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Adapting to the Shifting Intent of Search Queries∗ Umar Syed† Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 usyed@cis.upenn.edu Aleksandrs Slivkins Microsoft Research Mountain View, CA 94043 slivkins@microsoft.com Nina Mishra Microsoft Research Mountain View, CA 94043 ninam@microsoft.com Abstract Search engines today present results that are often oblivious to recent shifts in intent. For example, the meaning of the query ‘independence day’ shifts in early July to a US holiday and to a movie around the time of the box office release. While no studies exactly quantify the magnitude of intent-shifting traffic, studies suggest that news events, seasonal topics, pop culture, etc account for 1/2 the search queries. This paper shows that the signals a search engine receives can be used to both determine that a shift in intent happened, as well as find a result that is now more relevant. We present a meta-algorithm that marries a classifier with a bandit algorithm to achieve regret that depends logarithmically on the number of query impressions, under certain assumptions. We provide strong evidence that this regret is close to the best achievable. Finally, via a series of experiments, we demonstrate that our algorithm outperforms prior approaches, particularly as the amount of intent-shifting traffic increases. 1 Introduction Search engines typically use a ranking function to order results. The function scores a document by the extent to which it matches the query, and documents are ordered according to this score. This function is fixed in the sense that it does not change from one query to another and also does not change over time. For queries such as ‘michael jackson’ traditional ranking functions that value features such as high page rank will not work since documents new to the web will not have accrued sufficient inlinks. Thus, a search engine’s ranking function should not be fixed; different results should surface depending on the temporal context. Intuitively, a query is “intent-shifting” if the most desired search result(s) change over time. More concretely, a query’s intent has shifted if the click distribution over search results at some time differs from the click distribution at a later time. For the query ‘tomato’ on the heels of a tomato salmonella outbreak, the probability a user clicks on a news story describing the outbreak increases while the probability a user clicks on the Wikipedia entry for tomatoes rapidly decreases. There are studies that suggest that queries likely to be intent-shifting — such as pop culture, news events, trends, and seasonal topics queries — constitute roughly half of the search queries that a search engine receives [10]. The goal of this paper is to devise an algorithm that quickly adapts search results to shifts in user intent. Ideally, for every query and every point in time, we would like to display the search result that users are most likely to click. Since traditional ranking features like PageRank [4] change slowly over time, and may be misleading if user intent has shifted very recently, we want to use just the observed click behavior of users to decide which search results to display. ∗Full version of this paper [20] is available on arxiv.org. In the present version, all proofs are omitted. †This work was done while the author was an intern at Microsoft Research and a student in the Department of Computer Science, Princeton University. 1 There are many signals a search engine can use to detect when the intent of a query shifts. Query features such as as volume, abandonment rate, reformulation rate, occurrence in news articles, and the age of matching documents can all be used to build a classifier which, given a query, determines whether the intent has shifted. We refer to these features as the context, and an occassion when a shift in intent occurs as an event. One major challenge in building an event classifier is obtaining training data. For most query and date combinations (e.g. ‘tomato, 06/09/2008’), it will be difficult even for a human labeler to recall in hindsight whether an event related to the query occurred on that date. In this paper, we propose a novel solution that learns from unlabeled contexts and user click activity. Contributions. We describe a new algorithm that leverages the information contained in contexts. Our algorithm is really a meta-algorithm that combines a bandit algorithm designed for the eventfree setting with an online classification algorithm. The classifier uses the contexts to predict when events occur, and the bandit algorithm “starts over” on positive predictions. The bandit algorithm provides feedback to the classifier by checking, soon after each of the classifier’s positive predictions, whether the optimal search result actually changed. The key technical hurdle in proving a regret bound is handling events that happen during the “checking” phase. For suitable choices of the bandit and classifier subroutines, the regret incurred by our metaalgorithm is (under certain mild assumptions) at most O(k + dF)( n ∆log T), where k is the number of events, dF is a certain measure of the complexity of the concept class F used by the classifier, n is the number of possible search results, ∆is the “minimum suboptimality” of any search result (defined formally in Section 2), and T is the total number of impressions. This regret bound has a very weak dependence on T, which is highly desirable for search engines that receive much traffic. The context turns out to be crucial for achieving logarithmic dependence on T. Indeed, we show that any bandit algorithm that ignores context suffers regret Ω( √ T), even when there is only one event. Unlike many lower bounds for bandit problems, our lower bound holds even when ∆is a constant independent of T. We also show that assuming a logarithmic dependence on T, the dependence on k and dF is essentially optimal. For empirical evaluation, we ideally need access to the traffic of a real search engine so that search results can be adapted based on real-time click activity. Since we did not have access to live traffic, we instead conduct a series of synthetic experiments. The experiments show that if there are no events then the well-studied UCB1 algorithm [2] performs the best. However, when many different queries experience events, the performance of our algorithm significantly outperforms prior techniques. 2 Problem Formulation and Preliminaries We view the problem of deciding which search results to display in response to user click behavior as a bandit problem, a well-known type of sequential decision problem. For a given query q, the task is to determine, at each round t ∈{1, . . . , T} that q is issued by a user to our search engine, a single result it ∈{1, . . . , n} to display.1 This result is clicked by the user with probability pt(it). A bandit algorithm A chooses it using only observed information from previous rounds, i.e., all previously displayed results and received clicks. The performance of an algorithm A is measured by its regret: R(A) ≜E hPT t=1 pt(i∗ t ) −pt(it) i , where an optimal result i∗ t = arg maxi pt(i) is one with maximum click probability, and the expectation is taken over the randomness in the clicks and the internal randomization of the algorithm. Note our unusually strong definition of regret: we are competing against the best result on every round. We call an event any round t where pt−1 ̸= pt. It is reasonable to assume that the number of events k ≪T, since we believe that abrupt shifts in user intent are relatively rare. Most existing bandit algorithms make no attempt to predict when events will occur, and consequently suffer regret Ω( √ T). On the other hand, a typical search engine receives many signals that can be used to predict events, such as bursts in query reformulation, average age of retrieved document, etc. 1For simplicity, we focus on the task of returning a single result, and not a list of results. Techniques from [19] may be adopted to find a good list of results. 2 We assume that our bandit algorithm receives a context xt ∈X at each round t, and that there exists a function f ∈F, in some known concept class F, such that f(xt) = +1 if an event occurs at round t, and f(xt) = −1 otherwise.2 In other words, f is an event oracle. At each round t, an eventful bandit algorithm must choose a result it using only observed information from previous rounds, i.e., all previously displayed results and received clicks, plus all contexts up to round t. In order to develop an efficient eventful bandit algorithm, we make an additional key assumption: At least one optimal result before an event is significantly suboptimal after the event. More precisely, we assume there exists a minimum shift ǫS > 0 such that, whenever an event occurs at round t, we have pt(i∗ t−1) < pt(i∗ t ) −ǫS for at least one previously optimal search result i∗ t−1. For our problem setting, this assumption is relatively mild: the events we are interested in tend to have a rather dramatic effect on the optimal search results. Moreover, our bounds are parameterized by ∆= mint mini̸=i∗ t pt(i∗ t ) −pt(i), the minimum suboptimality of any suboptimal result. 3 Related Work While there has been a substantial amount of work on ranking algorithms [11, 5, 13, 8, 6], all of these results assume that there is a fixed ranking function to learn, not one that shifts over time. Online bandit algorithms (see [7] for background) have been considered in the context of ranking. For instance, Radlinski et al [19] showed how to compose several instantiations of a bandit algorithm to produce a ranked list of search results. Pandey et al [18] showed that bandit algorithms can be effective in serving advertisements to search engine users. These approaches also assume a stationary inference problem. Even though existing bandit work does not address our problem, there are two key algorithms that we do use in our work. The UCB1 algorithm [2] assumes fixed click probabilities and has regret at most O( n ∆log T). The EXP3.S algorithm [3] assumes that click probabilities can change on every round and has regret at most O(k p nT log(nT)) for arbitrary pt’s. Note that the dependence of EXP3.S on T is substantially stronger. The “contextual bandits” problem setting [21, 17, 12, 16, 14] is similar to ours. A key difference is that the context received in each round is assumed to contain information about the identity of an optimal result i∗ t , a considerably stronger assumption than we make. Our context includes only side information such as volume of the query, but we never actually receive information about the identity of the optimal result. A different approach is to build a statistical model of user click behavior. This approach has been applied to the problem of serving news articles on the web. Diaz [9] used a regularized logistic model to determine when to surface news results for a query. Agarwal et al [1] used several models, including a dynamic linear growth curve model. There has also been work on detecting bursts in data streams. For example, Kleinberg [15] describes a state-based model for inferring stages of burstiness. The goal of our work is not to detect bursts, but rather to predict shifts in intent. In a recent concurrent and independent work, Yu et al [22] studied bandit problems with “piecewisestationary” distributions, a notion that closely resembles our definition of events. However, they make different assumptions than we do about the information a bandit algorithm can observe. Expressed in the language of our problem setting, they assume that from time-to-time a bandit algorithm receives information about how users would have responded to search results that are never actually displayed. For us, this assumption is clearly inappropriate. 4 Bandit with Classifier Our algorithm is called BWC, or “Bandit with Classifier”. The high-level idea is to use a bandit algorithm such as UCB1, restart it every time the classifier predicts an event, and use subsequent rounds to generate feedback for the classifier. We will present our algorithm in a modular way, as a meta-algorithm which uses the following two components: classifier and bandit. In 2In some of our analysis, we require that contexts be restricted to a strict subset of X; the value of f outside this subset will technically be null. 3 each round, classifier inputs a context xt and outputs a “positive” or “negative” prediction of whether an event has happened in this round. Also, it may input labeled samples of the form (x, l), where x is a context and l is a boolean label, which it uses for training. Algorithm bandit is a bandit algorithm that is tuned for the event-free runs and provides the following additional functionality: after each round t of execution, it outputs the t-th round guess: a pair (G+, G−), where G+ and G−are subsets of arms that it estimates to be optimal and suboptimal, respectively.3 Since both classifier and bandit make predictions (about events and arms, respectively), for clarity we use the term “guess” exclusively to refer to predictions made by bandit, and reserve the term “prediction” for classifier. The algorithm operates as follows. It runs in phases of two alternating types: odd phases are called “testing” phases, and even phases are called “adapting” phases. The first round of phase j is denoted tj. In each phase we run a fresh instance of bandit. Each testing phase lasts for L rounds, where L is a parameter. Each adapting phase j ends as soon as classifier predicts “positive”; the round t when this happens is round tj+1. Phase j is called full if it lasts at least L rounds. For a full phase j, let (G+ j , G− j ) be the L-th round guess in this phase. After each testing phase j, we generate a boolean prediction l of whether there was an event in the first round thereof. Specifically, letting i be the most recent full phase before j, we set ltj = false if and only if G+ i ∩G− j ̸= ∅. If ltj is false, the labeled sample (xtj, ltj) is fed back to the classifier. Note that classifier never receives true-labeled samples. Pseudocode for BWC is given in Algorithm 1. Disregarding the interleaved testing phases for the moment, BWC restarts bandit whenever classifier predicts “positive”, optimistically assuming that the prediction is correct. By our assumption that events cause some optimal arm to become significantly suboptimal (see Section 2), an incorrect prediction should result in G+ i ∩G− j ̸= ∅, where i is a phase before the putative event, and j is a phase after it. However, to ensure that the estimates Gi and Gj are reliable, we require that phases i and j are full. And to ensure that the full phases closest to a putative event are not too far from it, we insert a full testing phase every other phase. Algorithm 1 BWC Algorithm 1: Given: Parameter L, a (L, ǫS)-testable bandit, and a safe classifier. 2: for phase j = 1, 2, . . . do 3: Initialize bandit. Let tj be current round. 4: if j is odd then 5: for round t = tj . . . tj + L do 6: Select arm it according to bandit. 7: Observe pt(it) and update bandit. 8: Let i be the most recent full phase before j. 9: If G+ i ∩G− j ̸= ∅let ltj = false and pass training example (xtj, ltj) to classifier. 10: else 11: for round t = tj, tj + 1, . . . do 12: Select arm it according to bandit. 13: Observe pt(it) and update bandit; pass context xt to classifier. 14: if classifier predicts “positive” then 15: Terminate inner for loop. Let S be the set of all contexts which correspond to an event. When the classifier receives a context x and predicts a “positive”, this prediction is called a true positive if x ∈S, and a false positive otherwise. Likewise, when the classifier predicts a “negative”, the prediction is called a true negative if x ̸∈S, and a false negative otherwise. The sample (x, l) is correctly labeled if l = (x ∈S). We make the following two assumptions. First, classifier is safe for a given concept class: if it inputs only correctly labeled samples, it never outputs a false negative. Second, bandit is (L, ǫ)-testable, in the following sense. Consider an event-free run of bandit, and let (G+, G−) be its L-th round guess. Then with probability at least 1 −T −2, each optimal arm lies in G+ but not in G−, and any arm that is at least ǫ-suboptimal lies in G−but not in G+. So an (L, ǫ)-testable 3Following established convention, we call the options available to a bandit algorithm “arms”. In our setting, each arm corresponds to a search result. 4 bandit algorithm is one that, after L rounds, has a good guess of which arms are optimal and which are at least ǫ-suboptimal. For correctness, we require bandit to be (L, ǫS)-testable, where ǫS is the minimum shift. The performance of bandit is quantified via its event-free regret, i.e. regret on the event-free runs. Likewise, for correctness we need classifier to be safe; we quantify its performance via the maximum possible number of false positives, in the precise sense defined below. We assume that the state of classifier is updated only if it receives a labeled sample, and consider a game in which in each round t, classifier receives a context xt ̸∈S, outputs a (false) positive, and receives a (correctly) labeled sample (x, false). For a given context set X and a given concept class F, let the FP-complexity of the classifier be the maximal possible number of rounds in such a game, where the maximum is taken over all event oracles f ∈F and all possible sequences {xt}. Put simply, the FP-complexity of classifier is the maximum number of consecutive false positives it can make when given correctly labeled examples. We will discuss efficient implementations of a safe classifier and a (L, ǫ)-testable bandit in Sections 5 and Section 6, respectively. We present provable guarantees for BWC in a modular way, in terms of FP-complexity, event-free regret, and the number of events. The main technical difficulty in the analysis is that the correct operation of the components of BWC — classifier and bandit — is interdependent. In particular, one challenge is to handle events that occur during the first L rounds of a phase; these events may potentially “contaminate” the L-th round guesses and cause incorrect feedback to classifier. Theorem 1. Consider an instance of the eventful bandit problem with number of rounds T, n arms, k events and minimum shift ǫS. Consider algorithm BWC with parameter L and components classifier and bandit such that for this problem instance, classifier is safe, and bandit is (L, ǫS)-testable. If any two events are at least 2L rounds apart, then the regret of BWC is R(T) ≤(2k + d) R0(T) + (k + d) R0(L) + kL. (1) where d is the FP-complexity of the classifier and R0(·) is the event-free regret of bandit. Remarks. The proof is available in the full version [20]. In our implementations of bandit, L = Θ( n ǫS log T) suffices. In the +kL term in (1), the k can be replaced by the number of testing phases that contain both a false positive in round 1 of the phase and an actual event later in the phase; this number can potentially be much smaller than k. 5 Safe Classifier We seek a classifier that is safe for a given concept class F and has low FP-complexity. We present a classifier whose FP-complexity is bounded in terms of the following property of F: Definition 1. Define the safe function SF : 2X →2X of F as follows: x ∈SF(N) if and only if there is no concept f ∈F such that: f(y) = −1 for all y ∈N and f(x) = +1. The diameter of F, denoted dF, is equal to the length of the longest sequence x1, . . . , xm ∈X such that xt /∈ SF({x1, . . . , xt−1}) for all t = 1, . . . , m. So if N contains only true negatives, then SF(N) contains only true negatives. This property suggests that SF can be used to construct a safe classifier SafeCl, which operates as follows: It maintains a set of false-labeled examples N, initially empty. When input an unlabeled context x, SafeCl outputs a positive prediction if and only if x /∈SF(N). After making a positive prediction, SafeCl inputs a labeled example (x, l). If l = false, then x is added to N; otherwise x is discarded. Clearly, SafeCl is a safe classifer. In the full version [20], we show that the FP-complexity of SafeCl is at most the diameter dF, which is to be expected: FP-complexity is a property of a classifier, and diameter is the completely analogous property for SF. Moreover, we give examples of common concept classes with efficiently computable safe functions. For example, if F is the space of hyperplanes with “margin” at least δ (probably the most commonly-used concept class in machine learning), then SF(N) is the convex hull of the examples in N, extended in all directions by a δ. By using SafeCl as our classifier, we introduce dF into the regret bound of bwc, and this quantity can be large. However, in Section 7 we show that the regret of any algorithm must depend on dF, unless it depends strongly on the number of rounds T. 5 6 Testable Bandit Algorithms In this section we will consider the stochastic n-armed bandit problem. We are looking for (L, ǫ)testable algorithms with low regret. The L will need to be sufficiently large, on the order of Ω(nǫ−2). A natural candidate would be algorithm UCB1 from [2] which does very well on regret. Unfortunately, it does not come with a guarantee of (L, ǫ)-testability. One simple fix is to choose at random between arms in the first L rounds, use these samples to form the best guess, in a straightforward way, and then run UCB1. However, in the first L rounds this algorithm incurs regret of Ω(L), which is very suboptimal. For instance, for UCB1 the regret would be R(L) ≤O(min( n ∆log L, √nL log L)). In this section, we develop an algorithm which has the same regret bound as UCB1, and is (L, ǫ)testable. We state this result more generally, in terms of estimating expected payoffs; we believe it may be of independent interest. The (L, ǫ)-testability is then an easy corollary. Since our analysis in this section is for the event-free setting, we can drop the subscript t from much of our notation. Let p(u) denote the (time-invariant) expected payoff of arm u. Let p∗= maxu p(u), and let ∆(u) = p∗−p(u) be the “suboptimality” of arm u. For round t, let µt(u) be the sample average of arm u, and let nt(u) be the number of times arm u has been played. We will use a slightly modified algorithm UCB1 from [2], with a significantly extended analysis. Recall that in each round t algorithm UCB1 chooses an arm u with the highest index It(u) = µt(u)+rt(u), where rt(u) = p 8 log(t)/nt(u) is a term that we’ll call the confidence radius whose meaning is that |p(u) −µt(u)| ≤rt(u) with high probability. For our purposes here it is instructive to re-write the index as It(u) = µt(u) + α rt(u) for some parameter α. Also, to better bound the early failure probability we will re-define the confidence radius as rt(u) = p 8 log(t0 + t)/nt(u) for some parameter t0. We will denote this parameterized version by UCB1(α, t0). Essentially, the original analysis of UCB1 in [2] carries over; we omit the details. Our contribution concerns estimating the ∆(u)’s. We estimate the maximal expected reward p∗via the sample average of an arm that has been played most often. More precisely, in order to bound the failure probability we consider a arm that has been played most often in the last t/2 rounds. For a given round t let vt be one such arm (ties broken arbitrarily), and let ∆t(u) = µt(vt) −µt(u) will be our estimate of ∆(u). We express the “quality” of this estimate as follows: Theorem 2. Consider the stochastic n-armed bandits problem. Suppose algorithm UCB1(6, t0) has been played for t steps, and t + t0 ≥32. Then with probability at least 1 −(t0 + t)−2 for any arm u we have |∆(u) −∆t(u)| < 1 4∆(u) + δ(t) (2) where δ(t) = O( p n t log(t + t0)). Remark. Either we know that ∆(u) is small, or we can approximate it up to a constant factor. Specifically, if δ(t) < 1 2 ∆t(u) then ∆(u) ≤2 ∆t(u) ≤5 ∆(u) else ∆(u) ≤4δ(t). Let us convert UCB1(6, T) into an (L, ǫ)-testable algorithm, as long as L ≥Ω( n ǫ2 log T). The t-th round best guess (G+ t , G− t ) is defined as G+ t = {u : ∆t(u) ≤ǫ/4} and G− t = {u : ∆t(u) > ǫ/2}. Then the resulting algorithm is (L, ǫ)-testable assuming that δ(L) ≤ǫ/4, where δ(t) is from Theorem 2. The proof is in the full version [20]. 7 Upper and Lower Bounds Plugging the classifier from Section 5 and the bandit algorithm from Section 6 into the metaalgorithm from Section 4, we obtain the following numerical guarantee. Theorem 3. Consider an instance S of the eventful bandit problem with with number of rounds T, n arms and k events, minimum shift ǫS, minimum suboptimality ∆, and concept class diameter dF. Assume that any two events are at least 2L rounds apart, where L = Θ( n ǫ2 S log T). Consider the BWC algorithm with parameter L and components classifier and bandit as presented, respectively, in Section 5 and Section 6. Then the regret of BWC is R(T) ≤ (3k + 2dF) n ∆+ k n ǫ2 S (log T). 6 While the linear dependence on n in this bound may seem large, note that without additional assumptions, regret must be linear in n, since each arm must be pulled at least once. In an actual search engine application, the arms can be restricted to, say, the top ten results that match the query. We now state two lower bounds about eventful bandit problems; the proofs are in the full version [20]. Theorem 4 shows that in order to achieve regret that is logarithmic in the number of rounds, a context-aware algorithm is necessary, assuming there is at least one event. Incidentally, this lowerbound can be easily extended to prove that, in our model, no algorithm can achieve logarithmic regret when an event oracle f is not contained in the concept class F. Theorem 4. Consider the eventful bandit problem with number of rounds T, two arms, minimum shift ǫS and minimum suboptimality ∆, where ǫS = ∆= ǫ, for an arbitrary ǫ ∈(0, 1 2). For any context-ignoring bandit algorithm A, there exists a problem instance with a single event such that regret RA(T) ≥Ω(ǫ √ T). Theorem 5 proves that in Theorem 3, linear dependence on k + dF is essentially unavoidable. If we desire a regret bound that has logarithmic dependence on the number of rounds, then a linear dependence on k + dF is necessary. Theorem 5. Consider the eventful bandit problem with number of rounds T and concept class diameter dF. Let A be an eventful bandit algorithm. Then there exists a problem instance with n arms, k events, minimum shift ǫS, minimum suboptimality ∆, where ǫS = ∆= ǫ, for any given values of k ≥1, n ≥3, and ǫ ∈(0, 1 4), such that RA(T) ≥Ω(k n ǫ ) log(T/k). Moreover, there exists a problem instance with two arms, a single event, event threshold Θ(1) and minimum suboptimality Θ(1) such that regret RA(T) ≥Ω(max(T 1/3, dF)) log T. 8 Experiments To truly demonstrate the benefits of BWC requires real-time manipulation of search results. Since we did not have the means to deploy a system that monitors click/skip activity and correspondingly alters search results with live users, we describe a collection of experiments on synthetically generated data. We begin with a head-to-head comparison of BWC versus a baseline UCB1 algorithm and show that BWC’s performance improves substantially upon UCB1. Next, we compare the performance of these algorithms as we vary the fraction of intent-shifting queries: as the fraction increases, BWC’s performance improves even further upon prior approaches. Finally, we compare the performance as we vary the number of features. While our theoretical results suggest that regret grows with the number of features in the context space, in our experiments, we surprisingly find that BWC is robust to higher dimensional feature spaces. Setup: We synthetically generate data as follows. We assume that there are 100 queries where the total number of times these queries are posed is 3M. Each query has five search results for a user to select from. If a query does not experience any events — i.e., it is not “intent-shifting” — then the optimal search result is fixed over time; otherwise the optimal search result may change. Only 10% of the queries are intent-shifting, with at most 10 events per such query. Due to the random nature with which data is generated, regret is reported as an average over 10 runs. The event oracle is an axis-parallel rectangle anchored at the origin, where points inside the box are negative and points outside the box are positive. Thus, if there are two features, say query volume and query abandonment rate, an event occurs if and only if both the volume and abandonment rate exceed certain thresholds. Bandit with Classifier (BWC): Figure 1(a) shows the average cumulative regret over time of three algorithms. Our baseline comparison is UCB1 which assumes that the best search result is fixed throughout. In addition, we compare to an algorithm we call ORA, which uses the event oracle to reset UCB1 whenever an event occurs. We also compared to EXP3.S, but its performance was dramatically worse and thus we have not included it in the figure. In the early stages of the experiment before any intent-shifting event has happened, UCB1 performs the best. BWC’s safe classifier makes many mistakes in the beginning and consequently pays the price of believing that each query is experiencing an event when in fact it is not. As time progresses, BWC’s classifier makes fewer mistakes, and consequently knows when to reset UCB1 more accu7 0 0.5 1 1.5 2 2.5 3 x 10 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 4 Time (impressions) Cumulative regret ORA BWC UCB1 0 1/8 1/4 3/8 1/2 ORA 17.2 22.8 30.4 33.8 39.5 BWC 17.8 24.6 39.9 46.7 99.4 UCB1 17.2 34.1 114.9 84.2 140.0 EXP3.S 78.4 123.7 180.2 197.6 243.1 10 20 30 40 ORA 21.9 23.2 21.9 22.8 BWC 23.1 24.4 22.9 23.7 UCB1 32.3 33.5 31.1 37.4 EXP3.S 111.6 109.4 112.5 121.3 Figure 1: (a) (Left) BWC’s cumulative regret compared to UCB1 and ORA (UCB1 with an oracle indicating the exact locations of the intent-shifting event) (b) (Right, Top Table) Final regret (in thousands) as the fraction of intent-shifting queries varies. With more intent-shifting queries, BWC’s advantage over prior approaches improves. (c) (Right, Bottom Table) Final regret (in thousands) as the number of features grows. rately. UCB1 alone ignores the context entirely and thus incurs substantially larger cumulative regret by the end. Fraction of Intent-Shifting Queries: In the next experiment, we varied the fraction of intentshifting queries. Figure 1(b) shows the result of changing the distribution from 0, 1/8, 1/4, 3/8 and 1/2 intent-shifting queries. If there are no intent-shifting queries, then UCB1’s regret is the best. We expect this outcome since BWC’s classifier, because it is safe, initially assumes that all queries are intent-shifting and thus needs time to learn that in fact no queries are intent-shifting. On the other hand, BWC’s regret dominates the other approaches, especially as the fraction of intent-shifting queries grows. EXP3.S’s performance is quite poor in this experiment – even when all queries are intent-shifting. The reason is that even when a query is intent-shifting, there are at most 10 intentshifting events, i.e., each query’s intent is not shifting all the time. With more intent-shifting queries, the expectation is that regret monotonically increases. In general, this seems to be true in our experiment. There is however a decrease in regret going from 1/4 to 3/8 intent-shifting queries. We believe that this is due to the fact that each query has at most 10 intentshifting events spread uniformly and it is possible that there were fewer events with potentially smaller shifts in intent in those runs. In other words, the standard deviation of the regret is large. Over the ten 3/8 intent-shifting runs for ORA, BWC, UCB1 and EXP3.S, the standard deviation was roughly 1K, 10K, 12K and 6K respectively. Number of Features: Finally, we comment on the performance of our approach as the number of features grows. Our theoretical results suggest that BWC’s performance should deteriorate as the number of features grows. Surprisingly, BWC’s performance is consistently close to the Oracle’s. In Figure 1(b), we show the cumulative regret after 3M impressions as the dimensionality of the context vector grows from 10 to 40 features. BWC’s regret is consistently close to ORA as the number of features grows. On the other hand, UCB1’s regret though competitive is worse than BWC, while EXP3.S’s performance is across the board poor. Note that both UCB1 and EXP3.S’s regret is completely independent of the number of features. The standard deviation of the regret over the 10 runs is substantially lower than the previous experiment. For example, over 10 features, the standard deviation was 355, 1K, 5K, 4K for ORA, BWC, UCB1 and EXP3.S, respectively. 9 Future Work The main question left for future work is testing this approach in a real setting. Since gaining access to live traffic is difficult, it would be interesting to find ways to rewind the search logs to simulate live traffic. Acknowledgements. We thank Rakesh Agrawal, Alan Halverson, Krishnaram Kenthapadi, Robert Kleinberg, Robert Schapire and Yogi Sharma for their helpful comments and suggestions. 8 References [1] Deepak Agarwal, Bee-Chung Chen, Pradheep Elango, Nitin Motgi, Seung-Taek Park, Raghu Ramakrishnan, Scott Roy, and Joe Zachariah. Online models for content optimization. In 22nd Advances in Neural Information Processing Systems (NIPS), 2008. [2] Peter Auer, Nicol`o Cesa-Bianchi, and Paul Fischer. Finite-time analysis of the multiarmed bandit problem. Machine Learning, 47(2-3):235–256, 2002. [3] Peter Auer, Nicol`o Cesa-Bianchi, Yoav Freund, and Robert E. Schapire. The nonstochastic multiarmed bandit problem. SIAM J. Comput., 32(1):48–77, 2002. [4] Sergey Brin and Lawrence Page. The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems, 30(1–7):107–117, 1998. [5] Christopher J. C. Burges, Tal Shaked, Erin Renshaw, Ari Lazier, Matt Deeds, Nicole Hamilton, and Gregory N. Hullender. Learning to rank using gradient descent. In 22nd Intl. Conf. on Machine Learning (ICML), 2005. [6] Zhe Cao, Tao Qin, Tie-Yan Liu, Ming-Feng Tsai, and Hang Li. Learning to rank: from pairwise approach to listwise approach. In 24th Intl. Conf. on Machine Learning (ICML), 2007. [7] Nicol`o Cesa-Bianchi and G´abor Lugosi. Prediction, learning, and games. Cambridge University Press, 2006. [8] William W. Cohen, Robert E. Schapire, and Yoram Singer. Learning to order things. J. of Artificial Intelligence Research, 10:243–270, 1999. [9] Fernando Diaz. Integration of news content into web results. In 2nd Intl. Conf. on Web Search and Data Mining, pages 182–191, 2009. [10] D. Fallows. Search engine users. Pew Internet and American Life Project, 2005. [11] Yoav Freund, Raj Iyer, Robert E. Schapire, and Yoram Singer. An efficient boosting algorithm for combining preferences. J. of Machine Learning Research, 4:933–969, 2003. [12] Elad Hazan and Nimrod Megiddo. Online Learning with Prior Knowledge. In 20th Conference on Learning Theory (COLT), pages 499–513, 2007. [13] Thorsten Joachims. Optimizing search engines using clickthrough data. In 8th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining (KDD), 2002. [14] Sham M. Kakade, Shai Shalev-Shwartz, and Ambuj Tewari. Efficient bandit algorithms for online multiclass prediction. In 25th Intl. Conf. on Machine Learning (ICML), 2008. [15] Jon M. Kleinberg. Bursty and hierarchical structure in streams. In 8th ACM SIGKDD Intl. Conf. on Knowledge Discovery and Data Mining (KDD), 2002. [16] John Langford and Tong Zhang. The epoch-greedy algorithm for multi-armed bandits with side information. In 21st Advances in Neural Information Processing Systems (NIPS), 2007. [17] Sandeep Pandey, Deepak Agarwal, Deepayan Chakrabarti, and Vanja Josifovski. Bandits for Taxonomies: A Model-based Approach. In SIAM Intl. Conf. on Data Mining (SDM), 2007. [18] Sandeep Pandey, Deepayan Chakrabarti, and Deepak Agarwal. Multi-armed Bandit Problems with Dependent Arms. In 24th Intl. Conf. on Machine Learning (ICML), 2007. [19] Filip Radlinski, Robert Kleinberg, and Thorsten Joachims. Learning diverse rankings with multi-armed bandits. In 25th Intl. Conf. on Machine Learning (ICML), 2008. [20] Umar Syed, Aleksandrs Slivkins, and Nina Mishra. Adapting to the shifting intent of search queries. Technical report. Available from arXiv. [21] Chih-Chun Wang, Sanjeev R. Kulkarni, and H. Vincent Poor. Bandit problems with side observations. IEEE Trans. on Automatic Control, 50(3):338355, 2005. [22] Jia Yuan Yu and Shie Mannor. Piecewise-stationary bandit problems with side observations. In 26th Intl. Conf. on Machine Learning (ICML), 2009. 9
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Predicting the Optimal Spacing of Study: A Multiscale Context Model of Memory Michael C. Mozer⋆, Harold Pashler†, Nicholas Cepeda◦, Robert Lindsey⋆, & Ed Vul‡ ⋆Dept. of Computer Science, University of Colorado †Dept. of Psychology, UCSD ◦Dept. of Psychology, York University ‡Dept. of Brain and Cognitive Sciences, MIT Abstract When individuals learn facts (e.g., foreign language vocabulary) over multiple study sessions, the temporal spacing of study has a significant impact on memory retention. Behavioral experiments have shown a nonmonotonic relationship between spacing and retention: short or long intervals between study sessions yield lower cued-recall accuracy than intermediate intervals. Appropriate spacing of study can double retention on educationally relevant time scales. We introduce a Multiscale Context Model (MCM) that is able to predict the influence of a particular study schedule on retention for specific material. MCM’s prediction is based on empirical data characterizing forgetting of the material following a single study session. MCM is a synthesis of two existing memory models (Staddon, Chelaru, & Higa, 2002; Raaijmakers, 2003). On the surface, these models are unrelated and incompatible, but we show they share a core feature that allows them to be integrated. MCM can determine study schedules that maximize the durability of learning, and has implications for education and training. MCM can be cast either as a neural network with inputs that fluctuate over time, or as a cascade of leaky integrators. MCM is intriguingly similar to a Bayesian multiscale model of memory (Kording, Tenenbaum, & Shadmehr, 2007), yet MCM is better able to account for human declarative memory. 1 Introduction Students often face the task of memorizing facts such as foreign language vocabulary or state capitals. To retain such information for a long time, students are advised not to cram their study, but rather to study over multiple, well-spaced sessions. This advice is based on a memory phenomenon known as the distributed practice or spacing effect (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006). The spacing effect is typically studied via a controlled experimental paradigm in which participants are asked to study unfamiliar paired associates (e.g., English-Japanese vocabulary) in two sessions. The time between sessions, known as the intersession interval or ISI, is manipulated across participants. Some time after the second study session, a cued-recall test is administered to the participants, e.g., “What is ‘rabbit’ in Japanese?” The lag between second session and the test is known as the retention interval or RI. Recall accuracy as a function of ISI follows a characteristic curve. The solid line of Figure 1a sketches this curve, which we will refer to as the spacing function. The left edge of the graph corresponds to massed practice, when session two immediately follows session one. Recall accuracy rises dramatically as the ISI increases, reaches a peak, and falls off gradually. The ISI corresponding to the peak—the optimal ISI—depends strongly on RI: a meta-analysis by Cepeda et al. (2006) sug1 % recall ISI forgetting function spacing function (a) (b) m1 m2 m3 m4 mN pool 1 pool 2 pool 3 pool 4 pool N ... ... Figure 1: (a) The spacing function (solid line) depicts recall at test following two study sessions separated by a given ISI; the forgetting function (dashed line) depicts recall as a function of the lag between study and test. (b) A sketch of the Multiscale Context Model. gests a power-law relationship. The optimal ISI almost certainly depends on the specific materials being studied and the manner of study as well. For educationally relevant RIs on the order of weeks and months, the effect of spacing can be tremendous: optimal spacing can double retention over massed practice (Cepeda et al., in press). The spacing function is related to another observable measure of retention, the forgetting function, which characterizes recall accuracy following a single study session as a function of the lag between study and test. For example, suppose participants in the experiment described above learned material in study session 1, and were then tested on the material immediately prior to study session 2. As the ISI increased, session 1 memories would decay. This decay is shown in the dashed line of Figure 1a. Typical forgetting functions follow a generalized power-law decay, of the form P(recall) = A(1 + Bt)−C, where A, B, and C are constants, and t is the study-test lag (Wixted & Carpenter, 2007). Our goal is to develop a model of long-term memory that characterizes the memory-trace strength of items learned over two or more sessions. The model predicts recall accuracy as a function of the RI, taking into account the study schedule—the ISI or set of ISIs determining the spacing of study sessions. We would like to use this model to prescribe the optimal study schedule. The spacing effect is among the best known phenomena in cognitive psychology, and many theoretical explanations have been suggested. Two well developed computational models of human memory have been elaborated to explain the spacing effect (Pavlik & Anderson, 2005; Raaijmakers, 2003). These models are necessarily complex: the brain contains multiple, interacting memory systems whose decay and interference characteristics depend on the specific content being stored and its relationship to other content. Consequently, these computational theories are fairly flexible and can provide reasonable post-hoc fits to spacing effect data, but we question their predictive value. Rather than developing a general theory of memory, we introduce a model that specifically predicts the shape of the spacing function. Because the spacing function depends not only on the RI, but also on the nature of the material being learned, and the manner and amount of study, the model requires empirical constraints. We propose a novel approach to obtaining a predictive model: we collect behavioral data to determine the forgetting function for the specific material being learned. We then use the forgetting function, which is based on a single study session, to predict the spacing function, which is based on two or more study sessions. Such a predictive model has significant implications for education and training. The model can be used to search for the ISI or set of ISIs that maximizes expected recall accuracy for a fixed RI. Although the required RI is not known in practical settings, one can instead optimize over RI as a random variable with an assumed distribution. 2 Accounts of the spacing effect We review two existing theories proposed to explain the spacing effect, and then propose a synthesis of these theories. The two theories appear to be unrelated and mutually exclusive on the surface, but in fact share a core unifying feature. In contrast to most modeling work appearing in the NIPS volumes, our model is cast at Marr’s implementation level, not at the level of a computational theory. However, after introducing our model and showing its predictive power, we discuss an intriguingly similar Bayesian theory of memory adaptation (Kording et al., 2007). Although our model has a 2 strong correspondence with the Bayesian model, their points of difference seem to be crucial for predicting behavioral phenomena of human declarative memory. 2.1 Encoding-variability theories One class of theories proposed to explain the spacing effect focuses on the notion of encoding variability. According to these theories, when an item is studied, a memory trace is formed that incorporates the current psychological context. Psychological context includes conditions of study, internal state of the learner, and recent experiences of the learner. Retrieval of a stored item depends at least in part on the similarity of the contexts at the study and test. If psychological context is assumed to fluctuate randomly over time, two study sessions close together in time will have similar contexts. Consequently, at the time of a recall test, either both study contexts will match the test context or neither will. Increasing the ISI can thus prove advantageous because the test context will have higher likelihood of matching one study context or the other. Greater contextual variation enhances memory on this account by making for less redundancy in the underlying memory traces. However, increasing the ISI also incurs a retrieval cost because random drift makes the first-study context increasingly less likely to match the test context. The optimal ISI depends on the tradeoff between the retrieval benefit and cost at test. Raaijmakers (2003) developed an encoding variability theory by incorporating time-varying contextual drift into the well-known Search of Associative Memory (SAM) model (Raaijmakers & Shiffrin, 1981), and explained a range of data from the spacing literature. In this model, the contextual state is characterized by a high-dimensional binary vector. Each element of the vector indicates the presence or absence of a particular contextual feature. The contextual state evolves according to a stochastic process in which features flip from absent to present at rate π01 and from present to absent at rate π10. If the context is sampled at two points in time with lag ∆t, the probability that a contextual feature will be present at both times is P(feature present at time t and t + ∆t) = β2 + β(1 −β) exp(−∆t/τ), (1) where τ ≡1/(π01 +π10) and β ≡π01τ is the expected proportion of features present at any instant. To assist in understanding the mechanisms of SAM, we find it useful to recast the model as a neural network. The input layer to this neural net is a pool of binary valued neurons that represent the contextual state at the current time; the output layer consists of a set of memory elements, one per item to be stored. To simplify notation throughout this paper, we’ll describe this model and all others in terms of a single-item memory, allowing us to avoid an explicit index term for the item being stored or retrieved. The memory element for the item under consideration has an activation level, m, which is a linear function of the context unit activities: m = P j wjcj, where cj is the binary activation level of context unit j and wj is the strength of connection from context j. The probability of retrieval of the item is assumed to be monotonically related to m. When an item is studied, its connection strengths are adjusted according to a Hebbian learning rule with an upper limit on the connection strength: ∆wj = min(1 −wj, cj ˆm), (2) where ˆm = 1 if the item was just presented for study, or 0 otherwise. When an item is studied, the weights for all contextual features present at the time of study will be strengthened. Later retrieval is more likely if the context at test matches the context at study: the memory element receives a contribution only when an input is active and its connection strength is nonzero. Thus, after a single study and lag ∆t, retrieval probability is directly related to Equation 1. When an item has been studied twice, retrieval will be more robust if the two study opportunities strengthen different weights, which occurs when the ISI is large and the contextual states do not overlap significantly. One other feature of SAM is crucial for explaining spacing-effect data. After an item has been studied at least once, SAM assumes that the memory trace resulting from further study is influenced by whether the item is accessible to retrieval at the time of study. Specifically, SAM assumes that the weights have effectively decayed to zero if recall fails. Other memory models similarly claim that memory traces are weaker if an item is inaccessible to retrieval at the time of study (e.g., Pavlik & Anderson, 2005), which we label as the retrieval-dependent update assumption. We have described the key components of SAM that explain the spacing effect, but the model has additional complexity, including a short-term memory store, inter-item interference, and additional 3 context based on associativity and explicit cues. Even with all this machinery, SAM has a serious limitation. Spacing effects occur on many time scales (Cepeda et al., 2006). SAM can explain effects on any one time scale (e.g., hours), but the same model cannot explain spacing effects on a different time scale (e.g., months). The reason is essentially that the exponential decay in context overlap bounds the time scale at which the model operates. 2.2 Predictive-utility theories We now turn to another class of theories that has been proposed to explain the spacing effect. These theories, which we will refer to as predictive-utility theories, are premised on the assumption that memory is limited in capacity and/or is imperfect and allows intrusions. To achieve optimal performance, memories should therefore be erased if they are not likely to be needed in the future. Anderson and Milson (1989) proposed a rational analysis of memory from which they estimated the future need probability of a stored trace. When an item is studied multiple times with a given ISI, the rational analysis suggests that the need probability drops off rapidly following the last study once an interval of time greater than the ISI has passed. Consequently, increasing the ISI should lead to a more persistent memory trace. Although this analysis yields a reasonable qualitative match to spacing-effect data, no attempt was made to make quantitative predictions. The notion of predictive utility is embedded in the multiple time-scale or MTS model of Staddon et al. (2002). In MTS, each item to be stored is represented by a dedicated cascade of N leaky integrators. The activation of integrator i, xi, decays over time according to: xi(t + ∆t) = xi(t) exp(−∆t/τi), (3) where τi is the decay time constant. The probability of retrieving the item is related to the total trace strength, sN, where sk = Pk j=1 xj. The integrators are ordered from shortest to longest time constant, i.e., τi < τi+1 for all i. When an item is studied, the integrators receive a bump in activity according to a cascaded error-correction update, ∆xi = ϵ max(0, 1 −si−1), (4) which is based on the idea that an integrator at some time scale τi receives a boost only if integrators at shorter time scales fail to represent the item at the time it is studied. The constant ϵ is a step size. When an item is repeatedly presented for study with short ISIs, the trace can successfully be represented by the integrators with short time constants, and consequently, the trace will decay rapidly. Increasing the spacing shifts the representation to integrators with slower decay rates. MTS was designed to explain rate-sensitive habituation data from the animal learning literature: the fact that recovery following spaced stimuli is slower than following massed. We tried fitting MTS to human-memory data and were unable to obtain quantitatively accurate fits. 3 The multiscale context model (MCM) SAM and MTS are motivated by quite different considerations, and appear to be unrelated mechanisms. Nonetheless, they share a fundamental property: both suppose an exponential decay of internal representations over time (compare Equations 1 and 3). When we establish a correspondence between the mechanisms in SAM and MTS that produce exponential decay, we obtain a synthesis of the two models that incorporates features of each. Essentially, we take from SAM the notion of contextual drift and retrieval-dependent update, and from MTS the multiscale representation and the cascaded error-correction memory update, and we obtain a new model which we call the Multiscale Context Model or MCM. MCM can be described as a neural network whose input layer consists of N pools of time-varying context units. Units in pool i operate with time constant τi. The relative size of pool i is γi. MCM is thus like SAM with multiple pools of context units. MCM can also be described in terms of N leaky integrators, where integrator i has time constant τi and activity scaled by γi. MCM is thus like MTS with the addition of scaling factors. Before formally describing MCM, we detour to explain the choice of the parameters {τi} and {γi}. As the reader might infer from our description of SAM and MTS, these parameters characterize memory decay, extending Equation 3 such that the total trace strength at time t is defined as: sN(t) = N X i=1 γi exp(−t τi )xi(0). 4 If xi(0) = 1 for all i—which is the integrator activity following the first study in MTS—the trace strength as a function of time is a mixture of exponentials. To match the form of human forgetting (Figure 1), this mixture must approximate a power function. We can show that a generalized power function can be exactly expressed as an infinite mixture of exponentials: A(1 + Bt)−C = A Z ∞ 0 Inv-Gamma(τ; C, 1) exp(Bt τ )dτ, where Inv-Gamma(τ; C, 1) is the inverse-gamma probability density function with shape parameter C and scale 1, and the equality is valid for t ≥0 and C > 0. We have identified several finite mixture-of-exponential formulations that empirically yield an extremely good approximation to arbitrary power functions over ten orders of magnitude. The formulation we prefer defines τi and γi in terms of four primitive parameters: τi = µνi and γi = ωξi/ N X j=1 ξj. (5) With ν > 1 and ξ < 1, the higher-order components (i.e., larger indices) represent exponentially longer time scales with exponentially smaller weighting. As a result, truncating higher-order mixture components has little impact on the approximation on shorter time scales. Consequently, we simply need to pick a value of N that allows for a representation of many orders of magnitude of time. Given N and human forgetting data collected in an experiment, we can search for the parameters {µ, ν, ω, ξ} that obtain a least squares fit to the data. Given the human forgetting function function, then, we can completely determine the {τi} and {γi}. In all simulation results we report, we fixed N = 100, although equivalent results are obtained for N = 50 or N = 200. 3.1 Casting MCM as a cascade of leaky integrators Assume that—as in MTS—a dedicated set of N leaky integrators hold the memory of each item to be learned. Let xi denote the activity of integrator i associated with the item, and let si be the average strength of the first i integrators, weighted by the {γj} terms: si = 1 Γi i X j=1 γjxj, where Γi = i X j=1 γj. The recall probability is simply related to the net strength of the item: P(recall) = min(1, sN). When an item is studied, its integrators receive a boost in activity. Integrator i receives a boost that depends on how close the average strength of the first i integrators is to full strength, i.e., ∆xi = ϵ(1 −si) (6) where ϵ is a step size. We adopt the retrieval-dependent update assumption of SAM, and fix ϵ = 1 for an item that is unsuccessfully recalled at the time of study, and ϵ = ϵr > 1 for an item that is successfully recalled. This description of MCM is identical to MTS except the following. (1) MTS weighs all integrators equally when combining the individual integrator activities. MCM uses a γ-weighted average. (2) MTS provides no guidance in setting the τ and γ constants; MCM constrains these parameters based on the human forgetting function. (3) The integrator update magnitude is retrieval dependent, as in SAM. (4) The MCM update rule (Equation 6) is based on si, whereas the MTS rule (Equation 4) is based on si−1. This modification is motivated by the neural net formulation of MCM, in which using si allows the update to be interpreted as performing gradient ascent in prediction ability. 3.2 Casting MCM as a neural network The neural net conceptualization of MCM is depicted in Figure 1b. The input layer is like that of SAM with the context units arranged in N pools, with γi being the relative size of pool i. The activity of unit j in pool i is denoted cij. The context units are binary valued and units in pool i flip with time constant τi. On average a fraction β are on at any time. (β has no effect on the model’s predictions, and is cancelled out in the formulation that follows.) 5 As depicted in Figure 1b, the model also includes a set of N memory elements for each item to be learned. Memory elements are in one-to-one correspondence with context pools. Activation of memory element i, denoted mi, indicates strength of retrieval for the item based on context pools 1...i. The activation function is cascaded such that memory element i receives input from context units in pool i as well as memory element i −1: mi = mi−1 + X j wijcij + b, where wij is the connection weight from context unit j to memory element i, m0 ≡0, and b = −β/(1−β) is a bias weight. The bias simply serves to offset spurious activity reaching the memory elements, activity that is unrelated to the fact that the item was previously studied and stored. The larger the fraction of context units that are on at any time (β), the more spurious activation there will be that needs to be cancelled out. The probability of recalling the item is related to the activity of memory element N: P(recall) = min(1, mN). When the item is studied, the weights from context units in pool i are adjusted according to an update rule that performs gradient descent in an error measure Ei = ei2, where ei = 1 −mi/Γi. This error is minimized when the memory element i reaches activation level Γi (defined earlier as the proportion of units in the entire context pool that contributes to activity at stage i). The weight update that performs gradient descent in Ei is ∆wij = ϵ Nβ(1 −β)eicij, (7) where ϵ is a learning rate and the denominator of the first term is a normalization constant which can be folded into the learning rate. As in SAM, ϵ is assumed to be contingent on retrieval success at the start of the study trial, in the manner we described previously. What is the motivation for minimizing the prediction error at every stage, versus minimizing the prediction error just at the final stage, EN? To answer this question, note that there are two consequences of minimizing the error Ei to zero for any i. First, reducing Ei will also likely serve to reduce El for all l > i. Second, achieving this objective will allow the {wl,j,k : l > i} to all be set to zero without any effect on the memory. Essentially, there is no need to store information for a longer time scale than it is needed. This description of MCM is identical to SAM except: (1) SAM has a single temporal scale of representation; MCM has a multiscale representation. (2) SAM’s memory update rule can be interpreted as Hebbian learning; MCM’s update can be interpreted as error-correction learning. 3.3 Relating leaky integrator and neural net characterizations of MCM To make contact with MTS, we have described MCM as a cascade of leaky integrators, and to make contact with SAM, we have described MCM as a neural net. One can easily verify that the leakyintegrator and neural-net descriptions of MCM are equivalent via the following correspondence between variables of the two models, where E[.] denotes the expectation over context representations: si = E[mi]/Γi and xi = P j E[wijcij] + b Nβ(1 −β) . 4 Simulations Cepeda and colleagues (Cepeda, Vul, Rohrer, Wixted, & Pashler, 2008; Cepeda et al., in press) have recently conducted well-controlled experimental manipulations of spacing involving RIs on educationally relevant time scales of days to months. Most research in the spacing literature involves brief RIs, on the scale of minutes to an hour, and methodological concerns have been raised with the few well-known studies involving longer RIs (Cepeda et al., 2006). In Cepeda’s experiments, participants study a set of paired associates over two sessions. In the first session, participants are trained until they reach a performance criterion, ensuring that the material has been successfully encoded. At the start of the second session, participants are tested via a cued-recall paradigm, and then are given a fixed number of study passes through all the pairs. Following a specified RI, a final cued-recall test is administered. Recall accuracy at the start of the second session provides the basic forgetting function, and recall accuracy at test provides the spacing function. 6 0 1 2 4 7 14 0 20 40 60 80 100 ISI (days) % recall (a) RI = 10 days 17 28 84 168 0 20 40 60 80 100 ISI (days) (b) RI = 168 days 17 28 84 168 0 20 40 60 80 100 ISI (days) (c) RI = 168 days 1 7 14 21 35 70 105 0 20 40 60 80 100 ISI (days) % recall (d) RIs = 7, 35, 70, 350 days 1 7 14 21 35 70 105 0 20 40 60 80 100 ISI (days) % recall (e) RIs = 7, 35, 70, 350 days Figure 2: Modeling and experimental data of (Cepeda et al., in press) (a) Experiment 1 (SwahiliEnglish), (b) Experiment 2a (obscure facts), and (c) Experiment 2b (object names). The four RI conditions of Cepeda et al. (2008) are modeled using (d) MCM and (e) the Bayesian multiscale model of Kording et al. (2007). In panel (e), the peaks of the model’s spacing functions are indicated by the triangle pointers. For each experiment, we optimized MCM’s parameters, {µ, ν, ω, ξ}, to obtain a least squares fit to the forgetting function. These four model parameters determine the time constants and weighting coefficients of the mixture-of-exponentials approximation to the forgetting function (Equation 5). The model has only one other free parameter, ϵr, the magnitude of update on a trial when an item is successfully recalled (see Equation 6). We chose ϵr = 9 for all experiments, based on hand tuning the parameter to fit the first experiment reported here. With ϵr, MCM is fully constrained and can make strong predictions regarding the spacing function. Figure 2 shows MCM’s predictions of Cepeda’s experiments. Panels a-c show the forgetting function data for the experiments (open blue squares connected by dotted lines), MCM’s post-hoc fit to the forgetting function (solid blue line), the spacing function data (solid green points connected by dotted lines), and MCM’s parameter-free prediction of the spacing function (solid green line). The individual panels show the ISIs studied and the RI. For each experiment, MCM’s prediction of the peak of the spacing function is entirely consistent with the data, and for the most part, MCM’s quantiative predictions are excellent. (In panel c, MCM’s predictions are about 20% too low across the range of ISIs.) Interestingly, the experiments in panels b and c explored identical ISIs and RIs with two different types of material. With the coarse range of ISIs explored, the authors of these experiments concluded that the peak ISI was the same independent of the material (28 days). MCM suggests a different peak for the two sets of material, a prediction that can be evaluated empirically. (It would be extremely surprising to psychologists if the peak were in general independent of the material, as content effects pervade the memory literature.) Panel d presents the results of a complex study involving a single set of items studied with 11 different ISIs, ranging from minutes to months, and four RIs, ranging from a week to nearly a year. We omit the fit to the forgetting function to avoid cluttering the graph. The data and model predictions 7 −5 −4 −3 −2 −1 0 1 2 3 −6 −5 −4 −3 −2 −1 0 1 2 log10(RI) log10(optimal ISI) human data MCM MCM regression Figure 3: A meta-analysis of the literature by Cepeda et al. (2006). Each red circle represents a single spacing experiment in which the ISI was varied for a given RI. The optimal ISI obtained in the experiment is plotted against the RI on a log-log scale. (Note that the data are intrinsically noisy because experiments typically examine only a small set of ISIs, from which the ’optimum’ is chosen.) The X’s represent the mean from 1000 replications of MCM for a given RI with randomly drawn parameter settings (i.e., random forgetting functions), and the dashed line is the best regression fit to the X’s. Both the experimental data and MCM show a power law relationship between optimal ISI and RI. are color coded by RI, with higher recall accuracy for shorter RIs. MCM predicts the spacing functions with absolutely spectacular precision, considering the predictions are fully constrained and parameter free. Moreover, MCM anticipates the peaks of the spacing functions, with the curvature of the peak decreasing with the RI, and the optimal ISI increasing with the RI. In addition to these results, MCM also predicts the probability of recall at test conditional on successful or unsuccessful recall during the test at the start of the second study session. As explained in Figure 3, MCM obtains a sensible parameter-free fit to a meta-analysis of the experimental literature by Cepeda et al. (2006). Finally, MCM is able to post-hoc fit classic studies from the spacing literature (for which forgetting functions are not available). 5 Discussion MCM’s blind prediction of 7 different spacing functions is remarkable considering that the domain’s complexity (the content, manner and amount of study) is reduced to four parameters, which are fully determined by the forgetting function. Obtaining empirical forgetting functions is straightforward. Obtaining empirical evidence to optimize study schedules, especially when more than two sessions are involved, is nearly infeasible. MCM thus offers a significant practical tool for educators in devising study schedules. Optimizing study schedules with MCM is straightfoward, and particularly useful considering that MCM can optimize not only for a known RI but for RI as a random variable. MCM arose from two existing models, MTS and SAM, and all three models are characterized at Marr’s implementation or algorithmic levels, not at the level of a computational theory. Kording et al. (2007) have proposed a Bayesian memory model which has intriguing similarities to MCM, and has the potential of serving as the complementary computational theory. The model is a Kalman filter (KF) with internal state variables that decay exponentially at different rates. The state predicts the appearance of an item in the temporal stream of experience. The dynamics of MCM can be exactly mapped onto the KF, with τ related to the decay of a variable, and γ to its internal noise level. However, the KF model has a very different update rule, based on the Kalman gain. We have tried to fit experimental data with the KF model, but have not been satisfied with the outcome. For example, Figure 2e shows a least-squares fit to the six free parameters of the KF model to the Cepeda et al. (2008) data. (Two parameters determine the range of time scales; two specify internal and observation noise levels; and two perform an affine transform from internal memory strength to recall probability.) In terms of sum-squared error, the model shows a reasonable fit, but the model clearly misses the peaks of the spacing functions, and in fact predicts a peak that is independent of RI. Notably, the KF model is a post-hoc fit to the spacing functions, whereas MCM produces a true prediction of the spacing functions, i.e., parameters of MCM are determined without peeking at the spacing function. Exploring many parameterizations of the KF model, we find that the model generally predicts decreasing or constant optimal ISIs as a function of the RI. In contrast, MCM necessarily produces an increasing optimal ISI as a function of the RI, consistent with all behavioral data. It remains an important and intriguing challenge to unify MCM and the KF model; each has something to offer the other. 8 References Anderson, J. R., & Milson, R. (1989). Human memory: An adaptive perspective. Psych. Rev., 96, 703–719. Cepeda, N. J., Coburn, N., Rohrer, D., Wixted, J. T., Mozer, M. C., & Pashler, H. (in press). Optimizing distributed practice: Theoretical analysis and practical implications. Journal of Experimental Psychology. Cepeda, N. J., Pashler, H., Vul, E., Wixted, J. T., & Rohrer, D. (2006). Distributed practice in verbal recall tasks: A review and quantitative synthesis. Psychological Bulletin, 132, 354–380. Cepeda, N. J., Vul, E., Rohrer, D., Wixted, J. T., & Pashler, H. (2008). Spacing effects in learning: A temporal ridgeline of optimal retention. Psychological Science, 19, 1095–1102. Kording, K. P., Tenenbaum, J. B., & Shadmehr, R. (2007). The dynamics of memory as a consequence of optimal adaptation to a changing body. Nature Neuroscience, 10, 779–786. Pavlik, P. I., & Anderson, J. R. (2005). Practice and forgetting effects on vocabulary memory: An activation-based model of the spacing effect. Cognitive Science, 29(4), 559-586. Raaijmakers, J. G. W. (2003). Spacing and repetition effects in human memory: application of the SAM model. Cognitive Science, 27, 431–452. Raaijmakers, J. G. W., & Shiffrin, R. M. (1981). Search of associative memory. Psych. Rev., 88, 93–134. Staddon, J. E. R., Chelaru, I. M., & Higa, J. J. (2002). Habituation, memory and the brain: The dynamics of interval timing. Behavioural Processes, 57, 71-88. Wixted, J. T., & Carpenter, S. K. (2007). The Wickelgren power law and the Ebbinghaus savings function. Psychological Science, 18, 133–134. 9
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A Game-Theoretic Approach to Hypergraph Clustering Samuel Rota Bul`o Marcello Pelillo University of Venice, Italy {srotabul,pelillo}@dsi.unive.it Abstract Hypergraph clustering refers to the process of extracting maximally coherent groups from a set of objects using high-order (rather than pairwise) similarities. Traditional approaches to this problem are based on the idea of partitioning the input data into a user-defined number of classes, thereby obtaining the clusters as a by-product of the partitioning process. In this paper, we provide a radically different perspective to the problem. In contrast to the classical approach, we attempt to provide a meaningful formalization of the very notion of a cluster and we show that game theory offers an attractive and unexplored perspective that serves well our purpose. Specifically, we show that the hypergraph clustering problem can be naturally cast into a non-cooperative multi-player “clustering game”, whereby the notion of a cluster is equivalent to a classical game-theoretic equilibrium concept. From the computational viewpoint, we show that the problem of finding the equilibria of our clustering game is equivalent to locally optimizing a polynomial function over the standard simplex, and we provide a discrete-time dynamics to perform this optimization. Experiments are presented which show the superiority of our approach over state-of-the-art hypergraph clustering techniques. 1 Introduction Clustering is the problem of organizing a set of objects into groups, or clusters, in a way as to have similar objects grouped together and dissimilar ones assigned to different groups, according to some similarity measure. Unfortunately, there is no universally accepted formal definition of the notion of a cluster, but it is generally agreed that, informally, a cluster should correspond to a set of objects satisfying two conditions: an internal coherency condition, which asks that the objects belonging to the cluster have high mutual similarities, and an external incoherency condition, which states that the overall cluster internal coherency decreases by adding to it any external object. Objects similarities are typically expressed as pairwise relations, but in some applications higherorder relations are more appropriate, and approximating them in terms of pairwise interactions can lead to substantial loss of information. Consider for instance the problem of clustering a given set of d-dimensional Euclidean points into lines. As every pair of data points trivially defines a line, there does not exist a meaningful pairwise measure of similarity for this problem. However, it makes perfect sense to define similarity measures over triplets of points that indicate how close they are to being collinear. Clearly, this example can be generalized to any problem of model-based point pattern clustering, where the deviation of a set of points from the model provides a measure of their dissimilarity. The problem of clustering objects using high-order similarities is usually referred to as the hypergraph clustering problem. In the machine learning community, there has been increasing interest around this problem. Zien and co-authors [24] propose two approaches called “clique expansion” and “star expansion”, respectively. Both approaches transform the similarity hypergraph into an edge-weighted graph, whose edge-weights are a function of the hypergraph’s original weights. This way they are able to tackle 1 the problem with standard pairwise clustering algorithms. Bolla [6] defines a Laplacian matrix for an unweighted hypergraph and establishes a link between the spectral properties of this matrix and the hypergraph’s minimum cut. Rodr`ıguez [16] achieves similar results by transforming the hypergraph into a graph according to “clique expansion” and shows a relationship between the spectral properties of a Laplacian of the resulting matrix and the cost of minimum partitions of the hypergraph. Zhou and co-authors [23] generalize their earlier work on regularization on graphs and define a hypergraph normalized cut criterion for a k-partition of the vertices, which can be achieved by finding the second smallest eigenvector of a normalized Laplacian. This approach generalizes the well-known “Normalized cut” pairwise clustering algorithm [19]. Finally, in [2] we find another work based on the idea of applying a spectral graph partitioning algorithm on an edge-weighted graph, which approximates the original (edge-weighted) hypergraph. It is worth noting that the approaches mentioned above are devised for dealing with higher-order relations, but can all be reduced to standard pairwise clustering approaches [1]. A different formulation is introduced in [18], where the clustering problem with higher-order (super-symmetric) similarities is cast into a nonnegative factorization of the closest hyper-stochastic version of the input affinity tensor. All the afore-mentioned approaches to hypergraph clustering are partition-based. Indeed, clusters are not modeled and sought directly, but they are obtained as a by-product of the partition of the input data into a fixed number of classes. This renders these approaches vulnerable to applications where the number of classes is not known in advance, or where data is affected by clutter elements which do not belong to any cluster (as in figure/ground separation problems). Additionally, by partitioning, clusters are necessarily disjoint sets, although it is in many cases natural to have overlapping clusters, e.g., two intersecting lines have the point in the intersection belonging to both lines. In this paper, following [14, 20] we offer a radically different perspective to the hypergraph clustering problem. Instead of insisting on the idea of determining a partition of the input data, and hence obtaining the clusters as a by-product of the partitioning process, we reverse the terms of the problem and attempt instead to derive a rigorous formulation of the very notion of a cluster. This allows one, in principle, to deal with more general problems where clusters may overlap and/or outliers may get unassigned. We found that game theory offers a very elegant and general mathematical framework that serves well our purposes. The basic idea behind our approach is that the hypergraph clustering problem can be considered as a multi-player non-cooperative “clustering game”. Within this context, the notion of a cluster turns out to be equivalent to a classical equilibrium concept from (evolutionary) game theory, as the latter reflects both the internal and external cluster conditions alluded to before. We also show that there exists a correspondence between these equilibria and the local solutions of a polynomial, linearly-constrained, optimization problem, and provide an algorithm for finding them. Experiments on two standard hypergraph clustering problems show the superiority of the proposed approach over state-of-the-art hypergraph clustering techniques. 2 Basic notions from evolutionary game theory Evolutionary game theory studies models of strategic interactions (called games) among large numbers of anonymous agents. A game can be formalized as a triplet Γ = (P, S, π), where P = {1, . . ., k} is the set of players involved in the game, S = {1, . . . , n} is the set of pure strategies (in the terminology of game-theory) available to each player and π : Sk →R is the payoff function, which assigns a payoff to each strategy profile, i.e., the (ordered) set of pure strategies played by the individuals. The payoff function π is assumed to be invariant to permutations of the strategy profile. It is worth noting that in general games, each player may have its own set of strategies and own payoff function. For a comprehensive introduction to evolutionary game theory we refer to [22]. By undertaking an evolutionary setting we assume to have a large population of non-rational agents, which are randomly matched to play a game Γ = (P, S, π). Agents are considered non-rational, because each of them initially chooses a strategy from S, which will be always played when selected for the game. An agent, who selected strategy i ∈S, is called i-strategist. Evolution in the population takes place, because we assume that there exists a selection mechanism, which, by analogy with a Darwinian process, spreads the fittest strategies in the population to the detriment of the weakest one, which will in turn be driven to extinction. We will see later in this work a formalization of such a selection mechanism. 2 The state of the population at a given time t can be represented as a n-dimensional vector x(t), where xi(t) represents the fraction of i-strategists in the population at time t. The set of all possible states describing a population is given by ∆= ( x ∈Rn : X i∈S xi = 1 and xi ≥0 for all i ∈S ) , which is called standard simplex. In the sequel we will drop the time reference from the population state, where not necessary. Moreover, we denote with σ(x) the support of x ∈∆, i.e., the set of strategies still alive in population x ∈∆: σ(x) = {i ∈S : xi > 0}. If y(i) ∈∆is the probability distribution identifying which strategy the ith player will adopt if drawn to play the game Γ, then the average payoff obtained by the agents can be computed as u(y(1), . . . , y(k)) = X (s1,...,sk)∈Sk π(s1, . . . , sk) k Y j=1 y(j) sj . (1) Note that (1) is invariant to any permutation of the input probability vectors. Assuming that the agents are randomly and independently drawn from a population x ∈∆to play the game Γ, the population average payoff is given by u(xk), where xk is a shortcut for x, . . . , x repeated k times. Furthermore, the average payoff that an i-strategist obtains in a population x ∈∆ is given by u(ei, xk−1), where ei ∈∆is a vector with xi = 1 and zero elsewhere. An important notion in game theory is that of equilibrium [22]. A population x ∈∆is in equilibrium when the distribution of strategies will not change anymore, which intuitively happens when every individual in the population obtains the same average payoff and no strategy can thus prevail on the other ones. Formally, x ∈∆is a Nash equilibrium if u(ei, xk−1) ≤u(xk) , for all i ∈S . (2) In other words, every agent in the population performs at most as well as the population average payoff. Due to the multi-linearity of u, a consequence of (2) is that u(ei, xk−1) = u(xk) , for all i ∈σ(x) , (3) i.e., all the agents that survived the evolution obtain the same average payoff, which coincides with the population average payoff. A key concept pertaining to evolutionary game theory is that of an evolutionary stable strategy [7, 22]. Such a strategy is robust to evolutionary pressure in an exact sense. Assume that in a population x ∈∆, a small share ǫ of mutant agents appears, whose distribution of strategies is y ∈∆. The resulting postentry population is given by wǫ = (1 −ǫ)x + ǫy. Biological intuition suggests that evolutionary forces select against mutant individuals if and only if the average payoff of a mutant agent in the postentry population is lower than that of an individual from the original population, i.e., u(y, wk−1 ǫ ) < u(x, wk−1 ǫ ) . (4) A population x ∈∆is evolutionary stable (or an ESS) if inequality (4) holds for any distribution of mutant agents y ∈∆\ {x}, granted the population share of mutants ǫ is sufficiently small (see, [22] for pairwise contests and [7] for n-wise contests). An alternative, but equivalent, characterization of ESSs involves a leveled notion of evolutionary stable strategies [7]. We say that x ∈∆is an ESS of level j against y ∈∆, if there exists j ∈ {0, . . . , k −1} such that both conditions u(yj+1, xk−j−1) < u(yj, xk−j) , (5) u(yi+1, xk−i−1) = u(yi, xk−i) , for all 0 ≤i < j , (6) are satisfied. Clearly, x ∈∆is an ESS if it satisfies a condition of this form for every y ∈∆\ {x}. It is straightforward to see that any ESS is a Nash equilibrium [22, 7]. An ESS, which satisfies conditions (6) with j never more than J, will be called an ESS of level J. Note that for the generic case most of the preceding conditions will be superfluous, i.e., only ESSs of level 0 or 1 are required [7]. Hence, in the sequel, we will consider only ESSs of level 1. It is not difficult to verify that any ESS (of level 1) x ∈∆satisfies u(wk ǫ ) < u(xk) , (7) for all y ∈∆\ {x} and small enough values of ǫ. 3 3 The hypergraph clustering game The hypergraph clustering problem can be described by an edge-weighted hypergraph. Formally, an edge-weighted hypergraph is a triplet H = (V, E, s), where V = {1, . . ., n} is a finite set of vertices, E ⊆P(V ) \ {∅} is the set of (hyper-)edges (here, P(V ) is the power set of V ) and s : E →R is a weight function which associates a real value with each edge. Note that negative weights are allowed too. Although hypergraphs may have edges of varying cardinality, we will focus on a particular class of hypergraphs, called k-graphs, whose edges have all fixed cardinality k ≥2. In this paper, we cast the hypergraph clustering problem into a game, called (hypergraph) clustering game, which will be played in an evolutionary setting. Clusters are then derived from the analysis of the ESSs of the clustering game. Specifically, given a k-graph H = (V, E, s) modeling a hypergraph clustering problem, where V = {1, . . ., n} is the set of objects to cluster and s is the similarity function over the set of objects in E, we can build a game involving k players, each of them having the same set of (pure) strategies, namely the set of objects to cluster V . Under this setting, a population x ∈∆of agents playing a clustering game represents in fact a cluster, where xi is the probability for object i to be part of it. Indeed, any cluster can be modeled as a probability distribution over the set of objects to cluster. The payoff function of the clustering game is defined in a way as to favour the evolution of agents supporting highly coherent objects. Intuitively, this is accomplished by rewarding the k players in proportion to the similarity that the k played objects have. Hence, assuming (v1, . . . , vk) ∈V k to be the tuple of objects selected by k players, the payoff function can be simply defined as π(v1, . . . , vk) = 1 k!s ({v1, . . . , vk}) if {v1, . . . , vk} ∈E , 0 else , (8) where the term 1/k! has been introduced for technical reasons. Given a population x ∈∆playing the clustering game, we have that the average population payoff u(xk) measures the cluster’s internal coherency as the average similarity of the objects forming the cluster, whereas the average payoff u(ei, xk−1) of an agent supporting object i ∈V in population x, measures the average similarity of object i with respect to the cluster. An ESS of a clustering game incorporates the properties of internal coherency and external incoherency of a cluster: internal coherency: since ESSs are Nash equilibria, from (3), it follows that every object contributing to the cluster, i.e., every object in σ(x), has the same average similarity with respect to the cluster, which in turn corresponds to the cluster’s overall average similarity. Hence, the cluster is internally coherent; external incoherency: from (2), every object external to the cluster, i.e., every object in V \ σ(x), has an average similarity which does not exceed the cluster’s overall average similarity. There may still be cases where the average similarity of an external object is the same as that of an internal object, mining the cluster’s external incoherency. However, since x is an ESS, from (7) we see that whenever we try to extend a cluster with small shares of external objects, the cluster’s overall average similarity drops. This guarantees the external incoherency property of a cluster to be also satisfied. Finally, it is worth noting that this theory generalizes the dominant-sets clustering framework which has recently been introduced in [14]. Indeed, ESSs of pairwise clustering games, i.e. clustering games defined on graphs, correspond to the dominant-set clusters [20, 17]. This is an additional evidence that ESSs are meaningful notions of cluster. 4 Evolution towards a cluster In this section we will show that the ESSs of a clustering game are in one-to-one correspondence with (strict) local solution of a non-linear optimization program. In order to find ESSs, we will also provide a dynamics due to Baum and Eagon, which generalizes the replicator dynamics [22]. Let H = (V, E, s) be a hypergraph clustering problem and Γ = (P, V, π) be the corresponding clustering game. Consider the following non-linear optimization problem: maximize f(x) = X e∈E s(e) Y i∈e xi , subject to x ∈∆. (9) 4 It is simple to see that any first-order Karush-Kuhn-Tucker (KKT) point x ∈∆of program (9) [13] is a Nash equilibrium of Γ. Indeed, by the KKT conditions there exist µi ≥0, i ∈S, and λ ∈R such that for all i ∈S, ∇f(x)i + µi −λ = 1 k u(ei, xk−1) + µi −λ = 0 and µixi = 0 , where ∇is the gradient operator. From this it follows straightforwardly that u(ei, xk−1) ≤u(xk) for all i ∈S. Moreover, it turns out that any strict local maximizer x ∈∆of (9) is an ESS of Γ. Indeed, by definition, a strict local maximizer of this program satisfies u(zk) = f(z) < f(x) = u(xk), for any z ∈∆\ {x} close enough to x, which is in turn equivalent to (7) for sufficiently small values of ǫ. The problem of extracting ESSs of our hypergraph clustering game can thus be cast into the problem of finding strict local solutions of (9). We will address this optimization task using a result due to Baum and Eagon [3], who introduced a class of nonlinear transformations in probability domain. Theorem 1 (Baum-Eagon). Let P(x) be a homogeneous polynomial in the variables xi with nonnegative coefficients, and let x ∈∆. Define the mapping z = M(x) as follows: zi = xi∂iP(x) . n X j=1 xj∂jP(x), i = 1, . . . , n. (10) Then P(M(x)) > P(x), unless M(x) = x. In other words M is a growth transformation for the polynomial P. The Baum-Eagon inequality provides an effective iterative means for maximizing polynomial functions in probability domains, and in fact it has served as the basis for various statistical estimation techniques developed within the theory of probabilistic functions of Markov chains [4]. It was also employed for the solution of relaxation labelling processes [15]. Since f(x) is a homogeneous polynomial in the variables xi, we can use the transformation of Theorem 1 in order to find a local solution x ∈∆of (9), which in turn provides us with an ESS of the hypergraph clustering game. By taking the support of x, we have a cluster under our framework. The complexity of finding a cluster is thus O(ρ|E|), where |E| is the number of edges of the hypergraph describing the clustering problem and ρ is the average number of iteration needed to converge. Note that ρ never exceeded 100 in our experiments. In order to obtain the clustering, in principle, we have to find the ESSs of the clustering game. This is a non-trivial, although still feasible, task [21], which we leave as a future extension of this work. By now, we adopt a naive peeling-off strategy for our cluster extraction procedure. Namely, we iteratively find a cluster and remove it from the set of objects, and we repeat this process on the remaining objects until a desired number of clusters have been extracted. The set of extracted ESSs with this procedure does not technically correspond to the ESSs of the original game, but to ESSs of sub-games of it. The cost of this approximation is that we unfortunately loose (by now) the possibility of having overlapping clusters. 5 Experiments In this section we present two types of experiments. The first one addresses the problem of line clustering, while the second one addresses the problem of illuminant-invariant face clustering. We tested our approach against Clique Averaging algorithm (CAVERAGE), since it was the best performing approach in [2] on the same type of experiments. Specifically, CAVERAGE outperformed Clique Expansion [10] combined with Normalized cuts, Gibson’s Algorithm under sum and product model [9], kHMeTiS [11] and Cascading RANSAC [2]. We also compare against Super-symmetric Non-negative Tensor Factorization (SNTF) [18], because it is the only approach, other than ours, which does not approximate the hypergraph to a graph. Since both CAVERAGE and SNTF, as opposed to our method, require the number of classes K to be specified, we run them with values of K ∈{K∗−1, K∗, K∗+ 1} among which the optimal one (K∗) is present. This allows us to verify the robustness of the approaches under wrong values of K, which may occur in general as the optimal number of clusters is not known in advance. 5 −1.5 −1 −0.5 0 0.5 1 1.5 2 −1.5 −1 −0.5 0 0.5 1 1.5 (a) Example of three lines with σ = 0.04. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.04 0.08 F-measure σ HoCluGame Cav. K=2 Cav. K=3 Cav. K=4 Sntf K=2 Sntf K=3 Sntf K=4 (b) Three lines. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.04 0.08 F-measure σ HoCluGame Cav. K=3 Cav. K=4 Cav. K=5 Sntf K=3 Sntf K=4 Sntf K=5 (c) Four lines. 0 0.2 0.4 0.6 0.8 1 1.2 0 0.01 0.02 0.04 0.08 F-measure σ HoCluGame Cav. K=4 Cav. K=5 Cav. K=6 Sntf K=4 Sntf K=5 Snft K=6 (d) Five lines. Figure 1: Results on clustering 3, 4 and 5 lines perturbed with increasing levels of Gaussian noise (σ = 0, 0.01, 0.02, 0.04, 0.08). We executed the experiments on a AMD Sempron 3Ghz computer with 1Gb RAM. Moreover, we evaluated the quality of a clustering by computing the average F-measure of each cluster in the ground-truth with the most compatible one in the obtained solution (according to a one-to-one correspondence). 5.1 Line clustering We consider the problem of clustering lines in spaces of dimension greater than two, i.e., given a set of points in Rd, the task is to find sets of collinear points. Pairwise measures of similarity are useless and at least three points are needed. The dissimilarity measure on triplets of points is given by their mean distance to the best fitting line. If d(i, j, k) is the dissimilarity of points {i, j, k}, the similarity function is given by s({i, j, k}) = exp(−d(i, j, k)2/σ2), where σ is a scaling parameter, which has been optimally selected for all the approaches according to a small test set. We conducted two experiments, in order to assess the robustness of the approaches to both local and global noise. Local noise refers to a Gaussian perturbation applied to the points of a line, while global noise consists of random outlier points. A first experiment consists in clustering 3, 4 and 5 lines generated in the 5-dimensional space [−2, 2]5. Each line consists of 20 points, which have been perturbed according to 5 increasing levels of Gaussian noise, namely σ = 0, 0.01, 0.02, 0.04, 0.08. With this setting there are no outliers and every point should be assigned to a line (e.g., see Figure 1(a)). Figure 1(b) shows the results obtained with three lines. We reported, for each noise level, the mean and the standard deviation of the average F-measures obtained by the algorithms on 30 randomly generated instances. Note that, if the optimal K is used, CAVERAGE and SNTF perform well and the influence of local noise is minimal. This behavior intuitively makes sense under moderate perturbations, because if the approaches correctly partitioned the data without noise, it is unlikely that the result will change by slightly perturbing them. Our approach however achieves good performances as well, although we can notice that with the highest noise level, the performance slightly drops. This is due to the fact that points deviating too much from the overall cluster average collinearity will be excluded as they undermine the cluster’s internal coherency. Hence, some perturbed points will be considered outliers. Nevertheless, it is worth noting that by underestimating the optimal number of classes both CAVERAGE and SNTF exhibit a drastic performance drop, whereas the influence of overestimations 6 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −2 −1.5 −1 −0.5 0 0.5 1 1.5 First line Second line Outliers (a) Example of two lines with 40 outliers. 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 40 F-measure σ HoCluGame Cav. K=2 Cav. K=3 Cav. K=4 Sntf K=2 Sntf K=3 Sntf K=4 (b) Two lines. 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 40 F-measure σ HoCluGame Cav. K=2 Cav. K=3 Cav. K=4 Sntf K=2 Sntf K=3 Sntf K=4 (c) Three lines. 0 0.2 0.4 0.6 0.8 1 1.2 0 10 20 40 F-measure σ HoCluGame Cav. K=3 Cav. K=4 Cav. K=5 Sntf K=3 Sntf K=4 Sntf K=5 (d) Four lines. Figure 2: Results on clustering 2, 3 and 4 lines with an increasing number of outliers (0, 10, 20, 40). has a lower impact on the two partition-based algorithms. By increasing the number of lines involved in the experiment from three to four (Figure 1(c)) and to five (Figure 1(d)) the scenario remains almost the same for our approach and SNTF, while we can notice a slight decrease of CAVERAGE’s performance. The second experiment consists in clustering 2, 3 and 4 slightly perturbed lines (with fixed local noise σ = 0.01) generated in the 5-dimensional space [−2, 2]5. Again, each line consists of 20 points. This time however we added also global noise, i.e., 0, 10, 20 and 40 random points as outliers (e.g., see Figure 2(a)). Figure 2(b) shows the results obtained with two lines. Here, the supremacy of our approach over partition-based ones is clear. Indeed, our method is not influenced by outliers and therefore it performs almost perfectly, whereas CAVERAGE and SNTF perform well only without outliers and with the optimal K. It is interesting to notice that, as outliers are introduced, CAVERAGE and SNTF perform better with K > 2. Indeed, the only way to get rid of outliers is to group them in additional clusters. However, since outliers are not mutually similar and intuitively they do not form a cluster, we have that the performance of CAVERAGE and SNTF drop drastically as the number of outliers increases. Finally, by increasing the number of lines from two to three (Figure 2(c)) and to four (Figure 2(d)), the performance of CAVERAGE and SNTF get worse, while our approach still achieves good results. 5.2 Illuminant-invariant face clustering In [5] it has been shown that images of a Lambertian object illuminated by a point light source lie in a three dimensional subspace. According to this result, if we assume that four images of a face form the columns of a matrix then d = s2 4/(s2 1 + · · · + s2 4) provides us with a measure of dissimilarity, where si is the ith singular value of this matrix [2]. We use this dissimilarity measure for the face clustering problem and we consider as dataset the Yale Face Database B and its extended version [8, 12]. In total we have faces of 38 individuals, each under 64 different illumination conditions. We compared our approach against CAVERAGE and SNTF on subsets of this face dataset. Specifically, we considered cases where we have faces from 4 and 5 random individuals (10 faces per individual), and with and without outliers. The case with outliers consists in 10 additional faces each from a different individual. For each of those combinations, we created 10 random subsets. Similarly to the case of line clustering, we run CAVERAGE and SNTF with values of K ∈{K∗−1, K∗, K∗+ 1}, where K∗is the optimal one. 7 n. of classes: 4 5 n. of outliers: 0 10 0 10 CAVERAGE K=3 0.63±0.11 0.59±0.07 CAVERAGE K=4 0.96±0.06 0.84±0.07 0.56±0.14 0.58±0.07 CAVERAGE K=5 0.91±0.06 0.79±0.05 0.85±0.12 0.83±0.06 CAVERAGE K=6 0.84±0.09 0.82±0.06 SNTF K=3 0.62±0.12 0.58±0.10 SNTF K=4 0.87±0.07 0.81±0.08 0.61±0.13 0.59±0.09 SNTF K=5 0.82±0.09 0.76±0.09 0.86±0.12 0.80±0.07 SNTF K=6 0.85±0.08 0.79±0.11 HoCluGame 0.95±0.03 0.94±0.02 0.95±0.05 0.94±0.02 Table 1: Experiments on illuminant-invariant face clustering. In Table 1 we report the average F-measures (mean and standard deviation) obtained by the three approaches. The results are consistent with those obtained in the case of line clustering with the exception of SNTF, which performs worse than the other approaches on this real-world application. CAVERAGE and our algorithm perform comparably well when clustering 4 individuals without outliers. However, our approach turns out to be more robust in every other tested case, i.e., when the number of classes increases and when outliers are introduced. Indeed, CAVERAGE’s performance decreases, while our approach yields the same good results. In both the experiments of line and face clustering the execution times of our approach were higher than those of CAVERAGE, but considerably lower than SNTF. The main reason why CAVERAGE run faster is that our approach and SNTF work directly on the hypergraph without resorting to pairwise relations, which is indeed what CAVERAGE does. Further, we mention that our code was not optimized to improve speed and all the approaches were run without any sampling policy. 6 Discussion In this paper, we offered a game-theoretic perspective to the hypergraph clustering problem. Within our framework the clustering problem is viewed as a multi-player non-cooperative game, and classical equilibrium notions from evolutionary game theory turn out to provide a natural formalization of the notion of a cluster. We showed that the problem of finding these equilibria (clusters) is equivalent to solving a polynomial optimization problem with linear constraints, which we solve using an algorithm based on the Baum-Eagon inequality. An advantage of our approach over traditional techniques is the independence from the number of clusters, which is indeed an intrinsic characteristic of the input data, and the robustness against outliers, which is especially useful when solving figureground-like grouping problems. We also mention, as a potential positive feature of the proposed approach, the possibility of finding overlapping clusters (e.g., along the lines presented in [21]), although in this paper we have not explicitly dealt with this problem. The experimental results show the superiority of our approach with respect to the state of the art in terms of quality of solution. We are currently studying alternatives to the plain Baum-Eagon dynamics in order to improve efficiency. Acknowledgments. We acknowledge financial support from the FET programme within EU FP7, under the SIMBAD project (contract 213250). We also thank Sameer Agarwal and Ron Zass for providing us with the code of their algorithms. References [1] S. Agarwal, K. Branson, and S. Belongie. Higher order learning with graphs. In Int. Conf. on Mach. Learning, volume 148, pages 17–24, 2006. [2] S. Agarwal, J. Lim, L. Zelnik-Manor, P. Perona, D. Kriegman, and S. Belongie. Beyond pairwise clustering. In IEEE Conf. Computer Vision and Patt. Recogn., volume 2, pages 838– 845, 2005. [3] L. E. Baum and J. A. Eagon. An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. Amer. Math. Soc., 73:360–363, 1967. 8 [4] L. E. Baum, T. Petrie, G. Soules, and N. Weiss. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. Ann. Math. Statistics, 41:164– 171, 1970. [5] P. Belhumeur and D. Kriegman. What is the set of images of an object under all possible lighting conditions. Int. J. Comput. Vision, 28(3):245–260, 1998. [6] M. Bolla. Spectral, euclidean representations and clusterings of hypergraphs. Discr. Math., 117:19–39, 1993. [7] M. Broom., C. Cannings, and G. T. Vickers. Multi-player matrix games. Bull. Math. Biology, 59(5):931–952, 1997. [8] A. S. Georghiades., P. N. Belhumeur, and D. J. Kriegman. From few to many: illumination cone models for face recognition under variable lighting and pose. IEEE Trans. Pattern Anal. Machine Intell., 23(6):643–660, 2001. [9] D. Gibson, J. M. Kleinberg, and P. Raghavan. VLDB, chapter Clustering categoral data: An approach based on dynamical systems., pages 311–322. Morgan Kaufmann Publishers Inc., 1998. [10] T. Hu and K. Moerder. Multiterminal flows in hypergraphs. In T. Hu and E. S. Kuh, editors, VLSI circuit layout: theory and design, pages 87–93. 1985. [11] G. Karypis and V. Kumar. Multilevel k-way hypergraph partitioning. VLSI Design, 11(3):285– 300, 2000. [12] K. C. Lee, J. Ho, and D. Kriegman. Acquiring linear subspaces for face recognition under variable lighting. IEEE Trans. Pattern Anal. Machine Intell., 27(5):684–698, 2005. [13] D. G. Luenberger. Linear and nonlinear programming. Addison Wesley, 1984. [14] M. Pavan and M. Pelillo. Dominant sets and pairwise clustering. IEEE Trans. Pattern Anal. Machine Intell., 29(1):167–172, 2007. [15] M. Pelillo. The dynamics of nonlinear relaxation labeling processes. J. Math. Imag. and Vision, 7(4):309–323, 1997. [16] J. Rodr`ıguez. On the Laplacian spectrum and walk-regular hypergraphs. Linear and Multilinear Algebra, 51:285–297, 2003. [17] S. Rota Bul`o. A game-theoretic framework for similarity-based data clustering. PhD thesis, University of Venice, 2009. [18] A. Shashua, R. Zass, and T. Hazan. Multi-way clustering using super-symmetric non-negative tensor factorization. In Europ. Conf. on Comp. Vision, volume 3954, pages 595–608, 2006. [19] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. Pattern Anal. Machine Intell., 22:888–905, 2000. [20] A. Torsello, S. Rota Bul`o, and M. Pelillo. Grouping with asymmetric affinities: a gametheoretic perspective. In IEEE Conf. Computer Vision and Patt. Recogn., pages 292–299, 2006. [21] A. Torsello, S. Rota Bul`o, and M. Pelillo. Beyond partitions: allowing overlapping groups in pairwise clustering. In Int. Conf. Patt. Recogn., 2008. [22] J. W. Weibull. Evolutionary game theory. Cambridge University Press, 1995. [23] D. Zhou, J. Huang, and B. Sch¨olkopf. Learning with hypergraphs: clustering, classification, embedding. In Adv. in Neur. Inf. Processing Systems, volume 19, pages 1601–1608, 2006. [24] J. Y. Zien, M. D. F. Schlag, and P. K. Chan. Multilevel spectral hypergraph partitioning with arbitrary vertex sizes. IEEE Trans. on Comp.-Aided Design of Integr. Circ. and Systems, 18:1389–1399, 1999. 9
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Entropic Graph Regularization in Non-Parametric Semi-Supervised Classification Amarnag Subramanya & Jeff Bilmes Department of Electrical Engineering, University of Washington, Seattle. {asubram,bilmes}@ee.washington.edu Abstract We prove certain theoretical properties of a graph-regularized transductive learning objective that is based on minimizing a Kullback-Leibler divergence based loss. These include showing that the iterative alternating minimization procedure used to minimize the objective converges to the correct solution and deriving a test for convergence. We also propose a graph node ordering algorithm that is cache cognizant and leads to a linear speedup in parallel computations. This ensures that the algorithm scales to large data sets. By making use of empirical evaluation on the TIMIT and Switchboard I corpora, we show this approach is able to outperform other state-of-the-art SSL approaches. In one instance, we solve a problem on a 120 million node graph. 1 Introduction The process of training classifiers with small amounts of labeled data and relatively large amounts of unlabeled data is known as semi-supervised learning (SSL). In many applications, such as speech recognition, annotating training data is time-consuming, tedious and error-prone. SSL lends itself as a useful technique in such situations as one only needs to annotate small amounts of data for training models. For a survey of SSL algorithms, see [1, 2]. In this paper we focus on graph-based SSL [1]. Here one assumes that the labeled and unlabeled samples are embedded within a low-dimensional manifold expressed by a graph — each data sample is represented by a vertex within a weighted graph with the weights providing a measure of similarity between vertices. Some graph-based SSL approaches perform random walks on the graph for inference [3, 4] while others optimize a loss function based on smoothness constraints derived from the graph [5, 6, 7, 8]. Graph-based SSL algorithms are inherently non-parametric, transductive and discriminative [2]. The results of the benchmark SSL evaluations in chapter 21 of [1] show that graph-based algorithms are in general better than other SSL algorithms. Most of the current graph-based SSL algorithms have a number of shortcomings – (a) in many cases, such as [6, 9], a two class problem is assumed; this necessitates the use of sub-optimal extensions like one vs. rest to solve multi-class problems, (b) most graph-based SSL algorithms (exceptions include [7, 8]) attempt to minimize squared error which is not optimal for classification problems [10], and (c) there is a lack of principled approaches to integrating class prior information into graph-based SSL algorithms. Approaches such as class mass normalization and label bidding are used as a postprocessing step rather than being tightly integrated with the inference. To address some of the above issues, we proposed a new graph-based SSL algorithm based on minimizing a Kullback-Leibler divergence (KLD) based loss in [11]. Some of the advantages of this approach include, straightforward extension to multi-class problems, ability to handle label uncertainty and integrate priors. We also showed that this objective can be minimized using alternating minimization (AM), and can outperform other state-of-the-art SSL algorithms for document classification. Another criticism of previous work in graph-based SSL (and SSL in general) is the lack of algorithms that scale to very large data sets. SSL is based on the premise that unlabeled data is easily 1 obtained, and adding large quantities of unlabeled data leads to improved performance. Thus practical scalability (e.g., parallelization), is very important in SSL algorithms. [12, 13] discuss the application of TSVMs to large-scale problems. [14] suggests an algorithm for improving the induction speed in the case of graph-based algorithms. [15] solves a graph transduction problem with 650,000 samples. To the best of our knowledge, the largest graph-based problem solved to date had about 900,000 samples (includes both labeled and unlabeled data) [16]. Clearly, this is a fraction of the amount of unlabeled data at our disposal. For example, on the Internet alone we create 1.6 billion blog posts, 60 billion emails, 2 million photos and 200,000 videos every day [17]. The goal of this paper is to provide theoretical analysis of our algorithm proposed in [11] and also show how it can be scaled to very large problems. We first prove that AM on our KLD based objective converges to the true optimum. We also provide a test for convergence and discuss some theoretical connections between the two SSL objectives proposed in [11]. In addition, we propose a graph node ordering algorithm that is cache cognizant and makes obtaining a linear speedup with a parallel implementation more likely. As a result, the algorithms are able to scale to very large datasets. The node ordering algorithm is quite general and can be applied to graph-based SSL algorithm such as [5, 11]. In one instance, we solve a SSL problem over a graph with 120 million vertices. We use the phone classification problem to demonstrate the scalability of the algorithm. We believe that speech recognition is an ideal application for SSL and in particular graph-based SSL for several reasons: (a) human speech is produced by a small number of articulators and thus amenable to representation in a low-dimensional manifold [18]; (b) annotating speech data is time-consuming and costly; and (c) the data sets tend to be very large. 2 Graph-based SSL Let Dl = {(xi, ri)}l i=1 be the set of labeled samples, Du = {xi}l+u i=l+1, the set of unlabeled samples and D ≜{Dl, Du}. Here ri is an encoding of the labeled data and will be explained shortly. We are interested in solving the transductive learning problem, i.e., given D, the task is to predict the labels of the samples in Du. The first step in most graph-based SSL algorithms is the construction of an undirected weighted graph G = (V, E), where the vertices (nodes) V = {1, . . . , m}, m = l + u, are the data points in D and the edges E ⊆V × V . Let Vl and Vu be the set of labeled and unlabeled vertices respectively. G may be represented via a symmetric matrix W which is referred to as the weight or affinity matrix. There are many ways of constructing the graph (see section 6.2 in [2]). In this paper, we use symmetric k-nearest neighbor (NN) graphs — that is, we first form wij ≜[W]ij = sim(xi, xj) and then make this graph sparse by setting wij = 0 unless i is one of j’s k nearest neighbors or j is one of i’s k nearest neighbors. It is assumed that sim(x, y) = sim(y, x). Let N(i) be the set of neighbors of vertex i. Choosing the correct similarity measure and |N(i)| are crucial steps in the success of any graph-based SSL algorithm as it determines the graph [2]. For each i ∈V and j ∈Vl, define probability measures pi and rj respectively over the measurable space (Y, Y). Here Y is the σ-field of measurable subsets of Y and Y ⊂N (the set of natural numbers) is the space of classifier outputs. Thus |Y| = 2 yields binary classification while |Y| > 2 implies multi-class. As we only consider classification problems here, pi and ri are multinomial distributions, pi(y) is the probability that xi belongs to class y and the classification result is given by argmaxy pi(y). {rj}, j ∈Vl encodes the labels of the supervised portion of the training data. If the labels are known with certainty, then rj is a “one-hot” vector (with the single 1 at the appropriate position in the vector). rj is also capable of representing cases where the label is uncertain, i.e., for example when the labels are in the form of a distribution (possibly derived from normalizing scores representing confidence). It is important to distinguish between the classical multi-label problem and the use of uncertainty in rj. If it is the case that rj(¯y1), rj(¯y2) > 0, ¯y1 ̸= ¯y2, it does not imply that the input xj possesses two output labels ¯y1 and ¯y2. Rather, rj represents our belief in the various values of the output. As pi, ri are probability measures, they lie within a |Y|-dimensional probability simplex which we represent using △|Y| and so pi, ri ∈△|Y| (henceforth denoted as △). Also let p ≜(p1, . . . , pm) ∈△m≜△× . . . × △(m times) and r ≜(r1, . . . , rl) ∈△l. Consider the optimization problem proposed in [11] where p∗= min p∈△m C1(p) and C1(p) = l X i=1 DKL ri||pi + µ m X i=1 X j∈N(i) wijDKL pi||pj −ν m X i=1 H(pi). 2 Here H(p) = −P y p(y) log p(y) is the Shannon entropy of p and DKL(p||q) is the KLD between measures p and q and is given by DKL(p||q) = P y p(y) log p(y) q(y). If µ, ν, wij ≥0, ∀i, j then C1(p) is convex [19]. (µ, ν) are hyper-parameters whose choice we discuss in Section 5. The first term in C1 penalizes the solution pi i ∈{1, . . . , l}, when it is far away from the labeled training data Dl, but it does not insist that pi = ri, as allowing for deviations from ri can help especially with noisy labels [20] or when the graph is extremely dense in certain regions. The second term of C1 penalizes a lack of consistency with the geometry of the data, i.e., a graph regularizer. If wij is large, we prefer a solution in which pi and pj are close in the KLD sense. The last term encourages each pi to be close to the uniform distribution if not preferred to the contrary by the first two terms. This acts as a guard against degenerate solutions commonly encountered in graph-based SSL [6], e.g., in cases where a sub-graph is not connected to any labeled vertex. We conjecture that by maximizing the entropy of each pi, the classifier has a better chance of producing high entropy results in graph regions of low confidence (e.g. close to the decision boundary and/or low density regions). To recap, C1 makes use of the manifold assumption, is naturally multi-class and able to encode label uncertainty. As C1 is convex in p with linear constraints, we have a convex programming problem. However, a closed form solution does not exist and so standard numerical optimization approaches such as interior point methods (IPM) or method of multipliers (MOM) can be used to solve the problem. But, each of these approaches have their own shortcomings and are rather cumbersome to implement (e.g. an implementation of MOM to solve this problem would have 7 extraneous parameters). Thus, in [11], we proposed the use of AM for minimizing C1. We will address the question of whether AM is superior to IPMs or MOMs for minimizing C1 shortly. Consider a problem of minimizing d(p, q) over p ∈P, q ∈Q. Sometimes solving this problem directly is hard and in such cases AM lends itself as a valuable tool for efficient optimization. It is an iterative process in which p(n) = argminp∈P d(p, q(n−1)) and q(n+1) = argminq∈Q d(p(n), q). The Expectation-Maximization (EM) [21] algorithm is an example of AM. C1 is not amenable to optimization using AM and so we have proposed a modified version of the objective where (p∗, q∗) = min p,q∈△m C2(p, q) and C2(p, q) = l X i=1 DKL ri||qi + µ m X i=1 X j∈N ′(i) w′ ijDKL pi||qj −ν m X i=1 H(pi). In the above, a third measure qi, ∀i ∈V is defined over the measurable space (Y, Y), W ′ = W + αIn, N ′(i) = {{i} ∪N(i)} and α ≥0. Here the qi’s play a similar role as the pi’s and can potentially be used to obtain a final classification result (argmaxy qi(y)), but α, which is a hyper-parameter, plays an important role in ensuring that pi and qi are close ∀i. It should be at least intuitively clear that as α gets large, the reformulated objective (C2) apparently approaches the original objective (C1). Our results from [11] suggest that setting α = 2 ensures that p∗= q∗(more on this in the next section). It is important to highlight that C2(p, q) is itself still a valid SSL criterion. While the first term encourages qi for the labeled vertices to be close to the labels, ri, the last term encourages higher entropy pi’s. The second term, in addition to acting as a graph regularizer, also acts as glue between the p’s and q’s. The update equations for solving C2(p, q) are given by p(n) i (y) = exp{ µ γi P j w′ ij log q(n−1) j (y)} P y exp{ µ γi P j w′ ij log q(n−1) j (y)} and q(n) i (y) = ri(y)δ(i ≤l) + µ P j w ′ jip(n) j (y) δ(i ≤l) + µ P j w ′ ji where γi = ν + µ P j w ′ ij. Intuitively, discrete probability measures are being propagated between vertices along edges, so we refer to this algorithm as measure propagation (MP). When AM is used to solve an optimization problem, a closed form solution to each of the steps of the AM is desired but not always guaranteed [7]. It can be seen that solving C2 using AM has a single additional hyper-parameter while other approaches such as MOM can have as many as 7. Further, as we show in section 4, the AM update equations can be easily parallelized. We briefly comment on the relationship to previous work. As noted in section 1, a majority of the previous graph-based SSL algorithms are based on minimizing squared error [6, 5]. While 3 these objectives are convex and in some cases admit closed-form (i.e., non-iterative) solutions, they require inverting a matrix of size m × m. Thus in the case of very large data sets (e.g., like the one we consider in section 5), it might not be feasible to use this approach. Therefore, an iterative update is employed in practice. Also, squared-error is only optimal under a Gaussian loss model and thus more suitable for regression rather than classification problems. Squared-loss penalizes absolute error, while KLD, on the other-hand penalizes relative error (pages 226 and 235 of [10]). Henceforth, we refer to a multi-class extension of algorithm 11.2 in [20] as SQ-Loss. The Information Regularization (IR) [7] approach and subsequently the algorithm of Tsuda [8] use KLD based objectives and utilize AM to solve the problem. However these algorithms are motivated from a different perspective. In fact, as stated above, one of the steps of the AM procedure in the case of IR does not admit a closed form solution. In addition, neither IR nor the work of Tsuda use an entropy regularizer, which, as our results will show, leads to improved performance. While the two steps of the AM procedure in the case of Tsuda’s work have closed form solutions and the approach is applicable to hyper-graphs, one of the updates (equation 13 in [8]) is a special of the update for p(n) i . For more connections to previous approaches, see Section 4 in [11]. 3 Optimal Convergence of AM on C2 We show that AM on C2 converges to the minimum of C2, and that there exists a finite α such that the optimal solutions of C1 and C2 are identical. Therefore, C2 is a perfect tractable surrogate for C1. In general, AM is not always guaranteed to converge to the correct solution. For example, consider minimizing f(x, y) = x2 + 3xy + y2 over x, y ∈R where f(x, y) is unbounded below (consider y = −x). But AM says that (x∗, y∗) = (0, 0) which is incorrect (see [22] for more examples). For AM to converge to the correct solution certain conditions must be satisfied. These might include topological properties of the optimization problem [23, 24] or certain geometrical properties [25]. The latter is referred to as the Information Geometry approach where the 5-points property (5pp) [25] plays an important role in determining convergence and is the method of choice here. Theorem 3.1 (Convergence of AM on C2). If p(n) = argmin p∈△m C2(p, q(n−1)), q(n) = argmin q∈△m C2(p(n), q) and q(0) i (y) > 0 ∀y ∈Y, ∀i then (a) C2(p, q) + C2(p, p(0)) ≥C2(p, q(1)) + C2(p(1), q(1)) for all p, q ∈△m, and (b) lim n→∞C2(p(n), q(n)) = infp,q∈△m C2(p, q). Proof Sketch: (a) is the 5-pp for C2(p, q). 5-pp holds if the 3-points (3-pp) and 4-points (4-pp) properties hold. In order to show 3-pp, let f(t) ≜C2(p(t), q(0)) where p(t) = (1−t)p+tp(1), 0 < t ≤1. Next we use the fact that the first-order Taylor’s approximation underestimates a convex function to upper bound the gradient of f(t) w.r.t t. We then pass this to the limit as t →1 and use the monotone convergence theorem to exchange the limit and the summation. This gives the 3-pp. The proof for 4-pp follows in a similar manner. (b) follows as a result of Theorem 3 in [25]. □ Theorem 3.2 (Test for Convergence). If {(p(n), q(n))}∞ n=1 is generated by AM of C2(p, q) and C2(p∗, q∗) ≜infp,q∈△m C2(p, q) then C2(p(n), q(n)) −C2(p∗, q∗) ≤ m X i=1 δ(i ≤l) + di βi where βi ≜log sup y q(n) i (y) q(n−1) i (y) , dj = X i w′ ij. Proof Sketch: As the 5-pp holds for all p, q ∈△m, it also holds for p = p∗and q = q∗. We use fact that E(f(Z)) ≤supz f(z) where Z is a random variable and f(·) is an arbitrary function. □ The above means that AM on C2 converges to its optimal value. We also have the following theorems that show the existence of a finite lower-bound on α such that the optimum of C1 and C2 are the same. Lemma 3.3. If C2(p, q; w′ ii = 0) is C2 when the diagonal elements of the affinity matrix are all zero then we have that min p,q∈△m C2(p, q; w′ ii = 0) ≤min p∈△m C1(p) 4 Theorem 3.4. Given any A, B, S ∈△m (i.e., A = [a1, . . . , an] , B = [b1, . . . , bn] , S = [s1, . . . , sn]) such that ai(y), bi(y), si(y) > 0, ∀i, y and A ̸= B (i.e., not all ai(y) = bi(y)) then there exists a finite α such that C2(A, B) ≥C2(S, S) = C1(S) Theorem 3.5 (Equality of Solutions of C1 and C2). Let ˆp = argmin p∈△m C1(p) and (p∗ ˜α, q∗ ˜α) = argmin p,q∈△m C2(p, q; ˜α) for an arbitrary α = ˜α > 0 where p∗ ˜α = (p∗ 1;˜α, · · · , p∗ m;˜α) and q∗ ˜α = (q∗ 1;˜α, · · · , q∗ m;˜α). Then there exists a finite ˆα such that at convergence of AM, we have that ˆp = p∗ ˆα = q∗ ˆα. Further, it is the case that if p∗ ˜α ̸= q∗ ˜α, then ˆα ≥C1(ˆp) −C2(p∗, q∗; α = 0) µ Pn i=1 DKL(p∗ i;˜α||q∗ i;˜α) . And if p∗ ˜α = q∗ ˜α then ˆα ≥˜α. 4 Parallelism and Scalability to Large Datasets One big advantage of AM on C2 over optimizing C1 directly is that it is naturally amenable to a parallel implementation, and is also amenable to further optimizations (see below) that yield a near linear speedup. Consider the update equations of Section 2. We see that one set of measures is held fixed while the other set is updated without any required communication amongst set members, so there is no write contention. This immediately yields a T ≥1-threaded implementation where the graph is evenly T-partitioned and each thread operates over only a size m/T = (l + u)/T subset of the graph nodes. We constructed a 10-NN graph using the standard TIMIT training and development sets (see section 5). The graph had 1.4 million vertices. We ran a timing test on a 16 core symmetric multiprocessor with 128GB of RAM, each core operating at 1.6GHz. We varied the number T of threads from 1 (single-threaded) up to 16, in each case running 3 iterations of AM (i.e., 3 each of p and q updates). Each experiment was repeated 10 times, and we measured the minimum CPU time over these 10 runs (total CPU time only was taken into account). The speedup for T threads is typically defined as the ratio of time taken for single thread to time taken for T threads. The solid (black) line in figure 1(a) represents the ideal case (a linear speedup), i.e., when using T threads results in a speedup of T. The pointed (green) line shows the actual speedup of the above procedure, typically less than ideal due to inter-process communication and poor shared L1 and/or L2 microprocessor cache interaction. When T ≤4, the speedup (green) is close to ideal, but for increasing T the performance diminishes away from the ideal case. Our contention is that the sub-linear speedup is due to the poor cache cognizance of the algorithm. At a given point in time, suppose thread t ∈{1, . . . , T} is operating on node it. The collective set of neighbors that are being used by these T threads are {∪T t=1N(it)} and this, along with nodes ∪T t=1{it} (and all memory for the associated measures), constitute the current working set. The working set should be made as small as possible to increase the chance it will fit in the microprocessor caches, but this becomes decreasingly likely as T increases since the working set is monotonically increasing with T. Our goal, therefore, is for the nodes that are being simultaneously operated on to have a large amount of neighbor overlap thus minimizing the working set size. Viewed as an optimization problem, we must find a partition (V1, V2, . . . , Vm/T ) of V that minimizes maxj∈{1,...,m/T } | ∪v∈Vj N(v)|. With such a partition, we may also order the subsets so that the neighbors of Vi would have maximal overlap with the neighbors of Vi+1. We then schedule the T nodes in Vj to run simultaneously, and schedule the Vj sets successively. Of course, the time to produce such a partition cannot dominate the time to run the algorithm itself. Therefore, we propose a simple fast node ordering procedure (Algorithm 1) that can be run once before the parallelization begins. The algorithm orders the nodes such that successive nodes are likely to have a high amount of neighbor overlap with each other and, by transitivity, with nearby nodes in the ordering. It does this by, given a node v, choosing another node v′ from amongst v’s neighbors’ neighbors (meaning the neighbors of v’s neighbors) that has the highest neighbor overlap. We need not search all V nodes for this, since anything other than v’s neighbors’ neighbors 5 Algorithm 1 Graph Ordering Algorithm Select an arbitrary node v. while there are any unselected nodes remaining do Let N(v) be the set of neighbors, and N 2(v) be the set of neighbors’ neighbors, of v. Select a currently unselected v′ ∈N 2(v) such that |N(v) ∩N(v′)| is maximized. If the intersection is empty, select an arbitrary unselected v′. v ←v′. end while 2 4 6 8 10 12 14 16 2 4 6 8 10 12 14 16 Number of Threads speed−up Linear Speed−Up Re−Ordered Graph Original Graph 2 4 6 8 10 12 14 16 3.5 4 4.5 5 5.5 6 6.5 Number of Threads log(CPU Time) After Re−Ordering Before Re−Ordering Figure 1: (a) speedup vs. number of threads for the TIMIT graph (see section 5). The process was run on a 128GB, 16 core machine with each core at 1.6GHz. (b) The actual CPU times in seconds on a log scale vs. number of threads for with and without ordering cases. has no overlap with the neighbors of v. Given such an ordering, the tth thread operates on nodes {t, t + m/T, t + 2m/T, . . . }. If the threads proceed synchronously (which we do not enforce) the set of nodes being processed at any time instant are {1 + jm/T, 2 + jm/T, . . . , T + jm/T}. This assignment is beneficial not only for maximizing the set of neighbors being simultaneously used, but also for successive chunks of T nodes since once a chunk of T nodes have been processed, it is likely that many of the neighbors of the next chunk of T nodes will already have been pre-fetched into the caches. With the graph represented as an adjacency list, and sets of neighbor indices sorted, our algorithm is O(mk3) in time and linear in memory since the intersection between two sorted lists may be computed in O(k) time. This is sometimes better than O(m log m) for cases where k3 < log m, true for very large m. We ordered the TIMIT graph nodes, and ran timing tests as explained above. To be fair, the time required for node ordering is charged against every run. The results are shown in figure 1(a) (pointed red line) where the results are much closer to ideal, and there are no obvious diminishing returns like in the unordered case. Running times are given in figure 1(b). Moreover, the ordered case showed better performance even for a single thread T = 1 (CPU time of 539s vs. 565s for ordered vs. unordered respectively, on 3 iterations of AM). We conclude this section by noting that (a) re-ordering may be considered a pre-processing (offline) step, (b) the SQ-Loss algorithm may also be implemented in a multi-threaded manner and this is supported by our implementation, (c) our re-ordering algorithm is general and fast and can be used for any graph-based algorithm where the iterative updates for a given node are a function of its neighbors (i.e., the updates are harmonic w.r.t. the graph [5]), and (d) while the focus here was on parallelization across different processors on a symmetric multiprocessor, this would also apply for distributed processing across a network with a shared network disk. 5 Results In this section we present results on two popular phone classification tasks. We use SQ-Loss as the competing graph-based algorithm and compare its performance against that of MP because (a) SQ-Loss has been shown to outperform its other variants, such as, label propagation [4] and the harmonic function algorithm [5], (b) SQ-Loss scales easily to very large data sets unlike approaches like spectral graph transduction [6], and (c) SQ-Loss gives similar performance as other algorithms that minimize squared error such as manifold regularization [20]. 6 0 5 10 15 20 25 30 46 48 50 52 54 56 58 60 62 64 Percentage of TIMIT Training Set Used Phone Accuracy MP MP (ν = 0) MLP SQ−Loss 0 20 40 60 80 100 34 36 38 40 42 44 46 Percentage of SWB Training Data Phone Recognition Accuracy MP SQ−Loss Figure 2: Phone accuracy on the TIMIT test set (a,left) and phone accuracy vs. amount of SWB training data (b,right). With all SWB data added, the graph has 120 million nodes. TIMIT Phone Classification: TIMIT is a corpus of read speech that includes time-aligned phonetic transcriptions. As a result, it has been popular in the speech community for evaluating supervised phone classification algorithms [26]. Here, we use it to evaluate SSL algorithms by using fractions of the standard TIMIT training set, i.e., simulating the case when only small amounts of data are labeled. We constructed a symmetrized 10-NN graph (Gtimit) over the TIMIT training and development sets (minimum graph degree is 10). The graph had about 1.4 million vertices. We used sim(xi, xj) = exp{−(xi −xj)T Σ−1(xi −xj)} where Σ is the covariance matrix computed over the entire training set. In order to obtain the features, xi, we first extracted mel-frequency cepstral coefficients (MFCC) along with deltas in the manner described in [27]. As phone classification performance is improved with context information, each xi was constructed using a 7 frame context window. We follow the standard practice of building models to classify 48 phones (|Y| = 48) and then mapping down to 39 phones for scoring [26]. We compare the performance of MP against MP with no entropy regularization (ν = 0), SQ-Loss, and a supervised state-of-the-art L2 regularized multi-layered perceptron (MLP) [10]. The hyperparameters in each case, i.e., number of hidden units and regularization weight in case of MLP, µ and ν in the case of MP and SQ-Loss, were tuned on the development set. For the MP and SQLoss, the hyper-parameters were tuned over the following sets µ ∈{1e–8, 1e–4, 0.01, 0.1} and ν ∈{1e–8, 1e–6, 1e–4, 0.01, 0.1}. We found that setting α = 1 in the case of MP ensured that p = q at convergence. As both MP and SQ-Loss are transductive, in order to measure performance on an independent test set, we induce the labels using the Nadaraya-Watson estimator (see section 6.4 in [2]) with 50 NNs using the similarity measure defined above. Figure 2(a) shows the phone classification results on the NIST Core test set (independent of the development set). We varied the number of labeled examples by sampling a fraction f of the TIMIT training set. We show results for f ∈{0.005, 0.05, 0.1, 0.25, 0.3}. In all cases, for MP and SQLoss, we use the same graph Gtimit, but the set of labeled vertices changes based on f. In all cases the MLP was trained fully-supervised. We only show results on the test set, but the results on the development set showed similar trends. It can be seen that (i) using an entropy regularizer leads to much improved results in MP, (ii) as expected, the MLP being fully-supervised, performs poorly compared to the semi-supervised approaches, and most importantly, (iii) MP significantly outperforms all other approaches. We believe that MP outperforms SQ-Loss as the loss function in the case of MP is better suited for classification. We also found that for larger values of f (e.g., at f = 1), the performances of MLP and MP did not differ significantly. But those are more representative of the supervised training scenarios which is not the focus here. Switchboard-I Phone Classification: Switchboard-I (SWB) is a collection of about 2,400 two-sided telephone conversations among 543 speakers [28]. SWB is often used for the training of large vocabulary speech recognizers. The corpus is annotated at the word-level. In addition, less reliable phone level annotations generated in an automatic manner by a speech recognizer with a non-zero error rate are also available [29]. The Switchboard Transcription Project (STP) [30] was undertaken to accurately annotate SWB at the phonetic and syllable levels. As a result of the arduous and costly nature of this transcription task, only 75 minutes (out of 320 hours) of speech segments selected from different SWB conversations were annotated at the phone level and about 150 minutes annotated at the syllable level. Having such annotations for all of SWB could be useful for speech processing in general, so this is an ideal real-world task for SSL. 7 We make use of only the phonetic labels ignoring the syllable annotations. Our goal is to phonetically annotate SWB in STP style while treating STP as labeled data, and in the process show that our aforementioned parallelism efforts scale to extremely large datasets. We extracted features xi from the conversations by first windowing them using a Hamming window of size 25ms at 100Hz. We then extracted 13 perceptual linear prediction (PLP) coefficients from these windowed features and appended both deltas and double-deltas resulting in a 39 dimensional feature vector. As with TIMIT, we are interested in phone classification and we use a 7 frame context window to generate xi, stepping successive context windows by 10ms as is standard in speech recognition. We randomly split the 75 minute phonetically annotated part of STP into three sets, one each for training, development and testing containing 70%, 10% and 20% of the data respectively (the size of the development set is considerably smaller than the size of the training set). This procedure was repeated 10 times (i.e. we generated 10 different training, development and test sets by random sampling). In each case, we trained a phone classifier using the training set, tuned the hyper-parameters on the development set and evaluated the performance on the test set. In the following, we refer to SWB that is not a part of STP as SWB-STP. We added the unlabeled SWB-STP data in stages. The percentage, s, included, 0%, 2%, 5%, 10%, 25%, 40%, 60%, and 100% of SWB-STP. We ran both MP and SQ-Loss in each case. When s =100%, there were about 120 million nodes in the graph! Due to the large size m = 120M of the dataset, it was not possible to generate the graph using the conventional brute-force search which is O(m2). Nearest neighbor search is a well-researched problem with many approximate solutions [31]. Here we make use of the Approximate Nearest Neighbor (ANN) library (see http://www.cs.umd.edu/˜mount/ANN/) [32]. It constructs a modified version of a kd-tree which is then used to query the NNs. The query process requires that one specify an error term, ϵ, and guarantees that (d(xi, N(xi))/d(xi, Nϵ(xi))) ≤1 + ϵ. where N(xi) is a function that returns the actual NN of xi while Nϵ(xi) returns the approximate NN. We constructed graphs using the STP data and s% of (unlabeled) SWB-STP data. For all the experiments here we used a symmetrized 10-NN graph and ϵ = 2.0. The labeled and unlabeled points in the graph changed based on training, development and test sets used. In each case, we ran both the MP and SQ-Loss objectives. For each set, we ran a search over µ ∈{1e–8, 1e–4, 0.01, 0.1} and ν ∈{1e–8, 1e–6, 1e–4, 0.01, 0.1} for both the approaches. The best value of the hyper-parameters were chosen based on the performance on the development set and the same value was used to measure the accuracy on the test set. The mean phone accuracy over the different test sets (and the standard deviations) are shown in figure 2(b) for the different values of s. It can be seen that MP outperforms SQ-Loss in all cases. Equally importantly, we see that the performance on the STP data improves with the addition of increasing amounts of unlabeled data. References [1] O. Chapelle, B. Scholkopf, and A. Zien, Semi-Supervised Learning. MIT Press, 2007. [2] X. Zhu, “Semi-supervised learning literature survey,” tech. rep., Computer Sciences, University of Wisconsin-Madison, 2005. [3] M. Szummer and T. Jaakkola, “Partially labeled classification with Markov random walks,” in Advances in Neural Information Processing Systems, vol. 14, 2001. [4] X. Zhu and Z. Ghahramani, “Learning from labeled and unlabeled data with label propagation,” tech. rep., Carnegie Mellon University, 2002. [5] X. Zhu, Z. Ghahramani, and J. Lafferty, “Semi-supervised learning using gaussian fields and harmonic functions,” in Proc. of the International Conference on Machine Learning (ICML), 2003. [6] T. Joachims, “Transductive learning via spectral graph partitioning,” in Proc. of the International Conference on Machine Learning (ICML), 2003. [7] A. Corduneanu and T. Jaakkola, “On information regularization,” in Uncertainty in Artificial Intelligence, 2003. [8] K. Tsuda, “Propagating distributions on a hypergraph by dual information regularization,” in Proceedings of the 22nd International Conference on Machine Learning, 2005. [9] M. Belkin, P. Niyogi, and V. Sindhwani, “On manifold regularization,” in Proc. of the Conference on Artificial Intelligence and Statistics (AISTATS), 2005. [10] C. Bishop, ed., Neural Networks for Pattern Recognition. Oxford University Press, 1995. 8 [11] A. Subramanya and J. Bilmes, “Soft-supervised text classification,” in EMNLP, 2008. [12] R. Collobert, F. Sinz, J. Weston, L. Bottou, and T. Joachims, “Large scale transductive svms,” Journal of Machine Learning Research, 2006. [13] V. Sindhwani and S. S. Keerthi, “Large scale semi-supervised linear svms,” in SIGIR ’06: Proceedings of the 29th annual international ACM SIGIR, 2006. [14] O. Delalleau, Y. Bengio, and N. L. Roux, “Efficient non-parametric function induction in semi-supervised learning,” in Proc. of the Conference on Artificial Intelligence and Statistics (AISTATS), 2005. [15] M. Karlen, J. Weston, A. Erkan, and R. Collobert, “Large scale manifold transduction,” in International Conference on Machine Learning, ICML, 2008. [16] I. W. Tsang and J. T. Kwok, “Large-scale sparsified manifold regularization,” in Advances in Neural Information Processing Systems (NIPS) 19, 2006. [17] A. Tomkins, “Keynote speech.” CIKM Workshop on Search and Social Media, 2008. [18] A. Jansen and P. Niyogi, “Semi-supervised learning of speech sounds,” in Interspeech, 2007. [19] T. M. Cover and J. A. Thomas, Elements of Information Theory. Wiley Series in Telecommunications, New York: Wiley, 1991. [20] Y. Bengio, O. Delalleau, and N. L. Roux, Semi-Supervised Learning, ch. Label Propogation and Quadratic Criterion. MIT Press, 2007. [21] Dempster, Laird, and Rubin, “Maximum likelihood from incomplete data via the em algorithm,” Journal of the Royal Statistical Society, Series B, vol. 39, no. 1, pp. 1–38, 1977. [22] T. Abatzoglou and B. O. Donnell, “Minimization by coordinate descent,” Journal of Optimization Theory and Applications, 1982. [23] W. Zangwill, Nonlinear Programming: a Unified Approach. Englewood Cliffs: N.J.: Prentice-Hall International Series in Management, 1969. [24] C. F. J. Wu, “On the convergence properties of the EM algorithm,” The Annals of Statistics, vol. 11, no. 1, pp. 95–103, 1983. [25] I. Csiszar and G. Tusnady, “Information Geometry and Alternating Minimization Procedures,” Statistics and Decisions, 1984. [26] A. K. Halberstadt and J. R. Glass, “Heterogeneous acoustic measurements for phonetic classification,” in Proc. Eurospeech ’97, (Rhodes, Greece), pp. 401–404, 1997. [27] K. F. Lee and H. Hon, “Speaker independant phone recognition using hidden markov models,” IEEE Transactions on Acoustics, Speech and Signal Processing, vol. 37, no. 11, 1989. [28] J. Godfrey, E. Holliman, and J. McDaniel, “Switchboard: Telephone speech corpus for research and development,” in Proceedings of ICASSP, vol. 1, (San Francisco, California), pp. 517–520, March 1992. [29] N. Deshmukh, A. Ganapathiraju, A. Gleeson, J. Hamaker, and J. Picone, “Resegmentation of switchboard,” in Proceedings of ICSLP, (Sydney, Australia), pp. 1543–1546, November 1998. [30] S. Greenberg, “The Switchboard transcription project,” tech. rep., The Johns Hopkins University (CLSP) Summer Research Workshop, 1995. [31] J. Friedman, J. Bentley, and R. Finkel, “An algorithm for finding best matches in logarithmic expected time,” ACM Transaction on Mathematical Software, vol. 3, 1977. [32] S. Arya and D. M. Mount, “Approximate nearest neighbor queries in fixed dimensions,” in ACM-SIAM Symp. on Discrete Algorithms (SODA), 1993. 9
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Dirichlet-Bernoulli Alignment: A Generative Model for Multi-Class Multi-Label Multi-Instance Corpora Shuang-Hong Yang College of Computing Georgia Tech shy@gatech.edu Hongyuan Zha College of Computing Georgia Tech zha@cc.gatech.edu Bao-Gang Hu NLPR & LIAMA Chinese Academy of Sciences hubg@nlpr.ia.ac.cn Abstract We propose Dirichlet-Bernoulli Alignment (DBA), a generative model for corpora in which each pattern (e.g., a document) contains a set of instances (e.g., paragraphs in the document) and belongs to multiple classes. By casting predefined classes as latent Dirichlet variables (i.e., instance level labels), and modeling the multi-label of each pattern as Bernoulli variables conditioned on the weighted empirical average of topic assignments, DBA automatically aligns the latent topics discovered from data to human-defined classes. DBA is useful for both pattern classification and instance disambiguation, which are tested on text classification and named entity disambiguation in web search queries respectively. 1 Introduction We consider multi-class, multi-label and multi-instance classification (M3C), a task of learning decision rules from corpora in which each pattern consists of multiple instances1 and is associated with multiple classes. M3C finds its application in many fields: For example, in web page classification, a web page (pattern) typically comprises of different entities (instances) (e.g., texts, pictures and videos) and is usually associated with several different topics (e.g., finance, sports and politics). In such tasks, a pattern usually consists of a set of instances, and the possible instances may be too diverse in nature (e.g., of different structures or types, described by different features) to be represented in a universal space. What makes the problem more complicated and challenging is that the pattern is usually ambiguous, i.e., it can belong to several different classes simultaneously. Traditional classification algorithms are typically incapable of handling such complications. Even for corpora consisting of relatively homogenous data, treating the tasks as M3C might still be advantageous since it enables us to explore the inner structures and the ambiguity of the data simultaneously. For example, in text classification, a document usually comprises several separate semantic parts (e.g., paragraphs), and several different topics are evolving along these parts. Since the class-labels are often only locally tied to the document (e.g., paragraphs are often far more topicfocused than the whole document), base the classification on the whole document would incur too much noise and in turn harm the performance. In addition, treating the task as M3C also offers a natural way to track the topic evolution along paragraphs, a task that is otherwise difficult to handle. M3C also arises naturally when the acquisition of labeled data is expensive. For example, in scene classification, a picture usually contains several objects (e.g., cat, desk, man) belonging to several different classes (e.g., animal, furniture, human). Ideal annotation requires a skilled expert to specify both the exact location and class label of each object in the image, which, though not completely impossible, involves too much human efforts especially for large image repositories. The annotation burden would be greatly relieved if each image is labeled as a whole (e.g., a caption indicating what is in the image), which, however, requires the learning system to be capable to handle M3C tasks. 1A “pattern” or “example” is a typical sample in a data collection and an“instance” is a part of a “pattern”. 1 Recently, the Latent Dirichlet Allocation (LDA, [4]) model has been established for automatic extraction of topical structures from large repository of documents. LDA is a highly-modularized probabilistic model with various variations and extensions (e.g., [2, 3]). By modeling a document as a mixture over topics, LDA allows each document to be associated with multiple topics with different proportions, and thus provides a promising way to capture the heterogeneity/ambiguity in the data. However, the topics discovered by LDA are implicit (i.e., each topic is expressed as a distribution over words, comprehensible interpretation of which requires human expertise), and cannot be easily aligned to the topics of human interests. In addition, the standard LDA does not model the multi-instance structure of a pattern. Hence, LDA and its like cannot be directly applied to M3C. In this paper, by taking advantage of the LDA building blocks, we present a new probabilistic generative model for multi-class, multi-label and multi-instance corpora, referred to as Dirichlet-Bernoulli Alignment (DBA). DBA assumes a tree-structure about the data, i.e., each multi-labeled pattern is a bag of single-labeled instances. In DBA, each pattern is modeled as a mixture over the set of predefined classes, an instance is then generated independently conditioned on a sampled class-label, and the label of a pattern is generated from a Bernoulli distribution conditioned on all the sampled labels used for generating its instances. DBA is essentially a topic model similar to LDA except that (1) an instance rather than a single feature is generated conditioned on each sampled topic; and (2) instead of using implicit topics for dimensionality reduction as in LDA, DBA casts each class as an explicit topic to gain discriminative power from the data. Through likelihood maximization, DBA automatically aligns the topics discovered from the data to the predefined classes of our interests. DBA can be naturally tailored to M3C tasks for both pattern classification and instance disambiguation. In this paper, we apply the DBA model to text classification tasks and an interesting real-world problem, i.e., named entity disambiguation for web search queries. The experiments confirm the usefulness of the proposed DBA model. The rest parts of this paper is organized as follows. Section 2 briefly reviews some related topics and Section 3 presents the formal description of the corpora used in M3C and the basic assumptions of our model. Section 4 introduces the detailed DBA model. In Section 5, we establish algorithms for inference and parameter estimation for DBA. And in Section 6, we apply the DBA model to text classification and query disambiguation tasks. Finally, Section 7 presents concluding remarks. 2 Related Works Traditional classification largely focuses on a single-label single-instance framework (i.e., i.i.d patterns, associated with exclusive/disjoint classes). However, the real-world is more like a web of (sub-)patterns connected with a web of classes that they belong to. Clearly, M3C reflects more of the reality. Recently, two partial solutions, i.e., multi-instance classification (MIC) [7, 11, 1] and multi-label classification (MLC) [10, 8, 5] were investigated. MIC assumes that each pattern consists of multiple instances but belongs to a single class, whereas MLC studies single-instance pattern associated with multiple classes. Although both MLC and MIC have drawn increasing attentions in the literature, neither of them can handle the cases where multi-instance and multi-label are simultaneously present. Perhaps the first work investigating M3C is [13], in which the authors proposed an indirect solution, i.e., to convert an M3C task into several MIC or MLC sub-tasks each of which is then divided into single-label and single-instance classification problems and solved by discriminative algorithms such as AdaBoost or SVM. A practical challenge of this approach is its complexity, i.e, the number of sub-tasks can be huge, making the training data extremely sparse for each subclassifier and the computation cost unacceptably high in both training and testing. Recently, Cour et al proposed a discriminative framework [6] based on convex surrogate loss minimization for classifying ambiguously labeled images; and Xu et al established a hybrid generative/discriminative approach (i.e., a heuristically regularized LDA classifier) [12] to mining named entity from web search click-through data. In this paper, we present a generative approach for M3C. Our proposed DBA model can be viewed as a supervised version of topic models. A widely used topic model for categorical data is the LDA model [4]. By modeling a pattern as a random mixture over latent topics and a topic as a Multinomial distribution over features in a dictionary, LDA is effective in discovering implicit topics from a corpus. The supervised LDA (sLDA) model [2], by linking the empirical topics to the label of each pattern, is able to learn classifiers using Generalized Linear Models. However, both LDA and sLDA are in essence dimensionality reduction techniques, and cannot be employed directly for the M3C tasks. 2 X c c x x x ... pattern class instance c (a) z θ f a y B N M L (b) Figure 1: (a): Tree structure of a multi-class multi-label multi-instance corpus. (b):A graphic representation of the DBA model with multinomial bag-of-feature instance model. 3 Problem Formalization Intuitively, we can think of a pattern as a document, an instance as a paragraph, and a feature as a word. In M3C, we are interested in inferring class labels for both the document and its paragraphs. Formally, let X ⊂RD denote the instance space (e.g., a vector space), Y = {1, 2, . . ., C} (C > 2) the set of class labels, and F = {f1, f2, . . . , fD} the dictionary of features. A multi-class, multilabel multi-instance corpus D consists of a set of input patterns {Xn}n=1,2,...,N along with the corresponding labels {Yn}n=1,2,...,N, where each pattern Xn = {xmn}m=1,2,...,Mn contains a set of instances xmn ∈X, and Yn ⊂Y consists of a set of class labels. The goal of M3C is to find a decision rule Y = ϕ(X) : 2X →2Y, where 2A denotes the power set of a set A. For simplicity, we make the following assumptions. Assumption 1 [Exchangeability]: A corpus is a bag of patterns, and each pattern is a bag of instances. Assumption 2 [Distinguishablity]: Each pattern can belong to several classes, but each instance belongs to a single class. These assumptions are equivalent to assuming a tree structure for the corpus (Figure 1(a)). 4 Dirichlet-Bernoulli Alignment In this section, we present Dirichlet-Bernoulli Alignment (DBA), a probabilistic generative model for the multi-class, multi-label and multi-instance corpus described in Section 3. In DBA, each pattern X in a corpus D is assumed to be generated by the following process: 1. Sample θ∼Dir(a). 2. For each of the M instances in X: ⊲Choose a class z ∼Mult(θ); ⊲Generate an instance x ∼p(x|z, B); 3. Generate the label y∼p(y|z1:M,λ). We assume the total number of predefined classes, C, is known and fixed. In DBA, a = [a1, . . . , aC]⊤with ac ⩾0, c = 1, . . . , C, is a C-vector prior parameter for a Dirichlet distribution Dir(a), which is defined in the (C-1)-simplex: θc ⩾0, PC c=1 θc = 1. z is a class indicator, i.e., a binary C-vector with the 1-of-C code: zc = 1 if the c-th class is chosen, and ∀i ̸= c, zi = 0. y = [y1, . . . , yC]⊤is also a binary C-vector with yc = 1 if the pattern X belongs to the c-th class and yc = 0 otherwise. In this paper, we assume the label of a pattern is generated by a cost-sensitive voting process according to the labels of the instances in it, which is intuitively reasonable. As a result, yc (c = 1, . . . , C) is generated from a Bernoulli distribution, i.e., p(yc|πc) = (πc)yc(1−πc)(1−yc), where π is a probability vector based on a weighted empirical average of the Dirichlet realization λ⊤¯z, ¯z = [¯z1, . . . , ¯zC]⊤ is the average of z1, . . . , zM: ¯zc = 1 M PM m=1 zmc. For example, π can be a Dirichlet distribution π∼Dir(λ1¯z1, . . . , λC ¯zC). In this paper, we use a logistic model: p(yc = 1|¯z, λ) = exp(λc¯zc) 1 + exp(λc¯zc). (1) 3 In practice, the set of possible instances can be quite diverse, such as pictures, texts, music and videos on a web page. Without loss of generality, we follow the convention of topic models to assume that each instance x is a bag of discrete features {f1, f2, . . . , fL} and use a multinomial distribution2: p(x|z, B) = p({f1, . . . , fL}|z, B) ∝bx1 c1bx2 c2 . . . bxD cD|zc=1, where L is the total number of feature occurrences in x (e.g., the length of a paragraph), B = [b1, . . . , bD] is a C × D-matrix with the (c, d)-th entry bcd = p(fd = 1|zc = 1) and xd is the frequency of fd in x. The joint probability is then given by: p(X, y, Z, θ|a, B, λ) = p(θ|a) M Y m=1 p(zm|θ) L Y l=1 p(fml|B, zm) ! p(y|¯z, λ). (2) The graphical model for DBA is depicted in Figure 1(b). We can see that DBA has a diagram very similar to that of sLDA (Figure 1 in [2]). The key differences are: (1) Instead of using implicit topics for dimensionality reduction as in sLDA, DBA casts the predefined classes as explicit topics to discover the discriminative properties from the data; (2) A bag-of-feature instance rather than a single feature is generated conditioned on each sampled topic (class); (3) DBA models a multi-class, multi-label multi-instance corpus and can be applied directly to M3C, i.e., the classification of each pattern as well as the instances within it. 5 Parameter Estimation and Inference Both parameter estimation and inferential tasks in DBA involve intractable computation of marginal probabilities. We use variational methods to approximate those distributions. 5.1 Variational Approximations We use the following fully-factorized variational distribution to approximate the posterior distribution of the latent variables: q(Z, θ|γ, Φ) = q(θ|γ) M Y m=1 q(zm|φm) = Γ(PC c=1 γc) QC c=1 Γ(γc) C Y c=1 θγc−1 c M Y m=1 φzmc mc ! , (3) where γ and Φ=[φ1,...,φM] are variational parameters for a pattern X. We have: log P(X, y|a, B, λ) = log Z θ X Z p(X, y, Z, θ|a, B, λ)dθ =L(γ, Φ) + KL(q(Z, θ|γ, Φ)||p(Z, θ|a, B, λ)) ≈max γ,Φ L(γ, Φ), (4) where KL(q(x)||p(x)) = R x q(x) log q(x) p(x)dx is the Kullback-Leibler (KL) divergence between two distributions p and q, and L(·) is the variational lower bound for the log-likelihood: L(γ, Φ) = log Z θ X Z q(Z, θ|γ, Φ) log p(X, y, Z, θ|a, B, λ) q(Z, θ|γ, Φ) dθ = Eq[log p(θ|a)] + M X m=1 Eq[log p(zm|θ)] + M X m=1 Eq[log p(xm|B, zm)] + Eq[log p(y|¯z, λ)] + Hq. (5) 2This is only a simple special case instance model for DBA. It is quite straightforward to substitute other instance models such as Gaussian, Poisson and other more complicated models like Gaussian mixtures. 4 The first two terms and the fifth term (the entropy of the variational distribution) in the right-hand side of Eq.(5) are identical to the corresponding terms in sLDA [2]. The third term, i.e., the variational expectation of the log likelihood for instance observations is: M X m=1 Eq[log p(xm|B, zm)] = M X m=1 C X c=1 D X d=1 φmcxmd log bcd. (6) The forth term in the righthand side of Eq.(5) corresponds to the expected log likelihood of observing the labels given the topic assignments: Eq[log p(y|¯z, λ)] = 1 M M X m=1 C X c=1 (yc −1 2)λcφmc − C X c=1 Eq[log(exp λc¯zc 2 + exp −λc¯zc 2 )]. (7) We bound the second term above by using the lower bound for logistic function [9]: −log(exp λc¯zc 2 + exp −λc¯zc 2 ) ⩾−log(1 + exp(−ξc)) −ξc 2 + ςc(λ2 c ¯z2 c −ξ2 c) ≈−log(1 + exp(−ξc)) −ξc 2 + 2ςc(λc¯zcξc −ξ2 c), (8) where ξ=[ξ1, . . . , ξC]⊤are variational parameters, ςc = 1 4ξc tanh( ξc 2 ), and the second order residue term is omitted since the lower bound is exact when ξc = −λc¯zc. Obtaining an approximate posterior distribution for the latent variables is then reduced to optimizing the objective max L(q) or min KL(q||p) with respect to the variational parameters. By using Lagrange multipliers, we can easily derive the optimal condition which can be achieved by iteratively updating the variational parameters according to the following formulas: φmc ∝ D Y d=1 (bcd)xmd exp Ψ(γc) + λc 2M [2yc −1 + tanh(ξc 2 )] , γc = ac + M X m=1 φmc, ξc = −λc 1 M M X m=1 φmc, (9) where Ψ(·) is the digamma function. Note that instead of only one feature contributing to φmc as in LDA, all the features appearing in an instance are now responsible for contributing. This property tends to make DBA more robust to data sparsity. Also, DBA makes use of the supervision information with a term PC c=1 λc¯zc(2yc −1) in the variational likelihood bound L. As L is optimized, this term is equivalent to maximizing the likelihood of sampling the classes to which the pattern belongs: {max λc PM m=1 zmc, if yc = 1} and simultaneously minimizing the likelihood of sampling the classes to which the pattern does not belong: {min λc PM m=1 zmc, if yc = 0}. Here λc (-λc) acts like a utility (cost) of assigning X to the c-th class. As a result, it tends to align the Dirichlet topics discovered from the data to the class labels (Bernoulli observations) y. This is why we coin the name Dirichlet-Bernoulli Alignment. 5.2 Parameter Estimation The maximum likelihood parameter estimation of DBA relies on the variational approximation procedure. Given a corpus D = {(Xn, yn)}n=1,...,N, the MLE can be formulated as: a∗, B∗, λ∗= arg max log P(D|a, B, λ) = arg max a,B,λ N X n=1 max γn,Φn L(γn, Φn|a, B, λ). (10) 5 Table 1: Characteristic of the data sets. Data Set #Train #Test D C |Y |avg #(|Y | > 1) Mavg Mmin Mmax Text 1200 679 500 10 1.4 721 (38.4%) 8.2 1 36 Query 300 100 2000 101 1.4 99 (24.8%) 65 3 731 acq corn crude earn grain interest money ship trade wheat overall 70 80 90 100 DBA MNB MIMLSVM MIMLBoost Figure 2: Accuracies(%) of DBA, MNB, MIMLSVM, and MIMLBoost for text classification. The two-layer optimization in Eq.(10) involves two groups of parameters corresponding to the DBA model and its variational approximation, respectively. Optimizing alternatively between these two groups leads to a Variational Expectation Maximization (VEM) algorithm similar to the one used in LDA, where the E-step corresponds to the variational approximation for each pattern in the corpus. And the M-step in turn maximizes the objective in Eq.(6) w.r.t. the model parameters. These two steps are repeated alternatively until convergence. 5.3 Inference DBA involves three types of inferential tasks. The first task is to infer the latent variables for a given pattern, which is straightforward after the variational approximation. The second task, pattern classification, addresses prediction of labels for a new pattern X: p(yc = 1|X; a, B, λ) ≈ exp(λc ¯φc)/(1 + exp(λc ¯φc)), where ¯φc = 1 M PM m=1 φmc and the term λc 2M [2yc −1 + tanh( ξc 2 )] is removed when updating φ in Eq.(9). The third task, instance disambiguation, finds labels for each instances within a pattern: p(zm|X, y) = R θ p(zm, θ|X, y)dθ ≈q(zm|φm), that is, p(zmc = 1|X, y) = φmc. 6 Experiments In this section, we conduct extensive experiments to test the DBA model as it is applied to pattern classification and instance disambiguation respectively. We first apply DBA to text classification and compare its performance with state-of-the-art M3C algorithms. Then the instance disambiguation performance of DBA is tested on a novel real-world task, i.e., named entity disambiguation for web search queries. Table 1 shows the information of the data sets used in our experiments. 6.1 Text Classification This experiment is conducted on the ModApte split of the Reuters-21578 text collection, which contains 10788 documents belonging to the most popular 10 classes. We use the top 500 words with the highest document frequency as features, and represent each document as a pattern with each of its paragraphs being an instance in order to exploit the semantic structure of documents explicitly. After eliminating the documents that have empty label set or less than 20 features, we obtain a subset of 1879 documents, among which 721 documents (about 38.4%) have multiple labels. The average number of labels per document is 1.4±0.6 and the average number of instances (paragraphs) per pattern (document) is 8.2±4.8. The data set is further randomly partitioned into a subset of 1200 documents for training and the rest for testing. For comparison, we also test two state-of-the-art M3C algorithms, the MIMLSVM and MIMLBoost [13], and use the Multinomial Na¨ıve Bayes (MNB) classifier trained on the vector space model of the whole documents as the baseline. For a fair comparison, linear kernel is used in both MIMLSVM and MIMLBoost and all the hyper-parameters are tuned by 5-fold cross validation prior to training. We use the Hamming-Accuracy [13] to evaluate the results, for DBA and MNB, the label is estimated by: y = δ(p(y = 1|X) ⩾t), where the cut-off probability threshold is also selected based on 5-fold cross validation. Each experiment is repeated for 5 random runs and the average results are reported by a bar chart as depicted in Figure 2. We can see that: (1) for most classes, the three 6 Table 2: Accuracy@N (N = 1, 2, 3) and micro-averaged and macro-averaged F-measures of DBA, MNB and SVM based disambiguation methods. Method A@1 Gain A@2 Gain A@3 Gain Fmicro Gain Fmacro Gain MNB-TF 0.4154 30.4% 0.4913 25.7% 0.5168 25.4% 0.4154 30.4% 0.3144 47.0% MNB-TF-IDF 0.4177 29.6% 0.4918 25.6% 0.5176 25.2% 0.4177 29.6% 0.2988 54.7% SVM-TF 0.4927 9.9% NA NA 0.4927 9.9% 0.3720 24.2% SVM-TF-IDF 0.4912 10.2% NA NA 0.4912 10.2% 0.3670 25.0% DBA 0.5415 0.6175 0.6482 0.5415 0.4622 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 class precision DBA MNBTF MNBTFIDF SVMTF SVMTFIDF 0 20 40 60 80 100 0 0.2 0.4 0.6 0.8 1 class recall DBA MNBTF MNBTFIDF SVMTF SVMTFIDF Figure 3: Precision and Recall scores for each of 101 classes by using DBA, MNB and SVM based methods. M3C algorithms outperform the MNB baseline; (2) the performance of DBA is at least comparable with MIMLBoost and MIMLSVM. For most classes and overall, DBA performs the best, whereas for some classes, MIMLBoost and MIMLSVM perform even slightly worse than MNB. A possible reason might be: if the documents are very short, splitting them might introduce severe data sparseness and in turn harms the performance. We also observe that DBA is much more efficient than MIMLBoost and MIMLSVM. For training, DBA takes 42 mins on average, in contrast to 557 minutes (MIMLSVM) and 806 minutes (MIMLBoost). 6.2 Named Entity Disambiguation Query ambiguity is a fundamental obstacle for search engine to capture users’ search intentions. In this section, we employ DBA to disambiguate the named entities in web search queries. This is a very challenging problem because queries are usually very short (2 to 3 words on average), noisy (e.g., misspellings, abbreviations, less grammatical structure) and topic-distracted. A single namedentity query Q can be viewed as a combination of a single named entity e and a set of context words w (the remaining text in Q). By differentiating the possible meanings of the named entity in a query and identifying the most possible one, entity disambiguation can help search engines to capture the precise information need of the user and in turn improve search by responding with the truly most relevant documents. For example, when a user inputs “When are the casting calls for Harry Potter in USA?”, the system should be able to identify that the ambiguous named entity “Harry Potter” (i.e., it can be a movie, a book or a game) really refers to a movie in this specific query. We treat the ambiguity of e as a hidden class z over e and make use of the query log as a data source for mining the relationship among e, w and z. In particular, the query log can be viewed as a multi-class, multi-label and multi-instance corpus {(Xn, Yn)}n=1,2,...,N, in which each pattern X corresponds to a named-entity e and is characterized by a set of instances {xm}m=1,2,...,M corresponding to all the contexts {wm}m=1,2,...,M that co-occur with e in queries, and the label Y contains all the ambiguities of e. Our data was based on a snapshot of answers.yahoo.com crawled in early 2008, containing 216563 queries from 101 classes. We manually collect 400 named entities and label them according to the labels of their co-occurring queries in Yahoo! CQA. A randomly chosen subset of 300 entities are used as training data and the other 100 are used for testing. We compare our DBA based method with baselines including Multinomial Na¨ıve Bayes classifier using TF (MNB-TF) or TF-IDF (MNBTFIDF) as word attributes, and SVM classifier using TF (SVM-TF) or TFIDF (SVM-TF-IDF). For SVM, a similar scheme as MIMLSVM is used for learning M3C classifiers. Table 2 demonstrates the Accuracy@N (N = 1, 2, 3) as well as micro-averaged and macro-average F-measure scores of each disambiguation approach3. All the results are obtained through 5-fold cross-validation. From the table, we observe that DBA achieves significantly better performance than all the other methods. In particular, for Accuracy@1 scores, DBA can achieve a gain of about 3Since SVM only outputs hard class assignments, there is no Accuracy@2,3 for SVM based methods. 7 30% relative to two MNB methods, and about 10% relative to two SVM methods; for macro-average F-measures, DBA can achieve a gain of about 50% over MNB methods, and about 25% over SVM methods. As a reference, in Figure 3, we also illustrate the sorted precision and recall scores for each of the 101 classes. We can see that, DBA slightly outperforms the baselines in terms of precision, and significantly performs better in terms of the recall scores. In particular, for recall, DBA can achieve a gain of more than 50% relative to MNB and SVM baselines. 7 Concluding Remarks Multi-class, multi-label and multi-instance classification (M3C) is encounteredin many applications. Even for task that is not explicitly an M3C problem, it might still be advantageous to treat it as M3C so as to better explore its inner structures and effectively handle the ambiguities. M3C also naturally arises from the difficulty of acquiring finely-labeled data. In this paper, we have proposed a probabilistic generative model for M3C corpora. The proposed DBA model is useful for both pattern classification and instance disambiguation, as has been tested respectively in text classification and named-entity disambiguation tasks. An interesting observation in practice is that, although there might be a large number of classes/topics, usually a pattern is only associated with a very limited number of them. In our experiment, we found that substantial improvement could be achieved by simply enforcing label sparsity, e.g., by using LASSO style regularization. In future, we will investigate such “Label Parsimoniousness” in a principled way. Another meaningful investigation would be to explicitly capture or explore the class correlations by using, for example, the Logistic Normal distribution [3] rather than Dirichlet. Acknowledgments Hongyuan Zha is supported by NSF #DMS-0736328 and grant from Microsoft. Bao-Gang Hu is supported by NSFC #60275025 and the MOST of China grant #2007DFC10740. References [1] Andrews S. and Hofmann T. (2003) Multiple Instance Learning via Disjunctive Programming Boosting, In Advances in Neural Information Processing Systems 17 (NIPS’03), MIT Press. [2] Blei D. and McAuliffe J. (2007) Supervised topic models. In Advances in Neural Information Processing Systems 21 (NIPS’07), MIT Press. [3] Blei D. and Lafferty J. (2007) A correlated topic model of Science. Annals of Applied Statistics. Vol. 1, No. 1, pp. 17–35, 2007. [4] Blei D., Ng A. and Jordan M. (2003) Latent Dirichlet Allocation. Journal of Machine Learning Research, Vol. 3, pp.993–1022, Jan. 2003, MIT Press. [5] Boutell M. R., Luo J., Shen X. and Brown C. M. (2004) Learning Multi-Label Scene Classification. Pattern Recognition, 37(9), pp.1757–1771, 2004. [6] Cour T., Sapp B., Jordan C. and Taskar B. (2009) Learning from Ambiguously Labeled Images, In the 23rd IEEE Conference on Computer Vision and Pattern Recognition (CVPR’09). [7] Dietterich T. G., Lathrop R. H., Lozano-Perez T. (1997) Solving the Multiple-Instance Problem with Axis-Parallel Rectangles. Artificial Intelligence Journal, Vol. 89, pp.31–71, Jan.1997. [8] Ghamrawi N. and McCallum A. (2005) Collective Multi-Label Classification, In ACM International Conference On Information And Knowledge Management (CIKM’05), pp.195–200. [9] Jaakkola, T. and Jordan M. I. (2000). Bayesian parameter estimation via variational methods. Statistics and Computing, Vol 10, Issue 1, pp. 25–37. [10] Ueda N. and Saito K. (2002) Parametric Mixture Models For Multi-Labeled Text. In Advances in Neural Information Processing Systems 15 (NIPS’02). [11] Viola P., Platt J. and Zhang C. (2006). Multiple Instance Boosting For Object Detection. In Advances in Neural Information Processing Systems 20 (NIPS’06), pp.1419–1426, MIT Press. [12] Xu G., Yang S.-H. and Li H. (2009) Named Entity Mining from Click-Through Data Using Weakly Supervised LDA, In ACM Knowledge Discovery and Data Mining (KDD’09). [13] Zhou Z.-H. and Zhang M.-L. (2006) Multi-Instance Multi-Label Learning with Application to Scene Classification, In Advances in Neural Information Processing Systems 20 (NIPS’06). 8
