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A Method for Inferring Label Sampling Mechanisms in Semi-Supervised Learning Saharon Rosset Data Analytics Research Group IBM T.J. Watson Research Center Yorktown Heights, NY 10598 srosset@us.ibm.com Ji Zhu Department of Statistics University of Michigan Ann Arbor, MI 48109 jizhu@umich.edu Hui Zou Department of Statistics Stanford University Stanford, CA 94305 hzou@stat.stanford.com Trevor Hastie Department of Statistics Stanford University Stanford, CA 94305 hastie@stanford.edu Abstract We consider the situation in semi-supervised learning, where the “label sampling” mechanism stochastically depends on the true response (as well as potentially on the features). We suggest a method of moments for estimating this stochastic dependence using the unlabeled data. This is potentially useful for two distinct purposes: a. As an input to a supervised learning procedure which can be used to “de-bias” its results using labeled data only and b. As a potentially interesting learning task in itself. We present several examples to illustrate the practical usefulness of our method. 1 Introduction In semi-supervised learning, we assume we have a sample (xi, yi, si)n i=1, of i.i.d. draws from a joint distribution on (X, Y, S), where:1 • xi ∈Rp are p-vectors of features. • yi is a label, or response (yi ∈R for regression, yi ∈{0, 1} for 2-class classification). • si ∈{0, 1} is a “labeling indicator”, that is yi is observed if and only if si = 1, while xi is observed for all i. In this paper we consider the interesting case of semi-supervised learning, where the probability of observing the response depends on the data through the true response, as well as 1Our notation here differs somewhat from many semi-supervised learning papers, where the unlabeled part of the sample is separated from the labeled part and sometimes called “test set”. potentially through the features. Our goal is to model this unknown dependence: l(x, y) = Pr(S = 1|x, y) (1) Note that the dependence on y (which is unobserved when S = 0) prevents us from using standard supervised modeling approaches to learn l. We show here that we can use the whole data-set (labeled+unlabeled data) to obtain estimates of this probability distribution within a parametric family of distributions, without needing to “impute” the unobserved responses.2 We believe this setup is of significant practical interest. Here are a couple of examples of realistic situations: 1. The problem of learning from positive examples and unlabeled data is of significant interest in document topic learning [4, 6, 8]. Consider a generalization of that problem, where we observe a sample of positive and negative examples and unlabeled data, but we believe that the positive and negative labels are supplied with different probabilities (in the document learning example, positive examples are typically more likely to be labeled than negative ones, which are much more abundant). These probabilities may also not be uniform within each class, and depend on the features as well. Our methods allow us to infer these labeling probabilities by utilizing the unlabeled data. 2. Consider a satisfaction survey, where clients of a company are requested to report their level of satisfaction, but they can choose whether or not they do so. It is reasonable to assume that their willingness to report their satisfaction depends on their actual satisfaction level. Using our methods, we can infer the dependence of the reporting probability on the actual satisfaction by utilizing the unlabeled data, i.e., the customers who declined to respond. Being able to infer the labeling mechanism is important for two distinct reasons. First, it may be useful for “de-biasing” the results of supervised learning, which uses only the labeled examples. The generic approach for achieving this is to use “inverse sampling” weights (i.e. weigh labeled examples by 1/l(x, y)). The us of this for maximum likelihood estimation is well established in the literature as a method for correcting sampling bias (of which semi-supervised learning is an example) [10]. We can also use the learned mechanism to post-adjust the probabilities from a probability estimation methods such as logistic regression to attain “unbiasedness” and consistency [11]. Second, understanding the labeling mechanism may be an interesting and useful learning task in itself. Consider, for example, the “satisfaction survey” scenario described above. Understanding the way in which satisfaction affects the customers’ willingness to respond to the survey can be used to get a better picture of overall satisfaction and to design better future surveys, regardless of any supervised learning task which models the actual satisfaction. Our approach is described in section 2, and is based on a method of moments. Observe that for every function of the features g(x), we can get an unbiased estimate of its mean as 1 n Pn i=1 g(xi). We show that if we know the underlying label sampling mechanism l(x, y) we can get a different unbiased estimate of Eg(x), which uses only the labeled examples, weighted by 1/l(x, y). We suggest inferring the unknown function l(x, y) by requiring that we get identical estimates of Eg(x) using both approaches. We illustrate our method’s implementation on the California Housing data-set in section 3. In section 4 we review related work in the machine learning and statistics literature, and we conclude with a discussion in section 5. 2The importance of this is that we are required to hypothesize and fit a conditional probability model for l(x, y) only, as opposed to the full probability model for (S, X, Y ) required for, say, EM. 2 The method Let g(x) be any function of our features. We construct two different unbiased estimates of Eg(x), one based on all n data points and one based on labeled examples only, assuming P(S = 1|x, y) is known. Then, our method uses the equality in expectation of the two estimates to infer P(S = 1|x, y). Specifically, consider g(x) and also: f(x, y, s) = ½ g(x) p(S=1|x,y) if s = 1 (y observed) 0 otherwise (2) Then: Theorem 1 Assume P(S = 1|x, y) > 0 , ∀x, y. Then: E(g(X)) = E(f(X, Y, S)) . Proof: E(f(X, Y, S)) = Z X,Y,S f(x, y, s)dP(x, y, s) = = Z X g(x) Z Y P(S = 1|x, y) P(S = 1|x, y)dP(y|x)dP(x) = = Z X g(x)dP(x) = Eg(X) □ The empirical interpretation of this expectation result is: 1 n n X i=1 f(xi, yi, si) = 1 n X i:si=1 g(xi) P(S = 1|xi, yi) ≈Eg(x) ≈1 n n X i=1 g(xi) (3) which can be interpreted as relating an estimate of Eg(x) based on the complete data on the right, to the one based on labeled data only, which requires weighting that is inversely proportional to the probability of labeling, to compensate for ignoring the unlabeled data. (3) is the fundamental result we use for our purpose, leading to a “method of moments” approach to estimating l(x, y) = P(S = 1|x, y), as follows: • Assume that l(x, y) = pθ(x, y) , θ ∈Rk belongs to a parametric family with k parameters. • Select k different functions g1(x), ..., gk(x), and define f1, ..., fk correspondingly according to (2). • Demand equality of the leftmost and rightmost sums in (3) for each of g1, ..., gk, and solve the resulting k equations to get an estimate of θ. Many practical and theoretical considerations arise when we consider what “good” choices of the representative functions g1(x), ..., gk(x) may be. Qualitatively we would like to accomplish the standard desirable properties of inverse problems: uniqueness, stability and robustness. We want the resulting equations to have a unique “correct” solution. We want our functions to have low variance so the inaccuracy in (3) is minimal, and we want them to be “different” from each other to get a stable solution in the k-dimensional space. It is of course much more difficult to give concrete quantitative criteria for selecting the functions in practical situations. What we can do in practice is evaluate how stable the results we get are. We return to this topics in more detail in section 5. A second set of considerations in selecting these functions is the computational one: can we even solve the resulting inverse problems with a reasonable computational effort? In general, solving systems of more than one nonlinear equation is a very hard problem. We also need to consider the possibility of non-unique solutions. These questions are sometimes inter-related with the choice of gk(x). Suppose we wish to solve a set of non-linear equations for θ: hk(θ) = X si=1 gk(xi) pθ(xi, yi) − X i gk(xi) = 0, k = 1, . . . , K (4) The solution of (4) is similar to arg min h(θ) = arg min X m hk(θ)2 (5) Notice that every solution to (4) minimizes (5), but there may be local minima of (5) that are not solutions to (4). Hence simply applying a Newton-Raphson method to (5) is not a good idea: if we have a sufficiently good initial guess about the solution, the NewtonRaphson method converges quadratically fast; however, it can also fail to converge, if the root does not exist nearby. In practice, we can combine the Newton-Raphson method with a line search strategy that makes sure h(θ) is reduced at each iteration (the Newton step is always a descent direction of h(θ)). While this method can still occasionally fail by landing on a local minimum of h(θ), this is quite rare in practice [1]. The remedy is usually to try a new starting point. Other global algorithms based on the so called model-trust region approach are also used in practice. These methods have a reputation for robustness even when starting far from the desired zero or minimum [2]. In some cases we can employ simpler methods, since the equations we get can be manipulated algebraically to give more “friendly” formulations. We show two examples in the next sub-section. 2.1 Examples of simplified calculations We consider two situations where we can use algebra to simplify the solution of the equations our method gives. The first is the obvious application to two-class classification, where the label sampling mechanism depends on the class label only. Our method then reduces to the one suggested by [11]. The second is a more involved regression scenario, with a logistic dependence between the sampling probability and the actual label. First, consider a two-class classification scenario, where the sampling mechanism is independent of x: P(S = 1|x, y) = ½ p1 if y = 1 p0 if y = 0 Then we need two functions of x to “de-bias” our classes. One natural choice is g(x) = 1, which implies we are simply trying to invert the sampling probabilities. Assume we use one of the features g(x) = xj as our second function. Plugging these into (3) we get that to find p0, p1 we should solve: n = #{yi = 1 observed} ˆp1 + #{yi = 0 observed} ˆp0 X i xij = P si=1,yi=1 xij ˆp1 + P si=1,yi=0 xij ˆp0 which we can solve analytically to get: ˆp1 = r1n0 −r0n1 rn0 −r0n ˆp0 = r1n0 −r0n1 r1n −rn1 where nk = #{yi = k observed} , rk = P si=1,yi=k xij , k = 0, 1 As a second, more involved, example, consider a regression situation (like the satisfaction survey mentioned in the introduction), where we assume the probability of observing the response has a linear-logistic dependence on the actual response (again we assume for simplicity independence on x, although dependence on x poses no theoretical complications): P(S = 1|x, y) = exp(a + by) 1 + exp(a + by) = logit−1(a + by) (6) with a, b unknown parameters. Using the same two g functions as above gives us the slightly less friendly set of equations: n = X si=1 1 logit−1(ˆa + ˆbyi) X i xij = X si=1 xij logit−1(ˆa + ˆbyi) which with some algebra we can re-write as: 0 = X si=1 exp(−ˆbyi)(¯x0j −xij) (7) exp(ˆa)m0 = X si=1 exp(−ˆbyi) (8) where ¯x0j is the empirical mean of the j’th feature over unlabeled examples and m0 is the number of unlabeled examples. We do not have an analytic solution for these equations. However, the decomposition they offer allows us to solve them by searching first over b to solve (7), then plugging the result into (8) to get an estimate of a. In the next section we use this solution strategy on a real-data example. 3 Illustration on the California Housing data-set To illustrate our method, we take a fully labeled regression data-set and hide some of the labels based on a logistic transformation of the response, then examine the performance of our method in recovering the sampling mechanism and improving resulting prediction through de-biasing. We use the California Housing data-set [9], collected based on US Census data. It contains 20640 observations about log( median house price) throughout California regions. The eight features are: median income, housing median age, total rooms, total bedrooms, population, households, latitude and longitude. We use 3/4 of the data for modeling and leave 1/4 aside for evaluation. Of the training data, we hide some of the labels stochastically, based on the “label sampling” model: P(S = 1|y) = logit−1(1.5(y −¯y) −0.5) (9) this scheme results in having 6027 labeled training examples, 9372 training examples with the labels removed and 5241 test examples. We use equations (7,8) to estimate ˆa,ˆb based on each one of the 8 features. Figure 1 and Table 1 show the results of our analysis. In Figure 1 we display the value of P si=1 exp(−byi)(¯x0j −xj) for a range of possible values for b. We observe that all features give us 0 crossing around the correct value of 1.5. In Table 1 we give details of the 8 models estimated by a search strategy as follows: 0 1 2 3 −5000 0 5000 0 1 2 3 −4000 −2000 0 2000 4000 6000 0 1 2 3 −1 0 1 2 3 x 10 6 0 1 2 3 0 2 4 x 10 5 0 1 2 3 0 5 10 x 10 5 0 1 2 3 0 2 4 x 10 5 0 1 2 3 −3000 −2000 −1000 0 1000 0 1 2 3 −1000 −500 0 500 Figure 1: Value of RHS of (7) (vertical axis) vs value of b (horizontal axis) for the 8 different features. The correct value is b = 1.5, and so we expect to observe “zero crossings” around that value, which we indeed observe on all 8 graphs. • Find ˆb by minimizing | P si=1 exp(−byi)(¯x0j −xij)| over the range b ∈[0, 3]. • Find ˆa by plugging ˆb from above into (8). The table also shows the results of using these estimates to “de-bias” the prediction model, i.e. once we have ˆa,ˆb we use them to calculate ˆP(S = 1|y) and use the inverse of these estimated probabilities as weights in a least squares analysis of the labeled data. The table compares the predictive performance of the resulting models on the 1/4 evaluation set (5241 observations) to that of the model built using labeled data only with no weighting and that of the model built using the labeled data and the “correct” weighting based on our knowledge of the true a, b. Most of the 8 features give reasonable estimates, and the prediction models built using the resulting weighting schemes perform comparably to the one built using the “correct” weights. They generally attain MSE about 20% smaller than that of the non-weighted model built without regard to the label sampling mechanism. The stability of the resulting estimates is related to the “reasonableness” of the selected g(x) functions. To illustrate that, we also tried the function g(x) = x3 · x4 · x5/(x1 · x2) (still in combination with the identity function, so we can use (7,8)). The resulting estimates were ˆb = 3.03, ˆa = 0.074. Clearly these numbers are way outside the reasonable range of the results in Table 1. This is to be expected as this choice of g(x) gives a function with very long tails. Thus, a few “outliers” dominate the two estimates of E(g(x)) in (3) which we use to estimate a, b. 4 Related work The surge of interest in semi-supervised learning in recent years has been mainly in the context of text classification ([4, 6, 8] are several examples of many). There is also a Table 1: Summary of estimates of sampling mechanism using each of 8 features Feature b a Prediction MSE median income 1.52 -0.519 0.1148 housing median age 1.18 -0.559 0.1164 total rooms 1.58 -0.508 0.1147 total bedrooms 1.64 -0.497 0.1146 population 1.7 -0.484 0.1146 households 1.63 -0.499 0.1146 latitude 1.55 -0.514 0.1147 longitude 1.33 -0.545 0.1155 (no weighting) N/A N/A 0.1354 (true sampling model) 1.5 -0.5 0.1148 wealth of statistical literature on missing data and biased sampling (e.g. [3, 7, 10]) where methods have been developed that can be directly or indirectly applied to semi-supervised learning. Here we briefly survey some of the interesting and popular approaches and relate them to our method. The EM approach to text classification is advocated by [8]. Some ad-hoc two-step variants are surveyed by [6]. They consists of iterating between completing class labels and estimating the classification model. The main caveat of all these methods is that they ignore the sampling mechanism, and thus implicitly assume it cancels out in the likelihood function — i.e., that the sampling is random and that l(x, y) is fixed. It is possible, in principle, to remove this assumption, but that would significantly increase the complexity of the algorithms, as it would require specifying a likelihood model for the sampling mechanism and adding its parameters to the estimation procedure. The methods described by [7] and discussed below take this approach. The use of re-weighted loss to account for unknown sampling mechanisms is suggested by [4, 11]. Although they differ significantly in the details, both of these can account for label-dependent sampling in two-class classification. They do not offer solutions for other modeling tasks or for feature-dependent sampling, which our approach covers. In the missing data literature, [7] (chapter 15) and references therein offer several methods for handling “nonignorable nonresponse”. These are all based on assuming complete probability models for (X, Y, S) and designing EM algorithms for the resulting problem. An interesting example is the bivariate normal stochastic ensemble model, originally suggested by [3]. In our notation, they assume that there is an additional fully unobserved “response” zi, and that yi is observed if and only if zi > 0. They also assume that yi, zi are bivariate normal, depending on the features xi, that is: µ yi zi ¶ ∼N ·µ xiβ1 xiβ2 ¶ , µ σ2 ρσ2 ρσ2 1 ¶¸ this leads to a complex, but manageable, EM algorithm for inferring the sampling mechanism and fitting the actual model at once. The main shortcoming of this approach, as we see it, is in the need to specify a complete and realistic joint probability model engulfing both the sampling mechanism and the response function. This precludes completely the use of non-probabilistic methods for the response model part (like trees or kernel methods), and seems to involve significant computational complications if we stray from normal distributions. 5 Discussion The method we suggest in this paper allows for the separate and unbiased estimation of label-sampling mechanisms, even when they stochastically depend on the partially unobserved labels. We view this “de-coupling” of the sampling mechanism estimation from the actual modeling task at hand as an important and potentially very useful tool, both because it creates a new, interesting learning task and because the results of the sampling model can be used to “de-bias” any black-box modeling tool for the supervised learning task through inverse weighting (or sampling, if the chosen tool does not take weights). Our method of moments suffers from the same problems all such methods (and inverse problems in general) share, namely the uncertainty about the stability and validity of the results. It is difficult to develop general theory for stable solutions to inverse problems. What we can do in practice is attempt to validate the estimates we get. We have already seen one approach for doing this in section 3, where we used multiple choices for g(x) and compared the resulting estimates of the parameters determining l(x, y). Even if we had not known the true values of a and b, the fact that we got similar estimates using different features would reassure us that these estimates were reliable, and give us an idea of their uncertainty. A second approach for evaluating the resulting estimates could be to use bootstrap sampling, which can be used to calculate bootstrap confidence intervals of the parameter estimates. The computational issues also need to be tackled if our method is to be applicable for large scale problems with complex sampling mechanisms. We have suggested a general methodology in section 2, and some ad-hoc solutions for special cases in section 2.1. However we feel that a lot more can be done to develop efficient and widely applicable methods for solving the moment equations. Acknowledgments We thank John Langford and Tong Zhang for useful discussions. References [1] Acton, F. (1990) Numerical Methods That Work. Washington: Math. Assoc. of America. [2] Dennis, J. & Schnabel, R. (1983) Numerical Methods for Unconstrained Optimization and Nonlinear Equations. New Jersey: Prentice-Hall. [3] Heckman, J.I. (1976). The common structure of statistical models for truncation, sample selection and limited dependent variables, and a simple estimator for such models. Annals of Economic and Social Measurement 5:475-492. [4] Lee, W.S. & Liu, B. (2003). Learning with Positive and Unlabeled Examples Using Weighted Logistic Regression. ICML-03 [5] Lin, Y., Lee, Y. & Wahba, G. (2000). Support vector machines for classification in nonstandard situations. Machine Learning, 46:191-202. [6] Liu, B., Dai, Y., Li, X., Lee, W.S. & Yu, P. (2003). Building Text Classifiers Using Positive and Unlabeled Examples. Proceedings ICDM-03 [7] Little, R. & Rubin, D. (2002). Statistical Analysis with Missing Data, 2nd Ed. . Wiley & Sons. [8] Nigam, K., McCallum , A., Thrun, S. & Mitchell, T. (2000) Text Classification from Labeled and Unlabeled Documents using EM. Machine Learning 39(2/3):103-134. [9] Pace, R.K. & Barry, R. (1997). Sparse Spatial Autoregressions. Stat. & Prob. Let., 33 291-297. [10] Vardi, Y. (1985). Empirical Distributions in Selection Bias Models. Annals of Statistics, 13. [11] Zou, H., Zhu, J. & Hastie, T. (2004). Automatic Bayes Carpentary in Semi-Supervised Classification. Unpublished.
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Semi-Markov Conditional Random Fields for Information Extraction Sunita Sarawagi Indian Institute of Technology Bombay, India sunita@iitb.ac.in William W. Cohen Center for Automated Learning & Discovery Carnegie Mellon University wcohen@cs.cmu.edu 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 s′, 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 CRFs and Semi-CRFs 2.1 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 = ⟨f 1, . . . , f K⟩, 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 measurements, and let F(x, y) = P|x| 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 Pr(y|x, W) = 1 Z(x)eW·F(x,y) (1) where W is a weight vector over the components of F, and Z(x) = P y′ eW·F(x,y′). To extend this to the semi-Markov case, let s = ⟨s1, . . . , sp⟩denote a segmentation of x, where segment sj = ⟨tj, uj, yj⟩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 = ⟨(1, 1, O), (2, 2, O), (3, 3, O), (4, 4, O), (5, 6, I), (7, 7, O), (8, 9, I)⟩, corresponding to the label sequence y = ⟨O, O, O, O, I, I, O, I, I⟩. We now assume a vector g of segment feature functions g = ⟨g1, . . . , gK⟩, each of which maps a triple (j, x, s) to a measurement gk(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 gk 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 gk(j, x, s) can be rewritten as gk(j, x, s) = g′k(yj, yj−1, x, tj, uj) (2) for an appropriately defined g′k. In the rest of the paper, we will drop the g′ notation and use g for both versions of the segment-level feature functions. A semi-CRF is then an estimator of the form Pr(s|x, W) = 1 Z(x)eW·G(x,s) (3) where again W is a weight vector for G and Z(x) = P s′ eW·G(x,s′). 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 argmax s Pr(s|x, W) = argmax sW · G(x, s) = argmax sW · X j g(yj, yj−1, x, tj, uj) 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, s′) for any s′ ∈si:y. Omitting the subscripts, the following recursive calculation implements a semi-Markov analog of the usual Viterbi algorithm: V (i, y) = ( maxy′,d=1...L V (i −d, y′) + W · g(y, y′, x, i −d + 1, i) if i > 0 0 if i = 0 −∞ if i < 0 (4) 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 µ be the average length of all segments with label I. Now consider the segment feature gk1(j, x, s) = (dj −µ)2 · [[yj = I]]. After training, the contribution of this feature toward Pr(s|x) associated with a length-d entity will be proportional to ewk·(d−µ)2—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 gk2(j, x, y) = dj · [[yj = I]]. In contrast to the Gaussian-length feature above, gk2 is “equivalent to” a local feature function f(i, x, y) = [[yi = I]], in the following sense: for every triple x, y, s, where y is the tags for s, P j gk2(j, x, s) = P i f(i, s, y). Thus a semi-CRF model based on the single feature gk2 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 = {(xℓ, sℓ)}N ℓ=1. Following the notation of Sha and Pereira [16], we express the loglikelihood over the training sequences as L(W) = X ℓ log Pr(sℓ|xℓ, W) = X ℓ (W · G(xℓ, sℓ) −log ZW(xℓ)) (5) 1Assuming 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: ∇L(W) = X ℓ G(xℓ, sℓ) − P s′ G(s′, xℓ)eW·G(xℓ,s′) ZW(xℓ) (6) = X ℓ G(xℓ, sℓ) −EPr(s′|W)G(xℓ, s′) (7) The first set of terms are easy to compute. However, to compute the the normalizer, ZW(xℓ), and the expected value of the features under the current weight vector, EPr(s′|W)G(xℓ, s′), we must use the Markov property of G to construct a dynamic programming algorithm, similar for forward-backward. We thus define α(i, y) as the value of P s′∈si:y eW·G(s′,x) where again si:y denotes all segmentations from 1 to i ending at i and labeled y. For i > 0, this can be expressed recursively as α(i, y) = L X d=1 X y′∈Y α(i −d, y′)eW·g(y,y′,x,i−d+1,i) with the base cases defined as α(0, y) = 1 and α(i, y) = 0 for i < 0. The value of ZW(x) can then be written as ZW(x) = P y α(|x|, y). A similar approach can be used to compute the expectation P s′ G(xℓ, s′)eW·G(xℓ,s′). For the k-th component of G, let ηk(i, y) be the value of the sum P s′∈si:y Gk(s′, xℓ)eW·G(xℓ,s′), restricted to the part of the segmentation ending at position i. The following recursion2 can then be used to compute ηk(i, y): ηk(i, y) = PL d=1 P y′∈Y(ηk(i−d, y′) + α(i−d, y′)gk(y, y′, x, i−d + 1, i))eW·g(y,y′,x,i−d+1,i) Finally we let EPr(s′|W)Gk(s′, x) = 1 ZW(x) P y ηk(|x|, y). 3 Experiments with NER data 3.1 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. 2As in the forward-backward algorithm for chain CRFs [16], space requirements here can be reduced from ML|Y| to M|Y|, where M is the length of the sequence, by pre-computing an appropriate set of β 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 ⟨xtj . . . xuj⟩, a dictionary feature is defined as gD,sim(j, x, s) = argmax u∈Dsim(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∈Dsim(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 3F1 is defined as 2*precision*recall/(precision+recall.) 0 10 20 30 40 50 60 70 80 90 100 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 F1 span accuracy Fraction of available training data Address_State CRF/4 SemiCRF+int CRF/4+dict SemiCRF+int+dict 65 70 75 80 85 90 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 F1 span accuracy Fraction of available training data Address_City CRF/4 SemiCRF+int CRF/4+dict SemiCRF+int+dict 65 70 75 80 85 90 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 F1 span accuracy Fraction of available training data Email_Person CRF/4 SemiCRF+int CRF/4+dict SemiCRF+int+dict 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 +internal dict +external dict +both dictionaries F1 F1 ∆base F1 ∆base F1 ∆base ∆extern CRF/1 state 20.8 44.5 113.9 69.2 232.7 55.2 165.4 -67.3 title 28.5 3.8 -86.7 38.6 35.4 19.9 -30.2 -65.6 person 67.6 48.0 -29.0 81.4 20.4 64.7 -4.3 -24.7 city 70.3 60.0 -14.7 80.4 14.4 69.8 -0.7 -15.1 company 51.4 16.5 -67.9 55.3 7.6 15.6 -69.6 -77.2 CRF/4 state 15.0 25.4 69.3 46.8 212.0 43.1 187.3 -24.7 title 23.7 7.9 -66.7 36.4 53.6 14.6 -38.4 -92.0 person 70.9 64.5 -9.0 82.5 16.4 74.8 5.5 -10.9 city 73.2 70.6 -3.6 80.8 10.4 76.3 4.2 -6.1 company 54.8 20.6 -62.4 61.2 11.7 25.1 -54.2 -65.9 semi-CRF state 25.6 35.5 38.7 62.7 144.9 65.2 154.7 9.8 title 33.8 37.5 10.9 41.1 21.5 40.2 18.9 -2.5 person 72.2 74.8 3.6 82.8 14.7 83.7 15.9 1.2 city 75.9 75.3 -0.8 84.0 10.7 83.6 10.1 -0.5 company 60.2 59.7 -0.8 60.9 1.2 60.9 1.2 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 4I.e., the both-dictionary version when external dictionaries are available, and the internaldictionary only version otherwise. 5Order-L CRFs were implemented by replacing the label set Y with YL. We limited experiments to L ≤3 for computational reasons. CRF/1 CRF/4 semi-CRF L = 1 L = 2 L = 3 L = 1 L = 2 L = 3 Address State 20.8 20.1 19.2 15.0 16.4 16.4 25.6 Address City 70.3 71.0 71.2 73.2 73.9 73.7 75.9 Email persons 67.6 63.7 66.7 70.9 70.7 70.4 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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Adaptive Manifold Learning Jing Wang, Zhenyue Zhang Department of Mathematics Zhejiang University, Yuquan Campus, Hangzhou, 310027, P. R. China wroaring@sohu.com zyzhang@zju.edu.cn Hongyuan Zha Department of Computer Science Pennsylvania State University University Park, PA 16802 zha@cse.psu.edu Abstract Recently, there have been several advances in the machine learning and pattern recognition communities for developing manifold learning algorithms to construct nonlinear low-dimensional manifolds from sample data points embedded in high-dimensional spaces. In this paper, we develop algorithms that address two key issues in manifold learning: 1) the adaptive selection of the neighborhood sizes; and 2) better fitting the local geometric structure to account for the variations in the curvature of the manifold and its interplay with the sampling density of the data set. We also illustrate the effectiveness of our methods on some synthetic data sets. 1 Introduction Recently, there have been advances in the machine learning community for developing effective and efficient algorithms for constructing nonlinear low-dimensional manifolds from sample data points embedded in high-dimensional spaces, emphasizing simple algorithmic implementation and avoiding optimization problems prone to local minima. The proposed algorithms include Isomap [6], locally linear embedding (LLE) [3] and its variations, manifold charting [1], hessian LLE [2] and local tangent space alignment (LTSA) [7], and they have been successfully applied in several computer vision and pattern recognition problems. Several drawbacks and possible extensions of the algorithms have been pointed out in [4, 7] and the focus of this paper is to address two key issues in manifold learning: 1) how to adaptively select the neighborhood sizes in the k-nearest neighbor computation to construct the local connectivity; and 2) how to account for the variations in the curvature of the manifold and its interplay with the sampling density of the data set. We will discuss those two issues in the context of local tangent space alignment (LTSA) [7], a variation of locally linear embedding (LLE) [3] (see also [5],[1]). We believe the basic ideas we proposed can be similarly applied to other manifold learning algorithms. We first outline the basic steps of LTSA and illustrate its failure modes using two simple examples. Given a data set X = [x1, . . . , xN] with xi ∈Rm, sampled (possibly with noise) from a d-dimensional manifold (d < m), LTSA proceeds in the following steps. 1) LOCAL NEIGHBORHOOD CONSTRUCTION. For each xi, i = 1, . . . , N, determine a set Xi = [xi1, . . . , xiki] of its neighbors (ki nearest neighbors, for example). −1 0 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 0 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 k = 4 −2 0 2 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 k = 6 −2 0 2 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 k = 8 −5 0 5 0 1 2 3 4 5 6 7 8 9 10 −20 0 20 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 −20 0 20 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 −20 0 20 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 Figure 1: The data sets (first column) and computed coordinates τi by LTSA vs. the centered arc-length coordinates Top row: Example 1. Bottom row: Example 2. 2) LOCAL LINEAR FITTING. Compute an orthonormal basis Qi for the d-dimensional tangent space of the manifold at xi, and the orthogonal projection of each xij to the tangent space: θ(i) j = QT i (xij −¯xi) where ¯xi is the mean of the neighbors. 3) LOCAL COORDINATES ALIGNMENT. Align the N local projections Θi = [θ(i) 1 , · · · , θ(i) ki ], i = 1, . . . , N, to obtain the global coordinates τ1, . . . , τN. Such an alignment is achieved by minimizing the global reconstruction error X i ∥Ei∥2 2 ≡ X i ∥Ti(I −1 ki eeT ) −LiΘi∥2 2 (1.1) over all possible Li ∈Rd×d and row-orthonormal T = [τ1, . . . , τN] ∈Rd×N, where Ti = [τi1, . . . , τiki ] with the index set {i1, . . . , iki} determined by the neighborhood of each xi, and e is a vector of all ones. Two strategies are commonly used for selecting the local neighborhood size ki: one is k nearest neighborhood ( k-NN with a constant k for all the sample points) and the other is ϵneighborhood [3, 6]. The effectiveness of the manifold learning algorithms including LTSA depends on the manner of how the nearby neighborhoods overlap with each other and the variation of the curvature of the manifold and its interplay with the sampling density [4]. We illustrate those issues with two simple examples. Example 1. We sample data points from a half unit circle xi = [cos(ti), sin(ti)]T , i = 1 . . . , N. It is easy to see that ti represent the arc-length of the circle. We choose ti ∈[0, π] according to ti+1 −ti = 0.1(0.001 + | cos(ti)|) starting at t1 = 0, and set N = 152 so that tN ≤π and tN+1 > π. Clearly, the half circle has unit curvature everywhere. This is an example of highly-varying sampling density. Example 2. The date set is generated as xi = [ti, 10e−t2 i ]T , i = 1 . . . , N, where ti ∈ [−6, 6] are uniformly distributed. The curvature of the 1-D curve at parameter value t is given by cg(t) = 20|1 −2t2|e−t2 (1 + 40t2e−2t2)3/2 which changes from mint cg(t) = 0 to maxt cg(t) = 20 over t ∈[−6, 6]. We set N = 180. This is an example of highly-varying curvature. For the above two data sets, LTSA with constant k-NN strategy fails for any reasonable k we have tested. So does LTSA with constant ϵ-neighborhoods. In the first column of Figure 1, we plot these two data sets. The computed coordinates by LTSA with constant kneighborhoods are plotted against the centered arc-length coordinates for a selected range of k (ideally, the plots should display points on a straight line of slops ±π/4). 2 Adaptive Neighborhood Selection In this section, we propose a neighborhood contraction and expansion algorithm for adaptively selecting ki at each sample point xi. We assume that the data are generated from a parameterized manifold, xi = f(τi), i = 1, . . . , N, where f : Ω⊂Rd →Rm. If f is smooth enough, using first-order Taylor expansion at a fixed τ, for a neighboring ¯τ, we have f(¯τ) = f(τ) + Jf(τ) · (¯τ −τ) + ϵ(τ, ¯τ), (2.2) where Jf(τ) ∈Rm×d is the Jacobi matrix of f at τ and ϵ(τ, ¯τ) represents the error term determined by the Hessian of f , ∥ϵ(τ, ¯τ)∥≈cf(τ)∥¯τ −τ∥2 2, where cf(τ) ≥0 represents the curvature of the manifold at τ. Setting τ = τi and ¯τ = τij gives xij = xi + Jf(τi) · (τij −τi) + ϵ(τi, τij). (2.3) A point xij can be regarded as a neighbor of xi with respect to the tangent space spanned by the columns of Jf(τi) if ∥τij −τi∥2 is small and ∥ϵ(τi, τij)∥2 ≪∥Jf(τi) · (τij −τi)∥2. The above conditions, however, are difficult to verify in practice since we do not know Jf(τi). To get around this problem, consider an orthogonal basis matrix Qi of the tangent space spanned by the columns of Jf(τi) which can be approximately computed by the SVD of Xi −¯xieT , where ¯xi is the mean of the neighbors xij = f(τij), j = 1, . . . , ki. Note that ¯xi = 1 ki ki X j=1 xij = xi + Jf(τi) · (¯τi −τi) + ¯ϵi, where ¯ϵi is the mean of ϵ(τi, τi1), . . . , ϵ(τi, τik1 ). Eliminating xi in (2.3) by the representation above yields xij = ¯xi + Jf(τi) · (τij −¯τi) + ϵ(i) j with ϵ(i) j = ϵ(τi, τij) −¯ϵi. Let θ(i) j = QT i (xij −¯xi), we have xij = ¯xi + Qiθ(i) j + ϵ(i) j . Thus, xij can be selected as a neighbor of xi if the orthogonal projection θ(i) j is small and ∥ϵ(i) j ∥2 = ∥xij −¯xi −Qiθ(i) j ∥2 ≪∥Qiθ(i) j ∥2 = ∥θ(i) j ∥2. (2.4) Assume all the xij satisfy the above inequality, then we should approximately have ∥(I −QiQT i )(Xi −x0eT )∥F ≤η∥QT i (Xi −x0eT )∥F (2.5) We will use (2.5) as a criterion for adaptive neighbor selection, starting with a K-NN at each sample point xi with a large enough initial K and deleting points one by one until (2.5) holds. This process will terminate when the neighborhood size equals d + k0 for some small k0 and (2.5) is not true. In that case, we may need to reselect a k-NN that minimizes the ratio ∥(I−QiQT i )(Xi−¯xieT )∥F ∥QT i (Xi−¯xieT )∥F as the neighborhood set as is detailed below. NEIGHBORHOOD CONTRACTION. C0. Determine the initial K and K-NN neighborhood X(K) i = [xi1, . . . , xiK] for xi, ordered in non-decreasing distances to xi, ∥xi1 −xi∥≤∥xi2 −xi∥≤. . . ≤∥xiK −xi∥. Set k = K. C1. Let ¯x(k) i be the column mean of X(k) i . Compute the orthogonal basis matrix Q(k) i , the d largest singular vectors of X(k) i −¯x(k) i eT . Set Θ(k) i = (Q(k) i )T (X(k) i − ¯x(k) i eT ). C2. If ∥X(k) i −¯x(k) i eT −Q(k) i Θ(k) i ∥F < η∥Θ(k) i ∥F , then set Xi = X(k) i , Θi = Θ(k) i , and terminate. C3. If k > d+k0, then delete the last column of X(k) i to obtain X(k−1) i , set k := k−1, and go to step C1, otherwise, go to step C4. C4. Let k = arc mind+k0≤j≤K ∥X(j) i −¯x(j) i eT −Q(j) i Θ(j) i ∥F ∥Θ(j) i ∥F , and set Xi = X(k) i , Θi = Θ(k) i . Step C4 means that if there is no k-NN (k ≥d + k0) satisfying (2.5), then the contracted neighborhood Xi should be one that minimizes ∥Xi−¯xieT −QiΘi∥F ∥Θi∥F . Once the contraction step is done we can still add back some of unselected xij to increase the overlap of nearby neighborhoods while still keep (2.5) intact. In fact, we can add xij if ∥xij −¯xi −Qiθj∥≤η∥θj∥which is demonstrated in the following result (we refer to [8] for the proof). Theorem 2.1 Let Xi = [xi1, . . . , xik] satisfy (2.5). Furthermore, we assume ∥xij −x0 −Qiθ(i) j ∥≤η∥θ(i) j ∥, j = k + 1, . . . , k + p, (2.6) where θ(i) j = QT i (xij −x0). Denote by ˜xi the column mean of the expanded matrix ˜Xi = [Xi, xik+1, . . . xik+p]. Then for the left-singular vector matrix ˜Qi corresponding to the d largest singular values of ˜Xi −˜xieT , ∥(I −˜Qi ˜QT i )( ˜Xi −˜xieT )∥F ≤η ∥˜QT i ( ˜Xi −˜xieT )∥F + √p k + p∥ k+p X j=k+1 θ(i) j ∥2 . The above result shows that if the mean of the projections θ(i) j of the expanding neighbors is small and/or the number of the expanding points are relatively small, then approximately, ∥(I −˜Qi ˜QT i )( ˜Xi −˜xieT )∥F ≤η∥˜QT i ( ˜Xi −˜xieT )∥F . NEIGHBORHOOD EXPANSION. E0. Set ki to be the column number of Xi obtained by the neighborhood contracting step. For j = ki + 1, . . . , K, compute θ(i) j = QT i (xij −¯xi). E1. Denote by Ji the index subset of j’s, ki < j ≤K, such that ∥(I −QiQT i )(xij − ¯xi)∥2 ≤∥θ(i) j ∥2. Expand Xi by adding xij, j ∈Ji. Example 3. We construct the data points as xi = [sin(ti), cos(ti), 0.02ti]T , i = 1, . . . , N, with ti ∈[0, 4π] uniformly distributed, which is plotted in the top-left panel in Figure 2. −1 0 1 −1 0 1 0 0.2 0.4 −10 0 10 −0.1 −0.05 0 0.05 0.1 (c) k=9 −10 0 10 −0.1 −0.05 0 0.05 0.1 (b) k=8 −10 0 10 −0.2 0 0.2 0.4 0.6 0.8 (a) k=7 −10 0 10 −0.05 0 0.05 0.1 0.15 (e) k=15 −10 0 10 −0.05 0 0.05 0.1 0.15 (f) k=30 −10 0 10 −0.15 −0.1 −0.05 0 0.05 (g) k=35 −10 0 10 −0.1 −0.05 0 0.05 0.1 (d) k=30 Figure 2: Plots of the data sets (top left), the computed coordinates τi by LTSA vs. the centered arc-length coordinates (a ∼c), the computed coordinates τi by LTSA with neighborhood C contraction vs the centered arc-length coordinates (e ∼g), and the computed coordinates τi by LTSA with neighborhood contraction and expansion vs. the centered arc-length coordinates (bottom left) LTSA with constant k-NN fails for any k: small k leads to lack of necessary overlap among the neighborhoods while for large k, the computed tangent space can not represent the local geometry well. In (a ∼c) of Figure 2, we plot the coordinates computed by LTSA vs. the arc-length of the curve. Contracting the neighborhoods without expansion also results in bad results, because of small sizes of the resulting neighborhoods, see (e ∼g) of Figure 2. Panel (d) of Figure 2 gives an excellent result computed by LTSA with both neighborhood contraction and expansion. We want mention that our adaptive strategies also work well for noisy data sets, we refer the readers to [8] for some examples. 3 Alignment incorporating variations of manifold curvature Let Xi = [xi1, . . . , xiki ] consists of the neighbors determined by the contraction and expansion steps in the above section. In (1.1), we can show that the size of the error term ∥Ei∥2 depends on the size of the curvature of manifold at sample point xi [8]. To make the minimization in (1.1) more uniform, we need to factor out the effect of the variations of the curvature. To this end, we pose the following minimization problem, min T,{Li} X i 1 ki ∥(Ti(I −1 ki eeT ) −LiΘi)D−1 i ∥2 2, (3.7) where Di = diag(φ(θ(i) 1 ), . . . , φ(θ(i) ki )), and φ(θ(i) j ) is proportional to the curvature of the manifold at the parameter value θi, the computation of which will be discussed below. For fixed T, the optimal Li is given by Li = Ti(Iki −1 ki eeT )Θ+ i = TiΘ+ i . Substituting it into (3.7), we have the reduced minimization problem min T X i 1 ki ∥Ti(Iki −1 ki eeT −Θ+ i Θi)D−1 i ∥2 2 Imposing the normalization condition TT T = I, a solution to the minimization problem above is given by the d eigenvectors corresponding to the second to (d + 1)st smallest eigenvalues of the following matrix B ≡(SW) diag(D2 1/k1, . . . , D2 n/kn)(SW)T , where W = (Iki −1 ki eeT )(Iki −Θ+ i Θi). Second-order analysis of the error term in (1.1) shows that we can set φi(θ(i) j ) = γ + cf(τi)∥θ(i) j ∥2 with a small positive constant γ to ensure φi(θ(i) j ) > 0, and cf(τi) ≥0 represents the mean of curvatures cf(τi, τij) for all neighbors of xi. Let Qi denote the orthonormal matrix of the largest d right singular vectors of Xi(I − 1 ki eeT ). We can approximately compute cf(τi) as follows. cf(τi) ≈ 1 ki −1 ki X ℓ=2 arccos(σmin(QT i Qiℓ)) ∥θℓ∥2 . where σmin(·) is the smallest singular value of a matrix. Then the diagonal weights φ(θi) can be computed as φi(θ(i) j ) = η + ∥θj∥2 2 ki −1 ki X ℓ=2 arccos(σmin(QT i Qiℓ)) ∥θℓ∥2 . With the above preparation, we are now ready to present the adaptive LTSA algorithm. Given a data set X = [x1, . . . , xN], the approach consists of the following steps: Step 1. Determining the neighborhood Xi = [xi1, . . . , xiki ] for each xi, i = 1, . . . , N, using the neighborhood contraction/expansion steps in Section 2. Step 2. Compute the truncated SVD, say QiΣiV T i of Xi(I −1 ki eeT ) with d columns in both Qi and Vi, the projections θ(i) ℓ = QT i (xiℓ−¯xi) with the mean ¯xi of the neighbors, and denote Θi = [θ(i) 1 , . . . , θ(i) ki ]. Step 3. Estimate the curvatures as follows. For each i = 1, . . . , N, ci = 1 ki −1 ki−1 X ℓ=2 arccos(σmin(QT i Qiℓ)) ∥θ(i) ℓ∥2 , Step 4. Construct alignment matrix. For i = 1, . . . , N, set Wi = Iki−[ 1 √ki e, Vi][ 1 √ki e, Vi]T , Di = γI+ diag(ci∥θ(i) 1 ∥2 2, . . . , ci∥θ(i) ki ∥2 2), where γ is a small constant number (usually we set γ = 1.0−6). Set initial B = 0. Update B iteratively by B(Ii, Ii) := B(Ii, Ii) + WiD−1 i D−1 i W T i /ki, i = 1, . . . , N. Step 5. Align global coordinates. Compute the d + 1 smallest eigen-vectors of B and pick up the eigenvector [u2, . . . , ud+1] matrix corresponding to the 2nd to d + 1st smallest eigenvalues, and set T = [u2, . . . , ud+1]T . 4 Experimental Results In this section, we present several numerical examples to illustrate the performance of the adaptive LTSA algorithm. The test data sets include curves in 2D/3D Euclidean spaces. −5 0 5 0 2 4 6 8 10 −20 0 20 −0.05 0 0.05 0.1 0.15 k=4 −20 0 20 −0.05 0 0.05 0.1 0.15 k=6 −20 0 20 −0.15 −0.1 −0.05 0 0.05 k=8 −20 0 20 −0.1 −0.05 0 0.05 0.1 0.15 k=16 −1 0 1 0 0.2 0.4 0.6 0.8 1 −2 0 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −2 0 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −2 0 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −2 0 2 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 Figure 3: The computed coordinates τi by LTSA taking into account curvature and variable size of neighborhood. First we apply the adaptive LTSA to the date sets shown in Examples 1 and 2. Adaptive LTSA with different starting k’s works every well. See Figure 3. It shows that for these tow data sets, the adaptive LTSA is not sensitive to the choice of the starting k or the variations in sampling densities and manifold curvatures. Next, we consider the swiss-roll surface defined by f(s, t) = [s cos(s), t, s sin(s)]T . It is easy to see that Jf(s, t)T Jf(s, t) = diag(1 + s2, 1). Denoting s = s(r) the inverse transformation of r = r(s) defined by r(s) = s Z 0 p 1 + α2 dα = 1 2(s p 1 + s2 + arcsinh(s)), the swiss-roll surface can be parameterized as ˆf(r, t) = [s(t) cos(s(r)), t, s(r) sin(s(r))]T and ˆf is isometric with respect to (r, t). In the left figure of Figure 4, we show there is a distortion between the computed coordinates by LTSA with the best-fit neighborhood size (bottom left) and the generating coordinates (r, t)T (top right). In the right panel of the bottom row of the left figure of Figure 4, we plot the computed coordinates by the adaptive LTSA with initial neighborhood size k = 30. (In fact, the adaptive LTSA is insensitive to k and we will get similar results with a larger or smaller initial k). We can see that the computed coordinates by the adaptive LTSA can recover the generating coordinates well without much distortion. Finally we applied both LTSA and the adaptive LTSA to a 2D manifold with 3 peaks embedded in a 100 dimensional space. The data points are generated as follows. First we generate N = 2000 3D points, yi = (ti, si, h(ti, si))T , where ti and si randomly distributed in the interval [−1.5, 1.5] and h(t, s) is defined by h(t, s) = e−20t2−20s2 −e−10t2−10(s+1)2 −e−10(1+t)2−10s2. Then we embed the 3D points into a 100D space by xQ i = Qyi, xH i = Hyi, where Q ∈R100×3 is a random orthonormal matrix resulting in an orthogonal transformation and H ∈R100×3 a matrix with its singular values uniformly distributed in (0, 1) resulting in an affine transformation. In the top row of the right figure of Figure 4, we plot the −10 −5 0 5 10 −1 0 1 −5 0 5 10 swiss role 0 10 20 30 40 50 −1 −0.5 0 0.5 1 Generating Coordinate −0.04 −0.02 0 0.02 0.04 −0.04 −0.02 0 0.02 0.04 0.06 −0.02 0 0.02 0.04 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 −0.1 −0.05 0 0.05 −0.1 −0.05 0 0.05 0.1 (a) −0.05 0 0.05 0.1 −0.08 −0.06 −0.04 −0.02 0 0.02 0.04 (b) −0.05 0 0.05 −0.05 0 0.05 (c) −0.06 −0.04 −0.02 0 0.02 0.04 −0.06 −0.04 −0.02 0 0.02 0.04 0.06 (d) Figure 4: Left figure: 3D swiss-roll and the generating coordinates (top row), computed 2D coordinates by LTSA with the best neighborhood size k = 15 (bottom left) and computed 2D coordinates by adaptive LTSA (bottom right). Right figure: coordinates computed by LTSA for the orthogonally embedded 100D data set {xQ i } (a) and the affinely embedded 100D data set {xH i } (b), and the coordinates computed by the adaptive LTSA for {xQ i } (c) and {xH i } (d). computed coordinates by LTSA for xQ i (shown in (a)) and xH i (shown in (b)) with best-fit neighborhood size k = 15. We can see the deformations (stretching and compression) are quite prominent. In the bottom row of the right figure of Figure 4, we plot the computed coordinates by the adaptive LTSA for xQ i (shown in (c)) and xH i (shown in (d)) with initial neighborhood size k = 15. It is clear that the adaptive LTSA gives a much better result. References [1] M. Brand. Charting a manifold. Advances in Neural Information Processing Systems, 15, MIT Press, 2003. [2] D. Donoho and C. Grimes. Hessian Eigenmaps: new tools for nonlinear dimensionality reduction. Proceedings of National Academy of Science, 5591-5596, 2003. [3] S. Roweis and L. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290: 2323–2326, 2000. [4] L. Saul and S. Roweis. Think globally, fit locally: unsupervised learning of nonlinear manifolds. Journal of Machine Learning Research, 4:119-155, 2003. [5] E. Teh and S. Roweis. Automatic Alignment of Local Representations. Advances in Neural Information Processing Systems, 15, MIT Press, 2003. [6] J. Tenenbaum, V. De Silva and J. Langford. A global geometric framework for nonlinear dimension reduction. Science, 290:2319–2323, 2000. [7] Z. Zhang and H. Zha. Principal Manifolds and Nonlinear Dimensionality Reduction via Tangent Space Alignment. SIAM J. Scientific Computing, 26:313–338, 2004. [8] J. Wang, Z. Zhang and H. Zha. Adaptive Manifold Learning. Technical Report CSE04-21, Dept. CSE, Pennsylvania State University, 2004.
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Euclidean Embedding of Co-occurrence Data Amir Globerson1 Gal Chechik2 Fernando Pereira3 Naftali Tishby1 1 School of computer Science and Engineering, Interdisciplinary Center for Neural Computation The Hebrew University Jerusalem, 91904, Israel 2 Computer Science Department, Stanford University, Stanford, CA 94305, USA 3 Department of Computer and Information Science, University of Pennsylvania, Philadelphia, PA 19104, USA Abstract Embedding algorithms search for low dimensional structure in complex data, but most algorithms only handle objects of a single type for which pairwise distances are specified. This paper describes a method for embedding objects of different types, such as images and text, into a single common Euclidean space based on their co-occurrence statistics. The joint distributions are modeled as exponentials of Euclidean distances in the low-dimensional embedding space, which links the problem to convex optimization over positive semidefinite matrices. The local structure of our embedding corresponds to the statistical correlations via random walks in the Euclidean space. We quantify the performance of our method on two text datasets, and show that it consistently and significantly outperforms standard methods of statistical correspondence modeling, such as multidimensional scaling and correspondence analysis. 1 Introduction Embeddings of objects in a low-dimensional space are an important tool in unsupervised learning and in preprocessing data for supervised learning algorithms. They are especially valuable for exploratory data analysis and visualization by providing easily interpretable representations of the relationships among objects. Most current embedding techniques build low dimensional mappings that preserve certain relationships among objects and differ in the relationships they choose to preserve, which range from pairwise distances in multidimensional scaling (MDS) [4] to neighborhood structure in locally linear embedding [12]. All these methods operate on objects of a single type endowed with a measure of similarity or dissimilarity. However, real-world data often involve objects of several very different types without a natural measure of similarity. For example, typical web pages or scientific papers contain varied data types such as text, diagrams, images, and equations. A measure of similarity between words and pictures is difficult to define objectively. Defining a useful measure of similarity is even difficult for some homogeneous data types, such as pictures or sounds, where the physical properties (pitch and frequency in sounds, color and luminosity distribution in images) do not directly reflect the semantic properties we are interested in. The current paper addresses this problem by creating embeddings from statistical associations. The idea is to find a Euclidean embedding in low dimension that represents the empirical co-occurrence statistics of two variables. We focus on modeling the conditional probability of one variable given the other, since in the data we analyze (documents and words, authors and terms) there is a clear asymmetry which suggests a conditional model. Joint models based on similar principles can be devised in a similar fashion, and may be more appropriate for symmetric data. We name our method CODE for Co-Occurrence Data Embedding. Our cognitive notions are often built through statistical associations between different information sources. Here we assume that those associations can be represented in a lowdimensional space. For example, pictures which frequently appear with a given text are expected to have some common, locally low-dimensional characteristic that allows them to be mapped to adjacent points. We can thus rely on co-occurrences to embed different entity types, such as words and pictures, genes and expression arrays, into the same subspace. Once this embedding is achieved it also naturally defines a measure of similarity between entities of the same kind (such as images), induced by their other corresponding modality (such as text), providing a meaningful similarity measure between images. Embedding of heterogeneous objects is performed in statistics using correspondence analysis (CA), a variant of canonical correlation analysis for count data [8]. These are related to Euclidean distances when the embeddings are constrained to be normalized. However, as we show below, removing this constraint has great benefits for real data. Statistical embedding of same-type objects was recently studied by Hinton and Roweis [9]. Their approach is similar to ours in that it assumes that distances induce probabilistic relations between objects. However, we do not assume that distances are given in advance, but instead we derive them from the empirical co-occurrence data. The Parametric Embedding method [11], which also appears in the current proceedings, is formally similar to our method but is used in the setting of supervised classification. 2 Problem Formulation Let X and Y be two categorical variables with an empirical distribution ¯p(x, y). No additional assumptions are made on the values of X and Y or their relationships. We wish to model the statistical dependence between X and Y through an intermediate Euclidean space Rd and mappings ⃗φ : X →Rd and ⃗ψ : Y →Rd. These mappings should reflect the dependence between X and Y in the sense that the distance between each ⃗φ(x) and ⃗ψ(y) determines their co-occurrence statistics. We focus in this manuscript on modeling the conditional distribution p(y|x)1, and define a model which relates conditional probabilities to distances by p(y|x) = ¯p(y) Z(x)e−d2 x,y ∀x ∈X, ∀y ∈Y (1) where d2 x,y ≡|⃗φ(x)−⃗ψ(y)|2 = Pd k=1(φk(x)−ψk(y))2 is the Euclidean distance between ⃗φ(x) and ⃗ψ(y) and Z(x) is the partition function for each value of x. This partition function equals Z(x) = P y ¯p(y)e−d2 x,y and is thus the empirical mean of the exponentiated distances from x (therefore Z(x) ≤1). This model directly relates the ratio p(y|x) ¯p(y) to the distance between the embedded x and y. The ratio decays exponentially with the distance, thus for any x, a closer y will have 1We have studied several other models of the joint rather than the conditional distribution. These differ by the way the marginals are modeled and will be described elsewhere Figure 1: Embedding of X, Y into the same d-dimensional space. a higher interaction ratio. As a result of the fast decay, the closest objects dominate the distribution. The model of Eq. 1 can also be described as the result of a random walk in the low-dimensional space illustrated in Figure 1. When y has a uniform marginal, the probability p(y|x) corresponds to a random walk from x to y, with transition probability inversely related to distance. We now turn to the task of learning ⃗φ, ⃗ψ from an empirical distribution ¯p(x, y). It is natural in this case to maximize the likelihood (up to constants depending on ¯p(y)) max ⃗φ, ⃗ψ l(⃗φ, ⃗ψ) = − X x,y ¯p(x, y)d2 x,y − X x ¯p(x) log Z(x) , (2) where ¯p(x, y) denotes the empirical distribution over X, Y . As in other cases, maximizing the likelihood is also equivalent to minimizing the DKL between the empirical and the model’s distributions. The likelihood is composed of two terms. The first is (minus) the mean distance between x and y. This will be maximized when all distances are zero. This trivial solution is avoided because of the regularization term P x ¯p(x) log Z(x), which acts to increase distances between x and y points. The next section discusses the relation of this target function to that of Canonical Correlation Analysis [10]. To characterize the maxima of the likelihood we differentiate it with respect to the embeddings of individual objects ⃗φ(x), ⃗ψ(y), and obtain the following gradients ∂l(⃗φ, ⃗ψ) ∂⃗φ(x) = 2¯p(x) ⟨⃗ψ(y)⟩¯p(y|x) −⟨⃗ψ(y)⟩p(y|x) (3) ∂l(⃗φ, ⃗ψ) ∂⃗ψ(y) = 2p(y) ⃗ψ(y) −⟨⃗φ(x)⟩p(x|y) −2¯p(y) ⃗ψ(y) −⟨⃗φ(x)⟩¯p(x|y) , where p(y) = P x p(y|x)¯p(x). Equating these gradients to zero, the ⃗φ(x) gradient yields ⟨⃗ψ(y)⟩p(y|x) = ⟨⃗ψ(y)⟩¯p(y|x). This characterization is similar to the one seen in maximum entropy learning. Since p(y|x) will have significant values for Y values such that ψ(y) is close to φ(x), this condition implies that the expected location of a neighbor of φ(x) is the same under the empirical and model distributions. To find the optimal ⃗φ, ⃗ψ for a given embedding dimension d, we used a conjugate gradient ascent algorithm with random restarts. In section 4 we describe a different approach to this optimization problem. 3 Relation to Other Methods Embedding the rows and columns of a contingency table into a low dimensional Euclidean space is related to statistical methods for the analysis of heterogeneous data. Fisher [6] described a method for mapping X and Y into φ(x), ψ(y) such that the correlation coefficient between φ(x), ψ(y) is maximized. His method is in fact the discrete analogue of the more widely known Canonical correlation analysis (CCA) [10]. Another closely related method is Correspondence analysis [8], which uses a different normalization scheme, and aims to model χ2 distances between rows and columns of ¯p(x, y). The goal of all the above methods is to maximize the correlation coefficient between the embeddings of X and Y . We now discuss their relation to our distance based method. First, note that the correlation coefficient is invariant under affine transformations and we can thus focus on centered solutions with a unity covariance matrix (⟨φ(x)⟩= 0 and COV (φ(x)) = COV (ψ(y)) = I) solutions. In this case, the correlation coefficient is given by the following expression (we focus on d = 1 for simplicity) ρ(φ(x), ψ(y)) = X x,y ¯p(x, y)φ(x)ψ(y) = −1 2 X x,y ¯p(x, y)d2 x,y + 1 . (4) Maximizing the correlation is therefore equivalent to minimizing the mean distance across all pairs. This clarifies the relation between CCA and our method: Both methods aim to minimize the average distance between X and Y embeddings. However, CCA forces both embeddings to be centered and with a unity covariance matrix, whereas our method introduces a global regularization term related to the partition function. Our method is additionally related to exponential models of contingency tables, where the counts are approximated by a normalized exponent of a low rank matrix [7]. The current approach can be understood as a constrained version of these models where the expression in the exponent is constrained to have a geometric interpretation. A well-known geometric oriented embedding method is multidimensional scaling (MDS) [4], whose standard version applies to same-type objects with predefined distances. MDS embedding of heterogeneous entities was studied in the context of modeling ranking data (see [4] section 7.3). These models, however, focus on specific properties of ordinal data and therefore result in optimization principles and algorithms different from our probabilistic interpretation. 4 Semidefinite Representation The optimal embeddings ⃗φ, ⃗ψ may be found using unconstrained optimization techniques. However, the Euclidean distances used in the embedding space also allow us to reformulate the problem as constrained convex optimization over the cone of positive semidefinite (PSD) matrices [14]. We start by showing that for embeddings with dimension d = |X|+|Y |, maximizing (2) is equivalent to minimizing a certain convex non-linear function over PSD matrices. Consider the matrix A whose columns are all the embedded vectors ⃗φ and ⃗ψ. The matrix G ≡AT A is the Gram matrix of the dot products between embedding vectors. It is thus a symmetric PSD matrix of rank ≤d. The converse is also true: any PSD matrix of rank ≤d can be factorized as AT A, where A is an embedding matrix of dimension d. The distance between two columns in A is linearly related to the Gram matrix via d2 ij = gii + gjj −2gij. Since the likelihood function depends only on the distances between points in X and in Y , we can write the optimization problem in (2) as min G X x ¯p(x) log X y ¯p(y)e−d2 xy + X x,y ¯p(x, y)d2 xy (5) Subject to G ⪰0 , rank(G) ≤d, d2 xy = gxx + gyy −2gxy where gxy denotes the element in G corresponding to specific values of x, y. Thus, our problem is equivalent to optimizing a nonlinear objective over the set of PSD matrices of a constrained rank. The minimized function is convex, since it is the sum of a linear function of G and functions log P exp of an affine expression in G, which are also convex (see Geometric Programming section in [2]). Moreover, when G has full rank, the set of constraints is also convex. We conclude that when the embedding dimension is of size d = |X| + |Y | the optimization problem of Eq. (5) is convex. Thus there are no local minima, and solutions can be found efficiently. The PSD formulation allows us to add non-trivial constraints. Consider, for example, constraining the p(y) marginal to its empirical values, i.e. P x p(y|x)¯p(x) = ¯p(y). To introduce this as a convex constraint we take two steps. First, we note that we can relax the constraint that distributions normalize to one, and require that they normalize to less than one. This is achieved by replacing log Z(x) with a free variable a(x) and writing the problem as follows (we omit the dependence of d2 xy on G for brevity) min G X x ¯p(x)a(x) + X x,y ¯p(x, y)d2 xy (6) Subject to G ⪰0 , rank(G) ≤d, log X y ¯p(y)e−d2 xy−a(x) ≤0 ∀x It can be shown that the optimum of 6 will be obtained for solutions normalized to one, and it thus coincides with the optimum of 5. The constraint P x p(y|x)¯p(x) = ¯p(y) can now be relaxed to the inequality P x ¯p(y)¯p(x)e−d2 xy−a(x) ≤¯p(y), which defines a convex set. Again, the optimum will be obtained when the constraint is satisfied with equality. Embedding into a low dimension requires constraining the rank, but this is difficult since the problem is no longer convex in the general case. One approach to obtaining low rank solutions is to optimize over a full rank G and then project it into a lower dimension via spectral decomposition as in [14] or classical MDS. However, in the current problem, this was found to be ineffective. Instead, we penalize high-rank solutions by adding the trace of G [5] weighted by a positive factor, λ, to the objective function in (5). Small values of Tr(G) are expected to correspond to sparse eigenvalue sets and thus penalize high rank solutions. This approach was tested on subsets of the databases described in section 5 and yielded similar results to those of the gradient based algorithm. We believe that PSD algorithms may turn out to be more efficient in cases where relatively high dimensional embeddings are sought. Furthermore, under the PSD formulation it is easy to introduce additional constraints, for example on distances between subsets of points (as in [14]), and on marginals of the distribution. 5 Applications We tested our approach on a variety of applications. Here we present embedding of words and documents and authors and documents. To provide quantitative assessment of the performance of our method, that goes beyond visual inspection, we apply it to problems where some underlying structures are known in advance. The known structures are only used for performance measurement and not during learning. (a) (b) (c) (d) AA NS BI VS VM VB LT CS IM AP SP CN ET bound bayesian convergence support regression loss classifiers gamma bounds machines bayes risk polynomial nips regularization variational marginal bootstrap papers response cells cell activity frequency stimulus temporal motion position spatial stimuli receptive eye head movement channels scene movements perception recorded eeg formation detector dominance receptor rat biol policy actions agent game policies documents mdp agents rewards dirichlet Figure 2: CODE Embedding of 2483 documents and 2000 words from the NIPS database (the 2000 most frequent words, excluding the first 100, were used). The left panel shows document embeddings for NIPS 15-17, with colors to indicate the document topic. Other panels show embedded words and documents for the areas specified by rectangles. Figure (b) shows the border region between algorithms and architecture (AA) and learning theory (LT) (bottom rectangle in (a)). Figure (c) shows the border region between neuroscience (NS) and biological vision (VB) (upper rectangle in (a)). Figure (d) shows mainly control and navigation (CN) documents (left rectangle in (a)). (a) (b) (c) (d) pac sv regularized shawe rational corollary proposition smola dual ranking hyperplane generalisation svms vapnik lemma norm lambda regularization proof kernels machines margin loss Shawe−Taylor Scholkopf Opper Meir Bartlett Vapnik bellman vertex player plan mdps games rewards singh agents mdp policies planning game agent actions policy Singh Thrun Moore Tesauro Barto Gordon Sutton Dietterich conductance pyramidal iiii neurosci oscillatory msec retinal ocular dendritic retina inhibition inhibitory auditory cortical cortex Koch Mel Li Baird Pouget Bower Figure 3: CODE Embedding of 2000 words and 250 authors from the NIPS database (the 250 authors with highest word counts were chosen; words were selected as in Figure 2). Left panel shows embeddings for authors (red crosses) and words (blue dots). Other panels show embedded authors (only first 100 shown) and words for the areas specified by rectangles. They can be seen to correspond to learning theory, control and neuroscience (from left to right). 5.1 NIPS Database Embedding algorithms may be used to study the structure of document databases. Here we used the NIPS 0-12 database supplied by Roweis 2, and augmented it with data from NIPS volumes 13-17 3. The last three volumes also contain an indicator of the document’s topic (AA for algorithms and architecture, LT for learning theory, NS for neuroscience etc.). We first used CODE to embed documents and words into R2. The results are shown in Figure 2. It can be seen that documents with similar topics are mapped next to each other (e.g. AA near LT and NS near Biological Vision). Furthermore, words characterize the topics of their neighboring documents. Next, we used the data to generate an authors-words matrix (as in the Roweis database). We could now embed authors and words into R2, by using CODE to model p(word|author). The results are shown in Figure 3. It can be seen that authors are indeed mapped next to terms relevant to their work, and that authors dealing with similar domains are also mapped together. This illustrates how co-occurrence of words and authors may be used to induce a metric on authors alone. 2See http://www.cs.toronto.edu/∼roweis/data.html 3Data available at http://robotics.stanford.edu/∼gal/ (a) (b) (c) 1 10 100 1000 0.5 0.6 0.7 0.8 0.9 1 N nearest neighbors purity CODE IsoMap CA MDS SVD CODE IsoM CA MDS SVD 0 1 doc−doc measure 0 1 Newsgroup Sets doc−word measure CODE CA Figure 4: (a) Document purity measure for the embedding of newsgroups crypt, electronics and med, as a function of neighborhood size. (b) The doc−doc measure averaged over 7 newsgroup sets. For each set, the maximum performance was normalized to one. Embedding dimension is 2. Sets are atheism, graphics, crypt; ms-windows, graphics; ibm.pc.hw, ms-windows; crypt, electronics; crypt, electronics, med; crypt, electronics, med, space; politics.mideast, politics.misc. (c) The word −doc measure for CODE and CA algorithms, for 7 newsgroup sets. Embedding dimension is 2. 5.2 Information Retrieval To obtain a more quantitative estimate of performance, we applied CODE to the 20 newsgroups corpus, preprocessed as described in [3]. This corpus consists of 20 groups, each with 1000 documents. We first removed the 100 most frequent words, and then selected the next k most frequent words for different values of k (see below). The resulting words and documents were embedded with CODE, Correspondence Analysis (CA), SVD, IsoMap and classical MDS 4. CODE was used to model the distribution of words given documents p(w|d). All methods were tested under several normalization schemes, including document sum normalization and TFIDF. Results were consistent over all normalization schemes. An embedding of words and documents is expected to map documents with similar semantics together, and to map words close to documents which are related to the meaning of the word. We next test how our embeddings performs with respect to these requirements. To represent the meaning of a document we use its corresponding newsgroup. Note that this information is used only for evaluation and not in constructing the embedding itself. To measure how well similar documents are mapped together we define a purity measure, which we denote doc −doc. For each embedded document, we measure the fraction of its neighbors that are from the same newsgroup. This is repeated for all neighborhood sizes, and averaged over all sizes and documents. To measure how documents are related to their neighboring words, we use a measured denoted by word−doc. For each document d we look at its n nearest words and calculate their probability under the document’s newsgroup, normalized by their prior. This is repeated for neighborhood sizes smaller than 100 and averaged over documents . The word −doc measure was only compared with CA, since this is the only method that provides joint embeddings. Figure 4 compares the performance of CODE with that of the other methods with respect to the doc −doc and word −doc measures. CODE can be seen to outperform all other methods on both measures. 4CA embedding followed the standard procedure described in [8]. IsoMap implementation was provided by the IsoMap authors [13]. We tested both an SVD over the count matrix and SVD over log of the count plus one, only the latter is described here because it was better than the former. For MDS, the distances between objects were calculated as the dot product between their count vectors (we also tested Euclidean distances) 6 Discussion We presented a method for embedding objects of different types into the same low dimension Euclidean space. This embedding can be used to reveal low dimensional structures when distance measures between objects are unknown. Furthermore, the embedding induces a meaningful metric also between objects of the same type, which could be used, for example, to embed images based on accompanying text, and derive the semantic distance between images. Co-occurrence embedding should not be restricted to pairs of variables, but can be extended to multivariate joint distributions, when these are available. It can also be augmented to use distances between same-type objects when these are known. An important question in embedding objects is whether the embedding is unique. In other words, can there be two non isometric embeddings which are obtained at the optimum of the problem. This question is related to the rigidity and uniqueness of embeddings on graphs, specifically complete bipartite graphs in our case. A theorem of Bolker and Roth [1] asserts that for such graphs with at least 5 vertices on each side, embeddings are rigid, i.e. they cannot be continuously transformed. This suggests that the CODE embeddings for |X|, |Y | ≥5 are unique (at least locally) for d ≤3. We focused here on geometric models for conditional distributions. While in some cases, such a modeling choice is more natural in others joint models may be more appropriate. In this context it will be interesting to consider models of the form p(x, y) ∝p(x)p(y)e−d2 x,y where p(x), p(y) are the marginals of p(x, y). Maximum likelihood in these models is a non-trivial constrained optimization problem, and may be approached using the semidefinite representation outlined here. References [1] E.D. Bolker. and B. Roth. When is a bipartite graph a rigid framework? Pacific J. Math., 90:27–44, 1980. [2] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge Univ. Press, 2004. [3] G. Chechik and N. Tishby. Extracting relevant structures with side information. In S. Becker, S. Thrun, and K. Obermayer, editors, NIPS 15, 2002. [4] T. Cox and M. Cox. Multidimensional Scaling. Chapman and Hall, London, 1984. [5] M. Fazel, H. Hindi, and S. P. Boyd. A rank minimization heuristic with application to minimum order system approximation. In Proc. of the American Control Conference, 2001. [6] R.A. Fisher. The percision of discriminant functions. Ann. Eugen. Lond., 10:422–429, 1940. [7] A. Globerson and N. Tishby. Sufficient dimensionality reduction. Journal of Machine Learning Research, 3:1307–1331, 2003. [8] M.J. Greenacre. Theory and applications of correspondence analysis. Academic Press, 1984. [9] G. Hinton and S.T. Roweis. Stochastic neighbor embedding. In NIPS 15, 2002. [10] H. Hotelling. The most predictable criterion. Journal of Educational Psych., 26:139–142, 1935. [11] T. Iwata, K. Saito, N. Ueda, S. Stromsten, T. Griffiths, and J. Tenenbaum. Parametric embedding for class visualization. In NIPS 18, 2004. [12] S. T. Roweis and L. K. Saul. Nonlinear dimensionality reduction by locally linear embedding. Science, 290:2323–2326, 2000. [13] J.B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290:2319–2323, 2000. [14] K. Q. Weinberger and L. K. Saul. Unsupervised learning of image manifolds by semidefinite programming. In CVPR, 2004.
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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 ptodd@mpib-berlin.mpg.de dieckmann@mpib-berlin.mpg.de 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 One-Reason Decision Making and Ordered Search 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 Simple approaches to constructing cue search orders 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 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. High memory load, complete reordering High memory load, local reordering Low memory load, local reordering Validity: reorders cues based on their current validity Tally: reorders cues by number of correct minus incorrect decisions made so far Associative/delta rule: reorders cues by learned association strength 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 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 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 ǻwi = 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 Simulation Study of Simple Ordering Rules 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 Figure 2: Mean offline frugality of order learning rules 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 Discussion 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] Bröder, 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] Bröder, 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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Coarticulation in Markov Decision Processes Khashayar Rohanimanesh Department of Computer Science University of Massachusetts Amherst, MA 01003 khash@cs.umass.edu Robert Platt Department of Computer Science University of Massachusetts Amherst, MA 01003 rplatt@cs.umass.edu Sridhar Mahadevan Department of Computer Science University of Massachusetts Amherst, MA 01003 mahadeva@cs.umass.edu Roderic Grupen Department of Computer Science University of Massachusetts Amherst, MA 01003 grupen@cs.umass.edu Abstract We investigate an approach for simultaneously committing to multiple activities, each modeled as a temporally extended action in a semi-Markov decision process (SMDP). For each activity we define a set of admissible solutions consisting of the redundant set of optimal policies, and those policies that ascend the optimal statevalue function associated with them. A plan is then generated by merging them in such a way that the solutions to the subordinate activities are realized in the set of admissible solutions satisfying the superior activities. We present our theoretical results and empirically evaluate our approach in a simulated domain. 1 Introduction Many real-world planning problems involve concurrent optimization of a set of prioritized subgoals of the problem by dynamically merging a set of (previously learned) policies optimizing the subgoals. A familiar example of this type of problem would be a driving task which may involve subgoals such as safely navigating the car, talking on the cell phone, and drinking coffee, with the first subgoal taking precedence over the others. In general this is a challenging problem, since activities often have conflicting objectives and compete for limited amount of resources in the system. We refer to the behavior of an agent that simultaneously commits to multiple objectives as Coarticulation, inspired by the coarticulation phenomenon in speech. In this paper we investigate a framework based on semi-Markov decision processes (SMDPs) for studying this problem. We assume that the agent has access to a set of learned activities modeled by a set of SMDP controllers ζ = {C1, C2, . . . , Cn} each achieving a subgoal ωi from a set of subgoals Ω= {ω1, ω2, . . . , ωn}. We further assume that the agent-environment interaction is an episodic task where at the beginning of each episode a subset of subgoals ω ⊆Ωare introduced to the agent, where subgoals are ranked according to some priority ranking system. The agent is to devise a global policy by merging the policies associated with the controllers into a global policy that simultaneously commits to them according to their degree of significance. In general optimal policies of controllers do not offer flexibility required for the merging process. Thus for every controller we also compute a set of admissible suboptimal policies that reflect the degree of flexibility we can afford in it. Given a controller, an admissible policy is either an optimal policy, or it is a policy that ascends the optimal state-value function associated with the controller (i.e., in average leads to states with higher values), and is not too offfrom the optimal policy. To illustrate this idea, consider Figure 1(a) that shows a two dimensional S a b c d C C C1 2 S c a b (a) (b) Figure 1: (a) actions a, b, and c are ascending on the state-value function associated with the controller C, while action d is descending; (b) action a and c ascend the state-value function C1 and C2 respectively, while they descend on the state-value function of the other controller. However action b ascends the state-value function of both controllers. state-value function. Regions with darker colors represents states with higher values. Assume that the agent is currently in state marked s. The arrows show the direction of state transition as a result of executing different actions, namely actions a, b, c, and d. The first three actions lead the agent to states with higher values, in other words they ascend the state-value function, while action d descends it. Figure 1(b) shows how introducing admissible policies enables simultaneously solving multiple subgoals. In this figure, action a and c are optimal in controllers C1 and C2 respectively, but they both descend the state-value function of the other controller. However if we allow actions such as action b, we are guaranteed to ascend both value functions, with a slight degeneracy in optimality. Most of the related work in the context of MDPs assume that the subprocesses modeling the activities are additive utility independent [1, 2] and do not address concurrent planning with temporal activities. In contrast we focus on problems that involve temporal abstraction where the overall utility function may be expressed as a non-linear function of sub-utility functions that have different priorities. Our approach is also similar in spirit to the redundancy utilization formalism in robotics [4, 3, 6]. Most of these ideas, however, have been investigated in continuous domains and have not been extended to discrete domains. In contrast we focus on discrete domains modeled as MDPs. In this paper we formally introduce the framework of redundant controllers in terms of the set of admissible policies associated with them and present an algorithm for merging such policies given a coarticulation task. We also present a set of theoretical results analyzing various properties of such controllers, and also the performance of the policy merging algorithm. The theoretical results are complemented by an experimental study that illustrates the trade-offs between the degree of flexibility of controllers and the performance of the policy generated by the merging process. 2 Redundant Controllers In this section we introduce the framework of redundant controllers and formally define the set of admissible policies in them. For modeling controllers, we use the concept of subgoal options [7]. A subgoal option can be viewed as a closed loop controller that achieves a subgoal of some kind. Formally, a subgoal option of an MDP M = ⟨S, A, P, R⟩is defined by a tuple C = ⟨MC, I, β⟩. The MDP MC = ⟨SC, AC, PC, RC⟩is the option MDP induced by the option C in which SC ⊆S, AC ⊆A, PC is the transition probability function induced by P, and RC is chosen to reflect the subgoal of the option. The policy component of such options are the solutions to the option MDP MC associated with them. For generality, throughout this paper we refer to subgoal options simply as controllers. For theoretical reasons, in this paper we assume that each controller optimizes a minimum cost-to-goal problem. An MDP M modeling a minimum cost-to-goal problem includes a set of goal states SG ⊂S. We also represent the set of non-goal states by ¯SG = S −SG. Every action in a non-goal state incurs some negative reward and the agent receives a reward of zero in goal states. A controller C is a minimum cost-to-goal controller, if MC optimizes a minimum cost-to-goal problem. The controller also terminates with probability one in every goal state. We are now ready to formally introduce the concept of ascending policies in an MDP: Definition 1: Given an MDP M = ⟨S, A, P, R⟩, a function L : S →IR, and a deterministic policy π : S →A, let ρπ(s) = Es′∼Pπ(s) s {L(s′)} −L(s), where Es′∼Pπ(s) s {.} is the expectation with respect to the distribution over next states given the current state and the policy π. Then π is ascending on L, if for every state s (except for the goal states if the MDP models a minimum cost-to-goal problem) we have ρπ(s) > 0. For an ascending policy π on a function L, function ρ : S →IR+ gives a strictly positive value that measures how much the policy π ascends on L in state s. A deterministic policy π is descending on L, if for some state s, ρπ(s) < 0. In general we would like to study how a given policy behaves with respect to the optimal value function in a problem. Thus we choose the function L to be the optimal state value function (i.e., V∗). The above condition can be interpreted as follows: we are interested in policies that in average lead to states with higher values, or in other words ascend the state-value function surface. Note that Definition 1 is closely related to the Lyapunov functions introduced in [5]. The minimum and maximum rate at which an ascending policy in average ascends V ∗are given by: Definition 2: Assume that the policy π is ascending on the optimal state value function V∗. Then π ascends on V∗with a factor at least α, if for all non-goal states s ∈¯SG, ρπ(s) ≥α > 0. We also define the guaranteed expected ascend rate of π as: κπ = mins∈¯ SG ρπ(s). The maximum possible achievable expected ascend rate of π is also given by ηπ = maxs∈¯ SG ρπ(s). One problem with ascending policies is that Definition 1 ignores the immediate reward which the agent receives. For example it could be the case that as a result of executing an ascending policy, the agent transitions to some state with a higher value, but receives a huge negative reward. This can be counterbalanced by adding a second condition that keeps the ascending policies close to the optimal policy: Definition 3: Given a minimum cost-to-goal problem modeled by an MDP M = ⟨S, A, P, R⟩, a deterministic policy π is ϵ-ascending on M if: (1) π is ascending on V∗, and (2) ϵ is the maximum value in the interval (0, 1] such that ∀s ∈S we have Q∗(s, π(s)) ≥1 ϵ V∗(s). Here, ϵ measures how close the ascending policy π is to the optimal policy. For any ϵ, the second condition assures that: ∀s ∈S, Q∗(s, π(s)) ∈[ 1 ϵ V∗(s), V∗(s)] (note that because M models a minimum cost-to-goal problem, all values are negative). Naturally we often prefer policies that are ϵ-ascending for ϵ values close to 1. In section 3 we derive a lower bound on ϵ such that no policy for values smaller than this bound is ascending on V∗(in other words ϵ cannot be arbitrarily small). Similarly, a deterministic policy π is called ϵ-ascending on C, if π is ϵ-ascending on MC. Next, we introduce the framework of redundant controllers: Definition 4: A minimum cost-to-goal controller C is an ϵ-redundant controller if there exist multiple deterministic policies that are either optimal, or ϵ-ascending on C. We represent the set of such admissible policies by χϵ C. Also, the minimum ascend rate of C is defined as: ¯κ = minπ∈χϵ C κπ, where κπ is the ascend rate of a policy π ∈χϵ C (see Definition 2). We can compute the ϵ-redundant set of policies for a controller C as follows. Using the reward model, state transition model, V∗and Q∗, in every state s ∈S, we compute the set of actions that are ϵ-ascending on C represented by Aϵ C(s) = {a ∈ A|a = π(s), π ∈χϵ C}, that satisfy both conditions of Definition 2. Next, we present an algorithm for merging policies associated with a set of prioritized redundant controllers that run in parallel. For specifying the order of priority relation among the controllers we use the expression Cj ◁Ci, where the relation “◁” expresses the subject-to relation (taken from [3]). This equation should read: controller Cj subject-to controller Ci. A priority ranking system is then specified by a set of relations {Cj ◁Ci}. Without loss of generality we assume that the controllers are prioritized based on the following ranking system: {Cj ◁Ci|i < j}. Algorithm MergeController summarizes the policy merging process. In this algoAlgorithm 1 Function MergeController(s, C1, C3, . . . , Cm) 1: Input: current state s; the set of controllers Ci; the redundant-sets Aϵi Ci(s) for every controller Ci. 2: Initialize: Λ1(s) = Aϵ1 C1(s). 3: For i = 2, 3, . . . , n perform: Λi(s) = {a|a ∈ Aϵi Ci(s) ∧a ∈ Λf(i)(s)} where f(i) = max j < i such that Λj(s) ̸= ∅(initially f(1) = 1). 4: Return an action a ∈Λf(n+1)(s). rithm, Λi(s) represents the ordered intersection of the redundant-sets Aϵj Cj up to the controller Ci (i.e., 1 ≤j ≤i) constrained by the order of priority. In other words, each set Λi(s) contains a set of actions in state s that are all ϵi-ascending with respect to the superior controllers C1, C2, . . . , Ci. Due to the limited amount of redundancy in the system, it is possible that the system may not be able to commit to some of the subordinate controllers. This happens when none of the actions with respect to some controller Cj (i.e., a ∈Aϵj Cj(s)) are ϵ-ascending with respect to the superior controllers. In this case the algorithm skips the controller Cj, and continues the search in the redundant-sets of the remaining subordinate controllers. The complexity of the above algorithm consists of the following costs: (1) cost of computing the redundant-sets Aϵi Ci for a controller which is linear in the number of states and actions: O(|S| |A|), (2) cost of performing Algorithm MergeController in every state s, which is O((m −1) |A|2), where m is the number of subgoals. In the next section, we theoretically analyze redundant controllers and the performance of the policy merging algorithm in various situations. 3 Theoretical Results In this section we present some of our theoretical results characterizing ϵ-redundant controllers, in terms of the bounds on the number of time steps it takes for a controller to complete its task, and the performance of the policy merging algorithm. For lack of space, we have left out the proofs and refer the readers to [8]. In section 2 we stated that there is a lower bound on ϵ such that there exist no ϵ-ascending policy for values smaller than this bound. In the first theorem we compute this lower bound: Theorem 1 Let M = ⟨S, A, P, R⟩be a minimum cost-to-goal MDP and let π be an ϵ-ascending policy defined on M. Then ϵ is bounded by ϵ > |V∗ max| |V∗ min| , where V∗ min = mins∈¯ SG V∗(s) and V∗ max = maxs∈¯ SG V∗(s). Such a lower bound characterizes the maximum flexibility we can afford in a redundant controller and gives us an insight on the range of ϵ values that we can choose for it. In the second theorem we derive an upper bound on the expected number of steps that a minimum cost-to-goal controller takes to complete when executing an ϵ-ascending policy: Theorem 2 Let C be an ϵ-ascending minimum cost-to-goal controller and let s denote the current state of the controller. Then any ϵ-ascending policy π on C will terminate the controller in some goal state with probability one. Furthermore, termination occurs in average in at most ⌈−V∗(s) κπ ⌉steps, where κπ is the guaranteed expected ascend rate of the policy π. This result assures that the controller arrives in a goal state and will achieve its goal in a bounded number of steps. We use this result when studying performance of running multiple redundant controllers in parallel. Next, we study how concurrent execution of two controllers using Algorithm MergeController impacts each controller (this result can be trivially extended to the case when a set of m > 2 controllers are executed concurrently): Theorem 3 Given an MDP M = ⟨S, A, P, R⟩, and any two minimum cost-to-goal redundant controllers {C1, C2} defined over M, the policy π obtained by Algorithm MergeController based on the ranking system {C2 ◁C1} is ϵ1-ascending on C1(s). Moreover, if ∀s ∈S, Aϵ1 C1(s) ∩Aϵ2 C2(s) ̸= ∅, policy π will be ascending on both controllers with the ascend rate at least κπ = min{κπ1, κπ2}. This theorem states that merging policies of two controllers using Algorithm MergeController would generate a policy that remains ϵ1-ascending on the superior controller. In other words it does not negatively impact the superior controller. In the next theorem, we establish bounds on the expected number of steps that it takes for the policy obtained by Algorithm MergeController to achieve a set of prioritized subgoals ω = {ω1, . . . , ωm} by concurrently executing the associated controllers {C1, . . . , Cm}: Theorem 4 Assume ζ = {C1, C2, . . . , Cm} is a set of minimum cost-to-goal ϵiredundant (i = 1, . . . , m) controllers defined over MDP M. Let the policy π denote the policy obtained by Algorithm MergeController based on the ranking system {Cj ◁Ci|i < j}. Let µζ(s) denote the expected number of steps for the policy π for achieving all the subgoals {ω1, ω2, . . . , ωm} associated with the set of controllers, assuming that the current state of the system is s. Then the following expression holds: max i ⌈−V∗ i (s) ηπ i ⌉≤µζ(s) ≤ X h∈H P(h) m X i=1 ⌈−V∗ i (h(i)) ¯κi ⌉ (1) where ηπ i is the maximum possible achievable expected ascend rate for the controller Ci (see Definition 2), H is the set of sequences h = ⟨s, g1, g2, . . . , gm⟩in which gi is a goal state in controller Ci (i.e., gi ∈SGi). The probability distribution P(h) = PC1 sg1 Qm i=2 PCi gi−1gi over sequences h ∈H gives the probability of executing the set of controllers in sequence based on the order of priority starting in state s, and observing the goal state sequence ⟨g1, . . . , gm⟩. Based on Theorem 3, when Algorithm MergeController always finds a policy π that optimizes all controllers (i.e., ∀s ∈S, ∩m i=1Aϵi Ci(s) ̸= ∅), policy π will ascend on all controllers. Thus in average the total time for all controllers to terminate equals the time required for a controller that takes the most time to complete which has the lower bound of maxi⌈−V∗ i (s) ηπ(s) ⌉. The worst case happens when the policy π generated by Algorithm MergeController can not optimize more than one controller at a time. In this case π always optimizes the controller with the highest priority until its termination, then optimizes the second highest priority controller and continues this process to the end in a sequential manner. The right hand side of the inequality given by Equation 1 gives an upper bound for the expected time required for all controllers to complete when they are executed sequentially. The above theorem implicitly states that when Algorithm MergeController generates a policy that in average commits to more than one subgoal it potentially takes less number of steps to achieve all the subgoals, compared to a policy that sequentially achieves them according to their degree of significance. 4 Experiments In this section we present our experimental results analyzing redundant controllers and the policy merging algorithm described in section 2. Figure 2(a) shows a 10 × 10 grid world where an agent is to visit a set of prioritized locations marked by G1, . . . , Gm (in this example m = 4). The agent’s goal is to achieve all of the subgoals by focusing on superior subgoals and coarticulating with the subordinate ones. Intuitively, when the agent is navigating to some subgoal Gi of higher priority, if some subgoal of lower priority Gj is en route to Gi, or not too offfrom the optimal path to Gi, the agent may choose to visit Gj. We model this problem by an MDP G G G G 1 2 3 4 G1 G1 G1 (a) (b) (c) (d) Figure 2: (a) A 10 × 10 grid world where an agent is to visit a set of prioritized subgoal locations; (b) The optimal policy associated with the subgoal G1; (c) The ϵ-ascending policy for ϵ = 0.95; (d) The ϵ-ascending policy for ϵ = 0.90. M = ⟨S, A, R, P⟩, where S is the set of states consisting of 100 locations in the room, and A is the set of actions consisting of eight stochastic navigation actions (four actions in the compass direction, and four diagonal actions). Each action moves the agent in the corresponding direction with probability p and fails with probability (1 −p) (in all of the experiments we used success probability p = 0.9). Upon failure the agent is randomly placed in one of the eight-neighboring locations with equal probability. If a movement would take the agent into a wall, then the agent will remain in the same location. The agent also receives a reward of −1 for every action executed. We assume that the gent has access to a set of controllers C1, . . . , Cm, associated with the set of subgoal locations G1, . . . , Gm. A controller Ci is a minimum cost-to-goal subgoal option Ci = ⟨MCi, I, β⟩, where MCi = M, the initiation set I includes any locations except for the subgoal location, and β forces the option to terminate only in the subgoal location. Figures 2(b)-(d) show examples of admissible policies for subgoal G1: Figure 2(b) shows the optimal policy of the controller C1 (navigating the agent to the location G1). Figures 2(c) and 2(d) show the ϵ-redundant policies for ϵ = 0.95 and ϵ = 0.90 respectively. Note that by reducing ϵ, we obtain a larger set of admissible policies although less optimal. We use two different planning methods: (1) sequential planning, where we achieve the subgoals sequentially by executing the controllers one at a time according to the order of priority of subgoals, (2) concurrent planning, where we use Algorithm MergeController for merging the policies associated with the controllers. In the first set of experiments, we fix the number of subgoals. At the beginning of each episode the agent is placed in a random location, and a fixed number of subgoals (in our experiments m = 4) are randomly selected. Next, the set of admissible policies (using ϵ = 0.9) for every subgoal is computed. Figure 3(a) shows the performance of both planning methods, for every starting location in terms of number of steps for completing the overall task. The concurrent planning method consistently outperforms the sequential planning in all starting locations. Next, for the 16 18 20 22 24 26 28 30 0 20 40 60 80 100 Average (steps) State Concurrent Sequential 19 20 21 22 23 24 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 Average (steps) Epsilon Concurrent (a) (b) Figure 3: (a) Performance of both planning methods in terms of the average number of steps in every starting state; (b) Performance of the concurrent method for different values of ϵ. same task, we measure how the performance of the concurrent method varies by varying ϵ, when computing the set of ϵ-ascending policies for every subgoal. Figure 3(b) shows the performance of the concurrent method and Figure 4(a) shows the average number of subgoals coarticulated by the agent – averaged over all states – for different values of ϵ. We varied ϵ from 0.6 to 1.0 using 0.05 intervals. All of these results are also averaged over 100 episodes, each consisting of 10 trials. Note that for ϵ = 1, the only admissible policy is the optimal policy and thus it does not offer much flexibility with respect to the other subgoals. This can be seen in Figure 3(b) in which the policy generated by the merging algorithm for ϵ = 1.0 has the minimum commitment to the other subgoals. As we reduce ϵ, we obtain a larger set of admissible policies, thus we observe improvement in the performance. However, the more we reduce ϵ, the less optimal admissible policies we obtain. Thus the performance degrades (here we can observe it for the values below ϵ = 0.85). Figure 4(a) also shows by relaxing optimality (reducing ϵ), the policy generated by the merging algorithm commits to more subgoals simultaneously. In the final set of experiments, we fixed ϵ to 0.9 and varied the number of subgoals from m = 2 to m = 50 (all of these results are averaged over 100 episodes, each consisting of 10 trials). Figure 4(b) shows the performance of both planning methods. It can be observed that the concurrent method consistently outperforms the sequential method by increasing the number of subgoals (top curve shows the performance of the sequential method and bottom curve shows that of concurrent method). This is because when there are many subgoals, the concurrent planning 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Number of subgoals committed Epsilon Concurrent 0 20 40 60 80 100 120 140 160 180 0 5 10 15 20 25 30 35 40 45 50 Average (steps) Number of subgoals Concurrent Sequential (a) (b) Figure 4: (a) Average number of subgoals coarticulated using the concurrent planning method for different values of ϵ; (b) Performance of the planning methods in terms of the average number of steps in every starting state. method is able to visit multiple subgoals of lower priority en route the primary subgoals, thus it can save more time. 5 Concluding Remarks There are a number of questions and open issues that remain to be addressed and many interesting directions in which this work can be extended. In many problems, the strict order of priority of subtasks may be violated: in some situations we may want to be sub-optimal with respect to the superior subtasks in order to improve the overall performance. One other interesting direction is to study situations when actions are structured. We are currently investigating compact representation of the set of admissible policies by exploiting the structure of actions. Acknowledgements This research is supported in part by a grant from the National Science Foundation #ECS-0218125. References [1] C. Boutilier, R. Brafman, and C. Geib. Prioritized goal decomposition of Markov decision processes: Towards a synthesis of classical and decision theoretic planning. In Martha Pollack, editor, Proceedings of the Fifteenth International Joint Conference on Artificial Intelligence, pages 1156–1163, San Francisco, 1997. Morgan Kaufmann. [2] C. Guestrin and G. Gordon. Distributed planning in hierarchical factored mdps. In In the Proceedings of the Eighteenth Conference on Uncertainty in Artificial Intelligence, pages 197 – 206, Edmonton, Canada, 2002. [3] M. Huber. A Hybrid Architecture for Adaptive Robot Control. PhD thesis, University of Massachusetts, Amherst, 2000. [4] Y. Nakamura. Advanced robotics: redundancy and optimization. Addison-Wesley Pub. Co., 1991. [5] Theodore J. Perkins and Andrew G. Barto. Lyapunov-constrained action sets for reinforcement learning. In Proc. 18th International Conf. on Machine Learning, pages 409–416. Morgan Kaufmann, San Francisco, CA, 2001. [6] R. Platt, A. Fagg, and R. Grupen. Nullspace composition of control laws for grasping. In the Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2002. [7] D. Precup. Temporal Abstraction in Reinforcement Learning. PhD thesis, Department of Computer Science, University of Massachusetts, Amherst., 2000. [8] K. Rohanimanesh, R. Platt, S. Mahadevan, and R. Grupen. A framework for coarticulation in markov decision processes. Technical Report 04-33, (www.cs.umass.edu/~khash/coarticulation04. pdf), Department of Computer Science, University of Massachusetts, Amherst, Massachusetts, USA., 2004.
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Discrete profile alignment via constrained information bottleneck Sean O’Rourke∗ seano@cs.ucsd.edu Gal Chechik† gal@stanford.edu Robin Friedman∗ rcfriedm@ucsd.edu Eleazar Eskin∗ eeskin@cs.ucsd.edu Abstract Amino acid profiles, which capture position-specific mutation probabilities, are a richer encoding of biological sequences than the individual sequences themselves. However, profile comparisons are much more computationally expensive than discrete symbol comparisons, making profiles impractical for many large datasets. Furthermore, because they are such a rich representation, profiles can be difficult to visualize. To overcome these problems, we propose a discretization for profiles using an expanded alphabet representing not just individual amino acids, but common profiles. By using an extension of information bottleneck (IB) incorporating constraints and priors on the class distributions, we find an informationally optimal alphabet. This discretization yields a concise, informative textual representation for profile sequences. Also alignments between these sequences, while nearly as accurate as the full profileprofile alignments, can be computed almost as quickly as those between individual or consensus sequences. A full pairwise alignment of SwissProt would take years using profiles, but less than 3 days using a discrete IB encoding, illustrating how discrete encoding can expand the range of sequence problems to which profile information can be applied. 1 Introduction One of the most powerful techniques in protein analysis is the comparison of a target amino acid sequence with phylogenetically related or homologous proteins. Such comparisons give insight into which portions of the protein are important by revealing the parts that were conserved through natural selection. While mutations in non-functional regions may be harmless, mutations in functional regions are often lethal. For this reason, functional regions of a protein tend to be conserved between organisms while non-functional regions diverge. ∗Department of Computer Science and Engineering, University of California San Diego †Department of Computer Science, Stanford University Many of the state-of-the-art protein analysis techniques incorporate homologous sequences by representing a set of homologous sequences as a probabilistic profile, a sequence of the marginal distributions of amino acids at each position in the sequence. For example, Yona et al.[10] uses profiles to align distant homologues from the SCOP database[3]; the resulting alignments are similar to results from structural alignments, and tend to reflect both secondary and tertiary protein structure. The PHD algorithm[5] uses profiles purely for structure prediction. PSI–BLAST[6] uses them to refine database searches. Although profiles provide a lot of information about the sequence, the use of profiles comes at a steep price. While extremely efficient string algorithms exist for aligning protein sequences (Smith-Waterman[8]) and performing database queries (BLAST[6]), these algorithms operate on strings and are not immediately applicable to profile alignment or profile database queries. While profile-based methods can be substantially more accurate than sequence-based ones, they can require at least an order of magnitude more computation time, since substitution penalties must be calculated by computing distances between probability distributions. This makes profiles impractical for use with large bioinformatics databases like SwissProt, which recently passed 150,000 sequences. Another drawback of profile as compared to string representations is that it is much more difficult to visually interpret a sequence of 20 dimensional vectors than a sequence of letters. Discretizing the profiles addresses both of these problems. First, once a profile is represented using a discrete alphabet, alignment and database search can be performed using the efficient string algorithms developed for sequences. For example, when aligning sequences of 1000 elements, runtime decreases from 20 seconds for profiles to 2 for discrete sequences. Second, by representing each class as a letter, discretized profiles can be presented in plain text like the original or consensus sequences, while conveying more information about the underlying profiles. This makes them more accurate than consensus sequences, and more dense than sequence logos (see figure 1). To make this representation intuitive, we want the discretization not only to minimize information loss, but also to reflect biologically meaningful categories by forming a superset of the standard 20-character amino acid alphabet. For example, we use “A” and “a” for strongly- and weakly-conserved Alanine. This formulation demands two types of constraints: similarities of the centroids to predefined values, and specific structural similarities between strongly- and weakly-conserved variants. We show below how these constraints can be added to the original IB formalism. In this paper, we present a new discrete representation of proteins that takes into account information from homologues. The main idea behind our approach is to compress the space of probabilistic profiles in a data-dependent manner by clustering the actual profiles and representing them by a small alphabet of distributions. Since this discretization removes some of the information carried by the full profiles, we cluster the distribution in a way that is directly targeted at minimizing the information loss. This is achieved using a variant of Information Bottleneck (IB)[9], a distributional clustering approach for informationally optimal discretization. We apply our algorithm to a subset of MEROPS[4], a database of peptidases organized structurally by family and clan, and analyze the results in terms of both information loss and alignment quality. We show that multivariate IB in particular preserves much of the information in the original profiles using a small number of classes. Furthermore, optimal alignments for profile sequences encoded with these classes are much closer to the original profile-profile alignments than are alignments between the seed proteins. IB discretization is therefore an attractive way to gain some of the additional sensitivity of profiles with less computational cost. 0.0 0.0 0.0 0.09 0.34 0.23 0.12 0.0 0.0 0.0 0.0 0.0 0.0 0.04 0.01 0.01 0.03 0.0 0.0 0.0 0.0 0.0 1.0 0.01 0.05 0.14 0.09 0.0 1.0 0.0 0.0 0.0 0.0 0.38 0.04 0.00 0.04 0.0 0.0 0.0 0.0 0.0 0.0 0.06 0.00 0.08 0.04 0.0 0.0 1.0 0.0 0.0 0.0 0.00 0.06 0.01 0.03 1.0 0.0 0.0 0.0 0.0 0.0 0.02 0.00 0.04 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.00 0.03 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.04 0.01 0.01 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.01 0.00 0.09 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.00 0.03 0.00 0.0 0.0 0.0 0.5 1.0 0.0 0.05 0.05 0.01 0.01 0.0 0.0 0.0 0.0 0.0 0.0 0.02 0.00 0.23 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.04 0.05 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.04 0.01 0.00 0.00 0.0 0.0 0.0 0.5 0.0 0.0 0.16 0.10 0.06 0.29 0.0 0.0 0.0 0.0 0.0 0.0 0.02 0.10 0.05 0.20 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.14 0.03 0.04 0.0 0.0 0.0 0.0 0.0 0.0 0.00 0.00 0.00 0.00 0.0 0.0 0.0 0.0 0.0 0.0 0.01 0.00 0.04 0.04 0.0 0.0 0.0 (a) ASN AN AD R Q K N F A S E E Q N D G T S V A Y H T S F D P A Y V F E L D A T S AG ADAF (b) P00790 Seq.: ---EAPT--Consensus Seq.: NNDEAASGDF IB Seq.: NNDeaptGDF (c) Figure 1: (a) Profile, (b) sequence logo[2], and (c) textual representations for part of an alignment of Pepsin A precursor P00790, showing IB’s concision compared to profiles and logos, and its precision compared to single sequences. 2 Information Bottleneck Information Bottleneck [9] is an information theoretic approach for distributional clustering. Given a joint distribution p(X, Y ) of two random variables X and Y , the goal is to obtain a compressed representation C of X, while preserving the information about Y . The two goals of compression and information preservation are quantified by the same measure of mutual information I(X; Y ) = P x,y p(x, y) log p(x,y) p(x)p(y) and the problem is therefore defined as the constrained optimization problem minp(c|x):I(C;Y )>K I(C; X) where K is a constraint on the level of information preserved about Y , and the problem should also obey the constraints p(y|c) = P x p(y|x)p(x|c) and p(y) = P x p(y|x)p(x). This constrained optimization can be reformulated using Lagrange multipliers, and turned into a tradeoffoptimization function with Lagrange multiplier β: min p(c|x) L def = I(C; X) −βI(C; Y ) (1) As an unsupervised learning technique, IB aims to characterize the set of solutions for the complete spectrum of constraint values K. This set of solutions is identical to the set of solutions of the tradeoffoptimization problem obtained for the spectrum of β values. When X is discrete, its natural compression is fuzzy clustering. In this case, the problem is not convex and cannot be guaranteed to contain a single global minimum. Fortunately, its solutions can be characterized analytically by a set of self consistent equations. These self consistent equations can then be used in an iterative algorithm that is guaranteed to converge to a local minimum. While the optimal solutions of the IB functional are in general soft clusters, in practice, hard cluster solutions are sometimes more easily interpreted. A series of algorithms was developed for hard IB, including an algorithm that can be viewed as a one-step look-ahead sequential version of K-Means [7]. To apply IB to the problem of profiles discretization discussed here, X is a given set of probabilistic profiles obtained from a set of aligned sequences and Y is the set of 20 amino acids. 2.1 Constraints on centroids’ semantics The application studied in this paper differs from standard IB applications in that we are interested in obtaining a representation that is both efficient and biologically meaningful. This requires that we add two kinds of constraints on clusters’ distributions, discussed below. First, some clusters’ meanings are naturally determined by limiting them to correspond to the common 20-letter alphabet used to describe amino acids. From the point of view of distributions over amino acids, each of these symbols is used today as the delta function distribution which is fully concentrated on a single amino acid. For the goal of finding an efficient representation, we require the centroids to be close to these delta distributions. More generally, we require the centroids to be close to some predefined values ˆci, thus adding constraints to the IB target function of the form DKL[p(y|ˆci)||p(y|ci)] < Ki for each constrained centroid. While solving the constrained optimization problem is difficult, the corresponding tradeoffoptimization problem can be made very similar to standard IB. With the additional constraints, the IB trade-offoptimization problem becomes min p(c|x) L′ ≡I(C; X) −βI(C; Y ) + β X ci∈C β(ci)DKL[p(y|ˆci)||p(y|ci)] . (2) We now use the following identity X x,c p(x, c)DKL[p(y|x)||p(y|c)] = X x p(x) X y p(y|x) log p(y|x) − X c p(c) X y log p(y|c) X x p(y|x)p(x|c) = −H(Y |X) + H(Y |C) = I(X; Y ) −I(Y ; C) to rewrite the IB functional of Eq. (1) as L = I(C; X) + β X c∈C X x∈X p(x, c)DKL[p(y|x)||p(y|c)] −βI(X; Y ) When P β(ci) ≤1 we can similarly rewrite Eq. (2) as L′ = I(C; X) + β X x∈X p(x) X ci∈C p(ci|x)DKL[p(y|x)||p(y|ci)] (3) +β X ci∈C β(ci)DKL[p(y|ˆci)||p(y|ci)] −βI(X; Y ) = I(C; X) + β X x′∈X′ p(x′) X ci∈C p(ci|x′)DKL[p(y|x′)||p(y|ci)] −βI(X; Y ) The optimization problem therefore becomes equivalent to the original IB problem, but with a modified set of samples x ∈X′, containing X plus additional “pseudocounts” or biases. This is similar to the inclusion of priors in Bayesian estimation. Formulated this way, the biases can be easily incorporated in standard IB algorithms by adding additional pseudo-counts x′ with prior probability p(x′) = βi(c). 2.2 Constraints on relations between centroids We want our discretization to capture correlations between strongly- and weaklyconserved variants of the same symbol. This can be done with standard IB using separate classes for the alternatives. However, since the distributions of other amino acids in these two variants are likely to be related, it is preferable to define a single shared prior for both variants, and to learn a model capturing their correlation. Friedman et al.[1] describe multivariate information bottleneck (mIB), an extension of information bottleneck to joint distributions over several correlated input and cluster variables. For profile discretization, we define two compression variables connected as in Friedman’s “parallel IB”: an amino acid class C ∈{A, C, . . .} with an associated prior, and a strength S ∈{0, 1}. Since this model correlates strong and weak variants of each category, it requires fewer priors than simple IB. It also has fewer parameters: a multivariate model with ns strengths and nc classes has as many categories as a univariate one with nc′ = nsnc classes, but has only ns+nc−2 free parameters for each x, instead of nsnc −1. 3 Results To test our method, we apply it to data from MEROPS[4]. Proteins within the same family typically contain high-confidence alignments, those from different families in the same clan less so. For each protein, we generate a profile from alignments obtained from PSI–BLAST with standard parameters, and compute IB classes from a large subset of these profiles using the priors described below. Finally, we encode and align pairs of profiles using the learned classes, comparing the results to those obtained both with the full profiles and with just the original sequences. For univariate IB, we have used four types of priors reflecting biases on stability, physical properties, and observed substitution frequencies: (1) Strongly conserved classes, in which a single symbol is seen with S% probability. These are the only priors used for multivariate IB. (2) Weakly conserved classes, in which a single symbol occurs with W% probability; (S−W)% of the remaining probability mass is distributed among symbols with non-negative log-odds of substitution. (3) Physical trait classes, in which all symbols with the same hydrophobicity, charge, polarity, or aromaticity occur uniformly S% of the time. (4) A uniform class, in which all symbols occur with their background probabilities. The choice of S and W depends upon both the data and one’s prior notions of “strong” and “weak” conservation. Unbiased IB on a large subset of MEROPS with several different numbers of unbiased categories yielded a mean frequency approaching 0.7 for the most common symbol in the 20 most sharply-distributed classes (0.59±0.13 for |C| = 52; 0.66±0.12 for |C| = 80; 0.70±0.09 for |C| = 100). Similarly, the next 20 classes have a mean most-likely-symbol frequency around 0.4. These numbers can be seen as lower bounds on S and W. We therefore chose S = 0.8 and W = 0.5, reflecting a bias toward stronger definitions of conservation than those inferred from the data. 3.1 Iterative vs. Sequential IB Slonim[7] compares several IB algorithms, concluding that best hard clustering results are obtained with a sequential method (sIB), in which elements are first assigned to a fixed number of clusters and then individually moved from cluster to cluster while calculating a 1-step lookahead score, until the score converges. While sIB is more efficient than exhaustive bottom-up clustering, it neglects information about the best potential candidates to be assigned to a cluster, yielding slow convergence. Furthermore updates are expensive, since each requires recomputing the class centroids. Therefore instead of sIB, we use iterative IB (iIB) with hard clustering, which only recomputes the centroids after performing all updates. This reduces R D C Y M Q L F P E N G T V SA W M E R H D V F Q P N I A G S K L Y TC V R Q S A T K G P E ND M H V R S I L N K A P Q T G DE H D Q K N G R M V T S Y I E L P AF P M V Q K F T R L I D N A E SG M L V G F P Y W D I T S K E R A N QH H M R A Y F D Q E P G N K S T L VI W V P M I L H N Q D F A T G R S EK K G R Q D Y W C S F T A P M V IL T N F W V L E I R D Q KM V L I A K F D H E Y S T Q R GN K R D G T V L Q E A SP W S P I H G F T V M L K A N E C RQ Y N M A P E C L I D G T V Q K SR I H L R Q K G F E D P N T AS R Q P M L F C D V G K E S AT P D T G R M N H E F A Q L IV H C R F P M E D S Q K I G A N Y T V LW P H T C N M K D A G E S Q I V R L FY H Y M N D R Q F E P K IT G V S LA M H R E Y P F Q I N K L V D A T G S C H M Y R K P I Q V F G N E A L T SD M H F Y V G N R IT P Q K L A D SE M R Y Q E K P N T S A ID V L GF H M R P E K Q N F Y A V D IT L SG W M C F Y D P N IV T R E G Q S K L A H C M H Y F R D P N S A T Q E V G K LI C M W H F Y D S Q N P T A V IG R E LK H E Q N C D W K R Y M P T S F G AIVL P D E Q N G R V IA F K T S LM M F Y H I E Q L V P A R D G K T SN Y F M H N I R D K Q E A V G L T SP F H M P Y R V T A G N S D IK E LQ C Y M F H I N D V T E P L A K Q S GR H C R I L Y F V E Q P K D A N G TS M R F Y K N E Q P ID V G A L ST R Q H C Y D N K G F P E M S T A LIV H M P Q R E K IV F D T L A N G SY H Y P M G Q R A T F D N IV S K E L M H C W E R D A Y Q K IN F V T G L S P M R H E K Y A W Q C N ID F G P L T VS W C M H Q N P A Y E G S F T V ID R K L H M C W Y F P Q K N R A E ID T V G L S Y H W M F D R I N P T Q V K A G E S L M H K F P A N Q R W E ID T V Y L G S M H K P W F Q D E N Y R A T IV S G L H M R Y F C W P K N E Q A ID L T V G S M H N Y P Q E S F IT D A G R K V L R W K C M E Q A Y N F D P IT V L G S H M Y F N S R Q I D P A T V G K E W L C R M K E Q Y F N D A T IP V G L S Y M H D R F L Q P E V T N S GA H G M T R Q L K W E D I V P F A S N YC Q M K G S P A N E V R I TD G M P N D T V S Q R I K L AE W S A P D T H Y V Q I R M N K E L GF P Q V F L K D I N E A R S M TG V F L P M G K W N T D Y R I S E A QH R D A Q M E Y F G N S K P T L VI W P M L H N F E D G Q T A R SK W R G Q Y D F C S K T A M P V IL Y G W N V L I E R Q D KM P A V K I E D F L T S Y H R Q GN K D G R T V L Q E A SP G S Y P M L V K N W T E C H A RQ W N Y P C A M E D G Q T V I K L SR V M C I H K Q F R L G E D P N T AS R C L P M D V G K A S ET P M T F A R G L H N I E QV Q H E C T S P R N G D M K F A I Y V LW W P D A T N G K H C M E S I V Q R L FY H Y M N R D Q E F P K G IT S V LA M W V E F D P R I Q Y G N L H A T S KC H A T Q E K S G PND L H I V T R S A K N P Q G DE G Q N H K R D E V L Y S IP T AF H M R P E K Q N F A Y V D L T ISG W C H M Y F R D S N A T Q P E V G L KI M W H F P D Y N Q IT V S L A E G RK H E Q N K D R C W P M Y T S F G AIVL M Y F I H L V A E P Q D R G T KSN Y H F M N R Q D K E G IA S L T VP H F M P R G Y S N V A T D K IE LQ H C R I L F V E Y Q P K D N A G TS F K Y E N Q P IV D G A S LT R Q D C P H Y N G K E F M S T L AIV M H P Q R E K I L D F T V G A N SY C M D F Y P G N IV E R Q T S L H K A W H R I M V Y K P Q E G L F N T A SD M F Y H I K G V L N R Q A T P DSE W M Q E R Y K P T S N A IV D L G F Y I F C H V N D T K L E A S Q P RG P R N Q IV T A K S FML C W M H E D Q IV R A T G F N Y S K L P R H E Y K Q C W I A G F P N L D V TS H P M Y G Q T R F A N V S D IK LE Y H F P M Q R I D A E K N T L VSG W C M H Y Q N E P A G F D T S IV K RL M R K E P F Y I Q N L V A T D C G S H Y R F W M N C K P E Q D T A V IL S G H Y M F N S D Q I R V A T P G W K E L R M K E Q Y F N A D IT P V G L S H M K F P R A N Q E W ID Y L V T S G Figure 2: Stretched sequence logos for categories found by iIB (top) and sIB (bottom), ordered by primary symbol and decreasing information. the convergence time from several hours to around ten minutes. Since Slonim argues that sIB outperforms soft iIB in part because sIB’s discrete steps allow it to escape local optima, we expect hard iIB to have similar behavior. To test this, we applied three complete sIB iterations initialized with categories from multivariate iIB. sIB decreased the loss L by only about 3 percent (from 0.380 to 0.368), with most of this gain occurring in the first iteration. Also, the resulting categories were mostly conserved up to exchanging labels, suggesting that hard iIB finds categories similar sIB ones (see figure 2). 3.2 Information Loss and Alignments One measure of the quality of the resulting clusters is the amount of information about Y lost through discretization, I(Y ; X) −I(Y ; C). Figure (3b) shows the effect on information loss of varying the prior weight w with three sets of priors: 20 strongly conserved symbols and one background; these plus 20 weakly conserved symbols; and these plus 10 categories for physical characteristics. As expected, both decreasing the number of categories and increasing the number or weight of priors increases information loss. However, with a fixed number of free categories, information loss is nearly independent of prior strength, suggesting that our priors correspond to actual regularities in the data. Finally, note that despite having fewer free parameters than the univariate models, mIB’s achieves comparable performance, suggesting that our decomposition into conserved class and degree of conservation is reasonable. Since we are ultimately using these classes in alignments, the true cost of discretization is best measured by the amount of change between profile and IB alignments, and the significance of this change. The latter is important because the best path can be very sensitive to small changes in the sequences or scoring matrix; if two radically different alignments have similar scores, neither is clearly “correct”. We can represent an alignment as a pair of index-insertion sequences, one for each profile sequence to be aligned (e.g. “1,2, , ,3,...” versus “1, ,2, ,3,...”). The edit distance between these sequences for two alignments then measures how much they differ. However, even when this distance is large, the difference between two alignments may not be significant if both choices’ scores are nearly the same. That is, if the optimal profile alignment’s score is only slightly lower than the optimal IB class alignment’s score as computed with the original profiles, either might be correct. Figure 4 shows at left both the edit distance and score change per length between profile alignments and those using IB classes, mIB classes, and the original sequences with the BLOSUM62 scoring matrix. To compare the profile and sequence alignments, profiles corresponding to gaps in the original sequences are replaced 1 4 16 64 400 800 1600 Time!(s) Length (a) Profile-profile IB-profile 2e-5 * L^2 + 0.1 3e-3 * L - 0.1 0.38 0.42 0.46 0.2 0.4 0.6 0.8 I(Y;X)!-!I(Y;C) w (b) multivariate 21/52 priors 41/52 priors 51/52 priors Figure 3: (a) Running times for profile-profile versus IB-profile alignment, showing speedups of 3.5-12.5x for pairwise global alignment. (b)I(Y ; X) −I(Y ; C) as a function of w for different groups of priors. The information loss for 52 categories without priors is 0.359, for 10, 0.474. Edit distance Score change Same Superfamily mIB 0.154 ± 0.182 0.086 ± 0.166 IB 0.170 ± 0.189 0.107 ± 0.198 BLOSUM 0.390 ± 0.065 Same Clan mIB 0.124 ± 0.209 0.019 ± 0.029 IB 0.147 ± 0.232 0.022 ± 0.037 BLOSUM 0.360 ± 0.062 Figure 4: Left: alignment differences for IB models and sequence alignment, within and between superfamilies. Right: ROC curve for same/different superfamily classification by alignment score. by gaps, and resulting pairs of aligned gaps in the profile-profile alignment are removed. We consider both sequences from the same family and those from other families in the same clan, the former being more similar than the latter, and therefore having better alignments. Assuming the profile-profile alignment is closest to the “true” alignment, iIB alignment significantly outperforms sequence alignment in both cases, with mIB showing a slight additional improvement. At right is the ROC curve for detecting superfamily relationships between profiles from different families based on alignment scores, showing that while IB fares worse than profiles, simple sequences perform essentially at chance. Finally, figure 3a compares the performance of profile and IB alignment for different sequence lengths. To use a profile alphabet for novel alignments, we must map each input profile to the closest IB class. To be consistent with Yona[10], we use the Jensen-Shannon (JS) distance with mixing coefficient 0.5 rather than the KL distance optimized in creating the categories. Aligning two sequences of lengths n and m requires computing the |C|(n+m) JS-distances between each profile and each category, a significant improvement over the mn distance computations required for profile-profile alignment when |C| ≪min(m,n) 2 . Our results show that JS distance computations dominate running time, since IB alignment time scales linearly with the input size, while profile alignment scales quadratically, yielding an order of magnitude improvement for typical 500- to 1000-base-pair sequences. 4 Discussion We have described a discrete approximation to amino acid profiles, based on minimizing information loss, that allows profile information to be used for alignment and search without additional computational cost compared to simple sequence alignment. Alignments of sequences encoded with a modest number of classes correspond to the original profile alignments significantly better than alignments of the original sequences. In addition to minimizing information loss, the classes can be constrained to correspond to the standard amino acid representation, yielding an intuitive, compact textual form for profile information. Our model is useful in three ways: (1) it makes it possible to apply existing fast discrete algorithms to arbitrary continuous sequences; (2) it models rich conditional distribution structures; and (3) its models can incorporate a variety of class constraints. We can extend our approach in each of these directions. For example, adjacent positions are highly correlated: the average entropy of a single profile is 0.99, versus 1.23 for an adjacent pair. Therefore pairs can be represented more compactly than the cross-product of a single-position alphabet. More generally, we can encode arbitrary conserved regions and still treat them symbolically for alignment and search. Other extensions include incorporating structural information in the input representation; assigning structural significance to the resulting categories; and learning multivariate IB’s underlying model’s structure. References [1] Nir Friedman, Ori Mosenzon, Noam Slonim, and Naftali Tishby. Multivariate information bottleneck. In Uncertainty in Artificial Intelligence: Proceedings of the Seventeenth Conference (UAI-2001), pages 152–161, San Francisco, CA, 2001. Morgan Kaufmann Publishers. [2] Crooks GE, Hon G, Chandonia JM, and Brenner SE. WebLogo: a sequence logo generator. Genome Research, in press, 2004. [3] A. G. Murzin, S. E. Brenner, T. Hubbard, and C. Chothia. SCOP: a structural classification of proteins database for the investigation of sequences and structures. J. Mol. Biol., 247:536–40, 1995. [4] N.D. Rawlings, D.P. Tolle, and A.J. Barrett. MEROPS: the peptidase database. Nucleic Acids Res, 32 Database issue:D160–4, 2004. [5] B. Rost and C. Sander. Prediction of protein secondary structure at better than 70% accuracy. J. Mol. Bio., 232:584–99, 1993. [6] Altschul SF, Gish W, Miller W, Myers EW, and Lipman DJ. Basic local alignment search tool. J Mol Biol, 215(3):403–10, October 1990. [7] Noam Slonim. The Information Bottleneck: Theory and Applications. PhD thesis, Hebrew University, Jerusalem, Israel, 2002. [8] T. F. Smith and M. S. Waterman. Identification of common molecular subsequences. Journal of Molecular Biology, 147:195–197, 1981. [9] Naftali Tishby, Fernando C. Pereira, and William Bialek. The information bottleneck method. In Proc. of the 37-th Annual Allerton Conference on Communication, Control and Computing, pages 368–77, 1999. [10] Golan Yona and Michael Levitt. Within the twilight zone: A sensitive profileprofile comparison tool based on information theory. Journal of Molecular Biology, 315:1257–75, 2002.
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Integrating Topics and Syntax Thomas L. Griffiths Mark Steyvers gruffydd@mit.edu msteyver@uci.edu Massachusetts Institute of Technology University of California, Irvine Cambridge, MA 02139 Irvine, CA 92614 David M. Blei Joshua B. Tenenbaum blei@cs.berkeley.edu jbt@mit.edu University of California, Berkeley Massachusetts Institute of Technology Berkeley, CA 94720 Cambridge, MA 02139 Abstract Statistical approaches to language learning typically focus on either short-range syntactic dependencies or long-range semantic dependencies between words. We present a generative model that uses both kinds of dependencies, and can be used to simultaneously find syntactic classes and semantic topics despite having no representation of syntax or semantics beyond statistical dependency. This model is competitive on tasks like part-of-speech tagging and document classification with models that exclusively use short- and long-range dependencies respectively. 1 Introduction A word can appear in a sentence for two reasons: because it serves a syntactic function, or because it provides semantic content. Words that play different roles are treated differently in human language processing: function and content words produce different patterns of brain activity [1], and have different developmental trends [2]. So, how might a language learner discover the syntactic and semantic classes of words? Cognitive scientists have shown that unsupervised statistical methods can be used to identify syntactic classes [3] and to extract a representation of semantic content [4], but none of these methods captures the interaction between function and content words, or even recognizes that these roles are distinct. In this paper, we explore how statistical learning, with no prior knowledge of either syntax or semantics, can discover the difference between function and content words and simultaneously organize words into syntactic classes and semantic topics. Our approach relies on the different kinds of dependencies between words produced by syntactic and semantic constraints. Syntactic constraints result in relatively short-range dependencies, spanning several words within the limits of a sentence. Semantic constraints result in long-range dependencies: different sentences in the same document are likely to have similar content, and use similar words. We present a model that can capture the interaction between short- and long-range dependencies. This model is a generative model for text in which a hidden Markov model (HMM) determines when to emit a word from a topic model. The different capacities of the two components of the model result in a factorization of a sentence into function words, handled by the HMM, and content words, handled by the topic model. Each component divides words into finer groups according to a different criterion: the function words are divided into syntactic classes, and the content words are divided into semantic topics. This model can be used to extract clean syntactic and semantic classes and to identify the role that words play in a document. It is also competitive in quantitative tasks, such as part-of-speech tagging and document classification, with models specialized to detect short- and long-range dependencies respectively. The plan of the paper is as follows. First, we introduce the approach, considering the general question of how syntactic and semantic generative models might be combined, and arguing that a composite model is necessary to capture the different roles that words can play in a document. We then define a generative model of this form, and describe a Markov chain Monte Carlo algorithm for inference in this model. Finally, we present results illustrating the quality of the recovered syntactic classes and semantic topics. 2 Combining syntactic and semantic generative models A probabilistic generative model specifies a simple stochastic procedure by which data might be generated, usually making reference to unobserved random variables that express latent structure. Once defined, this procedure can be inverted using statistical inference, computing distributions over latent variables conditioned on a dataset. Such an approach is appropriate for modeling language, where words are generated from the latent structure of the speaker’s intentions, and is widely used in statistical natural language processing [5]. Probabilistic models of language are typically developed to capture either short-range or long-range dependencies between words. HMMs and probabilistic context-free grammars [5] generate documents purely based on syntactic relations among unobserved word classes, while “bag-of-words” models like naive Bayes or topic models [6] generate documents based on semantic correlations between words, independent of word order. By considering only one of the factors influencing the words that appear in documents, these models assume that all words should be assessed on a single criterion: the posterior distribution for an HMM will group nouns together, as they play the same syntactic role even though they vary across contexts, and the posterior distribution for a topic model will assign determiners to topics, even though they bear little semantic content. A major advantage of generative models is modularity. A generative model for text specifies a probability distribution over words in terms of other probability distributions over words, and different models are thus easily combined. We can produce a model that expresses both the short- and long-range dependencies of words by combining two models that are each sensitive to one kind of dependency. However, the form of combination must be chosen carefully. In a mixture of syntactic and semantic models, each word would exhibit either short-range or long-range dependencies, while in a product of models (e.g. [7]), each word would exhibit both short-range and long-range dependencies. Consideration of the structure of language reveals that neither of these models is appropriate. In fact, only a subset of words – the content words – exhibit long-range semantic dependencies, while all words obey short-range syntactic dependencies. This asymmetry can be captured in a composite model, where we replace one of the probability distributions over words used in the syntactic model with the semantic model. This allows the syntactic model to choose when to emit a content word, and the semantic model to choose which word to emit. 2.1 A composite model We will explore a simple composite model, in which the syntactic component is an HMM and the semantic component is a topic model. The graphical model for this composite is shown in Figure 1(a). The model is defined in terms of three sets of variables: a sequence of words w = {w1, . . . , wn}, with each wi being one of W words, a sequence of topic assignments z = {z1, . . . zn}, with each zi being one of T topics, and a sequence of classes c = {c1, . . . , cn}, with each ci being one of C classes. One class, say ci = 1, is designated the “semantic” class. The zth topic is associated with a distribution over words 3 network trained with svm images network neural networks output ... image images object objects ... support vector svm ... kernel in with for on ... used trained obtained described ... (a) (b) image output network for used images kernel with obtained described with objects 0.2 0.5 0.4 0.1 0.8 0.9 0.7 z z z z θ 1 4 2 3 c c c c 1 4 2 3 w w w w 1 4 2 neural Figure 1: The composite model. (a) Graphical model. (b) Generating phrases. φ(z), each class c ̸= 1 is associated with a distribution over words φ(c), each document d has a distribution over topics θ(d), and transitions between classes ci−1 and ci follow a distribution π(si−1). A document is generated via the following procedure: 1. Sample θ(d) from a Dirichlet(α) prior 2. For each word wi in document d (a) Draw zi from θ(d) (b) Draw ci from π(ci−1) (c) If ci = 1, then draw wi from φ(zi), else draw wi from φ(ci) Figure 1(b) provides an intuitive representation of how phrases are generated by the composite model. The figure shows a three class HMM. Two classes are simple multinomial distributions over words. The third is a topic model, containing three topics. Transitions between classes are shown with arrows, annotated with transition probabilities. The topics in the semantic class also have probabilities, used to choose a topic when the HMM transitions to the semantic class. Phrases are generated by following a path through the model, choosing a word from the distribution associated with each syntactic class, and a topic followed by a word from the distribution associated with that topic for the semantic class. Sentences with the same syntax but different content would be generated if the topic distribution were different. The generative model thus acts like it is playing a game of Madlibs: the semantic component provides a list of topical words (shown in black) which are slotted into templates generated by the syntactic component (shown in gray). 2.2 Inference The EM algorithm can be applied to the graphical model shown in Figure 1, treating the document distributions θ, the topics and classes φ, and the transition probabilities π as parameters. However, EM produces poor results with topic models, which have many parameters and many local maxima. Consequently, recent work has focused on approximate inference algorithms [6, 8]. We will use Markov chain Monte Carlo (MCMC; see [9]) to perform full Bayesian inference in this model, sampling from a posterior distribution over assignments of words to classes and topics. We assume that the document-specific distributions over topics, θ, are drawn from a Dirichlet(α) distribution, the topic distributions φ(z) are drawn from a Dirichlet(β) distribution, the rows of the transition matrix for the HMM are drawn from a Dirichlet(γ) distribution, the class distributions φ(c) a re drawn from a Dirichlet(δ) distribution, and all Dirichlet distributions are symmetric. We use Gibbs sampling to draw iteratively a topic assignment zi and class assignment ci for each word wi in the corpus (see [8, 9]). Given the words w, the class assignments c, the other topic assignments z−i, and the hyperparameters, each zi is drawn from: P(zi|z−i, c, w) ∝ P(zi|z−i) P(wi|z, c, w−i) ∝ ( n(di) zi + α (n(di) zi + α) n (zi) wi +β n (zi) · +W β ci ̸= 1 ci = 1 where n(di) zi is the number of words in document di assigned to topic zi, n(zi) wi is the number of words assigned to topic zi that are the same as wi, and all counts include only words for which ci = 1 and exclude case i. We have obtained these conditional distributions by using the conjugacy of the Dirichlet and multinomial distributions to integrate out the parameters θ, φ. Similarly conditioned on the other variables, each ci is drawn from: P(ci|c−i, z, w) ∝ P(wi|c, z, w−i) P(ci|c−i) ∝ n (ci) wi +δ n (ci) · +W δ (n (ci−1) ci +γ)(n (ci) ci+1+I(ci−1=ci)·I(ci=ci+1)+γ) n (ci) · +I(ci−1=ci)+Cγ ci ̸= 1 n (zi) wi +β n (zi) · +W β (n (ci−1) ci +γ)(n (ci) ci+1+I(ci−1=ci)·I(ci=ci+1)+γ) n (ci) · +I(ci−1=ci)+Cγ ci = 1 where n(zi) wi is as before, n(ci) wi is the number of words assigned to class ci that are the same as wi, excluding case i, and n(ci−1) ci is the number of transitions from class ci−1 to class ci, and all counts of transitions exclude transitions both to and from ci. I(·) is an indicator function, taking the value 1 when its argument is true, and 0 otherwise. Increasing the order of the HMM introduces additional terms into P(ci|c−i), but does not otherwise affect sampling. 3 Results We tested the models on the Brown corpus and a concatenation of the Brown and TASA corpora. The Brown corpus [10] consists of D = 500 documents and n = 1, 137, 466 word tokens, with part-of-speech tags for each token. The TASA corpus is an untagged collection of educational materials consisting of D = 37, 651 documents and n = 12, 190, 931 word tokens. Words appearing in fewer than 5 documents were replaced with an asterisk, but punctuation was included. The combined vocabulary was of size W = 37, 202. We dedicated one HMM class to sentence start/end markers {.,?,!}. In addition to running the composite model with T = 200 and C = 20, we examined two special cases: T = 200, C = 2, being a model where the only HMM classes are the start/end and semantic classes, and thus equivalent to Latent Dirichlet Allocation (LDA; [6]); and T = 1, C = 20, being an HMM in which the semantic class distribution does not vary across documents, and simply has a different hyperparameter from the other classes. On the Brown corpus, we ran samplers for LDA and 1st, 2nd, and 3rd order HMM and composite models, with three chains of 4000 iterations each, taking samples at a lag of 100 iterations after a burn-in of 2000 iterations. On Brown+TASA, we ran a single chain for 4000 iterations for LDA and the 3rd order HMM and composite models. We used a Gaussian Metropolis proposal to sample the hyperparameters, taking 5 draws of each hyperparameter for each Gibbs sweep. 3.1 Syntactic classes and semantic topics The two components of the model are sensitive to different kinds of dependency among words. The HMM is sensitive to short-range dependencies that are constant across documents, and the topic model is sensitive to long-range dependencies that vary across documents. As a consequence, the HMM allocates words that vary across contexts to the semantic class, where they are differentiated into topics. The results of the algorithm, taken from the 4000th iteration of a 3rd order composite model on Brown+TASA, are shown in Figure 2. The model cleanly separates words that play syntactic and semantic roles, in sharp contrast to the results of the LDA model, also shown in the figure, where all words are forced into topics. The syntactic categories include prepositions, pronouns, past-tense verbs, and punctuation. While one state of the HMM, shown in the eighth column of the figure, emits common nouns, the majority of nouns are assigned to the semantic class. The designation of words as syntactic or semantic depends upon the corpus. For comparison, we applied a 3rd order composite model with 100 topics and 50 classes to a set the the the the the a the the the blood , , of a the , , , , and and , of of of a a of of of to , , a of in body a in in in in and and game heart in land and to water in drink ball and trees to classes picture is story alcohol and in tree farmers government film and is to team to with for a image matter to bottle to is on farm state lens are as in play blood forest farmers government light water story drugs ball heart trees land state eye matter stories drug game pressure forests crops federal lens molecules poem alcohol team body land farm public image liquid characters people * lungs soil food local mirror particles poetry drinking baseball oxygen areas people act eyes gas character person players vessels park farming states glass solid author effects football arteries wildlife wheat national object substance poems marijuana player * area farms laws objects temperature life body field breathing rain corn department lenses changes poet use basketball the in he * be said can time , a for it new have made would way ; his to you other see used will years ( this on they first make came could day : their with i same do went may part ) these at she great know found had number your by we good get called must kind her from there small go do place my as this little take have some into who old find did Figure 2: Upper: Topics extracted by the LDA model. Lower: Topics and classes from the composite model. Each column represents a single topic/class, and words appear in order of probability in that topic/class. Since some classes give almost all probability to only a few words, a list is terminated when the words account for 90% of the probability mass. of D = 1713 NIPS papers from volumes 0-12. We used the full text, from the Abstract to the Acknowledgments or References section, excluding section headers. This resulted in n = 4, 312, 614 word tokens. We replaced all words appearing in fewer than 3 papers with an asterisk, leading to W = 17, 268 types. We used the same sampling scheme as Brown+TASA. A selection of topics and classes from the 4000th iteration are shown in Figure 3. Words that might convey semantic information in another setting, such as “model”, “algorithm”, or “network”, form part of the syntax of NIPS: the consistent use of these words across documents leads them to be incorporated into the syntactic component. 3.2 Identifying function and content words Identifying function and content words requires using information about both syntactic class and semantic context. In a machine learning paper, the word “control” might be an innocuous verb, or an important part of the content of a paper. Likewise, “graph” could refer to a figure, or indicate content related to graph theory. Tagging classes might indicate that “control” appears as a verb rather than a noun, but deciding that “graph” refers to a figure requires using information about the content of the rest of the document. The factorization of words between the HMM and LDA components provides a simple means of assessing the role that a given word plays in a document: evaluating the posterior probability of assignment to the LDA component. The results of using this procedure to identify content words in sentences excerpted from NIPS papers are shown in Figure 4. Probabilities were evaluated by averaging over assignments from all 20 samples, and take into account the semantic context of the whole document. As a result of combining shortand long-range dependencies, the model is able to pick out the words in each sentence that concern the content of the document. Selecting the words that have high probability of image data state membrane chip experts kernel network images gaussian policy synaptic analog expert support neural object mixture value cell neuron gating vector networks objects likelihood function * digital hme svm output feature posterior action current synapse architecture kernels input recognition prior reinforcement dendritic neural mixture # training views distribution learning potential hardware learning space inputs # em classes neuron weight mixtures function weights pixel bayesian optimal conductance # function machines # visual parameters * channels vlsi gate set outputs in is see used model networks however # with was show trained algorithm values also * for has note obtained system results then i on becomes consider described case models thus x from denotes assume given problem parameters therefore t at being present found network units first n using remains need presented method data here into represents propose defined approach functions now c over exists describe generated paper problems hence r within seems suggest shown process algorithms finally p Figure 3: Topics and classes from the composite model on the NIPS corpus. 1. In contrast to this approach, we study here how the overall network activity can control single cell parameters such as input resistance, as well as time and space constants, parameters that are crucial for excitability and spariotemporal (sic) integration. The integrated architecture in this paper combines feed forward control and error feedback adaptive control using neural networks. 2. In other words, for our proof of convergence, we require the softassign algorithm to return a doubly stochastic matrix as *sinkhorn theorem guarantees that it will instead of a matrix which is merely close to being doubly stochastic based on some reasonable metric. The aim is to construct a portfolio with a maximal expected return for a given risk level and time horizon while simultaneously obeying *institutional or *legally required constraints. 3. The left graph is the standard experiment the right from a training with # samples. The graph G is called the *guest graph, and H is called the host graph. Figure 4: Function and content words in the NIPS corpus. Graylevel indicates posterior probability of assignment to LDA component, with black being highest. The boxed word appears as a function word and a content word in one element of each pair of sentences. Asterisked words had low frequency, and were treated as a single word type by the model. being assigned to syntactic HMM classes produces templates for writing NIPS papers, into which content words can be inserted. For example, replacing the content words that the model identifies in the second sentence with content words appropriate to the topic of the present paper, we could write: The integrated architecture in this paper combines simple probabilistic syntax and topic-based semantics using generative models. 3.3 Marginal probabilities We assessed the marginal probability of the data under each model, P(w), using the harmonic mean of the likelihoods over the last 2000 iterations of sampling, a standard method for evaluating Bayes factors via MCMC [11]. This probability takes into account the complexity of the models, as more complex models are penalized by integrating over a latent space with larger regions of low probability. The results are shown in Figure 5. LDA outperforms the HMM on the Brown corpus, but the HMM out-performs LDA on the larger Brown+TASA corpus. The composite model provided the best account of both corpora, 1st 2nd 3rd 1st 2nd 3rd −6e+06 −5.5e+06 −5e+06 −4.5e+06 −4e+06 LDA HMM Composite Brown Marginal likelihood 1st 2nd 3rd 1st 2nd 3rd −8e+07 −7e+07 −6e+07 −5e+07 −4e+07 LDA HMM Composite Brown+TASA Marginal likelihood Figure 5: Log marginal probabilities of each corpus under different models. Labels on horizontal axis indicate the order of the HMM. 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 1st 2nd 3rd 0 0.2 0.4 0.6 Adjusted Rand Index Brown Brown+TASA Brown Brown+TASA All tags Top 10 DC HMM Composite 0 0.2 0.4 0.6 0.8 1000 most frequent words Adjusted Rand Index HMM Composite Figure 6: Part-of-speech tagging for HMM, composite, and distributional clustering (DC). being able to use whichever kind of dependency information was most predictive. Using a higher-order transition matrix for either the HMM or the composite model produced little improvement in marginal likelihood for the Brown corpus, but the 3rd order models performed best on Brown+TASA. 3.4 Part-of-speech tagging Part-of-speech tagging – identifying the syntactic class of a word – is a standard task in computational linguistics. Most unsupervised tagging methods use a lexicon that identifies the possible classes for different words. This simplifies the problem, as most words belong to a single class. However, genuinely unsupervised recovery of parts-of-speech has been used to assess statistical models of language learning, such as distributional clustering [3]. We assessed tagging performance on the Brown corpus, using two tagsets. One set consisted of all Brown tags, excluding those for sentence markers, leaving a total of 297 tags. The other set collapsed these tags into ten high-level designations: adjective, adverb, conjunction, determiner, foreign, noun, preposition, pronoun, punctuation, and verb. We evaluated tagging performance using the Adjusted Rand Index [12] to measure the concordance between the tags and the class assignments of the HMM and composite models in the 4000th iteration. The Adjusted Rand Index ranges from −1 to 1, with an expectation of 0. Results are shown in Figure 6. Both models produced class assignments that were strongly concordant with part-of-speech, although the HMM gave a slightly better match to the full tagset, and the composite model gave a closer match to the top-level tags. This is partly because all words that vary strongly in frequency across contexts get assigned to the semantic class in the composite model, so it misses some of the fine-grained distinctions expressed in the full tagset. Both the HMM and the composite model performed better than the distributional clustering method described in [3], which was used to form the 1000 most frequent words in Brown into 19 clusters. Figure 6 compares this clustering with the classes for those words from the HMM and composite models trained on Brown. 3.5 Document classification The 500 documents in the Brown corpus are classified into 15 groups, such as editorial journalism and romance fiction. We assessed the quality of the topics recovered by the LDA and composite models by training a naive Bayes classifier on the topic vectors produced by the two models. We computed classification accuracy using 10-fold cross validation for the 4000th iteration from a single chain. The two models perform similarly. Baseline accuracy, choosing classes according to the prior, was 0.09. Trained on Brown, the LDA model gave a mean accuracy of 0.51(0.07), where the number in parentheses is the standard error. The 1st, 2nd, and 3rd order composite models gave 0.45(0.07), 0.41(0.07), 0.42(0.08) respectively. Trained on Brown+TASA, the LDA model gave 0.54(0.04), while the 1st. 2nd, and 3rd order composite models gave 0.48(0.06), 0.48(0.05), 0.46(0.08) respectively. The slightly lower accuracy of the composite model may result from having fewer data in which to find correlations: it only sees the words allocated to the semantic component, which account for approximately 20% of the words in the corpus. 4 Conclusion The composite model we have described captures the interaction between short- and longrange dependencies between words. As a consequence, the posterior distribution over the latent variables in this model picks out syntactic classes and semantic topics and identifies the role that words play in documents. The model is competitive in part-of-speech tagging and classification with models that specialize in short- and long-range dependencies respectively. Clearly, such a model does not do justice to the depth of syntactic or semantic structure, or their interaction. However, it illustrates how a sensitivity to different kinds of statistical dependency might be sufficient for the first stages of language acquisition, discovering the syntactic and semantic building blocks that form the basis for learning more sophisticated representations. Acknowledgements. The TASA corpus appears courtesy of Tom Landauer and Touchstone Applied Science Associates, and the NIPS corpus was provided by Sam Roweis. This work was supported by the DARPA CALO program and NTT Communication Science Laboratories. References [1] H. J. Neville, D. L. Mills, and D. S. Lawson. Fractionating language: Different neural subsytems with different sensitive periods. Cerebral Cortex, 2:244–258, 1992. [2] R. Brown. A first language. Harvard University Press, Cambridge, MA, 1973. [3] M. Redington, N. Chater, and S. Finch. Distributional information: A powerful cue for acquiring syntactic categories. Cognitive Science, 22:425–469, 1998. [4] T. K. Landauer and S. T. Dumais. A solution to Plato’s problem: the Latent Semantic Analysis theory of acquisition, induction, and representation of knowledge. Psychological Review, 104:211–240, 1997. [5] C. Manning and H. Sch¨utze. Foundations of statistical natural language processing. MIT Press, Cambridge, MA, 1999. [6] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet Allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [7] N. Coccaro and D. Jurafsky. Towards better integration of semantic predictors in statistical language modeling. In Proceedings of ICSLP-98, volume 6, pages 2403–2406, 1998. [8] T. L. Griffiths and M. Steyvers. Finding scientific topics. Proceedings of the National Academy of Science, 101:5228–5235, 2004. [9] W.R. Gilks, S. Richardson, and D. J. Spiegelhalter, editors. Markov Chain Monte Carlo in Practice. Chapman and Hall, Suffolk, 1996. [10] H. Kucera and W. N. Francis. Computational analysis of present-day American English. Brown University Press, Providence, RI, 1967. [11] R. E. Kass and A. E. Rafferty. Bayes factors. Journal of the American Statistical Association, 90:773–795, 1995. [12] L. Hubert and P. Arabie. Comparing partitions. Journal of Classification, 2:193–218, 1985.
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Object Classification from a Single Example Utilizing Class Relevance Metrics Michael Fink Interdisciplinary Center for Neural Computation The Hebrew University, Jerusalem 91904, Israel fink@huji.ac.il Abstract We describe a framework for learning an object classifier from a single example. This goal is achieved by emphasizing the relevant dimensions for classification using available examples of related classes. Learning to accurately classify objects from a single training example is often unfeasible due to overfitting effects. However, if the instance representation provides that the distance between each two instances of the same class is smaller than the distance between any two instances from different classes, then a nearest neighbor classifier could achieve perfect performance with a single training example. We therefore suggest a two stage strategy. First, learn a metric over the instances that achieves the distance criterion mentioned above, from available examples of other related classes. Then, using the single examples, define a nearest neighbor classifier where distance is evaluated by the learned class relevance metric. Finding a metric that emphasizes the relevant dimensions for classification might not be possible when restricted to linear projections. We therefore make use of a kernel based metric learning algorithm. Our setting encodes object instances as sets of locality based descriptors and adopts an appropriate image kernel for the class relevance metric learning. The proposed framework for learning from a single example is demonstrated in a synthetic setting and on a character classification task. 1 Introduction We describe a framework for learning to accurately discriminate between two target classes of objects (e.g. platypuses and opossums) using a single image of each class. In general, learning to accurately classify object images from a single training example is unfeasible due to overfitting effects of high dimensional data. However, if a certain distance function over the instances guarantees that all within-class distances are smaller than any betweenclass distance, then nearest neighbor classification could achieve perfect performance with a single training example. We therefore suggest a two stage method. First, learn from available examples of other related classes (like beavers, skunks and marmots), a metric over the instance space that satisfies the distance criterion mentioned above. Then, define a nearest neighbor classifier based on the single examples. This nearest neighbor classifier calculates distance using the class relevance metric. The difficulty in achieving robust object classification emerges from the instance variety of object appearance. This variability results from both class relevant and class non-relevant dimensions. For example, adding a stroke crossing the digit 7, adds variability due to a class relevant dimension (better discriminating 7’s from 1’s), while italic writing adds variability in a class irrelevant dimension. Often certain non-relevant dimensions could be avoided by the designer’s method of representation (e.g. incorporating translation invariance). Since such guiding heuristics may be absent or misleading, object classification systems often use numerous positive examples for training, in an attempt to manage within class variability. We are guided by the observation that in many settings providing an extended training set of certain classes might be costly or impossible due to scarcity of examples, thus motivating methods that suffice with few training examples. Categories’ appearance variety seems to inherently entail severe overfitting effects when only a small sample is available for training. In the extreme case of learning from a single example it appears that the effects of overfitting might prevent any robust category generalization. These overfitting effects tend to exacerbate as a function of the representation dimensionality. In the spirit of the learning to learn literature [17], we try to overcome the difficulties that entail training from a single example by using available examples from several other related objects. Recently, it has been demonstrated that objects share distribution densities on deformation transforms [13], shape or appearance [6]; and that objects could be detected by a common set of reusable features [1, 18]. We suggest that in many visual tasks it is natural to assume that one common set of constraints characterized a common set of relevant and non-relevant dimensions shared by a specific family of related classes [10]. Our paper is organized as follows. In Sec. 2 we start by formalizing the task of training from a single example. Sec. 3 describes a kernel over sets of local features. We then describe in Sec. 4 a kernel based method for learning a pseudo-metric that is capable of emphasizing the relevant dimensions and diminishing the overfitting effects of non-relevant dimensions. By projecting the single examples using this class relevance pseudo-metric, learning from a single example becomes feasible. Our experimental implementation described in Sec. 5, adopts shape context descriptors [3] of Latin letters to demonstrate the feasibility of learning from a single example. We conclude with a discussion on the scope and limitations of the proposed method. 2 Problem Setting Let X be our object instance space and let u and v indicate two classes defined over X. Our goal is to generate a classifier h(x) which discriminates between instances of the two object classes u and v. Formally, h : X →{u, v} so that ∀x in class u, h(x) = u and ∀x in class v, h(x) = v. We adopt a local features representation for encoding object images. Thus, every x in our instance space is characterized by the set {li j, pi j}k j=1 where li j is a locality based descriptor calculated at location pi j of image i 1. We assume that li j is encoded as a vector of length n and that the same number of locations k are selected from each image2. Thus any x in our instance space X is defined by an n × k matrix. Our method uses a single instance from classes u and v as well as instances from other related classes. We denote by q the total number of classes. An example is formally defined as a pair (x, y) where x ∈X is an instance and y ∈{1, . . . , q} is the index of the instance’s class. The proposed setting postulates that two sets are provided for training h(x): 1pi j might be selected from image i either randomly, or by a specialized interest point detector. 2This assumption could be relaxed as demonstrated in [16, 19] • A single example of class u, (x, u) and a single example of class v, (x, v) • An extended sample {(x1, y1), . . . , (xm, ym)} of m >> 1 examples where xi ∈X and yi /∈{u, v} for all 1 ≤i ≤m. We say that a set of classes is γ > 0 separated with respect to a distance function d if for any pair of examples belonging to the same class {(x1, c), (x′ 1, c)}, the distance d(x1, x′ 1) is smaller than the distance between any pair of examples from different classes {(x2, e), (x′ 2, g)} by at least γ: d(x1, x′ 1) ≤d(x2, x′ 2) −γ . Recall that our goal is to generate a classifier h(x) which discriminates between instances of the two object classes u and v. In general, learning from a single example is prone to overfitting, yet if a set of classes is γ separated, a single example is sufficient for a nearest neighbor classifier to achieve perfect performance. Therefore our proposed framework is composed of two stages: 1. Learn from the extended sample a distance function d that achieves γ separation on classes y /∈{u, v}. 2. Learn a nearest neighbor classifier h(x) from the single examples, where the classifier employs d for evaluating distances. From the theory of large margin classifiers we know that if a classifier achieves a large margin separation on an i.i.d. sample then it is likely to generalize well. We informally state that analogously, if we find a distance function d such that q −2 classes that form the extended sample are separated by a large γ with respect to d, with high probability classes u and v should exhibit the separation characteristic as well. If these assumptions hold and d indeed induces γ separation on classes u and v, then a nearest neighbor classifier would generalize well from a single training example of the target classes. It should be noted that when training from a single example nearest neighbor, max margin and naive Bayes algorithms, all yield the same classification rule. For simplicity we choose to focus on a nearest neighbor formulation. We will later show how the distance d might be parameterized by measuring Euclidian distance, after applying a linear projection W to the original instance space. Classifying instances in the original instance space by comparing them to the target classes’ single examples x and x′, leads to overfitting. In contrast, our approach projects the instance space by W and only then applies a nearest neighbor distance measurement to the projected single examples Wx and Wx′. Our method relies on the distance d, parameterized by W, to achieve γ separation on classes u and v. In certain problems it is not possible to achieve γ separation by using a distance function which is based on a linear transformation of the instance space. We therefore propose to initially map the instance space X into an implicit feature space defined by a Mercer kernel [20]. 3 A Principal Angles Image Kernel We dedicate this section to describe a Mercer kernel between sets of locality based image features {li j, pi j}k j=1 encoded as n × k matrices. Although potentially advantageous in many applications, one shortcoming in adopting locality based feature descriptors lays in the vagueness of matching two sets of corresponding locations pi j, pi′ j′ selected from different object images i and i′ (see Fig. 1). Recently attempts have been made to tackle this problem [19], we choose to follow [20] by adopting the principal angles kernel approach that implicitly maps x of size n × k to a significantly higher n k -dimensional feature space φ(x) ∈F. The principal angles kernel is formally defined as: K(xi, xi′) = φ(xi)φ(xi′) = det(Q⊤ i Qi′)2 5 10 15 20 25 30 35 40 10 20 30 40 50 60 5 10 15 20 25 30 35 40 10 20 30 40 50 60 5 10 15 20 25 30 35 40 10 20 30 40 50 60 Figure 1: The 40 columns in each matrix encode 60-dimentional descriptors (detailed in Sec. 5) of three instances of the letter e. Although the objects are similar, the random sequence of sampling locations pi j entails column permutation, leading to apparently different matrices. Ignoring selection permutation by reshaping the matrices as vectors would further obscure the relevant similarity. A kernel applied to matrices that is invariant to column permutation can circumvent this problem. The columns of Qi and Qi′ are each an orthonormal basis resulting from a QR decomposition of the instances xi and xi′ respectively. One advantage of the principal angels kernel emerges from its invariance to column permutations of the instance matrices xi and xi′, thus circumventing the location matching problem stated above. Extensions of the principal angles kernel that have the additional capacity to incorporate knowledge on the accurate location matching, might enhance the kernel’s descriptive power [16]. 4 Learning a Class Relevance Pseudo-Metric In this section we describe the two stage framework for learning from a single example to accurately classify classes u and v. We focus on transferring information from the extended sample of classes y /∈{u, v} in the form of a learned pseudo-metric over X. For sake of clarity we will start by temporarily referring to the instance space X as a vector space, but later return to our original definition of instances in X as being matrices which columns encode a selected set of locality based descriptors {li j, pi j}k j=1. A pseudo-metric is a function d : X × X →R, which satisfies three requirements, (i) d(x, x′) ≥0, (ii) d(x, x′) = d(x′, x), and (iii) d(x1, x2) + d(x2, x3) ≥d(x1, x3). Following [14], we restrict ourselves to learning pseudo-metrics of the form dA(x, x′) ≡ q (x −x′)⊤A(x −x′) , where A ⪰0 is a symmetric positive semi-definite (PSD) matrix. Since A is PSD, there exists a matrix W such that (x −x′)⊤A(x −x′) = ∥Wx −Wx′∥2 2 . Therefore, dA(x, x′) is the Euclidean distance between the image of x and x′ due to a linear transformation W. We now restate our goal as that of using the extended sample of classes y /∈{u, v} in order to find a linear projection W that achieves γ separation by emphasizing the relevant dimensions for classification and diminishing the overfitting effects of non-relevant dimensions. Several linear methods exist for finding a class relevance projection [2, 9], some of which have a kernel based variant [12]. Our method of choice, proposed by [14], is an online algorithm characterized by its capacity to efficiently handle high dimensional input spaces. In addition the method’s margin based approach is directly aimed at achieving our γ separation goal. We convert the online algorithm for finding A to our batch setting by averaging the resulting A over the algorithm’s τ iterations [4]. Fig. 2 demonstrates how a class relevance pseudo-metric enables training a nearest neighbor classifier from a single example of two classes in a synthetic two dimensional setting. Figure 2: A synthetic sample of six obliquely oriented classes in a two dimensional space (left). A class relevance metric is calculated from the (m = 200) examples of the four classes y /∈{u, v} marked in gray. The examples of the target classes u and v, indicated in black, are not used in calculating the metric. After learning the pseudo-metric, all the instances of the six classes are projected to the class relevance space. Here distance measurements are performed between the four classes y /∈{u, v}. The results are displayed as a color coded distance matrix (center-top). Throughout the paper distance matrix indices are ordered by class so γ separated classes should appear as block diagonal matrices. Although not participating in calculating the pseudo-metric, classes u and v exhibit γ separation (center-bottom). After the class relevance projection, a nearest neighbor classifier will generalize well from any single example of classes u and v (right). In the primal setting of the pseudo-metric learning, we temporarily addressed our instances x as vectors, thus enabling subtraction and dot product operations. These operations have no clear interpretation when applied to our representation of objects as sets of locality based descriptors {li j, pi j}k j=1. However the adopted pseudo-metric learning algorithm has a dual version, where interface to the data is limited to inner products. In the dual mode A is implicitly represented by a set of support examples {xj}τ j=1 and by learning two sets of scalar coefficients {βh}f h=1 and {ρj,h}(τ,f) (j,h)=(1,1). Thus, applying the dual representation of the pseudo-metric, distances between instances x and x′ could be calculated by: dA(x, x′)2 = f X h=1 βh τ X j=1 ρj,h [ K(xj, x) −K(xj, x′) −K(x′ j, x) + K(x′ j, x′) ] 2 dA(x, x′)2 in the above equation is therefore evaluated by calling upon the principal angles kernel previously described in Sec. 3. Fig. 3 demonstrates how a class relevance pseudometric enables training from a single example in a classification problem, where nonlinear projection of the instance space is required for achieving a γ margin. 5 Experiments Sets of six lowercase Latin letters (i.e. e, n, t, f, h and c) are selected as target classes for our experiment (see examples in Fig. 4). The Latin character database [7] includes 60 examples of each letter. Two representations are examined. The first is a pixel based representation resulting from column-wise encoding the raw 36 × 36 pixel images as a vector of length 1296. Our second representation adopts the shape context descriptors for object encoding. This representation relies on a set of 40 locations pj randomly sampled from the object contour. The descriptor of each location pj is based on a 60-bin histogram (5 radius × 12 orientation bins) summing the number of ”lit” pixels falling in each specific radius and orientation bin (using pj as the origin). Each example in our instance space is therefore encoded as a 60 × 40 matrix. Three shape context descriptors are depicted in Fig. 4. Shape Figure 3: A synthetic sample of six co-centric classes in a two dimensional space (left). Two class relevance metrics are calculated from the examples (m = 200) of the four classes y /∈{u, v} marked in gray using either a linear or a second degree polynomial kernel. The examples of the target classes u and v, indicated in black, are not used in calculating the metrics. After learning both metrics, all the instances of the six classes are projected using both class relevance metrics. Then distance measurements are performed between the four classes y /∈{u, v}. The resulting linear distance matrix (center-top) and polynomial distance matrix (right-top) seem qualitatively different. Classes u and v, not participating in calculating the pseudo-metric, exhibit γ separation only when using an appropriate kernel (right-bottom). A linear kernel cannot accommodate γ separation between co-centric classes (center-bottom). context descriptors have proven to be robust in many classification tasks [3] and avoid the common reliance on a detection of (the often elusive) interest points. In many writing systems letters tend to share a common underlying set of class relevant and non-relevant dimensions (Fig. 5-left). We therefore expect that letters should be a good candidate for exhibiting that a class relevance pseudo-metric achieving a large margin γ, would induce the distance separation characteristic on two additional letter classes in the same system. We randomly select a single example of two letters (i.e. e and n) for training and save the remaining examples for testing. A nearest neighbor classifier is defined by the two examples, in order to assess baseline performance of training from a single example. A linear kernel is applied for the pixel based representation while the principal angles kernel is used for the shape context representation. Performance is assessed by averaging the generalization accuracy (on the unseen testing examples) over 900 repetitions of random letter selection. Baseline results for the shape context and pixel representations are depicted in Fig. 5 A and C, respectively (letter references to Fig. 5 appear on the right bar plot). We now make use of the 60 examples of each of the remaining letters (i.e. t, f, h and c) in order to learn a distance over letters. The dual formulation of the pseudo-metric learning algorithm (described in Sec. 4) is implemented and run for 1000 iterations over random pairs selected from the 240 training examples (m = 4 classes × 60 examples). The same 900 example pairs used in the baseline testing are now projected using the letter metric. It is observed that the learned pseudo-metric approximates the separation goal on the two unseen target classes u and v (center plot of Fig. 5). A nearest neighbor classifier is then trained using the projected examples (Wx,Wx′) from class u and v. Performance is assessed as in the baseline test. Results for the shape context based representation are presented in Fig. 5B while performance of the pixel based representation is depicted in Fig. 5E. When training from a single example the lower dimensional pixel representation (of size 1296) displays less of an overfitting effect than the shape context representation paired with a principal angles kernel (implicitly mapped by the kernel from size 60 × 40 to size 60 40 ). This effect could be seen when comparing Fig. 5D and Fig. 5A. It is not surprising that although some dimensions in the high dimensional shape context feature represen5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 2 4 6 8 10 12 1 2 3 4 5 log(r) θπ/6 2 4 6 8 10 12 1 2 3 4 5 log(r) θπ/6 2 4 6 8 10 12 1 2 3 4 5 log(r) θπ/6 Figure 4: Examples of six character classes used in the letter classification experiment (left). The context descriptor at location p is based on a 60-bin histogram (5 radius × 12 orientation bins) of all surrounding pixels, using p as the origin. Three examples of the letter e, depicted with the histogram bin boundaries (top) and three derived shape context histograms plotted as log(radius) × orientation bins (bottom). Note the similarity of the two shape context descriptors sampled from analogous locations on two different examples of the letter e (two bottom-center plots). The shape context of a descriptor sampled from a distant location is evidently different (right). A B C D E F 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Figure 5: Letters in many writing systems, like uppercase Latin, tend to share a common underlying set of class relevant and non-relevant dimensions (left plot adapted from [5]). A class relevance pseudo-metric was calculated from four letters (i.e. t, f, h and c). The central plot depicts the distance matrix of the two target letters (i.e. e and n) after the class relevance pseudo-metric projection. The right plot presents average accuracy of classifiers trained on a single example of lowercase letters (i.e. e and n) in the following conditions: A. Shape Context Representation B. Shape Context Representation after class relevance projection C. Shape Context Representation after a projection derived from uppercase letters D. Pixel Representation E. Pixel Representation after class relevance projection F. Pixel Representation after a projection derived from uppercase letters. tation might exhibit superior performance in classification, increasing the representation dimensionality introduces numerous non-relevant dimensions, thus causing the substantial overfitting effects displayed at Fig. 5A. However, it appears that by projecting the single examples using the class relevance pseudo-metric, the class relevant dimensions are emphasized and hindering effects of the non-relevant dimensions are diminished (displayed at Fig. 5B). It should be noted that a simple linear pseudo-metric projection cannot achieve the desired margin on the extended sample, and therefore seems not to generalize well from the single trial training stage. This phenomenon is manifested by the decrease in performance when linearly projecting the pixel based representation (Fig. 5E). Our second experiment is aimed at examining the underlying assumptions of the proposed method. Following the same setting as in the first experiment we randomly selected two lowercase Latin letters for the single trial training task, while applying a pseudo-metric projection derived from uppercase Latin letters. It is observed that utilizing a less relevant pseudo-metric attenuates the benefit in the setting based on the shape context representation paired with the principal angles kernel (Fig. 5C). In the linear pixel based setting projecting lowercase letters to the uppercase relevance directions significantly deteriorates performance (Fig. 5F), possibly due to deemphasizing the lowercase characterizing curves. 6 Discussion We proposed a two stage method for classifying object images using a single example. Our approach, first attempts to learn from available examples of other related classes, a class relevance metric where all within class distances are smaller than between class distances. We then, define a nearest neighbor classifier for the two target classes, using the class relevance metric. Our high dimensional representation applied a principal angles kernel [20] to sets of local shape descriptors [3]. We demonstrated that the increased representational dimension aggravates overfitting when learning from a single example. However, by learning the class relevance metric from available examples of related objects, relevant dimensions for classification are emphasized and the overfitting effects of irrelevant dimensions are diminished. Our technique thereby generates a highly accurate classifier from only a single example of the target classes. Varying the choice of local feature descriptors [11, 15], and enhancing the image kernel [16] might further improve the proposed method’s generalization capacity in other object classification settings. We assume that our examples represent a set of classes that originate from a common set of constraints, thus imposing that the classes tend to agree on the relevance and non-relevance of different dimensions. Our assumption holds well for objects like textual characters [5]. It has been recently demonstrated that feature selection mechanisms can enable real-world object detection by a common set of shared features [18, 8]. These mechanisms are closely related to our framework when considering the common features as a subset of directions in our class relevance pseudo-metric. We therefore aim our current research at learning to classify more challenging objects. References [1] S. Krempp, D. Geman and Y. Amit. Sequential learning of reusable parts for object detection. Technical report, CS Johns Hopkins, 2002. [2] A. Bar-Hillel, T. Hertz, N. Shental and D. Weinshall. Learning Distance Functions Using Equivalence Relations. Proc ICML03, 2003. [3] S. Belongie, J. Malik and J. Puzicha. Matching Shapes. Proc. ICCV, 2001. [4] N. Cesa-Bianchi, A. Conconi, and C. Gentile. On the generalization ability of on-line learning algorithms. IEEE Transactions on Information Theory. To appear , 2004. [5] M.A. Chanagizi and S. Shimojo. Complexity and redundancy of writing systems, and implications for letter perception. under review, 2004. [6] L. Fei-Fei, R. Fergus and P. Perona. Learning generative visual models from few training examples. CVPR04 Workshop on Generative-Model Based Vision, 2004. [7] M. Fink. A Latin Character Database. www.cs.huji.ac.il/∼fink, 2004. [8] M. Fink and K. Levi. Encoding Reusable Perceptual Features Enables Learning Future Categories from Few Examples. Tech Report CS HUJI , 2004. [9] K. Fukunaga. Statistical Pattern Recognition. San Diego: Academic Press 2nd Ed., 1990. [10] K. Levi and M. Fink. Learning From a Small Number of Training Examples by Exploiting Object Categories. LCVPR04 workshop on Learning in Computer Vision, 2004. [11] D. G. Lowe. Object recognition from local scale-invariant features. Proc. ICCV99, 1999. [12] S. Mika, G. Ratsch, J. Weston, B. Scholkopf and K. R. Muller. Fisher Discriminant Analysis with Kernels. Neural Networks for Signal Processing IX, 1999. [13] E. Miller, N. Matsakis and P. Viola. Learning from One Example through Shared Densities on Transforms. Proc. CVPR00(1), 2000. [14] S. Shalev, Y. Singer and A. Ng. Online and Batch Learning of Pseudo-Metrics. Proc. ICML04, 2004. [15] M. J. Swain and D. H. Ballard. Color Indexing. IJCV 7(1), 1991. [16] A. Shashua and T. Hazan. Threading Kernel Functions: Localized vs. Holistic Representations and the Family of Kernels over Sets of Vectors with Varying Cardinality. NIPS04 under review. [17] S. Thrun and L. Pratt. Learning to Learn. Kluwer Academic Publishers, 1997. [18] A. Torralba, K. Murphy and W. Freeman. Sharing features: efficient boosting procedures for multiclass object detection. Proc. CVPR04, 2004. [19] C.Wallraven, B.Caputo and A.Graf Recognition with Local features kernel recipe. ICCV, 2003. [20] L. Wolf and A. Shashua. Learning over sets using kernel principal angles. JML 4, 2003.
