luch of the
effort in studying the encoding of sensory signals in the nervous system thus reduces
to probing the relation between these stimulus parameters and the resulting firing
rate. Generalizations to time-varying firing rates, especially in response to periodic
signals, have also been explored.
The firing rate is a continuous function of time which measures the probability
per unit time that the cell will generate a spike. The rate is thus by definition
an average quantity; it is not a property of a single spike train. The rate can
be estimated, in principle, by averaging over a large ensemble of redundant cells,
37
38
Bialek, Rieke, van Steveninck and Warland
or by averaging responses of a single cell over repeated presentations of the same
stimulus. This latter approach dominates the experimental study of spiking neurons.
Measurements of firing rate rely on some form of redundancy - either the spatial
redundancy of identical cells or the temporal redundancy of repeated stimuli. It is
simply not clear that such redundancy exists in real sensory systems under natural
stimulus conditions. In the absence of redundancy a characterization of neural
responses in terms of firing rate is oflittle relevance to the signal processing problems
faced by the organism. To say that "information is coded in firing rates" is of no
use unless one can explain how the organism could estimate these firing rates by
observing the spike trains of its own neurons.
We believe that none of the existing approaches2 to neural coding addresses the basic
problem of real-time signal processing with neural spike trains: The organism must
extract information about continuously varying stimulus waveforms using only the
discrete sequences of spikes. Real-time signal processing with neural spike trains
thus involves some sort of interpolation between the spikes that allows the organism
to estimate a continuous function of time.
The most basic problem of real-time signal processing is to decode the spike train and
recover an estimate of the stimulus waveform itself. Clearly if we can accomplish
this task then we can begin to understand how spike trains can be manipulated to
perform more complex computations; we can also address the quantitative issues
outlined in the Introduction. Because of the need to interpolate between spikes,
such decoding is not a simple matter of inverting the conventional stimulus-response
(rate) relations. In fact it is not obvious a priori that true decoding is even possible.
One approach to the decoding problem is to construct models of the encoding
process, and proceed analytically to develop algorithms for decoding within the
context of the model (Bialek & Zee, 1990). Using the results of this approach we
can predict that linear filtering will, under some conditions, be an effective decoding
algorithm, and we can determine the form of the filter itself. In this paper we have
a more limited goal, namely to see if the class of decoding algorithms identified
by Bialek and Zee is applicable to a real neuron. To this end we will treat the
structure of the decoding filter as unknown, and find the "best" filter under given
experimental conditions.
We imagine building a set of (generally non-linear) filters {Fn} which operate on
the spike train to produce an estimate of the stimulus. If the spikes arrive at times
{td, we write our estimate of the signal as a generalized convolution,
(1)
i
i,j
2Higher moments of the conditional probability P[{t i}ls(r)], such as the inter-spike interval
distribution (Perkel & Bullock, 1968) are also average properties, not properties of single spike
trains, and hence may not be relevant to real-time signal processing. White-noise methods (Marmarelis & Marmarelis, 1978) result in models which predict the time-varying firing rate in response
to arbitrary input waveforms and thus suffer the same limitations as other rate-based approaches.
Reading a Neural Code
How good are the reconstructions? We separate systematic and random errors by
introducing a frequency dependent gain g(w) such that (ls(w)l) g(w) (lsut(w)l).
The resulting gain is approximately unity through a reasonable bandwidth. Further,
the distribution of deviations between the stimulus and reconstruction is approxirr.ately Gaussian. The absence of systematic errors suggests that non-linearities
in the reconstruction filter are unlikely to help. Indeed, the contribution from the
st.. ~ond order term in Eq. (1) to the reconstructions is negligible.
=
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Figure 2: Spectral density of displacement noise from our reconstruction (upper
curve). By multiplying the displacement noise level by a bandwidth, we obtain the
square of the angular resolution of HI for a step displacement. For a reasonable
bandwidth the resolution is much less than the photoreceptor spacing, 1.350 "hyperacuity." Also shown is the limit to the resolution of small displacements set
by noise in the photoreceptor array (lower curve).
We identify the noise at frequency w as the difference between the stimulus and
the normalized reconstruction, n(w) = s(w) - g(w )Sed (w). \Ve then compute the
dpectral density (noise power per unit bandwidth) of the displacement noise (Fig 2).
The noise level achieved in HI is astonishing; with a one second integration time an
observer of the spike train in HI could judge the amplitude of a low frequency dither
to 0.01? - more than one hundred times less than the photoreceptor spacing! If the
fiY'f neural circuitry is noiseless, the fundamental limits to displacement resolution
39
40
Bialek, Rieke, van Steveninck and Warland
stimulus,
J
dw
Fl(T) =
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S(
T).
2. Minimize X 2 with respect to purely causal functions. This may be done analytically, or numerically by expanding F 1 ( T) in a complete set of functions
which vanish at negative times, then minimizing X2 by varying the coefficients
of the expansion. In this method we must explicitly introduce a delay time
which measures the lag between the true stimulus and our reconstruction.
We use the filter generated from the first method (which is the best possible linear
filter) to check the filter generated by the second method. Fig. 1 illustrates reconstructions using these two methods. The filters themselves are also shown in the
figure; we see that both methods give essentially the same answer.
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Figure 1: First order reconstruction se,,( T) using method 1 (solid line). The
st.imulus is shown here as a dotted line for comparison. The reconstruction shown
is for a segment of the spike train which was not used in the filter calculations. The
spike train is shown at the bottom of the figure, where the negative spikes are from
the "other eye" (cf. footnote 3). Both stimulus and reconstruction are smoothed
with a 5 msec half-wid th Gaussian filter. The filters calculating using both methods
are shown on the right.
Reading a Neural Code
We define the optimal filter to be that which minimizes X 2 = f dtls(t) - sest(t)1 2 ,
where s(t) is the true stimulus, and the integration is over the duration of the
experiment.
To insure that the filters we calculate allow real-time decoding, we require that the
filters be causal, for example FI(T < 0) = O. But the occurrence of a spike at
t' conveys information about the stimulus at a time t < t', so we must delay our
estimate of the stimulus by some time Tdelay > t' - t. In general we gain more
information by increasing the delay, so we face a tradeoff: Longer waiting times
allow us to gain more information but introduce longer reaction times to important
stimuli. This tradeoff is exactly the tradeoff faced by the organism in reacting to
external stimuli based on noisy and incomplete information.
3
Movelnent detection in the blowfly visual system
We apply our methods in experiments on a single wide field, movement-sensitive
neuron (H 1) in the visual system of the blowfly Calliphora erythrocephela. Flies
and other insects exhibit visually guided flight; during chasing behavior course
corrections can occur on time scales as short as 30 msec (Land & Collett, 1974). H1
appears to be an obligatory link in this control loop, encoding wide field horizontal
movements (Hausen, 1984). Given that the maximum firing rate in H1 is 100200 Hz, behavioral decisions must be based on the information carried by just a few
spikes from this neuron. Further, the horizontal motion detection system consists
of only a handful of neurons, so the fly has no opportunity to compute average
responses (or firing rates).
In the experiments described here, the fly is looking at a rigidly moving random pattern (de Ruyter van Steveninck, 1986). The pattern is presented on an oscilloscope,
and moved horizontally every 500 J-lsec in discrete steps chosen from an ensemble
which approximates Gaussian white noise. This time scale is short enough that we
can consider the resulting stimulus waveform s(t) to be the instantaneous angular
velocity. We record the spike arrival times {til extracellularly from the H1 neuron. 3
4
First order reconstructions
To reconstruct the stimulus waveform requires that we find the filter FI which
minimizes X2. We do this in two different ways:
1. Disregard the constraint that the filter be causal. In this case we can write
an explicit formula for the optimal filter in terms of the spike trains and the
3 There is one further caveat to the experiment. The firing rate in HI is increased for back-tofront motion and is decreased for front-to-back motion; the dynamic range is much greater in the
excitatory direction. The fiy, however, achieves high sensitivity in both directions by combining
information from both eyes. Because front-to-back motion in one eye corresponds to back-to-front
motion in the other eye, we can simulate the two eye case while recording from only one HI cell
by using an antisymmetric stimulus waveform. We combine the information coded in the spike
trains corresponding to the two "polarities" of the stimulus to obtain the information available
from both HI neurons.
41
42
Bialek, Rieke, van Steveninck and Warland
are set by noise in the photoreceptor array. We have calculated these limits in the
case where the displacements are small, which is true in our experiments at high
frequencies. In comparing these limits with the results in HI it is crucial that the
photoreceptor signal and noise characteristics (de Ruyter van Steveninck, 1986) are
measured under the same conditions as the HI experiments analyzed here. It is
clear from Fig. 2 that HI approaches the theoretical limit to its performance. We
emphasize that the noise spectrum in Fig. 2 is not a hypothetical measure of neural
performance. Rather it is the real noise level achieved in our reconstructions. As
far as we know this is the first instance in which the equivalent spectral noise level
of a spiking neuron has been measured.
To explore the tradeoff between the quality and delay of the reconstruction we
measure the cross-correlation of the smoothed stimulus with the reconstructions
calculated using method 2 above for delays of 10-70 msec. For a delay of 10 msec
the reconstruction carries essentially no information; this is expected since a delay of 10 msec is close to the intrinsic delay for phototransduction. As the delay
is increased the reconstructions improve, and this improvement saturates for delays greater than 40 msec, close to the behavioral reaction time of 30 msec - the
structure of the code is well matched to the behavioral decision task facing the
organism.
5
Conclusions
Learning how to read the neural code has allowed us to quantify the information
carried in the spike train independent of assumptions regarding the structure of
the code. In addition, our analysis gives some hopefully more general insights into
neural coding and computation:
1. The continuously varying movement signal encoded in the firing of H1 can be reconstructed by an astonishingly simple linear filter. If neurons summed their inputs
and marked the crossing of thresholds (as in many popular models), such reconstructions would be impossible; the threshold crossings are massively ambiguous
indicators of the signal waveform. We have carried out similar studies on a standard model neuron (the FitzHugh-Nagumo model), and find results similar to those
in the HI experiments. From the model neuron studies it appears that the linear
representation of signals in spike trains is a general property of neurons, at least in
a limited regime of their dynamics. In the near future we hope to investigate this
statement in other sensory systems.
2. The reconstruction is dominated by a "window" of - 40 msec during which
at most a few spikes are fired. Because so few spikes are important, it does not
make sense to talk about the "firing rate" - estimating the rate vs. time from
observations of the spike train is at least as hard as estimating the stimulus itself!
3. The quality of the reconstructions can be improved by accepting longer delays, but
this improvement saturates at - 30 - 40 msec, in good agreement with behavioral
decision times.
Reading a Neural Code
4. Having decoded the neural signal we obtain a meaningful estimate of the noise
level in the system and the information content of the code. H1 accomplishes a realtime version of hyper acuity, corresponding to a noise level near the limits imposed
by the quality of the sensory input. It appears that this system is close to achieving
optimal real-time signal processing.
5. From measurements of the fault tolerance of the code we can place requirements
on the noise levels in neural circuits using the information coded in H1. One of the
standard objections to discussions of "spike timing" as a mechanism of coding is
that there are no biologically plausible mechanisms which can make precise measurements of spike arrival times. We have tested the required timing precision by
introducing timing errors into the spike train and characterizing the resulting reconstructions. Remarkably the code is "fault tolerant," the reconstructions degrading
only slightly when we add timing errors of several msec.
Finally, we wish to emphasize our own surprise that it is so simple to recover time
dependent signals from neural spike trains. The filters we have constructed are not
very complicated, and they are linear. These results suggest that the representation
of time-dependent sensory data in the nervous system is much simpler than we migh t
have expected. We suggest that, correspondingly, simpler models of sensory signal
processing may be appropriate.
6
Acknowledglnents
We thank W. J. Bruno, M. Crair, L. Kruglyak, J. P. Miller, W. G. Owen, A. Zee,
and G. Zweig for many helpful discussions. This work was supported by the National Science Foundation through a Presidential Young Investigator Award to WB,
supplemented by funds from Cray Research and Sun Microsystems, and through a
Graduate Fellowship to FR. DW was supported in part by the System'S and Integrative Biology Training Program of the National Institutes of Health. Initial work was
supported by the Netherlands Organization for Pure Scientific Research (ZWO).
7
References
W. Bialek and A. Zee. J. Stat. Phys., in press, 1990.
K. Hausen. In M. Ali, editor, Photoreception and Vision in Invertebrates. Plenum
Press, New York and London, 1984.
M. Land and T. Collett. J. Compo Physiol., 89:331, 1974.
P. Marmarelis and V. Marmarelis. Analysis of Physiological Systems. The White
Noise Approach. Plenum Press, New York, 1978.
D. Perkel and T. Bullock. Neurosciences. Res. Prog. Bull., 6:221, 1968.
R. R. de Ruyter van Steveninck and W. Bialek. Proc. R. Soc. Lond. B, 234:379,
1988.
R. R. de Ruyter van Steveninck. Real-time Performance of a Movement-sensitive
Neuron in the Blowfly Visual System. Rijksuniversiteit Groningen, Groningen,
Netherlands, 1986.
43
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1,894 | 2,720 | Trait selection for assessing beef meat quality
using non-linear SVM
J.J. del Coz, G. F. Bay?on, J. D??ez,
O. Luaces, A. Bahamonde
Artificial Intelligence Center
University of Oviedo at Gij?on
[email protected]
?
Carlos Sanudo
Facultad de Veterinaria
University of Zaragoza
[email protected]
Abstract
In this paper we show that it is possible to model sensory impressions
of consumers about beef meat. This is not a straightforward task; the
reason is that when we are aiming to induce a function that maps object
descriptions into ratings, we must consider that consumers? ratings are
just a way to express their preferences about the products presented in
the same testing session. Therefore, we had to use a special purpose
SVM polynomial kernel. The training data set used collects the ratings of
panels of experts and consumers; the meat was provided by 103 bovines
of 7 Spanish breeds with different carcass weights and aging periods.
Additionally, to gain insight into consumer preferences, we used feature
subset selection tools. The result is that aging is the most important trait
for improving consumers? appreciation of beef meat.
1
Introduction
The quality of beef meat is appreciated through sensory impressions, and therefore its
assessment is very subjective. However, it is known that there are objective traits very important for the final properties of beef meat; this includes the breed and feeding of animals,
weight of carcasses, and aging of meat after slaughter. To discover the influence of these
and other attributes, we have applied Machine Learning tools to the results of an experience reported in [8]. In the experience, 103 bovines of 7 Spanish breeds were slaughtered
to obtain two kinds of carcasses, light and standard [5]; the meat was prepared with 3 aging
periods, 1, 7, and 21 days. Finally, the meat was consumed by a group, called panel, of 11
experts, and assessed by a panel of untrained consumers.
The conceptual framework used for the study reported in this paper was the analysis of
sensory data. In general, this kind of analysis is used for food industries in order to adapt
their productive processes to improve the acceptability of their specialties. They need to
discover the relationship between descriptions of their products and consumers? sensory
degree of satisfaction. An excellent survey of the use of sensory data analysis in the food
industry can be found in [15, 2]; for a Machine Learning perspective, see [3, 9, 6].
The role played by each panel, experts and consumers, is very clear. So, the experts? panel
is made up of a usually small group of trained people who rate several traits of products such
as fibrosis, flavor, odor, etc. . . The most essential property of expert panelists, in addition
to their discriminatory capacity, is their own coherence, but not necessarily the uniformity
of the group. Experts? panel can be viewed as a bundle of sophisticated sensors whose
ratings are used to describe each product, in addition to other objective traits. On the other
hand, the group of untrained consumers (C) are asked to rate their degree of acceptance or
satisfaction about the tested products on a given scale. Usually, this panel is organized in
a set of testing sessions, where a group of potential consumers assess some instances from
a sample E of the tested product. Frequently, each consumer only participates in a small
number (sometimes only one) of testing sessions, usually in the same day.
In general, the success of sensory analysis relies on the capability to identify, with a precise
description, a kind of product that should be reproducible as many times as we need to be
tested for as many consumers as possible. Therefore, the study of beef meat sensory quality
is very difficult. The main reason is that there are important individual differences in each
piece of meat, and the repeatability of tests can be only partially ensured. Notice that from
each animal there are only a limited amount of similar pieces of meat, and thus we can only
provide pieces of a given breed, weight, and aging period. Additionally, it is worthy noting
that the cost of acquisition of this kind of sensory data is very high.
The paper is organized as follows: in the next section we present an approach to deal with
testing sessions explicitly. The overall idea is to look for a preference or ranking function
able to reproduce the implicit ordering of products given by consumers instead of trying
to predict the exact value of consumer ratings; such function must return higher values
to those products with higher ratings. In Section 3 we show how some state of the art
FSS methods designed for SVM (Support Vector Machines) with non-linear kernels can be
adapted to preference learning. Finally, at the end of the paper, we return to the data set
of beef meat to show how it is possible to explain consumer behavior, and to interpret the
relevance of meat traits in this context.
2
Learning from sensory data
A straightforward approach to handle sensory data can be based on regression, where sensory descriptions of each object x ? E are endowed with the degree of satisfaction r(x)
for each consumer (or the average of a group of consumers). However, this approach does
not faithfully captures people?s preferences [7, 6]: consumers? ratings actually express a
relative ordering, so there is a kind of batch effect that often biases their ratings. Thus, a
product could obtain a higher (lower) rating depending on if it is assessed together with
worse (better) products. Therefore, information about batches tested by consumers in each
rating session is a very important issue. On the other hand, more traditional approaches,
such as testing some statistical hypotheses [16, 15, 2] require all available food products in
sample E to be assessed by the set of consumers C, a requisite very difficult to fulfill.
In this paper we use an approach to sensory data analysis based on learning consumers?
preferences, see [11, 14, 1], where training examples are represented by preference judgments, i.e. pairs of vectors (v, u) indicating that, for someone, object v is preferable to
object u. We will show that this approach can induce more useful knowledge than other
approaches, like regression based methods. The main reason is due to the fact that preference judgments sets can represent more relevant information to discover consumers? preferences.