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Adaptive Design Optimization in Experiments with People Daniel R. Cavagnaro Department of Psychology Ohio State University cavagnaro.2@osu.edu Mark A. Pitt Department of Psychology Ohio State University pitt.2@osu.edu Jay I. Myung Department of Psychology Ohio State University myung.1@osu.edu Abstract In cognitive science, empirical data collected from participants are the arbiters in model selection. Model discrimination thus depends on designing maximally informative experiments. It has been shown that adaptive design optimization (ADO) allows one to discriminate models as efficiently as possible in simulation experiments. In this paper we use ADO in a series of experiments with people to discriminate the Power, Exponential, and Hyperbolic models of memory retention, which has been a long-standing problem in cognitive science, providing an ideal setting in which to test the application of ADO for addressing questions about human cognition. Using an optimality criterion based on mutual information, ADO is able to find designs that are maximally likely to increase our certainty about the true model upon observation of the experiment outcomes. Results demonstrate the usefulness of ADO and also reveal some challenges in its implementation. 1 Introduction For better or worse, human memory is not perfect, causing us to forget. Over a century of research on memory has consistently shown that a person’s ability to remember information just learned (e.g., from studying a list of words), drops precipitously for a short time immediately after learning, but then quickly decelerates, leveling off to a very low rate as more and more time elapses. The simplicity of this data pattern has led to the introduction of a number of models to describe the rate at which information is retained in memory. Years of experimentation with humans (and animals) have resulted in a handful of models proving to be superior to the rest of the field, but also proving to be increasingly difficult to discriminate [1, 2]. Three strong competitors are the power model (POW), the exponential model (EXP), and the hyperbolic model (HYP). Their equations are given in Table 1. Despite the best efforts of researchers to design studies that were intended to discriminate among them, the results have not yielded decisive evidence that favors one model, let alone consistency across studies. [2, 3]. In these and other studies, well-established methods were used to increase the power of an experiment, and thus improve model discriminability. They included testing large numbers of participants to reduce measurement error, testing memory at more retention intervals (i.e., the time between the end of the study phase and when memory is probed) after the study phase (e.g., 8 instead of 5) so as to obtain a more accurate description of the rate of retention, and replicating the experiment using a range of different tasks or participant populations. 1 Model Equation Power (POW) p = a(t + 1)−b Exponential (EXP) p = ae−bt Hyperbolic (HYP) p = a 1+bt Table 1: Three quantitative models of memory retention. In each equation, the symbol p (0 < p < 1) denotes the predicted probability of correct recall as a function of time interval t with model parameters a and b. In the present study, we used Bayesian adaptive design optimization (ADO) [4, 5, 6, 7] on groups of people to achieve the same goal. Specifically, a retention experiment was repeated four times on groups of people, and the set of retention intervals at which memory was probed was optimized for each repetition using data collected in prior repetitions. The models in Table 1 were compared. Because model predictions can differ significantly across retention intervals, our intent was to exploit this information to the fullest using ADO, with the aim of providing some clarity on the form of the retention function in humans. While previous studies have demonstrated the potential of ADO to discriminate retention functions in computer simulations [4, 5], this is the first study to utilize the methodology in experiments with people. Although seemingly trivial, the application of such a methodology comes with challenges that can severely restrict its usefulness. Success in applying ADO to a relatively simple design is a necessary first step in assessing its ability to aid in model discrimination and its broader applicability. We begin by reviewing ADO. This is followed by a series of retention experiments using the algorithm. We conclude with discussions of the implications of the empirical findings and of the benefits and challenges of using ADO in laboratory experiments. 2 Adaptive optimal design 2.1 Bayesian framework Before data collection can even begin in an experiment, many choices about its design must be made. In particular, design parameters such as the sample size, the number of treatments (i.e., conditions or levels of the independent variable) to study, and the proportion of observations to be allocated to each treatment group must be chosen. These choices impact not only the statistical value of the results, but also the cost of the experiment. For example, basic statistics tells us that increasing the sample size would increase the statistical power of the experiment, but it would also increase its cost (e.g., number of participants, amount of testing). An optimal experimental design is one that maximizes the informativeness of the experiment, while being cost effective for the experimenter. A principled approach to the problem of finding optimal experimental designs can be found in the framework of Bayesian decision theory [8]. In this framework, each potential design is treated as a gamble whose payoff is determined by the outcome of an experiment carried out with that design. The idea is to estimate the utilities of hypothetical experiments carried out with each design, so that an “expected utility” of each design can be computed. This is done by considering every possible observation that could be obtained from an experiment with each design, and then evaluating the relative likelihoods and statistical values of these observations. The design with the highest expected utility value is then chosen as the optimal design. In the case of adaptively designed experiments, in which testing proceeds over the course of several stages (i.e., periods of data collection), the information gained from all prior stages can be used to improve the design at the current stage. Thus, the problem to be solved in adaptive design optimization (ADO) is to identify the most informative design at each stage of the experiment, taking into account the results of all previous stages, so that one can infer the underlying model and its parameter values in as few steps as possible. 2 Formally, ADO for model discrimination entails finding an optimal design at each stage that maximizes a utility function U(d) d∗= argmax d {U(d)} (1) with the utility function defined as U(d) = K X m=1 p(m) Z Z u(d, θm, y) p(y|θm, d) p(θm) dy dθm, (2) where m = {1, 2, . . . , K} is one of a set of K models being considered, d is a design, y is the outcome of an experiment with design d under model m, and θm is a parameterization of model m. We refer to the function u(d, θm, y) in Equation (2) as the local utility of the design d. It measures the utility of a hypothetical experiment carried out with design d when the data generating model is m, the parameters of the model takes the value θm, and the outcome y is observed. Thus, U(d) represents the expected value of the local utility function, where the expectation is taken over (1) all models under consideration, (2) the full parameter space of each model, and (3) all possible observations given a particular model-parameter pair, with respect to the model prior probability p(m), the parameter prior distribution p(θm), and the sampling distribution p(y|θm, d), respectively. 2.2 Mutual information utility function Selection of a utility function that adequately captures the goals of the experiment is an integral, often crucial, part of design optimization. For the goal of discriminating among competing models, one reasonable choice would be a utility function based on a statistical model selection criterion, such as sum-of-squares error (SSE) or minimum description length (MDL) [MDL 9] as shown by [10]. Another reasonable choice would be a utility function based on the expected Bayes factor between pairs of competing models [11]. Both of these approaches rely on pairwise model comparisons, which can be problematic when there are more than two models under consideration. Here, we use an information theoretic utility function based on mutual information [12]. It is an ideal measure for quantifying the value of an experiment design because it quantifies the reduction in uncertainty about one variable that is provided by knowledge of the value of another random variable. Formally, the mutual information of a pair of random variables P and Q, taking values in X, is given by I(P; Q) = H(P) −H(P|Q) (3) where H(P) = −P x∈X p(x) log p(x) is the entropy of P, and H(P|Q) = P x∈X p(x)H(P|Q = x) is the conditional entropy of P given Q. A high mutual information indicates a large reduction in uncertainty about P due to knowledge of Q. For example, if the distributions of P and Q were perfectly correlated, meaning that knowledge of Q allowed perfect prediction of P, then the conditional distribution would be degenerate, having entropy zero. Thus, the mutual information of P and Q would be H(P), meaning that all of the entropy of P was eliminated through knowledge of Q. Mutual information is symmetric in the sense that I(P; Q) = I(Q; P). Mutual information can be implemented as an optimality criterion in ADO for model discrimination of each stage s (= 1, 2, . . .) of experimentation in the following way. (For simplicity, we omit the subscript s in the equations below.) Let M be a random variable defined over a model set {1, 2, . . . , K}, representing uncertainty about the true model, and let Y be a random variable denoting an experiment outcome. Hence Prob.(M = m) = p(m) is the prior probability of model m, and Prob.(Y = y|d) = PK m=1 p(y|d, m) p(m), where p(y|d, m) = R p(y|θm, d)p(θm) dθm, is the associated prior over experimental outcomes given design d. Then I(M; Y |d) = H(M)−H(M|Y, d) measures the decrease in uncertainty about which model drives the process under investigation given the outcome of an experiment with design d. Since H(M) is independent of the design d, maximizing I(M; Y |d) on each stage of ADO is equivalent to minimizing H(M|Y, d), which is the expected posterior entropy of M given d. Implementing this ADO criterion requires identification of an appropriate local utility function u(d, θm, y) in Equation (2); specifically, a function whose expectation over models, parameters, and observations is I(M; Y |d). Such a function can be found by writing I(M; Y |d) = K X m=1 p(m) Z Z p(y|θm, d) p(θm) log p(m|y, d) p(m) dy dθm (4) 3 from whence it follows that setting u(d, θm, y) = log p(m|y,d) p(m) yields U(d) = I(M; Y |d). Thus, the local utility of a design for a given model and experiment outcome is the log ratio of the posterior probability to the prior probability of that model. Put another way, the above utility function prescribes that a design that increases our certainty about the model upon the observation of an outcome is more valued than a design that does not. A highly desirable property of this utility function is that it is suitable for comparing more than two models, because it does not rely on pairwise comparisons of the models under consideration. Further, as noted by [5], it can be seen as a natural extension of the Bayes factor for comparing more than two models. To see this, notice that the local utility function can be rewritten, applying Bayes rule, as u(d, θm, y) = −log PK k=1 p(k) p(y|k) p(y|m), 2.3 Computational methods Finding optimal designs for discriminating nonlinear models, such as POW, EXP and HYP, is a nontrivial task, as the computation requires simultaneous optimization and high-dimensional integration. For a solution, we draw on a recent breakthrough in stochastic optimization [13]. The basic idea is to recast the problem as a probability density simulation in which the optimal design corresponds to the mode of the distribution. This allows one to find the optimal design without having to evaluate the integration and optimization directly. The density is simulated by Markov Chain Monte-Carlo [14], and the mode is sought by gradually ”sharpening” the distribution with a simulated annealing procedure [15]. Details of the algorithm can be found in [10, 16]. The model and parameter priors are updated at each stage s = {1, 2, . . .} of experimentation. Upon the specific outcome zs observed at stage s of an actual experiment carried out with design ds, the model and parameter priors to be used to find an optimal design at the next stage are updated via Bayes rule and Bayes factor calculation [e.g., 17] as ps+1(θm) = p(zs|θm, ds) ps(θm) R p(zs|θm, ds) ps(θm) dθm (5) ps+1(m) = p0(m) PK k=1 p0(k) BF(k,m)(zs)ps(θ) (6) where BF(k,m)(zs)ps(θ) denotes the Bayes factor defined as the ratio of the marginal likelihood of model k to that of model m given the realized outcome zs, where the marginal likelihoods are computed with the updated priors from the preceding stage. The above updating scheme is applied successively at each stage of experimentation, after an initialization with equal model priors p(s=0)(m) = 1/K and a parameter prior p(s=0)(θm). 3 Discriminating retention models using ADO Retention experiments with people were performed using ADO to discriminate the three retention models in Table 1. The number of retention intervals was fixed at three, and ADO was used to optimize the experiment with respect to the selection of the specific retention intervals. The methodology paralleled very closely that of Experiment 1 from [3, 18]. Details of the implementation are described next. 3.1 Experiment methodology A variant of the Brown-Peterson task [19, 20] was used. In each trial, a target list of six words was randomly drawn from a pool of high frequency, monosyllabic nouns. These words were presented on a computer screen at a rate of two words per second, and served as the material that participants (undergraduates) had to remember. Five seconds of rehearsal followed, after which the target list was hidden and distractor words were presented, one at a time at a rate of one word per second, for the duration of the retention interval. Participants had to say each distractor word out loud as it appeared on the computer screen. The purpose of the distractor task was to occupy the participant’s verbal memory in order to prevent additional rehearsal of the target list during the retention interval. The distractor words were drawn from a separate pool of 2000 monosyllabic nouns, verbs, and 4 0 10 20 30 40 0. 2 0. 4 0. 6 0. 8 1 Proportion correct Experiment 1 Retention interval 0 10 20 30 40 0. 2 0. 4 0. 6 0. 8 1 Experiment 2 Retention interval 0 10 20 30 40 0. 2 0. 4 0. 6 0. 8 1 Retention interval Experiment 4 0 10 20 30 40 0. 2 0. 4 0. 6 0. 8 1 Retention interval Experiment 3 POW EXP HYP POW EXP HYP POW EXP HYP POW EXP HYP Figure 1: Best fits of POW, EXP, and HYP at the conclusion of each experiment. Each data point represents the observed proportion of correct responses out of 54 trials from one participant. The level of noise is consistent with the assumption of binomial error. The clustering of retention intervals around the regions where the best fitting models are visually discernable hints at the tendency for ADO to favor points at which the predictions of the models are most distinct. adjectives. At the conclusion of the retention interval, participants were given up to 60 seconds for free recall of the words (typed responses) from the target list. A word was counted as being remembered only if it was typed correctly. We used a method of moments [e.g. 21] to construct informative prior distributions for the model parameters. Independent Beta distributions were constructed to match the mean and variance of the best fitting parameters for the individual participant data from Experiment 1 from [3, 18]. We conducted four replications of the experiment to assess consistency across participants. Each experiment was carried out across five ADO stages using a different participant at each stage (20 participants total). At the first stage of an experiment, an optimal set of three retention intervals, each between 1 and 40 seconds, was computed using the ADO algorithm based on the priors at that stage. There were nine trials at each time interval per stage, yielding 54 Bernoulli observations at each of the three retention intervals. At the end of a stage, priors were updated before beginning the next stage. For example, the prior for stage 2 of experiment 1 was obtained by updating the prior for stage 1 of experiment 1 based on the results obtained in stage 1 in experiment 1. There was no sharing of information between experiments. 3.2 Results and analysis Before presenting the Bayesian analysis, we begin with a brief preliminary analysis in order to highlight a few points about the quality of the data. Figure 1 depicts the raw data from each of the four experiments, along with the best fitting parameterization of each model. These graphs reveal two important points. First, the noise level in the measure of memory (number of correct responses) is high, but not inconsistent with the assumption of binomial variance. Moreover, the variation does not excede that in [3], the data from which our prior distributions were constructed. Second, the retention intervals chosen by ADO are spread across their full range (1 to 40 seconds), but they are especially clustered around the regions where the best fitting models are most discernable visually (e.g., 5-15, 35-40). This hints at the tendency for ADO to favor retention intervals at which the models are most distinct given current beliefs about their parameterizations. A simple comparison of the fits of each model does not reveal a clear-cut winner. The fits are bad and often similar across experiments. This is not surprising since such an analysis does not take into account the noise in the models, nor does it take into account the complexity of the models. Both are addressed in the following, Bayesian analysis. When comparing three or more models, it can be useful to consider the probability of each model m relative to all other models under consideration, given the data y [22, 23]. Formally, this is given by p(m) = p(m|y) PK k=1 p(k|y) (7) 5 POW EXP HYP Experiment 1 0.093 0.525 0.382 Experiment 2 0.886 0.029 0.085 Experiment 3 0.151 0.343 0.507 Experiment 4 0.996 0.001 0.003 Table 2: Relative posterior probabilities of each model at the conclusion of each experiment. Experiments 2 and 4 provide strong evidence in favor of POW, while experiments 1 and 3 are inconclusive, neither favoring, nor ruling out, any model. which is simply a reformulation of Equation (6). Table 2 lists these relative posterior probabilities for each of the three models at the conclusion of each of the four experiments. Scanning across the table, two patterns are visible in the data. In Experiments 2 and 4, the data clearly favor the power model. The posterior probabilities of the power model (0.886 and 0.992, respectively) greatly exceed those for the other two models. Using the Bayes factor as a measure of support for a model, comparisons of POW over EXP and POW over HYP yield values of 30.6 and 10.4. This can be interpreted as strong evidence for POW as the correct model according to the scale given by Jeffreys (1961). Conclusions from Experiment 4 are even stronger. With Bayes factors of 336 for POW over EXP and 992 for POW over HYP, the evidence is decisively in support of the power model. The results in the other two experiments are equivocal. In contrast to Experiments 2 and 4, POW has the lowest posterior probability in both Experiments 1 and 3 (0.093 and 0.151, respectively). EXP has the highest probability in Experiment 1 (0.525), and HYP has the highest in Experiment 3 (0.507). When Bayes Factors are computed between models, not only is there is no decisive winner, but the evidence is not strong enough to rule out any model. For example, in Experiment 1, EXP over POW, the largest difference in posterior probability, yields a Bayes Factor of only 5.6. The corresponding comparison in Experiment 3, HYP over POW, yields a value of 3.3. Inspection of the model predictions at consecutive stages of an experiment provides insight into the workings of the ADO algorithm, and provides visual confirmation that the algorithm chooses time points that are intended to be maximally discriminating. Figure 2 contains the predictions of each of the three models for the first two stages of Experiments 2 and 3. The columns of density plots corresponding to stage 1 show the predictions for each model based on the prior parameter distributions. Based on these predictions, the ADO algorithm finds an optimal set of retention intervals to be 1 second, 7 seconds, and 12 seconds. It is easy to see that POW predicts a much steeper decline in retention for these three retention intervals than do EXP and HYP. Upon observing the number of correct responses at each of those intervals in stage 1 (depicted by the blue dots in the graphs), the algorithm computes the posterior likelihood of each model. In experiment 2, for example, the observed numbers of correct responses for that participant lie in regions that are much more likely under POW than under EXP or HYP, hence the posterior probability of POW is increased from 0.333 to 0.584 after stage 1, whereas the posteriors for EXP and HYP are decreased to 0.107 and 0.309, respectively. The data from stage 1 of experiment 3 similarly favor POW. At the start of stage 2, the parameter priors are updated based on the results from stage 1, hence the ranges of likely outcomes for each model are much narrower than they were in stage 1, and concentrated around the results from stage 1. Based on these updated parameter priors, the ADO algorithm finds 1 second, 11 seconds, and 36 seconds to be an optimal set of retention intervals to test in stage 2 of Experiment 2, and 1 second, 9 seconds, and 35 seconds to be an optimal set of retention intervals to test in stage 2 of Experiment 3. The difference between these two designs reflects the difference between the updated beliefs about the models, which can be seen by comparing the stage2 density plots for the respective experiments in Figure 2. As hoped for with ADO, testing in stage 2 produced results that begin to discriminate the models. What is somewhat surprising, however, is that the results favor different models, with POW having the highest probability (0.911) in Experiment 2, and HYP (0.566) in Experiment 3. The reason for this is the very different patterns of data produced by the participants in the two experiments. The participant in Experiment 2 remembered more words overall than the participant in Experiment 3, especially at the longest retention interval. These two factors together combine to yield very different posterior probabilities across models. 6 0 9 18 27 36 45 54 Stage 1 (POW ) Correct responses 0 9 18 27 36 45 54 (EXP) Correct responses 0 10 20 30 40 0 9 18 27 36 45 54 Retention interval (HYP) Correct responses Stage 2 0 10 20 30 40 Retention interval 0 9 18 27 36 45 54 (POW ) Correct responses Stage 1 0 9 18 27 36 45 54 (EXP) Correct responses 0 10 20 30 40 0 9 18 27 36 45 54 Retention interval (HYP) Correct responses Stage 2 0 10 20 30 40 Retention interval Experiment 2 Experiment 3 p(POW)=.098 p(POW)=.543 p(HYP)=.284 p(HYP)=.566 p(POW)=.911 p(POW)=.584 p(EXP)=.107 p(EXP)=.010 p(HYP)=.309 p(HYP)=.079 p(EXP)=.336 p(EXP)=.173 ) ) Figure 2: Predictions of POW, EXP and HYP based on the prior parameter distributions in the first two stages of Experiments 2 and 3. Darker colors indicate higher probabilities. Light blue dots mark the observations at the given stage, and dark blue dots mark observations from previous stages. Relative posterior model probabilities based on all observations up to the current stage are given in the lower left corner of each plot. 4 Discussion The results of the current study demonstrate that ADO can work as advertised. Over a series of testing stages, the algorithm updated the experiment’s design (with new retention intervals) on the basis of participant data to determine the form of the retention function, yielding final posterior probabilities in Experiments 2 and 4 that unambiguously favor the power model. Like Wixted and Ebbesen (1991), these results champion the power model, and they do so much more definitively than any experiment that we know of. The failure to replicate these results in Experiments 1 and 3 tempers such strong conclusions about the superiority of the power model and can raise doubts about the usefulness of ADO. The data in Figure 2 (also Figure 1) hint at a likely reason for the observed inconsistencies: participant variability. In Figure 2, the variability in performance at stage 2 of Experiments 2 and 3 is very large, near the upper limit of what one would expect from binomial noise. If the variability in the data were to exceed the variability predicted by the models, then the more extreme data points could be incorrectly interpreted as evidence in favor of the wrong model, rather than being attributed to the intrinsic noise in the true model. Moreover, even when the noise is taken into account accurately, ADO does not guarantee that an experiment will generate data that discriminates the models; it merely sets up ideal conditions for that to occur. It is up to the participants to provide discriminating data points. The inconsistencies across experiments reveal one of the challenges of using ADO. It is designed to be highly sensitive to participant performance, and this sensitivity can also be a weakness under certain conditions. If the variability noted above is uninteresting noise, then by testing the same participant at each stage (a within-subject design), we should be able to reduce the problem. On the other hand, the inconclusiveness of the data in Experiments 1 and 3 may point to a more interesting 7 possibility: a minority of participants may retain information at a rate that is best described by an exponential or hyperbolic function. Such individual differences would be identifiable with the use of a within-subject design. As with any search-based methodology, the application of ADO requires a number of decisions to be made. Although there are too many to cover here, we conclude the paper by touching on the most important ones. When running an experiment with ADO, any model that is expected to be a serious competitor should be included in the analysis from the start of experimentation. In the present study, we considered three retention functions with strong theoretical motivations, which have outperformed others in previous experiments [2, 3]. The current methodology does not preclude considering a larger set of models (the only practical limitations are computing time and the patience of the experimenter). However, once that set of models is decided, the designs chosen by ADO are optimal for discriminating those –and only those– models. Thus, the designs that we found and the data we have collected in these experiments are not necessarily optimal for discriminating between, say, a power model and a logarithmic model. Therefore, ADO is best used as a tool for confirmatory rather than exploratory analyses. That is, it is best suited for situations in which the field of potential models can be narrowed to a few of the strongest competitors. Another important choice to be made before using ADO is which prior distributions to use. Using informative priors is very helpful but not necessarily essential to implementing ADO. Since the parameter distributions are updated sequentially, the data will quickly trump all but the most pathological prior distributions. Therefore, using a different prior distribution should not affect the conclusions of the sequential experiment. The ideal approach would to use an informative prior that accurately reflects individual perfomance. In the absence of reliable information from which to construct such a prior, any vague prior that does not give appreciably different densities to those regions of the parameter space where there is a reasonable fit would do [22]. However, constructing such priors can be difficult due to the nonlinearity of the models. Finally, in the current study, we applied ADO to just one property of the experiment design: the lengths of the retention intervals. This leaves several other design variables open to subjective manipulation. Two such variables that are crucial to the timely and successful completion of the experiment are the number of retention intervals, and the number of trials allotted to each interval. In theory, one could allot all of the trials in each stage to just one interval.1 In practice, however, this approach would require more stages, and consequently more participants, to collect observations at the same number of intervals as an approach that allotted trials to multiple intervals in each stage. Such an approach could be disadvantageous if observations at several different intervals were essential for discriminating the models under consideration. On the other hand, increasing the number of intervals at which to test in each stage greatly increases the complexity of the design space, thus increasing the length of the computation needed to find an optimal design. Extending the ADO algorithm to address these multiple design variables simultaneously would be a useful contribution. 5 Conclusion In the current study, ADO was successfully applied in a laboratory experiment with people, the purpose of which was to discriminate models of memory retention. The knowledge learned from its application contributes to our understanding of human memory. Although challenges remain in the implementation of ADO, the present success is an encouraging sign. The goals of future work include applying ADO to more complex experimental designs and to other research questions in cognitive science (e.g., numerical representation in children). 1Testing at one interval per stage is not possible with a utility function based on statistical model selection criteria, such as MDL, which require computation of the maximum likelihood estimate [10]. However, it can be done with a utility function based on mutual information [5]. 8 References [1] D. J. Navarro, M. A. Pitt, and I. J. Myung. Assessing the distinguishability of models and the informativeness of data. Cognitive Psychology, 49:47–84, 2004. [2] D. C. Rubin and A. E. Wenzel. One hundred years of forgetting: A quantitative description of retention. Psychological Review, 103(4):734–760, 1996. [3] J. T. Wixted and E. B. Ebbesen. On the form of forgetting. Psychological Science, 2(6):409– 415, 1991. [4] D. R. Cavagnaro, J. I Myung, M. A. Pitt, and Y. Tang. Better data with fewer participants and trials: improving experiment efficiency with adaptive design optimization. In N. A. Taatgen and H. Van Rijn, editors, Proceedings of the 31st Annual Conference of the Cognitive Science Society, pages 93–98. Cognitive Science Society, 2009. [5] D. R. Cavagnaro, J. I. Myung, M. A. Pitt, and J. V. Kujala. Adaptive design optimization: A mutual information based approach to model discrimination in cognitive science. Neural Computation, 2009. In press. [6] J. V. Kujala and T. J. Lukka. Bayesian adaptive estimation: The next dimension. Journal of Mathematical Psychology, 50(4):369–389, 2006. [7] J. Lewi, R. Butera, and L. Paninski. Sequential optimal design of neurophysiology experiments. Neural Computation, 21:619–687, 2009. [8] K. Chaloner and I. Verdinelli. Bayesian experimental design: A review. Statistical Science, 10(3):273–304, 1995. [9] P. Gr¨unwald. A tutorial introduction to the minimum description length principle. In P. Gr¨unwald, I. J. Myung, and M. A. Pitt, editors, Advances in Minimum Description Length: Theory and Applications. The M.I.T. Press, 2005. [10] J. I. Myung and M. A. Pitt. Optimal experimental design for model discrimination. Psychological Review, in press. [11] A. Heavens, T. Kitching, and L. Verde. On model selection forecasting, dark energy and modified gravity. Monthly Notices of the Royal Astronomical Society, 380(3):1029–1035, 2007. [12] T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., 1991. [13] P. M¨uller, B. Sanso, and M. De Iorio. Optimal bayesian design by inhomogeneous markov chain simulation. Journal of the American Statistical Association, 99(467):788–798, 2004. [14] W. R. Gilks, S. Richardson, and D. Spiegelhalter. Markov Chain Monte Carlo in Practice. Chapman & Hall, 1996. [15] S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi. Optimization by simulated annealing. Science, 220:671–680, 1983. [16] B. Amzal, F. Y. Bois, E. Parent, and C. P. Robert. Bayesian-optimal design via interacting particle systems. Journal of the American Statistical Association, 101(474):773–785, 2006. [17] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian Data Analysis. Chapman & Hall, 2004. [18] J. T. Wixted and E. B. Ebbesen. Genuine power curves in forgetting: A quantitative analysis of individual subject forgetting functions. Memory & cognition, 25(5):731–739, 1997. [19] J. A. Brown. Some tests of the decay theory of immediate memory. Quarterly Journal of Experimental Psychology, 10:12–21, 1958. [20] L. R. Peterson and M. J. Peterson. Short-term retention of individual verbal items. Journal of Experimental Psychology, 58:193–198, 1959. [21] S. D. Guikema. Formulating informative, data-based priors for failure probability estimation in reliability analysis. Reliability Engineering & System Safety, 92:490–502, 2007. [22] M. D. Lee. A bayesian analysis of retention functions. Journal of Mathematical Psychology, 48:310–321, 2004. [23] H. P. Carlin and T. A. Louis. Bayes and empirical Bayes methods for data analysis, 2nd ed. Chapman & Hall, 2000. 9
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Riffled Independence for Ranked Data Jonathan Huang, Carlos Guestrin School of Computer Science, Carnegie Mellon University {jch1,guestrin}@cs.cmu.edu Abstract Representing distributions over permutations can be a daunting task due to the fact that the number of permutations of n objects scales factorially in n. One recent way that has been used to reduce storage complexity has been to exploit probabilistic independence, but as we argue, full independence assumptions impose strong sparsity constraints on distributions and are unsuitable for modeling rankings. We identify a novel class of independence structures, called riffled independence, which encompasses a more expressive family of distributions while retaining many of the properties necessary for performing efficient inference and reducing sample complexity. In riffled independence, one draws two permutations independently, then performs the riffle shuffle, common in card games, to combine the two permutations to form a single permutation. In ranking, riffled independence corresponds to ranking disjoint sets of objects independently, then interleaving those rankings. We provide a formal introduction and present algorithms for using riffled independence within Fourier-theoretic frameworks which have been explored by a number of recent papers. 1 Introduction Distributions over permutations play an important role in applications such as multi-object tracking, visual feature matching, and ranking. In tracking, for example, permutations represent joint assignments of individual identities to track positions, and in ranking, permutations represent the preference orderings of a list of items. Representing distributions over permutations is a notoriously difficult problem since there are n! permutations, and standard representations, such as graphical models, are ineffective due to the mutual exclusivity constraints typically associated with permutations. The quest for exploitable problem structure has led researchers to consider a number of possibilities including distribution sparsity [17, 9], exponential family parameterizations [15, 5, 14, 16], algebraic/Fourier structure [13, 12, 6, 7], and probabilistic independence [8]. While sparse distributions have been successfully applied in certain tracking domains, we argue that they are less suitable in ranking problems where it might be necessary to model indifference over a number of objects. In contrast, Fourier methods handle smooth distributions well but are not easily scalable without making aggressive independence assumptions [8]. In this paper, we argue that while probabilistic independence might be useful in tracking, it is a poor assumption in ranking. We propose a novel generalization of independence, called riffled independence, which we argue to be far more suitable for modeling distributions over rankings, and develop algorithms for working with riffled independence in the Fourier domain. Our major contributions are as follows. • We introduce an intuitive generalization of independence on permutations, which we call riffled independence, and show it to be a more appropriate notion of independence for ranked data, offering possibilities for efficient inference and reduced sample complexity. • We introduce a novel family of distributions, called biased riffle shuffles, that are useful for riffled independence and propose an algorithm for computing its Fourier transform. • We provide algorithms that can be used in the Fourier-theoretic framework of [13, 8, 7] for joining riffle independent factors (RiffleJoin), and for teasing apart the riffle independent factors from a joint (RiffleSplit), and provide theoretical and empirical evidence that our algorithms perform well. 1 j i P(σ:σ(j)=i) 2 4 6 2 4 6 (a) j i P(σ:σ(j)=i), Y={1,2,3} 2 4 6 2 4 6 (b) j i P(σ:σ(j)=i), Y={2,4,5} 2 4 6 2 4 6 (c) j i P(σ:σ(j)=i), Y={1,2,5} 2 4 6 2 4 6 (d) j i P(σ:σ(j)=i), Y={4,5,6} 2 4 6 2 4 6 (e) Figure 1: Example first-order matrices with X = {1, 2, 3}, ¯ X = {4, 5, 6} independent, where black means h(σ : σ(j) = i) = 0. In each case, there is some 3-subset Y which X is constrained to map to with probability one. By rearranging rows, one sees that independence imposes block-diagonal structure on the matrices. 2 Distributions on permutations and independence relations In the context of ranking, a permutation σ = [σ1, . . . , σn] represents a one-to-one mapping from n objects to n ranks, where, by σj = i (or σ(j) = i), we mean that the jth object is assigned rank i under σ. If we are ranking a list of fruits/vegetables enumerated as (1) Artichoke, (2) Broccoli, (3) Cherry, and (4) Dates, then the permutation σ = [σA σB σC σD] = [2 3 1 4] ranks Cherry first, Artichoke second, Broccoli third, Dates last. The set of permutations of {1, . . . , n} forms a group with respect to function composition called the symmetric group (written Sn). We write τσ to denote the permutation resulting from τ composed with σ (thus [τσ](j) = τ(σ(j))). A distribution h(σ), defined over Sn, can be viewed as a joint distribution over the n variables σ = (σ1, . . . , σn) (where σj ∈{1, . . . , n}), subject to mutual exclusivity constraints ensuring that objects i and j never map to the same rank (h(σi = σj) = 0 whenever i ̸= j). Since there are n! permutations, it is intractable to represent entire distributions and one can hope only to maintain compact summaries. There have been a variety of methods proposed for summarizing distributions over permutations ranging from older ad-hoc methods such as maintaining k-best hypotheses [17] to the more recent Fourier-based methods which maintain a set of low-order summary statistics [18, 2, 11, 7]. The firstorder summary, for example, stores a marginal probability of the form h(σ : σ(j) = i) for every pair (i, j) and thus requires storing a matrix of only O(n2) numbers. For example, we might store the probability that apples are ranked first. More generally, one might store the sth-order marginals, which are marginal probabilities of s-tuples. The second-order marginals, for example, take the form h(σ : σ(k, ℓ) = (i, j)), and require O(n4) storage. Low-order marginals correspond, in a certain sense, to the low-frequency Fourier coefficients of a distribution over permutations. For example, the first-order matrix of h(σ) can be reconstructed exactly from O(n2) of the lowest frequency Fourier coefficients of h(σ), and the second-order matrix from O(n4) of the lowest frequency Fourier coefficients. In general, one requires O(n2s) coefficients to exactly reconstruct sth-order marginals, which quickly becomes intractable for moderately large n. To scale to larger problems, Huang et al. [8] demonstrated that, by exploiting probabilistic independence, one could dramatically improve the scalability of Fourier-based methods, e.g., for tracking problems, since confusion in data association only occurs over small independent subgroups of objects in many problems. Probabilistic independence on permutations. Probabilistic independence assumptions on the symmetric group can simply be stated as follows. Consider a distribution h defined over Sn. Let X be a p-subset of {1, . . . , n}, say, {1, . . . , p} and let ¯X be its complement ({p + 1, . . . , n}) with size q = n −p. We say that σX = (σ1, σ2, . . . , σp) and σ ¯ X = (σp+1, . . . , σn) are independent if h(σ) = f(σ1, σ2, . . . , σp) · g(σp+1, . . . , σn). Storing the parameters for the above distribution requires keeping O(p! + q!) probabilities instead of the much larger O(n!) size required for general distributions. Of course, O(p! + q!) can still be quite large. Typically, one decomposes the distribution recursively and stores factors exactly for small enough factors, or compresses factors using Fourier coefficients (but using higher frequency terms than what would be possible without the independence assumption). In order to exploit probabilistic independence in the Fourier domain, Huang et al. [8] proposed algorithms for joining factors and splitting distributions into independent components in the Fourier domain. Restrictive first-order conditions. Despite its utility for many tracking problems, however, we argue that the independence assumption on permutations implies a rather restrictive constraint on distributions, rendering independence highly unrealistic in ranking applications. In particular, using the mutual exclusivity property, it can be shown [8] that, if σX and σ ¯ X are independent, then for some fixed p-subset Y ⊂{1, . . . , n}, σX is a permutation of elements in Y and σ ¯ X is a permutation of its complement, ¯Y , with probability 1. Continuing with our vegetable/fruit example with n = 4, 2 if the vegetables and fruits rankings, σveg = [σA σB] and σfruit = [σC σD], are known to be independent, then for Y = {1, 2}, the vegetables are ranked first and second with probability one, and the fruits are ranked third and last with probability one. Huang et al. [8] refer to this as the first-order condition because of the block structure imposed upon first-order marginals (see Fig. 1). In sports tracking, the first-order condition might say, quite reasonably, that there is potential identity confusion within tracks for the red team and within tracks for the blue team but no confusion between the two teams. In our ranking example however, the first-order condition forces the probability of any vegetable being in third place to be zero, even though both vegetables will, in general, have nonzero marginal probability of being in second place, which seems quite unrealistic. In the next section, we overcome the restrictive first-order condition with the more flexible notion of riffled independence. 3 Going beyond full independence: Riffled independence The riffle (or dovetail) shuffle [1] is perhaps the most popular method of card shuffling, in which one cuts a deck of n cards into two piles, X = {1, . . . , p} and ¯X = {p + 1, . . . , n}, of sizes p and q = n −p, respectively, and successively drops the cards, one by one, so that the piles are interleaved (see Fig. 2) into one deck again. Inspired by riffle shuffles, we present a novel relaxation of full independence, which we call riffled independence. Rankings that are riffle independent are formed by independently selecting Figure 2: Riffle shuffling a standard deck of cards. rankings for two disjoint subsets of objects, then interleaving the rankings using a riffle shuffle to form a ranking over all objects. For example, we might first ‘cut the deck’ into two piles, vegetables (X) and fruits ( ¯X), independently decide that Broccoli is preferred over Artichoke (σB < σA) and that Dates is preferred over Cherry (σD < σC), then interleave the fruit and vegetable rankings to form σB < σD < σA < σC (i.e. σ = [3 1 4 2]). Intuitively, riffled independence models complex relationships within each of set X and ¯X while allowing correlations between sets to be modeled only by a constrained form of shuffling. Riffle shuffling distributions. Mathematically, shuffles are modeled as random walks on Sn. The ranking σ′ after a shuffle is generated from the ranking prior to that shuffle, σ, by drawing a permutation, τ from a shuffling distribution m(τ), and setting σ′ = τσ. Given the distribution P over σ, we can find the distribution h′(σ′) after the shuffle via convolution: h′(σ′) = [m ∗h] (σ′) = P {σ,τ : σ′=τσ} m(τ)h(σ). Note that we use the ∗symbol to denote the convolution operation. The question is, what are the shuffling distributions m that correspond to riffle shuffles? To answer this question, we use the distinguishing property of the riffle shuffle, that, after cutting the deck into two piles of size p and q = n−p, the relative ranking relations within each pile are preserved. Thus, if the ith card lies above the jth in one of the piles, then after shuffling, the ith card remains above the jth. In our example, relative rank preservation says that if Artichoke is preferred over Broccoli prior to shuffling, it is preferred over Broccoli after shuffling. Any allowable riffle shuffling distribution must therefore assign zero probability to permutations which do not preserve relative ranking relations. The set of permutations which do preserve these relations have a simple description. Definition 1 (Riffle shuffling distribution). Define the set of (p, q)-interleavings as: Ωp,q ≡{τY = [Y(1) Y(2) . . . Y(p) ¯Y(1) ¯Y(2) . . . ¯Y(q)] : Y ⊂{1, . . . , n}, |Y | = p} ⊂Sn, n = p + q, where Y(1) represents the smallest element of Y , Y(2) the second smallest, etc. A distribution mp,q on Sn is called a riffle shuffling distribution if it assigns nonzero probability only to elements in Ωp,q. The (p, q)-interleavings can be shown to preserve the relative ranking relations within each of the subsets X = {1, . . . , p} and ¯X = {p + 1, . . . , n} upon multiplication. In our vegetable/fruits example, we have n = 4, p = 2, and so the collection of subsets of size p are: { {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} } , and the set of (2, 2)-interleavings is given by: Ωp,q = {[1 2 3 4], [1 3 2 4], [1 4 2 3], [2 3 1 4], [2 4 1 3], [3 4 1 2]}. Note that |Ωp,q| = n p = n q = 4!/(2!2!) = 6. One possible riffle shuffling distribution on S4 might, for example, assign uniform probability (munif 2,2 (σ) = 1/6) to each permutation in Ω2,2 and zero probability to everything else, reflecting indifference between vegetables and fruits. We now formally define our generalization of independence where a distribution which fully factors independently is allowed to undergo a single riffle shuffle. 3 Definition 2 (Riffled independence). The subsets X = {1, . . . , p} and ¯X = {p + 1, . . . , n} are said to be riffle independent if h = mp,q ∗(f(σp) · g(σq)), with respect to some riffle shuffling distribution mp,q and distributions f, g, respectively. We denote riffled independence by: h = f ⊥mp,q g, and refer to f, g as riffled factors. To draw from h, one independently draws a permutation σp, of cards {1, . . . , p}, a permutation σq, of cards {p + 1, . . . , n}, and a (p, q)-interleaving τY , then shuffles to obtain σ = τY [σp σq]. In our example, the rankings σp = [2 1] (Broccoli preferred to Artichoke) and σq = [4 3] (Cherry preferred to Dates) are selected, then shuffled (multiplied by τ{1,3} = [1 3 2 4]) to obtain σ = [3 1 4 2]. We remark that setting mp,q to be the delta distribution on any of the (p, q)-interleavings in Ωp,q recovers the definition of ordinary probabilistic independence, and thus riffled independence is a strict generalization thereof. Just as in the full independence regime, where the distributions f and g are marginal distributions of rankings of X and ¯X, in the riffled independence regime, they can be thought of as marginal distributions of the relative rankings of X and ¯X. Biased riffle shuffles. There is, in the general case, a significant increase in storage required for riffled independence over full independence. In addition to the O(p! + q!) storage required for distributions f and g, we now require O( n p ) storage for the nonzero terms of the riffle shuffling distribution mp,q. Instead of representing all possible riffle shuffling distributions, however, we now introduce a family of useful riffle shuffling distributions which can be described using only a handful of parameters. The simplest riffle shuffling distribution is the uniform riffle shuffle, munif p,q , which assigns uniform probability to all (p, q)-interleavings and zero probability to all other elements in Sn. Used in the context of riffled independence, munif p,q models potentially complex relations within X and ¯X, but only captures the simplest possible correlations across subsets. We might, for example, have complex preference relations amongst vegetables and amongst fruits, but be completely indifferent with respect to the subsets, vegetables and fruits, as a whole. DRAWRIFFLEUNIF(p, q, n) // (p + q = n) 1 with prob q/n // drop from right pile 2 σ−←DRAWRIFFLEUNIF(p, q −1, n −1) 3 foreach i do σ(i) ← σ−(i) if i < n n if i = n 4 otherwise // drop from left pile 5 σ−←DRAWRIFFLEUNIF(p −1, q, n −1) 6 foreach i do 7 σ(i) ← 8 < : σ−(i) if i < p n if i = p σ−(i −1) if i > p return σ 8 Algorithm 1: Recurrence for drawing σ ∼ munif p,q (Base case: return σ = [1] if n = 1). There is a simple recursive method for uniformly drawing (p, q)-interleavings. Starting with a deck of n cards cut into a left pile ({1, . . . , p}) and a right pile ({p + 1, . . . , n}), pick one of the piles with probability proportional to its size (p/n for the left pile, q/n for the right) and drop the bottommost card, thus mapping either card p or card n to rank n. Then recurse on the n−1 remaining undropped cards, drawing a (p −1, q)-interleaving if the right pile was picked, or a (p, q −1)-interleaving if the left pile was picked. See Alg. 1. It is natural to consider generalizations where one is preferentially biased towards dropping cards from the left hand over the right hand (or vice-versa). We model this bias using a simple one-parameter family of distributions in which cards from the left and right piles drop with probability proportional to αp and (1 −α)q, respectively, instead of p and q. We will refer to α as the bias parameter, and the family of distributions parameterized by α as the biased riffle shuffles.1 In the context of rankings, biased riffle shuffles provide a simple model for expressing groupwise preferences (or indifference) for an entire subset X over ¯X or vice-versa. The bias parameter α can be thought of as a knob controlling the preference for one subset over the other, and might reflect, for example, a preference for fruits over vegetables, or perhaps indifference between the two subsets. Setting α = 0 or 1 recovers the full independence assumption, preferring objects in X (vegetables) over objects in ¯X (fruits) with probability one (or vice-versa), and setting α = .5, recovers the uniform riffle shuffle (see Fig. 3). Finally, there are a number of straightforward generalizations of the biased riffle shuffle that one can use to realize richer distributions. For example, α might depend on the number of cards that have been dropped from each pile (allowing perhaps, for distributions to prefer crunchy fruits over crunchy vegetables, but soft vegetables over soft fruits). 1The recurrence in Alg. 1 has appeared in various forms in literature [1]. We are the first to (1) use the recurrence to Fourier transform mp,q, and to (2) consider biased versions. The biased riffle shuffles in [4] are not similar to our biased riffle shuffles. See Appendix for details. 4 j i P(σ:σ(j)=i), α=00 5 10 15 20 5 10 15 20 (a) j i P(σ:σ(j)=i), α=1.50e−01 5 10 15 20 5 10 15 20 (b) j i P(σ:σ(j)=i), α=5.00e−01 5 10 15 20 5 10 15 20 (c) j i P(σ:σ(j)=i), α=8.50e−01 5 10 15 20 5 10 15 20 (d) j i P(σ:σ(j)=i), α=01 5 10 15 20 5 10 15 20 (e) Figure 3: First-order matrices with a deck of 20 cards, X = {1, . . . , 10}, ¯ X = {11, . . . , 20}, riffle independent and various settings of α. Note that nonzero blocks ‘bleed’ into zero regions (compare to Fig. 1). Setting α = 0 or 1 recovers full independence, where a subset of objects is preferred over the other with probability one. 4 Between independence and conditional independence We have presented riffle independent distributions as fully independent distributions which have been convolved by a certain class of shuffling distributions. In this section, we provide an alternative view of riffled independence based on conditional independence, showing that the notion of riffled independence lies somewhere between full and conditional independence. In Section 3, we formed a ranking by first independently drawing permutations πp and πq, of object sets {1, . . . , p} (vegetables) and {p + 1, . . . , n} (fruits), respectively, drawing a (p, q)-interleaving (i.e., a relative ranking permutation, τY ∈Ωp,q), and shuffling to form σ = τY [πp πq]. Thus, an object i ∈{1, . . . , p} is ranked in position τY (πp(i)) after shuffling (and an object j ∈{p + 1, . . . , n} is ranked in position τY (πq(j))). An equivalent way to form the same σ, however, is to first draw an interleaving τY ∈Ωp,q, then, conditioned on the choice of Y , draw independent permutations of the sets Y and ¯Y . In our example, we might first draw the (2,2)-interleaving [1 3 2 4] (so that after shuffling, we would obtain σV eg < σF ruit < σV eg < σF ruit). Then we would draw a permutation of the vegetable ranks (Y = {1, 3}), say, [3 1], and a permutation of the fruit ranks ( ¯Y = {2, 4}), [4 2], to obtain a final ranking over all items: σ = [3 1 4 2], or σB < σD < σA < σC. It is tempting to think that riffled independence is exactly the conditional independence assumption, in which case the distribution would factor as h(σ) = h(Y ) · h(σX|Y ) · h(σ ¯ X|Y ). The general case of conditional independence, however, has O( n p (p! + q! + 1)) parameters, while riffled independence requires only O( n p + p! + q!) parameters. We now provide a simple correspondence between the conditional independence view of riffled independence presented in this section to the shuffle theoretic definition from Section 3 (Def. 2). Define the map φ, which, given a permutation of Y (or ¯Y ), returns the permutation in σp ∈Sp (or Sq) such that [σp]i is the rank of [σX]i relative to the set Y . For example, if the permutation of the vegetable ranks is σX = [3 1] (with Artichoke ranked third, Broccoli first), then φ(σX) = [2 1] since, relative to the set of vegetables, Artichoke is ranked second, and Broccoli first. Proposition 3. Consider a riffle independent h = f ⊥mp,q g. For each σ ∈Sn, h factors as h(σ) = h(Y ) · h(σX|Y ) · h(σ ¯ X|Y ), with h(Y ) = m(τY ), h(σX|Y ) = f(φ(σX)), and h(σ ¯ X) = g(φ(σ ¯ X)). Proposition 3 is useful because it shows that the probability of a single ranking can be computed without summing over the entire symmetric group (a convolution)— a fact that might not be obvious from Definition 2. The factorization h(σ) = m(τY )f(φ(σX))g(φ(σ ¯ X)) also suggests that riffled independence behaves essentially like full independence (without the first-order condition), where, in addition to the independent variables σX and σ ¯ X, we also independently randomize over the subset Y . An immediate consequence is thatjust as in the full independence regime, conditioning operations on certain observations and MAP (maximum a posteriori) assignment problems decompose according to riffled independence structure. Proposition 4 (Probabilistic inference decompositions). Consider riffle independent prior and likelihood functions, hprior and hlike, on Sn which factor as: hprior = fprior ⊥mprior gprior and hlike = flike ⊥mlike glike, respectively. The posterior distribution under Bayes rule can be written as the riffle independent distribution: hpost ∝(fprior ⊙flike) ⊥mprior⊙mlike (gprior ⊙glike),where the ⊙symbol denotes the pointwise product operation. A similar result allows us to also perform MAP assignments by maximizing each of the distributions mp,q, f and g, independently and combining the results. As a corollary, it follows that conditioning on simple pairwise ranking likelihood functions (that depend only on whether object i is preferred to object j) decomposes along riffled independence structures. 5 RIFFLEJOIN( bf, bg) 1 bh′ = JOIN( bf, bg) ; 2 foreach frequency level i do 3 bhi ← h [ mα p,q i i · bh′i ; 4 return bh ; 5 Algorithm 2: Pseudocode for RiffleJoin RIFFLESPLIT(bh) 1 foreach frequency level i do 2 bh′i ← ˆ bmunif p,q ˜T i · bhi ; 3 [ bf, bg] ←SPLIT( bh′) ; 4 Normalize ˆf and ˆg; 5 return ˆf, ˆg; 6 Algorithm 3: Pseudocode for RiffleSplit 5 Fourier domain algorithms: RiffleJoin and RiffleSplit In this section, we present two algorithms for working with riffled independence in the Fourier theoretic framework of [13, 8, 7] — one algorithm for merging riffled factors to form a joint distribution (RiffleJoin), and one for extracting riffled factors from a joint (RiffleSplit). We begin with a brief introduction to Fourier theoretic inference on permutations (see [11, 7] for a detailed exposition). Unlike its analog on the real line, the Fourier transform of a function on Sn takes the form of a collection of Fourier coefficient matrices ordered with respect to frequency. Discussing the analog of frequency for functions on Sn, is beyond the scope of our paper, and, given a distribution h, we simply index the Fourier coefficient matrices of h as bh0, bh1, . . . , bhK ordered with respect to some measure of increasing complexity. We use bh to denote the complete collection of Fourier coefficient matrices. One rough way to understand this complexity, as mentioned in Section 2, is by the fact that the low-frequency Fourier coefficient matrices of a distribution can be used to reconstruct low-order marginals. For example, the first-order matrix of marginals of h can always be reconstructed from the matrices ˆh0 and ˆh1. As on the real line, many of the familiar properties of the Fourier transform continue to hold. The following are several basic properties used in this paper: Proposition 5 (Properties of the Fourier transform, see [2]). Consider any f, g : Sn →R. • (Linearity) For any α, β ∈R, [ \ αf + βg]i = α bfi + βbgi holds at all frequency levels i. • (Convolution) The Fourier transform of a convolution is a product of Fourier transforms: [[ f ∗g]i = bfi · bgi, for each frequency level i, where the operation · is matrix multiplication. • (Normalization) The first coefficient matrix, ˆf0, is a scalar and equals P σ∈Sn f(σ). A number of papers in recent years ([13, 6, 8, 7]) have considered approximating distributions over permutations using a truncated (bandlimited) set of Fourier coefficients and have proposed inference algorithms that operate on these Fourier coefficient matrices. For example, one can perform generic marginalization, Markov chain prediction, and conditioning operations using only Fourier coefficients without ever having to perform an inverse Fourier transform. Huang et al. [8] introduced Fourier domain algorithms, Join and Split, for combining independent factors to form joints and for extracting the factors from a joint distribution, respectively. In this section, we provide generalizations of the algorithms in [8] that we call RiffleJoin and RiffleSplit. We will assume that X = {1, . . . , p}, ¯X = {p + 1, . . . , n} and that we are given a riffle independent distribution h : Sn →R (h = f ⊥mp,q g). We also, for the purposes of this section, assume that the parameters for the distribution mp,q are known, though it will not matter for the RiffleSplit algorithm. Although we begin each of the following discussions as if all of the Fourier coefficients are provided, we will be especially interested in algorithms that work well in cases where only a truncated set of Fourier coefficients are present, and where h is only approximately riffle independent. RiffleJoin. Given the Fourier coefficients of f, g, and m, we can compute the Fourier coefficients of h using Definition 2 by applying the Join algorithm from [8] and the Convolution Theorem (Prop. 5), which tells us that the Fourier transform of a convolution can be written as a pointwise product of Fourier transforms. To compute the ˆhλ, our RiffleJoin algorithm simply calls the Join algorithm on bf and bg, and convolves the result by bm (see Alg. 2). In general, it may be intractable to Fourier transform the riffle shuffling distribution mp,q. However, for the class of biased riffle shuffles from Section 3, one can efficiently compute the low-frequency terms of d mαp,q by employing the recurrence relation in Alg. 1. In particular, Alg. 1 expresses a biased riffle shuffle on Sn as a linear combination of biased riffle shuffles on Sn−1. By invoking linearity of the Fourier transform (Prop. 5), one can efficiently compute d mαp,q via a dynamic programming approach. To the best of our knowledge, we are the first to compute the Fourier transform of riffle shuffling distributions. 6 RiffleSplit. Given the Fourier coefficients of the riffle independent distribution h, we would like to tease apart the riffle factors f and g. From the RiffleJoin algorithm, we saw that for each frequency level i, ˆhi = [ d mp,q]i·[ d f · g]i. The first solution to the splitting problem that might occur is to perform a deconvolution by multiplying each bhi term by the inverse of the matrix [ d mp,q]i (to form [ d mp,q]−1 i · bhi) and call the Split algorithm from [8] on the result. Unfortunately, the matrix [ d mp,q]i is, in general, non-invertible. Instead, our RiffleSplit algorithm left-multiplies each bhi term by bmunif p,q T i , which can be shown to be equivalent to convolving the distribution h by the ‘dual shuffle’, m∗, defined as m∗(σ) = munif p,q (σ−1). While convolving by m∗does not produce a distribution that factors independently, the Split algorithm from [8] can still be shown to recover the Fourier transforms ˆf and ˆg: Theorem 6. If h = f ⊥mp,q g, then RiffleSplit (Alg. 3) (with ˆh as input), returns ˆf and ˆg exactly. As with RiffleJoin, it is necessary Fourier transform munif p,q , which we can again accomplish via the recurrence in Alg. 1. One must also normalize the output of Split to sum to one via Prop. 5. Theoretical guarantees. We now briefly summarize several results which show how, (1) our algorithms perform when called with a truncated set of Fourier coefficients, and (2) when RiffleSplit is called on a distribution which is only approximately riffle independent. Theorem 7. Given enough Fourier terms to reconstruct the kth-order marginals of f and g, RiffleJoin returns enough Fourier terms to exactly reconstruct the kth-order marginals of h. Likewise, given enough Fourier terms to reconstruct the kth-order marginals of h, RiffleSplit returns enough Fourier terms to exactly reconstruct the kth-order marginals of both f and g. Theorem 8. Let h be any distribution on Sn and mp,q any riffle shuffling distribution on Sn. If [ bf ′, bg′] = RIFFLESPLIT(bh), then (f ′, g′) is the minimizer of the problem: minimizef,g DKL(h||f ⊥mp,q g), (subject to: P σp f(σp) = 1, P σq g(σq) = 1), where DKL is the Kullback-Leibler divergence. 6 Experiments In this section, we validate our algorithms and show that riffled independence exists in real data. APA dataset. The APA dataset [3] is a collection of 5738 ballots from a 1980 presidential election of the American Psychological Association where members ordered five candidates from favorite to least favorite. We first perform an exhaustive search for subsets X and ¯X that are closest to riffle independent (with respect to DKL), and find that candidate 2 is nearly riffle independent of the remaining candidates. In Fig. 4(a) we plot the true vote distribution and the best approximation by a distribution in which candidate 2 is riffle independent of the rest. For comparison, we plot the result of splitting off candidate 3 instead of candidate 2, which one can see to be an inferior approximation. The APA, as described by Diaconis [3], is divided into “academicians and clinicians who are on uneasy terms”. In 1980, candidates {1, 3} and {4, 5} fell on opposite ends of this political spectrum with candidate 2 being somewhat independent. Diaconis conjectured that voters choose one group over the other, and then choose within. We are now able to verify his conjecture in a riffled independence sense. After removing candidate 2 from the distribution, we perform a search within candidates {1, 3, 4, 5} to again find nearly riffle independent subsets. We find that X = {1, 3} and ¯X = {4, 5} are very nearly riffle independent and thus are able to verify that candidate sets {2}, {1, 3}, {4, 5} are indeed grouped in a riffle independent sense in the APA data. Finally since there are two opposing groups within the APA, the riffle shuffling distribution for sets {1, 3} and {4, 5} is not well approximated by a biased riffle shuffle. Instead, we fit a mixture of two biased riffle shuffles to the data and found the bias parameters of the mixture components to be α1 ≈.67 and α2 ≈.17, indicating that the two components oppose each other (since α1 and α2 lie on either side of .5). Sushi dataset. The sushi dataset [10] consists of 5000 full rankings of ten types of sushi. Compared to the APA data, it has more objects, but fewer examples. We divided the data into training and test sets and estimated the true distribution in three ways: (1) directly from samples,(2) using a riffle independent distribution (split evenly into two groups of five) with the optimal shuffling distribution m, and (3) with a biased riffle shuffle (and optimal bias α). Fig. 4(b) plots testset log-likelihood as a function of training set size — we see that riffle independence assumptions can help significantly to lower the sample complexity of learning. Biased riffle shuffles, as can be seen, 7 10 20 30 40 50 60 70 80 90 100 110 120 APA ranking distribution true distribution Remove candidate {3} (DKL=0.1878) Remove candidate {2} (DKL=0.0398) (a) Purple line: approximation to vote distribution when candidate 2 is riffle independent; Blue line: approximation when candidate 3 is riffle independent. 50 100 200 400 800 1600 3200 −9000 −8500 −8000 −7500 Training set size Log−likelihood of held−out test set Full model Riffle Independent w/optimal m Biased riffle independent w/optimal α (b) Average log-likelihood of held out test examples from the Sushi dataset 2 4 6 8 10 0 0.05 0.1 0.15 0.2 0.25 0.3 Ranks i = 1 (favorite) through 10 (least favorite) Probability of Uni/Sea Urchin in rank i Estimated from 1000 examples Estimated from 100 examples Biased riffle indep. approx. (c) First-order probabilities of Uni (sea urchin) (Sushi dataset) rankings. 100 200 300 400 500 600 0 5 10 15 20 25 Sample sizes KL divergence from truth (20 trials) Split algorithm RiffleSplit algorithm (d) Estimating a riffle independent distribution using various sample sizes 10 20 30 40 0 20 40 60 80 100 n, p=n/2 Elapsed time in second 1st order (O(n2) terms) 2nd order (O(n4) terms) 3rd order (O(n4) terms) (e) Running time plot of RiffleJoin Figure 4: Experiments are a useful learning bias with very small samples. As an illustration, see Fig. 4(c) which shows the first-order marginals of Uni (Sea Urchin) rankings, and the biased riffle approximation. Approximation accuracy. To understand the behavior of RiffleSplit in approximately riffle independent situations, we draw sample sets of varying sizes from a riffle independent distribution on S8 (with bias parameter α = .25) and use RiffleSplit to estimate the riffle factors from the empirical distribution. In Fig. 4(d), we plot the KL-divergence between the true distribution and that obtained by applying RiffleJoin to the estimated riffle factors. With small sample sizes (far less than 8!), we are able to recover accurate approximations despite the fact that the empirical distributions are not exactly riffle independent. For comparison, we ran the experiment using the Split algorithm [8] to recover the riffle factors. Somewhat surprisingly, one can show (see Appendix) that Split also recovers the riffle factors, albeit without the optimality guarantee that we have shown for Rifflesplit (Theorem 8) and therefore requires far more samples to reliably approximate h. Running times. In general, the complexity of Split is cubic (O(d3)) in the dimension of each Fourier coefficient matrix [8]. The complexity of RiffleJoin/RiffleSplit is O(n2d3), in the worst case when p ∼O(n). If we precompute the Fourier coefficients of mp,q, (which requires O(n2d3)) for each coefficient matrix, then the complexity of RiffleSplit is also O(d3). In Fig. 4(e), we plot running times of RiffleJoin (no precomputation) as a function of n (setting p = ⌈n/2⌉) scaling up to n = 40. 7 Future Directions and Conclusions There are many open questions. For example, several papers note that graphical models cannot compactly represent distributions over permutations due to mutual exclusivity. An interesting question which our paper opens, is whether it is possible to use something similar to graphical models by substituting conditional generalizations of riffled independence for ordinary conditional independence. Other possibilities include going beyond the algebraic approach and studying riffled independence in non-Fourier frameworks and developing statistical (riffled) independence tests. In summary, we have introduced riffled independence and discussed how to exploit such structure in a Fourier-theoretic framework. Riffled independence is a new tool for analyzing ranked data and has the potential to offer novel insights into datasets both new and old. We believe that it will lead to the development of fast inference and low sample complexity learning algorithms. Acknowledgements This work is supported in part by the ONR under MURI N000140710747, and the Young Investigator Program grant N00014-08-1-0752. We thank K. El-Arini for feedback on an initial draft. 8 References [1] D. Bayer and P. Diaconis. Trailing the dovetail shuffle to its lair. The Annals of Probability, 1992. [2] P. Diaconis. Group Representations in Probability and Statistics. IMS Lecture Notes, 1988. [3] Persi Diaconis. A generalization of spectral analysis with application to ranked data. The Annals of Statistics, 17(3):949–979, 1989. [4] J. Fulman. The combinatorics of biased riffle shuffles. Combinatorica, 18(2):173–184, 1998. [5] D. P. Helmbold and M. K. Warmuth. Learning permutations with exponential weights. In COLT, 2007. [6] J. Huang, C. Guestrin, and L. Guibas. Efficient inference for distributions on permutations. In NIPS, 2007. [7] J. Huang, C. Guestrin, and L. Guibas. Fourier theoretic probabilistic inference over permutations. JMLR, 10, 2009. [8] J. Huang, C. Guestrin, X. Jiang, and L. Guibas. Exploiting probabilistic independence for permutations. In AISTATS, 2009. [9] S. Jagabathula and D. Shah. Inferring rankings under constrained sensing. In NIPS, 2008. [10] Toshihiro Kamishima. Nantonac collaborative filtering: recommendation based on order responses. In KDD, pages 583–588, 2003. [11] R. Kondor. Group Theoretical Methods in Machine Learning. PhD thesis, Columbia University, 2008. [12] R. Kondor and K. M. Borgwardt. The skew spectrum of graphs. In ICML, pages 496–503, 2008. [13] R. Kondor, A. Howard, and T. Jebara. Multi-object tracking with representations of the symmetric group. In AISTATS, 2007. [14] G. Lebanon and Y. Mao. Non-parametric modeling of partially ranked data. In NIPS, 2008. [15] M. Meila, K. Phadnis, A. Patterson, and J. Bilmes. Consensus ranking under the exponential model. Technical Report 515, University of Washington, Statistics Department, April 2007. [16] J. Petterson, T. Caetano, J. McAuley, and J. Yu. Exponential family graph matching and ranking. CoRR, abs/0904.2623, 2009. [17] D.B. Reid. An algorithm for tracking multiple targets. IEEE Trans. on Automatic Control, 6:843–854, 1979. [18] J. Shin, N. Lee, S. Thrun, and L. Guibas. Lazy inference on object identities in wireless sensor networks. In IPSN, 2005. 9
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A Neural Implementation of the Kalman Filter Robert C. Wilson Department of Psychology Princeton University Princeton, NJ 08540 rcw2@princeton.edu Leif H. Finkel Department of Bioengineering University of Pennsylvania Philadelphia, PA 19103 Abstract Recent experimental evidence suggests that the brain is capable of approximating Bayesian inference in the face of noisy input stimuli. Despite this progress, the neural underpinnings of this computation are still poorly understood. In this paper we focus on the Bayesian filtering of stochastic time series and introduce a novel neural network, derived from a line attractor architecture, whose dynamics map directly onto those of the Kalman filter in the limit of small prediction error. When the prediction error is large we show that the network responds robustly to changepoints in a way that is qualitatively compatible with the optimal Bayesian model. The model suggests ways in which probability distributions are encoded in the brain and makes a number of testable experimental predictions. 1 Introduction There is a growing body of experimental evidence consistent with the idea that animals are somehow able to represent, manipulate and, ultimately, make decisions based on, probability distributions. While still unproven, this idea has obvious appeal to theorists as a principled way in which to understand neural computation. A key question is how such Bayesian computations could be performed by neural networks. Several authors have proposed models addressing aspects of this issue [15, 10, 9, 19, 2, 3, 16, 4, 11, 18, 17, 7, 6, 8], but as yet, there is no conclusive experimental evidence in favour of any one and the question remains open. Here we focus on the problem of tracking a randomly moving, one-dimensional stimulus in a noisy environment. We develop a neural network whose dynamics can be shown to approximate those of a one-dimensional Kalman filter, the Bayesian model when all the distributions are Gaussian. Where the approximation breaks down, for large prediction errors, the network performs something akin to outlier or change detection and this ‘failure’ suggests ways in which the network can be extended to deal with more complex, non-Gaussian distributions over multiple dimensions. Our approach rests on the modification of the line attractor network of Zhang [26]. In particular, we make three changes to Zhang’s network, modifying the activation rule, the weights and the inputs in such a way that the network’s dynamics map exactly onto those of the Kalman filter when the prediction error is small. Crucially, these modifications result in a network that is no longer a line attractor and thus no longer suffers from many of the limitations of these networks. 2 Review of the one-dimensional Kalman filter For clarity of exposition and to define notation, we briefly review the equations behind the onedimensional Kalman filter. In particular, we focus on tracking the true location of an object, x(t), over time, t, based on noisy observations of its position z(t) = x(t) + nz(t), where nz(t) is zero mean Gaussian random noise with standard deviation σz(t), and a model of its dynamics, x(t+1) = 1 x(t) + v(t) + nv(t), where v(t) is the velocity signal and nv(t) is a Gaussian noise term with zero mean and standard deviation σv(t). Assuming that σz(t), σv(t) and v(t) are all known, then the Kalman filter’s estimate of the position, ˆx(t), can be computed via the following three equations ¯x(t + 1) = ˆx(t) + v(t) (1) 1 ˆσx(t + 1)2 = 1 ˆσx(t)2 + σv(t)2 + 1 σz(t + 1)2 (2) ˆx(t + 1) = ¯x(t + 1) + ˆσx(t + 1)2 σz(t + 1)2 [z(t + 1) −¯x(t + 1)] (3) In equation 1 the model computes a prediction, ¯x(t+1), for the position at time t+1; equation 2 updates the model’s uncertainty, ˆσx(t+1), in its estimate; and equation 3 updates the model’s estimate of position, ˆx(t + 1), based on this uncertainty and the prediction error [z(t + 1) −¯x(t + 1)]. 3 The neural network The network is a modification of Zhang’s line attractor model of head direction cells [26]. We use rate neurons and describe the state of the network at time t with the membrane potential vector, u(t), where each component of u(t) denotes the membrane potential of a single neuron. In discrete time, the update equation is then u(t + 1) = wJf [u(t)] + I(t + 1) (4) where w scales the strength of the weights, J is the connectivity matrix, f[·] is the activation rule that maps membrane potential onto firing rate, and I(t + 1) is the input at time t + 1. As in [26], we set J = Jsym +γ(t)Jasym such that the the connections are made up of a mixture of symmetric, Jsym, and asymmetric components, Jasym (defined as spatial derivative of Jsym), with mixing strength γ(t) that can vary over time. Although the results presented here do not depend strongly on the exact forms of Jsym and Jasym, for concreteness we use the following expressions Jsym ij = Kw exp cos 2π(i−j) N −1 σ2w −c ; Jasym ij = −2π Nσ2w sin 2π(i −j) N Jsym ij (5) where N is the number of neurons in the network and σw, Kw and c are constants that determine the width and excitatory and inhibitory connection strengths respectively. To approximate the Kalman filter, the activation function must implement divisive inhibition [14, 13] f[u] = [u]+ S + µ P i[ui]+ (6) where [u]+ denotes recitification of u; µ determines the strength of the divisive feedback and S determines the gain when there is no previous activity in the network. When w = 1, γ(t) = 0 and I(t) = 0, the network is a line attractor over a wide range of Kw, σw, c, S and µ, having a continuum of fixed points (as N →∞). Each fixed point has the same shape, taking the form of a smooth membrane potential profile, U(x) = Jsymf [U(x)], centered at location, x, in the network. When γ(t) ̸= 0, the bump of activity can be made to move over time (without losing its shape) [26] and hence, so long as γ(t) = v(t), implement the prediction step of the Kalman filter (equation 1). That is, if the bump at time t is centered at ˆx(t), i.e. u(t) = U(ˆx(t)), then at time t + 1 it is centered at ¯x(t + 1) = ˆx(t) + γ(t), i.e. u(t + 1) = U(ˆx(t) + γ(t)) = U(¯x(t + 1)). Thus, in this configuration, the network can already implement the first step of the Kalman filter through its recurrent connectivity. The next two steps, equations 2 and 3, however, remain inaccessible as the network has no way of encoding uncertainty and it is unclear how it will deal with external inputs. 4 Relation to Kalman filter - small prediction error case In this section we outline how the neural network dynamics can be mapped onto those of a Kalman filter. In the interests of space we focus only on the main points of the derivation, leaving the full working to the supplementary material. 2 Our approach is to analyze the network in terms of U, which, for clarity, we define here to be the fixed point membrane potential profile of the network when w = 1, γ(t) = 0, I(t) = 0, S = S0 and µ = µ0. Thus, the results described here are independent of the exact form of U so long as it is a smooth, non-uniform profile over the network. We begin by making the assumption that both the input, I(t), and the network membrane potential, u(t), take the form of scaled versions U, with the former encoding the noisy observations, z(t), and the latter encoding the network’s estimate of position, ˆx(t), i.e., I(t) = A(t)U(z(t)) and u(t) = α(t)U(ˆx(t)) (7) Substituting this ansatz for membrane potential into the left hand side of equation 4 gives LHS = α(t + 1)U (ˆx(t + 1)) (8) and into the right hand side of equation 4 gives RHS = wJf [α(t)U (ˆx(t))] | {z } recurrent input + A(t + 1)U (z(t + 1)) | {z } external input (9) For the ansatz to be self-consistent we require that RHS can be written in the same form as LHS. We now show that this is the case. As in the previous section, the recurrent input, implements the prediction step of the Kalman filter, which, after a little algebra (see supplementary material), allows us to write RHS ≈CU (¯x(t + 1)) | {z } prediction + A(t + 1)U (z(t + 1)) | {z } external input (10) With the variable C defined as C = 1 S w(S0+µ0I) 1 α(t) + µI w(S0+µ0I) (11) where I = P i [Ui(ˆx(t))]+. If we now suppose that the prediction error [z(t + 1) −¯x(t + 1)] is small, then we can linearize around the prediction, ¯x(t + 1), to get (see supplementary material) RHS ≈[C + A(t + 1)] U ¯x(t + 1) + A(t + 1) A(t + 1) + C [z(t + 1) −¯x(t + 1)] (12) which is of the same form as equation 8 and thus the ansatz holds. More specifically, equating terms in equations 8 and 12, we can write down expressions for α(t + 1) and ˆx(t + 1) α(t + 1) ≈C + A(t + 1) = 1 S w(S0+µ0I) 1 α(t) + µI w(S0+µ0I) + A(t + 1) (13) ˆx(t + 1) ≈¯x(t) + A(t + 1) α(t + 1) [z(t + 1) −x(t + 1)] (14) which, if we define w such that S w(S0 + µ0I) = 1 i.e. w = S S0 + µ0I (15) are identical to equations 2 and 3 so long as (a) α(t) ∝ 1 ˆσx(t)2 (b) A(t) ∝ 1 σz(t)2 (c) µI S ∝σv(t)2 (16) Thus the network dynamics, when the prediction error is small, map directly onto the Kalman filter equations. This is our main result. 3 neuron # time step 0 20 40 60 80 100 20 40 60 80 100 neuron # time step 0 20 40 60 80 100 20 40 60 80 100 0 20 40 60 80 100 20 30 40 50 60 position time step 0 20 40 60 80 100 0 2 4 6 time step σx(t) A B C D Figure 1: Comparison of noiseless network dynamics with dynamics of the Kalman Filter for small prediction errors. 4.1 Implications Reciprocal code for uncertainty in input and estimate Equation 16a provides a link between the strength of activity in the network and the overall uncertainty in the estimate of the Kalman filter, ˆσx(t), with uncertainty decreasing as the activity increases. A similar relation is also implied for the uncertainty in the observations, σz(t), where equation 16b suggests that this should be reflected in the magnitude of the input, A(t). Interestingly, such a scaling, without a corresponding narrowing of tuning curves, is seen in the brain [20, 5, 2]. Code for velocity signal As with Zhang’s line attractor network [26], the mean of the velocity signal, v(t) is encoded into the recurrent connections of the network, with the degree of asymmetry in the weights, γ(t), proportional to the speed. Such hard coding of the velocity signal represents a limitation of the model, as we would like to be able to deal with arbitrary, time varying speeds. However, this kind of change could be implemented by pre-synaptic inhibition [24] or by using a ‘double-ring’ network similar to [25]. Equation 16c implies that the variance of the velocity signal, σv(t), is encoded in the strength of the divisive feedback, µ (assuming constant S). This is very different from Zhang’s model, that has no concept of uncertainty and is also very different from the traditional view of divisive inhibition that sees it as a mechanism for gain control [14, 13]. The network is no longer a line attractor This can be seen by considering the fixed point values of the scale factor, α(t), when the input current, I(t) = 0. Requiring α(t + 1) = α(t) = α∗in equation 13 gives values for these fixed points as α∗= 0 and α∗= S0 + µ0I µI w −S µI (17) This second solution is exactly zero when w satisfies equation 15, hence the network only has one fixed point corresponding to the all zero state and is not a line attractor. This is a key result as it removes all of the constraints required for line attractor dynamics such as infinite precision in the weights and lack of noise in the network and thus the network is much more biologically plausible. 4.2 An example In figure 1 we demonstrate the ability of the network to approximate the dynamics of a onedimensional Kalman filter. The input, shown in figure 1A, is a noiseless bump of current centered 4 neuron # time step 0 20 40 60 80 100 20 40 60 80 100 neuron # time step 0 20 40 60 80 100 20 40 60 80 100 0 20 40 60 80 100 20 30 40 50 60 position time step 0 20 40 60 80 100 0 0.5 1 1.5 2 time step σx(t) A B C D Figure 2: Response of the network when presented with a noisy moving bump input. at the position of the observation, z(t). The observation noise has standard deviation σz(t) = 5, the speed v(t) = 0.5 for 1 ≤t < 50 and v(t) = −0.5 for 50 ≤t < 100 and the standard deviation of the random walk dynamics, σv(t) = 0.2. In accordance with equation 16b, the height of each bump is scaled by 1/σz(t)2. In figure 1B we plot the output activity of the network over time. Darker shades correspond to higher firing rates. We assume that the network gets the correct velocity signal, i.e. γ(t) = v(t) and µ is set such that equation 16c holds. The other parameters are set to Kw = 1, σw = 0.2, c = 0.05, S = S0 = 1 and µ0 = 1 which gives I = 5.47. As can be seen from the plot, the amount of activity in the network steadily grows from zero over time to an asymptotic value, corresponding to the network’s increasing certainty in its predictions. The position of the bump of activity in the network is also much less jittery than the input bump, reflecting a certain amount of smoothing. In figure 1C we compare the positions of the input bumps (gray dots) with the position of the network bump (black line) and the output of the equivalent Kalman filter (red line). The network clearly tracks the Kalman filter estimate extremely well. The same is true for the network’s estimate of the uncertainty, computed as 1/ p α(t) and shown as the black line in figure 1D, which tracks the Kalman filter uncertainty (red line) almost exactly. 5 Effect of input noise We now consider the effect of noise on the ability of the network to implement a Kalman filter. In particular we consider noise in the input signal, which for this simple one layer network is equivalent to having noise in the update equation. For brevity, we only present the main results along with the results of simulations, leaving more detailed analysis to the supplementary material. Specifically, we consider input signals where the only source of noise is in the input current i.e. there is no additional jitter in the position of the bump as there was in the noiseless case, thus we write I(t) = A(t)U (x(t)) + ϵ(t) (18) where ϵ(t) is some noise vector. The main effect of the noise is that it perturbs the effective position of the input bump. This can be modeled by extracting the maximum likelihood estimate of the input position given the noisy input and then using this position as the input to the equivalent Kalman filter. Because of the noise, this extracted position is not, in general, the same as the noiseless input position and for zero mean Gaussian noise with covariance Σ, the variance of the perturbation, σz(t), 5 0 0.5 1 1.5 2 2.5 3 0 2 4 6 α σnoise 0 0.5 1 1.5 2 2.5 3 0 2 4 6 8 10 σnoise ERMS vs KF A B Figure 3: Effect of noise magnitude on performance of network. is approximately given by σz(t) ≈ 1 A(t) r 2 U′T Σ−1U′ (19) Now, for the network to approximate a Kalman filter, equation 16b must hold which means that we require the magnitude of the covariance matrix to scale in proportion to the strength of the input signal, A(t), i.e. Σ ∝A(t). Interestingly this relation is true for Poisson noise, the type of noise that is found all over the brain. In figure 2 we demonstrate the ability of the network to approximate a Kalman filter. In panel A we show the input current which is a moving bump of activity corrupted by independent Gaussian noise of standard deviation σnoise = 0.23, or about two thirds of the maximum height of the fixed point bump, U. This is a high noise setting and it is hard to see the bump location by eye. The network dramatically cleans up this input signal (figure 2B) and the output activity, although still noisy, reflects the position of the underlying stimulus much more faithfully than the input. (Note that the colour scales in A and B are different). In panel C we compare the position of the output bump in the network (black line) with that of the equivalent Kalman filter. To do this we first fit the noisy input bump at each time to obtain input positions z(t) shown as gray dots. Then using σz = 2.23 computed using equation 19 we can compute the estimates of the equivalent Kalman filter (thick red line). which closely match those of the network (black line). Similarly, there is good agreement between the two estimates of the uncertainty, ˆσx(t), panel D (black line - network, red line - Kalman filter). 5.1 Performance of the network as a function of noise magnitude The noise not only affects the position of the input bump but also, in a slightly more subtle manner, causes a gradual decline in the ability of the network to emulate a Kalman filter. The reason for this (outlined in more detail in the supplementary material) is that the output bump scale factor, α, decreases as a function of the noise variance, σnoise. This effect is illustrated in figure 3A where we plot the steady state value of α (for constant input strength, A(t)) as a function of σnoise. The average results of simulations on 100 neurons are shown as the red dots, while the black line represents the results of the theory in the supplementary material. The reason for the decline in α as σnoise goes up is that, because of the rectifying non-linearity in the activation rule, increasing σnoise increases the amount of noisy activity in the network. Because of inhibition (both divisive and subtractive) in the network, this ‘noisy activity’ competes with the bump activity and decreases it - thus reducing α. This decrease in α results in a change in the Kalman gain of the network, by equation 14, making it different from that of the equivalent Kalman filter, thus degrading the network’s performance. We quantify this difference in figure 3B where we plot the root mean squared error (in units of neural position) between the network and the equivalent Kalman filter as a function of σnoise. As before, the results of simulations are shown as red dots and the theory (outlined in the supplementary material) is the black line. To give some sense for the scale on this plot, the horizontal blue line corresponds to the maximum height of the (noise free) input bump. Thus we may conclude that the performance of the network and the theory are robust up to fairly large values of σnoise. 6 time step neuron # 0 20 40 60 80 100 20 40 60 80 100 time step neuron # 0 20 40 60 80 100 20 40 60 80 100 0 20 40 60 80 100 0 20 40 60 80 100 time step position 0 20 40 60 80 100 0 1 2 3 time step αi(t) A B C D Figure 4: Response of the network to changepoints. 6 Response to changepoints (and outliers) - large prediction error case We now consider the dynamics of the network when the prediction error is large. By large we mean that the prediction error is greater than the width of the bump of activity in the network. Such a big discrepancy could be caused by an outlier or a changepoint, i.e. a sustained large and abrupt change in the input position at a random time. In the interests of space we focus only on the latter case and such an input, with a changepoint at t = 50, is shown in figure 4A. In figure 4B we show the network’s response to this stimulus. As before, prior to the change, there is a single bump of activity whose position approximates that of a Kalman filter. However, after the changepoint, the network maintains two bumps of activity for several time steps. One at the original position, that shrinks over time and essentially predicts where the input would be if the change had not occurred, and a second, that grows over time, at the location of the input after the changepoint. Thus in the period immediately after the changepoint, the network can be thought of as encoding two separate and competing hypotheses about the position of the stimulus, one corresponding to the case where no change has occurred, and the other, the case where a change occurred at t = 50. In figure 4C we compare the position of the bump(s) in the network (black dots whose size reflects the size of each bump) to the output from the Kalman filter (red line). Before the changepoint, the two agree well, but after the change, the Kalman filter becomes suboptimal, taking a long time to move to the new position. The network, however, by maintaining two hypotheses reacts much better. Finally, in figure 4D we plot the scale factor, αi(t), of each bump as computed from the simulations (black dots) and from the approximate analytic solution described in the supplementary material (red line for bump at 30, blue line for bump at 80). As can be seen, there is good agreement between theory and simulation, with the largest discrepancy occurring for small values of the scale factor. Thus, when confronted with a changepoint, the network no longer approximates a Kalman filter and instead maintains two competing hypotheses in a way that is qualitatively similar to that of the run-length distribution in [1]. This is an extremely interesting result and hints at ways in which more complex distributions may be encoded in these type of networks. 7 Discussion 7.1 Relation to previous work Of the papers mentioned in the introduction, two are of particular relevance to the current work. In the first, [8], the authors considered a neural implementation of the Kalman filter using line 7 attractors. Although this work, at first glance, seems similar to what is presented here, there are several major differences, the main one being that our network is not a line attractor at all, while the results in [8] rely on this property. Also, in [8], the Kalman gain is changed manually, where as in our case it adjusts automatically (equations 13 and 14), and the form of non-linearity is different. Probabilistic population coding [16, 4] is more closely related to model presented here. Combined with divisive normalization, these networks can implement a Kalman filter exactly, while the model presented here can ‘only’ approximate one. While this may seem like a limitation of our network, we see it as an advantage as the breakdown of the approximation leads to a more robust response to outliers and changepoints than a pure Kalman filter. 7.2 Extension beyond one-dimensional Gaussians A major limitation of the current model is that it only applies to one-dimensional Gaussian tracking - clearly an unreasonable restriction for the brain. One possible way around this limitation is hinted at by the response of the network in the changepoint case where we saw two, largely independent bumps of activity in the network. This ability to encode multiple ‘particles’ in the network may allow networks of this kind to implement something like the dynamics of a particle filter [12] that can approximate the inference process for non-linear and non-Gaussian systems. Such a possibility is an intriguing idea for future work. 7.3 Experimental predictions The model makes at least two easily testable predictions about the response of head direction cells [21, 22, 23] in rats. The first comes by considering the response of the neurons in the ‘dark’. Assuming that all bearing cues can indeed be eliminated, by setting A(t) = 0 in equation 13, we expect the activity of the neurons to fall off as 1/t and that the shape of the tuning curves will remain approximately constant. Note that this prediction is vastly different from the behaviour of a line attractor, where we would not expect the level of activity to fall off at all in the dark. Another, slightly more ambitious experiment would involve perturbing the reliability of one of the landmark cues. In particular, one could imagine a training phase, where the position of one landmark is jittered over time, such that each time the rat encounters it it is at a slightly different heading. In the test case, all other, reliable, landmark cues would be removed and the response of head direction cells measured in response to presentation of the unreliable cue alone. The prediction of the model is that this would reduce the strength of the input, A, which in turn reduces the level of activity in the head direction cells, α. In particular, if σz is the jitter of the unreliable landmark, then we expect α to scale as 1/σ2 z. This prediction is very different from that of a line attractor which would predict a constant level of activity regardless of the reliability of the landmark cues. 8 Conclusions In this paper we have introduced a novel neural network model whose dynamics map directly onto those of a one-dimensional Kalman filter when the prediction error is small. This property is robust to noise and when the prediction error is large, such as for changepoints, the output of the network diverges from that of the Kalman filter, but in a way that is both interesting and useful. Finally, the model makes two easily testable experimental predictions about head direction cells. Acknowledgements We would like to thank the anonymous reviewers for their very helpful comments on this work. References [1] R.P. Adams and D.J.C. MacKay. Bayesian online changepoint detection. Technical report, University of Cambridge, Cambridge, UK, 2007. [2] J. S. Anderson, I. Lampl, D. C. Gillespie, and D. Ferster. The contribution of noise to contrast invariance of orientation tuning in cat visual cortex. Science, 290:1968–1972, 2000. 8 [3] M. J. Barber, J. W. Clark, and C. H. Anderson. Neural representation of probabilistic information. Neural Computation, 15:1843–1864, 2003. [4] J. Beck, W. J. Ma, P. E. Latham, and A. Pouget. Probabilistic population codes and the exponential family of distributions. Progress in Brain Research, 165:509–519, 2007. [5] K. H. Britten, M. N. Shadlen, W. T. Newsome, and J. A. Movshon. Response of neurons in macaque mt to stochastic motion signals. Visual Neuroscience, 10(1157-1169), 1993. [6] S. Deneve. Bayesian spiking neurons i: Inference. Neural Computation, 20:91–117, 2008. [7] S. Deneve. Bayesian spiking neurons ii: Learning. Neural Computation, 20:118–145, 2008. [8] S. Deneve, J.-R. Duhammel, and A. Pouget. Optimal sensorimotor integration in recurrent cortical networks: a neural implementation of kalman filters. Journal of Neuroscience, 27(21):5744–5756, 2007. [9] S. Deneve, P. E. Latham, and A. Pouget. Reading population codes: a neural implementation of ideal observers. Nature Neuroscience, 2(8):740–745, 1999. [10] S. Deneve, P. E. Latham, and A. Pouget. Efficient computation and cue integration with noisy population codes. Nature Neuroscience, 4(8):826–831, 2001. [11] J. I. Gold and M. N. Shadlen. Representation of a perceptual decision in developing oculomotor commands. Nature, 404(390-394), 2000. [12] N. J. Gordon, D. J. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-gaussian bayesian state estimation. IEE-Proceedings-F, 140:107–113, 1993. [13] D. J. Heeger. Modeling simple cell direction selectivity with normalized half-squared, linear operators. Journal of Neurophysiology, 70:1885–1897, 1993. [14] D. J. Heeger. Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9:181–198, 1993. [15] P. E. Latham, S. Deneve, and A. Pouget. Optimal computation with attractor networks. Journal of Physiology Paris, 97(683-694), 2003. [16] W. J. Ma, J. M. Beck, P. E. Latham, and A. Pouget. Bayesian inference with probabilistic population codes. Nature Neuroscience, 9(11):1432–1438, 2006. [17] R. P. N. Rao. Bayesian computation in recurrent neural circuits. Neural Computation, 16:1–38, 2004. [18] R. P. N. Rao. Hierarchical bayesian inference in networks of spiking neurons. In Advances in Neural Information Processing Systems, volume 17, 2005. [19] M. Sahani and P. Dayan. Doubly distributional population codes: simultaneous representation of uncertainty and multiplicity. Neural Computation, 15:2255–2279, 2003. [20] G. Sclar and R. D. Freeman. Orientation selectivity in the cat’s striate cortex is invariant with stimulus contrast. Experimental Brain Research, 46:457–461, 1982. [21] J. S. Taube, R. U. Muller, and J. B. Ranck. Head-direction cells recorded from postsubiculum in freely moving rats. i. description and quantitative analysis. Journal of Neuroscience, 10(2):420–435, 1990. [22] J. S. Taube, R. U. Muller, and J. B. Ranck. Head-direction cells recorded from postsubiculum in freely moving rats. ii. effects of environmental manipulations. Journal of Neuroscience, 10(2):436–447, 1990. [23] S. I. Wiener and J. S. Taube. Head direction cells and the neural mechanisms of spatial orientation. MIT Press, 2005. [24] L.-G. Wu and P. Saggau. Presynaptic inhibition of elicited neurotransmitter release. Trends in Neuroscience, 20:204–212, 1997. [25] X. Xie, R. H. Hahnloser, and H. S. Seung. Double-ring network model of the head-direction system. Physical Review E, 66:0419021–0419029, 2002. [26] K. Zhang. Representation of spatial orientation by the intrinsic dynamics of the head-direction cell ensemble: a theory. Journal of Neuroscience, 16(6):2112–2126, 1996. 9
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An LP View of the M-best MAP problem Menachem Fromer Amir Globerson School of Computer Science and Engineering The Hebrew University of Jerusalem {fromer,gamir}@cs.huji.ac.il Abstract We consider the problem of finding the M assignments with maximum probability in a probabilistic graphical model. We show how this problem can be formulated as a linear program (LP) on a particular polytope. We prove that, for tree graphs (and junction trees in general), this polytope has a particularly simple form and differs from the marginal polytope in a single inequality constraint. We use this characterization to provide an approximation scheme for non-tree graphs, by using the set of spanning trees over such graphs. The method we present puts the M-best inference problem in the context of LP relaxations, which have recently received considerable attention and have proven useful in solving difficult inference problems. We show empirically that our method often finds the provably exact M best configurations for problems of high tree-width. A common task in probabilistic modeling is finding the assignment with maximum probability given a model. This is often referred to as the MAP (maximum a-posteriori) problem. Of particular interest is the case of MAP in graphical models, i.e., models where the probability factors into a product over small subsets of variables. For general models, this is an NP-hard problem [11], and thus approximation algorithms are required. Of those, the class of LP based relaxations has recently received considerable attention [3, 5, 18]. In fact, it has been shown that some problems (e.g., fixed backbone protein design) can be solved exactly via sequences of increasingly tighter LP relaxations [13]. In many applications, one is interested not only in the MAP assignment but also in the M maximum probability assignments [19]. For example, in a protein design problem, we might be interested in the M amino acid sequences that are most stable on a given backbone structure [2]. In cases where the MAP problem is tractable, one can devise tractable algorithms for the M best problem [8, 19]. Specifically, for low tree-width graphs, this can be done via a variant of max-product [19]. However, when finding MAPs is not tractable, it is much less clear how to approximate the M best case. One possible approach is to use loopy max-product to obtain approximate max-marginals and use those to approximate the M best solutions [19]. However, this is largely a heuristic and does not provide any guarantees in terms of optimality certificates or bounds on the optimal values. LP approximations to MAP do enjoy such guarantees. Specifically, they provide upper bounds on the MAP value and optimality certificates. Furthermore, they often work for graphs with large tree-width [13]. The goal of the current work is to leverage the power of LP relaxations to the M best case. We begin by focusing on the problem of finding the second best solution. We show how it can be formulated as an LP over a polytope we call the “assignment-excluding marginal polytope”. In the general case, this polytope may require an exponential number of inequalities, but we prove that when the graph is a tree it has a very compact representation. We proceed to use this result to obtain approximations to the second best problem, and show how these can be tightened in various ways. Next, we show how M best assignments can be found by relying on algorithms for 1 second best assignments, and thus our results for the second best case can be used to devise an approximation algorithm for the M best problem. We conclude by applying our method to several models, showing that it often finds the exact M best assignments. 1 The M-best MAP problem and its LP formulation Consider a function on n variables defined as: f(x1, . . . , xn; θ) = X ij∈E θij(xi, xj) + X i∈V θi(xi) (1) where V and E are the vertices and nodes of a graph G with n nodes. We shall be interested in the M assignments with largest f(x; θ) value.1 Denote these by x(1), . . . , x(M), so that x(1) is the assignment that maximizes f(x; θ), x(2) is the 2nd best assignment, etc. The MAP problem (i.e., finding x(1)) can be formulated as an LP as follows [15]. Let µ be a vector of distributions that includes {µij(xi, xj)}ij∈E over edge variables and {µi(xi)}i∈V over nodes. The set of µ that arise from some joint distribution is known as the marginal polytope [15] and is denoted by M(G). Formally: M(G) = {µ | ∃p(x) ∈∆s.t. p(xi, xj) = µij(xi, xj) , p(xi) = µi(xi)} . where ∆is the set of distributions on x. The MAP problem can then be shown to be equivalent to the following LP:2 max x f(x; θ) = max µ∈M(G) µ · θ , (2) It can be shown that this LP always has a maximizing µ that is a vertex of M(G) and is integral. Furthermore, this µ corresponds to the MAP assignment x(1). Although the number of variables in this LP is only O(|E|+|V |), the difficulty comes from an exponential number of linear inequalities generally required to describe the marginal polytope M(G). We shall find it useful to define a mapping between assignments x and integral vertices of the polytope. Given an integral vertex v ∈M(G), define x(v) to be the assignment that maximizes vi(xi). And, given an assignment z define v(z) to be the integral vertex in M(G) corresponding to the assignment z. Thus the LP in Eq. 2 will be maximized by v(x(1)). One simple outer bound of the marginal polytope is the local polytope ML(G), which only enforces pairwise constraints between variables: ML(G) = µ ≥0 X xj µij(xi, xj) = µi(xi), X xi µij(xi, xj) = µj(xj), X xi µi(xi) = 1 (3) The LP relaxation is then to maximize µ·θ where µ ∈ML(G). For tree structured graphs, ML(G) = M(G) [15] and thus the LP relaxation yields the exact MAP x(1). 2 An LP Formulation for the 2nd-best MAP Assume we found the MAP assignment x(1) and are now interested in finding x(2). Is there a simple LP whose solution yields x(2)? We begin by focusing on the case where G is a tree so that the local LP relaxation is exact. We first treat the case of a connected tree. To construct an LP whose solution is x(2), a natural approach is to use the LP for x(1) (i.e., the LP in Eq. 2) but somehow eliminate the solution x(1) using additional constraints. This, however, is somewhat trickier than it sounds. The key difficulty is that the new constraints should not generate fractional vertices, so that the resulting LP is still exact. We begin by defining the polytope over which we need to optimize in order to obtain x(2). 1This is equivalent to finding the maximum probability assignments for a model p(x) ∝ef(x;θ). 2We use the notation µ · θ = P ij∈E P xi,xj µij(xi, xj)θij(xi, xj) + P i P xi µi(xi)θi(xi) 2 Definition 1. The assignment-excluding marginal polytope is defined as: ˆ M(G, z) = {µ | ∃p(x) ∈∆s.t. p(z) = 0, p(xi, xj) = µij(xi, xj), p(xi) = µi(xi)} . (4) ˆ M(G, z) is simply the convex hull of all (integral) vectors v(x) for x ̸= z. The following result shows that optimizing over ˆ M(G, x(1)) will yield the second best solution x(2), so that we refer to ˆ M(G, x(1)) as the second-best marginal polytope. Lemma 1. The 2nd best solution is obtained via the following LP: maxx̸=x(1) f(x; θ) = maxµ∈ˆ M(G,x(1)) µ · θ. Furthermore, the µ that maximizes the LP on the right is integral and corresponds to the second-best MAP assignment x(2). The proof is similar to that of Eq. 2: instead of optimizing over x, we optimize over distributions p(x), while enforcing that p(x(1)) = 0 so that x(1) is excluded from the maximization. The key question which we now address is how to obtain a simple characterization of ˆ M(G, z). Intuitively, it would seems that ˆ M(G, z) should be “similar” to M(G), such that it can be described as M(G) plus some constraints that “block” the assignment z. To illustrate the difficulty in finding such “blocking” constraints, consider the following constraint, originally suggested by Santos [10]: P i µi(zi) ≤n −1. This inequality is not satisfied by µ = v(z) since v(z) attains the value n for the LHS of the above. Furthermore, for any x ̸= z and µ = v(x), the LHS would be n−1 or less. Thus, this inequality separates v(z) from all other integral vertices. One might conclude that we can define ˆ M(G, z) by adding this inequality to M(G). The difficulty is that the resulting polytope has fractional vertices,3 and maximizing over it won’t generally yield an integral solution. It turns out that there is a different inequality that does yield an exact characterization of ˆ M(G, z) when G is a tree. We now define this inequality and state our main theorem. Definition 2. Consider the functional I(µ, z) (which is linear in µ): I(µ, z) = X i (1 −di)µi(zi) + X ij∈E µij(zi, zj) (5) where di is the degree of node i in the tree graph G. Theorem 1. Adding the single inequality I(µ, z) ≤0 to M(G) yields ˆ M(G, z). ˆ M(G, z) = {µ | µ ∈M(G), I(µ, z) ≤0 } (6) The theorem is proved in the appendix. Taken together with Lemma 1, it implies that x(2) may be obtained via an LP that is very similar to the MAP-LP, but has an additional constraint. We note the interesting similarity between I(µ, z) and the Bethe entropy [20]. The only difference is that in Bethe, µi, µij are replaced by H(Xi), H(Xi, Xj) respectively.4 The theorem also generalizes to the case where G is not a tree, but we have a junction tree for G. In this case, the theorem still holds if we define a generalized I(µ, z) inequality as: X S∈S (1 −dS)µS(zS) + X C∈C µC(zC) ≤0 (7) where C and S are the junction tree cliques and their separators, respectively, and dS is the number of cliques that intersect on separator S. In this case, the marginal polytope should enforce consistency between marginals µC(zC) and their separators µS(zS). However, such a characterization requires variables whose cardinality is exponential in the tree-width and is thus tractable only for graphs of low tree-width. In the next section, we address approximations for general graphs. A corresponding result exists for the case when G is a forest. In this case, the inequality in Eq. 6 is modified to: I(µ, z) ≤|P| −1, where |P| denotes the number of connected components of G. Interestingly, for a graph without edges, this gives the Santos inequality. 3Consider the case of a single edge between 2 nodes where the MAP assignment is (0, 0). Adding the inequality µ1(0) + µ2(0) ≤1 produces the fractional vertex (0.5, 0.5). 4The connection to Bethe can be more clearly understood from a duality-based proof of Theorem 1. We will cover this in an extended version of the manuscript. 3 3 2nd best LPs for general graphs - Spanning tree inequalities When the graph G is not a tree, the marginal polytope M(G) generally requires an exponential number of inequalities. However, as mentioned above, it does have an exact description in terms of marginals over cliques and separators of a junction tree. Given such marginals on junction tree cliques, we also have an exact characterization of ˆ M(G, z) via the constraint in Eq. 7. However, in general, we cannot afford to be exponential in tree-width. Thus a common strategy [15] is to replace M(G) with an outer bound that enforces consistency between marginals on overlapping sets of variables. The simplest example is ML(G) in Eq. 3. In what follows, we describe an outer-bound approximation scheme for ˆ M(G, z). We use ML(G) as the approximation for M(G) (more generally ML(G) can enforce consistency between any set of small regions, e.g., triplets). When G is not a tree, the linear constraint in Eq. 6 will no longer suffice to derive ˆ M(G, z). Moreover, direct application of the inequality will incorrectly remove some integral vertices. An alternative approach is to add inequalities that separate v(z) from the other integral vertices. This will serve to eliminate more and more fractional vertices, and if enough constraints are added, this may result in an integral solution. One obvious family of such constraints are those corresponding to spanning trees in G and have the form of Eq. 5. Definition 3. Consider any T that is a spanning tree of G. Define the functional IT (µ, z): IT (µ, z) = X i (1 −dT i )µi(zi) + X ij∈T µij(zi, zj) (8) where dT i is the degree of i in T . We refer to IT (µ, z) ≤0 as a spanning tree inequality. For any sub-tree T of G, the corresponding spanning tree inequality separates the vertex v(z) from the other vertices. This can be shown via similar arguments as in the proof of Theorem 1. Note, however, that the resulting polytope may still have fractional vertices. The above argument shows that any spanning tree provides a separating inequality for ˆ M(G, z). In principle, we would like to use as many such inequalities as possible. Definition 4. The spanning tree assignment-excluding marginal polytope is defined as: ˆ MST L (G, z) = µ | µ ∈ML(G), ∀tree T ⊆E IT (µ, z) ≤0 (9) where the ST notation indicates the inclusion of all spanning tree inequalities for G.5 Thus, we would actually like to perform the following optimization problem: max µ∈ˆ MST L (G,z) µ·θ as an approximation to optimization over ˆ M(G, z); i.e., we seek the optimal µ subject to all spanning tree inequalities for G with the ambition that this µ be integral and thus provide the non-z MAP assignment, with a certificate of optimality. Although the number of spanning trees is exponential in n, it turns out that all spanning inequalities can be used in practice. One way to achieve this is via a cutting plane algorithm [12] that finds the most violated spanning tree inequality and adds it to the LP. To implement this efficiently, we note that for a particular µ and a spanning tree T , the value of IT (µ, z) can be decomposed into a sum over the edges in T (and a T -independent constant): IT (µ, z) = X ij∈T h µij(zi, zj) −µi(zi) −µj(zj) i + X i µi(zi) (10) The tree maximizing the above is the maximum-weight spanning tree with edge-weights wij = µij(zi, zj) −µi(zi) −µj(zj). It can thus be found efficiently. The cutting plane algorithm proceeds as follows. We start by adding an arbitrary spanning tree. Then, as long as the optimal µ is fractional, we find the spanning tree inequality that µ most violates (where this is implemented via the maximum-weight spanning tree). This constraint will necessarily remove µ from the polytope. If there are no violated inequalities 5Note that ˆ M(G, z) ⊆ˆ MST L (G, z) ⊂ML(G). 4 but µ is still fractional, then spanning tree inequalities do not suffice to find an integral solution (but see below on hypertree constraints to add in this case). In practice, we found that only a relatively small number of inequalities are needed to successfully yield an integral solution, or determine that all such inequalities are already satisfied. An alternative approach for solving the all spanning-tree problem is to work via the dual. The dual variables roughly correspond to points in the spanning tree polytope [16], optimization over which can be done in polynomial time, e.g., via the ellipsoid algorithm. We do not pursue this here since the cutting plane algorithm performed well in our experiments. As mentioned earlier, we can exactly characterize ˆ M(G, z) using Eq. 7, albeit at a cost exponential in the tree-width of the graph. A practical compromise would be to use inequalities over clique trees of G, where the cliques are relatively small, e.g., triplets. The corresponding constraint (Eq. 7 with the small cliques and their separators) will necessarily separate v(z) from the other integral vertices. Finding the maximally violated such inequality is an NP-hard problem, equivalent to a prize collecting Steiner tree problem, but recent work has found that such problems are often exactly solvable in practice [7]. It thus might be practical to include all such trees as constraints using a cutting plane algorithm. 4 From 2nd-best to M-best Thus far, we only dealt with the 2nd best case. As we show now, it turns out that the 2nd-best formalism can be used to devise an algorithm for M best. We begin by describing an algorithm for the exact M best and then show how it can be used to approximate those via the approximations for 2nd best described above. Fig. 1 describes our scheme, which we call Partitioning for Enumerating Solutions (or PES) for solving the M best problem. The scheme is general and only assumes that MAP-“like” problems can be solved. It is inspired by several pre-existing M best solution schemes [4, 6, 8, 19] but differs from them in highlighting the role of finding a second best solution within a given subspace. for m ←1 to M do if m = 1 then Run MAP solver to obtain the best assignment: x(1) ≡arg max f(x; θ) CONSTRAINTS1 ←∅ else k ←− arg max k′∈{1,...,m−1} f(y(k′); θ) // sub-space containing mth best assignment x(m) ←y(k) // mth best assignment // A variable choice that distinguishes x(m) from x(k): (v, a) ←any member of the set {(i, x(m) i ) : x(m) i ̸= x(k) i } CONSTRAINTSm ←CONSTRAINTSk ∪{xv = a} // Eliminate x(k) (as MAP) from subspace m CONSTRAINTSk ←CONSTRAINTSk ∪{xv ̸= a} // Eliminate x(m) (as 2nd-best) from subspace k y(k) ←CalcNextBestSolution(CONSTRAINTSk, x(k)) end y(m) ←CalcNextBestSolution(CONSTRAINTSm, x(m)) end return {x(m)}M m=1 /* Find next best solution in sub-space defined by CONSTRAINTS */ Function CalcNextBestSolution(CONSTRAINTS, x(∗)) // x(∗) is the MAP in the sub-space defined by CONSTRAINTS: Run MAP solver to obtain the second-best solution: y ≡ arg max x̸=x(∗),CONSTRAINTS f(x; θ), and return y. end Figure 1: Pseudocode for the PES algorithm. The modus operandi of the PES algorithm is to efficiently partition the search space while systematically excluding all previously determined assignments. Significantly, any MAP 5 Attractive Grids Mixed Grids Hard Protein SCP Ranks Run-times Ranks Run-times Ranks Run-times S N B 0 50 S N B 0 0.5 1 S+R N+R B+R 0 50 S+R N+R B+R 0 0.5 1 S+R B B+R 0 50 S+R B B+R 0 0.5 1 Figure 2: Number of best ranks and normalized run-times for the attractive and mixed grids, and the more difficult protein SCP problems. S, N, and B denote the STRIPES, Nilsson, and BMMF algorithms. Algorithms marked with +R denote that regions of variables were added for those runs. solver can be plugged into it, on the condition that it is capable of solving the arg max in the CalcNextBestSolution subroutine. The correctness of PES can be shown by observing that at the M th stage, all previous best solutions are excluded from the optimization and no other assignment is excluded. Of note, this simple partitioning scheme is possible due to the observation that the first-best and second-best MAP assignments must differ in the assignment of at least one variable in the graph. The main computational step of the PES algorithm is to maximize f(x; θ) subject to x ̸= x(∗) and x ∈CONSTRAINTS (see the CalcNextBestSolution subroutine). The CONSTRAINTS set merely enforces that some of the coordinates of x are either equal to or different from specified values.6 Within the LP, these can be enforced by setting µi(xi = a) = 1 or µi(xi = a) = 0. It can be shown that if one optimizes µ · θ with these constraints and µ ∈ ˆ M(G, x(∗)), the solution is integral. Thus, the only element requiring approximation in the general case is the description of ˆ M(G, x(∗)). We choose as this approximation the polytope ˆ MST L (G, x(∗)) in Eq. 9. We call the resulting approximation algorithm Spanning TRee Inequalities and Partitioning for Enumerating Solutions, or STRIPES. In the next section, we evaluate this scheme experimentally. 5 Experiments We compared the performance of STRIPES to the BMMF algorithm [19] and the Lawler/Nilsson algorithm [6, 8]. Nilsson’s algorithm is equivalent to PES where the 2nd best assignment is obtained from maximizations within O(n) partitions, so that its runtime is O(n) times the cost of finding a single MAP. Here we approximated each MAP with its LP relaxation (as in STRIPES), so that both STRIPES and Nilsson come with certificates of optimality when their LP solutions are integral. BMMF relies on loopy BP to approximate the M best solutions.7 We used M = 50 in all experiments. To compare the algorithms, we pooled all their solutions, noting the 50 top probabilities, and then counted the fraction of these that any particular algorithm found (its solution rank). For run-time comparisons, we normalized the times by the longest-running algorithm for each example. We begin by considering pairwise MRFs on binary grid graphs of size 10 × 10. In the first experiment, we used an Ising model with attractive (submodular) potentials, a setting in which the pairwise LP relaxation is exact [14]. For each grid edge ij, we randomly chose Jij ∈[0, 0.5], and local potentials were randomized in the range ±0.5. The results for 25 graphs are shown in Fig. 2. Both the STRIPES and Nilsson algorithms obtained the 50 optimal solutions (as learned from their optimality certificates), while BMMF clearly fared less well for some of the graphs. While the STRIPES algorithm took < 0.5 to 2 minutes to run, the Nilsson algorithm took around 13 minutes. On the other hand, BMMF was quicker, taking around 10 seconds per run, while failing to find a significant portion of the top solutions. Overall, the STRIPES algorithm was required to employ up to 19 spanning tree inequalities per calculation of second-best solution. 6This is very different from the second best constraint, since setting x1 = 1 blocks all assignments with this value, as opposed to setting x = 1 which blocks only the assignment with all ones. 7For BMMF, we used the C implementation at http://www.cs.huji.ac.il/~talyam/ inference.html. The LPs for STRIPES and Nilsson were solved using CPLEX. 6 Next, we studied Ising models with mixed interaction potentials (with Jij and the local potentials randomly chosen in [−0.5, 0.5]). For almost all of the 25 models, all three algorithms were not able to successfully find the top solutions. Thus, we added regions of triplets (two for every grid face) to tighten the LP relaxation (for STRIPES and Nilsson) and to perform GBP instead of BP (for BMMF). This resulted in STRIPES and Nilsson always provably finding the optimal solutions, and BMMF mostly finding these solutions (Fig. 2). For these more difficult grids, however, STRIPES was the fastest of the algorithms, taking 0.5 - 5 minutes. On the other hand, the Nilsson and BMMF algorithms took 18 minutes and 2.5 7 minutes, respectively. STRIPES added up to 23 spanning tree inequalities per iteration. The protein side-chain prediction (SCP) problem is to to predict the placement of amino acid side-chains given a protein backbone [2, 18]. Minimization of a protein energy function corresponds to finding a MAP assignment for a pairwise MRF [19]. We employed the dataset of [18] (up to 45 states per variable, mean approximate tree-width 50), running all algorithms to calculate the optimal side-chain configurations. For 315 of 370 problems in the dataset, the first MAP solution was obtained directly as a result of the LP relaxation having an integral solution (“easy” problems). STRIPES provably found the subsequent top 50 solutions within 4.5 hours for all but one of these cases (up to 8 spanning trees per calculation), and BMMF found the same 50 solutions for each case within 0.5 hours; note that only STRIPES provides a certificate of optimality for these solutions. On the other hand, only for 146 of the 315 problems was the Nilsson method able to complete within five days; thus, we do not compare its performance here. For the remaining 55 (“hard”) problems (Fig. 2), we added problem-specific triplet regions using the MPLP algorithm [13]. We then ran the STRIPES algorithm to find the optimal solutions. Surprisingly, it was able to exactly find the 50 top solutions for all cases, using up to 4 standard spanning tree inequalities per second-best calculation. The STRIPES run-times for these problems ranged from 6 minutes to 23 hours. On the other hand, whether running BMMF without these regions (BP) or with the regions (GBP), it did not perform as well as STRIPES in terms of the number of high-ranking solutions or its speed. To summarize, STRIPES provably found the top 50 solutions for 369 of the 370 protein SCP problems. 6 Conclusion In this work, we present a novel combinatorial object ˆ M(G, z) and show its utility in obtaining the M best MAP assignments. We provide a simple characterization of it for tree structured graphs, and show how it can be used for approximations in non-tree graphs. As with the marginal polytope, many interesting questions arise about the properties of ˆ M(G, z). For example, in which non-tree cases can we provide a compact characterization (e.g., as for the cut-polytope for planar graphs [1]). Another compelling question is in which problems the spanning tree inequalities are provably optimal. An interesting generalization of our method is to predict diverse solutions satisfying some local measure of “distance” from each other, e.g., as in [2]. Here we studied the polytope that results from excluding one assignment. An intriguing question is to characterize the polytope that excludes M assignments. We have found that it does not simply correspond to adding M constraints I(µ, zi) ≤0 for i = 1, . . . , M, so its geometry is apparently more complicated than that of ˆ M(G, z). Here we used LP solvers to solve for µ. Such generic solvers could be slow for large-scale problems. However, in recent years, specialized algorithms have been suggested for solving MAP-LP relaxations [3, 5, 9, 17]. These use the special form of the constraints to obtain local-updates and more scalable algorithms. We intend to apply these schemes to our method. Finally, our empirical results show that our method indeed leverages the power of LP relaxations and yields exact M best optimal solutions for problems with large tree-width. Acknowledgements We thank Nati Linial for his helpful discussions and Chen Yanover and Talya Meltzer for their insight and help in running BMMF. We also thank the anonymous reviewers for their useful advice. 7 A Proof of Theorem 1 Recall that for any µ ∈M(G), there exists a probability density p(x) s.t. µ = P x p(x)v(x). Denote pµ(z) as the minimal value of p(z) among all p(x) that give µ. We prove that pµ(z) = max(0, I(µ, z)), from which the theorem follows (since pµ(z) = 0 iffµ ∈ˆ M(G, z)). The proof is by induction on n. For n = 1, the node has degree 0, so I(µ, z) = µ1(z1). Clearly, pµ(z) = µ1(z1), so pµ(z) = I(µ, z). For n > 1, there must exist a leaf in G (assume that its index is n and its neighbor’s is n −1). Denote ˆG as the tree obtained by removing node n and its edge with n −1. For any assignment x, denote ˆx as the corresponding sub-assignment for the first n −1 variables. Also, any µ can be derived by adding appropriate coordinates to a unique ˆµ ∈M( ˆG). For an integral vertex µ = v(x), denote its projected ˆµ as ˆv(ˆx). Denote by ˆI(ˆµ, ˆz) the functional in Eq. 5 applied to ˆG. For any µ and its projected ˆµ, it can be seen that: I(µ, z) = ˆI(ˆµ, ˆz) −α (11) where we define α = P xn̸=zn µn−1,n(zn−1, xn) (so 0 ≤α ≤1). The inductive assumption gives a ˆp(ˆx) that has marginals ˆµ and also ˆp(ˆz) = max(0, I(ˆµ, ˆz)). We next use ˆp(ˆx) to construct a p(x) that has marginals µ and the desired minimal pµ(z). Consider three cases: I. I(µ, z) ≤0 and ˆI(ˆµ, ˆz) ≤0. From the inductive assumption, ˆpˆµ(ˆz) = 0, so we define: p(x) = ˆp(ˆx)µn−1,n(xn−1, xn) µn−1(xn−1) (12) which indeed marginalizes to µ, and p(z) = 0 so that pµ(z) = 0 as required. If µn−1(xn−1) = 0, then ˆp(ˆx) is necessarily 0, in which case we define p(x) = 0. Note that this construction is identical to that used in proving that ML(G) = M(G) for a tree graph G. II. I(µ, z) > 0. Based on Eq. 11 and α ≥0, we have ˆI(ˆµ, ˆz) > 0. Applying the inductive assumption to ˆµ, we obtain ˆI(ˆµ, ˆz) = ˆpˆµ(ˆz) > 0. Now, define p(x) so that p(z) = I(µ, z): xl, l ≤n −2 δ(xn−1 = zn−1) δ(xn = zn) p(x) no constraint 0 no constraint As in Eq. 12 ∃l xl ̸= zl 1 0 0 1 ˆp(ˆx) ∀l xl = zl 1 0 µn−1,n(zn−1, xn) 1 I(µ, z) Simple algebra shows that p(x) is non-negative and has µ as marginals. We now show that p(z) is minimal. Based on the inductive assumption and Eq. 11, it can easily be shown that I(v(z), z) = 1, I(v(x), z) ≤0 for x ̸= z. For any p(x) s.t. µ = P x p(x)v(x), from linearity, I(µ, z) = p(z) + P x̸=z p(x)I(v(x), z) ≤p(z) (since I(v(x), z) ≤0 for x ̸= z). Since the p(z) we define achieves this lower bound, it is clearly minimal. III. I(µ, z) ≤0 but ˆI(ˆµ, ˆz) > 0. Applying the inductive assumption to ˆµ, we see that ˆpˆµ(ˆz) = ˆI(ˆµ, ˆz) > 0; Eq. 11 implies α −ˆI(ˆµ, ˆz) ≥0. Define β = µn−1(zn−1) −ˆpˆµ(ˆz), which is non-negative since µn−1(zn−1) = ˆµn−1(ˆzn−1) and ˆp marginalizes to ˆµ. Define p(x) as: xl, l ≤n −2 δ(xn−1 = zn−1) δ(xn = zn) p(x) no constraint 0 no constraint As in Eq. 12 ∃l xl ̸= zl 1 0 ˆp(ˆx) µn−1,n(zn−1,xn) β α−ˆI(ˆµ,ˆz) α 1 ˆp(ˆx) µn−1,n(zn−1,zn) β ∀l xl = zl 1 0 ˆI(ˆµ, ˆz) µn−1,n(zn−1,xn) α 1 0 which indeed marginalizes to µ, and p(z) = 0 so that pµ(z) = 0, as required. 8 References [1] F. Barahona. On cuts and matchings in planar graphs. Math. Program., 60(1):53–68, 1993. [2] M. Fromer and C. Yanover. Accurate prediction for atomic-level protein design and its application in diversifying the near-optimal sequence space. Proteins: Structure, Function, and Bioinformatics, 75:682–705, 2009. [3] A. Globerson and T. Jaakkola. Fixing max-product: Convergent message passing algorithms for MAP LP-relaxations. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 21. MIT Press, Cambridge, MA, 2007. [4] E. Kloppmann, G. M. Ullmann, and T. Becker. An extended dead-end elimination algorithm to determine gap-free lists of low energy states. Journal of Comp. Chem., 28:2325–2335, 2007. [5] N. Komodakis and N. Paragios. Beyond loose LP-relaxations: Optimizing MRFs by repairing cycles. In D. Forsyth, P. Torr, and A. Zisserman, editors, ECCV, pages 806–820, Heidelberg, Germany, 2008. Springer. [6] E. L. Lawler. A procedure for computing the K best solutions to discrete optimization problems and its application to the shortest path problem. Management Science, 18(7):401–405, 1972. [7] I. Ljubic, R. Weiskircher, U. Pferschy, G. W. Klau, P. Mutzel, and M. Fischetti. An algorithmic framework for the exact solution of the prize-collecting steiner tree problem. Mathematical Programming, 105:427–449, Feb 2006. [8] D. Nilsson. An efficient algorithm for finding the M most probable configurations in probabilistic expert systems. Statistics and Computing, 8:159–173, Jun 1998. [9] P. Ravikumar, A. Agarwal, and M. Wainwright. Message-passing for graph-structured linear programs: proximal projections, convergence and rounding schemes. In Proc. of the 25th international conference on Machine learning, pages 800–807, New York, NY, USA, 2008. ACM. [10] E. Santos. On the generation of alternative explanations with implications for belief revision. In Proc. of the 7th Annual Conference on Uncertainty in Artificial Intelligence, 1991. [11] Y. Shimony. Finding the MAPs for belief networks is NP-hard. Aritifical Intelligence, 68(2):399–410, 1994. [12] D. Sontag and T. Jaakkola. New outer bounds on the marginal polytope. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1393–1400. MIT Press, Cambridge, MA, 2007. [13] D. Sontag, T. Meltzer, A. Globerson, T. Jaakkola, and Y. Weiss. Tightening LP relaxations for MAP using message passing. In Proc. of the 24th Annual Conference on Uncertainty in Artificial Intelligence, pages 503–510, 2008. [14] B. Taskar, S. Lacoste-Julien, and M. I. Jordan. Structured prediction, dual extragradient and bregman projections. J. Mach. Learn. Res., 7:1627–1653, 2006. [15] M. Wainwright and M. Jordan. Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn., 1(1-2):1–305, 2008. [16] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition function. IEEE Transactions on Information Theory, 51(7):2313–2335, 2005. [17] T. Werner. A linear programming approach to max-sum problem: A review. IEEE Trans. Pattern Anal. Mach. Intell., 29(7):1165–1179, 2007. [18] C. Yanover, T. Meltzer, and Y. Weiss. Linear programming relaxations and belief propagation – an empirical study. Journal of Machine Learning Research, 7:1887–1907, 2006. [19] C. Yanover and Y. Weiss. Finding the M most probable configurations using loopy belief propagation. In Advances in Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004. [20] J. Yedidia, W. W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. on Information Theory, 51(7):2282– 2312, 2005. 9
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Speeding up Magnetic Resonance Image Acquisition by Bayesian Multi-Slice Adaptive Compressed Sensing Matthias W. Seeger Saarland University and Max Planck Institute for Informatics Campus E1.4, 66123 Saarbr¨ucken, Germany mseeger@mmci.uni-saarland.de Abstract We show how to sequentially optimize magnetic resonance imaging measurement designs over stacks of neighbouring image slices, by performing convex variational inference on a large scale non-Gaussian linear dynamical system, tracking dominating directions of posterior covariance without imposing any factorization constraints. Our approach can be scaled up to high-resolution images by reductions to numerical mathematics primitives and parallelization on several levels. In a first study, designs are found that improve significantly on others chosen independently for each slice or drawn at random. 1 Introduction Magnetic resonance imaging (MRI) [10, 6] is a very flexible imaging modality. Inflicting no harm on patients, it is used for an ever-growing number of diagnoses in health-care. Its most serious limitation is acquisition speed, being based on a serial idea (gradient encoding) with limited scope for parallelization. Fourier (aka. k-space) coefficients are sampled along smooth trajectories (phase encodes), many of which are needed for reconstructions of sufficient quality [17, 1]. Long scan times lead to patient annoyance, grave errors due to movement, and high running costs. The Nyquist sampling theorem [2] fundamentally limits traditional linear image reconstruction, but with modern 3D MRI scenarios, dense sampling is not practical anymore. Acquisition is accelerated to some extent in parallel MRI1, by using receive coil arrays [19, 9]: the sensitivity profiles of different coils provide part of the localization normally done by more phase steps. A different idea is to use (nonlinear) sparse image reconstruction, with which the Nyquist limit can be undercut robustly for images, emphasized recently as compressed sensing [5, 3]. While sparse reconstruction has been used for MRI [28, 12], we address the more fundamental question of how to optimize the sampling design for sparse reconstruction over a specific real-world signal class (MR images) in an adaptive manner, avoiding strong assumptions such as exact, randomly distributed sparsity that do not hold for real images [23]. Our approach is in line with recent endeavours to extend MRI capabilities and reduce its cost, by complementing expensive, serial hardware with easily parallelizable digital computations. We extend the framework of [24], the first approximate Bayesian method for MRI sampling optimization applicable at resolutions of clinical interest. Their approach falls short of real MRI practice on a number of points. They considered single image slices only, while stacks2 of neighbouring 1While parallel MRI is becoming the standard, its use is not straightforward. The sensitivity maps are unknown up front, depend partly on what is scanned, and their reliable estimation can be difficult. 2“Stack-of-slices” acquisition along the z axis works by transmitting a narrow-band excitation pulse while applying a magnetic field gradient linear in z. If the echo time (between excitation and readout) is shorter than 1 slices are typically acquired. Reconstruction can be improved significantly by taking the strong statistical dependence between pixels of nearby slices into account [14, 26, 18]. Design optimization is a joint problem as well: using the same acquisition pattern for neighbouring slices is clearly redundant. Second, the latent image was modelled as real-valued in [24], while in reality it is a complex-valued signal. To our knowledge, the few directly comparable approaches rely on “trialand-error” exploration [12, 16, 27], requiring substantially more human expert interventions and real MRI measurements, whose high costs our goal-directed method aims to minimize. Our extension to stacks of slices requires new technology. Global Gaussian covariances have to be approximated, a straightforward extension of which to many slices is out of the question. We show how to use approximate Kalman smoothing, implementing message passing by the Lanczos algorithm, which has not been done in machine learning before (see [20, 25] for similar proposals to oceanography problems). Our technique is complementary to mean field variational inference approximations (“variational Bayes”), where most correlations are ruled out a priori. We track the dominating posterior covariance directions inside our method, allowing them to change during optimization. While our double loop approach may be technically more demanding to implement, relaxation as well as algorithm are characterized much better (convex problem; algorithm reducing to standard computational primitives), running orders of magnitude faster. Beyond MRI, applications could be to Bayesian inference over video streams, or to computational photography [11]. Our approach is parallelizable on several levels. This property is essential to even start projecting such applications: on the scale demanded by modern MRI applications, with practitioners being used to view images directly after acquisition, little else but highly parallelizable approaches are viable. Large scale variational inference is reviewed and extended to complex-valued data in Section 2, lifted to non-Gaussian linear dynamical systems in Section 3, and the experimental design extension is given in Section 4. Results of a preliminary study on data from a Siemens 3T scanner are provided in Section 5, using a serial implementation. 2 Large Scale Sparse Inference Our motivation is to improve MR image reconstruction, not by finding a better estimation technique, but by sampling data more economically. A latent MR image slice u ∈Cn (n pixels) is measured by a design matrix X ∈Cm×n: y = Xu + ε (ε ∼N(0, σ2I) models noise). For Cartesian MRI, X = IS,·Fn, Fn the 2D fast Fourier transform, S ⊂{1, . . . , n} the sampling pattern (which partitions into complete columns or rows: phase encodes, the atomic units of the design). Sparse reconstruction works by encoding super-Gaussian image statistics in a non-Gaussian prior, then finding the posterior mode (MAP estimation): a convex quadratic program for the model employed here. To improve the measurement design X itself, posterior information beyond (and independent of) its mode is required, chiefly posterior covariances. We briefly review [24], extending it to complex-valued u. The super-Gaussian image prior P(u) is adapted by placing potentials on absolute values |sj|, the posterior has the form P(u|y) ∝N(y|Xu, σ2I) Yq j=1 e−τj|sj/σ|, s = Bu ∈Cq. Here, B is a sparsity transform [24]. We use the C →R2 embedding, s = (sj), sj ∈R2, and norm potentials e−τj∥sj/σ∥. Two main ideas lead to [24]. First, inference is relaxed to an optimization problem by lower-bounding the log partition function [7] (intuitively, each Laplace potential e−τj∥sj/σ∥is lower-bounded by a Gaussian-form potential of variance γj > 0), leading to φ(γ) = log |A| + h(γ) + minu R(u, γ), R := σ−2 ∥y −Xu∥2 + sT Γ−1s , γ = (γj), (1) h(γ) = (τ 2)T γ. This procedure implies a Gaussian approximation Q(u|y) = N(u|u∗, σ2A−1) to P(u|y), with A = XHX + BT Γ−1B and u∗= u∗(γ). The complex extension is formally similar to [24] (π there is γ−1 here): Γ := (diag γ)⊗I2 = diag(γ1, γ1, γ2, . . . )T , B := Borig⊗I2, Borig the real-valued sparsity transform. Q(u|y) is fitted to P(u|y) by minγ≻0 φ: a convex problem [24]. Used within an automatic decision architecture, convexity and robustness of inference become assets that are more important than smaller bias after a lot of human expert attention. the repeat time (between phase encodes), several slices are acquired in an interleaved fashion, separated by slice gaps to avoid crosstalk [17]. 2 Second, φ(γ) can be minimized very efficiently by a double loop algorithm [24]. The computationally intensive log |A| term is concave in γ−1. Upper-bounding it tangentially by the affine zT (γ−1) −g∗(z) at outer loop (OL) update points, the resulting φz ≥φ decouples and is minimized much more efficiently in inner loops (ILs). minγ≻0 φz leaves us with min u n φz(u) = σ−2∥y −Xu∥2 + 2 X j h∗ j(|sj|) o , h∗ j(|sj|) := τj(zj + (|sj|/σ)2)1/2, (2) a penalized least squares problem. At convergence, u∗= EQ[u|y], γj ←(zj + |s∗,j/σ|2)1/2/τj. We can use iteratively reweighted least squares (IRLS), each step of which needs a linear system to be solved of the structure of A. Refitting z (OL updates) is much harder: z ←(I ⊗ 1T ) diag−1(BA−1BT ) = (I ⊗1T )(σ−2VarQ[sj|y]). In terms of Gaussian (Markov) random fields, the inner optimization needs posterior mean computations only, while OL updates require bulk Gaussian variances [21, 15]. The reason why the double loop algorithm is much faster than previous approaches is that only few variance computations are required. The extension to complexvalued u is non-trivial only when it comes to IRLS search direction computations (see Appendix). Given multi-slice data (Xt, yt), t = 1, . . . , T, we can use an undirected hidden Markov model over image slices u = (ut) ∈CnT . By the stack-of-slices methodology, the likelihood potentials P(yt|ut) are independent, and P(ut) from above serves as single-node potential, based on st = But. If st→:= ut −ut+1, the dependence between neighbouring slices is captured by additional Laplace coupling potentials Qn i=1 e−τc,i|(st→)i/σ|. The variational parameters γt at each node are complemented by coupling parameters γt→∈Rn +. The Gaussian Q(u|y), y = (yt), has the same form as above with a huge A ∈CnT ×nT . Inheriting the Markov structure, it is a Gaussian linear dynamical system (LDS) with very high-dimensional states. How will an efficient extension of the double loop algorithm look like? The IL criterion φz should be coupled between neighbouring slices, by way of potentials on st→. OL updates are more difficult to lift: we have to approximate marginal variances in a Gaussian LDS. We will do this by Kalman smoothing, approximating inversion in message computations (conversion from natural to moment parameters) by the Lanczos algorithm. The central role of Gaussian covariance for approximating non-Gaussian posteriors has not been emphasized much in machine learning, where if Bayesian computations are intractable, simpler “variational Bayesian” concepts are routinely used, imposing factorization constraints on the posterior up front. While such constraints can be adjusted in light of the data, this is difficult and typically not done. Factorization assumptions are a double-edged sword: they radically simplify implementations, but result in non-convex algorithms, and half of the problem is left undone. Our approach offers an alternative: by using Lanczos on Q(u|y), we retain precisely the maximum-covariance directions of intermediate fits to the posterior, without running into combinatorial or non-convex problems. Finally, we place more varied sparsity penalties on the in-plane dimensions [24] than on the third one. This is justified by voxels typically being larger and spaced with a gap in the third dimension, with partial volume effects reducing sparsity. Moreover, a non-local sparsity transform along the third dimension would destroy the Markovian structure essential for efficient computation. 3 Approximate Inference over Multiple Slices We aim to extend the single slice method of [24] to the hidden Markov extension, thereby reusing code whenever possible. The variational criterion is (1) with h(γ) = X t ht(γt) + I{t<T }ht→(γt→), R = X t Rt + I{t<T }Rt→, Γt→:= (diag γt→) ⊗I2, Rt = σ−2 ∥yt −Xtut∥2 + sT t Γ−1 t st , Rt→= σ−2sT t→Γ−1 t→st→. The coupling term log |A| is upper-bounded (φ ≤φz), so that the IL criterion φz(u) is the sum of terms φt,zt(ut), φt→,zt→(st→). Problems of the form minu φz, jointly convex with couplings between neighbours, are routinely addressed in parallel convex optimization. In order to update ut, we consider its neighbours ut−1, ut+1 fixed, massaging φt,zt(ut) + φ(t−1)→,z(t−1)→(s(t−1)→) + φt→,zt→(st→) into the form of [24]: ˜B = (BT , I, I)T , ˜s = (sT t , (ut −ut−1)T , (ut −ut+1)T )T , ˜u = ut. These updates can be run asynchronously in parallel, sending ut to neighbours after every few IRLS steps. 3 For OL updates, we have to compute zt = σ−2(I ⊗1T )VarQ[st|y] and zt→= σ−2(I ⊗ 1T )VarQ[st→|y], where Q(u|y) is a Gaussian LDS (fixed γ). To output a global criterion value, an estimate of log |A| is required as well. We use the two-filter Kalman information smoother, which entails passing Gaussian-form messages along the chain in both directions. Once all messages are available, marginal (co)variances are computed at each node in parallel. Shift Q(u|y) to zero mean (EQ[u|y] = u∗is found in the IL). Denoting N U(A) = N U(u|A) := e−(1/2)σ−2uT Au, Q(u|y) consists of single node potentials Φt(ut) = N U(At) and pair potentials Φt→(st→) = N U(Γ−1 t→), where At := XH t Xt + BT Γ−1 t B. Defining messages Mt→(ut) = N U( ˜At→), M←t(ut) = N U( ˜A←t), the usual message propagation equation is Mt→(ut) ∝ R M(t−1)→(ut−1)Φ(t−1)→(s(t−1)→)dut−1Φt(ut), so that ˜At→= At + M( ˜A(t−1)→, Γ(t−1)→), M( ˜A, Γ) := Γ−1 −Γ−1( ˜A + Γ−1)−1Γ−1. (3) In the same way, ˜A←t = At + M( ˜A←(t+1), Γt→). Denote Mt→:= M( ˜At→, Γt→), M←t := M( ˜A←t, Γ(t−1)→). Once all messages have been computed, the node marginal Q(ut|y) has precision matrix ˜At := At + M(t−1)→+ M←(t+1). If Ψ := (δ1 −δ2) ⊗I, the precision matrix of Q(ut, ut+1|y) is diag( ˜At→, ˜A←(t+1)) + ΨΓ−1 t→ΨT , and st→= ΨT (uT t , uT t+1)T . CovQ[st→|y] can be written in terms of ˜A −1 t+1 and Mt→. Finally, by tracking normalization constants: log |A| = P t<˜t log | ˜At→+ Γ−1 t→| + P t>˜t log | ˜A←t + Γ−1 (t−1)→| + log | ˜A˜t| for any ˜t. In practice, we average over ˜t. The algorithm is sketched in Algorithm 1. Algorithm 1 Double loop variational inference algorithm repeat if first iteration then Default-initialize z ∝1, u = 0. else Run Kalman smoothing to determine Mt→, and (in parallel) M←t. Determine node variances zt, pair variances zt→, and log |A| from messages. Refit upper bound φz to φ (tangent at γ). Initialize u = u∗(previous solution). end if repeat Distributed IRLS to minimize minγ φz w.r.t. u. Each local update of ut entails solving a linear system (conjugate gradients). until u∗= argminu φz converged Update γj = (zj + |s∗,j/σ|2)1/2/τj. until outer loop converged For reconstruction, we run parallel MAP estimation. Following [12], we smooth out the nondifferentiable l1 penalty by |sj/σ| ≈(ε + |sj/σ|2)1/2 for very small ε > 0, then use nonlinear conjugate gradients with Armijo line search. Nodes return with ∇utφz at the line minimum ut, the next search direction is centrally determined and distributed (just a scalar has to be transferred). This is not the same as centralized CG: line searches are distributed and not done on the global criterion. We briefly comment on how to approximate Kalman message passing by way of the Lanczos algorithm [8], full details are given in [22]. Gaussian (Markov) random field practitioners will appreciate the difficulties: there is no locally connected MRF structure, and the Q(u|y) are highly nonstationary, being fitted to a posterior with non-Gaussian statistics (edges in the image, etc). Message passing requires the inversion of a precision matrix A. The idea behind Lanczos approximations is PCA: if A ≈UΛU T , Λ the l ≪n smallest eigenvalues, UΛ−1U T is the PCA approximation of A−1. With matrices A of certain spectral decay, this representation can be approximated by Lanczos (see [24, 22] for details). For a low rank PCA approximation of ˜At→, Mt→has the same rank (see Appendix), which allows to run Gaussian message passing tractably. In a parallel implementation, the forward and backward filter passes run in parallel, passing low rank messages (the rank km of these should be smaller than the rank kc for subsequent marginal covariance computations). On a lower level, both matrix-vector multiplications with Xt (FFT) and reorthogonalizations required during the Lanczos algorithm can easily be parallelized on commodity graphics hardware. 4 4 Sampling Optimization by Bayesian Experimental Design With our multi-slice variational inference algorithm in place, we address sampling optimization by Bayesian sequential experimental design, following [24]. At slice t, the information gain score ∆(X∗) := log |I +X∗CovQ[ut|y]XT ∗| is computed for a fixed number of phase encode candidates X∗∈Cd×n not yet in Xt, the score maximizer is appended, and a novel measurement is acquired (for the maximizer only). ∆(X∗) depends primarily on the marginal posterior covariance matrix CovQ[ut|y], computed by Gaussian message passing just as variances in OL updates above (while a single value ∆(X∗) can be estimated more efficiently, the dominating eigendirections of the global covariance matrix seem necessary to approximate many score values for different candidates X∗). Once messages have been passed, scores can be computed in parallel at different nodes. A purely sequential approach, extending one design Xt by one encode in each round, is not tractable. In practice, we extend several node designs Xt in each round (a fixed subset Cit ⊂{1, . . . , T}; “it” the round number). Typically, Cit repeats cyclically. This is approximate, since candidates are scored independently at each node. Certainly, Cit should not contain neighbouring nodes. In the interleaved stack-of-slices methodology, scan time is determined by the largest factor Xt (number of rows), so we strive for balanced designs here. To sum up, our adaptive design optimization algorithm starts with an initial variational inference phase for a start-up design (low frequencies only), then runs through a fixed number of design rounds. Each round starts with Gaussian message passing, based on which scores are computed at nodes t ∈Cit, new measurements are acquired, and designs Xt are extended. Finally, variational inference is run for the extended model, using a small number of OL iterations (only one in our experiments). Time can be saved by basing the first OL update on the same messages and node marginal covariances than the design score computations (neglecting their change through new phase encodes). 5 Experiments We present experimental results, comparing designs found by our Bayesian joint design optimization method against alternative choices on real MRI data. We use the model of Section 2, with the prior previously used in [24] (potentials of strength τa on wavelet coefficients, of strength τr on Cartesian finite differences). While the MRI signal u is complex-valued, phase contributions are mostly erroneous, and reconstruction as well as design optimization are improved by multiplying a further term Q i e−(τi/σ)|ℑ(ui)| into each single node prior potential, easily incorporated into the generic setup by appending I ⊗δT 2 to B. We focus on Cartesian MRI (phase encodes are complete columns3 in k-space): a more clinically relevant setting than spiral sampling treated in [24]. We use data of resolution 64×64 (in-plane) to test our approach with a serial implementation. While this is not a resolution of clinical relevance, a truly parallel implementation is required in order to run our method at resolutions 256 × 256 or beyond: an important point for future work. 5.1 Quality of Lanczos Variance Approximations We begin with experiments to analyze the errors in Lanczos variance approximations. Recall from [24] that variances are underestimated. We work with a single slice of resolution 64 × 64, using a design X of 30 phase encodes, running a single common OL iteration (default-initialized z), comparing different ways of continuing from there: exact z computations (Cholesky decomposition of A) versus Lanczos approximations with different numbers of steps k. Results are in Figure 1. While the relative approximation errors are rather large uniformly, there is a clear structure to them: the largest (and also the very smallest) true values zj are approximated significantly more accurately than smaller true values. This structure can be used to motivate why, in the presence of large errors over all coefficients, our inference still works well for sparse linear models, indeed in some cases better than if exact computations are used (Figure 1, upper right). The spectrum of A shows a roughly linear decay, so that the largest and smallest eigenvalues (and eigenvectors) are well-approximated 3Our data are sagittal head scans, where the frequency encode direction (along which oversampling is possible at no extra cost) is typically chosen vertically (the longer anatomic axis). 5 1000 2000 3000 4000 5000 6000 7000 8000 0 0.5 1 1.5 2 2.5 3 Spectrum of A 1 2 3 4 5 6 7 1.88 1.9 1.92 1.94 1.96 1.98 2 2.02 2.04 Outer loop iteration exact k=1500 k=750 k=500 k=100 Figure 1: Lanczos approximations of Gaussian variances, at beginning of second OL iteration, 64 × 64 data (upper left). Spectral decay of inverse covariance matrix A roughly linear (upper middle). l2 reconstruction error of posterior mean estimate after subsequent OL iterations, for exact variance computation vs. k = 250, 500, 750, 1500 Lanczos steps (upper right). Lower panel: Relative accuracy zj 7→zk,j/zj at beginning of second OL iteration, separately for “a” sites (on wavelet coefficients; red), “r” sites (on derivatives; blue), and “i” sites (on ℑ(u); green). by Lanczos, while the middle part of the spectrum is not penetrated. Contributions to the largest values zj come dominatingly from small eigenvalues (large eigenvalues of A−1), explaining their smaller relative error. On the other hand, smaller values zj are strongly underestimated (zk,j ≪zj), which means that the selective shrinkage effect underlying sparse linear models (shrink most coefficients strongly, but some not at all) is strengthened by these systematic errors. Finally, the IL penalties are τj(zj + |sj/σ|2)1/2, enforcing sparsity more strongly for smaller zj. Therefore, Lanczos approximation errors lead to strengthened sparsity in subsequent ILs, but least so for sites with largest true zj. As an educated guess, this effect might even compensate for the fact that Laplace potentials may not be sparse enough for natural images. 5.2 Joint Design Optimization We use sagittal head scan data of resolution 64 × 64 in-plane, 32 slices, acquired on a Siemens 3T scanner (phase direction anterior-posterior), see [22] for further details. We consider joint and independent MAP reconstruction (for the latter, we run nonlinear CG separately for each slice), for a number of different design choices: {Xt} optimized jointly by our method here [op-jt]; each Xt optimized separately, by running the complex variant of [24] on slice ut [op-sp]; Xt = X for all t, with X optimized on the most detailed slice (number 16, Figure 2, row 2 middle) [op-eq]; and encodes of each Xt drawn at random, from the density proposed in [12] [rd], respecting the typical spectral decay of images [4] (all designs contain the 8 lowest-frequency encodes). Results for rd are averaged over ten repetitions. For all setups but op-eq, Xt are different across t. Hyperparameters are adjusted based on MAP reconstruction results for a fixed design picked ad hoc (τa = τr = 0.01, τi = 0.1 in-plane; τc = 0.08 between slices), then used for all design optimization and MAP reconstruction runs. We run the op-jt optimization with an odd-even schedule {Cit} (all odd (even) t ∈0, . . . , T −1 for odd (even) “it”); results for two other schedules of period four come out very similar, but require more running time. For variational inference, we run 6 OL iterations in the initial, 1 OL iteration in each design round, with up to 30 IL steps (ILs in design rounds typically converged in 2–3 steps). The rank parameters (number of Lanczos steps)4 were km = 100, kc = 250 (here, ut has ˜n = 8192 real coefficients). Results are given in Figure 2. First, across all designs, joint MAP reconstruction improves significantly upon independent MAP reconstruction. This improvement is strongest by far for op-jt (see Figure 2, rows 3,4), which for joint reconstruction improves on all other variants significantly, especially with 16–30 phase 4We repeated op-jt partly with km = 250, with very similar MAP reconstruction errors for the final designs, but significantly longer run time. 6 10 15 20 25 30 35 40 45 1 2 3 4 5 6 7 Number phase encodes L2 reconstruction error op−jt op−sp op−eq rd(avg) 10 15 20 25 30 35 40 45 1 2 3 4 5 6 7 Number phase encodes L2 reconstruction error op−jt op−sp op−eq rd(avg) Figure 2: Top row: l2 reconstruction errors ∥|ˆuMAP| −|utrue|∥of MAP reconstruction for different measurement designs. Left: joint MAP reconstruction; right: independent MAP reconstruction of each slice. op-jt: {Xt} optimized jointly; op-sp: Xt optimized separately for each slice; op-eq: Xt = X, optimized on slice 16; rd: Xt variable density drawn at random (averaged over 10 repetitions). Rows 2–4: Images for op-jt (25 encodes), slices 15–17. Row 2: true images (range 0–0.35). Row 3: errors joint MAP. Row 4: errors indep. MAP (range 0–0.08). encodes, where scan time is reduced by a factor 2–4 (Nyquist sampling requires 64 phase encodes). op-eq does worst in this domain: with a model of dependencies between slices in place, it pays 7 off to choose different Xt for each slice. rnd does best from about 35 phase encodes on. While this suboptimal behaviour of our optimization will be analyzed more closely in future work, it is our experience so far that the gain in using greedy sequential Bayesian design optimization over simpler choices is generally largest below 1/2 Nyquist. 6 Conclusions We showed how to implement MRI sampling optimization by Bayesian sequential experimental design, jointly over a stack of neighbouring slices, extending the single slice technique of [24]. Restricting ourselves to undersampling of Cartesian encodes, our method can be applied in practice whenever dense Cartesian sampling is well under control (sequence modification is limited to skipping encodes). We exploit the hidden Markov structure of the model by way of a Lanczos approximation of Kalman smoothing. While the latter has been proposed for spatial statistics applications [20, 25], it has not been used for non-Gaussian approximate inference before, nor in the context of sparsity-favouring image models or non-linear experimental design. Our method is a general alternative to structured variational mean field approximations typically used for non-Gaussian dynamical systems, in that dominating covariances are tracked a posteriori, rather than eliminating most of them a priori through factorization assumptions. In a first study, we obtain encouraging results in the range below 1/2 Nyquist. In future work, we will develop a truly parallel implementation, with which higher resolutions can be processed. We are considering extensions of our design optimization technology to 3D MRI5 and to parallel MRI with receiver coil arrays [19, 9], whose combination with k-space undersampling can be substantially more powerful than each acceleration technique on its own [13]. Appendix For norm potentials, h∗ j(sj) = h∗ j(∥sj∥), and the Hessians to solve for IRLS Newton directions do not have the form of A anymore. In order to understand this, note that we do not use complex calculus here: s 7→|s| is not complex differentiable at any s ∈C. Rather, we use the C →R2 embedding, then standard real-valued optimization for variables twice the size. If θj := (h∗ j)′, ρj := (h∗ j)′′ at ∥sj∦= 0, then using ∇sj∥sj∥= sj/∥sj∥, we have ∇∇sjh∗ j = ρjI2 + κ2 j(∥sj∥2I2 − sjsT j ), κj := (θj/∥sj∥−ρj)1/2/∥sj∥. Since ∥sj∥2I2 −sjsT j = νsj(νsj)T , ν := δ2δT 1 −δ1δT 2 , the Hessian is XHX + BH(s)BT . If ˆs := ((diag κ) ⊗ν)s, then for any v ∈R2q: H(s)v = ((diag ρ)⊗I2)v+((diag w)⊗I2)ˆs, where wj := vT j ˆsj, j = 1, . . . , q, which shows how to compute Hessian matrix-vector multiplications, thus to implement IRLS steps in the complex-valued case. Recall that messages are passed, alternating between ˜At→and Mt→matrices. For a PCA approximation ˜At→≈Qt→Tt→QT t→, Qt→∈R˜n×km orthonormal, Tt→tridiagonal (obtained by running km Lanczos steps for ˜At→), low rank algebra gives Mt→= M( ˜At→, Γ−1 t→) = Qt→ T −1 t→+ QT t→Γt→Qt→ −1 QT t→= Vt→V T t→, Vt→∈R˜n×km, computed in O(n k2 m) by way of a Cholesky decomposition. Now, ˜A(t+1)→= At+1 + Vt→V T t→ becomes the precision matrix for the next Lanczos run: MVMs have additional complexity of O(n km). Given all messages, node covariances are PCA-approximated by running Lanczos on At + V(t−1)→V T (t−1)→+ V←(t+1)V T ←(t+1) for kc iterations. Pair variances VarQ[st→|y] are estimated by running Lanczos on vectors of size 2˜n (say for kc/2 iterations; the precision matrix is given in Section 3). More details are given in [22]. Acknowledgments This work is partly funded by the Excellence Initiative of the German research foundation (DFG). It is part of an ongoing collaboration with Rolf Pohmann, Hannes Nickisch and Bernhard Sch¨olkopf, MPI for Biological Cybernetics, T¨ubingen, where data for this study has been acquired. 5In 3D MRI, image volumes are acquired without slice selection, using phase encoding along two dimensions. There are no unmeasured slice gaps and voxels are isotropic, but scan time is much longer. 8 References [1] M.A. Bernstein, K.F. King, and X.J. Zhou. Handbook of MRI Pulse Sequences. Elsevier Academic Press, 1st edition, 2004. [2] R. Bracewell. The Fourier Transform and Its Applications. McGraw-Hill, 3rd edition, 1999. [3] E. Cand`es, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theo., 52(2):489–509, 2006. [4] H. Chang, Y. Weiss, and W. Freeman. Informative sensing. Technical Report 0901.4275v1 [cs.IT], ArXiv, 2009. [5] D. Donoho. Compressed sensing. IEEE Trans. Inf. Theo., 52(4):1289–1306, 2006. [6] A. Garroway, P. Grannell, and P. Mansfield. Image formation in NMR by a selective irradiative pulse. J. Phys. C: Solid State Phys., 7:L457–L462, 1974. [7] M. Girolami. A variational method for learning sparse and overcomplete representations. N. Comp., 13:2517–2532, 2001. [8] G. Golub and C. Van Loan. Matrix Computations. Johns Hopkins University Press, 3rd edition, 1996. [9] M. A. Griswold, P. M. Jakob, R. M. Heidemann, M. Nittka, V. Jellus, J. Wang, B. Kiefer, and A. Haase. Generalized autocalibrating partially parallel acquisitions (GRAPPA). Magn. Reson. Med., 47(6):1202– 10, 2002. [10] P. Lauterbur. Image formation by induced local interactions: Examples employing nuclear magnetic resonance. Nature, 242:190–191, 1973. [11] A. Levin, W. Freeman, and F. Durand. Understanding camera trade-offs through a Bayesian analysis of light field projections. In European Conference on Computer Vision, LNCS 5305, pages 88–101. Springer, 2008. [12] M. Lustig, D. Donoho, and J. Pauly. Sparse MRI: The application of compressed sensing for rapid MR imaging. Magn. Reson. Med., 85(6):1182–1195, 2007. [13] M. Lustig and J. Pauly. SPIR-iT: Iterative self consistent parallel imaging reconstruction from arbitrary k-space. Magn. Reson. Med., 2009. In print. [14] B. Madore, G. Glover, and N. Pelc. Unalising by Fourier-encoding the overlaps using the temporal dimension (UNFOLD), applied to cardiac imaging and fMRI. Magn. Reson. Med., 42:813–828, 1999. [15] D. Malioutov, J. Johnson, and A. Willsky. Low-rank variance estimation in large-scale GMRF models. In ICASSP, 2006. [16] G. Marseille, R. de Beer, M. Fuderer, A. Mehlkopf, and D. van Ormondt. Nonuniform phase-encode distributions for MRI scan time reduction. J. Magn. Reson. B, 111(1):70–75, 1996. [17] D. McRobbie, E. Moore, M. Graves, and M. Prince. MRI: From Picture to Proton. Cambridge University Press, 2nd edition, 2007. [18] C. Mistretta, O. Wieben, J. Velikina, W. Block, J. Perry, Y. Wu, K. Johnson, and Y. Wu. Highly constrained backprojection for time-resolved MRI. Magn. Reson. Med., 55:30–40, 2006. [19] K. Pruessmann, M. Weiger, M. Scheidegger, and P. Boesiger. SENSE: Sensitivity encoding for fast MRI. Magn. Reson. Med., 42:952–962, 1999. [20] M. Schneider and A. Willsky. Krylov subspace algorithms for space-time oceanography data assimilation. In IEEE International Geoscience and Remote Sensing Symposium, 2000. [21] M. Schneider and A. Willsky. Krylov subspace estimation. SIAM J. Comp., 22(5):1840–1864, 2001. [22] M. Seeger. Speeding up magnetic resonance image acquisition by Bayesian multi-slice adaptive compressed sensing. Supplemental Appendix, 2010. [23] M. Seeger and H. Nickisch. Compressed sensing and Bayesian experimental design. In ICML 25, 2008. [24] M. Seeger, H. Nickisch, R. Pohmann, and B. Sch¨olkopf. Bayesian experimental design of magnetic resonance imaging sequences. In NIPS 21, pages 1441–1448, 2009. [25] D. Treebushny and H. Madsen. On the construction of a reduced rank square-root Kalman filter for efficient uncertainty propagation. Future Gener. Comput. Syst., 21(7):1047–1055, 2005. [26] J. Tsao, P. Boesinger, and K. Pruessmann. k-t BLAST and k-t SENSE: Dynamic MRI with high frame rate exploting spatiotemporal correlations. Magn. Reson. Med., 50:1031–1042, 2003. [27] F. Wajer. Non-Cartesian MRI Scan Time Reduction through Sparse Sampling. PhD thesis, Delft University of Technology, 2001. [28] J. Weaver, Y. Xu, D. Healy, and L. Cromwell. Filtering noise from images with wavelet transforms. Magn. Reson. Med., 21(2):288–295, 1991. 9