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A Large Deviation Bound for the Area Under the ROC Curve Shivani Agarwal∗, Thore Graepel†, Ralf Herbrich† and Dan Roth∗ ∗Dept. of Computer Science University of Illinois Urbana, IL 61801, USA {sagarwal,danr}@cs.uiuc.edu †Microsoft Research 7 JJ Thomson Avenue Cambridge CB3 0FB, UK {thoreg,rherb}@microsoft.com Abstract The area under the ROC curve (AUC) has been advocated as an evaluation criterion for the bipartite ranking problem. We study large deviation properties of the AUC; in particular, we derive a distribution-free large deviation bound for the AUC which serves to bound the expected accuracy of a ranking function in terms of its empirical AUC on an independent test sequence. A comparison of our result with a corresponding large deviation result for the classification error rate suggests that the test sample size required to obtain an ϵ-accurate estimate of the expected accuracy of a ranking function with δ-confidence is larger than that required to obtain an ϵ-accurate estimate of the expected error rate of a classification function with the same confidence. A simple application of the union bound allows the large deviation bound to be extended to learned ranking functions chosen from finite function classes. 1 Introduction In many learning problems, the goal is not simply to classify objects into one of a fixed number of classes; instead, a ranking of objects is desired. This is the case, for example, in information retrieval problems, where one is interested in retrieving documents from some database that are ‘relevant’ to a given query or topic. In such problems, one wants to return to the user a list of documents that contains relevant documents at the top and irrelevant documents at the bottom; in other words, one wants a ranking of the documents such that relevant documents are ranked higher than irrelevant documents. The problem of ranking has been studied from a learning perspective under a variety of settings [2, 8, 4, 7]. Here we consider the setting in which objects come from two categories, positive and negative; the learner is given examples of objects labeled as positive or negative, and the goal is to learn a ranking in which positive objects are ranked higher than negative ones. This captures, for example, the information retrieval problem described above; in this case, the training examples consist of documents labeled as relevant (positive) or irrelevant (negative). This form of ranking problem corresponds to the ‘bipartite feedback’ case of [7]; for this reason, we refer to it as the bipartite ranking problem. Formally, the setting of the bipartite ranking problem is similar to that of the binary classification problem. In both problems, there is an instance space X and a set of two class labels Y = {−1, +1}. One is given a finite sequence of labeled training examples S = ((x1, y1), . . . , (xM, yM)) ∈(X × Y)M, and the goal is to learn a function based on this training sequence. However, the form of the function to be learned in the two problems is different. In classification, one seeks a binary-valued function h : X→Y that predicts the class of a new instance in X. On the other hand, in ranking, one seeks a real-valued function f : X →R that induces a ranking over X; an instance that is assigned a higher value by f is ranked higher than one that is assigned a lower value by f. The area under the ROC curve (AUC) has recently gained some attention as an evaluation criterion for the bipartite ranking [3]. Given a ranking function f : X→R and a finite data sequence T = ((x1, y1), . . . , (xN, yN)) ∈(X × Y)N containing m positive and n negative examples, the AUC of f with respect to T, denoted ˆA(f; T), can be expressed as the following Wilcoxon-Mann-Whitney statistic [3]: ˆA(f; T) = 1 mn X {i:yi=+1} X {j:yj=−1} I{f(xi)>f(xj)} + 1 2I{f(xi)=f(xj)} , (1) where I{·} denotes the indicator variable whose value is one if its argument is true and zero otherwise. The AUC of f with respect to T is thus simply the fraction of positivenegative pairs in T that are ranked correctly by f, assuming that ties are broken uniformly at random.1 The AUC is an empirical quantity that evaluates a ranking function with respect to a particular data sequence. What does the empirical AUC tell us about the expected performance of a ranking function on future examples? This is the question we consider. The question has two parts, both of which are important for machine learning practice. First, what can be said about the expected performance of a ranking function based on its empirical AUC on an independent test sequence? Second, what can be said about the expected performance of a learned ranking function based on its empirical AUC on the training sequence from which it is learned? We address the first question in this paper; the second question is addressed in [1]. We start by defining the expected ranking accuracy of a ranking function (analogous to the expected error rate of a classification function) in Section 2. Section 3 contains our large deviation result, which serves to bound the expected accuracy of a ranking function in terms of its empirical AUC on an independent test sequence. Our conceptual approach in deriving the large deviation result for the AUC is similar to that of [9], in which large deviation properties of the average precision were considered. Section 4 compares our bound to a corresponding large deviation bound for the classification error rate. A simple application of the union bound allows the large deviation bound to be extended to learned ranking functions chosen from finite function classes; this is described in Section 5. 2 Expected Ranking Accuracy We begin by introducing some notation. As in classification, we shall assume that all examples are drawn randomly and independently according to some (unknown) underlying distribution D over X × Y. The notation D+1 and D−1 will be used to denote the class-conditional distributions DX|Y =+1 and DX|Y =−1, respectively. We shall find it convenient to decompose a data sequence T = ((x1, y1), . . . , (xN, yN)) ∈(X × Y)N into two components, TX = (x1, . . . , xN) ∈X N and TY = (y1, . . . , yN) ∈YN. Several of our results will involve the conditional distribution DTX|TY =y for some label sequence y = (y1, . . . , yN) ∈YN; this distribution is simply Dy1 × . . . × DyN .2 As a final note of 1In [3], a slightly simpler form of the Wilcoxon-Mann-Whitney statistic is used, which does not account for ties. 2Note that, since the AUC of a ranking function f with respect to a data sequence T ∈(X ×Y)N is independent of the ordering of examples in the sequence, our results involving the conditional distribution DTX|TY =y for some label sequence y = (y1, . . . , yN) ∈YN depend only on the number m of positive labels in y and the number n of negative labels in y. We state our results in terms of the distribution DTX|TY =y ≡Dy1 × . . . × DyN only because this is more general than Dm +1 × Dn −1. convention, we use T ∈(X × Y)N to denote a general data sequence (e.g., an independent test sequence), and S ∈(X × Y)M to denote a training sequence. Definition 1 (Expected ranking accuracy). Let f : X→R be a ranking function on X. Define the expected ranking accuracy (or simply ranking accuracy) of f, denoted by A(f), as follows: A(f) = EX∼D+1,X′∼D−1 n I{f(X)>f(X′)} + 1 2I{f(X)=f(X′)} o . The ranking accuracy A(f) defined above is simply the probability that an instance drawn randomly according to D+1 will be ranked higher by f than an instance drawn randomly according to D−1, assuming that ties are broken uniformly at random. The following simple lemma shows that the empirical AUC of a ranking function f is an unbiased estimator of the expected ranking accuracy of f: Lemma 1. Let f : X→R be a ranking function on X, and let y = (y1, . . . , yN) ∈YN be any finite label sequence. Then ETX|TY =y n ˆA(f; T) o = A(f) . Proof. Let m be the number of positive labels in y, and n the number of negative labels in y. Then from the definition of the AUC (Eq. (1)) and linearity of expectation, we have ETX|TY =y n ˆA(f; T) o = 1 mn X {i:yi=+1} X {j:yj=−1} EXi∼D+1,Xj∼D−1 n I{f(Xi)>f(Xj)} + 1 2I{f(Xi)=f(Xj)} o = 1 mn X {i:yi=+1} X {j:yj=−1} A(f) = A(f) . ⊓⊔ 3 Large Deviation Bound We are interested in bounding the probability that the empirical AUC of a ranking function f with respect to a (random) test sequence T will have a large deviation from its expected ranking accuracy. In other words, we are interested in bounding probabilities of the form P n ˆA(f; T) −A(f) ≥ϵ o for given ϵ > 0. Our main tool in deriving such a large deviation bound will be the following powerful concentration inequality of McDiarmid [10], which bounds the deviation of any function of a sample for which a single change in the sample has limited effect: Theorem 1 (McDiarmid, 1989). Let X1, . . . , XN be independent random variables with Xk taking values in a set Ak for each k. Let φ : (A1 × · · · × AN) →R be such that sup xi∈Ai,x′ k∈Ak |φ(x1, . . . , xN) −φ(x1, . . . , xk−1, x′ k, xk+1, . . . , xN)| ≤ ck . Then for any ϵ > 0, P {|φ(X1, . . . , XN) −E{φ(X1, . . . , XN)}| ≥ϵ} ≤ 2e−2ϵ2/ P N k=1 c2 k . Note that when X1, . . . , XN are independent bounded random variables with Xk ∈[ak, bk] with probability one and φ(X1, . . . , XN) = PN k=1 Xk, McDiarmid’s inequality (with ck = bk −ak) reduces to Hoeffding’s inequality. Next we define the following quantity which appears in several of our results: Definition 2 (Positive skew). Let y = (y1, . . . , yN) ∈YN be a finite label sequence of length N ∈N. Define the positive skew of y, denoted by ρ(y), as follows: ρ(y) = 1 N X {i:yi=+1} 1 . The following can be viewed as the main result of this paper. We note that our results are all distribution-free, in the sense that they hold for any distribution D over X × Y. Theorem 2. Let f : X→R be a fixed ranking function on X and let y = (y1, . . . , yN) ∈ YN be any label sequence of length N ∈N. Then for any ϵ > 0, PTX|TY =y n ˆA(f; T) −A(f) ≥ϵ o ≤ 2e−2ρ(y)(1−ρ(y))Nϵ2 . Proof. Let m be the number of positive labels in y, and n the number of negative labels in y. We can view TX = (X1, . . . , XN) ∈X N as a random vector; given the label sequence y, the random variables X1, . . . , XN are independent, with each Xk taking values in X. Now, define φ : X N→R as follows: φ (x1, . . . , xN) = ˆA (f; ((x1, y1), . . . , (xN, yN))) . Then, for each k such that yk = +1, we have the following for all xi, x′ k ∈X: φ(x1, . . . , xN) −φ(x1, . . . , xk−1, x′ k, xk+1 . . . , xN) = 1 mn X {j:yj=−1} I{f(xk)>f(xj)} + 1 2I{f(xk)=f(xj)} − I{f(x′ k)>f(xj)} + 1 2I{f(x′ k)=f(xj)} ! ≤ 1 mnn = 1 m . Similarly, for each k such that yk = −1, one can show for all xi, x′ k ∈X: φ(x1, . . . , xN) −φ(x1, . . . , xk−1, x′ k, xk+1 . . . , xN) ≤ 1 n . Thus, taking ck = 1/m for k such that yk = +1 and ck = 1/n for k such that yk = −1, and applying McDiarmid’s theorem, we get for any ϵ > 0, PTX|TY =y n ˆA(f; T) −ETX|TY =y n ˆA(f; T) o ≥ϵ o ≤ 2e−2ϵ2/(m( 1 m )2+n( 1 n )2) . (2) Now, from Lemma 1, ETX|TY =y n ˆA(f; T) o = A(f) . Also, we have 1 m( 1 m)2 + n( 1 n)2 = 1 1 m + 1 n = mn m + n = ρ(y)(1 −ρ(y))N . Substituting the above in Eq. (2) gives the desired result. ⊓⊔ We note that the result of Theorem 2 can be strengthened so that the conditioning is only on the numbers m and n of positive and negative labels, and not on the specific label vector y.3 From Theorem 2, we can derive a confidence interval interpretation of the bound that gives, for any 0 < δ ≤1, a confidence interval based on the empirical AUC of a ranking function (on a random test sequence) which is likely to contain the true ranking accuracy with probability at least 1 −δ. More specifically, we have: Corollary 1. Let f : X→R be a fixed ranking function on X and let y = (y1, . . . , yN) ∈ YN be any label sequence of length N ∈N. Then for any 0 < δ ≤1, PTX|TY =y ( ˆA(f; T) −A(f) ≥ s ln 2 δ 2ρ(y)(1 −ρ(y))N ) ≤ δ . Proof. This follows directly from Theorem 2 by setting 2e−2ρ(y)(1−ρ(y))Nϵ2 = δ and solving for ϵ. ⊓⊔ Theorem 2 also allows us to obtain an expression for a test sample size that is sufficient to obtain, for 0 < ϵ, δ ≤1, an ϵ-accurate estimate of the ranking accuracy with δ-confidence: Corollary 2. Let f : X→R be a fixed ranking function on X and let 0 < ϵ, δ ≤1. Let y = (y1, . . . , yN) ∈YN be any label sequence of length N ∈N. If N ≥ ln 2 δ 2ρ(y)(1 −ρ(y))ϵ2 , then PTX|TY =y n ˆA(f; T) −A(f) ≥ϵ o ≤ δ . Proof. This follows directly from Theorem 2 by setting 2e−2ρ(y)(1−ρ(y))Nϵ2 ≤δ and solving for N. ⊓⊔ Figure 1 illustrates the dependence of the above expression for the sufficient test sample size on the the accuracy parameter ϵ and positive skew ρ(y) for different values of δ. The confidence interval of Corollary 1 can in fact be generalized to remove the conditioning on the label vector completely: Theorem 3. Let f : X→R be a fixed ranking function on X and let N ∈N. Then for any 0 < δ ≤1, PT ∼DN ( ˆA(f; T) −A(f) ≥ s ln 2 δ 2ρ(TY )(1 −ρ(TY ))N ) ≤ δ . Proof. For T ∈(X × Y)N and 0 < δ ≤1, define the proposition Φ(T, δ) ≡ ( ˆA(f; T) −A(f) ≥ s ln 2 δ 2ρ(TY )(1 −ρ(TY ))N ) . Then for any 0 < δ ≤1, we have PT {Φ(T, δ)} = ET IΦ(T,δ) = ETY n ETX|TY =y IΦ(T,δ) o = ETY n PTX|TY =y {Φ(T, δ)} o ≤ ETY {δ} (by Corollary 1) = δ . ⊓⊔ 3Our thanks to an anonymous reviewer for pointing this out. Figure 1: The test sample size N (based on Corollary 2) sufficient to obtain an ϵ-accurate estimate of the ranking accuracy with δ-confidence, for various values of the positive skew ρ ≡ρ(y) for some label sequence y, for (left) δ = 0.01 and (right) δ = 0.001. Note that the above ‘trick’ works only once we have gone to a confidence interval; an attempt to generalize the bound of Theorem 2 in a similar way gives an expression in which the final expectation is not easy to evaluate. Interestingly, the above proof does not even require a factorized distribution DTY since it is built on a result for any fixed label sequence y. We note that the above technique could also be applied to generalize the results of [9] in a similar manner. 4 Comparison with Large Deviation Bound for Error Rate Our use of McDiarmid’s inequality in deriving the large deviation bound for the AUC of a ranking function is analogous to the use of Hoeffding’s inequality in deriving a large deviation bound for the error rate of a classification function. (e.g., see [6, Chapter 8]). The need for the more general inequality of McDiarmid in our derivations arises from the fact that the empirical AUC, unlike the empirical error rate, cannot be expressed as a sum of independent random variables. Given a classification function h : X→Y, let L(h) denote the expected error rate of h: L(h) = EXY ∼D I{h(X)̸=Y } . Similarly, given a classification function h : X→Y and a finite data sequence T = ((x1, y1), . . . , (xN, yN)) ∈(X × Y)N, let ˆL(h; T) denote the empirical error rate of h with respect to T: ˆL(h; T) = 1 N N X i=1 I{h(xi)̸=yi} . Then the large deviation bound obtained via Hoeffding’s inequality for the classification error rate states that for a fixed classification function h : X→Y and for any N ∈N, ϵ > 0, PT ∼DN n ˆL(h; T) −L(h) ≥ϵ o ≤ 2e−2Nϵ2 . (3) Comparing Eq. (3) to the bound of Theorem 2, we see that the AUC bound differs from the error rate bound by a factor of ρ(y)(1 −ρ(y)) in the exponent. This difference translates into a 1/(ρ(y)(1 −ρ(y))) factor difference in the resulting sample size bounds: given 0 < ϵ, δ ≤1, the test sample size sufficient to obtain an ϵ-accurate estimate of the expected accuracy of a ranking function with δ-confidence is 1/(ρ(y)(1−ρ(y))) times larger than the corresponding test sample size sufficient to obtain an ϵ-accurate estimate of the expected error rate of a classification function with the same confidence. For ρ(y) = 1/2, this means a sample size larger by a factor of 4; as the positive skew ρ(y) departs from 1/2, the factor grows larger (see Figure 2). Figure 2: The test sample size bound for the AUC, for positive skew ρ ≡ρ(y) for some label sequence y, is larger than the corresponding test sample size bound for the classification error rate by a factor of 1/(ρ(1 −ρ)). 5 Bound for Learned Ranking Functions Chosen from Finite Classes The large deviation result of Theorem 2 bounds the expected accuracy of a ranking function in terms of its empirical AUC on an independent test sequence. A simple application of the union bound allows the result to be extended to bound the expected accuracy of a learned ranking function in terms of its empirical AUC on the training sequence from which it is learned, in the case when the learned ranking function is chosen from a finite function class. More specifically, we have: Theorem 4. Let F be a finite class of real-valued functions on X and let fS ∈F denote the ranking function chosen by a learning algorithm based on the training sequence S. Let y = (y1, . . . , yM) ∈YM be any label sequence of length M ∈N. Then for any ϵ > 0, PSX|SY =y n ˆA(fS; S) −A(fS) ≥ϵ o ≤ 2|F|e−2ρ(y)(1−ρ(y))Mϵ2 . Proof. For any ϵ > 0, we have PSX|SY =y n ˆA(fS; S) −A(fS) ≥ϵ o ≤ PSX|SY =y max f∈F ˆA(f; S) −A(f) ≥ϵ ≤ X f∈F PSX|SY =y n ˆA(f; S) −A(f) ≥ϵ o (by the union bound) ≤ 2|F|e−2ρ(y)(1−ρ(y))Mϵ2 (by Theorem 2) . ⊓⊔ As before, we can derive from Theorem 4 expressions for confidence intervals and sufficient training sample size. We give these here without proof: Corollary 3. Under the assumptions of Theorem 4, for any 0 < δ ≤1, PSX|SY =y ( ˆA(fS; S) −A(fS) ≥ s ln |F| + ln 2 δ 2ρ(y)(1 −ρ(y))M ) ≤ δ . Corollary 4. Under the assumptions of Theorem 4, for any 0 < ϵ, δ ≤1, if M ≥ 1 2ρ(y)(1 −ρ(y))ϵ2 ln |F| + ln 2 δ , then PSX|SY =y n ˆA(fS; S) −A(fS) ≥ϵ o ≤ δ . Theorem 5. Let F be a finite class of real-valued functions on X and let fS ∈F denote the ranking function chosen by a learning algorithm based on the training sequence S. Let M ∈N. Then for any 0 < δ ≤1, PS∼DM ( ˆA(fS; S) −A(fS) ≥ s ln |F| + ln 2 δ 2ρ(SY )(1 −ρ(SY ))M ) ≤ δ . 6 Conclusion We have derived a distribution-free large deviation bound for the area under the ROC curve (AUC), a quantity used as an evaluation criterion for the bipartite ranking problem. Our result parallels the classical large deviation result for the classification error rate obtained via Hoeffding’s inequality. Since the AUC cannot be expressed as a sum of independent random variables, a more powerful inequality of McDiarmid was required. A comparison with the corresponding large deviation result for the error rate suggests that, in the distributionfree setting, the test sample size required to obtain an ϵ-accurate estimate of the expected accuracy of a ranking function with δ-confidence is larger than the test sample size required to obtain a similar estimate of the expected error rate of a classification function. A simple application of the union bound allows the large deviation bound to be extended to learned ranking functions chosen from finite function classes. A possible route for deriving an alternative large deviation bound for the AUC could be via the theory of U-statistics; the AUC can be expressed as a two-sample U-statistic, and therefore it may be possible to apply specialized results from U-statistic theory (see, for example, [5]) to the AUC. References [1] S. Agarwal, S. Har-Peled, and D. Roth. A uniform convergence bound for the area under the ROC curve. In Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics, 2005. [2] W. W. Cohen, R. E. Schapire, and Y. Singer. Learning to order things. Journal of Artificial Intelligence Research, 10:243–270, 1999. [3] C. Cortes and M. Mohri. AUC optimization vs. error rate minimization. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, Advances in Neural Information Processing Systems 16, 2004. [4] K. Crammer and Y. Singer. Pranking with ranking. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems 14, 2002. [5] V. H. de la P˜ena and E. Gin´e. Decoupling: From Dependence to Independence. Springer-Verlag, New York, 1999. [6] L. Devroye, L. Gy¨orfi, and G. Lugosi. A Probabilistic Theory of Pattern Recognition. SpringerVerlag, New York, 1996. [7] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research, 4:933–969, 2003. [8] R. Herbrich, T. Graepel, and K. Obermayer. Large margin rank boundaries for ordinal regression. Advances in Large Margin Classifiers, pages 115–132, 2000. [9] S. I. Hill, H. Zaragoza, R. Herbrich, and P. J. W. Rayner. Average precision and the problem of generalisation. In Proceedings of the ACM SIGIR Workshop on Mathematical and Formal Methods in Information Retrieval, 2002. [10] C. McDiarmid. On the method of bounded differences. In Surveys in Combinatorics 1989, pages 148–188. Cambridge University Press, 1989.
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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 rlin1@uiuc.edu dross@cs.toronto.edu jlim1@uiuc.edu myang@honda-ri.com 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 problem is formulated with a recursive equation: p(st|Ot) = kp(ot|st) Z 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 + µ + ϵ (2) where W is a n × m projection matrix associating y and x, µ is the mean of y, and ϵ 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 ϵ is zero mean Gaussian noise, ϵ ∼N(0, σ2In). Since x and ϵ are both Gaussians, it follows that y is also a Gaussian distribution, y ∼N(µ, C), where C = WW T + σ2I and In is an n-dimensional identity matrix. Together with (2), we have a generative observation model: p(ot|st) = p(yt|W, µ, ϵ) ∼N(yt|µ, WW T + σ2In) (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 samples Y = {y1, . . . , yN}, the covariance matrix of Y is denoted as S = 1 N P(y − µ)(y −µ)T . Let {λi|i = 1, . . . , N} be the eigenvalues of S arranged in descending order, i.e., λi ≥λj if i < j. Also, define the diagonal matrix Σm = diag(λ1, . . . , λm), and let Um be the eigenvectors that corresponds to the eigenvalues in Σm. Tipping and Bishop [10] show that the the maximum likelihood estimate of µ, W and ϵ can be obtained by µ = 1 N N X i=1 yi, W = Um(Σm −σ2Im)1/2R, σ2 = 1 n −m n X i=m+1 λi (4) 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, µ, and σ2 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 L(W, µ, σ2) = −1 2 N log 2π + log |C| + yT C−1y (5) where y = y −µ. Neglecting the constant terms, the log-probability is determined by yT C−1y. Together with C = WW T + σ2In and (4), it follows that yT C−1y = yT UmΣ−1 m U T my + 1 σ2 yT (In −UmU T m)y (6) Here yT UmΣ−1 m U T my is the Mahalanobis distance of y in the subspace spanned by Um, and yT (In −UmU T m)y is the shortest distance from y to this subspace spanned by Um. Usually σ is set to a small value, and consequently the probability will be determined solely by the distance to the subspace. However, the choice of σ is not trivial. From (6), if the σ 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 σ is even more critical in situations where the appearance changes dynamically and requires σ 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 Σm is complicated since it involves the computations of eigenvalues and eigenvectors. Here we use a forgetting factor γ to put more weights on the most recent data. Denote the newly arrived samples at time t as Y = {y1, . . . , yM}, and assume the mean µ is fixed, U t m and Σt m can be obtained by performing singular value decomposition (SVD) on [√γUm,t−1(Σm,t−1)1/2| p (1 −γ)Y ] (7) where Y = [y1−µ, . . ., yM −µ]. Σ1/2 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 µ constantly changes, the above update procedure can not be applied. We recently proposed a method [8] to compute SVD with correct updated mean in which Σ1/2 m,t and Um,t can be obtained by computing SVD on h√γUm,t−1(Σm,t−1)1/2 p (1 −γ)Y p (1 −γ)γ(µt−1 −µY ) i (8) where Y = [y1 −µY , . . . , yM −µY ] and µY = 1 M PM i=1 yi. This formulation is similar to the SVD computation with the fixed mean case, and the same incremental SVD algorithm can be used to compute Σ1/2 m,t and Um,t with an extra term shown in (8). Computing and updating σ is more difficult than the form in (8). In the previous section, we show that an inaccurate value of σ will severely affect probability estimates. In order to have an accurate estimate of σ 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 σ based on (4), we will start with a very small value of σ 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, λi in (4) and consequently σ may not be accurately updated if the number of drawn samples is insufficient. These constraints lead us to develop a method that adaptively adjusts σ 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 {si t−1|i = 1, ..., k} in ot−1 that have large p(ot−1|si t−1), but the low posterior p(si t−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 ′ = {y1, . . . , yk} where yi is the appearance vector collected in ot−1 based on state parameter si t−1, we want to find a linear projection V ∗that projects Y ′ onto a subspace such that the likelihood of Y ′ in the subspace is minimized. Let V be a p × n matrix and since p(y|W, µ, σ) is a Gaussian, p(V y|V, W, µ, σ) ∼N(V µ, V CV T ) is a also Gaussian. The log likelihood is computed by L(V, W, µ, σ) = −k 2 p log(2π) + log |V CV T | + tr((V CV T )−1V S′V T ) (9) where S′ = 1 k Pk i=1(yi −µ)(yi −µ)T . To facilitate the following analysis we first assume V projects Y ′ to a 1-D space, i.e., p = 1 and V = vT , and thus L(V, W, µ, σ) = −k 2 log(2π) + log |vT Cv| + vT S′v vT Cv (10) Note that vT Cv is the variance of the object samples in the projected space, and we need to impose a constraint, e.g., vtCv = 1, to ensure that the minimum likelihood solution of v does not increase the variance in the projected space. Let vT Cv = 1, the optimization problem becomes v∗= arg max {v|vT Cv=1} vT S′v = arg max v vT S′v vT Cv (11) 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 µ (with variance vT Cv = 1), while keeping negative samples in Y ′ away from µ. The optimal value of v is the generalized eigenvector of S′ and C that corresponds to largest eigenvalue. In a general case, it follows that V ∗= arg max {V CV T =I} |V S′V T | = arg max V |V S′V T | |V CV T | (12) where V ∗can be obtained by solving a generalized eigenvalue problem of S′ 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′. In section 3.1.2, we have shown the procedures to update C. The same procedures can be used to update S′. Let µY ′ = 1 k Pk i=1 yi and SY ′ = 1 k Pk i=1(yi −µY ′)(yi −µY ′)T , S′ = 1 k k X i=1 (yi −µ)(yi −µ)T = Sy + (µ −µY ′)(µ −µY ′)T (13) Given S′ and C, V is computed by solving a generalized eigenvalue problem. If we decompose S′ = AT A and C = BT B, then we can find V more efficiently using generalized singular value decomposition. Denote UY ′ and ΣY ′ as the SVD of SY ′, it follows that by letting A = [UY ′Σ1/2 Y ′ | (µ −µY ′)]T and B = [UmΣ1/2 m |σ2I]T , we obtain S′ = AT A and C = BT B. As is detailed in [4] , V can be computed by first performing a QR factorization: A B = QA QB R (14) and computing the singular value decomposition of QA QA = UADAV T A (15) , we then obtain V = R−1VA. 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, θ, w, h), that parameterizes the windows position (x, y), orientation (θ) 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, Σs) (16) where Σs 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 yi t as the appearance vector of ot, and Yt = {y1 t , . . . , yN t } 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) = κp(yi t|V, W, µ, σ)p(st = ci|s∗ t−1) (17) where κ is a constant. Therefore, s∗ t = arg maxci∈St p(st = ci|Ot). Once s∗ t is determined, the corresponding observation y∗ t will be a new example to update W and µ. Appearance vectors yi t with large p(yi t|V, W, µ, σ) 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 µ, 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 µ 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 σ. In our tracking the system, we adaptively adjust σ according to Σm in W. We set σ be a fixed fraction of the smallest eigenvalues in Σm. 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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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. a b e c d f g {a, b, e} {b, e, c} {b, c, d} {e, c, f} {c, f, g} {b, e} {b, c} {c, e} {f, c} 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 α and β are non-adjacent vertices of G then a set of vertices S ⊆V \ {α, β} is called an (α, β)-separator if α and β are in distinct components of G[V \ S]. S is a minimal (α, β)-separator if no proper subset of S is an (α, β)-separator. S is said to be a minimal separator if S is a minimal (α, β)-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 ViVj ∈S (where Vi, Vj ∈K and hence are cliques of G), then Vi ∩Vj ̸= φ. We label each edge ViVj ∈S with the set Vij = Vi ∩Vj, which is a non-empty complete separator in G. The removal of any link ViVj ∈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 ViVj ∈S, then the removal a b d e c e c f g {a, b, e} {b, e, c} {b, c, d} {e, c, f} {c, f, g} {b, e} {b, c} {f, c} 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 |Vij| = k, then in any subgraph H of G, H[Vij] must be one of the 2( k 2) possible subgraphs of G[Vij]. So, if k is sufficiently small (so 2( k 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∥PH) 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∥PH). Hence, finding a optimal subgraphical model is equivalent to finding a subgraph H for which D(PG∥PH) 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∥PH) = D(PG(i)∥PH(i)) + D(PG(j)∥PH(j)) − D(PG[Vij]∥PH[Vij]). Proof. Since H is a subgraph of G, and Vij is a separator of G, Vij must also be a separator of H. Therefore, PH {Xv}v∈V = PH(i)({Xv}v∈V (i))·PH(j)({Xv}v∈V (j)) PH[Vij ]({Xv}v∈Vij ) . The result 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∥PH) 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 e; cf; g abe d; bc; e dbc ebc cef cfg 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 ViVj ∈S be any link in T. Then removal of the link ViVj 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 ViVj ∈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 Finding optimal subgraphical models 3.1 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 ̸= w and {z, w} ̸= {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 k 2 · 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 k 2 · 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 ̸= 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 · k 2 ∈O(k2) ∈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 optimal one. Since there are |E(G)| d such sets, this can be done in polynomial 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 separator Vij, let CM = n (l, r, c, s, A) : l + r −c = d, s a d bit string, A ∈ E(G[Vij]) c o . The constraint represented 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 clique, then the total number of such constraints is bounded by 2d · 2d · ( k 2) d ∈O(k2d) 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]∥PH[M]); table[M, cM] ←table[Ml, cl] + table[Mr, cr] −loss; else table[M, cM] ←D(PG[M]∥PH[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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Modeling Conversational Dynamics as a Mixed-Memory Markov Process Tanzeem Choudhury Intel Research tanzeem.choudhury@intel.com Abstract Sumit Basu Microsoft Research sumitb@microsoft.com In this work, we quantitatively investigate the ways in which a given person influences the joint turn-taking behavior in a conversation. After collecting an auditory database of social interactions among a group of twenty-three people via wearable sensors (66 hours of data each over two weeks), we apply speech and conversation detection methods to the auditory streams. These methods automatically locate the conversations, determine their participants, and mark which participant was speaking when. We then model the joint turn-taking behavior as a Mixed-Memory Markov Model [1] that combines the statistics of the individual subjects' self-transitions and the partners' cross-transitions. The mixture parameters in this model describe how much each person's individual behavior contributes to the joint turn-taking behavior of the pair. By estimating these parameters, we thus estimate how much influence each participant has in determining the joint turntaking behavior. We show how this measure correlates significantly with betweenness centrality [2], an independent measure of an individual's importance in a social network. This result suggests that our estimate of conversational influence is predictive of social influence. 1 Introduction People's relationships are largely determined by their social interactions, and the nature of their conversations plays a large part in defining those interactions. There is a long history of work in the social sciences aimed at understanding the interactions between individuals and the influences they have on each others' behavior. However, existing studies of social network interactions have either been restricted to online communities, where unambiguous measurements about how people interact can be obtained, or have been forced to rely on questionnaires or diaries to get data on face-to-face interactions. Survey-based methods are error prone and impractical to scale up. Studies show that self-reports correspond poorly to communication behavior as recorded by independent observers [3]. In contrast, we have used wearable sensors and recent advances in speech processing techniques to automatically gather information about conversations: when they occurred, who was involved, and who was speaking when. Our goal was then to see if we could examine the influence a given speaker had on the turn-taking behavior of her conversational partners. Specifically, we wanted to see if we could better explain the turn-taking transitions observed in a given conversation between subjects i and} by combining the transitions typical to i and those typical toj. We could then interpret the contribution from i as her influence on the joint turn-taking behavior. In this paper, we first describe how we extract speech and conversation information from the raw sensor data, and how we can use this to estimate the underlying social network. We then detail how we use a Mixed-Memory Markov Model to combine the individuals' statistics. Finally, we show the performance of our method on our collected data and how it correlates well with other metrics of social influence. 2 Sensing and Modeling Face-to-face Communication Networks Although people heavily rely on email, telephone, and other virtual means of communication, high complexity information is primarily exchanged through face-toface interaction [4]. Prior work on sensing face-to-face networks have been based on proximity measures [5],[6], a weak approximation of the actual communication network. Our focus is to model the network based on conversations that take place within a community. To do this, we need to gather data from real-world interactions. We thus used an experiment conducted at MIT [7] in which 23 people agreed to wear the sociometer, a wearable data acquisition board [7],[8]. The device stored audio information from a single microphone at 8 KHz. During the experiment the users wore the device both indoors and outdoors for six hours a day for 11 days. The participants were a mix of students, facuity, and administrative support staff who were distributed across different floors of a laboratory building and across different research groups. 3 Speech and Conversation Detection Given the set of auditory streams of each subject, we now have the problem of detecting who is speaking when and to whom they are speaking. We break this problem into two parts: voicing/speech detection and conversation detection. 3.1 Voicing and Speech Detection To detect the speech, we use the linked-HMM model for VOlClllg and speech detection presented in [9]. This structure models the speech as two layers (see Figure 1); the lower level hidden state represents whether the current frame of audio is voiced or unvoiced (i.e., whether the audio in the frame has a harmonic structure, as in a vowel), while the second level represents whether we are in a speech or nonspeech segment. The principle behind the model is that while there are many voiced sounds in our environment (car horns, tones, computer sounds, etc.), the dynamics of voiced/unvoiced transitions provide a unique signature for human speech; the higher level is able to capture this dynamics since the lower level's transitions are dependent on this variable. speech layer (S[t) = {O, I}) voicing layer (V[t) = {O,l}) observation layer (3 features) Figure 1: Graphical model for the voicing and speech detector. To apply this model to data, the 8 kHz audio is split into 256-sample frames (32 milliseconds) with a 128-sample overlap. Three features are then computed: the non-initial maximum of the noisy autocorrelation, the number of autocorrelation peaks, and the spectral entropy. The features were modeled as a Gaussian with diagonal covariance. The model was then trained on 8000 frames of fully labeled data. We chose this model because of its robustness to noise and distance from the microphone: even at 20 feet away more than 90% of voiced frames were detected with negligible false alarms (see [9]). The results from this model are the binary sequences v[t} and s[t} signifying whether the frame is voiced and whether it is in a speech segment for all frames of the audio. 3.2 Conversation Detection Once the voicing and speech segments are identified, we are sti II left with the problem of determining who was talking with whom and when. To approach this, we use the method of conversation detection described in [10]. The basic idea is simple: since the speech detection method described above is robust to distance, the voicing segments v[t} of all the participants in the conversation will be picked up by the detector in all of the streams (this is referred to as a "mixed stream" in [10]). We can then examine the mutual information of the binary voicing estimates between each person as a matching measure. Since both voicing streams will be nearly identical, the mutual information should peak when the two participants are either involved in a conversation or are overhearing a conversation from a nearby group. However, we have the added complication that the streams are only roughly aligned in time. Thus, we also need to consider a range of time shifts between the streams. We can express the alignment measure a[k] for an offset of k between the two voicing streams as follows: " p(v,[t]=i,v,[t-l]=j) a[k] = l(vJt], v,[t - k]) = L." p(vJt] = i, v, [t - k] = j) log --.:...--'--'-'~----=-=---=---....::...:....i.j p(vJt]=i)p(v,[t-k]=j) where i and j take on values {O, l} for unvoiced and voiced states respectively. The distributions for p(v\, vJ and its marginals are estimated over a window of one minute (T=3750 frames). To see how well this measure performs, we examine an example pair of subjects who had one five-minute conversation over the course of half an hour. The streams are correctly aligned at k=0, and by examining the value of ark} over a large range we can investigate its utility for conversation detection and for aligning the auditory streams (see Figure 2). The peaks are both strong and unique to the correct alignment (k=0), implying that this is indeed a good measure for detecting conversations and aligning the audio in our setup. By choosing the optimal threshold via the ROC curve, we can achieve 100% detection with no false alarms using time windows T of one minute. Figure 2: Values of ark] over ranges: 1.6 seconds, 2.5 minutes, and 11 minutes. For each minute of data in each speaker's stream, we computed ark] for k ranging over +/- 30 seconds with T=3750 for each of the other 22 subjects in the study. While we can now be confident that this will detect most of the conversations between the subjects, since the speech segments from all the participants are being picked up by all of their microphones (and those of others within earshot), there is still the problem of determining who is speaking when. Fortunately, this is fairly straightforward. Since the microphones for each subject are pre-calibrated to have approximately equal energy response, we can classify each voicing segment among the speakers by integrating the audio energy over the segment and choosing the argmax over subjects. It is still possible that the resulting subject does not correspond to the actual speaker (she could simply be the one nearest to a nonsubject who is speaking), we determine an overall threshold below which the assignment to the speaker is rejected. Both of these methods are further detailed in [10]. For this work, we rejected all conversations with more than two participants or those that were simply overheard by the subjects. Finally, we tested the overall performance of our method by comparing with a hand-labeling of conversation occurrence and length from four subjects over 2 days (48 hours of data) and found an 87% agreement with the hand labeling. Note that the actual performance may have been better than this, as the labelers did miss some conversations. 3.3 The Turn-Taking Signal S; Finally, given the location of the conversations and who is speaking when, we can create a new signal for each subject i, S;, defined over five-second blocks, which is 1 when the subject is holding the turn and 0 otherwise. We define the holder of the turn as whoever has produced more speech during the five-second block. Thus, within a given conversation between subjects i and j , the turn-taking signals are complements of each other, i.e., Si = -,SJ . I I 4 Estimating the Social Network Structure Once we have detected the pairwise conversations we can identify the communication that occurs within the community and map the links between individuals. The link structure is calculated from the total number of conversations each subject has with others: interactions with another person that account for less than 5% of the subject's total interactions are removed from the graph. To get an intuitive picture of the interaction pattern within the group, we visualize the network diagram by performing multi-dimensional scaling (MDS) on the geodesic distances (number of hops) between the people (Figure 3). The nodes are colored according to the physical closeness of the subjects' office locations. From this we see that people whose offices are in the same general space seem to be close in the communication space as well. Figure 3: Estimated network of subjects 5 Modeling the Influence of Turn-taking Behavior in Conversations When we talk to other people we are influenced by their style of interaction. Sometimes this influence is strong and sometimes insignificant - we are interested in finding a way to quantify this effect. We probably all know people who have a strong effect on our natural interaction style when we talk to them, causing us to change our style as a result. For example, consider someone who never seems to stop talking once it is her turn. She may end up imposing her style on us, and we may consequently end up not having enough of a chance to talk, whereas in most other circumstances we tend to be an active and equal participant. In our case, we can model this effect via the signals we have already gathered. Let us consider the influence subject} has on subject i. We can compute i's average self-transition table, peS: I S;_I) , via simple counts over all conversations for subject i (excluding those with i). Similarly, we can compute j's average cross-transition table, p(Stk I Sf- I)' over all subjects k (excluding i) with which} had conversations. The question now is, for a given conversation between i and}, how much does} 's average cross-transition help explain peS: I S;_I ' Sf- I) ? We can formalize this contribution via the Mixed-Memory Markov Model of Saul and Jordan [1]. The basic idea of this model was to approximate a high-dimensional conditional probability table of one variable conditioned on many others as a convex combination of the pairwise conditional tables. For a general set of N interacting Markov chains in the form of a Coupled Markov Model [11], we can write this approximation as: peS; I sLI,··· , St~ l) = IaijP(S; I Sf- I) j For our case of a two chain (two person) model the transition probabilities will be the following: peS: I S,'_, , S,2_,) = a IlP(S,' I S,'_,) + a 12P(S,k I S,2_, ) p(S,2 I S,'_, , S,2_,) = a 2,P(S,k I S,'_,) + a 22P(S,2 I S,~, ) This is very similar to the original Mixed-Memory Model, though the transition tables are estimated over all other subjects k excluding the partner as described above. Also, since the a ij sum to one over j, in this case a ll = 1- a '2 . We thus have a single parameter, a'2' which describes the contribution of p(Stk I St2_1) to explaining P(S~ I SLl,St~I)' i.e., the contribution of subject 2's average turn-taking behavior on her interactions with subject 1. 5.1 Learning the influence parameters To find the a ij values, we would like to maximize the likelihood of the data. Since we have already estimated the relevant conditional probability tables, we can do this via constrained gradient ascent, where we ensure that a ij>O [12]. Let us first examine how the likelihood function simplifies for the Mixed-Markov model: Converting this expression to log likelihood and removing terms that are not relevant to maximization over a ij yields: Now we reparametrize for the normality constraint with fJij = a ij and fJ;N = 1- LfJij , remove the terms not relevant to chain i, and take the derivatives: a peS; I S,~ ,) - pes; I S,~ ,) afJij (.) = ~ LfJ;kP(S; I S,~, )+(I- LfJ;k )P(S; I S,~, ) We can show that the likelihood is convex in the a ij ' so we are guaranteed to achieve the global maximum by climbing the gradient. More details of this formulation are given in [12],[7]. 5.2 Aggregate Influence over Multiple Conversations In order to evaluate whether this model provides additional benefit over using a given subject's self-transition statistics alone, we estimated the reduction in KL divergence by using the mixture of interactions vs. using the self-transition model. We found that by using the mixture model we were able to reduce the KL divergence between a subject's average self-transition statistics and the observed transitions by 32% on average. However, in the mixture model we have added extra degrees of freedom, and hence tested whether the better fit was statistically significant by using the F-test. The resulting p-value was less than 0.01 , implying that the mixture model is a significantly better fit to the data. In order to find a single influence parameter for each person, we took a subset of 80 conversations and aggregated all the pairwise influences each subject had on all her conversational partners. In order to compute this aggregate value, there is an additional aspect about a ij we need to consider. If the subject's self-transition matrix and the complement of the partner's cross-transition matrix are very similar, the influence scores are indeterminate, since for a given interaction S; = -,s: : i.e., we would essentially be trying to find the best way to linearly combine two identical transition matrices. We thus weight the contribution to the aggregate influence estimate for each individual Ai by the relevant I-divergence (symmetrized KL divergence) for each conversational partner: Ai = L J(P(S: I-,SL,) II peS: I S:_,))aki kEpartners The upper panel of Figure 4 shows the aggregated influence values for the subset of subjects contained in the set of eighty conversations analyzed. 6 Link between Conversational Dynamics and Social Role Betweenness centrality is a measure frequently used in social network analysis to characterize importance in the social network. For a given person i, it is defined as being proportional to the number of pairs of people (j,k) for which that person lies along the shortest path in the network between j and k. It is thus used to estimate how much control an individual has over the interaction of others, since it is a count of how often she is a "gateway" between others. People with high betweenness are often perceived as leaders [2]. We computed the betweenness centrality for the subjects from the 80 conversations using the network structure we estimated in Section 3. We then discovered an interesting and statistically significant correlation between a person's aggregate influence score and her betweenness centrality -- it appears that a person's interaction style is indicative of her role within the community based on the centrality measure. Figure 4 shows the weighted influence values along with the centrality scores. Note that ID 8 (the experiment coordinator) is somewhat of an outlier -- a plausible explanation for this can be that during the data collection ID 8 went and talked to many of the subjects, which is not her usual behavior. This resulted in her having artificially high centrality (based on link structure) but not high influence based on her interaction style. We computed the statistical correlation between the influence values and the centrality scores, both including and excluding the outlier subject ID 8. The correlation excluding ID 8 was 0.90 (p-value < 0.0004, rank correlation 0.92) and including ID 8 it was 0.48 (p-value <0.07, rank correlation 0.65). The two measures, namely influence and centrality, are highly correlated, and this correlation is statistically significant when we exclude ID 8, who was the coordinator of the project and whose centrality is likely to be artificially large. 7 Conclusion We have developed a model for quantitatively representing the influence of a given person j's turn-taking behavior on the joint-turn taking behavior with person i. On real-world data gathered from wearable sensors, we have estimated the relevant component statistics about turn taking behavior via robust speech processing techniques, and have shown how we can use the Mixed-Memory Markov formalism to estimate the behavioral influence. Finally, we have shown a strong correlation between a person's aggregate influence value and her betweenness centrality score. This implies that our estimate of conversational influence may be indicative of importance within the social network. 0.25 " ~ 0.2 > l! 0.15 ~ £ 0.1 0.05 o ~ 0.2 ~ ~0. 15 e i 0.1 o 0.05 Aggregate Influence Values 10 11 12 13 14 BelweenneS5 CenlralHy Scores Figure 4: Aggregate influence values and corresponding centrality scores. 8 References [1] Saul, L.K. and M. Jordan. "Mixed Memory Markov Models." Machine Learning, 1999.37: p. 75-85. [2] Freeman, L.c., "A Set of Measures of Centrality Based on Betweenness." Sociometry, 1977.40: p. 35-41. [3] Bernard, H.R., et aI., "The Problem of Informant Accuracy: the Validity of Retrospective data." Annual Review of Anthropology, 1984. 13: p. pp. 495-517. [4] Allen, T., Architecture and Communication Among Product Development Engineers. 1997, Sloan School of Management, MIT: Cambridge. p. pp. 1-35. [5] Want, R., et aI., "The Active Badge Location System." ACM Transactions on Information Systems, 1992.10: p. 91-102. [6] Borovoy, R., Folk Computing: Designing Technology to Support Face-to-Face Community Building. Doctoral Thesis in Media Arts and Sciences. MIT, 2001. [7] Choudhury, T., Sensing and Modeling Human Networks, Doctoral Thesis in Media Arts and Sciences. MIT. Cambridge, MA, 2003. [8] Gerasimov, V., T. Selker, and W. Bender, Sensing and Effecting Environment with Extremity Computing Devices. Motorola Offspring, 2002. 1(1). [9] Basu, S. "A Two-Layer Model for Voicing and Speech Detection." in Int 'l Conference on Acoustics, Speech, and Signal Processing (ICASSP). 2003. [10]Basu, S., Conversation Scene Analysis. Doctoral Thesis in Electrical Engineering and Computer Science. MIT. Cambridge, MA 2002. [11]Brand, M., "Coupled Hidden Markov Models for Modeling Interacting Processes." MIT Media Lab Vision & Modeling Tech Report, 1996. [12]Basu, S., T. Choudhury, and B. Clarkson. "Learning Human Interactions with the Influence Model." MIT Media Lab Vision and Modeling Tech Report #539. June, 2001.
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Generalization Error Bounds for Collaborative Prediction with Low-Rank Matrices Nathan Srebro Department of Computer Science University of Toronto Toronto, ON, Canada nati@cs.toronto.edu Noga Alon School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel nogaa@tau.ac.il Tommi S. Jaakkola Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology Cambridge, MA, USA tommi@csail.mit.edu 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 1In 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 ⇔ target matrix Y random training set ⇔ random set S of observed entries hypothesis ⇔ predicted matrix X training error ⇔ observed discrepancy DS(X; Y ) generalization error ⇔ true discrepancy D(X; Y ) Figure 1: Correspondence with post-hoc bounds on the generalization error for standard feature-based prediction tasks bounds on the true average overall error D(X; Y ) = 1 nm Pn i=1 Pm a=1 loss(Xia; Yia) of the predictions X in terms of the average error over the observed entries DS(X; Y ) = 1 |S| P ia∈S loss(Xia; Yia), without making any assumptions on the true nature of the preferences 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) = 1YiaXia≤0 (1) Theorem 1. For any matrix Y ∈{±1}n×m, n, m > 2, δ > 0 and integer k, with probability at least 1 −δ 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: ∀X,rank X<kD±(X; Y ) < D± S(X; Y ) + s k(n + m) log 16em k −log δ 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: Pr S D±(X; Y ) ≥D± S(X; Y ) + ϵ ≤e−2|S|ϵ2 (2) 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 If Xia > 0 0 If Xia = 0 −1 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± S(X; Y )+ϵ for each of these f(n, m, k) random variables we have: Pr S ∃X,rank X≤kD±(X; Y ) ≥D± S(X; Y ) + s log f(n, m, k) −log δ 2|S| ! ≤δ (3) by using (2) and setting ϵ = q log f(n,m,k)−log δ 2|S| . The proof of Theorem 1 rests on bounding 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. 2All 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 f(n, m, k) ≤minh (8⌈nm/h⌉)(n+m)k+h+m, 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 ΠiPi, i.e. the set of points in which all the m polynomials are non-zero: C = {x ∈Rq|∀iPi(x) ̸= 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: c(C) ≤2(2d)q q X i=0 2i r i ≤ 4edr q q 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 = UV ′ 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: Xia = k X α=1 UiαVaα Applying Corollary 3 we obtain: Lemma 4. f(n, m, k) ≤ 8e·2·nm k(n+m) k(n+m) ≤(16em/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: Lemma 5. For m > k2, f(n, m, k) ≥m 1 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 UV ′, where U varies over all n × k matrices. Focusing only on +/−sign configurations (no zeros in UV ′), each row of sign UV ′ is a homogeneous linear classification of the rows of V , i.e. of m vectors in general position in Rk. There are exactly 2 Pk−1 i=0 m i possible homogeneous linear classifications of m vectors in general position in Rk, and so these many options for each row of sign UV ′. We can therefore bound: f(n, m, k) ≥f(n, V, k) ≥ 2 k−1 X i=0 m i !n ≥ m k−1 n ≥ m k−1 n(k−1) = m 1 2 (k−1)n 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 ϵ-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 k(n+m). Proof. We take a similar approach to that of Lemma 4, writing rank-k matrices as a product X = UV ′ 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 α=1 UiαVaα −Tia 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, δ > 0 and integer k, with probability at least 1 −δ over choosing a subset S of entries in Y uniformly among all subsets of |S| entries: ∀X,rank X<kD(X; Y ) < DS(X; Y ) + 6 v u u tk(n + m) log 16em k log M|S| k(n+m) −log δ |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 α, β > 0 such that β(2xB + 1) > α2xA and β2yB < α(2yA + 1). It follows that for any A < B and any α, β > 0, on an initial segment (possibly empty) of N we have βgB ≤βfB ≤αgA ≤αfA while on the rest of N we have αgA ≤αfA < βgB ≤βfB. In particular, any pair of 3We use A to refer both to a positive integer and the finite set it maps to. functions (βfA, αfB) or (βfA, αgB) or (βgA, αgB) in F that are not associated with the same subset (i.e. A ̸= B), cross each other at most once. This holds also when α or β 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 ϵ .= 1 2 min1≤i≤q,x∈RPi(x)̸=0 |Pi(x)| > 0. Now consider the 2q polynomials P + i (x) = Pi(x) + ϵ and P − i (x) = Pi(x) −ϵ and C′ = x ∈Rq|∀iP + i (x) ̸= 0, P − i (x) ̸= 0 . Different points in R (representing all sign configurations) lie in different connected components of C′. Invoking Theorem 2 on C′ establishes Corollary 3. The count in Corollary 3 differentiates between positive, negative and zero signs. However, we are only concerned with the positivity of YiaXia (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|∀iP + i (x) ̸= 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). Applying Corollary 10 on the nm degree-two polynomials Yia Pk α=1 UiαVaα 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. 4A 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 ϵ > 0: Pr S ∃f∈FEX,Y [loss(f(X), Y )] > 1 n n X i=1 loss(f(Xi), Yi) + ϵ ! < 4e(d + 1) 32eM ϵ d e−ϵ2n 32 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. 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Incremental Learning for Visual Tracking Jongwoo Lim† David Ross‡ Ruei-Sung Lin† Ming-Hsuan Yang∗ † University of Illinois ‡ University of Toronto ∗Honda Research Institute jlim1@uiuc.edu dross@cs.toronto.edu rlin1@uiuc.edu myang@honda-ri.com 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 = UΣV ⊤. When a d×m matrix E of new observations is available, the R-SVD algorithm efficiently computes the SVD of the matrix A ′ = (A|E) = U ′Σ ′V ′⊤ based on the SVD of A as follows: 1. Apply QR decomposition to and get orthonormal basis ˜E of E, and U ′ = (U| ˜E). 2. Let V ′ = V 0 0 Im where Im is an m × m identity matrix. It follows then, Σ ′ = U ′⊤A ′V ′ = U ⊤ ˜ E⊤ (A|E) V 0 0 Im = U ⊤AV U ⊤E ˜ E⊤AV ˜ E⊤E = Σ U ⊤E 0 ˜ E⊤E . 3. Compute the SVD of Σ ′ = ˜U ˜Σ ˜V ⊤and the SVD of A ′ is A ′ = U ′( ˜U ˜Σ ˜V ⊤)V ′⊤= (U ′ ˜U)˜Σ( ˜V ⊤V ′⊤). 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 respectively, then Sr = Sp + Sq + nm n+m(¯Iq −¯Ip)(¯Iq −¯Ip)⊤. Proof: By definition, ¯Ir = n n+m¯Ip + m n+m¯Iq, ¯Ip −¯Ir = m n+m(¯Ip −¯Iq); ¯Iq −¯Ir = n n+m(¯Iq −¯Ip) and, Sr = Pn i=1(Ii −¯Ir)(Ii −¯Ir)⊤+ Pn+m i=n+1(Ii −¯Ir)(Ii −¯Ir)⊤ = Pn i=1(Ii −¯Ip + ¯Ip −¯Ir)(Ii −¯Ip + ¯Ip −¯Ir)⊤+ Pn+m i=m+1(Ii −¯Iq + ¯Iq −¯Ir)(Ii −¯Iq + ¯Iq −¯Ir)⊤ = Sp + n(¯Ip −¯Ir)(¯Ip −¯Ir)⊤+ Sq + m(¯Iq −¯Ir)(¯Iq −¯Ir)⊤ = Sp + nm2 (n+m)2 (¯Ip −¯Iq)(¯Ip −¯Iq)⊤+ Sq + n2m (n+m)2 (¯Ip −¯Iq)(¯Ip −¯Iq)⊤ = Sp + Sq + nm n+m(¯Ip −¯Iq)(¯Ip −¯Iq)⊤ □ Let ˆIp = {I1 −¯Ip, . . . , In −¯Ip}, ˆIq = {In+1 −¯Iq, . . . , In+m −¯Iq}, and ˆIr = {I1 − ¯Ir, . . . , In+m −¯Ir}, and the SVD of ˆIr = UrΣrV ⊤ r . Let ˜E = ˆIq| q nm n+m(¯Ip −¯Iq) , and use Proposition 1, Sr = (ˆIp| ˜E)(ˆIp| ˜E)⊤. Therefore, we compute SVD on (ˆIp| ˜E) to 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., UpΣpV ⊤ p and new data Iq, we can compute the the mean ¯Ir and the SVD of Ir, i.e., UrΣrV ⊤ r easily: 1. Compute ¯Ir = n n+m¯Ip + m n+m¯Iq, and ˜E = Iq −¯Ir 1(1×m) | q nm n+m(¯Ip −¯Iq) . 2. Compute R-SVD with (UpΣpV ⊤ p ) and ˜E to obtain (UrΣrV ⊤ r ). 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., A ′ = (fA |E) = (U(fΣ)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 It = {I1, . . . , It}. we aim to estimate the value of the hidden state variable Xt. Using Bayes’ theorem, we have p(Xt| It) ∝p(It|Xt) Z 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, θt, st, αt, φt) where xt, yt, θt, st, αt, φt, 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, Ψ) where Ψ is a diagonal covariance matrix whose elements are the corresponding variances of affine parameters, i.e., σ2 x, σ2 y, σ2 θ, σ2 s, σ2 α, σ2 φ. 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 µ. 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 ; µ, UU ⊤+ εI) where I is an identity matrix, µ is the mean, and εI 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 −µ) −UU ⊤(It −µ)||2), is proportional to N(It; µ, UU ⊤+ εI) as ε →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 ; µ, UΣ−2U ⊤) where µ is the center of the subspace and Σ 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; µ, UU ⊤+ εI) N(It; µ, UΣ−2U ⊤) (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 [4], ρ(x, σ) = x2 σ2+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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Face Detection — Efficient and Rank Deficient Wolf Kienzle, G¨okhan Bakır, Matthias Franz and Bernhard Sch¨olkopf 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, {α1, . . . αm} the corresponding coefficients, k(·, ·) the kernel function and b the bias of the SVM solution. The decision rule for a test pattern X reads f(X) = sgn m X i=1 yiαik(Xi, X) + b ! . (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 Ψ = m X i=1 yiαik(Xi, ·) (2) can be approximated using a smaller, so-called reduced set (RS) {Z1, . . . Zm′} of size m′ < m, i.e. an approximation to Ψ of the form Ψ′ = m′ X i=1 βik(Zi, ·). (3) This speeds up the decision process by a factor of m/m′. To find such Ψ′, we fix a desired set size m′ and solve min ∥Ψ −Ψ′∥2 RKHS (4) for βi and Zi. Here, ∥· ∥RKHS denotes the Euclidian norm in the RKHS. The resulting RS decision function f ′ is then given by f ′(X) = sgn m′ X i=1 βik(Zi, X) + b . (5) In practice, βi, 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 H = r X i=1 siuivi ⊤ (10) 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 J = r X i=1 si [I ∗ui] ∗vi ⊤ (11) 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. 1In 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(m′ · h · w) to O(m′ · 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(γ(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 ∥x −z∥2 = ∥X −Z∥2 F , (14) whereas the dot product amounts to x⊤z = 1 2 ∥X∥2 F + ∥Z∥2 F −∥X −Z∥2 F , (15) where X and Z are the corresponding image patches and ∥· ∥F 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 Zi ←UiSiVi ⊤. (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 Ψ′ r = m′ X i=1 βik(Ui,rSi,rVi,r ⊤, ·) (17) 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 m′ and r and find Si,r, Ui,r, Vi,r and βi that minimize the approximation error ρ = ∥Ψ−Ψ′ r∥2 RKHS. The minimization problem is solved via 2In 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 ∥X∥2 F = 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 βi 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 = ∂ρ/∂Ui,r we compute the ’constrained’ gradient ˆG = G −Ui,rG⊤Ui,r, (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 A = t · ˆG⊤Ui,r −( ˆG⊤U⊥ i,r)⊤ ˆG⊤U⊥ i,r 0 ! (19) 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 σ = 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 Ψ (see equation (2)) during the experiments described below. 5.1 Rank deficient faces In order to see how m′ and r affect the accuracy of our approximations, we compute rank deficient reduced sets for m′ = 1 . . . 32 and r = 1 . . . 3 (the left array in Figure 1 illustrates the actual appearance of rank deficient RSVs for the m′ = 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 r=1 r=2 r=3 r=full m' = 6 = = = + + + + + r=1 r=2 r=3 r=ful + ... = m ' = 1 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 m′ = 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 m′ = 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 m′ there is no significant loss in accuracy compared to the unconstrained approximation (at least for m′ > 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 m′. 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 m′ = 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 m′ = 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 m′ = 3, r = 1 discards 91.5% of the false positives, whereas the m′ = 1, r = full can only reject 70.2%. At the same time, when scanning a 320 × 240 image3, 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 3For 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. 10 0 10 1 0.6 0.7 0.8 0.9 1 #RSVs (m') ROC score 10 2 10 3 10 4 0.6 0.7 0.8 0.9 1 ROC score #operations (m'⋅r⋅(h+w)) r=1 r=2 r=3 r=full r=1 r=2 r=3 r=full 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 (m′ = 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. m′ · 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) m′ = 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 m′ = 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. Sch¨olkopf. 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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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: shpigi@cs.huji.ac.il 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 of movement and neural recordings, designated by Yi, Oi . Yi = yi t ti end t=1 is a time series of movement state values and yi t ∈Rd is the movement state vector at time t in trial i. We are interested in reconstructing position, however, for better modeling, yi t may be a vector of position, velocity and acceleration (as is the case in Sec. 4). This trajectory is observed during model learning and is the inference target. Oi = {ot}ti end t=1 is a time series of neural spike counts and oi t ∈Rq is a vector of spike counts from q cortical units at time t. We wish to learn a function zi t = f Oi 1:t that is a good estimate (in a sense formalized in the sequel) of the movement yi t. 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 + Wφ (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. φ (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. Note that unfolding the recursion in Eq. (1) yields zt = Aty0 + Pt k=1 At−kWφ (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 zi t is defined as in Eq. (1) while in standard SVR, zt = Wφ (ot) (i.e. without the linear dynamics). Given a training set of fully observed trials Yi, Oi m i=1 we define the learning problem to be min W 1 2 ∥W∥2 + c m X i=1 ti end X t=1 d X s=1 zi t s − yi t s ε . (2) Where ∥W∥2 = P a,b (W)2 ab (is the Forbenius norm). The second term is a sum of training errors (in all trials, times and movement dimensions). | · |ε is the ε insensitive loss and is defined as |v|ε = max {0, |v| −ε}. 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 εs 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 ε-loss with pairs of slack variables. We then write a Lagrangian of the primal problem and replace zi t 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 max α,α∗ −1 2 (α∗−α)T G (α∗−α) + (α∗−α)T y −(α∗+ α)T ϵ s.t. α, α∗∈[0, c] . (3) Note that the above expression conforms to the dual form of SVR. Let ℓequal the size of the movement space (d), multiplied by the total number of time steps in all the training trajectories. α, α∗∈Rℓare vectors of Lagrange multipliers, y ∈Rℓis a column concatenation of all the training set movement trajectories y1 1 T · · · ym tm end T T , ϵ = [ε, . . . , ε]T ∈Rℓ and G ∈Rℓ×ℓis 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 G = G11 · · · G1m ... ... ... Gm1 · · · Gmm , Gij = Kij 11 · · · Kij 1tj ... ... ... Kij ti end1 · · · Kij ti endtj end , where block Gij refers to a pair of trials (i and j). Finally Each basic block, Kij tq refers to a pair of time steps t and q in trajectories i and j respectively. ti end, tj end are the time lengths of trials i and j. Basic blocks are defined as Kij tq = t X r=1 q X s=1 At−r kij rs Aq−sT , (4) where kij rs = k oi r, oj s is a (freely chosen) basic kernel between the two neural observations oi r and oj s at times r and s in trials i and j respectively. For an explanation of kernel operators we refer the reader to [14] and mention that the kernel operator can be viewed as computing φ oi r · φ oj s where φ 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 between neural patterns are measured. The resulting dual form of the tracker is zt = P k αkGtk 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 ti end = n, then calculation of each basic block would take Θ(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 block Ktq. First, define Btq = Pt k=1 kkqAt−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 (6) Ktq = AK(t−1)(q−1)AT + (Bqt)T + 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 + ktqI . Btq (and the entailed summation) is eliminated in exchange for a 2D dynamic program with initial conditions: K11 = k11I , K1q = K1(q−1)AT +k1qI , Kt1 = AK(t−1)1 +kt1I. Table 1: Mean R2, MAEε & MSE (across datasets, folds, hands and directions) for each algorithm. R2 MAEε MSE Algorithm pos. vel. accl. pos. vel. accl. pos. vel. accl. Kalman filter 0.64 0.58 0.30 0.40 0.15 0.37 0.78 0.27 1.16 DDT-linear 0.59 0.49 0.17 0.63 0.41 0.58 0.97 0.50 1.23 SVR-Spikernel 0.61 0.64 0.37 0.44 0.14 0.34 0.76 0.20 0.98 DDT-Spikernal 0.73 0.67 0.40 0.37 0.14 0.34 0.50 0.16 0.91 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Kalman filter, R2 Scores DDT−Spikernel, R2 Scores 0 0.2 0.4 0.6 0.8 1 DDT−linear, R2 Scores 0 0.2 0.4 0.6 0.8 1 SVR−Spikernel, R2 Scores left hand, X dir. left hand, Y dir. right hand, X dir. right hand, Y dir. 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: δi = X j α∗ j −αj Gij −yi ε − min αi,α∗ i ∈[0,c] X j α∗ j −αj Gij −yi ε . Then δi is the amount of ε-insensitive error that can be corrected for example i by keeping all α(∗) j̸=i constant and changing α(∗) i . Optimality is reached by iteratively choosing the example with the largest δi and changing its α(∗) i within the [0, c] limits to minimize the error for this example. 4 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]. DDT (Spikernel) DDT (Linear) 100% SVR (Spikernel) 100% Kalman Filter 88.1% 62.5% 78.12% 63.75% DDT (Spikernel) DDT (Linear) 99.4% SVR (Spikernel) 75% Kalman Filter 87.5% 96.3% 80.0% 86.8% DDT (Spikernel) DDT (Linear) 98.7% SVR (Spikernel) 78.7% Kalman Filter 91.88% 95.6% 86.3% 84.4% 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 hand separately, choosing the movement space to be [x, y, vx, vy, ax, ay]T for each of the hands (preliminary results using only [x, y, vx, vy]T 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 yi t, oi t 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 θ, 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 + η , ot = Hyt+ξ, where η, ξ 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 θ. This parameter was selected to produce best position results on the training set. The selected θ 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 R2 Position Velocity θ Acceleration MAE θ MSE θ θ # Support 6K 8K 10K 12K 14K Figure 3: Effect of θ on R2, MAEε ,MSE and number of support vectors. position X Y velocity acceleration Actual DDT−Spikernel SVR−Spikernel 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 ε insensitive Absolute Errors (MAEε) and Mean Square Errors (MSE). R2 is a standard performance measure, MAEε 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 θ scales its effect. When θ = 0 we get the standard SVR algorithm (without state dynamics). To illustrate the effect of θ 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 θ. It is clear that the value chosen to minimize position error is not optimal for minimizing velocity and acceleration errors. Another important effect of θ 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 ε-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 ε-insensitive ℓ1 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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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 chiru@csa.iisc.ernet.in Pannagadatta K. S. Department of Electrical Engineering Indian Institute of Science Bangalore, 560 012, India pannaga@ee.iisc.ernet.in Alexander J. Smola Machine Learning Program National ICT Australia and ANU Canberra, ACT 0200, Australia Alex.Smola@anu.edu.au 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 that we will require that with high probability any (˜xi, yi) pair, where ˜xi is drawn from a 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(⟨w, x⟩+ 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,ξ m X i=1 ξi (2a) subject to yi (⟨w, xi⟩+ b) ≥1 −ξi, ξi ≥0, ∥w∥≤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 whenever ξi ≥1, where ξi are the slack variables. The Euclidean norm of ∥w∥= p ⟨w, w⟩, 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]: minimize w,b,ξ 1 2∥w∥2 + C m X i=1 ξi (3a) subject to yi (⟨w, xi⟩+ b) ≥1 −ξi, ξi ≥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 (⟨w, xi⟩+ b) ≥1 −ξi} ≥κi, ξi ≥0, ∥w∥≤W ∀1 ≤i ≤m (4) Here we replaced the linear classification constraint by a probabilistic one, which is required to hold with probability κi ∈(0, 1]. This means that by choosing a value of κi close to 1 we can find a conservative classifier which will classify even very infrequent (xi, yi) pairs correctly. Hence κi provides robustness of the estimate with respect to deviating xi. It is clear that unless we impose further restrictions on Pi, it will be difficult to minimize the objective Pm i=1 ξi 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, Σi), i.e., xi is drawn from a Gaussian distribution with mean ¯xi and covariance Σi. We will not require that Σi 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 (⟨w, xi⟩+ b) ≥1 −ξi} ≥κi where xi ∼N(¯xi, Σi) (5a) ξi ≥0, ∥w∥≤W for all 1 ≤i ≤m (5b) The stochastic constraint can be restated as a deterministic optimization problem Pr zi −zi σzi ≥yib + ξi −1 −zi σzi ≤ κi (6) where zi := −yiw⊤xi is a normal random variable with mean ¯zi and variance σ2 zi := w⊤Σiw. Consequently (zi−¯zi)/σzi 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 φ(u) := 1 √ 2π Z u −∞ e−s2 2 ds. In summary, (6) is equivalent to the condition φ yib + ξi −1 −zi σzi ≥κi. which can be solved (since φ(u) is monotonic and invertible), for the argument of φ and obtain a condition on its argument yi(w⊤¯xi + b) ≥1 −ξi + γi p wT Σiw , γi = φ−1(κi) (7) We now proceed to deriving a mathematical programming formulation. 3.2 Second Order Cone Programming Formulation Depending on γi we can distinguish between three different cases. First consider the case where γi = 0 or κi = 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 γi < 0 or κi < 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 γi > 0 or κi > 0.5 second order cone constraint. In this case (7) describes a convex set in in w, b, ξi. We obtain the following optimization problem: minimize w,b,ξ m X i=1 ξi (8a) subject to yi(w⊤xi + b) ≥1 −ξi + γi∥Σ 1 2 i w∥and ξi ≥0 ∀1 ≤i ≤m (8b) ∥w∥≤W (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 ξi > 0, the hyperplane intersects with the constraint set B(xi, Σi, γi). 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 y = sgn(z) sgn(h −γ) where γ = φ−1(κ), z = ⟨w, xi⟩+ b √ w⊤Σw and h = |z|. (9) Here sgn(z) gives us the sign of the point in the center of the ellipsoid and (h −γ) 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 κ = 0.5, i.e., γ = 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 B(xi, Σi, γi) = {x : (x −xi)⊤Σ−1 i (x −xi) ≤γ2 i } (10) in conjunction with the constraint yi (⟨w, x⟩+ b) ≥1 −ξi for all x ∈Si (11) where Si = B(xi, Σi, γi) As before γi = φ−1(κi) for κi ≥0. In other words, we have ξi = 0 only when the hyperplane w⊤x + b = 0 does not intersect the ball B(xi, Σi, γi). 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, β) is a polynomial in β 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 (⟨w, xij⟩+ b) ≥1 −ξi 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 Set Si Optimization Problem Plain SVM[3] {xi} Quadratic Program Knowledge Based SVM[4] Polyhedral set Quadratic Program Invariances [5] trajectory of polynomial Semidefinite Program Normal Distribution B(xi, Σi, γi) Second Order Cone Program Convex Hull co{xij ∀1 ≤j ≤mi} 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 µy and covariance Σy. Hence we have the following two-stage procedure to deal with missing variables: • Estimate Σy, µy 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 µ and covariance Σ it follows that xm|xa ∼N(µm + ΣamΣ−1 aa (xa −µa), Σmm −Σ⊤ amΣ−1 aa Σam). (12) Here we decomposed µ, Σ according to (xa, xm) into µ = (µa, µm) and Σ = Σaa Σam Σ⊤ am Σmm . (13) Hence, knowing Σ, µ we can estimate the missing variables and determine their degree of uncertainty. One can show that [7] to obtain Σ, µ the EM algorithm reads as follows: 1. Initialize Σ, µ. 2. Estimate xm|xa for all observations using (12). 3. Recompute Σ, µ 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,ξ m X i=1 ξi (14) subject to yi (⟨w, xi⟩+ b) ≥1 −ξi ∀1 ≤i ≤ma yj(w⊤xj + b) ≥1 −ξj + φ−1(κj)∥Σ 1 2 j w∥ ∀ma + 1 ≤j ≤ma + mm ξi ≥0 ∀1 ≤i ≤ma + mm ∥w∥≤W The mean xj has two components; xaj has values available, while the imputed vector is given by ˆxmj, via (12). The matrix Σj 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 µ1 and Σ1 (see (13)) respectively. The training dataset of class 1 is to be used in the computation of µ1 and Σ1. Using the estimated µ1 and Σ1 compute h as defined in (9), and denote it by h1. Compute the label of µ1 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 µ2, Σ2, h2 respectively. Denote by y2 the label of µ2 as predicted by (1). We decide that the observation belongs to class with label yµ as yµ = y2 if h1 < h2 and yµ = 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 yµ, h1, h2 as in (15), the worst case prediction rule (9) can be modified as follows y = yµ sgn(h −γ) where γ = φ−1(κ) 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(= fN, 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 other variables is a gaussian, the mean(xj) and the covariance (ˆCj) can be estimated by the methods described in (4.1). The robust optimization problem was then solved for different values of κ. The parameter κj(= κ) is set to the same value for all the Nm datapoints. For each value of κ 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 κ = 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 0.7 0.8 0.9 1 0.4 robust nomwc robustwc 0.5 0.6 0.7 0.8 0.9 1 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 robust nomwc robustwc 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 0.5 robust nomwc robustwc 0.5 0.6 0.7 0.8 0.9 1 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 robust nomwc robustwc 0.5 0.6 0.7 0.8 0.9 1 0.2 0.3 0.4 robust nomwc robustwc 0.5 0.6 0.7 0.8 0.9 1 0.2 0.25 0.3 0.35 0.4 0.45 robust nomwc robustwc 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 κ 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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Multiple Relational Embedding Roland Memisevic Department of Computer Science University of Toronto roland@cs.toronto.edu Geoffrey Hinton Department of Computer Science University of Toronto hinton@cs.toronto.edu 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 knowledge 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 σ2 be some real-valued neighborhood variance or “kernel bandwidth”. SNE finds a low-dimensional representation for a set of input data points yi(i = 1, . . . , N) by first constructing a similarity matrix P with entries Pij := exp(−1 σ2 ∥yi −yj∥2) P k exp(−1 σ2 ∥yi −yk∥2) (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 Qij(X) := exp(−∥xi −xj∥2) P k exp(−∥xi −xk∥2). (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 jth 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 RcT Rc . In order to learn the transformations Rc from the data along with the set of latent representations X we consider the loss function E(X) = X c Ec(X), (3) where we define Ec(X) := 1 N X i,j P c ij log P c ij Qc ij ! and Qc ij := Qij(RcX). (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 λc. We found that the solutions are rather robust with regard to different sets of λc 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) ∂xl = 2 N X c X i (P c il + P c li −Qc li −Qc il) RcT Rc(xl −xi), (5) −1 −0.5 0 0.5 −1 −0.5 0 0.5 1 −0.5 0 0.5 −1 −0.5 0 0.5 −1 −0.5 0 0.5 1 −0.5 0 0.5 1 2 3 −6 −4 −2 0 2 REucl 1 2 3 −6 −4 −2 0 2 RClass 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) ∂Rc ll = 2 N Rc ll X i X j P c ij −Qc ij (xi l −xj l )2 , (6) where xi l 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 that contains homogeneous blocks of size 30 × 30 with entries ( 1 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 class1For training we set σ2 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 Ec(X) := 1 N c X i,j∈Ic P c ij log P c ij Qc ij ! . (7) 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 σ2 set as in the previous experiment), with corresponding index-sets ICoil and IUmist 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 1 2. Each of the corresponding index sets I1 and I2 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 Rclass. (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 2Training 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 σ2 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 σ2 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 3This 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. 4See previous footnote. −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 x4 x1 −0.2 −0.1 0 0.1 0.2 0.3 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 x2 x3 1 2 3 4 −5 0 5 REucl 1 2 3 4 −5 0 5 RLights 1 2 3 4 −5 0 5 RPose2 1 2 3 4 −5 0 5 RPose1 1 2 3 4 −5 0 5 RInk 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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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 mfigueroa@die.udec.cl 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 0.35µm 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 0.35µm 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: z(i) = bw0(i) + N X j=1 xj(i) wj(i) (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) + η xj(i) e(i) i = 0 . . . N, x0(i) = b (2) where η 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 (V + w , V − w ), 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 V + x , V − x ). The output is a differential analog current pair (I+ o , I− o ), 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 (P + e and P − e 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 0.35µm 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: z(i) = N X j=0 aj f x j xj(i) −dx j f w j wj(i) −dw j x0(i) = b (3) where aj represents the gain mismatch between multipliers, f x j and f w j are the nonlinearities applied to the inputs and weights (also mismatched across the perceptron), and dx j 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 f x j than to the linearity of f w j , 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 f w j remains a monotonic odd function [2]. Consequently, we sized the transistors in the Gilbert multiplier to maximize the linearity of f x j , but paid less attention (in order to minimize size and power) to f w j . 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) + ηj xj(i) e(i) i = 0 . . . N, x0(i) = b (4) where ηj 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 < ηmax < 1/λmax, where λmax is the maximal eigenvalue of the input’s correlation matrix and ηmax = maxj(ηj). 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 , P− x ). 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 , P− e ) such that: Pinc = (P + x AND P + e ) OR (P − x AND P − e ) (5) Pdec = (P + x AND P − e ) OR (P − x AND P + e ) (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. rb = −log2 0.5 RMS error output range , we find that for a 2µA output range, this error represents an 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 0.35µm CMOS process. The circuit uses 0.25mm2 of die area and dissipates 200µW. 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 10µA RMS, which corresponds to an output resolution of about 3 bits for a full range of 128µA differential. Calibrating the synapses for symmetric learning rates only improves the RMS error to 5µA (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 perceptron 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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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 Thomas.Natschlaeger@scch.at Nils Bertschinger Max Planck Institute for Mathematics in the Sciences D-04103 Leipzig, Germany bertschi@mis.mpg.de Robert Legenstein Institute for Theoretical Computer Science, TU Graz A-8010 Graz, Austria legi@igi.tu-graz.ac.at 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 timesteps neuron # 0 20 40 10 20 30 input timesteps network activity 0 20 40 timesteps 0 20 40 0.1 1 10 −0.6 −0.4 −0.2 0 0.2 0.4 σ2 µ ordered chaotic critical mean activity 0.4 0.6 0.8 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 µ and σ2 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 µ and σ2. 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 µ and variance σ2. Furthermore, the network is driven by an external input signal u(·) which is injected into each node. Hence, in summary, the update of the network state xt = (x1,t, . . . , xN,t) is given by xi,t = Θ(PN j=1 wij · xj,t−1 + ut−1) which is applied to all neurons in parallel and where Θ(h) = 1 if h ≥0 and Θ(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 µ ∈R. This implies that the network activity at = 1 N PN 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 σ2 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 Hamming distance dt = |{i : xi,t ̸= ˜xi,t}|/N between two states xt and ˜xt as well as the 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→∞⟨F t u⟩.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 ∗̸= 0 and the network is in the chaotic phase. Whether f ∗= 0 or f ∗̸= 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 α∗ 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 α∗< 1, α∗= 1 or α∗> 1 respectively. In [8] it is shown that the so called critical line α∗= 1 where the phase transition from ordered to chaotic behavior occurs is given by Pbf = K−1 X n=0 K −1 n a∗n(1−a∗)K−1−n(rQ(1, n, ¯u+1)+(1−r)Q(1, n, ¯u)) = 1 K (1) 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 1F t u denotes t-fold composition of the map Fu(·, ·) where in the k-th iteration the input uk is applied and ⟨·⟩denotes the average over all possible initial conditions and all input signals with a given statistics determined by ¯u and r. 2The actual single bit-flip probability Q depends on the number n of inputs which are 1 and the 0.1 1 10 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 σ2 µ K = 5 NM−Separation 0 0.02 0.04 0.06 0.08 0.1 0.1 1 10 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 σ2 µ K = 10 NM−Separation 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 Figure 2: NM-separation assumes high values on the critical line. The gray coded image shows the NM-separation in dependence on µ and σ2 for K denoted in the panels, r = 0.3, ¯u = −0.5 and b = 0.1. The solid line marks the critical values for µ and σ2. 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 xt 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 ̸= ˜ut}, i.e. the separation. In summary the three-dimensional map Su,˜u(dt, at, ˜at) := (dt+1, at+1, ˜at+1) = (s(dt, at, ˜at, ut, ˜ut), A(at, ut), A(˜at, ˜ut)) fully describes the time evolution of the Hamming distance and the network activities. Again we consider the steady state of the averaged Hamming distance s∗and the network activities a∗, ˜a∗, i.e. (s∗, a∗, ˜a∗) = limt→∞ St u,˜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 computaexternal input u and is given by Q(1, n, u) = R −u −∞φ(ξ, nµ, nσ2) 1 −Φ(−u −ξ, µ, σ2) dξ + R ∞ −u φ(ξ, nµ, nσ2)Φ(−u−ξ, µ, σ2)dξ where φ, Φ denote the Gaussian density and cumulative density respectively (see [8] for a detailed explanation). 3For each value of µ = −0.6 + k ∗0.01, k = 0 . . . 100 a search was conducted to find the value for σ2 such that α∗= 1. Numerical iterations of the function A were used to determine a∗. A B C 0.01 0.1 1 10 100 −0.6 −0.4 −0.2 0 0.2 0.4 σ2 µ 3bit parity (K = 5) MC (MI) 0 1 2 3 4 5 0.01 0.1 1 10 100 σ2 3bit parity (K = 10) MC (MI) 0 1 2 3 4 0.01 0.1 1 10 100 σ2 5bit random boolean functions mean MI 0 0.2 0.4 0.6 0.8 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 µ and σ2 for the delayed 3-bit parity task. Performance is measured as the memory capacity MC = P τ I(v, y(τ)) where I(v, y(τ)) is the mutual information between the classifier output v(·) and the target function y(τ) t = PARITY(ut−τ, ut−τ−1, ut−τ−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 NM-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, ̸= ˜v} is applied: i∗= limt→∞ P1 v,˜v=0 rv(1−r)1−vb|v−˜v|(1−b)1−|v−˜v| s(·, ·, ·, v, ˜v) ◦St u,u −f ∗. Hence a measure of the network mediated separation NMsep due to input differences is given by NMsep = s∗−f ∗−i∗ (2) In Fig. 2 the NM-separation resulting from an input difference of b = 0.1 is shown in dependence of the network parameters µ and σ2.4 Note that the NM-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 4For each value of µ = −0.6 + k ∗0.05, k = 0 . . . 20, 10 values for σ2 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 (µ, σ2) 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) = Θ(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 ∈RN, 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 ut′, t′ < 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 input statistics given by ¯u, r and a∗respectively) be equal to 1 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 single bit in its input is flipped. Formally P i bf of node i is given by the average (over the internal and external input statistics) of the following quantity: 1 K N X j=1,wij̸=0 Θ (wij(1 −2xj,t−1)(1 −2xi,t) −mi,t) (3) where mi,t = PN j=1 wijxj,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: wij(t + 1) = 1 1+ν · wij if P esti bf (t) > 1 K (1 + ν) · wij(t) if P esti bf (t) < 1 K (4) A timesteps neuron # 100 200 300 400 500 600 700 50 100 150 200 B C 0 100 200 300 400 500 600 700 0 0.5 1 1.5 timesteps K*Pbf 0 100 200 300 400 500 600 700 0 0.5 1 1.5 timesteps K*Pbf Figure 4: Self-organized criticality. A Time evolution of the network state xt 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, µ = 0 and initial σ2 = 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 σ2 = 0.01 in the ordered regime. where 0 < ν ≪1 is the learning rate and P esti bf (t) is a running average of the formula in Equ. (3) to estimate P i bf. 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. 45. 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, µ = 0 and σ2 = 100) to critical dynamics that respect the input signal. The lower row of Fig. 4 shows how the averaged estimated bit-flip probability 1 N PN i=1 P esti bf (t) approaches its critical value for the case of the above network and one that started in the ordered regime (K = 10, µ = 0, σ2 = 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 MC (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 σ2 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 5Here a learning rate of ν = 0.01 and an exponentially weighted running average with a time constant of 15 time steps were used. A B 0 500 1000 1500 2000 0 1 2 3 4 5 SOC steps MC [bits] K = 5, start σ2 = 100 (chaotic) 0 500 1000 1500 2000 0 1 2 3 4 SOC steps MC [bits] K = 10, start σ2 = 0.01 (ordered) Figure 5: Time evolution of the performance with activated SOC-rule. A The plot shows the memory capacity MC (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 MC 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 NM-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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Trait selection for assessing beef meat quality using non-linear SVM J.J. del Coz, G. F. Bay´on, J. D´ıez, O. Luaces, A. Bahamonde Artificial Intelligence Center University of Oviedo at Gij´on juanjo@aic.uniovi.es Carlos Sa˜nudo Facultad de Veterinaria University of Zaragoza csanudo@posta.unizar.es 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 PJ = {vj > 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 PJ, we can specify F by means of the restrictions F(vj, uj) > 0 and F(uj, vj) < 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) = ⟨φ(x), φ(y)⟩is a kernel function defined as the inner product of two objects represented in the feature space by their φ images. In the experiments reported in Section 4, we will employ a polynomial kernel, defining k(x, y) = (⟨x, y⟩+ c)g, with c = 1 and g = 2. Notice that, finally we can express the ranking function f in a non-linear form: f(x) = n X i=1 αizi(k(x(1) i , x) −k(x(2) i , x)) (4) 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 R1(i) = |∇i∥w∥2| = X k,j αkαjzkzj ∂K(s · xk, s · xj) ∂si , i = 1, . . . , d (5) 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) = (⟨x, y⟩+ c)g and a vector s such that ∀i, si = 1 we have that ∂k(s · x, s · y) ∂si = 2g(xiyi)(c + ⟨x, y⟩)g−1. (6) 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 R2(i) = X k zk · X j αjzjK(x(1),i j , x(2),i j , x(1),i k , x(2),i k ) (7) where xi 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 method to compute ˆd(g, t), an adjusted distance estimate between any hypothesis f and the true target classification function t. Therefore, the selected hypothesis is fk = arg min fl ˆd(fl, t). (8) The estimation of distance, ˆd, is computed by means of the expected disagreement in the 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 ADJ(fl, t) def = dT (fl, t) · max k<l dU(fk, fl) dT (fk, fl) (9) 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, dS(f, g) def = 1 |S| + 2 1 + X x∈S 1f(x)̸=g(x) ! (10) 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, similarity(ai, aj) = ⟨ai, aj⟩ ∥ai∥· ∥aj∥ (11) 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 Rmad Rmad Error Error Error Error tenderness 96.3% 97.8% 41.5% 43.1% 29.6% 19.4% flavor 99.3% 103.4% 43.8% 46.5% 32.7% 23.8% acceptance 94.0% 97.2% 38.4% 40.2% 31.9% 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 20.0% 50.0 21.8% 27.0 21.3% 37.5 flavor 25.0% 65.0 26.5% 33.5 26.1% 29.0 acceptance 24.7% 39.5 24.8% 30.0 25.3% 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´ıa (MCyT) and Feder grant TIC2001-3579. References [1] A. Bahamonde, G. F. Bay´on, J. D´ıez, 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´ıez, 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´ıez, 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´ın, 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´ıez. 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. Næs 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´ı. 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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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)Computer Science Div., (2)Dept. of Statistics University of California at Berkeley Berkeley CA 94720, USA {ywteh,jordan,blei}@cs.berkeley.edu (3)Dept. of Computer Science University of Toronto Toronto M5S 3G4, Canada beal@cs.toronto.edu 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 µ0 is drawn from some distribution, a set of K means are then drawn independently from a Gaussian with mean µ0, 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 (Θ, B) be a measurable space, with G0 a probability measure on the space, and let α0 be a positive real number. A Dirichlet process is the distribution of a random probability measure G over (Θ, B) such that, for any finite partition (A1, . . . , Ar) of Θ, the random vector (G(A1), . . . , G(Ar)) is distributed as a finite-dimensional Dirichlet distribution: (G(A1), . . . , G(Ar)) ∼Dir α0G0(A1), . . . , α0G0(Ar) . (1) We write G ∼DP(α0, G0) if G is a random probability measure distributed according to a DP. We call G0 the base measure of G, and α0 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(α0, G0), independently draw n latent factors from G: φi ∼G. Then, for each i = 1, . . . , n, draw xi ∼F(φi), for a distribution F. This setup is referred to as a DP mixture model. If the factors φi 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 φi 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 {φi}. 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: θk ∼G0 β′ k ∼Beta(1, α0) βk = β′ k Qk−1 l=1 (1 −β′ k). (2) Then with probability one the random measure defined by G = P∞ k=1 βkδθk is a sample from DP(α0, G0). The construction for β1, β2, . . . in (2) can be understood as taking a stick of unit length, and repeatedly breaking off segments of length βk. 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 θk as the parameter of the kth mixture component, with mixing proportion given by βk. 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 (Θ, B): one probability measure Gj for each group j, and a global probability measure G0. The global measure G0 is distributed as DP(γ, H), with H the base measure and γ the concentration parameter, while each Gj is conditionally independent given G0, with distribution Gj ∼DP(α0, G0). To complete the description of the HDP mixture model, we associate each xji with a factor φji, with distributions given by F(φji) and Gj respectively. The overall model is given in Figure 1 left, with conditional distributions: G0 | γ, H ∼DP(γ, H) Gj | α, G0 ∼DP(α0, G0) (3) φji | Gj ∼Gj xji | φji ∼F(φji) . (4) The stick-breaking construction (2) shows that a draw of G0 can be expressed as a weighted sum of point masses: G0 = P∞ k=1 βkδθk. 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 θ = (θk)∞ k=1, the atoms of Gj must also come from θ, hence we may write: G0 = P∞ k=1 βkδθk Gj = P∞ k=1 πjkδθk . (5) Identifying θk as the parameters of the kth mixture component, we see that each submodel corresponding to distinct groups share the same set of mixture components, but have differing mixing proportions, πj = (πjk)∞ k=1. Finally, it is useful to explicitly describe the relationships between the mixing proportions β and (πj)J j=1. Details are provided in [10]. Note that the weights πj are conditionally independent given β since each Gj is independent given G0. Applying (1) to finite partitions G2 G1 G3 x 1 φ 1i n 1i x2i n2 φ2i x n3 3i 3i φ H γ 0 α 0 α 0 α G0 φ13 φ18 φ14 φ12 φ15 φ16 φ17 φ22 φ21 φ34 φ33 32 φ φ35 φ36 φ31 φ27 φ28 φ25 ψ21 ψ13 ψ11 ψ31 ψ22 ψ23ψ32 ψ12 11 φ ψ24 ψ11 ψ13 ψ12 ψ21 ψ24 ψ23 φ26 φ23 24 φ ψ22 ψ32 ψ31 2θ 1θ 3θ group j=1 group j=2 group j=3 global 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 θ, we get πj ∼DP(α0, β), where we interpret β and πj as probability measures over the positive integers. Hence β is simply the putative mixing proportion over the groups. We may in fact obtain an explicit stick-breaking construction for the πj’s as well. Applying (1) to partitions ({1, . . . , k −1}, {k}, {k + 1, . . .}) of positive integers, we have: π′ jk ∼Beta α0βk, α0 1 −Pk l=1 βl πjk = π′ jk Qk−1 l=1 (1 −π′ jl) . (6) 4 The Chinese Restaurant Franchise We describe an alternative view of the HDP based directly upon the distribution a HDP induces on the samples φji, 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 φj1, . . . , φjnj as we marginalize out Gj. Recall that since Gj ∼DP(α0, G0) we can describe this distribution by describing how to generate φj1, . . . , φjnj using the CRP. Imagine nj customers (each corresponds to a φji) 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 α0. Suppose customer i sat at table tji. The conditional distributions are: tji | tj1, . . . , tji−1, α0 ∼ X t njt P t′ njt′ +α0 δt + α0 P t′ njt′ +α0 δtnew , (7) 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 φj1, . . . , φjnj. 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 ψjt from G0, and assign φji = ψjtji. Performing this process independently for each group j, we have now integrated out all the Gj’s, and have an assignment of each φji to a sample ψjtji from G0, with the partition structures given by CRPs. Notice now that all ψjt’s are simply i.i.d. draws from G0, which is again distributed according to DP(γ, H), so we may apply the same CRP partitioning process to the ψjt’s. Let the customer associated with ψjt sit at table kjt. We have: kjt | k11, . . . , k1n1, k21, . . . , kjt−1, γ ∼ X k mk P k′ mjk′ +γ δk + γ P k′ mk′+α0 δknew . (8) 10 20 30 40 50 60 70 80 90 100 110 120 750 800 850 900 950 1000 1050 Perplexity Number of LDA topics Perplexity on test abstacts of LDA and HDP mixture LDA HDP Mixture 61 62 63 64 65 66 67 68 69 70 71 72 73 0 5 10 15 Number of topics Number of samples Posterior over number of topics in HDP mixture 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 θk from H and assign ψjt = θkjt. This completes the generative process for the φji’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 (φji’s), who sit at tables (ψjt’s). Now each table is served a dish (θk’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 1Available 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. 2In 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. 3To ensure we are dealing with informative words in the documents, we culled stop words as well 0 10 20 30 40 50 60 70 80 2500 3000 3500 4000 4500 5000 5500 6000 Number of VS training documents Perplexity Average perplexity over NIPS sections of 3 models M1: additional sction ignored M2: flat, additional section M3: hierarchical, additional section 0 10 20 30 40 50 60 70 80 2500 3000 3500 4000 4500 5000 Number of VS training documents Perplexity Generalization from LT, AA, AP to VS LT AA AP 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. 4Though 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. CS NS LT AA IM SP AP CN task representation pattern processing trained representations three process unit patterns cells cell activity response neuron visual patterns pattern single fig 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 visual images video language image pixel acoustic delta lowpass flow approach based trained test layer features table classification rate paper ii tree pomdp observable strategy class stochastic history strategies density 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. ML, MAP, and variational Bayesian (VB) 0 5 10 15 20 25 30 0 10 20 30 40 50 Number of hidden states Perplexity Perplexity on test sentences of Alice ML MAP VB 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). [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 θ = (θk). We describe an inference procedure for the HDP mixture model based on Gibbs sampling t, k and θ given data items x. For the full derivation see [10]. Let f(·|θ) and h be the density functions for F(θ) and H respectively, n−i jt be the number of tji′’s equal to t except tji, and m−jt k be the number of kj′t′’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|θkjt) where for t = tnew we may sample kjtnew using (8), and θknew ∼H. The distribution is then: p(tji = t | t\tji, k, θ, x) ∝ α0f(xji|θkjt) if t = tnew n−i jt f(xji|θkjt) if t currently used. (9) Similarly the conditional distributions for kjt and θk are: p(kjt = k | t, k\kjt, θ, x) ∝ ( γ Q i:tji=t f(xji|θk) if k = knew m−t k Q i:tji=t f(xji|θk) if k currently used. (10) p(θk | t, k, θ\θk, x) ∝h(θk) Y ji:kjtji =k f(xji|θk) (11) where θknew ∼H. If H is conjugate to F(·) we have the option of integrating out θ. 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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Spike-Timing Dependent Plasticity and Mutual Information Maximization for a Spiking Neuron Model Taro Toyoizumi†‡, Jean-Pascal Pfister‡ Kazuyuki Aihara§ ∗, Wulfram Gerstner‡ † Department of Complexity Science and Engineering, The University of Tokyo, 153-8505 Tokyo, Japan ‡ Ecole Polytechnique F´ed´erale de Lausanne (EPFL), School of Computer and Communication Sciences and Brain-Mind Institute, 1015 Lausanne, Switzerland § Graduate School of Information Science and Technology, The University of Tokyo, 153-8505 Tokyo, Japan taro@sat.t.u-tokyo.ac.jp, jean-pascal.pfister@epfl.ch aihara@sat.t.u-tokyo.ac.jp, wulfram.gerstner@epfl.ch Abstract We derive an optimal learning rule in the sense of mutual information maximization for a spiking neuron model. Under the assumption of small fluctuations of the input, we find a spike-timing dependent plasticity (STDP) function which depends on the time course of excitatory postsynaptic potentials (EPSPs) and the autocorrelation function of the postsynaptic neuron. We show that the STDP function has both positive and negative phases. The positive phase is related to the shape of the EPSP while the negative phase is controlled by neuronal refractoriness. 1 Introduction Spike-timing dependent plasticity (STDP) has been intensively studied during the last decade both experimentally and theoretically (for reviews see [1, 2]). STDP is a variant of Hebbian learning that is sensitive not only to the spatial but also to the temporal correlations between pre- and postsynaptic neurons. While the exact time course of the STDP function varies between different types of neurons, the functional consequences of these differences are largely unknown. One line of modeling research takes a given STDP rule and analyzes the evolution of synaptic efficacies [3–5]. In this article, we take a different ∗Alternative address: ERATO Aihara Complexity Modeling Project, JST, 45-18 Oyama, Shibuyaku, 151-0065 Tokyo , Japan approach and start from first principles. More precisely, we ask what is the optimal synaptic update rule so as to maximize the mutual information between pre- and postsynaptic neurons. Previously information theoretical approaches to neural coding have been used to quantify the amount of information that a neuron or a neural network is able to encode or transmit [6–8]. In particular, algorithms based on the maximization of the mutual information between the output and the input of a network, also called infomax principle [9], have been used to detect the principal (or independent) components of the input signal, or to reduce the redundancy [10–12]. Although it is a matter of discussion whether neurons simply ’transmit’ information as opposed to classification or task-specific processing [13], strategies based on information maximization provide a reasonable starting point to construct neuronal networks in an unsupervised, but principled manner. Recently, using a rate neuron, Chechik applied information maximization to detect static input patterns from the output signal, and derived the optimal temporal learning window; the learning window has a positive part due to the effect of the postsynaptic potential and has flat negative parts with a length determined by the memory span [14]. In this paper, however, we employ a stochastic spiking neuron model to study not only the effect of postsynaptic potentials generated by synaptic input but also the effect of the refractory period of the postsynaptic neuron on the shape of the optimal learning window. We discuss the relation of mutual information and Fisher information for small input variance in Sec. 2. Optimization of the Fisher information by gradient ascent yields an optimal learning rule as shown in Sec. 3 2 Model assumptions 2.1 Neuron model The model we are considering is a stochastic neuron with refractoriness. The instantaneous firing rate ρ at time t depends on the membrane potential u(t) and refractoriness R(t): ρ(t) = g(βu(t))R(t), (1) where g(βu) = g0 log2[1+eβu] is a smoothed piecewise linear function with a scaling variable β and a constant g0 = 85Hz. The refractory variable is R(t) = (t−ˆt−τabs)2 τ 2 refr+(t−ˆt−τabs)2 Θ(t − ˆt −τabs) and depends on the time elapsed since the last firing time ˆt, the absolute refractory period τabs = 3 ms, and the time constant of relative refractoriness τrefr = 10 ms. The Heaviside step function Θ takes a value of 1 for positive arguments and zero otherwise. The postsynaptic potential depends on the input spike trains of N presynaptic neurons. A presynaptic spike of neuron i ∈{1, 2, . . . , N} emitted at time tf i evokes a postsynaptic potential with time course ϵ(t −tf i ). The total membrane potential is u(t) = N X i=1 wi X f ϵ(t −tf i ) = N X i=1 wi Z ϵ(s)xi(t −s)ds (2) where xi(t) = P f δ(t−tf i ) denotes the spike train of the presynaptic neuron i. The above model is a special case of the spike response model with escape noise [2]. For vanishing refractoriness τrefr →0 and τabs →0, the above model reduces to an inhomogeneous Poisson process. For a given set of presynaptic spikes in an interval [0, T], hence for a given time course of membrane potential {u(t)|t ∈[0, T]}, the model generates an output spike train y(t) = X f δ(t −tf) (3) with firing times {tf|f = 1, . . . , n} with a probability density P(y|u) = exp "Z T 0 (y(t) log ρ(t) −ρ(t)) dt # . (4) where ρ(t) is given by Eq. (1), i.e., ρ(t) = g(βu(t)) R(t). Since the refractory variable R depends on the firing time ˆt of the previous output spike, we sometimes write ρ(t|ˆt) instead of ρ(t) in order to make this dependence explicit. Equation (4) can then be re-expressed in terms of the survivor function S(t|ˆt) = e− R t ˆt ρ(s|ˆt)ds and the interval distribution Q(t|ˆt) = ρ(t|ˆt)S(t|ˆt) in a more transparent form: P(y|u) = n Y f=1 Q(tf|tf−1) S(T|tn), (5) where t0 = 0 and n is the number of postsynaptic spikes in [0, T]. In words, the probability that a specific output spike train y occurs can be calculated from the interspike intervals Q(tf|tf−1) and the probability that the neuron ‘survives’ from the last spike at time tn to time T without further firing. 2.2 Fisher information and mutual information Let us consider input spike trains with stationary statistics. These input spike trains generate an input potential u(t) with an average value u0 and standard deviation σ. Assuming a weak dependence of g on the membrane potential u, i.e., for small β, we expand g around g0 = g(0) to obtain g(βu(t)) = g0 + g′ 0βu(t) + g′′ 0[βu(t)]2/2 + O(β3) where g0 is the value of g in the absence of input and the next terms describe the influence of the input. Here and in the following, all calculations will be done to order β2. In the limit of small β, the mutual information is given by [15] I(Y ; X) = β2 2 Z T 0 dt Z T 0 dt′Σ(t −t′)J0(t −t′) + O(β3), (6) with the autocovariance function of the membrane potential Σ(t −t′) = ⟨∆u(t)∆u(t′)⟩X, (7) with ∆u(t) = u(t) −u0 and Fisher information J0(t −t′) = − * ∂2 log P(y|u) ∂βu(t)∂βu(t′) ¯¯¯¯ β=0 + Y |β=0 , (8) with ⟨·⟩Y |β=0 = R · P(y|β = 0)dy and ⟨·⟩X = R · P(x)dx. Note that the Fisher information (8) is to be evaluated at the constant g0, i.e., at the value βu = 0, whereas the autocovariance in Eq. (7) is calculated with respect to the mean membrane potentital u0 = ⟨u(t)⟩X which is in general different from zero. The derivation of (6) is based on the assumption that the variability of the output signal is small and g(βu) does not deviate much from g0, i.e., it corresponds to the regime of small signal-to-noise ratio. It is well known that the information capacity of the Gaussian channel is given by the log of the signal-to-noise ratio [16], and the mutual information is proportional to the signal-to-noise ratio when it is small. The relation between the Fisher information, the mutual information, and optimal tuning curves has previously been established in the regime of large signal-to-noise ratio [17]. We introduce the following notation: Let µ0 = ⟨y(t)⟩Y |β=0 = ⟨ρ(t)⟩Y |β=0 be the spontaneous firing rate in the absence of input and µ−1 0 ⟨y(t)y(t′)⟩Y |β=0 = δ(t −t′) + µ0[1 + φ(t −t′)] be the postsynaptic firing probability at time t given a postsynaptic spike at t′, i.e., the autocorrelation function of Y . From the theory of stationary renewal processes [2] µ0 = ·Z s Q0(s)ds ¸−1 , µ0[1 + φ(s)] = Q0(|s|) + Z Q0(s′)µ0[1 + φ(|s| −s′)] Θ(|s| −s′)ds′, (9) where Q0(s) = g0R(s)e−g0[(s−τabs)−τrefr arctan(s−τabs)/τrefr] is the interval distribution for constant g = g0. The interval distribution vanishes during the absolute refractory time τabs; cf. Fig. 1. (A) (B) 0 20 40 60 80 100 −0.01 0 0.01 0.02 0.03 0.04 0.05 PSfrag replacements s [ms] Q0(s) φ(s) 0 10 20 30 40 50 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 PSfrag replacements s [ms] Q0(s) φ(s) Figure 1: Interspike interval distribution Q0 and normalized autocorrelation function φ. The circles show numerical results, the solid line the theory. The Fisher information of (8) is calculated from (4) to be J0(t −t′) = δ(t −t′) µg′ 0 g0 ¶2 ⟨ρ0(t)⟩Y |β=0 (10) with the instantaneous firing rate ρ0(t) = g0R(t). Hence the mutual information is I(Y ; X) = β2 2 µg′ 0 g0 ¶2 Z T 0 dt µ0σ2 (11) = β2 2 µg′ 0 g0 ¶2 Tµ0σ2. (12) For an interpretation of Eq. (11) we note that σ2 = Σ(0) is the variance of the membrane potential and depends on the statistics of the presynaptic input whereas µ0 is the spontaneous firing rate which characterizes the output of the postsynaptic neuron. Hence, Equation (11) contains both pre- and postsynaptic factors. 3 Results: Optimal spike-timing dependent learning rule In the previous section we have calculated the mutual information between presynaptic input spike trains and the output of the postsynaptic neuron under the assumption of small fluctuations of g. The mutual information depends on parameters of the model neuron, in particular the synaptic weights that characterize the efficacy of the connections between pre- and postsynaptic neurons. In this section, we will optimize the mutual information by changing the synaptic weights in an appropriate fashion. To do so we will proceed in several steps. First, based on gradient ascent we derive a batch learning rule of synaptic weights that maximizes the mutual information. In a second step, we transform the batch rule into an online rule that reduces to the batch version when averaged. Finally, in subsection 3.2, we will see that the online learning rule shares properties with STDP, in particular a biphasic dependence upon the relative timing of pre- and postsynaptic spikes. 3.1 Learning rule for spiking model neuron In order to keep the analysis as simple as possible, we suppose that the input spike trains are independent Poisson trains, i.e., ⟨∆xi(t)∆xj(t′)⟩X = νiδ(t −t′)δij, where ∆xi(t) = xi(t)−νi with rate νi = ⟨xi(t)⟩X. Then we obtain the variance of the membrane potential σ2 = ⟨[∆u(t)]2⟩X = ϵ2 X j w2 jνj (13) with ϵ2 = R ϵ2(s)ds. Applying gradient ascent to (11) with an appropriate learning rate α, we obtain the batch learning rule of synaptic weights as ∆wi = α∂I(Y ; X) ∂wi ≈αβ2 2 µg′ 0 g0 ¶2 Z T 0 dt µ0 ∂σ2 ∂wi . (14) The derivative of µ0 with respect to wi vanishes, since µ0 is the spontaneous firing rate in the absence of input. We note that both µ0 and σ2 are defined by an ensemble averages, as is typical for a ‘batch’ rule. While there are many candidates of online learning rule that give (14) on average, we are interested in rules that depend directly on neuronal spikes rather than mean rates. To proceed it is useful to write σ2 = ⟨[∆u(t)]2⟩X with ∆u = P i wi∆ϵi(t) where ∆ϵi(t) = R ϵ(s)∆xi(t −s)ds. In this notation, one simple form of an online learning rule that depends on both the postsynaptic firing statistics and presynaptic autocorrelation is dwi dt = αβ2 µg′ 0 g0 ¶2 y(t)∆ϵi(t)∆u(t), (15) Hence weights are updated with each postsynaptic spike with an amplitude proportional to an online estimate of the membrane potential variance calculated as the product of ∆u and ∆ϵi. Indeed, to order β0, the input and the output spikes are independent; ⟨y(t)∆ϵi(t)∆u(t)⟩Y,X = ⟨y(t)⟩Y |β=0⟨∆ϵi(t)∆u(t)⟩X and the average of (15) leads back to (14). 3.2 STDP function as a spike-pair effect Application of the online learning rule (15) during a trial of duration T, yields a total change of the synaptic efficacy which depends on all the presynaptic spikes via the factor ∆ϵi; on the postsynaptic potential via the factor ∆u; and on the postsynaptic spike train y(t). In order to extract the spike pair effect evoked by a given presynaptic spike at tpre i and a postsynaptic spike at tpost, we average over x and y given the pair of spikes. The spike pair effect up to the second order of β is therefore described as ∆wi(tpost −tpre i ) = αβ2 µg′ 0 g0 ¶2 Z T 0 dt⟨y(t)⟩Y |tpost,β=0⟨∆ϵi(t)∆u(t)⟩X|tpre i , (16) where ⟨·⟩Y |tpost,β=0 = R dy · P(y|tpost, β = 0) and ⟨·⟩X|tpre i = R dx · P(x|tpre i ). Note that the leading factor of Eq. (16) is already of order β2, so that all other factors have to be evaluated to order β0. Suppressing all terms containing β, we obtain P(y|tpost, u) ≈P(y|tpost, β = 0) and from the Bayes formula P(x|tpre i , tpost) = P (tpost|x,tpre i ) ⟨P (tpost|x,tpre i )⟩X|tpre i P(x|tpre i ) ≈P(x|tpre i ). In order to see the contribution of tpre i and tpost, we think of separating the effects caused by spikes at tpre i , tpost from the mean weight evolution caused by all other spikes. Therefore we insert ⟨y(t)⟩Y |tpost,β=0 = δ(t−tpost)+µ0[1+φ(t−tpost)] and ⟨∆ϵi(t)∆u(t)⟩X|tpre i = wi[ϵ2(t −tpre) + ϵ2νi] into Eq. (16) and decompose ∆wi(tpost −tpre i ) into the following four terms: the drift term ∆w0 i = αβ2 ³ g′ 0 g0 ´2 Tµ0ϵ2wiνi of the batch learning (14) that does not depend on tpre i or tpost; the presynaptic component ∆wpre i = αβ2 ³ g′ 0 g0 ´2 µ0ϵ2wi that is triggered by the presynaptic spike at tpre i ; the postsynaptic component ∆wpost i = αβ2 ³ g′ 0 g0 ´2 h 1 + µ0 R T 0 φ(t −tpost)dt i ϵ2wiνi that is triggered by the postsynaptic spike at tpost; and the correlation component ∆wcorr i = αβ2 µg′ 0 g0 ¶2 wi " ϵ2(tpost −tpre i ) + µ0 Z T 0 φ(t −tpost)ϵ2(t −tpre i )dt # (17) that depends on the difference of the pre- and postsynaptic spike timing. (A) (B) (C) −50 −25 0 25 50 0 0.2 0.4 0.6 0.8 1 PSfrag replacements s [ms] tpost −tpre i [ms] ϵ2(s) µ0(φ ∗ϵ2)(s) ∆wcorr i −50 −25 0 25 50 −0.2 −0.15 −0.1 −0.05 0 0.05 PSfrag replacements s [ms] tpost −tpre i [ms] ϵ2(s) µ0(φ ∗ϵ2)(s) ∆wcorr i −50 −25 0 25 50 −5 0 5 10 15 x 10 −5 PSfrag replacements s [ms] tpost −tpre i [ms] ϵ2(s) µ0(φ ∗ϵ2)(s) ∆wcorr i Figure 2: (A) The effect from EPSP: the first term in the square bracket of (17). (B) The effect from refractoriness: the second term in the square bracket of (17). (C) Temporal learning window ∆wcorr i of (17). In the following, we choose a simple exponential EPSP ϵ(t) = Θ(s)e−s/τu with a time constant τu = 10 ms. The parameters are N = 100, νi = 40 Hz for all i, wi = (Nτuνi)−1, α = 1 and β = 0.1. Figure 2 shows ∆wcorr i of (17). The first term of (17) indicates the contribution of a presynaptic spike at tpre i to increase the online estimation of membrane potential variance at time tpost, whereas the second term represents the effect of the refractory period on postsynaptic firing intensity, i.e., the normalized autocorrelation function convolved with the presynaptic contribution term. Due to the averaging of ⟨·⟩Y |tpost,β=0 and ⟨·⟩X|tpre i in (16), this optimal temporal learning window is local in time; we do not need to impose a memory span [14] to restrict the negative part of the learning window. Figure 3 compares ∆wi of (16) with numerical simulations of (15). We note a good agreement between theory and simulation. We recall, that all calculations, and hence the STDP function of (17) are valid for small β, i.e., for small fluctuation of g. −50 −25 0 25 50 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 x 10 −4 PSfrag replacements tpost −tpre i [ms] tpost −tpre i [ms] ∆wi Figure 3: The comparison of the analytical result of (16) ( solid line ) and the numerical simulation of the online learning rule (15) ( circles ). For the simulation, the conditional average ⟨∆wi⟩X,Y |tpre i ,tpost is evaluated by integrating dwi dt over 200 ms around spike pairs with the given interval tpost −tpre i ; 4 Conclusion It is important for neurons especially in primary sensory systems to send information from previous processing circuits to neurons in other areas while capturing the essential features of its input. Mutual information is a natural quantity to be maximized from this perspective. We introduced an online learning rule for synaptic weights that increases information transmission for small input fluctuation. Introduction of the temporal properties of the target neuron enables us to analyze the temporal properties of the learning rule required to increase the mutual information. Consequently, the temporal learning window is given in terms of the time course of EPSPs and the autocorrelation function of the postsynaptic neuron. In particular, neuronal refractoriness plays a major role and yields the negative part of the learning window. Though we restrict our analysis here to excitatory synapses with independent spike trains, it is straightforward to generalize the approach to a mixture of excitatory and inhibitory neurons with weakly correlated spike trains as long as the synaptic weights are small enough. The analytically derived temporal learning window is similar to the experimentally observed bimodal STDP window [1]. Since the effective time course of EPSPs and the autocorrelation function of output spike trains vary from one part of the brain to another, it is important to compare those functions with the temporal learning window in biological settings. Acknowledgments T.T. is supported by the Japan Society for the Promotion of Science and a Grant-in-Aid for JSPS Fellows; J.-P.P. is supported by the Swiss National Science Foundation. We thank Y. Aviel for discussions. References [1] G. Bi and M. Poo. Synaptic modification of correlated activity: Hebb’s postulate revisited. Annu. Rev. Neurosci., 24:139–166, 2001. [2] W. Gerstner and W. M. Kistler. Spiking Neuron Models. Cambridge University Press, 2002. [3] R. Kempter, W. Gerstner, and J. L. van Hemmen. Hebbian learning and spiking neurons. Phys. Rev. E, 59:4498–4514, 1999. [4] W. Gerstner and W. M. Kistler. Mathematical formulations of hebbian learning. Biol. Cybern., 87:404–415, 2002. [5] R. G¨utig, R. Aharonov, S. Rotter, and H. Sompolinsky. Learning input correlations through nonlinear temporally asymmetric hebbian plasticity. J. Neurosci., 23(9):3697–3714, 2003. [6] R. B. Stein. The information capacity of nerve cells using a frequency code. Biophys. J., 7:797–826, 1967. [7] W. Bialek, F. Rieke, R. de Ruyter van Stevenick, and D. Warland. Reading a neural code. Science, 252:1854–1857, 1991. [8] F. Rieke, D. Warland, R. R. van Steveninck, and W. Bialek. Spikes. MIT Press, 1997. [9] R. Linsker. Self-organization in a perceptual network. Computer, 21:105–117, 1988. [10] J-P. Nadal and N. Parga. Nonlinear neurons in the low-noise limit: a factorial code maximizes information transfer. Network: Comput.Neural Syst., 5:565–581, 1994. [11] J-P Nadal, N. Brunel, and N Parga. Nonlinear feedforward networks with stochastic outputs: infomax implies redundancy reduction. Network: Comput. Neural Syst., 9:207–217, 1998. [12] A. J. Bell and T. Sejnowski. An information-maximization approach to blind separation and blind deconvolution. Neural Comput., 7(6):1004–1034, 1995. [13] J. J. Hopfield. Encoding for computation: recognizing brief dynamical patterns by exploiting effects of weak rhythms on action-potential timing. Proc. Natl. Acad. Sci. USA, 101(16):6255– 6260, 2004. [14] G. Checkik. Spike-timing-dependent plasticity and relevant mutual information maximization. Neural Comput., 15:1481–1510, 2003. [15] V. V. Prelov and E. C. van der Meulen. An asymptotic expression for the information and capacity of a multidimensional channel with weak input signals. IEEE. Trans. Inform. Theory, 39(5):1728–1735, 1993. [16] T. M. Cover and J. A. Thomas. Elements of Information Theory. New York: Wiley, 1991. [17] N. Brunel and J-P. Nadal. Mutual information, fisher information, and population coding. Neural Comput., 10:1731–1757, 1998.
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Approximately Efficient Online Mechanism Design David C. Parkes DEAS, Maxwell-Dworkin Harvard University parkes@eecs.harvard.edu Satinder Singh Comp. Science and Engin. University of Michigan baveja@umich.edu Dimah Yanovsky Harvard College yanovsky@fas.harvard.edu Abstract Online mechanism design (OMD) addresses the problem of sequential decision making in a stochastic environment with multiple self-interested agents. The goal in OMD is to make value-maximizing decisions despite this self-interest. In previous work we presented a Markov decision process (MDP)-based approach to OMD in large-scale problem domains. In practice the underlying MDP needed to solve OMD is too large and hence the mechanism must consider approximations. This raises the possibility that agents may be able to exploit the approximation for selfish gain. We adopt sparse-sampling-based MDP algorithms to implement ϵefficient policies, and retain truth-revelation as an approximate BayesianNash equilibrium. Our approach is empirically illustrated in the context of the dynamic allocation of WiFi connectivity to users in a coffeehouse. 1 Introduction Mechanism design (MD) is concerned with the problem of providing incentives to implement desired system-wide outcomes in systems with multiple self-interested agents. Agents are assumed to have private information, for example about their utility for different outcomes and about their ability to implement different outcomes, and act to maximize their own utility. The MD approach to achieving multiagent coordination supposes the existence of a center that can receive messages from agents and implement an outcome and collect payments from agents. The goal of MD is to implement an outcome with desired system-wide properties in a game-theoretic equilibrium. Classic mechanism design considers static systems in which all agents are present and a one-time decision is made about an outcome. Auctions, used in the context of resourceallocation problems, are a standard example. Online mechanism design [1] departs from this and allows agents to arrive and depart dynamically requiring decisions to be made with uncertainty about the future. Thus, an online mechanism makes a sequence of decisions without the benefit of hindsight about the valuations of the agents yet to arrive. Without the issue of incentives, the online MD problem is a classic sequential decision problem. In prior work [6], we showed that Markov decision processes (MDPs) can be used to define an online Vickrey-Clarke-Groves (VCG) mechanism [2] that makes truth-revelation by the agents (called incentive-compatibility) a Bayesian-Nash equilibrium [5] and implements a policy that maximizes the expected summed value of all agents. This online VCG model differs from the line of work in online auctions, introduced by Lavi and Nisan [4] in that it assumes that the center has a model and it handles a general decision space and any decision horizon. Computing the payments and allocations in the online VCG mechanism involves solving the MDP that defines the underlying centralized (ignoring self-interest) decision making problem. For large systems, the MDPs that need to be solved exactly become large and thus computationally infeasible. In this paper we consider the case where the underlying centralized MDPs are indeed too large and thus must be solved approximately, as will be the case in most real applications. Of course, there are several choices of methods for solving MDPs approximately. We show that the sparse-sampling algorithm due to Kearns et al. [3] is particularly well suited to online MD because it produces the needed local approximations to the optimal value and action efficiently. More challengingly, regardless of our choice the agents in the system can exploit their knowledge of the mechanism’s approximation algorithm to try and “cheat” the mechanism to further their own selfish interests. Our main contribution is to demonstrate that our new approximate online VCG mechanism has truth-revelation by the agents as an ϵ-Bayesian-Nash equilibrium (BNE). This approximate equilibrium supposes that each agent is indifferent to within an expected utility of ϵ, and will play a truthful strategy in bestresponse to truthful strategies of other agents if no other strategy can improve its utility by more than ϵ. For any ϵ, our online mechanism has a run-time that is independent of the number of states in the underlying MDP, provides an ϵ-BNE, and implements a policy with expected value within ϵ of the optimal policy’s value. Our approach is empirically illustrated in the context of the dynamic allocation of WiFi connectivity to users in a coffeehouse. We demonstrate the speed-up introduced with sparsesampling (compared with policy calculation via value-iteration), as well as the economic value of adopting an MDP-based approach over a simple fixed-price approach. 2 Preliminaries Here we formalize the multiagent sequential decision problem that defines the online mechanism design (OMD) problem. The approach is centralized. Each agent is asked to report its private information (for instance about its value and its capabilities) to a central planner or mechanism upon arrival. The mechanism implements a policy based on its view of the state of the world (as reported by agents). The policy defines actions in each state, and the assumption is that all agents acquiesce to the decisions of the planner. The mechanism also collects payments from agents, which can themselves depend on the reports of agents. Online Mechanism Design We consider a finite-horizon problem with a set T of time points and a sequence of decisions k = {k1, . . . , kT }, where kt ∈Kt and Kt is the set of feasible decisions in period t. Agent i ∈I arrives at time ai ∈T, departs at time li ∈T, and has value vi(k) ≥0 for a sequence of decisions k. By assumption, an agent has no value for decisions outside of interval [ai, li]. Agents also face payments, which can be collected after an agent’s departure. Collectively, information θi = (ai, li, vi) defines the type of agent i with θi ∈Θ. Agent types are sampled i.i.d. from a probability distribution f(θ), assumed known to the agents and to the central mechanism. Multiple agents can arrive and depart at the same time. Agent i, with type θi, receives utility ui(k, p; θi) = vi(k; θi) −p, for decisions k and payment p. Agents are modeled as expected-utility maximizers. Definition 1 (Online Mechanism Design) The OMD problem is to implement the sequence of decisions that maximizes the expected summed value across all agents in equilibrium, given self-interested agents with private information about valuations. In economic terms, an optimal (value-maximizing) policy is the allocatively-efficient, or simply the efficient policy. Note that nothing about the OMD models requires centralized execution of the joint plan. Rather, the agents themselves can have capabilities to perform actions and be asked to perform particular actions by the mechanism. The agents can also have private information about the actions that they are able to perform. Using MDPs to Solve Online Mechanism Design. In the MDP-based approach to solving the OMD problem the sequential decision problem is formalized as an MDP with the state at any time encapsulating both the current agent population and constraints on current decisions as reflected by decisions made previously. The reward function in the MDP is simply defined to correspond with the total reported value of all agents for all sequences of decisions. Given types θi ∈f(θ) we define an MDP, Mf, as follows. Define the state of the MDP at time t as the history-vector ht = (θ1, . . . , θt; k1, . . . , kt−1), to include the reported types up to and including period t and the decisions made up to and including period t −1.1 The set of all possible states at time t is denoted Ht. The set of all possible states across all time is H = ST +1 t=1 Ht. The set of decisions available in state ht is Kt(ht). Given a decision kt ∈ Kt(ht) in state ht, there is some probability distribution Prob(ht+1|ht, kt) over possible next states ht+1. In the setting of OMD, this probability distribution is determined by the uncertainty on new agent arrivals (as represented within f(θ)), together with departures and the impact of decision kt on state. The payoff function for the induced MDP is defined to reflect the goal of maximizing the total expected reward across all agents. In particular, payoff Ri(ht, kt) = vi(k≤t; θi) − vi(k≤t−1; θi) becomes available from agent i upon taking action kt in state ht. With this, we have Pτ t=1 Ri(ht, kt) = vi(k≤τ; θi), for all periods τ to provide the required correspondence with agent valuations. Let R(ht, kt) = P i Ri(ht, kt), denote the payoff obtained from all agents at time t. Given a (nonstationary) policy π = {π1, π2, . . . , πT } where πt : Ht →Kt, an MDP defines an MDP-value function V π as follows: V π(ht) is the expected value of the summed payoff obtained from state ht onwards under policy π, i.e., V π(ht) = Eπ{R(ht, π(ht)) + R(ht+1, π(ht+1)) + · · · + R(hT , π(hT ))}. An optimal policy π∗is one that maximizes the MDP-value of every state in H. The optimal MDP-value function V ∗can be computed by value-iteration, and is defined so that V ∗(h) = maxk∈Kt(h)[R(h, k) + P h′∈Ht+1Prob(h′|h, k)V ∗(h′)] for t = T − 1, T −2, . . . , 1 and all h ∈Ht, with V ∗(h ∈HT ) = maxk∈KT (h) R(h, k). Given the optimal MDP-value function, the optimal policy is derived as follows: for t < T, policy π∗(h ∈Ht) chooses action arg maxk∈Kt(h)[R(h, k) + P h′∈Ht+1Prob(h′|h, k)V ∗(h′)] and π∗(h ∈HT ) = arg maxk∈KT (h) R(h, k). Let ˆθ≤t′ denote reported types up to and including period t′. Let Ri ≤t′(ˆθ≤t′; π) denote the total reported reward to agent i up to and including period t′. The commitment period for agent i is defined as the first period, mi, for which ∀t ≥mi, Ri ≤mi(ˆθ≤mi; π) = Ri ≤t(ˆθ≤mi ∪θ′ >mi; π), for any types θ′ >mi still to arrive. This is the earliest period in which agent i’s total value is known with certainty. Let ht′(ˆθ≤t′; π) denote the state in period t′ given reports ˆθ≤t′. Let ˆθ≤t′\i = ˆθ≤t′ \ ˆθi. Definition 2 (Online VCG mechanism) Given history h ∈H, mechanism Mvcg = (Θ; π, pvcg) implements policy π and collects payment, pvcg i (ˆθ≤mi; π) = Ri ≤mi(ˆθ≤mi; π) − h V π(hˆai(ˆθ≤ˆai; π)) −V π(hˆai(ˆθ≤ˆai\i; π)) i (1) from agent i in some period t′ ≥mi. 1Using histories as state will make the state space very large. Often, there will be some function g for which g(h) is a sufficient statistic for all possible states h. We ignore this possibility here. Agent i’s payment is equal to its reported value discounted by the expected marginal value that it will contribute to the system as determined by the MDP-value function for the policy in its arrival period. The incentive-compatibility of the Online VCG mechanism requires that the MDP satisfies stalling which requires that the expected value from the optimal policy in every state in which an agent arrives is at least the expected value from following the optimal action in that state as though the agent had never arrived and then returning to the optimal policy. Clearly, property Kt(ht) ⊇Kt(ht \ θi) in any period t in which θi has just arrived is sufficient for stalling. Stalling is satisfied whenever an agent’s arrival does not force a change in action on a policy. Theorem 1 (Parkes & Singh [6]) An online VCG mechanism, Mvcg = (Θ; π∗, pvcg), based on an optimal policy π∗for a correct MDP model that satisfies stalling is BayesianNash incentive compatible and implements the optimal MDP policy. 3 Solving Very Large MDPs Approximately From Equation 1, it can be seen that making outcome and payment decisions at any point in time in an online VCG mechanism does not require a global value function or a global policy. Unlike most methods for approximately solving MDPs that compute global approximations, the sparse-sampling methods of Kearns et al. [3] compute approximate values and actions for a single state at a time. Furthermore, sparse-sampling methods provide approximation guarantees that will be important to establish equilibrium properties — they can compute an ϵ-approximation to the optimal value and action in a given state in time independent of the size of the state space (though polynomial in 1 ϵ and exponential in the time horizon). Thus, sparse-sampling methods are particularly suited to approximating online VCG and we adopt them here. The sparse-sampling algorithm uses the MDP model Mf as a generative model, i.e., as a black box from which a sample of the next-state and reward distributions for any given state-action pair can be obtained. Given a state s and an approximation parameter ϵ, it computes an ϵ-accurate estimate of the optimal value for s as follows. We make the parameterization on ϵ explicit by writing sparse-sampling(ϵ). The algorithm builds out a depth-T sampled tree in which each node is a state and each node’s children are obtained by sampling each action in that state m times (where m is chosen to guarantee an ϵ approximation to the optimal value of s), and each edge is labeled with the sample reward for that transition. The algorithm computes estimates of the optimal value for nodes in the tree working backwards from the leaves as follows. The leaf-nodes have zero value. The value of a node is the maximum over the values for all actions in that node. The value of an action in a node is the summed value of the m rewards on the m outgoing edges for that action plus the summed value of the m children of that node. The estimated optimal value of state s is the value of the root node of the tree. The estimated optimal action in state s is the action that leads to the largest value for the root node in the tree. Lemma 1 (Adapted from Kearns, Mansour & Ng [3]) The sparse-sampling(ϵ) algorithm, given access to a generative model for any n-action MDP M, takes as input any state s ∈S and any ϵ > 0, outputs an action, and satisfies the following two conditions: • (Running Time) The running time of the algorithm is O((nC)T ), where C = f ′(n, 1 ϵ , Rmax, T) and f ′ is a polynomial function of the approximation parameter 1 ϵ , the number of actions n, the largest expected reward in a state Rmax and the horizon T. In particular, the running time has no dependence on the number of states. • (Near-Optimality) The value function of the stochastic policy implemented by the sparse-sampling(ϵ) algorithm, denoted V ss, satisfies |V ∗(s) −V ss(s)| ≤ϵ simultaneously for all states s ∈S. It is straightforward to derive the following corollary from the proof of Lemma 1 in [3]. Corollary 1 The value function computed by the sparse-sampling(ϵ) algorithm, denoted ˆV ss, is near-optimal in expectation, i.e., |V ∗(s) −E{ ˆV ss(s)}| ≤ϵ simultaneously for all states s ∈S and where the expectation is over the randomness introduced by the sparsesampling(ϵ) algorithm. 4 Approximately Efficient Online Mechanism Design In this section, we define an approximate online VCG mechanism and consider the effect on incentives of substituting the sparse-sampling(ϵ) algorithm into the online VCG mechanism. We model agents as indifferent between decisions that differ by at most a utility of ϵ > 0, and consider an approximate Bayesian-Nash equilibrium. Let vi(θ; π) denote the final value to agent i after reports θ given policy π. Definition 3 (approximate BNE) Mechanism Mvcg = (Θ, π, pvcg) is ϵ-Bayesian-Nash incentive compatible if Eθ|θ≤t′ {vi(θ; π) −pvcg i (θ; π)} + ϵ ≥ Eθ|θ≤t′{vi(θ−i, ˆθi; π) −pvcg i (θ−i, ˆθi; π)}(2) where agent i with type θi arrives in period t′, and with the expectation taken over future types given current reports θ≤t′. In particular, when truth-telling is an ϵ-BNE we say that the mechanism is ϵ-BNE incentive compatible and no agent can improve its expected utility by more than ϵ > 0, for any type, as long as other agents are bidding truthfully. Equivalently, one can interpret an ϵ-BNE as an exact equilibrium for agents that face a computational cost of at least ϵ to compute the exact BNE. Definition 4 (approximate mechanism) A sparse-sampling(ϵ) based approximate online VCG mechanism, Mvcg(ϵ) = (Θ; ˜π, ˜pvcg), uses the sparse-sampling(ϵ) algorithm to implement stochastic policy ˜π and collects payment ˜pvcg i (ˆθ≤mi; ˜π) = Ri ≤mi(ˆθ≤mi; ˜π) − h ˆV ss(hˆai(ˆθ≤ˆai; ˜π)) −ˆV ss(hˆai(ˆθ≤ˆai\i; ˜π)) i from agent i in some period t′ ≥mi, for commitment period mi. Our proof of incentive-compatibility first demonstrates that an approximate delayed VCG mechanism [1, 6] is ϵ-BNE. With this, we demonstrate that the expected value of the payments in the approximate online VCG mechanism is within 3ϵ of the payments in the approximate delayed VCG mechanism. The delayed VCG mechanism makes the same decisions as the online VCG mechanism, except that payments are delayed until the final period and computed as: pDvcg i (ˆθ; π) = Ri ≤T (ˆθ; π) − h R≤T (ˆθ; π) −R≤T (ˆθ−i; π) i (3) where the discount is computed ex post, once the effect of an agent on the system value is known. In an approximate delayed-VCG mechanism, the role of the sparse-sampling algorithm is to implement an approximate policy, as well as counterfactual policies for the worlds θ−i without each agent i in turn. The total reported reward to agents ̸= i over this counterfactual series of states is used to define the payment to agent i. Lemma 2 Truthful bidding is an ϵ-Bayesian-Nash equilibrium in the sparse-sampling(ϵ) based approximate delayed-VCG mechanism. Proof: Let ˜π denote the approximate policy computed by the sparse-sampling algorithm. Assume agents ̸= i are truthful. Now, if agent i bids truthfully its expected utility is Eθ|θ≤ai{vi(θ; ˜π) + X j̸=i Rj ≤T (θ; ˜π) − X j̸=i Rj ≤T (θ−i; ˜π)} (4) where the expectation is over both the types yet to be reported and the randomness introduced by the sparse-sampling(ϵ) algorithm. Substituting R<ai(θ<ai; ˜π) + V ss(hai(θ≤ai; ˜π)) for the first two terms in Equation (4) and R<ai(θ<ai; ˜π) + V ss(hai(θ≤ai\i; ˜π)) for the third term, then its expected utility is at least V ∗(hai(θ≤ai; ˜π)) −V ss(hai(θ≤ai\i; ˜π)) −ϵ (5) because V ss(hai(θ≤ai; ˜π)) ≥V ∗(hai(θ≤ai; ˜π)) −ϵ by Lemma 1. Now, ignore term R≤T (θ−i; ˜π) in Equation (4), which is independent of agent i’s bid ˆθi, and consider the maximal expected utility to agent i from some non-truthful bid. The effect of ˆθi on the first two terms is indirect, through a change in the policy for periods ≥ai. An agent can change the policy only indirectly, by changing the center’s view of the state by misreporting its type. By definition, the agent can do no better than selecting optimal policy π∗, which is defined to maximize the expected value of the first two terms. Putting this together, the expected utility from ˆθi is at most V ∗(hai(θ≤ai; ˜π)) −V ss(hai(θ≤ai\i; ˜π)) and at most ϵ better than that from bidding truthfully. Theorem 2 Truthful bidding is an 4ϵ-Bayesian-Nash equilibrium in the sparsesampling(ϵ) based approximate online VCG mechanism. Proof: Assume agents ̸= i bid truthfully, and consider report ˆθi. Clearly, the policy implemented in the approximate online-VCG mechanism is the same as in the delayedVCG mechanism for all ˆθi. Left to show is that the expected value of the payments are within 3ϵ for all ˆθi. From this we conclude that the expected utility to agent i in the approximate-VCG mechanism is always within 3ϵ of that in the approximate delayed-VCG mechanism, and therefore 4ϵ-BNE by Lemma 2. The expected payment in the approximate online VCG mechanism is Eθ|θ≤ai{Ri ≤T (ˆθ; ˜π)} − h E{ ˆV ss(hˆai(ˆθ≤ˆai; ˜π)} −E{ ˆV ss(hˆai(ˆθ≤ˆai\i; ˜π)} i The value function computed by the sparse-sampling(ϵ) algorithm is a random variable to agent i at the time of bidding, and the second and third expectations are over the randomness introduced by the sparse-sampling(ϵ) algorithm. The first term is the same as in the payment in the approximate delayed-VCG mechanism. By Corollary 1, the value function estimated in the sparse-sampling(ϵ) is near-optimal in expectation and the total of the second two terms is at least V ∗(hˆai(ˆθ≤ˆai\i; π∗)) −V ∗(hˆai(ˆθ≤ˆai; π∗)) −2ϵ. Ignoring the first term in pDvcg i , the expected payment in the approximate delayed-VCG mechanism is no more than V ∗(hˆai(ˆθ≤ˆai\i; π∗)) −(V ∗(hˆai(ˆθ≤ˆai; π∗)) −ϵ) because of the near-optimality of the value function of the stochastic policy (Lemma 1). Putting this together, we have a maximum difference in expected payments of 3ϵ. Similar analysis yields a maximum difference of 3ϵ when an upper-bound is taken on the payment in the online VCG mechanism and compared with a lower-bound on the payment in the delayed mechanism. Theorem 3 For any parameter ϵ > 0, the sparse-sampling(ϵ) based approximate online VCG mechanism has ϵ-efficiency in an 4ϵ-BNE. 5 Empirical Evaluation of Approximate Online VCG The WiFi Problem. The WiFi problem considers a fixed number of channels C with a random number of agents (max A) that can arrive per period. The time horizon T = 50. Agents demand a single channel and arrive with per-unit value, distributed i.i.d. V = {v1, . . . , vk} and duration in the system, distributed i.i.d. D = {d1, . . . , dl}. The value model requires that any allocation to agent i must be for contiguous periods, and be made while the agent is present (i.e., during periods [t, ai + di], for arrival ai and duration di). An agent’s value for an allocation of duration x is vix where vi is its per-unit value. Let dmax denote the maximal possible allocated duration. We define the following MDP components: State space: We use the following compact, sufficient, statistic of history: a resource schedule is a (weakly non-decreasing) vector of length dmax that counts the number of channels available in the current period and next dmax −1 periods given previous actions (C channels are available after this); an agent vector of size (k × l) that provides a count of the number of agents present but not allocated for each possible per-unit value and each possible duration (the duration is automatically decremented when an agent remains in the system for a period without receiving an allocation); the time remaining until horizon T. Action space: The policy can postpone an agent allocation, or allocate an agent to a channel for the remaining duration of the agent’s time in the system if a channel is available, and the remaining duration is not greater than dmax. Payoff function: The reward at a time step is the summed value obtained from all agents for which an allocation is made in this time step. This is the total value such an agent will receive before it departs. Transition probabilities: The change in resource schedule, and in the agent vector that relates to agents currently present, is deterministic. The random new additions to the agent vector at each step are unaffected by the actions and also independent of time. Mechanisms. In the exact online VCG mechanism we compute an optimal policy, and optimal MDP values, offline using finite-horizon value iteration [7]. In the sparsesampling(ϵ) approach, we define a sampling tree depth L (perhaps < T) and sample each state m times. This limited sampling depth places a lower-bound on the best possible approximation accuracy from the sparse-sampling algorithm. We also employ agent pruning, with the agent vector in the state representation pruned to remove dominated agents: consider agent type with duration d and value v and remove all but C −N agents where N is the number of agents that either have remaining duration ≤d and value > v or duration < d and value ≥v. In computing payments we use factoring, and only determine VCG payments once for each type of agent to arrive. We compare performance with a simple fixed-price allocation scheme that given a particular problem, computes off-line a fixed number of periods and a fixed price (agents are queued and offered the price at random as resources become available) that yields the maximum expected total value. Results In the default model, we set C = 5, A = 5, define the set of values V = {1, 2, 3}, define the set of durations D = {1, 2, 6}, with lookahead L = 4 and sampling width m = 6. All results are averaged over at least 10 instances, and experiments were performed on a 3GHz P4, with 512 MB RAM. Value and revenue is normalized by the total value demanded by all agents, i.e. the value with infinite capacity.2 Looking first at economic properties, Figure 1(A) shows the effect of varying the number of agents from 2 to 12, comparing the value and revenue between the approximate online VCG mechanism and the fixed price mechanism. Notice that the MDP method dominates the price-based scheme for value, with a notable performance improvement over fixed price when demand is neither very low (no contention) nor very high (lots of competition). Revenue is also generally better from the MDP-based mechanism than in the fixed price scheme. Fig. 1(B) shows the similar effect of varying the number of channels from 3 to 10. Turning now to computational properties, Figure 1 (C) illustrates the effectiveness of sparse-sampling, and also agent pruning, sampled over 100 instances. The model is very 2This explains why the value appears to drop as we scale up the number of agents— the total available value is increasing but supply remains fixed. 2 4 6 8 10 12 20 40 60 80 Number of agents % value:mdp rev:mdp value:fixed rev:fixed 3 4 5 6 7 8 9 10 0 20 40 60 80 100 Number of channels % value:mdp rev:mdp value:fixed rev:fixed 2 4 6 8 10 88 90 92 94 96 98 Sampling Width % of Exact Value % of Exact Time 0 0.2 0.4 0.6 0.8 1.0 time:pruning time:no pruning value:pruning value:no pruning vs. #agents vs. #agents (no pruning) vs. #channels 2 4 6 8 10 12 0 100 200 300 400 500 600 Number of Agents Run time (s) time:pruning time:no pruning Figure 1: (A) Value and Revenue vs. Number of Agents. (B) Value and Revenue vs. Number of Channels. (C) Effect of Sampling Width. (D) Pruning speed-up. small: A = 2, C = 2, D = {1, 2, 3}, V = {1, 2, 3} and L = 4, to allow a comparison with the compute time for an optimal policy. The sparse-sampling method is already running in less than 1% of the time for optimal value-iteration (right-hand axis), with an accuracy as high as 96% of the optimal. Pruning provides an incremental speed-up, and actually improves accuracy, presumably by making better use of each sample. Figure 1 (D) shows that pruning is extremely useful computationally (in comparison with plain sparsesampling), for the default model parameters and as the number of agents is increased from 2 to 12. Pruning is effective, removing around 50% of agents (summed across all states in the lookahead tree) at 5 agents. Acknowledgments. David Parkes was funded by NSF grant IIS-0238147. Satinder Singh was funded by NSF grant CCF 0432027 and by a grant from DARPA’s IPTO program. References [1] Eric Friedman and David C. Parkes. Pricing WiFi at Starbucks– Issues in online mechanism design. In Fourth ACM Conf. on Electronic Commerce (EC’03), pages 240–241, 2003. [2] Matthew O. Jackson. Mechanism theory. In The Encyclopedia of Life Support Systems. EOLSS Publishers, 2000. [3] Michael Kearns, Yishay Mansour, and Andrew Y Ng. A sparse sampling algorithm for nearoptimal planning in large Markov Decision Processes. In Proc. 16th Int. Joint Conf. on Artificial Intelligence, pages 1324–1331, 1999. To appear in Special Issue of Machine Learning. [4] Ron Lavi and Noam Nisan. Competitive analysis of incentive compatible on-line auctions. In Proc. 2nd ACM Conf. on Electronic Commerce (EC-00), 2000. [5] Martin J Osborne and Ariel Rubinstein. A Course in Game Theory. MIT Press, 1994. [6] David C. Parkes and Satinder Singh. An MDP-based approach to Online Mechanism Design. In Proc. 17th Annual Conf. on Neural Information Processing Systems (NIPS’03), 2003. [7] M L Puterman. Markov decision processes: Discrete stochastic dynamic programming. John Wiley & Sons, New York, 1994.
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Comparing Beliefs, Surveys and Random Walks Erik Aurell SICS, Swedish Institute of Computer Science P.O. Box 1263, SE-164 29 Kista, Sweden and Dept. of Physics, KTH – Royal Institute of Technology AlbaNova – SCFAB SE-106 91 Stockholm, Sweden eaurell@sics.se Uri Gordon and Scott Kirkpatrick School of Engineering and Computer Science Hebrew University of Jerusalem 91904 Jerusalem, Israel {guri,kirk}@cs.huji.ac.il Abstract Survey propagation is a powerful technique from statistical physics that has been applied to solve the 3-SAT problem both in principle and in practice. We give, using only probability arguments, a common derivation of survey propagation, belief propagation and several interesting hybrid methods. We then present numerical experiments which use WSAT (a widely used random-walk based SAT solver) to quantify the complexity of the 3-SAT formulae as a function of their parameters, both as randomly generated and after simpli£cation, guided by survey propagation. Some properties of WSAT which have not previously been reported make it an ideal tool for this purpose – its mean cost is proportional to the number of variables in the formula (at a £xed ratio of clauses to variables) in the easy-SAT regime and slightly beyond, and its behavior in the hardSAT regime appears to re¤ect the underlying structure of the solution space that has been predicted by replica symmetry-breaking arguments. An analysis of the tradeoffs between the various methods of search for satisfying assignments shows WSAT to be far more powerful than has been appreciated, and suggests some interesting new directions for practical algorithm development. 1 Introduction Random 3-SAT is a classic problem in combinatorics, at the heart of computational complexity studies and a favorite testing ground for both exactly analyzable and heuristic solution methods which are then applied to a wide variety of problems in machine learning and arti£cial intelligence. It consists of a ensemble of randomly generated logical expressions, each depending on N Boolean variables xi, and constructed by taking the AND of M clauses. Each clause a consists of the OR of 3 “literals” yi,a. yi,a is taken to be either xi or ¬xi at random with equal probability, and the three values of the index i in each clause are distinct. Conversely, the neighborhood of a variable xi is Vi, the set of all clauses in which xi or ¬xi appear. For each such random formula, one asks whether there is some set of xi values for which the formula evaluates to be TRUE. The ratio α = M/N controls the dif£culty of this decision problem, and predicts the answer with high accuracy, at least as both N and M tend to in£nity, with their ratio held constant. At small α, solutions are easily found, while for suf£ciently large α there are almost certainly no satisfying con£gurations of the xi, and compact proofs of this fact can be constructed. Between these limits lies a complex, spin-glass-like phase transition, at which the cost of analyzing the problem with either exact or heuristic methods explodes. A recent series of papers drawing upon the statistical mechanics of disordered materials has not only clari£ed the nature of this transition, but also lead to a thousand-fold increase in the size of the concrete problems that can be solved [1, 2, 3] This paper provides a derivation of the new methods using nothing more complex than probabilities, suggests some generalizations, and reports numerical experiments that disentangle the contributions of the several component heuristics employed. For two related discussions, see [4, 5]. An iterative ”belief propagation” [6] (BP) algorithm for K-SAT can be derived to evaluate the probability, or ”belief,” that a variable will take the value TRUE in variable con£gurations that satisfy the formula considered. To calculate this, we £rst de£ne a message (”transport”) sent from a variable to a clause: • ti→a is the probability that variable xi satis£es clause a In the other direction, we de£ne a message (”in¤uence”) sent from a clause to a variable: • ia→i is the probability that clause a is satis£ed by another variable than xi In 3-SAT, where clause a depends on variables xi, xj and xk, BP gives the following iterative update equation for its in¤uence. i(l) a→i = t(l) j→a + t(l) k→a −t(l) j→at(l) k→a (1) The BP update equations for the transport ti→a involve the products of in¤uences acting on a variable from the clauses which surround xi, forming its ”cavity,” Vi, sorted by which literal (xi or ¬xi) appears in the clause: A0 i = Y b∈Vi, yi,b=¬xi ib→i and A1 i = Y b∈Vi, yi,b=xi ib→i (2) The update equations are then t(l) i→a = i(l−1) a→i A1 i i(l−1) a→i A1 i +A0 i if yi,a = ¬xi i(l−1) a→i A0 i i(l−1) a→i A0 i +A1 i if yi,a = xi (3) The superscripts (l) and (l −1) denote iteration. The probabilistic interpretation is the following: suppose we have i(l) b→i for all clauses b connected to variable i. Each of these clauses can either be satis£ed by another variable (with probability i(l) b→i), or not be satis£ed by another variable (with probability ³ 1 −i(l) b→i ´ ), and also be satis£ed by variable i itself. If we set variable xi to 0, then some clauses are satis£ed by xi, and some have to be satis£ed by other variables. The probability that they are all satis£ed is Q b̸=a,yi,b=xi i(l) b→i. Similarly, if xi is set to 1 then all these clauses b are satis£ed with probability Q b̸=a,yi,b=¬xi i(l) b→i. The products in (3) can therefore be interpreted as joint probabilities of independent events. Variable xi can be 0 or 1 in a solution if the clauses in which xi appears are either satis£ed directly by xi itself, or by other variables. Hence Prob(xi) = A0 i A0 i + A1 i and Prob(¬xi) = A1 i A0 i + A1 i (4) A BP-based decimation scheme results from £xing the variables with largest probability to be either true or false. We then recalculate the beliefs for the reduced formula, and repeat. To arrive at SP we introduce a modi£ed system of beliefs: every variable falls into one of three classes: TRUE in all solutions (1); FALSE in all solutions(0); and TRUE in some and FALSE in other solutions (free). The message from a clause to a variable (an in¤uence) is then the same as in BP above. Although we will again only need to keep track of one message from a variable to a clause (a transport), it is convenient to £rst introduce three ancillary messages: • ˆTi→a(1) is the probability that variable xi is true in clause a in all solutions • ˆTi→a(0) is the probability that variable xi is false in clause a in all solutions • ˆTi→a(free) is the probability that variable xi is true in clause a in some solutions and false in others. Note that there are here three transports for each directed link i →a, from a variable to a clause, in the graph. As in BP, these numbers will be functions of the in¤uences from clauses to variables in the preceeding update step. Taking again the incoming in¤uences independent, we have ˆT (l) i→a(free) ∝ Q b∈Vi\a i(l−1) b→i ˆT (l) i→a(0) + ˆT (l) i→a(free) ∝ Q b∈Vi\a,yi,b=xi i(l−1) b→i ˆT (l) i→a(1) + ˆT (l) i→a(free) ∝ Q b∈Vi\a,yi,b=¬xi i(l−1) b→i (5) The proportionality indicates that the probabilities are to be normalized. We see that the structure is quite similar to that in BP. But we can make it closer still by introducing ti→a with the same meaning as in BP. In SP it will then, as the case might be, be equal to to Ti→a(free) + Ti→a(0) or Ti→a(free) + Ti→a(1). That gives (compare (3)): t(l) i→a = i(l−1) a→i A1 i i(l−1) a→i A1 i +A0 i −A1 i A0 i if yi,a = ¬xi i(l−1) a→i A0 i i(l−1) a→i A0 i +A1 i −A1 i A0 i if yi,a = xi (6) The update equations for ti→a are the same in SP as in BP, ´.e. one uses (1) in SP as well. Similarly to (4), decimation now removes the most £xed variable, i.e. the one with the largest absolute value of (A0 i −A1 i )/(A0 i + A1 i −A1 i A0 i ). Given the complexity of the original derivation of SP [1, 2], it is remarkable that the SP scheme can be interpreted as a type of belief propagation in another belief system. And even more remarkable that the £nal iteration formulae differ so little. A modi£cation of SP which we will consider in the following is to interpolate between BP 3.5 3.6 3.7 3.8 3.9 4 4.1 4.2 4.3 4.4 0 0.2 0.4 0.6 0.8 1 1.2 α = M/N Fraction of sites remaining after decimation ρ=1.05 ρ=1 ρ=0.95 ρ=0 Figure 1: Dependence of decimation depth on the interpolation parameter ρ. (ρ = 0) and SP (ρ = 1) 1 by considering equations t(l) i→a i(l−1) a→i A1 i i(l−1) a→i A1 i +A0 i −ρA1 i A0 i if yi,a = ¬xi i(l−1) a→i A0 i i(l−1) a→i A0 i +A1 i −ρA1 i A0 i if yi,a = xi (7) We do not have an interpretation of the intermediate cases of ρ as belief systems. 2 The Phase Diagram of 3-SAT Early work on developing 3-SAT heuristics discovered that as α is increased, the problem changes from being easy to solve to extremely hard, then again relatively easy when the formulae are almost certainly UNSAT. It was natural to expect that a sharp phase boundary between SAT and UNSAT phases in the limit of large N accompanies this “easy-hard-easy” observed transition, and the £nite-size scaling results of [7] con£rmed this. Their work placed the transition at about α = 4.2. Monasson and Zecchina [8] soon showed, using the replica method from statistical mechanics, that the phase transition to be expected had unusual characteristics, including “frozen variables” and a highly nonuniform distribution of solutions, making search dif£cult. Recent technical advances have made it possible to use simpler cavity mean £eld methods to pinpoint the SAT/UNSAT boundary at α = 4.267 and suggest that the “hard-SAT” region in which the solution space becomes inhomogeneous begins at about α = 3.92. These calculations also predicted a speci£c solution structure (termed 1-RSB for “one step replica symmetry-breaking”) [1, 2] in which the satis£able con£gurations occur in large clusters, maximally separated from each other. Two types of frozen variables are predicted, one set which take the same value in all clusters and a second set whose value is £xed within a particular cluster. The remaining variables are “paramagnetic” and can take either value in some of the states of a given cluster. A careful analysis of the 1-RSB solution has subsequently shown that this extreme structure is only stable above α = 4.15. Between 3.92 and 4.15 a wider range of cluster sizes, and wide range of inter-cluster Hamming distances are expected.[9] As a result, we expect the values α = 3.9, 4.15 and 4.267 to separate regions in which the nature of the 3-SAT decision problem is distinctly different. 1This interpolation has also been considered and implemented by R. Zecchina and co-workers. “Survey-induced decimation” consists of using SP to determine the variable most likely to be frozen, then setting that variable to the indicated frozen value, simplifying the formula as a result, updating the SP calculation, and repeating the process. For α < 3.9 we expect SP to discover that all spins are free to take on more than one value in some ground state, so no spins will be decimated. Above 3.9, SP ideally should identify frozen spins until all that remain are paramagnetic. The depth of decimation, or fraction of spins reminaing when SP sees only paramagnetic spins, is thus an important characteristic. We show in Fig. 1 the fraction of spins remaining after survey-induced decimation for values of α from 3.85 to 4.35 in hundreds of formulae with N = 10, 000. The error bars show the standard deviation, which becomes quite large for large values of α. To the left of α = 4.2, on the descending part of the curves, SP reaches a paramagnetic state and halts. On the right, or ascending portion of the curves, SP stops by simply failing to converge. Fig 1 also shows how different the behavior of BP and the hybrids between BP and SP are in their decimation behavior. We studied BP (ρ = 0), underrelaxed SP (ρ = 0.95), SP, and overrelaxed SP (ρ = 1.05). BP and underrelaxed SP do not reach a paramagnetic state, but continue until the formula breaks apart into clauses that have no variables shared between them. We see in Fig. 1 that BP stops working at roughly α = 3.9, the point at which SP begins to operate. The underrelaxed SP behaves like BP, but can be used well into the RSB region. On the rising parts of all four curves in Fig 1, the scheme halted as the surveys ceased to converge. Overrelaxed SP in Fig. 1 may give reasonable recommendations for simpli£cation even on formulae which are likely to be UNSAT. 3 Some Background on WSAT Next we consider WSAT, the random walk-based search routine used to £nish the job of exhibiting a satisfying con£guration after SP (or some other decimation advisor) has simpli£ed the formula. The surprising power exhibited by SP has to some extent obscured the fact that WSAT is itself a very powerful tool for solving constraint satisfaction problems, and has been widely used for this. Its running time, expressed in the number of walk steps required for a successful search is also useful as an informal de£nition of the complexity of a logical formula. Its history goes back to Papadimitriou’s [10] observation that a subtly biased random walk would with high probability discover satisfying solutions in the simpler 2-SAT problem after, at worst, O(N 2) steps. His procedure was to start with an arbitary assignment of values to the binary variables, then reverse the sign of one variable at a time using the following random process: • select an unsatis£ed clause at random • select at random a variable that appears in the clause • reverse that variable This procedure, sometimes called RWalkSAT, works because changing the sign of a variable in an unsatis£ed clause always satis£es that clause and, at £rst, has no net effect on other clauses. It is much more powerful than was proven initially. Two recent papers [12, 13]. have argued analytically and shown experimentally that Rwalksat £nds satisfying con£gurations of the variables after a number of steps that is proportional to N for values of α up to roughly 2.7. after which this cost increases exponentially with N. The second trick in WSAT was introduced by Kautz and Selman [11]. They also choose an unsatis£ed clause at random, but then reverse one of the “best” variables, selected at random, where “best” is de£ned as causing the fewest satis£ed clauses to become unsatis£ed. For robustness, they mix this greedy move with random moves as used in RWalkSAT, recommending an equal mixture of the two types of moves. Barthel et al.[13] used these two moves in numerical experiments, but found little improvement over RWalkSAT. 1 2 3 4 10 0 10 2 10 4 Median Cost per variable α N=1000 N=2000 N=5000 N=10000 N=20000 0 1 2 3 4 5 10 −5 10 0 10 5 10 10 10 15 Variance of Cost per variable x N α N=1000 N=2000 N=5000 N=10000 N=20000 Figure 2: (a) Median of WSAT cost per variable in 3-SAT as a function of α. (b) Variance of WSAT cost, scaled by N. There is a third trick in the most often used variant of WSAT, introduced slightly later [14]. If any variable in the selected unsatis£ed clause can be reversed without causing any other clauses to become unsatis£ed, this “free” move is immediately accepted and no further exploration is required. Since we shall show that WSAT works well above α = 2.7, this third move apparently gives WSAT its extra power. Although these moves were chosen by the authors of WSAT after considerable experiment, we have no insight into why they should be the best choices. In Fig. 2a, we show the median number of random walk steps per variable taken by the standard version of WSAT to solve 3-SAT formulas at values of α ranging from 0.5 to 4.3 and for formulae of sizes ranging from N = 1000 to N = 20000. The cost of WSAT remains linear in N well above α = 3.9. WSAT cost distributions were collected on at least 1000 cases at each point. Since the distributions are asymmetric, with strong tails extending to higher cost, it is not obvious that WSAT cost is, in the statistical mechanics language, self-averaging, or concentrated about a well-de£ned mean value which dominates the distribution as N →∞. To test this, we calculated higher moments of the WSAT cost distribution and found that they scale with simple powers of N. For example, in Fig. 2b, we show that the variance of the WSAT cost per variable, scaled up by N, is a wellde£ned function of α up to almost 4.2. The third and fourth moments of the distribution (not shown) also are constant when multiplied by N and by N 2, respectively. The WSAT cost per variable is thus given by a distribution which concentrates with increasing N in exactly the way that a process governed by the usual laws of large numbers is expected to behave, even though the typical cost increases by six orders of magnitude as we move from the trivial cases to the critical regime. A detailed analysis of the cost distributions which we observed will be published elsewhere but we conclude that the median cost of solving 3-SAT using the WSAT random walk search, as well as the mean cost if that is well-de£ned, remains linear in N up to α = 4.15, coincidentally the onset of 1-RSB. In the 1-RSB regime, the WSAT cost per variable distributions shift to higher values as N increases, and an exponential increase in cost with N is likely. Is 4.15 really the endpoint for WSAT’s linearity, or will the search cost per variable converge at still larger values of N which we could not study? We de£ne a rough estimate of Nonset(α) by study of the cumulative distributions of WSAT cost as the value of N for a given α above which the distributions cross at a £xed percentile. Plotting log(Nonset) against log(4.15 −α) in Fig. 3, we £nd strong indication that 4.15 is indeed an asymptote for WSAT. 100 1000 10000 100000 0.01 0.1 1 10 N onset (4.15 - M/N) Onset for linear WSAT cost per variable 3.4 3.6 3.8 4 4.2 10 0 10 5 Median WalkSat Cost α N=1000 N=2000 N=5000 N=10000 N=20000 Figure 3: Size N at which WSAT cost is linear in N as function of 4.15 −α. Figure 4: WSAT cost, before and after SP-guided decimation. 4 Practical Aspects of SP + WSAT The power of SP comes from its use to guide decimation by identifying spins which can be frozen while minimally reducing the number of solutions that can be constructed. To assess the complexity of the reduced formulae that decimation guided in this way produces we compare, in Fig. 4, the median number of WSAT steps required to £nd a satisfying con£guration of the variables before and after decimation. To a rough approximation, we can say that SP caps the cost of £nding a solution to what it would be at the entry to the critical regime. There are two factors, the reduction in the number of variables that have to be searched, and the reduction of the distance the random walk must traverse when it is restricted to a single cluster of solutions. In Fig. 2c the solid lines show the WSAT costs divided by N, the original number of variables in each formula. If we instead divide the WSAT cost after decimation by the number of variables remaining, the complexity measure that we obtain is only a factor of two larger, as shown by the dotted lines. The relative cost of running WSAT without bene£t of decimation is 3-4 decades larger. We measured the actual compute time consumed in survey propagation and in WSAT. For this we used the Zecchina group’s version 1.3 survey propagation code, and the copy of WSAT (H. Kautz’s release 35, see [15]) that they have also employed. All programs were run on a Pentium IV Xeon 3GHz dual processor server with 4GB of memory, and only one processor busy. We compare timings from runs on the same 100 formulas with N = 10000 and α = 4.1 and 4.2 (the formulas are simply extended slightly for the second case). In the £rst case, the 100 formulas were solved using WSAT alone in 921 seconds. Using SP to guide decimation one variable at a time, with the survey updates performed locally around each modi£ed variable, the same 100 formulas required 6218 seconds to solve, of which only 31 sec was spent in WSAT. When we increase alpha to 4.2, the situation is reversed. Running WSAT on 100 formulas with N = 10000 required 27771 seconds on the same servers, and would have taken even longer if about half of the runs had not been stopped by a cutoff without producing a satisfying con£guration. In contrast, the same 100 formulas were solved by SP followed with WSAT in 10,420 sec, of which only 300 seconds were spent in WSAT. The cost of SP does not scale linearly with N, but appears to scale as N 2 in this regime. We solved 100 formulas with N = 20, 000 using SP followed by WSAT in 39643 seconds, of which 608 sec was spent in WSAT. The cost of running SP to decimate roughly half the spins has quadrupled, while the cost of the £nal WSAT runs remained proportional to N. Decimation must stop short of the paramagnetic state at the highest values of α, to avoid having SP fail to converge. In those cases we found that WSAT could sometimes £nd satisfying con£gurations if started slightly before this point. We also explored partial decimation as a means of reducing the cost of WSAT just below the 1-RSB regime, but found that decimation of small fractions of the variables caused the WSAT running times to be highly unpredictable, in many cases increasing strongly. As a result, partial decimation does not seem to be a useful approach. 5 Conclusions and future work The SP and related algorithms are quite new, so programming improvements may modify the practical conclusions of the previous section. However, a more immediate target for future work could be the WSAT algorithms. Further directing its random choices to incorporate the insights gained from BP and SP might make it an effective algorithm even closer to the SAT/UNSAT transition. Acknowledgments We have enjoyed discussions of this work with members of the replica and cavity theory community, especially Riccardo Zecchina, Alfredo Braunstein, Marc Mezard, Remi Monasson and Andrea Montanari. This work was performed in the framework of EU/FP6 Integrated Project EVERGROW (www.evergrow.org), and in part during a Thematic Institute supported by the EXYSTENCE EU/FP5 network of excellence. E.A. acknowledges support from the Swedish Science Council. S.K. and U.G. are partially supported by a US-Israeli Binational Science Foundation grant. References [1] M´ezard M., Parisi G. & Zecchina R.. (2002) Analytic and Algorithmic Solutions of Random Satis£ability Problems. Science, 297:812-815 [2] M´ezard M. & Zecchina R. (2002) The random K-satis£ability problem: from an analytic solution to an ef£cient algorithm. Phys. Rev. E 66: 056126. [3] Braunstein A., Mezard M. & Zecchina R., “Survey propagation: an algorithm for satis£ability”, arXiv:cs.CC/0212002 (2002). [4] Parisi G. (2003), On the probabilistic approach to the random satis£ability problem, Proc. SAT 2003 and arXiv:cs:CC/0308010v1 . [5] Braunstein A. and Zecchina R., (2004) Survey Propagation as Local Equilibrium Equations. arXiv:cond-mat/0312483 v5. [6] Pearl J. (1988) Probabilistic Reasoning in Intelligent Systems, 2nd Edition, Kauffmann. [7] Kirkpatrick S. & Selman B. (1994) Critical Behaviour in the Sati£ability of Random Boolean Expressions. Science 264: 1297-1301. [8] Monasson R. & Zecchina R. (1997) Statistical mechanics of the random K-Sat problem. Phys. Rev. E 56: 1357–1361. [9] Montanari A., Parisi G. & Ricci-Tersenghi F. (2003) Instability of one-step replica-symmetricbroken phase in satis£ability problems. cond-mat/0308147. [10] Papadimitriou C.H. (1991). In FOCS 1991, p. 163. [11] Selman B. & Kautz H.A. (1993) In Proc. AAAI-93 26, pp. 46-51. [12] Semerjian G. & Monasson R. (2003). Phys Rev E 67: 066103. [13] Barthel W., Hartmann A.K. & Weigt M. (2003). Phys. Rev. E 67: 066104. [14] Selman B., Kautz K. & Cohen B. (1996) Local Search Strategies for Satis£ability Testing. DIMACS Series in Discrete Mathematics and Theoretical Computer Science 26. [15] http://www.cs.washington.edu/homes/kautz/walksat/
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Theories Of Access Consciousness Michael D. Colagrosso Department of Computer Science Colorado School of Mines Golden, CO 80401 USA mcolagro@mines.edu Michael C. Mozer Institute of Cognitive Science University of Colorado Boulder, CO 80309 USA mozer@colorado.edu 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. (a) 3 Y1 Y2 Y3 X2 X Y0 X1 (b) 0 0.2 0.4 0.6 0.8 1 Time P(Output) O Q 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 −β)δij + βP(Y (0) = yi), where β is the diffusion constant and δij 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 this distribution as P(X(t) = xi|Y (t) = yj) ∼1 + P k γikαkj, where αij indicates the frequency of experience with the association between states xi and yj, and γik specifies the similarity between states xi and xk. (Although the representation of states is localist, the γ 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, αij, 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: ∆αij = η maxt P(X(t) = xi)P(Y (t) = yj), where η 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, Xr(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 conditions 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 γik = 0.95; all other similarity coefficients are zero. The association frequency, α, for valid associations in the perceptual pathway was 22, and the response pathway 18. Other parameters were βp = .05, βr = .01, and η = 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 5 10 15 20 25 30 35 40 0 Percent Correct Naming Control Objects Prime Objects SHAPE: POSITION: Same Same Same Different Different Same Different Different Second Control First Block Second Block 5 10 15 20 25 30 35 40 0 Percent Correct Naming Control Objects Prime Objects SHAPE: POSITION: Same Same Same Different Different Same Different Different Second Control First Block Second Block 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) = θδij + (1 −θ)/|Y r|, and an analogous distribution for P(F(t) = fi|Y r(t) = yj). The free parameter θ determines how veridically naming and forced-choice actions reflect response-pathway output. Over a range of θ, θ < 1, the model obtains forced-choice performance near chance on the first presentation when the naming response is incorrect. For example, with θ = 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 (a) 0 0.1 0.2 0.3 0.4 0.5 0.6 1000 50 Probability Time (msec) object, first presentation object, third presentation different object (b) N(t) F(t) Yr(t) Figure 3: (a) Activation of the perceptual representation in PIT as a function of processing iterations on the first (thin solid line) and third (thick solid line) presentations of target. (b) Bayes net for performing 4AFC conditional on incorrect naming response. 1 targets hulip-type tumor-type orphans 0 0.1 0.2 0.3 0.4 Experiment Model Effective Valence 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 significant 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 γ = 0.837. No similarity was assumed in the semantic state representation; consistent with the previous simulation, βp = .05, βs = .01, βj = .01, and η = .01. At the outset of the simulation, α 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 ∆αij 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 sufficient 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 sufficient to support an overt naming response, yet it may be sufficient 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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Conditional Models of Identity Uncertainty with Application to Noun Coreference Andrew McCallum† †Department of Computer Science University of Massachusetts Amherst Amherst, MA 01003 USA mccallum@cs.umass.edu Ben Wellner†∗ ∗The MITRE Corporation 202 Burlington Road Bedford, MA 01730 USA wellner@cs.umass.edu Abstract Coreference analysis, also known as record linkage or identity uncertainty, is a difficult and important problem in natural language processing, databases, citation matching and many other tasks. This paper introduces several discriminative, conditional-probability models for coreference analysis, all examples of undirected graphical models. Unlike many historical approaches to coreference, the models presented here are relational—they do not assume that pairwise coreference decisions should be made independently from each other. Unlike other relational models of coreference that are generative, the conditional model here can incorporate a great variety of features of the input without having to be concerned about their dependencies—paralleling the advantages of conditional random fields over hidden Markov models. We present positive results on noun phrase coreference in two standard text data sets. 1 Introduction In many domains—including computer vision, databases and natural language processing—we find multiple views, descriptions, or names for the same underlying object. Correctly resolving these references is a necessary precursor to further processing and understanding of the data. In computer vision, solving object correspondence is necessary for counting or tracking. In databases, performing record linkage or de-duplication creates a clean set of data that can be accurately mined. In natural language processing, coreference analysis finds the nouns, pronouns and phrases that refer to the same entity, enabling the extraction of relations among entities as well as more complex propositions. Consider, for example, the text in a news article that discusses the entities George Bush, Colin Powell, and Donald Rumsfeld. The article contains multiple mentions of Colin Powell by different strings—“Secretary of State Colin Powell,” “he,” “Mr. Powell,” “the Secretary”—and also refers to the other two entities with sometimes overlapping strings. The coreference task is to use the content and context of all the mentions to determine how many entities are in the article, and which mention corresponds to which entity. This task is most frequently solved by examining individual pair-wise distance measures between mentions independently of each other. For example, database record-linkage and citation reference matching has been performed by learning a pairwise distance metric between records, and setting a distance threshold below which records are merged (Monge & Elkan, 1997; McCallum et al., 2000; Bilenko & Mooney, 2002; Cohen & Richman, 2002). Coreference in NLP has also been performed with distance thresholds or pairwise classifiers (McCarthy & Lehnert, 1995; Ge et al., 1998; Soon et al., 2001; Ng & Cardie, 2002). But these distance measures are inherently noisy and the answer to one pair-wise coreference decision may not be independent of another. For example, if we measure the distance between all of the three possible pairs among three mentions, two of the distances may be below threshold, but one above—an inconsistency due to noise and imperfect measurement. For example, “Mr. Powell” may be correctly coresolved with “Powell,” but particular grammatical circumstances may make the model incorrectly believe that “Powell” is coreferent with a nearby occurrence of “she.” Inconsistencies might be better resolved if the coreference decisions are made in dependent relation to each other, and in a way that accounts for the values of the multiple distances, instead of a threshold on single pairs independently. Recently Pasula et al. (2003) have proposed a formal, relational approach to the problem of identity uncertainty using a type of Bayesian network called a Relational Probabilistic Model (Friedman et al., 1999). A great strength of this model is that it explicitly captures the dependence among multiple coreference decisions. However, it is a generative model of the entities, mentions and all their features, and thus has difficulty using many features that are highly overlapping, non-independent, at varying levels of granularity, and with long-range dependencies. For example, we might wish to use features that capture the phrases, words and character n-grams in the mentions, the appearance of keywords anywhere in the document, the parse-tree of the current, preceding and following sentences, as well as 2-d layout information. To produce accurate generative probability distributions, the dependencies between these features should be captured in the model; but doing so can lead to extremely complex models in which parameter estimation is nearly impossible. Similar issues arise in sequence modeling problems. In this area significant recent success has been achieved by replacing a generative model—hidden Markov models—with a conditional model—conditional random fields (CRFs) (Lafferty et al., 2001). CRFs have reduced part-of-speech tagging errors by 50% on out-of-vocabulary words in comparison with HMMs (Ibid.), matched champion noun phrase segmentation results (Sha & Pereira, 2003), and significantly improved extraction of named entities (McCallum & Li, 2003), citation data (Peng & McCallum, 2004), and the segmentation of tables in government reports (Pinto et al., 2003). Relational Markov networks (Taskar et al., 2002) are similar models, and have been shown to significantly improve classification of Web pages. This paper introduces three conditional undirected graphical models for identity uncertainty. The models condition on the mentions, and generate the coreference decisions, (and in some cases also generate attributes of the entities). In the first most general model, the dependency structure is unrestricted, and the number of underlying entities explicitly appears in the model structure. The second and third models have no structural dependence on the number of entities, and fall into a class of Markov random fields in which inference corresponds to graph partitioning (Boykov et al., 1999). After introducing the first two models as background generalizations, we show experimental results using the third, most specific model on a noun coreference problem in two different standard newswire text domains: broadcast news stories from the DARPA Automatic Content Extraction (ACE) program, and newswire articles from the MUC-6 corpus. In both domains we take advantage of the ability to use arbitrary, overlapping features of the input, including multiple grammatical features, string equality, substring, and acronym matches. Using the same features, in comparison with an alternative natural language processing technique, we reduce error by 33% and 28% in the two domains on proper nouns and by 10% on all nouns in the MUC-6 data. 2 Three Conditional Models of Identity Uncertainty We now describe three possible configurations for conditional models of identity uncertainty, each progressively simpler and more specific than its predecessor. All three are based on conditionally-trained, undirected graphical models. Undirected graphical models, also known as Markov networks or Markov random fields, are a type of probabilistic model that excels at capturing interdependent data in which causality among attributes is not apparent. We begin by introducing notation for mentions, entities and attributes of entities, then in the following subsections describe the likelihood, inference and estimation procedures for the specific undirected graphical models. Let E = (E1, ...Em) be a collection of classes or “entities”. Let X = (X1, ...Xn) be a collection of random variables over observations or “mentions”; and let Y = (Y1, ...Yn) be a collection of random variables over integer identifiers, unique to each entity, specifying to which entity a mention refers. Thus the y’s are integers ranging from 1 to m, and if Yi = Yj, then mention Xi is said to refer to the same underlying entity as Xj. For example, some particular entity e4, U.S. Secretary of State, Colin L. Powell, may be mentioned multiple times in a news article that also contains mentions of other entities: x6 may be “Colin Powell”; x9 may be “he”; x17 may be “the Secretary of State.” In this case, the unique integer identifier for this entity, e4, is 4, and y6 = y9 = y17 = 4. Furthermore, entities may have attributes. Let A be a random variable over the collection of all attributes for all entities. Borrowing the notation of Relational Markov Networks (Taskar et al., 2002), we write the random variable over the attributes of entity Es as Es.A = {Es.A1, Es.A2, Es.A3, ...}. For example, these three attributes may be gender, birth year, and surname. Continuing the above example, then e4.a1 = MALE, e4.a2 = 1937, and e4.a3 = Powell. One can interpret the attributes as the values that should appear in the fields of a database record for the given entity. Attributes such as surname may take on one of the finite number of values that appear in the mentions of the data set. We may examine many features of the mentions, x, but since a conditional model doesn’t generate them, we don’t need random variable notation for them. Separate measured features of the mentions and entity-assignments, y, are captured in different feature functions, f(·), over cliques in the graphical model. Although the functions may be real-valued, typically they are binary. The parameters of the model are associated with these different feature functions. Details and example feature functions and parameterizations are given for the three specific models below. The task is then to find the most likely collection of entity-assignments, y, (and optionally also the most likely entity attributes, a), given a collection of mentions and their context, x. A generative probabilistic model of identity uncertainty is trained to maximize P(Y, A, X). A conditional probabilistic model of identity uncertainty is instead trained to maximize P(Y, A|X), or simply P(Y|X). 2.1 Model 1: Groups of nodes for entities First we consider an extremely general undirected graphical model in which there is a node for the mentions, x,1 a node for the entity-assignment of each mention, y, and a node for each of the attributes of each entity, e.a. These nodes are connected by edges in some unspecified structure, where an edge indicates that the values of the two connected random variables are dependent on each the other. 1Even though there are many mentions in x, because we are not generating them, we can represent them as a single node. This helps show that feature functions can ask arbitrary questions about various large and small subsets of the mentions and their context. We will still use xi to refer to the content and context of the ith mention. The parameters of the model are defined over cliques in this graph. Typically the parameters on many different cliques would be tied in patterns that reflect the nature of the repeated relational structure in the data. Patterns of tied parameters are common in many graphical models, including HMMs and other finite state machines (Lafferty et al., 2001), where they are tied across different positions in the input sequence, and by more complex patterns based on SQL-like queries, as in Markov Relational Networks (Taskar et al., 2002). Following the nomenclature of the later, these parameter-tying-patterns are called clique templates; each particular instance a template in the graph we call a hit. For example, one clique template may specify a pattern consisting of two mentions, their entity-assignment nodes, and an entity’s surname attribute node. The hits would consist of all possible combinations of such nodes. Multiple feature functions could then be run over each hit. One feature function might have value 1 if, for example, both mentions were assigned to the same entity as the surname node, and if the surname value appears as a substring in both mention strings (and value 0 otherwise). The Hammersley-Clifford theorem stipulates that the probability of a particular set of values on the random variables in an undirected graphical model is a product of potential functions over cliques of the graph. Our cliques will be the hits, h = {h, ...}, resulting from a set of clique templates, t = {t, ...}. In typical fashion, we will write the probability distribution in exponential form, with each potential function calculated as a dot-product of feature functions, f, and learned parameters, λ, P(y, a|x) = 1 Zx exp X t∈t X ht∈ht X l λlfl(y, a, x : ht) ! , where (y, a, x : ht) indicates the subset of the entity-assignment, attribute, and mention nodes selected by the clique template hit ht; and Zx is a normalizer to make the probabilities over all y sum to one (also known as the partition function). The parameters, λ, can be learned by maximum likelihood from labeled training data. Calculating the partition function is problematic because there are a very large number of possible y’s and a’s. Loopy belief propagation or Gibbs sampling sampling have been used successfully in other similar situations, e.g. (Taskar et al., 2002). However, note that both loopy belief propagation and Gibbs sampling only work over a graph with fixed structure. But in our problem the number of entities (and thus number of attribute nodes, and the domain of the entity-assignment nodes) is unknown. Inference in these models must determine for us the highest-probability number of entities. In related work on a generative probabilistic model of identity uncertainty, Pasula et al. (2003), solve this problem by alternating rounds of Metropolis-Hastings sampling on a given model structure with rounds of Metropolis-Hastings to explore the space of new graph structures. 2.2 Model 2: Nodes for mention pairs, with attributes on mentions To avoid the need to change the graphical model structure during inference, we now remove any parts of the graph that depend on the number of entities, m: (1) The per-mention entity-assignment nodes, Yi, are random variables whose domain is over the integers 0 through m; we remove these nodes, replacing them with binary-valued random variables, Yij, over each pair of mentions, (Xi, Xj) (indicating whether or not the two mentions are coreferent); although it is not strictly necessary, we also restrict the clique templates to operate over no more than two mentions (for efficiency). (2) The per-entity attribute nodes A are removed and replaced with attribute nodes associated with each mention; we write xi.a for the set of attributes on mention xi. Even though the clique templates are now restricted to pairs of mentions, this does not imply that pairwise coreference decisions are made independently of each other—they are still highly dependent. Many pairs will overlap with each other, and constraints will flow through these overlaps. This point is reiterated with an example in the next subsection. Notice, however, that it is possible for the model as thus far described to assign non-zero probability to an inconsistent set of entity-assignments, y. For example, we may have an “inconsistent triangle” of coreference decisions in which yij and yjk are 1, while yik is 0. We can enforce the impossibility of all inconsistent configurations by adding inconsistencychecking functions f∗(yij, yjk, yik) for all mention triples, with the corresponding λ∗’s fixed at negative infinity—thus assigning zero probability to them. (Note that this is simply a notational trick; in practice the inference implementation simply avoids any configurations of y that are inconsistent—a check that is simple to perform.) Thus we have P(y, a|x) = 1 Zx exp X i,j,l λlfl(xi, xj, yij, xi.a, xj.a) + X i,j,k λ∗f∗(yij, yjk, yik) . We can also enforce consistency among the attributes of coreferent mentions by similar means. There are many widely-used techniques for efficiently and drastically reducing the number of pair-wise comparisons, e.g. (Monge & Elkan, 1997; McCallum et al., 2000). In this case, we could also restrict fl(xi, xj, yij) ≡0, ∀yij = 0. 2.3 Model 3: Nodes for mention pairs, graph partitioning with learned distance When gathering attributes of entities is not necessary, we can avoid the extra complication of attributes by removing them from the model. What results is a straightforward, yet highly expressive, discriminatively-trained, undirected graphical model that can use rich feature sets and relational inference to solve identity uncertainty tasks. Determining the most likely number of entities falls naturally out of inference. The model is P(y|x) = 1 Zx exp X i,j,l λlfl(xi, xj, yij) + X i,j,k λ∗f∗(yij, yjk, yik) . (1) Recently there has been increasing interest in study of the equivalence between graph partitioning algorithms and inference in certain kinds of undirected graphical models, e.g. (Boykov et al., 1999). This graphical model is an example of such a case. With some thought, one can straightforwardly see that finding the highest probability coreference solution, y⋆= arg maxy P(y|x), exactly corresponds to finding the graph partitioning of a (different) graph in which the mentions are the nodes and the edge weights are the (log) clique potentials on the pair of nodes ⟨xi, xj⟩involved in their edge: P l λlfl(xi, xj, yij), where fl(xi, xj, 1) = −fl(xi, xj, 0), and edge weights range from −∞to +∞. Unlike classic mincut/maxflow binary partitioning, here the number of partitions (corresponding to entities) is unknown, but a single optimal number of partitions exists; negative edge weights encourage more partitions. Graph partitioning with negative edge weights is NP-hard, but it has a history of good approximations, and several efficient algorithms to choose from. Our current experiments use an instantiation of the minimizing-disagreements Correlational Clustering algorithm in (Bansal et al., 2002). This approach is a simple yet effective partitioning scheme. It works by measuring the degree of inconsistency incurred by including a node in a partition, and making repairs. We refer the reader to Bansal et al. (2002) for further details. The resulting solution does not make pairwise coreference decisions independently of each other. It has a significant “relational” nature because the assignment of a node to a partition (or, mention to an entity) depends not just on a single low distance measurement to one other node, but on its low distance measurement to all nodes in the partition (and furthermore on its high distance measurement to all nodes of all other partitions). For example, the “Mr. Powell”/“Powell”/“she” problem discussed in the introduction would be prevented by this model because, although the distance between “Powell” and “she” might grammatically look low, the distance from “she” to another member of the same partition, (“Mr. Powell”) is very high. Interestingly, in our model, the distance measure between nodes is learned from labeled training data. That is, we use data, D, in which the correct coreference partitions are known in order to learn a distance metric such that, when the same data is clustered, the correct partitions emerge. This is accomplished by maximum likelihood—adjusting the weights, λ, to maximize the product of Equation 1 over all instances ⟨x, y⟩in the training set. Fortunately this objective function is concave—it has a single global maximum— and there are several applicable optimization methods to choose from, including gradient ascent, stochastic gradient ascent and conjugate gradient; all simply require the derivative of the objective function. The derivative of the log-likelihood, L, is ∂L ∂λl = X ⟨x,y⟩∈D X i,j,l fl(xi, xj, yij) − X y′ PΛ(y′|x) X i,j,l fl(xi, xj, y′ ij) , where PΛ(y′|x) is defined by Equation 1, using the current set of λ parameters, Λ, and P y′ is a sum over all possible partitionings. The number of possible partitionings is exponential in the number of mentions, so for any reasonably-sized problem, we obviously must resort to approximate inference for the second expectation. A simple option is stochastic gradient ascent in the form of a voted perceptron (Collins, 2002). Here we calculate the gradient for a single training instance at a time, and rather than use a full expectation in the second line, simply using the single most likely (or nearly most likely) partitioning as found by a graph partitioning algorithm, and make progressively smaller steps in the direction of these gradients while cycling through the instances, ⟨x, y⟩in the training data. Neither the full sum, P y′, or the partition function, Zx, need to be calculated in this case. Further details are given in (Collins, 2002). 3 Experiments with Noun Coreference We present experimental results on natural language noun phrase coreference using Model 3 applied to two applicable data sets: the DARPA MUC-6 corpus, and a set of 117 stories from the broadcast news portion of the DARPA ACE data set. Both data sets have annotated coreferences. We pre-process both data sets with the Brill part-of-speech tagger. We compare our Model 3 against two other techniques representing typical approaches to the problem of identity uncertainty. The first is single-link clustering with a threshold, (single-link-threshold), which is universally used in database record-linkage and citation reference matching (Monge & Elkan, 1997; Bilenko & Mooney, 2002; McCallum et al., 2000; Cohen & Richman, 2002). It forms partitions by simply collapsing the spanning trees of all mentions with pairwise distances below some threshold. For each experiment, the threshold was selected by cross validation. The second technique, which we call best-previous-match, has been used in natural language processing applications (Morton, 1997; Ge et al., 1998; Ng & Cardie, 2002). It works by scanning linearly through a document, and associating each mention with its best-matching predecessor—best as measured with a single pairwise distance. In our experiments, both single-link-threshold and best-previous-match implementations use a distance measure based on a binary maximum entropy classifier—matching the practice of Morton (1997) and Cohen and Richman (2002). We use an identical feature set for all techniques, including our Method 3. The features, typical of those used in many other NLP coreference systems, are modeled after those in Ng and Cardie (2002). They include tests for string and substring matches, acronym matches, parse-derived head-word matches, gender, WORDNET subsumption, sentence distance, distance in the parse tree; etc., and are detailed in an accompanying technical report. They are quite non-independent, and operate at multiple levels of granularity. Table 1 shows standard MUCACE MUC-6 MUC-6 (Proper) (Proper) (All) best-previous-match 90.98 88.83 70.41 single-link-threshold 91.65 88.90 60.83 Model 3 93.96 91.59 73.42 Table 1: F1 results on three data sets. style F1 scores for three experiments. In the first two experiments, we consider only proper nouns, and perform five-fold cross validation. In the third experiment, we perform the standard MUC evaluation, including all nouns—pronouns, common and proper—and use the standard 30/30 document train/test split; furthermore, as in Harabagiu et al. (2001), we consider only mentions that have a coreferent. Model 3 out-performs both the single-link-threshold and the best-previousmatch techniques, reducing error by 28% over single-link-threshold on the ACE proper noun data, by 24% on the MUC-6 proper noun data, and by 10% over the best-previousmatch technique on the full MUC-6 task. All differences from Model 3 are statistically significant. Historically, these data sets have been heavily studied, and even small gains have been celebrated. Our overall results on MUC-6 are slightly better (with unknown statistical significance) than the best published results of which we are aware with a matching experimental design, Harabagiu et al. (2001), who reach 72.3% using the same training and test data. 4 Related Work and Conclusions There has been much related work on identity uncertainty in various specific fields. Traditional work in de-duplication for databases or reference-matching for citations measures the distance between two records by some metric, and then collapses all records at a distance below a threshold, e.g. (Monge & Elkan, 1997; McCallum et al., 2000). This method is not relational, that is, it does not account for the inter-dependent relations among multiple decisions to collapse. Most recent work in the area has focused on learning the distance metric (Bilenko & Mooney, 2002; Cohen & Richman, 2002) not the clustering method. Natural language processing has had similar emphasis and lack of emphasis respectively. Pairwise coreference learned distance measures have used decision trees (McCarthy & Lehnert, 1995; Ng & Cardie, 2002), SVMs (Zelenko et al., 2003), maximum entropy classifiers (Morton, 1997), and generative probabilistic models (Ge et al., 1998). But all use thresholds on a single pairwise distance, or the maximum of a single pairwise distance to determine if or where a coreferent merge should occur. Pasula et al. (2003) introduce a generative probability model for identity uncertainty based on Probabilistic Relational Networks networks. Our work is an attempt to gain some of the same advantages that CRFs have over HMMs by creating conditional models of identity uncertainty. The models presented here, as instances of conditionally-trained undirected graphical models, are also instances of relational Markov networks (Taskar et al., 2002) and conditional Random fields (Lafferty et al., 2001). Taskar et al. (2002) briefly discuss clustering of dyadic data, such as people and their movie preferences, but not identity uncertainty or inference by graph partitioning. Identity uncertainty is a significant problem in many fields. In natural language processing, it is not only especially difficult, but also extremely important, since improved coreference resolution is one of the chief barriers to effective data mining of text data. Natural language data is a domain that has particularly benefited from rich and overlapping feature representations—representations that lend themselves better to conditional probability models than generative ones (Lafferty et al., 2001; Collins, 2002; Morton, 1997). Hence our interest in conditional models of identity uncertainty. Acknowledgments We thank Andrew Ng, Jon Kleinberg, David Karger, Avrim Blum and Fernando Pereira for helpful and insightful discussions. This work was supported in part by the Center for Intelligent Information Retrieval and in part by SPAWARSYSCEN-SD grant numbers N66001-99-1-8912 and N66001-021-8903, and DARPA under contract number F30602-01-2-0566 and in part by the National Science Foundation under NSF grant #IIS-0326249 and in part by the Defense Advanced Research Projec ts Agency (DARPA), through the Department of the Interior, NBC, Acquisition Services Division, under contract number NBCHD030010. References Bansal, N., Chawala, S., & Blum, A. (2002). Correlation clustering. The 43rd Annual Symposium on Foundations of Computer Science (FOCS) (pp. 238–247). Bilenko, M., & Mooney, R. J. (2002). Learning to combine trained distance metrics for duplicate detection in databases (Technical Report Technical Report AI 02-296). Artificial Intelligence Laboratory, University of Texas at Austin, Austin, TX. Boykov, Y., Veksler, O., & Zabih, R. (1999). Fast approximate energy minimization via graph cuts. ICCV (1) (pp. 377–384). Cohen, W., & Richman, J. (2002). Learning to match and cluster entity names. Proceedings of KDD-2002, 8th International Conference on Knowledge Discovery and Data Mining. Collins, M. (2002). Discriminative training methods for hidden markov models: Theory and experiments with perceptron algorithms. Friedman, N., Getoor, L., Koller, D., & Pfeffer, A. (1999). Learning probabilistic relational models. IJCAI (pp. 1300–1309). Ge, N., Hale, J., & Charniak, E. (1998). A statistical approach to anaphora resolution. Proceedings of the Sixth Workshop on Very Large Corpora (pp. 161–171). Harabagiu, S., Bunescu, R., & Maiorano, S. (2001). Text and knowledge mining for coreference resolution. Proceedings of the 2nd Meeting of the North American Chapter of the Association of Computational Linguistics (NAACL-2001) (pp. 55–62). Lafferty, J., McCallum, A., & Pereira, F. (2001). Conditional random fields: Probabilistic models for segmenting and labeling sequence data. Proc. ICML (pp. 282–289). McCallum, A., & Li, W. (2003). Early results for named entity recognition with conditional random fields, feature induction and web-enhanced lexicons. Seventh Conference on Natural Language Learning (CoNLL). McCallum, A., Nigam, K., & Ungar, L. H. (2000). Efficient clustering of high-dimensional data sets with application to reference matching. Knowledge Discovery and Data Mining (pp. 169–178). McCarthy, J. F., & Lehnert, W. G. (1995). Using decision trees for coreference resolution. IJCAI (pp. 1050–1055). Monge, A. E., & Elkan, C. (1997). An efficient domain-independent algorithm for detecting approximately duplicate database records. Research Issues on Data Mining and Knowledge Discovery. Morton, T. (1997). Coreference for NLP applications. Proceedings ACL. Ng, V., & Cardie, C. (2002). Improving machine learning approaches to coreference resolution. Fortieth Anniversary Meeting of the Association for Computational Linguistics (ACL-02). Pasula, H., Marthi, B., Milch, B., Russell, S., & Shpitser, I. (2003). Identity uncertainty and citation matching. Advances in Neural Information Processing (NIPS). Peng, F., & McCallum, A. (2004). Accurate information extraction from research papers using conditional random fields. Proceedings of Human Language Technology Conference and North American Chapter of the Association for Computational Linguistics (HLT-NAACL). Pinto, D., McCallum, A., Lee, X., & Croft, W. B. (2003). Table extraction using conditional random fields. Proceedings of the 26th ACM SIGIR. Sha, F., & Pereira, F. (2003). Shallow parsing with conditional random fields (Technical Report CIS TR MS-CIS-02-35). University of Pennsylvania. Soon, W. M., Ng, H. T., & Lim, D. C. Y. (2001). A machine learning approach to coreference resolution of noun phrases. Computational Linguistics, 27, 521–544. Taskar, B., Abbeel, P., & Koller, D. (2002). Discriminative probabilistic models for relational data. Eighteenth Conference on Uncertainty in Artificial Intelligence (UAI02). Zelenko, D., Aone, C., & Richardella, A. (2003). Kernel methods for relation extraction. Journal of Machine Learning Research (submitted).