2.1
A formal framework to learn consumer preferences
In order to learn our preference problems, we will try to find a real ranking function f that
maximizes the probability of having f (v) > f (u) whenever v is preferable to u [11, 14, 1].
Our input data is made up of a set of ratings (ri (x) : x ? Ei ) for i ? C. To avoid the batch
effect, we will create a preference judgment set P J = {v j > uj : j = 1, . . . , n} suitable
for our needs just considering all pairs (v, u) such that objects v and u were presented in
the same session to a given consumer i, and ri (v) > ri (u).
Thus, following the approach introduced in [11], we look for a function F : Rd ? Rd ? R
such that
?x, y ? Rd , F (x, y) > 0 ? F (x, 0) > F (y, 0).
(1)
Then, the ranking function f : Rd ? R can be simply defined by f (x) = F (x, 0).
As we have already constructed a set of preference judgments P J, we can specify F by
means of the restrictions
F (v j , uj ) > 0 and F (uj , v j ) < 0, ?j = 1, . . . , n.
(2)
Therefore, we have a binary classification problem that can be solved using SVM. We
follow the same steps as Herbrich et al. in [11], and define a kernel K as follows
K(x1 , x2 , x3 , x4 ) = k(x1 , x3 ) ? k(x1 , x4 ) ? k(x2 , x3 ) + k(x2 , x4 )
(3)
where k(x, y) = h?(x), ?(y)i is a kernel function defined as the inner product of two
objects represented in the feature space by their ? images. In the experiments reported in
Section 4, we will employ a polynomial kernel, defining k(x, y) = (hx, yi + c)g , with
c = 1 and g = 2. Notice that, finally we can express the ranking function f in a non-linear
form:
n
X
(1)
(2)
?i zi (k(xi , x) ? k(xi , x))
(4)
f (x) =
i=1
3
Feature subset selection methods in a non-linear environment
When dealing with sensory data, it is important to know not only which classifier is the
best and how accurate it is, but also which features are relevant for the tastes of consumers.
Producers can focus on these features to improve the quality of the final product. Additionaly, reductions on the number of features often lead to a cheaper data acquisition labour,
making these systems suitable for industrial operation [9].
There are many feature subset selection methods applied to SVM classification. If our
goal is to find a linear separator, RFE (Recursive Feature Elimination) [10] will be a good
choice. It is a ranking method that returns an ordering of the features. RFE iteratively
removes the less useful feature. This process is repeated until there are no more features.
Thus, we obtain an ordered sequence of features.
Following the main idea of RFE, we have used two methods capable of ordering features in
non-linear scenarios. We must also point that, in this case, preference learning data sets are
formed by pairs of objects (v, u), and each object in the pair has the same set of features.
Thus, we must modify the ranking methods so they can deal with the duplicated features.
3.1
Ranking features for non-linear preference learning
Method 1.- This method orders the list of features according to their influence in the
variations of the weights. It is a gradient-like method, introduced in [17], and found to be
a generalization of RFE to the non-linear case. It removes in each iteration the feature that
minimizes the ranking value
X
?K(s ? xk , s ? xj )
R1 (i) = |?i kwk2 | =
?k ?j zk zj
(5)
, i = 1, . . . , d
?si
k,j
where s is a scaling factor used to simplify the computation of partial derivatives. Due to
the fact that we are working on a preference learning problem, we need 4 copies of the
scaling factor. In this formula, for a polynomial kernel k(x, y) = (hx, yi + c)g and a
vector s such that ?i, si = 1 we have that
?k(s ? x, s ? y)
= 2g(xi yi )(c + hx, yi)g?1 .
(6)
?si
Method 2.- This method, introduced in [4], works in an iterative way; removing each
time the feature which minimizes the loss of predictive performance. When using this
method for preference learning with the kernel of equation (3) the ranking criterion can be
expressed as
?
?
X
X
(1),i
(2),i
(1),i
(2),i
R2 (i) = ?
zk ?
?j zj K(xj , xj , xk , xk )?
(7)
k
j
i
where x denotes a vector describing an object where the value for the i-th feature was
replaced by its mean value. Notice that a higher value of R2 (i), that is, a higher accuracy
on the training set when replacing feature i-th, means a lower relevance of that feature.
Therefore, we will remove the feature yielding the highest ranking value, as opposite to the
ranking method described previously.
3.2
Model selection on an ordered sequence of feature subsets
Once we have an ordering of the features, we must select the subset Fi which maximizes
the generalization performance of the system. The most common choice for a model selection method is cross-validation (CV), but its efficiency and high variance [1] lead us to
try another kind of methods. We have used ADJ (ADJusted distance estimate)[19]. This
is a metric-based method that selects one from a nested sequence of complexity-increasing
models. We construct a sequence of subsets F1 ? F2 ? . . . ? Fd , where Fi represents the
subset containing only the i most relevant features. Then we can create a nested sequence
of models fi , each one of these induced by SVM from the corresponding Fi .
The key idea is the definition of a metric on the space of hypothesis. Thus, given two
different hypothesis f and g, their distance is calculated as the expected disagreement in
their predictions. Given that these distances can only be approximated, ADJ establish a
? t), an adjusted distance estimate between any hypothesis f and
method to compute d(g,
the true target classification function t. Therefore, the selected hypothesis is
? l , t).
fk = arg min d(f
(8)
fl
? is computed by means of the expected disagreement in the
The estimation of distance, d,
predictions in a couple of sets: the training set T , and a set U of unlabeled examples, that is,
a set of cases sampled from the same distribution of T but for which the pretended correct
output is not given. The ADJ estimation is given by
dU (fk , fl )
def
(9)
ADJ(fl , t) = dT (fl , t) ? max
k<l dT (fk , fl )
where, for a given subset of examples S, dS (f, g) is the expected disagreement of hypothesis f and g in S. To avoid the impossibility of using the previous equation when there are
zero disagreements in T for two hypotheses we use the Laplace correction to the probability
estimation; thus,
!
X
1
def
dS (f, g) =
1+
1f (x)6=g(x)
(10)
|S| + 2
x?S
In general, it is not straightforward to obtain a set of unlabeled examples. However, for
learning preferences, we can easily build the set of unlabeled examples from a set of preference judgments formed by pairs of real objects randomly selected from the original preference judgment pairs.
3.3
Summarizing the data: dealing with redundancy
As we have previously pointed out, sensory data include ratings of experts for different
characteristics of the products, as well as physical and chemical features directly measured
on them. It is not infrequent to find out that some of these features are highly correlated;
some experts may have similar opinions about a certain feature, and similarities among
several chemical and physical features may be possible as well. In order to take advantage of these peculiarity, we have developed a simple redundancy filter, RF. It is meant
to be applied before any feature subset selection method, allowing us to discover intrinsic
redundancies in the data and to reduce the number of features used.
RF is an iterative process where each step gives rise to a new description of the original data
set. The two most similar features are replaced by a new one whose values are computed as
the average of them. Considering two given features ai and aj as (column) vectors whose
dimension is the number of examples in the data set, the similarity can be estimated by
means of their cosine, that is,
hai , aj i
similarity(ai , aj ) =
(11)
kai k ? kaj k
Applying this method we obtain a sequence of different descriptions of the original data set,
each one with one feature less than the previous. To select an adequate description in terms
of prediction accuracy, we use again ADJ. The selected description can be considered a
summarized version of the original data set to be processed by the feature subset selection
methods previously described.
4
Experimental results
In this section we show the experimental results obtained when we applied the tools described in previous sections to the beef meat data base [8]. Each piece of meat was described by 147 attributes: weight of the animal, breed (7 boolean attributes), aging, 6 physical attributes describing its texture and 12 sensory traits rated by 11 different experts (132
ratings). The meat comes from 103 bovines of 7 Spanish breeds (from 13 to 16 animals
of each breed); animals were slaughtered in order to obtain 54 light and 49 standard carcasses, uniformly distributed across breeds. In each rating session, 4 or 5 pieces of meat
were tested and a group of consumers were asked to rate only three different aspects: tenderness, flavor and acceptance. These three data sets have over 2420 preference judgments.
All the results shown in this section have been obtained by a 10-fold cross-validation.
4.1
Preference learning vs. regression
First, we performed a comparison between preference learning and regression methods.
We have experimented with a simple linear regression and with a well reputed regression
algorithm: Cubist, a commercial product from RuleQuest Research. To interpret regression
results we used the relative mean absolute deviation (Rmad), which is the quotient between
the mean absolute distance of the function learned by the regression method and the mean
absolute distance of the constant predictor that returns the mean value in all cases.
On the other hand, we can obtain some preference judgments from the ratings of the sessions comparing the rating of each product with the rest, one by one, and constructing the
Table 1: Regression and preference learning scores on beef meat data sets.
Regression
Preferences
Linear
Cubist Linear Cubist SVMl
SVMp
Error
Error
Error
Error
Rmad
Rmad
tenderness
flavor
acceptance
96.3%
99.3%
94.0%
97.8%
103.4%
97.2%
41.5%
43.8%
38.4%
43.1%
46.5%
40.2%
29.6%
32.7%
31.9%
19.4%
23.8%
22.1%
Avg.
96.5%
99.5%
41.2%
43.3%
31.4%
21.8%
corresponding pair. To learn from preference judgment data sets we used SVMlight [13]
with linear and polynomial kernels. In this case, the errors have a straightforward meaning as misclassifications; so in order to allow a fair comparison between regression and
preference learning approaches, we also tested regression models on preference judgments
test sets, calculating their misclassifications. The scores achieved on the three data sets
described previously, are shown in Table 1.
We can observe that regression methods are unable to learn any useful knowledge: their
relative mean absolute deviation (Rmad) is near 100% in all cases, that is, regression models usually perform equal than the constant predictor forecasting the mean value. From a
practical point of view, these results mean that raw consumers? ratings can not be used to
measure the overall sensory opinions. Even when these regression models are tested on
preference judgment sets the percentage of misclassifications is over 40%, clearly higher
than those obtained when using the preference learning approach. SVM-based methods can
reduce these errors down to an average near 30% with a linear kernel (SVMl ), and near 20%
if the kernel is a polynomial of degree 2 (SVMp ). This improvement shows that non-linear
kernels can explain consumers preferences better.
4.2
Feature selection
We used the FSS tools to gain insight into consumer preferences. For the sake of simplicity,
in what follows FSS1 and FSS2 will denote the feature subset selectors that use ranking
Method 1 and Method 2 respectively. The learner used in these experiments was SVMp
because it was the most accurate in previous tests (see Table 1). Given the size of the
three data sets it is almost impractical to use FSS1 and FSS2 due to its computational cost,
unless a previous reduction in the number of features can be achieved; therefore, in both
cases we used RF as a previous filter. Additionally, to improve the overall speed, features
were removed in chunks of five. In all cases we used ADJ to choose among the subsets of
features. We can see in Table 2 that FSS1 , FSS2 , and RF considerably reduce the number
of features at the expense of accuracy: it slightly decreases when we use the RF filter with
respect to the accuracy obtained on the original data set by SVMp ; it also decreases when
using FSS1 and FSS2 after RF.
The most useful result obtained from feature selection is the ranking list of traits. We
concentrate our study in tenderness and acceptance categories because they are more interesting from the point of view of beef meat producers. So, in acceptance data set, the three
most useful traits are: aging, breed and fibrosis. Some research works in the beef meat field
corroborate the importance of these characteristics [18, 12]. Specially, aging is crucial to
improve consumer acceptance. On the other hand, fibrosis is closely related with tenderness: the less fibrosis, the more tenderness. Usually many consumers identify tenderness
with acceptance, in the sense that a higher tenderness yields to a higher acceptance; then,
fibrosis and acceptance are inversely related. With respect to the breed trait, two of the
Table 2: Percentage of misclassifications and the number of selected features when polynomial kernel (SVMp ) and FSS methods are used. The three original data sets have 147
features.
RF
RF+FSS1
RF+FSS2
Error #Att. Error #Att. Error #Att.
tenderness
flavor
acceptance
20.0%
25.0%
24.7%
50.0
65.0
39.5
21.8%
26.5%
24.8%
27.0
33.5
30.0
21.3%
26.1%
25.3%
37.5
29.0
26.7
Avg.
23.2%
51.5
24.4%
30.2
24.2%
31.1
seven possible values, retinta and asturiana breeds [8], have more influence than others in
the preference function that describes consumer acceptance; for example, meat from retinta
animals seems to be the most appreciated by consumers.
In tenderness data set, the most useful attributes are: aging, fibrosis, residue and odor
intensity. Aging and fibrosis appear again, showing the relationship between acceptance
and tenderness. Residue depends on fibrosis, so it is not a surprise to find it in the list.
Apparently, odor intensity is not so related to tenderness, but it is closely related to aging.
5
Conclusions
We have shown that an approach based on nonlinear SVMs can be useful to model consumer preferences about beef meat. The polynomial model obtained and the FSS tools used
allow us to emphasize the relevance of meat traits previously described in the literature of
the field. However, the novelty of our approach is that we can algorithmically deduce the
expressions of relevance.
The sensory data base available probably tries to cover too many aspects of beef meat
affecting to its sensory quality. Therefore, it is not possible to obtain more detailed conclusions from the polynomial model. Nevertheless, the experience reported in this paper can
be very useful for the design of future experiments involving specific traits of beef meat
quality.
Acknowledgements
The research reported in this paper has been supported in part under Spanish Ministerio de
Ciencia y Tecnolog??a (MCyT) and Feder grant TIC2001-3579.
References
[1] A. Bahamonde, G. F. Bay?on, J. D??ez, J. R. Quevedo, O. Luaces, J. J. del Coz, J. Alonso, and
F. Goyache. Feature subset selection for learning preferences: A case study. In Proceedings
of the International Conference on Machine Learning, Banff, Alberta (Canada), July 2004.
Morgan Kaufmann.
[2] D. Buck, I. Wakeling, K. Greenhoff, and A. Hasted. Predicting paired preferences from sensory
data. Food quality and preference, 12:481?487, 2001.
[3] D. Corney. Designing food with bayesian belief networks. In Proceedings of the International
Conference on Adaptive Computing in engineering Design and Manufacture, pages 83?94,
2002.
[4] S. Degroeve, B. De Baets, Y. Van de Peer, and P. Rouz?e. Feature subset selection for splice site
prediction. Bioinformatics, 18(2):75?83, 2002.
[5] J. D??ez, A. Bahamonde, J. Alonso, S. L?opez, J. del Coz, J. Quevedo, J. Ranilla, O. Luaces,
?
I. Alvarez,
L. Royo, and F. Goyache. Artificial intelligence techniques point out differences
in classification performance between light and standard bovine carcasses. Meat Science,
64(3):249?258, 2003.
[6] J. D??ez, G. F. Bay?on, J. R. Quevedo, J. J. del Coz, O. Luaces, J. Alonso, and A. Bahamonde.
Discovering relevancies in very difficult regression problems: applications to sensory data analysis. In Proceedings of the European Conference on Artificial Intelligence (ECAI ?04), Valencia, Spain, 2004.
[7] S. Dumais, K. Bharat, T. Joachims, and A. Weigend, editors. Workshop on implicit measures
of user interests and preferences. In ACM SIGIR Conference, Toronto, Canada, 2003.
[8] M. Gil, X. Serra, M. Gispert, M. Oliver, C. Sa?nudo, B. Panea, J. Olleta, M. Campo, M. Oliv?an,
K. Osoro, M. Garcia-Cachan, R. Cruz-Sagredo, M. Izquierdo, M. Espejo, M. Mart??n, and
J. Piedrafita. The effect of breed-production systems on the myosin heavy chain 1, the biochemical characteristics and the colour variables of longissimus thoracis from seven spanish
beef cattle breeds. Meat Science, 58:181?188, 2001.
[9] F. Goyache, A. Bahamonde, J. Alonso, S. L?opez, del Coz J.J., J. Quevedo, J. Ranilla, O. Luaces,
I. Alvarez, L. Royo, and J. D??ez. The usefulness of artificial intelligence techniques to assess
subjective quality of products in the food industry. Trends in Food Science and Technology,
12(10):370?381, 2001.
[10] I. Guyon, J. Weston, S. Barnhill, and V. Vapnik. Gene selection for cancer classification using
support vector machines. Machine Learning, 46(1?3):389?422, 2002.
[11] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. In A. Smola, P. Bartlett, B. Scholkopf, and D. Schuurmans, editors, Advances in Large
Margin Classifiers, pages 115?132. MIT Press, Cambridge, MA, 2000.
[12] L. Jeremiah and L. Gibson. The effects of postmortem product handling and aging time on beef
palatability. Food Research International, 36:929?941, 2003.
[13] T. Joachims. Making large-scale support vector machines learning practical. In A. S.
B. Sch?olkopf, C. Burges, editor, Advances in Kernel Methods: Support Vector Machines. MIT
Press, Cambridge, MA, 1998.
[14] T. Joachims. Optimizing search engines using clickthrough data. In Proceedings of the ACM
Conference on Knowledge Discovery and Data Mining (KDD), 2002.
[15] J. Murray, C. Delahunty, and I. Baxter. Descriptive sensory analysis: past, present and future.
Food Research International, 36:461?471, 2001.
[16] T. N?s and E. Risvik. Multivariate analysis of data in sensory science. Elsevier, 1996.
[17] A. Rakotomamonjy. Variable selection using SVM-based criteria. Journal of Machine Learning
Research, 3:1357?1370, 2003.
[18] C. Sa?nudo, E. S. Macie, J. L. Olleta, M. Villaroel, B. Panea, and P. Albert??. The effects of
slaughter weight, breed type and ageing time on beef meat quality using two different texture
devices. Meat Science, 66:925?932, 2004.
[19] D. Schuurmans and F. Southey. Metric-based methods for adaptive model selection and regularization. Machine Learning, 48:51?84, 2002.
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1,895 | 2,721 | Multiple Alignment of Continuous Time Series
Jennifer Listgarten? , Radford M. Neal? , Sam T. Roweis? and Andrew Emili?