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Toward Provably Correct Feature Selection in Arbitrary Domains Dimitris Margaritis Department of Computer Science Iowa State University Ames, IA 50010, USA dmarg@cs.iastate.edu Abstract In this paper we address the problem of provably correct feature selection in arbitrary domains. An optimal solution to the problem is a Markov boundary, which is a minimal set of features that make the probability distribution of a target variable conditionally invariant to the state of all other features in the domain. While numerous algorithms for this problem have been proposed, their theoretical correctness and practical behavior under arbitrary probability distributions is unclear. We address this by introducing the Markov Boundary Theorem that precisely characterizes the properties of an ideal Markov boundary, and use it to develop algorithms that learn a more general boundary that can capture complex interactions that only appear when the values of multiple features are considered together. We introduce two algorithms: an exact, provably correct one as well a more practical randomized anytime version, and show that they perform well on artificial as well as benchmark and real-world data sets. Throughout the paper we make minimal assumptions that consist of only a general set of axioms that hold for every probability distribution, which gives these algorithms universal applicability. 1 Introduction and Motivation The problem of feature selection has a long history due to its significance in a wide range of important problems, from early ones like pattern recognition to recent ones such as text categorization, gene expression analysis and others. In such domains, using all available features may be prohibitively expensive, unnecessarily wasteful, and may lead to poor generalization performance, especially in the presence of irrelevant or redundant features. Thus, selecting a subset of features of the domain for use in subsequent application of machine learning algorithms has become a standard preprocessing step. A typical task of these algorithms is learning a classifier: Given a number of input features and a quantity of interest, called the target variable, choose a member of a family of classifiers that can predict the target variable’s value as well as possible. Another task is understanding the domain and the quantities that interact with the target quantity. Many algorithms have been proposed for feature selection. Unfortunately, little attention has been paid to the issue of their behavior under a variety of application domains that can be encountered in practice. In particular, it is known that many can fail under certain probability distributions such as ones that contain a (near) parity function [1], which contain interactions that only appear when the values of multiple features are considered together. There is therefore an acute need for algorithms that are widely applicable and can be theoretically proven to work under any probability distribution. In this paper we present two such algorithms, an exact and a more practical randomized approximate one. We use the observation (first made in Koller and Sahami [2]) that an optimal solution to the problem is a Markov boundary, defined to be a minimal set of features that make the probability distribution of a target variable conditionally invariant to the state of all other features in the domain (a more precise definition is given later in Section 3) and present a family of algorithms for learning 1 the Markov boundary of a target variable in arbitrary domains. We first introduce a theorem that exactly characterizes the minimal set of features necessary for probabilistically isolating a variable, and then relax this definition to derive a family of algorithms that learn a parameterized approximation of the ideal boundary that are provably correct under a minimal set of assumptions, including a set of axioms that hold for any probability distribution. In the following section we present work on feature selection, followed by notation and definitions in Section 3. We subsequently introduce an important theorem and the aforementioned parameterized family of algorithms in Sections 4 and 5 respectively, including a practical anytime version. We evaluate these algorithms in Section 6 and conclude in Section 7. 2 Related Work Numerous algorithms have been proposed for feature selection. At the highest level algorithms can be classified as filter, wrapper, or embedded methods. Filter methods work without consulting the classifier (if any) that will make use of their output i.e., the resulting set of selected features. They therefore have typically wider applicability since they are not tied to any particular classifier family. In contrast, wrappers make the classifier an integral part of their operation, repeatedly invoking it to evaluate each of a sequence of feature subsets, and selecting the subset that results in minimum estimated classification error (for that particular classifier). Finally, embedded algorithms are classifier-learning algorithms that perform feature selection implicitly during their operation e.g., decision tree learners. Early work was motivated by the problem of pattern recognition which inherently contains a large number of features (pixels, regions, signal responses at multiple frequencies etc.). Narendra and Fukunaga [3] first cast feature selection as a problem of maximization of an objective function over the set of features to use, and proposed a number of search approaches including forward selection and backward elimination. Later work by machine learning researchers includes the FOCUS algorithm of Almuallim and Dietterich [4], which is a filter method for deterministic, noise-free domains. The RELIEF algorithm [5] instead uses a randomized selection of data points to update a weight assigned to each feature, selecting the features whose weight exceeds a given threshold. A large number of additional algorithms have appeared in the literature, too many to list here—good surveys are included in Dash and Liu [6]; Guyon and Elisseeff [1]; Liu and Motoda [7]. An important concept for feature subset selection is relevance. Several notions of relevance are discussed in a number of important papers such as Blum and Langley [8]; Kohavi and John [9]. The argument that the problem of feature selection can be cast as the problem of Markov blanket discovery was first made convincingly in Koller and Sahami [2], who also presented an algorithm for learning an approximate Markov blanket using mutual information. Other algorithms include the GS algorithm [10], originally developed for learning of the structure of a Bayesian network of a domain, and extensions to it [11] including the recent MMMB algorithm [12]. Meinshausen and B¨uhlmann [13] recently proposed an optimal theoretical solution to the problem of learning the neighborhood of a Markov network when the distribution of the domain can be assumed to be a multidimensional Gaussian i.e., linear relations among features with Gaussian noise. This assumption implies that the Composition axiom holds in the domain (see Pearl [14] for a definition of Composition); the difference with our work is that we address here the problem in general domains where it may not necessarily hold. 3 Notation and Preliminaries In this section we present notation, fundamental definitions and axioms that will be subsequently used in the rest of the paper. We use the term “feature” and “variable” interchangeably, and denote variables by capital letters (X, Y etc.) and sets of variables by bold letters (S, T etc.). We denote the set of all variables/features in the domain (the “universe”) by U. All algorithms presented are independence-based, learning the Markov boundary of a given target variable using the truth value of a number of conditional independence statements. The use of conditional independence for feature selection subsumes many other criteria proposed in the literature. In particular, the use of classification accuracy of the target variable can be seen as a special case of testing for its conditional independence with some of its predictor variables (conditional on the subset selected at any given moment). A benefit of using conditional independence is that, while classification error estimates depend on the classifier family used, conditional independence does not. In addition, algorithms utilizing conditional independence for feature selection are applicable to all domain types, 2 e.g., discrete, ordinal, continuous with non-linear or arbitrary non-degenerate associations or mixed domains, as long as a reliable estimate of probabilistic independence is available. We denote probabilistic independence by the symbol “ ⊥⊥” i.e., (X⊥⊥Y | Z) denotes the fact that the variables in set X are (jointly) conditionally independent from those in set Y given the values of the variables in set Z; (X ̸⊥⊥Y | Z) denotes their conditional dependence. We assume the existence of a probabilistic independence query oracle that is available to answer any query of the form (X, Y | Z), corresponding to the question “Is the set of variables in X independent of the variables in Y given the value of the variables in Z?” (This is similar to the approach of learning from statistical queries of Kearns and Vazirani [15].) In practice however, such an oracle does not exist, but can be approximated by a statistical independence test on a data set. Many tests of independence have appeared and studied extensively in the statistical literature over the last century; in this work we use the χ2 (chi-square) test of independence [16]. A Markov blanket of variable X is a set of variables such that, after fixing (by “knowing”) the value of all of its members, the set of remaining variables in the domain, taken together as a single setvalued variable, are statistically independent of X. More precisely, we have the following definition. Definition 1. A set of variables S ⊆U is called a Markov blanket of variable X if and only if (X⊥⊥U −S −{X} | S). Intuitively, a Markov blanket S of X captures all the information in the remaining domain variables U −S −{X} that can affect the probability distribution of X, making their value redundant as far as X is concerned (given S). The blanket therefore captures the essence of the feature selection problem for target variable X: By completely “shielding” X, a Markov blanket precludes the existence of any possible information about X that can come from variables not in the blanket, making it an ideal solution to the feature selection problem. A minimal Markov blanket is called a Markov boundary. Definition 2. A set of variables S ⊆U −{X} is called a Markov boundary of variable X if it is a minimal Markov blanket of X i.e., none of its proper subsets is a Markov blanket. Pearl [14] proved that that the axioms of Symmetry, Decomposition, Weak Union, and Intersection are sufficient to guarantee a unique Markov boundary. These are shown below together with the axiom of Contraction. (Symmetry) (X⊥⊥Y | Z) =⇒ (Y⊥⊥X | Z) (Decomposition) (X⊥⊥Y ∪W | Z) =⇒ (X⊥⊥Y | Z) ∧(X⊥⊥W | Z) (Weak Union) (X⊥⊥Y ∪W | Z) =⇒ (X⊥⊥Y | Z ∪W) (1) (Contraction) (X⊥⊥Y | Z) ∧(X⊥⊥W | Y ∪Z) =⇒ (X⊥⊥Y ∪W | Z) (Intersection) (X⊥⊥Y | Z ∪W) ∧(X⊥⊥W | Z ∪Y) =⇒ (X⊥⊥Y ∪W | Z) The Symmetry, Decomposition, Contraction and Weak Union axioms are very general: they are necessary axioms for the probabilistic definition of independence i.e., they hold in every probability distribution, as their proofs are based on the axioms of probability theory. Intersection is not universal but it holds in distributions that are positive, i.e., any value combination of the domain variables has a non-zero probability of occurring. 4 The Markov Boundary Theorem According to Definition 2, a Markov boundary is a minimal Markov blanket. We first introduce a theorem that provides an alternative, equivalent definition of the concept of Markov boundary that we will relax later in the paper to produce a more general boundary definition. Theorem 1 (Markov Boundary Theorem). Assuming that the Decomposition and Contraction axioms hold, S ⊆U −{X} is a Markov boundary of variable X ∈U if and only if ∀T ⊆U −{X}, n T ⊆U −S ⇐⇒(X⊥⊥T | S −T) o . (2) A detailed proof cannot be included here due to space constraints but a proof sketch appears in Appendix A. According to the above theorem, a Markov boundary S partitions the powerset of U −{X} into two parts: (a) set P1 that contains all subsets of U −S, and (b) set P2 containing the remaining subsets. All sets in P1 are conditionally independent of X, and all sets in P2 are conditionally dependent with X. Intuitively, the two directions of the logical equivalence relation of Eq. (2) correspond to the concept of Markov blanket and its minimality i.e., the equation ∀T ⊆U −{X}, n T ⊆U −S =⇒(X⊥⊥T | S −T) o 3 Algorithm 1 The abstract GS(m)(X) algorithm. Returns an m-Markov boundary of X. 1: S ←∅ 2: /* Growing phase. */ 3: for all Y ⊆U −S −{X} such that 1 ≤|Y| ≤m do 4: if (X ̸⊥⊥Y | S) then 5: S ←S ∪Y 6: goto line 3 /* Restart loop. */ 7: /* Shrinking phase. */ 8: for all Y ∈S do 9: if (X⊥⊥Y | S −{Y }) then 10: S ←S −{Y } 11: goto line 8 /* Restart loop. */ 12: return S or, equivalently, (∀T ⊆U −S −{X}, (X⊥⊥T | S)) (as T and S are disjoint) corresponds to the definition of Markov blanket, as it includes T = U −S −{X}. In the opposite direction, the contrapositive form is ∀T ⊆U −{X}, n T ̸⊆U −S =⇒(X ̸⊥⊥T | S −T) o . This corresponds to the concept of minimality of the Markov boundary: It states that all sets that contain a part of S cannot be independent of X given the remainder of S. Informally, this is because if there existed some set T that contained a non-empty subset T′ of S such that (X⊥⊥T | S −T), then one would be able to shrink S by T′ (by the property of Contraction) and therefore S would not be minimal (more details in Appendix A). 5 A Family of Algorithms for Arbitrary Domains Theorem 1 defines conditions that precisely characterize a Markov boundary and thus can be thought of as an alternative definition of a boundary. By relaxing these conditions we can produce a more general definition. In particular, an m-Markov boundary is defined as follows. Definition 3. A set of variables S ⊆U −{X} of a domain U is called an m-Markov boundary of variable X ∈U if and only if ∀T ⊆U −{X} such that |T| ≤m, n T ⊆U −S ⇐⇒(X⊥⊥T | S −T) o . We call the parameter m of an m-Markov boundary the Markov boundary margin. Intuitively, an m-boundary S guarantees that (a) all subsets of its complement (excluding X) of size m or smaller are independent of X given S, and (b) all sets T of size m or smaller that are not subsets of its complement are dependent of X given the part of S that is not contained in T. This definition is a special case of the properties of a boundary stated in Theorem 1, with each set T mentioned in the theorem now restricted to having size m or smaller. For m = n −1, where n = |U|, the condition |T| ≤m is always satisfied and can be omitted; in this case the definition of an (n −1)-Markov boundary results in exactly Eq. (2) of Theorem 1. We now present an algorithm called GS(m), shown in Algorithm 1, that provably correctly learns an m-boundary of a target variable X. GS(m) operates in two phases, a growing and a shrinking phase (hence the acronym). During the growing phase it examines sets of variables of size up to m, where m is a user-specified parameter. During the shrinking phase, single variables are examined for conditional independence and possible removal from S (examining sets in the shrinking phase is not necessary for provably correct operation—see Appendix B). The orders of examination of the sets for possible addition and deletion from the candidate boundary are left intentionally unspecified in Algorithm 1—one can therefore view it as an abstract representative of a family of algorithms, with each member specifying one such ordering. All members of this family are m-correct, as the proof of correctness does not depend on the ordering. In practice numerous choices for the ordering exist; one possibility is to examine the sets in the growing phase in order of increasing set size and, for each such size, in order of decreasing conditional mutual information I(X, Y, S) between X and Y given S. The rationale for this heuristic choice is that (usually) tests with smaller conditional sets tend to be more reliable, and sorting by mutual information tends to lessen the chance of adding false members of the Markov boundary. We used this implementation in all our experiments, presented later in Section 6. Intuitively, the margin represents a trade-off between sample and computational complexity and completeness: For m = n −1 = |U| −1, the algorithm returns a Markov boundary in unrestricted 4 Algorithm 2 The RGS(m,k)(X) algorithm, a randomized anytime version of the GS(m) algorithm, utilizing k random subsets for the growing phase. 1: S ←∅ 2: /* Growing phase. */ 3: repeat 4: Schanged ←false 5: Y ←subset of U −S −{X} of size 1 ≤|Y| ≤m of maximum dependence out of k random subsets 6: if (X ̸⊥⊥Y | S) then 7: S ←S ∪Y 8: Schanged ←true 9: until Schanged = false 10: /* Shrinking phase. */ 11: for all Y ∈S do 12: if (X⊥⊥Y | S −{Y }) then 13: S ←S −{Y } 14: goto line 11 /* Restart loop. */ 15: return S (arbitrary) domains. For 1 ≤m < n −1, GS(m) may recover the correct boundary depending on characteristics of the domain. For example, it will recover the correct boundary in domains containing embedded parity functions such that the number of variables involved in every k-bit parity function is m + 1 or less i.e., if k ≤m + 1 (parity functions are corner cases in the space of probability distributions that are known to be hard to learn [17]). The proof of m-correctness of GS(m) is included in Appendix B. Note that it is based on Theorem 1 and the universal axioms of Eqs. (1) only i.e., Intersection is not needed, and thus it is widely applicable (to any domain). A Practical Randomized Anytime Version While GS(m) is provably correct even in difficult domains such as those that contain parity functions, it may be impractical with a large number of features as its asymptotic complexity is O(nm). We therefore also we here provide a more practical randomized version called RGS(m,k) (Randomized GS(m)), shown in Algorithm 2. The RGS(m,k) algorithm has an additional parameter k that limits its computational requirements: instead of exhaustively examining all possible subsets of (U−S−{X}) (as GS(m) does), it instead samples k subsets from the set of all possible subsets of (U −S −{X}), where k is user-specified. It is therefore a randomized algorithm that becomes equivalent to GS(m) given a large enough k. Many possibilities for the method of random selection of the subsets exist; in our experiments we select a subset Y = {Yi} (1 ≤|Y| ≤m) with probability proportional to P|Y| i=1(1/p(X, Yi | S)), where p(X, Yi | S) is the p-value of the corresponding (univariate) test between X and Yi given S, which has a low computational cost. The RGS(m,k) algorithm is useful in situations where the amount of time to produce an answer may be limited and/or the limit unknown beforehand: it is easy to show that the growing phase of GS(m) produces an an upper-bound of the m-boundary of X. Therefore, the RGS(m,k) algorithm, if interrupted, will return an approximation of this upper bound. Moreover, if there exists time for the shrinking phase to be executed (which conducts a number of tests linear in n and is thus fast), extraneous variables will be removed and a minimal blanket (boundary) approximation will be returned. These features make it an anytime algorithm, which is a more appropriate choice for situations where critical events may occur that require the interruption of computation, e.g., during the planning phase of a robot, which may be interrupted at any time due to an urgent external event that requires a decision to be made based on the present state’s feature values. 6 Experiments We evaluated the GS(m) and the RGS(m,k) algorithms on synthetic as well as real-world and benchmark data sets. We first systematically examined the performance on the task of recovering near-parity functions, which are known to be hard to learn [17]. We compared GS(m) and RGS(m,k) with respect to accuracy of recovery of the original boundary as well as computational cost. We generated domains of sizes ranging from 10 to 100 variables, of which 4 variables (X1 to X4) were related through a near-parity relation with bit probability 0.60 and various degrees of noise. The remaining independent variables (X5 to Xn) act as “dis5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 F1 measure Noise probability F1 measure of GS(m ), RGS(m, k ) and RELIEVED vs. noise level 50 variables, true Markov boundary size = 3 Bernoulli probability = 0.6, 1000 data points GS(1) GS(3) RGS(1, 1000) RGS(3, 1000) Relieved, threshold = 0.001 Relieved, threshold = 0.03 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 RELIEVED(threshold = 0.03) average isolation measure GS(m = 3) average isolation measure Probabilistic isolation performance of GS(m) and RELIEVED Real-world and benchmark data sets Data set Balance scale Balloons Car evaluation Credit screening Monks Nursery Tic-tac-toe Breast cancer Chess Audiology 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 RGS(m = 3, k = 300) average isolation measure GS(m = 3) average isolation measure Probabilistic isolation performance of GS(m) and RGS(m ,k) Real-world and benchmark data sets Data set Balance scale Balloons Car evaluation Credit screening Monks Nursery Tic-tac-toe Breast cancer Chess Audiology Figure 2: Left: F1 measure of GS(m), RGS(m,k) and RELIEVED under increasing amounts of noise. Middle: Probabilistic isolation performance comparison between GS(3) and RELIEVED on real-world and benchmark data sets. Right: Same for GS(3) and RGS(3,1000). tractors” and had randomly assigned probabilities i.e., the correct boundary of X1 is B1 = {X2, X3, X4}. In such domains, learning the boundary of X1 is difficult because of the large number of distractors and because each Xi ∈B1 is independent of X1 given any proper subset of B1 −{Xi} (they only become dependent when including all of them in the conditioning set). 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 10 20 30 40 50 60 70 80 90 100 F1-measure Number of domain variables F1-measure of GS(m ) and RGS(m, k ) vs. domain size True Markov boundary size = 3, 1000 data points Bernoulli probability = 0.6, noise probability = 0.1 GS(1) GS(2) GS(3) RGS(1, 1000) RGS(2, 1000) RGS(3, 1000) 0.01 0.1 1 10 100 1000 10000 10 20 30 40 50 60 70 80 90 100 Execution time (sec) Number of domain variables Running time of GS(m ) and RGS(m, k ) vs. domain size True Markov boundary size = 3, 1000 data points Bernoulli probability = 0.6, noise probability = 0.1 GS(1) GS(2) GS(3) RGS(1, 1000) RGS(2, 1000) RGS(3, 1000) Figure 1: GS(m) and RGS(m,k) performance with respect to domain size (number of variables). Top: F1 measure, reflecting accuracy. Bottom: Execution time in seconds (log scale). To measure an algorithm’s feature selection performance, accuracy (fraction of variables correctly included or excluded) is inappropriate as the accuracy of trivial algorithms such as returning the empty set will tend to 1 as n increases. Precision and recall are therefore more appropriate, with precision defined as the fraction of features returned that are in the correct boundary (3 features for X1), and recall as the fraction of the features present in the correct boundary that are returned by the algorithm. A convenient and frequently used measure that combines precision and recall is the F1 measure, defined as the harmonic mean of precision and recall [18]. In Fig. 1 (top) we report 95% confidence intervals for the F1 measure and execution time of GS(m) (margins m = 1 to 3) and RGS(m,k) (margins 1 to 3 and k = 1000 random subsets), using 20 data sets containing 10 to 100 variables, with the target variable X1 was perturbed (inverted) by noise with 10% probability. As can be seen, the RGS(m,k) and GS(m) using the same value for margin perform comparably with respect to F1, up to their 95% confidence intervals. With respect to execution time however RGS(m,k) exhibits much greater scalability (Fig. 1 bottom, log scale); for example, it executes in about 10 seconds on average in domains containing 100 variables, while GS(m) executes in 1,000 seconds on average for this domain size. We also compared GS(m) and RGS(m,k) to RELIEF [5], a well-known algorithm for feature selection that is known to be able to recover parity functions in certain cases [5]. RELIEF learns a weight for each variable and compares it to a threshold τ to decide on its inclusion in the set of relevant variables. As it has been reported [9] that RELIEF can exhibit large variance due to randomization that is necessary only for very large data sets, we instead used a deterministic variant called RELIEVED [9], whose behavior corresponds to RELIEF at the limit of infinite execution time. We calculated the F1 measure for GS(m), RGS(m,k) and RELIEVED in the presence of varying amounts of noise, with noise probability ranging from 0 (no noise) to 0.4. We used domains containing 50 variables, as GS(m) becomes computationally demanding in larger domains. In Figure 2 (left) we show the performance of GS(m) and RGS(m,k) for m equal to 1 and 3, k = 1000 and RELIEVED for thresholds τ = 0.01 and 0.03 for various amounts of noise on the target variable. Again, each experiment was repeated 20 times to generate 95% confidence intervals. We can observe that even though m = 1 (equivalent to the GS algorithm) performs poorly, increasing the margin m makes it more likely to recover the correct Markov boundary, and GS(3) (m = 3) recovers the exact blanket even with few (1,000) data points. RELIEVED does comparably to GS(3) for little noise and for a large threshold, 6 but appears to deteriorate for more noisy domains. As we can see it is difficult to choose the “right” threshold for RELIEVED—better performing τ at low noise can become worse in noisy environments; in particular, small τ tend to include irrelevant variables while large τ tend to miss actual members. We also evaluated GS(m), RGS(m,k), and RELIEVED on benchmark and real-world data sets from the UCI Machine Learning repository. As the true Markov boundary for these is impossible to know, we used as performance measure a measure of probabilistic isolation by the Markov boundary returned of subsets outside the boundary. For each domain variable X, we measured the independence of subsets Y of size 1, 2 and 3 given the blanket S of X returned by GS(3) and RELIEVED for τ = 0.03 (as this value seemed to do better in the previous set of experiments), as measured by the average p-value of the χ2 test between X and Y given S (with p-values of 0 and 1 indicating ideal dependence and independence, respectively). Due to the large number of subsets outside the boundary when the boundary is small, we limited the estimation of isolation performance to 2,000 subsets per variable. We plot the results in Figure 2 (middle and right). Each point represents a variable in the corresponding data set. Points under the diagonal indicate better probabilistic isolation performance for that variable for GS(3) compared to RELIEVED (middle plot) or to RGS(3,1000) (right plot). To obtain a statistically significant comparison, we used the non-parametric Wilcoxon paired signed-rank test, which indicated that GS(3) RGS(3,1000) are statistically equivalent to each other, while both outperformed RELIEVED at the 99.99% significance level (α < 10−7). 7 Conclusion In this paper we presented algorithms for the problem of feature selection in unrestricted (arbitrary distribution) domains that may contain complex interactions that only appear when the values of multiple features are considered together. We introduced two algorithms: an exact, provably correct one as well a more practical randomized anytime version, and evaluated them on on artificial, benchmark and real-world data, demonstrating that they perform well, even in the presence of noise. We also introduced the Markov Boundary Theorem that precisely characterizes the properties of a boundary, and used it to prove m-correctness of the exact family of algorithms presented. We made minimal assumptions that consist of only a general set of axioms that hold for every probability distribution, giving our algorithms universal applicability. Appendix A: Proof sketch of the Markov Boundary Theorem Proof sketch. (=⇒direction) We need to prove that if S is a Markov boundary of X then (a) for every set T ⊆U −S −{X}, (X⊥⊥T | S −T), and (b) for every set T′ ̸⊆U −S that does not contain X, (X ̸⊥⊥T′ | S −T′). Case (a) is immediate from the definition of the boundary and the Decomposition theorem. Case (b) can be proven by contradiction: Assuming the independence of T′ that contains a non-empty part T′ 1 in S and a part T′ 2 in U −S, we get (from Decomposition) (X⊥⊥T′ 1 | S −T′ 1). We can then use Contraction to show that the set S −T′ 1 satisfies the independence property of a Markov boundary, i.e., that (X⊥⊥U −(S −T′ 1) −{X} | S −T′ 1), which contradicts the assumption that S is a boundary (and thus minimal). (⇐= direction) We need to prove that if Eq. (2) holds, then S is a minimal Markov blanket. The proof that S is a blanket is immediate. We can prove minimality by contradiction: Assume S = S1 ∪S2 with S1 a blanket and S2 ̸= ∅i.e., S1 is a blanket strictly smaller than S. Then (X⊥⊥S2 | S1) = (X⊥⊥S2 | S −S2). However, since S2 ̸⊆U −S, from Eq. (2) we get (X ̸⊥⊥S2 | S −S2), which is a contradiction. Appendix B: Proof of m-Correctness of GS(m) Let the value of the set S at the end of the growing phase be SG, its value at the end of the shrinking phase SS, and their difference S∆= SG −SS. The following two observations are immediate. Observation 1. For every Y ⊆U −SG −{X} such that 1 ≤|Y| ≤m, (X⊥⊥Y | SG). Observation 2. For every Y ∈SS, (X ̸⊥⊥Y | SS −{Y }). Lemma 2. Consider variables Y1, Y2, . . . , Yt for some t ≥1 and let Y = {Yj}t j=1. Assuming that Contraction holds, if (X⊥⊥Yi | S −{Yj}i j=1) for all i = 1, . . . , t, then (X⊥⊥Y | S −Y). Proof. By induction on Yj, j = 1, 2, . . . , t, using Contraction to decrease the conditioning set S down to S −{Yj}i j=1 for all i = 1, 2, . . . , t. Since Y = {Yj}t j=1, we immediately obtain the desired relation (X⊥⊥Y | S −Y). 7 Lemma 2 can be used to show that the variables found individually independent of X during the shrinking phase are actually jointly independent of X, given the final set SS. Let S∆= {Y1, Y2, . . . , Yt} be the set of variables removed (in that order) from SG to form the final set SS i.e., S∆= SG −SS. Using the above lemma, the following is immediate. Corollary 3. Assuming that the Contraction axiom holds, (X⊥⊥S∆| SS). Lemma 4. If the Contraction, Decomposition and Weak Union axioms hold, then for every set T ⊆U −SG −{X} such that (X⊥⊥T | SG), (X⊥⊥T ∪(SG −SS) | SS). (3) Furthermore SS is minimal i.e., there does not exist a subset of SS for which Eq. (3) is true. Proof. From Corollary 3, (X⊥⊥S∆| SS). Also, by the hypothesis, (X⊥⊥T | SG) = (X⊥⊥T | SS ∪S∆), where S∆= SG −SS as usual. From these two relations and Contraction we obtain (X⊥⊥T ∪S∆| SS). To prove minimality, let us assume that SS ̸= ∅(if SS = ∅then it is already minimal). We prove by contradiction: Assume that there exists a set S′ ⊂SS such that (X⊥⊥T ∪(SG −S′) | S′). Let W = SS −S′ ̸= ∅. Note that W and S′ are disjoint. We have that SS ⊆SS ∪S∆ =⇒ SS −S′ ⊆SS ∪S∆−S′ ⊆T ∪(SS ∪S∆−S′) =⇒ W ⊆T ∪(SS ∪S∆−S′) = T ∪(SG −S′) • Since (X⊥⊥T ∪(SG −S′) | S′) and W ⊆T ∪(SS ∪S∆−S′), from Decomposition we get (X⊥⊥W | S′). • From (X⊥⊥W | S′) and Weak Union we have that for every Y ∈W, (X⊥⊥Y | S′ ∪ (W −{Y })). • Since S′ and W are disjoint and since Y ∈W, Y ̸∈S′. Applying the set equality (A−B)∪C = (A∪B)−(A−C) to S′ ∪(W−{Y }) we obtain S′ ∪W−({Y }−S′) = SS −{Y }. • Therefore, ∀Y ∈W, (X⊥⊥Y | SS −{Y }). However, at the end of the shrinking phase, all variables Y in SS (and therefore in W, as W ⊆SS) have been evaluated for independence and found dependent (Observation 2). Thus, since W ̸= ∅, there exists at least one Y such that (X ̸⊥⊥Y | SS −{Y }), producing a contradiction. Theorem 5. Assuming that the Contraction, Decomposition, and Weak Union axioms hold, Algorithm 1 is m-correct with respect to X. Proof. We use the Markov Boundary Theorem. We first prove that ∀T ⊆U −{X} such that |T| ≤m, n T ⊆U −SS =⇒(X⊥⊥T | SS −T) o or, equivalently, ∀T ⊆U −SS −{X} such that |T| ≤m, (X⊥⊥T | SS). Since U −SS −{X} = S∆∪(U −SG −{X}), S∆and U −SG −{X} are disjoint, there are three kinds of sets of size m or less to consider: (i) all sets T ⊆S∆, (ii) all sets T ⊆U −SG −{X}, and (iii) all sets (if any) T = T′ ∪T′′, T′ ∩T′′ = ∅, that have a non-empty part T′ ⊆S∆and a non-empty part T′′ ⊆U −SG −{X}. (i) From Corollary 3, (X⊥⊥S∆| SS). Therefore, from Decomposition, for any set T ⊆S∆, (X⊥⊥T | SS). (ii) By Observation 1, for every set T ⊆U −SG −{X} such that |T| ≤m, (X⊥⊥T | SG). By Lemma 4 we get (X⊥⊥T ∪S∆| SS), from which we obtain (X⊥⊥T | SS) by Decomposition. (iii) Since |T| ≤m, we have that |T′′| ≤m. Since T′′ ⊆U −SG −{X}, by Observation 1, (X⊥⊥T′′ | SG). Therefore, by Lemma 4, (X⊥⊥T′′ ∪S∆| SS). Since T′ ⊆S∆⇒ T′′ ∪T′ ⊆T′′ ∪S∆, by Decomposition to obtain (X⊥⊥T′′ ∪T′ | SS) = (X⊥⊥T | SS). To complete the proof we need to prove that ∀T ⊆U −{X} such that |T| ≤m, n T ̸⊆U −SS =⇒(X ̸⊥⊥T | SS −T) o . Let T = T1 ∪T2, with T1 ⊆SS and T2 ⊆U −SS. Since T ̸⊆U −SS, T1 contains at least one variable Y ∈SS. From Observation 2, (X ̸⊥⊥Y | SS −{Y }). From this and (the contrapositive of) Weak Union, we get (X ̸⊥⊥{Y }∪(T1 −{Y }) | SS −{Y }−(T1 −{Y })) = (X ̸⊥⊥T1 | SS −T1). From (the contrapositive of) Decomposition we get (X ̸⊥⊥T1 ∪T2 | SS −T1) = (X ̸⊥⊥T | SS −T1), which is equal to (X ̸⊥⊥T | SS −T1 −T2) = (X ̸⊥⊥T | SS −T) as SS and T2 are disjoint. 8 References [1] Isabelle Guyon and Andr´e Elisseeff. An introduction to variable and feature selection. Journal of Machine Learning Research, 3:1157–1182, 2003. [2] Daphne Koller and Mehran Sahami. Toward optimal feature selection. In Proceedings of the Tenth International Conference on Machine Learning (ICML), pages 284–292, 1996. [3] P. M. Narendra and K. Fukunaga. A branch and bound algorithm for feature subset selection. IEEE Transactions on Computers, C-26(9):917–922, 1977. [4] H. Almuallim and T. G. Dietterich. Learning with many irrelevant features. In Proceedings of the National Conference on the Americal Association for Artifical Intelligence (AAAI), 1991. [5] K. Kira and L. A. Rendell. The feature selection problem: Traditional methods and a new algorithm. In Proceedings of the National Conference on the Americal Association for Artifical Intelligence (AAAI), pages 129–134, 1992. [6] M. Dash and H. Liu. Feature selection for classification. Intelligent Data Analysis, 1(3): 131–156, 1997. [7] Huan Liu and Hiroshi Motoda, editors. Feature Extraction, Construction and Selection: A Data Mining Perspective, volume 453 of The Springer International Series in Engineering and Computer Science. 1998. [8] Avrim Blum and Pat Langley. Selection of relevant features and examples in machine learning. Artificial Intelligence, 97(1-2):245–271, 1997. [9] R. Kohavi and G. H. John. Wrappers for feature subset selection. Artificial Intelligence, 97 (1-2):273–324, 1997. [10] Dimitris Margaritis and Sebastian Thrun. Bayesian network induction via local neighborhoods. In Advances in Neural Information Processing Systems 12 (NIPS), 2000. [11] I. Tsamardinos, C. Aliferis, and A. Statnikov. Algorithms for large scale Markov blanket discovery. In Proceedings of the 16th International FLAIRS Conference, 2003. [12] I. Tsamardinos, C. Aliferis, and A. Statnikov. Time and sample efficient discovery of Markov blankets and direct causal relations. In Proceedings of the 9th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pages 673–678, 2003. [13] N. Meinshausen and P. B¨uhlmann. High-dimensional graphs and variable selection with the Lasso. Annals of Statistics, 34:1436–1462, 2006. [14] Judea Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. 1988. [15] Michael Kearns and Umesh V. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994. [16] A. Agresti. Categorical Data Analysis. John Wiley and Sons, 1990. [17] M. Kearns. Efficient noise-tolerant learning from statistical queries. J. ACM, 45(6):983–1006, 1998. [18] C. J. van Rijsbergen. Information Retrieval. Butterworth-Heinemann, London, 1979. 9
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3D Object Recognition with Deep Belief Nets Vinod Nair and Geoffrey E. Hinton Department of Computer Science, University of Toronto 10 King’s College Road, Toronto, M5S 3G5 Canada {vnair,hinton}@cs.toronto.edu Abstract We introduce a new type of top-level model for Deep Belief Nets and evaluate it on a 3D object recognition task. The top-level model is a third-order Boltzmann machine, trained using a hybrid algorithm that combines both generative and discriminative gradients. Performance is evaluated on the NORB database (normalized-uniform version), which contains stereo-pair images of objects under different lighting conditions and viewpoints. Our model achieves 6.5% error on the test set, which is close to the best published result for NORB (5.9%) using a convolutional neural net that has built-in knowledge of translation invariance. It substantially outperforms shallow models such as SVMs (11.6%). DBNs are especially suited for semi-supervised learning, and to demonstrate this we consider a modified version of the NORB recognition task in which additional unlabeled images are created by applying small translations to the images in the database. With the extra unlabeled data (and the same amount of labeled data as before), our model achieves 5.2% error. 1 Introduction Recent work on deep belief nets (DBNs) [10], [13] has shown that it is possible to learn multiple layers of non-linear features that are useful for object classification without requiring labeled data. The features are trained one layer at a time as a restricted Boltzmann machine (RBM) using contrastive divergence (CD) [4], or as some form of autoencoder [20], [16], and the feature activations learned by one module become the data for training the next module. After a pre-training phase that learns layers of features which are good at modeling the statistical structure in a set of unlabeled images, supervised backpropagation can be used to fine-tune the features for classification [7]. Alternatively, classification can be performed by learning a top layer of features that models the joint density of the class labels and the highest layer of unsupervised features [6]. These unsupervised features (plus the class labels) then become the penultimate layer of the deep belief net [6]. Early work on deep belief nets was evaluated using the MNIST dataset of handwritten digits [6] which has the advantage that a few million parameters are adequate for modeling most of the structure in the domain. For 3D object classification, however, many more parameters are probably required to allow a deep belief net with no prior knowledge of spatial structure to capture all of the variations caused by lighting and viewpoint. It is not yet clear how well deep belief nets perform at 3D object classification when compared with shallow techniques such as SVM’s [19], [3] or deep discriminative techniques like convolutional neural networks [11]. In this paper, we describe a better type of top-level model for deep belief nets that is trained using a combination of generative and discriminative gradients [5], [8], [9]. We evaluate the model on NORB [12], which is a carefully designed object recognition task that requires 1 hj lk vi (a) hj lk vi (b) h1 h2 l1 v1 v2 l2 (c) h1 h2 l1 l22 v1 v2 (d) W112 W212 W122 W222 W111 W211 h dd 111 211 W121 W221 Nv visible units Nh hidden units (e) Figure 1: The Third-Order Restricted Boltzmann Machine. (a) Every clique in the model contains a visible unit, hidden unit, and label unit. (b) Our shorthand notation for representing the clique in (a). (c) A model with two of each unit type. There is one clique for every possible triplet of units created by selecting one of each type. The “restricted” architecture precludes cliques with multiple units of the same type. (d) Our shorthand notation for representing the model in (c). (e) The 3D tensor of parameters for the model in (c). The architecture is the same as that of an implicit mixture of RBMs [14], but the inference and learning algorithms have changed. generalization to novel object instances under varying lighting conditions and viewpoints. Our model significantly outperforms SVM’s, and it also outperforms convolutional neural nets when given additional unlabeled data produced by small translations of the training images. We use restricted Boltzmann machines trained with one-step contrastive divergence as our basic module for learning layers of features. These are fully described elsewhere [6], [1] and the reader is referred to those sources for details. 2 A Third-Order RBM as the Top-Level Model Until now, the only top-level model that has been considered for a DBN is an RBM with two types of observed units (one for the label, another for the penultimate feature vector). We now consider an alternative model for the top-level joint distribution in which the class label multiplicatively interacts with both the penultimate layer units and the hidden units to determine the energy of a full configuration. It is a Boltzmann machine with three-way cliques [17], each containing a penultimate layer unit vi, a hidden unit hj, and a label unit lk. See figure 1 for a summary of the architecture. Note that the parameters now form a 3D tensor, instead of a matrix as in the earlier, bipartite model. Consider the case where the components of v and h are stochastic binary units, and l is a discrete variable with K states represented by 1-of-K encoding. The model can be defined in terms of its energy function E(v, h, l) = − X i,j,k Wijkvihjlk, (1) where Wijk is a learnable scalar parameter. (We omit bias terms from all expressions for clarity.) The probability of a full configuration {v, h, l} is then P(v, h, l) = exp(−E(v, h, l)) Z , (2) where Z = P v′,h′,l′ exp(−E(v′, h′, l′)) is the partition function. Marginalizing over h gives the distribution over v and l alone. 2 The main difference between the new top-level model and the earlier one is that now the class label multiplicatively modulates how the visible and hidden units contribute to the energy of a full configuration. If the label’s kth unit is 1 (and the rest are 0), then the kth slice of the tensor determines the energy function. In the case of soft activations (i.e. more than one label has non-zero probability), a weighted blend of the tensor’s slices specifies the energy function. The earlier top-level (RBM) model limits the label’s effect to changing the biases into the hidden units, which modifies only how the hidden units contribute to the energy of a full configuration. There is no direct interaction between the label and the visible units. Introducing direct interactions among all three sets of variables allows the model to learn features that are dedicated to each class. This is a useful property when the object classes have substantially different appearances that require very different features to describe. Unlike an RBM, the model structure is not bipartite, but it is still “restricted” in the sense that there are no direct connections between two units of the same type. 2.1 Inference The distributions that we would like to be able to infer are P(l|v) (to classify an input), and P(v, l|h) and P(h|v, l) (for CD learning). Fortunately, all three distributions are tractable to sample from exactly. The simplest case is P(h|v, l). Once l is observed, the model reduces to an RBM whose parameters are the kth slice of the 3D parameter tensor. As a result P(h|v, l) is a factorized distribution that can be sampled exactly. For a restricted third-order model with Nv visible units, Nh hidden units and Nl class labels, the distribution P(l|v) can be exactly computed in O(NvNhNl) time. This result follows from two observations: 1) setting lk = 1 reduces the model to an RBM defined by the kth slice of the tensor, and 2) the negative log probability of v, up to an additive constant, under this RBM is the free energy: Fk(v) = − Nh X j=1 log(1 + exp( Nv X i=1 Wijkvi)). (3) The idea is to first compute Fk(v) for each setting of the label, and then convert them to a discrete distribution by taking the softmax of the negative free energies: P(lk = 1|v) = exp(−Fk(v)) PNl k=1 exp(−Fk(v)) . (4) Equation 3 requires O(NvNh) computation, which is repeated Nl times for a total of O(NvNhNl) computation. We can use the same method to compute P(l|h). Simply switch the role of v and h in equation 3 to compute the free energy of h under the kth RBM. (This is possible since the model is symmetric with respect to v and h.) Then convert the resulting Nl free energies to the probabilities P(lk = 1|h) with the softmax function. Now it becomes possible to exactly sample P(v, l|h) by first sampling ˜l ∼P(l|h). Suppose ˜lk = 1. Then the model reduces to its kth-slice RBM from which ˜v ∼P(v|h, ˜lk = 1) can be easily sampled. The final result {˜v,˜l} is an unbiased sample from P(v, l|h). 2.2 Learning Given a set of N labeled training cases {(v1, l1), (v2, l2), ..., (vN, lN)} , we want to learn the 3D parameter tensor W for the restricted third-order model. When trained as the top-level model of a DBN, the visible vector v is a penultimate layer feature vector. We can also train the model directly on images as a shallow model, in which case v is an image (in row vector form). In both cases the label l represents the Nl object categories using 1-of-Nl encoding. For the same reasons as in the case of an RBM, maximum likelihood learning is intractable here as well, so we rely on Contrastive Divergence learning instead. CD was originally formulated in the context of the RBM and its bipartite architecture, but here we extend it to the non-bipartite architecture of the third-order model. 3 An unbiased estimate of the maximum likelihood gradient can be computed by running a Markov chain that alternatively samples P(h|v, l) and P(v, l|h) until it reaches equilibrium. Contrastive divergence uses the parameter updates given by three half-steps of this chain, with the chain initialized from a training case (rather than a random state). As explained in section 2.1, both of these distributions are easy to sample from. The steps for computing the CD parameter updates are summarized below: Contrastive divergence learning of P(v, l): 1. Given a labeled training pair {v+, l+ k = 1}, sample h+ ∼P(h|v+, l+ k = 1). 2. Compute the outer product D+ k = v+h+T . 3. Sample {v−, l−} ∼P(v, l|h+). Let m be the index of the component of l−set to 1. 4. Sample h−∼P(h|v−, l− m = 1). 5. Compute the outer product D− m = v−h−T . Let W·,·,k denote the Nh ×Nv matrix of parameters corresponding to the kth slice along the label dimension of the 3D tensor. Then the CD update for W·,·,k is: ∆W·,·,k = D+ k −D− k , (5) W·,·,k ←W·,·,k + η∆W·,·,k, (6) where η is a learning rate parameter. Typically, the updates computed from a “mini-batch” of training cases (a small subset of the entire training set) are averaged together into one update and then applied to the parameters. 3 Combining Gradients for Generative and Discriminative Models In practice the Markov chain used in the learning of P(v, l) can suffer from slow mixing. In particular, the label l−generated in step 3 above is unlikely to be different from the true label l+ of the training case used in step 1. Empirically, the chain has a tendency to stay “stuck” on the same state for the label variable because in the positive phase the hidden activities are inferred with the label clamped to its true value. So the hidden activities contain information about the true label, which gives it an advantage over the other labels. Consider the extreme case where we initialize the Markov chain with a training pair {v+, l+ k = 1} and the label variable never changes from its initial state during the chain’s entire run. In effect, the model that ends up being learned is a class-conditional generative distribution P(v|lk = 1), represented by the kth slice RBM. The parameter updates are identical to those for training Nl independent RBMs, one per class, with only the training cases of each class being used to learn the RBM for that class. Note that this is very different from the model in section 2: here the energy functions implemented by the class-conditional RBMs are learned independently and their energy units are not commensurate with each other. Alternatively, we can optimize the same set of parameters to represent yet another distribution, P(l|v). The advantage in this case is that the exact gradient needed for maximum likelihood learning, ∂logP(l|v)/∂W, can be computed in O(NvNhNl) time. The gradient expression can be derived with some straightforward differentiation of equation 4. The disadvantage is that it cannot make use of unlabeled data. Also, as the results show, learning a purely discriminative model at the top level of a DBN gives much worse performance. However, now a new way of learning P(v, l) becomes apparent: we can optimize the parameters by using a weighted sum of the gradients for log P(v|l) and log P(l|v). As explained below, this approach 1) avoids the slow mixing of the CD learning for P(v, l), and 2) allows learning with both labeled and unlabeled data. It resembles pseudo-likelihood in how it optimizes the two conditional distributions in place of the joint distribution, except here one of the conditionals (P(v|l)) is still learned only approximately. In our experiments, a model trained with this hybrid learning algorithm has the highest classification accuracy, beating both a generative model trained using CD as well as a purely discriminative model. 4 The main steps of the algorithm are listed below. Hybrid learning algorithm for P(v, l): Let {v+, l+ k = 1} be a labeled training case. Generative update: CD learning of P(v|l) 1. Sample h+ ∼P(h|v+, l+ k = 1). 2. Compute the outer product D+ k = v+h+T . 3. Sample v−∼P(v|h+, l+ k = 1). 4. Sample h−∼P(h|v−, l+ k = 1). 5. Compute the outer product D− k = v−h−T . 6. Compute update ∆W g ·,·,k = D+ k −D− k . Discriminative update: ML learning of P(l|v) 1. Compute log P(lc = 1|v+) for c ∈{1, ..., Nl}. 2. Using the result from step 1 and the true label l+ k = 1, compute the update ∆W d ·,·,k = ∂log P(l|v)/∂W·,·,c for c ∈{1, ..., Nl}. The two types of update for the cth slice of the tensor W·,·,c are then combined by a weighted sum: W·,·,c ←W·,·,c + η(∆W g ·,·,c + λ∆W d ·,·,c), (7) where λ is a parameter that sets the relative weighting of the generative and discriminative updates, and η is the learning rate. As before, the updates from a mini-batch of training cases can be averaged together and applied as a single update to the parameters. In experiments, we set λ by trying different values and evaluating classification accuracy on a validation set. Note that the generative part in the above algorithm is simply CD learning of the RBM for the kth class. The earlier problem of slow mixing does not appear in the hybrid algorithm because the chain in the generative part does not involve sampling the label. Semi-supervised learning: The hybrid learning algorithm can also make use of unlabeled training cases by treating their labels as missing inputs. The model first infers the missing label by sampling P(l|vu) for an unlabeled training case vu. The generative update is then computed by treating the inferred label as the true label. (The discriminative update will always be zero in this case.) Therefore the unlabeled training cases contribute an extra generative term to the parameter update. 4 Sparsity Discriminative performance is improved by using binary features that are only rarely active. Sparse activities are achieved by specifying a desired probability of being active, p << 1, and then adding an additional penalty term that encourages an exponentially decaying average, q, of the actual probability of being active to be close to p. The natural error measure to use is the cross entropy between the desired and actual distributions: p log q +(1−p) log(1−q). For logistic units this has a simple derivative of p−q with respect to the total input to a unit. This derivative is used to adjust both the bias and the incoming weights of each hidden unit. We tried various values for p and 0.1 worked well. In addition to specifying p it is necessary to specify how fast the estimate of q decays. We used qnew = 0.9 ∗qold + 0.1 ∗qcurrent where qcurrent is the average probability of activation for the current mini-batch of 100 training cases. It is also necessary to specify how strong the penalty term should be, but this is easy to set empirically. We multiply the penalty gradient by a coefficient that is chosen to ensure that, on average, q is close to p but there is still significant variation among the q values for different hidden units. This prevents the penalty term from dominating the learning. One 5 added advantage of this sparseness penalty is that it revives any hidden units whose average activities are much lower than p. 5 Evaluating DBNs on the NORB Object Recognition Task 5.1 NORB Database For a detailed description see [12]. The five object classes in NORB are animals, humans, planes, trucks, and cars. The dataset comes in two different versions, normalized-uniform and jittered-cluttered. In this paper we use the normalized-uniform version, which has objects centred in the images with a uniform background. There are 10 instances of each object class, imaged under 6 illuminations and 162 viewpoints (18 azimuths × 9 elevations). The instances are split into two disjoint sets (pre-specified in the database) of five each to define the training and test sets, both containing 24,300 cases. So at test time a trained model has to recognize unseen instances of the same object classes. Pre-processing: A single training (and test) case is a stereo-pair of grayscale images, each of size 96×96. To speed up experiments, we reduce dimensionality by using a “foveal” image representation. The central 64 × 64 portion of an image is kept at its original resolution. The remaining 16 pixel-wide ring around it is compressed by replacing non-overlapping square blocks of pixels with the average value of a block. We split the ring into four smaller ones: the outermost ring has 8 × 8 blocks, followed by a ring of 4 × 4 blocks, and finally two innermost rings of 2 × 2 blocks. The foveal representation reduces the dimensionality of a stereo-pair from 18432 to 8976. All our models treat the stereo-pair images as 8976dimensional vectors1. 5.2 Training Details Model architecture: The two main decisions to make when training DBNs are the number of hidden layers to greedily pre-train and the number of hidden units to use in each layer. To simplify the experiments we constrain the number of hidden units to be the same at all layers (including the top-level model). We have tried hidden layer sizes of 2000, 4000, and 8000 units. We have also tried models with two, one, or no greedily pre-trained hidden layers. To avoid clutter, only the results for the best settings of these two parameters are given. The best classification results are given by the DBN with one greedily pre-trained sparse hidden layer of 4000 units (regardless of the type of top-level model). A DBN trained on the pre-processed input with one greedily pre-trained layer of 4000 hidden units and a third-order model on top of it, also with 4000 hidden units, has roughly 116 million learnable parameters in total. This is roughly two orders of magnitude more parameters than some of the early DBNs trained on the MNIST images [6], [10]. Training such a model in Matlab on an Intel Xeon 3GHz machine takes almost two weeks. See a recent paper by Raina et al. [15] that uses GPUs to train a deep model with roughly the same number of parameters much more quickly. We put Gaussian units at the lowest (pixel) layer of the DBN, which have been shown to be effective for modelling grayscale images [7]. See [7], [21] for details about Gaussian units. 6 Results The results are presented in three parts: part 1 compares deep models to shallow ones, all trained using CD. Part 2 compares CD to the hybrid learning algorithm for training the top-level model of a DBN. Part 3 compares DBNs trained with and without unlabeled data, using either CD or the hybrid algorithm at the top level. For comparison, here are some published results for discriminative models on normalized-uniform NORB (without any pre-processing) [2], [12]: logistic regression 19.6%, kNN (k=1) 18.4%, Gaussian kernel SVM 11.6%, convolutional neural net 6.0%, convolutional net + SVM hybrid 5.9%. 1Knowledge about image topology is used only along the (mostly empty) borders, and not in the central portion that actually contains the object. 6 6.1 Deep vs. Shallow Models Trained with CD We consider here DBNs with one greedily pre-trained layer and a top-level model that contains the greedily pretrained features as its “visible” layer. The corresponding shallow version trains the top-level model directly on the pixels (using Gaussian visible units), with no pre-trained layers in between. Using CD as the learning algorithm (for both greedy pretraining and at the top-level) with the two types of top-level models gives us four possibilities to compare. The test error rates for these four models(see table 1) show that one greedily pre-trained layer reduces the error substantially, even without any subsequent fine-tuning of the pre-trained layer. Model RBM with Third-order label unit RBM Shallow 22.8% 20.8% Deep 11.9% 7.6% Table 1: NORB test set error rates for deep and shallow models trained using CD with two types of top-level models. The third-order RBM outperforms the standard RBM top-level model when they both have the same number of hidden units, but a better comparison might be to match the number of parameters by increasing the hidden layer size of the standard RBM model by five times (i.e. 20000 hidden units). We have tried training such an RBM, but the error rate is worse than the RBM with 4000 hidden units. 6.2 Hybrid vs. CD Learning for the Top-level Model We now compare the two alternatives for training the top-level model of a DBN. There are four possible combinations of top-level models and learning algorithms, and table 2 lists their error rates. All these DBNs share the same greedily pre-trained first layer – only the top-level model differs among them. Learning RBM with Third-order algorithm label unit RBM CD 11.9% 7.6% Hybrid 10.4% 6.5% Table 2: NORB test set error rates for top-level models trained using CD and the hybrid learning algorithms. The lower error rates of hybrid learning are partly due to its ability to avoid the poor mixing of the label variable when CD is used to learn the joint density P(v, l) and partly due to its greater emphasis on discrimination (but with strong regularization provided by also learning P(v|l)). 6.3 Semi-supervised vs. Supervised Learning In this final part, we create additional images from the original NORB training set by applying global translations of 2, 4, and 6 pixels in eight directions (two horizontal, two vertical and four diagonal directions) to the original stereo-pair images2. These “jittered” images are treated as extra unlabeled training cases that are combined with the original labeled cases to form a much larger training set. Note that we could have assigned the jittered images the same class label as their source images. By treating them as unlabeled, the goal is to test whether improving the unsupervised, generative part of the learning alone can improve discriminative performance. There are two ways to use unlabeled data: 1. Use it for greedy pre-training of the lower layers only, and then train the top-level model as before, with only labeled data and the hybrid algorithm. 2The same translation is applied to both images in the stereo-pair. 7 2. Use it for learning the top-level model as well, now with the semi-supervised variant of the hybrid algorithm at the top-level. Table 3 lists the results for both options. Top-level model Unlabeled Unlabeled (hyrbid learning jitter for jitter at the Error only) pre-training top-level? lower layer? RBM with No No 10.4% label unit Yes No 9.0% Third-order No No 6.5% model Yes No 5.3% Yes Yes 5.2% Table 3: NORB test set error rates for DBNs trained with and without unlabeled data, and using the hybrid learning algorithm at the top-level. The key conclusion from table 3 is that simply using more unlabeled training data in the unsupervised, greedy pre-training phase alone can significantly improve the classification accuracy of the DBN. It allows a third-order top-level model to reduce its error from 6.5% to 5.3%, which beats the current best published result for normalized-uniform NORB without using any extra labeled data. Using more unlabeled data also at the top level further improves accuracy, but only slightly, to 5.2%. Now consider a discriminative model at the top, representing the distribution P(l|v). Unlike in the generative case, the exact gradient of the log-likelihood is tractable to compute. Table 4 shows the results of some discriminative models. These models use the same greedily pre-trained lower layer, learned with unlabeled jitter. They differ in how the top-level parameters are initialized, and whether they use the jittered images as extra labeled cases for learning P(l|v). Initialization Use jittered of top-level images as Error parameters labeled? Random No 13.4% Random Yes 7.1% Model with 5.2% error Yes 5.0% from table 3 Table 4: NORB test set error rates for discriminative third-order models at the top level. We compare training the discriminative toplevel model “from scratch” (random initialization) versus initializing its parameters to those of a generative model learned by the hybrid algorithm. We also compare the effect of using the jittered images as extra labeled cases. As mentioned before, it is possible to assign the jittered images the same labels as the original NORB images they are generated from, which expands the labeled training set by 25 times. The bottom two rows of table 4 compare a discriminative third-order model initialized with and without pre-training. Pre-trained initialization (5.0%) significantly improves accuracy over random initialization (7.1%). But note that discriminative training only makes a small additional improvement (5.2% to 5.0%) over the accuracy of the pre-trained model itself. 7 Conclusions Our results make a strong case for the use of generative modeling in object recognition. The main two points are: 1) Unsupervised, greedy, generative learning can extract an image representation that supports more accurate object recognition than the raw pixel representation. 2) Including P(v|l) in the objective function for training the top-level model results in better classification accuracy than using P(l|v) alone. In future work we plan to factorize the third-order Boltzmann machine as described in [18] so that some of the top-level features can be shared across classes. 8 References [1] Y. Bengio, P. Lamblin, P. Popovici, and H. Larochelle. Greedy Layer-Wise Training of Deep Networks. In NIPS, 2006. [2] Y. Bengio and Y. LeCun. Scaling learning algorithms towards AI. In Large-Scale Kernel Machines, 2007. [3] D. DeCoste and B. Scholkopf. Training Invariant Support Vector Machines. Machine Learning, 46:161–190, 2002. [4] G. E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14(8):1711–1800, 2002. [5] G. E. Hinton. To Recognize Shapes, First Learn to Generate Images. Technical Report UTML TR 2006-04, Dept. of Computer Science, University of Toronto, 2006. [6] G. E. Hinton, S. Osindero, and Y. Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18:1527–1554, 2006. [7] G. E. Hinton and R. Salakhutdinov. Reducing the dimensionality of data with neural networks. Science, 313:504–507, 2006. [8] M. Kelm, C. Pal, and A. McCallum. Combining Generative and Discriminative Methods for Pixel Classification with Multi-Conditional Learning. In ICPR, 2006. [9] H. Larochelle and Y. Bengio. Classification Using Discriminative Restricted Boltzmann Machines. In ICML, pages 536–543, 2008. [10] H. Larochelle, D. Erhan, A. Courville, J. Bergstra, and Y. Bengio. An empirical evaluation of deep architectures on problems with many factors of variation. In ICML, pages 473–480, 2007. [11] 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. [12] Y. LeCun, F. J. Huang, and L. Bottou. Learning methods for generic object recognition with invariance to pose and lighting. In CVPR, Washington, D.C., 2004. [13] H. Lee, R. Grosse, R. Ranganath, and A. Ng. Convolutional Deep Belief Networks for Scalable Unsupervised Learning of Hierarchical Representations. In ICML, 2009. [14] V. Nair and G. E. Hinton. Implicit mixtures of restricted boltzmann machines. In Neural information processing systems, 2008. [15] R. Raina, A. Madhavan, and A. Ng. Large-scale Deep Unsupervised Learning using Graphics Processors. In ICML, 2009. [16] Marc’Aurelio Ranzato, Fu-Jie Huang, Y-Lan Boureau, and Yann LeCun. Unsupervised learning of invariant feature hierarchies with applications to object recognition. In Proc. Computer Vision and Pattern Recognition Conference (CVPR’07). IEEE Press, 2007. [17] T. J. Sejnowski. Higher-order Boltzmann Machines. In AIP Conference Proceedings, pages 398–403, 1987. [18] G. Taylor and G. E. Hinton. Factored Conditional Restricted Boltzmann Machines for Modeling Motion Style. In ICML, 2009. [19] V. Vapnik. Statistical Learning Theory. John Wiley and Sons, 1998. [20] P. Vincent, H. Larochelle, Y. Bengio, and P. A. Manzagol. Extracting and Composing Robust Features with Denoising Autoencoders. In ICML, 2008. [21] M. Welling, M. Rosen-Zvi, and G. E. Hinton. Exponential family harmoniums with an application to information retrieval. In NIPS 17, 2005. 9
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Improving Existing Fault Recovery Policies Guy Shani Department of Information Systems Engineering Ben Gurion University, Beer-Sheva, Israel shanigu@bgu.ac.il Christopher Meek Microsoft Research One Microsoft Way, Redmond, WA meek@microsoft.com Abstract An automated recovery system is a key component in a large data center. Such a system typically employs a hand-made controller created by an expert. While such controllers capture many important aspects of the recovery process, they are often not systematically optimized to reduce costs such as server downtime. In this paper we describe a passive policy learning approach for improving existing recovery policies without exploration. We explain how to use data gathered from the interactions of the hand-made controller with the system, to create an improved controller. We suggest learning an indefinite horizon Partially Observable Markov Decision Process, a model for decision making under uncertainty, and solve it using a point-based algorithm. We describe the complete process, starting with data gathering, model learning, model checking procedures, and computing a policy. 1 Introduction Many companies that provide large scale online services, such as banking services, e-mail services, or search engines, use large server farms, often containing tens of thousands of computers in order to support fast computation with low latency. Occasionally, these computers may experience failures, due to software, or hardware problems. Often, these errors can be fixed automatically through actions such as rebooting or re-imaging of the computer [6]. In such large systems it is prohibitively costly to have a technician decide on a repair action for each observed problem. Therefore, these systems often use some automatic repair policy or controller to choose appropriate repair actions. These repair policies typically receive failure messages from the system. For example, Isard [6] suggests using a set of watchdogs — computers that probe other computers to test some attribute. Messages from the watchdogs are then typically aggregated into a small set of notifications, such as “Software Error” or “Hardware Error”. The repair policy receives notifications and decides which actions can fix the observed problems. In many cases such policies are created by a human experts based on their experience and knowledge of the process. While human-made controllers often exhibit a reasonable performance, they are not automatically optimized to reduce costs. Thus, in many cases, it is possible to create a better controller, that would improve the performance of the system. A natural choice for modeling such systems is to model each machine as a Partially Observable Markov Decision Process (POMDP) [8] — a well known model for decision making under uncertainty [12]. Given the POMDP parameters, we can compute a policy that optimizes repair costs, but learning the POMDP parameters may be difficult. Most researchers that use POMDPs therefore assume that the parameters are known. Alternatively, Reinforcement Learning (RL) [14] offers a wide range of techniques for learning optimized controllers through interactions with the environment, often avoiding the need for an explicit model. These techniques are typically used in an online learning setting, and require that the agent will explore all possible state-action pairs. In the case of the management of large data centers, where inappropriate actions may result in considerable increased costs, it is unlikely that the learning process would be allowed to try every 1 combination of state and action. It is therefore unclear how standard RL techniques can be used in this setting. On the other hand, many systems log the interactions of the existing hand-made controller with the environment, accumulating significant data. Typically, the controller will not be designed to perform exploration, and we cannot expect such logs to contain sufficient data to train standard RL techniques. In this paper we introduce a passive policy learning approach, that uses only available information without exploration, to improve an existing repair policy. We adopt the indefinite-horizon POMDP formalization [4], and use the existing controller’s logs to learn the unkown model parameters, using an EM algorithm (an adapted Baum-Welch [1, 2, 15] algorithm). We suggest a model-checking phase, providing supporting evidence for the quality of the learned model, which may be crucial to help the system administrators decide whether the learned model is appropriate. We proceed to compute a policy for our learned model, that can then be used in the data center instead of the original hand-made controller. We experiment with a synthetic, yet realistic, simulation of machine failures, showing how the policy of the learned POMDP performs close to optimal, and outperforms a set of simpler techniques that learn a policy directly in history space. We discuss the limitations of our method, mainly the dependency on a reasonable hand-made controller in order to learn good models. Many other real world applications, such as assembly lines, medical diagnosis systems, and failure detection and recovery systems, are also controlled by hand-made controllers. While in this paper we focus on recovery from failures, our approach may be applicable to other similar domains. 2 Properties of the Error Recovery Problem In this section we describe aspects of the error recovery problem and a POMDP model for the problem. Key aspects of the problem include the nature of repair actions and costs, machine failure, failure detection, and control policies. Key aspects of repair actions include: (1) actions may succeed or fail stochastically. (2) These actions often provide an escalating behavior. We label actions using increasing levels, where problems fixed by an action at level i, are also fixed by any action of level j > i. Probabilistically, this would mean that if j > i then pr(healthy|aj, e) ≥pr(healthy|ai, e) for any error e. (3) Action costs are typically escalating, where lower level actions that fix minor problems are relatively cheap, while higher level actions are more expensive. In many real world systems this escalation is exponential. For example, restarting a service takes 5 seconds, rebooting a machine takes approximately 10 minutes, while re-imaging the machine takes about 2 hours. Another stochastic feature of this problem is the inexact failure detection. It is not uncommon for a watchdog to report an error for a machine that is fully operational, or to report an “healthy” status for a machine that experiences a failure. In this domain, machines are identical and independent. Typically computers in service farms share the same configuration and execute independent programs, attempting, for example, to answer independent queries to a search engine. It is therefore unlikely, if not impossible, for errors to propagate from one machine to another. In view of the escalating nature of actions and costs, a natural choice for a policy is an escalation policy. Such policies choose a starting level based on the first observation, and execute an action at that level. In many cases, due to the non-deterministic success of repair actions, each action is tried several times. After the controller decides that the action at the current level cannot fix the problem, the controller escalates to the next action level. Such policies have several hand tuned decisions. For example, the number of retries of an action before an escalation occurs, and the entry level given an observation. We can hope that these features, at least, could be optimized by a learning algorithm. System administrators typically collect logs of the hand-made controller execution, for maintenance purposes. These logs represent a valuable source of data about the system behavior that can be used to learn a policy. We would like to use this knowledge to construct an improved policy that will perform better than the original policy. Formally, we assume that we receive as input a log L of repair sessions. Each repair session is a sequence l = o0, a1, o1, ..., onl, starting with an error notification, followed by a set of repair actions and observations until the problem is fixed. In 2 some cases, sessions end with the machine declared as “dead”, but in practice a technician is called for these machines, repairing or replacing them. Therefore, we can assume that all sessions end successfully in the healthy state. 2.1 A POMDP for Error Recovery Given the problem features above, a natural choice is to model each machine independently as a partially observable Markov decision process (POMDP) with common parameters. We define a cost-based POMDP through a tuple < S, A, tr, C, Ω, O > where S is a set of states. In our case, we adopt a factored representation, where s =< e0, ..., en > where ei ∈{0, 1} indicates whether error i exists. That is, states are sets of failures, or errors of a machine, such as software error or a hardware failure. We also add a special state sH =< 0, ..., 0 > — the healthy state. A is a set of actions, such as rebooting a machine or re-imaging it. tr(s, a, s′) is a state transition function, specifying the probabilities of moving between states. We restrict our transition function such that tr(s, a, s′) > 0 iff ∀i if si = 0 then s′ i = 0. That is, an action may only fix an error, not generate new errors. C(s, a) is a cost function, assigning a cost to each state-action pair. Often, costs can be measured as the time (minutes) for executing the action. For example, a reboot may take 15 minutes, while re-imaging takes 2 hours. Ωis a set of possible observations. For us, observations are messages from the watchdogs, such as a notification of a hard disk failure, or a service reporting an error, and notifications about the success or failure of an action. O(a, s′, o) is an observation function, assigning a probability to each observation pr(o|a, s′). In a POMDP the true state is not directly observable and we thus maintain a belief state b ∈B — a probability distribution over states, where b(s) is the probability that the system is at state s. We assume that every repair session starts with an error observation, typically provided by one of the watchdogs. We therefore define bo 0 — the prior distribution over states given an initial observation o. We will also maintain a probability distribution pr0(o) over initial observations. While this probability distribution is not used in model learning, it is useful for evaluating the quality of policies through trials. It is convenient to define a policy for a POMDP as a mapping from belief states to actions π : B → A. Our goal is to find an optimal policy that brings the machine to the healthy state with the minimal cost. One method for computing a policy is through a value function, V , assigning a value to each belief state b. Such a value function can be expressed as a set of |S| dimensional vectors known as α-vectors, i.e., V = {α1, ..., αn}. Then, αb = minα∈V α · b is the optimal α-vector for belief state b, and V (b) = b · αb is the value that the value function V assigns to b, where α · b = P i αibi is the standard vector inner product. By associating an action a(α) which each vector, a policy π : B →A can be defined through π(b) = a(αb). While exact value iteration, through complete updates of the belief space, does not scale beyond small toy examples, Pineau et al. [10] suggest to update the value function by creating a single α-vector that is optimized for a specific belief state. Such methods, known as point-based value iteration, compute a value function over a finite set of belief states, resulting in a finite size value function. Perseus [13] is an especially fast point-based solver that incrementally updates a value function over a randomly chosen set of belief points, ensuring that at each iteration, the value for each belief state is improved, while maintaining a compact value function representation. We adopt here the indefinite horizon POMDP framework [4], which we consider to be most appropriate for failure recovery. In this framework the POMDP has a single special action aT , available in any state, that terminates the repair session. In our case, the action is to call a technician, deterministically repairing the machine, but with a huge cost. For example, Isard [6] estimates that a technician will fix a computer within 2 weeks. Executing aT in sH incurs no cost. Using indefinite horizon it is easy to define a lower bound on the value function using aT , and execute any point-based algorithm, such as the Perseus algorithm that we use. 3 3 Learning Policies from System Logs In this section we propose two alternatives for computing a recovery policy given the logs. We begin with a simple, model-free, history-based policy computation. Then, we suggest a more sophisticated method that learns the POMDP model parameters, and then uses the POMDP to compute a policy. 3.1 Model-Free Learning of Q-values The optimal policy for a POMDP can be expressed as a mapping from action-observation histories to actions. Histories are directly observable, allowing us to use the standard Q function terminology, where Q(h, a) is the expected cost of executing action a with history h and continuing the session until it terminates. This approach is known as model-free, because (e.g.) the parameters of a POMDP are never learned, and has some attractive properties, because histories are directly observable, and do not require any assumption about the unobserved state space. As opposed to standard Q-learning, where the Q function is learned while interacting with the environment, we use the system log L to compute Q: Cost(li) = |l| X j=i+1 C(aj) (1) Q(h, a) = P l∈L δ(h + a, l)Cost(l|h|) P l∈L δ(h + a, l) (2) where li is a suffix of l starting at action ai, C(a) is the cost of action a, h + a is the history h with the action a appended at its end, and δ(h, l) = 1 if h is a prefix of l and 0 otherwise. The Q function is hence the average cost until repair of executing the action a in history h, under the policy that generated L. Learning a Q function is much faster than learning the POMDP parameters, requiring only a single pass over the training sequences in the system log. Given the learned Q function, we can define the following policy: πQ(h) = min a Q(h, a) (3) One obvious problem of learning a direct mapping from history to actions is that such policies do not generalize — if a history sequence was not observed in the logs, then we cannot evaluate the expected cost until the error is repaired. An approach that generalizes better is to use a finite history window of size k, discarding all the observations and action occurring more than k steps ago. For example, when k = 1 the result is a completely reactive Q function, computing Q(o, a) using the last observation only. 3.2 Model-Based Policy Learning While we assume that the behavior of a machine can be captured perfectly using a POMDP as described above, in practice we cannot expect the parameters of the POMDP to be known a-priori. In practice, the only parameters that are known are the set of possible repair actions and the set of possible observations, but even the number of possible errors is not initially known, let alone the probability of repair or observation. Given the log of repair sessions, we can use a learning algorithm to learn the parameters of the POMDP. In this paper we choose to use an adapted Baum-Welch algorithm [1, 2, 15], an EM algorithm originally developed for computing the parameters of Hidden Markov Models (HMMs). The Baum-Welch algorithm takes as input the number of states (the number of possible errors) and a set of training sequences. Then, using the forward-backward procedure, the parameters of the POMDP are computed, attempting to maximize the likelihood of the data (the observation sequences). After the POMDP parameters have been learned, we execute Perseus [13] to compute a policy. While training the model parameters, it is important to test likelihood on a held out set of sequences that are not used in training, in order to ensure that the resulting model does not over-fit the data. We hence split the input sequences into a train set (80%) and test set (20%). We check the likelihood of the test set after each forward-backward iteration, and stop the training when the likelihood of the test set does not improve. 4 3.2.1 Model Checking When employing automatic learning methods to create an improved policy, it is important to provide evidence for the quality of the learned models. Such evidence can be helpful for the system administrators in order to make a decision whether to replace the existing policy with a new policy. Using an imperfect learner such as Baum-Welch does not guarantee that the resulting model indeed maximizes the likelihood of the observations given the policy, even for the same policy that was used to generate the training data. Also, the loss function used for learning the model ignores action costs, thus ignoring an important aspect of the problem. For these reasons, it is possible that the resulting model will describe the domain poorly. After the model has been learned, however, we can use the average cost to provide evidence for the validity of the model. Such a process can help us determine whether these shortcomings of the learning process have indeed resulted in an inappropriate model. This phase is usually known as model checking (see, e.g. [3]). As opposed to the Q-learning approach, learning a generative model (the POMDP) allows us check how similar the learned model is to the original model. We say that two POMDPs M1 =< S1, A, tr1, C1, Ω, O1 > and M2 =< S2, A, tr2, C2, Ω, O2 > are indistinguishable if for each policy π : H →A, E[P t Ct|M1, π] = E[P t Ct|M2, π]. That is, the models are indistinguishable if any policy has the same expected accumulated cost when executed in both models. Many policies cannot be evaluated on the real system because we cannot tolerate damaging policies. We can, however, compare the performance of the original, hand-made policy, on the system and on the learned POMDP model. We hence focus the model checking phase on comparing the expected cost of the hand-made policy predicted be the learned model to the true expected cost on the real system. To estimate the expected cost in the real system, we use the average cost of the sessions in the logs. To estimate the expected cost of the policy on the learned POMDP we execute a set of trials, each simulating a repair session, using the learned parameters of the POMDP to govern the trial advancement (observation emissions given the history and action). We can then use the two expected cost estimates as a measure of closeness. For example, if the predicted cost of the policy over the learned POMDP is more than 20% away from the true expected cost, we may deduce that the learned mode does not properly capture the system dynamics. While checking the models under a single policy cannot ensure that the models are identical, it can detect whether the model is defective. If the learned model produces a substantially different expectation over the cost of a policy than the real system, we know that the model is corrupted prior to executing its optimal policy on the real system. After ensuring that the original policy performs similarity on the real system and on the learned model, we can also evaluate the performance of the computed policy on the learned model. Thus, we can compare the quality of the new policy to the existing one, helping us to understand the potential cost reduction of the new policy. 4 Empirical Evaluation In this section we provide an empirical evaluation to demonstrate that our methods can improve an existing policy. We created a simulator of recovery sessions. In the simulator we assume that a machine can be in one of n error states, or in healthy state, we also assume n possible repair actions, and m possible observations. We assume that each action was designed to fix a single error state, and set the number of errors to be the number of repair actions. We set pr(sH|ei, aj) = 0.7 + 0.3 · j−i j if j ≥i and 0 otherwise, simulating the escalation power of repair actions. We set C(s, ai) = 4i and C(sH, aT ) = 0, simulating the exponential growth of costs in the real AutoPilot system [6], and the zero downtime caused by terminating the session in the healthy state. For observations, we compute the relative severity of an error ei in the observation space µi = i∗m n , and then set pr(oj|ei, a) = κe−(i−µi)2 2 / √ 2π, where κ is a normalizing factor, and j ∈[µi −1, µi + 1]. We execute a hand-made escalation policy with 3 retries (see Section 2) over the simulator and gather a log of repair sequences. Each repair sequence begins with selecting an error uniformly, and executing the policy until the error is fixed. Then, we use the logs in order to learn a Q function 5 Table 1: Average cost of recovery policies in simulation, with increasing model size. Results are averaged over 10 executions, and the worst standard error across all recovery policies is reported in the last column. Problem parameters Original Optimal Policies learned from logs |E| |O| |L| πE, S πM∗, S πM, S Q, S Q1, S Q3, S Q5, S SE 2 2 10000 21.6 17.3 17.3 17.3 18.0 17.4 17.3 < 0.2 4 2 10000 220.3 167.7 172.3 193.6 174.6 179.6 190.8 < 3 4 4 10000 221.6 136.8 141.5 197.8 239.5 163.6 178.5 < 2.5 8 4 50000 29070 15047 20592 52636 29611 24611 27951 < 250 8 8 50000 28978 15693 18303 54585 61071 26808 27038 < 275 over the complete history, finite history window Q functions with k = 1, 3, and a POMDP model. For the POMDP model, we initialize the number of states to the number of repair actions, initialize transition uniformly, and observation randomly, and execute the Baum-Welch algorithm. We also constructed a maximum-likelihood POMDP model, by initializing the state space, transition, and observation function using the true state labels (the simulated errors), and executing Baum-Welch afterwards. This initialization simulates the result of a “perfect learner” that does not suffer from the local maximum problems of Baum-Welch. In the tables below we use S for simulator, M for the learned model, and M ∗for the model initialized by the true error labels. For policies, we use πE for the escalation policy, πM for the policy computed by Perseus on the learned model, and πM ∗for the Perseus policy over the ‘perfect learner’ model. For the history-based Q functions, we use Q to denote the function computed for the complete history, and Qi denotes a policy over history suffixes of length i. A column header π, S denotes the estimated cost of executing π on the simulator, and π, M denotes the estimated cost of executing π on the model M. We also report the standard error in the estimations. 4.1 Results We begin with showing the improvement of the policy of the learned models over the original escalation policies. As Table 1 demonstrates, learning a POMDP model and computing its policy always result in a substantial reduction in costs. The M ∗model, initialized using the true error labels, provides an upper bound on the best performance gain that can be achieved using our approach. We can see that in many cases, the result of the learned model is very closed to this upper bound. The Q functions over histories did well on the smallest domains, but not on larger domains. The worst performance is of the reactive Q function (Q1) over the latest observation. In the smaller domains Q learning, especially with a history window of 3 (Q3) does fairly well, but in the larger domains all history-based policies do not perform well. We now take a look at the results of the model checking technique. As we explained above, a model checking phase, comparing the expected cost of a policy on both the real system and the learned model, can provide evidence as to the validity of the learned model. Indeed, as we see in Table 2, the learned models predict an expected cost that is within 3% of the real expected cost. To further validate our learned models, we also compare the expected cost of the policies computed from the models (M and M ∗) over the model and the simulator. Again, we can see that the predicted costs are very close to the real costs of these policies. As expected, the M ∗predicted costs are within measurement error of the true costs of the policy on the real system. 4.2 Discussion The experimental results provide a few interesting insights. First, when the observation space is rich enough to capture all the errors, the learned model is very close to the optimal one. When we use fewer observations, the quality of the learned model is reduced (further from M ∗), but the policy that is learned still significantly outperforms the original escalation policy. It is important to note that the hand-made escalation policy that we use is very natural for domains with actions that have escalating costs and effects. Also, the number of actions and errors that we use is similar to these used by current controllers of repair services [6]. As such, the improvements 6 Table 2: Comparing expected cost of policies on the learned model and the simulator for model checking. Results are averaged over 10 executions, and the worst standard error across all recovery policies is reported in the last column. Problem parameters Escalation policy Optimal model Learned model |E| |O| |L| πE, M ∗ πE, M πE, S πM∗, M ∗ πM∗, S πM, M πM, S SE 2 2 10000 21.6 22.3 21.6 17.3 17.3 17.6 17.3 < 0.2 4 2 10000 219.5 227.2 220.4 165.5 167.7 173.5 172.4 < 3 4 4 10000 221.1 225.4 221.6 137.4 136.8 138.7 141.5 < 2.5 8 4 50000 28985 29152 29070 16461 15047 21104 20592 < 250 8 8 50000 28870 29104 28978 15630 15693 17052 18303 < 275 that we achieve over this hand-made policy hint that similar gains can be made over the real system. Our results show an improvement of 20%−40%, increasing with the size of the domain. As driving down the costs of data centers is crucial for the success of such systems, increasing performance by 20% is a substantial contribution. The history-based approaches did well only on the smallest domains. This is because for a history based value function, we have to evaluate any action at every history. As the input policy does not explore, the set of resulting histories does not provide enough coverage of the history space. For example, if the current repair policy only escalates, the history-based approach will never observe a higher level action followed by a low level action, and cannot evaluate its expected cost. Finite history windows increase coverage, by reducing the number of possible histories to a finite scale. Thus, finite-history window provide some generalization power. Indeed, the finite history window methods (except for the reactive policy) improve upon the original escalation policy in some cases. We note, though, that we used a very simple finite history model, and more complicated approaches, such as variable length history windows [9, 11] may provide better results. Our model checking technique indicates that none of the models that were learned, even when the number of observations was smaller than the number of errors, was defective. This is not particulary surprising, as the input data originated from a simulator that operated under the assumptions of the model. This is unlikely to happen in the real system, where any modeling assumptions compromise some aspect of the real world. Still, in the real world this technique will allow us to test, before changing the existing policy, whether the assumptions are close to the truth, and whether the model is reasonable. This is a crucial step in making the decision to replace an existing, imperfect yet operative policy, with a new one. It is unclear how to run a similar model checking phase over the history-based approach. In improving unexploring policies, we must assume that many possible histories will not be observed. However, a complete model definition must set a value for every possibility. In learning such cases, it is important to set default values for these unknown model parameters. In our case, it is best to be pessimistic about these parameters, that is, to overestimate the cost of repair. It is therefore safe to assume that action a will not fix error e if we never observed a to fix e in the logs, except for the terminating action aT . 5 Related Work Using decision-theoretic techniques for troubleshooting and recovery dates back to Heckerman et al. [5], who employed Bayesian networks for troubleshooting, and a myopic approximation for recovery. Heckerman et al. assume that the parameters of the Bayesian network are given as input, and training it using the unlabeled data that the logs contain is difficult. This Bayesian network approach is also not designed for sequential data. Partially Observable Markov Decision Processes were previously suggested for modeling automated recovery from errors. Most notably, Littman et al. [8] suggests the CSFR model which is similar to our POMDP formalization, except for a deterministic observation function, the escalation of actions, and the terminating action. They then proceed to define a belief state in this model, which is a set of possible error states, and a Q-function Q(b, a) over beliefs. The Q-function is computed using standard value iteration. As these assumptions reduce the partial observability, the resulting 7 Q function can produce good policies. Littman et al. assume that the model is either given, or that Q-learning can be executed online, using an exploration strategy, both of which are not applicable in our case. Also, as we argue above, in our case a Q function produces substantially inferior policies because of its lack of generalization power in partially observable domains. Another, more recent, example of a recovery approach based on POMDPs was suggested by Joshi et al. [7]. Similar to Littman et al., Joshi et al. focus on the problem of fault recovery in networks, which adds a layer of difficulty because we can no longer assume that machines are independent, as often faults cascade through the network. Joshi et al. also assume that the parameters of the model, such as the probability that a watchdog will detect each failure, and the effects of actions on failures, are known a-priori. They then suggest a one step lookahead repair strategy, and a multi-step lookahead, that uses a value function over a belief space similar to the Littman et al. belief space. Bayer-Zubek and Dietterich [16] use a set of examples, similar to our logs, to learn a policy for disease diagnosis. They formalize the problem as an MDP, assuming that test results are discrete and exact, and use AO∗search, while computing the needed probabilities using the example set. They did not address the problem of missing data in the example set that arises from a non-exploring policy. Indeed, in the medical diagnosis case, one may argue that trying an action sequence that was never tried by a human doctor may result in an unreasonable risk of harm to the patient, and that therefore the system should not consider such policies. 6 Conclusion We have presented an approach to improving imperfect repair policies through learning a POMDP model of the problem. Our method takes as input a log of interaction of the existing controller with the system, learns a POMDP model, and computes a policy for the POMDP that can be used in the the real system. The advantage of our method is that it does not require the existing controller to actively explore the effects of actions in all conditions, which may result in unacceptable costs in the real system. On the other hand, our approach may not converge to an optimal policy. We experiment with a synthetic, yet realistic, example of a hand-made escalation policy, where actions are ordered by increasing cost, and any action is repeated a number of times. We show how the policy of the learned model significantly improves the original escalation policy. In the future we intend to use the improved policies to manage repairs in a real data center within the AutoPilot system [6]. The first step would be to “flight” candidate policies to evaluate their performance in the real system. Our current method is a single shot improvement, and an interesting next step is to create an incremental improvement process, where new policies constantly improve the existing one. In this setting, it would be interesting to explore bounded exploration, an exploration technique that puts a bound on the risk of the strategy. There are a number of interesting theoretical questions about our passive policy learning method and about passive policy learning in general. First, for what families of initial policies and system dynamics would a passive policy learning method be expected to yields an improvement in expected costs. Second, what families of initial policies and systems dynamics would a passive policy learning method be expected to yield the optimal policy. Third, how would one characterize when iteratively applying a passive policy learning method would yield expected improvements in expected costs. Finally, while this paper focuses on the important failure recovery problem, our methods may be applicable to a wide range of similar systems, such as assembly line management, and medical diagnosis systems, that currently employ hand-made imperfect controllers. References [1] Leonard E. Baum, Ted Petrie, George Soules, and Norman Weiss. A maximization technique occurring in the statistical analysis of probabilistic functions of Markov chains. The Annals of Mathematical Statistics, 41(1):164–171, 1970. [2] Lonnie Chrisman. Reinforcement learning with perceptual aliasing: The perceptual distinctions approach. In In Proceedings of the Tenth National Conference on Artificial Intelligence, pages 183–188. AAAI Press, 1992. 8 [3] Andrew Gelman, John B. Carlin, Hal S. Stern, and Donald B. Rubin. Bayesian Data Analysis. Chapman and Hall, 1996. [4] Eric A. Hansen. Indefinite-horizon POMDPs with action-based termination. In AAAI, pages 1237–1242, 2007. [5] David Heckerman, John S. Breese, and Koos Rommelse. Decision-theoretic troubleshooting. Commun. ACM, 38(3):49–57, 1995. [6] Michael Isard. Autopilot: automatic data center management. Operating Systems Review, 41(2):60–67, 2007. [7] Kaustubh R. Joshi, William H. Sanders, Matti A. Hiltunen, and Richard D. Schlichting. Automatic modeldriven recovery in distributed systems. In SRDS, pages 25–38, 2005. [8] Michael L. Littman and Nishkam Ravi. An instance-based state representation for network repair. In in Proceedings of the Nineteenth National Conference on Artificial Intelligence (AAAI, pages 287–292, 2004. [9] Andrew Kachites Mccallum. Reinforcement learning with selective perception and hidden state. PhD thesis, 1996. Supervisor-Ballard, Dana. [10] Joelle Pineau, Geoffrey Gordon, and Sebastian Thrun. Point-based value iteration: An anytime algorithm for POMDPs. In International Joint Conference on Artificial Intelligence (IJCAI), pages 1025 – 1032, August 2003. [11] Guy Shani and Ronen I. Brafman. Resolving perceptual aliasing in the presence of noisy sensors. In NIPS, 2004. [12] R. D. Smallwood and E. J. Sondik. The optimal control of partially observable Markov decision processes over a finite horizon. Operations Research, 21:1071–1098, 1973. [13] Matthijs T. J. Spaan and Nikos Vlassis. Perseus: Randomized point-based value iteration for POMDPs. Journal of Artificial Intelligence Research, 24:195–220, 2005. [14] Richard S. Sutton and Andrew Barto. Reinforcement Learning: An Introduction. MIT Press, 1998. [15] Daan Wierstra and Marco Wiering. Utile distinction hidden Markov models. In ICML ’04: Proceedings of the twenty-first international conference on Machine learning, page 108, New York, NY, USA, 2004. ACM. [16] Valentina Bayer Zubek and Thomas G. Dietterich. Integrating learning from examples into the search for diagnostic policies. J. Artif. Intell. Res. (JAIR), 24:263–303, 2005. 9
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Convex Relaxation of Mixture Regression with Efficient Algorithms Novi Quadrianto, Tib´erio S. Caetano, John Lim NICTA - Australian National University Canberra, Australia {firstname.lastname}@nicta.com.au Dale Schuurmans University of Alberta Edmonton, Canada dale@cs.ualberta.ca Abstract We develop a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Although our relaxation involves a semidefinite matrix variable, we reformulate the problem to eliminate the need for general semidefinite programming. In particular, we provide two reformulations that admit fast algorithms. The first is a max-min spectral reformulation exploiting quasi-Newton descent. The second is a min-min reformulation consisting of fast alternating steps of closed-form updates. We evaluate the methods against Expectation-Maximization in a real problem of motion segmentation from video data. 1 Introduction Regression is a foundational problem in machine learning and statistics. In practice, however, data is often better modeled by a mixture of regressors, as demonstrated by the prominence of mixture regression in a number of application areas. Gaffney and Smyth [1], for example, use mixture regression to cluster trajectories, i.e. sets of short sequences of data such as cyclone or object movements in video sequences as a function of time. Each trajectory is believed to have been generated from one of a number of components, where each component is associated with a regression model. Finney et al. [2] have employed an identical mixture regression model in the context of planning: regression functions are strategies for a given planning problem. Elsewhere, the mixture of regressors model has been shown to be useful in addressing covariate shift, i.e. the situation where the distribution of the training set used for modeling does not match the distribution of the test set in which the model will be used. Storkey and Sugiyama [3] model the covariate shift process in a mixture regression setting by assuming a shift in the mixing proportions of the components. In each of these problems, one must estimate k distinct latent regression functions; that is, estimate functions whose values correspond to the mean of response variables, under the assumption that the response variable is generated by a mixture of k components. This estimation problem can be easily tackled if it is known to which component each response variable belongs (yielding k independent regression problems). However in general the component of a given observation is not known and is modeled as a latent variable. A commonly adopted approach for maximum-likelihood estimation with latent variables (in this case, component membership for each response variable) is Expectation-Maximization (EM) [4]. Essentially, EM iterates inference over the hidden variables and parameter estimation of the resulting decoupled models until a local optimum is reached. We are not aware of any approach to maximum likelihood estimation of a mixture of regression models that is not based on the non-convex marginal likelihood objective of EM. In this paper we present a convex relaxation of maximum a posteriori estimation of a mixture of regression models. Recently, convex relaxations have gained considerable attention in machine learning (c.f. [5, 6]). By exploiting convex duality, we reformulate a relaxation of mixture regression as a semidefinite program. To achieve a scalable approach, however, we propose two reformulations that admit fast algorithms. The first is a max-min optimization problem which can be solved by iterations of quasi-Newton steps and eigenvector computations. The second is a min-min optimization problem solvable by iterations of closed-form solutions. We present experimental results comparing our methods against EM, both in synthetic problems and real computer vision problems, and show some benefits of a convex approach over a local solution method. 1 Related work Goldfeld and Quandt [7] introduced a mixture regression model with two components called switching regressions. The problem is re-cast into a single composite regression equation by introducing a switching variable. A consistent estimator is then produced by a continuous relaxation of this switching variable. An EM algorithm for switching regressions was first presented by Hosmer [8]. Sp¨ath [9] introduced a problem called clusterwise linear regression, consisting of finding a k-partition of the data such that a least squares regression criterion within those partitions becomes a minimum. A non-probabilistic algorithm similar to k-means was proposed. Subsequently, the general k-partition case employing EM was developed (c.f. [10, 11, 1]) and extended to various situations including the use of variable length trajectory data and to non-parametric regression models. In the extreme, each individual could have its specific regression model but coupled at higher level with a mixture on regression parameters [12]. An EM algorithm is again employed to handle hidden data, in this case group membership of parameters. The Hierarchical Mixtures of Experts [13] model also shares some similarity to mixture regression in that gating networks which contain mixtures of generalized linear models are defined. In principle, our algorithmic advances can be applied to many of these formulations. 2 The Model Notation In the following we use the uppercase letters (X, Π, Ψ) to denote matrices and the lowercase letters (x, y, w, π, ψ, c) to denote vectors. We use t to denote the sample size, n to denote the dimensionality of the data and k to denote the number of mixture components. Λ(a) denotes a diagonal matrix whose diagonal is equal to vector a, and diag(A) is a vector equal to the diagonal of matrix A. Finally, we let 1 denote the vector of all ones, use ⊙to denote Hadamard (componentwise) matrix product, and use ⊗to denote Kronecker product. We are given a matrix of regressors X ∈Rt×n and a vector of regressands y ∈Rt×1 where the response variable y is generated by a mixture of k components, but we do not know which component of the mixture generates each response yi. We therefore use the matrix Π ∈{0, 1}t×k, Π1 = 1, to denote the hidden assignment of mixture labels to each observation: Πij = 1 iff observation i has mixture label j. We use xi to denote the ith row of X (i.e. observation i as a row vector), πi to denote the ith row of Π and yi to denote the ith element of y. We assume a linear generative model for yi on a feature representation ψi = πi ⊗xi, under i.i.d. sampling yi|xi, πi = ψiw + ϵi, ϵi ∼N(0, σ2), (1) where w ∈R(n×k)×1 is the vector of stacked parameter vectors of the components. We therefore have the likelihood p(yi|xi, πi; w) = 1 √ 2πσ2 exp −1 2σ2 (ψiw −yi)2 (2) for a single observation i (recalling that ψi depends on both xi and πi). We further impose a Gaussian prior on w for capacity control. Also, one may want to constrain the size of the largest mixture component. For that purpose one could constrain the solutions Π such that max(diag(ΠT Π)) ≤γt, where γt is an upper bound on the size of the largest component (γ is an upper bound on the proportion of the largest component). Combining these assumptions and adopting matrix notation we obtain the optimization problem: minimize the negative log-posterior of the entire sample min Π,w "X i A(ψi, w) −1 σ2 yT Ψw + 1 2σ2 yT y + α 2 wT w # , where (3) A(ψi, w) = 1 2σ2 wT ψT i ψiw + 1 2 log(2πσ2). (4) Here Ψ is the matrix whose rows are the vectors ψi = πi ⊗xi. Since X is observed, note that the optimization only runs over Π in Ψ. The constraint max(diag(ΠT Π)) ≤γt may also be added. Eliminating constant terms, our final task will be to solve min Π,w 1 2σ2 wT ΨT Ψw −1 σ2 yT Ψw + α 2 wT w . (5) Although marginally convex on w, this objective is not jointly convex on w and Π (and involves non-convex constraints on Π owing to its discreteness). The lack of joint convexity makes the optimization difficult. The typical approach in such situations is to use an alternating descent strategy, such as EM. Instead, in the following we develop a convex relaxation for problem (5). 2 3 Semidefinite Relaxation To obtain a convex relaxation we proceed in three steps. First, we dualize the first term in (5). Lemma 1 Define A(Ψw) := 1 2σ2 wT ΨT Ψw. Then the Fenchel dual of A(Ψw) is A∗(c) = 1 2σ2cT c, and therefore A(Ψw) = maxc cT Ψw −1 2σ2cT c. Proof From the definition of Fenchel dual we have A∗(u) := maxw uT w − 1 2σ2 wT ΨT Ψw. Differentiating with respect to w and equating to zero we obtain u = 1 σ2 ΨT Ψw. Therefore u is only realizable if there exists a c such that u = ΨT c. Solving for A∗(c) we obtain A∗(c) = 1 2σ2cT c, and therefore by definition of Fenchel duality A(Ψw) = maxc cT Ψw −1 2σ2cT c. A second Lemma is required to further establish the relaxation: Lemma 2 The following set inclusion holds {ΠΠT : Π ∈{0, 1}t×k, Π1 = 1, max(diag(ΠT Π)) ≤γt} (6) ⊆ {M : M ∈Rt×t, tr M = t, γtI ≽M ≽0}. (7) Proof Let ΠΠT be an element of the first set. First notice that [ΠΠT ]ij ∈{0, 1} since Π ∈{0, 1}t×k and Π1 = 1 together imply that Π has a single 1 per row (and the rest are zeros). In particular [ΠΠT ]ii = 1 for all i, i.e. tr M = t. Finally, note that (ΠΠT )Π = Π(ΠT Π) where ΠT Π is a diagonal matrix and therefore its diagonal elements are the eigenvalues of ΠΠT and in particular max(diag(ΠT Π)) ≤γt means that the largest possible eigenvalue of ΠΠT is γt, which implies γtI ≽ΠΠT . Since ΠΠT is by construction positive semidefinite, we have γtI ≽ΠΠT ≽0. Therefore ΠΠT is also a member of the second set. The above two lemmas allow us to state our first main result below. Theorem 3 The following convex optimization problem min M:tr M=t,γtI≽M≽0 max c −1 2σ2cT c −1 2α y σ2 −c T M ⊙XXT y σ2 −c (8) is a relaxation of (5) only in the sense that domain (6) is replaced by domain (7). Proof We first use Lemma 1 in order to rewrite the objective (5) and obtain min Π,w max c cT Ψw −1 2σ2cT c −1 σ2 yT Ψw + α 2 wT w . (9) Second, using the distributivity of the (max, +) semi-ring, the maxc can be pulled out and we then use Sion’s minimax theorem [14], which allows us to interchange maxc with minw min Π max c min w cT Ψw −1 2σ2cT c −1 σ2 yT Ψw + α 2 wT w , (10) and we can solve for w first, obtaining w = 1 αΨT y σ2 −c . (11) Substituting (11) in the objective of (10) results in min Π max c −1 2σ2cT c −1 2α y σ2 −c T ΨΨT y σ2 −c . (12) We now note the critical fact that Ψ only shows up in the expression ΨΨT which, from the definition ψi = πi⊗xi, is seen to be equivalent to ΠΠT ⊙XXT . Therefore the minimization over Π effectively takes place over ΠΠT (since X is observed), and we have that (12) can be rewritten as min ΠΠT max c −1 2σ2cT c −1 2α y σ2 −c T ΠΠT ⊙XXT y σ2 −c . (13) So far no relaxation has taken place. By finally replacing the constraint (6) with constraint (7) from Lemma 2, we obtain the claimed semidefinite relaxation. 3 4 Max-Min Reformulation By upper bounding the inner maximization in (8) and applying a Schur complement, problem (8) can be re-expressed as a semidefinite program. Unfortunately, such a formulation is computationally expensive to solve, requiring O(t6) for typical interior-point methods. Instead, we can reformulate problem (8) to allow for a fast algorithmic approach, without the introduction of any additional relaxation. The basis of our development is the following classical result. Theorem 4 ([15]) Let V ∈Rt×t, V = V T have eigenvalues λ1 ≥λ2 ≥· · · ≥λt. Let P be the matrix whose columns are the normalized eigenvectors of V, i.e. P T V P = Λ((λ1, . . . , λt)). Let q ∈{1, . . . , t} and Pq be the matrix comprised by the top q eigenvectors of P. Then max M:tr(M)=q,I≽M≽0 tr MV T = q X i=1 λi and (14) argmax M:tr(M)=q,I≽M≽0 tr MV T ∋PqP T q . (15) Proof See [15] for a proof of a slightly more general result (Theorem 3.4). We will now show how the optimization on M for problem (8) can be cast in the terms of Theorem 4. This will turn out to be critical for the efficiency of the optimization procedure, since Theorem 4 describes a purely spectral optimization routine, which is far more efficient (O(t3)) than standard interior-point methods used for semidefinite programming (O(t6)). Proposition 5 Define ¯y := y σ2 . The following optimization problem max c −1 2σ2cT c −1 2α max M:tr M=t,γtI≽M≽0 tr(M(XXT ⊙(¯y −c)(¯y −c)T )) (16) is equivalent to optimization problem (8). Proof By Sion’s minimax theorem [14], minM and maxc in (8) can be interchanged max c min M:tr M=t,γtI≽M≽0 −1 2σ2cT c −1 2α (¯y −c)T M ⊙XXT (¯y −c) (17) which, by distributivity of the (min, +) semi-ring, is equivalent to max c −1 2σ2cT c + 1 2α min M:tr M=t,γtI≽M≽0 −(¯y −c)T M ⊙XXT (¯y −c) . (18) Now, define K := XXT . The objective of the minimization in (18) can then be written as −(¯y −c)T (M ⊙K)(¯y −c) = −tr (M ⊙K) (¯y −c)(¯y −c)T (19) = − X ij (MijKij) (¯y −c)(¯y −c)T ij = − X ij Mij Kij (¯y −c)(¯y −c)T ij (20) = −tr(M(K ⊙(¯y −c)(¯y −c)T )) = −tr(M(XXT ⊙(¯y −c)(¯y −c)T )). (21) Finally, by writing minM −f(M) as −maxM f(M), we obtain the claim. We can now exploit the result in Theorem 4 for the purpose of our optimization problem. Proposition 6 Let q = {u : u = max{1, . . . , t}, u ≤γ−1}. The following optimization problem max c −1 2σ2cT c − t 2αq max ¯ M:tr ¯ M=q,I≽¯ M≽0 tr( ¯ M(XXT ⊙(¯y −c)(¯y −c)T )) (22) is equivalent to optimization problem (16). 4 Algorithm 1 1: Input: γ, σ, α, XXT 2: Output: (c∗, M ∗) 3: Initialize c = 0 4: repeat 5: Solve for maximum value in inner maximization of (22) using (14) 6: Solve outer maximization in (22) using nonsmooth BFGS [16], obtain new c 7: until c has converged (c = c∗) 8: At c∗, solve for the maximizer(s) Pq in the inner maximization of (22) using (15) 9: if Pq is unique then 10: return M ∗= PqP T q break 11: else 12: Assemble top l eigenvectors in Pl 13: Solve (24) 14: return M ∗= PlΛ(λ∗)P T l 15: end if Proof The only differences between (16) and (22) are (i) the factor t/q in the second term of (22) and (ii) the constraints {M : tr M = t, γtI ≽M ≽0} in (16) versus {M : tr M = q, I ≽M ≽0} in (22). These differences are simply the result of a proper rescaling of M. If we define ¯ M := (q/t)M, then I ≽¯ M ≽0 since q ≤γ−1. We then have tr ¯ M = q. The result follows. And finally we have the second main result Theorem 7 Optimization problem (22) is equivalent to optimization problem (8). Proof The equivalence follows directly from Propositions 5 and 6. Note that, crucially, the objective in (22) is concave in c. Our strategy is now clear. Instead of solving (8), which demands O(t6) operations, we instead solve (22), which has as inner optimization a max eigenvalue problem, demanding only O(t3) operations. In the next section we describe an algorithm to jointly optimize for M and c in (22), which will essentially consist of alternating the efficient spectral solution over M with a subgradient optimization over c. 4.1 Max-Min Algorithm Algorithm 1 describes how we solve optimization problem (22). The idea of the algorithm is the following. First, having noted that (22) is concave in c, we can simply initialize c arbitrarily and pursue a fast subgradient ascent algorithm (e.g. such as nonsmooth BFGS [16]). So at each step we solve the eigenvalue problem and recompute a subgradient, until convergence to c∗. We then need to recover M ∗such that (c∗, M ∗) is a saddle point (note that problem (22) is concave in c and convex in M). For that purpose we use (15). If M ∗= PqP T q is such that Pq is unique, then we are done and the labeling solution of mixture membership is M ∗(subject to roundoff). If Pq is not unique, then we have multiplicity of eigenvalues and we need to proceed as follows. Define Pl = [p1 . . . pq . . . pl], l > q, where each of the additional l −q eigenvectors has an associated eigenvalue which is equal to the eigenvalue of some of the previous q eigenvectors. We then have that at the saddle point there must exist a diagonal matrix Λ such that M ∗= PlΛP T l , subject to Λ ≽0 and tr Λ = q (if this were not the case there would be an ascent direction in c∗, contradicting the hypothesis that c∗is optimal). To find such a Λ and therefore recover the correct M, we need to enforce that we are at the optimal c (c∗), i.e. we must have
d dc −1 2σ2cT c − q 2αt max M:tr M=q,I≽M≽0 tr(M(XXT ⊙(¯y −c)(¯y −c)T ))
2 2 = 0 (23) Such condition can be pursued by minimizing the above norm, which gives a quadratic program min λ≥0,λT 1=q
σ2c∗+ q αt PlΛ(λ)P T l ⊙XXT (c∗−¯y)
2 2 (24) We can then recover the final solution (subject to roundoff) by M ∗= PlΛ(λ∗)P T l , where λ∗is the optimizer of (24). The optimal value of (24) should be very close to zero (since it’s the norm of the derivative at point c∗). The pseudocode for the algorithm appears in Algorithm 1. 5 Algorithm 2 1: Input: γ, σ, α, XXT 2: Output: (c∗, M ∗) 3: Initialize M = Λ((1/(γt), . . . , 1/(γt))) 4: repeat 5: Solve for minimum value in inner minimization of (25), obtain A 6: Solve outer minimization in (25) given SVD of A using Theorem 4.1 of [18], obtain new M 7: until M has converged (M = M ∗) 8: Recover c∗= −1 σ2 diag(X(A∗)T ) 5 Min-Min Reformulation Although the max-min formulation appears satisfactory, the recent literature on multitask learning [17, 18] has developed an alternate strategy for bypassing general semidefinite programming. Specifically, work in this area lead to convex optimization problems expressed jointly over two matrix variables where each step is an alternating min-min descent that can be executed in closed-form or by a very fast algorithm. Although it is not immediately apparent that this algorithmic strategy is applicable to the problem at hand, with some further reformulation of (8) we discover that in fact the same min-min algorithmic approach can be applied to our mixture of regression problem. Theorem 8 The following optimization problem min {M:I⪰M⪰0,tr M=1/γ} min A 1 σ2 yT diag(XAT ) + 1 2σ2 diag(XAT )T diag(XAT ) + α 2γt tr(AT M −1A) (25) is equivalent to optimization problem (8). Proof min {M:I⪰M⪰0,tr M=1/γ} max c −σ2 2 cT c −γt 2α(c −¯y)T (M ⊙XXT )(c −¯y) (26) = min {M:I⪰M⪰0,tr M=1/γ} max {c,C:C=Λ(c−¯y)X} −σ2 2 cT c −γt 2α tr(CT MC) (27) = min {M:I⪰M⪰0,tr M=1/γ} min A max c,C −σ2 2 cT c −γt 2α tr(CT MC) + tr(AT C) −tr(AT Λ(c −¯y)X) (28) We can then solve for c and C, obtaining c = −1 σ2 diag(XAT ) and C = α γtM −1A. Substituting those two variables into (28) proves the claim. 5.1 Min-Min Algorithm The problem (25) is jointly convex in A and M [14] and Algorithm 2 describes how to solve it. It is important to note that although each iteration in Algorithm 2 is efficient, many iterations are required to reach a desired tolerance, since it is only first-order convergent. It is observed in our experiments that the concave-convex max-min approach in Algorithm 1 is more efficient simply because it has the same iteration cost but exploits a quasi-Newton descent in the outer optimization, which converges faster. Remark 9 In practice, similarly to [17], a regularizer on M is added to avoid singularity, resulting in the following regularized objective function, min {M:I⪰M⪰0,tr M=1/γ} min A 1 σ2 yT diag(XAT ) + 1 2σ2 diag(XAT )T diag(XAT ) + α 2γt tr(AT M −1A) + ϵ tr(M −1). (29) The problem is still jointly convex in M and A. 6 6 Experiments Our primary objective in formulating this convex approach to mixture regression is to tackle a difficult problem in video analysis (see below). However, to initially evaluate the different approaches we conducted some synthetic experiments. We generated 30 synthetic data points according to yi = (πi ⊗xi)w + ϵi, with xi ∈R, ϵi ∼N(0, 1) and w ∈U(0, 1). The response variable yi is assumed to be generated from a mixture of 5 components. We compared the quality of the relaxation in (22) to EM. Max-min algorithm is used in this experiment. For EM, 100 random restarts was used to help avoid poor local optima. The experiment is repeated 10 times. The error rates are 0.347 ± 0.086 and 0.280 ± 0.063 for EM and convex relaxation, respectively. The visualization of the recovered membership for one of the runs is given in Figure 1. This demonstrates that the relaxation can retain much of the structure of the problem. 6.1 Vision Experiment In a dynamic scene, various static and moving objects are viewed by a possibly moving observer. For example, consider a moving, hand-held camera filming a scene of several cars driving down the road. Each car has a separate motion, and even the static objects, such as trees, appear to move in the video due to the self-motion of the camera. The task of segmenting each object according to its motion, estimating the parameters of each motion, and recovering the structure of the scene is known as the multibody structure and motion problem. This is a missing variable problem. If the motions have been segmented correctly, it is easy to estimate the parameters of each motion. Naturally, models employing EM have been proposed to tackle such problems (c.f. [19, 20]). From epipolar geometry, given a pair of corresponding points pi and qi from two images (pi, qi ∈ R3×1), we have the epipolar equation qT i Fpi = 0. The fundamental matrix F encapsulates information about the translation and rotation relative to the scene points between the positions of the camera where the two images were captured, as well as the camera calibration parameters such as its focal length. In a static scene, where only the camera is moving, there is only one fundamental matrix, which arises from the camera self-motion. However, if some of the scene points are moving independently under multiple different motions, there are several fundamental matrices. If there are k motion groups, the epipolar equation can be expressed in term of the multibody fundamental matrix [21], i.e. Qk j=1(qT i Fjpi) = 0. An algebraic method was proposed to recover this matrix via Generalized PCA [21]. An alternative approach, which we follow here, is by Li [22], who casts the problem as a mixture of fundamental matrices, i.e. qT i (Pk j=1 πijFj)pi = 0 where the membership variable πij = 1 when image point i belongs to motion group j, and zero otherwise. Furthermore, since qT i Fpi = 0 is bilinear in the image points, we can rewrite it to be xT i wj = 0, with the column vectors xi = [qx i px i qx i py i qx i pπ i .... qπ i pπ i ]T and w = vec(F T j ). Thus, we will end up with the following linear equation: Pk j=1 πijxT i wj = 0. The weight vector wj for motion group j can be recovered easily if the indicator variable πij is known. We are interested in assessing the effectiveness of EM-based and convex relaxation-based methods for this multibody structure and motion problem. We used the Hopkins 155 dataset [23]. The experimental results are summarized in Table 1. All hyperparameters (EM: α and σ; Convex relaxation: α, σ, and γ) were tuned and the best performances for each learning algorithm are reported. The EM algorithm was run with 100 random restarts to help avoid poor local optima. In terms of computation time, the max-min runs comparably to the EM algorithm, while min-min runs in the order of 3 to 4 times slower. As an illustration, on a Pentium 4 3.6 GHz machine, the elapsed time (in seconds) for two cranes dataset is 16.880, 23.536, and 60.003 for EM, max-min and min-min, respectively. Rounding for the convex versions was done by k-means, which introduces some differences in the final results for both algorithms. Noticeably, both max-min and min-min outperform the EM algorithm. Visualizations of the motion segmentation on two cranes, three cars, and cars2 07 datasets are given in Figure 2 (for kanatani2 and articulated please refer to Appendix). 7 Conclusion The mixture regression problem is pervasive in many applications and known approaches for parameter estimation rely on variants of EM, which naturally have issues with local minima. In this paper we introduced a semidefinite relaxation for the mixture regression problem, thus obtaining a convex formulation which does not suffer from local minima. In addition we showed how to avoid the 7 use of expensive interior-point methods typically needed to solve semidefinite programs. This was achieved by introducing two reformulations amenable to the use of faster algorithms. Experimental results with synthetic data as well as with real computer vision data suggest the proposed methods can substantially improve on EM while one of the methods in addition has comparable runtimes. Table 1: Error rate on several datasets from the Hopkins 155 Data set m EM Max-Min Convex Min-Min Convex three cars 173 0.0532 0.0289 0.0347 kanatani2 63 0.0000 0.0000 0.0000 cars2 07 212 0.3396 0.2642 0.2594 two cranes 94 0.0532 0.0213 0.0106 articulated 150 0.0000 0.0000 0.0000 (a) Ground Truth (b) EM (c) Convex Relaxation Figure 1: Recovered membership on synthetic data with EM and convex relaxation. 30 data points are generated according to yi = (πi ⊗xi)w + ϵi, with xi ∈R, ϵi ∼N(0, 1) and w ∈U(0, 1). (a) Ground Truth (b) EM (c) Max-Min Convex (d) Min-Min Convex (e) Ground Truth (f) EM (g) Max-Min Convex (h) Min-Min Convex (i) Ground Truth (j) EM (k) Max-Min Convex (l) Min-Min Convex Figure 2: Resulting motion segmentations produced by the various techniques on the Hopkins 155 dataset. 2(a)-2(d): two cranes, 2(e)-2(h): three cars, and 2(i)-2(l): cars2 07. In two cranes (first row), EM produces more segmentation errors at the left crane. In three cars (second row), the max-min method gives the least segmentation error (at the front side of the middle car) and EM produces more segmentation errors at the front side of the left car. The contrast of EM and convex methods is apparent for cars2 07 (third row): the convex methods segment correctly the static grass field object, while EM makes mistakes. Further, the min-min method can almost perfectly segment the car in the middle of the scene from the static tree background. 8 References [1] S. Gaffney and P. Smyth. Trajectory clustering with mixtures of regression models. In ACM SIGKDD, volume 62, pages 63–72, 1999. [2] S. Finney, L. Kaelbling, and T. Lozano-Perez. Predicting partial paths from planning problem parameters. In Proceedings of Robotics: Science and Systems, Atlanta, GA, USA, June 2007. [3] A. J. Storkey and M. Sugiyama. Mixture regression for covariate shift. In Sch¨olkopf, editor, Advances in Neural Information Processing Systems 19, pages 1337–1344, 2007. [4] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society. Series B (Methodological), 39(1):1–38, 1977. [5] T. De Bie, N. Cristianini, P. Bennett, and E. Parrado-hern¨andez. Fast sdp relaxations of graph cut clustering, transduction, and other combinatorial problems. JMLR, 7:1409–1436, 2006. [6] Y. Guo and D. Schuurmans. Convex relaxations for latent variable training. In Platt et al., editor, Advances in Neural Information Processing Systems 20, pages 601–608, 2008. [7] S. M. Goldfeld and R.E. Quandt. Nonlinear methods in econometrics. Amsterdam: NorthHolland Publishing Co., 1972. [8] D. W. Hosmer. Maximum likelihood estimates of the parameters of a mixture of two regression lines. Communications in Statistics, 3(10):995–1006, 1974. [9] H. Sp¨ath. Algorithm 39: clusterwise linear regression. Computing, 22:367–373, 1979. [10] W.S. DeSarbo and W.L. Cron. A maximum likelihood methodology for clusterwise linear regression. Journal of Classification, 5(1):249–282, 1988. [11] P.N. Jones and G.J. McLachlan. Fitting finite mixtures models in a regression context. Austral. J. Statistics, 34(2):233–240, 1992. [12] S. Gaffney and P. Smyth. Curve clustering with random effects regression mixtures. In AISTATS, 2003. [13] M.I. Jordan and R.A. Jacobs. Hierarchical mixtures of experts and the em algorithm. Neural computation, 6:181–214, 1994. [14] S. Boyd and L. Vandenberghe. Convex Optimization. Cambridge University Press, 2004. [15] M. Overton and R. Womersley. Optimality conditions and duality theory for minimizing sums of the largest eigenvalues of symmetric matrices. Mathematical Programming, 62:321–357, 1993. [16] J. Yu, S.V.N. Vishwanathan, S. G¨unter, and N. Schraudolph. A quasi-Newton approach to nonsmooth convex optimization. In ICML, 2008. [17] A. Argyriou, T. Evgeniou, and M. Pontil. Convex multi-task feature learning. Machine Learning, 73:243–272, 2008. [18] J. Chen, L. Tang, J. Liu, and J. Ye. A convex formulation for learning shared structures from multiple tasks. In ICML, 2009. [19] N.Vasconcelos and A. Lippman. Empirical bayesian em-based motion segmentation. In CVPR, 1997. [20] P. Torr. Geometric motion segmentation and model selection. Philosophical Trans. of the Royal Society of London, 356(1740):1321–1340, 1998. [21] R. Vidal, Y. Ma, S. Soatto, and S. Sastry. Two-view multibody structure from motion. IJCV, 68(1):7–25, 2006. [22] H. Li. Two-view motion segmentation from linear programming relaxation. In CVPR, 2007. [23] http://www.vision.jhu.edu/data/hopkins155/. 9
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Replacing supervised classification learning by Slow Feature Analysis in spiking neural networks Stefan Klampfl, Wolfgang Maass Institute for Theoretical Computer Science Graz University of Technology A-8010 Graz, Austria {klampfl,maass}@igi.tugraz.at Abstract It is open how neurons in the brain are able to learn without supervision to discriminate between spatio-temporal firing patterns of presynaptic neurons. We show that a known unsupervised learning algorithm, Slow Feature Analysis (SFA), is able to acquire the classification capability of Fisher’s Linear Discriminant (FLD), a powerful algorithm for supervised learning, if temporally adjacent samples are likely to be from the same class. We also demonstrate that it enables linear readout neurons of cortical microcircuits to learn the detection of repeating firing patterns within a stream of spike trains with the same firing statistics, as well as discrimination of spoken digits, in an unsupervised manner. 1 Introduction Since the presence of supervision in biological learning mechanisms is rare, organisms often have to rely on the ability of these mechanisms to extract statistical regularities from their environment. Recent neurobiological experiments [1] have suggested that the brain uses some type of slowness objective to learn the categorization of external objects without a supervisor. Slow Feature Analysis (SFA) [2] could be a possible mechanism for that. We establish a relationship between the unsupervised SFA learning method and a commonly used method for supervised classification learning: Fisher’s Linear Discriminant (FLD) [3]. More precisely, we show that SFA approximates the classification capability of FLD by replacing the supervisor with the simple heuristics that two temporally adjacent samples in the input time series are likely to be from the same class. Furthermore, we demonstrate in simulations of a cortical microcircuit model that SFA could also be an important ingredient in extracting temporally stable information from trajectories of network states and that it supports the idea of “anytime” computing, i.e., it provides information about the stimulus identity not only at the end of a trajectory of network states, but already much earlier. This paper is structured as follows. We start in section 2 with brief recaps of the definitions of SFA and FLD. We discuss the relationship between these methods for unsupervised and supervised learning in section 3, and investigate the application of SFA to trajectories in section 4. In section 5 we report results of computer simulations of several SFA readouts of a cortical microcircuit model. Section 6 concludes with a discussion. 2 Basic Definitions 2.1 Slow Feature Analysis (SFA) Slow Feature Analysis (SFA) [2] is an unsupervised learning algorithm that extracts the slowest components yi from a multi-dimensional input time series x by minimizing the temporal variation 1 ∆(yi) of the output signal yi, which is defined in [2] as the average of its squared temporal derivative. Thus the objective is to minimize min ∆(yi) := ⟨˙yi 2⟩t. (1) The notation ⟨·⟩t denotes averaging over time, and ˙y is the time derivative of y. The additional constraints of zero mean (⟨yi⟩t = 0) and unit variance (⟨y2 i ⟩t = 1) avoid the trivial constant solution yi(t) ≡0. If multiple slow features are extracted, a third constraint (⟨yiyj⟩t = 0, ∀j < i) ensures that they are decorrelated and ordered by decreasing slowness, i.e., y1 is the slowest feature extracted, y2 the second slowest feature, and so on. In other words, SFA finds those functions gi out of a certain predefined function space that produce the slowest possible outputs yi = gi(x) under these constraints. This optimization problem is hard to solve in the general case [4], but if we assume that the time series x has zero mean (⟨x⟩t = 0) and if we only allow linear functions y = wT x the problem simplifies to the objective min JSF A(w) := wT ⟨˙x ˙xT ⟩tw wT ⟨xxT ⟩tw. (2) The matrix ⟨xxT ⟩t is the covariance matrix of the input time series and ⟨˙x ˙xT ⟩t denotes the covariance matrix of time derivatives (or time differences, for discrete time) of the input time series. The weight vector w which minimizes (2) is the solution to the generalized eigenvalue problem ⟨˙x ˙xT ⟩tw = λ⟨xxT ⟩tw (3) corresponding to the smallest eigenvalue λ. To make use of a larger function space one typically considers linear combinations y = wT z of fixed nonlinear expansions z = h(x) and performs the optimization (2) in this high-dimensional space. 2.2 Fisher’s Linear Discriminant (FLD) Fisher’s Linear Discriminant (FLD) [3], on the other hand, is a supervised learning method, since it is applied to labeled training examples ⟨x, c⟩, where c ∈{1, . . ., C} is the class to which this example x belongs. The goal is to find a weight vector w so that the ability to predict the class of x from the value of wT x is maximized. FLD searches for that projection direction w which maximizes the separation between classes while at the same time minimizing the variance within classes, thereby minimizing the class overlap of the projected values: max JF LD(w) := wT SBw wT SW w. (4) For C point sets Sc, each with Nc elements and means µc, SB is the between-class covariance matrix given by the separation of the class means, SB = P c Nc(µc −µ)(µc −µ)T , and SW is the within-class covariance matrix given by SW = P c P x∈Sc(x −µc)(x −µc)T . Again, the vector w optimizing (4) can be viewed as the solution to a generalized eigenvalue problem, SBw = λSW w, (5) corresponding to the largest eigenvalue λ. 3 SFA can acquire the classification capability of FLD SFA and FLD receive different data types as inputs: unlabeled time series for SFA, in contrast to labeled single data points for the FLD. Therefore, in order to apply the unsupervised SFA learning algorithm to the same classification problem as the supervised FLD, we have to convert the labeled training samples into a time series of unlabeled data points that can serve as an input to the SFA algorithm1. In the following we investigate the relationship between the weight vectors found by both methods for a particular way of time series generation. 1A first link between SFA and pattern recognition has been established in [5]. There the optimization is performed over all possible pattern pairs of the same class. However, it might often be implausible to have access to such an artificial time series, e.g., from the perspective of a readout neuron that receives input on-thefly. We take a different approach and apply the standard SFA algorithm to a time series consisting of randomly selected patterns of the classification problem, where the class at each time step is switched with a certain probability. 2 We consider a classification problem with C classes, i.e., assume we are given point sets S1, S2, . . . , SC ⊂Rn. Let Nc be the number of points in Sc and let N = PC c=1 Nc be the total number of points. In order to create a time series xt out of these point sets we define a Markov model with C states S = {1, 2, . . ., C}, one for each class, and choose at each time step t = 1, . . . , T a random point from the class that corresponds to the current state in the Markov model. We define the transition probability from state i ∈S to state j ∈S as Pij = ( a · Nj N if i ̸= j, 1 −P k̸=j Pik if i = j, (6) with some appropriate constant a > 0. The stationary distribution of this Markov model is π = (N1/N, N2/N, . . . , NC/N). We choose the initial distribution p0 = π, i.e., at any time t the probability that point xt is chosen from class c is Nc/N. For this particular way of generating the time series from the original classification problem we can express the matrices ⟨xxT ⟩t and ⟨˙x ˙xT ⟩t of the SFA objective (2) in terms of the within-class and between-class scatter matrices of the FLD (4), SW and SB, in the following way [6]: ⟨xxT ⟩t = 1 N SW + 1 N SB (7) ⟨˙x ˙xT ⟩t = 2 N SW + a · 2 N SB (8) Note that only ⟨˙x ˙xT ⟩t depends on a, whereas ⟨xxT ⟩t does not. For small a we can neglect the effect of SB on ⟨˙x ˙xT ⟩t in (8). In this case the time series consists mainly of transitions within a class, whereas switching between the two classes is relatively rare. Therefore the covariance of time derivatives is mostly determined by the within-class scatter of the two point sets, and both matrices become approximately proportional: ⟨˙x ˙xT ⟩t ≈2/N · SW . Moreover, if we assume that SW (and therefore ⟨˙x ˙xT ⟩t) is positive definite, we can rewrite the SFA objective (2) as min JSF A(w) ⇔max 1 JSF A(w) ⇔max wT ⟨xxT ⟩tw wT ⟨˙x ˙xT ⟩tw ⇔max 1 2 + 1 2 · wT SBw wT SW w ⇔max JF LD(w). (9) That is, the weight vector that optimizes the SFA objective (2) also optimizes the FLD objective (4). For C > 2 this equivalence can be seen by recalling the definition of SFA as a generalized eigenvalue problem (3) and inserting (7) and (8): ⟨˙x ˙xT ⟩tW = ⟨xxT ⟩tWΛ SBW = SW W 2Λ−1 −E , (10) where W = (w1, . . . , wn) is the matrix of generalized eigenvectors and Λ = diag(λ1, . . . , λn) is the diagonal matrix of generalized eigenvalues. The last line of (10) is just the formulation of FLD as a generalized eigenvalue problem (5). More precisely, the eigenvectors of the SFA problem are also eigenvectors of the FLD problem. Note that the eigenvalues correspond by λF LD i = 2/λSF A i −1, which means the order of eigenvalues is reversed (λSF A i > 0). Thus, the subspace spanned by the slowest features is the same that optimizes separability in terms of Fisher’s Discriminant, and the slowest feature is the weight vector which achieves maximal separation. Figure 1A demonstrates this relationship on a sample two-class problem in two dimensions for the special case of N1 = N2 = N/2. In this case at each time the class is switched with probability p = a/2 or is left unchanged with probability 1 −p. We interpret the weight vectors found by both methods as normal vectors of hyperplanes in the input space, which we place simply onto the mean value µ of all training data points (i.e., the hyperplanes are defined as wT x = θ with θ = wT µ). One sees that the weight vector found by the application of SFA to the training time series xt generated with p = 0.2 is approximately equal to the weight vector resulting from FLD on the initial sets of training points. This demonstrates that SFA has extracted the class of the points as the slowest varying feature by finding a direction that separates both classes. 3 Figure 1: Relationship between SFA and FLD for a two-class problem in 2D. (A) Sample point sets with 250 points for each class. The dashed line indicates a hyperplane corresponding to the weight vector wF LD resulting from the application of FLD to the two-class problem. The black solid line shows a hyperplane for the weight vector wSF A resulting from SFA applied to the time series generated from these training points as described in the text (T = 5000, p = 0.2). The dotted line displays an additional SFA hyperplane resulting from a time series generated with p = 0.45. All hyperplanes are placed onto the mean value of all training points. (B) Dependence of the error between the weight vectors found by FLD and SFA on the switching probability p. This error is defined as the average angle between the weight vectors obtained on 100 randomly chosen classification problems. Error bars denote the standard error of the mean. Figure 1B quantifies the deviation of the weight vector resulting from the application of SFA to the time series from the one found by FLD on the original points. We use the average angle between both weight vectors as an error measure. It can be seen that if p is low, i.e., transitions between classes are rare compared to transitions within a class, the angle between the vectors is small and SFA approximates FLD very well. The angle increases moderately with increasing p; even with higher values of p (up to 0.45) the approximation is reasonable and a good classification by the slowest feature can be achieved (see dotted hyperplane in Figure 1A). As soon as p reaches a value of about 0.5, the error grows almost immediately to the maximal value of 90◦. For p = 0.5 (a = 1) points are chosen independently of their class, making the matrices ⟨˙x ˙xT ⟩t and ⟨xxT ⟩t proportional. This means that every possible vector w is a solution to the generalized eigenvalue problem (3), resulting in an average angle of about 45◦. 4 Application to trajectories of training examples In the previous section we have shown that SFA approximates the classification capability of FLD if the probability is low that two successive points in the input time series to SFA are from different classes. Apart from this temporal structure induced by the class information, however, these samples are chosen independently at each time step. In this section we investigate how the SFA objective changes when the input time series consists of a sequence of trajectories of samples instead of individual points only. First, we consider a time series xt consisting of multiple repetitions of a fixed predefined trajectory ˜t, which is embedded into noise input consisting of a random number of points drawn from the same distribution as the trajectory points, but independently at each time step. It is easy to show [6] that for such a time series the SFA objective (2) reduces to finding the eigenvector of the matrix ˜Σt corresponding to the largest eigenvalue. ˜Σt is the covariance matrix of the trajectory ˜t with ˜t delayed by one time step, i.e., it measures the temporal covariances (hence the index t) of ˜t with time lag 1. Since the transitions between two successive points of the trajectory ˜t occur much more often in the time series xt than transitions between any other possible pair of points, SFA has to respond as smoothly as possible (i.e., maximize the temporal correlations) during ˜t in order to produce the 4 slowest possible output. This means that SFA is able to detect repetitions of ˜t by responding during such instances with a distinctive shape. Next, we consider a classification problem given by C sets of trajectories, T1, T2, . . . , TC ⊂(Rn) ˜T , i.e., the elements of each set Tc are sequences of ˜T n-dimensional points. We generate a time series according to the same Markov model as described in the previous section, except that we do not choose individual points at each time step, rather we generate a sequence of trajectories. For this time series we can express the matrices ⟨xxT ⟩t and ⟨˙x ˙xT ⟩t in terms of the within-class and between-class scatter of the individual points of the trajectories in Tc, analogously to (7) and (8) [6]. While the expression for ⟨xxT ⟩t is unchanged the temporal correlations induced by the use of trajectories however have an effect on the covariance of temporal differences ⟨˙x ˙xT ⟩t. First, this matrix additionally depends on the temporal covariance ˜Σt with time lag 1 of all available trajectories in all sets Tc. Second, the effective switching probability is reduced by a factor of 1/ ˜T. Whenever a trajectory is selected, ˜T points from the same class are presented in succession. This means that even for a small switching probability2 the objective of SFA cannot be solely reduced to the FLD objective, but rather that there is a trade-off between the tendency to separate trajectories of different classes (as explained by the relation between SB and SW ) and the tendency to produce smooth responses during individual trajectories (determined by the temporal covariance matrix ˜Σt): min JSF A(w) = wT ⟨˙x ˙xT ⟩tw wT ⟨xxT ⟩tw ≈1 N · wT SW w wT ⟨xxT ⟩tw −˜p · wT ˜Σtw wT ⟨xxT ⟩tw, (11) where N is here the total number of points in all trajectories and ˜p is the fraction of transitions between two successive points of the time series that belong to the same trajectory. The weight vector w which minimizes the first term in (11) is equal to the weight vector found by the application of FLD to the classification problem of the individual trajectory points (note that SB enters (11) through ⟨xxT ⟩t, cf. eq. (9)). The weight vector which maximizes the second term is the one which produces the slowest possible response during individual trajectories. If the separation between the trajectory classes is large compared to the temporal correlations (i.e., the first term in (11) dominates for the resulting w) the slowest feature will be similar to the weight vector found by FLD on the corresponding classification problem. On the other hand, as the temporal correlations of the trajectories increase, i.e., the trajectories themselves become smoother, the slowest feature will tend to favor exploiting this temporal structure of the trajectories over the separation of different classes (in this case, (11) is dominated by the second term for the resulting w). 5 Application to linear readouts of a cortical microcircuit model In the following we discuss several computer simulations of a cortical microcircuit of spiking neurons that demonstrate the theoretical arguments given in the previous section. We trained a number of linear SFA readouts3 on a sequence of trajectories of network states, each of which is defined by the low-pass filtered spike trains of the neurons in the circuit. Such recurrent circuits typically provide a temporal integration of the input stream and project it nonlinearly into a high-dimensional space [7], thereby boosting the expressive power of the subsequent linear SFA readouts. Note, however, that the optimization (2) implicitly performs an additional whitening of the circuit response. As a model for a cortical microcircuit model we use the laminar circuit from [8] consisting of 560 spiking neurons organized into layers 2/3, 4, and 5, with layer-specific connection probabilities obtained from experimental data [9, 10]. In the first experiment we investigated the ability of SFA to detect a repeating firing pattern within noise input of the same firing statistics. We recorded circuit trajectories in response to 200 repetitions of a fixed spike pattern which are embedded into a continuous Poisson input stream of the same rate. We then trained linear SFA readouts on this sequence of circuit trajectories (we used an exponential 2In fact, for sufficiently long trajectories the SFA objective becomes effectively independent of the switching probability. 3We interpret the linear combination defined by each slow feature as the weight vector of a hypothetical linear readout. 5 Figure 2: Detecting embedded spike patterns. (A) From top to bottom: sample stimulus sequence, response spike trains of the network, and slowest features. The stimulus consists of 10 channels and is defined by repetitions of a fixed spike pattern (dark gray) which are embedded into random Poisson input of the same rate. The pattern has a length of 250ms and is made up by Poisson spike trains of rate 20Hz. The period between two repetitions is drawn uniformly between 100ms and 500ms. The response spike trains of the laminar circuit of [8] are shown separated into layers 2/3, 4, and 5. The numbers of neurons in the layers are indicated on the left, but only the response of every 12th neuron is plotted. Shown are the 5 slowest features, y1 to y5, for the network response shown above. The dashed lines indicate values of 0. (B) Phase plots of low-pass filtered versions (leaky integration, τ = 100ms) of individual slow features in response to a test sequence of 50 embedded patterns plotted against each other (black: traces during the pattern, gray: during random Poisson input). filter with τ = 30ms and a sample time of 1ms). The period of Poisson input in between two such patterns was randomly chosen. At first glance there is no clear difference in Figure 2A between the raw SFA responses during periods of pattern presentations and during phases of noise input due to the same firing statistics. However, we found that on average the slow feature responses during noise input are zero, whereas a characteristic response remains during pattern presentations. This effect is predicted by the theoretical arguments in section 4. It can be seen in phase plots of traces that are obtained by a leaky integration of the slowest features in response to a test sequence of 50 embedded patterns (see Figure 2B) that the slow features span a subspace where the response during pattern presentations can be nicely separated from the response during noise input. That is, by simple threshold operations on the low-pass filtered versions of the slowest features one can in principle detect the presence of patterns within the continuous input stream. Furthermore, this extracted information is not only available after a pattern has been presented, but already during the presentation of the pattern, which supports the idea of “anytime” computing. In the second experiment we tested whether SFA is able to discriminate two classes of trajectories as described in section 4. We performed a speech recognition task using the dataset considered originally in [11] and later in the context of biological circuits in [7, 12, 13]. This isolated spoken digits dataset consists of the audio signals recorded from 5 speakers pronouncing the digits “zero”, “one”, ..., “nine” in ten different utterances (trials) each. We preprocessed the raw audio files with a model of the cochlea [14] and converted the resulting analog cochleagrams into 20 spike trains (using the algorithm in [15]) that serve as input to our microcircuit model (see Figure 3A). We tried to dis6 Figure 3: SFA applied to digit recognition of a single speaker. (A) From top to bottom: cochleagrams, input spike trains, response spike trains of the network, and traces of different linear readouts. Each cochleagram has 86 channels with analog values between 0 and 1 (white, near 1; black, near 0). Stimulus spike trains are shown for two different utterances of the given digit (black and gray, the black spike times correspond to the cochleagram shown above). The response spike trains of the laminar circuit from [8] are shown separated into layers 2/3, 4, and 5. The number of neurons in each layer is indicated on the left, but only the response of every 12th neuron is plotted. The responses to the two stimulus spike trains in the panel above are shown superimposed with the corresponding color. Each readout trace corresponds to a weighted sum (Σ) of network states of the black responses in the panel above. The trace of the slowest feature (“SF1”, see B) is compared to traces of readouts trained by FLD and SVM with linear kernel to discriminate at any time between the network states of the two classes. All weight vectors are normalized to length 1. The dotted line denotes the threshold of the respective linear classifier. (B) Response of the 5 slowest features y1 to y5 of the previously learned SFA in response to trajectories of the three test utterances of each class not used for training (black, digit “one”; gray, digit “two”). The slowness index η = T/2π p ∆(y) [2] is calculated from these output signals. The angle α denotes the deviation of the projection direction of the respective feature from the direction found by FLD. The thick curves in the shaded area display the mean SFA responses over all three test trajectories for each class. (C) Phase plots of individual slow features plotted against each other (thin lines: individual responses, thick lines: mean response over all test trajectories). criminate between trajectories in response to inputs corresponding to utterances of digits “one” and “two”, of a single speaker. We kept three utterances of each digit for testing and generated from the remaining training samples a sequence of 100 input samples, recorded for each sample the response of the circuit, and concatenated the resulting trajectories in time. Note that here we did not switch the classes of two successive trajectories with a certain probability because, as explained in the previous section, for long trajectories the SFA response is independent of this switching probability. Rather, we trained linear SFA readouts on a completely random trajectory sequence. 7 Figure 3B shows the 5 slowest features, y1 to y5, ordered by decreasing slowness in response to the trajectories corresponding to the three remaining test utterances for each class, digit “one” and digit “two”. In this example, already the slowest feature y1 extracts the class of the input patterns almost perfectly: it responds with positive values for trajectories in response to utterances of digit “two” and with negative values for utterances of digit “one”. This property of the extracted features, to respond differently for different stimulus classes, is called the What-information [2]. The second slowest feature y2, on the other hand, responds with shapes whose sign is independent of the pattern identity. One can say that, in principle, y2 encodes simply the presence of and the location within a response. This is a typical example of a representation of Where-information [2], i.e., the “pattern location” regardless of the identity of the pattern. The other slow features y3 to y5 do not extract either What- or Where-information explicitly, but rather a mixed version of both. As a measure for the discriminative capability of a specific SFA response, i.e., its quality as a possible classifier, we measured the angle between the projection direction corresponding to this slow feature and the direction of the FLD. It can be seen in Figure 3B that the slowest feature y1 is closest to the FLD. Hence, according to (11), this constitutes an example where the separation between classes dominates, but is already significantly influenced by the temporal correlations of the circuit trajectories. Figure 3C shows phase plots of these slow features shown in Figure 3B plotted against each other. In the three plots involving feature y1 it can be seen that the directions of the response vector (i.e., the vector composed of the slow feature values at a particular point in time) cluster at class-specific angles, which is characteristic for What-information. On the other hand, these phase plots tend to form loops in phase space (instead of just straight lines from the origin), where each point on this loop corresponds to a position within the trajectory. This is a typical property of Where-information. Similar responses have been theoretically predicted in [4] and found in simulations of a hierarchical (nonlinear) SFA network trained with a sequence of one-dimensional trajectories [2]. This experiment demonstrates that SFA extracts information about the spoken digit in an unsupervised manner by projecting the circuit trajectories onto a subspace where they are nicely separable so that they can easily be classified by later processing stages. Moreover, this information is provided not only at the end of a specific trajectory, but is made available already much earlier. After sufficient training, the slowest feature y1 in Figure 3B responds with positive or negative values indicating the stimulus class almost during the whole duration of of the network trajectory. This again supports the idea of “anytime” computing. It can be seen in the bottom panel of Figure 3A that the slowest feature, which is obtained in an unsupervised manner, achieves a good separation between the two test trajectories, comparable to the supervised methods of FLD and Support Vector Machine (SVM) [16] with linear kernel. 6 Discussion The results of our paper show that Slow Feature Analysis is in fact a very powerful tool, which is able to approximate the classification capability that results from supervised classification learning. Its elegant formulation as a generalized eigenvalue problem has allowed us to establish a relationship to the supervised method of Fisher’s Linear Discriminant (FLD). A more detailed discussion of this relationship, including complete derivations, can be found in [6]. If temporal contiguous points in the time series are likely to belong to the same class, SFA is able to extract the class as a slowly varying feature in an unsupervised manner. This ability is of particular interest in the context of biologically realistic neural circuits because it could enable readout neurons to extract from the trajectories of network states information about the stimulus – without any “teacher”, whose existence is highly dubious in the brain. We have shown in computer simulations of a cortical microcircuit model that linear readouts trained with SFA are able to detect specific spike patterns within a stream of spike trains with the same firing statistics and to discriminate between different spoken digits. Moreover, SFA provides in these tasks an “anytime” classification capability. Acknowledgments We would like to thank Henning Sprekeler and Laurenz Wiskott for stimulating discussions. This paper was written under partial support by the Austrian Science Fund FWF project # S9102-N13 and project # FP6-015879 (FACETS), project # FP7-216593 (SECO) and project # FP7-231267 (ORGANIC) of the European Union. 8 References [1] N. Li and J. J. DiCarlo. Unsupervised natural experience rapidly alters invariant object representation in visual cortex. Science, 321:1502–1507, 2008. [2] L. Wiskott and T. J. Sejnowski. Slow feature analysis: unsupervised learning of invariances. Neural Computation, 14(4):715–770, 2002. [3] R. A. Fisher. The use of multiple measurements in taxonomic problems. Annuals of Eugenics, 7:179–188, 1936. [4] L. Wiskott. Slow feature analysis: A theoretical analysis of optimal free responses. Neural Computation, 15(9):2147–2177, 2003. [5] P. Berkes. Pattern recognition with slow feature analysis. Cognitive Sciences EPrint Archive (CogPrint) 4104, February 2005. http://cogprints.org/4104/. [6] S. Klampfland W. Maass. A theoretical basis for emergent pattern discrimination in neural systems through slow feature extraction. Submitted for publication, 2009. [7] 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):2531–2560, 2002. [8] S. H¨ausler and W. Maass. A statistical analysis of information processing properties of lamina-specific cortical microcircuit models. Cerebral Cortex, 17(1):149–162, 2007. [9] A. Gupta, Y. Wang, and H. Markram. Organizing principles for a diversity of GABAergic interneurons and synapses in the neocortex. Science, 287:273–278, 2000. [10] A. M. Thomson, D. C. West, Y. Wang, and A. P. Bannister. Synaptic connections and small circuits involving excitatory and inhibitory neurons in layers 2–5 of adult rat and cat neocortex: triple intracellular recordings and biocytin labelling in vitro. Cerebral Cortex, 12(9):936–953, 2002. [11] J. J. Hopfield and C. D. Brody. What is a moment? Transient synchrony as a collective mechanism for spatio-temporal integration. Proc. Nat. Acad. Sci. USA, 98(3):1282–1287, 2001. [12] D. Verstraeten, B. Schrauwen, D. Stroobandt, and J. Van Campenhout. Isolated word recognition with the liquid state machine: a case study. Inf. Process. Lett., 95(6):521–528, 2005. [13] R. Legenstein, D. Pecevski, and W. Maass. A learning theory for reward-modulated spike-timingdependent plasticity with application to biofeedback. PLoS Computational Biology, 4(10):1–27, 2008. [14] R. F. Lyon. A computational model of filtering, detection, and compression in the cochlea. In Proc. IEEE Int. Conf. Acoustics Speech and Signal Processing, pages 1282–1285, May 1982. [15] B. Schrauwen and J. V. Campenhout. BSA, a fast and accurate spike train encoding scheme. In Proceedings of the International Joint Conference on Neural Networks, 2003. [16] B. Sch¨olkopf and A. J. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. 9
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Hierarchical Learning of Dimensional Biases in Human Categorization Katherine Heller Department of Engineering University of Cambridge Cambridge CB2 1PZ heller@gatsby.ucl.ac.uk Adam Sanborn Gatsby Computational Neuroscience Unit University College London London WC1N 3AR asanborn@gatsby.ucl.ac.uk Nick Chater Cognitive, Perceptual and Brain Sciences University College London London WC1E 0AP n.chater@ucl.ac.uk Abstract Existing models of categorization typically represent to-be-classified items as points in a multidimensional space. While from a mathematical point of view, an infinite number of basis sets can be used to represent points in this space, the choice of basis set is psychologically crucial. People generally choose the same basis dimensions – and have a strong preference to generalize along the axes of these dimensions, but not “diagonally”. What makes some choices of dimension special? We explore the idea that the dimensions used by people echo the natural variation in the environment. Specifically, we present a rational model that does not assume dimensions, but learns the same type of dimensional generalizations that people display. This bias is shaped by exposing the model to many categories with a structure hypothesized to be like those which children encounter. The learning behaviour of the model captures the developmental shift from roughly “isotropic” for children to the axis-aligned generalization that adults show. 1 Introduction Given only a few examples of a particular category, people have strong expectations as to which new examples also belong to that same category. These expectations provide important insights into how objects are mentally represented. One basic insight into mental representations is that objects that have similar observed properties will be expected to belong to the same category, and that expectation decreases as the Euclidean distance between the properties of the objects increases [1, 2]. The Euclidean distance between observed properties is only part of the story however. Dimensions also play a strong role in our expectations of categories. People do not always generalize isotropically: the direction of generalizations turns out to be centrally important. Specifically, people generalize along the dimensions, such as size, color, or shape – dimensions that are termed separable. In contrast, dimensions such as hue and saturation, which show isotropic generalization are termed integral [3] . An illustration of the importance of separable dimensions is found in the time to learn categories. If dimensions did not play a strong role in generalization, then rotating a category structure in a parameter space of separable dimensions should not influence how easily it can be learned. To the contrary, rotating a pair of categories 45 degrees [3, 4] makes it more difficult to learn to 1 discriminate between them. Similarity rating results also show strong trends of judging objects to be more similar if they match along separable dimensions [3, 5]. The tendency to generalize categories along separable dimensions is learned over development. On dimensions such as size and color, children produce generalizations that are more isotropic than adults [6]. Interestingly, the developmental transition between isotropic and dimensionally biased generalizations is gradual [7]. What privileges separable dimensions? And why are they acquired over development? One possibility is that there is corresponding variation in real-world categories, and this provides a bias that learners carry over to laboratory experiments. For example, Rosch et al. [1] identified shape as a key constant in categories, and we can find categories that are constant along other separable dimensions as well. For instance, categories of materials such as gold, wood, and ice all display a characteristic color while being relatively unconstrained as to the shapes and sizes that they take. Size is often constrained in artifacts such as books and cars, while color can vary across a very wide range. Models of categorization are able to account for both the isotropic and dimension-based components of generalization. Classic models of categorization, such as the exemplar and prototype model, account for these using different mechanisms [8, 9, 10]. Rational models of categorization have accounted for dimensional biases by assuming that the shapes of categories are aligned with the axes that people use for generalization [11, 12, 13]. Neither the classic models nor rational models have investigated how people learn to use the particular dimension basis that they do. This paper presents a model that learns the dimensional basis that people use for generalization. We connect these biases with a hypothesis about the structure of categories in the environment and demonstrate how exposure to these categories during development results in human dimensional biases. In the next section, we review models of categorization and how they have accounted for dimensional biases. Next, we review current nonparametric Bayesian models of categorization, which all require that the dimensions be hand-coded. Next, we introduce a new prior for categorization models that starts without pre-specified dimensions and learns to generalize new categories in the same way that previous categories varied. We show that without the use of pre-specified dimensions, we are able to produce generalizations that fit human data. We demonstrate that training the model on reasonable category structures produces generalization behavior that mimics that of human subjects at various ages. In addition, our trained model predicts the challenging effect of violations of the triangle inequality for similiarity judgments. 2 Modeling Dimensional Biases in Categorization Models of categorization can be divided into generative and discriminative models – we will focus on generative models here and leave discriminative models for the discussion. Generative models of categorization, such as the prototype [8] and exemplar models [9, 10], assume that people learn category distributions, not just rules for discriminating between categories. In order to make a judgment of whether a new item belongs to one category or another, a comparison is made of the new item to the already existing categories, using Bayes rule with a uniform prior on the category labels, P(cn = i|xn, xn−1, cn−1) = P(xn|cn = i, xn−1, cn−1) P j P(xn|cn = j, xn−1, cn−1) (1) where xn is the nth item and cn = j assigns that item to category j. The remaining items are collected in the vector xn−1 and the known labels for these items are cn−1. For the prototype and exemplar models, the likelihood of an item belonging to a category is based on the weighted Minkowski power metric1, X i ⇣X d w(d)$$$x(d) n −R(d) i $$$ r⌘1 r (2) 1For an exemplar model, R(d) i is each example in xn−1, while for the prototype model, it is the single average of xn−1. 2 which computes the absolute value of the power metric between the new example xn and the category representation Ri for category i on a dimension d. Integral dimensions are modeled with r = 2, which results in a Euclidean distance metric. The Euclidean metric has the special property that changing the basis set for the dimensions of the space does not affect the distances. Any other choice of r means that the distances are affected by the basis set, and thus it must be chosen to match human judgments. Separable dimensions are modeled with either r = 1, the city-block metric, or r < 1, which no longer obeys the triangle equality [5]. Dimensional biases are also modeled in categorization by modifying the dimension weights for each dimension, w(d). In effect, the weights stretch or shrink the space of stimuli along each dimension so that some items are closer than others. These dimension weights are assumed to correspond to attention. To model learning of categories, it is often necessary to provide non-zero weights to only a few features early in learning and gradually shift to uniform weights late in learning [14]. These generative models of categorization have been developed to account for the different types of dimensional biases that are displayed by people, but they lack means for learning the dimensions themselves. Extensions to these classical models learn the dimension weights [15, 16], but can only learn the weights for pre-specified dimensions. If the chosen basis set did not match that used by people, then the models would be very poor descriptions of human dimensional biases. A stronger notion of between-category learning is required. 3 Rational Models of Categorization Rational models of categorization view categorization behavior as the solution to a problem posed by the environment: how best to to generalize properties from one object to another. Both exemplar and prototype models can be viewed as restricted versions of rational models of categorization, which also allow interpolations between these two extreme views of representation. Anderson [11] proposed a rational model of categorization which modeled the stimuli in a task as a mixture of clusters. This model treated category labels as features, performing unsupervised learning. The model was extended to supervised learning so each category is a mixture [17], P(x`|x`−1, s`−1) = K X k=1 P(s` = k|s`−1)P(x`|s` = k, x`−1, s`−1) (3) where x` is the newest example in a category i and x`−1 are the other members of category i. x` is a mixture over a set of K components with the prior probability of x` belonging to a component depending on the component membership of the other examples s`−1. Instead of a single component or a component for each previous item, the mixture model has the flexibility to choose an intermediate number of components. To make full use of this flexibility, Anderson used a nonparametric Chinese Restaurant Process (CRP) prior on the mixing weights, which allows the flexibility of having an unspecified and potentially infinite number of components (i.e., clusters) in our mixture model. The mixing proportions in a CRP are based on the number of items already included in the cluster, P(s` = k|s`−1) = ( Mk i−1+↵ if Mk > 0 (i.e., k is old) ↵ i−1+↵ if Mk = 0 (i.e., k is new) (4) where Mj is the number of objects assigned to component k, and ↵is the dispersion parameter. Using Equation 4, the set of assignments s`−1 is built up as a simple sequential stochastic process [18] in which the order of the observations is unimportant [19]. The likelihood of belonging to a component depends on the other members of the cluster. In the case of continuous data, the components were modeled as Gaussian distributions, P(x`|s` = k, x`−1, s`−1) = Y d Z µ(d) Z ⌃(d) N(x(d) ` ; µ(d), ⌃(d))P(⌃(d))P(µ(d)|⌃(d)) (5) 3 where the mean and variance of each Gaussian distribution is given by µ(d) and ⌃(d) respectively. The prior for the mean was assumed to be Gaussian given the variance and the prior for the variance was a inverse-gamma distribution. The likelihood distribution for this model assumes a fixed basis set of dimensions, which must align with the separable dimensions to produce dimensional biases in generalization. 4 A Prior for Dimensional Learning The rational model presented above assumes a certain basis set of dimensions, and the likelihood distributions are aligned with these dimensions. To allow the learning of the basis set, we first need multivariate versions of the prior distributions over the mean and variance parameters. For the mean parameter, we will use a multivariate Gaussian distribution and for the covariance matrix, we will use the multivariate generalization of the inverse-gamma distribution, the inverse-Wishart distribution. The inverse-Wishart distribution has its mode at ⌃ m+D+1, where ⌃is the mean covariance matrix parameter, m is the degrees of freedom, and D is the number of dimensions of the stimulus. A covariance matrix is always diagonal under some rotated version of the initial basis set. This new basis set gives the possible dimensional biases for this cluster. However, using Gaussian distributions for each cluster, with a unimodal prior on the covariance matrix, greatly limits the patterns of generalizations that can be produced. For a diagonal covariance matrix, strong generalization along a particular dimension would be produced if the covariance matrix has a high variance along that dimension, but low variances along the remaining dimensions. Thus, this model can learn to strongly generalize along one dimension, but people often make strong generalizations along multiple dimensions [5], such as in Equation 2 when r < 1. A unimodal prior on covariance matrices cannot produce this behavior, so we use a mixture of inverse Wishart distributions as a prior for covariance matrices, p(⌃k|uk, Φ) = J X j=1 p(uk = j|uk−1)p(⌃k|Φj, uk = j) (6) where ⌃k is the covariance parameter for the kth component. For simplicity, the component parameters ⌃k are assumed i.i.d. given their class. Φj are the parameters of component j which reflect the expected covariances generated by the jth inverse-Wishart distribution in the mixture. uk = j is the assignment of parameters ⌃k to component Φj and the set of all other component assignments is uk−1. Φ and µk are the sets of all Φj and µk. The means of categories k have Gaussian priors, which depend on ⌃k, but are otherwise independent of each other. As before, we will use a nonparametric CRP prior over the mixture weights uk. We now have two infinite mixtures: one that allows a category to be composed of a mixture of clusters, and one that allows the prior for the covariance matrices to be composed of a mixture of inverse-Wishart distributions. The final piece of the model is to specify p(Φ). We use another inverse-Wishart prior, but with an identity matrix for the mean parameter, so as not to bias the Φj components toward a particular dimension basis set. Figure 1 gives a schematic depiction of the model. 5 Learning the Prior The categories we learn during development often vary along separable dimensions – and people are sensitive to this variability. The linguistic classification of nouns helps to identify categories that are fixed on one separable dimension and variable on others. Nouns can be classified into count nouns and mass nouns. Count nouns refer to objects that are discrete, such as books, shirts, and cars. Mass nouns are those that refer to objects that appear in continuous quantities, such as grass, steel, and milk. These two types of nouns show an interesting regularity: count nouns are often relatively similar in size but vary greatly in color, while mass nouns are often relatively fixed in color but vary greatly in size. Smith [7] tested the development of children’s dimensionality biases. In this study, experimenters showed participants six green circles that varied in shade and size. The discriminability judgments of adults to scale the parameters of the stimuli, so that one step in color caused the same gain in discriminability as one step in size. Participants were asked to group the stimuli into clusters according 4 Figure 1: Schematic illustration of the hierarchical prior over covariance matrices. The top-level prior is a covariance matrix (shown as the equiprobability curves of a Gaussian) that is not biased towards any dimension. The mid level priors Φj are drawn from an inverse-Wishart distribution centered on the top-level prior. The Φj components are used as priors for the covariance matrices for clusters. The plot on the right shows some schematic examples of natural categories that tend to vary along either color or size. The covariance matrices for these clusters are drawn from an inverse-Wishart prior using one of the Φj components. to their preferences, only being told that they should group the “ones that go together”. The partitions of the stimuli into clusters that participants produced tended toward three informative patterns, shown in Figure 2. The Overall Similarity pattern ignores dimension and appears to result from isotropic similarity. The One-dimensional Similarity pattern is more biased towards generalizing along separable dimensions than the Overall Similarity pattern. The strongest dimensional biases are shown by the One-Dimensional Identity pattern, with the dimensional match overriding the close isotropic similarity between neighboring stimuli. Children aged 3 years, 4 years, 5 years and adults participated in this experiment. There were ten participants in each age group, participants clustered eight problems each, and all dimension-aligned orientations of the stimuli were tested. Figure 2 shows the developmental trend of each of the informative clustering patterns. The tendency to cluster according to Overal Similarity decreased with age, reflecting a reduced influence of isotropic similarity. Clustering according One-dimensional Similarity increased from 3-year-olds to 5-year-olds, but adults produced few of these patterns. The percentage of One-dimensional Identity clusterings increased with age, and was the dominant response for adults, supporting the idea that strong dimensional biases are learned. We trained our model with clusters that were aligned with the dimensions of size and color. Half of the clusters varied strongly in shape and weakly in size, while the other half varied strongly in size and weakly in shape. The larger standard deviation of the distribution that generated the training stimuli was somewhat smaller than the largest distance between stimuli in the Smith experiment, while the smaller standard deviation in the distribution that generated the training stimuli was much smaller than the smallest distance between Smith stimuli. The two dispersion parameters were set to 1, the degrees of freedom for all inverse-Wishart distributions were set to the number of dimensions plus 1, and 0.01 was used for the scale factor for the mean parameters of the inverseWishart distributions2. Inference in the model was done as a combination of the Gibbs sampling and Metropolis-Hastings algorithms. The assignments of data points to clusters in each class were Gibbs sampled conditioned on the cluster assignments to inverse-Wishart components and the parameters of those components, Φj. Following a complete pass of the assignments of data points to clusters, we then Gibbs sampled the assignments of the cluster covariance parameters ⌃k to components of the inverse-Wishart 2The general pattern of the results was only weakly dependent on the parameter settings, but unsupervised learning of the clusters required a small value of the scale factor. 5 Figure 2: Experiment 2 of Smith [7]. In a free categorization task, the stimuli marked by dots in the top row were grouped by participants. The three critical partitions are shown as circles in the top row of plots. The top bar graph displays the developmental trends for each of the critical partitions. The bottom bar graph displays the trend as the model is trained on a larger number of axis-aligned clusters. mixture prior. After a pass of this mid-level sampling, we resampled Φj, the parameters of the inverse-Wishart components, and the prior expected means of each cluster. This sampling was done using Metropolis-Hastings, with the non-symmetric proposals made from a separate inverse-Wishart distribution. A large finite Dirichlet distribution was used to approximate p(U). Given the learned Φ and uk, the predicted probabilities for the Smith experiment were computed exactly. The predictions of our model as a result of training are shown in Figure 2. The model was trained on 0, 2, 4, 8, and 16 axis-aligned clusters in an unsupervised fashion. For the all three patterns, the model shows the same developmental trajectory as human data. Overall Similarity decreases with the number of trained categories, One-dimensional Similarity increases and then decreases, and One-dimensional Identity patterns are overwhelmingly produced by the fully trained model. The probabilities plotted in the figure are the predicted posterior of only the partitions that exactly matched the informative patterns, out of all 203 possible partitions, showing that the patterns in Figure 2 dominated the model’s predictions as they dominated the participants’ responses in the free categorization task. 6 Generalization Gradients Standard models of categorization, such as the prototype or exemplar model, have a variety of mechanisms for producing the dimensional biases seen in experiments with adults. We propose a very different explanation for these dimensional biases. In this section we plot generalization gradients, 6 Figure 3: Generalization gradients of the exemplar model and the posterior predictive distribution of the model presented in this paper. The dots are the stimuli. which provide a good feel for how the priors we propose match with the mechanisms used in earlier models across a variety of conditions. Generalizations of single items are studied by collecting similarity ratings. In this task, participants judge the similarity of two items. In standard models of categorization, similarity ratings are modeled mainly by the exponent in the Minkowski power metric (Equation 2). For rational models, similarity ratings can be modeled as the posterior predictive probability of one item, given the second item [20]. The first two columns of Figure 3 give a comparison between the exemplar model and the model we propose for similarity ratings. The central dot is a particular stimulus and the color gradient shows the predicted similarity ratings of all other stimuli. For integral dimensions, a Euclidean metric (r = 2) is used in the exemplar model, which the model we propose matches if it has not been trained on dimension-aligned categories. For separable categories, the exemplar model usually uses a city-block metric (r = 1) [10]. However, experimental evidence shows that dimensions have an even stronger effect than predicted by a city-block metric. In experiments to test violations of the triangle equality, Tversky and Gati [5] showed that the best fitting exponent for similarity data is often r < 1. The model we propose can produce this type of similarity prediction by using a prior that is a mixture of covariance matrices, in which each component of the mixture generalizes strongly along one dimension. In a category of one item, which is the case when making similarity judgments with the posterior predictive distribution, it is uncertain which covariance component best describes the category. This uncertainty results in a generalization gradient that imitates an exponent of r < 1 using Gaussian distributions. As a result, our proposed model predicts violations of the triangle inequality if it has been trained on a set of clusters in which some vary strongly along one dimension and others vary strongly along another dimension. A comparison between this generalization gradient and the exemplar model is shown in the second column of Figure 3. The second mechanism for dimensional biases in standard models of categorization is selective attention. Selective attention is used to describe biases that occur in categorization experiments, when many items are trained in each category. These biases are implemented in the exemplar model as weights along each dimension, and early in learning there are usually large weights on a small number of separable dimensions [14, 21]. Our proposed model does not have a mechanism for selective attention, but provides a rational explanation for this effect in terms of the strong sampling assumption [13]. If two items are assumed to come from the same cluster, then generalization tends to be along a single dimension that has varied during training (third column of Figure 3). However, if two items are inferred to belong to different clusters, then the generalization gradient corresponds to additive similarity without selective attention (fourth column of Figure 3). We have shown that the model we have proposed can reproduce the key generalization gradients of the exemplar and prototype models. The important difference between our model of dimensional 7 biases and these standard categorization models is that we learn basis set for dimensional biases, assuming these dimensions have proven to be useful for predicting category structure in the past. Other models must have these dimensions pre-specified. To show that our model is not biased towards a particular basis set, we rotated the training stimuli 45 degrees in space. The resulting posterior predictive distributions in Figure 3 extendend in the same direction as the rotated training categories varied. 7 Discussion The approach to dimensional biases we have outlined in this paper provides a single explanation for dimensional biases, in contrast to standard models of categorization, such as exemplar and prototype models. These standard models of categorization assume two distinct mechanisms for producing dimensional biases: a Minkowski metric exponent, and attentional weights for each dimension. In our approach, biases in both similarity judgments and categorization experiments are produced by learning covariance matrices that are shared between clusters. For similarity judgments, the single item does not give information about which covariance mixture component was used to generate it. This uncertainty produces similarity judgments that would be best fit with an Minkowski exponent of r < 1. For category judgments, the alignment of the items along a dimension allows the generating covariance mixture component to be inferred, so the judgments will show a bias like that of attentional weights to the dimensions. The difference between tasks drives the different types of dimensional biases in our approach. We propose that people learn more complex cross-category information than most previous approaches do. Attention to dimensions is learned in connectionist models of categorization by finding the best single set of weights for each dimension in a basis set [15, 16], or by cross-category learning in a Bayesian approach [22]. A more flexible approach is used in associative models of categorization, which allow for different patterns of generalizations for different items. One associative model used a Hopfield network to predict different generalizations for solid and non-solid objects [23]. A hierarchical Bayesian model with very similar properties to this associative model motivated this result from cross-category learning [24]. The key difference between all these models and our proposal is that they use only a single strong dimensional bias for each item, while we use multiple latent strong dimensional biases for each item, which is needed for modeling both similarity and categorization dimensional biases with a single explanation. The only previous approach we are aware of that learns such complex cross-category information is a Bayesian rule-based model of categorization [25]. The main advantage of our approach over many other models of categorization is that we learn the basis set of dimensions that can display dimensional biases. Our model learns the basis the same way people do, from categories in the environment (as opposed to fitting to human similarity or category judgements). We begin with a feature space of stimuli in which physically similar items are near to each other. Using a version of the Transformed Dirichlet Process [26], a close relation to the Hierarchical Dirichlet Process previously proposed as a unifying model of categorization [17], a mixture of covariance matrices are learned from environmentally plausible training data. Most other models of categorization, including exemplar models [10], prototype models [8], rule-based discriminative models [27], as well as hierarchical Bayesian models for learning features [24, 22] and Bayesian rule-based models [25] all must have a pre-specified basis set. 8 Summary and Conclusions People generalize categories in two ways: they generalize to stimuli with parameters near to the category and generalize to stimuli that match along separable dimensions. Existing models of categorization must assume the dimensions to produce human-like generalization performance. Our model learns these dimensions from the data: starting with an unbiased prior, the dimensions that categories vary along are learned to be dimensions important for generalization. After training the model with categories intended to mirror those learned during development, our model reproduces the trajectory of generalization biases as children grow into adults. Using this type of approach, we hope to better tie models of human generalization to the natural world to which we belong. 8 References [1] E. Rosch, C. B. Mervis, W. D. Gray, D. M. Johnson, and P. Boyes-Braem. Basic objects in natural categories. Cognitive Psychology, 8:382–439, 1976. [2] R. N. Shepard. Toward a universal law of generalization for psychological science. Science, 237:1317– 1323, 1987. [3] W. R. Garner. The processing of information and structure. Erlbaum, Hillsdale, NJ, 1974. [4] J. K. Kruschke. Human category learning: implications for backpropagation models. Connection Science, 5:3–36, 1993. [5] A. Tversky and I. Gati. Similarity, separability and the triangular inequality. Psychological Review, 93:3–22, 1982. [6] L. B. Smith and Kemler D. G. Developmental trends in free classification: Evidence for a new conceptualization of perceptual development. Journal of Experimental Child Psychology, 24:279–298, 1977. [7] L. B. Smith. A model of perceptual classification in children and adults. Psychological Review, 96:125– 144, 1989. [8] S. K. Reed. Pattern recognition and categorization. Cognitive Psychology, 3:393–407, 1972. [9] D. L. Medin and M. M. Schaffer. Context theory of classification learning. Psychological Review, 85:207– 238, 1978. [10] R. M. Nosofsky. Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 115:39–57, 1986. [11] J. R. Anderson. The adaptive nature of human categorization. Psychological Review, 98(3):409–429, 1991. [12] D. J. Navarro. From natural kinds to complex categories. In Proceedings of CogSci, pages 621–626, Mahwah, NJ, 2006. Lawrence Erlbaum. [13] J. B. Tenenbaum and T. L. Griffiths. Generalization, similarity, and Bayesian inference. Behavioral and Brain Sciences, 24:629–641, 2001. [14] M. K. Johansen and T. J. Palmeri. Are there representational shifts in category learning? Cognitive Psychology, 45:482–553, 2002. [15] John K. Kruschke. Alcove: An exemplar-based connectionist model of category learning. Psychological Review, 99:22–44, 1992. [16] B. C. Love, D. L. Medin, and T. M. Gureckis. SUSTAIN: A network model of category learning. Psychological Review, 111:309–332, 2004. [17] T. L. Griffiths, K. R. Canini, A. N. Sanborn, and D. J. Navarro. Unifying rational models of categorization via the hierarchical dirichlet process. In R. Sun and N. Miyake, editors, Proceedings CogSci, 2007. [18] D. Blackwell and J. MacQueen. Ferguson distributions via Polya urn schemes. The Annals of Statistics, 1:353–355, 1973. [19] 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. [20] T. L. Griffiths, M. Steyvers, and J. B. Tenenbaum. Topics in semantic representation. Psychological Review, 114:211–244, 2007. [21] R. M. Nosofsky and S. R. Zaki. Exemplar and prototype models revisted: response strategies, selective attention, and stimulus generalization. Journal of Experimental Psychology: Learning, Memory, and Cognition, 28:924–940, 2002. [22] A. Perfors and J.B. Tenenbaum. Learning to learn categories. In Proceedings of CogSci, 2009. [23] E. Colunga and L. B. Smith. From the lexicon to expectations about kinds: a role for associative learning. Psychological Review, 112, 2005. [24] C. Kemp, A. Perfors, and J. B. Tenenbaum. Learning overhypotheses with hierarchical Bayesian models. Developmental Science, 10:307–321, 2007. [25] N. D. Goodman, J. B. Tenenbaum, J. Feldman, and T. L. Griffiths. A rational analysis of rule-based concept learning. Cognitive Science, 32:108–154, 2008. [26] E. Sudderth, A. Torralba, W. Freeman, and A. Willsky. Describing visual scenes using transformed dirichlet processes. In Neural Information Processing Systems NIPS, 2005. [27] R. M. Nosofsky and T. J. Palmeri. A rule-plus-exception model for classifying objects in continuousdimension spaces. Psychonomic Bulletin & Review, 5:345–369, 1998. 9