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Outlier Detection with One-class Kernel Fisher Discriminants Volker Roth ETH Zurich, Institute of Computational Science Hirschengraben 84, CH-8092 Zurich vroth@inf.ethz.ch 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 ν-SVM classifier with RBF kernel function is used. The parameter ν 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′ = (X, −X)⊤denote the augmented (2n × d) data matrix which also contains the negative samples −xi. Without loss of generality we assume that the sample mean µ+ ≡P i xi > 0, so that the sample means of the positive data and the negative data differ: µ+ ̸= µ−. Let us now assume that our data are realizations of a normally distributed random variable in d dimensions: X ∼Nd(µ, Σ). Denoting by Xc the centered data matrix, the estimator for Σ takes the form ˆΣ ≡W = (1/n)Xc⊤Xc. The LDA solution β∗maximizes the between-class scatter β⊤ ∗Bβ∗with B = µ+µ⊤ + + µ−µ⊤ −under the constraint on the within-class scatter β⊤ ∗Wβ∗= 1. Note that in our special case with X′ = (X, −X)⊤the usual pooled within-class matrix W simply reduces to the above defined W = (1/n)Xc⊤Xc. Denoting by y′ = (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 functional: ˆβ = arg minβ ∥y′−X′β∥2. In [3] a slightly more general form of the problem is described where the above functional is minimized under a constrained on β, which in the simplest case amounts to adding a term γβ⊤β to the functional. Such a ridge regression model assumes a penalized total covariance of the form T = (1/(2n)) · X ′⊤X′ + γI = (1/n) · X⊤X + γI. Defining an n-vector of ones y = (1, . . . , 1)⊤, the solution vector ˆβ reads ˆβ = (X′⊤X′ + γI)−1X′⊤y′ = (X⊤X + γI)−1X⊤y. (1) According to [3], an appropriate scaling factor is defied in terms of the quantity s2 = (1/n) · y⊤ˆy = (1/n) · y⊤X ˆβ, which leads us to the correctly scaled LDA vector β∗= s−1(1 −s2)−1/2 ˆβ that fulfills the normalization condition β⊤ ∗Wβ∗= 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 µ+ is the sum of the squared Euclidean distance in the projected space and an orthogonal distance term: D(x, µ+) = (β⊤ ∗x −m+)2 + D⊥with D⊥= −(1 −s2)(β⊤ ∗x)2 + x⊤T −1x. (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 β, the true density model additionally takes into account the distances in the orthogonal subspace. Since the assumption X ∼Nd(µ, Σ) is very restrictive, we propose to relax it by assuming that we have found a suitable transformation of our input data φ : Rd 7→Rp, x 7→φ(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 = ΦΦ⊤and the expansion coefficients α: ˆα = (K + γI)−1y. (3) From [11] it follows that the mapped vectors can be represented in Rn as φ(x) = K−1/2k(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: D(x, µ+) = (α⊤ ∗k(x) −m+)2 −(1 −s2)(α⊤ ∗k(x))2 + nΩ(x), (4) where Ω(x) = φ⊤(x)(Φ⊤Φ + γI)−1φ(x) = k⊤(x)(K + γI)−1K−1k(x), m+ = s(1 −s2)−1/2, s2 = (1/n) · y⊤ˆy = (1/n) · y⊤K ˆα, and α∗= s−1(1 −s2)−1/2 ˆα. 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 γ 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(µ, Σ). Then ∆≡(X − µ)⊤Σ−1(X −µ) follows a chi-square (χ2) 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 + γI)−1X⊤), [8], which for a kernel model translates into df = trace(K(K + γI)−1). The intuitive interpretation of the quantity df is the following: denoting by V the matrix of eigenvectors of K and by {λi}n i=1 the corresponding eigenvalues, the fitted values ˆy read ˆy = V diag {δi = λi/(λi + γ)} V ⊤y. (5) It follows that compared to the unpenalized case, where all eigenvectors vi are constantly weighted by 1, the contribution of the i-th eigenvector vi is down-weighted by a factor δi/1 = δi. If the ordered eigenvalues decrease rapidly, however, the values δi 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 Ω(x) = k⊤(x)(K +γI)−1K−1k(x) = k⊤V diag δ′ i = ((λi + γ)λi)−1 V ⊤k(x). (6) Compared to the unpenalized case (the contribution of vi is weighted by λ−2 i ), the contribution of vi is down-weighted by the same factor δ′ i/λ−2 i = δi. 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 χ2-distribution with df degrees of freedom. A graphical way to test this hypothesis is to plot the observed quantiles against the theoretical χ2 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 pi the cumulative proportion before each ∆(i) given by pi = (i −1/2)/n. Let further zi = F −1pi denote the theoretical quantile at position pi, where F is the cumulative χ2-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]) σ(∆(i)) = b · (χ2(zi))−1p (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 −ε)% envelope around the fit is then defined as ∆(i) ± zε/2σ(∆(i)) where zε/2 is the 1 −(1 −ε)/2 quantile of the standard normal distribution. The choice of the confidence level ε 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 ε 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, ε 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 σ. For selecting all free parameters in (4), we are, thus, left with the problem of selecting θ = (σ, γ)⊤. The idea is now to select θ 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(φ(xi)|·) = p(φ(xj)|·). Denoting by Xn = {xi}n i=1 a sample from p(x) from which the kernel matrix K is built, a natural input space model is pn(x|Xn, θ) = Z−1 exp{−1 2D(x; Xn, θ)}, with Z = R Rd pn(x|Xn, θ) dx, (8) where D(x; Xn, θ) 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(−∥xi −xj∥2/σ). As σ →0 , pn(x|Xn, θ) converges to a Parzen window with vanishing kernel width: pn(x|Xn, θ) →1 n Pn i=1 δ(x −xi). A formal proof is omitted due to space limitations. The basic ingredients of the proof are: (i) In the limit σ →0 the expansion coefficients approach ˆα →1/(1 + γ)1. Thus, ˆy = K ˆα →1/(1 + γ)1 and s2 →1/(1 + γ). (ii) D(x; σ, γ) →C(x) < ∞, if x ∈ {xi}n i=1, and D(x; σ, γ) →∞, else. Finally pn(x|Xn, σ, γ) →1 n Pn i=1 δ(x −xi). Note that in the limit σ →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 σ →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 χ2 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 σ such that n grows at a higher rate as the effective degrees of freedom df: Lemma 3. Let k(xi, xj) = exp(−∥xi −xj∥2/σ) and pn(x|Xn, σ, γ) defined by (8). If σ ≤1 decays like O(n−1/2), and for fixed γ ≤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) the eigenvalues λ′ i of (1/n)K converge to ¯λi as n →∞, (ii) the eigenvalue spectrum of a Gaussian RBF kernel decays at an exponential-quadratic rate: ¯λi ∝exp(−σi2), (iii) for n sufficiently large, it holds that Pn i=1 1/[1+(γ/n) exp(n−1/2σi2)] ≤n1/2σ−1 log(n/γ) (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(−∥xi −xj∥2/σ). In a first step, the free model parameters (σ, γ) 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 (σ, γ). In order to simplify the selection process we kept γ = 10−4 fixed and varied only σ. 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 σ 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 (σ too small), or decreases to zero in the underfitting regime (σ 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 out an eigenvalue decomposition of K. Both the the effective degrees of freedom df = P i λi/(λi + γ) and the Mahalanobis distances in eq. (4) can then by derived easily ∆ ∆ 5 10 15 20 25 30 10 20 30 40 50 5 10 15 20 25 30 10 20 30 40 (i) (i) 2 χ quantiles quantiles 2 χ σ log( ) effective degrees of freedom test log−likelihood −600 −550 −500 −450 −400 −350 −300 −250 −200 6 7 8 9 10 11 12 13 15 10 5 Figure 2: Middle panel: Selecting the kernel width σ 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). 5 10 15 20 25 30 10 20 30 40 50 5 10 15 20 25 30 10 20 30 40 50 ∆(i) χ 2 quantiles 99% 99.99% 99% 99.99% 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 χ2-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. M¨uller, 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. M¨uller, 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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Kernel Projection Machine: a New Tool for Pattern Recognition∗ Gilles Blanchard Fraunhofer First (IDA), K´ekul´estr. 7, D-12489 Berlin, Germany blanchar@first.fhg.de Pascal Massart D´epartement de Math´ematiques, Universit´e Paris-Sud, Bat. 425, F-91405 Orsay, France Pascal.Massart@math.u-psud.fr R´egis Vert LRI, Universit´e Paris-Sud, Bat. 490, F-91405 Orsay, France Masagroup 24 Bd de l’Hopital, F-75005 Paris, France Regis.Vert@lri.fr Laurent Zwald D´epartement de Math´ematiques, Universit´e Paris-Sud, Bat. 425, F-91405 Orsay, France Laurent.Zwald@math.u-psud.fr Abstract This paper investigates the effect of Kernel Principal Component Analysis (KPCA) within the classification framework, essentially the regularization properties of this dimensionality reduction method. KPCA has been previously used as a pre-processing step before applying an SVM but we point out that this method is somewhat redundant from a regularization point of view and we propose a new algorithm called Kernel Projection Machine to avoid this redundancy, based on an analogy with the statistical framework of regression for a Gaussian white noise model. Preliminary experimental results show that this algorithm reaches the same performances as an SVM. 1 Introduction Let (xi, yi)i=1...n be n given realizations of a random variable (X, Y ) living in X × {−1; 1}. Let P denote the marginal distribution of X. The xi’s are often referred to as inputs (or patterns), and the yi’s as labels. Pattern recognition is concerned with finding a classifier, i.e. a function that assigns a label to any new input x ∈X and that makes as few prediction errors as possible. It is often the case with real world data that the dimension of the patterns is very large, and some of the components carry more noise than information. In such cases, reducing the dimension of the data before running a classification algorithm on it sounds reasonable. One of the most famous methods for this kind of pre-processing is PCA, and its kernelized version (KPCA), introduced in the pioneering work of Sch¨olkopf, Smola and M¨uller [8]. ∗This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778. Now, whether the quality of a given classification algorithm can be significantly improved by using such pre-processed data still remains an open question. Some experiments have already been carried out to investigate the use of KPCA for classification purposes, and numerical results are reported in [8]. The authors considered the USPS handwritten digit database and reported the test error rates achieved by the linear SVM trained on the data pre-processed with KPCA: the conclusion was that the larger the number of principal components, the better the performance. In other words, the KPCA step was useless or even counterproductive. This conclusion might be explained by a redundancy arising in their experiments: there is actually a double regularization, the first corresponding to the dimensionality reduction achieved by KPCA, and the other to the regularization achieved by the SVM. With that in mind it does not seem so surprising that KPCA does not help in that case: whatever the dimensionality reduction, the SVM anyway achieves a (possibly strong) regularization. Still, de-noising the data using KPCA seems relevant. The aforementioned experiments suggest that KPCA should be used together with a classification algorithm that is not regularized (e.g. a simple empirical risk minimizer): in that case, it should be expected that the KPCA is by itself sufficient to achieve regularization, the choice of the dimension being guided by adequate model selection. In this paper, we propose a new algorithm, called the Kernel Projection Machine (KPM), that implements this idea: an optimal dimension is sought so as to minimize the test error of the resulting classifier. A nice property is that the training labels are used to select the optimal dimension – optimal means that the resulting D-dimensional representation of the data contains the right amount of information needed to classify the inputs. To sum up, the KPM can be seen as a dimensionality-reduction-based classification method that takes into account the labels for the dimensionality reduction step. This paper is organized as follows: Section 2 gives some statistical background on regularized method vs. projection methods. Its goal is to explain the motivation and the “Gaussian intuition” that lies behind the KPM algorithm from a statistical point of view. Section 3 explicitly gives the details of the algorithm; experiments and results, which should be considered preliminary, are reported in Section 4. 2 Motivations for the Kernel Projection Machine 2.1 The Gaussian Intuition: a Statistician’s Perspective Regularization methods have been used for quite a long time in non parametric statistics since the pioneering works of Grace Wahba in the eighties (see [10] for a review). Even if the classification context has its own specificity and offers new challenges (especially when the explanatory variables live in a high dimensional Euclidean space), it is good to remember what is the essence of regularization in the simplest non parametric statistical framework: the Gaussian white noise. So let us assume that one observes a noisy signal dY (x) = s(x)dx+ 1 √ndw(x) , Y (0) = 0 on [0,1] where dw(x) denotes standard white noise. To the reader not familiar with this model, it should be considered as nothing more but an idealization of the well-known fixed design regression problem Yi = s(i/n) + εi for i = 1, . . . , n, where εi ∼N(0, 1), where the goal is to recover the regression function s. (The white noise model is actually simpler to study from a mathematical point of view). The least square criterion is defined as γn(f) = ∥f∥2 −2 Z 1 0 f(x)dY (x) for every f ∈L2([0, 1]). Given a Mercer kernel k on [0, 1]×[0, 1], the regularization least square procedure proposes to minimize γn(f) + ζn∥f∥Hk (1) where (ζn) is a conveniently chosen sequence and Hk denotes the RKHS induced by k. This procedure can indeed be viewed as a model selection procedure since minimizing γn(f) + ζn∥f∥Hk amounts to minimizing inf ∥f∥≤R γn(f) + ζnR2 over R > 0. In other words, regularization aims at selecting the “best” RKHS ball {f, ∥f∥≤R} to represent our data. At this stage, it is interesting to realize that the balls in the RKHS space can be viewed as ellipsoids in the original Hilbert space L2([0, 1]). Indeed, let (φi)∞ i=1 be some orthonormal basis of eigenfunctions for the compact and self adjoint operator Tk : f −→ Z 1 0 k(x, y)f(x)dx Then, setting βj = R 1 0 f(x)φj(x)dx one has ∥f∥2 Hk = P∞ j=1 β2 j λj where (λj)j≥1 denotes the non increasing sequence of eigenvalues corresponding to (φj)j≥1. Hence {f, ∥f∥Hk ≤R} = ∞ X j=1 βjφj ; ∞ X j=1 β2 j λj ≤R2 . Now, due to the approximation properties of the finite dimensional spaces {φj, j ≤D}, D ∈N∗with respect to the ellipsoids, one can think of penalized finite dimensional projection as an alternative method to regularization. More precisely, if bsD denotes the projection estimator on ⟨φj, j ≤D⟩, i.e. bsD = PD j=1 R φjdY φj and one considers the penalized selection criterion bD = argmin D [γn(bsD) + 2D n ] then, it is proved in [1] that the selected estimator bs b D obeys to the following oracle inequality E[∥s −bs b D∥2] ≤C inf D≥1 E∥s −bsD∥2 where C is some absolute constant. The nice thing is that whenever s belongs to some ellipsoid E(c) = ∞ X j=1 βjφj : ∞ X j=1 β2 j c2 j ≤1 where (cj)j≥1 is a decreasing sequence tending to 0 as j →∞, then inf D≥1 E ∥s −bsD∥2 = inf D≥1 inf t∈SD ∥s −t∥2 + D n ≤inf D≥1 c2 D + D n As shown in [5] infD≥1[c2 D + D n ] is (up to some absolute constant) of the order of magnitude of the minimax risk over E(c). As a consequence, the estimator bs b D is simultaneously minimax over the collection of all ellipsoids E(c), which in particular includes the collection {E( √ λR), R > 0}. To conclude and summarize, from a statistical performance point of view, what we can expect from a regularized estimator bs (i.e. a minimizer of (1)) is that a convenient device of ζn ensures that bs is simultaneously minimax over the collection of ellipsoids {E( √ λR), R > 0}, (at least as far as asymptotic rates of convergence are concerned ). The alternative estimator bs b D actually achieves this goal and even better since it is also adaptive over the collection of all ellipsoids and not only the family {E( √ λR), R > 0}. 2.2 Extension to a general classification framework In this section we go back to classification framework as described in the introduction. First of all, it has been noted by several authors ([6],[9]) that the SVM can be seen as a regularized estimation method, where the regularizer is the squared norm of the function in Hk. Precisely, the SVM algorithm solves the following unconstrained optimization problem: min f∈Hb k 1 n n X i=1 (1 −yif(xi))+ + λ∥f∥2 Hk , (2) where Hb k = {f(x) + b, f ∈Hk, b ∈R}. The above regularization can be viewed as a model selection process over RKHS balls, similarly to the previous section. Now, the line of ideas developed there suggests that it might actually be a better idea to consider a sequence of finite-dimensional estimators. Additionally, it has been shown in [4] that the regularization term of the SVM is actually too strong. We therefore transpose the ideas of previous Gaussian case to the classification framework. Consider a Mercer kernel k defined on X × X and Let Tk denote the operator associated with kernel k in the following way Tk : f(.) ∈L2(X) 7−→ Z X k(x, .)f(x)dP(x) ∈L2(X) Let φ1, φ2, . . . denote the eigenvectors of Tk, ordered by decreasing associated eigenvalues (λi)1≥1. For each integer D, the subspace FD defined by FD = span{11, φ1, . . . , φD} (where 11 denotes the constant function equal to 1) corresponds to a subspace of Hb k associated with kernel k, and Hb k = S∞ D=1 FD. Instead of selecting the “best” ball in the RKHS, as the SVM does, we consider the analogue of the projection estimator bsD: ˆfD = arg min f∈FD n X i=1 (1 −yif(xi))+ (3) that is, more explicitly, ˆfD(.) = D X j=1 β∗ j φj(.) + b∗ with (β∗, b∗) = arg min (β∈RD,b∈R) n X i=1 1 −yi D X j=1 βjφj(xi) + b + (4) An appropriate D can then be chosen using an adequate model selection procedure such as penalization; we do not address this point in detail in the present work but it is of course the next step to be taken. Unfortunately, since the underlying probability P is unknown, neither are the eigenfunctions φ1, . . ., and it is therefore not possible to implement this procedure directly. We thus resort to considering empirical quantities as will be explained in more detail in section 3. Essentially, the unknown vectorial space spanned by the first eigenfunctions of Tk is replaced by the space spanned by the first eigenvectors of the normalized kernel Gram matrix 1 n(k(xi, xj))1≤i,j≤n. At this point we can see the relation appear with Kernel PCA. We next precise this relation and give an interpretation of the resulting algorithm in terms of dimensionality reduction. 2.3 Link with Kernel Principal Component Analysis Principal Component Analysis (PCA), and its non-linear variant, KPCA are widely used algorithms in data analysis. They extract from the input data space a basis (vi)i≥1 which is, in some sense, adapted to the data by looking for directions where the variance is maximized. They are often used as a pre-processing on the data in order to reduce the dimensionality or to perform de-noising. As will be made more explicit in the next section, the Kernel Projection Machine consists in replacing the ideal projection estimator defined by (3) by bfD = argmin f∈SD 1 n n X i=1 (1 −yif(Xi))+ where SD is the space of dimension D chosen by the first D principal components chosen by KPCA in feature space. Hence, roughly speaking, in the KPM, the SVM penalization is replaced by dimensionality reduction. Choosing D amounts to selecting the optimal D-dimensional representation of our data for the classification task, in other words to extracting the information that is needed for this task by model selection taking into account the relevance of the directions for the classification task. To conclude, the KPM is a method of dimensionality reduction that takes into account the labels of the training data to choose the “best” dimension. 3 The Kernel Projection Machine Algorithm In this section, the empirical (and computable) version of the KPM algorithm is derived from the previous theoretical arguments. In practice the true eigenfunctions of the kernel operator are not computable. But since only the values of functions φ1, . . . , φD at points x1, . . . , xn are needed for minimizing the empirical risk over FD, the eigenvectors of the kernel matrix K = (k(xi, xj))1≤i,j≤n will be enough for our purpose. Indeed, it is well known in numerical analysis (see [2]) that the eigenvectors of the kernel matrix approximate the eigenfunctions of the kernel operator. This result has been pointed out in [7] in a more probabilistic language. More precisely, if V1, . . . , VD denote the D first eigenvectors of K with associated eigenvalues bλ1 ≥bλ2 ≥. . . ≥bλD, then for each Vi Vi = V (1) i , . . . , V (n) i ≈(φi(x1), . . . , φi(xn)) (5) Hence, considering Equation (4), the empirical version of the algorithm described above will first consist of solving, for each dimension D, the following optimization problem: (β∗, b∗) = arg min β∈RD,b∈R n X i=1 1 −yi D X j=1 βjV (i) j + b + (6) Then the solution should be ˆfD(.) = D X j=1 β∗ j φj(.) + b∗. (7) Once again the true functions φj’s are unknown. At this stage, we can do an expansion of the solution in terms of the kernel similarly to the SVM algorithm, in the following way: ˆfD(.) = n X i=1 α∗ i k(xi, .) + b∗ (8) 0 5 10 15 20 25 30 0.3 0.32 0.34 0.36 0.38 0.4 0.42 0.44 0.46 0 2 4 6 8 10 12 14 16 18 20 0.32 0.325 0.33 0.335 0.34 0.345 0.35 Figure 1: Left: KPM risk (solid) and empirical risk (dashed) versus dimension D. Right: SVM risk and empirical risk versus C. Both on dataset ’flare-solar’. Narrowing both expressions ( 7) and ( 8) at points x1, . . . , xn leads the following equation: β∗ 1V1 + . . . + β∗ DVD = Kα∗ (9) which has a straightforward solution: α∗= PD j=1 β∗ j bλj Vj (provided the D first eigenvalues are all strictly positive). Now the KPM algorithm can be summed up as follows: 1. given data x1, . . . , xn ∈X and a positive kernel k defined on X × X, compute the kernel matrix K and its eigenvectors V1, . . . , Vn together with its eigenvalues in decreasing order bλ1 ≥bλ2 ≥. . . ≥bλn. 2. for each dimension D such that bλD > 0 solve the linear optimization problem (β∗, b∗) = arg min β,b,ξ n X i=1 ξi (10) under constraints ∀i = 1 . . . n, ξi ≥0 , yi D X j=1 βjV (i) j + b ≥1 −ξi . (11) Next, compute α∗= D X j=1 β∗ j bλj Vj and ˆfD(.) = Pn i=1 α∗ i k(xi, .) + b∗ 3. The last step is a model selection problem: choose a dimension ˆD for which ˆf ˆ D performs well. We do not address directly this point here; one can think of applying cross-validation, or to penalize the empirical loss by a penalty function depending on the dimension. 4 Experiments The KPM was implemented in Matlab using the free library GLPK for solving the linear optimization problem. Since the algorithm involves the eigendecomposition of the kernel matrix, only small datasets have been considered for the moment. In order to assess the performance of the KPM, we carried out experiments on benchmark datasets available on Gunnar R¨atsch’s web site [3]. Several state-of-art algorithms have already been applied to those datasets, among which the SVM. All results are reported on the web site. To get a valid comparison with the SVM, on each classification task, we used Table 1: Test errors of the KPM on several benchmark datasets, compared with SVM, using G.R¨atsch’s parameter selection procedure (see text). As an indication the best of the six results presented in [3] are also reported. KPM (selected D) SVM Best of 6 Banana 10.73 ± 0.42 15 11.53 ± 0.66 10.73 ± 0.43 Breast Cancer 26.51 ± 4.75 24 26.04 ± 4.74 24.77 ± 4.63 Diabetis 23.37 ± 1.92 11 23.53 ± 1.73 23.21 ± 1.63 Flare Solar 32.43 ± 1.85 6 32.43 ± 1.82 32.43 ± 1.82 German 23.59 ± 2.15 14 23.61 ± 2.07 23.61 ± 2.07 Heart 16.89 ± 3.53 10 15.95 ± 3.26 15.95 ± 3.26 Table 2: Test errors of the KPM on several benchmark datasets, compared with SVM, using standard 5-fold cross-validation on each realization. KPM SVM Banana 11.14 ±0.73 10.69 ± 0.67 Breast Cancer 26.55±4.43 26.68 ± 5.23 Diabetis 24.14 ±1.86 23.79 ± 2.01 Flare Solar 32.70±1.97 32.62 ± 1.86 German 23.82±2.23 23.79 ± 2.12 Heart 17.59±3.30 16.23 ± 3.18 the same kernel parameters as those used for SVM, so as to work with exactly the same geometry. There is a subtle, but important point arising here. In the SVM performance reported by G. R¨atsch, the regularization parameter C was first determined by cross-validation on the first 5 realizations of each dataset; then the median of these values was taken as a fixed value for the other realizations. This was done apparently for saving computation time, but this might lead to an over-optimistic estimation of the performances since in some sense some extraneous information is then available to the algorithm and the variation due to the choice of C is reduced to almost zero. We first tried to mimic this methodology by applying it, in our case, to the choice of D itself (the median of 5 D values obtained by cross-validation on the first realizations was then used on the other realizations). One might then argue that this way we are selecting a parameter by this method instead of a meta-parameter for the SVM, so that the comparison is unfair. However, this distinction being loose, this a rather moot point. To avoid this kind of debate and obtain fair results, we decided to re-run the SVM tests by selecting systematically the regularization parameter by a 5-fold cross-validation on each training set, and for our method, apply the same procedure to select D. Note that there is still extraneous information in the choice of the kernel parameters, but at least it is the same for both algorithms. Results relative to the first methodology are reported in table 1, and those relative the second one are reported in table 2. The globally worst performances exhibited in the second table show that the first procedure may indeed be too optimistic. It is to be mentionned that the parameter C of the SVM was systematically sought on a grid of only 100 values, ranging from 0 to three times the optimal value given in [3]. Hence those experimental results are to be considered as preliminary, and in no way they should be used to establish a significant difference between the performances of the KPM and the SVM. Interestingly, the graphic on the left in Figure 4 shows that our procedure is very different from the one of [8]: when D is very large, our risk increases (leading to the existence of a minimum) while the risk of [8] always decreases with D. 5 Conclusion and discussion To summarize, one can see the KPM as an alternative to the regularization of the SVM: regularization using the RKHS norm can be replaced by finite dimensional projection. Moreover, this algorithm performs KPCA towards classification and thus offers a criterion to decide what is the right order of expansion for the KPCA. Dimensionality reduction can thus be used for classification but it is important to keep in mind that it behaves like a regularizer. Hence, it is clearly useless to plug it in a classification algorithm that is already regularized: the effect of the dimensionality reduction may be canceled as noted by [8]. Our experiments explicitly show the regularizing effect of KPCA: no other smoothness control has been added in our algorithm and still, it gives performances comparable to the one of SVM provided the dimension D is picked correctly. We only considered here selection of D by cross-validation; other methods such as penalization will be studied in future works. Moreover, with this algorithm, we obtain a D-dimensional representation of our data which is optimal for the classification task. Thus KPM can be see as a de-noising method who takes into account the labels. This version of the KPM only considers one kernel and thus one vectorial space by dimension. A more advanced version of this algorithm is to consider several kernels and thus choose among a bigger family of spaces. This family then contains more than one space by dimension and will allow to directly compare the performance of different kernels on a given task, thus improving efficiency for the dimensionality reduction while taking into account the labels. References [1] P. Massart A. Barron, L. Birg´e. Risk bounds for model selection via penalization. Proba.Theory Relat.Fields, 113:301–413, 1999. [2] Baker. The numerical treatment of integral equations. Oxford:Clarendon Press, 1977. [3] http://ida.first.gmd.de/˜raetsch/data/benchmarks.htm. Benchmark repository used in several Boosting, KFD and SVM papers. [4] G. Blanchard, O. Bousquet, and P.Massart. Statistical performance of support vector machines. Manuscript, 2004. [5] D.L. Donoho, R.C. Liu, and B. MacGibbon. Minimax risk over hyperrectangles, and implications. Ann. Statist. 18,1416-1437, 1990. [6] T. Evgeniou, M. Pontil, and T. Poggio. Regularization networks and support vector machines. In A. J. Smola, P. L. Bartlett, B. Sch¨olkopf, and D. Schuurmans, editors, Advances in Large Margin Classifiers, pages 171–203, Cambridge, MA, 2000. MIT Press. [7] V. Koltchinskii. Asymptotics of spectral projections of some random matrices approximating integral operators. Progress in Probability, 43:191–227, 1998. [8] B. Sch¨olkopf, A. J. Smola, and K.-R. M¨uller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998. [9] A. J. Smola and B. Sch¨olkopf. On a kernel-based method for pattern recognition, regression, approximation and operator inversion. Algorithmica, 22:211–231, 1998. [10] G. Wahba. Spline Models for Observational Data, volume 59 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1990.
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Seeing through water Alexei A. Efros∗ School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213, U.S.A. efros@cs.cmu.edu Volkan Isler, Jianbo Shi and Mirk´o Visontai Dept. of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 {isleri,jshi,mirko}@cis.upenn.edu Abstract We consider the problem of recovering an underwater image distorted by surface waves. A large amount of video data of the distorted image is acquired. The problem is posed in terms of finding an undistorted image patch at each spatial location. This challenging reconstruction task can be formulated as a manifold learning problem, such that the center of the manifold is the image of the undistorted patch. To compute the center, we present a new technique to estimate global distances on the manifold. Our technique achieves robustness through convex flow computations and solves the “leakage” problem inherent in recent manifold embedding techniques. 1 Introduction Consider the following problem. A pool of water is observed by a stationary video camera mounted above the pool and looking straight down. There are waves on the surface of the water and all the camera sees is a series of distorted images of the bottom of the pool, e.g. Figure 1. The aim is to use these images to recover the undistorted image of the pool floor – as if the water was perfectly still. Besides obvious applications in ocean optics and underwater imaging [1], variants of this problem also arise in several other fields, including astronomy (overcoming atmospheric distortions) and structure-from-motion (learning the appearance of a deforming object). Most approaches to solve this problem try to model the distortions explicitly. In order to do this, it is critical not only to have a good parametric model of the distortion process, but also to be able to reliably extract features from the data to fit the parameters. As such, this approach is only feasible in well understood, highly controlled domains. On the opposite side of the spectrum is a very simple method used in underwater imaging: simply, average the data temporally. Although this method performs surprisingly well in many situations, it fails when the structure of the target image is too fine with respect to the amplitude of the wave (Figure 2). In this paper we propose to look at this difficult problem from a more statistical angle. We will exploit a very simple observation: if we watch a particular spot on the image plane, most of the time the picture projected there will be distorted. But once in a while, when the water just happens to be locally flat at that point, we will be looking straight down and seeing exactly the right spot on the ground. If we can recognize when this happens ∗Authors in alphabetical order. Figure 1: Fifteen consecutive frames from the video. The experimental setup involved: a transparent bucket of water, the cover of a vision textbook “Computer Vision/A Modern Approach”. Figure 2: Ground truth image and reconstruction results using mean and median and snap the right picture at each spatial location, then recovering the desired ground truth image would be simply a matter of stitching these correct observations together. In other words, the question that we will be exploring in this paper is not where to look, but when! 2 Problem setup Let us first examine the physical setup of our problem. There is a “ground truth” image G on the bottom of the pool. Overhead, a stationary camera pointing downwards is recording a video stream V . In the absence of any distortion V (x, y, t) = G(x, y) at any time t. However, the water surface refracts in accordance with Snell’s Law. Let us consider what the camera is seeing at a particular point x on the CCD array, as shown in Figure 3(c) (assume 1D for simplicity). If the normal to the water surface directly underneath x is pointing straight up, there is no refraction and V (x) = G(x). However, if the normal is tilted by angle θ1, light will bend by the amount θ2 = θ1 −sin−1 ( 1 1.33 sin θ1), so the camera point V (x) will see the light projected from G(x + dx) on the ground plane. It is easy to see that the relationship between the tilt of the normal to the surface θ1 and the displacement dx is approximately linear (dx ≈0.25θ1h using small angle approximation, where h is the height of the water). This means that, in 2D, what the camera will be seeing over time at point V (x, y, t) are points on the ground plane sampled from a disk centered at G(x, y) and with radius related to the height of the water and the overall roughness of the water surface. A similar relationship holds in the inverse direction as well: a point G(x, y) will be imaged on a disk centered around V (x, y). What about the distribution of these sample points? According to Cox-Munk Law [2], the surface normals of rough water are distributed approximately as a Gaussian centered around the vertical, assuming a large surface area and stationary waves. Our own experiments, conducted by hand-tracking (Figure 3b), confirm that the distribution, though not exactly Gaussian, is definitely unimodal and smooth. Up to now, we only concerned ourselves with infinitesimally small points on the image or the ground plane. However, in practice, we must have something that we can compute with. Therefore, we will make an assumption that the surface of the water can be locally approximated by a planar patch. This means that everything that was true for points is now true for local image patches (up to a small affine distortion). 3 Tracking via embedding From the description outlined above, one possible solution emerges. If the distribution of a particular ground point on the image plane is unimodal, then one could track feature points in the video sequence over time. Computing their mean positions over the entire video will give an estimate of their true positions on the ground plane. Unfortunately, tracking over long periods of time is difficult even under favorable conditions, whereas our data is so fast (undersampled) and noisy that reliable tracking is out of the question (Figure 3(c)). However, since we have a lot of data, we can substitute smoothness in time with smoothness in similarity – for a given patch we are more likely to find a patch similar to it somewhere in time, and will have a better chance to track the transition between them. An alternative to tracking the patches directly (which amounts to holding the ground patch G(x, y) fixed and centering the image patches V (x+dxt, y+dyt) on top of it in each frame), is to fix the image patch V (x, y) in space and observe the patches from G(x + dxt, y + dyt) appearing in this window. We know that this set of patches comes from a disk on the ground plane centered around patch G(x, y) – our goal. If the disk was small enough compared to the size of the patch, we could just cluster the patches together, e.g. by using translational EM [3]. Unfortunately, the disk can be rather large, containing patches with no overlap at all, thus making only the local similarity comparisons possible. However, notice that our set of patches lies on a low-dimensional manifold; in fact we know precisely which manifold – it’s the disk on the ground plane centered at G(x, y)! So, if we could use the local patch similarities to find an embedding of the patches in V (x, y, t) on this manifold, the center of the embedding will hold our desired patch G(x, y). The problem of embedding the patches based on local similarity is related to the recent work in manifold learning [4, 5]. Basic ingredients of the embedding algorithms are: defining a distance measure between points, and finding an energy function that optimally places them in the embedding space. The distance can be defined as all-pairs distance matrix, or as distance from a particular reference node. In both cases, we want the distance function to satisfy some constraints to model the underlying physical problem. The local similarity measure for our problem turned out to be particularly unreliable, so none of the previous manifold learning techniques were adequate for our purposes. In the following section we will describe our own, robust method for computing a global distance function and finding the right embedding and eventually the center of it. θ1 θ2 N h Surface G(x) G(x + dx) (a) (b) (c) Figure 3: (a) Snell’s Law (b)-(c) Tracking points of the bottom of the pool: (b) the tracked position forms a distribution close to a Gaussian, (c): a vertical line of the image shown at different time instances (horizontal axis). The discontinuity caused by rapid changes makes the tracking infeasible. 4 What is the right distance function? Let I = {I1, . . . , In} be the set of patches, where It = V (x, y, t) and x = [xmin, xmax], y = [ymin, ymax] are the patch pixel coordinates. Our goal is to find a center patch to represent the set I. To achieve this goal, we need a distance function d : I × I →IR such that d(Ii, Ij) < d(Ii, Ik) implies that Ij is more similar to Ii than Ik. Once we have such a measure, the center can be found by computing: I∗= arg min Ii∈I X Ij∈I d(Ii, Ij) (1) Unfortunately, the measurable distance functions, such as Normalized Cross Correlation (NCC) are only local. A common approach is to design a global distance function using the measurable local distances and transitivity [6, 4]. This is equivalent to designing a global distance function of the form: d(Ii, Ij) = dlocal(Ii, Ij), if dlocal(Ii, Ij) ≤τ dtransitive(Ii, Ij), otherwise. (2) where dlocal is a local distance function, τ is a user-specified threshold and dtransitive is a global, transitive distance function which utilizes dlocal. The underlying assumption here is that the members of I lie on a constraint space (or manifold) S. Hence, a local similarity function such as NCC can be used to measure local distances on the manifold. An important research question in machine learning is to extend the local measurements into global ones, i.e. to design dtransitive above. One method for designing such a transitive distance function is to build a graph G = (V, E) whose vertices correspond to the members of I. The local distance measure is used to place edges which connect only very similar members of I. Afterwards, the length of pairwise shortest paths are used to estimate the true distances on the manifold S. For example, this method forms the basis of the well-known Isomap method [4]. Unfortunately, estimating the distance dtransitive(·, ·) using shortest path computations is not robust to errors in the local distances – which are very common. Consider a patch that contains the letter A and another one that contains the letter B. Since they are different letters, we expect that these patches would be quite distant on the manifold S. However, among the A patches there will inevitably be a very blurry A that would look quite similar to a very blurry B producing an erroneous local distance measurement. When the transitive global distances are computed using shortest paths, a single erroneous edge will singlehandedly cause all the A patches to be much closer to all the B patches, short-circuiting the graph and completely distorting all the distances. Such errors lead to the leakage problem in estimating the global distances of patches. This problem is illustrated in Figure 4. In this example, our underlying manifold S is a triangle. Suppose our local distance function erroneously estimates an edge between the corners of the triangle as shown in the figure. After the erroneous edge is inserted, the shortest paths from the top of the triangle leak through this edge. Therefore, the shortest path distances will fail to reflect the true distance on the manifold. 5 Solving the leakage problem Recall that our goal is to find the center of our data set as defined in Equation 1. Note that, in order to compute the center we do not need all pairwise distances. All we need is the quantity dI(Ii) = P Ij∈I d(Ii, Ij) for all Ii. The leakage problem occurs when we compute the values dI(Ii) using the shortest path metric. In this case, even a single erroneous edge may reduce the shortest paths from many different patches to Ii – changing the value of dI(Ii) drastically. Intuitively, in order to prevent the leakage problem we must prevent edges from getting involved in many shortest path computations to the same node (i.e. leaking edges). We can formalize this notion by casting the computation as a network flow problem. Let G = (V, E) be our graph representation such that for each patch Ii ∈I, there is a vertex vi ∈V . The edge set E is built as follows: there is an edge (vi, vj) if dlocal(Ii, Ij) is less than a threshold. The weight of the edge (vi, vj) is equal to dlocal(Ii, Ij). To compute the value dI(Ii), we build a flow network whose vertex set is also V . All vertices in V −{vi} are sources, pushing unit flow into the network. The vertex vi is a sink with infinite capacity. The arcs of the flow network are chosen using the edge set E. For each edge (vj, vk) ∈E we add the arcs vj →vk and vk →vj. Both arcs have infinite capacity and the cost of pushing one unit of flow on either arc is equal to the weight of (vj, vk), as shown in Figure 4 left (top and bottom). It can easily be seen that the minimum cost flow in this network is equal to dI(Ii). Let us call this network which is used to compute dI(Ii) as NW(Ii). The crucial factor in designing such a flow network is choosing the right cost and capacity. Computing the minimum cost flow on NW(Ii) not only gives us dI(Ii) but also allows us to compute how many times an edge is involved in the distance computation: the amount of flow through an edge is exactly the number of times that edge is used for the shortest path computations. This is illustrated in Figure 4 (box A) where d1 units of cost is charged for each unit of flow through the edge (u, w). Therefore, if we prevent too much flow going through an edge, we can prevent the leakage problem. Error v v d/∞ A: Shortest Path B: Convex Flow C: Shortest Path with Capacity Error c1 c1 c1 + c2 d1 u u u u v w w w w d1/c1 d1/c1 d2/c2 d1/∞ d3/∞ ∞ Figure 4: The leakage problem. Left: Equivalence of shortest path leakage and uncapacitated flow leakage problem. Bottom-middle: After the erroneous edge is inserted, the shortest paths from the top of the triangle to vertex v go through this edge. Boxes A-C:Alternatives for charging a unit of flow between nodes u and w. The horizontal axis of the plots is the amount of flow and the vertical axis is the cost. Box A: Linear flow. The cost of a unit of flow is d1 Box B: Convex flow. Multiple edges are introduced between two nodes, with fixed capacity, and convexly increasing costs. The cost of a unit of flow increases from d1 to d2 and then to d3 as the amount of flow from u to w increases. Box C: Linear flow with capacity. The cost is d1 until a capacity of c1 is achieved and becomes infinite afterwards. One might think that the leakage problem can simply be avoided by imposing capacity constraints on the arcs of the flow network (Figure 4, box C). Unfortunately, this is not very easy. Observe that in the minimum cost flow solution of the network NW(Ii), the amount of flow on the arcs will increase as the arcs get closer to Ii. Therefore, when we are setting up the network NW(Ii), we must adaptively increase the capacities of arcs “closer” to the sink vi – otherwise, there will be no feasible solution. As the structure of the graph G gets complicated, specifying this notion of closeness becomes a subtle issue. Further, the structure of the underlying space S could be such that some arcs in G must indeed carry a lot of flow. Therefore imposing capacities on the arcs requires understanding the underlying structure of the graph G as well as the space S – which is in fact the problem we are trying to solve! Our proposed solution to the leakage problem uses the notion of a convex flow. We do not impose a capacity on the arcs. Instead, we impose a convex cost function on the arcs such that the cost of pushing unit flow on arc a increases as the total amount of flow through a increases. See Figure 4, box B. This can be achieved by transforming the network NW(Ii) to a new network NW ′(Ii). The transformation is achieved by applying the following operation on each arc in NW(Ii): Let a be an arc from u to w in NW(Ii). In NW ′(Ii), we replace a by k arcs a1, . . . , ak. The costs of these arcs are chosen to be uniformly increasing so that cost(a1) < cost(a2) < . . . < cost(ak). The capacity of arc ak is infinite. The weights and capacities of the other arcs are chosen to reflect the steepness of the desired convexity (Figure 4, box B). The network shown in the figure yields the following function for the cost of pushing x units of flow through the arc: cost(x) = ( d1x, if 0 ≤x ≤c1 d1c1 + d2(x −c1), if c1 ≤x ≤c2 d1c1 + d2(c2 −c1) + d3(x −c1 −c2), if c2 ≤x (3) The advantage of this convex flow computation is twofold. It does not require putting thresholds on the arcs a-priori. It is always feasible to have as much flow on a single arc as required. However, the minimum cost flow will avoid the leakage problem because it will be costly to use an erroneous edge to carry the flow from many different patches. 5.1 Fixing the leakage in Isomap As noted earlier, the Isomap method [4] uses the shortest path measurements to estimate a distance matrix M. Afterwards, M is used to find an embedding of the manifold S via MDS. As expected, this method also suffers from the leakage problem as demonstrated in Figure 5. The top-left image in Figure 5 shows our ground truth. In the middle row, we present an embedding of these graphs computed using Isomap which uses the shortest path length as the global distance measure. As illustrated in these figures, even though isomap does a good job in embedding the ground truth when there are no errors, the embedding (or manifold) collapses after we insert the erroneous edges. In contrast, when we use the convex-flow based technique to estimate the distances, we recover the true embedding – even in the presence of erroneous edges (Figure 5 bottom row). 6 Results In our experiments we used 800 image frames to reconstruct the ground truth image. We fixed 30 × 30 size patches in each frame at the same location (see top of Figure 7 for two sets of examples), and for every location we found the center. The middle row of Figure 7 shows embeddings of the patches computed using the distance derived from the convex flow. The transition path and the morphing from selected patches (A,B,C) to the center patch (F) is shown at the bottom. The embedding plot on the left is considered an easier case, with a Gaussian-like embedding (the graph is denser close to the center) and smooth transitions between the patches in a transition path. The plot to the right shows a more difficult example, when the embedding has no longer a Gaussian shape, but rather a triangular one. Also note that the transitions can have jumps connecting non-similar patches which are distant in the embedding space. The two extremes of the triangle represent the blurry patches, which are so numerous and Ground Truth −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C Isomap [4] −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C Convex flow −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C −0.6 −0.4 −0.2 0 0.2 0.4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 A B C Figure 5: Top row: Ground truth. After sampling points from a triangular disk, a kNN graph is constructed to provide a local measure for the embedding (left). Additional erroneous edges AC and CB are added to perturb the local measure (middle, right). Middle row: Isomap embedding. Isomap recovers the manifold for the error-free cases (left). However, all-pairs shortest path can “leak” through AC and CB, resulting a significant change in the embedding. Bottom row: Convex flow embedding. Convex flow penalized too many paths going through the same edge – correcting the leakage problem. The resulting embedding is more resistant to perturbations in the kNN graph. very similar to each other, so that they are no longer treated as noise or outliers. This results in ‘folding in’ the embedding and thus, moving estimated the center towards the blurry patches. To solve this problem, we introduced additional two centers, which ideally would represent the blurry patches, allowing the third center to move to the ground truth. Once we have found the centers for all patches we stitched them together to form the complete reconstructed image. In case of three centers, we use overlapping patches and dynamic programming to determine the best stitching. Figure 6 shows the reconstruction Figure 6: Comparison of reconstruction results of different methods using the first 800 frames, top: patches stitched together which are closest to mean (left) and median (right), bottom: our results using a single (left) and three (right) centers result of our algorithm compared to simple methods of taking the mean/median of the patches and finding the closest patch to them. The bottom row shows our result for a single and for three center patches. The better performance of the latter suggests that the two new centers relieve the correct center from the blurry patches. For a graph with n vertices and m edges, the minimum cost flow computation takes O(m log n(m + n log n)) time, therefore finding the center I∗of one set of patches can be done in O(mn log n(m + n log n)) time. Our flow computation is based on the min-cost max-flow implementation by Goldberg [7]. The convex function used in our experiments was as described in Equation 3 with parameters d1 = 1, c1 = 1, d2 = 5, c2 = 9, d3 = 50. A B C F A1 A2 B1 B2 C1 C2 FA FB FC F F A F B F C FA A1 FA A2 FB B1 FB B2 FC C1 FC C2 Figure 7: Top row: sample patches (two different locations) from 800 frames, Middle row: Convex flow embedding, showing the transition paths. Bottom row: corresponding patches (A, B, C, A1, A2, B1, B2, C1, C2) and the morphing of them to the centers F F, FA, FB, FC respectively 7 Conclusion In this paper, we studied the problem of recovering an underwater image from a video sequence. Because of the surface waves, the sequence consists of distorted versions of the image to be recovered. The novelty of our work is in the formulation of the reconstruction problem as a manifold embedding problem. Our contribution also includes a new technique, based on convex flows, to recover global distances on the manifold in a robust fashion. This technique solves the leakage problem inherent in recent embedding methods. References [1] Lev S. Dolin, Alexander G. Luchinin, and Dmitry G. Turlaev. Correction of an underwater object image distorted by surface waves. In International Conference on Current Problems in Optics of Natural Waters, pages 24–34, St. Petersburg, Russia, 2003. [2] Charles Cox and Walter H. Munk. Slopes of the sea surface deduced from photographs of sun glitter. Scripps Inst. of Oceanogr. Bull., 6(9):401–479, 1956. [3] Brendan Frey and Nebojsa Jojic. Learning mixture models of images and inferring spatial transformations using the em algorithm. In IEEE Conference on Computer Vision and Pattern Recognition, pages 416–422, Fort Collins, June 1999. [4] Joshua B. Tenenbaum, Vin de Silva, and John C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, pages 2319–2323, Dec 22 2000. [5] Sam Roweis and Lawrence Saul. Nonlinear dimeansionality reduction by locally linear embedding. Science, 290(5500):2323–2326, Dec 22 2000. [6] Bernd Fischer, Volker Roth, and Joachim M. Buhmann. Clustering with the connectivity kernel. In Advances in Neural Information Processing Systems 16. MIT Press, 2004. [7] Andrew V. Goldberg. An efficient implementation of a scaling minimum-cost flow algorithm. Journal of Algorithms, 22:1–29, 1997.
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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 m f b T 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 π. 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 π. The observed image x is generated by adding noise β 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 π, both of which are to be inferred from the image set. The elements of π 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 Tπ. Given the transformed mask prior, the conditional distribution for the kth mask pixel is P(mk = 1 | π, T) = (Tπ)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, β) = N(xk | (Tf)k, β−1 f )mkN(xk | bk, β−1 b )1−mk (2) where β = (βf, βb) 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 β is a broad Gamma distribution. The prior on each element of π 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 π and also with respect to βf and βb. 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 π is a Beta distribution Q(πk) = β(πk | a′, b′). 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) = ⟨log P({x1, . . . , xN}, H)⟩∼Q(Hi) + const. (3) m f b T x N 2 b b 2 f f − ) 1 log( log π π 2 Tf Tf − ) 1 log( log π π T T −m m 1 β β log − − − ) 1( * ) 1( 2 1 m x m b b β β − m x m f f β β 2 1 * − − − − 2 2 1 2 2 1 ) ( log ) ( log Tf x b x f f b b β β β β − − − − − 2 2 1 2 2 1 ) ( * ) ( * ) 1( Tf x m b x m −m m 1 − − − m T m T 1 1 1 − − − m T x m T f f 1 2 1 1 ) * ( β β 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 ⟨.⟩∼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 | µ, γ−1), the parameter vector of the factor Q(b) = N(b | µ′, γ′−1) is updated from the messages received at b using natural param. vector z }| { µ′γ′ −1 2γ′ = prior z }| { µγ −1 2γ + N X i=1 received messages z }| { ⟨βbi(1 −mi) ∗xi⟩ −1 2 ⟨βbi(1 −mi)⟩ . (4) 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 = TxyTrsTa. (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, log Q(Txy) = ⟨m⟩. (Txy ⟨TrsTa log π⟩) + ⟨1 −m⟩. (Txy ⟨TrsTa log(1 −π)⟩) + βf ⟨m⟩. x ∗Txy ⟨TrsTaf⟩−1 2Txy TrsTaf 2 + zxy (7) log Q(Trs) = T−1 xy m . (Trs ⟨Ta log π⟩) + T−1 xy (1 −m) . (Trs ⟨Ta log(1 −π)⟩) + βf T−1 xy (m ∗x) . (Trs ⟨Taf⟩) −1 2βf T−1 xy m Trs Taf 2 + zrs(8) where zxy and zrs 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) = δ(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 Fa = T−1 rs T−1 xy m . (T⋆ a ⟨log π⟩) + T−1 rs T−1 xy (1 −m) . (T⋆ a ⟨log(1 −π)⟩) + βf T−1 rs T−1 xy (m ∗x) . (T⋆ a ⟨f⟩) −1 2βf T−1 rs T−1 xy m T⋆ a f 2 . (10) 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 π 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 ⟨π⟩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(π) 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 π is greater than 0.5. 4.1 Using bottom-up information to improve inference Given that π is centred, we can significantly improve convergence by using bottom-up information about the translation of the object. For example, the inferred mask mi 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 | ⟨mi⟩) 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 α ≈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 seAppearance & 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 π 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 = sif + (1 −si)T−1 i−1xi−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 si is found to have an expected value ≈0 for all frames (except the first). Using the tracked appearance 1Videos 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, ⟨si⟩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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Schema Learning: Experience-Based Construction of Predictive Action Models Michael P. Holmes College of Computing Georgia Institute of Technology Atlanta, GA 30332-0280 mph@cc.gatech.edu Charles Lee Isbell, Jr. College of Computing Georgia Institute of Technology Atlanta, GA 30332-0280 isbell@cc.gatech.edu Abstract Schema learning is a way to discover probabilistic, constructivist, predictive action models (schemas) from experience. It includes methods for finding and using hidden state to make predictions more accurate. We extend the original schema mechanism [1] to handle arbitrary discrete-valued sensors, improve the original learning criteria to handle POMDP domains, and better maintain hidden state by using schema predictions. These extensions show large improvement over the original schema mechanism in several rewardless POMDPs, and achieve very low prediction error in a difficult speech modeling task. Further, we compare extended schema learning to the recently introduced predictive state representations [2], and find their predictions of next-step action effects to be approximately equal in accuracy. This work lays the foundation for a schema-based system of integrated learning and planning. 1 Introduction Schema learning1 is a data-driven, constructivist approach for discovering probabilistic action models in dynamic controlled systems. Schemas, as described by Drescher [1], are probabilistic units of cause and effect reminiscent of STRIPS operators [3]. A schema predicts how specific sensor values will change as different actions are executed from within particular sensory contexts. The learning mechanism also discovers hidden state features in order to make schema predictions more accurate. In this work we have generalized and extended Drescher’s original mechanism to learn more accurate predictions by using improved criteria both for discovery and refinement of schemas as well as for creation and maintenance of hidden state. While Drescher’s work included mechanisms for action selection, here we focus exclusively on the problem of learning schemas and hidden state to accurately model the world. In several benchmark POMDPs, we show that our extended schema learner produces significantly better action models than the original. We also show that the extended learner performs well on a complex, noisy speech modeling task, and that its prediction accuracy is approximately equal to that of predictive state representations [2] on a set of POMDPs, with faster convergence. 1This use of the term schema derives from Piaget’s usage in the 1950s; it bears no relation to database schemas or other uses of the term. 2 Schema Learning Schema learning is a process of constructing probabilistic action models of the environment so that the effects of agent actions can be predicted. Formally, a schema learner is fitted with a set of sensors S = {s1, s2, . . .} and a set of actions A = {a1, a2, . . .} through which it can perceive and manipulate the environment. Sensor values are discrete: sj i means that si has value j. As it observes the effects of its actions on the environment, the learner constructs predictive units of sensorimotor cause and effect called schemas. A schema C ai −→R essentially says, “If I take action ai in situation C, I will see result R.” Schemas thus have three components: (1) the context C = {c1, c2, . . . , cn} , which is a set of sensor conditions ci ≡sk j that must hold for the schema to be applicable, (2) the action that is taken, and (3) the result, which is a set of sensor conditions R = {r1, r2, . . . , rm} predicted to follow the action. A schema is said to be applicable if its context conditions are satisfied, activated if it is applicable and its action is taken, and to succeed if it is activated and the predicted result is observed. Schema quality is measured by reliability, which is the probability that activation culminates in success: Rel(C ai −→R) = prob(Rt+1|Ct, ai(t)). Note that schemas are not rules telling an agent what to do; rather, they are descriptions of what will happen if the agent takes a particular action in a specific circumstance. Also note that schema learning has no predefined states such as those found in a POMDP or HMM; the set of sensor readings is the state. Because one schema’s result can set up another schema’s context, schemas fit naturally into a planning paradigm in which they are chained from the current situation to reach sensor-defined goals. 2.1 Discovery and Refinement Schema learning comprises two basic phases: discovery, in which context-free action/result schemas are found, and refinement, in which context is added to increase reliability. In discovery, statistics track the influence of each action ai on each sensor condition sj r. Drescher’s original schema mechanism accommodated only binary-valued sensors, but we have generalized it to allow a heterogeneous set of sensors that take on arbitrary discrete values. In the present work, we assume that the effects of actions are observed on the subsequent timestep, which leads to the following criterion for discovering action effects: count(at, sj r(t+1)) > θd, (1) where θd is a noise-filtering threshold. If this criterion is met, the learner constructs a schema ∅ ai −→sj r , where the empty set, ∅, means that the schema is applicable in any situation. This works in a POMDP because it means that executing ai in some state has caused sensor sr to give observation j, implying that such a transition exists in the underlying (but unknown) system model. The presumption is that we can later learn what sensory context makes this transition reliable. Drescher’s original discovery criterion generalizes in the non-binary case to: prob(sj r(t+1)|at) prob(sj r(t+1)|at) > θod, (2) where θod > 1 and at means a was not taken at time t. Experiments in worlds of known structure show that this criterion misses many true action effects. When a schema is discovered, it has no context, so its reliability may be low if the effect occurs only in particular situations. Schemas therefore begin to look for context conditions Criterion Extended Schema Learner Original Schema Learner Discovery count(at, sj r(t+1)) > θd prob(sj r(t+1)|at) prob(sj r(t+1)|at) > θod Binary sensors only Refinement Rel(C ∪{sj c} ai −→R) Rel(C ai −→R) > θ Rel(C ∪{sj c} ai −→R) Rel(C ai −→R) > θ Annealed threshold Static threshold Binary sensors only Synthetic Item Creation 0 < Rel(C ai −→R) < θ 0 < Rel(C ai −→R) < θ No context refinement possible Schema is locally consistent Synthetic Item Maintenance Predicted by other schemas Average duration Table 1: Comparison of extended and original schema learners. that increase reliability. The criterion for adding sj c to the context of C ai −→R is: Rel(C ∪{sj c} ai −→R) Rel(C ai −→R) > θc, (3) where θc > 1. In practice we have found it necessary to anneal θc to avoid adding spurious context. Once the criterion is met, a child schema C′ ai −→R is formed, where C′ = C∪sj c. 2.2 Synthetic Items In addition to basic discovery and refinement of schemas, a schema learner also discovers hidden state. Consider the case where no context conditions are found to make a schema reliable. There must be unperceived environmental factors on which the schema’s reliability depends (see [4]). The schema learner therefore creates a new binary-valued virtual sensor, called a synthetic item, to represent the presence of conditions in the environment that allow the schema to succeed. This addresses the state aliasing problem by splitting the state space into two parts, one where the schema succeeds, and one where it does not. Synthetic items are said to reify the host schemas whose success conditions they represent; they have value 1 if the host schema would succeed if activated, and value 0 otherwise. Upon creation, a synthetic item begins to act as a normal sensor, with one exception: the agent has no way of directly perceiving its value. Creation and state maintenance criteria thus emerge as the main problems associated with synthetic items. Drescher originally posited two conditions for the creation of a synthetic item: (1) a schema must be unreliable, and (2) the schema must be locally consistent, meaning that if it succeeds once, it has a high probability of succeeding again if activated soon afterward. The second of these conditions formalizes the assumption that a well-behaved environment has persistence and does not tend to radically change from moment to moment. This was motivated by the desire to capture Piagetian “conservation phenomena.” While well-motivated, we have found that the second condition is simply too restrictive. Our criterion for creating synthetic items is 0 < Rel(C ai −→R) < θr, subject to the constraint that the statistics governing possible additional context conditions have converged. When this criterion is met, a synthetic item is created and is thenceforth treated as a normal sensor, able to be incorporated into the contexts and results of other schemas. A newly created synthetic item is grounded: it represents whatever conditions in the world allow the host schema to succeed when activated. Thus, upon activation of the host schema, we retroactively know the state of the synthetic item at the time of activation (1 if the schema succeeded, 0 otherwise). Because the synthetic item is treated as a sensor, we can Figure 1: Benchmark problems. (left) The flip system. All transitions are deterministic. (right) The float/reset system. Dashed lines represent float transitions that happen with probability 0.5, while solid lines represent deterministic reset transitions. discover which previous actions led to each synthetic item state, and the synthetic item can come to be included as a result condition in new schemas. Once we have reliable schemas that predict the state of a synthetic item, we can begin to know its state non-retroactively, without having to activate the host schema. The synthetic item’s state can potentially be known just as well as that of the regular sensors, and its addition expands the state representation in just such a way as to make sensory predictions more reliable. Predicted synthetic item state implicitly summarizes the relevant preceding history: it indicates that one of the schemas that predicts it was just activated. If the predicting schema also has a synthetic item in its context, an additional step of history is implied. Such chaining allows synthetic items to summarize arbitrary amounts of history without explicitly remembering any of it. This use of schemas to predict synthetic item state is in contrast to [1], which relied on the average duration of synthetic item states in order to predict them. Table 1 compares our extended schema learning criteria with Drescher’s original criteria. 3 Empirical Evaluation In order to test the advantages of the extended learning criteria, we compared four versions of schema learning. The first two were basic learners that made no use of synthetic items, but discovered and refined schemas using our extended criteria in one case, and the direct generalizations of Drescher’s original criteria in the other. The second pair added the extended and original synthetic item mechanisms, respectively, to the first pair. Our first experimental domains are based on those used in [5]. They have a mixture of transient and persistent hidden state and, though small, are non-trivial.2 The flip system is shown on the left in Figure 1; it features deterministic transitions, hidden state, and a null action that confounds simplistic history approaches to handling hidden state. The float/reset system is illustrated on the right side of Figure 1; it features both deterministic and stochastic transitions, as well as a more complicated hidden state structure. Finally, we use a modified float/reset system in which the f action from the two right-most states leads deterministically to their left neighbor; this reveals more about the hidden state structure. To test predictive power, each schema learner, upon taking an action, uses the most reliable of all activated schemas to predict what the next value of each sensor will be. If there is no activation of a reliable schema to predict the value of a particular sensor, its value is predicted to stay constant. Error is measured as the fraction of incorrect predictions. In these experiments, actions were chosen uniformly at random, and learning was allowed to continue throughout.3 No learning parameters are changed over time; schemas stop being created when discovery and refinement criteria cease to generate them. Figure 2 shows the performance in each domain, while Table 2 summarizes the average error. 2E.g. [5] showed that flip is non-trivial because it cannot be modeled exactly by k-Markov models, and its EM-trained POMDP representations require far more than the minimum number of states. 3Note that because a prediction is made before each observation, the observation does not contribute to the learning upon which its predicted value is based. 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.1 0.2 0.3 0.4 0.5 0.6 PREDICTION ERROR TIMESTEP extended extended baseline original original baseline flip 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.1 0.2 0.3 0.4 0.5 0.6 TIMESTEP PREDICTION ERROR extended extended baseline original original baseline float/reset 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.1 0.2 0.3 0.4 0.5 0.6 PREDICTION ERROR TIMESTEP extended extended baseline original original baseline modified float/reset 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 0 0.1 0.2 0.3 0.4 0.5 0.6 TIMESTEP PREDICTION ERROR weather predictor 2−context schema learner 3−context schema learner speech modeling Figure 2: Prediction error in several domains. The x-axis represents timesteps and the y-axis represents error. Each point represents average error over 100 timesteps. In the speech modeling graph, learning is stopped after approximately 4300 timesteps (shown by the vertical line), after which no schemas are added, though reliabilities continue to be updated. Learner flip float/reset modified f/r Extended 0.020 0.136 0.00716 Extended baseline 0.331 0.136 0.128 Original 0.426 0.140 0.299 Original baseline 0.399 0.139 0.315 Table 2: Average error. Calculated over 10 independent runs of 10,000 timesteps each. 3.1 Speech Modeling The Japanese vowel dataset [6] contains time-series recordings of nine Japanese speakers uttering the ae vowel combination 54-118 times. Each data point consists of 12 continuousvalued cepstral coefficients, which we transform into 12 sensors with five discrete values each. The data is noisy and the dynamics are non-stationary between speakers. Each utterance is divided in half, with the first half treated as the action of speaking a and the latter half as e. In order to more quickly adapt to discontinuity resulting from changes in speaker, reliability was calculated using an exponential weighting of more recent observations; each relevant probability p was updated according to: pt+1 = αpt + (1 −α) 1 if event occurred at time t 0 otherwise . (4) The parameter α is set equal to the current prediction accuracy so that decreased accuracy causes faster adaptation. Several modifications were necessary for tractability: (1) schemas whose reliability fell below a threshold of their parents’ reliability were removed, (2) context sizes were, on separate experimental runs, restricted to two and three items, and (3) the synthetic item mechanisms were deactivated. Figure 2 displays results for this learner compared to a baseline weather predictor.4 3.2 Analysis In each benchmark problem, the learners drop to minimum error after no more than 1000 timesteps. Large divergence in the curves corresponds to the creation of synthetic items and the discovery of schemas that predict synthetic item state. Small divergence corresponds to differences in discovery and refinement criteria. In flip and modified float/reset, the extended schema learner reaches zero error, having a complete model of the hidden state, and outperforms all other learners, while the extended basic version outperforms both original learners. In float/reset, all learners perform approximately equally, reflecting the fact that, given the hidden stochasticity of this system, the best schema for action r is one that, without reference to synthetic items, gives a prediction of 1. Surprisingly, the original learner never significantly outperformed its baseline, and even performed worse than the baseline in flip. This is accounted for by the duration-based maintenance of synthetic items, which causes the original learner to maintain transient synthetic item state longer than it should. Prediction-based synthetic item maintenance overcomes this limitation. The speech modeling results show that schema learning can induce high-quality action models in a complex, noisy domain. With a maximum of three context conditions, it averaged only 1.2% error while learning, and 1.6% after learning stopped, a large improvement over the 30.3% error of the baseline weather predictor. Note that allowing three instead of two context conditions dropped the error from 4.6% to 1.2% and from 9.0% to 1.6% in the training and testing phases, respectively, demonstrating the importance of incremental specialization of schemas through context refinement. All together, these results show that our extended schema learner produces better action models than the original, and can handle more complex domains. Synthetic items are seen to effectively model hidden state, and prediction-based maintenance of synthetic item state is shown to be more accurate than duration-based maintenance in POMDPs. Discovery of schemas is improved by our criterion, missing fewer legitimate schemas, and therefore producing more accurate predictions. Refinement using the annealed generalization of the original criterion performs correctly with a lower false positive rate. 4 Comparison to Predictive State Representations Predictive state representations (PSRs; [2]), like schema learning, are based on grounded, sensorimotor predictions that uncover hidden state. Instead of schemas, PSRs rely on the notion of tests. A test q is a series of alternating actions and observations a0o0a1o1 . . . anon. In a PSR, the environment state is represented as the probabilities that each of a set of core tests would yield its observations if its actions were executed. These probabilities are updated at each timestep by combining the current state with the new action/observation pair. In this way, the PSR implicitly contains a sufficient history-based statistic for prediction, and should overcome aliasing relative to immediate observations. [2] shows that linear PSRs are at least as compact and general as POMDPs, while [5] shows that PSRs can learn to accurately maintain their state in several POMDP problems. A schema is similar to a one-step PSR test, and schema reliability roughly corresponds to the probability of a PSR test. Schemas differ, however, in that they only specify context and result incrementally, incorporating incremental history via synthetic items, while PSR tests incorporate the complete history and full observations (i.e. all sensor readings at once) into 4A weather predictor always predicts that values will stay the same as they are presently. Problem PSR Schema Learner Difference Schema Learning Steps flip 0 0 0 10, 000 float/reset 0.11496 0.13369 0.01873 10, 000 network 0.04693 0.06457 0.01764 10, 000 paint 0.20152 0.21051 0.00899 30, 000 Table 3: Prediction error for PSRs and schema learning on several POMDPs. Error is averaged over 10 epochs of 10,000 timesteps each. Performance differs by less than 2% in every case. a test probability. A multi-step test can say more about the current state than a schema, but is not as useful for regression planning because there is no way to extract the probability that a particular one of its observations will be obtained. Thus, PSRs are more useful as Markovian state for reinforcement learning, while schemas are useful for explicit planning. Note that synthetic items and PSR core test probabilities both attempt to capture a sufficient history statistic without explicitly maintaining history. This suggests a deeper connection between the two approaches, but the relationship has yet to be formalized. We compared the predictive performance of PSRs with that of schema learning on some of the POMDPs from [5]. One-step PSR core tests can be used to predict observations: as an action is taken, the probability of each observation is the probability of the one-step core test that uses the current action and terminates in that observation. We choose the most probable observation as the PSR prediction. This allows us to evaluate PSR predictions using the same error measure (fraction of incorrect predictions) as in schema learning.5 In our experiments, the extended schema learner was first allowed to learn until it reached an asymptotic minimum error (no longer than 30,000 steps). Learning was then deactivated, and the schema learner and PSR each made predictions over a series of randomly chosen actions. Table 3 presents the average performance for each approach. Learning PSR parameters required 1-10 million timesteps [5], while schema learning used no more than 30,000 steps. Also, learning PSR parameters required access to the underlying POMDP [5], whereas schema learning relies solely on sensorimotor information. 5 Related Work Aside from PSRs, schema learning is also similar to older work in learning planning operators, most notably that of Wang [7], Gil [8], and Shen [9]. These approaches use observations to learn classical, deterministic STRIPS-like operators in predicate logic environments. Unlike schema learning, they make the strong assumption that the environment does not produce noisy observations. Wang and Gil further assume no perceptual aliasing. Other work in this area has attempted to handle noise, but only in the problem of context refinement. Benson [10] gives his learner prior knowledge about action effects, and the learner finds conditions to make the effects reliable with some tolerance for noise. One advantage of Benson’s formalism is that his operators are durational, rather than atomic over a single timestep. Balac et al. [11] use regression trees to find regions of noisy, continuous sensor space that cause a specified action to vary in the degree of its effect. Finally, Shen [9] and McCallum [12] have mechanisms for handling state aliasing. Shen uses differences in successful and failed predictions to identify pieces of history that reveal hidden state. His approach, however, is completely noise intolerant. McCallum’s UTree algorithm selectively adds pieces of history in order to maximize prediction of reward. 5Unfortunately, not all the POMDPs from [5] had one-step core tests to cover the probability of every observation given every action. We restricted our comparisons to the four systems that had at least two actions for which the probability of all next-step observations could be determined. This bears a strong resemblance to the history represented by chains of synthetic items, a connection that should be explored more fully. Synthetic items, however, are for general sensor prediction, which contrasts with UTree’s task-specific focus on reward prediction. Schema learning, PSRs, and the UTree algorithm are all highly related in this sense of selectively tracking history information to improve predictive performance. 6 Discussion and Future Work We have shown that our extended schema learner produces accurate action models for a variety of POMDP systems and for a complex speech modeling task. The extended schema learner performs substantially better than the original, and compares favorably in predictive power to PSRs while appearing to learn much faster. Building probabilistic goal-regression planning on top of the schemas is a logical next step; however, to succeed with real-world planning problems, we believe that we need to extend the learning mechanism in several ways. For example, the schema learner must explicitly handle actions whose effects occur over an extended duration instead of after one timestep. The learner should also be able to directly handle continuous-valued sensors. Finally, the current mechanism has no means of abstracting similar schemas, e.g., to reduce x1 1 a−→x2 1 and x2 1 a−→x3 1 to xp 1 a−→xp+1 1 . Acknowledgements Thanks to Satinder Singh and Michael R. James for providing POMDP PSR parameters. References [1] G. Drescher. Made-up minds: a constructivist approach to artificial intelligence. MIT Press, 1991. [2] M. L. Littman, R. S. Sutton, and S. Singh. Predictive representations of state. In Advances in Neural Information Processing Systems, pages 1555–1561. MIT Press, 2002. [3] R. E. Fikes and N. J. Nilsson. STRIPS: a new approach to the application of theorem proving to problem solving. Artificial Intelligence, 2:189–208, 1971. [4] C. T. Morrison, T. Oates, and G. King. Grounding the unobservable in the observable: the role and representation of hidden state in concept formation and refinement. In AAAI Spring Symposium on Learning Grounded Representations, pages 45–49. AAAI Press, 2001. [5] S. Singh, M. L. Littman, N. K. Jong, D. Pardoe, and P. Stone. Learning predictive state representations. In International Conference on Machine Learning, pages 712–719. AAAI Press, 2003. [6] M. Kudo, J. Toyama, and M. Shimbo. Multidimensional curve classification using passingthrough regions. Pattern Recognition Letters, 20(11–13):1103–1111, 1999. [7] X. Wang. Learning by observation and practice: An incremental approach for planning operator acquisition. In International Conference on Machine Learning, pages 549–557. AAAI Press, 1995. [8] Y. Gil. Learning by experimentation: Incremental refinement of incomplete planning domains. In International Conference on Machine Learning, pages 87–95. AAAI Press, 1994. [9] W.-M. Shen. Discovery as autonomous learning from the environment. Machine Learning, 12:143–165, 1993. [10] Scott Benson. Inductive learning of reactive action models. In International Conference on Machine Learning, pages 47–54. AAAI Press, 1995. [11] N. Balac, D. M. Gaines, and D. Fisher. Using regression trees to learn action models. In IEEE Systems, Man and Cybernetics Conference, 2000. [12] A. W. McCallum. Reinforcement Learning with Selective Perception and Hidden State. PhD thesis, University of Rochester, 1995.
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Machine Learning Applied to Perception: Decision-Images for Gender Classification Felix A. Wichmann and Arnulf B. A. Graf Max Planck Institute for Biological Cybernetics T¨ubingen, Germany felix.wichmann@tuebingen.mpg.de Eero P. Simoncelli Howard Hughes Medical Institute Center for Neural Science New York University, USA Heinrich H. B¨ulthoff and Bernhard Sch¨olkopf Max Planck Institute for Biological Cybernetics T¨ubingen, Germany Abstract We study gender discrimination of human faces using a combination of psychophysical classification and discrimination experiments together with methods from machine learning. We reduce the dimensionality of a set of face images using principal component analysis, and then train a set of linear classifiers on this reduced representation (linear support vector machines (SVMs), relevance vector machines (RVMs), Fisher linear discriminant (FLD), and prototype (prot) classifiers) using human classification data. Because we combine a linear preprocessor with linear classifiers, the entire system acts as a linear classifier, allowing us to visualise the decision-image corresponding to the normal vector of the separating hyperplanes (SH) of each classifier. We predict that the female-tomaleness transition along the normal vector for classifiers closely mimicking human classification (SVM and RVM [1]) should be faster than the transition along any other direction. A psychophysical discrimination experiment using the decision images as stimuli is consistent with this prediction. 1 Introduction One of the central problems in vision science is to identify the features used by human subjects to classify visual stimuli. We combine machine learning and psychophysical techniques to gain insight into the algorithms used by human subjects during visual classification of faces. Comparing gender classification performance of humans to that of machines has attracted considerable attention in the past [2, 3, 4, 5]. The main novel aspect of our study is to analyse the machine algorithms to make inferences about the features used by human subjects, thus providing an alternative to psychophysical feature extraction techniques such as the “bubbles” [6] or the noise classification image [7] techniques. In this “machine-learning-psychophysics research” we first we train machine learning classifiers on the responses (labels) of human subjects to re-create the human decision boundaries by learning machines. Then we look for correlations between machine classifiers and several characteristics of subjects’ responses to the stimuli—proportion correct, reaction times (RT) and confidence ratings. Ideally this allows us to find preprocessor-classifier pairings that are closely aligned with the algorithm employed by the human brain for the task at hand. Thereafter we analyse properties of the machine closest to the human—in our case support vector machines (SVMs), and to slightly lesser degree, relevance vector machines (RVMs)—and make predictions about human behaviour based on machine properties. In the current study we extract a decision-image containing the information relevant for classification by the machine classifiers. The decision-image ⃗W is the image corresponding to a vector ⃗w orthogonal to the SH of the classifier. The decision-image has the same dimensionality as the (input-) images—in our case 256 × 256—whereas the normal vector lives in the (reduced dimensionality) space after preprocessing—in our case in 200 × 1 after Principal Component Analysis (PCA). Second, we use ⃗w of the classifiers to generate novel stimuli by adding (or subtracting) various “amounts” (λ⃗w) to a genderless face in PCA space. The novel stimuli, images, I(λ) are generated as I(λ) = PCA−1λ ⃗w ∥⃗w∥. We predict that the female-to-maleness transition along the vectors normal to the SHs, ⃗wSVM and ⃗wRVM, should be significantly faster than those along the normal vectors of machine classifiers that do not correlate as well with human subjects. A psychophysical gender discrimination experiment confirms our predictions: the female-to-maleness axis of the SVM and, to a smaller extent, RVM, are more closely aligned with the human female-tomaleness axis than those of the prototype (Prot) and a Fisher linear discriminant (FLD) classifier. 2 Preprocessing and Machine Learning Methods We preprocessed the faces using PCA. PCA is a good preprocessor in the current context since we have previously shown that in PCA-space strong correlations exist between man and machine [1]. Second, there is evidence that the PCA representation may be biologically-plausible [8]. The face stimuli were taken from the gender-balanced Max Planck Institute (MPI) face database1 composed of 200 greyscale 256 × 256-pixel frontal views of human faces, yielding a data matrix X ∈R200×2562. For the gender discrimination task we adhere to the following convention for the class labels: y = −1 for females and y = +1 for males. We consider no dimensionality reduction and keep all 200 components of the PCA. This implies that the reconstruction of the data from the PCA analysis is perfect and we can write: E = ¯XBT ⇔¯X = EB where E ∈R200×200 is the matrix of the encodings (each row is a PCA vector in the space of reduced dimensionality), B ∈R200×2562 is the orthogonal basis matrix and ¯X the centered data matrix. The combination of the encoding matrix E with the true class labels y of the MPI database yields the true dataset, whereas its combination with the class labels yest by the subjects yields the subject dataset. To model classification in human subjects we use methods from supervised machine learning. In particular, we consider linear classifiers where classification is done using a SH defined by its normal vector ⃗w and offset b. Furthermore the normal vector ⃗w of our classifiers can then be written as a linear combination of the input patterns ⃗xi with suitable coefficients αi as ⃗w = P i αi⃗xi. We define the distance of a pattern to the SH as δ(⃗x) = ⟨⃗w|⃗x⟩+b ∥⃗w∥ . Note that in our experiments the ⃗xi are the PCA coefficients of the images, that is ⃗xi ∈R200, whereas the images themselves are in R2562. For the subject dataset we chose the mean values of ⃗w, b and ⃗w± over all subjects. 1The MPI face database is located at http://faces.kyb.tuebingen.mpg.de 2.1 Machine Classifiers The Support Vector Machine (SVM, [9, 10]) is a state-of-the-art maximum margin algorithm based on statistical learning theory. SVMs have an intuitive geometrical interpretation: they classify by maximizing the margin separating both classes while minimizing the classification error. The Relevance Vector Machine (RVM, [11]) is a probabilistic Bayesian classifier. It optimises the expansion coefficients of a SV-style decision function using a hyperprior which favours sparse solutions. Common classifiers in neuroscience, cognitive science and psychology are variants of the Prototype classifier (Prot, [12]). Their popularity is due to their simplicity: they classify according to the nearest mean-of-class prototype; in the simplest form all dimensions are weighted equally but variants exist that weight the dimensions inversely proportional the class variance along the dimensions. As we cannot estimate class variance along all 200 dimensions from only 200 stimuli, we chose to implement the simplest Prot with equal weight along all dimensions. The Fisher linear discriminant classifier (FLD, [13]) finds a direction in the dataset which allows best linear separation of the two classes. This direction is then used as the normal vector of the separating hyperplane. In fact, FLD is arguably a more principled whitened variant of the Prot classifier: Its weight vector can be written as ⃗w = S−1 W (⃗µ+−⃗µ−), where S−1 W is the within class covariance matrix of the two classes, and µ± are the class means. Consequently, if we disregard the constant offset b, we can write the decision function as ⟨⃗w|⃗x⟩= ⟨S−1 W (⃗µ+ −⃗µ−)|⃗x⟩= ⟨S−1/2 W (⃗µ+ −⃗µ−)|S−1/2 W ⃗x⟩, which is a prototype classifier using the prototypes ⃗µ± after whitening the space with S−1/2 W . 2.2 Decision-Images and Generalised Portraits We combine the linear preprocessor (PCA) ¯X = EB and the linear classifier (SVM, RVM, Prot, FLD) y(⃗x) = ⟨⃗w|⃗x⟩+ b to yield a linear classification system: ⃗y = ⃗wT ET +⃗b where ⃗b = b⃗1. We define the decision-image as the vector ⃗W effectively used for classification as: ⃗y = ⃗W T ¯XT +⃗b. We then have ⃗wT ET = ⃗W T ¯XT ⇔⃗wT B−T ¯XT = ⃗W T ¯XT where B−1 is the pseudo-inverse of B. For the last condition, we obtain a definition of the decisionimage ⃗W = B−1 ⃗w ∈R2562. In the case of PCA where B−1 = BT , we simply have ⃗W = BT ⃗w. Figure 1 shows the decision-images ⃗W for the four classifiers, SVM, RVM, Prot and FLD. The decision-images in the first row are those obtained if the classifiers are trained on the true dataset; those in the second row if trained on the subject dataset, marked on the right hand side of the figure by “true data” and “subj data”, respectively. Decision-images are represented by a vector pointing to the positive class and can thus be expected to have male attributes (the negative of it looks female). Both dark and light regions are more important for classification than the grey regions. Inspection of the decision-images is instructive. For the prototype learner, the eye and beard regions are most important. SVM, RVM and FLD have somewhat more “holistic” decision-images. Equally instructive is the comparison of the optimal decision-images of the machine classifiers in row one (0 to 1% classification error for SVM, RVM and FLD) and those trained on the subject labels in row two (the average subject error is 16 % when classifying the faces; the machines attempt to re-create the decision boundaries of the subjects and thus show similar mis-classification errors). The decision-images for the subject dataset are slightly more “face-like” and less holistic than those obtained using the true labels; the eye and mouth regions are more strongly emphasised. This trend is true across all classifiers. This suggest that human subjects base their gender classification strongly on the eye and mouth regions of the face—clearly a sub-optimal strategy as revealed by the more holistic true dataset SVM, RVM and FLD decision-images. A decision-image thus represents a way to extract the visual cues and features used by human subjects during visual classification without using a priori assumptions or knowledge about the task at hand. SVM RVM Prot FLD trained on → W true data → W subj data Figure 1: Decision-images ⃗W for each classifier for both the true and the subject dataset; all images are rescaled to [0, 1] and their means set to 128 for illustration purposes (different scalers for different images). We can also define generalised portraits2 ⃗W±. The generalised portraits ⃗W± can be seen as “summary” faces in each class reflecting the decision rule of the classifier. They can be viewed as an extension of the concept of a prototype: they are the prototype of the faces the classifier bases its decision on. We note that ⃗w can be written as: ⃗w = P i αi⃗xi = P i| sign(αi)=+1 αi⃗xi −P i| sign(αi)=−1 |αi|⃗xi. This allows to define the generalized portraits as ⃗W± which are computed by inverting the PCA transformation on the patterns ⃗w± = P i| sign(αi)=±1 αi⃗xi P i| sign(αi)=±1 αi . The vector ⃗w± is constrained to be in the convex hull of the respective data in order to yield a “viewable” portrait. The generalised portraits for the SVM, RVM and FLD together with the Prot, where the prototype is the same as the generalised portrait, are shown in figure 2. We also note that ⃗w can be written as ⃗w = P i αi⃗xi = P i| sign(αi)=+1 αi⃗xi −P i| sign(αi)=−1 |αi|⃗xi. The generalised portraits can be associated with the correct class: ⃗W+ are males whereas ⃗W−are females. The SVM and the FLD use patterns close to the SH for classification and hence their decision-images appear androgynous, whereas Prot and RVM tend to use patterns distant from the SH resulting in more female and male generalised portraits. Comparison of the optimal, true, generalised portraits to those based on the subject labels shows that classification has become more difficult: generalised portraits have moved closer to each other in gender space, narrowing the distance between the classes and thereby diminishing the gender typicality of the generalised portraits for all classifiers. 3 Human Gender Discrimination along the Decision-Image Axes The decision-images introduced in section 2.2 are based purely on machine learning, albeit on labels provided by human subjects in the case of the subject dataset. Our previous paper [1] reported that the subjects’ responses to the faces—proportion correct, reaction times 2This term was introduced by [14] with the idea in mind that when trained on a set of portraits of members of a family, one would obtain a “generalized” portrait which captures the essential features of the family as a superposition of all family members. SVM RVM Prot FLD trained on → W+ true data → W− true data → W+ subj data → W− subj data Figure 2: Generalised portraits ⃗W± for each classifier for both the true and the subject dataset; all images are rescaled to [0, 1] and their means set to 128 for illustration purposes (different scalers for different images). [Unfortunately the downsampling (low-pass filtering) of the faces necessary to fit them in the figure makes all the faces somewhat more androgynous than they are viewed at full resolution.] (RT) and confidence ratings—correlated very well with the distance of the stimuli to their separating hyperplane (SH) for support and relevance vector machines (SVMs, RVMs) but not for simple prototype (Prot) classifier. If these correlations really implied that SVM and RVM capture some crucial aspects of human internal face representation the following prediction must hold: already for small |λ| ISVM(λ) and IRVM(λ) should look male/female whereas |λ| IProt(λ) and IFLD(λ) should only be perceptually male/female for larger |λ|. In other words: the female-to-maleness axis of SVM and RVM should be closely aligned to those of our subjects whereas that is not expected to be the case for FLD and Prot. 3.1 Psychophysical Methods Four observers—one of the authors (FAW) with extensive psychophysical training and three na¨ıve subjects paid for their participation—took part in a standard, spatial (left versus right) two-alternative forced-choice (2AFC) discrimination experiment. Subjects were presented with two faces I(−λ) and I(λ) and had to indicate which face looked more male. Stimuli were presented against the mean luminance (50 cd/m2) of a carefully linearised Clinton Monoray CRT driven by a Cambridge Research Systems VSG 2/5 display controller. Neither male nor female faces changed the mean luminance. Subjects viewed the screen binocularly with their head stabilised by a headrest. The temporal envelope of stimulus presentation was a modified Hanning window (a raised cosine function with rise and fall times of 500 ms and a plateau time of 1000 ms). The probability of the female face being presented on the left was 0.5 on each trial and observers indicated whether they threshold elevation re. SVM threshold elevation re. SVM 1 2 3 b. FAW @75% correct @90% correct RVM Prot FLD RVM Prot FLD 1 c. FJ @75% correct @90% correct RVM Prot FLD RVM Prot FLD 1 e. KT @75% correct @90% correct RVM Prot FLD RVM Prot FLD 1 d. HM @75% correct @90% correct RVM Prot FLD RVM Prot FLD 1.4 1.8 0.6 1.4 1.8 0.6 1.4 1.8 0.6 RVM Prot FLD RVM Prot FLD 0. 5 1 1. 5 2 f. pooled @75% correct @90% correct threshold elevation re. SVM length of normalised decision image vector λ W / ||W|| proportion correct gender identification 0.05 0.0 9 0.4 0.8 1.4 2.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 a. FAW FLD Prot RVM SVM Figure 3: a. Shows raw data and fitted psychometric functions for one observer (FAW). b–e. For each of four observers the threshold elevation for the RVM, Prot and FLD decision-image relative to that of the SVM; results are shown for both 75 and 90% correct together with 68%-CIs. f. Same as in b–e but pooled across observers. thought the left or right face was female by touching the corresponding location on a Elo TouchSystems touch-screen immediately in front of the display; no feedback was provided. Trials were run in blocks of 256 in which eight repetitions of eight stimulus levels (±λ1 . . . ± λ8) for each of the four classifiers were randomly intermixed. The na¨ıve subjects required approximately 2000 trials before their performance stabilised; thereafter they did another five to six blocks of 256 trials. All results presented below are based on the trials after training; all training trials were discarded. 3.2 Results and Discussion Figure 3a shows the raw data and fitted psychometric functions for one of the observers. Proportion correct gender identification on the y-axis is plotted against λ on the x-axis on semi-logarithmic coordinates. Psychometric functions were fitted using the psignifit toolbox for Matlab which implements the constrained maximum-likelihood method described in [15]. 68%-confidence intervals (CIs), indicated by horizontal lines at 75 and 90-% correct in figure 3a, were estimated by the BCa bootstrap method also implemented in psignifit [16]. The raw data appear noisy because each data point is based on only eight trials. However, none of fitted psychometric functions failed various Monte Carlo based goodness-of-fit tests [15]. To summarise the data we extracted the λ required for two performance levels (“thresholds”), 75 and 90% correct, together with their corresponding 68%-CIs. Figure 3b– e shows the thresholds for all four observers normalised by λSVM (the “threshold elevation” re. SVM). Thus values larger than 1.0 for RVM, Prot and FLD indicate that more of the corresponding decision-images had to be added for the human observers to be able to discriminate females from males. In figure 3f we pool the data across observers as the main trend, poorer performance for Prot and FLD compared to SVM and RVM, is apparent for all four observers. The difference between SVM and RVM is small; going along the direction of both Prot and FLD, however, results in a much ”slower” transition from female-tomaleness. The psychophysical data are very clear: all observers require a larger λ for Prot and FLD; the length ratio ranges from 1.2 to nearly 3.0, and averages to around 1.7 across observers. In the pooled data all the differences are statistically significant but even at the individual subject level all differences are significant at the 90% performance level, and five of eight are significant at the 75% performance level. It thus appears that SVM and RVM capture more of the psychological face-space of our human observers than Prot and FLD. From our results we cannot exclude the possibility that some other direction might have yielded even steeper psychometric functions, i.e. faster female-to-maleness transitions, but we can conclude that the decision-images of SVM and RVM are closer to the decision-images used by human subjects than those of Prot and FLD. This is exactly as predicted by the correlations between proportion correct, RTs and confidence ratings versus distance to the hyperplane reported in [1]—high correlations for SVM and RVM, low correlations for Prot. 4 Summary and Conclusions We studied classification and discrimination of human faces both psychophysically as well as using methods from machine learning. The combination of linear preprocessor (PCA) and classifier (SVM, RVM, Prot and FLD) allowed us to visualise the decision-images of a classifier corresponding to the vector normal to the SH of the classifier. Decision-images can be used to determine the regions of the stimuli most useful for classification simply by analysing the distribution of light and dark regions in the decision-image. In addition we defined the generalised portraits to be the prototypes of all faces used by the classifier to obtain its classification. For the SVM this is the weighted average of all the support vectors (SVs), for the RVM the weighted average of all the relevance vectors (RVs), and for the Prot it is the prototype itself. The generalised portraits are, like the decision-images, another useful visualisation of the categorisation algorithm of the machine classifier. However, the central result of our paper is the corroboration of the machine-learningpsychophysics research methodology. In the machine-learning-psychophysics research we substitute a very hard to analyse complex system (the human brain) by a reasonably complex system (learning machine) that is complex enough to capture essentials of our human subjects’ behaviour but is nonetheless amenable to close analysis. From the analysis of the machines we then derive predictions for human subjects which we subsequently test psychophysically. Given the success in predicting the steepness of the female-to-male transition of the ⃗wSVM -axis we believe that the decision-image ⃗WSVM captures some of the essential characteristics of the human decision algorithm. Acknowledgements The authors would like to thank Bruce Henning, Frank J¨akel, Ulrike von Luxburg and Christian Wallraven for helpful comments and suggestions. In addition we thank Frank J¨akel for supplying us with the code to run the touch-screen experiment. References [1] A.B.A. Graf and F.A. Wichmann. Insights from machine learning applied to human visual classification. In Advances in Neural Information Processing Systems 16. MIT Press, 2004. [2] M.S. Gray, D.T. Lawrence, B.A. Golomb, and T.S. Sejnowski. A perceptron reveals the face of sex. Neural Computation, 7(6):1160–1164, 1995. [3] P.J.B. Hancock, V. Bruce, and A.M. Burton. A comparison of two computer-based face recognition systems with human perceptions of faces. Vision Research, 38:2277–2288, 1998. [4] A.J. O’Toole, P.J. Phillips, Y. Cheng, B. Ross, and H.A. Wild. Face recognition algorithms as models of human face processing. In Proceedings of the 4th IEEE International Conference on Automatic Face and Gesture Recognition, 2000. [5] B. Moghaddam and M.-H. Yang. Learning gender with support faces. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(5):707–711, 2002. [6] F. Gosselin and P.G. Schyns. Bubbles: a technique to reveal the use of information in recognition tasks. Vision Research, 41:2261–2271, 2001. [7] A.J. Ahumada Jr. Classification image weights and internal noise level estimation. Journal of Vision, 2:121–131, 2002. [8] M. Turk and A. Pentland. Eigenfaces for recognition. Journal of Cognitive Neuroscience, 3(1), 1991. [9] V.N. Vapnik. The Nature of Statistical Learning Theory. Springer, second edition, 2000. [10] B. Sch¨olkopf and A.J. Smola. Learning with Kernels. MIT Press, 2002. [11] M.E. Tipping. Sparse Bayesian learning and the relevance vector machine. Journal of Machine Learning Research, 1:211–214, 2001. [12] S.K. Reed. Pattern recognition and categorization. Cognitive Psychology, 3:382–407, 1972. [13] R. A. Fisher. The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7(2):179–188, 1936. [14] V. Vapnik and A. Lerner. Pattern recognition using generalized portrait method. Automation and Remote Control, 24:774–780, 1963. [15] F.A. Wichmann and N.J. Hill. The psychometric function: I. fitting, sampling and goodness-offit. Perception and Psychophysics, 63(8):1293–1313, 2001. [16] F.A. Wichmann and N.J. Hill. The psychometric function: II. bootstrap-based confidence intervals and sampling. Perception and Psychophysics, 63(8):1314–1329, 2001.
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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 malik@eecs.berkeley.edu 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 Introduction 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 12.5º 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 10º 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. fixate, initiate trial maintain fixation (300ms) release fixation, view object freely (1200ms) Which shape is a match? fixate, initiate trial maintain fixation (300ms) release fixation, view object freely (1200ms) Which shape is a match? 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 toπ , 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 measured as eccentricity E, the visual angle between any point and the fovea. If ) , ( y x x = r is the location of a point in an image and ) , ( y x f f f = r is the fixation (i.e. foveal) location in the image then the eccentricity is f x E r r − = , measured in units of visual degrees. The effective resolution of orientation discrimination falls with increasing eccentricity as where r(E) is an effective radius over which the visual system spatially pools information and F ) ( 2 E E FPH + = ) (E r PH =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 ) , ( i i i y x x = r relative to the current fixation f r , i.e. f x E i r r − = . To define the histogram more precisely we will introduce the neighborhood set Ni of all indices j corresponding to edgelets within radius ri of ei : { } i j i i r x x t s j all N ≤ − = .. , with number of neighborhood edgelets |Ni|. The (normalized) histogram centered at edgelet ei is then defined as r r ∑ ∈ = i j N j z z i iz N h , 1 δ , 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 by k=1, 2, …, K (for K total fixations). The kth fixation location is denoted by ) , ( ) ( k y k x k f f f = r . The quantities ri , Ni and hiz depend on fixation location and so to make this dependence explicit we will augment them with superscripts as r , , ) (k i ) (k i N and h . ) (k iz 2 y x, δ is the Kronecker delta function, defined to equal 1 if y x = and 0 if y x ≠ . 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: , where {h represents all histogram components z at every edgelet e ) , , , ) ( ) 2 ( ) 1( 2 } {h } {h } | {h , ... z , z P(z K iz iz iz N i K } k iz ) ( ) (k i for fixation f r . This exact distribution is intractable, so we will use a simple approximation. We assume the distribution factors over individual edgelets: ∏ = = N i i i K iz iz iz N i ) (z g }) {h } {h } | {h , ... z , z P(z 1 ) ( ) 2 ( ) 1 ( 2 , , , K where gi(zi) is the marginal distribution of orientation zi. Determining these marginal distributions is still difficult even with the factorization assumption, so we will make an additional approximation: ∏ = = K k k iz i i i h Z ) (z g 1 ) ( 1 , where Zi is a suitable normalization factor. This approximation corresponds to treating 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 g ) (k iz h i(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 Given the past K fixations, the next fixation ) 1 ( + K f r is chosen to minimize the model entropy of the edgelet orientations. In other words, ) 1 ( + K f r is chosen to minimize ) 1 (K+ ) 2 ( ] , , , [ ) ( ) 1( 2 ) 1 ( }) {h } {h } | {h , ... z , z P(z entropy f H iz iz iz N i K+ = K r , where the entropy of a distribution P(x) is defined as ∑ − x x P x P ) ( log ) ( . In practice, we minimize the entropy by evaluating it across a set of candidate locations ) 1 ( + K f r 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). R S I R S I R S I R S I R S I R S I Visual Angle (deg) Location Error Sequence Error R S I R S I R S I R S I R S I R S I Visual Angle (deg) Location Error Sequence Error 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 2-4º and rarely exceed 15º [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 Discussion 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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Learning Hyper-Features for Visual Identification Andras Ferencz Erik G. Learned-Miller Jitendra Malik Computer Science Division, EECS University of California at Berkeley Berkeley, CA 94720 Abstract We address the problem of identifying specific instances of a class (cars) from a set of images all belonging to that class. Although we cannot build a model for any particular instance (as we may be provided with only one “training” example of it), we can use information extracted from observing other members of the class. We pose this task as a learning problem, in which the learner is given image pairs, labeled as matching or not, and must discover which image features are most consistent for matching instances and discriminative for mismatches. We explore a patch based representation, where we model the distributions of similarity measurements defined on the patches. Finally, we describe an algorithm that selects the most salient patches based on a mutual information criterion. This algorithm performs identification well for our challenging dataset of car images, after matching only a few, well chosen patches. 1 Introduction Figure 1 shows six cars: the two leftmost cars were captured by one camera; the right four cars were seen later by another camera from a different angle. The goal is to determine which images, if any, show the same vehicle. We call this task visual identification. Most existing identification systems are aimed at biometric applications such as identifying fingerprints or faces. While object recognition is used loosely for several problems (including this one), we differentiate visual identification, where the challenge is distinguishing between visually similar objects of one category (e.g. faces, cars), and categorization where Figure 1: The Identification Problem: Which of these cars are the same? The two cars on the left, photographed from camera 1, also drive past camera 2. Which of the four images on the right, taken by camera 2, match the cars on the left? Solving this problem will enable applications such as wide area tracking of cars with a sparse set of cameras [2, 9]. Figure 2: Detecting and warping car images into alignment: Our identification algorithm assumes that a detection process has found members of the class and approximately aligned them to a canonical view. For our data set, detection is performed by a blob tracker. A projective warp to align the sides is computed by calibrating the pose of the camera to the road and finding the wheels of the vehicle. Note that this is only a rough approximation (the two warped images, center and right, are far from perfectly aligned) that helps to simplify our patch descriptors and positional bookkeeping. the algorithm must group together objects that belong to the same category but may be visually diverse[1, 5, 10, 13]. Identification is also distinct from “object localization,” where the goal is locating a specific object in scenes in which distractors have little similarity to the target object [6].1 One characteristic of the identification problem is that the algorithm typically only receives one positive example of each query class (e.g. a single image of a specific car), before having to classify other images as the “same” or “different”. Given this lack of a class specific training set, we cannot use standard supervised feature selection and classification methods such as [12, 13, 14]. One possible solution to this problem is to try to pick universally good features, such as corners [4, 6], for detecting salient points. However, such features are likely to be suboptimal as they are not category specific. Another possibility is to hand-select good features for the task, such as the distance between the eyes for face identification. Here we present an identification framework that attempts to be more general. The core idea is to use a training set of other image pairs from the category (in our case cars), labeled as matching or not, to learn what characterizes features that are informative in distinguishing one instance from another (i.e. consistent for matching instances and dissimilar for mismatches). Our algorithm, given a single novel query image, can build a “same” vs. “different” classifier by: (1) examining a set of candidate features (local image patches) on the query image (2) selecting a small number of them that are likely to be the most informative for this query class and (3) estimating a function for scoring the match for each selected feature. Note that a different set of features (patches) will be selected for each unique query. The paper is organized as follows. In Section 2, we describe our decision framework including the decomposition of an image pair into bi-patches, which give local indications of match or mismatch, and introduce the appearance distance between the two halves as a discriminative statistic of bi-patches. This model is then refined in Section 3 by conditioning the distance distributions on hyper-features such as patch location, contrast, and dominant orientation. A patch saliency measure based on the estimated distance distributions is introduced in Section 3.4. In Section 4, we extend our model to include another comparison statistic, the difference in patch position between images. Finally, in Section 5, we conclude and show that comparing a small number of well-chosen patches produces performance nearly as good as matching a dense sampling of them. 2 Matching Patches We seek to determine whether a new query image IL (the “Left” image) represents the same vehicle as any of our previously seen database images IR (the “Right” image). We assume that these images are known to contain vehicles, have been brought into rough correspondence (in our data set, through a projective transformation that aligns the sides of the car) and have been scaled to approximately 200 pixels in length (see Figure 2 for details). 1There is evidence that this distinction exists in the human visual system. Some findings suggest that the fusiform face area is specialized for identification of instances from familiar categories[11]. Figure 3: Patch Matching: The left (query) image is sampled (red dots) by patches encoded as oriented filter channels (for labeled patch 2, this encoding is shown). Each patch is matched to the best point in the database image of the same car by maximizing the appearance similarity between the patches (the similarity score is indicated by the size and color of the dots, where larger and redder is more similar). Three bi-patches are labeled. Although the classification result for this pair of images should be “same”(C = 1), notice that some bi-patches are better predictors of this result than others (the similarity score of 2 & 3 is much better than for patch 1). Our goal is to be able to predict the distribution of P(d|C = 1) and P(d|C = 0) for each patch accurately based on the appearance and position of the patch in the query image (for the 3 patches, our predictions are shown on the right). 2.1 Image Patch Features Our strategy is to break up the whole image comparison problem into multiple local matching problems, where we encode a small patch F L j (1 ≤j ≤n) of the query image IL and compare each piece separately [12, 14]. As the exact choice of features, their encoding and comparison metric is not crucial to our technique, we chose a fairly simple representation that was general enough to use in a wide variety of settings, but informative enough to capture the details of objects (given the subtle variation that can distinguish two different cars, features such as [6] were found not to be precise enough for this task). Specifically, we apply a first derivative Gaussian odd-symmetric filter to the patch at four orientations (horizontal, vertical, and two diagonal), giving four signed numbers per pixel. To compare a query patch F L j to an area of the right image F R j , we encode both patches as 4 × 252 length vectors (4 orientations per pixel) and compute the normalized correlation (dj = 1 −CorrCoef(F L j , F R j )) between these vectors. As the two car images are in rough alignment, we need only to search a small area of IR to find the best corresponding patch F R j - i.e. the one that minimizes dj. We will refer to such a matched left and right patch pair F L j , F R j , together with the derived distance dj, as a bi-patch Fj. 2.2 The Decision Rule We pose the task of deciding if the a database image IR is the same as a query image IL as a decision rule R = P(C = 1|IL, IR) P(C = 0|IL, IR) = P(IL, IR|C = 1)P(C = 1) P(IL, IR|C = 0)P(C = 0) > λ. (1) where λ is chosen to balance the cost of the two types of decision errors. The priors are assumed to be known.2 Specifically, for the remaining equations in this paper, the priors are assumed to be equal, and hence are dropped from subsequent equations. With our image decomposition into patches, the posteriors from Eq. (1) will be approximated using the bipatches F1, ..., Fn as P(C|IL, IR) ≈P(C|F1, ..., Fm) ∝P(F1, ..., Fm|C). Furthermore, in this paper, we will assume a naive Bayes model in which, conditioned on C, the bipatches are assumed to be independent. That is, R = P(IL, IR|C = 1) P(IL, IR|C = 0) ≈P(F1, ..., Fm|C = 1) P(F1, ..., Fm|C = 0) = m Y j=1 P(Fj|C = 1) P(Fj|C = 0). (2) 2For our application, dynamic models of traffic flow can supply the prior on P(C). In practice, we compute the log of this likelihood ratio, where each patch contributes an additive term (denoted LLRi for patch i). Modeling the likelihoods in this ratio (P(Fj|C)) is the central focus of this paper. 2.3 Uniform Appearance Model Figure 4: Identification using appearance differences: The bottom curve shows the precision vs. recall for non-patch based direct comparison of rectified images. (An ideal precision-recall curve would reach the top right corner.) Notice that all three patch based models outperform this method. The three top curves show results for various models of dj from Sections 2.3 (Baseline), 3.1 (Discrete), and 3.2 & 3.3 (Continuous). The regression model outperforms the uniform one significantly it reduces the error in precision by close to 50% for most values of recall below 90%. The most straightforward way to estimate P(Fj|C) is to assume that the appearance difference dj captures all of the information Fj about the probability of a match (i.e. C and Fj are independent given dj), and that all of dj’s from all patches are identically distributed. Thus the decision rule, Eqn. 1, becomes R ≈ m Y j=1 P(dj|C = 1) P(dj|C = 0) > λ. (3) The two conditional distributions, P(dj |C ∈{0, 1}), are estimated as normalized histograms from all bi-patches matched within the training data.3 For each value of λ, we evaluate Eqn.(3) to classify each test pair as matching or not, producing a precision-recall curve. Figure 4 compares this patch-based model to a direct image comparison method.4 Notice that even this naive patch-based technique significantly outperforms the global matching. 3 Refining the Appearance Distributions with Hyper-Features The most significant weakness of the above model is the assumption that the dj’s from different bi-patches should be identically distributed (observe the 3 labeled patches in Figure 3). When a training set of “same” (C = 1) and “different” (C = 0) images is available for a specific query image, estimating these distributions directly for each patch is straightforward. How can we estimate a distribution for P(dj|C = 1), where F L j is a patch from a new query image, when we only have that single positive example of F L j ? The intuitive answer: by finding analogous patches in the training set of labeled (same/different) image pairs. However, since the space of all possible patches (appearance & position, ℜ25∗25+2) is very large, the chance of having seen a very similar patch to F L j in the training set is small. In the next sections we present two approaches both of which rely on projecting F L j into a much lower dimensional space by extracting meaningful features from its position and appearance (the hyper-features). 3.1 Non-Parametric Model with Discrete Hyper-Features First we attempted a non-parametric approach, where we model the joint distribution of dj and a few hyper-features (e.g. the x and y coordinate of the patch F L j , 3Data consisted of 175 pairs (88 training, 87 test pairs) of matching car images (C=1) from two cameras located on the same side of the street one block apart. Within training and testing sets, about 4000 pairs of mismatched cars (C=0) were formed from non-corresponding images, one from each camera. All comparisons were performed on grayscale (not color) images. 4The global image comparison method used here as a baseline technique uses normalized correlation on a combination of intensity and filter channels, and attempts to overcome slight misalignment. Figure 5: Fitting a GLM to the Γ distribution: we demonstrate our approach by fitting a gamma distribution, through the latent variables Θ = (µ, γ), to the y position of the patches. Here we allowed µ and σ to be a 3rd degree polynomial function of y (i.e. Z = [y3, y2, y, 1]T). The centerleft square shows, on each row, a distribution of d conditioned on the y position of the left patch (F L) for each bi-patch, for training data taken from matching vehicles. The center-right square shows the same distributions for mismatched data. The height of histogram distributions is color-coded, dark red indicating higher density. The central curve shows the polynomial fit to the conditional means, while the outer curves show the ±σ range. For reference, we include a partial image of a car whose y-coordinate is aligned with the center images. On the right, we show two histogram plots, each corresponding to one row of the center images (a small range of y corresponding to the black arrows). The resulting gamma distributions are superimposed on the histograms. i.e. Z = [x, y]). The distribution is modeled “non-parametrically” (similar to Section 2.3) using an N-dimensional normalized histogram where each dimension (d,x, and y) has been quantized into several bins. In this model P(C|Fj) ≈P(C|dj, yj, xj) ∝ P(dj|yj, xj, C)P(yj, xj|C)P(C) ∝P(dj|yj, xj, C), where the last formula follows from the assumption of equal priors (P(C) = 0.5) and the independence of (yj, xj) and C. The Discrete Hyper-Features curve in Figure 4 shows the performance gain from conditioning on these positional hyper-features. 3.2 Parametric Model with Continuous Hyper-Features The drawback of using a non-parametric model for the distributions is that the amount of data needed to populate the histograms grows exponentially with the number of dimensions. In order to add additional appearance-based hyper-features, such as contrast, oriented edge energy, etc., we moved to a smooth parametric representation for both the distribution of dj and the model by which the the hyper-features influence this distribution. Specifically, we model the distributions P(dj|C = 1) and P(dj|C = 0) as gamma distributions (notated Γ()) parameterized by the mean and shape parameter θ = {µ, γ} (see the right panel of Figure 5 for examples of the Γ() fitting the empirical distributions). The smooth variation of θ with respect to the hyper-features can be modeled using a generalized linear model (GLM). Ordinary (least-squares) linear models assume that the data is normally distributed with constant variance. GLMs are extensions to ordinary linear models that can fit data which is not normally distributed and where the dispersion parameter also depends on the covariates (see [7] for more information on GLMs). Our goal is to fit gamma distributions to the distributions of d values for various patches by maximizing the probability density of data under gamma distributions whose parameters are simple polynomial functions of the hyper-features. Consider a set X1, ..., Xk of hyperfeatures such as position, contrast, and brightness of a patch. Let Z = [Z1, ..., Zl]T be a vector of l pre-chosen functions of those hyper-features, like squares, cubes, cross terms, or simply copies of the variables themselves. Then each bi-patch distance distribution has the form P(d|X1, X2, ..., Xk, C) = Γ(d; αµ C · Z, αγ C · Z), (4) where the second and third arguments to Γ() are mean and shape parameters.5 Each α (there are four of these: αµ C=0, αγ C=0, αµ C=1, αγ C=1) is a vector of parameters of length l 5For the GLM, we use the identity link function for both µ and γ. While the identity is not the canonical link function for µ, its advantage is that our ML optimization can be initialized by that weights each hyper-feature monomial Zi. The α’s are adapted to maximize the joint data likelihood over all patches for C = 0 or C = 1 withing the training set. These ideas are illustrated in detail in Figure 5. 3.3 Automatic Selection of Hyper-Features In this section we describe the automatic determination of Z. Recall that in our GLM model we assumed a linear relationship between Z and µ, γ. This allows us to use standard feature selection techniques, such as Least Angle Regression (LARS)[3], to choose a few (around 10) hyper-features from a large set of candidates,6 such as: (a) the x and y positions of F L, (b) the intensity and contrast within F L and the average intensity of the entire vehicle, (c) the average energy in each of the 8 oriented filter channels, and (d) derived quantities from the above (e.g. square, cubic, and cross terms). LARS was then asked to choose Z from these features. Once Z is set, we proceed as in Section 3.2. Running an automatic feature selection technique on this large set of possible conditioning features gives us a principled method of reducing the complexity of our model. Reducing the complexity is important not only to speed up computation, but also to mitigate the risk of over-fitting to the training set. The top curve in Figure 4 shows results when Z includes the first 10 features found by LARS. Even with such a naive set of features to choose from, the performance of the system improves significantly. 3.4 Estimating the Saliency of a Patch From the distributions P(dj|C = 0) and P(dj|C = 1) computed separately for each patch, it is also possible to estimate the saliency of the patch, i.e. the amount of information about our decision variable C we are likely to gain should we compute the best corresponding F R j . Intuitively, if the distribution of Dj is very different for C = 0 and C = 1, then the amount of information gained by matching patch j is likely to be large (see the 3 distributions on the right of Figure 3). To emphasize the fact that the distribution P(dj|C) is a fixed function of F L j , given the learned hyper-feature weights α, we slightly abuse notation and refer to the random variable from which dj is sampled as F L j . With this notation, computing the mutual information between F L j and C gives us a measure of the expected information gain from a patch with particular hyper-features: I(F L j ; C) = H(F L j ) −H(F L j |C). Here H() is Shannon entropy. The key fact to notice is that this measure can be computed just from the estimated distributions over dj (which, in turn, were estimated from the hyperfeatures of F L j ) before the patch has been matched. This allows us to match only those patches that are likely to be informative, leading to significant computational savings. 4 Modeling Appearance and Position Differences In the last section, we only considered the similarity of two matching patches that make up a bi-patch in terms of the appearance of the patches (dj). Recall that for each left patch F L j , a matching right patch F R j is found by searching for the most similar patch in some large neighborhood around the expected location for the match. In this section, we show how to model the change in position, rj, of the match relative to its expected location, and how this, when combined with the appearance model, improves the matching performance. solving an ordinary least squares problem. We experimentally compared it to the canonical inverse link (µ = (αµ C T ∗Z)−1), but observed no noticeable change in performance on our data set. 6In order to use LARS (or most other feature selection methods) “out of the box”, we use regression based on an L2 loss function. While this is not optimal for non-normal data, from experiments we have verified that it is a reasonable approximation for the feature selection step. Figure 6: Results: The LEFT plot shows precision vs. recall curves for models of r. The results for δx and δy are shown separately (as there are often more horizontal than vertical features on cars, δy is better). Re-estimating parameters of the global alignment, W (affine fit), significantly improves the curves. Finally, performance is improved by combining position with appearance (“Complete” curve) compared to using appearance alone. The CENTER pair of images show a correct match, with the patch centers indicated by circles. The color of the circles in the top image indicates MI j, in bottom image LLRj. Our patch selection algorithm chooses the top patches based on MI where subsequent patches are penalized for overlapping with earlier ones (neighborhood suppression). The top 10 “left” patches chosen are marked with arrows connecting them to the corresponding “right” patches. Notice that these are concentrated in informative regions. The RIGHT plot quantifies this observation: the curves show 3 different methods of choosing the order of patches - random order, MI and MI with neighborhood suppression. Notice that this top curve with 3 patches does as well as the direct comparison method. All 3 methods converge above 50 patches. Let rj = (δxj, δyj) be the difference in position between the coordinates of F L j and F R j within the standardized coordinate frames. Generally, we expect rj ≈0 if the two images portray the same object (C = 1). The estimate for R, incorporating the information from both d and r becomes R ≈ m Y j=1 P(rj|dj, Zj, C = 1)P(dj|Zj, C = 1) P(rj|dj, Zj, C = 0)P(dj|Zj, C = 0), (5) where Zj again refers to a set of hyper-features. Here we focus on the first factor, where the distribution of rj given C is dependent on the appearance and position of the left patch (F L j , through the hyper-features Zj) and on the similarity in appearance (dj). The intuition for the dependence on dj is that for the C = 1 case, we expect rj to be smaller on average when a good appearance match (small dj) was found. Following our approach for dj, we model the distribution of rj as a 0 mean normal distribution, N(0, Σ) , where Σ (we use a diagonal covariance) is a function of Zj,dj. The parameterization of (Zj,dj) is found through feature selection, while the weights for the linear function are obtained by maximizing the likelihood of rj over the training data. To address initial misalignment, we select a small number of patches, match them, and compute a global affine alignment between the images. We subsequently score each match relative to this global alignment. The bottom four curves of Figure 6 show that fitting an affine model first significantly improves the positional signal. While position seems to be less informative than appearance, the complete model, which combines appearance and position (Eq. 5), outperforms appearance alone. 5 Conclusion The center and right sides of Figure 6 show our ability to select the most informative patches using the estimated mutual information I(F L j , C) of each patch. To prevent spatially overlapping patches from being chosen, we added a penalty factor to the mutual information score that penalizes patches that are very close to other chosen patches (MI with neighborhood suppression). To give a numerical indication of the performance, we note that with only 10 patches, given a 1-to-87 forced choice problem, our algorithm chooses the correct matching image 93% of the time. A different approach to a learning problem that is similar to ours can be found in [5, 8], which describe methods for learning character or object categories from few training examples. These works approach this problem by learning distributions on shared factors [8] or priors on parameters of fixed distributions for a category [5] where the training data consists of images from other categories. We, on the other hand, abandon the notion of building a model with a fixed form for an object from a single example. Instead, we take a discriminative approach and model the statistical properties of image patch differences conditioned on properties of the patch. These learned conditional distributions allow us to evaluate, for each feature, the amount of information potentially gained by matching it to the other image.7 Acknowledgments This work was partially funded by DARPA under the Combat Zones That See project. References [1] Y. Amit and D. Geman. A computational model for visual selection. Neural Computation, 11(7), 1999. [2] D. Beymer, P. McLauchlan, B. Coifman, and J. Malik. A real-time computer vision system for measuring traffic parameters. CVPR, 1997. [3] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32(2):407–499, 2004. [4] T. Kadir and M. Brady. Scale, saliency and image description. International Journal of Computer Vision, 45(2):83–105, 2001. [5] F. Li, R. Fergus, and P. Perona. A Bayesian approach to unsupervised one-shot learning of object categories. In ICCV, 2003. [6] D. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60(2):91–110, 2004. [7] P. McCullagh and J. A. Nelder. Generalized Linear Models. Chapman and Hall, 1989. [8] E. Miller, N. Matsakis, and P. Viola. Learning from one example through shared densities on transforms. In CVPR, 2000. [9] H. Pasula, S. Russell, M. Ostland, and Y. Ritov. Tracking many objects with many sensors. IJCAI, 1999. [10] H. Schneiderman and T. Kanade. A statistical approach to 3d object detection applied to faces and cars. CVPR, 2000. [11] M. Tarr and I. Gauthier. FFA: A flexible fusiform area for subordinate-level visual processing automatized by expertise. Nature Neuroscience, 3(8):764–769, 2000. [12] M. Vidal-Naquet and S. Ullman. Object recognition with informative features and linear classification. In International Conference on Computer Vision, 2003. [13] P. Viola and M. Jones. Rapid object detection using a boosted cascade of simple features. In CVPR, 2001. [14] M. Weber, M. Welling, and P. Perona. Unsupervised learning of models for recognition. ECCV, 2000. 7Answer to Figure 1: top left matches bottom center; bottom left matches bottom right. For our algorithm, matching these images was not a challenge.
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Learning first-order Markov models for control Pieter Abbeel Computer Science Department Stanford University Stanford, CA 94305 Andrew Y. Ng Computer Science Department Stanford University Stanford, CA 94305 Abstract First-order Markov models have been successfully applied to many problems, for example in modeling sequential data using Markov chains, and modeling control problems using the Markov decision processes (MDP) formalism. If a first-order Markov model’s parameters are estimated from data, the standard maximum likelihood estimator considers only the first-order (single-step) transitions. But for many problems, the firstorder conditional independence assumptions are not satisfied, and as a result the higher order transition probabilities may be poorly approximated. Motivated by the problem of learning an MDP’s parameters for control, we propose an algorithm for learning a first-order Markov model that explicitly takes into account higher order interactions during training. Our algorithm uses an optimization criterion different from maximum likelihood, and allows us to learn models that capture longer range effects, but without giving up the benefits of using first-order Markov models. Our experimental results also show the new algorithm outperforming conventional maximum likelihood estimation in a number of control problems where the MDP’s parameters are estimated from data. 1 Introduction First-order Markov models have enjoyed numerous successes in many sequence modeling and in many control tasks, and are now a workhorse of machine learning.1 Indeed, even in control problems in which the system is suspected to have hidden state and thus be non-Markov, a fully observed Markov decision process (MDP) model is often favored over partially observable Markov decision process (POMDP) models, since it is significantly easier to solve MDPs than POMDPs to obtain a controller. [5] When the parameters of a Markov model are not known a priori, they are often estimated from data using maximum likelihood (ML) (and perhaps smoothing). However, in many applications the dynamics are not truly first-order Markov, and the ML criterion may lead to poor modeling performance. In particular, we will show that the ML model fitting criterion explicitly considers only the first-order (one-step) transitions. If the dynamics are truly governed by a first-order system, then the longer-range interactions would also be well modeled. But if the system is not first-order, then interactions on longer time scales are often poorly approximated by a model fit using maximum likelihood. In reinforcement learning and control tasks where the goal is to maximize our long-term expected rewards, the predictive accuracy of a model on long time scales can have a significant impact on the attained performance. 1To simplify the exposition, in this paper we will consider only first-order Markov models. However, the problems we describe in this paper also arise with higher order models and with more structured models (such as dynamic Bayesian networks [4, 10] and mixed memory Markov models [8, 14]), and it is straightforward to extend our methods and algorithms to these models. As a specific motivating example, consider a system whose dynamics are governed by a random walk on the integers. Letting St denote the state at time t, we initialize the system to S0 = 0, and let St = St−1 + εt, where the increments εt ∈{−1, +1} are equally likely to be −1 or +1. Writing St in terms of only the εt’s, we have St = ε1 + · · · + εt. Thus, if the increments are independent, we have Var(ST ) = T. However if the increments are perfectly correlated (so ε1 = ε2 = · · · with probability 1), then Var(ST ) = T 2. So, depending on the correlation between the increments, the expected value E[|ST |] can be either O( √ T) or O(T). Further, regardless of the true correlation in the data, using maximum likelihood (ML) to estimate the model parameters from training data would return the same model with E[|ST |] = O( √ T). To see how these effects can lead to poor performance on a control task, consider learning to control a vehicle (such as a car or a helicopter) under disturbances εt due to very strong winds. The influence of the disturbances on the vehicle’s position over one time step may be small, but if the disturbances εt are highly correlated, their cumulative effect over time can be substantial. If our model completely ignores these correlations, we may overestimate our ability to control the vehicle (thinking our variance in position is O(T) rather than O(T 2)), and try to follow overly narrow/dangerous paths. Our motivation also has parallels in the debate on using discriminative vs. generative algorithms for supervised learning. There, the consensus (assuming there is ample training data) seems to be that it is usually better to directly minimize the loss with respect to the ultimate performance measure, rather than an intermediate loss function such as the likelihood of the training data. (See, e.g., [16, 9].) This is because the model (no matter how complicated) is almost always not completely “correct” for the problem data. By analogy, when modeling a dynamical system for a control task, we are interested in having a model that accurately predicts the performance of different control policies—so that it can be used to select a good policy—and not in maximizing the likelihood of the observed sequence data. In related work, robust control offers an alternative family of methods for accounting for model inaccuracies, specifically by finding controllers that work well for a large class of models. (E.g., [13, 17, 3].) Also, in applied control, some practitioners manually adjust their model’s parameters (particularly the model’s noise variance parameters) to obtain a model which captures the variability of the system’s dynamics. Our work can be viewed as proposing an algorithm that gives a more structured approach to estimating the “right” variance parameters. The issue of time scales has also been addressed in hierarchical reinforcement learning (e.g., [2, 15, 11]), but most of this work has focused on speeding up exploration and planning rather than on accurately modeling non-Markovian dynamics. The rest of this paper is organized as follows. We define our notation in Section 2, then formulate the model learning problem ignoring actions in Section 3, and propose a learning algorithm in Section 4. In Section 5, we extend our algorithm to incorporate actions. Section 6 presents experimental results, and Section 7 concludes. 2 Preliminaries If x ∈Rn, then xi denotes the i-th element of x. Also, let j:k = [j j+1 j+2 · · · k−1 k]T . For any k-dimensional vector of indices I ∈Nk, we denote by xI the k-dimensional vector with the subset of x’s entries whose indices are in I. For example, if x = [0.0 0.1 0.2 0.3 0.4 0.5]T , then x0:2 = [0.0 0.1 0.2]T . A finite-state decision process (DP) is a tuple (S, A, T, γ, D, R), where S is a finite set of states; A is a finite set of actions; T = {P(St+1 = s′|S0:t = s0:t, A0:t = a0:t)} is a set of state transition probabilities (here, P(St+1 = s′|S0:t = s0:t, A0:t = a0:t) is the probability of being in a state s′ ∈S at time t + 1 after having taken actions a0:t ∈At+1 in states s0:t ∈St+1 at times 0 : t); γ ∈[0, 1) is a discount factor; D is the initial state distribution, from which the initial state s0 is drawn; and R : S 7→R is the reward function. We assume all rewards are bounded in absolute value by Rmax. A DP is not necessarily Markov. A policy π is a mapping from states to probability distributions over actions. Let V π(s) = E[P∞ t=0 γtR(st)|π, s0 = s] be the usual value function for π. Then the utility of π is U(π) = Es0∼D[V π(s0)] = E[P∞ t=0 γtR(st)|π] = P∞ t=0 γt P st P(St = st|π)R(st). The second expectation above is with respect to the random state sequence s0, s1, . . . drawn by starting from s0 ∼D, picking actions according to π and transitioning according to P. Throughout this paper, Pˆθ will denote some estimate of the transition probabilities. We denote by ˆU(π) the utility of the policy π in an MDP whose first-order transition probabilities are given by Pˆθ (and similarly ˆV π the value function in the same MDP). Thus, we have2 ˆU(π) = ˆEs0∼D[ ˆV π(s0)] = ˆE[P∞ t=0 γtR(st)|π] = P∞ t=0 γt P st Pˆθ(St = st|π)R(st). Note that if |U(π) −ˆU(π)| ≤ε for all π, then finding the optimal policy in the estimated MDP that uses parameters Pˆθ (using value iteration or any other algorithm) will give a policy whose utility is within 2ε of the optimal utility. [6] For stochastic processes without decisions/actions, we will use the same notation but drop the conditioning on π. Often we will also abbreviate P(St = st) by P(st). 3 Problem Formulation To simplify our exposition, we will begin by considering stochastic processes that do not have decisions/actions. Section 5 will discuss how actions can be incorporated into the model. We first consider how well ˆV (s0) approximates V (s0). We have | ˆV (s0) −V (s0)| = ∞ X t=0 γt X st Pˆθ(st|s0)R(st) − ∞ X t=0 γt X st P(st|s0)R(st) ≤ Rmax ∞ X t=0 γt X st Pˆθ(st|s0) −P(st|s0) . (1) So, to ensure that ˆV (s0) is an accurate estimate of V (s0), we would like the parameters ˆθ of the model to minimize the right hand side of (1). The term P st Pˆθ(st|s0) −P(st|s0) is exactly (twice) the variational distance between the two conditional distributions Pˆθ(·|s0) and P(·|s0). Unfortunately P is not known when learning from data. We only get to observe state sequences sampled according to P. This makes Eqn. (1) a difficult criterion to optimize. However, it is well known that the variational distance is upper bounded by a function of the KL-divergence. (See, e.g., [1].) The KL-divergence between P and Pˆθ can be estimated (up to a constant) as the log-likelihood of a sample. So, given a training sequence s0:T sampled from P, we propose to estimate the transition probabilities Pˆθ by ˆθ = arg max θ T −1 X t=0 T −t X k=1 γk log Pθ(st+k|st). (2) Note the difference between this and the standard maximum likelihood (ML) estimate. Since we are using a model that is parameterized as a first-order Markov model, the probability of the data under the model is given by Pθ(s0, . . . , sT ) = Pθ(sT |sT −1)Pθ(sT −1|sT −2) . . . Pθ(s1|s0)D(s0) (where D is the initial state distribution). By definition, maximum likelihood (ML) chooses the parameters θ that maximize the probability of the observed data. Taking logs of the probability above, (and ignoring D(s0), which is usually parameterized separately), we find that the ML estimate is given by ˆθ = arg max θ T −1 X t=0 log Pθ(st+1|st). (3) 2Since Pˆθ is a first-order model, it explicitly parameterizes only Pˆθ(St+1 = st+1|St = st, At = at). We use Pˆθ(St = st|π) to denote the probability that St = st in an MDP with one-step transition probabilities Pˆθ(St+1 = st+1|St = st, At = at) and initial state distribution D when acting according to the policy π. S S S S 0 1 2 3 S S S S S S 0 1 1 2 2 3 S S S S S S S S 0 1 2 3 2 1 2 S S 0 1 3 (a) (b) (c) Figure 1: (a) A length four training sequence. (b) ML estimation for a first-order Markov model optimizes the likelihood of the second node given the first node in each of the length two subsequences. (c) Our objective (Eqn. 2) also includes the likelihood of the last node given the first node in each of these three longer subsequences of the data. (White nodes represent unobserved variables, shaded nodes represent observed variables.) All the terms above are of the form Pθ(st+1|st). Thus, the ML estimator explicitly considers, and tries to model well, only the observed one-step transitions. In Figure 1 we use Bayesian network notation to illustrate the difference between the two objectives for a training sequence of length four. Figure 1(a) shows the training sequence, which can have arbitrary dependencies. Maximum likelihood (ML) estimation maximizes fML(θ) = log Pθ(s1|s0) + log Pθ(s2|s1) + log Pθ(s3|s2). Figure 1(b) illustrates the interactions modeled by ML. Ignoring γ for now, for this example our objective (Eqn. 2) is fML(θ) + log Pθ(s2|s0) + log Pθ(s3|s1) + log Pθ(s3|s0). Thus, it takes into account both the interactions in Figure 1(b) as well as the longer-range ones in Figure 1(c). 4 Algorithm We now present an EM algorithm for optimizing the objective in Eqn. (2) for a first-order Markov model.3 Our algorithm is derived using the method of [7]. (See the Appendix for details.) The algorithm iterates between the following two steps: • E-step: Compute expected counts – ∀i, j ∈S, set stats(j, i) = 0 – ∀t : 0 ≤t ≤T −1, ∀k : 1 ≤k ≤T −t, ∀l : 0 ≤l ≤k −1, ∀i, j ∈S stats(j, i) + = γkPˆθ(St+l+1 = j, St+l = i|St = st, St+k = st+k) • M-step: Re-estimate model parameters Update ˆθ such that ∀i, j ∈S, Pˆθ(j|i) = stats(j, i)/ P k∈S stats(k, i) Prior to starting EM, the transition probabilities Pˆθ can be initialized with the first-order transition counts (i.e., the ML estimate of the parameters), possibly with smoothing.4 Let us now consider more carefully the computation done in the E-step for one specific pair of values for t and k (corresponding to one term log Pθ(st+k|st) in Eqn. 2). For k ≥2, as in the forward-backward algorithm for HMMs (see, e.g., [12, 10]), the pairwise marginals can be computed by a forward propagation (computing the forward messages), a backward propagation (computing the backward messages), and then combining the forward and backward messages.5 Forward and backward messages are computed recursively: for l = 1 to k −1, ∀i ∈S m→t+l(i) = P j∈S m→t+l−1(j)Pˆθ(i|j), (4) for l = k −1 down to 1, ∀i ∈S mt+l←(i) = P j∈S mt+l+1←(j)Pˆθ(j|i), (5) 3Using higher order Markov models or more structured models (such as dynamic Bayesian networks [4, 10] or mixed memory Markov models [8, 14]) offer no special difficulties, though the notation becomes more involved and the inference (in the E-step) might become more expensive. 4A parameter Pˆθ(j|i) initialized to zero will remain zero throughout successive iterations of EM. If this is undesirable, then smoothing could be used to eliminate zero initial values. 5Note that the special case k = 1 (and thus l = 0) does not require inference. In this case we simply have Pˆθ(St+1 = j, St = i|St = st, St+1 = st+1) = 1{i = st}1{j = st+1}. where we initialize m→t(i) = 1{i = st}, and mt+k←(i) = 1{i = st+k}. The pairwise marginals can be computed by combining the forward and backward messages: Pˆθ(St+l+1 = j, St+l = i|St = st, St+k = st+k) = m→t+l(i)Pˆθ(j|i)mt+l+1←(j). (6) For the term log Pθ(st+k|st), we end up performing 2(k −1) message computations, and combining messages into pairwise marginals k −1 times. Doing this for all terms in the objective results in O(T 3) message computations and O(T 3) computations of pairwise marginals from these messages. In practice, the objective (2) can be approximated by considering only the terms in the summation with k ≤H, where H is some time horizon.6 In this case, the computational complexity is reduced to O(TH2). 4.1 Computational Savings The following observation leads to substantial savings in the number of message computations. The forward messages computed for the term log Pθ(st+k|st) depend only on the value of st. So the forward messages computed for the terms {log Pθ(st+k|st)}H k=1 are the same as the forward messages computed just for the term log Pθ(st+H|st). A similar observation holds for the backward messages. As a result, we need to compute only O(TH) messages (as opposed to O(TH2) in the naive algorithm). The following observation leads to further, (even more substantial) savings. Consider two terms in the objective log Pθ(st1+k|st1) and log Pθ(st2+k|st2). If st1 = st2 and st1+k = st2+k, then both terms will have exactly the same pairwise marginals and contribution to the expected counts. So expected counts have to be computed only once for every triple i, j, k for which (St = i, St+k = j) occurs in the training data. As a consequence, the running time for each iteration (once we have made an initial pass over the data to count the number of occurrences of the triples) is only O(|S|2H2), which is independent of the size of the training data. 5 Incorporating actions In decision processes, actions influence the state transition probabilities. To generate training data, suppose we choose an exploration policy and take actions in the DP using this policy. Given the resulting training data, and generalizing Eqn. (2) to incorporate actions, our estimator now becomes ˆθ = arg max θ T −1 X t=0 T −t X k=1 γk log Pθ(st+k|st, at:t+k−1). (7) The EM algorithm is straightforwardly extended to this setting, by conditioning on the actions during the E-step, and updating state-action transition probabilities Pθ(j|i, a) in the M-step. As before, forward messages need to be computed only once for each value of t, and backward messages only once for each value of t + k. However achieving the more substantial savings, as described in the second paragraph of Section 4.1, is now more difficult. In particular, now the contribution of a triple i, j, k (one for which (St = i, St+k = j) occurs in the training data) depends on the action sequence at:t+k−1. The number of possible sequences of actions at:t+k−1 grows exponentially with k. If, however, we use a deterministic exploration policy to generate the training data (more specifically, one in which the action taken is a deterministic function of the current state), then we can again obtain these computational advantages: Counts of the number of occurrences of the triples described previously are now again a sufficient statistic. However, a single deterministic exploration policy, by definition, cannot explore all state-action pairs. Thus, we will instead use a combination of several deterministic exploration policies, which jointly can explore all state-action pairs. In this case, the running time for the E-step becomes O(|S|2H2|Π|), where |Π| is the number of different deterministic exploration policies used. (See Section 6.2 for an example.) 6Because of the discount term γk in the objective (2), one can safely truncate the summation over k after about O(1/(1 −γ)) terms without incurring too much error. S G A B 0 0.2 0.4 0.6 0.8 −80 −70 −60 −50 −40 −30 Correlation level for noise Utility new algorithm maximum likelihood 0.7 0.75 0.8 0.85 0.9 0.95 −800 −600 −400 −200 Correlation level between arrivals Utility new algorithm maximum likelihood (a) (b) (c) Figure 2: (a) Grid-world. (b) Grid-world experimental results, showing the utilities of policies obtained from the MDP estimated using ML (dash-dot line), and utilities of policies obtained from the MDP estimated using our objective (solid line). Results shown are means over 5 independent trials, and the error bars show one standard error for the mean. The horizontal axis (correlation level for noise) corresponds to the parameter q in the experiment description. (c) Queue experiment, showing utilities obtained using ML (dash-dot line), and using our algorithm (solid line). Results shown are means over 5 independent trials, and the error bars show one standard error for the mean. The horizontal axis (correlation level between arrivals) corresponds to the parameter b in the experiment description. (Shown in color, where available.) 6 Experiments In this section, we empirically study the performance of model fitting using our proposed algorithm, and compare it to the performance of ordinary ML estimation. 6.1 Shortest vs. safest path Consider an agent acting for 100 time steps in the grid-world in Figure 2(a). The initial state is marked by S, and the absorbing goal state by G. The reward is -500 for the gray squares, and -1 elsewhere. This DP has four actions that (try to) move in each of the four compass directions, and succeed with probability 1 −p. If an action is not successful, then the agent’s position transitions to one of the neighboring squares. Similar to our example in Section 1, the random transitions (resulting from unsuccessful actions) may be correlated over time. In this problem, if there is no noise (p = 0), the optimal policy is to follow one of the shortest paths to the goal that do not pass through gray squares, such as path A. For higher noise levels, the optimal policy is to stay as far away as possible from the gray squares, and try to follow a longer path such as B to the goal.7 At intermediate noise levels, the optimal policy is strongly dependent on how correlated the noise is between successive time steps. The larger the correlation, the more dangerous path A becomes (for reasons similar to the random walk example in Section 1). In our experiments, we compare the behavior of our algorithm and ML estimation with different levels of noise correlation.8 Figure 2(b) shows the utilities obtained by the two different models, under different degrees of correlation in the noise. The two algorithms perform comparably when the correlation is weak, but our method outperforms ML when there is strong correlation. Empirically, when the noise correlation is high, our algorithm seems to be fitting a first-order model with a larger “effective” noise level. When the resulting estimated MDP is solved, this gives more cautious policies, such as ones more inclined to choose path B over A. In contrast, the ML estimate performs poorly in this problem because it tends to underestimate how far sideways the agent tends to move due to the noise (cf. the example in Section 1). 7For very high noise levels (e.g. p = 0.99) the optimal policy is qualitatively different again. 8Experimental details: The noise is governed by an (unobserved) Markov chain with four states corresponding to the four compass directions. If an action at time t is not successful, the agent moves in the direction corresponding to the state of this Markov chain. On each step, the Markov chain stays in the current state with probability q, and transitions with probability 1 −q uniformly to any of the four states. Our experiments are carried out varying q from 0 (low noise correlation) to 0.9 (strong noise correlation). A 200,000 length state-action sequence for the grid-world, generated using a random exploration policy, was used for model fitting, and a constant noise level p = 0.3 was used in the experiments. Given a learned MDP model, value iteration was used to find the optimal policy for it. To reduce computation, we only included the terms of the objective (Eqn. 7) for which k = 10. 6.2 Queue We consider a service queue in which the average arrival rate is p. Thus, p = P(a customer arrives in one time step). Also, for each action i, let qi denote the service rate under that action (thus, qi = P(a customer is served in one time step|action = i)). In our problem, there are three service rates q0 < q1 < q2 with respective rewards 0, −1, −10. The maximum queue size is 20, and the reward for any state of the queue is 0, except when the queue becomes full, which results in a reward of -1000. The service rates are q0 = 0, q1 = p and q2 = 0.75. So the inexpensive service rate q1 is sufficient to keep up with arrivals on average. However, even though the average arrival rate is p, the arrivals come in “bursts,” and even the high service rate q2 is insufficient to keep the queue small during the bursts of many consecutive arrivals.9 Experimental results on the queue are shown in Figure 2(c). We plot the utilities obtained using each of the two algorithms for high arrival correlations. (Both algorithms perform essentially identically at lower correlation levels.) We see that the policies obtained with our algorithm consistently outperform those obtained using maximum likelihood to fit the model parameters. As expected, the difference is more pronounced for higher correlation levels, i.e., when the true model is less well approximated by a first-order model. For learning the model parameters, we used three deterministic exploration policies, each corresponding to always taking one of the three actions. Thus, we could use the more efficient version of the algorithm described in the second paragraph of Section 4.1 and at the end of Section 5. A single EM iteration for the experiments on the queue took 6 minutes for the original version of the algorithm, but took only 3 seconds for the more efficient version; this represents more than a 100-fold speedup. 7 Conclusions We proposed a method for learning a first-order Markov model that captures the system’s dynamics on longer time scales than a single time step. In our experiments, this method was also shown to outperform the standard maximum likelihood model. In other experiments, we have also successfully applied these ideas to modeling the dynamics of an autonomous RC car. (Details will be presented in a forthcoming paper.) References [1] T. M. Cover and J. A. Thomas. Elements of Information Theory. Wiley, 1991. [2] T. G. Dietterich. Hierarchical reinforcement learning with the MAXQ value function decomposition. JAIR, 2000. [3] P. Gahinet, A. Nemirovski, A. Laub, and M. Chilali. LMI Control Toolbox. Natick, MA, 1995. [4] Z. Ghahramani. Learning dynamic Bayesian networks. In Adaptive Processing of Sequences and Data Structures, pages 168–197. Springer-Verlag, 1998. 9Experimental details: The true process has two different (hidden) modes for arrivals. The first mode has a very low arrival rate, and the second mode has a very high arrival rate. We denote the steady state distribution over the two modes by (φ1, φ2). (I.e., the system spends a fraction φ1 of the time in the low arrival rate mode, and a fraction φ2 = 1 −φ1 of the time in high arrival rate mode.) Given the steady state distribution, the state transition matrix [a 1 −a; 1 −b b] has only one remaining degree of freedom, which (essentially) controls how often the system switches between the two modes. (Here, a [resp. b] is the probability, if we are in the slow [resp. fast] mode, of staying in the same mode the next time step.) More specifically, assuming φ1 > φ2, we have b ∈[0, 1], a = 1 −(1 −b)φ2/φ1. The larger b is, the more slowly the system switches between modes. Our experiments used φ1 = 0.8, φ2 = 0.2, P(arrival|mode 1) = 0.01, P(arrival|mode 2) = 0.99. This means b = 0.2 gives independent arrival modes for consecutive time steps. In our experiments, q0 = 0, and q1 was equal to the average arrival rate p = φ1P(arrival|mode 1) + φ2P(arrival|mode 2). Note that the highest service rate q2(= 0.75) is lower than the fast mode’s arrival rate. Training data was generated using 8000 simulations of 25 time steps each, in which the queue length is initialized randomly, and the same (randomly chosen) action is taken on all 25 time steps. To reduce computational requirements, we only included the terms of the objective (Eqn. 7) for which k = 20. We used a discount factor γ = .95 and approximated utilities by truncating at a finite horizon of 100. Note that although we explain the queuing process by arrival/departure rates, the algorithm learns full transition matrices for each action, and not only the arrival/departure rates. [5] L. P. Kaelbling, M. L. Littman, and A. R. Cassandra. Planning and acting in partially observable stochastic domains. Artificial Intelligence, 101, 1998. [6] M. Kearns, Y. Mansour, and A. Y. Ng. Approximate planning in large POMDPs via reusable trajectories. In NIPS 12, 1999. [7] R. Neal and G. Hinton. A view of the EM algorithm that justifies incremental, sparse, and other variants. In Learning in Graphical Models, pages 355–368. MIT Press, 1999. [8] H. Ney, U. Essen, and R. Kneser. On structuring probabilistic dependencies in stochastic language modeling. Computer Speech and Language, 8, 1994. [9] A. Y. Ng and M. I. Jordan. On discriminative vs. generative classifiers: A comparison of logistic regression and naive Bayes. In NIPS 14, 2002. [10] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kauffman, 1988. [11] D. Precup, R. S. Sutton, and S. Singh. Theoretical results on reinforcement learning with temporally abstract options. In Proc. ECML, 1998. [12] L. R. Rabiner. A tutorial on hidden Markov models and selected applications in speech recognition. Proceedings of the IEEE, 77, 1989. [13] J. K. Satia and R. L. Lave. Markov decision processes with uncertain transition probabilities. Operations Research, 1973. [14] L. K. Saul and M. I. Jordan. Mixed memory Markov models: decomposing complex stochastic processes as mixtures of simpler ones. Machine Learning, 37, 1999. [15] R. S. Sutton. TD models: Modeling the world at a mixture of time scales. In Proc. ICML, 1995. [16] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, 1998. [17] C. C. White and H. K. Eldeib. Markov decision processes with imprecise transition probabilities. Operations Research, 1994. Appendix: Derivation of EM algorithm This Appendix derives the EM algorithm that optimizes Eqn. (7). The derivation is based on [7]’s method. Note that because of discounting, the objective is slightly different from the standard setting of learning the parameters of a Markov chain with unobserved variables in the training data. Since we are using a first-order model, we have Pˆθ(st+k|st, at:t+k−1) = P St+1:t+k−1 Pˆθ(st+k|St+k−1, at+k−1)Pˆθ(St+k−1|St+k−2, at+k−2) . . . Pˆθ(St+1|st, at). Here, the summation is over all possible state sequences St+1:t+k−1. So we have PT −1 t=0 PT −t k=1 γk log Pˆθ(st+k|st, at:t+k−1) = PT −1 t=0 γ log Pˆθ(st+1|st, at) + PT −1 t=0 PT −t k=2 γk log P St+1:t+k−1 Qt,k(St+1:t+k−1) Qt,k(St+1:t+k−1) Pˆθ(st+k|St+k−1, at+k−1)Pˆθ(St+k−1|St+k−2, at+k−2) . . . Pˆθ(St+1|st, at) ≥ PT −1 t=0 γ log Pˆθ(st+1|st, at) + PT −1 t=0 PT −t k=2 γkQt,k(St+1:t+k−1) log Pˆ θ(st+k|St+k−1,at+k−1)Pˆ θ(St+k−1|St+k−2,at+k−2)...Pˆ θ(St+1|st,at) Qt,k(St+1:t+k−1) . (8) Here, Qt,k is a probability distribution, and the inequality follows from Jensen’s inequality and the concavity of log(·). As in [7], the EM algorithm optimizes Eqn. (8) by alternately optimizing with respect to the distributions Qt,k (E-step), and the transition probabilities Pˆθ(·|·, ·) (M-step). Optimizing with respect to the Qt,k variables (E-step) is achieved by setting Qt,k(St+1:t+k−1) = Pˆθ(St+1, . . . , St+k−1|St = st, St+k = st+k, At:t+k−1 = at:t+k−1). (9) Optimizing with respect to the transition probabilities Pˆθ(·|·, ·) (M-step) for Qt,k fixed as in Eqn. (9) is done by updating ˆθ to ˆθnew such that ∀i, j ∈S, ∀a ∈A we have that Pˆθnew(j|i, a) = stats(j, i, a)/ P k∈S stats(k, i, a), where stats(j, i, a) = PT −1 t=0 PT −t k=1 Pk−1 l=0 γkPˆθ(St+l+1 = j, St+l = i|St = st, St+k = st+k, At:t+k−1 = at:t+k−1)1{at+l = a}. Note that only the pairwise marginals Pˆθ(St+l+1, St+l|St, St+k, At:t+k−1) are needed in the M-step, and so it is sufficient to compute only these when optimizing with respect to the Qt,k variables in the E-step.
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Parametric Embedding for Class Visualization Tomoharu Iwata, Kazumi Saito, Naonori Ueda NTT Communication Science Laboratories NTT Corporation 2-4 Hikaridai Seika-Cho Soraku-gun Kyoto, 619-0237 JAPAN {iwata,saito,ueda}@cslab.kecl.ntt.co.jp Sean Stromsten, Thomas L. Griffiths, Joshua B. Tenenbaum Department of Brain and Cognitive Sciences Massachusetts Institute of Technology {sean s,gruffydd,jbt}@mit.edu Abstract In this paper, we propose a new method, Parametric Embedding (PE), for visualizing the posteriors estimated over a mixture model. PE simultaneously embeds both objects and their classes in a low-dimensional space. PE takes as input a set of class posterior vectors for given data points, and tries to preserve the posterior structure in an embedding space by minimizing a sum of Kullback-Leibler divergences, under the assumption that samples are generated by a Gaussian mixture with equal covariances in the embedding space. PE has many potential uses depending on the source of the input data, providing insight into the classifier’s behavior in supervised, semi-supervised and unsupervised settings. The PE algorithm has a computational advantage over conventional embedding methods based on pairwise object relations since its complexity scales with the product of the number of objects and the number of classes. We demonstrate PE by visualizing supervised categorization of web pages, semi-supervised categorization of digits, and the relations of words and latent topics found by an unsupervised algorithm, Latent Dirichlet Allocation. 1 Introduction Recently there has been great interest in algorithms for constructing low-dimensional feature-space embeddings of high-dimensional data sets. These algorithms seek to capture some aspect of the data set’s intrinsic structure in a low-dimensional representation that is easier to visualize or more efficient to process by other learning algorithms. Typical embedding algorithms take as input a matrix of data coordinates in a high-dimensional ambient space (e.g., PCA [5]), or a matrix of metric relations between pairs of data points (MDS [7], Isomap [6], SNE [4]). The algorithms generally attempt to map all and only nearby input points onto nearby points in the output embedding. Here we consider a different sort of embedding problem with two sets of points X = {x1, . . . , xN} and C = {c1, . . . , cK}, which we call “objects” (X) and “classes” (C). The input consists of conditional probabilities p(ck|xn) associating each object xn with each class ck. Many kinds of data take this form: for a classification problem, C may be the set of classes, and p(ck|xn) the posterior distribution over these classes for each object xn; in a marketing context, C might be a set of products and p(ck|xn) the probabilistic preferences of a consumer; or in language modeling, C might be a set of semantic topics, and p(ck|xn) the distribution over topics for a particular document, as produced by a method like Latent Dirichlet Allocation (LDA) [1]. Typically, the number of classes is much smaller than the number of objects, K << N. We seek a low-dimensional embedding of both objects and classes such that the distance between object n and class k is monotonically related to the probability p(ck|xn). This embedding simultaneously represents not only the relations between objects and classes, but also the relations within the set of objects and within the set of classes – each defined in terms of relations to points in the other set. That is, objects that tend to be associated with the same classes should be embedded nearby, as should classes that tend to have the same objects associated with them. Our primary goals are visualization and structure discovery, so we typically work with two- or three-dimensional embeddings. Object-class embeddings have many potential uses, depending on the source of the input data. If p(ck|xn) represents the posterior probabilities from a supervised Bayesian classifier, an object-class embedding provides insight into the behavior of the classifier: how well separated the classes are, where the errors cluster, whether there are clusters of objects that “slip through a crack” between two classes, which objects are not well captured by any class, and which classes are intrinsically most confusable with each other. Answers to these questions could be useful for improved classifier design. The probabilities p(ck|xn) may also be the product of unsupervised or semi-supervised learning, where the classes ck represent components in a generative mixture model. Then an object-class embedding shows how well the intrinsic structure of the objects (and, in a semi-supervised setting, any given labels) accords with the clustering assumptions of the mixture model. Our specific formulation of the embedding problem assumes that each class ck can be represented by a spherical Gaussian distribution in the embedding space, so that the embedding as a whole represents a simple Gaussian mixture model for each object xn. We seek an embedding that matches the posterior probabilities for each object under this Gaussian mixture model to the input probabilities p(ck|xn). Minimizing the Kullback-Leibler (KL) divergence between these two posterior distributions leads to an efficient algorithm, which we call Parametric Embedding (PE). PE can be seen as a generalization of stochastic neighbor embedding (SNE). SNE corresponds to a special case of PE where the objects and classes are identical sets. In SNE, the class posterior probabilities p(ck|xn) are replaced by the probability p(xm|xn) of object xn under a Gaussian distribution centered on xm. When the inputs (posterior probabilities) to PE come from an unsupervised mixture model, PE performs unsupervised dimensionality reduction just like SNE. However, it has several advantages over SNE and other methods for embedding a single set of data points based on their pairwise relations (e.g., MDS, Isomap). It can be applied in supervised or semi-supervised modes, when class labels are available. Because its computational complexity scales with NK, the product of the number of objects and the number of classes, it can be applied efficiently to data sets with very many objects (as long as the number of classes remains small). In this sense, PE is closely related to landmark MDS (LMDS) [2], if we equate classes with landmarks, objects with data points, and −log p(ck|xn) with the squared distances input to LMDS. However, LMDS lacks a probabilistic semantics and is only suitable for unsupervised settings. Lastly, even if hard classifications are not available, it is often the relations of the objects to the classes, rather than to each other, that we are interested in. After describing the mathematical formulation and optimization procedures used in PE (Section 2), we present applications to visualizing the structure of several kinds of class posteriors. In section 3, we look at supervised classifiers of hand-labeled web pages. In section 4, we examine semi-supervised classifiers of handwritten digits. Lastly, in section 5, we apply PE to an unsupervised probabilistic topics model, treating latent topics as classes, and words as objects. PE handles these datasets easily, in the last producing an embedding for over 26,000 objects in a little over a minute (on a 2GHz Pentium computer). 2 Parametric Embedding method Given as input conditional probabilities p(ck|xn), PE seeks an embedding of objects with coordinates rn and classes with coordinates φk, such that p(ck|xn) is approximated as closely as possible by the posterior probabilities from a unit-variance spherical Gaussian mixture model in the embedding space: p(ck|rn) = p(ck) exp(−1 2 ∥rn −φk ∥2) PK l=1 p(cl) exp(−1 2 ∥rn −φl ∥2) . (1) Here ∥· ∥is the Euclidean norm in the embedding space. When the conditional probabilities p(ck|xn) arise as posterior probabilities from a mixture model, we will also typically be given priors p(ck) as input; otherwise the p(ck) terms above may be assumed equal. It is natural to measure the degree of correspondence between input probabilities and embedding-space probabilities using a sum of KL divergences for each object: PN n=1 KL(p(ck|xn)||p(ck|rn)). Minimizing this sum w.r.t. {p(ck|rn))} is equivalent to minimizing the objective function E({rn}, {φk}) = − N X n=1 K X k=1 p(ck|xn) log p(ck|rn). (2) Since this minimization problem cannot be solved analytically, we employ a coordinate descent method. We initialize {φk}, and we iteratively minimize E w.r.t. to {φk} or {rn} while fixing the other set of parameters, until E converges. Derivatives of E are: ∂E ∂rn = K X k=1 αn,k(rn −φk) and ∂E ∂φk = N X n=1 αn,k(φk −rn), (3) where αn,k = p(ck|xn) −p(ck|rn). These learning rules have an intuitive interpretation (analogous to those in SNE) as a sum of forces pulling or pushing rn (φk) depending on the sign of αn,k. Importantly, the Hessian of E w.r.t. {rn} is a semi-positive definite matrix: ∂2E ∂rn∂r′n = K X k=1 p(ck|rn)φkφ′ k − K X k=1 p(ck|rn)φk ! K X k=1 p(ck|rn)φk !′ (4) since the r.h.s. of (4) is exactly a covariance matrix. Thus we can find the globally optimal solution for {rn} given {φk}.1 The computational complexity of PE is O(NK), which is much more efficient than that of pairwise (dis)similarity-based methods with O(N 2) computations (such as SNE, MDS, or Isomap). 1In our experiments, we found that optimization proceeded more smoothly with a regularized objective function, J = E + ηr PN n=1 ∥rn ∥2 +ηφ PK k=1 ∥φk ∥2, where ηr, ηφ > 0. 3 Analyzing supervised classifiers on web data In this section, we show how PE can be used to visualize the structure of labeled data (web pages) in a supervised classification task. We also compare PE with two conventional methods, MDS [7] and Fisher linear discriminant analysis (FLDA) [3]. MDS seeks a lowdimensional embedding that preserves the input distances between objects. It does not normally use class labels for data points, although below we discuss a way to apply MDS to label probabilities that arise in classification. FLDA, in contrast, naturally uses labeled data in constructing a low-dimensional embedding. It seeks a a linear projection of the objects’ coordinates in a high-dimensional ambient space that maximizes between-class variance and minimizes within-class variance. The set of objects comprised 5500 human-classified web pages: 500 pages sampled from each of 11 top level classes in Japanese directories of Open Directory (http://dmoz.org/). Pages with less than 50 words, or which occurred under multiple categories, were eliminated. A Naive Bayes (NB) classifier was trained on the full data (represented as word frequency vectors). Posterior probabilities p(ck|xn) were calculated for classifying each object (web page), assuming its true class label was unknown. These probabilities, as well as estimated priors p(ck), form the input to PE. Fig.1(a) shows the output of PE, which captures many features of this data set and classification algorithm. Pages belonging to the same class tend to cluster well in the embedding, which makes sense given the large sample of labeled data. Related categories are located nearby: e.g., sports and health, or computers and online-shopping. Well-separated clusters correspond to classes (e.g. sports) that are easily distinguished from others. Conversely, regional pages are dispersed, indicating that they are not easily classified. Distinctive pages are evident as well: a few pages that are scattered among the objects of another category might be misclassified. Pages located between clusters are likely to be categorized in multiple classes; arcs between two classes show subsets of objects that distribute their probability among those two classes and no others. Fig.1(b) shows the result of MDS applied to cosine distances between web pages. No labeled information is used (only word frequency vectors for the pages), and consequently no class structure is visible. Fig.1(c) shows the result of FLDA. To stabilize the calculation, FLDA was applied only after word frequencies were smoothed via SVD. FLDA uses label information, and clusters together the objects in each class better than MDS does. However, most clusters are highly overlapping, and the separation of classes is much poorer than with PE. This seems to be a consequence of FLDA’s restriction to purely linear projections, which cannot, in general, separate all of the classes. Fig.1(d) shows another way of embedding the data using MDS, but this time applied to Euclidean distances in the (K −1)−dimensional space of posterior distributions p(ck|xn). Pages belonging to the same class are definitely more clustered in this mode, but still the clusters are highly overlapping and provide little insight into the classifier’s behavior. This version of MDS uses the same inputs as PE, rather than any high-dimensional word frequency vectors, but its computations are not explicitly probabilistic. The superior results of PE (Fig.1(a)) illustrate the advantage of optimizing an appropriate probabilistic objective function. 4 Application to semi-supervised classification The utility of PE for analyzing classifier performance may best be illustrated in a semisupervised setting, with a large unlabeled set of objects and a smaller set of labeled objects. We fit a probabilistic classifier based on the labeled objects, and we would like to visualize the behavior of the classifier applied to the unlabeled objects, in a way that suggests how (a) PE (b) MDS (word frequencies) (c) FLDA (d) MDS (posteriors) Figure 1: The visualizations of categorized web pages. Each of the 5500 web pages is show by a particle with shape indicating the page’s class. accurate the classifier is likely to be and what kinds of errors it is likely to make. We constructed a simple probabilistic classifier for 2558 handwritten digits (classes 0-4) from the MNIST database. The classifier was based on a mixture model for the density of each class, defined by selecting either 10 or 100 digits uniformly at random from each class and centering a fixed-covariance Gaussian (in pixel space) on each of these examples – essentially a soft nearest-neighbor method. The posterior distribution over this classifier for all 2558 digits was submitted as input to PE. The resulting embeddings allow us to predict the classifiers’ patterns of confusions, calculated based on the true labels for all 2558 objects. Fig. 2 shows embeddings for both 10 labels/class and 100 labels/class. In both cases we see five clouds of points corresponding to the five classes. The clouds are elongated and oriented roughly towards a common center, forming a star shape (also seen to some extent in our other applications). Objects that concentrate their probability on only one class will lie as far from the center of the plot as possible – ideally, even farther than the mean of their class, because this maximizes their . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . . . . . . .. . . . .. . . . . . . . . . ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . .. . . . . . . . . .. . . . . . . . . . . .. . . . 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. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . ... .. . . . . . . . . . . xxxxxxxxxx xxxxxxxxxx xxx xx xxxxx x x xxxxx xxx xxx x xxxxxx 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 330 0 26 36 3 117 557 134 144 117 5 4 278 17 5 26 0 49 294 6 1 2 1 2 404 Estimated class True class (a) PE with 10 labels/class . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . .. . . . . .. . . . ... 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. . . . .. . .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . . . . . . .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ... . . . . . . . . . . . . ... ... . . . . . . . . . . x x xx x x xxxx xx x x x x x x x x x x x x x x x x x xx x x x x x x xxx x x x x x x x x x x x x xx x xxx x xxx x x x x x xxx xx x x xx xxx xx xxxx xx xx x x x xx x x x x xx x xx x x x x x xxxxx x xxxxxx xxxxxxx x xx xxx x x xxxxxx xxx x xx xxx xxx xx xx x x x x x x xxxxxxx x x x x x xx x x xx xx x xx xxxx xxxxxx xxxx x x x x x xxxxxxx x x x xx x xx x x x xxxxxxx x xx x x x x x xxxxx x xxxx xx xx x x x xx x x x x x x x xx xx x x x x x xxx x x xxx x xx x x x x x x x x x xxx x xx x x x x x x x x x x x xx x x x x x x x x x xx x x x x x x x x x x x x x xx x x x x xx x x x x xxx x x x xxx x x x x x x xxx x x x x x x x x x x x x x x xx x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x x xx x xx x xx x x x xx x x x x x x x x xx x x x x x xx x x x x xx x x xx xx x x x xx x xx x x x x xxxx x x x xx x xx x xxx x x x xx xx x x xx x xx 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 471 0 8 2 1 8 559 74 34 47 0 1 388 2 0 0 2 16 455 1 0 1 2 0 486 Estimated class True class (b) PE with 100 labels/class Figure 2: Parametric embeddings for handwritten digit classification. Each dot represents the coordinates rn of one image. Boxed numbers represent the class means φk. ×’s show labeled examples used to train the classifier. Images of several unlabeled digits are shown for each class. posterior probability on that class. Moving towards the center of the plot, objects become increasingly confused with other classes. Relative to using only 10 labels/class, using 100 labels yields clusters that are more distinct, reflecting better between-class discrimination. Also, the labeled examples are more evenly spread through each cluster, reflecting more faithful within-class models and less overfitting. In both cases, the ‘1’ class is much closer than any other to the center of the plot, reflecting the fact that instances of other classes tend to be mistaken for ‘1’s. Instances of other classes near the ‘1’ center also tend to look rather “one-like” – thinner and more elongated. The dense cluster of points just outside the mean for ‘1’ reflects the fact that ‘1’s are rarely mistaken for other digits. In Fig. 2(a), the ‘0’ and ‘3’ distributions are particularly overlapping, reflecting that those two digits are most readily confused with each other (apart from 1). The ‘webbing’ between the diffuse ‘2’ arm and the tighter ‘3’ arm reflects the large number of ‘2’s taken for ‘3’s. In Fig. 2(b), that ‘webbing’ persists, consistent with the observation that (again, apart from many mistaken responses of 1) the confusion of ‘2’s for ‘3’s is the only large-scale error these larger data permit. 5 Application to unsupervised latent class models In the applications above, PE was applied to visualize the structure of classes based at least to some degree on labeled examples. The algorithm can also be used in a completely unsupervised setting, to visualize the structure of a probabilistic generative model based on latent classes. Here we illustrate this application of PE by visualizing a semantic space of word meanings: objects correspond to words, and classes correspond to topics in a latent Dirichlet allocation (LDA) model [1] fit to a large (>37,000 documents, >12,000,000 word tokens) corpus of educational materials for first grade to college (TASA). The problem of mapping a large vocabulary is particularly challenging, and, with over 26,000 objects (word types), prohibitively expensive for pairwise methods. Again, PE solves for the configuration shown in about a minute. In LDA (not to be confused with FLDA above), each topic defines a probability distribution ... . ....... . . ............ . . . . . . .... . .............. .. ....... . . ... .. ... ... ...... . . ....... . . .. . . .. .. .. .... . . . . . . . . . .. . .. .. . ... . .... . . .. . . ... . . ... . . . . . . . . . . . . .. . ....... . . ... . . ... .. . . . . . .. .... ........ . .. . . . . . . . . ....... . . . . .. . . . . . ... . . .... . . .. . .... . ..... . . . .. . . .. . . . ...... . . . . ......... . ..... . . .... . ........... . ...... .. . .. .. .......... . . ... . ......... . .......... .. . .. . . . . .. . . . . ......... . ................ . . . ........... . . . . .. . . ..... . ..... . ......... . . .............. . ... . ... . ... . ... . . .. . ............ . . . .... ...... . .. . . . . .. .. . . . ... . ................................... . . . . . . . ... . .. . ....... . ..... . ...... . ... . ............... . .... . . ....... ............... . . . . ....... . . .. . . ... . ... ........... . ..... . . .... . . . . ... . ... . ....... . . ..... .. ... .. . ........ . .. . . . ........... . . . . . . ... . ... .. ..... . . . . .. . ........ . ........... . ........... .. .. . .. . . .... .... ...... . . ...... . .................................... . . . . ........... . . . ... . . . . .. .... . . . ....... . .. . ... . ............... .... .... . . ...... . . . ....... . ..................... . . . . . . ................... . .. . .. . .. . . . . ......... . ...... . . .. . . . . . . .......... . . . . . . .. . . . . . . . . . . . ........ . ........ .. ...... . . . ................. .. . . ... . . . .... . ..... . . .... . . . . ..... . . . ....... . .. . ........ .. . ......... . .......... . ......... . . . ..... . . . . . . . . ........ . . . .. . . . ... . . ......... . . . .... . . . . . ..... . ... . ...... . ..... . . . . ........ . . ..... . ................. . . . ......... . . .. .. . . . . .. ... . . . . . . . ... . . . . . . . . . . . .. . . . ........ .. . .. .. . . .. .... . .......... .. ....... ... . ..... . .. . ...... . ... . . . .. . ... . .... . . .. . ...... .... . . ......... . . ...... . . . . . ...... .. .............. . . .. . . ........... . .. . ..... .. . . ... . ...... . ...... . . . . . . ...... . . . ... . ... . ... . .. . . .. . ........ . . . . .................... . ....... . . . . . ... . . . . . .. . .. . .. . .. . . . .. . ............... . ... .. .... . . . ...... . ................... . ...... . . . .... . .. . ...... . .. . . ...... ................ . ........... . ... . ...... . ......... . .... . ............... . .... . ..... . . . . . . . .... . . . .. . .. . ....... . . .... . ... . ......... . ................. . ... . .. . .. . ........................ . . . ..... . .................... . . . . .... . .. . ....................... . .... . . . . . . . . . . . . . . .......... . ... . .. .. . . . .................... . . . . ... . ... . .... .. . .. . ........ . ...... . . . ..... . .. . .... . . ................. . . .. . .............. . ........ . .... . ....... . . . ...... . . .. . .. . . . . . .... . .. . . . .... . . ....... .... . . . .... . . . ..... . ................ . ............................................. . .... .. ...... . .. . ........ . ........ . ... . ............... . ... . . . .. . ........ . . . ........... . . . . . .. . . . .. . . .. . ........... . . .. ................ . . ........ . .... . ............... . .. . .. .... . . . . . . . . . . . . . ..... .. . . ........ . ... . ... . ................ . .. . .. . .. . . ..... . ........ . . . ........ .. ... . .. ..... . . .. ................ . ...... . .............. . ....... .. .... . ........... . ... ........ . . . ......... . . . ............... . . . .... . . . ........ . ..... . . ... . . . ...................... . ......... . ...... . ....... . . ..... . . ........ . ...... . . ...... . . .......... . . ... .. ..... . . ........ . ...... . . . ... . ........ . ....... 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... . . . . . . . ... ................. . . . . .. . . . . .... . . ........ ... .. . . .. . . ... . ... . ........ . .................. . . . . . .. ... . ...... . .. . . . ... . . .......... . ......... . ..... . ... . . . . . . . . . .............. .. .. .. ... . . ................. . . . .. . . . ..... . . . ....... ... . .. .. . . . . . .. . . . . . . . . ..... . .. .. . . .......... . .... . .... . .. . . . . . .. . . .. . ... . . . ..... . .... . . . . ... ...... . ... . . . .... . . ... . . .. . . . . . . . .. .. ..... . .. .......... .... . . . . . ... .............. . .... . ...... . . . . . . .............. . .. . . . . .... . .. . ..... . ............ . .. . ..... . .......... . ..... . . . . . . . . . . .... . .. . ...... . .. . ..... ... . . ........... .. . . . . . ............. .. .. . . . . .. . . . .... . ... . ........ . .... . ....... .. . .. . . . . . . . . . ....... .. .. . ... . ........ . . . . . . . . . . ... . . .... . ... . . . .... . ..... . . . . .. . . . 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AUTISTIC CARPENTRY STATEWIDE IMPLEMENTED COVALENTLY ADSORPTION ACTIVATED ALKENE HUMAN SCIENTIFIC DISCOVERY HYPOTHESIS PRICE MONEY TAX INSURANCE FOLD DOME PERMEABLE DRILLED SCHOLARSHIP DIFFICULTY ADMINISTRATION CONTENT MIXTURE COMPOSITION DISSOLVES FRACTIONAL PROBLEM ORIGIN ORDER ESTIMATE DUE OWES PROPERTY INTEREST STONE ERA SHAPED DENSE GROUP PHASE CHEMISTRY PHENOMENON DETECTION ESSENTIALS ENABLE DEPOSITS BILLIONS RISE COLLECTS STRATIFIED AREAS UPPER PATIENCE EFFORTS CONCEPTS APPARATUS SELECTIONVISION REPRESENTED GUIDED RETAINED ABSENT FIRE REQUIRES THOUSANDS ACCUMULATION DROPPED SUBSTANTIALLY COMPREHENSIVE EXPOSED AVERAGE FOUND AGENTS CALLED DESTROYED BLENDS philosophy/history of science training/education geology banking/insurance chemistry Figure 3: Parametric embedding for word meanings and topics based on posterior distributions from an LDA model. Each dot represents the coordinates rn of one word. Large phrases indicate the positions of topic means φk (with topics labeled intuitively). Examples of words that belong to one or more topics are also shown. over word types that can occur in a document. This model can be inverted to give the probability that topic ck was responsible for generating word xn; these probabilities p(ck|xn) provide the input needed to construct a space of word and topic meanings in PE. More specifically, we fit a 50-topic LDA model to the TASA corpus. Then, for each word type, we computed its posterior distribution restricted to a subset of 5 topics, and input these conditional probabilities to PE (with N = 26, 243, K = 5). Fig. 3 shows the resulting embedding. As with the embeddings in Figs. 1 and 2, the topics are arranged roughly in a star shape, with a tight cluster of points at each corner of the star corresponding to words that place almost all of their probability mass on that topic. Semantically, the words in these extreme clusters often (though not always) have a fairly specialized meaning particular to the nearest topic. Moving towards the center of the plot, words take on increasingly general meanings. This embedding shows other structures not visible in previous figures: in particular, dense curves of points connecting every pair of clusters. This pattern reflects the characteristic probabilistic structure of topic models of semantics: in addition to the clusters of words that associate with just one topic, there are many words that associate with just two topics, or just three, and so on. The dense curves in Fig. 3 show that for any pair of topics in this corpus, there exists a substantial subset of words that associate with just those topics. For words with probability sharply concentrated on two topics, points along these curves minimize the sum of the KL and regularization terms. This kind of distribution is intrinsically high-dimensional and cannot be captured with complete fidelity in any 2-dimensional embedding. As shown by the examples labeled in Fig. 3, points along the curves connecting two apparently unrelated topics often have multiple meanings or senses that join them to each topic: ‘deposit’ has both a geological and a financial sense, ‘phase’ has both an everyday and a chemical sense, and so on. 6 Conclusions We have proposed a probabilistic embedding method, PE, that embeds objects and classes simultaneously. PE takes as input a probability distribution for objects over classes, or more generally of one set of points over another set, and attempts to fit that distribution with a simple class-conditional parametric mixture in the embedding space. Computationally, PE is inexpensive relative to methods based on similarities or distances between all pairs of objects, and converges quickly on many thousands of data points. The visualization results of PE shed light on features of both the data set and the classification model used to generate the input conditional probabilities, as shown in applications to classified web pages, partially classified digits, and the latent topics discovered by an unsupervised method, LDA. PE may also prove useful for similarity-preserving dimension reduction, where the high-dimensional model is not of primary interest, or more generally, in analysis of large conditional probability tables that arise in a range of applied domains. As an example of an application we have not yet explored, purchases, web-surfing histories, and other preference data naturally form distributions over items or categories of items. Conversely, items define distributions over people or categories thereof. Instances of such dyadic data abound–restaurants and patrons, readers and books, authors and publications, species and foods...–with patterns that might be visualized. PE provides a tractable, principled, and effective visualization method for large volumes of such data, for which pairwise methods are not appropriate. Acknowledgments This work was supported by a grant from the NTT Communication Sciences Laboratories. References [1] D. Blei, A. Ng and M. Jordan. Latent dirichlet allocation. NIPS 15, 2002. [2] V. de Silva, J. B. Tenenbaum. Global versus local methods in nonlinear dimensionality reduction. NIPS 15, pp. 705-712, 2002. [3] R. Fisher. The use of multiple measurements in taxonomic problem. Annuals of Eugenics 7, pp.179–188, 1950. [4] G. Hinton and S. Roweis. Stochastic neighbor embedding. NIPS 15, 2002. [5] I.T. Joliffe. Principal Component Analysis. Springer, 1980. [6] J. Tenenbaum, V. de Silva and J. Langford. A global geometric framework for nonlinear dimensionality reduction. Science 290 pp. 2319–2323, 2000. [7] W. Torgerson. Theory and Methods of Scaling. New York, Wiley, 1958.