Department of Computer Science, ? Banting and Best Department of Medical Research
and Program in Proteomics and Bioinformatics
University of Toronto, Toronto, Ontario, M5S 3G4
{jenn,radford,roweis}@cs.toronto.edu, [email protected]
?
Abstract
Multiple realizations of continuous-valued time series from a stochastic
process often contain systematic variations in rate and amplitude. To
leverage the information contained in such noisy replicate sets, we need
to align them in an appropriate way (for example, to allow the data to be
properly combined by adaptive averaging). We present the Continuous
Profile Model (CPM), a generative model in which each observed time
series is a non-uniformly subsampled version of a single latent trace, to
which local rescaling and additive noise are applied. After unsupervised
training, the learned trace represents a canonical, high resolution fusion
of all the replicates. As well, an alignment in time and scale of each
observation to this trace can be found by inference in the model. We
apply CPM to successfully align speech signals from multiple speakers
and sets of Liquid Chromatography-Mass Spectrometry proteomic data.
1
A Profile Model for Continuous Data
When observing multiple time series generated by a noisy, stochastic process, large systematic sources of variability are often present. For example, within a set of nominally
replicate time series, the time axes can be variously shifted, compressed and expanded,
in complex, non-linear ways. Additionally, in some circumstances, the scale of the measured data can vary systematically from one replicate to the next, and even within a given
replicate.
We propose a Continuous Profile Model (CPM) for simultaneously analyzing a set of such
time series. In this model, each time series is generated as a noisy transformation of a
single latent trace. The latent trace is an underlying, noiseless representation of the set
of replicated, observable time series. Output time series are generated from this model
by moving through a sequence of hidden states in a Markovian manner and emitting an
observable value at each step, as in an HMM. Each hidden state corresponds to a particular
location in the latent trace, and the emitted value from the state depends on the value of
the latent trace at that position. To account for changes in the amplitude of the signals
across and within replicates, the latent time states are augmented by a set of scale states,
which control how the emission signal will be scaled relative to the value of the latent trace.
During training, the latent trace is learned, as well as the transition probabilities controlling
the Markovian evolution of the scale and time states and the overall noise level of the
observed data. After training, the latent trace learned by the model represents a higher
resolution ?fusion? of the experimental replicates. Figure 1 illustrate the model in action.
Unaligned, Linear Warp Alignment and CPM Alignment
Amplitude
40
30
20
10
0
50
Amplitude
40
30
20
10
Amplitude
0
30
20
10
0
Time
a)
b)
Figure 1: a) Top: ten replicated speech energy signals as described in Section 4), Middle:
same signals, aligned using a linear warp with an offset, Bottom: aligned with CPM (the
learned latent trace is also shown in cyan). b) Speech waveforms corresponding to energy
signals in a), Top: unaligned originals, Bottom: aligned using CPM.
2
Defining the Continuous Profile Model (CPM)
The CPM is generative model for a set of K time series, ~xk = (xk1 , xk2 , ..., xkN k ). The
temporal sampling rate within each ~xk need not be uniform, nor must it be the same across
the different ~xk . Constraints on the variability of the sampling rate are discussed at the end
of this section. For notational convenience, we henceforth assume N k = N for all k, but
this is not a requirement of the model.
The CPM is set up as follows: We assume that there is a latent trace, ~z = (z1 , z2 , ..., zM ), a
canonical representation of the set of noisy input replicate time series. Any given observed
time series in the set is modeled as a non-uniformly subsampled version of the latent trace
to which local scale transformations have been applied. Ideally, M would be infinite, or at
least very large relative to N so that any experimental data could be mapped precisely to the
correct underlying trace point. Aside from the computational impracticalities this would
pose, great care to avoid overfitting would have to be taken. Thus in practice, we have used
M = (2 + )N (double the resolution, plus some slack on each end) in our experiments
and found this to be sufficient with < 0.2. Because the resolution of the latent trace is
higher than that of the observed time series, experimental time can be made effectively to
speed up or slow down by advancing along the latent trace in larger or smaller jumps.
The subsampling and local scaling used during the generation of each observed time series are determined by a sequence of hidden state variables. Let the state sequence for
observation k be ~? k . Each state in the state sequence maps to a time state/scale state pair:
?ik ? {?ik , ?ki }. Time states belong to the integer set (1..M ); scale states belong to an
ordered set (?1 ..?Q ). (In our experiments we have used Q=7, evenly spaced scales in
logarithmic space). States, ?ik , and observation values, xki , are related by the emission
probability distribution: A?ik (xki |~z) ? p(xki |?ik , ~z, ?, uk ) ? N (xki ; z?ik ?ki uk , ?), where ?
is the noise level of the observed data, N (a; b, c) denotes a Gaussian probability density
for a with mean b and standard deviation c. The uk are real-valued scale parameters, one
per observed time series, that correct for any overall scale difference between time series k
and the latent trace.
To fully specify our model we also need to define the state transition probabilities. We
define the transitions between time states and between scale states separately, so that
T?ki?1 ,?i ? p(?i |?i?1 ) = p(?i |?i?1 )pk (?i |?i?1 ). The constraint that time must move
forward, cannot stand still, and that it can jump ahead no more than J? time states is enforced. (In our experiments we used J? = 3.) As well, we only allow scale state transitions
between neighbouring scale states so that the local scale cannot jump arbitrarily. These
constraints keep the number of legal transitions to a tractable computational size and work
well in practice. Each observed time series has its own time transition probability distribution to account for experiment-specific patterns. Both the time and scale transition
probability distributions are given by multinomials:
?
?
dk1 , if a ? b = 1
?
?
?dk , if a ? b = 2
?
?
? 2
k
p (?i = a|?i?1 = b) = ...
?
?
?
?dkJ? , if a ? b = J?
?
?
?0,
otherwise
p(?i = a|?i?1
?
s0 , if D(a, b) = 0
?
?
?
s1 , if D(a, b) = 1
= b) =
?s1 , if D(a, b) = ?1
?
?
0, otherwise
where D(a, b) = 1 means that a is one scale state larger than b, and D(a, b) = ?1 means
that a is one scale state smaller than b, and D(a, b) = 0 means that a = b. The distributions
PJ ? k
are constrained by: i=1
di = 1 and 2s1 + s0 = 1.
J? determines the maximum allowable instantaneous speedup of one portion of a time
series relative to another portion, within the same series or across different series. However,
the length of time for which any series can move so rapidly is constrained by the length of
the latent trace; thus the maximum overall ratio in speeds achievable by the model between
any two entire time series is given by min(J? , M
N ).
After training, one may examine either the latent trace or the alignment of each observable
time series to the latent trace. Such alignments can be achieved by several methods, including use of the Viterbi algorithm to find the highest likelihood path through the hidden
states [1], or sampling from the posterior over hidden state sequences. We found Viterbi
alignments to work well in the experiments below; samples from the posterior looked quite
similar.
3
Training with the Expectation-Maximization (EM) Algorithm
As with HMMs, training with the EM algorithm (often referred to as Baum-Welch in the
context of HMMs [1]), is a natural choice. In our model the E-Step is computed exactly
using the Forward-Backward algorithm [1], which provides the posterior probability over
states for each time point of every observed time series, ?sk (i) ? p(?i = s|~x) and also
the pairwise state posteriors, ?s,t (i) ? p(?i?1 = s, ?i = t|~xk ). The algorithm is modified
only in that the emission probabilities depend on the latent trace as described in Section 2.
The M-Step consists of a series of analytical updates to the various parameters as detailed
below.
Given the latent trace (and the emission and state transition probabilities), the complete log
likelihood of K observed time series, ~xk , is given by Lp ? L + P. L is the likelihood term
arising in a (conditional) HMM model, and can be obtained from the Forward-Backward
algorithm. It is composed of the emission and state transition terms. P is the log prior
(or penalty term), regularizing various aspects of the model parameters as explained below.
These two terms are:
!
K
N
N
X
X
X
k
k
L?
log T?i?1 ,?i
log p(?1 ) +
log A?i (xi |~z) +
(1)
P ? ??
i=2
i=1
k=1
?
?1
X
(zj+1 ? zj )2 +
j=1
K
X
log D(dkv |{?vk }) + log D(sv |{?v0 }),
(2)
k=1
where p(?1 ) are priors over the initial states. The first term in Equation 2 is a smoothing
penalty on the latent trace, with ? controlling the amount of smoothing. ?vk and ?v0 are
Dirichlet hyperprior parameters for the time and scale state transition probability distributions respectively. These ensure that all non-zero transition probabilities remain non-zero.
For the time state transitions, v ? {1, J? } and ?vk corresponds to the pseudo-count data for
the parameters d1 , d2 . . . dJ? . For the scale state transitions, v ? {0, 1} and ?vk corresponds
to the pseudo-count data for the parameters s0 and s1 .
Letting S be the total number of possible states, that is, the number of elements in the
cross-product of possible time states and possible scale states, the expected complete log
likelihood is:
p
<L >? =P +
K X
S
X
k
?sk (1) log T0,s
+
k=1 s=1
... +
K X
S
X
K X
S X
N
X
?sk (i) log As (xki |~z) + . . .
k=1 s=1 i=1
S X
N
X
k
k
?s,s
0 (i) log Ts,s0
k=1 s=1 s0 =1 i=2
k
k
using the notation T0,s
? p(?1 = s), and where ?sk (i) and ?s,s
0 (i) are the posteriors over
states as defined above. Taking derivatives of this quantity with respect to each of the
parameters and finding the critical points provides us with the M-Step update equations. In
updating the latent trace ~z we obtain a system of M simultaneous equations, for j = 1..M :
K
X
? <Lp>?
=0=
?zj
X
N
X
?sk (i)?s uk
k=1 {s|?s =j} i=1
(xki ? zj uk ?s )
? ?(4zj ? 2zj?1 ? 2zj+1 )
?2
For the cases j = 1, N , the terms 2zj?1 and 2zj+1 , respectively, drop out. Considering
all such equations we obtain a system of M equations in M unknowns. Each equation
depends only linearly on three variables from the latent trace. Thus the solution is easily
obtained numerically by solving a tridiagonal linear system.
Analytic updates for ? 2 and uk are given by:
2
? =
PS
s=1
PN
i=1
?sk (i)(xki ? z?s uk ?s )2
,
N
k
u =
PS
PN k
k
s=1 z?s ?s
i=1 ?s (i)xi
PS
P
N
2
k
s=1 (z?s ?s )
i=1 ?s (i)
Lastly, updates for the scale and state transition probability distributions are given by:
dkv
= PJ ?
j=1
s v = P1
j=0
?vk +
?jk +
?j0 +
?j0 +
PS
s=1
PJ ?
j=1
PN k
{s0 |?s0 ??s =v}
i=2 ?s,s00 (i)
PS P
PN k
s=1
{s0 |?s0 ??s =j}
i=2 ?s,s00 (i)
P
PK P S
k=1
PK
k=1
s=1
PS
s=1
PN
k
i=2 ?s,s00 (i)
P
PN k
{s00 ?H(s,1),H(s,0)}
i=2 ?s,s00 (i)
P
{s00 ?H(s,v)}
where H(s, j) ? {s0 |s0 is exactly j scale states away from s}.
Note that we do
not normalize the Dirichlets, and omit the traditional minus one in the exponent:
QJ?
Q1
0
k
D(dkv |{?vk }) = v=1
(dkv )?v and D(sv |{?v0 }) = v=0 (sv )?v .
The M-Step updates uk , ?, and ~z are coupled. Thus we arbitrarily pick an order to update
them and as one is updated, its new values are used in the updates for the coupled parameter
updates that follow it. In our experiments we updated in the following order: ?, ~z, u k . The
other two parameters, dkv and sv , are completely decoupled.
4
Experiments with Laboratory and Speech Data
We have applied the CPM model to analyze several Liquid Chromatography - Mass Spectrometry (LC-MS) data sets from an experimental biology laboratory. Mass spectrometry
technology is currently being developed to advance the field of proteomics [2, 3]. A mass
spectrometer takes a sample as input, for example, human blood serum, and produces a
measure of the abundance of molecules that have particular mass/charge ratios. In proteomics the molecules in question are small protein fragments. From the pattern of abundance values one can hope to infer which proteins are present and in what quantity. For
protein mixtures that are very complex, such as blood serum, a sample preparation step
is used to physically separate parts of the sample on the basis of some property of the
molecules, for example, hydrophobicity. This separation spreads out the parts over time
so that at each unique time point a less complex mixture is fed into the mass spectrometer.
The result is a two-dimensional time series spectrum with mass/charge on one axis and
time of input to the mass spectrometer on the other. In our experiments we collapsed the
data at each time point to one dimension by summing together abundance values over all
mass/charge values. This one-dimensional data is referred to as the Total Ion Count (TIC).
We discuss alternatives to this in the last section. After alignment of the TICs, we assessed
the alignment of the LC-MS data by looking at both the TIC alignments, and also the corresponding two-dimensional alignments of the non-collapsed data, which is where the true
information lies.
The first data set was a set of 13 replicates, each using protein extracted from lysed E. coli
cells. Proteins were digested and subjected to capillary-scale LC-MS coupled on-line to an
ion trap mass spectrometer. First we trained the model with no smoothing (i.e., ? = 0) on
the 13 replicates. This provided nice alignments when viewed in both the TIC space and
the full two-dimensional space. Next we used leave-one-out cross-validation on six of the
replicates in order to choose a suitable value for ?. Because the uk and dkv are time series
specific, we ran a restricted EM on the hold-out case to learn these parameters, holding the
other parameters fixed at the values found from learning on the training set. Sixteen values
of ? over five orders of magnitude, and also zero, were used. Note that we did not include
the regularization likelihood term in the calculations of hold-out likelihood. One of the
non-zero values was found to be optimal (statistically significant at a p=0.05 level using a
paired sample t-test to compare it to no smoothing). Visually, there did not appear to be
a difference between no regularization and the optimal value of ?, in either the TIC space
or the full two-dimensional space. Figure 2 shows the alignments applied to the TICs and
also the two-dimensional data, using the optimal value of ?.
Unaligned and Aligned Time Series
8
x 10
Replicate 5
8
x 10
9
Latent Trace
Aligned Experimental Time Series
Original Time Series
8
8
7
6
6
Amplitude
Amplitude
10
4
2
5
4
3
8
0 x 10
Latent Space Amplitude
2
1
6
0
100
Residual
200
300
400
500
600
700
800
4
3
2
2
Time Jump From Previous State
1
Scale States
0
100
200
300
400
500
Time
600
700
800
200
400
Latent Time
a)
b)
c)
d)
600
800
Figure 2: Figure 2: a) Top: 13 Replicate pre-processed TICs as described in Section 4),
Bottom: same as top, but aligned with CPM (the learned latent trace is also shown). b) The
fifth TIC replicate aligned to the learned latent trace (inset shows the original, unaligned).
Below are three strips showing, from top-to-bottom, i) the error residual, ii) the number
of time states moved between every two states in the Viterbi alignment, and iii) the local
scaling applied at each point in the alignment. c) A portion of the two-dimensional LC-MS
data from replicates two (in red) and four (in green). d) Same as c), but after alignment (the
same one dimensional alignment was applied to every Mass/Charge value). Marker lines
labeled A to F show how time in c) was mapped to latent time using the Viterbi alignment.
We also trained our model on five different sets of LC-MS data, each consisting of human
blood serum. We used no smoothing and found the results visually similar in quality to the
first data set.
To ensure convergence to a good local optimum and to speed up training, we pre-processed
the LC-MS data set by coarsely aligning and scaling each time series as follows: We 1)
translated each time series so that the center of mass of each time series was aligned to the
median center of mass over all time series, 2) scaled the abundance values such that the
sum of abundance values in each time series was equal to the median sum of abundance
values over all time series.
We also used our model to align 10 speech signals, each an utterance of the same sentence
spoken by a different speaker. The short-time energy (using a 30ms Hanning window) was
computed every 8ms for each utterance and the resulting vectors were used as the input
to CPM for alignment. The smoothing parameter ? was set to zero. For comparison, we
also performed a linear warping of time with an offset. (i.e. each signal was translated so
as to start at the same time, and the length of each signal was stretched or compressed so
as to each occupy the same amount of time). Figure 1 shows the successful alignment of
the speech signals by CPM and also the (unsuccessful) linear warp. Audio for this example can be heard at http://www.cs.toronto.edu/?jenn/alignmentStudy,
which also contains some supplemental figures for the paper.
Initialization for EM training was performed as follows: ? was set to 15% of the difference
between the maximum and minimum values of the first time series. The latent trace was
initialized to be the first observed time series, with Gaussian, zero-mean noise added, with
standard deviation equal to ?. This was then upsampled by a factor of two by repeating
every value twice in a row. The additional slack at either end of the latent trace was set to
be the minimum value seen in the given time series. The uk were each set to one and the
multinomial scale and state transition probabilities were set to be uniform.
5
Related Algorithms and Models
Our proposed CPM has many similarities to Input/Output HMMs (IOHMMs), also called
Conditional HMMs [4]. IOHMMs extend standard HMMs [1] by conditioning the emission
and transition probabilities on an observed input sequence. Each component of the output
sequence corresponds to a particular component of the input. Training of an IOHMM
is supervised ? a mapping from an observed input sequence to output target sequence
is learned. Our CPM also requires input and thus is also a type of conditional HMM.
However, the input is unobserved (but crucially it is shared between all replicates) and
hence learning is unsupervised in the CPM model. One could also take the alternative view
that the CPM is simply an HMM with an extra set of parameters, the latent trace, that affect
the emission probabilities and which are learned by the model.
The CPM is similar in spirit to Profile HMMs which have been used with great success
for discrete, multiple sequence alignment, modeling of protein families and their conserved structures, gene finding [5], among others. Profile HMM are HMMs augmented
by constrained-transition ?Delete? and ?Insert? states, with the former emitting no observations. Multiple sequences are provided to the Profile HMM during training and a summary
of their shared statistical properties is contained in the resulting model. The development
of Profile HMMs has provided a robust, statistical framework for reasoning about sets of
related discrete sequence data. We put forth the CPM as a continuous data, conditional
analogue.