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Polynomial Semantic Indexing Bing Bai(1) Jason Weston(1)(2) David Grangier(1) Ronan Collobert(1) Kunihiko Sadamasa(1) Yanjun Qi(1) Corinna Cortes(2) Mehryar Mohri(2)(3) (1)NEC Labs America, Princeton, NJ {bbai, dgrangier, collober, kunihiko, yanjun}@nec-labs.com (2) Google Research, New York, NY {jweston, corinna, mohri}@google.com (3) NYU Courant Institute, New York, NY mohri@cs.nyu.edu Abstract We present a class of nonlinear (polynomial) models that are discriminatively trained to directly map from the word content in a query-document or documentdocument pair to a ranking score. Dealing with polynomial models on word features is computationally challenging. We propose a low-rank (but diagonal preserving) representation of our polynomial models to induce feasible memory and computation requirements. We provide an empirical study on retrieval tasks based on Wikipedia documents, where we obtain state-of-the-art performance while providing realistically scalable methods. 1 Introduction Ranking text documents given a text-based query is one of the key tasks in information retrieval. A typical solution is to: (i) embed the problem in a feature space, e.g. model queries and target documents using a vector representation; and then (ii) choose (or learn) a similarity metric that operates in this vector space. Ranking is then performed by sorting the documents based on their similarity score with the query. A classical vector space model, see e.g. [24], uses weighted word counts (e.g. via tf-idf) as the feature space, and the cosine similarity for ranking. In this case, the model is chosen by hand and no machine learning is involved. This type of model often performs remarkably well, but suffers from the fact that only exact matches of words between query and target texts contribute to the similarity score. That is, words are considered to be independent, which is clearly a false assumption. Latent Semantic Indexing [8], and related methods such as pLSA and LDA [18, 2], are unsupervised methods that choose a low dimensional feature representation of “latent concepts” where words are no longer independent. They are trained with reconstruction objectives, either based on mean squared error (LSI) or likelihood (pLSA, LDA). These models, being unsupervised, are still agnostic to the particular task of interest. More recently, supervised models for ranking texts have been proposed that can be trained on a supervised signal (i.e., labeled data) to provide a ranking of a database of documents given a query. For example, if one has click-through data yielding query-target relationships, one can use this to train these models to perform well on this task. Or, if one is interested in finding documents related to a given query document, one can use known hyperlinks to learn a model that performs well on this task. Many of these models have typically relied on optimizing over only a few hand-constructed features, e.g. based on existing vector space models such as tf-idf, the title, URL, PageRank and other information [20, 5]. In this work, we investigate an orthogonal research direction, as we analyze supervised methods that are based on words only. Such models are both more flexible, e.g. can be used for tasks such as cross-language retrieval, and can still be used in conjunction with 1 other features explored in previous work for further gains. At least one recent work, called Hash Kernels [25], has been proposed that does construct a word-feature based model in a learning-to-rank context. In this article we define a class of nonlinear (polynomial) models that can capture higher order relationships between words. Our nonlinear representation of the words results in a very high dimensional feature space. To deal with this space we propose low rank (but diagonal preserving) representations of our polynomial models to induce feasible memory and computation requirements, resulting in a method that both exhibits strong performance and is tractable to train and test. We show experimentally on retrieval tasks derived from Wikipedia that our method strongly outperforms other word based models, including tf-idf vector space models, LSI, query expansion, margin rank perceptrons and Hash Kernels. The rest of this article is as follows. In Section 2, we describe our method, Section 3 discusses prior work, and Section 4 describes the experimental study of our method. 2 Polynomial Semantic Indexing Let us denote the set of documents in the corpus as {dt}ℓ t=1 ⊂RD and a query text as q ∈RD, where D is the dictionary size, and the jth dimension of a vector indicates the frequency of occurrence of the jth word, e.g. using the tf-idf weighting and then normalizing to unit length. Given a query q and a document d we wish to learn a (nonlinear) function f(q, d) that returns a score measuring the relevance of d given q. Let us first consider the naive approach of concatenating (q, d) into a single vector and using f(q, d) = w⊤[q, d] as a linear ranking model. This clearly does not learn anything useful as it would result in the same document ordering for any query, given fixed parameters w. However, considering a polynomial model: f(q, d) = w⊤Φk([q, d]) where Φk(·) is a feature map that considers all possible k-degree terms: Φk(x1, . . . , xD) = ⟨xi1 . . . xik : 1 ≤i1 . . . ik ≤D⟩ does render a useful discriminative model. For example for degree k = 2 we obtain: f(q, d) = X ij w1 ijqiqj + X ij w2 ijdiqj + X ij w3 ijdidj where w has been rewritten as w1 ∈RD×D, w2 ∈RD×D and w3 ∈RD×D. The ranking order of documents d given a fixed query q is independent of w1 and the value of the term with w3 is independent of the query, so in the following we will consider models containing only terms with both q and d. In particular, we will consider the following degree k = 2 model: f 2(q, d) = D X i,j=1 Wijqidj = q⊤Wd (1) where W ∈RD×D, and the degree k = 3 model: f 3(q, d) = D X i,j,k=1 Wijkqidjdk + f 2(q, d). (2) Note that if W is an identity matrix in equation (1), we obtain the cosine similarity with tf-idf weighting. When other weights are nonzero this model can capture synonymy and polysemy as it looks at all possible cross terms, which can be tuned directly for the task of interest during training, e.g. the value of Wij corresponding to related words e.g. the word “jagger” in the query and “stones” in the target could be given a large value during training. The degree k = 3 model goes one stage further and can upweight Wijk for the triple “jagger”, “stones” and “rolling” and can downweight the triple “jagger”, “gem” and “stones”. Note that we do not necessarily require preprocessing methods such as stemming here since these models can already match words with common stems (if 2 it is useful for the task). Note also that in equation (2) we could have just as easily have considered pairs of words in the query (rather than the document) as well. Unfortunately, using such polynomial models is clearly infeasible for several reasons. Firstly, it will hardly be possible to fit W in memory for realistic tasks. If the dictionary size is D = 30000, then, for k = 2 this requires 3.4GB of RAM (assuming floats), and if the dictionary size is 2.5 Million (as it will be in our experiments in Section 4) this amounts to 14.5TB. For k = 3 this is even worse. Besides memory requirements, the huge number of parameters can of course also affect the generalization ability of this model. We thus propose a low-rank (but diagonal preserving) approximation of these models which will lead to capacity control, faster computation speed and smaller memory footprint. For k = 2 we propose to replace W with W, where W ij = (U ⊤V )ij + Iij = X l UliVlj + Iij. Plugging this into equation (1) yields: f 2 LR(q, d) = q⊤(U ⊤V + I)d, (3) = N X i=1 (Uq)i(V d)i + q⊤d. (4) Here, U and V are N ×D matrices. Before looking at higher degree polynomials, let us first analyze this case. This induces a N-dimensional “latent concept” space in a way similar to LSI. However, this is different in several ways: • First, and most importantly, we advocate training from a supervised signal using preference relations (ranking constraints). • Further, U and V differ so it does not assume the query and target document should be embedded in the same way. This can hence model when the query text distribution is very different to the document text distribution, e.g. queries are often short and have different word occurrence and co-occurrence statistics. In the extreme case in cross language retrieval query and target texts are in different languages yet are naturally modeled in this setup. • Finally, the addition of the identity term means this model automatically learns the tradeoff between using the low dimensional space and a classical vector space model. This is important because the diagonal of the W matrix gives the specificity of picking out when a word co-occurs in both documents (indeed, setting W = I is equivalent to cosine similarity using tf-idf). The matrix I is full rank and therefore cannot be approximated with the low-rank model U ⊤V , so our model combines both terms in the approximation. However, the efficiency and memory footprint are as favorable as LSI. Typically, one caches the N-dimensional representation for each document to use at query time. For higher degree polynomials, e.g. k = 3 one can perform a similar approximation. Indeed, Wijk is approximated with W ijk = X l UliVljYlk where U, V and Y are N × D. When adding the diagonal preserving term and the lower order terms from the k = 2 polynomial, we obtain f 3 LR(q, d) = N X i=1 (Uq)i(V d)i(Y d)i + f 2 LR(q, d). Clearly, we can approximate any degree k polynomial using a product of k linear embeddings in such a scheme. Note that at test time one can again cache the N-dimensional representation for each document by computing the product between the V and Y terms and are then still left with only N multiplications per document for the embedding term at query time. Interestingly, one can view this model as a “product of experts”: the document is projected twice i.e. by two experts V and Y and the training will force them to focus on different aspects. 3 2.1 Training Training such models could take many forms. In this paper we will adopt the typical “learning to rank” setup [17, 20]. Suppose we are given a set of tuples R (labeled data), where each tuple contains a query q, a relevant document d+ and an non-relevant (or lower ranked) document d−. We would like to choose W such that f(q, d+) > f(q, d−), that is d+ should be ranked higher than d−. We thus employ the margin ranking loss [17] which has already been used in several IR methods before [20, 5, 14], and minimize: X (q,d+,d−)∈R max(0, 1 −f(q, d+) + f(q, d−)). (5) We train this using stochastic gradient descent, (see, e.g. [5]): iteratively, one picks a random tuple and makes a gradient step for that tuple. We choose the (fixed) learning rate which minimizes the training error. Convergence (or early stopping) is assessed with a validation set. Stochastic training is highly scalable and is easy to implement for our model. For example, for k = 2, one makes the following updates: U ←U + λV (d+ −d−)q⊤, if 1 −f 2 LR(q, d+) + f 2 LR(q, d−) > 0 V ←V + λUq(d+ −d−)⊤, if 1 −f 2 LR(q, d+) + f 2 LR(q, d−) > 0. Clearly, it is important to exploit the sparsity of q and d when calculating these updates. In our experiments we initialized the matrices U and V randomly using a normal distribution with mean zero and standard deviation one. The gradients for k = 3 are similar. Note that researchers have also explored optimizing various alternative loss functions other than the ranking loss including optimizing normalized discounted cumulative gain (NDCG) and mean average precision (MAP) [5, 4, 6, 28]. In fact, one could use those optimization strategies to train our models instead of optimizing the ranking loss. One could also just as easily use them in unsupervised learning, such as in LSI, as well, e.g. by stochastic gradient descent on the reconstruction error. 3 Prior Work Joachims et al. [20] trained a SVM with hand-designed features based on the title, body, search engines rankings and the URL. Burges et al. [5] proposed a neural network method using a similar set of features (569 in total). As described before, in contrast, we limited ourselves to body text (not using title, URL, etc.) and trained on millions of features based on these words. The authors of [15] used a model similar to the naive full rank model (1), but for the task of image retrieval, and [13] also used a related (regression-based) method for advert placement. These techniques are implemented in related software to these two publications, PAMIR1 and Vowpal Wabbit2. When the memory usage is too large, the latter bins the features randomly into a reduced space (hence with random collisions), a technique called Hash Kernels [25]. In all cases, the task of document retrieval, and the use of low-rank approximation or polynomial features is not studied. The current work generalizes and extends the Supervised Semantic Indexing approach [1] to general polynomial models. Another related area of research is in distance metric learning [27, 19, 12]. Methods like LMNN [27] also learn a model similar to the naive full rank model (1), i.e. with the full matrix W (but not with our improvements of this model that make it tractable for word features). They impose the constraint during the optimization that W be a positive semidefinite matrix. Their method has considerable computational cost. For example, even after considerable optimization of the algorithm, it still takes 3.5 hours to train on 60,000 examples and 169 features (a pre-processed version of MNIST). This would hence not be scalable for large-scale text ranking experiments. Nevertheless, [7] compared LMNN [27], LEGO [19] and MCML [12] to a stochastic gradient method with a full matrix W (identical to the model (1)) on a small image ranking task and reported in fact that the stochastic method provides both improved results and efficiency. Our method, on the other hand, both outperforms models like (1) and is feasible for word features, when (1) is not. 1http://www.idiap.ch/pamir/ 2http://hunch.net/˜vw/ 4 A tf-idf vector space model and LSI [8] are two standard baselines we will also compare to. We already mentioned pLSA [18] and LDA [2]; both have scalability problems and are not reported to generally outperform LSA and TF-IDF [11]. Query Expansion, often referred to as blind relevance feedback, is another way to deal with synonyms, but requires manual tuning and does not always yield a consistent improvement [29]. Several authors [23, 21] have proposed interesting nonlinear versions of unsupervised LSI using neural networks and showed they outperform LSI or pLSA. However, in the case of [23] we note their method is rather slow, and a dictionary size of only 2000 was used. A supervised method for LDA (sLDA) [3] has also been proposed where a set of auxiliary labels are trained on jointly with the unsupervised task. This provides supervision at the document level (via a class label or regression value) which is not a task of learning to rank, whereas here we study supervision at the (query,documents) level. The authors of [10] proposed “Explicit Semantic Analysis” which represents the meaning of texts in a high-dimensional space of concepts by building a feature space derived from the human-organized knowledge from an encyclopedia, e.g. Wikipedia. In the new space, cosine similarity is applied. Our method could be applied to such feature representations so that they are not agnostic to a particular supervised task as well. As we will also evaluate our model over cross-language retrieval, we also briefly mention methods previously applied to this problem. These include first applying machine translation and then a conventional retrieval method such as LSI [16], a direct method of applying LSI for this task called CL-LSI [9], or using Kernel Canonical Correlation Analysis, KCCA [26]. While the latter is a strongly performing method, it also suffers from scalability problems. 4 Experimental Study Learning a model of term correlations over a large vocabulary is a considerable challenge that requires a large amount of training data. Standard retrieval datasets like TREC3 or LETOR [22] contain only a few hundred training queries, and are hence too small for that purpose. Moreover, some datasets only provide pre-processed features like tf, idf or BM25, and not the actual words. Click-through from web search engines could provide valuable supervision. However, such data is not publicly available. We hence conducted experiments on Wikipedia and used links within Wikipedia to build a largescale ranking task. We considered several tasks: document-document and query-document retrieval described in Section 4.1, and cross-language document-document retrieval described in Section 4.2. In these experiments we compared our approach, Polynomial Semantic Indexing (PSI), to the following methods: tf-idf + cosine similarity (TFIDF), Query Expansion (QE), LSI4, αLSI + (1 −α) TFIDF, and the margin ranking perceptron and Hash Kernels with hash size h using model (1). Query Expansion involves applying TFIDF and then adding mean vector β PE i=1 dri of the top E retrieved documents multiplied by a weighting β to the query, and applying TFIDF again. For all methods, hyperparameters such as the embedding dimension N ∈{50, 100, 200, 500, 1000}, h ∈{1M, 3M, 6M}, α, β and E were chosen using a validation set. For each method, we measured the ranking loss (the percentage of tuples in R that are incorrectly ordered), precision P(n) at position n = 10 (P@10) and the mean average precision (MAP), as well as their standard deviations. For computational reasons, MAP and P@10 were measured by averaging over a fixed set of 1000 test queries, and the true test links and random subsets of 10,000 documents were used as the database, rather than the whole testing set. The ranking loss is measured using 100,000 testing tuples. 4.1 Document Retrieval We considered a set of 1,828,645 English Wikipedia documents as a database, and split the 24,667,286 links randomly into two portions, 70% for training (plus validation) and 30% for test3http://trec.nist.gov/ 4We use the SVDLIBC software http://tedlab.mit.edu/˜dr/svdlibc/ and the cosine distance in the latent concept space. 5 Table 1: Document-document ranking results on Wikipedia (limited dictionary size of 30,000 words). Polynomial Semantic Indexing (PSI) outperforms all baselines, and performs better with higher degree k = 3. Algorithm Rank-Loss MAP P@10 TFIDF 1.62% 0.329±0.010 0.163±0.006 QE 1.62% 0.330±0.010 0.163±0.006 LSI 4.79% 0.158±0.006 0.098±0.005 αLSI + (1 −α)TFIDF 1.28% 0.346±0.011 0.170±0.007 Margin Ranking Perceptron using (1) 0.41% 0.477±0.011 0.212±0.007 PSI (k = 2) 0.30% 0.517±0.011 0.229±0.007 PSI (k = 3) 0.14% 0.539±0.011 0.236±0.007 Table 2: Empirical results for document-document ranking on Wikipedia (unlimited dictionary size). Algorithm Rank-Loss MAP P@10 TFIDF 0.842% 0.432±0.012 0.1933±0.007 QE 0.842% 0.432±0.012 0.1933±0.007 αLSI + (1 −α)TFIDF 0.721% 0.433±0.012 0.193±0.007 Hash Kernels using (1) 0.347% 0.485±0.011 0.215±0.007 PSI (k = 2) 0.158% 0.547±0.012 0.239±0.008 PSI (k = 3) 0.099% 0.590±0.012 0.249±0.008 Table 3: Empirical results for document-document ranking in two train/test setups: partitioning into train+test sets of links, or into train+test sets of documents with no cross-links (limited dictionary size of 30,000 words). The two setups yield. similar results. Algorithm Testing Setup Rank-Loss MAP P@10 PSI (k = 2) Partitioned links 0.407% 0.506±0.012 0.225±0.007 PSI (k = 2) Partitioned docs+links 0.401% 0.503±0.010 0.225±0.006 Table 4: Empirical results for query-document ranking on Wikipedia where query has n keywords (this experiment uses a limited dictionary size of 30,000 words). For each n we measure the ranking loss, MAP and P@10 metrics. n = 5 n = 10 n = 20 Algorithm Rank MAP P@10 Rank MAP P@10 Rank MAP P@10 TFIDF 21.6% 0.047 0.023 14.0% 0.083 0.035 9.14% 0.128 0.054 αLSI + (1 −α)TFIDF 14.2% 0.049 0.023 9.73% 0.089 0.037 6.36% 0.133 0.059 PSI (k = 2) 4.37% 0.166 0.083 2.91% 0.229 0.100 1.80% 0.302 0.130 ing.5 We then considered the following task: given a query document q, rank the other documents such that if q links to d then d is highly ranked. In our first experiment we constrained all methods to use only the top 30,000 most frequent words. This allowed us to compare to a margin ranking perceptron using model (1) which would otherwise not fit in memory. For our approach, Polynomial Semantic Indexing (PSI), we report results for degrees k = 2 and k = 3. Results on the test set are given in Table 1. Both variants of our method PSI strongly outperform the existing techniques. The margin rank perceptron using (1) can be seen as a full rank version of PSI for k = 2 (with W unconstrained) but is outperformed by its lowrank counterpart – probably because it has too much capacity. Degree k = 3 outperforms k = 2, indicating that the higher order nonlinearities captured provide better ranking scores. For LSI and PSI embedding dimension N = 200 worked best, but other values gave similar results. In terms of other techniques, LSI is slightly better than TFIDF but QE in this case does not improve much over TFIDF, perhaps because of the difficulty of this task (there may too often be many irrelevant documents in the top E documents initially retrieved for QE to help). 5We removed links to calendar years as they provide little information while being very frequent. 6 Table 5: The closest five words in the document embedding space to some example query words. kitten cat cats animals species dogs vet veterinarian veterinary medicine animals animal ibm computer company technology software data nyc york new manhattan city brooklyn c++ programming windows mac unix linux xbox console game games microsoft windows beatles mccartney lennon song band harrison britney spears album music pop her In our second experiment we no longer constrained methods to a fixed dictionary size, so all 2.5 million words are used. In this setting we compare to Hash Kernels which can deal with these dictionary sizes. The results, given in Table 2 show the same trends, indicating that the dictionary size restriction in the previous experiment did not bias the results in favor of any one algorithm. Note also that as a page has on average just over 3 test set links to other pages, the maximum P@10 one can achieve in this case is 0.31. In some cases, one might be worried that our experimental setup has split training and testing data only by partitioning the links, but not the documents, hence performance of our model when new unseen documents are added to the database might be in question. We therefore also tested an experimental setup where the test set of documents is completely separate from the training set of documents, by completely removing all training set links between training and testing documents. In fact, this does not alter the performance significantly, as shown in Table 3. Query-Document Ranking So far, our evaluation uses whole Wikipedia articles as queries. One might wonder if the reported improvements also hold in a setup where queries consist of only a few keywords. We thus also tested our approach in this setup. We used the same setup as before but we constructed queries by keeping only n random words from query documents in an attempt to mimic a “keyword search”. Table 4 reports the results for keyword queries of length n = 5, 10 and 20. PSI yields similar improvements as in the document-document retrieval case over the baselines. Word Embedding The document embedding V d in equation (3) (similarly for the query embedding Uq) can be viewed as V d = P i V·idi, in which each column V·i is the embedding of the word di. It is natural that semantically similar words are more likely to have similar embeddings. Table 5 shows a few examples. The first column contains query words, on the right are the 5 words with smallest Euclidean distance in the embedded space. We can see that they are quite relevant. 4.2 Cross Language Document Retrieval Cross Language Retrieval [16] is the task of retrieving documents in a target language E given a query in a different source language F. For example, Google provides such a service6. This is an interesting case for word-based learning to rank models which can naturally deal with this task without the need for machine translation as they directly learn the correspondence between the two languages from bi-lingual labeled data in the form of tuples R. The use of a non-symmetric lowrank model like (3) also naturally suits this task (however in this case adding the identity does not make sense). We therefore also provide a case study in this setting. We thus considered the same set of 1,828,645 English Wikipedia documents and a set of 846,582 Japanese Wikipedia documents, where 135,737 of the documents are known to be about the same concept as a corresponding English page (this information can be found in the wiki mark-up provided in a Wikipedia dump.) For example, the page about “Microsoft” can be found in both English and Japanese, and they are cross-referenced. These pairs are referred to as “mates” in the literature (see, e.g. [9]). We then consider a cross language retrieval task that is analogous to the task in Section 4.1: given a Japanese query document qJap that is the mate of the English document qEng, rank the English 6http://translate.google.com/translate_s 7 Table 6: Cross-lingual Japanese document-English document ranking (limited dictionary size of 30,000 words). Algorithm Rank-Loss MAP P@10 TFIDFEngEng(Google translated queries) 4.78% 0.319±0.009 0.259±0.008 TFIDFEngEng(ATLAS word-based translation) 8.27% 0.115±0.005 0.103±0.005 TFIDFEngEng(ATLAS translated queries) 4.83% 0.290±0.008 0.243±0.008 LSIEngEng(ATLAS translated queries) 7.54% 0.169±0.007 0.150±0.007 αLSIEngEng(ATLAS)+(1 −α)TFIDFEngEng(ATLAS) 3.71% 0.300±0.008 0.253±0.008 CL-LSIJapEng 9.29% 0.190±0.007 0.161±0.007 αCL-LSIJapEng+(1 −α)TFIDFEngEng(ATLAS) 3.31% 0.275±0.009 0.212±0.008 PSIEngEng(ATLAS) 1.72% 0.399±0.009 0.325±0.009 PSIJapEng 0.96% 0.438±0.009 0.351±0.009 αPSIJapEng + (1 −α)TFIDFEngEng(ATLAS) 0.75% 0.493±0.009 0.377±0.009 αPSIJapEng + (1 −α)PSIEngEng(ATLAS) 0.63% 0.524±0.009 0.386±0.009 documents so that the documents linked to qEng appear above the others. The document qEng is removed and not considered during training or testing. The dataset is split into train/test as before. The first type of baseline we considered is based on machine translation. We used a machine translation tool on the Japanese query, and then applied TFIDF or LSI. We considered three methods of machine translation: Google’s API7 or Fujitsu’s ATLAS8 was used to translate each query document, or we translated each word in the Japanese dictionary using ATLAS and then applied this word-based translation to a query. We also compared to CL-LSI [9] trained on all 90,000 Jap-Eng pairs from the training set. For PSI, we considered two cases: (i) apply the ATLAS machine translation tool first, and then use PSI trained on the task in Section 4.1, e.g. the model given in equation (3) (PSIEngEng), which was trained on English queries and English target documents; or (ii) train PSI directly with Japanese queries and English target documents using the model using (3) without the identity, which we call PSIJapEng. We use degree k = 2 for PSI (trying k = 3 would have been interesting, but we have not performed this experiment). The results are given in Table 6. The dictionary size was again limited to the 30,000 most frequent words in both languages for ease of comparison with CL-LSI. TFIDF using the three translation methods gave relatively similar results. Using LSI or CLLSI slightly improved these results, depending on the metric. Machine translation followed by PSIEngEng outperformed all these methods, however the direct PSIJapEng which required no machine translation tool at all, improved results even further. We conjecture that this is because translation mistakes generate noisy features which PSIJapEng circumvents. However, we also considered combining PSIJapEng with TFIDF or PSIEngEng using a mixing parameter α and this provided further gains at the expense of requiring a machine translation tool. Note that many cross-lingual experiments, e.g. [9], typically measure the performance of finding a “mate”, the same document in another language, whereas our experiment tries to model a querybased retrieval task. We also performed an experiment in the mate-finding setting. In this case, PSI achieves a ranking error of 0.53%, and CL-LSI achieves 0.81%. 5 Conclusion We described a versatile, powerful set of discriminatively trained models for document ranking based on polynomial features over words, which was made feasible with a low-rank (but diagonal preserving) approximation. Many generalizations are possible: adding more features into our model, using other choices of loss function and exploring the use of the same models for tasks other than document retrieval, for example applying these models to ranking images rather than text, or to classification, rather than ranking, tasks. 7http://code.google.com/p/google-api-translate-java/ 8http://www.fujitsu.com/global/services/software/translation/atlas/ 8 References [1] B. Bai, J. Weston, R. Collobert, and D. Grangier. Supervised Semantic Indexing. In European Conference on Information Retrieval, pages 761–765, 2009. [2] D. Blei, A. Ng, and M. Jordan. Latent dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, 2003. [3] D. M. Blei and J. D. McAuliffe. Supervised topic models. In NIPS, pages 121–128, 2007. [4] C. Burges, R. Ragno, and Q. Le. Learning to Rank with Nonsmooth Cost Functions. In NIPS, pages 193–200, 2007. [5] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender. Learning to Rank Using Gradient Descent. In ICML, pages 89–96, 2005. [6] Z. Cao, T. Qin, T. Liu, M. Tsai, and H. Li. Learning to rank: from pairwise approach to listwise approach. In ICML, pages 129–136, 2007. [7] G. Chechik, V. Sharma, U. Shalit, and S. Bengio. Large scale online learning of image similarity through ranking. In (Snowbird) Learning Workshop, 2009. [8] S. Deerwester, S. Dumais, G. Furnas, T. Landauer, and R. Harshman. Indexing by latent semantic analysis. JASIS, 41(6):391–407, 1990. [9] S. Dumais, T. Letsche, M. Littman, and T. Landauer. Automatic cross-language retrieval using latent semantic indexing. In AAAI Spring Symposium on Cross-Language Text and Speech Retrieval, pages 15–21, 1997. [10] E. Gabrilovich and S. Markovitch. Computing semantic relatedness using wikipedia-based explicit semantic analysis. In IJCAI, pages 1606–1611, 2007. [11] P. Gehler, A. Holub, and M. Welling. The rate adapting poisson (rap) model for information retrieval and object recognition. In ICML, pages 337–344, 2006. [12] A. Globerson and S. Roweis. Visualizing pairwise similarity via semidefinite programming. In AISTATS, 2007. [13] S. Goel, J. Langford, and A. Strehl. Predictive indexing for fast search. In NIPS, pages 505–512, 2008. [14] D. Grangier and S. Bengio. Inferring document similarity from hyperlinks. In CIKM, pages 359–360, 2005. [15] D. Grangier and S. Bengio. A discriminative kernel-based approach to rank images from text queries. IEEE Transactions on Pattern Analysis and Machine Intelligence., 30(8):1371–1384, 2008. [16] G. Grefenstette. Cross-Language Information Retrieval. Kluwer, Norwell, MA, USA, 1998. [17] R. Herbrich, T. Graepel, and K. Obermayer. Advances in Large Margin Classifiers, chapter Large margin rank boundaries for ordinal regression. MIT Press, Cambridge, MA, 2000. [18] T. Hofmann. Probabilistic latent semantic indexing. In SIGIR, pages 50–57, 1999. [19] P. Jain, B. Kulis, I. S. Dhillon, and K. Grauman. Online metric learning and fast similarity search. In NIPS, pages 761–768, 2008. [20] T. Joachims. Optimizing search engines using clickthrough data. In SIGKDD, pages 133–142, 2002. [21] M. Keller and S. Bengio. A Neural Network for Text Representation. In International Conference on Artificial Neural Networks, 2005. IDIAP-RR 05-12. [22] T. Liu, J. Xu, T. Qin, W. Xiong, and H. Li. Letor: Benchmark dataset for research on learning to rank for information retrieval. In Proceedings of SIGIR 2007 Workshop on Learning to Rank, 2007. [23] R. Salakhutdinov and G. Hinton. Semantic Hashing. Proceedings of the SIGIR Workshop on Information Retrieval and Applications of Graphical Models, 2007. [24] G. Salton and M. McGill. Introduction to Modern Information Retrieval. McGraw-Hill, 1986. [25] Q. Shi, J. Petterson, G. Dror, J. Langford, A. Smola, A. Strehl, and V. Vishwanathan. Hash Kernels. In AISTATS, 2009. [26] A. Vinokourov, J. Shawe-Taylor, and N. Cristianini. Inferring a Semantic Representation of Text via Cross-Language Correlation Analysis. In NIPS, pages 1497–1504, 2003. [27] K. Weinberger and L. Saul. Fast solvers and efficient implementations for distance metric learning. In ICML, pages 1160–1167, 2008. [28] Y. Yue, T. Finley, F. Radlinski, and T. Joachims. A support vector method for optimizing average precision. In SIGIR, pages 271–278, 2007. [29] L. Zighelnic and O. Kurland. Query-drift prevention for robust query expansion. In SIGIR, pages 825– 826, 2008. 9
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Neural Implementation of Hierarchical Bayesian Inference by Importance Sampling Lei Shi Helen Wills Neuroscience Institute University of California, Berkeley Berkeley, CA 94720 lshi@berkeley.edu Thomas L. Griffiths Department of Psychology University of California, Berkeley Berkeley, CA 94720 tom griffiths@berkeley.edu Abstract The goal of perception is to infer the hidden states in the hierarchical process by which sensory data are generated. Human behavior is consistent with the optimal statistical solution to this problem in many tasks, including cue combination and orientation detection. Understanding the neural mechanisms underlying this behavior is of particular importance, since probabilistic computations are notoriously challenging. Here we propose a simple mechanism for Bayesian inference which involves averaging over a few feature detection neurons which fire at a rate determined by their similarity to a sensory stimulus. This mechanism is based on a Monte Carlo method known as importance sampling, commonly used in computer science and statistics. Moreover, a simple extension to recursive importance sampling can be used to perform hierarchical Bayesian inference. We identify a scheme for implementing importance sampling with spiking neurons, and show that this scheme can account for human behavior in cue combination and the oblique effect. 1 Introduction Living creatures occupy an environment full of uncertainty due to noisy sensory inputs, incomplete observations, and hidden variables. One of the goals of the nervous system is to infer the states of the world given these limited data and make decisions accordingly. This task involves combining prior knowledge with current data [1], and integrating cues from multiple sensory modalities [2]. Studies of human psychophysics and animal behavior suggest that the brain is capable of solving these problems in a way that is consistent with optimal Bayesian statistical inference [1, 2, 3, 4]. Moreover, complex brain functions such as visual information processing involves multiple brain areas [5]. Hierarchical Bayesian inference has been proposed as a computational framework for modeling such processes [6]. Identifying neural mechanisms that could support hierarchical Bayesian inference is important, since probabilistic computations can be extremely challenging. Just representing and updating distributions over large numbers of hypotheses is computationally expensive. Much effort has recently been devoted towards proposing possible mechanisms based on known neuronal properties. One prominent approach to explaining how the brain uses population activities for probabilistic computations has been done in the “Bayesian decoding” framework [7]. In this framework, it is assumed that the firing rate of a population of neurons, r, can be converted to a probability distribution over stimuli, p(s|r), by applying Bayesian inference, where the likelihood p(r|s) reflects the probability of that firing pattern given the stimulus s. A firing pattern thus encodes a distribution over stimuli, which can be recovered through Bayesian decoding. The problem of performing probabilistic computations then reduces to identifying a set of operations on firing rates r that result in probabilistically correct operations on the resulting distributions p(s|r). For example, 1 [8] showed that when the likelihood p(r|s) is an exponential family distribution with linear sufficient statistics, adding two sets of firing rates is equivalent to multiplying probability distributions. In this paper, we take a different approach, allowing a population of neurons to encode a probability distribution directly. Rather than relying on a separate decoding operation, we assume that the activity of each neuron translates directly to the weight given to the optimal stimulus for that neuron in the corresponding probability distribution. We show how this scheme can be used to perform Bayesian inference, and how simple extensions of this basic idea make it possible to combine sources of information and to propagate uncertainty through multiple layers of random variables. In particular, we focus on one Monte Carlo method, namely importance sampling with the prior as a surrogate, and show how recursive importance sampling approximates hierarchical Bayesian inference. 2 Bayesian inference and importance sampling Given a noisy observation x, we can recover the true stimulus x∗by using Bayes’ rule to compute the posterior distribution p(x∗|x) = p(x∗)p(x|x∗) R x∗p(x∗)p(x|x∗)dx∗ (1) where p(x∗) is the prior distribution over stimulus values, and p(x|x∗) is the likelihood, indicating the probability of the observation x if the true stimulus value is x∗. A good guess for the value of x∗is the expectation of x∗given x. In general, we are often interested in the expectation of some function f(x∗) over the posterior distribution p(x∗|x), E[f(x∗)|x]. The choice of f(x∗) depends on the task. For example, in noise reduction where x∗itself is of interest, we can take f(x∗) = x∗. However, evaluating expectations over the posterior distribution can be challenging: it requires computing a posterior distribution and often a multidimensional integration. The expectation E[f(x∗)|x] can be approximated using a Monte Carlo method known as importance sampling. In its general form, importance sampling approximates the expectation by using a set of samples from some surrogate distribution q(x∗) and assigning those samples weights proportional to the ratio p(x∗|x)/q(x∗). E[f(x∗)|x] = Z f(x∗)p(x∗|x) q(x∗) q(x∗)dx∗≃1 M M X i=1 f(x∗ i )p(x∗ i |x) q(x∗ i ) x∗ i ∼q(x∗) (2) If we choose q(x∗) to be the prior p(x∗), the weights reduce to the likelihood p(x|x∗), giving E[f(x∗)|x] ≃ 1 M M X i=1 f(x∗ i )p(x∗ i |x) p(x∗ i ) = 1 M M X i=1 f(x∗ i ) p(x, x∗ i ) p(x∗ i )p(x) = 1 M M X i=1 f(x∗ i )p(x|x∗ i ) p(x) = 1 M PM i=1 f(x∗ i )p(x|x∗ i ) R p(x|x∗)p(x∗)dx∗ ≃ X x∗ i f(x∗ i ) p(x|x∗ i ) P x∗ i p(x|x∗ i ) x∗ i ∼p(x∗) (3) Thus, importance sampling provides a simple and efficient way to perform Bayesian inference, approximating the posterior distribution with samples from the prior weighted by the likelihood. Recent work also has suggested that importance sampling might provide a psychological mechanism for performing probabilistic inference, drawing on its connection to exemplar models [9]. 3 Possible neural implementations of importance sampling The key components of an importance sampler can be realized in the brain if: 1) there are feature detection neurons with preferred stimulus tuning curves proportional to the likelihood p(x|x∗ i ); 2) the frequency of these feature detection neurons is determined by the prior p(x∗); and 3) divisive normalization can be realized by some biological mechanism. In this section, we first describe a radial basis function network implementing importance sampling, then discuss the feasibility of three assumptions mentioned above. The model is then extended to networks of spiking neurons. 3.1 Radial basis function (RBF) networks Radial basis function (RBF) networks are a multi-layer neural network architecture in which the hidden units are parameterized by locations in a latent space x∗ i . On presentation of a stimulus x, 2 SX p(Sx|xi*) p(Sx|xn*) p(Sx|x1*) ∑ ∑ ∑ E[f1(x)|Sx]≈ f1(x1*) f2(x1*) f1(xi*) f1(xn*) f2(xi*) f2(xn*) lateral normalization ∑ f1(xi*)p(Sx|xi*) ∑ ∑ p(Sx|xi*) ∑ f2(xi*)p(Sx|xi*) ∑ ∑ p(Sx|xi*) stimulus RBF neurons output neurons inhibitory neuron xi* ~ p(x) E[f2(x)|)|Sx]≈ ]≈ Figure 1: Importance sampler realized by radial basis function network. For details see Section 3.1. these hidden units are activated according to a function that depends only on the distance ||x −x∗ i ||, e.g., exp(−|x −x∗ i |2/2σ2), similar to the tuning curve of a neuron. RBF networks are popular because they have a simple structure with a clear interpretation and are easy to train. Using RBF networks to model the brain is not a new idea – similar models have been proposed for pattern recognition [10] and as psychological accounts of human category learning [11]. Implementing importance sampling with RBF networks is straightforward. A RBF neuron is recruited for a stimulus value x∗ i drawn from the prior (Fig. 1). The neuron’s synapses are organized so that its tuning curve is proportional to p(x|x∗ i ). For a Gaussian likelihood, the peak firing rate would be reached at preferred stimulus x = x∗ i and diminishes as ||x −x∗ i || increases. The ith RBF neuron makes a synaptic connection to output neuron j with strength fj(x∗ i ), where fj is a function of interest. The output units also receive input from an inhibitory neuron that sums over all RBF neurons’ activities. Such an RBF network produces output exactly in the form of Eq. 3, with the activation of the output units corresponding to E[fj(x∗)|x]. Training RBF networks is practical for neural implementation. Unlike the multi-layer perceptron that usually requires global training of the weights, RBF networks are typically trained in two stages. First, the radial basis functions are determined using unsupervised learning, and then, weights to the outputs are learned using supervised methods. The first stage is even easier in our formulation, because RBF neurons simply represent samples from the prior, independent of the second stage later in development. Moreover, the performance of RBF networks is relatively insensitive to the precise form of the radial basis functions [12], providing some robustness to differences between the Bayesian likelihood p(x|x∗ i ) and the activation function in the network. RBF networks also produce sparse coding, because localized radial basis likelihood functions mean only a few units will be significantly activated for a given input x. 3.2 Tuning curves, priors and divisive normalization We now examine the neural correlates of the three components in RBF model. First, responses of cortical neurons to stimuli are often characterized by receptive fields and tuning curves, where receptive fields specify the domain within a stimulus feature space that modify neuron’s response and tuning curves detail how neuron’s responses change with different feature values. A typical tuning curve (like orientation tuning in V1 simple cells) has a bell-shape that peaks at the neuron’s preferred stimulus parameter and diminishes as parameter diverges. These neurons are effectively measure the likelihood p(x|x∗ i ), where x∗ i is the preferred stimulus. Second, importance sampling requires neurons with preferred stimuli x∗ i to appear with frequency proportional to the prior distribution p(x∗). This can be realized if the number of neurons representing x∗is roughly proportional to p(x∗). While systematic study of distribution of neurons over their preferred stimuli is technically challenging, there are cases where this assumption seems to hold. For example, research on the ”oblique effect” supports the idea that the distribution of orientation tuning curves in V1 is proportional to the prior. Electrophysiology [13], optical imaging [14] and 3 fMRI studies [15] have found that there are more V1 neurons tuned to cardinal orientations than to oblique orientations. These findings are in agreement with the prior distribution of orientations of lines in the visual environment. Other evidence comes from motor areas. Repetitive stimulation of a finger expands its corresponding cortical representation in somatosensory area [16], suggesting more neurons are recruited to represent this stimulus. Alternatively, recruiting neurons x∗ i according to the prior distribution can be implemented by modulating feature detection neurons’ firing rates. This strategy also seems to be used by the brain: studies in parietal cortex [17] and superior colliculus [18] show that increased prior probability at a particular location results in stronger firing for neurons with receptive fields at that location. Third, divisive normalization is a critical component in many neural models, notably in the study of attention modulation [19, 20]. It has been suggested that biophysical mechanisms such as shunting inhibition and synaptic depression might account for normalization and gain control [10, 21, 22]. Moreover, local interneurons [23] act as modulator for pooled inhibitory inputs and are good candidates for performing normalization. Our study makes no specific claims about the underlying biophysical processes, but gains support from the literature suggesting that there are plausible neural mechanisms for performing divisive normalization. 3.3 Importance sampling by Poisson spiking neurons Neurons communicate mostly by spikes rather than continuous membrane potential signals. Poisson spiking neurons play an important role in other analyses of systems for representing probabilities [8]. Poisson spiking neurons can also be used to perform importance sampling if we have an ensemble of neurons with firing rates λi proportional to p(x|x∗ i ), with the values of x∗ i drawn from the prior. To show this we need a property of Poisson distributions: if yi ∼Poisson(λi) and Y = P i yi, then Y ∼Poisson(P i λi) and (y1, y2, . . . , ym|Y = n) ∼Multinomial(n, λi/ P i λi). This further implies that E(yi/Y |Y = n) = λi/ P i λi. Assume a neuron tuned to stimulus x∗ i emits spikes ri ∼Poisson(c · p(x|x∗ i )), where c is any positive constant. An average of a function f(x∗ i ) using the number of spikes produced by the corresponding neurons yields P i f(x∗ i )ri/ P i ri, whose expectation is E "X i f(x∗ i ) ri P j rj # = X i f(x∗ i )E " ri P j rj # = X i f(x∗ i ) cλi P j cλj = P i f(x∗ i )p(x|x∗ i ) P i p(x|x∗ i ) (4) which is thus an unbiased estimator of the importance sampling approximation to the posterior expectation. The variance of this estimator decreases as population activity n = P i ri increases because var[ri/n] ∼1/n. Thus, Poisson spiking neurons, if plugged into an RBF network, can perform importance sampling and give similar results to “neurons” with analog output, as we confirm later in the paper through simulations. 4 Hierarchical Bayesian inference and multi-layer importance sampling Inference tasks solved by the brain often involve more than one random variable, with complex dependency structures between those variables. For example, visual information process in primates involves dozens of subcortical areas that interconnect in a hierarchical structure containing two major pathways [5]. Hierarchical Bayesian inference has been proposed as a solution to this problem, with particle filtering and belief propagation as possible algorithms implemented by the brain [6]. However, few studies have proposed neural models that are capable of performing hierarchical Bayesian inference (although see [24]). We show how a multi-layer neural network can perform such computations using importance samplers (Fig. 1) as building blocks. 4.1 Generative models and Hierarchical Bayesian inference Generative models describe the causal process by which data are generated, assigning a probability distribution to each step in that process. To understand brain function, it is often helpful to identify the generative model that determines how stimuli to the brain Sx are generated. The brain then has to reverse the generative model to recover the latent variables expressed in the data (see Fig. 2). The direction of inference is thus the opposite of the direction in which the data are generated. 4 Z Y X Sx generative model Z Y X Sx inference process p(yj|zk) p(xi|yj) p(Sx|xi) pp(zk|y|yj) p(yj|x|xi) p(xi|S|Sx) Figure 2: A hierarchical Bayesian model. The generative model specifies how each variable is generated (in circles), while inference reverses this process (in boxes). Sx is the stimulus presented to the nervous system, while X, Y , and Z are latent variables at increasing levels of abstraction. In the case of a hierarchical Bayesian model, as shown in Fig. 2, the quantity of interest is the posterior expectation of some function f(z) of a high-level latent variable Z given stimulus Sx, E[f(z)|Sx] = R f(z)p(z|Sx) dz. After repeatedly using the importance sampling trick (see Eq. 5), this hierarchical Bayesian inference problem can decomposed into three importance samplers with values x∗ i ,y∗ j and z∗ k drawn from the prior. E[f(z)|S ] x = f (z) p(z|y)[ p(y|x)p(x|S x ) dx] dy dz ≈ f (z) p(z|y) i p(y|x * i )p(S x |x * i ) i p(S x |x * i ) dy dz = f (z) i p(z|y)p(y|x * i ) dy p(S x |x * i ) i p(S x |x * i ) dz ≈ f (z) i j p(z|y* j )p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) dz = j [ f (z)p(z|y* j ) dz] i p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) ≈ j k f (z * k )p(y* j |z * k ) k p(y* j |z * k ) i p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) = k f(z * k ) j p(y* j |z * k ) k p(y* j |z * k ) i p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) importance sampling importance sampling importance sampling zk yj xi x * i ~ p(x) y * j ~ p(y) z * k ~ p(z) (5) This result relies on recursively applying importance sampling to the integral, with each recursion resulting in an approximation to the posterior distribution of another random variable. This recursive importance sampling scheme can be used in a variety of graphical models. For example, tracking a stimulus over time is a natural extension where an additional observation is added at each level of the generative model. We evaluate this scheme in several generative models in Section 5. 4.2 Neural implementation of the multi-layer importance sampler The decomposition of hierarchical inference into recursive importance sampling (Eq. 5) gives rise to a multi-layer neural network implementation (see Fig. 3a). The input layer X is similar to that in Fig. 1, composed of feature detection neurons with output proportional to the likelihood p(Sx|x∗ i ). Their output, after presynaptic normalization, is fed into a layer corresponding to the Y variables, with synaptic weights p(x∗ i |y∗ j ) P j p(x∗ i |y∗ j ). The response of neuron y∗ j , summing over synaptic inputs, approximates p(y∗ j |Sx). Similarly, the response of z∗ k ≈p(z∗ k|Sx), and the activities of these neurons are pooled to compute E[f(z)|Sx]. Note that, at each level, x∗ i ,y∗ j and z∗ k are sampled from prior distributions. Posterior expectations involving any random variable can be computed because the neuron activities at each level approximate the posterior density. A single pool of neurons can also feed activation to multiple higher levels. Using the visual system as an example (Fig. 3b), such a multi-layer importance sampling scheme could be used to account for hierarchical inference in divergent pathways by projecting a set of V2 cells to both MT and V4 areas with corresponding synaptic weights. 5 x2* ∑ lateral normalization GRBF neurons xi*~p(x) ∑ ∑ ∑ xn* y1*=∑i ∑ ∑ x1* yj*=∑i ym*=∑i xi* z1*=∑j zk*=∑j zN*=∑j ∑ yj*~p(y) zk*~p(z) synaptic weight: activity of yj: activity of xi: p(S x |x * i ) i p(S x |x * i ) i p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) activity of zk: j p(y* j |z * k ) k p(y* j |z * k ) i p(x * i |y* j ) j p(x * i |y* j ) p(S x |x * i ) i p(S x |x * i ) synaptic weight: p(y* j |z * k ) k p(y* j |z * k ) p(x * i |y* j ) j p(x * i |y* j ) ∑ f( Z1*) f( Zk*) f( ZN*) (a) (b) V1 MT V4 V2 E[f(z)|Sx] synaptic weight: p(V* 1,i |V * 2,j) j synaptic weight: p(x * i |y* j ) j p(x * i |y* j ) p(V* 1,i |V * 2,j) p(V* 2,j |V * 4,k) j p(V* 2,j |V * 4,k) p(V* 2,j |MT* m) (V*2,j|MT* m) p ) ‘Where’ pathway ‘What’ pathway Figure 3: a) Multi-layer importance sampler for hierarchical Bayesian inference. b) Possible implementation in dorsal-ventral visual inference pathways, with multiple higher levels receiving input from one lower level. Note that the arrow directions in the figure are direction of inference, which is opposite to that of its generative model. 5 Simulations In this section we examine how well the mechanisms introduced in the previous sections account for human behavioral data for two perceptual phenomena: cue combination and the oblique effect. 5.1 Haptic-visual cue combination When sensory cues come from multiple modalities, the nervous system is able to combine those cues optimally in the way dictated by Bayesian statistics [2]. Fig. 4a shows the setup of an experiment where a subject measures the height of a bar through haptic and visual inputs. The object’s visual input is manipulated so that the visual cues can be inconsistent with haptic cues and visual noise can be adjusted to different levels, i.e. visual cue follows xV ∼N(SV , σ2 V ) and haptic cue follows xH ∼N(SH, σ2 H), where SV , SH, σ2 V are controlled parameters. The upper panel of Fig. 4d shows the percentage of trials that participants report the comparison stimulus (consistent visual/haptic cues from 45-65mm) is larger than the standard stimulus (inconsistent visual/haptic cues, SV = 60mm and SH = 50mm). With the increase of visual noise, haptic input accounts for larger weights in decision making and the percentage curve is shifted towards SH, consistent with Bayesian statistics. Several studies have suggested that this form of cue combination could be implemented by population coding [2, 8]. In particular, [8] made an interesting observation that, for Poisson-like spiking neurons, summing firing activities of two populations is the optimal strategy. This model is under the Bayesian decoding framework and requires construction of the network so that these two populations of neurons have exactly the same number of neurons and precise one-to-one connection between two populations, with the connected pair of neurons having exactly the same tuning curves. We present an alternative solution based on importance sampling that encodes the probability distribution by a population of neurons directly. The importance sampling solution approximates the posterior expectation of the bar’s height x∗ C given SV and SH. Sensory inputs are channeled in through xV and xH (Fig.4b). Because sensory input varies in a small range (45-65mm in [2]), we assume priors p(xC), p(xV ) and p(xH) are uniform. It is straightforward to approximate posterior p(xV |SV ) using importance sampling: p(xV = x∗ V |SV ) = E[1(xV = x∗ V )|SV ] ≈ p(SV |x∗ V ) P i p(SV |x∗ V,i) ≈ rV P i rV,i x∗ V,i ∼p(xV ) (6) where rV,i ∼Poisson[c·p(SV |x∗ V,i)] is the number of spikes emitted by neuron x∗ V,i. A similar strategy applies to p(xH|SH). The posterior p(xC|SV , SH), however, is not trivial since multiplication 6 CRT Stereo glasses Opaque mirror Forcefeedback devices Visual and haptic scene Noise: 3 cm equals 100% Visual height Haptic height Width 3-cm depth step M.O. Ernst and M.S. Banks Nature (2002) (a) Experiment setting (d) Visual–haptic discrimination SV SH Normalized comparison height (mm) 0 0.25 0.50 0.75 1.00 0% 67% 133% 200% 5 4 5 5 5 6 Noise level Visual–haptic Proportion of trials perceived as 'taller' human behavior (Ernst et. al. 2002) 0 0.25 0.50 0.75 1.00 SV SH 5 4 5 5 5 6 simulation (b) Generative model of cue combination xC xV xH SV SH (c) Importance sampling from visual−haptic examples 50 55 60 Visual input (mm) 50 55 60 Haptic input (mm) p(xV,xH|xC,k*) {xC,k*} p(xV,xH) {xV,i*} {xH, j*} Figure 4: (a) Experimental setup [2]. (b) Generative model. SV and SH are the sensory stimuli, XV and XH the values along the visual and haptic dimensions, and XC the combined estimate of object height. (c) Illustration of importance sampling using two sensory arrays {x∗ V,i}, {x∗ H,j}. The transparent ellipses indicate the tuning curves of high level neurons centered on values x∗ C,k over xV and xH. The big ellipse represents the manipulated input with inconsistent sensory input and different variance structure. Bars at the center of opaque ellipses indicate the relative firing rates of xC neurons, proportional to p(x∗ C,k|SV , SH). (d) Human data and simulation results. of spike trains is needed. p(xC = x∗ C|SV , SH) = Z 1(xC = x∗ C)p(xC|xV , xH)p(xV |SV )p(xH|SH) dxV dxH ≈ X i X j Z 1(xC = x∗ C)p(xC|xV , xH) rV,i P i rV,i rH,j P j rH,j (7) Fortunately, the experiment gives an important constraint, namely subjects were not aware of the manipulation of visual input. Thus, the values x∗ C,k employed in the computation are sampled from normal perceptual conditions, namely consistent visual and haptic inputs (xV = xH) and normal variance structure (transparent ellipses in Fig.4c, on the diagonal). Therefore, the random variables {xV , xH} effectively become one variable xV,H and values of x∗ V,H,i are composed of samples drawn from xV and xH independently. Applying importance sampling, p(xC = x∗ C|SV , SH) ≈ P i p(x∗ V,i|x∗ C)rV,i + P j p(x∗ H,j|x∗ C)rH,j P i rV,i + P j rH,j (8) E[x∗ C|SV , SH] ≈ X k x∗ C,krC,k/ X k rC,k (9) where rC,k ∼Poisson(c · p(x∗ C,k|SV , SH)) and x∗ C,k ∼p(xC). Compared with Eq. 6, inputs x∗ V,i and x∗ H,j are treaded as from one population in Eq 8. rV,i and rH,j are weighted differently only because of different observation noise. Eq. 9 is applicable for manipulated sensory input (in Fig. 4c, the ellipse off the diagonal). The simulation results (for an average of 500 trials) are shown in the lower panel of Fig.4d, compared with human data in the upper panel. There are two parameters, noise levels σV and σH, are optimized to fit within-modality discrimination data (see [2] Fig. 3a). {x∗ V,i},{x∗ H,j} and {x∗ C,k} consist of 20 independently drawn examples each, and the total firing rate of each set of neurons is limited to 30. The simulations produce a close match to human behavior. 5.2 The oblique effect The oblique effect describes the phenomenon that people show greater sensitivity to bars with horizontal or vertical (0o/90o) orientations than “oblique” orientations. Fig. 5a shows an experimental setup where subjects exhibited higher sensitivity in detecting the direction of rotation of a bar when the reference bar to which it was compared was in one of these cardinal orientations. Fig. 5b shows the generative model for this detection problem. The top-level binary variable D randomly chooses a direction of rotation. Conditioning on D, the amplitude of rotation ∆θ is generated from a truncated 7 45 90 135 180 0 1 2 45 90 135 180 0 1 2 (b) Generative model 0 o 90 o 180 o p(clockwise)? reference bar test bar (a) Oblique effect Relative detection sensitivity adopted from Furmanski & Engel (2000) D ∆θ θ r Sθ p(D=1) = p(D=-1) = 0.5 Clockwise or counterclockwise? ∆θ | D ~ NT(D) (0,σ∆θ) 2 ∆θ ~ N (0,σ∆θ) 2 θ | ∆θ, r ~ N (∆θ+r ,σθ ) 2 θ ~ { Sθ | θ ~ N (θ ,σS ) 2 ∆θ Uni([0, pi]) or (1-k)/2[N(0, σθ )+N(pi/2, σθ )] +k Uni([0, pi]) (c) Oblique effect and prior 0 90 180 prior 0 90 180 prior Orientation Relative detection sensitivity Figure 5: (a) Orientation detection experiment. The oblique effect is shown in lower panel, being greater sensitivity to orientation near the cardinal directions. (b) Generative model. (c) The oblique effect emerges from our model, but depends on having the correct prior p(θ). normal distribution (NT (D), being restricted to ∆θ > 0 if D = 1 and ∆θ < 0 otherwise). When combined with the angle of the reference bar r (shaded in the graphical model, since it is known), ∆θ generates the orientation of a test bar θ, and θ further generates the observation Sθ, both with normal distributions with variance σθ and σSθ respectively. The oblique effect has been shown to be closely related to the number of V1 neurons that tuned to different orientations [25]. Many studies have found more V1 neurons tuned to cardinal orientations than other orientations [13, 14, 15]. Moreover, the uneven distribution of feature detection neurons is consistent with the idea that these neurons might be sampled proportional to the prior: more horizontal and vertical segments exist in the natural visual environment of humans. Importance sampling provides a direct test of the hypothesis that preferential distribution of V1 neurons around 0o/90o can cause the oblique effect, which becomes a question of whether the oblique effect depends on the use of a prior p(θ) with this distribution. The quantity of interest is: p(D = 1|Sθ, r) ≈ X j′ X i p(θ∗ i |∆θ∗ j′, r) P j p(θ∗ i |∆θ∗ j , r) p(Sθ|θ∗ i ) P i p(Sθ|θ∗ i ) (10) where j′ indexes all ∆θ∗> 0. If p(D = 1|Sθ, r) > 0.5, then we should assign D = 1. Fig. 5c shows that detection sensitivity is uncorrelated with orientations if we take a uniform prior p(θ), but exhibits the oblique effect under a prior that prefers cardinal directions. In both cases, 40 neurons are used to represent each of ∆θ∗ i and θ∗ i , and results are averaged over 100 trials. Sensitivity is measured by percentage correct in inference. Due to the qualitative nature of this simulation, model parameters are not tuned to fit experiment data. 6 Conclusion Understanding how the brain solves the problem of Bayesian inference is a significant challenge for computational neuroscience. In this paper, we have explored the potential of a class of solutions that draw on ideas from computer science, statistics, and psychology. We have shown that a small number of feature detection neurons whose tuning curves represent a small set of typical examples from sensory experience is sufficient to perform some basic forms of Bayesian inference. Moreover, our theoretical analysis shows that this mechanism corresponds to a Monte Carlo sampling method, i.e. importance sampling. The basic idea behind this approach – storing examples and activating them based on similarity – is at the heart of a variety of psychological models, and straightforward to implement either in traditional neural network architectures like radial basis function networks, circuits of Poisson spiking neurons, or associative memory models. The nervous system is constantly reorganizing to capture the ever-changing structure of our environment. Components of the importance sampler, such as the tuning curves and their synaptic strengths, need to be updated to match the distributions in the environment. Understanding how the brain might solve this daunting problem is a key question for future research. Acknowledgments. Supported by the Air Force Office of Scientific Research (grant FA9550-07-1-0351). 8 References [1] K. K¨ording and D. M. Wolpert. Bayesian integration in sensorimotor learning. Nature, 427:244–247, 2004. [2] M. O. Ernst and M. S. Banks. Humans integrate visual and haptic information in a statistically optimal fashion. Nature, 415(6870):429–433, 2002. [3] A. Stocker and E. Simoncelli. A bayesian model of conditioned perception. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1409– 1416. MIT Press, Cambridge, MA, 2008. [4] A. P. Blaisdell, K. Sawa, K. J. Leising, and M. R. Waldmann. Causal reasoning in rats. Science, 311(5763):1020–1022, 2006. [5] D. C. Van Essen, C. H. Anderson, and D. J. Felleman. Information processing in the primate visual system: an integrated systems perspective. Science, 255(5043):419–423, 1992 Jan 24. [6] T. S. Lee and D. Mumford. Hierarchical bayesian inference in the visual cortex. J.Opt.Soc.Am.A Opt.Image Sci.Vis., 20(7):1434–1448, 2003. [7] R. S. Zemel, P. Dayan, and A. Pouget. Probabilistic interpretation of population codes. Neural Comput, 10(2):403–430, 1998. [8] W. J. Ma, J. M. Beck, P. E. Latham, and A. Pouget. Bayesian inference with probabilistic population codes. Nat.Neurosci., 9(11):1432–1438, 2006. [9] L. Shi, N. H. Feldman, and T. L. Griffiths. Performing bayesian inference with exemplar models. In Proceedings of the 30th Annual Conference of the Cognitive Science Society, 2008. [10] M. Kouh and T. Poggio. A canonical neural circuit for cortical nonlinear operations. Neural Comput, 20(6):1427–1451, 2008. [11] J. K. Kruschke. Alcove: An exemplar-based connectionist model of category learning. Psychological Review, 99:22–44, 1992. [12] M. J. D. Powell. Radial basis functions for multivariable interpolation: a review. Clarendon Press, New York, NY, USA, 1987. [13] R. L. De Valois, E. W. Yund, and N. Hepler. The orientation and direction selectivity of cells in macaque visual cortex. Vision Res, 22(5):531–544, 1982. [14] D. M. Coppola, L. E. White, D. Fitzpatrick, and D. Purves. Unequal representation of cardinal and oblique contours in ferret visual cortex. Proc Natl Acad Sci U S A, 95(5):2621–2623, 1998 Mar 3. [15] C. S. Furmanski and S. A. Engel. An oblique effect in human primary visual cortex. Nat Neurosci, 3(6):535–536, 2000. [16] A. Hodzic, R. Veit, A. A. Karim, M. Erb, and B. Godde. Improvement and decline in tactile discrimination behavior after cortical plasticity induced by passive tactile coactivation. J Neurosci, 24(2):442–446, 2004. [17] M. L. Platt and P. W. Glimcher. Neural correlates of decision variables in parietal cortex. Nature, 400:233– 238, 1999. [18] M. A. Basso and R. H. Wurtz. Modulation of neuronal activity by target uncertainty. Nature, 389(6646):66–69, 1997. [19] J. H. Reynolds and D. J. Heeger. The normalization model of attention. Neuron, 61(2):168–185, 2009 Jan 29. [20] J. Lee and J. H. R. Maunsell. A normalization model of attentional modulation of single unit responses. PLoS ONE, 4(2):e4651, 2009. [21] S. J. Mitchell and R. A. Silver. Shunting inhibition modulates neuronal gain during synaptic excitation. Neuron, 38(3):433–445, 2003. [22] J. S. Rothman, L. Cathala, V. Steuber, and R A. Silver. Synaptic depression enables neuronal gain control. Nature, 457(7232):1015–1018, 2009 Feb 19. [23] H. Markram, M. Toledo-Rodriguez, Y. Wang, A. Gupta, G. Silberberg, and C. Wu. Interneurons of the neocortical inhibitory system. Nat Rev Neurosci, 5(10):793–807, 2004 Oct. [24] K. Friston. Hierarchical models in the brain. PLoS Comput Biol, 4(11):e1000211, 2008 Nov. [25] G. A. Orban, E. Vandenbussche, and R. Vogels. Human orientation discrimination tested with long stimuli. Vision Res, 24(2):121–128, 1984. 9
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Discriminative Network Models of Schizophrenia Guillermo A. Cecchi, Irina Rish IBM T. J. Watson Research Center Yorktown Heights, NY, USA Benjamin Thyreau Neurospin CEA, Saclay, France Bertrand Thirion INRIA Saclay, France Marion Plaze INSERM - CEA - Univ. Paris Sud Research Unit U.797 Neuroimaging & Psychiatry SHFJ & Neurospin, Orsay, France Marie-Laure Paillere-Martinot AP-HP, Adolescent Psychopathology and Medicine Dept., Maison de Solenn, Cochin Hospital, University Paris Descartes F-75014 Paris, France Catherine Martelli Departement de Psychiatrie et d’Addictologie Centre Hospitalier Paul Brousse Villejuif, France Jean-Luc Martinot INSERM - CEA - Univ. Paris Sud Research Unit U.797 Neuroimaging & Psychiatry SHFJ & Neurospin, Orsay, France Jean-Baptiste Poline Neurospin CEA, Saclay, France Abstract Schizophrenia is a complex psychiatric disorder that has eluded a characterization in terms of local abnormalities of brain activity, and is hypothesized to affect the collective, “emergent” working of the brain. We propose a novel data-driven approach to capture emergent features using functional brain networks [4] extracted from fMRI data, and demonstrate its advantage over traditional region-of-interest (ROI) and local, task-specific linear activation analyzes. Our results suggest that schizophrenia is indeed associated with disruption of global brain properties related to its functioning as a network, which cannot be explained by alteration of local activation patterns. Moreover, further exploitation of interactions by sparse Markov Random Field classifiers shows clear gain over linear methods, such as Gaussian Naive Bayes and SVM, allowing to reach 86% accuracy (over 50% baseline - random guess), which is quite remarkable given that it is based on a single fMRI experiment using a simple auditory task. 1 Introduction It has been long recognized that extracting an informative set of application-specific features from the raw data is essential in practical applications of machine learning, and often contributes even more to the success of learning than the choice of a particular classifier. In biological applications, such as brain image analysis, proper feature extraction is particularly important since the primary objective of such studies is to gain a scientific insight rather than to learn a “black-box” predictor; thus, the focus shifts towards the discovery of predictive patterns, or “biomarkers”, forming a basis for interpretable predictive models. Conversely, biological knowledge can drive the definition of features and lead to more powerful classification. The objective of this work is to identify biomarkers predictive of schizophrenia based on fMRI data collected for both schizophrenic and non-schizophrenic subjects performing a simple auditory task in the scanner [14]. Unlike some other brain disorders (e.g., stroke or Parkinsons disease), schizophrenia appears to be “delocalized”, i.e. difficult to attribute to a dysfunction of some par1 ticular brain areas1. The failure to identify specific areas, as well as the controversy over which localized mechanisms are responsible for the symptoms associated with schizophrenia, have led us amongst others [7, 1, 10] to hypothesize that this disease may be better understood as a disruption of the emergent, collective properties of normal brain states, which can be better captured by functional networks [4], based on inter-voxel correlation strength, as opposed (or limited) to activation failures localized to specific, task-dependent areas. To test this hypothesis, we measured diverse topological features of the functional networks and compared them across the normal subjects and schizophrenic patients groups. Specifically, we decided to ask the following questions: (1) What specific effects does schizophrenia have on the functional connectivity of brain networks? (2) Does schizophrenia affect functional connectivity in ways that are congruent with the effect it has on area-specific, task-dependent activations? (3) Is it possible to use functional connectivity to improve the classification accuracy of schizophrenic patients? In answer to these questions, we will show that degree maps, which assign to each voxel the number of its neighbors in a network, identify spatially clustered groups of voxels with statistically significant group (i.e. normal vs. schizophrenic) differences; moreover, these highly significant voxel subsets are quite stable over different data subsets. In contrast, standard linear activation maps commonly used in fMRI analysis show much weaker group differences as well as stability. Moreover, degree maps yield very informative features, allowing for up to 86% classification accuracy (with 50% baseline), as opposed to standard local voxel activations. The best accuracy is achieved by further exploiting non-local interactions with probabilistic graphical models such as Markov Random Fields, as opposed to linear classifiers. Finally, we demonstrate that traditional approaches based on a direct comparison of the correlation at the level of relevant regions of interest (ROIs) or using a functional parcellation technique [17], do not reveal any statistically significant differences between the groups. Indeed, a more data-driven approach that exploits properties of voxel-level networks appears to be necessary in order to achieve high discriminative power. 2 Background and Related Work In Functional Magnetic Resonance Imaging (fMRI), a MR scanner non-invasively records a subject’s blood-oxygenation-level dependent (BOLD) signal, known to be correlated with neural activity, as a subject performs a task of interest (e.g., viewing a picture or reading a sentence). Such scans produce a sequence of 3D images, where each image typically has on the order of 10,000-100,000 subvolumes, or voxels, and the sequence typically contains a few hundreds of time points, or TRs (time repetitions). Standard fMRI analysis approaches, such as the General Linear Model (GLM) [9], examine mass-univariate relationships between each voxel and the stimulus in order to build so-called statistical parametric maps that associate each voxel with some statistics that reflects its relationship to the stimulus. Commonly used activation maps depict the “activity” level of each voxel determined by the linear correlation of its time course with the stimulus (see Supplemental Material for details). Clearly, such univariate analysis can miss important information contained in the interactions among voxels. Indeed, as it was shown in [8], highly predictive models of mental states can be built from voxels with sub-maximal activation. Recently, applying multivariate predictive methods to fMRI became an active area of research, focused on predicting “mental states” from fMRI data [11, 13, 2]. However, our focus herein is not just predictive modeling, but rather discovery of interpretable features with high discriminative power. Also, our problem is much more high-dimensional, since each sample (e.g., schizophrenic vs. non-schizophrenic) corresponds to a sequence of 3D images over about 400 time points, rather than to a single 3D image as in [11, 13, 2]. While the importance of modeling brain connectivity and interactions became widely recognized in the current fMRI-analysis literature [6, 19, 16], practical applications of the proposed approaches such as dynamic causal modeling [6], dynamic Bays nets [19], or structural equations [16] were 1This is often referred to as the disconnection hypothesis [5, 15], and can be traced back to the early research on schizophrenia: in 1906, Wernicke [18] was the first one to postulate that anatomical disruption of association fiber tracts is at the roots of psychosis; in fact, the term schizophrenia was introduced by Bleuler [3] in 1911, and was meant to describe the separation (splitting) of different mental functions. 2 ROI name (x,y,z) position Anatomical position 1 ’Temporal mid L’ -44,-48,4 Left temporal 2 ’Temporal mid et sup L’ -56,-36,0 Middle and superior left temporal 3 ’Frontal inf L’ -40,28,0 Left Inferior frontal 4 ’cuneus L’ -12,-72,24 Left cuneus 5 ’Temporal sup et mid L’ -52,-16,-8 Middle and superior left temporal 6 ’Angular L’ -44,-48,32 Left angular gyrus 7 ’Temporal sup R’ 40,-64,24 Right superior temporal 8 ’Angular R’ 40,-64,24 Right angular gyrus 9 ’Cingulum post R’ 4,-32,24 Right posterior cingulum 10 ’ACC’ 0,20,30 Anterior cingulated cortex Figure 1: Regions of Interest and their location on standard brain. usually limited to interactions analysis among just a few (e.g., less than 15) known brain regions believed to be relevant to the task or phenomenon of interest. In this paper, we demonstrate that such model-based region-of-interest (ROI) analysis may fail to reveal informative interactions which, nevertheless, become visible at the finer-grain voxel level when using a purely data-driven, networkbased approach [4]. Moreover, while recent publications have already indicated that functional networks in the schizophrenic brain display disrupted topological properties, we demonstrate, for the first time, that (1) specific topological properties (e.g. voxel degrees) of functional networks can help to construct highly-predictive schizophrenia classifiers that generalize well and (2) functional network differences cannot be attributed to alteration of local activation patterns, a hypothesis that was not ruled out by the results of [1, 10] and similar work. 3 Experimental Setup The present study is a reanalysis of image datasets previously acquired according to the methodology described in [14]. Two groups of 12 subjects each were submitted to the same experimental paradigm involving language: schizophrenic patients and age-matched normal controls (same experiment was performed with a third group of alcoholic patients, yielding similar results - see Suppl. Materials for details). The studies had been performed after approval of the local ethics committee and all subjects were studied after they gave written informed consent. The task is based on auditory stimuli; subjects listen to emotionally neutral sentences either in native (French) or foreign language. Average length (3.5 sec mean) or pitch of both kinds of sentences is normalized. In order to catch attention of subjects, each trial begins with a short (200 ms) auditory tone, followed by the actual sentence. The subject’s attention is asserted through a simple validation task: after each played sentences, a short pause of 750 ms is followed by a 500 ms two-syllable auditory cue, which belongs to the previous sentence or not, to which the subject must answer to by yes (the cue is part of the previous sentence) or no with push-buttons, when the language of the sentence was his own. For each subject, two fMRI acquisition runs are acquired, each of which consisted of 420-scans (from which the first 4 are discarded to eliminate T1 effect). A full fMRI run contains 96 trials, with 32 sentences in French (native), 32 sentences in foreign languages, and 32 silence interval controls. Data were spatially realigned and warped into the MNI template and smoothed (FWHM of 5mm) using SPM5 (www.fil.ucl.ac.uk); also, standard SPM5 motion correction was performed. Several subjects were excluded from the consideration due to excessive head motion in the scanner, leaving us with 11 schizophrenic and 11 healthy subjects, i.e. the total of 44 samples (there were two samples per subject, corresponding to the two runs of the experiment). Each sample associated with roughly 53,000 voxels (after removing out-of-brain voxels from the original 53 × 63 × 46 image), over 420 time points (TRs), i.e. with more than 22,000,000 voxels/variables. Thus, some kind of dimensionality reduction and/or feature extraction is necessary prior to learning a predictive model. 4 Methods We explored two different data analysis approaches aimed at discovery of discriminative patterns: (1) model-driven approaches based on prior knowledge about the regions of interest (ROI) that are believed to be relevant to schizophrenia, or model-based functional clustering, and (2) data-driven approaches based on various features extracted from the fMRI data, such as standard activation maps and a set of topological features derived from functional networks. 4.1 Model-Driven Approach using ROI First, we decided to test whether the interactions between several known regions of interest (ROIs) would contain enough discriminative information about schizophrenic versus normal subjects. Ten 3 regions of interests (ROI) were defined using previous literature on schizophrenia and language studies, including inferior, middle and superior left temporal cortex, left inferior temporal cortex, left cuneus, left angular gyrus, right superior temporal, right angular gyrus, right posterior cingulum, and anterior cingular cortex (Figure 1). Each region was defined as a sphere of 12mm diameter centered on the x,y,z coordinates of the corresponding ROI. Because predefined regions of interest may be based on too much a priori knowledge and miss important areas, we also ran a more exploratory analysis. A second set of 600 ROI’s was defined automatically using a parcellation algorithm [17] that estimates, for each subject, a collection of regions based on task-based functional signal similarity and position in the MNI space. Time series were extracted as the spatial mean over each ROI, leading to 10 time series per subject for the predefined ROIs and 600 for the parcellation technique. The connectivity measures were of two kinds. First, the correlation coefficient was computed along time between ROIs blindly with respect to the experimental paradigm. Additionally, we computed a psycho-physiological interaction (PPI), by contrasting the correlation coefficient weighted by experimental conditions (i.e. correlation weighted by the ”Language French” condition versus correlation weighted by ”Control” condition after convolution with a standard hemodynamic response function). Those connectivity measures were then tested for significance using standards non parametric tests between groups (Wilcoxon signed-rank test) with corrected p-values for multiple comparisons. 4.2 Data-driven Approach: Feature Extraction Topological Features and Degree Maps. In order to continue investigating possible disruptions of global brain functioning associated with schizophrenia, we decided to explore lower-level (as compared to ROI-level) functional brain networks [4] constructed at the voxel level: (1) pair-wise Pearson correlation coefficients are computed among all pairs of time-series (vi(t), vj(t)) where vi(i) corresponds to the BOLD signal of i-th voxel; (2) an edge between a pair of voxels (i, j) is included in the network if the correlation between vi and vj exceeds a specified threshold (herein, we used the same threshold of c(Pearson)=0.7 for all voxel pairs). For each subject, and each run, a separate functional network was constructed. Next, we measured a number of its topological features, including the degree distribution, mean degree, the size of the largest connected subgraph (giant component), and so on (see the supplemental material for the full list). Besides global topological features, we also computed a series of degree maps based on the individual voxel degree in functional network: (1) full degree maps, where the value assigned to each voxel is the total number of links in the corresponding network node, (2) long-distance degree maps, where the value is the number of links making non-local connections (5 voxels apart or more), and (3) inter-hemispheric degree maps, where only links reaching across the brain hemispheres are considered when computing each voxel’s degree. Activation maps. To find out whether local task-dependent linear activations alone could possibly explain the differences between the schizophrenic and normal brains, we used as a baseline set of features based on the standard voxel activation maps. For each subject, and for each run, activation maps, as well as their differences, or activation contrast maps, were obtained using several regressors based on the language task, as described in the supplemental material (for simplicity, we will refer to all such maps as activation maps). The activation values of each voxel were subsequently used as features in the classification task. Similarly to degree maps, we also computed a global feature, mean-activation (mean-t-val)), by taking the mean absolute value of the voxel’s t-statistics. Both activation and degree maps for each sample were also normalized, i.e. divided by their maximal value for the given sample. 4.3 Classification Approaches First, off-the-shelf methods such Gaussian Naive Bayes (GNB) and Support Vector Machines (SVM) were used in order to compare the discriminative power of different sets of features described above. Moreover, we decided to further investigate our hypothesis that interactions among voxels contain highly discriminative information, and compare those linear classifiers against probabilistic graphical models that explicitly model such interactions. Specifically, we learn a classifier based on a sparse Gaussian Markov Random Field (MRF) model [12], which leads to a convex problem with unique optimal solution, and can be solved efficiently; herein, we used the COVSEL procedure [12]. The weight on the l1-regularization penalty serves as a tuning parameter of the classifier, allowing to control the sparsity of the model, as described below. 4 Sparse Gaussian MRF classifier. Let X = {X1, ..., Xp} be a set of p random variables (e.g., voxels), and let G = (V, E) be an undirected graphical model (Markov Network, or MRF) representing conditional independence structure of the joint distribution P(X). The set of vertices V = {1, ..., p} is in the one-to-one correspondence with the set X. There is no edge between Xi and Xj if and only if the two variables are conditionally independent given all remaining variables. Let x = (x1, ..., xp) denote a random assignment to X. We will assume a multivariate Gaussian probability density p(x) = (2π)−p/2 det(C) 1 2 e−1 2 xT Cx, where C = Σ−1 is the inverse covariance matrix, and the variables are normalized to have zero mean. Let x1, ..., xn be a set of n i.i.d. samples from this distribution, and let S = 1 n Pn i=1 xT i xi denote the empirical covariance matrix. Missing edges in the above graphical model correspond to zero entries in the inverse covariance matrix C, and thus the problem of learning the structure for the above probabilistic graphical model is equivalent to the problem of learning the zero-pattern of the inverse-covariance matrix 2. A popular approach is to use l1-norm regularization that is known to promote sparse solutions, while still allowing (unlike non-convex lq-norm regularization with 0 < q < 1) for efficient optimization. From the Bayesian point of view, this is equivalent to assuming that the parameters of the inverse covariance matrix C = Σ−1 are independent random variables Cij following the Laplace distributions p(Cij) = λij 2 e−λij|Cij−αij| with zero location parameters (means) αij and equal scale parameters λij = λ. Then p(C) = Qp i=1 Qp j=1 p(Cij) = (λ/2)p2e−λ||C||1, where ||C||1 = P ij |Cij| is the (vector) l1-norm of C. Assume a fixed parameter λ, our objective is to find arg maxC≻0 p(C|X), where X is the n × p data matrix, or equivalently, since p(C|X) = P(X, C)/p(X) and p(X) does not include C, to find arg maxC≻0 P(X, C), over positive definite matrices C. This yields the following optimization problem considered, for example, in [12] max C≻0 ln det(C) −tr(SC) −λ||C||1 where det(A) and tr(A) denote the determinant and the trace (the sum of the diagonal elements) of a matrix A, respectively. For the classification task, we estimate on the training data the Gaussian conditional density p(x|y) (i.e. the (inverse) covariance matrix parameter) for each class Y = {0, 1} (schizophrenic vs non-schizophrenic), and then choose the most-likely class label arg maxc p(x|c)P(c) for each unlabeled test sample x. Variable Selection: We used variable selection as a preprocessing step before applying a particular classifier, in order to (1) reduce the computational complexity of classification (especially for sparse MRF, which, unlike GNB and SVM, could not be directly applied to over 50,000 variables), (2) reduce noise and (3) identify relatively small predictive subsets of voxels. We applied a simple filter-based approach, selecting a subset of top-ranked voxels, where the ranking criterion used p-values resulting from the paired t-test, with the null-hypothesis being that the voxel values corresponding to schizophrenic and non-schizophrenic subjects came from distributions with equal means. The variables were ranked in the ascending order of their p-values (lower p = higher confidence in between-group differences), and classification results on top k voxels will be presented for a range of k values. Evaluation via Cross-validation. We used leave-one-subject-out rather than leave-one-sample-out cross-validation, since the two runs (two samples) for each subject are clearly not i.i.d. and must be handled together to avoid biases towards overly-optimistic results. 5 Results Model-driven ROI analysis. First, we observed that correlations (blind to experimental paradigm) between regions and within subjects were very strong and significant (p-value of 0.05, corrected for the number of comparisons) when tested against 0 for all subjects (mean correlation > 0.8 for every group). However, these inter-region correlations do not seem to differ significantly between the groups. The parcellation technique led to some smaller p-values, but also to a stricter correction for multiple comparison and no correlation was close to the corrected threshold. Concerning the psycho-physiological interaction, results were closer to significance, but did not survive multiple comparisons. In conclusion, we could not detect significant differences between the schizophrenic patient data and normal subjects in either the BOLD signal correlation or the interaction between the signal and the main experimental contrast (native language versus silence). 2Note that the inverse of the empirical covariance matrix, even if it exists, does not typically contain exact zeros. Therefore, an explicit sparsity constraint is usually added to the estimation process. 5 10 0 10 1 10 2 10 3 10 4 10 5 10 −9 10 −8 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 k/N p value P values and FDR correction 0.05* k/N activation 1 FrenchNative−Silence activation 6 FrenchNative degree (full) degree (long−distance) degree (inter−hemispheric) (a) (b) Figure 2: (a) FDR-corrected 2-sample t-test results for (normalized) degree maps, where the null hypothesis at each voxel assumes no difference between the schizophrenic vs normal groups. Red/yellow denotes the areas of low p-values passing FDR correction at α = 0.05 level (i.e., 5% false-positive rate). Note that the mean (normalized) degree at those voxels was always (significantly) higher for normals than for schizophrenics. (b) Direct comparison of voxel p-values and FDR threshold: p-values sorted in ascending order; FDR test select voxels with p < α · k/N (α - false-positive rate, k - the index of a p-value in the sorted sequence, N - the total number of voxels). Degree maps yield a large number (1033, 924 and 508 voxels in full, long-distance and inter-hemispheric degree maps, respectively) of highly-significant (very low) p-values, staying far below the FDR cut-off line, while only a few voxels survive FDR in case of activation maps: 7 and 2 voxels in activation maps 1 (contrast “FrenchNative - Silence”) and 6 (“FrenchNative”), respectively (the rest of the activation maps do not survive the FDR correction at all). Data-driven analysis: topological vs activation features. Empirical results are consistent with our hypothesis that schizophrenia disrupts the normal structure of functional networks in a way that is not derived from alterations in the activation; moreover, they demonstrate that topological properties are highly predictive, consistently outperforming predictions based on activations. 1. Voxel-wise statistical analysis. Degree maps show much stronger statistical differences between the schizophrenic vs. non-schizophrenic groups than the activation maps. Figure 2 show the 2-sample t-test results for the full degree map and the activation maps, after False-Discovery Rate (FDR) correction for multiple comparisons (standard in fMRI analysis), at α = 0.05 level (i.e., 5% false-positive rate). While the degree map (Figure 2a) shows statistically significant differences bilaterally in auditory areas (specifically, normal group has higher degrees than schizophrenic group), the activation maps show almost no significant differences at all: practically no voxels there survived the FDR correction (Figure 2b. This suggests that (a) the differences in the collective behavior cannot be explained by differences in the linear task-related response, and that (b) topology of voxel-interaction networks is more informative than task-related activations, suggesting an abnormal degree distribution for schizophrenic patients that appear to lack hubs in auditory cortex, i.e., have significantly lower (normalized) voxel degrees in that area than the normal group (possibly due to a more even spread of degrees in schizophrenic vs. normal networks). Moreover, degree maps demonstrate much higher stability than activation maps with respect to selecting a subset of top ranked voxels over different subsets of data. Figure 3a shows that degree maps have up to almost 70% top-ranked voxels in common over different training data sets when using the leaveone-subject out cross-validation, while activation maps have below 50% voxels in common between different selected subsets. This property of degree vs activation features is particularly important for interpretability of predictive modeling. 2. Inter-hemispheric degree distributions. A closer look at the degree distributions reveals that a large percentage of the differential connectivity appears to be due to long-distance, inter-hemispheric links. Figure 3a compares (normalized) histograms, for schizophrenic (red) versus normal (blue) groups, of the fraction of inter-hemispheric connections over the total number of connections, computed for each subject within the group. The schizophrenic group shows a significant bias towards low relative inter-hemispheric connectivity. A t-test analysis of the distributions indicates that differences are statistically significant (p=2.5x10-2). Moreover, it is evident that a major contributor to the high degree difference discussed before is the presence of a large number of inter-hemispheric connections in the normal group, which is lacking in schizophrenic group. Furthermore, we selected a bilateral regions of interest (ROI’s) corresponding to left and right Brodmann Area 22 (roughly, the clusters in Figure 2a), such that the linear activation for these ROI’s was not significantly different between the groups, even in the uncorrected case. For each subject, the link between the left and 6 0 1000 2000 3000 4000 5000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 % voxels in common # of top−ranked voxels selected Stability of top−ranked voxel subset degree(full) degree (long distance) degree(inter−hemispheric) activation 1 (and 3) activation 2 (and 4) activation 5 activation 6 activation 7 activation 8 0 0.1 0.2 0 0.2 0.4 0.6 Relative link density Histograms over samples 0 5 10 15 x 10 −4 0 0.2 0.4 0.6 0.8 1 Relative link density Histogram over samples (a) (b) (c) Figure 3: (a) Stability of feature subset selection over CV folds, i.e. the percent of voxels in common among the subsets of k top variables selected at all CV folds. (b) Disruption of global inter-hemispheric connectivity. For each subject, we compute the fraction of inter-hemispheric connections over the total number of connections, and plot a normalized histogram over all subjects in a particular group (normal - blue, schizophrenic red). (c) Disruption of task-dependent inter-hemispheric connectivity between specific ROIs (Brodmann Area 22 selected bilaterally). The ROIs were defined by a 9 mm radius ball centered at [x=-42, y=-24, z=3] and [x=42, y=-24, z=3]. Feature (GNB SVM MRF(0.01) degree (D) 27.5% 27.5% 27.5% clustering coeff. (C) 30.0% 42.5% 45.0% geodesic dist. (G) 67.5% 45.0% 45.0% mean activation (A) 40.0% 45% 72.5% D + A 27.5% 27.5% 32.5% C + A 27.5% 45.0% 55.0% G + A 45.0% 45.0% 72.5% G +D +C 37.5% 27.5% 27.5% G+D+C+A 30.0% 27.5% 32.5% Feature Error False Pos False Neg degree (full) 16% 27% 5% degree (long-distance) 21% 32% 9% degree (inter-hemis) 32% 46% 18% activation 1 (and 3) 54% 29% 82% activation 2 (and 4) 50% 55% 45% activation 5 43% 18% 68% activation 6 36% 27% 46% activation 7 32% 18% 46% activation 8 30% 23% 37% (a) (b) Table 1: Classification errors using (a) global features and (b) activation and degree maps (using SVM on the complete set of voxels (i.e., without voxel subset selection). right ROIs was computed as the fraction of ROI-to-ROI connections over all connections; Figure 3c shows the normalized histograms. Clearly, the normal group displays a high density of interhemispheric connections, which are significantly disrupted in the schizophrenic group (p=3.7x107). This provides a strong indication that the group differences in connectivity cannot be explained by differences in local activation. 3. Global features. For each global feature (full list in Suppl. Mat.) we computed its mean for each group and p-value produced by the t-test, as well as the classification accuracies using our classifiers. While more details are presented in the supplemental material, we outline here the main observations: while mean activation (we used map 8, the best performer for SVM on the full set of voxels - see Table1b) had an relatively low p-value of 5.5 × 10−4, as compared to less significant p = 5.3 × 10−2 for mean-degree, the predictive power of the latter, alone or in combination with some other features, was the best among global features reaching 27.5% in schizophrenic vs normal classification (Table 1a), while mean activation yielded more than 40% error with all classifiers. 4. Classification results using degree vs. activation maps. While mean-degree indicates the presence of discriminative information in voxel degrees, its generalization ability, though the best among global features and their combinations, is relatively poor. However, voxel-level degree maps turned out to be excellent predictive features, often outperforming activation features by far. Table 1b compares prediction made by SVM on complete maps (without voxel subset selection): both full and long-distance degree maps greatly outperform all activation maps, achieving 16% error vs. above 30% for even the best-performing activation map 8. Next, in Figure 4, we compare the predictive power of different maps when using all three classifiers: Support Vector Machines (SVM), Gaussian Naive Bayes (GNB) and sparse Gaussian Markov Random Field (MRF), on the subsets of k top-ranked voxels, for a variety of k values. We used the best-performing activation map 8 from the Table above, as well as maps 1 and 6 (that survived FDR); map 6 was also outperforming other activation maps in low-voxel regime. To avoid clutter, we only plot the two best-performing degree maps out of three (i.e., full and long-distance ones). For sparse MRF, we experimented with a variety of λ values, ranging from 0.0001 to 10, and present the best results. We can see that: (a) Degree maps frequently outperform activation maps, for all classifiers we used; the differences are 7 10 1 10 2 10 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 classification error Gaussian Naive Bayes schizophrenic vs normal K top voxels (ttest) activation 1 FrenchNative − Silence activation 6 FrenchNative activation 8 Silence degree (long−distance) degree (full) 10 1 10 2 10 3 10 4 10 5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 K top voxels (ttest) Support Vector Machine: schizophrenic vs normal activation 1 FrenchNative − Silence activation 6 FrenchNative activation 8 Silence degree (long−distance) degree (full) 50 100 150 200 250 300 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Markov Random Field: schizophrenic vs normal K top voxels (ttest) activation 1 FrenchNative − Silence activation 6 FrenchNative activation 8 Silence degree (long−distance) degree (full) 50 100 150 200 250 300 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 MRF vs GNB vs SVM: schizophrenic vs normal K top voxels (ttest) MRF (0.1): degree (long−distance) GNB: degree (long−distance) SVM:degree (long−distance) (a) (b) (c) (d) Figure 4: Classification results comparing (a) GNB, (b) SVM and (c) sparse MRF on degree versus activation contrast maps; (d) all three classifiers compared on long-distance degree maps (best-performing for MRF). particularly noticeable when the number of selected voxels is relatively low. The most significant differences are observed for SVM in low-voxel (approx. < 500) and full-map regimes, as well as for MRF classifiers: it is remarkable that degree maps can achieve an impressively low error of 14% with only 100 most significant voxels, while even the best activation map 6 requires more than 200-300 to get just below 30% error; the other activation maps perform much worse, often above 30-40% error, or even just at the chance level. (b) Full and long-distance degree maps perform quite similarly, with long-distance map achieving the best result (14% error) using MRFs. (c) Among the activation maps only, while the map 8 (“Silence”) outperforms others on the full set of voxels using SVM, its behavior in low-voxel regime is quite poor (always above 30-35% error); instead, map 6 (“FrenchNative”) achieves best performance among activation maps in this regime3. (d) MRF classifiers clearly outperform SVM and GNB, possibly due to their ability to capture inter-voxel relationships that are highly discriminative between the two classes (see Figure 4d). 6 Summary The contributions of this paper are two-fold. From a machine-learning and fMRI analysis perspective, we (a) introduced a novel feature-construction approach based on topological properties of functional networks, that is generally applicable to any multivariate-timeseries classification problems, and can outperform standard linear activation approaches in fMRI analysis field, (b) demonstrated advantages of this data-driven approach over prior-knowledge-based (ROI) approaches, and (c) demonstrated advantages of network-based classifiers (Markov Random Fields) over linear models (SVM, Naive Bayes) on fMRI data, suggesting to exploit voxel interactions in fMRI analyzes (i.e., treat brain as a network). From neuroscience perspective, we provided strong support for the hypothesis that schizophrenia is associated with the disruption of global, emergent brain properties which cannot be explained just by alteration of local activation patterns. Moreover, while prior art is mainly focused on exploring the differences between the functional and anatomical networks of schizophrenic patients versus healthy subjects [10, 1], this work, to our knowledge, is the first attempt to explore the generalization ability of predictive models of schizophrenia built on network features. Finally, a word of caution. Note that the schizophrenia patients studied here have been selected for their prominent, persistent, and pharmaco-resistant auditory hallucinations [14], which might have increased their clinical homogeneity. However, the patient group is not representative of the full spectrum of the disease, and thus our conclusions may not necessarily apply to all schizophrenia patients, due to the clinical characteristics and size of the studied samples. Acknowledgements We would like to thank Rahul Garg for his help with the data preprocessing and many stimulating discussions that contributed to the ideas of this paper, and Drs. Andr´e Galinowski, Thierry Gallarda, and Frank Bellivier who recruited and clinically rated the patients. We also would like to thank INSERM as promotor of the MR data acquired (project RBM 01 −26). 3We also observed that performing normalization really helped activation maps, since otherwise their performance could get much worse, especially with MRFs - we provide those results in supplemental material. 8 References [1] D.S. Bassett, E.T. Bullmore, B.A. Verchinski, V.S. Mattay, D.R. Weinberger, and A. MeyerLindenberg. Hierarchical organization of human cortical networks in health and schizophrenia. J Neuroscience, 28(37):9239–9248, 2008. [2] A. Battle, G. Chechik, and D. Koller. Temporal and cross-subject probabilistic models for fmri prediction tasks. In B. Sch¨olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19, pages 121–128. MIT Press, Cambridge, MA, 2007. [3] E. Bleuler. Dementia Praecox or the Group of Schizophrenias. International Universities Press, New York, NY, 1911. [4] V.M. Eguiluz, D.R. Chialvo, G.A. Cecchi, M. Baliki, and A.V. Apkarian. Scale-free functional brain networks. Physical Review Letters, 94(018102), 2005. [5] K.J. Friston and C.D. Frith. Schizophrenia: A Disconnection Syndrome? Clinical Neuroscience, (3):89–97, 1995. [6] K.J. Friston, L. Harrison, and W.D. Penny. Dynamic Causal Modelling. Neuroimage, 19(4):1273–1302, Aug 2003. [7] A.G. Garrity, G. D. Pearlson, K. McKiernan, D. Lloyd, K.A. Kiehl, and V.D. Calhoun. Aberrant “Default Mode” Functional Connectivity in Schizophrenia. Am J Psychiatry, 164:450– 457, March 2007. [8] J.V. Haxby, M.I. Gobbini, M.L. Furey, A.Ishai, J.L. Schouten, and P. Pietrini. Distributed and Overlapping Representations of Faces and Objects in Ventral Temporal Cortex. Science, 293(5539):2425–2430, 2001. [9] K. J. Friston et al. Statistical parametric maps in functional imaging - a general linear approach. Human Brain Mapping, 2:189–210, 1995. [10] Y. Liu, M. Liang, Y. Zhou, Y. He, Y. Hao, M. Song, C. Yu, H. Liu, Z. Liu, and T. Jiang. Disrupted Small-World Networks in Schizophrenia. Brain, 131:945–961, February 2008. [11] T.M. Mitchell, R. Hutchinson, R.S. Niculescu, F. Pereira, X. Wang, M. Just, and S. Newman. Learning to Decode Cognitive States from Brain Images. Machine Learning, 57:145–175, 2004. [12] O.Banerjee, L. El Ghaoui, and A. d’Aspremont. Model selection through sparse maximum likelihood estimation for multivariate gaussian or binary data. Journal of Machine Learning Research, 9:485–516, March 2008. [13] F. Pereira and G. Gordon. The Support Vector Decomposition Machine. In ICML2006, pages 689–696, 2006. [14] M. Plaze, D. Bartrs-Faz, JL Martinot, D. Januel, F. Bellivier, R. De Beaurepaire, S. Chanraud, J. Andoh, JP Lefaucheur, E. Artiges, C. Pallier, and ML Paillere-Martinot. Left superior temporal gyrus activation during sentence perception negatively correlates with auditory hallucination severity in schizophrenia patients. Schizophrenia Research, 87(1-3):109–115, 2006. [15] K.E. Stephan, K.J. Friston, and C.D. Frith. Dysconnection in Schizophrenia: From Abnormal Synaptic Plasticity to Failures of Self-monitoring. Schizophrenia Bulletin, 35(3):509–527, 2009. [16] A. J. Storkey, E. Simonotto, H. Whalley, S. Lawrie, L. Murray, and D. McGonigle. Learning structural equation models for fmri. In Advances in Neural Information Processing Systems 19, pages 1329–1336. 2007. [17] B. Thirion, G. Flandin, P. Pinel, A. Roche, P. Ciuciu, and J.-B. Poline. Dealing with the shortcomings of spatial normalization: Multi-subject parcellation of fmri datasets. Human Brain Mapping, 27(8):678–693, 2006. [18] C. Wernicke. Grundrisse der psychiatrie. Thieme, 1906. [19] L. Zhang, D. Samaras, N. Alia-Klein, N. Volkow, and R. Goldstein. Modeling neuronal interactivity using dynamic bayesian networks. In Advances in Neural Information Processing Systems 18, pages 1593–1600. 2006. 9
|
2009
|
123
|
3,619
|
Sparsistent Learning of Varying-coefficient Models with Structural Changes Mladen Kolar, Le Song and Eric P. Xing ∗ School of Computer Science, Carnegie Mellon University {mkolar,lesong,epxing}@cs.cmu.edu Abstract To estimate the changing structure of a varying-coefficient varying-structure (VCVS) model remains an important and open problem in dynamic system modelling, which includes learning trajectories of stock prices, or uncovering the topology of an evolving gene network. In this paper, we investigate sparsistent learning of a sub-family of this model — piecewise constant VCVS models. We analyze two main issues in this problem: inferring time points where structural changes occur and estimating model structure (i.e., model selection) on each of the constant segments. We propose a two-stage adaptive procedure, which first identifies jump points of structural changes and then identifies relevant covariates to a response on each of the segments. We provide an asymptotic analysis of the procedure, showing that with the increasing sample size, number of structural changes, and number of variables, the true model can be consistently selected. We demonstrate the performance of the method on synthetic data and apply it to the brain computer interface dataset. We also consider how this applies to structure estimation of time-varying probabilistic graphical models. 1 Introduction Consider the following regression model: Yi = X′ iβ(ti) + ǫi, i = 1, . . . , n, (1) where the design variables Xi ∈Rp are i.i.d. zero mean random variables sampled at some conditions indexed by i = 1, . . . , n, such as the prices of a set of stocks at time i, or the signals from some sensors deployed at location i; the noise ǫ1, . . . , ǫn are i.i.d. Gaussian variables with variance σ2 independent of the design variables; and β(ti) = (β1(ti), . . . , βp(ti))′ : [0, 1] 7→Rp is a vector of unknown coefficient functions. Since the coefficient vector is a function of the conditions rather than a constant, such a model is called a varying-coefficient model [12]. Varying-coefficient models are a non-parametric extension to the linear regression models, which unlike other non-parametric models, assume that there is a linear relationship (generalizable to log-linear relationship) between the feature variables and the output variable, albeit a changing one. The model given in Eq. (1) has the flexibility of a non-parametric model and the interpretability of an ordinary linear regression. Varying-coefficient models were popularized in the work of [9] and [16]. Since then, they have been applied to a variety of domains, including multidimensional regression, longitudinal and functional data analysis, and modeling problems in econometrics and finance, to model and predict time- or space- varying response to multidimensional inputs (see e.g. [12] for an overview.) One can easily imagine a more general form of such a model applicable to these domains, where both the coefficient value and the model structure change with values of other variables. We refer to this class of models as varying-coefficient varying-structure (VCVS) models. The more challenging problem of structure recovery (or model selection) under VCVS has started to catch attention very recently [1, 24]. ∗LS is supported by a Ray and Stephenie Lane Research Fellowship. EPX is supported by grant ONR N000140910758, NSF DBI-0640543, NSF DBI-0546594, NSF IIS-0713379 and an Alfred P. Sloan Research Fellowship. We also thank Za¨ıd Harchaoui for useful discussions. 1 −0.5 0 0.5 β1(t) −0.5 0 0.5 β2(t) … 0 0.2 0.4 0.6 0.8 1 −0.5 0 0.5 βp(t) Time t (i/n) (a) Y ... β2 βp β1 (b) −0.5 0 0.5 β1(t) −0.5 0 0.5 β2(t) … 0 0.2 0.4 0.6 0.8 1 −0.5 0 0.5 βp(t) Time t (i/n) (c) −0.5 0 0.5 β1(t) −0.5 0 0.5 β2(t) … 0 0.2 0.4 0.6 0.8 1 −0.5 0 0.5 βp(t) Time t (i/n) (d) Figure 1: (a) Illustration of an VCVS as varying functions of time. The interval [0, 1] is partitioned into {0, 0.25, 0.4, 0.7, 1}, which defines blocks on which the coefficient functions are constant. At different blocks only covariates with non-zero coefficient affect the response, e.g. on the interval B2 = (0.25, 0.4) covariates X2 and Xp do not affect response. (b) Schematic representation of the covariates affecting the response during the second block in panel (a), which is reminiscent of neighborhood selection in graph structure learning. (c) and (d) Application of VCVS for graph structure estimation (see Section 7) of non-piecewise constant evolving graphs. Coefficients defining neighborhoods of different nodes can change on different partitions. In this paper, we analyze VCVS as functions of time, and the main goal is to estimate the dynamic structure and jump points of the unknown vector function β(t). To be more specific, we consider the case where the function β(t) is time-varying, but piecewise constant (see Fig. 1), i.e., there exists a partition T = {T1 = 0 < T2 < . . . < TB = 1}, 1 < B ≤n, of the time interval (scaled to) [0, 1], such that β(t) = γj, t ∈[Tj−1, Tj) for some constant vectors γj ∈Rp, j = 1, . . . , B. We refer to points T1, . . . , TB as jump points. Furthermore, we assume that at each time point ti only a few covariates affect the response, i.e., the vector β(ti) is sparse. A good estimation procedure would be able to identify the correct partition of the interval [0, 1] so that within each segment the coefficient function is constant. In addition, the procedure can identify active coefficients and their values within each segment, i.e., the time-varying structure of the model. This estimation problem is particularly important in applications where one needs to uncover dynamic relational information or model structures from time series data. For example, one may want to infer at chosen time points the (changing) set of stocks that are predictive of a particular stock one has been holding from a time series of all stock prices; or to understand the evolving circuitry of gene regulation at different growth stages of an organism that determines the activity of a target gene based on other regulative genes, based on time series of microarray data. Another important problem is to identify structural changes in fields such as signal processing, EEG segmentation and analysis of seismic signals. In all these problems, the goal is not to estimate the optimum value of β(t) for predicting Y , but to consistently uncover the zero and non-zero patterns in β(t) at time points of interest that reveal the changing structure of the model. In this paper, we provide a new algorithm to achieve this goal, and a theoretical analysis that proves the asymptotic consistency of our algorithm. Our problem is remotely related to, but very different from, earlier works on linear regression models with structural changes [4], and the problem of change-point detection (e.g. [19]), which can also be analyzed in the framework of varying-coefficient models. A number of existing methods are available to identify only one structural change in the data; in order to identify multiple changes these methods can be applied sequentially on smaller intervals that are assumed to harbor only one change [14]. Another common approach is to assume that there are K changes and use Dynamic Programming to estimate them [4]. In this paper, we propose and analyze a penalized least squares approach, which automatically adapts to the unknown number of structural changes present in the data and performs the variable selection on each of the constant regions. 2 Preliminaries For a varying-coefficient regression model described in Eq. (1) with structural changes, a reasonable estimator of the time-varying structure can be obtained by minimizing the so-called TESLA (temporally smoothed L1-regularized regression) loss proposed in [1]: (for simplicity we suppress the sample-size notation n in the regularization constants λn = {λn 1, λn 2}, but it should be clear that their values depend on n) ˆβ(t1; λ), . . . , ˆβ(tn; λ) = arg min β n X i=1 (Yi −X′ iβ(ti))2 + 2λ1 n X i=1 ||β(ti)||1 + 2λ2 p X k=1 ||βk||TV , (2) where ||·||1 denotes the ℓ1 norm, and ||·||TV denotes a total variation norm: ||βk||TV = Pn i=2 |βk(ti) −βk(ti−1)|. From the analysis of [20], it is known that each component function 2 βk can be chosen as a piecewise constant and right continuous function, i.e., βk is a spline function, with potential jump points at observation times ti, i = 1, . . . , n. In this particular case, the total variation penalty defined above allows us to conceptualize βk as a vector in Rn, whose components βk,i ≡βk(ti) correspond to function values at ti, i = 1, . . . , n, but not as a function [0, 1] 7→R. We continue to use the vector representation through the rest of the paper as it will simplify the notation. The estimation problem defined in Eq. (2) has a few appealing properties. The objective function on the right-hand-side is convex and there exists a solution ˆβ, which can be found efficiently using a standard convex optimization package. Furthermore, the penalty terms in Eq. (2) are constructed in a way to perform model selection. Observe that ℓ1 penalty encourages sparsity of the signal at each time point and enables a selection over the relevant coefficients; whereas the total variation penalty is used to partition the interval [0, 1] so that ˆβk is constant within each segment. However, there are also some drawbacks of the procedure, as shown in Lemma 1 below. Let’s start with some notational clarifications. Let X denote the design matrix, input observation Xi at time i corresponds to the i-th row in X. For simplicity, we assume throughout the paper that X are normalized to have unit length columns, i.e., each dimension has unit Euclidean norm. Let Bj, j = 1, . . . , B, denote the set of time points that fall into the interval [Tj−1, Tj); when the meaning is clear from the context, we also use Bj as a shorthand of this interval. For example, XBj and YBj represent the submatrix of X and subvector of Y , respectively, that include elements only corresponding to time points within interval Bj. For a given solution ˆβ to Eq. (2), there exists a block partition ˆT = { ˆT1, . . . , ˆT ˆ B} of [0, 1] (possibly a trivial one) and unique vectors ˆγj ∈Rp, j = 1, . . . , ˆB, such that ˆβk,i = ˆγj,k for ti ∈ˆBj. The set of relevant covariates during inverval Bj, i.e., the support of vector γj, is denoted as SBj = {k | γj,k ̸= 0}. Likewise we define ˆS ˆ Bj over ˆγj. By construction, no consecutive vectors ˆγj and ˆγj+1 are identical. Note that both the number of partitions ˆB = | ˆT |, and the elements in the partition ˆT , are random quantities. The following lemma characterizes the vectors ˆγj using the subgradient equation of Eq. (2). Lemma 1 Let ˆγj and ˆBj, j = 1, . . . , ˆB be vectors and segments obtained from a minimizer of Eq. (2). Then each ˆγj can be found as a solution to the subgradient equation: X′ ˆ BjX ˆ Bj ˆγj −X′ ˆ BjY ˆ Bj + λ1| ˆBj|ˆs(1) j + λ2ˆs(TV) j = 0, (3) where ˆs(1) j ∈∂||ˆγj||1 = sign(γj), (4) by convention sign(0) ∈[−1, 1], and ˆs(TV) j ∈Rp such that ˆs(TV) 1,k = −1 if ˆγ2,k −ˆγ1,k > 0 1 if ˆγ2,k −ˆγ1,k < 0 , ˆs(TV) ˆ B,k = 1 if ˆγ ˆ B,k −ˆγ ˆ B−1,k > 0 −1 if ˆγ ˆ B,k −ˆγ ˆ B−1,k < 0 (5) and, for 1 < j < ˆB, ˆs(TV) j,k = ( 2 if ˆγj+1,k −ˆγj,k > 0, ˆγj,k −ˆγj−1,k < 0 −2 if ˆγj+1,k −ˆγj,k < 0, ˆγj,k −ˆγj−1,k > 0 0 if (ˆγj,k −ˆγj−1,k)(ˆγj+1,k −ˆγj,k) = 1. (6) Lemma 1 does not provide a practical way to estimate ˆβTV, but it does characterize a solution. From Eq. (3) we can see that the coefficients in each of the estimated blocks are biased by two terms coming from the ℓ1 and ||·||TV penalties. The larger the estimated segments, the smaller the relative influence of the bias from the total variation, while the magnitude of the bias introduced by the ℓ1 penalty is uniform across different segments. The additional bias coming from the total variation penalty was also noted in the problem of signal denoising [23]. In the next section, we introduce a two step procedure which alleviate this effect. 3 A two-step procedure for estimating time-varying structures In this section, we propose a new algorithm for estimating the time-varying structure of the varyingcoefficient model in Eq. (1), which does not suffer from the bias introduced by minimizing the objective in Eq. (2). The algorithm is a two-step procedure summarized as follows: 3 1. Estimate the block partition ˆT , on which the coefficient vector is constant within each block. This can be obtained by minimizing the following objective: n X i=1 (Yi −X′ iβ(ti))2 + 2λ2 p X k=1 ||βk||TV , (7) which we refer to as a temporal difference (TD) regression for reasons that will be clear shortly. We will employ a TD-transformation to Eq. (7) and turn it into an ℓ1-regularized regression problem, and solve it using the randomized Lasso. Details of the algorithm and how to extract ˆT from the TD-estimate will be given shortly. 2. For each block of the partition, ˆBj, 1 ≤j ≤ˆB, estimate ˆγj by minimizing the Lasso objective within the block: ˆγj = argmin γ∈Rp X ti∈ˆ Bj (Yi −X′ iγ)2 + 2λ1 ||γ||1 . (8) We name this procedure TDB-Lasso (or TDBL), after the two steps (TD randomized Lasso, and Lasso within Blocks) given above. The advantage of the TDB-Lasso compared to a minimizer of Eq. (2) comes from decoupling the interactions between the ℓ1 and TV penalties (note that the two procedures result in different estimates). Now we discuss step 1 in detail; step 2 is straightforward using a standard Lasso toolbox. To obtain a consistent estimate of ˆT from the TD-regression in Eq. (7), we can transform Eq. (7) into an equivalent ℓ1 penalized regression problem, which allows us to cast the ˆT estimation problem as a feature selection problem. Let β† k,i denote the temporal difference between the regression coefficients corresponding to the same covariate k at successive time points ti−1 and ti: β† k,i ≡βk(ti) −βk(ti−1), k = 1, . . . , p, i = 1, . . . , n with βk(t0) = 0, by convention. It can be shown that the model in Eq. (1) can be expressed as Y † = X†β† + ǫ†, where Y † ∈Rn is a transformed vector of the TDs of responses, i.e., each element Y † i ≡Yi −Yi−1; X† = (X† 1, . . . , X† p) ∈Rn×np is the transformed design matrix with lower triangular matrices X† k ∈Rn×n corresponding to TD features computed from the covariates; ǫ† ∈Rn is the transformed TD-error vector; and β† ∈Rnp is a vector obtained by stacking TD-coefficient vectors β† k. (See Appendix for more details of the transformation.) Note that the elements of the vector ǫ† are not i.i.d. any more. Using the transformation above, the estimation problem defined on objective Eq. (7) can be expressed in the following matrix form: ˆβ† = argmin β∈Rnp Y † −X†β† 2 2 + 2λ2 β† 1 . (9) This transformation was proposed in [8] in the context of one-dimensional signal denoising, however, we are interested in the estimation of jump points in the context of time-varying coefficient model. The estimator defined in Eq. (9) is not robust with respect to small perturbations of data, i.e., small changes of variables Xi or Yi would result in a different ˆT . To deal with the problem of robustness, we employed the stability selection procedure of [22] (see also the bootstrap Lasso [2], however, we have decided to use the stability selection because of the weaker assumptions). The stability selection approach to estimating the jump-points is comprised of two main components: i) simulating multiple datasets using bootstrap, and ii) using the randomized Lasso outlined in Algorithm 1 (see also Appendix) to solve (9). While the bootstrap step improves the robustness of the estimator, the randomized Lasso weakens the conditions under which the estimator ˆβ† selects exactly the true features. Let {ˆβ† b, ˆ J † b }M b=1 represent the set of estimates and their supports (i.e., index of non-zero elements) obtained by minimizing (9) for each of the M bootstrapped datasets. We obtain a stable estimate of the support by selecting variables that appear in multiple supports ˆ J τ = {k | PM b=1 1I{k ∈ˆ J † b } M ≥τ}, (10) which is then used to obtain the block partition estimate ˆT . The parameter τ is a tuning parameter that controls the number of falsely identified jump points. 4 Algorithm 1 Randomized Lasso Input: Dataset {Xi, Yi}n i=1 Xi ∈Rp, penalty parameter λ, weakness parameter α ∈(0, 1] Output: Estimate ˆβ ∈Rp, support ˆS 1: Choose randomly p weights {Wk}p k=1 from interval [α, 1] 2: ˆβ = argminβ∈Rp Pn i=1(Yi −Xiβ)2 + 2λ Pp k=1 |βk| Wk 3: ˆS = {k | ˆβk ̸= 0} 4 Theoretical analysis We provide a theoretical analysis of TDB-Lasso, and show that under certain conditions both the jump points and structure of VCVS can be consistently estimated. Proofs are deferred to Appendix. 4.1 Estimating jump points We first address the issue of estimating jump points by analyzing the transformed TD-regression problem Eq. (9) and its feature selection properties. The feature selection using ℓ1 penalization has been analyzed intensively over the past few years and we can adapt some of the existing results to the problem at hand. To prove that all the jump points are included in ˆ J τ, we first state a sparse eigenvalue condition on the design (e.g. [6]). The minimal and maximal sparse eigenvalue, for matrix X ∈Rn×p, are defined as ϕmin(k, X) := inf a∈Rp,||a||0≤k ||Xa||2 ||a||2 , ϕmax(k, X) := sup a∈Rp,||a||0≤k ||Xa||2 ||a||2 , k ≤p. (11) Note that in Eq. (11) eigenvalues are computed over submatrices of size k (i.e., due to the constraint on a by the ||·||0 norm). We can now express the sparse eigenvalues condition on the design. A1: Let J † be the true support of β† and J = |J †|. There exist some C > 1 and κ ≥10 such that ϕmax(CJ2, X†) ϕ3/2 min(CJ2, X†) < √ C/κ. (12) This condition guarantees a correlation structure between TD-transformed covariates that allows for detection of the jump points. Comparing to the irrepresentible condition [30, 21, 27], necessary for the ordinary Lasso to perform feature selection, condition A1 is much weaker [22] and is sufficient for the randomized Lasso to select the relevant feature with high probability (see also [26]). Theorem 1 Let A1 be satisfied; and let the weakness α be given as α2 = νϕmin(CJ2, X†)/(CJ2), for any ν ∈(7/κ, 1/ √ 2). If the minimum size of the jump is bounded away from zero as min k∈J † |β† k| ≥0.3(CJ)3/2λmin, (13) where λmin = 2σ†( √ CJ + 1) q log np n and σ†2 ≥V ar(Y † i ), for np > 10 and J ≥7, there exists some δ = δJ ∈(0, 1) such that for all τ ≥1 −δ, the collection of the estimated jump points ˆ J τ satisfies, P( ˆ J τ = J †) ≥1 −5/np. (14) Remark: Note that Theorem 1 gives conditions under which we can recover every jump point in every covariates. In particular, there are no assumptions on the number of covariates that change values at a jump point. Assuming that multiple covariates change their values at a jump point, we could further relax the condition on the minimal size of a jump given in Eq. (13). It was also pointed to us that the framework of [18] may be a more natural way to estimate jump points. 4.2 Identifying correct covariates Now we address the issue of selecting the relevant features for every estimated segment. Under the conditions of Theorem 1, correct jump points will be detected with probability arbitrarily close to 1. That means under the assumption A1, we can run the regular Lasso on each of the estimated segments to select the relevant features therein. We will assume that the mutual coherence condition [10] holds for each segment Bj. Let Σj = 1 |Bj| P i∈Bj X′ iXi, with σj kl = (Σj)k,l. 5 A2: We assume there is a constant 0 < d ≤1 such that P max k∈SBj ,l̸=k ( |σj kl| ≤ d SBj )! = 1. (15) The assumption A2 is a mild version of the mutual coherence condition used in [7], which is necessary for identification of the relevant covariates in each segment. Let ˆγj, k = 1, . . . , ˆBn denote the Lasso estimates for each segment obtained by minimizing (8). Theorem 2 Let A2 be satisfied. Also, assume that the conditions of Theorem 1 are satisfied. Let K = max1≤j≤B ||γj||0 be the upper bound on the number of features in segments and let L be an upper bound on elements of X. Let ρ = min1≤j≤B |Bj| denote the number of samples in the smallest segment. Then for a sequence δ = δn →0, λ1 ≥4Lσ s ln 2Kp δ ρ ∨8Lln 4Kp δ ρ and min 1≤j≤B min k∈SBj |γj,k| ≥2λ1, we have lim n→∞P( ˆB = B) = 1, (16) lim n→∞max 1≤j≤B P(||ˆγj −γj||1 = 0) = 1, (17) lim n→∞min 1≤j≤B P( ˆSBj = SBj) = 1. (18) Theorem 2 states that asymptotically, the two stage procedure estimates the correct model, i.e., it selects the correct jump points and for each segment between two jump points it is able to select the correct covariates. Furthermore, we can conclude that the procedure is consistent. 5 Practical considerations As in standard Lasso, the regularization parameters in TDB-Lasso need to be tuned appropriately to attain correct structural recovery. The TD regression procedure requires three parameters: the penalty parameter λ2, cut-off parameter τ, and weakness parameter α. From our empirical experience, the recovered set of jump points ˆT vary very little with respect to these parameters in a wide range. The result of Theorem 1 is valid as long as λ2 is larger than λmin given in the statement of the theorem. Theorem 1 in [22] gives a way to select the cutoff τ while controlling the number of falsely included jump points. Note that this relieves users from carefully choosing the range of parameter λ2, which is challenging. The weakness parameter can be chosen in quite a large interval (see Appendix on the randomized Lasso) and we report our results using the values α = 0.6. In the second step of the algorithm, the ordinary Lasso minimizes Eq. (8) on each estimated segment to select relevant variables, which requires a choice of the penalty parameter λ1. We do so by minimizing the BIC criterion [25]. In practice, one cannot verify assumptions A1 and A2 on real datasets. In cases where the assumptions are violated, the resulting set of estimated jump points is larger than the true set T , e.g. the points close to the true jump points get included into the resulting estimate ˆT . We propose to use an ad hoc heuristic to refine the initially selected set of jump points. A commonly used procedure for estimation of linear regression models with structural changes [3] is a dynamic programming method that considers a possible structural change at every location ti, i = 1, . . . , n, with a computational complexity of O(n2) (see also [15]). We modify this method to consider jump points only in the estimated set ˆT and thus considerably reducing the computational complexity to O(| ˆT |2), since | ˆT | ≪n. The algorithm effectively chooses a subset ˜T ⊆ˆT of size ˆB that minimizes the BIC objective. 6 Experiments on Synthetic Data We compared the TDB-Lasso on synthetic data with commonly used methods for estimating VCVS models. The synthetic data was generated as follows. We varied the sample size from n = 100 6 200 400 0 0.5 1 1.5 MREE Sample size 200 400 0 0.5 1 Corr. zeros Sample size 200 400 0 0.5 1 Precision Sample size 200 400 0 0.5 1 Recall Sample size 200 400 0 0.5 1 F1 Sample size Kernel + l1/l2 Kernel + l1 l1 + TV TDB−Lasso Figure 2: Comparison results of different estimation procedures on a synthetic dataset. to 500 time points, and fixed the number of covariates is fixed to p = 20. The block partition was generated randomly and consists of ten blocks with minimum length set to 10 time points. In each of the block, only 5 covariates out of 20 affected the response. Their values were uniformly at random drawn from [−1, −0.1]∪[0.1, 1]. With this configuration, a dataset was created by randomly drawing Xi ∼N(0, Ip), ǫi ∼N(0, 1.52) and computing Yi = Xiβ(ti) + ǫi for i = 1, . . . , n. For each sample size, we independently generated 100 datasets and report results averaged over them. A simple local regression method [13], which is commonly used for estimation in varying coefficient models, was used as the simplest baseline for comparing the relative performance of estimation. Our first competitor is an extension of the baseline, which uses the following estimator [28]: min β∈Rp×n n X i′=1 n X i=1 (Yi −X′ iβi′)2Kh(ti′ −ti) + p X j=1 λj v u u t n X i′=1 β2 i′,j, (19) where Kh(·) = 1 hK(·/h) is the kernel function. We will call this method “Kernel ℓ1/ℓ2”. Another competitor uses the ℓ1 penalized local regression independently at each time point, which leads to the following estimator of β(t), min β∈Rp n X i=1 (Yi −X′ iβ)2Kh(ti −t) + p X j=1 λj|βj|. (20) We call this method “Kernel ℓ1”. The difference between the two methods is that “Kernel ℓ1/ℓ2” biases certain covariates toward zero at every time point, based on global information; whereas “Kernel ℓ1” biases covariates toward zero only based on local information. The final competitor is chosen to be the minimizer of Eq. (2) [1], which we call “ℓ1 + TV”. The bandwidth parameter for “Kernel ℓ1” and “Kernel ℓ1/ℓ2” is chosen using a generalized cross validation of a non-penalized estimator. The penalty parameters λj are chosen according to the BIC criterion [28]. For the “ℓ1 + TV” method, we optimize the BIC criterion over a two-dimensional grid of values for λ1 and λ2. We report the relative estimation error, REE = 100 × Pn i=1 Pp j=1 | ˆβi,j−β∗ i,j| Pn i=1 Pp j=1 | ˜βi,j−β∗ i,j|, where ˜β is the baseline local linear estimator, as a measure of estimation accuracy. To asses the performance of the model selection, we report precision, recall and their harmonic mean F1 measure when estimating the relevant covariates at each time point and the percentage of correctly identified irrelevant covariates. From the experimental results, summarized in Fig. 2, we can see that the TDB-Lasso succeeds in recovering the true model as the sample size increases. It also estimates the coefficient values with better accuracy than the other methods. It worth noting that the “Kernel + ℓ1” performs better than the “Kernel + ℓ1/ℓ2” approach, which is due to the violation of the assumptions made in [28]. The “ℓ1 + TV” performs better than the local linear regression approaches, however, the method gets very slow for the larger values of the sample size and it requires selecting two tuning parameters, which makes it quite difficult to use. We conjecture that the “ℓ1 +TV” and TDB-Lasso have similar asymptotic properties with respect to model selection, however, from our numerical experiments we can see that for finite sample data, the TDB-Lasso performs better. 7 Application to Time-varying Graph Structure Estimation An interesting application of the TDB-Lasso is in structural estimation of time-varying undirected graphical models [1, 17]. A graph structure estimation can be posed as a neighborhood selection 7 problem, in which neighbors of each node are estimated independently. Neighborhood selection in the time-varying Gaussian graphical models (GGM) is equivalent to model selection in VCVS, where value of one node is regressed to the rest of nodes. The regression problem for each node can be solved using the TDB-Lasso. Graphs estimated in this way will have neighborhoods of each node that are constant on a partition, but the graph as a whole changes more flexibly (Fig. 1b-d). t=1.00s Subject aa t=2.00s t=3.00s Figure 3: Brain interactions for the subject ’aa’ when presented with visual cues of the class 1 The graph structure estimation using the TDBLasso is demonstrated on a real dataset of electroencephalogram (EEG) measurements. We use the brain computer interface (BCI) dataset IVa from [11] in which the EEG data is collected from 5 subjects, who were given visual cues based on which they were required to imagine right hand or right foot for 3.5s. The measurement was performed when the visual cues were presented on the screen (280 times), intermitted by periods of random length in which the subject could relax. We use the down-sampled data at 100Hz. Fig. 3 gives a visualization of the brain interactions over the time of the experiment for the subject ’aa’ while presented with visual cues for the class 1 (right hand). Estimated graphs of interactions between different parts of the brain for other subjects and classes are given in Appendix due to the space limit. We also want to study whether the estimated time-varying network are discriminative features for classifying the type of imaginations in the EEG signal. For this purpose, we perform unsupervised clustering of EEG signals using the time-varying networks and study whether the grouping correspond to the true grouping according to imagination label. We estimate a time-varying GGM using the TDB-Lasso for each visual cue and cluster the graphs using the spectral K-means clustering [29] (using a linear kernel on the coefficients to measure similarity). Each cluster is labeled according to the majority of points it contains. Finally, each cue if classified based on labels of the time-points that it contains. Table 1 summarizes the classification accuracy for each subject based on K = 4 clusters (K was chosen as a cutoff point, when there was little decrease in K-means objective). We compare this approach to a case when GGMs with a static structure are estimated [5]. Note that the supervised classifiers with special EEG features are able to achieve much higher classification accuracy, however, our approach does not use any labeled data and can be seen as an exploratory step. We also used TDB-Lasso for estimating the time-varying gene networks from microarray data time series data, but due to space limit, results will be reported later in a biological paper. Table 1: Classification accuracies based on learned brain interactions. Subject aa al av aw ay TDB-Lasso 0.69 0.80 0.59 0.67 0.83 Static 0.58 0.63 0.54 0.57 0.61 8 Discussion We have developed the TDB-Lasso procedure, a novel approach for model selection and variable estimation in the varying-coefficient varying-structure models with piecewise constant functions. The VCVS models form a flexible nonparametric class of models that retain interpretability of parametric models. Due to their flexibility, important classical problems, such as linear regression with structural changes and change point detection, and some more recent problems, like structure estimation of varying graphical models, can be modeled within this class of models. The TDB-Lasso compares favorably to other commonly used [28] or latest [1] techniques for estimation in this class of models, which was demonstrated on the synthetic data. The model selection properties of the TDB-Lasso, demonstrated on the synthetic data, are also supported by the theoretical analysis. Furthermore, we demonstrate a way of applying the TDB-Lasso for graph estimation on a real dataset. Application of the TDB-Lasso procedure goes beyond the linear varying coefficient regression models. A direct extension is to generalized varying-coefficient models g(m(Xi, ti)) = X′ iβ(ti), i = 1, . . . , n, where g(·) is a given link function and m(Xi, ti) = E[Y |X = Xi, t = ti] is the conditional mean. Estimation in generalized varying-coefficient models proceeds by changing the squared loss in Eq. (7) and Eq. (8) to a different appropriate loss function. The generalized varyingcoefficient models can be used to estimate the time-varying structure of discrete Markov Random Fields, again by performing the neighborhood selection. 8 References [1] Amr Ahmed and Eric P. Xing. Tesla: Recovering time-varying networks of dependencies in social and biological studies. Proceeding of the National Academy of Science, 2009. [2] Francis R. Bach. Bolasso: model consistent lasso estimation through the bootstrap. In William W. Cohen, Andrew McCallum, and Sam T. Roweis, editors, ICML, volume 307 of ACM International Conference Proceeding Series, pages 33–40. ACM, 2008. [3] J Bai and P Perron. Computation and analysis of multiple structural change models. Journal of Applied Econometrics, (18):1–22, 2003. [4] Jushan Bai and Pierre Perron. Estimating and testing linear models with multiple structural changes. Econometrica, 66(1):47–78, January 1998. [5] O. Banerjee, L. El Ghaoui, and A. d’Aspremont. Model selection through sparse maximum likelihood estimation. J. Mach. Learn. Res., 9:485–516, 2008. [6] P. Bickel, Y. Ritov, and A. Tsybakov. Simultaneous analysis of lasso and dantzig selector. Ann. of Stat. [7] Florentina Bunea. Honest variable selection in linear and logistic regression models via ℓ1 and ℓ1 + ℓ2 penalization. Electronic Journal of Statistics, 2:1153, 2008. [8] Scott S. Chen, David L. Donoho, and Michael A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal on Scientific Computing, 20(1):33–61, 1999. [9] William S. Cleveland, Eric Grosse, and William M. Shyu. Local regression models. In John M. Chambers and Trevor J. Hastie, editors, Statistical Models in S, pages 309–376, 1991. [10] David L. Donoho, Michael Elad, and Vladimir N. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory, 52:6–18, 2006. [11] G. Dornhege, B. Blankertz, G. Curio, and K. M¨uller. Boosting bit rates in non-invasive EEG single-trial classifications by feature combination and multi-class paradigms. IEEE Trans. Biomed. Eng., 51:993– 1002, 2004. [12] Jianqing Fan and Qiwei Yao. Nonlinear Time Series: Nonparametric and Parametric Methods. (Springer Series in Statistics). Springer, August 2005. [13] Jianqing Fan and Wenyang Zhang. Statistical estimation in varying-coefficient models. The Annals of Statistics, 27:1491–1518, 2000. [14] Za¨ıd Harchaoui, Francis Bach, and ´Eric Moulines. Kernel change-point analysis. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing Systems 21. 2009. [15] Za¨ıd Harchaoui and C´eline Levy-Leduc. Catching change-points with lasso. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 617– 624. MIT Press, Cambridge, MA, 2008. [16] Trevor Hastie and Robert Tibshirani. Varying-coefficient models. Journal of the Royal Statistical Society. Series B (Methodological), 55(4):757–796, 1993. [17] Mladen Kolar, Le Song, and Eric Xing. Estimating time-varying networks. In arXiv:0812.5087, 2008. [18] Marc Lavielle and Eric Moulines. Least-squares estimation of an unknown number of shifts in a time series. Journal of Time Series Analysis, 21(1):33–59, 2000. [19] E. Lebarbier. Detecting multiple change-points in the mean of gaussian process by model selection. Signal Process., 85(4):717–736, 2005. [20] E. Mammen and S. van de Geer. Locally adaptive regression splines. Ann. of Stat., 25(1):387–413, 1997. [21] N. Meinshausen and P. B¨uhlmann. High-dimensional graphs and variable selection with the lasso. Annals of Statistics, 34:1436, 2006. [22] Nicolai Meinshausen and Peter B¨uhlmann. Stability selection. Preprint, 2008. [23] Alessandro Rinaldo. Properties and refinements of the fused lasso. Preprint, 2008. [24] Le Song, Mladen Kolar, and Eric P. Xing. Keller: Estimating time-evolving interactions between genes. In Proceedings of the 16th International Conference on Intelligent Systems for Molecular Biology, 2009. [25] Robert Tibshirani, Michael Saunders, Saharon Rosset, Ji Zhu, and Keith Knight. Sparsity and smoothness via the fused lasso. Journal Of The Royal Statistical Society Series B, 67(1):91–108, 2005. [26] S. A. van de Geer and P. Buhlmann. On the conditions used to prove oracle results for the lasso, 2009. [27] M. J. Wainwright. Sharp thresholds for high-dimensional and noisy recovery of sparsity. Preprint, 2006. [28] H. Wang and Y. Xia. Shrinkage estimation of the varying coefficient model. Manuscript, 2008. [29] H Zha, C Ding, M Gu, X He, and H Simon. Spectral relaxation for k-means clustering. pages 1057–1064. MIT Press, 2001. [30] P. Zhao and B. Yu. On model selection consistency of lasso. J. Mach. Learn. Res., 7:2541–2563, 2006. 9
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Dual Averaging Method for Regularized Stochastic Learning and Online Optimization Lin Xiao Microsoft Research, Redmond, WA 98052 lin.xiao@microsoft.com Abstract We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as ℓ1-norm for promoting sparsity. We develop a new online algorithm, the regularized dual averaging (RDA) method, that can explicitly exploit the regularization structure in an online setting. In particular, at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient. Computational experiments show that the RDA method can be very effective for sparse online learning with ℓ1-regularization. 1 Introduction In machine learning, online algorithms operate by repetitively drawing random examples, one at a time, and adjusting the learning variables using simple calculations that are usually based on the single example only. The low computational complexity (per iteration) of online algorithms is often associated with their slow convergence and low accuracy in solving the underlying optimization problems. As argued in [1, 2], the combined low complexity and low accuracy, together with other tradeoffs in statistical learning theory, still make online algorithms a favorite choice for solving largescale learning problems. Nevertheless, traditional online algorithms, such as stochastic gradient descent (SGD), has limited capability of exploiting problem structure in solving regularized learning problems. As a result, their low accuracy often makes it hard to obtain the desired regularization effects, e.g., sparsity under ℓ1-regularization. In this paper, we develop a new online algorithm, the regularized dual averaging (RDA) method, that can explicitly exploit the regularization structure in an online setting. We first describe the two types of problems addressed by the RDA method. 1.1 Regularized stochastic learning The regularized stochastic learning problems we consider are of the following form: minimize 푤 { 휙(푤) ≜E푧푓(푤, 푧) + Ψ(푤) } (1) where 푤∈R푛is the optimization variable (called weights in many learning problems), 푧= (푥, 푦) is an input-output pair drawn from an (unknown) underlying distribution, 푓(푤, 푧) is the loss function of using 푤and 푥to predict 푦, and Ψ(푤) is a regularization term. We assume 푓(푤, 푧) is convex in 푤 for each 푧, and Ψ(푤) is a closed convex function. Examples of the loss function 푓(푤, 푧) include: ∙Least-squares: 푥∈R푛, 푦∈R, and 푓(푤, (푥, 푦)) = (푦−푤푇푥)2. ∙Hinge loss: 푥∈R푛, 푦∈{+1, −1}, and 푓(푤, (푥, 푦)) = max{0, 1 −푦(푤푇푥)}. ∙Logistic regression: 푥∈R푛, 푦∈{+1, −1}, and 푓(푤, (푥, 푦))=log ( 1+ exp ( −푦(푤푇푥) )) . 1 Examples of the regularization term Ψ(푤) include: ∙ℓ1-regularization: Ψ(푤) = 휆∥푤∥1 with 휆> 0. With ℓ1-regularization, we hope to get a relatively sparse solution, i.e., with many entries of 푤being zeroes. ∙ℓ2-regularization: Ψ(푤) = (휎/2)∥푤∥2 2, for some 휎> 0. ∙Convex constraints: Ψ(푤) is the indicator function of a closed convex set 퐶, i.e., Ψ(푤) = 0 if 푤∈퐶and +∞otherwise. In this paper, we focus on online algorithms that process samples sequentially as they become available. Suppose at time 푡, we have the most up-to-date weight 푤푡. Whenever 푧푡is available, we can evaluate the loss 푓(푤푡, 푧푡), and a subgradient 푔푡∈∂푓(푤푡, 푧푡) (here ∂푓(푤, 푧) denotes the subdifferential of 푓with respect to 푤). Then we compute the new weight 푤푡+1 based on these information. For solving the problem (1), the standard stochastic gradient descent (SGD) method takes the form 푤푡+1 = 푤푡−훼푡(푔푡+ 휉푡) , (2) where 훼푡is an appropriate stepsize, and 휉푡is a subgradient of Ψ at 푤푡. The SGD method has been very popular in the machine learning community due to its capability of scaling with large data sets and good generalization performance observed in practice (e.g., [3, 4]). Nevertheless, a main drawback of the SGD method is its lack of capability in exploiting problem structure, especially for regularized learning problems. As a result, their low accuracy (compared with interior-point method for batch optimization) often makes it hard to obtain the desired regularization effect. An important example and motivation for this paper is ℓ1-regularized stochastic learning, where Ψ(푤) = 휆∥푤∥1. Even with relatively big 휆, the SGD method (2) usually does not generate sparse solutions because only in very rare cases two float numbers add up to zero. Various methods for rounding or truncating the solutions are proposed to generate sparse solutions (e.g., [5]). Inspired by recently developed first-order methods for optimizing composite functions [6, 7, 8], the regularized dual averaging (RDA) method we develop exploits the full regularization structure at each online iteration. In other words, at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the whole regularization term, not just its subgradients. For many practical learning problems, we actually are able to find a closed-form solution for the auxiliary optimization problem at each iteration. This means that the computational complexity per iteration is 푂(푛), the same as the SGD method. Moreover, the RDA method converges to the optimal solution of (1) with the optimal rate 푂(1/ √ 푡). If the the regularization function Ψ(푤) is strongly convex, we have the better rate 푂(ln 푡/푡) by setting appropriate parameters in the algorithm. 1.2 Regularized online optimization In online optimization (e.g., [9]), we make a sequence of decision 푤푡, for 푡= 1, 2, 3, . . .. At each time 푡, a previously unknown cost function 푓푡is revealed, and we encounter a loss 푓푡(푤푡). We assume that the functions 푓푡are convex for all 푡≥1. The goal of an online algorithm is to ensure that the total cost up to each time 푡, ∑푡 휏=1 푓푡(푤푡), is not much larger than min푤 ∑푡 휏=1 푓푡(푤), the smallest total cost of any fixed decision 푤from hindsight. The difference between these two cost is called the regret of the online algorithm. Applications of online optimization include online prediction of time series and sequential investment (e.g. [10]). In regularized online optimization, we add to each cost function a convex regularization function Ψ(푤). For any fixed decision variable 푤, consider the regret 푅푡(푤) ≜ 푡 ∑ 휏=1 ( 푓휏(푤휏) + Ψ(푤휏) ) − 푡 ∑ 휏=1 ( 푓휏(푤) + Ψ(푤) ) . (3) The RDA method we develop can also be used to solve the above regularized online optimization problem, and it has an 푂( √ 푡) regret bound. Again, if the regularization term Ψ(푤) is strongly convex, the regret bound is 푂(ln 푡). However, the main advantage of the RDA method, compared with other online algorithms, is its explicit regularization effect at each iteration. 2 Algorithm 1 Regularized dual averaging (RDA) method input: ∙a strongly convex function ℎ(푤) with modulus 1 on domΨ, and 푤0 ∈R푛, such that 푤0 = arg min 푤ℎ(푤) ∈Arg min 푤Ψ(푤). (4) ∙a pre-determined nonnegative and nondecreasing sequence 훽푡for 푡≥1. initialize: 푤1 = 푤0, ¯푔0 = 0. for 푡= 1, 2, 3, . . . do 1. Given the function 푓푡, compute a subgradient 푔푡∈∂푓푡(푤푡). 2. Update the average subgradient ¯푔푡: ¯푔푡= 푡−1 푡 ¯푔푡−1 + 1 푡푔푡 (5) 3. Compute the next iterate 푤푡+1: 푤푡+1 = arg min 푤 { ⟨¯푔푡, 푤⟩+ Ψ(푤) + 훽푡 푡ℎ(푤) } (6) end for 2 Regularized dual averaging method In this section, we present the generic RDA method (Algorithm 1) for solving regularized stochastic learning and online optimization problems, and give some concrete examples. To unify notation, we write 푓(푤, 푧푡) as 푓푡(푤) for stochastic learning problems. The RDA method uses an auxiliary strongly convex function ℎ(푤). A function ℎis called strongly convex with respect to a norm ∥⋅∥if there exists a constant 휎> 0 such that ℎ(훼푤+ (1 −훼)푢) ≤훼ℎ(푤) + (1 −훼)ℎ(푢) −휎 2 훼(1 −훼)∥푤−푢∥2, (7) for all 푤, 푢∈domℎ. The constant 휎is called the convexity parameter, or the modulus of strong convexity. In equation (4), Arg min푤Ψ(푤) denotes the convex set of minimizers of Ψ. In Algorithm 1, step 1 is to compute a subgradient of 푓푡at 푤푡, which is standard for all (sub)gradientbased methods. Step 2 is the online version of computing average gradient ¯푔푡(dual average). In step 3, we assume that the functions Ψ and ℎare simple, meaning that the minimization problem in (6) can be solved with litter effort, especially if we are able to find a closed-form solution for 푤푡+1. This assumption seems to be restrictive. But the following examples show that this indeed is the case for many important learning problems in practice. If the regularization function Ψ(푤) has convexity parameter 휎= 0 (i.e., it is not strongly convex), we can choose a parameter 훾> 0 and use the sequence 훽푡= 훾 √ 푡, 푡= 1, 2, 3, . . . (8) to obtain an 푂(1/ √ 푡) convergence rate for stochastic learning, or an 푂( √ 푡) regret bound for online optimization. The formal convergence theorems are given in Sections 3. Here are some examples: ∙Nesterov’s dual averaging method. Let Ψ(푤) be the indicator function of a close convex set 퐶. This recovers the method of [11]: 푤푡+1 = arg min푤∈퐶 { ⟨¯푔푡, 푤⟩+ (훾/ √ 푡)ℎ(푤) } . ∙ℓ1-regularization: Ψ(푤) = 휆∥푤∥1 for some 휆> 0. In this case, let 푤0 = 0 and ℎ(푤) = 1 2∥푤∥2 2 + 휌∥푤∥1, where 휌≥0 is a sparsity enhancing parameter. The solution to (6) can be found as 푤(푖) 푡+1 = ⎧ ⎨ ⎩ 0 if ¯푔(푖) 푡 ≤휆RDA 푡 , − √ 푡 훾 ( ¯푔(푖) 푡 −휆RDA 푡 sign ( ¯푔(푖) 푡 )) otherwise, 푖= 1, . . . , 푛, (9) where 휆RDA 푡 = 휆+ 휌/ √ 푡. Notice that the truncating threshold 휆푡is at least as large as 휆. This is the main difference of our method from related work, see Section 4. 3 If the regularization function Ψ(푤) has convexity parameter 휎> 0, we can use any nonnegative, nondecreasing sequence {훽푡}푡≥1 that is dominated by ln 푡, to obtain an 푂(ln 푡/ √ 푡) convergence rate for stochastic learning, or an 푂(ln 푡) regret bound for online optimization (see Section 3). For simplicity, in the following examples, we use 훽푡= 0 for all 푡≥1, and we do not need ℎ(푤). ∙Mixed ℓ1/ℓ2 2-regularization. Let Ψ(푤) = 휆∥푤∥1 + (휎/2)∥푤∥2 2 with 휆, 휎> 0. Then 푤(푖) 푡+1 = ⎧ ⎨ ⎩ 0 if ¯푔(푖) 푡 ≤휆, −1 휎 ( ¯푔(푖) 푡 −휆sign ( ¯푔(푖) 푡 )) otherwise, 푖= 1, . . . , 푛. Of course, setting 휆= 0 gives the algorithm for pure ℓ2 2-regularization. ∙Kullback-Leibler (KL) divergence regularization: Ψ(푤) = 휎퐷KL(푤∥푝), where 푤lies in the standard simplex, 푝is a given probability distribution, and 퐷KL(푤∥푝) ≜ 푛 ∑ 푖=1 푤(푖) ln (푤(푖) 푝(푖) ) . Note that 퐷KL(푤∥푝) is strongly convex with respect to ∥푤∥1 with modulus 1 (e.g., [12]). In this case, 푤(푖) 푡+1 = 1 푍푡+1 푝(푖) exp ( −1 휎¯푔(푖) 푡 ) , where 푍푡+1 is a normalization parameter such that ∑푛 푖=1 푤(푖) 푡+1 = 1. 3 Regret bounds and convergence rates We first give bounds on the regret 푅푡(푤) defined in (3), when the RDA method is used for solving regularized online optimization problem. To simplify notations, we define the following sequence: Δ푡≜(훽0 −훽1)ℎ(푤2) + 훽푡퐷2 + 퐿2 2 푡−1 ∑ 휏=0 1 휎휏+ 훽휏 , 푡= 1, 2, 3, . . . , (10) where 퐷and 퐿are some given constants, 휎is the convexity parameter of the regularization function Ψ(푤), and {훽휏}푡 휏=1 is the input sequence to the RDA method, which is nonnegative and nondecreasing. Notice that we just introduced an extra parameter 훽0. We require 훽0 > 0 to avoid blowup of the first term (when 휏= 0) in the summation in (10). This parameter does not appear in Algorithm 1, instead, it is solely for the convenience of convergence analysis. In fact, whenever 훽1 > 0, we can set 훽0 = 훽1, so that the term (훽0 −훽1)ℎ(푤2) vanishes. We also note that 푤2 is determined at the end of the step 푡= 1, so Δ1 is well defined. Finally, for any given constant 퐷> 0, we define ℱ퐷≜ { 푤∈domΨ ℎ(푤) ≤퐷2} . Theorem 1 Let the sequences {푤휏}푡 휏=1 and {푔휏}푡 휏=1 be generated by Algorithm 1. Assume there is a constant 퐿such that ∥푔푡∥∗≤퐿for all 푡≥1, where ∥⋅∥∗is the dual norm of ∥⋅∥. Then for any 푡≥1 and any 푤∈ℱ퐷, we have 푅푡(푤) ≤Δ푡. (11) The proof of this theorem is given in the longer version of this paper [13]. Here we give some direct consequences based on concrete choices of algorithmic parameters. If the regularization function Ψ(푤) has convexity parameter 휎= 0, then the sequence {훽푡}푡≥1 defined in (8) together with 훽0 = 훽1 lead to Δ푡= 훾 √ 푡퐷2 + 퐿2 2훾 ( 1 + 푡−1 ∑ 휏=1 1 √휏 ) ≤훾 √ 푡퐷2 + 퐿2 2훾 ( 1 + ( 2 √ 푡−2 )) ≤ ( 훾퐷2 + 퐿2 훾 ) √ 푡. The best 훾that minimizes the above bound is 훾★= 퐿/퐷, which leads to 푅푡(푤) ≤2퐿퐷 √ 푡. (12) 4 If the regularization function Ψ(푤) is strongly convex, i.e., with a convexity parameter 휎> 0, then any nonnegative, nondecreasing sequence that is dominated by ln 푡will give an 푂(ln 푡) regret bound. We can simply choose ℎ(푤) = (1/휎)Ψ(푤) whenever needed. Here are several possibities: ∙Positive constant sequences. For simplicity, let 훽푡= 휎for 푡≥1 and 훽0 = 훽1. In this case, Δ푡= 휎퐷2 + 퐿2 2휎 푡 ∑ 휏=1 1 휏≤휎퐷2 + 퐿2 2휎(1 + ln 푡). ∙The logrithmic sequence. Let 훽푡= 휎(1 + ln 푡) for 푡≥1, and 훽0 = 휎. In this case, Δ푡= 휎(1 + ln 푡)퐷2 + 퐿2 2휎 ( 1 + 푡−1 ∑ 휏=1 1 휏+ 1 + ln 휏 ) ≤ ( 휎퐷2 + 퐿2 2휎 ) (1 + ln 푡). ∙The zero sequence 훽푡= 0 for 푡≥1, with 훽0 = 휎. Using ℎ(푤) = (1/휎)Ψ(푤), we have Δ푡≤Ψ(푤2) + 퐿2 2휎 ( 1 + 푡 ∑ 휏=1 1 휏 ) ≤퐿2 2휎(6 + ln 푡), where we used Ψ(푤2) ≤2퐿2/휎, as proved in [13]. This bound does not depend on 퐷. When Algorithm 1 is used to solve regularized stochastic learning problems, we have the following: Theorem 2 Assume there exists an optimal solution 푤★to the problem (1) that satisfies ℎ(푤★) ≤퐷2 for some 퐷> 0, and there is an 퐿> 0 such that E ∥푔∥2 ∗≤퐿2 for all 푔∈∂푓(푤, 푧) and 푤∈domΨ. Then for any 푡≥1, we have E 휙( ¯푤푡) −휙(푤★) ≤Δ푡 푡, where ¯푤푡= 1 푡 푡 ∑ 휏=1 푤휏. The proof of Theorem 2 is given in [13]. Further analysis for the cases 휎= 0 and 휎> 0 are the same as before. We only need to divide every regret bound by 푡to obtain the convergence rate. 4 Related work There have been several recent work that address online algorithms for regularized learning problems, especially with ℓ1-regularization; see, e.g., [14, 15, 16, 5, 17]. In particular, a forwardbackward splitting method (FOBOS) is studied in [17] for solving the same problems we consider. In an online setting, each iteration of the FOBOS method can be written as 푤푡+1 = arg min 푤 {1 2 ∥푤−(푤푡−훼푡푔푡)∥2 + 훼푡Ψ(푤) } , (13) where 훼푡is set to be 푂(1/ √ 푡) if Ψ(푤) has convexity parameter 휎= 0, and 푂(1/푡) if 휎> 0. The RDA method and FOBOS use very different weights on the regularization term Ψ(푤): RDA in (6) uses the original Ψ(푤) without any scaling, while FOBOS scales Ψ(푤) by a diminishing stepsize 훼푡. The difference is more clear in the special case of ℓ1-regularization, i.e., when Ψ(푤) = 휆∥푤∥1. For this purpose, we consider the Truncated Gradient (TG) method proposed in [5]. The TG method truncates the solutions obtained by the standard SGD method with an integer period 퐾≥1. More specifically, each component of 푤푡is updated as 푤(푖) 푡+1 = { trnc ( 푤(푖) 푡 −훼푡푔(푖) 푡, 휆TG 푡 , 휃 ) if mod(푡, 퐾) = 0, 푤(푖) 푡 −훼푡푔(푖) 푡 otherwise. (14) where 휆TG 푡 = 훼푡휆퐾, the function mod(푡, 퐾) means the remainder on division of 푡by 퐾, and trnc(휔, 휆TG 푡 , 휃) = ⎧ ⎨ ⎩ 0 if ∣휔∣≤휆TG 푡 , 휔−휆TG 푡 sign(휔) if 휆TG 푡 < ∣휔∣≤휃, 휔 if ∣휔∣> 휃. When 퐾= 1 and 휃= +∞, the TG method is the same as the FOBOS method (13). Now comparing the truncation thresholds 휆TG 푡 and 휆RDA 푡 used in (9): with 훼푡= 푂(1/ √ 푡), we have 휆TG 푡 = 푂(1/ √ 푡)휆RDA 푡 . Therefore, the RDA method can generate much more sparse solutions. This is confirmed by our computational experiments in Section 5. 5 휆= 0.01 휆= 0.03 휆= 0.1 휆= 0.3 휆= 1 휆= 3 휆= 10 SGD TG RDA IPM SGD TG RDA 푤푇 푤푇 푤푇 푤★ ¯푤푇 ¯푤푇 ¯푤푇 Figure 1: Sparsity patterns of the weight 푤푇and the average weight ¯푤푇for classifying the digits 6 and 7 when varying the regularization parameter 휆from 0.01 to 10. The background gray represents the value zero, bright spots represent positive values and dark spots represent negative values. 5 Computational experiments We provide computational experiments for the ℓ1-RDA method using the MNIST dataset of handwritten digits [18]. Each image from the dataset is represented by a 28 × 28 gray-scale pixel-map, for a total of 784 features. Each of the 10 digits has roughly 6,000 training examples and 1,000 testing examples. No preprocessing of the data is employed. We use ℓ1-regularized logistic regression to do binary classification on each of the 45 pairs of digits. In the experiments, we compare the ℓ1-RDA method (9) with the SGD method (2) and the TG/FOBOS method (14) with 휃= ∞. These three online algorithms have similar convergence rate and the same order of computational complexity per iteration. We also compare them with the batch optimization approach, using an efficient interior-point method (IPM) developed by [19]. Each pair of digits have about 12,000 training examples and 2,000 testing examples. We use online algorithms to go through the (randomly permuted) data only once, therefore the algorithms stop at 푇= 12,000. We vary the regularization parameter 휆from 0.01 to 10. As a reference, the maximum 휆for the batch optimization case [19] is mostly in the range of 30−50 (beyond which the optimal weights are all zeros). In the ℓ1-RDA method (9), we use 훾= 5,000, and set 휌= 0 for basic regularization, or 휌= 0.005 (effectively 훾휌= 25) for enhanced regularization effect. The tradeoffs in choosing these parameters are further investigated in [13]. For the SGD and TG methods, we use a constant stepsize 훼= (1/훾) √ 2/푇. When 훾= 퐿/퐷, which gives the best convergence bound (12) 6 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 0 2000 4000 6000 8000 10000 12000 0 100 200 300 400 500 600 SGD SGD TG TG RDA RDA Number of samples 푡 Number of samples 푡 NNZs when 휆=0.1 NNZs when 휆=10 Left: 퐾= 1 for TG, 휌= 0 for RDA Right: 퐾= 10 for TG, 훾휌= 25 for RDA Figure 2: Number of non-zeros (NNZs) in 푤(푡) for the three online algorithms (classifying 6 and 7). for the RDA method, the corresponding 훼= (퐷/퐿) √ 2/푇also gives the best convergence rate for the SGD method (e.g., [20]). In the TG method, the truncation period is set to 퐾= 1 for basic regularization, or 퐾= 10 for enhanced regularization effect, as suggested in [5]. Figure 1 shows the sparsity patterns of the solutions 푤푇and ¯푤푇for classifying the digits 6 and 7. Both the TG and RDA methods were run with parameters for enhanced ℓ1-regularization: 퐾= 10 for TG and 훾휌= 25 for RDA. The sparsity patterns obtained by the RDA method are most close to the batch optimization results solved by IPM, especially for larger 휆. Figure 2 plots the number of non-zeros (NNZs) in 푤(푡) for different online algorithms. Only the RDA method and TG with 퐾= 1 give explicit zero weights at every step. In order to count the NNZs in all other cases, we set a small threshold for rounding the weights to zero. Considering that the magnitudes of the largest weights in Figure 1 are mostly on the order of 10−3, we set 10−5 as the threshold and verified that rounding elements less than 10−5 to zero does not affect the testing errors. Note that we do not truncate the weights for RDA and TG with 퐾= 1 further, even if some of their components are below 10−5. It can be seen that the RDA method maintains a much more sparse 푤(푡) than the other two online algorithms. While the TG method generate more sparse solutions than the SGD method when 휆is large, the NNZs in 푤(푡) oscillates with a very big range. In contrast, the RDA method demonstrate a much more smooth variation in the NNZs. Figure 3 illustrates the tradeoffs between sparsity and testing error rates for classifying 6 and 7. Since the performance of the online algorithms vary when the training data are given in different permutations, we run them on 100 randomly permuted sequences of the same training set, and plot the means and standard deviations shown as error bars. For the SGD and TG methods, the testing error rates of 푤푇vary a lot for different random sequences. In contrast, the RDA method demonstrates very robust performance (small standard deviations) for 푤푇, even though the theorems only give performance bound for the averaged weight ¯푤푇. Note that ¯푤푇obtained by SGD and TG have much smaller error rates than those of RDA and batch optimization, especially for larger 휆. The explanation is that these lower error rates are obtained with much more nonzero features. Figure 4 shows summary of classification results for all the 45 pairs of digits. For clarity of presentation, here we only plot results of the ℓ1-RDA method and batch optimization using IPM. (The NNZs obtained by SGD and TG are mostly above the limit of the vertical axes, which is set at 200). We see that, overall, the solutions obtained by the ℓ1-RDA method demonstrate very similar tradeoffs between sparsity and testing error rates as rendered by the batch optimization solutions. 7 0.01 0.1 1 10 0.1 1 10 0.01 0.1 1 10 0 1 2 3 4 0.01 0.1 1 10 0 200 400 600 0.01 0.1 1 10 0 200 400 600 0.01 0.1 1 10 0.1 1 10 0.01 0.1 1 10 0 1 2 3 4 0.01 0.1 1 10 0 200 400 600 0.01 0.1 1 10 0 200 400 600 휆 휆 휆 휆 Error rates (%) NNZs SGD SGD SGD SGD TG (퐾=1) TG (퐾=1) RDA (휌= 0) RDA (휌= 0) TG (퐾=10) TG (퐾=10) RDA (훾휌=25) RDA (훾휌=25) IPM IPM IPM IPM Last weight 푤푇 Last weight 푤푇 Average weight ¯푤푇 Average weight ¯푤푇 Figure 3: Tradeoffs between testing error rates and NNZs in solutions (for classifying 6 and 7). 0.1 1 10 0 0.5 1 0.1 1 10 0 2 4 0.1 1 10 0 2 0.1 1 10 0 0.5 1 0.1 1 10 0 5 0.1 1 10 0 2 4 0.1 1 10 0 2 0.1 1 10 0 2 0.1 1 10 0 1 2 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 2 4 0.1 1 10 0 1 2 0.1 1 10 0 5 0.1 1 10 0 1 2 0.1 1 10 0 2 4 0.1 1 10 0 5 0.1 1 10 0 2 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 10 0.1 1 10 0 5 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 10 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 10 0.1 1 10 0 5 0 50 100 150 200 0.1 1 10 0 5 10 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 10 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 10 0.1 1 10 0 5 10 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 5 0.1 1 10 0 5 0 50 100 150 200 0.1 1 10 0 5 0.1 1 10 0 5 10 0 50 100 150 200 0.1 1 10 0 5 10 0 50 100 150 200 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Figure 4: Binary classification for all 45 pairs of digits. The images in the lower-left triangular area show sparsity patterns of 푤푇with 휆= 1, obtained by the ℓ1-RDA with 훾휌= 25. The plots in the upper-right triangular area show tradeoffs between sparsity and testing error rates, by varying 휆 from 0.1 to 10. The solid circles and solid squares show error rates and NNZs in 푤푇, respectively, using IPM for batch optimization. The hollow circles and hollow squares show error rates and NNZs of 푤푇, respectively, using the ℓ1-RDA method. The vertical bars centered at hollow circles and squares show standard deviations by running on 100 random permutations of the training data. 8 References [1] L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 161–168. MIT Press, Cambridge, MA, 2008. [2] S. Shalev-Shwartz and N. Srebro. SVM optimization: Inverse dependence on training set size. In Proceedings of the 25th International Conference on Machine Learning (ICML), 2008. [3] L. Bottou and Y. LeCun. Large scale online learning. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, Advances in Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004. [4] T. Zhang. Solving large scale linear prediction problems using stochastic gradient descent algorithms. In Proceedings of the 21st International Conference on Machine Learning (ICML), Banff, Alberta, Canada, 2004. [5] J. Langford, L. Li, and T. Zhang. Sparse online learning via truncated gradient. Journal of Machine Learning Research, 10:777–801, 2009. [6] Yu. Nesterov. Gradient methods for minimizing composiite objective function. CORE Discussion Paper 2007/76, Catholic University of Louvain, Center for Operations Research and Econometrics, 2007. [7] P. Tseng. On accelerated proximal gradient methods for convex-concave optimization. Submitted to SIAM Journal on Optimization, 2008. [8] A. Beck and M. Teboulle. A fast iterative shrinkage-threshold algorithm for linear inverse problems. Technical report, Technion, 2008. To appear in SIAM Journal on Image Sciences. [9] M. Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th International Conference on Machine Learning (ICML), pages 928– 936, Washington DC, 2003. [10] N. Cesa-Bianchi and G. Lugosi. Predictioin, Learning, and Games. Cambridge University Press, 2006. [11] Yu. Nesterov. Primal-dual subgradient methods for convex problems. Mathematical Programming, 120(1):221–259, 2009. Appeared early as CORE discussion paper 2005/67, Catholic University of Louvain, Center for Operations Research and Econometrics. [12] Yu. Nesterov. Smooth minimization of nonsmooth functions. Mathematical Programming, 103:127–152, 2005. [13] L. Xiao. Dual averaging method for regularized stochastic learning and online optimization. Technical Report MSR-TR-2009-100, Microsoft Research, 2009. [14] J. Duchi, S. Shalev-Shwartz, Y. Singer, and T. Chandra. Efficient projections onto the ℓ1ball for learning in high dimensions. In Proceedings of the 25th International Conference on Machine Learning (ICML), pages 272–279, 2008. [15] P. Carbonetto, M. Schmidt, and N. De Freitas. An interior-point stochastic approximation method and an 푙1-regularized delta rule. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing Systems 21, pages 233–240. MIT Press, 2009. [16] S. Balakrishnan and D. Madigan. Algorithms for sparse linear classifiers in the massive data setting. Journal of Machine Learning Research, 9:313–337, 2008. [17] J. Duchi and Y. Singer. Efficient learning using forward-backward splitting. In Proceedings of Neural Information Processing Systems, December 2009. [18] 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. Dataset available at http://yann.lecun.com/exdb/mnist. [19] K. Koh, S.-J. Kim, and S. Boyd. An interior-point method for large-scale ℓ1-regularized logistic regression. Journal of Machine Learning Research, 8:1519–1555, 2007. [20] A. Nemirovski, A. Juditsky, G. Lan, and A. Shapiro. Robust stochastic approximation approach to stochastic programming. SIAM Journal on Optimization, 19(4):1574–1609, 2009. 9