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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 johann@cs.tu-berlin.de, oby@cs.tu-berlin.de 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 dataset1. 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 ̸∈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 Gc = {Gi, i ∈O}; Gi = {k ∈O : 0 < (k−i)2 ≤c}. This way, G1 describes the first order neighborhood system (4 neighbors), G2 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 1Centre 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 κ pixels sites U = {u1, ..., uκ} of an unlabeled image, for which a set of measurements ˜X = {˜xi, ∀i ∈[1..κ]} 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 xi are mapped by φ(x), and n slack variables ξi the primal objective function of a C-SVM can be formulated as min w∈H,ξ∈Rn 1 2
w
2 + C n n X i=1 ξi , (1) subject to yi(wT φ(xi) + b) ≤1 −ξi, ξi ≥0, C > 0, i = 1, ..., n. In order to classify a new object h with unknown label, the following decision rule is evaluated: f(xh) = sgn m X i=1 αiyiK(xh, xi) + b , (2) 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) + λ · K2(yGh, yGj), (3) where λ 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 q ⟨yGh|yGj⟩, (4) where ⟨...|...⟩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 Gc h of order c we obtain K2(yGh, yGj) = 1 q X |s|<√c,s̸=0 yh+s · yj+s (5) 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 κ 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 τ as yh(τ), ∀h. At each step τ new labels are assigned according to yh(τ) = sgn m X j=1 αj · yv(j) · K(xh, xv(j), yGh(τ −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 τ = 0 the results from a conventional support-vector machine (λ = 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 τ, as ch(j) = αj · yv(j) · K1(xh, xv(j)) equation (6) becomes yh(τ) = sgn m X j=1 λαjyv(j)K2(yGh(τ −1), yGv(j)) + ch(j) + b , ∀h. (7) Putting in the linear kernel from equation (5), we get yh(τ) = sgn m X j=1 αjyv(j) λ q X |s|<√c,s̸=0 [yh+s(τ −1) · yv(j)+s] + ch(j) + b , ∀h. (8) Interchanging the sums, using the definitions wh,k = λ q Pm j=1 αjyv(j)yv(j)+(k−h) : k ∈Gh 0 : k /∈Gh (9) and θh = − m X j=1 ch(j) + b , (10) we obtain yh(τ) = sgn X k yk(τ −1) · wh,k −θh , ∀h. (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 wh,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(τ) = yh(τ −1), ∀h, or an attractor cycle yh(τ) = yh(ρ), ρ < τ −1, ∀h. The network described by eq. (11) corresponds to a diluted network of κ 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 Gh. The measurement xh only influences the individual unit threshold θh 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 λ, 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 λ = 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 Ksym 2 of kernel K2. For the pixel grid we can define the inversion operation as l + t →l −t, t ∈ N2, ∀l + t ∈Gl, and denote the inverse of a by ¯a. Taking the inverse of the vector yGl, 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 Ksym 2 (yGh, yGj) = 1 q (⟨yGh|yGj⟩+ ⟨yGh|¯yGj⟩). (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 ⟨yGh|yGj⟩= ⟨¯yGh|¯yGj⟩= ⟨yGj|yGh⟩= ⟨¯yGj|¯yGh⟩and ⟨¯yGh|yGj⟩= ⟨yGh|¯yGj⟩= ⟨yGj|¯yGh⟩= ⟨¯yGj|yGh⟩. Therefore, Ksym 2 (yGh, yGj) = 1 q (⟨yGh|yGj⟩+ ⟨yGh|¯yGj⟩) = 1 q (⟨yGj|yGh⟩+ ⟨¯yGj|yGh⟩) = 1 q (⟨yGj|yGh⟩+ ⟨yGj|¯yGh⟩) = Ksym 2 (yGj, yGh) □. Putting kernel (12) into eq.(7) and defining wsym h,k = λ q Pm j=1 αjyv(j)(yv(j)+(k−h) + yv(j)−(k−h)) : k ∈Gh 0 : k /∈Gh (13) we get yh(τ) = sgn X k yk(τ −1) · wsym h,k −θh , ∀h. (14) Since the network weights wsym h,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 σ at optimal hyper-parameters C, S and λ. algorithm log2 C log2 S λ R[%] σ[%] SVM 4 0.5 0 2.23 0.05 STSVM 4 0.5 1.2 1.96 0.06 TSVM 2 0.5 1.4 1.86 0.05 3 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 Ksym 2 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(−∥x1 −x2∥2/S2) 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 λ) 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 λ, fig.1 shows 10 typical test images and their segmentations achieved by an STSVM at different values of λ for fixed S and C. For increasing λ the label images become less noisy, and at λ = 0.4 most artifacts have disappeared. This is caused by the increasing weight put on the label configuration via kernel Ksym 2 . Increasing λ 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 = 26, S = 22. (a) λ = 0, (b) λ = 0.1, (c) λ = 0.2, (d) λ = 0.3, (e) λ = 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 Sch¨olkopf, 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 Sch¨olkopf, 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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New Criteria and a New Algorithm for Learning in Multi-Agent Systems Rob Powers Computer Science Department Stanford University Stanford, CA 94305 powers@cs.stanford.edu Yoav Shoham Computer Science Department Stanford University Stanford, CA 94305 shoham@cs.stanford.edu 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 →ℜis 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 Defect Cooperate 3, 3 0, 4 Defect 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 1The 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 ǫ > 0 and δ > 0 there exist a T0, polynomial in 1 ǫ , 1 δ , k, and b, such that for any number of rounds t > T0 the algorithm achieves the following payoff guarantees with probability at least 1 −δ. Targeted Optimality: When the opponent is a member of the selected set of opponents, the average payoff is at least VBR−ǫ, 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 VselfP lay −ǫ, where VselfP 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 −ǫ, with Vsecurity defined as maxπ1∈Π1 minπ2∈Π2 EV (π1, π2).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. BRǫ(π) ←arg max x∈X(π,ǫ) (EOV (x, π)), 4 where X(π, ǫ) = {y ∈Π1 : EV (y, π) ≥max z∈Π1(EV (z, π)) −ǫ} 2An 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. 3Throughout the paper, we use EV (π1, π2) to indicate the expected payoff to a player for playing strategy π1 against an opponent playing π2 and EOV (π1, π2) as the expected payoff the opponent achieves. Π1 and Π2 are the sets of mixed strategies for the agent and its opponent respectively. 4Note that BR0(π) is a member of the standard set of best response strategies to π. 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. BullyMixed ←arg max x∈X (EOV (x, BR(x))), where X = {y ∈Π1 : EV (y, BR0(y)) = max z∈Π1(EV (z, BR0(z)))} The maximin strategy is defined as Maximin ←arg max π1∈Π1 min π2∈Π2 EV (π1, π2) 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 (BullyMixed, BR0(BullyMixed)), and dt2 t1 represents the distribution of opponent actions for the period from round t1 to round t2. Set strategy = BullyMixed for τ1 time steps Play strategy for τ2 time steps if (strategy == BullyMixed AND AvgV alueH < VBully −ǫ1) With probability, p, set strategy = BRǫ2(dt 0) Play strategy if ||dτ1 0 −dt t−τ1|| < ǫ3 Set bestStrategy = BRǫ2(dt 0) else if (strategy == BullyMixed AND AvgV alueH > VBully −ǫ1) Set bestStrategy = BullyMixed else Set bestStrategy = BestResponse while not end of game if avgV aluet−τ0 < Vsecurity −ǫ0 Play maximin strategy for τ3 time steps else Play bestStrategy for τ3 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 ǫ)3 1 δ . 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. 0.100 0.200 0.300 0.400 0.500 MiniMax BullyMixed Bully Random MetaStrategy StochFP StochIGA WoLF-PHC LocalQ Hyper-Q JointQ-Max MetaStrategy StochFP StochIGA WoLF-PHC BullyMixed MiniMax 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. 70% 75% 80% 85% 90% 95% 100% BattleOfTheSexes BertrandOligopoly Chicken CollaborationGame CoordinationGame CovariantGame DispersionGame GameWActionPar GrabTheDollar HawkAndDove MajorityVoting MatchingPennies MinimumEffort PrisonersDilemma RandomCompound RandomGame RandomZeroSum Rochambeau ShapleysGame TravelersDilemma TwoByTwoGame MetaStrategy StochFP StochIGA WoLF-PHC BullyMixed MiniMax 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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The cerebellum chip: an analog VLSI implementation of a cerebellar model of classical conditioning Constanze Hofstötter, Manuel Gil, Kynan Eng, Giacomo Indiveri, Matti Mintz, Jörg Kramer* and Paul F. M. J. Verschure Institute of Neuroinformatics University/ETH Zurich CH-8057 Zurich, Switzerland pfmjv@ini.phys.ethz.ch 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. 1 Introduction 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 *Jörg 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; Hofstötter 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 The model circuit and aVLSI implementation 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 (Hofstötter 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 1.6µm 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 Hofstötter 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 Simulated conditioning experiments 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 (Hofstötter 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 Robot associative learning experiments 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 Discussion 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 References 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. Hofstötter, 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 Hofstötter, 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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Pictorial Structures for Molecular Modeling: Interpreting Density Maps Frank DiMaio, Jude Shavlik George Phillips Department of Computer Sciences Department of Biochemistry University of Wisconsin-Madison University of Wisconsin-Madison {dimaio,shavlik}@cs.wisc.edu phillips@biochem.wisc.edu Abstract X-ray crystallography is currently the most common way protein structures are elucidated. One of the most time-consuming steps in the crystallographic process is interpretation of the electron density map, a task that involves finding patterns in a three-dimensional picture of a protein. This paper describes DEFT (DEFormable Template), an algorithm using pictorial structures to build a flexible protein model from the protein's amino-acid sequence. Matching this pictorial structure into the density map is a way of automating density-map interpretation. Also described are several extensions to the pictorial structure matching algorithm necessary for this automated interpretation. DEFT is tested on a set of density maps ranging from 2 to 4Å resolution, producing rootmean-squared errors ranging from 1.38 to 1.84Å. 1 Introduction An important question in molecular biology is what is the structure of a particular protein? Knowledge of a protein’s unique conformation provides insight into the mechanisms by which a protein acts. However, no algorithm exists that accurately maps sequence to structure, and one is forced to use "wet" laboratory methods to elucidate the structure of proteins. The most common such method is x-ray crystallography, a rather tedious process in which x-rays are shot through a crystal of purified protein, producing a pattern of spots (or reflections) which is processed, yielding an electron density map. The density map is analogous to a threedimensional image of the protein. The final step of x-ray crystallography – referred to as interpreting the map – involves fitting a complete molecular model (that is, the position of each atom) of the protein into the map. Interpretation is typically performed by a crystallographer using a time-consuming manual process. With large research efforts being put into high-throughput structural genomics, accelerating this process is important. We investigate speeding the process of x-ray crystallography by automating this time-consuming step. When interpreting a density map, the amino-acid sequence of the protein is known in advance, giving the complete topology of the protein. However, the intractably large conformational space of a protein – with hundreds of amino acids and thousands of atoms – makes automated map interpretation challenging. A few groups have attempted automatic interpretation, with varying success [1,2,3,4]. Confounding the problem are several sources of error that make automated interpretation extremely difficult. The primary source of difficulty is due to the crystal only diffracting to a certain extent, eliminating higher frequency components of the density map. This produces an overall blurring effect evident in the density map. This blurring is quantified as the resolution of the density map and is illustrated in Figure 1. Noise inherent in data collection further complicates interpretation. Given minimal noise and sufficiently good resolution – about 2.3Å or less – automated density map interpretation is essentially solved [1]. However, in poorer quality maps, interpretation is difficult and inaccurate, and other automated approaches have failed. The remainder of the paper describes DEFT (DEFormable Template), our computational framework for building a flexible three-dimensional model of a molecule, which is then used to locate patterns in the electron density map. 2 Pictorial structures Pictorial structures model classes of objects as a single flexible template. The template represents the object class as a collection of parts linked in a graph structure. Each edge defines a relationship between the two parts it connects. For example, a pictorial structure for a face may include the parts "left eye" and "right eye." Edges connecting these parts could enforce the constraint that the left eye is adjacent to the right eye. A dynamic programming (DP) matching algorithm of Felzenszwalb and Huttenlocher (hereafter referred to as the F-H matching algorithm) [5] allows pictorial structures to be quickly matched into a twodimensional image. The matching algorithm finds the globally optimal position and orientation of each part in the pictorial structure, assuming conditional independence on the position of each part given its neighbors. Formally, we represent the pictorial structure as a graph G = (V,E), V = {v1,v2,…,vn} the set of parts, and edge eij ∈ E connecting neighboring parts vi and vj if an explicit dependency exists between the configurations of the corresponding parts. Each part vi is assigned a configuration li describing the part's position and orientation in the image. We assume Markov independence: the probability distribution over a part's configurations is conditionally independent of every other part's configuration, given the configuration of all the part's neighbors in the graph. We assign each edge a deformation cost dij(li,lj), and each part a "mismatch" cost mi(li,I). These functions are the negative log likelihoods of a part (or pair of parts) taking a specified configuration, given the pictorial structure model. The matching algorithm places the model into the image using maximum-likelihood. That is, it finds the configuration L of parts in model Θ in image I maximizing ( ) ( ) ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟⎠ ⎞ ⎜⎝ ⎛ ⋅ ⎟⎠ ⎞ ⎜⎝ ⎛ = ∑ ∑ ∈ ∈ Θ Θ ∝ Θ E V I I ) , ( , m exp , m exp 1 ) ( ) , ( ) , ( j v iv iv i i i i l l Z L P L I P I L P (1) Figure 1: This graphic illustrates density map quality at various resolutions. All resolutions depict the same alpha helix structure 1Å 2Å 3Å 4Å 5Å By monotonicity of exponentiation, this minimizes ( ) ( ) ∑ ∑ ∈ ∈ + E V I ) , ( , d , m j v iv iv j i ij i i l l l . The F-H matching algorithm places several additional limitations on the pictorial structure. The object's graph must be tree structured (cyclic constraints are not allowed), and the deformation cost function must take the form ) ( ) ( j i l l ji ij T T − , where Tij and Tji are arbitrary functions and ||·|| is some norm (e.g. Euclidian distance). 3 Building a flexible atomic model Given a three-dimensional map containing a large molecule and the topology (i.e., for proteins, the amino-acid sequence) of that molecule, our task is to determine the Cartesian coordinates in the 3D density map of each atom in the molecule. Figure 2 shows a sample interpreted density map. DEFT finds the coordinates of all atoms simultaneously by first building a pictorial structure corresponding to the protein, then using F-H matching to optimally place the model into the density map. This section describes DEFT's deformation cost function and matching cost function. DEFT's deformation cost is related to the probability of observing a particular configuration of a molecule. Ideally, this function is proportional to the inverse of the molecule's potential function, since configurations with lower potential energy are more likely observed in nature. However, this potential is quite complicated and cannot be accurately approximated in a tree-structured pictorial structure graph. Our solution is to only consider the relationships between covalently bonded atoms. DEFT constructs a pictorial structure graph where vertices correspond to nonhydrogen atoms, and edges correspond to the covalent bonds joining atoms. The cost function each edge defines maintain invariants – interatomic distance and bond angles – while allowing free rotation around the bond. Given the protein's amino acid sequence, model construction, illustrated in Figure 3, is trivial. Each part's configuration is defined by six parameters: three translational, three rotational (Euler angles α, β, and γ ). For the cost function, we define a new connection type in the pictorial structure framework, the screw-joint, shown in Figure 4. The screw-joint's cost function is mathematically specified in terms of a directed version of the pictorial structure's undirected graph. Since the graph is constrained by the fast matching algorithm to take a tree structure, we arbitrarily pick a root node and point every edge toward this root. We now define the screw joint in terms of a parent and a child. We rotate the child such that its z axis is coincident with the vector from child to parent, and allow each part in the model (that is, each atom) to freely rotate about its local z axis. The ideal geometry between child and parent is then described by three parameters stored at each edge, xij = (xij, yij, zij). These three parameters define the optimal translation between parent and child, in the coordinate system of the parent (which in turn is defined such that its z-axis corresponds to the axis connecting it to its parent). Figure 2. An "interpreted" density map. The right figure shows the arrangement of atoms that generated the observed density. Figure 3. An example of the construction of a pictorial structure model given an amino acid. N C α C C β O O N Cα Cβ C In using these to construct the cost function dij, we define the function Tij, which maps a parent vi's configuration li into the configuration lj of that parent's ideal child vj. Given parameters xij on the edge between vi and vj, the function is defined ( ) j j j j j j i i i i i i z y x z y x γ β α γ β α , , , , , , , , , , = ij T (2) with i j α α = , ( ) z y x j ′ − ′ + ′ = , atan2 2 2 β , ( ) x y j ′ ′ + = , atan2 2 π γ , and 〉′ ′ ′ 〈 + 〉 〈 = 〉 〈 z y x z y x z y x i i i j j j , , , , , , where (x', y', z') is rotation of the bond parameters (xij, yij, zij) to world coordinates. That is, T T ) , , ( ) , , ( ij ij ij z y x z y x i i i γ, β, α R = ′ ′ ′ with i i i γ, β , α R the rotation matrix corresponding to Euler angles (αi, βi, γi). The expressions for βj and γj define the optimal orientation of each child: +z coincident with the axis that connects child and parent. The F-H matching algorithm requires our cost function to take a particular form, specifically, it must be some norm. The screw-joint model sets the deformation cost between parent vi and child vj to the distance between child configuration lj and Tij(li), the ideal child configuration given parent configuration li (Tji in equation (2) is simply the identity function). We use the 1-norm weighted in each dimension, ( ). ) ( ) ( ) ( ) , atan( 2 ) ( ) , atan( ) ( ) ( ) ( ) , ( 2 2 z z z y y y x x x w x y z y x w w l l l l j i j i j i translate ij i j j i orient ij j i rotate ij j i j i ′ − − + ′ − − + ′ − − + ⎟⎠ ⎞ ⎜⎝ ⎛ ′ ′ + − − + ′ − ′ + ′ + − + − = − = π γ γ β β α α ij ij T d (3) In the above equation, rotate ij w is the cost of rotating about a bond, orient ij w is the cost of rotating around any other axis, and translate ij w is the cost of translating in x, y or z. DEFT's screw-joint model sets rotate ij w to 0, and orient ij w and translate ij w to +100. DEFT's match-cost function implementation is based upon Cowtan's fffear algorithm [4]. This algorithm quickly and efficiently calculates the mean squared distance between a weighted 3D template of density and a region in a density map. Given a learned template and a corresponding weight function, fffear uses a Fourier convolution to determine the maximum likelihood that the weighted template generated a region of density in the density map. For each non-hydrogen atom in the protein, we create a target template corresponding to a neighborhood around that particular atom, using a training set of crystallographer-solved structures. We build a separate template for each atom type – e.g., the β-carbon (2nd sidechain carbon) of leucine and the backbone oxygen of serine – producing 171 different templates in total. A part's m function is the fffearcomputed mismatch score of that part's template over all positions and orientations. Once we construct the model, parameters – including the optimal orientation xij corresponding to each edge, and the template for each part – are learned by training Figure 4: Showing the screw-joint connection between two parts in the model. In the directed version of the MRF, vi is the parent of vj. By definition, vj is oriented such that its local z-axis is coincident with it's ideal bond orientation T) , , ( ij ij ij z y x = ij xv in vi. Bond parameters ij xv are learned by DEFT. vi vj (x',y',z') (βi,γi) (βj,γj) (xi,yi,zi) (xj,yj,zj) αj αi the model on a set of crystallographer-determined structures. Learning the orientation parameters is fairly simple: for each atom we define canonic coordinates (where +z corresponds to the axis of rotation). For each child, we record the distance r and orientation (θ,φ) in the canonic coordinate frame. We average over all atoms of a given type in our training set – e.g., over all leucine β-carbon’s – to determine average parameters ravg, θavg, and φavg. Converting these averages from spherical to Cartesian coordinates gives the ideal orientation parameters xij. A similarly-defined canonic coordinate frame is employed when learning the model templates; in this case, DEFT's learning algorithm computes target and weight templates based on the average and inverse variance over the training set, respectively. Figure 5 shows an overview of the learning process. Implementation used Cowtan's Clipper library. For each part in the model, DEFT searches through a six-dimensional conformation space (x,y,z,α,β,γ), breaking each dimension into a number of discrete bins. The translational parameters x, y, and z are sampled over a region in the unit cell. Rotational space is uniformly sampled using an algorithm described by Mitchell [6]. 4 Model Enhancements Upon initial testing, the pictorial-structure matching algorithm performs rather poorly at the density-map interpretation task. Consequently, we added two routines – a collision-detection routine, and an improved template-matching routine – to DEFT's pictorial-structure matching implementation. Both enhancements can be applied to the general pictorial structure algorithm, and are not specific to DEFT. 4.1 Collision Detection Our closer investigation revealed that much of the algorithm's poor performance is due to distant chains colliding. Since DEFT only models covalent bonds, the matching algorithm sometimes returns a structure with non-bonded atoms impossibly close together. These collisions were a problem in DEFT's initial implementation. Figure 6 shows such a collision (later corrected by the algorithm). Given a candidate solution, it is straightforward to test for spatial collisions: we simply test if any two atoms in the structure are impossibly (physically) close together. If a collision occurs in a candidate, DEFT perturbs the structure. Though Figure 5: An overview of the parameter-learning process. For each atom of a given type – here alanine Cα – we rotate the atom into a canonic orientation. We then average over every atom of that type to get a template and average bond geometry. O N N O C Cα N Cβ fffear Target Template Map Averaged Bond Geometry Standard Orientation N C-1 Cα C Cβ O N+1 Alanine Cα C Cα N Cβ r = 1.53 θ = 0.0° φ = -19.3° r = 1.51 θ = 118.4° φ = -19.7° the optimal match is no longer returned, this approach works well in practice. If two atoms are both aligned to the same space in the most probable conformation, it seems quite likely that one of the atoms belongs there. Thus, DEFT handles collisions by assuming that at least one of the two colliding branches is correct. When a collision occurs, DEFT finds the closest branch point above the colliding nodes – that is, the root y of the minimum subtree containing all colliding nodes. DEFT considers each child xi of this root, matching the subtree rooted at xi, keeping the remainder of the tree fixed. The change in score for each perturbed branch is recorded, and the one with the smallest score increase is the one DEFT keeps. Table 1 describes the collision-avoidance algorithm. In the case that the colliding node is due to a chain wrapping around on itself (and not two branches running into one another), the root y is defined as the colliding node nearest to the top of the tree. Everything below y is matched anew while the remainder of the structure is fixed. 4.2 Improved template matching In our original implementation, DEFT learned a template by averaging over each of the 171 atom types. For example, for each of the 12 (non-hydrogen) atoms in the amino-acid tyrosine we build a single template – producing 12 tyrosine templates in total. Not only is this inefficient, requiring DEFT to match redundant templates against the unsolved density map, but also for some atoms in flexible sidechains, averaging blurs density contributions from atoms more than a bond away from the target, losing valuable information about an atom's neighborhood. DEFT improves the template-matching algorithm by modeling the templates using a mixture of Gaussians, a generative model where each template is modeled using a mixture of basis templates. Each basis template is simply the mean of a cluster of templates. Cluster assignments are learned iteratively using the EM algorithm. In each iteration of the algorithm we compute the a priori likelihood of each image being generated by a particular cluster mean (the E step). Then we use these probabilities to update the cluster means (the M step). After convergence, we use each cluster mean (and weight) as an fffear search target. Given: An illegal pictorial structure configuration L = {l1,l2,…,ln} Return: A legal perturbation L' Algorithm: X ← all nodes in L illegally close to some other node y ← root of smallest subtree containing all nodes in X for each child xi of y Li ← optimal position of subtree rooted at xi fixing remainder of tree scorei ← score(Li) – score(subtree of L rooted at xi) imin ← arg min (scorei) L' ← replace subtree rooted at xi in L with Limin return L' Table 1. DEFT's collision handing routine. Figure 6. This illustrates the collision avoidance algorithm. On the left is a collision (the predicted molecule is in the darker color). The amino acid's sidechain is placed coincident with the backbone. On the right, collision avoidance finds the right structure. 5 Experimental Studies We tested DEFT on a set of proteins provided by the Phillips lab at the University of Wisconsin. The set consists of four different proteins, all around 2.0Å in resolution. With all four proteins, reflections and experimentally-determined initial phases were provided, allowing us to build four relatively poor-quality density maps. To test our algorithm with poor-quality data, we down-sampled each of the maps to 2.5, 3 and 4Å by removing higher-resolution reflections and recomputed the density. These down-sampled maps are physically identical to maps natively constructed at this resolution. Each structure had been solved by crystallographers. For this paper, our experiments are conducted under the assumption that the mainchain atoms of the protein were known to within some error factor. This assumption is fair; approaches exist for mainchain tracing in density maps [7]. DEFT simply walks along the mainchain, placing atoms one residue at a time (considering each residue independently). We split our dataset into a training set of about 1000 residues and a test set of about 100 residues (from a protein not in the training set). Using the training set we built a set of templates for matching using fffear. The templates extended to a 6Å radius around each atom at 0.5Å sampling. Two sets of templates were built and subsequently matched: a large set of 171 produced by averaging all training set templates for each atom type, and a smaller set of 24 learned through by the EM algorithm. We ran DEFT's pictorial structure matching algorithm using both sets of templates, with and without the collision detection code. Although placing individual atoms into the sidechain is fairly quick, taking less than six hours for a 200-residue protein, computing fffear match scores is very CPUdemanding. For each of our 171 templates, fffear takes 3-5 CPU-hours to compute the match score at each location in the image, for a total of one CPU-month to match templates into each protein! Fortunately the task is trivially parallelized; we regularly do computations on over 100 computers simultaneously. The results of all tests are summarized in Figure 7. Using individual-atom templates and the collision detection code, the all-atom RMS deviation varied from 1.38Å at 2Å resolution to 1.84Å at 4Å. Using the EM-based clusters as templates produced slight or no improvement. However, much less work is required; only 24 templates need to be matched to the image instead of 171 individual-atom templates. Finally, it was promising that collision detection leads to significant error reduction. It is interesting to note that individually using the improved templates and using the collision avoidance both improved the search results; however, using both together was a bit worse than with collision detection alone. More research is needed to get a synergy between the two enhancements. Further investigation is also needed balancing between the number and templates and template size. The match cost function is a critically important part of DEFT and improvements there will have the most profound impact on the overall error. 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 2A 2.5A 3A 4A Density Map Resolution Test Protein RMS Deviation base improved templates only collision detection + improved templates collision detection only Figure 7. Testset error under four strategies. 6 Conclusions and future work DEFT has applied the F-H pictorial structure matching algorithm to the task of interpreting electron density maps. In the process, we extended the F-H algorithm in three key ways. In order to model atoms rotating in 3D, we designed another joint type, the screw joint. We also developed extensions to deal with spatial collisions of parts in the model, and implemented a slightly-improved template construction routine. Both enhancements can be applied to pictorial-structure matching in general, and are not specific to the task presented here. DEFT attempts to bridge the gap between two types of model-fitting approaches for interpreting electron density maps. Several techniques [1,2,3] do a good job placing individual atoms, but all fail around 2.5-3Å resolution. On the other hand, fffear [4] has had success finding rigid elements in very poor resolution maps, but is unable to locate highly flexible “loops”. Our work extends the resolution threshold at which individual atoms can be identified in electron density maps. DEFT's flexible model combines weakly-matching image templates to locate individual atoms from maps where individual atoms have been blurred away. No other approach has investigated sidechain refinement in structures of this poor resolution. We next plan to use DEFT as the refinement phase complementing a coarser method. Rather than model the configuration of each individual atom, instead treat each amino acid as a single part in the flexible template, only modeling rotations along the backbone. Then, our current algorithm could place each individual atom. A different optimization algorithm that handles cycles in the pictorial structure graph would better handle collisions (allowing edges between non-bonded atoms). In recent work [8], loopy belief propagation [9] has been used with some success (though with no optimality guarantee). We plan to explore the use of belief propagation in pictorial-structure matching, adding edges in the graph to avoid collisions. Finally, the pictorial-structure framework upon which DEFT is built seems quite robust; we believe the accuracy of our approach can be substantially improved through implementation improvements, allowing finer grid spacing and larger fffear ML templates. The flexible molecular template we have described has the potential to produce an atomic model in a map where individual atoms may not be visible, through the power of combining weakly matching image templates. DEFT could prove important in high-throughput protein-structure determination. Acknowledgments This work supported by NLM Grant 1T15 LM007359-01, NLM Grant 1R01 LM07050-01, and NIH Grant P50 GM64598. References [1] A. Perrakis, T. Sixma, K. Wilson, & V. Lamzin (1997). wARP: improvement and extension of crystallographic phases. Acta Cryst. D53:448-455. [2] D. Levitt (2001). A new software routine that automates the fitting of protein X-ray crystallographic electron density maps. Acta Cryst. D57:1013-1019. [3] T. Ioerger, T. Holton, J. Christopher, & J. Sacchettini (1999). TEXTAL: a pattern recognition system for interpreting electron density maps. Proc. ISMB:130-137. [4] K. Cowtan (2001). Fast fourier feature recognition. Acta Cryst. D57:1435-1444. [5] P. Felzenszwalb & D. Huttenlocher (2000). Efficient matching of pictorial structures. Proc. CVPR. pp. 66-73. [6] J. Mitchell (2002). Uniform distributions of 3D rotations. Unpublished Document. [7] J. Greer (1974). Three-dimensional pattern recognition. J. Mol. Biol. 82:279-301. [8] E. Sudderth, M. Mandel, W. Freeman & A Willsky (2005). Distributed occlusion reasoning for tracking with nonparametric belief propagation. NIPS. [9] D. Koller, U. Lerner & D. Angelov (1999). A general algorithm for approximate inference and its application to hybrid Bayes nets. UAI. 15:324-333.
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Support Vector Classification with Input Data Uncertainty Jinbo Bi Computer-Aided Diagnosis & Therapy Group Siemens Medical Solutions, Inc. Malvern, PA 19355 jinbo.bi@siemens.com Tong Zhang IBM T. J. Watson Research Center Yorktown Heights, NY 10598 tzhang@watson.ibm.com Abstract This paper investigates a new learning model in which the input data is corrupted with noise. We present a general statistical framework to tackle this problem. Based on the statistical reasoning, we propose a novel formulation of support vector classification, which allows uncertainty in input data. We derive an intuitive geometric interpretation of the proposed formulation, and develop algorithms to efficiently solve it. Empirical results are included to show that the newly formed method is superior to the standard SVM for problems with noisy input. 1 Introduction In the traditional formulation of supervised learning, we seek a predictor that maps input x to output y. The predictor is constructed from a set of training examples {(xi, yi)}. A hidden underlying assumption is that errors are confined to the output y. That is, the input data are not corrupted with noise; or even when noise is present in the data, its effect is ignored in the learning formulation. However, for many applications, this assumption is unrealistic. Sampling errors, modeling errors and instrument errors may preclude the possibility of knowing the input data exactly. For example, in the problem of classifying sentences from speech recognition outputs for call-routing applications, the speech recognition system may make errors so that the observed text is corrupted with noise. In image classification applications, some features may rely on image processing outputs that introduce errors. Hence classification problems based on the observed text or image features have noisy inputs. Moreover, many systems can provide estimates for the reliability of their outputs, which measure how uncertain each element of the outputs is. This confidence information, typically ignored in the traditional learning formulations, can be useful and should be considered in the learning formulation. A plausible approach for dealing with noisy input is to use the standard learning formulation without modeling the underlying input uncertainty. If we assume that the same noise is observed both in the training data and in the test data, then the noise will cause similar effects in the training and testing phases. Based on this (non-rigorous) reasoning, one can argue that the issue of input noise may be ignored. However, we show in this paper that by modeling input uncertainty, we can obtain more accurate predictors. 2 Statistical models for prediction problems with uncertain input Consider (xi, yi), where xi is corrupted with noise. Let x′ i be the original uncorrupted input. We consider the following data generating process: first (x′ i, yi) is generated according to a distribution p(x′ i, yi|θ), where θ is an unknown parameter that should be estimated from the data; next, given (x′ i, yi), we assume that xi is generated from x′ i (but independent of yi) according to a distribution p(xi|θ′, σi, x′ i), where θ′ is another possibly unknown parameter, and σi is a known parameter which is our estimate of the uncertainty (e.g. variance) for xi. The joint probability of (x′ i, xi, yi) can be written as: p(x′ i, xi, yi) = p(x′ i, yi|θ)p(xi|θ′, σi, x′ i). The joint probability of (xi, yi) is obtained by integrating out the unobserved quantity x′ i: p(xi, yi) = Z p(x′ i, yi|θ)p(xi|θ′, σi, x′ i)dx′ i. This model can be considered as a mixture model where each mixture component corresponds to a possible true input x′ i not observed. In this framework, the unknown parameter (θ, θ′) can be estimated from the data using the maximum-likelihood estimate as: max θ,θ′ X i ln p(xi, yi|θ, θ′) = max θ,θ′ X i ln Z p(x′ i, yi|θ)p(xi|θ′, σi, x′ i)dx′ i. (1) Although this is a principled approach under our data generation process, due to the integration over the unknown true input x′ i, it often leads to a very complicated formulation which is difficult to solve. Moreover, it is not straight-forward to extend the method to nonprobability formulations such as support vector machines. Therefore we shall consider an alternative that is computationally more tractable and easier to generalize. The method we employ in this paper can be regarded as an approximation to (1), often used in engineering applications as a heuristics for mixture estimation. In this method, we simply regard each x′ i as a parameter of the probability model, so the maximum-likelihood becomes: max θ,θ′ X i ln sup x′ i [p(x′ i, yi|θ)p(xi|θ′, σi, x′ i)]. (2) If our probability model is correctly specified, then (1) is the preferred formulation. However in practice, we may not know the exact p(xi|θ′, σi, x′ i) (for example, we may not be able to estimate the level of uncertainty σi accurately). Therefore in practice, under mis-specified probability models, (1) is not necessarily always a better method. Intuitively (1) and (2) have similar effects since large values of p(x′ i, yi|θ)p(xi|θ′, σi, x′ i) dominate the summation in R p(x′ i, yi|θ)p(xi|θ′, σi, x′ i)dx′ i. That is, both methods prefer a parameter configuration such that the product p(x′ i, yi|θ)p(xi|θ′, σi, x′ i) is large for some x′ i. If an observation xi is contaminated with large noise so that p(xi|θ′, σi, x′ i) has a flat shape, then we can pick a x′ i that is very different from xi which predicts yi well. On the other hand, if an observation xi is contaminated with very small noise, then (1) and (2) penalize a parameter θ such that p(xi, yi|θ) is small. This has the effect of ignoring data that are very uncertain and relying on data that are less contaminated. In the literature, there are two types of statistical models: generative models and discriminative models (conditional models). We focus on discriminative modeling in this paper since it usually leads to better prediction performance. In discriminative modeling, we assume that p(x′ i, yi|θ) has a form p(x′ i, yi|θ) = p(x′ i)p(yi|θ, x′ i). As an example, we consider regression problems with Gaussian noise: p(x′ i, yi|θ) ∼p(x′ i) exp −(θT x′ i −yi)2 2σ2 , p(xi|θ′, σi, x′ i) ∼exp −∥xi −x′ i∥2 2σ2 i . The method in (2) becomes θ = arg min θ X i inf x′ i (θT x′ i −yi)2 2σ2 + ∥xi −x′ i∥2 2σ2 i . (3) This formulation is closely related (but not identical) to the so-called total least squares (TLS) method [6, 5]. The motivation for total least squares is the same as what we consider in this paper: input data are contaminated with noise. Unlike the statistical modeling approach we adopted in this section, the total least squares algorithm is derived from a numerical computation point of view. The resulting formulation is similar to (3), but its solution can be conveniently described by a matrix SVD decomposition. The method has been widely applied in engineering applications, and is known to give better performance than the standard least squares method for problems with uncertain inputs. In our framework, we can regard (3) as the underlying statistical model for total least squares. For binary classification where yi ∈{±1}, we consider logistic conditional probability model for yi, while still assume Gaussian noise in the input: p(x′ i, yi|θ) ∼p(x′ i) 1 1 + exp(−θT x′ iyi), p(xi|θ′, σi, x′ i) ∼exp −∥xi −x′ i∥2 2σ2 i . Similar to the total least squares method (3), we obtain the following formulation from (2): θ = arg min θ X i inf x′ i ln(1 + e−θT x′ iyi) + ∥xi −x′ i∥2 2σ2 i . (4) A well-known disadvantage of logistic model for binary classification is that it does not model deterministic conditional probability (that is, p(y = 1|x) = 0, 1) very well. This problem can be remedied using the support vector machine formulation, which has attractive intuitive geometric interpretations for linearly separable problems. Although in this section a statistical modeling approach is used to gain useful insights, we will focus on support vector machines in the rest of the paper. 3 Total support vector classification Our formulation of support vector classification with uncertain input data is motivated by the total least squares regression method that can be derived from the statistical model (3). We thus call the proposed algorithm total support vector classification (TSVC) algorithm. We assume that inputs are subject to an additive noise, i.e., x′ i = xi + ∆xi where noise ∆xi follows certain distribution. Bounded and ellipsoidal uncertainties are often discussed in the TLS context [7], and resulting approaches find many real-life applications. Hence instead of assuming Gaussian noise as in (3) and (4), we consider a simple bounded uncertainty model ∥∆xi∥≤δi with uniform priors. The bound δi has a similar effect of the standard deviation σi in the Gaussian noise model. However, under the bounded uncertainty model, the squared penalty term ||xi −x′ i||2/2σ2 i is replaced by a constraint ∥∆xi∥≤δi. Another reason for us to use the bounded uncertainty noise model is that the resulting formulation has a more intuitive geometric interpretation (see Section 4). SVMs construct classifiers based on separating hyperplanes {x : wT x+b = 0}. Hence the parameter θ in (3) and (4) is replaced by a weight vector w and a bias b. In the separable case, TSVC solves the following problem: min w,b,∆xi,i=1,··· ,ℓ 1 2∥w∥2 subject to yi wT (xi + ∆xi) + b ≥1, ∥∆xi∥≤δi, i = 1, · · · , ℓ. (5) For non-separable problems, we follow the standard practice of introducing slack variables ξi, one for each data point. In the resulting formulation, we simply replace the square loss in (3) or the logistic loss in (4) by the margin-based hinge-loss ξ = max{0, 1−y(wT x+b)}, which is used in the standard SVC. min w,b,ξ,∆xi,i=1,··· ,ℓ C Pℓ i=1 ξi + 1 2∥w∥2 subject to yi wT (xi + ∆xi) + b ≥1 −ξi, ξi ≥0, i = 1, · · · , ℓ, ∥∆xi∥≤δi, i = 1, · · · , ℓ. (6) Note that we introduced the standard Tikhonov regularization term 1 2∥w∥2 2 as usually employed in SVMs. The effect is similar to a Gaussian prior in (3) and (4) with the Bayesian MAP (maximum a posterior) estimator. One can regard (6) as a regularized instance of (2) with a non-probabilistic SVM discriminative loss criterion. Problems with corrupted inputs are more difficult than problems with no input uncertainty. Even if there is a large margin separator for the original uncorrupted inputs, the observed noisy data may become non-separable. By modifying the noisy input data as in (6), we reconstruct an easier problem, for which we may find a good linear separator. Moreover, by modeling noise in the input data, TSVC becomes less sensitive to data points that are very uncertain since we can find a choice of ∆xi such that xi+∆xi is far from the decision boundary and will not be a support vector. This is illustrated later in Figure 1 (right). TSVC thus constructs classifiers by focusing on the more trust-worthy data that are less uncertain. 4 Geometric interpretation Further investigation reveals an intuitive geometric interpretation for TSVC which allows users to easily grasp the fundamentals of this new formulation. We first derive the following fact that when the optimal ˆw is obtained, the optimal ∆ˆxi can be represented in terms of ˆw. If w is fixed in problem (6), optimizing problem (6) is equivalent to minimizing P ξi over ∆xi. The following lemma characterizes the solution. Lemma 1. For any given hyperplane (w, b), the solution ∆ˆxi of problem (6) is ∆ˆxi = yiδi w ∥w∥, i = 1, · · · , ℓ. Proof. Since the noise vector ∆xi only affects ξi and does not have impact on other slack variables ξj, j ̸= i. The minimization of P ξi can be decoupled into ℓsubproblems of minimizing each ξi = max{0, 1 −yi(wT (xi + ∆xi) + b)} = max{0, 1 −yi(wT xi + b) −yiwT ∆xi} over its corresponding ∆xi. By the Cauchy-Schwarz inequality, we have |yiwT ∆xi| ≤∥w∥·∥∆xi∥with equality if and only if ∆xi = cw for some scalar c. Since ∆xi is bounded by δi, the optimal ∆ˆxi = yiδi w ∥w∥and the minimal ˆξi = max{0, 1 − yi(wT xi + b) −δi∥w∥}. Define Sw(X) = {xi + yiδi w ∥w∥, i = 1, · · · , ℓ}. Then Sw(X) is a set of points that are obtained by shifting the original points labeled +1 along w and points labeled −1 along −w, respectively, to its individual uncertainty boundary. These shifted points are illustrated in Figure 1(middle) as filled points. Theorem 1. The optimal hyperplane ( ˆw,ˆb) obtained by TSVC (5) separates S ˆw(X) with the maximal margin. The optimal hyperplane ( ˆw,ˆb) obtained by TSVC (6) separates S ˆw(X) with the maximal soft margin. Proof. 1. If there exists any w such that Sw(X) is linearly separable, we can solve problem (5) to obtain the largest separation margin. Let ( ˆw,ˆb, ∆ˆxi) be optimal to problem (5). Note that solving problem (5) is equivalent to max ρ subject to constraints yi(wT (xi+ w w Figure 1: The separating hyperplanes obtained (left) by standard SVC and (middle) by total SVC (6). The margin can be magnified by taking into account uncertainties. Right: TSVC solution is less sensitive to outliers with large noise. ∆xi) + b) ≥ρ and ∥w∥= 1, so the optimal ρ = 1 ∥ˆw∥[8]. To have the greatest ρ, we want to max yi( ˆwT (xi + ∆xi) + ˆb) for all i’s over ∆xi. Hence following similar arguments in Lemma 1, we have |yi ˆwT ∆xi| ≤∥ˆw∥∥∆xi∥= δi∥ˆw∥and when ∆ˆxi = yiδi ˆw ∥ˆw∥, the “equal” sign holds. 2. If no w exists to separate Sw(X) or even when such a w exists, we may solve problem (6) to achieve the best compromise between the training error and the margin size. Let ˆw be optimal to problem (6). By Lemma 1, the optimal ∆ˆxi = yiδi ˆw ∥ˆw∥. According to the above analysis, we can convert problems (5) and (6) to a problem in variables w, b, ξ, as opposed to optimizing over both (w, b, ξ) and ∆xi, i = 1, · · · , ℓ. For example, the linearly non-separable problem (6) becomes min w,b,ξ C Pℓ i=1 ξi + 1 2∥w∥2 subject to yi wT xi + b + δi∥w∥≥1 −ξi, ξi ≥0, i = 1, · · · , ℓ. (7) Solving problem (7) yields an optimal solution to problem (6), and problem (7) can be interpreted as finding (w, b) to separate Sw(X) with the maximal soft margin. The similar argument holds true for the linearly separable case. 5 Solving and kernelizing TSVC TSVC problem (6) can be recast to a second-order cone program (SOCP) as usually done in TLS or Robust LS methods [7, 4]. However, directly implementing this SOCP will be computationally quite expensive. Moreover, the SOCP formulation involves a large amount of redundant variables, so a typical SOCP solver will take much longer time to achieve an optimal solution. We propose a simple iterative approach as follows based on alternating optimization method [1]. Algorithm 1 Initialize ∆xi = 0, repeat the following two steps until a termination criterion is met: 1. Fix ∆xi, i = 1, · · · , ℓto the current value, solve problem (6) for w, b, and ξ. 2. Fix w, b to the current value, solve problem (6) for ∆xi, i = 1, · · · , ℓ, and ξ. The first step of Algorithm 1 solves no more than a standard SVM by treating xi + ∆xi as the training examples. Similar to how SVMs are usually optimized, we can solve the dual SVM formulation [8] for ˆw,ˆb. The second step of Algorithm 1 solves a problem which has been discussed in Lemma 1. No optimization solver is needed. The solution ∆xi of the second step has a closed form in terms of the fixed w. 5.1 TSVC with linear functions When only linear functions are considered, an alternative exists to solve problem (6) other than Algorithm 1. As analyzed in [5, 3], Tikhonov regularization min C P ξi + 1 2∥w∥2 has an important equivalent formulation as min P ξi, subject to ∥w∥≤γ where γ is a positive constant. It can be shown that if γ ≤∥w∗∥where w∗is the solution to problem (6) with 1 2∥w∥2 removed, then the solution for the constraint problem is identical to the solution of the Tikhonov regularization problem for an appropriately chosen C. Furthermore, at optimality, the constraint ∥w∥≤γ is active, which means ∥ˆw∥= γ. Hence TSVC problem (7) can be converted to a simple SOCP with the constraint ∥w∥≤γ or a quadratically constrained quadratic program (QCQP) as follows if equivalently using ∥w∥2 ≤γ2. min w,b,ξ Pℓ i=1 ξi subject to yi wT xi + b + γδi ≥1 −ξi, ξi ≥0, i = 1, · · · , ℓ, ∥w∥2 ≤γ2. (8) This QCQP produces exactly the same solution as problem (6) but is much easier to implement than (6) since it contains much less variables. By duality analysis similarly adopted in [3], problem (8) has a dual formulation in dual variables α as follows min α γ qPℓ i,j=1 αiαjyiyjxT i xj −Pℓ i=1(1 −γδi)αi subject to Pℓ i=1 αiyi = 0, 0 ≤αi ≤1, i = 1, · · · , ℓ. (9) 5.2 TSVC with kernels By using a kernel function k, the input vector xi is mapped to Φ(xi) in a usually high dimensional feature space. The uncertainty in the input data introduces uncertainties for images Φ(xi) in the feature space. TSVC can be generalized to construct separating hyperplanes in the feature space using the images of input vectors and the mapped uncertainties. One possible generalization of TSVC is to assume the images are still subject to an additive noise and the uncertainty model in the feature space can be represented as ∥∆Φ(xi)∥≤δi. Then following the similar analysis in Sections 4 and 5.1, we obtain a problem same as (8) only with xi replaced by Φ(xi) and ∆xi replaced by ∆Φ(xi), which can be easily kernelized by solving its dual formulation (9) with inner products xT i xj replaced by k(xi, xj). It is more realistic, however, that we are only able to estimate uncertainties in the input space as bounded spheres ∥∆xi∥≤δi. When the uncertainty sphere is mapped to the feature space, the mapped uncertainty region may correspond to an irregular shape in the feature space, which brings difficulties to the optimization of TSVC. We thus propose an approximation strategy for Algorithm 1 based on the first order Taylor expansion of k. A kernel function k(x, z) takes two arguments x and z. When we fix one of the arguments, for example z, k can be viewed as a function of the other argument x. The first order Taylor expansion of k with respect to x is k(xi +∆x, ·) = k(xi, ·)+∆xT k′(xi, ·) where k′(xi, ·) is the gradient of k with respect to x at point xi. Solving the dual SVM formulation in step 1 of Algorithm 1 with ∆xj fixed to ∆¯xj yields a solution ( ¯w = P j yj ¯αjΦ(xj +∆¯xj),¯b) and thus a predictor f(x) = P j yj ¯αjk(x, xj + ∆¯xj) + ¯b. In step 2, we set (w, b) to ( ¯w,¯b) and minimize P ξi over ∆xi, which as we discussed in Lemma 1, amounts to minimizing each ξi = max{0, 1 −yi(P j yj ¯αjk(xi + ∆xi, xj + ∆¯xj) + b)} over ∆xi. Applying the Taylor expansion yields yi P j yj ¯αjk(xi + ∆xi, xj + ∆¯xj) + b = yi P j yj ¯αjk(xi, xj + ∆¯xj) + b + yi∆xT i P j yj ¯αjk′(xi, xj + ∆¯xj). Table 1: Average test error percentages of TSVC and standard SVC algorithms on synthetic problems (left and middle ) and digits classification problems (right). Synthetic linear target Synthetic quadratic target Digits ℓ 20 30 50 100 150 20 30 50 100 150 100 500 SVC 8.9 7.8 5.5 2.9 2.1 9.9 7.5 6.7 3.2 2.8 24.35 18.91 TSVC 6.1 5.2 3.8 2.1 1.6 7.9 6.1 4.4 2.8 2.4 23.00 16.10 The optimal ∆xi = yiδi vi ∥vi∥where vi = P yj ¯αjk′(xi, xj+∆¯xj) by the Cauchy-Schwarz inequality. A closed-form approximate solution for the second step is thus acquired. 6 Experiments Two sets of simulations were performed, one on synthetic datasets and one on NIST handwritten digits, to validate the proposed TSVC algorithm. We used the commercial optimization package ILOG CPLEX 9.0 to solve problems (8), (9) and the standard SVC dual problem that is part of Algorithm 1. In the experiments with synthetic data in 2 dimensions, we generated ℓ(=20, 30, 50, 100, 150) training examples xi from the uniform distribution on [−5, 5]2. Two binary classification problems were created with target separating functions as x1−x2 = 0 and x2 1+x2 2 = 9, respectively. We used TSVC with linear functions for the first problem and TSVC with the quadratic kernel (xT i xj)2 for the second problem. The input vectors xi were contaminated by Gaussian noise with mean [0,0] and covariance matrix Σ = σiI where σi was randomly chosen from [0.1, 0.8]. The matrix I denotes the 2 × 2 identity matrix. To produce an outlier effect, we randomly chose 0.1ℓexamples from the first 0.2ℓexamples after examples were ordered in an ascending order of their distances to the target boundary. For these 0.1ℓexamples, noise was generated using a larger σ randomly drawn from [0.5, 2]. Models obtained by the standard SVC and TSVC were tested on a test set of 10000 examples that were generated from the same distribution and target functions but without contamination. We performed 50 trials for each experimental setting. The misclassification error rates averaged over the 50 trials are reported in Table 1. TSVC performed overall better than SVC. Two representative modeling results of ℓ= 50 are also visually depicted in Figure 2. −4 −3 −2 −1 0 1 2 3 4 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 −5 0 5 −5 −4 −3 −2 −1 0 1 2 3 4 5 Figure 2: Results obtained by TSVC (solid lines) and standard SVC (dash lines) for the problem with (left) a linear target function and the problem with (right) a quadratic target function. The true target functions are illustrated using dash-dot lines. The NIST database of handwritten digits does not contain any uncertainty information originally. We created uncertainties by image distortions. Different types of distortions can present in real-life data. We simulated it only by rotating images. We used ℓ(=100, 500) digits from the beginning of the database in training and 2000 digits from the end of the database in test. We discriminated between odd numbers and even numbers. The angle of rotation for each digit was randomly chosen from [−8o, 8o]. The uncertainty upper bounds δi can be regarded as tuning parameters. We simply set all δi = δ. The data was preprocessed in the following way: training examples were centered to have mean 0 and scaled to have standard deviation 1. The test data was preprocessed using the mean and standard deviation of training examples. We performed 50 trials with TSVC and SVC using the linear kernel, which means we need to solve problem (9). Results are reported in Table 1 and the tuned parameter δ was 1.38 for ℓ= 100 and 1.43 for ℓ= 500. We conjecture that TSVC performance can be further improved if we obtain an estimate of δi. 7 Discussions We investigated a new learning model in which the observed input is corrupted with noise. Based on a probability modeling approach, we derived a general statistical formulation where unobserved input is modeled as a hidden mixture component. Under this framework, we were able to develop estimation methods that take input uncertainty into consideration. Motivated by this probability modeling approach, we proposed a new SVM classification formulation that handles input uncertainty. This formulation has an intuitive geometric interpretation. Moreover, we presented simple numerical algorithms which can be used to solve the resulting formulation efficiently. Two empirical examples, one artificial and one with real data, were used to illustrate that the new method is superior to the standard SVM for problems with noisy input data. A related approach, with a different focus, is presented in [2]. Our work attempts to recover the original classifier from the corrupted training data, and hence we evaluated the performance on clean test data. In our statistical modeling framework, rigorously speaking, the input uncertainty of test-data should be handled by a mixture model (or a voted classifier under the noisy input distribution). The formulation in [2] was designed to separate the training data under the worst input noise configuration instead of the most likely configuration in our case. The purpose is to directly handle test input uncertainty with a single linear classifier under the worst possible error setting. The relationship and advantages of these different approaches require further investigation. References [1] J. Bezdek and R. Hathaway. Convergence of alternating optimization. Neural, Parallel Sci. Comput., 11:351–368, 2003. [2] C. Bhattacharyya, K.S. Pannagadatta, and A. J. Smola. A second order cone programming formulation for classifying missing data. In NIPS, Vol 17, 2005. [3] J. Bi and V. N. Vapnik. Learning with rigorous support vector machines. In M. Warmuth and B. Sch¨olkopf, editors, Proceedings of the 16th Annual Conference on Learning Theory, pages 35–42, Menlo Park, CA, 2003. AAAI Press. [4] L. El Ghaoui and H. Lebret. Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18:1035–1064, 1997. [5] G. H. Golub, P. C. Hansen, and D. P. O’Leary. Tikhonov regularization and total least squares. SIAM Journal on Numerical Analysis, 30:185–194, 1999. [6] G. H. Golub and C. F. Van Loan. An analysis of the total least squares problem. SIAM Journal on Numerical Analysis, 17:883–893, 1980. [7] S. Van Huffel and J. Vandewalle. The Total Least Squares Problem: Computational Aspects and Analysis, in Frontiers in Applied Mathematics 9. SIAM Press, Philadelphia, PA, 1991. [8] V. N. Vapnik. Statistical Learning Theory. John Wiley & Sons, Inc., New York, 1998.
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Planning for Markov Decision Processes with Sparse Stochasticity Maxim Likhachev Geoff Gordon Sebastian Thrun School of Computer Science School of Computer Science Dept. of Computer Science Carnegie Mellon University Carnegie Mellon University Stanford University Pittsburgh, PA 15213 Pittsburgh, PA 15213 Stanford CA 94305 maxim+@cs.cmu.edu ggordon@cs.cmu.edu thrun@stanford.edu Abstract Planning algorithms designed for deterministic worlds, such as A* search, usually run much faster than algorithms designed for worlds with uncertain action outcomes, such as value iteration. Real-world planning problems often exhibit uncertainty, which forces us to use the slower algorithms to solve them. Many real-world planning problems exhibit sparse uncertainty: there are long sequences of deterministic actions which accomplish tasks like moving sensor platforms into place, interspersed with a small number of sensing actions which have uncertain outcomes. In this paper we describe a new planning algorithm, called MCP (short for MDP Compression Planning), which combines A* search with value iteration for solving Stochastic Shortest Path problem in MDPs with sparse stochasticity. We present experiments which show that MCP can run substantially faster than competing planners in domains with sparse uncertainty; these experiments are based on a simulation of a ground robot cooperating with a helicopter to fill in a partial map and move to a goal location. In deterministic planning problems, optimal paths are acyclic: no state is visited more than once. Because of this property, algorithms like A* search can guarantee that they visit each state in the state space no more than once. By visiting the states in an appropriate order, it is possible to ensure that we know the exact value of all of a state’s possible successors before we visit that state; so, the first time we visit a state we can compute its correct value. By contrast, if actions have uncertain outcomes, optimal paths may contain cycles: some states will be visited two or more times with positive probability. Because of these cycles, there is no way to order states so that we determine the values of a state’s successors before we visit the state itself. Instead, the only way to compute state values is to solve a set of simultaneous equations. In problems with sparse stochasticity, only a small fraction of all states have uncertain outcomes. It is these few states that cause all of the cycles: while a deterministic state s may participate in a cycle, the only way it can do so is if one of its successors has an action with a stochastic outcome (and only if this stochastic action can lead to a predecessor of s). In such problems, we would like to build a smaller MDP which contains only states which are related to stochastic actions. We will call such an MDP a compressed MDP, and we will call its states distinguished states. We could then run fast algorithms like A* search to plan paths between distinguished states, and reserve slower algorithms like value iteration for deciding how to deal with stochastic outcomes. (a) Segbot (b) Robotic helicopter (d) Planning map (e) Execution simulation (c) 3D Map Figure 1: Robot-Helicopter Coordination There are two problems with such a strategy. First, there can be a large number of states which are related to stochastic actions, and so it may be impractical to enumerate all of them and make them all distinguished states; we would prefer instead to distinguish only states which are likely to be encountered while executing some policy which we are considering. Second, there can be a large number of ways to get from one distinguished state to another: edges in the compressed MDP correspond to sequences of actions in the original MDP. If we knew the values of all of the distinguished states exactly, then we could use A* search to generate optimal paths between them, but since we do not we cannot. In this paper, we will describe an algorithm which incrementally builds a compressed MDP using a sequence of deterministic searches. It adds states and edges to the compressed MDP only by encountering them along trajectories; so, it never adds irrelevant states or edges to the compressed MDP. Trajectories are generated by deterministic search, and so undistinguished states are treated only with fast algorithms. Bellman errors in the values for distinguished states show us where to try additional trajectories, and help us build the relevant parts of the compressed MDP as quickly as possible. 1 Robot-Helicopter Coordination Problem The motivation for our research was the problem of coordinating a ground robot and a helicopter. The ground robot needs to plan a path from its current location to a goal, but has only partial knowledge of the surrounding terrain. The helicopter can aid the ground robot by flying to and sensing places in the map. Figure 1(a) shows our ground robot, a converted Segway with a SICK laser rangefinder. Figure 1(b) shows the helicopter, also with a SICK. Figure 1(c) shows a 3D map of the environment in which the robot operates. The 3D map is post-processed to produce a discretized 2D environment (Figure 1(d)). Several places in the map are unknown, either because the robot has not visited them or because their status may have changed (e.g, a car may occupy a driveway). Such places are shown in Figure 1(d) as white squares. The elevation of each white square is proportional to the probability that there is an obstacle there; we assume independence between unknown squares. The robot must take the unknown locations into account when planning for its route. It may plan a path through these locations, but it risks having to turn back if its way is blocked. Alternately, the robot can ask the helicopter to fly to any of these places and sense them. We assign a cost to running the robot, and a somewhat higher cost to running the helicopter. The planning task is to minimize the expected overall cost of running the robot and the helicopter while getting the robot to its destination and the helicopter back to its home base. Figure 1(e) shows a snapshot of the robot and helicopter executing a policy. Designing a good policy for the robot and helicopter is a POMDP planning problem; unfortunately POMDPs are in general difficult to solve (PSPACE-complete [7]). In the POMDP representation, a state is the position of the robot, the current location of the helicopter (a point on a line segment from one of the unknown places to another unknown place or the home base), and the true status of each unknown location. The positions of the robot and the helicopter are observable, so that the only hidden variables are whether each unknown place is occupied. The number of states is (# of robot locations)×(# of helicopter locations)×2# of unknown places. So, the number of states is exponential in the number of unknown places and therefore quickly becomes very large. We approach the problem by planning in the belief state space, that is, the space of probability distributions over world states. This problem is a continuous-state MDP; in this belief MDP, our state consists of the ground robot’s location, the helicopter’s location, and a probability of occupancy for each unknown location. We will discretize the continuous probability variables by breaking the interval [0, 1] into several chunks; so, the number of belief states is exponential in the number of unknown places, and classical algorithms such as value iteration are infeasible even on small problems. If sensors are perfect, this domain is acyclic: after we sense a square we know its true state forever after. On the other hand, imperfect sensors can lead to cycles: new sensor data can contradict older sensor data and lead to increased uncertainty. With or without sensor noise, our belief state MDP differs from general MDPs because its stochastic transitions are sparse: large portions of the policy (while the robot and helicopter are traveling between unknown locations) are deterministic. The algorithm we propose in this paper takes advantage of this property of the problem, as we explain in the next section. 2 The Algorithm Our algorithm can be broken into two levels. At a high level, it constructs a compressed MDP, denoted M c, which contains only the start, the goal, and some states which are outcomes of stochastic actions. At a lower level, it repeatedly runs deterministic searches to find new information to put into M c. This information includes newly-discovered stochastic actions and their outcomes; better deterministic paths from one place to another; and more accurate value estimates similar to Bellman backups. The deterministic searches can use an admissible heuristic h to focus their effort, so we can often avoid putting many irrelevant actions into M c. Because M c will often be much smaller than M, we can afford to run stochastic planning algorithms like value iteration on it. On the other hand, the information we get by planning in M c will improve the heuristic values that we use in our deterministic searches; so, the deterministic searches will tend to visit only relevant portions of the state space. 2.1 Constructing and Solving a Compressed MDP Each action in the compressed MDP represents several consecutive actions in M: if we see a sequence of states and actions s1, a1, s2, a2, . . . , sk, ak where a1 through ak−1 are deterministic but ak is stochastic, then we can represent it in M c with a single action a, available at s1, whose outcome distribution is P(s′ | sk, ak) and whose cost is c(s1, a, s′) = k−1 X i=1 c(si, ai, si+1) + c(sk, ak, s′) (See Figure 2(a) for an example of such a compressed action.) In addition, if we see a sequence of deterministic actions ending in sgoal, say s1, a1, s2, a2, . . . , sk, ak, sk+1 = sgoal, we can define a compressed action which goes from s1 to sgoal at cost Pk i=1 c(si, ai, si+1). We can label each compressed action that starts at s with (s, s′, a) (where a = null if s′ = sgoal). Among all compressed actions starting at s and ending at (s′, a) there is (at least) one with lowest expected cost; we will call such an action an optimal compression of (s, s′, a). Write Astoch for the set of all pairs (s, a) such that action a when taken from state s has more than one possible outcome, and include as well (sgoal, null). Write Sstoch for the states which are possible outcomes of the actions in Astoch, and include sstart as well. If we include in our compressed MDP an optimal compression of (s, s′, a) for every s ∈Sstoch and every (s′, a) ∈Astoch, the result is what we call the full compressed MDP; an example is in Figure 2(b). If we solve the full compressed MDP, the value of each state will be the same as the value of the corresponding state in M. However, we do not need to do that much work: (a) action compression (b) full MDP compression (c) incremental MDP compression Figure 2: MDP compression Main() 01 initialize M c with sstart and sgoal and set their v-values to 0; 02 while (∃s ∈M c s.t. RHS(s) −v(s) > δ and s belongs to the current greedy policy) 03 select spivot to be any such state s; 04 [v; vlim] = Search(spivot); 05 v(spivot) = v; 06 set the cost c(spivot, ¯a, sgoal) of the limit action ¯a from spivot to vlim; 07 optionally run some algorithm satisfying req. A for a bounded amount of time to improve the value function in M c; Figure 3: MCP main loop many states and actions in the full compressed MDP are irrelevant since they do not appear in the optimal policy from sstart to sgoal. So, the goal of the MCP algorithm will be to construct only the relevant part of the compressed MDP by building M c incrementally. Figure 2(c) shows the incremental construction of a compressed MDP which contains all of the stochastic states and actions along an optimal policy in M. The pseudocode for MCP is given in Figure 3. It begins by initializing M c to contain only sstart and sgoal, and it sets v(sstart) = v(sgoal) = 0. It maintains the invariant that 0 ≤v(s) ≤v∗(s) for all s. On each iteration, MCP looks at the Bellman error of each of the states in M c. The Bellman error is v(s) −RHS(s), where RHS(s) = min a∈A(s) RHS(s, a) RHS(s, a) = Es′∈succ(s,a)(c(s, a, s′) + v(s′)) By convention the min of an empty set is ∞, so an s which does not have any compressed actions yet is considered to have infinite RHS. MCP selects a state with negative Bellman error, spivot, and starts a search at that state. (We note that there exist many possible ways to select spivot; for example, we can choose the state with the largest negative Bellman error, or the largest error when weighted by state visitation probabilities in the best policy in M c.) The goal of this search is to find a new compressed action a such that its RHS-value can provide a new lower bound on v∗(spivot). This action can either decrease the current RHS(spivot) (if a seems to be a better action in terms of the current v-values of action outcomes) or prove that the current RHS(spivot) is valid. Since v(s′) ≤v∗(s′), one way to guarantee that RHS(spivot, a) ≤v∗(spivot) is to compute an optimal compression of (spivot, s, a) for all s, a, then choose the one with the smallest RHS. A more sophisticated strategy is to use an A∗search with appropriate safeguards to make sure we never overestimate the value of a stochastic action. MCP, however, uses a modified A∗search which we will describe in the next section. As the search finds new compressed actions, it adds them and their outcomes to M c. It is allowed to initialize newly-added states to any admissible values. When the search returns, MCP sets v(spivot) to the returned value. This value is at least as large as RHS(spivot). Consequently, Bellman error for spivot becomes non-negative. In addition to the compressed action and the updated value, the search algorithm returns a “limit value” vlim(spivot). These limit values allow MCP to run a standard MDP planning algorithm on M c to improve its v(s) estimates. MCP can use any planning algorithm which guarantees that, for any s, it will not lower v(s) and will not increase v(s) beyond the smaller of vlim(s) and RHS(s) (Requirement A). For example, we could insert a fake “limit action” into M c which goes directly from spivot to sgoal at cost vlim(spivot) (as we do on line 06 in Figure 3), then run value iteration for a fixed amount of time, selecting for each backup a state with negative Bellman error. After updating M c from the result of the search and any optional planning, MCP begins again by looking for another state with a negative Bellman error. It repeats this process until there are no negative Bellman errors larger than δ. For small enough δ, this property guarantees that we will be able to find a good policy (see section 2.3). 2.2 Searching the MDP Efficiently The top level algorithm (Figure 3) repeatedly invokes a search method for finding trajectories from spivot to sgoal. In order for the overall algorithm to work correctly, there are several properties that the search must satisfy. First, the estimate v that search returns for the expected cost of spivot should always be admissible. That is, 0 ≤v ≤v∗(spivot) (Property 1). Second, the estimate v should be no less than the one-step lookahead value of spivot in M c. That is, v ≥RHS(spivot) (Property 2). This property ensures that search either increases the value of spivot or finds additional (or improved) compressed actions. The third and final property is for the vlim value, and it is only important if MCP uses its optional planning step (line 07). The property is that v ≤vlim ≤v∗(spivot) (Property 3). Here v∗(spivot) denotes the minimum expected cost of starting at spivot, picking a compressed action not in M c, and acting optimally from then on. (Note that v∗can be larger than v∗if the optimal compressed action is already part of M c.) Property 3 uses v∗rather than v∗since the latter is not known while it is possible to compute a lower bound on the former efficiently (see below). One could adapt A* search to satisfy at least Properties 1 and 2 by assuming that we can control the outcome of stochastic actions. However, this sort of search is highly optimistic and can bias the search towards improbable trajectories. Also, it can only use heuristics which are even more optimistic than it is: that is, h must be admissible with respect to the optimistic assumption of controlled outcomes. We therefore present a version of A*, called MCP-search (Figure 4), that is more efficient for our purposes. MCP-search finds the correct expected value for the first stochastic action it encounters on any given trajectory, and is therefore far less optimistic. And, MCP-search only requires heuristic values to be admissible with respect to v∗values, h(s) ≤v∗(s). Finally, MCP-search speeds up repetitive searches by improving heuristic values based on previous searches. A* maintains a priority queue, OPEN, of states which it plans to expand. The OPEN queue is sorted by f(s) = g(s)+h(s), so that A* always expands next a state which appears to be on the shortest path from start to goal. During each expansion a state s is removed from OPEN and all the g-values of s’s successors are updated; if g(s′) is decreased for some state s′, A* inserts s′ into OPEN. A* terminates as soon as the goal state is expanded. We use the variant of A* with pathmax [5] to use efficiently heuristics that do not satisfy the triangle inequality. MCP is similar to A∗, but the OPEN list can also contain state-action pairs {s, a} where a is a stochastic action (line 31). Plain states are represented in OPEN as {s, null}. Just ImproveHeuristic(s) 01 if s ∈M c then h(s) = max(h(s), v(s)); 02 improve heuristic h(s) further if possible using fbest and g(s) from previous iterations; procedure fvalue({s, a}) 03 if s = null return ∞; 04 else if a = null return g(s) + h(s); 05 else return g(s) + max(h(s), Es′∈Succ(s,a){c(s, a, s′) + h(s′)}); CheckInitialize(s) 06 if s was accessed last in some previous search iteration 07 ImproveHeuristic(s); 08 if s was not yet initialized in the current search iteration 09 g(s) = ∞; InsertUpdateCompAction(spivot, s, a) 10 reconstruct the path from spivot to s; 11 insert compressed action (spivot, s, a) into A(spivot) (or update the cost if a cheaper path was found) 12 for each outcome u of a that was not in M c previously 13 set v(u) to h(u) or any other value less than or equal to v∗(u); 14 set the cost c(u, ¯a, sgoal) of the limit action ¯a from u to v(u); procedure Search(spivot) 15 CheckInitialize(sgoal), CheckInitialize(spivot); 16 g(spivot) = 0; 17 OPEN = {{spivot, null}}; 18 {sbest, abest} = {null, null}, fbest = ∞; 19 while(g(sgoal) > min{s,a}∈OPEN(fvalue({s, a})) AND fbest + θ > min{s,a}∈OPEN(fvalue({s, a}))) 20 remove {s, a} with the smallest fvalue({s, a}) from OPEN breaking ties towards the pairs with a = null; 21 if a = null //expand state s 22 for each s′ ∈Succ(s) 23 CheckInitialize(s′); 24 for each deterministic a′ ∈A(s) 25 s′ = Succ(s, a′); 26 h(s′) = max(h(s′), h(s) −c(s, a′, s′)); 27 if g(s′) > g(s) + c(s, a′, s′) 28 g(s′) = g(s) + c(s, a′, s′); 29 insert/update {s′, null} into OPEN with fvalue({s′, null}); 30 for each stochastic a′ ∈A(s) 31 insert/update {s, a′} into OPEN with fvalue({s, a′}); 32 else //encode stochastic action a from state s as a compressed action from spivot 33 InsertUpdateCompAction(spivot, s, a); 34 if fbest > fvalue({s, a}) then {sbest, abest} = {s, a}, fbest = fvalue({s, a}); 35 if (g(sgoal) ≤min{s,a}∈OPEN(fvalue({s, a})) AND OPEN ̸= ∅) 36 reconstruct the path from spivot to sgoal; 37 update/insert into A(spivot) a deterministic action a leading to sgoal; 38 if fbest ≥g(sgoal) then {sbest, abest} = {sgoal, null}, fbest = g(sgoal); 39 return [fbest; min{s,a}∈OPEN(fvalue({s, a}))]; Figure 4: MCP-search Algorithm like A*, MCP-search expands elements in the order of increasing f-values, but it breaks ties towards elements encoding plain states (line 20). The f-value of {s, a} is defined as g(s) + max[h(s), Es′∈Succ(s,a)(c(s, a, s′) + h(s′))] (line 05). This f-value is a lower bound on the cost of a policy that goes from sstart to sgoal by first executing a series of deterministic actions until action a is executed from state s. This bound is as tight as possible given our heuristic values. State expansion (lines 21-31) is very similar to A∗. When the search removes from OPEN a state-action pair {s, a} with a ̸= null, it adds a compressed action to M c (line 33). It also adds a compressed action if there is an optimal deterministic path to sgoal (line 37). fbest tracks the minimum f-value of all the compressed actions found. As a result, fbest ≤v∗(spivot) and is used as a new estimate for v(spivot). The limit value vlim(spivot) is obtained by continuing the search until the minimum f-value of elements in OPEN approaches fbest + θ for some θ ≥0 (line 19). This minimum f-value then provides a lower bound on v∗(spivot). To speed up repetitive searches, MCP-search improves the heuristic of every state that it encounters for the first time in the current search iteration (lines 01 and 02). Line 01 uses the fact that v(s) from M c is a lower bound on v∗(s). Line 02 uses the fact that fbest−g(s) is a lower bound on v∗(s) at the end of each previous call to Search; for more details see [4]. 2.3 Theoretical Properties of the Algorithm We now present several theorems about our algorithm. The proofs of these and other theorems can be found in [4]. The first theorem states the main properties of MCP-search. Theorem 1 The search function terminates and the following holds for the values it returns: (a) if sbest ̸= null then v∗(spivot) ≥fbest ≥E{c(spivot, abest, s′) + v(s′)} (b) if sbest = null then v∗(spivot) = fbest = ∞ (c) fbest ≤min{s,a}∈OPEN(fvalue({s, a})) ≤v∗(spivot). If neither sgoal nor any state-action pairs were expanded, then sbest = null and (b) says that there is no policy from spivot that has a finite expected cost. Using the above theorem it is easy to show that MCP-search satisfies Properties 1, 2 and 3, considering that fbest is returned as variable v and min{s,a}∈OPEN(fvalue({s, a})) is returned as variable vlim in the main loop of the MCP algorithm (Figure 3). Property 1 follows directly from (a) and (b) and the fact that costs are strictly positive and v-values are non-negative. Property 2 also follows trivially from (a) and (b). Property 3 follows from (c). Given these properties the next theorem states the correctness of the outer MCP algorithm (in the theorem πc greedy denotes a greedy policy that always chooses an action that looks best based on its cost and the v-values of its immediate successors). Theorem 2 Given a deterministic search algorithm which satisfies Properties 1–3, the MCP algorithm will terminate. Upon termination, for every state s ∈M c ∩πc greedy we have RHS(s) −δ ≤v(s) ≤v∗(s). Given the above theorem one can show that for 0 ≤δ < cmin (where cmin is the smallest expected action cost in our MDP) the expected cost of executing πc greedy from sstart is at most cmin cmin−δv∗(sstart). Picking δ ≥cmin is not guaranteed to result in a proper policy, even though Theorem 2 continues to hold. 3 Experimental Study We have evaluated the MCP algorithm on the robot-helicopter coordination problem described in section 1. To obtain an admissible heuristic, we first compute a value function for every possible configuration of obstacles. Then we weight the value functions by the probabilities of their obstacle configurations, sum them, and add the cost of moving the helicopter back to its base if it is not already there. This procedure results in optimistic cost estimates because it pretends that the robot will find out the obstacle locations immediately instead of having to wait to observe them. The results of our experiments are shown in Figure 5. We have compared MCP against three algorithms: RTDP [1], LAO* [2] and value iteration on reachable states (VI). RTDP can cope with large size MDPs by focussing its planning efforts along simulated execution trajectories. LAO* uses heuristics to prune away irrelevant states, then repeatedly performs dynamic programming on the states in its current partial policy. We have implemented LAO* so that it reduces to AO* [6] when environments are acyclic (e.g., the robot-helicopter problem with perfect sensing). VI was only able to run on the problems with perfect sensing since the number of reachable states was too large for the others. The results support the claim that MCP can solve large problems with sparse stochasticity. For the problem with perfect sensing, on average MCP was able to plan 9.5 times faster than LAO*, 7.5 times faster than RTDP, and 8.5 times faster than VI. On average for these problems, MCP computed values for 58633 states while M c grew to 396 states, and MCP encountered 3740 stochastic transitions (to give a sense of the degree of stochasticity). The main cost of MCP was in its deterministic search subroutine; this fact suggests that we might benefit from anytime search techniques such as ARA* [3]. The results for the problems with imperfect sensing show that, as the number and density of uncertain outcomes increases, the advantage of MCP decreases. For these problems MCP was able to solve environments 10.2 times faster than LAO* but only 2.2 times faster than RTDP. On average MCP computed values for 127,442 states, while the size of M c was 3,713 states, and 24,052 stochastic transitions were encountered. Figure 5: Experimental results. The top row: the robot-helicopter coordination problem with perfect sensors. The bottom row: the robot-helicopter coordination problem with sensor noise. Left column: running times (in secs) for each algorithm grouped by environments. Middle column: the number of backups for each algorithm grouped by environments. Right column: an estimate of the expected cost of an optimal policy (v(sstart)) vs. running time (in secs) for experiment (k) in the top row and experiment (e) in the bottom row. Algorithms in the bar plots (left to right): MCP, LAO*, RTDP and VI (VI is only shown for problems with perfect sensing). The characteristics of the environments are given in the second and third rows under each of the bar plot. The second row indicates how many cells the 2D plane is discretized into, and the third row indicates the number of initially unknown cells in the environment. 4 Discussion The MCP algorithm incrementally builds a compressed MDP using a sequence of deterministic searches. Our experimental results suggest that MCP is advantageous for problems with sparse stochasticity. In particular, MCP has allowed us to scale to larger environments than were otherwise possible for the robot-helicopter coordination problem. Acknowledgements This research was supported by DARPA’s MARS program. All conclusions are our own. References [1] S. Bradtke A. Barto and S. Singh. Learning to act using real-time dynamic programming. Artificial Intelligence, 72:81–138, 1995. [2] E. Hansen and S. Zilberstein. LAO*: A heuristic search algorithm that finds solutions with loops. Artificial Intelligence, 129:35–62, 2001. [3] M. Likhachev, G. Gordon, and S. Thrun. ARA*: Anytime A* with provable bounds on sub-optimality. In Advances in Neural Information Processing Systems (NIPS) 16. Cambridge, MA: MIT Press, 2003. [4] M. Likhachev, G. Gordon, and S. Thrun. MCP: Formal analysis. Technical report, Carnegie Mellon University, Pittsburgh, PA, 2004. [5] L. Mero. A heuristic search algorithm with modifiable estimate. Artificial Intelligence, 23:13–27, 1984. [6] N. Nilsson. Principles of Artificial Intelligence. Palo Alto, CA: Tioga Publishing, 1980. [7] C. H. Papadimitriou and J. N. Tsitsiklis. The complexity of Markov decision processses. Mathematics of Operations Research, 12(3):441–450, 1987.
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Expectation Consistent Free Energies for Approximate Inference Manfred Opper ISIS School of Electronics and Computer Science University of Southampton SO17 1BJ, United Kingdom mo@ecs.soton.ac.uk Ole Winther Informatics and Mathematical Modelling Technical University of Denmark DK-2800 Lyngby, Denmark owi@imm.dtu.dk 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 Z f(x) , (1) for a vector of random variables x = (x1, x2, . . . , xN) with the partition function Z = R 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 exp λT g(x) ≡exp PK j=1 λjgj(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 λ 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 f1 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 p(x) = Y α Ψα(xα) exp X i<j xiJijxj , (3) where the xα denote tractable potentials defined on disjoint subsets of variables xα. In order to have a non-trivial problem, the Ψα should be a non-Gaussian function. One may approximate p(x) by a factorizing distribution, thereby replacing f2(x) ≡ exp P i<j xiJijxj by some function which factorizes in the components xi. Alternatively, one can consider replacing f1(x) = Q i Ψi(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 xi, 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 KL(q, p) = Z dx q(x) ln q(x) p(x) (4) 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 ⟨g(x)⟩q. This will be helpful later to enable the communication between approximations. We define the Gibbs Free Energy G(µµµ) as G(µµµ) = min q {KL(q, p) | ⟨g(x)⟩q = µµµ} −ln Z . (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 minimizing G(µµµ) with respect to the µµµ. min µµµ G(µµµ) = −ln Z and ⟨g⟩= argmin µµµ G(µµµ) . (6) We will next give a short summary of properties of the Gibbs free energy (GFE). The optimizing density in (5) is given by q(x) = f(x) Z(λ) exp λT g(x) , (7) with a normalizing partition function Z(λ). The set of Lagrange parameters λ = λ(µµµ) is chosen such that the conditions ⟨g(x)⟩q = µµµ are fulfilled, i.e. λ satisfies ∂ln Z(λ) ∂λ = µµµ . (8) Inserting the optimizing distribution eq. (7) into the definition of the Gibbs free energy eq. (5), we get the explicit expression: G(µµµ) = −ln Z(λ(µµµ)) + λT (µµµ)µµµ = max λ n −ln Z(λ) + λTµµµ o , (9) i.e., G is the Legendre transform or dual of −ln Z(λ). Hence, G is a convex function of its arguments and ∂G(µµµ) ∂µµµ = λ. G(µµµ) can be used to generate moments, e.g. ∂2G(µµµ) ∂µµµ∂µµµT = ∂λ ∂µµµT = ∂µµµT ∂λ −1 = ⟨g(x)gT (x)⟩−⟨g(x)⟩⟨g(x)⟩T −1 , (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 dG(µµµ, t) dt = −∂ln Z(λ, t) ∂t + µµµ −∂ln Z(λ, t) ∂λ dλT dt = −∂ln Z(λ, t) ∂t . (11) The important message is that we only need to take the explicit t dependence into account, i.e. we can keep λ 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. Completely factorized, i.e. p(x) = Q i ψi(xi). For simplicity we will consider biased binary variables: Ψi(xi) = [δ(xi +1)+δ(xi −1)]eθixi and fix the first moments m = ⟨x⟩. Denoting the conjugate Lagrange parameters by γ: G(m) = X i Gi(mi) with Gi(mi) = max γi {−ln Zi(γi) + miγi} (12) with Zi(γi) = R dxi Ψi(ξ)eγixi = 2 cosh(γi + θi). 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 = ⟨xixj⟩: G(m, {Mij}(ij)∈G) = X (ij)∈G Gij(mi, mj, Mij) + X i (1 −ni)Gi(mi) , (13) 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 µµµ = (m, M) with all first moments m and an arbitrary subset of second moments M for a Gaussian model Ψi(xi) ∝exp[aixi −bi 2 x2 i ] and p(x) given by eq. (3). We introduce conjugate variables γ and −Λ/2. γ can be eliminated analytically, whereas we get a log-determinant maximization problem for Λ: G(m, M) = −1 2mT Jm −mT a + 1 2 X i Miibi (14) + max Λ 1 2 ln det(Λ −J) −1 2 Tr Λ(M −mmT ) . 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) = 1 Zt f1(x)f2(x, t) and q(x|t) = 1 Zq(λ, t)f1(x)f2(x, t) exp λT g(x) (15) Gq(µµµ, t) = max λ n −ln Zq(λ, t) + λTµµµ o . (16) Using eq. (11), and the fact that ∂ln Z(λ,t) ∂t = D d ln f2(x,t) dt E q(x|t) we derive the following exact identity for the free energy G(µµµ, t) Gq(µµµ, 1) −Gq(µµµ, 0) = − Z 1 0 dt d ln f2(x, t) dt q(x|t) . (17) to relate the Gibbs free energy of the intractable model Gq(µµµ) = G(µµµ, t = 1) and tractable model G(µµµ, t = 0). An simple approximation is obtained for the case f2(x, t) = [f2(x)]t, when the expectation over q(x|t) is replaced by the expectation over the tractable q(x|0), ie G(µµµ) ≈G(µµµ, 0) − Z 1 0 dt d ln f2(x, t) dt q(x|0) = G(µµµ, 0) −⟨ln f2(x)⟩q(x|0) . (18) 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 r(x|t) = 1 Zr(λ, t)f2(x, t) exp λT g(x) , (19) where the parameters λ will be chosen in such a way as to guarantee consistency for the expectations of g, i.e. ⟨g(x)⟩r(x|t) = µµµ. Using eq. (19) in place of q(x|t) in eq. (17), we get the approximation Gq(µµµ, 1) −Gq(µµµ, 0) ≈ Z 1 0 dt d ln f2(x, t) dt r(x|t) = Gr(µµµ, 1) −Gr(µµµ, 0) , (20) 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 Gq(µµµ, 1) ≈Gq(µµµ, 0) + Gr(µµµ, 1) −Gr(µµµ, 0) ≡GEC(µµµ) . (21) To simplify notation, we will write Gq ≡Gq(µµµ, 0), Gr ≡Gr(µµµ, 1) and Gs ≡Gr(µµµ, 0) in 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 approximation using a Gaussian distributions. The corresponding free energies G(µµµ, 1) and r(µµµ, 0) appearing in eq. (21) are thus found using the Gaussian free energy eq. (14) with J and J = 0 GEC(m, M) = Gq(m, M, 0) −1 2mT Jm (22) + max Λ 1 2 ln det(Λ −J) −1 2 Tr Λ(M −mmT ) −max Λ 1 2 ln det Λ −1 2 Tr Λ(M −mmT ) , where the free energy Gq(m, M, 0) will depend explicitly upon the potentials Ψα(xα). 6 Free energy minimization algorithms In our approach, inference is based on the minimization of the free energy with respect to its arguments µµµ. While the exact free energy is by construction a convex function in µµµ, 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 GEC is not a simple explicit functions of µµµ 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 bound is derived by linearizing the concave term −Gs(µµµ) at the present iteration µµµ∗, i.e. using Gs(µµµ) ≥Glbound s (µµµ) = −C∗+µµµT λ∗ s , with C∗≡ln Zq(λ∗ s) and λ∗ s = λs(µµµ∗). Since 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 λ’s GEC(µµµ) ≤ Gq(µµµ) + Gr(µµµ) −µµµT λ∗ s + C∗ = min µµµ max λq,λr −ln Zq(λq) −ln Zr(λr) + µµµT (λq + λr −λ∗ s) + C∗ = max λq,λr {−ln Zq(λq) −ln Zr(λr)|λq + λr = λ∗ s} + C∗ = max λr {−ln Zq(λ∗ s −λr) −ln Zr(λr) + C∗} . (23) This can be summarized in the following double loop algorithm which is guaranteed not to increase our free energy approximation. 1. Outer loop: For fixed old value µµµ∗, bound the concave term −Gs(µµµ) by −Glbound s (µµµ) go get the convex upper bound to GEC(µµµ). 2. Inner loop: Solve the concave maximization problem max λr L with L = −ln Zq(λ∗ s −λr) −ln Zr(λr) . (24) Inserting the solution into µµµ(λr) = ⟨g(x)⟩r gives the new value µµµ∗for µµµ. 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 θi are drawn from a uniform distribution θi ∼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, +2dcoup] with dcoup > 0. We compute the average one-norm error on the marginals: P 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 ψ(x) in eq. (3) which contains all couplings on a spanning tree. For Gq, 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 GEC = Table 1: The average one-norm error on marginals for the Wainwright-Jordan set-up. Problem type Method SP LD EC fac EC struct Graph Coupling dcoup Mean Mean Mean Mean Repulsive 0.25 0.037 0.020 0.003 0.0017 Repulsive 0.50 0.071 0.018 0.031 0.0143 Full Mixed 0.25 0.004 0.020 0.002 0.0013 Mixed 0.50 0.055 0.021 0.022 0.0151 Attractive 0.06 0.024 0.027 0.004 0.0031 Attractive 0.12 0.435 0.033 0.117 0.0211 Repulsive 1.0 0.294 0.047 0.153 0.0031 Repulsive 2.0 0.342 0.041 0.198 0.0021 Grid Mixed 1.0 0.014 0.016 0.011 0.0018 Mixed 2.0 0.095 0.038 0.082 0.0068 Attractive 1.0 0.440 0.047 0.125 0.0028 Attractive 2.0 0.520 0.042 0.177 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. 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Using Machine Learning to Break Visual Human Interaction Proofs (HIPs) Kumar Chellapilla Patrice Y. Simard Microsoft Research Microsoft Research One Microsoft Way One Microsoft Way Redmond, WA 98052 Redmond, WA 98052 kumarc@microsoft.com patrice@microsoft.com Abstract Machine learning is often used to automatically solve human tasks. In this paper, we look for tasks where machine learning algorithms are not as good as humans with the hope of gaining insight into their current limitations. We studied various Human Interactive Proofs (HIPs) on the market, because they are systems designed to tell computers and humans apart by posing challenges presumably too hard for computers. We found that most HIPs are pure recognition tasks which can easily be broken using machine learning. The harder HIPs use a combination of segmentation and recognition tasks. From this observation, we found that building segmentation tasks is the most effective way to confuse machine learning algorithms. This has enabled us to build effective HIPs (which we deployed in MSN Passport), as well as design challenging segmentation tasks for machine learning algorithms. 1 Introduction The OCR problem for high resolution printed text has virtually been solved 10 years ago [1]. On the other hand, cursive handwriting recognition today is still too poor for most people to rely on. Is there a fundamental difference between these two seemingly similar problems? To shed more light on this question, we study problems that have been designed to be difficult for computers. The hope is that we will get some insight on what the stumbling blocks are for machine learning and devise appropriate tests to further understand their similarities and differences. Work on distinguishing computers from humans traces back to the original Turing Test [2] which asks that a human distinguish between another human and a machine by asking questions of both. Recent interest has turned to developing systems that allow a computer to distinguish between another computer and a human. These systems enable the construction of automatic filters that can be used to prevent automated scripts from utilizing services intended for humans [4]. Such systems have been termed Human Interactive Proofs (HIPs) [3] or Completely Automated Public Turing Tests to Tell Computers and Humans Apart (CAPTCHAs) [4]. An overview of the work in this area can be found in [5]. Construction of HIPs that are of practical value is difficult because it is not sufficient to develop challenges at which humans are somewhat more successful than machines. This is because the cost of failure for an automatic attacker is minimal compared to the cost of failure for humans. Ideally a HIP should be solved by humans more than 80% of the time, while an automatic script with reasonable resource use should succeed less than 0.01% of the time. This latter ratio (1 in 10,000) is a function of the cost of an automatic trial divided by the cost of having a human perform the attack. This constraint of generating tasks that are failed 99.99% of the time by all automated algorithms has generated various solutions which can easily be sampled on the internet. Seven different HIPs, namely, Mailblocks, MSN (before April 28th, 2004), Ticketmaster, Yahoo, Yahoo v2 (after Sept’04), Register, and Google, will be given as examples in the next section. We will show in Section 3 that machinelearning-based attacks are far more successful than 1 in 10,000. Yet, some of these HIPs are harder than others and could be made even harder by identifying the recognition and segmentation parts, and emphasizing the latter. Section 4 presents examples of more difficult HIPs which are much more respectable challenges for machine learning, and yet surprisingly easy for humans. The final section discusses a (known) weakness of machine learning algorithms and suggests designing simple artificial datasets for studying this weakness. 2 Examples of HIPs The HIPs explored in this paper are made of characters (or symbols) rendered to an image and presented to the user. Solving the HIP requires identifying all characters in the correct order. The following HIPs can be sampled from the web: Mailblocks: While signing up for free email service with mailblocks (www.mailblocks.com), you will find HIP challenges of the type: MSN: While signing up for free e-mail with MSN Hotmail (www.hotmail.com), you will find HIP challenges of the type: Register.com: While requesting a whois lookup for a domain at www.register.com, you will HIP challenges of the type: Yahoo!/EZ-Gimpy (CMU): While signing up for free e-mail service with Yahoo! (www.yahoo.com), you will receive HIP challenges of the type: Yahoo! (version 2): Starting in August 2004, Yahoo! introduced their second generation HIP. Three examples are presented below: Ticketmaster: While looking for concert tickets at www.ticketmaster.com, you will receive HIP challenges of the type: Google/Gmail: While signing up for free e-mail with Gmail at www.google.com, one will receive HIP challenges of the type: While solutions to Yahoo HIPs are common English words, those for ticketmaster and Google do not necessarily belong to the English dictionary. They appear to have been created using a phonetic generator [8]. 3 Using machine learning to break HIPs Breaking HIPs is not new. Mori and Malik [7] have successfully broken the EZGimpy (92% success) and Gimpy (33% success) HIPs from CMU. Our approach aims at an automatic process for solving multiple HIPs with minimum human intervention, using machine learning. In this paper, our main goal is to learn more about the common strengths and weaknesses of these HIPs rather than to prove that we can break any one HIP in particular with the highest possible success rate. We have results for six different HIPs: EZ-Gimpy/Yahoo, Yahoo v2, mailblocks, register, ticketmaster, and Google. To simplify our study, we will not be using language models in our attempt to break HIPs. For example, there are only about 600 words in the EZ-Gimpy dictionary [7], which means that a random guess attack would get a success rate of 1 in 600 (more than enough to break the HIP, i.e., greater than 0.01% success). HIPs become harder when no language model is used. Similarly, when a HIP uses a language model to generate challenges, success rate of attacks can be significantly improved by incorporating the language model. Further, since the language model is not common to all HIPs studied, it was not used in this paper. Our generic method for breaking all of these HIPs is to write a custom algorithm to locate the characters, and then use machine learning for recognition. Surprisingly, segmentation, or finding the characters, is simple for many HIPs which makes the process of breaking the HIP particularly easy. Gimpy uses a single constant predictable color (black) for letters even though the background color changes. We quickly realized that once the segmentation problem is solved, solving the HIP becomes a pure recognition problem, and it can trivially be solved using machine learning. Our recognition engine is based on neural networks [6][9]. It yielded a 0.4% error rate on the MNIST database, uses little memory, and is very fast for recognition (important for breaking HIPs). For each HIP, we have a segmentation step, followed by a recognition step. It should be stressed that we are not trying to solve every HIP of a given type i.e., our goal is not 100% success rate, but something efficient that can achieve much better than 0.01%. In each of the following experiments, 2500 HIPs were hand labeled and used as follows (a) recognition (1600 for training, 200 for validation, and 200 for testing), and (b) segmentation (500 for testing segmentation). For each of the five HIPs, a convolution neural network, identical to the one described in [6], was trained and tested on gray level character images centered on the guessed character positions (see below). The trained neural network became the recognizer. 3.1 Mailblocks To solve the HIP, we select the red channel, binarize and erode it, extract the largest connected components (CCs), and breakup CCs that are too large into two or three adjacent CCs. Further, vertically overlapping half character size CCs are merged. The resulting rough segmentation works most of the time. Here is an example: For instance, in the example above, the NN would be trained, and tested on the following images: … The end-to-end success rate is 88.8% for segmentation, 95.9% for recognition (given correct segmentation), and (0.888)*(0.959)7 = 66.2% total. Note that most of the errors come from segmentation, even though this is where all the custom programming was invested. 3.2 Register The procedure to solve HIPs is very similar. The image was smoothed, binarized, and the largest 5 connected components were identified. Two examples are presented below: The end-to-end success rate is 95.4% for segmentation, 87.1% for recognition (given correct segmentation), and (0.954)*(0.871)5 = 47.8% total. 3.3 Yahoo/EZ-Gimpy Unlike the mailblocks and register HIPs, the Yahoo/EZ-Gimpy HIPs are richer in that a variety of backgrounds and clutter are possible. Though some amount of text warping is present, the text color, size, and font have low variability. Three simple segmentation algorithms were designed with associated rules to identify which algorithm to use. The goal was to keep these simple yet effective: a) No mesh: Convert to grayscale image, threshold to black and white, select large CCs with sizes close to HIP char sizes. One example: b) Black mesh: Convert to grayscale image, threshold to black and white, remove vertical and horizontal line pixels that don’t have neighboring pixels, select large CCs with sizes close to HIP char sizes. One example: c) White mesh: Convert to grayscale image, threshold to black and white, add black pixels (in white line locations) if there exist neighboring pixels, select large CCs with sizes close to HIP char sizes. One example: Tests for black and white meshes were performed to determine which segmentation algorithm to use. The end-to-end success rate was 56.2% for segmentation (38.2% came from a), 11.8% from b), and 6.2% from c), 90.3% for recognition (given correct segmentation), and (0.562)*(0.903)4.8 = 34.4% total. The average length of a Yahoo HIP solution is 4.8 characters. 3.4 Ticketmaster The procedure that solved the Yahoo HIP is fairly successful at solving some of the ticket master HIPs. These HIPs are characterized by cris-crossing lines at random angles clustered around 0, 45, 90, and 135 degrees. A multipronged attack as in the Yahoo case (section 3.3) has potential. In the interests of simplicity, a single attack was developed: Convert to grayscale, threshold to black and white, up-sample image, dilate first then erode, select large CCs with sizes close to HIP char sizes. One example: The dilate-erode combination causes the lines to be removed (along with any thin objects) but retains solid thick characters. This single attack is successful in achieving an end-to-end success rate of 16.6% for segmentation, the recognition rate was 82.3% (in spite of interfering lines), and (0.166)*(0.823)6.23 = 4.9% total. The average HIP solution length is 6.23 characters. 3.5 Yahoo version 2 The second generation HIP from Yahoo had several changes: a) it did not use words from a dictionary or even use a phonetic generator, b) it uses only black and white colors, c) uses both letters and digits, and d) uses connected lines and arcs as clutter. The HIP is somewhat similar to the MSN/Passport HIP which does not use a dictionary, uses two colors, uses letters and digits, and background and foreground arcs as clutter. Unlike the MSN/Passport HIP, several different fonts are used. A single segmentation attack was developed: Remove 6 pixel border, up-sample, dilate first then erode, select large CCs with sizes close to HIP char sizes. The attack is practically identical to that used for the ticketmaster HIP with different preprocessing stages and slightly modified parameters. Two examples: This single attack is successful in achieving an end-to-end success rate of 58.4% for segmentation, the recognition rate was 95.2%, and (0.584)*(0.952)5 = 45.7% total. The average HIP solution length is 5 characters. 3.6 Google/GMail The Google HIP is unique in that it uses only image warp as a means of distorting the characters. Similar to the MSN/Passport and Yahoo version 2 HIPs, it is also two color. The HIP characters are arranged closed to one another (they often touch) and follow a curved baseline. The following very simple attack was used to segment Google HIPs: Convert to grayscale, up-sample, threshold and separate connected components. a) b) This very simple attack gives an end-to-end success rate of 10.2% for segmentation, the recognition rate was 89.3%, giving (0.102)*(0.893)6.5 = 4.89% total probability of breaking a HIP. Average Google HIP solution length is 6.5 characters. This can be significantly improved upon by judicious use of dilate-erode attack. A direct application doesn’t do as well as it did on the ticketmaster and yahoo HIPs (because of the shear and warp of the baseline of the word). More successful and complicated attacks might estimate and counter the shear and warp of the baseline to achieve better success rates. 4 Lessons learned from breaking HIPs From the previous section, it is clear that most of the errors come from incorrect segmentations, even though most of the development time is spent devising custom segmentation schemes. This observation raises the following questions: Why is segmentation a hard problem? Can we devise harder HIPs and datasets? Can we build an automatic segmentor? Can we compare classification algorithms based on how useful they are for segmentation? 4.1 The segmentation problem As a review, segmentation is difficult for the following reasons: 1. Segmentation is computationally expensive. In order to find valid patterns, a recognizer must attempt recognition at many different candidate locations. 2. The segmentation function is complex. To segment successfully, the system must learn to identify which patterns are valid among the set of all possible valid and non-valid patterns. This task is intrinsically more difficult than classification because the space of input is considerably larger. Unlike the space of valid patterns, the space of non-valid patterns is typically too vast to sample. This is a problem for many learning algorithms which yield too many false positives when presented non-valid patterns. 3. Identifying valid characters among a set of valid and invalid candidates is a combinatorial problem. For example, correctly identifying which 8 characters among 20 candidates (assuming 12 false positives), has a 1 in 125,970 (20 choose 8) chances of success by random guessing. 4.2 Building better/harder HIPs We can use what we have learned to build better HIPs. For instance the HIP below was designed to make segmentation difficult and a similar version has been deployed by MSN Passport for hotmail registrations (www.hotmail.com): The idea is that the additional arcs are themselves good candidates for false characters. The previous segmentation attacks would fail on this HIP. Furthermore, simple change of fonts, distortions, or arc types would require extensive work for the attacker to adjust to. We believe HIPs that emphasize the segmentation problem, such as the above example, are much stronger than the HIPs we examined in this paper, which rely on recognition being difficult. Pushing this to the extreme, we can easily generate the following HIPs: Despite the apparent difficulty of these HIPs, humans are surprisingly good at solving these, indicating that humans are far better than computers at segmentation. This approach of adding several competing false positives can in principle be used to automatically create difficult segmentation problems or benchmarks to test classification algorithms. 4.3 Building an automatic segmentor To build an automatic segmentor, we could use the following procedure. Label characters based on their correct position and train a recognizer. Apply the trained recognizer at all locations in the HIP image. Collect all candidate characters identified with high confidence by the recognizer. Compute the probability of each combination of candidates (going from left to right), and output the solution string with the highest probability. This is better illustrated with an example. Consider the following HIP (to the right). The trained neural network has these maps (warm colors indicate recognition) that show that K, Y, and so on are correctly identified. However, the maps for 7 and 9 show several false positives. In general, we would get the following color coded map for all the different candidates: HIP K Y B 7 9 With a threshold of 0.5 on the network’s outputs, the map obtained is: We note that there are several false positives for each true positive. The number of false positives per true positive character was found to be between 1 and 4, giving a 1 in C(16,8) = 12,870 to 1 in C(32,8) = 10,518,300 random chance of guessing the correct segmentation for the HIP characters. These numbers can be improved upon by constraining solution strings to flow sequentially from left to right and by restricting overlap. For each combination, we compute a probability by multiplying the 8 probabilities of the classifier for each position. The combination with the highest probability is the one proposed by the classifier. We do not have results for such an automatic segmentor at this time. It is interesting to note that with such a method a classifier that is robust to false positives would do far better than one that is not. This suggests another axis for comparing classifiers. 5 Conclusion In this paper, we have successfully applied machine learning to the problem of solving HIPs. We have learned that decomposing the HIP problem into segmentation and recognition greatly simplifies analysis. Recognition on even unprocessed images (given segmentation is a solved) can be done automatically using neural networks. Segmentation, on the other hand, is the difficulty differentiator between weaker and stronger HIPs and requires custom intervention for each HIP. We have used this observation to design new HIPs and new tests for machine learning algorithms with the hope of improving them. Acknowledgements We would like to acknowledge Chau Luu and Eric Meltzer for their help with labeling and segmenting various HIPs. We would also like to acknowledge Josh Benaloh and Cem Paya for stimulating discussions on HIP security. References [1] Baird HS (1992), “Anatomy of a versatile page reader,” IEEE Pro., v.80, pp. 1059-1065. [2] Turing AM (1950), “Computing Machinery and Intelligence,” Mind, 59:236, pp. 433-460. [3] First Workshop on Human Interactive Proofs, Palo Alto, CA, January 2002. [4] Von Ahn L, Blum M, and Langford J, The Captcha Project. http://www.captcha.net [5] Baird HS and Popat K (2002) “Human Interactive Proofs and Document Image Analysis,” Proc. IAPR 2002 Workshop on Document Analysis Systerms, Princeton, NJ. [6] Simard PY, Steinkraus D, and Platt J, (2003) “Best Practice for Convolutional Neural Networks Applied to Visual Document Analysis,” in International Conference on Document Analysis and Recognition (ICDAR), pp. 958-962, IEEE Computer Society, Los Alamitos. [7] Mori G, Malik J (2003), “Recognizing Objects in Adversarial Clutter: Breaking a Visual CAPTCHA,” Proc. of the Computer Vision and Pattern Recognition (CVPR) Conference, IEEE Computer Society, vol.1, pages:I-134 - I-141, June 18-20, 2003 [8] Chew, M. and Baird, H. S. (2003), “BaffleText: a Human Interactive Proof,” Proc., 10th IS&T/SPIE Document Recognition & Retrieval Conf., Santa Clara, CA, Jan. 22. [9] LeCun Y, Bottou L, Bengio Y, and Haffner P, “Gradient-based learning applied to document recognition,’ Proceedings of the IEEE, Nov. 1998.
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Making Latin Manuscripts Searchable using gHMM’s Jaety Edwards Yee Whye Teh David Forsyth Roger Bock Michael Maire Grace Vesom {jaety,ywteh,daf,bock,mmaire}@cs.berkeley.edu 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 Ω. Let Tc 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, σcij) for each pixel in b – draw all pixels above and below b from background gaussian N(µ0, σ0) (See 2.2 for greater detail on pixel emission model) • move to column w + 1 and repeat until we enter the end state Ω. 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 matching chars substitutions insertions deletions Perfect transcription 21019 0 0 0 unigram 14603 5487 534 773 bigram 15572 4597 541 718 trigram 15788 4410 507 695 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. ϵ1 and ϵ2 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. bigram b u r t o r a d v o s v e m o o r u a t u p r o l o g r trigram f o r a t o r a d v o s v e n i o o r n a t u p r o l o g i Editorial translation Orator ad vos venio ornatu prologi: unigram b u r t o r a d u o s u e m o o r n a t u p r o l o g r 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. bg Ω a b bg Path 1 Path 2 ε1 ε2 1 − ε1 1 − ε2 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 50 100 150 200 250 300 350 400 450 500 550 arbitror pater etiam nisi factum primum siet vero illi inter hic michi ibi qui tu 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 nisi siet vero illi inter hic michi ibi qui tu 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: michi: Quando nec gnatu’ neque hic mi quicquam obtemperant, mihi: Habet, ut consumat nunc quom nil obsint doli; 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”). interrogaverunt sunt 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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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 millert@cs.berkeley.edu 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. F2 F1 Null F1 F2 Null All possible name/face assignments N1 N2 Null F1 F2 Null N2 N1 Null N1 N2 Null N1 N2 F2 Null N1 N2 F1 F2 Null N1 N2 Null F1 F2 Null N1 N2 Null F1 F2 Null Null Null F1 British Columbia 2010 Bid Corporation John Furlong 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 Josek balloting at an IOC gathering in Prague. REUTERS/Petr President and Chief Operating Officer of the Vancouver, (rear) smiles while celebrating with compatriots their F1 = F2 = N1 = John Furlong, N2 = Winter Olympics, 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 − al Qaeda after − Null before − U.S. House after − Andrew Fastow after − Jennifer Capriati after − President George W. before − Marcel Avram after − Michael Jackson before − Julia Vakulenko before − Vice President Dick Cheney before − James Ivory after − Naomi Watts before − U.S. Joint after − Null after − Jon Voight after − Heidi Klum before − Ric Pipino before − U.S. Open before − CEO Summit after − Martha Stewart before − Angelina Jolie before − James Bond after − David Nalbandian after − Pierce Brosnan 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: N name, context pictured D face_u face_n Fu Fn 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 from P(face|name) (Fn = P pictured). 5. Generate Fu = F −Fn other faces from P(face). The parameters necessary for the EM process are P(face|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(face) the probability that a face is generated without a name. Angelina Jolie U.S. Open Donald Fehr Elizabeth Dole Abraham Lincoln Angelina Jolie empty U.S. Open Elizabeth Dole Anastasia Myskina Abraham Lincoln Donald Fehr Anastasia Myskina empty Without language model 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 α P(picturedα|cα)P(fσ(α)|nα) Y β (1 −P(picturedβ|cβ)) Y γ P(fγ) 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, α indexes into the names that are pictured, σ(α) indexes into the faces assigned to the pictured names, β indexes into the names that are not pictured and γ 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 iϵpics X jϵass (Pijlog(L(xi, aj)) 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(face|name) and P(pictured|context) using soft counts. 2.2 Modeling the appearance of faces – P(face|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(face|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 EM MM Appearance Model, No Lang Model 56% 67% Appearance Model + Lang Model 72% 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 Nystr¨om 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 labels correct IN correct OUT correct Baseline 67% 100% 0% EM Labeling with Language Model 76% 95% 56% MM Labeling with Language Model 84% 87% 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(face|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¨om 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 Nystr¨om Method to Speed up Kernel Machines,” Advances in Neural Information Processing Systems, Vol 13, pp. 682-688, 2001.
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Nonparametric Transforms of Graph Kernels for Semi-Supervised Learning Xiaojin Zhu† Jaz Kandola‡ Zoubin Ghahramani‡† John Lafferty† †School of Computer Science ‡Gatsby Computational Neuroscience Unit Carnegie Mellon University University College London 5000 Forbes Avenue 17 Queen Square Pittsburgh, PA 15213 USA 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, and hence do not study its construction. Let D be a diagonal matrix where dii = P j wij is the degree of node i. This allows us to define the combinatorial graph Laplacian as L = D −W (the normalized Laplacian ˜L = D−1/2LD−1/2 can be used as well). We denote L’s eigensystem by {λi, φi}, so that L = Pn i=1 λiφiφ⊤ 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 eigenvalue λ corresponds to a smoother eigenvector φ over the graph; that is, the value P ij wij(φ(i) −φ(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 λ into r(λ), where the spectral transformation r is a non-negative and decreasing function1 K = n X i=1 r(λi) φiφ⊤ i (1) Note that it may be that r reverses the order of the eigenvalues, so that smooth φi’s have larger eigenvalues in K. A “soft labeling” function f = P ciφi in a kernel machine has a penalty term in the RKHS norm given by Ω(||f||2 K) = Ω(P c2 i /r(λi)). 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. For example, the diffusion kernel [5] corresponds to r(λ) = exp(−σ2 2 λ) and the Gaussian field kernel [10] corresponds to r(λ) = 1 λ+ϵ. Cross validation has been used to find the hyperparameters σ or ϵ 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 = φiφ⊤ i , i = 1 · · · n be the outer product matrices of the eigenvectors. The semisupervised kernel K is a linear combination K = Pn i=1 µiKi, where µi ≥0. We formulate the problem of finding the spectral transformation as one that finds the interpolation coefficients {r(λi) = µi} 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 1 2x⊤P0x + q⊤ 0 x + r0 (2) subject to 1 2x⊤Pix + q⊤ i x + ri ≤0 i = 1 · · · m (3) Ax = b (4) 1We use a slightly different notation where r is the inverse of that in [7]. where Pi ∈Sn +, i = 1, . . . , m, where Sn + 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: ˆA(Ktr, T) = ⟨Ktr, T⟩F p ⟨Ktr, Ktr⟩F ⟨T, T⟩F (5) where Ktr is the kernel matrix restricted to the training points, ⟨M, N⟩F denotes the Frobenius product between two square matrices ⟨M, N⟩F = P ij mijnij = Tr(MN ⊤), 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 µi ≥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 µi’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 µi = r(λi) to encourage smooth functions over the graph. This motivates the set of order constraints µi ≥µi+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 ˆA(Ktr, T) (7) subject to K = Pn i=1 µiKi (8) µi ≥0 (9) trace(K) = 1 (10) µi ≥µi+1, i = 1 · · · n −1 (11) where T is the training target matrix, Ki = φiφ⊤ i and φi’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 µi ≥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 maxµ vec(T)⊤Mµ (12) subject to ||Mµ|| ≤1 (13) µi ≥0 (14) µi ≥µi+1, i = 1 · · · n −1 (15) The objective function is linear in µ, 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 φ1 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 µ1 ≥µ2 on the constant bias term. Instead we let µ1 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: µi ≥µi+1, i = 1 · · · n −1, and φi 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 φi 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 σ and ϵ 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 λi. The maximal-alignment kernels ignore λi altogether. The order constrained semi-supervised kernels only use the order of λi 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 µi’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 µi 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 λi (the original eigenvalues of the Laplacian). The mean (thick lines) and ±1 standard deviation (dotted lines) of only the top 50 µi’s are plotted for clarity. The µi 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 µ1 because of the order constraint. This seems to be disadvantageous — the spectral transformation tries to balance it out by increasing the value of other µi’s so that the constant K1’s relative influence is smaller. On the other hand the improved order constrained kernels (black) allow µ1 to be small. As a result the rest µi’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. 2The hyperparameters σ2 and ϵ are learned with the fminbnd() function in Matlab to maximize kernel alignment. 3Results 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 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank µ scaled PC vs. MAC Improved order Order Max−align Gaussian field Diffusion 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank µ scaled Religion vs. Atheism Improved order Order Max−align Gaussian field Diffusion 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank µ scaled Ten Digits (10 classes) Improved order Order Max−align Gaussian field Diffusion 0 5 10 15 20 25 30 35 40 45 50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank µ scaled ISOLET (26 classes) Improved order Order Max−align Gaussian field Diffusion 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. semi-supervised kernels standard kernels Training Improved Order Gaussian Diffusion Max-align RBF Linear Quadratic set size Order Field pc-mac 10 87.0 ± 5.0 84.9 ± 7.2 56.4 ± 6.2 57.8 ±11.5 71.1 ± 9.7 51.6 ± 3.4 63.0 ± 5.1 62.3 ± 4.2 0.71 ( 1) 0.57 ( 1) 0.32 0.35 0.90 ( 1) 0.11 0.30 0.25 30 90.3 ± 1.3 89.6 ± 2.3 76.4 ± 6.1 79.6 ±11.2 85.4 ± 3.9 62.6 ± 9.6 71.8 ± 5.5 71.2 ± 5.3 0.68 ( 8) 0.49 ( 8) 0.19 0.23 0.74 ( 6) 0.03 0.18 0.13 50 91.3 ± 0.9 90.5 ± 1.7 81.1 ± 4.6 87.5 ± 2.8 88.4 ± 2.1 67.8 ± 9.0 77.6 ± 4.8 75.7 ± 5.4 0.64 (31) 0.46 (31) 0.16 0.20 0.68 (25) 0.02 0.14 0.10 70 91.5 ± 0.6 90.8 ± 1.3 84.6 ± 2.1 90.5 ± 1.2 89.6 ± 1.6 74.7 ± 7.4 80.2 ± 4.6 74.3 ± 8.7 0.63 (70) 0.46 (56) 0.14 0.19 0.66 (59) 0.01 0.12 0.08 90 91.5 ± 0.6 91.3 ± 1.3 86.3 ± 2.3 91.3 ± 1.1 90.3 ± 1.0 79.0 ± 6.4 82.5 ± 4.2 79.1 ± 7.3 0.63 (108) 0.45 (98) 0.13 0.18 0.65 (84) 0.01 0.11 0.08 religion-atheism 10 72.8 ±11.2 70.9 ±10.9 55.2 ± 5.8 60.9 ±10.7 60.7 ± 7.5 55.8 ± 5.8 60.1 ± 7.0 61.2 ± 4.8 0.50 ( 1) 0.42 ( 1) 0.31 0.31 0.85 ( 1) 0.13 0.30 0.26 30 84.2 ± 2.4 83.0 ± 2.9 71.2 ± 6.3 80.3 ± 5.1 74.4 ± 5.4 63.4 ± 6.5 63.7 ± 8.3 70.1 ± 6.3 0.38 ( 8) 0.31 ( 6) 0.20 0.22 0.60 ( 7) 0.05 0.18 0.15 50 84.5 ± 2.3 83.5 ± 2.5 80.4 ± 4.1 83.5 ± 2.7 77.4 ± 6.1 69.3 ± 6.5 69.4 ± 7.0 70.7 ± 8.5 0.31 (28) 0.26 (23) 0.17 0.20 0.48 (27) 0.04 0.15 0.11 70 85.7 ± 1.4 85.3 ± 1.6 83.0 ± 2.9 85.4 ± 1.8 82.3 ± 3.0 73.1 ± 5.8 75.7 ± 6.0 71.0 ±10.0 0.29 (55) 0.25 (42) 0.16 0.19 0.43 (51) 0.03 0.13 0.10 90 86.6 ± 1.3 86.4 ± 1.5 84.5 ± 2.1 86.2 ± 1.6 82.8 ± 2.6 77.7 ± 5.1 74.6 ± 7.6 70.0 ±11.5 0.27 (86) 0.24 (92) 0.15 0.18 0.40 (85) 0.02 0.12 0.09 one-two 10 96.2 ± 2.7 90.6 ±14.0 58.2 ±17.6 59.4 ±18.9 85.4 ±11.5 78.7 ±14.3 85.1 ± 5.7 85.7 ± 4.8 0.87 ( 2) 0.66 ( 1) 0.43 0.53 0.95 ( 1) 0.38 0.26 0.30 20 96.4 ± 2.8 93.9 ± 8.7 87.0 ±16.0 83.2 ±19.8 94.5 ± 1.6 90.4 ± 4.6 86.0 ± 9.4 90.9 ± 3.7 0.87 ( 3) 0.64 ( 4) 0.38 0.50 0.90 ( 3) 0.33 0.22 0.25 30 98.2 ± 2.1 97.2 ± 2.5 98.1 ± 2.2 98.1 ± 2.7 96.4 ± 2.1 93.6 ± 3.1 89.6 ± 5.9 92.9 ± 2.8 0.84 ( 8) 0.61 ( 7) 0.35 0.47 0.86 ( 6) 0.30 0.17 0.24 40 98.3 ± 1.9 96.5 ± 2.4 98.9 ± 1.8 99.1 ± 1.4 96.3 ± 2.3 94.0 ± 2.7 91.6 ± 6.3 94.9 ± 2.0 0.84 (13) 0.61 (15) 0.36 0.48 0.86 (11) 0.29 0.18 0.21 50 98.4 ± 1.9 95.6 ± 9.0 99.4 ± 0.5 99.6 ± 0.3 96.6 ± 2.3 96.1 ± 2.4 93.0 ± 3.6 95.8 ± 2.3 0.83 (31) 0.60 (37) 0.35 0.46 0.84 (25) 0.28 0.17 0.20 Ten digits (10 classes) 50 76.6 ± 4.3 71.5 ± 5.0 41.4 ± 6.8 49.8 ± 6.3 70.3 ± 5.2 57.0 ± 4.0 50.2 ± 9.0 66.3 ± 3.7 0.47 (26) 0.21 (26) 0.15 0.16 0.51 (25) -0.62 -0.50 -0.25 100 84.8 ± 2.6 83.4 ± 2.6 63.7 ± 3.5 72.5 ± 3.3 80.7 ± 2.6 69.4 ± 1.9 56.0 ± 7.8 77.2 ± 2.3 0.47 (124) 0.17 (98) 0.12 0.13 0.49 (100) -0.64 -0.52 -0.29 150 86.5 ± 1.7 86.4 ± 1.3 75.1 ± 3.0 80.4 ± 2.1 84.5 ± 1.9 75.2 ± 1.4 56.2 ± 7.2 81.4 ± 2.2 0.48 (310) 0.18 (255) 0.11 0.13 0.50 (244) -0.66 -0.53 -0.31 200 88.1 ± 1.3 88.0 ± 1.3 80.4 ± 2.5 84.4 ± 1.6 86.0 ± 1.5 78.3 ± 1.3 60.8 ± 7.3 84.3 ± 1.7 0.47 (708) 0.16 (477) 0.10 0.11 0.49 (523) -0.65 -0.54 -0.33 250 89.1 ± 1.1 89.3 ± 1.0 84.6 ± 1.4 87.2 ± 1.3 87.2 ± 1.3 80.4 ± 1.4 61.3 ± 7.6 85.7 ± 1.3 0.47 (942) 0.16 (873) 0.10 0.11 0.49 (706) -0.65 -0.54 -0.33 isolet (26 classes) 50 56.0 ± 3.5 42.0 ± 5.2 41.2 ± 2.9 29.0 ± 2.7 50.1 ± 3.7 28.7 ± 2.0 30.0 ± 2.7 23.7 ± 2.4 0.27 (26) 0.13 (25) 0.03 0.11 0.31 (24) -0.89 -0.80 -0.65 100 64.6 ± 2.1 59.0 ± 3.6 58.5 ± 2.9 47.4 ± 2.7 63.2 ± 1.9 46.3 ± 2.4 46.6 ± 2.7 42.0 ± 2.9 0.26 (105) 0.10 (127) -0.02 0.08 0.29 (102) -0.90 -0.82 -0.69 150 67.6 ± 2.6 65.2 ± 3.0 65.4 ± 2.6 57.2 ± 2.7 67.9 ± 2.5 57.6 ± 1.5 57.3 ± 1.8 53.8 ± 2.2 0.26 (249) 0.09 (280) -0.05 0.07 0.27 (221) -0.90 -0.83 -0.70 200 71.0 ± 1.8 70.9 ± 2.3 70.6 ± 1.9 64.8 ± 2.1 72.3 ± 1.7 63.9 ± 1.6 64.2 ± 2.0 60.5 ± 1.6 0.26 (441) 0.08 (570) -0.07 0.06 0.27 (423) -0.91 -0.83 -0.72 250 71.8 ± 2.3 73.6 ± 1.5 73.7 ± 1.2 69.8 ± 1.5 74.2 ± 1.5 68.8 ± 1.5 69.5 ± 1.7 66.2 ± 1.4 0.26 (709) 0.08 (836) -0.07 0.06 0.27 (665) -0.91 -0.84 -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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Sparse Coding of Natural Images Using an Overcomplete Set of Limited Capacity Units Eizaburo Doi Center for the Neural Basis of Cognition Carnegie Mellon University Pittsburgh, PA 15213 edoi@cnbc.cmu.edu Michael S. Lewicki Center for the Neural Basis of Cognition Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213 lewicki@cnbc.cmu.edu Abstract It has been suggested that the primary goal of the sensory system is to represent input in such a way as to reduce the high degree of redundancy. Given a noisy neural representation, however, solely reducing redundancy is not desirable, since redundancy is the only clue to reduce the effects of noise. Here we propose a model that best balances redundancy reduction and redundant representation. Like previous models, our model accounts for the localized and oriented structure of simple cells, but it also predicts a different organization for the population. With noisy, limited-capacity units, the optimal representation becomes an overcomplete, multi-scale representation, which, compared to previous models, is in closer agreement with physiological data. These results offer a new perspective on the expansion of the number of neurons from retina to V1 and provide a theoretical model of incorporating useful redundancy into efficient neural representations. 1 Introduction Efficient coding theory posits that one of the primary goals of sensory coding is to eliminate redundancy from raw sensory signals, ideally representing the input by a set of statistically independent features [1]. Models for learning efficient codes, such as sparse coding [2] or ICA [3], predict the localized, oriented, and band-pass characteristics of simple cells. In this framework, units are assumed to be non-redundant and so the number of units should be identical to the dimensionality of the data. Redundancy, however, can be beneficial if it is used to compensate for inherent noise in the system [4]. The models above assume that the system noise is low and negligible so that redundancy in the representation is not necessary. This is equivalent to assuming that the representational capacity of individual units is unlimited. Real neurons, however, have limited capacity [5], and this should place constraints on how a neural population can best encode a sensory signal. In fact, there are important characteristics of simple cells, such as the multi-scale representation, that cannot be explained by efficient coding theory. The aim of this study is to evaluate how the optimal representation changes when the system is constrained by limited capacity units. We propose a model that best balances redundancy reduction and redundant representation given the limited capacity units. In contrast to the efficient coding models, it is possible to have a larger number of units than the intrinsic dimensionality of the data. This further allows to introduce redundancy in the population, enabling precise reconstruction using the imprecise representation of a single unit. 2 Model Encoding We assume that the encoding is a linear transform of the input x, followed by the additive channel noise n ∼N(0, σ2 nI), r = Wx + n (1) = u + n, (2) where rows of W are referred to as the analysis vectors, r is the representation, and u is the signal component of the representation. We will refer to u as coefficients because it is a set of clean coefficients associated with the synthesis vectors in the decoding process, as described below. We define channel noise level as follows, (channel noise level) = σ2 n σ2 t × 100 [%] (3) where σ2 t is a constant target value of the coefficient variance. It is the inverse of the signalto-noise ratio in the representation, and therefore, we can control the information capacity of a single unit by varying the channel noise variance. Note that in the previous models [2, 3, 6] there is no channel noise; therefore r = u, where the signal-to-noise ratio of the representation is infinite. Decoding The decoding process is assumed to be a linear transform of the representation, ˆx = Ar, (4) where the columns of A are referred to as the synthesis vectors1, and ˆx is the reconstruction of the input. The reconstruction error e is then expressed as e = x −ˆx (5) = (I −AW) x −An. (6) Note that no assumption on the reconstruction error is made, because eq. 4 is not a probabilistic data generative model, in contrast to the previous approaches [2, 6]. Representation desiderata We assume a two-fold goal for the representation. The first is to preserve input information a given noisy, limited information capacity unit. The second is to make the representation 1In the noiseless and complete case, they are equivalent to the basis functions [2, 3]. In our setting, however, they are in general no longer basis functions. To make this clear, we call A and W as synthesis and analysis vectors. (a) 0% Ch.Noise (b) 20% Ch.Noise (c) 80% Ch.Noise (d) 8x overcomp. Synthesis Analysis Figure 1: Optimal codes for toy problems. Data (shown with small dots) is generated with two i.i.d. Laplacians mixed via non-orthogonal basis functions (shown by gray bars). The optimal synthesis vectors (top row) and analysis vectors (bottom row) are shown as black bars. Plots of synthesis vectors are scaled for visibility. (a-c) shows the complete code with 0, 20, and 80% channel noise level. (d) shows the case of 80% channel noise using an 8x overcomplete code. Reconstruction error is (a) 0.0%, (b) 13.6%, (c) 32.2%, (d) 6.8%. as sparse as possible, which yields an efficient code. The cost function to be minimized is therefore defined as follows, C(A, W) = (reconstruction error) −λ1(sparseness) + λ2(fixed variance) (7) = ⟨∥e∥2⟩−λ1 M X i=1 ⟨ln p(ui)⟩+ λ2 M X i=1 ln ⟨u2 i ⟩ σ2 t 2 , (8) where ⟨·⟩represents an ensemble average over the samples, and M is the number of units. The sparseness is measured by the loglikelihood of a sparse prior p as in the previous models [2, 3, 6]. The third, fixed variance term penalizes the case in which the coefficient variance of the i-th unit ⟨u2 i ⟩deviates from its target value σ2 t . It serves to fix the signalto-noise ratio in the representation, yielding a fixed information capacity. Without this term, the coefficient variance could become trivially large so that the signal-to-noise ratio is high, yielding smaller reconstruction error; or, the variance becomes small to satisfy only the sparseness constraint, which is not desirable either. Note that in order to introduce redundancy in the representation, we do not assume statistical independence of the coefficients. The second term in eq. 8 measures the sparseness of coefficients individually but it does not impose their statistical independence. We illustrate it with toy problems in Figure 1. If there is no channel noise, the optimal complete (1x) code is identical to the ICA solution (a), since it gives the most sparse, non-Gaussian solution with minimal error. As the channel noise increases (b and c), sparseness is compromised for minimizing the reconstruction error by choosing correlated, redundant representation. In an extreme case where the channel noise is high enough, the two units are almost completely redundant (c). It should be noted that in such a case two vectors represent the direction of the first principal component of the data. In addition to de-emphasizing sparseness, there is another way to introduce redundancy in the representation. Since the goal of the representation is not the separation of independent sources, we can set an arbitrarily large number of units in the representation. When the information capacity of a single unit is limited, the capacity of a population can be made large by increasing the number of units. As shown in Figure 1c-d, the reconstruction error decreases as we increase the degree of overcompleteness. Note that the optimal overcomplete code is not simply a duplication of the complete code. Learning rule The optimal code can be learned by the gradient descent of the cost function (eq. 8) with respect to A and W, ∆A ∝ (I −AW) xxT WT −σ2 nA, (9) ∆W ∝ AT (I −AW) xxT +λ1 ∂ln(u) ∂u xT −λ2diag ln[⟨u2⟩/σ2 t ] ⟨u2⟩ W xxT . (10) In the limit of zero channel noise in the square case (e.g., Figure 1a) the solution is at the equilibrium when W = A−1 (see eq. 9), where the learning rule becomes similar to the standard ICA (except the 3rd term in eq. 10). In all other cases, there is no reason to believe that W = A−1, if it exists, minimizes the cost function. This is the reason why we need to optimize A and W individually. 3 Optimal representations for natural images We examined optimal codes for natural image patches using the proposed model. The training data is 8x8 pixel image patches, sampled from a data set of 62 natural images [7]. The data is not preprocessed except for the subtraction of DC components [8]. Accordingly, the intrinsic dimensionality of the data is 63, and an N-times overcomplete code consists of N×63 units. The training set is sequentially updated during the learning and the order is randomized to prevent any local structure in the sequence. A typical number of image patches in a training is 5 × 106. Here we first descirbe how the presence of channel noise changes the optimal code in the complete case. Next, we examine the optimal code at different degree of overcompleteness given a high channel noise level. 3.1 Optimal code at different channel noise level We varied the channel noise level as 10, 20, 40, and 80%. As shown in Figure 2, learned synthesis and analysis vectors look somewhat similar to ICA (only 10 and 80% are shown for clarity). The comparison to the receptive fields of simple cells should be made with the analysis vectors [9, 10, 7]. They show localized and oriented structures and are well fitted by the Gabor function, indicating the similarity to simple cells in V1. Now, an additional characteristic to the Gabor-like structure is that the spatial-frequency tuning of the analysis vectors shifts towards lower spatial-frequencies as the channel noise increases (Figure 2d). The learned code is expected to be robust to the channel noise. The reconstruction error with respect to the data variance turned out to be 6.5, 10.1, 15.7, and 23.8% for 10, 20, 40, and 80% of channel noise level, respectively. The noise reduction is significant considering the fact that any whitened representation including ICA should generate the reconstruction error of exactly the same amount of the channel noise level2. For the learned ICA code shown in Figure 2a, the reconstruction error was 82.7% when 80% channel noise was applied. 2Since the mean squared error is expressed as ⟨∥e∥2⟩= σ2 n · Tr(AAT ) = σ2 n · Tr(⟨xxT ⟩) = σ2 n·(data variance), where W is whitening filters, A(≡W−1) is their corresponding basis functions. We used eq. (6) and ⟨xxT ⟩= AW⟨xxT ⟩WT AT = AAT . 0 0.5 1 1.5 0 10 20 30 40 50 Spatial frequency Count [#] ICA 10% 80% Synthesis Analysis (a) ICA (b) 10% (c) 80% (d) Figure 2: Optimal complete code at different channel noise level. (a-c) Optimized synthesis and analysis vectors. (a) ICA. (b) Proposed model at 10% channel noise level. (c) Proposed model at 80% channel noise level. Here 40 vectors out of 63 are shown. (d) Distribution of spatial-frequency tuning of the analysis vectors in the condition of (a)-(c). The robustness to channel noise can be explained by the shift of the representation towards lower spatial-frequencies. We analyzed the reconstruction error by projecting it to the principal axes of the data. Figure 3a shows the error spectrum of the code for 80% channel noise, along with the data spectrum (the percentage of the data variance along the principal axes). Note that the data variance of natural images is mostly explained by the first principal components, which correspond to lower spatial-frequencies. In the proposed model, the ratio of the error to the data variance is relatively small around the first principal components. It can be seen much clearer in Figure 3b, where the reconstruction percentage at each principal component is replotted. The reconstruction is more precise for more significant principal components (i.e., smaller index), and it drops down to zero for minor components. For comparison, we analyzed the error for ICA code, where the synthesis and analysis vectors are optimized without channel noise and its robustness to channel noise is tested with 80% channel noise level. As shown in Figure 3, ICA reconstructs every component equally irrespective of their very different data variance3, therefore the percentage of reconstruction is flat. The proposed model can be robust to channel noise by primarily representing the principal components. Note that such a biased reconstruction depends on the channel noise level. In Figure 3b we also shows the reconstruction spectrum with 10% channel noise using the code for 10% channel noise level. Compared to the 80% case, the model comes to reconstruct the data at relatively minor components as well. It means that the model can represent finer information if the information capacity of a single unit is large enough. Such a shift of representation is also demonstrated with the toy probems in Figure 1a-c. 3.2 Optimal code at different degree of overcompleteness Now we examine how the optimal representation changes with the different number of available units. We fixed the channel noise level at 80% and vary the degree of overcompleteness as 1x, 2x, 4x, and 8x. Learned vectors for 8x are shown in Figure 4a, and those for 3Since the error spectrum for a whitened representation is expressed as (Ete)2 = σ2 n · Diag(ET ⟨xxT ⟩E) = σ2 n · Diag(D) = σ2 n · (data spectrum), where EDET = ⟨xxT ⟩is the eigen value decomposition of the data covariance matrix. 100 101 101 100 101 102 Index of Principal Components Variance [%] 10 20 30 40 50 60 0 20 40 60 80 100 Index of Principal Components Reconstruction [%] ICA 80% DAT ICA 80% 10% 8x (a) (b) Figure 3: Error analysis. (a) Power spectrum of the data (‘DAT’) and the reconstruction error with 80% channel noise. ‘80%’ is the error of the 1x code for 80% channel noise level. ‘ICA’ is the error of the ICA code. (b) Percentage of reconstruction at each principal component. In addition to the conditions in (a), we also show the following (see text). ‘10%’: 1x code for 10% channel noise level. The error is measured with 10% channel noise. ‘8x’: 8x code for 80% channel noise level. The error is measured with 80% channel noise. 1x are in Figure 2c. Compared to the 1x case, where the synthesis and analysis vectors look uniform in shape, the 8x code shows more diversity. To be precise, as illustrated in Figure 4b, the spatial-frequency tuning of the analysis vectors becomes more broadly distribued and cover a larger region as the degree of overcompleteness increases. Physiological data at the central fovea shows that the spatial-frequency tuning of V1 simple cells spans three [11] or two [12] octaves. Models for efficient coding, especially ICA which provides the most efficient code, do not reproduce such a multi-scale representation; instead, the resulting analysis vectors tune only to the highest spatial-frequency (Figure 2a; [3, 9, 10, 7]). It is important that the proposed model generates a broader tuning distribution under the presence of the channel noise and with the high degree of overcompleteness. An important property of the proposed model is that the reconstruction error decreases as the degree of overcompleteness increases. The resulting error is 23.8, 15.5, 9.7, and 6.2% for 1x, 2x, 4x, and 8x code. The noise analysis shows that the model comes to represent minor components as the degree of overcompleteness increases (Figure 3b). There is an interesting similarity between the error spectra of 8x code for 80% channel noise and 1x code for 10% channel noise. It is suggested that the population of units can represent the same amount and the same kind of information using N times larger number of units if the information capacity of a single unit is decreased with N times larger channel noise level. 4 Discussion A multi-scale representation is known to provide an approximately efficient representation, although it is not optimal as there are known statistical dependencies between scales [13]. We conjecture these residual dependences may be one reason why previous efficient coding models could not yield a broad multi-scale representation. In contrast, the proposed model can introduce useful redundancies in the representation, which is consistent with the emergence of a multi-scale representation. Although it can generate a broader distribution of the spatial-frequency tuning, in these experiments, it covers only about one octave, not two or three octaves as in the physiological data [11, 12]. This issue still remains to be explained. 0 0.5 1 1.5 0 1 2 3 x102 Spatial frequency Count [#] 1x 2x 4x 8x Synthesis Analysis (a) 8x overcomplete w/ 80% ch. noise (b) Figure 4: Optimal overcomplete code. (a) Optimized 8x overcomplete code for 80% channel noise level. Here only 176 out of 504 functions are shown. The functions are sorted according to the spatial-frequency tuning of the analysis vectors. (b) Distribution of spatialfrequency tuning of the analysis vectors at different degree of overcompleteness. Another important characteristic of simple cells is the fact that the more numerous cells are tuned to the lower spatial-frequencies [11, 12]. An explanation of it is that the high data-variance components should be highly oversampled so that the reconstruction erorr is minimized given the limited precision of a single unit [12]. As we described earlier, such a biased representation for the high variance components is observed in our model (Figure 3b). However, the distribution of the spatial-frequency tuning of the analysis vectors does not correspond to this trend; instead, it is bell-shaped (Figure 4b). This apparent inconsistency might be resolved by considering the synthesis vectors, because the reconstruction error is determined by both synthesis and analysis vectors. A related work is the Atick & Redlich’s model for retinal ganglion cells [14]. It also utilizes redundancy in the representation but to compensate for sensory noise rather than channel noise; therefore, the two models explain different phenomena. Another related work is Olshausen & Field’s sparse coding model for simple cells [2], but this again looks at the effects of sensory noise (note that if the sensory noise is neglegible this algorithm does not learn a sparse representation, while the proposed model is appropriate for this condition; of course such a condition might be unrealistic). Now, given a photopic environment where the sensory noise can reasonably be regarded to be small [14], it should rather be important to examine how the constraint of noisy, limited information capacity units changes the representation. It is reported that the information capacity is significantly decreased from photoreceptors to spiking neurons [15], which supports our approach. In spite of its significance, to our knowledge the influence of channel noise on the representation had not been explored. 5 Conclusion We propose a model that both utilizes redundancy in the representation in order to compensate for the limited precision of a single unit and reduces unnecessary redundancy in order to yield an efficient code. The noisy, overcomplete code for natural images generates a distributed spatial-frequency tuning in addition to the Gabor-like analysis vectors, showing a closer agreement with the physiological data than the previous efficient coding models. The information capacity of a representation may be constrained either by the intrinsic noise in a single unit or by the number of units. In either case, the proposed model can adapt the parameters to primarily represent the high-variance, coarse information, yielding a robust representation to channel noise. As the limitation is relaxed by decreasing the channel noise level or by increasing the number of units, the model comes to represent low-variance, fine information. References [1] H. B. Barlow. Possible principles underlying the transformation of sensory messages. In W. A. Rosenblith, editor, Sensory communication, pages 217–234. MIT Press, MA, 1961. [2] B. A. Olshausen and D. J. Field. Sparse coding with an overcomplete basis set: A strategy employed by V1? Vision Research, 37:3311–3325, 1997. [3] A. J. Bell and T. J. Sejnowski. The independent components of natural scenes are edge filters. Vision Research, 37:3327–3338, 1997. [4] H. B. Barlow. Redundancy reduction revisited. Network: Comput. Neural Syst., 12:241–253, 2001. [5] A. Borst and F. E. Theunissen. Information theory and neural coding. Nature Neuroscience, 2:947–957, 1999. [6] M. S. Lewicki and B. A. Olshausen. Probabilistic framework for the adaptation and comparison of image codes. J. Opt. Soc. Am. A, 16:1587–1601, 1999. [7] E. Doi, T. Inui, T.-W. Lee, T. Wachtler, and T. J. Sejnowski. Spatiochromatic receptive field properties derived from information-theoretic analyses of cone mosaic responses to natural scenes. Neural Computation, 15:397–417, 2003. [8] A. Hyvarinen, J. Karhunen, and E. Oja. Independent Component Analysis. John Wiley & Sons, NY, 2001. [9] J. H. van Hateren and A. van der Schaaf. Independent component filters of natural images compared with simple cells in primary visual cortex. Proc. R. Soc. Lond. B, 265:359–366, 1998. [10] D. L. Ringach. Spatial structure and symmetry of simple-cell receptive fields in macaque primary visual cortex. Journal of Neurophysiology, 88:455–463, 2002. [11] R. L. De Valois, D. G. Albrecht, and L. G. Thorell. Spatial frequency selectivity of cells in macaque visual cortex. Vision Research, 22(545-559), 1982. [12] C. H. Anderson and G. C. DeAngelis. Population codes and signal to noise ratios in primary visual cortex. In Society for Neuroscience Abstract, page 822.3, 2004. [13] E. P. Simoncelli. Modeling the joint statistics of images in the wavelet domain. In Proc. SPIE 44th Annual Meeting, pages 188–195, Denver, Colorado, 1999. [14] J. J. Atick and A. N. Redlich. What does the retina know about natural scenes? Neural Computation, 4:196–210, 1992. [15] S. B. Laughlin and R. R. de Ruyter van Steveninck. The rate of information transfer at gradedpotential synapses. Nature, 379(15):642–645, 1996.
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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 m.montemurro@manchester.ac.uk Stefano Panzeri ∗ The University of Manchester Faculty of Life Sciences Moffat Building PO Box 88, Manchester M60 1QD, UK s.panzeri@manchester.ac.uk 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 τ. 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 σf, and response modulation m: f (k)(s) = me −(s−s(k))2 2σf 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 σf indicate coarse coding, whereas small values of σf 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(ˆs) = 1 (2π)D/2σD p e −(ˆs−ˆs0)2 2σ2p (2) In Eq. (2) the parameter ˆs0 represents the the center of the distribution, thus being the most represented preferred stimulus in the population. (we set, without loss of generality, ˆs0 = 0.) The parameter σp controls the spread of stimulus preferences of the afferent neuronal population: a small value of σp indicates that a large fraction of the population have similar stimulus preferences, and a large value of σp 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 σf and the spread of stimulus preferences σp 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 σp 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 σf 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 σp 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 σf of the cortical inputs to the V1 target neuron is that of other V1 neurons that project to it. This σf 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 σp is 20o [2] and σf is ≈17o. When considering a target neuron in ferret visual cortex and motion direction tuning, the spread of tuning of its inputs σp 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 σp and σf appear to be approximately of the same size and their ratio σf/σp 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 σf/σp 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]: ILL(S, R) = dsP(s) r P(r|s) log P(r|s) P(r) (3) 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’ sρ). 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]: ILL(S, R) = H(S) −D 2 ln (2πe) + 1 2 ds P(s) ln (|J (s)|) (4) 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: Ji,j(s) = − r P(r|s) ∂2 ∂si sj log P(r|s) , (5) 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(ˆs). After performing the integral over the distribution of preferred stimuli, we arrived at the following result for the elements of the Fisher information matrix: Ji,j(s) = Nτm σ2p σD−2 (1 + σ2) D 2 +2 δi,j + σ2 (δi,j + ξiξj) e − ξ2 2(1+σ2) (6) where we have introduced the following short-hand notation σf/σp →σ and s/σp →ξ; δi,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=1 λi = α(ξ)D(1 + σ2)D−1 1 + σ2(1 + ξ2) (7) where α(ξ) is given by: α(ξ) = Nτm σ2p σD−2 (1 + σ2) D 2 +1 e − ξ2 2(1+σ2) (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 σf and σp as a function of the number of encoded stimulus features D 1. We investigated this by fixing σp and then varying the ration σf/σp over a wide range. Results (obtained for σp = 1 but representative of a wide σf 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 σf that maximizes the information for all D ≥2. Interestingly, for D ≥2 the range 0.7 ≤σf/σp ≤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 σf/σp ≈1, in that the tail of the D = 1 information curve is avoided in that region. Thus, the range of values of σf and σp found in visual cortex allows efficient synaptic information transmission over a wide range of number of stimulus features encoded by the neuron. 1We found (data not shown) that other parameters such as m and τ, had a weak or null effect on the optimal configuration; see [17] for a D = 1 example in a different context. 0 2 4 6 8 σf/σp ILL(S,R) D=5 D=1 Figure 1: Mutual labeled-line information as a function of the ratio of tuning curve width and stimulus preference spread σf/σp. 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 σf/σp found in visual cortex, or very close to it (see text) . Parameters are as follows: sρ = 2, rmax = 50Hz, τ = 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 ρ equal to the sum of the individual activities: ρ = rk. Thus, the mutual information that can be extracted by any decoder that only pools it inputs is given by the following expression: IP (S, R) = dsP(s) ρ P(ρ|s) log P(ρ|s) P(ρ) (9) where P(ρ|s) and P(ρ) are the the stimulus-conditional and stimulus-unconditional probability of observing a pooled population response ρ on a single trial. The probability P(ρ|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 σf and σp 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 0 1 2 3 4 σf/σp IP(S,R) D=1 D=3 Figure 2: Pooled mutual information as a function of the ratio of tuning curve width and stimulus preference spread σf/σp. The maxima are inside the range of experimental values of σf/σp 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: sρ = 2, rmax = 50 Hz,τ = 10ms. neuronal decoder can still be reached in the visual cortical range 0.7 ≤σf σp ≤1.1. This finding shows that the range of values of σf and σp 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. [2] T. Yousef, T. Bonhoeffer, D-S. Kim, U. T. Eysel, ´E. T´oth, and Z. F. Kisv´arday. Orientation 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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Implicit Wiener Series for Higher-Order Image Analysis Matthias O. Franz Bernhard Sch¨olkopf Max-Planck-Institut f¨ur biologische Kybernetik Spemannstr. 38, D-72076 T¨ubingen, Germany mof;bs@tuebingen.mpg.de Abstract The computation of classical higher-order statistics such as higher-order moments or spectra is difficult for images due to the huge number of terms to be estimated and interpreted. We propose an alternative approach in which multiplicative pixel interactions are described by a series of Wiener functionals. Since the functionals are estimated implicitly via polynomial kernels, the combinatorial explosion associated with the classical higher-order statistics is avoided. First results show that image structures such as lines or corners can be predicted correctly, and that pixel interactions up to the order of five play an important role in natural images. Most of the interesting structure in a natural image is characterized by its higher-order statistics. Arbitrarily oriented lines and edges, for instance, cannot be described by the usual pairwise statistics such as the power spectrum or the autocorrelation function: From knowing the intensity of one point on a line alone, we cannot predict its neighbouring intensities. This would require knowledge of a second point on the line, i.e., we have to consider some third-order statistics which describe the interactions between triplets of points. Analogously, the prediction of a corner neighbourhood needs at least fourth-order statistics, and so on. In terms of Fourier analysis, higher-order image structures such as edges or corners are described by phase alignments, i.e. phase correlations between several Fourier components of the image. Classically, harmonic phase interactions are measured by higher-order spectra [4]. Unfortunately, the estimation of these spectra for high-dimensional signals such as images involves the estimation and interpretation of a huge number of terms. For instance, a sixth-order spectrum of a 16×16 sized image contains roughly 1012 coefficients, about 1010 of which would have to be estimated independently if all symmetries in the spectrum are considered. First attempts at estimating the higher-order structure of natural images were therefore restricted to global measures such as skewness or kurtosis [8], or to submanifolds of fourth-order spectra [9]. Here, we propose an alternative approach that models the interactions of image points in a series of Wiener functionals. A Wiener functional of order n captures those image components that can be predicted from the multiplicative interaction of n image points. In contrast to higher-order spectra or moments, the estimation of a Wiener model does not require the estimation of an excessive number of terms since it can be computed implicitly via polynomial kernels. This allows us to decompose an image into components that are characterized by interactions of a given order. In the next section, we introduce the Wiener expansion and discuss its capability of modeling higher-order pixel interactions. The implicit estimation method is described in Sect. 2, followed by some examples of use in Sect. 3. We conclude in Sect. 4 by briefly discussing the results and possible improvements. 1 Modeling pixel interactions with Wiener functionals For our analysis, we adopt a prediction framework: Given a d × d neighbourhood of an image pixel, we want to predict its gray value from the gray values of the neighbours. We are particularly interested to which extent interactions of different orders contribute to the overall prediction. Our basic assumption is that the dependency of the central pixel value y on its neighbours xi, i = 1, . . . , m = d2 −1 can be modeled as a series y = H0[x] + H1[x] + H2[x] + · · · + Hn[x] + · · · (1) of discrete Volterra functionals H0[x] = h0 = const. and Hn[x] = Xm i1=1 · · · Xm in=1 h(n) i1...inxi1 . . . xin. (2) Here, we have stacked the grayvalues of the neighbourhood into the vector x = (x1, . . . , xm)⊤∈Rm. The discrete nth-order Volterra functional is, accordingly, a linear combination of all ordered nth-order monomials of the elements of x with mn coefficients h(n) i1...in. Volterra functionals provide a controlled way of introducing multiplicative interactions of image points since a functional of order n contains all products of the input of order n. In terms of higher-order statistics, this means that we can control the order of the statistics used since an nth-order Volterra series leads to dependencies between maximally n + 1 pixels. Unfortunately, Volterra functionals are not orthogonal to each other, i.e., depending on the input distribution, a functional of order n generally leads to additional lower-order interactions. As a result, the output of the functional will contain components that are proportional to that of some lower-order monomials. For instance, the output of a second-order Volterra functional for Gaussian input generally has a mean different from zero [5]. If one wants to estimate the zeroeth-order component of an image (i.e., the constant component created without pixel interactions) the constant component created by the second-order interactions needs to be subtracted. For general Volterra series, this correction can be achieved by decomposing it into a new series y = G0[x] + G1[x] + · · · + Gn[x] + · · · of functionals Gn[x] that are uncorrelated, i.e., orthogonal with respect to the input. The resulting Wiener functionals1 Gn[x] are linear combinations of Volterra functionals up to order n. They are computed from the original Volterra series by a procedure akin to Gram-Schmidt orthogonalization [5]. It can be shown that any Wiener expansion of finite degree minimizes the mean squared error between the true system output and its Volterra series model [5]. The orthogonality condition ensures that a Wiener functional of order n captures only the component of the image created by the multiplicative interaction of n pixels. In contrast to general Volterra functionals, a Wiener functional is orthogonal to all monomials of lower order [5]. So far, we have not gained anything compared to classical estimation of higher-order moments or spectra: an nth-order Volterra functional contains the same number of terms as 1Strictly speaking, the term Wiener functional is reserved for orthogonal Volterra functionals with respect to Gaussian input. Here, the term will be used for orthogonalized Volterra functionals with arbitrary input distributions. the corresponding n + 1-order spectrum, and a Wiener functional of the same order has an even higher number of coefficients as it consists also of lower-order Volterra functionals. In the next section, we will introduce an implicit representation of the Wiener series using polynomial kernels which allows for an efficient computation of the Wiener functionals. 2 Estimating Wiener series by regression in RKHS Volterra series as linear functionals in RKHS. The nth-order Volterra functional is a weighted sum of all nth-order monomials of the input vector x. We can interpret the evaluation of this functional for a given input x as a map φn defined for n = 0, 1, 2, . . . as φ0(x) = 1 and φn(x) = (xn 1, xn−1 1 x2, . . . , x1xn−1 2 , xn 2, . . . , xn m) (3) such that φn maps the input x ∈Rm into a vector φn(x) ∈Fn = Rmn containing all mn ordered monomials of degree n. Using φn, we can write the nth-order Volterra functional in Eq. (2) as a scalar product in Fn, Hn[x] = η⊤ n φn(x), (4) with the coefficients stacked into the vector ηn = (h(n) 1,1,..1, h(n) 1,2,..1, h(n) 1,3,..1, . . . )⊤∈Fn. The same idea can be applied to the entire pth-order Volterra series. By stacking the maps φn into a single map φ(p)(x) = (φ0(x), φ1(x), . . . , φp(x))⊤, one obtains a mapping from Rm into F(p) = R×Rm ×Rm2 ×. . . Rmp = RM with dimensionality M = 1−mp+1 1−m . The entire pth-order Volterra series can be written as a scalar product in F(p) Xp n=0 Hn[x] = (η(p))⊤φ(p)(x) (5) with η(p) ∈F(p). Below, we will show how we can express η(p) as an expansion in terms of the training points. This will dramatically reduce the number of parameters we have to estimate. This procedure works because the space Fn of nth-order monomials has a very special property: it has the structure of a reproducing kernel Hilbert space (RKHS). As a consequence, the dot product in Fn can be computed by evaluating a positive definite kernel function kn(x1, x2). For monomials, one can easily show that (e.g., [6]) φn(x1)⊤φn(x2) = (x⊤ 1 x2)n =: kn(x1, x2). (6) Since F(p) is generated as a direct sum of the single spaces Fn, the associated scalar product is simply the sum of the scalar products in the Fn: φ(p)(x1)⊤φ(p)(x2) = Xp n=0(x⊤ 1 x2)n = k(p)(x1, x2). (7) Thus, we have shown that the discretized Volterra series can be expressed as a linear functional in a RKHS2. Linear regression in RKHS. For our prediction problem (1), the RKHS property of the Volterra series leads to an efficient solution which is in part due to the so called representer theorem (e.g., [6]). It states the following: suppose we are given N observations 2A similar approach has been taken by [1] using the inhomogeneous polynomial kernel k(p) inh(x1, x2) = (1 + x⊤ 1 x2)p. This kernel implies a map φinh into the same space of monomials, but it weights the degrees of the monomials differently as can be seen by applying the binomial theorem. (x1, y1), . . . , (xN, yN) of the function (1) and an arbitrary cost function c, Ωis a nondecreasing function on R>0 and ∥.∥F is the norm of the RKHS associated with the kernel k. If we minimize an objective function c((x1, y1, f(x1)), . . . , (xN, yN, f(xN))) + Ω(∥f∥F), (8) over all functions in the RKHS, then an optimal solution3 can be expressed as f(x) = XN j=1 ajk(x, xj), aj ∈R. (9) In other words, although we optimized over the entire RKHS including functions which are defined for arbitrary input points, it turns out that we can always express the solution in terms of the observations xj only. Hence the optimization problem over the extremely large number of coefficients η(p) in Eq. (5) is transformed into one over N variables aj. Let us consider the special case where the cost function is the mean squared error, c((x1, y1, f(x1)), . . . , (xN, yN, f(xN))) = 1 N PN j=1(f(xj) −yj)2, and the regularizer Ωis zero4. The solution for a = (a1, . . . , aN) is readily computed by setting the derivative of (8) with respect to the vector a equal to zero; it takes the form a = K−1y with the Gram matrix defined as Kij = k(xi, xj), hence5 y = f(x) = a⊤z(x) = y⊤K−1z(x), (10) where z(x) = (k(x, x1), k(x, x2), . . . k(x, xN))⊤∈RN. Implicit Wiener series estimation. As we stated above, the pth-degree Wiener expansion is the pth-order Volterra series that minimizes the squared error. This can be put into the regression framework: since any finite Volterra series can be represented as a linear functional in the corresponding RKHS, we can find the pth-order Volterra series that minimizes the squared error by linear regression. This, by definition, must be the pth-degree Wiener series since no other Volterra series has this property6. From Eqn. (10), we obtain the following expressions for the implicit Wiener series G0[x] = 1 N y⊤1, Xp n=0 Gn[x] = Xp n=0 Hn[x] = y⊤K−1 p z(p)(x) (11) where the Gram matrix Kp and the coefficient vector z(p)(x) are computed using the kernel from Eq. (7) and 1 = (1, 1, . . . )⊤∈RN. Note that the Wiener series is represented only implicitly since we are using the RKHS representation as a sum of scalar products with the training points. Thus, we can avoid the “curse of dimensionality”, i.e., there is no need to compute the possibly large number of coefficients explicitly. The explicit Volterra and Wiener expansions can be recovered from Eq. (11) by collecting all terms containing monomials of the desired order and summing them up. The individual nth-order Volterra functionals in a Wiener series of degree p > 0 are given implicitly by Hn[x] = y⊤K−1 p zn(x) (12) with zn(x) = ((x⊤ 1 x)n, (x⊤ 2 x)n, . . . , (x⊤ Nx)n)⊤. For p = 0 the only term is the constant zero-order Volterra functional H0[x] = G0[x]. The coefficient vector ηn = (h(n) 1,1,...1, h(n) 1,2,...1, h(n) 1,3,...1, . . . )⊤of the explicit Volterra functional is obtained as ηn = Φ⊤ n K−1 p y (13) 3for conditions on uniqueness of the solution, see [6] 4Note that this is different from the regularized approach used by [1]. If Ωis not zero, the resulting Volterra series are different from the Wiener series since they are not orthogonal with respect to the input. 5If K is not invertible, K−1 denotes the pseudo-inverse of K. 6assuming symmetrized Volterra kernels which can be obtained from any Volterra expanson. using the design matrix Φn = (φn(x1)⊤, φn(x1)⊤, . . . , φn(x1)⊤)⊤. The individual Wiener functionals can only be recovered by applying the regression procedure twice. If we are interested in the nth-degree Wiener functional, we have to compute the solution for the kernels k(n)(x1, x2) and k(n−1)(x1, x2). The Wiener functional for n > 0 is then obtained from the difference of the two results as Gn[x] = Xn i=0 Gi[x] − Xn−1 i=0 Gi[x] = y⊤h K−1 n z(n)(x) −K−1 n−1 z(n−1)(x) i . (14) The corresponding ith-order Volterra functionals of the nth-degree Wiener functional are computed analogously to Eqns. (12) and (13) [3]. Orthogonality. The resulting Wiener functionals must fulfill the orthogonality condition which in its strictest form states that a pth-degree Wiener functional must be orthogonal to all monomials in the input of lower order. Formally, we will prove the following Theorem 1 The functionals obtained from Eq. (14) fulfill the orthogonality condition E [m(x)Gp[x]] = 0 (15) where E denotes the expectation over the input distribution and m(x) an arbitrary ithorder monomial with i < p. We will show that this a consequence of the least squares fit of any linear expansion in a set of basis functions of the form y = PM j=1 γjϕj(x). In the case of the Wiener and Volterra expansions, the basis functions ϕj(x) are monomials of the components of x. We denote the error of the expansion as e(x) = y −PM j=1 γjϕj(xi). The minimum of the expected quadratic loss L with respect to the expansion coefficient γk is given by ∂L ∂γk = ∂ ∂γk E∥e(x)∥2 = −2E [ϕk(x)e(x)] = 0. (16) This means that, for an expansion in a set of basis functions minimizing the squared error, the error is orthogonal to all basis functions used in the expansion. Now let us assume we know the Wiener series expansion (which minimizes the mean squared error) of a system up to degree p −1. The approximation error is given by the sum of the higher-order Wiener functionals e(x) = P∞ n=p Gn[x], so Gp[x] is part of the error. As a consequence of the linearity of the expectation, Eq. (16) implies X∞ n=p E [ϕk(x)Gn[x]] = 0 and X∞ n=p+1 E [ϕk(x)Gn[x]] = 0 (17) for any φk of order less than p. The difference of both equations yields E [ϕk(x)Gp[x]] = 0, so that Gp[x] must be orthogonal to any of the lower order basis functions, namely to all monomials with order smaller than p. □ 3 Experiments Toy examples. In our first experiment, we check whether our intuitions about higher-order statistics described in the introduction are captured by the proposed method. In particular, we expect that arbitrarily oriented lines can only be predicted using third-order statistics. As a consequence, we should need at least a second-order Wiener functional to predict lines correctly. Our first test image (size 80 × 110, upper row in Fig. 1) contains only lines of varying orientations. Choosing a 5 × 5 neighbourhood, we predicted the central pixel using (11). 1st-order reconstruction 2nd-order reconstruction 3rd-order reconstruction original image 1st-order component 2nd-order component 3rd-order component mse = 583.7 mse = 0.006 mse = 0 0th-order component/ reconstruction mse = 1317 mse = 37.4 mse = 0.001 mse = 1845 mse = 334.9 mse = 19.0 Figure 1: Higher-order components of toy images. The image components of different orders are created by the corresponding Wiener functionals. They are added up to obtain the different orders of reconstruction. Note that the constant 0-order component and reconstruction are identical. The reconstruction error (mse) is given as the mean squared error between the true grey values of the image and the reconstruction. Although the linear first-order model seems to reconstruct the lines, this is actually not true since the linear model just smoothes over the image (note its large reconstruction error). A correct prediction is only obtained by adding a second-order component to the model. The third-order component is only significant at crossings, corners and line endings. Models of orders 0 . . . 3 were learned from the image by extracting the maximal training set of 76×106 patches of size 5×57. The corresponding image components of order 0 to 3 were computed according to (14). Note the different components generated by the Wiener functionals can also be negative. In Fig. 1, they are scaled to the gray values [0..255]. The behaviour of the models conforms to our intuition: the linear model cannot capture the line structure of the image thus leading to a large reconstruction error which drops to nearly zero when a second-order model is used. The additional small correction achieved by the third-order model is mainly due to discretization effects. Similar to lines, we expect that we need at least a third-order model to predict crossings or corners correctly. This is confirmed by the second and third test image shown in the corresponding row in Fig. 1. Note that the third-order component is only significant at crossings, corners and line endings. The fourth- and fifth-order terms (not shown) have only negligible contributions. The fact that the reconstruction error does not drop to zero for the third image is caused by the line endings which cannot be predicted to a higher accuracy than one pixel. Application to natural images. Are there further predictable structures in natural images that are not due to lines, crossings or corners? This can be investigated by applying our method to a set of natural images (an example of size 80 × 110 is depicted in Fig. 2). Our 7In contrast to the usual setting in machine learning, training and test set are identical in our case since we are not interested in generalization to other images, but in analyzing the higher-order components of the image at hand. original image 0th-order component/ reconstruction 1st-order component 2nd-order component 1st-order reconstruction mse = 1070 2nd-order reconstruction mse = 957.4 3rd-order component 4th-order component 5th-order component 4th-order reconstruction mse = 98.5 5th-order reconstruction mse = 18.5 3rd-order reconstruction mse = 414.6 6th-order component 7th-order component 8th-order component 7th-order reconstruction mse = 1.32 8th-order reconstruction mse = 0.41 6th-order reconstruction mse = 4.98 Figure 2: Higher-order components and reconstructions of a photograph. Interactions up to the fifth order play an important role. Note that significant components become sparser with increasing model order. results on a set of 10 natural images of size 50 × 70 show an an approximately exponential decay of the reconstruction error when more and more higher-order terms are added to the reconstruction (Fig. 3). Interestingly, terms up to order 5 still play a significant role, although the image regions with a significant component become sparser with increasing model order (see Fig. 2). Note that the nonlinear terms reduce the reconstruction error to almost 0. This suggests a high degree of higher-order redundancy in natural images that cannot be exploited by the usual linear prediction models. 4 Conclusion The implicit estimation of Wiener functionals via polynomial kernels opens up new possibilities for the estimation of higher-order image statistics. Compared to the classical methods such as higher-order spectra, moments or cumulants, our approach avoids the combinatorial explosion caused by the exponential increase of the number of terms to be estimated and interpreted. When put into a predictive framework, multiplicative pixel interactions of different orders are easily visualized and conform to the intuitive notions about image structures such as edges, lines, crossings or corners. There is no one-to-one mapping between the classical higher-order statistics and multiplicative pixel interactions. Any nonlinear Wiener functional, for instance, creates infinitely many correlations or cumulants of higher order, and often also of lower order. On the other 0 1 2 3 4 5 6 7 0 100 200 300 400 500 600 700 model order mse Figure 3: Mean square reconstruction error of models of different order for a set of 10 natural images. hand, a Wiener functional of order n produces only harmonic phase interactions up to order n+1, but sometimes also of lower orders. Thus, when one analyzes a classical statistic of a given order, one often cannot determine by which order of pixel interaction it was created. In contrast, our method is able to isolate image components that are created by a single order of interaction. Although of preliminary nature, our results on natural images suggest an important role of statistics up to the fifth order. Most of the currently used low-level feature detectors such as edge or corner detectors maximally use third-order interactions. The investigation of fourth- or higher-order features is a field that might lead to new insights into the nature and role of higher-order image structures. As often observed in the literature (e.g. [2][7]), our results seem to confirm that a large proportion of the redundancy in natural images is contained in the higher-order pixel interactions. Before any further conclusions can be drawn, however, our study needs to be extended in several directions: 1. A representative image database has to be analyzed. The images must be carefully calibrated since nonlinear statistics can be highly calibrationsensitive. In addition, the contribution of image noise has to be investigated. 2. Currently, only images up to 9000 pixels can be analyzed due to the matrix inversion required by Eq. 11. To accomodate for larger images, our method has to be reformulated in an iterative algorithm. 3. So far, we only considered 5 × 5-patches. To systematically investigate patch size effects, the analysis has to be conducted in a multi-scale framework. References [1] T. J. Dodd and R. F. Harrison. A new solution to Volterra series estimation. In CD-Rom Proc. 2002 IFAC World Congress, 2002. [2] D. J. Field. What is the goal of sensory coding? Neural Computation, 6:559 – 601, 1994. [3] M. O. Franz and B. Sch¨olkopf. Implicit Wiener series. Technical Report 114, Max-PlanckInstitut f¨ur biologische Kybernetik, T¨ubingen, June 2003. [4] C. L. Nikias and A. P. Petropulu. Higher-order spectra analysis. Prentice Hall, Englewood Cliffs, NJ, 1993. [5] M. Schetzen. The Volterra and Wiener theories of nonlinear systems. Krieger, Malabar, 1989. [6] B. Sch¨olkopf and A. J. Smola. Learning with kernels. MIT Press, Cambridge, MA, 2002. [7] O. Schwartz and E. P. Simoncelli. Natural signal statistics and sensory gain control. Nature Neurosc., 4(8):819 – 825, 2001. [8] M. G. A. Thomson. Higher-order structure in natural scenes. J. Opt.Soc. Am. A, 16(7):1549 – 1553, 1999. [9] M. G. A. Thomson. Beats, kurtosis and visual coding. Network: Compt. Neural Syst., 12:271 – 287, 2001.