Many algorithms currently used for aligning continuous time series data are variations of
Dynamic Time Warping (DTW) [6], a dynamic programming based approach which originated in the speech recognition community as a robust distance measure between two time
series. DTW works on pairs of time series, aligning one time series to a specified reference
time series. DTW does not take in to account systematic variations in the amplitude of the
signal. Our CPM can be viewed as a rich and robust extension of DTW that can be applied
to many time series in parallel and which automatically uncovers the underlying template
of the data.
6
Discussion and Conclusion
We have introduced a generative model for sets of continuous, time series data. By training
this model one can leverage information contained in noisy, replicated experimental data,
and obtain a single, superior resolution ?fusion? of the data. We demonstrated successful
use of this model on real data, but note that it could be applied to a wide range of problems
involving time signals, for example, alignment of gene expression time profiles, alignment
of temporal physiological signals, alignment of motion capture data, to name but a few.
Certain assumptions of the model presented here may be violated under different experimental conditions. For example, the Gaussian emission probabilities treat errors in
large amplitudes in the same absolute terms as in smaller amplitudes, whereas in reality, it may be that the error scales with signal amplitude. Similarly, the penalty term
P? ?1
?? j=1 (zj+1 ? zj )2 does not scale with the amplitude; this might result in the model
arbitrarily preferring a lower amplitude latent trace. (However, in practice, we did not find
this to be a problem.)
One immediate and straight-forward extension to the model would be to allow the data at
each time point to be a multi-dimensional feature vector rather than a scalar value. This
could easily be realized by allowing the emission probabilities to be multi-dimensional. In
this way a richer set of information could be used: either the raw, multi-dimensional feature
vector, or some transformation of the feature vectors, for example, Principal Components
Analysis. The rest of the model would be unchanged and each feature vector would move
as a coherent piece. However, it might also be useful to allow different dimensions of the
feature vector to be aligned differently. For example, with the LC-MS data, this might
mean allowing different mass/charge peptides to be aligned differently at each time point.
However, in its full generality, such a task would be extremely computational intense.
A perhaps more interesting extension is to allow the model to work with non-replicate data.
For example, suppose one had a set of LC-MS experiments from a set of cancer patients,
and also a set from normal persons. It would be desirable to align the whole set of time
series and also to have the model tease out the differences between them. One approach
is to consider the model to be semi-supervised - the model is told the class membership
of each training example. Then each class is assigned its own latent trace, and a penalty
is introduced for any disagreements between the latent traces. Care needs to be taken to
ensure that the penalty plateaus after a certain amount of disagreement between latent trace
points, so that parts of the latent trace which are truly different are able to whole-heartedly
disagree. Assuming that the time resolution in the observed time series is sufficiently high,
one might also want to encourage the amount of disagreement over time to be Markovian.
That is, if the previous time point disagreed with the other latent traces, then the current
point should be more likely to disagree.
References
[1] Alan B. Poritz. Hidden markov models: A guided tour. In Proceedings of the IEEE Conference
on Acoustics, Speech and Signal Processing (ICASSP), pages 7?13. Morgan Kaufmann, 1988.
[2] Ruedi Aebersold and Matthias Mann. Mass spectrometry-based proteomics. Nature, 422:198?
207, 2003.
[3] P. Kearney and P. Thibault. Bioinformatics meets proteomics - bridging the gap between mass
spectrometry data analysis and cell biology. Journal of Bioinformatics and Computational Biology, 1:183?200, 2003.
[4] Yoshua Bengio and Paolo Frasconi. An input output HMM architecture. In G. Tesauro, D. Touretzky, and T. Leen, editors, Advances in Neural Information Processing Systems, volume 7, pages
427?434. The MIT Press, 1995.
[5] Richard Durbin, Sean R. Eddy, Anders Krogh, and Graeme Mitchison. Biological Sequence
Analysis: Probabilistic Models of Proteins and Nucleic Acids. Cambridge Univ. Press, 2000.
Durbin.
[6] H. Sakoe and S.Chiba. Dynamic programming algorithm for spoken word recognition. Readings
in Speech Recognition, pages 159?165, 1990.
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1,896 | 2,722 | Learning, Regularization and Ill-Posed Inverse
Problems
Lorenzo Rosasco
DISI, Universit`a di Genova
Genova, I
[email protected]
Ernesto De Vito
Dipartimento di Matematica
Universit`a di Modena
and INFN, Sezione di Genova
Genova, I
[email protected]
Andrea Caponnetto
DISI, Universit`a di Genova
Genova, I
[email protected]
Umberto De Giovannini
DISI, Universit`a di Genova
Genova, I
[email protected]
Francesca Odone
DISI, Universit`a di Genova
Genova, I
[email protected]
Abstract
Many works have shown that strong connections relate learning from examples to regularization techniques for ill-posed inverse problems. Nevertheless by now there was no formal evidence neither that learning from
examples could be seen as an inverse problem nor that theoretical results
in learning theory could be independently derived using tools from regularization theory. In this paper we provide a positive answer to both
questions. Indeed, considering the square loss, we translate the learning
problem in the language of regularization theory and show that consistency results and optimal regularization parameter choice can be derived
by the discretization of the corresponding inverse problem.
1
Introduction
The main goal of learning from examples is to infer an estimator, given a finite sample
of data drawn according to a fixed but unknown probabilistic input-output relation. The
desired property of the selected estimator is to perform well on new data, i.e. it should generalize. The fundamental works of Vapnik and further developments [16], [8], [5], show
that the key to obtain a meaningful solution to the above problem is to control the complexity of the solution space. Interestingly, as noted by [12], [8], [2], this is the idea underlying
regularization techniques for ill-posed inverse problems [15], [7]. In such a context to
avoid undesired oscillating behavior of the solution we have to restrict the solution space.
Not surprisingly the form of the algorithms proposed in both theories is strikingly similar.
Anyway a careful analysis shows that a rigorous connection between learning and regularization for inverse problem is not straightforward. In this paper we consider the square loss
and show that the problem of learning can be translated into a convenient inverse problem
and consistency results can be derived in a general setting. When a generic loss is considered the analysis becomes immediately more complicated.
Some previous works on this subject considered the special case in which the elements of
the input space are fixed and not probabilistically drawn [11], [9]. Some weaker results
in the same spirit of those presented in this paper can be found in [13] where anyway the
connections with inverse problems is not discussed. Finally, our analysis is close to the
idea of stochastic inverse problems discussed in [16]. It follows the plan of the paper. After recalling the main concepts and notation of learning and inverse problems, in section
4 we develop a formal connection between the two theories. In section 5 the main results
are stated and discussed. Finally in section 6 we conclude with some remarks and open
problems.
2
Learning from examples
We briefly recall some basic concepts of learning theory [16], [8]. In the framework of
learning, there are two sets of variables: the input space X, compact subset of R n , and
the output space Y , compact subset of R. The relation between the input x ? X and the
output y ? Y is described by a probability distribution ?(x, y) = ?(x)?(y|x) on X ? Y .
The distribution ? is known only through a sample z = (x, y) = ((x1 , y1 ), . . . , (x` , y` )),
called training set, drawn i.i.d. according to ?. The goal of learning is, given the sample
z, to find a function fz : X ? R such that fz (x) is an estimate of the output y when the
new input x is given. The function fz is called estimator and the rule that, given a sample
z, provides us with fz is called learning algorithm.
Given a measurable function f : X ? R, the ability of f to describe the distribution ? is
measured by its expected risk defined as
Z
I[f ] =
(f (x) ? y)2 d?(x, y).
X?Y
The regression function
g(x) =
Z
y d?(y|x),
Y
is the minimizer of the expected risk over the set of all measurable functions and always
exists since Y is compact. Usually, the regression function cannot be reconstructed exactly
since we are given only a finite, possibly small, set of examples z.
To overcome this problem, in the regularized least squares algorithm an hypothesis space
H is fixed, and, given ? > 0, an estimator fz ? is defined as the solution of the regularized
least squares problem,
`
min{
f ?H
1X
2
(f (xi ) ? yi )2 + ? kf kH }.
` i=1
(1)
The regularization parameter ? has to be chosen depending on the available data, ? =
?(`, z), in such a way that, for every > 0
?(`,z)
lim P I[fz
] ? inf I[f ] ? = 0.
(2)
`?+?
f ?H
We note that in general inf f ?H I[f ] is larger that I[g] and represents a sort of irreducible
error associated with the choice of the space H. The above convergence in probability is
usually called consistency of the algorithm [16] [14].
3
Ill-Posed Inverse Problems and Regularization
In this section we give a very brief account of linear inverse problems and regularization
theory [15], [7]. Let H and K be two Hilbert spaces and A : H ? K a linear bounded
operator. Consider the equation
Af = g?
(3)
where g? , g ? K and kg ? g? kK ? ?. Here g represents the exact, unknown data and g? the
available, noisy data. Finding the function f satisfying the above equation, given A and g ? ,
is the linear inverse problem associated to Eq. (3). The above problem is, in general, illposed, that is, the Uniqueness can be restored introducing the Moore-Penrose generalized
inverse f ? = A? g defined as the minimum norm solution of the problem
2
min kAf ? gkK .
f ?H
(4)
However the operator A? is usually not bounded so, in order to ensure a continuous dependence of the solution on the data, the following Tikhonov regularization scheme can be
considered1
2
2
min{kAf ? g? kK + ? kf kH },
(5)
f ?H
whose unique minimizer is given by
f?? = (A? A + ?I)?1 A? g? ,
(6)
where A? denotes the adjoint of A.
A crucial step in the above algorithm is the choice of the regularization parameter ? =
?(?, g? ), as a function of the noise level ? and the data g? , in such a way that
?(?,g? )
lim
f?
? f ?
= 0,
(7)
??0
H
?(?,g )
?
that is, the regularized solution f?
converges to the generalized solution f ? = A? g
?
(f exists if and only if P g ? Range(A), where P is the projection on the closure of the
range of A and, in that case, Af ? = P g) when the noise ? goes to zero.
The similarity between regularized least squares algorithm (1) and Tikhonov regularization (5) is apparent. However, several difficulties emerge. First, to treat the problem of
learning in the setting of ill-posed inverse problems we have to define a direct problem by
means of a suitable operator A. Second, in the context of learning, it is not clear the nature
of the noise ?. Finally we have to clarify the relation between consistency (2) and the kind
of convergence expressed by (7). In the following sections we will show a possible way to
tackle these problems.
4
Learning as an Inverse Problem
We can now show how the problem of learning can be rephrased in a framework close to
the one presented in the previous section.
We assume that hypothesis space H is a reproducing kernel Hilbert space [1] with a continuous kernel K : X ? X ? R. If x ? X, we let Kx (s) = K(s, x), and, if ? is the marginal
distribution of ? on X, we define the bounded linear operator A : H ? L2 (X, ?) as
(Af )(x) = hf, Kx iH = f (x),
1
In the framework of inverse problems, many other regularization procedures are introduced [7].
For simplicity we only treat the Tikhonov regularization.
that is, A is the canonical injection of H in L2 (X, ?). In particular, for all f ? H, the
expected risk becomes,
2
I[f ] = kAf ? gkL2 (X,?) + I[g],
where g is the regression function [2]. The above equation clarifies that if the expected
risk admits a minimizer fH on the hypothesis space H, then it is exactly the generalized
solution2 f ? = A? g of the problem
Af = g.
(8)
Moreover, given a training set z = (x, y), we get a discretized version Ax : H ? E` of A,
that is
(Ax f )i = hf, Kxi iH = f (xi ),
`
`
where E = R is the finite dimensional euclidean space endowed with the scalar product
`
1X
0
hy, y iE` =
yi yi0 .
` i=1
It is straightforward to check that
`
1X
2
(f (xi ) ? yi )2 = kAx f ? ykE` ,
` i=1
so that the estimator fz ? given by the regularized least squares algorithm is the regularized
solution of the discrete problem
Ax f = y.
(9)
At this point it is useful to remark the following two facts. First, in learning from examples
we are not interested into finding an approximation of the generalized solution of the discretized problem (9), but we want to find a stable approximation of the solution of the exact
problem (8) (compare with [9]). Second, we notice that in learning theory the consistency
property (2) involves the control of the quantity
2
2
I[fz ? ] ? inf I[f ] = kAf ? gkL2 (X,?) ? inf kAf ? gkL2 (X,?) .
f ?H
f ?H
If P is the projection on the closure of the range of A, the definition of P gives
2
I[fz ? ] ? inf I[f ] =
Afz ? ? P g
2
f ?H
L (X,?)
(10)
(11)
(the above equality stronlgy depends on the fact that the loss function is the square loss). In
the inverse problem setting, the square root of the above quantity is called the residue of the
solution fz ? . Hence, consistency
is
controlled by the residue of the estimator, instead of
?
?
the reconstruction error
fz ? f
(as in inverse problems). In particular, consistency
H
is a weaker condition than the one required by (7) and does not require the existence of the
generalized solution fH .
5
Regularization, Stochastic Noise and Consistency
To apply the framework of ill-posed inverse problems of Section 3 to the formulation of
learning proposed above, we note that the operator Ax in the discretized problem (9) differs
from the operator A in the exact problem (8) and a measure of the difference between A x
and A is required. Moreover, the noisy data y ? E` and the exact data g ? L2 (X, ?)
belong to different spaces, so that the notion of noise has to be modified. Given the above
premise our derivation of consistency results is developed in two steps: we first study the
residue of the solution by means of a measure of the noise due to discretization and then we
show a possible way to give a probabilistic evaluation of the noise previously introduced.
2
The fact that fH is the minimal norm solution of (4) is ensured by the assumption that the support
of the measure ? is X, since in this case the operator A is injective.
5.1
Bounding the Residue of the Regularized Solution
We recall that the regularized solutions of problems (9) and (8) are given by
fz?
=
(A?x Ax + ?I)?1 A?x y,
f?
=
(A? A + ?I)?1 A? g.
The above equations show that fz? and f ? depend only on A?x Ax and A? A which are
operators from H into H and on A?x y and A? g which are elements of H, so that the space
E` disappears. This observation suggests that noise levels could be kA?x Ax ? A? AkL(H)
and kA?x y ? A? gkH , where k?kL(H) is the uniform operator norm. To this purpose, for
every ? = (?1 , ?2 ) ? R2+ we define the collection of training sets.
U? = {z ? (X ? Y )` | kA?x y ? A? gkH ? ?1 , kA?x Ax ? A? AkL(H) ? ?2 , ` ? N}
and we let M = sup{|y| | y ? Y }. The next theorem is the central result of the paper.
Theorem 1 If ? > 0, the following inequalities hold
1. for any training set z ? U?
M?
?1
?
2
+ ?
Afz ? P g
L2 (X,?) ?
Af ? ? P g
L2 (X,?) ?
4?
2 ?
2. if P g ? Range(A), for any training set z ? U? ,
?
fz ? f ?
?
f ? ? f ?
? M ?32 + ?1
H
H
?
2? 2
Moreover if we choose ? = ?(?, z) in such a way that
?
?
? lim??0 supz?U? ?(?,2z)
?1
lim??0 supz?U? ?(?,z)
?
?2
? lim??0 sup
z?U?
?(?,z)
then
lim sup
Afz?(?,z) ? P g
??0 z?U?
= 0
= 0
= 0
L2 (X,?)
(12)
= 0.
(13)
We omit the complete proof and refer to [3]. Briefly, the idea is to note that
?
Afz ? P g
L2 (X,?) ?
Af ? ? P g
L2 (X,?)
1
?
Afz? ? Af ?
L2 (X,?) =
(A? A) 2 (fz? ? f ? )
H
where the last equation follows by polar decomposition of the operator A. Moreover a
simple algebraic computation gives
fz? ?f ? = (A? A+?I)?1 (A? A?A?x Ax )(A?x Ax +?I)?1 A?x y+(A? A+?I)?1 (A?x y?A? g)
where the relevant quantities for definition of the noise appear.
The first item in the above proposition quantifies the difference between the residues of
the regularized solutions of the exact and discretized problems in terms of the noise level
? = (?1 , ?2 ). As mentioned before this is exactly the kind of result needed to derive
consistency. On the other hand the last part of the proposition gives sufficient conditions
on the parameter ? to ensure convergence of the residue to zero as the level noise decreases.
The above results were obtained introducing the collection U? of training sets compatible
with a certain noise level ?. It is left to quantify the noise level corresponding to a training
set of cardinality `. This will be achieved in a probabilistic setting in the next section.
5.2
Stochastic Evaluation of the Noise
In this section we estimate the discretization noise ? = (?1 , ?2 ).
p
Theorem 2 Let 1 , 2 > 0 and ? = supx?X K(x, x), then
M?
?2
?
?
?
?
?
?
P kA g ? Ax ykH ?
+ 1 , kA A ? Ax Ax kL(H) ?
+ 2
`
`
2 `
1
2 `
2
? 1 ? e? 2?2 M 2 ? e? 2?4
(14)
The proof is given in [3] and it is based on McDiarmid inequality [10] applied to the random
variables
F (z) = kAx ? y ? A? gkH G(z) = kAx ? Ax ? A? AkL(H) .
Other estimates of the noise ? can be given using, for example, union bounds and Hoeffding?s inequality. Anyway rather then providing a tight analysis our concern was to find an
natural, explicit and easy to prove estimate of ?.
5.3
Consistency and Regularization Parameter Choice
Combining Theorems 1 and 2, we easily derive the following corollary.
Corollary 1 Given 0 < ? < 1, with probability greater that 1 ? ?,
?
Afz ? ? P g
? Af ? P g L2 (X,?)
2
L (X,?)
r
4
?M
1
?
? +
? ?
1 + log
?
2 `
? 2?
(15)
for all ? > 0.
Recalling (10) and (11) it is straightforward to check that the above inequality can be easily
restated in the usual learning notation, in fact we obtain
?2
?
r
?