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Analysis of SVM with Indefinite Kernels Yiming Ying† , Colin Campbell† and Mark Girolami‡ †Department of Engineering Mathematics, University of Bristol, Bristol BS8 1TR, United Kingdom ‡Department of Computer Science, University of Glasgow, S.A.W. Building, G12 8QQ, United Kingdom Abstract The recent introduction of indefinite SVM by Luss and d’Aspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indefinite kernel matrix and the proxy positive semi-definite one. Our elementary result greatly facilitates the application of gradient-based algorithms. Based on our analysis, we further develop Nesterov’s smooth optimization approach [17, 18] for indefinite SVM which has an optimal convergence rate for smooth problems. Experiments on various benchmark datasets validate our analysis and demonstrate the efficiency of our proposed algorithms. 1 Introduction Kernel methods [5, 24] such as Support Vector Machines (SVM) have recently attracted much attention due to their good generalization performance and appealing optimization approaches. The basic idea of kernel methods is to map the data into a high dimensional (even infinite-dimensional) feature space through a kernel function. The kernel function over samples forms a similarity kernel matrix which is usually required to be positive semi-definite (PSD). The PSD property of the similarity matrix ensures that the SVM can be efficiently solved by a convex quadratic programming. However, many potential kernel matrices could be non-positive semi-definite. Such cases are quite common in applications such as the sigmoid kernel [14] for various values of the hyper-parameters, hyperbolic tangent kernels [25], and the protein sequence similarity measures derived from SmithWaterman and BLAST score [23]. The problem of learning with a non-PSD similarity matrix (indefinite kernel) has recently attracted considerable attention [4, 8, 9, 14, 20, 21, 26]. One widely used method is to convert the indefinite kernel matrix into a PSD one by using the spectrum transformation. The denoise method neglects the negative eigenvalues [8, 21], flip [8] takes the absolute value of all eigenvalues, shift [22] shifts eigenvalues to be positive by adding a positive constant, and the diffusion method [11] takes the exponentials of eigenvalues. One can also see [26] for a detailed coverage. However, useful information in the data could be lost in the above spectral transformations since they are separated from the process of training classifiers. In [9], the classification problem with indefinite kernels is regarded as the minimization of the distance between convex hulls in the pseudo-Euclidean space. In [20], general Reproducing Kernel Kreˇın spaces (RKKS) with indefinite kernels are introduced which allows a general representer theorem and regularization formulations. Luss and d’Aspremont [15] recently proposed a regularized formulation for SVM classification with indefinite kernel. Training a SVM with an indefinite kernel was viewed as a learning the kernel 1 matrix problem [13] i.e. learning a proxy PSD kernel matrix to approximate the indefinite one. Without realizing that the objective function is differentiable, the authors quadratically smoothed the objective function, and then formulated two approximate algorithms including the projected gradient method and the analytic center cutting plane method. In this paper we follow the formulation of SVM with indefinite kernels proposed in [15]. We mainly establish the differentiability of the objective function (see its precise definition in equation (3)) and prove that it is, indeed, differentiable with Lipschitz continuous gradient. This elementary result suggests there is no need to smooth the objective function which greatly facilitates the application of gradient-based algorithms. The main idea behind our analysis is from its saddle (min-max) representation which involves a penalty term in the form of Frobenius norm of matrices, measuring the distance between the indefinite kernel matrix and the proxy PSD one. This penalty term can be regarded as a Moreau-Yosida regularization term [12] to smooth out the objective function. The paper is organized as follows. In Section 2, we review the formulation of indefinite SVM classification presented in [15]. Our main contribution is outlined in Section 3. There, we first show that the objective function of interest is continuously differentiable and its gradient function can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. Based on our analysis, in Section 4 we propose a simplified formulation of the projected gradient method presented in [15] and show that it has a convergence rate of O(1/k) where k is the iteration number. We further develop Nesterov’s smooth optimization approach [17, 18] for indefinite SVM which has an optimal convergence rate of O(1/k2) for smooth problems. In Section 5, our analysis and proposed optimization approaches are validated by experiments on various benchmark data sets. 2 Indefinite SVM Classification In this section we review the regularized formulation of indefinite SVM presented in [15]. To this end, we introduce some notation. Let Nn = {1, 2, . . . , n} for any n ∈N and Sn be the space of all n × n symmetric matrices. If A ∈Sn is positive semi-definite, we write it as A ⪰0. The cone of PSD matrices is denoted by Sn +. For any A, B ∈Rn×n, ⟨A, B⟩F := Tr(A⊤B) where Tr(·) denotes the trace of a matrix. Finally, the Frobenius norm over the vector space Sn is denoted, for any A ∈Sn, by ∥A∥F := (Tr(A⊤A)) 1 2 . The standard Euclidean norm and inner product are respectively denoted by ∥· ∥and ⟨·, ·⟩. Let a set of training samples be given by inputs x = {xi ∈Rd : i ∈Nn} and outputs y = {yi ∈ {±1} : i ∈Nn}. Suppose that K is a positive semi-definite kernel matrix (proxy kernel matrix) on inputs x. Let matrix Y = diag(y), vector e be an n-dimensional vector of all ones and C be a positive trade-off parameter. Then, the dual formulation of 1-norm soft margin SVM [5, 24] is given by maxα α⊤e −1 2α⊤Y KY α s.t. α⊤y = 0, 0 ≤α ≤C. Since we assume that K is positive semi-definite, the above problem is a standard convex quadratic program [2] and a global solution can be efficiently obtained by, e.g., the primal-dual interior method. Suppose now we are only given an indefinite kernel matrix K0 ∈Sn. Luss and d’Aspremont [15] proposed the following max-min approach to simultaneously learn a proxy PSD kernel matrix K for the indefinite matrix K0 and the SVM classification: minK maxα α⊤e −1 2α⊤Y KY α + ρ∥K −K0∥2 F s.t. α⊤y = 0, 0 ≤α ≤C, K ⪰0. (1) Let Q1 = {α ∈Rn : α⊤y = 0, 0 ≤α ≤C} and L(α, K) = α⊤e −1 2α⊤Y KY α + ρ∥K −K0∥2 F . By the min-max theorem [2], problem (1) is equivalent to max α∈Q1 min K∈Sn + L(α, K). (2) For simplicity, we refer to the following function defined by f(α) = min K∈Sn + L(α, K) (3) as the objective function. It is obviously concave since f is the minimum of a sequence of concave functions. We also call the associated function L(α, K) the saddle representation of the objective function f. 2 For fixed α ∈Q1, the optimization K(α) = arg minK⪰0 L(α, K) is equivalent to a projection to the semi-definite cone Sn +. Indeed, it was shown in [15] that the optimal solution is given by K(α) = (K0 + Y αα⊤Y/(4ρ))+ (4) where, for any matrix A ∈Sn, the notation A+ denotes the positive part of A by simply setting its negative eigenvalues to zero. The optimal solution (α∗, K∗) ∈Q1 × Sn + to the above min-max problem is a saddle point of L(α, K) (see e.g. [2]), i.e. for any α ∈Q1, K ∈Sn + there holds L(α, K∗) ≤L(α∗, K∗) ≤L(α∗, K). For a matrix A ∈Sn, denote its maximum eigenvalue by λmax(A). The next lemma tells us that the optimal solution K∗belongs to a bounded domain in Sn +. Lemma 1. Problem (2) is equivalent to the formulation maxα∈Q1 minK∈Q2 L(α, K) and the objective function can be defined by f(α) = min K∈Q2 L(α, K) (5) where Q2 := n K ∈Sn + : λmax(K) ≤λmax(K0) + nC2 4ρ o . Proof. By the saddle theorem [2], we have L(α∗, K∗) = minK∈Q2 L(α∗, K). Combining this with equation (4) yields that K∗= K(α∗) = (K0 + Y α∗(α∗)⊤Y/(4ρ))+. We can easily see λmax(K∗) ≤λmax(K0 + Y α∗(α∗)⊤Y/(4ρ) ≤λmax(K0) + λmax(Y α∗(α∗)⊤Y/(4ρ) ¢ ≤ λmax(K0) + ∥α∗∥2 4ρ , where the second to last inequality uses the property of maximum eigenvalues (e.g. [10, Page 201]) i.e. λmax(A + B) ≤λmax(A) + λmax(B) for any A, B ∈Sn. Note that 0 ≤α∗≤C, ∥α∗∥2 ≤nC2. Combining this with the above inequality yields the desired lemma. It is worthy of mentioning that it was shown in [18, Theorem 1] that a function g has a Lipschitz continuous gradient if it enjoys a special structure: g(α) = min{⟨Aα, K⟩+ γd(K) : K ∈Q} where Q is a closed convex subset in a certain vector space and d(·) is a strongly convex function, and, most importantly, A is a linear operator. Since the variable α appeared in a quadratic form, i.e. α⊤Y KY α, in the objective function defined by (5), it can not be written as the above special form, and hence the theorem there can not be applied to our case. 3 Differentiability of the Objective Function The following lemma outlines a very useful characterization of differentiable properties of the optimal value function [3, Theorem 4.1], essentially due to Danskin [7]. Lemma 2. Let X be a metric space and U be a normed space. Suppose that for all x ∈X the function L(α, ·) is differentiable, L(α, x) and ∂αL(α, x), the derivative of L(·, x), are continuous on X × U and let Q be a compact subset of X. Define the optimal value function as f(α) = infx∈Q L(α, x). The optimal value function is directionally differentiable. Furthermore, if for α ∈ U, L(α, ·) has a unique minimizer x(α) over Q then f is differentiable at α and the gradient of f is given by ∇f(α) = ∂αL(α, x(α)). Applying the above lemma to the objective function f defined by equation (5), we have: Theorem 1. The objective function f defined by (3) (equivalently by (5)) is differentiable and its gradient is given by ∇f(α) = e −Y (K0 + Y αα⊤Y/(4ρ))+Y α. (6) Proof. We apply Lemma 2 with X = Sn and Q = Q2 ⊆Sn, U = Q1 and x = K. To this end, we first prove the uniqueness of K(α). Suppose there are two minimizers K1, K2 for problem arg minK∈Sn + L(α, K). By the first order optimality condition, for the minimizer K1, we have that ⟨∂KL(α, K1), K2 −K1⟩F ≥0. Considering the minimizer K2, we also have ⟨∂KL(α, K2), K1 − K2⟩F ≥0. Noting that ∂KL(α, K) = −1 2Y αα⊤Y + 2ρ(K −K0) and adding the above two firstorder optimaility inequalities together, we have −∥K2−K1∥2 F ≥0 which means that K1 = K2, and hence completes the proof of the uniqueness of K(α). Now the desired result follows directly from Lemma 2 by noting that the derivative of L w.r.t. the first argument ∂αL(α, K) = e −Y KY α. 3 Indeed, we can go further to establish the Lipschitz continuity of ∇f based on the strongly convex property of L(α, ·). To this end, we first establish a useful lemma. Lemma 3. For any α1, α2 ∈ Q1, there holds ∥(K0 + Y α1α⊤ 1 Y/(4ρ))+ −(K0 + Y α2α⊤ 2 Y/(4ρ))+∥F ≤(∥α1∥+ ∥α2∥)∥α1 −α2∥/(4ρ). Proof. Let ∂KL(α, ·) denote the gradient w.r.t. K. Now, consider the minimization problem arg minK∈Q2 L(α, K). By the first order optimality conditions, for any K ∈Q2 there holds ⟨∂KL(α, K(α)), K −K(α)⟩F ≥0. Applying the above inequality twice implies that ⟨∂KL(α1, K(α1)), K(α2) −K(α1)⟩F ≥0, and ⟨∂KL(α2, K(α2)), K(α1) −K(α2)⟩F ≥0. Consequently, ⟨∂KL(α1, K(α1)) −∂KL(α2, K(α2)), K(α2) −K(α1)⟩F ≥0. Substituting the fact that ∂KL(α, K) = −1 2Y αα⊤Y + 2ρ(K −K0) into the above equation, we have 4ρ∥K(α1) − K(α2)∥2 F ≤⟨Y (α2α⊤ 2 −α1α⊤ 1 )Y, K(α2) −K(α1)⟩F ≤∥Y (α2α⊤ 2 −α1α⊤ 1 )Y ∥F ∥K(α2) − K(α1)∥F . Consequently, ∥K(α1) −K(α2)∥F ≤∥Y (α2α⊤ 2 −α1α⊤ 1 )Y ∥F 4ρ ≤∥(α2α⊤ 2 −α1α⊤ 1 )∥F 4ρ (7) where the last inequality follows from the fact that Y is an orthonormal matrix since yi ∈{±1} and Y = diag(y1, . . . , yn). Note that ∥α2α⊤ 2 −α1α⊤ 1 ∥F = ∥(α2 −α1)α⊤ 2 −α1(α1 −α2)⊤∥F ≤ (∥α1∥+∥α2∥)∥α1−α2∥. Putting this back into inequality (7) completes the proof of the lemma. It is interesting to point out that the above lemma can be alternatively established by delicate techniques in matrix analysis. To see this, recall that a spectral function G : Sn →Sn is defined by applying a real-valued function g to the eigenvalues of its argument i.e. for any K ∈Sn with eigen-decomposition K = Udiag(λ1, . . . , λn)U ⊤, G(K) := Udiag(g(λ1), . . . , g(λn))U ⊤. The perturbation inequality in matrix analysis [1, Lemma VII.5.5] shows that if g is Lipschitz continuous with Lipschitz constant κ then ∥G(K1) −G(K2)∥F ≤κ∥K1 −K2∥F , ∀K1, K2 ∈Sn. Applying the above inequality with g(t) = max(0, t) and K1 = K0 + Y α1α⊤ 1 Y/(4ρ) and K2 = K0 + Y α2α⊤ 2 Y/(4ρ) implies equation (7), and hence Lemma 3. However, we prefer the original proof presented for Lemma 3 since it explains more clearly how the strong convexity of the regularization term ∥K −K0∥2 F plays a critical role in the analysis. From the above lemma, we can establish the Lipschitz continuity of the gradient of the objective function. Theorem 2. The gradient of the objective function given by (6) is Lipschitz continuous with Lipschitz constant L = λmax(K0)+ nC2 ρ i.e. for any α1, α2 ∈Q1 the following inequality holds ∥∇f(α1)− ∇f(α2)∥≤ £ λmax(K0)) + nC2/ρ ¤ ∥α1 −α2∥. Proof. For any α1, α2 ∈Q1, from representation of ∇f in Theorem 1 the term ∥∇f(α1)−∇f(α2)∥ can be bounded by n ∥Y £ (K0 + Y α1α⊤ 1 Y/(4ρ))+ −(K0 + Y α2α⊤ 2 Y/(4ρ))+ ¤ Y α1∥ o + n ∥Y (K0 + Y α2α⊤ 2 Y/(4ρ))+Y (α2 −α1)∥ o . (8) Now it remains to estimate the two terms within parentheses on the right-hand side of inequality (8). Let’s begin with the first one by applying Lemma 3. ∥Y ¡ (K0 + Y α1α⊤ 1 Y/(4ρ))+ −(K0 + Y α2α⊤ 2 Y/(4ρ))+ ¢ Y α1∥ ≤∥Y ¡ (K0 + Y α1α⊤ 1 Y/(4ρ))+ −(K0 + Y α2α⊤ 2 Y/(4ρ))+ ¢ Y ∥F ∥α1∥ ≤∥ ¡ K0 + Y α1α⊤ 1 Y/(4ρ) ¢ + − ¡ K0 + Y α2α⊤ 2 Y/(4ρ) ¢ +∥F ∥α1∥ ≤∥α1∥(∥α1∥+ ∥α2∥) ∥α1 −α2∥/(4ρ) ≤nC2 2ρ ∥α1 −α2∥. (9) where the second inequality follows from the fact that Y is an orthonormal matrix. For the second term on the right-hand side of inequality (8), we apply the fact proved in Theorem 1 that K(α) ∈Q2 for any α ∈Q1. Indeed, ∥Y (K0 + Y α2α⊤ 2 Y/(4ρ))+Y (α2 −α1)∥≤ λmax ³ Y (K0 + Y α2α⊤ 2 Y/(4ρ))+Y ´ ∥α2 −α1∥≤λmax ³ (K0 + Y α2α⊤ 2 Y/(4ρ))+ ´ ∥α2 −α1∥≤ h λmax(K0) + nC2 4ρ i ∥α1 −α2∥. Putting this equation and (9) back into equality (8) completes the proof of Theorem 2. 4 Simplified Projected Gradient Method (SPGM) 1. Choose γ ≥λmax(K0) + nC2 ρ . Let ε > 0, α0 ∈Q1 be given and set k = 0. 2. Compute ∇f(αk) = e −Y ¡ K0 + Y αkα⊤ k Y/(4ρ) ¢ + Y αk . 3. αk+1 = PQ1 (αk + ∇f(αk)/γ) . 4. Set k ←k + 1. Go to step 2 until the stopping criterion less than ε. Table 1: Pseudo-code of projected gradient method 4 Smooth Optimization Algorithms This section is based on the theoretical analysis above, mainly Theorem 2. We first outline a simplified version of the projected gradient method proposed in [15] and show it has a convergence rate of O(1/k) where k is the iteration number. We can further develop a smooth optimization approach [17, 18] for indefinite SVM (5). This scheme has an optimal convergence rate O(1/k2) for smooth problems which has been applied to various problems, e.g. [6]. 4.1 Simplified Projected Gradient Method In [15], the objective function was smoothed by adding a quadratic term (see details in Section 3 there) and then they proposed a projected gradient algorithm to solve this approximation problem. Using the explicit gradient representation in Theorem 1 we formulate its simplified version in Table 1 where the projection PQ1 : Rn →Q1 is defined, for any β ∈Rn, by PQ1(β) = arg min α∈Q1 ∥α −β∥2. (10) Indeed, from Theorem 2 we can further obtain the following result by developing the techniques in Sections 2.1.5, 2.2.3 and 2.2.4 of [18]. Lemma 4. Let γ ≥ h λmax(K0) + nC2 ρ i and {αk : k ∈N} be given by the simplified projected gradient method in Table 1. For any α ∈Q1, the following inequality holds f(αk+1) ≥f(α) + γ⟨αk −αk+1, α −αk⟩+ γ 2 ∥αk −αk+1∥2. Proof. We know from Theorem 2 that ∇f is Lipschitz continuous with Lipschitz constant L = λmax(K0) + nC2 ρ , then we have f(α) −f(αk) −⟨∇f(αk), α −αk⟩= R 1 0 ⟨∇f(θα + (1 −θ)αk) − ∇f(αk), α −αk⟩dθ ≥−L R 1 0 (1 −θ)∥α −αk∥2dθ ≥−γ 2 ∥α −αk∥2. Applying this inequality with α = αk+1 implies that −f(αk) −⟨∇f(αk), αk+1 −αk⟩≥−f(αk+1) −γ 2 ∥αk+1 −αk∥2. (11) Let φ(α) = −f(αk) −∇f(αk)(α −αk) + γ 2 ∥α −αk∥2 which implies that αk+1 = arg minα∈Q1 φ(α). Then, by the first-order optimality condition over αk+1 there holds, for any α ∈Q1, ⟨∇φ(αk), α−αk+1⟩≥0, i.e. −⟨∇f(αk), α−αk+1⟩≥γ⟨αk+1 −αk, αk+1 −α⟩. Adding this equation and (11) together yields that −f(αk) −⟨∇f(αk), α −αk⟩≥−f(αk+1) + γ⟨αk − αk+1, α−αk⟩+ γ 2 ∥αk−αk+1∥2. Also, since −f is convex, −f(α) ≥−f(αk)−⟨∇f(αk), α−αk⟩. Combining this with the above inequality finishes the proof of the lemma. Theorem 3. Let γ ≥ h λmax(K0) + nC2 ρ i and the iteration sequence {αk : k ∈N} be given by the simplified projected gradient method in Table 1. Then, we have that f(αk+1) ≥f(αk) + γ 2 ∥αk+1 −αk∥2, (12) Moreover, max α∈Q1 f(α) −f(αk) ≤γ 2k ∥α0 −α∗∥2 (13) where α∗is an optimal solution of problem maxα∈Q1 f(α). 5 Nesterov’s Smooth Optimization Method (SMM) 1. Let ε > 0, k = 0 and initialize α0 ∈Q1 and let L = λmax(K0)) + nC2/ρ. 2. Compute ∇f(αk) = e −Y ¡ K0 + Y αkα⊤ k Y/(4ρ) ¢ + Y αk . 3. Compute γk = PQ1 (αk + ∇f(αk)/L) . 4. Compute βk = PQ1 ³ α0 + Pk i=0(i + 1)∇f(αk)/(2L) ´ . 5. Set αk+1 = 2 k+3βk + k+1 k+3γk . 6. Set k ←k + 1. Go to step 2 until the stopping criterion less than ε. Table 2: Pseudo-code of first-order Nesterov’s smooth optimization method Proof. Applying Lemma 4 with α = αk yields inequality (12). To prove inequality (13), we first apply Lemma 4 with α = α∗to get that, for any i, maxα∈Q1 f(α) −f(αi) ≤−γ⟨αi −αi+1, α∗− αi⟩−γ 2 ∥αi −αi+1∥2 = γ 2 ∥α∗−αi∥2 −γ 2 ∥α∗−αi+1∥2. Adding them over i from 0 and k −1 and also, noting from (12) that {maxα∈Q1 f(α) −f(αk) : k ∈N} is decreasing, we have that k (maxα∈Q1 f(α) −f(αk)) ≤Pk−1 i=0 (maxα∈Q1 f(α) −f(αi+1)) ≤ γ 2 ∥α∗−α0∥2. This completes the proof of the theorem. From the above theorem, the sequence {f(αk) : k ∈N} is monotonically increasing and the iteration complexity of SPGM is O(L/ε) for finding an ε-optimal solution. 4.2 Nesterov’s Smooth Optimization Method In [18, 17], Nesterov proposed an efficient smooth optimization method for solving convex programming problems of the form min x∈U g(x) where g is a convex function with Lipschitz continuous gradient, and U is a closed convex set in Rn. Specifically, suppose there exists L > 0 such that ∥∇g(x) −∇g(x′)∥≤L∥x −x′∥, ∀x, x′ ∈U. The smooth optimization approach needs to introduce a proxy-function d(x) associated with the set U. It is assumed to be continuous and strongly convex on U with convexity parameter σ > 0. Let x0 = arg minx∈U d(x). Without loss of generality, assume that d(x0) = 0. Thus, strong convexity of d means that , for any x ∈U, d(x) ≥1 2σ∥x−x0∥2. Then, a specific first-order smooth optimization scheme detailed in [18] can be then applied to the function g with convergence rate in O( p L/ε). The first-order method needs to define a proxy-function associated with Q1. Here, we define the proxy-function by d(α) = 1 2∥α −α0∥2 with α0 ∈Q1. The Lipschitz constant of −f is established in Theorem 2 given by L = λmax(K0) + nC2/ρ. Translating the first-order Nesterov’s scheme [18, Section 3] to our problem (5), we can get the smooth optimization algorithm for indefinite SVM, see its pseudo-code in Table 2. One can see [17] for its variants with general step sizes. The effectiveness of the first-order Nesterov’s algorithm largely depends on the Steps 2, 3 and 4 outlined in Table 2. By Theorem 1, the computation of ∇f(αk) in Step 2 needs an eigen-decomposition. Steps 3 and 4 are the projection problem (10) by replacing β respectively by αk + ∇f(αk)/L and α0 + Pk i=0(i + 1)∇f(αi)/(2L). The convergence of this optimal method was shown in [18]: maxα∈Q1 f(α) −f(γk) ≤4L∥α0−α∗∥2 (k+1)(k+2) where α∗is one of the optimal solutions. It is worthy of pointing out that either {f(αk) : k ∈N} or {f(γk) : k ∈N} may not monotonically increase, however it can be made to monotonically increase by a simple modification of the algorithm [18]. In addition, the above estimation of the Lipschitz constant L could be loose in reality and one could further accelerate the algorithm by using a line search scheme [16]. 4.3 Related Work and Complexity Discussion We list the theoretical time complexity of algorithms to run Indefinite SVM. It is worth noting that the number of iterations to reach a target precision of ε means that −f(αk) −minα∈Q1 −f(α) = maxα∈Q1 f(α) −f(αk) ≤ε. However, this does not mean the dual gap as used in [15] is less than ε. In [15], the objective function is smoothed by adding a quadratic term and then they further 6 proposed a projected gradient algorithm and analytic center cutting plane method (ACCPM)1. As proved in Theorem 3, the number of iterations of the projected gradient method is usually O(L/ε). In each iteration, the main complexity cost O(n3) is from the eigen-decomposition. Hence, the overall complexity of SPGM is O(n3L/ε). As discussed in [15], ACCPM has an overall complexity is O(n4 log(1/ε)2) for finding an ε-optimal solution. However, this method needs to use interior methods at each iteration which would be slow for large scale datasets. Chen and Ye [4] reformulated indefinite SVM as an appealing semi-infinite quadratically constrained linear programming (SIQCLP) without applying extra smoothing techniques. There, the algorithm iteratively solves a linear programming with a finite number of quadratic constraints. The iteration complexity of semi-infinite linear programming is usually O(1/ε3). In each iteration, one needs to find maximum violation constraints which involves eigen-decomposition of complexity O(n3). Hence, the overall complexity is of O(n3/ε3). The main limitation of this approach is that one needs to save the subset of increasing quadratically constrained conditions indexed by n × n matrices and iteratively solve a quadratically constrained linear programming (QCLP). The QCLP sub-problem can be solved by general software packages, e.g. Mosek (http://www.mosek.com/), which is generally slow in our experience. This tends to make the algorithm inefficient during the iteration process, although pruning techniques were proposed to avoid too many quadratically constrained conditions. Based on our theoretical results (Theorem 2), Nesterov’s smooth optimization method can be applied. The complexity of this smooth optimization method (SMM) mainly relies on the eigenvalue decomposition on Step 2 listed in Table 2 which costs O(n3). Step 3 and 4 are projections onto the convex region Q1 which costs O(n log n) as pointed out in [15]. The first-order smooth optimization approach [17, 18] has iteration complexity O( p L/ε) for finding an ε-optimal solution. Consequently, the overall complexity is O(n3p L/ε). Hence, from theoretical comparison the complexity of smoothing optimization is better than the simplified projected gradient method (SPGM) and SIQCLP. Compared with ACCPM, SMM has better dependence on the sample number n but with a worse precision i.e. worse dependence on ε. 5 Experimental Validation We run our proposed smooth optimization approach and simplified projected gradient method on various datasets to validate our analysis. The experiments are done on several benchmark data sets from the UCI repository [19] including Sonar, Ionosphere, Heart, Pima Indians Diabetes, Breast Cancer, and USPS with digits 3 and 5. For USPS dataset, we randomly select 600 samples for each digit. All the results reported are based on 10 random training/test partition with ratio 4/1. In each data split, as in [4] we first generate a Gaussian kernel matrix K with the hyper-parameter determined by cross-validation on the training data using LIBSVM and then construct indefinite matrices by adding a small noisy matrix i.e. K0 := K −0.1 bE. Here, the noisy matrix bE = (E + E′)/2 where E is randomly generated by zero mean and identity covariance matrix. For all methods, the parameters C and ρ for Indefinite SVM are tuned by cross-validation and we terminate the algorithm if the relative change of the objective value is less than 10−6. In Table 3, we report the average test set accuracy (%) and CPU time (seconds) across different algorithms: smooth optimization method (SMM), simplified projected gradient method (SPGM), analytic center cutting plane method (ACCPM), and semi-infinite quadratically constrained linear programming (SIQCLP). For the QCLP sub-problem in the SIQCLP method, we use Mosek software package (http://www.mosek.com/). We can see that test accuracies are statistically the same across different algorithms, which validates our analysis on the objective function. In particular, we observe that SMM is consistently more efficient than other methods, especially for a large number of training samples. SIQCLP needs much more time since, in each iteration, it needs to solves a quadratically constrained linear programming. In Figure 1, we plot the objective values versus iteration on Sonar and Diabetes for SMM, SPGM, and ACCPM. The SIQCLP approach is not included here since its objective value is not based on the iteration w.r.t. the variable α which does not directly yield an increasing iteration sequence of objective values in contrast to those of the other three algorithms. From Figure 1, we can see that SMM converges faster than SPGM which is consistent with the complexity analysis. The convergence of ACCPM is quite similar to SMM, especially for 1MATLAB codes are available in http://www.princeton.edu/ rluss/IndefiniteSVM.htm 7 Data Size λmin λmax SMM SPGM ACCPM SIQCLP Sonar 208 −1.38 21.47 76.34% 76.34% 75.12% 76.09% 0.74s 5.12s 3.20s 244.55s Ionosphere 351 -2.08 101.34 93.14% 93.43% 93.54% 93.54% 5.47s 28.93s 22.73s 455.81s Heart 270 -1.98 178.03 79.81% 79.44% 79.25% 79.25% 3.54s 12.05s 11.96s 689.17s Diabetes 768 -3.44 539.12 70.00% 69.86% 70.52% 69.73% 39.93s 345.48s 678.85s 3134.31s Breast-cancer 683 -2.87 290.41 95.93% 96.02% 96.02% 95.40% 5.71s 50.13s 212.96s 4610.82s USPS-35 1200 −3.72 112.65 96.33% 96.33% 96.04% 95.54% 23.22s 236.00s 3713.05s 5199.17s Table 3: Average test set accuracy (%) and CPU time in seconds (s) of different algorithms where λmax(λmin) denotes the average maximum (minimum) eigenvalues of the indefinite kernel matrix over training samples. 20 40 60 80 100 120 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5 x 10 4 Iteration Objectve value SMM SPGM ACCPM 20 40 60 80 100 −600 −500 −400 −300 −200 −100 0 100 Iteration SMM SPGM ACCPM Figure 1: Objective value versus iteration: Sonar (left) and Diabetes (right). Curves: SMM (blue), SPGM (red) and ACCPM (black) small-sized datasets which coincides with the complexity analysis in Section 4.3 since it generally has a high precision. However, ACCPM needs more time in each iteration than SMM and this observation becomes more apparent for the relatively large datasets shown in the time comparison of Table 3. 6 Conclusion In this paper we analyzed the regularization formulation for training SVM with indefinite kernels proposed by Luss and d’Aspremont [15]. We show that the objective function of interest is continuously differentiable with Lipschitz continuous gradient. Our elementary analysis greatly facilitates the application of gradient-based methods. We formulated a simplified version of the projected gradient method presented in [15] and showed that it has a convergence rate of O(1/k). We further developed Nesterov’s smooth optimization method [17, 18] for Indefinite SVM which has an optimal convergence rate of O(1/k2) for smooth problems. Experiments on various datasets validate our analysis and the efficiency of our proposed optimization approach. In future, we are planning to further accelerate the algorithm by using a line search scheme [16]. We are also applying this method to real biological datasets such as protein sequence analysis using sequence alignment measures. Acknowledgements This work is supported by EPSRC grant EP/E027296/1. 8 References [1] R. Bhatia. Matrix analysis. Graduate texts in Mathematics. Springer, 1997. [2] S. Boyd and L. Vandenberghe. Convex optimization. Cambridge University Press, 2004. [3] J. F. Bonnans and A. Shapiro. Optimization problems with perturbation: A guided tour. SIAM Review, 40: 202–227, 1998. [4] J. Chen and J. Ye. Training SVM with Indefinite Kernels. ICML, 2008. [5] N. Cristianini and J. Shawe-Taylor. An introduction to support vector machines and other kernel-based learning methods. Cambridge University Press, 2000. [6] A. d’Aspremont, O. Banerjee and L. El Ghaoui. First-order methods for sparse covariance selection. SIAM Journal on Matrix Analysis and its Applications, 30: 56–66, 2007. [7] J.M. Danskin. The theory of max-min and its applications to weapons allocation problems, Springer-Verlag, New York, 1967. [8] T. Graepel, R. Herbrich, P. Bollmann-Sdorra, and K. Obermayer. Classification on pairwise proximity data. NIPS, 1998. [9] B. Haasdonk. Feature space interpretation of SVMs with indefinite kernels. IEEE Transactions on Pattern Analysis and Machine Intelligence, 27: 482–492, 2005. [10] R. A. Horn and C. R. Johnson. Topics in Matrix Analysis. Cambridge University Press, 1991. [11] R. I. Kondor and J. Laffferty. Diffusion kernels on graphs and other discrete input spaces. ICML, 2002. [12] C. Lemar´echal and C. Sagastiz´abal. Practical aspects of the Moreau-Yosida regularization: theoretical preliminaries. SIAM Journal on Optimization, 7: 367–385, 1997. [13] G. R. G. Lanckriet, N. Cristianini, N., P. L. Bartlett, L. E. Ghaoui, and M. I. Jordan. Learning the kernel matrix with semidefinite programming. J. of Machine Learning Research, 5: 27– 72, 2004. [14] H.-T. Lin and C. J. Lin. A study on sigmoid kernels for SVM and the training of non-psd kernels by smo-type methods. Technical Report, National Taiwan University, 2003. [15] R. Luss and A. d’Aspremont. Support vector machine classification with indefinite kernels. NIPS, 2007. [16] A. Nemirovski. Efficient methods in convex programming. Lecture Notes, 1994. [17] Y. Nesterov. Introductory Lectures on Convex Optimization: A Basic Course. Springer, 2003. [18] Y. Nesterov. Smooth minimization of non-smooth functions. Mathematical Programming, 103:127–152, 2005. [19] D. Newman, S. Hettich, C. Blake, and C. Merz. UCI repository of machine learning datasets. 1998. [20] C. S. Ong, X. Mary, S. Canu, and A. J. Smola. Learning with non-positive kernels. ICML, 2004. [21] E. Pekalska, P. Paclik, and R. P. W. Duin. A generalized kernel approach to dissimilaritybased classification. J. of Machine Learning Research, 2: 175–211, 2002. [22] V. Roth, J. Laub, M. Kawanabe, and J. M. Buhmann. Optimal cluster preserving embedding of nonmetric proximity data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25:1540–1551, 2003. [23] H. Saigo, J.P.Vert and N. Ueda, and T. Akutsu. Protein homology detection using string alignment kernels. Bioinformatics, 20: 1682–1689., 2004. [24] B. Sch¨olkopf, and A.J. Smola. Learning with kernels: Support vector machines, regularization, optimization, and beyond. The MIT Press, 2001. [25] A. J. Smola, Z. L. ´O´vari, and R. C. Williamson. Regularization with dot-product kernels. NIPS, 2000. [26] G. Wu, Z. Zhang, and E. Y. Chang. An analysis of transformation on non-positive semidefinite similarity matrix for kernel machines. Technical Report, UCSB, 2005. 9
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Quantification and the language of thought Charles Kemp Department of Psychology Carnegie Mellon University ckemp@cmu.edu Abstract Many researchers have suggested that the psychological complexity of a concept is related to the length of its representation in a language of thought. As yet, however, there are few concrete proposals about the nature of this language. This paper makes one such proposal: the language of thought allows first order quantification (quantification over objects) more readily than second-order quantification (quantification over features). To support this proposal we present behavioral results from a concept learning study inspired by the work of Shepard, Hovland and Jenkins. Humans can learn and think about many kinds of concepts, including natural kinds such as elephant and water and nominal kinds such as grandmother and prime number. Understanding the mental representations that support these abilities is a central challenge for cognitive science. This paper proposes that quantification plays a role in conceptual representation—for example, an animal X qualifies as a predator if there is some animal Y such that X hunts Y . The concepts we consider are much simpler than real-world examples such as predator, but even simple laboratory studies can provide important clues about the nature of mental representation. Our approach to mental representation is based on the language of thought hypothesis [1]. As pursued here, the hypothesis proposes that mental representations are constructed in a compositional language of some kind, and that the psychological complexity of a concept is closely related to the length of its representation in this language [2, 3, 4]. Following previous researchers [2, 4], we operationalize the psychological complexity of a concept in terms of the ease with which it is learned and remembered. Given these working assumptions, the remaining challenge is to specify the representational resources provided by the language of thought. Some previous studies have relied on propositional logic as a representation language [2, 5], but we believe that the resources of predicate logic are needed to capture the structure of many human concepts. In particular, we suggest that the language of thought can accommodate relations, functions, and quantification, and focus here on the role of quantification. Our primary proposal is that quantification is supported by the language of thought, but that quantification over objects is psychologically more natural than quantification over features. To test this idea we compare concept learning in two domains which are very similar except for one critical difference: one domain allows quantification over objects, and the other allows quantification over features. We consider several logical languages that can be used to formulate concepts in both domains, and find that learning times are best predicted by a language that supports quantification over objects but not features. Our work illustrates how theories of mental representation can be informed by comparing concept learning across two or more domains. Existing studies work with a range of domains, and it is useful to consider a “conceptual universe” that includes these possibilities along with many others that have not yet been studied. Table 1 charts a small fragment of this universe, and the penultimate column shows example stimuli that will be familiar from previous studies of concept learning. Previous studies have made important contributions by choosing a single domain in Table 1 and explaining 1 why some concepts within this domain are easier to learn than others [2, 4, 6, 7, 8, 9]. Comparisons across domains can also provide important information about learning and mental representation, and we illustrate this claim by comparing learning times across Domains 3 and 4. The next section introduces the conceptual universe in Table 1 in more detail. We then present a formal approach to concept learning that relies on a logical language and compare three candidate languages. Language OQ (for object quantification) supports quantification over objects but not features, language FQ (for feature quantification) supports quantification over features but not objects, and language OQ + FQ supports quantification over both objects and features. We use these languages to predict learning times across Domains 3 and 4, and present an experiment which suggests that language OQ comes closest to the language of thought. 1 The conceptual universe Table 1 provides an organizing framework for thinking about the many domains in which learning can occur. The table includes 8 domains, each of which is defined by specifying some number of objects, features, and relations, and by specifying the range of each feature and each relation. We refer to the elements in each domain as items, and the penultimate column of Table 1 shows items from each domain. The first row shows a domain commonly used by studies of Boolean concept learning. Each item in this domain includes a single object a and specifies whether that object has value v1 (small) or v2 (large) on feature F (size), value v3 (white) or v4 (gray) on feature G (color), and value v5 (vertical) or v6 (horizontal) on feature H (texture). Domain 2 also includes three features, but now each item includes three objects and each feature applies to only one of the objects. For example, feature H (texture) applies to only the third object in the domain (i.e. the third square on each card). Domain 3 is similar to Domain 1, but now the three features can be aligned— for any given item each feature will be absent (value 0) or present. The example in Table 1 uses three features (boundary, dots, and slash) that can each be added to an unadorned gray square. Domain 4 is similar to Domain 2, but again the feature values can be aligned, and the feature for each object will be absent (value 0) or present. Domains 5 and 6 are similar to domains 2 and 4 respectively, but each one includes relations rather than features. In Domain 6, for example, the relation R assigns value 0 (absent) or value 1 (present) to each undirected pair of objects. The first six domains in Table 1 are all variants of Domain 1, which is the domain typically used by studies of Boolean concept learning. Focusing on six related domains helps to establish some of the dimensions along which domains can differ, but the final two domains in Table 1 show some of the many alternative possibilities. Domain 7 includes two categorical features, each of which takes three rather than two values. Domain 8 is similar to Domain 6, but now the number of objects is 6 rather than 3 and relation R is directed rather than undirected. To mention just a handful of possibilities which do not appear in Table 1, domains may also have categorical features that are ordered (e.g. a size feature that takes values small, medium, and large), continuous valued features or relations, relations with more than two places, and objects that contain sub-objects or parts. Several learning problems can be formulated within any given domain. The most basic is to learn a single item—for example, a single item from Domain 8 [4]. A second problem is to learn a class of items—for example, a class that includes four of the items in Domain 1 and excludes the remaining four [6]. Learning an item class can be formalized as learning a unary predicate defined over items, and a natural extension is to consider predicates with two or more arguments. For example, problems of the form A is to B as C is to ? can be formulated as problems where the task is to learn a binary relation analogous(·, ·) given the single example analogous(A, B). Here, however, we focus on the task of learning item classes or unary predicates. Since we focus on the role of quantification, we will work with domains where quantification is appropriate. Quantification over objects is natural in cases like Domain 4 where the feature values for all objects can be aligned. Note, for example, that the statement “every object has its feature” picks out the final example item in Domain 4 but that no such statement is possible in Domain 2. Quantification over features is natural in cases like Domain 3 where the ranges of each feature can be aligned. For example, “object a has all three features” picks out the final example item in Domain 3 but no such statement is possible in Domain 1. We therefore focus on Domains 3 and 4, and explore the problem of learning item classes in each domain. 2 Domain specification # Objects O Features Relations Example Items Ref. 1 {a} F : O →{v1, v2} — , , , , , , , [2, 6, 7, 10, 11] G : O →{v3, v4} H : O →{v5, v6} 2 {a, b, c} F : a →{v1, v2} — . . . , [6] G : b →{v3, v4} H : c →{v5, v6} 3 {a} F : O →{0, v1} — , , , , , , , [12] G : O →{0, v2} H : O →{0, v3} 4 {a, b, c} F : a →{0, v1} — . . . , [6] G : b →{0, v2} H : c →{0, v3} 5 {a, b, c} — R : (a, b) →{v1, v2} , , , , , , , S : (a, c) →{v3, v4} T : (b, c) →{v5, v6} 6 {a, b, c} — R : O × O →{0, 1} , , , , , , , [13] 7 {a} — , , , , , , , , [8, 9] F : O →{v1, v2, v3} G : O →{v4, v5, v6} 8 {a, b, c, d, e, f} — R : O × O →{0, 1} , . . . , , , , , , , [4] Table 1: The conceptual universe. Eight domains are shown, and each one is defined by a set of objects, a set of features, and a set of relations. We call the members of each domain items, and an item is created by specifying the extension of each feature and relation in the domain. The six domains above the double lines are closely related to the work of Shepard et al. [6]. Each one includes eight items which differ along three dimensions. These dimensions, however, emerge from different underlying representations in the six cases. 3 4 (III) (b) (a) 10 (VI) 9 (V) 8 (V) 7 (V) 6 (IV) 1 (I) 2 (II) 3 (III) 5 (IV) 000 011 101 110 001 010 100 111 Figure 1: (a) A stimulus lattice for domains (e.g. Domains 3, 4, and 6) that can be encoded as a triple of binary values where 0 represents “absent” and 1 represents “present.” (b) If the order of the values in the triple is not significant, there are 10 distinct ways to partition the lattice into two classes of four items. The SHJ type for each partition is shown in parentheses. Domains 3 and 4 both include 8 items each and we will consider classes that include exactly four of these items. Each item in these domains can be represented as a triple of binary values, where 0 indicates that a feature is absent and value 1 indicates that a feature is present. Each triple represents the values of the three features (Domain 3) or the feature values for the three objects (Domain 4). By representing each domain in this way, we have effectively adopted domain specifications that are simplifications of those shown in Table 1. Domain 3 is represented using three features of the form F, G, H : O →{0, 1}, and Domain 4 is represented using a single feature of the form F : O →{0, 1}. Simplifications of this kind are possible because the features in each domain can be aligned—notice that no corresponding simplifications are possible for Domains 1 and 2. The eight binary triples in each domain can be organized into the lattice shown in Figure 1a. Here we consider all ways to partition the vertices of the lattice into two groups of four. If partitions that differ only up to a permutation of the features (Domain 3) or objects (Domain 4) are grouped into equivalence classes, there are ten of these classes, and a representative of each is shown in Figure 1b. Previous researchers [6] have pointed out that the stimuli in Domain 1 can be organized into a cube similar to Figure 1a, and that there are six ways to partition these stimuli into two groups of four up to permutations of the features and permutations of the range of each feature. We refer to these equivalence classes as the six Shepard-Hovland-Jenkins types (or SHJ types), and each partition in Figure 1b is labeled with its corresponding SHJ type label. Note, for example, that partitions 3 and 4 are both examples of SHJ type III. For us, partitions 3 and 4 are distinct since items 000 (all absent) and 111 (all present) are uniquely identifiable, and partition 3 assigns these items to different classes but partition 4 does not. Previous researchers have considered differences between some of the first six domains in Table 1. Shepard et al. [6] ran experiments using compact stimuli (Domain 1) and distributed stimuli (Domains 2 and 4), and observed the same difficulty ranking of the six SHJ types in all cases. Their work, however, does not acknowledge that Domain 4 leads to 10 distinct types rather than 6, and therefore fails to address issues such as the relative complexities of concepts 5 and 6 in Figure 1. Social psychologists [13, 14] have studied Domain 6 and found that learning patterns depart from the standard SHJ order—in particular, that SHJ type VI (Concept 10 in Figure 1) is simpler than types III, IV and V. This finding has been used to support the claim that social learning relies on a domain-specific principle of structural balance [14]. We will see, however, that the relative simplicity of type VI in domains like 4 and 6 is consistent with a domain-general account based on representational economy. 2 A representation length approach to concept learning The conceptual universe in Table 1 calls for an account of learning that can apply across many domains. One candidate is the representation length approach, which proposes that concepts are mentally represented in a language of thought, and that the subjective complexity of a concept is 4 determined by the length of its representation in this language [4]. We consider the case where a concept corresponds to a class of items, and explore the idea that these concepts are mentally represented in a logical language. More formally, a concept is represented as a logical sentence, and the concept includes all models of this sentence, or all items that make the sentence true. The predictions of this representation length approach depend critically on the language chosen. Here we consider three languages—an object quantification language OQ that supports quantification over objects, a feature quantification language FQ that supports quantification over features, and a language OQ + FQ that supports quantification over both objects and features. Language OQ is based on a standard logical language known as predicate logic with equality. The language includes symbols representing objects (e.g. a and b), and features (e.g. F and G) and these symbols can be combined to create literals that indicate that an object does (Fa) or does not have a certain feature (Fa′). Literals can be combined using two connectives: AND (FaGa) and OR (Fa + Ga). The language includes two quantifiers—for all (∀) and there exists (∃)—and allows quantification over objects (e.g. ∀xFx, where x is a variable that ranges over all objects in the domain). Finally, language OQ includes equality and inequality relations (= and ̸=) which can be used to compare objects and object variables (e.g. =xa or ̸=xy). Table 2 shows several sentences formulated in language OQ. Suppose that the OQ complexity of each sentence is defined as the number of basic propositions it contains, where a basic proposition can be a positive or negative literal (Fa or Fa′) or an equality or inequality statement (=xa or ̸=xy). Equivalently, the complexity of a sentence is the total number of ANDs plus the total number of ORs plus one. This measure is equivalent by design to Feldman’s [2] notion of Boolean complexity when applied to a sentence without quantification. The complexity values in Table 2 show minimal complexity values for each concept in Domains 3 and 4. Table 2 also shows a single sentence that achieves each of these complexity values, although some concepts admit multiple sentences of minimal complexity. The complexity values in Table 2 were computed using an “enumerate then combine” approach. We began by enumerating a set of sentences according to criteria described in the next paragraph. Each sentence has an extension that specifies which items in the domain are consistent with the sentence. Given the extensions of all sentences generated during the enumeration phase, the combination phase considered all possible ways to combine these extensions using conjunctions or disjunctions. The procedure terminated once extensions corresponding to all of the concepts in the domain had been found. Although the number of possible sentences grows rapidly as the complexity of these sentences increases, the number of extensions is fixed and relatively small (28 for domains of size 8). The combination phase is tractable since sentences with the same extension can be grouped into a single equivalence class. The enumeration phase considered all formulae which had at most two quantifiers and which had a complexity value lower than four. For example, this phase did not include the formula ∃x∃y∃z ̸=yz F′ xFyFz (too many quantifiers) or the formula ∀x∃y ̸=xy Fy(Fx + Gx + Hx) (complexity too high). Despite these restrictions, we believe that the complexity values in Table 2 are identical to the values that would be obtained if we had considered all possible sentences. Language FQ is similar to OQ but allows quantification over features rather than objects. For example, FQ includes the statement ∀Q Qa, where Q is a variable that ranges over all features in the domain. Language FQ also allows features and feature variables to be compared for equality or inequality (e.g. =QF or ̸=QR). Since FQ and OQ are closely related, it follows that the FQ complexity values for Domains 3 and 4 are identical to the OQ complexity values for Domains 4 and 3. For example, FQ can express concept 5 in Domain 3 as ∀Q∃R ̸=QR Ra. We can combine OQ and FQ to create a language OQ + FQ that allows quantification over both objects and features. Allowing both kinds of quantification leads to identical complexity values for Domains 3 and 4. Language OQ + FQ can express each of the formulae for Domain 4 in Table 2, and these formulae can be converted into corresponding formulae for Domain 3 by translating each instance of object quantification into an instance of feature quantification. Logicians distinguish between first-order logic, which allows quantification over objects but not predicates, and second-order logic, which allows quantification over objects and predicates. The difference between languages OQ and OQ + FQ is superficially similar to the difference between first-order and second-order logic, but does not cut to the heart of this matter. Since language 5 # Domain 3 C Domain 4 C 1 Ga 1 Fb 1 2 Fa′Ha′ + FaHa 4 Fa′Fc′ + FaFc 4 3 Fa′Ga + FaHa 4 Fa′Fb + FaFc 4 4 Fa′Ga′ + FaHa 4 Fa′Fb′ + FaFc 4 5 Ga(Fa + Ha) + FaHa 5 ∀x∃y ̸=xy Fy 2 6 Ga′(Fa + Ha) + FaHa 5 (∀xFx) + Fb′∃yFy 3 7 Ga′(Fa + Ha) + FaGaHa 6 (∀xFx) + Fb′(Fa′ + Fc′) 4 8 Ha(Fa′ + Ga) + FaGa′Ha′ 6 Fc(Fa′ + Fb) + FaFb′Fc′ 6 9 Fa(Ga + Ha) + Fa′Ga′Ha′ 6 (∀xFx′) + Fa(Fb + Fc) 4 10 Ga′(FaHa′ + Fa′Ha) + Ga(Fa′Ha′ + FaHa) 10 (∀xFx) + ∃y∀zFy(=zy +Fz′) 4 Table 2: Complexity values C and corresponding formulae for language OQ. Boolean complexity predicts complexity values for both domains that are identical to the OQ complexity values shown here for Domain 3. Language FQ predicts complexity values for Domains 3 and 4 that are identical to the OQ values for Domains 4 and 3 respectively. Language OQ+FQ predicts complexity values for both domains that are identical to the OQ complexity values for Domain 4. OQ + FQ only supports quantification over a pre-specified set of features, it is equivalent to a typed first order logic that includes types for objects and features [15]. Future studies, however, can explore the cognitive relevance of higher-order logic as developed by logicians. 3 Experiment Now that we have introduced languages OQ, FQ and OQ + FQ our theoretical proposals can be sharply formulated. We suggest that quantification over objects plays an important role in mental representations, and predict that OQ complexity will account better for human learning than Boolean complexity. We also propose that quantification over objects is more natural than quantification over features, and predict that OQ complexity will account better for human learning than both FQ complexity and OQ + FQ complexity. We tested these predictions by designing an experiment where participants learned concepts from Domains 3 and 4. Method. 20 adults participated for course credit. Each participant was assigned to Domain 3 or Domain 4 and learned all ten concepts from that domain. The items used for each domain were the cards shown in Table 1. Note, for example, that each Domain 3 card showed one square, and that each Domain 4 card showed three squares. These items are based on stimuli developed by Sakamoto and Love [12]. The experiment was carried out using a custom built graphical interface. For each learning problem in each domain, all eight items were simultaneously presented on the screen, and participants were able to drag them around and organize them however they liked. Each problem had three phases. During the learning phase, the four items belonging to the current concept had red boundaries, and the remaining four items had blue boundaries. During the memory phase, these colored boundaries were removed, and participants were asked to sort the items into the red group and the blue group. If they made an error they returned to the learning phase, and could retake the test whenever they were ready. During the description phase, participants were asked to provide a written description of the two groups of cards. The color assignments (red or blue) were randomized across participants— in other words, the “red groups” learned by some participants were identical to the “blue groups” learned by others. The order in which participants learned the 10 concepts was also randomized. Model predictions. The OQ complexity values for the ten concepts in each domain are shown in Table 2 and plotted in Figure 2a. The complexity values in Figure 2a have been normalized so that they sum to one within each domain, and the differences of these normalized scores are shown in the final row of Figure 2a. The two largest bars in the difference plot indicate that Concepts 10 and 5 are predicted to be easier to learn in Domain 4 than in Domain 3. Language OQ can express 6 Domain 4 Domain 3 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 1 2 3 4 5 6 7 8 9 10 −0.05 0 0.05 0.1 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 1 2 3 4 5 6 7 8 9 10 −0.05 0 0.05 0.1 a) b) Difference Learning time OQ complexity Figure 2: Normalized OQ complexity values and normalized learning times for the 10 concepts in Domains 3 and 4. statements like “either 1 or 3 objects have F” (Concept 10 in Domain 4), or “2 or more objects have F” (Concept 5 in Domain 4). Since quantification over features is not permitted, however, analogous statements (e.g. “object a has either 1 or 3 features”) cannot be formulated in Domain 3. Concept 10 corresponds to SHJ type VI, which often emerges as the most difficult concept in studies of Boolean concept learning. Our model therefore predicts that the standard ordering of the SHJ types will not apply in Domain 4. Our model also predicts that concepts assigned to the same SHJ type will have different complexities. In Domain 4 the model predicts that Concept 6 will be harder to learn than Concept 5 (both are examples of SHJ type IV), and that Concept 8 will be harder to learn than Concepts 7 or 9 (all three are examples of SHJ type V). Results. The computer interface recorded the amount of time participants spent on the learning phase for each concept. Domain 3 was a little more difficult than Domain 4 overall: on average, Domain 3 participants took 557 seconds and Domain 4 participants took 467 seconds to learn the 10 concepts. For all remaining analyses, we consider learning times that are normalized to sum to 1 for each participant. Figure 2b shows the mean values for these normalized times, and indicates the relative difficulties of the concepts within each condition. The difference plot in Figure 2b supports the two main predictions identified previously. Concepts 10 and 5 are the cases that differ most across the domains, and both concepts are easier to learn in Domain 3 than Domain 4. As predicted, Concept 5 is substantially easier than Concept 6 in Domain 4 even though both correspond to the same SHJ type. Concepts 7 through 9 also correspond to the same SHJ type, and the data for Domain 4 suggest that Concept 8 is the most difficult of the three, although the difference between Concepts 8 and 7 is not especially large. Four sets of complexity predictions are plotted against the human data in Figure 3. Boolean complexity and OQ complexity make identical predictions about Domain 3, and OQ complexity and OQ + FQ complexity make identical predictions about Domain 4. Only OQ complexity, however, accounts for the results observed in both domains. The concept descriptions generated by participants provide additional evidence that there are psychologically important differences between Domains 3 and 4. If the descriptions for concepts 5 and 10 are combined, 18 out of 20 responses in Domain 4 referred to quantification or counting. One representative description of Concept 5 stated that “red has multiple filled” and that “blue has one filled or none.” Only 3 of 20 responses in Domain 3 mentioned quantification. One representative description of Concept 5 stated that “red = multiple features” and that “blue = only one feature.” 7 0 0.1 0.2 0.1 0.2 r=0.84 0 0.1 0.2 0.1 0.2 r=0.27 0 0.1 0.2 0.1 0.2 r=0.84 0 0.1 0.2 0.1 0.2 r=0.26 0 0.1 0.2 0.1 0.2 r=0.26 0 0.1 0.2 0.1 0.2 r=0.83 0 0.1 0.2 0.1 0.2 r=0.83 0 0.1 0.2 0.1 0.2 r=0.27 Learning time Learning time (Domain 4) (Domain 3) OQ + FQ complexity Boolean complexity OQ complexity FQ complexity Figure 3: Normalized learning times for each domain plotted against normalized complexity values predicted by four languages: Boolean logic, OQ, FQ and OQ + FQ. These results suggest that people can count or quantify over features, but that it is psychologically more natural to quantify over objects rather than features. Although we have focused on three specific languages, the results in Figure 2b can be used to evaluate alternative proposals about the language of thought. One such alternative is an extension of Language OQ that allows feature values to be compared for equality. This extended language supports concise representations of Concept 2 in both Domain 3 (Fa = Ha) and Domain 4 (Fa = Fc), and predicts that Concept 2 will be easier to learn than all other concepts except Concept 1. Note, however, that this prediction is not compatible with the data in Figure 2b. Other languages might also be considered, but we know of no simple language that will account for our data better than OQ. 4 Conclusion Comparing concept learning across qualitatively different domains can provide valuable information about the nature of mental representation. We compared two domains that that are similar in many respects, but that differ according to whether they include a single object (Domain 3) or multiple objects (Domain 4). Quantification over objects is possible in Domain 4 but not Domain 3, and this difference helps to explain the different learning patterns we observed across the two domains. Our results suggest that concept representations can incorporate quantification, and that quantifying over objects is more natural than quantifying over features. The model predictions we reported are based on a language (OQ) that is a generic version of first order logic with equality. Our results therefore suggest that some of the languages commonly considered by logicians (e.g. first order logic with equality) may indeed capture some aspects of the “laws of thought” [16]. A simple language like OQ offers a convenient way to explore the role of quantification, but this language will need to be refined and extended in order to provide a more accurate account of mental representation. For example, a comprehensive account of the language of thought will need to support quantification over features in some cases, but might be formulated so that quantification over features is typically more costly than quantification over objects. Many possible representation languages can be imagined and a large amount of empirical data will be needed to identify the language that comes closest to the language of thought. Many relevant studies have already been conducted [2, 6, 8, 9, 13, 17], but there are vast regions of the conceptual universe (Table 1) that remain to be explored. Navigating this universe is likely to involve several challenges, but web-based experiments [18, 19] may allow it to be explored at a depth and scale that are currently unprecedented. Characterizing the language of thought is undoubtedly a long term project, but modern methods of data collection may support rapid progress towards this goal. Acknowledgments I thank Maureen Satyshur for running the experiment. This work was supported in part by NSF grant CDI-0835797. 8 References [1] J. A. Fodor. The language of thought. Harvard University Press, Cambridge, 1975. [2] J. Feldman. Minimization of Boolean complexity in human concept learning. Nature, 407: 630–633, 2000. [3] D. Fass and J. Feldman. Categorization under complexity: A unified MDL account of human learning of regular and irregular categories. In S. Thrun S. Becker and K. Obermayer, editors, Advances in Neural Information Processing Systems 15, pages 35–34. MIT Press, Cambridge, MA, 2003. [4] C. Kemp, N. D. Goodman, and J. B. Tenenbaum. Learning and using relational theories. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 753–760. MIT Press, Cambridge, MA, 2008. [5] N. D. Goodman, J. B. Tenenbaum, J. Feldman, and T. L. Griffiths. A rational analysis of rule-based concept learning. Cognitive Science, 32(1):108–154, 2008. [6] R. N. Shepard, C. I. Hovland, and H. M. Jenkins. Learning and memorization of classifications. Psychological Monographs, 75(13), 1961. Whole No. 517. [7] R. M. Nosofsky, M. Gluck, T. J. Palmeri, S. C. McKinley, and P. Glauthier. Comparing models of rule-based classification learning: A replication and extension of Shepard, Hovland, and Jenkins (1961). Memory and Cognition, 22:352–369, 1994. [8] M. D. Lee and D. J. Navarro. Extending the ALCOVE model of category learning to featural stimulus domains. Psychonomic Bulletin and Review, 9(1):43–58, 2002. [9] C. D. Aitkin and J. Feldman. Subjective complexity of categories defined over three-valued features. In R. Sun and N. Miyake, editors, Proceedings of the 28th Annual Conference of the Cognitive Science Society, pages 961–966. Psychology Press, New York, 2006. [10] F. Mathy and J. Bradmetz. A theory of the graceful complexification of concepts and their learnability. Current Psychology of Cognition, 22(1):41–82, 2004. [11] R. Vigo. A note on the complexity of Boolean concepts. Journal of Mathematical Psychology, 50:501–510, 2006. [12] Y. Sakamoto and B. C. Love. Schematic influences on category learning and recognition memory. Journal of Experimental Psychology: General, 133(4):534–553, 2004. [13] W. H. Crockett. Balance, agreement and positivity in the cognition of small social structures. In Advances in Experimental Social Psychology, Vol 15, pages 1–57. Academic Press, 1982. [14] N. B. Cottrell. Heider’s structural balance principle as a conceptual rule. Journal of Personality and Social Psychology, 31(4):713–720, 1975. [15] H. B. Enderton. A mathematical introduction to logic. Academic Press, New York, 1972. [16] G. Boole. An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities. 1854. [17] B. C. Love and A. B. Markman. The nonindependence of stimulus properties in human category learning. Memory and Cognition, 31(5):790–799, 2003. [18] L. von Ahn. Games with a purpose. Computer, 39(6):92–94, 2006. [19] R. Snow, B. O’Connor, D. Jurafsky, and A. Ng. Cheap and fast–but is it good? Evaluating non-expert annotations for natural language tasks. In Proceedings of the 2008 Conference on empirical methods in natural language processing, pages 254–263. Association for Computational Linguistics, 2008. 9
|
2009
|
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|
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Exploring Functional Connectivity of the Human Brain using Multivariate Information Analysis Barry Chai1∗ Dirk B. Walther2∗ Diane M. Beck2,3† Li Fei-Fei1† 1Computer Science Department, Stanford University, Stanford, CA 94305 2Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801 3Psychology Department, University of Illinois at Urbana-Champaign, Champaign, IL 61820 {bwchai,feifeili}@cs.stanford.edu {walther,dmbeck}@illinois.edu Abstract In this study, we present a new method for establishing fMRI pattern-based functional connectivity between brain regions by estimating their multivariate mutual information. Recent advances in the numerical approximation of highdimensional probability distributions allow us to successfully estimate mutual information from scarce fMRI data. We also show that selecting voxels based on the multivariate mutual information of local activity patterns with respect to ground truth labels leads to higher decoding accuracy than established voxel selection methods. We validate our approach with a 6-way scene categorization fMRI experiment. Multivariate information analysis is able to find strong information sharing between PPA and RSC, consistent with existing neuroscience studies on scenes. Furthermore, an exploratory whole-brain analysis uncovered other brain regions that share information with the PPA-RSC scene network. 1 Introduction To understand how the brain represents and processes information we must account for two complementary properties: information is represented in a distributed fashion, and brain regions are strongly interconnected. Although heralded as a tool to address these issues, functional magnetic resonance imaging (fMRI) initially fell short of achieving these goals because of limitations of traditional analysis methods, which treat voxels as independent. Multi-voxel pattern analysis (MVPA) has revolutionized fMRI analysis by accounting for distributed patterns of activity rather than absolute activation levels. The analysis of functional connectivity, however, is so far mostly limited to comparing the time courses of individual voxels. To overcome these limitations we demonstrate a new method of pattern-based functional connectivity analysis based on mutual information of sets of voxels. Furthermore, we show that selecting voxels based on the mutual information of local activity with respect to ground truth outperforms other voxel selection methods. We apply our new analysis methods to the decoding of natural scene categories from the human brain. Human observers are able to quickly and efficiently perceive the content of natural scenes [15, 26]. It was recently shown by [23] that activity patterns in the parahippocampal place area (PPA), the retrosplenial cortex (RSC), the lateral occipital complex (LOC), and, to some degree, primary visual cortex (V1) contain information about the categories of natural scenes. To truly understand how the brain categorizes natural scenes, however, it is necessary to grasp the interactions between these regions of interest (ROIs). Our new technique for pattern-based functional connectivity enables us to uncover shared scene category-specific information among the ROIs. When configured for exploratory whole-brain analysis, the technique allows us to discover other brain regions that may be involved in natural scene categorization. Mutual information is appropriate for fMRI analysis if one considers fMRI data as a noisy communication channel in the sense of Shannon’s information theory [19]; the information contained ∗Barry Chai and Dirk B. Walther contributed equally to this work. †Diane M. Beck and Li Fei-Fei contributed equally to this work. 1 in a population of neurons must be communicated through hemodynamic changes and concomitant changes in magnetization which can be measured as the blood-oxygen level dependent (BOLD) fMRI signal, then proceed through several layers of data processing, culminating in a single time varying value in a particular voxel. While this noisy communication concept has been embraced by the brain machine interface community [25], information theory has, thus far, been less utilized in the fMRI analysis community (see [8] for exceptions). This may be partly due to the numerical difficulties in estimating the probability distributions necessary for computing mutual information. This problem is exacerbated when patterns of voxels are considered. In this case distributions of higher dimensionality need to be estimated from preciously few data points. Recent developments in information theory, however, help us overcome these hurdles. In Section 2 we review these theoretical advances and adapt them for our dual purpose of voxel selection and pattern-based functional connectivity analysis. Following a discussion of related work in Section 3, in Section 4 we apply our new methods to fMRI data from an experiment on distinguishing natural scene categories in the human brain. We lay conclude the paper in Section 5. 2 Multivariate mutual information for fMRI data Information theory was originally formulated for discrete variables. In order to adapt the theory to continuous random variables, the underlying probability distribution needs to be estimated from the sampled data points. Previous work such as [7, 18] have used fixed bin-size histogram or Parzen window methods for this purpose. However, these methods do not generalize to high-dimensional data. Recently, Perez-Cruz has shown that a k-nearest-neighbor (kNN) approach to estimating information theoretic measures converges to the true information theoretic measures asymptotically with finite k, even in higher dimensional spaces [16]. In this section we adapt this strategy to estimate multi-voxel mutual information. 2.1 Nearest-neighbor mutual information estimate In information theory, the randomness of a probability distribution is measured by its entropy. For a discrete random variable x, entropy can be calculated as H(x) = − n X i=1 p(xi) log p(xi). (1) Mutual information is intuitively defined as the reduction of the entropy of the random variable x by the entropy of x after y is known: I(x, y) = H(x) −H(x|y). (2) The separation into entropies allow us to calculate mutual information for multivariate data. Random variables x, y can be of arbitrary dimensions. As shown in [24], using kNN estimation, entropies and conditional entropies can be defined as H(x) = −1 n n X i=1 log pk(xi), (3) H(x|y) = −1 n n X i=1 log pk(xi, yi) pk(yi) , (4) where the summation is over n data points, each represented by xi. pk(xi) is the kNN density estimated at xi. pk(xi) is defined as pk(xi) = k n −1 Γ(d/2 + 1) πd/2 1 rk(xi)d . (5) where Γ is the gamma function, d is the dimensionality of xi and rk(xi) is the Euclidean distance from xi to the kth nearest training point. pk(xi) is the probability density function at xi, which is a set of voxel values for a given category task(or label) in the context of our fMRI experiment. 2 2.2 fMRI multivariate information analysis In previous work, such as [7], information theory has been used as a measure for functional connectivity of one voxel to another voxel. While such analysis is valuable for exploring connections in the brain, it does not fully leverage the information stored in the local pattern of voxels. In this section we propose a framework for multivariate information analysis of fMRI data for dual purposes: voxel selection and functional connectivity. 2.2.1 Voxel selection based on mutual information with respect to ground truth label For voxel selection we are interested in finding a subset of voxels that are highly informative for discriminating between the ground truth labels in the experiment. This is a useful step that serves two purposes. From a machine learning perspective, reducing the dimensionality of the brain image data can boost classifier performance and reduce classifier variance. From a neuroscience perspective, the locations of highly informative voxels identify functional regions involved in the experiment. To achieve both of these goals we use a multivariate mutual information measure to analyze a localized pattern of M voxels. This local analysis windows is moved across the brain image. At each location we estimate the mutual information shared between the pattern of M voxels and the experiment label. In our experiments we choose M = 7 to evaluate the smallest symmetrical pattern around a center voxel, which consists of the center voxel and its 6 face-connected neighbors. Mutual information between voxels V and labels L is defined as I(V, L) = H(V ) + H(L) −H(V, L). (6) Using equation 1 the entropies can be calculated by I(V, L) = −1 n n X i=1 log pk(Vi) −1 n n X i=1 log pk(Li) + 1 n n X i=1 log pk(Vi, Li), (7) where n is the number of data-points observed, Li is the experiment label for ith data point, Vi is a 7-dimensional random variable, Vi = (vi1,vi2,vi3,vi4,vi5,vi6, vi7) with each entry corresponding to one of 7 voxels’ values at data point i. Equation 7 can be used to compute the mutual information of localized set of voxels Vi with respect to their ground truth label Li. We can then perform voxel selection by selecting the locations of highest mutual information. This is useful as a preprocessing step before applying any machine learning algorithms and as well as a way to spatially map out the informative voxels with respect to the task. 2.2.2 Functional connectivity by shared information between distributed voxel patterns Two distributed brain regions can be modeled as a communication channel. Measuring the mutual information across the two regions provides an intuitive measure for their functional connectivity. The voxel values observed in each region can be regarded as observed data from an underlying probability distribution – the distribution that characterizes the functional region under the experiment condition. Previous approaches have analyzed shared information in a univariate way, computing the mutual information between two voxels. However such univariate information analysis disregards the information stored in the local patterns of voxels. In this work we present a multivariate information analysis that estimates shared information between two sets of voxels that leverages the information stored in the local patterns: I(V, S|L) = H(V |L) + H(S|L) −H(V, S|L), (8) where V and S are random variables for sets of 7 voxels. L is the experiment label. Using equations 3 and 4 this can be written as I(V, S|L) = −1 n n X i=1 log pk(Vi, Li) pk(Li) −1 n n X i=1 log pk(Si, Li) pk(Li) + 1 n n X i=1 log pk(Vi, Si, Li) pk(Li) . (9) Equation 9 allows us to measure the functional connectivity between two distributed sets of voxels V and S by computing the mutual information between the two sets of voxels conditioned on the experiment task label L. We show in our experiments (sec 4.4) that by using this measurement, our algorithm can uncover meaningful functional connectivity patterns among regions of the brain. 3 0 200 400 600 800 1000 1200 1400 0 0.1 0.3 0.4 0.5 Number of voxels Decoding accuracy most discr. random chance level 1 6 Mut. Info. (7 vox.) Mut. Info. (1 vox.) most active Figure 1: Comparison of decoding accuracy3 between MI voxel selection and other standard voxel selection methods(refer to section 4.2). The single voxel MI approach surpasses most discr.1 voxel selection but performs on par with most active2 voxel selection. Using a pattern of 7 voxels, the MI7D approach achieves the highest decoding accuracy. At 600 voxels, MI7D decoding accuracy3 is significantly higher than most active with pvalue < 0.05. At 1250 voxels, MI7D decoding accuracy is significantly higher than MI1D with p-value < 0.01 (This figure must be viewed in color) 3 Related work Statistical relationships between different parts of the brain, referred to as functional connectivity, have been computed with a number of different methods. The methods can be broadly classified as either data-driven or model-based [10]. In data-driven approaches, no specific hypothesis of connectivity is used, but large networks of brain regions are discovered based purely on the data. Most commonly, this is achieved with a dimensionality-reduction procedure such as principal component analysis (PCA) or independent component analysis (ICA). Originally applied to the analysis of PET data [5], PCA has also been applied to fMRI data (see [12]). ICA has been gained interest for the investigation of the so-called default network in the brain at rest [11]. Model-based approaches test a prior hypothesis about the statistical relations between a seed voxel and a target voxel. By fixing the seed voxel and moving the target voxel all over the brain, a connectivity map with respect to the seed voxel can be generated. The statistical relation of the two voxels is usually modeled assuming temporal dependence between voxels in methods such as: cross-correlation [2], coherence [21], Granger causality [1], or transfer entropy [20]. These methods compare the time courses of individual voxels. Following the same principal idea, we model functional connectivity based on the mutual information between sets of seed and target voxels to leverage the spatial information contained in activity patterns among voxels rather than the temporal information between two voxels. fMRI has a higher spatial resolution than temporal resolution. We design our mutual information connectivity measure to exploit this property of fMRI data. Yao et al. [26] have also explored pattern-based functional connectivity by modeling the interactions between distributed sets of voxels with a generative model. We take a simpler approach by using only the multivariate information measure which allows us to explore for unknown connections in the whole brain in a searchlight manner. In recent years it has become apparent that patterns of fMRI activity hold more detailed information about experimental conditions than the activation levels of individual voxels [6]. It is therefore 1Most discri. – Most discriminative voxels are those showing the largest difference in activity between any pair of scene categories. 2Most active – Most active voxels are those showing the largest difference in activity between the fixation condition and viewing images of any category. 4 0 0.17 MI L R Figure 2: Locations of voxels with high 7D mutual information with respect to scene category label. The known functional areas that respond to scenes and visual stimuli such as PPA, RSC, V1 are all selected, which also explains the high decoding accuracy using the selected voxels. The brain maps shown above are based on group analysis over 5 subjects superimposed on an MNI standard brain. (This figure must be viewed in color) cogent to also consider the information contained in voxel patterns for the analysis of functional connectivity. We achieve this by computing the mutual information of a pattern of locally connected voxels at the seed location with a pattern at the target location. As with the univariate functional connectivity analysis, this multivariate version also allows us to test hypotheses about connectivity of brain regions as well as generate connectivity maps. Because of the large number of voxels in the brain (many thousands, depending on resolution), multivariate techniques usually require some kind of feature selection or dimensionality reduction. This can be achieved by focusing on pre-defined ROIs, or by selecting voxels form the brain based on some statistical criteria [3, 14]. Here we show that using mutual information of individual voxels with respect to ground truth for voxel selection works at the same level as these previous methods, but that mutual information of patterns of voxels with respect to ground truth outperforms all of the univariate methods we tested. Information theory has been applied to fMRI data in the context of brain machine interfaces [25], to generate activation maps [8], for effective connectivity in patients [7], and for image registration [17]. However, to our knowledge this is the first application to both voxel selection and functional connectivity based on multivariate activity patterns. 4 Experiments 4.1 Data For the experiments described in this section we use the data from the fMRI experiment on natural scene categories by [23]. Briefly, five participants passively viewed color images belonging to six categories of natural scenes (beaches, buildings, forests, highways, industry, and mountains). Stimuli were arranged into blocks of 10 images from the same natural scene category. Each image was displayed sequentially for 1.6 seconds. A run was composed of 6 blocks, one for each natural scene category, interleaved with 12 s fixation periods. Images were presented upright inverted on alternating runs, with each inverted run preserving the image and category order used in the preceding upright run. A session contained 12 such runs, and the order of categories was randomized across blocks. Each subject performed two blocks with a total of 24 runs. In total we have 1192 data points per subject across all 6 categories. The data obtained from the authors in [23] contains only localizers for V1, PPA, RSC, LOC, FFA areas. Thus we limit our seed areas to these ROIs. 4.2 Voxel selection The goal of voxel selection is to identify the most relevant voxels for the experiment task out of the tens of thousands of voxels in the entire brain. A quantitative evaluation of voxel selection is the decoding accuracy3 of the selected voxels, which measures how well can the selected voxels predict the viewing condition from the neural responses. Fig.1 compares our mutual information-based voxel selection method to other voxel selection methods. Decoding accuracy3 using univariate kNN mutual information is comparable to most active1 5 lPPA rPPA lRSC rRSC 0 0.5 1 1.5 2 Log (6) 3 MI MI 7D MI 1D * 2 * ** ** ** (a) Comparing 7D and 1D mutual information within-ROI connections lPPA rPPA lRSC rRSC V1 0.95** 0.84** 1.1** 0.71** 0.85** 1.16** 0.75* (b) 7D MI between-ROI connections Figure 3: a) Within-ROI MI values for 7D and 1D mutual information, b) Schematic showing the significant ROI connections found using 7D mutual information analysis. The network shows strong connections between PPA and RSC, both ipsilaterally and contralaterally. ∗∗p < 10−6, ∗p < 0.01 voxel selection. Multivariate information measure is able to select the more informative voxels by considering a local pattern of voxels jointly, leading to a boost in decoding accuracy3. To further understand why multivariate mutual information boosts the decoding accuracy3, we can look at the spatial locations of the informative voxels selected by multivariate information analysis shown in Fig.2. The most informative voxels selected correspond to known functional regions for scenes. In this figure we see the scene areas V1, RSC, PPA, LOC that were also identified in [23]. Interestingly, our automatic voxel selection achieves a higher decoding accuracy3 than the ROIs selected by localizer in [23]. This may suggest that the multivariate information voxel selection is a better segmentation of the relevant ROIs than the localizer runs. 4.3 Functional connectivity of ROIs In the previous section, we have shown that multivariate information can effectively select informative voxels for classification. In this section, we first illustrate the increased sensitivity of a multivariate assessment of functional connectivity within known ROIs. Then we use multivariate information to explore connections between ROIs. A good comparison for the functional connectivity measure is the within-ROI connectivity. Voxels within the same ROI should exhibit high functional connectivity with each other. In Fig.3a we compared our 7D measures with equivalent one dimensional measures using within-ROI connectivity. To this end we randomly selected 15 seed and 15 target locations within each ROI, making sure that seed and target patterns have no voxels in common. Then we computed mutual information between all seed and all target locations, either using individual voxels (1D case) or patterns of seven voxels (7D case). Fig.3a shows the mean of the mutual information values for these two cases in each ROI. In all ROIs, we find that multivariate information measure(7D) is significantly higher than the univariate measure(1D), suggesting that a pattern-based mutual information has a higher fidelity than univariate-based mutual information in mapping out functional connections. After having established that 7D mutual information significantly outperforms 1D mutual information we proceed to calculate the between-ROI connectivity for scene areas V1, left/right PPA, and left/right RSC using 7D mutual information as shown in Fig.3b. Between-ROI connectivity is de3Decoding accuracy is obtained with a leave-two-runs-out cross-validation on the our scene data. In each fold two runs from viewing the same images upright and inverted are left out as test data. Voxel selection is performed on the training runs using k = n/2, where n is the number of training examples in each category. Using selected voxels, a linear SVM classifier is trained on the upright runs with C = 0.02 as in [23]. In testing we use majority voting on the SVM prediction labels to vote for the most likely scene label for each block of data. Decoding accuracy is the average of cross-validation accuracy over the 5 subjects. 6 0.37 0.51 rRSC left Precuneus lPPA rPPA lRSC rRSC left Medial Frontal Gyrus lPPA rPPA right Inferior Frontal Gyrus MI Z = -13 mm Z = 23 mm Y = 52 mm L R Figure 4: Connectivity map seeding from left PPA. Talairach coordinates defined in [22] are shown as the Z and Y coordinates for axial and coronal slices respectively. The intensity of the maps shows the MI values(This figure must be viewed in color) left Medial Frontal Gyrus lRSC rRSC lPPA rPPA left Cuneus Z = 26 mm Y = 43 mm Z = 29 mm left Medial Frontal Gyrus left Cuneus right Cuneus right Precuneus - 4 overlap - 3 overlap - 2 ovlerap L R Figure 5: Overlap analysis showing areas where overlap occurs with the strongest connections from more than two scene network ROIs. Talairach coordinates defined in [22] are shown as the Z and Y coordinates for axial and coronal slices respectively. The color code indicates the amount of overlap.(This figure must be viewed in color) fined similarly as within-ROI connectivity except that seed and target locations are the chosen in different ROIs. A number of aspects of the connections mapped out with MI7D analysis agree with neuroscience findings. First, it is expected that PPA and RSC should be strongly connected as part of a scene network. Moreover, since V1 is the input to the cortical visual system, it is also likely that it should share information with at least one member of the scene network, which in this case was the right PPA. One novel finding from this analysis is that all of the strongest connections we discovered included right RSC. In particular, right RSC shares strong connections with left RSC, right PPA and left PPA, suggesting that right RSC may play a particularly important role in distinguishing natural scene categories. More work will be needed to verify this hypothesis. To summarize, we have verified that multivariate information analysis can reliably map out connections within and between ROIs known to be involved in processing natural scene categories. In the next section we show how the same analysis can be extended to uncover other ROIs that share information with this scene network in our scene classification experiment. 4.4 Functional connectivity - whole brain analysis While it is valuable to confirm existing hypotheses about areas that represent scene categories, it is also interesting to uncover new brain areas that might be related to scene categorization. In this section, we show that we can use our multivariate information analysis approach to explore other areas outside of the known ROIs that form strong connections with the known ROIs. For each of the functional areas in the scene network, we can explore other areas connected to it. As in section 3 we measure functional connectivity as multivariate mutual information between the seed and candidate target areas. We fix the seed area to an ROI defined by a localizer. The candidate area moves around the brain, at each location measuring the mutual information with respect to the seed area. 7 4.4.1 Confirming known connections Fig.4 shows an example of the connectivity map seeding from left PPA. Each highlighted location in the connectivity map shows its connectivity to left PPA as measured by the multivariate information. As shown in Fig.4, both left and right PPA are highlighted, confirming their bilateral connection. Furthermore, we see strong connections between left PPA and left and right RSC. A minimum cluster size of 13 is used to threshold the connectivity map. The minimum cluster size is determined by AlphaSim in AFNI [4]. Notice in Fig.4 that the highest MI in the whole-brain analysis has MI of 0.51 whereas the within-ROI MI of left PPA in Fig.3b has a value of 1.5. The decrease in MI is due to the smoothing of connectivity maps when we combine them across subjects. 4.4.2 Discovering new connections Besides confirming known regions of the scene network, our connectivity maps allow us to explore other brain areas that might be related to the scene network. In Fig.4 we not only observe known scene network ROIs but additional areas such as the right Inferior Frontal Gyrus, left Medial Frontal Gyrus, and left Precuneus. Interestingly, the Inferior Frontal Gyrus, typically associated with language processing [13], also showed up in a searchlight analysis for decoding accuracy in [23]. So far we have examined how the rest of the brain connects to one ROI in the scene network, specifically we used left PPA as the example. However, to further strongly establish which regions are functionally connected in regards to distinguishing scene category, we asked which brain areas are strongly connected to two or more of the scene network ROIs. Areas that connect to more than one of the scene network ROIs are particularly interesting, because having multiple connections strengthens evidence that they play a significant role in distinguishing scene categories. To investigate this question, we generate one connectivity map for each of the 4 scene network ROIs, similar to Fig.4. We take the areas with the top 5 percent highest mutual information in each of the 4 maps and overlap them. Fig.5 shows this overlap analysis. Similar to the previous analysis, the overlap analysis highlights all 4 known areas of the scene network. Interestingly, this analysis shows that right RSC and right PPA are connected with more regions of the scene network than left RSC and PPA. This suggests that perhaps there is a laterality effect in the scene network that could be investigated in future studies. Furthermore, we can also explore areas outside of the scene network with the overlap analysis. In Fig.5, left/right Cuneus and right Precuneus, highlighted in orange, exhibit strong connections with 3/4 of the scene network ROIs. Left Medial Frontal Gyrus is strongly connected to 2/4 of the scene network ROIs. These exploratory areas also point to interesting future investigations for scene category studies. 5 Conclusion In this paper we have introduced a new method for evaluating the mutual information that patterns of fMRI voxels share with the ground truth labels of the experiment and with patterns of voxels elsewhere in the brain. When used as a voxel selection method for subsequent decoding of viewed natural scene category, mutual information of patterns of voxels with respect to the ground truth label is superior to mutual information of individual voxels. We have shown that mutual information of voxel patterns in two ROIs is a more sensitive measure of task-specific functional connectivity analysis than mutual information of individual voxels. We have identified a network of regions consisting of left and right PPA and left and right RSC that share information about the category of a natural scene viewed by the subject. Connectivity maps generated with this method have identified left medial frontal gyrus, left/right cuenus, and right precuneus as sharing scene-specific information with PPA and RSC. This could stimulate interesting future work such as estimating mutual information for an even larger set of voxels and understanding the exploratory areas highlighted by this analysis. Although we confined our experiments to data from a scene category task, all the analysis proposed here could be used for other tasks in other domains. Acknoledgements This work is funded by National Institutes of Health Grant 1 R01 EY019429 (to L.F.-F., D.M.B., D.B.W.), a Beckman Postdoctoral Fellowship (to D.B.W.), a Microsoft Research New Faculty Fellowship (to L.F.-F.), and the Frank Moss Gift Fund (to L.F-F.). The authors would like to thank Todd Coleman and Fernando Perez-Cruz for the helpful discussions on entropy estimation. 8 References [1] Granger, C. W. J. Investigating causal relations by econometric models and cross-spectral methods. Econometrica 37, 424-438, 1969 [2] J. Cao and K. Worsley The geometry of correlation fields with an application to functional connectivity of the brain. Ann. Appl. Probab, 9:1021C1057, 1998. [3] D. Cox and R. Savoy Functional magnetic resonance imaging (fMRI) “brain reading”: Detecting and classifying distributed patterns of fMRI activity in human visual cortex. NeuroImage, 19(2):261C270, 2003. [4] RW Cox. AFNI: Software for analysis and visualization of functional magnetic resonance neuroimages. Computers and Biomedical Research, 29:162-173, 1996. [5] K. J. Friston, C. D. Frith, P. F. Liddle, and R. S. Frackowiak. Functional connectivity: the principalcomponent analysis of large (PET) data sets. J Cereb Blood Flow Metab, 13(1):5C14, January 1993. [6] J. V. Haxby, M. I. Gobbini, M. L. Furey, A. Ishai, J. L. Schouten, and P. Pietrini. Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293(5539):2425C30, 2001. Journal Article United States. [7] H. Hinrichs, H. Heinze, and M. Schoenfeld. Causal visual interactions as revealed by an information theoretic measure and fMRI. NeuroImage, 31(3):1051 C 1060, 2006. [8] A. T. John, J. W. F. Iii, W. M. W. Iii, J. Kim, and A. S.‘Willsky. Analysis of functional MRI data using mutual information, 1999. [9] Fei-Fei, L., Iyer, A., Koch, C., Perona, P. (2007). What do we perceive in a glance of a real-world scene? Journal of Vision,7(1):10, 1-29, http://journalofvision.org/7/1/10/,doi:10.1167/7.1.10. [10] K. Li, L. Guo, J. Nie, G. Li, and T. Liu. Review of methods for functional brain connectivity detection using fMRI. Computerized medical imaging and graphics: The Official Journal of the Computerized Medical Imaging Society, 33(2):131C139, March 2009. [11] M. J. Mckeown, S. Makeig, G. G. Brown, T.-P. Jung, S. S. Kindermann, R. S. Kindermann, A. J. Bell, and T. J. Sejnowski. Analysis of fMRI data by blind separation into independent spatial components. Human Brain Mapping, 6:160C188, 1998. [12] A. Meyer-Baese, A. Wismueller, and O. Lange. Comparison of two exploratory data analysis methods for fMRI: unsupervised clustering versus independent component analysis. Information Technology in Biomedicine, IEEE Transactions on, 8(3):387C398, Sept. 2004. [13] Geschwind N. (1970) The organization of language and the brain. Science 170:940 944.. [14] D. Neill, A. Moore, F. Pereira, and T. Mitchell. Detecting significant multidimensional spatial clusters. In Proceedings of Neural Information Processing Systems, 2004. [15] M. Potter. Short-term conceptual memory for pictures. Journal of Experimental Psychology: Human Learning and Memory, 2(5):509C522, 1976. [16] F. Perez-Cruz. Estimation of information theoretic measures for continuous random variables. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, NIPS, pages 1257C1264. MIT Press, 2008. [17] Pluim, J. P. W. and Maintz, J. B. A. and Viergever, M. A. Mutual-information based registration of medical images: a survey. IEEE Trans Med Imaging, 2003, 22:986C1004. [18] X. U. Rui, C. H. E. N. Yen-Wei, T. A. N. G. Song-Yuan, S. Morikawa, and Y. Kurumi. Parzen-window based normalized mutual information for medical image registration. IEICE Transactions on Information and Systems, E91-D(1):132C144, January 2008. [19] C. E. Shannon. A Mathematical Theory of Communication. CSLI Publications, 1948. [20] T. Schreiber. Measuring Information Transfer. Physical Review Letters, vol. 85, no. 2, pp. 461+, July 2000. [21] F. T. Sun, L. M. Miller, and M. DEsposito. Measuring interregional functional connectivity using coherence and partial coherence analyses of fmri data. NeuroImage, 21(2):647 C 658, 2004. [22] J. Talairach and P. Tournoux. Co-planar Stereotaxic Atlas of the Human Brain: 3-Dimensional Proportional System - an Approach to Cerebral Imaging. Thieme Medical Publishers, New York, 1988 [23] D. Walther, E. Caddigan, L. Fei-Fei*, and D. Beck* (2009), Natural scene categories revealed in distributed patterns of activity in the human brain. The Journal of Neuroscience, 29(34):1057310581.(*indicates equal contribution) [24] Q. Wang, S. Kulkarni, and S. Verdu. Divergence estimation of continuous distributions based on datadependent partitions. Information Theory, IEEE Transactions on, 51(9):3064C3074, Sept. 2005. [25] B. D. Ward and Y. Mazaheri. Information transfer rate in fMRI experiments measured using mutual information theory. Journal of Neuroscience Methods, 167(1):22 C 30, 2008. Brain-Computer Interfaces (BCIs). [26] B. Yao, D.B. Walther, D.M. Beck*, L. Fei-Fei*. Hierarchical Mixture of Classification Experts Uncovers Interactions between Brain Regions. NIPS, 2009. (* indicates equal contribution) 9
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Semi-supervised Learning using Sparse Eigenfunction Bases Kaushik Sinha Dept. of Computer Science and Engineering Ohio State University Columbus, OH 43210 sinhak@cse.ohio-state.edu Mikhail Belkin Dept. of Computer Science and Engineering Ohio State University Columbus, OH 43210 mbelkin@cse.ohio-state.edu Abstract We present a new framework for semi-supervised learning with sparse eigenfunction bases of kernel matrices. It turns out that when the data has clustered, that is, when the high density regions are sufficiently separated by low density valleys, each high density area corresponds to a unique representative eigenvector. Linear combination of such eigenvectors (or, more precisely, of their Nystrom extensions) provide good candidates for good classification functions when the cluster assumption holds. By first choosing an appropriate basis of these eigenvectors from unlabeled data and then using labeled data with Lasso to select a classifier in the span of these eigenvectors, we obtain a classifier, which has a very sparse representation in this basis. Importantly, the sparsity corresponds naturally to the cluster assumption. Experimental results on a number of real-world data-sets show that our method is competitive with the state of the art semi-supervised learning algorithms and outperforms the natural base-line algorithm (Lasso in the Kernel PCA basis). 1 Introduction Semi-supervised learning, i.e., learning from both labeled and unlabeled data has received considerable attention in recent years due to its potential in reducing the need for expensive labeled data. However, to make effective use of unlabeled examples one needs to make some assumptions about the connection between the process generating the data and the process of assigning labels. There are two important assumptions popular in semi-supervised learning community the “cluster assumption” [CWS02] and the “manifold assumption” [BNS06] as well as a number of model-based methods, such as Naive Bayes [HTF03]. In particular, the cluster assumption can be interpreted as saying that two points are likely to have the same class labels if they can be connected by a path passing through a high density area. In other words two high density areas with different class labels must be separated by a low density valley. In this paper, we develop a framework for semi-supervised learning when the cluster assumption holds. Specifically, we show that when the high density areas are sufficiently separated, a few appropriately chosen eigenfunctions of a convolution operator (which is the continuous counterpart of the kernel matrix) represents the high density areas reasonably well. Under the ideal conditions each high density area can be represented by a single unique eigenfunction called the “representative” eigenfunction. If the cluster assumption holds, each high density area will correspond to just one class label and thus a sparse linear combination of these representative eigenfunctions would be a good classifier. Moreover, the basis of such eigenfunctions can be learned using only the unlabeled data by constructing the Nystrom extension of the eigenvectors of an appropriate kernel matrix. Thus, given unlabeled data we construct the basis of eigenfunctions and then apply L1 penalized optimization procedure Lasso [Tib96] to fit a sparse linear combination of the basis elements to 1 the labeled data. We provide a detailed theoretical analysis of the algorithm and show that it is comparable to the state-of-the-art on several common UCI datasets. The rest of the paper is organized as follows. In section 2 we provide the proposed framework for semi-supervised learning and describe the algorithm. In section 3 we provide an analysis of this algorithm to show that it can consistently identify the correct model. In section 4 we provide experimental results on synthetic and real datasets and finally we conclude with a discussion in section 5. 2 Semi-supervised Learning Framework 2.1 Outline of the Idea In this section we present a framework for semi-supervised learning under the cluster assumption. Specifically we will assume that (i) data distribution has natural clusters separated by regions of low density and (ii) the label assignment conforms to these clusters. The recent work of [SBY08a, SBY08b] shows that if the (unlabeled) data is clustered, then for each high density region there is a unique (representative) eigenfunction of a convolution operator, which takes positive values for points in the chosen cluster and whose values are close to zero everywhere else (no sign change). Moreover, it can be shown (e.g., [RBV08]) that these eigenfunctions can be approximated from the eigenvectors of a kernel matrix obtained from the unlabeled data. Thus, if the cluster assumption holds we expect each cluster to have exactly one label assignment. Therefore eigenfunctions corresponding to these clusters should produce a natural sparse basis for constructing a classification function. This suggests the following learning strategy: 1. From unlabeled and labeled data obtain the eigenvectors of the Gaussian kernel matrix. 2. From these eigenvectors select a subset of candidate eigenvectors without sign change. 3. Using the labeled data, apply Lasso (sparse linear regression) in the constructed basis to obtain a classifier. 4. Using the Nystrom extension (see [BPV03]), extend the eigenvectors to obtain the classification function defined everywhere. Connection to Kernel PCA ( [SSM98]). We note that our method is related to KPCA, where data is projected onto the space spanned by the top few eigenvectors of the kernel matrix and classification or regression task can be performed in that projected space. The important difference is that we choose a subset of the eigenvectors in accordance to the cluster assumption. We note that the method simply using the KPCA basis does not seem to benefit from unlabeled data and, in fact, cannot outperform the standard fully supervised SVM classifier. On the other hand, our algorithm using a basis subselection procedure shows results comparable to the state of the art. This is due to two reasons. We will see that each cluster in the data corresponds to its unique representative eigenvector of the kernel matrix. However, this eigenvector may not be among the top eigenvectors and may thus be omitted when applying KPCA. Alternatively, if the representative eigenvector is included, it will be included with a number of other uninformative eigenvectors resulting in poor performance due to overfitting. We now proceed with the detailed discussion of our algorithm and its analysis. 2.2 Algorithm The focus of our discussion will be binary classification in the semi-supervised setting. Given l labeled examples {(xi, yi)}l i=1 sampled from an underlying joint probability distribution PX,Y, X ⊂Rd, Y = {−1, 1}, where xis are the data points, yis are their corresponding labels and u unlabeled examples {xi}l+u i=l+1 drawn iid from the marginal distribution PX , we choose a Gaussian kernel k(x, z) = exp −∥x−z∥2 2ω2 with kernel bandwidth ω to construct the kernel matrix K where Kij = 1 uk(zi, zj). Let (λi, vi)u i=1 be the eigenvalue-eigenvector pair of K sorted by the non-increasing eigenvalues. It has been shown ([SBY08a, SBY08b]) that when data distribution PX 2 has clusters, for each high density region there is a unique representative eigenfunction of a convolution operator that takes positive values around the chosen cluster and is close to zero everywhere else. Moreover these eigenfunctions can be approximated from the eigenvectors of a kernel matrix obtained from the unlabeled data ([RBV08]), thus for each high density region there is a unique representative eigenvector of the kernel matrix that takes only positive or negative values in the chosen cluster and is nearly zero everywhere else (no sign change). If the cluster assumption holds, i.e., each high density region corresponds to a portion of a pure class, then the classifier can be naturally expressed as a linear combination of the representative eigenfunctions. representative eigenvector basis and a linear combination of the representative eigenvectors will be a reasonable candidate for a good classification function. However, identifying representative eigenvectors is not very trivial because in real life depending on the separation between high density clusters the representative eigenvectors can have no sign change up to some small precision ϵ > 0. Specifically, we say that a vector e = (e1, e2, ..., en) ∈Rn has no sign change up to precision ϵ if either ∀i ei > −ϵ or ∀i ei < ϵ. Let Nϵ be the set of indices of all eigenvectors that have no sign change up to precision ϵ. If ϵ is chosen properly, Nϵ will contain representative eigenvectors (note that the set Nϵ and the set {1, 2, ..., |Nϵ|} are not necessarily the same). Thus, instead of identifying the representative eigenvectors, we carefully select a small set containing the representative eigenvectors. Our goal is to learn a linear combination of the eigenvectors P i∈Nϵ βivi which minimizes classification error on the labeled examples and the coefficients corresponding to non-representative eigenvectors are zeros. Thus, the task is more of model selection or sparse approximation. Standard approach to get a sparse solution is to minimize a convex loss function V on the labeled examples and apply a L1 penalty (on βis). If we select V to be square loss function, we end up solving the L1 penalized least square or so called Lasso [Tib96], whose consistency property was studied in [ZY06]. Thus we would seek a solution of the form arg min β (y −Ψβ)T (y −Ψβ) + λ||β||L1 (1) which is a convex optimization problem, where Ψ is the l × |Nϵ| design matrix whose ith column is the first l elements of vNϵ(i), y ∈Rl is the label vector, β is the vector of coefficients and λ is a regularization parameter. Note that solving the above problem is equivalent to solving arg min β (y −Ψβ)T (y −Ψβ) s.t. X i∈Nϵ |βi| ≤t (2) because for any given λ ∈[0, ∞), there exists a t ≥0 such that the two problems have the same solution, and vice versa [Tib96]. We will denote the solution of Equation 2, by ˆβ. To obtain a classification function which is defined everywhere, we use the Nystrom extension of the ith eigenvector defined as ψi(x) = 1 λi √l+u Pl+u j=1 vi(xj)k(x, xj). Let the set T contains indices of all nonzero ˆβis. Using Nystrom extension, classification function is given by, f(x) = P i∈T ˆβiψi(x) = Pl+u i=1 Wik(xi, x), where, W ∈Ru is a weight vector whose ith element is given by Wi = X j∈T ˆβjvj(xi) λj √u (3) and can be computed while training. Algorithm for Semi-supervised Learning Input: {(xi, yi)}l i=1, {xi}l+u i=l+1 Parameters: ω, t, ϵ 1. Construct kernel matrix K from l + u unlabeled examples {xi}l+u i=1. 2. Select set Nϵ containing indices of the eigenvectors with no sign change up to precision ϵ. 3. Construct design matrix Ψ whose ith column is top l rows of vNϵ(i). 4. Solve Equation 2 to get ˆβ and calculate weight vector W using Equation 3. 5. Given a test point x, predict its label as y = sign (Pu i=1 k(xi, x)Wi) 3 3 Analysis of the Algorithm The main purpose of the analysis is, (i) to estimate the amount of separation required among the high density regions which ensures that each high density region can be well represented by a unique (representative) eigenfunction, (ii) to estimate the number of unlabeled examples required so that eigenvectors of kernel matrix can approximate the eigenfunctions of a convolution operator (defined below) and (iii) to show that using few labeled examples Lasso can consistently identify the correct model consisting of linear combination of representative eigenvectors. Before starting the actual analysis, we first note that the continuous counterpart of the Gram matrix is a convolution operator LK : L2(X, PX ) →L2(X, PX ) defined by, (LKf)(x) = Z X k(x, z)f(z)dPX (z) (4) The eigenfunctions of the symmetric positive definite operator LK will be denoted by φL i . Next, we briefly discuss the effectiveness of model selection using Lasso (established by [ZY06]) which will be required for our analysis. Let ˆβl(λ) be the solution of Equation 1 for a chosen regularization parameter λ. In [ZY06] a concept of sign consistency was introduced which states that Lasso is sign consistent if, as l tends to infinity, signs of ˆβl(λ) matches with the signs of β∗with probability 1, where β∗is the coefficients of the correct model. Note that since we are expecting a sparse model, matching zeros of ˆβl(λ) to the zeros of β∗is not enough, but in addition, matching the signs of the non zero coefficients ensures that the true model will be selected. Next, without loss of generality assume β∗= (β∗ 1, · · · , β∗ q, β∗ q+1, · · · , β∗ |Nϵ|) has only first q terms non-zero, i.e., only q predictors describe the model and rest of the predictors are irrelevant in describing the model. Now let us write the first q and |Nϵ| −q columns of Ψ as Ψ(1) and Ψ(2) respectively. Let C = 1 l ΨT Ψ. Note that, for a random design matrix, sign consistency is equivalent to irrepresentable condition (see [ZY06]). When β∗is unknown, in order to ensure that irrepresentable condition holds for all possible signs, it requires that L1 norm of the regression coefficients corresponding to the irrelevant predictors to be less than 1, which can be written as µΨ = maxψu j ∈Ψ(2) ΨT (1)Ψ(1) T ΨT (1)ψu j 1 < 1. The requirement µΨ < 1 is not new and have also appeared in the context of noisy or noiseless sparse recovery of signal [Tro04, Wai06, Zha08]. Note that Lasso is sign consistent if irrepresentable condition holds and the sufficient condition needed for irrepresentable condition to hold is given by the following result, Theorem 3.1. [ZY06] Suppose β∗has q nonzero entries. Let the matrix C′ be normalized version of C such that C′ ij = Cij Cii and maxi,j,i̸=j |C′ ij| ≤ c 2q−1 for a constant 0 ≤c < 1, then strong irrepresentable condition holds. Our main result in the following shows that this sufficient condition is satisfied with high probability requiring relatively few labeled examples, as a result the correct model is identified consistently, which in turn describes a good classification function. Theorem 3.2. Let q be the minimum number of columns of the design matrix Ψ ∈Rl×|Nϵ|, constructed from l labeled examples, that describes the sparse model. Then for any 0 < δ < 1, if the number of unlabeled examples u satisfies u > 2048q2 log( 2 δ) g2 Nmaxλ2 Nmax , then with probability greater than 1 −δ 2 −4 exp − lλ2 Nmax 50q2 , maxi̸=j |C′ ij| < 1 2q−1. where λNmax is the N th max (to be defined later) largest eigenvalue of LK and gNmax is the N th max eigengap. Note that in our framework, unlabeled examples help polynomially fast in estimating the eigenfunctions while labeled examples help exponentially fast in identifying the sparse model consisting of representative eigenfunctions. Interestingly, in semi-supervised learning setting, similar role of labeled and unlabeled examples (in reducing classification error) has been reported in literature [CC96, RV95, SB07, SNZ08]. 3.1 Brief Overview of the Analysis As a first step of our analysis, in section 3.2, we estimate the separation requirement among the high density regions which ensures that each high density region (class) can be well represented by a unique eigenfunction. This allows us to express the classification task in this eigenfunction 4 basis where we look for a classification function consisting of linear combination of representative eigenfunctions only and thus relate the problem to sparse approximation from the model selection point of view, which is a well studied field [Wai06, ZH06, CP07]. As a second step in section 3.3, using perturbation results from [RBV08], we estimate the number of unlabeled examples required to ensure that Nystrom extensions of eigenvectors of K approximate the eigenfunctions of the convolution operator LK reasonably well with high probability. Finally, as a third step in section 3.4, we establish a concentration inequality, which along with result from the second step 2, ensures that as more and more labeled examples are used to fit the eigenfunctions basis to the data, the probability that Lasso identifies correct model consisting of representative eigenfunctions increases exponentially fast. 3.2 Separation Requirement To motivate our discussion we consider binary classification problem where the marginal density can be considered as a mixture model where each class has its own probability density function, p1(x), p2(x) and corresponding mixing weights π1, π2 respectively. Thus, the density of the mixture is p(x) = π1p1(x) + π2p2(x). We will use the following results from [SBY08a] specifying the behavior of the eigenfunction of LK corresponding to the largest eigenvalue. Theorem 3.3. [SBY08a] The top eigenfunction φL 0 (x) of LK corresponding to the largest eigenvalue λ0, (1) is the only eigenfunction with no sign change, (2) has multiplicity one, (3) is non zero on the support of the underlying density, (4) satisfies |φL 0 (x)| ≤ 1 λ0 qR k2(x, z)p(z)dz (Tail decay property), where p is the underlying probability density function. Note that the last (tail decay) property above is not restricted to the top eigenfunction alone but is satisfied by all eigenfunctions of LK. Now, consider applying LK to the three cases when the underlying probability distributions are p1, p2 and p. The largest eigenvalues and corresponding eigenfunctions in the above three cases are λ1 0, λ2 0, λ0 and φL,1 0 , φL,2 0 , φL 0 respectively. To show explicit dependency on the underlying probability distribution, we will denote the corresponding operators as Lp1 K , Lp2 K and Lp K respectively. Clearly, Lp K = π1Lp1 K + π2Lp2 K . Then we can write, Lp KφL,1 0 (x) = R k(x, z)φL,1 0 (z)p(z)dz = π1λ1 0 φL,1 0 + T1(x) where, T1(x) = π2 π1λ1 0 R k(x, z)φL,1 0 (z)p2(z)dz. In a similar way we can write, Lp KφL,2 0 (x) = π2λ2 0 φL,2 0 + T2(x) where, T2(x) = π1 π2λ2 0 R k(x, z)φL,2 0 (z)p1(z)dz. Thus, when T1(x) and T2(x) are small enough then φL,1 0 and φL,2 0 are eigenfunctions of Lp K with corresponding eigenvalues π1λ1 0 and π2λ2 0 respectively. Note that “separation condition” requirement refers to T1(x), T2(x) being small, so that eigenfunctions corresponding to the largest eigenvalues of convolution operator when applied to individual high density bumps are preserved in the case when convolution operator is applied to the mixture. Clearly, we can not expect T1(x), T2(x) to arbitrarily small if there is sufficient overlap between p1 and p2. Thus, we will restrict ourselves to the following class of probability distributions for each individual class which has reasonably fast tail decay. Assumption 1. For any 1/2 < η < 1, let M(η, R) be the class of probability distributions such that its density function p satisfies 1) R R p(x)d(x) = η where R is the minimum volume ball around the mean of the distribution. 2) For any positive t > 0, smaller than the radius of R, and for any point z ∈X \ R with dist(z, R) ≥t, the volume S = {x ∈(X \ R) ∩B(z, 3t/ √ 2)} has total probability mass R S p(x)dx ≤C1η exp −dist2(z,R) t2 for some C1 > 0. where the distance between a point x and set D is defined as dist(x, D) = inf y∈D ||x −y||. With a little abuse of notation we will use p ∈M(η, R) to mean that p is the probability density function of a member of M(η, R). Now a rough estimate of separation requirement can be given by the following lemma. Lemma 3.1. Let p1 ∈M(η, R1) and p2 ∈M(η, R2) and let the minimum distance between R1, R2 be ∆. If ∆= Ω∗ ω √ d then T1(x) and T2(x) can be made arbitrarily small for all x ∈X. The estimate of ∆in the above lemma, where we hide the log factor by Ω∗, is by no means tight, nevertheless, it shows that separation requirement refers to existence of a low density valley between 5 two high density regions each corresponding to one of the classes. This separation requirement is roughly of the same order required to learn mixture of Gaussians [Das99]. Note that, provided separation requirement is satisfied, φL,1 0 and φL,2 0 are not necessarily the top two eigenfunctions of LK corresponding to the two largest eigenvalues but can be quite far down the spectrum of Lp K depending on the mixing weights π1, π2. Next, the following lemma suggests that we can say more about the eigenfunction corresponding to the largest eigenvalue. Lemma 3.2. For any e 1+e < η < 1, let q ∈M(η, R). If φL 0 is the eigenfunction of Lq K corresponding to the largest eigenvalue λ0 then there exists a C1 > 0 such that 1) For all x ∈X \ R, |φL 0 (x)| ≤ √ (C1+η) λ0 exp −dist2(x,R) 2ω2 2) For all z ∈R and x ∈X \ R, |φL 0 (z)| ≥|φL 0 (x)| Thus for each class, top eigenfunction corresponding to the largest eigenvalue represents high density region reasonably well, outside high density region is has lower absolute value and decays exponentially fast. 3.3 Finite Sample Results We start with the following assumption. Assumption 2. The Nmax largest eigenvalues of LK and K, where Nmax = maxi{i : i ∈Nϵ}, are simple and bounded away from zero. Note that Nystrom extension ψis are eigenfunctions of an operator LK,H : H →H , where H is the unique RKHS defined by the chosen Gaussian kernel and all the eigenvalues of K are also eigenvalues of LK,H ([RBV08]). There are two implications of Assumption 2. The first one is due to the bounded away from zero part, which ensures that if we restrict to ψi ∈H corresponding to the largest Nmax eigenvalues, then each of them is square integrable hence belongs to L2(X, PX ). The second implication due to the simple part, ensures that eigenfunctions corresponding to the Nmax largest eigenvalues are uniquely defined and so are the orthogonal projections on to them. Note that if any eigenvalue has multiplicity greater than one then the corresponding eigenspace is well defined but not the individual eigenfunctions. Thus, Assumption 2 enables us to compare how close each ψi is to some other function in L2(X, PX ) in L2(X, PX ) norm sense. Let gNmax be the N th max eigengap when eigenvalues of LK are sorted in non increasing order. Then we have the following results. Lemma 3.3. Suppose Assumption 2 holds and the top Nmax eigenvalues of LK and K are sorted in the decreasing order. Then for any 0 < δ < 1 and for any i ∈Nϵ, with probability at least (1 −δ), ∥ψi −φL i ∥L2(X,PX ) = 2 gNmax q 2 log(2/δ) uλi Corollary 3.1. Under the above conditions, for any 0 < δ < 1 and for any i, j ∈Nϵ, with probability at least (1 −δ) the following holds, 1) ⟨ψi, ψj⟩L2(X,PX ) ≤ 8 log(2/δ) g2 Nmax √ λiλj 1 u + √ 8 log(2/δ) gNmax 1 √λi + 1 √ λj 1 √u 2) 1 − r 8 log(2/δ) g2 Nmaxλi 1 √u ≤∥ψu i ∥L2(X,PX ) ≤1 + r 8 log(2/δ) g2 Nmaxλi 1 √u 3.4 Concentration Results Having established that {ψi}i∈Nϵ approximate the top Nϵ eigenfunctions of LK reasonably well, next, we need to consider what happens when we restrict each of the ψis to finite labeled examples. Note that the design matrix Ψ ∈Rl×|Nϵ| is constructed by restricting the {ψj}j∈Nϵ to l labeled data points {xi}l i=1 such that the ith column of Ψ is ψNϵ(i)(x1), ψNϵ(i)(x2), · · · , ψNϵ(i)(xl) T ∈Rl. Now consider the |Nϵ| × |Nϵ| matrix C = 1 l ΨT Ψ where, Cij = 1 l Pl k=1 ψNϵ(i)(xk)ψNϵ(j)(xk). First, applying Hoeffding’s inequality we establish, Lemma 3.4. For all i, j ∈Nϵ and ϵ1 > 0 the following two facts hold. P 1 l Pl k=1[ψi(xk)]2 −E [ψi(X)]2 ≥ϵ1 ≤2 exp −lϵ2 1λ2 i 2 P 1 l Pl k=1 ψi(xk)ψj(xk) −E (ψi(X)ψj(X)) ≥ϵ1 ≤2 exp −lϵ2 1λiλj 2 Next, consider the |Nϵ| × |Nϵ| normalized matrix C′ where C′ ij = Cij Cii and C′ ii = 1. To ensure that Lasso will consistently choose the correct model we need to show (see Theorem 3.1) that 6 maxi̸=j |C′ ij| < 1 2q−1 with high probability. Applying the above concentration result and finite sample results yields Theorem 3.2. 4 Experimental Results 4.1 Toy Dataset Here we present a synthetic example in 2-D. Consider a binary classification problem where the positive examples are generated from a Gaussian distribution with mean (0, 0) and covariance matrix [2 0; 0 2] and the negative examples are generated from a mixture of Gaussians having means and covariance matrices (5, 5), [2 1; 1 2] and (7, 7), [1.5 0; 0 1.5] respectively. The corresponding mixing weights are 0.4, 0.3 and 0.3 respectively. Left panel in Figure 1 shows the probability density of the mixture in blue and representative eigenfunctions of each class in green and magenta respectively using 1000 examples (positive and negative) drawn from this mixture. It is clear that each representative eigenfunction represents high density area of a particular class reasonably well. So intuitively a linear combination of them will represent a good decision function. In fact, the right panel of Fig 1 shows the regularization path for L1 penalized least square regression with 20 labeled examples. The bold green and magenta lines shows the coefficient values for the representative eigenfunctions for different values of regularization parameter t. As can be seen, regularization parameter t can be so chosen that the decision function will consist of a linear combination of representative eigenfunctions only. Note that these representative eigenfunctions need not be the top two eigenfunctions corresponding to the largest eigenvalues. −10 −5 0 5 10 15 −20 0 20 −0.15 −0.1 −0.05 0 0.05 y Probability density for the mixture and representative eigenfunctions x Density / Eigenfunctions 0 10 20 30 40 50 60 −20 −10 0 10 20 t Coefficients Regularization path Figure 1: Left panel: Probability density of the mixture in blue and representative eigenfunctions in green and magenta. Right panel: Regularization path. Bold lines correspond to regularization path associated with representative eigenfunctions. 4.2 UCI Datasets In this set of experiment we tested the effectiveness of our algorithm (we call it SSL SEB) on some common UCI datasets. We compared our algorithm with state of the art semi-supervised learning (manifold regularization) method Laplacian SVM (LapSVM) [BNS06], fully supervised SVM and also two other kernel sparse regression methods. In KPCA+L1 we selected top |Nϵ| eigenvectors, and applied L1 regularization, in KPCA F+L1 we selected the top 20 (fixed) eigenvectors of Ku and applied L1 regularization1, where as in KPCA max+L1 we selected top max eigenvectors, and applied L1 regularization, where max is the maximum index of set of eigenvectors in Nϵ, that is the index of the lowest eigenvector, chosen by our method. For both SVM and LapSVM we used RBF kernel. In each experiment a specified number of examples (l) were randomly chosen and labeled and the rest (u) were treated as unlabeled test set. Such random splitting was performed 30 times and the average is reported. The results are reported in Table 1. As can be seen, for small number of labeled examples our method convincingly outperform SVM and is comparable to LapSVM. The result also suggests that instead of selecting top few eigenvectors, as is normally done in KPCA, selecting them by our method and then applying L1 regularization yields better result. In particular, in case of IONOSPHERE and BREAST-CANCER data sets top |Nϵ| (5 and 3 respectively) eigenvectors do not contain the representative ones. As a result in these two cases KPCA+L1 performs very poorly. Table 2 shows that the solution obtained by our method is very sparse, where average sparsity is the average number of non-zero coefficients. We note that our method does not work equally well for all datasets, and has generally higher variability than LapSVM. 1We also selected 100 top eigenvectors and applied L1 penalty but it gave worse result. 7 DATA SET IONOSPHERE HEART WINE BREAST-CANCER VOTING d=33, l+u=351 d=13, l+u=303 d=13, l+u=178 d=30, l+u=569 d=16, l+u=435 # Labeled Data l=10 l=20 l=30 l=10 l=20 l=30 l=10 l=20 l=5 l=10 l=10 l=15 SSL SEB 78.26 85.84 87.25 75.45 77.34 79.92 93.01 98.95 96.68 98.66 86.85 87.84 ±13.56 ±10.61 ±4.16 ±6.14 ±6.04 ±1.18 ±8.49 ±8.49 ±3.43 ±2.86 ±6.21 ±3.82 KPCA+L1 65.15 65.66 69.57 66.82 70.36 75.16 93.47 98.75 70.26 73.95 86.85 87.84 ±8.82 ±9.81 ±9.89 ±7.94 ±8.41 ±6.68 ±10.06 ±3.89 ±14.43 ±13.68 ±6.21 ±3.82 KPCA F+L1 64.92 67.43 69.43 60.91 67.32 71.46 79.82 87.32 63.04 81.44 71.78 77.38 ±10.13 ±11.68 ±11.26 ±7.33 ±7.01 ±5.91 ±10.29 ±8.56 ±12.29 ±13.12 ±12.65 ±10.43 KPC max+L1 59.76 64.73 66.89 57.26 60.16 63.36 84.62 89.96 59.32 73.95 71.78 77.38 ±10.23 ±11.62 ±12.45 ±5.16 ±6.69 ±6.15 ±9.63 ±9.26 ±15.18 ±8.97 ±12.65 ±10.43 SVM 65.16 72.09 79.8 64.61 73.16 76.55 83.98 88.12 72.83 97.32 81.53 88.51 ±10.87 ±10.04 ±9.94 ±11.63 ±5.95 ±4.29 ±10.25 ±11.68 ±17.56 ±8.65 ±16.05 ±5.88 LapSVM 71.17 77.18 81.32 74.91 75.33 77.43 98.33 97.67 98.95 99.72 89.52 89.97 ±7.33 ±4.07 ±3.81 ±5.55 ±6.08 ±3.14 ±5.33 ±1.57 ±2.32 ±1.42 ±1.43 ±1.26 Table 1: Classification Accuracies for different UCI datasets DATA SET IONOSPHERE HEART WINE BREAST-CANCER VOTING SSL SEB 2.83 / 5 4.63 / 9 3.52 / 6 2.10 / 3 2.02 / 3 KPCA+L1 3.23 / 5 5.84 / 9 3.8 / 6 2.78 / 3 2.02/ 3 KPCA F+L1 6.05 / 20 8.11 / 20 6.12 / 20 4.70 / 20 3.05 / 20 KPC max+L1 6.85 / 23 16.42 / 78 6.07 /16 10.81 / 57 2.02 / 3 Table 2: Average sparsity of our method for different UCI datasets. The notation A/B represents average sparsity A and number of eigenvectors (|Nϵ| or 20). 4.3 Handwritten Digit Recognition In this set of experiments we applied our method to the 45 binary classification problems that arise in pairwise classification of handwritten digits and compare its performance with LapSVM. For each pairwise classification problem, in each trial, 500 images of each digit in the USPS training set were chosen uniformly at random out of which 20 images were labeled and the rest were set aside for testing. This trial was repeated 10 times. For the LapSVM we set the regularization terms and the kernel as reported by [BNS06] for a similar set of experiments, namely we set γAl = 0.005, γIl (u+l)2 = 0.045 and chose a polynomial kernel of degree 3. The results are shown2 in Figure2. As can be seen our method is comparable to LapSVM. 0 5 10 15 20 25 30 35 40 45 0 5 10 15 20 All 45 two class classifications for USPS dataset Test error rate (%) SSL_SEB LapSVM Figure 2: Classification results for USPS dataset We also performed multi-class classification on USPS dataset. In particular, we chose all the images of digits 3, 4 and 5 from USPS training data set (there were 1866 in total) and randomly labeled 10 images from each class. Rest of the 1836 images were set aside for testing. Average prediction accuracy of LapSVM, after repeating this procedure 20 times, was 90.14% as compared to 87.53% of our method. 5 Conclusion In this paper we have presented a framework for spectral semi-supervised learning based on the cluster assumption. We showed that the cluster assumption is equivalent to the classifier being sparse in a certain appropriately chosen basis and demonstrated how such basis can be computed using only unlabeled data. We have provided theoretical analysis of the resulting algorithm and given experimental results demonstrating that the resulting algorithm has performance comparable to the state-of-the-art for a number of data sets and dramatically outperforms the natural baseline of KPCA + Lasso. 2It turned out that the cases where our method performed very poorly, the respective distances between the means of corresponding two classes were very small. 8 References [BNS06] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples. Journal of Machine Learning Research, 7:2399–2434, 2006. [BPV03] Y. Bengio, J-F. Paiement, and P. Vincent. Out-of-sample Extensions for LLE, Isomap, MDS, Eigenmaps and Spectral Clustering. In NIPS. 2003. [CC96] V. Castelli and T. M. Cover. The Relative Value of Labeled and Unlabeled Samples in Pattern Recognition with Unknown Mixing Parameters. IEEE Transactions on Information Theory, 42(6):2102–2117, 1996. [CP07] E. J. Candes and Y. Plan. Near Ideal Model Selection by ℓ1 Minimization, eprint arxiv:0801.0345. 2007. [CWS02] O. Chapelle, J. Weston, and B. Scholkopf. Cluster Kernels for Semi-supervised Learning. In NIPS. 2002. [Das99] S. Dasgupta. Learning Mixture of Gaussians. In 40th Annual Symposium on Foundations of Computer Science, 1999. [HTF03] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning Data Mining, Inference and Prediction. Springer, 2003. [RBV08] L. Rosasco, M. Belkin, and E. De Vito. Perturbation Results for Learning Empirical Opertors. Technical Report TR-2008-052, Massachusetts Institute of Technology, Cambridge, MA, August 2008. [RV95] J. Ratsaby and S. Venkatesh. Learning From a Mixture of Labeled and Unlabeled Examples with Parametric Side Information. In COLT. 1995. [SB07] K. Sinha and M. Belkin. The Value of Labeled and Unlabeled Examples when the Model is Imperfect. In NIPS. 2007. [SBY08a] T. Shi, M. Belkin, and B. Yu. Data Spectroscopy: Eigenspace of Convolution Operators and Clustering. Technical report, Dept. of Statistics, Ohio State University, 2008. [SBY08b] T. Shi, M. Belkin, and B. Yu. Data Spectroscopy: Learning Mixture Models using Eigenspaces of Convolution Operators. In ICML. 2008. [SNZ08] A. Singh, R. D. Nowak, and X. Zhu. Unlabeled Data: Now it Helps Now it Doesn’t. In NIPS. 2008. [SSM98] Bernhard Scholkopf, A. Smola, and Klaus-Robert Muller. Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10:1299–1319, 1998. [Tib96] R. Tibshirani. Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society, Series B, 58:267–288, 1996. [Tro04] J. A. Tropp. Greed is Good: Algorithmic Result for Sparse Approximation. IEEE Trans. Info. Theory, 50(10):2231–2242, 2004. [Wai06] M. Wainwright. Sharp Thresholds for Noisy and High-dimensional Recovery of Sparsity using ℓ1-constrained Quadratic Programming. Technical Report TR-709, Dept. of Statistics, U. C. Berkeley, September 2006. [ZH06] C. Zhang and J. Huang. Model Selection Consistency of Lasso in High Dimensional Linear Regression. Technical report, Dept. of Statistics, Rutgers University, 2006. [Zha08] T. Zhang. On consistency of feature selection using greedy least square regression. Journal of Machine Learning Research, 2008. [ZY06] P. Zhao and B. Yu. On Model Selection Consistency of Lasso. Journal of Machine Learning Research, 7:2541–2563, 2006. 9
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Free energy score-space Alessandro Perina1,3, Marco Cristani1,2, Umberto Castellani1 Vittorio Murino1,2 and Nebojsa Jojic3 {alessandro.perina, marco.cristani, umberto.castellani, vittorio.murino}@univr.it jojic@microsoft.com 1 Department of Computer Science, University of Verona, Italy 2 IIT, Italian Institute of Technology, Genova, Italy 3 Microsoft Research, Redmond, WA Abstract A score function induced by a generative model of the data can provide a feature vector of a fixed dimension for each data sample. Data samples themselves may be of differing lengths (e.g., speech segments, or other sequence data), but as a score function is based on the properties of the data generation process, it produces a fixed-length vector in a highly informative space, typically referred to as a “score space”. Discriminative classifiers have been shown to achieve higher performance in appropriately chosen score spaces than is achievable by either the corresponding generative likelihood-based classifiers, or the discriminative classifiers using standard feature extractors. In this paper, we present a novel score space that exploits the free energy associated with a generative model. The resulting free energy score space (FESS) takes into account latent structure of the data at various levels, and can be trivially shown to lead to classification performance that at least matches the performance of the free energy classifier based on the same generative model, and the same factorization of the posterior. We also show that in several typical vision and computational biology applications the classifiers optimized in FESS outperform the corresponding pure generative approaches, as well as a number of previous approaches to combining discriminating and generative models. 1 Introduction The complementary nature of discriminative and generative approaches to machine learning [20] has motivated lots of research on the ways in which these can be combined [5, 12, 15, 18, 9, 24, 27]. One recipe for such integration uses “generative score-spaces.” Using the notation of [24], such spaces can be built from data by considering for each observed sequence x = (x1, . . . , xk, . . . , xK) of observations xk ∈ℜd, k = 1, . . . , K, a family of generative models P = {P (x|θi)} parameterized by θi. The observed sequence x is mapped to the fixed-length score vector ϕf ˆ F (x), ϕf ˆ F (x) = ϕ ˆ F f({Pi (x|θi))}), (1) where f is the function of the set of probability densities under the different models, and ˆF is some operator applied to it. For instance, in case of the Fisher score [9], f is the log likelihood, and the operator ˆF produces the first order derivatives with respect to parameters, whereas in [24] other derivatives are also included. Another example is the TOP kernel [27] for which the function f is the posterior log-odds and ˆF is again the gradient operator. In these cases, the generative score-space approaches help to distill the relationship between a model parameter θi and the particular data sample. After the mapping, a score-space metric must 1 be defined in order to employ discriminative approaches. A number of nice properties for these mappings, and especially for Fisher score, can be derived under the assumption that the test data indeed follows the generative model used for the score computation. However, the generative score spaces build upon the choice of one (or few) out of many possible generative models, as well as the parameters fit to a limited amount of data. In practice, these models can therefore suffer from improper parametrization of the probability density function, local minima, over-fitting add under-training problems. Consider, for instance, the situation where the assumed model over high dimensional data is a mixture of n diagonal Gaussians with a given small and fixed variance, and a uniform prior over the components. The only free parameters are therefore the Gaussian centers, and let us assume that training data is best captured with these centers all lying on (or close to) a hypersphere with a radius sufficiently larger than the Gaussians’ deviation. An especially surprising and inconvenient outlier in this case would be a test data point that falls close to the center of the hypersphere, as the derivatives of its log likelihood with respect to these parameters (Gaussian centers) evaluated at the estimate could be very low when the number of components n in the mixture is large, because the derivatives are scaled by the uniform posterior 1/n. But, this makes such a test point insufficiently distinguishable from the test points that actually satisfy the model perfectly by falling directly into one of the Gaussian centers. If the model parameters are extended to include the prior distribution over mixture components, then derivatives with respect to these parameters would help disambiguate these points. In this paper, we propose a novel score space which focuses on how well the data point fits different parts of the generative model, rather than on derivatives with respect to the model parameters. We start with the variational free energy as a lower bound on the negative log-likelihood of the data, as this affords us with two advantages. First of all, the variational free energy can be computed for an arbitrary structure of the posterior distribution, allowing us to deal with generative models with many latent variables and complex structure without compromising tractability, as was previously done for inference in generative models. Second, a variational approximation of the posterior typically provides an additive decomposition of the free energy, providing many terms that can be used as features. These terms/features are divided into two categories: the “entropy set” of terms that express uncertainty in the posterior distribution, and the “cross-entropy set” describing the quality of the fit of the data to different parts of the model according to the posterior distribution. We find the resulting score space to be highly informative for discriminative learning. In particular, we tested our approach on three computational biology problems (promoter recognition, exons/introns classification, and homology detection), as well as vision problems (scene/object recognition). The results compare favorably with the state-of-the-art from recent literature. The rest of the paper is organized as follows. The next section describes the proposed framework in more detail. In Sec. 3, we show that the proposed generative score space leads to better classification performances than the related generative counterpart. Some simple extensions are described in Sec. 4, and used in the experiments in Sec. 5. 2 FESS: Free Energy Score Space A generative model defines the distribution P(h, x|θ) = QT t=1 P(h(t), x(t)|θ) over a set of observations x = {x(t)}T t=1, each with associated hidden variables h(t), for a given set of model parameters θ shared across all observations. In addition, to model the posterior distribution P(h|x), we also define a family of distributions Q from which we need to select a variational distribution Q(h) that best fits the model and the data. Assuming i.i.d data, the family Q can be simplified to include only distributions of the form Q(h) = QT t=1 q(h(t)). The free energy [19, 11] is a function of the data, parameters of the posterior Q(h), and the parameters of the model P, defined as FQ = KL(Q, P(h|x, θ)) −log P(x|θ) = X h Q(h) log Q(h) P(h, x|θ) (2) The free energy bounds the log likelihood, FQ ≤−log P(x) and the equality is attained only if Q is expressive enough to capture the true posterior distribution, as the free energy is minimized when Q(h) = P(h|x). Constraining Q to belong to a simplified family of distributions Q, however, 2 provides computational advantages for dealing with intractable models P. Examples of distribution families used for approximation are the fully-factorized mean field form [13], or the structured variational approximation [7], where some dependencies among the hidden variables are kept. Minimization of FQ as a proxy for negative log likelihood is usually achieved by alternating optimization of with respect to Q and θ, a special case of which – when Q is fully expressive – is the EM algorithm. Different choices of Q provide different types of compromise between the accuracy and computational complexity. For some models, accurate inference of some of the latent variables may require excessive computation even though the results of the inference can be correctly reinterpreted by studying the posterior Q from a simpler family and observing the symmetries of the model, or by reparametrizing the model (see for example [1]). In what follows, we will develop a technique that uses the parts of the free energy to infer the mapping of the data to a class variable with an increased accuracy despite possible imperfections of the data fit, whether this imperfection is due to the approximations and errors in the model or the posterior. Having obtained an estimate of parameters ˆθ that fit the given i.i.d. data we can rearrange the free energy (Eq.2) as FQ = X t Ft Q, and Ft Q = X h(t) q(h(t)|ˆθ) · log q(h(t)|ˆθ) − X h(t) q(h(t)|ˆθ) · log P(h(t), x(t)|ˆθ) (3) The second term in the equation above is the cross-entropy term and it quantifies how well the data point fits the model, assuming that hidden variables follow the estimated posterior distribution. This posterior distribution is fit to minimize the free energy; the first term in 3 is the entropy and quantifies the uncertainty in this fit. If Q and P factorize, then each of these two terms further breaks into a sum of individual terms, each quantifying the aspects of the fit of the data point with respect to different parts of the model. For example, if the generative model is described by a Bayesian network, the joint distribution can be written as P(v(t) = Q n P(v(t) n |PAn), where v(t) = {x(t), h(t)} denotes the set of all variables (hidden or visible) and PAn are the parents of the n −th of these variables, i.e., v(t) n . The cross-entropy term in the equation above further decomposes into X [v(t) 1 ] q(v(t) 1 ∪PA1|ˆθ) · log P(v(t) 1 |PA1, ˆθ) + · · · + X [v(t) N ] q(v(t) N ∪PAN|ˆθ) · log P(v(t) N |PAN, ˆθ) (4) For each discrete hidden variable v(t) n , the appropriate terms above can be further broken down into individual terms in the summation over the Dn possible configurations of the variable, e.g., q(v(t) n = 1, ∪PAn|ˆθ)·log P(v(t) n = 1|PAn, ˆθ)+· · ·+ q(v(t) n = Dn, ∪PAn|ˆθ)·log P(v(t) n = Dn|PAn, ˆθ) (5) In a similar fashion, the entropy term can also be decomposed further into a sum of terms as dictated by the factorization of the family Q. Therefore, the free energy for a single sample t can be expressed as the sum Ft Q = X i f t i,ˆθ (6) where all the free energy pieces f t i,ˆθ derive from the finest decomposition (5) or (4). The terms f t i,ˆθ describe how the data point fits possible configurations of the hidden variables in different parts of the model. Such information can be encapsulated in a score space that we call free energy score space or simply FESS. For example, in the case of a binary classification problem, given the generative models for the two classes, we can define as F(Q,ˆθ)(x(t)) the mapping of x(t) to a vector of scores f with respect to a particular model with its estimated parameters, and a particular choice of the posterior family Q for each of the classes, and then concatenate the scores. Therefore, using the notation from [24] the free 3 energy score operator ϕF ESS ˆ F (x(t)) is defined as ϕF ESS ˆ F : x(t) → h F(Q1,ˆθ1)(x(t)); F(Q2,ˆθ2)(x(t)) i where F(Qc,ˆθc) = [. . . , f t i, ˆ θc, . . . ]T , c = 1, 2 (7) If the posterior families are fully expressive, then the MAP estimate based on the generative models for the two classes can be obtained from this mapping by simply summing the appropriate terms to obtain the log likelihood difference, as the free energy equals the negative log likelihood. However, the mapping also allows for the parts of the model fit to play uneven roles in classification after an additional step of discriminative training. In this case the data points do not have to fit either model well in order to be correctly classified. Furthermore, even in the extreme case where one model provides a higher likelihood than the other for the data from both classes (e.g., because the models are not nested, and likelihoods cannot be directly compared), the mapping may still provide an abstraction from which another step of discriminative training can benefit. The additional step of training a discriminative model allows for mining the similarities among the data points in terms of the path through different hidden variables that has to be followed in their generation. These similarities may be informative even if the generative process is imperfect. Obviously, (7) can be generalized to include multiple models (or the use of a single model) and/or multiple posterior approximations, either for two-class or multi-class classification problems. 3 Free energy score space classification dominates the MAP classification We use here the terminology introduced in [27], under which FESS would be considered a modeldependent feature extractor, as different generative models lead to different feature vectors [25]. The family of feature extractors ϕ ˆ F : X →ℜd maps the input data x ∈X in a space of fixed dimension derived from a plug-in estimate λ, in our case the generative model with parameters ˆθ from which the features are extracted. Given some observations x and the corresponding class labels y ∈{−1, +1} following the joint probability P(x, y|θ∗), a generative model can be trained to provide an estimate ˆθ ̸= θ∗, where θ∗ are the true parameters. As most kernels (e.g. Fisher and TOP) are commonly used in combination with linear classifiers such as linear SVMs, [27] proposes as a starting point for evaluating the performance of a feature extractor the classification error of a linear classifier wT · ϕ ˆ F (x) + b in the feature space ℜd, where w ∈ℜd and b ∈ℜ. Assuming that w and b are chosen by an optimal learning algorithm on a sufficiently large training dataset, and that the test set follows the same distribution with parameter θ∗, the classification error R(ϕ ˆ F ) can be shown to tend to R(ϕ ˆ F ) = min w,b Ex,yΦ[−y(wT · ϕ ˆ F (x) + b)] (8) where Φ[a] is an indicator function which is 1 when a > 0, and 0 otherwise, and Ex,y denotes the expectation with respect to the true distribution P(x, y|θ∗). The Fisher kernel (FK) classifier can perform at least as well as its plug-in estimate if the parameters of a linear classifier are properly determined [9, 27], R(ϕF K ˆ F ) ≤Ex,tΦ[−y(P(y = +1|x, ˆθ) −1 2)] = R(λ) (9) where λ represents the generative model used as plug-in estimate. This property also trivially holds for our method, where ϕ ˆ F (x(t)) = ϕF ESS ˆ F (x(t)), because the free energy can be expressed as a linear combination of the elements of ϕ . In fact, the minimum free energy test (and the maximum likelihood rule when Q is fully expressive) can be defined on ϕ derived from the generative models with parameters ˆθ+1 for one class and ˆθ−1 for another as ˆy = min y {Ft (Q,ˆθ+1), Ft (Q,ˆθ−1)} = Φ h 1T F(Q,ˆθ+1)(x(t)) −1T F(Q,ˆθ−1)(x(t)) i (10) The extension to a multiclass classification is straightforward. When the family Q is expressive enough to capture the true posterior distribution, then free energy reduces to negative log likelihood, 4 and the free energy test reduces to ML classification. In other cases, likelihood computation is intractable, and free energy test is used instead of the likelihood ratio test. It is straightforward to prove that a kernel classifier that works in FESS is asymptotically at least as good as the MAP labelling based on the generative models for the two classes since generative classification is a special case of our framework. Lemma 3.1 For ϕF ESS ˆ F (x(t)) derived as above with its first M1 elements being the components of the free energy for one model, and the remaining M2 for the second, a linear classifier employing ϕF ESS ˆ F will, asymptotically (with enough data), provide classification error which is at least as low as RQ(λ) achieved using the free energy test above. R(ϕF ESS ˆ F ) ≤Ex,tΦ · −y(P(y = +1|x, ˆθ) −1 2) ¸ = RQ(λ) Proof R(ϕF ESS ˆ F ) = min w,b Ex,yΦ[−y(wT · ϕF ESS ˆ F (x) + b)] ≤Ex,yΦ[−y(wT · ϕF ESS ˆ F (x) + b)] ∀w, b R(ϕF ESS ˆ F ) ≤ Ex,yΦ[−y(wT g · ϕF ESS ˆ F (x) + bg)] for wg = [ M1 times z }| { +1, · · · , +1, M2 times z }| { −1, · · · , −1]T , bg = 0 R(ϕF ESS ˆ F ) ≤ RQ(λ) (11) □ Furthermore, when the family Q is expressive enough to capture the true posterior distribution, the free energy test is equivalent to maximum likelihood (ML) classification, RQ(λ) = R(λ). The dominance of the Fisher and Top kernels [9, 27] over their plug-in holds for FESS too, and the same plug-in (the likelihood under a generative model) may be used when this is tractable. However, if the computation of the likelihood (and the kernels derived from it) is intractable, then the free energy test as well as the kernel methods based on FESS that will outperform this test, can both still be used with an appropriate family of variational distributions Q. 4 Controlling the length of the feature vector In some generative models, especially sequence models, the number of hidden variables may change from one data point to the next. In speech processing, for instance, hidden Markov models (HMM) [23] may have to model utterances x(t) 1 , . . . , x(t) K(t) of different sequence lengths K(t). As each element in the sequence has an associated hidden variable, the hidden state sequences s(t) 1 , . . . , s(t) K(t) are also of variable lengths. The parameters θ of this model include the prior state distribution π, the state transition probability matrix A = a{ij}, and the emission probabilities B = b{iv}. Exact inference is tractable in HMMs and so we can use the exact posterior (EX) distribution to formulate the free energy and the free energy minimization is equivalent to the usual Baum-Welch training algorithm [17] and FEX = −log P(x). The free energy of each sample xt is Ft EX = X [s] q(s(t) 1 ) log q(s(t) 1 ) + X [s] K(t)−1 X k=1 q(s(t) k , s(t) k+1) log q(s(t) k , s(t) k+1) − X [s] q(s(t) 1 ) log πs(t) 1 − X [s] K(t)−1 X k=1 q(s(t) k , s(t) k+1) log a{s(t) k ,s(t) k+1} − X [s] K(t) X k=1 q(s(t) k ) log b{s(t) k ,x(t) k } (12) Depending on how this is broken into terms fi, we could get feature vectors whose dimension depends on the length of the sample K(t). To solve this problem, we first note that a standard approach to dealing with utterances of different lengths is to normalize the likelihood by the sequence length, and this approach is also used for defining other score spaces. If, before the application of the score operator, we simply evaluate the sums over k in the free energy and divide each by K(t), we obtain a fixed number of terms independent of the sequence length. This results in a length-normalized score space nFESS, where the granularity of the decomposition of the free energy is dramatically reduced. 5 1 2 3 4 5 6 0.05 0.1 0.15 0.2 Regularization log(C) Error Rate 1 2 3 4 5 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Regularization log(C) K=4 T=7 T=8 T=9 T=10 T=11 hidden Markov model FESS TOP Kernel (TK) Fisher-SVM (FK) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 FESS - EX FESS - MF FPS FK SAM ( λΗΜΜ ) PSI−BLAST Number of families median RFP A) C) FESS B) Error Rate 0.15 Markov model λHMM Figure 1: A) SVM error rates for nFESS and probability product kernels [10] using Markov models (we reported only their best result) and hidden Markov models as plug-ins. T represents the parameters used in the kernel of [10], and K is the order of the Markov chain. The results are arranged along the x axis by the regularization constant used in SVM training. B) Comparison with results obtained using FK and TK score spaces. C) Comparison of the five homology detection methods in Experiment 3. Y axis represents the total number of families for which a given method exceeds a median RFP score on the X axis. In general, even for fixed-length data points and arbitrary generative models, we do not need to create large feature vectors corresponding to the finest level of granularity described in (5), or for that matter the slightly coarser level of granularity in (4). Some of the terms in these equations can be grouped and summed up to ensure for shorter feature vectors, if this is warranted by the application. The longer the feature vector, the finer is the level of detail with which the generative process for the data sample is represented, but more data is needed for the training of the discriminative classifier. Domain knowledge can often be used to reduce the complexity of the representation by summing appropriate terms without sacrificing the amount of useful information packed in the feature vectors. Such control of the feature vector length does not negate the previously discussed advantages of the classification in the free energy score space compared with the straightforward application of free energy, likelihood, or in case of sequence models, length-normalized likelihood tests. 5 Experiments We evaluated our approach on four standard datasets and compared its performance with the classification results provided by the datasets’ creators, those estimated using the plug-in estimate λ, and those obtained using the Fisher (FK) and TOP (TK) kernel [9, 27] derived from the plug-ins. Support vector machines (SVMs) with RBF kernel were used as discriminative classifiers in all the score spaces, as this technique was previously identified as most potent for dealing with variablelength sequences [25]. As plug-ins, or generative models/likelihoods λ, for the three score spaces we compare across experiments, we used hidden Markov models (HMMs)[23] in Experiments 1-3 and latent Dirichlet allocation (LDA)[4] in Experiment 4. For each experiment, comparisons are based on the same validation procedure used in the appropriate original papers that introduced the datasets. For both FK and FESS, in each experiment we trained a single generative model (HMM or LDA, depending on the experiment). For all HMM models, the length-normalization with associated summation over the sequence as described in the previous section was used in the construction of the free energy score space. The model complexity, e.g., the number of states for the HMM were chosen by cross-validation on the training set. Experiment 1: E. coli promoter gene sequences. The first analyzed dataset consists of the E. coli promoter gene sequences (DNA) with associated imperfect domain theory [26]. The standard task on this dataset is to recognize promoters in strings of nucleotides (A, G, T, or C). A promoter is a genetic region which facilitates the transcription of gene located nearby. The input features are 57 sequential DNA nucleotides. Results, obtained using leave-one-out (LOO) validation, are 6 reported in Table 1 and illustrate that FESS represents well the fixed size genetic sequences, leading to superior performance over other score spaces as well as over the plug-in λHMM. E.Coli λHMM FESS nFESS FK TK Accuracy 67,34% 94,33% 85,80% 79,20% 85,30% Table 1: Promoter classification results. Experiment 2: Introns/Exons classification in HS3D data set. The HS3D data set 1[10] contains labelled intron and exon sequences of nucleotides. The task here is to distinguish between the two types of gene sequences that can both vary in length (from dozens of nucleotides to tens of thousands of nucleotides). For the sake of comparison, we adopted the same experimental setting of [10]. In Fig.1-A (top right), we reported the results obtained in [10] (overall error rate, OER, 7,5%), the results obtained using the HMM model (λHMM, OER 27,59%) together with the results obtained by our method (OER 6,12%). In Fig. 1-B (bottom right), we compared our method also with FK (OER 10,06%) and TK (OER 12,82%) kernels. Experiment 3: Homology detection in SCOP 1.53. We tested the ability of FESS to classify protein domains into superfamilies in the Structural Classification of Proteins (SCOP)2 version 1.53. The sequences in the database were selected from the Astral database, based on the E-value threshold of 10−25 for removing similar sequences from it. In the end, 4352 distinct sequences were grouped into families and superfamilies. For each family, the protein domains within the family are considered positive test examples, and the protein domains outside the family, but within the same superfamily, are taken as positive training examples. The data set yields 54 families containing at least 10 family members (positive test) and 5 superfamily members outside of the family (positive train) for a total of 54 One-Vs-All problems. The experimental setup is similar to that used in [8], except for one important difference: in the current experiments, the positive training sets do not include additional protein sequences extracted from a large, unlabelled database. Therefore, the recognition tasks performed here are more difficult than those in [8]. In order to measure the quality of the ranking, we used the median RFP score [8] which is the fraction of negative test sequences that score as high as or better than the median-scoring positive sequence. We used SVM decision values as score. We find that FESS outperforms task-specific algorithms (PSI-Blast [2] and SAM [14]) as well as the Fisher score (FK,[8]) with statistical significance with p-values of 5.1e-9, 8.3e-7, 1.1e-5, respectively. There is no statistical difference between our results FESS and those based on FPS [3]. In particular, the poor performance of [8] is explained by the under-training of HMMs [6]. The FESS representation proved to be much less sensitive to the training problems. We repeated the test using two different choices of Q: the approximate mean field factorization and the exact posterior (FESS-MF and FESS-EX, respectively, in Fig.1-C). Interestingly, the performance was also robust with respect to these choices. Experiment 4: Scene/object recognition. Our final set of experiments used the data from the Graz dataset3, as well as the dataset proposed in [21]. In both tests, we used Latent Dirichlet allocation (LDA) [4] as the generative model. The free energy for LDA is derived in [4]. To serve as words in the model, we extracted SIFT features from 16x16 pixel windows computed over a grid with spacing of 8 pixels. These features were mapped to 175 codewords (W = 175). We varied the number of topics to explore the effectiveness of different techniques. Graz dataset has two object classes, bikes (373 images) and persons (460 images), in addition to a background class (270 images)4. The range of scales and poses at which exemplars are presented is highly diverse, e.g., a “person” image may show a pedestrian at a certain distance, a side view of a complete body, or just a closeup of a head. We performed two-class detection (object vs. background) using an experimental setup consistent with [16, 22]. We generated ROC curves by thresholding raw SVM output, and report here the ROC equal error rate averaged over ten runs. The results are shown in Table 2. The standard deviation of the classification rate is quite high as the images in the database have very different complexities, and the performance for any single run is 1www.sci.unisannio.it/docenti/rampone 2http://scop.mrc-lmb.cam.ac.uk/scop/ 3http://www.emt.tugraz.at/ pinz/data/GRAZ 02/ 4The car class is ignored as in [16] 7 Graz dataset FESS - Z=15 FESS - Z=30 FESS - Z=45 [16] [22] Bikes 86,1% (1,8) 86,5% (2,0) 89,1% (2,3) 86,3% (2,5) 86,5% People 83,1% (3,1) 82,9% (2,8) 84,4% (2,0) 82,3% (3,1) 80,8% Scenes dataset λLDA FESS FK [21] [16] Natural 63,93% 95,21% 90,10% 89,00% 84,51% Artificial 67,21% 94,38% 90,32% 89,00% 89,43% Table 2: Classification rates for object/scene recognition tasks. The deviation is shown in brackets. Our approach tends to be robust to the choice of the number Z of topics, and so in scene recognition experiments, we report only the result for Z=40. highly dependent on the composition of the training set. We also tested our approach on the scene recognition task using the datasets of [21], composed of two (Natural and Artificial scenes) datasets, each with 4 different classes. The results are reported in Table 2 where for the first time we employed Fisher-LDA in a vision application. Although this new technique outperformed state of the art, once again, FESS outperforms both this result and other state-of-the-art discriminative methods [21, 16]. 6 Conclusions In this paper, we present a novel generative score space, FESS, exploiting variational free energy terms as features. The additive free energy terms arise naturally as a consequence of the factorization of the model P and the posterior Q. We show that the use of these terms as features in discriminative classification leads to more robust results than the use of the Fisher scores, which are based on the derivatives of the log likelihood of the data with respect to the model parameters. As was previously observed, we find that the Fisher score space suffers from the so called “wrap-around” problem, where very different data points may map to the same derivative, an example of which was discussed in the introduction. The free energy terms, on the other hand, quantify the data fit in different parts of the model, and seem to be informative even when the model is imperfect. This indicates that the re-scaling of these terms, which the subsequent discriminative training provides, leads to improved modelling of the data in some way. Scaling a term in the free energy composition, e.g., the term P h q(h) log p(x|h), by a constant w is equivalent to raising the appropriate conditional distribution to the power w. This is indeed reminiscent of some previous approaches to correcting generative modelling problems. In speech applications, for example, it is a standard practice to raise the observation likelihood in HMMs to a power less than 1, before inference is performed on the test sample, as the acoustic signal would otherwise overwhelm the hidden process modelling the language constraints [28]. This problem arises from the approximations in the acoustic model. For instance, a high-dimensional acoustic observation is often modelled as following a diagonal Gaussian distribution, thus assuming independent noise in the elements of the signal, even though the true acoustics of speech is far more constrained. This results in over-accounting for the variation in the observed acoustic signal, and to correct for this in practice, the log probability of the observation given the hidden variable is scaled down. The technique described here proposes a way to automatically infer the best scaling, but it also goes a step further in allowing for such corrections at all levels of the model hierarchy, and even for specific configurations of hidden variables. Furthermore, the use of kernel methods provides for nonlinear corrections, as well. This extremely simple technique was shown here to work remarkably well, outperforming previous score space approaches as well as the state of the art in multiple applications. It is possible to extend the ideas here to other types of model/data energy. For example, the free energy approximated in different ways is used in [1] to construct various inference algorithms for a single scene parsing task. It may also be effective, for example, to use the terms in the Bethe free energy linked to different belief propagation messages to construct the feature vectors. Finally, although we find that FESS outperforms the previously studied score spaces that depend on the derivatives, i.e. where ˆF is a derivative with respect to θ, the use of this derivative in (7) is, of course, possible. This allows for the construction of kernels similar to FK and TK, but derived from intractable generative models as we show in Experiment 4 (FK in Table 2) on latent Dirichlet allocation. Acknowledgements We acknowledge financial support from the FET programme within the EU FP7, under the SIMBAD project (contract 213250). 8 References [1] B. Frey and N. Jojic. A Comparison of Algorithms for Inference and Learning in Probabilistic Graphical Models Transactions on pattern analysis and machine intelligence, 1392:1416–27, 2005. [2] S. F. Altschul, W. Gish, W. Miller, E. W. Myers, and D. J. Lipman. Basic local alignment search tool. J Mol Biol, 215(3):403–410, October 1990. [3] T. L. Bailey and W. N. Grundy. Classifying proteins by family using the product of correlated p-values. In Proceedings of the Third Annual International Conference on Computational Molecular Biology, pages 10–14. ACM, 1999. [4] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent dirichlet allocation. J. Mach. Learn. Res., 3:993–1022, 2003. [5] G. Bouchard and B. Triggs. The tradeoff between generative and discriminative classifiers. In IASC International Symposium on Computational Statistics, pages 721–728, Prague, August 2004. [6] K. Tsuda B.Schlkopf and J. Vert. Kernel Methods in Computational Biology. The MIT Press, 2004. [7] Z. Ghahramani. On structured variational approximations. Technical Report CRG-TR-97-1, 1997. [8] T. Jaakkola, M. Diekhaus, and D. Haussler. Using the fisher kernel method to detect remote protein homologies. 7th Intell. Sys. Mol. Biol., pages 149–158, 1999. [9] T. Jaakkola and D. Haussler. Exploiting generative models in discriminative classifiers. Nips, 1998. [10] T. Jebara, R. Kondor, A. Howard, K. Bennett, and N. Cesa-bianchi. Probability product kernels. Journal of Machine Learning Research, 5:819–844, 2004. [11] M.I. Jordan, Z. Ghahramani, T. Jaakkola, and L.K. Saul. An introduction to variational methods for graphical models. Machine Learning, 37(2):183–233, 1999. [12] S. Kapadia. Discriminative Training of Hidden Markov Models. PhD thesis, 1998. [13] H. Kappen and W. Wiegerinck. Mean field theory for graphical models, 2001. [14] K. Karplus, C. Barrett, and R. Hughey. Hidden markov models for detecting remote protein homologies. Bioinformatics, 14:846–856, 1999. [15] J. A. Lasserre, C. M. Bishop, and T. P. Minka. Principled hybrids of generative and discriminative models. In Cvpr, pages 87–94, 2006. [16] S. Lazebnik, C. Schmid, and J. Ponce. Beyond bags of features: Spatial pyramid matching for recognizing natural scene categories. Cvpr, 2:2169–2178, 2006. [17] D. MacKay. Ensemble learning for Hidden Markov Models, 1997. Unpublished. Department of Physics, University of Cambridge. [18] A. Mccallum, C. Pal, G. Druck, and X. Wang. Multi-conditional learning: Generative/discriminative training for clustering and classification. In In Proceedings of the 21st National Conference on Artificial Intelligence, pages 433–439, 2006. [19] R. M. Neal and G. E. Hinton. A view of the em algorithm that justifies incremental, sparse, and other variants. pages 355–368, 1999. [20] A. Y. Ng and M. I. Jordan. On discriminative vs. generative classifiers: A comparison of logistic regression and naive bayes. In T. G. Dietterich, S. Becker, and Z. Ghahramani, editors, NIPS, Cambridge, MA, 2002. MIT Press. [21] A. Oliva and A. Torralba. Modeling the shape of the scene: A holistic representation of the spatial envelope. International Journal of Computer Vision, 42:145–175, 2001. [22] A. Opelt, M. Fussenegger, A. Pinz, and P. Auer. Weak hypotheses and boosting for generic object detection and recognition. In Eccv, volume 2, pages 71–84, 2004. [23] L.R. Rabiner. A tutorial on Hidden Markov Models and selected applications in speech recognition. Proc. of IEEE, 77(2):257–286, 1989. [24] N. Smith and M. Gales. Speech recognition using SVMs. In Nips, pages 1197–1204. MIT Press, 2002. [25] N. Smith and M. Gales. Using svms to classify variable length speech patterns. Technical Report CUED/F-INGENF/TR.412, University of Cambridge, UK, 2002. [26] G. G. Towell, J. W. Shavlik, and M. O. Noordewier. Refinement of approximate domain theories by knowledge-based neural networks. In In Proceedings of the Eighth National Conference on Artificial Intelligence, pages 861–866, 1990. [27] K. Tsuda, M. Kawanabe, G. R¨atsch, S. Sonnenburg, and K. R. M¨uller. A new discriminative kernel from probabilistic models. Neural Comput., 14(10):2397–2414, 2002. [28] L. Deng and D. O’Shaughnessy, Speech Processing - A Dynamic and Optimization-Oriented Approach Marcel Dekker Inc., June 2003 9