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Following Curved Regularized Optimization Solution Paths Saharon Rosset IBM T.J. Watson Research Center Yorktown Heights, NY 10598 srosset@us.ibm.com Abstract Regularization plays a central role in the analysis of modern data, where non-regularized fitting is likely to lead to over-fitted models, useless for both prediction and interpretation. We consider the design of incremental algorithms which follow paths of regularized solutions, as the regularization varies. These approaches often result in methods which are both efficient and highly flexible. We suggest a general path-following algorithm based on second-order approximations, prove that under mild conditions it remains “very close” to the path of optimal solutions and illustrate it with examples. 1 Introduction Given a data sample (xi, yi)n i=1 (with xi ∈Rp and yi ∈R for regression, yi ∈{±1} for classification), the generic regularized optimization problem calls for fitting models to the data while controlling complexity by solving a penalized fitting problem: ˆβ(λ) = arg min β X i C(yi, β′xi) + λJ(β) (1) where C is a convex loss function and J is a convex model complexity penalty (typically taken to be the lq norm of β, with q ≥1).1 Many commonly used supervised learning methods can be cast in this form, including regularized 1-norm and 2-norm support vector machines [13, 4], regularized linear and logistic regression (i.e. Ridge regression, lasso and their logistic equivalents) and more. In [8] we show that boosting can also be described as approximate regularized optimization, with an l1-norm penalty. Detailed discussion of the considerations in selecting penalty and loss functions for regularized fitting is outside the scope of this paper. In general, there are two main areas we need to consider in this selection: 1. Statistical considerations: robustness (which affects selection of loss), sparsity (l1-norm penalty encourages sparse solutions) and identifiability are among the questions we should 1We assume a linear model in (1), but this is much less limiting than it seems, as the model can be linear in basis expansions of the original predictors, and so our approach covers Kernel methods, wavelets, boosting and more keep in mind when selecting our formulation. 2. Computational considerations: we should be able to solve the problems we pose with the computational resources at our disposal. Kernel methods and boosting are examples of computational tricks that allow us to solve very high dimensional problems – exactly or approximately – with a relatively small cost. In this paper we suggest a new computational approach. Once we have settled on a loss and penalty, we are still faced with the problem of selecting a “good” regularization parameter λ, in terms of prediction performance. A common approach is to solve (1) for several values of λ, then use holdout data (or theoretical approaches, like AIC or SRM) to select a good value. However, if we view the regularized optimization problem as a family of problems, parameterized by the regularization parameter λ, it allows us to define the “path” of optimal solutions {ˆβ(λ) : 0 ≤λ ≤∞}, which is a 1-dimensional curve through Rp. Path following methods attempt to utilize the mathematical properties of this curve to devise efficient procedures for “following” it and generating the full set of regularized solutions with a (relatively) small computational cost. As it turns out, there is a family of well known and interesting regularized problems for which efficient exact path following algorithms can be devised. These include the lasso [3], 1- and 2-norm support vector machines [13, 4] and many others [9]. The main property of these problems which makes them amenable to such methods is the piecewise linearity of the regularized solution path in Rp. See [9] for detailed exposition of these properties and the resulting algorithms. However, the path following idea can stretch beyond these exact piecewise linear algorithms. The “first order” approach is to use gradient-based approaches. In [8] we have described boosting as an approximate gradient-based algorithm for following l1-norm regularized solution paths. [6] suggest a gradient descent algorithm for finding an optimal solution for a fixed value of λ and are seemingly unaware that the path they are going through is of independent interest as it consists of approximate (alas very approximate) solutions to l1-regularized problems. Gradient-based methods, however, can only follow regularized paths under strict and non-testable conditions, and theoretical “closeness” results to the optimal path are extremely difficult to prove for them (see [8] for details). In this paper, we suggest a general second-order algorithm for following “curved” regularized solution paths (i.e. ones that cannot be followed exactly by piecewise-linear algorithms). It consists of iteratively changing the regularization parameter, while making a single Newton step at every iteration towards the optimal penalized solution, for the current value of λ. We prove that if both the loss and penalty are “nice” (in terms of bounds on their relevant derivatives in the relevant region), then the algorithm is guaranteed to stay “very close” to the true optimal path, where “very close” is defined as: If the change in the regularization parameter at every iteration is ϵ, then the solution path we generate is guaranteed to be within O(ϵ2) from the true path of penalized optimal solutions In section 2 we present the algorithm, and we then illustrate it on l1- and l2-regularized logistic regression in section 3. Section 4 is devoted to a formal statement and proof outline of our main result. We discuss possible extensions and future work in section 5. 2 Path following algorithm We assume throughout that the loss function C is twice differentiable. Assume for now also that the penalty J is twice differentiable (this assumption does not apply to the l1norm penalty which is of great interest and we address this point later). The key to our method are the normal equations for (1): ∇C(ˆβ(λ)) + λ∇J(ˆβ(λ)) = 0 (2) Our algorithm iteratively constructs an approximate solution β(ϵ) t by taking “small” Newton-Raphson steps trying to maintain (2) as the regularization changes. Our main result in this paper is to show, both empirically and theoretically, that for small ϵ, the difference ∥β(ϵ) t −ˆβ(λ0 + ϵ · t)∥is small, and thus that our method successfully tracks the path of optimal solutions to (1). Algorithm 1 gives a formal description of our quadratic tracking method. We start from a solution to (1) for some fixed λ0 (e.g. ˆβ(0), the non-regularized solution). At each iteration we increase λ by ϵ and take a single Newton-Raphson step towards the solution to (2) with the new λ value in step 2(b). Algorithm 1 Approximate incremental quadratic algorithm for regularized optimization 1. Set β(ϵ) 0 = ˆβ(λ0), set t = 0. 2. While (λt < λmax) (a) λt+1 = λt + ϵ (b) β(ϵ) t+1 = β(ϵ) t − h ∇2C(β(ϵ) t ) + λt+1∇2J(β(ϵ) t ) i−1 h ∇C(β(ϵ) t ) + λt+1∇J(β(ϵ) t ) i (c) t = t + 1 2.1 The l1-norm penalty The l1-norm penalty, J(β) = ∥β∥1, is of special interest because of its favorable statistical properties (e.g. [2]) and its widespread use in popular methods, such as the lasso [10] and 1-norm SVM [13]. However it is not differentiable and so our algorithm does not apply to l1-penalized problems directly. To understand how we can generalize Algorithm 1 to this situation, we need to consider the Karush-Kuhn-Tucker (KKT) conditions for optimality of the optimization problem implied by (1). It is easy to verify that the normal equations (2) can be replaced by the following KKT-based condition for l1-norm penalty: |∇C(ˆβ(λ))j| < λ ⇒ˆβ(λ)j = 0 (3) ˆβ(λ)j ̸= 0 ⇒|∇C(ˆβ(λ))j| = λ (4) these conditions hold for any differentiable loss and tell us that at each point on the path we have a set A of non-0 coefficients which corresponds to the variables whose current “generalized correlation” |∇C(ˆβ(λ))j| is maximal and equal to λ. All variables with smaller generalized correlation have 0 coefficient at the optimal penalized solution for this λ. Note that the l1-norm penalty is twice differentiable everywhere except at 0. So if we carefully manage the set of non-0 coefficients according to these KKT conditions, we can still apply our algorithm in the lower-dimensional subspace spanned by non-0 coefficients only. Thus we get Algorithm 2, which employs the Newton approach of Algorithm 1 for twice differentiable penalty, limited to the sub-space of “active” coefficients denoted by A. It adds to Algorithm 1 updates for the “add variable to active set” and “remove variable from active set” events, when a variable becomes “highly correlated” as defined in (4) and when a coefficient hits 0 , respectively. 2 Algorithm 2 Approximate incremental quadratic algorithm for regularized optimization with lasso penalty 1. Set β(ϵ) 0 = ˆβ(λ0), set t = 0, set A = {j : ˆβ(λ0)j ̸= 0}. 2. While (λt < λmax) (a) λt+1 = λt + ϵ (b) β(ϵ) t+1 = β(ϵ) t − h ∇2C(β(ϵ) t )A i−1 · h ∇C(β(ϵ) t )A + λt+1sgn(β(ϵ) t )A i (c) A = A ∪{j /∈A : ∇C(β(ϵ) t+1)j > λt+1} (d) A = A −{j ∈A : |β(ϵ) t+1,j| < δ} (e) t = t + 1 2.2 Computational considerations For a fixed λ0 and λmax, Algorithms 1 and 2 take O(1/ϵ) steps. At each iteration they need to calculate the Hessians of both the loss and the penalty at a typical computational cost of O(n · p2); invert the resulting p × p matrix at a cost of O(p3); and perform the gradient calculation and multiplication, which are o(n · p2) and so do not affect the complexity calculation. Since we implicitly assume throughout that n ≥p, we get overall complexity of O(n · p2/ϵ). The choice of ϵ represents a tradeoff between computational complexity and accuracy (in section 4 we present theoretical results on the relationship between ϵ and the accuracy of the path approximation we get). In practice, our algorithm is practical for problems with up to several hundred predictors and several thousand observations. See the example in section 3. It is interesting to compare this calculation to the obvious alternative, which is to solve O(1/ϵ) regularized problems (1) separately, using a Newton-Raphson approach, resulting in the same complexity (assuming the number of Newton-Raphson iterations for finding each solution is bounded). There are several reasons why our approach is preferable: • The number of iterations until convergence of Newton-Raphson may be large even if it does converge. Our algorithm guarantees we stay very close to the optimal solution path with a single Newton step at each new value of λ. • Empirically we observe that in some cases our algorithm is able to follow the path while direct solution for some values of λ fails to converge. We assume this is related to various numeric properties of the specific problems being solved. • For the interesting case of l1-norm penalty and a “curved” loss function (like logistic log-likelihood), there is no direct Newton-Raphson algorithm. Re-formulating the problem into differentiable form requires doubling the dimensionality. Using our Algorithm 2, we can still utilize the same Newton method, with significant computational savings when many coefficients are 0 and we work in a lowerdimensional subspace. 2When a coefficient hits 0 it not only hits a non-differentiability point in the penalty, it also ceases to be maximally correlated as defined in (4). A detailed proof of this fact and the rest of the “accounting” approach can be found in [9] On the flip side, our results in section 4 below indicate that to guarantee successful tracking we require ϵ to be small, meaning the number of steps we do in the algorithm may be significantly larger than the number of distinct problems we would typically solve to select λ using a non-path approach. 2.3 Connection to path following methods from numerical analysis There is extensive literature on path-following methods for solution paths of general parametric problems. A good survey is given in [1]. In this context, our method can be described as a “predictor-corrector” method with a redundant first order predictor step. That is, the corrector step starts from the previous approximate solution. These methods are recognized as attractive options when the functions defining the path (in our case, the combination of loss and penalty) are “smooth” and “far from linear”. These conditions for efficacy of our approach are reflected in the regularity conditions for the closeness result in Section 4. 3 Example: l2- and l1-penalized logistic regression Regularized logistic regression has been successfully used as a classification and probability estimation approach [11, 12]. We first illustrate applying our quadratic method to this regularized problem using a small subset of the “spam” data-set, available from the UCI repository (http://www.ics.uci.edu/˜mlearn/MLRepository.html) which allows us to present some detailed diagnostics. Next, we apply it to the full “spam” data-set, to demonstrate its time complexity on bigger problems. We first choose five variables and 300 observations and track the solution paths to two regularized logistic regression problems with the l2-norm and the l1-norm penalties: ˆβ(λ) = arg min β log(1 + exp{−yiβ′xi}) + λ∥β∥2 2 (5) ˆβ(λ) = arg min β log(1 + exp{−yiβ′xi}) + λ∥β∥1 (6) Figure 1 shows the solution paths β(ϵ)(t) generated by running Algorithms 1 and 2 on this data using ϵ = 0.02 and starting at λ = 0, i.e. from the non-regularized logistic regression solution. The interesting graphs for our purpose are the ones on the right. They represent the “optimality gap”: et = ∇C(β(ϵ) t ) ∇J(β(ϵ) t ) + ϵ · t where the division is done componentwise (and so the five curves in each plot correspond to the five variables we are using). Note that the optimal solution ˆβ(tϵ) is uniquely defined by the fact that (2) holds and therefore the “optimality gap” is equal to zero componentwise at ˆβ(tϵ). By convexity and regularity of the loss and the penalty, there is a correspondence between small values of e and small distance ∥β(ϵ)(t)−ˆβ(tϵ)∥. In our example we observe that the components of e seem to be bounded in a small region around 0 for both paths (note the small scale of the y axis in both plots — the maximal error is under 10−3). We conclude that on this simple example our method tracks the optimal solution paths well, both for the l1- and l2-regularized problems. The plots on the left show the actual coefficient paths — the curve in R5 is shown as five coefficient traces in R, each corresponding to one variable, with the non-regularized solution (identical for both problems) on the extreme left. Next, we run our algorithm on the full “spam” data-set, containing p = 57 predictors and n = 4601 observations. For both the l1- and l2-penalized paths we used 0 10 20 30 40 −0.5 0 0.5 1 1.5 2 2.5 λ βε(λ/ε) 0 10 20 30 40 −2 0 2 4 x 10 −4 λ ∇ C / ∇ J + λ 0 10 20 30 40 −0.5 0 0.5 1 1.5 2 2.5 λ βε(λ/ε) 0 10 20 30 40 −2 0 2 4 x 10 −4 λ ∇ C / ∇ J + λ Figure 1: Solution paths (left) and optimality criterion (right) for l1 penalized logistic regression (top) and l2 penalized logistic regression (bottom). These result from running Algorithms 2 and 1, respectively, using ϵ = 0.02 and starting from the non-regularized logistic regression solution (i.e. λ = 0) λ0 = 0, λmax = 50, ϵ = 0.02, and the whole path was generated in under 5 minutes using a Matlab implementation on an IBM T-30 Laptop. Like in the small scale example, the “optimality criterion” was uniformly small throughout the two paths, with none of its 57 components exceeding 10−3 at any point. 4 Theoretical closeness result In this section we prove that our algorithm can track the path of true solutions to (1). We show that under regularity conditions on the loss and penalty (which hold for all the candidates we have examined), if we run Algorithm 1 with a specific step size ϵ, then we remain within O(ϵ2) of the true path of optimal regularized solutions. Theorem 1 Assume λ0 > 0, then for ϵ small enough and under regularity conditions on the derivatives of C and J, ∀0 < c < λmax −λ0 , ∥β(ϵ)(c/ϵ) −ˆβ(λ0 + c)∥= O(ϵ2) So there is a uniform bound O(ϵ2) on the error which does not depend on c. Proof We give the details of the proof in Appendix A of [7]. Here we give a brief review of the main steps. Similar to section 3 we define the “optimality gap”: ¯¯¯¯¯(∇C(β(ϵ) t ) ∇J(β(ϵ) t ) )j + λt ¯¯¯¯¯ = etj (7) Also define a “regularity constant” M, which depends on λ0 and the first, second and third derivatives of the loss and penalty. The proof is presented as a succession of lemmas: Lemma 2 Let u1 = M · p · ϵ2, ut = M(ut−1 + √p · ϵ)2, then: ∥et∥2 ≤ut This lemma gives a recursive expression bounding the error in the optimality gap (7) as the algorithm proceeds. The proof is based on separate Taylor expansions of the numerator and denominator of the ratio ∇C ∇J in the optimality gap and some tedious algebra. Lemma 3 If √pϵM ≤1/4 then ut ↗ 1 2M −√p · ϵ − √ 1−4√p·ϵM 2M = O(ϵ2) , ∀t This lemma shows that the recursive bound translates to a uniform O(ϵ2) bound, if ϵ is small enough. The proof consists of analytically finding the fixed point of the increasing series ut. Lemma 4 Under regularity conditions on the penalty and loss functions in the neighborhood of the solutions to (1), the O(ϵ2) uniform bound of lemma 3 translates to an O(ϵ2) uniform bound on ∥β(ϵ)(c/ϵ) −ˆβ(λ0 + c)∥ Finally, this lemma translates the optimality gap bound to an actual closeness result. This is proven via a Lipschitz argument. 4.1 Required regularity conditions Regularity in the loss and the penalty is required in the definition of the regularity constant M and in the translation of the O(ϵ2) bound on the “optimality gap” into one on the distance from the path in lemma 4. The exact derivation of the regularity conditions is highly technical and lengthy. They require us to bound the norm of third derivative “hyper-matrices” for the loss and the penalty as well as the norms of various functions of the gradients and Hessians of both (the boundedness is required only in the neighborhood of the optimal path where our approximate path can venture, obviously). We also need to have λ0 > 0 and λmax < ∞. Refer to Appendix A of [7] for details. Assuming that λ0 > 0 and λmax < ∞ these conditions hold for every interesting example we have encountered, including: • Ridge regression and the lasso (that is, l2- and l1- regularized squared error loss). • l1- and l2-penalized logistic regression. Also Poisson regression and other exponential family models. • l1- and l2-penalized exponential loss. Note that in our practical examples above we have started from λ0 = 0 and our method still worked well. We observe in figure 1 that the tracking algorithm indeed suffers the biggest inaccuracy for the small values of λ, but manages to “self correct” as λ increases. 5 Extensions We have described our method in the context of linear models for supervised learning. There are several natural extensions and enhancements to consider. Basis expansions and Kernel methods Our approach obviously applies, as is, to models that are linear in basis expansions of the original variables (like wavelets or kernel methods) as long as p < n is preserved. However, the method can easily be applied to high (including infinite) dimensional kernel versions of regularized models where RKHS theory applies. We know that the solution path is fully within the span of the representer functions, that is the columns of the Kernel matrix. With a kernel matrix K with columns k1, ..., kn and the standard l2-norm penalty, the regularized problem becomes: ˆα(λ) = arg min α X i C(yi, α′ki) + λα′Kα so the penalty now also contains the Kernel matrix, but this poses no complications in using Algorithm 1. The only consideration we need to keep in mind is the computational one, as our complexity is O(n3/ϵ). So our method is fully applicable and practical for kernel methods, as long as the number of observations, and the resulting kernel matrix, are not too large (up to several hundreds). Unsupervised learning There is no reason to limit the applicability of this approach to supervised learning. Thus, for example, adaptive density estimation using negative log-likelihood as a loss can be regularized and the solution path be tracked using our algorithm. Computational tricks The computational complexity of our algorithm limits its applicability to large problems. To improve its scalability we primarily need to reduce the effort in the Hessian calculation and inversion. The obvious suggestion here would be to keep the Hessian part of step 2(b) in Algorithm 1 fixed for many iterations and change the gradient part only, then update the Hessian occasionally. The clear disadvantage would be that the “closeness” guarantees would no longer hold. We have not tried this in practice but believe it is worth pursuing. Acknowledgements. The author thanks Noureddine El Karoui for help with the proof and Jerome Friedman, Giles Hooker, Trevor Hastie and Ji Zhu for helpful discussions. References [1] Allgower, E. L. & Georg, K. (1993). Continuation and path following. Acta Numer., 2:164 [2] Donoho, D., Johnstone, I., Kerkyachairan, G. & Picard, D. (1995). Wavelet shrinkage: Asymptopia? Annals of Statistics [3] Efron, B., Hastie, T., Johnstone, I. & Tibshirani, R.(2004). Least Angle Regression. Annals of Statistics . [4] Hastie, T., Rosset, S., Tibshirani, R. & Zhu, J. (2004). The Entire Regularization Path for the Support Vector Machine. Journal of Machine Learning Research, 5(Oct):1391–1415. [5] Hastie, T., Tibshirani, R. & Friedman, J. (2001). Elements of Stat. Learning. Springer-Verlag [6] Kim, Y & Kim, J. (2004) Gradient LASSO for feature selection. ICML-04, to appear. [7] Rosset, S. (2003). Topics in Regularization and Boosting. PhD thesis, dept. of Statistics, Stanford University. http://www-stat.stanford.edu/˜saharon/papers/PhDThesis.pdf [8] Rosset, S., Zhu, J. & Hastie,T. (2003). Boosting as a regularized path to a maximum margin classifier. Journal of Machine Learning Research, 5(Aug):941-973. [9] Rosset, S. & Zhu, J. (2003). Piecewise linear regularized solution paths. Submitted. [10] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. JRSSB [11] Wahba, G., Gu, C., Wang, Y. & Chappell, R. (1995) Soft Classification, a.k.a. Risk Estimation, via Penalized Log Likelihood and Smoothing Spline Analysis of Variance. In D.H. Wolpert, editor, The Mathematics of Generalization. [12] Zhu, J. & Hastie, T. (2003). Classification of Gene Microarrays by Penalized Logistic Regression. Biostatistics, to appear. [13] Zhu, J., Hastie, T., Rosset, S. & Tibshirani, R. (2004). 1-norm support vector machines. Neural Information Processing Systems, 16.
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Learning, Regularization and Ill-Posed Inverse Problems Lorenzo Rosasco DISI, Universit`a di Genova Genova, I rosasco@disi.unige.it Andrea Caponnetto DISI, Universit`a di Genova Genova, I caponnetto@disi.unige.it Ernesto De Vito Dipartimento di Matematica Universit`a di Modena and INFN, Sezione di Genova Genova, I devito@unimo.it Umberto De Giovannini DISI, Universit`a di Genova Genova, I umberto.degiovannini@fastwebnet.it Francesca Odone DISI, Universit`a di Genova Genova, I odone@disi.unige.it 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 Rn, 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 ρ(x, y) = ν(x)ρ(y|x) on X × Y . The distribution ρ is known only through a sample z = (x, y) = ((x1, y1), . . . , (xℓ, yℓ)), called training set, drawn i.i.d. according to ρ. 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 ρ is measured by its expected risk defined as I[f] = Z X×Y (f(x) −y)2 dρ(x, y). The regression function g(x) = Z Y y dρ(y|x), 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 λ > 0, an estimator fz λ is defined as the solution of the regularized least squares problem, min f∈H{1 ℓ ℓ X i=1 (f(xi) −yi)2 + λ ∥f∥2 H}. (1) The regularization parameter λ has to be chosen depending on the available data, λ = λ(ℓ, z), in such a way that, for every ϵ > 0 lim ℓ→+∞P I[fz λ(ℓ,z)] −inf f∈H I[f] ≥ϵ = 0. (2) We note that in general inff∈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 = gδ (3) where gδ, g ∈K and ∥g −gδ∥K ≤δ. Here g represents the exact, unknown data and gδ the available, noisy data. Finding the function f satisfying the above equation, given A and gδ, 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 min f∈H ∥Af −g∥2 K . (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 min f∈H{∥Af −gδ∥2 K + λ ∥f∥2 H}, (5) whose unique minimizer is given by f λ δ = (A∗A + λI)−1A∗gδ, (6) where A∗denotes the adjoint of A. A crucial step in the above algorithm is the choice of the regularization parameter λ = λ(δ, gδ), as a function of the noise level δ and the data gδ, in such a way that lim δ→0
f λ(δ,gδ) δ −f †
H = 0, (7) that is, the regularized solution f λ(δ,gδ) δ converges to the generalized solution f † = A†g (f † exists if and only if Pg ∈Range(A), where P is the projection on the closure of the range of A and, in that case, Af † = Pg) when the noise δ 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 δ. 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 ν is the marginal distribution of ρ on X, we define the bounded linear operator A : H →L2(X, ν) as (Af)(x) = ⟨f, Kx⟩H = f(x), 1In 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, ν). In particular, for all f ∈H, the expected risk becomes, I[f] = ∥Af −g∥2 L2(X,ν) + 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 →Eℓof A, that is (Axf)i = ⟨f, Kxi⟩H = f(xi), where Eℓ= Rℓis the finite dimensional euclidean space endowed with the scalar product ⟨y, y′⟩Eℓ= 1 ℓ ℓ X i=1 yiy′ i. It is straightforward to check that 1 ℓ ℓ X i=1 (f(xi) −yi)2 = ∥Axf −y∥2 Eℓ, so that the estimator fz λ given by the regularized least squares algorithm is the regularized solution of the discrete problem Axf = 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 I[fz λ] −inf f∈H I[f] = ∥Af −g∥2 L2(X,ν) −inf f∈H ∥Af −g∥2 L2(X,ν) . (10) If P is the projection on the closure of the range of A, the definition of P gives I[fz λ] −inf f∈H I[f] =
Afz λ −Pg
2 L2(X,ν) (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 λ. Hence, consistency is controlled by the residue of the estimator, instead of the reconstruction error
fz λ −f †
H (as in inverse problems). In particular, consistency 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 Ax and A is required. Moreover, the noisy data y ∈Eℓand the exact data g ∈L2(X, ν) 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. 2The fact that fH is the minimal norm solution of (4) is ensured by the assumption that the support of the measure ν 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 f λ z = (A∗ xAx + λI)−1A∗ xy, f λ = (A∗A + λI)−1A∗g. The above equations show that f λ z and f λ depend only on A∗ xAx and A∗A which are operators from H into H and on A∗ xy and A∗g which are elements of H, so that the space Eℓdisappears. This observation suggests that noise levels could be ∥A∗ xAx −A∗A∥L(H) and ∥A∗ xy −A∗g∥H, where ∥·∥L(H) is the uniform operator norm. To this purpose, for every δ = (δ1, δ2) ∈R2 + we define the collection of training sets. Uδ = {z ∈(X × Y )ℓ| ∥A∗ xy −A∗g∥H ≤δ1, ∥A∗ xAx −A∗A∥L(H) ≤δ2, ℓ∈N} and we let M = sup{|y| | y ∈Y }. The next theorem is the central result of the paper. Theorem 1 If λ > 0, the following inequalities hold 1. for any training set z ∈Uδ
Af λ z −Pg
L2(X,ν) −
Af λ −Pg
L2(X,ν) ≤Mδ2 4λ + δ1 2 √ λ 2. if Pg ∈Range(A), for any training set z ∈Uδ,
f λ z −f †
H −
f λ −f †
H ≤Mδ2 2λ 3 2 + δ1 λ Moreover if we choose λ = λ(δ, z) in such a way that limδ→0 supz∈Uδ λ(δ, z) = 0 limδ→0 supz∈Uδ δ1 2 λ(δ,z) = 0 limδ→0 supz∈Uδ δ2 λ(δ,z) = 0 (12) then lim δ→0 sup z∈Uδ
Af λ(δ,z) z −Pg
L2(X,ν) = 0. (13) We omit the complete proof and refer to [3]. Briefly, the idea is to note that
Af λ z −Pg
L2(X,ν) −
Af λ −Pg
L2(X,ν) ≤
Af λ z −Af λ
L2(X,ν) =
(A∗A) 1 2 (f λ z −f λ)
H where the last equation follows by polar decomposition of the operator A. Moreover a simple algebraic computation gives f λ z −f λ = (A∗A+λI)−1(A∗A−A∗ xAx)(A∗ xAx+λI)−1A∗ xy+(A∗A+λI)−1(A∗ xy−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 δ = (δ1, δ2). 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 λ to ensure convergence of the residue to zero as the level noise decreases. The above results were obtained introducing the collection Uδ of training sets compatible with a certain noise level δ. It is left to quantify the noise level corresponding to a training set of cardinality ℓ. 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 δ = (δ1, δ2). Theorem 2 Let ϵ1, ϵ2 > 0 and κ = supx∈X p K(x, x), then P ∥A∗g −Ax ∗y∥H ≤Mκ √ ℓ + ϵ1, ∥A∗A −Ax ∗Ax∥L(H) ≤κ2 √ ℓ + ϵ2 ≥1 −e− ϵ2 1ℓ 2κ2M2 −e− ϵ2 2ℓ 2κ4 (14) The proof is given in [3] and it is based on McDiarmid inequality [10] applied to the random variables F(z) = ∥Ax ∗y −A∗g∥H G(z) = ∥Ax ∗Ax −A∗A∥L(H). Other estimates of the noise δ 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 δ. 5.3 Consistency and Regularization Parameter Choice Combining Theorems 1 and 2, we easily derive the following corollary. Corollary 1 Given 0 < η < 1, with probability greater that 1 −η,
Afz λ −Pg
L2(X,ν) −
Af λ −Pg
L2(X,ν) ≤κM 2 √ ℓ 1 √ λ + κ 2λ 1 + log r4 η (15) for all λ > 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 I[fz λ] ≤ κL 2 √ ℓ 1 √ λ + κ 2λ 1 + log r4 η | {z } sample error +
Af λ −Pg
L2(X,ν) | {z } approximation error 2 + inf f∈H I[f] | {z } irreducible error . In the above inequality the first term plays the role of sample error. If we choose the regularization parameter so that λ = λ(ℓ, z) = O( 1 ℓb ), with 0 < b < 1 2 the sample error converges in probability to zero with order O q 1 ℓ1−2b when ℓ→∞. On the other hand the second term represents the approximation error and it is possible to show, using standard results from spectral theory, that it vanishes as λ 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 λ 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 ρ (see for example [2]). It can be shown that if the explicit dependence of the approximation error on λ is not available we cannot determine an optimal a priori (data independent) dependency λ = λ(ℓ) for the regularization parameter. Nevertheless a posteriori (data dependent) choices λ = λ(ℓ, 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 ℓand η, but does not depend λ, 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 λ and by the noise levels ∥A∗ xAx −A∗A∥L(H) and ∥A∗ xy −A∗g∥H, whereas Theorem 2, which is of probabilistic nature, relates the noise levels to the number of data ℓ and the confidence level η. 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 (LKf)(x) = Z X K(x, s)f(s)dν(s) and O is the operator defined by (Of)(x) = Z Y V ′(y, f(x))dρ(y|x). 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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Density Level Detection is Classification Ingo Steinwart, Don Hush and Clint Scovel Modeling, Algorithms and Informatics Group, CCS-3 Los Alamos National Laboratory {ingo,dhush,jcs}@lanl.gov Abstract We show that anomaly detection can be interpreted as a binary classification problem. Using this interpretation we propose a support vector machine (SVM) for anomaly detection. We then present some theoretical results which include consistency and learning rates. Finally, we experimentally compare our SVM with the standard one-class SVM. 1 Introduction One of the most common ways to define anomalies is by saying that anomalies are not concentrated (see e.g. [1, 2]). To make this precise let Q be our unknown data-generating distribution on the input space X. Furthermore, to describe the concentration of Q we need a known reference distribution µ on X. Let us assume that Q has a density h with respect to µ. Then, the sets {h > ρ}, ρ > 0, describe the concentration of Q. Consequently, to define anomalies in terms of the concentration we only have to fix a threshold level ρ > 0, so that an x ∈X is considered to be anomalous whenever x ∈{h ≤ρ}. Therefore our goal is to find the density level set {h ≤ρ}, or equivalently, the ρ-level set {h > ρ}. Note that there is also a modification of this problem where µ is not known but can be sampled from. We will see that our proposed method can handle both problems. Finding density level sets is an old problem in statistics which also has some interesting applications (see e.g. [3, 4, 5, 6]) other than anomaly detection. Furthermore, a mathematical framework similar to classical PAC-learning has been proposed in [7]. Despite this effort, no efficient algorithm is known, which is a) consistent, i.e. it always finds the level set of interest asymptotically, and b) learns with fast rates under realistic assumptions on h and µ. In this work we propose such an algorithm which is based on an SVM approach. Let us now introduce some mathematical notions. We begin with emphasizing that—as in many other papers (see e.g. [5] and [6])—we always assume µ({h = ρ}) = 0. Now, let T = (x1, . . . , xn) ∈Xn be a training set which is i.i.d. according to Q. Then, a density level detection algorithm constructs a function fT : X →R such that the set {fT > 0} is an estimate of the ρ-level set {h > ρ} of interest. Since in general {fT > 0} does not exactly coincide with {h > ρ} we need a performance measure which describes how well {fT > 0} approximates the set {h > ρ}. Probably the best known performance measure (see e.g. [6, 7] and the references therein) for measurable functions f : X →R is Sµ,h,ρ(f) := µ {f > 0} △{h > ρ} , where △denotes the symmetric difference. Obviously, the smaller Sµ,h,ρ(f) is, the more {f > 0} coincides with the ρ-level set of h, and a function f minimizes Sµ,h,ρ if and only if {f > 0} is µ-almost surely identical to {h > ρ}. Furthermore, for a sequence of functions fn : X →R with Sµ,h,ρ(fn) →0 we easily see that sign fn →1{h>ρ} both µ-almost and Q-almost surely if 1A denotes the indicator function of a set A. Finally, it is important to note, that the performance measure Sµ,h,ρ is somehow natural in that it is insensitive to µ-zero sets. 2 Detecting density levels is a classification problem In this section we show how the density level detection (DLD) problem can be formulated as a binary classification problem. To this end we write Y := {−1, 1} and define: Definition 2.1 Let µ and Q be probability measures on X and s ∈(0, 1). Then the probability measure Q ⊖s µ on X × Y is defined by Q ⊖s µ (A) := sEx∼Q1A(x, 1) + (1 −s)Ex∼µ1A(x, −1) for all measurable A ⊂X × Y . Here we used the shorthand 1A(x, y) := 1A((x, y)). Obviously, the measure P := Q⊖sµ can be associated with a binary classification problem in which positive samples are drawn from Q and negative samples are drawn from µ. Inspired by this interpretation let us recall that the binary classification risk for a measurable function f : X →R and a distribution P on X × Y is defined by RP (f) = P {(x, y) : sign f(x) ̸= y} , where we define sign t := 1 if t > 0 and sign t = −1 otherwise. Furthermore, we denote the Bayes risk of P by RP := inf{RP (f) f : X →R measurable}. We will show that learning with respect to Sµ,h,ρ is equivalent to learning with respect to RP (.). To this end we begin with the following easy to prove but fundamental proposition: Proposition 2.2 Let µ and Q be probability measures on X such that Q has a density h with respect to µ, and let s ∈(0, 1). Then the marginal distribution of P := Q ⊖s µ on X is PX = sQ + (1 −s)µ. Furthermore, we PX-a.s. have P(y = 1|x) = sh(x) sh(x) + 1 −s . Note that the above formula for PX implies that the µ-zero sets of X are exactly the PXzero sets of X. Furthermore, Proposition 2.2 shows that every distribution P := Q ⊖s µ with dQ := hdµ and s ∈(0, 1) determines a triple (µ, h, ρ) with ρ := (1 −s)/s and vice-versa. In the following we therefore use the shorthand SP (f) := Sµ,h,ρ(f). Let us now compare RP (.) with SP (.). To this end we first observe that h(x) > ρ = 1−s s is equivalent to sh(x) sh(x)+1−s > 1 2. By Proposition 2.2 the latter is µ-almost surely equivalent to η(x) := P(y = 1|x) > 1/2 and hence µ({η > 1/2} △{h > ρ}) = 0. Now recall, that binary classification aims to discriminate {η > 1/2} from {η < 1/2}. Thus it is no surprise that RP (.) can serve as a performance measure as the following theorem shows: Theorem 2.3 Let µ and Q be distributions on X such that Q has a density h with respect to µ. Let ρ > 0 satisfy µ({h = ρ}) = 0. We write s := 1 1+ρ and define P := Q⊖sµ. Then for all sequences (fn) of measurable functions fn : X →R the following are equivalent: i) SP (fn) →0. ii) RP (fn) →RP . In particular, for measurable f : X →R we have SP (f) = 0 if and only if RP (f) = RP . Proof: For n ∈N we define En := {fn > 0} △{h > ρ}. Since µ({h > ρ} △{η > 1 2}) = 0 it is easy to see that the classification risk of fn can be computed by RP (fn) = RP + Z En |2η −1|dPX . (1) Now, {|2η−1| = 0} is a µ-zero set and hence a PX-zero set. This implies that the measures |2η −1|dPX and PX are absolutely continuous with respect to each other. Furthermore, we have already observed after Proposition 2.2 that PX and µ are absolutely continuous with respect to each other. Now, the assertion follows from SP (fn) = µ(En). Theorem 2.3 shows that instead of using SP (.) as a performance measure for the DLD problem one can alternatively use the classification risk RP (.). Therefore, we will establish some basic properties of this performance measure in the following. To this end we write I(y, t) := 1(−∞,0](yt), y ∈Y and t ∈R, for the standard classification loss function. With this notation we can easily compute RP (f): Proposition 2.4 Let µ and Q be probability measures on X. For ρ > 0 we write s := 1 1+ρ and define P := Q ⊖s µ. Then for all measurable f : X →R we have RP (f) = 1 1 + ρEQI(1, sign f) + ρ 1 + ρEµI(−1, sign f) . It is interesting that the classification risk RP (.) is strongly connected with another approach for the DLD problem which is based on the so-called excess mass (see e.g. [4], [5], [6], and the references therein). To be more precise let us first recall that the excess mass of a measurable function f : X →R is defined by EP (f) := Q({f > 0}) −ρµ({f > 0}) , where Q, ρ and µ have the usual meaning. The following proposition, that shows that RP (.) and EP (.) are essentially the same, can be easily checked: Proposition 2.5 Let µ and Q be probability measures on X. For ρ > 0 we write s := 1 1+ρ and define P := Q ⊖s µ. Then for all measurable f : X →R we have EP (f) = 1 −(1 + ρ)RP (f) . If Q is an empirical measure based on a training set T in the definition of EP (.) we obtain the empirical excess mass which we denote by ET (.). Then given a function class F the (empirical) excess mass approach chooses a function fT ∈F which maximizes ET (.) within F. Since the above proposition shows ET (f) = 1 −1 n n X i=1 I(1, sign f(xi)) −ρEµI(−1, sign f) . we see that this approach is actually a type of empirical risk minimization (ERM). In the above mentioned papers the analysis of the excess mass approach needs an additional assumption on the behaviour of h around the level ρ. Since this condition can be used to establish a quantified version of Theorem 2.3 we will recall it now. Definition 2.6 Let µ be a distribution on X and h : X →[0, ∞) be a measurable function with R hdµ = 1, i.e. h is a density with respect to µ. For ρ > 0 and 0 ≤q ≤∞we say that h is of ρ-exponent q if there exists a constant C > 0 such that for all sufficiently small t > 0 we have µ {|h −ρ| ≤t} ≤Ctq . (2) Condition (2) was first considered in [5, Thm. 3.6.]. This paper also provides an example of a class of densities on Rd, d ≥2, which has exponent q = 1. Later, Tsybakov [6, p. 956] used (2) for the analysis of a DLD method which is based on a localized version of the empirical excess mass approach. Surprisingly, (2) is satisfied if and only if P := Q ⊖s µ has Tsybakov exponent q in the sense of [8], i.e. PX |2η −1| ≤t ≤C · tq (3) for some constant C > 0 and all sufficiently small t > 0 (see the proof of Theorem 2.7 for (2) ⇒(3) and [9] for the other direction). Recall that recently (3) has played a crucial role for establishing learning rates faster than n−1 2 for ERM algorithms and SVM’s (see e.g. [10] and [8]). Remarkably, it was already observed in [11], that the classification problem can be analyzed by methods originally developed for the DLD problem. However, to our best knowledge the exact relation between the DLD problem and binary classification has not been presented, yet. In particular, it has not been observed yet, that this relation opens a non-heuristic way to use classification methods for the DLD problem as we will demonstrate by example in the next section. Let us now use the ρ-exponent to establish inequalities between SP (.) and RP (.): Theorem 2.7 Let ρ > 0 and µ and Q be probability measures on X such that Q has a density h with respect to µ. For s := 1 1+ρ we write P := Q ⊖s µ. Then we have i) If h is bounded there is a c > 0 such that for all measurable f : X →R we have RP (f) −RP ≤c SP (f) . ii) If h has ρ-exponent q there is a c > 0 such that for all measurable f : X →R we have SP (f) ≤c RP (f) −RP q 1+q . Sketch of the proof: The first assertion directly follows from (1) and Proposition 2.2. For the second assertion we first show (2) ⇒(3). To this end we observe that for 0 < t < 1 2 we have Q |h −ρ| ≤t ≤(1 + ρ)µ |h −ρ| ≤t . Thus there exists a ˜C > 0 such that PX {|h −ρ| ≤t} ≤˜Ctq for all 0 < t < 1 2. Furthermore, |2η −1| = h−ρ h+ρ implies |2η −1| ≤t = n1 −t 1 + tρ ≤h ≤1 + t 1 −tρ o , whenever 0 < t < 1 2. Let us now define tl := 2t 1+t and tr := 2t 1−t. This gives 1 −tl = 1−t 1+t and 1 + tr = 1+t 1−t. Furthermore, we obviously also have tl ≤tr. Therefore we find n1 −t 1 + tρ ≤h ≤1 + t 1 −tρ o ⊂ |h −ρ| ≤trρ , which shows (3). Now the assertion follows from [10, Prop. 1]. 3 A support vector machine for density level detection One of the benefits of interpreting the DLD problem as a classification problem is that we can construct an SVM for the DLD problem. To this end let k : X × X →R be a positive definite kernel with reproducing kernel Hilbert space (RKHS) H. Furthermore, let µ be a known probability measure on X and l : Y × R →[0, ∞) be the hinge loss function, i.e. l(y, t) := max{0, 1 −yt}, y ∈Y , t ∈R. Then for a training set T = (x1, . . . , xn) ∈ Xn, a regularization parameter λ > 0, and ρ > 0 our SVM for the DLD problem chooses a pair (fT,µ,λ, bT,µ,λ) ∈H × R which minimizes λ∥f∥2 H + 1 (1 + ρ)n n X i=1 l(1, f(xi) + b) + ρ 1 + ρEx∼µl(−1, f(x) + b) (4) in H × R. The corresponding decision function of this SVM is fT,µ,λ + bT,µ,λ : X →R. Although the measure µ is known, almost always the expectation Ex∼µl(−1, f(x)) can be only numerically approximated by using finitely many function evaluations of f. Unfortunately, since the hinge loss is not differentiable we do not know a deterministic method to choose these function evaluations efficiently. Therefore in the following we will use points T ′ := (z1, . . . , zm) which are randomly sampled from µ in order to approximate Ex∼µl(−1, f(x)). We denote the corresponding approximate solution of (4) by (fT,T ′,λ, bT,T ′,λ). Since this modification of (4) is identical to the standard SVM formulation besides the weighting factors in front of the empirical l-risk terms we do not discuss algorithmic issues. However note that this approach simultaneously addresses the original “µ is known” and the modified “µ can be sampled from” problems described in the introduction. Furthermore it is also closely related to some heuristic methods for anomaly detection that are based on artificial samples (see [9] for more information). The fact that the SVM for DLD essentially coincides with the standard L1-SVM also allows us to modify many known results for these algorithms. For simplicity we will only state results for the Gaussian RBF kernel with width 1/σ, i.e. k(x, x′) = exp(−σ2∥x −x′∥2 2), x, x′ ∈Rd, and the case m = n. More general results can be found in [12, 9]. We begin with a consistency result with respect to the performance measure RP (.). Recall that by Theorem 2.3 this is equivalent to consistency with respect to SP (.): Theorem 3.1 Let X ⊂Rd be compact and k be the Gaussian RBF kernel with width 1/σ on X. Furthermore, let ρ > 0, and µ and Q be distributions on X such that Q has a density h with respect to µ. For s := 1 1+ρ we write P := Q ⊖s µ. Then for all positive sequences (λn) with λn →0 and nλ1+δ n →∞for some δ > 0, and for all ε > 0 we have lim n→∞(Q ⊗µ)n (T, T ′) ∈(X × X)n : RP (fT,T ′,λ + bT,T ′,λ) > RP + ε = 0 . Sketch of the proof: Let us introduce the shorthand ν = Q ⊗µ for the product measure of Q and µ. Moreover, for a measurable function f : X →R we define the function g ◦f : X × X →R by g ◦f(x, x′) := 1 1 + ρl(1, f(x)) + ρ 1 + ρl(−1, f(x′)) , x, x′ ∈X. Furthermore, we write l ◦f(x, y) := l(y, f(x)), x ∈X, y ∈Y . Then it is easy to check that we always have Eνg ◦f = EP l ◦f. Analogously, we see ET ⊗T ′g ◦f = ET ⊖sT ′l ◦f if T ⊗T ′ denotes the product measure of the empirical measures based on T and T ′. Now, using Hoeffding’s inequality for ν it is easy to establish a concentration inequality in the sense of [13, Lem. III.5]. The rest of the proof is analogous to the steps in [13] since these steps are independent of the specific structure of the data-generating measure. In general, we cannot obtain convergence rates in the above theorem without assuming specific conditions on h, ρ, and µ. We will now present such a condition which can be used to establish rates. To this end we write τx := d(x, {h > ρ}) if x ∈{h < ρ} d(x, {h < ρ}) if x ∈{h ≥ρ} , where d(x, A) denotes the Euclidian distance between x and a set A. Now we define: Definition 3.2 Let µ be a distribution on X ⊂Rd and h : X →[0, ∞) be a measurable function with R hdµ = 1, i.e. h is a density with respect to µ. For ρ > 0 and 0 < α ≤∞ we say that h has geometric ρ-exponent α if Z X τ −αd x |h −ρ|dµ < ∞. Since {h > ρ} and {h ≤ρ} are the classes which have to be discriminated when interpreting the DLD problem as a classification problem it is easy to check by Proposition 2.2 that h has geometric ρ-exponent α if and only if for P := Q ⊖s µ we have (x 7→τ −1 x ) ∈Lαd(|2η −1|dPX). The latter is a sufficient condition for P to have geometric noise exponent α in the sense of [8]. We can now state our result on learning rates which is proved in [12]. Theorem 3.3 Let X be the closed unit ball of the Euclidian space Rd, and µ and Q be distributions on X such that dQ = hdµ. For fixed ρ > 0 assume that the density h has both ρ-exponent 0 < q ≤∞and geometric ρ-exponent 0 < α < ∞. We define λn := ( n−α+1 2α+1 if α ≤q+2 2q n− 2(α+1)(q+1) 2α(q+2)+3q+4 otherwise , and σn := λ − 1 (α+1)d n in both cases. For s := 1 1+ρ we write P := Q ⊖s µ. Then for all ε > 0 there exists a constant C > 0 such that for all x ≥1 and all n ≥1 the SVM using λn and Gaussian RBF kernel with width 1/σn satisfies (Q ⊗µ)n (T, T ′) ∈(X × X)n : RP (fT,T ′,λ + bT,T ′,λ) > RP + Cx2n− α 2α+1 +ε ≤e−x if α ≤q+2 2q and (Q⊗µ)n (T, T ′) ∈X2n : RP (fT,T ′,λ+bT,T ′,λ) > RP +Cx2n− 2α(q+1) 2α(q+2)+3q+4 +ε ≤e−x otherwise. If α = ∞the latter holds if σn = σ is a constant with σ > 2 √ d. Remark 3.4 With the help of Theorem 2.7 we immediately obtain rates with respect to the performance measure SP (.). It turns out that these rates are very similar to those in [5] and [6] for the empirical excess mass approach. 4 Experiments We present experimental results for anomaly detection problems where the set X is a subset of Rd. Two SVM type learning algorithms are used to produce functions f which declare the set {x : f(x) < 0} anomalous. These algorithms are compared based on their risk RP (f). The data in each problem is partitioned into three pairs of sets; the training sets (T, T ′), the validation sets (V, V ′) and the test sets (W, W ′). The sets T, V and W contain samples drawn from Q and the sets T ′,V ′ and W ′ contain samples drawn from µ. The training and validation sets are used to design f and the test sets are used to estimate its performance by computing an empirical version of RP (f) that we denote R(W,W ′)(f). The first learning algorithm is the density level detection support vector machine (DLD– SVM) with Gaussian RBF kernel described in the previous section. With λ and σ2 fixed and the expected value Ex∼µl(−1, f(x) + b) in (4) replaced with an empirical estimate based on T ′ this formulation can be solved using, for example, the C–SVC option in the LIBSVM software [14] by setting C = 1 and setting the class weights to w1 = 1/ |T|(1 + ρ) and w−1 = ρ/ |T ′|(1 + ρ) . The regularization parameters λ and σ2 are chosen to (approximately) minimize the empirical risk R(V,V ′)(f) on the validation sets. This is accomplished by employing a grid search over λ and a combined grid/iterative search over σ2. In particular, for each fixed grid value of λ we seek a minimizer over σ2 by evaluating the validation risk at a coarse grid of σ2 values and then performing a Golden search over the interval defined by the two σ2 values on either side of the coarse grid minimum. As the overall search proceeds the (λ, σ2) pair with the smallest validation risk is retained. The second learning algorithm is the one–class support vector machine (1CLASS–SVM) introduced by Sch¨olkopf et al. [15]. Due to its “one–class” nature this method does not use the set T ′ in the production of f. Again we employ the Gaussian RBF kernel with parameter σ2. The one–class algorithm in Sch¨olkopf et al. contains a parameter ν which controls the size of the set {x ∈T : f(x) ≤0} (and therefore controls the measure Q(f ≤0) through generalization). To make a valid comparison with the DLD–SVM we determine ν automatically as a function of ρ. In particular both ν and σ2 are chosen to (approximately) minimize the validation risk using the search procedure described above for the DLD–SVM where the grid search for λ is replaced by a Golden search (over [0, 1]) for ν. Data for the first experiment are generated as follows. Samples of the random variable x ∼Q are generated by transforming samples of the random variable u that is uniformly distributed over [0, 1]27. The transform is x = Au where A is a 10–by–27 matrix whose rows contain between m = 2 and m = 5 non-zero entries with value 1/m. Thus the support of Q is the hypercube [0, 1]10 and Q is concentrated about its centers. Partial overlap in the nonzero entries across the rows of A guarantee that the components of x are partially correlated. We chose µ to be the uniform distribution over [0, 1]10. Data for the second experiment are identical to the first except that the vector (0, 0, 0, 0, 0, 0, 0, 0, 0, 1) is added to the samples of x with probability 0.5. This gives a bi-modal distribution Q and since the support of the last component is extended to [0, 2] the corresponding component of µ is also extended to this range. The training and validation set sizes are |T| = 1000, |T ′| = 2000, |V | = 500, and |V ′| = 2000. The test set sizes |W| = 100, 000 and |W ′| = 100, 000 are large enough to provide very accurate estimates of risk. The λ grid for the DLD–SVM method consists of 15 values ranging from 10−7 to 1 and the coarse σ2 grid for the DLD–SVM and 1CLASS–SVM methods consists of 9 values that range from 10−3 to 102. The learning algorithms are applied for values of ρ ranging from 10−2 to 102. Figure 1(a) plots the risk R(W,W ′) versus ρ for the two learning algorithms. In both experiments the performance of DLD–SVM is superior to 1CLASS–SVM at smaller values of ρ. The difference in the bi–modal case is substantial. Comparisons for larger sizes of |T| and |V | yield similar results, but at smaller sample sizes the superiority of DLD–SVM is even more pronounced. This is significant because ρ ≫1 appears to have little utility in the general anomaly detection problem since it defines anomalies in regions where the concentration of Q is much larger than the concentration of µ, which is contrary to our premise that anomalies are not concentrated. The third experiment considers a real world application in cybersecurity. The goal is to monitor the network traffic of a computer and determine when it exhibits anomalous behavior. The data for this experiment was collected from an active computer in a normal working environment over the course of 16 months. Twelve features were computed over each 1 hour time frame to give a total of 11664 12–dimensional feature vectors. These features are normalized to the range [0, 1] and treated as samples from Q. We chose µ to be the uniform distribution over [0, 1]12. The training, validation and test set sizes are |T| = 4000, |T ′| = 10000, |V | = 2000, |V ′| = 100, 000, |W| = 5664 and |W ′| = 100, 000. The λ 0 0.02 0.04 0.06 0.08 0.1 0.01 0.1 1 10 100 PSfrag replacements DLD-SVM 1CLASS-SVM DLD-SVM-1 DLD-SVM-2 1CLASS-SVM-1 1CLASS-SVM-2 ρ R(W,W ′) (a) Experiments 1 & 2 0 0.0005 0.001 0.0015 0.002 0.0025 0.01 0.1 1 10 100 PSfrag replacements DLD-SVM 1CLASS-SVM DLD-SVM-1 DLD-SVM-2 1CLASS-SVM-1 1CLASS-SVM-2 ρ R(W,W ′) (b) Cybersecurity Experiment Figure 1: Comparison of DLD–SVM and 1CLASS–SVM. The curves with extension -1 and -2 in Figure 1(a) correspond to experiments 1 and 2 respectively. grid for the DLD–SVM method consists of 7 values ranging from 10−7 to 10−1 and the coarse σ2 grid for the DLD–SVM and 1CLASS–SVM methods consists of 6 values that range from 10−3 to 102. The learning algorithms are applied for values of ρ ranging from 0.05 to 50.0. Figure 1(b) plots the risk R(W,W ′) versus ρ for the two learning algorithms. The performance of DLD–SVM is superior to 1CLASS–SVM at all values of ρ. References [1] B.D. Ripley. Pattern recognition and neural networks. Cambridge Univ. Press, 1996. [2] B. Sch¨olkopf and A.J. Smola. Learning with Kernels. MIT Press, 2002. [3] J.A. Hartigan. Clustering Algorithms. Wiley, New York, 1975. [4] J.A. Hartigan. Estimation of a convex density contour in 2 dimensions. J. Amer. Statist. Assoc., 82:267–270, 1987. [5] W. Polonik. Measuring mass concentrations and estimating density contour clusters—an excess mass aproach. Ann. Stat., 23:855–881, 1995. [6] A.B. Tsybakov. On nonparametric estimation of density level sets. Ann. Statist., 25:948–969, 1997. [7] S. Ben-David and M. Lindenbaum. Learning distributions by their density levels: a paradigm for learning without a teacher. J. Comput. System Sci., 55:171–182, 1997. [8] C. Scovel and I. Steinwart. Fast rates for support vector machines. Ann. Statist., submitted, 2003. http://www.c3.lanl.gov/˜ingo/publications/ann-03.ps. [9] I. Steinwart, D. Hush, and C. Scovel. A classification framework for anomaly detection. Technical report, Los Alamos National Laboratory, 2004. [10] A.B. Tsybakov. Optimal aggregation of classifiers in statistical learning. Ann. Statist., 32:135– 166, 2004. [11] E. Mammen and A. Tsybakov. Smooth discrimination analysis. Ann. Statist., 27:1808–1829, 1999. [12] C. Scovel, D. Hush, and I. Steinwart. Learning rates for support vector machines for density level detection. Technical report, Los Alamos National Laboratory, 2004. [13] I. Steinwart. Consistency of support vector machines and other regularized kernel machines. IEEE Trans. Inform. Theory, to appear, 2005. [14] Chih-Chung Chang and Chih-Jen Lin. LIBSVM: a library for support vector machines, 2004. [15] B. Sch¨olkopf, J.C. Platt, J. Shawe-Taylor, and A.J. Smola. Estimating the support of a highdimensional distribution. Neural Computation, 13:1443–1471, 2001.
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Learning syntactic patterns for automatic hypernym discovery Rion Snow Computer Science Department Stanford University Stanford, CA 94305 rion@cs.stanford.edu Daniel Jurafsky Linguistics Department Stanford University Stanford, CA 94305 jurafsky@stanford.edu Andrew Y. Ng Computer Science Department Stanford University Stanford, CA 94305 ang@cs.stanford.edu 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 NPX”, or “NP X and other NP Y ”, often implies that NP X is a hyponym of NP Y , i.e., that NPX 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,CATEGORY1:RELATION:CATEGORY2, 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 PREP 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]. ... authors such -N:pre:PreDet as -N:mod:Prep Herrick -Prep:pcomp-n:N Shakespeare -Prep:pcomp-n:N and -N:punc:U -N:conj:N Figure 1: MINIPAR dependency tree example with transform NP X and other NP Y : (and,U:PUNC:N),-N:CONJ:N, (other,A:MOD:N) NP X or other NP Y : (or,U:PUNC:N),-N:CONJ:N, (other,A:MOD:N) NP Y such as NP X: N:PCOMP-N:PREP,such as,such as,PREP:MOD:N Such NP Y as NP X: N:PCOMP-N:PREP,as,as,PREP:MOD:N,(such,PREDET:PRE:N) NP Y including NP X: N:OBJ:V,include,include,V:I:C,dummy node,dummy node,C:REL:N NP Y , especially NP X: -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, -PREP: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 1The corpus contains articles from the Associated Press, Wall Street Journal, and Los Angeles Times, drawn from the TIPSTER 1, 2, 3, and TREC 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. 2We access WordNet 2.0 via Jason Rennie’s WordNet::QueryData interface. 3A 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). Figure 2: Hypernym pre/re for all features 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Hypernym Classifiers on WordNet-labeled dev set Recall Precision Logistic Regression (Buckets) Logistic Regression (Binary) Hearst Patterns And/Or Other Pattern 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: N:PCOMP-N:PREP,like,like,PREP:MOD:N NP Y called NP X: N:DESC:V,call,call,V:VREL:N NP X is a NP Y : N:S:VBE,be,be,-VBE:PRED:N NP X, a NP Y (appositive): N:APPO:N Table 2: Dependency path representations of other high-scoring patterns 4Redundant 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. 5We 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): 0.3480 Best Logistic Regression (Binary): 0.3200 Best Multinomial Naive Bayes: 0.3175 Best Complement Naive Bayes: 0.3024 Hearst Patterns: 0.1500 “And/Or Other” Pattern: 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: 0.6405 Distributional Similarity Vector Space Model: 0.3327 Thresholded Conjunction Pattern Classifier: 0.2857 Best WordNet Classifier: 0.2630 Table 4: Summary of maximum F-scores on hand-labeled coordinate pairs 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Coordinate term classifiers on hand-labeled test set Recall Precision Interannotator Agreement Distributional Similarity Conjunct Pattern WordNet Figure 4: Coordinate classifiers on hand-labeled test set 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Recall Precision Hypernym Classifiers on hand-labeled test set Interannotator Agreement TREC+Wikipedia TREC, Hybrid TREC, Hypernym-only WordNet Classifiers Hearst Patterns And/Or Other Pattern 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(ni < H nj), 6We 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: 0.8318 TREC+Wikipedia Hypernym-only Classifier (Logistic Regression): 0.3592 TREC Hybrid Linear Interpolation Hypernym/Coordinate Model: 0.3268 TREC Hypernym-only Classifier (Logistic Regression): 0.2714 Best WordNet Classifier: 0.2339 Hearst Patterns Classifier: 0.1417 “And/Or Other” Pattern Classifier: 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 ∼ C nj), the probability that nouns ni and nj are coordinate terms, i.e., that they share a common hypernym ancestor at some level. Defining the probability produced by our best hypernym-only classifier as Pold(ni < H nj), and a probability obtained by normalizing the similarity score from our coordinate classifier as P(ni ∼ C nj), we apply a simple 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 Pnew(ni < H nk) ∝λ1Pold(ni < H nk) + λ2 P j P(ni ∼ C nj)Pold(nj < H nk) 7 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. 7We constrain our parameters λ1, λ2 such that λ1+λ2 = 1; we set these parameters using 10-fold cross-validation on our hand-labeled test set. For our final evaluation we use λ1 = 0.7. 8We 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. 9There 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 Count Example Pair NE: Person 7 “John F. Kennedy / president”, “Marlin Fitzwater / spokesman” NE: Place 7 “Diamond Bar / city”, “France / place” NE: Company 2 “American Can / company”, “Simmons / company” NE: Other 1 “Is Elvis Alive / book” Not Named Entity: 9 “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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Joint Tracking of Pose, Expression, and Texture using Conditionally Gaussian Filters Tim K. Marks John Hershey Department of Cognitive Science University of California San Diego La Jolla, CA 92093-0515 tkmarks@cogsci.ucsd.edu hershey@microsoft.com J. Cooper Roddey Javier R. Movellan Institute for Neural Computation University of California San Diego La Jolla, CA 92093-0523 cooper@sccn.ucsd.edu movellan@mplab.ucsd.edu Abstract We present a generative model and stochastic filtering algorithm for simultaneous tracking of 3D position and orientation, non-rigid motion, object texture, and background texture using a single camera. We show that the solution to this problem is formally equivalent to stochastic filtering of conditionally Gaussian processes, a problem for which well known approaches exist [3, 8]. We propose an approach based on Monte Carlo sampling of the nonlinear component of the process (object motion) and exact filtering of the object and background textures given the sampled motion. The smoothness of image sequences in time and space is exploited by using Laplace’s method to generate proposal distributions for importance sampling [7]. The resulting inference algorithm encompasses both optic flow and template-based tracking as special cases, and elucidates the conditions under which these methods are optimal. We demonstrate an application of the system to 3D non-rigid face tracking. 1 Background Recent algorithms track morphable objects by solving optic flow equations, subject to the constraint that the tracked points belong to an object whose non-rigid deformations are linear combinations of a set of basic shapes [10, 2, 11]. These algorithms require precise initialization of the object pose and tend to drift out of alignment on long video sequences. We present G-flow, a generative model and stochastic filtering formulation of tracking that address the problems of initialization and error recovery in a principled manner. We define a non-rigid object by the 3D locations of n vertices. The object is a linear combination of k fixed morph bases, with coefficients c = [c1, c2, · · · , ck]T . The fixed 3 × k matrix hi contains the position of the ith vertex in all k morph bases. The transformation from object-centered to image coordinates consists of a rotation, weak perspective projection, and translation. Thus xi, the 2D location of the ith vertex on the image plane, is xi = grhic + l, (1) where r is the 3 × 3 rotation matrix, l is the 2 × 1 translation vector, and g = 1 0 0 0 1 0 is the projection matrix. The object pose, ut, comprises both the rigid motion parameters and the morph parameters at time t: ut = {r(t), l(t), c(t)}. (2) 1.1 Optic flow Let yt represent the current image, and let xi(ut) index the image pixel that is rendered by the ith object vertex when the object assumes pose ut. Suppose that we know ut−1, the pose at time t −1, and we want to find ut, the pose at time t. This problem can be solved by minimizing the following form with respect to ut: ˆut = argmin ut 1 2 n X i=1 [yt(xi(ut)) −yt−1(xi(ut−1))]2 . (3) In the special case in which the xi(ut) are neighboring points that move with the same 2D displacement, this reduces to the standard Lucas-Kanade optic flow algorithm [9, 1]. Recent work [10, 2, 11] has shown that in the general case, this optimization problem can be solved efficiently using the Gauss-Newton method. We will take advantage of this fact to develop an efficient stochastic inference algorithm within the framework of G-flow. Notational conventions Unless otherwise stated, capital letters are used for random variables, small letters for specific values taken by random variables, and Greek letters for fixed model parameters. Subscripted colons indicate sequences: e.g., X1:t = X1 · · · Xt. The term In stands for the n × n identity matrix, E for expected value, V ar for the covariance matrix, and V ar−1 for the inverse of the covariance matrix (precision matrix). 2 The Generative Model for G-Flow Figure 1: Left: a(Ut) determines which texel (color at a vertex of the object model or a pixel of the background model) is responsible for rendering each image pixel. Right: G-flow video generation model: At time t, the object’s 3D pose, Ut, is used to project the object texture, Vt, into 2D. This projection is combined with the background texture, Bt, to generate the observed image, Yt. We model the image sequence Y as a stochastic process generated by three hidden causes, U, V , and B, as shown in the graphical model (Figure 1, right). The m × 1 random vector Yt represents the m-pixel image at time t. The n × 1 random vector Vt and the m × 1 random vector Bt represent the n-texel object texture and the m-texel background texture, respectively. As illustrated in Figure 1, left, the object pose, Ut, determines onto which image pixels the object and background texels project at time t. This is formulated using the projection function a(Ut). For a given pose, ut, the projection a(ut) is a block matrix, a(ut) def= av(ut) ab(ut) . Here av(ut), the object projection function, is an m × n matrix of 0s and 1s that tells onto which image pixel each object vertex projects; e.g., a 1 at row j, column i it means that the ith object point projects onto image pixel j. Matrix ab plays the same role for background pixels. Assuming the foreground mapping is one-toone, we let ab = Im−av(ut)av(ut)T , expressing the simple occlusion constraint that every image pixel is rendered by object or background, but not both. In the G-flow generative model: Yt = a(Ut) Vt Bt + Wt Wt ∼N(0, σwIm), σw > 0 Ut ∼p(ut | ut−1) Vt = Vt−1 + Zv t−1 Zv t−1 ∼N(0, Ψv), Ψv is diagonal Bt = Bt−1 + Zb t−1 Zb t−1 ∼N(0, Ψb), Ψb is diagonal (4) where p(ut | ut−1) is the pose transition distribution, and Zv, Zb, W are independent of each other, of the initial conditions, and over time. The form of the pose distribution is left unspecified since the algorithm proposed here does not require the pose distribution or the pose dynamics to be Gaussian. For the initial conditions, we require that the variance of V1 and the variance of B1 are both diagonal. Non-rigid 3D tracking is a difficult nonlinear filtering problem because changing the pose has a nonlinear effect on the image pixels. Fortunately, the problem has a rich structure that we can exploit: under the G-flow model, video generation is a conditionally Gaussian process [3, 6, 4, 5]. If the specific values taken by the pose sequence, u1:t, were known, then the texture processes, V and B, and the image process, Y , would be jointly Gaussian. This suggests the following scheme: we could use particle filtering to obtain a distribution of pose experts (each expert corresponds to a highly probable sample of pose, u1:t). For each expert we could then use Kalman filtering equations to infer the posterior distribution of texture given the observed images. This method is known in the statistics community as a Monte Carlo filtering solution for conditionally Gaussian processes [3, 4], and in the machine learning community as Rao-Blackwellized particle filtering [6, 5]. We found that in addition to Rao-Blackwellization, it was also critical to use Laplace’s method to generate the proposal distributions for importance sampling [7]. In the context of G-flow, we accomplished this by performing an optic flow-like optimization, using an efficient algorithm similar to those in [10, 2]. 3 Inference Our goal is to find an expression for the filtering distribution, p(ut, vt, bt | y1:t). Using the law of total probability, we have the following equation for the filtering distribution: p(ut, vt, bt | y1:t) = Z p(ut, vt, bt | u1:t−1, y1:t) | {z } Opinion of expert p(u1:t−1 | y1:t) | {z } Credibility of expert du1:t−1 (5) We can think of the integral in (5) as a sum over a distribution of experts, where each expert corresponds to a single pose history, u1:t−1. Based on its hypothesis about pose history, each expert has an opinion about the current pose of the object, Ut, and the texture maps of the object and background, Vt and Bt. Each expert also has a credibility, a scalar that measures how well the expert’s opinion matches the observed image yt. Thus, (5) can be interpreted as follows: The filtering distribution at time t is obtained by integrating over the entire ensemble of experts the opinion of each expert weighted by that expert’s credibility. The opinion distribution of expert u1:t−1 can be factorized into the expert’s opinion about the pose Ut times the conditional distribution of texture Vt, Bt given pose: p(ut, vt, bt | u1:t−1, y1:t) | {z } Opinion of expert = p(ut | u1:t−1, y1:t) | {z } Pose Opinion p(vt, bt | u1:t, y1:t) | {z } Texture Opinion given pose (6) The rest of this section explains how we evaluate each term in (5) and (6). We cover the distribution of texture given pose in 3.1, pose opinion in 3.2, and credibility in 3.3. 3.1 Texture opinion given pose The distribution of Vt and Bt given the pose history u1:t is Gaussian with mean and covariance that can be obtained using the Kalman filter estimation equations: V ar−1(Vt, Bt | u1:t, y1:t) = V ar−1(Vt, Bt | u1:t−1, y1:t−1) + a(ut)T σ−1 w a(ut) (7) E(Vt, Bt | u1:t, y1:t) = V ar(Vt, Bt | u1:t, y1:t) × V ar−1(Vt, Bt | u1:t−1, y1:t−1)E(Vt, Bt | u1:t−1, y1:t−1) + a(ut)T σ−1 w yt (8) This requires p(Vt, Bt|u1:t−1, y1:t−1), which we get from the Kalman prediction equations: E(Vt, Bt | u1:t−1, y1:t−1) = E(Vt−1, Bt−1 | u1:t−1, y1:t−1) (9) V ar(Vt, Bt | u1:t−1, y1:t−1) = V ar(Vt−1, Bt−1 | u1:t−1, y1:t−1) + Ψv 0 0 Ψb (10) In (9), the expected value E(Vt, Bt | u1:t−1, y1:t−1) consists of texture maps (templates) for the object and background. In (10), V ar(Vt, Bt | u1:t−1, y1:t−1) represents the degree of uncertainty about each texel in these texture maps. Since this is a diagonal matrix, we can refer to the mean and variance of each texel individually. For the ith texel in the object texture map, we use the following notation: µv t (i) def= ith element of E(Vt | u1:t−1, y1:t−1) σv t (i) def= (i, i)th element of V ar(Vt | u1:t−1, y1:t−1) Similarly, define µb t(j) and σb t(j) as the mean and variance of the jth texel in the background texture map. (This notation leaves the dependency on u1:t−1 and y1:t−1 implicit.) 3.2 Pose opinion Based on its current texture template (derived from the history of poses and images up to time t−1) and the new image yt, each expert u1:t−1 has a pose opinion, p(ut|u1:t−1, y1:t), a probability distribution representing that expert’s beliefs about the pose at time t. Since the effect of ut on the likelihood function is nonlinear, we will not attempt to find an analytical solution for the pose opinion distribution. However, due to the spatio-temporal smoothness of video signals, it is possible to estimate the peak and variance of an expert’s pose opinion. 3.2.1 Estimating the peak of an expert’s pose opinion We want to estimate ˆut(u1:t−1), the value of ut that maximizes the pose opinion. Since p(ut | u1:t−1, y1:t) = p(y1:t−1 | u1:t−1) p(y1:t | u1:t−1) p(ut | ut−1) p(yt | u1:t, y1:t−1), (11) ˆut(u1:t−1) def= argmax ut p(ut | u1:t−1, y1:t) = argmax ut p(ut | ut−1) p(yt | u1:t, y1:t−1). (12) We now need an expression for the final term in (12), the predictive distribution p(yt | u1:t, y1:t−1). By integrating out the hidden texture variables from p(yt, vt, bt | u1:t, y1:t−1), and using the conditional independence relationships defined by the graphical model (Figure 1, right), we can derive: log p(yt | u1:t, y1:t−1) = −m 2 log 2π −1 2 log |V ar(Yt | u1:t, y1:t−1)| −1 2 n X i=1 (yt(xi(ut)) −µv t (i))2 σv t (i) + σw −1 2 X j̸∈X(ut) (yt(j) −µb t(j))2 σb t(j) + σw , (13) where xi(ut) is the image pixel rendered by the ith object vertex when the object assumes pose ut, and X(ut) is the set of all image pixels rendered by the object under pose ut. Combining (12) and (13), we can derive ˆut(u1:t−1) = argmin ut −log p(ut | ut−1) (14) + 1 2 n X i=1 [yt(xi(ut)) −µv t (i)]2 σv t (i) + σw | {z } Foreground term −[yt(xi(ut)) −µb t(xi(ut))]2 σb t(xi(ut)) + σw −log[σb t(xi(ut)) + σw] | {z } Background terms ! Note the similarity between (14) and constrained optic flow (3). For example, focus on the foreground term in (14) and ignore the weights in the denominator. The previous image yt−1 from (3) has been replaced by µv t (·), the estimated object texture based on the images and poses up to time t −1. As in optic flow, we can find the pose estimate ˆut(u1:t−1) efficiently using the Gauss-Newton method. 3.2.2 Estimating the distribution of an expert’s pose opinion We estimate the distribution of an expert’s pose opinion using a combination of Laplace’s method and importance sampling. Suppose at time t −1 we are given a sample of experts indexed by d, each endowed with a pose sequence u(d) 1:t−1, a weight w(d) t−1, and the means and variances of Gaussian distributions for object and background texture. For each expert u(d) 1:t−1, we use (14) to compute ˆu(d) t , the peak of the pose distribution at time t according to that expert. Define ˆσ(d) t as the inverse Hessian matrix of (14) at this peak, the Laplace estimate of the covariance matrix of the expert’s opinion. We then generate a set of s independent samples {u(d,e) t : e = 1, · · · , s} from a Gaussian distribution with mean ˆu(d) t and variance proportional to ˆσ(d) t , g(·|ˆu(d) t , αˆσ(d) t ), where the parameter α > 0 determines the sharpness of the sampling distribution. (Note that letting α →0 would be equivalent to simply setting the new pose equal to the peak of the pose opinion, u(d,e) t = ˆu(d) t .) To find the parameters of this Gaussian proposal distribution, we use the Gauss-Newton method, ignoring the second of the two background terms in (14). (This term is not ignored in the importance sampling step.) To refine our estimate of the pose opinion we use importance sampling. We assign each sample from the proposal distribution an importance weight wt(d, e) that is proportional to the ratio between the posterior distribution and the proposal distribution: ˆp(ut | u(d) 1:t−1, y1:t) = s X e=1 δ(ut −u(d,e) t ) wt(d, e) Ps f=1 wt(d, f) (15) wt(d, e) = p(u(d,e) t | u(d) t−1)p(yt | u(d) 1:t−1, u(d,e) t , y1:t−1) g(u(d,e) t | ˆu(d) t , αˆσ(d) t ) (16) The numerator of (16) is proportional to p(u(d,e) t |u(d) 1:t−1, y1:t) by (12), and the denominator of (16) is the sampling distribution. 3.3 Estimating an expert’s credibility The credibility of the dth expert, p(u(d) 1:t−1 | y1:t), is proportional to the product of a prior term and a likelihood term: p(u(d) 1:t−1 | y1:t) = p(u(d) 1:t−1 | y1:t−1)p(yt | u(d) 1:t−1, y1:t−1) p(yt | y1:t−1) . (17) Regarding the likelihood, p(yt|u1:t−1, y1:t−1) = Z p(yt, ut|u1:t−1, y1:t−1)dut = Z p(yt|u1:t, y1:t−1)p(ut|ut−1)dut (18) We already generated a set of samples {u(d,e) t : e = 1, · · · , s} that estimate the pose opinion of the dth expert, p(ut | u(d) 1:t−1, y1:t). We can now use these samples to estimate the likelihood for the dth expert: p(yt | u(d) 1:t−1, y1:t−1) = Z p(yt | u(d) 1:t−1, ut, y1:t−1)p(ut | u(d) t−1)dut (19) = Z p(yt | u(d) 1:t−1, ut, y1:t−1)g(ut | ˆu(d) t , αˆσ(d) t ) p(ut | u(d) t−1) g(ut | ˆu(d) t , αˆσ(d) t ) dut ≈ Ps e=1 wt(d, e) s 3.4 Updating the filtering distribution Once we have calculated the opinion and credibility of each expert u1:t−1, we evaluate the integral in (5) as a weighted sum over experts. The credibilities of all of the experts are normalized to sum to 1. New experts u1:t (children) are created from the old experts u1:t−1 (parents) by appending a pose ut to the parent’s history of poses u1:t−1. Every expert in the new generation is created as follows: One parent is chosen to sire the child. The probability of being chosen is proportional to the parent’s credibility. The child’s value of ut is chosen at random from its parent’s pose opinion (the weighted samples described in Section 3.2.2). 4 Relation to Optic Flow and Template Matching In basic template-matching, the same time-invariant texture map is used to track every frame in the video sequence. Optic flow can be thought of as template-matching with a template that is completely reset at each frame for use in the subsequent frame. In most cases, optimal inference under G-flow involves a combination of optic flow-based and template-based tracking, in which the texture template gradually evolves as new images are presented. Pure optic flow and template-matching emerge as special cases. Optic Flow as a Special Case Suppose that the pose transition probability p(ut | ut−1) is uninformative, that the background is uninformative, that every texel in the initial object texture map has equal variance, V ar(V1) = κIn, and that the texture transition uncertainty is very high, Ψv →diag(∞). Using (7), (8), and (10), it follows that: µv t (i) = [av(ut−1)]T yt−1 = yt−1(xi(ut−1)) , (20) i.e., the object texture map at time t is determined by the pixels from image yt−1 that according to pose ut−1 were rendered by the object. As a result, (14) reduces to: ˆut(u1:t−1) = argmin ut 1 2 n X i=1 yt(xi(ut)) −yt−1(xi(ut−1)) 2 (21) which is identical to (3). Thus constrained optic flow [10, 2, 11] is simply a special case of optimal inference under G-flow, with a single expert and with sampling parameter α →0. The key assumption that Ψv →diag(∞) means that the object’s texture is very different in adjacent frames. However, optic flow is typically applied in situations in which the object’s texture in adjacent frames is similar. The optimal solution in such situations calls not for optic flow, but for a texture map that integrates information across multiple frames. Template Matching as a Special Case Suppose the initial texture map is known precisely, V ar(V1) = 0, and the texture transition uncertainty is very low, Ψv →0. By (7), (8), and (10), it follows that µv t (i) = µv t−1(i) = µv 1(i), i.e., the texture map does not change over time, but remains fixed at its initial value (it is a texture template). Then (14) becomes: ˆut(u1:t−1) = argmin ut n X i=1 yt(xi(ut)) −µv 1(i) 2 (22) where µv 1(i) is the ith texel of the fixed texture template. This is the error function minimized by standard template-matching algorithms. The key assumption that Ψv →0 means the object’s texture is constant from each frame to the next, which is rarely true in real data. G-flow provides a principled way to relax this unrealistic assumption of template methods. General Case In general, if the background is uninformative, then minimizing (14) results in a weighted combination of optic flow and template matching, with the weight of each approach depending on the current level of certainty about the object template. In addition, when there is useful information in the background, G-flow infers a model of the background which is used to improve tracking. Figure 2: G-flow tracking an outdoor video. Results are shown for frames 1, 81, and 620. 5 Simulations We collected a video (30 frames/sec) of a subject in an outdoor setting who made a variety of facial expressions while moving her head. A later motion-capture session was used to create a 3D morphable model of her face, consisting of a set of 5 morph bases (k = 5). Twenty experts were initialized randomly near the correct pose on frame 1 of the video and propagated using G-flow inference (assuming an uninformative background). See http://mplab.ucsd.edu for video. Figure 2 shows the distribution of experts for three frames. In each frame, every expert has a hypothesis about the pose (translation, rotation, scale, and morph coefficients). The 38 points in the model are projected into the image according to each expert’s pose, yielding 760 red dots in each frame. In each frame, the mean of the experts gives a single hypothesis about the 3D non-rigid deformation of the face (lower right) as well as the rigid pose of the face (rotated 3D axes, lower left). Notice G-flow’s ability to recover from error: bad initial hypotheses are weeded out, leaving only good hypotheses. To compare G-flow’s performance versus deterministic constrained optic flow algorithms such as [10, 2, 11] , we used both G-flow and the method from [2] to track the same video sequence. We ran each tracker several times, introducing small errors in the starting pose. Figure 3: Average error over time for G-flow (green) and for deterministic optic flow [2] (blue). Results were averaged over 16 runs (deterministic algorithm) or 4 runs (G-flow) and smoothed. As ground truth, the 2D locations of 6 points were hand-labeled in every 20th frame. The error at every 20th frame was calculated as the distance from these labeled locations to the inferred (tracked) locations, averaged across several runs. Figure 3 compares this tracking error as a function of time for the deterministic constrained optic flow algorithm and for a 20-expert version of the G-flow tracking algorithm. Notice that the deterministic system has a tendency to drift (increase in error) over time, whereas G-flow can recover from drift. Acknowledgments Tim K. Marks was supported by NSF grant IIS-0223052 and NSF grant DGE-0333451 to GWC. John Hershey was supported by the UCDIMI grant D00-10084. J. Cooper Roddey was supported by the Swartz Foundation. Javier R. Movellan was supported by NSF grants IIS-0086107, IIS-0220141, and IIS-0223052, and by the UCDIMI grant D00-10084. References [1] Simon Baker and Iain Matthews. Lucas-kanade 20 years on: A unifying framework. International Journal of Computer Vision, 56(3):221–255, 2002. [2] M. Brand. Flexible flow for 3D nonrigid tracking and shape recovery. In CVPR, volume 1, pages 315–322, 2001. [3] H. Chen, P. Kumar, and J. van Schuppen. On Kalman filtering for conditionally gaussian systems with random matrices. Syst. Contr. Lett., 13:397–404, 1989. [4] R. Chen and J. Liu. Mixture Kalman filters. J. R. Statist. Soc. B, 62:493–508, 2000. [5] A. Doucet and C. Andrieu. Particle filtering for partially observed gaussian state space models. J. R. Statist. Soc. B, 64:827–838, 2002. [6] A. Doucet, N. de Freitas, K. Murphy, and S. Russell. Rao-blackwellised particle filtering for dynamic bayesian networks. In 16th Conference on Uncertainty in AI, pages 176–183, 2000. [7] A. Doucet, S. J. Godsill, and C. Andrieu. On sequential monte carlo sampling methods for bayesian filtering. Statistics and Computing, 10:197–208, 2000. [8] Zoubin Ghahramani and Geoffrey E. Hinton. Variational learning for switching state-space models. Neural Computation, 12(4):831–864, 2000. [9] B. Lucas and T. Kanade. An iterative image registration technique with an application to stereo vision. In Proceedings of the International Joint Conference on Artificial Intelligence, 1981. [10] L. Torresani, D. Yang, G. Alexander, and C. Bregler. Tracking and modeling non-rigid objects with rank constraints. In CVPR, pages 493–500, 2001. [11] Lorenzo Torresani, Aaron Hertzmann, and Christoph Bregler. Learning non-rigid 3d shape from 2d motion. In Advances in Neural Information Processing Systems 16. MIT Press, 2004.
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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, andrew.emili@utoronto.ca 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. 0 10 20 30 40 Amplitude Unaligned, Linear Warp Alignment and CPM Alignment 0 10 20 30 Amplitude Time 0 10 20 30 40 50 Amplitude 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 = (xk 1, xk 2, ..., xk Nk). 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 + ϵ)N (double the resolution, plus some slack on each end) in our experiments and found this to be sufficient with ϵ < 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 ⃗πk. Each state in the state sequence maps to a time state/scale state pair: πk i →{τ k i , φk i }. Time states belong to the integer set (1..M); scale states belong to an ordered set (φ1..φQ). (In our experiments we have used Q=7, evenly spaced scales in logarithmic space). States, πk i , and observation values, xk i , are related by the emission probability distribution: Aπk i (xk i |⃗z) ≡p(xk i |πk i ,⃗z, σ, uk) ≡N(xk i ; zτ k i φk i uk, σ), where σ 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 k πi−1,πi ≡p(πi|πi−1) = p(φi|φi−1)pk(τi|τi−1). The constraint that time must move forward, cannot stand still, and that it can jump ahead no more than Jτ time states is enforced. (In our experiments we used Jτ = 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: pk(τi = a|τi−1 = b) = dk 1, if a −b = 1 dk 2, if a −b = 2 ... dk Jτ , if a −b = Jτ 0, otherwise p(φi = a|φi−1 = b) = s0, if D(a, b) = 0 s1, if D(a, b) = 1 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 are constrained by: PJτ i=1 dk i = 1 and 2s1 + s0 = 1. Jτ 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(Jτ, 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, γk s (i) ≡p(πi = s|⃗x) and also the pairwise state posteriors, ξs,t(i) ≡p(πi−1 = s, πi = 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: L ≡ K X k=1 log p(π1) + N X i=1 log Aπi(xk i |⃗z) + N X i=2 log T k πi−1,πi ! (1) P ≡−λ τ−1 X j=1 (zj+1 −zj)2 + K X k=1 log D(dk v|{ηk v}) + log D(sv|{η′ v}), (2) where p(π1) are priors over the initial states. The first term in Equation 2 is a smoothing penalty on the latent trace, with λ controlling the amount of smoothing. ηk v and η′ v 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, Jτ} and ηk v corresponds to the pseudo-count data for the parameters d1, d2 ...dJτ . For the scale state transitions, v ∈{0, 1} and ηk v 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: <Lp>π=P + K X k=1 S X s=1 γk s (1) log T k 0,s + K X k=1 S X s=1 N X i=1 γk s (i) log As(xk i |⃗z) + . . . . . . + K X k=1 S X s=1 S X s′=1 N X i=2 ξk s,s′(i) log T k s,s′ using the notation T k 0,s ≡p(π1 = s), and where γk s (i) and ξk s,s′(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: ∂<Lp>π ∂zj = 0 = K X k=1 X {s|τs=j} N X i=1 γk s (i)φsuk (xk i −zjukφs) σ2 −λ(4zj−2zj−1−2zj+1) 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 σ2 and uk are given by: σ2 = PS s=1 PN i=1 γk s (i)(xk i −zτsukφs)2 N , uk = PS s=1 zτsφs PN i=1 γk s (i)xk i PS s=1(zτsφs)2 PN i=1 γks (i) Lastly, updates for the scale and state transition probability distributions are given by: dk v = ηk v + PS s=1 P {s′|τs′−τs=v} PN i=2 ξk s,s′′(i) PJτ j=1 ηk j + PJτ j=1 PS s=1 P {s′|τs′−τs=j} PN i=2 ξk s,s′′(i) sv = η′ j + PK k=1 PS s=1 P {s′′∈H(s,v)} PN i=2 ξk s,s′′(i) P1 j=0 η′ j + PK k=1 PS s=1 P {s′′∈H(s,1),H(s,0)} PN i=2 ξk s,s′′(i) where H(s, j) ≡ {s′|s′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: D(dk v|{ηk v}) = QJτ v=1(dk v)ηk v and D(sv|{η′ v}) = Q1 v=0(sv)η′ v. The M-Step updates uk, σ, 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: σ, ⃗z, uk. The other two parameters, dk v 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., λ = 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 λ. Because the uk and dk v 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 λ 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 λ, 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 λ. 0 2 4 6 8 10 x 10 8 Amplitude Unaligned and Aligned Time Series 100 200 300 400 500 600 700 800 0 2 4 6 x 10 8 Latent Space Amplitude Time 100 200 300 400 500 600 700 800 0 1 2 3 4 5 6 7 8 9 x 10 8 Replicate 5 Amplitude Latent Trace Aligned Experimental Time Series Original Time Series 200 400 600 800 Latent Time Scale States Residual 1 2 3 Time Jump From Previous State a) b) c) d) 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 λ 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: σ 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 σ. 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 −λ Pτ−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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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´ınez-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 (Ω, Σ, 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 conv(H) := N X j=1 λjhj : N X j=1 |λj| ≤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 Ph or Pnh 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 ℓbe a nonnegative decreasing convex function on R such that ℓ(u) ≥1 for u ≤0. We will denote (ℓ• f)(x, y) := ℓ(yf(x)). The quantity P(ℓ• f) = Z (ℓ• f)dP = Eℓ(Y f(X)) is called the risk of f with respect to the loss ℓ, or the ℓ-risk of f. We will call a function f0 := N X j=1 λ0 jhj ∈conv(H) an ℓ-otimal aggregate classifier if it minimizes the ℓ-risk over conv(H). Similarly to the excess risk, one can define the excess ℓ-risk of f as Eℓ(f) := P(ℓ• f) − inf g∈conv(H) P(ℓ• g). Despite the fact that we concentrate in what follows on optimizing the excess ℓ-risk (ℓ-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 ℓ-risk has to be replaced by the corresponding empirical risk minimization problem Pn(ℓ• f) = 1 n n X i=1 ℓ Yjf(Xj) −→min, f ∈conv(H), whose solution ˆf := PN j=1 ˆλjhj is called an empirical ℓ-optimal aggregate classifier. We will show that if f0, ˆf are ”sparse” (i.e., λ0 j, ˆλj are small for most of the values of j), then the excess ℓ-risk of the empirical ℓ-optimal aggregate classifier is small and, moreover, the coefficients of ˆf are close to the coefficients of f0 in ℓ1-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 ̸∈J} are independent and, in addition, Ehj(X) = 0, j ̸∈J, then, using Jensen’s inequality, it is easy to check that in an ℓ-optimal aggregate classifier f0 one can take λ0 j = 0, j ̸∈J. We will define a measure of sparsity of a function f := PN j=1 λjhj ∈conv(H) that is somewhat akin to sparsity charactersitics considered in [5, 6]. For 0 ≤d ≤N, let ∆(f; d) := min X j̸∈J |λj| : J ⊂{1, . . . , N}, card(J) = d and let βn(d) := d log(Nn/d) n . Define dn(f) := min d : 1 ≤d ≤N, p βn(d) ≥∆(d) . Of course, if there exists J ⊂{1, . . . , N} such that λj = 0 for all j ̸∈J and card(J) = d, then dn(f) ≤d. We will also need the following measure of linear independence of functions in H : γ(d) := γ(H; d) = inf J⊂{1,...,N},card(J)=d inf P j∈J |αj|=1
X j∈J αjhj
L2(P ) −1 . Finally, we need some standard conditions on the loss function ℓ(as, for instance, in [4]). Assume that ℓis Lipschitz on [−1, 1] with some constant L, |ℓ(u) −ℓ(v)| ≤L|u − v|, u, v ∈[−1, 1], and the following condition on the convexity modulus of ℓholds with Λ ≤L : ℓ(u) + ℓ(v) 2 −ℓ u + v 2 ≥Λ|u −v|2, u, v ∈[−1, 1]. In fact, ℓ(u) is often replaced by a function ℓ(uM) with a large enough M (in other words, the ℓ-risk is minimized over Mconv(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 P Eℓ( ˆf) ≥K1 L2 Λ βn(dn( ˆf)) ^ r log N n + t n ≤e−t and P N X j=1 |ˆλj −λ0 j| ≥K2 L Λγ(dn( ˆf) + dn(f0)) r βn(dn( ˆf) + dn(f0)) + t n ≤e−t. 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( ˆf) and the second on dn( ˆf), 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 ℓ-risk over the convex hull does not guarantee that ˆf is sparse even if f0 is. To address this problem, we also studied another approach based on minimization of the penalized empirical ℓ-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 ℓ(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 1A 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 assign to the training examples the initial weights wj = e−Yj f(Xj ) Z , where Z is a standard 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 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 35 5 10 15 20 25 30 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 10 15 20 25 30 35 5 10 15 20 25 30 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: 0 right brainstem: 0 left cerebellum: 0.15 right cerebellum: 0.16 left parietal: 0.02 right parietal: 0.06 left temporal: 0.03 right temporal: 0.15 left occipital: 0.29 right occipital: 0.15 left subcortical: 0 right subcortical: 0 left frontal: 0 right frontal: 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 σ = 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 γ(d). If γ(d) is too large for d = dn(f0) ∨dn( ˆf), 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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Hierarchical Clustering of a Mixture Model Jacob Goldberger Sam Roweis Department of Computer Science, University of Toronto {jacob,roweis}@cs.toronto.edu Abstract In this paper we propose an efficient algorithm for reducing a large mixture of Gaussians into a smaller mixture while still preserving the component structure of the original model; this is achieved by clustering (grouping) the components. The method minimizes a new, easily computed distance measure between two Gaussian mixtures that can be motivated from a suitable stochastic model and the iterations of the algorithm use only the model parameters, avoiding the need for explicit resampling of datapoints. We demonstrate the method by performing hierarchical clustering of scenery images and handwritten digits. 1 Introduction The Gaussian mixture model (MoG) is a flexible and powerful parametric framework for unsupervised data grouping. Mixture models, however, are often involved in other learning processes whose goals extend beyond simple density estimation to hierarchical clustering, grouping of discrete categories or model simplification. In many such situations we need to group the Gaussians components and re-represent each group by a new single Gaussian density. This grouping results in a compact representation of the original mixture of many Gaussians that respects the original component structure in the sense that no original component is split in the reduced representation. We can view the problem of Gaussian component clustering as general data-point clustering with side information that points belonging to the same original Gaussian component should end up in the same final cluster. Several algorithms that perform clustering of data points given such constraints were recently proposed [11, 5, 12]. In this study we extend these approaches to model-based rather than datapoint based settings. Of course, one could always generate data by sampling from the model, enforcing the constraint that any two samples generated by the same mixture component must end up in the same final cluster. We show that if we already have a parametric representation of the constraint via the MoG density, there is no need for an explicit sampling phase to generate representative datapoints and their associated constraints. In other situations we want to collapse a MoG into a mixture of fewer components in order to reduce computation complexity. One example is statistical inference in switching dynamic linear models, where performing exact inference with a MoG prior causes the number of Gaussian components representing the current belief to grow exponentially in time. One common solution to this problem is grouping the Gaussians according to their common history in recent timesteps and collapsing Gaussians grouped together into a single Gaussian [1]. Such a reduction, however, is not based on the parameters of the Gaussians. Other instances in which collapsing MoGs is relevant are variants of particle filtering [10], non-parametric belief propagation [7] and fault detection in dynamical systems [3]. A straight-forward solution for these situations is first to produce samples from the original MoG and then to apply the EM algorithm to learn a reduced model; however this is computationally inefficient and does not preserve the component structure of the original mixture. 2 The Clustering Algorithm We assume that we are given a mixture density f composed of k d-dimensional Gaussian components: f(y) = k X i=1 αiN(y; µi, Σi) = k X i=1 αifi(y) (1) We want to cluster the components of f into a reduced mixture of m < k components. If we denote the set of all (d-dimensional) Gaussian mixture models with at most m components by MoG(m), one way to formalize the goal of clustering is to say that we wish to find the element g of MoG(m) “closest” to f under some distance measure. A common proximity criterion is the cross-entropy from f to g, i.e. ˆg = arg ming KL(f||g) = arg maxg R f log g, where KL() is the Kullback-Leibler divergence and the minimization is performed over all g in MoG(m). This criterion leads to an intractable optimization problem; there is not even a closed-form expression for the KL-divergence between two MoGs let alone an analytic minimizer of its second argument. Furthermore, minimizing a KL-based criterion does not preserving the original component structure of f. Instead, we introduce the following new distance measure between f = Pk i=1 αifi and g = Pm j=1 βjgj: d(f, g) = k X i=1 αi m min j=1 KL(fi||gj) (2) which can be intuitively thought of as the cost of coding data generated by f under the model g, if all points generated by component i of f must be coded under a single component of g. Unlike the KL-divergence between two MoGs, this distance can be analytically computed. In particular, each term is a KL-divergence between two Gaussian distributions N(µ1, Σ1) and N(µ2, Σ2) which is given by: 1 2(log |Σ2| |Σ1| + Tr(Σ−1 2 Σ1) + (µ1 −µ2)T Σ−1 2 (µ1 −µ2) −d). Under this distance, the optimal reduced MoG representation ˆg is the solution to the minimization of (2) over MoG(m): ˆg = arg ming d(f, g). Although the minimization ranges over all the MoG(m), we prove that the optimal density ˆg is a MoG obtained from grouping the components of f into clusters and collapsing all Gaussians within a cluster into a single Gaussian. There is no closed-form solution for the minimization; rather, we propose an iterative algorithm to obtain a locally optimal solution. Denote the set of all the mk mappings from {1, ..., k} to {1, ..., m} by S. For each π ∈S and g ∈MoG(m) define: d(f, g, π) = k X i=1 αiKL(fi||gπ(i)). (3) For a given g ∈MoG(m), we associate a matching function πg ∈S: πg(i) = arg m min j=1 KL(fi||gj) i = 1, ..., k (4) It can be easily verified that: d(f, g) = d(f, g, πg) = min π∈S d(f, g, π) (5) i.e. πg is the optimal mapping between the components of f and g. Using (5) to define our main optimization we obtain the optimal reduced model as a solution of the following double minimization problem: ˆg = arg min g min π∈S d(f, g, π) (6) For m > 1 the double minimization (6) can not be solved analytically. Instead, we can use alternating minimization to obtain a local minimum. Given a matching function π ∈S, we define gπ ∈MoG(m) as follows. For each j such that π−1(j) is non empty, define the following MoG density: f π j = P i∈π−1(j) αifi P i∈π−1(j) αi (7) The mean and variance of the set f π j , denoted by µ′ j and Σ′ j, are: µ′ j = 1 βj X i∈π−1(j) αiµi, Σ′ j = 1 βj X i∈π−1(j) αi Σi + (µi −µ′ j)(µi −µ′ j)T where βj = P i∈π−1(j) αi. Let gπ j = N(µ′ j, Σ′ j) be the Gaussian distribution obtained by collapsing the set f π j into a single Gaussian. It satisfies: gπ j = N(µ′ j, Σ′ j) = arg min g KL(f π j ||g) = arg min g d(f π j , g) such that the minimization is performed over all the d-dimensional Gaussian densities. Denote the collapsed version of f according to π by gπ, i.e.: gπ = m X j=1 βjgπ j (8) Lemma 1: Given a MoG f and a matching function π ∈S, gπ is the unique minimum point of d(f, g, π). More precisely, d(f, gπ, π) ≤d(f, g, π) for all g ∈ MoG(m), and if d(f, gπ, π) = d(f, g, π) then gπ j = gj for all j = 1, .., m such that gπ j and gj are the Gaussian components of gπ and g respectively. Proof: Denote c = Pk i=1 αi R fi log fi (a constant independent of g). c −d(f, g, π) = k X i=1 αi Z fi log(gπ(i)) = m X j=1 X i∈π−1(j) αi Z fi log(gj) = m X j=1 βj Z f π j log(gj) = m X j=1 βj Z gπ j log(gj) The Jensen inequality yields: ≤ m X j=1 βj Z gπ j log(gπ j ) = m X j=1 βj Z f π j log(gπ j ) = k X i=1 αi Z fi log(gπ π(i)) = c −d(f, gπ, π) The equality R f π j log(gj) = R gπ j log(gj) is due to the fact that log(gj) is a quadratic expression and the first two moments of f π j and its collapsed version gπ j are equal. Jensen’s inequality is saturated if and only if for all j = 1, .., m (such that π−1(j) is not empty) the Gaussian densities gj and gπ j are equal. 2 Using Lemma 1 we obtain a closed form description of a single iteration of the alternating minimization algorithm, which can be viewed as a type of K-means operating at the meta-level of model parameters: πg = arg min π d(f, g, π) (REGROUP) gπ = arg min g d(f, g, π) (REFIT) Above, πg(i) = arg minj KL(fi||gj) and gπ is computed using (8). The iterative algorithm monotonically decreases the distance measure d(f, g). Hence, since S is finite, the algorithm converges to a local minimum point after finite number of iterations. The next theorem ensures that once the iterative algorithm converges we obtain a clustering of the MoG components. Definition 1: A MoG g ∈MoG(m) is an m-mixture collapsed version of f if there exists a matching function π ∈S such that g is obtained by collapsing f according to π, .i.e. g = gπ. Theorem 1: If applying a single iteration (expressions (regroup) and (refit)) to a function g ∈MoG(m) does not decrease the distance function (2), then necessarily g is a collapsed version of f. Proof: Let g ∈MoG(m) and let π be a matching function such that d(f, g) = d(f, g, π). Let gπ be a collapsed version of f according to π. The MoG gπ is obtained as a result of applying a single iteration to g. Let g be composed of the following Gaussians {g1, ..., gm} and similarly let gπ = {gπ 1 , ..., gπ m}. According to Lemma 1, d(f, g) = d(f, g, π) ≥ d(f, gπ, π) ≥d(f, gπ). Assume that a single iteration does not decrease the distance, i.e. d(f, g) = d(f, gπ). Hence d(f, g, π) = d(f, gπ, π). According to Lemma 1, this implies that gj = gπ j for all j = 1, ..., m. Therefore g is a collapsed version of f. 2 Theorem 1 implies that each local minimum of the propose iterative algorithm is a collapsed version of f. Given the optimal matching function π, the last step of the algorithm is to set the weights of the reduced representation. βπ j = P {i|π(i)=j} αi. These weights are automatically obtained via the collapsing process. 3 Experimental Results In this section we evaluate the performance of our semi-supervised clustering algorithm and compare it to the standard “flat” clustering approach that does not respect the original component structure. We have applied both methods to clustering handwritten digits and natural scene images. In each case, a set of objects is organized in predefined categories. For each category c we learn from a labeled training set a Gaussian distribution f(x|c). A prior distribution over the categories p(c) can be also extracted from the labeled training set. The goal is to cluster the objects into a small number of clusters (fewer than the number of class labels). The standard (flat) approach is to apply an unsupervised clustering to entire collection of original objects, ignoring their class labels. Alternatively we can utilize the given categorization as side-information in order to obtain an improved reduced clustering which also respects the original labels, thus inducing a hierarchical structure. B BN
Class A Class B Figure 1: (top) Means of 10 models of digit classes. (bottom) Means of two clusters after our algorithm has grouped 0,2,3,5,6,8 and 1,4,7,9. method cls 0 1 2 3 4 5 6 7 8 9 this Class A 100 4 99 99 3 99 99 0 94 1 paper Class B 0 96 1 1 98 2 1 100 6 99 unsupervised Class 1 93 16 93 87 22 66 96 16 23 25 EM Class 2 7 85 7 14 78 34 4 84 77 76 Table 1: Clustering results showing the purity of a 2-cluster reduced model learned from a training set of handwritten digits in 10 original classes. For each true label, the percentage of cases (from an unseen test set) falling into each of the two reduced classes is shown. The top two lines show the purity of assignments provided by our clustering algorithm; the bottom two lines show assignments from a flat unsupervised fitting of a two component mixture. Our first experiment used a database of handwritten digits. Each example is represented by a 8×8 grayscale pixel image; 700 cases are used to learn a 64-dimensional full covariance Gaussian distribution for each class. In the next step we want to divide the digits into two natural clusters, while taking into account their original 10-way structure. We applied our semi-supervised algorithm to reduce the mixture of 10 Gaussians into a mixture of two Gaussians. The minimal distance (2) is obtained when the ten digits are divided into the two groups {0, 2, 3, 5, 6, 8} and {1, 4, 7, 9}. The means of the two resulting clusters are shown in Figure 1. To evaluate the purity of this clustering, the reduced MoG was used to label a test set consists of 4000 previously unseen examples. The binary labels on the test set are obtained by comparing the likelihood of the two components in the reduced mixture. Table 1 (top) presents, for each digit, the percentage of images that were affiliated with each of the two clusters. Alternatively we can apply a standard EM algorithm to learn by maximum likelihood a flat mixture of 2 Gaussians directly from the 7000 training examples, without utilizing their class labels. Table 1 (bottom) shows the results of such an unsupervised clustering, evaluated on the same test set. Although the likelihood of the unsupervised mixture model was significantly better than the semi-supervised model, both on train and test data-sets it is obvious that the purity of the clusters it learns is much worse since it is not preserving the hierarchical class structure. Comparing the top and bottom of Table 1, we can see that using the side information we obtain a clustering of the digit data-base which is much more correlated with categorization of the set into ten digits than the unsupervised procedure. In a second experiment, we evaluate the performance of our proposed algorithm on image category models. The database used consists of 1460 images selectively handpicked from the COREL database to create 16 categories. The images within each category have similar color spatial layout, and are labeled with a high-level semantic 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 # of clusters mutual information clustering results semi−supervised unsupervised A C B D Figure 2: Hierarchical clustering of natural image categories. (left) Mutual information between reduced cluster index and original class. (right) Sample images from the sets A,B,C,D learned by hierarchical clustering. description (e.g. fields, sunset). For each pixel we extract a five-dimensional feature vector (3 color features and x,y position). From all the pixels that are belonging to the same category we learn a single Gaussian. We have clustered the image categories into k = 2, ..., 6 sets using our algorithm and compared the results to unsupervised clustering obtained from an EM procedure that learned a mixture of k Gaussians. In order to evaluate the quality of the clustering in terms of correlation with the category information we computed the mutual information (MI) between the clustering result (into k clusters) and the category affiliation of the images in a test set. A high value of mutual information indicates a strong resemblance between the content of the learned clusters and the hand-picked image categories. It can be verified from the results summarized in Figure 2 that, as we can expect, the MI in the case of semi-supervised clustering is consistently larger than the MI in the case of completely unsupervised clustering. A semi-supervised clustering of the image database yields clusters that are based on both low-level features and a high level available categorization. Sampled images from clustering into 4 sets presented in Figure 2. 4 A Stochastic Model for the Proposed Distance In this section we describe a stochastic process that induces a likelihood function which coincides with the distance measure d(f, g) presented in section 2. Suppose we are given two MoGs: f(y) = k X i=1 αifi(y) = k X i=1 αiN(y; µi, Σi) , g(y) = m X j=1 βjgj(y) = m X j=1 βjN(y; µ′ j, Σ′ j) Consider an iid sample set of size n, drawn from f(y). The samples can be arranged in k blocks according to the Gaussian component that was selected to produce the sample. Assume that ni samples were drawn from the i-th component fi and denote these samples by yi = {yi1, ..., yini}. Next, we compute the likelihood of the sample set according to the model g; but under the constraint that samples within the same block must be assigned to the same mixture component of g. In other words, instead of having a hidden variable for each sample point we shall have one for each sample block. The likelihood of the sample set yn according to the MoG g under this constraint is: Ln(g) = g(y1, ..., yk) = k Y i=1 m X j=1 βj ni Y t=1 N(yit; µ′ j, Σ′ j) The main result is that as the number of points sampled grows large, the expected negative log likelihood becomes equal to the distance d(f, g) under the measure proposed above: Theorem 2: For each g ∈MoG(m) lim n→∞ 1 n log Ln(g) = c −d(f, g) (9) such that c = P αi R fi log fi does not depend on g. Surprisingly, as noted earlier the mixture weights βj do not appear in the asymptotic likelihood function of the generative model presented in this section. Proof: To prove the theorem we shall use the following lemma: Lemma 2: Let {xjn} j = 1, .., m be a set of m sequences of real positive numbers such that xjn → xj and let {βj} be a set of positive numbers. Then 1 n log P j βj(xjn)n →maxj log xj [This can be shown as follows: Let a = arg maxj xj. Then for n sufficiently large, βa(xan)n ≤P j βj(xjn)n ≤mβa(xan)n. Hence log xa ≤ limn→∞1 n log P j βj(xjn)n ≤log xa.] The points {yi1, ..., yini} are independently sampled from the Gaussian distribution fi. Therefore, the law of large numbers implies: 1 ni log Qni t=1 N(yit; µ′ j, Σ′ j) →R fi log gj. Hence, substituting: xjni = (Qni t=1 N(yit; µ′ j, Σ′ j)) 1 ni →exp(R fi log gj) = xj in Lemma 2, we obtain: 1 ni log Pm j=1 βj Qni t=1 N(yit; µ′ j, Σ′ j) →maxm j=1 R fi log gj In a similar manner, the law of large numbers, applied to the discrete distribution (α1, ..., αk), yields ni n →αi. Hence 1 n log Ln(g) = 1 n log g(y1, ..., yk) = Pk i=1 ni n · 1 ni log Pm j=1 βj Qni t=1 N(yit; µ′ j, Σ′ j) → Pk i=1 αi maxm j=1 R fi log gj = c −Pk i=1 αi minm j=1 KL(fi||gj) = c −d(f, g) 2 5 Relations to Previous Approaches and Conclusions Other authors have recently investigated the learning of Gaussian mixture models using various pieces of side information or constraints. Shental et al. [5] utilized the generative model described in the previous section and the EM algorithm derived from it, to learn a MoG from data set endowed with equivalence constraints that enforce equivalent points to be assigned to the same cluster. Vasconcelos and Lippman [9] proposed a similar EM based clustering algorithm for constructing mixture hierarchies using a finite set of virtual samples. Given the generative model presented above, we can apply the EM algorithm to learn the (locally) maximum likelihood parameters of the reduced MoG model g(y). This EM-based approach, however, is not precisely suitable for our component clustering problem. The EM update rule for the weights of the reduced mixture density is based only on the number of the original components that are clustered into a single component without taking into account the relative weights [9]. The problem discussed in this study is also related to the Information-Bottleneck (IB) principle [8]. In the case of mixture of histograms f = Pk i=1 αifi , the IB principle yields the following iterative algorithm for finding a clustering of a mixture of histograms g = Pm j=1 βjgj(y): wij = βje−λKL(fi||gj) P l βle−λKL(fi||gl) , βj = X i wijαi , gj = P i wijαifi P i wijαi (10) Assuming that the number of the (virtual) samples tends to ∞, we can derive, in a manner similar to the Gaussian case, a grouping algorithm for a mixture of histograms. Slonim and Weiss [6] showed that the clustering algorithm in this case can be either motivated from the EM algorithm applied to a suitable generative model [4] or from the (hard decision version) of the IB principle [8]. However, when we want to represent the clustering result as a mixture density there is a difference in the resulting mixture coefficient between the EM and the IB based algorithms. Unlike the IB updating equation (10) of the coefficients wij , the EM update equation is based only on the number of components that are collapsed into a single Gaussian. In the case of mixture of Gaussians, applying the IB principle results only in a partitioning of the original components but does not deliver a reduced representation in the form of a smaller mixture [2]. If we modify gj in equation (10) by collapsing the mixture gj into a single Gaussian we obtain a soft version of our algorithm. Setting the Lagrange multiplier λ to ∞we recover exactly the algorithm described in Section 2. To conclude, we have presented an efficient Gaussian component clustering algorithm that can be used for object category clustering and for MoG collapsing. We have shown that our method optimizes the distance measure between two MoG that we proposed. In this study we have assumed that the desired number of clusters is given as part of the problem setup, but if this is not the case, standard methods for model selection can be applied. References [1] Y. Bar-Shalom and X. Li. Estimation and tracking: principles, techniques and software. Artech House, 1993. [2] S. Gordon, H. Greenspan, and J. Goldberger. Applying the information bottleneck principle to unsupervised clustering of discrete and continuous image representations. In ICCV, 2003. [3] U. Lerner, R. Parr, D. Koller, and G. Biswas. Bayesian fault detection and diagnosis in dynamic systems. In AAAI/IAAI, pp. 531–537, 2000. [4] J. Puzicha, T. Hofmann, and J. Buhmann. Histogram clustering for unsupervised segmentation and image retrieval. Pattern Recognition Letters, 20(9):899–909, 1999. [5] N. Shental, A. Bar-Hillel, T. Hertz, and D. Weinshall. Computing gaussian mixture models with em using equivalence constraints. In Proc. of Neural Information Processing Systems, 2003. [6] N. Slonim and Y. Weiss. Maximum likelihood and the information bottleneck. In Proc. of Neural Information Processing Systems, 2003. [7] E. Sudderth, A. Ihler, W. Freeman, and A. Wilsky. Non-parametric belief propagation. In CVPR, 2003. [8] N. Tishby, F. Pereira, and W. Bialek. The information bottleneck method. In Proc. of the 37-th Annual Allerton Conference on Communication, Control and Computing, pages 368–377, 1999. [9] N. Vasconcelos and A. Lippman. Learning mixture hierarchies. In Proc. of Neural Information Processing Systems, 1998. [10] J. Vermaak, A. A. Doucet, and P. Perez. Maintaining multi-modality through mixture tracking. In Int. Conf. on Computer Vision, 2003. [11] K. Wagstaff, C. Cardie, S. Rogers, and S. Schroell. Constraind k-means clustering with background knowledge. In Proc. Int. Conf. on Machine Learning, 2001. [12] E.P. Xing, A. Y. Ng, M.I. Jordan, and S. Russell. Distance learning metric. In Proc. of Neural Information Processing Systems, 2003.
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Hierarchical Bayesian Inference in Networks of Spiking Neurons Rajesh P. N. Rao Department of Computer Science and Engineering University of Washington, Seattle, WA 98195 rao@cs.washington.edu 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: τ dvi dt = −vi + X j wijIj + X j uijv′ j (1) where τ 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, v′ j 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: vi(t + 1) = vi(t) + ϵ(−vi(t) + X j wijIj(t)) + X j uijv′ j(t)) (2) i.e. vi(t + 1) = ϵ X j wijIj(t) + X j u′ ijv′ j(t) (3) where ϵ is the integration rate, u′ ii = 1 + ϵ(uii −1) and for i ̸= j, u′ ij = ϵuij. 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: vi(t + 1) = f X j wijIj(t) + g X j u′ ijv′ j(t) (4) 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 vi 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 θ(t) represent the hidden state of a Markov model at time t with transition probabilities given by P(θ(t) = θi|θ(t −1) = θj) = P(θt i|θt−1 j ) for i, j = 1 . . . N. Let I(t) be the observable output governed by the probabilities P(I(t)|θ(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: mt,t+1 i = P(I(t)|θt i) X j P(θt i|θt−1 j )mt−1,t j (6) If m0,1 i = P(θi) (the prior distribution over states), then it is easy to show using Bayes rule that mt,t+1 i = P(θt i, I(t), . . . , I(1)). If the probabilities are normalized at each update step: mt,t+1 i = P(I(t)|θt i) X j P(θt i|θt−1 j )mt−1,t j /nt−1,t (7) where nt−1,t = P j mt−1,t j , then the message becomes equal to the posterior probability of the state and current input, given all past inputs: mt,t+1 i = P(θt i, I(t)|I(t −1), . . . , I(1)) (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 (9) f X j wijIj(t) = log P(I(t)|θt i) (10) g X j u′ ijv′ j(t) = log( X j P(θt i|θt−1 j )mt−1,t j /nt−1,t) (11) In this model, the dendritic filtering functions f and g approximate the logarithm function1, the synaptic currents Ij(t) and v′ j(t) are approximated by the corresponding instantaneous firing rates, and the recurrent synaptic weights u′ ij encode the transition probabilities P(θt i|θt−1 j ). Normalization by nt−1,t is implemented by subtracting log nt−1,t using inhibition. 1An 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+1 θ θt I(t) I(t) t+1 t B t+1 θ θt I(t+1) I(t) θt t+1 θ 1 1 2 2 C I(t+1) I(t) A D t t+1 Figure 1: Graphical Models and their Neural Implementation. (A) Single-level dynamic graphical model. Each circle represents a node denoting the state variable θt which can take on values θ1, . . . , θN. (B) Recurrent network for implementing on-line belief propagation for the graphical model in (A). Each circle represents a neuron encoding a state θi. 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 i + T (12) 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 (13) = elog mt,t+1 i = mt,t+1 i (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 particular level and between levels: mt,t+1 1,i (message from state i to other states at level 1 from time step t to t + 1), mt 1→2,i (“feedforward” message from state i at level 1 sent to level 2 at time t), mt,t+1 2,i (message from state i to other states at level 2 from time step t to t + 1), and mt 2→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: mt 1→2,i = X j X k P(θt 1,k|θt 2,i, θt−1 1,j )mt−1,t 1,j P(I(t)|θt 1,k) (15) mt 2→1,i = X j P(θt 2,i|θt−1 2,j )mt−1,t 2,j (16) mt,t+1 1,i = P(I(t)|θt 1,i) X j X k P(θt 1,i|θt 2,j, θt−1 1,k )mt 2→1,jmt−1,t 1,k (17) mt,t+1 2,i = mt 1→2,i X j P(θt 2,i|θt−1 2,j )mt−1,t 2,j (18) 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 θ1,i as its preferred state while neuron i in level 2 encodes θ2,i. We also assume specific feedforward and feedback neurons for computing and conveying mt 1→2,i and mt 2→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: vi(t + 1) = f X j wijIj(t) + g X j X k u′ ijkv′ j(t)xk(t) (19) 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 P(θt 1,k|θt 2,i, θt−1 1,j ) were chosen such that for θt 2 = left side, the transition probabilities for 0 5 10 15 20 25 30 0 0.1 0.2 0.3 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 5 10 15 20 25 30 0 0.1 0.2 0.3 0.4 0.5 θt+1 θt Leftward w1 w15 12 10 8 Neuron Rightward Motion Leftward Motion Rightward Motion Leftward Motion A B Rightward Spatial Location (pixels) 15 1 30 15 1 30 0 0.1 0.2 0.3 20 30 10 1 C D 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(θt+1 i |θt j) 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 θt 1 coding for the right side were set to values close to zero (and vice versa, for θt 2 = 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. An0 5 10 15 20 15 20 25 30 0 5 10 15 20 10 15 20 25 30 0 5 10 15 20 15 20 25 30 Pair Att Away Pair Att Ref Spikes/second Spikes/second Spikes/second Time steps from stim onset Time steps from stim onset Time steps from stim onset Ref Att Away A B C D 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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Efficient Kernel Discriminant Analysis via QR Decomposition Tao Xiong Department of ECE University of Minnesota txiong@ece.umn.edu Jieping Ye Department of CSE University of Minnesota jieping@cs.umn.edu Qi Li Department of CIS University of Delaware qili@cis.udel.edu Vladimir Cherkassky Department of ECE University of Minnesota cherkass@ece.umn.edu Ravi Janardan Department of CSE University of Minnesota janardan@cs.umn.edu 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 {Πi}c i=1, and the size of the ith class |Πi| = ni. Define between-class, within-class, and total scatter matrices Sb, Sw, and St respectively, as follows [3]: Sb = HbHt b, Sw = HwHt w, and St = HtHt t, where Hb = [√N1(m1 − m), · · · , √Nc(mc −m)] ∈Rd×c, Hw = A −[m1et 1, · · · , mcet c] ∈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 GtG=I trace(GtSbG). (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 1KDA/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 ←Ht bQ; 4. Z ←Ht tQ; 5. B ←Y tY ; /*Reduced between-class scatter matrix*/ 6. T ←ZtZ; /*Reduced total scatter matrix*/ 7. Compute the c eigenvectors φi of (T + µIc)−1B with decreasing eigenvalues; 8. G ←QV , where V = [φ1, · · · , φc]. 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(GtSbG) = trace(V tGtSbGV ), 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 GtSbG = V t(QtSbQ)V , GtSwG = V t(QtSwQ)V , and GtStG = V t(QtStQ)V , the original problem of finding optimal G is equivalent to finding V , with B = QtSbQ, W = QtSwQ, and T = QtStQ 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 + µIc)−1B or (T + µIc)−1B (note T = B + W), for some positive constant µ [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 Φ be a mapping to the feature space and Φ(A) be the data matrix in the feature space. Then, the centroid matrix CΦ in the feature space is CΦ = mΦ 1 , · · · , mΦ c = 1 n1 i∈Π1 Φ(ai), · · · , 1 nc i∈Πc Φ(ai) . (2) The global centroid in the feature space can be computed as mΦ = 1 n i nimΦ i . To maximize between-class distance in the feature space, as discussed in Section 2, we perform QR decomposition on CΦ, i.e., CΦ = QΦRΦ. A key observation is that RΦ can be computed as (CΦ)tCΦ = (RΦ)tRΦ by applying the Cholesky decomposition on (CΦ)tCΦ [4]. Note that CΦ = AΦM, where AΦ = Φ(A) = [Φ(a1) . . . Φ(an)], and the ith column of M is (0, · · · , 0, 1 ni , · · · , 1 ni , 0, · · · , 0)t. Let K be the kernel matrix with K(i, j) = ⟨Φ(ai), Φ(aj)⟩. Then (CΦ)tCΦ = M tKM. (3) Algorithm 2: KDA/QR /* Stage I: */ 1. Construct kernel matrix K; 2. Compute (CΦ)tCΦ = M t(KM) as in Eq. (3); 3. Compute RΦ from the Cholesky Decomposition of (CΦ)tCΦ; /* Stage II: */ 4. Y Φ ←N tM tKM(RΦ)−1; 5. ZΦ ←EtKM(RΦ)−1; 6. BΦ ←(Y Φ)tY Φ; 7. T Φ ←(ZΦ)tZΦ; 8. Compute the c eigenvectors φΦ i of (T Φ + µIc)−1BΦ, with decreasing eigenvalues; 9. V Φ ←[φΦ 1 , φΦ 2 , · · · , φΦ c ]; 10. GΦ ←CΦ(RΦ)−1V Φ; With the computed RΦ, QΦ = CΦ(RΦ)−1. The matrices Y Φ, ZΦ, BΦ, and W Φ in the feature space (corresponding to the second stage in LDA/QR) can be computed as follows. In the feature space, we have HΦ b = CΦN, where the ith column of N is ((0, · · · , √ni, · · · 0)t − √ni n (n1, · · · , nc)t. It follows that Y Φ = (HΦ b )tQΦ = N t(CΦ)tCΦ(RΦ)−1 = N tM tKM(RΦ)−1. Similarly, HΦ t = AΦE and ZΦ = (HΦ t )tQΦ = Et(AΦ)tCΦ(RΦ)−1 = Et(AΦ)tAΦM(RΦ)−1 = EtKM(RΦ)−1, where E = I −1 neet. Since SΦ b = HΦ b (HΦ b )t and SΦ t = HΦ t (HΦ t )t, we have BΦ = (QΦ)tSΦ b QΦ = (QΦ)tHΦ b (HΦ b )tQΦ = (Y Φ)tY Φ, T Φ = (QΦ)tSΦ t QΦ = (QΦ)tHΦ t (HΦ t )tQΦ = (ZΦ)tZΦ. We proceed by computing the c eigenvectors {φΦ i }c i=1 of (T Φ + µIc)−1BΦ. Define V Φ = [φΦ 1 , φΦ 2 , · · · , φΦ c ]. The final transformation matrix can be computed as GΦ = QΦV Φ = CΦ(RΦ)−1V Φ. (4) For a given data point z, its projection by GΦ is (GΦ)tΦ(z) = (V Φ)t((RΦ)−1)t(CΦ)tΦ(z) = (V Φ)t((RΦ)−1)tM tKtz, where Ktz ∈ IRn and Ktz(i) = ⟨Φ(ai), Φ(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(n2d). The computation of (CΦ)tCΦ 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 tKM is already computed in Line 2. In Line 5, the computation of ZΦ = EtKM(RΦ)−1 = (I −1 neet)KM(RΦ)−1 = KM(RΦ)−1 −1 n e (etKM)(RΦ)−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(n2d). 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 (CΦ)tCΦ 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. The key to AKDA/QR is the efficient computation of (CΦ)tCΦ, where CΦ = [mΦ 1 , · · · , mΦ c ] and mΦ j = 1 nj i∈Πj Φ(ai). AKDA/QR aims to find x∗ j in the original space such that Φ(x∗ j) approximates mΦ j . Mathematically, the optimal x∗ j can be computed by solving the following optimization problem: min xj∈Rd ∥Φ(xj) −1 nj i∈Πj Φ(ai)∥2 for j = 1, · · · , c. (5) 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 −y∥2/σ), where σ is the bandwidth parameter. The optimization problem in (5) is convex if for each j = 1, · · · , c and for all i ∈Πj, ∥ 2 σ (xj −ai)∥≤1 (6) Proof. It is easy to check that, for the Gaussian kernel, the optimization problem in (5) reduces to: min xj∈Rd f(xj) for j = 1, · · · , c, (7) where f(x) = i∈Πj fi(x) and fi(x) = −exp(−∥x −ai∥2/σ). The Hessian matrix of fi(x) is H(fi) = 2 σexp(−∥x −ai∥2/σ)(I −2 σ(x −ai)(x −ai)t). It is easy to show that if ∥ 2 σ(x −ai)∥≤1, for all i ∈Πj, 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, σ 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 σ, 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 approximate solution of the optimization problem in (7). Experiments show that choosing x∗ j = 1 nj i∈Πj 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 CΦ can be approximated by CΦ ≈[Φ(x∗ 1) . . . Φ(x∗ c)] (≡ˆCΦ) (8) and ( ˆCΦ)t ˆCΦ = ˆK, (9) Algorithm 3: AKDA/QR /* Stage I: */ 1. Compute x∗ j = 1 nj i∈Πj ai, for j = 1, · · · , c; 2. Construct kernel matrix ˆK as in Eq. (9); 3. Compute ˆRΦ from the Cholesky Decomposition of ˆK; /* Stage II: */ 4. ˆY Φ ←N t ˆK( ˆRΦ)−1; 5. ˆZΦ ←Et ˆKtc( ˆRΦ)−1; 6. ˆBΦ ←( ˆY Φ)t ˆY Φ; 7. ˆT Φ ←( ˆZΦ)t ˆZΦ; 8. Compute the c eigenvectors ˆφΦ i of ( ˆT Φ + µIc)−1 ˆBΦ, with decreasing eigenvalues; 9. ˆV Φ ←[ˆφΦ 1 , ˆφΦ 2 , · · · , ˆφΦ c ]; 10. ˆGΦ ←ˆCΦ( ˆRΦ)−1 ˆV Φ; PCA LDA/QR GDA KDA/QR AKDA/QR time O(n2d) O(ndc) O(n2d + n3) O(n2d) O(ndc) space O(nd) O(nc) O(n2) O(n2) 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. where ˆK(i, j) = ⟨Φ(x∗ i ), Φ(x∗ j)⟩and ˆK ∈Rc×c. The Cholesky decomposition of ˆK will give us ˆRΦ by ˆK = ( ˆRΦ)t ˆRΦ. It follows that ˆHΦ b = ˆCΦN, and ˆY Φ = N t ˆK( ˆRΦ)−1. Similarly, ˆZΦ = Et ˆKtc( ˆRΦ)−1, where N and E are defined as in Section 3, and ˆKtc(i, j) = ⟨Φ(ai), Φ(x∗ 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 It takes O(dn) in Line 1. The construction of the matrix ˆK in Line 2 takes O(c2d). The Cholesky Decomposition in Line 3 takes O(c3) [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: σ for the kernel function and µ for the regularization. Experiments show that choosing σ = 100000 and µ = 0.15 for KDA/QR, and σ = 100000 and µ = 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 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1 3 4 5 6 7 8 Accuracy Number of training samples per class PCA LDA/QR KDA/QR AKDA/QR 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 3 4 5 6 7 8 Accuracy Number of training samples per class PCA LDA/QR KDA/QR AKDA/QR 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 PCA LDA/QR GDA KDA/QR AKDA/QR 3 0.8611 0.8561 0.8782 0.9132 0.9118 4 0.8938 0.9083 0.9270 0.9321 0.9300 5 0.9320 0.9385 0.9535 0.9625 0.9615 6 0.9512 0.9444 0.9668 0.9737 0.9744 7 0.9633 0.9692 0.9750 0.9825 0.9815 8 0.9713 0.9713 0.9938 0.9875 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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Semi-parametric exponential family PCA Sajama Alon Orlitsky Department of Electrical and Computer Engineering University of California at San Diego, La Jolla, CA 92093 sajama@ucsd.edu and alon@ece.ucsd.edu 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(θ) and a conditional or component distribution P(x|θ), i.e., P(x) = R P(θ)P(x|θ)dθ. 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(θ) 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 θ. 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|θ) to be any member of the exponential family of distributions. Use of exponential family distributions for P(x|θ) 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 x1, . . . , xn are outcomes of iid draws of a random variable whose distribution P(x) = R P(θ)P(x|θ)dθ is determined by the latent distribution P(θ) and the conditional distribution P(x|θ). This can also be viewed as a mixture density with P(θ) being the mixing distribution, the mixture components labelled by θ and P(x|θ) being the component distribution corresponding to θ. 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|θ). Conditional distribution P(x|θ): We assume that P(θ) adequately models the dependencies among the components of x and hence that the components of x are independent when conditioned upon θ, i.e., P(x|θ) = ΠjP(xj|θj), where xj and θj are the j’th components of x and θ. 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|θ) = log P0(x) + xθ −G(θ) where θ is called the natural parameter and G(θ) is a function that ensures that the probabilities sum to one. An important property of this family is that the mean µ of a distribution and its natural parameter θ are related through a monotone invertible, nonlinear function µ = G′(θ) = g(θ). 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(θ): Like previous latent variable methods, including PCA, we constrain the latent variable θ to an ℓ-dimensional Euclidean subspace of Rd 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 θ can take is to write it as an invertible linear transformation of another random variable which takes values a ∈Rℓ, θ = aV + b where V is an ℓ× d rotation matrix and b is a d-dimensional displacement vector. Hence any distribution PΘ(θ) satisfying the low dimensional constraints can be represented using a triple (P(a), V, b), where P(a) is a distribution over Rℓ. 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 θ (or the corresponding a related to θ by θ = aV + b) that generated it, since the conditional distribution P(x|θ) 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 ℓ-dimensional space, any of its estimators like the posterior mean or mode (MAP estimate) can be used to represent x in ℓdimensions. 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 θ on (V, b) that is closest according to the appropriate Bregman distance (it can be shown that there is a unique such θopt 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|θk), 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, Π, σ} and let D be any direction parallel to the plane (V, b) along which the latent distribution P(θ) has non-zero variance. Since Gaussian noise with variance σ is added to this latent distribution to obtain P(x), variance of P(x) along D will be greater than σ. The variance of P(x) along any direction perpendicular to (V, b) will be exactly σ. 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 ℓ-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 Π = (π1, . . . , πc). Associated with each mixture component (indexed by k) are parameter vectors θk and ak which are related by θk = akV + 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, ˆzik = E(zik) = πkP(xi/θk) Pc m=1 πmP(xi/θm) ; ˜xkj = Pn i=1 ˆzikxij Pn i=1 ˆzik In the M-step we update Π, V , b, and ak in the following manner πk = Pn i=1 ˆzik Pn i=1 Pc m=1 zim = Pn i=1 ˆzik n ai is updated by adding δai calculated using (V ΩiV ′)δai = GRi ; [Ωi]qq = ∂g(θiq) ∂θiq ; [GRi]l1 = d X j=1 (˜xij −g(θij))Vlj Here the function g(θ) 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 δvs calculated using (A′ΩsA)δvs = GRs ; [Ωs]kk = ∂g(θks) ∂θks ; [GRs]l1 = c X k′=1 (˜xk′s −g(θk′s))Ak′l Each component of vector b, bs, is updated by adding δbs calculated using Hsδbs = GRs ; Hs = c X k′=1 ∂g(θk′s) ∂θk′s ; GRs = c X k′=1 (˜xk′s −g(θk′s)) 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 πk < C1, then drop the k’th component • Nearby components : If max i |P(xi|θk1)−P(x|θk2)| < C2, then drop either k1’th or k2’th component The value of C1 should be Θ(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(cdℓ2). For the Gaussian case, the E-step only takes O(cℓn) 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|θj). 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 ℓ, an alternative method is to pick ℓwhich 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 ℓwould 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 Rℓand 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 γ0 be the ‘true’ parameter from which observed samples are drawn. Let C0 be the set of all parameters γ corresponding to the ‘true’ distribution F(x/γ0) (i.e., C0 = {γ : F(x/γ) = F(x/γ0) ∀x}). Let ˆγn be an estimator of γ based on n observed samples of X and let ˆΓ be the quotient topological space obtained from Γ obtained by identifying the set C0 to a point ˆγ0. De£nition The sequence of estimators {ˆγn, n = 1, . . . , ∞} is said to be strongly consistent in the sense of Redner if limm→∞ˆγn = ˆγ0 almost surely. Theorem If P(a) is assumed to be zero outside a bounded subset of Rℓ, 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. DENSITY ISOTROPIC PPCA SP-PCA FULL GAUSSIAN ℓ=1 ℓ=2 ℓ=3 ℓ=1 ℓ=2 ℓ=3 GAUSSIAN ERROR 50.39 38.03 34.71 34.76 36.85 30.99 28.54 343.83 4 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 ℓ(the projected space dimension) starting with the zero dimensional model of an isotropic Gaussian. For a £xed lower dimension ℓ, 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 ℓ= 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. −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 (a) PCA −50 −40 −30 −20 −10 0 10 20 −40 −30 −20 −10 0 10 20 (b) Expontl. PCA −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (c) GTM −60 −40 −20 0 20 40 60 80 100 −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. −4 −3 −2 −1 0 1 2 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 1 (a) PCA −20 −15 −10 −5 0 5 10 15 20 25 30 0 2 4 6 8 10 12 14 (b) Exponential PCA −200 −150 −100 −50 0 −150 −100 −50 0 50 (c) SP-PCA −200 −150 −100 −50 0 −150 −100 −50 0 50 (d) Test data - SP-PCA −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (e) Bernoulli GTM −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 (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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Self-Tuning Spectral Clustering Lihi Zelnik-Manor Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125, USA lihi@vision.caltech.edu Pietro Perona Department of Electrical Engineering California Institute of Technology Pasadena, CA 91125, USA perona@vision.caltech.edu http://www.vision.caltech.edu/lihi/Demos/SelfTuningClustering.html Abstract We study a number of open issues in spectral clustering: (i) Selecting the appropriate scale of analysis, (ii) Handling multi-scale data, (iii) Clustering with irregular background clutter, and, (iv) Finding automatically the number of groups. We first propose that a ‘local’ scale should be used to compute the affinity between each pair of points. This local scaling leads to better clustering especially when the data includes multiple scales and when the clusters are placed within a cluttered background. We further suggest exploiting the structure of the eigenvectors to infer automatically the number of groups. This leads to a new algorithm in which the final randomly initialized k-means stage is eliminated. 1 Introduction Clustering is one of the building blocks of modern data analysis. Two commonly used methods are K-means and learning a mixture-model using EM. These methods, which are based on estimating explicit models of the data, provide high quality results when the data is organized according to the assumed models. However, when it is arranged in more complex and unknown shapes, these methods tend to fail. An alternative clustering approach, which was shown to handle such structured data is spectral clustering. It does not require estimating an explicit model of data distribution, rather a spectral analysis of the matrix of point-to-point similarities. A first set of papers suggested the method based on a set of heuristics (e.g., [8, 9]). A second generation provided a level of theoretical analysis, and suggested improved algorithms (e.g., [6, 10, 5, 4, 3]). There are still open issues: (i) Selection of the appropriate scale in which the data is to be analyzed, (ii) Clustering data that is distributed according to different scales, (iii) Clustering with irregular background clutter, and, (iv) Estimating automatically the number of groups. We show here that it is possible to address these issues and propose ideas to tune the parameters automatically according to the data. 1.1 Notation and the Ng-Jordan-Weiss (NJW) Algorithm The analysis and approaches suggested in this paper build on observations presented in [5]. For completeness of the text we first briefly review their algorithm. Given a set of n points S = {s1, . . . , sn} in Rl cluster them into C clusters as follows: 1. Form the affinity matrix A ∈Rn×n defined by Aij = exp ( −d2(si,sj) σ2 ) for i ̸= j and Aii = 0, where d(si, sj) is some distance function, often just the Euclidean σ = 0.041235 σ = 0.054409 σ = 0.035897 σ = 0.03125 σ = 0.015625 σ = 0.35355 σ = 1 Figure 1: Spectral clustering without local scaling (using the NJW algorithm.) Top row: When the data incorporates multiple scales standard spectral clustering fails. Note, that the optimal σ for each example (displayed on each figure) turned out to be different. Bottom row: Clustering results for the top-left point-set with different values of σ. This highlights the high impact σ has on the clustering quality. In all the examples, the number of groups was set manually. The data points were normalized to occupy the [−1, 1]2 space. distance between the vectors si and sj. σ is a scale parameter which is further discussed in Section 2. 2. Define D to be a diagonal matrix with Dii = n j=1 Aij and construct the normalized affinity matrix L = D−1/2AD−1/2. 3. Manually select a desired number of groups C. 4. Find x1, . . . , xC, the C largest eigenvectors of L, and form the matrix X = [x1, . . . , xC] ∈Rn×C. 5. Re-normalize the rows of X to have unit length yielding Y ∈Rn×C, such that Yij = Xij/( j X2 ij)1/2. 6. Treat each row of Y as a point in RC and cluster via k-means. 7. Assign the original point si to cluster c if and only if the corresponding row i of the matrix Y was assigned to cluster c. In Section 2 we analyze the effect of σ on the clustering and suggest a method for setting it automatically. We show that this allows handling multi-scale data and background clutter. In Section 3 we suggest a scheme for finding automatically the number of groups C. Our new spectral clustering algorithm is summarized in Section 4. We conclude with a discussion in Section 5. 2 Local Scaling As was suggested by [6] the scaling parameter is some measure of when two points are considered similar. This provides an intuitive way for selecting possible values for σ. The selection of σ is commonly done manually. Ng et al. [5] suggested selecting σ automatically by running their clustering algorithm repeatedly for a number of values of σ and selecting the one which provides least distorted clusters of the rows of Y . This increases significantly the computation time. Additionally, the range of values to be tested still has to be set manually. Moreover, when the input data includes clusters with different local statistics there may not be a singe value of σ that works well for all the data. Figure 1 illustrates the high impact σ has on clustering. When the data contains multiple scales, even using the optimal σ fails to provide good clustering (see examples at the right of top row). (a) (b) (c) Figure 2: The effect of local scaling. (a) Input data points. A tight cluster resides within a background cluster. (b) The affinity between each point and its surrounding neighbors is indicated by the thickness of the line connecting them. The affinities across clusters are larger than the affinities within the background cluster. (c) The corresponding visualization of affinities after local scaling. The affinities across clusters are now significantly lower than the affinities within any single cluster. Introducing Local Scaling: Instead of selecting a single scaling parameter σ we propose to calculate a local scaling parameter σi for each data point si. The distance from si to sj as ‘seen’ by si is d(si, sj)/σi while the converse is d(sj, si)/σj. Therefore the square distance d2 of the earlier papers may be generalized as d(si, sj)d(sj, si)/σiσj = d2(si, sj)/σiσj The affinity between a pair of points can thus be written as: ˆAij = exp −d2(si, sj) σiσj (1) Using a specific scaling parameter for each point allows self-tuning of the point-to-point distances according to the local statistics of the neighborhoods surrounding points i and j. The selection of the local scale σi can be done by studying the local statistics of the neighborhood of point si. A simple choice, which is used for the experiments in this paper, is: σi = d(si, sK) (2) where sK is the K’th neighbor of point si. The selection of K is independent of scale and is a function of the data dimension of the embedding space. Nevertheless, in all our experiments (both on synthetic data and on images) we used a single value of K = 7, which gave good results even for high-dimensional data (the experiments with high-dimensional data were left out due to lack of space). Figure 2 provides a visualization of the effect of the suggested local scaling. Since the data resides in multiple scales (one cluster is tight and the other is sparse) the standard approach to estimating affinities fails to capture the data structure (see Figure 2.b). Local scaling automatically finds the two scales and results in high affinities within clusters and low affinities across clusters (see Figure 2.c). This is the information required for separation. We tested the power of local scaling by clustering the data set of Figure 1, plus four additional examples. We modified the Ng-Jordan-Weiss algorithm reviewed in Section 1.1 substituting the locally scaled affinity matrix ˆA (of Eq. (1)) instead of A. Results are shown in Figure 3. In spite of the multiple scales and the various types of structure, the groups now match the intuitive solution. 3 Estimating the Number of Clusters Having defined a scheme to set the scale parameter automatically we are left with one more free parameter: the number of clusters. This parameter is usually set manually and Figure 3: Our clustering results. Using the algorithm summarized in Section 4. The number of groups was found automatically. 2 4 6 8 10 0.95 0.96 0.97 0.98 0.99 1 2 4 6 8 10 0.95 0.96 0.97 0.98 0.99 1 2 4 6 8 10 0.95 0.96 0.97 0.98 0.99 1 2 4 6 8 10 0.95 0.96 0.97 0.98 0.99 1 Figure 4: Eigenvalues. The first 10 eigenvalues of L corresponding to the top row data sets of Figure 3. not much research has been done as to how might one set it automatically. In this section we suggest an approach to discovering the number of clusters. The suggested scheme turns out to lead to a new spatial clustering algorithm. 3.1 The Intuitive Solution: Analyzing the Eigenvalues One possible approach to try and discover the number of groups is to analyze the eigenvalues of the affinity matrix. The analysis given in [5] shows that the first (highest magnitude) eigenvalue of L (see Section 1.1) will be a repeated eigenvalue of magnitude 1 with multiplicity equal to the number of groups C. This implies one could estimate C by counting the number of eigenvalues equaling 1. Examining the eigenvalues of our locally scaled matrix, corresponding to clean data-sets, indeed shows that the multiplicity of eigenvalue 1 equals the number of groups. However, if the groups are not clearly separated, once noise is introduced, the values start to deviate from 1, thus the criterion of choice becomes tricky. An alternative approach would be to search for a drop in the magnitude of the eigenvalues (this was pursued to some extent by Polito and Perona in [7]). This approach, however, lacks a theoretical justification. The eigenvalues of L are the union of the eigenvalues of the sub-matrices corresponding to each cluster. This implies the eigenvalues depend on the structure of the individual clusters and thus no assumptions can be placed on their values. In particular, the gap between the C’th eigenvalue and the next one can be either small or large. Figure 4 shows the first 10 eigenvalues corresponding to the top row examples of Figure 3. It highlights the different patterns of distribution of eigenvalues for different data sets. 3.2 A Better Approach: Analyzing the Eigenvectors We thus suggest an alternative approach which relies on the structure of the eigenvectors. After sorting L according to clusters, in the “ideal” case (i.e., when L is strictly block diagonal with blocks L(c), c = 1, . . . , C), its eigenvalues and eigenvectors are the union of the eigenvalues and eigenvectors of its blocks padded appropriately with zeros (see [6, 5]). As long as the eigenvalues of the blocks are different each eigenvector will have non-zero values only in entries corresponding to a single block/cluster: ˆX = x(1) −→0 −→0 −→0 · · · −→0 −→0 −→0 x(C) n×C where x(c) is an eigenvector of the sub-matrix L(c) corresponding to cluster c. However, as was shown above, the eigenvalue 1 is bound to be a repeated eigenvalue with multiplicity equal to the number of groups C. Thus, the eigensolver could just as easily have picked any other set of orthogonal vectors spanning the same subspace as ˆX’s columns. That is, ˆX could have been replaced by X = ˆXR for any orthogonal matrix R ∈RC×C. This, however, implies that even if the eigensolver provided us the rotated set of vectors, we are still guaranteed that there exists a rotation ˆR such that each row in the matrix X ˆR has a single non-zero entry. Since the eigenvectors of L are the union of the eigenvectors of its individual blocks (padded with zeros), taking more than the first C eigenvectors will result in more than one non-zero entry in some of the rows. Taking fewer eigenvectors we do not have a full basis spanning the subspace, thus depending on the initial X there might or might not exist such a rotation. Note, that these observations are independent of the difference in magnitude between the eigenvalues. We use these observations to predict the number of groups. For each possible group number C we recover the rotation which best aligns X’s columns with the canonical coordinate system. Let Z ∈Rn×C be the matrix obtained after rotating the eigenvector matrix X, i.e., Z = XR and denote Mi = maxj Zij. We wish to recover the rotation R for which in every row in Z there will be at most one non-zero entry. We thus define a cost function: J = n i=1 C j=1 Z2 ij M 2 i (3) Minimizing this cost function over all possible rotations will provide the best alignment with the canonical coordinate system. This is done using the gradient descent scheme described in Appendix A. The number of groups is taken as the one providing the minimal cost (if several group numbers yield practically the same minimal cost, the largest of those is selected). The search over the group number can be performed incrementally saving computation time. We start by aligning the top two eigenvectors (as well as possible). Then, at each step of the search (up to the maximal group number), we add a single eigenvector to the already rotated ones. This can be viewed as taking the alignment result of the previous group number as an initialization to the current one. The alignment of this new set of eigenvectors is extremely fast (typically a few iterations) since the initialization is good. The overall run time of this incremental procedure is just slightly longer than aligning all the eigenvectors in a non-incremental way. Using this scheme to estimate the number of groups on the data set of Figure 3 provided a correct result for all but one (for the right-most dataset at the bottom row we predicted 2 clusters instead of 3). Corresponding plots of the alignment quality for different group numbers are shown in Figure 5. Yu and Shi [11] suggested rotating normalized eigenvectors to obtain an optimal segmentation. Their method iterates between non-maximum suppression (i.e., setting Mi = 1 and Zij = 0 otherwise) and using SVD to recover the rotation which best aligns the columns of X with those of Z. In our experiments we noticed that this iterative method can easily get stuck in local minima and thus does not reliably find the optimal alignment and the group number. Another related approach is that suggested by Kannan et al. [3] who assigned points to clusters according to the maximal entry in the corresponding row of the eigenvector matrix. This works well when there are no repeated eigenvalues as then the eigenvectors 2 4 6 8 10 0 0.05 0.1 0.15 0.2 2 4 6 8 10 0 0.02 0.04 0.06 0.08 2 4 6 8 10 0 0.05 0.1 0.15 0.2 2 4 6 8 10 0 0.02 0.04 0.06 0.08 Figure 5: Selecting Group Number. The alignment cost (of Eq. (3)) for varying group numbers corresponding to the top row data sets of Figure 3. The selected group number marked by a red circle, corresponds to the largest group number providing minimal cost (costs up to 0.01% apart were considered as same value). corresponding to different clusters are not intermixed. Kannan et al. used a non-normalized affinity matrix thus were not certain to obtain a repeated eigenvalue, however, this could easily happen and then the clustering would fail. 4 A New Algorithm Our proposed method for estimating the number of groups automatically has two desirable by-products: (i) After aligning with the canonical coordinate system, one can use non-maximum suppression on the rows of Z, thus eliminating the final iterative k-means process, which often requires around 100 iterations and depends highly on its initialization. (ii) Since the final clustering can be conducted by non-maximum suppression, we obtain clustering results for all the inspected group numbers at a tiny additional cost. When the data is highly noisy, one can still employ k-means, or better, EM, to cluster the rows of Z. However, since the data is now aligned with the canonical coordinate scheme we can obtain by non-maximum suppression an excellent initialization so very few iterations suffice. We summarize our suggested algorithm: Algorithm: Given a set of points S = {s1, . . . , sn} in Rl that we want to cluster: 1. Compute the local scale σi for each point si ∈S using Eq. (2). 2. Form the locally scaled affinity matrix ˆA ∈Rn×n where ˆAij is defined according to Eq. (1) for i ̸= j and ˆAii = 0. 3. Define D to be a diagonal matrix with Dii = n j=1 ˆAij and construct the normalized affinity matrix L = D−1/2 ˆAD−1/2. 4. Find x1, . . . , xC the C largest eigenvectors of L and form the matrix X = [x1, . . . , xC] ∈Rn×C, where C is the largest possible group number. 5. Recover the rotation R which best aligns X’s columns with the canonical coordinate system using the incremental gradient descent scheme (see also Appendix A). 6. Grade the cost of the alignment for each group number, up to C, according to Eq. (3). 7. Set the final group number Cbest to be the largest group number with minimal alignment cost. 8. Take the alignment result Z of the top Cbest eigenvectors and assign the original point si to cluster c if and only if maxj(Z2 ij) = Z2 ic. 9. If highly noisy data, use the previous step result to initialize k-means, or EM, clustering on the rows of Z. We tested the quality of this algorithm on real data. Figure 6 shows intensity based image segmentation results. The number of groups and the corresponding segmentation were obtained automatically. In this case same quality of results were obtained using non-scaled affinities, however, this required manual setting of both σ (different values for different images) and the number of groups, whereas our result required no parameter settings. Figure 6: Automatic image segmentation. Fully automatic intensity based image segmentation results using our algorithm. More experiments and results on real data sets can be found on our web-page http://www.vision.caltech.edu/lihi/Demos/SelfTuningClustering.html 5 Discussion & Conclusions Spectral clustering practitioners know that selecting good parameters to tune the clustering process is an art requiring skill and patience. Automating spectral clustering was the main motivation for this study. The key ideas we introduced are three: (a) using a local scale, rather than a global one, (b) estimating the scale from the data, and (c) rotating the eigenvectors to create the maximally sparse representation. We proposed an automated spectral clustering algorithm based on these ideas: it computes automatically the scale and the number of groups and it can handle multi-scale data which are problematic for previous approaches. Some of the choices we made in our implementation were motivated by simplicity and are perfectible. For instance, the local scale σ might be better estimated by a method which relies on more informative local statistics. Another example: the cost function in Eq. (3) is reasonable, but by no means the only possibility (e.g. the sum of the entropy of the rows Zi might be used instead). Acknowledgments: Finally, we wish to thank Yair Weiss for providing us his code for spectral clustering. This research was supported by the MURI award number SA3318 and by the Center of Neuromorphic Systems Engineering award number EEC-9402726. References [1] G. H. Golub and C. F. Van Loan “Matrix Computation”, John Hopkins University Press, 1991, Second Edition. [2] V. K. Goyal and M. Vetterli “Block Transform by Stochastic Gradient Descent” IEEE Digital Signal Processing Workshop, 1999, Bryce Canyon, UT, Aug. 1998 [3] R. Kannan, S. Vempala and V.Vetta “On Spectral Clustering – Good, Bad and Spectral” In Proceedings of the 41st Annual Symposium on Foundations of Computer Sceince, 2000. [4] M. Meila and J. Shi “Learning Segmentation by Random Walks” In Advances in Neural Information Processing Systems 13, 2001 [5] A. Ng, M. Jordan and Y. Weiss “On spectral clustering: Analysis and an algorithm” In Advances in Neural Information Processing Systems 14, 2001 [6] P. Perona and W. T. Freeman “A Factorization Approach to Grouping” Proceedings of the 5th European Conference on Computer Vision, Volume I, pp. 655–670 1998. [7] M. Polito and P. Perona “Grouping and dimensionality reduction by locally linear embedding” Advances in Neural Information Processing Systems 14, 2002 [8] G.L. Scott and H.C. Longuet-Higgins “Feature grouping by ‘relocalisation’ of eigenvectors of the proximity matrix” In Proc. British Machine Vision Conference, Oxford, UK, pages 103–108, 1990. [9] J. Shi and J. Malik “Normalized Cuts and Image Segmentation” IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888-905, August 2000. [10] Y. Weiss “Segmentation Using Eigenvectors: A Unifying View” International Conference on Computer Vision, pp.975–982,September,1999. [11] S. X. Yu and J. Shi “Multiclass Spectral Clustering” International Conference on Computer Vision, Nice, France, pp.11–17,October,2003. A Recovering the Aligning Rotation To find the best alignment for a set of eigenvectors we adopt a gradient descent scheme similar to that suggested in [2]. There, Givens rotations where used to recover a rotation which diagonalizes a symmetric matrix by minimizing a cost function which measures the diagonality of the matrix. Similarly, here, we define a cost function which measures the alignment quality of a set of vectors and prove that the gradient descent, using Givens rotations, converges. The cost function we wish to minimize is that of Eq. (3). Let mi = j such that Zij = Zimi = Mi. Note, that the indices mi of the maximal entries of the rows of X might be different than those of the optimal Z. A simple non-maximum supression on the rows of X can provide a wrong result. Using the gradient descent scheme allows to increase the cost corresponding to part of the rows as long as the overall cost is reduced, thus enabling changing the indices mi. Similar to [2] we wish to represent the rotation matrix R in terms of the smallest possible number of parameters. Let ˜Gi,j,θ denote a Givens rotation [1] of θ radians (counterclockwise) in the (i, j) coordinate plane. It is sufficient to consider Givens rotations so that i < j, thus we can use a convenient index re-mapping Gk,θ = ˜Gi,j,θ, where (i, j) is the kth entry of a lexicographical list of (i, j) ∈{1, 2, . . . , C}2 pairs with i < j. Hence, finding the aligning rotation amounts to minimizing the cost function J over Θ ∈[−π/2, π/2)K. The update rule for Θ is: Θk+1 = Θk −α ∇J|Θ=Θk where α ∈R+ is the step size. We next compute the gradient of J and bounds on α for stability. For convenience we will further adopt the notation convention of [2]. Let U(a,b) = Ga,θaGa+1,θa+1 · · · Gb,θb where U(a,b) = I if b < a, Uk = U(k,k), and Vk = ∂ ∂θk Uk. Define A(k), 1 ≤k ≤K, element wise by A(k) ij = ∂Zij ∂θk . Since Z = XR we obtain A(k) = XU(1,k−1)VkU(k+1,K). We can now compute ∇J element wise: ∂J ∂θk = n i=1 C j=1 ∂ ∂θk Z2 ij M 2 i −1 = 2 n i=1 C j=1 Zij M 2 i A(k) ij −Z2 ij M 3 i ∂Mi ∂θk Due to lack of space we cannot describe in full detail the complete convergence proof. We thus refer the reader to [2] where it is shown that convergence is obtained when 1 −αFkl lie in the unit circle, where Fkl = ∂2J ∂θl∂θk Θ=0. Note, that at Θ = 0 we have Zij = 0 for j ̸= mi, Zimi = Mi, and ∂Mi ∂θk = ∂Zimi ∂θk = A(k) imi (i.e., near Θ = 0 the maximal entry for each row does not change its index). Deriving thus gives ∂2J ∂θl∂θk ij|Θ=0 = 2 n i=1 j̸=mi 1 M 2 i A(k) ij A(l) ij . Further substituting in the values for A(k) ij |Θ=0 yields: Fkl = ∂2J ∂θl∂θk ij|Θ=0 =
2#i s.t. mi = ik or mi = jk if k = l 0 otherwise where (ik, jk) is the pair (i, j) corresponding to the index k in the index re-mapping discussed above. Hence, by setting α small enough we get that 1 −αFkl lie in the unit circle and convergence is guaranteed.
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The Entire Regularization Path for the Support Vector Machine Trevor Hastie Department of Statistics Stanford University Stanford, CA 94305, USA hastie@stanford.edu Saharon Rosset IBM Watson Research Center P.O. Box 218 Yorktown Heights, N.Y. 10598 srosset@us.ibm.com Robert Tibshirani Department of Statistics Stanford University Stanford, CA 94305, USA tibs@stanford.edu Ji Zhu Department of Statistics University of Michigan Ann Arbor, MI 48109-1092 jizhu@umich.edu 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 min β0,β 1 2||β||2 + C n i=1 ξi, (1) subject to, for each i: yi(β0 + xT i β) ≥ 1 −ξi. Here the ξi 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 β0,β n i=1 [1 −yi(β0 + βT xi)]+ + λ 2 ||β||2. (2) The regularization parameter λ in (2) corresponds to 1/C, with C in (1). 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 (λ ↓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 f(x) = β0 + n i=1 θiK(x, xi), we solve [5] min β0,θ n i=1 [1 −yi(β0 + n j=1 θiK(xi, xj))] + λ 2 n j=1 n j′=1 θjθj′K(xj, x′ j). (3) Often the regularization parameter C (or λ) 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(−γ||x −x′||2). Figure 1 shows the test error as a function of C for these data, using four different values for γ. Here we see a dramatic range in the correct choice for C (or λ = 1/C). When γ = 5, the most regularized model is called for; when γ = 0.1, the least regularized. 1e−01 1e+01 1e+03 0.20 0.25 0.30 0.35 1e−01 1e+01 1e+03 1e−01 1e+01 1e+03 1e−01 1e+01 1e+03 Test Error Test Error Curves − SVM with Radial Kernel γ = 5 γ = 1 γ = 0.5 γ = 0.1 C = 1/λ Figure 1: Test error curves for the mixture example, using four different values for the radial kernel parameter γ. 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 [β0(C), β(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 ˆβ(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): min β,β0 n i=1 ξi + λ 2 βT β subject to 1 −yif(xi) ≤ξi; ξi ≥0; f(x) = β0 + βT x. (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 i=1 ξi + λ 2 βT β + n i=1 αi(1 −yif(xi) −ξi) − n i=1 γiξi (5) and set the derivatives to zero. This gives ∂ ∂β : β = 1 λ n i=1 αiyixi (6) ∂ ∂β0 : n i=1 yiαi = 0, (7) along with the KKT conditions αi(1 −yif(xi) −ξi) = 0 (8) γiξi = 0 (9) 1 −αi −γi = 0 (10) We see that 0 ≤αi ≤1, with αi = 1 when ξi > 0 (which is when yif(xi) < 1). Also when yif(xi) > 1, ξi = 0 since no cost is incurred, and αi = 0. When yif(xi) = 1, αi can lie between 0 and 1. The usual Lagrange multipliers associated with the solution to (1) are α′ i = αi/λ = Cαi. We prefer our formulation here since our αi ∈[0, 1], and this simplifies the definition of the paths we define. We wish to find the entire solution path for all values of λ ≥0. Our basic idea is as follows. We start with λ large and decrease it toward zero, keeping track of all the events that occur along the way. As λ decreases, ||β|| increases, and hence the width of the margin decreases. As this width decreases, points move from being inside to outside their margins. Their corresponding αi change from αi = 1 when they are inside their margin (yif(xi) < 1) to αi = 0 when they are outside their margin (yif(xi) > 1). By continuity, points must linger on the margin (yif(xi) = 1) while their αi decrease from 1 to 0. We will see that the αi(λ) trajectories are piecewise-linear in λ, 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 αi(λ) are piecewise linear in λ, then both α′ i(C) = Cαi(C) and β(C) are piecewise linear in C. It turns out that β0(C) is also piecewise linear in C. Our algorithm keeps track of the following sets: • M = {i : yif(xi) = 1, 0 ≤αi ≤1}, M for Margin • I = {i : yif(xi) < 1, αi = 1}, I for Inside the margin • O = {i : yif(xi) > 1, αi = 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 λ, ||β|| is small, and the the margin is very wide, all points are in O, and hence αi = 1∀i. From (6) this means the orientation of β is fixed until the αi change. The margin narrows as λ 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 β0. 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) = β0 + g(x), and the only change that occurs is that (6) is changed to g(xi) = 1 λ n j=1 αjyjK(xi, xj), i = 1, . . . , n, (11) or θj(λ) = αjyj/λ 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 α ∈[0, 1]. 2. A point from I has just entered M, with its value of αi initially 1. 3. A point from O has reentered M, with its value of αi 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 αi must change from 0 to 1 or vice versa. Since all points in M have yif(xi) = 1, we can establish a path for their αi. We use the subscript ℓto index the sets above immediately after the ℓth event has occurred. Suppose |Mℓ| = m, and let αℓ i, βℓ 0 and λℓbe the values of these parameters at the point of entry. Likewise f ℓis the function at this point. For convenience we define α0 = λβ0, and hence αℓ 0 = λℓβℓ 0. Since f(x) = 1 λ n j=1 yjαjK(x, xj) + α0 , (12) for λℓ> λ > λℓ+1 we can write f(x) = f(x) −λℓ λ f ℓ(x) + λℓ λ f ℓ(x) = 1 λ j∈Mℓ (αj −αℓ j)yjK(x, xj) + (α0 −αℓ 0) + λℓf ℓ(x) . (13) The second line follows because all the observations in Iℓhave their αi = 1, and those in Oℓhave their αi = 0, for this range of λ. Since each of the m points xi ∈Mℓare to stay on the margin, we have that 1 λ j∈Mℓ (αj −αℓ j)yiyjK(xi, xj) + yi(α0 −αℓ 0) + λℓ = 1, ∀i ∈Mℓ. (14) Writing δj = αℓ j −αj, from (14) we have j∈Mℓ δjyiyjK(xi, xj) + yiδ0 = λℓ−λ, ∀i ∈Mℓ. (15) Furthermore, since at all times n i=1 yiαi = 0, we have that j∈Mℓ yjδj = 0. (16) Equations (15) and (16) constitute m + 1 linear equations in m + 1 unknowns δj, and can be solved. The δj and hence αj will change linearly in λ, until the next event occurs: αj = αℓ j −(λℓ−λ)bj, j ∈{0} ∪Mℓ. (17) See [7] for more precise details on solving these equations. From (13) we have f(x) = λℓ λ
f ℓ(x) −hℓ(x) + hℓ(x), (18) where hℓ(x) = j∈Mℓ yjbjK(x, xj) + b0 (19) Thus the function itself changes in a piecewise-inverse manner in λ. Finding λℓ+1 The paths continue until one of the following events occur: 1. One of the αi for i ∈Mℓreaches a boundary (0 or 1). For each i the value of λ for which this occurs is easily established. 2. One of the points in Iℓor Oℓattains yif(xi) = 1. By examining these conditions, we can establish the largest λ < λℓfor which an event occurs, and hence establish λℓ+1 and update the sets. Termination In the separable case, we terminate when I becomes empty. At this point, all the ξi in (4) are zero, and further movement increases the norm of β unnecessarily. In the non-separable case, λ runs all the way down to zero. For this to happen without f “blowing up” in (18), we must have fℓ−hℓ= 0, and hence the boundary and margins remain fixed at a point where i ξi is as small as possible, and the margin is as wide as possible subject to this constraint. 3.1 Computational Complexity At any update event ℓalong the path of our algorithm, the main computational burden is solving the system of equations of size mℓ= |Mℓ|. While this normally involves O(m3 ℓ) computations, since Mℓ+1 differs from Mℓby typically one observation, inverse updating can reduce the computations to O(m2 ℓ). The computation of hℓ(xi) in (19) requires O(nmℓ) computations. Beyond that, several checks of cost O(n) are needed to evaluate the next move. 1e−04 1e−02 1e+00 0 20 40 60 80 100 111 11111 1111111 11111 1111111111111111 11111111111111111111111111111111111111111111111 11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11111111111111111111111111111 111111 1111111111111111111111111111111111111 111111111 111111111111111 11111 11 111 11 1 111111111 111 111 111 111111 111 11 1 111 11 11 11 111 1 11 11 111 1 1 11 11 1 1 1 11 111 1111111 1 1 1 1 1 11111111 111 11 1 111 111 1 1 1111 111 111 11 1 1 1111 1 11 111 1 11 111111 11 1 11111111 1 111 1 1 1 1 1 1 1 1 2222222222222222222222222222222222222222 22222222222222222222222 22222222222222222222222222222222 22222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 2222222222222222222222222222222222222222222222222222222222 2222222222 2222222222222222 2222222222 222 2 2 2 22222 222222 22 22222 2 22 222222 222 2 2 22222222 22222222 22222222222 2222222 22222 22222222222222 2222 22222222222222222 2 2222222 22222 2222 2222 222222 22222 222222 22222222 2 22222 2222222 2 2222 22 222222 2222 22222222222 2222222 222 22 22 22222 222 22 2 222222 22 2 22222222222 2222 222222 2 22 222222222222 2 222 22 2222222222 22 22222 2222222 222 2222 222 222 2 2222 22222 22 222 22 22 222222 2222 3333333333333333333333333333333333333333333333333333333 333333333333 333333333333333333333333 333333333333333333333 33333333333333333333333333333333 3 333333333333333333 33333333333333333333333 333 3333333333333 333 333 33333 3 333333333333333 333333 3333333333 333 333333333333 3333333333333 3 333 3 3 3 333333333 3 3333333333333333333333 33333 33333333 333333333 33333333 3 3 3 33 33333333333 3333333 333 33333 33 333 33 333 33 3333 33333 3 33 3 33 3333333 333 3333333333333333 3 333 3333333333333333333333 333 3 3 3 3 33 33333 3 3333 333 33 333 333 333333 3 333 3 33 3 33 333333333333 333 3 3333 33 333 33 3 333 333 333333 33 3 33 333 3 3333 3333 3 4 444444444444444444 44444444444 44444 44444444444444444 444444 4444 4444 4 44444444 44 4444444 44 4 4 44 44444 4 4 4 444 4 44 4444 4 4 44444444 44 444444 4 44 4 4444 4 44 4444 4 4 44444 44444 444444 444 444444 444 4 44444 444444 4 4444 4444 444 4 44 444444444 444 44 44 4 4 4 4 44 44444 4 444 4444 444444 44 4 44444 444 44444 44 4444444444444 44 444444 444444 44444 4444 4 444 44444 444 4 4444 44 4444 4 4 4 44 4444 444444444444 44444 4444 4444 4444 44 4 44 4444 44 4 44444 444 4 444 444 4 4 4444 4444 44444 4 γ = 0.1 γ = 0.5 γ = 1 γ = 5 λ Margin Size 111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 11 0 50 100 150 200 1e−15 1e−11 1e−07 1e−03 1e+01 222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222 2222222222222222 2 3 33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333 333333333333 3333333333333333333333333333333333333333333 3 4 4 4 44 444 44444 4444444444444 444 44 4444444 44444 44444444444444444 4444444444444 4444 4444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444444 4 4 γ = 0.1 γ = 0.5 γ = 1 γ = 5 Sequence Number Eigenvalue Figure 2: [Left] The margin sizes |Mℓ| as a function of λ, for different values of the radial-kernel parameter γ. The vertical lines show the positions used to compare the times with libsvm. [Right] The eigenvalues (on the log scale) for the kernel matrices Kγ corresponding to the four values of γ.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 γ, the effective dimension of the space is truncated. Although we have no hard results, our experience so far suggests that the total number Λ 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 Mℓis m, this suggests the total computational burden is O(cn2m + 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, γ = 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 λℓin 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 γ 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 γ 5 1 0.5 0.1 Training Errors 0 12 21 33 Effective Rank 200 177 143 76 Table 1: The number of minimal training errors for different values of the radial kernel scale parameter γ, for the mixture simulation example. Also shown is the effective rank of the 200 × 200 Gram matrix Kγ. 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 γ. Small γ implies wide, flat kernels, and a suppression of wiggly, “rough” functions. Writing (3) in matrix form, min β0,θ L[y, Kθ] + λ 2 θT Kθ, (20) we reparametrize using the eigen-decomposition of K = UDUT . Let Kθ = Uθ∗where θ∗= DUT θ. Then (20) becomes min β0,θ∗L[y, Uθ∗] + λ 2 θ∗T D−1θ∗. (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 γ. 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 λ [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 αi 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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The Laplacian PDF Distance: A Cost Function for Clustering in a Kernel Feature Space Robert Jenssen1∗, Deniz Erdogmus2, Jose Principe2, Torbjørn Eltoft1 1Department of Physics, University of Tromsø, Norway 2Computational 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: robertj@phys.uit.no 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 Φ : Rd →F, x →Φ(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 Φ(x1), . . . , Φ(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. ∀ψ ∈L2(C) : Z C k(x, y)ψ(x)ψ(y)dxdy ≥0, (2) then there exists a space F and a mapping Φ : Rd →F, such that by Mercer’s theorem [4] k(x, y) = ⟨Φ(x), Φ(y)⟩= NF X i=1 λiφi(x)φi(y), (3) where ⟨·, ·⟩denotes an inner product, the φi’s are the orthonormal eigenfunctions of the kernel and NF ≤∞[3]. In this case Φ(x) = [ p λ1φ1(x), p λ2φ2(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 = Λ, where the columns of E contains the eigenvectors of K and Λ is a diagonal matrix containing the non-negative eigenvalues ˜λ1, . . . , ˜λN, ˜λ1 ≥ · · · ≥˜λN. In [5], it was shown that the eigenfunctions and eigenvalues of (4) can be approximated as φj(xi) ≈ √ Neji, λj ≈ ˜ λj N , where eji denotes the ith element of the jth eigenvector. Hence, the mapping (4), can be approximated as Φ(xi) ≈[ q ˜λ1e1i, . . . , q ˜λNeNi]T . (5) Thus, the mapping is based on the eigenspectrum of K. The feature space data set may be represented in matrix form as ΦN×N = [Φ(x1), . . . , Φ(xN)]. Hence, Φ = Λ 1 2 ET . It may be desirable to truncate the mapping (5) to C-dimensions. Thus, only the C first rows of Φ are kept, yielding ˆΦ. It is well-known that ˆK = ˆΦ T ˆΦ is the best rank-C approximation to K wrt. the Frobenius norm [6]. The most widely used Mercer kernel is the radial-basis-function (RBF) k(x, y) = exp −||x −y||2 2σ2 . (6) 3 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] ˆf(x) = 1 N N X i=1 Wσ2(x, xi), (7) where Wσ2 is the Parzen window, or kernel, and σ2 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 Wσ2(x, xi) = 1 (2πσ2) d 2 exp −||x −xi||2 2σ2 , (8) 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 ˆh(x) = 1 N N X i=1 v(xi)Wσ2(x, xi). (9) This estimator is asymptotically unbiased, which can be shown as follows Ef ( 1 N N X i=1 v(xi)Wσ2(x, xi) ) = Z v(z)f(z)Wσ2(x, z)dz = [v(x)f(x)] ∗Wσ2(x), (10) where Ef(·) denotes expectation with respect to the density f(x). In the limit as N →∞and σ(N) →0, we have lim N→∞ σ(N)→0 [v(x)f(x)] ∗Wσ2(x) = v(x)f(x). (11) 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 σ(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: σopt = σX 4N −1(2d + 1)−1 1 d+4 , (12) where d is the dimensionality of the data and σ2 X = d−1 P i ΣXii, where ΣXii 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 data set. Now define the f −1 weighted inner product between p(x) and q(x) as ⟨p, q⟩f ≡ R p(x)q(x)f −1(x)dx. In such an inner product space, the Cauchy-Schwarz inequality holds, that is, ⟨p, q⟩2 f ≤⟨p, p⟩f ⟨q, q⟩f. Based on this discussion, an information theoretic distance measure between the two pdfs can be expressed as DL = −log ⟨p, q⟩f q ⟨p, p⟩f ⟨q, q⟩f ≥0. (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 q(x), the density of C2. Let h(x) = f −1 2 (x)p(x) and g(x) = f −1 2 (x)q(x). Hence, we may write L = R h(x)g(x)dx qR h2(x)dx R g2(x)dx . (14) We estimate h(x) and g(x) by the generalized Parzen kernel estimators, as follows ˆh(x) = 1 N1 N1 X i=1 f −1 2 (xi)Wσ2(x, xi), ˆg(x) = 1 N2 N2 X j=1 f −1 2 (xj)Wσ2(x, xj). (15) The approach taken, is to substitute these estimators into (14), to obtain Z h(x)g(x)dx ≈ Z 1 N1 N1 X i=1 f −1 2 (xi)Wσ2(x, xi) 1 N2 N2 X j=1 f −1 2 (xj)Wσ2(x, xj) = 1 N1N2 N1,N2 X i,j=1 f −1 2 (xi)f −1 2 (xj) Z Wσ2(x, xi)Wσ2(x, xj)dx = 1 N1N2 N1,N2 X i,j=1 f −1 2 (xi)f −1 2 (xj)W2σ2(xi, xj), (16) where in the last step, the convolution theorem for Gaussians has been employed. Similarly, we have Z h2(x)dx ≈ 1 N 2 1 N1,N1 X i,i′=1 f −1 2 (xi)f −1 2 (xi′)W2σ2(xi, xi′), (17) Z g2(x)dx ≈ 1 N 2 2 N2,N2 X j,j′=1 f −1 2 (xj)f −1 2 (xj′)W2σ2(xj, xj′). (18) Now we define the matrix Kf, such that Kfij = Kf(xi, xj) = f −1 2 (xi)f −1 2 (xj)K(xi, xj), (19) where K(xi, xj) = W2σ2(xi, xj) for i, j = 1, . . . , N and N = N1 + N2. As a consequence, (14) can be re-written as follows L = PN1,N2 i,j=1 Kf(xi, xj) qPN1,N1 i,i′=1 Kf(xi, xi′) PN2,N2 j,j′=1 Kf(xj, xj′) (20) 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 ψ1, . . . , ψN N X i=1 N X j=1 ψiψjK(xi, xj) ≥0, (21) in analogy to (3). Moreover, this means that N X i=1 N X j=1 ψiψjf −1 2 (xi)f −1 2 (xj)K(xi, xj) = N X i=1 N X j=1 ψiψjKf(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) = ⟨Φf(xi), Φf(xj)⟩. Consequently, (20) can be written as follows L = PN1,N2 i,j=1 ⟨Φf(xi), Φf(xj)⟩ qPN1,N1 i,i′=1 ⟨Φf(xi), Φf(xi′)⟩PN2,N2 j,j′=1 ⟨Φf(xj), Φf(xj′)⟩ = D 1 N1 PN1 i=1 Φf(xi), 1 N2 PN2 j=1 Φf(xj) E rD 1 N1 PN1 i=1 Φf(xi), 1 N1 PN1 i′=1 Φf(xi′) E D 1 N2 PN2 j=1 Φf(xj), 1 N2 PN2 j′=1 Φf(xj′) E = m1f , m2f ||m1f ||||m2f || = cos ̸ (m1f , m2f ), (23) where mif = 1 Ni PNi l=1 Φf(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. Note that f 1 2 (xi) can be estimated from the data by the traditional Parzen pdf estimator as follows f 1 2 (xi) = v u u t 1 N N X l=1 Wσ2 f (xi, xl) = p di. (24) Define the matrix D = diag(d1, . . . , dN). Then Kf can be expressed as Kf = D−1 2 KD−1 2 . (25) Quite interestingly, for σ2 f = 2σ2, 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 σopt = 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 X i=1 X j̸=i ⟨pi, pj⟩f C q ⟨pi, pi⟩f ⟨pj, pj⟩f . (26) 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 1It 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 λi to 1 −λi. (a) Data set (b) Parzen pdf estimate 0 0 (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) = P1p(x)+P2q(x), were Pi, i = 1, 2, are the priors. Assume that the two clusters are well separated, such that for xi ∈C1, f(xi) ≈P1p(xi), while for xi ∈C2, f(xi) ≈P2q(xi). Let us examine the numerator of (14) in this case. It can be approximated as R p(x)q(x) f(x) dx ≈ Z C1 p(x)q(x) f(x) dx + Z C2 p(x)q(x) f(x) dx ≈1 P1 Z C1 q(x)dx + 1 P2 Z C2 p(x)dx. (27) By performing a similar calculation for the denominator of (14), it can be shown to be approximately equal to 1 √P1P1 . Hence, the Laplacian pdf distance can be written as a risk function, given by L ≈ p P1P2 1 P1 Z C1 q(x)dx + 1 P2 Z C2 p(x)dx . (28) Note that if P1 = P2 = 1 2, then L = 2Pe, where Pe is the probability of error when assigning data points to the two clusters, that is Pe = P1 Z C1 q(x)dx + P2 Z C2 p(x)dx. (29) Thus, in this case, minimizing L is equivalent to minimizing Pe. However, in the case that P1 ̸= P2, (28) has an even more interesting interpretation. In that situation, it can be seen that the two integrals in the expressions (28) and (29) are weighted exactly oppositely. For example, if P1 is close to one, L ≈ R C2 p(x)dx, while Pe ≈ R C1 q(x)dx. Thus, the Laplacian pdf distance emphasizes to cluster the most unlikely 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. This work was partially supported by NSF grant ECS0300340. 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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Experts in a Markov Decision Process Eyal Even-Dar Computer Science Tel-Aviv University evend@post.tau.ac.il Sham M. Kakade Computer and Information Science University of Pennsylvania skakade@linc.cis.upenn.edu Yishay Mansour ∗ Computer Science Tel-Aviv University mansour@post.tau.ac.il Abstract We consider an MDP setting in which the reward function is allowed to change during each time step of play (possibly in an adversarial manner), yet the dynamics remain fixed. Similar to the experts setting, we address the question of how well can an agent do when compared to the reward achieved under the best stationary policy over time. We provide efficient algorithms, which have regret bounds with no dependence on the size of state space. Instead, these bounds depend only on a certain horizon time of the process and logarithmically on the number of actions. We also show that in the case that the dynamics change over time, the problem becomes computationally hard. 1 Introduction There is an inherent tension between the objectives in an expert setting and those in a reinforcement learning setting. In the experts problem, during every round a learner chooses one of n decision making experts and incurs the loss of the chosen expert. The setting is typically an adversarial one, where Nature provides the examples to a learner. The standard objective here is a myopic, backwards looking one — in retrospect, we desire that our performance is not much worse than had we chosen any single expert on the sequence of examples provided by Nature. In contrast, a reinforcement learning setting typically makes the much stronger assumption of a fixed environment, typically a Markov decision process (MDP), and the forward looking objective is to maximize some measure of the future reward with respect to this fixed environment. The motivation of this work is to understand how to efficiently incorporate the benefits of existing experts algorithms into a more adversarial reinforcement learning setting, where certain aspects of the environment could change over time. A naive way to implement an experts algorithm is to simply associate an expert with each fixed policy. The running time of such algorithms is polynomial in the number of experts and the regret (the difference from the optimal reward) is logarithmic in the number of experts. For our setting the number of policies is huge, namely #actions#states, which renders the naive experts approach computationally infeasible. Furthermore, straightforward applications of standard regret algorithms produce regret bounds which are logarithmic in the number of policies, so they have linear dependence ∗This work was supported in part by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778, by a grant from the Israel Science Foundation and an IBM faculty award. This publication only reflects the authors’ views. on the number of states. We might hope for a more effective regret bound which has no dependence on the size of state space (which is typically large). The setting we consider is one in which the dynamics of the environment are known to the learner, but the reward function can change over time. We assume that after each time step the learner has complete knowledge of the previous reward functions (over the entire environment), but does not know the future reward functions. As a motivating example one can consider taking a long road-trip over some period of time T. The dynamics, namely the roads, are fixed, but the road conditions may change frequently. By listening to the radio, one can get (effectively) instant updates of the road and traffic conditions. Here, the task is to minimize the cost during the period of time T. Note that at each time step we select one road segment, suffer a certain delay, and need to plan ahead with respect to our current position. This example is similar to an adversarial shortest path problem considered in Kalai and Vempala [2003]. In fact Kalai and Vempala [2003], address the computational difficulty of handling a large number of experts under certain linear assumptions on the reward functions. However, their algorithm is not directly applicable to our setting, due to the fact that in our setting, decisions must be made with respect to the current state of the agent (and the reward could be changing frequently), while in their setting the decisions are only made with respect to a single state. McMahan et al. [2003] also considered a similar setting — they also assume that the reward function is chosen by an adversary and that the dynamics are fixed. However, they assume that the cost functions come from a finite set (but are not observable) and the goal is to find a min-max solution for the related stochastic game. In this work, we provide efficient ways to incorporate existing best experts algorithms into the MDP setting. Furthermore, our loss bounds (compared to the best constant policy) have no dependence on the number of states and depend only on on a certain horizon time of the environment and log(#actions). There are two sensible extensions of our setting. The first is where we allow Nature to change the dynamics of the environment over time. Here, we show that it becomes NP-Hard to develop a low regret algorithm even for oblivious adversary. The second extension is to consider one in which the agent only observes the rewards for the states it actually visits (a generalization of the multi-arm bandits problem). We leave this interesting direction for future work. 2 The Setting We consider an MDP with state space S; initial state distribution d1 over S; action space A; state transition probabilities {Psa(·)} (here, Psa is the next-state distribution on taking action a in state s); and a sequence of reward functions r1, r2, . . . rT , where rt is the (bounded) reward function at time step t mapping S × A into [0, 1]. The goal is to maximize the sum of undiscounted rewards over a T step horizon. We assume the agent has complete knowledge of the transition model P, but at time t, the agent only knows the past reward functions r1, r2, ... rt−1. Hence, an algorithm A is a mapping from S and the previous reward functions r1, . . . rt−1 to a probability distribution over actions, so A(a|s, r1, . . . rt−1) is the probability of taking action a at time t. We define the return of an algorithm A as: Vr1,r2,...rT (A) = 1 T E " T X t=1 rt(st, at) d1, A # where at ∼A(a|st, r1, . . . rt−1) and st is the random variable which represents the state at time t, starting from initial state s1 ∼d1 and following actions a1, a2, . . . at−1. Note that we keep track of the expectation and not of a specific trajectory (and our algorithm specifies a distribution over actions at every state and at every time step t). Ideally, we would like to find an A which achieves a large reward Vr1,...rT (A) regardless of how the adversary chooses the reward functions. In general, this of course is not possible, and, as in the standard experts setting, we desire that our algorithm competes favorably against the best fixed stationary policy π(a|s) in hindsight. 3 An MDP Experts Algorithm 3.1 Preliminaries Before we provide our algorithm a few definitions are in order. For every stationary policy π(a|s), we define P π to be the transition matrix induced by π, where the component [P π]s,s′ is the transition probability from s to s′ under π. Also, define dπ,t to be the state distribution at time t when following π, ie dπ,t = d1(P π)t where we are treating d1 as a row vector here. Assumption 1 (Mixing) We assume the transition model over states, as determined by π, has a well defined stationary distribution, which we call dπ. More formally, for every initial state s, dπ,t converges to dπ as t tends to infinity and dπP π = dπ. Furthermore, this implies there exists some τ such that for all policies π, and distributions d and d′, ∥dP π −d′P π∥1 ≤e−1/τ∥d −d′∥1 where ∥x∥1 denotes the l1 norm of a vector x. We refer to τ as the mixing time and assume that τ > 1. The parameter τ provides a bound on the planning horizon timescale, since it implies that every policy achieves close to its average reward in O(τ) steps 1. This parameter also governs how long it effectively takes to switch from one policy to another (after time O(τ) steps there is little information in the state distribution about the previous policy). This assumption allows us to define the average reward of policy π in an MDP with reward function r as: ηr(π) = Es∼dπ,a∼π(a|s)[r(s, a)] and the value, Qπ,r(s, a), is defined as Qπ,r(s, a) ≡E " ∞ X t=1 (r(st, at) −ηr(π)) s1 = s, a1 = a, π # where and st and at are the state and actions at time t, after starting from state s1 = s then deviating with an immediate action of a1 = a and following π onwards. We slightly abuse notation by writing Qπ,r(s, π′) = Ea∼π′(a|s)[Qπ,r(s, a)]. These values satisfy the well known recurrence equation: Qπ,r(s, a) = r(s, a) −ηr(π) + Es′∼Psa[Qπ(s′, π)] (1) where Qπ(s′, π) is the next state value (without deviation). 1If this timescale is unreasonably large for some specific MDP, then one could artificially impose some horizon time and attempt to compete with those policies which mix in this horizon time, as done Kearns and Singh [1998]. If π∗is an optimal policy (with respect to ηr), then, as usual, we define Q∗ r(s, a) to be the value of the optimal policy, ie Q∗ r(s, a) = Qπ∗,r(s, a). We now provide two useful lemmas. It is straightforward to see that the previous assumption implies a rate of convergence to the stationary distribution that is O(τ), for all policies. The following lemma states this more precisely. Lemma 2 For all policies π, ∥dπ,t −dπ∥1 ≤2e−t/τ . Proof. Since π is stationary, we have dπP π = dπ, and so ∥dπ,t −dπ∥1 = ∥dπ,t−1P π −dπP π∥1 ≤∥dπ,t−1 −dπ∥1e−1/τ which implies ∥dπ,t −dπ∥1 ≤∥d1 −dπ∥1e−t/τ. The claim now follows since, for all distributions d and d′, ∥d −d′∥1 ≤2. □ The following derives a bound on the Q values as a function of the mixing time. Lemma 3 For all reward functions r, Qπ,r(s, a) ≤3τ . Proof. First, let us bound Qπ,r(s, π), where π is used on the first step. For all t, including t = 1, let dπ,s,t be the state distribution at time t starting from state s and following π. Hence, we have Qπ,r(s, π) = ∞ X t=1 Es′∼dπ,s,t,a∼π[r(s′, a)] −ηr(π)) ≤ ∞ X t=1 Es′∼dπ,a∼π[r(s′, a)] −ηr(π) + 2e−t/τ = ∞ X t=1 2e−t/τ ≤ Z ∞ 0 2e−t/τ = 2τ Using the recurrence relation for the values, we know Qπ,r(s, a) could be at most 1 more than the above. The result follows since 1 + 2τ ≤3τ □ 3.2 The Algorithm Now we provide our main result showing how to use any generic experts algorithm in our setting. We associate each state with an experts algorithm, and the expert for each state is responsible for choosing the actions at that state. The immediate question is what loss function should we feed to each expert. It turns out Qπt,rt is appropriate. We now assume that our experts algorithm achieves a performance comparable to the best constant action. Assumption 4 (Black Box Experts) We assume access to an optimized best expert algorithm which guarantees that for any sequence of loss functions c1, c2, . . . cT over actions A, the algorithm selects a distribution qt over A (using only the previous loss functions c1, c2, . . . ct−1) such that T X t=1 Ea∼qt[ct(a)] ≤ T X t=1 ct(a) + M p T log |A|, where ∥ct(a)∥≤M. Furthermore, we also assume that decision distributions do not change quickly: ∥qt −qt+1∥1 ≤ r log |A| t These assumptions are satisfied by the multiplicative weights algorithms. For instance, the algorithm in Freund and Schapire [1999] is such that the for each decision a, | log qt(a) − log qt+1(a)| changes by O( q log |A| t ), which implies the weaker l1 condition above. In our setting, we have an experts algorithm associated with every state s, which is fed the loss function Qπt,rt(s, ·) at time t. The above assumption then guarantees that at every state s for every action a we have that T X t=1 Qπt,rt(s, πt) ≤ T X t=1 Qπt,rt(s, a) + 3τ p T log |A| since the loss function Qπt,rt is bounded by 3τ, and that |πt(·|s) −πt+1(·|s)|1 ≤ r log |A| t As we shall see, it is important that this ’slow change’ condition be satisfied. Intuitively, our experts algorithms will be using a similar policy for significantly long periods of time. Also note that since the experts algorithms are associated with each state and each of the N experts chooses decisions out of A actions, the algorithm is efficient (polynomial in N and A, assuming that that the black box uses a reasonable experts algorithm). We now state our main theorem. Theorem 5 Let A be the MDP experts algorithm. Then for all reward functions r1, r2, . . . rT and for all stationary policies π, Vr1,r2,...rT (A) ≥Vr1,r2,...rT (π) −8τ 2 r log |A| T −3τ r log |A| T −4τ T As expected, the regret goes to 0 at the rate O(1/ √ T), as is the case with experts algorithms. Importantly, note that the bound does not depend on the size of the state space. 3.3 The Analysis The analysis is naturally divided into two parts. First, we analyze the performance of the algorithm in an idealized setting, where the algorithm instantaneously obtains the average reward of its current policy at each step. Then we take into account the slow change of the policies to show that the actual performance is similar to the instantaneous performance. An Idealized Setting: Let us examine the case in which at each time t, when the algorithm uses πt, it immediately obtains reward ηrt(πt). The following theorem compares the performance of our algorithms to that of a fixed constant policy in this setting. Theorem 6 For all sequences r1, r2, . . . rT , the MDP experts algorithm have the following performance bound. For all π, T X t=1 ηrt(πt) ≥ T X t=1 ηrt(π) −3τ p T log |A| where π1, π2, . . . πT is the sequence of policies generated by A in response to r1, r2, . . . rT . Next we provide a technical lemma, which is a variant of a result in Kakade [2003] Lemma 7 For all policies π and π′, ηr(π′) −ηr(π) = Es∼dπ′[Qπ,r(s, π′) −Qπ,r(s, π)] Proof. Note that by definition of stationarity, if the state distribution is at dπ′, then the next state distribution is also dπ′ if π′ is followed. More formally, if s ∼dπ′, a ∼π′(a|s), and s′ ∼Psa, then s′ ∼dπ′. Using this and equation 1, we have: Es∼dπ′[Qπ,r(s, π′)] = Es∼dπ′,a∼π′[Qπ,r(s, a)] = Es∼dπ′,a∼π′[r(s, a) −ηr(π) + Es′∼Psa[Qπ(s′, π)] = Es∼dπ′,a∼π′[r(s, a) −ηr(π)] + Es∼dπ′[Qπ(s, π)] = ηr(π′) −ηr(π) + Es∼dπ′[Qπ(s, π)] Rearranging terms leads to the result. □ The lemma shows why our choice to feed each experts algorithm Qπt,rt was appropriate. Now we complete the proof of the above theorem. Proof. Using the assumed regret in assumption 4, T X t=1 ηrt(π) − T X t=1 ηrt(πt) = T X t=1 Es∼dπ[Qπt,rt(s, π) −Qπt,rt(s, πt)] = Es∼dπ[ T X t=1 Qπt,rt(s, π) −Qπt,rt(s, πt)] ≤ Es∼dπ[3τ p T log A] = 3τ p T log A where we used the fact that dπ does not depend on the time in the second step. □ Taking Mixing Into Account: This subsection relates the values V to the sums of average reward used in the idealized setting. Theorem 8 For all sequences r1, r2, . . . rT and for all A |Vr1,r2,...rT (A) −1 T T X t=1 ηrt(πt)| ≤4τ 2 r log |A| T + 2τ T where π1, π2, . . . πT is the sequence of policies generated by A in response to r1, r2, . . . rT . Since the above holds for all A (including those A which are the constant policy π), then combining this with Theorem 6 (once with A and once with π) completes the proof of Theorem 5. We now prove the above. The following simple lemma is useful and we omit the proof. It shows how close are the next state distributions when following πt rather than πt+1. Lemma 9 Let π and π′ be such that ∥π(·|s)−π′(·|s)∥1 ≤ǫ. Then for any state distribution d, we have ∥dP π −dP π′∥1 ≤ε. Analogous to the definition of dπ,t, we define dA,t dA,t = Pr[st = s|d1, A] which is the probability that the state at time t is s given that A has been followed. Lemma 10 Let π1, π2, . . . πT be the sequence of policies generated by A in response to r1, r2, . . . rT . We have ∥dA,t −dπt∥1 ≤2τ 2 r log |A| t + 2e−t/τ Proof. Let k ≤t. Using our experts assumption, it is straightforward to see that that the change in the policy over k steps is |πk(·|s) −πt(·|s)|1 ≤(t −k) p log |A|/t. Using this with dA,k = dA,k−1P(πk) and dπtP πt = dπt, we have ∥dA,k −dπt∥1 = ∥dA,k−1P πk −dπt∥1 ≤ ∥dA,k−1P πt −dπt∥1 + ∥dA,k−1P πk −dA,k−1P πt∥1 ≤ ∥dA,k−1P πt −dπtP πt∥1 + 2(t −k) p log |A|/t ≤ e−1/τ∥dA,k−1 −dπt∥1 + 2(t −k) p log |A|/t where we have used the last lemma in the third step and our contraction assumption 1 in the second to last step. Recursing on the above equation leads to: ∥dA,t −dπt∥ ≤ 2 p log |A|/t 2 X k=t (t −k)e−(t−k)/τ + e−t/τ∥d1 −dπt∥ ≤ 2 p log |A|/t ∞ X k=1 ke−k/τ + 2e−t/τ The sum is bounded by an integral from 0 to ∞, which evaluates to τ 2. □ We are now ready to complete the proof of Theorem 8. Proof. By definition of V , Vr1,r2,...rT (A) = 1 T T X t=1 Es∼dA,t,a∼πt[rt(s, a)] ≤ 1 T T X t=1 Es∼dπt,a∼πt[rt(s, a)] + 1 T T X t=1 ∥dA,t −dπt∥1 ≤ 1 T T X t=1 ηrt(πt) + 1 T T X t=1 2τ 2 r log |A| t + 2e−t/τ ! ≤ 1 T T X t=1 ηrt(πt) + 4τ 2 r log |A| T + 2τ T where we have bounded the sums by integration in the second to last step. A symmetric argument leads to the result. □ 4 A More Adversarial Setting In this section we explore a different setting, the changing dynamics model. Here, in each timestep t, an oblivious adversary is allowed to choose both the reward function rt and the transition model Pt — the model that determines the transitions to be used at timestep t. After each timestep, the agent receives complete knowledge of both rt and Pt. Furthermore, we assume that Pt is deterministic, so we do not concern ourselves with mixing issues. In this setting, we have the following hardness result. We let R∗ t (M) be the optimal average reward obtained by a stationary policy for times [1, t]. Theorem 11 In the changing dynamics model, if there exists a polynomial time online algorithm (polynomial in the problem parameters) such that, for any MDP, has an expected average reward larger than (0.875 + ε)R∗ t (M), for some ε > 0 and t, then P = NP. The following lemma is useful in the proof and uses the fact that it is hard to approximate MAX3SAT within any factor better than 0.875 (Hastad [2001]). Lemma 12 Computing a stationary policy in the changing dynamics model with average reward larger than (0.875 + ε)R∗(M), for some ε > 0, is NP-Hard. Proof: We prove it by reduction from 3-SAT. Suppose that the 3-SAT formula, φ has m clauses, C1, . . . , Cm, and n literals, x1, . . . , xn then we reduce it to MDP with n + 1 states,s1, . . . sn, sn+1, two actions in each state, 0, 1 and fixed dynamic for 3m steps which will be described later. We prove that a policy with average reward p/3 translates to an assignment that satisfies p fraction of φ and vice versa. Next we describe the dynamics. Suppose that C1 is (x1 ∨¬x2 ∨x7) and C2 is (x4 ∨¬x1 ∨x7). The initial state is s1 and the reward for action 0 is 0 and the agent moves to state s2, for action 1 the reward is 1 and it moves to state sn+1. In the second timestep the reward in sn+1 is 0 for every action and the agents stay in it; in state s2 if the agent performs action 0 then it obtains reward 1 and move to state sn+1 otherwise it obtains reward 0 and moves to state s7. In the next timestep the reward in sn+1 is 0 for every action and the agents moves to x4, the reward in s7 is 1 for action 1 and zero for action 0 and moves to s4 for both actions. The rest of the construction is done identically. Note that time interval [3(ℓ−1) + 1, 3ℓ] corresponds to Cℓand that the reward obtained in this interval is at most 1. We note that φ has an assignment y1, . . . , yn where yi = {0, 1} that satisfies p fraction of it, if and only if π which takes action yi in si has average reward p/3. We prove it by looking on each interval separately and noting that if a reward 1 is obtained then there is an action a that we take in one of the states which has reward 1 but this action corresponds to a satisfying assignment for this clause. □ We are now ready to prove Theorem 11. Proof: In this proof we make few changes from the construction given in Lemma 12. We allow the same clause to repeat few times, and its dynamics are described in n steps and not in 3 steps, where in the k step we move from sk to sk+1 and obtains 0 reward, unless the action ”satisfies” the chosen clause, if it satisfies then we obtain an immediate reward 1, move to sn+1 and stay there for n −k −1 steps. After n steps the adversary chooses uniformly at random the next clause. In the analysis we define the n steps related to a clause as an iteration. The strategy defined by the algorithm at the k iteration is the probability assigned to action 0/1 at state sℓjust before arriving to sℓ. Note that the strategy at each iteration is actually a stationary policy for M. Thus the strategy in each iteration defines an assignment for the formula. We also note that before an iteration the expected reward of the optimal stationary policy in the iteration is k/(nm), where k is the maximal number of satisfiable clauses and there are m clauses, and we have E[R∗(M)] = k/(nm). If we choose at random an iteration, then the strategy defined in that iteration has an expected reward which is larger than (0.875 + ε)R∗(M), which implies that we can satisfy more than 0.875 fraction of satisfiable clauses, but this is impossible unless P = NP. □ References Y. Freund and R. Schapire. Adaptive game playing using multiplicative weights. Games and Economic Behavior, 29:79–103, 1999. J. Hastad. Some optimal inapproximability results. J. ACM, 48(4):798–859, 2001. S. Kakade. On the Sample Complexity of Reinforcement Learning. PhD thesis, University College London, 2003. A. Kalai and S. Vempala. Efficient algorithms for on-line optimization. Proceedings of COLT, 2003. M. Kearns and S. Singh. Near-optimal reinforcement learning in polynomial time. Proceedings of ICML, 1998. H. McMahan, G. Gordon, and A. Blum. Planning in the presence of cost functions controlled by an adversary. In In the 20th ICML, 2003.