? ?L
4
1
?
?
?
? +
I[fz ? ] ? ? ?
1 + log
+
Af ? ? P g
L2 (X,?) ? + inf I[f ] .
f ?H
?
?
?2 `
? 2?
|
{z
}
| {z }
|
{z
}
sample error
approximation error
irreducible error
In the above inequality the first term plays the role of sample error. If we choose the
regularization parameter so that ? = ?(`, z) = O( `1b ), with 0 < b < 12 the sample error
q
1
converges in probability to zero with order O
when ` ? ?. On the other
1?2b
`
hand the second term represents the approximation error and it is possible to show, using
standard results from spectral theory, that it vanishes as ? goes to zero [7]. Finally, the last
term represents the minimum attainable risk once the hypothesis space H has been chosen.
From the above observations it is clear that consistency is ensured once the parameter ? is
chosen according to the aforementioned conditions. Nonetheless to provide convergence
rates it is necessary to control the convergence rate of the approximation error. Unfortunately it is well known that this can be accomplished only making some assumptions on
the underlying probability distribution ? (see for example [2]). It can be shown that if the
explicit dependence of the approximation error on ? is not available we cannot determine
an optimal a priori (data independent) dependency ? = ?(`) for the regularization parameter. Nevertheless a posteriori (data dependent) choices ? = ?(`, z) can be considered to
automatically achieve optimal convergence rate [5], [6]. With respect to this last fact we
notice that the set of samples such that inequality (14) holds depends on ` and ?, but does
not depend ?, so that we can consider without any further effort a posteriori parameter
choices (compare with [4], [5]).
Finally, we notice that the estimate (15) is the result of two different procedures: Theorem 1, which is of functional type, gives the dependence of the bound by the regularization
parameter ? and by the noise levels kA?x Ax ? A? AkL(H) and kA?x y ? A? gkH , whereas
Theorem 2, which is of probabilistic nature, relates the noise levels to the number of data `
and the confidence level ?.
6
Conclusions
In this paper we defined a direct and inverse problem suitable for the learning problem and
derived consistency results for the regularized least squares algorithm. Though our analysis
formally explains the connections between learning theory and linear inverse problems, its
main limit is that we considered only the square loss. We briefly sketch how the arguments
presented in the paper extend to general loss functions. For sake of simplicity we consider
a differentiable loss function V . It is easy to see that the minimizer fH of the expected risk
satisfies the following equation
SfH = 0
(16)
where S = LK ? O and LK is the integral operator with kernel K, that is
Z
(LK f )(x) =
K(x, s)f (s)d?(s)
X
and O is the operator defined by
(Of )(x) =
Z
V 0 (y, f (x))d?(y|x).
Y
If we consider a generic differentiable loss the operator O and hence S is non linear, and
estimating fH is an ill-posed non linear inverse problem. It is well known that the theory
for this kind of problems is much less developed than the corresponding theory for linear
problems. Moreover, since, in general, I[f ] does not define a metric, it is not so clear the
relation between the expected risk and the residue. It appears evident that the attempt to
extend our results to a wider class of loss function is not straightforward. A possible way to
tackle the problem, further developing our analysis, might pass through the exploitation of
a natural convexity assumption on the loss function. Future work also aims to derive tighter
probabilistic bounds on the noise using recently proposed data dependent techniques.
Acknowledgments
We would like to thank M.Bertero, C. De Mol, M. Piana, T. Poggio, G. Talenti, A. Verri
for useful discussions and suggestions. This research has been partially funded by the
INFM Project MAIA, the FIRB Project ASTA2 and the IST Programme of the European
Community, under the PASCAL Network of Excellence, IST-2002-506778.
References
[1] N. Aronszajn. Theory of reproducing kernels. Trans. Amer. Math. Soc., 68:337?404,
1950.
[2] Felipe Cucker and Steve Smale. On the mathematical foundations of learning. Bull.
Amer. Math. Soc. (N.S.), 39(1):1?49 (electronic), 2002.
[3] E. De Vito, A. Caponnetto, and L. Rosasco.
Discretization error analysis
for Tikhonov regularization. submitted to Inverse Problem, 2004. available
http://www.disi.unige.it/person/RosascoL/publications/discre iop.pdf.
[4] E. De Vito, A. Caponnetto, and L. Rosasco. Model selection for regularized leastsquares algorithm in learning theory. to appear on Journal Machine Learning Research, 2004.
[5] L. Devroye, L. Gy?orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition.
Number 31 in Applications of mathematics. Springer, New York, 1996.
[6] Schock E. and Sergei V. Pereverzev. On the adaptive selection of the parameter in
regularization of ill-posed problems. Technical report, University of Kaiserslautern,
august 200r.
[7] Heinz W. Engl, Martin Hanke, and Andreas Neubauer. Regularization of inverse problems, volume 375 of Mathematics and its Applications. Kluwer Academic Publishers
Group, Dordrecht, 1996.
[8] Theodoros Evgeniou, Massimiliano Pontil, and Tomaso Poggio. Regularization networks and support vector machines. Adv. Comput. Math., 13(1):1?50, 2000.
[9] Vera Kurkova. Supervised learning as an inverse problem. Technical Report 960,
Institute of Computer Science, Academy of Sciences of the Czech Republic, April
2004.
[10] Colin McDiarmid. On the method of bounded differences. In Surveys in combinatorics, 1989 (Norwich, 1989), volume 141 of London Math. Soc. Lecture Note Ser.,
pages 148?188. Cambridge Univ. Press, Cambridge, 1989.
[11] S. Mukherjee, T. Niyogi, P.and Poggio, and R. Rifkin. Statistical learning: Stability is
sufficient for generalization and necessary and sufficient for consistency of empirical
risk minimization. Technical Report CBCL Paper 223, Massachusetts Institute of
Technology, january revision 2004.
[12] T. Poggio and Girosi F. Networks for approximation and learning. Proc. IEEE,
78:1481?1497, 1990.
[13] Cynthia Rudin. A different type of convergence for statistical learning algorithms.
Technical report, Program in Applied and Computational Mathematics Princeton University, 2004.
[14] I. Steinwart. Consistency of support vector machines and other regularized kernel
machines. IEEE Transaction on Information Theory, 2004. (accepted).
[15] Andrey N. Tikhonov and Vasiliy Y. Arsenin. Solutions of ill-posed problems. V. H.
Winston & Sons, Washington, D.C.: John Wiley & Sons, New York, 1977. Translated
from the Russian, Preface by translation editor Fritz John, Scripta Series in Mathematics.
[16] Vladimir N. Vapnik. Statistical learning theory. Adaptive and Learning Systems
for Signal Processing, Communications, and Control. John Wiley & Sons Inc., New
York, 1998. A Wiley-Interscience Publication.
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1,897 | 2,723 | Resolving Perceptual Aliasing In The Presence
Of Noisy Sensors?
Ronen I. Brafman & Guy Shani
Department of Computer Science
Ben-Gurion University
Beer-Sheva 84105, Israel
{brafman, shanigu}@cs.bgu.ac.il
Abstract
Agents learning to act in a partially observable domain may need to
overcome the problem of perceptual aliasing ? i.e., different states that
appear similar but require different responses. This problem is exacerbated when the agent?s sensors are noisy, i.e., sensors may produce different observations in the same state. We show that many well-known
reinforcement learning methods designed to deal with perceptual aliasing, such as Utile Suffix Memory, finite size history windows, eligibility
traces, and memory bits, do not handle noisy sensors well. We suggest
a new algorithm, Noisy Utile Suffix Memory (NUSM), based on USM,
that uses a weighted classification of observed trajectories. We compare
NUSM to the above methods and show it to be more robust to noise.
1 Introduction
Consider an agent situated in a partially observable domain: It executes an action; the action
may change the state of the world; this change is reflected, in turn, by the agent?s sensors;
the action may have some associated cost, and the new state may have some associated
reward or penalty. Thus, the agent?s interaction with this environment is characterized by a
sequence of action-observation-reward steps, known as instances [7]. In this paper we are
interested in agents with imperfect and noisy sensors that learn to act in such environments
without any prior information about the underlying set of world-states and the world?s
dynamics, only information about their sensor?s capabilities. This is a known variant of
reinforcement learning (RL) in partially observable domains [1].
As the agent works with observations, rather than states, two possible problems arise: The
agent may observe too much data which requires computationally intensive filtering ? a
problem we do not discuss. Or, sensors may supply insufficient data to identify the current
state of the world based on the current observation. This leads to a phenomena known as
perceptual aliasing [2], where the same observation is obtained in distinct states requiring
different actions. For example, in Figure 1(a) the states marked with X are perceptually
aliased. Various RL techniques were developed to handle this problem.
?
Partially supported by the Lynn and William Frankel Center for Computer Sciences.
(a)
(b)
Figure 1: Two maze domains. If only wall configuration is sensed, states marked with the
same letter (X or Y) are perceptually aliased.
The problem of resolving the perceptual aliasing is exacerbated when the agent?s sensors
are not deterministic. For example, if walls can sometimes be reported where none exist, or
if a wall sometimes goes undetected. The performance of existing techniques for handling
RL domains with perceptually aliased states, such as finite size history windows, eligibility
traces, and internal memory quickly degrades as the level of noise increases.
In this paper, we introduce the Noisy Utile Suffix Memory (MUSM) algorithm, which
builds on McCallum?s Utile Suffix Memory (USM) algorithm [7]. We compare the performance of MUSM to USM and other existing methods on the above two standard maze
domains, and show that it is more robust to noise.
2 Background
We briefly review a number of algorithms for resolving the problem of perceptual aliasing.
We assume familiarity with basic RL techniques, and in particular, Q-learning, SARSA,
and eligibility traces (see Sutton and Barto [12] for more details).
The simplest way to handle perceptual aliasing is to ignore it by using the observation
space as the state space, defining a memory-less policy. This approach works sometimes,
although it generally results in poor performance. Jaakkola et al. [4] suggest stochastic
memory-less policies, and Williams and Singh [13] implement an online version of their
algorithm and showed that it approximates an optimal solution with 50% accuracy.
Eligibility traces can be viewed as a type of short-term memory, as they update the recently
visited state-action couplings. Therefore, they can be used to augment the ability to handle
partial observability. Loch and Singh [6] explore problems where a memory-less optimal
policy exists and demonstrated that Sarsa(?) can learn an optimal policy for such domains.
Finite-size history methods are a natural extension to memory-less policies. Instead of
identifying a state with the last observation, we can use the last k observations. The size of
the history window (k) is a fixed parameter, identical for all observation sequences. Lock
and Singh [6] show Sarsa(?) using fixed history window to learn a good policy in domains
where short-term memory optimal policies exist.
An arbitrarily predefined history window cannot generally solve perceptual aliasing: an
agent cannot be expected to know in advance how long should it remember the actionobservation-reward trajectories. Usually, some areas of the state space require the agent to
remember more, while in other locations a reactive policy is sufficient. A better solution is
to learn online the history length needed to decide on the best action in the current location.
McCallum [7] extensively handles those issues in his dissertation. We review McCallums?
Utile Suffix Memory (USM) algorithm in Section 3.
Another possible approach to handle perceptual aliasing is to augment the observations
with some internal memory [10]. The agents? state s is composed of both the current
observation o and the internal memory state m. The agents? actions are enhanced with
actions that modify the memory state by flipping one of the bits. The agent uses some
standard learning mechanism to learn the proper action, including actions that change the
memory state. The algorithm is modified so that no cost or decay is applied to the actions
that modify the memory. This approach is better than using finite or variable history length
because meaningful events can occur arbitrarily far in the past. Keeping all the possible
trajectories from the event onwards until the outcome is observed might cost too much, and
McCallums? techniques are unable to group those trajectories together to deduce the proper
action. Peshkin et al. [10] demonstrated the memory-bits approach to converge, but did not
show it to be superior to any other algorithm.
Other approaches include the use of finite-state automata (FSA) [8] (which can be viewed
as a special case of the memory-bits approach); the use of neural networks for internal
memory [5, 3]; and constructing and solving a POMDP model [7, 2]. The new emerging
technique of Predictive State Representations (PSR) [11] may also provide a good way
to online learn a model of the environment. We note that most researchers tested their
algorithms on environments with very little noise, and do not examine the effect of noise
on their performance.
3 Utile Suffix Memory
Instance-based state identification [7] resolves perceptual aliasing with variable length short
term memory. An Instance is a tuple Tt = hTt?1 , at?1 , ot , rt i ? the individual observed
raw experience. Algorithms of this family keep all the observed raw data (sequences of
instances), and use it to identify matching subsequences. The algorithm assumes that if the
suffix of two sequences is similar both were likely generated in the same world state.
Utile Suffix Memory creates a tree structure, based on the well known suffix trees for string
operations. This tree maintains the raw experiences and identifies matching suffixes. The
root of the tree is an unlabelled node, holding all available instances. Each immediate child
of the root is labelled with one of the observations encountered during the test. A node
holds all the instances Tt = hTt?1 , at?1 , ot , rt i whose final observation ot matches the
observation in the node?s label. At the next level, instances are split based on the last action
of the instance at . Then, we split again based on (the next to last) observation ot?1 , etc.
All nodes act as buckets, grouping together instances that have matching history suffixes of
a certain length. Leaves take the role of states, holding Q-values and updating them. The
deeper a leaf is in the tree, the more history the instances in this leaf share.
The tree is built on-line during the test run. To add a new instance to the tree, we examine
its precept, and follow the path to the child node labeled by that precept. We then look at
the action before this precept and move to the node labeled by that action, then branch on
the precept prior to that action and so forth, until a leaf is reached.
Identifying the proper depth for a certain leaf is a major issue, and we shall present a number of improvements to McCallum?s methods. Leaves should be split if their descendants
show a statistical difference in expected future discounted reward associated with the same
action. We split instances in a node if knowing where the agent came from helps predict
future discounted rewards. Thus, the tree must keep what McCallum calls fringes, i.e.,
subtrees below the ?official? leaves.
For better performance, McCallum did not compare the nodes in the fringes to their siblings, only to their ancestor ?official? leaf. He also did not compare values from all actions
executed from the fringe, only the action that has the highest Q-value in the leaf (the policy
action of that leaf). To compare the populations of expected discounted future rewards from
the two nodes (the fringe and the ?official? leaf), he used the Kolmogorov-Smirnov (KS)
test ? a non-parametric statistical test used to find whether two populations were generated by the same distribution. If the test reported that a difference was found between the
expected discounted future rewards after executing the policy action, the leaf was split, the
fringe node would become the new leaf, and the tree would be expanded to create deeper
fringes. Figure 3 presents an example of a possible USM tree, without fringe nodes.
Figure 2: A possible USM suffix tree generated by the maze in Figure 1(a). Below is a
sequence of instances demonstrating how some instances are clustered into the tree leaves.
Instead of comparing the fringe node to its ancestor ? official? leaf, we found it computationally possible to compare the siblings of the fringe, avoiding the problem that the same
instance appears in both distributions. McCallum compared only the expected discounted
future rewards from executing the policy action, where we compare all the values following
all actions executed after any of the instances in the fringe. McCallum used the KS test,
where we choose to use the more robust randomization test [14] that works well with small
sets of instances. McCallum also considered only fringe nodes of certain depth, given as
a parameter to the algorithm, where we choose to create fringe nodes as deep as possible,
until the number of instances in the node diminish below some threshold (we use a value
of 10 in our experiments).
The expected future discounted reward of instance Ti is defined by:
Q(Ti ) = ri + ?U (L(Ti+1 ))
(1)
where L(Ti ) is the leaf associated with instance Ti and U (s) = maxa (Q(s, a)).
After inserting new instances to the tree, we update Q-values in the leaves using:
P
Ti ?T (s,a) ri
R(s, a) =
|T (s, a)|
|?Ti ? T (s, a), L(Ti+1 ) = s0 |
P r(s0 |s, a) =
|T (s, a)|
X
P r(s0 |s, a)U (s0 )
Q(s, a) = R(s, a) + ?
(2)
(3)
(4)
s0
We use s and s0 to denote the leaves of the tree, as in an optimal tree configuration for a
problem the leaves of the tree define the sates of the underlying MDP. The above equations
therefore correspond to a single step of the value iteration algorithm used in MDPs.
Now that the Q-values have been updated, the agent chooses the next action to perform
based on the Q-values in the leaf corresponding to the current instance Tt :
at+1 = argmaxa Q(L(Tt ), a)
(5)
McCallum uses the fringes of the tree for a smart exploration strategy. In our implementation we use a simple ?-greedy technique for exploration.
4 Adding Noise to Utile Suffix Memory
There are two types of noise in perception. The system makes different observations at the
same location (false negative), or it makes identical observations at different locations (false
positive). USM handles false positives by differentiating identical current observations
using the agents? history. Knowing that the agent came from different locations, helps it
realize that it is in two different locations, though the observations look the same.
USM does not handle false negative perceptions well. When the agent is at the same state,
but receives different observations, it is unable to learn from the noisy observation and thus
wastes much of the available information. Our Noisy Utile Suffix Memory (NUSM) is
designed to overcome this weakness.
It is reasonable to assume that the agent has some sensor model defining pr(o|s) ? the
probability that the agent will observe o in world state s. We can use the sensor model to
augment USM with the ability to learn from noisy instances. In our experiments we assume
the agent has n boolean sensors with an accuracy probability pi . A single observation is
composed of n output values o = h?1 , ..., ?n i when ?i ? 0, 1. The probability that an
observation o = h?1o , ..., ?1o i came from an actual world state s = h?1s , ..., ?ns i is therefore:
pr(o|s) =
n
Y
(6)
?i
i=0
?i =
?
pi
(1 ? pi )
?i0 = ?is
?i0 6= ?is
(7)
USM inserts each instance into a single path, ending at one leaf. Using any sensor model
we can insert the new instance Tt = hTt?1 , at?1 , ot , rt i into several paths with different
weights. When inserting Tt with weight w into an action node at depth k (with its children
labeled by observations) we will insert the instance into every child node c, with weight
w ? pr(ot?k?1 |label(c)). When inserting Tt with weight w into an observation node at
depth k (with its children labeled by actions) we will insert the instance only into the child
c labeled by at?k?1 with the same weight w. Weights of instances are stored in each node
with the instance as ws (Tt ) ? the weight of instance Tt in node s.