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On Learning Rotations Raman Arora University of Wisconsin-Madison Department of Electrical and Computer Engineering 1415 Engineering Drive, Madison, WI 53706 rmnarora@u.washington.edu Abstract An algorithm is presented for online learning of rotations. The proposed algorithm involves matrix exponentiated gradient updates and is motivated by the von Neumann divergence. The multiplicative updates are exponentiated skew-symmetric matrices which comprise the Lie algebra of the rotation group. The orthonormality and unit determinant of the matrix parameter are preserved using matrix logarithms and exponentials and the algorithm lends itself to intuitive interpretation in terms of the differential geometry of the manifold associated with the rotation group. A complexity reduction result is presented that exploits the eigenstructure of the matrix updates to simplify matrix exponentiation to a quadratic form. 1 Introduction The problem of learning rotations finds application in many areas of signal processing and machine learning. It is an important problem since many problems can be reduced to that of learning rotations; for instance Euclidean motion in Rn−1 is simply rotation in Rn. A conformal embedding was presented in [1] that extends rotations to a representation for all Euclidean transformations. Furthermore, the rotation group provides a universal representation for all Lie groups. This was established in [2] by showing that any Lie algebra can be expressed as a bivector algebra. Since the Lie algebra describes the structure of the associated Lie group completely, any Lie group can be represented as rotation group. The batch version of the problem was originally posed as the problem of estimating the attitude of satellites by Wahba in 1965 [3]. In psychometrics, it was presented as the orthogonal Procrustes problem [4]. It has been studied in various forms over the last few decades and finds application in many areas of computer vision [5, 6, 7], face recognition [8], robotics [9, 10], crystallography[11] and physics [12]. While the batch version of the problem is well understood, the online learning of rotations from vector instances is challenging since the manifold associated with the rotation group is a curved space and it is not possible to form updates that are linear combinations of rotations [13]. The set of rotations about the origin in n-dimensional Euclidean space forms a compact Lie group, SO(n), under the operation of composition. The manifold associated with the n-dimensional rotation group is the unit sphere Sn−1 in n dimensional Euclidean space. 1.1 Related Work The online version of learning rotations was posed as an open problem by Smith and Warmuth [13]. Online learning algorithms were recently presented for some matrix groups. In [14], an online algorithm was proposed for learning density matrix parameters and was extended in [15] to the problem of learning subspaces of low rank. However, the extension of these algorithms to learning rotations will require repeated projection and approximation [13]. Adaptive algorithms were also 1 studied in [16] for optimization under unitary matrix constraint. The proposed methods are steepest descent methods on Riemannian manifolds. 1.2 Our Approach This paper presents an online algorithm for learning rotations that utilizes the Bregman matrix divergence with respect to the quantum relative entropy (also known as von Neumann divergence) as a distance measure between two rotation matrices. The resulting algorithm has matrix-exponentiated gradient (MEG) updates [14]. The key ingredients of our approach are (a) von Neumann Divergence between rotation matrices [17], (b) squared error loss function and (c) matrix exponentiated gradient (MEG) updates. Any Lie group is also a smooth manifold and the updates in the proposed algorithm have an intuitive interpretation in terms of the differential topology of the associated manifold. We also utilize various elementary Lie algebra concepts to provide intuitive interpretation of the updates. The development in the paper closely follows that of the matrix exponentiated gradient (MEG) updates in [14] for density matrix parameters. The form of the updates are similar to steepest descent methods of [16], but are derived for learning rotations from vector instances using an information-theoretic approach. The MEG updates are reduced to a quadratic form in the Lie algebra element corresponding to the gradient of loss function on the rotation group. The paper is organized as follows. The problem is formulated in Section 2. Section 3 presents mathematical preliminaries in differential geometry and Bregman matrix divergence. The matrix exponentiated gradient updates are developed in Section 4. The MEG updates are simplified in Section 5. Experimental results are discussed in Section 6. 2 Problem Statement Let xt be a stream of instances of n-dimensional unit vectors. Let R∗be an unknown n×n rotation matrix that acts on xt to give the rotated vector yt = R∗xt. The matrix ˆRt denotes the estimate of R∗at instance t and ˆyt = ˆRt xt represents the prediction for the rotated vector yt. The loss incurred due to error in prediction is Lt( ˆRt) = d(ˆyt, yt), where d(·, ·) is a distance function. The estimate of the rotation needs to be updated based on the loss incurred at every instance and the objective is to develop an algorithm for learning R∗that has a bounded regret. We seek adaptive updates that solve the following optimization problem at each step, ˆRt+1 = arg min R ∆F (R, ˆRt) + η Lt(R), (1) where ˆRt is the estimated rotation matrix at instance t, η is the learning rate or the step-size and ∆F is a matrix divergence that measures the discrepancy between matrices. This is a typical problem formulation in online learning where the objective comprises a loss function and a divergence term. The parameter η balances the trade-off between the two conflicting goals at each update: incurring small loss on the new data versus confidence in the estimate from the previously observed data. Minimizing the weighted objective therefore results in smooth updates as well as minimizes the loss function. In this paper, the updates are smoothed using the von Neumann divergence which is defined for matrices as ∆F (R, ˆRt) = tr(R log R −R log ˆRt −R + ˆRt), (2) where tr(A) is the trace of the matrix A. The search is over all R ∈SO(n), i.e. over all n × n matrices such that RT R = I, RRT = I and det(R) = 1. 3 Mathematical Preliminaries This section reviews some basic definitions and concepts in linear algebra and differential geometry that are utilized for the development of the updates in the next section. 2 3.1 Matrix Calculus Given a real-valued matrix function F : Rn×n →R, the gradient of the function with respect to the matrix R ∈Rn×n is defined to be the matrix [18], ∇RF(R) = ∂F ∂R11 · · · ∂F ∂R1n ... ... ... ∂F ∂Rn1 · · · ∂F ∂Rnn . (3) Some of the matrix derivatives that are used later in the paper are following: for a constant matrix Γ ∈Rn×n, 1. ∇R tr(ΓRRT ) = (Γ + ΓT )R, 2. ∇R det(R) = det(R)(R−1)T , 3. ∇R(y −Rx)T (y −Rx) = −2(y −Rx)xT . A related concept in differential geometry is that of the space of vectors tangent to a group at the identity element of the group. This is defined to be the Lie algebra associated with the group. It is a convenient way of describing the infinitesimal structure of a topological group about the identity element and completely determines the associated group. The utility of the Lie algebra is due to the fact that it is a vector space and thus it is much easier to work with it than with the linear group. A real n × n matrix A is in the Lie algebra of the rotation group SO(n) if and only if it is a skewsymmetric matrix (i.e. AT = −A). Furthermore, for any matrix A in the Lie algebra of SO(n), exp(ηA) is a one-parameter subgroup of the rotation group, parametrized by η ∈R [19]. The matrix exponential and logarithm play an important role in relating a matrix Lie group G and the associated Lie algebra g. The exponential of a matrix R ∈Rn×n is given by the following series, exp(R) = I + R + 1 2!R2 + 1 3!R2 + · · · (4) Given an element A ∈g, the matrix exponential exp(A) is the corresponding element in the group. The matrix logarithm log (R) is defined to be the inverse of the matrix exponential: it maps from the Lie group G into the Lie algebra g. The matrix logarithm is a well-defined map since the exponential map is a local diffeomorphism between a neighborhood of the zero matrix and a neighborhood of the identity matrix [19, 20]. 3.2 Riemannian Gradient Consider a real-valued differentiable function, Lt : SO(n) →R, defined on the rotation group. The Riemannian gradient ˜∇RLt of the function Lt on the Lie group SO(n) evaluated at the rotation matrix R and translated to the identity (to get a Lie algebra element) is given as [16] ˜∇RLt = ∇RLt RT −R ∇T RLt, (5) where ∇RLt is the matrix derivative of the cost function in the Euclidean space defined in (3) at matrix R. 3.3 Von Neumann Divergence In any online learning problem, the choice of divergence between the parameters dictates the resulting updates. This paper utilizes the von Neumann divergence which is a special case of the Bregman divergence and measures discrepancy between two matrices. Let F be convex differentiable function defined on a subset of Rn×n with the gradient f(R) = ∇RF(R). The Bregman divergence between two matrices R1 and R2 is defined as ∆F (R1, R2) := F(R1) −F(R2) −tr((R1 −R2)f(R2)T ). (6) The gradient of Bregman divergence with respect to R1 is given as, ∇R1∆F (R1, R2) = f(R1) −f(R2). (7) 3 Choosing the function F in the definition of Bregman divergence to be the von Neumann entropy, given as F(R) = tr(R log R −R)), obtain the von Neumann divergence [14, 17]: ∆F (R1, R2) = Tr(R1 log R1 −R1 log R2 −R1 + R2). (8) Finally, the gradient of the von Neumann entropy was shown to be f(R) = ∇RF(R) = log R in [14]. Consequently, the gradient of the von Neumann divergence can be expressed as ∇R1∆F (R1, R2) = log (R1) −log (R2). (9) 4 Online Algorithm The problem of online learning of rotations can be expressed as the optimization problem ˆRt+1 = arg min R ∆F (R, ˆRt) + ηLt(R) s.t. RT R = I, RRT = I det(R) = 1 (10) where ˆRt is the estimate of the rotation matrix at time instance t and Lt is the loss incurred in the prediction of yt. The proposed adaptive updates are matrix exponentiated gradient (MEG) updates given as ˆRt+1 = exp log ˆRt −η skew ˆRT t ∇RLt( ˆRt) ! , (11) where ∇RLt( ˆRt) is the gradient of the cost function in the Euclidean space with respect to the rotation matrix R and skew (·) is the skew-symmetrization operator on the matrices, skew (A) = A −AT . The updates seem intuitive, given the following elementary facts about the Lie algebraic structure of the rotation group: (a) the gradient of loss function gives geodesic direction and velocity vector on the unit sphere, (b) a skew-symmetric matrix is an element of Lie algebra [19, 20], (c) the matrix logarithm maps a rotation matrix to the corresponding Lie algebra element, (d) composition of two elements of Lie algebra yields another Lie algebra element and (e) the matrix exponential maps a Lie algebra element to corresponding rotation matrix. The loss function is defined to be the squared error loss function and therefore the gradient of the loss function is given by the matrix ∇RLt( ˆRt) = 2(ˆyt −yt)xT t . This results in the online updates ˆRt+1 = exp log ˆRt −2η skew ˆRT t (ˆyt −yt)xT t ! , = ˆRt exp −2η skew ˆRT t (ˆyt −yt)xT t ! . (12) 4.1 Updates Motivated by von-Neumann Divergence The optimization problem in (10) is solved using the method of Lagrange multipliers. First observe that the constraints RT R = I and RRT = I are redundant since one implies the other. Introducing the Lagrangian multiplier matrix Γ for the orthonormality constraint and Lagrangian multiplier λ for the unity determinant constraint, the objective function can be written as J (R, Γ, λ) = ∆F (R, ˆRt) + ηLt(R) + tr(Γ(RRT −I)) + λ(det(R) −1). (13) Taking the gradient on both sides of equation with respect to the matrix R, get ∇R J (R, Γ, λ) = ∇R ∆F (R, ˆRt) + η ˜∇R Lt(R) +(Γ + ΓT )R + λ det(R)(R−1)T , (14) 4 using the matrix derivatives from Section 3.1 and the Riemannian gradient for the loss function from eqn. (5). Putting ∇R J (R, Γ, λ) = 0 and using the fact that ∇R∆F (R, ˆRt) = f(R) −f( ˆRt), get 0 = f(R) −f( ˆRt) + η skew ˆRT t ∇RLt(R) + (Γ + ΓT )R + λ det(R)(R−1)T . (15) Given that f is a bijective map, write R = f −1 f( ˆRt) −η skew ˆRT t ∇RLt(R) −(Γ + ΓT )R −λ det(R)(R−1)T . (16) Since the objective is convex, it is sufficient to produce a choice of Lagrange multipliers that enforces the rotation constraint. Choosing λ = det(R)−1 and Γ = −(1/2) R−1T R−1 yields the following implicit update ˆRt+1 = exp log ˆRt −η skew ˆRT t ∇RLt( ˆRt+1) ! . (17) As noted by Tsuda et. al. in [14], the implicit updates of the form above are usually not solvable in closed form. However, by approximating ∇RLt( ˆRt+1) with ∇RLt( ˆRt) (as in [21, 14]), we obtain an explicit update ˆRt+1 = exp log ˆRt −η skew ˆRT t ∇RLt( ˆRt) ! . (18) The next result ensures the closure property for the matrix exponentiated gradient updates in the equation above. In other words, the estimates for the rotation matrix do not steer away from the manifold associated with the rotation group. Therefore, if ˆR0 ∈SO(n) then ˆRt+1 ∈SO(n). Lemma 1. If ˆRt ∈SO(n) then ˆRt+1 given by the updates in (18) is a rotation matrix in SO(n). Proof. Using the properties of matrix logarithm and matrix exponential, express (18) as ˆRt+1 = ˆRt exp(−ηS), (19) where S = ˆRT t ∇RLt(R) −∇T RLt(R) ˆRt is an n × n dimensional skew-symmetric matrix with trace zero. Then ˆRT t+1 ˆRt+1 = ˆRt e−ηST ˆRt e−ηS , = e−ηST ˆRT t ˆRt e−ηS , = e−ηST e−ηS , = eη(−ST −S) = eη(S−S) = I, where we used the facts that ˆRt ∈SO(n), eST = eST , ST = −S and that e0 = I. Similarly, ˆRt+1 ˆRT t+1 = I. Finally, note that det( ˆRt+1) = det( ˆRt e−ηS) = det( ˆRt) · det(e−ηS) = e−η Tr (S), since determinant of exponential of a matrix is equal to the exponential of the trace of the matrix. And since S is a trace zero matrix, det( ˆRt+1) = 1. 4.2 Differential Geometrical Interpretation The resulting updates in (18) have nice interpretation in terms of the differential geometry of the rotation group. The gradient of the cost function, ∇RLt( ˆRt), in the Euclidean space gives a tangent direction at the current estimate of the rotation matrix. The Riemannian gradient is computed as ∇RLt( ˆRt) −ˆRt ∇T RLt( ˆRt) ˆRt. The Riemannian gradient at the identity element of the group is obtained by de-rotation by ˆRt, giving ˜∇RLt( ˆRt), as in (5). The gradient corresponds to an element of the Lie algebra, so(n), of the rotation group. The exponential map gives the corresponding rotation matrix which is the multiplicative update to the estimate of the rotation matrix at the previous instance. 5 5 Complexity Reduction of MEG Updates The matrix exponentiated gradient updates ensure that the estimates for the rotation matrix stay on the manifold associated with the rotation group at each iteration. However, with the matrix exponentiation at each step, the updates are computationally intensive and in fact the computational complexity of the updates is comparable to other approaches that would require repeated approximation and projection on to the manifold. This section discusses a fundamental complexity reduction result to establish a simpler update by exploiting the eigen-structure of the update matrix. First observe that the matrix in the exponential in eqn. (12) (for the case of squared error loss function) can be written as S = −2η skew ˆRT t (ˆyt −yt)xT t , = −2η skew ˆRT t ( ˆRtxt −R∗xt)xT t , = −2η skew xtxT t −ˆRT t R∗xtxT t , = 2η ˆRT t R∗xtxT t −xtxT t RT ∗ˆRt , = AT X −XA, (20) where X ≡xtxT t and A ≡2ηRT ∗ˆRt. Each term in the matrix S is a rank-one matrix (due to pre and post-multiplication with xtxT t , respectively). Thus S is at most rank-two. Since S is skewsymmetric, it has (at most) two eigenvalues in a complex conjugate pair ±jλ (and n −2 zero eigenvalues) [22], which allows the following simplification. Lemma 2. The matrix exponentiated gradient updates in eqn. (12) are equivalent to the following updates, ˆRt+1 = ˆRt I + sin(λ) λ S + 1 −cos(λ) λ2 S2 ! , (21) where λ = 2η q 1 − yT t ˆyt 2 and S is the skew-symmetric matrix given in eqn. (20) with eigenvalues ±jλ. Note that yt, ˆyt are unit vectors in Rn and therefore λ is real-valued. The proof of the complexity reduction follows easily from a generalization of the Rodrigues’ formula for computing matrix exponentials for skew-symmetric matrix. The proof is not presented here due to space constraints but the interested reader is referred to [23, 24]. Owing to the result above the matrix exponential reduces to a simple quadratic form involving an element from the Lie algebra of the rotation group. The pseudocode is given in Algorithm 1. Choose η Initialize R1 = I for t = 1, 2, . . . do Obtain an instance of unit vector xt ∈Rn; Predict the rotated vector ˆyt = ˆRt xt; Receive the true rotated vector yt = R∗xt; Incur the loss Lt( ˆRt) = |yt −ˆyt|2; Compute the matrix S = 2η ˆRT t ytxT t −xtyT t ˆRt ; Compute the eigenvalues λ = 2η q 1 − yT t ˆyt 2; Update the rotation matrix ˆRt+1 = ˆRt I + sin(λ) λ S + 1−cos(λ) λ2 S2 end Algorithm 1: Pseudocode for Learning rotations using Matrix Exponentiated Gradient updates 6 6 Experimental Results This section presents experimental results with the proposed algorithm for online learning of rotations. The performance of the algorithm is evaluated in terms of the Frobenius norm of the difference of the true rotation matrix and the estimate. Figure 1 shows the error plot with respect to time. The unknown rotation is a 12 × 12 dimensional matrix and changes randomly every 200 instances. The trajectories are averaged over 1000 random simulations. It is clear from the plot that the estimation error decays rapidly to zero and estimates of the rotation matrices are exact. 0 200 400 600 800 1000 1200 0 1 2 3 4 5 Estimation error versus time − SO(12) Time (instance index) Estimation Error Frobenius norm Spectral norm Figure 1: Online learning of rotations: Estimate of unknown rotation is updated every time new instance of rotation is observed. The true rotation matrix is randomly changing at regular interval (N=200). The error in Frobenius norm is plotted against the instance index. The online algorithm is also found robust to small amount of additive white Gaussian noise in observations of the true rotated vectors, i.e. the observations are now given as yt = R∗xt + α wt, where α determines the signal to noise ratio. The performance of the algorithm is studied with various noisy conditions. Figure 2 shows error plots with respect to time for various noisy conditions in R20. The Frobenius norm error decays quickly to a noise floor determined by the SNR as well as the step size η. In the simulations in Fig. 2 the step size was decreased gradually over time. It is not clear immediately how to pick the optimal step size but a classic step size adaptation rule or Armijo rule may be followed [25, 16]. The tracking performance of the online algorithm is compared with the batch version. In Figure 3, the unknown rotation R∗∈SO(30) changes slightly after every 30 instances. The smoothly changing rotation is induced by composing R∗matrix with a matrix Rδ every thirty iterations. The matrix Rδ is composed of 3 × 3 block-diagonal matrices, each corresponding to rotation about the X-axis in 3D space by π/360 radians. The batch version stores the last 30 instances in an 30 × 30 matrix X and corresponding rotated vectors in matrix Y. The estimate of the unknown rotation is given as YX−1. The batch version achieves zero error only at time instances when all the data in X, Y correspond to the same rotation whereas the online version consistently achieves a low error and tracks the changing rotation. It is clear from the simulations that the Frobenius norm decreases at each iteration. It is easy to show this global stability of the updates proposed here in noise-free scenario [24]. The proposed algorithm was also applied to learning and tracking the rotations of 3D objects. Videos showing experimental results with the 3D Stanford bunny [26] are posted online at [27]. 7 Conclusion In this paper, we have presented an online algorithm for learning rotations. The algorithm was motivated using the von Neumann divergence and squared error loss function and the updates were 7 0 100 200 300 400 500 600 0 1 2 3 4 5 Time (instance index) Estimation Error (Frobenius norm) Avg. Estimation error (Frobenius norm) vs time − SO(20) 0 5× 10−4 1x10−3 1.5x10−3 2x10−3 Figure 2: Average error plotted against instance index for various noise levels. 100 200 300 400 500 600 0 1 2 3 4 5 Tracking rotations in SO(30) Time (instance index) Error (Frobenius norm) Batch version Online algorithm Figure 3: Comparing the performance of tracking rotations for the batch version versus the online algorithm. The rotation matrix changes smoothly every M = 30 instances. developed in the Lie algebra of the rotation group. The resulting matrix exponentiated gradient updates were reduced to a simple quadratic form. The estimation performance of the proposed algorithm was studied under various scenarios. Some of the future directions include identifying alternative loss functions that exploit the spherical geometry as well as identifying regret bounds for the proposed updates. Acknowledgements: The author would like to thank W. A. Sethares, M. R. Gupta and A. B. Frigyik for helpful discussions and feedback on early drafts of the paper. References [1] Rich Wareham, Jonathan Cameron, and Joan Lasenby, “Applications of conformal geometric algebra in computer vision and graphics,” in IWMM/GIAE, 2004, pp. 329–349. [2] C. Doran, D. Hestenes, F. Sommen, and N. Van Acker, “Lie groups as spin groups,” Journal of Mathematical Physics, vol. 34, no. 8, pp. 36423669, August 1993. [3] Grace Wahba, “Problem 65-1, a least squares estimate of satellite attitude,” SIAM Review, vol. 7, no. 3, July 1965. 8 [4] P. Schonemann, “A generalized solution of the orthogonal Procrustes problem,” Psychometrika, vol. 31, no. 1, pp. 3642–3669, March 1966. [5] P. Besl and N. McKay, “A method for registration of 3D shapes,” . IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 14, pp. 239–256, 1992. [6] Hannes Edvardson and ¨Orjan Smedby, “Compact and efficient 3D shape description through radial function approximation,” Computer Methods and Programs in Biomedicine, vol. 72, no. 2, pp. 89–97, 2003. [7] D.W. Eggert, A. Lorusso, and R.B. Fisher, “Estimating 3D rigid body transformations: a comparison of four major algorithms,” Machine Vision and Applications, Springer, vol. 9, no. 5-6, Mar 1997. [8] R. Sala Llonch, E. Kokiopoulou, I. Tosic, and P. Frossard, “3D face recognition with sparse spherical representations,” Preprint, Elsiever, 2009. [9] Ameesh Makadia and Kostas Daniilidis, “Rotation recovery from spherical images without correspondences,” IEEE Trans. Pattern Analysis Machine Intelligence, vol. 28, no. 7, pp. 1170–1175, 2006. [10] Raman Arora and Harish Parthasarathy, “Navigation using a spherical camera,” in International Conference on Pattern Recognition (ICPR), Tampa, Florida, Dec 2008. [11] Philip R. Evans, “Rotations and rotation matrices,” Acta Cryst., vol. D57, pp. 1355–1359, 2001. [12] Richard L. Liboff, Introductory Quantum Mechanics, Addison-Wesley, 2002. [13] Adam Smith and Manfred Warmuth, “Learning rotations,” in Conference on Learning Theory (COLT), Finland, Jun 2008. [14] Koji Tsuda, Gunnar Ratsch, and Manfred K Warmuth, “Matrix exponentiated gradient updates for on-line learning and Bregman projection,” Journal of Machine Learning Research, vol. 6, Jun 2005. [15] Manfred K Warmuth, “Winnowing subspaces,” in Proc. 24th Int. Conf. on Machine Learning, 2007. [16] T.E. Abrudan, J. Eriksson, and V. Koivunen, “Steepest descent algorithms for optimization under unitary matrix constraint,” Signal Processing, IEEE Transactions on, vol. 56, no. 3, pp. 1134–1147, March 2008. [17] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge, 2000. [18] Kaare Brandt Petersen and Michael Syskind Pedersen, “The matrix cookbook,” http:// matrixcookbook.com, November 14, 2008. [19] Michael Artin, Algebra, Prentice Hall, 1991. [20] John A. Thorpe, Elementary topics in Differential Geometry, Springer-Verlag, 1994. [21] J. Kivinen andM. K.Warmuth, “Exponentiated gradient versus gradient descent for linear predictors,” Information and Computation, vol. 132, no. 1, pp. 1–63, Jan 1997. [22] L. J. Butler, Applications of Matrix Theory to Approximation Theory, MS Thesis, Texas Tech University, Aug. 1999. [23] J. Gallier and D. Xu, “Computing exponentials of skew-symmetric matrices and logarithms of orthogonal matrices,” International Journal of Robotics and Automation, vol. 17, no. 4, 2002. [24] Raman Arora, Group theoretical methods in signal processing: learning similarities, transformation and invariants, Ph.D. thesis, University of Wisconsin-Madison, Madison, August 2009. [25] E. Polak, Optimization: Algorithms and Consistent Approximations, Springer-Verlag, 1997. [26] Stanford University Computer Graphics Laboratory, “The Stanford 3D scanning repository,” http://graphics.stanford.edu/data/. [27] Raman Arora, “Tracking rotations of 3D Stanford bunny,” http://www.cae.wisc.edu/ ˜sethares/links/raman/LearnROT/vids.html, 2009. 9
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A Gaussian Tree Approximation for Integer Least-Squares Jacob Goldberger School of Engineering Bar-Ilan University goldbej@eng.biu.ac.il Amir Leshem School of Engineering Bar-Ilan University leshema@eng.biu.ac.il Abstract This paper proposes a new algorithm for the linear least squares problem where the unknown variables are constrained to be in a finite set. The factor graph that corresponds to this problem is very loopy; in fact, it is a complete graph. Hence, applying the Belief Propagation (BP) algorithm yields very poor results. The algorithm described here is based on an optimal tree approximation of the Gaussian density of the unconstrained linear system. It is shown that even though the approximation is not directly applied to the exact discrete distribution, applying the BP algorithm to the modified factor graph outperforms current methods in terms of both performance and complexity. The improved performance of the proposed algorithm is demonstrated on the problem of MIMO detection. 1 Introduction Finding the linear least squares fit to data is a well-known problem, with applications in almost every field of science. When there are no restrictions on the variables, the problem has a closed form solution. In many cases, a-priori knowledge on the values of the variables is available. One example is the existence of priors, which leads to Bayesian estimators. Another example of great interest in many applications is when the variables are constrained to a discrete finite set. This problem has many diverse applications such as decoding of multi-input-multi-output (MIMO) digital communication systems, GPS system ambiguity resolution [15] and many lattice problems in computer science, such as finding the closest vector in a lattice to a given point in Rn [1], and vector subset sum problems which have applications in cryptography [11]. In contrast to the continuous linear least squares problem, this problem is known to be NP hard. This paper concentrates on the MIMO application. It should be noted, however, that the proposed method is general and can be applied to any integer linear least-square problem. A multiple-inputmultiple-output (MIMO) is a communication system with n transmit antennas and m receive antennas. The tap gain from transmit antenna i to receive antenna j is denoted by Hij. In each use of the MIMO channel a vector x = (x1, ..., xn) ⊤is independently selected from a finite set of points A according to the data to be transmitted, so that x ∈An. A standard example of a finite set A in MIMO communication is A = {−1, 1} or more generally A = {±1, ±3, ..., ±(2k+1)}. The received vector y is given by: y = Hx + ǫ (1) The vector ǫ is an additive noise in which the noise components are assumed to be zero mean, statistically independent Gaussians with a known variance σ2I. The m×n matrix H is assumed to be known. (In the MIMO application we further assume that H comprises iid elements drawn from a normal distribution of unit variance.) The MIMO detection problem consists of finding the unknown transmitted vector x given H and y. The task, therefore, boils down to solving a linear system in which the unknowns are constrained to a discrete finite set. Since the noise ǫ is assumed 1 to be additive Gaussian, the optimal maximum likelihood (ML) solution is: ˆx = arg min x∈An ∥Hx −y∥2 (2) However, going over all the |A|n vectors is unfeasible when either n or |A| are large. A simple sub-optimal solution is based on a linear decision that ignores the finite set constraint: z = (H ⊤H)−1H ⊤y (3) and then, neglecting the correlation between the symbols, finding the closest point in A for each symbol independently: ˆxi = arg min a∈A |zi −a| (4) This scheme performs poorly due to its inability to handle ill-conditioned realizations of the matrix H. Somewhat better performance can be obtained by using a minimum mean square error (MMSE) Bayesian estimation on the continuous linear system. Let e be the variance of a uniform distribution over the members of A. We can partially incorporate the information that x ∈An by using the prior Gaussian distribution x ∼N(0, eI). The MMSE estimation becomes: E(x|y) = (H ⊤H + σ2 e I)−1H ⊤y (5) and then the finite-set solution is obtained by finding the closest lattice point in each component independently. A vast improvement over the linear approaches described above can be achieved by using sequential decoding: • Apply MMSE (5) and choose the most reliable symbol, i.e. the symbol that corresponds to the column with the minimal norm of the matrix: (H ⊤H + σ2 e I)−1H ⊤ • Make a discrete symbol decision for the most reliable symbol ˆxi. Eliminate the detected symbol: P j̸=i hjxj = y−hiˆxi (hj is the j-th column of H) to obtain a new smaller linear system. Go to the first step to detect the next symbol. This algorithm, known as MMSE-SIC [5], has the best performance for this family of linear-based algorithms but the price is higher complexity. These linear type algorithms can also easily provide probabilistic (soft-decision) estimates for each symbol. However, there is still a significant gap between the detection performance of the MMSE-SIC algorithm and the performance of the optimal ML detector. Many alternative structures have been proposed to approach ML detection performance. For example, sphere decoding algorithm (an efficient way to go over all the possible solutions) [7], approaches using the sequential Monte Carlo framework [3] and methods based on semidefinite relaxation [17] have been implemented. Although the detection schemes listed above reduce computational complexity compared to the exhaustive search of ML solution, sphere decoding is still exponential in the average case [9] and the semidefinite relaxation is a high-degree polynomial. Thus, there is still a need for low complexity detection algorithms that can achieve good performance. This study attempts to solve the integer least-squares problem using the Belief Propagation (BP) paradigm. It is well-known (see e.g. [14]) that a straightforward implementation of the BP algorithm to the MIMO detection problem yields very poor results since there are a large number of short cycles in the underlying factor graph. In this study we introduce a novel approach to utilize the BP paradigm for MIMO detection. The proposed variant of the BP algorithm is both computationally efficient and achieves near optimal results. 2 The Loopy Belief Propagation Approach Given the constrained linear system y = Hx+ǫ, and a uniform prior distribution on x, the posterior probability function of the discrete random vector x given y is: p(x|y) ∝exp(−1 2σ2 ∥Hx −y∥2) , x ∈An (6) 2 The notation ∝stands for equality up to a normalization constant. Observing that ∥Hx −y∥2 is a quadratic expression, it can be easily verified that p(x|y) is factorized into a product of two- and single-variable potentials: p(x1, .., xn|y) ∝ Y i ψi(xi) Y i<j ψij(xi, xj) (7) such that ψi(xi) = exp(−1 2σ2 y ⊤hixi) , ψij(xi, xj) = exp(−1 σ2 h ⊤ i hjxixj) (8) where hi is the i-th column of the matrix H. Since the obtained factors are simply a function of pairs, we obtain a Markov Random Field (MRF) representation [18]. In the MIMO application the (known) matrix H is randomly selected and therefore, the MRF graph is usually a completely connected graph. In a loop-free MRF graph the max-product variant of the BP algorithm always converges to the most likely configuration (which corresponds to ML decoding in our case). For loop-free graphs, BP is essentially a distributed variant of dynamic programming. The BP message update equations only involve passing messages between neighboring nodes. Computationally, it is thus straightforward to apply the same local message updates in graphs with cycles. In most such models, however, this loopy BP algorithm will not compute exact marginal distributions; hence, there is almost no theoretical justification for applying the BP algorithm. (One exception is that, for Gaussian graphs, if BP converges, then the means are correct [16]). However, the BP algorithm applied to loopy graphs has been found to have outstanding empirical success in many applications, e.g., in decoding LDPC codes [6]. The performance of BP in this application may be attributed to the sparsity of the graphs. The cycles in the graph are long, hence the graph have tree-like properties, so that messages are approximately independent and inference may be performed as though the graph was loop-free. The BP algorithm has also been used successfully in image processing and computer vision (e.g. [4]) where the image is represented using a grid-structured MRF that is based on local connections between neighboring nodes. However, when the graph is not sparse, and is not based on local grid connections, loopy BP almost always fails to converge. Unlike the sparse graphs of LDPC codes, or grid graphs in computer vision applications, the MRF graphs of MIMO channels are completely connected graphs and therefore the associated detection performance is poor. This has prevented the BP from being an asset for the MIMO problem. Fig. 1 shows an example of a MIMO real-valued system based on an 8 × 8 matrix and A = {−1, 1} (see the experiment section for a detailed description of the simulation set-up). As can be seen in Fig. 1, the BP decoder based on the MRF representation (7) has very poor results. Standard techniques to stabilize the BP iterations such as damping the message updates do not help here. Even applying more advanced versions of BP (e.g. Generalized BP and Expectation Propagation) to inference problems on complete MRF graphs yields poor results [12]. The problem here is not in the optimization method but in the cost function that needs to be modified yield a good approximate solution. There have been several recent attempts to apply BP to the MIMO detection problem with good results (e.g. [8, 10]). However in the methods proposed in [8] and [10] the factorization of the probability function is done in such a way that each factor corresponds to a single linear equation. This leads to a partition of the probability function into factors each of which is a function of all the unknown variables. This leads to exponential computational complexity in computing the BP messages. Shental et. al [14] analyzed the case where the matrix H is relatively sparse (and has a grid structure). They showed that even under this restricted assumption the BP still does not perform well. As an alternative method they proposed the generalized belief propagation (GBP) algorithm that does work well on the sparse matrix if the algorithm regions are carefully chosen. There are situations where the sparsity assumption makes sense (e.g. 2D intersymbol interference (ISI) channels). However, in the MIMO channel model we assume that the channel matrix elements are iid and Gaussian; hence we cannot assume that the channel matrix H is sparse. 3 0 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 ES/N0 SER BP MMSE MMSE−SIC ML Figure 1: Decoding results for 8 × 8 system, A = {−1, 1}. 3 The Tree Approximation of the Gaussian Density Our approach is based on an approximation of the exact probability function: p(x1, .., xn|y) ∝exp(−1 2σ2 ∥Hx −y∥2) , x ∈An (9) that enables a successful implementation of the Belief Propagation paradigm. Since the BP algorithm is optimal on loop-free factor graphs (trees) a reasonable approach is finding an optimal tree approximation of the exact distribution (9). Chow and Liu [2] proposed a method to find a tree approximation of a given distribution that has the minimum Kullback-Leibler distance to the actual distribution. They showed that the optimal tree can be learned efficiently via a maximum spanning tree whose edge weights correspond to the mutual information between the two variables corresponding to the edges endpoints. The problem is that the Chow-Liu algorithm is based on the (2-dimensional) marginal distributions. However, finding the marginal distribution of the probability function (9) is, unfortunately, NP hard and it is (equivalent to) our final target. To overcome this obstacle, our approach is based on applying the Chow-Liu algorithm on the distribution corresponding to the unconstrained linear system. This distribution is Gaussian and therefore it is straightforward in this case to compute the (2-dimensional) marginal distributions. Given the Gaussian tree approximation, the next step of our approach is to apply the finite-set constraint and utilize the Gaussian tree distribution to form a discrete loop free approximation of p(x|y) which can be efficiently globally maximized using the BP algorithm. To motivate this approach we first apply a simplified version to derive the linear solution (4) described in Section 2. Let z(y) = (H ⊤H)−1H ⊤y be the least-squares estimator (3) and C = σ2(H ⊤H)−1 is its variance. It can be easily verified that p(x|y) (9) can be written as: p(x|y) ∝f(x; z, C) = exp(−1 2(z −x) ⊤C−1(z −x)) (10) where f(x; z, C) is a Gaussian density with mean z and covariance matrix C (to simplify notation we ignore hereafter the constant coefficient of the Gaussian densities). Now, instead of marginalizing the true distribution p(x|y), which is an NP hard problem, we approximate it by the product of the marginals of the Gaussian density f(x; z, C): f(x; z, C) ≈ Y i f(xi; zi, Cii) = exp(−(zi −xi)2 2Cii ) (11) From the Gaussian approximation (11) we can extract a discrete approximation: ˆp(xi = a|y) ∝f(xi; zi, Cii) = exp(−(zi −a)2 2Cii ) , a ∈A (12) 4 Input: A constrained linear LS problem: Hx + ǫ = y, a noise level σ2 and a finite symbol set A. Goal: Find (approx. to) arg minx∈An ∥Hx −y∥2 Algorithm: • Compute z = (H ⊤H + σ2 e I)−1H ⊤y and C = σ2(H ⊤H + σ2 e I)−1. • Denote: f(xi; z, C) = exp(−1 2 (xi −zi)2 Cii ) f(xi|xj; z, C) = exp(−1 2 ((xi −zi) −Cij/Cjj(xj −zj))2 Cii −C2 ij/Cjj ) • compute maximum spanning tree of the n-node graph where the weight of the i-j edge is the square of the correlation coefficient: ρ2 ij = C2 ij/(CiiCjj) Assume the tree is rooted at node ‘1’ and denote the parent of node i by p(i). • Apply BP on the loop free MRF: ˆp(x1, ..., xn|y) ∝f(x1; z, C) n Y i=2 f(xi|xp(i); z, C) x1, ..., xn ∈A to find the (approx. to the) most likely configuration. Figure 2: The Gaussian Tree Approximation (GTA) Algorithm. Taking the most likely symbol we obtain the sub-optimal linear solution (4). Motivated by the simple product-of-marginals approximation described above, we suggest approximating the discrete distribution p(x|y) via a tree-based approximation of the Gaussian distribution f(x; z, C). Although the Chow-Liu algorithm was originally stated for discrete distributions, one can easily verify that it also applies for the Gaussian case. Let I(xi; xj) = log Cii + log Cjj −log Cii Cij Cji Cjj = −log(1 −ρ2 ij) be the mutual information of xi and xj based on the Gaussian distribution f(x; z, C), where ρij is the correlation coefficient between xi and xj. Let ˆf(x) be the optimal Chow-Liu tree approximation of f(x; z, C). We can assume, without loss of generality, that ˆf(x) is rooted at x1. ˆf(x) is a loopfree Gaussian distribution on x1, ..., xn, i.e. ˆf(x) = f(x1; z, C) n Y i=2 f(xi|xp(i); z, C) , x ∈Rn (13) where p(i) is the ‘parent’ of the i-th node in the optimal tree. The Chow-Liu algorithm guarantees that ˆf(x) is the optimal Gaussian tree approximation of f(x; z, C) in the sense that the KL divergence D(f|| ˆf) is minimal among all the Gauss-Markov distributions on Rn. We note in passing that applying a monotonic function on the graph weights does not change the topology of the optimal tree. Hence to find the optimal tree we can use the weights ρ2 ij instead of −log(1−ρ2 ij). The optimal tree, therefore is one that maximizes the sum of the square correlation coefficients between adjacent nodes. 5 Our approximation approach is, therefore, based on replacing the true distribution p(x|y) with the following approximation: ˆp(x1, ..., xn|y) ∝ˆf(x) = f(x1; z, C) n Y i=2 f(xi|xp(i); z, C) , x ∈An (14) The probability function ˆp(x|y) is a loop free factor graph. Hence the BP algorithm can be applied to find its most likely configuration. An optimal BP schedule requires passing a message once for each direction of each edge. The BP messages are first sent from leaf variables to the root and then back to the leaves. We demonstrate empirically in the experiment section that the optimal solution of ˆp(x|y) is indeed nearly optimal for p(x|y). The MMSE Bayesian approach (5) is known to be better than the linear based solution (4). In a similar way we can consider a Bayesian version of the proposed Gaussian tree approximation. We can partially incorporate the information that x ∈An by using the prior Gaussian distribution x ∼N(0, eI) such that e = 1 |A| P a∈A a2. This yields the posterior Gaussian distribution: f(x|y)(x|y) ∝ exp(−1 2e∥x∥2 − 1 2σ2 ∥Hx −y∥2) (15) ∝ exp(−1 2(x −E(x|y)) ⊤(H ⊤H + σ2 e I)(x −E(x|y)) such that E(x|y) = (H ⊤H + σ2 e I)−1H ⊤y. We can apply the Chow-Liu tree approximation on the Gaussian distribution (15) to obtain a ‘Bayesian’ Gaussian tree approximation for p(x|y). One can expect that this yields is a better approximation of the discrete distribution p(x|y) than the tree distribution that is based on the unconstrained distribution f(x; z, C) since it partially includes the finite-set constraint. We show in Section 4 that the Bayesian version indeed yields better results. To summarize, our solution to the constrained least squares problem is based on applying BP on a Gaussian tree approximation of the Bayesian version of the continuous least-square case. We dub this method “The Gaussian-Tree-Approximation (GTA) Algorithm”. The GTA algorithm is summarized in Fig. 3. We next compute the complexity of the GTA algorithm. The complexity of computing the covariance matrix (H ⊤H + σ2 e I)−1 is O(n3), the complexity of the Chow-Liu algorithm (based on Prim’s algorithm for finding the minimum spanning tree) is O(n2) and the complexity of the BP algorithm is O(|A|2n). 4 Experimental Results In this section we provide simulation results for the GTA algorithm over various MIMO systems. We assume a frame length of 100, i.e. the channel matrix H is constant for 100 channel uses. The channel matrix comprised iid elements drawn from a zero-mean normal distribution of unit variance. We used 104 realizations of the channel matrix. This resulted in 106 vector messages. The performance of the proposed algorithm is shown as a function of the variance of the additive noise σ2. The signal-to-noise ratio (SNR) is defined as 10 log10(Es/N0) where Es/N0 = ne σ2 (n is the number of variables, σ2 is the variance of the Gaussian additive noise, and e is the variance of the uniform distribution over the discrete set A). Fig. 3 shows the symbol error rate (SER) versus SNR for a 10×10, |A| = 8, MIMO system and for a 20×20, |A| = 4, MIMO system. Note that the algorithm was applied in Fig. 3 to a real world practical application (MIMO communication) using real world parameters. Unlike other areas (e.g computer vision, bioinformatics) here the real world performance analysis is based on extensive simulations of the communication channel. Note that a 20 × 20 fully connected MRF is not a small problem and unlike the Potts model that is defined on a grid MRF, the BP and it variants do not work here. The performance of the GTA method was compared to the MMSE and the MMSESIC algorithms (see Section 2). The GTA algorithm differs from these algorithms in two ways. The first is a Markovian approximation of f(x; z, C) instead of a product of independent densities. The second aspect is utilizing the optimal tree. To clarify the contribution of each component we modified the GTA algorithm by replaced the Chow-Liu optimal tree by the tree 1 →2 →3, ..., →n. We call this method the ‘Line-Tree’. As can be seen from Fig. 3, using the optimal tree is crucial 6 to obtain improved results. Fig. 3b also shows results of the non-Bayesian variant of the GTA algorithm. As can be seen, the Bayesian version yields better results. In Fig. 3a the two versions yield the same results. It can be seen that the performance of the GTA algorithm is significantly better than the MMSE-SIC (and its computational complexity is much smaller). 10 15 20 25 30 35 40 45 50 10 −4 10 −3 10 −2 10 −1 10 0 ES/N0 SER MMSE Line−Tree MMSE−SIC GTA ML 10 15 20 25 30 35 40 45 50 10 −4 10 −3 10 −2 10 −1 10 0 ES/N0 SER MMSE Line−Tree MMSE−SIC non−Bayesian−GTA GTA ML (a) 10 × 10, |A| = 8 (b) 20 × 20, |A| = 4 Figure 3: Comparative results of MMSE, MMSE-SIC and variants of the GTA. 0 20 40 60 80 100 0.1 0.15 0.2 0.25 0.3 0.35 n SER MMSE MMSE−SIC GTA−MP GTA−SP 0 20 40 60 80 100 10 −4 10 −3 10 −2 10 −1 10 0 n SER MMSE MMSE−SIC GTA (MP/SP) (a) noise level: σ2 = 2.5 (b) noise level:σ2 = 0.25 Figure 4: Comparative results of MMSE, MMSE-SIC and the GTA approximation followed by the sum-product and max-product variants of the BP algorithm. The alphabet size is |A| = 8 and the results are shown as a function of the matrix size n × n. Fig. 4 depicts comparative performance results as a function of n, the size of the linear system. The alphabet size in all the experiments was |A| = 8 and as in Fig. 3 each experiment was repeated 104×102 times. The performance of the GTA method was compared to the MMSE and the MMSESIC algorithms (see Section 2). In Fig. 4a the noise variance was set to σ2 = 2.5 and in Fig. 4b to σ2 = 0.25. In all cases the GTA was found to be better than the MMSE-SIC. The GTA algorithm is based on an optimal Gaussian tree approximation followed by a BP algorithm. There are two variants of the BP, namely the max-product (MP) and the sum-product (SP). Since the performance is measured in symbol error-rate and not frame error-rate the SP should yield improved results. Note that if the exact distribution was loop-free then SP would obviously be the optimal method when the error is measured in number of symbols. However, since the BP is applied to an approximated distribution the superiority of the SP is not straightforward. When the noise level is relatively high the sum-product version is better than the max-product. When the noise level is lower there is no significant difference between the two BP variants. Note that from an algorithmic point of view, the MP unlike the SP, can be easily computed in the log domain. 7 5 Conclusion Solving integer linear least squares problems is an important issue in many fields. We proposed a novel technique based on the principle of a tree approximation of the Gaussian distribution that corresponds to the continuous linear problem. The proposed method improved performance compared to all other polynomial algorithms for solving the problem as demonstrated in simulations. As far as we know this is the first successful attempt to apply the BP paradigm to completely connected MRF. A main concept in the GTA model is the interplay between discrete and Gaussian models. Such hybrid ideas can be considered also for discrete inference problems other than least-squares. One example is the work of Opper and Winther who applied an iterative algorithm using a model which is seen as discrete and Gaussian in turn to address Ising model problems [13]. Although the focus of this paper is on an approach based on tree approximation, more complicated approximations such as multi-parent trees have potential to improve performance and can potentially provide a smooth performance-complexity trade-off. Although the proposed method yields improved results, the tree approximation we applied nay not be the best one (finding the best tree for the integer constrained linear problem is NP hard). It is left for future research to search for a better discrete tree approximation for the constrained linear least squares problem. References [1] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger. Closest point search in lattices. IEEE Transactions on Information Theory, 48(8):2201–2214, 2002. [2] C. K. Chow and C. N. Liu. Approximating discrete probability distributions with dependence trees. IEEE Trans. on Info. Theory, pages 462–467, 1968. [3] B. Dong, X. Wang, and A. Doucet. A new class of soft MIMO demodulation algorithms. IEEE Trans. Signal Processing, pages 2752–63, 2003. [4] P. F. Felzenszwalb and D. P. Huttenlocher. Efficient belief propagation for early vision. International Journal of Computer Vision, pages 41–54, 2006. [5] G.J. Foschini. Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas. Bell Labs Technical Journal, 1(2):41–59, 1996. [6] R. G. Gallager. Low density parity check codes. IRE Trans. Inform.Theory, pages 21–28, 1962. [7] B. M. Hochwald and S. ten Brink. Achieving near-capacity on a multiple antenna channel. IEEE Trans. Commun., pages 389–399, 2003. [8] J. Hu and T. M. Duman. Graph-based detector for BLAST architecture. IEEE Communications Society ICC, 2007. [9] J. Jalden and B. Ottersten. An exponential lower bound on the expected complexity of sphere decoding. IEEE Intl. Conf. Acoustic, Speech, Signal Processing, 2004. [10] M. Kaynak, T. Duman, and E. Kurtas. Belief propagation over SISO/MIMO frequency selective fading channels. IEEE Transactions on Wireless Communications, pages 2001–5, 2007. [11] J. C. Lagarias and A. M. Odlyzko. Solving low-density subset sum problems. J. ACM, 32(1):229–246, 1985. [12] T. Minka and Y. Qi. Tree-structured approximations by expectation propagation. Advances in Neural Information Processing Systems, 2004. [13] M. Opper and O. Winther. Expectation consistent approximate inference. Journal of Machine Learning Research, pages 2177–2204, 2005. [14] O. Shental, N. Shental, S. Shamai (Shitz), I. Kanter, A.J. Weiss, and Y. Weiss. Discrete-input twodimensional gaussian channels with memory: Estimation and information rates via graphical models and statistical mechanics. Information Theory, IEEE Transactions on, pages 1500–1513, 2008. [15] P.J.G. Teunissen. Success probability of integer GPS ambiguity rounding and bootstrapping. Journal of Geodesy, 72:606–612, 1998. [16] Y. Weiss and W.T. Freeman. Correctness of belief propagation in Gaussian graphical models of arbitrary topology. Neural Computation, pages 2173–2200, 2001. [17] A. Wiesel, Y. C. Eldar, and S. Shamai. Semidefinite relaxation for detection of 16-QAM signaling in MIMO channels. IEEE Signal Processing Letters, 2005. [18] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Understanding belief propagation and its generalizations. IJCAI, 2001. 8
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Nonlinear directed acyclic structure learning with weakly additive noise models Robert E. Tillman Carnegie Mellon University Pittsburgh, PA rtillman@cmu.edu Arthur Gretton Carnegie Mellon University, MPI for Biological Cybernetics Pittsburgh, PA arthur.gretton@gmail.com Peter Spirtes Carnegie Mellon University Pittsburgh, PA ps7z@andrew.cmu.edu Abstract The recently proposed additive noise model has advantages over previous directed structure learning approaches since it (i) does not assume linearity or Gaussianity and (ii) can discover a unique DAG rather than its Markov equivalence class. However, for certain distributions, e.g. linear Gaussians, the additive noise model is invertible and thus not useful for structure learning, and it was originally proposed for the two variable case with a multivariate extension which requires enumerating all possible DAGs. We introduce weakly additive noise models, which extends this framework to cases where the additive noise model is invertible and when additive noise is not present. We then provide an algorithm that learns an equivalence class for such models from data, by combining a PC style search using recent advances in kernel measures of conditional dependence with local searches for additive noise models in substructures of the Markov equivalence class. This results in a more computationally efficient approach that is useful for arbitrary distributions even when additive noise models are invertible. 1 Introduction Learning probabilistic graphical models from data serves two primary purposes: (i) finding compact representations of probability distributions to make inference efficient and (ii) modeling unknown data generating mechanisms and predicting causal relationships. Until recently, most constraint-based and score-based algorithms for learning directed graphical models from continuous data required assuming relationships between variables are linear with Gaussian noise. While this assumption may be appropriate in many contexts, there are well known contexts, such as fMRI images, where variables have nonlinear dependencies and data do not tend towards Gaussianity. A second major limitation of the traditional algorithms is they cannot identify a unique structure; they reduce the set of possible structures to an equivalence class which entail the same Markov properties. The recently proposed additive noise model [1] for structure learning addresses both limitations; by taking advantage of observed nonlinearity and non-Gaussianity, a unique directed acyclic structure can be identified in many contexts. However, it too suffers from limitations: (i) for certain distributions, e.g. linear Gaussians, the model is invertible and not useful for structure learning; (ii) it was originally proposed for two variables with a multivariate extension that requires enumerating all possible DAGs, which is super-exponential in the number of variables. In this paper, we address the limitations of the additive noise model. We introduce weakly additive noise models, which have the advantages of additive noise models, but are still useful when the additive noise model is invertible and in most cases when additive noise is not present. Weakly additive noise models allow us to express greater uncertainty about the 1 data generating mechanism, but can still identify a unique structure or a smaller equivalence class in most cases. We also provide an algorithm for learning an equivalence class for such models from data that is more computationally efficient in the more than two variables case. Section 2 reviews the appropriate background; section 3 introduces weakly additive noise models; section 4 describes our learning algorithm; section 5 discusses some related research; section 6 presents some experimental results; finally, section 7 offers conclusions.. 2 Background Let G = ⟨V, E⟩be a directed acyclic graph (DAG), where V denotes the set of vertices and Eij ∈E denotes a directed edge Vi →Vj. Vi is a parent of Vj and Vj is a child of Vi. For Vi ∈V, PaVi G denotes the parents of Vi and ChVi G denotes the children of Vi. The degree of Vi is the number of edges with an endpoint at Vi. A v-structure is a triple ⟨Vi, Vj, Vk⟩⊆V such that {Vi, Vk} ⊆PaVj G . A v-structure is immoral, or an immorality, if Eik /∈E and Eki /∈E. A joint distribution P over variables corresponding to nodes in V is Markov with respect to G if PP(V) = Y Vi∈V PP Vi | PaVi G . P is faithful to G if every conditional independence true in P is entailed by the above factorization. A partially directed acyclic graph (PDAG) H for G is a mixed graph, i.e. consisting of directed and undirected edges, representing all DAGs Markov equivalent to G, i.e. DAGs entailing exactly the same conditional independencies. If Vi →Vj is a directed edge in H, then all DAGs Markov equivalent to G have this directed edge; if Vi −Vj is an undirected edge in H, then some DAGs that are Markov equivalent to G have the directed edge Vi →Vj while others have the directed edge Vi ←Vj. The PC algorithm is a well known constraint-based, or conditional independence based, structure learning algorithm. It is an improved greedy version of the SGS [2] and IC [3] algorithms, shown below. Instead of searching all subsets of V\{Vi, Vj} for an S such Input : Observed data for variables in V Output: PDAG G over nodes V G ←the complete undirected graph over the variables in V 1 For {Vi, Vj} ⊆V, if ∃S ⊆V\{Vi, Vj}, such that Vi ⊥⊥Vj | S, remove the Vi −Vj edge 2 For {Vi, Vj, Vk} ⊆V such that Vi −Vj and Vj −Vk remain as edges, but Vi −Vk does 3 not remain, if ∄S ⊆V\{Vi, Vj, Vk}, such that Vi ⊥⊥Vk | {S ∪Vj}, orient Vi →Vj ←Vk Orient edges to prevent additional immoralities and cycles using the Meek rules [4] 4 Algorithm 1: SGS/IC algorithm that Vi ⊥⊥Vj | S, PC (i) initially sets S = ∅for all {Vi, Vj} pairs, (ii) checks to see if any edges can be removed based on the results of conditional independence tests with these S sets, and (iii) iteratively increases the cardinality of S considered until ∄Vk ∈V with degree greater than |S|. S is only considered if it is a subset of nodes connected to Vi or Vj at the current iteration. PC learns the correct PDAG in the large sample limit when the Markov, faithfulness, and causal sufficiency (that there are no unmeasured common causes of two or more measured variables) assumptions hold [2]. The partial correlation based Fisher Z-transformation test, which assumes linear Gaussian distributions, is used for conditional independence testing with continuous variables. The statistical advantage of PC is it limits the number of tests performed, particularly those with large conditioning sets. This also yields a computational advantage since the number of possible tests is exponential in |V|. The recently proposed additive noise model approach to structure learning [1] assumes only that each variable can be represented as a (possibly nonlinear) function f of its parents plus additive noise ǫ with some arbitrary distribution, and that the noise components are mutually independent, i.e. P(ǫ1, . . . , ǫn) = n Y i=1 P(ǫi). Consider the two variable case where X →Y is the true DAG, X = ǫX, Y = sin(πX) + ǫY , ǫX ∼Unif(−1, 1), and ǫY ∼ Unif(−1, 1). If we regress Y on X (nonparametrically), the forward model, figure 1a, and 2 −1 −0.5 0 0.5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 X Y −2−1.5−1−0.5 0 0.5 1 1.5 2 −1 −0.5 0 0.5 1 Y X −1 −0.5 0 0.5 1 −8 −6 −4 −2 0 2 4 6 8 10 X Z −8 −6 −4 −2 0 2 4 6 8 10 −1 −0.5 0 0.5 1 Z X (a) (b) (c) (d) Figure 1: Nonparametric regressions with data overlayed for (a) Y regressed on X, (b) X regressed on Y , (c) Z regressed on X, and (d) X regressed on Z regress X on Y , the backward model, figure 1b, we observe the residuals ˆǫY ⊥⊥X and ˆǫX /⊥⊥Y . This provides a criterion for distinguishing X →Y from X ←Y in many cases, but there are counterexamples such as the linear Gaussian case, where the forward model is invertible so we find ˆǫY ⊥⊥X and ˆǫX ⊥⊥Y . [1, 5] show, however, that whenever f is nonlinear, the forward model is noninvertible, and when f is linear, the forward model is only invertible when ǫ is Gaussian and a few other special cases. Another limitation of this approach is that it is not closed under marginalization of intermediary variables when f is nonlinear, e.g. for X →Y →Z with X = ǫX, Y = X3 + ǫY , Z = Y 3 + ǫZ, ǫX ∼Unif(−1, 1), ǫY ∼Unif(−1, 1), and ǫZ ∼Unif(0, 1), observing only X and Z, figures 1c and 1d, causes us to reject both the forward and backward models. [5] shows this method can be generalized to more variables. To test whether a DAG is compatible with the data, we regress each variable on its parents and test whether the resulting residuals are mutually independent. This procedure is impractical even for a few variables, however, since the number of possible DAGs grows super-exponentially with the number of variables, e.g. there are ≈4.2×1018 DAGs with 10 nodes. Since we do not assume linearity or Gaussianity in this framework, a sufficiently powerful nonparametric independence test must be used. Typically, the Hilbert Schmidt Independence Criterion [6] is used, which we now define. Let X be a random variable with domain X. A Hilbert space HX of functions from X to R is a reproducing kernel Hilbert space (RKHS) if for some kernel k(·, ·) (the reproducing kernel for HX ), for every f(·) ∈HX and x ∈X, the inner product ⟨f(·), k(x, ·)⟩HX = f(x). We may treat k(x, ·) as a mapping of x to the feature space HX . For x, x′ ∈X, ⟨k(x, ·), k(x′, ·)⟩HX = k(x, x′), so we can compute inner products efficiently in this high dimensional space. The Moore-Aronszajn theorem shows that all symmetric positive definite kernels (most popular kernels) are reproducing kernels that uniquely define corresponding RKHSs [7]. Let Y be a random variable with domain Y and l(·, ·) the reproducing kernel for HY. We define the mean map µX and cross covariance CXY as follows, using ⊗to denote the tensor product. µX = EX[k(x, ·)] CXY = ([k(x, ·) −µX] ⊗[l(y, ·) −µY ]) If the kernels are characteristic, e.g. Gaussian and Laplace kernels, the mean map is injective [8, 9, 10] so distinct probability distributions have different mean maps. The Hilbert Schmidt Independence Criteria (HSIC) HXY = ∥CXY ∥2 HS measures the dependence of X and Y , where ∥· ∥HS denotes the Hilbert Schmidt norm. [9] shows HXY = 0 if and only if X ⊥⊥Y for characteristic kernels. For m paired i.i.d. samples, let K and L be Gram matrices for k(·, ·) and l(·, ·), i.e. kij = k(xi, xj). For H = IN −1 N 1N1T N, let ˜K = HKH and ˜L = HLH be centered Gram matrices. ˆHXY = 1 m2 tr ˜K ˜L , where tr denotes the trace, is an empirical estimator for HXY [6]. To determine the threshold of a level-α statistical test, we can use the permutation approach (where we compute ˆHXY for multiple random assignments of the Y samples to X, and use the 1 −α quantile of the resulting empirical distribution over ˆHXY ), or a Gamma approximation to the null distribution of mˆHXY (see [6] for details). 3 Weakly additive noise models We now extend the additive noise model framework to account for cases where additive noise models are invertible and cases where additive noise may not be present. 3 Definition 3.1. ψ = D Vi, PaVi G E is a local additive noise model for a distribution P over V that is Markov to a DAG G = ⟨V, E⟩if Vi = f PaVi G + ǫ is an additive noise model. Definition 3.2. A weakly additive noise model M = ⟨G, Ψ⟩for a distribution P over V is a DAG G = ⟨V, E⟩and set of local additive noise models Ψ, such that P is Markov to G, ψ ∈Ψ if and only if ψ is a local additive noise model for P, and ∀ D Vi, PaVi G E ∈Ψ, ∄Vj ∈PaVi G such that there exists some graph G′ (not necessarily related to P) such that Vi ∈PaVj G′ and D Vj, PaVj G′ E is a local additive noise model for P. When we assume a data generating process has a weakly additive noise model representation, we assume only that there are no cases where X →Y can be written X = f(Y ) + ǫX, but not Y = f(X) + ǫY . In other words, the data cannot appear as though it admits an additive noise model representation, but only in the incorrect direction. This representation is still appropriate when additive noise models are invertible, and when additive noise is not present: such cases only lead to weakly additive noise models which express greater underdetermination of the true data generating process. We now define the notion of distribution-equivalence for weakly additive noise models. Definition 3.3. A weakly additive noise model M = ⟨G, Ψ⟩is distribution-equivalent to N = ⟨G′, Ψ′⟩if and only if G and G′ are Markov equivalent and ψ ∈Ψ if and only if ψ ∈Ψ′. Distribution-equivalence defines what can be discovered about the true data generating mechanism using observational data. We now define a new structure to partition data generating processes which instantiate distribution-equivalent weakly additive noise models. Definition 3.4. A weakly additive noise partially directed acyclic graph (WAN-PDAG) for M = ⟨G, Ψ⟩is a mixed graph H = ⟨V, E⟩such that for {Vi, Vj} ⊆V, 1. Vi →Vj is a directed edge in H if and only if Vi →Vj is a directed edge in G and in all G′ such that N = ⟨G′, Ψ′⟩is distribution-equivalent to M 2. Vi −Vj is an undirected edge in H if and only if Vi →Vj is a directed edge in G and there exists a G′ and N = ⟨G′, Ψ′⟩distribution-equivalent to M such that Vi ←Vj is a directed edge in G′ We now get the following results. Lemma 3.1. Let M = ⟨G, Ψ⟩be a weakly additive noise model, D Vi, PaVi G E ∈Ψ, and N = ⟨G′, Ψ′⟩be distribution equivalent to M. Then PaVi G = PaVi G′ and ChVi G = ChVi G′. Proof. Since M and N are distribution-equivalent, PaVi G = PaVi G′. Thus, ChVi G = ChVi G′ Theorem 3.1. The WAN-PDAG for M = ⟨G, Ψ⟩is constructed by (i) adding all directed and undirected edges in the PDAG instantiated by M, (ii) ∀ D Vi, PaVi G E ∈Ψ, directing all Vj ∈PaVi G as Vj →Vi and all Vk ∈ChVi G as Vi →Vk, and (iii) applying the extended Meek rules [4], treating orientions made using Ψ as background knowledge. Proof. (i) This is correct because of Markov equivalence [2]. (ii) This is correct by lemma 3.1. (iii) These rules are correct and complete [4]. WAN-PDAGs can used to identify the same information about the data generating mechanism as additive noise models, when additive noise models are identifiable, but provide a more powerful representation of uncertainty and can be used to discover more information when additive noise models are unidentifiable. The next section describes an efficient algorithm for learning WAN-PDAGs from data. 4 4 The Kernel PC (kPC) algorithm We now describe the Kernel PC (kPC) algorithm1, which consists of two stages: (i) a constraint-based search using the PC algorithm with a nonparametric conditional independence test (the Fisher Z test is inappropriate since we want to allow nonlinearity and non-Gaussianity) to identify the Markov equivalence class and (ii) a “PC-style” search for noninvertible additive noise models in submodels of the Markov equivalence class. In the first stage, we use a kernel-based conditional dependence measure similar to HSIC [9] (see also [11, Section 2.2] for a related quantity with a different normalization). For a conditioning variable Z with centered Gram matrix ˜ M for a reproducing kernel m(·, ·), we define the conditional cross covariance CXY |Z = C ¨ XZC−1 ZZCZ ¨Y , where ¨X = (X, Z) and ¨Y = (Y, Z). Let HXY |Z = ∥CXY |Z∥2 HS. It follows from [9, Theorem 3] that HXY |Z = 0 if and only if X ⊥⊥Y |Z when kernels are characteristic. [9] provides the empirical estimator: ˆHXY |Z = 1 m2 tr( ˜K ˜L −2 ˜K ˜ M( ˜ M + ǫIN)−2 ˜ M ˜L + ˜K ˜ M( ˜ M + ǫIN)−2 ˜ M ˜L ˜ M( ˜ M + ǫIN)−2 ˜ M) The null distribution of ˆHXY |Z is unknown and difficult to derive so we must use the permutation approach described in section 2. This is not straightforward since permuting X or Y while leaving Z fixed changes the marginal distribution of X given Z or Y given Z. We thus (making analogy to the discrete case) must cluster Z and then permute elements only within clusters for the permutation test, as in [12]. This first stage is not computational efficient, however, since each evaluation of ˆHXY |Z is naively O N 3 and we need to evaluate ˆHXY |Z approximately 1000 times for each permutation test. Fortunately, we see from [13, Appendix C] that the eigenspectra of Gram matrices for Gaussian kernels decay very rapidly, so low rank approximations of these matrices can be obtained even when using a very conservative threshold. We implemented the incomplete Cholesky factorization [14], which can be used to obtain an m × p matrix G, where p ≪m, and an m × m permutation matrix P such that K ≈PGG⊤P ⊤, where K is an m × m Gram matrix. A clever implementation after replacing Gram matrices in ˆHXY |Z with their incomplete Cholesky factorizations and using an appropriate equivalence to invert G⊤G + ǫIp for ˜ M instead of GG⊤+ ǫIm results in a straightforward O mp3 operation. Unfortunately, this is not numerically stable unless a relatively large regularizer ǫ is chosen or only a small number of columns are used in the incomplete Cholesky factorizations. A more stable (and faster) approach is to obtain incomplete Cholesky factorizations GX, GY , and GZ with permutation matrices PX, PY , and PZ, and then obtain the thin SVDs for HPXGX, HPY GY , and HPZGZ, e.g HPG = USV , where U is m × p, S is the p × p diagonal matrix of singular values, and V is p × p. Now define matrices ¯SX, ¯SY , and ¯SZ and ¯GX, ¯GY , and ¯GZ as follows: ¯sX ii = sX ii 2 ¯sY ii = sY ii 2 ¯sZ ii = sZ ii 2 sZ ii 2 + ǫ ¯GX = U X ¯SXU X⊤ ¯GY = U Y ¯SY U Y ⊤ ¯GZ = U Z ¯SZU Z⊤ We can compute ˆHXY |Z = 1 m2 tr ¯GX ¯GY −2 ¯GX ¯GZ ¯GY + ¯GX ¯GZ ¯GY ¯GZ stably and efficiently in O mp3 by choosing an appropriate associative ordering of matrix multiplications. Figure 2 shows that this method leads to a significant increase in speed when used with a permutation test for conditional independence without significantly affecting the empirically observed type I error rate for a level-.05 test. In the second stage, we look for additive noise models in submodels of the Markov equivalence class because (i) it may be more efficient to do so and require fewer tests since orientations implied by an additive noise model may imply further orientations and (ii) we 1MATLAB code may be obtained from http://www.andrew.cmu.edu/∼rtillman/kpc 5 200 400 600 800 1000 0 5 10 15 20 Sample Size Time (minutes) Naive Incomplete Cholesky + SVD 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Empirical Type I Error Rate Naive Incomplete Cholesky + SVD Figure 2: Runtime and Empirical Type I Error Rate. Results are over the generation of 20 3-node DAGs for which X ⊥⊥Y |Z and the generating distribution was Gaussian. may find more orientations by considering submodels, e.g. if all relations are linear and only one variable has a non-Gaussian noise term. The basic strategy used is a“PC-style” greedy search where we look for undirected edges in the current mixed graph (starting with the PDAG resulting from the first stage) adjacent to the fewest other undirected edges. If these edges can be oriented using additive noise models, we make the implied orientations, apply the extended Meek rules, and then iterate until no more edges can be oriented. Algorithm 2 provides pseudocode. Let G = ⟨V, E⟩be the resulting PDAG and ∀Vi ∈V, let UVi G denote the nodes connected to Vi in G by an undirected edge. We get the following results. Input : PDAG G = ⟨V, E⟩ Output: WAN-PDAG G = ⟨V, E⟩ s ←1 1 while max Vi∈V UVi G ≥s do 2 foreach Vi ∈V such that UVi G = s or UVi G < s and UVi G was updated do 3 s′ ←s 4 while s′ > 0 do 5 foreach S ⊆UVi G such that |S| = s′ and ∀Sk ∈S, orienting Sk →Vi, does 6 not create an immorality do Nonparametrically regress Vi on PaVi G ∪S and compute the residual ˆǫiS 7 if ˆǫiS ⊥⊥S and ∄Vj ∈S and S′ ⊆UVj G such that. regressing Vj on 8 PaG Vj ∪S′ ∪Vi results in the residual ˆǫjS′∪{Vi} ⊥⊥S′ ∪{Vi} then ∀Sk ∈S, orient Sk →Vi, and ∀Ul ∈UVi G \S orient Vi →Ul 9 Apply the extended Meek rules 10 ∀Vm ∈V, update UVm G , set s′ = 1, and break 11 end 12 end 13 s′ ←s′ −1; 14 end 15 end 16 s ←s + 1 17 end 18 Algorithm 2: Second Stage of kPC Lemma 4.1. If an edge is oriented in the second stage of kPC, it is implied by a noninvertible local additive noise model. Proof. If the condition at line 8 is true then D Vi, PaVi G ∪S E is a noninvertible local additive noise model. All Ul ∈UVi G \S must be children of Vi by lemma 3.1. 6 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Precision kPC PC GES LiNGAM 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Precision kPC PC GES LiNGAM 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Precision kPC PC GES LiNGAM 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Recall kPC PC GES LiNGAM 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Recall kPC PC GES LiNGAM 200 400 600 800 1000 0 0.2 0.4 0.6 0.8 1 Sample Size Recall kPC PC GES LiNGAM Linear Gaussian Linear Non-Gaussian Nonlinear Non-Gaussian Figure 3: Precision and Recall Lemma 4.2. Suppose ψ = ⟨Vi, W⟩is a noninvertible local additive noise model. Then kPC will make all orientations implied by ψ. Proof. Let ˜S = W\PaG Vi for PaG Vi at the current iteration. kPC must terminate with s > |˜S| since |˜S| ≤|UVi G | so S = ˜S at some iteration. Since D Vi, PaVi G ∪˜S E is a noninvertible local additive noise model, line 8 is satisfied so all edges connected to Vi are oriented. Theorem 4.1. Assume data is generated according to some weakly additive noise model M = ⟨G, Ψ⟩. Then kPC will return the WAN-PDAG instantiated by M assuming perfect conditional independence information, Markov, faithfulness, and causal sufficiency. Proof. The PC algorithm is correct and complete with respect to conditional independence [2]. Orientations made with respect to additive noise models are correct by lemma 4.1 and all such orientations that can be made are made by lemma 4.2. The Meek rules, which are correct and complete [4], are invoked after each orientation made with respect to additive noise models so they are invoked after all such orientations are made. 5 Related research kPC is similar in spirit to the PC-LiNGAM structure learning algorithm [15], which assumes dependencies are linear with either Gaussian or non-Gaussian noise. PC-LiNGAM combines the PC algorithm with LiNGAM to learn structures referred to as ngDAGs. KCL [11] is a heuristic search for a mixed graph that uses the same kernel-based dependence measures as kPC (while not determining significance threhsholds via a hypothesis test), but does not take advantage of additive noise models. [16] provides a more efficient algorithm for learning additive noise models, by first finding a causal ordering after doing a series of high dimensional regressions and HSIC independence tests and then pruning the resulting DAG implied by this ordering. Finally, [17] proposes a two-stage procedure for learning additive noise models from data that is similar to kPC, but requires the additive noise model assumptions in the first stage where the Markov equivalence class is identified. 6 Experimental results To evaluate kPC, we generated 20 random 7-nodes DAGs using the MCMC algorithm in [18] and sampled 1000 data points from each DAG under three conditions: linear dependencies 7 I LOCC LACC LIFG LIPL LMTG I LOCC LACC LIFG LIPL LMTG kPC iMAGES Figure 4: Structures learned by kPC and iMAGES with Gaussian noise, linear dependencies with non-Gaussian noise, and nonlinear dependencies with non-Gaussian noise. We generated non-Gaussian noise using the same procedure as [19] and used polynomial and trigonometric functions for nonlinear dependencies. We compared kPC to PC, the score-based GES with the BIC-score [20], and the ICA-based LiNGAM [19], which assumes linear dependencies and non-Gaussian noise. We applied two metrics in measuring performance vs sample size: precision, i.e. proportion of directed edges in the resulting graph that are in the true DAG, and recall, i.e. proportion of directed edges in the true DAG that are in the resulting graph. Figure 3 reports the results. In the linear Gaussian case, we see PC shows slightly better performance than kPC in precision, which is unsurprising since PC assumes linear Gaussian distributions. Only LiNGAM shows better recall, but worse precision. LiNGAM performs significantly better than the other algorithms in the linear non-Gaussian case. kPC performs about the same as PC in precision and recall, which again is unsurprising since previous simulation results have shown that nonlinearity, but not non-Gaussianity can significantly affect the performance of PC. In the nonlinear non-Gaussian case, kPC performs slightly better than PC in precision. We note, however, that in some of these cases the performance of kPC was significantly better.2 We also ran kPC on data from an fMRI experiment that is analyzed in [21] where nonlinear dependencies can be observed. Figure 4 shows the structure that kPC learned, where each of the nodes corresponds to a particular brain region. This structure is the same as the one learned by the (GES-style) iMAGES algorithm in [21] except for the absence of one edge. However, iMAGES required background knowledge to direct the edges. kPC successfully found the same directed edges without using any background knowledge. Domain experts in neuroscience have confirmed the plausibility of the observed relationships. 7 Conclusion We introduced weakly additive noise models, which extend the additive noise model framework to cases such as the linear Gaussian, where the additive noise model is invertible and thus unidentifiable, as well as cases where additive noise is not present. The weakly additive noise framework allows us to identify a unique DAG when the additive noise model assumptions hold, and a structure that is at least as specific as a PDAG (possibly still a unique DAG) when some additive noise assumptions fail. We defined equivalence classes for such models and introduced the kPC algorithm for learning these equivalence classes from data. Finally, we found that the algorithm performed well on both synthetic and real data. Acknowledgements We thank Dominik Janzing and Bernhard Sch¨olkopf for helpful comments. RET was funded by a grant from the James S. McDonnel Foundation. AG was funded by DARPA IPTO FA8750-09-1-0141, ONR MURI N000140710747, and ARO MURI W911NF0810242. 2When simulating nonlinear data, we must be careful to ensure that variances do not blow up and result in data for which no finite sample method can show adequate performance. This has the unfortunate side effect that the nonlinear data generated may be well approximated using linear methods. Future research will consider more sophisticated methods for simulating data that is more appropriate when comparing kPC to linear methods. 8 References [1] P. O. Hoyer, D. Janzing, J. M. Mooij, J. Peters, and B. Sch¨olkopf. Nonlinear causal discovery with additive noise models. In Advances in Neural Information Processing Systems 21, 2009. [2] P. Spirtes, C. Glymour, and R. Scheines. Causation, Prediction, and Search. 2nd edition, 2000. [3] J. Pearl. Causality: Models, Reasoning, and Inference. 2000. [4] C. Meek. Causal inference and causal explanation with background knowledge. In Proceedings of the 11th Conference on Uncertainty in Artificial Intelligence, 1995. [5] K. Zhang and A. Hyv¨arinen. On the identifiability of the post-nonlinear causal model. In Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, 2009. [6] A. Gretton, K. Fukumizu, C. H. Teo, L. Song, B. Sch¨olkopf, and A. J. Smola. A kernel statistical test of independence. In Advances in Neural Information Processing Systems 20, 2008. [7] Nachman Aronszajn. Theory of reproducing kernels. Transactions of the American Mathematical Society, 68(3):337404, 1950. [8] A. Gretton, K. Borgwardt, M. Rasch, B. Sch¨olkopf, and A. Smola. A kernel method for the two-sample-problem. In Advances in Neural Information Processing Systems 19, 2007. [9] K. Fukumizu, A. Gretton, X. Sun, and B. Sch¨olkopf. Kernel measures of conditional dependence. In Advances in Neural Information Processing Systems 20, 2008. [10] B. Sriperumbudur, A. Gretton, K. Fukumizu, G. Lanckriet, and B. Sch¨olkopf. Injective hilbert space embeddings of probability measures. In Proceedings of the 21st Annual Conference on Learning Theory, 2008. [11] X. Sun, D. Janzing, B. Scholk¨opf, and K. Fukumizu. A kernel-based causal learning algorithm. In Proceedings of the 24th International Conference on Machine Learning, 2007. [12] X. Sun. Causal inference from statistical data. PhD thesis, Max Plank Institute for Biological Cybernetics, 2008. [13] F. R. Bach and M. I. Jordan. Kernel independent component analysis. Journal of Machine Learning Research, 3:1–48, 2002. [14] S. Fine and K. Scheinberg. Efficient SVM training using low-rank kernel representations. Journal of Machine Learning Research, 2:243–264, 2001. [15] P. O. Hoyer, A. Hyv¨arinen, R. Scheines, P. Spirtes, J. Ramsey, G. Lacerda, and S. Shimizu. Causal discovery of linear acyclic models with arbitrary distributions. In Proceedings of the 24th Conference on Uncertainty in Artificial Intelligence, 2008. [16] J. M. Mooij, D. Janzing, J. Peters, and B. Scholk¨opf. Regression by dependence minimization and its application to causal inference in additive noise models. In Proceedings of the 26th International Conference on Machine Learning, 2009. [17] K. Zhang and A. Hyv¨arinen. Acyclic causality discovery with additive noise: An information-theoretical perspective. In Proceedings of the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases 2009, 2009. [18] G. Melan¸con, I. Dutour, and M. Bousquet-M´elou. Random generation of dags for graph drawing. Technical Report INS-R0005, Centre for Mathematics and Computer Sciences, 2000. [19] S. Shimizu, P. Hoyer, A. Hyv¨arinen, and A. Kerminen. A linear non-gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7:1003–2030, 2006. [20] D. M. Chickering. Optimal structure identification with greedy search. Journal of Machine Learning Research, 3:507–554, 2002. [21] J. D. Ramsey, S. J. Hanson, C. Hanson, Y. O. Halchenko, R. A. Poldrack, and C. Glymour. Six problems for causal inference from fMRI. NeuroImage, 2009. In press. 9
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FACTORIE: Probabilistic Programming via Imperatively Defined Factor Graphs Andrew McCallum, Karl Schultz, Sameer Singh Department of Computer Science University of Massachusetts Amherst Amherst, MA 01003 {mccallum, kschultz, sameer}@cs.umass.edu Abstract Discriminatively trained undirected graphical models have had wide empirical success, and there has been increasing interest in toolkits that ease their application to complex relational data. The power in relational models is in their repeated structure and tied parameters; at issue is how to define these structures in a powerful and flexible way. Rather than using a declarative language, such as SQL or first-order logic, we advocate using an imperative language to express various aspects of model structure, inference, and learning. By combining the traditional, declarative, statistical semantics of factor graphs with imperative definitions of their construction and operation, we allow the user to mix declarative and procedural domain knowledge, and also gain significant efficiencies. We have implemented such imperatively defined factor graphs in a system we call FACTORIE, a software library for an object-oriented, strongly-typed, functional language. In experimental comparisons to Markov Logic Networks on joint segmentation and coreference, we find our approach to be 3-15 times faster while reducing error by 20-25%—achieving a new state of the art. 1 Introduction Conditional random fields [1], or discriminatively trained undirected graphical models, have become the tool of choice for addressing many important tasks across bioinformatics, natural language processing, robotics, and many other fields [2, 3, 4]. While relatively simple structures such as linear chains, grids, or fully-connected affinity graphs have been employed successfully in many contexts, there has been increasing interest in more complex relational structure—capturing more arbitrary dependencies among sets of variables, in repeated patterns—and interest in models whose variablefactor structure changes during inference, as in parse trees and identity uncertainty. Implementing such complex models from scratch in traditional programming languages is difficult and error-prone, and hence there has been several efforts to provide a high-level language in which models can be specified and run. For generative, directed graphical models these include BLOG [5], IBAL [6], and Church [7]. For conditional, undirected graphical models, these include Relational Markov Networks (RMNs) using SQL [8], and Markov Logic Networks (MLNs) using first-order logic [9]. Regarding logic, for many years there has been considerable effort in integrating first-order logic and probability [9, 10, 11, 12, 13]. However, we contend that in many of these proposed combinations, the ‘logic’ aspect is not crucial to the ultimate goal of accurate and expressive modeling. The power of relational factor graphs is in their repeated relational structure and tied parameters. First-order logic is one way to specify this repeated structure, but it is less than ideal because of its focus on boolean outcomes and inability to easily and efficiently express relations such as graph reachability and set size comparison. Logical inference is used in some of these systems, such as PRISM [12], but in others, such as Markov Logic [9], it is largely replaced by probabilistic inference. 1 This paper proposes an approach to probabilistic programming that preserves the declarative statistical semantics of factor graphs, while at the same time leveraging imperative constructs (pieces of procedural programming) to greatly aid both efficiency and natural intuition in specifying model structure, inference, and learning, as detailed below. Our approach thus supports users in combining both declarative and procedural knowledge. Rather than first-order logic, model authors have access to a Turing complete language when writing their model specification. The point, however, is not merely to have greater formal expressiveness; it is ease-of-use and efficiency. We term our approach imperatively defined factor graphs (IDFs). Below we develop this approach in the context of Markov chain Monte-Carlo inference, and define four key imperative constructs— arguing that they provide a natural interface to central operations in factor graph construction and inference. These imperative constructs (1) define the structure connecting variables and factors, (2) coordinate variable values, (3) map the variables neighboring a factor to sufficient statistics, and (4) propose jumps from one possible world to another. A model written as an IDF is a factor graph, with all the traditional semantics of factors, variables, possible worlds, scores, and partition functions; we are simply providing an extremely flexible language for their succinct specification, which also enables efficient inference and learning. Our first embodiment of the approach is the system we call FACTORIE (loosely named for “Factor graphs, Imperative, Extensible”, see http://factorie.cs.umass.edu) strongly-typed, functional programming language Scala [14]. The choice of Scala stems from key inherent advantages of the language itself, plus its full interoperability with Java, and recent growing usage in the machine learning community. By providing a library and direct access to a full programming language (as opposed to our own, new “little language”), the model authors have familiar and extensive resources for implementing the procedural aspects of the design, as well as the ability to beneficially mix data pre-processing, evaluation, and other book-keeping code in the same files as the probabilistic model specification. Furthermore, FACTORIE is object-oriented in that variables and factor templates are objects, supporting inheritance, polymophism, composition, and encapsulation. The contributions of this paper are introducing the novel IDF methodology for specifying factor graphs, and successfully demonstrating it on a non-trivial task. We present experimental results applying FACTORIE to the substantial task of joint inference in segmentation and coreference of research paper citations, surpassing previous state-of-the-art results. In comparison to Markov Logic (Alchemy) on the same data, we achieve a 20-25% reduction in error, and do so 3-15 times faster. 2 Imperatively Defined Factor Graphs A factor graph G is a bipartite graph over factors and variables defining a probability distribution over a set of target variables y, optionally conditioned on observed variables x. A factor Ψi computes a scalar value over the subset of variables that are its neighbors in the graph. Often this realvalued function is defined as the exponential of the dot product over sufficient statistics {fik(xi, yi)} and parameters {θik}, where k ∈{1 . . . Ki} and Ki is the number of parameters for factor Ψi. Factor graphs often use parameter tying, i.e. the same parameters for several factors. A factor template Tj consists of parameters {θjk}, sufficient statistic functions {fjk}, and a description of an arbitrary relationship between variables, yielding a set of satisfying tuples {(xi, yi)}. For each of these variable tuples (xi, yi) ∈Tj that fulfills the relationship, the factor template instantiates a factor that shares {θjk} and {fjk} with all other instantiations of Tj. Let T be the set of factor templates. In this case the probability distribution is defined: p(y|x) = 1 Z(x) Y Tj∈T Y (xi,yi)∈Tj exp Kj X k=1 θjkfjk(xi, yi) . As in all relational factor graphs, our language supports variables and factor template definitions. In our case the variables—which can be binary, categorical, ordinal, real, etc—are typed objects in the object-oriented language, and can be sub-classed. Relations between variables can be represented directly as members (instance variables) in these variable objects, rather than as indices into global tables. In addition we allow for new variable types to be programmed by model authors via polymorphism. For example, the user can easily create new variable types such as a set-valued variable type, 2 Figure 1: Example of variable classes for a linear chain and a coreference model. class Token(str:String) extends EnumVariable(str) class Label(str:String, val token:Token) extends EnumVariable(str) with VarInSeq class Mention(val string:String) extends PrimitiveVariable[Entity] class Entity extends SetVariable[Mention] { var canonical:String = "" def add(m:Mention, d:DiffList) = { super.add(m,d); m.set(this,d) canonical = recomputeCanonical(members) } def remove(m:Mention, d:DiffList) = { super.remove(m,d); m.set(null,d) canonical = recomputeCanonical(members) } } representing a group of unique values, as well as traits augmenting variables to represent sequences of elements with left and right neighbors. Typically, IDF programming consists of two distinct stages: defining the data representation, then defining the factors for scoring. This separation offers great flexibility. In the first stage the model author implements infrastructure for storing a possible world—variables, their relations and values. Somewhat surprisingly, authors can do this with a mind-set and style they would employ for deterministic programming, including usage of standard data structures such as linked lists, hash tables and objects embedded in other objects. In some cases authors must provide API functions for “undoing” and “redoing” changes to variables that will be tracked by MCMC, but in most cases such functionality is already provided by the library’s wide variety of variable object implementations. For example, in a linear-chain CRF model, a variable containing a word token can be declared as the Token class shown in Figure 1.1 A variable for labels can be declared similarly, with the addition that each Label2 object has an instance variable that points to its corresponding Token. The second stage of our linear-chain CRF implementation is described in Section 2.2. Consider also the task of entity resolution in which we have a set of Mentions to be co-referenced into Entities. A Mention contains its string form, but its value as a random variable is the Entity to which it is currently assigned. An Entity is a set-valued variable—the set of Mentions assigned to it; it holds and maintains a canonical string form representative of all its Mentions (see Figure 13). The add/remove methods are explained in section 2.3. 2.1 Inference and Imperative Constraint Preservation For inference, we rely on MCMC to achieve efficiency with models that not only have large treewidth but an exponentially-sized unrolled network, as is common with complex relational data [15, 9, 5]. The key is to avoid unrolling the network over multiple hypotheses, and to represent only one variable-value configuration at a time. As in BLOG [5], MCMC steps can adjust model structure as necessary, and with each step the FACTORIE library automatically builds a DiffList—a compact object containing the variables changed by the step, as well as undo and redo capabilities. Calculating the factor graph’s ‘score’ for a step only requires DiffList variables, their factors, and neighboring variables, as described in Section 2.4. In fact, unlike BLOG and BLAISE [16], we build inference and learning entirely on DiffList scores and never need to score the entire model. This enables efficient reasoning about observed data larger than memory, or models in which the number of factors is a high-degree polynomial of the number of variables. A key component of many MCMC inference procedures is the proposal distribution that proposes changes to the current configuration. This is a natural place for injecting prior knowledge about coordination of variable values and various structural changes. In fact, in some cases we can avoid 1Objects of class EnumVariable hold variables with a value selected from a finite enumerated set. 2In Scala var/val indicates a variable declaration; trait VarInSeq provides methods for obtaining next and prev labels in a sequence. 3In Scala def indicates a function definition where the value returned is the last line-of-code in the function; members is the set of variables in the superclass SetVariable. 3 Figure 2: Examples of FACTORIE factor templates. Some error-checking code is elided for brevity. val crfTemplate = new TemplateWithDotStatistics3[Label,Label,Token] { def unroll1 (label:Label) = Factor(label, label.next, label.token) def unroll2 (label:Label) = Factor(label.prev, label, label.prev.token) def unroll3 (token:Token) = throw new Error("Token values shouldn’t change") } val depParseTemplate = new Template1[Node] with DotStatistics2[Word,Word] { def unroll1(n:Node) = n.selfAndDescendants def statistics(n:Node) = Stat(n.word, closestVerb(n).word) def closestVerb(n:Node) = if (isVerb(n.word)) n else closestVerb(n.parent) } val corefTemplate = new Template2[Mention,Entity] with DotStatistics1[Bool] { def unroll1 (m:Mention) = Factor(m, m.entity) def unroll2 (e:Entity) = for (mention <- e.mentions) yield Factor(mention, e) def statistics(m:Mention,e:Entity) = Bool(distance(m.string,e.canonical)<0.5) } val logicTemplate1 = Forany[Person] { p => p.smokes −−> p.cancer } val logicTemplate2 = Forany[Person] { p => p.friends.smokes <−−> p.smokes } expensive deterministic factors altogether with property-preserving proposal functions [17]. For example, coreference transitivity can be efficiently enforced by proper initialization and a transitivitypreserving proposal function; projectivity in dependency parsers can be enforced similarly. We term this imperative constraint preservation. In FACTORIE proposal distributions may be implemented by the model author. Alternatively, the FACTORIE library provides several default inference methods, including Gibbs sampling, as well as default proposers for many variable classes. 2.2 Imperative Structure Definition At the heart of model structure definition is the pattern of connectivity between variables and factors, and the DiffList must have extremely efficient access to this. Unlike BLOG, which uses a complex, highly-indexed data structure that must be updated during inference, we instead specify this connectivity imperatively: factor template objects have methods (e.g., unroll1, unroll2, etc., one for each factor argument) that find the factor’s other variable neighbors given a single variable from the DiffList. This is typically accomplished using a simple data structure that is already available as part of the natural representation of the data, (e.g., as would be used by a non-probabilistic programmer). The unroll method then constructs a Factor with these neighbors as arguments, and returns it. The unroll method may optionally return multiple Factors in response to a single changed variable. Note that this approach also efficiently supports a model structure that varies conditioned on variable values, because the unroll methods can examine and perform calculations on these values. Thus we now have the second stage of FACTORIE programming, in which the model author implements the factor templates that define the factors which score possible worlds. In our linearchain CRF example, the factor between two succesive Labels and a Token might be declared as crfTemplate in Figure 2. Here unroll1 simply uses the token instance variable of each Label to find the corresponding third argument to the factor. This simple example does not, however, show the true expressive power of imperative structure definition. Consider instead a model for dependency parsing (with similarly defined Word and Node variables). In the same Figure, depParsingTemplate defines a template for factors that measure compatibility between a word and its closest verb as measured through parse tree connectivity. Such arbitrary-depth graph search is awkward in firstorder logic, yet it is a simple one-line recursive method in FACTORIE. The statistics method is described below in Section 2.4. Consider also the coreference template measuring the compatibility between a Mention and the canonical representation of its assigned Entity. In response to a moved Mention, unroll1 returns a factor between the Mention and its newly assigned Entity. In response to a changed Entity, unroll2 returns a list of factors between itself all its member Mentions. It is inherent that sometimes different unroll methods will construct multiple copies of the same factor; they are automatically deduplicated by the FACTORIE library. Syntactic sugar for extended first-order logic primitives is also provided, and these can be mixed with imperative constructs; see the bottom of Figure 2 for two small examples. Specifying templates in FACTORIE can certainly be more verbose when not restricted to first-order logic; in this case we trade off some brevity for flexibility. 4 2.3 Imperative Variable Coordination Variables’ value-assignment methods can be overriden to automatically change other variable values in coordination with the assignment—an often-desirable encapsulation of domain knowledge we term imperative variable coordination. For example, in response to a named entity label change, a coreference mention can have its string value automatically adjusted, rather than relying on MCMC inference to stumble upon this self-evident coordination. In Figure 1, Entity does a basic form of coordination by re-calculating its canonical string representation whenever a Mention is added or removed from its set. The ability to use prior knowledge for imperative variable coordination also allows the designer to define the feasible region for the sampling. In the proposal function, users make changes by calling value-assignment functions, and any changes made automatically through coordinating variables are appended to the DiffList. Since a factor template’s contribution to the overall score will not change unless its neighboring variables have changed, once we know every variable that has changed we can efficiently score the proposal. 2.4 Imperative Variable-Statistics Mapping In a somewhat unconventional use of functional mapping, we support a separation between factor neighbors and sufficient statistics. Neighbors are variables touching the factor whose changes imply that the factor needs to be re-scored. Sufficient statistics are the minimal set of variable values that determine the score contribution of the factor. These are usually the same; however, by allowing a function to perform the mapping, we provide an extremely powerful yet simple way to allow model designers to represent their data in natural ways, and concern themselves separately with how to parameterize them. For example, the two neighbors of a skip-edge factor [18] may each have cardinality equal to the number of named entities types, but we may only care to have the skip-edge factor enforce whether or not they match. We term this imperative variable-statistics mapping. Consider corefTemplate in Figure 2, the neighbors of the template are ⟨Mention, Entity⟩pairs. However, the sufficient statistic is simply a Boolean based on the “distance” of the unrolled Mention from the canonical value of the Entity. This allows the template to separate the natural representation of possible worlds from the sufficient statistics needed to score its factors. Note that these sufficient statistics can be calculated as arbitrary functions of the unrolled Mention and the Entity. The models described in Section 3 use a number of factors whose sufficient statistics derive from the domains of its neighbors as well as those with arbitrary feature functions based on their neighbors. An MCMC proposal is scored as follows. First, a sample is generated from the proposal distribution, placing an initial set of variables in the DiffList. Next the value-assignment method is called for each of the variables on the DiffList, and via imperative variable coordination other variables may be added to the DiffList. Given the set of variables that have changed, FACTORIE iterates over each one and calls the unroll function for factor templates matching the variable’s type. This dynamically provides the relevant structure of the graph via imperative structure definition, resulting in a set of factors that should be re-scored. The neighbors of each returned factor are given to the template’s statistics function, and the sufficient statistics are used to generate the factor’s score using the template’s current parameter vector. These scores are summed, producing the final score for the MCMC step. 2.5 Learning Maximum likelihood parameter estimation traditionally involves finding the gradient, however for complex models this can be prohibitively expensive since it requires the inference of marginal distributions over factors. Alternatively some have proposed online methods, such as perceptron, which avoids the need for marginals however still requires full decoding which can also be computationally expensive. We avoid both of these issues by using sample-rank [19]. This is a parameter estimation method that learns a ranking over all possible configurations by observing the difference between scores of proposed MCMC jumps. Parameter changes are made when the model’s ranking of a proposed jump disagrees with a ranking determined by labeled truth. When there is such a disagreement, a perceptron-style update to active parameters is performed by finding all factors whose score has changed (i.e., factors with a neighbor in the DiffList). The active parameters are indexed by the 5 sufficient statistics of these factors. Sample-rank is described in detail in [20]. As with inference, learning is efficient because it uses the DiffList and the imperative constructs described earlier. 3 Joint Segmentation and Coreference Tasks involving multiple information extraction steps are traditionally solved using a pipeline architecture, in which the output predictions of one stage are input to the next stage. This architecture is susceptible to cascading of errors from one stage to the next. To minimize this error, there has been significant interest in joint inference over multiple steps of an information processing pipeline [21, 22, 23]. Full joint inference usually results in exponentially large models for which learning and inference become intractable. One widely studied joint-inference task in information extraction is segmentation and coreference of research paper citation strings [21, 23, 24]. This involves segmenting citation strings into author, title and venue fields (segmentation), and clustering the citations that refer to the same underlying paper entity (coreference). Previous results have shown that joint inference reduces error [21], and this task provides a good testbed for probabilistic programming. We now describe an IDF for the task. For more details, see [24]. 3.1 Variables and Proposal Distribution As in the example given in Section 2, a Mention represents a citation and is a random variable that takes a single Entity as its value. An Entity is a set-valued variable containing Mention variables. This representation eliminates the need for an explicit transitivity constraint, since a Mention can hold only one Entity value, and this value is coordinated with the Entity’s set-value. Variables for segmentation consist of Tokens, Labels and Fields. Each Token represents an observed word in a citation. Each Token has a corresponding Label which is an unobserved variable that can take one of four values: author, title, venue or none. There are three Field variables associated with each Mention, one for each field type (author, venue or title), that store the contiguous block of Tokens representing the Field; Labels and Fields are coordinated. This alternate representation of segmentation provides flexibility in specifying factor templates over predicted Fields. The proposal function for coreference randomly selects a Mention, and with probability 0.8 moves it to a random existing cluster, otherwise to a new singleton cluster. The proposal function for segmentation selects a random Field and grows or shrinks it by a random amount. When jointly performing both tasks, one of the proposal functions is randomly selected. The value-assignment function for the Field ensures that the Labels corresponding to the affected Tokens are correctly set when a Field is changed. This is an example of imperative variable coordination. 3.2 Factor Templates Segmentation Templates: Segmentation templates examine only Field, Label and Token variables, i.e. not using information from coreference predictions. These factor templates are IDF translations of the Markov logic rules described in [21]. There is a template between every Token and its Label. Markov dependencies are captured by a template that examines successive Labels as well as the Token of the earlier Label. The sufficient statistics for these factors are the tuples created from the neighbors of the factor: e.g., the values of two Labels and one Token. We also have a factor template examining every Field with features based on the presence of numbers, dates, and punctuation. This takes advantage of variable-statistics mapping. Coreference Templates: The isolated coreference factor templates use only Mention variables. They consist of two factor templates that share the same sufficient statistics, but have separate weight vectors and different ways of unrolling the graph. An Affinity factor is created for all pairs of Mentions that are coreferent, while a Repulsion factor is created for all pairs that are not coreferent. The features of these templates correspond to the SimilarTitle and SimilarVenue first-order features in [21]. We also add SimilarDate and DissimilarDate features that look at the “date-like” tokens. Joint Templates: To allow the tasks to influence each other, factor templates are added that are unrolled during both segmentation and coreference sampling. Thus these factor templates neighbor Mentions, Fields, and Labels, and use the segmentation predictions for coreference, and viceversa. We add templates for the JntInfCandidates rule from [21]. We create this factor template 6 Table 1: Cora coreference and segmentation results Coreference Segmentation F1 Prec/Recall F1 Cluster Rec. Author Title Venue Total Fellegi-Sunter 78.0/97.7 86.7 62.7 n/a n/a n/a n/a Isolated MLN 94.3/97.0 95.6 78.1 99.3 97.3 98.2 98.2 Joint MLN 94.3/97.0 95.6 75.2 99.5 97.6 98.3 98.4 Isolated IDF 97.09/95.42 96.22 86.01 99.35 97.63 98.58 98.51 Joint IDF 95.34/98.25 96.71 94.62 99.42 97.99 98.78 98.72 such that (m, m′) are unrolled only if they are in the same Entity. The neighbors include Label and Mention. Affinity and Repulsion factor templates are also created between pairs of Fields of the same type; for Affinity the Fields belong to coreferent mention pairs, and for Repulsion they belong to a pair of mentions that are not coreferent. The features of these templates denote similarity between field strings, namely: StringMatch, SubString, Prefix/SuffixMatch, TokenIntersectSize, etc. One notable difference between the JntInfCandidate and joint Affinity/Repulsion templates is the possible number of instantiations. JntInfCandidates can be calculated during preprocessing as there are O(nm2) of these (where n is the maximum mention length, and m is the number of mentions). However, preprocessing joint Affinity/Repulsion templates is intractable as the number of such factors is O(m2n4). We are able to deal with such a large set of possible factor instantiations due to the interplay of structure definition, variable-statistics mapping, and on-the-fly feature calculation. Our model also contains a number of factor templates that cannot be easily captured by first-order logic. For example consider StringMatch and SubString between two fields. For arbitrary length strings these features require the model designer to specify convoluted logic rules. The rules are even less intuitive when considering a feature based on more complex calculations such as StringEditDistance. It is conceivable to preprocess and store all instantiations of these features, but in practice this is intractable. Thus on-the-fly feature calculation within FACTORIE is employed to remain tractable. 4 Experimental Results The joint segmentation and coreference model described above is applied to the Cora dataset[25].4 The dataset contains 1295 total mentions in 134 clusters, with a total of 36487 tokens. Isolated training consists of 5 loops of 100,000 samples each, and 300,000 samples for inference. For the joint task we run training for 5 loops of 250,000 samples each, with 750,000 samples for inference. We average the results of 10 runs of three-fold cross validation, with the same folds as [21]. Segmentation is evaluated on token precision, recall and F1. For coreference, pairwise coreference decisions are evaluated. The fraction of clusters that are correctly predicted (cluster recall) is also calculated. In Table 1, we see both our isolated and joint models outperform the previous state-of-the-art results of [21] on both tasks. We see a 25.23% error reduction in pairwise coreference F1, and a 20.0% error reduction of tokenwise segmentation F1 when comparing to the joint MLN. The improvements of joint over isolated IDF are statistically significant at 1% using the T-test. The experiments run very quickly, which can be attributed to sample-rank and the application of variable coordination and structure definition of the models as described earlier. Each of the isolated tasks finishes initialization, training and evaluation within 3 minutes, while the joint task takes 18 minutes. The running time for the MLNs reported in [21] are between 50-90 minutes for learning and inference. Thus we can see that IDFs provide a significant boost in efficiency by avoiding the need to unroll or score the entire graph. Note also that the timing result from [21] is for a model that did not enforce transitivity constraints on the coreference predictions. Adding transitivity constraints dramatically increases running time [26], whereas the IDF supports transitivity implicitly. 4Available at http://alchemy.cs.washington.edu/papers/poon07 7 5 Related Work Over the years there have been many efforts to build graphical models toolkits. Many of them are useful as teaching aids, such as the Bayes Net Toolbox and Probabilistic Modeling Toolkit (PMTK)[27] (both in Matlab), but do not scale up to substantial real problems. There has been growing interest in building systems that can perform as workhorses, doing real work on large data. For example, Infer.NET (CSoft) [28] is intended to be deployed in a number of Microsoft products, and has been applied to problems in computer vision. Like IDFs it is embedded in a pre-existing programming language, rather than embodying its own new “little language,” and its users have commented positively about this facet. Unlike IDFs it is designed for messaging-passing inference, and must unroll the graphical model before inference, creating factors to represent all possible worlds, which makes it unsuitable for our applications. The very recent language Figaro [29] is also implemented as a library. Like FACTORIE it is implemented in Scala, and provides an object-oriented framework for models; unlike FACTORIE it tightly intertwines data representation and scoring, and it is not designed for changing model structure during inference; it also does not yet support learning. BLOG [5] and some of its derivatives can also scale to substantial data sets, and, like IDFs, are designed for graphical models that cannot be fully unrolled. Unlike IDFs, BLOG, as well as IBAL [6] and Church [7], are designed for generative models, though Church can also represent conditional, undirected models. We are most interested in supporting advanced discriminative models of the type that have been successful for natural language processing, computer vision, bioinformatics, and elsewhere. Note that FACTORIE also supports generative models; for example latent Dirichlet allocation can be coded in about 15 lines. Two systems focussing on discriminatively-trained relational models are relational Markov networks (RMNs) [8], and Markov logic networks (MLNs, with Alchemy as its most popular implementation). To define repeated relational structure and parameter tying, both use declarative languages: RMNs use SQL and MLNs use first-order logic. By contrast, as discussed above, IDFs are in essence an experiment in taking an imperative approach. There has, however, been both historical and recently growing interest in using imperative programming languages for defining learning systems and probabilistic models. For example, work on theory refinement [30] viewed domain theories as “statements in a procedural programming language, rather than the common view of a domain theory being a collection of declarative Prolog statements.” More recently, IBAL [6] and Church [7] are both fundamentally programs that describe the generative storyline for the data. IDFs, of course, share the combination of imperative programming with probabilistic modeling, but IDFs have their semantics defined by undirected factor graphs, and are typically discriminatively trained. 6 Conclusion In this paper we have described imperatively defined factor graphs (IDFs), a framework to support efficient learning and inference in large factor graphs of changing structure. We preserve the traditional, declarative, statistical semantics of factor graphs while allowing imperative definitions of the model structure and operation. This allows model authors to combine both declarative and procedural domain knowledge, while also obtaining significantly more efficient inference and learning than declarative approaches. We have shown state-of-the-art results in citation matching that highlight the advantages afforded by IDFs for both accuracy and speed. Acknowledgments This work was supported in part by NSF medium IIS-0803847; the Central Intelligence Agency, the National Security Agency and National Science Foundation under NSF grant IIS-0326249; SRI International subcontract #27-001338 and ARFL prime contract #FA8750-09-C-0181; Army prime contract number W911NF-07-1-0216 and University of Pennsylvania subaward number 103548106. Any opinions, findings and conclusions or recommendations expressed in this material are the authors’ and do not necessarily reflect those of the sponsor. 8 References [1] John D. Lafferty, Andrew McCallum, and Fernando Pereira. Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Int Conf on Machine Learning (ICML), 2001. [2] Charles Sutton and Andrew McCallum. An introduction to conditional random fields for relational learning. In Introduction to Statistical Relational Learning. 2007. [3] A. Bernal, K. Crammer, A. Hatzigeorgiou, and F. Pereira. Global discriminative learning for higheraccuracy computation gene prediction. In PloS Computational Biology, 2007. [4] A. Quottoni, M. Collins, and T. Darrell. Conditional random fields for object recognition. In NIPS, 2004. [5] Brian Milch. Probabilistic Models with Unknown Objects. PhD thesis, University of California, Berkeley, 2006. [6] Avi Pfeffer. IBAL: A probabilistic rational programming language. In IJCAI, pages 733–740, 2001. [7] Noah D. Goodman, Vikash K. Mansinghka, Daniel Roy, Keith Bonawitz, and Joshua B. Tenenbaum. Church: a language for generative models. In Uncertainty in Artificial Intelligence (UAI), 2008. [8] Ben Taskar, Abbeel Pieter, and Daphne Koller. Discriminative probabilistic models for relational data. In Uncertainty in Artificial Intelligence (UAI), 2002. [9] Matthew Richardson and Pedro Domingos. Markov logic networks. Machine Learning, 62(1-2), 2006. [10] David Poole. Probabilistic horn abduction and bayesian networks. Artificial Intelligence, 64, 1993. [11] Stephen Muggleton and Luc DeRaedt. Inductive logic programming theory and methods. In Journal of Logic Programming, 1994. [12] Taisuke Sato and Yoshitaka Kameya. PRISM: a language for symbolic-statistical modeling. In International Joint Conference on Artificial Intelligence (IJCAI), 1997. [13] Luc De Raedt and Kristian Kersting. Probabilistic logic learning. SIGKDD Explorations: MultiRelational Data Mining, 2003. [14] Martin Odersky. An Overview of the Scala Programming Language (second edition). Technical Report IC/2006/001, EPFL Lausanne, Switzerland, 2006. [15] Aron Culotta and Andrew McCallum. Tractable learning and inference with high-order representations. In ICML WS on Open Problems in Statistical Relational Learning, 2006. [16] Keith A. Bonawitz. Composable Probabilistic Inference with Blaise. PhD thesis, MIT, 2008. [17] Aron Culotta. Learning and inference in weighted logic with application to natural language processing. PhD thesis, University of Massachusetts, 2008. [18] Charles Sutton and Andrew McCallum. Collective segmentation and labeling of distant entities in information extraction. Technical Report TR#04-49, University of Massachusetts, July 2004. [19] Aron Culotta, Michael Wick, and Andrew McCallum. First-order probabilistic models for coreference resolution. In NAACL: Human Language Technologies (NAACL/HLT), 2007. [20] Khashayar Rohanimanesh, Michael Wick, and Andrew McCallum. Inference and learning in large factor graphs with a rank based objective. Technical Report UM-CS-2009-08, University of Massachusetts, Amherst, 2009. [21] Hoifung Poon and Pedro Domingos. Joint inference in information extraction. In AAAI, 2007. [22] Vasin Punyakanok, Dan Roth, and Wen-tau Yih. The necessity of syntactic parsing for semantic role labeling. In International Joint Conf on Artificial Intelligence (IJCAI), pages 1117–1123, 2005. [23] Ben Wellner, Andrew McCallum, Fuchun Peng, and Michael Hay. An integrated, conditional model of information extraction and coreference with application to citation matching. In AUAI, 2004. [24] Sameer Singh, Karl Schultz, and Andrew McCallum. Bi-directional joint inference for entity resolution and segmentation using imperatively-defined factor graphs. In ECML PKDD, pages 414–429, 2009. [25] Andrew McCallum, Kamal Nigam, Jason Rennie, and Kristie Seymore. A machine learning approach to building domain-specific search engines. In Int Joint Conf on Artificial Intelligence (IJCAI), 1999. [26] Hoifung Poon, Pedro Domingos, and Marc Sumner. A general method for reducing the complexity of relational inference and its application to MCMC. In AAAI, 2008. [27] Kevin Murphy and Matt Dunham. PMTK: Probabilistic modeling toolkit. In Neural Information Processing Systems (NIPS) Workshop on Probabilistic Programming, 2008. [28] John Winn and Tom Minka. Infer.NET/CSoft, 2008. http://research.microsoft.com/mlp/ml/Infer/Csoft.htm. [29] Avi Pfeffer. Figaro: An Object-Oriented Probabilistic Programming Language. Technical report, Charles River Analytics, 2009. [30] Richard Maclin and Jude W. Shavlik. Creating advice-taking reinforcement learners. Machine Learning, 22, 1996. 9
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Exponential Family Graph Matching and Ranking James Petterson, Tib´erio S. Caetano, Julian J. McAuley and Jin Yu NICTA, Australian National University Canberra, Australia Abstract We present a method for learning max-weight matching predictors in bipartite graphs. The method consists of performing maximum a posteriori estimation in exponential families with sufficient statistics that encode permutations and data features. Although inference is in general hard, we show that for one very relevant application–document ranking–exact inference is efficient. For general model instances, an appropriate sampler is readily available. Contrary to existing max-margin matching models, our approach is statistically consistent and, in addition, experiments with increasing sample sizes indicate superior improvement over such models. We apply the method to graph matching in computer vision as well as to a standard benchmark dataset for learning document ranking, in which we obtain state-of-the-art results, in particular improving on max-margin variants. The drawback of this method with respect to max-margin alternatives is its runtime for large graphs, which is comparatively high. 1 Introduction The Maximum-Weight Bipartite Matching Problem (henceforth ‘matching problem’) is a fundamental problem in combinatorial optimization [22]. This is the problem of finding the ‘heaviest’ perfect match in a weighted bipartite graph. An exact optimal solution can be found in cubic time by standard methods such as the Hungarian algorithm. This problem is of practical interest because it can nicely model real-world applications. For example, in computer vision the crucial problem of finding a correspondence between sets of image features is often modeled as a matching problem [2, 4]. Ranking algorithms can be based on a matching framework [13], as can clustering algorithms [8]. When modeling a problem as one of matching, one central question is the choice of the weight matrix. The problem is that in real applications we typically observe edge feature vectors, not edge weights. Consider a concrete example in computer vision: it is difficult to tell what the ‘similarity score’ is between two image feature points, but it is straightforward to extract feature vectors (e.g. SIFT) associated with those points. In this setting, it is natural to ask whether we could parameterize the features, and use labeled matches in order to estimate the parameters such that, given graphs with ‘similar’ features, their resulting max-weight matches are also ‘similar’. This idea of ‘parameterizing algorithms’ and then optimizing for agreement with data is called structured estimation [27, 29]. [27] and [4] describe max-margin structured estimation formalisms for this problem. Max-margin structured estimators are appealing in that they try to minimize the loss that one really cares about (‘structured losses’, of which the Hamming loss is an example). However structured losses are typically piecewise constant in the parameters, which eliminates any hope of using smooth optimization directly. Max-margin estimators instead minimize a surrogate loss which is easier to optimize, namely a convex upper bound on the structured loss [29]. In practice the results are often good, but known convex relaxations produce estimators which are statistically inconsistent [18], i.e., the algorithm in general fails to obtain the best attainable model in the limit of infinite training data. The inconsistency of multiclass support vector machines is a well-known issue in the literature that has received careful examination recently [16, 15]. 1 Motivated by the inconsistency issues of max-margin structured estimators as well as by the wellknown benefits of having a full probabilistic model, in this paper we present a maximum a posteriori (MAP) estimator for the matching problem. The observed data are the edge feature vectors and the labeled matches provided for training. We then maximize the conditional posterior probability of matches given the observed data. We build an exponential family model where the sufficient statistics are such that the mode of the distribution (the prediction) is the solution of a max-weight matching problem. The resulting partition function is ]P-complete to compute exactly. However, we show that for learning to rank applications the model instance is tractable. We then compare the performance of our model instance against a large number of state-of-the-art ranking methods, including DORM [13], an approach that only differs from our model instance by using max-margin instead of a MAP formulation. We show very competitive results on standard document ranking datasets, and in particular we show that our model performs better than or on par with DORM. For intractable model instances, we show that the problem can be approximately solved using sampling and we provide experiments from the computer vision domain. However the fastest suitable sampler is still quite slow for large models, in which case max-margin matching estimators like those of [4] and [27] are likely to be preferable even in spite of their potential inferior accuracy. 2 Background 2.1 Structured Prediction In recent years, great attention has been devoted in Machine Learning to so-called structured predictors, which are predictors of the kind g✓: X 7! Y, (1) where X is an arbitrary input space and Y is an arbitrary discrete space, typically exponentially large. Y may be, for example, a space of matrices, trees, graphs, sequences, strings, matches, etc. This structured nature of Y is what structured prediction refers to. In the setting of this paper, X is the set of vector-weighted bipartite graphs (i.e., each edge has a feature vector associated with it), and Y is the set of perfect matches induced by X. If N graphs are available, along with corresponding annotated matches (i.e., a set {(xn, yn)}N n=1), our task will be to estimate ✓such that when we apply the predictor g✓to a new graph it produces a match that is similar to matches of similar graphs from the annotated set. Structured learning or structured estimation refers to the process of estimating a vector ✓for predictor g✓when data {(x1, y1), . . . , (xN, yN)} 2 (X ⇥Y)N are available. Structured prediction for input x means computing y = g(x; ✓) using the estimated ✓. Two generic estimation strategies have been popular in producing structured predictors. One is based on max-margin estimators [29, 27], and the other on maximum-likelihood (ML) or MAP estimators in exponential family models [12]. The first approach is a generalization of support vector machines to the case where the set Y is structured. However the resulting estimators are known to be inconsistent in general: in the limit of infinite training data the algorithm fails to recover the best model in the model class [18, 16, 15]. McAllester recently provided an interesting analysis on this issue, where he proposed new upper bounds whose minimization results in consistent estimators, but no such bounds are convex [18]. The other approach uses ML or MAP estimation in conditional exponential families with ‘structured’ sufficient statistics, such as in probabilistic graphical models, where they are decomposed over the cliques of the graph (in which case they are called Conditional Random Fields, or CRFs [12]). In the case of tractable graphical models, dynamic programming can be used to efficiently perform inference. ML and MAP estimators in exponential families not only amount to solving an unconstrained and convex optimization problem; in addition they are statistically consistent. The main problem with these types of models is that often the partition function is intractable. This has motivated the use of max-margin methods in many scenarios where such intractability arises. 2.2 The Matching Problem Consider a weighted bipartite graph with m nodes in each part, G = (V, E, w), where V is the set of vertices, E is the set of edges and w : E 7! R is a set of real-valued weights associated with the edges. G can be simply represented by a matrix (wij) where the entry wij is the weight of the edge ij. Consider also a bijection y : {1, 2, . . . , m} 7! {1, 2, . . . , m}, i.e., a permutation. Then the matching problem consists of computing 2 to attain the graph G = (V, E, w). See Figure 1 for an illustration. Gx G i j i j xij wij = hxij, ✓i Figure 1. Left: Illustration of an input vector-weighted bipartite graph Gx with 3 ⇥3 edges. There is a vector xe associated to each edge e (for clarity only xij is shown, corresponding to the solid edge). Right: weighted bipartite graph G obtained by evaluating Gx on the learned vector ✓(again only edge ij is shown). More formally, assume that a training set {X, Y } = {(x1, y1), . . . , (xN, yN)} is available, for n = 1, 2, . . . , N (where xn := (xn 11, xn 12 . . . , xn M(n)M(n))). Here M(n) is the number of nodes in each part of the vector-weighted bipartite graph xn. We then parameterize xij as wiy(i) = f(xiy(i); ✓), and the goal is to find the ✓which maximizes the posterior likelihood of the observed data. We will assume f to be bilinear, i.e. f(xiy(i); ✓) = ⌦ xiy(i), ✓ ↵ . 3.2. Exponential Family Model We assume an exponential family model, where the probability model is p(y|x; ✓) = exp (hφ(x, y), ✓i −g(x; ✓)), (3) where g(x; ✓) = log X y exp hφ(x, y), ✓i (4) is the log-partition function, which is a convex and differentiable function of ✓(Wainwright & Jordan, 2003). The prediction in this model is the most likely y, i.e. y⇤= argmax y p(y|x; ✓) = argmax y hφ(x, y), ✓i (5) and ML estimation amounts to maximizing the conditional likelihood of a sample {X, Y }, i.e. computing argmax✓p(Y |X; ✓). In practice we will in general introduce a prior on ✓and perform MAP estimation: ✓⇤= argmax ✓ p(Y |X; ✓)p(✓) = argmax ✓ p(✓|Y, X). (6) Assuming iid sampling, we have p(Y |X; ✓) = QN n=1 p(yn|xn; ✓). Therefore, p(✓|Y, X) / p(✓) N Y n=1 exp (hφ(xn, yn), ✓i − = exp log p(✓) + N X n=1 (hφ(xn, yn), ✓i −g(x We impose a Gaussian prior on ✓. Instead o ing the posterior we can instead minimize th log-posterior `(Y |X; ✓), which becomes our tion (we suppress the constant term): `(Y |X; ✓) =λ 2 k✓k2 + 1 N N X n=1 (g(xn; ✓) −hφ(x where λ is a regularization constant. `(Y convex function of ✓since the log-partitio g(✓) is a convex function of ✓(Wainwright 2003) and the other terms are clearly conve 3.3. Feature Parameterization The critical observation now is that we equ lution of the matching problem (2) to the of the exponential family model (5), i.e. P hφ(x, y), ✓i. Since our goal is to parame tures of individual pairs of nodes (so as t the weight of an edge), the most natural m φ(x, y) = M X i=1 xiy(i), which gives wiy(i) = ⌦ xiy(i), ✓ ↵ , i.e. linear in both x and ✓(see Figure 1, r specific form for xij will be discussed in t mental section. In light of (10), (2) now cle a prediction of the best match for Gx under ✓. 4. Learning the Model 4.1. Basics We need to solve ✓⇤= argmin✓`(Y |X; ✓). ` a convex and di↵erentiable function of ✓(W & Jordan, 2003), therefore gradient descen the global optimum. In order to compute r✓ we need to compute r✓g(✓). It is a stand of exponential families that the gradient o partition function is the expectation of th statistics: r✓g(x; ✓) = Ey⇠p(y|x;✓)[φ(x, y)]. Figure 1: Left: Illustration of an input vector-weighted bipartite graph Gx with 3 ⇥3 edges. There is a vector xe associated with each edge e (for clarity only xij is shown, corresponding to the solid edge). Right: weighted bipartite graph G obtained by evaluating Gx on the learned vector ✓(again only edge ij is shown). y⇤= argmax y m X i=1 wiy(i). (2) This is a well-studied problem; it is tractable and can be solved in O(m3) time [22]. This model can be used to match features in images [4], improve classification algorithms [8] and rank documents [13], to cite a few applications. The typical setting consists of engineering the score matrix wij according to domain knowledge and subsequently solving the combinatorial problem. 3 The Model 3.1 Basic Goal In this paper we assume that the weights wij are instead to be estimated from training data. More precisely, the weight wij associated with the edge ij in a graph will be the result of an appropriate composition of a feature vector xij (observed) and a parameter vector ✓(estimated from training data). Therefore, in practice, our input is a vector-weighted bipartite graph Gx = (V, E, x) (x : E 7! Rn), which is ‘evaluated’ at a particular ✓(obtained from previous training) so as to attain the graph G = (V, E, w). See Figure 1 for an illustration. More formally, assume that a training set {X, Y } = {(xn, yn)}N n=1 is available, where xn := (xn 11, xn 12 . . . , xn M(n)M(n)). Here M(n) is the number of nodes in each part of the vector-weighted bipartite graph xn. We then parameterize xij as wiy(i) = f(xiy(i); ✓), and the goal is to find the ✓which maximizes the posterior probability of the observed data. We will assume f to be bilinear, i.e., f(xiy(i); ✓) = ⌦ xiy(i), ✓ ↵ . 3.2 Exponential Family Model We assume an exponential family model, where the probability model is p(y|x; ✓) = exp (hφ(x, y), ✓i −g(x; ✓)), where (3) g(x; ✓) = log X y exp hφ(x, y), ✓i (4) is the log-partition function, which is a convex and differentiable function of ✓[31]. The prediction in this model is the most likely y, i.e., y⇤= argmax y p(y|x; ✓) = argmax y hφ(x, y), ✓i (5) and ML estimation amounts to maximizing the conditional likelihood of the training set {X, Y }, i.e., computing argmax✓p(Y |X; ✓). In practice we will in general introduce a prior on ✓and perform MAP estimation: ✓⇤= argmax ✓ p(Y |X; ✓)p(✓) = argmax ✓ p(✓|Y, X). (6) Assuming iid sampling, we have p(Y |X; ✓) = QN n=1 p(yn|xn; ✓). Therefore, p(✓|Y, X) / exp log p(✓) + N X n=1 (hφ(xn, yn), ✓i −g(xn; ✓)) ! . (7) 3 We impose a Gaussian prior on ✓. Instead of maximizing the posterior we can instead minimize the negative log-posterior `(Y |X; ✓), which becomes our loss function (we suppress the constant term): `(Y |X; ✓) =λ 2 k✓k2 + 1 N N X n=1 (g(xn; ✓) −hφ(xn, yn), ✓i) (8) where λ is a regularization constant. `(Y |X; ✓) is a convex function of ✓since the log-partition function g(✓) is a convex function of ✓[31] and the other terms are clearly convex in ✓. 3.3 Feature Parameterization The critical observation now is that we equate the solution of the matching problem (2) to the prediction of the exponential family model (5), i.e., P i wiy(i) = hφ(x, y), ✓i. Since our goal is to parameterize features of individual pairs of nodes (so as to produce the weight of an edge), the most natural model is φ(x, y) = M X i=1 xiy(i), which gives (9) wiy(i) = ⌦ xiy(i), ✓ ↵ , (10) i.e., linear in both x and ✓(see Figure 1, right). The specific form for xij will be discussed in the experimental section. In light of (10), (2) now clearly means a prediction of the best match for Gx under the model ✓. 4 Learning the Model 4.1 Basics We need to solve ✓⇤= argmin✓`(Y |X; ✓). `(Y |X; ✓) is a convex and differentiable function of ✓[31], therefore gradient descent will find the global optimum. In order to compute r✓`(Y |X; ✓), we need to compute r✓g(✓). It is a standard result of exponential families that the gradient of the log-partition function is the expectation of the sufficient statistics: r✓g(x; ✓) = Ey⇠p(y|x;✓)[φ(x, y)]. (11) Therefore in order to perform gradient descent we need to compute the above expectation. Opening the above expression gives Ey⇠p(y|x;✓)[φ(x, y)] = X y φ(x, y)p(y|x; ✓) (12) = 1 Z(x; ✓) X y φ(x, y) M Y i=1 exp( ⌦ xiy(i), ✓ ↵ ), (13) which reveals that the partition function Z(x; ✓) needs to be computed. The partition function is: Z(x; ✓) = X y M Y i=1 exp( ⌦ xiy(i), ✓ ↵ ) | {z } =:Biy(i) . (14) Note that the above is the expression for the permanent of matrix B [19]. The permanent is similar in definition to the determinant, the difference being that for the latter sgn(y) comes before the product. However, unlike the determinant, which is computable efficiently and exactly by standard linear algebra manipulations, computing the permanent is a ]P-complete problem [30]. Therefore we have no realistic hope of computing (11) exactly for general problems. 4.2 Exact Expectation The exact partition function itself can be efficiently computed for up to about M = 30 using the O(M2M) algorithm by Ryser [25]. However for arbitrary expectations we are not aware of any exact algorithm which is more efficient than full enumeration (which would constrain tractability to very small graphs). However we will see that even in the case of very small graphs we find a very important application: learning to rank. In our experiments, we successfully apply a tractable instance of our model to benchmark document ranking datasets, obtaining very competitive results. For larger graphs, we have alternative options as indicated below. 4 4.3 Approximate Expectation If we have a situation in which the set of feasible permutations is too large to be fully enumerated efficiently, we need to resort to some approximation for the expectation of the sufficient statistics. The best solution we are aware of is one by Huber and Law, who recently presented an algorithm to approximate the permanent of dense non-negative matrices [10]. The algorithm works by producing exact samples from the distribution of perfect matches on weighted bipartite graphs. This is in precisely the same form as the distribution we have here, p(y|x; ✓) [10]. We can use this algorithm for applications that involve larger graphs.We generate K samples from the distribution p(y|x; ✓), and directly approximate (12) with a Monte Carlo estimate Ey⇠p(y|x;✓)[φ(x, y)] ⇡1 K K X i=1 φ(x, yi). (15) In our experiments, we apply this algorithm to an image matching application. 5 Experiments 5.1 Ranking Here we apply the general matching model introduced in previous sections to the task of learning to rank. Ranking is a fundamental problem with applications in diverse areas such as document retrieval, recommender systems, product rating and others. Early learning to rank methods applied a pairwise approach, where pairs of documents were used as instances in learning [7, 6, 3]. Recently there has been interest in listwise approaches, where document lists are used as instances, as in our method. In this paper we focus, without loss of generality, on document ranking. We are given a set of queries {qk} and, for each query qk, a list of D(k) documents {dk 1, . . . , dk D(k)} with corresponding ratings {rk 1, . . . , rk D(k)} (assigned by a human editor), measuring the relevance degree of each document with respect to query qk. A rating or relevance degree is usually a nominal value in the list {1, . . . , R}, where R is typically between 2 and 5. We are also given, for every retrieved document dk i , a joint feature vector k i for that document and the query qk. Training At training time, we model each query qk as a vector-weighted bipartite graph (Figure 1) where the nodes on one side correspond to a subset of cardinality M of all D(k) documents retrieved by the query, and the nodes on the other side correspond to all possible ranking positions for these documents (1, . . . , M). The subset itself is chosen randomly, provided at least one exemplar document of every rating is present. Therefore M must be such that M ≥R. The process is then repeated in a bootstrap manner: we resample (with replacement) from the set of documents {dk 1, . . . , dk D(k)}, M documents at a time (conditioned on the fact that at least one exemplar of every rating is present, but otherwise randomly). This effectively boosts the number of training examples since each query qk ends up being selected many times, each time with a different subset of M documents from the original set of D(k) documents. In the following we drop the query index k to examine a single query. Here we follow the construction used in [13] to map matching problems to ranking problems (indeed the only difference between our ranking model and that of [13] is that they use a max-margin estimator and we use MAP in an exponential family.) Our edge feature vector xij will be the product of the feature vector i associated with document i, and a scalar cj (the choice of which will be explained below) associated with ranking position j xij = icj. (16) i is dataset specific (see details below). From (10) and (16), we have wij = cj h i, ✓i, and training proceeds as explained in Section 4. Testing At test time, we are given a query q and its corresponding list of D associated documents. We then have to solve the prediction problem, i.e., y⇤= argmax y D X i=1 ⌦ xiy(i), ✓ ↵ = argmax y D X i=1 cy(i) h i, ✓i . (17) 5 1 2 3 4 5 6 7 8 9 10 0.35 0.4 0.45 0.5 0.55 0.6 0.65 k NDCG TD2004 RankMatch (Our Method), M=2 DORM RankBoost RankSVM FRank ListNet AdaRank−MAP AdaRank−NDCG QBRank IsoRank SortNet 20 hiddens MAP SortNet 20 hiddens P@10 StructRank C−CRF 1 2 3 4 5 6 7 8 9 10 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 k NDCG TD2003 RankMatch (Our Method), M=2 (all) DORM RankBoost RankSVM FRank ListNet AdaRank−MAP AdaRank−NDCG QBRank IsoRank SortNet 10 hiddens MAP SortNet 10 hiddens P@10 1 2 3 4 5 6 7 8 9 10 0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 0.58 k NDCG OHSUMED RankMatch (Our Method), M=3 DORM RankBoost RankSVM FRank ListNet AdaRank−MAP AdaRank−NDCG QBRank IsoRank StructRank C−CRF Figure 2: Results of NDCG@k for state-of-the-art methods on TD2004 (left), TD2003 (middle) and OHSUMED (right). This is best viewed in color. We now notice that if the scalar cj = c(j), where c is a non-increasing function of rank position j, then (17) can be solved simply by sorting the values of h i, ✓i in decreasing order.1 In other words, the matching problem becomes one of ranking the values h i, ✓i. Inference in our model is therefore very fast (linear time).2 In this setting it makes sense to interpret the quantity h i, ✓i as a score of document di for query q. This leaves open the question of which non-increasing function c should be used. We do not solve this problem in this paper, and instead choose a fixed c. In theory it is possible to optimize over c during learning, but in that case the optimization problem would no longer be convex. We describe the results of our method on LETOR 2.0 [14], a publicly available benchmark data collection for comparing learning to rank algorithms. It is comprised of three data sets: OHSUMED, TD2003 and TD2004. Data sets OHSUMED contains features extracted from query-document pairs in the OHSUMED collection, a subset of MEDLINE, a database of medical publications. It contains 106 queries. For each query there are a number of associated documents, with relevance degrees judged by humans on three levels: definitely, possibly or not relevant. Each query-document pair is associated with a 25 dimensional feature vector, i. The total number of query-document pairs is 16,140. TD2003 and TD2004 contain features extracted from the topic distillation tasks of TREC 2003 and TREC 2004, with 50 and 75 queries, respectively. Again, for each query there are a number of associated documents, with relevance degrees judged by humans, but in this case only two levels are provided: relevant or not relevant. Each query-document pair is associated with a 44 dimensional feature vector, i. The total number of query-document pairs is 49,171 for TD2003 and 74,170 for TD2004. All datasets are already partitioned for 5-fold cross-validation. See [14] for more details. Evaluation Metrics In order to measure the effectiveness of our method we use the normalized discount cumulative gain (NDCG) measure [11] at rank position k, which is defined as NDCG@k = 1 Z k X j=1 2r(j) −1 log(1 + j), (18) where r(j) is the relevance of the jth document in the list, and Z is a normalization constant so that a perfect ranking yields an NDCG score of 1. 1If r(v) denotes the vector of ranks of entries of vector v, then ha, ⇡(b)i is maximized by the permutation ⇡⇤such that r(a) = r(⇡⇤(b)), a theorem due to Polya, Littlewood, Hardy and Blackwell [26]. 2Sorting the top k items of a list of D items takes O(k log k + D) time [17]. 6 Table 1: Training times (per observation, in seconds, Intel Core2 2.4GHz) for the exponential model and max-margin. Runtimes for M = 3, 4, 5 are from the ranking experiments, computed by full enumeration; M = 20 corresponds to the image matching experiments, which use the sampler from [10]. A problem of size 20 cannot be practically solved by full enumeration. M exponential model max margin 3 0.0006661 0.0008965 4 0.0011277 0.0016086 5 0.0030187 0.0015328 20 36.0300000 0.9334556 External Parameters The regularization constant λ is chosen by 5-fold cross-validation, with the partition provided by the LETOR package. All experiments are repeated 5 times to account for the randomness of the sampling of the training data. We use c(j) = M −j on all experiments. Optimization To optimize (8) we use a standard BFGS Quasi-Newton method with a backtracking line search, as described in [21]. Results For the first experiment training was done on subsets sampled as described above, where for each query qk we sampled 0.4 · D(k) · M subsets, therefore increasing the number of samples linearly with M. For TD2003 we also trained with all possible subsets (M = 2(all) in the plots). In Figure 2 we plot the results of our method (named RankMatch), for M = R, compared to those achieved by a number of state-of-the-art methods which have published NDCG scores in at least two of the datasets: RankBoost [6], RankSVM [7], FRank [28], ListNet [5], AdaRank [32], QBRank [34], IsoRank [33], SortNet [24], StructRank [9] and C-CRF [23]. We also included a plot of our implementation of DORM [13], using precisely the same resampling methodology and data for a fair comparison. RankMatch performs among the best methods on both TD2004 and OHSUMED, while on TD2003 it performs poorly (for low k) or fairly well (for high k). We notice that there are four methods which only report results in two of the three datasets: the two SortNet versions are only reported on TD2003 and TD2004, while StructRank and C-CRF are only reported on TD2004 and OHSUMED. RankMatch compares similarly with SortNet and StructRank on TD2004, similarly to C-CRF and StructRank on OHSUMED and similarly to the two versions of SortNet on TD2003. This exhausts all the comparisons against the methods which have results reported in only two datasets. A fairer comparison could be made if these methods had their performance published for the respective missing dataset. When compared to the methods which report results in all datasets, RankMatch entirely dominates their performance on TD2004 and is second only to IsoRank on OHSUMED. These results should be interpreted cautiously; [20] presents an interesting discussion about issues with these datasets. Also, benchmarking of ranking algorithms is still in its infancy and we don’t yet have publicly available code for all of the competitive methods. We expect this situation to change in the near future so that we are able to compare them on a fair and transparent basis. Consistency In a second experiment we trained RankMatch with different training subset sizes, starting with 0.03·D(k)·M and going up to 1.0·D(k)·M. Once again, we repeated the experiments with DORM using precisely the same training subsets. The purpose here is to see whether we observe a practical advantage of our method with increasing sample size, since statistical consistency only provides an asymptotic indication. The results are plotted in Figure 3-right, where we can see that, as more training data is available, RankMatch improves more saliently than DORM. Runtime The runtime of our algorithm is competitive with that of max-margin for small graphs, such as those that arise from the ranking application. For larger graphs, the use of the sampling algorithm will result in much slower runtimes than those typically obtained in the max-margin framework. This is certainly the benefit of the max-margin matching formulations of [4, 13]: it is much faster for large graphs. Table 1 shows the runtimes for graphs of different sizes, for both estimators. 5.2 Image Matching For our computer vision application we used a silhouette image from the Mythological Creatures 2D database3. We randomly selected 20 points on the silhouette as our interest points and applied 3http://tosca.cs.technion.ac.il 7 shear to the image creating 200 different images. We then randomly selected N pairs of images for training, N for validation and 500 for testing, and trained our model to match the interest points in the pairs – that is, given two images with corresponding points, we computed descriptors for each pair i, j of points (one from each image) and learned ✓such that the solution to the matching problem (2) with the weights set to wij = hxij, ✓i best matches the expected solution that a human would manually provide. In this setup, xij = | i − j|2, where | · | denotes the elementwise difference (19) and i is the Shape Context feature vector [1] for point i. For a graph of this size computing the exact expectation is not feasible, so we used the sampling method described in Section 4.3. Once again, the regularization constant λ was chosen by crossvalidation. Given the fact that the MAP estimator is consistent while the max-margin estimator is not, one is tempted to investigate the practical performance of both estimators as the sample size grows. However, since consistency is only an asymptotic property, and also since the Hamming loss is not the criterion optimized by either estimator, this does not imply a better large-sample performance of MAP in real experiments. In any case, we present results with varying training set sizes in Figure 3-left. The max-margin method is that of [4]. After a sufficiently large training set size, our model seems to enjoy a slight advantage. 0 50 100 150 200 250 300 350 400 450 500 0.08 0.1 0.12 0.14 0.16 0.18 0.2 number of training pairs error exponential model max margin 10 −1 10 0 0.53 0.535 0.54 0.545 0.55 0.555 0.56 0.565 0.57 sample size (x M D) NDCG−1 OHSUMED RankMatch DORM rning image matching. Left: hamming loss for different numbers of training pairs (test to 500 pairs). Right: an example match from the test set (blue are correct and red ches). mmender systems, product rating and others. We are going to focus on web page em we are given a set of queries {qk} and, for each query qk, a list of D(k) documents )} with corresponding ratings {rk 1, . . . , rk D(k)} (assigned by a human editor), measurnce degree of each document with respect to query qk. A rating or relevance degree is inal value in the list {1, . . . , R}, where R is typically between 2 and 5. We are also ry retrieved document dk i , a joint feature vector k i for that document and the query raining time, we model each query qk as a vector-weighted bipartite graph (Figure Figure 3: Performance with increasing sample size. Left: Hamming loss for different numbers of training pairs in the image matching problem (test set size fixed to 500 pairs). Right: results of NDCG@1 on the ranking dataset OHSUMED. This evidence is in agreement with the fact that our estimator is consistent, while max-margin is not. 6 Conclusion and Discussion We presented a method for learning max-weight bipartite matching predictors, and applied it extensively to well-known document ranking datasets, obtaining state-of-the-art results. We also illustrated–with an image matching application–that larger problems can also be solved, albeit slowly, with a recently developed sampler. The method has a number of convenient features. First, it consists of performing maximum-a-posteriori estimation in an exponential family model, which results in a simple unconstrained convex optimization problem solvable by standard algorithms such as BFGS. Second, the estimator is not only statistically consistent but also in practice it seems to benefit more from increasing sample sizes than its max-margin alternative. Finally, being fully probabilistic, the model can be easily integrated as a module in a Bayesian framework, for example. The main direction for future research consists of finding more efficient ways to solve large problems. This will most likely arise from appropriate exploitation of data sparsity in the permutation group. References [1] Belongie, S., & Malik, J (2000). Matching with shape contexts. CBAIVL00. [2] Belongie, S., Malik, J., & Puzicha, J. (2002). Shape matching and object recognition using shape contexts. IEEE Trans. on PAMI, 24, 509–521. [3] Burges, C. J. C., Shaked, T., Renshaw, E., Lazier, A., Deeds, M., Hamilton, N. & Hulldender, G. (2005). Learning to rank using gradient descent. ICML. 8 [4] Caetano, T. S., Cheng, L., Le, Q. V., & Smola, A. J. (2009). Learning graph matching. IEEE Trans. on PAMI, 31, 1048–1058. [5] Cao, Z., Qin, T., Liu, T.-Y., Tsai, M.-F., & Li, H. (2007). Learning to rank: from pairwise approach to listwise approach. ICML [6] Freund, Y., Iyer, R., Schapire, R. E., & Singer, Y. (2003). An efficient boosting algorithm for combining preferences. J. Mach. Learn. Res., 4, 933–969. [7] Herbrich, A., Graepel, T., & Obermayer, K. (2000). Large margin rank boundaries for ordinal regression. In Advances in Large Margin Classifiers. [8] Huang, B., & Jebara, T. (2007). Loopy belief propagation for bipartite maximum weight b-matching. AISTATS. [9] Huang, J. C., & Frey, B. J. (2008). Structured ranking learning using cumulative distribution networks. In NIPS. [10] Huber, M., & Law, J. (2008). Fast approximation of the permanent for very dense problems. SODA. [11] Jarvelin, K., & Kekalainen, J. (2002). Cumulated gain-based evaluation of ir techniques. ACM Transactions on Information Systems, 20, 2002. [12] Lafferty, J. D., McCallum, A., & Pereira, F. (2001). Conditional random fields: Probabilistic modeling for segmenting and labeling sequence data. ICML. [13] Le, Q., & Smola, A. (2007). Direct optimization of ranking measures. http://arxiv.org/abs/0704.3359. [14] Liu, T.-Y., Xu, J., Qin, T., Xiong, W., & Li, H. (2007). Letor: Benchmark dataset for research on learning to rank for information retrieval. LR4IR. [15] Liu, Y. & Shen, X. (2005) Multicategory -learning and support vector machine: Computational tools. J. Computational and Graphical Statistics, 14, 219–236. [16] Liu, Y. & Shen, X. (2006) Multicategory -learning. JASA, 101, 500–509. [17] Martinez, C. (2004). Partial quicksort. SIAM. [18] McAllester, D. (2007). Generalization bounds and consistency for structured labeling. Predicting Structured Data. [19] Minc, H. (1978). Permanents. Addison-Wesley. [20] Minka, T., & Robertson, S. (2008). Selection bias in the letor datasets. LR4IR. [21] Nocedal, J., & Wright, S. J. (1999). Numerical optimization. Springer Series in Operations Research. Springer. [22] Papadimitriou, C. H., & Steiglitz, K. (1982). Combinatorial optimization: Algorithms and complexity. New Jersey: Prentice-Hall. [23] Qin, T., Liu, T.-Y., Zhang, X.-D., Wang, D.-S., & Li, H. (2009). Global ranking using continuous conditional random fields. NIPS. [24] Rigutini, L., Papini, T., Maggini, M., & Scarselli, F. (2008). Sortnet: Learning to rank by a neural-based sorting algorithm. LR4IR. [25] Ryser, H. J. (1963). Combinatorial mathematics. The Carus Mathematical Monographs, No. 14, Mathematical Association of America. [26] Sherman, S. (1951). On a Theorem of Hardy, Littlewood, Polya, and Blackwell. Proceedings of the National Academy of Sciences, 37, 826–831. [27] Taskar, B. (2004). Learning structured prediction models: a large-margin approach. Doctoral dissertation, Stanford University. [28] Tsai, M., Liu, T., Qin, T., Chen, H., & Ma, W. (2007). Frank: A ranking method with fidelity loss. SIGIR. [29] Tsochantaridis, I., Joachims, T., Hofmann, T., & Altun, Y. (2005). Large margin methods for structured and interdependent output variables. JMLR, 6, 1453–1484. [30] Valiant, L. G. (1979). The complexity of computing the permanent. Theor. Comput. Sci. (pp. 189–201). [31] Wainwright, M. J., & Jordan, M. I. (2003). Graphical models, exponential families, and variational inference (Technical Report 649). UC Berkeley, Department of Statistics. [32] Xu, J., & Li, H. (2007). Adarank: a boosting algorithm for information retrieval. SIGIR. [33] Zheng, Z., Zha, H., & Sun, G. (2008a). Query-level learning to rank using isotonic regression. LR4IR. [34] Zheng, Z., Zha, H., Zhang, T., Chapelle, O., Chen, K., & Sun, G. (2008b). A general boosting method and its application to learning ranking functions for web search. NIPS. 9
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Extending Phase Mechanism to Differential Motion Opponency for Motion Pop-Out Yicong Meng and Bertram E. Shi Department of Electronic and Computer Engineering Hong Kong University of Science and Technology Clear Water Bay, Kowloon, Hong Kong {eeyicong, eebert}@ust.hk Abstract We extend the concept of phase tuning, a ubiquitous mechanism among sensory neurons including motion and disparity selective neurons, to the motion contrast detection. We demonstrate that the motion contrast can be detected by phase shifts between motion neuronal responses in different spatial regions. By constructing the differential motion opponency in response to motions in two different spatial regions, varying motion contrasts can be detected, where similar motion is detected by zero phase shifts and differences in motion by non-zero phase shifts. The model can exhibit either enhancement or suppression of responses by either different or similar motion in the surrounding. A primary advantage of the model is that the responses are selective to relative motion instead of absolute motion, which could model neurons found in neurophysiological experiments responsible for motion pop-out detection. 1 Introduction Motion discontinuity or motion contrast is an important cue for the pop-out of salient moving objects from contextual backgrounds. Although the neural mechanism underlying the motion pop-out detection is still unknown, the center-surround receptive field (RF) organization is considered as a physiological basis responsible for the pop-out detection. The center-surround RF structure is simple and ubiquitous in cortical cells especially in neurons processing motion and color information. Nakayama and Loomis [1] have predicted the existence of motion selective neurons with antagonistic center-surround receptive field organization in 1974. Recent physiological experiments [2][3] show that neurons with center-surround RFs have been found in both middle temporal (MT) and medial superior temporal (MST) areas related to motion processing. This antagonistic mechanism has been suggested to detect motion segmentation [4], figure/ground segregation [5] and the differentiation of object motion from ego-motion [6]. There are many related works [7]-[12] on motion pop-out detection. Some works [7]-[9] are based on spatio-temporal filtering outputs, but motion neurons are not fully interacted by either only inhibiting similar motion [7] or only enhancing opposite motion [8]. Heeger, et al. [7] proposed a center-surround operator to eliminate the response dependence upon rotational motions. But the Heeger's model only shows a complete center-surround interaction for moving directions. With respect to the surrounding speed effects, the neuronal responses are suppressed by the same speed with the center motion but not enhanced by other speeds. Similar problem existed in [8], which only modeled the suppression of neuronal responses in the classical receptive field (CRF) by similar motions in surrounding regions. Physiological experiments [10][11] show that many neurons in visual cortex are sensitive to the motion contrast rather than depend upon the absolute direction and speed of the object motion. Although pooling over motion neurons tuned to different velocities can eliminate the dependence upon absolute velocities, it is computationally inefficient and still can't give full interactions of both suppression and enhancement by similar and opposite surrounding motions. The model proposed by Dellen, et al. [12] computed differential motion responses directly from complex cells in V1 and didn't utilize responses from direction selective neurons. In this paper, we propose an opponency model which directly responds to differential motions by utilizing the phase shift mechanism. Phase tuning is a ubiquitous mechanism in sensory information processing, including motion, disparity and depth detection. Disparity selective neurons in the visual cortex have been found to detect disparities by adjusting the phase shift between the receptive field organizations in the left and right eyes [13][14]. Motion sensitive cells have been modeled in the similar way as the disparity energy neurons and detect image motions by utilizing the phase shift between the real and imaginary parts of temporal complex valued responses, which are comparable to images to the left and right eyes [15]. Therefore, the differential motion can be modeled by exploring the similarity between images from different spatial regions and from different eyes. The remainder of this paper is organized as following. Section 2 illustrates the phase shift motion energy neurons which estimate image velocities by the phase tuning in the imaginary path of the temporal receptive field responses. In section 3, we extend the concept of phase tuning to the construction of differential motion opponency. The phase difference determines the preferred velocity difference between adjacent areas in retinal images. Section 4 investigates properties of motion pop-out detection by the proposed motion opponency model. Finally, in section 5, we relate our proposed model to the neural mechanism of motion integration and motion segmentation in motion related areas and suggest a possible interpretation for adaptive center-surround interactions observed in biological experiments. 2 Phase Shift Motion Energy Neurons Adelson and Bergen [16] proposed the motion energy model for visual motion perception by measuring spatio-temporal orientations of image sequences in space and time. The motion energy model posits that the responses of direction-selective V1 complex cells can be computed by a combination of two linear spatio-temporal filtering stages, followed by squaring and summation. The motion energy model was extended in [15] to be phase tuned by splitting the complex valued temporal responses into real and imaginary paths and adding a phase shift on the imaginary path. Figure 1(a) demonstrates the schematic diagram of the phase shift motion energy model. Here we assume an input image sequence in two-dimensional space (x, y) and time t. The separable spatio-temporal receptive field ensures the cascade implementation of RF with spatial and temporal filters. Due to the requirement of the causal temporal RF, the phase shift motion energy model didn’t adopt the Gabor filter like the spatial RF. The phase shift spatio-temporal RF is modeled with a complex valued function ( ) ( ) ( ) , , , , f x y t g x y h t = ⋅ Φ , where the spatial and temporal RFs are denoted by ( ) , g x y and ( ) , h t Φ respectively, ( ) ( ) ( ) ( ) ( ) ( ) ( ) real imag , , | 0, exp , exp x y g x y x y j x j y h t h t j h t = Ω + Ω Φ = + Φ C N (1) and C is the covariance matrix of the spatial Gaussian envelope and Φ is the phase tuning of the motion energy neuron. The real and imaginary profiles of the temporal receptive field are Gamma modulated sinusoidal functions with quadrature phases, ( ) ( ) ( ) ( ) ( ) ( ) real imag | , cos | , sin t t h t t t h t t t α τ α τ = Ω = Ω G G (2) The envelopes for complex exponentials are functions of Gaussian and Gamma distributions, ( ) 2 2 2 2 1 , | 0, exp 2 2 2 x y x y x y x y πσ σ σ σ ⎛ ⎞ = − − ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ C N (3) 1 1 ( | , ) exp ( ) ( ) t t t u t α α α τ τ α τ − ⎛ ⎞ = − ⎜ ⎟ Γ ⎝ ⎠ G (4) where ( ) α Γ is the gamma function and ( ) u t is the unit step function. The parameters α and τ determine the temporal RF size. As derived in [15], the motion energy at location (x, y) can be computed by ( ) ( ) E , , cos v x y S P Φ = + Ψ −Φ (5) where ( ) 2 2 real imag * real imag * real imag 2 arg S V V P V V V V = + = Ψ = (6) and complex valued responses in real and imaginary paths are obtained as, ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) real real , , imag imag , , , , , , , , , , , , V x y t g h I x y t d d d V x y t g h I x y t d d d ξ ζ η ξ ζ η ξ ζ η ξ ζ η ξ ζ η ξ ζ η ξ ζ η ξ ζ η = − − − = − − − ∫∫∫ ∫∫∫ (7) The superscript * represents the complex conjugation and the phase shift parameter Φ controls the spatio-temporal orientation tuning. To avoid clutter, the spatial location variables x and y for S, P, Ψ, Vreal and Vimag are not explicitly shown in Eq. (5) and (6). Figure 1(b) demonstrates the even and odd profiles of the spatio-temporal RF tuned to a particular phase shift. je Φ ( ) V Φ real V imag V real( ) h t ( , ) g x y ( , ) g x y imag( ) h t (a) (b) (·)2 (·)2 (·)2 (·)2 (·)2 (·)2 ( ) max v E Φ ( ) min v E Φ ( ) 0 v E max je Φ min je Φ 0 je c K je Θ s K max je Φ min je Φ 0 je (·)2 (·)2 (·)2 (·)2 (·)2 (·)2 M M M M M M M M M M M M M M M M ( ) v EΔ Θ , , x y Φ ∫∫∫ ( ) , s w x y ( ) , s w x y ( ) , s w x y ( ) , c w x y ( ) , c w x y ( ) , c w x y , , x y Φ ∫∫∫ (c) Figure 1. (a) shows the diagram of the phase shift motion energy model adapted from [15]. (b) draws the spatiotemporal representation of the phase shift motion energy neuron with the real and imaginary receptive field demonstrated by the two left pictures. (c) illustrates the construction of differential motion opponency with a phase difference Θ from two populations of phase shift motion energy neurons in two spatial areas c and s. To avoid clutter, the space location (x, y) is not explicitly shown in phase tuned motion energies. 3 Extending Phase Mechanism to Differential Motion Opponency Based on the above phase shift motion energy model, the local image velocity at each spatial location can be represented by a phase shift which leads to the peak response across a population of motion energy neurons. Across regions of different motions, there are clear discontinuities on the estimated velocity map. The motion discontinuities can be detected by edge detectors on the velocity map to segment different motions. However, this algorithm for motion discontinuities detection can’t discriminate between the object motion and uniform motions in contextual backgrounds. Here we propose a phase mechanism to detect differential motions inspired by the disparity energy model and adopt the center-surround inhibition mechanism to pop out the object motion from contextual background motions. The motion differences between different spatial locations can be modeled in the similar way as the disparity model. The motion energies from two neighboring locations are considered as the retinal images to the left and right eyes. Thus, we can construct a differential motion opponency by placing two populations of phase shift motion energy neurons at different spatial locations and the energy ( ) v EΔ Θ of the opponency is the squared modulus of the averaged phase shift motion energies over space and phase, ( ) ( ) ( ) 2 E , , , , | v v E x y w x y dxdyd Δ Θ = Φ ⋅ Φ Θ Φ ∫∫∫ (8) where ( ) , , w x y Θ is the profile for differential motion opponency and v Δ is the velocity difference between the two spatial regions defined by the kernel ( ) , , w x y Θ . Since ( ) , , w x y Θ is intended to implement the functional role of spatial interactions, it is desired to be a separable function in space and phase domain and can be modeled by phase tuned summation of two spatial kernels, ( ) ( ) ( ) , , | , , j j j c s w x y w x y e e w x y Φ Θ+ Φ Φ Θ = + (9) where ( ) , c w x y and ( ) , s w x y are Gaussian kernels of different spatial sizes c σ and s σ , and Θ is the phase difference representing velocity difference between two spatial regions c and s. Substituting Eq. (9) into Eq. (8), the differential motion energy can be reformulated as ( ) 2 j v c s E K e K Θ Δ Θ = + (10) Θ ൏0 Θ 0 (a) (b) Figure 2. Two types of differential motion opponency constructions of (a) center-surrounding interaction and (b) left-right interaction. Among cells in area MT with surrounding modulations, 25% of cells are with the antagonistic RF structure in the top row and another 50% of cells have the integrative RF structure as shown in the bottom row. where ( ) ( ) ( ) ( ) ( ) ( ) , , , , , , , , exp , , , exp , c v c c x y s v s s x y K E x y j w x y dxdyd K E x y j w x y dxdyd Φ Φ = Φ Φ Φ = Φ Φ Φ ∫∫∫ ∫∫∫ (11) ( ) , , , v c E x y Φ and ( ) , , , v s E x y Φ are phase shift motion energies at location (x, y) and with phase shift Φ. Utilizing the results in Eq. (5) and (6), Eq. (10) and (11) generate similar results, ( ) ( ) opp opp opp E cos v S P Δ Θ = + Θ −Θ (12) where ( ) 2 2 opp * opp * opp 2 arg c s c s c s S K K P K K K K = + = Θ = (13) According to above derivations, by varying the phase shift Θ between –π and π, the relative motion energy of the differential motion opponency can be modeled as population responses across a population of phase tuned motion opponencies. The response is completely specified by three parameters opp S , opp P and opp Θ . The schematic diagram of this opponency is illustrated in Figure 1(c). The differential motion opponency is constituted by three stages. At the first stage, a population of phase shift motion energy neurons is applied to be selective to different velocities. At the second stage, motion energies from the first stage are weighted by kernels tuned to different spatial locations and phase shifts respectively for both spatial regions and two single differential motion signals in region c and region s are achieved by integrating responses from these two regions over space and phase tuning. Finally, the differential motion energy is computed by the squared modulus of the summation of the integrated motion signal in region c and phase shifted motion signal in region s. The subscripts c and s represent two interacted spatial regions which are not limited to the center and surround regions. The opponency could also be constructed by the neighboring left and right Left Velocity Right Velocity -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 Left Velocity Right Velocity -3 -2 -1 0 1 2 3 -3 -2 -1 0 1 2 3 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 (a) (b) Figure 3. (a) Phase map and (b) peak magnitude map are obtained from stimuli of two patches of random dots moving with different velocities. The two patches of stimuli are statistically independent but share the same spatial properties: dot size of 2 pixels, dot density of 10% and dot coherence level of 100%. The phase tuned population of motion energy neurons are applied to each patch of random dots with RF parameters: Ωt = 2π/8, Ωt = 2π/16, σx = 5 and τ = 5.5. For each combination of velocities from left and right patches, averaged phase shifts over space and time are computed and so do the magnitudes of peak responses. The unit for velocities is pixels per frame. spatial regions. Figure 2 shows two types of structures for the differential motion opponency. In [17], the authors demonstrates that among cells in area MT with surrounding modulations, 25% of cells are with the antagonistic RF structure as shown in Figure 2(a) and another 50% of cells have the integrative RF structure as shown in Figure 2(b). The velocity difference tuning of the opponency is determined by the phase shift parameter Θ combined with parameters of spatial and temporal frequencies for motion energy neurons. The larger phase shift magnitude prefers the bigger velocity difference. This phase tuning of velocity difference is consistent with the phase tuning of motion energy neurons. Figure 3 shows the phase map obtained by using random dots stimuli with different velocities on two spatial patches (left and right patches with sizes of 128 pixels ൈ 128 pixels). Along the diagonal line, velocities from left and right patches are equal to each other and therefore phase estimates are zeros along this line. Deviated from the diagonal line to upper-left and lower-right, the phase magnitudes increase while positive phases indicate larger left velocities and negative phases indicate larger right velocities. The phase tuning can give a good classification of velocity differences. 4 Validation of Differential Motion Opponency Out derivation and analysis above show that the phase shift between two neighboring spatial regions is a good indicator for motion difference between these two regions. In this section, we validate the proposed differential motion opponency by two sets of experiments, which show effects of both surrounding directions and speeds on the center motion. Inhibitive interaction, Θ =െπ/2 Excitatory interaction, Θ =0 0 pi/2 pi 3pi/2 2pi 0 0.4 0.8 1.2 1.6 2 Surrouding Direction Responses Inhibitory 0 pi/2 pi 3pi/2 2pi 0 0.4 0.8 1.2 1.6 2 Surrouding Direction Responses Excitatory (a) (b) Model by Petkov et al. [8] Model by Heeger et al. [7] 0 pi/2 pi 3pi/2 2pi 0 0.4 0.8 1.2 1.6 2 Surrouding Direction Responses Inhibitory 0 pi/2 pi 3pi/2 2pi 0 0.4 0.8 1.2 1.6 2 Surrouding Direction Responses Inhibitory (c) (d) Figure 4. Demonstrations of center-surround differential motion opponency, where (a) show the excitation of opposite directions outside the CRF and (b) show the inhibition by surrounding motions in same directions. The center-surround inhibition models by Petkov, et al. [8] and Heeger, et al. [7] are shown in (c) and (d). Responses above 1 indicate enhancement and responses below 1 indicate suppressions. Physiological experiments [2][3] have demonstrated that the neuronal activities in the classical receptive field are suppressed by responses outside the CRF to stimuli with similar motions including both directions and speeds on the center and surrounding regions. On the contrary, visual stimuli of opposite directions or quite different speeds outside the CRF enhance the responses in the CRF. In their experiments, they used a set of stimuli of random dots moving at different velocities, where there are small patches of moving random dots on the center. We tested the properties of the proposed opponency model for motion difference measurement by using similar random dots stimuli. The random dots on background move with different speeds and in different direction but have the same statistical parameters: dot size of 2 pixels, dot density of 10% and motion coherence level of 100%. The small random dots patches are placed on the center of background stimuli to stimulate the neurons in the CRF. These small patches share the same statistical parameters with background random dots but move with a constant velocity of 1 pixel per frame. Figure 4 shows results for the enhanced and suppressed responses in the CRF with varying surrounding directions. The phase shift motion energy neurons had the same spatial and temporal frequencies and the same receptive field sizes, and were selective to vertical orientations. The preferred spatial frequency was 2π/16 radian per pixel and the temporal frequency was 2π/16 radian per frame. The sizes of RF in horizontal and vertical directions were respectively 5 pixels and 10 pixels, corresponding to a spatial bandwidth of 1.96 octaves. The time constant τ was 5.5 frames which resulted in a temporal bandwidth of 1.96 octaves. As shown in Figure 4 (a) and (b), the surrounding motion of opposite direction gives the largest response to the motion in the CRF for the inhibitory interaction and the smallest response for the excitatory interaction. Results demonstrated in Figure 4 are consistent with physiological results reported in [3]. In Born’s paper, inhibitory cells show response enhancement and excitatory cells show response suppression when surrounding motions are in opposite directions. The 3-dB bandwidth for the surrounding moving direction is about 135 degrees for the physiological experiments while the bandwidth is about 180 degrees for the simulation results in our proposed model. Models proposed by Petkov, et al. [8] and Heeger, et al. [7] also show clear inhibition between opposite motions. The Petkov’s model achieves the surrounding suppression for each point in ( ) , , x y t space by the subtraction between responses from that point and its surroundings and followed by a half-wave rectification, ( ) ( ) ( ) , , , , , , , , , v v v E x y t E x y t S x y t θ θ θ α + = − ⋅ % (14) -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2 Center Speed Responses Inhibitory -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2 Center Speed Responses Inhibitory (a) (b) Figure 5. The insensitivity of the proposed opponency model to absolute center and surrounding velocities is demonstrated in (a), where responses are enhanced for all center velocities from -2 to 2 pixels per frame. In (b), the model by Heeger, et al. [7] only shows enhancement when the center speed matches the preferred speed of 1.2 pixel per frame. Similarly, responses above 1 indicate enhancement and below 1 indicate suppressions. In both curves, the velocity differences between center and surrounding regions are maintained as a constant of 3 pixels per frame. where ( ) , , , v E x y t θ is the motion energy at location (x,y) and time t for a given preferred speed v and orientation θ, ( ) , , , vS x y t θ is the average motion energy in the surrounding of point (x, y, t), ( ) , , , v E x y t θ % is the suppressed motion energy and the factor α controls the inhibition strength. The inhibition term is computed by weighted motion energy ( ) ( ) ( ) , , , , , , , , , v v v S x y t E x y t w x y t θ θ θ = ∗ (15) where ( ) , , , v w x y t θ is the surround weighting function. The Heeger’s model constructs the center-surround motion opponent by computing the weighted sum of responses from motion selective cells, ( ) ( ) ( ) ( ) , , , , , , , , , v v v x y R t x y E x y t E x y t θ θ θ β − ⎡ ⎤ = − ⎣ ⎦ ∑ (16) where ( ) ,x y β is a center-surround weighting function and the motion energy at each point should be normalized across all cells with different tuning properties. As shown in Figure 4 (c) and (d) for results of Petkov’s and Heeger’s models, we replace the conventional frequency tuned motion energy neuron with our proposed phase tuned neuron. The model by Petkov, et al. [8] is generally suppressive and only reproduces less suppression for opposite motions, which is inconsistent with results from [3]. The model by Heeger, et al. [7] has similar properties with our proposed model with respect to both excitatory and inhibitory interactions. To evaluate the sensitivity of the proposed opponency model to velocity differences, we did simulations by using similar stimuli with the above experiment in Figure 4 but maintaining a constant velocity difference of 3 pixels per frame between the center and surrounding random dot patches. As shown in Figure 5, by varying the velocities of random dots on the center region, we found that responses by the proposed model are always enhanced independent upon absolute velocities of center stimuli, but responses by the Heeger’s model achieve the enhancement at a center velocity of 1.2 pixels per frame and maintain suppressed at other speeds. 5 Discussion We proposed a new biologically plausible model of the differential motion opponency to model the spatial interaction property of motion energy neurons. The proposed opponency model is motivated by the phase tuning mechanism of disparity energy neurons which infers the disparity information from the phase difference between complex valued responses to left and right retinal images. Hence, the two neighboring spatial areas can be considered as left and right images and the motion difference between these two spatial regions is detected by the phase difference between the complex valued responses at these two regions. Our experimental results demonstrate a consistent conclusion with physiological experiments that motions of opposite directions and different speeds outside the CRF can show both inhibitive and excitatory effects on the CRF responses. The inhibitive interaction helps to segment the moving object from backgrounds when fed back to low-level features such as edges, orientations and color information. Except providing a unifying phase mechanism in understanding neurons with different functional roles at different brain areas, the proposed opponency model could possibly provide a way to understand the motion integration and motion segmentation. Integration and segmentation are two opposite motion perception tasks but co-exist to constitute two fundamental types of motion processing. Segmentation is achieved by discriminating motion signals from different objects, which is thought to be due to the antagonistic interaction between center and surrounding RFs. Integration is obtained by utilizing the enhancing function of surrounding areas to CRF areas. Both types of processing have been found in motion related areas including area MT and MST. Tadin, et al. [18] have found that motion segmentation dominants at high stimulus contrast and gives the way to motion integration at low stimulus contrast. Huang, et al. [19] suggests that the surrounding modulation is adaptive according to the visual stimulus such as contrasts and noise levels. Since our proposed opponency model determines the functional role of neurons by only the phase shift parameter, this makes the proposed model to be an ideal candidate model for the adaptive surrounding modulation with the phase tuning between two spatial regions. References [1]. K. Nakayama and J. M. Loomis, “Optical velocity patterns, velocity-sensitive neurons, and space perception: A hypothesis,” Perception, vol. 3, 63-80, 1974. [2]. K. Tanaka, K. Hikosaka, H. Saito, M. Yukie, Y. Fukada and E. Iwai, “Analysis of local and wide-field movements in the superior temporal visual areas of the macaque monkey,” Journal of Neuroscience, vol. 6, pp. 134-144, 1986. [3]. R. T. Born and R. B. H. Tootell, “Segregation of global and local motion processing in primate middle temporal visual area,” Nature, vol. 357, pp. 497-499, 1992. [4]. J. Allman, F. Miezin and E. McGuinness, “Stimulus specific responses from beyond the classical receptive field: Neurophysiological mechanisms for local-global comparisions in visual neurons,” Annual Review Neuroscience, vol. 8, pp. 407-430, 1985. [5]. V. A. F. Lamme, “The neurophysiology of figure-ground segregation in primary visual cortex,” Journal of Neuroscience, vol. 15, pp. 1605-1615, 1995. [6]. D. C. Bradley and R. A. Andersen, “Center-surround antagonism based on disparity in primate area MT,” Journal of Neuroscience, vol. 18, pp. 7552-65, 1998. [7]. D. J. Heeger, A. D. Jepson and E. P. Simoncelli, “Recovering observer translation with center-surround operators,” Proc IEEE Workshop on Visual Motion, pp. 95-100, Oct 1991. [8]. N. Petkov and E. Subramanian, “Motion detection, noise reduction, texture suppression, and contour enhancement by spatiotemporal Gabor filters with surround inhibition,” Biological Cybernetics, vol. 97, pp. 423-439, 2007. [9]. M. Escobar and P. Kornprobst, “Action recognition with a Bio-inspired feedforward motion processing model: the richness of center-surround interactions,” ECCV '08: Proceedings of the 10th European Conference on Computer Vision, pp. 186-199, Marseille, France, 2008. [10]. B. J. Frost and K. Nakayama, “Single visual neurons code opposing motion independent of direction,” Science, vol. 200, pp. 744-745, 1983. [11]. A. Cao and P. H. Schiller, “Neural responses to relative speed in the primary visual cortex of rhesus monkey,” Visual Neuroscience, vol. 20, pp. 77-84, 2003. [12]. B. K. Dellen, J. W. Clark and R. Wessel, “Computing relative motion with complex cells,” Visual Neuroscience, vol. 22, pp. 225-236, 2005. [13]. I. Ohzawa, G. C. Deangelis and R. D. Freeman, “Encoding of binocular disparity by complex cells in the cat’s visual cortex,” Journal of Neurophysiology, vol. 77, pp. 2879-2909, 1997. [14]. D. J. Fleet, H. Wagner and D. J. Heeger, “Neural Encoding of binocular disparity: energy model, position shifts and phase shifts,” Vision Research, vol. 26, pp. 1839-1857, 1996. [15]. Y. C. Meng and B. E. Shi, “Normalized Phase Shift Motion Energy Neuron Populations for Image Velocity Estimation,” International Joint Conference on Neural Network, Atlanta, GA, June 14-19, 2009. [16]. E. H. Adelson and J. R. Bergen, “Spatiotemporal energy models for the perception of motion,” J. Opt. Soc. Am. A Opt. Image Sci. Vis., vol. 2, pp. 284-299, 1985. [17]. D. K. Xiao, S. Raiguel, V. Marcar, J. Koenderink and G. A. Orban, “The spatial distribution of the antagonistic surround of MT/V5,” Cereb Cortex, vol. 7, pp. 662-677, 1997. [18]. D. Tadin, J. S. Lappin, L. A. Gilroy and R. Blake, “Perceptual consequences of centre-surround antagonism in visual motion processing,” Nature, vol. 424, pp. 312-315, 2003. [19]. X. Huang, T. D. Albright and G. R. Stoner, “Adaptive surround modulation in cortical area MT,” Neuron, vol. 53, pp. 761-770, 2007.