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Neighbourhood Components Analysis Jacob Goldberger, Sam Roweis, Geoff Hinton, Ruslan Salakhutdinov Department of Computer Science, University of Toronto {jacob,roweis,hinton,rsalakhu}@cs.toronto.edu Abstract In this paper we propose a novel method for learning a Mahalanobis distance measure to be used in the KNN classification algorithm. The algorithm directly maximizes a stochastic variant of the leave-one-out KNN score on the training set. It can also learn a low-dimensional linear embedding of labeled data that can be used for data visualization and fast classification. Unlike other methods, our classification model is non-parametric, making no assumptions about the shape of the class distributions or the boundaries between them. The performance of the method is demonstrated on several data sets, both for metric learning and linear dimensionality reduction. 1 Introduction Nearest neighbor (KNN) is an extremely simple yet surprisingly effective method for classification. Its appeal stems from the fact that its decision surfaces are nonlinear, there is only a single integer parameter (which is easily tuned with cross-validation), and the expected quality of predictions improves automatically as the amount of training data increases. These advantages, shared by many non-parametric methods, reflect the fact that although the final classification machine has quite high capacity (since it accesses the entire reservoir of training data at test time), the trivial learning procedure rarely causes overfitting itself. However, KNN suffers from two very serious drawbacks. The first is computational, since it must store and search through the entire training set in order to classify a single test point. (Storage can potentially be reduced by “editing” or “thinning” the training data; and in low dimensional input spaces, the search problem can be mitigated by employing data structures such as KD-trees or ball-trees[4].) The second is a modeling issue: how should the distance metric used to define the “nearest” neighbours of a test point be defined? In this paper, we attack both of these difficulties by learning a quadratic distance metric which optimizes the expected leave-one-out classification error on the training data when used with a stochastic neighbour selection rule. Furthermore, we can force the learned distance metric to be low rank, thus substantially reducing storage and search costs at test time. 2 Stochastic Nearest Neighbours for Distance Metric Learning We begin with a labeled data set consisting of n real-valued input vectors x1, . . . , xn in RD and corresponding class labels c1, ..., cn. We want to find a distance metric that maximizes the performance of nearest neighbour classification. Ideally, we would like to optimize performance on future test data, but since we do not know the true data distribution we instead attempt to optimize leave-one-out (LOO) performance on the training data. In what follows, we restrict ourselves to learning Mahalanobis (quadratic) distance metrics, which can always be represented by symmetric positive semi-definite matrices. We estimate such metrics through their inverse square roots, by learning a linear transformation of the input space such that in the transformed space, KNN performs well. If we denote the transformation by a matrix A we are effectively learning a metric Q = A⊤A such that d(x, y) = (x −y)⊤Q(x −y) = (Ax −Ay)⊤(Ax −Ay). The actual leave-one-out classification error of KNN is quite a discontinuous function of the transformation A, since an infinitesimal change in A may change the neighbour graph and thus affect LOO classification performance by a finite amount. Instead, we adopt a more well behaved measure of nearest neighbour performance, by introducing a differentiable cost function based on stochastic (“soft”) neighbour assignments in the transformed space. In particular, each point i selects another point j as its neighbour with some probability pij, and inherits its class label from the point it selects. We define the pij using a softmax over Euclidean distances in the transformed space: pij = exp(−∥Axi −Axj∥2) P k̸=i exp(−∥Axi −Axk∥2) , pii = 0 (1) Under this stochastic selection rule, we can compute the probability pi that point i will be correctly classified (denote the set of points in the same class as i by Ci = {j|ci = cj}): pi = X j∈Ci pij (2) The objective we maximize is the expected number of points correctly classified under this scheme: f(A) = X i X j∈Ci pij = X i pi (3) Differentiating f with respect to the transformation matrix A yields a gradient rule which we can use for learning (denote xij = xi −xj): ∂f ∂A = −2A X i X j∈Ci pij(xijx ⊤ ij − X k pikxikx ⊤ ik) (4) Reordering the terms we obtain a more efficiently computed expression: ∂f ∂A = 2A X i pi X k pikxikx ⊤ ik − X j∈Ci pijxijx ⊤ ij (5) Our algorithm – which we dub Neighbourhood Components Analysis (NCA)– is extremely simple: maximize the above objective (3) using a gradient based optimizer such as deltabar-delta or conjugate gradients. Of course, since the cost function above is not convex, some care must be taken to avoid local maxima during training. However, unlike many other objective functions (where good optima are not necessarily deep but rather broad) it has been our experience that the larger we can drive f during training the better our test performance will be. In other words, we have never observed an “overtraining” effect. Notice that by learning the overall scale of A as well as the relative directions of its rows we are also effectively learning a real-valued estimate of the optimal number of neighbours (K). This estimate appears as the effective perplexity of the distributions pij. If the learning procedure wants to reduce the effective perplexity (consult fewer neighbours) it can scale up A uniformly; similarly by scaling down all the entries in A it can increase the perplexity of and effectively average over more neighbours during the stochastic selection. Maximizing the objective function f(A) is equivalent to minimizing the L1 norm between the true class distribution (having probability one on the true class) and the stochastic class distribution induced by pij via A. A natural alternative distance is the KL-divergence which induces the following objective function: g(A) = X i log( X j∈Ci pij) = X i log(pi) (6) Maximizing this objective would correspond to maximizing the probability of obtaining a perfect (error free) classification of the entire training set. The gradient of g(A) is even simpler than that of f(A): ∂g ∂A = 2A X i X k pikxikx ⊤ ik − P j∈Ci pijxijx⊤ ij P j∈Ci pij ! (7) We have experimented with optimizing this cost function as well, and found both the transformations learned and the performance results on training and testing data to be very similar to those obtained with the original cost function. To speed up the gradient computation, the sums that appear in equations (5) and (7) over the data points and over the neigbours of each point, can be truncated (one because we can do stochastic gradient rather than exact gradient and the other because pij drops off quickly). 3 Low Rank Distance Metrics and Nonsquare Projection Often it is useful to reduce the dimensionality of input data, either for computational savings or for regularization of a subsequent learning algorithm. Linear dimensionality reduction techniques (which apply a linear operator to the original data in order to arrive at the reduced representation) are popular because they are both fast and themselves relatively immune to overfitting. Because they implement only affine maps, linear projections also preserve some essential topology of the original data. Many approaches exist for linear dimensionality reduction, ranging from purely unsupervised approaches (such as factor analysis, principal components analysis and independent components analysis) to methods which make use of class labels in addition to input features such as linear discriminant analysis (LDA)[3] possibly combined with relevant components analysis (RCA)[1]. By restricting A to be a nonsquare matrix of size d×D, NCA can also do linear dimensionality reduction. In this case, the learned metric will be low rank, and the transformed inputs will lie in Rd. (Since the transformation is linear, without loss of generality we only consider the case d ≤D. ) By making such a restriction, we can potentially reap many further benefits beyond the already convenient method for learning a KNN distance metric. In particular, by choosing d ≪D we can vastly reduce the storage and search-time requirements of KNN. Selecting d = 2 or d = 3 we can also compute useful low dimensional visualizations on labeled datasets, using only a linear projection. The algorithm is exactly the same: optimize the cost function (3) using gradient descent on a nonsquare A. Our method requires no matrix inversions and assumes no parametric model (Gaussian or otherwise) for the class distributions or the boundaries between them. For now, the dimensionality of the reduced representation (the number of rows in A) must be set by the user. By using an highly rectangular A so that d ≪D, we can significantly reduce the computational load of KNN at the expense of restricting the allowable metrics to be those of rank at most d. To achieve this, we apply the NCA learning algorithm to find the optimal transformation A, and then we store only the projections of the training points yn = Axn (as well as their labels). At test time, we classify a new point xtest by first computing its projection ytest = Axtest and then doing KNN classification on ytest using the yn and a simple Euclidean metric. If d is relatively small (say less than 10), we can preprocess the yn by building a KD-tree or a ball-tree to further increase the speed of search at test time. The storage requirements of this method are O(dN) + Dd compared with O(DN) for KNN in the original input space. 4 Experiments in Metric Learning and Dimensionality Reduction We have evaluated the NCA algorithm against standard distance metrics for KNN and other methods for linear dimensionality reduction. In our experiments, we have used 6 data sets (5 from the UC Irvine repository). We compared the NCA transformation obtained from optimizing f (for square A) on the training set with the default Euclidean distance A = I, the “whitening” transformation , A = Σ−1 2 (where Σ is the sample data covariance matrix), and the RCA [1] transformation A = Σ −1 2 w (where Σw is the average of the within-class covariance matrices). We also investigated the behaviour of NCA when A is restricted to be diagonal, allowing only axis aligned Mahalanobis measures. Figure 1 shows that the training and (more importantly) testing performance of NCA is consistently the same as or better than that of other Mahalanobis distance measures for KNN, despite the relative simplicity of the NCA objective function and the fact that the distance metric being learned is nothing more than a positive definite matrix A⊤A. We have also investigated the use of linear dimensionality reduction using NCA (with nonsquare A) for visualization as well as reduced-complexity classification on several datasets. In figure 2 we show 4 examples of 2-D visualization. First, we generated a synthetic threedimensional dataset (shown in top row of figure 2) which consists of 5 classes (shown by different colors). In two dimensions, the classes are distributed in concentric circles, while the third dimension is just Gaussian noise, uncorrelated with the other dimensions or the class label. If the noise variance is large enough, the projection found by PCA is forced to include the noise (as shown on the top left of figure 2). (A full rank Euclidean metric would also be misled by this dimension.) The classes are not convex and cannot be linearly separated, hence the results obtained from LDA will be inappropriate (as shown in figure 2). In contrast, NCA adaptively finds the best projection without assuming any parametric structure in the low dimensional representation. We have also applied NCA to the UCI “wine” dataset, which consists of 178 points labeled into 3 classes and to a database of gray-scale images of faces consisting of 18 classes (each a separate individual) and 560 dimensions (image size is 20×28). The face dataset consists of 1800 images (100 for each person). Finally, we applied our algorithm to a subset of the USPS dataset of handwritten digit images, consisting of the first five digit classes (“one” through “five”). The grayscale images were downsampled to 8 × 8 pixel resolution resulting in 64 dimensions. As can be seen in figure 2 when a two-dimensional projection is used, the classes are consistently much better separated by the NCA transformation than by either PCA (which is unsupervised) or LDA (which has access to the class labels). Of course, the NCA transformation is still only a linear projection, just optimized with a cost function which explicitly encourages local separation. To further quantify the projection results we can apply a nearest-neighbor classification in the projected space. Using the same projection learned at training time, we project the training set and all future test points and perform KNN in the low-dimensional space using the Euclidean measure. The results under the PCA, LDA, LDA followed by RCA and NCA transformations (using K=1) appear in figure 1. The NCA projection consistently gives superior performance in this highly constrained low0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 distance metric learning − training bal ion iris wine hous digit NCA diag−NCA RCA whitened Euclidean 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 distance metric learning − testing bal ion iris wine hous digit NCA diag−NCA RCA whitened Euclidean 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank 2 transformation − training bal ion iris wine hous digit NCA LDA+RCA LDA PCA 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 rank 2 transformation − testing bal ion iris wine hous digit NCA LDA+RCA LDA PCA Figure 1: KNN classification accuracy (left train, right test) on UCI datasets balance, ionosphere, iris, wine and housing and on the USPS handwritten digits. Results are averages over 40 realizations of splitting each dataset into training (70%) and testing (30%) subsets (for USPS 200 images for each of the 10 digit classes were used for training and 500 for testing). Top panels show distance metric learning (square A) and bottom panels show linear dimensionality reduction down to d = 2. rank KNN setting. In summary, we have found that when labeled data is available, NCA performs better both in terms of classification performance in the projected representation and in terms of visualization of class separation as compared to the standard methods of PCA and LDA. 5 Extensions to Continuous Labels and Semi-Supervised Learning Although we have focused here on discrete classes, linear transformations and fully supervised learning, many extensions of this basic idea are possible. Clearly, a nonlinear transformation function A(·) could be learned using any architecture (such as a multilayer perceptron) trainable by gradient methods. Furthermore, it is possible to extend the classification framework presented above to the case of a real valued (continuous) supervision signal by defining the set of “correct matches” Ci for point i to be those points j having similar (continuous) targets. This naturally leads to the idea of “soft matches”, in which the objective function becomes a sum over all pairs, each weighted by their agreement according to the targets. Learning under such an objective can still proceed even in settings where the targets are not explicitly provided as long as information identifying close pairs PCA LDA NCA Figure 2: Dataset visualization results of PCA, LDA and NCA applied to (from top) the “concentric rings”, “wine”, “faces” and “digits” datasets. The data are reduced from their original dimensionalities (D=3,D=13,D=560,D=256 respectively) to the d=2 dimensions show. Figure 3: The two dimensional outputs of the neural network on a set of test cases. On the left, each point is shown using a line segment that has the same orientation as the input face. On the right, the same points are shown again with the size of the circle representing the size of the face. is available. Such semi-supervised tasks often arise in domains with strong spatial or temporal continuity constraints on the supervision, e.g. in a video of a person’s face we may assume that pose, and expression vary slowly in time even if no individual frames are ever labeled explicitly with numerical pose or expression values. To illustrate this, we generate pairs of faces in the following way: First we choose two faces at random from the FERET-B dataset (5000 isolated faces that have a standard orientation and scale). The first face is rotated by an angle uniformly distributed between ±45o and scaled to have a height uniformly distributed between 25 and 35 pixels. The second face (which is of a different person) is given the same rotation and scaling but with Gaussian noise of ±1.22o and ±1.5 pixels. The pair is given a weight, wab, which is the probability density of the added noise divided by its maximum possible value. We then trained a neural network with one hidden layer of 100 logistic units to map from the 35×35 pixel intensities of a face to a point, y, in a 2-D output space. Backpropagation was used to minimize the cost function in Eq. 8 which encourages the faces in a pair to be placed close together: Cost = − X pair(a,b) wab log exp(−||ya −yb||2) P c,d exp(−||yc −yd||2) ! (8) where c and d are indices over all of the faces, not just the ones that form a pair. Four example faces are shown to the right; horizontally the pairs agree and vertically they do not. Figure 3 above shows that the feedforward neural network discovered polar coordinates without the user having to decide how to represent scale and orientation in the output space. 6 Relationships to Other Methods and Conclusions Several papers recently addressed the problem of learning Mahalanobis distance functions given labeled data or at least side-information of the form of equivalence constraints. Two related methods are RCA [1] and a convex optimization based algorithm [7]. RCA is implicitly assuming a Gaussian distribution for each class (so it can be described using only the first two moments of the class-conditional distribution). Xing et. al attempt to find a transformation which minimizes all pairwise squared distances between points in the same class; this implicitly assumes that classes form a single compact connected set. For highly multimodal class distributions this cost function will be severely penalized. Lowe[6] proposed a method similar to ours but used a more limited idea for learning a nearest neighbour distance metric. In his approach, the metric is constrained to be diagonal (as well, it is somewhat redundantly parameterized), and the objective function corresponds to the average squared error between the true class distribution and the predicted distribution, which is not entirely appropriate in a more probabilistic setting. In parallel there has been work on learning low rank transformations for fast classification and visualization. The classic LDA algorithm[3] is optimal if all class distributions are Gaussian with a single shared covariance; this assumption, however is rarely true. LDA also suffers from a small sample size problem when dealing with high-dimensional data when the within-class scatter matrix is nearly singular[2]. Recent variants of LDA (e.g. [5], [2]) make the transformation more robust to outliers and to numerical instability when not enough datapoints are available. (This problem does not exist in our method since there is no need for a matrix inversion.) In general, there are two classes of regularization assumption that are common in linear methods for classification. The first is a strong parametric assumption about the structure of the class distributions (typically enforcing connected or even convex structure); the second is an assumption about the decision boundary (typically enforcing a hyperplane). Our method makes neither of these assumptions, relying instead on the strong regularization imposed by restricting ourselves to a linear transformation of the original inputs. Future research on the NCA model will investigate using local estimates of K as derived from the entropy of the distributions pij; the possible use of a stochastic classification rule at test time; and more systematic comparisons between the objective functions f and g. To conclude, we have introduced a novel non-parametric learning method — NCA — that handles the tasks of distance learning and dimensionality reduction in a unified manner. Although much recent effort has focused on non-linear methods, we feel that linear embedding has still not fully fulfilled its potential for either visualization or learning. Acknowledgments Thanks to David Heckerman and Paul Viola for suggesting that we investigate the alternative cost g(A) and the case of diagonal A. References [1] A. Bar-Hillel, T. Hertz, N. Shental, and D. Weinshall. Learning distance functions using equivalence relation. In International Conference on Machine Learning, 2003. [2] L. Chen, H. Liao, M. Ko, J. Lin, and G. Yu. A new lda-based face recognition system which can solve the small sample size problem. In Pattern Recognition, pages 1713–1726, 2000. [3] R. A. Fisher. The use of multiple measurements in taxonomic problems. In Annual of Eugenic, pages 179–188, 1936. [4] J. Friedman, J.bentley, and R. Finkel. An algorithm for finding best matches in logarithmic expected time. In ACM, 1977. [5] Y. Koren and L. Carmel. Robust linear dimensionality reduction. In IEEE Trans. Vis. and Comp. Graph., pages 459–470, 2004. [6] D. Lowe. Similarity metric learning for a variable kernel classifier. In Neural Computation, pages 72–85, 1995. [7] E.P. Xing, A. Y. Ng, M.I. Jordan, and S. Russell. Distance learning metric. In Proc. of Neural Information Processing Systems, 2003.
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Learning Gaussian Process Kernels via Hierarchical Bayes Anton Schwaighofer Fraunhofer FIRST Intelligent Data Analysis (IDA) Kekul´estrasse 7, 12489 Berlin anton@first.fhg.de Volker Tresp, Kai Yu Siemens Corporate Technology Information and Communications 81730 Munich, Germany {volker.tresp,kai.yu}@siemens.com Abstract We present a novel method for learning with Gaussian process regression in a hierarchical Bayesian framework. In a first step, kernel matrices on a fixed set of input points are learned from data using a simple and efficient EM algorithm. This step is nonparametric, in that it does not require a parametric form of covariance function. In a second step, kernel functions are fitted to approximate the learned covariance matrix using a generalized Nystr¨om method, which results in a complex, data driven kernel. We evaluate our approach as a recommendation engine for art images, where the proposed hierarchical Bayesian method leads to excellent prediction performance. 1 Introduction In many real-world application domains, the available training data sets are quite small, which makes learning and model selection difficult. For example, in the user preference modelling problem we will consider later, learning a preference model would amount to fitting a model based on only 20 samples of a user’s preference data. Fortunately, there are situations where individual data sets are small, but data from similar scenarios can be obtained. Returning to the example of preference modelling, data for many different users are typically available. This data stems from clearly separate individuals, but we can expect that models can borrow strength from data of users with similar tastes. Typically, such problems have been handled by either mixed effects models or hierarchical Bayesian modelling. In this paper we present a novel approach to hierarchical Bayesian modelling in the context of Gaussian process regression, with an application to recommender systems. Here, hierarchical Bayesian modelling essentially means to learn the mean and covariance function of the Gaussian process. In a first step, a common collaborative kernel matrix is learned from the data via a simple and efficient EM algorithm. This circumvents the problem of kernel design, as no parametric form of kernel function is required here. Thus, this form of learning a covariance matrix is also suited for problems with complex covariance structure (e.g. nonstationarity). A portion of the learned covariance matrix can be explained by the input features and, thus, generalized to new objects via a content-based kernel smoother. Thus, in a second step, we generalize the covariance matrix (learned by the EM-algorithm) to new items using a generalized Nystr¨om method. The result is a complex content-based kernel which itself is a weighted superposition of simple smoothing kernels. This second part could also be applied to other situations where one needs to extrapolate a covariance matrix on a finite set (e.g. a graph) to a continuous input space, as, for example, required in induction for semi-supervised learning [14]. The paper is organized as follows. Sec. 2 casts Gaussian process regression in a hierarchical Bayesian framework, and shows the EM updates to learn the covariance matrix in the first step. Extrapolating the covariance matrix is shown in Sec. 3. We illustrate the function of the EM-learning on a toy example in Sec. 4, before applying the proposed methods as a recommender system for images in Sec. 4.1. 1.1 Previous Work In statistics, modelling data from related scenarios is typically done via mixed effects models or hierarchical Bayesian (HB) modelling [6]. In HB, parameters of models for individual scenarios (e.g. users in recommender systems) are assumed to be drawn from a common (hyper)prior distribution, allowing the individual models to interact and regularize each other. Recent examples of HB modelling in machine learning include [1, 2]. In other contexts, this learning framework is called multi-task learning [4]. Multi-task learning with Gaussian processes has been suggested by [8], yet with the rather stringent assumption that one has observations on the same set of points in each individual scenario. Based on sparse approximations of GPs, a more general GP multi-task learner with parametric covariance functions has been presented in [7]. In contrast, the approach presented in this paper only considers covariance matrices (and is thus non-parametric) in the first step. Only in a second extrapolation step, kernel smoothing leads to predictions based on a covariance function that is a data-driven combination of simple kernel functions. 2 Learning GP Kernel Matrices via EM The learning task we are concerned with can be stated as follows: The data are observations from M different scenarios. In the i.th scenario, we have observations yi = (yi 1, . . . , yi N i) on a total of N i points, Xi = {xi 1, . . . , xi Ni}. In order to analyze this data in a hierarchical Bayesian way, we assume that the data for each scenario is a noisy sample of a Gaussian process (GP) with unknown mean and covariance function. We assume that mean and covariance function are shared across different scenarios.1 In the first modelling step presented in this section, we consider transductive learning (“labelling a partially labelled data set”), that is, we are interested in the model’s behavior only on points X, with X = SM i=1 Xi and cardinality N = |X|. This situation is relevant for most collaborative filtering applications. Thus, test points are the unlabelled points in each scenario. This reduces the whole “infinite dimensional” Gaussian process to its finite dimensional projection on points X, which is an N-variate Gaussian distribution with covariance matrix K and mean vector m. For the EM algorithm to work, we also require that there is some overlap between scenarios, that is, Xi ∩Xj ̸= ∅for some i, j. Coming back to the user modelling problem mentioned above, this means that at least some items have been rated by more than one user. Thus, our first modelling step focusses on directly learning the covariance matrix K and 1Alternative HB approaches for collaborative filtering, like that discussed in [5], assume that model weights are drawn from a shared Gaussian distribution. m from the data via an efficient EM algorithm. This may be of particular help in problems where one would need to specify a complex (e.g. nonstationary) covariance function. Following the hierarchical Bayesian assumption, the data observed in each scenario is thus a partial sample from N(y | m, K + σ21), where 1 denotes the unit matrix. The joint model is simply p(m, K) M Y i=1 p(yi | f i)p(f i | m, K), (1) where p(m, K) denotes the prior distribution for mean and covariance. We assume a Gaussian likelihood p(yi | f i) with diagonal covariance matrix σ21. 2.1 EM Learning For the above hierarchical Bayesian model, Eq. (1), the marginal likelihood becomes p(m, K) M Y i=1 Z p(yi | f i)p(f i | m, K) df i. (2) To obtain simple and stable solutions when estimating m and K from the data, we consider point estimates of the parameters m and K, based on a penalized likelihood approach with conjugate priors.2 The conjugate prior for mean m and covariance K of a multivariate Gaussian is the so-called Normal-Wishart distribution [6], which decomposes into the product of an inverse Wishart distribution for K and a Normal distribution for m, p(m, K) = N(m | ν, η−1K)Wi−1(K|α, U). (3) That is, the prior for the Gram matrix K is given by an inverse Wishart distribution with scalar parameter α > 1/2(N −1) and U being a symmetric positive-definite matrix. Given the covariance matrix K, m is Gaussian distributed with mean ν and covariance η−1K, where η is a positive scalar. The parameters can be interpreted in terms of an equivalent data set for the mean (this data set has size A, with A = ν, and mean µ = ν) and a data set for the covariance that has size B, with α = (B + N)/2, and covariance S, U = (B/2)S. In order to write down the EM algorithm in a compact way, we denote by I(i) the set of indices of those data points that have been observed in the i.th scenario, that is I(i) = {j | j ∈{1, . . . , N} and xj ∈Xi}. Keep in mind that in most applications of interest N i ≪N such that most targets are missing in training. KI(i),I(i) denotes the square submatrix of K that corresponds to points I(i), that is, the covariance matrix for points in the i.th scenario. By K·,I(i) we denote the covariance matrix of all N points versus those in the i.th scenario. 2.1.1 E-step In the E-step, one first computes ˜f i, the expected value of functional values on all N points for each scenario i. The expected value is given by the standard equations for the predictive mean of Gaussian process models, where the covariance functions are replaced by corresponding sub-matrices of the current estimate for K: ˜f i = K·,I(i)(KI(i),I(i) + σ21)−1(yi −mI(i)) + m, i = 1, . . . , M. (4) Also, covariances between all pairs of points are estimated, based on the predictive covariance for the GP models: (⊤denotes matrix transpose) ˜Ci = K −K·,I(i)(KI(i),I(i) + σ21)−1K⊤ ·,I(i), i = 1, . . . , M. (5) 2An efficient EM-based solution for the case σ2 = 0 is also given by [9]. 2.1.2 M-step In the M-step, the vector of mean values m, the covariance matrix K and the noise variance σ2 are being updated. Denoting the updated quantities by m′, K′, and (σ2)′, we get m′ = 1 M + A Aµ + M X i=1 ˜f i ! K′ = 1 M + B A(m′ −µ)(m′ −µ)⊤+ BS + M X i=1 (˜f i −m′)(˜f i −m′)⊤+ ˜Ci! (σ2)′ = 1 N M X i=1 ∥yi −˜f i I(i)∥2 + trace ˜Ci I(i),I(i) ! . An intuitive explanation of the M-step is as follows: The new mean m′ is a weighted combination of the prior mean, weighted by the equivalent sample size, and the predictive mean. The covariance update is a sum of four terms. The first term is typically irrelevant, it is a result of the coupling of the Gaussian and the inverse Wishart prior distributions via K. The second term contains the prior covariance matrix, again weighted by the equivalent sample size. As the third term, we get the empirical covariance, based on the estimated and measured functional values f i. Finally, the fourth term gives a correction term to compensate for the fact that the functional values f i are only estimates, thus the empirical covariance will be too small. 3 Learning the Covariance Function via Generalized Nystr¨om Using the EM algorithm described in Sec. 2.1, one can easily and efficiently learn a covariance matrix K and mean vector m from data obtained in different related scenarios. Once K is found, predictions within the set X can easily be made, by appealing to the same equations used in the EM algorithm (Eq. (4) for the predictive mean and Eq. (5) for the covariance). This would, for example, be of interest in a collaborative filtering application with a fixed set of items. In this section we describe how the covariance can be generalized to new inputs z ̸∈X. Note that, in all of the EM algorithm, the content features xi j do not contribute at all. In order to generalize the learned covariance matrix, we employ a kernel smoother with an auxiliary kernel function r(·, ·) that takes a pair of content features as input. As a constraint, we need to guarantee that the derived kernel is positive definite, such that straightforward interpolation schemes cannot readily be applied. Thus our strategy is to interpolate the eigenvectors of K instead and subsequently derive a positive definite kernel. This approach is related to the Nystr¨om method, which is primarily a method for extrapolating eigenfunctions that are only known at a discrete set of points. In contrast to Nystr¨om, the extrapolating smoothing kernel is not known in our setting and we employ a generic smoothing kernel r(·, ·) instead [12]. Let K = UΛU T be the eigendecomposition of covariance matrix K, with a diagonal matrix of eigenvalues Λ and orthonormal eigenvectors U. With V = UΛ1/2, the columns of V are scaled eigenvectors. We now approximate the i-th scaled eigenvector vi by a Gaussian process with covariance function r(·, ·) and obtain as an approximation of the scaled eigenfunction φi(w) = N X j=1 r(w, xj)bi,j (6) with weights bi = (bi,1, . . . , bi,N)⊤= (R + λI)−1vi. R denotes the Gram matrix for the smoothing kernel on all N points. An additional regularization term λI is introduced to stabilize the inverse. Based on the approximate scaled eigenfunctions, the resulting kernel function is simply l(w, z) = X i φi(w)φi(z) = r(w)⊤(R + λI)−1K(R + λI)−1r(z). (7) with r(w)⊤= (r(x1, w), . . . , r(xN, w)). R (resp. L) are the Gram matrices at the training data points X for kernel function r (resp. l) . λ is a tuning parameter that determines which proportion of K is explained by the content kernel. With λ = 0, L = K is reproduced which means that all of K can be explained by the content kernel. With λ →∞ then l(w, z) →0 and no portion of K is explained by the content kernel.3 Also, note that the eigenvectors are only required in the derivation, and do not need to be calculated when evaluating the kernel.4 Similarly, one can build a kernel smoother to extrapolate from the mean vector m to an approximate mean function ˆm(·). The prediction for a new object v in scenario i thus becomes f i(v) = ˆm(v) + X j∈I(i) l(v, xj) βi j (8) with weights β given by βi = (KI(i),I(i) + σ2I)−1(yi −mI(i)). It is important to note l has a much richer structure than the auxiliary kernel r. By expanding the expression for l, one can see that l amounts to a data-dependent covariance function that can be written as a superposition of kernels r, l(v, w) = N X i=1 r(xi, v)aw j , (9) with input dependent weights aw = (R + λI)−1K(R + λI)−1rw. 4 Experiments We first illustrate the process of covariance matrix learning on a small toy example: Data is generated by sampling from a Gaussian process with the nonstationary “neural network covariance function” [11]. Independent Gaussian noise of variance 10−4 is added. Input points X are 100 randomly placed points in the interval [−1, 1]. We consider M = 20 scenarios, where each scenario has observations on a random subset Xi of X, with N i ≈ 0.1N. In Fig. 1(a), each scenario corresponds to one “noisy line” of points. Using the EM-based covariance matrix learning (Sec. 2.1) on this data, the nonstationarity of the data does no longer pose problems, as Fig. 1 illustrates. The (stationary) covariance matrix shown in Fig. 1(c) was used both as the initial value for K and for the prior covariance S in Eq. (3). While the learned covariance matrix Fig. 1(d) does not fully match the true covariance, it clearly captures the nonstationary effects. 4.1 A Recommendation Engine As a testbed for the proposed methods, we consider an information filtering task. The goal is to predict individual users’ preferences for a large collection of art images5, where 3Note that, also if the true interpolating kernel was known, i.e., r = k, and with λ = 0, we obtain l(w, z) = k(w, z)K−1k(w, z) which is the approximate kernel obtained with Nystr¨om. 4A related form of kernel matrix extrapolation has been recently proposed by [10]. 5http://honolulu.dbs.informatik.uni-muenchen.de:8080/paintings/index.jsp (a) Training data (b) True covariance matrix (c) Initial covariance matrix (d) Covariance matrix learned via EM Figure 1: Example to illustrate covariance matrix learning via EM. The data shown in (a) was drawn from a Gaussian process with a nonstationary “neural network” covariance function. When initialized with the stationary matrix shown in (c), EM learning resulted in the covariance matrix shown in (d). Comparing the learned matrix (d) with the true matrix (b) shows that the nonstationary structure is captured well each user rated a random subset out of a total of 642 paintings, with ratings “like” (+1), “dislike”(−1), or “not sure” (0). In total, ratings from M = 190 users were collected, where each user had rated 89 paintings on average. Each image is also described by a 275dimensional feature vector (containing correlogram, color moments, and wavelet texture). Fig. 2(a) shows ROC curves for collaborative filtering when preferences of unrated items within the set of 642 images are predicted. Here, our transductive approach (Eq. (4), “GP with EM covariance”) is compared with a collaborative approach using Pearson correlation [3] (“Collaborative Filtering”) and an alternative nonparametric hierarchical Bayesian approach [13] (“Hybrid Filter”). All algorithms are evaluated in a 10-fold cross validation scheme (repeated 10 times), where we assume that ratings for 20 items are known for each test user. Based on the 20 known ratings, predictions can be made for all unrated items. We obtain an ROC curve by computing sensitivity and specificity for the proportion of truly liked paintings among the N top ranked paintings, averaged over N. The figure shows that our approach is considerably better than collaborative filtering with Pearson correlation and even gains a (yet small) advantage over the hybrid filtering technique. Note that the EM algorithm converged6 very quickly, requiring about 4–6 EM steps to learn the covariance matrix K. Also, we found that the performance is rather insensitive with respect to the hyperparameters, that is, the choice of µ, S and the equivalent sample sizes A and B. Fig. 2(b) shows ROC curves for the inductive setting where predictions for items outside 6S was set by learning a standard parametric GPR model from the preference data of one randomly chosen user, setting kernel parameters via marginal likelihood, and using this model to generate a full covariance matrix for all points. (a) Transductive methods (b) Inductive methods Figure 2: ROC curves of different methods for predicting user preferences for art images the training set are to be made (sometimes referred to as the “new item problem”). Shown is the performance obtained with the generalized Nystr¨om method ( Eq. (8), “GP with Generalized Nystr¨om”)7, and when predicting user preferences from image features via an SVM with squared exponential kernel (“SVM content-based filtering”). It is apparent that the new approach with the learned kernel is superior to the standard SVM approach. Still, the overall performance of the inductive approach is quite limited. The low-level content features are only very poor indicators for the high level concept “liking an art image”, and inductive approaches in general need to rely on content-dependent collaborative filtering. The purely content-independent collaborative effect, which is exploited in the transductive setting, cannot be generalized to new items. The purely content-independent collaborative effect can be viewed as correlated noise in our model. 5 Summary and Conclusions This article introduced a novel method of learning Gaussian process covariance functions from multi-task learning problems, using a hierarchical Bayesian framework. In the hierarchical framework, the GP models for individual scenarios borrow strength from each other via a common prior for mean and covariance. The learning task was solved in two steps: First, an EM algorithm was used to learn the shared mean vector and covariance matrix on a fixed set of points. In a second step, the learned covariance matrix was generalized to new points via a generalized form of Nystr¨om method. Our initial experiments, where we use the method as a recommender system for art images, showed very promising results. Also, in our approach, a clear distinction is made between content-dependent and content-independent collaborative filtering. We expect that our approach will be even more effective in applications where the content features are more powerful (e.g. in recommender systems for textual items such as news articles), and allow a even better prediction of user preferences. Acknowledgements This work was supported in part by the IST Programme of the European Union, under the PASCAL Network of Excellence (EU # 506778). 7To obtain the kernel r, we fitted GP user preference models for a few randomly chosen users, with individual ARD weights for each input dimension in a squared exponential kernel. ARD weights for r are taken to be the medians of the fitted ARD weights. References [1] Bakker, B. and Heskes, T. Task clustering and gating for bayesian multitask learning. Journal of Machine Learning Research, 4:83–99, 2003. [2] Blei, D. M., Ng, A. Y., and Jordan, M. I. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [3] Breese, J. S., Heckerman, D., and Kadie, C. Empirical analysis of predictive algorithms for collaborative filtering. Tech. Rep. MSR-TR-98-12, Microsoft Research, 1998. [4] Caruana, R. Multitask learning. Machine Learning, 28(1):41–75, 1997. [5] Chapelle, O. and Harchaoui, Z. A machine learning approach to conjoint analysis. In L. Saul, Y. Weiss, and L. Bottou, eds., Neural Information Processing Systems 17. MIT Press, 2005. [6] Gelman, A., Carlin, J., Stern, H., and Rubin, D. Bayesian Data Analysis. CRCPress, 1995. [7] Lawrence, N. D. and Platt, J. C. Learning to learn with the informative vector machine. In R. Greiner and D. Schuurmans, eds., Proceedings of ICML04. Morgan Kaufmann, 2004. [8] Minka, T. P. and Picard, R. W. Learning how to learn is learning with point sets, 1999. Unpublished manuscript. Revised 1999. [9] Schafer, J. L. Analysis of Incomplete Multivariate Data. Chapman&Hall, 1997. [10] Vishwanathan, S., Guttman, O., Borgwardt, K. M., and Smola, A. Kernel extrapolation, 2005. Unpublished manuscript. [11] Williams, C. K. Computation with infinite neural networks. Neural Computation, 10(5):1203– 1216, 1998. [12] Williams, C. K. I. and Seeger, M. Using the nystr¨om method to speed up kernel machines. In T. K. Leen, T. G. Dietterich, and V. Tresp, eds., Advances in Neural Information Processing Systems 13, pp. 682–688. MIT Press, 2001. [13] Yu, K., Schwaighofer, A., Tresp, V., Ma, W.-Y., and Zhang, H. Collaborative ensemble learning: Combining collaborative and content-based information filtering via hierarchical Bayes. In C. Meek and U. Kjærulff, eds., Proceedings of UAI 2003, pp. 616–623, 2003. [14] Zhu, X., Ghahramani, Z., and Lafferty, J. Semi-supervised learning using Gaussian fields and harmonic functions. In Proceedings of ICML03. Morgan Kaufmann, 2003. Appendix To derive an EM algorithm for Eq. (2), we treat the functional values f i in each scenario i as the unknown variables. In each EM iteration t, the parameters to be estimated are θ(t) = {m(t), K(t), σ2(t)}. In the E-step, the sufficient statistics are computed, E M X i=1 f i | yi, θ(t) = M X i=1 ˜f i,(t) (10) E M X i=1 f i(f i)⊤| yi, θ(t) = M X i=1 ˜f i,(t)(˜f i,(t))⊤+ ˜Ci (11) with ˜f i and ˜Ci defined in Eq. (4) and (5). In the M-step, the parameters θ are re-estimated as θ(t+1) = arg maxθ Q(θ | θ(t)), with Q(θ | θ(t)) = E h lp(θ | f, y) | y, θ(t)i , (12) where lp stands for the penalized log-likelihood of the complete data, lp(θ | f, y) = log Wi−1(K | α, β) + log N(m | ν, η−1K)+ + M X i=1 log N(˜f i | m, K) + M X i=1 log N(yi I(i) | ˜f i I(i), σ21) (13) Updated parameters are obtained by setting the partial derivatives of Q(θ | θ(t)) to zero.
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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 In q = {i = (i1, i2, . . . , iq) : 1 ≤i1 < . . . < iq ≤n, ii ∈N} and likewise Im q . Then the compound matrix of order q is defined as [Cq(A)]i,j := det(A(ik, jl))q k,l=1 where i ∈In q and j ∈Im q . (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 [Af](x) = Z Y kA(x, y)f(y)dy. (2) 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 IX q = {x ∈Xq : x1 ≤. . . ≤xq} and likewise IY q . Then the compound kernel of order q is defined as k[q](x, y) := det(k(xk, yl))q k,l=1 where x ∈IX q and y ∈IY q . (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). (4) To recover Theorem 2 from Theorem 7 set X = [1..m], Y = [1..n] and Z = [1..l]. 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 ψ(A)⊤ψ(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⊤TB (5) k(A, B) = det SA⊤TB . (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 = VSV ⊤ S and T = V ⊤ T VT . By virtue of the commutativity of the trace we have 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⊤TB and k(A, B) = det Cq SA⊤TB satisfy Mercer’s condition. Proof We exploit the factorization S = VSV ⊤ S , T = V ⊤ T VT and apply Theorem 7. This yields Cq(SA⊤TB) = 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]: D(A, µ) := ∞ X q=1 µq q! tr Cq(A). (7) This series converges for all µ ∈C and it is an entire function of µ. 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 µ > 0. Then the following kernel satisfies Mercer’s condition: k(A, B) := D(A⊤B, µ) where µ > 0. (8) D(A⊤B, µ) is a weighted combination of the kernels discussed above. The exponential down-weighting via 1 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 computational task even in the finite dimensional case. If A, B ∈Rm×n, the matrix Cq(A⊤B) has n 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(q3nq) 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 det Cq(A) = (det A)( n−1 q−1). (9) and consequently k(A, B) = det Cq[SA⊤TB] = (det[SA⊤TB])( n−1 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 λ1, . . . , λm be its eigenvalues. Then tr Cq(A) can be computed by the following recursion: tr Cq(A) = 1 q q X j=1 (−1)j+1 ¯Cq−j(A) ¯Cj(A) where ¯Cq(A) = n X j=1 λq j. (10) Proof We begin by writing A in its Jordan normal form as A = PDP −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⊤TB 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 ̸= 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 = Pxt + wt where wt ∼N(0, R) (12a) xt = Qxt−1 + vt where vt ∼N(0, S). (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 1Eq. (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−λt. In this case the Binet-Cauchy kernel between systems X and X′ becomes tr Cq(S Traj(X) T Traj(X′)⊤) = ∞ X i=1 e−λty⊤ t y′ t. (13) Since yt, y′ t are random variables, we also need to take expectations over wt, vt, w′ t, v′ t. Some tedious yet straightforward algebra [14] allows us to compute (13) as follows: k(X, X′) = x⊤ 0 M1x′ 0 + 1 eλ −1 tr [SM2 + R] , (14) where M1, M2 satisfy the Sylvester equations: M1 = e−λQ⊤P ⊤P ′Q′ + e−λQ⊤M1Q′ and M2 = P ⊤P ′ + e−λQ⊤M2Q′. (15) Such kernels can be computed in O(n3) 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) = QXRX 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(SQXTQ⊤ X′) = det(QXQ⊤ X′). (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. 2Martin [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. 3To 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 = WD−1xt, 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 one uses x(t) = exp(−˜Lt)x(0), where ˜L is the normalized graph Laplacian [9]. 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 = τ. This amounts to choosing T = 1 and a S which is zero except for a diagonal entry at τ. • 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). −5 −4 −3 −2 −1 0 1 2 3 4 −4 −3 −2 −1 0 1 2 Figure 1: LLE embeddings of 480 random clips from Kill Bill 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. This corresponds to clips which are temporally 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. 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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Nonlinear Blind Source Separation by Integrating Independent Component Analysis and Slow Feature Analysis Tobias Blaschke Institute for Theoretical Biology Humboldt University Berlin Invalidenstraße 43, D-10115 Berlin, Germany t.blaschke@biologie.hu-berlin.de Laurenz Wiskott Institute for Theoretical Biology Humboldt University Berlin Invalidenstraße 43, D-10115 Berlin, Germany l.wiskott@biologie.hu-berlin.de Abstract In contrast to the equivalence of linear blind source separation and linear independent component analysis it is not possible to recover the original source signal from some unknown nonlinear transformations of the sources using only the independence assumption. Integrating the objectives of statistical independence and temporal slowness removes this indeterminacy leading to a new method for nonlinear blind source separation. The principle of temporal slowness is adopted from slow feature analysis, an unsupervised method to extract slowly varying features from a given observed vectorial signal. The performance of the algorithm is demonstrated on nonlinearly mixed speech data. 1 Introduction Unlike in the linear case the nonlinear Blind Source Separation (BSS) problem can not be solved solely based on the principle of statistical independence [1, 2]. Performing nonlinear BSS with Independent Component Analysis (ICA) requires additional information about the underlying sources or to regularize the nonlinearities. Since source signal components are usually more slowly varying than any nonlinear mixture of them we consider to require the estimated sources to be as slowly varying as possible. This can be achieved by incorporating ideas from Slow Feature Analysis (SFA) [3] into ICA. After a short introduction to linear BSS, nonlinear BSS, and SFA we will show a way how to combine SFA and ICA to obtain an algorithm that solves the nonlinear BSS problem. 2 Linear Blind Source Separation Let x(t) = [x1 (t) , . . . , xN (t)]T be a linear mixture of a source signal s(t) = [s1 (t) , . . . , sN (t)]T and defined by x (t) = As (t) , (1) with an invertible N × N mixing matrix A. Finding a mapping u (t) = QWx (t) (2) such that the components of u are mutually statistically independent is called Independent Component Analysis (ICA). The mapping is often divided into a whitening mapping W, resulting in uncorrelated signal components yi with unit variance and a successive orthogonal transformation Q, because one can show [4] that after whitening an orthogonal transformation is sufficient to obtain independence. It is well known that ICA solves the linear BSS problem [4]. There exists a variety of algorithms performing ICA and therefore BSS (see e.g. [5, 6, 7]). Here we focus on a method using only second-order statistics introduced by Molgedey and Schuster [8]. The method consists of optimizing an objective function subject to minimization, which can be written as ΨICA (Q) = N X α,β=1 α̸=β C(u) αβ (τ) 2 = N X α,β=1 α̸=β N X γ,δ=1 QαγQβδC(y) γδ (τ) 2 , (3) operating on the already whitened signal y. C(y) γδ (τ) is an entry of a symmetrized time delayed covariance matrix defined by C(y) (τ) = D y (t) y (t + τ)T + y (t + τ) y (t)T E , (4) and C(u) (τ) is defined correspondingly. Qαβ denotes an entry of Q. Minimization of ΨICA can be understood intuitively as finding an orthogonal matrix Q that diagonalizes the covariance matrix with time delay τ. Since, because of the whitening, the instantaneous covariance matrix is already diagonal this results in signal components that are decorrelated instantaneously and at a given time delay τ. This can be sufficient to achieve statistical independence [9]. 2.1 Nonlinear BSS and ICA An obvious extension to the linear mixing model (1) has the form x (t) = F (s (t)) , (5) with a function F (· ) RN →RM that maps N-dimensional source vectors s onto Mdimensional signal vectors x. The components xi of the observable are a nonlinear mixture of the sources and like in the linear case source signal components si are assumed to be mutually statistically independent. Extracting the source signal is in general only possible if F (· ) is an invertible function, which we will assume from now on. The equivalence of BSS and ICA in the linear case does in general not hold for a nonlinear function F (· ) [1, 2]. To solve the nonlinear BSS problem additional constraints on the mixture or the estimated signals are needed to bridge the gap between ICA and BSS. Here we propose a new way to achieve this by adding a slowness objective to the independence objective of pure ICA. Assume for example a sinusoidal signal component xi = sin (2πt) and a second component that is the square of the first xj = x2 i = 0.5 (1 −cos (4πt)) is given. The second component is more quickly varying due to the frequency doubling induced by the squaring. Typically nonlinear mixtures of signal components are more quickly varying than the original components. To extract the right source components one should therefore prefer the slowly varying ones. The concept of slowness is used in our approach to nonlinear BSS by combining an ICA part that provides the independence of the estimated source signal components with a part that prefers slowly varying signals over more quickly varying ones. In the next section we will give a short introduction to Slow Feature Analysis (SFA) building the basis of the second part of our method. 3 Slow Feature Analysis Assume a vectorial input signal x(t) = [x1(t), . . . , xM(t)]T is given. The objective of SFA is to find an in general nonlinear input-output function u (t) = g (x (t)) with g (x (t)) = [g1 (x (t)) , . . . , gR (x (t))]T such that the ui (t) are varying as slowly as possible. This can be achieved by successively minimizing the objective function ∆(ui) := ˙u2 i (6) for each ui under the constraints ⟨ui⟩ = 0 (zero mean), (7) u2 i = 1 (unit variance), (8) ⟨uiuj⟩ = 0 ∀j < i (decorrelation and order). (9) Constraints (7) and (8) ensure that the solution will not be the trivial solution ui = const. Constraint (9) provides uncorrelated output signal components and thus guarantees that different components carry different information. Intuitively we are searching for signal components ui that have on average a small slope. Interestingly Slow Feature Analysis (SFA) can be reformulated with an objective function similar to second-order ICA, subject to maximization [10], ΨSFA (Q) = M X α=1 C(u) αα (τ) 2 = M X α=1 M X β,γ=1 QαβQαγC(y) βγ (τ) 2 . (10) To understand (10) intuitively we notice that slowly varying signal components are easier to predict, and should therefore have strong auto correlations in time. Thus, maximizing the time delayed variances produces slowly varying signal components. 4 Independent Slow Feature Analysis If we combine ICA and SFA we obtain a method we refer to as Independent Slow Feature Analysis (ISFA) that recovers independent components out of a nonlinear mixture using a combination of SFA and second-order ICA. As already explained, second-order ICA tends to make the output components independent and SFA tends to make them slow. Since we are dealing with a nonlinear mixture we first compute a nonlinearly expanded signal z = h (x) with h (· ) RM →RL being typically monomials up to a given degree, e.g. an expansion with monomials up to second degree can be written as h (x (t)) = [x1, . . . , xN, x1x1, x1x2, . . . , xMxM]T −hT 0 (11) when given an M-dimensional signal x. The constant vector hT 0 is used to make the expanded signal mean free. In a second step z is whitened to obtain y = Wz. Thirdly we apply linear ICA combined with linear SFA on y in order to find the estimated source signal u. Because of the whitening we know that ISFA, like ICA and SFA, is solved by finding an orthogonal L × L matrix Q. We write the estimated source signal u as v = u ˜u = Qy = QWz = QWh (x) , (12) where we introduced ˜u, since R, the dimension of the estimated source signal u, is usually much smaller than L, the dimension of the expanded signal. While the ui are statistically independent and slowly varying the components ˜ui are more quickly varying and may be statistically dependent on each other as well as on the selected components. To summarize, we have an M dimensional input x an L dimensional nonlinearly expanded and whitened y and an R dimensional estimated source signal u. ISFA searches an R dimensional subspace such that the ui are independent and slowly varying. This is achieved at the expense of all ˜ui. 4.1 Objective function To recover R source signal components ui i = 1, . . . , R out of an L-dimensional expanded and whitened signal y the objective reads ΨISFA (u1, . . . , uR; τ) = bICA R X α,β=1, α̸=β C(u) αβ (τ) 2 −bSFA R X α=1 C(u) αα (τ) 2 , (13) where we simply combine the ICA objective (3) and SFA objective (10) weighted by the factors bICA and bSFA, respectively. Note that the ICA objective is usually applied to the linear case to unmix the linear whitened mixture y whereas here it is used on the nonlinearly expanded whitened signal y = Wz. ISFA tries to minimize ΨISFA which is the reason why the SFA part has a negative sign. 4.2 Optimization Procedure From (12) we know that C(u) (τ) in (13) depends on the orthogonal matrix Q. There are several ways to find the orthogonal matrix that minimizes the objective function. Here we apply successive Givens rotations to obtain Q. A Givens rotation Qµν is a rotation around the origin within the plane of two selected components µ and ν and has the matrix form Qµν αβ := cos(φ) for (α, β) ∈{(µ, µ) , (ν, ν)} −sin(φ) for (α, β) ∈{(µ, ν)} sin(φ) for (α, β) ∈{(ν, µ)} δαβ otherwise (14) with Kronecker symbol δαβ and rotation angle φ. Any orthogonal L × L matrix such as Q can be written as a product of L(L−1) 2 (or more) Givens rotation matrices Qµν (for the rotation part) and a diagonal matrix with elements ±1 (for the reflection part). Since reflections do not matter in our case we only consider the Givens rotations as is often used in second-order ICA algorithms (see e.g. [11]). We can therefore write the objective as a function of a Givens rotation Qµν as ΨISFA (Qµν) = bICA R X α,β=1, α̸=β L X γ,δ=1 Qµν αγQµν βδC(y) γδ (τ) 2 − bSFA R X α=1 L X β,γ=1 Qµν αβQµν αγC(y) βγ (τ) 2 . (15) Assume we want to minimize ΨISFA for a given R, where R denotes the number of signal components we want to extract. Applying a Givens rotation Qµν we have to distinguish three cases. • Case 1: Both axes uµ and uν lie inside the subspace spanned by the first R axes (µ, ν ≤R). The sum over all squared cross correlations of all signal components that lie outside the subspace is constant as well as those of all signal components inside the subspace. There is no interaction between inside and outside, in fact the objective function is exactly the objective for an ICA algorithm based on secondorder statistics e.g. TDSEP or SOBI [12, 13]. In [10] it has been shown that this is equivalent to SFA in the case of a single time delay. • Case 2: Only one axis, w.l.o.g. uµ, lies inside the subspace, the other, uν, outside (µ ≤R < ν). Since one axis of the rotation plane lies outside the subspace, uµ in the objective function can be optimized at the expense of ˜uν outside the subspace. A rotation of π/2, for instance, would simply exchange components uµ and uν. This gives the possibility to find the slowest and most independent components in the whole space spanned by all ui and ˜uj (i = 1, . . . , R, j = R + 1, . . . , L) in contrast to Case 1 where the minimum is searched within the subspace spanned by the R components in the objective function. • Case 3: Both axes lie outside the subspace (R < µ, ν): A Givens rotation with the two rotation axes outside the relevant subspace does not affect the objective function and can therefore be disregarded. It can be shown that like in [14] the objective function (15) as a function of φ can always be written in the form Ψµν ISFA (φ) = A0 + A2 cos (2φ + φ2) + A4 cos (4φ + φ4) , (16) where the second term on the right hand side vanishes for Case 1. There exists a single minimum (if w.l.o.g. φ ∈ −π 2 , π 2 ) that can easily be calculated (see e.g.[14]). The derivation of (16) involves various trigonometric identities and, because of its length, is documented elsewhere1. It is important to notice that the rotation planes of the Givens rotations are selected from the whole L-dimensional space whereas the objective function only uses information of correlations among the first R signal components ui. Successive application of Givens rotations Qµν leads to the final rotation matrix Q which is in the ideal case such that QT C(y) (τ) Q = C(v) (τ) has a diagonal R × R submatrix C(u) (τ), but it is not clear if the final minimum is also the global one. However, in various simulations no local minima have been found. 4.3 Incremental Extracting of Independent Components It is possible to find the number of independent source signal components R by successively increasing the number of components to be extracted. In each step the objective function (13) is optimized for a fixed R. First a single signal component is extracted (R = 1) and then an additional one (R = 2) etc. The algorithm is stopped when no additional signal component can be extracted. As a stopping criterion every suitable measure of independence can be applied; we used the sum over squared cross-cumulants of fourth order. In our artificial examples this value is typically small for independent components, and increases by two orders of magnitudes if the number of components to be extracted is greater than the number of original source signal components. 1http://itb.biologie.hu-berlin.de/~blaschke 5 Simulation Here we show a simple example, with two nonlinearly mixed signal components as shown in Figure 1. The mixture is defined by x1 (t) = (s1 (t) + 1) sin (πs2 (t)) , x2 (t) = (s1 (t) + 1) cos (πs2 (t)) . (17) We used the ISFA algorithm with different nonlinearities (see Tab. 1). Already a nonlinear expansion with monomials up to degree three was sufficient to give good results in extracting the original source signal (see Fig. 1). In all cases ISFA did find exactly two independent signal components. A linear BSS method failed completely to find a good unmixing matrix. 6 Conclusion We have shown that connecting the ideas of slow feature analysis and independent component analysis into ISFA is a possible way to solve the nonlinear blind source separation problem. SFA enforces the independent components of ICA to be slowly varying which seems to be a good way to discriminate between the original and nonlinearly distorted source signal components. A simple simulation showed that ISFA is able to extract the original source signal out of a nonlinear mixture. Furthermore ISFA can predict the number of source signal components via an incremental optimization scheme. Acknowledgments This work has been supported by the Volkswagen Foundation through a grant to LW for a junior research group. References [1] A. Hyvärinen and P. Pajunen. Nonlinear independent component analysis: existence and uniqueness results. Neural Networks, 12(3):429–439, 1999. [2] C. Jutten and J. Karhunen. Advances in nonlinear blind source separation. In Proc. of the 4th Int. Symposium on Independent Component Analysis and Blind Signal Separation, Nara, Japan, (ICA 2003), pages 245–256, 2003. [3] Laurenz Wiskott and Terrence Sejnowski. Slow feature analysis: Unsupervised learning of invariances. Neural Computation, 14(4):715–770, 2002. Table 1: Correlation coefficients of extracted (u1 and u2) and original (s1 and s2) source signal components linear degree 2 degree 3 degree 4 u1 u2 u1 u2 u1 u2 u1 u2 s1 -0.803 -0.544 -0.001 -0.978 0.001 0.995 0.002 0.995 s2 0.332 0.517 -0.988 -0.001 -0.995 0.001 -0.996 0.000 Correlation coefficients of extracted (u1 and u2) and original (s1 and s2) source signal components for linear ICA (first column) and ISFA with different nonlinearities (monomials up to degree 2, 3, and 4). Using monomials up to degree 3 in the nonlinear expansion step already suffices to extract the original source signal. Note that the source signal can only be estimated up to permutation and scaling, resulting in different signs and permutations of the two estimated source signal components. s1 s2 (a) x1 x2 (b) u1 u2 (c) Figure 1: Waveforms and Scatter-plots of (a) the original source signal components si, (b) the nonlinear mixture, and (c) recovered components with nonlinear ISFA (ui). As a nonlinearity we used all monomials up to degree 4. [4] P. Comon. Independent component analysis, a new concept? Signal Processing, 36(3):287–314, 1994. Special Issue on Higher-Order Statistics. [5] J.-F. Cardoso and A. Souloumiac. Blind beamforming for non Gaussian signals. IEE Proceedings-F, 140:362–370, 1993. [6] T.-W. Lee, M. Girolami, and T.J. Sejnowski. Independent component analysis using an extended Infomax algorithm for mixed sub-Gaussian and super-Gaussian sources. Neural Computation, 11(2):409–433, 1999. [7] A. Hyvärinen. Fast and robust fixed-point algorithms for independent component analysis. IEEE Transactions on Neural Networks, 10(3):626–634, 1999. [8] L. Molgedey and G. Schuster. Separation of a mixture of independent signals using time delayed correlations. Physical Review Letters, 72(23):3634–3637, 1994. [9] Lang Tong, Ruey-wen Liu, Victor C. Soon, and Yih-Fang Huang. Indeterminacy and identifiability of blind identification. IEEE Transactions on Circuits and Systems, 38(5):499–509, may 1991. [10] T. Blaschke, L. Wiskott, and P. Berkes. What is the relation between independent component analysis and slow feature analysis? (in preparation), 2004. [11] Jean-François Cardoso and Antoine Souloumiac. Jacobi angles for simultaneous diagonalization. SIAM J. Mat. Anal. Appl., 17(1):161–164, 1996. [12] A. Ziehe and K.-R. Müller. TDSEP – an efficient algorithm for blind separation using time structure. In Proc. of the 8th Int. Conference on Artificial Neural Networks (ICANN’98), pages 675 – 680, Berlin, 1998. Springer Verlag. [13] Adel Belouchrani, Karim Abed Meraim, Jean-François Cardoso, and Éric Moulines. A blind source separation technique based on second order statistics. IEEE Transactions on Signal Processing, 45(2):434–44, 1997. [14] T. Blaschke and L. Wiskott. CuBICA: Independent component analysis by simultaneous third- and fourth-order cumulant diagonalization. IEEE Transactions on Signal Processing, 52(5):1250–1256, 2004.
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Solitaire: Man Versus Machine Xiang Yan∗ Persi Diaconis∗ Paat Rusmevichientong† Benjamin Van Roy∗ ∗Stanford University {xyan,persi.diaconis,bvr}@stanford.edu †Cornell University paatrus@orie.cornell.edu Abstract In this paper, we use the rollout method for policy improvement to analyze a version of Klondike solitaire. This version, sometimes called thoughtful solitaire, has all cards revealed to the player, but then follows the usual Klondike rules. A strategy that we establish, using iterated rollouts, wins about twice as many games on average as an expert human player does. 1 Introduction Though proposed more than fifty years ago [1, 7], the effectiveness of the policy improvement algorithm remains a mystery. For discounted or average reward Markov decision problems with n states and two possible actions per state, the tightest known worst-case upper bound in terms of n on the number of iterations taken to find an optimal policy is O(2n/n) [9]. This is also the tightest known upper bound for deterministic Markov decision problems. It is surprising, however, that there are no known examples of Markov decision problems with two possible actions per state for which more than n + 2 iterations are required. A more intriguing fact is that even for problems with a large number of states – say, in the millions – an optimal policy is often delivered after only half a dozen or so iterations. In problems where n is enormous – say, a googol – this may appear to be a moot point because each iteration requires Ω(n) compute time. In particular, a policy is represented by a table with one action per state and each iteration improves the policy by updating each entry of this table. In such large problems, one might resort to a suboptimal heuristic policy, taking the form of an algorithm that accepts a state as input and generates an action as output. An interesting recent development in dynamic programming is the rollout method. Pioneered by Tesauro and Galperin [13, 2], the rollout method leverages the policy improvement concept to amplify the performance of any given heuristic. Unlike the conventional policy improvement algorithm, which computes an optimal policy off-line so that it may later be used in decision-making, the rollout method performs its computations on-line at the time when a decision is to be made. When making a decision, rather than applying the heuristic policy directly, the rollout method computes an action that would result from an iteration of policy improvement applied to the heuristic policy. This does not require Ω(n) compute time since only one entry of the table is computed. The way in which actions are generated by the rollout method may be considered an alternative heuristic that improves on the original. One might consider applying the rollout method to this new heuristic. Another heuristic would result, again with improved performance. Iterated a sufficient number of times, this process would lead to an optimal policy. However, iterating is usually not an option. Computational requirements grow exponentially in the number of iterations, and the first iteration, which improves on the original heuristic, is already computationally intensive. For this reason, prior applications of the rollout method have involved only one iteration [3, 4, 5, 6, 8, 11, 12, 13]. For example, in the interesting study of Backgammon by Tesauro and Galperin [13], moves were generated in five to ten seconds by the rollout method running on configurations of sixteen to thirtytwo nodes in a network of IBM SP1 and SP2 parallel-RISC supercomputers with parallel speedup efficiencies of 90%. A second iteration of the rollout method would have been infeasible – requiring about six orders of magnitude more time per move. In this paper, we apply the rollout method to a version of solitaire, modeled as a deterministic Markov decision problem with over 52! states. Determinism drastically reduces computational requirements, making it possible to consider iterated rollouts1. With five iterations, a game, implemented in Java, takes about one hour and forty-five minutes on average on a SUN Blade 2000 machine with two 900MHz CPUs, and the probability of winning exceeds that of a human expert by about a factor of two. Our study represents an important contribution both to the study of the rollout method and to the study of solitaire. 2 Solitaire It is one of the embarrassments of applied mathematics that we cannot determine the odds of winning the common game of solitaire. Many people play this game every day, yet simple questions such as What is the chance of winning? How does this chance depend on the version I play? What is a good strategy? remain beyond mathematical analysis. According to Parlett [10], solitaire came into existence when fortune-telling with cards gained popularity in the eighteenth century. Many variations of solitaire exist today, such as Klondike, Freecell, and Carpet. Popularized by Microsoft Windows, Klondike has probably become the most widely played. Klondike is played with a standard deck of cards: there are four suits (Spades, Clubs, Hearts, and Diamonds) each made up of thirteen cards ranked 1 through 13: Ace, 2, 3, ..., 10, Jack, Queen, and King. During the game, each card resides in one of thirteen stacks2 : the pile, the talon, four suit stacks and seven build stacks. Each suit stack corresponds to a particular suit and build stacks are labeled 1 through 7. At the beginning of the game, cards are dealt so that there is one card in the first build stack, two cards in the second build stack, ..., and seven cards in the seventh build stack. The top card on each of the seven build stacks is turned face-up while the rest of the cards in the build stacks face down. The other twenty-four cards, forming the pile, face down as well. The talon is initially empty. The goal of the game is to move all cards into the suit stacks, aces first, then two’s, and so on, with each suit stack evolving as an ordered increasing arrangement of cards of the same suit. The figure below shows a typical mid-game configuration. 1Backgammon is stochastic because play is influenced by the roll of dice. 2In some solitaire literature, stacks are referred to as piles. We will study a version of solitaire in which the identity of each card at each position is revealed to the player at the beginning of the game but the usual Klondike rules still apply. This version is played by a number of serious solitaire players as a much more difficult version than standard Klondike. Parlett [10] offers further discussion. We call this game thoughtful solitaire and now spell out the rules. On each turn, the player can move cards from one stack to another in the following manner: • Face-up cards of a build stack, called a card block, can be moved to the top of another build stack provided that the build stack to which the block is being moved accepts the block. Note that all face-up cards on the source stack must be moved together. After the move, these cards would then become the top cards of the stack to which they are moved, and their ordering is preserved. The card originally immediately beneath the card block, now the top card in its stack, is turned faceup. In the event that all cards in the source stack are moved, the player has an empty stack. 3 • The top face-up card of a build stack can be moved to the top of a suit stack, provided that the suit stack accepts the card. • The top card of a suit stack can be moved to the top of a build stack, provided that the build stack accepts the card. • If the pile is not empty, a move can deal its top three cards to the talon, which maintains its cards in a first-in-last-out order. If the pile becomes empty, the player can redeal all the cards on the talon back to the pile in one card move. A redeal preserves the ordering of cards. The game allows an unlimited number of redeals. • A card on the top of the talon can be moved to the top of a build stack or a suit stack, provided that the stack to which the card is being moved accepts the card. 3It would seem to some that since the identity of all cards is revealed to the player, whether a card is face-up or face-down is irrelevant. We retain this property of cards as it is still important in describing the rules and formulating our strategy. • A build stack can only accept an incoming card block if the top card on the build stack is adjacent to and braided with the bottom card of the block. A card is adjacent to another card of rank r if it is of rank r + 1. A card is braided with a card of suit s if its suit is of a color different from s. Additionally, if a build stack is empty, it can only accept a card block whose bottom card is a King. • A suit stack can only accept an incoming card of its corresponding suit. If a suit stack is empty, it can only accept an Ace. If it is not empty, the incoming card must be adjacent to the current top card of the suit stack. As stated earlier, the objective is to end up with all cards on suit stacks. If this event occurs, the game is won. 3 Expert Play We were introduced to thoughtful solitaire by a senior American mathematician (former president of the American Mathematical Society and indeed a famous combinatorialist) who had spent a number of years studying the game. He finds this version of solitaire much more thought-provoking and challenging than the standard Klondike. For instance, while the latter is usually played quickly, our esteemed expert averages about 20 minutes for each game of thoughtful solitaire. He carefully played and recorded 2,000 games, achieving a win rate of 36.6%. With this background, it is natural to wonder how well an optimal player can perform at thoughtful solitaire. As we will illustrate, our best strategy offers a win rate of about 70%. 4 Machine Play We have developed two strategies that play thoughtful solitaire. Both are based on the following general procedure: 1. Identify the set of legal moves. 2. Select and execute a legal move. 3. If all cards are on suit stacks, declare victory and terminate. 4. If the new card configuration repeats a previous one, declare loss and terminate 4. 5. Repeat procedure. The only nontrivial task in this procedure is selection from the legal moves. We will first describe a heuristic strategy for selecting a legal move based on a card configuration. Afterwards, we will discuss the use of rollouts. 4.1 A Heuristic Strategy Our heuristic strategy is based on part of the Microsoft Windows Klondike scoring system: • The player starts the game with an initial score of 0. 4One straight-forward way to determine if a card configuration has previously occurred is to store all encountered card configurations. Instead of doing so, however, we notice that there are three kinds of moves that could lead us into an infinite loop: pile-talon moves, moves that could juggle a card block between two build stacks, and moves that could juggle a card block between a build stack and a suit stack. Hence, to simplify our strategy, we disable the second kind of moves. Our heuristic will also practically disable the third kind. For the first kind, we record if any card move other than a pile-talon move has occurred since the last redeal. If not, we detect an infinite loop and declare loss. • Whenever a card is moved from a build stack to a suit stack, the player gains 5 points. • Whenever a card is moved from the talon to a build stack, the player gains 5 points. • Whenever a card is moved from a suit stack to a build stack, the player loses 10 points. In our heuristic strategy, we assign a score to each card move based on the above scoring system. We assign the score zero to any moves not covered by the above rules. When selecting a move, we choose among those that maximize the score. Intuitively, this heuristic seems reasonable. The player has incentive to move cards from the talon to a build stack and from a build stack to a suit stack. One important element that the heuristic fails to capture, however, is what move to make when multiple moves maximize the score. Such decisions – especially during the early phases of a game – are crucial. To select among moves that maximize score, we break the tie by assigning the following priorities: • If the card move is from a build stack to another build stack, one of the following two assignments of priority occurs: – If the move turns an originally face-down card face-up, we assign this move a priority of k + 1, where k is the number of originally face-down cards on the source stack before the move takes place. – If the move empties a stack, we assign this move a priority of 1. • If the card move is from the talon to a build stack, one of the following three assignments of priority occurs: – If the card being moved is not a King, we assign the move priority 1. – If the card being moved is a King and its matching Queen is in the pile, in the talon, in a suit stack, or is face-up in a build stack, we assign the move priority 1. – If the card being moved is a King and its matching Queen is face-down in a build stack, we assign the move priority -1. • For card moves not covered by the description above, we assign them a priority of 0. In addition to introducing priorities, we modify the Windows Klondike scoring system further by adding the following change: in a card move, if the card being moved is a King and its matching Queen is face-down in a build stack, we assign the move a score of 0. Note that given our assignment of scores and priorities, we practically disable card moves from a suit stack to a build stack. Because such moves have a negative score and a card move from the pile to the talon or from the talon to the pile has zero score and is almost always available, our strategy would always choose the pile-talon move over the moves from a suit stack to a build stack. In the case when multiple moves equal in priority maximize the score, we randomly select a move among them. The introduction of priority improves our original game-playing strategy in two ways: when we encounter a situation where we can move either one of two blocks on two separate build stacks atop the top card of a third build stack, we prefer moving the block whose stack has more face-down cards. Intuitively, such a move would strive to balance the number of face-down cards in stacks. Our experiments show that this heuristic significantly improves success rate. The second way in which our prioritization scheme helps is that we are more deliberate in which King to select to enter an empty build stack. For instance, consider a situation where the King of Hearts and the King of Spades, both on the pile, are vying for an empty build stack and there is a face-up Queen of Diamonds on a build stack. We should certainly move the King of Spades to the empty build stack so that the Queen of Diamonds can be moved on top of it. Whereas our prioritization warrants such consideration, our original heuristic does not. 4.2 Rollouts Consider a strategy h that maps a card configuration x to a legal move h(x). What we described in the previous section was one example of a strategy h. In this section, we will discuss the rollout method as a procedure for amplifying the performance of any strategy. Given a strategy h, this procedure generates an improved strategy h′, called a rollout strategy. This idea was originally proposed by Tesauro and Galperin [13] and builds on the policy improvement algorithm of dynamic programming [1, 7]. Given a card configuration x. A strategy h would make a move h(x). A rollout strategy would make a move h′(x), determined as follows: 1. For each legal move a, simulate the remainder of the game, taking move a and then employing strategy h thereafter. 2. If any of these simulations leads to victory, choose one of them randomly and let h′(x) be the corresponding move a5. 3. If none of the simulations lead to victory, let h′(x) = h(x). We can then iterate this procedure to generate a further improved strategy h′′ that is a rollout strategy relative to h′. It is easy to prove that after a finite number of such iterations, we would arrive at an optimal strategy [2]. However, the computation time required grows exponentially in the number of iterations, so this may not be practical. Nevertheless, one might try a few iterations and hope that this offers the bulk of the mileage. 5 Results We implemented in Java the heuristic strategy and the procedure for computing rollout strategies. Simulation results are provided in the following table and chart. We randomly generated a large number of games and played them with our algorithms in an effort to approximate the success probability with the percentage of games actually won. To determine a sufficient number of games to simulate, we used the Central Limit Theorem to compute the confidence bounds on success probability for each algorithm with a confidence level of 99%. For the original heuristic and 1 through 3 rollout iterations, we managed to achieve confidence bounds of [-1.4%, 1.4%]. For 4 and 5 rollout iterations, due to time constraints, we simulated fewer games and obtained weaker confidence bounds. Interestingly, however, after 5 rollout iterations, the resulting strategy wins almost twice as frequently as our esteemed mathematician. 5Note that at this stage, we could record all moves made in this simulation and declare victory. That is how our program is implemented. However, we leave step 2 as stated for the sake of clarity in presentation. Player Success Games Average Time 99% Confidence Rate Played Per Game Bounds Human expert 36.6% 2,000 20 minutes ±2.78% heuristic 13.05% 10,000 .021 seconds ±.882% 1 rollout 31.20% 10,000 .67 seconds ±1.20% 2 rollouts 47.60% 10,000 7.13 seconds ±1.30% 3 rollouts 56.83% 10,000 1 minute 36 seconds ±1.30% 4 rollouts 60.51% 1,000 18 minutes 7 seconds ±4.00% 5 rollouts 70.20% 200 1 hour 45 minutes ±8.34% 6 Future Challenges One limitation of our rollout method lies in its recursive nature. Although it is clearly formulated and hence easily implemented in software, the algorithm does not provide a simple and explicit strategy for human players to make decisions. One possible direction for further exploration would be to compute a value function, mapping the state of the game to an estimate of whether or not the game can be won. Certainly, this function could not be represented exactly, but we could try approximating it in terms of a linear combination of features of the game state, as is common in the approximate dynamic programming literature [2]. We have also attempted proving an upper bound for the success rate of thoughtful solitaire by enumerating sets of initial card configurations that would force loss. Currently, the tightest upper bound we can rigorously prove is 98.81%. Speed optimization of our software implementation is under way. If the success rate bound is improved and we are able to run additional rollout iterations, we may produce a verifiable near-optimal strategy for thoughtful solitaire. Acknowlegment This material is based upon work supported by the National Science Foundation under Grant ECS-9985229. References [1] R. Bellman. Applied Dynamic Programming. Princeton University Press, 1957. [2] D. Bertsekas and J.N. Tsitsiklis. Neuro-Dynamic Programming. Athena Scientific, 1996. [3] D. P. Bertsekas, J. N. Tsitsiklis, and C. Wu, Rollout Algorithms for Combinatorial Optimization. Journal of Heuristics, 3:245-262, 1997. [4] D. P. Bertsekas and D. A. Casta˜non. Rollout Algorithms for Stochastic Scheduling Problems. Journal of Heuristics, 5:89-108, 1999. [5] D. Bertsimas and R. Demir. An Approximate Dynamic Programming Approach to Multi-dimensional Knapsack Problems. Management Science, 4:550-565, 2002. [6] D. Bertsimas and I. Popescu. Revenue Management in a Dynamic Network Environment. Transportation Science, 37:257-277, 2003. [7] R. Howard. Dynamic Programming and Markov Processes. MIT Press, 1960. [8] A. McGovern, E. Moss, and A. Barto. Building a Basic Block Instruction Scheduler Using Reinforcement Learning and Rollouts. Machine Learning, 49:141-160, 2002. [9] Y. Mansour and S. Singh. On the Complexity of Policy Iteration. In Fifteenth Conference on Uncertainty in Artificial Intelligence, 1999. [10] D. Parlett. A History of Card Games. Oxford University Press, 1991. [11] N. Secomandi. Analysis of a Rollout Approach to Sequencing Problems with Stochastic Routing Applications. Journal of Heuristics, 9:321-352, 2003. [12] N. Secomandi. A Rollout Policy for the Vehicle Routing Problem with Stochastic Demands. Operations Research, 49:796-802, 2001. [13] G. Tesauro and G. Galperin. On-line Policy Improvement Using Monte-Carlo Search. In Advances in Neural Information Processing Systems, 9:1068-1074, 1996.
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A Machine Learning Approach to Conjoint Analysis Olivier Chapelle, Za¨ıd 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 ({x1 k, . . . , xp k}, yk) indicating that among the p products {x1 k, . . . , xp k}, the yth k was preferred. Without noise, this is expressed mathematically by u(xyk k ) ≥u(xb k), ∀b ̸= 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 profile is described through a attributes with l1, ..., la levels each, via a vector of length m = Pa 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 1e.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 α (representing the aggregated partworths) and covariance Σ (encoding population’s heterogeneity), 2. The covariance matrix Σ has an invert Wishart prior, and α 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 P(x∗|wi) = exp(wi · x∗) Pp b=1 exp(wi · xb). (1) 2.2 Model estimation We describe now the standard way of estimating α, w ≡(w1, . . . , wn) and Σ 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 (α, w and Σ) is drawn in turn from its posterior distribution the two others being fixed. Sampling for α and Σ is straightforward, whereas sampling from P(w|α, Σ, Y ) ∝P(Y |w). P(w|α, Σ) is achieved with the Metropolis-Hastings algorithm. When convergence is reached, the sampling goes on and finally outputs the empirical expectation of α, w and Σ. 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. 2Technical papers of Sawtooth software [11], the world leading company for conjoint analysis softwares, provide very useful and extensive references. 3that we will call HB-Sampled or HB-S in the rest of the paper. To find α, w and Σ which maximize P(α, w, Σ|Y ), let us use Bayes’ rule, P(α, w, Σ|Y ) ∝ P(Y |α, w, Σ) · P(w|α, Σ) · P(α|Σ) · P(Σ) ∝ P(Y |w) · P(w|α, Σ) · P(Σ) (2) Maximizing (2) with respect to Σ yields ΣMAP = I+Cw n+d , with Cw being the “covariance” matrix of the wi centered at α: Cw = P(wi −α)(wi −α)⊤. Putting back this value in (2), we get −log P(α, w, ΣMAP|Y ) = −log P(Y |w) + log |I + Cw(α)| + 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 not the second term. For this reason, we propose to change log |I + Cw| by trace(Cw) = P ||wi −α||2 (this would be a valid approximation if trace(Cw) ≪1). With this new prior on w, the rhs of (3) becomes W(α, w) = n X i=1 −log P(Yi|wi) + ||wi −α||2. (4) As in equation (3), this objective function is minimized with respect to α when α 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 α, minimize (4) with respect to each of the wi independently. 2. For a given w, set α 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 α, 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 Σ to the identity matrix. One might instead consider Σ 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 yth k product from the basket {x1 k, . . . , xp k}: w·xyk k ≥w·xb k, ∀b ̸= 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 · (x1 k −xb k) ≥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. 4which is consistent with the L2-loss measuring deviations of wi-s from α. 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 ξkb: Minimize w2 + C Pq k=1 Pp b=2 ξ2 kb subject to w · (x1 k −xb k) ≥1 −ξkb. 3.2 Estimation of individual utilities It was proposed in [3] to train one SVM per consumer to get wi and to compute the individual partworths by regularizing with the aggregated partworths w = 1 n Pn i=1 wi: w∗ i = wi+w 2 . 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 x1 k, . . . , xp k) : ( Minimize w2 i + C qi P k∈Qi Pp b=2 ξ2 kb + ˜ C P j̸=i qj P k/∈Qi Pp b=2 ξ2 kb subject to wi · (x1 k −xb k) ≥1 −ξkb, ∀k, ∀b ≥2 . Here the ratio C ˜ C determines the trade-off between the individual scale and the aggregated one.5 For C ˜ C = 1, the population is modeled as if it were homogeneous, i.e. all partworths wi are equal. For C ˜ C ≫1, the individual partworths are computed independently, without 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. 5C ≥˜C In this way, directions for which the xj, j ∈Qi contain information are estimated 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 (−β, −β/3, β/3, β) (for each attribute) and with a diagonal covariance matrix σ2I. Each answer is a choice from the basket of products, sampled from the discrete logit-type distribution (1). Hence when β (called the magnitude6) is large, the consumer will choose with high probability the product with the highest utility, whereas when β is small, the answers will be less reliable. The ratio σ2/β 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 6as in [3], we tested High Magnitude (β = 3) and Low Magnitude (β = 0.5). 7It was either set to σ2 = 3β or σ2 = 0.5β, respectively High and Low Heterogeneity cases. 8We limited ourselves to datasets in which respondents were asked to choose 1 product among a basket at each question. 9see [4] for more details on the numerical features of the datasets. 10One could use EM-based methods to deal with such missing training choice data. 11When this procedure boiled down to unreasonable number of questions for hold-out evaluation of our algorithms, we simply removed the corresponding individuals. 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 C13, 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 might be that hyperparameters (C, ˜C) were optimized once for the whole population. This 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 Het HB-S HB-MAP Logistic Im.SV L L 0.90 0.83 0.84 0.86 L H 0.95 0.91 1.16 0.90 H L 0.44 0.40 0.43 0.41 H H 0.72 0.68 0.82 0.67 Table 2: Hit rate performances on real datasets. Im.SV HB-MAP HB-S Dat12 0.16 0.16 0.17 Dat22 0.15 0.13 0.15 Im.SV HB-MAP HB-S Dat15 0.52 0.45 0.48 Dat25 0.58 0.47 0.51 Im.SV HB-MAP HB-S Dat13 0.37 0.24 0.25 Dat23 0.34 0.33 0.33 Dat33 0.35 0.28 0.24 Dat43 0.35 0.31 0.28 Im.SV HB-MAP HB-S Dat14 0.33 0.36 0.35 Dat24 0.33 0.36 0.28 Dat34 0.45 0.40 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. 12since individual choice data are Immersed in the rest of the population choice data, via the optimization objective 13We observed that the value of the constant ˜C was irrelevant, and that only the ratio C/ ˜C mattered. 14Indeed 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 Het Random Orthogonal Adaptive L L 0.66 0.61 0.66 L H 0.62 0.56 0.56 H L 0.31 0.29 0.24 H H 0.49 0.45 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. 15Since 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. 16Indeed noisy answers are neither informative nor reliable for selecting the next question. 17aka 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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Intrinsically Motivated Reinforcement Learning Satinder Singh Computer Science & Eng. University of Michigan baveja@umich.edu Andrew G. Barto Dept. of Computer Science University of Massachusetts barto@cs.umass.edu Nuttapong Chentanez Computer Science & Eng. University of Michigan nchentan@umich.edu Abstract Psychologists call behavior intrinsically motivated when it is engaged in for its own sake rather than as a step toward solving a specific problem of clear practical value. But what we learn during intrinsically motivated behavior is essential for our development as competent autonomous entities able to efficiently solve a wide range of practical problems as they arise. In this paper we present initial results from a computational study of intrinsically motivated reinforcement learning aimed at allowing artificial agents to construct and extend hierarchies of reusable skills that are needed for competent autonomy. 1 Introduction Psychologists distinguish between extrinsic motivation, which means being moved to do something because of some specific rewarding outcome, and intrinsic motivation, which refers to being moved to do something because it is inherently enjoyable. Intrinsic motivation leads organisms to engage in exploration, play, and other behavior driven by curiosity in the absence of explicit reward. These activities favor the development of broad competence rather than being directed to more externally-directed goals (e.g., ref. [14]). In contrast, machine learning algorithms are typically applied to single problems and so do not cope flexibly with new problems as they arise over extended periods of time. Although the acquisition of competence may not be driven by specific problems, this competence is routinely enlisted to solve many different specific problems over the agent’s lifetime. The skills making up general competence act as the “building blocks” out of which an agent can form solutions to new problems as they arise. Instead of facing each new challenge by trying to create a solution out of low-level primitives, it can focus on combining and adjusting its higher-level skills. In animals, this greatly increases the efficiency of learning to solve new problems, and our main objective is to achieve a similar efficiency in our machine learning algorithms and architectures. This paper presents an elaboration of the reinforcement learning (RL) framework [11] that encompasses the autonomous development of skill hierarchies through intrinsically motivated reinforcement learning. We illustrate its ability to allow an agent to learn broad competence in a simple “playroom” environment. In a related paper [1], we provide more extensive background for this approach, whereas here the focus is more on algorithmic details. Lack of space prevents us from providing a comprehensive background to the many ideas to which our approach is connected. Many researchers have argued for this kind of developmental approach in which an agent undergoes an extended developmental period during which collections of reusable skills are autonomously learned that will be useful for a wide range of later challenges (e.g., [4, 13]). The previous machine learning research most closely related is that of Schmidhuber (e.g., [8]) on confidence-based curiosity and the ideas of exploration and shaping bonuses [6, 10], although our definition of intrinsic reward differs from these. The most direct inspiration behind the experiment reported in this paper, comes from neuroscience. The neuromodulator dopamine has long been associated with reward learning [9]. Recent studies [2, 3] have focused on the idea that dopamine not only plays a critical role in the extrinsic motivational control of behaviors aimed at harvesting explicit rewards, but also in the intrinsic motivational control of behaviors associated with novelty and exploration. For instance, salient, novel sensory stimuli inspire the same sort of phasic activity of dopamine cells as unpredicted rewards. However, this activation extinguishes more or less quickly as the stimuli become familiar. This may underlie the fact that novelty itself has rewarding characteristics [7]. These connections are key components of our approach to intrinsically motivated RL. 2 Reinforcement Learning of Skills According to the “standard” view of RL (e.g., [11]) the agent-environment interaction is envisioned as the interaction between a controller (the agent) and the controlled system (the environment), with a specialized reward signal coming from a “critic” in the environment that evaluates (usually with a scalar reward value) the agent’s behavior (Fig. 1A). The agent learns to improve its skill in controlling the environment in the sense of learning how to increase the total amount of reward it receives over time from the critic. A Agent Environment States Actions Rewards Critic B Agent Internal Environment Rewards Critic External Environment Sensations States Decisions Actions Agent Internal Environment Rewards Critic States Decisions "Organism" Figure 1: Agent-Environment Interaction in RL. A: The usual view. B: An elaboration. Sutton and Barto [11] point out that one should not identify this RL agent with an entire animal or robot. An an animal’s reward signals are determined by processes within its brain that monitor not only external state but also the animal’s internal state. The critic is in an animal’s head. Fig. 1B makes this more explicit by “factoring” the environment of Fig. 1A into an external environment and an internal environment, the latter of which contains the critic which determines primary reward. This scheme still includes cases in which reward is essentially an external stimulus (e.g., a pat on the head or a word of praise). These are simply stimuli transduced by the internal environment so as to generate the appropriate level of primary reward. The usual practice in applying RL algorithms is to formulate the problem one wants the agent to learn how to solve (e.g., win at backgammon) and define a reward function specially tailored for this problem (e.g., reward = 1 on a win, reward = 0 on a loss). Sometimes considerable ingenuity is required to craft an appropriate reward function. The point of departure for our approach is to note that the internal environment contains, among other things, the organism’s motivational system, which needs to be a sophisticated system that should not have to be redesigned for different problems. Handcrafting a different specialpurpose motivational system (as in the usual RL practice) should be largely unnecessary. Skills—Autonomous mental development should result in a collection of reusable skills. But what do we mean by a skill? Our approach to skills builds on the theory of options [12]. Briefly, an option is something like a subroutine. It consists of 1) an option policy that directs the agent’s behavior for a subset of the environment states, 2) an initiation set consisting of all the states in which the option can be initiated, and 3) a termination condition, which specifies the conditions under which the option terminates. It is important to note that an option is not a sequence of actions; it is a closed-loop control rule, meaning that it is responsive to on-going state changes. Furthermore, because options can invoke other options as actions, hierarchical skills and algorithms for learning them naturally emerge from the conception of skills as options. Theoretically, when options are added to the set of admissible agent actions, the usual Markov decision process (MDP) formulation of RL extends to semi-Markov decision processes (SMDPs), with the one-step actions now becoming the “primitive actions.” All of the theory and algorithms applicable to SMDPs can be appropriated for decision making and learning with options [12]. Two components of the the options framework are especially important for our approach: 1. Option Models: An option model is a probabilistic description of the effects of executing an option. As a function of an environment state where the option is initiated, it gives the probability with which the option will terminate at any other state, and it gives the total amount of reward expected over the option’s execution. Option models can be learned from experience (usually only approximately) using standard methods. Option models allow stochastic planning methods to be extended to handle planning at higher levels of abstraction. 2. Intra-option Learning Methods: These methods allow the policies of many options to be updated simultaneously during an agent’s interaction with the environment. If an option could have produced a primitive action in a given state, its policy can be updated on the basis of the observed consequences even though it was not directing the agent’s behavior at the time. In most of the work with options, the set of options must be provided by the system designer. While an option’s policy can be improved through learning, each option has to be predefined by providing its initiation set, termination condition, and the reward function that evaluates its performance. Many researchers have recognized the desirability of automatically creating options, and several approaches have recently been proposed (e.g., [5]). For the most part, these methods extract options from the learning system’s attempts to solve a particular problem, whereas our approach creates options outside of the context of solving any particular problem. Developing Hierarchical Collections of Skills—Children accumulate skills while they engage in intrinsically motivated behavior, e.g., while at play. When they notice that something they can do reliably results in an interesting consequence, they remember this in a form that will allow them to bring this consequence about if they wish to do so at a future time when they think it might contribute to a specific goal. Moreover, they improve the efficiency with which they bring about this interesting consequence with repetition, before they become bored and move on to something else. We claim that the concepts of an option and an option model are exactly appropriate to model this type of behavior. Indeed, one of our main contributions is a (preliminary) demonstration of this claim. 3 Intrinsically Motivated RL Our main departure from the usual application of RL is that our agent maintains a knowledge base of skills that it learns using intrinsic rewards. In most other regards, our extended RL framework is based on putting together learning and planning algorithms for Loop forever Current state st, current primitive action at, current option ot, extrinsic reward re t , intrinsic reward ri t Obtain next state st+1 //— Deal with special case if next state is salient If st+1 is a salient event e If option for e, oe, does not exist in O (skill-KB) Create option oe in skill-KB; Add st to Ioe // initialize initiation set Set βoe(st+1) = 1 // set termination probability //— set intrinsic reward value ri t+1 = τ[1 −P oe(st+1|st)] // τ is a constant multiplier else ri t+1 = 0 //— Update all option models For each option o ̸= oe in skill-KB (O) If st+1 ∈Io, then add st to Io // grow initiation set If at is greedy action for o in state st //— update option transition probability model P o(x|st) α←[γ(1 −βo(st+1)P o(x|st+1) + γβo(st+1)δst+1x] //— update option reward model Ro(st) α←[re t + γ(1 −βo(st+1))Ro(st+1)] //— Q-learning update of behavior action-value function QB(st, at) α←[re t + ri t + γ maxa∈A∪O QB(st+1, a)] //— SMDP-planning update of behavior action-value function For each option o in skill-KB QB(st, o) α←[Ro(st) + P x∈S P o(x|st) maxa∈A∪O QB(x, a)] //— Update option action-value functions For each option o ∈O such that st ∈Io Qo(st, at) α←[re t + γ (βo(st+1) × terminal value for option o) +γ(1 −βo(st+1)) × maxa∈A∪O Qo(st+1, a)] For each option o′ ∈O such that st ∈Io′ and o ̸= o′ Qo(st, o′) α←Ro′(st) + P x∈S P o′(x|st)[βo(x) × terminal val for option o +((1 −βo(x)) × maxa∈A∪O Qo(x, a))] Choose at+1 using ϵ-greedy policy w.r.to QB // — Choose next action //— Determine next extrinsic reward Set re t+1 to the extrinsic reward for transition st, at →st+1 Set st ←st+1; at ←at+1; re t ←re t+1; ri t ←ri t+1 Figure 2: Learning Algorithm. Extrinsic reward is denoted re while intrinsic reward is denoted ri. Equations of the form x α←[y] are short for x ←(1−α)x+α[y]. The behavior action value function QB is updated using a combination of Q-learning and SMDP planning. Throughout γ is a discount factor and α is the step-size. The option action value functions Qo are updated using intra-option Q-learning. Note that the intrinsic reward is only used in updating QB and not any of the Qo. options [12]. Behavior The agent behaves in its environment according to an ϵ-greedy policy with respect to an action-value function QB that is learned using a mix of Q-learning and SMDP planning as described in Fig. 2. Initially only the primitive actions are available to the agent. Over time, skills represented internally as options and their models also become available to the agent as action choices. Thus, QB maps states s and actions a (both primitive and options) to the expected long-term utility of taking that action a in state s. Salient Events In our current implementation we assume that the agent has intrinsic or hardwired notions of interesting or “salient” events in its environment. For example, in the playroom environment we present shortly, the agent finds changes in light and sound intensity to be salient. These are intended to be independent of any specific task and likely to be applicable to many environments. Reward In addition to the usual extrinsic rewards there are occasional intrinsic rewards generated by the agent’s critic (see Fig. 1B). In this implementation, the agent’s intrinsic reward is generated in a way suggested by the novelty response of dopamine neurons. The intrinsic reward for each salient event is proportional to the error in the prediction of the salient event according to the learned option model for that event (see Fig. 2 for detail). Skill-KB The agent maintains a knowledge base of skills that it has learned in its environment. Initially this may be empty. The first time a salient event occurs, say light turned on, structures to learn an option that achieves that salient event (turn-light-on option) are created in the skill-KB. In addition, structures to learn an option model are also created. So for option o, Qo maps states s and actions a (again, both primitive and options) to the long-term utility of taking action a in state s. The option for a salient event terminates with probability one in any state that achieves that event and never terminates in any other state. The initiation set, Io, for an option o is incrementally expanded to includes states that lead to states in the current initiation set. Learning The details of the learning algorithm are presented in Fig. 2. 4 Playroom Domain: Empirical Results We implemented intrinsically motivated RL (of Fig. 2) in a simple artificial “playroom” domain shown in Fig. 3A. In the playroom are a number of objects: a light switch, a ball, a bell, two movable blocks that are also buttons for turning music on and off, as well as a toy monkey that can make sounds. The agent has an eye, a hand, and a visual marker (seen as a cross hair in the figure). The agent’s sensors tell it what objects (if any) are under the eye, hand and marker. At any time step, the agent has the following actions available to it: 1) move eye to hand, 2) move eye to marker, 3) move eye one step north, south, east or west, 4) move eye to random object, 5) move hand to eye, and 6) move marker to eye. In addition, if both the eye and and hand are on some object, then natural operations suggested by the object become available, e.g., if both the hand and the eye are on the light switch, then the action of flicking the light switch becomes available, and if both the hand and eye are on the ball, then the action of kicking the ball becomes available (which when pushed, moves in a straight line to the marker). The objects in the playroom all have potentially interesting characteristics. The bell rings once and moves to a random adjacent square if the ball is kicked into it. The light switch controls the lighting in the room. The colors of any of the blocks in the room are only visible if the light is on, otherwise they appear similarly gray. The blue block if pressed turns music on, while the red block if pressed turns music off. Either block can be pushed and as a result moves to a random adjacent square. The toy monkey makes frightened sounds if simultaneously the room is dark and the music is on and the bell is rung. These objects were designed to have varying degrees of difficulty to engage. For example, to get the monkey to cry out requires the agent to do the following sequence of actions: 1) get its eye to the light switch, 2) move hand to eye, 3) push the light switch to turn the light on, 4) find the blue block with its eye, 5) move the hand to the eye, 6) press the blue block to turn music on, 7) find the light switch with its eye, 8) move hand to eye, 9) press light switch to turn light off, 10) find the bell with its eye, 11) move the marker to the eye, 12) find the ball with its eye, 13) move its hand to the ball, and 14) kick the ball to make the bell ring. Notice that if the agent has already learned how to turn the light on and off, how to turn music on, and how to make the bell ring, then those learned skills would be of obvious use in simplifying this process of engaging the toy monkey. A B C 0 0.5 1 1.5 2 2.5 x 10 7 0 20 40 60 80 100 120 Number of Actions Average # of Actions to Salient Event Performance of Learned Options Toy Monkey On Music On Sound On Light On 0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000 Number of extrinsic rewards Number of steps between extrinsic rewards Effect of Intrinsically Motivated Learning Extrinsic Reward Only Intrinsic & Extrinsic Rewards Figure 3: A. Playroom domain. B. Speed of learning of various skills. C. The effect of intrinsically motivated learning when extrinsic reward is present. See text for details For this simple example, changes in light and sound intensity are considered salient by the playroom agent. Because the initial action value function, QB, is uninformative, the agent starts by exploring its environment randomly. Each first encounter with a salient event initiates the learning of an option and an option model for that salient event. For example, the first time the agent happens to turn the light on, it initiates the data structures necessary for learning and storing the light-on option. As the agent moves around the environment, all the options (initiated so far) and their models are simultaneously updated using intra-option learning. As shown in Fig. 2, the intrinsic reward is used to update QB. As a result, when the agent encounters an unpredicted salient event a few times, its updated action value function drives it to repeatedly attempt to achieve that salient event. There are two interesting side effects of this: 1) as the agent tries to repeatedly achieve the salient event, learning improves both its policy for doing so and its option-model that predicts the salient event, and 2) as its option policy and option model improve, the intrinsic reward diminishes and the agent gets “bored” with the associated salient event and moves on. Of course, the option policy and model become accurate in states the agent encounters frequently. Occasionally, the agent encounters the salient event in a state (set of sensor readings) that it has not encountered before, and it generates intrinsic reward again (it is “surprised”). A summary of results is presented in Fig. 4. Each panel of the figure is for a distinct salient event. The graph in each panel shows both the time steps at which the event occurs as well as the intrinsic reward associated by the agent to each occurrence. Each occurrence is denoted by a vertical bar whose height denotes the amount of associated intrinsic reward. Note that as one goes from top to bottom in this figure, the salient events become harder to achieve and, in fact, become more hierarchical. Indeed, the lowest one for turning on the monkey noise (Non) needs light on, music on, light off, sound on in sequence. A number of interesting results can be observed in this figure. First note that the salient events that are simpler to achieve occur earlier in time. For example, Lon (light turning on) and Loff (light turning off) are the simplest salient events, and the agent makes these happen quite early. The agent tries them a large number of times before getting bored and moving on to other salient events. The reward obtained for each of these events diminishes after repeated exposure to the event. Thus, automatically, the skill of achieving the simpler events are learned before those for the more complex events. Figure 4: Results from the playroom domain. Each panel depicts the occurrences of salient events as well as the associated intrinsic rewards. See text for details. Of course, the events keep happening despite their diminished capacity to reward because they are needed to achieve the more complex events. Consequently, the agent continues to turn the light on and off even after it has learned this skill because this is a step along the way toward turning on the music, as well as along the way toward turning on the monkey noise. Finally note that the more complex skills are learned relatively quickly once the required sub-skills are in place, as one can see by the few rewards the agent receives for them. The agent is able to bootstrap and build upon the options it has already learned for the simpler events. We confirmed the hierarchical nature of the learned options by inspecting the greedy policies for the more complex options like Non and Noff. The fact that all the options are successfully learned is also seen in Fig. 3B in which we show how long it takes to bring about the events at different points in the agent’s experience (there is an upper cutoff of 120 steps). This figure also shows that the simpler skills are learned earlier than the more complex ones. An agent having a collection of skills learned through intrinsic reward can learn a wide variety of extrinsically rewarded tasks more easily than an agent lacking these skills. To illustrate, we looked at a playroom task in which extrinsic reward was available only if the agent succeeded in making the monkey cry out. This requires the 14 steps described above. This is difficult for an agent to learn if only the extrinsic reward is available, but much easier if the agent can use intrinsic reward to learn a collection of skills, some of which are relevant to the overall task. Fig. 3C compares the performance of two agents in this task. Each starts out with no knowledge of task, but one employs the intrinsic reward mechanism we have discussed above. The extrinsic reward is always available, but only when the monkey cries out. The figure, which shows the average of 100 repetitions of the experiment, clearly shows the advantage of learning with intrinsic reward. Discussion One of the key aspects of the Playroom example was that intrinsic reward was generated only by unexpected salient events. But this is only one of the simplest possibilities and has many limitations. It cannot account for what makes many forms of exploration and manipulation “interesting.” In the future, we intend to implement computational analogs of other forms of intrinsic motivation as suggested in the psychological, statistical, and neuroscience literatures. Despite the “toy” nature of this domain, these results are among the most sophisticated we have seen involving intrinsically motivated learning. Moreover, they were achieved quite directly by combining a collection of existing RL algorithms for learning options and option-models with a simple notion of intrinsic reward. The idea of intrinsic motivation for artificial agents is certainly not new, but we hope to have shown that the elaboration of the formal RL framework in the direction we have pursued, together with the use of recentlydeveloped hierarchical RL algorithms, provides a fruitful basis for developing competently autonomous agents. Acknowledgement Satinder Singh and Nuttapong Chentanez were funded by NSF grant CCF 0432027 and by a grant from DARPA’s IPTO program. Andrew Barto was funded by NSF grant CCF 0432143 and by a grant from DARPA’s IPTO program. References [1] A. G. Barto, S. Singh, and N. Chentanez. Intrinsically motivated learning of hierarchical collections of skills. In Proceedings of the 3rd International Conference on Developmental Learning (ICDL ’04), LaJolla CA, 2004. [2] P. Dayan and B. W. Balleine. Reward, motivation and reinforcement learning. Neuron, 36:285– 298, 2002. [3] S. Kakade and P. Dayan. Dopamine: Generalization and bonuses. Neural Networks, 15:549– 559, 2002. [4] F. Kaplan and P.-Y. Oudeyer. Motivational principles for visual know-how development. In C. G. Prince, L. Berthouze, H. Kozima, D. Bullock, G. Stojanov, and C. Balkenius, editors, Proceedings of the Third International Workshop on Epigenetic Robotics : Modeling Cognitive Development in Robotic Systems, pages 73–80, Edinburgh, Scotland, 2003. Lund University Cognitive Studies. [5] A. McGovern. Autonomous Discovery of Temporal Abstractions from Interaction with An Environment. PhD thesis, University of Massachusetts, 2002. [6] A. Ng, D. Harada, and S. Russell. Policy invariance under reward transformations: Theory and application to reward shaping. In Proceedings of the Sixteenth ICML. Morgan Kaufmann, 1999. [7] P. Reed, C. Mitchell, and T. Nokes. Intrinsic reinforcing properties of putatively neutral stimuli in an instrumental two-lever discrimination task. Animal Learning and Behavior, 24:38–45, 1996. [8] J. Schmidhuber. A possibility for implementing curiosity and boredom in model-building neural controllers. In From Animals to Animats: Proceedings of the First International Conference on Simulation of Adaptive Behavior, pages 222–227, Cambridge, MA, 1991. MIT Press. [9] W. Schultz. Predictive reward signal of dopamine neurons. Journal of Neurophysiology, 80:1– 27, 1998. [10] R. S. Sutton. Integrated modeling and control based on reinforcement learning and dynamic programming. In Proceedings of NIPS, pages 471–478, San Mateo, CA, 1991. [11] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [12] R. S. 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. [13] J. Wang, J. McClelland, A. Pentland, O. Sporns, I. Stockman, M. Sur, and E. Thelen. Autonomous mental develoopment by robots and animals. Science, 291:599–600, 2001. [14] R. W. White. Motivation reconsidered: The concept of competence. Psychological Review, 66:297–333, 1959.
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Joint Probabilistic Curve Clustering and Alignment Scott Gaffney and Padhraic Smyth School of Information and Computer Science University of California, Irvine, CA 92697-3425 {sgaffney,smyth}@ics.uci.edu Abstract Clustering and prediction of sets of curves is an important problem in many areas of science and engineering. It is often the case that curves tend to be misaligned from each other in a continuous manner, either in space (across the measurements) or in time. We develop a probabilistic framework that allows for joint clustering and continuous alignment of sets of curves in curve space (as opposed to a fixed-dimensional featurevector space). The proposed methodology integrates new probabilistic alignment models with model-based curve clustering algorithms. The probabilistic approach allows for the derivation of consistent EM learning algorithms for the joint clustering-alignment problem. Experimental results are shown for alignment of human growth data, and joint clustering and alignment of gene expression time-course data. 1 Introduction We introduce a novel methodology for the clustering and prediction of sets of smoothly varying curves while jointly allowing for the learning of sets of continuous curve transformations. Our approach is to formulate models for both the clustering and alignment sub-problems and integrate them into a unified probabilistic framework that allows for the derivation of consistent learning algorithms. The alignment sub-problem is handled with the introduction of a novel curve alignment procedure employing model priors over the set of possible alignments leading to the derivation of EM learning algorithms that formalize the so-called Procrustes approach for curve data [1]. These alignment models are then integrated into a finite mixture model setting in which the clustering is carried out. We make use of both polynomial and spline regression mixture models to complete the joint clustering-alignment framework. The following simple illustrative example demonstrates the importance of jointly handling the clustering-alignment problem as opposed to treating alignment and clustering separately. Figure 1(a) shows a simulated set of curves which have been subjected to random translations in time. The underlying generative model contains three clusters each described by a cubic polynomial (not shown). Figure 1(b) shows the output of the proposed joint EM algorithm introduced in this paper, where curves have been simultaneously aligned and clustered. The algorithm recovers the hidden labels and alignments nearperfectly in this case. On the other hand, Figure 1(c) shows the result of first clustering −5 0 5 10 15 −200 −100 0 100 Time Y−axis −5 0 5 10 15 −200 −100 0 100 Time Y−axis −5 0 5 10 15 −200 −100 0 100 Time Y−axis −5 0 5 10 15 −200 −100 0 100 Time Y−axis Figure 1: Comparison of joint EM and sequential clustering-alignment: (a, top-left) unlabelled simulated data with hidden alignments; (b, top-right) solution recovered by joint EM; (c, bottom-left) partial solution after clustering first, and (d, bottom-right) final solution after aligning clustered data in (c). the unaligned data in Figure 1(b), while Figure 1(d) shows the final result of aligning each of the found clusters individually. The sequential approach results in significant misclassification and incorrect alignment demonstrating that a two-stage approach can be quite suboptimal when compared to a joint clustering-alignment methodology. (Similar results, not shown, are obtained when the curves are first aligned and then clustered—see [2] for full details.) There has been little prior work on the specific problem of joint curve clustering and alignment, but there is related work in other areas. For example, clustering of gene-expression time profiles with mixtures of splines was addressed in [3]. However, alignment was only considered as a post-processing step to compare cluster results among related datasets. In image analysis, the transformed mixture of Gaussians (TMG) model uses a probabilistic framework and an EM algorithm to jointly learn clustering and alignment of image patches subject to various forms of linear transformations [4]. However, this model only considers sets of transformations in discrete pixel space, whereas we are focused on curve modelling that allows for arbitrary continuous alignment in time and space. Another branch of work in image analysis focuses on the problem of estimating correspondences of points across images [5] (or vertices across graphs [6]), using EM or deterministic annealing algorithms. The results we describe here differ primarily in that (a) we focus specifically on sets of curves rather than image data (generally making the problem more tractable), (b) we focus on clustering and alignment rather than just alignment, (c) we allow continuous affine transformations in time and measurement space, and (d) we have a fully generative probabilistic framework allowing for (for example) the incorporation of informative priors on transformations if such prior information exists. In earlier related work we developed general techniques for curve clustering (e.g., [7]) and also proposed techniques for transformation-invariant curve clustering with discrete time alignment and Gaussian mixture models for curves [8, 9]. In this paper we provide a much more general framework that allows for continuous alignment in both time and measurement space for a general class of “cluster shape” models, including polynomials and splines. 2 Joint clustering and alignment It is useful to represent curves as variable-length vectors. In this case, y i is a curve that consists of a sequence of ni observations or measurements. The j-th measurement of y i is denoted by yij and is usually taken to be univariate (the generalization to multivariate observations is straightforward). The associated covariate of y i is written as xi in the same manner. xi is often thought of as time so that xij gives the time at which yij was observed. Regression mixture models can be effectively used to cluster this type of curve data [10]. In the standard setup, yi is modelled using a normal (Gaussian) regression model in which yi = Xiβ+ϵi, where β is a (p+1)×1 coefficient vector, ϵi is a zero-mean Gaussian noise variable, and Xi is the regression matrix. The form of Xi depends on the type of regression model employed. For polynomial regression, X i is often associated with the standard Vandermonde matrix; and for spline regression, X i takes the form of a spline-basis matrix (see, e.g., [7] for more details). The mixture model is completed by repeating this model over K clusters and indexing the parameters by k so that, for example, y i = Xiβk + ϵi gives the regression model for yi under the k-th cluster. B-splines [11] are particularly efficient for computational purposes due to the blockdiagonal basis matrices that result. Using B-splines, the curve point y ij can be represented as the linear combination yij = B′ ij c, in which the vector Bij gives the vector of B-spline basis functions evaluated at xij, and c gives the spline coefficient vector [2]. The full curve yi can then be written compactly as yi = Bi c in which the spline basis matrix takes the form Bi = [Bi1 · · · Bini]′. Spline regression models can be easily integrated into the regression mixture model framework by equating the regression matrix X i with the spline basis matrix Bi. In what follows, we use the more general notation X i in favor of the more specific Bi. 2.1 Joint model definition The joint clustering-alignment model definition is based on a regression mixture model that has been augmented with up to four individual random transformation parameters or variables (ai, bi, ci, di). The ai and bi allow for scaling and translation in time, while the ci and di allow for scaling and translation in measurement space. The model definition takes the form yi = ciaixi −biβk + di + ϵi, (1) in which aixi −bi represents the regression matrix Xi (either spline or polynomial) evaluated at the transformed time aixi −bi. Below we use the matrix X i to denote aixi − bi when parsimony is required. It is assumed that ϵi is a zero-mean Gaussian vector with covariance σ2 kI. The conditional density pk(yi|ai, bi, ci, di) = N(yi|ciaixi −biβk + di, σ2 kI) (2) gives the probability density of yi when all the transformation parameters (as well as cluster membership) are known. (Note that the density on the left is implicitly conditioned on an appropriate set of parameters—this is always assumed in what follows.) In general, the values for the transformation parameters are unknown. Treating this as a standard hidden-data problem, it is useful to think of each of the transformation parameters as random variables that are curve-specific but with “population-level” prior probability distributions. In this way, the transformation parameters and the model parameters can be learned simultaneously in an efficient manner using EM. 2.2 Transformation priors Priors are attached to each of the transformation variables in such a way that the identity transformation is the most likely transformation. A useful prior for this is the Gaussian density N(µ, σ2) with mean µ and variance σ2. The time transformation priors are specified as ai ∼N (1, r2 k), bi ∼N(0, s2 k), (3) and the measurement space priors are given as ci ∼N(1, u2 k) , di ∼N(0, v2 k). (4) Note that the identity transformation is indeed the most likely. All of the variance parameters are cluster-specific in general; however, any subset of these parameters can be “tied” across clusters if desired in a specific application. Note that these priors technically allow for negative scaling in time and in measurement space. In practice this is typically not a problem, though one can easily specify other priors (e.g., log-normal) to strictly disallow this possibility. It should be noted that each of the prior variance parameters are learned from the data in the ensuing EM algorithm. We do not make use of hyperpriors for these prior parameters; however, it is straightforward to extend the method to allow hyperpriors if desired. 2.3 Full probability model The joint density of yi and the set of transformation variables Φi = {ai, bi, ci, di} can be written succinctly as pk(yi, Φi) = pk(yi|Φi)pk(Φi), (5) where pk(Φi) = N(ai|1, r2 k)N (bi|0, s2 k)N (ci|1, u2 k)N(di|0, v2 k). The space transformation parameters can be integrated-out of (5) resulting in the marginal of y i conditioned only on the time transformation parameters. This conditional marginal takes the form pk(yi|ai, bi) = pk(yi, ci, di|ai, bi) dci, ddi = N(yi|X iβk, Uik + Vk −σ2 kI), (6) with Uik = u2 kX iβkβ′ kX ′ i + σ2 kI and Vk = v2 k11′ + σ2 kI. The unconditional (though, still cluster-dependent) marginal for yi cannot be computed analytically since ai, bi cannot be analytically integrated-out. Instead, we use numerical Monte Carlo integration for this task. The resulting unconditional marginal for y i can be approximated by pk(yi) = pk(yi|ai, bi)pk(ai)pk(bi) dai dbi ≈ 1 M m pk(yi|a(m) i , b(m) i ), (7) where the M Monte Carlo samples are taken according to a(m) i ∼N(1, r2 k), and b(m) i ∼N(0, s2 k), for m = 1, . . . , M. (8) A mixture results when cluster membership is unknown: p(yi) = k αkpk(yi). (9) The log-likelihood of all n curves Y = {yi} follows directly from this approximation and takes the form log p(Y ) ≈ i log mk αkpk(yi|a(m) i , b(m) i ) −n log M. (10) 2.4 EM algorithm We derive an EM algorithm that simultaneously allows the learning of both the model parameters and the transformation variables Φ with time-complexity that is linear in the total number of data points N = i ni. First, let zi give the cluster membership for curve yi. Now, regard the transformation variables {Φi} as well as the cluster memberships {zi} as being hidden. The complete-data log-likelihood function is defined as the joint loglikelihood of Y and the hidden data {Φi, zi}. This can be written as the sum over all n curves of the log of the product of αzi and the cluster-dependent joint density in (5). This function takes the form Lc = i log αzipzi(yi|Φi) pzi(Φi). (11) In the E-step, the posterior p(Φi, zi|yi) is calculated and then used to take the posterior expectation of Equation (11). This expectation is then used in the M-step to calculate the re-estimation equations for updating the model parameters {β k, σ2 k, r2 k, s2 k, u2 k, v2 k}. 2.5 E-step The posterior p(Φi, zi|yi) can be factorized as pzi(Φ|yi)p(zi|yi). The second factor is the membership probability wik that yi was generated by cluster k. It can be rewritten as p(zi = k|yi) ∝pk(yi) and evaluated using Equation (7). The first factor requires a bit more work. Further factoring reveals that pzi(Φ|yi) = pzi(ci, di|ai, bi, yi)pzi(ai, bi|yi). The new first factor pzi(ci, di|ai, bi, yi) can be solved for exactly by noting that it is proportional to a bivariate normal distribution for each z i [2]. The new second factor pzi(ai, bi|yi) cannot, in general, be solved for analytically, so instead we use an approximation. The fact that posterior densities tend towards highly peaked Gaussian densities has been widely noted (e.g, [12]) and leads to the normal approximation of posterior densities. To make the approximation here, the vector (ˆa ik,ˆbik) representing the multi-dimensional mode of pk(ai, bi|yi), the covariance matrix V (k) aibi for (ˆaik,ˆbik), and the separate variances Vaik, Vbik must be found. These can readily be estimated using a Nelder-Mead optimization method. Experiments have shown this approximation works well across a variety of experimental and real-world data sets [2]. The above calculations of the posterior p(Φi, zi|yi) allow the posterior expectation of the complete-data log-likelihood in Equation (11) to be solved for. This expectation results in the so-called Q-function which is maximized in the M-step. Although the derivation is quite complex, the Q-function can be calculated exactly for polynomial regression [2]; for spline regression, the basis functions do not afford an exact formula for the solution of the Q-function. However, in the spline case, removal of a few problematic variance terms gives an efficient approximation (the interested reader is referred to [2] for more details). 2.6 M-step The M-step is straightforward since most of the hard work is done in the E-step. The Qfunction is maximized over the set of parameters {β k, σ2 k, r2 k, s2 k, u2 k, v2 k} for 1 ≤k ≤K. The derived solutions are as follows: ˆr2 k = 1 i wik i wik ˆa2 ik + Vaik , ˆs2 k = 1 i wik i wik ˆb2 ik + Vbik , ˆu2 k = 1 i wik i wik ˆc2 ik + Vcik , ˆv2 k = 1 i wik i wik ˆd2 ik + Vdik , 10 12 14 16 18 −6 −4 −2 0 2 4 Age Height acceleration 8 10 12 14 16 18 −6 −4 −2 0 2 4 Age Height acceleration Figure 2: Curves measuring the height acceleration for 39 boys; (left) smoothed versions of raw observations, (right) automatically aligned curves. ˆβk = i wikˆc2 ik ˆX ′ ik ˆX ik + Vxxi −1 i wikˆcik ˆX ′ ik(yi −ˆdik) + V′xiyi −V′xcd1 , and ˆσ2 k = 1 i wikni i wik yi −ˆcik ˆX ikβ −ˆdik 2 −2y′ iVxiˆβk + ˆβ ′ kVxxi ˆβk + 2ˆβ ′ kVxcd1 + niVdik , where ˆX ik = ˆaikxi −ˆbik, and Vxxi, Vxi, Vxcd are special “variance” matrices whose components are functions of the posterior expectations of Φ calculated in the E-step (the exact forms of these matrices can be found in [2]). 3 Experimental results and conclusions The results of a simple demonstration of EM-based alignment (using splines and the learning algorithm of the previous section, but with no clustering) are shown in Figure 2. In the left plot are a set of smoothed curves representing the acceleration of height for each of 39 boys whose heights were measured at 29 observation times over the ages of 1 to 18 [1]. Notice that the curves share a similar shape but seem to be misaligned in time due to individual growth dynamics. The right plot shows the same acceleration curves after processing from our spline alignment model using quartic splines with 8 uniformly spaced knots allowing for a maximum time translation of 2 units. The x-axis in this plot can be seen as canonical (or “average”) age. The aligned curves in the right plot of Figure 2 represent the average behavior in a much clearer way. For example, it appears there is an interval of 2.5 years from peak (age 12.5) to trough (age 15) that describes the average cycle that all boys go through. The results demonstrate that it is common for important features of curves to be randomly translated in time and that it is possible to use the data to recover these underlying hidden transformations using our alignment models. Next we briefly present an application of the joint clustering-alignment model to the problem of gene expression clustering. We analyze the alpha arrest data described in [13] that captures gene expression levels at 7 minute intervals for two consecutive cell cycles (totaling 17 measurements per gene). Clustering is often used in gene expression analysis to reveal groups of genes with similar profiles that may be physically related to the same underlying biological process (e.g., [13]). It is well-known that time-delays play an impor0 5 10 15 −2 −1 0 1 2 Canonical time Expression 0 5 10 15 −2 −1 0 1 2 Time Expression 0 5 10 15 −2 −1 0 1 2 Canonical time Expression 0 5 10 15 −2 −1 0 1 2 Time Expression 0 5 10 15 −2 −1 0 1 2 Canonical time Expression 0 5 10 15 −2 −1 0 1 2 Time Expression Figure 3: Three clusters for the time translation alignment model (left) and the nonalignment model (right). tant role in gene regulation, and thus, curves measured over time which represent the same process may often be misaligned from each other. [14]. Since these gene expression data are already normalized, we did not allow for transformations in measurement space. We only allowed for translations in time since experts do not expect scaling in time to be a factor in these data. For the curve model, cubic splines with 6 uniformly spaced knots across the interval from −4 to 21 were chosen, allowing for a maximum time translation of 4 units. Due to limited space, we present a single case of comparison between a standard spline regression mixture model (SRM) and an SRM that jointly allows for time translations. Ten random starts of EM were allowed for each algorithm with the highest likelihood model selected for comparison for each algorithm. It is common to assume that there are five distinct clusters of genes in these data; as such we set K = 5 for each algorithm [13]. Three of the resulting clusters from the two methods are shown in Figure 3. The left column of the figure shows the output from the joint clustering-alignment model, while the right column shows the output from the standard cluster model. It is apparent that the time-aligned clusters represent the mean behavior more accurately. The overall cluster variance is much lower than in the non-aligned clustering. The results also demonstrate the appearance of cluster-dependent alignment effects. Out-of-sample experiments (not shown here) show that the joint model produces better predictive models than the standard clustering method. Experimental results on a variety of other data sets are provided in [2], including applications to clustering of cyclone trajectories. 4 Conclusions We proposed a general probabilistic framework for joint clustering and alignment of sets of curves. The experimental results indicate that the approach provides a new and useful tool for curve analysis in the face of underlying hidden transformations. The resulting EM-based learning algorithms have time-complexity that is linear in the number of measurements—in contrast, many existing curve alignment algorithms themselves are O(n2) (e.g., dynamic time warping) without regard to clustering. The incorporation of splines gives the method an overall non-parametric freedom which leads to general applicability. Acknowledgements This material is based upon work supported by the National Science Foundation under grants No. SCI-0225642 and IIS-0431085. References [1] J.O. Ramsay and B. W. Silverman. Functional Data Analysis. Springer-Verlag, New York, NY, 1997. [2] Scott J. Gaffney. Probabilistic Curve-Aligned Clustering and Prediction with Regression Mixture Models. Ph.D. Dissertation, University of California, Irvine, 2004. [3] Z. Bar-Joseph et al. A new approach to analyzing gene expression time series data. Journal of Computational Biology, 10(3):341–356, 2003. [4] B. J. Frey and N. Jojic. Transformation-invariant clustering using the EM algorithm. IEEE Trans. PAMI, 25(1):1–17, January 2003. [5] H. Chui, J. Zhang, and A. Rangarajan. Unsupervised learning of an atlas from unlabeled pointsets. IEEE Trans. PAMI, 26(2):160–172, February 2004. [6] A. D. J. Cross and E. R. Hancock. Graph matching with a dual-step EM algorithm. IEEE Trans. PAMI, 20(11):1236–1253, November 1998. [7] S. J. Gaffney and P. Smyth. Curve clustering with random effects regression mixtures. In C. M. Bishop and B. J. Frey, editors, Proc. Ninth Inter. Workshop on Artificial Intelligence and Stats, Key West, FL, January 3–6 2003. [8] D. Chudova, S. J. Gaffney, and P. J. Smyth. Probabilistic models for joint clustering and timewarping of multi-dimensional curves. In Proc. of the Nineteenth Conference on Uncertainty in Artificial Intelligence (UAI-2003), Acapulco, Mexico, August 7–10, 2003. [9] D. Chudova, S. J. Gaffney, E. Mjolsness, and P. J. Smyth. Translation-invariant mixture models for curve clustering. In Proc. Ninth ACM SIGKDD Inter. Conf. on Knowledge Discovery and Data Mining, Washington D.C., August 24–27, New York, 2003. ACM Press. [10] S. Gaffney and P. Smyth. Trajectory clustering with mixtures of regression models. In Surajit Chaudhuri and David Madigan, editors, Proc. Fifth ACM SIGKDD Inter. Conf. on Knowledge Discovery and Data Mining, August 15–18, pages 63–72, N.Y., 1999. ACM Press. [11] P. H. C. Eilers. and B. D. Marx. Flexible smoothing with B-splines and penalties. Statistical Science, 11(2):89–121, 1996. [12] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Chapman & Hall, New York, NY, 1995. [13] P. T. Spellman et al. Comprehensive identification of cell cycle-regulated genes of the yeast Saccharomyces cerevisiae by microarray hybridization. Molec. Bio. Cell, 9(12):3273–3297, December 1998. [14] J. Aach and G. M. Church. Aligning gene expression time series with time warping algorithms. Bioinformatics, 17(6):495–508, 2001.
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Triangle Fixing Algorithms for the Metric Nearness Problem Inderjit S. Dhillon Suvrit Sra Dept. of Computer Sciences The Univ. of Texas at Austin Austin, TX 78712. {inderjit,suvrit}@cs.utexas.edu Joel A. Tropp Dept. of Mathematics The Univ. of Michigan at Ann Arbor Ann Arbor, MI, 48109. jtropp@umich.edu Abstract Various problems in machine learning, databases, and statistics involve pairwise distances among a set of objects. It is often desirable for these distances to satisfy the properties of a metric, especially the triangle inequality. Applications where metric data is useful include clustering, classification, metric-based indexing, and approximation algorithms for various graph problems. This paper presents the Metric Nearness Problem: Given a dissimilarity matrix, find the “nearest” matrix of distances that satisfy the triangle inequalities. For ℓp nearness measures, this paper develops efficient triangle fixing algorithms that compute globally optimal solutions by exploiting the inherent structure of the problem. Empirically, the algorithms have time and storage costs that are linear in the number of triangle constraints. The methods can also be easily parallelized for additional speed. 1 Introduction Imagine that a lazy graduate student has been asked to measure the pairwise distances among a group of objects in a metric space. He does not complete the experiment, and he must figure out the remaining numbers before his adviser returns from her conference. Obviously, all the distances need to be consistent, but the student does not know very much about the space in which the objects are embedded. One way to solve his problem is to find the “nearest” complete set of distances that satisfy the triangle inequalities. This procedure respects the measurements that have already been taken while forcing the missing numbers to behave like distances. More charitably, suppose that the student has finished the experiment, but—measurements being what they are—the numbers do not satisfy the triangle inequality. The student knows that they must represent distances, so he would like to massage the data so that it corresponds with his a priori knowledge. Once again, the solution seems to require the “nearest” set of distances that satisfy the triangle inequalities. Matrix nearness problems [6] offer a natural framework for developing this idea. If there are n points, we may collect the measurements into an n × n symmetric matrix whose (j, k) entry represents the dissimilarity between the j-th and k-th points. Then, we seek to approximate this matrix by another whose entries satisfy the triangle inequalities. That is, mik ≤mij + mjk for every triple (i, j, k). Any such matrix will represent the distances among n points in some metric space. We calculate approximation error with a distortion measure that depends on how the corrected matrix should relate to the input matrix. For example, one might prefer to change a few entries significantly or to change all the entries a little. We call the problem of approximating general dissimilarity data by metric data the Metric Nearness (MN) Problem. This simply stated problem has not previously been studied, although the literature does contain some related topics (see Section 1.1). This paper presents a formulation of the Metric Nearness Problem (Section 2), and it shows that every locally optimal solution is globally optimal. To solve the problem we present triangle-fixing algorithms that take advantage of its structure to produce globally optimal solutions. It can be computationally prohibitive, both in time and storage, to solve the MN problem without these efficiencies. 1.1 Related Work The Metric Nearness (MN) problem is novel, but the literature contains some related work. The most relevant research appears in a recent paper of Roth et al. [11]. They observe that machine learning applications often require metric data, and they propose a technique for metrizing dissimilarity data. Their method, constant-shift embedding, increases all the dissimilarities by an equal amount to produce a set of Euclidean distances (i.e., a set of numbers that can be realized as the pairwise distances among an ensemble of points in a Euclidean space). The size of the translation depends on the data, so the relative and absolute changes to the dissimilarity values can be large. Our approach to metrizing data is completely different. We seek a consistent set of distances that deviates as little as possible from the original measurements. In our approach, the resulting set of distances can arise from an arbitrary metric space; we do not restrict our attention to obtaining Euclidean distances. In consequence, we expect metric nearness to provide superior denoising. Moreover, our techniques can also learn distances that are missing entirely. There is at least one other method for inferring a metric. An article of Xing et al. [12] proposes a technique for learning a Mahalanobis distance for data in Rs. That is, a metric dist(x, y) = p (x −y)T G(x −y), where G is an s × s positive semi-definite matrix. The user specifies that various pairs of points are similar or dissimilar. Then the matrix G is computed by minimizing the total squared distances between similar points while forcing the total distances between dissimilar points to exceed one. The article provides explicit algorithms for the cases where G is diagonal and where G is an arbitrary positive semi-definite matrix. In comparison, the metric nearness problem is not restricted to Mahalanobis distances; it can learn a general discrete metric. It also allows us to use specific distance measurements and to indicate our confidence in those measurements (by means of a weight matrix), rather than forcing a binary choice of “similar” or “dissimilar.” The Metric Nearness Problem may appear similar to metric Multi-Dimensional Scaling (MDS) [8], but we emphasize that the two problems are distinct. The MDS problem endeavors to find an ensemble of points in a prescribed metric space (usually a Euclidean space) such that the distances between these points are close to the set of input distances. In contrast, the MN problem does not seek to find an embedding. In fact MN does not impose any hypotheses on the underlying space other than requiring it to be a metric space. The outline of rest of the paper is as follows. Section 2 formally describes the MN problem. In Section 3, we present algorithms that allow us to solve MN problems with ℓp nearness measures. Some applications and experimental results follow in Section 4. Section 5 discusses our results, some interesting connections, and possibilities for future research. 2 The Metric Nearness Problem We begin with some basic definitions. We define a dissimilarity matrix to be a nonnegative, symmetric matrix with zero diagonal. Meanwhile, a distance matrix is defined to be a dissimilarity matrix whose entries satisfy the triangle inequalities. That is, M is a distance matrix if and only if it is a dissimilarity matrix and mik ≤mij + mjk for every triple of distinct indices (i, j, k). Distance matrices arise from measuring the distances among n points in a pseudo-metric space (i.e., two distinct points can lie at zero distance from each other). A distance matrix contains N = n (n −1)/2 free parameters, so we denote the collection of all distance matrices by MN. The set MN is a closed, convex cone. The metric nearness problem requests a distance matrix M that is closest to a given dissimilarity matrix D with respect to some measure of “closeness.” In this work, we restrict our attention to closeness measures that arise from norms. Specifically, we seek a distance matrix M so that, M ∈ argmin X∈MN
W ⊙ X −D
, (2.1) where ∥· ∥is a norm, W is a symmetric non-negative weight matrix, and ‘⊙’ denotes the elementwise (Hadamard) product of two matrices. The weight matrix reflects our confidence in the entries of D. When each dij represents a measurement with variance σ2 ij, we might set wij = 1/σ2 ij. If an entry of D is missing, one can set the corresponding weight to zero. Theorem 2.1. The function X 7→
W ⊙ X −D
always attains its minimum on MN. Moreover, every local minimum is a global minimum. If, in addition, the norm is strictly convex and the weight matrix has no zeros or infinities off its diagonal, then there is a unique global minimum. Proof. The main task is to show that the objective function has no directions of recession, so it must attain a finite minimum on MN. Details appear in [4]. It is possible to use any norm in the metric nearness problem. We further restrict our attention to the ℓp norms. The associated Metric Nearness Problems are min X∈MN X j̸=k wjk (xjk −djk) p 1/p for 1 ≤p < ∞, and (2.2) min X∈MN max j̸=k wjk (xjk −djk) for p = ∞. (2.3) Note that the ℓp norms are strictly convex for 1 < p < ∞, and therefore the solution to (2.2) is unique. There is a basic intuition for choosing p. The ℓ1 norm gives the absolute sum of the (weighted) changes to the input matrix, while the ℓ∞only reflects the maximum absolute change. The other ℓp norms interpolate between these extremes. Therefore, a small value of p typically results in a solution that makes a few large changes to the original data, while a large value of p typically yields a solution with many small changes. 3 Algorithms This section describes efficient algorithms for solving the Metric Nearness Problems (2.2) and (2.3). For ease of exposition, we assume all weights to equal one. At first, it may appear that one should use quadratic programming (QP) software when p = 2, linear programming (LP) software when p = 1 or p = ∞, and convex programming software for the remaining p. It turns out that the time and storage requirements of this approach can be prohibitive. An efficient algorithm must exploit the structure of the triangle inequalities. In this paper, we develop one such approach, which may be viewed as a triangle-fixing algorithm. This method examines each triple of points in turn and optimally enforces any triangle inequality that fails. (The definition of “optimal” depends on the ℓp nearness measure.) By introducing appropriate corrections, we can ensure that this iterative algorithm converges to a globally optimal solution of MN. Notation. We must introduce some additional notation before proceeding. To each matrix X of dissimilarities or distances, we associate the vector x formed by stacking the columns of the lower triangle, left to right. We use xij to refer to the (i, j) entry of the matrix as well as the corresponding component of the vector. Define a constraint matrix A so that M is a distance matrix if and only if Am ≤0. Note that each row of A contains three nonzero entries, +1, −1, and −1. 3.1 MN for the ℓ2 norm We first develop a triangle-fixing algorithm for solving (2.2) with respect to the ℓ2 norm. This case turns out to be the simplest and most illuminating case. It also plays a pivotal role in the algorithms for the ℓ1 and ℓ∞MN problems. Given a dissimilarity vector d, we wish to find its orthogonal projection m onto the cone MN. Let us introduce an auxiliary variable e = m −d that represents the changes to the original distances. We also define b = −Ad. The negative entries of b indicate how much each triangle inequality is violated. The problem becomes mine ∥e∥2, subject to Ae ≤b. (3.1) After finding the minimizer e⋆, we can use the relation m⋆= d+e⋆to recover the optimal distance vector. Here is our approach. We initialize the vector of changes to zero (e = 0), and then we begin to cycle through the triangles. Suppose that the (i, j, k) triangle inequality is violated, i.e., eij −ejk −eki > bijk. We wish to remedy this violation by making an ℓ2-minimal adjustment of eij, ejk, and eki. In other words, the vector e is projected orthogonally onto the constraint set {e′ : e′ ij −e′ jk −e′ ki ≤bijk}. This is tantamount to solving mine′ 1 2 (e′ ij −eij)2 + (e′ jk −ejk)2 + (e′ ki −eki)2) , subject to e′ ij −e′ jk −e′ ki = bijk. (3.2) It is easy to check that the solution is given by e′ ij ←eij −µijk, e′ jk ←ejk + µijk, and e′ ki ←eki + µijk, (3.3) where µijk = 1 3(eij −ejk −eki −bijk) > 0. Only three components of the vector e need to be updated. The updates in (3.3) show that the largest edge weight in the triangle is decreased, while the other two edge weights are increased. In turn, we fix each violated triangle inequality using (3.3). We must also introduce a correction term to guide the algorithm to the global minimum. The corrections have a simple interpretation in terms of the dual of the minimization problem (3.1). Each dual variable corresponds to the violation in a single triangle inequality, and each individual correction results in a decrease in the violation. We continue until no triangle receives a significant update. Algorithm 3.1 displays the complete iterative scheme that performs triangle fixing along with appropriate corrections. Algorithm 3.1: Triangle Fixing For ℓ2 norm. TRIANGLE FIXING(D, ϵ) Input: Input dissimilarity matrix D, tolerance ϵ Output: M = argminX∈MN ∥X −D∥2. for 1 ≤i < j < k ≤n (zijk, zjki, zkij) ←0 {Initialize correction terms} for 1 ≤i < j ≤n eij ←0 {Initial error values for each dissimilarity dij} δ ←1 + ϵ {Parameter for testing convergence} while (δ > ϵ) {convergence test} foreach triangle (i, j, k) b ←dki + djk −dij µ ←1 3(eij −ejk −eki −b) (⋆) θ ←min{−µ, zijk} {Stay within half-space of constraint} eij ←eij −θ, ejk ←ejk + θ, eki ←eki + θ (⋆⋆) zijk ←zijk −θ {Update correction term} end foreach δ ←sum of changes in the e values end while return M = D + E Remark: Algorithm 3.1 is an efficient adaptation of Bregman’s method [1]. By itself, Bregman’s method would suffer the same storage and computation costs as a general convex optimization algorithm. Our triangle fixing operations allow us to compactly represent and compute the intermediate variables required to solve the problem. The correctness and convergence properties of Algorithm 3.1 follow from those of Bregman’s method. Furthermore, our algorithms are very easy to implement. 3.2 MN for the ℓ1 and ℓ∞norms The basic triangle fixing algorithm succeeds only when the norm used in (2.2) is strictly convex. Hence, it cannot be applied directly to the ℓ1 and ℓ∞cases. These require a more sophisticated approach. First, observe that the problem of minimizing the ℓ1 norm of the changes can be written as an LP: min e,f 0T e + 1T f subject to Ae ≤b, −e −f ≤0, e −f ≤0. (3.4) The auxiliary variable f can be interpreted as the absolute value of e. Similarly, minimizing the ℓ∞norm of the changes can be accomplished with the LP min e,ζ 0T e + ζ subject to Ae ≤b, −e −ζ1 ≤0, e −ζ1 ≤0. (3.5) We interpret ζ = ∥e∥∞. Solving these linear programs using standard software can be prohibitively expensive because of the large number of constraints. Moreover, the solutions are not unique because the ℓ1 and ℓ∞norms are not strictly convex. Instead, we replace the LP by a quadratic program (QP) that is strictly convex and returns the solution of the LP that has minimum ℓ2-norm. For the ℓ1 case, we have the following result. Theorem 3.1 (ℓ1 Metric Nearness). Let z = [e; f] and c = [0; 1] be partitioned conformally. If (3.4) has a solution, then there exists a λ0 > 0, such that for all λ ≤λ0, argmin z∈Z ∥z + λ−1c∥2 = argmin z∈Z⋆ ∥z∥2, (3.6) where Z is the feasible set for (3.4) and Z⋆is the set of optimal solutions to (3.4). The minimizer of (3.6) is unique. Theorem 3.1 follows from a result of Mangasarian [9, Theorem 2.1-a-i]. A similar theorem may be stated for the ℓ∞case. The QP (3.6) can be solved using an augmented triangle-fixing algorithm since the majority of the constraints in (3.6) are triangle inequalities. As in the ℓ2 case, the triangle constraints are enforced using (3.3). Each remaining constraint is enforced by computing an orthogonal projection onto the corresponding halfspace. We refer the reader to [5] for the details. 3.3 MN for ℓp norms (1 < p < ∞) Next, we explain how to use triangle fixing to solve the MN problem for the remaining ℓp norms, 1 < p < ∞. The computational costs are somewhat higher because the algorithm requires solving a nonlinear equation. The problem may be phrased as mine 1 p ∥e∥p p subject to Ae ≤b. (3.7) To enforce a triangle constraint optimally in the ℓp norm, we need to compute a projection of the vector e onto the constraint set. Define ϕ(x) = 1 p ∥x∥p p, and note that (∇ϕ(x))i = sgn(xi) |xi|p−1. The projection of e onto the (i, j, k) violating constraint is the solution of mine′ ϕ(e′) −ϕ(e) −⟨∇ϕ(e), e′ −e⟩ subject to aT ijke′ = bijk, where aijk is the row of the constraint matrix corresponding to the triangle inequality (i, j, k). The projection may be determined by solving ∇ϕ(e′) = ∇ϕ(e) + µijk aijk so that aT ijke′ = bijk. (3.8) Since aijk has only three nonzero entries, we see that e only needs to be updated in three components. Therefore, in Algorithm 3.1 we may replace (⋆) by an appropriate numerical computation of the parameter µijk and replace (⋆⋆) by the computation of the new value of e. Further details are available in [5]. 4 Applications and Experiments Replacing a general graph (dissimilarity matrix) by a metric graph (distance matrix) can enable us to use efficient approximation algorithms for NP-Hard graph problems (MAXCUT clustering) that have guaranteed error for metric data, for example, see [7]. The error from MN will carry over to the graph problem, while retaining the bounds on total error incurred. As an example, constant factor approximation algorithms for MAX-CUT exist for metric graphs [3], and can be used for clustering applications. See [4] for more details. Applications that use dissimilarity values, such as clustering, classification, searching, and indexing, could potentially be sped up if the data is metric. MN is a natural candidate for enforcing metric properties on the data to permit these speedups. We were originally motivated to formulate and solve MN by a problem that arose in connection with biological databases [13]. This problem involves approximating mPAM matrices, which are a derivative of mutation probability matrices [2] that arise in protein sequencing. They represent a certain measure of dissimilarity for an application in protein sequencing. Owing to the manner in which these matrices are formed, they tend not to be distance matrices. Query operations in biological databases have the potential to be dramatically sped up if the data were metric (using a metric based indexing scheme). Thus, one approach is to find the nearest distance matrix to each mPAM matrix and use that approximation in the metric based indexing scheme. We approximated various mPAM matrices by their nearest distance matrices. The relative errors of the approximations ∥D −M∥/∥D∥are reported in Table 1. Table 1: Relative errors for mPAM dataset (ℓ1, ℓ2, ℓ∞nearness, respectively) Dataset ∥D−M∥1 ∥D∥1 ∥D−M∥2 ∥D∥2 ∥D−M∥∞ ∥D∥∞ mPAM50 0.339 0.402 0.278 mPAM100 0.142 0.231 0.206 mPAM150 0.054 0.121 0.151 mPAM250 0.004 0.025 0.042 mPAM300 0.002 0.017 0.056 4.1 Experiments The MN problem has an input of size N = n(n −1)/2, and the number of constraints is roughly N 3/2. We ran experiments to ascertain the empirical behavior of the algorithm. Figure 1 shows log–log plots of the running time of our algorithms for solving the ℓ1 1 2 3 4 5 6 7 8 −6 −4 −2 0 2 4 6 8 Log(N) −− N is the input size Log(Running time in seconds) Log−Log plot showing runtime behavior of l1 MN y=1.6x−6.3 Running Time 7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 Log(N) −− where N is the input size Log(Running time in seconds) Log−Log plot of running time for l2 MN y=1.5x − 6.1 Running time Figure 1: Running time for ℓ1 and ℓ2 norm solutions (plots have different scales). and ℓ2 Metric Nearness Problems. Note that the time cost appears to be O(N 3/2), which is linear in the number of constraints. The results plotted in the figure were obtained by executing the algorithms on random dissimilarity matrices. The procedure was halted when the distance values changed less than 10−3 from one iteration to the next. For both problems, the results were obtained with a simple MATLAB implementation. Nevertheless, this basic version outperforms MATLAB’s optimization package by one or two orders of magnitude (depending on the problem), while numerically achieving similar results. A more sophisticated (C or parallel) implementation could improve the running time even more, which would allow us to study larger problems. 5 Discussion In this paper, we have introduced the Metric Nearness problem, and we have developed algorithms for solving it for ℓp nearness measures. The algorithms proceed by fixing violated triangles in turn, while introducing correction terms to guide the algorithm to the global optimum. Our experiments suggest that the algorithms require O(N 3/2) time, where N is the total number of distances, so it is linear in the number of constraints. An open problem is to obtain an algorithm with better computational complexity. Metric Nearness is a rich problem. It can be shown that a special case (allowing only decreases in the dissimilarities) is identical with the All Pairs Shortest Path problem [10]. Thus one may check whether the N distances satisfy metric properties in O(APSP) time. However, we are not aware if this is a lower bound. It is also possible to incorporate other types of linear and convex constraints into the Metric Nearness Problem. Some other possibilities include putting box constraints on the distances (l ≤m ≤u), allowing λ triangle inequalities (mij ≤λ1mik +λ2mkj), or enforcing order constraints (dij < dkl implies mij < mkl). We plan to further investigate the application of MN to other problems in data mining, machine learning, and database query retrieval. Acknowledgments This research was supported by NSF grant CCF-0431257, NSF Career Award ACI0093404, and NSF-ITR award IIS-0325116. References [1] Y. Censor and S. A. Zenios. Parallel Optimization: Theory, Algorithms, and Applications. Numerical Mathematics and Scientific Computation. OUP, 1997. [2] M. O. Dayhoff, R. M. Schwarz, and B. C. Orcutt. A model of evolutionary change in proteins. Atlas of Protein Sequence and Structure, 5(Suppl. 3), 1978. [3] W. F. de la Vega and C. Kenyon. A randomized approximation scheme for Metric MAX-CUT. J. Comput. Sys. and Sci., 63:531–541, 2001. [4] I. S. Dhillon, S. Sra, and J. A. Tropp. The Metric Nearness Problems with Applications. Tech. Rep. TR-03-23, Comp. Sci. Univ. of Texas at Austin, 2003. [5] I. S. Dhillon, S. Sra, and J. A. Tropp. Triangle Fixing Algorithms for the Metric Nearness Problem. Tech. Rep. TR-04-22, Comp. Sci., Univ. of Texas at Austin, 2004. [6] N. J. Higham. Matrix nearness problems and applications. In M. J. C. Gower and S. Barnett, editors, Applications of Matrix Theory, pages 1–27. Oxford University Press, 1989. [7] P. Indyk. Sublinear time algorithms for metric space problems. In 31st Symposium on Theory of Computing, pages 428–434, 1999. [8] J. B. Kruskal and M. Wish. Multidimensional Scaling. Number 07-011. Sage Publications, 1978. Series: Quantitative Applications in the Social Sciences. [9] O. L. Mangasarian. Normal solutions of linear programs. Mathematical Programming Study, 22:206–216, 1984. [10] C. G. Plaxton. Personal Communication, 2003–2004. [11] V. Roth, J. Laub, J. M. Buhmann, and K.-R. M¨uller. Going metric: Denoising pariwise data. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems (NIPS) 15, 2003. [12] E. P. Xing, A. Y. Ng, M. I. Jordan, and S. Russell. Distance metric learning, with application to clustering with side constraints. In S. Becker, S. Thrun, and K. Obermayer, editors, Advances in Neural Information Processing Systems (NIPS) 15, 2003. [13] W. Xu and D. P. Miranker. A metric model of amino acid substitution. Bioinformatics, 20(0):1–8, 2004.
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Economic Properties of Social Networks Sham M. Kakade Michael Kearns Luis E. Ortiz Robin Pemantle Siddharth Suri University of Pennsylvania Philadelphia, PA 19104 Abstract We examine the marriage of recent probabilistic generative models for social networks with classical frameworks from mathematical economics. We are particularly interested in how the statistical structure of such networks influences global economic quantities such as price variation. Our findings are a mixture of formal analysis, simulation, and experiments on an international trade data set from the United Nations. 1 Introduction There is a long history of research in economics on mathematical models for exchange markets, and the existence and properties of their equilibria. The work of Arrow and Debreu [1954], who established equilibrium existence in a very general commodities exchange model, was certainly one of the high points of this continuing line of inquiry. The origins of the field go back at least to Fisher [1891]. While there has been relatively recent interest in network models for interaction in economics (see Jackson [2003] for a good review), it was only quite recently that a network or graph-theoretic model that generalizes the classical Arrow-Debreu and Fisher models was introduced (Kakade et al. [2004]). In this model, the edges in a network over individual consumers (for example) represent those pairs of consumers that can engage in direct trade. As such, the model captures the many real-world settings that can give rise to limitations on the trading partners of individuals (regulatory restrictions, social connections, embargoes, and so on). In addition, variations in the price of a good can arise due to the topology of the network: certain individuals may be relatively favored or cursed by their position in the graph. In a parallel development over the last decade or so, there has been an explosion of interest in what is broadly called social network theory — the study of apparently “universal” properties of natural networks (such as small diameter, local clustering of edges, and heavytailed distribution of degree), and statistical generative models that explain such properties. When viewed as economic networks, the assumptions of individual rationality in these works are usually either non-existent, or quite weak, compared to the Arrow-Debreu or Fisher models. In this paper we examine classical economic exchange models in the modern light of social network theory. We are particularly interested in the interaction between the statistical structure of the underlying network and the variation in prices at equilibrium. We quantify the intuition that increased levels of connectivity in the network result in the equalization of prices, and establish that certain generative models (such as the the preferential attachment model of network formation (Barabasi and Albert [1999]) are capable of explaining the heavy-tailed distribution of wealth first observed by Pareto. Closely related work to ours is that of Kranton and Minehart [2001], which also considers networks of buyers and sellers, though they focus more on the economics of network formation. Many of our results are based on a powerful new local approximation method for global equilibrium prices: we show that in the preferential attachment model, prices computed from only local regions of a network yield strikingly good estimates of the global prices. We exploit this method theoretically and computationally. Our study concludes with an application of our model to United Nations international trade data. 2 Market Economies on Networks We first describe the standard Fisher model, which consists of a set of consumers and a set of goods. We assume that there are gj units of good j in the market, and that each good j is be sold at some price pj. Each consumer i has a cash endowment ei, to be used to purchase goods in a manner that maximizes the consumers’ utility. In this paper we make the wellstudied assumption that the utility function of each consumer is linear in the amount of goods consumed (see Gale [1960]), and leave the more general case to future research. Let uij ≥0 denote the utility derived by i on obtaining a single unit of good j. If i consumes xij amount of good j, then the utility i derives is P j uijxij. A set of prices {pj} and consumption plans {xij} constitutes an equilibrium if the following two conditions hold: 1. The market clears, i.e. supply equals demand. More formally, for each j, P i xij = gj. 2. For each consumer i, their consumption plan {xij}j is optimal. By this we mean that the consumption plan maximizes the linear utility function of i, subject to the constraint that the total cost of the goods purchased by i is not more than the endowment ei. It turns out that such an equilibrium always exists if each good j has a consumer which derives nonzero utility for good j — that is, uij > 0 for some i (see Gale [1960]). Furthermore, the equilibrium prices are unique. We now consider the graphical Fisher model, so named because of the introduction of a graph-theoretic or network structure to exchange. In the basic Fisher model, we implicitly assume that all goods are available in a centralized exchange, and all consumers have equal access to these goods. In the graphical Fisher model, we desire to capture the fact that each good may have multiple vendors or sellers, and that individual buyers may have access only to some, but not all, of these sellers. There are innumerable settings where such asymmetries arise. Examples include the fact that consumers generally purchase their groceries from local markets, that social connections play a major role in business transactions, and that securities regulations prevent certain pairs of parties from engaging in stock trades. Without loss of generality, we assume that each seller j sells only one of the available goods. (Each good may have multiple competing sellers.) Let G be a bipartite graph, where buyers and sellers are represented as vertices, and all edges are between a buyerseller pair. The semantics of the graph are as follows: if there is an edge from buyer i to seller j, then buyer i is permitted to purchase from seller j. Note that if buyer i is connected to two sellers of the same good, he will always choose to purchase from the cheaper source, since his utility is identical for both sellers (they sell the same good). The graphical Fisher model is a special case of a more general and recently introduced framework (Kakade et al. [2004]). One of the most interesting features of this model is the fact that at equilibrium, significant price variations can appear solely due to structural properties of the underlying network. We now describe some generative models of economies. 3 Generative Models for Social Networks For simplicity, in the sequel we will consider economies in which the numbers of buyers and sellers are equal. We will also restrict attention to the case in which all sellers sell the same good1. The simplest generative model for the bipartite graph G might be the random graph, in which each edge between a buyer i and a seller j is included independently with probability p. This is simply the bipartite version of the classical Erdos-Renyi model (Bollobas [2001]). Many researchers have sought more realistic models of social network formation, in order to explain observed phenomena such as heavy-tailed degree distributions. We now describe a slight variant of the preferential attachment model (see Mitzenmacher [2003]) for the case of a bipartite graph. We start with a graph in which one buyer is connected to one seller. At each time step, we add one buyer and one seller as follows. With probability α, the buyer is connected to a seller in the existing graph uniformly at random; and with probability 1 −α, the buyer is connected to a seller chosen in proportion to the degree of the seller (preferential attachment). Simultaneously, a seller is attached in a symmetric manner: with probability α the seller is connected to a buyer chosen uniformly at random, and with probability 1 −α the seller is connected under preferential attachment. The parameter α in this model thus allows us to move between a pure preferential attachment model (α = 0), and a model closer to classical random graph theory (α = 1), in which new parties are connected to random extant parties2. Note that the above model always produces trees, since the degree of a new party is always 1 upon its introduction to the graph. We thus will also consider a variant of this model in which at each time step, a new seller is still attached to exactly one extant buyer, while each new buyer is connected to ν > 1 extant sellers. The procedure for edge selection is as outlined above, with the modification that the ν new edges of the buyer are added without replacement — meaning that we resample so that each buyer gets attached to exactly ν distinct sellers. In a forthcoming long version, we provide results on the statistics of these networks. The main purpose of the introduction of ν is to have a model capable of generating highly cyclical (non-tree) networks, while having just a single parameter that can “tune” the asymmetry between the (number of) opportunities for buyers and sellers. There are also economic motivations: it is natural to imagine that new sellers of the good arise only upon obtaining their first customer, but that new buyers arrive already aware of several alternative sellers. In the sequel, we shall refer to the generative model just described as the bipartite (α, ν)model. We will use n to denote the number of buyers and the number of sellers, so the network has 2n vertices. Figure 1 and its caption provide an example of a network generated by this model, along with a discussion of its equilibrium properties. 4 Economics of the Network: Theory We now summarize our theoretical findings. The proofs will be provided in a forthcoming long version. We first present a rather intuitive “frontier” theorem, which implies a scheme in which we can find upper and lower bounds on the equilibrium prices using only local computations. To state the theorem we require some definitions. First, note that any subset V ′ of buyers and sellers defines a natural induced economy, where the induced graph G′ 1From a mathematical and computational standpoint, this restriction is rather weak: when considered in the graphical setting, it already contains the setting of multiple goods with binary utility values, since additional goods can be encoded in the network structure. 2We note that α = 1 still does not exactly produce the Erdos-Renyi model due to the incremental nature of the network generation: early buyers and sellers are still more likely to have higher degree. B0 S0: 1.50 B1 B2 B3 B4 B5 B9 B10 B11 B12 B15 B16 B17 B19 S1: 1.50 B6 B7 B8 S2: 1.00 B14 S3: 1.00 S4: 1.00 B18 S5: 1.50 S6: 0.67 S7: 1.50 S8: 1.00 S9: 0.75 S10: 1.00 B13 S11: 1.00 S12: 1.00 S13: 1.00 S14: 0.75 S15: 0.67 S16: 0.67 S17: 1.00 S18: 0.75 S19: 0.75 Figure 1: Sample network generated by the bipartite (α = 0, ν = 2)-model. Buyers and sellers are labeled by ‘B’ or ‘S’ respectively, followed by an index indicating the time step at which they were introduced to the network. The solid edges in the figure show the exchange subgraph —those pairs of buyers and sellers who actually exchange currency and goods at equilibrium. The dotted edges are edges of the network that are unused at equilibrium because they represent inferior prices for the buyers, while the dashed edges are edges of the network that have competitive prices, but are unused at equilibrium due to the specific consumption plan required for market clearance. Each seller is labeled with the price they charge at equilibrium. The example exhibits non-trivial price variation (from 2.00 down to 0.33 per unit good). Note that while there appears to be a correlation between seller degree and price, it is far from a deterministic relation, a topic we shall examine later. consists of all edges between buyers and sellers in V ′ that are also in G. We say that G′ has a buyer (respectively, seller) frontier if on every (simple) path in G from a node in V ′ to a node outside of V ′, the last node in V ′ on this path is a buyer (respectively, seller). Theorem 1 (Frontier Bound) If V ′ has a subgraph G′ with a seller (respectively, buyer) frontier, then the equilibrium price of any good j in the induced economy on V ′ is a lower bound (respectively, upper bound) on the equilibrium price of j in G. Theorem 1 implies a simple price upper bound: the price commanded by any seller j is bounded by its degree d. Although the same upper bound can be seen from first principles, it is instructive to apply Theorem 1. Let G′ be the immediate neighborhood of j (which is j and its d buyers); then the equilibrium price in G′ is just d, since all d buyers are forced to buy from seller j. This provides an upper bound since G′ has a buyer frontier. Since it can be shown that the degree distribution obeys a power law in the bipartite (α, ν)-model, we have an upper bound on the cumulative price distribution. We use β = (1 −α)ν/(1 + ν). Theorem 2 In the bipartite (α, ν)-model, the proportion of sellers with price greater than w is O(w−1/β). For example, if α = 0 (pure preferential attachment) and ν = 1, the proportion falls off as 1/w2. We do not yet have such a closed-form lower bound on the cumulative price distribution. However, as we shall see in Section 5, the price distributions seen in large simulation results do indeed show power-law behavior. Interestingly, this occurs despite the fact that degree is a poor predictor of individual seller price. 10 −1 10 0 10 1 10 2 10 −4 10 −3 10 −2 10 −1 10 0 Degree/Wealth Cumulative of Degree/Wealth 50 100 150 200 250 10 −3 10 −2 10 −1 10 0 N Average Error k=1 k=2 k=3 k=4 10 1 10 2 10 0 10 1 10 2 10 3 N Maximum to Minimum Wealth ν=1 ν=2 ν=3 ν=4 0 0.2 0.4 0.6 0.8 1 10 0 10 1 10 2 10 3 alpha Maximum to Minimum Wealth Figure 2: See text for descriptions. Another quantity of interest is what we might call price variation — the ratio of the price of the richest seller to the poorest seller. The following theorem addresses this. Theorem 3 In the bipartite (α, ν)-model, if α(ν2 + 1) < 1, then the ratio of the maximum price to the minimum price scales with number of buyers n as Ω(n 2−α(ν2+1) 1+ν ). For the simplest case in which α = 0 and ν = 1, this lower bound is just Ω(n). We conclude our theoretical results with a remark on the price variation in the Erdos-Renyi (random graph) model. First, let us present a condition for there to be no price variation. Theorem 4 A necessary and sufficient condition for there to be no price variation, ie for all prices to be equal to 1, is that for all sets of vertices S, |N(S)| ≥|S|, where N(S) is the set of vertices connected by an edge to some vertex in S. This can be viewed as an extremely weak version of standard expansion properties wellstudied in graph theory and theoretical computer science — rather than demanding that neighbor sets be strictly larger, we simply ask that they not be smaller. One can further show that for large n, the probability that a random graph (for any edge probability p > 0) obeys this weak expansion property approaches 1. In other words, in the Erdos-Renyi model, there is no variation in price — a stark contrast to the preferential attachment results. 5 Economics of the Network: Simulations We now present a number of studies on simulated networks (generated according to the bipartite (α, ν)-model). Equilibrium computations were done using the algorithm of Devanur et al. [2002] (or via the application of this algorithm to local subgraphs). We note that it was only the recent development of this algorithm and related ones that made possible the simulations described here (involving hundreds of buyers and sellers in highly cyclical graphs). However, even the speed of this algorithm limits our experiments to networks with n = 250 if we wish to run repeated trials to reduce variance. Many of our results suggest that the local approximation schemes discussed below may be far more effective. Price and Degree Distributions: The first (leftmost) panel of Figure 2 shows empirical cumulative price and degree distributions on a loglog scale, averaged over 25 networks drawn according to the bipartite (α = 0.4, ν = 1)-model with n = 250. The cumulative degree distribution is shown as a dotted line, where the y-axis represents the fraction of the sellers with degree greater than or equal to d, and the degree d is plotted on the x-axis. Similarly, the solid curve plots the fraction of sellers with price greater than some value w, where the price w is shown on the x-axis. The thin sold line has our theoretically predicted slope of −1 β = −3.33, which shows that degree distribution is quite consistent with our expectations, at least in the tails. Though a natural conjecture from the plots is that the price of a seller is essentially determined by its degree, below we will see that the degree is a rather poor predictor of an individual seller price, while more complex (but still local) properties are extremely accurate predictors. Perhaps the most interesting finding is that the tail of the price distribution looks linear, i.e. it also exhibits power law behavior. Our theory provided an upper bound, which is precisely the cumulative degree distribution. We do not yet have a formal lower bound. This plot (and other experiments we have done) further confirm the robustness of the power law behavior in the tail, for α < 1 and ν = 1. As discussed in the Introduction, Pareto’s original observation was that the wealth (which corresponds to seller price in our model) distribution in societies obey a power law, which has been born out in many studies on western economies. Since Pareto’s original observation, there have been too many explanations of this phenomena to recount here. However, to our knowledge, all of these explanations are more dynamic in nature (eg a dynamical system of wealth exchange) and don’t capture microscopic properties of individual rationality. Here we have power law wealth distribution arising from the combination of certain natural statistical properties of the network, and classical theories of economic equilibrium. Bounds via Local Computations: Recall that Theorem 1 suggests a scheme by which we can do only local computations to approximate the global equilibrium price for any seller. More precisely, for some seller j, consider the subgraph which contains all nodes that are within distance k of j. In our bipartite setting, for k odd, this subgraph has a buyer frontier, and for k even, this subgraph has a seller frontier, since we start from a seller. Hence, the equilibrium computation on the odd k (respectively, even k) subgraph will provide an upper (respectively, lower) bound. This provides an heuristic in which one can examine the equilibrium properties of small regions of the graph, without having to do expensive global equilibrium computations. The effectiveness of this heuristic will of course depend on how fast the upper and lower bounds tighten. In general, it is possible to create specific graphs in which these bounds are arbitrarily poor until k is large enough to encompass the entire graph. As we shall see, the performance of this heuristic is dramatically better in the bipartite (α, ν)-model. The second panel in Figure 2 shows how rapidly the local equilibrium computations converge to the true global equilibrium prices as a function of k, and also how this convergence is influenced by n. In these experiments, graphs were generated by the bipartite (α = 0, ν = 1) model. The value of n is given on the x-axis; the average errors (over 5 trials for each value of k and n) in the local equilibrium computations are given on the y-axis; and there is a separate plot for each of 4 values for k. It appears that for each value of k, the quality of approximation obtained has either mild or no dependence on n. Furthermore, the regular spacing of the four plots on the logarithmic scaling of the y-axis establishes the fact that the error of the local approximations is decaying exponentially with increased k — indeed, by examining only neighborhoods of 3 steps from a seller in an economy of hundreds, we are already able to compute approximations to global equilibrium prices with errors in the second decimal place. Since the diameter for n = 250 was often about 17, this local graph is considerably smaller than the global. However, for the crudest approximation k = 1, which corresponds exactly to using seller degree as a proxy for price, we can see that this performs rather poorly. Computationally, we found that the time required to do all 250 local computations for k = 3 was about 60% less than the global computation, and would result in presumably greater savings at much larger values of n. Parameter Dependencies: We now provide a brief examination of how price variation depends on the parameters of the bipartite (α, ν)-model. We first experimentally evaluate the lower bounds provided in Theorem 3. The third panel of Figure 2 shows the maximum to minimum price as function of n (averaged over 25 trials) on a loglog scale. Each line is for a fixed value of ν, and the values of ν range form 1 to 4 (α = 0). Recall from Theorem 3, our lower bound on the ratio is Ω(n 2 1+ν ) (using α = 0). We conjecture that this is tight, and, if so, the slopes of lines (in the loglog plot) should be 2 1+ν , which would be (1, 0.67, 0.5, 0.4). The estimated slopes are somewhat close: (1.02, 0.71, 0.57, 0.53). The overall message is that for small values of ν, price variation increases rapidly with the economy size n in preferential attachment. The rightmost panel of Figure 2 is a scatter plot of α vs. the maximum to minimum price in a graph (where n = 250) . Here, each point represents the maximum to minimum price ratio in a specific network generated by our model. The circles are for economies generated with ν = 1 and the x’s are for economies generated with ν = 3. Here we see that in general, increasing α dramatically decreases price variation (note that the price ratio is plotted on a log scale). This justifies the intuition that as α is increased, more “economic equality” is introduced in the form of less preferential bias in the formation of new edges. Furthermore, the data for ν = 1 shows much larger variation, suggesting that a larger value of ν also has the effect of equalizing buyer opportunities and therefore prices. 6 An Experimental Illustration on International Trade Data We conclude with a brief experiment exemplifying some of the ideas discussed so far. The statistics division of the United Nations makes available extensive data sets detailing the amounts of trade between major sovereign nations (see http://unstats.un.org/unsd/comtrade). We used a data set indicating, for each pair of nations, the total amount of trade in U.S. dollars between that pair in the year 2002. For our purposes, we would like to extract a discrete network structure from this numerical data. There are many reasonable ways this could be done; here we describe just one. For each of the 70 largest nations (in terms of total trade), we include connections from that nation to each of its top k trading partners, for some integer k > 1. We are thus including the more “important” edges for each nation. Note that each nation will have degree at least k, but as we shall see, some nations will have much higher degree, since they frequently occur as a top k partner of other nations. To further cast this extracted network into the bipartite setting we have been considering, we ran many trials in which each nation is randomly assigned a role as either a buyer or seller (which are symmetric roles), and then computed the equilibrium prices of the resulting network economy. We have thus deliberately created an experiment in which the only economic asymmetries are those determined by the undirected network structure. The leftmost panel of Figure 3 show results for 1000 trials under the choice k = 3. The upper plot shows the average equilibrium price for each nation, where the nations have been sorted by this average price. We can immediately see that there is dramatic price variation due to the network structure; while many nations suffer equilibrium prices well under $1, the most topologically favored nations command prices of $4.42 (U.S.), $4.01 (Germany), $3.67 (Italy), $3.16 (France), $2.27 (Japan), and $2.09 (Netherlands). The lower plot of the leftmost panel shows a scatterplot of a nation’s degree (x-axis) and its average equilibrium price (y-axis). We see that while there is generally a monotonic relationship, at smaller degree values there can be significant price variation (on the order of $0.50). The center panel of Figure 3 shows identical plots for the choice k = 10. As suggested by the theory and simulations, increasing the overall connectivity of each party radically reduces price variation, with the highest price being just $1.10 and the lowest just under $1. Interestingly, the identities of the nations commanding the highest prices (in order, U.S., France, Switzerland, Germany, Italy, Spain, Netherlands) overlaps significantly with the k = 3 case, suggesting a certain robustness in the relative economic status predicted by the model. The lower plot shows that the relationship between degree and price divides the population into “have” (degree above 10) and “have not” (degree below 10) components. The preponderance of European nations among the top prices suggests our final experi0 10 20 30 40 50 60 70 80 0 1 2 3 4 5 price rank price UN data network, top 3 links, full set of nations 0 5 10 15 20 25 0 1 2 3 4 5 average degree average price 0 10 20 30 40 50 60 70 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 price rank price UN data network, top 10 links, full set of nations 0 5 10 15 20 25 30 35 0.9 0.95 1 1.05 1.1 1.15 average degree average price 0 5 10 15 20 25 30 35 40 0 2 4 6 8 price rank price UN data network, top 3 links, EU collapsed nation set 0 2 4 6 8 10 12 14 0 2 4 6 8 average degree average price Figure 3: See text for descriptions. ment, in which we modified the k = 3 network by merging the 15 current members of the European Union (E.U.) into a single economic nation. This merged vertex has much higher degree than any of its original constituents and can be viewed as an (extremely) idealized experiment in the economic power that might be wielded by a truly unified Europe. The rightmost panel of Figure 3 provides the results, where we show the relative prices and the degree-price scatterplot for the 35 largest nations. The top prices are now commanded by the E.U. ($7.18), U.S. ($4.50), Japan ($2.96), Turkey ($1.32), and Singapore ($1.22). The scatterplot shows a clear example in which the highest degree (held by the U.S.) does not command the highest price. Acknowledgments We are grateful to Tejas Iyer and Vijay Vazirani for providing their software implementing the Devanur et al. [2002] algorithm. Siddharth Suri acknowledges the support of NIH grant T32HG0046. Robin Pemantle acknowledges the support of NSF grant DMS-0103635. References Kenneth J. Arrow and Gerard Debreu. Existence of an equilibrium for a competitive economy. Econometrica, 22(3):265–290, July 1954. A. Barabasi and R. Albert. Emergence of scaling in random networks. Science, 286:509–512, 1999. B. Bollobas. Random Graphs. Cambridge University Press, 2001. Nikhil R. Devanur, Christos H. Papadimitriou, Amin Saberi, and Vijay V. Vazirani. Market equilibrium via a primal-dual-type algorithm. In FOCS, 2002. Irving Fisher. PhD thesis, Yale University, 1891. D. Gale. Theory of Linear Economic Models. McGraw Hill, N.Y., 1960. Matthew Jackson. A survey of models of network formation: Stability and efficiency. In Group Formation in Economics: Networks, Clubs and Coalitions. Cambridge University Press, 2003. S. Kakade, M. Kearns, and L. Ortiz. Graphical economics. COLT, 2004. R. Kranton and D. Minehart. A theory of buyer-seller networks. American Economic Review, 2001. M. Mitzenmacher. A brief history of generative models for power law and lognormal distributions. Internet Mathematics, 1, 2003.