We can now rewrite Equation 2 and Equation 3:
P
R(s, a) =
0
T ?T (s,a) ri
Pi
P r(s |s, a) =
Ti ?T (s,a)
P
? ws (Ti )
ws (Ti )
T ,L(Ti+1 )=s0
Pi
Ti ?T (s,a)
ws (Ti )
ws (Ti )
(8)
(9)
The noisy instances are used only for updating the Q-values in the leaves. The test for
splitting is still calculated using identical sequences only. The tree structure of USM and
NUSM is hence identical, but NUSM learns a better policy. Conducted experiments indicate that using the noisy sequences for deciding when to split leaves provides a slight gain
in collected rewards, but the constructed tree is much larger, resulting in a considerable hit
to performance.
NUSM learns noisy sequences better. When a state corresponding to a noisy sequence is
observed, even though the noise in it might make it rare, NUSM can still use data from real
sequences to decide which action is useful for this state.
5 Experimental Results
To test our algorithms we used two maze worlds, seen in Figure 1(a) and Figure 1(b),
identical to the worlds McCallum used to show the performance of the USM algorithm. In
both cases some of the world states are perceptually aliased and the algorithm should learn
to identify the real world states. The agent in our experiments has four sensors allowing it
to sense an immediate wall above, below, to the left, and to the right of its current location.
Sensors have a boolean output that has a probability ? of being accurate. The probability
of all sensors providing the correct output is ?4 . In both mazes there is a single location
that grants the agent a reward of 10. Upon receiving that reward the agent is transformed to
any of the perceptually aliased states of the maze randomly. If the agent bumps into a wall
it pays a cost (a negative reward) of 1. For every move the agent pays a cost of 0.1.
We compared the performance of applying finite size history windows to Q-learning and
Sarsa, eligibility traces, memory bits, USM and NUSM on the two worlds. In the tables
below, QL2 denotes using Q-learning with a history window of size 2, and S2 denotes using
Sarsa with a window size of 2. S(?) denotes the Sarsa(?) algorithm. Adding the superfix 1
denotes adding 1 bit of internal memory. For example, S(?)12 denotes using Sarsa(?) with a
history window of 2 and 1 internal memory bit. The columns and rows marked ? 2 present
the average variance over methods (bottom row) and ? values (rightmost column). In the
NUSM column, in brackets, is the improvement NUSM gains over USM.
As we are only interested in the effect of noisy sensors, the maze examples we use do
not demonstrate the advantages of the various algorithms; USM?s ability to automatically
compute the needed trajectory length in different locations and the internal memory ability
to remember events that occurred arbitrary far in the past are unneeded since our examples
require the agent to look only at the last 2 instances in every perceptually aliased location.
We ran each algorithm for 50000 steps learning a policy as explained above, and calculated
the average reward over the last 5000 iterations only to avoid the difference in convergence
time. We ran experiments with varying values of ? (accuracy of the sensors) ranging from
1.00 (sensor output is without noise) to 0.85 (overall output accuracy of 0.522). Reported
results are averaged over 10 different executions of each algorithm. We also ran experiments for Sarsa and Q-learning with only the immediate observation, which yielded poor
results as expected, and for history window of size 3 and 4 which resulted in lower performance than history window of size 2 for all algorithms (and were therefore removed from
the tables). Additional memory bits did not improve performance either. In our Sarsa(?)
implementation we used ? = 0.9.
?
QL2
S2
S(?)1
S(?)2
S(?)3
S(?)11
S(?)12
USM
NUSM
?2
1.00
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.88
0.87
0.86
0.85
?2
1.51
1.46
1.42
1.36
1.28
1.24
1.11
1.05
0.96
0.88
0.83
0.74
0.61
0.58
0.40
0.45
0.01
1.54
1.47
1.42
1.38
1.26
1.23
1.10
1.03
0.88
0.82
0.73
0.60
0.59
0.50
0.37
0.35
0.015
0.32
0.33
0.32
0.41
0.38
0.43
0.47
0.42
0.47
0.47
0.46
0.48
0.42
0.48
0.46
0.45
0.004
1.53
1.47
1.36
1.31
1.29
1.21
1.16
1.13
1.10
1.06
1.02
0.95
0.90
0.85
0.79
0.75
0.004
0.98
0.98
0.83
0.64
0.58
0.55
0.45
0.43
0.33
0.29
0.22
0.23
0.14
0.12
0.07
0.05
0.022
1.27
1.34
1.41
1.35
1.24
1.26
0.89
0.94
0.92
0.74
0.77
0.80
0.66
0.63
0.55
0.47
0.061
1.53
1.45
1.40
1.35
1.27
1.22
1.12
1.07
1.04
0.94
0.92
0.87
0.83
0.78
0.76
0.68
0.005
1.56
1.49
1.42
1.38
1.35
1.30
1.18
1.16
1.12
0.96
0.99
0.93
0.84
0.71
0.57
0.47
0.02
1.57(+0%)
1.54(+3%)
1.44(+1%)
1.43(+4%)
1.40(+4%)
1.35(+4%)
1.29(+9%)
1.29(+11%)
1.20(+7%)
1.12(+17%)
1.07(+8%)
1.04(+12%)
1.01(+20%)
0.98(+38%)
0.92(+61%)
0.87(+85%)
0.003
0.033
0.016
0.008
0.007
0.014
0.009
0.025
0.023
0.014
0.022
0.013
0.015
0.013
0.011
0.021
0.018
Table 1: Average reward as function of sensor accuracy, for the maze in Figure 1(a).
?
QL2
S2
S(?)1
S(?)2
S(?)3
S(?)11
S(?)12
USM
NUSM
?2
1.00
0.99
0.98
0.97
0.96
0.95
0.94
0.93
0.92
0.91
0.90
0.89
0.87
0.86
0.85
?2
1.42
1.40
1.33
1.30
1.26
1.19
1.09
1.05
0.94
0.85
0.69
0.64
0.44
0.30
0.08
0.008
1.46
1.41
1.35
1.29
1.25
1.16
1.05
0.94
0.84
0.80
0.72
0.58
0.42
0.21
0.10
0.01
0.23
0.24
0.25
0.26
0.24
0.21
0.14
0.12
0.12
0.09
0.10
0.06
0.06
0.04
0.01
0.003
1.53
1.44
1.35
1.22
1.06
1.00
0.93
0.82
0.72
0.77
0.65
0.58
0.47
0.26
0.29
0.009
1.49
1.34
1.24
1.15
1.06
0.90
0.85
0.76
0.64
0.47
0.42
0.24
0.16
0.09
0.09
0.006
1.47
1.24
0.94
0.90
0.43
0.33
0.30
0.39
0.30
0.24
0.23
0.24
0.16
0.13
0.18
0.071
1.54
1.43
1.34
1.21
1.12
1.03
0.93
0.77
0.66
0.48
0.48
0.40
0.31
0.20
0.16
0.016
1.75
1.57
1.43
1.40
1.28
1.23
1.09
1.09
1.02
0.93
0.87
0.81
0.72
0.68
0.61
0.011
1.72(-2%)
1.61(+3%)
1.46(+2%)
1.40(0%)
1.31(+2%)
1.26(+2%)
1.14(+5%)
1.09(+0%)
1.03(+1%)
0.96(+3%)
0.91(+5%)
0.92(+14%)
0.82(+14%)
0.81(+19%)
0.75(+23%)
0.006
0.004
0.027
0.034
0.032
0.015
0.015
0.011
0.018
0.011
0.013
0.013
0.010
0.012
0.015
0.008
Table 2: Average reward as function of sensor accuracy, for the maze in Figure 1(b).
4/
6/0%
$YHUDJHUHZDUG
6/
860
1860
,QVWDQFHV
Figure 3: Convergence rates for the maze in Figure 1(a) when sensor accuracy is 0.9.
As we can see, when sensor output is only slightly noisy, all algorithms perform reasonably;
NUSM performs the best, but differences are minor. This is because when sensors supply
perfect output, resolving the perceptual aliasing results in a fully observable environment
which Q-learning and Sarsa can solve optimally. When noise increases, the ability of
NUSM to use similar suffixes of trajectories, results in a noticeable performance gain over
other algorithms. The only algorithm that competes with NUSM is Sarsa(?) with a history
window of 2. The ability of Sarsa(?) to perform well in partially observable domains have
been noted by Lock and Singh [6]1 , but we note here that the performance of Sarsa(?) relies
heavily on the proper definition of the required history window size. When the history
window differs slightly from the required size, the performance hit is substantial, as we can
see in the two adjacent columns.
NUSM is more expensive computationally than USM and takes longer to converge. In
Figure 3, we can see that it still converges reasonably fast. Moreover, each NUSM iteration
takes about 5.6 milliseconds, when a USM iteration takes 3.1 milliseconds with the same
1
Lock and Singh also recommend the use of replacing traces but we found that using accumulating
traces produced better performance.
accuracy of ? = 0.85 and a similar number of nodes (10, 229 for NUSM and 10, 349 for
USM ? including fringe nodes), making NUSM reasonable for online learning.
Finally, Both USM and NUSM attempt to disambiguate the perceptual aliasing and create
a fully observable MDP. Yet, it is better to model the world directly as partially observable
using a Partially Observable Markov Decision Process (POMDP). POMDP policies explicitly address the problem of incomplete knowledge, taking into account the ability of certain
actions to reduce uncertainty without immediately generating useful rewards. Nikovski [9]
used McCallum?s Nearest Sequence Memory (NSM), a predecessor of USM to generate
and solve a POMDP from the observed instances. They, however, considered environments with little noise. We implemented their algorithms and obtained poor results in the
presence of noise in our domains, probably due to the use of NSM for state identification.
6 Conclusions
We show that some RL algorithms, including finite size history windows, memory bits
and USM, that resolve perceptual aliasing, provide poor results in the presence of noisy
sensors. We provided some insights as to why McCallums? USM algorithm does not handle
well noisy input from the agents? sensors and introduce NUSM ? an extension to USM
that learns from noisy sequences and handles environments where sensors provide noisy
output better. As noise arises, NUSM works better than other algorithms used for handling
domains with perceptual aliasing.
References
[1] A. R. Cassandra, L. P. Kaelbling, and M. L. Littman. Acting optimally in partially observable
stochastic domains. In AAAI?94, pages 1023?1028, 1994.
[2] L. Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions
approach. In AAAI?02, pages 183?188, 1992.
[3] S. Hochreiter and J. Schmidhuber. Long short-term memory. Neural Computation, 9(8):1735?
1780, 1997.
[4] T. Jaakkola, S. P. Singh, and M. I. Jordan. Reinforcement learning algorithm for partially
observable Markov decision problems. In NIPS?95, volume 7, pages 345?352, 1995.
[5] L.-J. Lin and T. M. Mitchell. Memory approaches to reinforcement learning in non-markovian
domains. Technical Report CMU-CS-92-138, 1992.
[6] J. Loch and S. Singh. Using eligibility traces to find the best memoryless policy in partially
observable Markov decision processes. In ICML?98, pages 323?331, 1998.
[7] A. K. McCallum. Reinforcement Learning with Selective Perception and Hidden State. PhD
thesis, University of Rochester, 1996.
[8] N. Meuleau, L. Peshkin, K. Kim, and L. P. Kaelbling. Learning finite-state controllers for
partially observable environments. In UAI?99, pages 427?436, 1999.
[9] D. Nikovski. State-Aggregation Algorithms for Learning Probabilistic Models for Robot Control. PhD thesis, Carnegie Mellon University, 2002.
[10] L. Peshkin, N. Meuleau, and L. P. Kaelbling. Learning policies with external memory. In
ICML?99, pages 307?314, 1999.
[11] S. Singh, M. L. Littman, and R. S. Sutton. Predictive representations of state. In NIPS 2001,
pages 1555?1561, December 2001.
[12] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, 1998.
[13] J. K. Williams and S. Singh. Experimental results on learning stochastic memoryless policies
for partially observable markov decision processes. In NIPS, 1998.
[14] A. Yeh. More accurate tests for the statistical significance of result differences. In 18th Int.
Conf. on Computational Linguistics, pages 947?953, 2000.
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1,898 | 2,724 | Kernel Methods for Implicit Surface Modeling
?
Bernhard Sch?olkopf? , Joachim Giesen+? & Simon Spalinger+
Max Planck Institute for Biological Cybernetics, 72076 Tu? bingen, Germany
[email protected]
+
Department of Computer Science, ETH Zu? rich, Switzerland
[email protected],[email protected]
Abstract
We describe methods for computing an implicit model of a hypersurface
that is given only by a finite sampling. The methods work by mapping
the sample points into a reproducing kernel Hilbert space and then determining regions in terms of hyperplanes.
1
Introduction
Suppose we are given a finite sampling (in machine learning terms, training data)
x1 , . . . , xm ? X , where the domain X is some hypersurface in Euclidean space Rd . The
case d = 3 is especially interesting since these days there are many devices, e.g., laser
range scanners, that allow the acquisition of point data from the boundary surfaces of
solids. For further processing it is often necessary to transform this data into a continuous model. Today the most popular approach is to add connectivity information to the data
by transforming them into a triangle mesh (see [4] for an example of such a transformation
algorithm). But recently also implicit models, where the surface is modeled as the zero set
of some sufficiently smooth function, gained some popularity [1]. They bear resemblance
to level set methods used in computer vision [6]. One advantage of implicit models is that
they easily allow the derivation of higher order differential quantities such as curvatures.
Another advantage is that an inside-outside test, i.e., testing whether a query point lies on
the bounded or unbounded side of the surface, boils down to determining the sign of a
function-evaluation at the query point. Inside-outside tests are important when one wants
to intersect two solids.
The goal of this paper is, loosely speaking, to find a function which takes the value zero on
a surface which
(1) contains the training data and
(2) is a ?reasonable? implicit model of X .
To capture properties of its shape even in the above general case, we need to exploit some
structure on X . In line with a sizeable amount of recent work on kernel methods [11], we
assume that this structure is given by a (positive definite) kernel, i.e., a real valued function
?
Partially supported by the Swiss National Science Foundation under the project ?Non-linear
manifold learning?.
o
Figure 1: In the 2-D toy example depicted,
the hyperplane hw, ?(x)i = ? separates all
but one of the points from the origin. The outlier ?(x) is associated with a slack variable ?,
which is penalized in the objective function
(4). The distance from the outlier to the hyperplane is ?/kwk; the distance between hyperplane and origin is ?/kwk. The latter implies that a small kwk corresponds to a large
margin of separation from the origin.
o
o
o
.
o
o
o
?/ ||w||
w
? (x) o
o
?/ ||w||
o
k on X ? X which can be expressed as
k(x, x0 ) = h?(x), ?(x0 )i
(1)
for some map ? into a Hilbert space H. The space H is the reproducing kernel Hilbert
space (RKHS) associated with k, and ? is called its feature map. A popular example, in
the case where X is a normed space, is the Gaussian (where ? > 0)
kx ? x0 k2
0
.
(2)
k(x, x ) = exp ?
2 ?2
The advantage of using a positive definite kernel as a similarity measure is that it allows us
to construct geometric algorithms in Hilbert spaces.
2
Single-Class SVMs
Single-class SVMs were introduced [8, 10] to estimate quantiles C ? {x ? X |f (x) ?
[?, ?[} of an unknown distribution P on X using kernel expansions. Here,
X
f (x) =
?i k(xi , x) ? ?,
(3)
i
where x1 , . . . , xm ? X are unlabeled data generated i.i.d. according to P . The single-class
SVM approximately computes the smallest set C ? C containing a specified fraction of
all training examples, where smallness is measured in terms of the norm in the RKHS H
associated with k, and C is the family of sets corresponding to half-spaces in H. Depending
on the kernel, this notion of smallness will coincide with the intuitive idea that the quantile
estimate should not only contain a specified fraction of the training points, but it should
also be sufficiently smooth so that the same is approximately true for previously unseen
points sampled from P .
Let us briefly describe the main ideas of the approach. The training points are mapped into
H using the feature map ? associated with k, and then it is attempted to separate them from
the origin with a large margin by solving the following quadratic program: for ? ? (0, 1], 1
1
1 X
minimize
kwk2 +
?i ? ?
(4)
2
?m i
w?H,??Rm ,??R
subject to
hw, ?(xi )i ? ? ? ?i , ?i ? 0.
(5)
Since non-zero slack variables ?i are penalized in the objective function, we can expect that
if w and ? solve this problem, then the decision function, f (x) = sgn (hw, ?(x)i ? ?) will
1
Here and below, bold face greek character denote vectors, e.g., ? = (? 1 , . . . , ?m )> , and indices
i, j by default run over 1, . . . , m.
Figure 2: Models computed with a single class SVM using a Gaussian kernel (2). The
three examples differ in the value chosen for ? in the kernel - a large value (0.224 times the
diameter of the hemisphere) in the left figure and a small value (0.062 times the diameter
of the hemisphere) in the middle and right figure. In the right figure also non-zero slack
variables (outliers) were allowed. Note that that the outliers in the right figure correspond
to a sharp feature (non-smoothness) in the original surface.
equal 1 for most examples xi contained in the training set,2 while the regularization term
kwk will still be small. For an illustration, see Figure 1. The trade-off between these two
goals is controlled by a parameter ?.
One can show that the solution takes the form
f (x) = sgn
X
!
?i k(xi , x) ? ? ,
i
where the ?i are computed by solving the dual problem,
1X
?i ?j k(xi , xj )
minimize
m
??R
2 ij
subject to
0 ? ?i ?
X
1
and
?i = 1.
?m
i
(6)
(7)
(8)
Note that according to (8), the training examples contribute with nonnegative weights ? i ?