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Periodic Step-Size Adaptation for Single-Pass On-line Learning Chun-Nan Hsu1,2,∗, Yu-Ming Chang1, Han-Shen Huang1 and Yuh-Jye Lee3 1Institute of Information Science, Academia Sinica, Taipei 115, Taiwan 2USC/Information Sciences Institute, Marina del Rey, CA 90292, USA 3Department of Computer Science and Information Engineering, National Taiwan University of Science and Technology, Taipei 106, Taiwan ∗chunnan@isi.edu Abstract It has been established that the second-order stochastic gradient descent (2SGD) method can potentially achieve generalization performance as well as empirical optimum in a single pass (i.e., epoch) through the training examples. However, 2SGD requires computing the inverse of the Hessian matrix of the loss function, which is prohibitively expensive. This paper presents Periodic Step-size Adaptation (PSA), which approximates the Jacobian matrix of the mapping function and explores a linear relation between the Jacobian and Hessian to approximate the Hessian periodically and achieve near-optimal results in experiments on a wide variety of models and tasks. 1 Introduction On-line learning has been studied for decades. Early works concentrate on minimizing the required number of model corrections made by the algorithm through a single pass of training examples. More recently, on-line learning is considered as a solution of large scale learning mainly because of its fast convergence property. New on-line learning algorithms for large scale learning, such as SMD [1] and EG [2], are designed to learn incrementally to achieve fast convergence. They usually still require several passes (or epochs) through the training examples to converge at a satisfying model. However, the real bottleneck of large scale learning is I/O time. Reading a large data set from disk to memory usually takes much longer than CPU time spent in learning. Therefore, the study of on-line learning should focus more on single-pass performance. That is, after processing all available training examples once, the learned model should generalize as well as possible so that used training example can really be removed from memory to minimize disk I/O time. In natural learning, single-pass learning is also interesting because it allows for continual learning from unlimited training examples under the constraint of limited storage, resembling a nature learner. Previously, many authors, including [3] and [4], have established that given a sufficiently large set of training examples, 2SGD can potentially achieve generalization performance as well as empirical optimum in a single pass through the training examples. However, 2SGD requires computing the inverse of the Hessian matrix of the loss function, which is prohibitively expensive. Many attempts to approximate the Hessian have been made. For example, one may consider to modify L-BFGS [5] for online settings. L-BFGS relies on line search. But in online settings, we only have the surface of the loss function given one training example, as opposed to all in batch settings. The search direction obtained by line search on such a surface rarely leads to empirical optimum. A review of similar attempts can be found in Bottou’s tutorial [6], where he suggested that none is actually sufficient to achieve theoretical single-pass performance in practice. This paper presents a new 2SGD method, called Periodic Step-size Adaptation (PSA). PSA approximates the Jacobian matrix of the mapping function and explores a linear relation between the Jacobian and Hessian to approximate the Hessian 1 periodically. The per-iteration time-complexity of PSA is linear to the number of nonzero dimensions of the data. We analyze the accuracy of the approximation and derive the asymptotic rate of convergence for PSA. Experimental results show that for a wide variety of models and tasks, PSA is always very close to empirical optimum in a single-pass. Experimental results also show that PSA can run much faster compared to state-of-the-art algorithms. 2 Aitken’s Acceleration Let w ∈ℝ푑be a 푑-dimensional weight vector of a model. A machine learning problem can be formulated as a fixed-point iteration that solves the equation w = ℳ(w), where ℳis a mapping ℳ: ℝ푑→ℝ푑, until w∗= ℳ(w∗). Assume that the mapping ℳis differentiable. Then we can apply Aitken’s acceleration, which attempts to extrapolate to the local optimum in one step, to accelerate the convergence of the mapping: w∗= w(푡) + (I −J)−1(ℳ(w(푡)) −w(푡)), (1) where J := ℳ′(w∗) is the Jacobian of the mapping ℳat w∗. When 휆푖:= eig(J) ∈(−1, 1), the mapping ℳis guaranteed to converge. That is, when 푡→∞, w(푡) →w∗. It is usually difficult to compute J for even a simple machine learning model. To alleviate this issue, we can approximate J with the estimates of its 푖-th eigenvalue 휆푖by 훾(푡) 푖 := ℳ(w(푡))푖−w(푡) 푖 w(푡) 푖 −w(푡−1) 푖 , ∀푖, (2) and extrapolate at each dimension 푖by: w(푡+1) 푖 = w(푡) 푖 + (1 −훾(푡) 푖)−1(ℳ(w(푡))푖−w(푡) 푖) . (3) In practice, Aitken’s acceleration alternates a step for preparing 훾(푡) and a step for the extrapolation. That is, when 푡is an even number, ℳis used to obtain w(푡+1). Otherwise, the extrapolation (3) is used. A benefit of the above approximation is that the cost for performing an extrapolation is 푂(푑), linear in terms of the dimension. 3 Periodic Step-Size Adaptation When ℳis a gradient descent update rule, that is, ℳ(w) ←w −휂g(w; D), where 휂is a scalar step size, D is the entire set of training examples, and g(w; D) is the gradient of a loss function to be minimized, Aitken’s acceleration is equivalent to Newton’s method, because J = ℳ′(w) = I −휂H(w; D), (4) (I −J)−1 = 1 휂H(w; D)−1, and ℳ(w) −w = w −휂g(w; D) −w = −휂g(w; D), where H(w; D) = 푔′(w; D), the Hessian matrix of the loss function, and the extrapolation given in (1) becomes w = w + (I −J)−1(ℳ(w) −w) = w −1 휂H−1휂g = w −H−1g. In this case, Aitken’s acceleration enjoys the same local quadratic convergence as Newton’s method. This can also be extended to a SGD update rule: w(푡+1) ←w(푡) −휂∙g(w(푡); B(푡)), where the mini-batch B ⊆D, ∣B∣≪∣D∣, is a randomly selected small subset of D. A genuine on-line learner usually has ∣B∣= 1. We consider a positive vector-valued step-size 휂∈ℝ푑 + and “∙” denotes component-wise (Hadamard) product of two vectors. Again, by exploiting (4), since eig(I −diag(휂)H) = eig(ℳ′) = eig(J) ≈훾, where 훾is an estimated eigenvalue of J as given in (2), when H is a symmetric matrix, its eigenvalue is given by eig(J) = 1 −휂푖eig(H) ⇒eig(H) = 1 −eig(J) 휂푖 . 2 Therefore, we can update the step size component-wise by eig(H−1) = 휂푖 1 −eig(J) ≈ 휂푖 1 −훾푖 ⇒휂(푡+1) 푖 ∝ 휂(푡) 푖 1 −훾(푡) 푖 . (5) Since the mapping ℳin SGD involves the gradient g(w(푡); B(푡)) of a randomly selected training example B(푡), ℳis itself a random variable. It is unlikely that we can obtain a reliable eigenvalue estimation at each single iteration. To increase stationary of the mapping, we take advantage of the law of large numbers and aggregate consecutive SGD mappings into a new mapping ℳ푏= ℳ(ℳ(. . . ℳ(w) . . .)) | {z } 푏 , which reduces the variance of gradient estimation by 1 푏, compared to the plain SGD mapping ℳ. The approximation is valid because w(푡+푖), 푖= 0, . . . , 푏−1 are approximately fixed when 휂is sufficiently small [7]. We can proceed to estimate the eigenvalues of ℳ푏from w(푡), w(푡+푏) and w(푡+2푏) by applying (2) for each component 푖: ¯훾푏 푖= w(푡+2푏) 푖 −w(푡+푏) 푖 w(푡+푏) 푖 −w(푡) 푖 . (6) We note that our aggregate mapping ℳ푏is different from a mapping that takes 푏mini-batches as the input in a single iteration. Their difference is similar to that between batch and stochastic gradient descent. Aggregate mappings have 푏chances to adjust its search direction, while mappings that use 푏mini-batches together only have one. With the estimated eigenvalues, we can present the complete update rule to adjust the step size vector 휂. To ensure that the estimated values of eig(J) ∈(−1, 1) and to ensure numerical stability, we introduce a positive constant 휅< 1 as the upper bound of ∣¯훾푏 푖∣. Let u denote the constrained ¯훾푏. Its components are given by 푢푖:= sgn(¯훾푏 푖) min(∣¯훾푏 푖∣, 휅), ∀푖. (7) Then we can update the step size every 2푏iterations based on u by: 휂(푡+2푏+1) = v ∙휂(푡+2푏), (8) where v is a discount factor with components defined by 푣푖:= 푚+ 푢푖 푚+ 휅+ 푛, ∀푖. (9) The discount factor is derived from (5) and the fact that when 푢< 1, 1 1−푢> 푒푢≈1 + 푢to ensure numerical stability, with 푚and 푛controlling the range. Let 훼be the maximum value and 훽be the minimum value of 푣푖. We can obtain 푚and 푛by solving 훽≤푣푖≤훼for all 푖. Since −휅≤푢푖≤휅, we have 푣푖= 훼when 푢푖= 휅and 푣푖= 훽when 푢푖= −휅. Solving these equations yields: 푚= 훼+ 훽 훼−훽휅and 푛= 2(1 −훼) 훼−훽휅. (10) For example, if we want to set 훼= 0.9999 and 훽= 0.99, then 푚and 푛will be 201휅and 0.0202휅, respectively. Setting 0 < 훽< 훼≤1 ensures that the step size is decreasing and approaches zero so that SGD can be guaranteed to converge [7]. Algorithm 1 shows the PSA algorithm. In a nutshell, PSA applies SGD with a fixed step size and periodically updates the step size by approximating Jacobian of the aggregated mapping. The complexity per iteration is 푂( 푑 푏) because the cost of eigenvalue estimation given in (6) is 2푑and it is required for every 2푏iterations. That is, PSA updates 휂after learning from 2푏⋅B examples. 3 Algorithm 1 The PSA Algorithm 1: Given: 훼, 훽, 휅< 1 and 푏 2: Initialize 휃(0) and 휂(0); 푡←0; 푚←훼+훽 훼−훽휅and 푛←2(1−훼) 훼−훽휅 ⊳Equation (10) 3: repeat 4: Choose a small batch B(푡) uniformly at random from the set of training examples D 5: update 휃(푡+1) ←휃(푡) −휂∙g(휃(푡); B(푡)) ⊳SGD update 6: if (푡+ 1) mod 2푏= 0 then ⊳Update 휂 7: update ¯훾푏 푖←휃(푡+2푏) 푖 −휃(푡+푏) 푖 휃(푡+푏) 푖 −휃(푡) 푖 ⊳Equation (6) 8: For all 푖, update 푢푖←sgn(¯훾푏 푖) min(∣¯훾푏 푖∣, 휅) ⊳Equation (7) 9: For all 푖, update 푣푖← 푚+푢푖 푚+휅+푛 ⊳Equation (9) 10: update 휂(푡+1) ←v ∙휂(푡) ⊳Equation (8) 11: else 12: 휂(푡+1) ←휂(푡) 13: end if 14: 푡←푡+ 1 15: until Convergence 4 Analysis of PSA We analyze the accuracy of 훾(푡) 푖 as an eigenvalue estimate as follows. Let eigen decomposition J = QΛQ−1 and u푖be column vectors of Q and v푇 푖be row vectors of Q−1. Then we have J푡= 푑 ∑ 푗=1 휆푡 푗u푗v푇 푗, where 휆푗is the 푗-th eigenvalue of J. By applying Taylor’s expansion to ℳ, we have w(푡) −w∗ ≈ J푡(w(0) −w∗) w(푡−1) −w∗ ≈ J푡−1(w(0) −w∗) ⇒Δ(푡) = w(푡) −w(푡−1) ≈ J푡J−1(J −I)(w(0) −w∗) ⇒Δ(푡+1) = w(푡+1) −w(푡) ≈ 푑 ∑ 푗=1 휆푗휆푡 푗u푗v푇 푗J−1(J −I)(w(0) −w∗) Now let 휔푖푗:= e푇 푖u푗v푇 푗J−1(J −I)(w(0) −w∗), where e푖is the 푖-th column of I. Let Δ푖be the 푖-th element of Δ and 휆푗푖be the largest eigenvalue of J such that 휔푖푗∕= 0. Then 훾푖≡Δ(푡+1) 푖 Δ(푡) 푖 = ∑푑 푗=1 휆푡+1 푗 휔푖푗 ∑푑 푗=1 휆푡 푗휔푖푗 = 휆푗푖+ ∑ 푗∕=푗푖(휆푗/휆푗푖)푡휆푗휔푖푗/휔푖푗푖 1 + ∑ 푗∕=푗푖(휆푗/휆푗푖)푡휔푖푗/휔푖푗푖 . Therefore, we can conclude that ∙훾푖→휆푗푖as 푡→∞because ∀푖, if 휔푖푗∕= 0 then 휆푗/휆푗푖≤1. 휆푗푖≡푅푖is the 푖-th componentwise rate of convergence. ∙훾푖= 휆푖if J is a diagonal matrix. In this case, our approximation is exact. This happens when there are high percentages of missing data for a Bayesian network model trained by EM [8] and when features are uncorrelated for training a conditional random field model [9]. ∙훾푖is the average of eigenvalues weighted by 휆푡 푗휔푖푗. Since 휔푖푗is usually the largest when 푖= 푗, we have 훾푖≈휆푖. 4 When we have the least possible step size 휂(푡+1) = 훽휂(푡) for all 푡mod 2푏= 0 in PSA, the expectation of w(푡) obtained by PSA can be shown to be: 퐸(w(푡)) = w∗+ 푡∏ 푘=1 ( 퐼−휂(0)훽⌊푘 푏⌋H(w∗; D) ) (w(0) −w∗) = w∗+ S(푡)(w(0) −w∗). The rate of convergence is governed by the largest eigenvalue of S(푡). We now derive a bound of this eigenvalue. Theorem 1 Let 휆ℎbe the least eigenvalue of H(w∗; D). The asymptotic rate of convergence of PSA is bounded by eig(S(푡)) ≤exp {−휂(0)휆ℎ푏 1 −훽 } . Proof We can show that eig(S(푡)) = 푡∏ 푘=1 ( 1 −휂(0)훽⌊푘 푏⌋휆ℎ ) ≤ exp { − 푡 ∑ 푘=1 휂(0)휆ℎ훽⌊푘 푏⌋ } = exp { −휂(0)휆ℎ 푡 ∑ 푘=1 훽⌊푘 푏⌋ } because for any 0 ≤푎푗< 1, 1 −푎푗≤푒−푎푗, 0 ≤ 푛 ∏ 푗=1 (1 −푎푗) ≤ 푛 ∏ 푗=1 푒−푎푗= 푒−∑푛 푗=1 푎푗. Now, since 푡 ∑ 푘=1 훽⌊푘 푏⌋≈ ⎛ ⎝ ⌊푡 푏⌋ ∑ 푙=0 푏훽푙 ⎞ ⎠= 푏 ⌊푡 푏⌋ ∑ 푙=0 훽푙−→ 푏 1 −훽when 푡→∞, we have eig(S(푡)) ≤exp { −휂(0)휆ℎ 푡 ∑ 푘=1 훽⌊푘 푏⌋ } →exp {−휂(0)휆ℎ푏 1 −훽 } when 푡→∞. □ Though this analysis suggests that for rapid convergence to 휃∗, we should assign 훽≈1 with a large 푏and 휂(0), it is based on a worst-case scenario and thus insufficient as a practical guideline for parameter assignment. In practice, we fix (훼, 훽, 휅) = (0.9999, 0.99, 0.9) and tune 푏as follows. When the training set size ∣D∣≫2000, set 푏in the order of 0.5∣D∣/1000 is usually sufficient. This setting implies that the step size will be adjusted per ∣D∣/1000 examples. In fact, when 푏 is in the same order, PSA performs similarly. Consider the following three settings: (푏, 훼, 훽) = (10, 0.9999, 0.99), (100, 0.999, 0.9) or (1, 0.99999, 0.999). They all yield nearly identical singlepass F-scores for the BaseNP task (see Section 5). The first setting was used in this paper. To see why this is the case, consider the decreasing factor 푣푖(see (8) and (9)), which will be confined within the interval (훼, 훽). Assume that 푣푖is selected at ransom uniformly, then the mean of 푣푖= 0.995 when (훼, 훽) = (0.9999, 0.99) and 휂푖will be decreased by a factor of 0.995 on average in each PSA update. When 푏= 10, PSA will update 휂푖per 20 examples. After learning from 200 examples, PSA will decrease 휂푖10 times by a combined factor of 0.9511. Similarly, we can obtain that the factors for the other two settings are 0.95 and 0.9512, respectively, nearly identical. 5 5 Experimental Results Table 1 shows the tasks chosen for our comparison. The tasks for CRF have been used in competitions and the performance was measured by F-score. Weight for CRF reported here is Number of features provided by CRF++. Target provides the empirical optimal performance achieved by batch learners. If PSA accurately approximates 2SGD, then its single-pass performance should be very close to Target. The target F-score for BioNLP/NLPBA is not >85% as reported in [1] because it was due to a bug that included true labels as a feature 1. Table 1: Tasks for the experiments. Task Model Training Test Tag/Class Weight Target Base NP CRF 8936 2012 3 1015662 94.0% [10] Chunking CRF 8936 2012 23 7448606 93.6% [11] BioNLP/NLPBA CRF 18546 3856 11 5977675 70.0% [12] BioCreative 2 CRF 15000 5000 3 10242972 86.5% [13] LS FD LSVM 2734900 2734900 2 900 3.26% LS OCR LSVM 1750000 1750000 2 1156 23.94% MNIST [14] CNN 60000 10000 10 134066 0.99% 5.1 Conditional Random Field We compared PSA with plain SGD and SMD [1] to evaluate PSA’s performance for training conditional random fields (CRF). We implemented PSA by replacing the L-BFGS optimizer in CRF++ [11]. For SMD, we used the implementation available in the public domain 2. Our SGD implementation for CRF is from Bottou 3. All the above implementations are revisions of CRF++. Finally, we ran the original CRF++ with default settings to obtain the performance results of LBFGS. We simply used the original parameter settings for SGD and SMD as given in the literature. For PSA, we used 휅= 0.9, (훼, 훽) = (0.9999, 0.99), 푏= 10, and 휂(0) 푖 = 0.1, ∀푖. The batch size is one for all tasks. These parameters were determined by using a small subset from CoNLL 2000 baseNP and we simply used them for all tasks. All of the experiments reported here for CRF were ran on an Intel Q6600 Fedora 8 i686 PC with 4G RAM. Table 2 compares SGD variants in terms of the execution time and F-scores achieved after processing the training examples for a single pass. Since the loss function in CRF training is convex, the convergence results of L-BFGS can be considered as the empirical minimum. The results show that single-pass F-scores achieved by PSA are about as good as the empirical minima, suggesting that PSA has effectively approximated Hessian in CRF training. Fig. 1 shows the learning curves in terms of the CPU time. Though as expected, plain SGD is the fastest, it is remarkable that PSA is faster than SMD for all tasks. SMD is supposed to have an edge here because the mini-batch size for SMD was set to 6 or 8, as specified in [1], while PSA used one for all tasks. But PSA is still faster than SMD partly because PSA can take advantage of the sparsity trick as plain SGD [15]. 5.2 Linear SVM We also evaluated PSA’s single-pass performance for training linear SVM. It is straightforward to apply PSA as a primal optimizer for linear SVM. We used two very large data sets: FD (face detection) and OCR (see Table 1), from the Pascal large-scale learning challenge in 2008 and compared the performance of PSA with the state-of-the-art linear SVM solvers: Liblinear 1.33 [16], the winner of the challenge, and SvmSgd, from Bottou’s SGD web site. They have been shown to outperform many well-known linear SVM solvers, such as SVM-perf [17] and Pegasos [15]. 1Thanks to Shing-Kit Chan of the Chinese University of Hong Kong for pointing that out. 2Available under LGPL from the following URL: http://sml.nicta.com.au/code/crfsmd/. 3http://leon.bottou.org/projects/sgd. 6 Base NP Chunking BioNLP/NLPBA BioCreative 2 Method (pass) time F-score time F-score time F-score time F-score SGD (1) 1.15 92.42 13.04 92.26 12.23 66.37 3.18 34.33 SMD (1) 41.50 91.81 350.00 91.89 522.00 66.53 497.71 69.04 PSA (1) 16.30 93.31 160.00 93.16 206.00 69.41 191.61 80.79 L-BFGS (batch) 221.17 93.91 8694.40 93.78 20130.00 70.30 1601.50 86.82 Table 2: CPU time in seconds and F-scores achieved after a single pass of CRF training. 0 50 100 150 200 90 90.5 91 91.5 92 92.5 93 93.5 94 94.5 BaseNP Time(sec) F−score PSA SMD SGD L−BFGS 0 200 400 600 800 1000 1200 90 90.5 91 91.5 92 92.5 93 93.5 94 94.5 Chunking Time(sec) F−score PSA SMD SGD L−BFGS 0 100 200 300 400 500 600 700 800 20 30 40 50 60 70 80 NLPBA04 Time(sec) F−score PSA SMD SGD L−BFGS 0 100 200 300 400 500 30 40 50 60 70 80 90 BioCreative 2 GM Task Time(sec) F−score PSA SMD SGD L−BFGS Figure 1: Comparison of CPU time; Horizontal lines indicate target F-scores. We selected L2-regularized logistic regression as the loss function for PSA and Liblinear because it is twice differentiable. The weight 퐶of the margin error term was set to one. We kept SvmSgd intact. The experiment was run on an Open-SUSE Linux machine with Intel Xeon E7320 CPU (2.13GHz) and 64GB RAM. Table 3 shows the results. Again, PSA achieves the best single-pass accuracy for both tasks. Its test accuracies are very close to that of converged Liblinear. PSA takes much less time than the other two solvers. PSA (1) is faster than SvmSgd (1) for SVM because SvmSgd uses the sparsity trick [15], which speeds up training for sparse data, but otherwise may slow down. Both data sets we used turn out to be dense, i.e., with no zero features. We implemented PSA with the sparsity trick for CRF only but not for SVM and CNN. LS FD LS OCR Method (pass) accuracy time accuracy time Liblinear converge 96.74 4648.49 76.06 4454.42 Liblinear (1) 91.43 290.58 74.33 398.00 SvmSgd (20) 93.78 1135.67 SvmSgd (10) 93.77 567.68 73.71 473.35 SvmSgd (1) 93.60 56.78 73.76 46.96 PSA (1) 95.10 30.65 75.68 25.33 Table 3: Test accuracy rates and elapsed CPU time in seconds by various linear SVM solvers. 7 The parameter settings for PSA are basically the same as those for CRF but with a large period 푏= 1250 for FD and 500 for OCR. For FD, the worst accuracy by PSA is 94.66% with 푏between 250 to 2000. For OCR, the worst is 75.20% with 푏between 100 to 1000, suggesting that PSA is not very sensitive to parameter settings. 5.3 Convolutional Neural Network Approximating Hessian is particularly challenging when the loss function is non-convex. We tested PSA in such a setting by applying PSA to train a large convolutional neural network for the original 10-class MNIST task (see Table 1). We tried to duplicate the implementation of LeNet described in [18] in C++. Our implementation, referred to as “LeNet-S”, is a simplified variant of LeNet-5. The differences include that the sub-sampling layers in LeNet-S picks only the upper-left value from a 2 × 2 area and abandons the other three. LeNet-S used more maps (50 vs. 16) in the third layer and less nodes (120 vs. 100) in the fifth layer, due to the difference in the previous sub-sampling layer. Finally, we did not implement the Gaussian connections in the last layer. We trained LeNet-S by plain SGD and PSA. The initial 휂for SGD was 0.7 and decreased by 3 percent per pass. For PSA, we used 휅= 0.9, (훼, 훽) = (0.99999, 0.999), 푏= 10, 휂(0) 푖 = 0.5, ∀푖, and the mini-batch size is one for all tasks. We also adapted a trick given in [19] which advises that step sizes in the lower layers should be larger than in the higher layer. Following their trick, the initial step sizes for the first and the third layers were 5 and √ 2.5 times as large as those for the other layers, respectively. The experiments were ran on an Intel Q6600 Fedora 8 i686 PC with 4G RAM. Table 4 shows the results. To obtain the empirical optimal error rate of our LeNet-S model, we ran plain SGD with sufficient passes and obtained 0.99% error rate at convergence, slightly higher than LeNet-5’s 0.95% [18]. Single-pass performance of PSA with the layer trick is within one percentage point to the target. Starting from an initial weight closer to the optimum helped improving PSA’s performance further. We ran SGD 100 passes with randomly selected 10K training examples then re-started training with PSA using the rest 50K training examples for a single pass. Though PSA did achieve a better error rate, this is infeasible because it took 4492 seconds to run SGD 100 passes. Finally, though not directly comparable, we also report the performance of TONGA given in [20] as a reference. TONGA is a 2SGD method based on natural gradient. Method (pass) time error Method (pass) time error SGD (1) 266.77 2.36 PSA w/o layer trick (1) 311.95 2.31 SGD (140) 37336.20 0.99 PSA w/ layer trick (1) 311.00 1.97 TONGA (n/a) 500.00 2.00 PSA re-start (1) 253.72 1.90 Table 4: CPU time in seconds and percentage test error rates for various neural network trainers. 6 Conclusions It has been shown that given a sufficiently large training set, a single pass of 2SGD generalizes as well as the empirical optimum. Our results show that PSA provides a practical solution to accomplish near optimal performance of 2SGD as predicted theoretically for a variety of large scale models and tasks with a reasonably low cost per iteration compared to competing 2SGD methods. The benefit of 2SGD with PSA over plain SGD becomes clearer when the scale of the tasks are increasingly large. For non-convex neural network tasks, since the curvature of the error surface is so complex, it is still very challenging for an eigenvalue approximation method like PSA. A complete version of this paper will appear as [21]. Source codes of PSA are available at http://aiia.iis.sinica.edu.tw. References [1] S.V.N. Vishwanathan, Nicol N. Schraudolph, Mark W. Schmidt, and Kevin P. Murphy. Accelerated training of conditional random fields with stochastic gradient methods. In Proceedings of the 23rd International Conference on Machine Learning (ICML’06), Pittsburgh, PA, USA, June 2006. 8 [2] Michael Collins, Amir Globerson, Terry Koo, Xavier Carreras, and Peter L. Bartlett. Exponentiated gradient algorithms for conditional random fields and max-margin markov networks. Journal of Machine Learning Research, 9:1775–1822, August 2008. [3] Noboru Murata and Shun-Ichi Amari. Statistical analysis of learning dynamics. Signal Processing, 74(1):3–28, April 1999. [4] L´eon Bottou and Yann LeCun. On-line learning for very large data sets. Applied Stochastic Models in Business and Industry, 21(2):137–151, 2005. [5] Jorge Nocedal and Stephen J. Wright. Numerical Optimization. Springer, 1999. [6] L´eon Bottou. The tradeoffs of large-scale learning. Tutorial, the 21st Annual Conference on Neural Information Processing Systems (NIPS 2007), Vancouver, BC, Canada, December 2007. http://leon.bottou.org/talks/largescale. [7] Albert Benveniste, Michel Metivier, and Pierre Priouret. Adaptive Algorithms and Stochastic Approximations. Springer-Verlag, 1990. [8] Chun-Nan Hsu, Han-Shen Huang, and Bo-Hou Yang. Global and componentwise extrapolation for accelerating data mining from large incomplete data sets with the EM algorithm. In Proceedings of the Sixth IEEE International Conference on Data Mining (ICDM’06), pages 265–274, Hong Kong, China, December 2006. [9] Han-Shen Huang, Bo-Hou Yang, Yu-Ming Chang, and Chun-Nan Hsu. Global and componentwise extrapolations for accelerating training of Bayesian networks and conditional random fields. Data Mining and Knowledge Discovery, 19(1):58–91, 2009. [10] Fei Sha and Fernando Pereira. Shallow parsing with conditional random fields. In Proceedings of Human Language Technology, the North American Chapter of the Association for Computational Linguistics (NAACL’03), pages 213–220, 2003. [11] Taku Kudo. CRF++: Yet another CRF toolkit, 2006. Available under LGPL from the following URL: http://crfpp.sourceforge.net/. [12] Burr Settles. Biomedical named entity recognition using conditional random fields and novel feature sets. In Proceedings of the Joint Workshop on Natural Language Processing in Biomedicine and its Applications (JNLPBA-2004), pages 104–107, 2004. [13] Cheng-Ju Kuo, Yu-Ming Chang, Han-Shen Huang, Kuan-Ting Lin, Bo-Hou Yang, Yu-Shi Lin, Chun-Nan Hsu, and I-Fang Chung. Rich feature set, unification of bidirectional parsing and dictionary filtering for high f-score gene mention tagging. In Proceedings of the Second BioCreative Challenge Evaluation Workshop, pages 105–107, 2007. [14] Yann LeCun and Corinna Cortes. The MNIST database of handwritten digits, 1998. http://yann.lecun.com/exdb/mnist/. [15] Shai Shalev-Shwartz, Yoram Singer, and Nathan Srebro. Pegasos: Primal Estimated subGrAdient SOlver for SVM. In ICML’07: Proceedings of the 24th international conference on Machine learning, pages 807–814, New York, NY, USA, 2007. ACM Press. [16] Chih-Chung Chang and Chih-Jen Lin. LIBSVM: a library for support vector machines, 2001. Software available at http://www.csie.ntu.edu.tw/∼cjlin/libsvm. [17] Thorsten Joachims. Training linear SVMs in linear time. In Proceedings of the 12th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD’06), pages 217–226, New York, NY, USA, 2006. ACM. [18] Yann LeCun, L´eon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998. [19] Yann LeCun, Leon Bottou, Genevieve B. Orr, and Klaus-Robert Muller. Efficient backprop. In G. Orr and Muller K., editors, Neural Networks: Tricks of the trade. Springer, 1998. [20] Nicolas LeRoux, Pierre-Antoine Manzagol, and Yoshua Bengio. Topmoumoute online natural gradient algorithm. In Advances in Neural Information Processing Systems, 20 (NIPS 2007), Cambridge, MA, USA, 2008. MIT Press. [21] Chun-Nan Hsu, Yu-Ming Chang, Han-Shen Huang, and Yuh-Jye Lee. Periodic step-size adaptation in second-order gradient descent for single-pass on-line structured learning. To appear in Mchine Learning, Special Issue on Structured Prediction. DOI: 10.1007/s10994-009-5142-6, 2009. 9
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Construction of Nonparametric Bayesian Models from Parametric Bayes Equations Peter Orbanz University of Cambridge and ETH Zurich p.orbanz@eng.cam.ac.uk Abstract We consider the general problem of constructing nonparametric Bayesian models on infinite-dimensional random objects, such as functions, infinite graphs or infinite permutations. The problem has generated much interest in machine learning, where it is treated heuristically, but has not been studied in full generality in nonparametric Bayesian statistics, which tends to focus on models over probability distributions. Our approach applies a standard tool of stochastic process theory, the construction of stochastic processes from their finite-dimensional marginal distributions. The main contribution of the paper is a generalization of the classic Kolmogorov extension theorem to conditional probabilities. This extension allows a rigorous construction of nonparametric Bayesian models from systems of finitedimensional, parametric Bayes equations. Using this approach, we show (i) how existence of a conjugate posterior for the nonparametric model can be guaranteed by choosing conjugate finite-dimensional models in the construction, (ii) how the mapping to the posterior parameters of the nonparametric model can be explicitly determined, and (iii) that the construction of conjugate models in essence requires the finite-dimensional models to be in the exponential family. As an application of our constructive framework, we derive a model on infinite permutations, the nonparametric Bayesian analogue of a model recently proposed for the analysis of rank data. 1 Introduction Nonparametric Bayesian models are now widely used in machine learning. Common models, in particular the Gaussian process (GP) and the Dirichlet process (DP), were originally imported from statistics, but the nonparametric Bayesian idea has since been adapted to the needs of machine learning. As a result, the scope of Bayesian nonparametrics has expanded significantly: Whereas traditional nonparametric Bayesian statistics mostly focuses on models on probability distributions, machine learning researchers are interested in a variety of infinite-dimensional objects, such as functions, kernels, or infinite graphs. Initially, existing DP and GP approaches were modified and combined to derive new models, including the Infinite Hidden Markov Model [2] or the Hierarchical Dirichlet Process [15]. More recently, novel stochastic process models have been defined from scratch, such as the Indian Buffet Process (IBP) [8] and the Mondrian Process [13]. This paper studies the construction of new nonparametric Bayesian models from finite-dimensional distributions: To construct a model on a given type of infinite-dimensional object (for example, an infinite graph), we start out from available probability models on the finite-dimensional counterparts (probability models on finite graphs), and translate them into a model on infinite-dimensional objects using methods of stochastic process theory. We then ask whether interesting statistical properties of the finite-dimensional models used in the constructions, such as conjugacy of priors and posteriors, carry over to the stochastic process model. 1 In general, the term nonparametric Bayesian model refers to a Bayesian model on an infinitedimensional parameter space. Unlike parametric models, for which the number of parameters is constantly bounded w.r.t. sample size, nonparametric models allow the number of parameters to grow with the number of observations. To accommodate a variable and asymptotically unbounded number of parameters within a single parameter space, the dimension of the space has to be infinite, and nonparametric models can be defined as statistical models with infinite-dimensional parameter spaces [17]. For a given sample of finite size, the model will typically select a finite subset of the available parameters to explain the observations. A Bayesian nonparametric model places a prior distribution on the infinite-dimensional parameter space. Many nonparametric Bayesian models are defined in terms of their finite-dimensional marginals: For example, the Gaussian process and Dirichlet process are characterized by the fact that their finite-dimensional marginals are, respectively, Gaussian and Dirichlet distributions [11, 5]. The probability-theoretic construction result underlying such definitions is the Kolmogorov extension theorem [1], described in Sec. 2 below. In stochastic process theory, the theorem is used to study the properties of a process in terms of its marginals, and hence by studying the properties of finitedimensional distributions. Can the statistical properties of a nonparametric Bayesian model, i.e. of a parameterized family of distributions, be treated in a similar manner, by considering the model’s marginals? For example, can a nonparametric Bayesian model be guaranteed to be conjugate if the marginals used in its construction are conjugate? Techniques such as the Kolmogorov theorem construct individual distributions, whereas statistical properties are properties of parameterized families of distributions. In Bayesian estimation, such families take the form of conditional probabilities. The treatment of the statistical properties of nonparametric Bayesian models in terms of finite-dimensional Bayes equations therefore requires an extension result similar to the Kolmogorov theorem that is applicable to conditional distributions. The main contribution of this paper is to provide such a result. We present an analogue of the Kolmogorov theorem for conditional probabilities, which permits the direct construction of conditional stochastic process models on countable-dimensional spaces from finite-dimensional conditional probabilities. Application to conjugate models shows how a conjugate nonparametric Bayesian model can be constructed from conjugate finite-dimensional Bayes equations – including the mapping to the posterior parameters. The converse is also true: To construct a conjugate nonparametric Bayesian model, the finite-dimensional models used in the construction all have to be conjugate. The construction of stochastic process models from exponential family marginals is almost generic: The model is completely described by the mapping to the posterior parameters, which has a generic form as a function of the infinite-dimensional counterpart of the model’s sufficient statistic. We discuss how existing models fit into the framework, and derive the nonparametric Bayesian version of a model on infinite permutations suggested by [9]. By essentially providing a construction recipe for conjugate models of countable dimension, our theoretical results have clear practical implications for the derivation of novel nonparametric Bayesian models. 2 Formal Setup and Notation Infinite-dimensional probability models cannot generally be described with densities and therefore require some basic notions of measure-theoretic probability. In this paper, required concepts will be measures on product spaces and abstract conditional probabilities (see e.g. [3] or [1] for general introductions). Randomness is described by means of an abstract probability space (Ω, A, P). Here, Ωis a space of points ω, which represent atomic random events, A is a σ-algebra of events on Ω, and P a probability measure defined on the σ-algebra. A random variable is a measurable mapping from Ωinto some space of observed values, such as X : Ω→Ωx. The distribution of X is the image measure PX := X(P) = P ◦X−1. Roughly speaking, the events ω ∈Ωrepresent abstract states of nature, i.e. knowing the value of ω completely describes all probabilistic aspects of the model universe, and all random aspects are described by the probability measure P. However, Ω, A and P are never known explicitly, but rather constitute the modeling assumption that any explicitly known distribution PX is derived from one and the same probability measure P through some random variable X. Multiple dimensions of random variables are formalized by product spaces. We will generally deal with an infinite-dimensional space such as ΩE x, were E is an infinite index set and ΩE x is the E2 fold product of Ωx with itself. The set of finite subsets of E will be denoted F(E), such that ΩI x with I ∈F(E) is a finite-dimensional subspace of ΩE x. Each product space ΩI x is equipped with the product Borel σ-algebra BI x. Random variables with values on these spaces have product structure, such as XI = N i∈I X{i}. Note that this does not imply that the corresponding measure P I X := XI(P) is a product measure; the individual components of XI may be dependent. The elements of the infinite-dimensional product space ΩE x can be thought of as functions of the form E →Ωx. For example, the space RR contains all real-valued functions on the line. Product spaces ΩI x ⊂ΩJ x of different dimensions are linked by a projection operator πJI, which restricts a vector xJ ∈ΩJ x to xI, the subset of entries of xJ that are indexed by I ⊂J. For a set AI ⊂ΩI x, the preimage π-1 JI AI under projection is called a cylinder set with base AI. The projection operator can be applied to measures as [πJIP J X] := P J X ◦π-1 JI , so for an I-dimensional event AI ∈BI x, we have [πJIP J X](AI) = P J X(π-1 JI AI). In other words, a probability is assigned to the I-dimensional set AI by applying the J-dimensional measure P J X to the cylinder with base AI. The projection of a measure is just its marginal, that is, [πJIP J X] is the marginal of the measure P J X on the lowerdimensional subspace ΩI x. We denote observation variables (data) by XI, parameters by ΘI and hyperparameters by ΨI. The corresponding measures and spaces are indexed accordingly, as PX, PΘ, Ωθ etc. The likelihoods and posteriors that occur in Bayesian estimation are conditional probability distributions. Since densities are not generally applicable in infinite-dimensional spaces, the formulation of Bayesian models on such spaces draws on the abstract conditional probabilities of measure-theoretic probability, which are derived from Kolmogorov’s implicit formulation of conditional expectations [3]. We will write e.g. PX(X|Θ) for the conditional probability of X given Θ. For the reader familiar with the theory, we note that all spaces considered here are Borel spaces, such that regular versions of conditionals always exist, and we hence assume all conditionals to be regular conditional probabilities (Markov kernels). Introducing abstract conditional probabilities here is far beyond the possible scope of this paper. A reader not familiar with the theory should simply read PX(X|Θ) as a conditional distribution, but take into account that these abstract objects are only uniquely defined almost everywhere. That is, the probability PX(X|Θ = θ) can be changed arbitrarily for those values of θ within some set of exceptions, provided that this set has measure zero. While not essential for understanding most of our results, this fact is the principal reason that limits the results to countable dimensions. Example: GP. Assume that P E X(XE|ΘE) is to represent a Gaussian process with fixed covariance function. Then XE is function-valued, and if for example E := R+ and Ωx := R, the product space ΩE x = RR+ contains all functions xE of the form xE : R+ →R. Each axis label i ∈E in the product space is a point on the real line, and a finite index set I ∈F(E) is a finite collection of points I = (i1, . . . , im). The projection πEIxE of a function in ΩE x is then the vector xI := (xE(i1), . . . , xE(im)) of function values at the points in I. The parameter variable ΘE represents the mean function of the process, and so we would choose ΩE θ := ΩE x = RR+. Example: DP. If P E X(XE|ΘE) is a Dirichlet process, the variable XE takes values xE in the set of probability measures over a given domain, such as R. A probability measure on R (with its Borel algebra B(R)) is in particular a set function B(R) →[0, 1], so we could choose E = B(R) and Ωx = [0, 1]. The parameters of a Dirichlet process DP(α, G0) are a scalar concentration parameter α ∈R+, and a probability measure G0 with the same domain as the randomly drawn measure xE. The parameter space would therefore be chosen as R+ × [0, 1]B(R). 2.1 Construction of Stochastic Processes from their Marginals Suppose that a family P I X of probability measures are the finite-dimensional marginals of an infinitedimensional measure P E X (a “stochastic process”). Each measure P I X lives on the finite-dimensional subspace ΩI x of ΩE x. As marginals of one and the same measure, the measures must be marginals of each other as well: P I X = P J X ◦π-1 JI whenever I ⊂J . (1) Any family of probability measures satisfying (1) is called a projective family. The marginals of a stochastic process measure are always projective. A famous theorem by Kolmogorov states that the converse is also true: Any projective family on the finite-dimensional subspaces of an infinitedimensional product space ΩE x uniquely defines a stochastic process on the space ΩE x [1]. The only assumption required is that the “axes” Ωx of the product space are so-called Polish spaces, i.e. 3 topological spaces that are complete, separable and metrizable. Examples include Euclidean spaces, separable Banach or Hilbert spaces, countable discrete spaces, and countable products of spaces that are themselves Polish. Theorem 1 (Kolmogorov Extension Theorem). Let E be an arbitrary infinite set. Let Ωx be a Polish space, and let {P I X|I ∈F(E)} be a family of probability measures on the spaces (ΩI x, BI x). If the family is projective, there exists a uniquely defined probability measure P E X on ΩE x with the measures P I X as its marginals. The infinite-dimensional measure P E X constructed in Theorem 1 is called the projective limit of the family P I X. Intuitively, the theorem is a regularity result: The marginals determine the values of P E X on a subset of events (namely on those events involving only a finite subset of the random variables, which are just the cylinder sets with finite-dimensional base). The theorem then states that a probability measure is such a regular object that knowledge of these values determines the measure completely, in a similar manner as continuous functions on the line are completely determined by their values on a countable dense subset. The statement of the Kolmogorov theorem is deceptive in its generality: It holds for any index set E, but if E is not countable, the constructed measure P E X is essentially useless – even though the theorem still holds, and the measure is still uniquely defined. The problem is that the measure P E X, as a set function, is not defined on the space ΩE x, but on the σ-algebra BE x (the product σ-algebra on ΩE x). If E is uncountable, this σ-algebra is too coarse to resolve events of interest1. In particular, it does not contain the singletons (one-point sets), such that the measure P E X is incapable of assigning a probability to an event of the form {XE = xE}. 3 Extension of Conditional and Bayesian Models According to the Kolmogorov extension theorem, the properties of a stochastic process can be analyzed by studying its marginals. Can we, analogously, use a set of finite-dimensional Bayes equations to represent a nonparametric Bayesian model? The components of a Bayesian model are conditional distributions. Even though these conditionals are probability measures for (almost) each value of the condition variable, the Kolmogorov theorem cannot simply be applied to extend conditional models: Conditional probabilities are functions of two arguments, and have to satisfy a measurability requirement in the second argument (the condition). Application of the extension theorem to each value of the condition need not yield a proper conditional distribution on the infinite-dimensional space, as it disregards the properties of the second argument. But since the second argument takes the role of a parameter in statistical estimation, these properties determine the statistical properties of the model, such as sufficiency, identifiability, or conjugacy. In order to analyze the properties of an infinite-dimensional Bayesian model in terms of finite-dimensional marginals, we need a theorem that establishes a correspondence between the finite-dimensional and infinite-dimensional conditional distributions. Though a number of extension theorems based on conditional distributions is available in the literature, these results focus on the construction of sequential stochastic processes from a sequence of conditionals (see [10] for an overview). Theorem 2 below provides a result that, like the Kolmogorov theorem, is applicable on product spaces. To formulate the result, the projector used to define the marginals has to be generalized from measures to conditionals. The natural way to do so is the following: If P J X(XJ|ΘJ) is a conditional probability on the product space ΩJ, and I ⊂J, define [πJIP J X]( . |ΘJ) := P J X(π-1 JI . |ΘJ) . (2) This definition is consistent with that of the projector above, in the sense that it coincides with the standard projector applied to the measure P J X( . |ΘJ = θJ) for any fixed value θJ of the parameter. As with projective families of measures, we then define projective families of conditional probabilities. Definition 1 (Conditionally Projective Probability Models). Let P I X(XI|ΘI) be a family of regular conditional probabilities on product spaces ΩI x, for all I ∈F(E). The family will be called conditionally projective if [πJIP J X]( . |ΘJ) =a.e. P I X( . |ΘI) whenever I ⊂J. As conditional probabilities are unique almost everywhere, the equality is only required to hold almost everywhere as well. In the jargon of abstract conditional probabilities, the definition requires 1This problem is unfortunately often neglected in the statistics literature, and measures in uncountable dimensions are “constructed” by means of the extension theorem (such as in the original paper [5] on the Dirichlet process). See e.g. [1] for theoretical background, and [7] for a rigorous construction of the DP. 4 that P I X( . |ΘI) is a version of the projection of P J X( . |ΘJ). Theorem 2 states that a conditional probability on a countably-dimensional product space is uniquely defined (up to a.e.-equivalence) by a conditionally projective family of marginals. In particular, if we can define a parametric model on each finite-dimensional space ΩI x for I ∈F(E) such that these models are conditionally projective, the models determine an infinite-dimensional parametric model (a “nonparametric” model) on the overall space ΩE x. Theorem 2 (Extension of Conditional Probabilities). Let E be a countable index set. Let P I X(XI|ΘI) be a family of regular conditional probabilities on the product space ΩI x. Then if the family is conditionally projective, there exists a regular conditional probability P E X (XE|CE) on the infinitedimensional space ΩE x with the P I X(XI|ΘI) as its conditional marginals. P E X (XE|CE) is measurable with respect to the σ-algebra CE := σ(∪I∈F(E)σ(ΘI)). In particular, if the parameter variables satisfy πJIΘJ = ΘI, then P E X (XE|CE) can be interpreted as the conditional probability P E X (XE|ΘE) with ΘE := N i∈E Θ{i}. Proof Sketch2. We first apply the Kolmogorov theorem separately for each setting of the parameters that makes the measures P I X(XI|ΘI = θI) projective. For any given ω ∈Ω(the abstract probability space), projectiveness holds if θI = ΘI(ω) for all I ∈F(E). However, for any conditionally projective family, there is a set N ⊂Ωof possible exceptions (for which projectiveness need not hold), due to the fact that conditional probabilities and conditional projections are only unique almost everywhere. Using the countability of the dimension set E, we can argue that N is always a null set; the resulting set of constructed infinite-dimensional measures is still a valid candidate for a regular conditional probability. We then show that if this set of measures is assembled into a function of the parameter, it satisfies the measurability conditions of a regular conditional probability: We first use the properties of the marginals to show measurability on the subset of events which are preimages under projection of finite-dimensional events (the cylinder sets), and then use the π-λ theorem [3] to extend measurability to all events. 4 Conjugacy The posterior of a Dirichlet process is again a Dirichlet process, and the posterior parameters can be computed as a function of the data and the prior parameters. This property is known as conjugacy, in analogy to conjugacy in parametric Bayesian models, and makes Dirichlet process inference tractable. Virtually all known nonparametric Bayesian models, including Gaussian processes, P´olya trees, and neutral-to-the-right processes are conjugate [16]. In the Bayesian and exponential family literature, conjugacy is often defined as “closure under sampling”, i.e. for a given likelihood and a given class of priors, the posterior is again an element of the prior class [12]. This definition does not imply tractability of the posterior: In particular, the set of all probability measures (used as priors) is conjugate for any possible likelihood, but obviously this does not facilitate computation of the posterior. In the following, we call a prior and a likelihood of a Bayesian model conjugate if the posterior (i) is parameterized and (ii) there is a measurable mapping T from the data x and the prior parameter ψ to the parameter ψ′ = T(x, ψ) which specifies the corresponding posterior. In the definition below, the conditional probability k represents the parametric form of the posterior. The definition is applicable to “nonparametric” models, in which case the parameter simply becomes infinite-dimensional. Definition 2 (Conjugacy and Posterior Index). Let PX(X|Θ) and PΘ(Θ|Ψ) be regular conditional probabilities. Let PΘ(Θ|X, Ψ) be the posterior of the model PX(X|Θ) under prior PΘ(Θ|Ψ). Model and prior are called conjugate if there exists a regular conditional probability k : Bθ × Ωt →[0, 1], parameterized on a measurable Polish space (Ωt, Bt), and a measurable map T : Ωx × Ωψ →Ωt, such that PΘ(A|X = x, Ψ = ψ) = k(A, T(x, ψ)) for all A ∈Bθ . (3) The mapping T is called the posterior index of the model. The definition becomes trivial for Ωt = Ωx × Ωψ and T chosen as the identity mapping; it is meaningful if T is reasonably simple to evaluate, and its complexity does not increase with sample size. Theorem 3 below shows that, under suitable conditions, the structure of the posterior index carries 2Complete proofs for both theorems in this paper are provided as supplementary material. 5 over to the projective limit model: If the finite-dimensional marginals admit a tractable posterior index, then so does the projective limit model. Example. (Posterior Indices in Exponential Families) Suppose that PX(X|Θ) is an exponential family model with sufficient statistic S and density p(x|θ) = exp(⟨S(x), θ⟩−γ(x)−φ(θ)). Choose PΘ(Θ|Ψ) as the “natural conjugate prior” with parameters ψ = (α, y). Its density, w.r.t. a suitable measure νΘ on parameter space, is of the form q(θ|α, y) = K(α, y)−1 exp(⟨θ, y⟩−αφ(θ)). The posterior PΘ(Θ|X, Ψ) is conjugate in the sense of Def. 2, and its density is q(θ|α + 1, y + S(x)). The probability kernel k is given by k(A, (t1, t2)) := R A q(θ|t1, t2)dνΘ(θ), and the posterior index is T(x, (α, y)) := (α + 1, y + S(x)). The main result of this section is Theorem 3, which explains how conjugacy carries over from the finite-dimensional to the infinite-dimensional case, and vice versa. Both extension theorems discussed so far require a projection condition on the measures and models involved. A similar condition is now required for the mappings T I: The preimages T I,-1 of the posterior indices T I must commute with the preimage under projection, (πEI ◦T E)-1 = (T I ◦πEI)-1 for all I ∈F(E) . (4) The posterior indices of all well-known exponential family models, such as Gaussians and Dirichlets, satisfy this condition. The following theorem states that (i) stochastic process Bayesian models that are constructed from conjugate marginals are conjugate if the projection equation (4) is satisfied, and that (ii) such conjugate models can only be constructed from conjugate marginals. Theorem 3 (Functional Conjugacy of Projective Limit Models). Let E be a countable index set and ΩE x and ΩE θ be Polish product spaces. Assume that there is a Bayesian model on each finitedimensional subspace ΩI x, such that the families of all priors, all observation models and all posteriors are conditionally projective. Let P E Θ(ΘE), P E X (XE|ΘE) and P E Θ(ΘE|XE) denote the respective projective limits. Then P E Θ(ΘE|XE) is a posterior for the infinite-dimensional Bayesian model defined by P E X (XE|ΘE) with prior P E Θ(ΘE), and the following holds: (i) Assume that each finite-dimensional posterior P I Θ(ΘI|XI) is conjugate w.r.t. its respective Bayesian model, with posterior index T I and probability kernel kI. Then if there is a measurable mapping T : ΩE x →ΩE t satisfying the projection condition (4), the projective limit posterior P E Θ(ΘE|XE) is conjugate with posterior index T. (ii) Conversely, if the infinite-dimensional posterior P E Θ(ΘE|XE) is conjugate with posterior index T E and probability kernel kE, then each marginal posterior P I Θ(ΘI|XI) is conjugate, with posterior index T I := πEI ◦T E ◦π-1 EI. The corresponding probability kernels kI are given by kI(AI, tI) := kE(π-1 EIAI, t) for any t ∈π-1 EItI . (5) The theorem is not stated here in full generality, but under two simplifying assumptions: We have omitted the use of hyperparameters, such that the posterior indices depend only on the data, and all involved spaces (observation space, parameter space etc) are assumed to have the same dimension for each Bayesian model. Generalizing the theorem beyond both assumptions is technically not difficult, but the additional parameters and notation for book-keeping on dimensions reduce readability. Proof Sketch2. Part (i): We define a candidate for the probability kernel kE representing the projective limit posterior, and then verify that it makes the model conjugate when combined with the mapping T given by assumption. To do so, we first construct the conditional probabilities P I Θ(ΘI|T I), show that they form a conditionally projective family, and take their conditional projective limit using Theorem 2. This projective limit is used as a candidate for kE. To show that kE indeed represents the posterior, we show that the two coincide on the cylinder sets (events which are preimages under projection of finite-dimensional events). From this, equality for all events follows by the Caratheodory theorem [1]. Part (ii): We only have to verify that the mappings T I and probability kernels kI indeed satisfy the definition of conjugacy, which is a straightforward computation. 5 Construction of Nonparametric Bayesian Models Theorem 3(ii) states that conjugate models have conjugate marginals. Since, in the finitedimensional case, conjugate Bayesian models are essentially limited to exponential families and 6 their natural conjugate priors3, a consequence of the theorem is that we can only expect a nonparametric Bayesian model to be conjugate if it is constructed from exponential family marginals – assuming that the construction is based on a product space approach. When an exponential family model and its conjugate prior are used in the construction, the form of the resulting model becomes generic: The posterior index T of a conjugate exponential family Bayesian model is always given by the sufficient statistic S in the form T(x, (α, y)) := (α + 1, y + S(x)). Addition commutes with projection, and hence the posterior indices T I of a family of such models over all dimensions I ∈F(E) satisfy the projection condition (4) if and only if the same condition is satisfied by the sufficient statistics SI of the marginals. Accordingly, the infinite-dimensional posterior index T E in Theorem 3 exists if and only if there is an infinite-dimensional “extension” SE of the sufficient statistics SI satisfying (4). If that is the case, T E(xE, (α, yE)) := (α + 1, yE + SE(xE)) is a posterior index for the infinite-dimensional projective limit model. In the case of countable dimensions, Theorem 3 therefore implies a construction recipe for nonparametric Bayesian models from exponential family marginals; constructing the model boils down to checking whether the models selected as finite-dimensional marginals are conditionally projective, and whether the sufficient statistics satisfy the projection condition. An example construction, for a model on infinite permutations, is given in below. The following table summarizes some stochastic process models from the conjugate extension point of view: Marginals (d-dim) Projective limit model Observations (limit) Bernoulli/Beta Beta process; IBP Binary arrays Multin./Dirichlet DP; CRP Discrete distributions Gaussian/Gaussian GP/GP (continuous) functions Mallows/conjugate Example below Bijections N →N A Construction Example. The analysis of preference data, in which preferences are represented as permutations, has motivated the definition of distributions on permutations of an infinite number of items [9]. A finite permutation on r items always implies a question such as “rank your favorite movies out of r movies”. A nonparametric approach can generalize the question to “rank your favorite movies”. Meila and Bao [9] derived a model on infinite permutations, that is, on bijections of the set N. We construct a nonparametric Bayesian model on bijections, with a likelihood component P E X(XE|ΘE) equivalent to the model of Meila and Bao. Choice of marginals. The finite-dimensional marginals are probability models of rankings of a finite number of items, introduced by Fligner and Verducci [6]. For permutations τ ∈Sr of length r, the model is defined by the exponential family density p(τ|σ, θ) := Z(θ)−1 exp( S(τσ−1), θ ), where the sufficient statistic is the vector Sr(τ) := (S1(τ), . . . , Sr(τ)) with components Sj(τ) := Pr l=j+1 I{τ −1(j) > τ −1(l)}. Roughly speaking, the model is a location-scale model, and the permutation σ defines the distribution’s mean. If all entries of θ are chosen identical as some constant, this constant acts as a concentration parameter, and the scalar product is equivalent to the Kendall metric on permutations. This metric measures distance between permutations as the minimum number of adjacent transpositions (i.e. swaps of neighboring entries) required to transform one permutation into the other. If the entries of θ differ, they can be regarded as weights specifying the relevance of each position in the ranking [6]. Definition of marginals. In the product space context, each finite set I ∈F(E) of axis labels is a set of items to be permuted, and the marginal P I Θ(τ I|σI, θI) is a model on the corresponding finite permutation group SI on the elements of I. The sufficient statistics SI maps each permutation to a vector of integers, and thus embeds the group SI into RI. The mapping is one-to-one [6]. Projections, i.e. restrictions, on the group mean deletion of elements. A permutation τ J is restricted to a subset I ⊂J of indices by deleting all items indexed by J \ I, producing the restriction τ J|I. We overload notation and write πJI for both the restriction in the group SI and axes-parallel projection in the Euclidean space RI, into which the sufficient statistic SI embeds SI. It follows from the definition of SI that, whenever πJIτ J = τ I, then πJISJ(τ J) = SI(τ I). In other words, πJI ◦SJ = SI ◦πJI, which is a stronger form of the projection condition SJ,-1 ◦π-1 JI = π-1 JI ◦SI,-1 given in Eq. 4. We will define a nonparametric Bayesian model that puts a prior on the infinite-dimensional analogue 3Mixtures of conjugate priors are conjugate in the sense of closure under sampling [4], but the posterior index in Def. 2 has to be evaluated for each mixture component individually. An example of a conjugate model not in the exponential family is the uniform distribution on [0, θ] with a Pareto prior [12]. 7 of θ, i.e. on the weight function θE. For I ∈F(N), the marginal of the likelihood component is given by the density pI(τ I|σI, θI) := ZI(θI)−1 exp( SI(τ I(σI)−1), θI ). The corresponding natural conjugate prior on θI has density qI(θI|α, yI) ∝exp(⟨θI, yI⟩−α log ZI(θI)). Since the model is an exponential family model, the posterior index is of the form T I((α, yI), τ I) = (α + 1, yI + SI(τ I)), and since SI is projective in the sense of Eq. 4, so is T I. The prior and likelihood densities above define two families P I(XI|ΘI) and P I(ΘI|Ψ) of measures over all finite dimensions I ∈F(E). It is reasonably straightforward to show that both families are conditionally projective, and so is the family of the corresponding posteriors. Each therefore has a projective limit, and the projective limit of the posteriors is the posterior of the projective limit P E(XE|ΘE) under prior P E(ΘE). Posterior index. The posterior index of the infinite-dimensional model can be derived by means of Theorem 3: To get rid of the hyperparameters, we first fix a value ψE := (α, yE) of the infinite-dimensional hyperparameter, and only consider the corresponding infinite-dimensional prior P E Θ(ΘE|ΨE = ψE), with its marginals P I Θ(ΘI|ΨI = πEIψE). Now define a function SE on the bijections of N as follows. For each bijection τ : N →N, and each j ∈N, set SE j(τ) := P∞ l=j+1 I{τ −1(j) > τ −1(l)}. Since τ −1(j) is a finite number for any j ∈N, the indicator function is non-zero only for a finite number of indices l, such that the entries of SE are always finite. Then SE satisfies the projection condition SE,-1 ◦π-1 EI = π-1 EISI,-1 for all I ∈F(E). As candidate posterior index, we define the function T E((α, yE), τ E) = (α + 1, yE + SE(τ E)) for yE ∈ΩN θ . Then T E also satisfies the projection condition (4) for any I ∈F(E). By Theorem 3, this makes T E a posterior index for the projective limit model. 6 Discussion and Conclusion We have shown how nonparametric Bayesian models can be constructed from finite-dimensional Bayes equations, and how conjugacy properties of the finite-dimensional models carry over to the infinite-dimensional, nonparametric case. We also have argued that conjugate nonparametric Bayesian models arise from exponential families. A number of interesting questions could not be addressed within the scope of this paper, including (1) the extension to model properties other than conjugacy and (2) the generalization to uncountable dimensions. For example, a model property which is closely related to conjugacy is sufficiency [14]. In this case, we would ask whether the existence of sufficient statistics for the finite-dimensional marginals implies the existence of a sufficient statistic for the nonparametric Bayesian model, and whether the infinite-dimensional sufficient statistic can be explicitly constructed. Second, the results presented here are restricted to the case of countable dimensions. This restriction is inconvenient, since the natural product space representations of, for example, Gaussian and Dirichlet processes on the real line have uncountable dimensions. The GP (on continuous functions) and the DP are within the scope of our results, as both can be derived by means of countable-dimensional surrogate constructions: Since continuous functions on R are completely determined by their values on Q, a GP can be constructed on the countable-dimensional product space RQ. Analogous constructions have been proposed for the DP [7]. The drawback of this approach is that the actual random draw is just a partial version of the object of interest, and formally has to be completed e.g. into a continuous function or a probability measure after it is sampled. On the other hand, uncountable product space constructions are subject to all the subtleties of stochastic process theory, many of which do not occur in countable dimensions. The application of construction methods to conditional probabilities also becomes more complicated (roughly speaking, the point-wise application of the Kolmogorov theorem in the proof of Theorem 2 is not possible if the dimension is uncountable). Product space constructions are by far not the only way to define nonparametric Bayesian models. A P´olya tree model [7], for example, is much more intuitive to construct by means of a binary partition argument than from marginals in product space. As far as characterization results, such as which models can be conjugate, are concerned, our results are still applicable, since the set of Poly´a trees can be embedded into a product space. However, the marginals may then not be the marginals in terms of which we “naturally” think about the model. Nonetheless, we have hopefully demonstrated that the theoretical results are applicable for the construction of an interesting and practical range of nonparametric Bayesian models. Acknowledgments. I am grateful to Joachim M. Buhmann, Zoubin Ghaharamani, Finale DoshiVelez and the reviewers for helpful comments. This work was in part supported by EPSRC grant EP/F028628/1. 8 References [1] H. Bauer. Probability Theory. W. de Gruyter, 1996. [2] M. J. Beal, Z. Ghahramani, and C. E. Rasmussen. The infinite hidden Markov model. In Advances in Neural Information Processing Systems, 2001. [3] P. Billingsley. Probability and measure, 1995. [4] S. R. Dalal and W. J. Hall. Approximating priors by mixtures of natural conjugate priors. Annals of Statistics, 45(2):278–286, 1983. [5] T. S. Ferguson. A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1(2), 1973. [6] M. A. Fligner and J. S. Verducci. Distance based ranking models. Journal of the Royal Statistical Society B, 48(3):359–369, 1986. [7] J. K. Ghosh and R. V. Ramamoorthi. Bayesian Nonparametrics. Springer, 2002. [8] T. L. Griffiths and Z. Ghahramani. Infinite latent feature models and the Indian buffet process. In Advances in Neural Information Processing Systems, 2005. [9] M. Meil˘a and L. Bao. Estimation and clustering with infinite rankings. In Uncertainty in Artificial Intelligence, 2008. [10] M. M. Rao. Conditional Measures and Applications. Chapman & Hall, second edition, 2005. [11] C. E. Rasmussen and C. K. I. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [12] C. P. Robert. The Bayesian Choice. Springer, 1994. [13] D. M. Roy and Y. W. Teh. The Mondrian process. In Advances in Neural Information Processing Systems, 2009. [14] M. J. Schervish. Theory of Statistics. Springer, 1995. [15] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Journal of the American Statistical Association, (476):1566–1581, 2006. [16] S. G. Walker, P. Damien, P. W. Laud, and A. F. M. Smith. Bayesian nonparametric inference for random distributions and related functions. Journal of the Royal Statistical Society B, 61(3):485–527, 1999. [17] L. Wasserman. All of Nonparametric Statistics. Springer, 2006. 9
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Adaptive Regularization for Transductive Support Vector Machine Zenglin Xu †‡ † Cluster MMCI Saarland Univ. & MPI INF Saarbrucken, Germany zlxu@mpi-inf.mpg.de Rong Jin Computer Sci. & Eng. Michigan State Univ. East Lansing, MI, U.S. rongjin@cse.msu.edu Jianke Zhu Computer Vision Lab ETH Zurich Zurich, Switzerland zhuji@vision.ee.ethz.ch Irwin King‡ Michael R. Lyu‡ ‡ Computer Science & Engineering The Chinese Univ. of Hong Kong Shatin, N.T., Hong Kong {king,lyu}@cse.cuhk.edu.hk Zhirong Yang Information & Computer Science Helsinki Univ. of Technology Espoo, Finland zhirong.yang@tkk.fi Abstract We discuss the framework of Transductive Support Vector Machine (TSVM) from the perspective of the regularization strength induced by the unlabeled data. In this framework, SVM and TSVM can be regarded as a learning machine without regularization and one with full regularization from the unlabeled data, respectively. Therefore, to supplement this framework of the regularization strength, it is necessary to introduce data-dependant partial regularization. To this end, we reformulate TSVM into a form with controllable regularization strength, which includes SVM and TSVM as special cases. Furthermore, we introduce a method of adaptive regularization that is data dependant and is based on the smoothness assumption. Experiments on a set of benchmark data sets indicate the promising results of the proposed work compared with state-of-the-art TSVM algorithms. 1 Introduction Semi-supervised learning has attracted a lot of research focus in recently years. Most of the existing approaches can be roughly divided into two categories: (1) the clustering-based methods [12, 4, 8, 17] assume that most of the data, including both the labeled ones and the unlabeled ones, should be far away from the decision boundary of the target classes; (2) the manifold-based methods make the assumption that most of data lie on a low-dimensional manifold in the input space, which include Label Propagation [21], Graph Cuts [2], Spectral Kernels [9, 22], Spectral Graph Transducer [11], and Manifold Regularization [1]. The comprehensive study on semi-supervised learning techniques can be found in the recent surveys [23, 3]. Although semi-supervised learning wins success in many real-world applications, there still remains two major unsolved challenges. One is whether the unlabeled data can help the classification, and the other is what is the relation between the clustering assumption and the manifold assumption. As for the first challenge, Singh et al. [16] provided a finite sample analysis on the usefulness of unlabeled data based on the cluster assumption. They show that unlabeled data may be useful for improving the error bounds of supervised learning methods when the margin between different classes satisfies some conditions. However, in the real-world problems, it is hard to identify the conditions that unlabeled data can help. On the other hand, it is interesting to explore the relation between the low density assumption and the manifold assumption. Narayanan et al. [14] implied that the cut-size of the graph partition converges to the weighted volume of the boundary which separates the two regions of the domain for a fixed partition. This makes a step forward for exploring the connection between graph-based partitioning and the idea surrounding the low density assumption. Unfortunately, this approach cannot be generalized uniformly over all partitions. Lafferty and Wasserman [13] revisited the assumptions of semi-supervised learning from the perspective of minimax theory, and suggested that the manifold assumption is stronger than the smoothness assumption for regression. Till now, the underlying relationships between the cluster assumption and the manifold assumption are still undisclosed. Specifically, it is unclear that in what kind of situation the clustering assumption or the manifold assumption should be adopted. In this paper, we address these current limitations by a unified solution from the perspective of the regularization strength of the unlabeled data. Taking Transductive Support Vector Machine (TSVM) as an example, we suggest an framework that introduces the regularization strength of the unlabeled data when estimating the decision boundary. Therefore, we can obtain a spectrum of models by varying the regularization strength of unlabeled data which corresponds to changing the models from supervised SVM to Transductive SVM. To select the optimal model under the proposed framework, we employ the manifold regularization assumption that enables the prediction function to be smooth over the data space. Further, the optimal function is a linear combination of supervised models, weakly semi-supervised models, and semi-supervised models. Additionally, it provides an effective approach towards combining the cluster assumption and the manifold assumption in semi-supervised learning. The rest of this paper is organized as follows. In Section 2, we review the background of Transductive SVM. In Section 3, we first present a framework of models with different regularization strength, followed by an integrating approach based on manifold regularization. In Section 4, we report the experimental results on a series of benchmark data sets. Section 5 concludes the paper. 2 Related Work on TSVM Before presenting the formulation of TSVM, we first describe the notations used in this paper. Let X = (x1, . . . , xn) denote the entire data set, including both the labeled examples and the unlabeled ones. We assume that the first l examples within X are labeled and the next n −l examples are unlabeled. We denote the unknown labels by yu = (yu l+1, . . . , yu n). TSVM [12] maximizes the margin in the presence of unlabeled data and keeps the boundary traversing through low density regions while respecting labels in the input space. Under the maximum-margin framework, TSVM aims to find the classification model with the maximum classification margin for both labeled and unlabeled examples, which amounts to solve the following optimization problem: min w∈Rn,yu∈Rn−ℓ,ξ∈Rn 1 2∥w∥K + C l X i=1 ξi + C∗ n X i=l+1 ξi (1) s. t. yiw⊤φ(xi) ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, yu i w⊤φ(xi) ≥1 −ξi, ξi ≥0, l + 1 ≤i ≤n, where C and C∗are the trade-offparameters between the complexity of the function w and the margin errors. Moreover, the prediction function can be formulated as f(x) = w⊤φ(x). Note that we remove the bias term in the above formulation, since it can be taken into account by introducing a constant element into the input pattern alternatively. As in [19] and [20], we can rewrite (1) into the following optimization problem: min f,ξ 1 2f ⊤K−1f + C l X i=1 ξi + C∗ n X i=l+1 ξi (2) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, |fi| ≥1 −ξi, ξi ≥0, l + 1 ≤i ≤n. The optimization problem held in TSVM is a non-linear non-convex optimization [6]. During past several years, researchers have devoted a significant amount of research efforts to solving this critical problem. A branch-and-bound method [5] was developed to search for the optimal solution, which is only limited to solve the problem with a small number of examples due to involving the heavy computational cost. To apply TSVM for large-scale problems, Joachims [12] proposed a label-switching-retraining procedure to speed up the optimization procedure. Later, the hinge loss in TSVM is replaced by a smooth loss function, and a gradient descent method is used to find the decision boundary in a region of low density [4]. In addition, there are some iterative methods, such as deterministic annealing [15], concaveconvex procedure (CCCP) [8], and convex relaxation method [19, 18]. Despite the success of TSVM, the unlabeled data not necessarily improve classification accuracy. To better utilize the unlabeled data, unlike existing TSVM approaches, we propose a framework that tries to control the regularization strength of the unlabeled data. To do this, we intend to learn the optimal regularization strength configuration from the combination of a spectrum of models: supervised, weakly-supervised, and semi-supervised. 3 TSVM: A Regularization View For the sake of illustration, we first study a model that does not penalize on the classification errors of unlabeled data. Note that the penalization on the margin errors of unlabeled data can be included if needed. Therefore, we have the following form of TSVM that can be derived through the duality: min f,ξ 1 2f ⊤K−1f + C l X i=1 ξi (3) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, f 2 i ≥1, l + 1 ≤i ≤n. 3.1 Full Regularization of Unlabeled Data In order to adjust the strength of the regularization raised from the unlabeled examples, we introduce a coefficient ρ ≥0, and modify the above problem (3) as below: min f,ξ 1 2f ⊤K−1f + C l X i=1 ξi (4) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, f 2 i ≥ρ, l + 1 ≤i ≤n. Obviously, it is the standard TSVM for ρ = 1. In particular, the larger the ρ is, the stronger the regularization of unlabeled data is. It is also important to note that we only take into account the classification errors on the labeled examples in the above equation. Namely, we only denote ξi for each labeled example. Further, we write f = (fl; fu) where fl = (f1, . . . , fl) and fu = (fl+1, . . . , fn) represent the prediction for the labeled and the unlabeled examples, respectively. According to the inverse lemma of the block matrix, we can write K−1 as follows: K−1 = M−1 l −K−1 l,l Kl,uM−1 u −M−1 u Ku,lK−1 l,l M−1 u , where Ml = Kl,l −Kl,uK−1 u,uKu,l, Mu = Ku,u −Ku,lK−1 l,l Kl,u. Thus, the term f ⊤K−1f is computed as f ⊤K−1f = f ⊤ l M−1 l fl + f ⊤ u M−1 u fu −2f ⊤ l K−1 l,l Kl,uM−1 u fu. When the unlabeled data are loosely correlated to the labeled data, namely when most of the elements within Ku,l are small, this leads to Mu ≈Ku. We refer to this case as “weakly unsupervised learning”. Using the above equations, we rewrite TSVM as follows: min fl,fu,ξ 1 2f ⊤ l M−1 l fl + C l X i=1 ξi + ω(fl, ρ) (5) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, where ω(fl, ρ) is a regularization function for fl and it is the result of the following optimization problem: min fu 1 2f ⊤ u M−1 u fu −f ⊤ l K−1 l,l Kl,uM−1 u fu (6) s. t. [f u i ]2 ≥ρ, l + 1 ≤i ≤n. To understand the regularization function ω(fl, ρ), we first compute the dual of the problem (6) by the Lagrangian function: L = 1 2f ⊤ u M−1 u fu −f ⊤ l K−1 l,l Kl,uM−1 u fu − nu X i=1 1 2λi([f u i ]2 −ρ) = 1 2f ⊤ u (M−1 u −D(λ))fu −f ⊤ l K−1 l,l Kl,uM−1 u fu + ρ 2λ⊤e, where D(λ) = diag(λ1, . . . , λn−l) and e denotes a vector with all elements being one. As the derivatives vanish for optimality, we have fu = (M−1 u −D(λ))−1M−1 u Ku,lK−1 l,l fl = (I −MuD(λ))−1Ku,lK−1 l,l fl, where I is an identity matrix. Replacing fu in (6) with the above equation, we have the following dual problem: max λ −1 2f ⊤ l K−1 l,l Kl,u(Mu −MuD(λ)Mu)−1Ku,lK−1 l,l fl + ρλ⊤e (7) s. t. M−1 u ⪰D(λ), λi ≥0, i = 1, . . . , n −l. The above formulation allows us to understand how the parameter ρ controls the strength of regularization from the unlabeled data. In the following, we will show that a series of learning models can be derived through assigning various values for the coefficient ρ. 3.2 No Regularization from Unlabeled Data First, we study the case of ρ = 0. We have the following theorem to illustrate the relationship between the dual problem (7) and the supervised SVM. Theorem 1 When ρ = 0, the optimization problem is reduced to the standard supervised SVM. Proof 1 It is not difficult to see that the optimal solution to (7) is λ = 0. As a result, ω(fl, ρ) becomes ω(fl, ρ = 0) = −1 2flK−1 l,l Kl,uM−1 u Ku,lK−1 l,l fl Substituting ω(fl, ρ) in (5) with the formulation above, the overall optimization problem becomes min fl,ξ 1 2f ⊤ l (M−1 l −K−1 l,l Kl,uM−1 u Ku,lK−1 l,l )fl + C l X i=1 ξi s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l. According to the matrix inverse lemma, we calculate M−1 l as below: M−1 l = (Kl,l −Kl,uK−1 u,uKu,l)−1 = K−1 l,l + K−1 l,l Kl,u(Ku,u −Ku,lK−1 l,l Kl,u)−1Ku,lK−1 l,l = K−1 l,l + K−1 l,l Kl,uM−1 u Ku,lK−1 l,l . Finally, the optimization problem is simplified as min fl,ξ 1 2f ⊤ l K−1 l,l fl + C l X i=1 ξi (8) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l. Clearly, the above optimization is identical to the standard supervised SVM. Hence, the unlabeled data are not employed to regularize the decision boundary when ρ = 0. 3.3 Partial Regularization of Unlabeled Data Second, we consider the case when ρ is small. According to (7), we expect λ to be small when ρ is small. As a result, we can approximate (Mu −MuD(λ)Mu)−1 as follows: (Mu −MuD(λ)Mu)−1 ≈M−1 u + D(λ). Consequently, we can write ω(fl, ρ) as follows: ω(fl, ρ) = −1 2f ⊤ l K−1 l,l Kl,uM−1 u Ku,lK−1 l,l fl + φ(fl, ρ), (9) where φ(fl, ρ) is the output of the following optimization problem max λ ρλ⊤e −1 2f ⊤ l K−1 l,l Kl,uD(λ)Ku,lK−1 l,l fl s. t. M−1 u ⪰D(λ), λi ≥0, i = 1, . . . , n −l. We can simplify the above problem by approximating M−1 u ⪰D(λ) as λi ≤[σ1(Mu)]−1, i = 1, . . . , n −l, where σ1(Mu) represents the maximum eigenvalue of matrix Mu. The resulting simplified problem becomes max λ ρ 2λ⊤e −1 2f ⊤ l K−1 l,l Kl,uD(λ)Ku,lK−1 l,l fl s. t. 0 ≤λi ≤[σ1(Mu)]−1, 1 ≤i ≤n −l. As the above problem is a linear programming problem, the solution for λ can be computed as: λi = 0 [Ku,lK−1 l,l fl]2 i > ρ, σ(Mu)−1 [Ku,lK−1 l,l fl]2 i ≤ρ. From the above formulation, we find that ρ plays the role of a threshold of selecting the unlabeled examples. Since [Ku,lK−1 l,l fl]i can be regarded as the approximation for the ith unlabeled example, the above formulation can be interpreted in the way that only the unlabeled examples with low prediction confidence will be selected for regularizing the decision boundary. Moreover, all the unlabeled examples with high prediction confidence will be ignored. From the above discussions, we can conclude that ρ determines the regularization strength of unlabeled examples. Then, we rewrite the overall optimization problem as below: min fl,ξ max λ 1 2f ⊤ l K−1 l,l fl + C l X i=1 ξi −1 2f ⊤ l K−1 l,l Kl,uD(λ)Ku,lK−1 l,l fl (10) s. t. yifi ≥1 −ξi, ξi ≥0, 1 ≤i ≤l, 0 ≤λi ≤[σ1(Mu)]−1, 1 ≤i ≤n −l. This is a min-max optimization problem and thus the global optimal solution can be guaranteed. To obtain the optimal solution, we employ an alternating optimization procedure, which iteratively computes the values of fl and λ. To account for the penalty on the margin error from the unlabeled data, we just need to add an extra constraint of λi ≤2C for i = 1, . . . , n −l. By varying the parameter ρ from 0 to 1, we can indeed obtain a series of transductive models for SVM. When ρ is small, we call the corresponding optimization problem as weakly semisupervised learning. Therefore, it is important to find an appropriate ρ which adapts for the input data. However, as the data distribution is usually unknown, it is very challenging to directly estimate an optimal regularization strength parameter ρ. Instead, we try to explore an alternative approach to select an appropriate ρ by combining the prediction functions. Due to the large cost in calculating the inverse of kernel matrices, one can solve the dual problems according to the Representer theorem. 3.4 Adaptive Regularization As stated in previous sections, ρ determines the regularization strength of the unlabeled data. We now try to adapt the parameter ρ according to the unlabeled data information. Specifically, we intend to implicitly select the best ρ from a given list, i.e., Υ = {ρ1, . . . , ρm} where ρ1 = 0 and ρm = 1. This is equivalent to selecting the optimal f from a list of prediction functions, i.e., F = {f1, . . . , fm}. Motivated from the ensemble technique for semi-supervised learning [7], we assume that the optimal f comes from a linear combination of the base functions {fi}. We then have: f = m X i=1 θifi, m X i=1 θi = 1, θi ≥0, i = 1, . . . , m. where θi is the weight of the prediction function fi and θ ∈Rm. One can also involve a priori to θi. For example, if we have more confidences on the semi-supervised classifier, we can introduce a constraint like θm ≥0.5. It is important to note that the learning functions in ensemble methods [7] are usually weak learners, while in our approach, the learning functions are strong learners with different degrees of regularization. In the following, we study how to set the regularization strength adaptive to data. Since TSVM naturally follows the cluster assumption of semi-supervised learning, in order to complement the cluster assumption, we adopt another principle in semi-supervised learning, i.e., the manifold assumption. From the point of view of manifold assumption in semisupervised learning, the prediction function f should be smooth on unlabeled data. To this end, the approach of manifold regularization is widely adopted as a smoothing term in semi-supervised learning literatures, e.g., [1, 10]. In the following, we will employ the manifold regularization principle for selecting the regularization strength. The manifold regularization is mainly based on a graph G =< V, E > derived from the whole data space X, where V = {xi}n i=1 is the vertex set, and E denotes the edges linking pairs of nodes. In general, a graph is built in the following four steps: (1) constructing adjacency graph; (2) calculating the weights on edges; (3) computing the adjacency matrix W; (4) obtaining the graph Laplacian by L = diag(Pn j=1 Wij) −W. Then, we denote the manifold regularization term as f ⊤Lf. For simplicity, we denote the predicted values of function fi on the data X as fi, such that fi = ([fi]1, . . . , [fi]n). F = (f1, . . . , fm)⊤is used to represent the set of the prediction values of all prediction functions. Finally, We have the following minimization problem: min θ 1 2η(θ⊤F)L(F⊤θ) −y⊤ ℓ(F⊤ ℓθ) (11) s. t. θ⊤e = 1, θi ≥0, i = 1, . . . , m, where the second term, y⊤ ℓ(F⊤ ℓθ), is used to strengthen the confidence on the prediction over the labeled data. η is a trade-offparameter. The above optimization problem is a simple quadratic programming problem, which can be solved very efficiently. It is important to note that the above optimization problem is less sensitive to the graph structure than Laplacian SVM as used in [1], since the basic learning functions are all strong learners. It also saves a huge amount of efforts in estimating the parameters compared with Laplacian SVM. The above approach indeed provides a practical approach towards a combination of both the cluster assumption and the manifold assumption. It is empirically suggested that combining these two assumptions helps to improve the prediction accuracy of semi-supervised learning according to the survey paper on semi-supervised SVM [6]. Moreover, when ρ = 0, supervised models are incorporated in the framework. Thus the usefulness of unlabeled in naturally considered by the regularization. This therefore provides a practical solution to the problems described in Section 1. 4 Experiment In this section, we give details of our implementation and discuss the results on several benchmark data sets for our proposed approach. To conduct a comprehensive evaluation, we employ several well-known datasets as the testbed. As summarized in Table 1, three image data sets and five text data sets are selected from the recent book (www.kyb.tuebingen. mpg.de/ssl-book/) and the literature (www.cs.uchicago.edu/~vikass/). Table 1: Datasets used in our experiments. d represents the data dimensionality, and n denotes the total number of examples. Data set n d Data set n d usps 1500 241 digit1 1500 241 coil 1500 241 ibm vs rest 1500 11960 pcmac 1946 7511 page 1051 3000 link 1051 1800 pagelink 1051 4800 For simplicity, our proposed adaptive regularization approach is denoted as ARTSVM. To evaluate it, we conduct an extensive comparison with several state-of-the-art approaches, including the label-switching-retraining algorithm in SVM-Light [12], CCCP [8], and ∇TSVM [4]. We employ SVM as the baseline method. In our experiments, we repeat all the algorithms 20 times for each dataset. In each run, 10% of the data are randomly selected as the training data and the remaining data are used as the unlabeled data. The value of C in all algorithms are selected from [1, 10, 100, 1000] using cross-validation. The set of ρ is set to [0, 0.01, 0.05, 0.1, 1] and η is fixed to 0.001. As stated in Section 3.4, ARTSVM is less sensitive to the graph structure. Thus, we adopt a simple way to construct the graph: for each data, the number of neighbors is set to 20 and binary weighting is employed. In ARTSVM, the supervised, weakly semi-supervised, and semi-supervised algorithms are based on implementation in LibSVM (www.csie.ntu.edu. tw/~cjlin/libsvm/), MOSEK (www.mosek.org), and ∇TSVM (www.kyb.tuebingen.mpg. de/bs/people/chapelle/lds/), respectively. For the comparison algorithms, we adopt the original authors’ own implementations. Table 2 summarizes the classification accuracy and the standard deviations of the proposed ARTSVM method and other competing methods. We can draw several observations from the results. First of all, we can clearly see that our proposed algorithm performs significantly better than the baseline SVM method across all the data sets. Note that some very large deviations in SVM are mainly because the labeled data and the unlabeled data may have quite different distributions after the random sampling. On the other hand, the unlabeled data capture the underlying distribution and help to correct such random error. Comparing ARTSVM with other TSVM algorithms, we observe that ARTSVM achieves the best performance in most cases. For example, for the digital image data sets, especially digit1, supervised learning usually works well and the advantages of TSVM are very limited. However, the proposed ARTSVM outperforms both the supervised and other semisupervised algorithms. This indicates that the appropriate regularization from the unlabel data improves the classification performance. Table 2: The classification performance of Transductive SVMs on benchmark data sets. Data Set ARTSVM ∇TSVM SVM CCCP SVM-light usps 81.30±4.04 79.44±3.63 79.23±8.60 80.48±3.20 78.16±4.41 digit1 82.10±2.11 80.55±1.94 81.70±5.61 80.69±2.97 77.53±4.24 coil 81.70±2.10 79.84±1.88 78.98±8.07 80.15±2.90 79.03±2.84 ibm vs rest 78.04±1.44 76.83±2.11 72.90±2.32 77.52±1.51 73.99±5.18 pcmac 95.50±0.88 95.42±0.95 92.57±0.82 94.86±1.09 91.42±7.24 page 94.65±1.19 94.78±1.83 75.22±17.38 94.47±1.67 93.98±2.60 link 94.27±0.97 93.56±1.58 40.79±3.63 92.60±2.10 92.18±2.45 pagelink 97.31±0.68 96.53±1.84 89.41±3.12 95.97±2.22 94.89±1.81 5 Conclusion This paper presents a novel framework for semi-supervised learning from the perspective of the regularization strength from the unlabeled data. In particular, for Transductive SVM, we show that SVM and TSVM can be incorporated as special cases within this framework. In more detail, the loss on the unlabeled data can essentially be regarded as an additional regularizer for the decision boundary in TSVM. To control the regularization strength, we introduce an alternative method of data-dependant regularization based on the principle of manifold regularization. Empirical studies on benchmark data sets demonstrate that the proposed framework is more effective than the previous transductive algorithms and purely supervised methods. For future work, we plan to design a controlling strategy that is adaptive to data from the perspective of low density assumption and manifold regularization of semi-supervised learning. Finally, it is desirable to integrate the low density assumption and manifold regularization into a unified framework. Acknowledgement The work was supported by the National Science Foundation (IIS-0643494), National Institute of Health (1R01GM079688-01), Research Grants Council of Hong Kong (CUHK4158/08E and CUHK4128/08E), and MSRA (FY09-RES-OPP-103). It is also affiliated with the MS-CUHK Joint Lab for Human-centric Computing & Interface Technologies. References [1] Mikhail Belkin, Partha Niyogi, and Vikas Sindhwani. Manifold regularization: A geometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research, 7:2399–2434, 2006. [2] Avrim Blum and Shuchi Chawla. Learning from labeled and unlabeled data using graph mincuts. In ICML ’01: Proceedings of the 18th international conference on Machine learning, pages 19–26. Morgan Kaufmann, San Francisco, CA, 2001. [3] O. Chapelle, B. Sch¨olkopf, and A. Zien, editors. Semi-Supervised Learning. MIT Press, Cambridge, MA, 2006. [4] O. Chapelle and A. Zien. Semi-supervised classification by low density separation. In Proceedings of the Tenth International Workshop on Artificial Intelligence and Statistics, pages 57–64, 2005. [5] Olivier Chapelle, Vikas Sindhwani, and Sathiya Keerthi. Branch and bound for semi-supervised support vector machines. In B. Sch¨olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19. MIT Press, Cambridge, MA, 2007. [6] Olivier Chapelle, Vikas Sindhwani, and Sathiya S. Keerthi. Optimization techniques for semisupervised support vector machines. Journal of Machine Learning Research, 9:203–233, 2008. [7] Ke Chen and Shihai Wang. Regularized boost for semi-supervised learning. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 281–288. MIT Press, Cambridge, MA, 2008. [8] Ronan Collobert, Fabian Sinz, Jason Weston, and L´eon Bottou. Large scale transductive SVMs. Journal of Machine Learning Reseaerch, 7:1687–1712, 2006. [9] S. C. H. Hoi, M. R. Lyu, and E. Y. Chang. Learning the unified kernel machines for classification. In Proceedings of Twentith International Conference on Knowledge Discovery and Data Mining (KDD-2006), pages 187–196, New York, NY, USA, 2006. ACM Press. [10] Steven C. H. Hoi, Rong Jin, and Michael R. Lyu. Learning nonparametric kernel matrices from pairwise constraints. In ICML ’07: Proceedings of the 24th international conference on Machine learning, pages 361–368, New York, NY, USA, 2007. ACM. [11] T. Joachims. Transductive learning via spectral graph partitioning. In ICML ’03: Proceedings of the 20th international conference on Machine learning, pages 290–297, 2003. [12] Thorsten Joachims. Transductive inference for text classification using support vector machines. In ICML ’99: Proceedings of the 16th international conference on Machine learning, pages 200–209, San Francisco, CA, USA, 1999. Morgan Kaufmann Publishers Inc. [13] John Lafferty and Larry Wasserman. Statistical analysis of semi-supervised regression. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 801–808. MIT Press, Cambridge, MA, 2008. [14] Hariharan Narayanan, Mikhail Belkin, and Partha Niyogi. On the relation between low density separation, spectral clustering and graph cuts. In B. Sch¨olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Information Processing Systems 19, pages 1025–1032. MIT Press, Cambridge, MA, 2007. [15] Vikas Sindhwani, S. Sathiya Keerthi, and Olivier Chapelle. Deterministic annealing for semisupervised kernel machines. In ICML ’06: Proceedings of the 23rd international conference on Machine learning, pages 841–848, New York, NY, USA, 2006. ACM Press. [16] Aarti Singh, Robert Nowak, and Xiaojin Zhu. Unlabeled data: Now it helps, now it doesn’t. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing Systems 21, pages 1513–1520. 2009. [17] Junhui Wang, Xiaotong Shen, and Wei Pan. On efficient large margin semisupervised learning: Method and theory. Journal of Machine Learning Research, 10:719–742, 2009. [18] Linli Xu and Dale Schuurmans. Unsupervised and semi-supervised multi-class support vector machines. In AAAI, pages 904–910, 2005. [19] Zenglin Xu, Rong Jin, Jianke Zhu, Irwin King, and Michael R. Lyu. Efficient convex relaxation for transductive support vector machine. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1641–1648. MIT Press, Cambridge, MA, 2008. [20] T. Zhang and R. Ando. Analysis of spectral kernel design based semi-supervised learning. In Y. Weiss, B. Sch¨olkopf, and J. Platt, editors, Advances in Neural Information Processing Systems (NIPS 18), pages 1601–1608. MIT Press, Cambridge, MA, 2006. [21] Dengyong Zhou, Olivier Bousquet, Thomas Navin Lal, Jason Weston, and Bernhard Sch¨olkopf. Learning with local and global consistency. In Sebastian Thrun, Lawrence Saul, and Bernhard Sch¨olkopf, editors, Advances in Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004. [22] X. Zhu, J. Kandola, Z. Ghahramani, and J. Lafferty. Nonparametric transforms of graph kernels for semi-supervised learning. In Advances in Neural Information Processing Systems (NIPS 17), pages 1641–1648, Cambridge, MA, 2005. MIT Press. [23] Xiaojin Zhu. Semi-supervised learning literature survey. Technical report, Computer Sciences, University of Wisconsin-Madison, 2005.
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Subject independent EEG-based BCI decoding Siamac Fazli Cristian Grozea M´arton Dan´oczy Florin Popescu Benjamin Blankertz Klaus-Robert M¨uller Abstract In the quest to make Brain Computer Interfacing (BCI) more usable, dry electrodes have emerged that get rid of the initial 30 minutes required for placing an electrode cap. Another time consuming step is the required individualized adaptation to the BCI user, which involves another 30 minutes calibration for assessing a subject’s brain signature. In this paper we aim to also remove this calibration proceedure from BCI setup time by means of machine learning. In particular, we harvest a large database of EEG BCI motor imagination recordings (83 subjects) for constructing a library of subject-specific spatio-temporal filters and derive a subject independent BCI classifier. Our offline results indicate that BCI-na¨ıve users could start real-time BCI use with no prior calibration at only a very moderate performance loss. 1 Introduction The last years in BCI research have seen drastically reduced training and calibration times due to the use of machine learning and adaptive signal processing techniques (see [9] and references therein) and novel dry electrodes [18]. Initial BCI systems were based on operant conditioning and could easily require months of training on the subject side before it was possible to use them [1, 10]. Second generation BCI systems require to record a brief calibration session during which a subject assumes a fixed number of brain states, say, movement imagination and after which the subject-specific spatio-temporal filters (e.g. [6]) are inferred along with individualized classifiers [9]. Recently, first steps to transfer a BCI user’s filters and classifiers between sessions was studied [14] and a further online-study confirmed that indeed such transfer is possible without significant performance loss [16]. In the present paper we even will go one step further in this spirit and propose a subject-independent zero-training BCI that enables both experienced and novice BCI subjects to use BCI immediately without calibration. Our offline study applies a number of state of the art learning methods (e.g. SVM, Lasso etc.) in order to optimally construct such one-size-fits-all classifiers from a vast number of redundant features, here a large filter bank available from 83 BCI users. The use of sparsifying techniques specifically tell us what are the interesting aspects in EEG that are predictive to future BCI users. As expected, we find that a distribution of different alpha band features in combination with a number of characteristic common spatial patterns (CSPs) is highly predictive for all users. What is found as the outcome of a machine learning experiment can also be viewed as a compact quantitative description of the characteristic variability between individuals in the large subject group. Note that it is not the best subjects that characterize the variance necessary for a subject independent algorithm, rather the spread over existing physiology is to be represented concisely. Clearly, our proceedure may also be of use appart from BCI in other scientific fields, where complex characteristic features need to be homogenized into one overall inference model. The paper first provides an overview of the data used, then the ensemble learning algorithm is outlined, consisting of the procedure for building the 1 filters, the classifiers and the gating function, where we apply various machine learning methods. Interestingly we are able to successfully classify trials of novel subjects with zero training suffering only a small loss in performance. Finally we put our results into perspective. 2 Available Data and Experiments We used 83 BCI datasets (sessions), each consisting of 150 trials from 83 individual subjects. Each trial consists of one of two predefined movement imaginations, being left and right hand, i.e. data was chosen such that it relies only on these 2 classes, although originally three classes were cued during the calibration session, being left hand (L), right hand (R) and foot (F). 45 EEG channels, which are in accordance with the 10-20 system, were identified to be common in all sessions considered. The data were recorded while subjects were immobile, seated on a comfortable chair with arm rests. The cues for performing a movement imagination were given by visual stimuli, and occurred every 4.5-6 seconds in random order. Each trial was referenced by a 3 second long time-window starting at 500 msec after the presentation of the cue. Individual experiments consisted of three different training paradigms. The first two training paradigms consisted of visual cues in form of a letter or an arrow, respectively. In the third training paradigm the subject was instructed to follow a moving target on the screen. Within this target the edges lit up to indicate the type of movement imagination required. The experimental proceedure was designed to closely follow [3]. Electromyogram (EMG) on both forearms and the foot were recorded as well as electrooculogram (EOG) to ensure there were no real movements of the arms and that the movements of the eyes were not correlated to the required mental tasks. 3 Generation of the Ensemble The ensemble consists of a large redundant set of subject-dependent common spatial pattern filters (CSP cf. [6]) and their matching classifiers (LDA). Each dataset is first preprocessed by 18 predefined temporal filters (i.e. band-pass filters) in parallel (see upper panel of Figure 1). A corresponding spatial filter and linear classifier is obtained for every dataset and temporal filter. Each resulting CSP-LDA couple can be interpreted as a potential basis function. Finding an appropriate weighting for the classifier outputs of these basis functions is of paramount importance for the accurate prediction. We employed different forms of regression and classification in order to find an optimal weighting for predicting the movement imagination data of unseen subjects[2, 4]. This processing was done by leave-one-subject-out cross-validation, i.e. the session of a particular subject was removed, the algorithm trained on the remaining trials (of the other subjects) and then applied to this subject’s data (see lower panel of Figure 1). 3.1 Temporal Filters The µ-rhythm (9-14 Hz) and synchronized components in the β-band (16-22 Hz) are macroscopic idle rhythms that prevail over the postcentral somatosensory cortex and precentral motor cortex, when a given subject is at rest. Imaginations of movements as well as actual movements are known to suppress these idle rhythms contralaterally. However, there are not only subject-specific differences of the most discriminative frequency range of the mentioned idle-rhythms, but also session differences thereof. We identified 18 neurophysiologically relevant temporal filters, of which 12 lie within the µ-band, 3 in the β-band, two in between µ- and β-band and one broadband 7 −30Hz. In all following performance related tables we used the percentage of misclassified trials, or 0-1 loss. 3.2 Spatial Filters and Classifiers CSP is a popular algorithm for calculating spatial filters, used for detecting event-related (de)synchronization (ERD/ERS), and is considered to be the gold-standard of ERD-based BCI systems [13, 19, 6]. The CSP algorithm maximizes the variance of right hand trials, while simultaneously minimizing the variance for left hand trials. Given the two covariance matrices Σ1 and Σ2, of size channels x concatenated timepoints, the CSP algorithm returns the matrices W and D. W is a matrix of projections, where the i-th row has a relative variance of di for trials of class 1 and a relative 2 Figure 1: 2 Flowcharts of the ensemble method. The red patches in the top panel illustrate the inactive nodes of the ensemble after sparsification. variance of 1 −di for trials of class 2. D is a diagonal matrix with entries di ∈[0, 1], with length n, the number of channels: WΣ1W T = D and WΣ2W T = I −D (1) Best discrimination is provided by filters with very high (emphazising one class) or very low eigenvalues (emphazising the other class), we therefore chose to only include projections with the highest 2 and corresponding lowest 2 eigenvalues for our analysis. We use Linear Discriminant Analysis (LDA) [5], each time filtered session corresponds to a CSP set and to a matched LDA. 3.3 Final gating function The final gating function combines the outputs of the individual ensemble members to a single one. This can be realized in many ways. For a number of ensemble methods the mean has proven to be a surprisingly good choice [17]. As a baseline for our ensemble we simply averaged all outputs of our individual classifiers. This result is given as mean in Table 2. Classification We employ various classification methods such as k Nearest Neighbor (kNN), Linear Discriminant Analysis (LDA), Support Vector Machine (SVM) and a Linear Programming Machine (LPM) [12]. Quadratic regression with ℓ1 regularization argmin w(k) ij X x∈X\Xk (hk(x) −y(x))2 + α v u u t B X i=1 X j∈S\Sk X x∈X\Xk cij(x)2 B X i=1 X j∈S\Sk |w(k) ij | + |b| (2) hk(x) = B X i=1 X j∈S\Sk w(k) ij cij(x) −b (3) where cij(x) ∈[−∞; ∞] is the continuous classifier output, before thresholding, obtained from the session j by applying the bandpass filter i, B is the number of frequency bands, S the complete set 3 Figure 2: Feature selection during cross-validation: white dashes mark the features kept after regularization for the prediction of the data of each subject. The numbers on the vertical axis represent the subject index as well as the Error Rate (%). The red line depicts the baseline error of individual subjects (classical auto-band CSP). Features as well as baseline errors are sorted by the error magnitude of the self-prediction. Note that some of the features are useful in predicting the data of most other subjects, while some are rarely or never used. of sessions, X the complete data set, Sk the set of sessions of subject k, Xk the dataset for subject k, y(x) is the class label of trial x and wk ij in equation (3) are the weights given to the LDA outputs. The hyperparameter α in equation (2) was varied on a logarithmic scale and multiplied by a dataset scaling factor which accounted for fluctuations in voting population distribution and size for each subject. The dataset scaling factor is computed using cij(x), for all x ∈X \ Xk. For computational efficiency reasons the hyperparameter was tuned on a small random subset of subjects whose labels are to be predicted from data obtained from other subjects such that the resulting test/train error ratio was minimal, which in turn affected the choice (leave in/out) of classifiers among the 83x18 candidates. The ℓ1 regularized regression with this choice of α was then applied to all subjects, with results (in terms of feature sparsification) shown in Figure 2. In fact the exemplary CSP patterns shown in the lower part of the Figure exhibit neurophysiologically meaningful activation in motorcortical areas. The most predictive subjects show smooth monopolar patterns, while subjects with a higher self-prediction loss slowly move from dipolar to rather ragged maps. From the point of view of approximation even the latter make sense for capturing the overall ensemble variance. The implementation of the regressions were performed using CVX, a package for specifying and solving convex programs [11]. We coupled an ℓ2 loss with an ℓ1 penalty term on a linear voting scheme ensemble. Least Squares Regression Is a special case of equation (2), with α = 0. 3.4 Validation The subject-specific CSP-based classification methods with automatically, subject-dependent tuned temporal filters (termed reference methods) are validated by an 8-fold cross-validation, splitting the data chronologically. The chronological splitting for cross-validation is a common practice in EEG classification, since the non-stationarity of the data is thus preserved [9]. To validate the quality of the ensemble learning we employed a leave-one-subject out crossvalidation (LOSO-CV) procedure, i.e. for predicting the labels of a particular subject we only use data from other subjects. 4 Results Overall performance of the reference methods, other baseline methods and of the ensemble method is presented in Table 2. Reference method performances of subject-specific CSP-based classification are presented with heuristically tuned frequency bands [6]. Furthermore we considered much simpler (zero-training) methods as a control. Laplacian stands for the power difference in two Laplace filtered channels (C3 vs. C4) and simple band-power stands for the power difference of the same two 4 classification regression % of data kNN LDA LPM SVM LSR LSR-ℓ1 10 31.3 45.3 37.3 31.3 46.0 30.7 20 32.0 40.0 38.0 28.7 42.0 31.3 30 32.7 38.7 37.3 33.1 38.0 30.0 40 32.7 36.0 37.9 31.3 36.7 29.3 Table 1: Main results of various machine learning algorithms. approach machine learning classical zero training training method mean kNN LDA LPM SVM LSR LSR-ℓ1 Lap BP CSP # <25% 31 30 18 14 29 19 36 24 11 39 25%-tile 17.3 17.3 27.3 26.7 18.7 26.0 16.0 22.0 31.3 11.9 median 30.7 31.3 36.0 37.3 28.7 36.7 29.3 34.7 38.7 25.9 75%-tile 41.3 42.0 43.3 44.0 41.3 44.0 40.7 45.3 45.3 41.4 Table 2: Comparing ML results to various baselines. channels without any spatial filtering. For the simple zero-training methods we chose a broad-band filter of 7−30Hz, since it is the least restrictive and scored one of the best performances on a subject level. The bias b in equation (3) can be tuned broadly for all sessions or corrected individually by session, and implemented for online experiments in multiple ways [16, 20, 15]. In our case we chose to adapt b without label information, but operating under the assumption that class frequency is balanced. We therefore simply subtracted the mean over all trials of a given session. Table 1 shows a comparison of the various classification schemes. We evaluate the performance on a given percentage of the training data in order to observe information gain as a function of datapoints. Clearly the two best ML techniques are on par with subject-dependent CSP classifiers and outperform the simple zero-training methods (not shown in Table 1 but in Table 2) by far. While SVM scores the best median loss over all subjects (see Table 1), L1 regularized regression scored better results for well performing BCI subjects (Figure 3 column 1, row 3). In Figure 3 and Table 2 we furthermore show the results of the L1 regularized regression and SVM versus the auto-band reference method (zero-training versus subject-dependent training) as well as vs. the simple zero-training methods Laplace and band-power. Figure 4 shows all individual temporal filters used to generate the ensemble, where their color codes for the frequency they were used to predict labels of previously unseen data. As to be expected mostly µ-band related temporal filters were selected. Contrary to what one may expect, features that generalize well to other subjects’ data do not exclusively come from BCI subjects with low self-prediction errors (see white dashes in Figure 2), in fact there are some features of weak performing subjects that are necessary to capture all variance of the ensemble. However there is a strong correlation between subjects with a low self-prediction loss and the generalizability of their features to predicting other subjects, as can be seen on the right part of Figure 4. 4.1 Focusing on a particular subject In order to give an intuition of how the ensemble works in detail we will focus on a particular subject. We chose to use the subject with the lowest reference method cross-validation error (10%). Given the non-linearity in the band-power estimation (see Figure 1) it is impossible to picture the resulting ensemble spatial filter exactly. However, by averaging the chosen CSP filters with the weightings, obtained by the ensemble and multiplying them by their LDA classifier weight, we get an approximation: PENS = B X i=1 X j∈S\Sk wijWijCij (4) where wij is the weight matrix, resulting from the ℓ1 regularized regression, given in equations (2) and (3), Wij the CSP filter, corresponding to temporal filter i and subject j and Cij the LDA weights (B in Figure 5). For the case of classical auto-band CSP this simply reduces to PCSP = WC (A in Figure 5). Another way to exemplify the ensemble performance is to refer to a transfer function. By injecting a sinusoid with a frequency within the corresponding band-pass filter into a given channel 5 0 10 20 30 40 50 0 10 20 30 40 50 L1 regularized regression subject−dependent CSP 0 10 20 30 40 50 0 10 20 30 40 50 L1 regularized regression Laplace C3−C4 0 10 20 30 40 50 0 10 20 30 40 50 L1 regularized regression simple band−power C3−C4 0 10 20 30 40 50 0 10 20 30 40 50 SVM subject−dependent CSP 0 10 20 30 40 50 0 10 20 30 40 50 SVM Laplace C3−C4 0 10 20 30 40 50 0 10 20 30 40 50 SVM simple band−power C3−C4 0 10 20 30 40 50 0 10 20 30 40 50 SVM L1 regularized regression 0 10 20 30 40 50 0 10 20 30 40 50 ensemble mean L1 regularized regression 0 10 20 30 40 50 0 10 20 30 40 50 SVM ensemble mean Figure 3: Compares the two best-scoring machine learning methods ℓ1-regularized regression and Support Vector Machine to subject-dependent CSP and other simple zero-training approaches. The axis show the classification loss in percent. frequency [Hz] 7−30 8−15 7−14 10−15 9−14 8−13 7−12 12−18 9−1212−15 18−24 8−1111−14 16−22 7−1010−13 14−20 26−35 5 10 15 20 25 30 35 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 50 100 150 200 250 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 correlation coefficient: −0.78 cross−validation loss [%] Number of active features Figure 4: On the left: The used temporal filters and in color-code their contribution to the final L1 regularized regression classification (the scale is normalized from 0 to 1). Cleary µ-band temporal filters between 10 −13Hz are most predictive. On the right: Number of features used vs. selfpredicted cross-validation. A high self-prediction can be seen to yield a large number of features that are predictable for the whole ensemble. 6 and processing it by the four CSP filters, estimating the bandpower of the resulting signal and finally combining the four outputs by the LDA classifier, we obtain a response for the particular channel, where the sinusoid was injected. Repeating this procedure for each channel results in a response matrix. This procedure can be applied for a single CSP/LDA pair, however we may also repeat the given method for as many times as features were chosen for a given subject by the ensemble and hence obtain an accurate description of how the ensemble processes the given EEG data. The resulting response matrices are displayed in panel C of Figure 5. While the subject-specific pattern (classical) looks less focused and more diverse the general pattern matches the one obtained by the ensemble. A third way of visualizing how the ensemble works, we show the primary projections of the CSP filters that were given the 6 highest weights by the ensemble on the left panel (F) and the distribution of all weights in panel D. The spatial positions of highest channel weightings differ slightly for each of the CSP filters given, however the maxima of the projection matrices are clearly positioned around the primary motor cortex. 5 Conclusion On the path of bringing BCI technology from the lab to a more practical every day situation, it becomes indispensable to reduce the setup time which is nowadays more than one hour towards less than one minute. While dry electrodes provide a first step to avoid the time for placing the cap, calibration still remained and it is here where we contribute by dispensing with calibration sessions. Our present study is an offline analysis providing a positive answer to the question whether a subject independent classifier could become reality for a BCI-na¨ıve user. We have taken great care in this work to exclude data from a given subject when predicting his/her performance by using the previously described LOSOCV. In contrast with previous work on ensemble approaches to BCI classification based on simple majority voting and Adaboost [21, 8] that have utilized only a limited dataset, we have profitted greatly by a large body of high quality experimental data accumulated over the years. This has enabled us to choose by means of machine learning technology a very sparse set of voting classifiers which performed as well as standard, state-of-the-art subject calibrated methods. ℓ1 regularized regression in this case performed better than other methods (such as majority voting) which we have also tested. Note that, interestingly, the chosen features (see Figure 2), do not exclusively come from the best performing subjects, in fact some average performer was also selected. However most white dashes are present in the left half, i.e. most subjects with high autoband reference method performance were selected. Interestingly some subjects with very high BCI performance are not selected at all, while others generalize well in the sense that their model are able to predict other subject’s data. No single frequency band dominated classification accuracy – see Figure 4. Therefore, the regularization must have selected diverse features. Nevertheless, as can be seen in panel G of Figure 5 there is significant redundancy between classifiers in the ensemble. Our approach of finding a sparse solution reduces the dimensionality of the chosen features significantly. For very able subjects our zero-training method exhibits a slight performance decrease, which however will not prevent them from performing successfully in BCI. The sparsification of classifiers, in this case, also leads to potential insight into neurophysiological processes. It identifies relevant cortical locations and frequency bands of neuronal population activity which are in agreement with general neuroscientific knowledge. While this work concentrated on zero training classification and not brain activity interpretation, a much closer look is warranted. Movement imagination detection is not only determined by the cortical representation of the limb whose control is being imagined (in this case the arm) but also by differentially located cortical regions involved in movement planning (frontal), execution (fronto-parietal) and sensory feedback (occipito-parietal). Patterns relevant to BCI detection appear in all these areas and while dominant discriminant frequencies are in the α range, higher frequencies appear in our ensemble, albeit in combination with less focused patterns. So what we have found from our machine learning algorithm should be interpreted as representing the characteristic neurophysiological variation a large subject group, which in itself is a highly relevant topic that goes beyond the scope of this technical study. Future online studies will be needed to add further experimental evidence in support of our findings. We plan to adopt the ensemble approach in combination with a recently developed EEG cap having dry electrodes [18] and thus to be able to reduce the required preparation time for setting up a running BCI system to essentially zero. The generic ensemble classifier derived here is also an excellent starting point for a subsequent coadaptive learning procedure in the spirit of [7]. 7 Figure 5: A: primary projections for classical auto-band CSP. B: linearly averaged CSP’s from the ensemble. C: transfer function for classical auto-band and ensemble CSP’s. D: weightings of 28 ensemble members, the six highest components are shown in F. E: linear average ensemble temporal filter (red), heuristic (blue). F: primary projections of the 6 ensemble members that received highest weights. G: Broad-band version of the ensemble for a single subject. The outputs of all basis classifiers are applied to each trial of one subject. The top row (broad) gives the label, the second row (broad) gives the output of the classical auto-band CSP, and each of the following rows (thin) gives the outputs of the individual classifiers of other subjects. The individual classifier outputs are sorted by their correlation coefficient with respect to the class labels. The trials (columns) are sorted by true labels with primary key and by mean ensemble output as a secondary key. The row at the bottom gives the sign of the average ensemble output. 8 References [1] N. Birbaumer, N. Ghanayim, T. Hinterberger, I. Iversen, B. Kotchoubey, A. K¨ubler, J. Perelmouter, E. Taub, and H. Flor. A spelling device for the paralysed. Nature, 398:297–298, 1999. [2] B. Blankertz, G. Curio, and K.-R. M¨uller. Classifying single trial EEG: Towards brain computer interfacing. In T. G. Diettrich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Inf. Proc. Systems (NIPS 01), volume 14, pages 157–164, 2002. [3] B. Blankertz, G. Dornhege, M. Krauledat, K.-R. M¨uller, V. Kunzmann, F. Losch, and G. Curio. The Berlin Brain-Computer Interface: EEG-based communication without subject training. IEEE Trans Neural Syst Rehabil Eng, 14:147–152, 2006. [4] B. Blankertz, G. Dornhege, S. Lemm, M. Krauledat, G. Curio, and K.-R. M¨uller. The Berlin BrainComputer Interface: Machine learning based detection of user specific brain states. Journal of Universal Computer Science, 12:2006, 2006. [5] B. Blankertz, G. Dornhege, C. Sch¨afer, R. Krepki, J. Kohlmorgen, K.-R. M¨uller, V. Kunzmann, F. Losch, and G. Curio. Boosting bit rates and error detection for the classification of fast-paced motor commands based on single-trial EEG analysis. IEEE Trans. Neural Sys. Rehab. Eng., 11(2):127–131, 2003. [6] B. Blankertz, R. Tomioka, S. Lemm, M. Kawanabe, and K.-R. M¨uller. Optimizing spatial filters for robust EEG single-trial analysis. IEEE Signal Proc Magazine, 25(1):41–56, 2008. [7] Benjamin Blankertz and Carmen Vidaurre. Towards a cure for bci illiteracy: Machine-learning based co-adaptive learning. BMC Neuroscience, 10, 2009. [8] R. Boostani and M. H. Moradi. A new approach in the BCI research based on fractal dimension as feature and adaboost as classifier. J. Neural Eng., 1:212–217, 2004. [9] G. Dornhege, J.del R. Mill´an, T. Hinterberger, D. McFarland, and K.-R. M¨uller, editors. Toward BrainComputer Interfacing. Cambridge, MA: MIT Press, 2007. [10] T. Elbert, B. Rockstroh, W. Lutzenberger, and N. Birbaumer. Biofeedback of slow cortial potentials. I. Electroencephalogr. Clin. Neurophysiol., 48:293–301, 1980. [11] M. Grant and S. Boyd. CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/ boyd/cvx, 2008. [12] Trevor Hastie, Robert Tibshirani, and Jerome Friedman. The Elements of Statistical Learning, Second Edition: Data Mining, Inference, and Prediction (Springer Series in Statistics). Springer New York, 2 edition, 2001. [13] Z.J. Koles and A. C. K. Soong. EEG source localization: implementing the spatio-temporal decomposition approach. Electroencephalogr. Clin Neurophysiol, 107:343–352, 1998. [14] M. Krauledat, M. Schr¨oder, B. Blankertz, and K.-R. M¨uller. Reducing calibration time for brain-computer interfaces: A clustering approach. In B. Sch¨olkopf, J. Platt, and T. Hoffman, editors, Advances in Neural Inf. Proc. Systems (NIPS 07), volume 19, pages 753–760, 2007. [15] M. Krauledat, P. Shenoy, B. Blankertz, R.P.N. Rao, and K.-R. M¨uller. Adaptation in csp-based BCI systems. In Toward Brain-Computer Interfacing, pages 305–309. MIT Press, 2007. [16] M. Krauledat, M. Tangermann, B. Blankertz, and K.-R. M¨uller. Towards zero training for brain-computer interfacing. PLoS ONE, 3:e2967, 2008. [17] R. Polikar. Ensemble based systems in decision making. IEEE Circuits and Systems Magazine, 6(3):21– 45, 2006. [18] F. Popescu, S. Fazli, Y. Badower, B. Blankertz, and K.-R. M¨uller. Single trial classification of motor imagination using 6 dry EEG electrodes. PLoS ONE, 2:e637, 2007. [19] H. Ramoser, J. M¨uller-Gerkin, and G. Pfurtscheller. Optimal spatial filtering of single trial EEG during imagined hand movement. IEEE Trans. Rehab. Eng, 8(4):441–446, 2000. [20] P. Shenoy, M. Krauledat, B. Blankertz, R.P.N. Rao, and K.-R. M¨uller. Towards adaptive classification for BCI. Journal of Neural Engineering, 3(1):R13–R23, 2006. [21] S. Wang, Z. Lin, and C. Zhang. Network boosting for BCI applications. Book Series Lecture Notes in Computer Science, 3735:386–388, 2005. 9
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Multi-label Prediction via Sparse Infinite CCA Piyush Rai and Hal Daum´e III School of Computing, University of Utah {piyush,hal}@cs.utah.edu Abstract Canonical Correlation Analysis (CCA) is a useful technique for modeling dependencies between two (or more) sets of variables. Building upon the recently suggested probabilistic interpretation of CCA, we propose a nonparametric, fully Bayesian framework that can automatically select the number of correlation components, and effectively capture the sparsity underlying the projections. In addition, given (partially) labeled data, our algorithm can also be used as a (semi)supervised dimensionality reduction technique, and can be applied to learn useful predictive features in the context of learning a set of related tasks. Experimental results demonstrate the efficacy of the proposed approach for both CCA as a stand-alone problem, and when applied to multi-label prediction. 1 Introduction Learning with examples having multiple labels is an important problem in machine learning and data mining. Such problems are encountered in a variety of application domains. For example, in text classification, a document (e.g., a newswire story) can be associated with multiple categories. Likewise, in bio-informatics, a gene or protein usually performs several functions. All these settings suggest a common underlying problem: predicting multivariate responses. When the responses come from a discrete set, the problem is termed as multi-label classification. The aforementioned setting is a special case of multitask learning [6] when predicting each label is a task and all the tasks share a common source of input. An important characteristics of these problems is that the labels are not independent of each other but actually often have significant correlations with each other. A na¨ıve approach to learn in such settings is to train a separate classifier for each label. However, such an approach ignores the label correlations and leads to sub-optimal performance [20]. In this paper, we show how Canonical Correlation Analysis (CCA) [11] can be used to exploit label relatedness, learning multiple prediction problems simultaneously. CCA is a useful technique for modeling dependencies between two (or more) sets of variables. One important application of CCA is in supervised dimensionality reduction, albeit in the more general setting where each example has several labels. In this setting, CCA on input-output pair (X, Y) can be used to project inputs X to a low-dimensional space directed by label information Y. This makes CCA an ideal candidate for extracting useful predictive features from data in the context of multi-label prediction problems. The classical CCA formulation, however, has certain inherent limitations. It is non-probabilistic which means that it cannot deal with missing data, and precludes a Bayesian treatment which can be important if the dataset size is small. An even more crucial issue is choosing the number of correlation components, which is traditionally dealt with by using cross-validation, or model-selection [21]. Another issue is the potential sparsity [18] of the underlying projections that is ignored by the standard CCA formulation. Building upon the recently suggested probabilistic interpretation of CCA [3], we propose a nonparametric, fully Bayesian framework that can deal with each of these issues. In particular, the proposed model can automatically select the number of correlation components, and effectively capture the 1 sparsity underlying the projections. Our framework is based on the Indian Buffet Process [9], a nonparametric Bayesian model to discover latent feature representation of a set of observations. In addition, our probabilistic model allows dealing with missing data and, in the supervised dimensionality reduction case, can incorporate additional unlabeled data one may have access to, making our CCA algorithm work in a semi-supervised setting. Thus, apart from being a general, nonparametric, fully Bayesian solution to the CCA problem, our framework can be readily applied for learning useful predictive features from labeled (or partially labeled) data in the context of learning a set of related tasks. This paper is organized as follows. Section 2 introduces the CCA problem and its recently proposed probabilistic interpretation. In section 3, we describe our general framework for infinite CCA. Section 4 gives a concrete example of an application (multi-label learning) where the proposed approach can be applied. In particular, we describe a fully supervised setting (when the test data is not available at the time of training), and a semi-supervised setting with partial labels (when we have access to test data at the time of training). We describe our experiments in section 5, and discuss related work in section 6 drawing connections of the proposed method with previously proposed ones for this problem. . 2 Canonical Correlation Analysis Canonical correlation analysis (CCA) is a useful technique for modeling the relationships among a set of variables. CCA computes a low-dimensional shared embedding of a set of variables such that the correlations among the variables is maximized in the embedded space. More formally, given a pair of variables x ∈RD1 and y ∈RD2, CCA seeks to find linear projections ux and uy such that the variables are maximally correlated in the projected space. The correlation coefficient between the two variables in the embedded space is given by ρ = uT x xyT uy q (uTx xxT ux)(uTy yyT uy) Since the correlation is not affected by rescaling of the projections ux and uy, CCA is posed as a constrained optimization problem. max ux,uy uT x xyT uy, subject to : uT x xxT ux = 1, uT y yyT uy = 1 It can be shown that the above formulation is equivalent to solving the following generalized eigenvalue problem: 0 Σxy Σyx 0 ux uy = ρ Σxx 0 0 Σyy ux uy where Σ denotes the covariance matrix of size D × D (where D = D1 + D2) obtained from the data samples X = [x1, . . . , xn] and Y = [y1, . . . , yn]. 2.1 Probabilistic CCA Bach and Jordan [3] gave a probabilistic interpretation of CCA by posing it as a latent variable model. To see this, let x and y be two random vectors of size D1 and D2. Let us now consider the following latent variable model z ∼ Nor(0, IK), min{D1, D2} ≥K x ∼ Nor(µx + Wxz, Ψx), Wx ∈RD1×K, Ψx ⪰0 y ∼ Nor(µy + Wyz, Ψy), Wy ∈RD2×K, Ψy ⪰0 Equivalently, we can also write the above as [x; y] ∼Nor(µ + Wz, Ψ) 2 where µ = [µx; µy], W = [Wx; Wy], and Ψ is a block-diagonal matrix consisting of Ψx and Ψx on its diagonals. [.; .] denotes row-wise concatenation. The latent variable z is shared between x and y. Bach and Jordan [3] showed that, given the maximum likelihood solution for the model parameters, the expectations E(z|x) and E(z|y) of the latent variable z lie in the same subspace that classical CCA finds, thereby establishing the equivalence between the above probabilistic model and CCA. The probabilistic interpretation opens doors to several extension of the basic setup proposed in [3] which suggested a maximum likelihood approach for parameter estimation. However, it still assumes an apriori fixed number of canonical correlation components. In addition, another important issue is the sparsity of the underlying projection matrix which is usually ignored. 3 The Infinite Canonical Correlation Analysis Model Recall that the CCA problem can be defined as [x; y] ∼Nor(Wz, Ψ) (assuming centered data). A crucial issue in the CCA model is choosing the number of canonical correlation components which is set to a fixed value in classical CCA (and even in the probabilistic extensions of CCA). In the Bayesian formulation of CCA, one can use the Automatic Relevance Determination (ARD) prior [5] on the projection matrix W that gives a way to select this number. However, it would be more appropriate to have a principled way to automatically figure out this number based on the data. We propose a nonparametric Bayesian model that selects the number of canonical correlation components automatically. More specifically, we use the Indian Buffet Process [9] (IBP) as a nonparametric prior on the projection matrix W. The IBP prior allows W to have an unbounded number of columns which gives a way to automatically determine the dimensionality K of the latent space associated with Z. 3.1 The Indian Buffet Process The Indian Buffet Process [9] defines a distribution over infinite binary matrices, originally motivated by the need to model the latent feature structure of a given set of observations. The IBP has been a model of choice in variety of non-parametric Bayesian approaches, such as for factorial structure learning, learning causal structures, modeling dyadic data, modeling overlapping clusters, and several others [9]. In the latent feature model, each observation can be thought of as being explained by a set of latent features. Given an N × D matrix X of N observations having D features each, we can consider a decomposition of the form X = ZA + E where Z is an N × K binary feature-assignment matrix describing which features are present in each observation. Zn,k is 1 if feature k is present in observation n, and is otherwise 0. A is a K × D matrix of feature scores, and the matrix E consists of observation specific noise. A crucial issue in such models is the choosing the number K of latent features. The standard formulation of IBP lets us define a prior over the binary matrix Z such that it can have an unbounded number of columns and thus can be a suitable prior in problems dealing with such structures. The IBP derivation starts by defining a finite model for K many columns of a N × K binary matrix. P(Z) = K Y k=1 α K Γ(mk + α K )Γ(P −mk −1) Γ(P + 1 + α K ) (1) Here mk = P i Zik. In the limiting case, as K →∞, it as was shown in [9] that the binary matrix Z generated by IBP is equivalent to one produced by a sequential generative process. This equivalence can be best understood by a culinary analogy of customers coming to an Indian restaurant and selecting dishes from an infinite array of dishes. In this analogy, customers represent observations and dishes represent latent features. Customer 1 selects Poisson(α) dishes to begin with. Thereafter, each incoming customer n selects an existing dish k with a probability mk/N, where mk denotes how many previous customers chose that particular dish. The customer n then goes on further to additionally select Poisson(α/N) new dishes. This process generates a binary matrix Z with rows representing customer and columns representing dishes. Many real world datasets have a sparseness 3 Figure 1: The graphical model depicts the fully supervised case when all variables X and Y are observed. The semisupervised case can have X and/or Y consisting of missing values as well. The graphical model structure remains the same property which means that each observation depends only on a subset of all the K latent features. This means that the binary matrix Z is expected to be reasonably sparse for many datasets. This makes IBP a suitable choice for also capturing the underlying sparsity in addition to automatically discovering the number of latent features. 3.2 The Infinite CCA Model In our proposed framework, the matrix W consisting of canonical correlation vectors is modeled using an IBP prior. However since W can be real-valued and the IBP prior is defined only for binary matrices, we represent the (D1 + D2) × K matrix W as (B ⊙V), where B = [Bx; By] is a (D1 + D2) × K binary matrix, V = [Vx; Vy] is a (D1 + D2) × K real-valued matrix, and ⊙denotes their element-wise (Hadamard) product. We place an IBP prior on B that automatically determines K, and a Gaussian prior on V. Note that B and V have the same number of columns. Under this model, two random vectors x and y can be modeled as x = (Bx ⊙Vx)z + Ex and y = (By ⊙Vy)z + Ey. Here z is shared between x and y, and Ex and Ey are observation specific noise. In the full model, X = [x1, . . . , xN] is D1 × N matrix consisting of N samples of D1 dimensions each, and Y = [y1, . . . , yN] is another matrix consisting of N samples of D2 dimensions each. Here is the generative story for our basic model: B ∼ IBP(α) V ∼ Nor(0, σ2 vI), σv ∼IG(a, b) Z ∼ Nor(0, I) [X; Y] ∼ Nor(B ⊙V)Z, Ψ), where Ψ is a diagonal matrix of size D × D where D = (D1 + D2), with each diagonal entry having an inverse-Gamma prior.. Since our model is probabilistic, it can also deal the problem when X or Y have missing entries. This is particularly important in the case of supervised dimensionality reduction (i.e., X consisting of inputs and Y associated responses) when the labels for some of the inputs are unknown, making it a model for semi-supervised dimensionality reduction with partially labeled data. In addition, placing the IBP prior on the projection matrix W (via the binary matrix B) also helps in capturing the sparsity in W (see results section for evidence). 3.3 Inference We take a fully Bayesian approach by treating everything at latent variables and computing the posterior distributions over them. We use Gibbs sampling with a few Metropolis-Hastings steps to do inference in this model. 4 In what follows, D denotes the data [X; Y], B = [Bx; By], and V = [Vx; Vy] Sampling B: Sampling the binary IBP matrix B consists of sampling existing dishes, proposing new dishes and accepting or rejecting them based on the acceptance ratio in the associated M-H step. For sampling existing dishes, an entry in B is set as 1 according to p(Bik = 1|D, B−ik, V, Z, Ψ) ∝ m−i,k D p(D|B, V, F, Ψ) whereas it is set as 0 according to p(Bik = 0|D, B−ik, V, Z, Ψ) ∝ D−m−i,k D p(D|B, V, Z, Ψ). m−i,k = P j̸=i Bjk is how many other customers chose dish k. For sampling new dishes, we use an M-H step where we simultaneously propose η = (Knew, V new, Znew) where Knew ∼Poisson(α/D). We accept the proposal with an acceptance probability given by a = min{1, p(rest|η∗) p(rest|η) }. Here, p(rest|η) is the probability of the data given parameters η. We propose V new from its prior (Gaussian) but, for faster mixing, we propose Znew from its posterior. Sampling V: We sample the real-valued matrix V from its posterior p(Vi,k|D, B, Z, Ψ) ∝ Nor(Vi,k|µi,k, Σi,k), where Σi,k = (PN n=1 Z2 k,n Ψi + 1 σ2 v )−1 and µi,k = Σi,k(PN n=1 Ak,nD∗ i,k)Ψ−1 i . We define D∗ i,k = Di,n −PK l=1,l̸=k(Bi,lVi,l)Zl,n. The hyperparameter σv on V has an inversegamma prior and posterior also has the same form. Note that the number of columns in V is the same as number of columns in the IBP matrix B. Sampling Z: We sample for Z from its posterior p(Z|D, B, V, Ψ) ∝Nor(Z|µ, Σ) where µ = WT(WWT + Ψ)−1D and Σ = I −WT(WWT + Ψ)−1W, where W = B ⊙V. Note that, in our sampling scheme, we considered the matrices Bx and By as simply parts of the big IBP matrix B, and sampled them together using a single IBP draw. However, one could also sample them separately as two separate IBP matrices for Bx and By. This would require different IBP draws for sampling Bx and By with some modification of the existing Gibbs sampler. Different IBP draws could result in different number of nonzero columns in Bx and By. To deal with this issue, one could sample Bx (say having Kx nonzero columns) and By (say having Ky nonzero columns) first, introduce extra dummy columns (|Kx−Ky| in number) in the matrix having smaller number of nonzero columns, and then set all such columns to zero. The effective K for each iteration of the Gibbs sampler would be max{Kx, Ky}. A similar scheme could also be followed for the corresponding real-valued matrices Vx and Vy, sampling them in conjunction with Bx and By respectively. 4 Multitask Learning using Infinite CCA Having set up the framework for infinite CCA, we now describe its applicability for the problem of multitask learning. In particular, we consider the setting when each example is associated with multiple labels. Here predicting each individual label becomes a task to be learned. Although one can individually learn a separate model for each task, doing this would ignore the label correlations. This makes borrowing the information across tasks crucial, making it imperative to share the statistical strength across all the task. With this motivation, we apply our infinite CCA model to capture the label correlations and to learn better predictive features from the data by projective it to a subspace directed by label information. It has been empirically and theoretically [25] shown that incorporating label information in dimensionality reduction indeed leads to better projections if the final goal is prediction. More concretely, let X = [x1, . . . , xN] be an D × N matrix of predictor variables, and Y = [y1, . . . , yN] be an M × N matrix of the responses variables (i.e., the labels) with each yi being an M × 1 vector of responses for input xi. The labels can take real (for regression) or categorical (for classification) values. The infinite CCA model is applied on the pair X and Y which is akin to doing supervised dimensionality reduction for the inputs X. Note that the generalized eigenvalue problem posed in such a supervised setting of CCA consists of cross-covariance matrix ΣXY and label covariance matrix ΣY Y . Therefore the projection takes into account both the input-output correlations and the label correlations. Such a subspace therefore is expected to consist of much better predictive features than one obtained by a na¨ıve feature extraction approach such as simple PCA that completely ignores the label information, or approaches like Linear Discriminant Analysis (LDA) that do take into account label information but ignore label correlations. 5 Multitask learning using the infinite CCA model can be done in two settings: supervised and semisupervised depending on whether or not the inputs of test data are involved in learning the shared subspace Z. 4.1 Fully supervised setting In the supervised setting, CCA is done on labeled data (X, Y) to give a single shared subspace Z ∈RK×N that is good across all tasks. A model is then learned in the Z subspace to learn M task parameters {θm} ∈RK×1 where m ∈{1, . . . , M}. Each of the parameters θm is then used to predict the labels for the test data of task m. However that since the test data is still D dimensional, we need to either separately project it down onto the K dimensional subspace and do predictions in this subspace, or “inflate” each task parameter back to D dimensions by applying the projection matrix Wx and do predictions in the original D dimensional space. The first option requires using the fact that P(Z|Xte) ∝P(Xte|Z)P(Z) which is a Gaussian Nor(µZ|X, ΣZ|X) with µZ|X = (WT x ΨxWx + I)−1WT x Xte and ΣZ|X = (WT x ΨxWx + I)−1. With the second option, we can inflate each learned task parameter back to D dimensions by applying the projection matrix Wx. We choose the second option for the experiments. We call this fully supervised setting as model-1. 4.2 A Semi-supervised setting In the semi-supervised setting, we combine training data and test data (with unknown labels) as X = [Xtr, Xte] and Y = [Ytr, Yte] where the labels Yte are unknown. The infinite CCA model is then applied on the pair (X, Y) and the parts of Y consisting of Yte are treated as a latent variables to be imputed. With this model, we get the embeddings also for the test data and thus training and testing both take place in the K dimensional subspace, unlike model-1 in which training is done in K dimensional subspace and prediction are made in the original D dimensional subspace. We call this semi-supervised setting as model-2. 5 Experiments Here we report our experimental results on several synthetic and real world datasets. We first show our results with the infinite CCA as a stand alone algorithm for CCA by using it on a synthetic dataset demonstrating its effectiveness in capturing the canonical correlations. We then also report our experiments on applying the infinite CCA model to the problem of multitask learning on two real world datasets. 5.1 Infinite CCA results on synthetic data In the first experiment, we demonstrate the effectiveness of our proposed infinite CCA model in discovering the correct number of canonical correlation components, and in capturing the sparsity pattern underlying the projection matrix. For this, we generated two datasets of dimensions 25 and 10 respectively, with each having 100 samples. For this synthetic dataset, we knew the ground truth (i.e., the number of components, and the underlying sparsity of projection matrix). In particular, the dataset had 4 correlation components with a 63% sparsity in the true projection matrix. We then ran both classical CCA and infinite CCA algorithm on this dataset. Looking at all the correlations discovered by classical CCA, we found that it discovered 8 components having significant correlations, whereas our model correctly discovered exactly 4 components in the first place (we extract the MAP samples for W and Z output by our Gibbs sampler). Thus on this small dataset, standard CCA indeed seems to be finding spurious correlations, indicating a case of overfitting (the overfitting problem of classical CCA was also observed in [15] when comparing Bayesian versus classical CCA). Furthermore, as expected, the projection matrix inferred by the classical CCA had no exact zero entries and even after thresholding significantly small absolute values to zero, the uncovered sparsity was only about 25%. On the other hand, the projection matrix inferred by the infinite CCA model had 57% exact zero entries and 62% zero entries after thresholding very small values, thereby demonstrating its effectiveness in also capturing the sparsity patterns. 6 Model Yeast Scene Acc F1-macro F1-micro AUC Acc F1-macro F1-micro AUC Full 0.5583 0.3132 0.3929 0.5054 0.7565 0.3445 0.3527 0.6339 PCA 0.5612 0.3144 0.4648 0.5026 0.7233 0.2857 0.2734 0.6103 CCA 0.5441 0.2888 0.3923 0.5135 0.7496 0.3342 0.3406 0.6346 Model-1 0.5842 0.3327 0.4402 0.5232 0.7533 0.3630 0.3732 0.6517 Model-2 0.6156 0.3463 0.4954 0.5386 0.7664 0.3742 0.3825 0.6686 Table 1: Results on the multi-label classification task. Bold face indicates the best performance. Model-1 and Model-2 scores are averaged over 10 runs with different initializations. 5.2 Infinite CCA applied to multi-label prediction In the second experiment, we use infinite CCA model to learn a set of related task in the context of multi-label prediction. For our experiments, we use two real-world multi-label datasets (Yeast and Scene) from the UCI repository. The Yeast dataset consists of 1500 training and 917 test examples, each having 103 features. The number of labels (or tasks) per example is 14. The Scene dataset consists of 1211 training and 1196 test examples, each having 294 features. The number of labels per example for this dataset is 6. We compare the following models for our experiments. • Full: Train separate classifiers (SVM) on the full feature set for each task. • PCA: Apply PCA on training and test data and then train separate classifiers for each task in the low dimensional subspace. This baseline ignores the label information while learning the low dimensional subspace. • CCA: Apply classical CCA on training data to extract the shared subspace, learn separate model (i.e., task parameters) for each task in this subspace, project the task parameters back to the original D dimensional feature space by applying the projection Wx, and do predictions on the test data in this feature pace. • Model-1: Use our supervised infinite CCA model to learn the shared subspace using only the training data (see section 4.1). • Model-2: Use our semi-supervised infinite CCA model to simultaneously learn the shared subspace for both training and test data (see section 4.2). The performance metrics used are overall accuracy, F1-Macro, F1-Micro, and AUC (Area Under ROC Curve). For PCA and CCA, we chose K that gives the best performance, whereas this parameter was learned automatically for both of our proposed models. The results are shown in Table-1. As we can see, both the proposed models do better than the other baselines. Of the two proposed model, we see that model-2 does better in most cases suggesting that it is useful to incorporate the test data while learning the projections. This is possible in our probabilistic model since we could treat the unknown Y’s of the test data as latent variables to be imputed while doing the Gibbs sampling. We note here that our results are with cases where we only had access to small number of related task (yeast has 14, scene has 6). We expect the performance improvements to be even more significant when the number of (related) tasks is high. 6 Related Work A number of approaches have been proposed in the recent past for the problem of supervised dimensionality reduction of multi-label data. The few approaches that exist include Partial Least Squares [2], multi-label informed latent semantic indexing [24], and multi-label dimensionality reduction using dependence maximization (MDDM) [26]. None of these, however, deal with the case when the data is only partially labeled. Somewhat similar in spirit to our approach is the work on supervised probabilistic PCA [25] that extends probabilistic PCA to the setting when we also have access to labels. However, it assumes a fixed number of components and does not take into account sparsity of the projections. 7 The CCA based approach to supervised dimensionality reduction is more closely related to the notion of dimension reduction for regression (DRR) which is formally defined as finding a low dimensional representation z ∈RK of inputs x ∈RD (K ≪D) for predicting multivariate outputs y ∈RM. An important notion in DRR is that of sufficient dimensionality reduction (SDR) [10, 8] which states that given z, x and y are conditionally independent, i.e., x ⊥⊥y|z. As we can see in the graphical model shown in figure-1, the probabilistic interpretation of CCA yields the same condition with X and Y being conditionally independent given Z. Among the DRR based approaches to dimensionality reduction for real-valued multilabel data, Covariance Operator Inverse Regression (COIR) exploits the covariance structures of both the inputs and outputs [14]. Please see [14] for more details on the connection between COIR and CCA. Besides the DRR based approaches, the problem of extracting useful features from data, particularly with the goal of making predictions, has also been considered in other settings. The information bottleneck (IB) method [19] is one such example. Given input-output pairs (X, Y), the information bottleneck method aims to obtain a compressed representation T of X that can account for Y. IB achieves this using a single tradeoff parameter to represent the tradeoff between the complexity of the representation of X, measured by I(X; T), and the accuracy of this representation, measured by I(T; Y), where I(.; .) denotes the mutual information between two variables. In another recent work [13], a joint learning framework is proposed which performs dimensionality reduction and multi-label classification simultaneously. In the context of CCA as a stand-alone problem, sparsity is another important issue. In particular, sparsity improves model interpretation and has been gaining lots of attention recently. Existing works on sparsity in CCA include the double barrelled lasso which is based on a convex least squares approach [17], and CCA as a sparse solution to the generalized eigenvalue problem [18] which is based on constraining the cardinality of the solution to the generalized eigenvalue problem to obtain a sparse solution. Another recent solution is based on a direct greedy approach which bounds the correlation at each stage [22]. The probabilistic approaches to CCA include the works of [15] and [1], both of which use an automatic relevance determination (ARD) prior [5] to determine the number of relevant components, which is a rather ad-hoc way of doing this. In contrast, a nonparametric Bayesian alternative proposed here is a more principled to determine the number of components. We note that the sparse factor analysis model proposed in [16] actually falls out as a special case of our proposed infinite CCA model if one of the datasets (X or Y) is absent. Besides, the sparse factor analysis model is limited to factor analysis whereas the proposed model can be seen as an infinite generalization of both an unsupervised problem (sparse CCA), and (semi)supervised problem (dimensionality reduction using CCA with full or partial label information), with the latter being especially relevant for multitask learning in the presence of multiple labels. Finally, multitask learning has been tackled using a variety of different approaches, primarily depending on what notion of task relatedness is assumed. Some of the examples include tasks generated from an IID space [4], and learning multiple tasks using a hierarchical prior over the task space [23, 7], among others. In this work, we consider multi-label prediction in particular, based on the premise that that a set of such related tasks share an underlying low-dimensional feature space [12] that captures the task relatedness. 7 Conclusion We have presented a nonparametric Bayesian model for the Canonical Correlation Analysis problem to discover the dependencies between a set of variables. In particular, our model does not assume a fixed number of correlation components and this number is determined automatically based only on the data. In addition, our model enjoys sparsity making the model more interpretable. The probabilistic nature of our model also allows dealing with missing data. Finally, we also demonstrate the model’s applicability to the problem of multi-label learning where our model, directed by label information, can be used to automatically extract useful predictive features from the data. Acknowledgements We thank the anonymous reviewers for helpful comments. This work was partially supported by NSF grant IIS-0712764. 8 References [1] C. Archambeau and F. Bach. Sparse probabilistic projections. In Neural Information Processing Systems 21, 2008. [2] J. Arenas-Garc´ıa, K. B. Petersen, and L. K. Hansen. Sparse kernel orthonormalized pls for feature extraction in large data sets. In Neural Information Processing Systems 19, 2006. [3] F. R. Bach and M. I. Jordan. A Probabilistic Interpretation of Canonical Correlation Analysis. In Technical Report 688, Dept. of Statistics. University of California, 2005. [4] J. Baxter. A Model of Inductive Bias Learning. Journal of Artificial Intelligence Research, 12:149–198, 2000. [5] C. M. Bishop. Bayesian PCA. In Neural Information Processing Systems 11, Cambridge, MA, USA, 1999. MIT Press. [6] R. Caruana. Multitask Learning. Machine Learning, 28(1):41–75, 1997. [7] H. Daum´e III. Bayesian Multitask Learning with Latent Hierarchies. In Conference on Uncertainty in Artificial Intelligence, Montreal, Canada, 2009. [8] K. Fukumizu, F. R. Bach, and M. I. Jordan. Dimensionality reduction for supervised learning with reproducing kernel hilbert spaces. J. Mach. Learn. Res., 5:73–99, 2004. [9] Z. Ghahramani, T. L. Griffiths, and P. Sollich. Bayesian Nonparametric Latent Feature Models. In Bayesian Statistics 8. Oxford University Press, 2007. [10] A. Globerson and N. Tishby. Sufficient dimensionality reduction. J. Mach. Learn. Res., 3:1307–1331, 2003. [11] H. Hotelling. Relations Between Two Sets of Variables. Biometrika, pages 321–377, 1936. [12] S. Ji, L. Tang, S. Yu, and J. Ye. Extracting Shared Subspace for Multi-label Classification. 2008. [13] S. Ji and J. Ye. Linear dimensionality reduction for multi-label classification. In Twenty-first International Joint Conference on Artificial Intelligence, 2009. [14] M. Kim and V. Pavlovic. Covariance operator based dimensionality reduction with extension to semisupervised settings. In Twelfth International Conference on Artificial Intelligence and Statistics, Florida USA, 2009. [15] A. Klami and S. Kaski. Local dependent components. In ICML ’07: Proceedings of the 24th international conference on Machine learning, 2007. [16] P. Rai and H. Daum´e III. The infinite hierarchical factor regression model. In Neural Information Processing Systems 21, 2008. [17] D. Hardoon J. Shawe-Taylor. The Double-Barrelled LASSO (Sparse Canonical Correlation Analysis). In Workshop on Learning from Multiple Sources (NIPS), 2008. [18] B. Sriperumbudur, D. Torres, and G. Lanckriet. The Sparse Eigenvalue Problem. In arXiv:0901.1504v1, 2009. [19] N. Tishby, F. C. 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. [20] N. Ueda and K. Saito. Parametric Mixture Models for Multi-labeled Text. Advances in Neural Information Processing Systems, pages 737–744, 2003. [21] C. Wang. Variational Bayesian approach to Canonical Correlation Analysis. In IEEE Transactions on Neural Networks, 2007. [22] A. Wiesel, M. Kliger, and A. Hero. A Greedy Approach to Sparse Canonical Correlation Analysis. In arXiv:0801.2748, 2008. [23] Y. Xue, X. Liao, L. Carin, and B. Krishnapuram. Multi-task Learning for Classification with Dirichlet Process Priors. The Journal of Machine Learning Research, 8:35–63, 2007. [24] K. Yu, S. Yu, and V. Tresp. Multi-label Informed Latent Semantic Indexing. In Proceedings of the 28th annual international ACM SIGIR conference on Research and development in information retrieval, pages 258–265. ACM New York, NY, USA, 2005. [25] S. Yu, K. Yu, V. Tresp, H. Kriegel, and M. Wu. Supervised Probabilistic Principal Component Analysis. In KDD ’06: Proceedings of the 12th ACM SIGKDD international conference on Knowledge discovery and data mining, 2006. [26] Y. Zhang Z. H. Zhou. Multi-Label Dimensionality Reduction via Dependence Maximization. In Proceedings of the Twenty-Third AAAI Conference on Artificial Intelligence, AAAI 2008, pages 1503–1505, 2008. 9