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Beat Tracking the Graphical Model Way Dustin Lang Nando de Freitas Department of Computer Science University of British Columbia Vancouver, BC {dalang, nando}@cs.ubc.ca Abstract We present a graphical model for beat tracking in recorded music. Using a probabilistic graphical model allows us to incorporate local information and global smoothness constraints in a principled manner. We evaluate our model on a set of varied and difficult examples, and achieve impressive results. By using a fast dual-tree algorithm for graphical model inference, our system runs in less time than the duration of the music being processed. 1 Introduction This paper describes our approach to the beat tracking problem. Dixon describes beats as follows: “much music has as its rhythmic basis a series of pulses, spaced approximately equally in time, relative to which the timing of all musical events can be described. This phenomenon is called the beat, and the individual pulses are also called beats”[1]. Given a piece of recorded music (an MP3 file, for example), we wish to produce a set of beats that correspond to the beats perceived by human listeners. The set of beats of a song can be characterised by the trajectories through time of the tempo and phase offset. Tempo is typically measured in beats per minute (BPM), and describes the frequency of beats. The phase offset determines the time offset of the beat. When tapping a foot in time to music, tempo is the rate of foot tapping and phase offset is the time at which the tap occurs. The beat tracking problem, in its general form, is quite difficult. Music is often ambiguous; different human listeners can perceive the beat differently. There are often several beat tracks that could be considered correct. Human perception of the beat is influenced both by ‘local’ and contextual information; the beat can continue through several seconds of silence in the middle of a song. We see the beat tracking problem as not only an interesting problem in its own right, but as one aspect of the larger problem of machine analysis of music. Given beat tracks for a number of songs, we could extract descriptions of the rhythm and use these features for clustering or searching in music collections. We could also use the rhythm information to do structural analysis of songs - for example, to find repeating sections. In addition, we note that beat tracking produces a description of the time scale of a song; knowledge of the tempo of a song would be one way to achieve time-invariance in a symbolic description. Finally, we note that beat tracking tells us where the important parts of a song are; the beats (and major divisions of the beats) are good sampling points for other music-analysis problems such as note detection. 2 Related Work Many researchers have investigated the beat tracking problem; we present only a brief overview here. Scheirer [2] presents a system, based on psychoacoustical observations, in which a bank of resonators compete to explain the processed audio input. The system is tested on a difficult set of examples, and considerable success is reported. The most common problem is a lack of global consistency in the results - the system switches between locally optimal solutions. Goto [3] has described several systems for beat tracking. He takes a very pragmatic view of the problem, and introduces a number of assumptions that allow good results in a limited domain - pop music in 4/4 time with roughly constant tempo, where bass or snare drums keep the beat according to drum patterns known a priori, or where chord changes occur at particular times within the measure. Cemgil and Kappen [4] phrase the beat tracking problem in probabilistic terms, and we adapt their model as our local observation model. They use MIDI-like (event-based) input rather than audio, so the results are not easily comparable to our system. 3 Graphical Model In formulating our model for beat tracking, we assume that the tempo is nearly constant over short periods of time, and usually varies smoothly. We expect the phase to be continuous. This allows us to use the simple graphical model shown in Figure 1. We break the song into a set of frames of two seconds; each frame is a node in the graphical model. We expect the tempo to be constant within each frame, and the tempo and phase offset parameters to vary smoothly between frames. PSfrag replacements X1 X2 X3 XF Y1 Y2 Y3 YF ψ ψ ψ φ φ φ φ Figure 1: Our graphical model for beat tracking. The hidden state X is composed of the state variables tempo and phase offset. The observations Y are the features extracted by our audio signal processing. The potential function φ describes the compatibility of the observations with the state, while the potential function ψ describes the smoothness between neighbouring states. In this undirected probabilistic graphical model, the potential function φ describes the compatibility of the state variables X = {T, P} composed of tempo T and phase offset P with the local observations Y . The potential function ψ describes the smoothness constraints between frames. The observation Y comes from processing the audio signal, which is described in Section 5. The φ function comes from domain knowledge and is described in Section 4. This model allows us to trade off local fit and global smoothness in a principled manner. By using an undirected model, we allow contextual information to flow both forward and backward in time. In such models, belief propagation (BP) [5] allows us to compute the marginal probabilities of the state variables in each frame. Alternatively, maximum belief propagation (max-BP) allows a joint maximum a posteriori (MAP) set of state variables to be determined. That is, given a song, we generate the observations Yi, i = 1 . . . F, (where F is the number of frames in the song) and seek a set of states Xi that maximize the joint product P(X, Y ) = 1 Z F Y i=1 φ(Yi, Xi) F −1 Y i=1 ψ(Xi, Xi+1) . Our smoothness function ψ is the product of tempo and phase smoothness components ψT and ψP. For the tempo component, we use a Gaussian on the log of tempo. For the phase offset component, we want the phases to agree at a particular point in time: the boundary between the two frames (nodes), tb. We find the phase θ of tb predicted by the parameters in each frame, and place a Gaussian prior on the distance between points on the unit circle with these phases: ψ(X1, X2 | tb) = ψT(T1, T2) ψP(T1, P1, T2, P2 | tb) = N(log T1 −log T2, σ2 T) N((cos θ1 −cos θ2, sin θ1 −sin θ2), σ2 P) where θi = 2πTitb −Pi and N(x, σ2) is a zero-mean Gaussian with variance σ2. We set σT = 0.1 and σP = 0.1π. The qualitative results seem to be fairly stable as a function of these smoothness parameters. 4 Domain Knowledge In this section, we describe the derivation of our local potential function (also known as the observation model) φ(Yi, Xi). Our model is an adaptation of the work of [4], which was developed for use with MIDI input. Their model is designed so that it “prefers simpler [musical] notations”. The beat is divided into a fixed number of bins (some power of two), and each note is assigned to the nearest bin. The probability of observing a note at a coarse subdivision of the beat is greater than at a finer subdivision. More precisely, a note that is quantized to the bin at beat number k has probability p(k) ∝exp(−λ d(k)), where d(k) is the number of digits in the binary representation of the number k mod 1. Since we use recorded music rather than MIDI, we must perform signal processing to extract features from the raw data. This process produces a signal that has considerably more uncertainty than the discrete events of MIDI data, so we adjust the model. We add the constraint that features should be observed near some quantization point, which we express by centering a Gaussian around each of the quantization points. The variance of this Gaussian, σ2 Q is in units of beats, so we arrive at the periodic template function b(t), shown in Figure 2. We have set the number of bins to 8, λ to one, and σQ = 0.025. The template function b(t) expresses our belief about the distribution of musical events within the beat. By shifting and scaling b(t), we can describe the expected distribution of notes in time for different tempos and phase offsets: b(t | T, P) = b Tt −P 2π . Our signal processing (described below) yields a discrete set of events that are meant to correspond to musical events. Events occur at a particular time t and have a ‘strength’ or ‘energy’ E. Given a set of discrete events Y = {ti, Ei}, i = 1 . . . M, and state variables X = {T, P}, we take the probability that the events were drawn from the expected distribution b(t | T, P): φ(Y , X) = φ({t, E}, {T, P}) = M Y i=1 b(ti | T, P)Ei . PSfrag replacements Time (units of beats) Note Probability 0 1/8 1/4 3/8 1/2 5/8 3/4 7/8 1 Figure 2: One period of our template function b(t), which gives the expected distribution of notes within a beat. Given tempo and phase offset values, we stretch and shift this function to get the expected distribution of notes in time. This is a multinomial probability function in the continuous limit (as the bin size becomes zero). Note that φ is a positive, unnormalized potential function. 5 Signal Processing Our signal processing stage is meant to extract features that approximate musical events (drum beats, piano notes, guitar strums, etc.) from the raw audio signal. As discussed above, we produce a set of events composed of time and ‘strength’ values, where the strength describes our certainty that an event occurred. We assume that musical events are characterised by brief, rapid increases in energy in the audio signal. This is certainly the case for percussive instruments such as drums and piano, and will often be the case for string and woodwind instruments and for voices. This assumption breaks for sounds that fade in smoothly rather than ‘spikily’. We begin by taking the short-time Fourier transform (STFT) of the signal: we slide a 50 millisecond Hann window over the signal in steps of 10 milliseconds, take the Fourier transform, and extract the energy spectrum. Following a suggestion by [2], we pass the energy spectrum through a bank of five filters that sum the energy in different portions of the spectrum. We take the logarithm of the summed energies to get a ‘loudness’ signal. Next, we convolve each of the five resulting energy signals with a filter that detects positivegoing edges. This can be considered a ‘loudness gain’ signal. Finally, we find the maxima within 50 ms neighbourhoods. The result is a set of points that describe the energy gain signal in each band, with emphasis on the maxima. These are the features Y that we use in our local probability model φ. 6 Fast Inference To find a maximum a posteriori (MAP) set of state variables that best explain a set of observations, we need to optimize a 2F-dimensional, continuous, non-linear, non-Gaussian function that has many local extrema. F is the number of frames in the song, so is on the order of the length of the song in seconds - typically in the hundreds. This is clearly difficult. We present two approximation strategies. In the first strategy, we convert the continuous state space into a uniform discrete grid and run discrete belief propagation. In the second strategy, we run a particle filter in the forward direction, then use the particles as ‘grid’ points and run discrete belief propagation as per [6]. Since the landscape we are optimizing has many local maxima, we must use a fine discretization grid (for the first strategy) or a large number of particles (for the second strategy). The message-passing stage in discrete belief propagation takes O(N 2) if performed naively, where N is the number of discretized states (or particles) per frame. We use a dualtree recursion strategy as proposed in [7] and extended to maximum a posteriori inference in [8]. With this approach, the computation becomes feasible. As an aside, we note that if we wish to compute the smoothed marginal probabilities rather than the MAP set of parameters, then we can use standard discrete belief propagation or particle smoothing. In both cases, the naive cost in O(N 2), but by using the Fast Gauss Transform[9] the cost becomes O(N). This is possible because our smoothness potential ψ is a low-dimensional Gaussian. For the results presented here, we discretize the state space into NT = 90 tempo values and NP = 50 phase offset values for the belief propagation version. We distribute the tempo values uniformly on a log scale between 40 and 150 BPM, and distribute the phase offsets uniformly. For the particle filter version, we use NT × NP = 4500 particles. With these values, our Matlab and C implementation runs at faster than real time (the duration of the song) on a standard desktop computer. 7 Results A standard corpus of labelled ground truth data for the beat-tracking problem does not exist. Therefore, we labelled a relatively small number of songs for evaluation of our algorithm, by listening to the songs and pressing a key at each perceived beat. We sought out examples that we thought would be difficult, and we attempted to avoid the methods of [10]. Ideally, we would have several human listeners label each song, since this would help to capture the ambiguity inherent in the problem. However, this would be quite time-consuming. One can imagine several methods for speeding up the process of generating ground truth labellings and of cleaning up the noisy results generated by humans. For example, a human labelling of a short segment of the song could be automatically extrapolated to the remainder of the song, using energy spikes in the audio signal to fine-tune the placement of beats. However, by generating ground truth using assumptions similar to those embodied in the models we intend to test, we risk invalidating the results. We instead opted to use ‘raw’ human-labelled songs. There is no standard evaluation metric for beat tracking. We use the ρ function presented by Cemgil et al [11] and used by Dixon [1] in his analysis: ρ(S, T) = 100 (NS + NT )/2 NS X i=1 max j∈T exp −(Si −Tj)2 2σ2 where S and T are the ground-truth and proposed beat times, and σ is set to 40 milliseconds. A ρ value near 100 means that each predicted beat in close to a true beat, while a value near zero means that each predicted beat is far from a true beat. We have focused on finding a globally-optimum beat track rather than precisely locating each beat. We could likely improve the ρ values of our results by fine-tuning each predicted beat, for example by finding nearby energy peaks, though we have not done this in the results presented here. Table 1 shows a summary of our results. Note the wide range of genres and the choice of songs with features that we thought would make beat tracking difficult. This includes all our results (not just the ones that look good). The first columns list the name of the song and the reason we included it. The third column lists the qualitative performance of the fixed grid version: double means our algorithm produced a beat track twice as fast as ground truth, half means we tracked at half speed, and sync means we produced a syncopated (π phase error) beat track. A blank entry means our algorithm produced the correct beat track. A star (⋆) means that our result incorrectly switches phase or tempo. The ρ values are after compensating for the qualitative error (if any). The fifth column shows a histogram of the absolute phase error (0 to π); this is also Song Comment BP Perf. BP ρ Phase Err PF Perf. PF ρ Phase Err Glenn Gould / Bach Goldberg Var’ns 1982 / Var’n 1 Classical piano 88 86 Jeno Jand´o / Bach WTC / Fuga 2 (C Minor) Piano; rubato at end 77 77 Kronos Quartet / Caravan / Aaj Ki Raat Modern string quartet 75 71 Maurice Ravel / Piano Concertos / G Major - Presto Classical orchestra ⋆sync 44 ⋆sync 50 Miles Davis / Kind Of Blue / So What (edit) Jazz instrumental half 61 half 59 Miles Davis / Kind Of Blue / Blue In Green Jazz instrumental 57 59 Holly Cole / Temptation / Jersey Girl Jazz vocal 78 77 Don Ross / Passion Session / Michael Michael Michael Solo guitar ⋆threehalf 40 ⋆threehalf 42 Don Ross / Huron Street / Luci Watusi Solo guitar 70 69 Tracy Chapman / For You Guitar and voice double 59 double 61 Ben Harper / Fight For Your Mind / Oppression Acoustic 70 68 Great Big Sea / Up / Chemical Worker’s Song Newfoundland folk 79 78 Buena Vista Social Club / Chan Chan Cuban 72 72 Beatles / 1967-1970 / Lucy In The Sky With Diamonds Changes time signature ⋆ 42 ⋆ 41 U2 / Joshua Tree / Where The Streets Have No Name (edit) Rock 82 82 Cake / Fashion Nugget / I Will Survive Rock sync 81 sync 80 Sublime / Second-Hand Smoke / Thanx Dub (excerpt) Reggae 79 79 Rancid / ... And Out Come The Wolves / Old Friend Punk half 82 half 79 Green Day / Dookie / When I Come Around Pop-punk 75 74 Tortoise / TNT / A Simple Way To Go Faster Than Light... Organic electronica double 79 double 79 Pole / 2 / Stadt Ambient electronica 71 71 Underworld / A Hundred Days Off / MoMove Electronica 79 79 Ravi Shankar / The Sounds Of India / Bhimpalsi (edit) Solo sitar 71 67 Pitamaha: Music From Bali / Puri Bagus, Bamboo (excerpt) Indonesian gamelan 86 sync 89 Gamelan Sekar Jaya / Byomantara (excerpt) Indonesian gamelan 89 88 Table 1: The songs used in our evaluation. See the text for explanation. 85 90 95 100 105 Tempo (BPM) 50 100 150 200 250 Time (s) Smoothed ground truth Predicted Smoothed ground truth Raw ground truth Smoothed ground truth Predicted Figure 3: Tempo tracks for Cake / I Will Survive. Center: ‘raw’ ground-truth tempo (instantaneous tempo estimate based on the time between adjacent beats) and smoothed ground truth (by averaging). Left: fixed-grid version result. Right: particle filter result. after correcting for qualitative error. The remaining columns contain the same items for the particle filter version. Out of 25 examples, the fixed grid version produces the correct answer in 17 cases, tracks at double speed in two cases, half speed in two cases, syncopated in one case, and in three cases produces a track that (incorrectly) switches tempo or phase. The particle filter version produces 16 correct answers, two double-speed, two half-speed, two syncopated, and the same three ‘switching’ tracks. An example of a successful tempo track is shown in Figure 3. The result for Lucy In The Sky With Diamonds (one of the ‘switching’ results) is worth examination. The song switches time signature between 3/4 and 4/4 time a total of five times; see Figure 4. Our results follow the time signature change the first three times. On the fourth change (from 4/4 to 3/4), it tracks at 2/3 the ground truth rate instead. We note an interesting effect when we examine the final message that is passed during belief propagation. This message tells us the maximum probability of a sequence that ends with each state. The global maximum corresponds to the beat track shown in the left plot. The local maximum near 50 BPM corresponds to an alternate solution in which, rather than tracking the quarter notes, we produce one beat per measure; this track is quite plausible. Indeed, the ‘true’ track is difficult for human listeners. Note also that there is also a local maximum near 100 BPM but phase-shifted a half beat. This is the solution in which the beats are syncopated from the true result. 8 Conclusions and Further Work We present a graphical model for beat tracking and evaluate it on a set of varied and difficult examples. We achieve good results that are comparable with those reported by other researchers, although direct comparisons are impossible without a shared data set. There are several advantages to formulating the problem in a probabilistic setting. The beat tracking problem has inherent ambiguity and multiple interpretations are often plausible. With a probabilistic model, we can produce several candidate solutions with different probabilities. This is particularly useful for situations in which beat tracking is one element in a larger machine listening application. Probabilistic graphical models allow flexible and powerful handling of uncertainty, and allow local and contextual information to interact in a principled manner. Additional domain knowledge and constraints can be added in a clean and principled way. The adoption of an efficient dual tree recursion for graphical models 20 40 60 80 100 120 140 160 180 200 60 65 70 75 80 85 90 95 100 Time (s) Tempo (BPM) 0 0.2 0.4 0.6 0.8 40 50 75 100 140 Phase offset Tempo (BPM) 1/2 Ground Truth 2/3 Ground Truth Predicted Figure 4: Left: Tempo tracks for Lucy In The Sky With Diamonds. The vertical lines mark times at which the time signature changes between 3/4 and 4/4. Right: the last maxmessage computed during belief propagation. Bright means high probability. The global maximum corresponds to the tempo track shown. Note the local maximum around 50 BPM, which corresponds to an alternate feasible result. See the text for discussion. [7, 8] enables us to carry out inference in real time. We would like to investigate several modifications of our model and inference methods. Longer-range tempo smoothness constraints as suggested by [11] could be useful. The extraction of MAP sets of parameters for several qualitatively different solutions would help to express the ambiguity of the problem. The particle filter could also be changed. At present, we first perform a full particle filtering sweep and then run max-BP. Taking into account the quality of the partial MAP solutions during particle filtering might allow superior results by directing more particles toward regions of the state space that are likely to contain the final MAP solution. Since we know that our probability terrain is multimodal, a mixture particle filter would be useful [12]. References [1] S Dixon. An empirical comparison of tempo trackers. Technical Report TR-2001-21, Austrian Research Institute for Artificial Intelligence, Vienna, Austria, 2001. [2] E D Scheirer. Tempo and beat analysis of acoustic musical signals. J. Acoust. Soc. Am., 103(1):588–601, Jan 1998. [3] M Goto. An audio-based real-time beat tracking system for music with or without drum-sounds. Journal of New Music Research, 30(2):159–171, 2001. [4] A T Cemgil and H J Kappen. Monte Carlo methods for tempo tracking and rhythm quantization. Journal of Artificial Intelligence Research, 18(1):45–81, 2003. [5] J Pearl. Probabilistic reasoning in intelligent systems: networks of plausible inference. MorganKaufmann, 1988. [6] S J Godsill, A Doucet, and M West. Maximum a posteriori sequence estimation using Monte Carlo particle filters. Ann. Inst. Stat. Math., 53(1):82–96, March 2001. [7] A G Gray and A W Moore. ‘N-Body’ problems in statistical learning. In Advances in Neural Information Processing Systems 4, pages 521–527, 2000. [8] M Klaas, D Lang, and N de Freitas. Fast maximum a posteriori inference in monte carlo state space. In AI-STATS, 2005. [9] L Greengard and J Strain. The fast Gauss transform. SIAM Journal of Scientific Statistical Computing, 12(1):79–94, 1991. [10] D LaLoudouana and M B Tarare. Data set selection. Presented at NIPS Workshop, 2002. [11] A T Cemgil, B Kappen, P Desain, and H Honing. On tempo tracking: Tempogram representation and Kalman filtering. Journal of New Music Research, 28(4):259–273, 2001. [12] J Vermaak, A Doucet, and Patrick P´erez. 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Large-Scale Prediction of Disulphide Bond Connectivity Pierre Baldi Jianlin Cheng Schoolof Information and Computer Science University of California, Irvine Irvine, CA 92697-3425 {pfbaldi,jianlinc}@ics.uci.edu Alessandro Vullo Computer Science Department University College Dublin Dublin, Ireland alessandro.vullo@ucd.ie Abstract The formation of disulphide bridges among cysteines is an important feature of protein structures. Here we develop new methods for the prediction of disulphide bond connectivity. We first build a large curated data set of proteins containing disulphide bridges and then use 2-Dimensional Recursive Neural Networks to predict bonding probabilities between cysteine pairs. These probabilities in turn lead to a weighted graph matching problem that can be addressed efficiently. We show how the method consistently achieves better results than previous approaches on the same validation data. In addition, the method can easily cope with chains with arbitrary numbers of bonded cysteines. Therefore, it overcomes one of the major limitations of previous approaches restricting predictions to chains containing no more than 10 oxidized cysteines. The method can be applied both to situations where the bonded state of each cysteine is known or unknown, in which case bonded state can be predicted with 85% precision and 90% recall. The method also yields an estimate for the total number of disulphide bridges in each chain. 1 Introduction The formation of covalent links among cysteine (Cys) residues with disulphide bridges is an important and unique feature of protein folding and structure. Simulations [1], experiments in protein engineering [15, 8, 14], theoretical studies [7, 18], and even evolutionary models [9] stress the importance of disulphide bonds in stabilizing the native state of proteins. Disulphide bridges may link distant portions of a protein sequence, providing strong structural constraints in the form of long-range interactions. Thus prediction/knowledge of the disulphide connectivity of a protein is important and provides essential insights into its structure and possibly also into its function and evolution. Only recently has the problem of predicting disulphide bridges received increased attention. In the current literature, this problems is typically split into three subproblems: (1) prediction of whether a protein chain contains intra-chain disulphide bridges or not; (2) prediction of the intra-chain bonded/non-bonded state of individual cysteines; and (3) prediction of intra-chain disulphide bridges, i.e. of the actual pairings between bonded cysteines (see Fig.1). In this paper, we address the problem of intra-chain connectivity prediction, and AVITGACERDLQCGKGTCCAVSLWIKSVRVCTPVGTSGEDCHPASHKIPFSGQRKMHHTCPCAPNLACVQTSPKKFKCLSK Figure 1: Structure (top) and connectivity pattern (bottom) of intestinal toxin 1, PDB code 1IMT. Disulphide bonds in the structure are shown as thick lines. specifically the solution of problem (3) alone, and of problems (2) and (3) simultaneously. Existing approaches to connectivity prediction use stochastic global optimization [10], combinatorial optimization [13] and machine learning techniques [11, 17]. The method in [10] represents the set of potential disulphide bridges in a sequence as a complete weighted undirected graph. Vertices are oxidized cysteines and edges are labeled by the strength of interaction (contact potential) in the associated pair of cysteines. A simulated annealing approach is first used to find an optimal set of weights. After a complete labeled graph is obtained, candidate bridges are then located by finding the maximum weight perfect matching1. The method in [17] attempts to solve the problem using a different machine learning approach. Candidate connectivity patterns are modelled as undirected graphs. A recursive neural network architecture is trained to score candidate patterns according to a similarity metric with respect to correct graphs. Vertices of the graphs are labeled by fixed-size vectors corresponding to multiple alignment profiles in a local window around each cysteine. During prediction, the score computed by the network is used to exhaustively search the space of candidate graphs. This method, tested on the same data as in [11], achieved the best results. Unfortunately, for computational reasons, both this method and the previous one can only deal with sequences containing a limited number of bonds (K ≤5). A different approach to predicting disulphide bridges is reported in [13], where finding disulphide bridges is part of a more general protocol aimed at predicting the topology of β-sheets in proteins. Residue-to-residue contacts (including Cys-Cys bridges) are predicted by solving a series of integer linear programming problems in which customized hydrophobic contact energies must be maximized. This method cannot be compared with the other approaches because the authors report validation results only for two relatively short polypeptides with few bonds (2 and 3). In this paper we use 2-Dimensional Recursive Neural Network (2D-RNN, [4]) to predict disulphide connectivity in proteins starting from their primary sequence and its homologues. The output of 2D-RNN are the pairwise probabilities of the existence of a bridge between any pair of cysteines. Candidate disulphide connectivities are predicted by finding the maximum weight perfect matching. The proposed framework represents a significant improvement in disulphide connectivity prediction for several reasons. First, we show how the method consistently achieves better results than all previous approaches on the same validation data. Second, our architecture can easily cope with chains with arbitrary number 1A perfect matching of a graph (V, E) is a subset E′ ⊆E such that each vertex v ∈V is met by only one edge in E′. Output Plane Input Plane 4 Hidden Planes NE NW SW SE (a) SE i,j-1 NW i,j I i,j SE i,j SW i,j NE i,j NW i+1,j SE i-1,j NE i,j-1 NE i+1,j NW i,j+1 SW i-1,j SW i,j+1 O i,j (b) Figure 2: (a) General layout of a 2D-RNN for processing two-dimensional objects such as disulphide contacts, with nodes regularly arranged in one input plane, one output plane, and four hidden planes. In each plane, nodes are arranged on a square lattice. The hidden planes contain directed edges associated with the square lattices. All the edges of the square lattice in each hidden plane are oriented in the direction of one of the four possible cardinal corners: NE, NW, SW, SE. Additional directed edges run vertically in column from the input plane to each hidden plane, and from each hidden plane to the output plane. (b) Connections within a vertical column (i, j) of the directed graph. Iij represents the input, Oij the output, and NEij represents the hidden variable in the North-East hidden plane. of bonded cysteines. Therefore, it overcomes the limitation of previous approaches which restrict predictions to chains with no more than 10 oxidized cysteines. Third, our methods can be applied both to situations where the bonded state of each cysteine is known or unknown. And finally, once trained, our system is very rapid and can be used on a high-throughput scale. 2 Methods Algorithms To predict disulphide connectivity patterns, we use the 2D-RNN approach described in [4], whereby a suitable Bayesian network is recast, for computational effectiveness, in terms of recursive neural networks, where local conditional probability tables in the underlying directed graph are replaced by deterministic relationships between a variable and its parent node variables. These functions are parameterized by neural networks using appropriate weight sharing as described below. Here the underlying directed graph for disulphide connectivity has six 2D-layers: input, output, and four hidden layers (Figure 2(a)). Vertical connections, within an (i, j) column, run from input to hidden and output layers, and from hidden layers to output (Figure 2(b)). In each one of the four hidden planes, square lattice connections are oriented towards one of the four cardinal corners. Detailed motivation for these architectures can be found in [4] and a mathematical analysis of their relationships to Bayesian networks in [5]. The essential point is that they combine the flexibility of graphical models with the deterministic propagation and learning speed of artificial neural networks. Unlike traditional neural networks with fixed-size input, these architectures can process inputs of variable structure and length, and allow lateral propagation of contextual information over considerable length scales. In a disulphide contact map prediction, the (i, j) output represents the probability of whether the i-th and j-th cysteines in the sequence are linked by a disulphide bridge or not. This prediction depends directly on the (i, j) input and the four-hidden units in the same column, associated with omni-directional contextual propagation in the hidden planes. Hence, using weight sharing across different columns, the model can be summarized by 5 distinct neural networks in the form Oij = NO(Iij, HNW i,j , HNE i,j , HSW i,j , HSE i,j ) HNE i,j = NNE(Ii,j, HNE i−1,j, HNE i,j−1) HNW i,j = NNW (Ii,j, HNW i+1,j, HNW i,j−1) HSW i,j = NSW (Ii,j, HSW i+1,j, HSW i,j+1) HSE i,j = NSE(Ii,j, HSE i−1,j, HSE i,j+1) (1) where N represents NN parameterization. Learning can proceed by gradient descent (backpropagation) due to the acyclic nature of the underlying graph. The input information is based on the sequence itself or rather the corresponding profile derived by multiple alignment methods to leverage evolutionary information, possibly augmented with secondary structure and solvent accessibility information derived from the PDB files and/or our SCRATCH suite of predictors [16, 3, 4]. For a sequence of length N and containing M cysteines, the output layer contains M × M units. The input and hidden layer can scale like N × N if the full sequence is used, or like M × M if only fixed-size windows around each cysteine are used, as in the experiments reported here. The results reported here are obtained using local windows of size 5 around each cysteine, as in [17]. The input of each position within a window is the normalized frequency of all 20 amino acids at that position in the multiple alignment generated by aligning the sequence with the sequences in the NR database using the PSI-BLAST program as described, for instance, in [16]. Gaps are treated as one additional amino acid. For each (i, j) location an extra input is added to represent the absolute linear distance between the two corresponding cysteines. Finally, it is essential to remark that the same 2D-RNN approach can be trained and applied here in two different modes. In the first mode, we can assume that the bonded state of the individual cysteines is known, for instance through the use of a specialized predictor for bonded/non-bonded states. Then if the sequence contains M cysteines, 2K (2K ≤M) of which are intra-chain disulphide bonded, the prediction of the connectivity can focus on the 2K bonded cysteines exclusively and ignore the remaining M −2K cysteines that are not bonded. In the second mode, we can try to solve both prediction problems–bond state and connectivity–at the same time by focusing on all cysteines in a given sequence. In both cases, the output is an array of pairwise probabilities from which the connectivity pattern graph must be inferred. In the first case, the total number of bonds or edges in the connectivity graph is known (K). In the second case, the total number of edges must be inferred. In section 3, we show that sum of all probabilities across the output array can be used to estimate the number of disulphide contacts. Data Preparation In order to assess our method, two data sets of known disulphide connectivities were compiled from the Swiss-Prot archive [2]. First, we considered the same selection of sequences as adopted in [11, 17] and taken from the Swiss-Prot database release no. 39 (October 2000). Additionally, we collected and filtered a more recent selection of chains extracted from the latest available Swiss-Prot archive, version 41.19 (August 2003). In the following, we refer to these two data sets as SP39 and SP41, respectively. SP41 was compiled with the same filtering procedure used for SP39. Specifically, only chains whose structure is deposited in the Protein Data Bank PDB [6] were retained. We filtered out proteins with disulphide bonds assigned tentatively or disulphide bonds inferred by similarity. We finally ended up with 966 chains, each with a number of disulphide bonds in the range of 1 to 24. As previously pointed out, our methodology is not limited by the number of disulphide bonds, hence we were able to retain and test the algorithm on the whole filtered set of non-trivial chains. This set consists of 712 sequences, each containing at least two bridges (K ≥2)–the case K = 1 being trivial when the bonded state is known. By comparison, SP39 includes 446 chains with no more than 5 bridges; SP41 additionally includes 266 sequences and 112 of these have more than 10 oxidized cysteines. In order to avoid biases during the assessment procedure and to perform k-fold cross validation, SP41 was partitioned in ten different subsets, with the constraint that sequence similarity between two different subsets be less or equal to 30%. This is similar to the criteria adopted in [17, 10], where SP39 was splitted into four subsets. Graph Matching to Derive Connectivity from Output Probabilities In the case where the bonded state of the cysteines is known, one has a graph with 2K nodes, one for each bonded cysteine. The weight associated with each edge is the probability that the corresponding bridge exists, as computed by the predictor. The problem is then to find a connectivity pattern with K edges and maximum weight, where each cysteine is paired uniquely with another cysteine. The maximum weight matching algorithm of Gabow [12] is used to chosen paired cysteines (edges), whose time complexity is cubic O(V 3) = O(K3), where V is the number of vertices and linear O(V ) = O(K) space complexity beyond the storage of the graph. Note that because the number of bonded cysteines in general is not very large, it is also possible in many cases to use an exhaustive search of all possible combinations. Indeed, the number of combinations is 1 × 3 × 5 × . . . (2K −1) which yields 945 connectivity patterns in the case of 10 bonded cysteines. The case where the bonded state of the cysteines is not known is slightly more involved and the Gabow algorithm cannot be applied directly since the graph has M nodes but, if some of the cysteines are not bonded, only a subset of 2K < M nodes participate in the final maximum weighted matching. Alternatively, we use a greedy algorithm to derive the connectivity pattern using the estimate of the total number of bonds. First, we order the edges in decreasing order of probabilities. Then we pick the edge with the highest probability. Then we pick the next edge with highest probability that is not incident to the first edge and so forth, until K edges have been selected. Because this greedy procedure is not guaranteed to find the global optimum, we find it useful to make it a little more robust by repeating L times. In each run i = 1, . . . , L, the first edge selected is the i-th most probable edge. In other words the different runs differ by the choice of the first edge, noting that in practice the optimal solution always contain one of the top L edges. This procedure works well in practice because the edges with largest probabilities tend to occur in the final pattern. For L reasonably large, the optimal connectivity pattern can usually be found. We have compared this method with Gabow’s algorithm in the case where the bonding state is known and observed that when L = 6, this greedy heuristic yields results that are as good as those obtained with Gabow’s algorithm which, in this case, is guaranteed to find a global optimum. The results reported here are obtained using the greedy procedure with L = 6. The advantage of the greedy algorithm is its low O(LM 2) complexity time. It is important to note that this method ends up by producing a prediction of both the connectivity pattern and of the bonding state of each cysteine. 3 Results Disulphide Connectivity Prediction for Bonded Cysteines Here we assume that the bonding state is known. We train 2D-RNN architectures using the SP39 data set to compare with other published results. We evaluate the performance using the precision P (P=TP/(TP+FP) with TP = true positives and FP = false positives) and recall R (R=TP/(TP+FN) with FN = false negatives). As shown in Table 1, in all but one case the results are superior to what has been previK Pair Precision Pattern Precision 2 0.74* (0.73) 0.74* (0.73) 3 0.61* (0.51) 0.51* (0.41) 4 0.44* (0.37) 0.27* (0.24) 5 0.41* (0.30) 0.11 (0.13) 2 . . . 5 0.56* (0.49) 0.49* (0.44) Table 1: Disulphide connectivity prediction with 2D-RNN assuming the bonding state is known. Last row reports performance on all test chains. * denote levels of precision that exceeds previously reported best results in the literature [17] (in parentheses). Figure 3: Correlation between number of bonded cysteines (2K) and qP i̸=j Oi,j log M. ously reported in the literature [17, 11]. In some cases, results are substantially better. For instance, in the case of 3 bonded cysteines, the precision reaches 0.61 and 0.51 at the pair and pattern levels, whereas the best similar results reported in the literature are 0.51 (pair) and 0.41 (pattern). Estimation of the Number K of Bonds Analysis of the prediction results shows that there is a relationship between the sum of all the probabilities,P i̸=j Oi,j, in the graph (or the output layer of the 2D-RNN) and the total number of bonded cysteines (2K). For instance, on one of the cross-validation training sets, the correlation coefficient between 2K and P i̸=j Oi,j is 0.7, the correlation coefficient between 2K and M is 0.68, and the correlation coefficient between 2K and qP i̸=j Oi,j log M is 0.72. As shown in Figure 3, there is a reasonably linear relationship between the total number 2K of bonded cysteines and the product qP i̸=j Oi,j log M , where M is the total number of cysteines in the sequence being considered. The slope and y-intercept for the line are respectively 0.66 and 3.01 on one training data set. Using this, we estimate the total number of bonded cysteines using linear regression and rounding off, making sure that the total number is even and does not exceed the total number of cysteines in the sequence. In the following experiments, the regression equation for predicting K is solved separately based on each cross-validation training set. K Pair Recall Pair Precision Pattern Precision 2 0.59 0.49 0.40 3 0.50 0.45 0.32 4 0.36 0.37 0.15 5 0.28 0.31 0.03 Table 2: Prediction of disulphide connectivity pattern with 2D-RNN on all the cysteines, without assuming knowledge of the bonding state. Disulphide Connectivity Prediction from Scratch In the last set of experiments, we do not assume any knowledge of the bonding state and apply the 2D-RNN approach to all the cysteines (both bonded and not bonded) in each sequence. We predict the number of bonds, the bonding state, and connectivity pattern using one predictor. Experiments are run both on SP39 (4-fold cross validation) and SP41 (10-fold cross validation). For lack of space, we cannot report all the results but, for example, precision and recall for SP39 are given in Table 2 for 2 ≤K ≤5. Table 3 shows the kind of results that are obtained when the method is applied to sequences with more than K = 5 bonds in SP41. The pair precision remains quite good, although the results can be noisy for certain values because there are not many such examples in the data. Finally, the precision of bonded state prediction is 0.85, and the recall of bonded state prediction is 0.9. The precision and recall of bond number prediction is 0.68. The average absolute difference between true bond and predicted bond number is 0.42. The average absolute difference between true bond number and wrongly predicted bond number is 1.3. K 6 7 8 9 10 11 12 15 16 17 18 19 24 Precision 0.41 0.40 0.34 0.37 0.5 0.4 0.17 0.37 0.57 0.40 0.56 0.42 0.24 Table 3: Prediction of disulphide connectivity pattern with 2D-RNN on all the cysteines, without assuming knowledge of the bonding state and when the number of bridges K exceeds 5. 4 Conclusion We have presented a complete system for disulphide connectivity prediction in cysteinerich proteins. Assuming knowledge of cysteine bonding state, the method outperforms existing approaches on the same validation data. The results also show that the 2D-RNN method achieves good recall and accuracy on the prediction of connectivity pattern even when the bonding state of individual cysteines is not known. Differently from previous approaches, our method can be applied to chains with K > 5 bonds and yields good, cooperative, predictions of the total number of bonds, as well as of the bonding states and bond locations. Training can take days but once trained predictions can be carried on a proteomic or protein engineering scale. Several improvements are currently in progress including (a) developing a classifier to discriminate protein chains that do not contain any disulphide bridges, using kernel methods; (b) assessing the effect on prediction of additional input information, such as secondary structure and solvent accessibility; (c) leveraging the predicted cysteine contacts in 3D protein structure prediction; and (d) curating a new larger training set. The current version of our disulphide prediction server DIpro (which includes step (a)) is available through: http://www.igb.uci.edu/servers/psss.html. Acknowledgments Work supported by an NIH grant, an NSF MRI grant, a grant from the University of California Systemwide Biotechnology Research and Education Program, and by the Institute for Genomics and Bioinformatics at UCI. References [1] V.I. Abkevich and E.I. Shankhnovich. What can disulfide bonds tell us about protein energetics, function and folding: simulations and bioinformatics analysis. J. Math. Biol., 300:975–985, 2000. [2] A. Bairoch and R. Apweiler. The SWISS-PROT protein sequence database and its supplement TrEMBL. Nucleic Acids Res., 28:45–48, 2000. [3] P. Baldi and G. Pollastri. Machine learning structural and functional proteomics. IEEE Intelligent Systems. Special Issue on Intelligent Systems in Biology, 17(2), 2002. [4] P. Baldi and G. Pollastri. The principled design of large-scale recursive neural network architectures–dag-rnns and the protein structure prediction problem. Journal of Machine Learning Research, 4:575–602, 2003. [5] P. Baldi and M. Rosen-Zvi. On the relationship between deterministic and probabilistic directed graphical models. 2004. Submitted. [6] H. M. Berman, J. Westbrook, Z. Feng, G. Gilliland, T. N. Bhat, H. Weissig, I. N. Shindyalov, and P. E. Bourne. The Protein Data Bank. Nucl. Acids Res., 28:235–242, 2000. [7] S. Betz. Disulfide bonds and the stability of globular proteins. Proteins, Struct., Function Genet., 21:167–195, 1993. [8] J. Clarke and A.R. Fersht. Engineered disulfide bonds as probes of the folding pathway of barnase - increasing stability of proteins against the rate of denaturation. Biochemistry, 32:4322– 4329, 1993. [9] L. Demetrius. Thermodynamics and kinetics of protein folding: an evolutionary perpective. J. Theor. Biol., 217:397–411, 2000. [10] P. Fariselli and R. Casadio. Prediction of disulfide connectivity in proteins. Bioinformatics, 17:957–964, 2001. [11] P. Fariselli, P. L. Martelli, and R. Casadio. A neural network-based method for predicting the disulfide connectivity in proteins. In E. Damiani et al., editors, Knowledge based intelligent information engineering systems and allied technologies (KES 2002), volume 1, pages 464– 468. IOS Press, 2002. [12] H.N. Gabow. An efficient implementation of Edmond’s algorithm for maximum weight matching on graphs. Journal of the ACM, 23(2):221–234, 1976. [13] J.L. Klepeis and C.A. Floudas. Prediction of β-sheet topology and disulfide bridges in polypeptides. J. Comput. Chem., 24:191–208, 2003. [14] T.A. Klink, K.J. Woycechosky, K.M. Taylor, and R.T. Raines. Contribution of disulfide bonds to the conformational stability and catalytic activity of ribonuclease A. Eur. J. Biochem., 267:566– 572, 2000. [15] M. Matsumura et al. Substantial increase of protein stability by multiple disulfide bonds. Nature, 342:291–293, 1989. [16] G. Pollastri, D. Przybylski, B. Rost, and P. Baldi. Improving the prediction of protein secondary structure in three and eight classes using recurrent neural networks and profiles. Proteins, 47:228–235, 2002. [17] A. Vullo and P. Frasconi. Disulfide connectivity prediction using recursive neural networks and evolutionary information. Bioinformatics, 20:653–659, 2004. [18] W.J. Wedemeyer, E. Welkler, M. Narayan, and H.A. Scheraga. Disulfide bonds and proteinfolding. Biochemistry, 39:4207–4216, 2000.
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An Application of Boosting to Graph Classification Taku Kudo, Eisaku Maeda NTT Communication Science Laboratories. 2-4 Hikaridai, Seika-cho, Soraku, Kyoto, Japan {taku,maeda}@cslab.kecl.ntt.co.jp Yuji Matsumoto Nara Institute of Science and Technology. 8916-5 Takayama-cho, Ikoma, Nara, Japan matsu@is.naist.jp Abstract This paper presents an application of Boosting for classifying labeled graphs, general structures for modeling a number of real-world data, such as chemical compounds, natural language texts, and bio sequences. The proposal consists of i) decision stumps that use subgraph as features, and ii) a Boosting algorithm in which subgraph-based decision stumps are used as weak learners. We also discuss the relation between our algorithm and SVMs with convolution kernels. Two experiments using natural language data and chemical compounds show that our method achieves comparable or even better performance than SVMs with convolution kernels as well as improves the testing efficiency. 1 Introduction Most machine learning (ML) algorithms assume that given instances are represented in numerical vectors. However, much real-world data is not represented as numerical vectors, but as more complicated structures, such as sequences, trees, or graphs. Examples include biological sequences (e.g., DNA and RNA), chemical compounds, natural language texts, and semi-structured data (e.g., XML and HTML documents). Kernel methods, such as support vector machines (SVMs) [11], provide an elegant solution to handling such structured data. In this approach, instances are implicitly mapped into a high-dimensional space, where information about their similarities (inner-products) is only used for constructing a hyperplane for classification. Recently, a number of kernels have been proposed for such structured data, such as sequences [7], trees [2, 5], and graphs [6]. Most are based on the idea that a feature vector is implicitly composed of the counts of substructures (e.g., subsequences, subtrees, subpaths, or subgraphs). Although kernel methods show remarkable performance, their implicit definitions of feature space make it difficult to know what kind of features (substructures) are relevant or which features are used in classifications. To use ML algorithms for data mining or as knowledge discovery tools, they must output a list of relevant features (substructures). This information may be useful not only for a detailed analysis of individual data but for the human decision-making process. In this paper, we present a new machine learning algorithm for classifying labeled graphs that has the following characteristics: 1) It performs learning and classification using the Figure 1: Labeled connected graphs and subgraph relation structural information of a given graph. 2) It uses a set of all subgraphs (bag-of-subgraphs) as a feature set without any constraints, which is essentially the same idea as a convolution kernel [4]. 3) Even though the size of the candidate feature set becomes quite large, it automatically selects a compact and relevant feature set based on Boosting. 2 Classifier for Graphs We first assume that an instance is represented in a labeled graph. The focused problem can be formalized as a general problem called the graph classification problem. The graph classification problem is to induce a mapping f(x) : X →{±1}, from given training examples T = {⟨xi, yi⟩}L i=1, where xi ∈X is a labeled graph and yi ∈{±1} is a class label associated with the training data. We here focus on the problem of binary classification. The important characteristic is that input example xi is represented not as a numerical feature vector but as a labeled graph. 2.1 Preliminaries In this paper we focus on undirected, labeled, and connected graphs, since we can easily extend our algorithm to directed or unlabeled graphs with minor modifications. Let us introduce a labeled connected graph (or simply a labeled graph), its definitions and notations. Definition 1 Labeled Connected Graph A labeled graph is represented in a 4-tuple G = (V, E, L, l), where V is a set of vertices, E ⊆V × V is a set of edges, L is a set of labels, and l : V ∪E →L is a mapping that assigns labels to the vertices and the edges. A labeled connected graph is a labeled graph such that there is a path between any pair of verticies. Definition 2 Subgraph Let G′ = (V ′, E′, L′, l′) and G = (V, E, L, l) be labeled connected graphs. G′ matches G, or G′ is a subgraph of G (G′ ⊆G) if the following conditions are satisfied: (1) V ′ ⊆V , (2) E′ ⊆E, (3) L′ ⊆L, and (4) l′ = l. If G′ is a subgraph of G, then G is a supergraph of G′. Figure 1 shows an example of a labeled graph and its subgraph and non-subgraph. 2.2 Decision Stumps Decision stumps are simple classifiers in which the final decision is made by a single hypothesis or feature. Boostexter [10] uses word-based decision stumps for text classification. To classify graphs, we define the subgraph-based decision stumps as follows. Definition 3 Decision Stumps for Graphs Let t and x be labeled graphs and y be a class label (y ∈{±1}). A decision stump classifier for graphs is given by h⟨t,y⟩(x) def = y t ⊆x −y otherwise. The parameter for classification is a tuple ⟨t, y⟩, hereafter referred to as a rule of decision stumps. The decision stumps are trained to find a rule ⟨ˆt, ˆy⟩that minimizes the error rate for the given training data T = {⟨xi, yi⟩}L i=1: ⟨ˆt, ˆy⟩= argmin t∈F,y∈{±1} 1 L L X i=1 I(yi ̸= h⟨t,y⟩(xi)) = argmin t∈F,y∈{±1} 1 2L L X i=1 (1 −yih⟨t,y⟩(xi)), (1) where F is a set of candidate graphs or a feature set (i.e., F = SL i=1{t|t ⊆xi}) and I(·) is the indicator function. The gain function for a rule ⟨t, y⟩is defined as gain(⟨t, y⟩) def = L X i=1 yih⟨t,y⟩(xi). (2) Using the gain, the search problem (1) becomes equivalent to the problem: ⟨ˆt, ˆy⟩= argmaxt∈F,y∈{±1} gain(⟨t, y⟩). In this paper, we use gain instead of error rate for clarity. 2.3 Applying Boosting The decision stump classifiers are too inaccurate to be applied to real applications, since the final decision relies on the existence of a single graph. However, accuracies can be boosted by the Boosting algorithm [3, 10]. Boosting repeatedly calls a given weak learner and finally produces a hypothesis f, which is a linear combination of K hypotheses produced by the weak learners, i,e.: f(x) = sgn(PK k=1 αkh⟨tk,yk⟩(x)). A weak learner is built at each iteration k with different distributions or weights d(k) = (d(k) i , . . . , d(k) L ) on the training data, where PL i=1 d(k) i = 1, d(k) i ≥0. The weights are calculated to concentrate more on hard examples than easy examples. To use decision stumps as the weak learner of Boosting, we redefine the gain function (2) as: gain(⟨t, y⟩) def = L X i=1 yidih⟨t,y⟩(xi). (3) In this paper, we use the AdaBoost algorithm, the original and the best known algorithm among many variants of Boosting. However, it is trivial to fit our decision stumps to other boosting algorithms, such as Arc-GV [1] and Boosting with soft margins [8]. 3 Efficient Computation In this section, we introduce an efficient and practical algorithm to find the optimal rule ⟨ˆt, ˆy⟩from given training data. This problem is formally defined as follows. Problem 1 Find Optimal Rule Let T = {⟨x1, y1, d1⟩, . . . , ⟨xL, yL, dL⟩} be training data where xi is a labeled graph, yi ∈{±1} is a class label associated with xi and di (PL i=1 di = 1, di ≥0) is a normalized weight assigned to xi. Given T, find the optimal rule ⟨ˆt, ˆy⟩that maximizes the gain, i.e., ⟨ˆt, ˆy⟩= argmaxt∈F,y∈{±1} diyih⟨t,y⟩, where F = SL i=1{t|t ⊆xi}. The most naive and exhaustive method in which we first enumerate all subgraphs F and then calculate the gains for all subgraphs is usually impractical, since the number of subgraphs is exponential to its size. We thus adopt an alternative strategy to avoid such exhaustive enumerations. The method to find the optimal rule is modeled as a variant of branch-and-bound algorithm and will be summarized as the following strategies: 1) Define Figure 2: Example of DFS Code Tree for a graph a canonical search space in which a whole set of subgraphs can be enumerated. 2) Find the optimal rule by traversing this search space. 3) Prune the search space by proposing a criteria for the upper bound of the gain. We will describe these steps more precisely in the next subsections. 3.1 Efficient Enumeration of Graphs Yan et al. proposed an efficient depth-first search algorithm to enumerate all subgraphs from a given graph [12]. The key idea of their algorithm is a DFS (depth first search) code, a lexicographic order to the sequence of edges. The search tree given by the DFS code is called a DFS Code Tree. Leaving the details to [12], the order of the DFS code is defined by the lexicographic order of labels as well as the topology of graphs. Figure 2 illustrates an example of a DFS Code Tree. Each node in this tree is represented in a 5-tuple [i, j, vi, eij, vj], where eij, vi and vj are the labels of i−j edge, i-th vertex, and j-th vertex respectively. By performing a pre-order search of the DFS Code Tree, we can obtain all the subgraphs of a graph in order of their DFS code. However, one cannot avoid isomorphic enumerations even giving pre-order traverse, since one graph can have several DFS codes in a DFS Code Tree. So, canonical DFS code (minimum DFS code) is defined as its first code in the pre-order search of the DFS Code Tree. Yan et al. show that two graphs G and G′ are isomorphic if and only if minimum DFS codes for the two graphs min(G) and min(G′) are the same. We can thus ignore non-minimum DFS codes in subgraph enumerations. In other words, in depth-first traverse, we can prune a node with DFS code c, if c is not minimum. The isomorphic graph represented in minimum code has already been enumerated in the depth-first traverse. For example, in Figure 2, if G1 is identical to G0, G0 has been discovered before the node for G1 is reached. This property allows us to avoid an explicit isomorphic test of the two graphs. 3.2 Upper bound of gain DFS Code Tree defines a canonical search space in which one can enumerate all subgraphs from a given set of graphs. We consider an upper bound of the gain that allows pruning of subspace in this canonical search space. The following lemma gives a convenient method of computing a tight upper bound on gain(⟨t′, y⟩) for any supergraph t′ of t. Lemma 1 Upper bound of the gain: µ(t) For any t′ ⊇t and y ∈{±1}, the gain of ⟨t′, y⟩is bounded by µ(t) (i.e., gain(⟨t′y⟩) ≤µ(t)), where µ(t) is given by µ(t) def = max 2 X {i|yi=+1,t⊆xi} di − L X i=1 yi · di, 2 X {i|yi=−1,t⊆xi} di + L X i=1 yi · di . Proof 1 gain(⟨t′, y⟩) = L X i=1 diyih⟨t′,y⟩(xi) = L X i=1 diyi · y · (2I(t′ ⊆xi) −1), where I(·) is the indicator function. If we focus on the case y = +1, then gain(⟨t′, +1⟩) = 2 X {i|t′⊆xi} yidi − L X i=1 yi · di ≤2 X {i|yi=+1,t′⊆xi} di − L X i=1 yi · di ≤ 2 X {i|yi=+1,t⊆xi} di − L X i=1 yi · di, since |{i|yi = +1, t′ ⊆xi}| ≤|{i|yi = +1, t ⊆xi}| for any t′ ⊇t. Similarly, gain(⟨t′, −1⟩) ≤ 2 X {i|yi=−1,t⊆xi} di + L X i=1 yi · di. Thus, for any t′ ⊇t and y ∈{±1}, gain(⟨t′, y⟩) ≤µ(t). 2 We can efficiently prune the DFS Code Tree using the upper bound of gain u(t). During pre-order traverse in a DFS Code Tree, we always maintain the temporally suboptimal gain τ among all the gains calculated previously. If µ(t) < τ, the gain of any supergraph t′ ⊇t is no greater than τ, and therefore we can safely prune the search space spanned from the subgraph t. If µ(t) ≥τ, then we cannot prune this space since a supergraph t′ ⊇t might exist such that gain(t′) ≥τ. 3.3 Efficient Computation in Boosting At each Boosting iteration, the suboptimal value τ is reset to 0. However, if we can calculate a tighter upper bound in advance, the search space can be pruned more effectively. For this purpose, a cache is used to maintain all rules found in the previous iterations. Suboptimal value τ is calculated by selecting one rule from the cache that maximizes the gain of the current distribution. This idea is based on our observation that a rule in the cache tends to be reused as the number of Boosting iterations increases. Furthermore, we also maintain the search space built by a DFS Code Tree as long as memory allows. This cache reduces duplicated constructions of a DFS Code Tree at each Boosting iteration. 4 Connection to Convolution Kernel Recent studies [1, 9, 8] have shown that both Boosting and SVMs [11] work according to similar strategies: constructing an optimal hypothesis that maximizes the smallest margin between positive and negative examples. The difference between the two algorithms is the metric of margin; the margin of Boosting is measured in l1-norm, while that of SVMs is measured in l2-norm. We describe how maximum margin properties are translated in the two algorithms. AdaBoost and Arc-GV asymptotically solve the following linear program, [1, 9, 8], max w∈IRJ,ρ∈IR+ ρ; s.t. yi J X j=1 wjhj(xi) ≥ρ, ||w||1 = 1 (4) where J is the number of hypotheses. Note that in the case of decision stumps for graphs, J = |{±1} × F| = 2|F|. SVMs, on the other hand, solve the following quadratic optimization problem [11]: 1 max w∈IRJ,ρ∈IR+ ρ; s.t. yi · (w · Φ(xi)) ≥ρ, ||w||2 = 1. (5) 1For simplicity, we omit the bias term (b) and the extension of Soft Margin. The function Φ(x) maps the original input example x into a J-dimensional feature vector (i.e., Φ(x) ∈IRJ). The l2-norm margin gives the separating hyperplane expressed by dotproducts in feature space. The feature space in SVMs is thus expressed implicitly by using a Marcer kernel function, which is a generalized dot-product between two objects, (i.e., K(x1, x2) = Φ(x1) · Φ(x2)). The best known kernel for modeling structured data is a convolution kernel [4] (e.g., string kernel [7] and tree kernel [2, 5]), which argues that a feature vector is implicitly composed of the counts of substructures. 2 The implicit mapping defined by the convolution kernel is given as: Φ(x) = (#(t1 ⊆x), . . . , #(t|F| ⊆x)), where tj ∈F and #(u) is the cardinality of u. Noticing that a decision stump can be expressed as h⟨t,y⟩(x) = y·(2I(t ⊆ x) −1), we see that the constraints or feature space of Boosting with substructure-based decision stumps are essentially the same as those of SVMs with the convolution kernel 3. The critical difference is the definition of margin: Boosting uses l1-norm, and SVMs use l2-norm. The difference between them can be explained by sparseness. It is well known that the solution or separating hyperplane of SVMs is expressed in a linear combination of training examples using coefficients λ, (i.e., w = PL i=1 λiΦ(xi)) [11]. Maximizing l2-norm margin gives a sparse solution in the example space, (i.e., most of λi becomes 0). Examples having non-zero coefficients are called support vectors that form the final solution. Boosting, in contrast, performs the computation explicitly in feature space. The concept behind Boosting is that only a few hypotheses are needed to express the final solution. l1-norm margin realizes such a property [8]. Boosting thus finds a sparse solution in the feature space. The accuracies of these two methods depend on the given training data. However, we argue that Boosting has the following practical advantages. First, sparse hypotheses allow the construction of an efficient classification algorithm. The complexity of SVMs with tree kernel is O(l|n1||n2|), where n1 and n2 are trees, and l is the number of support vectors, which is too heavy to be applied to real applications. Boosting, in contrast, performs faster since the complexity depends only on a small number of decision stumps. Second, sparse hypotheses are useful in practice as they provide “transparent” models with which we can analyze how the model performs or what kind of features are useful. It is difficult to give such analysis with kernel methods since they define feature space implicitly. 5 Experiments and Discussion To evaluate our algorithm, we employed two experiments using two real-world data. (1) Cellphone review classification (REV) The goal of this task is to classify reviews for cellphones as positive or negative. 5,741 sentences were collected from an Web-BBS discussion about cellphones in which users were directed to submit positive reviews separately from negative reviews. Each sentence is represented in a word-based dependency tree using a Japanese dependency parser CaboCha4. (2) Toxicology prediction of chemical compounds (PTC) The task is to classify chemical compounds by carcinogenicity. We used the PTC data set5 consisting of 417 compounds with 4 types of test animals: male mouse (MM), female 2Strictly speaking, graph kernel [6] is not a convolution kernel because it is not based on the count of subgraphs, but on random walks in a graph. 3The difference between decision stumps and the convolution kernels is that the former uses a binary feature denoting the existence (or absence) of each substructure, whereas the latter uses the cardinality of each substructure. However, it makes little difference since a given graph is often sparse and the cardinality of substructures will be approximated by their existence. 4http://chasen.naist.jp/˜ taku/software/cabocha/ 5http://www.predictive-toxicology.org/ptc/ Table 1: Classification F-scores of the REV and PTC tasks REV PTC MM FM MR FR Boosting BOL-based Decision Stumps 76.6 47.0 52.9 42.7 26.9 Subgraph-based Decision Stumps 79.0 48.9 52.5 55.1 48.5 SVMs BOL Kernel 77.2 40.9 39.9 43.9 21.8 Tree/Graph Kernel 79.4 42.3 34.1 53.2 25.9 mouse (FM), male rat (MR) and female rat (FR). Each compound is assigned one of the following labels: {EE,IS,E,CE,SE,P,NE,N}. We here assume that CE,SE, and P are “positive” and that NE and NN are “negative”, which is exactly the same setting as [6]. We thus have four binary classifiers (MM/FM/MR/FR) in this data set. We compared the performance of our Boosting algorithm and support vector machines with tree kernel [2, 5] (for REV) and graph kernel [6] (for PTC) according to their F-score in 5-fold cross validation. Table 1 summarizes the best results of REV and PCT task, varying the hyperparameters of Boosting and SVMs (e.g., maximum iteration of Boosting, soft margin parameter of SVMs, and termination probability of random walks in graph kernel [6]). We also show the results with bag-of-label (BOL) features as a baseline. In most tasks and categories, ML algorithms with structural features outperform the baseline systems (BOL). These results support our first intuition that structural features are important for the classification of structured data, such as natural language texts and chemical compounds. Comparing our Boosting algorithm with SVMs using tree kernel, no significant difference can be found the REV data set. However, in the PTC task, our method outperforms SVMs using graph kernel on the categories MM, FM, and FR at a statistically significant level. Furthermore, the number of active features (subgraphs) used in Boosting is much smaller than those of SVMs. With our methods, about 1800 and 50 features (subgraphs) are used in the REV and PTC tasks respectively, while the potential number of features is quite large. Even giving all subgraphs as feature candidates, Boosting selects a small and highly relevant subset of features. Figure 3 show an example of extracted support features (subgraphs) in the REV and PTC task respectively. In the REV task, features reflecting the domain knowledge (cellphone reviews) are extracted: 1) “want to use ”→positive, 2) “hard to use”→negative, 3) “recharging time is short” →positive, 4) “recharging time is long” →negative. These features are interesting because we cannot determine the correct label (positive/negative) only using such bag-of-label features as “charging,” “short,” or “long.” In the PTC task, similar structures show different behavior. For instance, Trihalomethanes (TTHMs), wellknown carcinogenic substances (e.g., chloroform, bromodichloromethane, and chlorodibromomethane), contain the common substructure H-C-Cl (Fig. 3(a)). However, TTHMs do not contain the similar but different structure H-C(C)-Cl (Fig. 3(b)). Such structural information is useful for analyzing how the system classifies the input data in a category and what kind of features are used in the classification. We cannot examine such analysis in kernel methods, since they define their feature space implicitly. The reason why graph kernel shows poor performance on the PTC data set is that it cannot identify subtle difference between two graphs because it is based on a random walks in a graph. For example, kernel dot-product between the similar but different structures 3(c) and 3(d) becomes quite large, although they show different behavior. To classify chemical compounds by their functions, the system must be capable of capturing subtle differences among given graphs. The testing speed of our Boosting algorithm is also much faster than SVMs with tree/graph Figure 3: Support features and their weights kernels. In the REV task, the speed of Boosting and SVMs are 0.135 sec./1,149 instances and 57.91 sec./1,149 instances respectively6. Our method is significantly faster than SVMs with tree/graph kernels without a discernible loss of accuracy. 6 Conclusions In this paper, we focused on an algorithm for the classification of labeled graphs. The proposal consists of i) decision stumps that use subtrees as features, and ii) a Boosting algorithm in which subgraph-based decision stumps are applied as the weak learners. Two experiments are employed to confirm the importance of subgraph features. In addition, we experimentally show that our Boosting algorithm is accurate and efficient for classification tasks involving discrete structural features. References [1] Leo Breiman. Prediction games and arching algoritms. Neural Computation, 11(7):1493 – 1518, 1999. [2] Michael Collins and Nigel Duffy. Convolution kernels for natural language. In NIPS 14, Vol.1, pages 625–632, 2001. [3] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. Journal of Computer and System Sicences, 55(1):119–139, 1996. [4] David Haussler. Convolution kernels on discrete structures. Technical report, UC Santa Cruz (UCS-CRL-99-10), 1999. [5] Hisashi Kashima and Teruo Koyanagi. Svm kernels for semi-structured data. In Proc. of ICML, pages 291–298, 2002. [6] Hisashi Kashima, Koji Tsuda, and Akihiro Inokuchi. Marginalized kernels between labeled graphs. In Proc. of ICML, pages 321–328, 2003. [7] Huma Lodhi, Craig Saunders, John Shawe-Taylor, Nello Cristianini, and Chris Watkins. Text classification using string kernels. Journal of Machine Learning Research, 2, 2002. [8] Gunnar. R¨atsch, Takashi. Onoda, and Klaus-Robert M¨uller. Soft margins for AdaBoost. Machine Learning, 42(3):287–320, 2001. [9] Robert E. Schapire, Yoav Freund, Peter Bartlett, and Wee Sun Lee. Boosting the margin: a new explanation for the effectiveness of voting methods. In Proc. of ICML, pages 322–330, 1997. [10] Robert E. Schapire and Yoram Singer. BoosTexter: A boosting-based system for text categorization. Machine Learning, 39(2/3):135–168, 2000. [11] Vladimir N. Vapnik. Statistical Learning Theory. Wiley-Interscience, 1998. [12] Xifeng Yan and Jiawei Han. gspan: Graph-based substructure pattern mining. In Proc. of ICDM, pages 721–724, 2002. 6We tested the performances on Linux with XEON 2.4Ghz dual processors.
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Synergistic Face Detection and Pose Estimation with Energy-Based Models Margarita Osadchy NEC Labs America Princeton NJ 08540 rita@osadchy.net Matthew L. Miller NEC Labs America Princeton NJ 08540 mlm@nec-labs.com Yann Le Cun The Courant Institute New York University yann@cs.nyu.edu 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 = θ, 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. Fi(θ) = cos(θ −αi); i = 1, 2, 3; θ = [−π 2 , π 2 ] (1) 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: θ = arctan P3 i=1 Gi(X) cos(αi) P3 i=1 Gi(X) sin(αi) (2) The point on the manifold closest to G(X) is just F(θ). 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(θ, φ) = cos(θ −αi) cos(φ −βj); 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: cc = P ij Gij(X) cos(αi) cos(βj); cs = P ij Gij(X) cos(αi) sin(βj) sc = P ij Gij(X) sin(αi) cos(βj); ss = P ij Gij(X) sin(αi) sin(βj) θ = 0.5(atan2(cs + sc, cc −ss) + atan2(sc −cs, cc + ss)) φ = 0.5(atan2(cs + sc, cc −ss) −atan2(sc −cs, cc + ss)) (4) 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′, X) for any pose Z′, and EW (1, Z′, X) ≫EW (1, Z, X) for any pose Z′ ̸= 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) = ∥GW (X) −F(Z)∥ (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 ∥GW (X) −F(Z)∥+ (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, ∥GW (X) −F(Z)∥, 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: L(W) = 1 |S1| X i∈S1 L1(W, Zi, Xi) + 1 |S0| X i∈S0 L0(W, Xi) (8) Figure 1: Architecture of the Minimum Energy Machine. where S1is the set of training faces, S0the set of non-faces, L1(W, Zi, Xi) and L0(W, Xi) 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 (Xi, Zi, 1), will make EW (1, Zi, Xi) < EW (Y, Z, Xi) for Y ̸= Y i or Z ̸= Zi. To satisfy this, it is sufficient to make EW (1, Zi, Xi) < EW (0, Z, Xi). For a particular negative training sample (Xi, 0), minimizing the loss should make EW (1, Z, Xi) > EW (0, Z, Xi) = T for any Z. To satisfy this, it is sufficient to make EW (1, Z, Xi) > T. Let W be the current parameter value, and W ′ 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 ′(1, Z, X) < EW (1, Z, X) For a non-face example (X, 1), we must have: EW ′(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 is of the form: δW = W ′ −W = −ηA ∂L ∂W . where A is a judiciously chosen symmetric positive semi-definite matrix, and η is a small positive constant. For Y = 1 (face): An update step will change the parameter by δW = −ηA ∂EW (1,Z,X)2 ∂W = −2ηEW (1, Z, X)A ∂EW (1,Z,X) ∂W . To first order (for small values of η), the resulting change in EW (1, Z, X) is given by: ∂EW (1, Z, X) ∂W T δW = −2ηEW (1, Z, X) ∂EW (1, Z, X) ∂W T A∂EW (1, Z, X) ∂W < 0 because EW (1, Z, X) > 0 (it’s a distance), and the quadratic form is positive. Therefore EW ′(1, Z, X) < EW (1, Z, X). 1Although face samples whose pose is unknown can easily be accommodated, we will not discuss this possibility here. 0 2 4 6 8 10 12 14 16 18 20 50 55 60 65 70 75 80 85 90 95 100 False positive rate Percentage of faces detected Detection only Pose + detection 0 5 10 15 20 25 30 50 55 60 65 70 75 80 85 90 95 100 Yaw-error tolerance (degrees) Percentage of yaws correctly estimated Pose only Pose + detection 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 δW = −ηA ∂K exp[−E(1,Z,X)] ∂W = ηK exp[−EW (1, Z, X)] ∂EW (1,Z,X) ∂W . To first order (for small values of η), the resulting change in EW (1, Z, X) is given by: ∂EW (1, Z, X) ∂W T δW = ηK exp[−EW (1, Z, X)] ∂EW (1, Z, X) ∂W T A∂EW (1, Z, X) ∂W > 0 Therefore EW ′(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 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 50 55 60 65 70 75 80 85 90 95 100 False positives per image Percentage of faces detected Profile Frontal Rotated in plane 0 5 10 15 20 25 30 50 55 60 65 70 75 80 85 90 95 100 Pose-error tolerance (degrees) Percentage of poses correctly estimated Yaw In-plane rotation 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 → TILTED PROFILE MIT+CMU False positives per image → 4.42 26.90 .47 3.36 .50 1.28 Our detector 90% 97% 67% 83% 83% 88% Jones & Viola [5] (tilted) 90% 95% x x Jones & Viola [5] (profile) x 70% 83% x Rowley et al [11] 89% 96% x x Schneiderman & Kanade [13] x 86% 93% 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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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 adjusting 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 ξi(t) of agent i at time t (ξi(t) = ±1), the payoff is −ξi(t)G(A(t)), where A(t) ≡P i ξi(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 α (α = 1, . . . , s −1) of agent i with respect to her sth policy have random biases ωiα. 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 ρ ≡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 σ 2/N of the population making decision +, and is given by σ2 N ≡N 4 ⟨[Aµ∗(t)(t) −⟨Aµ∗(t)(t)⟩t]2⟩t, (1) where ⟨ ⟩t denotes time average at the steady state. Its dependence on the diversity is 10 −6 10 −2 10 2 10 6 ρ 10 −4 10 0 10 4 σ 2/N N = 127 N = 511 N = 2047 N = 8191 N = 32767 10 2 10 4 10 6 ρ 10 −4 10 −3 10 −2 σ 2/N N = 127 N = 511 N = 2047 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 ρ ∼N −1 and σ2/N ∼N, the behavior of the variance is dominated by the scaling relation σ2/N ∼ρ−1 for ρ ∼1. To interpret this result, we describe the macroscopic dynamics of the system by de£ning the D-dimensional vector Aµ(t), which is the sum of the decisions of all agents responding to history µ of their policies, normalized by N. While only one of the D components corresponds to the historical state µ∗(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 Aµ(t) when µ is the historical state µ∗(t). We let Sαβ(ω) be the number of agents holding policies α and β (with α < β), and the bias of α with respect to β is ω. If the cumulative payoff of policy α at time t is Ωα(t), then the agents holding policies α and β make decisions according to policy α if ω + Ωα(t) −Ωβ(t) > 0, and policy β otherwise. Hence ω + Ωα(t) −Ωβ(t) is referred to as the preference of α with respect to β. At time t, the cumulative payoff of policy α changes from Ωα(t) to Ωα(t) −ξµ αsignAµ(t), where ξµ α is the decision of policy α at state µ. Only the £ckle agents, that is, those agents with preferences on the verge of switching signs, contribute to the change in Aµ(t), namely, ω + Ωα(t) −Ωβ(t) = ±1 and ξµ α −ξµ β = ±2signAµ(t). Hence we have Aµ(t + 1) −Aµ(t) = −signAµ(t) 2 N X α<β X r=±1 Sαβ(r −Ωα(t) + Ωβ(t)) ×δ(ξµ α −ξµ β −2rsignAµ(t)) (2) where δ(n) = 1 if n = 0, and 0 otherwise. In the region where D ≪ln N, we have P 8 3 Q 1 Prob = R 8 1 Prob = Prob = Prob = 8 3 A− 8 A+ −3001 −1 3001 preference 0.0 0.2 0.4 0.6 average number of agents 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 ρ = N = 1, 023 and averaged over 100,000 samples of initial conditions. Sαβ(ω) ≫1, and Eq. (2) is self-averaging. Following the derivation in [5], we arrive at Aµ(t + 1) = Aµ(t) −signAµ(t) r 2 πRδ(µ −µ∗(t)). (3) Equation (3) shows that the dynamics proceeds in the direction which reduces the magnitude of the population vector, each time by a step of size p 2/πR. 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 the zero value. As a result, each state is con£ned in a D-dimensional hypercube of size p 2/πR, irrespective of the initial position of the population vector. This con£nement enables us to compute the variance of the decisions, given by σ2/N = f(ρ)/2πρ, where f(ρ) is a smooth function of ρ, which approaches (1 −1/4D)/3 for ρ ≫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 σ2/N as ρ−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), σ2/N deviates above the scaling with ρ−1 when ρ ∼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 ξµ α = µ and its complementary policy ξµ β = −µ, depends on the preference ω+Ωα−Ωβ, 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 ω + Ωα −Ωβ = −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 ω + Ωα −Ωβ 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 ρ ∼N, Aµ(t + 1) −Aµ(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 at time t, which is Poisson distributed with a mean of N/ √ 2πR = ∆/2, where ∆≡N p 2/πR is the average step size. However, since the attractor is formed by steps which reverse the sign of Aµ, 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 ∆Aµ > 0 for all µ). Assuming that all states of the phase space are equally likely to be accessed, we have Patt(∆A) = P 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 along each axis Aµ. The sign reversal condition implies that Patt(∆A, A) ∝PPoi(∆A) Q µ Θ[−Aµ(Aµ + ∆Aµ)], where Θ(x) is the step function of x, and PPoi(∆A) is the Poisson distribution of step sizes, yielding Patt(∆A) ∝PPoi(∆A) Q µ ∆Aµ. We note that the extra factors of ∆Aµ favor larger step sizes. Thus, the attractor averages ⟨(∆A±)2⟩att are given by ⟨(∆A±)2⟩att = ⟨(∆A±)2∆A+∆A−⟩Poi ⟨∆A+∆A−⟩Poi . (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 ξ+ α −ξ+ β = −2r and ξ− α −ξ− β = 2r respectively. Among the agents contributing to ∆A+, the extra requirement of ξ− α −ξ− β = 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, ⟨∆A+∆A−⟩Poi = 4 N2 P a0,a+,a− e−∆ 8 a0! ¡ ∆ 8 ¢a0 e−3∆ 8 a+! ¡ 3∆ 8 ¢a+ e−3∆ 8 a−! ¡ 3∆ 8 ¢a− (a0 + a+)(a0 + a−). (5) Together with similar expressions of the numerator in Eq. (4), we obtain ⟨(∆A±)2⟩att = 2∆3 + 15∆2 + 20∆+ 4 N 2(2∆+ 1) . (6) The attractor states are given by Aµ = mµ/N and mµ/N −∆Aµ, where mµ = 1, 3, . . . , N∆Aµ −1. This yields a variance of σ2 N = 7⟨(N∆A+)2⟩att + 7⟨(N∆A−)2⟩att −8 192N , (7) which gives, on combining with Eq. (6), σ2 N = 14∆3 + 105∆2 + 132∆+ 24 96N(2∆+ 1) . (8) When the diversity is low, ∆≫1, and Eq. (8) reduces to σ2/N = 7/48πρ, agreeing with the scaling result of the previous section. When ρ ∼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), σ2/N further deviates above the predictions of kinetic sampling when ρ ≫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 ω + Ωα −Ωβ = ±1. Since there are fewer and fewer £ckle agents in the limit ρ ≫N, one would expect that a single agent of this type would dominate the game dynamics, and σ2/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 α can be written as Ωα + ξ+ α at P, Ωα, . . . , Ωα −kξ− α at Q and, in the attractor of period 4, repeating the sequence of Ωα −kξ− α −ξ− α at R, Ωα −kξ− α at Q, Ωα −kξ− α + ξ+ α at P, and Ωα −kξ− α at Q. The movement of the cumulative payoffs can be conveniently represented by writing Ωα = P µ kµξµ α, where kµ denotes the number of wins minus losses of decision 1 at state µ 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 ⟨1 2[(∆A+)2+(∆A−)2]⟩att = X j Pj ÃP i=0,1⟨1 2[(∆A+)2 + (∆A−)2]∆A+∆A−⟩i,j P i=0,1⟨∆A+∆A−⟩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 ⟨∆A+∆A−⟩for one-wait 0 400 800 1200 Time −2000 −1000 0 1000 2000 Preference 1st agent 2nd agent QQ QQ QQ PQ QR QP 1 − k + k r 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 ⟨∆A+∆A−⟩1,arm = 4 N2 P∞ k=1⟨[1 −δ(ai)δ(aturn,1)δ(acum)]δ(aturn,1) Qk r=1 δ(await,r)δ(aturn,2)(a−+ a0)(acum + aturn,2 + a0)⟩Poi = 4 N2 1 1−e−∆ 2 n e−∆ 2 h 12 ¡ ∆ 8 ¢2 + ∆ 8 i −e−7∆ 8 h 4 ¡ ∆ 8 ¢2 + ∆ 8 io . (10) 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 Table 1: The number of £ckle agents in the steps of one wait. Label Steps No. of agents Poisson averages PQ Ωα + ξ+ α →Ωα ai + aturn,1 + acum ⟨ai⟩= ∆/8, ⟨aturn,1⟩= ∆/8, ⟨acum⟩= ∆/4. QQ1 Ωα →Ωα −ξ− α await,1 + aturn,1 ⟨await,1⟩= 3∆/8. QQr Ωα −(r −1)ξ− α await,r ⟨await,r⟩= ∆/2, →Ωα −rξ− α (2 ≤r ≤k −1). QQk Ωα −(k −1)ξ− α await,k + aturn,2 ⟨await,k⟩= 3∆/8, →Ωα −kξ− α ⟨aturn,2⟩= ∆/8. QR Ωα −kξ− α → a−+ a0 ⟨a−⟩= 3∆/8, Ωα −(k + 1)ξ− α ⟨a0⟩= ∆/8. QP Ωα −kξ− α → acum + aturn,2 + a0 Ωα −kξ− α + ξ+ α 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. Weiß 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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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 Introduction 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. Au X = (1) ) ( ) ( iu P u P ∏ = (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 2nd-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. ( ) q u c u P λ λ / exp ) | ( − = (3) Bv = ] log[λ (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 1st 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. ( ) ∏∑∏ ∏∑ ∏ − = = Π Λ = Π Λ N n K k M m m k m n m k k N n K k k n k N n n u u P u P U P , , , exp 2 1 ) | ( ) , | ( ) , | ( λ λ π λ π r r r (5) ) ,,, , ( , 2 , 1, M n n n n u u u u = r : n-th data sample, ) ,,, ,,, , ( 2 1 N i u u u u U r r r r = ) ,..., , ( , 2 , 1, M k k k k λ λ λ λ = r : Variance of k-th Laplacian distribution, ) ,,, ,,, , ( 2 1 K k λ λ λ λ r r r r = Λ k π : probability of Laplacian distribution k, ) ,,, ( 1 K π π = Π and 1 = ∑k k π 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), ( ) ∏∏ ∏ ∏ − ⋅ = Π Λ = Π Λ N n K k m m k m n n k z m k z k N n n u z Z u P Z U P n k n k , , , exp 2 1 ) , | , ( ) , | , ( λ λ π r (6) n kz : Hidden binary random variable, 1 if n-th data sample is generated from k-th Laplacian, 0 other wise. ( ( ) n kz Z = and 1 = ∑ k n kz for all n = 1…N) 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). ∑ ∑ − + = Π Λ k n m m k m n m k n k k n k u z z Z U P , , , , ) 2 1 log( ) log( ) , | , ( log λ λ π (7) { } { } ∑ ∑ − + = Π Λ k n m m k m n m k k n k u z E Z U P E , , , , ) 2 1 log( ) log( ) , | , ( log λ λ π (8) The expectation in Eq.(8) can be evaluated, if we are given the data U and estimated parameters Λ and Π. For Λ and Π, EM algorithm uses current estimation Λ’ and Π’. { } { } ∏ ∏ ∑ − = ⋅ − Π Λ = Π Λ Π Λ = Π Λ = = Π Λ = = Π Λ = Π Λ ≡ = M m m k m n m k k n M m k m k m n m k n n n k n k n n n k z n n k n k n k n k u c u u P u P z P z u P u z P u z P z U z E z E n k )' exp( ' 2 ' 1 ' )' exp( ' 2 1 )' ,' | ( 1 )' ,' | ( )' ,' |1 ( )' ,' ,1 | ( )' ,' , |1 ( )' ,' , | ( ' ,' , | , , , , , , 1 0 λ λ π π λ λ (9) Where the normalization constant can be computed as ( ) ∑∏ ∑ = = − = Π Λ Π Λ = Π Λ = K k M m m k m n m k k K k n k n k n n n u z P z u P u P c 1 1 , , , ) exp( 2 1 )' ,' | ( )' ,' , | ( )' ,' | ( λ λ π (10) The EM algorithm works by maximizing Eq.(8), given the expectation computed from Eq.(9,10). Eq.(9,10) can be computed using Λ’ and Π’ 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 Λ and Π. First, we can maximize Eq.(8) with respect to Λ, by setting the derivative as 0. { } { } { } { } ∑ ∑ ∑ ⋅ = ⇒ = + − = ∂ Π Λ ∂ n n k n m n n k m k n m k m n m k n k m k z E u z E u z E Z U P E , , 2 , , , , 0 ) ( 1 ) , | , ( log λ λ λ λ (11) Second, for maximization of Eq.(8) with respect to Π, we can rewrite Eq.(8) as below. { } { } ∑ + = Π Λ ' , ' ' ) log( ) , | , ( log k n k n kz E C Z U P E π (12) As we see, the derivative of Eq.(12) with respect to Π cannot be 0. Instead, we need to use Lagrange multiplier method for maximization. A Lagrange function can be defined as Eq.(14) where ρ is a Lagrange multiplier. { } )1 ( ) log( ) , ( ' ' ' , ' ' − + − = Π ∑ ∑ k k k n k n kz E L π ρ π ρ (13) By setting the derivative of Eq.(13) to be 0 with respect to ρ and Π, we can simply get the maximization solution with respect to Π. We just show the solution in Eq.(14). { } { } = ⇒ = Π ∂ Π ∂ = ∂ Π ∂ ∑∑ ∑ k n n k n n k k z E z E L L / 0 ) , ( ,0 ) , ( π ρ ρ ρ (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). 1. Initialize K k 1 = π , { } e u E m m k + = , λ (e is small random noise) 2. Calculate the Expectation by { } { } ∏ − = Π Λ ≡ M m m k m n m k k n n k n k u c U z E z E )' exp( ' 2 ' 1 ' ,' , | , , , λ λ π 3. Maximize the log likelihood given the Expectation { } { } ⋅ ← ∑ ∑ n n k n m n n k m k z E u z E / , , λ , { } { } ← ∑∑ ∑ k n n k n n k k z E z E / π 4. If (converged) stop, otherwise repeat from step 2. 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 1st 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 2nd 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 non- linearly 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. n Au X + = (15) ( )) ( log ) , | ( log ) , | ( log ˆ u P A u X P argmax A X u P argmax u u u + = = (16) 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) Figure 7: SNR and SSIM for 3 different algorithms (signal variance = 1.0) 4 Discussion 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 high- level structures, which are highly correlated with the underlying physical properties 0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 0.8 1 SSIM Index Noise variance ICA MAP(Mixture prior) ICA MAP(Laplacian prior) W iener Noisy Image 0 0.5 1 1.5 2 2.5 2 4 6 8 10 12 14 16 SNR Noise variance ICA MAP(Mixture prior) ICA MAP(Laplacian prior) Wiener 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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Dependent Gaussian Processes Phillip Boyle and Marcus Frean School of Mathematical and Computing Sciences Victoria University of Wellington, Wellington, New Zealand {pkboyle,marcus}@mcs.vuw.ac.nz Abstract Gaussian processes are usually parameterised in terms of their covariance functions. However, this makes it difficult to deal with multiple outputs, because ensuring that the covariance matrix is positive definite is problematic. An alternative formulation is to treat Gaussian processes as white noise sources convolved with smoothing kernels, and to parameterise the kernel instead. Using this, we extend Gaussian processes to handle multiple, coupled outputs. 1 Introduction Gaussian process regression has many desirable properties, such as ease of obtaining and expressing uncertainty in predictions, the ability to capture a wide variety of behaviour through a simple parameterisation, and a natural Bayesian interpretation [15, 4, 9]. Because of this they have been suggested as replacements for supervised neural networks in non-linear regression [8, 18], extended to handle classification tasks [11, 17, 6], and used in a variety of other ways (e.g. [16, 14]). A Gaussian process (GP), as a set of jointly Gaussian random variables, is completely characterised by a covariance matrix with entries determined by a covariance function. Traditionally, such models have been specified by parameterising the covariance function (i.e. a function specifying the covariance of output values given any two input vectors). In general this needs to be a positive definite function to ensure positive definiteness of the covariance matrix. Most GP implementations model only a single output variable. Attempts to handle multiple outputs generally involve using an independent model for each output - a method known as multi-kriging [18] - but such models cannot capture the structure in outputs that covary. As an example, consider the two tightly coupled outputs shown at the top of Figure 2, in which one output is simply a shifted version of the other. Here we have detailed knowledge of output 1, but sampling of output 2 is sparse. A model that treats the outputs as independent cannot exploit their obvious similarity - intuitively, we should make predictions about output 2 using what we learn from both output 1 and 2. Joint predictions are possible (e.g. co-kriging [3]) but are problematic in that it is not clear how covariance functions should be defined [5]. Although there are many known positive definite autocovariance functions (e.g. Gaussians and many others [1, 9]), it is difficult to define cross-covariance functions that result in positive definite covariance matrices. Contrast this to neural network modelling, where the handling of multiple outputs is routine. An alternative to directly parameterising covariance functions is to treat GPs as the outputs of stable linear filters. For a linear filter, the output in response to an input x(t) is y(t) = h(t) ⋆x(t) = R ∞ −∞h(t −τ)x(τ)dτ, where h(t) defines the impulse response of the filter and ⋆denotes convolution. Provided the linear filter is stable and x(t) is Gaussian white noise, the output process y(t) is necessarily a Gaussian process. It is also possible to characterise p-dimensional stable linear filters, with M-inputs and N-outputs, by a set of M × N impulse responses. In general, the resulting N outputs are dependent Gaussian processes. Now we can model multiple dependent outputs by parameterising the set of impulse responses for a multiple output linear filter, and inferring the parameter values from data that we observe. Instead of specifying and parameterising positive definite covariance functions, we now specify and parameterise impulse responses. The only restriction is that the filter be linear and stable, and this is achieved by requiring the impulse responses to be absolutely integrable. Constructing GPs by stimulating linear filters with Gaussian noise is equivalent to constructing GPs through kernel convolutions. A Gaussian process V (s) can be constructed over a region S by convolving a continuous white noise process X(s) with a smoothing kernel h(s), V (s) = h(s) ⋆X(s) for s ∈S, [7]. To this can be added a second white noise source, representing measurement uncertainty, and together this gives a model for observations Y . This view of GPs is shown in graphical form in Figure 1(a). The convolution approach has been used to formulate flexible nonstationary covariance functions [13, 12]. Furthermore, this idea can be extended to model multiple dependent output processes by assuming a single common latent process [7]. For example, two dependent processes V1(s) and V2(s) are constructed from a shared dependence on X(s) for s ∈S0, as follows V1(s) = Z S0∪S1 h1(s −λ)X(λ)dλ and V2(s) = Z S0∪S2 h2(s −λ)X(λ)dλ where S = S0 ∪S1 ∪S2 is a union of disjoint subspaces. V1(s) is dependent on X(s), s ∈ S1 but not X(s), s ∈S2. Similarly, V2(s) is dependent on X(s), s ∈S2 but not X(s), s ∈ S1. This allows V1(s) and V2(s) to possess independent components. In this paper, we model multiple outputs somewhat differently to [7]. Instead of assuming a single latent process defined over a union of subspaces, we assume multiple latent processes, each defined over ℜp. Some outputs may be dependent through a shared reliance on common latent processes, and some outputs may possess unique, independent features through a connection to a latent process that affects no other output. 2 Two Dependent Outputs Consider two outputs Y1(s) and Y2(s) over a region ℜp, where s ∈ℜp. We have N1 observations of output 1 and N2 observations of output 2, giving us data D1 = {s1,i , y1,i}N1 i=1 and D2 = {s2,i , y2,i}N2 i=1. We wish to learn a model from the combined data D = {D1, D2} in order to predict Y1(s′) or Y2(s′), for s′ ∈ℜp. As shown in Figure 1(b), we can model each output as the linear sum of three stationary Gaussian processes. One of these (V ) arises from a noise source unique to that output, under convolution with a kernel h. A second (U) is similar, but arises from a separate noise source X0 that influences both outputs (although via different kernels, k). The third is additive noise as before. Thus we have Yi(s) = Ui(s) + Vi(s) + Wi(s), where Wi(s) is a stationary Gaussian white noise process with variance, σ2 i , X0(s), X1(s) and X2(s) are independent stationary Gaussian white noise processes, U1(s), U2(s), V1(s) and V2(s) are Gaussian processes given by Ui(s) = ki(s) ⋆X0(s) and Vi(s) = hi(s) ⋆Xi(s). Figure 1: (a) Gaussian process prior for a single output. The output Y is the sum of two Gaussian white noise processes, one of which has been convolved (⋆) with a kernel (h). (b) The model for two dependent outputs Y1 and Y2. All of X0, X1, X2 and the “noise” contributions are independent Gaussian white noise sources. Notice that if X0 is forced to zero Y1 and Y2 become independent processes as in (a) - we use this as a control model. The k1, k2, h1, h2 are parameterised Gaussian kernels where k1(s) = v1 exp −1 2sT A1s , k2(s) = v2 exp −1 2(s −µ)T A2(s −µ) , and hi(s) = wi exp −1 2sT Bis . Note that k2(s) is offset from zero by µ to allow modelling of outputs that are coupled and translated relative to one another. We wish to derive the set of functions CY ij (d) that define the autocovariance (i = j) and cross-covariance (i ̸= j) between the outputs i and j, for a given separation d between arbitrary inputs sa and sb. By solving a convolution integral, CY ij (d) can be expressed in a closed form [2], and is fully determined by the parameters of the Gaussian kernels and the noise variances σ2 1 and σ2 2 as follows: CY 11(d) = CU 11(d) + CV 11(d) + δabσ2 1 CY 12(d) = CU 12(d) CY 22(d) = CU 22(d) + CV 22(d) + δabσ2 2 CY 21(d) = CU 21(d) where CU ii (d) = π p 2 v2 i p |Ai| exp −1 4 dT Aid CU 12(d) = (2π) p 2 v1v2 p |A1 + A2| exp −1 2(d −µ)T Σ(d −µ) CU 21(d) = (2π) p 2 v1v2 p |A1 + A2| exp −1 2(d + µ)T Σ(d + µ) = CU 12(−d) CV ii (d) = π p 2 w2 i p |Bi| exp −1 4 dT Bid with Σ = A1(A1 + A2)−1A2 = A2(A1 + A2)−1A1. Given CY ij (d) then, we can construct the covariance matrices C11, C12, C21, and C22 as follows Cij = CY ij (si,1 −sj,1) · · · CY ij (si,1 −sj,Nj) ... ... ... CY ij (si,Ni −sj,1) · · · CY ij (si,Ni −sj,Nj) (1) Together these define the positive definite symmetric covariance matrix C for the combined output data D: C = C11 C12 C21 C22 (2) We define a set of hyperparameters Θ that parameterise {v1, v2, w1, w2, A1, A2, B1, B2, µ, σ1, σ2}. Now, we can calculate the likelihood L = −1 2 log C −1 2 yT C−1y −N1 + N2 2 log 2π where yT = [y1,1 · · · y1,N1 y2,1 · · · y2,N2] and C is a function of Θ and D. Learning a model now corresponds to either maximising the likelihood L, or maximising the posterior probability P(Θ | D). Alternatively, we can simulate the predictive distribution for y by taking samples from the joint P(y, Θ | D), using Markov Chain Monte Carlo methods [10]. The predictive distribution at a point s′ on output i given Θ and D is Gaussian with mean ˆy′ and variance σ2 ˆy′ given by ˆy′ = kT C−1y and σ2 ˆy′ = κ −kT C−1k where κ = CY ii(0) = v2 i + w2 i + σ2 i and k = CY i1(s′ −s1,1) . . . CY i1(s′ −s1,N1) CY i2(s′ −s2,1) . . . CY i2(s′ −s2,N2) T 2.1 Example 1 - Strongly dependent outputs over 1d input space Consider two outputs, observed over a 1d input space. Let Ai = exp(fi), Bi = exp(gi), and σi = exp(βi). Our hyperparameters are Θ = {v1, v2, w1, w2, f1, f2, g1, g2, µ, β1, β2} where each element of Θ is a scalar. As in [2] we set Gaussian priors over Θ. We generated N = 48 data points by taking N1 = 32 samples from output 1 and N2 = 16 samples from output 2. The samples from output 1 were linearly spaced in the interval [−1, 1] and those from output 2 were uniformly spaced in the region [−1, −0.15]∪[0.65, 1]. All samples were taken under additive Gaussian noise, σ = 0.025. To build our model, we maximised P(Θ|D) ∝P(D | Θ) P(Θ) using a multistart conjugate gradient algorithm, with 5 starts, sampling from P(Θ) for initial conditions. The resulting dependent model is shown in Figure 2 along with an independent (control) model with no coupling (see Figure 1). Observe that the dependent model has learned the coupling and translation between the outputs, and has filled in output 2 where samples are missing. The control model cannot achieve such infilling as it is consists of two independent Gaussian processes. 2.2 Example 2 - Strongly dependent outputs over 2d input space Consider two outputs, observed over a 2d input space. Let Ai = 1 α2 i I Bi = 1 τ 2 i I where I is the identity matrix. Furthermore, let σi = exp(βi). In this toy example, we set µ = 0, so our hyperparameters become Θ = {v1, v2, w1, w2, α1, α2, τ1, τ2β1, β2} where each element of Θ is a scalar. Again, we set Gaussian priors over Θ. −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 True function Model mean Output 1 − independent model −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Output 2 − independent model −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Output 1 − dependent model −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 Output 2 − dependent model Figure 2: Strongly dependent outputs where output 2 is simply a translated version of output 1, with independent Gaussian noise, σ = 0.025. The solid lines represent the model, the dotted lines are the true function, and the dots are samples. The shaded regions represent 1σ error bars for the model prediction. (top) Independent model of the two outputs. (bottom) Dependent model. We generated 117 data points by taking 81 samples from output 1 and 36 samples from output 2. Both sets of samples formed uniform lattices over the region [−0.9, 0.9]⊗[−0.9, 0.9] and were taken with additive Gaussian noise, σ = 0.025. To build our model, we maximised P(Θ|D) as before. The dependent model is shown in Figure 3 along with an independent control model. The dependent model has filled in output 2 where samples are missing. Again, the control model cannot achieve such in-filling as it is consists of two independent Gaussian processes. 3 Time Series Forecasting Consider the observation of multiple time series, where some of the series lead or predict the others. We simulated a set of three time series for 100 steps each (figure 4) where series 3 was positively coupled to a lagged version of series 1 (lag = 0.5) and negatively coupled to a lagged version of series 2 (lag = 0.6). Given the 300 observations, we built a dependent GP model of the three time series and compared them with independent GP models. The dependent GP model incorporated a prior belief that series 3 was coupled to series 1 and 2, with the lags unknown. The independent GP model assumed no coupling between its outputs, and consisted of three independent GP models. We queried the models for forecasts of the future 10 values of series 3. It is clear from figure 4 that the dependent GP model does a far better job at forecasting the dependent series 3. The independent model becomes inaccurate after just a few time steps into the future. This inaccuracy is expected as knowledge of series 1 and 2 is required to accurately predict series 3. The Figure 3: Strongly dependent outputs where output 2 is simply a copy of output 1, with independent Gaussian noise. (top) Independent model of the two outputs. (bottom) Dependent model. Output 1 is modelled well by both models. Output 2 is modelled well only by the dependent model dependent GP model performs well as it has learned that series 3 is positively coupled to a lagged version of series 1 and negatively coupled to a lagged version of series 2. 4 Multiple Outputs and Non-stationary Kernels The convolution framework described here for constructing GPs can be extended to build models capable of modelling N-outputs, each defined over a p-dimensional input space. In general, we can define a model where we assume M-independent Gaussian white noise processes X1(s) . . . XM(s), N-outputs U1(s) . . . UN(s), and M × N kernels {{kmn(s)}M m=1}N n=1 where s ∈ℜp. The autocovariance (i = j) and cross-covariance (i ̸= j) functions between output processes i and j become CU ij(d) = M X m=1 Z ℜp kmi(s)kmj(s + d)ds (3) and the matrix defined by equation 2 is extended in the obvious way. The kernels used in (3) need not be Gaussian, and need not be spatially invariant, or stationary. We require kernels that are absolutely integrable, R ∞ −∞. . . R ∞ −∞|k(s)|dps < ∞. This provides a large degree of flexibility, and is an easy condition to uphold. It would seem that an absolutely integrable kernel would be easier to define and parameterise than a positive definite function. On the other hand, we require a closed form of CY ij (d) and this may not be attainable for some non-Gaussian kernels. −3 −2 −1 0 1 2 Series 1 −3 −2 −1 0 1 2 Series 2 0 1 2 3 4 5 6 7 8 −3 −2 −1 0 1 2 Series 3 Figure 4: Three coupled time series, where series 1 and series 2 predict series 3. Forecasting for series 3 begins after 100 time steps where t = 7.8. The dependent model forecast is shown with a solid line, and the independent (control) forecast is shown with a broken line. The dependent model does a far better job at forecasting the next 10 steps of series 3 (black dots). 5 Conclusion We have shown how the Gaussian Process framework can be extended to multiple output variables without assuming them to be independent. Multiple processes can be handled by inferring convolution kernels instead of covariance functions. This makes it easy to construct the required positive definite covariance matrices for covarying outputs. One application of this work is to learn the spatial translations between outputs. However the framework developed here is more general than this, as it can model data that arises from multiple sources, only some of which are shared. Our examples show the infilling of sparsely sampled regions that becomes possible in a model that permits coupling between outputs. Another application is the forecasting of dependent time series. Our example shows how learning couplings between multiple time series may aid in forecasting, particularly when the series to be forecast is dependent on previous or current values of other series. Dependent Gaussian processes should be particularly valuable in cases where one output is expensive to sample, but covaries strongly with a second that is cheap. By inferring both the coupling and the independent aspects of the data, the cheap observations can be used as a proxy for the expensive ones. References [1] ABRAHAMSEN, P. A review of gaussian random fields and correlation functions. Tech. Rep. 917, Norwegian Computing Center, Box 114, Blindern, N-0314 Oslo, Norway, 1997. [2] BOYLE, P., AND FREAN, M. Multiple-output gaussian process regression. Tech. rep., Victoria University of Wellington, 2004. [3] CRESSIE, N. Statistics for Spatial Data. Wiley, 1993. [4] GIBBS, M. Bayesian Gaussian Processes for Classification and Regression. PhD thesis, University of Cambridge, Cambridge, U.K., 1997. [5] GIBBS, M., AND MACKAY, D. J. Efficient implementation of gaussian processes. www.inference.phy.cam.ac.uk/mackay/abstracts/gpros.html, 1996. [6] GIBBS, M. N., AND MACKAY, D. J. Variational gaussian process classifiers. IEEE Trans. on Neural Networks 11, 6 (2000), 1458–1464. [7] HIGDON, D. Space and space-time modelling using process convolutions. In Quantitative methods for current environmental issues (2002), C. Anderson, V. Barnett, P. Chatwin, and A. El-Shaarawi, Eds., Springer Verlag, pp. 37–56. [8] MACKAY, D. J. Gaussian processes: A replacement for supervised neural networks? In NIPS97 Tutorial, 1997. [9] MACKAY, D. J. Information theory, inference, and learning algorithms. Cambridge University Press, 2003. [10] NEAL, R. Probabilistic inference using markov chain monte carlo methods. Tech. Report CRG-TR-93-1, Dept. of Computer Science, Univ. of Toronto, 1993. [11] NEAL, R. Monte carlo implementation of gaussian process models for bayesian regression and classification. Tech. Rep. CRG-TR-97-2, Dept. of Computer Science, Univ. of Toronto, 1997. [12] PACIOREK, C. Nonstationary Gaussian processes for regression and spatial modelling. PhD thesis, Carnegie Mellon University, Pittsburgh, Pennsylvania, U.S.A., 2003. [13] PACIOREK, C., AND SCHERVISH, M. Nonstationary covariance functions for gaussian process regression. Submitted to NIPS, 2004. [14] RASMUSSEN, C., AND KUSS, M. Gaussian processes in reinforcement learning. In Advances in Neural Information Processing Systems (2004), vol. 16. [15] RASMUSSEN, C. E. Evaluation of Gaussian Processes and other methods for Non-Linear Regression. PhD thesis, Graduate Department of Computer Science, University of Toronto, 1996. [16] TIPPING, M. E., AND BISHOP, C. M. Bayesian image super-resolution. In Advances in Neural Information Processing Systems (2002), S. Becker S., Thrun and K. Obermayer, Eds., vol. 15, pp. 1303 – 1310. [17] WILLIAMS, C. K., AND BARBER, D. Bayesian classification with gaussian processes. IEEE trans. Pattern Analysis and Machine Intelligence 20, 12 (1998), 1342 – 1351. [18] WILLIAMS, C. K., AND RASMUSSEN, C. E. Gaussian processes for regression. In Advances in Neural Information Processing Systems (1996), D. Touretzsky, M. Mozer, and M. Hasselmo, Eds., vol. 8.
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Synchronization of neural networks by mutual learning and its application to cryptography Einat Klein Department of Physics Bar-Ilan University Ramat-Gan, 52900 Israel Rachel Mislovaty Department of Physics Bar-Ilan University Ramat-Gan, 52900 Israel Ido Kanter Department of Physics Bar-Ilan University Ramat-Gan, 52900 Israel Andreas Ruttor Institut f¨ur Theoretische Physik, Universit¨at W¨urzbur Am Hubland 97074 W¨urzburg, Germany Wolfgang Kinzel Institut f¨ur Theoretische Physik, Universit¨at W¨urzbur Am Hubland 97074 W¨urzburg, Germany Abstract Two neural networks that are trained on their mutual output synchronize to an identical time dependant weight vector. This novel phenomenon can be used for creation of a secure cryptographic secret-key using a public channel. Several models for this cryptographic system have been suggested, and have been tested for their security under different sophisticated attack strategies. The most promising models are networks that involve chaos synchronization. The synchronization process of mutual learning is described analytically using statistical physics methods. 1 Introduction Neural networks learn from examples. This concept has extensively been investigated using models and methods of statistical mechanics [1, 2]. A ”teacher” network is presenting input/output pairs of high dimensional data, and a ”student” network is being trained on these data. Training means, that synaptic weights adapt by simple rules to the i/o pairs. When the networks — teacher as well as student — have N weights, the training process needs of the order of N examples to obtain generalization abilities. This means, that after the training phase the student has achieved some overlap to the teacher, their weight vectors are correlated. As a consequence, the student can classify an input pattern which does not belong to the training set. The average classification error decreases with the number of training examples. Training can be performed in two different modes: Batch and on-line training. In the first case all examples are stored and used to minimize the total training error. In the second case only one new example is used per time step and then destroyed. Therefore on-line training may be considered as a dynamic process: at each time step the teacher creates a new example which the student uses to change its weights by a tiny amount. In fact, for random input vectors and in the limit N →∞, learning and generalization can be described by ordinary differential equations for a few order parameters [3]. w σ σ w x Figure 1: Two perceptrons receive an identical input x and learn their mutual output bits σ. On-line training is a dynamic process where the examples are generated by a static network - the teacher. The student tries to move towards the teacher. However, the student network itself can generate examples on which it is trained. What happens if two neural networks learn from each other? In the following section an analytic solution is presented [6], which shows a novel phenomenon: synchronization by mutual learning. The biological consequences of this phenomenon are not explored, yet, but we found an interesting application in cryptography: secure generation of a secret key over a public channel. In the field of cryptography, one is interested in methods to transmit secret messages between two partners A and B. An attacker E who is able to listen to the communication should not be able to recover the secret message. In 1976, Diffie and Hellmann found a method based on number theory for creating a secret key over a public channel accessible to any attacker[7]. Here we show how neural networks can produce a common secret key by exchanging bits over a public channel and by learning from each other. 2 Mutual Learning We start by presenting the process of mutual learning for a simple network: Two perceptrons receive a common random input vector x and change their weights w according to their mutual bit σ, as sketched in Fig. 1. The output bit σ of a single perceptron is given by the equation σ = sign(w · x) (1) x is an N-dimensional input vector with components which are drawn from a Gaussian with mean 0 and variance 1. w is a N-dimensional weight vector with continuous components which are normalized, w · w = 1 (2) The initial state is a random choice of the components wA/B i , i = 1, ...N for the two weight vectors wA and wB. At each training step a common random input vector is presented to the two networks which generate two output bits σA and σB according to (1). Now the weight vectors are updated by the perceptron learning rule [3]: wA(t + 1) = wA(t) + η N xσB Θ(−σAσB) wB(t + 1) = wB(t) + η N xσA Θ(−σAσB) (3) Θ(x) is the step function. Hence, only if the two perceptrons disagree a training step is performed with a learning rate η. After each step (3), the two weight vectors have to be normalized. In the limit N →∞, the overlap R(t) = wA(t) · wB(t) (4) 0 0.5 1 1.5 2 η −1 −0.5 0 0.5 1 cos(θ) theory simulation ηc cos(θ)c Figure 2: Final overlap R between two perceptrons as a function of learning rate η. Above a critical rate ηc the time dependent networks are synchronized. From Ref. [6] has been calculated analytically [6]. The number of training steps t is scaled as α = t/N, and R(α) follows the equation dR dα = (R + 1) Ãr 2 π η(1 −R) −η2 ϕ π ! (5) where ϕ is the angle between the two weight vectors wA and wB, i.e. R = cos ϕ. This equation has fixed points R = 1, R = −1, and η √ 2π = 1 −cos ϕ ϕ (6) Fig. 2 shows the attractive fixed point of (5) as a function of the learning rate η. For small values of η the two networks relax to a state of a mutual agreement, R →1 for η →0. With increasing learning rate η the angle between the two weight vectors increases up to ϕ = 133◦for η →ηc ∼= 1.816 (7) Above the critical rate ηc the networks relax to a state of complete disagreement, ϕ = 180◦, R = −1. The two weight vectors are antiparallel to each other, wA = −wB. As a consequence, the analytic solution shows, well supported by numerical simulations for N = 100, that two neural networks can synchronize to each other by mutual learning. Both networks are trained to the examples generated by their partner and finally obtain an antiparallel alignment. Even after synchronization the networks keep moving, the motion is a kind of random walk on an N-dimensional hypersphere producing a rather complex bit sequence of output bits σA = −σB [8]. 3 Random walk in weight space We want to apply synchronization of neural networks to cryptography. In the previous section we have seen that the weight vectors of two perceptrons learning from each other can synchronize. The new idea is to use the common weights wA = −wB as a key for encryption [11]. But two issues have to be solved yet: (i) Can an external observer, recording the exchange of bits, calculate the final wA(t) ? The essence of using mutual learning as an encryption tool is the fact that while the parties preform a mutual process in which they react towards one another, the attacker preforms a learning process, in which the ’teacher’ does not react towards him. (ii) Does this phenomenon exist for discrete weights? Since communication is usually based on bit sequences, this is an important practical issue. Both issues are discussed below. Synchronization occurs for normalized weights, unnormalized ones do not synchronize [6]. Therefore, for discrete weights, we introduce a restriction in the space of possible vectors and limit the components wA/B i to 2L + 1 different values, wA/B i ∈{−L, −L + 1, ..., L −1, L} (8) In order to obtain synchronization to a parallel – instead of an antiparallel – state wA = wB, we modify the learning rule (3) to: wA(t + 1) = wA(t) −xσAΘ(σAσB) wB(t + 1) = wB(t) −xσBΘ(σAσB) (9) Now the components of the random input vector x are binary xi ∈{+1, −1}. If the two networks produce an identical output bit σA = σB, then their weights move one step in the direction of −xiσA. But the weights should remain in the interval (8), therefore if any component moves out of this interval, |wi| = L+1, it is set back to the boundary wi = ±L. Each component of the weight vectors performs a kind of random walk with reflecting boundary. Two corresponding components wA i and wB i receive the same random number ±1. After each hit at the boundary the distance |wA i −wB i | is reduced until it has reached zero. For two perceptrons with a N-dimensional weight space we have two ensembles of N random walks on the interval {−L, ..., L}. We expect that after some characteristic time scale τ = O(L2) the probability of two random walks being in different states decreases as P(t) ∼P(0)e−t/τ. Hence the total synchronization time should be given by N · P(t) ≃ 1 which gives tsync ∼τ ln N. In fact, our simulations show the synchronization time increases logarithmically with N. 4 Mutual Learning in the Tree Parity Machine A single perceptron transmits too much information. An attacker, who knows the set of input/output pairs, can derive the weights of the two partners. On one hand, the information should be hidden so that the attacker does not calculate the weights, but on the other hand enough information should be transmitted so that the two partners can synchronize. We found that multilayer networks with hidden units may be candidates for such a task [11]. More precisely, we consider a Tree Parity Machine(TPM), with three hidden units as shown in Fig. 3. 13 1 2 1 1 2 ... N 2 1 ... N 2 2 N ... 1 1 Figure 3: A tree parity machine with K = 3 Each hidden unit is a perceptron (1) with discrete weights (8). The output bit τ of the total network is the product of the three bits of the hidden units τ A = σA 1 σA 2 σA 3 τ B = σB 1 σB 2 σB 3 (10) At each training step the two machines A and B receive identical input vectors x1, x2, x3. The training algorithm is the following: Only if the two output bits are identical, τ A = τ B, the weights can be changed. In this case, only the hidden unit σi which is identical to τ changes its weights using the Hebbian rule wA i (t + 1) = wA i (t) −xiτ A (11) The partner as well as any attacker does not know which one of the K weight vectors is updated. The partners A and B react to their mutual output and move signals τ A and τ B, whereas an attacker can only receive these signals but not influence the partners with its own output bit. This is the essential mechanism which allows synchronization but prohibits learning. Nevertheless, advanced attackers use different heuristics to accelerate their synchronization, as described in the next section. 5 Attackers The following are possible attack strategies, which were suggested by Shamir et al.[12]: The Genetic Attack, in which a large population of attackers is trained, and every new time step each attacker is multiplied to cover the 2K−1 possible internal representations of {σi} for the current output τ. As dynamics proceeds successful attackers stay while the unsuccessful are removed. The Probabilistic Attack, in which the attacker tries to follow the probability of every weight element by calculating the distribution of the local field of every input and using the output, which is publicly known. The Naive Attacker, in which the attacker imitates one of the parties. More successful is the Flipping Attack strategy, in which the attacker imitates one of the parties, but in steps in which his output disagrees with the imitated party’s output, he negates (”flips”) the sign of one of his hidden units. The unit most likely to be wrong is the one with the minimal absolute value of the local field, therefore that is the unit which is flipped. While the synchronization time increases with L2[15], the probability of finding a successful flipping-attacker decreases exponentially with L, P ∝e−yL as seen in Figure 4. Therefore, for large L values the system is secure[15]. Every time step, the parties either appraoch each other (”attractive step” or drift apart (”repulsive step”). Close to synchronization the probability for a repulsive step in the mutual learning between A and B scales like (ϵ)2, while in the dynamic learning between the naive attacker C and A it scales like ϵ, where we define ϵ = Prob ¡ σC i ̸= σA i ¢ [18]. It has been shown that among a group of Ising vector students which perform learning, and have an overlap R with the teacher, the best student is the center of mass vector (which was shown to be an Ising vector as well) which has an overlap Rcm ∝ √ R , for R ∈[0 : 1][19]. Therefore letting a group of attackers cooperate throughout the process may be to their advantage. The most successful attack strategy, the “Majority Flipping Attacker” uses a group of attackers as a cooperating group rather than as individuals. When updating the weights, instead of each attacker being updated according to its own result, all are updated according to the majority’s result. This “team-work” approach improves the attacker’s performance. When using the majority scheme, the probability for a successful attacker seems to approach a constant value ∼0.5 independent of L. 0.001 0.01 0.1 1 0 2 4 6 8 10 12 L P Flipping attack Majority-Flipping attack P = 1.55 exp( -0.4335 L ) Figure 4: The attacker’s success probability P as a function of L, for the flipping attack and the majority-flipping attack, with N=1000, M=100, averaged over 1000 samples. To avoid fluctuations, we define the attacker successful if he found out 98% of the weights 6 Analytical description The semi-analytical description of this process gives us further insight to the synchronization process of mutual and dynamic learning. The study of discrete networks requires different methods of analysis than those used for the continuous case. We found that instead of examining the evolution of R and Q, we must examine (2L + 1) × (2L +1) parameters, which describe the mutual learning process. By writing a Markovian process that describes the development of these parameters, one gains an insight into the learning procedure. Thus we define a (2L + 1) × (2L + 1) matrix, Fµ, in which the state of the machines in the time step µ is represented. The elements of F, are fqr, where q, r = −L, ... −1, 0, 1, ...L. The element fqr represents the fraction of components in a weight vector in which the A’s components are equal to q and the matching components in d unit B are equal to r. Hence, the overlap between the two units as well as their norms are defined through this matrix, R = L X q,r=−L qrfqr, QA = L X q=−L q2fqrQB = L X r=−L r2fqr (12) The updating of matrix elements is described as follows: for the elements with q and r which are not on the boundary, (q ̸= ±L and r ̸= ±L) the update can be written in a simple manner, f + q,r = θ (pα −ϵ) fq,r + θ (ϵ −pα) µ1 2fq+1,r−1 + 1 2fq−1,r+1 ¶ . (13) Our results indicate that the order parameters are not self-averaged quantities [16]. Several runs with the same N, results in different curves for the order parameters as a function of the number of steps, see Figure 5. This explains the non-zero variance of ρ as a results of the fluctuations in the local fields induced by the input even in the thermodynamic limit. 7 Combining neural networks and chaos synchronization Two chaotic system starting from different initial conditions can be synchronized by different kinds of couplings between them. This chaotic synchronization can been used in neural 0 5 10 15 20 # steps -1 -0.8 -0.6 -0.4 -0.2 0 <ρ> 0 20 40 60 80 100 # steps -1 -0.8 -0.6 -0.4 -0.2 0 <ρ> Figure 5: The averaged overlap ⟨ρ⟩and its standard deviation as a function of the number of steps as found from the analytical results (solid line) and simulation results (circles) of mutual learning in TPMs. Inset: analytical results (solid line) and simulation results (circles) results for the perceptron, with L = 1 and N = 104. cryptography to enhance the cryptographic systems and to improve their security. A model which combines a TPM and logistic maps and is hereby presented, was shown to be more secure than the TPM discussed above. Other models which use mutual synchronization of networks whose dynamics are those of the Lorenz system are now under research and seem very promising. In the following system we combine neural networks with logistic maps: Both partners A and B use their neural networks as input for the logistic maps which generate the output bits to be learned. By mutually learning these bits, the two neural networks approach each other and produce an identical signal to the chaotic maps which – in turn – synchronize as well, therefore accelerating the synchronization of the neural nets. Previously, the output bit of each hidden unit was the sign of the local field[11]. Now we combine the PM with chaotic synchronization by feeding the local fields into logistic maps: sk(t + 1) = λ(1 −β)sk(t)(1 −sk(t)) + β 2 ˜hk(t) (14) Here ˜h denotes a transformed local field which is shifted and normalized to fit into the interval [0, 2]. For β = 0 one has the usual quadratic iteration which produces K chaotic series sk(t) when the parameter λ is chosen correspondingly; here we use λ = 3.95. For 0 < β < 1 the logistic maps are coupled to the fields of the hidden units. It has been shown that such a coupling leads to chaotic synchronization[17]: If two identical maps with different initial conditions are coupled to a common external signal they synchronize when the coupling strength is large enough, β > βc. The security of key generation increases as the system approaches the critical point of chaotic synchronization. The probability of a successful attack decreases like exp(−yL) and it is possible that the exponent y diverges as the coupling constant between the neural nets and the chaotic maps is tuned to be critical. 8 Conclusions A new phenomenon has been observed: Synchronization by mutual learning. If the learning rate η is large enough, and if the weight vectors keep normalized, then the two networks relax to a parallel orientation. Their weight vectors still move like a random walk on a hypersphere, but each network has complete knowledge about its partner. It has been shown how this phenomenon can be used for cryptography. The two partners can create a common secret key over a public channel. The fact that the parties are learning mutually, gives them an advantage over the attacker who is learning one-way. In contrast to number theoretical methods the networks are very fast; essentially they are linear filters, the complexity to generate a key of length N scales with N (for sequential update of the weights). Yet sophisticated attackers which use ensembles of cooperating attackers have a good chance to synchronize. However, advanced algorithms for synchronization, which involve different types of chaotic synchronization seem to be more secure. Such models are subjects of active research, and only the future will tell whether the security of neural network cryptography can compete with number theoretical methods. References [1] J. Hertz, A. Krogh, and R. G. Palmer: Introduction to the Theory of Neural Computation, (Addison Wesley, Redwood City, 1991) [2] A. Engel, and C. Van den Broeck: Statistical Mechanics of Learning, (Cambridge University Press, 2001) [3] M. Biehl and N. Caticha: Statistical Mechanics of On-line Learning and Generalization, The Handbook of Brain Theory and Neural Networks, ed. by M. A. Arbib (MIT Press, Berlin 2001) [4] E. Eisenstein, I. Kanter, D.A. Kessler and W. Kinzel, Phys. Rev. Lett. 74, 6-9 (1995) [5] I. Kanter, D.A. Kessler, A. Priel and E. Eisenstein, Phys. Rev. Lett. 75, 2614-2617 (1995);L. Ein-Dor and I. Kanter, Phys. Rev. E 57, 6564 (1998);M. Schr¨oder and W. Kinzel, J. Phys. A 31, 9131-9147 (1998); A. Priel and I. Kanter, Europhys. Lett.(2000) [6] R. Metzler and W. Kinzel and I. Kanter, Phys. Rev. E 62, 2555 (2000) [7] D. R. Stinson, Cryptography: Theory and Practice (CRC Press 1995) [8] R. Metzler, W. Kinzel, L. Ein-Dor and I. Kanter, Phys. Rev. E 63, 056126 (2001) [9] M. Rosen-Zvi, I. Kanter and W. Kinzel, cond-mat/0202350 (2002) [10] R. Urbanczik, private communication [11] I. Kanter, W. Kinzel and E. Kanter, Europhys. Lett., 57, 141 (2002). [12] A.Klimov, A. Mityagin, A. Shamir, ASIACRYPT 2002 : 288-298. [13] W. Kinzel, R. Metzler and I. Kanter, J. Phys. A. 33 L141 (2000). [14] W. Kinzel, Contribution to Networks, ed. by H. G. Schuster and S. Bornholdt, to be published by Wiley VCH (2002). [15] R Mislovaty, Y. Perchenok, I. Kanter and W. Kinzel, Phys. Rev. E 66, 066102 (2002). [16] G. Reents and R. Urbanczik, Phys. Rev. Lett., 80, 5445 (1998). [17] R. Mislovaty, E. Klein, I. Kanter and W. Kinzel, Phys. Rev. Lett. 91, 118701 (2003). [18] M. Rosen-Zvi, E. Klein, I. Kanter and W. Kinzel, Phys. Rev. E 66 066135 (2002). [19] M. Copelli, M. Boutin, C. Van Der Broeck and B. Van Rompaey, Europhys. Lett., 46, 139 (1999).
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Semigroup Kernels on Finite Sets Marco Cuturi Computational Biology Group Ecole des Mines de Paris 35 rue Saint Honor´e 77300 Fontainebleau marco.cuturi@ensmp.fr Jean-Philippe Vert Computational Biology Group Ecole des Mines de Paris 35 rue Saint Honor´e 77300 Fontainebleau jean-philippe.vert@ensmp.fr 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 τ which maps any z ∈Z into a finite unordered list of elements of X, τ(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 τ-lists. Most recent approaches to compare two τ-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 τ(z) and τ(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 τ-lists comparison that studies the subspaces generated by points of τ(z) and τ(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 τ-lists, but the motivation remains the same as in [7]: if two τ-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 κ 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 µx with respect to a common measure (e.g. the Borel uniform measure), with finite entropy h(µ) def = − R X µ ln µ. Consider for example a Gaussian distribution with mean x and fixed variance. A natural way to represent an unordered list τ(z) = [x1, x2, ..., xn] ∈X n is by the density µτ = 1/n Pn i=1 µxi. In that case, a p.d. kernel k between unordered lists τ and τ ′ that only depends on their concatenation τ(z) · τ(z′) is equivalent to a p.d. kernel between densities µ and µ′ that only depends on µ+µ′. Hence we are looking for a p.d. kernel on the set P of probability densities of finite entropy of the form κ(µ, µ′) = ϕ(µ + µ′). An example of such a kernel is provided in the following proposition. Recall that a negative definite (n.d.) kernel on a set X is a symmetric function g : X2 →R that satisfies Pn i,j=1 cicjg(xi, xj) ≤0 for any n ∈N, (x1, . . . , xn) ∈Xn, and (c1 . . . , cn) ∈Rn with Pn 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 : µ, µ′ 7→h( µ+µ′ 2 ) is negative definite on P, making kh(µ, µ′) def = e−th( µ+µ′ 2 ) a p.d. kernel on P for any t > 0. We call kh the entropy kernel between two measures. The entropy kernel is already a satisfactory answer to our initial motivation to look at merger of points. Observe that if µx is a probability density around x, then µτ can often be thought of as an estimate of the distribution of the points in τ, and (µτ + µτ ′)/2 is an estimate of the distribution of the points enumerated in τ · τ ′. If the latter estimate has a small entropy we can guess that the points in τ and τ ′ 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 h( f+f ′ 2 ) = 1 2h(f + f ′) + ln 2 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 ϕ : S 7→R is called a positive definite (resp. negative definite) function on the semigroup (S, ◦) if (s, t) 7→ϕ(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 measures M b +(X) [8]. For µ ∈M b +(X), we note |µ| = µ(X) < +∞. For a Borel measurable function f ∈RX , we note µ[f] = R X fdµ. Endowed with the usual Abelian addition between measures, (M b +(X), +) is an Abelian semigroup. The reason to consider this semigroup is that there is a natural semigroup homomorphism between finite lists of points and elements of M b +(X) given by τ = [x1, ..., xn] 7→µτ = Pn i=1 µxi, where µx ∈M b +(X) is an arbitrary finite measure associated with each x ∈X. We discussed in section 2 the case where µx has a density, but more general measures are allowed, such as µx = δx, 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 into account (specially when X is finite), making us consider weighted measures µ = Pn i=1 ciµxi in the general case. We now state the main result of this section which characterizes bounded p.d. functions on the semigroup M b +(X), Theorem 1. A bounded real-valued function ϕ on M b +(X) such that ϕ(0) = 1 is p.d. if and only if it has an integral representation: ϕ(µ) = Z C+(X) e−µ[f]dν(f), where ν 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. Proof. (sketch) Endowed with the topology of weak convergence, M b +(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 on M b +(X) are exactly the functions of the form µ 7→exp(−µ[f]) where f ∈C+(X) by using the characterization of semicharacters on (R+, +) [8, Theorem 6.5.8] and the fact that atomic measures is a dense subset of M b +(X) [8, Theorem 2.3.5]. As a constructive application to this general representation theorem, let us consider the case µx = δx 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, namely of the form pθ(x) = exp(PN j=1 θjfj(x)−ψ(θ)) where θ = (θj)j=1..N ∈Θ ⊂RN is variable and ψ is a real-valued function defined on Θ to ensure normalization of the densities pθ. Considering a prior ω on the parameter space Θ 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. ˆθµ ∈Θ being the ML parameter associated with µ and noting pµ = pˆθµ, ϕω(µ) = e−|µ|h(pµ) Z Θ e−|µ|d(pµ||pθ)ω(dθ), is a p.d. kernel on the semigroup of measures, where d(p||q) = R supp(q) p ln p q is the Kullback-Leibler divergence between p and q. 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: ϕ(µ) ∼ |µ|→∞˜ϕ(µ) := e−|µ|h(pµ) µ2π |µ| ¶ N 2 . (1) The ML estimator being unaffected by the total weight |µ|, we have ˜ϕ(2µ) = ˜ϕ(µ)2( |µ| 4π ) N 2 which we use to renormalize our kernel on its diagonal: k(µ, µ′) = e−(|µ+µ′|)h(pµ+µ′) e−|µ|h(pµ)−|µ′|h(pµ′) à 2 p |µ||µ′| |µ| + |µ′| ! N 2 Two problems call now for a proper renormalization: First, if |µ′| ≪|µ| (which would be the case if τ describes far more elements than τ ′), the entropy h(pµ+µ′) will not take into account the elements enumerated in µ′. Second, the value taken by our p.d function ˜ϕ decreases exponentially with |µ| 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 |µ| ≫0. To take into account this tradeoff on the ideal range of |µ|, we rewrite the previous expression using a common width parameter β after having applied a renormalization on µ and µ′: kβ(µ, µ′) = k( β |µ|µ, β |µ′|µ′) = e −2β µ h(pµ′′)− h(pµ)+h(pµ′ ) 2 ¶ , (2) where µ′′ = µ |µ| + µ′ |µ′|. β should hence be big enough in practical applications to ensure the consistency of Laplace’s approximation and thus positive definiteness, while small enough to avoid diagonal dominance. We will now always suppose that our atomic measures are normalized, meaning that their total weight Pn 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,Σ ∼N(m, Σ) (where Σ is a k × k p.d. matrix) to define our densities. The ML parameters of a measure µ are in that case : ¯µ = Pn i=1 cixi and Σµ = Pn i=1 ci(xi −¯µ)(xi −¯µ)⊤. 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,Σ) = 1 2 ln ((2πe)n|Σ|). The value of the normalized kernel in (2) is then: kβ(µ, µ′) = Ãp |Σµ||Σµ′| |Σµ′′| !2β . 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 κ. 4 A kernel defined through regularized covariance operators Endowing X (now also considered 2-separable) with a p.d. kernel κ 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 Ξ, and its feature map by ξ : x 7→κ(x, ·). Ξ 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 µ, µ′ and µ′′ 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 Σµ, Σµ′, Σµ′′ 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 Sµ, Sµ′, Sµ′′ associated with each measure on Ξ, and compare them by measuring their respective dispersion in a similar way. We note for ξ ∈Ξ its dual ξ∗(namely the linear form Ξ →R s.t. ζ 7→ξ∗ζ = ⟨ξ, ζ⟩Ξ) and ||ξ||2 = ξ∗ξ. Let (ei)i∈N be a complete orthonormal base of Ξ (i.e. such that span(ei)i∈N = Ξ and e∗ i ej = δij). Given a family of positive real numbers (ti)i∈N, we note St,e the bilinear symmetric operator which maps ξ, ζ 7→ξ∗St,eζ where St,e = P i∈N tieie∗ i . For an atomic measure µ and noting ˜ξi def = (ξi −µ[ξ]) its n centered points in Ξ, the empirical covariance operator Sµ = Pn i=1 ciξiξ∗ i on Ξ can be described through such a diagonal representation by finding its principal eigenfunctions, namely orthogonal functions in Ξ which maximize the expected (w.r.t to µ) variance of the normalized dot-product hv(ξ) def = v∗ξ ||v|| here defined for any v of Ξ. Such functions can be obtained through the following recursive maximizations: vj = argmax v∈Ξ,v⊥{v1,...,vj−1} varµ(hv(ξ)) = argmax v∈Ξ,v⊥{v1,...,vj−1} 1 ||vj||2 n X i=1 civ∗ j ˜ξi. 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({˜ξi}i=1..n). Thus for each vj there exists a vector αj of Rn such that vj = Pd i=1 αj,i ˜ξi with ||vj||2 = α⊤ j ˜Kµαj where ˜Kµ = (In −1n,n∆c)Kµ(In −∆c1n,n) is the centered Gram matrix Kµ = [κ(xi, xj)]1≤i,j≤n of the points taken in the support of µ, 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 ill-defined, since any αj is determined up to the addition of any element of ker ˜Kµ. We thus restrict our parameters α to lie in E def = ker ˜K⊥ µ ⊂Rn to consider functions of positive squared norm, having now: αj = argmax α∈E:∀k<j,α⊤˜ Kαk=0 α⊤˜Kµ∆c ˜Kµα α⊤˜Kµα ¡ = varµ(hvj(ξ)) ¢ Both endomorphism ˜Kµ∆c ˜Kµ and ˜Kµ being symmetric positive definite on E (one can easily prove that ker ˜Kµ = ker ˜Kµ∆c ˜Kµ), the right-hand argument of the previous equation, known as the Rayleigh quotient of ˜Kµ∆c ˜Kµ by ˜Kµ, can be maximized through a Hermitian generalized eigenvalue decomposition. This computation yields a basis αj of E such that α⊤ j ˜Kµαi = 0 for i < j ≤dim(E), and with corresponding positive eigenvalues in decreasing order λ1, ..., λdim(E). Through vj = Pd i=1 αj,i ˜ξi and writing r = dim(E), this also yields an orthogonal basis (vj)i≤r of span{(˜ξi)i≤n}, which can be completed to span Ξ through a Gram-Schmidt orthonormalization process using the original basis (ei)i∈N. The orthonormal base corresponding to Sµ is thus (vi)i∈N, where the r first vectors are the original eigenvectors obtained through the previous maximization. Such a diagonal representation of Sµ takes the form Sµ = Sλ,v where λ = (λ1, ..., λr, 0, ...). This bilinear form is however degenerated on Ξ and facing the same problem encountered in [4, 6] we also propose to solve this issue through a regularization by adding a component η on every vector of the base, i.e. defining λη = (λ1 + η, ..., λr + η, η, ...) with η > 0, to propose a regularization of Sµ as: Sλη,v = r X i=1 (λi + η)viv∗ i + X i>r η viv∗ i . 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 introducing the quantity |St,e|d = d 2 ln(2πe)+ 1 2 Pd i=1 ln ti. Let us sum up ideas now and consider three normalized measures µ, µ′ and µ′′ = µ+µ′ 2 , which yield three different orthonormal bases vi, v′ i and v′′ i of Ξ and three different families of weights λη = (λi≤r+η, η, ...), λ′ η = (λ′ i≤r′ + η, η, ...) and λ′′ η = (λ′′ i≤r′′ + η, η, ...). 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 Ξ of arbitrary size d ≫r′′ ≥max(r, r′): kd,β(µ, µ′) = exp à −2β à |Sλ′′ η ,v′′|d − |Sλη,v|d + |Sλ′ η,v′|d 2 !! = qQr i=1 1 + λi η Qr′ i=1 1 + λ′ i η Qr′′ i=1 1 + λ′′ i η 2β , (3) 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 Ξ. 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 τ(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 β to be 1 2 which always yielded positive definite Gram matrices in practice. To define our RKHS Ξ we used both the linear kernel, κa((x1, y1), (x2, y2)) = (x1x2 + y1y2)/272 and the Gaussian kernel of width σ, namely κb((x1, y1), (x2, y2)) = e−(x1−x2)2+(y1−y2)2 272·2σ2 . The 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 (Σ1,1, Σ2,2),(Σ′ 1,1, Σ′ 2,2) and (Σ′′ 1,1, Σ′′ 2,2) measure the dispersion of our data for each of the three measures, yielding a kernel value of √ Σ1,1Σ2,2 Σ′ 1,1Σ′ 2,2 Σ′′ 1,1Σ′′ 2,2 equal to 0.382 in the case shown in figure 1. The linear kernel manages 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 Ξ. In this case two parameters (a) Σ1,1 = 0.0552 Σ′ 1,1 = 0.0441 Σ′′ 1,1 = 0.0497 Σ2,2 = 0.0013 Σ′ 2,2 = 0.0237 Σ′′ 2,2 = 0.0139 (b) λ1 = 0.276 λ′ 1 = 0.168 λ′′ 1 = 0.184 Figure 1: First Eigenfunction of three empirical measures µ1, µ0 and µ1+µ0 2 using the linear (a) and the Gaussian (b, with η = 0.01, σ = 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: σ (the width of κ) controls the range of the typical eigenvalues found in the spectrum of our regularized operators whereas η 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 η and σ based on preliminary attempts: η ∈10−2 × {0.1, 0.3, 0.5, 0.8, 1, 1.5, 2, 3, 5, 8, 10, 20} and σ ∈10−1×{0.5, 1, 1.2, 1.5, 1.8, 2, 2.5, 3}. For each kernel computed on the base of a (σ, η) 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 σ value of 0.12 with an η 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 τ 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 k(z, z′) = e−||z−z′|| 30·2σ2 . We obtained similar results with an optimal error rate of roughly 44.5% for σ ∈{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. 1see http://www.kyb.tuebingen.mpg.de/bs/people/spider/ 102 η σ 0.1 0.3 0.5 0.8 1 1.5 2 3 5 8 10 20 0.05 0.1 0.12 0.15 0.18 0.2 0.25 0.3 e < 19.5 % e < 22 % e < 22 % 10 15 20 25 30 15% 20% 25% 30% 35% 40% 45% 50% # points Averaged error rate (a) (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 η (regularization) and σ (width of the Gaussian kernel) have been tuned to different values. (b) Curve of the same error (with η = 0.01, σ = 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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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 bernd.fischer@inf.ethz.ch 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 AYSARDGFSHEK AYSAR DGFSHEK m/z m/z 2. MS A K AY EK AYS HEK 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 12C/13C isotope proportions of the approximately 30-80 C-atoms contained in a peptide. Fluctuations of the 13C 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 12C 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 (H2O), 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 α ∈A and the mass of an amino acid with M(α). 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 m′ p of the peptide P = (α1, . . . , αn) with αi ∈A is the sum of all amino acid masses plus a constant mass for the N- and C-termini. m′ p = constN + Pn i=1 M(αi) + 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 mp = Pn i=1 M(αi) . (1) 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 0s 1sA 2sA 3sA +s −s 1sR 2sR 3sR 71 sA 156 sR 1sV 2sV 3sV 99 sV Figure 3: The finite state machine of the Hidden Markov Model. For each amino acid α there is a list of M(α) states. The finite state automaton (fig. 3) has one initial state s0. For each amino acid α ∈A there exists a list of M(α) states sα 1 , . . . , sα M(α). Together with the end states s+ and s− the complete set of states is S = {s0} ∪ sα j | α ∈A, 1 ≤j ≤M(α) ∪{s+, s−} . (2) 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 sα 1 of a state list of one amino acid α, it has to pass through all other states within the specific list. Thus for the next M(α) steps the list corresponding to amino acid α is linearly traversed. If the automaton is in the last state sα M(α) of a list, it can reach the start states sα′ 1 of any other amino acid α′. The random variable for the state sequence is denoted by Y1, . . . , Ymp. The transition probabilities are a(s, t) = P {Yi+1 = t|Yi = s} = 1 ∀α ∈A, 1 ≤i < M(α) : s = sα i ∧t = sα i+1 rα ∀α ∈A, β ∈A : s = sβ m(β) ∧t = sα 1 0 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 α (1 ≤i < M(α) : s = sα 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 rα. The probabilities rα are the probabilities of occurrence of amino acid α. The transition probabilities a(s0, t) from the start state s0 are the occurrence probabilities of the amino acids. a(s0, t) = rα ∀α ∈A : t = sα 1 0 else (4) 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 a′(s, t). The dotted arrows in figure 3 show the transition probabilities a′(s, t) into the end states s+ and s−. a′(s, t) = 1 ∀α ∈A : s = sα M(α), t = s+ 1 ∀α ∈A, 1 ≤i < M(α) : s = sα i , t = s− 0 else (5) If the automaton is in the last state sα M(α) 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 −50 −40 −30 −20 −10 0 10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 H O 2 NH3 2H O 2 NH3 H O+ 2 CO+NH3 CO (a) −50 −40 −30 −20 −10 0 10 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 −50 −40 −30 −20 −10 0 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 NH3 H O 2 (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 (H2O) 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. m=mp s+ s0 m=0 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 bs,s′(xm) = P ¯Xm = xm = (x(m), x(mp −m)) | ¯Ym = (s, s′) (6) as the emission probabilities of ion counts. ¯Xm are the (coupled) random variables of the ion counts. The hidden variables for the state sequence are denoted by ¯Ym. This notion of coupled variables ¯Xm describes the transition 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 mp is P {X = x, Y = y | s+, mp} = a (s0, y1) a′ ymp, s+ · (7) · mp−1 2 Y m=1 bym,ymp−m(xm)a (ym, ym+1) a ymp−m, ymp−m+1 a y mp−1 2 , y mp−1 2 +1 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 s0 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 ˆmp = argmax mp P {X = x | s+, mp} = argmax mp X y P {X = x, Y = y | s+, mp} . (8) 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 ˆmp the maximum posterior estimate of the sequence is y∗= argmax y P {Y = y | X = x, s+, ˆmp} = argmax y P {X = x, Y = y | s+, ˆmp} . (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+, ˆmp} by the evidence P {X = x | s+, ˆmp} 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} (10) = P y1,...yp−1 P yq+1,...,ymp P y1, . . . , ymp, x | s+, mp P {x | s+, mp} (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=mp m=mp s+ s0 m=0 m=0 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. 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 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 recall 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 <−2 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Lutefisk −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Peaks −3 −2 −1 0 1 2 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 HMM (a) Lutefisk Peaks HMM 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Recall (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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Instance-Based Relevance Feedback for Image Retrieval Giorgio Giacinto and Fabio Roli Department of Electrical and Electronic Engineering University of Cagliari Piazza D’Armi, Cagliari – Italy 09121 {giacinto,roli}@diee.unica.it Abstract High retrieval precision in content-based image retrieval can be attained by adopting relevance feedback mechanisms. These mechanisms require that the user judges the quality of the results of the query by marking all the retrieved images as being either relevant or not. Then, the search engine exploits this information to adapt the search to better meet user’s needs. At present, the vast majority of proposed relevance feedback mechanisms are formulated in terms of search model that has to be optimized. Such an optimization involves the modification of some search parameters so that the nearest neighbor of the query vector contains the largest number of relevant images. In this paper, a different approach to relevance feedback is proposed. After the user provides the first feedback, following retrievals are not based on knn search, but on the computation of a relevance score for each image of the database. This score is computed as a function of two distances, namely the distance from the nearest non-relevant image and the distance from the nearest relevant one. Images are then ranked according to this score and the top k images are displayed. Reported results on three image data sets show that the proposed mechanism outperforms other state-of-the-art relevance feedback mechanisms. 1 Introduction A large number of content-based image retrieval (CBIR) systems rely on the vector representation of images in a multidimensional feature space representing low-level image characteristics, e.g., color, texture, shape, etc. [1]. Content-based queries are often expressed by visual examples in order to retrieve from the database the images that are “similar” to the examples. This kind of retrieval is often referred to as K nearest-neighbor retrieval. It is easy to see that the effectiveness of content-based image retrieval systems (CBIR) strongly depends on the choice of the set of visual features, on the choice of the “metric” used to model the user’s perception of image similarity, and on the choice of the image used to query the database [1]. Typically, if we allow different users to mark the images retrieved with a given query as relevant or non-relevant, different subsets of images will be marked as relevant. Accordingly, the need for mechanisms to adapt the CBIR system response based on some feedback from the user is widely recognized. It is interesting to note that while relevance feedback mechanisms have been first introduced in the information retrieval field [2], they are receiving more attention in the CBIR field (Huang). The vast majority of relevance feedback techniques proposed in the literature is based on modifying the values of the search parameters as to better represent the concept the user bears in mind. To this end, search parameters are computed as a function of the relevance values assigned by the user to all the images retrieved so far. As an example, relevance feedback is often formulated in terms of the modification of the query vector, and/or in terms of adaptive similarity metrics. [3]-[7]. Recently, pattern classification paradigms such as SVMs have been proposed [8]. Feedback is thus used to model the concept of relevant images and adjust the search consequently. Concept modeling may be difficult on account of the distribution of relevant images in the selected feature space. “Narrow domain” image databases allows extracting good features, so that images bearing similar concepts belong to compact clusters. On the other hand, “broad domain” databases, such as image collection used by graphic professionals, or those made up of images from the Internet, are more difficult to subdivide in cluster because of the high variability of concepts [1]. In these cases, it is worth extracting only low level, non-specialized features, and image retrieval is better formulated in terms of a search problem rather then concept modeling. The present paper aims at offering an original contribution in this direction. Rather then modeling the concept of “relevance” the user bears in mind, feedback is used to assign each image of the database a relevance score. Such a score depends only from two dissimilarities (distances) computed against the images already marked by the user: the dissimilarity from the set of relevant images, and the dissimilarity from the set of non-relevant images. Despite its computational simplicity, this mechanism allows outperforming state-of-the-art relevance feedback mechanisms both on “narrow domain” databases, and on “broad domain” databases. This paper is organized as follows. Section 2 illustrates the idea behind the proposed mechanism and provides the basic assumptions. Section 3 details the proposed relevance feedback mechanism. Results on three image data sets are presented in Section 4, where performances of other relevance feedback mechanisms are compared. Conclusions are drawn in Section 5. 2 Instance-based relevance estimation The proposed mechanism has been inspired by classification techniques based on the “nearest case” [9]-[10]. Nearest-case theory provided the mechanism to compute the dissimilarity of each image from the sets of relevant and non–relevant images. The ratio between the nearest relevant image and the nearest non-relevant image has been used to compute the degree of relevance of each image of the database [11]. The present section illustrates the rationale behind the use of the nearest-case paradigm. Let us assume that each image of the database has been represented by a number of low-level features, and that a (dis)similarity measure has been defined so that the proximity between pairs of images represents some kind of “conceptual” similarity. In other words, the chosen feature space and similarity metric is meaningful at least for a restricted number of users. A search in image databases is usually performed by retrieving the k most similar images with respect to a given query. The dimension of k is usually small, to avoid displaying a large number of images at a time. Typical values for k are between 10 and 20. However, as the “relevant” images that the user wishes to retrieve may not fit perfectly with the similarity metric designed for the search engine, the user may be interested in exploring other regions of the feature space. To this end, the user marks the subset of “relevant” images out of the k retrieved. Usually, such relevance feedback is used to perform a new k-nn search by modifying some search parameters, i.e., the position of the query point, the similarity metric, and other tuning parameters [1]-[7]. Recent works proposed the use of support vector machine to learn the distribution of relevant images [8]. These techniques require some assumption about the general form of the distribution of relevant images in the feature space. As it is difficult to make any assumption about such a distribution for broad domain databases, we propose to exploit the information about the relevance of the images retrieved so far in a nearest-neighbor fashion. Nearest-neighbor techniques, as used in statistical pattern recognition, case-based reasoning, or instance-based learning, are effective in all applications where it is difficult to produce a high-level generalization of a “class” of objects [9]-[10],[12][13]. Relevance learning in content base image retrieval may well fit into this definition, as it is difficult to provide a general model that can be adapted to represent different concepts of similarity. In addition, the number of available cases may be too small to estimate the optimal set of parameters for such a general model. On the other hand, it can be more effective to use each “relevant” image as well as each “non-relevant” image, as “cases” or “instances” against which the images of the database should be compared. Consequently, we assume that an image is as much as relevant as much as its dissimilarity from the nearest relevant image is small. Analogously, an image is as much as non-relevant as much as its dissimilarity from the nearest non-relevant image is small. 3 Relevance Score Computation According to previous section, each image of the database can be thus characterized by a “degree of relevance” and a “degree of non-relevance” according to the dissimilarities from the nearest relevant image, and from the nearest non-relevant image, respectively. However, it should be noted that these degrees should be treated differently because only “relevant” images represent a “concept” in the user’s mind, while “non-relevant” images may represent a number of other concepts different from user’s interest. In other words, while it is meaningful to treat the degree of relevance as a degree of membership to the class of relevant images, the same does not apply to the degree of non-relevance. For this reason, we propose to use the “degree of non-relevance” to weight the “degree of relevance”. Let us denote with R the subset of indexes j ∈ {1,...,k} related to the set of relevant images retrieved so far and the original query (that is relevant by default), and with NR the subset of indexes j ∈ (1,...,k} related to the set of non-relevant images retrieved so far. For each image I of the database, according to the nearest neighbor rule, let us compute the dissimilarity from the nearest image in R and the dissimilarity from the nearest image in NR. Let us denote these dissimilarities as dR(I) and dNR(I), respectively. The value of dR(I) can be clearly used to measure the degree of relevance of image I, assuming that small values of dR(I) are related to very relevant images. On the other hand, the hypothesis that image I is relevant to the user’s query can be supported by a high value of dNR(I). Accordingly, we defined the relevance score relevance I( ) = 1+ dR I( ) dN I( ) ! "# $ %& '1 (1) This formulation of the score can be easily explained in terms of a distanceweighted 2-nn estimation of the posterior probability that image I is relevant. The 2 nearest neighbors are made up of the nearest relevant image, and the nearest nonrelevant image, while the weights are computed as the inverse of the distance from the nearest neighbors. The relevance score computed according to equation (1) is then used to rank the images and the first k are presented to the user. 4 Experimental results In order to test the proposed method and compare it with other methods described in the literature, three image databases have been used: the MIT database, a database contained in the UCI repository, and a subset of the Corel database. These databases are currently used for assessing and comparing relevance feedback techniques [5],[7],[14]. The MIT database was collected by the MIT Media Lab (ftp://whitechapel.media.mit.edu/pub/VisTex). This database contains 40 texture images that have been manually classified into fifteen classes. Each of these images has been subdivided into sixteen non-overlapping images, obtaining a data set with 640 images. Sixteen Gabor filters were used to characterise these images, so that each image is represented by a 16-dimensional feature vector [14]. The database extracted from the UCI repository (http://www.cs.uci.edu/mlearn/MLRepository.html) consists of 2,310 outdoor images. The images are subdivided into seven data classes (brickface, sky, foliage, cement, window, path, and grass). Nineteen colour and spatial features characterise each image. (Details are reported in the UCI web site). The database extracted from the Corel collection is available at the KDD-UCI repository (http://kdd.ics.uci.edu/databases/CorelFeatures/CorelFeatures.data.html). We used a subset made up of 19513 images, manually subdivided into 43 classes. For each image, four sets of features were available at the web site. In this paper, we report the results related to the Color Moments (9 features), and the Co-occurrence Texture (16 features) feature sets For each dataset, the Euclidean distance metric has been used. A linear normalisation procedure has been performed, so that each feature takes values in the range between 0 and 1. For the first two databases, each image is used as a query, while for the Corel database, 500 images have been randomly extracted and used as query, so that all the 43 classes are represented. At each retrieval iteration, twenty images are returned. Relevance feedback is performed by marking images belonging to the same class of the query as relevant, and all other images as non-relevant. The user’s query itself is included in the set of relevant images. This experimental set up affords an objective comparison among different methods, and is currently used by many researchers [5],[7],[14]. Results are evaluated in term of the retrieval precision averaged over all the considered queries. The precision is measured as the fraction of relevant images contained in the 20 top retrieved images. As the first two databases are of the “narrow domain” type, while the third is of the “broad domain” type, this experimental set-up allowed a thorough testing of the proposed technique. For the sake of comparison, retrieval performances obtained with two methods recently described in the literature are also reported: MindReader [3] which modifies the query vector and the similarity metric on account of features relevance, and Bayes QS (Bayesian Query Shifting) which is based on query reformulation [7]. These two methods have been selected because they can be easily implemented, and their performances can be compared to those provided by a large number of relevance feedback techniques proposed in the CBIR literature (see for example results presented in [15]). It is worth noting that results presented in different papers cannot be directly compared to each other because they are not related to a common experimental set-up. However, as they are related to the same data sets with similar experimental set-up, a qualitative comparisons let us conclude that the performance of the two above techniques are quite close to other results in the literature. 4.1 Experiments w ith th e MIT database This database can be considered of the “narrow domain” type as it contains only images of textures of 40 different types. In addition, the selected feature space is very suited to measure texture similarity. Figure 1 show the performances of the proposed relevance feedback mechanism and those of the two techniques used for comparison. 75 80 85 90 95 100 0 rf 1 rf 2 rf 3 rf 4 rf 5 rf 6 rf 7 rf 8 rf Iter. Rel. Feedback % Precision Relevance Score Bayes QS MindReader Figure 1: Retrieval Performances for the MIT database in terms of average percentage retrieval precision. After the first feedback iteration (1rf in the graph), each relevance feedback mechanism is able to improve the average precision attained in the first retrieval by more than 10%, the proposed mechanism performing slightly better than MindReader. This is a desired behaviour as a user typically allows few iterations. However, if the user aims to better refine the search by additional feedback iteration, MindReader and Bayes QS are not able to exploit the additional information, as they provide no improvements after the second feedback iteration. On the other hand, the proposed mechanism provides further improvement in precision by increasing the number of iteration. These improvements are very small because the first feedback already provides a high precision value, near to 95%. 4.2 Experiments w ith th e UC I database This database too can be considered of the “narrow domain” type as the images clearly belong to one of the seven data classes, and features have been extracted accordingly. 90 92 94 96 98 100 0 rf 1 rf 2 rf 3 rf 4 rf 5 rf 6 rf 7 rf 8 rf Iter. Rel. Feedback % Precision Relevance Score Bayes QS MindReader Figure 2: Retrieval Performances for the UCI data set in terms of average percentage retrieval precision. Figure 2 show the performances attained on the UCI database. Retrieval precision is very high after the first extraction with no feedback. Nonetheless, each of the considered mechanism is able to exploit relevance feedback, Mindreader and Bayes QS providing a 6% improvement, while the proposed mechanism attains a 8% improvement. This example clearly shows the superiority of the proposed technique, as it attains a precision of 99% after the second iteration. Further iterations allow attaining a 100% precision. On the other hand, Bayes QS also exploits further feedback iteration attaining a precision of 98% after 7 iterations, while MindReader does not improve the precision attained after the first iteration. As the user typically allows very few feedback iterations, the proposed mechanism proved to be very suited for narrow domain databases as it allows attaining a precision close to 100%. 4.3 Experiments w ith th e Co rel database Figures 3 and 4 show the performances attained on two feature sets extracted from the Corel database. This database is of the “broad domain” type as images represent a very large number of concepts, and the selected feature sets represent conceptual similarity between pairs of images only partly. Reported results clearly show the superiority of the proposed mechanism. Let us note that the retrieval precision after the first k-nn search (0rf in the graphs) is quite small. This is a consequence of the difficulty of selecting a good feature space to represent conceptual similarity between pairs of images in a broad domain database. This difficulty is partially overcome by using MindReader or Bayes QS as they allow improving the retrieval precision by 10% to 15% according to the number of iteration allowed, and according to the selected feature space. Let us recall that both MindReader and Bayes QS perform a query movement in order to perform a k-nn query on a more promising region of the feature space. On the other hand, the proposed mechanism based on ranking all the images of the database according to a relevance score, not only provided higher precision after the first feedback, but also allow to improve significantly the retrieval precision as the number of iteration is increased. As the initial precision is quite small, a user may have more willingness to perform further iterations as the proposed mechanism allows retrieving new relevant images. Figure 3: Retrieval Performances for the Corel data set (Color Moments feature set) in terms of average percentage retrieval precision Figure 4: Retrieval Performances for the Corel data set (Co-occurrence Texture feature set) in terms of average percentage retrieval precision. 5 Conclusions In this paper, we proposed a novel relevance feedback technique for content-based image retrieval. While the vast majority of relevance feedback mechanisms aims at modeling user’s concept of relevance based on the available labeled samples, the proposed mechanism is based on ranking the images according to a relevance score depending on the dissimilarity from the nearest relevant and non-relevant images. The rationale behind our choice is the same of case-based reasoning, instance-based learning, and nearest-neighbor pattern classification. These techniques provide good performances when the number of available training samples is too small to use statistical techniques. This is the case of relevance feedback in CBIR, where the use of classification models should require a suitable formulation in order to avoid socalled “small sample” problems. Reported results clearly showed the superiority of the proposed mechanism especially when large databases made up of images related to many different concepts are searched. In addition, while many relevance feedback techniques require the tuning of some parameters, and exhibit high computational complexity, the proposed mechanism does not require any parameter tuning, and exhibit a low computational complexity, as a number of techniques are available to speed-up distance computations. Referen ces [1] Smeulders A.W.M., Worring M., Santini S., Gupta A., Jain R.: Content-based image retrieval at the end of the early years. IEEE Trans. on Pattern Analysis and Machine Intelligence 22(12) (2000) 1349-1380 [2] G. Salton and M.J. McGill, Introduction to modern information retrieval, New York, McGraw-Hill, 1988. [3] Ishikawa Y., Subramanys R., Faloutsos C.: MindReader: Querying databases through multiple examples. In Proceedings. of the 24th VLDB Conference (1998) 433-438 [4] Santini S., Jain R.: Integrated browsing and querying for image databases. IEEE Multimedia 7(3) (2000) 26-39 [5] Rui Y., Huang T.S.: Relevance Feedback Techniques in Image retrieval. In Lew M.S. (ed.): Principles of Visual Information Retrieval. Springer, London, (2001) 219-258 [6] Sclaroff S., La Cascia M., Sethi S., Taycher L.: Mix and Match Features in the ImageRover search engine. In Lew M.S. (ed.): Principles of Visual Information Retrieval. Springer-Verlag, London (2001) 219-258 [7] Giacinto G., Roli F.: Bayesian relevance feedback for content-based image retrieval. Pattern Recognition 37(7) (2004) 1499-1508 [8] Zhou X.S. and Huang T.S.: Relevance feedback in image retrieval: a comprehensive review, Multimedia Systems 8(6) (2003) 536-544 [9] Aha D.W., Kibler D., Albert M.K. Instance Based learning Algorithms. Machine Learning, 6, (1991) 37-66 [10] Althoff K-D. Case-Based Reasoning. In Chang S.K. (ed.) Handbook on Software Engineering and Knowledge Engineering, World Scientific (2001), 549-588. [11] Bloch I. Information Combination Operators for Data Fusion: A Comparative Review with Classification. IEEE Trans. on System, Man and Cybernetics - Part A, 26(1) (1996) 52-67 [12] Duda R.O., Hart P.E., and Stork D.G.: Pattern Classification. John Wiley and Sons, Inc., New York, 2001 [13] Hastie T., Tibshrirani R., and Friedman J.: The Elements of Statistical Learning. Springer, New York, 2001 [14] Peng J., Bhanu B., Qing S., Probabilistic feature relevance learning for content-based image retrieval, Computer Vision and Image Understanding 75 (1999) 150-164. [15] He J., Li M., Zhang H-J, Tong H., Zhang C, Mean Version Space: a New Active Learning Method for Content-Based Image Retrieval, Proc. of MIR 2004, New York, USA. (2004) 15-22.