0 to the solution (6). One can show that asymptotically, a fraction ? of all training examples
will have strictly positive weights, and the rest will be zero (the ??-property?).
In our application we are not primarily interested in a decision function itself but in the
boundaries of the regions in input space defined by the decision function. That is, we are
interested in f ?1 (0), where f is the kernel expansion (3) and the points x1 , . . . , xm ? X
are sampled from some unknown hypersurface X ? Rd . We want to consider f ?1 (0) as
a model for X . In the following we focus on the case d = 3. If we assume that the x i are
sampled without noise from X ? which for example is a reasonable assumption for data
obtained with a state of the art 3d laser scanning device ? we should set the slack variables
in (4) and (5) to zero. In the dual problem this results in removing the upper constraints
on the ?i in (8). Note that sample points with non-zero slack variable cannot be contained
in f ?1 (0). But also sample points whose image in feature space lies above the optimal
hyperplane are not contained in f ?1 (0) (see Figure 1) ? we will address this in the next
section. It turns out that it is useful in practice to allow non-zero slack variables, because
they prevent f ?1 (0) from decomposing into many connected components (see Figure 2 for
an illustration).
In our experience, one can ensure that the images of all sample points in feature space lie
close to (or on) the optimal hyperplane can be achieved by choosing ? in the Gaussian
2
We use the convention that sgn (z) equals 1 for z ? 0 and ?1 otherwise.
Figure 3: Two parallel hyperplanes hw, ?(x)i = ? +
? (?) enclosing all but two
of the points. The outlier
?(x(?) ) is associated with
a slack variable ? (?) , which
is penalized in the objective
function (9).
? (x *)o
o
.
?*/ ||w||
o
o
o
(?+?* )/||w||
(?+?)/ ||w||
w
? (x) o
o
?/ ||w||
o
o
kernel (2) such that the Gaussians in the kernel expansion (3) are highly localized. However, highly localized Gaussians are not well suited for interpolation ? the implicit surface
decomposes into several components. Allowing outliers mitigates the situation to a certain
extent. Another way to deal with the problem is to further restrict the optimal region in
feature space. In the following we will pursue the latter approach.
3
Slab SVMs
A richer class of solutions, where some of the weights can be negative, is obtained if we
change the geometric setup. In this case, we estimate a region which is a slab in the RKHS,
i.e., the area enclosed between two parallel hyperplanes (see Figure 3).
To this end, we consider the following modified program:3
minimize
w?H,? (?) ?Rm ,??R
subject to
and
1 X
1
kwk2 +
(?i + ?i? ) ? ?
2
?m i
(9)
? ? ?i ? hw, ?(xi )i ? ? ? ? ? + ?i?
(10)
(?)
?i
(11)
? 0.
Here, ? (?) are fixed parameters. Strictly speaking, one of them is redundant: one can show
that if we subtract some offset from both, then we obtain the same overall solution, with ?
changed by the same offset. Hence, we can generally set one of them to zero, say, ? = 0.
Below we summarize some relationships of this convex quadratic optimization problem to
known SV methods:
1. For ? = 0 and ? ? = ? (i.e., no upper constraint), we recover the single-class SVM
(4)?(5).
2. If we drop ? from the objective function and set ? = ??, ? ? = ? (for some fixed ? ? 0),
we obtain the ?-insensitive support vector regression algorithm [11], for a data set where
all output values y1 , . . . , ym are zero. Note that in this case, the solution is trivial, w = 0.
This shows that the ? in our objective function plays an important role.
P
3. For ? = ? ? = 0, the term i (?i + ?i? ) measures the distance of the point ?(xi ) from the
hyperplane hw, ?(xi )i ? ? = 0 (up to a scaling of kwk). If ? tends to zero, this term will
dominate the objective function. Hence, in this case, the solution will be a hyperplane that
approximates the data well in the sense that the points lie close to it in the RKHS norm.
3
Here and below, the superscript (?) simultaneously denotes the variables with and without asterisk, e.g., ? (?) is a shorthand for ? and ? ? .
From the following constraints and Lagrange multipliers
?i ? ? + hw, ?(xi )i ? ? ? 0,
?i ? 0
?i? + ? ? + ? ? hw, ?(xi )i ? 0,
?i? ? 0
(?)
?i
? 0,
(?)
?i
?0
(12)
(13)
(14)
4
we derive the Lagrangian dual optimization problem of (9) - (11):
X
X
1X
?i + ? ?
?i?
(?i ? ?i? )(?j ? ?j? )k(xi , xj ) ? ?
minimize
m
??R
2 ij
i
i
subject to
and
1
(?)
0 ? ?i ?
?m
X
(?i ? ?i? ) = 1,
(15)
(16)
(17)
i
Note that for ? = ? ? , we can simplify the optimization problem using the transformation
?new = ? ? ?? . For ? = ? ? = 0, we thus obtain the single-class SVM (7) with the
1
1
modified box constraint ? ?m
? ?inew ? ?m
.
The dual problem can be solved using standard quadratic programming packages. The
offset ? can be computed from the value of the corresponding variable in the double dual, or
using the Karush-Kuhn-Tucker (KKT) conditions, just as in other support vector methods.
Once this is done, we can evaluate for each test point x whether it satisfies ? ? hw, ?(x)i?
? ? ? ? . In other words, we have an implicit description of the region in input space that
corresponds to the region in between the two hyperplanes in the RKHS. For ? = ? ? , this
is a single hyperplane, corresponding to a hypersurface in input space.5 To compute this
surface we use the kernel expansion
X
hw, ?(x)i =
(?i ? ?i? )k(xi , x).
(18)
i
Support Vectors and Outliers In our discussion of single class SVMs for surface modeling we already mentioned that we aim for many support vectors (as we want most training
points to lie on the surface) and that outliers might represent features like certain singularities in the original hypersurface.
Here we analyze how the parameter ? influences the SVs and outliers. To this end, we
introduce the following shorthands for the sets of SV and outlier indices:
SV := {i | hw, ?(xi )i ? ? ? ? ? 0}
(19)
SV ? := {i | hw, ?(xi )i ? ? ? ? ? ? 0}
(20)
(?)
OL(?) := {i | ?i > 0}
(21)
It is clear from the primal optimization problem that for all i, ?i > 0 implies hw, ?(xi )i ?
? ? ? < 0 (and likewise, ?i? > 0 implies hw, ?(xi )i ? ? ? ? ? > 0), hence OL(?) ? SV (?) .
The difference of the SV and OL sets are those points that lie precisely on the boundaries
of the constraints.6 Below, |A| denotes the cardinality of the set A.
4
Note that due to (17), the dual solution is invariant with respect to the transformation ? (?) ?
? (?) + const. ? such a transformation only adds a constant to the objective function, leaving the
solution unaffected.
5
subject to suitable conditions on k
6
The present usage differs slightly from the standard definition of SVs, which are usually those
(?)
that satisfy ?i > 0. In our definition, SVs are those points where the constraints are active. How(?)
ever, the difference is marginal: (i) It follows from the KKT conditions that ? i > 0 implies that
the corresponding constraint is active. (ii) while it can happen in theory that a constraint is active and
(?)
nevertheless the corresponding ?i is zero, this almost never occurs in practice.
Proposition 1 The solution of (9)?(11) satisfies
|OL| |SV ? |
|SV | |OL? |
?
???
?
.
m
m
m
m
(22)
The proof is analogous to the one of the ??-property? for standard SVMs, cf. [8]. Due to
lack of space, we skip it, and instead merely add the following observations:
1. The above statements are not symmetric with respect to exchanging the quantities with
asterisks and their counterparts without asterisk. This is due to the sign of ? in the primal
objective function. If we used +? rather than ??, we would obtain almost the same dual,
the only difference being that the constraint (17) would have a ??1? on the right hand
side. In this case, the role of the quantities with and without asterisks would be reversed in
Proposition 1.
2. The ?-property of single class SVMs is obtained as the special case where OL ? =
SV ? = ?.
3. Essentially, if we require that the distribution has a density w.r.t. the Lebesgue measure,
and that k is analytic and non-constant (cf. [8, 9]), it can be shown that asymptotically, the
two inequalities in the proposition become equalities with probability 1.
Implementation On larger problems, solving the dual with standard QP solvers becomes
too expensive (scaling with m3 ). For this case, we can use decomposition methods. The
adaptation of known decomposition methods to the present case is straightforward, noticing
that the dual of the standard ?-SV regression algorithm [11] becomes almost identical to the
present dual if we set ? = (? ? ? ?)/2 and yi = ?(? ? + ?)/2 for all i. The only difference
is that in our case, there is a ?1? in (17), whereas in the SVR case, we would have a ?0?.
As a consequence, we have to change the initialization of the optimization algorithm to
ensure that we start with a feasible solution. As an optimizer, we used a modified version
of libSVM [2].
Experimental Results In all our experiments we used a Gaussian kernel (2). To render
the implicit surfaces, i.e., the zero-set f ?1 (0), we generated a triangle mesh that approximates it. To compute the mesh we used an adaptation of the marching cubes algorithm [5]
which is a standard technique to transform an implicitly given surfaces into a mesh. The
most costly operations in the marching cubes algorithm are evaluations of the kernel expansion (18). To reduce the number of these evaluations we implemented a surface following
technique that exploits the fact that we know quite some sample points on the surface,
namely the support vectors.7 Some results can be seen in Figure 4.
Our experiments indicate a nice geometric interpretation of negative coefficients ? i ? ?i? .
It seems that negative coefficients correspond to concavities in the original model. The
coefficients seem well suited to extract shape features from the sample point set, e.g., the
detection of singularities like sharp edges or feature lines ? which is an important topic in
computer graphics [7].
We also tried a multi-scale approach. In this approach at first a rough model is computed
from ten percent of the sample points using a slab SVM. For the remaining 90% of the
sample points we compute the residual values, i.e., we evaluate the kernel expansion (18)
at the sample points. Finally we use support vector regression (SVR) and the residual
values to derive a new kernel expansion (using a smaller kernel width) whose zero set we
use as our surface model. An example how this approach works can be seen in Figure 5.
7
In the experiments, both the SVM optimization and the marching cubes rendering took up to
about 2 hours.
Figure 4: First row: Computing a model of the Stanford bunny (35947 points) and of a golf
club (16864 points) with the slab SVM. The close up of the ears and nose of the bunny
shows the sample points colored according to the coefficients ?i ? ?i? . Dark gray points
have negative coefficients and light gray points positive ones. In the right figure we show
the bottom of the golf club model. The model on the left of this figure was computed with
a different method [4]. Note that with this method fine details like the figure three become
visible. Such details get leveled out by the limited resolution of the marching cubes method.
However the information about these details is preserved and detected in the SVM solution,
as can be seen from the color coding. Second row: In the left and in the middle figure we
show the results of the slab SVM method on the screwdriver model (27152 points) and
the dinosaur model (13990 points), respectively. In the right figure a color coding of the
coefficients for the rockerarm data set (40177 points) is shown. Note that we can extract
sharp features from this data set by filtering the coefficients according to some threshold.
Figure 5: First row: The multi-scale approach applied to a knot data set (10000 points).
The blobby support surface (left figure) was computed from 1000 randomly chosen sample
points with the slab SVM. In the middle we show a color coding of the residual values of
all sample points (cf. http://books.nips.cc for color images). In the right figure we show the
surface that we get after applying support vector regression using the residual values.
4
Discussion and Outlook
An approximate description of the data as the zero set of a function can be useful as a compact representation of the data. It could potentially also be employed in other tasks where
models of the data are useful, such as denoising and image super-resolution. We therefore
consider it worthwhile to explore the algorithmic aspects of implicit surface estimation in
more depth, including the study of regression based approaches.
Some acquisition devices do not only provide us with points from a surface embedded
in R3 , but also with the normals at these points. Using methods similar to the ones in
[3], it should be possible to integrate such additional information into our approach. We
expect that it will improve the quality of the computed models in the sense that even more
geometric details are preserved.
A feature of our approach is that its complexity depends only marginally on the dimension
of the input space (in our examples this was three). Thus the approach should work also
well for hypersurfaces in higher dimensional input spaces. From an applications point of
view hypersurfaces might not be as interesting as manifolds of higher co-dimension. It
would be interesting to see if our approach can be generalized to handle also this situation.
Acknowledgment We thank Chih-Jen Lin for help with libSVM. The bunny data were
taken from the Stanford 3d model repository. The screwdriver, dinosaur and rockerarm
data were taken from the homepage of Cyberware Inc. Thanks to Koby Crammer, Florian
Steinke, and Christian Walder for useful discussion.
References
[1] J. Carr, R. Beatson, J. Cherrie, T. Mitchell, W. Fright, B. McCallum, and T. Evans.
Reconstruction and representation of 3D objects with radial basis functions. In Proc.
28th Ann. Conf. Computer Graphics and Interactive Techniques, pages 67?76. 2001.
[2] C.-C. Chang and C.-J. Lin. LIBSVM: a library for support vector machines, 2001.
Software available at http://www.csie.ntu.edu.tw/?cjlin/libsvm.
[3] O. Chapelle and B. Sch?olkopf. Incorporating invariances in nonlinear SVMs. In T.G.
Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information
Processing Systems 14, Cambridge, MA, 2002. MIT Press.
[4] J. Giesen and M. John. Surface reconstruction based on a dynamical system. Computer Graphics Forum, 21(3):363?371, 2002.
[5] T. Lewiner, H. Lopes, A. Wilson, and G. Tavares. Efficient implementation of marching cubes cases with topological guarantee. Journal of Graphics Tools, 8:1?15, 2003.
[6] S. Osher and N. Paragios. Geometric Level Set Methods. Springer, New York, 2003.
[7] M. Pauly, R. Keiser, and M. Gross. Multi-scale feature extraction on point-sampled
surfaces. Computer Graphics Forum, 22(3):281?289, 2003.
[8] B. Sch?olkopf, J. Platt, J. Shawe-Taylor, A. J. Smola, and R. C. Williamson. Estimating
the support of a high-dimensional distribution. Neural Computation, 13:1443?1471,
2001.
[9] I. Steinwart. Sparseness of support vector machines?some asymptotically sharp
bounds. In S. Thrun, L. Saul, and B. Sch?olkopf, editors, Advances in Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004.
[10] D. M. J. Tax and R. P. W. Duin. Support vector data description. Machine Learning,
54:45?66, 2004.
[11] V. N. Vapnik. The Nature of Statistical Learning Theory. Springer Verlag, New York,
1995.
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evaluate:2 |
1,899 | 2,725 | A Machine Learning Approach
to Conjoint Analysis
Olivier Chapelle, Za??d Harchaoui
Max Planck Institute for Biological Cybernetics
Spemannstr. 38 - 72076 T?ubingen - Germany
{olivier.chapelle,zaid.harchaoui}@tuebingen.mpg.de
Abstract
Choice-based conjoint analysis builds models of consumer preferences
over products with answers gathered in questionnaires. Our main goal is
to bring tools from the machine learning community to solve this problem more efficiently. Thus, we propose two algorithms to quickly and
accurately estimate consumer preferences.
1
Introduction
Conjoint analysis (also called trade-off analysis) is one of the most popular marketing research technique used to determine which features a new product should have, by conjointly
measuring consumers trade-offs between discretized1 attributes. In this paper, we will focus on the choice-based conjoint analysis (CBC) framework [11] since it is both widely
used and realistic: at each question in the survey, the consumer is asked to choose one
product from several.
The preferences of a consumer are modeled via a utility function representing how much a
consumer likes a given product. The utility u(x) of a product x is assumed to be the sum
of the partial utilities (or partworths) for each attribute, i.e. linear: u(x) = w ? x. However,
instead of observing pairs (xl , yl ), the training samples are of the form ({x1k , . . . , xpk }, yk )
indicating that among the p products {x1k , . . . , xpk }, the ykth was preferred. Without noise,
this is expressed mathematically by u(xykk ) ? u(xbk ), ? b 6= yk .
Let us settle down the general framework of a regular conjoint analysis survey. We have a
population of n consumers available for the survey. The survey consists of a questionnaire
of q questions for each consumer, each asking to choose one product from a basket of
p. Each product profilePis described through a attributes with l1 , ..., la levels each, via a
a
vector of length m = s=1 ls , with 1 at positions of levels taken by each attribute and 0
elsewhere.
Marketing researchers are interested in estimating individual partworths in order to perform for instance a segmentation of the population afterwards. But traditional conjoint
estimation techniques are not reliable for this task since the number of parameters m to be
estimated is usually larger than the number of answers q available for each consumer. They
estimate instead the partworths on the whole population (aggregated partworths). Here we
1
e.g. if the discretized attribute is weight, the levels would be light/heavy.
aim to investigate this issue, for which machine learning can provide efficient tools. We
also address adaptive questionnaire design with active learning heuristics.
2
Hierarchical Bayes Analysis
The main idea of HB2 is to estimate the individual utility functions under the constraint that
their variance should not be too small. By doing so, the estimation problem is not ill-posed
and the lack of information for a consumer can be completed by the other ones.
2.1
Probabilistic model
In this section, we follow [11] for the description of the HB model and its implementation.
This method aims at estimating the individual linear utility functions ui (x) = wi ? x, for
1 ? i ? n. The probabilistic model is the following:
1. The individual partworths wi are drawn from a Gaussian distribution with mean ?
(representing the aggregated partworths) and covariance ? (encoding population?s
heterogeneity),
2. The covariance matrix ? has an invert Wishart prior, and ? has an (improper) flat
prior.
3. Given a set of products (x1 , . . . xp ), the probability that the consumer i chooses
the product x? is given by
exp(wi ? x? )
P (x? |wi ) = Pp
.
b=1 exp(wi ? xb )
2.2
(1)
Model estimation
We describe now the standard way of estimating ?, w ? (w1 , . . . , wn ) and ? based on
Gibbs sampling and then propose a much faster algorithm that approximates the maximum
a posteriori (MAP) solution.