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Learning Preferences for Multiclass Problems Fabio Aiolli Dept. of Computer Science University of Pisa, Italy aiolli@di.unipi.it Alessandro Sperduti Dept. of Pure and Applied Mathematics University of Padova, Italy sperduti@math.unipd.it Abstract Many interesting multiclass problems can be cast in the general framework of label ranking defined on a given set of classes. The evaluation for such a ranking is generally given in terms of the number of violated order constraints between classes. In this paper, we propose the Preference Learning Model as a unifying framework to model and solve a large class of multiclass problems in a large margin perspective. In addition, an original kernel-based method is proposed and evaluated on a ranking dataset with state-of-the-art results. 1 Introduction The presence of multiple classes in a learning domain introduces interesting tasks besides the one to select the most appropriate class for an object, the well-known (single-label) multiclass problem. Many others, including learning rankings, multi-label classification, hierarchical classification and ordinal regression, just to name a few, have not yet been sufficiently studied even though they should not be considered less important. One of the major problems when dealing with this large set of different settings is the lack of a single universal theory encompassing all of them. In this paper we focus on multiclass problems where labels are given as partial order constraints over the classes. Tasks naturally falling into this family include category ranking, which is the task to infer full orders over the classes, binary category ranking, which is the task to infer orders such that a given subset of classes are top-ranked, and any general (q-label) classification problem. Recently, efforts have been made in the direction to unify different ranking problems. In particular, in [5, 7] two frameworks have been proposed which aim at inducing a label ranking function from examples. Similarly, here we consider labels coded into sets of preference constraints, expressed as preference graphs over the set of classes. The multiclass problem is then reduced to learning a good set of scoring functions able to correctly rank the classes according to the constraints which are associated to the label of the examples. Each preference graph disagreeing with the obtained ranking function will count as an error. The primary contribution of this work is to try to make a further step towards the unification of different multiclass settings, and different models to solve them, by proposing the Preference Learning Model, a very general framework to model and study several kinds of multiclass problems. In addition, a kernel-based method particularly suited for this setting is proposed and evaluated in a binary category ranking task with very promising results. The Multiclass Setting Let Ωbe a set of classes, we consider a multiclass setting where data are supposed to be sampled according to a probability distribution D over X × Y, X ⊆Rd and an hypothesis space of functions F = {fΘ : X × Ω→R} with parameters Θ. Moreover, a cost function c(x, y|Θ) defines the cost suffered by a given hypothesis on a pattern x ∈X having label y ∈Y. A multiclass learning algorithm searches for a set of parameters Θ∗such to minimize the true cost, that is the expected value of the cost according to the true distribution of data, i.e. Rt[Θ] = E(x,y)∼D[c(x, y|Θ)]. The distribution D is typically unknown, while it is available a training set S = {(x1, y1), . . . , (xn, yn)} with examples drawn i.i.d. from D. An empirical approximation of the true cost, also referred to as the empirical cost, is defined by Re[Θ, S] = 1 n Pn i=1 c(xi, yi|Θ). 2 The Preference Learning Model In this section, starting from the general multiclass setting described above, we propose a general technique to solve a large family of multiclass settings. The basic idea is to ”code” labels of the original multiclass problem as sets of ranking constraints given as preference graphs. Then, we introduce the Preference Learning Model (PLM) for the induction of optimal scoring functions that uses those constraints as supervision. In the case of ranking-based multiclass settings, labels are given as partial orders over the classes (see [1] for a detailed taxonomy of multiclass learning problems). Moreover, as observed in [5], ranking problems can be generalized by considering labels given as preference graphs over a set of classes Ω= {ω1, . . . , ωm}, and trying to find a consistent ranking function fR : X →Π(Ω) where Π(Ω) is the set of permutations over Ω. More formally, considering a set Ω, a preference graph or ”p-graph” over Ωis a directed graph v = (N, A) where N ⊆Ωis the set of nodes and A is the set of arcs of the graph accessed by the function A(v). An arc a ∈A is associated with its starting node ωs = ωs(a) and its ending node ωe = ωe(a) and represents the information that the class ωs is preferred to, and should be ranked higher than, ωe. The set of p-graphs over Ωwill be denoted by G(Ω). Let be given a set of scoring functions f : X × Ω→R with parameters Θ working as predictors of the relevance of the associated class to given instances. A definition of a ranking function naturally follows by taking the permutation of elements in Ωcorresponding to the sorting of the values of these functions, i.e. fR(x|Θ) = argsortω∈Ωf(x, ω|Θ). We say that a preference arc a = (ωs, ωe) is consistent with a ranking hypothesis fR(x|Θ), and we write a ⊑fR(x|Θ), when f(x, ωs|Θ) ≥f(x, ωe|Θ) holds. Generalizing to graphs, a p-graph g is said to be consistent with an hypothesis fR(x|Θ), and we write g ⊑fR(x|Θ), if every arc compounding it is consistent, i.e. g ⊑fR(x|Θ) ⇔∀a ∈A(g), a ⊑fR(x|Θ). The PLM Mapping Let us start by considering the way a multiclass problem is transformed into a PLM problem. As seen before, to evaluate the quality of a ranking function fR(x|Θ) is necessary to specify the nature of a cost function c(x, y|Θ). Specifically, we consider cost definitions corresponding to associate penalties whenever uncorrect decisions are made (e.g. a classification error for classification problems or wrong ordering for ranking problems). To this end, as in [5], we consider a label mapping G : y 7→{g1(y), . . . , gqy(y)} where a set of subgraphs gi(y) ∈G(Ω) are associated to each label y ∈Y. The total cost suffered by a ranking hypothesis fR on the example x ∈X labeled y ∈Y is the number of p-graphs in G(y) not consistent with the ranking, i.e. c(x, y|Θ) = Pqy j=1[[gj(y) ̸⊑f(x|Θ)]], where [[b]] is 1 if the condition b holds, 0 otherwise. Let us describe three particular mappings proposed in [5] that seem worthwhile of note: (i) The identity mapping, denoted by GI, where the label is mapped on itself and every inconsistent graph will have a unitary cost, (ii) the disagreement mapping, denoted by Gd, where a simple (single-preference) subgraph is built for each arc in A(y), and (iii) the domination mapping, denoted by GD, where for each node ωr in y a subgraph consisting of ωr plus (a) (b) (c) (d) (e) (f) Figure 1: Examples of label mappings for 2-label classification (a-c) and ranking (d-f). the nodes of its outgoing set is built. To clarify, in Figure 1 a set of mapping examples are proposed. Considering Ω= {1, 2, 3, 4, 5}, in Figure 1-(a) the label y = [1, 2|3, 4, 5] for a 2-label classification setting is given. In particular, this corresponds to the mapping G(y) = GI(y) = y where a single wrong ranking of a class makes the predictor to pay a unit of cost. Similarly, in Figure 1-(b) the label mapping G(y) = GD(y) is presented for the same problem. Another variant is presented in Figure 1-(c) where the label mapping G(y) = Gd(y) is used and the target classes are independently evaluated and their errors cumulated. Note that all these graphs are subgraphs of the original label in 1-(a). As an additional example we consider the three cases depicted in the right hand side of Figure 1 that refer to a ranking problem with three classes Ω= {1, 2, 3}. In Figure 1-(d) the label y = [1|2|3] is given. As before, this also corresponds to the label mapping G(y) = GI(y). Two alternative cost definitions can be obtained by using the p-graphs (sets of basic preferences actually) depicted in Figure 1-(e) and 1-(f). Note that the cost functions in these cases are different. For example, assume fR(x|Θ) = [3|1|2], the p-graph in (e) induces a cost c(x, yb|Θ) = 2 while the p-graph in (f) induces a cost c(x, yc|Θ) = 1. The PLM Setting Once the label mapping G is fixed, the preference constraints of the original multiclass problem can be arranged into a set of preference constraints. Specifically, we consider the set V(S) = S (xi,yi)∈S V(xi, yi) where V(x, y) = {(x, gj(y))}j∈{1,..,qy} and each pair (x, g) ∈X × G(Ω) is a preference constraint. Note that the same instance can be replicated in V(S). This can happen, for example, when multiple ranking constraints are associated to the same example of the original multiclass problem. Because of this, in the following, we prefer to use a different notation for the instances in preference constraints to avoid confusion with training examples. Notions defined for the standard classification setting are easily extended to PLM. For a preference constraint (v, g) ∈V, the constraint error incurred by the ranking hypothesis fR(v|Θ) is given by δ(v, g|Θ) = [[g ̸⊑fR(v|Θ)]]. The empirical cost is then defined as the cost over the whole constraint set, i.e. Re[Θ, V] = PN i=1 δ(vi, gi|Θ). In addition, we define the margin of an hypothesis on a pattern v for a preference arc a = (ωs, ωe), expressing how well the preference is satisfied, as the difference between the scores of the two linked nodes, i.e. ρA(v, a|Θ) = f(v, ωs|Θ) −f(v, ωe|Θ). The margin for a pgraph constraint (v, g) is then defined as the minimum of the margin of the compounding preferences, ρG(v, g|Θ) = mina∈A(g) ρA(v, a|Θ), and gives a measure of how well the hypothesis fulfills a given preference constraint. Note that, consistently with the classification setting, the margin is greater than 0 if and only if g ⊑fR(v|Θ). Learning in PLM In the PLM we try to learn a ”simple” hypothesis able to minimize the empirical cost of the original multiclass problem or equivalently to satisfy the constraints in V(S) as much as possible. The learning setting of the PLM can be reduced to the following scheme. Given a set V of pairs (vi, gi) ∈X × G(Ω), i ∈{1, . . . , N}, N = Pn i=1 qyi, find a set of parameters for the ranking function fR(v|Θ) able to minimize a combination of a regularization and an empirical loss term, ˆΘ = arg minΘ{Re[Θ, V] + µR(Θ)} with µ a given constant. However, since the direct minimization of this functional is hard due to the non continuous form of the empirical error term, we use an upper-bound on the true empirical error. To this end, let be defined a monotonically decreasing loss function L such that L(ρ) ≥0 and L(0) = 1, then by defining a margin-based loss LC(v, g|Θ) = L (ρG(v, g|Θ)) = max a∈A(g) L (ρA(v, a|Θ)) (1) for a p-graph constraint (v, g) ∈V and recalling the margin definition, the condition δ(v, g|Θ) ≤LC(v, g|Θ) always holds thus obtaining Re[Θ, V] ≤PN i=1 LC(vi, gi|Θ). The problem of learning with multiple classes (up to constant factors) is then reduced to a minimization of a (possibly regularized) loss functional ˆΘ = arg min Θ {L(V|Θ) + µR(Θ)} (2) where L(V|Θ) = PN i=1 maxa∈A(gi) L(f(vi, ωs(a)|Θ) −f(vi, ωe(a)|Θ)). Method L(ρ) β-margin Perceptron [1 −β−1ρ]+ Logistic Regression log2(1 + exp(−ρ)) Soft margin [1 −ρ]+ Mod. Least Square [1 −ρ]2 + Exponential exp(−ρ) Many different choices can be made for the function L(·). Some well known examples are the ones given in the table at the left. Note that, if the function L(·) is convex with respect to the parameters Θ, the minimization of the functional in Eq. (2) will result quite easy given a convex regularization term. The only difficulty in this case is represented by the max term. A shortcoming to this problem would consist in upper-bounding the max with the sum operator, though this would probably lead to a quite row approximation of the indicator function when considering p-graphs with many arcs. It can be shown that a number of related works, e.g. [5, 7], after minor modifications, can be seen as PLM instances when using the sum approximation. Interestingly, PLM highlights that this approximation in fact corresponds to a change on the label mapping obtained by decomposing a complex preference graph into a set of binary preferences and thus changing the cost definition we are indeed minimizing. In this case, using either GD or Gd is not going to make any difference at all. Multiclass Prediction through PLM A multiclass prediction is a function H : X →Y mapping instances to their associated label. Let be given a label mapping defined as G(y) = {g1(y), . . . , gqy(y)}. Then, the PLM multiclass prediction is given as the label whose induced preference constraints mostly agree with the current hypothesis, i.e. H(x) = arg miny L(V(x, y)|Θ) where V(x, y) = {(x, gj(y))}j∈{1,..,qy}. It can be shown that many of the most effective methods used for learning with multiple classes, including output coding (ECOC, OvA, OvO), boosting, least squares methods and all the methods in [10, 3, 7, 5] fit into the PLM setting. This issue is better discussed in [1]. 3 Preference Learning with Kernel Machines In this section, we focus on a particular setting of the PLM framework consisting of a multivariate embedding h : X →Rs of linear functions parameterized by a set of vectors Wk ∈Rd, k ∈{1, . . . , s} accommodated in a matrix W ∈Rs×d, i.e. h(x) = [h1(x), . . . , hs(x)] = [⟨W1, x⟩, . . . , ⟨Ws, x⟩]. Furthermore, we consider the set of classes Ω= {ω1, . . . , ωm} and M ∈Rm×s a matrix of codes of length s with as many rows as classes. This matrix has the same role as the coding matrix in multiclass coding, e.g. in ECOC. Finally, the scoring function for a given class is computed as the dot product between the embedding function and the class code vector f(x, ωr|W, M) = ⟨h(x), Mr⟩= s X k=1 Mrk⟨Wk, x⟩ (3) Now, we are able to describe a kernel-based method for the effective solution of the PLM problem. In particular, we present the problem formulation and the associated optimization method for the task of learning the embedding function given fixed codes for the classes (embedding problem). Another worthwhile task consists in the optimization of the codes for the classes when the embedding function is kept fixed (coding problem), or even to perform a combination of the two (see for example [8]). A deeper study of the embeddingcoding version of PLM and a set of examples can be found in [1]. PLM Kesler’s Construction As a first step, we generalize the Kesler’s Construction originally defined for single-label classification (see [6]) to the PLM setting, thus showing that the embedding problem can be formulated as a binary classification problem in a higher dimensional space when new variables are appropriately defined. Specifically, consider the vector y(a) = (Mωs(a) −Mωe(a)) ∈Rs defined for every preference arc in a given preference constraint, that is a = (ωs, ωe) ∈A(g). For every instance vi and preference (ωs, ωe), the preference condition ρA(vi, a) ≥0 can be rewritten as ρA(vi, a) = f(vi, ωs) −f(vi, ωe) = ⟨y(a), h(vi)⟩ = Ps k=1 yk(a)⟨Wk, vi⟩ = Ps k=1⟨Wk, yk(a)vi⟩ = Ps k=1⟨Wk, [za i ]s k⟩ = ⟨W, za i ⟩≥0 (4) where [·]s k denotes the k-th chunk of a s-chunks vector, W ∈Rs·d is the vector obtained by sequentially arranging the vectors Wk, and za i = y(a) ⊗vi ∈Rs·d is the embedded vector made of the s chunks defined by [za i ]s k = yk(a)vi, k ∈{1, . . . , s}. From this derivation it turns out that each preference of a constraint in the set V can be viewed as an example of dimension s · d in a binary classification problem. Each pair (vi, gi) ∈V then generates a number of examples in this extended binary problem equal to the number of arcs of the p-graph gi for a total of PN i=1 |A(gi)| examples. In particular, the set Z = {za i } is linearly separable in the higher dimensional problem if and only if there exists a consistent solution for the original PLM problem. Very similar considerations, omitted for space reasons, could be given for the coding problem as well. The Kernel Preference Learning Optimization As pointed out before, the central task in PLM is to learn scoring functions in such a way to be as much as possible consistent with the set of constraints in V. This is done by finding a set of parameters minimizing a loss function that is an upper-bound on the empirical error function. For the embedding problem, instantiating the problem (2), and choosing the 2-norm of the parameters as regularizer, we obtain ˆW = arg minW 1 N PN i=1 LC(vi, gi|W, M) + µ||W||2 where, according to Eq.(1), the loss for each preference constraint is computed as the maximum between the losses of all the associated preferences, that is Li = maxa∈A(gi) L(⟨W, za i ⟩). When the constraint set in V contains basic preferences only (that is p-graphs consisting of a single arc ai = A(gi)), the optimization problem can be simplified into the minimization of a standard functional combining a loss function with a regularization term. Specifically, all the losses presented before can be used and, for many of them, it is possible to give a kernel-based solution. See [11] for a set of examples of loss functions and the formulation of the associated problem with kernels. The Kernel Preference Learning Machine For the general case of p-graphs possibly containing multiple arcs, we propose a kernel-based method (hereafter referred to as Kernel Preference Learning Machine or KPLM for brevity) for PLM optimization which adopts the loss max in Eq. (2). Borrowing the idea of soft-margin [9], for each preference arc, a linear loss is used giving an upper bound on the indicator function loss. Specifically, we use the SVM-like soft margin loss L(ρ) = [1 −ρ]+. Summarizing, we require a set of small norm predictors that fulfill the soft constraints of the problem. These requirements can be expressed by the following quadratic problem minW,ξ 1 2||W||2 + C PN i ξi subject to: ⟨W, za i ⟩≥1 −ξi, i ∈{1, .., N}, a ∈A(gi) ξi ≥0, i ∈{1, .., N} (5) Note that differently from the SVM formulation for the binary classification setting, here the slack variables ξi are associated to multiple examples, one for each preference arc in the p-graph. Moreover, the optimal value of the ξi corresponds to the loss value as defined by Li. As it is easily verifiable, this problem is convex and it can be solved in the usual way by resorting to the optimization of the Wolfe dual problem. Specifically, we have to find the saddle point (minimization w.r.t. to the primal variables {W, ξ} and maximization w.r.t. the dual variables {α, λ}) of the following Lagrangian: Q(W, ξ, α, λ) = 1 2||W||2 + C PN i ξi + PN i P a∈A(gi) αa i (1 −ξi −⟨W, za i ⟩) −PN i λiξi, s.t. αa i , λi ≥0 (6) By differentiating the Lagrangian with respect to the primal variables and imposing the optimality conditions we obtain the set of constraints that the variables have to fulfill in order to be an optimal solution ∂Q ∂W = W −PN i P a∈A(gi) αa i za i = 0 ⇔W = PN i P a∈A(gi) αa i za i ∂Q ∂ξi = C −P a∈A(gi) αa i −λi = 0 ⇔P a∈A(gi) αa i ≤C (7) Substituting conditions (7) in (6) and omitting constants that do not change the solution, the problem can be restated as maxα P i,a αa i −1 2 Ps k P i,ai P j,aj yk(ai)yk(aj)αai i αaj j ⟨vi, vj⟩ subject to: αa i ≥0, i ∈{1, .., N}, a ∈A(gi) P a αa i ≤C, i ∈{1, .., N} (8) Since Wk = P i,a yk(a)αa i vi = P i,a[Mωs(a) −Mωe(a)]s kαa i vi, k = 1, .., s, we obtain hk(x) = ⟨Wk, x⟩= P i,a[Mωs(a)−Mωe(a)]s kαa i ⟨vi, x⟩. Note that any kernel k(·, ·) can be substituted in place of the linear dot product ⟨, ⟩to allow for non-linear decision functions. Embedding Optimization The problem in (8) recalls the one obtained for single-label multiclass SVM [1, 2] and, in fact, its optimization can be performed in a similar way. Assuming a number of arcs for each preference constraint equal to q, the dual problem in (8) involves N · q variables leading to a very large scale problem. However, it can be noted that the independence of constraints among the different preference constraints allows for the separation of the variables in N disjoints sets of q variables each. The algorithm we propose for the optimization of the overall problem consists in iteratively selecting a preference constraint from the constraints set (a p-graph) and then optimizing with respect to the variables associated with it, that is one for each arc of the p-graph. From the convexity of the problem and the separation of the variables, since on each iteration we optimize on a different subset of variables, this guarantees that the optimal solution for the Lagrangian will be found when no new selections can lead to improvements. The graph to optimize at each step is selected on the basis of an heuristic selection strategy. Let the preference constraint (vi, gi) ∈V be selected at a given iteration, to enforce the constraint P a∈A(gi) αa i +λi = C, λi ≥0, two elements from the set of variables {αa i |a ∈ A(gi)} ∪{λi} will be optimized in pairs while keeping the solution inside the feasible region αa i ≥0. In particular, let χ1 and χ2 be the two selected variables, we restrict the updates to the form χ1 ←χ1−ν and χ2 ←χ2+ν with optimal choices for ν. The variables which most violate the constraints are iteratively selected until they reach optimality KKT conditions. For this, we have devised a KKT-based procedure which is able to select these variables in time linear with the number of classes. For space reasons we omit the details and we do not consider at all any implementation issue. Details and optimized versions of this basic algorithm can be found in [1]. Generalization of KPLM As a first immediate result we can give an upper-bound on the leave-one-out error by utilizing the sparsity of a KPLM solution, namely LOO ≤|V |/N, where V = {i ∈{1, . . . , N}| maxa∈A(gi) αa i > 0} is the set of support vectors. Another interesting result about the generalization ability of a KPLM is in the following theorem. Theorem 1 Consider a KPLM hypothesis Θ = (W, M) with Ps r=1 ||Wr||2 = 1 and ||Mr||2 ≤RM such that min(v,g)∈V ρG(v, g|Θ) ≥γ. Then, for any probability distribution D on X × Y with support in a ball of radius RX around the origin, with probability 1 −δ over n random examples S, the following bound for the true cost holds Rt[Θ] ≤2QA n 64R2 γ2 log enγ 8R2 log 32n γ2 + log 4 δ where ∀y ∈Y, qy ≤Q, |A(gr(y))| ≤A, r ∈{1, . . . , qy} and R = 2RMRX . Proof. Similar to that of Theorem 4.11 in [7] when noting that the size of examples in Z are upper-bounded by R = 2RMRX . 4 Experiments Experimental Setting We performed experiments on the ‘ModApte” split of Reuters21578 dataset. We selected the 10 most popular categories thus obtaining a reduced set of 6,490 training documents and a set of 2,545 test documents. The corpus was then preprocessed by discarding numbers and punctuation and converting letters to lowercase. We used a stop-list to remove very frequent words and stemming has been performed by means of Porter’s stemmer. Term weights are calculated according to the tf/idf function. Term selection was not considered thus obtaining a set of 28,006 distinct features. We evaluated our framework on the binary category ranking task induced by the original multi-label classification task, thus requiring rankings having target classes of the original multi-label problem on top. Five different well-known cost functions have been used. Let x be an instance having ranking label y. IErr is the cost function indicating a non-perfect ranking and corresponds to the identity mapping in Figure 1-(a). DErr is the cost defined as the number of relevant classes uncorrectly ranked by the algorithm and corresponds to the domination mapping in Figure 1-(b). dErr is the cost obtained counting the number of uncorrect rankings and corresponds to the disagreement mapping in Figure 1-(c). Other two well-known Information Retrieval (IR) based cost functions have been used. The OneErr cost function that is 1 whenever the top ranked class is not a relevant class and the average precision cost function, which is AvgP = 1 |y| P r∈y |{r′∈y:rank(x,r′)≤rank(x,r)}| rank(x,r) . Results The model evaluation has been performed by comparing three different label mappings for KPLM and the baseline MMP algorithm [4], a variant of the Perceptron algorithm for ranking problems, with respect to the above-mentioned ranking losses. We used the configuration which gave the best results in the experiments reported in [4]. KPLM has been implemented setting s = m and the standard basis vectors er ∈Rm as codes associated to the classes. A linear kernel k(x, y) = (⟨x, y⟩+1) was used. Model selection for the KPLM has been performed by means of a 5-fold cross validation for different values of the parameter C. The optimal parameters have been chosen as the ones minimizing the mean of the values of the loss (the one used for training) over the different folders. In Table 1 we report the obtained results. It is clear that KPLM definitely outperforms the MMP method. This is probably due to the use of margins in KPLM. Moreover, using identity and domination mappings seems to lead to models that outperform the ones obtained by using the disagreement mapping. Interestingly, this also happens when comparing with respect to its own corresponding cost. This can be due to a looser approximation (as a sum of approximations) of the true cost function. The same trend was confirmed by another set of experiments on artificial datasets that we are not able to report here due to space limitations. Method IErr % DErr % dErr % OneErr % AvgP % MMP 5.07 4.92 0.89 4.28 97.49 KPLM (GI) 3.77 3.66 0.55 3.10 98.25 KPLM (GD) 3.81 3.59 0.54 3.14 98.24 KPLM (Gd) 4.12 4.13 0.66 3.58 97.99 Table 1: Comparisons of ranking performance for different methods using different loss functions according to different evaluation metrics. Best results are shown in bold. 5 Conclusions and Future Work We have presented a common framework for the analysis of general multiclass problems and proposed a kernel-based method as an instance of this setting which has shown very good results on a binary category ranking task. Promising directions of research, that we are currently pursuing, include experimenting with coding optimization and considering to extend the current setting to on-line learning, interdependent labels (e.g. hierarchical or any other structured classification), ordinal regression problems, and classification with costs. References [1] F. Aiolli. Large Margin Multiclass Learning: Models and Algorithms. PhD thesis, Dept. of Computer Science, University of Pisa, 2004. http://www.di.unipi.it/˜ aiolli/thesis.ps. [2] F. Aiolli and A. Sperduti. Multi-prototype support vector machine. In Proceedings of International Joint Conference of Artificial Intelligence (IJCAI), 2003. [3] K. Crammer and Y. Singer. On the learnability and design of output codes for multiclass problems. In Proceedings of the Thirteenth Annual Conference on Computational Learning Theory, pages 35–46, 2000. [4] K. Crammer and Y. Singer. A new family of online algorithms for category ranking. Journal of Machine Learning Research, 2003. [5] O. Dekel, C.D. Manning, and Y. Singer. Log-linear models for label ranking. In Advances in Neural Information Processing Systems, 2003. [6] R.O. Duda, P.E. Hart, and D.G. Stork. Pattern Classification, chapter 5, page 266. Wiley, 2001. [7] S. Har Peled, D. Roth, and D. Zimak. Constraint classification: A new approach to multiclass classification. In Proceedings of the 13th International Conference on Algorithmic Learning Theory (ALT-02), 2002. [8] G. R¨atsch, A. Smola, and S. Mika. Adapting codes and embeddings for polychotomies. In Advances in Neural Information Processing Systems, 2002. [9] V. Vapnik. Statistical Learning Theory. Wiley, New York, NY, 1998. [10] J. Weston and C. Watkins. Multiclass support vector machines. In M. Verleysen, editor, Proceedings of ESANN99. D. Facto Press, 1999. [11] T. Zhang and F.J. Oles. Text categorization based on regularized linear classification methods. Information Retrieval, 1(4):5–31, 2001.
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Mass meta-analysis in Talairach space Finn ˚ Arup Nielsen Neurobiology Research Unit, Rigshospitalet Copenhagen, Denmark and Informatics and Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark fn@imm.dtu.dk Abstract We provide a method for mass meta-analysis in a neuroinformatics database containing stereotaxic Talairach coordinates from neuroimaging experiments. Database labels are used to group the individual experiments, e.g., according to cognitive function, and the consistent pattern of the experiments within the groups are determined. The method voxelizes each group of experiments via a kernel density estimation, forming probability density volumes. The values in the probability density volumes are compared to null-hypothesis distributions generated by resamplings from the entire unlabeled set of experiments, and the distances to the nullhypotheses are used to sort the voxels across groups of experiments. This allows for mass meta-analysis, with the construction of a list with the most prominent associations between brain areas and group labels. Furthermore, the method can be used for functional labeling of voxels. 1 Introduction Neuroimaging experimenters usually report their results in the form of 3dimensional coordinates in the standardized stereotaxic Talairach system [1]. Automated meta-analytic and information retrieval methods are enabled when such data are represented in databases such as the BrainMap DBJ ([2], www.brainmapdbj.org) or the Brede database [3]. Example methods include outlier detection [4] and identification of similar volumes [5]. Apart from the stereotaxic coordinates, the databases record details of the experimental situation, e.g., the behavioral domain and the scanning modality. In the Brede database the main annotation is the so-called “external components”1 which are heuristically organized in a simple ontology: A directed graph (more specifically, a causal network) with the most general components as the roots of the graph, e.g., 1External components might be thought of as “cognitive components” or simply “brain functions”, but they are more general, e.g., they also incorporate neuroreceptors. The components are called “external” since they are external variables to the brain image. WOEXT: 40 Pain WOEXT: 261 Thermal pain WOEXT: 41 Cold pain WOEXT: 69 Hot pain Figure 1: The external components around “thermal pain” with “pain” as the parent of “thermal pain” and “cold pain” and “hot pain” as children. “hot pain” is a child of “thermal pain” that in turn is a child of “pain” (see Figure 1). The simple ontology is setup from information typically found in the introduction section of scientific articles, and it is compared with the Medical Subject Headings ontology of the National Library of Medicine. The ontology is stored in a simple XML file. The Brede database is organized, like the BrainMap DBJ, on different levels with scientific papers on the highest level. Each scientific paper contains one or more “experiments”, which each in turn contains one or more locations. The individual experiments are typically labeled with an external component. The experiments that are labeled with the same external component form a group, and the distribution of locations within the group become relevant: If a specific external component is localized to a specific brain region, then the locations associated with the external component should cluster in Talairach space. We will describe a meta-analytic method that identifies important associations between external components and clustered Talairach coordinates. We have previously modeled the relation between Talairach coordinates and neuroanatomical terms [4, 6] and the method that we propose here can be seen as an extension describing the relationship between Talairach coordinates and, e.g., cognitive components. 2 Method The data from the Brede database [3] was used, which at the time contained data from 126 scientific article containing 391 experiments and 2734 locations. There were 380 external components. The locations referenced with respect to the MNI atlas were realigned to the Talairach atlas [7]. To form a vectorial representation, each location was voxelized by convolving the location l at position vl = [x, y, z]′ with a Gaussian kernel [4, 8, 9]. This constructed a probability density in Talairach space v p(v|l) = (2πσ2)−3/2 exp −(v −vl)′(v −vl) 2σ2 , (1) with the width σ fixed to 1 centimeter. To form a resulting probability density volume p(v|t) for an external component t the individual components from each location were multiplied by the appropriate priors and summed p(v|t) = X l,e p(v|l) P(l|e) P(e|t), (2) with P(l|e) = 0 if the l location did not appear in the e experiment and P(e|t) = 0 if the e experiment is not associated with the t external components. The precise normalization of these priors is an unresolved problem. A paper with many locations and experiments should not be allowed to dominate the results. This can be the case if all locations are given equal weight. On the other hand a paper which reports just a single coordinate should probably not be weighted as much as one with many experiments and locations: Few reported locations might be due to limited (statistical) power of the experiment. As a compromise between the two extremes we used the square root of the number of the locations within an experiment and the square root of the number of experiments within a paper for the prior P(l|e). The square root normalization is also an appropriate normalization in certain voting systems [10]. The second prior was uniform P(e|t) ∝1 for those experiments that were labeled with the t external component. The continuous volume were sampled at regular grid points to establish a vector wt for each external component wt ≡p(v|t). (3) Null-hypothesis distributions for the maximum statistics u across the voxels in the volume were built up by resampling: A number of experiments E was selected and E experiments were resampled, with replacement, from the entire set of 391 experiments, ignoring the grouping imposed by the external component labeling. The experiments were resampled without regard to the paper they originated from. The maximum across voxels was found as: ur(E) = max j [wr(j)] , (4) where j is an index over voxels and r is the resample index. With R resamplings we obtain a vector u(E) = [u1(E) . . . ur(E) . . . uR(E)] that is a function of the number of experiments and which forms an empirical distribution u(E). When the value wt,j of the j voxel of the t external component was compared with the distribution, a distance to the null-hypothesis can be generated dt,j = Prob [wt,j > u(Et)] , (5) where 1 −d is a statistical P-value and where Et is the number of experiment associated with the t external component. Thus the resampling allows us to convert the probability density value to a probability that is comparable across external components of different sizes. The maximum statistics deal automatically with the multiple comparison problem across voxels [11]. dt,j can be computed by counting the fraction of the resampled values ur that are below the value of wt,j. The resampling distribution can also be approximated and smoothed by modeling it with a non-linear function. In our case we used a standard two-layer feed-forward neural network with hyperbolic tangent hidden units [12, 13] modeling the function f(E, u) = atanh(2d −1) with a quadratic cost function. The non-linear function allows for a more compact representation of the empirical distribution of the resampled maximum statistics. As a final step, the probability volumes for the external components wt were thresholded on selected levels and isosurfaces generated in the distance volume for visualization. Connected voxels within the thresholded volume were found by region identification and the local maxima in the regions were determined. Functional labeling of specified voxels is also possible: The distances dt,j were collected in a (external component × voxel)-matrix D and the elements in the j column sorted. Lastly, the voxel were labeled with the top external component. Only the bottom nodes of the causal networks of external components are likely to be directly associated with experiments. To label the ancestors, the labels from 10 0 10 1 10 2 10 4 10 5 10 6 Number of experiments test statistics (max pdf) Randomization test statistics 0.5 0.75 0.9 0.95 0.99 Figure 2: The test statistics at various distances to the null-hypothesis (d = 1 −P) after 1000 resamplings. The distance is shown as a function of the number of experiments E in the resampling. their descendants were back propagated, e.g., a study explicitly labeled as “hot pain” were also be labeled as “thermal pain” and “pain”. Apart from this simple back propagation the hierarchical structure of the external components was not incorporated into the prior. 3 Results Figure 2 shows isolines in the cumulative distribution of the resampled maximum statistics u(E) as a function of the resampling set size (number of experiments) from E = 1 to E = 100. Since the vectorized volume is not normalized to form a probability density the curves are increasing with our selected normalization. Table 1 shows the result of sorting the maximum distances across voxel within the external components. Topping the list are external components associated with movement. The voxel with the largest distance is localized in v = (0, −8, 56) which most likely is due to movement studies activating the supplementary motor area. In the Brede database the mean is (6, −7, 55) for the locations in the right hemisphere labeled as supplementary motor area. Other voxels with a high distance for the movement external components are located in the primary motor area. A number of other entries on the list are associated with pain, with the main voxel at (0, 8, 32) in the right anterior cingulate. Other important areas are shown in Figure 3 with isosurfaces in the distance volume for the external component “pain” (WOEXT: 40). These are localized in the anterior cingulate, right and left insula and thalamus. Other external components high on the list are “audition” together with “voice” # d x y z Name (WOEXT) 1 1.00 0 −8 56 Localized movement (266) 2 1.00 0 −8 56 Motion, movement, locomotion (4) 3 1.00 0 8 32 Pain (40) 4 1.00 0 8 32 Thermal pain (261) 5 1.00 56 −16 0 Audition (14) 6 1.00 0 8 32 Temperature sensation (204) 7 1.00 0 8 32 Somesthesis (17) 8 0.99 0 −56 16 Memory retrieval (24) 9 0.99 0 8 32 Warm temperature sensation (207) 10 0.99 24 −8 −8 Unpleasantness (153) 11 0.99 56 −16 0 Voice (167) 12 0.99 0 −56 16 Memory (9) 13 0.99 24 −8 −8 Emotion (3) 14 0.99 0 −56 16 Long-term memory (112) 15 0.99 0 −56 16 Declarative memory (319) Table 1: The top 15 elements of the list, showing the external components that score the highest, the distance to the null-hypothesis d, and the associated Talairach x, y and z coordinates. The numbers in the parentheses are the Brede database identifiers for the external components (WOEXT). This list was generated with coarse 8 × 8 × 8mm3 voxels and using the non-linear model approximation for the cumulative distribution functions. appearing in left and right superior temporal gyrus, and memory emerging in the posterior cingulate area. Unpleasantness and emotion are high on the list due to, e.g., “fear” and “disgust” experiments that report activation in the right amygdala and nearby areas. An example of the functional labeling of a voxel appears in Table 2. The chosen voxel is (0, −56, 16) that appears in the posterior cingulate. Memory retrieval is the first on the list in accordance with Table 1. Many of the other external components on the list are also related to memory. 4 Discussion The Brede database contains many thermal pain experiments, and it causes high scores for voxels from external components such as “pain” and “thermal pain”. The four focal “brain activations” that appear in Figure 3 are localized in areas (anterior cingulate, insula and thalamus) that an expert reviewer has previously identified as important in pain [14]. Thus there is consistency between our automated metaanalytic technique and a “manual” expert review. Many experiments that report activation in the posterior cingulate area have been included in the Brede database, and this is probably why memory is especially associated with this area. A major review of 275 functional neuroimaging studies found that episodic memory retrieval is the cognitive function with highest association with the posterior cingulate [15], so our finding is again in alignment with an Figure 3: Plot of the important areas associated with the external component “pain”. The red opaque isosurface is on the level d = 0.95 in the distance volume while the gray transparent surface appears at d = 0.05. Yellow glyphs appear at the local maxima in the thresholded volume. The viewpoint is situated nearest to the left superior posterior corner of the brain. expert review. A number of the substantial associations between brain areas and external components are not surprising, e.g., audition associating with superior temporal gyrus. Our method has no inherent knowledge of what is already known, and thus not able distinguish novel associations from trivial. A down-side with the present method is that it requires the labeling of experiments during database entry and the construction of the hierarchy of the labels (Figure 1). Both are prone to “interpretation” and this is particularly a problem for complex cognitive functions. Our methodology, however, does not necessarily impose a single organization of the external components, and it is possible to rearrange these by defining another adjacency matrix for the graph. In Table 1 the brain areas are represented in terms of Talairach coordinates. It should be possible to convert these coordinates further to neuroanatomical terms # d Name (WOEXT) 1 0.99 Memory retrieval (24) 2 0.99 Memory (9) 3 0.99 Long-term memory (112) 4 0.99 Declarative memory (319) 5 0.99 Episodic memory (49) 6 0.96 Autobiographical memory (259) 7 0.94 Cognition (2) 8 0.94 Episodic memory retrieval (109) 9 0.58 Disease (79) 10 0.16 Recognition (190) 11 0.14 Psychiatric disorders (82) 12 0.14 Neurotic, stress and somatoform disorders (227) 13 0.11 Severe stress reactions and adjustment disorders (228) 14 0.09 Emotion (3) 15 0.02 Semantic memory (318) Table 2: Example of a functional label list of a voxel v = (0, −56, 16) in the posterior cingulate area. by using the models between coordinates and lobar anatomy that we previously have established [4, 6]. Functional labeling should allow us to build a complete functional atlas for the entire brain. The utility of this approach is, however, limited by the small size of the Brede database and its bias towards specific brain regions and external components. But such a functional atlas will serve as a neuroinformatic organizer for the increasing number of neuroimaging studies. Acknowledgment I am grateful to Matthew G. Liptrot for reading and commenting on the manuscript. Lars Kai Hansen is thanked for discussion, Andrew C. N. Chen for identifying some of the thermal pain studies and the Villum Kann Rasmussen Foundation for their generous support of the author. References [1] Jean Talairach and Pierre Tournoux. Co-planar Stereotaxic Atlas of the Human Brain. Thieme Medical Publisher Inc, New York, January 1988. [2] Peter T. Fox and Jack L. Lancaster. Mapping context and content: the BrainMap model. Nature Reviews Neuroscience, 3(4):319–321, April 2002. [3] Finn ˚Arup Nielsen. The Brede database: a small database for functional neuroimaging. NeuroImage, 19(2), June 2003. Presented at the 9th International Conference on Functional Mapping of the Human Brain, June 19–22, 2003, New York, NY. Available on CD-Rom. [4] Finn ˚Arup Nielsen and Lars Kai Hansen. Modeling of activation data in the BrainMapTM database: Detection of outliers. Human Brain Mapping, 15(3):146–156, March 2002. [5] Finn ˚Arup Nielsen and Lars Kai Hansen. Finding related functional neuroimaging volumes. Artificial Intelligence in Medicine, 30(2):141–151, February 2004. [6] Finn ˚Arup Nielsen and Lars Kai Hansen. Automatic anatomical labeling of Talairach coordinates and generation of volumes of interest via the BrainMap database. NeuroImage, 16(2), June 2002. Presented at the 8th International Conference on Functional Mapping of the Human Brain, June 2–6, 2002, Sendai, Japan. Available on CD-Rom. [7] Matthew Brett. The MNI brain and the Talairach atlas. http://www.mrccbu.cam.ac.uk/Imaging/mnispace.html, August 1999. Accessed 2003 March 17. [8] Peter E. Turkeltaub, Guinevere F. Eden, Karen M. Jones, and Thomas A. Zeffiro. Meta-analysis of the functional neuroanatomy of single-word reading: method and validation. NeuroImage, 16(3 part 1):765–780, July 2002. [9] J. M. Chein, K. Fissell, S. Jacobs, and Julie A. Fiez. Functional heterogeneity within Broca’s area during verbal working memory. Physiology & Behavior, 77(4-5):635–639, December 2002. [10] Lionel S. Penrose. The elementary statistics of majority voting. Journal of the Royal Statistical Society, 109:53–57, 1946. [11] Andrew. P. Holmes, R. C. Blair, J. D. G. Watson, and I. Ford. Non-parametric analysis of statistic images from functional mapping experiments. Journal of Cerebral Blood Flow and Metabolism, 16(1):7–22, January 1996. [12] Claus Svarer, Lars Kai Hansen, and Jan Larsen. On the design and evaluation of tapped-delay lines neural networks. In Proceedings of the IEEE International Conference on Neural Networks, San Francisco, California, USA, volume 1, pages 46–51, 1993. [13] Lars Kai Hansen, Finn ˚Arup Nielsen, Peter Toft, Matthew George Liptrot, Cyril Goutte, Stephen C. Strother, Nicholas Lange, Anders Gade, David A. Rottenberg, and Olaf B. Paulson. “lyngby” — a modeler’s Matlab toolbox for spatio-temporal analysis of functional neuroimages. NeuroImage, 9(6):S241, June 1999. [14] Martin Ingvar. Pain and functional imaging. Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 354(1387):1347–1358, July 1999. [15] Roberto Cabeza and Lars Nyberg. Imaging cognition II: An empirical review of 275 PET and fMRI studies. Journal of Cognitive Neuroscience, 12(1):1–47, January 2000.
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Result Analysis of the NIPS 2003 Feature Selection Challenge Isabelle Guyon ClopiNet Berkeley, CA 94708, USA isabelle@clopinet.com Steve Gunn School of Electronics and Computer Science University of Southampton, U.K. s.r.gunn@ecs.soton.ac.uk Asa Ben Hur Department of Genome Sciences University of Washington, USA asa@gs.washington.edu Gideon Dror Department of Computer Science Academic College of Tel-Aviv-Yaffo, Israel gideon@mta.ac.il Abstract The NIPS 2003 workshops included a feature selection competition organized by the authors. We provided participants with five datasets from different application domains and called for classification results using a minimal number of features. The competition took place over a period of 13 weeks and attracted 78 research groups. Participants were asked to make on-line submissions on the validation and test sets, with performance on the validation set being presented immediately to the participant and performance on the test set presented to the participants at the workshop. In total 1863 entries were made on the validation sets during the development period and 135 entries on all test sets for the final competition. The winners used a combination of Bayesian neural networks with ARD priors and Dirichlet diffusion trees. Other top entries used a variety of methods for feature selection, which combined filters and/or wrapper or embedded methods using Random Forests, kernel methods, or neural networks as a classification engine. The results of the benchmark (including the predictions made by the participants and the features they selected) and the scoring software are publicly available. The benchmark is available at www.nipsfsc.ecs.soton.ac.uk for post-challenge submissions to stimulate further research. 1 Introduction Recently, the quality of research in Machine Learning has been raised by the sustained data sharing efforts of the community. Data repositories include the well known UCI Machine Learning repository [13], and dozens of other sites [10]. Yet, this has not diminished the importance of organized competitions. In fact, the proliferation of datasets combined with the creativity of researchers in designing experiments makes it hardly possible to compare one paper with another [12]. A number of large conferences have regularly organized competitions (e.g. KDD, CAMDA, ICDAR, TREC, ICPR, and CASP). The NIPS workshops offer an ideal forum for organizing such competitions. In 2003, we organized a competition on the theme of feature selection, the results of which were presented at a workshop on feature extraction, which attracted 98 participants. We are presently preparing a book combining tutorial chapters and papers from the proceedings of that workshop [9]. In this paper, we present to the NIPS community a concise summary of our challenge design and the findings of the result analysis. 2 Benchmark design We formatted five datasets (Table 1) from various application domains. All datasets are two-class classification problems. The data were split into three subsets: a training set, a validation set, and a test set. All three subsets were made available at the beginning of the benchmark, on September 8, 2003. The class labels for the validation set and the test set were withheld. The identity of the datasets and of the features (some of which were random features artificially generated) were kept secret. The participants could submit prediction results on the validation set and get their performance results and ranking on-line for a period of 12 weeks. By December 1st, 2003, which marked the end of the development period, the participants had to turn in their results on the test set. Immediately after that, the validation set labels were revealed. On December 8th, 2003, the participants could make submissions of test set predictions, after having trained on both the training and the validation set. Some details on the benchmark design are provided in this Section. Challenge format We gave our benchmark the format of a challenge to stimulate participation. We made available an automatic web-based system to submit prediction results and get immediate feed-back, inspired by the system of the NIPS2000 and NIPS2001 unlabelled data competitions [4, 5]. However, unlike what had been done for these other competitions, we used a “validation set” to assess performance during the development period, and a separate “test set” for final scoring. During development participants could submit validation results on any of the five datasets proposed (not necessarily all). Competitors were required to submit results on all five test sets by the challenge deadline to be included in the final ranking. This avoided a common problem of “multiple track” benchmarks in which no conclusion can be drawn because too few participants enter all tracks. To promote collaboration between researchers, reduce the level of anxiety, and let people explore various strategies (e.g. “pure” methods and “hybrids”), we allowed participating groups to submit a total of five final entries on December 1st and five entries on December 8th. Our format was very successful: it attracted 78 research groups who competed for 13 weeks and made (submitted) a total of 1863 entries. Twenty groups were eligible for being ranked on December 1st (56 submissions1), and 16 groups on December 8th (36 submissions.) The feature selection benchmark web site at www.nipsfsc.ecs.soton.ac.uk remains available as a resource for researchers in the feature selection. 1After imposing a maximum of 5 submissions per group and eliminating some incomplete submissions, there remained 56 eligible submissions out of the 135 received. Table 1: NIPS 2003 challenge datasets. For each dataset we show the domain it was taken from, its type (dense, sparse, or sparse binary), the number of features, the percentage of probes, and the number of examples in the training, validation, and test sets. All problems are two-class classification problems. Dataset Domain Type #Fe %Pr #Tr #Val #Te Arcene Mass Spectrometry Dense 10000 30 100 100 700 Dexter Text classification Sparse 20000 50 300 300 2000 Dorothea Drug discovery Sparse binary 100000 50 800 350 800 Gisette Digit recognition Dense 5000 30 6000 1000 6500 Madelon Artificial Dense 500 96 2000 600 1800 The challenge datasets Until the late 90s most published papers on feature selection considered datasets with less than 40 features2 (see [1, 11] from a 1997 special issue on relevance for example). The situation has changed considerably in the past few years, and in the 2003 special issue we edited for JMLR including papers from the proceedings of the NIPS 2001 workshop [7], most papers explore domains with hundreds to tens of thousands of variables or features. The applications are driving this effort: bioinformatics, chemistry (drug design, cheminformatics), text processing, pattern recognition, speech processing, and machine vision provide machine learning problems in very high dimensional spaces, but often with comparably few examples. Feature selection is a particular way of tackling the problem of space dimensionality reduction. The necessary computing power to handle large datasets is now available in simple laptops, so there is a proliferation of solutions proposed for such feature selection problems. Yet, there does not seem to be an emerging unity of experimental design and algorithms. We formatted five datasets for the purpose of benchmarking variable selection algorithms (see Table 1.) The datasets were chosen to span a variety of domains and difficulties (the input variables are continuous or binary, sparse or dense; one dataset has unbalanced classes.) One dataset (Madelon) was artificially constructed to illustrate a particular difficulty: selecting a feature set when no feature is informative by itself. We chose datasets that had sufficiently many examples to create a large enough test set to obtain statistically significant results [6]. To prevent researchers familiar with the datasets to have an advantage, we concealed the identity of the datasets during the benchmark. We performed several preprocessing and data formatting steps, which contributed to disguising the origin of the datasets. In particular, we introduced a number of features called probes. The probes were drawn at random from a distribution resembling that of the real features, but carrying no information about the class labels. Such probes have a function in performance assessment: a good feature selection algorithm should eliminate most of the probes. The details of data preparation can be found in a technical memorandum [6]. 2In this paper, we do not make a distinction between features and variables. The benchmark addresses the problem of selecting input variables. Those may actually be features derived from the original variables through preprocessing. Table 2: We show the top entries sorted by their score (times 100), the balanced error rate in percent (BER) and corresponding rank in parenthesis, the area under the ROC curve times 100 (AUC) and corresponding rank in parenthesis, the percentage of features used (Fe), and the percentage of probes in the features selected (Pr). (a) December 1st 2003 challenge results. Method (Team) Score BER AUC Fe Pr BayesNN-DFT (Neal/Zhang) 88.0 6.84 (1) 97.22 (1) 80.3 47.8 BayesNN-DFT (Neal/Zhang) 86.2 6.87 (2) 97.21 (2) 80.3 47.8 BayesNN-small (Neal) 68.7 8.20 (3) 96.12 (5) 4.7 2.9 BayesNN-large (Neal) 59.6 8.21 (4) 96.36 (3) 60.3 28.5 RF+RLSC (Torkkola/Tuv) 59.3 9.07 (7) 90.93 (29) 22.5 17.5 final2 (Chen) 52.0 9.31 (9) 90.69 (31) 24.9 12.0 SVMBased3 (Zhili/Li) 41.8 9.21 (8) 93.60 (16) 29.5 21.7 SVMBased4 (Zhili/Li) 41.1 9.40 (10) 93.41 (18) 29.5 21.7 final1 (Chen) 40.4 10.38 (23) 89.62 (34) 6.2 6.1 transSVM2 (Zhili) 36.0 9.60 (13) 93.21 (20) 29.5 21.7 BayesNN-E (Neal) 29.5 8.43 (5) 96.30 (4) 96.8 56.7 Collection2 (Saffari) 28.0 10.03 (20) 89.97 (32) 7.7 10.6 Collection1 (Saffari) 20.7 10.06 (21) 89.94 (33) 32.3 25.5 (b) December 8th 2003 challenge results. Method (Team) Score BER AUC Fe Pr BayesNN-DFT (Neal/Zhang) 71.4 6.48 (1) 97.20 (1) 80.3 47.8 BayesNN-large (Neal) 66.3 7.27 (3) 96.98 (3) 60.3 28.5 BayesNN-small (Neal) 61.1 7.13 (2) 97.08 (2) 4.7 2.9 final 2-3 (Chen) 49.1 7.91 (8) 91.45 (25) 24.9 9.9 BayesNN-large (Neal) 49.1 7.83 (5) 96.78 (4) 60.3 28.5 final2-2 (Chen) 40.0 8.80 (17) 89.84 (29) 24.6 6.7 Ghostminer1 (Ghostminer) 37.1 7.89 (7) 92.11 (21) 80.6 36.1 RF+RLSC (Torkkola/Tuv) 35.4 8.04 (9) 91.96 (22) 22.4 17.5 Ghostminer2 (Ghostminer) 35.4 7.86 (6) 92.14 (20) 80.6 36.1 RF+RLSC (Torkkola/Tuv) 34.3 8.23 (12) 91.77 (23) 22.4 17.5 FS+SVM (Lal) 31.4 8.99 (19) 91.01 (27) 20.9 17.3 Ghostminer3 (Ghostminer) 26.3 8.24 (13) 91.76 (24) 80.6 36.1 CBAMethod3E (CBAGroup) 21.1 8.14 (10) 96.62 (5) 12.8 0.1 CBAMethod3E (CBAGroup) 21.1 8.14 (11) 96.62 (6) 12.8 0.1 Nameless (Navot/Bachrach) 12.0 7.78 (4) 96.43 (9) 32.3 16.2 Performance assessment Final submissions qualified for scoring if they included the class predictions for training, validation, and test sets for all five tasks proposed, and the list of features used. Optionally, classification confidence values could be provided. Performance was assessed using several metrics: • BER: The balanced error rate, that is the average of the error rate of the positive class and the error rate of the negative class. This metric was used because some datasets (particularly Dorothea) are unbalanced. • AUC: Area under the ROC curve. The ROC curve is obtained by varying a threshold on the discriminant values (outputs) of the classifier. The curve represents the fraction of true positive as a function of the fraction of false negative. For classifiers with binary outputs, BER=1-AUC. • Ffeat: Fraction of features selected. • Fprobe: Fraction of probes found in the feature set selected. We ranked the participants with the test set results using a score combining BER, Ffeat and Fprobe. Briefly: We used the McNemar test to determine whether classifier A is better than classifier B according to the BER with 5% risk yielding to a score of 1 (better), 0 (don’t know) or 1 (worse). Ties (zero score) were broken with Ffeat (if the relative difference in Ffeat was larger than 5%.) Remaining ties were broken with Fprobe. The overall score for each dataset is the sum of the pairwise comparison scores (normalized by the maximum achievable score, that is the number of submissions minus one.) The code is provided on the challenge website. The scores were averaged over the five datasets. Our scoring method favors accuracy over feature set compactness. Our benchmark design could not prevent participants from “cheating” in the following way. An entrant could “declare” a smaller feature subset than the one used to make predictions. To deter participants from cheating, we warned them that we would be performing a stage of verification. We performed several checks as detailed in [9] and did not find any entry that should be suspected of being fraudulent. 3 Challenge results The overall scores of the best entries are shown in Table 2. The main features of the methods of the participants listed in that table are summarized in Table 3. The analysis of this section also includes the survey of ten more top ranking participants. Winners The winners of the benchmark (both December 1st and 8th) are Radford Neal and Jianguo Zhang, with a combination of Bayesian neural networks [14] and Dirichlet diffusion trees [15]. Their achievements are significant since they win on the overall ranking with respect to our scoring metric, but also with respect to the balanced error rate (BER), the area under the ROC curve (AUC), and they have the smallest feature set among the top entries that have performance not statistically significantly worse. They are also the top entrants December 1st for Arcene and Dexter and December 1st and 8th for Dorothea. Two aspects of their approach were the same for all data sets: • They reduced the number of features used for classification to no more than a few hundred, either by selecting a subset of features using simple univariate significance tests, or by Principal Component Analysis (PCA) performed on all available labeled and unlabeled data. • They then applied a classification method based on Bayesian learning, using an Automatic Relevance Determination (ARD) prior that allows the model to determine which of these features are most relevant. Bayesian neural network learning with computation by Markov chain Monte Carlo (MCMC) is a well developed technology [14]. Dirichlet diffusion trees are a new Bayesian approach to density modeling and hierarchical clustering. As allowed by the challenge rules, the winners constructed these trees using both the training data and the unlabeled data in the validation and test sets. Classification was then performed with the k-nearest neighbors method, using the metric induced by the tree. Table 3: Methods employed by the challengers. The classifiers are grouped in four categories: N - neural network, K - SVM or other kernel method, T tree classifiers (none found in the top ranking methods), O - other. The feature selection engines (Fengine) are grouped in three categories: C - single variable criteria including correlation coefficients, T - tree classifiers or RF used as a filter E - Wrapper or embedded methods. The search methods are identified by: E embedded, R - feature ranking, B - backward elimination, S - more elaborated search. Team Classifier Fengine Fsearch Ensemble Transduction Neal/Zhang N/O C/E E Yes Yes Torkkola/Tuv K T R Yes No Chen/Lin K C/T/E R/E No No Zhili/Li K C/E E No Yes Saffari N C R Yes No Ghostminer K C/T B Yes No Lal et al K C R No No CBAGroup K C R No No Bachrach/Navot K/O E S No No Other methods employed We group methods into coarse categories to draw useful conclusions. Our findings include: Feature selection The winners and several top ranking challengers use a combination of filters and embedded methods3. Several high ranking participants obtain good results using only filters, even simple correlation coefficients. The second best entrants use Random Forests, an ensemble of tree classifiers, to perform feature selection [3].4 Search strategies are generally unsophisticated (simple feature ranking, forward selection or backward elimination.) Only 2 out of 19 in our survey used a more sophisticated search strategy. The selection criterion is usually based on cross-validation. A majority use K-fold, with K between 3 and 10. Only one group used “random probes” purposely introduced to track the fraction of falsely selected features. One group used the area under the ROC curve computed on the training set. Classifier Kernel methods [16] are most popular: 7/9 in Table 3 and 12/19 in the survey. Of the 12 kernel methods employed, 8 are SVMs. In spite of the high risk of overfitting, 7 of the 9 top groups using kernel methods found that Gaussian kernels gave them better results than the linear kernel on Arcene, Dexter, Dorothea, or Gisette (for Madelon all best ranking groups used a Gaussian kernel.) Ensemble methods Some groups relied on a committee of classifiers to make the final decision. The techniques to build such committee include sampling 3We distinguish embedded methods that have a feature selection mechanism built into the learning algorithm from wrappers, which perform feature selection by using the classifier as a black box. 4Random Forests (RF) are classification techniques with an embedded feature selection mechanism. The participants used the features generated by RF, but did not use RF for classification. from the posterior distribution using MCMC [14] and bagging [2]. Most groups that used ensemble methods reported improved accuracy. Transduction Since all the datasets were provided since the beginning of the benchmark (validation and test set deprived of their class labels), it was possible to make use of the unlabelled data as part of learning (sometimes referred to as transduction [17]). Only two groups took advantage of that, including the winners. Preprocessing Centering and scaling the features was the most common preprocessing used. Some methods required discretization of the features. One group normalized the patterns. Principal Componant Analysis (PCA) was used by several groups, including the winners, as a means of constructing features. 4 Conclusions and future work The challenge demonstrated both that feature selection can be performed effectively and that eliminating meaningless features is not critical to achieve good classification performance. By design, our datasets include many irrelevant “distracters” features, called “probes”. In contrast with redundant features, which may not be needed to improve accuracy but carry information, those distracters are “pure noise”. It is surprising that some of the best entries use all the features. Still, there is always another entry close in performance, which uses only a small fraction of the original features. The challenge outlined the power of filter methods. For many years, filter methods have dominated feature selection for computational reasons. It was understood that wrapper and embedded methods are more powerful, but too computationally expensive. Some of the top ranking entries use one or several filters as their only selection strategy. A filter as simple as the Pearson correlation coefficient proves to be very effective, even though it does not remove feature redundancy and therefore yields unnecessarily large feature subsets. Other entrants combined filters and embedded methods to further reduce the feature set and eliminate redundancies. Another important outcome is that non-linear classifiers do not necessarily overfit. Several challenge datasets included a very large number of features (up to 100,000) and only a few hundred examples. Therefore, only methods that avoid overfitting can succeed in such adverse aspect ratios. Not surprisingly, the winning entries use as classifies either ensemble methods or strongly regularized classifiers. More surprisingly, non-linear classifiers often outperform linear classifiers. Hence, with adequate regularization, non-linear classifiers do not overfit the data, even when the number of features exceeds the number of examples by orders of magnitude. Principal Component Analysis was successfully used by several researchers to reduce the dimension of input space down to a few hundred features, without any knowledge of the class labels. This was not harmful to the prediction performances and greatly reduced the computational load of the learning machines. The analysis of the challenge results revealed that hyperparameter selection may have played an important role in winning the challenge. Indeed, several groups were using the same classifier (e.g. an SVM) and reported significantly different results. We have started laying the basis of a new benchmark on the theme of model selection and hyperparameter selection [8]. Acknowledgments We are very thankful to the institutions that have contributed data: the National Cancer Institute (NCI), the Eastern Virginia Medical School (EVMS), the National Institute of Standards and Technology (NIST), DuPont Pharmaceuticals Research Laboratories, Reuters Ltd., and the Carnegie Group, Inc. We also thank the people who formatted the data and made them available: Thorsten Joachims, Yann Le Cun, and the KDD Cup 2001 organizers. We thank Olivier Chapelle for providing ideas and corrections. The workshop co-organizers and advisors Masoud Nikravesh, Kristin Bennett, Richard Caruana, and Andr´e Elisseeff, are gratefully acknowledged for their help, and advice, in particular with result dissemination. References [1] A. Blum and P. Langley. Selection of relevant features and examples in machine learning. Artificial Intelligence, 97(1-2):245–271, December 1997. [2] Leo Breiman. Bagging predictors. Machine Learning, 24(2):123–140, 1996. [3] Leo Breiman. Random forests. Machine Learning, 45(1):5–32, 2001. [4] S. Kremer, et al. NIPS 2000 unlabeled data competition. http://q.cis.uoguelph.ca/~skremer/Research/NIPS2000/, 2000. [5] S. Kremer, et al. NIPS 2001 unlabeled data competition. http://q.cis.uoguelph.ca/~skremer/Research/NIPS2001/, 2001. [6] I. Guyon. Design of experiments of the NIPS 2003 variable selection benchmark. http://www.nipsfsc.ecs.soton.ac.uk/papers/Datasets.pdf, 2003. [7] I. Guyon and A. Elisseeff. An introduction to variable and feature selection. JMLR, 3:1157–1182, March 2003. [8] I. Guyon and S. Gunn. Model selection and ensemble methods challenge in preparation http://clopinet.com/isabelle/projects/modelselect. [9] I. Guyon, S. Gunn, M. Nikravesh, and L. Zadeh, Editors. Feature Extraction, Foundations and Applications. Springer-Verlag, http://clopinet.com/isabelle/Projects/NIPS2003/call-for-papers.html, In preparation. See also on-line supplementary material: http://clopinet.com/isabelle/Projects/NIPS2003/analysis.html. [10] D. Kazakov, L. Popelinsky, and O. Stepankova. MLnet machine learning network on-line information service. In http://www.mlnet.org. [11] R. Kohavi and G. John. Wrappers for feature selection. Artificial Intelligence, 97(1-2):273–324, December 1997. [12] D. LaLoudouana and M. Bonouliqui Tarare. Data set selection. In NIPS02 http://www.jmlg.org/papers/laloudouana03.pdf, 2002. [13] P. M. Murphy and D. W. Aha. UCI repository of machine learning databases. In http://www.ics.uci.edu/~mlearn/MLRepository.html, 1994. [14] R. M. Neal. Bayesian Learning for Neural Networks. Number 118 in Lecture Notes in Statistics. Springer-Verlag, New York, 1996. [15] R. M. Neal. Defining priors for distributions using dirichlet diffusion trees. Technical Report 0104, Dept. of Statistics, University of Toronto, March 2001. [16] B. Schoelkopf and A. Smola. Learning with Kernels – Support Vector Machines, Regularization, Optimization and Beyond. MIT Press, Cambridge MA, 2002. [17] V. Vapnik. 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The power of feature clustering: An application to object detection Shai Avidan Mitsibishi Electric Research Labs 201 Broadway Cambridge, MA 02139 avidan@merl.com Moshe Butman Adyoron Intelligent Systems LTD. 34 Habarzel St. Tel-Aviv, Israel mosheb@adyoron.com Abstract We give a fast rejection scheme that is based on image segments and demonstrate it on the canonical example of face detection. However, instead of focusing on the detection step we focus on the rejection step and show that our method is simple and fast to be learned, thus making it an excellent pre-processing step to accelerate standard machine learning classifiers, such as neural-networks, Bayes classifiers or SVM. We decompose a collection of face images into regions of pixels with similar behavior over the image set. The relationships between the mean and variance of image segments are used to form a cascade of rejectors that can reject over 99.8% of image patches, thus only a small fraction of the image patches must be passed to a full-scale classifier. Moreover, the training time for our method is much less than an hour, on a standard PC. The shape of the features (i.e. image segments) we use is data-driven, they are very cheap to compute and they form a very low dimensional feature space in which exhaustive search for the best features is tractable. 1 Introduction This work is motivated by recent advances in object detection algorithms that use a cascade of rejectors to quickly detect objects in images. Instead of using a full fledged classifier on every image patch, a sequence of increasingly more complex rejectors is applied. Nonface image patches will be rejected early on in the cascade, while face image patches will survive the entire cascade and will be marked as a face. The work of Viola & Jones [15] demonstrated the advantages of such an approach. Other researchers suggested similar methods [4, 6, 12]. Common to all these methods is the realization that simple and fast classifiers are enough to reject large portions of the image, leaving more time to use more sophisticated, and time consuming, classifiers on the remaining regions of the image. All these “fast” methods must address three issues. First, is the feature space in which to work, second is a fast method to calculate the features from the raw image data and third is the feature selection algorithm to use. Early attempts assumed the feature space to be the space of pixel values. Elad et al. [4] suggest the maximum rejection criteria that chooses rejectors that maximize the rejection rate of each classifier. Keren et al. [6] use anti-face detectors by assuming normal distribution on the background. A different approach was suggested by Romdhani et al. [12], that constructed the full SVM classifier first and then approximated it with a sequence or support vector rejectors that were calculated using non-linear optimization. All the above mentioned method need to “touch” every pixel in an image patch at least once before they can reject the image patch. Viola & Jones [15], on the other hand, construct a huge feature space that consists of combined box regions that can be quickly computed from the raw pixel data using the “integral image” and use a sequential feature selection algorithm for feature selection. The rejectors are combined using a variant of AdaBoost [2]. Li et al [7] replaced the sequential forward searching algorithm with a float search algorithm (which can backtrack as well). An important advantage of the huge feature space advocated by Viola & Jones is that now image patches can be rejected with an extremely small number of operations and there is no need to “touch” every pixel in the image patch at least once. Many of these methods focus on developing fast classifiers that are often constructed in a greedy manner. This precludes classifiers that might demonstrate excellent classification results but are slower to compute, such as the methods suggested by Schneiderman et al. [8], Rowley et al. [13], Sung and Poggio [10] or Heisele et al [5]. Our method offers a way to accelerate “slow” classification methods by using a preprocessing rejection step. Our rejection scheme is fast to be trained and very effective in rejecting the vast majority of false patterns. On the canonical face detection example, it took our method much less than an hour to train and it was able to reject over 99.8% of the image patches, meaning that we can effectively accelerate standard classifiers by several orders of magnitude, without changing the classifier at all. Like other, “fast”, methods we use a cascade of rejectors, but we use a different type of filters and a different type of feature selection method. We take our features to be the approximated mean and variance of image segments, where every image segment consists of pixels that have similar behavior across the entire image set. As a result, our features are derived from the data and do not have to be hand crafted for the particular object of interest. In fact they do not even have to form contiguous regions. We use only a small number of representative pixels to calculate the approximated mean and variance, which makes our features very fast to compute during detection (in our experiments we found that our first rejector rejects almost 50% of all image patches, using just 8 pixels). Finally, the number of segments we use is quite small which makes it possible to exhaustively calculate all possible rejectors based on single, pairs and triplets of segments in order to find the best rejectors in every step of the cascade. This is in contrast to methods that construct a huge feature bank and use a greedy feature selection algorithm to choose “good” features from it. Taken together, our algorithm is fast to train and fast to test. In our experiments we train on a database that contains several thousands of face images and roughly half-a-million non-faces in less than an hour on an average PC and our rejection module runs at several frames per second. 2 Algorithm At the core of our algorithm is the realization that feature representation is a crucial ingredient in any classification system. For instance, the Viola-Jones box filters are extremely efficient to compute using the “integral image” but they form a large feature space, thus placing a heavy computational burden on the feature selection algorithm that follows. Moreover, empirically they show that the first feature selected by their method correspond to meaningful regions in the face. This suggests that it might be better to focus on features that correspond to coherent regions in the image. This leads to the idea of image segmentation, that breaks an ensemble of images into regions of pixels that exhibit similar temporal behavior. Given the image segmentation we take our features to be the mean and variance of each segment, giving us a very small feature space to work on (we chose to segment the face image into eight segments). Unfortunately, calculating the mean and variance of an image segment requires going over all the pixels in the segment, a time consuming process. However, since the segments represent similar-behaving pixels we found that we can approximate the calculation of the mean and variance of the entire segment using quite a small number of representative pixels. In our experiments, four pixels were enough to adequately represent segments that contain several tens of pixels. Now that we have a very small feature space to work with, and a fast way to extract features from raw pixels data we can exhaustively search for all possible combinations of single, pairs or triplets of features to find the best rejector in every stage. The remaining patterns should be passed to a standard classifier for final validation. 2.1 Image Segments Image segments were already presented in the past [1] for the problem of classification of objects such as faces or vehicles. We briefly repeat the presentation for the paper to be self-contained. An ensemble of scaled, cropped and aligned images of a given object (say faces) can be approximated by its leading principal components. This is done by stacking the images (in vector form) in a design matrix A and taking the leading eigenvectors of the covariance matrix C = 1 N AAT , where N is the number of images. The leading principal components are the leading eigenvectors of the covariance matrix C and they form a basis that approximates the space of all the columns of the design matrix A [11, 9]. But instead of looking at the columns of A look at the rows of A. Each row in A gives the intensity profile of a particular pixel, i.e., each row represents the intensity values that a particular pixel takes in the different images in the ensemble. If two pixels come from the same region of the face they are likely to have the same intensity values and hence have a strong temporal correlation. We wish to find this correlations and segment the image plane into regions of pixels that have similar temporal behavior. This approach broadly falls under the category of Factor Analysis [3] that seeks to find a low-dimensional representation that captures the correlations between features. Let Ax be the x-th row of the design matrix A. Then Ax is the intensity profile of pixel x (We address pixels with a single number because the images are represented in a scan-line vector form). That is, Ax is an N-dimensional vector (where N is the number of images) that holds the intensity values of pixel x in each image in the ensemble. Pixels x and y are temporally correlated if the dot product of rows Ax and Ay is approaching 1 and are temporally uncorrelated if the dot-product is approaching 0. Thus, to find temporally correlated pixels all we need to do is run a clustering algorithm on the rows of the design matrix A. In particular, we used the k-means algorithm on the rows of the matrix A but any method of Factor Analysis can be used. As a result, the image-plane is segmented into several (possibly non-continuous) segments of temporally correlated pixels. Experiments in the past [1] showed good classification results on different objects such as faces and vehicles. 2.2 Finding Representative Pixels Our algorithm works by comparing the mean and variance properties of one or more image segments. Unfortunately this requires touching every pixel in the image segment during test time, thus slowing the classification process considerably. Therefor, during train time we find a set of representative pixels that will be used during test time. Specifically, we approximate every segment in a face image with a small number of representative pixels Face segments 2 4 6 8 10 12 14 16 18 20 2 4 6 8 10 12 14 16 18 20 (a) (b) Figure 1: Face segmentation and representative pixels. (a) Face segmentation and representative pixels. The face segmentation was computed using 1400 faces, each segment is marked with a different color and the segments need not be contiguous. The crosses overlaid on the segments mark the representative pixels that were automatically selected by our method. (b) Histogram of the difference between an approximated mean and the exact mean of a particular segment (the light blue segment on the left). The histogram is peaked at zero, meaning that the representative pixels give a good approximation. that approximate the mean and variance of the entire image segment. Define µi(xj) to be the true mean of segment i of face j, and let ˆµi(xj) be its approximation, defined as ˆµi(xj) = Pk j=1 xj k where {xj}k j=1 are a subset of pixels in segment i of pattern j. We use a greedy algorithm that incrementally searches for the next representative pixel that minimize n X j=1 (ˆµi(xj)) −µi(xj))2 and add it to the collection of representative pixels of segment i. In practice we use four representative pixels per segment. The representative pixels computed this way are used for computing both the approximated mean and the approximated variance of every test pattern. Figure 1 show how well this approximation works in practice. Given the representative pixels, the approximated variance ˆσi(xj) of segment i of pattern j is given by: ˆσi(xj) = k X j=1 |xj −ˆµi(xj)| 2.3 The rejection cascade We construct a rejection cascade that can quickly reject image patches, with minimal computational load. Our feature space consist of the approximated mean and variance of the image segments. In our experiments we have 8 segments, each represented by its mean and variance, giving rise to a 16D feature space. This feature space is very fast to compute, as we need only four pixels to calculate the approximate mean and variance of the segment. Because the feature space is so small we can exhaustively search for all classifiers on single, pairs and triplets of segments. In addition this feature space gives enough information to reject texture-less regions without the need to normalize the mean or variance of the entire image patch. We next describe our rejectors in detail. 2.3.1 Feature rejectors Now, that we have segmented every image into several segments and approximated every segment with a small number of representative pixels, we can exhaustively search for the best combination of segments that will reject the largest number of non-face images. We repeat this process until the improvement in rejection is negligible. Given a training set of P positive examples (i.e. faces) and N negative examples we construct the following linear rejectors and adjust the parameter θ so that they will correctly classify d · P (we use d = 0.95) of the face images and save r, the number of negative examples they correctly rejected, as well as the parameter θ. 1. For each segment i, find a bound on its approximated mean. Formally, find θ s.t. ˆµi(x) > θ or ˆµi(x) < θ 2. For each segment i, find a bound on its approximated variance. Formally, find θ s.t. ˆσi(x) > θ or ˆσi(x) < θ 3. For each pair of segments i, j, find a bound on the difference between their approximated means. Formally, find θ s.t. ˆµi(x) −ˆµj(x) > θ or ˆµi(x) −ˆµj(x) < θ 4. For each pair of segments i, j, find a bound on the difference between their approximated variance. Formally, find θ s.t. ˆσi(x) −ˆσj(x) > θ or ˆσi(x) −ˆσj(x) < θ 5. For each triplet of segments i, j, k find a bound on the difference of the absolute difference of their approximated means. Formally, find θ s.t. |ˆµi(x) −ˆµj(x)| −|ˆµi(x) −ˆµk(x)| > θ This process is done only once to form a pool of rejectors. We do not re-train rejectors after selecting a particular rejector. 2.3.2 Training We form the cascade of rejectors from a large pattern vs. rejector binary table T, where each entry T(i, j) is 1 if rejector j rejects pattern i. Because the table is binary we can store every entry in a single bit and therefor a table of 513, 000 patterns and 664 rejectors can easily fit in the memory. We then use a greedy algorithm to pick the next rejector with the highest rejection score r. We repeat this process until r falls below some predefined threshold. 1. Sum each column and choose column (rejector) j with the highest sum. 2. For each entry T(i, j), in column j, that is equal to one, zero row i. 3. Go to step 1 The entire process is extremely fast and takes only several minutes, including I/O. The idea of creating a rejector pool in advance was independently suggested by [16] to accelerate the Viola-Jones training time. We obtain 50 rejectors using this method. Figure 2a shows the rejection rate of this cascade on a training set of 513, 000 images, as well as the number of arithmetic operations it takes. Note that roughly 50% of all patterns are rejected by the first rejector using only 12 operations. During testing we compute the approximated mean and variance only when they are needed and not before hand. 0 50 100 150 200 250 45 50 55 60 65 70 75 80 85 90 number of operations rejection rate Rejection rate 0 5 10 15 20 25 10 20 30 40 50 60 70 80 90 number of rejectors rejection rate Comparing different image segmentations random segments vertical segments horizontal segments image segments (a) (b) Figure 2: (a) Rejection rate on training set. The x-axis counts the number of arithmetic operations needed for rejection. The y-axis is the rejection rate on a training set of about half-a-million non-faces and about 1500 faces. Note that almost 50% of the false patterns are rejected with just 12 operations. Overall rejection rate of the feature rejectors on the training set is 88%, it drops to about 80% on the CMU+MIT database. (b) Rejection rate as a function of image segmentation method. We trained our system using four types of image segmentation and show the rejector. We compare our image segmentation approach against naive segmentation of the image plane into horizontal blocks, vertical blocks or random segmentation. In each case we trained a cascade of 21 rejectors and calculated their accumulative rejection rate on our training set. Clearly working with our image segments gives the best results. We wanted to confirm our intuition that indeed only meaningful regions in the image can produce such results and we therefor performed the following experiment. We segmented the pixels in the image using four different methods. (1) using our image segments (2) into 8 horizontal blocks (3) into 8 vertical blocks (4) into 8 randomly generated segments. Figure 2b show that image segments gives the best results, by far. The remaining false positive patterns are passed on to the next rejectors, as described next. 2.4 Texture-less region rejection We found that the feature rejectors defined in the previous section are doing poorly in rejecting texture-less regions. This is because we do not perform any sort of variance normalization on the image patch, a step that will slow us down. However, by now we have computed the approximated mean and variance of all the image segments and we can construct rejectors based on all of them to reject texture-less regions. In particular we construct the following two rejectors 1. Reject all image patches where the variance of all 8 approximated means falls below a threshold. Formally, find θ s.t. ˆσ(ˆµi(x)) < θ i = 1...8 2. Reject all image patches where the variance of all 8 approximated variances falls below a threshold. Formally, find θ s.t. ˆσ(ˆσi(x)) < θ i = 1...8 2.5 Linear classifier Finally, we construct a cascade of 10 linear rejectors, using all 16 features (i.e. the approximated means and variance of all 8 segments). (a) (b) Figure 3: Examples. We show examples from the CMU+MIT dataset. Our method correctly rejected over 99.8% of the image patches in the image, leaving only a handful of image patches to be tested by a “slow”, full scale classifier. 2.6 Multi-detection heuristic As noted by previous authors [15] face classifiers are insensitive to small changes in position and scale and therefor we adopt the heuristic that only four overlapping detections are declared a face. This help reduce the number of detected rectangles around and face, as well as reject some spurious false detections. 3 Experiments We have tested our rejection scheme on the standard CMU+MIT database [13]. We created a pyramid at increasing scales of 1.1 and scanned every scale for rectangles of size 20 × 20 in jumps of two pixels. We calculate the approximated mean and variance only when they are needed, to save time. Overall, our rejection scheme rejected over 99.8% of the image patches, while correctly detecting 93% of the faces. On average the feature rejectors rejected roughly 80% of all image patches, the textureless region rejectors rejected additional 10% of the image patches, the linear rejectors rejected additional 5% and the multi-detection heuristic rejected the remaining image patterns. The average rejection rate per image is over 99.8%. This is not enough for face detection, as there are roughly 615, 000 image patches per image in the CMU+MIT database, and our rejector cascade passes, on average, 870 false positive image patches, per image. This patterns will have to be passed to a full-scale classifier to be properly rejected. Figure 3 give some examples of our system. Note that the system correctly detects all the faces, while allowing a small number of false positives. We have also experimented with rescaling the features, instead of rescaling the image, but noted that the number of false positives increased by about 5% for every fixed detection rate we tried (All the results reported here use image pyramids). 4 Summary and Conclusions We presented a fast rejection scheme that is based on image segments and demonstrated it on the canonical example of face detection. Image segments are made of regions of pixels with similar behavior over the image set. The shape of the features (i.e. image segments) we use is data-driven and they are very cheap to compute The relationships between the mean and variance of image segments are used to form a cascade of rejectors that can reject over 99.8% of the image patches, thus only a small fraction of the image patches must be passed to a full-scale classifier. The training time for our method is much less than an hour, on a standard PC. We believe that our method can be used to accelerate standard machine learning algorithms that are too slow for object detection, by serving as a gate keeper that rejects most of the false patterns. References [1] Shai Avidan. EigenSegments: A spatio-temporal decomposition of an ensemble of image. In European Conference on Computer Vision (ECCV), May 2002, Copenhagen, Denmark. [2] Yoav Freund and Robert E. Schapire. A decision-theoretic generalization of on-line learning and an application to boosting. In Computational Learning Theory: Eurocolt 95, pages 2337. Springer-Verlag, 1995. [3] R. O. Duda and P. E. Hart. Pattern Classification and Scene Analysis. WileyInterscience publication, 1973. [4] M. Elad, Y. Hel-Or and R. Keshet. Rejection based classifier for face detection. Pattern Recognition Letters 23 (2002) 1459-1471. [5] B. Heisele, T. Serre, S. Mukherjee, and T. Poggio. Feature reduction and hierarchy of classifiers for fast object detection in video images. In Proc. CVPR, volume 2, pages 1824, 2001. [6] D. Keren, M. Osadchy, and C. Gotsman. Antifaces: A novel, fast method for image detection. IEEE Trans. on Pattern Analysis and Machine Intelligence, 23(7):747761, 2001. [7] S.Z. Li, L. Zhu, Z.Q. Zhang, A. Blake, H.J. Zhang and H. Shum. Statistical Learning of Multi-View Face Detection. In Proceedings of the 7th European Conference on Computer Vision, Copenhagen, Denmark, May 2002. [8] Henry Schneiderman and Takeo Kanade. A statistical model for 3d object detection applied to faces and cars. In IEEE Conference on Computer Vision and Pattern Recognition. IEEE, June 2000. [9] L. Sirovich and M. Kirby. Low-dimensional procedure for the characterization of human faces. In Journal of the Optical Society of America 4, 510-524. [10] K.-K. Sung and T. Poggio. Example-based Learning for View-Based Human Face Detection. In IEEE Transactions on Pattern Analysis and Machine Intelligence 20(1):3951, 1998. [11] M. Turk and A. Pentland. Eigenfaces for recognition. In Journal of Cognitive Neuroscience, vol. 3, no. 1, 1991. [12] S. Romdhani, P. Torr, B. Schoelkopf, and A. Blake. Computationally efficient face detection. In Proc. Intl. Conf. Computer Vision, pages 695700, 2001. [13] H. A. Rowley, S. Baluja, and T. Kanade. Neural network-based face detection. IEEE Trans. on Pattern Analysis and Machine Intelligence, 20(1):2338, 1998. [14] V. Vapnik. The Nature of Statistical Learning Theory. Springer, N.Y., 1995. [15] P. Viola and M. Jones. Rapid Object Detection using a Boosted Cascade of Simple Features. In IEEE Conference on Computer Vision and Pattern Recognition, Hawaii, 2001. [16] J. Wu, J. M. Rehg, and M. D. Mullin. Learning a Rare Event Detection Cascade by Direct Feature Selection. To appear in Advances in Neural Information Processing Systems 16 (NIPS*2003), MIT Press, 2004.
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Sub-Microwatt Analog VLSI Support Vector Machine for Pattern Classification and Sequence Estimation Shantanu Chakrabartty and Gert Cauwenberghs Department of Electrical and Computer Engineering Johns Hopkins University, Baltimore, MD 21218 {shantanu,gert}@jhu.edu Abstract An analog system-on-chip for kernel-based pattern classification and sequence estimation is presented. State transition probabilities conditioned on input data are generated by an integrated support vector machine. Dot product based kernels and support vector coefficients are implemented in analog programmable floating gate translinear circuits, and probabilities are propagated and normalized using sub-threshold current-mode circuits. A 14-input, 24-state, and 720-support vector forward decoding kernel machine is integrated on a 3mm×3mm chip in 0.5µm CMOS technology. Experiments with the processor trained for speaker verification and phoneme sequence estimation demonstrate real-time recognition accuracy at par with floating-point software, at sub-microwatt power. 1 Introduction The key to attaining autonomy in wireless sensory systems is to embed pattern recognition intelligence directly at the sensor interface. Severe power constraints in wireless integrated systems incur design optimization across device, circuit, architecture and system levels [1]. Although system-on-chip methodologies have been primarily digital, analog integrated systems are emerging as promising alternatives with higher energy efficiency and integration density, exploiting the analog sensory interface and computational primitives inherent in device physics [2]. Analog VLSI has been chosen, for instance, to implement Viterbi [3] and HMM-based [4] sequence decoding in communications and speech processing. Forward-Decoding Kernel Machines (FDKM) [5] provide an adaptive framework for general maximum a posteriori (MAP) sequence decoding, that avoid the need for backward recursion over the data in Viterbi and HMM-based sequence decoding [6]. At the core of FDKM is a support vector machine (SVM) [7] for large-margin trainable pattern classification, performing noise-robust regression of transition probabilities in forward sequence estimation. The achievable limits of FDKM power-consumption are determined by the number of support vectors (i.e., regression templates), which in turn are determined by the complexity of the discrimination task and the signal-to-noise ratio of the sensor interface [8]. xs x NORMALIZATION λi1 s 14 24x24 30x24 30x24 12 24 αj[n-1] αi[n] 24 MVM MVM SUPPORT VECTORS INPUT fi1(x) 24 FORWARD DECODING Pi1 Pi24 KERNEL K(x,xs) 24x24 Figure 1: FDKM system architecture. In this paper we describe an implementation of FDKM in silicon, for use in adaptive sequence detection and pattern recognition. The chip is fully configurable with parameters directly downloadable onto an array of floating-gate CMOS computational memory cells. By means of calibration and chip-in-loop training, the effect of mismatch and non-linearity in the analog implementation is significantly reduced. Section 2 reviews FDKM formulation and notations. Section 3 describes the schematic details of hardware implementation of FDKM. Section 4 presents results from experiments conducted with the fabricated chip and Section 5 concludes with future directions. 2 FDKM Sequence Decoding FDKM recognition and sequence decoding are formulated in the framework of MAP (maximum a posteriori) estimation, combining Markovian dynamics with kernel machines. The MAP forward decoder receives the sequence X[n] = {x[1], x[2], . . . , x[n]} and produces an estimate of conditional probability measure of state variables q[n] over all classes i ∈1, .., S, αi[n] = P(q[n] = i | X[n]). Unlike hidden Markov models, the states directly encode the symbols, and the observations x modulate transition probabilities between states [6]. Estimates of the posterior probability αi[n] are obtained from estimates of local transition probabilities using the forward-decoding procedure [6] αi[n] = S j=1 Pij[n] αj[n −1] (1) where Pij[n] = P(q[n] = i | q[n −1] = j, x[n]) denotes the probability of making a transition from class j at time n −1 to class i at time n, given the current observation vector x[n]. Forward decoding (1) expresses first order Markovian sequential dependence of state probabilities conditioned on the data. The transition probabilities Pij[n] in (1) attached to each outgoing state j are obtained by normalizing the SVM regression outputs fij(x): Pij[n] = [fij(x[n]) −zj[n]]+ (2) Vdd M3 M1 M2 Vtunn Vg ref A Vc Vtunn Vg Vc Iin C B Iout M4 (a) Vdd M5 M6 M7 M8 M9 Vbias x M10 (x.xs)2 λij sK(x, xs) (b) Figure 2: Schematic of the SVM stage. (a) Multiply accumulate cell and reference cell for the MVM blocks in Figure 1. (b) Combined input, kernel and MVM modules. where [.]+ = max(., 0). The normalization mechanism is subtractive rather than divisive, with normalization offset factor zj[n] obtained using a reverse-waterfilling criterion with respect to a probability margin γ [10], i [fij(x[n]) −zj[n]]+ = γ. (3) Besides improved robustness [8], the advantage of the subtractive normalization (3) is its amenability to current mode implementation as opposed to logistic normalization [11] which requires exponentiation of currents. The SVM outputs (margin variables) fij(x) are given by: fij(x) = N s λs ij K(x, xs) + bij (4) where K(·, ·) denotes a symmetric positive-definite kernel1 satisfying the Mercer condition, such as a Gaussian radial basis function or a polynomial spline [7], and xs[m], m = 1, .., N denote the support vectors. The parameters λs ij in (4) and the support vectors xs[m] are determined by training on a labeled training set using a recursive FDKM procedure described in [5]. 3 Hardware Implementation A second order polynomial kernel K(x, y) = (x.y)2 was chosen for convenience of implementation. This inner-product based architecture directly maps onto an analog computational array, where storage and computation share common circuit elements. The FDKM 1K(x, y) = Φ(x).Φ(y). The map Φ(·) need not be computed explicitly, as it only appears in inner-product form. Vdd Vdd Vdd Vdd fij[n] γ Vref M1 M2 M4 M6 M5 M9 M7 M8 Aij αj[n-1] αi[n] Pij[n] M3 Figure 3: Schematic of the margin propagation block. system architecture is shown in Figure 1. It consists of several SVM stages that generates state transition probabilities Pij[n] modulated by input data x[n], and a forward decoding block that performs maximum a posteriori (MAP) estimation of the state sequence αi[n]. 3.1 SVM Stage The SVM stage implements (4) to generate unnormalized probabilities. It consists of a kernel stage computing kernels K(xs, x) between input vector x and stored support vectors xs, and a coefficient stage linearly combining kernels using stored training parameters λs ij. Both kernel and coefficient blocks incorporate an analog matrix-vector multiplier (MVM) with embedded storage of support vectors and coefficients. A single multiply-accumulate cell, using floating-gate CMOS non-volative analog storage, is shown in Figure 2(a). The floating gate node voltages (Vg) of transistors M2 are programmed using hot-electron injection and tunneling [12]. The input stage comprising transistors M1, M3 and M4 forms a key component in the design of the array and sets the voltage at node A as a function of input current. By operating the array in weak-inversion, the output current through the floating gate element M2 in terms of the input stage floating gate potential Vgref and memory element floating gate potential Vg is given by Iout = Iine−κ(Vg−Vgref )/UT (5) as a product of two pseudo-currents, leading to single quadrant multiplier. Two observations can be directly made regarding Eqn. (5): 1. The input stage eliminates the effect of the bulk on the output current, making it a function of the reference floating gate voltage which can be easily programmed for the entire row. 2. The weight is differential in the floating gate voltages Vg −Vgref, allowing to increase or decrease the weight by hot electron injection only, without the need for repeated high-voltage tunneling. For instance, the leakage current in unused rows can be reduced significantly by programming the reference gate voltage to a high value, leading to power savings. The feedback transistor in the input stage M3 reduces the output impedance of node A given by ro ≈gd1/gm1gm2. This makes the array scalable as additional memory elements can be added to the node without pulling the voltage down. An added benefit of keeping the voltage at node A fixed is reduced variation in back gate parameter κ in the floating gate elements. The current from each memory element is summed on a low impedance node established by two diode connected transistors M7-M10. This partially compensates for large Early voltage effects implicit in floating gate transistors. (a) (b) Figure 4: Single input-output response of the SVM stage illustrating the square transfer function of the kernel block (log(Iout) vs. log(Iin)) where all the MVM elements are programmed for unity gain. (a) Before calibration showing mismatch between rows. (b) After pre-distortion compensation of input and output coefficients. The array of elements M2 with peripheral circuits as shown in Figure 2(a) thus implement a simple single quadrant matrix-vector multiplication module. The single quadrant operation is adequate for unsigned inputs, and hence unsigned support vectors. A simple squaring circuit M7-M10 is used to implement the non-linear kernel as shown in figure 2(b). The requirement on the type of non-linearity is not stringent and can be easily incorporated into the kernel in SVM training procedure [5]. The coefficient block consists of the same matrix-vector multiplier given in figure 2(a). For the general probability model given by (2) a single quadrant multiplication is sufficient to model any distribution. This can be easily verified by observing that the distribution (2) is invariant to uniform offset in the coefficients λs ij. 3.2 Forward Decoding Stage The forward recursion decoding is implemented by a modified version of the sum-product probability propagation circuit in [13], performing margin-based probability propagation according to (1). In contrast to divisive normalization that relies on the translinear principle using sub-threshold MOS or bipolar circuits in [13], the implementation of margin-based subtractive normalization shown in figure 3 [10] is device operation independent. The circuit consists of several normalization cells Aij along columns computing Pij = [fij − z]+ using transistors M1-M4. Transistors M5-M9 form a feedback loop that compares and stabilizes the circuit to the normalization criterion (3). The currents through transistors M4 are auto-normalized to the previous state value αj[n −1] to produce a new estimate of αi[n1] based on recursion (1). The delay in equation (1) is implemented using a logdomain filter and a fixed normalization current ensures that all output currents be properly scaled to stabilize the continuous-time feedback loop. 4 Experimental Results A 14-input, 24-state, and 24×30-support vector FDKM was integrated on a 3mm×3mm FDKM chip, fabricated in a 0.5µm CMOS process, and fully tested. Figure 5(c) shows the micrograph of the fabricated chip. Labeled training data pertaining to a certain task were used to train an SVM, and the training coefficients thus obtained were programmed onto the chip. Table 1: FDKM Chip Summary Technology Value Area 3mm×3mm Technology 0.5µ CMOS Supply Voltage 4 V System Parameters Floating Cell Count 28814 Number of Support Vectors 720 Input Dimension 14 Number of States 24 Power Consumption 80nW - 840nW Energy Efficiency 1.6pJ/MAC q1 q2 q3 q4 q5 q6 q8 q9 q10 q11 q12 q13 q7 x1 x2 x3 x4 x5 x6 x6 x5 x4 x3 x2 x1 (a) (b) (c) Figure 5: (a) Transition-based sequence detection in a 13-state Markov model. (b) Experimental recording of α7 = P(q7), detecting one of two recurring sequences in inputs x1 →x6 (x1, x3 and x5 shown). (c) Micrograph of the FDKM chip Programming of the trained coefficients was performed by programming respective cells M2 along with the corresponding input stage M1, so as to establish the desired ratio of currents. The values were established by continuing hot electron injection until the desired current was attained. During hot electron injection, the control gate Vc was adjusted to set the injection current to a constant level for stable injection. All cells in the kernel and coefficient modules of the SVM stage are random accessible for read, write and calibrate operations. The calibration procedure compensates for mismatch between different input/output paths by adapting the floating gate elements in the MVM cells. This is illustrated in Figure 4 where the measured square kernel transfer function is shown before and after calibration. The chip is fully reconfigurable and can perform different recognition tasks by programming different training parameters, as demonstrated through three examples below. Depending on the number of active support vectors and the absolute level of currents (in relation to decoding bandwidth), power dissipation is in the lower nanowatt to microwatt range. 0 5 10 15 20 25 65 70 75 80 85 90 95 100 False Positive (%) True Positive (%) Simulated Measured (a) (b) Figure 6: (a) Measured and simulated ROC curve for the speaker verification experiment. (b) Experimental phoneme recognition by FDKM chip. The state probability shown is for consonant /t/ in words “torn,” “rat,” and “error.” Two peaks are located as expected from the input sequence, shown on top. For the first set of experiments, parameters corresponding to a simple Markov chain shown in figure 5(a) were programmed onto the chip to differentiate between two given sequences of input features: one a sweep of active input components in rising order (x1 through x6), and the other in descending order (x6 through x1). The output of state q7 in the Markov chain is shown in figure 5(b). It can be clearly observed that state q7 “fires” only when a rising sequence of pulse trains arrives. The FDKM chip thereby demonstrates probability propagation similar to that in the architecture of [4]. The main difference is that the present architecture can be configured for detecting other, more complex sequences through programming and training. For the second set of experiments the FDKM chip was programmed to perform speaker verification using speech data from YOHO corpus. For training we chose 480 utterances corresponding to 10 separate speakers (101-110). For each of these utterances 12 mel-cepstra coefficients were computed for every 25ms frames. These coefficients were clustered using k-means clustering to obtain 50 clusters per speaker which were then used for training the SVM. For testing 480 utterances for those speakers were chosen, and confidence scores returned by the SVMs were integrated over all frames of an utterance to obtain a final decision. Verification results obtained from the chip demonstrate 97% true acceptance at 1% false positive rate, identical to the performance obtained through floating point software simulations as shown by the receiver operating characteristic shown in figure 6(a). The total power consumption for this task is only 840nW, demonstrating its suitability for autonomous sensor applications. A third set of experiment aimed at detecting phone utterances in human speech. Melcepstra coefficients of six phone utterances (/t/,/n/,/r/,/ow/,/ah/,/eh/) selected from the TIMIT corpus were transformed using singular value decomposition and thresholding. Even though the recognition was demonstrated for the reduced set of features, the chip operates internally with analog inputs. Figure 6(b) illustrates correct detection of phonemes as identified by the presence of phone /t/ at the expected time instances in the input sequence. 5 Discussion and Conclusion We designed an FDKM based sequence recognition system on silicon and demonstrated its performance on simple but general tasks. The chip is fully reconfigurable and different sequence recognition engines can be programmed using parameters obtained through SVM training. FDKM decoding is performed in real-time and is ideally suited for sequence recognition and verification problems involving speech features. All analog processing in the chip is performed by transistors operating in weak-inversion resulting in power dissipation in the nanowatt to microwatt range. Non-volatile storage of training parameters further reduces standby power dissipation. We also note that while low power dissipation is a virtue in many applications, increased power can be traded for increased bandwidth. For instance, the presented circuits could be adapted using heterojunction bipolar junction transistors in a SiGe process for ultra-high speed MAP decoding applications in digital communication, using essentially the same FDKM architecture as presented here. Acknowledgement: This work is supported by a grant from The Catalyst Foundation (http://www.catalyst-foundation.org), NSF IIS-0209289, ONR/DARPA N00014-00-C0315, and ONR N00014-99-1-0612. The chip was fabricated through the MOSIS service. References [1] Wang, A. and Chandrakasan, A.P, “Energy-Efficient DSPs for Wireless Sensor Networks,” IEEE Signal Proc. Mag., vol. 19 (4), pp. 68-78, July 2002. [2] Vittoz, E.A., “Low-Power Design: Ways to Approach the Limits,” Dig. 41st IEEE Int. Solid-State Circuits Conf. (ISSCC), San Francisco CA, 1994. [3] Shakiba, M.S, Johns, D.A, and Martin, K.W, “BiCMOS Circuits for Analog Viterbi Decoders,” IEEE Trans. Circuits and Systems II, vol. 45 (12), Dec. 1998. [4] Lazzaro, J, Wawrzynek, J, and Lippmann, R.P, “A Micropower Analog Circuit Implementation of Hidden Markov Model State Decoding,” IEEE J. Solid-State Circuits, vol. 32 (8), Aug. 1997. [5] Chakrabartty, S. and Cauwenberghs, G. “Forward Decoding Kernel Machines: A hybrid HMM/SVM Approach to Sequence Recognition,” IEEE Int. Conf. of Pattern Recognition: SVM workshop. (ICPR’2002), Niagara Falls, 2002. [6] Bourlard, H. and Morgan, N., Connectionist Speech Recognition: A Hybrid Approach, Kluwer Academic, 1994. [7] Vapnik, V. The Nature of Statistical Learning Theory, New York: Springer-Verlag, 1995. [8] Chakrabartty, S., and Cauwenberghs, G. “Power Dissipation Limits and Large Margin in Wireless Sensors,” Proc. IEEE Int. Symp. Circuits and Systems(ISCAS2003), vol. 4, 25-28, May 2003. [9] Bahl, L.R., Cocke J., Jelinek F. and Raviv J. “Optimal Decoding of Linear Codes for Minimizing Symbol Error Rate,” IEEE Transactions on Inform. Theory, vol. IT-20, pp. 284-287, 1974. [10] Chakrabartty, S., and Cauwenberghs, G. “Margin Propagation and Forward Decoding in Analog VLSI,” Proc. IEEE Int. Symp. Circuits and Systems(ISCAS2004), Vancouver Canada, May 23-26, 2004. [11] Jaakkola, T. and Haussler, D. “Probabilistic kernel regression models,” Proc. Seventh Int. Workshop Artificial Intelligence and Statistics , 1999. [12] C. Dorio,P. Hasler,B. Minch and C.A. Mead, “A Single-Transistor Silicon Synapse,” IEEE Trans. Electron Devices, vol. 43 (11), Nov. 1996. [13] H. Loeliger, F. Lustenberger, M. Helfenstein and F. Tarkoy, “Probability Propagation and Decoding in Analog VLSI,” IEEE Proc. ISIT, 1998.
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Maximum Margin Clustering Linli Xu∗† James Neufeld† Bryce Larson† Dale Schuurmans† ∗University of Waterloo †University of Alberta Abstract We propose a new method for clustering based on finding maximum margin hyperplanes through data. By reformulating the problem in terms of the implied equivalence relation matrix, we can pose the problem as a convex integer program. Although this still yields a difficult computational problem, the hard-clustering constraints can be relaxed to a soft-clustering formulation which can be feasibly solved with a semidefinite program. Since our clustering technique only depends on the data through the kernel matrix, we can easily achieve nonlinear clusterings in the same manner as spectral clustering. Experimental results show that our maximum margin clustering technique often obtains more accurate results than conventional clustering methods. The real benefit of our approach, however, is that it leads naturally to a semi-supervised training method for support vector machines. By maximizing the margin simultaneously on labeled and unlabeled training data, we achieve state of the art performance by using a single, integrated learning principle. 1 Introduction Clustering is one of the oldest forms of machine learning. Nevertheless, it has received a significant amount of renewed attention with the advent of nonlinear clustering methods based on kernels. Kernel based clustering methods continue to have a significant impact on recent work in machine learning [14, 13], computer vision [16], and bioinformatics [9]. Although many variations of kernel based clustering has been proposed in the literature, most of these techniques share a common “spectral clustering” framework that follows a generic recipe: one first builds the kernel (“affinity”) matrix, normalizes the kernel, performs dimensionality reduction, and finally clusters (partitions) the data based on the resulting representation [17]. In this paper, our primary focus will be on the final partitioning step where the actual clustering occurs. Once the data has been preprocessed and a kernel matrix has been constructed (and its rank possibly reduced), many variants have been suggested in the literature for determining the final partitioning of the data. The predominant strategies include using k-means clustering [14], minimizing various forms of graph cut cost [13] (relaxations of which amount to clustering based on eigenvectors [17]), and finding strongly connected components in a Markov chain defined by the normalized kernel [4]. Some other recent alternatives are correlation clustering [12] and support vector clustering [1]. What we believe is missing from this previous work however, is a simple connection to other types of machine learning, such as semisupervised and supervised learning. In fact, one of our motivations is to seek unifying machine learning principles that can be used to combine different types of learning problems in a common framework. For example, a useful goal for any clustering technique would be to find a way to integrate it seamlessly with a supervised learning technique, to obtain a principled form of semisupervised learning. A good example of this is [18], which proposes a general random field model based on a given kernel matrix. They then find a soft cluster assignment on unlabeled data that minimizes a joint loss with observed labels on supervised training data. Unfortunately, this technique actually requires labeled data to cluster the unlabeled data. Nevertheless, it is a useful approach. Our goal in this paper is to investigate another standard machine learning principle— maximum margin classification—and modify it for clustering, with the goal of achieving a simple, unified way of solving a variety of problems, including clustering and semisupervised learning. Although one might be skeptical that clustering based on large margin discriminants can perform well, we will see below that, combined with kernels, this strategy can often be more effective than conventional spectral clustering. Perhaps more significantly, it also immediately suggests a simple semisupervised training technique for support vector machines (SVMs) that appears to improve the state of the art. The remainder of this paper is organized as follows. After establishing the preliminary ideas and notation in Section 2, we tackle the problem of computing a maximum margin clustering for a given kernel matrix in Section 3. Although it is not obvious that this problem can be solved efficiently, we show that the optimal clustering problem can in fact be formulated as a convex integer program. We then propose a relaxation of this problem which yields a semidefinite program that can be used to efficiently compute a soft clustering. Section 4 gives our experimental results for clustering. Then, in Section 5 we extend our approach to semisupervised learning by incorporating additional labeled training data in a seamless way. We then present experimental results for semisupervised learning in Section 6 and conclude. 2 Preliminaries Since our main clustering idea is based on finding maximum margin separating hyperplanes, we first need to establish the background ideas from SVMs as well as establish the notation we will use. For SVM training, we assume we are given labeled training examples (x1, y1), ..., (xN, yN), where each example is assigned to one of two classes yi ∈{−1, +1}. The goal of an SVM of course is to find the linear discriminant fw,b(x) = w⊤φ(x) + b that maximizes the minimum misclassification margin γ∗ = max w,b,γ γ subject to yi(w⊤φ(xi) + b) ≥γ, ∀N i=1, ∥w∥2 = 1 (1) Here the Euclidean normalization constraint on w ensures that the Euclidean distance between the data and the separating hyperplane (in φ(x) space) determined by w∗, b∗is maximized. It is easy to show that this same w∗, b∗is a solution to the quadratic program γ∗−2 = min w,b ∥w∥2 subject to yi(w⊤φ(xi) + b) ≥1, ∀N i=1 (2) Importantly, the minimum value of this quadratic program, γ∗−2, is just the inverse square of the optimal solution value γ∗to (1) [10]. To cope with potentially inseparable data, one normally introduces slack variables to reduce the dependence on noisy examples. This leads to the so called soft margin SVM (and its dual) which is controlled by a tradeoff parameter C γ∗−2 = min w,b,ϵ ∥w∥2 + Cϵ⊤e subject to yi(w⊤φ(xi) + b) ≥1 −ϵi, ∀N i=1, ϵ ≥0 = max λ 2λ⊤e −⟨K ◦λλ⊤, yy⊤⟩ subject to 0 ≤λ ≤C, λ⊤y = 0 (3) The notation we use in this dual formulation requires some explanation, since we will use it below: Here K denotes the N ×N kernel matrix formed from the inner products of feature vectors Φ = [φ(x1), ..., φ(xN)] such that K = Φ⊤Φ. Thus kij = φ(xi)⊤φ(xj). The vector e denotes the vector of all 1 entries. We let A ◦B denote componentwise matrix multiplication, and let ⟨A, B⟩= P ij aijbij. Note that (3) is derived from the standard dual SVM by using the fact that λ⊤(K ◦yy⊤)λ = ⟨K ◦yy⊤, λλ⊤⟩= ⟨K ◦λλ⊤, yy⊤⟩. To summarize: for supervised maximum margin training, one takes a given set of labeled training data (x1, y1), ..., (xN, yN), forms the kernel matrix K on data inputs, forms the kernel matrix yy⊤on target outputs, sets the slack parameter C, and solves the quadratic program (3) to obtain the dual solution λ∗and the inverse square maximum margin value γ∗−2. Once these are obtained, one can then recover a classifier directly from λ∗[15]. Of course, our main interest initially is not to find a large margin classifier given labels on the data, but instead to find a labeling that results in a large margin classifier. 3 Maximum margin clustering The clustering principle we investigate is to find a labeling so that if one were to subsequently run an SVM, the margin obtained would be maximal over all possible labellings. That is, given data x1, .., xN, we wish to assign the data points to two classes yi ∈{−1, +1} so that the separation between the two classes is as wide as possible. Unsurprisingly, this is a hard computational problem. However, with some reformulation we can express it as a convex integer program, which suggests that there might be some hope of obtaining practical solutions. However, more usefully, we can relax the integer constraint to obtain a semidefinite program that yields soft cluster assignments which approximately maximize the margin. Therefore, one can obtain soft clusterings efficiently using widely available software. However, before proceeding with the main development, there are some preliminary issues we need to address. First, we clearly need to impose some sort of constraint on the class balance, since otherwise one could simply assign all the data points to the same class and obtain an unbounded margin. A related issue is that we would also like to avoid the problem of separating a single outlier (or very small group of outliers) from the rest of the data. Thus, to mitigate these effects we will impose a constraint that the difference in class sizes be bounded. This will turn out to be a natural constraint for semisupervised learning and is very easy to enforce. Second, we would like the clustering to behave gracefully on noisy data where the classes may in fact overlap, so we adopt the soft margin formulation of the maximum margin criterion. Third, although it is indeed possible to extend our approach to the multiclass case [5], the extension is not simple and for ease of presentation we focus on simple two class clustering in this paper. Finally, there is a small technical complication that arises with one of the SVM parameters: It turns out that an unfortunate nonconvexity problem arises when we include the use of the offset b in the underlying large margin classifier. We currently do not have a way to avoid this nonconvexity, and therefore we currently set b = 0 and therefore only consider homogeneous linear classifiers. The consequence of this restriction is that the constraint λ⊤y = 0 is removed from the dual SVM quadratic program (3). Although it would seem like this is a harsh restriction, the negative effects are mitigated by centering the data at the origin, which can always be imposed. Nevertheless, dropping this restriction remains an important question for future research. With these caveats in mind, we proceed to the main development. We wish to solve for a labeling y ∈{−1, +1}N that leads to a maximum (soft) margin. Straightforwardly, one could attempt to tackle this optimization problem by directly formulating min y∈{−1,+1}N γ∗−2(y) subject to −ℓ≤e⊤y ≤ℓ where γ∗−2(y) = max λ 2λ⊤e −⟨K ◦λλ⊤, yy⊤⟩ subject to 0 ≤λ ≤C Unfortunately, γ∗−2(y) is not a convex function of y, and this formulation does not lead to an effective algorithmic approach. In fact, to obtain an efficient technique for solving this problem we need two key insights. The first key step is to re-express this optimization, not directly in terms of the cluster labels y, but instead in terms of the label kernel matrix M = yy⊤. The main advantage of doing so is that the inverse soft margin γ∗−2 is in fact a convex function of M γ∗−2(M) = max λ 2λ⊤e −⟨K ◦λλ⊤, M⟩ subject to 0 ≤λ ≤C (4) The convexity of γ∗−2 with respect to M is easy to establish since this quantity is just a maximum over linear functions of M [3]. This observation parallels one of the key insights of [10], here applied to M instead of K. Unfortunately, even though we can pose a convex objective, it does not allow us to immediately solve our problem because we still have to relate M to y, and M = yy⊤is not a convex constraint. Thus, the main challenge is to find a way to constrain M to ensure M = yy⊤while respecting the class balance constraints −ℓ≤e⊤y ≤ℓ. One obvious way to enforce M = yy⊤would be to impose the constraint that rank(M) = 1, since combined with M ∈{−1, +1}N×N this forces M to have a decomposition yy⊤for some y ∈{−1, +1}N. Unfortunately, rank(M) = 1 is not a convex constraint on M [7]. Our second key idea is to realize that one can indirectly enforce the desired relationship M = yy⊤by imposing a different set of linear constraints on M. To do so, notice that any such M must encode an equivalence relation over the training points. That is, if M = yy⊤ for some y ∈{−1, +1}N then we must have mij = 1 if yi = yj −1 if yi ̸= yj Therefore to enforce the constraint M = yy⊤for y ∈{−1, +1}N it suffices to impose the set of constraints: (1) M encodes an equivalence relation, namely that it is transitive, reflexive and symmetric; (2) M has at most two equivalence classes; and (3) M has at least two equivalence classes. Fortunately we can enforce each of these requirements by imposing a set of linear constraints on M ∈{−1, +1}N×N respectively: L1: mii = 1; mij = mji; mik ≥mij + mjk −1; ∀ijk L2: mjk ≥−mij −mik −1; ∀ijk L3: P i mij ≤N −2; ∀j The result is that with only linear constraints on M we can enforce the condition M = yy⊤.1 Finally, we can enforce the class balance constraint −ℓ≤e⊤y ≤ℓby imposing the additional set of linear constraints: 1Interestingly, for M ∈{−1, +1}N×N the first two sets of linear constraints can be replaced by the compact set of convex constraints diag(M) = e, M ⪰0 [7, 11]. However, when we relax the integer constraint below, this equivalence is no longer true and we realize some benefit in keeping the linear equivalence relation constraints. L4: −ℓ≤P i mij ≤ℓ; ∀j which obviates L3. The combination of these two steps leads to our first main result: One can solve for a hard clustering y that maximizes the soft margin by solving a convex integer program. To accomplish this, one first solves for the equivalence relation matrix M in min M∈{−1,+1}N×N max λ 2λ⊤e −⟨K ◦λλ⊤, M⟩subject to 0≤λ≤C, L1, L2, L4 (5) Then, from the solution M ∗recover the optimal cluster assignment y∗simply by setting y∗to any column vector in M ∗. Unfortunately, the formulation (5) is still not practical because convex integer programming is still a hard computational problem. Therefore, we are compelled to take one further step and relax the integer constraint on M to obtain a convex optimization problem over a continuous parameter space min M∈[−1,+1]N×Nmax λ 2λ⊤e −⟨K ◦λλ⊤, M⟩subject to 0≤λ≤C, L1, L2, L4, M ⪰0 (6) This can be turned into an equivalent semidefinite program using essentially the same derivation as in [10], yielding min M,δ,µ,ν δ subject to L1, L2, L4, µ ≥0, ν ≥0, M ⪰0 M ◦K e + µ −ν (e + µ −ν)⊤ δ −2Cν⊤e ⪰0 (7) This gives us our second main result: To solve for a soft clustering y that approximately maximizes the soft margin, first solve the semidefinite program (7), and then from the solution matrix M ∗recover the soft cluster assignment y by setting y = √λ1v1, where λ1, v1 are the maximum eigenvalue and corresponding eigenvector of M ∗.2 4 Experimental results We implemented the maximum margin clustering algorithm based on the semidefinite programming formulation (7), using the SeDuMi library, and tested it on various data sets. In these experiments we compared the performance of our maximum margin clustering technique to the spectral clustering method of [14] as well as straightforward k-means clustering. Both maximum margin clustering and spectral clustering were run with the same radial basis function kernel and matching width parameters. In fact, in each case, we chose the best width parameter for spectral clustering by searching over a small set of five widths related to the scale of the problem. In addition, the slack parameter for maximum margin clustering was simply set to an arbitrary value.3 To assess clustering performance we first took a set of labeled data, removed the labels, ran the clustering algorithms, labeled each of the resulting clusters with the majority class according to the original training labels, and finally measured the number of misclassifications made by each clustering. Our first experiments were conducted on the synthetic data sets depicted in Figure 1. Table 1 shows that for the first three sets of data (Gaussians, Circles, AI) maximum margin and spectral clustering obtained identical small error rates, which were in turn significantly 2One could also employ randomized rounding to choose a hard class assignment y. 3It turns out that the slack parameter C did not have a significant effect on any of our preliminary investigations, so we just set it to C = 100 for all of the experiments reported here. smaller than those obtained by k-means. However, maximum margin clustering demonstrates a substantial advantage on the fourth data set (Joined Circles) over both spectral and k-means clustering. We also conducted clustering experiments on the real data sets, two of which are depicted in Figures 2 and 3: a database of images of handwritten digits of twos and threes (Figure 2), and a database of face images of two people (Figure 3). The last two columns of Table 1 show that maximum margin clustering obtains a slight advantage on the handwritten digits data, and a significant advantage on the faces data. 5 Semi-supervised learning Although the clustering results are reasonable, we have an additional goal of adapting the maximum margin approach to semisupervised learning. In this case, we assume we are given a labeled training set (x1, y1), ..., (xn, yn) as well as an unlabeled training set xn+1, ..., xN, and the goal is to combine the information in these two data sets to produce a more accurate classifier. In the context of large margin classifiers, many techniques have been proposed for incorporating unlabeled data in an SVM, most of which are intuitively based on ensuring that large margins are also preserved on the unlabeled training data [8, 2], just as in our case. However, none of these previous proposals have formulated a convex optimization procedure that was guaranteed to directly maximize the margin, as we propose in Section 3. For our procedure, extending the maximum margin clustering approach of Section 3 to semisupervised training is easy: We simply add constraints on the matrix M to force it to respect the observed equivalence relations among the labeled training data. In addition, we impose the constraint that each unlabeled example belongs to the same class as at least one labeled training example. These conditions can be enforced with the simple set of additional linear constraints S1: mij = yiyj for labeled examples i, j ∈{1, ..., n} S2: Pn i=1 mij ≥2 −n for unlabeled examples j ∈{n + 1, ..., N} Note that the observed training labels yi for i ∈{1, ..., n} are constants, and therefore the new constraints are still linear in the parameters of M that are being optimized. The resulting training procedure is similar to that of [6], with the addition of the constraints L1–L4, S2 which enforce two classes and facilitate the ability to perform clustering on the unlabeled examples. 6 Experimental results We tested our approach to semisupervised learning on various two class data sets from the UCI repository. We compared the performance of our technique to the semisupervised SVM technique of [8]. In each case, we evaluated the techniques transductively. That is, we split the data into a labeled and unlabeled part, held out the labels of the unlabeled portion, trained the semisupervised techniques, reclassified the unlabeled examples using the learned results, and measured the misclassification error on the held out labels. Here we see that the maximum margin approach based on semidefinite programming can often outperform the approach of [8]. Table 2 shows that our maximum margin method is effective at exploiting unlabeled data to improve the prediction of held out labels. In every case, it significantly reduces the error of plain SVM, and obtains the best overall performance of the semisupervised learning techniques we have investigated. 0.5 1 1.5 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2 −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 −0.5 0 0.5 1 1.5 2 2.5 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Figure 1: Four artificial data sets used in the clustering experiments. Each data set consists of eighty two-dimensional points. The points and stars show the two classes discovered by maximum margin clustering. Figure 2: A sampling of the handwritten digits (twos and threes). Each row shows a random sampling of images from a cluster discovered by maximum margin clustering. Maximum margin made very few misclassifications on this data set, as shown in Table 1. Figure 3: A sampling of the face data (two people). Each row shows a random sampling of images from a cluster discovered by maximum margin clustering. Maximum margin made no misclassifications on this data set, as shown in Table 1. Gaussians Circles A I Joined Circles Digits Faces Maximum Margin 1.25 0 0 1 3 0 Spectral Clustering 1.25 0 0 24 6 16.7 K-means 5 50 38.5 50 7 24.4 Table 1: Percentage misclassification errors of the various clustering algorithms on the various data sets. HWD 1-7 HWD 2-3 UCI Austra. UCI Flare UCI Vote UCI Diabet. Max Marg 3.3 4.7 32 34 14 35.55 Spec Clust 4.2 6.4 48.7 40.7 13.8 44.67 TSVM 4.6 5.4 38.7 33.3 17.5 35.89 SVM 4.5 10.9 37.5 37 20.4 39.44 Table 2: Percentage misclassification errors of the various semisupervised learning algorithms on the various data sets. SVM uses no unlabeled data. TSVM is due to [8]. 7 Conclusion We have proposed a simple, unified principle for clustering and semisupervised learning based on the maximum margin principle popularized by supervised SVMs. Interestingly, this criterion can be approximately optimized using an efficient semidefinite programming formulation. The results on both clustering and semisupervised learning are competitive with, and sometimes exceed the state of the art. Overall, margin maximization appears to be an effective way to achieve a unified approach to these different learning problems. For future work we plan to address the restrictions of the current method, including the ommission of an offset b and the restriction to two class problems. We note that a multiclass extension to our approach is possible, but it is complicated by the fact that it cannot be conveniently based on the standard multiclass SVM formulation of [5] Acknowledgements Research supported by the Alberta Ingenuity Centre for Machine Learning, NSERC, MITACS, IRIS and the Canada Research Chairs program. References [1] A. Ben-Hur, D. Horn, H. Siegelman, and V. Vapnik. Support vector clustering. In Journal of Machine Learning Research 2 (2001), 2001. [2] K. Bennett and A. Demiriz. Semi-supervised support vector machines. In Advances in Neural Information Processing Systems 11 (NIPS-98), 1998. [3] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge U. Press, 2004. [4] Chakra Chennubhotla and Allan Jepson. Eigencuts: Half-lives of eigenflows for spectral clustering. In In Advances in Neural Information Processing Systems, 2002, 2002. [5] K. Crammer and Y. Singer. On the algorithmic interpretation of multiclass kernel-based vector machines. Journal of Machine Learning Research, 2, 2001. [6] T. De Bie and N. Cristianini. Convex methods for transduction. In Advances in Neural Information Processing Systems 16 (NIPS-03), 2003. [7] C. Helmberg. Semidefinite programming for combinatorial optimization. Technical Report ZIB-Report ZR-00-34, Konrad-Zuse-Zentrum Berlin, 2000. [8] T. Joachims. Transductive inference for text classification using support vector machines. In International Conference on Machine Learning (ICML-99), 1999. [9] Y. Kluger, R. Basri, J. Chang, and M. Gerstein. Spectral biclustering of microarray cancer data: co-clustering genes and conditions. Genome Research, 13, 2003. [10] G. Lanckriet, N. Cristianini, P. Bartlett, L Ghaoui, and M. Jordan. Learning the kernel matrix with semidefinite programming. Journal of Machine Learning Research, 5, 2004. [11] M. Laurent and S. Poljak. On a positive semidefinite relaxation of the cut polytope. Linear Algebra and its Applications, 223/224, 1995. [12] S. Chawla N. Bansal, A. Blum. Correlation clustering. In Conference on Foundations of Computer Science (FOCS-02), 2002. [13] J. Kandola N. Cristianini, J. Shawe-Taylor. Spectral kernel methods for clustering. In In Advances in Neural Information Processing System, 2001, 2001. [14] A. Ng, M. Jordan, and Y Weiss. On spectral clustering: analysis and an algorithm. In Advances in Neural Information Processing Systems 14 (NIPS-01), 2001. [15] B. Schoelkopf and A. Smola. Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. MIT Press, 2002. [16] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans PAMI, 22(8), 2000. [17] Y. Weiss. Segmentation using eigenvectors: a unifying view. In International Conference on Computer Vision (ICCV-99), 1999. [18] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In International Conference on Machine Learning (ICML-03), 2003.
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On the Adaptive Properties of Decision Trees Clayton Scott Statistics Department Rice University Houston, TX 77005 cscott@rice.edu Robert Nowak Electrical and Computer Engineering University of Wisconsin Madison, WI 53706 nowak@engr.wisc.edu Abstract Decision trees are surprisingly adaptive in three important respects: They automatically (1) adapt to favorable conditions near the Bayes decision boundary; (2) focus on data distributed on lower dimensional manifolds; (3) reject irrelevant features. In this paper we examine a decision tree based on dyadic splits that adapts to each of these conditions to achieve minimax optimal rates of convergence. The proposed classifier is the first known to achieve these optimal rates while being practical and implementable. 1 Introduction This paper presents three adaptivity properties of decision trees that lead to faster rates of convergence for a broad range of pattern classification problems. These properties are: Noise Adaptivity: Decision trees can automatically adapt to the (unknown) regularity of the excess risk function in the neighborhood of the Bayes decision boundary. The regularity is quantified by a condition similar to Tsybakov’s noise condition [1]. Manifold Focus: When the distribution of features happens to have support on a lower dimensional manifold, decision trees can automatically detect and adapt their structure to the manifold. Thus decision trees learn the “effective” data dimension. Feature Rejection: If certain features are irrelevant (i.e., independent of the class labels), then decision trees can automatically ignore these features. Thus decision trees learn the “relevant” data dimension. Each of the above properties can be formalized and translated into a class of distributions with known minimax rates of convergence. Adaptivity is a highly desirable quality of classifiers since in practice the precise characteristics of the distribution are unknown. We show that dyadic decision trees achieve the (minimax) optimal rate (to within a log factor) without needing to know the specific parameters defining the class. Such trees are constructed by minimizing a complexity penalized empirical risk over an appropriate family of dyadic partitions. The complexity term is derived from a new generalization error bound for trees, inspired by [2]. This bound in turn leads to an oracle inequality from which the optimal rates are derived. Full proofs of all results are given in [11]. The restriction to dyadic splits is necessary to achieve a computationally tractable classifier. Our classifiers have computational complexity nearly linear in the training sample size. The same rates may be achieved by more general tree classifiers, but these require searches over prohibitively large families of partitions. Dyadic decision trees are thus preferred because they are simultaneously implementable, analyzable, and sufficiently flexible to achieve optimal rates. 1.1 Notation Let Z be a random variable taking values in a set Z, and let Zn = {Z1, . . . , Zn} be iid realizations of Z. Let PZ be the probability measure for Z, and let bPn be the empirical estimate of PZ based on Zn: bPn(B) = (1/n) Pn i=1 I{Zi∈B}, B ⊆Z, where I denotes the indicator function. In classification we take Z = X × Y, where X is the collection of feature vectors and Y is a finite set of class labels. Assume X = [0, 1]d, d ≥2, and Y = {0, 1}. A classifier is a measurable function f : [0, 1]d →{0, 1}. Each classifier f induces a set Bf = {(x, y) ∈Z | f(x) ̸= y}. Define the probability of error and empirical error (risk) of f by R(f) = PZ(Bf) and bRn(f) = bPn(Bf), respectively. The Bayes classifier f ∗achieves minimum probability of error and is given by f ∗(x) = I{η(x)>1/2}, where η(x) = PY |X(1 |x) is the posterior probability that the correct label is 1. The Bayes error is R(f ∗) and denoted R∗. The Bayes decision boundary, denoted ∂G∗, is the topological boundary of the Bayes decision set G∗= {x | f ∗(x) = 1}. 1.2 Rates of Convergence in Classification In this paper we study the rate at which EZn{R( bfn)} −R∗goes to zero as n →∞, where bfn is a classification learning rule, i.e., a rule for constructing a classifier from Zn. Yang [3] shows that for η(x) in certain smoothness classes minimax optimal rates are achieved by appropriate plug-in density estimates. Tsybakov and collaborators replace global restrictions on η by restrictions on η near ∂G∗. Faster rates are then possible, although existing optimal classifiers typically rely on ϵ-nets or otherwise non-implementable methods. [1, 4, 5]. Other authors have derived rates of convergence for existing practical classifiers, but these rates are suboptimal in the minimax sense considered here [6–8]. Our contribution is to demonstrate practical classifiers that adaptively attain minimax optimal rates for some of Tsybakov’s and other classes. 2 Dyadic Decision Trees A dyadic decision tree (DDT) is a decision tree that divides the input space by means of axis-orthogonal dyadic splits. More precisely, a dyadic decision tree T is specified by assigning an integer s(v) ∈{1, . . . , d} to each internal node v of T (corresponding to the coordinate/attribute that is split at that node), and a binary label 0 or 1 to each leaf node. The nodes of DDTs correspond to hyperrectangles (cells) in [0, 1]d (see Figure 1). Given a hyperrectangle A = Qd r=1[ar, br], let As,1 and As,2 denote the hyperrectangles formed by splitting A at its midpoint along coordinate s. Specifically, define As,1 = {x ∈A | xs ≤ (as + bs)/2} and As,2 = A\As,1. Each node of a DDT is associated with a cell according to the following rules: (1) The root node is associated with [0, 1]d; (2) If v is an internal node associated to the cell A, then the children of v are associated to As(v),1 and As(v),2. Let π(T) = {A1, . . . , Ak} denote the partition induced by T. Let j(A) denote the depth of A and note that λ(A) = 2−j(A) where λ is the Lebesgue measure on Rd. Define T to be the collection of all DDTs and A to be the collection of all cells corresponding to nodes Figure 1: A dyadic decision tree (right) with the associated recursive dyadic partition (left) when d = 2. Each internal node of the tree is labeled with an integer from 1 to d indicating the coordinate being split at that node. The leaf nodes are decorated with class labels. of trees in T . Let M be a dyadic integer, that is, M = 2L for some nonnegative integer L. Define TM to be the collection of all DDTs such that no terminal cell has a sidelength smaller than 2−L. In other words, no coordinate is split more than L times when traversing a path from the root to a leaf. We will consider classifiers of the form bTn = arg min T ∈TM bRn(T) + Φn(T) (1) where Φn is a “penalty” or regularization term specified below. An algorithm of Blanchard et al. [9] may be used to compute bTn in O(ndLd log(ndLd)) operations. For all of our theorems on rates of convergence below we have L = O(log n), in which case the computational cost is O(nd(log n)d+1). 3 Generalization Error Bounds for Trees In this section we state a uniform error bound and an oracle inequality for DDTs. These two results are extensions of our previous work on DDTs [10]. The bounding techniques are quite general and can be extended to larger (even uncountable) families of trees using VC theory, but for the sake of simplicity we confine the discussion to DDTs. Complete proofs may be found in [11]. Before stating these results, some additional notation is necessary. Let A ∈A, and define JAK = (2 + log2 d)j(A). JAK represents the number of bits needed to uniquely encode A and will be used to measure the complexity of a DDT having A as a leaf cell. These “codelengths” satisfy a Kraft inequality P A∈A 2−JAK ≤1. For a cell A ⊆[0, 1]d, define pA = PX(A) and ˆpA = (1/n) Pn i=1 I{Xi∈A}. Further define ˆp′ A = 4 max(ˆpA, (JAK log 2 + log n)/n) and p′ A = 4 max(pA, (JAK log 2 + log n)/(2n)). It can be shown that with high probability, pA ≤ˆp′ A and ˆpA ≤p′ A uniformly over all A ∈A [11]. The mutual boundedness of pA and ˆpA is a key to making our proposed classifier both computable on the one hand and analyzable on the other. Define the data-dependent penalty Φn(T) = X A∈π(T ) r 2ˆp′ A JAK log 2 + log(2n) n . (2) Our first main result is the following uniform error bound. Theorem 1. With probability at least 1 −2/n, R(T) ≤bRn(T) + Φn(T) for all T ∈T . (3) Traditional error bounds for trees involve a penalty proportional to p |T| log n/n, where |T| denotes the number of leaves in T (see [12] or the “naive” bound in [2]). The penalty in (2) assigns a different weight to each leaf of the tree depending on both the depth of the leaf and the fraction of data reaching the leaf. Indeed, for very deep leaves, little data will reach those nodes, and such leaves will contribute very little to the overall penalty. For example, we may bound ˆp′ A by p′ A with high probability, and if X has a bounded density, then p′ A decays like max{2−j, log n/n}, where j is the depth of A. Thus, as j increases, JAK grows additively with j, but ˆp′ A decays at a multiplicative rate. The upshot is that the penalty Φn(T) favors unbalanced trees. Intuitively, if two trees have the same size and empirical error, minimizing the penalized empirical risk with this new penalty will select the tree that is more unbalanced, whereas a traditional penalty based only on tree size would not distinguish the two. This has advantages for classification because unbalanced trees are what we expect when approximating a lower dimensional decision boundary. The derivation of (2) comes from applying standard concentration inequalities for sums of Bernoulli trials (most notably the relative Chernoff bound) in a spatially decomposed manner. Spatial decomposition allows the introduction of local probabilities pA to offset the complexity of each leaf node A. Our analysis is inspired by the work of Mansour and McAllester [2]. The uniform error bound of Theorem 1 can be converted (using standard techniques) into an oracle inequality that is the key to deriving rates of convergence for DDTs. Theorem 2. Let bTn be as in (1) with Φn as in (2). Define ˜Φn(T) = X A∈π(T ) r 8p′ A JAK log 2 + log(2n) n . Then EZn{R( bTn)} −R∗≤min T ∈T h R(T) −R∗+ 2˜Φn(T) i + O 1 n . (4) Note that with high probability, p′ A is an upper bound on ˆpA, and therefore ˜Φn upper bounds Φn. The use of ˜Φn instead of Φn in the oracle bound facilitates rate of convergence analysis. The oracle inequality essentially says that bTn performs nearly as well as the DDT chosen by an oracle to minimize R(T) −R∗. The right hand side of (4) bears the interpretation of a decomposition into approximation error (R(T) −R∗) and estimation error ˜Φn(T). 4 Rates of Convergence The classes of distributions we study are motivated by the work of Mammen and Tsybakov [4] and Tsybakov [1] which we now review. The classes are indexed by the smoothness γ of the Bayes decision boundary ∂G∗and a parameter κ that quantifies how “noisy” the distribution is near ∂G∗. We write an ≼bn when an = O(bn) and an ≍bn if both an ≼bn and bn ≼an. Let γ > 0, and take r = ⌈γ⌉−1 to be the largest integer not exceeding γ. Suppose b : [0, 1]d−1 →[0, 1] is r times differentiable, and let pb,s denote the Taylor polynomial of b of order r at the point s. For a constant c1 > 0, define Σ(γ, c1), the class of functions with H¨older regularity γ, to be the collection of all b such that |b(s′) −pb,s(s′)| ≤c1|s −s′|γ for all s, s′ ∈[0, 1]d−1. Using Tsybakov’s terminology, the Bayes decision set G∗is a boundary fragment of smoothness γ if G∗= epi(b) for some b ∈Σ(γ, c1). Here epi(b) = {(s, t) ∈[0, 1]d : b(s) ≤t} is the epigraph of b. In other words, for a boundary fragment, the last coordinate of ∂G∗is a H¨older-γ function of the first d −1 coordinates. Tsybakov also introduces a condition that characterizes the level of “noise” near ∂G∗in terms of a noise exponent κ, 1 ≤κ ≤∞. Let ∆(f1, f2) = {x ∈[0, 1]d : f1(x) ̸= f2(x)}. Let c2 > 0. A distribution satisfies Tsybakov’s noise condition with noise exponent κ and constant c2 if PX(∆(f, f ∗)) ≤c2(R(f) −R∗)1/κ for all f. (5) The case κ = 1 is the “low noise” case and corresponds to a jump of η(x) at the Bayes decision boundary. The case κ = ∞is the high noise case and imposes no constraint on the distribution (provided c2 ≥1). See [6] for further discussion. Define the class F = F(γ, κ) = F(γ, κ, c0, c1, c2) to be the collection of distributions of Z = (X, Y ) such that 0A For all measurable A ⊆[0, 1]d, PX(A) ≤c0λ(A) 1A G∗is a boundary fragment defined by b ∈Σ(γ, c1). 2A The margin condition is satisfied with noise exponent κ and constant c2. Introducing the parameter ρ = (d −1)/γ, Tsybakov [1] proved the lower bound inf bfn sup F h EZn{R( bfn)} −R∗i ≽n−κ/(2κ+ρ−1). (6) The inf is over all rules for constructing classifiers from training data. Theoretical rules that achieve this lower bound are studied by [1, 4, 5, 13]. Unfortunately, none of these works provide computationally efficient algorithms for implementing the proposed discrimination rules, and it is unlikely that practical algorithms exist for these rules. It is important to note that the lower bound in (6) is tight only when ρ < 1. To see this, fix ρ > 1. From the definition of F(γ, κ) we have F(γ, 1) ⊂F(γ, κ) for any κ > 1. As κ →∞, the right-hand side of (6) decreases. Therefore, the minimax rate for F(γ, κ) can be no faster than n−1/(1+ρ), which is the lower bound for F(γ, 1). In light of the above, Tsybakov’s noise condition does not improve the learning situation when ρ > 1. To achieve rates faster than n−1/(1+ρ) when ρ > 1, clearly an alternate assumption must be made. If the right-hand side of (6) is any indication, then the distributions responsible for slower rates are those with small κ. Thus, it would seem that we need a noise assumption that excludes those “low noise” distributions with small κ that cause slower rates when ρ > 1. Since recursive dyadic partitions can well-approximate G∗with smoothness γ ≤1, we are in the regime of ρ ≥(d −1)/γ ≥1. As motivated above, faster rates in this situation require an assumption that excludes low noise levels. We propose such an assumption. Like Tsybakov’s noise condition, our assumption is also defined in terms of constants κ ≥1 and c2 > 0. Because of limited space we are unable to fully present the modified noise condition, and we simply write 2B Low noise levels are excluded as defined in [11]. Effectively, 2B says that the inequality in (5) is reversed, not for all classifiers f, but only for those f that are the best DDT approximations to f ∗for each DDT resolutions parameter M. Using techniques presented in [13], we show in [11] that lower bounds of the form in (6) are valid when 2A is replaced by 2B. According to the results in the next section, these lower bounds are tight to within a log factor for ρ > 1. 5 Adaptive Rates for Dyadic Decision Trees All of our rate of convergence proofs use the oracle inequality in the same basic way. The objective is to find an “oracle tree” T ∗∈T such that both R(T ∗) −R∗and ˜Φn(T ∗) decay at the desired rate. This tree is roughly constructed as follows. First form a “regular” dyadic partition (the exact construction will depend on the specific problem) into cells of sidelength 1/m = 2−K, for a certain K ≤L. Then “prune back” all cells that do not intersect ∂G∗. Both approximation and estimation error may now be bounded using the given assumptions and elementary bounding methods. For example, R(T ∗) −R∗≼ (PZ(∆(T ∗, f ∗)))κ (by 2B) ≼(λ(∆(T ∗, f ∗)))κ (by 0A) ≼m−κ (by 1A). This example reveals how the noise exponent enters the picture to affect the approximation error. See [11] for complete proofs. 5.1 Noise Adaptive Classification Dyadic decision trees, selected according to the penalized empirical risk criterion discussed earlier, adapt to the unknown noise level to achieve faster rates as stated in Theorem 3 below. For now we focus on distributions with γ = 1 (ρ = d −1), i.e., Lipschitz decision boundaries (the case γ ̸= 1 is discussed in Section 5.4), and arbitrary noise parameter κ. The optimal rate for this class is n−κ/(2κ+d−2) [11]. We will see that DDTs can adaptively learn at a rate of (log n/n)κ/(2κ+d−2). In an effort to be more general and practical, we replace the boundary fragment condition 1A with a less restrictive assumption. Tysbakov and van de Geer [5] assume the Bayes decision set G∗is a boundary fragment, meaning it is known a priori that (a) one coordinate of ∂G∗is a function of the others, (b) that coordinate is known, and (c) class 1 corresponds to the region above ∂G∗. The following condition includes all piecewise Lipschitz decision boundaries, and allows ∂G∗to have arbitrary orientation and G∗to have multiple connected components. Let Pm denote the regular partition of [0, 1]d into hypercubes of sidelength 1/m where m is a dyadic integer (i.e., a power of 2). A distribution of Z satisfies the box-counting assumption with constant c1 > 0 if 1B For all dyadic integers m, ∂G∗intersects at most c1md−1 of the md cells in Pm. Condition 1A (γ = 1) implies 1B, (with a different c1) so the minimax rate under 0A, 1B, and 2B is no faster than n−κ/(2κ+d−2). Theorem 3. Let M ≍(n/ log n). Take bTn as in (1) with Φn as in (2). Then sup h EZn{R( bTn)} −R∗i ≼ log n n κ 2κ+d−2 . (7) where the sup is over all distributions such that 0A, 1B, and 2B hold. The complexity regularized DDT is adaptive in the sense that the noise exponent κ and constants c0, c1, c2 need not be known. bTn can always be constructed and in opportune circumstances the rate in (7) is achieved. 5.2 When the Data Lie on a Manifold For certain problems it may happen that the feature vectors lie on a manifold in the ambient space X. When this happens, dyadic decision trees automatically adapt to achieve faster rates of convergence. To recast assumptions 0A and 1B in terms of a data manifold1, we again use box-counting ideas. Let c0, c1 > 0 and 1 ≤d′ ≤d. The boundedness and regularity assumptions for a d′ dimensional manifold are given by 0B For all dyadic integers m and all A ∈Pm, PX(A) ≤c0m−d′. 1C For all dyadic integers m, ∂G∗passes through at most c1md′−1 of the md hypercubes in Pm. The minimax rate under these assumptions is n−1/d′. To see this, consider the mapping of features X′ = (X1, . . . , Xd′) ∈[0, 1]d′ to X = (X1, . . . , Xd′, 1/2, . . . , 1/2) ∈[0, 1]d. Then X lives on a d′ dimensional manifold, and clearly there can be no classifier achieving a rate faster than n−1/d′ uniformly over all such X, as this would lead to a classifier outperforming the minimax rate for X′. As the following theorem shows, DDTs can achieve this rate to within a log factor. Theorem 4. Let M ≍(n/ log n). Take bTn as in (1) with Φn as in (2). Then sup h EZn{R( bTn)} −R∗i ≼ log n n 1 d′ . (8) where the sup is over all distributions such that 0B and 1C hold. Again, bTn is adaptive in that it does not require knowledge d′, c0, or c1. 5.3 Irrelevant Features The “relevant” data dimension is the number of relevant features/attributes, meaning the number d′′ < d of features of X that are not independent of Y . By an argument like that in the previous section, the minimax rate under this assumption (and 0A and 1B) can be seen to be n−1/d′′. Once again, DDTs can achieve this rate to within a log factor. Theorem 5. Let M ≍(n/ log n). Take bTn as in (1) with Φn as in (2). sup h EZn{R( bTn)} −R∗i ≼ log n n 1 d′′ . (9) where the sup is over all distributions with relevant data dimension d′′ and such that 0A and 1B hold. As in the previous theorems, our learning rule is adaptive in the sense that it does not need to be told d′′ or which d′′ features are relevant. 1For simplicity, we eliminate the margin assumption here and in subsequent sections, although it could be easily incorporated to yield faster adaptive rates. 5.4 Adapting to Bayes Decision Boundary Smoothness Our results thus far apply to Tsybakov’s class with γ = 1. In [10] we show that DDTs with polynomial classifiers decorating the leaves can achieve faster rates for γ > 1. Combined with the analysis here, these rates can approach n−1 under appropriate noise assumptions. Unfortunately, the rates we obtain are suboptimal and the classifiers are not practical. For γ ≤1, free DDTs adaptively attain the minimax rate (within a log factor) of n−γ/(γ+d−1). Due to space limitations, this discussion is deferred to [11]. Finding practical classifiers that adapt to the optimal rate for γ > 1 is a current line of research. 6 Conclusion Dyadic decision trees adapt to a variation of Tsybakov’s noise condition, data manifold dimension and the number of relevant features to achieve minimax optimal rates of convergence (to within a log factor). DDTs are constructed by a computationally efficient penalized empirical risk minimization procedure based on a novel, spatially adaptive, datadependent penalty. Although we consider each condition separately so as to simplify the discussion, the conditions can be combined to yield a rate of (log n/n)κ/(2κ+d∗−2) where d∗is the dimension of the manifold supporting the relevant features. References [1] A. B. Tsybakov, “Optimal aggregation of classifiers in statistical learning,” Ann. Stat., vol. 32, no. 1, pp. 135–166, 2004. [2] Y. Mansour and D. McAllester, “Generalization bounds for decision trees,” in Proceedings of the Thirteenth Annual Conference on Computational Learning Theory, N. Cesa-Bianchi and S. Goldman, Eds., Palo Alto, CA, 2000, pp. 69–74. [3] Y. Yang, “Minimax nonparametric classification–Part I: Rates of convergence,” IEEE Trans. Inform. Theory, vol. 45, no. 7, pp. 2271–2284, 1999. [4] E. Mammen and A. B. Tsybakov, “Smooth discrimination analysis,” Ann. Stat., vol. 27, pp. 1808–1829, 1999. [5] A. B. Tsybakov and S. A. van de Geer, “Square root penalty: adaptation to the margin in classification and in edge estimation,” 2004, preprint. [6] P. Bartlett, M. Jordan, and J. McAuliffe, “Convexity, classification, and risk bounds,” Department of Statistics, U.C. Berkeley, Tech. Rep. 638, 2003, to appear in Journal of the American Statistical Association. [7] G. Blanchard, G. Lugosi, and N. Vayatis, “On the rate of convergence of regularized boosting classifiers,” J. Machine Learning Research, vol. 4, pp. 861–894, 2003. [8] J. C. Scovel and I. Steinwart, “Fast rates for support vector machines,” Los Alamos National Laboratory, Tech. Rep. LA-UR 03-9117, 2004. [9] G. Blanchard, C. Sch¨afer, and Y. Rozenholc, “Oracle bounds and exact algorithm for dyadic classification trees,” in Learning Theory: 17th Annual Conference on Learning Theory, COLT 2004, J. Shawe-Taylor and Y. Singer, Eds. Heidelberg: Springer-Verlag, 2004, pp. 378–392. [10] C. Scott and R. Nowak, “Near-minimax optimal classification with dyadic classification trees,” in Advances in Neural Information Processing Systems 16, S. Thrun, L. Saul, and B. Sch¨olkopf, Eds. Cambridge, MA: MIT Press, 2004. [11] ——, “Minimax optimal classification with dyadic decision trees,” Rice University, Tech. Rep. TREE0403, 2004. [Online]. Available: http://www.stat.rice.edu/∼cscott [12] A. Nobel, “Analysis of a complexity based pruning scheme for classification trees,” IEEE Trans. Inform. Theory, vol. 48, no. 8, pp. 2362–2368, 2002. [13] J.-Y. Audibert, “PAC-Bayesian statistical learning theory,” Ph.D. dissertation, Universit´e Paris 6, June 2004.
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Kernel Methods for Implicit Surface Modeling Bernhard Sch¨olkopf†, Joachim Giesen+∗& Simon Spalinger+ † Max Planck Institute for Biological Cybernetics, 72076 T¨ubingen, Germany bernhard.schoelkopf@tuebingen.mpg.de + Department of Computer Science, ETH Z¨urich, Switzerland giesen@inf.ethz.ch,spsimon@inf.ethz.ch 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 o o o w x Φ( ) o o o o o o ||w|| ρ/ ξ/||w|| . Figure 1: In the 2-D toy example depicted, the hyperplane ⟨w, Φ(x)⟩= ρ separates all but one of the points from the origin. The outlier Φ(x) is associated with a slack variable ξ, which is penalized in the objective function (4). The distance from the outlier to the hyperplane is ξ/∥w∥; the distance between hyperplane and origin is ρ/∥w∥. The latter implies that a small ∥w∥corresponds to a large margin of separation from the origin. k on X × X which can be expressed as k(x, x′) = ⟨Φ(x), Φ(x′)⟩ (1) for some map Φ into a Hilbert space H. The space H is the reproducing kernel Hilbert space (RKHS) associated with k, and Φ is called its feature map. A popular example, in the case where X is a normed space, is the Gaussian (where σ > 0) k(x, x′) = exp −∥x −x′∥2 2 σ2 . (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) ∈ [ρ, ∞[} of an unknown distribution P on X using kernel expansions. Here, f(x) = X i αik(xi, x) −ρ, (3) 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 Φ 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 ν ∈(0, 1],1 minimize w∈H,ξ∈Rm,ρ∈R 1 2∥w∥2 + 1 νm X i ξi −ρ (4) subject to ⟨w, Φ(xi)⟩≥ρ −ξi, ξi ≥0. (5) Since non-zero slack variables ξi are penalized in the objective function, we can expect that if w and ρ solve this problem, then the decision function, f(x) = sgn (⟨w, Φ(x)⟩−ρ) will 1Here and below, bold face greek character denote vectors, e.g., ξ = (ξ1, . . . , ξm)⊤, 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 σ 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 ∥w∥will still be small. For an illustration, see Figure 1. The trade-off between these two goals is controlled by a parameter ν. One can show that the solution takes the form f(x) = sgn X i αik(xi, x) −ρ ! , (6) where the αi are computed by solving the dual problem, minimize α∈Rm 1 2 X ij αiαjk(xi, xj) (7) subject to 0 ≤αi ≤ 1 νm and X i αi = 1. (8) Note that according to (8), the training examples contribute with nonnegative weights αi ≥ 0 to the solution (6). One can show that asymptotically, a fraction ν of all training examples will have strictly positive weights, and the rest will be zero (the “ν-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 xi 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 αi 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 σ in the Gaussian 2We use the convention that sgn (z) equals 1 for z ≥0 and −1 otherwise. o o o o w x Φ( ) o o o o o ||w|| (ρ+δ)/ ξ/||w|| . ξ / ||w|| * * x Φ( ) ||w|| (ρ+δ )/ * Figure 3: Two parallel hyperplanes ⟨w, Φ(x)⟩= ρ + δ(∗) enclosing all but two of the points. The outlier Φ(x(∗)) is associated with a slack variable ξ(∗), which is penalized in the objective function (9). 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,ξ (∗)∈Rm,ρ∈R 1 2∥w∥2 + 1 νm X i (ξi + ξ∗ i ) −ρ (9) subject to δ −ξi ≤⟨w, Φ(xi)⟩−ρ ≤δ∗+ ξ∗ i (10) and ξ(∗) i ≥0. (11) Here, δ(∗) 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 ρ changed by the same offset. Hence, we can generally set one of them to zero, say, δ = 0. Below we summarize some relationships of this convex quadratic optimization problem to known SV methods: 1. For δ = 0 and δ∗= ∞(i.e., no upper constraint), we recover the single-class SVM (4)–(5). 2. If we drop ρ from the objective function and set δ = −ε, δ∗= ε (for some fixed ε ≥0), we obtain the ε-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 ρ in our objective function plays an important role. 3. For δ = δ∗= 0, the term P i(ξi +ξ∗ i ) measures the distance of the point Φ(xi) from the hyperplane ⟨w, Φ(xi)⟩−ρ = 0 (up to a scaling of ∥w∥). If ν 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. 3Here and below, the superscript (∗) simultaneously denotes the variables with and without asterisk, e.g., ξ(∗) is a shorthand for ξ and ξ∗. From the following constraints and Lagrange multipliers ξi −δ + ⟨w, Φ(xi)⟩−ρ ≥0, αi ≥0 (12) ξ∗ i + δ∗+ ρ −⟨w, Φ(xi)⟩≥0, α∗ i ≥0 (13) ξ(∗) i ≥0, β(∗) i ≥0 (14) we derive the Lagrangian dual optimization problem of (9) - (11):4 minimize α∈Rm 1 2 X ij (αi −α∗ i )(αj −α∗ j)k(xi, xj) −δ X i αi + δ∗X i α∗ i (15) subject to 0 ≤α(∗) i ≤ 1 νm (16) and X i (αi −α∗ i ) = 1, (17) Note that for δ = δ∗, we can simplify the optimization problem using the transformation αnew = α −α∗. For δ = δ∗= 0, we thus obtain the single-class SVM (7) with the modified box constraint −1 νm ≤αnew i ≤ 1 νm. The dual problem can be solved using standard quadratic programming packages. The offset ρ 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 δ ≤⟨w, Φ(x)⟩− ρ ≤δ∗. 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 δ = δ∗, this is a single hyperplane, corresponding to a hypersurface in input space.5 To compute this surface we use the kernel expansion ⟨w, Φ(x)⟩= X i (αi −α∗ i )k(xi, x). (18) 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 ν influences the SVs and outliers. To this end, we introduce the following shorthands for the sets of SV and outlier indices: SV := {i | ⟨w, Φ(xi)⟩−ρ −δ ≤0} (19) SV ∗ := {i | ⟨w, Φ(xi)⟩−ρ −δ∗≥0} (20) OL(∗) := {i | ξ(∗) i > 0} (21) It is clear from the primal optimization problem that for all i, ξi > 0 implies ⟨w, Φ(xi)⟩− ρ −δ < 0 (and likewise, ξ∗ i > 0 implies ⟨w, Φ(xi)⟩−ρ −δ∗> 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. 4Note that due to (17), the dual solution is invariant with respect to the transformation δ(∗) → δ(∗) + const. — such a transformation only adds a constant to the objective function, leaving the solution unaffected. 5subject to suitable conditions on k 6The present usage differs slightly from the standard definition of SVs, which are usually those that satisfy α(∗) i > 0. In our definition, SVs are those points where the constraints are active. However, the difference is marginal: (i) It follows from the KKT conditions that α(∗) 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 α(∗) i is zero, this almost never occurs in practice. Proposition 1 The solution of (9)–(11) satisfies |SV | m −|OL∗| m ≥ν ≥|OL| m −|SV ∗| m . (22) The proof is analogous to the one of the “ν-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 ρ in the primal objective function. If we used +ρ rather than −ρ, 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 ν-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 ε-SV regression algorithm [11] becomes almost identical to the present dual if we set ε = (δ∗−δ)/2 and yi = −(δ∗+ δ)/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 αi −α∗ 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. 7In 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 αi −α∗ 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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Neural network computation by in vitro transcriptional circuits Jongmin Kim1, John J. Hopfield3, Erik Winfree2 Biology1, CNS and Computer Science2, California Institute of Technology. Molecular Biology3, Princeton University. {jongmin,winfree}@dna.caltech.edu, hopfield@princeton.edu 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 (A) RNAP RNase DNA switch activator transcript inhibitor (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: OFF ON OFF ON A + I D + A DA + I D + AI DA AI 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 DA kp →DA + R R kd →φ D αkp →D + R where 0 < α < 1 due to lack of activation and φ 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] dt = X β kβ Y j [Xj]rβ j (pβ i −rβ i ) where kβ is the rate constant, rβ i is the stoichiometry of species Xi as a reactant (typically 0 or 1), and pβ i is the stoichiometry of Xi as a product in reaction β. Analysis of our system is greatly simplified by the assumption that the binding reactions are fast and go to completion. We define Dtot as the sum of free and bound species:Dtot = [D] + [DA]. Similarly, Itot = [I]+[AI] and Atot = [A]+[DA]+[AI]. Then, [DA] depends on Dtot and ∆, where ∆= Atot −Itot. 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 < ∆< Dtot, as shown in Figure 2A. It is convenient to represent this nonlinearity using a piecewise-linear approximation of a sigmoidal function, specifically, σ(x) = |x+1|−|x−1| 2 . Thus, we can represent [DA] using σ and a rescaled ∆: [DA] = 1 2Dtot(1 + σ( ˆ∆)), where ˆ∆= 2∆ Dtot −1 is called the signal activity. At steady-state, kd[R] = kp[DA] + αkp[D]; thus, [R] = 1 2 kp kd Dtot((1 −α)σ( ˆ∆) + 1 + α) . 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, Dtot, 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. (A) ∆ tot D tot D [ DA ] (B) −1 1 1 −1 x σ (x) (C) [ R ] A tot (D) [ R ] A tot Figure 2: (A) [DA] as a function of ∆. (B) The sigmoid σ(x). (C,D) [R] as a function of Atot for three values of Dtot and Itot, 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 ON OFF ON j j j j j ij j ij j j A + I A I D A D ij + A D ij + j j A I D A + I 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 DijAj 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 By RNase DijAj kp →DijAj + Ai if sij = 1 Ij kd →φ Dij αkp →Dij + Ai Aj kd →φ DijAj kp →DijAj + Ii if sij = −1 AjIj kd →φ Dij αkp →Dij + Ii DijAj kd →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 change of RNA concentrations over time is easy to express with ∆i = Atot i −Itot i : d∆i dt = −kd · ∆i + kp X j sij([DijAj] + α[Dij]) . (1) 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]: τ dxi dt = −xi + X j wijσ(xj) + θi . (2) Equation 1 can be rewritten to use the same nonlinear activation function σ defined earlier. Let ˆ∆i = 2∆i Dtot ∗i −1 be a rescaled difference between activator and inhibitor concentrations, where Dtot ∗i is the load on Ai, i.e., the total concentration of all switches that bind to Ai: Dtot ∗i = P j Dtot ji and Dtot ij = [DijAj] + [Dij]. Then, we can derive the following rate equation, where ˆ∆i plays the role of unit i’s activity xi: 1 kd d ˆ∆i dt = −ˆ∆i + X j kp kd (1 −α)sij Dtot ij Dtot ∗i σ( ˆ∆j)+ X j kp kd (1 + α)sij Dtot ij Dtot ∗i −1 . (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 τ 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 θi 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 DiB 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 D0i which bind to Ai but have no output domain; this allows us to ensure that all Dtot ∗i are equal. Consequently, given any neural network with weights wij and thresholds θi, we can specify concentrations Dtot ij such that the biochemical network has identical dynamics, for some τ. 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, E + DijAj k+ ⇀ ↽ k−EDijAj kcat →E + DijAj + Ii/Ai . 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 = Etot 1 + L · kcat KM α · kp = Etot 1 + L · k′ cat K′ M kd = Etot d 1 + Ld · kd,cat Kd,M (4) where kcat and KM = (k−+ kcat)/k+ are the catalytic constant (enzyme’s speed) and Michaelis constant (enzyme’s affinity to target) of RNAP for the ON state switch, k′ cat and K′ M are for the OFF state switch, and kd,cat and Kd,M are the constants of RNase. Etot and Etot d are the concentrations of RNAP and RNase, respectively. L = P i,j [DijAj] KM + P i,j [Dij] K′ M is the load on RNAP and Ld = P i,j [Aj]+[Ij]+[AjIj]+[DijAj] Kd,M is the load on 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) = ¯xyz + ¯yz +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 Etot and Etot d 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 (A) f x y z (B) 1 x y z f 1 1 −1 1 1 −1 1 −2 −1 1 2 (C) 0 200 400 600 800 1000 −4 −2 0 2 4 6 time(sec) ∆i^ Figure 3: (A,B) A Boolean circuit and a neural network to compute f(x, y, z) = ¯xyz+¯yz+ 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. (A) (B) 0 200 400 600 −3 −2 −1 0 1 2 3 time(sec) ∆i^ 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 Dii 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 i dt = −Etot d 1 + Ld · kd,cat Kd,M Atot i + Etot 1 + L kcat KM [DiiAi] + k′ cat K′ M [Dii] . (5) The production rate of Ai depends on Atot i and on L, while the degradation rate of Ai depends on Atot i and on Ld, as shown in Figure 6A. For a winner-take-all network, an ON 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 δ above the rest (as a percentage). Analysis indicates that a less leaky system (small α) and sufficient differences in initial activator concentrations (large δ) 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. (A) 1 1 1 0.5 0.5 0.5 −1 −1 −1 −1 −1 −1 (B) 1 1 1 0.5 0.5 0.5 Figure 5: (A) A 3-unit WTA network with explicit mutual inhibition. (B) An equivalent biochemical network. (A) i dt i tot i I +D ii I tot tot A tot i d A tot L : high L : low (B) 5 10 15 5 10 15 20 25 30 δ(%) 1/α 0 0.2 0.4 0.6 0.8 1 (C) 0 5000 10000 15000 0 0.5 1 1.5 2 2.5 3 time(sec) [Ai] / M µ 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 α and δ. (C) Time-course with δ = 0.33% and α = 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(= 10−19J) per nucleotide incorporated, and 1 bit of information 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 1µM signal in 20µl) 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] McCulloch WS, Pitts W, Bull. Math. Biophys. 5 (1943), 115. [2] Monod J, Jacob F, Cold Spring Harb. Symp. Quant. Biol. 26 (1961), 389-401. [3] Hopfield JJ, Proc. Nat. Acad. Sci. USA 81 (1984), 3088-3092. [4] Hasty J, McMillen D, Issacs F, Collins JJ, Nat. Rev. Genet. 2 (2001), 268-279. [5] Elowitz MB, Leibler S, Nature 403 (2000), 335-338. [6] Gardner TS, Cantor CR, Collins JJ, Nature 403 (2000), 339-342. [7] Martin CT, Coleman JE, Biochemistry 26 (1987), 2690-2696. [8] Yurke B, Mills AP Jr., Genetic Programming and Evolvable Machines 4 (2003), 111-122. [9] Shea MA, Ackers GK, J. Mol. Biol. 181 (1985), 211-230. [10] Hammes GG, Thermodynamics and kinetics for the biological sciences, Wiley (2000). [11] Yuille AL, Gieger D, in The Handbook of Brain Theory and Neural Networks, Arbib MA, ed., MIT Press (1995), 1056-1060. [12] Tank DW, Hopfield JJ, IEEE Trans. on Circuits and Systems 33 (1986), 533-541. [13] Maass W, Neural Computation 12 (2000), 2519-2535. [14] Glass L, Kauffman SA, J. Theo. Biol. 39 (1973), 103-129. [15] Mjolsness E, Sharp DH, Reinitz J, J. Theo. Biol. 152 (1991), 429-453. [16] Buchler NE, Gerland U, Hwa T, Proc. Nat. Acad. Sci. USA 100 (2003), 5136-5141. [17] Bray D, Science 301 (2003), 1864-1865. [18] Reed RD, IEEE Trans. on Neural Networks, 4 (1993), 740-744. [19] Dirks R, Lin M, Winfree E, Pierce NA, Nucleic Acids Research 32 (2004), 1392-1403. [20] Lilley DM, Trends Biochem. Sci. 28 (2003), 495-501. [21] Nudler E, Mironov AS, Trends Biochem. Sci. 29 (2004), 11-17. [22] Mills AP Jr., Yurke B, Platzman PM, Biosystems 52 (1999), 175-180.
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Similarity and discrimination in classical conditioning: A latent variable account Aaron C. Courville*1,3, Nathaniel D. Daw4 and David S. Touretzky2,3 1Robotics Institute, 2Computer Science Department, 3Center for the Neural Basis of Cognition, Carnegie Mellon University, Pittsburgh, PA 15213 4Gatsby Computational Neuroscience Unit, University College London {aaronc,dst}@cs.cmu.edu; daw@gatsby.ucl.ac.uk 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 animals 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. 1In 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: P(Sj | m, wm, x) = (1 + exp(−(w(j) m )T x −wbias))−1, (1) where the weight vector w(j) m 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: P(R | Stim, m, wm, D) = X x P(R | m, wm, x)P(x | Stim, m, wm, D) (2) 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: P(R | Stim, m, D) = Z 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: P(R | Stim, D) = X m P(R | Stim, m, D)P(m | D), (4) 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 li, the number of links between the stimuli and each latent variable: P(m) = P(nx) Qnx i=1 P(li). We 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, 2The Laplace distribution is given by f(y) = 1 2be−|y−µ|/b. In our simulations µ = 0 and b = 2. As a prior, it encodes a bias for sparsity consistent with a preference for simpler model structures. A AB Control 0 0.2 0.4 0.6 0.8 1 P(R | Probe,D ) Probe Stimulus Trained Transfer Element 0 0.2 0.4 0.6 0.8 1 P(R | Probe,D ) Probe Stimulus R A B x1 (c) Overshadowing R A B C D x1 x2 (a) Overshadowing (b) Summation (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 acti0 5 10 15 20 25 30 0 0.2 0.4 0.6 0.8 1 Trial Blocks P(R | A, D ) P(R | B,C, D ) P(R | A,B,C, D ) 0 5 10 15 20 25 30 0 0.5 1 1.5 2 2.5 3 3.5 Trial Blocks Average number of latent variables (a) Overlap: learning curves (b) Model growth with training R A B C x1 R A B C x1 x2 R A B C x1 x2 x3 (c) 4 Trial Blocks (d) 10 Trial Blocks (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 favor 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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Boosting on manifolds: adaptive regularization of base classifiers Bal´azs K´egl and Ligen Wang Department of Computer Science and Operations Research, University of Montreal CP 6128 succ. Centre-Ville, Montr´eal, Canada H3C 3J7 {kegl|wanglige}@iro.umontreal.ca Abstract In this paper we propose to combine two powerful ideas, boosting and manifold learning. On the one hand, we improve ADABOOST by incorporating knowledge on the structure of the data into base classifier design and selection. On the other hand, we use ADABOOST’s efficient learning mechanism to significantly improve supervised and semi-supervised algorithms proposed in the context of manifold learning. Beside the specific manifold-based penalization, the resulting algorithm also accommodates the boosting of a large family of regularized learning algorithms. 1 Introduction ADABOOST [1] is one of the machine learning algorithms that have revolutionized pattern recognition technology in the last decade. The algorithm constructs a weighted linear combination of simple base classifiers in an iterative fashion. One of the remarkable properties of ADABOOST is that it is relatively immune to overfitting even after the training error has been driven to zero. However, it is now a common knowledge that ADABOOST can overfit if it is run long enough. The phenomenon is particularly pronounced on noisy data, so most of the effort to regularize ADABOOST has been devoted to make it tolerant to outliers by either “softening” the exponential cost function (e.g., [2]) or by explicitly detecting outliers and limiting their influence on the final classifier [3]. In this paper we propose a different approach based on complexity regularization. Rather than focusing on possibly noisy data points, we attempt to achieve regularization by favoring base classifiers that are smooth in a certain sense. The situation that motivated the algorithm is not when the data is noisy, rather when it has a certain structure that is ignored by ordinary ADABOOST. Consider, for example, the case when the data set is embedded in a high-dimensional space but concentrated around a low dimensional manifold (Figure 1(a)). ADABOOST will compare base classifiers based on solely their weighted errors so, implicitly, it will consider every base classifier having the same (usually low) complexity. On the other hand, intuitively, we may hope to achieve better generalization if we prefer base classifiers that “cut through” sparse regions to base classifiers that cut into “natural” clusters or cut the manifold several times. To formalize this intuition, we use the graph Laplacian regularizer proposed in connection to manifold learning [4] and spectral clustering [5] (Section 3). For binary base classifiers, this penalty is proportional to the number of edges of the neighborhood graph that the classifier cuts (Figure 1(b)). (a) (b) Figure 1: (a) Given the data, the vertical stump has a lower “effective” complexity than the horizontal stump. (b) The graph Laplacian penalty is proportional to the number of separated neighbors. To incorporate this adaptive penalization of base classifiers into ADABOOST, we will turn to the marginal ADABOOST algorithm [6] also known as arc-gv [7]. This algorithm can be interpreted as ADABOOST with an L1 weight decay on the base classifier coefficients with a weight decay coefficient θ. The algorithm has been used to maximize the hard margin on the data [7, 6] and also for regularization [3]. The coefficient θ is adaptive in all these applications: in [7] and [6] it depends on the hard margin and the weighted error, respectively, whereas in [3] it is different for every training point and it quantifies the “noisiness” of the points. The idea of this paper is to make θ dependent on the individual base classifiers, in particular, to set θ to the regularization penalty of the base classifier. First, with this choice, the objective of base learning becomes standard regularized error minimization so the proposed algorithm accommodates the boosting of a large family of regularized learning algorithms. Second, the coefficients of the base classifiers are lowered proportionally with their complexity, which can be interpreted as an adaptive weight decay. The formulation can be also justified by theoretical arguments which are sketched after the formal description of the algorithm in Section 2. Experimental results (Section 4) show that the regularized algorithm can improve generalization. Even when the improvement is not significant, the difference between the training error and the test error decreases significantly and the final classifier is much sparser than ADABOOST’s solution, both of which indicate reduced overfitting. Since the Laplacian penalty can be computed without knowing the labels, the algorithm can also be used for semi-supervised learning. Experiments in this context show that algorithm besignificantly the semi-supervised algorithm proposed in [4]. 2 The REGBOOST algorithm For the formal description, let the training data be Dn = (x1, y1), . . . , (xn, yn) where data points (xi, yi) are from the set Rd ×{−1, 1}. The algorithm maintains a weight distribution w(t) = w(t) 1 , . . . , w(t) n over the data points. The weights are initialized uniformly in line 1 (Figure 2), and are updated in each iteration in line 10. We suppose that we are given a base learner algorithm BASE Dn, w, P(·) that, in each iteration t, returns a base classifier h(t) coming from a subset of H = h : Rd 7→{−1, 1} . In ADABOOST, the goal of the base classifier is to minimize the weighted error ϵ = ϵ(t)(h) = n X i=1 w(t) i I {h(xi) ̸= yi} , 12 1The indicator function I{A} is 1 if its argument A is true and 0 otherwise. 2We will omit the iteration index (t) and the argument (h) where it does not cause confusion. REGBOOST Dn, BASE(·, ·, ·), P(·), λ, T 1 w ←(1/n, . . . , 1/n) 2 for t ←1 to T 3 h(t) ←BASE Dn, w(t), P(·) 4 γ(t) ← n X i=1 w(t) i h(t)(xi)yi ▷edge 5 θ(t) ←2λP(h(t)) ▷edge offset 6 α(t) ←1 2 ln 1 + γ(t) 1 −γ(t) · 1 −θ(t) 1 + θ(t) ▷base coefficient 7 if α(t) ≤0 ▷⇐⇒base error ≥(1 −θ(t))/2 8 return f (t−1)(·) = Pt−1 j=1 α(j)h(j)(·) 9 for i ←1 to n 10 w(t+1) i ←w(t) i exp −α(t)h(t)(xi)yi Pn j=1 w(t) j exp −α(t)h(t)(xj)yj 11 return f (T )(·) = PT t=1 α(t)h(t)(·) Figure 2: The pseudocode of the REGBOOST algorithm with binary base classifiers. Dn is the training data, BASE is the base learner, P is the penalty functional, λ is the penalty coefficient, and T is the number of iterations. which is equivalent to maximizing the edge γ = 1 −2ϵ = Pn i=1 w(t) i h(xi)yi. The goal of REGBOOST’s base learner is to minimize the penalized cost R1(h) = ϵ(h) + λP(h) = 1 2 −1 2(γ −θ), (1) where P : H 7→R is an arbitrary penalty functional or regularization operator, provided to REGBOOST and to the base learner, λ is the penalty coefficient, and θ = 2λP(h) is the edge offset. Intuitively, the edge γ quantifies by how much h is better than a random guess, while the edge offset θ indicates by how much h(t) must be better than a random guess. This means that for complex base classifiers (with large penalties), we require a better base classification than for simple classifiers. The main advantage of R1 is that it has the form of conventional regularized error minimization, so it accommodates the boosting of all learning algorithms that minimize an error functional of this form (e.g., neural networks with weight decay). However, the minimization of R1 is suboptimal from boosting’s point of view.3 If computationally possible, the base learner should minimize R2(h) = 2 s 1 −ϵ 1 + θ 1+θ ϵ 1 −θ 1−θ = s1 + γ 1 + θ 1+θ 1 −γ 1 −θ 1−θ . (2) 3This statement along with the formulae for R1, R2, and α(t) are explained formally after Theorem 1. After computing the edge and the edge offset in lines 4 and 5, the algorithm sets the coefficient α(t) of the base classifier h(t) to α(t) = 1 2 ln 1 + γ(t) 1 −γ(t) −1 2 ln 1 + θ(t) 1 −θ(t) . (3) In line 11, the algorithm returns the weighted average of the base classifiers f (T )(·) = PT t=1 α(t)h(t)(·) as the combined classifier, and uses the sign of f (T )(x) to classify x. The algorithm must terminate if α(t) ≤0 which is equivalent to γ(t) ≤θ(t) and to ϵ(t) ≥ (1−θ(t))/2.4 In this case, the algorithm returns the actual combined classifier in line 8. This means that either the capacity of the set of base classifiers is too small (γ(t) is small), or the penalty is too high (θ(t) is high), so we cannot find a new base classifier that would improve the combined classifier. Note that the algorithm is formally equivalent to ADABOOST if θ(t) ≡0 and to marginal ADABOOST if θ(t) ≡θ is constant. For the analysis of the algorithm, we first define the unnormalized margin achieved by f (T ) on (xi, yi) as ρi = f (T )(xi)yi, and the (normalized) margin as eρi = ρi ∥α∥1 = PT t=1 α(t)h(t)(xi)yi PT t=1 α(t) , (4) where ∥α∥1 = PT t=1 α(t) is the L1 norm of the coefficient vector. Let the average penalty or margin offset be defined as the average edge offset ¯θ = PT t=1 α(t)θ(t) PT t=1 α(t) . (5) The following theorem upper bounds the marginal training error L(¯θ)(f (T )) = 1 n n X i=1 I eρi < ¯θ (6) achieved by the combined classifier f (T ) that REGBOOST outputs. Theorem 1 Let θ(t) = 2λP(h(t)), let ¯θ and L(¯θ)(f (T )) be as defined in (5) and (6), respectively. Let w(t) i be the weight of training point (xi, yi) after the tth iteration (updated in line 10 in Figure 2), and let α(t) be the weight of the base regressor h(t)(·) (computed in line 6 in Figure 2). Then L(¯θ)(f (T )) ≤ T Y t=1 eθ(t)α(t) n X i=1 w(t) i e−α(t)h(t)(xi)yi △= T Y t=1 E(t) α(t), h(t) . (7) Proof. The proof is an extension of the proof of Theorem 5 in [8]. L(¯θ)(f (T )) = 1 n n X i=1 I ( ¯θ T X t=1 α(t) − T X t=1 α(t)h(t)(xi)yi ≥0 ) (8) ≤ 1 n n X i=1 e ¯θ PT t=1 α(t)−PT t=1 α(t)h(t)(xi)yi (9) = e ¯θ PT t=1 α(t) T Y t=1 n X j=1 w(t) j e−α(t)h(t)(xj)yj n X i=1 w(T +1) i . (10) 4Strictly speaking, α(t) = 0 could be allowed but in this case the α(t) would remain 0 forever so it makes no sense to continue. In (8) we used the definitions (6) and (4), the inequality (9) holds since ex ≥I{x ≥0}, and we obtained (10) by recursively applying line 10 in Figure 2. The theorem follows by the definition (5) and since Pn i=1 w(T +1) i = 1. □ First note that Theorem 1 explains the base objectives (1) and (2) and the base coefficient (3). The goal of REGBOOST is the greedy minimization of the exponential bound in (7), that is, in each iteration we attempt to minimize E(t) (α, h). Given h(t), E(t) α, h(t) is minimized by (3), and with this choice for α(t), R2(h) = E(t) α(t), h , so the base learner should attempt to minimize R2(h). If this is computationally impossible, we follow Mason et al.’s functional gradient descent approach [2], that is, we find h(t) by maximizing the negative gradient −∂E(t)(α,h) ∂α in α = 0. Since −∂E(t)(α,h) ∂α α=0 = γ −θ, this criterion is equivalent to the minimization of R1(h).5 Theorem 1 also suggests various interpretations of REGBOOST which indicate why it would indeed achieve regularization. First, by (9) it can be seen that REGBOOST directly minimizes 1 n n X i=1 exp −ρi + ¯θ∥α∥1 , which can be interpreted as an exponential cost on the unnormalized margin with an L1 weight decay. The weight decay coefficient ¯θ is proportional to the average complexity of the base classifiers. Second, Theorem 1 also indicates that REGBOOST indirectly minimizes the marginal error L(¯θ)(f (T )) (6) where the margin parameter ¯θ, again, is moving adaptively with the average complexity of the base classifiers. This explanation is supported by theoretical results that bound the generalization error in terms of the marginal error (e.g., Theorem 2 in [8]). The third explanation is based on results that show that the difference between the marginal error and the generalization error can be upper bounded in terms of the complexity of the base classifier class H (e.g., Theorem 4 in [9]). By imposing a non-zero penalty on the base classifiers, we can reduce the pool of admissible functions to those of which the edge γ is larger than the edge offset θ. Although the theoretical results do not apply directly, they support the empirical evidence (Section 4) that indicate that the reduction of the pool of admissible base classifiers and the sparsity of the combined classifier play an important role in decreasing the generalization error. Finally note that the algorithm can be easily extended to real-valued base classifiers along the lines of [10] and to regression by using the algorithm proposed in [11]. If base classifiers come from the set {h : Rd 7→R}, we can only use the base objective R1(h) (1), and the analytical solution (3) for the base coefficients α(t) must be replaced by a simple numerical minimization (line search) of E(t) α, h(t) .6 In the case of regression, the binary cost function I {h(x) ̸= y} should be replaced by an appropriate regression cost (e.g., quadratic), and the final regressor should be the weighted median of the base regressors instead of their weighted average. 3 The graph Laplacian regularizer The algorithm can be used with any regularized base learner that optimizes a penalized cost of the form (1). In this paper we apply a smoothness functional based on the graph 5Note that if θ is constant (ADABOOST or marginal ADABOOST), the minimization of R1(h) and R2(h) leads to the same solution, namely, to the base classifier that minimizes the weighted error ϵ. This is no more the case if θ depends on h. 6As a side remark, note that applying a non-zero (even constant) penalty θ would provide an alternative solution to the singularity problem (α(t) = ∞) in the abstaining base classifier model of [10]. Laplacian operator, proposed in a similar context by [4]. The advantage of this penalty is that it is relatively simple to compute for enumerable base classifiers (e.g., decision stumps or decision trees) and that it suits applications where the data exhibits a low dimensional manifold structure. Formally, let G = (V, E) be the neighborhood graph of the training set where the vertex set V = {x1, . . . , xn} is identical to the set of observations, and the edge set E contains pairs of “neighboring” vertices (xi, xj) such that either ∥xi −xj∥< r or xi (xj) is among the k nearest neighbors of xj (xi) where r or k is fixed. This graph plays a crucial role in several recently developed dimensionality reduction methods since it approximates the natural topology of the data if it is confined to a low-dimensional smooth manifold in the embedding space. To penalize base classifiers that cut through dense regions, we use the smoothness functional PL(h) = 1 2|W| n X i=1 n X j=i+1 h(xi) −h(xj) 2Wij, where W is the adjacency matrix of G, that is, Wij = I (xi, xj) ∈E , and 2|W| = 2 Pn i=1 Pn j=1 Wij is a normalizing factor so that 0 ≤PL(h) ≤1.7 For binary base classifiers, PL(h) is proportional to the number of separated neighbors, that is, the number of connected pairs that are classified differently by h. Let the diagonal matrix D defined by Dii = Pn j=1 Wij, and let L = D −W be the graph Laplacian of G. Then it is easy to see that 2|W|PL(h) = hLhT = ⟨h, Lh⟩= n X j=1 λi⟨h, ei⟩, where h = h(x1), . . . , h(xn) , and ei and λi are the (normalized) eigenvectors and eigenvalues of L, that is, Lei = λiei, ∥ei∥= 1. Since L is positive definite, all the eigenvalues are non-negative. The eigenvectors with the smallest eigenvalues can be considered as the “smoothest” functions on the neighborhood graph. Based on this observation, [4] proposed to learn a linear combination of a small number of the eigenvectors with the smallest eigenvalues. One problem of this approach is that the out-of-sample extension of the obtained classifier is non-trivial since the base functions are only known at the data points that participated in forming the neighborhood graph, so it can only be used in a semi-supervised settings (when unlabeled test points are known before the learning). Our approach is based on the same intuition, but instead of looking for a linear combination of the eigenvectors, we form a linear combination of known base functions and penalize them according to their smoothness on the underlying manifold. So, beside semi-supervised learning (explored in Section 4), our algorithm can also be used to classify out-of-sample test observations. The penalty functional can also be justified from the point of view of spectral clustering [5]. The eigenvectors of L with the smallest eigenvalues8 represent “natural” clusters in the data set, so PL(h) is small if h is aligned with these eigenvectors, and PL(h) is large if h splits the corresponding clusters. 7Another variant (that we did not explore in this paper) is to weight edges decreasingly with their lengths. 8Starting from the second smallest; the smallest is 0 and it corresponds to the constant function. Also note that spectral clustering usually uses the eigenvectors of the normalized Laplacian eL = D−1/2LD−1/2. Nevertheless, if the neighborhood graph is constructed by connecting a fixed number of nearest neighbors, Dii is approximately constant, so the eigenvectors of L and eL are approximately equal. 4 Experiments In this section we present experimental results on four UCI benchmark datasets. The results are preliminary in the sense that we only validated the penalty coefficient λ, and did not optimize the number of neighbors (set to k = 8) and the weighting scheme of the edges of the neighborhood graph (Wij = 0 or 1). We used decision stumps as base classifiers, 10-fold cross validation for estimating errors, and 5-fold cross validation for determining λ. The results (Figure 3(a)-(d) and Table 1) show that the REGBOOST consistently improves generalization. Although the improvement is within the standard deviation, the difference between the test and the training error decreases significantly in two of the four experiments, which indicates reduced overfitting. The final classifier is also significantly sparser after 1000 iterations (last two columns of Table 1). To measure how the penalty affects the base classifier pool, in each iteration we calculated the number of admissible base classifiers relative to the total number of stumps considered by ADABOOST. Figure 3(e) shows that, as expected, REGBOOST traverses only a (sometimes quite small) subset of the base classifier space. (a) 0 0.05 0.1 0.15 0.2 0.25 1 10 100 1000 t ionosphere training error (AdaBoost) test error (AdaBoost) training error (RegBoost) test error (RegBoost) (b) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 1 10 100 1000 t breast cancer training error (AdaBoost) test error (AdaBoost) training error (RegBoost) test error (RegBoost) (c) 0 0.1 0.2 0.3 0.4 0.5 0.6 1 10 100 1000 t sonar training error (AdaBoost) test error (AdaBoost) training error (RegBoost) test error (RegBoost) 0.1 0.15 0.2 0.25 0.3 0.35 1 10 100 1000 t pima indians diabetes training error (AdaBoost) test error (AdaBoost) training error (RegBoost) test error (RegBoost) (d) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 10 100 1000 t rate of admissible stumps ionosphere breast cancer sonar pima indians diabetes (e) 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 1 10 100 1000 t semi-supervised ionosphere training error (AdaBoost) test error (AdaBoost) training error (RegBoost) test error (RegBoost) (f) Figure 3: Learning curves. Test and training errors for the (a) ionosphere, (b) breast cancer, (c) sonar, and (d) Pima Indians diabetes data sets. (e) Rate of admissible stumps. (f) Test and training errors for the ionosphere data set with 100 labeled and 251 unlabeled data points. data set training error test error # of stumps ADAB REGB ADAB REGB ADAB REGB ionosphere 0% 0% 9.14% (7.1) 7.7% (6.0) 182 114 breast cancer 0% 2.44% 5.29% (3.5) 3.82% (3.7) 58 30 sonar 0% 0% 32.5% (19.8) 29.8% (18.8) 234 199 Pima Indians 10.9% 16.0% 25.3% (5.3) 23.3% (6.8) 175 91 Table 1: Errors rates and number of base classifiers after 1000 iterations. Since the Laplacian penalty can be computed without knowing the labels, the algorithm can also be used for semi-supervised learning. Figure 3(f) shows the results when only a subset of the training points are labeled. In this case, REGBOOST can use the combined data set to calculate the penalty, whereas both algorithms can use only the labeled points to determine the base errors. Figure 3(f) indicates that REGBOOST has a clear advantage here. REGBOOST is also far better than the semi-supervised algorithm proposed in [12] (their best test error using the same settings is 18%). 5 Conclusion In this paper we proposed to combine two powerful ideas, boosting and manifold learning. The algorithm can be used to boost any regularized base learner. Experimental results indicate that REGBOOST slightly improves ADABOOST by incorporating knowledge on the structure of the data into base classifier selection. REGBOOST also significantly improves a recently proposed semi-supervised algorithm based on the same regularizer. In the immediate future our goal is to conduct a larger scale experimental study in which we optimize all the parameters of the algorithm, and compare it not only to ADABOOST, but also to marginal ADABOOST, that is, REGBOOST with a constant penalty θ. Marginal ADABOOST might exhibit a similar behavior on the supervised task (sparsity, reduced number of admissible base classifiers), however, it can not be used to semi-supervised learning. We also plan to experiment with other penalties which are computationally less costly than the Laplacian penalty. References [1] Y. Freund and R. E. Schapire, “A decision-theoretic generalization of on-line learning and an application to boosting,”Journal of Computer and System Sciences, vol. 55, pp. 119–139, 1997. [2] L. Mason, P. Bartlett, J. Baxter, and M. Frean, “Boosting algorithms as gradient descent,” in Advances in Neural Information Processing Systems. 2000, vol. 12, pp. 512–518, The MIT Press. [3] G. R¨atsch, T. Onoda, and K.-R. M¨uller, “Soft margins for AdaBoost,” Machine Learning, vol. 42, no. 3, pp. 287–320, 2001. [4] M. Belkin and P. Niyogi, “Semi-supervised learning on Riemannian manifolds,” Machine Learning, to appear, 2004. [5] J. Shi and J. Malik, “Normalized cuts and image segmentation,” IEEE Transactions on Pettern Analysis and Machine Intelligence, vol. 22, no. 8, pp. 888–905, 2000. [6] G. R¨atsch and M. K. Warmuth, “Maximizing the margin with boosting,” in Proceedings of the 15th Conference on Computational Learning Theory, 2002. [7] L. Breiman, “Prediction games and arcing classifiers,” Neural Computation, vol. 11, pp. 1493– 1518, 1999. [8] R. E. Schapire, Y. Freund, P. Bartlett, and W. S. Lee, “Boosting the margin: a new explanation for the effectiveness of voting methods,” Annals of Statistics, vol. 26, no. 5, pp. 1651–1686, 1998. [9] A. Antos, B. K´egl, T. Linder, and G. Lugosi, “Data-dependent margin-based generalization bounds for classification,” Journal of Machine Learning Research, pp. 73–98, 2002. [10] R. E. Schapire and Y. Singer, “Improved boosting algorithms using confidence-rated predictions,” Machine Learning, vol. 37, no. 3, pp. 297–336, 1999. [11] B. K´egl, “Robust regression by boosting the median,” in Proceedings of the 16th Conference on Computational Learning Theory, Washington, D.C., 2003, pp. 258–272. [12] M. Belkin, I. Matveeva, and P. Niyogi, “Regression and regularization on large graphs,” in Proceedings of the 17th Conference on Computational Learning Theory, 2004.
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Surface Reconstruction using Learned Shape Models Jan Erik Solem School of Technology and Society Malm¨o University, Sweden jes@ts.mah.se Fredrik Kahl RSISE, Australian National University ACT 0200, Australia fredrik@maths.lth.se 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 = Wu + µ, (1) where W is a matrix of size d × q and µ 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 images1. 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)) + ϵ = f(T (Wu + µ)) + ϵ, (2) where ϵ ∼N(0, σ2I) for some scalar σ. 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}n i=1, the ML estimate for W and µ is obtained by minimizing: n X i=1 1 σ2 ||(t2D,i −f(Ti(Wui + µ))||2 + ||ui||2 , (3) over all unknowns. The standard deviation σ is estimated a priori from the data. Once the model parameters W and µ have been learned from examples, they are kept fix. In practice, to minimize (3) we alternatively optimize over (W, µ) and the latent variables {ui}n i 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. 1In 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 = Wglugl + µgl + ϵgl , where tgl is a vector containing the grey-level values of all the 2D image features and ϵgl 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 µgl 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 ugl. 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 Ω⊂R3. The time dependent surface Γ(t) is represented implicitly as the zero level set of a function φ(x, t) : Ω× R+ →R as Γ(t) = {x ; φ(x, t) = 0} , (4) where φ is defined such that φ(x, t) < 0 inside Γ and φ(x, t) > 0 outside. Using the definition above gives the outward unit normal n and the mean curvature κ as n = ∇φ |∇φ| and κ = ∇· ∇φ |∇φ| . (5) One important, frequently used example is the signed distance function, where the requirement |∇φ(x)| = 1 is imposed. The zero set of φ(x, t) represents Γ(t) at all times t. This means that φ(x(t), t) ≡0 for a point on the curve x(t) ∈Γ(t). Differentiating with respect to t gives φt + v · ∇φ = 0 ⇔φt + vn|∇φ| = 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 EP (Γ) = Z Γ d(x) dσ , (7) where Γ is the surface, dσ 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], φt = (∇d(x) · n + d(x)κ) |∇φ| , (8) where n is the surface normal and κ 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 γ be an apparent contour in an image parameterized as γ(s) : I ⊂R →R2. The back-projected cone, written in homogeneous coordinates, C(s) = c + λP +γ(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 = γ(s), let x∗denote the point closest to Γ (If there are several, then choose the one with smallest depth). The function φ is kept to be a signed distance function, i.e. |∇φ| = 1. This means that the point x∗is easily found by checking the values of φ along the line of sight given by (9). The set of these points, S′, will be a (possibly discontinuous) space curve γ∗(λ, s). This set is then added to the distance potential as d(x) = min(d(x), d′(x)) , (10) 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 EPrior(Γ) = Z Γ dPrior(x) dσ , (11) where dPrior(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 +αEPrior, where α ∈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 φt = [(∇d(x) + α∇dPrior(x)) · n + (d(x) + αdPrior(x))κ]|∇φ| . (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 qgl = 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 φ 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 α 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. 0 1 2 3 0 10 20 30 40 50 60 70 Histogram for distance to the surface for all 3D points Distance in voxel width mean median Person 1 0.978 0.788 Person 2 0.805 0.601 Person 3 0.861 0.843 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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