Gibbs sampling As far as we know, all implementations of HB rely on a variant of the
Gibbs sampling [11]. During one iteration, each of the three sets of variables (?, w and ?)
is drawn in turn from its posterior distribution the two others being fixed. Sampling for ?
and ? is straightforward, whereas sampling from P (w|?, ?, Y ) ? P (Y |w). P (w|?, ?)
is achieved with the Metropolis-Hastings algorithm.
When convergence is reached, the sampling goes on and finally outputs the empirical expectation of ?, w and ?. Although the results of this sampling-based implementation of
HB3 are impressive, practitioners complain about its computational burden.
Approximate MAP solution So far HB implementations make predictions by evaluating
(1) at the empirical mean of the samples, in contrast with the standard bayesian approach,
which would average the rhs of (1) over the different samples, given samples w from the
posterior. In order to alleviate the computational issues associated with Gibbs sampling,
we suggest to consider the maximum of the posterior distribution (maximum a posteriori,
MAP) rather than its mean.
2
Technical papers of Sawtooth software [11], the world leading company for conjoint analysis
softwares, provide very useful and extensive references.
3
that we will call HB-Sampled or HB-S in the rest of the paper.
To find ?, w and ? which maximize P (?, w, ?|Y ), let us use Bayes? rule,
P (?, w, ?|Y ) ? P (Y |?, w, ?) ? P (w|?, ?) ? P (?|?) ? P (?)
? P (Y |w) ? P (w|?, ?) ? P (?)
(2)
w
Maximizing (2) with respect to ? yields ?MAP = I+C
n+d , with Cw being the ?covariance?
P
matrix of the wi centered at ?: Cw = (wi ? ?)(wi ? ?)> . Putting back this value in
(2), we get
? log P (?, w, ?MAP |Y ) = ? log P (Y |w) + log |I + Cw (?)| + C,
(3)
where C is an irrelevant constant. Using the model (1), the first term in the rhs of (3) is
convex in w, but P
not the second term. For this reason, we propose to change log |I + Cw |
by trace(Cw ) =
||wi ? ?||2 (this would be a valid approximation if trace(Cw ) 1).
With this new prior on w, the rhs of (3) becomes
W (?, w) =
n
X
? log P (Yi |wi ) + ||wi ? ?||2 .
(4)
i=1
As in equation (3), this objective function is minimized with respect to ? when ? is equal to
the empirical mean of the wi . We thus suggest the following iterative scheme to minimize
the convex functional (4):
1. For a given ?, minimize (4) with respect to each of the wi independently.
2. For a given w, set ? to the empirical mean4 of the w.
Thanks to the convexity, this optimization problem can be solved very efficiently. A Newton approach in step 1, as well as in step 2 to speed-up the global convergence to a fixed
point ?, has been implemented. Only couple of steps in both cases are necessary to reach
convergence.
Remark The approximation from equation (3) to (4) might be too crude. After all it boils
down to setting ? to the identity matrix. One might instead consider ? as an hyperparameter and optimize it by maximizing the marginalized likelihood [14].
3
Conjoint Analysis with Support Vector Machines
Similarly to what has recently been proposed in [3], we are now investigating the use of
Support Vector Machines (SVM) [1, 12] to solve the conjoint estimation problem.
3.1
Soft margin formulation of conjoint estimation
Let us recall the learning problem. At the k-th question, the consumer chooses the ykth
product from the basket {x1k , . . . , xpk }: w ? xykk ? w ? xbk , ? b 6= yk . Our goal is to estimate
the individual partworths w, with the individual utility function now being u(x) = w ? x.
With a reordering of the products, we can actually suppose that yk = 1. Then the above
inequalities can be rewritten as a set of p ? 1 constraints:
w ? (x1k ? xbk ) ? 0, 2 ? b ? p.
(5)
Eq. (5) shows that the conjoint estimation problem can be cast as a classification problem
in the product-profiles differences space. From this point of view, it seems quite natural to
use state-of-the-art classifiers such as SVMs for this purpose.
4
which is consistent with the L2 -loss measuring deviations of wi -s from ?.
More specifically, we propose to train a L2 -soft margin classifier (see also [3] for a similar
approach) with only positive examples and with a hyperplane passing through the origin
(no bias), modelling the noise in the answers with slack variables ?kb :
Pq Pp
2
Minimize w2 + C k=1 b=2 ?kb
1
b
subject to w ? (xk ? xk ) ? 1 ? ?kb .
3.2
Estimation of individual utilities
It was proposed in [3] to train one SVM per consumer to get wi and to compute
Pn the
individual partworths by regularizing with the aggregated partworths w = n1 i=1 wi :
wi? = wi2+w .
Instead, to estimate the individual utility partworths wi , we suggest the following optimization problem (the set Qi contains the indices j such that the consumer i was asked to
choose between products x1k , . . . , xpk ) :
(
P
Pp
P
Pp
?
2
2
Minimize wi2 + qCi k?Qi b=2 ?kb
+ P C qj k?Q
/ i
b=2 ?kb
j6=i
subject to wi ? (x1k ? xbk ) ? 1 ? ?kb , ?k, ?b ? 2 .
C
Here the ratio C
? determines the trade-off between the individual scale and the aggregated
C
5
one. For C? = 1, the population is modeled as if it were homogeneous, i.e. all partworths
wi are equal. For C
? 1, the individual partworths are computed independently, without
C
taking into account aggregated partworths.
4
Related work
Ordinal regression Very recently [2] explores the so-called ordinal regression task for
ranking, and derive two techniques for hyperparameters learning and model selection in
a hierarchical bayesian framework, Laplace approximation and Expectation Propagation
respectively. Ordinal regression is similar yet distinct from conjoint estimation since training data are supposed to be rankings or ratings in contrast with conjoint estimation where
training data are choice-based. See [4] for more extensive bibliography.
Large margin classifiers Casting the preference problem in a classification framework,
leading to learning by convex optimization, was known for a long time in the psychometrics community. [5] pioneered the use of large margin classifiers for ranking tasks.
[3] introduced the kernel methods machinery for conjoint analysis on the individual scale.
Very recently [10] proposes an alternate method for dealing with heterogeneity in conjoint
analysis, which boils down to a very similar optimization to our HB-MAP approximation
objective function, but with large margin regularization and with minimum deviation from
the aggregated partworths.
Collaborative filtering Collaborative filtering exploits similarity between ratings across
a population. The goal is to predict a person?s rating on new products given the person?s
past ratings on similar products and the ratings of other people on all the products. Again
collaborative is designed for overlapping training samples for each consumer, and usually
rating/ranking training data, whereas conjoint estimation usually deals with different questionnaires for each consumer and choice-based training data.
5
? In this way, directions for which the xj , j ? Qi contain information are estimated
C ? C
accurately, whereas the others directions are estimated thanks to the answers of the other consumers.
5
Experiments
Artificial experiments We tested our algorithms on the same benchmarking artificial experimental setup used in [3, 16]. The simulated product profiles consist of 4 attributes, each
of them being discretized through 4 levels. A random design was used for the questionnaire. For each question, the consumer was asked to choose one product from a basket of 4.
A population of 100 consumers was simulated, each of them having to answer 4 questions.
Finally, the results presented below are averaged over 5 trials.
The 100 true consumer partworths were generated from a Gaussian distribution with mean
(??, ??/3, ?/3, ?) (for each attribute) and with a diagonal covariance matrix ? 2 I. Each
answer is a choice from the basket of products, sampled from the discrete logit-type distribution (1). Hence when ? (called the magnitude6 ) is large, the consumer will choose with
high probability the product with the highest utility, whereas when ? is small, the answers
will be less reliable. The ratio ? 2 /? controls the heterogeneity7 of the population.
Finally, as in [3], the performances are computed using the mean of the L2 distances between the true and estimated individual partworths (also called RMSE). Beforehand the
partworths are translated such that the mean on each attribute is 0 and normalized to 1.
Real experiments We tested our algorithms on disguised industrial datasets kindly provided by Sawtooth Software Inc., the world leading company in conjoint analysis softwares.
11 one-choice-based8 conjoint surveys datasets9 were used for real experiments below. The
number of attributes ranged from 3 to 6 (hence total number of levels from 13 to 28), the
size of the baskets, to pick one product from at each question, ranged from 2 to 5, and
the number of questions ranged from 6 to 15. The numbers of respondents ranged roughly
from 50 to 1200. Since here we did not address the issue of no choice options in question
answering, we removed10 questions where customers refused to choose a product from the
basket and chose the no-choice-option as an answer11 .
Finally, as in [16], the performances are computed using the hit rate, i.e. the misprediction
rate of the preferred product.
5.1
Analysis of HB-MAP
We compare in this section our implementation of the HB method described in Section 2,
that we call HB-MAP, to HB-S, the standard HB implementation.
The average training time for HB-S was 19 minutes (with 12000 iterations as suggested in
[11]), whereas our implementation based on the approximation of the MAP solution took
in average only 1.8 seconds. So our primary goal, i.e. to alleviate the sampling phase
complexity, was achieved since we got a speed-up factor of the order of 1000.
The accuracy does not seem to be significantly weakened by this new implementation.
Indeed, as shown in both Table 1 and Table 2, the performances achieved by HB-MAP
were surprisingly often as good as HB-S?s, and sometimes even a bit better. This might be
6
as in [3], we tested High Magnitude (? = 3) and Low Magnitude (? = 0.5).
It was either set to ? 2 = 3? or ? 2 = 0.5?, respectively High and Low Heterogeneity cases.
8
We limited ourselves to datasets in which respondents were asked to choose 1 product among a
basket at each question.
9
see [4] for more details on the numerical features of the datasets.
10
One could use EM-based methods to deal with such missing training choice data.
11
When this procedure boiled down to unreasonable number of questions for hold-out evaluation
of our algorithms, we simply removed the corresponding individuals.
7
explained by the fact that assuming that the covariance matrix is quasi-diagonal is a reasonable approximation, and that the mode of the posterior distribution is actually roughly close
to the mean, for the real datasets considered. Additionally it is likely that HB-S may have
demanded much more iterations for convergence to systematically behave more accurately
than HB-MAP as one would have normally expected.
5.2
Analysis of SVMs
We now turn to the SVM approach presented in section 3.2 that we call Im.SV12 . We did
not use a non-linear kernel in our experiments. Hence it was possible to minimize (3.2)
directly in the primal, instead of using the dual formulation as done usually. This turned
out to be faster since the number of constraints was, for our problem, larger than the number
of variables. The resulting mean training time was 4.7 seconds. The so-called chapspan,
span estimate of leave-one-out prediction error [17], was used to select a suitable value of
C 13 , since it gave a quasi-convex estimation on the regularization path.
The performances of Im.SV in Table 2, compared to the HB methods and logistic regression
[3] are very satisfactory in case of artificial experiments. In real experiments, Im.SV gives
overall quite satisfactory results, but sometimes disappointing ones in Table 2. One reason
? were optimized once for the whole population. This
might be that hyperparameters (C, C)
may also be due to the lack of robustness14 of Im.SV to heterogeneity in the number of
training samples for each consumer.
Table 1: Average RMSE between estimated and true individual partworths
Mag
L
L
H
H
Het
L
H
L
H
HB-S
0.90
0.95
0.44
0.72
HB-MAP
0.83
0.91
0.40
0.68
Logistic
0.84
1.16
0.43
0.82
Im.SV
0.86
0.90
0.41
0.67
Table 2: Hit rate performances on real datasets.
Dat12
Dat22
Dat13
Dat23
Dat33
Dat43
Im.SV
0.16
0.15
Im.SV
0.37
0.34
0.35
0.35
HB-MAP
0.16
0.13
HB-MAP
0.24
0.33
0.28
0.31
HB-S
0.17
0.15
HB-S
0.25
0.33
0.24
0.28
Dat15
Dat25
Im.SV
0.52
0.58
HB-MAP
0.45
0.47
HB-S
0.48
0.51
Dat14
Dat24
Dat34
Im.SV
0.33
0.33
0.45
HB-MAP
0.36
0.36
0.40
HB-S
0.35
0.28
0.25
Legend of Tables 1 and 2 The first two columns indicate the Magnitude and the
Heterogeneity (High or Low). p in Datmp is the number of products respondents are
asked to choose one from at each question.
12
since individual choice data are Immersed in the rest of the population choice data, via the
optimization objective
13
? was irrelevant, and that only the ratio C/C
? matWe observed that the value of the constant C
tered.
14
Indeed the no-choice data cleaning step might have lead to a strong unbalance to which Im.SV
is maybe much more sensitive than HB-MAP or HB-S.
6
Active learning
Motivation Traditional experimental designs are built by minimizing the variance of an
estimator (e.g. orthogonal designs [6]). However, they are sub-optimal because they do
not take into account the previous answers of the consumer. Therefore adaptive conjoint
analysis was proposed [11, 16] for adaptively designing questionnaires.
The adaptive design concept is often called active learning in machine learning, as the
algorithm can actively select questions whose responses are likely to be informative. In the
SVM context, a common and intuitive strategy is to select, as the next point to be labeled,
the nearest one from the decision boundary (see for instance [15]).
Experiments We implemented this heuristic for conjoint analysis by selecting for each
question a set of products whose estimated utilities are as close as possible15 . To compare
the different designs, we used the same artificial simulations as in section 5, but with 16
questions per consumer in order to fairly compare to the orthogonal design.
Table 3: Comparison of the RMSE achieved by different designs.
Mag
L
L
H
H
Het
L
H
L
H
Random
0.66
0.62
0.31
0.49
Orthogonal
0.61
0.56
0.29
0.45
Adaptive
0.66
0.56
0.24
0.34
Results in Table 3 show that active learning produced an adaptive design which seems
efficient, especially in the case of high magnitude, i.e. when the answers are not noisy16 .
7
Discussion
We may need to capture correlations between attributes to model interaction effects among
them. The polynomial kernel K(u, v) = (u.v + 1)d seems particularly relevant for such a
task. HB methods kernelization can be done in the framework presented in [7]. For large
margin methods [10, 3] give a way to use the kernel trick in the space of product-profiles
differences. Prior knowledge of product-profile structure [3] may also be incorporated in
the estimation process by using virtual examples [12].
[9] approach would allow us to improve our approximate MAP solution by learning a variational approximation of a non-isotropic diagonal covariance matrix.
A fully bayesian HB setting, i.e. with a maximum likelihood type II17 (ML II) step, in
contrast of sampling from the posterior, is known in the statistics community as bayesian
multinomial logistic regression. [18] use Laplace approximation to compute integration
over hyperparameters for multi-class classification, while [8] develop a variational approximation of the posterior distribution.
New insights on learning gaussian process regression in a HB framework have just been
given in [13], where a method combining an EM algorithm and a generalized Nystr?om approximation of covariance matrix is proposed, and could be incorporated in the HB-MAP
approximation above.
15
Since the bottom-line goal of the conjoint analysis is not really to estimate the partworths but
to design the ?optimal? product, adaptive design can also be helpful by focusing on products which
have a high estimated utility.
16
Indeed noisy answers are neither informative nor reliable for selecting the next question.
17
aka evidence maximization or hyperparameters learning
8
Conclusion
Choice-based conjoint analysis seems to be a very promising application field for machine
learning techniques. Further research include fully bayesian HB methods, extensions to
non-linear models as well as more elaborate and realistic active learning schemes.
Acknowledgments
The authors are very grateful to J. Qui?nonero-Candela and C. Rasmussen for fruitful discussions, and O. Toubia for providing us with his HB implementation. Many thanks to
Sawtooth Software Inc. for providing us with real conjoint analysis datasets.
References
[1] B. E. Boser, I. M. Guyon, and V. N. Vapnik. A training algorithm for optimal margin
classifiers. In Proc. 5th Annu. Workshop on Comput. Learning Theory, 1992.
[2] W. Chu and Z. Ghahramani. Gaussian processes for ordinal regression. Technical
report, University College London, 2004.
[3] T. Evgeniou, C. Boussios, and G. Zacharia. Generalized robust conjoint estimation.
Marketing Science, 25, 2005.
[4] Z. Harchaoui. Statistical learning approaches to conjoint estimation. Technical report,
Max Planck Institute for Biological Cybernetics, to appear.
[5] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal
regression. In Advances in Large Margin Classifiers. MIT Press, 2000.
[6] J. Huber and K. Zwerina. The importance of utility balance in efficient choice designs.
Journal of Marketing Research, 33, 1996.
[7] T. S. Jaakkola and D. Haussler. Probabilistic kernel regression models. In Artificial
Intelligence and Statistics, 1999.
[8] T. S. Jaakkola and M. I. Jordan. Bayesian logistic regression: a variational approach.
Statistics and Computing, 10:25?37, 2000.
[9] T. Jebara. Convex invariance learning. In Artificial Intelligence and Statistics, 2003.
[10] C. A. Micchelli and M. Pontil. Kernels for multi?task learning. In Advances in Neural
Information Processing Systems 17, 2005.
[11] Sawtooth Software.
Research paper series.
Available at
www.sawtoothsoftware.com/techpap.shtml#hbrel.
[12] B. Sch?olkopf and A. Smola. Learning with kernels. MIT Press, 2002.
[13] A. Schwaighofer, V. Tresp, and K. Yu. Hierarchical bayesian modelling with gaussian
processes. In Advances in Neural Information Processing Systems 17, 2005.
[14] M. Tipping. Bayesian inference: Principles and practice. In Advanced Lectures on
Machine Learning. Springer, 2004.
[15] S. Tong and D. Koller. Support vector machine active learning with applications to
text classification. In Journal of Machine Learning Research, volume 2, 2001.
[16] O. Toubia, J. R. Hauser, and D. I. Simester. Polyhedral methods for adaptive choicebased conjoint analysis. Journal of Marketing Research, 41(1):116?131, 2004.
[17] V. Vapnik and O. Chapelle. Bounds on error expectation for support vector machines.
Neural Computation, 12(9), 2000.
[18] C. K. I. Williams and D. Barber. Bayesian classification with gaussian processes.
IEEE Trans. Pattern Anal. Mach. Intell., 20, 1998.
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