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On the Convergence of the Concave-Convex Procedure Bharath K. Sriperumbudur Department of Electrical and Computer Engineering University of California, San Diego La Jolla, CA 92093 bharathsv@ucsd.edu Gert R. G. Lanckriet Department of Electrical and Computer Engineering University of California, San Diego La Jolla, CA 92093 gert@ece.ucsd.edu Abstract The concave-convex procedure (CCCP) is a majorization-minimization algorithm that solves d.c. (difference of convex functions) programs as a sequence of convex programs. In machine learning, CCCP is extensively used in many learning algorithms like sparse support vector machines (SVMs), transductive SVMs, sparse principal component analysis, etc. Though widely used in many applications, the convergence behavior of CCCP has not gotten a lot of specific attention. Yuille and Rangarajan analyzed its convergence in their original paper, however, we believe the analysis is not complete. Although the convergence of CCCP can be derived from the convergence of the d.c. algorithm (DCA), its proof is more specialized and technical than actually required for the specific case of CCCP. In this paper, we follow a different reasoning and show how Zangwill’s global convergence theory of iterative algorithms provides a natural framework to prove the convergence of CCCP, allowing a more elegant and simple proof. This underlines Zangwill’s theory as a powerful and general framework to deal with the convergence issues of iterative algorithms, after also being used to prove the convergence of algorithms like expectation-maximization, generalized alternating minimization, etc. In this paper, we provide a rigorous analysis of the convergence of CCCP by addressing these questions: (i) When does CCCP find a local minimum or a stationary point of the d.c. program under consideration? (ii) When does the sequence generated by CCCP converge? We also present an open problem on the issue of local convergence of CCCP. 1 Introduction The concave-convex procedure (CCCP) [30] is a majorization-minimization algorithm [15] that is popularly used to solve d.c. (difference of convex functions) programs of the form, min x f(x) s.t. ci(x) ≤0, i ∈[m], dj(x) = 0, j ∈[p], (1) where f(x) = u(x) −v(x) with u, v and ci being real-valued convex functions, dj being an affine function, all defined on Rn. Here, [m] := {1, . . . , m}. Suppose v is differentiable. The CCCP 1 algorithm is an iterative procedure that solves the following sequence of convex programs, x(l+1) ∈arg min x u(x) −xT ∇v(x(l)) s.t. ci(x) ≤0, i ∈[m], dj(x) = 0, j ∈[p]. (2) As can be seen from (2), the idea of CCCP is to linearize the concave part of f, which is −v, around a solution obtained in the current iterate so that u(x) −xT ∇v(x(l)) is convex in x, and therefore the non-convex program in (1) is solved as a sequence of convex programs as shown in (2). The original formulation of CCCP by Yuille and Rangarajan [30] deals with unconstrained and linearly constrained problems. However, the same formulation can be extended to handle any constraints (both convex and non-convex). CCCP has been extensively used in solving many nonconvex programs (of the form in (1)) that appear in machine learning. For example, [3] proposed a successive linear approximation (SLA) algorithm for feature selection in support vector machines, which can be seen as a special case of CCCP. Other applications where CCCP has been used include sparse principal component analysis [27], transductive SVMs [11, 5, 28], feature selection in SVMs [22], structured estimation [10], missing data problems in Gaussian processes and SVMs [26], etc. The algorithm in (2) starts at some random point x(0) ∈{x : ci(x) ≤0, i ∈[m]; dj(x) = 0, j ∈ [p]}, solves the program in (2) and therefore generates a sequence {x(l)}∞ l=0. The goal of this paper is to study the convergence of {x(l)}∞ l=0: (i) When does CCCP find a local minimum or a stationary point1 of the program in (1)? (ii) Does {x(l)}∞ l=0 converge? If so, to what and under what conditions? From a practical perspective, these questions are highly relevant, given that CCCP is widely applied in machine learning. In their original CCCP paper, Yuille and Rangarajan [30, Theorem 2] analyzed its convergence, but we believe the analysis is not complete. They showed that {x(l)}∞ l=0 satisfies the monotonic descent property, i.e., f(x(l+1)) ≤f(x(l)) and argued that this descent property ensures the convergence of {x(l)}∞ l=0 to a minimum or saddle point of the program in (1). However, a rigorous proof is not provided, to ensure that their claim holds for all u, v, {ci} and {dj}. Answering the previous questions, however, requires a rigorous proof of the convergence of CCCP that explicitly mentions the conditions under which it can happen. In the d.c. programming literature, Pham Dinh and Hoai An [8] proposed a primal-dual subdifferential method called DCA (d.c. algorithm) for solving a general d.c. program of the form min{u(x) −v(x) : x ∈Rn}, where it is assumed that u and v are proper lower semi-continuous convex functions, which form a larger class of functions than the class of differentiable functions. It can be shown that if v is differentiable, then DCA exactly reduces to CCCP. Unlike in CCCP, DCA involves constructing two sets of convex programs (called the primal and dual programs) and solving them iteratively in succession such that the solution of the primal is the initialization to the dual and vice-versa. See [8] for details. [8, Theorem 3] proves the convergence of DCA for general d.c. programs. The proof is specialized and technical. It fundamentally relies on d.c. duality, however, outlining the proof in any more detail requires a substantial discussion which would lead us too far here. In this work, we follow a fundamentally different approach and show that the convergence of CCCP, specifically, can be analyzed in a more simple and elegant way, by relying on Zangwill’s global convergence theory of iterative algorithms. We make some simple assumptions on the functions involved in (1), which are not too restrictive and therefore applicable to many practical situations. The tools employed in our proof are of completely different flavor than the ones used in the proof of DCA convergence: DCA convergence analysis exploits d.c. duality while we use the notion of point-to-set maps as introduced by Zangwill. Zangwill’s theory is a powerful and general framework to deal with the convergence issues of iterative algorithms. It has also been used to prove the convergence of the expectation-maximation (EM) algorithm [29], generalized alternating minimization algorithms [12], multiplicative updates in non-negative quadratic programming [25], etc. and is therefore a natural framework to analyze the convergence of CCCP in a more direct way. The paper is organized as follows. In Section 2, we provide a brief introduction to majorizationminimization (MM) algorithms and show that CCCP is obtained as a particular form of majorization1x∗is said to be a stationary point of a constrained optimization problem if it satisfies the corresponding Karush-Kuhn-Tucker (KKT) conditions. Assuming constraint qualification, KKT conditions are necessary for the local optimality of x∗. See [2, Section 11.3] for details. 2 minimization. The goal of this section is also to establish the literature on MM algorithms and show where CCCP fits in it. In Section 3, we present Zangwill’s theory of global convergence, which is a general framework to analyze the convergence behavior of iterative algorithms. This theory is used to address the global convergence of CCCP in Section 4. This involves analyzing the fixed points of the CCCP algorithm in (2) and then showing that the fixed points are the stationary points of the program in (1). The results in Section 4 are extended in Section 4.1 to analyze the convergence of the constrained concave-convex procedure that was proposed by [26] to deal with d.c. programs with d.c. constraints. We briefly discuss the local convergence issues of CCCP in Section 5 and conclude the section with an open question. 2 Majorization-minimization MM algorithms can be thought of as a generalization of the well-known EM algorithm [7]. The general principle behind MM algorithms was first enunciated by the numerical analysts, Ortega and Rheinboldt [23] in the context of line search methods. The MM principle appears in many places in statistical computation, including multidimensional scaling [6], robust regression [14], correspondence analysis [13], variable selection [16], sparse signal recovery [4], etc. We refer the interested reader to a tutorial on MM algorithms [15] and the references therein. The general idea of MM algorithms is as follows. Suppose we want to minimize f over Ω⊂Rn. The idea is to construct a majorization function g over Ω× Ωsuch that ½ f(x) ≤g(x, y), ∀x, y ∈Ω f(x) = g(x, x), ∀x ∈Ω . (3) Thus, g as a function of x is an upper bound on f and coincides with f at y. The majorization algorithm corresponding with this majorization function g updates x at iteration l by x(l+1) ∈arg min x∈Ωg(x, x(l)), (4) unless we already have x(l) ∈arg minx∈Ωg(x, x(l)), in which case the algorithm stops. The majorization function, g is usually constructed by using Jensen’s inequality for convex functions, the first-order Taylor approximation or the quadratic upper bound principle [1]. However, any other method can also be used to construct g as long as it satisfies (3). It is easy to show that the above iterative scheme decreases the value of f monotonically in each iteration, i.e., f(x(l+1)) ≤g(x(l+1), x(l)) ≤g(x(l), x(l)) = f(x(l)), (5) where the first inequality and the last equality follow from (3) while the sandwiched inequality follows from (4). Note that MM algorithms can be applied equally well to the maximization of f by simply reversing the inequality sign in (3) and changing the “min” to “max” in (4). In this case, the word MM refers to minorization-maximization, where the function g is called the minorization function. To put things in perspective, the EM algorithm can be obtained by constructing the minorization function g using Jensen’s inequality for concave functions. The construction of such a g is referred to as the E-step, while (4) with the “min” replaced by “max” is referred to as the M-step. The algorithm in (3) and (4) is also referred to as the auxiliary function method, e.g., for non-negative matrix factorization [18]. [17] studied this algorithm under the name optimization transfer while [19] referred to it as the SM algorithm, where “S” stands for the surrogate step (same as the majorization/minorization step) and “M” stands for the minimization/maximization step depending on the problem at hand. g is called the surrogate function. In the following example, we show that CCCP is an MM algorithm for a particular choice of the majorization function, g. Example 1 (Linear Majorization). Let us consider the optimization problem, minx∈Ωf(x) where f = u −v, with u and v both real-valued, convex, defined on Rn and v differentiable. Since v is convex, we have v(x) ≥v(y) + (x −y)T ∇v(y), ∀x, y ∈Ω. Therefore, f(x) ≤u(x) −v(y) −(x −y)T ∇v(y) =: g(x, y). (6) It is easy to verify that g is a majorization function of f. Therefore, we have x(l+1) ∈ arg min x∈Ωg(x, x(l)) = arg min x∈Ωu(x) −xT ∇v(x(l)). (7) 3 If Ωis a convex set, then the above procedure reduces to CCCP, which solves a sequence of convex programs. As mentioned before, CCCP is proposed for unconstrained and linearly constrained non-convex programs. This example shows that the same idea can be extended to any constraint set. Suppose u and v are strictly convex, then a strict descent can be achieved in (5) unless x(l+1) = x(l), i.e., if x(l+1) ̸= x(l), then f(x(l+1)) < g(x(l+1), x(l)) < g(x(l), x(l)) = f(x(l)). (8) The first strict inequality follows from (6). The strict convexity of u leads to the strict convexity of g and therefore g(x(l+1), x(l)) < g(x(l), x(l)) unless x(l+1) = x(l). 3 Global convergence theory of iterative algorithms For an iterative procedure like CCCP to be useful, it must converge to a local optimum or a stationary point from all or at least a significant number of initialization states and not exhibit other nonlinear system behaviors, such as divergence or oscillation. This behavior can be analyzed by using the global convergence theory of iterative algorithms developed by Zangwill [31]. Note that the word “global convergence” is a misnomer. We will clarify it below and also introduce some notation and terminology. To understand the convergence of an iterative procedure like CCCP, we need to understand the notion of a set-valued mapping, or point-to-set mapping, which is central to the theory of global convergence.2 A point-to-set map Ψ from a set X into a set Y is defined as Ψ : X →P(Y ), which assigns a subset of Y to each point of X, where P(Y ) denotes the power set of Y . We introduce few definitions related to the properties of point-to-set maps that will be used later. Suppose X and Y are two topological spaces. A point-to-set map Ψ is said to be closed at x0 ∈X if xk →x0 as k →∞, xk ∈X and yk →y0 as k →∞, yk ∈Ψ(xk), imply y0 ∈Ψ(x0). This concept of closure generalizes the concept of continuity for ordinary point-to-point mappings. A point-to-set map Ψ is said to be closed on S ⊂X if it is closed at every point of S. A fixed point of the map Ψ : X →P(X) is a point x for which {x} = Ψ(x), whereas a generalized fixed point of Ψ is a point for which x ∈Ψ(x). Ψ is said to be uniformly compact on X if there exists a compact set H independent of x such that Ψ(x) ⊂H for all x ∈X. Note that if X is compact, then Ψ is uniformly compact on X. Let φ : X →R be a continuous function. Ψ is said to be monotonic with respect to φ whenever y ∈Ψ(x) implies that φ(y) ≤φ(x). If, in addition, y ∈Ψ(x) and φ(y) = φ(x) imply that y = x, then we say that Ψ is strictly monotonic. Many iterative algorithms in mathematical programming can be described using the notion of pointto-set maps. Let X be a set and x0 ∈X a given point. Then an algorithm, A, with initial point x0 is a point-to-set map A : X →P(X) which generates a sequence {xk}∞ k=1 via the rule xk+1 ∈ A(xk), k = 0, 1, . . .. A is said to be globally convergent if for any chosen initial point x0, the sequence {xk}∞ k=0 generated by xk+1 ∈A(xk) (or a subsequence) converges to a point for which a necessary condition of optimality holds. The property of global convergence expresses, in a sense, the certainty that the algorithm works. It is very important to stress the fact that it does not imply (contrary to what the term might suggest) convergence to a global optimum for all initial points x0. With the above mentioned concepts, we now state Zangwill’s global convergence theorem [31, Convergence theorem A, page 91]. Theorem 2 ([31]). Let A : X →P(X) be a point-to-set map (an algorithm) that given a point x0 ∈X generates a sequence {xk}∞ k=0 through the iteration xk+1 ∈A(xk). Also let a solution set Γ ⊂X be given. Suppose (1) All points xk are in a compact set S ⊂X. (2) There is a continuous function φ : X →R such that: (a) x /∈Γ ⇒φ(y) < φ(x), ∀y ∈A(x), 2Note that depending on the objective and constraints, the minimizer of the CCCP algorithm in (2) need not be unique. Therefore, the algorithm takes x(l) as its input and returns a set of minimizers from which an element, x(l+1) is chosen. Hence the notion of point-to-set maps appear naturally in such iterative algorithms. 4 (b) x ∈Γ ⇒φ(y) ≤φ(x), ∀y ∈A(x). (3) A is closed at x if x /∈Γ. Then the limit of any convergent subsequence of {xk}∞ k=0 is in Γ. Furthermore, limk→∞φ(xk) = φ(x∗) for all limit points x∗. The general idea in showing the global convergence of an algorithm, A is to invoke Theorem 2 by appropriately defining φ and Γ. For an algorithm A that solves the minimization problem, min{f(x) : x ∈Ω}, the solution set, Γ is usually chosen to be the set of corresponding stationary points and φ can be chosen to be the objective function itself, i.e., f, if f is continuous. In Theorem 2, the convergence of φ(xk) to φ(x∗) does not automatically imply the convergence of xk to x∗. However, if A is strictly monotone with respect to φ, then Theorem 2 can be strengthened by using the following result due to Meyer [20, Theorem 3.1, Corollary 3.2]. Theorem 3 ([20]). Let A : X →P(X) be a point-to-set map such that A is uniformly compact, closed and strictly monotone on X, where X is a closed subset of Rn. If {xk}∞ k=0 is any sequence generated by A, then all limit points will be fixed points of A, φ(xk) →φ(x∗) =: φ∗as k →∞, where x∗is a fixed point, ∥xk+1 −xk∥→0, and either {xk}∞ k=0 converges or the set of limit points of {xk}∞ k=0 is connected. Define F(a) := {x ∈F : φ(x) = a} where F is the set of fixed points of A. If F(φ∗) is finite, then any sequence {xk}∞ k=0 generated by A converges to some x∗in F(φ∗). Both these results just use basic facts of analysis and are simple to prove and understand. Using these results on the global convergence of algorithms, [29] has studied the convergence properties of the EM algorithm, while [12] analyzed the convergence of generalized alternating minimization procedures. In the following section, we use these results to analyze the convergence of CCCP. 4 Convergence theorems for CCCP Let us consider the CCCP algorithm in (2) pertaining to the d.c. program in (1). Let Acccp be the point-to-set map, x(l+1) ∈Acccp(x(l)) such that Acccp(y) = arg min{u(x) −xT ∇v(y) : x ∈Ω}, (9) where Ω:= {x : ci(x) ≤0, i ∈[m], dj(x) = 0, j ∈[p]}. Let us assume that {ci} are differentiable convex functions defined on Rn. We now present the global convergence theorem for CCCP. Theorem 4 (Global convergence of CCCP−I). Let u and v be real-valued differentiable convex functions defined on Rn. Suppose ∇v is continuous. Let {x(l)}∞ l=0 be any sequence generated by Acccp defined by (9). Suppose Acccp is uniformly compact3 on Ωand Acccp(x) is nonempty for any x ∈Ω. Then, assuming suitable constraint qualification, all the limit points of {x(l)}∞ l=0 are stationary points of the d.c. program in (1). In addition liml→∞(u(x(l))−v(x(l))) = u(x∗)−v(x∗), where x∗is some stationary point of Acccp. Before we proceed with the proof of Theorem 4, we need a few additional results. The idea of the proof is to show that any generalized fixed point of Acccp is a stationary point of (1), which is shown below in Lemma 5, and then use Theorem 2 to analyze the generalized fixed points. Lemma 5. Suppose x∗is a generalized fixed point of Acccp and assume that constraints in (9) are qualified at x∗. Then, x∗is a stationary point of the program in (1). Proof. We have x∗∈Acccp(x∗) and the constraints in (9) are qualified at x∗. Then, there exists Lagrange multipliers {η∗ i }m i=1 ⊂R+ and {µ∗ j}p j=1 ⊂R such that the following KKT conditions hold: ∇u(x∗) −∇v(x∗) + Pm i=1 η∗ i ∇ci(x∗) + Pp j=1 µ∗ j∇dj(x∗) = 0, ci(x∗) ≤0, η∗ i ≥0, ci(x∗)η∗ i = 0, ∀i ∈[m] dj(x∗) = 0, µ∗ j ∈R, ∀j ∈[p]. (10) (10) is exactly the KKT conditions of (1) which are satisfied by (x∗, {η∗ i }, {µ∗ j}) and therefore, x∗ is a stationary point of (1). 3Assuming that for every x ∈Ω, the set H(x) := {y : u(y) −u(x) ≤v(y) −v(x), y ∈Acccp(Ω)} is bounded is also sufficient for the result to hold. 5 Before proving Theorem 4, we need a result to test the closure of Acccp. The following result from [12, Proposition 7] shows that the minimization of a continuous function forms a closed point-to-set map. A similar sufficient condition is also provided in [29, Equation 10]. Lemma 6 ([12]). Given a real-valued continuous function h on X × Y , define the point-to-set map Ψ : X →P(Y ) by Ψ(x) = arg min y′∈Y h(x, y′) = {y : h(x, y) ≤h(x, y′), ∀y′ ∈Y }. (11) Then, Ψ is closed at x if Ψ(x) is nonempty. We are now ready to prove Theorem 4. Proof of Theorem 4. The assumption of Acccp being uniformly compact on Ωensures that condition (1) in Theorem 2 is satisfied. Let Γ be the set of all generalized fixed points of Acccp and let φ = f = u −v. Because of the descent property in (5), condition (2) in Theorem 2 is satisfied. By our assumption on u and v, we have g(x, y) = u(x) −v(y) −(x −y)T ∇v(y) is continuous in x and y. Therefore, by Lemma 6, the assumption of non-emptiness of Acccp(x) for any x ∈Ωensures that Acccp is closed on Ωand so satisfies condition (3) in Theorem 2. Therefore, by Theorem 2, all the limit points of {x(l)}∞ l=0 are the generalized fixed points of Acccp and liml→∞(u(x(l)) − v(x(l))) = u(x∗) −v(x∗), where x∗is some generalized fixed point of Acccp. By Lemma 5, since the generalized fixed points of Acccp are stationary points of (1), the result follows. Remark 7. If Ωis compact, then Acccp is uniformly compact on Ω. In addition, since u is continuous on Ω, by the Weierstrass theorem4 [21], it is clear that Acccp(x) is nonempty for any x ∈Ωand therefore is also closed on Ω. This means, when Ωis compact, the result in Theorem 4 follows trivially from Theorem 2. In Theorem 4, we considered the generalized fixed points of Acccp. The disadvantage with this case is that it does not rule out “oscillatory” behavior [20]. To elaborate, we considered {x∗} ⊂ Acccp(x∗). For example, let Ω0 = {x1, x2} and let Acccp(x1) = Acccp(x2) = Ω0 and u(x1) − v(x1) = u(x2) −v(x2) = 0. Then the sequence {x1, x2, x1, x2, . . .} could be generated by Acccp, with the convergent subsequences converging to the generalized fixed points x1 and x2. Such an oscillatory behavior can be avoided if we allow Acccp to have fixed points instead of generalized fixed points. With appropriate assumptions on u and v, the following stronger result can be obtained on the convergence of CCCP through Theorem 3. Theorem 8 (Global convergence of CCCP−II). Let u and v be strictly convex, differentiable functions defined on Rn. Also assume ∇v be continuous. Let {x(l)}∞ l=0 be any sequence generated by Acccp defined by (9). Suppose Acccp is uniformly compact on Ωand Acccp(x) is nonempty for any x ∈Ω. Then, assuming suitable constraint qualification, all the limit points of {x(l)}∞ l=0 are stationary points of the d.c. program in (1), u(x(l)) −v(x(l)) →u(x∗) −v(x∗) =: f ∗ as l →∞, for some stationary point x∗, ∥x(l+1) −x(l)∥→0, and either {x(l)}∞ l=0 converges or the set of limit points of {x(l)}∞ l=0 is a connected and compact subset of S (f ∗), where S (a) := {x ∈S : u(x) −v(x) = a} and S is the set of stationary points of (1). If S (f ∗) is finite, then any sequence {x(l)}∞ l=0 generated by Acccp converges to some x∗in S (f ∗). Proof. Since u and v are strictly convex, the strict descent property in (8) holds and therefore Acccp is strictly monotonic with respect to f. Under the assumptions made about Acccp, Theorem 3 can be invoked, which says that all the limit points of {x(l)}∞ l=0 are fixed points of Acccp, which either converge or form a connected compact set. From Lemma 5, the set of fixed points of Acccp are already in the set of stationary points of (1) and the desired result follows from Theorem 3. Theorems 4 and 8 answer the questions that we raised in Section 1. These results explicitly provide sufficient conditions on u, v, {ci} and {dj} under which the CCCP algorithm finds a stationary point of (1) along with the convergence of the sequence generated by the algorithm. From Theorem 8, it should be clear that convergence of f(x(l)) to f ∗does not automatically imply the convergence of x(l) to x∗. The convergence in the latter sense requires more stringent conditions like the finiteness of the set of stationary points of (1) that assume the value of f ∗. 4Weierstrass theorem states: If f is a real continuous function on a compact set K ⊂Rn, then the problem min{f(x) : x ∈K} has an optimal solution x∗∈K. 6 4.1 Extensions So far, we have considered d.c. programs where the constraint set is convex. Let us consider a general d.c. program given by min x u0(x) −v0(x) s.t. ui(x) −vi(x) ≤0, i ∈[m], (12) where {ui}, {vi} are real-valued convex and differentiable functions defined on Rn. While dealing with kernel methods for missing variables, [26] encountered a problem of the form in (12) for which they proposed a constrained concave-convex procedure given by x(l+1) ∈arg min x u0(x) −bv0(x; x(l)) s.t. ui(x) −bvi(x; x(l)) ≤0, i ∈[m], (13) where bvi(x; x(l)) := vi(x(l)) + (x −x(l))T ∇vi(x(l)). Note that, similar to CCCP, the algorithm in (13) is a sequence of convex programs. Though [26, Theorem 1] have provided a convergence analysis for the algorithm in (13), it is however not complete due to the fact that the convergence of {x(l)}∞ l=0 is assumed. In this subsection, we provide its convergence analysis, following an approach similar to what we did for CCCP by considering a point-to-set map, Bccp associated with the iterative algorithm in (13), where x(l+1) ∈Bccp(x(l)). In Theorem 10, we provide the global convergence result for the constrained concave-convex procedure, which is an equivalent version of Theorem 4 for CCCP. We do not provide the stronger version of the result as in Theorem 8 as it can be obtained by assuming strict convexity of u0 and v0. Before proving Theorem 10, we need an equivalent version of Lemma 5 which we provide below. Lemma 9. Suppose x∗is a generalized fixed point of Bccp and assume that constraints in (13) are qualified at x∗. Then, x∗is a stationary point of the program in (12). Proof. Based on the assumptions x∗∈Bccp(x∗) and the constraint qualification at x∗in (13), there exist Lagrange multipliers {η∗ i }m i=1 ⊂R+ (for simplicity, we assume all the constraints to be inequality constraints) such that the following KKT conditions hold: ∇u0(x∗) + Pm i=1 η∗ i (∇ui(x∗) −∇vi(x∗)) = ∇v0(x∗), ui(x∗) −vi(x∗) ≤0, η∗ i ≥0, i ∈[m], (ui(x∗) −vi(x∗))η∗ i = 0, i ∈[m]. (14) which is exactly the KKT conditions for (12) satisfied by (x∗, {η∗ i }) and therefore, x∗is a stationary point of (12). Theorem 10 (Global convergence of constrained CCP). Let {ui}, {vi} be real-valued differentiable convex functions on Rn. Assume ∇v0 to be continuous. Let {x(l)}∞ l=0 be any sequence generated by Bccp defined in (13). Suppose Bccp is uniformly compact on Ω:= {x : ui(x) −vi(x) ≤0, i ∈ [m]} and Bccp(x) is nonempty for any x ∈Ω. Then, assuming suitable constraint qualification, all the limit points of {x(l)}∞ l=0 are stationary points of the d.c. program in (12). In addition liml→∞(u0(x(l)) −v0(x(l))) = u0(x∗) −v0(x∗), where x∗is some stationary point of Bccp. Proof. The proof is very similar to that of Theorem 4 wherein we check whether Bccp satisfies the conditions of Theorem 2 and then invoke Lemma 9. The assumptions mentioned in the statement of the theorem ensure that conditions (1) and (3) in Theorem 2 are satisfied. [26, Theorem 1] has proved the descent property, similar to that of (5), which simply follows from the linear majorization idea and therefore the descent property in condition (2) of Theorem 2 holds. Therefore, the result follows from Theorem 2 and Lemma 9. 5 On the local convergence of CCCP: An open problem The study so far has been devoted to the global convergence analysis of CCCP and the constrained concave-convex procedure. As mentioned before, we say an algorithm is globally convergent if for any chosen starting point, x0, the sequence {xk}∞ k=0 generated by xk+1 ∈A(xk) converges to a point for which a necessary condition of optimality holds. In the results so far, we have shown 7 that all the limit points of any sequence generated by CCCP (resp. its constrained version) are the stationary points (local extrema or saddle points) of the program in (1) (resp. (12)). Suppose, if x0 is chosen such that it lies in an ϵ-neighborhood around a local minima, x∗, then will the CCCP sequence converge to x∗? If so, what is the rate of convergence? This is the question of local convergence that needs to be addressed. [24] has studied the local convergence of bound optimization algorithms (of which CCCP is an example) to compare the rate of convergence of such methods to that of gradient and second-order methods. In their work, they considered the unconstrained version of CCCP with Acccp to be a pointto-point map that is differentiable. They showed that depending on the curvature of u and v, CCCP will exhibit either quasi-Newton behavior with fast, typically superlinear convergence or extremely slow, first-order convergence behavior. However, extending these results to the constrained setup as in (2) is not obvious. The following result due to Ostrowski which can be found in [23, Theorem 10.1.3] provides a way to study the local convergence of iterative algorithms. Proposition 11 (Ostrowski). Suppose that Ψ : U ⊂Rn →Rn has a fixed point x∗∈int(U) and Ψ is Fr´echet-differentiable at x∗. If the spectral radius of Ψ′(x∗) satisfies ρ(Ψ′(x∗)) < 1, and if x0 is sufficiently close to x∗, then the iterates {xk} defined by xk+1 = Ψ(xk) all lie in U and converge to x∗. Few remarks are in place regarding the usage of Proposition 11 to study the local convergence of CCCP. Note that Proposition 11 treats Ψ as a point-to-point map which can be obtained by choosing u and v to be strictly convex so that x(l+1) is the unique minimizer of (2). x∗in Proposition 11 can be chosen to be a local minimum. Therefore, the desired result of local convergence with at least linear rate of convergence is obtained if we show that ρ(Ψ′(x∗)) < 1. However, currently we are not aware of a way to compute the differential of Ψ and, moreover, to impose conditions on the functions in (2) so that Ψ is a differentiable map. This is an open question coming out of this work. On the other hand, the local convergence behavior of DCA has been proved for two important classes of d.c. programs: (i) the trust region subproblem [9] (minimization of a quadratic function over a Euclidean ball) and (ii) nonconvex quadratic programs [8]. We are not aware of local optimality results for general d.c. programs using DCA. 6 Conclusion & Discussion The concave-convex procedure (CCCP) is widely used in machine learning. In this work, we analyze its global convergence behavior by using results from the global convergence theory of iterative algorithms. We explicitly mention the conditions under which any sequence generated by CCCP converges to a stationary point of a d.c. program with convex constraints. The proposed approach allows an elegant and direct proof and is fundamentally different from the highly technical proof for the convergence of DCA, which implies convergence for CCCP. It illustrates the power and generality of Zangwill’s global convergence theory as a framework for proving the convergence of iterative algorithms. We also briefly discuss the local convergence of CCCP and present an open question, the settlement of which would address the local convergence behavior of CCCP. Acknowledgments The authors thank anonymous reviewers for their constructive comments. They wish to acknowledge support from the National Science Foundation (grant DMS-MSPA 0625409), the Fair Isaac Corporation and the University of California MICRO program. References [1] D. B¨ohning and B. G. Lindsay. Monotonicity of quadratic-approximation algorithms. Annals of the Institute of Statistical Mathematics, 40(4):641–663, 1988. [2] J. F. Bonnans, J. C. Gilbert, C. Lemar´echal, and C. A. Sagastiz´abal. Numerical Optimization: Theoretical and Practical Aspects. Springer-Verlag, 2006. [3] P. S. Bradley and O. L. Mangasarian. Feature selection via concave minimization and support vector machines. In Proc. 15th International Conf. on Machine Learning, pages 82–90. Morgan Kaufmann, San Francisco, CA, 1998. 8 [4] E. J. Candes, M. Wakin, and S. Boyd. Enhancing sparsity by reweighted ℓ1 minimization. J. Fourier Anal. Appl., 2007. To appear. [5] R. Collobert, F. Sinz, J. Weston, and L. Bottou. Large scale transductive SVMs. Journal of Machine Learning Research, 7:1687–1712, 2006. [6] J. deLeeuw. Applications of convex analysis to multidimensional scaling. In J. R. Barra, F. Brodeau, G. Romier, and B. Van Cutsem, editors, Recent advantages in Statistics, pages 133–146, Amsterdam, The Netherlands, 1977. North Holland Publishing Company. [7] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B, 39:1–38, 1977. [8] T. Pham Dinh and L. T. Hoai An. Convex analysis approach to d.c. programming: Theory, algorithms and applications. Acta Mathematica Vietnamica, 22(1):289–355, 1997. [9] T. Pham Dinh and L. T. Hoai An. D.c. optimization algorithms for solving the trust region subproblem. SIAM Journal of Optimization, 8:476–505, 1998. [10] C. B. Do, Q. V. Le, C. H. Teo, O. Chapelle, and A. J. Smola. Tighter bounds for structured estimation. In Advances in Neural Information Processing Systems 21, 2009. To appear. [11] G. Fung and O. L. Mangasarian. Semi-supervised support vector machines for unlabeled data classification. Optimization Methods and Software, 15:29–44, 2001. [12] A. Gunawardana and W. Byrne. Convergence theorems for generalized alternating minimization procedures. Journal of Machine Learning Research, 6:2049–2073, 2005. [13] W. J. Heiser. Correspondence analysis with least absolute residuals. Comput. Stat. Data Analysis, 5:337– 356, 1987. [14] P. J. Huber. Robust Statistics. John Wiley, New York, 1981. [15] D. R. Hunter and K. Lange. A tutorial on MM algorithms. The American Statistician, 58:30–37, 2004. [16] D. R. Hunter and R. Li. Variable selection using MM algorithms. Annals of Statistics, 33:1617–1642, 2005. [17] K. Lange, D. R. Hunter, and I. Yang. Optimization transfer using surrogate objective functions with discussion. Journal of Computational and Graphical Statistics, 9(1):1–59, 2000. [18] D. D. Lee and H. S. Seung. Algorithms for non-negative matrix factorization. In T.K. Leen, T.G. Dietterich, and V. Tresp, editors, Advances in Neural Information Processing Systems 13, pages 556–562. MIT Press, Cambridge, 2001. [19] X.-L. Meng. Discussion on “optimization transfer using surrogate objective functions”. Journal of Computational and Graphical Statistics, 9(1):35–43, 2000. [20] R. R. Meyer. Sufficient conditions for the convergence of monotonic mathematical programming algorithms. Journal of Computer and System Sciences, 12:108–121, 1976. [21] M. Minoux. Mathematical Programming: Theory and Algorithms. John Wiley & Sons Ltd., 1986. [22] J. Neumann, C. Schn¨orr, and G. Steidl. Combined SVM-based feature selection and classification. Machine Learning, 61:129–150, 2005. [23] J. M. Ortega and W. C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970. [24] R. Salakhutdinov, S. Roweis, and Z. Ghahramani. On the convergence of bound optimization algorithms. In Proc. 19th Conference in Uncertainty in Artificial Intelligence, pages 509–516, 2003. [25] F. Sha, Y. Lin, L. K. Saul, and D. D. Lee. Multiplicative updates for nonnegative quadratic programming. Neural Computation, 19:2004–2031, 2007. [26] A. J. Smola, S. V. N. Vishwanathan, and T. Hofmann. Kernel methods for missing variables. In Proc. of the Tenth International Workshop on Artificial Intelligence and Statistics, 2005. [27] B. K. Sriperumbudur, D. A. Torres, and G. R. G. Lanckriet. Sparse eigen methods by d.c. programming. In Proc. of the 24th Annual International Conference on Machine Learning, 2007. [28] L. Wang, X. Shen, and W. Pan. On transductive support vector machines. In J. Verducci, X. Shen, and J. Lafferty, editors, Prediction and Discovery. American Mathematical Society, 2007. [29] C. F. J. Wu. On the convergence properties of the EM algorithm. Annals of Statistics, 11(1):95–103, 1983. [30] A. L. Yuille and A. Rangarajan. The concave-convex procedure. Neural Computation, 15:915–936, 2003. [31] W. I. Zangwill. Nonlinear Programming: A Unified Approach. Prentice-Hall, Englewood Cliffs, N.J., 1969. 9
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Orthogonal Matching Pursuit from Noisy Measurements: A New Analysis∗ Alyson K. Fletcher University of California, Berkeley Berkeley, CA alyson@eecs.berkeley.edu Sundeep Rangan Qualcomm Technologies Bedminster, NJ srangan@qualcomm.com Abstract A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m = 4k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n →∞. This work strengthens this result by showing that a lower number of measurements, m = 2k log(n −k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies kmin ≤k ≤kmax but is unknown, m = 2kmax log(n −kmin) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m = 2k log(n −k) exactly matches the number of measurements required by the more complex lasso method for signal recovery in a similar SNR scaling. 1 Introduction Suppose x ∈Rn is a sparse vector, meaning its number of nonzero components k is smaller than n. The support of x is the locations of the nonzero entries and is sometimes called its sparsity pattern. A common sparse estimation problem is to infer the sparsity pattern of x from linear measurements of the form y = Ax + w, (1) where A ∈Rm×n is a known measurement matrix, y ∈Rm represents a vector of measurements and w ∈Rm is a vector of measurements errors (noise). Sparsity pattern detection and related sparse estimation problems are classical problems in nonlinear signal processing and arise in a variety of applications including wavelet-based image processing [1] and statistical model selection in linear regression [2]. There has also been considerable recent interest in sparsity pattern detection in the context of compressed sensing, which focuses on large random measurement matrices A [3–5]. It is this scenario with random measurements that will be analyzed here. Optimal subset recovery is NP-hard [6] and usually involves searches over all the n k possible support sets of x. Thus, most attention has focused on approximate methods for reconstruction. One simple and popular approximate algorithm is orthogonal matching pursuit (OMP) developed in [7–9]. OMP is a simple greedy method that identifies the location of one nonzero component of x at a time. A version of the algorithm will be described in detail below in Section 2. The best known ∗This work was supported in part by a University of California President’s Postdoctoral Fellowship and the Centre Bernoulli at ´Ecole Polytechnique F´ed´erale de Lausanne. 1 analysis of the performance of OMP for large random matrices is due to Tropp and Gilbert [10,11]. Among other results, Tropp and Gilbert show that when the number of measurements scales as m ≥(1 + δ)4k log(n) (2) for some δ > 0, A has i.i.d. Gaussian entries, and the measurements are noise-free (w = 0), the OMP method will recover the correct sparse pattern of x with a probability that approaches one as n and k →∞. Deterministic conditions on the matrix A that guarantee recovery of x by OMP are given in [12]. However, numerical experiments reported in [10] suggest that a smaller number of measurements than (2) may be sufficient for asymptotic recovery with OMP. Specifically, the experiments suggest that the constant 4 can be reduced to 2. Our main result, Theorem 1 below, proves this conjecture. Specifically, we show that the scaling in measurements m ≥(1 + δ)2k log(n −k) (3) is also sufficient for asymptotic reliable recovery with OMP provided both n −k and k →∞. The result goes further by allowing uncertainty in the sparsity level k. We also improve upon the Tropp–Gilbert analysis by accounting for the effect of the noise w. While the Tropp–Gilbert analysis requires that the measurements are noise-free, we show that the scaling (3) is also sufficient when there is noise w, provided the signal-to-noise ratio (SNR) goes to infinity. The main significance of the new scaling (3) is that it exactly matches the conditions for sparsity pattern recovery using the well-known lasso method. The lasso method, which will be described in detail in Section 4, is based on a convex relaxation of the optimal detection problem. The best analysis of the sparsity pattern recovery with lasso is due to Wainwright [13, 14]. He showed in [13] that under a similar high SNR assumption, the scaling (3) in number of measurements is both necessary and sufficient for asymptotic reliable sparsity pattern detection.1 Now, although the lasso method is often more complex than OMP, it is widely believed that lasso has superior performance [10]. Our results show that at least for sparsity pattern recovery with large Gaussian measurement matrices in high SNR, lasso and OMP have identical performance. Hence, the additional complexity of lasso for these problems is not warranted. Of course, neither lasso nor OMP is the best known approximate algorithm, and our intention is not to claim that OMP is optimal in any sense. For example, where there is no noise in the measurements, the lasso minimization (14) can be replaced by bx = arg min v∈Rn ∥v∥1, s.t. y = Av. A well-known analysis due to Donoho and Tanner [15] shows that, for i.i.d. Gaussian measurement matrices, this minimization will recover the correct vector with m ≍2k log(n/m) (4) when k ≪n. This scaling is fundamentally better than the scaling (3) achieved by OMP and lasso. There are also several variants of OMP that have shown improved performance. The CoSaMP algorithm of Needell and Tropp [16] and subspace pursuit algorithm of Dai and Milenkovic [17] achieve a scaling similar to (4). Other variants of OMP include the stagewise OMP [18] and regularized OMP [19]. Indeed with the recent interest in compressed sensing, there is now a wide range of promising algorithms available. We do not claim that OMP achieves the best performance in any sense. Rather, we simply intend to show that both OMP and lasso have similar performance in certain scenarios. Our proof of (3) follows along the same lines as Tropp and Gilbert’s proof of (2), but with two key differences. First, we account for the effect of the noise by separately considering its effect in the “true” subspace and its orthogonal complement. Second and more importantly, we provide a tighter bound on the maximum correlation of the incorrect vectors. Specifically, in each iteration of the 1Sufficient conditions under weaker conditions on the SNR are more subtle [14]: the scaling of SNR with n determines the sequences of regularization parameters for which asymptotic almost sure success is achieved, and the regularization parameter sequence affects the sufficient number of measurements. 2 OMP algorithm, there are n −k possible incorrect vectors that the algorithm can choose. Since the algorithm runs for k iterations, there are total of k(n −k) possible error events. The Tropp and Gilbert proof bounds the probability of these error events with a union bound, essentially treating them as statistically independent. However, here we show that energies on any one of the incorrect vectors across the k iterations are correlated. In fact, they are precisely described by samples on a certain normalized Brownian motion. Exploiting this correlation we show that the tail bound on error probability grows as n −k, not k(n −k), independent events. The outline of the remainder of this paper is as follows. Section 2 describes the OMP algorithm. Our main result, Theorem 1, is stated in Section 3. A comparison to lasso is provided in Section 4, and we suggest some future problems in Section 6. The proof of the main result is sketched in Section 7. 2 Orthogonal Matching Pursuit To describe the algorithm, suppose we wish to determine the vector x from a vector y of the form (1). Let Itrue = { j : xj ̸= 0 }, (5) which is the support of the vector x. The set Itrue will also be called the sparsity pattern. Let k = |Itrue|, which is the number of nonzero components of x. The OMP algorithm produces a sequence of estimates ˆI(t), t = 0, 1, 2, . . ., of the sparsity pattern Itrue, adding one index at a time. In the description below, let aj denote the jth column of A. Algorithm 1 (Orthogonal Matching Pursuit) Given a vector y ∈Rm, a measurement matrix A ∈Rm×n and threshold level µ > 0, compute an estimate ˆIOMP of the sparsity pattern of x as follows: 1. Initialize t = 0 and ˆI(t) = ∅. 2. Compute P(t), the projection operator onto the orthogonal complement of the span of {ai, i ∈ˆI(t)}. 3. For each j, compute ρ(t, j) = |a′ jP(t)y|2 ∥P(t)y∥2 , and let [ρ∗(t), i∗(t)] = max j=1,...,n ρ(t, j), (6) where ρ∗(t) is the value of the maximum and i∗(t) is an index which achieves the maximum. 4. If ρ∗(t) > µ, set ˆI(t + 1) = ˆI(t) ∪{i∗(t)}. Also, increment t = t + 1 and return to step 2. 5. Otherwise stop. The final estimate of the sparsity pattern is ˆIOMP = ˆI(t). Note that since P(t) is the projection onto the orthogonal complement of aj for all j ∈ˆI(t), P(t)aj = 0 for all j ∈ˆI(t). Hence, ρ(t, j) = 0 for all j ∈ˆI(t), and therefore the algorithm will not select the same vector twice. The algorithm above only provides an estimate, ˆIOMP, of the sparsity pattern of Itrue. Using ˆIOMP, one can estimate the vector x in a number of ways. For example, one can take the least-squares estimate, bx = arg min ∥y −Av∥2 (7) where the minimization is over all vectors v such vj = 0 for all j ̸∈ˆIOMP. The estimate bx is the projection of the noisy vector y onto the space spanned by the vectors ai with i in the sparsity pattern estimate ˆIOMP. However, this paper only analyzes the sparsity pattern estimate ˆIOMP itself, and not the vector estimate bx. 3 3 Asymptotic Analysis We analyze the OMP algorithm in the previous section under the following assumptions. Assumption 1 Consider a sequence of sparse recovery problems, indexed by the vector dimension n. For each n, let x ∈Rn be a deterministic vector and let k = k(n) be the number of nonzero components in x. Also assume: (a) The sparsity level, k = k(n) satisfies k(n) ∈[kmin(n), kmax(n)], (8) for some deterministic sequences kmin(n) and kmax(n) with kmin(n) →∞as n →∞ and kmax(n) < n/2 for all n. (b) The number of measurements m = m(n) is a deterministic sequence satisfying m ≥(1 + δ)2kmax log(n −kmin), (9) for some δ > 0. (c) The minimum component power x2 min satisfies lim n→∞kx2 min = ∞, (10) where xmin = min j∈Itrue |xj|, (11) is the magnitude of the smallest nonzero component of x. (d) The powers of the vectors ∥x∥2 satisfy lim n→∞ 1 (n −k)ǫ log 1 + ∥x∥2 = 0. (12) for all ǫ > 0. (e) The vector y is a random vector generated by (1) where A and w have i.i.d. Gaussian components with zero mean and variance of 1/m. Assumption 1(a) provides a range on the sparsity level, k. As we will see below in Section 5, bounds on this range are necessary for proper selection of the threshold level µ > 0. Assumption 1(b) is our the main scaling law on the number of measurements that we will show is sufficient for asymptotic reliable recovery. In the special case when k is known so that kmax = kmin = k, we obtain the simpler scaling law m ≥(1 + δ)2k log(n −k). (13) We have contrasted this scaling law with the Tropp–Gilbert scaling law (2) in Section 1. We will also compare it to the scaling law for lasso in Section 4. Assumption 1(c) is critical and places constraints on the smallest component magnitude. The importance of the smallest component magnitude in the detection of the sparsity pattern was first recognized by Wainwright [13,14,20]. Also, as discussed in [21], the condition requires that signalto-noise ratio (SNR) goes to infinity. Specifically, if we define the SNR as SNR = E∥Ax∥2 ∥w∥2 , then under Assumption 1(e), it can be easily checked that SNR = ∥x∥2. Since x has k nonzero components, ∥x∥2 ≥kx2 min, and therefore condition (10) requires that SNR →∞. For this reason, we will call our analysis of OMP a high-SNR analysis. The analysis of OMP with SNR that remains bounded above is an interesting open problem. 4 Assumption (d) is technical and simply requires that the SNR does not grow too quickly with n. Note that even if SNR = O(kα) for any α > 0, Assumption 1(d) will be satisfied. Assumption 1(e) states that our analysis concerns large Gaussian measurement matrices A and Gaussian noise w. Theorem 1 Under Assumption 1, there exists a sequence of threshold levels µ = µ(n) such that the OMP method in Algorithm 1 will asymptotically detect the correct sparsity pattern in that lim n→∞Pr ˆIOMP ̸= Itrue = 0. Moreover, the threshold levels µ can be selected simply as a function of kmin, kmax, n, m and δ. Theorem 1 provides our main result and shows that the scaling law (9) is sufficient for asymptotic recovery. 4 Comparison to Lasso Performance It is useful to compare the scaling law (13) to the number of measurements required by the widelyused lasso method described for example in [22]. The lasso method finds an estimate for the vector x in (1) by solving the quadratic program bx = arg min v∈Rn ∥y −Av∥2 + µ∥v∥1, (14) where µ > 0 is an algorithm parameter that trades off the prediction error with the sparsity of the solution. Lasso is sometimes referred to as basis pursuit denoising [23]. While the optimization (14) is convex, the running time of lasso is significantly longer than OMP unless A has some particular structure [10]. However, it is generally believed that lasso has superior performance. The best analysis of lasso for sparsity pattern recovery for large random matrices is due to Wainwright [13,14]. There, it is shown that with an i.i.d. Gaussian measurement matrix and white Gaussian noise, the condition (13) is necessary for asymptotic reliable detection of the sparsity pattern. In addition, under the condition (10) on the minimum component magnitude, the scaling (13) is also sufficient. We thus conclude that OMP requires an identical scaling in the number of measurements to lasso. Therefore, at least for sparsity pattern recovery from measurements with large random Gaussian measurement matrices and high SNR, there is no additional performance improvement with the more complex lasso method over OMP. 5 Threshold Selection and Stopping Conditions In many problems, the sparsity level k is not known a priori and must be detected as part of the estimation process. In OMP, the sparsity level of estimated vector is precisely the number of iterations conducted before the algorithm terminates. Thus, reliable sparsity level estimation requires a good stopping condition. When the measurements are noise-free and one is concerned only with exact signal recovery, the optimal stopping condition is simple: the algorithm should simply stop whenever there is no more error. That is ρ∗(t) = 0 in (6). However, with noise, selecting the correct stopping condition requires some care. The OMP method as described in Algorithm 1 uses a stopping condition based on testing if ρ∗(t) > µ for some threshold µ. One of the appealing features of Theorem 1 is that it provides a simple sufficient condition under which this threshold mechanism will detect the correct sparsity level. Specifically, Theorem 1 provides a range k ∈[kmin, kmax] under which there exists a threshold that the OMP algorithm will terminate in the correct number of iterations. The larger the number of measurements, m, the greater one can make the range [kmin, kmax]. The formula for the threshold level is given in (20). Of course, in practice, one may deliberately want to stop the OMP algorithm with fewer iterations than the “true” sparsity level. As the OMP method proceeds, the detection becomes less reliable and it is sometimes useful to stop the algorithm whenever there is a high chance of error. Stopping early 5 may miss some small components, but may result in an overall better estimate by not introducing too many erroneous components or components with too much noise. However, since our analysis is only concerned with exact sparsity pattern recovery, we do not consider this type of stopping condition. 6 Conclusions and Future Work We have provided an improved scaling law on the number of measurements for asymptotic reliable sparsity pattern detection with OMP. This scaling law exactly matches the scaling needed by lasso under similar conditions. However, much about the performance of OMP is still not fully understood. Most importantly, our analysis is limited to high SNR. It would be interesting to see if reasonable sufficient conditions can be derived for finite SNR as well. Also, our analysis has been restricted to exact sparsity pattern recovery. However, in many problems, especially with noise, it is not necessary to detect every component in the sparsity pattern. It would be useful if partial support recovery results such as [24–27] can be obtained for OMP. Finally, our main scaling law (9) is only sufficient. While numerical experiments in [10,28] suggest that this scaling is also necessary for vectors with equal magnitude, it is possible that OMP can perform better than the scaling law (9) when the component magnitudes have some variation; this is demonstrated numerically in [28]. The benefit of dynamic range in an OMP-like algorithm has also been observed in [29] and sparse Bayesian learning methods in [30,31]. 7 Proof Sketch for Theorem 1 7.1 Proof Outline Due to space considerations, we only sketch the proof; additional details are given in [28]. The main difficulty in analyzing OMP is the statistical dependencies between iterations in the OMP algorithm. Following along the lines of the Tropp–Gilbert proof in [10], we avoid these difficulties by considering the following “genie” algorithm. A similar alternate algorithm is analyzed in [29]. 1. Initialize t = 0 and Itrue(t) = ∅. 2. Compute Ptrue(t), the projection operator onto the orthogonal complement of the span of {ai, i ∈Itrue(t)}. 3. For all j = 1, . . . , n, compute ρtrue(t, j) = |a′ jPtrue(t)y|2 ∥Ptrue(t)y∥2 , (15) and let [ρ∗ true(t), i∗(t)] = max j∈Itrue ρtrue(t, j). (16) 4. If t < k, set Itrue(t + 1) = Itrue(t) ∪{i∗(t)}. Increment t = t + 1 and return to step 2. 5. Otherwise stop. The final estimate of the sparsity pattern is Itrue(k). This “genie” algorithm is identical to the regular OMP method in Algorithm 1, except that it runs for precisely k iterations as opposed to using a threshold µ for the stopping condition. Also, in the maximization in (16), the genie algorithm searches over only the correct indices j ∈Itrue. Hence, this genie algorithm can never select an incorrect index j ̸∈Itrue. Also, as in the regular OMP algorithm, the genie algorithm will never select the same vector twice for almost all vectors y. Therefore, after k iterations, the genie algorithm will have selected all the k indices in Itrue and terminate with correct sparsity pattern estimate Itrue(k) = Itrue with probability one. So, we need to show that true OMP algorithm behaves identically to the “genie” algorithm with high probability. 6 To this end, define the following two probabilities: pMD = Pr max t=0,...k−1 min j∈Itrue ρtrue(t, j) ≤µ (17) pFA = Pr max t=0,...k max j̸∈Itrue ρtrue(t, j) ≥µ (18) Both probabilities are implicitly functions of n. The first term, pMD, can be interpreted as a “missed detection” probability, since it corresponds to the event that the maximum correlation energy ρtrue(t, j) on the correct vectors j ∈Itrue falls below the threshold. We call the second term pFA the “false alarm” probability since it corresponds to the maximum energy on one of the “incorrect” indices j ̸∈Itrue exceeding the threshold. A simple induction argument shows that if there are no missed detections or false alarms, the true OMP algorithm will select the same vectors as the “genie” algorithm, and therefore recover the sparsity pattern. This shows that Pr ˆIOMP ̸= Itrue ≤pMD + pFA. So we need to show that there exists a sequence of thresholds µ = µ(n) > 0, such that pMD and pFA →0 as n →∞. To set this threshold, we select an ǫ > 0 such that 1 + δ 1 + ǫ ≥1 + ǫ, (19) where δ is from (9). Then, define the threshold level µ = µ(n) = 2(1 + ǫ) m log(n −kmin). (20) 7.2 Probability of Missed Detection The proof that pMD →0 is similar to that of Tropp and Gilbert’s proof in [10]. The key modification is to use (10) to show that the effect of the noise is asymptotically negligible so that for large n, y ≈Ax = Φxtrue. (21) This is done by separately considering the components of w in the span of the vectors aj for j ∈ Itrue and its orthogonal complement. One then follows the Tropp–Gilbert proof for the noise-free case to show that max j∈Itrue ρtrue(t, j) ≥1 k for large k. Hence, using (9) and (20) one can then show lim inf n→∞ max j∈Itrue 1 µρtrue(t, j) ≥1 + ǫ, which shows that pMD →0. 7.3 Probability of False Alarm This part is harder. Define z(t, j) = a′ jPtrue(t)y ∥Ptrue(t)y∥, so that ρtrue(t, j) = |z(t, j)|2. Now, Ptrue(t) and y are functions of w and aj for j ∈Itrue. Therefore, they are independent of aj for any j ̸∈Itrue. Also, since the vectors aj have i.i.d. Gaussian components with variance 1/m, conditional on Ptrue(t) and y, z(t, j) is normal with variance 1/m. Hence, mρtrue(t, j) is a chi-squared random variable with one degree of freedom. Now, there are k(n −k) values of ρtrue(t, k) for t = 1, . . . , k and j ̸∈Itrue. The Tropp–Gilbert proof bounds the maximum of these k(n −k) value by the standard tail bound max j̸∈Itrue max t=1,...,k ρtrue(t, j) ≤2 m log(k(n −k)) ≤2 m log(n2) = 4 m log(n). 7 To improve the bound in this proof, we exploit the fact that for any j, the values of z(t, j) are correlated. In fact, we show that the values z(t, j), t = 1, . . . , k are distributed identically to points on a normalized Brownian motion. Specifically, let W(s) be a standard linear Brownian motion and let S(s) be the normalized Brownian motion S(s) = 1 √sB(s), s > 0. (22) We then show that, for every j, there exists times s1, . . . , sk with 1 ≤s1 < · · · < sk ≤1 + ∥x∥2 such that the vector z(j) = [z(1, j), . . . , z(k, j)] is identically distributed to [S(s1), . . . , S(sj)]. Hence, max t=1,...,k |z(t, j)|2 = max t=1,...,k |S(sj)|2 ≤ sup s∈[1,1+∥x∥2] |S(s)|2. The right-hand side of the sample path can then be bounded by the reflection principle [32]. This yields an improved bound, max j̸∈Itrue max t=1,...,k ρtrue(t, j) ≤2 m log(n −k). Combining this with (20) shows lim inf n→∞ max j∈Itrue 1 µρtrue(t, j) ≥ 1 1 + ǫ, which shows that pFA →0. References [1] S. Mallat. A Wavelet Tour of Signal Processing. Academic Press, second edition, 1999. [2] A. Miller. Subset Selection in Regression. Number 95 in Monographs on Statistics and Applied Probability. Chapman & Hall/CRC, New York, second edition, 2002. [3] E. J. Cand`es, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 52(2):489– 509, February 2006. [4] D. L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289–1306, April 2006. [5] E. J. Cand`es and T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inform. Theory, 52(12):5406–5425, December 2006. [6] B. K. Natarajan. Sparse approximate solutions to linear systems. SIAM J. Computing, 24(2):227–234, April 1995. [7] S. Chen, S. A. Billings, and W. Luo. Orthogonal least squares methods and their application to non-linear system identification. Int. J. Control, 50(5):1873–1896, November 1989. [8] Y. C. Pati, R. Rezaiifar, and P. S. Krishnaprasad. Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition. In Conf. Rec. 27th Asilomar Conf. Sig., Sys., & Comput., volume 1, pages 40–44, Pacific Grove, CA, November 1993. [9] G. Davis, S. Mallat, and Z. Zhang. Adaptive time-frequency decomposition. Optical Eng., 37(7):2183–2191, July 1994. [10] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit. IEEE Trans. Inform. Theory, 53(12):4655–4666, December 2007. [11] J. A. Tropp and A. C. Gilbert. Signal recovery from random measurements via orthogonal matching pursuit: The Gaussian case. Appl. Comput. Math. 2007-01, California Inst. of Tech., August 2007. 8 [12] J. A. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theory, 50(10):2231–2242, October 2004. [13] M. J. Wainwright. Sharp thresholds for high-dimensional and noisy recovery of sparsity. Technical report, Univ. of California, Berkeley, Dept. of Statistics, May 2006. arXiv:math.ST/0605740 v1 30 May 2006. [14] M. J. Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery using ℓ1-constrained quadratic programming (lasso). IEEE Trans. Inform. Theory, 55(5):2183–2202, May 2009. [15] D. L. Donoho and J. Tanner. Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc., 22(1):1–53, January 2009. [16] D. Needell and J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Appl. Comput. Harm. Anal., 26(3):301–321, July 2008. [17] W. Dai and O. Milenkovic. Subspace pursuit for compressive sensing: Closing the gap between performance and complexity. arXiv:0803.0811v1 [cs.NA]., March 2008. [18] D. L. Donoho, Y. Tsaig, I. Drori, and J. L. Starck. Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit. preprint, March 2006. [19] D. Needell and R. Vershynin. Uniform uncertainty principle and signal recovery via regularized orthogonal matching pursuit. Found. Comput. Math., 9(3):317–334, June 2008. [20] M. J. Wainwright. Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting. Technical Report 725, Univ. of California, Berkeley, Dept. of Statistics, January 2007. [21] A. K. Fletcher, S. Rangan, and V. K. Goyal. Necessary and sufficient conditions for sparsity pattern recovery. IEEE Trans. Inform. Theory, 55(12), December 2009. To appear. Original submission available online [33]. [22] R. Tibshirani. Regression shrinkage and selection via the lasso. J. Royal Stat. Soc., Ser. B, 58(1):267–288, 1996. [23] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SIAM J. Sci. Comp., 20(1):33–61, 1999. [24] M. Akc¸akaya and V. Tarokh. Noisy compressive sampling limits in linear and sublinear regimes. In Proc. Conf. on Inform. Sci. & Sys., pages 1–4, Princeton, NJ, March 2008. [25] M. Akc¸akaya and V. Tarokh. Shannon theoretic limits on noisy compressive sampling. arXiv:0711.0366v1 [cs.IT]., November 2007. [26] G. Reeves. Sparse signal sampling using noisy linear projections. Technical Report UCB/EECS-2008-3, Univ. of California, Berkeley, Dept. of Elec. Eng. and Comp. Sci., January 2008. [27] S. Aeron, M. Zhao, and V. Saligrama. On sensing capacity of sensor networks for the class of linear observation, fixed SNR models. arXiv:0704.3434v3 [cs.IT]., June 2007. [28] A. K. Fletcher and S. Rangan. Sparse support recovery from random measurements with orthogonal matching pursuit. Manuscript available online at http://www.eecs.berkeley.edu/∼alyson/Publications/FletcherRangan OMP.pdf, October 2009. [29] A. K. Fletcher, S. Rangan, and V. K. Goyal. On–off random access channels: A compressed sensing framework. arXiv:0903.1022v1 [cs.IT]., March 2009. [30] Y. Jin and B. Rao. Performance limits of matching pursuit algorithms. In Proc. IEEE Int. Symp. Inform. Th., pages 2444–2448, Toronto, Canada, June 2008. [31] D. Wipf and B. Rao. Comparing the effects of different weight distributions on finding sparse representations. In Proc. Neural Information Process. Syst., Vancouver, Canada, December 2006. [32] I. Karatzas and S. E. Shreve. Brownian Motion and Stochastic Calculus. Springer-Verlag, New York, NY, 2nd edition, 1991. [33] A. K. Fletcher, S. Rangan, and V. K. Goyal. Necessary and sufficient conditions on sparsity pattern recovery. arXiv:0804.1839v1 [cs.IT]., April 2008. 9
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Noise Characterization, Modeling, and Reduction for In Vivo Neural Recording Zhi Yang1, Qi Zhao2, Edward Keefer3,4, and Wentai Liu1 1 University of California at Santa Cruz, 2 California Institute of Technology 3 UT Southwestern Medical Center, 4 Plexon Inc yangzhi@soe.ucsc.edu Abstract Studying signal and noise properties of recorded neural data is critical in developing more efficient algorithms to recover the encoded information. Important issues exist in this research including the variant spectrum spans of neural spikes that make it difficult to choose a globally optimal bandpass filter. Also, multiple sources produce aggregated noise that deviates from the conventional white Gaussian noise. In this work, the spectrum variability of spikes is addressed, based on which the concept of adaptive bandpass filter that fits the spectrum of individual spikes is proposed. Multiple noise sources have been studied through analytical models as well as empirical measurements. The dominant noise source is identified as neuron noise followed by interface noise of the electrode. This suggests that major efforts to reduce noise from electronics are not well spent. The measured noise from in vivo experiments shows a family of 1/f x spectrum that can be reduced using noise shaping techniques. In summary, the methods of adaptive bandpass filtering and noise shaping together result in several dB signal-to-noise ratio (SNR) enhancement. 1 Introduction Neurons in the brain communicate through the firing of action potentials. This process induces brief “voltage” spikes in the surrounding environment that can be recorded by electrodes. The recorded neural signal may come from single, or multiple neurons. While single neurons only require a detection algorithm to identify the firings, multiple neurons require the separation of superimposed activities to obtain individual neuron firings. This procedure, also known as spike sorting, is more complex than what is required for single neurons. Spike sorting has acquired general attention. Many algorithms have been reported in the literature [1–7], with each claiming an improved performance based on different data. Comparisons among different algorithms can be subjective and difficult. For example, benchmarks of in vivo recordings that thoroughly evaluate the performance of algorithms are unavailable. Also, synthesized sequences with benchmarks obtained through neuron models [8], isolated single neuron recordings [2], or simultaneous intra- and extra- cellular recordings [9] lack the in vivo recording environment. As a result, synthesized data provide useful but limited feedback on algorithms. This paper discusses a noise study, based on which SNR enhancement techniques are proposed. These techniques are applicable to an unspecified spike sorting algorithm. Specifically, a procedure of online estimating both individual spike and noise spectrum is first applied. Based on the estimation, a bandpass filter that fits the spectrum of the underlying spike is selected. This maximally reduces the broad band noise without sacrificing the signal integrity. In addition, a comprehensive study of multiple noise sources are performed through lumped circuit model as well as measurements. Experiments suggest that the dominant noise is not from recording electronics, 1 Figure 1: Block diagram of the proposed noise reduction procedures. thus de-emphasize the importance of low noise hardware design. More importantly, the measured noise generally shows a family of 1/f x spectrum, which can be reduced by using noise shaping techniques [10,11]. Figure 1 shows the proposed noise reduction procedures. The rest of this paper is organized as follows. Section 2 focuses on noise sources. Section 3 gives a Wiener kernel based adaptive bandpass filter. Section 4 describes a noise shaping technique that uses fractional order differentiation. Section 5 reports experiment results. Section 6 gives concluding remarks. 2 Noise Spectrum and Noise Model Recorded neural spikes are superimposed with noise that exhibit non-Gaussian characteristics and can be approximated as 1/f x noise. The frequency dependency of noise is contributed by multiple sources. Identified noise sources include 1/f α−neuron noise [12–14] (notations of 1/f x and 1/f α represent frequency dependencies of the total noise and neuron noise respectively), electrodeelectrolyte interface noise [15], tissue thermal noise, and electronic noise, which are illustrated in Figure 2 using a lumped circuit model. Except electrolyte bulk noise (4kTRb in Figure 2) that has a flattened spectrum, the rest show frequency dependency. Specifically, 1/f α−neuron noise is induced from distant neurons [12–14]. Numeric simulations based on simplified neuron models [12] suggest that α can vary a wide range depending on the parameters. For the electrode-electrolyte interface noise, non-faradaic type in particular, an effective resistance (Ree) is defined for modeling purposes. Ree generates noise that is attenuated quadratically to frequency in high frequency region by the interface capacitance (Cee). Electronic noise consists of two major components: thermal noise (∼kT/gm [16]) and flicker noise (or 1/f noise [16]). Flicker noise dominates at lower frequency range and is fabrication process dependent. Next, we will address the noise model that will later be used to develop noise removal techniques in Section 3 and Section 4, and verified by experiment results in section 5. 2.1 1/f α−Neuron Noise Background spiking activities of the vast distant neurons (e.g. spike, synaptic release [17–19]) overlap the spectrum of the recorded spike signal. They usually have small magnitudes and are noisily aggregated. Analytically, the background activities are described as Vneu = X i X k vi.neu(t −ti,k), (1) where Vneu represents the superimposed background activities of distant neurons; i and ti,k represent the object identification and its activation time respectively, and vi.neu is the spiking activity template of the ith object. Based on Eq. 1, the power spectrum of Vneu is P{Vneu} = X i X k |Xi(f)|2fi 2 < e2πjf(ti,k1+k−ti,k1) >, (2) where < > represents the average over the ensemble and over k1, P{} is the spectrum operation, Xi(f) is the fourier transform of vi.neu, and fi is the frequency of spiking activity vi.neu (the number of activations divided by a period of time). The spectrum of a delta function spike pulse 2 Figure 2: Noise illustration for extracellular spikes. train (P k < e2πjf(tk1+k−tk1) >), according to [12], features a lower frequency and exhibits a 1/f α frequency dependency. As this term multiplies |Xi(f)|2, the unresolved spiking activities of distant neurons contribute a spectrum of 1/f x within the signal spectrum. 2.2 Electrode Noise Assume the electrode-electrolyte interface is the non-faradaic type where charges such as electrons and ions, can not pass across the interface. In a typical in vivo recording environment that involves several different ionic particles, e.g. Na+, K+, ..., the current flux of any ith charged particle Ji(x) at location x assuming spatial concentration ni(x) is described by Nernst equation Ji(x) = −Di∇ni(x) + ni(x)υ −ziq kT Dini∇Φ(x), (3) where Di is the diffusion coefficient, Φ electrical potential, zi charge of the particle, q the charge of one electron, k the Boltzmann constant, T the temperature, and υ the convection coefficient. In a steady state, Ji(x) is zero with the boundary condition of maintaining about 1V voltage drop from metal to electrolyte. In such a case, the electrode interface can be modeled as a lumped resistor Ree in parallel with a lumped capacitor Cee. This naturally forms a lowpass filter for the interface noise. As a result, the induced noise from Ree at the input of the amplifier is Ne.e = 4kT Ree (Ree||jωCee||(Rb + jωCi))2 = 4kT Ree | 1 1/Ree + jωCee + 1/(Rb + 1 jωCi )|2. (4) Referring to the hypothesis that the amplifier input capacitance (Ci) is sufficiently small, introducing negligible waveform distortion, the integrated noise by electrode interface satisfies Z fc2 fc1 Ne.edf ≈ Z fc2 fc1 4kTRee |1 + 2πjfReeCee|2 df = 2kT πCee tan−12πReeCeef|f=fc2 f=fc1 < kT Cee . (5) Equation 5 suggests reducing electrode interface noise by increasing double layer capacitance (Cee). Without increasing the size of electrodes, carbon-nanotube (CNT) coating [20] can dramatically increase electrode surface area, thus, reducing the interface noise. Section 5 will compare conventional electrodes and CNT coated electrodes from a noise point of view. In regions away from the interface boundary, ∇ni(x) = 0 results in a flattened noise spectrum. Here we use a lumped bulk resistance Rb in series with the double-layer interface for modeling noise Ne.b = 4kTRb = 4kTχρtissue πrs , (6) where Rb is the bulk resistance, ρtissue is the electrolyte resistivity, rs is the radius of the electrode, and χ is a constant that relates to the electrode geometry. As given in [21], χ ≈0.5 for a plate electrode. 2.3 Electronic Noise Noise generated by electronics can be predicted by circuit design tools and validated through measurements. At the frequency of interest, there are two major components: thermal noise of transistors and flicker noise Nelectronic = Nc.thermal + Nc.flicker = γ 4kT gm + K CoxWL 1 f , (7) 3 where Nc.thermal is the circuit thermal noise, Nc.flicker the flicker noise, gm the transconductance of the amplifier (∂iout/∂vin), γ a circuit architecture dependent constant on the order of O(1), K a process-dependent constant on the order of 10−25V 2F [16], Cox the transistor gate capacitance density, and W and L the transistor width and length respectively. Given a design schematic, circuit thermal noise can be reduced by increasing transconductance (gm), which is to the first order linear to bias current thus power consumption. Flicker noise can be reduced using design techniques such as large size input transistors and chopper modulations [22]. By using advanced semiconductor technologies, also, power and area trade off to noise [16], and elegant design techniques like chopper modulation, current feedback [23], the state-of-the-art low noise neural amplifier can provide less than 2µV total noise [24]. Such design costs can be necessary and useful if electronics noise contributes significantly to the total noise. Otherwise, the over-designed noise specification may be used to trade off other specifications and potentially result in overall improved performance of the system. Section 5 will present experiments of evaluating noise contribution from different sources, which show that electronics are not the dominant noise source in our experiments. 2.4 Total Noise The noise sources as shown in Figure 2 include unresolved neuron activities (Nneu), electrodeelectrolyte interface noise (Ne.e), thermal noise from the electrolyte bulk (Ne.b) and active circuitry (Nc.thermal), and flicker noise (Nc.flicker). The noise spectrum is empirically fitted by N(f) = Nneu + Ne.e + Ne.b + Nc.thermal + Nc.flicker ≈N1 f x + N0, (8) where N1/f x and N0 represent the frequency dependent and flat terms, respectively. Equation 8 describes a combination of both colored noise (1/f x) and broad band noise, which can be reduced by using noise removal techniques. Section 3 presents an adaptive filtering scheme used to optimally attenuate the broad band noise. Section 4 presents a noise shaping technique used to improve the differentiation between signals and noise within the passband. 3 Adaptive Bandpass Filtering SNR is calculated by integrating both signal and noise spectrum. Intuitively, a passband, either too narrow or wide, introduces signal distortion or unwanted noise. Figure 5(b) plots the detected spikes from one single electrode with different widths and shows the difficulty of optimally sizing the passband. While a passband that only fits one spike template may introduce waveform distortion to spikes of other templates, a passband that covers every template will introduce more noise to spikes of every template. A possible solution is to adaptively assign a passband to each spike waveform such that each span will be just wide enough to cover the underlying waveform. This section presents the steps used in order to achieve this solution and includes spike detection, spectrum estimation, and filter generation. 3.1 Spike Detection In this work, spike detection is performed using a nonlinear energy operator (NEO) [25] that captures instantaneous high frequency and high magnitude activities. With a discrete time signal xi, i = ...1, 2, 3..., NEO outputs ψ(xi) = x2 i −xi+1xi−1. (9) The usefulness of NEO for spike detection can be explored by taking the expectation of Eq. 9 ψ(xi) = Rx(0) −Rx(2 △T) ≈ Z P(f, i)(1 −cos4πf △T)df, (10) where Rx is the auto correlation function, △T is the sampling interval, and P(f, i) is the estimated power spectrum density with window centered at sample xi. When the frequency of interest is much lower than the sampling frequency, 1 −cos2πfτ is approximately 2π2f 2τ 2. This emphasizes the high frequency power spectrum. Because spikes are high frequency activities by definition, NEO outputs a larger score when spikes are present. An example of NEO based spike detection is shown in Figure 4, where NEO improves the separation between spikes and the background activity. 4 0 5 10 15 x 10 5 −1.5 −1 −0.5 0 0.5 1 1.5 2 2.5 x 10 4 (a) 0 5 10 15 x 10 5 −2 −1 0 1 2 3 4 x 10 7 (b) Figure 3: Spike sequence and its corresponding NEO output. (a) Raw sequence of one channel. (b) The corresponding NEO output of the raw sequence in (a). 3.2 Corner Frequency Estimation Spectrum estimation of individual spikes is performed to select a corresponding bandpass filter that balances the spectrum distortion and noise. Knowing its ability to separate bandlimited signals from broad band noise, a Weiner filter [26] is used here to size the signal passband. In the frequency domain, denoting PXX and PNN as the signal and noise spectra, Weiner filter is W(f) = PXX(f) PXX(f) + PNN(f) = SNR(f) SNR(f) + 1. (11) Implementing a precise Weiner filter for each detected spike requires considerable computation, as well as a reliable estimation of the signal spectrum. In this work, we are interested in using one of a series of prepared bandpass filters Hi (i = 1, 2...n) that better matches the solved “optimal” Weiner filter arg min i Z |Hi(f) −W(f)|2df, (12) subjected to R [Hi(f) −W(f)]df = 0. 4 Noise Shaping The adaptive scheme presented in Section 3 tacitly assigns a matched frequency mask to individual spikes and balances noise and spectrum integrity. The remaining noise exhibits 1/f x frequency dependency according to Section 2. In this section, we focus on noise shaping techniques to further distinguish signal from noise. The fundamentals of noise shaping are straightforward. Instead of equally amplifying the spectrum, a noise shaping filter allocates more weight to high SNR regions while reducing weight at low SNR regions. This results in an increased ratio of the integrated signal power over the noise power. In general, there are a variety of noise shaping filters that can improve the integrated SNR [10]. In this work, we use a series of fractional derivative operation for noise shaping D(h(x)) = dph(x) dxp , (13) where h(x) is a general function, p is a positive number (can be integer or non-integer) that adjusts the degree of noise shaping; the larger the p, the more emphasis on high frequency spectrum. In Z domain, the realization of fractional derivative operation can be done using binomial series [27] H(z) = (1 −z−1)p = ∞ X n=0 h(n)z−k = 1 −pz−1 + ∞ X n=2 (−1)n p(p −1)...(p −n + 1) n! z−n, (14) where h(n) are the fractional derivative filter coefficients that converge to zero. The SNR gain in applying a fractional derivative filter H(f) is SNRgain = 10log R Ispike(f)|H(f)|2df R Inoise(f)|H(f)|2df −10log R Ispike(f)df R Inoise(f)df , (15) 5 0 1 2 3 4 5 −2.5 −1.25 0 1.25 2.5 Time, unit minute Recording, unit mV (a) 0 10 20 30 40 50 60 10 1 10 2 10 3 Time, unit minute Noise power, unit (uV)2 Noise measured using conventinal electrode Noise measured using CNT coated electrode (b) 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 −5 0 5 10 15 20 25 Frequency (kHz) Power/frequency (dB/Hz) 0 minute 15 minute 30 minute 45 minute (c) 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 −5 0 5 10 15 20 25 Frequency (kHz) Power/frequency (dB/Hz) 0 minute 15 minute 30 minute 45 minute (d) Figure 4: In vivo recording for identifying noise sources. (a) 5-minute recording segment capturing the decay of background activities. (b) Traces of the estimated noise vs. time are plotted. Black ■ curve represents the noise recorded from a custom tungsten electrode; red ▲curve represents the noise recorded from a CNT coated electrodes with the same size. (c), (d) Noise power spectrums estimated at the 0, 15, 30, 45 minutes after the drug injection. In (c) a conventional tungsten electrode is used. In (d), a CNT coated tungsten electrode of equal size is used for comparison. where Ispike(f) and Inoise(f) are power spectrums of spike and noise respectively. Numeric values of SNR gain depend on both data and p (the degree of noise shaping). In our experiments, we empirically choose p in a range of 0.5 to 2.5, where numerically calculated SNR gains using Eq. 15 of in vivo recordings are typically more than 3dB, which is consistent with [10]. 5 Experiment To verify the noise analysis presented in Section 2, an in vivo experiment is performed that uses two sharp tungsten electrodes separated by 125 µm to record the hippocampus neuronal activities of a rat. One of the electrodes is coated with carbon-nanotube (CNT), while the other is uncoated. After the electrodes have been placed, a euthanizing drug is injected. After 5 seconds of drug injection, the recording of the two electrodes start and last until to the time of death. The noise analysis results are summarized and presented in Figure 4. In Figure 4(a), a 5-minute segment that captures the decaying of background activities is plotted. In Figure 4(b), the estimated noise from 600Hz to 6KHz for both recording sites are plotted, where noise dramatically reduces (> 80%) after the drug takes effect. Initially, the CNT electrode records a comparatively larger noise (697µV 2) compared with the uncoated electrode (610µV 2). After a few minutes, the background noise recorded by the CNT electrode quickly reduces eventually reaching 37µV 2 that is about 1/3 of noise recorded by its counterpart (112µV 2), suggesting the noise floor of using the uncoated tungsten electrode (112µV 2) is set by the electrode. From these two plots, we can estimate that the neuron noise is around 500 ∼600µV 2, electrode interface noise is ∼80µV , while the sum of electronic noise and electrolyte bulk noise is less than 37µV 2 (only ∼5% of the total noise). Figure 4(c) displays the 1/f x noise spectrum recorded from the uncoated tungsten electrode (x = 1.8, 1.4, 1.0, 0.9, estimated at 0, 15, 30, 45 minutes after drug injection). Figure 4(d) displays 1/f x noise spectrum recorded from the CNT coated electrode (x = 2.1, 1.3, 0.9, 0.8, estimated at 0, 15, 30, 45 minutes after drug injection). 6 Table 1: Statistics of 1/f x noise spectrum from in vivo preparations. 1/f x x < 1 1 ≤x < 1.5 1.5 ≤x < 2 x ≥2 Number of Recordings 5 38 23 11 0 5 10 15 20 25 30 35 40 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 (a) 0 5 10 15 20 25 30 35 40 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 (b) −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 0.2 0.25 (c) −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 −0.35 −0.3 −0.25 −0.2 −0.15 −0.1 −0.05 0 0.05 0.1 0.15 (d) Figure 5: In vivo experiment of evaluating the proposed adaptive bandpass filter. (a) Detected spikes are aligned and superimposed. (b) Example waveforms that have distinguished widths are plotted. (c) Feature extraction results using PCA with a global bandpass filter (400Hz to 5KHz) are displayed. (d) Feature extraction results using PCA with adaptive bandpass filters are displayed showing a much improved cluster isolation compared to (c). In the second experiment, 77 recordings of in vivo preparations are used to explore the stochastic distribution of 1/f x noise spectrum. “x” is averaged at 1.5 with a standard deviation of 0.5 (1/f 1.5±0.5). The results are summarized in Table 1. The third experiment uses an in vivo recording from a behaving cat. This recording is used to compare the feature extraction results produced by a global bandpass filter (conventional one) and the proposed adaptive bandpass filter, discussed in Section 3. In Figure 5(a), detected spikes are superimposed, where “a thick waveform bundle” is observed. In Figure 5(b), example waveforms in Figure 5(a) that have different widths are shown. Clearly, these waveforms have noticeably different spectrum spans. In Figure 5(c), feature extraction results using PCA (a widely used feature extraction algorithm in spike sorting applications) with a global bandpass filter are displayed. As a comparison, feature extraction results using PCA with adaptive bandpass filters are displayed in Figure 5(d), where multiple clusters are differentiable in the feature space. In the fourth experiment, earth mover’s distance (EMD), as a cross-bin similarity measure that is robust to waveform misalignment [28], is applied to synthesized data for evaluation of the spike waveform separation before and after noise shaping. Assume VA(i), i = 1, 2..., VB(i), i = 1, 2... to be the spike waveform bundles from candidate neuron A and B. To estimate the spike variation of a candidate neuron, two waveforms are randomly picked from a same waveform bundle, and the distance between them is calculated using EMD. After repeating the procedure many times, the results are plotted as the black (waveforms from VA) and blue (waveforms from VA) traces in Figure 6. The x-axis indexes the trial and the y-axis is the EMD score. Black/blue traces describe the intra-cluster waveform variations of the two neurons under testing. To estimate the separation between candidate neuron A and B, we randomly pick two waveforms, one from VA and the other from VB, then compute the EMD between them. This procedure is repeated many times and the EMD vs. trial index is plotted as the red curve in Figure 6. Four pairs of candidate neurons are tested and shown in Figure 6(a)-(d). It can be observed from Figure 6 that the red curves are not well differentiated from the black/blue ones, which indicate that candidate neurons are not well separated. In Figure 6(e)-(h), we apply a similar procedure on the same four pairs of candidate neurons. The only difference from plots shown in Figure 6(a)-(d) is that the waveforms after noise shaping are used rather than their original counterparts. In Figure 6(e)-(h), the red curves separate from the black/blue traces, suggesting that the noise shaping filter improves waveform differentiations. In the fifth experiment, we apply different orders of noise shaping filters and the same feature extraction algorithm to evaluate the feature extraction results. The noise shaping technique is developed as a general tool that can be incorporated into an unspecified feature extraction algorithm. Here, we use PCA as an example. In Figure 7, 8 figures in the same row are results of the same sequence. Figures from left to right display the feature extraction results with different orders of noise shaping; from 0 (no noise shaping) to 3.5, and stepped by 0.5. All the tests are obtained after adaptive band7 0 50 100 150 200 250 300 0 5 10 15 20 25 30 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (a) 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 18 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (b) 0 50 100 150 200 250 300 0 2 4 6 8 10 12 14 16 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (c) 0 50 100 150 200 250 300 0 2 4 6 8 10 12 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (d) 0 50 100 150 200 250 300 0 1 2 3 4 5 6 7 8 9 10 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (e) 0 50 100 150 200 250 300 0 1 2 3 4 5 6 7 8 9 10 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (f) 0 50 100 150 200 250 300 0 1 2 3 4 5 6 IntraCluster1 Distance IntraCluster2 Distance InterClster1to2 Distance (g) 0 50 100 150 200 250 300 0 1 2 3 4 5 6 IntraCluster1 Distance IntraCluster2 Distance InterCluster1to2 Distance (h) Figure 6: EMD vs. trial index. Black and blue trace: EMDs for intra-cluster waveforms; red trace: EMDs for inter-cluster waveforms. (a)-(d) and (e)-(h) are results of 4 different pairs of neurons before and after noise shaping respectively. Traces in (a)-(d) and (e)-(h) have one-to-one correspondence. Noise level increases from (a) to (d). 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (a) (b) (c) (d) (e) (f) (g) (h) 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 (i) (j) (k) (l) (m) (n) (o) (p) Figure 7: Feature extraction results using PCA with different orders of noise shaping. Each row represents a different sequence. Each column represents a different order of noise shaping (p in dpf(x) dxp ), sweeping from 0 (without noise shaping) to 3.5 , stepped by 0.5. (a)-(h) are results of a synthesized sequence. (i)-(p) are results of an in vivo preparation. Clearly, (f) is better than (a); (m) is better than (i). pass filtering. The first sequence (a)-(h) is a synthesized one from public data base [2], the second sequence is recorded from an in vivo preparation. For both sequences, increased numbers of isolated clusters can be obtained by appropriately choosing the order of the noise shaping filter. 6 Conclusion In this paper, a study of multiple noise sources for in vivo neural recording is carried out. The dominant noise source is identified to be neuron noise followed by interface noise of the electrode. Overall, the noise exhibits a family of 1/f x spectrum. The concept of adaptive bandpass filter is proposed to reduce noise because it maintains the signal spectrum integrity while maximally reducing the broad band noise. To reduce the noise within the signal passband and improve waveform separation, a series of fractional order differentiator based noise shaping filters are proposed. The proposed noise removal techniques are generally applicable to an unspecified spike sorting algorithm. Experiment results from in vivo preparations, synthesized sequences, and comparative recordings using both conventional and CNT coated electrodes are reported, which verify the noise model and demonstrate the usefulness of the proposed noise removal techniques. 8 References [1] Lewicki MS. A review of methods for spike sorting: the detection and classification of neural action potentials. Network Comput Neural Syst. 1998;9:53–78. [2] Quian Quiroga R, Nadasdy Z, Ben-Shaul Y. Unsupervised spike detection and sorting with wavelets and superparamagnetic clustering. Neural Computation. 2004 Aug;16(8):1661–1687. [3] Bar-Hillel A, Spiro A, Stark E. Spike sorting: Bayesian clustering of non-stationary data. Advances in Neural Information Processing Systems 17. 2005;p. 105–112. [4] Zumsteg ZS, Kemere C, O’Driscoll S, Santhanam G, Ahmed RE, Shenoy KV, et al. Power feasibility of implantable digital spike sorting circuits for neural prosthetic systems. IEEE Trans Neural Syst Rehabil Eng. 2005 Sep;13(3):272–279. [5] Vargas-Irwin C, Donoghue JP. Automated spike sorting using density grid contour clustering and subtractive waveform decomposition. J Neurosci Methods. 2007;164(1). [6] Yang Z, Zhao Q, Liu W. Neural signal classification using a simplified feature set with nonparametric clustering. Neurocomputing, doi:101016/jneucom200907013, in press;. [7] Gasthaus J, Wood F, Dilan G, Teh YW. Dependent dirichlet process spike sorting. Advances in Neural Information Processing Systems 21. 2009;p. 497–504. [8] Smith LS, Mtetwa N. A tool for synthesizing spike trains with realistic interference. J Neurosci Methods. 2007 Jan;159(1):170–180. [9] Harris KD, Henze DA, Csicsvari J, Hirase H, Buzsaki G. Accuracy of tetrode spike separation as determined by simultaneous intracellular and extracellular measurements. J Neurophysiol. 2000;84:401–414. [10] Yang Z, Zhao Q, Liu W. Spike feature extraction using informative samples. Advances in Neural Information Processing Systems 21. 2009;p. 1865–1872. [11] Yang Z, Zhao Q, Liu W. Improving Spike Separation Using Waveform Derivative. Journal of Neural Engineering, doi: 101088/1741-2560/6/4/046006. 2009 Aug;6(4). [12] Davidsen J, Schuster HZ. Simple model for 1/f α noise. Phys Rev Lett 65, 026120(1)-026120(4). 2002;. [13] Yu Y, Romero R, Lee TS. Preference of sensory neural coding for 1/f signals. Phys Rev Lett 94, 108103(1)-108103(4). 2005;. [14] Bedard C, Kroger H, Destexhe A. Does the 1/f frequency scaling of brain reflect self-organized critical states? Phys Rev Lett 97, 118102(1)-118102(4). 2006;. [15] Hassibi A, Navid R, Dutton RW, Lee TH. Comprehensive study of noise processes in electrode electrolyte interfaces. J Appl Phys. 2004 July;96(2):1074–1082. [16] Razavi B. Design of Analog CMOS Integrated Circuits. Boston, MA:McGraw-Hill; 2001. [17] Keener J, Sneyd J. Mathematical Physiology. New York: Springer Verlag; 1998. [18] Manwani A, N Steinmetz P, Koch C. Channel noise in excitable neural membranes. Advances in Neural Information Processing Systems 12. 2000;p. 142–149. [19] Fall C, Marland E, Wagner J, Tyson J. Computational Cell Biology. New York: Springer Verlag; 2002. [20] Keefer EW, Botterman BR, Romero MI, Rossi AF, Gross GW. Carbon nanotube-coated electrodes improve brain readouts. Nat Nanotech. 2008;3:434–439. [21] Wiley JD, Webster JG. Analysis and control of the current distribution under circular dispersive. IEEE Trans Biomed Eng. 1982;29:381–385. [22] Denison T, Consoer K, Kelly A, Hachenburg A, Santa W. A 2.2µW 94nV/ √ Hz, chopper-stabilized instrumentation amplifier for EEG detection in chronic implants. IEEE ISSCC Dig Tech Papers. 2007 Feb;8(6). [23] Ferrari G, Gozzini F, Sampietro M. A current-sensitive front-end amplifier for nano biosensors with a 2MHz BW. IEEE ISSCC Dig Tech Papers. 2007 Feb;8(7). [24] Harrison RR. The design of integrated circuits to observe brain activity. Proc IEEE. 2008 July;96:1203– 1216. [25] Kaiser JF. On a simple algorithm to calculate the energy of a signal. In Proc IEEE Int Conf Acoustic Speech and Signal Processing. 1990;p. 381–384. [26] Vaseghi SV. Advanced Digital Signal Processing and Noise Reduction. Wiley-Teubner; 1996. [27] Hosking J. Fractional differencing. Biometrika. 1981 Jan;68:165–176. [28] Rubner Y. Perceptual metrics for image database navigation. In: Ph.D. dissertation, Stanford University; 1999. . 9
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Adaptive Regularization of Weight Vectors Koby Crammer Department of Electrical Enginering The Technion Haifa, 32000 Israel koby@ee.technion.ac.il Alex Kulesza Department of Computer and Information Science University of Pennsylvania Philadelphia, PA 19104 kulesza@cis.upenn.edu Mark Dredze Human Language Tech. Center of Excellence Johns Hopkins University Baltimore, MD 21211 mdredze@cs.jhu.edu Abstract We present AROW, a new online learning algorithm that combines several useful properties: large margin training, confidence weighting, and the capacity to handle non-separable data. AROW performs adaptive regularization of the prediction function upon seeing each new instance, allowing it to perform especially well in the presence of label noise. We derive a mistake bound, similar in form to the second order perceptron bound, that does not assume separability. We also relate our algorithm to recent confidence-weighted online learning techniques and show empirically that AROW achieves state-of-the-art performance and notable robustness in the case of non-separable data. 1 Introduction Online learning algorithms are fast, simple, make few statistical assumptions, and perform well in a wide variety of settings. Recent work has shown that parameter confidence information can be effectively used to guide online learning [2]. Confidence weighted (CW) learning, for example, maintains a Gaussian distribution over linear classifier hypotheses and uses it to control the direction and scale of parameter updates [6]. In addition to formal guarantees in the mistake-bound model [11], CW learning has achieved state-of-the-art performance on many tasks. However, the strict update criterion used by CW learning is very aggressive and can over-fit [5]. Approximate solutions can be used to regularize the update and improve results; however, current analyses of CW learning still assume that the data are separable. It is not immediately clear how to relax this assumption. In this paper we present a new online learning algorithm for binary classification that combines several attractive properties: large margin training, confidence weighting, and the capacity to handle non-separable data. The key to our approach is the adaptive regularization of the prediction function upon seeing each new instance, so we call this algorithm Adaptive Regularization of Weights (AROW). Because it adjusts its regularization for each example, AROW is robust to sudden changes in the classification function due to label noise. We derive a mistake bound, similar in form to the second order perceptron bound, that does not assume separability. We also provide empirical results demonstrating that AROW is competitive with state-of-the-art methods and improves upon them significantly in the presence of label noise. 2 Confidence Weighted Online Learning of Linear Classifiers Online algorithms operate in rounds. In round t the algorithm receives an instance xt ∈Rd and applies its current prediction rule to make a prediction ˆyt ∈Y. It then receives the true 1 label yt ∈Y and suffers a loss ℓ(yt, ˆyt). For binary classification we have Y = {−1, +1} and use the zero-one loss ℓ01(yt, ˆyt) = 0 if yt = ˆyt and 1 otherwise. Finally, the algorithm updates its prediction rule using (xt, yt) and proceeds to the next round. In this work we consider linear prediction rules parameterized by a weight vector w: ˆy = hw(x) = sign(w · x). Recently Dredze, Crammer and Pereira [6, 5] proposed an algorithmic framework for online learning of binary classification tasks called confidence weighted (CW) learning. CW learning captures the notion of confidence in a linear classifier by maintaining a Gaussian distribution over the weights with mean µ ∈Rd and covariance matrix Σ ∈Rd×d. The values µp and Σp,p, respectively, encode the learner’s knowledge of and confidence in the weight for feature p: the smaller Σp,p, the more confidence the learner has in the mean weight value µp. Covariance terms Σp,q capture interactions between weights. Conceptually, to classify an instance x, a CW classifier draws a parameter vector w ∼ N (µ, Σ) and predicts the label according to sign(w · x). In practice, however, it can be easier to simply use the average weight vector E [w] = µ to make predictions. This is similar to the approach taken by Bayes point machines [9], where a single weight vector is used to approximate a distribution. Furthermore, for binary classification, the prediction given by the mean weight vector turns out to be Bayes optimal. CW classifiers are trained according to a passive-aggressive rule [3] that adjusts the distribution at each round to ensure that the probability of a correct prediction is at least η ∈(0.5, 1]. This yields the update constraint Pr [yt (w · xt) ≥0] ≥η . Subject to this constraint, the algorithm makes the smallest possible change to the hypothesis weight distribution as measured using the KL divergence. This implies the following optimization problem for each round t: (µt, Σt) = min µ,Σ DKL N (µ, Σ) ∥N µt−1, Σt−1 s.t. Prw∼N(µ,Σ) [yt (w · xt) ≥0] ≥η Confidence-weighted algorithms have been shown to perform well in practice [5, 6], but they suffer from several problems. First, the update is quite aggressive, forcing the probability of predicting each example correctly to be at least η > 1/2 regardless of the cost to the objective. This may cause severe over-fitting when labels are noisy; indeed, current analyses of the CW algorithm [5] assume that the data are linearly separable. Second, they are designed for classification, and it is not clear how to extend them to alternative settings such as regression. This is in part because the constraint is written in discrete terms where the prediction is either correct or not. We deal with both of these issues, coping more effectively with label noise and generalizing the advantages of CW learning in an extensible way. 3 Adaptive Regularization Of Weights We identify two important properties of the CW update rule that contribute to its good performance but also make it sensitive to label noise. First, the mean parameters µ are guaranteed to correctly classify the current training example with margin following each update. This is because the probability constraint Pr [yt (w · xt) ≥0] ≥η can be written explicitly as yt (µ · xt) ≥φ p x⊤ t Σxt, where φ > 0 is a positive constant related to η. This aggressiveness yields rapid learning, but given an incorrectly labeled example, it can also force the learner to make a drastic and incorrect change to its parameters. Second, confidence, as measured by the inverse eigenvalues of Σ, increases monotonically with every update. While it is intuitive that our confidence should grow as we see more data, this also means that even incorrectly labeled examples causing wild parameter swings result in artificially increased confidence. In order to maintain the positives but reduce the negatives of these two properties, we isolate and soften them. As in CW learning, we maintain a Gaussian distribution over weight vectors with mean µ and covariance Σ; however, we recast the above characteristics of the CW constraint as regularizers, minimizing the following unconstrained objective on 2 each round: C (µ, Σ) = DKL N (µ, Σ) ∥N µt−1, Σt−1 + λ1ℓh2 (yt, µ · xt) + λ2x⊤ t Σxt , (1) where ℓh2 (yt, µ · xt) = (max{0, 1 −yt(µ · xt)})2 is the squared-hinge loss suffered using the weight vector µ to predict the output for input xt when the true output is yt. λ1, λ2 ≥0 are two tradeoffhyperparameters. For simplicity and compactness of notation, in the following we will assume that λ1 = λ2 = 1/(2r) for some r > 0. The objective balances three desires. First, the parameters should not change radically on each round, since the current parameters contain information about previous examples (first term). Second, the new mean parameters should predict the current example with low loss (second term). Finally, as we see more examples, our confidence in the parameters should generally grow (third term). Note that this objective is not simply the dualization of the CW constraint, but a new formulation inspired by the properties discussed above. Since the loss term depends on µ only via the inner-product µ·xt, we are able to prove a representer theorem (Sec. 4). While we use the squared-hinge loss for classification, different loss functions, as long as they are convex and differentiable in µ, yield algorithms for different settings.1 To solve the optimization in (1), we begin by writing the KL explicitly: C (µ, Σ) = 1 2 log det Σt−1 det Σ + 1 2Tr Σ−1 t−1Σ + 1 2 µt−1 −µ ⊤Σ−1 t−1 µt−1 −µ −d 2 + 1 2rℓh2 (yt, µ · xt) + 1 2rx⊤ t Σxt (2) We can decompose the result into two terms: C1(µ), depending only on µ, and C2(Σ), depending only on Σ. The updates to µ and Σ can therefore be performed independently. The squared-hinge loss yields a conservative (or passive) update for µ in which the mean parameters change only when the margin is too small, and we follow CW learning by enforcing a correspondingly conservative update for the confidence parameter Σ, updating it only when µ changes. This results in fewer updates and is easier to analyze. Our update thus proceeds in two stages. 1. Update the mean parameters: µt = arg min µ C1 (µ) (3) 2. If µt ̸= µt−1, update the confidence parameters: Σt = arg min Σ C2 (Σ) (4) We now develop the update equations for (3) and (4) explicitly, starting with the former. Taking the derivative of C (µ, Σ) with respect to µ and setting it to zero, we get µt = µt−1 −1 2r d dz ℓh2 (yt, z) |z=µt·xt Σt−1xt , (5) assuming Σt−1 is non-singular. Substituting the derivative of the squared-hinge loss in (5) and assuming 1 −yt (µt · xt) ≥0, we get µt = µt−1 + yt r (1 −yt (µt · xt)) Σt−1xt . (6) We solve for µt by taking the dot product of each side of the equality with xt and substituting back in (6) to obtain the rule µt = µt−1 + max 0, 1 −ytx⊤ t µt−1 x⊤ t Σt−1xt + r Σt−1ytxt . (7) It can be easily verified that (7) satisfies our assumption that 1 −yt (µt · xt) ≥0. 1It can be shown that the well known recursive least squares (RLS) regression algorithm [7] is a special case of AROW with the squared loss. 3 Input parameters r Initialize µ0 = 0 , Σ0 = I, For t = 1, . . . , T • Receive a training example xt ∈Rd • Compute margin and confidence mt = µt−1 · xt vt = x⊤ t Σt−1xt • Receive true label yt, and suffer loss ℓt = 1 if sign (mt) ̸= yt • If mtyt < 1, update using eqs. (7) & (9): µt = µt−1 + αtΣt−1ytxt Σt = Σt−1 −βtΣt−1xtx⊤ t Σt−1 βt = 1 x⊤ t Σt−1xt + r αt = max “ 0, 1 −ytx⊤ t µt−1 ” βt Output: Weight vector µT and confidence ΣT . Figure 1: The AROW algorithm for online binary classification. The update for the confidence parameters is made only if µt ̸= µt−1, that is, if 1 > ytx⊤ t µt−1. In this case, we compute the update of the confidence parameters by setting the derivative of C (µ, Σ) with respect to Σ to zero: Σ−1 t = Σ−1 t−1 + xtx⊤ t r (8) Using the Woodbury identity we can also rewrite the update for Σ in non-inverted form: Σt = Σt−1 −Σt−1xtx⊤ t Σt−1 r + x⊤ t Σt−1xt (9) Note that it follows directly from (8) and (9) that the eigenvalues of the confidence parameters are monotonically decreasing: Σt ⪯Σt−1; Σ−1 t ⪰Σ−1 t−1 . Pseudocode for AROW appears in Fig. 1. 4 Analysis We first show that AROW can be kernelized by stating the following representer theorem. Lemma 1 (Representer Theorem) Assume that Σ0 = I and µ0 = 0. The mean parameters µt and confidence parameters Σt produced by updating via (7) and (9) can be written as linear combinations of the input vectors (resp. outer products of the input vectors with themselves) with coefficients depending only on inner-products of input vectors. Proof sketch: By induction. The base case follows from the definitions of µ0 and Σ0, and the induction step follows algebraically from the update rules (7) and (9). We now prove a mistake bound for AROW. Denote by M (M = |M|) the set of example indices for which the algorithm makes a mistake, yt µt−1 · xt ≤0, and by U (U = |U|) the set of example indices for which there is an update but not a mistake, 0 < yt (µt · xt) ≤1. Other examples do not affect the behavior of the algorithm and can be ignored. Let XM = P t∈M xix⊤ i , XU = P t∈U xix⊤ i and XA = XM + XU. Theorem 2 For any reference weight vector u ∈Rd, the number of mistakes made by AROW (Fig. 1) is upper bounded by M ≤ q r ∥u∥2 + u⊤XAu s log det I + 1 r XA + U + X t∈M∪U gt −U , (10) where gt = max 0, 1 −ytu⊤xt . The proof depends on two lemmas; we omit the proof of the first for lack of space. 4 Lemma 3 Let ℓt = max 0, 1 −ytµ⊤ t−1xt and χt = x⊤ t Σt−1xt. Then, for every t ∈M∪U, u⊤Σ−1 t µt = u⊤Σ−1 t−1µt−1 + ytu⊤xt r µ⊤ t Σ−1 t µt = µ⊤ t−1Σ−1 t−1µt−1 + χt + r −ℓ2 tr r (χt + r) Lemma 4 Let T be the number of rounds. Then X t χtr r (χt + r) ≤log det Σ−1 T +1 . Proof: We compute the following quantity: x⊤ t Σtx⊤ t = x⊤ t Σt−1 −βtΣt−1xtx⊤ t Σt−1 xt = χt − χ2 t χt + r = χtr χt + r . Using Lemma D.1 from [2] we have that 1 r x⊤ t Σtx⊤ t = 1 −det Σ−1 t−1 det Σ−1 t . (11) Combining, we get X t χtr r (χt + r) = X t 1 −det Σ−1 t−1 det Σ−1 t ! ≤− X t log det Σ−1 t−1 det Σ−1 t ! ≤log det Σ−1 T +1 . We now prove Theorem 2. Proof: We iterate the first equality of Lemma 3 to get u⊤Σ−1 T µT = X t∈M∪U ytu⊤xt r ≥ X t∈M∪U 1 −gt r = M + U r −1 r X t∈M∪U gt . (12) We iterate the second equality to get µ⊤ T Σ−1 T µT = X t∈M∪U χt + r −ℓ2 tr r (χt + r) = X t∈M∪U χt r (χt + r) + X t∈M∪U 1 −ℓ2 t χt + r . (13) Using Lemma 4 we have that the first term of (13) is upper bounded by 1 r log det Σ−1 T . For the second term in (13) we consider two cases. First, if a mistake occurred on example t, then we have that yt xt · µt−1 ≤0 and ℓt ≥1, so 1−ℓ2 t ≤0. Second, if an the algorithm made an update (but no mistake) on example t, then 0 < yt xt · µt−1 ≤1 and ℓt ≥0, thus 1 −ℓ2 t ≤1. We therefore have X t∈M∪U 1 −ℓ2 t χt + r ≤ X t∈M 0 χt + r + X t∈U 1 χt + r = X t∈U 1 χt + r . (14) Combining and plugging into the Cauchy-Schwarz inequality u⊤Σ−1 T µT ≤ q u⊤Σ−1 T u q µ⊤ T Σ−1 T µT , we get M + U r −1 r X t∈M∪U gt ≤ q u⊤Σ−1 T u s 1 r log det Σ−1 T + X t∈U 1 χt + r . (15) Rearranging the terms and using the fact that χt ≥0 yields M ≤√r q u⊤Σ−1 T u q log det Σ−1 T + U + X t∈M∪U gt −U . 5 By definition, Σ−1 T =I + 1 r X t∈M∪U xix⊤ i =I + 1 r XA , so substituting and simplifying completes the proof: M ≤√r s u⊤ I + 1 r XA u s log det I + 1 r XA + U + X t∈M∪U gt −U = q r ∥u∥2 + u⊤XAu s log det I + 1 r XA + U + X t∈M∪U gt −U . A few comments are in order. First, the two square-root terms of the bound depend on r in opposite ways: the first is monotonically increasing, while the second is monotonically decreasing. One could expect to optimize the bound by minimizing over r. However, the bound also depends on r indirectly via other quantities (e.g. XA), so there is no direct way to do so. Second, if all the updates are associated with errors, that is, U = ∅, then the bound reduces to the bound of the second-order perceptron [2]. In general, however, the bounds are not comparable since each depends on the actual runtime behavior of its algorithm. 5 Empirical Evaluation We evaluate AROW on both synthetic and real data, including several popular datasets for document classification and optical character recognition (OCR). We compare with three baselines: Passive-Aggressive (PA), Second Order Perceptron (SOP)2 and ConfidenceWeighted (CW) learning3. Our synthetic data are as in [5], but we invert the labels on 10% of the training examples. (Note that evaluation is still done against the true labels.) Fig. 2(a) shows the online learning curves for both full and diagonalized versions of the algorithms on these noisy data. AROW improves over all competitors, and the full version outperforms the diagonal version. Note that CW-full performs worse than CW-diagonal, as has been observed previously for noisy data. We selected a variety of document classification datasets popular in the NLP community, summarized as follows. Amazon: Product reviews to be classified into domains (e.g., books or music) [6]. We created binary datasets by taking all pairs of the six domains (15 datasets). Feature extraction follows [1] (bigram counts). 20 Newsgroups: Approximately 20,000 newsgroup messages partitioned across 20 different newsgroups4. We binarized the corpus following [6] and used binary bag-of-words features (3 datasets). Each dataset has between 1850 and 1971 instances. Reuters (RCV1-v2/LYRL2004): Over 800,000 manually categorized newswire stories. We created binary classification tasks using pairs of labels following [6] (3 datasets). Details on document preparation and feature extraction are given by [10]. Sentiment: Product reviews to be classified as positive or negative. We used each Amazon product review domain as a sentiment classification task (6 datasets). Spam: We selected three task A users from the ECML/PKDD Challenge5, using bag-ofwords to classify each email as spam or ham (3 datasets). For OCR data we binarized two well known digit recognition datasets, MNIST6 and USPS, into 45 all-pairs problems. We also created ten one vs. all datasets from the MNIST data (100 datasets total). Each result for the text datasets was averaged over 10-fold cross-validation. The OCR experiments used the standard split into training and test sets. Hyperparameters (including 2For the real world (high dimensional) datasets, we must drop cross-feature confidence terms by projecting onto the set of diagonal matrices, following the approach of [6]. While this may reduce performance, we make the same approximation for all evaluated algorithms. 3We use the “variance” version developed in [6]. 4http://people.csail.mit.edu/jrennie/20Newsgroups/ 5http://ecmlpkdd2006.org/challenge.html 6http://yann.lecun.com/exdb/mnist/index.html 6 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 200 400 600 800 1000 1200 1400 1600 Instances Mistakes Perceptron PA SOP AROW−full AROW−diag CW−full CW−diag (a) synthetic data 0 2000 4000 6000 8000 10000 Instances 0 100 200 300 400 500 600 700 800 Mistakes PA CW AROW SOP 0 2000 4000 6000 8000 10000 Instances 0 500 1000 1500 2000 Mistakes PA CW AROW SOP (b) MNIST data Figure 2: Learning curves for AROW (full/diagonal) and baseline methods. (a) 5k synthetic training examples and 10k test examples (10% noise, 100 runs). (b) MNIST 3 vs. 5 binary classification task for different amounts of label noise (left: 0 noise, right: 10%). r for AROW) and the number of online iterations (up to 10) were optimized using a single randomized run. We used 2000 instances from each dataset unless otherwise noted above. In order to observe each algorithm’s ability to handle non-separable data, we performed each experiment using various levels of artifical label noise, generated by independently flipping each binary label with fixed probability. 5.1 Results and Discussion Noise level Algorithm 0.0 0.05 0.1 0.15 0.2 0.3 AROW 1.51 1.44 1.38 1.42 1.25 1.25 CW 1.63 1.87 1.95 2.08 2.42 2.76 PA 2.95 2.83 2.78 2.61 2.33 2.08 SOP 3.91 3.87 3.89 3.89 4.00 3.91 Table 1: Mean rank (out of 4, over all datasets) at different noise levels. A rank of 1 indicates that an algorithm outperformed all the others. Our experimental results are summarized in Table 1. AROW outperforms the baselines at all noise levels, but does especially well as noise increases. More detailed results for AROW and CW, the overall best performing baseline, are compared in Fig. 3. AROW and CW are comparable when there is no added noise, with AROW winning the majority of the time. As label noise increases (moving across the rows in Fig. 3) AROW holds up remarkably well. In almost every high noise evaluation, AROW improves over CW (as well as the other baselines, not shown). Fig. 2(b) shows the total number of mistakes (w.r.t. noise-free labels) made by each algorithm during training on the MNIST dataset for 0% and 10% noise. Though absolute performance suffers with noise, the gap between AROW and the baselines increases. To help interpret the results, we classify the algorithms evaluated here according to four characteristics: the use of large margin updates, confidence weighting, a design that accomodates non-separable data, and adaptive per-instance margin (Table 2). While all of these properties can be desirable in different situations, we would like to understand how they interact and achieve high performance while avoiding sensitivity to noise. Large ConfNonAdaptive Algorithm Margin idence Separable Margin PA Yes No Yes No SOP No Yes Yes No CW Yes Yes No Yes AROW Yes Yes Yes No Table 2: Online algorithm properties overview. Based on the results in Table 1, it is clear that the combination of confidence information and large margin learning is powerful when label noise is low. CW easily outperforms the other baselines in such situations, as it has been shown to do in previous work. However, as noise increases, the separability assumption inherent in CW appears to reduce its performance considerably. 7 0.75 0.80 0.85 0.90 0.95 1.00 CW 0.75 0.80 0.85 0.90 0.95 1.00 AROW 20news amazon reuters sentiment spam 0.5 0.6 0.7 0.8 0.9 1.0 CW 0.5 0.6 0.7 0.8 0.9 1.0 AROW 20news amazon reuters sentiment spam 0.5 0.6 0.7 0.8 0.9 1.0 CW 0.5 0.6 0.7 0.8 0.9 1.0 AROW 20news amazon reuters sentiment spam 0.90 0.92 0.94 0.96 0.98 1.00 CW 0.90 0.92 0.94 0.96 0.98 1.00 AROW USPS 1 vs. All USPS All Pairs MNIST 1 vs. All 0.5 0.6 0.7 0.8 0.9 1.0 CW 0.5 0.6 0.7 0.8 0.9 1.0 AROW USPS 1 vs. All USPS All Pairs MNIST 1 vs. All 0.5 0.6 0.7 0.8 0.9 1.0 CW 0.5 0.6 0.7 0.8 0.9 1.0 AROW USPS 1 vs. All USPS All Pairs MNIST 1 vs. All Figure 3: Accuracy on text (top) and OCR (bottom) binary classification. Plots compare performance between AROW and CW, the best performing baseline (Table 1). Markers above the line indicate superior AROW performance and below the line superior CW performance. Label noise increases from left to right: 0%, 10% and 30%. AROW improves relative to CW as noise increases. AROW, by combining the large margin and confidence weighting of CW with a soft update rule that accomodates non-separable data, matches CW’s performance in general while avoiding degradation under noise. AROW lacks the adaptive margin of CW, suggesting that this characteristic is not crucial to achieving strong performance. However, we leave open for future work the possibility that an algorithm with all four properties might have unique advantages. 6 Related and Future Work AROW is most similar to the second order perceptron [2]. The SOP performs the same type of update as AROW, but only when it makes an error. AROW, on the other hand, updates even when its prediction is correct if there is insufficient margin. Confidence weighted (CW) [6, 5] algorithms, by which AROW was inspired, update the mean and confidence parameters simultaneously, while AROW makes a decoupled update and softens the hard constraint of CW. The AROW algorithm can be seen as a variant of the PA-II algorithm from [3] where the regularization is modified according to the data. Hazan [8] describes a framework for gradient descent algorithms with logarithmic regret in which a quantity similar to Σt plays an important role. Our algorithm differs in several ways. First, Hazan [8] considers gradient algorithms, while we derive and analyze algorithms that directly solve an optimization problem. Second, we bound the loss directly, not the cumulative sum of regularization and loss. Third, the gradient algorithms perform a projection after making an update (not before) since the norm of the weight vector is kept bounded. Ongoing work includes the development and analysis of AROW style algorithms for other settings, including a multi-class version following the recent extension of CW to multi-class problems [4]. Our mistake bound can be extended to this case. Applying the ideas behind AROW to regression problems turns out to yield the well known recursive least squares (RLS) algorithm, for which AROW offers new bounds (omitted). Finally, while we used the confidence term x⊤ t Σxt in (1), we can replace this term with any differentiable, monotonically increasing function f(x⊤ t Σxt). This generalization may yield additional algorithms. 8 References [1] John Blitzer, Mark Dredze, and Fernando Pereira. Biographies, bollywood, boom-boxes and blenders: Domain adaptation for sentiment classification. In ACL, 2007. [2] Nicol´o Cesa-Bianchi, Alex Conconi, and Claudio Gentile. A second-order perceptron algorithm. Siam J. of Comm., 34, 2005. [3] Koby Crammer, Ofer Dekel, Joseph Keshet, Shai Shalev-Shwartz, and Yoram Singer. Online passive-aggressive algorithms. Journal of Machine Learning Research, 7:551– 585, 2006. [4] Koby Crammer, Mark Dredze, and Alex Kulesza. Multi-class confidence weighted algorithms. In Empirical Methods in Natural Language Processing (EMNLP), 2009. [5] Koby Crammer, Mark Dredze, and Fernando Pereira. Exact convex confidence-weighted learning. In Neural Information Processing Systems (NIPS), 2008. [6] Mark Dredze, Koby Crammer, and Fernando Pereira. Confidence-weighted linear classification. In International Conference on Machine Learning, 2008. [7] Simon Haykin. Adaptive Filter Theory. 1996. [8] Elad Hazan. Efficient algorithms for online convex optimization and their applications. PhD thesis, Princeton University, 2006. [9] Ralf Herbrich, Thore Graepel, and Colin Campbell. Bayes point machines. Journal of Machine Learning Research (JMLR), 1:245–279, 2001. [10] David D. Lewis, Yiming Yang, Tony G. Rose, and Fan Li. Rcv1: A new benchmark collection for text categorization research. JMLR, 5:361–397, 2004. [11] Nick Littlestone. Learning when irrelevant attributes abound: A new linear-threshold algorithm. Machine Learning, 2:285–318, 1988. 9
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Posterior vs. Parameter Sparsity in Latent Variable Models João V. Graça L2F INESC-ID Lisboa, Portugal Kuzman Ganchev Ben Taskar University of Pennsylvania Philadelphia, PA, USA Fernando Pereira Google Research Mountain View, CA, USA Abstract We address the problem of learning structured unsupervised models with moment sparsity typical in many natural language induction tasks. For example, in unsupervised part-of-speech (POS) induction using hidden Markov models, we introduce a bias for words to be labeled by a small number of tags. In order to express this bias of posterior sparsity as opposed to parametric sparsity, we extend the posterior regularization framework [7]. We evaluate our methods on three languages — English, Bulgarian and Portuguese — showing consistent and significant accuracy improvement over EM-trained HMMs, and HMMs with sparsity-inducing Dirichlet priors trained by variational EM. We increase accuracy with respect to EM by 2.3%-6.5% in a purely unsupervised setting as well as in a weaklysupervised setting where the closed-class words are provided. Finally, we show improvements using our method when using the induced clusters as features of a discriminative model in a semi-supervised setting. 1 Introduction Latent variable generative models are widely used in inducing meaningful representations from unlabeled data. Maximum likelihood estimation is a standard method for fitting such models, but in most cases we are not so interested in the likelihood of the data as in the distribution of the latent variables, which we hope will capture regularities of interest without direct supervision. In this paper we explore the problem of biasing such unsupervised models to favor a novel kind of sparsity that expresses our expectations about the role of the latent variables. Many important language processing tasks (tagging, parsing, named-entity classification) involve classifying events into a large number of possible classes, where each event type can have just a few classes. We extend the posterior regularization framework [7] to achieve that kind of posterior sparsity on the unlabeled training data. In unsupervised part-of-speech (POS) tagging, a well studied yet challenging problem, the new method consistently and significantly improves performance over a non-sparse baseline and over a variational Bayes baseline with a Dirichlet prior used to encourage sparsity [9, 4]. A common approach to unsupervised POS tagging is to train a hidden Markov model where the hidden states are the possible tags and the observations are word sequences. The model is typically trained with the expectation-maximization (EM) algorithm to maximize the likelihood of the observed sentences. Unfortunately, while supervised training of HMMs achieves relatively high accuracy, the unsupervised models tend to perform poorly. One well-known reason for this is that EM tends to allow each word to be generated by most hidden states some of the time. In reality, we would like most words to have a small number of possible tags. To solve this problem, several studies [14, 17, 6] investigated weakly-supervised approaches where the model is given the list of possible tags for each word. The task is then to disambiguate among the possible tags for each word type. Recent work has made use of smaller dictionaries, trying to model the set of possible tags for each word [18, 5], or use a small number of “prototypes” for each tag [8]. All these approaches initialize the model in a way that encourages sparsity by zeroing out impossible tags. Although this 1 has worked extremely well for the weakly-supervised case, we are interested in the setting where we have only high-level information about the model: we know that the distribution over the latent variables (such as POS tags) should be sparse. This has been explored in a Bayesian setting, where a prior is used to encourage sparsity in the model parameters [4, 9, 6]. This sparse prior, which prefers each tag to have few word types associated with it, indirectly achieves sparsity over the posteriors, meaning each word type should have few possible tags. Our method differs in that it encourages sparsity in the model posteriors, more directly encoding the desiderata. Additionally our method can be applied to log-linear models where sparsity in the parameters leads to dense posteriors. Sparsity at this level has already been suggested before under a very different model[18]. We use a first-order HMM as our model to compare the different training conditions: classical expectation-maximization (EM) training without modifications to encourage sparsity, the sparse prior used by [9] with variational Bayes EM (VEM), and our sparse posterior regularization (Sparse). We evaluate these methods on three languages, English, Bulgarian and Portuguese. We find that our method consistently improves performance with respect to both baselines in a completely unsupervised scenario, as well as in a weakly-supervised scenario where the tags of closed-class words are supplied. Interestingly, while VEM achieves a state size distribution (number of words assigned to hidden states) that is closer to the empirical tag distribution than EM and Sparse its state-token distribution is a worse match to the empirical tag-token distribution than the competing methods. Finally, we show that states assigned by the model are useful as features for a supervised POS tagger. 2 Posterior Regularization In order to express the desired preference for posterior sparsity, we use the posterior regularization (PR) framework [7], which incorporates side information into parameter estimation in the form of linear constraints on posterior expectations. This allows tractable learning and inference even when the constraints would be intractable to encode directly in the model, for instance to enforce that each hidden state in an HMM is used only once in expectation. Moreover, PR can represent prior knowledge that cannot be easily expressed as priors over model parameters, like the constraint used in this paper. PR can be seen as a penalty on the standard marginal likelihood objective, which we define first: Marginal Likelihood: L(θ) = bE[−log pθ(x)] = bE[−log X z pθ(z, x)] over the parameters θ, where bE is the empirical expectation over the unlabeled sample x, and z are the hidden states. This standard objective may be regularized with a parameter prior −log p(θ) = C(θ), for example a Dirichlet. Posterior information in PR is specified with sets Qx of distributions over the hidden variables z defined by linear constraints on feature expectations: Qx = {q(z | x) : Eq[f(x, z)] ≤b}. (1) The marginal log-likelihood of a model is then penalized with the KL-divergence between the desired distributions Qx and the model, KL(Qx ∥pθ(z|x)) = minq∈Qx KL(q(z) ∥pθ(z|x)). The revised learning objective minimizes: PR Objective: L(θ) + C(θ) + bE[KL(Qx ∥pθ(z|x))]. (2) Since the objective above is not convex in θ, PR estimation relies on an EM-like lower-bounding scheme for model fitting, where the E step computes a distribution q(z|x) over the latent variables and the M step minimizes negative marginal likelihood under q(z|x) plus parameter regularization: M-Step: min θ bE [Eq[−log pθ(x, z)]] + C(θ) (3) In a standard E step, q is the posterior over the model hidden variables given current θ: q(z|x) = pθ(z|x). However, in PR, q is a projection of the posteriors onto the constraint set Qx for each example x: arg min q KL(q(z|x) ∥pθ(z|x)) s.t. Eq[f(x, z)] ≤b. (4) 2 pθ λ qti ∝pθe−λti instance NN VB JJ DT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 instance NN VB JJ DT 0 1 2 3 4 5 6 7 8 9 10 instance NN VB JJ DT 0 0.2 0.4 0.6 0.8 1 Figure 1: An illustration of ℓ1/ℓ∞regularization. Left panel: initial tag distributions (columns) for 15 instances of a word. Middle panel: optimal regularization parameters λ, each row sums to σ = 20. Right panel: q concentrates the posteriors for all instances on the NN tag, reducing the ℓ1/ℓ∞norm from just under 4 to a little over 1. The new posteriors q(z|x) are used to compute sufficient statistics for this instance and hence to update the model’s parameters in the M step. The optimization problem in Equation 4 can be solved efficiently in dual form: arg min λ≥0 b⊤λ + log X z pθ(z|x) exp {−λ⊤f(x, z)}. (5) Given λ, the primal solution is q(z|x) = pθ(z|x) exp{−λ⊤f(x, z)}/Z, where Z is a normalization constant. There is one dual variable per expectation constraint, which can be optimized by projected gradient descent where gradient for λ is b −Eq[f(x, z)]. Gradient computation involves an expectation under q(z|x) that can be computed efficiently if the features f(x, z) factor in the same way as the model pθ(z|x) [7]. 3 Relaxing Posterior Regularization In this work, we modify PR so that instead of hard constraints on q(z | x), it allows the constraints to be relaxed at a cost specified by a penalty. This relaxation can allow combining multiple constraints without having to explicitly ensure that the constraint set remains non-empty. Additionally, it will be useful in dealing with the ℓ1/ℓ∞constraints we need. If those were incorporated as hard constraints, the dual objective would become non-differentiable, making the optimization (somewhat) more complicated. Using soft constraints, the non-differentiable portion of the dual objective turns into simplex constraints on the dual variables, allowing us to use an efficient projected gradient method. For soft constraints, Equation 4 is replaced by arg min q,b KL(q ∥p) + R(b) s. t. Eq[f(x, z)] ≤b (6) where b is the constraint vector, and R(b) penalizes overly lax constraints. For POS tagging, we will design R(b) to encourage each word type to be observed with a small number of POS tags in the projected posteriors q. The overall objective minimized can be shown to be: Soft PR Objective: arg min θ,q,b L(θ) + C(θ) + bE[KL(q ∥pθ) + R(b)] s. t. Eq[f(x, z)] ≤b. (7) 3.1 ℓ1/ℓ∞regularization We now choose the posterior constraint regularizer R(b) to encourage each word to be associated with only a few parts of speech. Let feature fwti have value 1 whenever the ith occurrence of word w has part of speech tag t. For every word w, we would like there to be only a few POS tags t such that there are occurrences i where t has nonzero probability. This can be achieved if it “costs” a lot to allow an occurrence of a word to take a tag, but once that happens, it should be “free” for other occurrences of the word to receive that same tag. More precisely, we would like the sum (ℓ1 norm) over tags t and word types w of the maxima (ℓ∞norm) of the expectation of taking tag t 3 over all occurrences of w to be small. Table 1 shows the value of the ℓ1/ℓ∞sparsity measure for three different corpora, comparing fully supervised HMM and fully unsupervised HMM learned with standard EM, with standard EM having 3-4 times larger value of ℓ1/ℓ∞than the supervised. This discrepancy is what our PR objective is attempting to eliminate. Formally, the E-step of our approach is expressed by the objective: min q,cwt KL(q ∥pθ) + σ X wt cwt s. t. Eq[fwti] ≤cwt (8) where σ is the strength of the regularization. Note that setting σ = 0 we are back to normal EM where q is the model posterior distribution. As σ →∞, the constraints force each occurrence of a word type to have the same posterior distribution, effectively reducing the mode to a 0th-order Markov chain in the E step. The dual of this objective has a very simple form (see supplementary material for derivation): max λ≥0 −log X z pθ(z) exp(−λ · f(z)) ! s. t. X i λwti ≤σ (9) where z ranges over assignments to the hidden tag variables for all of the occurrences in the training data, f(z) is the vector of fwti feature values for assignment z, λ is the vector of dual parameters λwti, and the primal parameters are q(z) ∝pθ(z) exp (−λ · f(z)). This can be computed by projected gradient, as described by Bertsekas [3]. Figure 1 illustrates how the ℓ1/ℓ∞norm operates on a toy example. For simplicity suppose we are only regularizing one word and our model pθ is just a product distribution over 15 instances of the word. The left panel in Figure 1 shows the posteriors under pθ. We would like to concentrate the posteriors on a small subset of rows. The center panel of the figure shows the λ values determined by Equation 9, and the right panel shows the projected distribution q, which concentrates most of the posterior on the bottom row. Note that we are not requiring the posteriors to be sparse, which would be equivalent to preferring that the distribution is peaked; rather, we want a word to concentrate its tag posterior on a few tags across all instances of the word. Indeed, most of the instances (columns) become less peaked than in the original posterior to allow posterior mass to be redistributed away from the outlier tags. Since they are more numerous than the outliers, they moved less. This also justifies only regularizing relatively frequent events in our model. 4 Bayesian Estimators Recent advances in inference methods for sparsifying Bayesian estimation have been applied to unsupervised POS tagging [4, 9, 6]. In the Bayesian setting, preference for sparsity is expressed as a prior distribution over model structures and parameters, rather than as constraints on feature posteriors. To compare these two approaches, in Section 5 we compare our method to a Bayesian approach proposed by Johnson [9], which relies on a Dirichlet prior to encourage sparsity in a firstorder HMM for POS tagging. The complete description of the model is: θi ∼ Dir(αi) P(ti|tt−1 = tag) ∼ Multi(θi) φi ∼ Dir(λi) P(wi|ti = tag) ∼ Multi(φi) Here, αi controls sparsity over the state transition matrix and λi controls the sparsity of state emission probabilities. Johnson [9] notes that αi does not influence the model that much. In contrast, as λi approaches zero, it encourages the model to have highly skewed P(wi|ti = tag) distributions, that is, each tag is encouraged to generate a few words with high probability, and the rest with very low probability. This is not exactly the constraint we would like to enforce: there are some POS tags that generate many different words with relatively high probability (for example, nouns and verbs), while each word is associated with a small number of tags. This difference is one possible explanation for the relatively worse performance of this prior compared to our method. Johnson [9] describes two approaches to learn the model parameters: a component-wise Gibbs sampling scheme (GS) and a variational Bayes (VB) approximation using a mean field. Since Johnson [9] found VB worked much better than GS, we use VB in our experiments. Additionally, VB is particularly simple to implement, consisting only a small modification to the M-Step of the EM algorithm. The Dirichlet prior hyper-parameters are added to the expected counts and passed through 4 a squashing function (exponential of the Digamma function) before being normalized. We refer the reader to the original paper for more detail (see also http://www.cog.brown.edu/~mj/ Publications.htm for a bug fix in the Digamma function implementation). 5 Experiments We now compare first-order HMMs trained using the three methods described earlier: the classical EM algorithm (EM), our ℓ1/ℓ∞posterior regularization based method (Sparse), and the model presented in Section 4 (VEM). Models were trained and tested on all available data of three corpora: the Wall Street Journal portion of the Penn treebank [13] using the reduced tag set of 17 tags [17] (PTB17); the Bosque subset of the Portuguese Floresta Sinta(c)tica Treebank [1] used for the ConLL X shared task on dependency parsing (PT-CoNLL); and the Bulgarian BulTreeBank [16] (BulTree) with the 12 coarse tags. We also report results on the full Penn treebank tag set in the supplementary materials. All words that occurred only once were replaced by the token “unk”. To measure model sparsity, we compute the average ℓ1/ℓ∞norm over words occurring more than 10 times (denoted ‘L1LMax’ in our figures). Table 1 gives statistics for each corpus as well as the sparsity for a first-order HMM trained using the labeled data and using standard EM with unlabeled data. Types Tokens Unk Tags Sup. ℓ1/ℓ∞ EM ℓ1/ℓ∞ PT-Conll 11293 206678 8.5% 22 1.14 4.57 BulTree 12177 174160 10% 12 1.04 3.51 PTB17 23768 950028 2% 17 1.23 3.97 Table 1: Corpus statistics. All words with only one occurrence where replaced by the ‘unk’ token. The third column shows the percentage of tokens replaced. Sup. ℓ1/ℓ∞is the value of the sparsity measure for a fully supervised HMM trained on all available data and EM ℓ1/ℓ∞is the value of the sparsity measure for a fully unsupervised HMM trained using standard EM on all available data. Following Gao and Johnson [4], the parameters were initialized with a “pseudo E step” as follows: we filled the expected count matrices with numbers 1 + X × U(0, 1), where U(0, 1) is a random number between 0 and 1 and X is a parameter. These matrices are then fed to the M step; the resulting “random” transition and emission probabilities are used for the first real E step. For VEM, X was set to 0.0001 (almost uniform) since this showed a significant improvement in performance. On the other hand, EM showed less sensitivity to initialization, and we used X = 1 which resulted in the best results. The models were trained for 200 iterations as longer runs did not significantly change the results (models converge before 100 iterations). For VEM we tested 4 different prior combinations, (all combinations of 10−1 and 10−3 for emission prior and transition prior), based on Johnson’s results [9]. As previously noted, changing the transition priors does not affect the Estimator PT-Conll BG PTB17 1-Many 1-1 1-Many 1-1 1-Many 1-1 EM 64.0(1.2) 40.4(3.0) 59.4(2.2) 42.0(3.0) 67.5(1.3) 46.4(2.6) VEM(10−1) 60.4(0.6) 51.1(2.3) 54.9(3.1) 46.4(3.0) 68.2(0.8)* 52.8(3.5) VEM(10−4) 63.2(1.0)* 48.1(2.2) 56.1(2.8) 43.3(1.7)* 67.3(0.8)* 49.6(4.3) Sparse (10) 68.5(1.3) 43.3(2.2) 65.1(1.0) 48.0(3.3) 69.5(1.6) 50.0(3.5) Sparse (32) 69.2(0.9) 43.2(2.9) 66.0(1.8) 48.7(2.2) 70.2(2.2) 49.5(2.0) Sparse (100) 68.3(2.1) 44.5(2.4) 65.9(1.6) 48.9(2.8) 68.7(1.1) 47.8(1.5)* Table 2: Average accuracy (standard deviation in parentheses) over 10 different runs (random seeds identical across models) for 200 iterations. 1-Many and 1-1 are the two hidden-state to POS mappings described in the text. All models are first order HMMs: EM trained using expectation maximization, VEM trained using variational EM observation priors shown in parentheses, Sparse trained using PR with the constraint strength (σ) in parentheses. Bold indicates the best value for each column. All results except those starred are significant (p=0.005) on a paired t-test against the EM model. 5 60 61 62 63 64 65 66 67 68 69 70 0 1 2 3 4 Sparse 100 Sparse 32 Sparse 10 EM VEM 10-1 VEM 10-3 40 41 42 43 44 45 46 47 48 49 50 51 52 0 1 2 3 4 Sparse 10 Sparse 32 Sparse 100 EM VEM 10-3 VEM 10-1 0 1 2 3 4 5 10 15 20 EM VEM 10-3 VEM 10-1 Sparse 32 True 1.4 1.6 1.8 2 2.2 2.4 EM VEM 0.001 VEM 0.1 Sparse 10 Sparse 32 Sparse 100 Max (a) (b) (c) (d) Figure 2: Detailed visualizations of the results on the PT-Conll corpus. (a) 1-many accuracy vs ℓ1/ℓ∞, (b) 1-1 accuracy vs ℓ1/ℓ∞, (c) tens of thousands of tokens assigned to hidden state vs rank, (d) mutual information in bits between gold tag distribution and hidden state distribution. results, so we only report results for different emission priors. Later work [4] considered a wider range of values but did not identify definitely better choices. Sparse was initialized with the parameters obtained by running EM for 30 iterations, followed by 170 iterations of the new training procedure. Predictions were obtained using posterior decoding since this consistently showed small improvements over Viterbi decoding. We evaluate the accuracy of the models using two established mappings between hidden states and POS tags: 1-Many maps each hidden state to the tag with which it co-occurs the most; 1-1 [8] greedily picks a tag for each state under the constraint of never using the same tag twice. This results in an approximation of the optimal 1-1 mapping. If the numbers of hidden states and tags are not the same, some hidden states will be unassigned (and hence always wrong) or some tags not used. In all our experiments the number of hidden states is the same as the number of POS tags. Table 2 shows the accuracy of the different methods averaged over 15 different random parameter initializations. Comparing the methods for each of the initialization points individually, our ℓ1/ℓ∞ regularization always outperforms EM baseline model on both metrics, and always outperforms VEM using 1-Many mapping, while for the 1-1 mapping our method outperforms VEM roughly half the time. The improvements are consistent for different constraint strength values. Figure 2 shows detailed visualizations of the behavior of the different methods on the PT-Conll corpus. The results for the other corpora are qualitatively similar, and are reported in the supplemental material. The left two plots show scatter graphs of accuracy with respect to ℓ1/ℓ∞value, where accuracy is measured with either the 1-many mapping (left) or 1-1 mapping (center). We see that Sparse is much better using the 1-many mapping and worse using the 1-1 mapping than VEM, even though they achieve similar ℓ1/ℓ∞. The third plot shows the number of tokens assigned to each hidden state at decoding time, in frequency rank order. While both EM and Sparse exhibit a fast decrease in the size of the states, VEM more closely matches the power law-like distribution achieved by the gold labels. This difference explains the improvement on the 1-1 mapping, where VEM is assigning larger size states to the most frequent tags. However, VEM achieves this power law distribution at the expense of the mutual information with the gold labels as we see in the rightmost plot. From all methods, VEM has the lowest mutual information, while Sparse has the highest. 5.1 Closed-class words We now consider the case where some supervision has been given in the form of a list of the closedclass words for the language, along with POS tags. Example closed classes are punctuation, pronouns, possessive markers, while open classes would include nouns, verbs, and adjectives. (See the supplemental materials for details.) We assume that we are given the POS tags of closed classes along with the words in each closed class. In the models, we set the emission probability from a closed-class tag to any word not in its class to zero. Also, any word appearing in a closed class is assumed to have zero probability of being generated by an open-class tag. This improves performance significantly for all languages, but our sparse training procedure is still able to outperform EM training significantly as shown in Table 3. Note, for these experiments we do not use an unknown word, since doing so for closed-class words would allow closed class tags to generate unknown words. 6 Estimator PT-Conll BulTree PTB-17 1-Many 1-1 1-Many 1-1 1-Many 1-1 EM 72.5(1.7) 52.6(4.2) 77.9(1.7) 65.4(2.8) 76.7(0.9) 61.1(1.8) Sparse (32) 75.3(1.2) 57.5(5.0) 82.4(1.2) 69.5(1.3) 78.0(1.6) 62.2(2.0) Table 3: Results with given closed-class tags, using posterior decoding, and projection at test time. PT-Conll BulTree PTB-17 65 70 75 80 10 20 30 40 50 60 70 80 90 100 Sparse 32 EM VEM none 55 60 65 70 75 80 85 10 20 30 40 50 60 70 80 90 100 Sparse 32 EM VEM none 65 70 75 80 85 90 10 20 30 40 50 60 70 80 90 100 Sparse 32 EM VEM none Figure 3: Accuracy of a supervised classifier when trained using the output of various unsupervised models as features. Vertical axis: accuracy, Horizontal axis: number of labeled sentences. 5.2 Supervised POS tagging As a further comparison of the models trained using the different methods, we use them to generate features for a supervised POS tagger. The basic supervised model has features for the identity of the current token as well as suffixes of length 2 and 3. We augment these features with the state identity for the current token, based on the automatically generated models. We train the supervised model using averaged perceptron for 20 iterations. For each unsupervised training procedure (EM, Sparse, VEM) we train 10 models using different random initializations and got 10 state identities per training method for each token. We then add these cluster identities as features to the supervised model. Figure 3 shows the average accuracy of the supervised model as we vary the type of unsupervised features. The average is taken over 10 random samples for the training set at each training set size. We can see from Figure 3 that using our method or EM always improves performance relative to the baseline features (labeled “none” in the figure). VEM always under performs EM and for larger amounts of training data, the VEM features appear not to be useful. This should not be surprising given that VEM has very low mutual information with the gold labeling. 6 Related Work Our learning method is very closely related to the work of Mann and McCallum [11, 12], who concurrently developed the idea of using penalties based on posterior expectations of features to guide learning. They call their method generalized expectation (GE) constraints or alternatively expectation regularization. In the original GE framework, the posteriors of the model are regularized directly. For equality constraints, our objective would become: arg max θ L(θ) −ED[R(Eθ[f])]. (10) Notice that there is no intermediate distribution q. For some kinds of constraints this objective is difficult to optimize in θ and in order to improve efficiency Bellare et al. [2] propose interpreting the PR framework as an approximation to the GE objective in Equation 10. They compare the two frameworks on several datasets and find that performance is similar, and we suspect that this would be true for the sparsity constraints also. Liang et al. [10] cast the problem of incorporating partial information about latent variables into a Bayesian framework using “measurements,” and they propose active learning for acquiring measurements to reduce uncertainty. 7 Recently, Ravi et al. [15] show promising results in weakly-supervised POS tagging, where a tag dictionary is provided. This method first searches, using integer programming, for the smallest grammar (in terms of unique transitions between tags) that explains the data. This sparse grammar and the dictionary are provided as input for training an unsupervised HMM. Results show that using a sparse grammar, hence enforcing sparsity over possible sparsity transitions leads to better results. This method is different from ours in the sense that our method focuses on learning the sparsity pattern they their method uses as input. 7 Conclusion We presented a new regularization method for unsupervised training of probabilistic models that favors a kind of sparsity that is pervasive in natural language processing. In the case of part-ofspeech induction, the preference can be summarized as “each word occurs as only a few different parts-of-speech,” but the approach is more general and could be applied to other tasks. For example, in grammar induction, we could favor models where only a small number of production rules have non-zero probability for each child non-terminal. Our method uses the posterior regularization framework to specify preferences about model posteriors directly, without having to say how these should be encoded in model parameters. This means that the sparse regularization penalty could be used for a log-linear model, where sparse parameters do not correspond to posterior sparsity. We evaluated the new regularization method on the task of unsupervised POS tagging, encoding the prior knowledge that each word should have a small set of tags as a mixed-norm penalty. We compared our method to a previously proposed Bayesian method (VEM) for encouraging sparsity of model parameters [9] and found that ours performs better in practice. We explain this advantage by noting that VEM encodes a preference that each POS tag should generate a few words, which goes in the wrong direction. In reality, in POS tagging (as in several other language processing task), a few event types (tags) (such the NN for POS tagging) generate the bulk of the word occurrences, but each word is only associated with a few tags. Even when some supervision was provided with through closed class lists, our regularizer still improved performance over the other methods. An analysis of sparsity shows that both VEM and Sparse achieve a similar posterior sparsity as measured by the ℓ1/ℓ∞metric. While VEM models better the empirical sizes of states (tags), the states it assigns have lower mutual information to the true tags, suggesting that parameter sparsity is not as good at generating good tag assignments. In contrast, Sparse’s sparsity seems to help build a model that contains more information about the correct tag assignments. Finally, we evaluated the worth of states assigned by unsupervised learning as features for supervised tagger training with small training sets. These features are shown to be useful in most conditions, especially those created by Sparse. The exceptions are some of the annotations provided by VEM which actually hinder the performance, confirming that its lower mutual information states are not so informative. In future work, we would like to evaluate the usefulness of these sparser annotations for downstream tasks, for example determining whether Sparse POS tags are better for unsupervised parsing. Finally, we would like to apply the ℓ1/ℓ∞posterior regularizer to other applications such as unsupervised grammar induction where we would like sparsity in production rules. Similarly, it would be interesting to use this to regularize a log-linear model, where parameter sparsity does not achieve the same goal. Acknowledgments J. V. Graça was supported by a fellowship from Fundação para a Ciência e Tecnologia (SFRH/ BD/ 27528/ 2006). K. Ganchev was supported by ARO MURI SUBTLE W911NF-07-1-0216 The authors would like to thank Mark Johnson and Jianfeng Gao for their help in reproducing the VEM results. 8 References [1] S. Afonso, E. Bick, R. Haber, and D. Santos. Floresta Sinta(c)tica: a treebank for Portuguese. In In Proc. LREC, pages 1698–1703, 2002. [2] K. Bellare, G. Druck, and A. McCallum. Alternating projections for learning with expectation constraints. In In Proc. UAI, 2009. [3] D.P. Bertsekas, M.L. Homer, D.A. Logan, and S.D. Patek. Nonlinear programming. Athena scientific, 1995. [4] Jianfeng Gao and Mark Johnson. A comparison of Bayesian estimators for unsupervised Hidden Markov Model POS taggers. In In Proc. EMNLP, pages 344–352, Honolulu, Hawaii, October 2008. ACL. [5] Y. Goldberg, M. Adler, and M. Elhadad. Em can find pretty good hmm pos-taggers (when given a good start). In Proc. ACL, pages 746–754, 2008. [6] S. Goldwater and T. Griffiths. A fully bayesian approach to unsupervised part-of-speech tagging. In In Proc. ACL, volume 45, page 744, 2007. [7] J. Graça, K. Ganchev, and B. Taskar. Expectation maximization and posterior constraints. In In Proc. NIPS. MIT Press, 2008. [8] A. Haghighi and D. Klein. Prototype-driven learning for sequence models. In In Proc. NAACL, pages 320–327, 2006. [9] M Johnson. Why doesn’t EM find good HMM POS-taggers. In In Proc. EMNLP-CoNLL, 2007. [10] P. Liang, M. I. Jordan, and D. Klein. Learning from measurements in exponential families. In In proc. ICML, 2009. [11] G. Mann and A. McCallum. Simple, robust, scalable semi-supervised learning via expectation regularization. In Proc. ICML, 2007. [12] G. Mann and A. McCallum. Generalized expectation criteria for semi-supervised learning of conditional random fields. In In Proc. ACL, pages 870 – 878, 2008. [13] M.P. Marcus, M.A. Marcinkiewicz, and B. Santorini. Building a large annotated corpus of English: The Penn Treebank. Computational linguistics, 19(2):313–330, 1993. [14] B. Merialdo. Tagging English text with a probabilistic model. Computational linguistics, 20(2):155–171, 1994. [15] Sujith Ravi and Kevin Knight. Minimized models for unsupervised part-of-speech tagging. In In Proc. ACL, 2009. [16] Kiril Simov, Petya Osenova, Milena Slavcheva, Sia Kolkovska, Elisaveta Balabanova, Dimitar Doikoff, Krassimira Ivanova, Alexander Simov, Er Simov, and Milen Kouylekov. Building a linguistically interpreted corpus of bulgarian: the bultreebank. In In Proc. LREC, page pages, 2002. [17] N.A. Smith and J. Eisner. Contrastive estimation: Training log-linear models on unlabeled data. In In Proc. ACL, pages 354–362, 2005. [18] K. Toutanova and M. Johnson. A Bayesian LDA-based model for semi-supervised part-ofspeech tagging. In Proc. NIPS, 20, 2007. 9
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1 Reconstruction of Sparse Circuits Using Multi-neuronal Excitation (RESCUME) Tao Hu and Dmitri B. Chklovskii Janelia Farm Research Campus, HHMI 19700 Helix Drive, Ashburn, VA 20147 hut, mitya@janelia.hhmi.org Abstract One of the central problems in neuroscience is reconstructing synaptic connectivity in neural circuits. Synapses onto a neuron can be probed by sequentially stimulating potentially pre-synaptic neurons while monitoring the membrane voltage of the post-synaptic neuron. Reconstructing a large neural circuit using such a “brute force” approach is rather time-consuming and inefficient because the connectivity in neural circuits is sparse. Instead, we propose to measure a post-synaptic neuron’s voltage while stimulating sequentially random subsets of multiple potentially pre-synaptic neurons. To reconstruct these synaptic connections from the recorded voltage we apply a decoding algorithm recently developed for compressive sensing. Compared to the brute force approach, our method promises significant time savings that grow with the size of the circuit. We use computer simulations to find optimal stimulation parameters and explore the feasibility of our reconstruction method under realistic experimental conditions including noise and non-linear synaptic integration. Multineuronal stimulation allows reconstructing synaptic connectivity just from the spiking activity of post-synaptic neurons, even when sub-threshold voltage is unavailable. By using calcium indicators, voltage-sensitive dyes, or multi-electrode arrays one could monitor activity of multiple postsynaptic neurons simultaneously, thus mapping their synaptic inputs in parallel, potentially reconstructing a complete neural circuit. 1 Introduction Understanding information processing in neural circuits requires systematic characterization of synaptic connectivity [1, 2]. The most direct way to measure synapses between a pair of neurons is to stimulate potentially pre-synaptic neuron while recording intra-cellularly from the potentially post-synaptic neuron [3-8]. This method can be scaled to reconstruct multiple synaptic connections onto one neuron by combining intracellular recordings from the postsynaptic neuron with photo-activation of pre-synaptic neurons using glutamate uncaging [913] or channelrhodopsin [14, 15], or with multi-electrode arrays [16, 17]. Neurons are sequentially stimulated to fire action potentials by scanning a laser beam (or electrode voltage) over a brain slice, while synaptic weights are measured by recording post-synaptic voltage. Although sequential excitation of single potentially pre-synaptic neurons could reveal connectivity, such a “brute force” approach is inefficient because the connectivity among neurons is sparse. Even among nearby neurons in the cerebral cortex, the probability of connection is only about ten percent [3-8]. Connection probability decays rapidly with the 2 distance between neurons and falls below one percent on the scale of a cortical column [3, 8]. Thus, most single-neuron stimulation trials would result in zero response making the brute force approach slow, especially for larger circuits. Another drawback of the brute force approach is that single-neuron stimulation cannot be combined efficiently with methods allowing parallel recording of neural activity, such as calcium imaging [18-22], voltage-sensitive dyes [23-25] or multi-electrode arrays [17, 26]. As these techniques do not reliably measure sub-threshold potential but report only spiking activity, they would reveal only the strongest connections that can drive a neuron to fire [2730]. Therefore, such combination would reveal only a small fraction of the circuit. We propose to circumvent the above limitations of the brute force approach by stimulating multiple potentially pre-synaptic neurons simultaneously and reconstructing individual connections by using a recently developed method called compressive sensing (CS) [31-35]. In each trial, we stimulate F neurons randomly chosen out of N potentially pre-synaptic neurons and measure post-synaptic activity. Although each measurement yields only a combined response to stimulated neurons, if synaptic inputs sum linearly in a post-synaptic neuron, one can reconstruct the weights of individual connections by using an optimization algorithm. Moreover, if the synaptic connections are sparse, i.e. only K << N potentially pre-synaptic neurons make synaptic connections onto a post-synaptic neuron, the required number of trials M ~ K log(N/K), which is much less than N [31-35]. The proposed method can be used even if only spiking activity is available. Because multiple neurons are driven to fire simultaneously, if several of them synapse on the post-synaptic neuron, they can induce one or more spikes in that neuron. As quantized spike counts carry less information than analog sub-threshold voltage recordings, reconstruction requires a larger number of trials. Yet, the method can be used to reconstruct a complete feedforward circuit from spike recordings. Reconstructing neural circuit with multi-neuronal excitation may be compared with mapping retinal ganglion cell receptive fields. Typically, photoreceptors are stimulated by white-noise checkerboard stimulus and the receptive field is obtained by Reverse Correlation (RC) in case of sub-threshold measurements or Spike-Triggered Average (STA) of the stimulus [36, 37]. Although CS may use the same stimulation protocol, for a limited number of trials, the reconstruction quality is superior to RC or STA. 2 Mapping synaptic inputs onto one neuron We start by formalizing the problem of mapping synaptic connections from a population of N potentially pre-synaptic neurons onto a single neuron, as exemplified by granule cells synapsing onto a Purkinje cell (Figure 1a). Our experimental protocol can be illustrated using linear algebra formalism, Figure 1b. We represent synaptic weights as components of a column vector x, where zeros represent non-existing connections. Each row in the stimulation matrix A represents a trial, ones indicating neurons driven to spike once and zeros indicating non-spiking neurons. The number of rows in the stimulation matrix A is equal to the number of trials M. The column vector y represents M measurements of membrane voltage obtained by an intra-cellular recording from the post-synaptic neuron: y = Ax. (1) In order to recover individual synaptic weights, Eq. (1) must be solved for x. RC (or STA) solution to this problem is x = (ATA)-1AT y, which minimizes (y-Ax)2 if M>N. In the case M<N, the corresponding expression x = AT(AAT)-1y is a solution to the following problem: min 2 2 1 N i l i x x = = ∑ , subject to y = Ax. Given prior knowledge that the connectivity is sparse, we propose to recover x by solving instead: min 0l x , subject to y = Ax, where 0l x is the l0-norm of x or the number of non-zero elements. Under certain conditions 3 [25-29], this solution can be obtained by minimizing the l1-norm: 1 1 N i l i x x = =∑ using linear programming [38] or by iterative greedy algorithms [39, 40]. In this paper, we used a particularly efficient Compressive Sampling Matching Pursuit (CoSaMP) algorithm [41, 42]. We simulate the proposed reconstruction method in silico by generating a neural network, simulating experimental measurements, and recovering synaptic weights. We draw unitless synaptic weights from a distribution derived from electrophysiological measurements [4, 5, 43, 44] containing a delta-function at zero and an exponential distribution with a unit mean (Figure 2a). We generate an M-by-N stimulation matrix A by setting F randomly chosen entries in each row to one, and the rest to zero. We compute the measurement vector y by multiplying A and x. Then, we use the CoSaMP algorithm to recover synaptic weights, xr, from A and y. Figure 2b compares a typical result of such reconstruction and a result of RC with originally generated non-zero synaptic weights x. Despite using fewer measurements than required in the brute force approach, CS achieves perfect reconstruction while RC yields a worse result [45]. y A x (a) (b) measurements M M N N neurons K connections synaptic Figure 1: Mapping synapses onto one neuron. a) A potentially post-synaptic neuron (red) receives synaptic connection (blue) from K neurons out of N potentially pre-synaptic neurons. b) Linear algebra representation of the experimental protocol. The column vector x contains synaptic weights from N potentially pre-synaptic neurons: K blue squares represent existing connections, white squares represent absent connections. The matrix A represents the sequence of stimulation: black squares in each row represent stimulated neurons in each trial. The column vector y contains measured membrane voltage in the red neuron. 0 100 200 300 400 500 1 2 3 Potential pre−synaptic neurons 1 0.98 0.96 0.94 0.92 0.08 0.06 0.04 0.02 Probability // Synaptic weights synaptic weights synaptic weights recoverd by CS synaptic weights recoverd by RC (a) (b) K = 30 non-zero synaptic weights N = 500 potential pre-synaptic neurons M = 200 trials F = 50 neurons stimulated per trial Figure 2: Reconstruction of synaptic weights onto one neuron. a) Synaptic weights are drawn from the empirically motivated probability distribution. b) Reconstruction by CS (red) coincides perfectly with generated synaptic weights (blue), achieving 60% improvement in the number of trials over the brute force approach. RC result (green) is significantly worse. 3 Minimum number of measurements as a function of network size and sparseness In order to understand intuitively why the number of trials can be less than the number of potential synapses, note that the minimum number of trials, i.e. information or entropy, is 4 given by the logarithm of the total number of possible connectivity patterns. If connections are binary, the number of different connectivity patterns onto a post-synaptic neuron from N neurons is 2N, and hence the minimum number of trials is N. However, prior knowledge that only K connections are present reduces the number of possible connectivity patterns from 2N to the binomial coefficient, CK N ~ (N/K)K. Thus, the number of trials dramatically reduces from N to K log(N/K) << N for a sparse circuit. In this section we search computationally for the minimum number of trials required for exact reconstruction as a function of the number of non-zero synaptic weights K out of N potentially pre-synaptic neurons. First, note that the number of trials depends on the number of stimulated neurons F. If F = 1 we revert to the brute force approach and the number of measurements is N, while for F = N, the measurements are redundant and no finite number suffices. As the minimum number of measurements is expected to scale as K logN, there must be an optimal F which makes each measurement most informative about x. To determine the optimal number of stimulated neurons F for given K and N, we search for the minimum number of trials M, which allows a perfect reconstruction of the synaptic connectivity x. For each F, we generate 50 synaptic weight vectors and attempt reconstruction from sequentially increasing numbers of trials. The value of M, at which all 50 recoveries are successful (up to computer round-off error), estimates the number of trial needed for reconstruction with probability higher than 98%. By repeating this procedure 50 times for each F, we estimate the mean and standard deviation of M. We find that, for given N and K, the minimum number of trials, M, as a function of the number of stimulated neurons, F, has a shallow minimum. As K decreases, the minimum shifts towards larger F because more neurons should be stimulated simultaneously for sparser x. For the explored range of simulation parameters, the minimum is located close to 0.1N. Next, we set F = 0.1N and explore how the minimum number of measurements required for exact reconstruction depends on K and N. Results of the simulations following the recipe described above are shown in Figure 3a. As expected, when x is sparse, M grows approximately linearly with K (Figure 3b), and logarithmically with N (Figure 3c). Number of potential connections (N) Number of actual connections (K) Number of necessary measurements (M) 150 250 400 650 1000 5 10 15 20 25 30 80 100 120 140 160 180 200 5 10 15 20 25 30 80 100 120 140 160 180 200 220 Number of actual connections (K) Number of measurements (M) N = 1000 10 2 10 3 120 130 140 150 160 170 180 190 200 210 220 Number of measurements (M) K = 30 Number of potential connections (N) (a) (b) (c) Figure 3: a) Minimum number of measurements M required for reconstruction as a function of the number of actual synapses, K, and the number of potential synapses, N. b) For given N, we find M ~ K. c) For given K, we find M ~ logN (note semi-logarithmic scale in c). 4 Robustness of reconstructions to noise and violation of simplifying assumptions To make our simulation more realistic we now take into account three possible sources of noise: 1) In reality, post-synaptic voltage on a given synapse varies from trial to trial [4, 5, 46-52], an effect we call synaptic noise. Such noise detrimentally affects reconstructions because each row of A is multiplied by a different instantiation of vector x. 2) Stimulation of neurons may be imprecise exciting a slightly different subset of neurons than intended and/or firing intended neurons multiple times. We call this effect stimulation noise. Such noise detrimentally affects reconstructions because, in its presence, the actual measurement matrix A is different from the one used for recovery. 3) A synapse may fail to release neurotransmitter with some probability. Naturally, in the presence of noise, reconstructions cannot be exact. We quantify the 5 reconstruction error by the normalized l2–error 2 2 / r l l x x x − , where 2 2 1( ) N r i ri l i x x x x = − = − ∑ . We plot normalized reconstruction error in brute force approach (M = N = 500 trials) as a function of noise, as well as CS and RC reconstruction errors (M = 200, 600 trials), Figure 4. For each noise source, the reconstruction error of the brute force approach can be achieved with 60% fewer trials by CS method for the above parameters (Figure 4). For the same number of trials, RC method performs worse. Naturally, the reconstruction error decreases with the number of trials. The reconstruction error is most sensitive to stimulation noise and least sensitive to synaptic noise. 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Synaptic failure probability 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Stimulation noise level 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Synaptic noise level Normalized reconstruction error ||x-xr||2/||x||2 RC: M=200 RC: M=600 Brute force method: M=500 CS: M=200 CS: M=600 (a) (b) (c) Figure 4: Impact of noise on the reconstruction quality for N = 500, K = 30, F = 50. a) Recovery error due to trial-to-trial variation in synaptic weight. The response y is calculated using the synaptic connectivity x perturbed by an additive Gaussian noise. The noise level is given by the coefficient of variation of synaptic weight. b) Recovery error due to stimulation noise. The matrix A used for recovery is obtained from the binary matrix used to calculate the measurement vector y by shifting, in each row, a fraction of ones specified by the noise level to random positions. c) Recovery error due to synaptic failures. The detrimental effect of the stimulation noise on the reconstruction can be eliminated by monitoring spiking activity of potentially pre-synaptic neurons. By using calcium imaging [18-22], voltage-sensitive dyes [23] or multi-electrode arrays [17, 26] one could record the actual stimulation matrix. Because most random matrices satisfy the reconstruction requirements [31, 34, 35], the actual stimulation matrix can be used for a successful recovery instead of the intended one. If neuronal activity can be monitored reliably, experiments can be done in a different mode altogether. Instead of stimulating designated neurons with high fidelity by using highly localized and intense light, one could stimulate all neurons with low probability. Random firing events can be detected and used in the recovery process. The light intensity can be tuned to stimulate the optimal number of neurons per trial. Next, we explore the sensitivity of the proposed reconstruction method to the violation of simplifying assumptions. First, whereas our simulation assumes that the actual number of connections, K, is known, in reality, connectivity sparseness is known a priori only approximately. Will this affect reconstruction results? In principle, CS does not require prior knowledge of K for reconstruction [31, 34, 35]. For the CoSaMP algorithm, however, it is important to provide value K larger than the actual value (Figure 5a). Then, the algorithm will find all the actual synaptic weights plus some extra non-zero weights, negligibly small when compared to actual ones. Thus, one can provide the algorithm with the value of K safely larger than the actual one and then threshold the reconstruction result according to the synaptic noise level. Second, whereas we assumed a linear summation of inputs [53], synaptic integration may be non-linear [54]. We model non-linearity by setting 2 l l y y y α = + , where yl represents linearly summed synaptic inputs. Results of simulations (Figure 5b) show that although nonlinearity can significantly degrade CS reconstruction quality, it still performs better than RC. 6 -0.15-0.12-0.09-0.06-0.03 0 0.03 0.06 0.09 0.12 0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Relative strength of the non-linear term α × mean(yl) Normalized reconstruction error ||x-xr||2/||x||2 CS RC (a) (b) 10 20 30 40 50 60 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 K fed to CoSaMP Normalized reconstrcution error ||x-xr||2/||x||2 Actual K = 30 Figure 5: Sensitivity of reconstruction error to the violation of simplifying assumptions for N = 500, K = 30, M = 200, F = 50. a) The quality of the reconstruction is not affected if the CoSaMP algorithm is fed with the value of K larger than actual. b) Reconstruction error computed in 100 realizations for each value of the quadratic term relative to the linear term. 5 Mapping synaptic inputs onto a neuronal population Until now, we considered reconstruction of synaptic inputs onto one neuron using subthreshold measurements of its membrane potential. In this section, we apply CS to reconstructing synaptic connections onto a population of potentially post-synaptic neurons. Because in CS the choice of stimulated neurons is non-adaptive, by recording from all potentially post-synaptic neurons in response to one sequence of trials one can reconstruct a complete feedforward network (Figure 6). y A x (a) (b) Ax p(y=1) 0 (c) 100 300 500 700 900 1100 1000 3000 5000 7000 9000 0 0.2 0.4 0.6 0.8 1 Number of trials (M) Normalized reconstruction error ||x/||x||2−xr/||xr||2||2 STA CS Number of spikes (d) Figure 6: Mapping of a complete feedforward network. a) Each post-synaptic neuron (red) receives synapses from a sparse subset of potentially pre-synaptic neurons (blue). b) Linear algebra representation of the experimental protocol. c) Probability of firing as a function of synaptic current. d) Comparison of CS and STA reconstruction error using spike trains for N = 500, K = 30 and F = 50. Although attractive, such parallelization raises several issues. First, patching a large number of neurons is unrealistic and, therefore, monitoring membrane potential requires using different methods, such as calcium imaging [18-22], voltage sensitive dyes [23-25] or multielectrode arrays [17, 26]. As these methods can report reliably only spiking activity, the measurement is not analog but discrete. Depending on the strength of summed synaptic inputs compared to the firing threshold, the postsynaptic neuron may be silent, fire once or multiple times. As a result, the measured response y is quantized by the integer number of spikes. Such quantized measurements are less informative than analog measurements of the sub-threshold membrane potential. In the extreme case of only two quantization levels, spike or no spike, each measurement contains only 1 bit of information. Therefore, to achieve reasonable reconstruction quality using quantized measurements, a larger number of trials M>>N is required. We simulate circuit reconstruction from spike recordings in silico as follows. First, we draw synaptic weights from an experimentally motivated distribution. Second, we generate a 7 random stimulation matrix and calculate the product Ax. Third, we linear half-wave rectify this product and use the result as the instantaneous firing rate for the Poisson spike generator (Figure 6c). We used a rectifying threshold that results in 10% of spiking trials as typically observed in experiments. Fourth, we reconstruct synaptic weights using STA and CS and compare the results with the generated weights. We calculated mean error over 100 realizations of the simulation protocol (Figure 6d). Due to the non-linear spike generating procedure, x can be recovered only up to a scaling factor. We propose to calibrate x with a few brute-force measurements of synaptic weights. Thus, in calculating the reconstruction error using l2 norm, we normalize both the generated and recovered synaptic weights. Such definition is equivalent to the angular error, which is often used to evaluate the performance of STA in mapping receptive field [37, 55]. Why is CS superior to STA for a given number of trials (Figure 6d)? Note that spikeless trials, which typically constitute a majority, also carry information about connectivity. While STA discards these trials, CS takes them into account. In particular, CoSaMP starts with the STA solution as zeroth iteration and improves on it by using the results of all trials and the sparseness prior. 6 Discussion We have demonstrated that sparse feedforward networks can be reconstructed by stimulating multiple potentially pre-synaptic neurons simultaneously and monitoring either subthreshold or spiking response of potentially post-synaptic neurons. When sub-threshold voltage is recorded, significantly fewer measurements are required than in the brute force approach. Although our method is sensitive to noise (with stimulation noise worse than synapse noise), it is no less robust than the brute force approach or RC. The proposed reconstruction method can also recover inputs onto a neuron from spike counts, albeit with more trials than from sub-threshold potential measurements. This is particularly useful when intra-cellular recordings are not feasible and only spiking can be detected reliably, for example, when mapping synaptic inputs onto multiple neurons in parallel. For a given number of trials, our method yields smaller error than STA. The proposed reconstruction method assumes linear summation of synaptic inputs (both excitatory and inhibitory) and is sensitive to non-linearity of synaptic integration. Therefore, it is most useful for studying connections onto neurons, in which synaptic integration is close to linear. On the other hand, multi-neuron stimulation is closer than single-neuron stimulation to the intrinsic activity in the live brain and can be used to study synaptic integration under realistic conditions. In contrast to circuit reconstruction using intrinsic neuronal activity [56, 57], our method relies on extrinsic stimulation of neurons. Can our method use intrinsic neuronal activity instead? We see two major drawbacks of such approach. First, activity of non-monitored presynaptic neurons may significantly distort reconstruction results. Thus, successful reconstruction would require monitoring all active pre-synaptic neurons, which is rather challenging. Second, reliable reconstruction is possible only when the activity of presynaptic neurons is uncorrelated. Yet, their activity may be correlated, for example, due to common input. We thank Ashok Veeraraghavan for introducing us to CS, Anthony Leonardo for making a retina dataset available for the analysis, Lou Scheffer and Hong Young Noh for commenting on the manuscript and anonymous reviewers for helpful suggestions. References [1] Luo, L., Callaway, E.M. & Svoboda, K. (2008) Genetic dissection of neural circuits. Neuron 57(5):634-660. [2] Helmstaedter, M., Briggman, K.L. & Denk, W. (2008) 3D structural imaging of the brain with photons and electrons. Current opinion in neurobiology 18(6):633-641. [3] Holmgren, C., Harkany, T., Svennenfors, B. & Zilberter, Y. (2003) Pyramidal cell communication within local networks in layer 2/3 of rat neocortex. Journal of Physiology 551:139-153. [4] Markram, H. (1997) A network of tufted layer 5 pyramidal neurons. Cerebral Cortex 7(6):523-533. 8 [5] Markram, H., Lubke, J., Frotscher, M., Roth, A. & Sakmann, B. (1997) Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex. Journal of Physiology 500(2):409-440. [6] Thomson, A.M. & Bannister, A.P. (2003) Interlaminar connections in the neocortex. Cerebral Cortex 13(1):5-14. [7] Thomson, A.M., West, D.C., Wang, Y. & Bannister, A.P. (2002) Synaptic connections and small circuits involving excitatory and inhibitory neurons in layers 2-5 of adult rat and cat neocortex: triple intracellular recordings and biocytin labelling in vitro. Cerebral Cortex 12(9):936-953. [8] Song, S., Sjostrom, P.J., Reigl, M., Nelson, S. & Chklovskii, D.B. (2005) Highly nonrandom features of synaptic connectivity in local cortical circuits. Plos Biology 3(3):e68. [9] Callaway, E.M. & Katz, L.C. (1993) Photostimulation using caged glutamate reveals functional circuitry in living brain slices. Proceedings of the National Academy of Sciences of the United States of America 90(16):7661-7665. [10] Dantzker, J.L. & Callaway, E.M. (2000) Laminar sources of synaptic input to cortical inhibitory interneurons and pyramidal neurons. Nature Neuroscience 3(7):701-707. [11] Shepherd, G.M. & Svoboda, K. (2005) Laminar and columnar organization of ascending excitatory projections to layer 2/3 pyramidal neurons in rat barrel cortex. Journal of Neuroscience 25(24):5670-5679. [12] Nikolenko, V., Poskanzer, K.E. & Yuste, R. (2007) Two-photon photostimulation and imaging of neural circuits. Nature Methods 4(11):943-950. [13] Shoham, S., O'connor, D.H., Sarkisov, D.V. & Wang, S.S. (2005) Rapid neurotransmitter uncaging in spatially defined patterns. Nature Methods 2(11):837-843. [14] Gradinaru, V., Thompson, K.R., Zhang, F., Mogri, M., Kay, K., Schneider, M.B. & Deisseroth, K. (2007) Targeting and readout strategies for fast optical neural control in vitro and in vivo. Journal of Neuroscience 27(52):14231-14238. [15] Petreanu, L., Huber, D., Sobczyk, A. & Svoboda, K. (2007) Channelrhodopsin-2-assisted circuit mapping of long-range callosal projections. Nature Neuroscience 10(5):663-668. [16] Na, L., Watson, B.O., Maclean, J.N., Yuste, R. & Shepard, K.L. (2008) A 256×256 CMOS Microelectrode Array for Extracellular Neural Stimulation of Acute Brain Slices. Solid-State Circuits Conference, 2008. ISSCC 2008. Digest of Technical Papers. IEEE International. [17] Fujisawa, S., Amarasingham, A., Harrison, M.T. & Buzsaki, G. (2008) Behavior-dependent shortterm assembly dynamics in the medial prefrontal cortex. Nature Neuroscience 11(7):823-833. [18] Ikegaya, Y., Aaron, G., Cossart, R., Aronov, D., Lampl, I., Ferster, D. & Yuste, R. (2004) Synfire chains and cortical songs: temporal modules of cortical activity. Science 304(5670):559-564. [19] Ohki, K., Chung, S., Ch'ng, Y.H., Kara, P. & Reid, R.C. (2005) Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature 433(7026):597-603. [20] Stosiek, C., Garaschuk, O., Holthoff, K. & Konnerth, A. (2003) In vivo two-photon calcium imaging of neuronal networks. Proceedings of the National Academy of Sciences of the United States of America 100(12):7319-7324. [21] Svoboda, K., Denk, W., Kleinfeld, D. & Tank, D.W. (1997) In vivo dendritic calcium dynamics in neocortical pyramidal neurons. Nature 385(6612):161-165. [22] Sasaki, T., Minamisawa, G., Takahashi, N., Matsuki, N. & Ikegaya, Y. (2009) Reverse optical trawling for synaptic connections in situ. Journal of Neurophysiology 102(1):636-643. [23] Zecevic, D., Djurisic, M., Cohen, L.B., Antic, S., Wachowiak, M., Falk, C.X. & Zochowski, M.R. (2003) Imaging nervous system activity with voltage-sensitive dyes. Current Protocols in Neuroscience Chapter 6:Unit 6.17. [24] Cacciatore, T.W., Brodfuehrer, P.D., Gonzalez, J.E., Jiang, T., Adams, S.R., Tsien, R.Y., Kristan, W.B., Jr. & Kleinfeld, D. (1999) Identification of neural circuits by imaging coherent electrical activity with FRET-based dyes. Neuron 23(3):449-459. [25] Taylor, A.L., Cottrell, G.W., Kleinfeld, D. & Kristan, W.B., Jr. (2003) Imaging reveals synaptic targets of a swim-terminating neuron in the leech CNS. Journal of Neuroscience 23(36):11402-11410. [26] Hutzler, M., Lambacher, A., Eversmann, B., Jenkner, M., Thewes, R. & Fromherz, P. (2006) High-resolution multitransistor array recording of electrical field potentials in cultured brain slices. Journal of Neurophysiology 96(3):1638-1645. [27] Egger, V., Feldmeyer, D. & Sakmann, B. (1999) Coincidence detection and changes of synaptic efficacy in spiny stellate neurons in rat barrel cortex. Nature Neuroscience 2(12):1098-1105. [28] Feldmeyer, D., Egger, V., Lubke, J. & Sakmann, B. (1999) Reliable synaptic connections between pairs of excitatory layer 4 neurones within a single 'barrel' of developing rat somatosensory cortex. Journal of Physiology 521:169-190. [29] Peterlin, Z.A., Kozloski, J., Mao, B.Q., Tsiola, A. & Yuste, R. (2000) Optical probing of neuronal circuits with calcium indicators. Proceedings of the National Academy of Sciences of the United States of America 97(7):3619-3624. 9 [30] Thomson, A.M., Deuchars, J. & West, D.C. (1993) Large, deep layer pyramid-pyramid single axon EPSPs in slices of rat motor cortex display paired pulse and frequency-dependent depression, mediated presynaptically and self-facilitation, mediated postsynaptically. Journal of Neurophysiology 70(6):2354-2369. [31] Baraniuk, R.G. (2007) Compressive sensing. Ieee Signal Processing Magazine 24(4):118-120. [32] Candes, E.J. (2008) Compressed Sensing. Twenty-Second Annual Conference on Neural Information Processing Systems, Tutorials. [33] Candes, E.J., Romberg, J.K. & Tao, T. (2006) Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics 59(8):1207-1223. [34] Candes, E.J. & Tao, T. (2006) Near-optimal signal recovery from random projections: Universal encoding strategies? Ieee Transactions on Information Theory 52(12):5406-5425. [35] Donoho, D.L. (2006) Compressed sensing. Ieee Transactions on Information Theory 52(4):12891306. [36] Ringach, D. & Shapley, R. (2004) Reverse Correlation in Neurophysiology. Cognitive Science 28:147-166. [37] Schwartz, O., Pillow, J.W., Rust, N.C. & Simoncelli, E.P. (2006) Spike-triggered neural characterization. Journal of Vision 6(4):484-507. [38] Candes, E.J. & Tao, T. (2005) Decoding by linear programming. Ieee Transactions on Information Theory 51(12):4203-4215. [39] Needell, D. & Vershynin, R. (2009) Uniform Uncertainty Principle and Signal Recovery via Regularized Orthogonal Matching Pursuit. Foundations of Computational Mathematics 9(3):317-334. [40] Tropp, J.A. & Gilbert, A.C. (2007) Signal recovery from random measurements via orthogonal matching pursuit. Ieee Transactions on Information Theory 53(12):4655-4666. [41] Dai, W. & Milenkovic, O. (2009) Subspace Pursuit for Compressive Sensing Signal Reconstruction. Ieee Transactions on Information Theory 55(5):2230-2249. [42] Needell, D. & Tropp, J.A. (2009) CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis 26(3):301-321. [43] Varshney, L.R., Sjostrom, P.J. & Chklovskii, D.B. (2006) Optimal information storage in noisy synapses under resource constraints. Neuron 52(3):409-423. [44] Brunel, N., Hakim, V., Isope, P., Nadal, J.P. & Barbour, B. (2004) Optimal information storage and the distribution of synaptic weights: perceptron versus Purkinje cell. Neuron 43(5):745-757. [45] Napoletani, D. & Sauer, T.D. (2008) Reconstructing the topology of sparsely connected dynamical networks. Physical Review E 77(2):026103. [46] Allen, C. & Stevens, C.F. (1994) An evaluation of causes for unreliability of synaptic transmission. Proceedings of the National Academy of Sciences of the United States of America 91(22):10380-10383. [47] Hessler, N.A., Shirke, A.M. & Malinow, R. (1993) The probability of transmitter release at a mammalian central synapse. Nature 366(6455):569-572. [48] Isope, P. & Barbour, B. (2002) Properties of unitary granule cell-->Purkinje cell synapses in adult rat cerebellar slices. Journal of Neuroscience 22(22):9668-9678. [49] Mason, A., Nicoll, A. & Stratford, K. (1991) Synaptic transmission between individual pyramidal neurons of the rat visual cortex in vitro. Journal of Neuroscience 11(1):72-84. [50] Raastad, M., Storm, J.F. & Andersen, P. (1992) Putative Single Quantum and Single Fibre Excitatory Postsynaptic Currents Show Similar Amplitude Range and Variability in Rat Hippocampal Slices. European Journal of Neuroscience 4(1):113-117. [51] Rosenmund, C., Clements, J.D. & Westbrook, G.L. (1993) Nonuniform probability of glutamate release at a hippocampal synapse. Science 262(5134):754-757. [52] Sayer, R.J., Friedlander, M.J. & Redman, S.J. (1990) The time course and amplitude of EPSPs evoked at synapses between pairs of CA3/CA1 neurons in the hippocampal slice. Journal of Neuroscience 10(3):826-836. [53] Cash, S. & Yuste, R. (1999) Linear summation of excitatory inputs by CA1 pyramidal neurons. Neuron 22(2):383-394. [54] Polsky, A., Mel, B.W. & Schiller, J. (2004) Computational subunits in thin dendrites of pyramidal cells. Nature Neuroscience 7(6):621-627. [55] Paninski, L. (2003) Convergence properties of three spike-triggered analysis techniques. Network: Computation in Neural Systems 14(3):437-464. [56] Okatan, M., Wilson, M.A. & Brown, E.N. (2005) Analyzing functional connectivity using a network likelihood model of ensemble neural spiking activity. Neural Computation 17(9):1927-1961. [57] Timme, M. (2007) Revealing network connectivity from response dynamics. Physical Review Letters 98(22):224101.
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Label Selection on Graphs Andrew Guillory Department of Computer Science University of Washington guillory@cs.washington.edu Jeff Bilmes Department of Electrical Engineering University of Washington bilmes@ee.washington.edu Abstract We investigate methods for selecting sets of labeled vertices for use in predicting the labels of vertices on a graph. We specifically study methods which choose a single batch of labeled vertices (i.e. offline, non sequential methods). In this setting, we find common graph smoothness assumptions directly motivate simple label selection methods with interesting theoretical guarantees. These methods bound prediction error in terms of the smoothness of the true labels with respect to the graph. Some of these bounds give new motivations for previously proposed algorithms, and some suggest new algorithms which we evaluate. We show improved performance over baseline methods on several real world data sets. 1 Introduction In this work we consider learning on a graph. Assume we have an undirected graph of n nodes given by a symmetric weight matrix W. The ith node in the graph has a label yi ∈{0, 1} stored in a vector of labels y ∈{0, 1}n. We want to predict all of y from the labels yL for a labeled subset L ⊂V = [n]. V is the set of all vertices. We use ˆy ∈{0, 1}n to denote our predicted labels. The number of incorrect predictions is ||y −ˆy||2. Graph-based learning is an interesting alternative to traditional feature-based learning. In many problems, graph representations are more natural than feature vector representations. When classifying web pages, for example, edge weights in the graph may incorporate information about hyperlinks. Even when the original data is represented as feature vectors, transforming the data into a graph (for example using a Gaussian kernel to compute weights between points) can be convenient for exploiting properties of a data set. In order to bound prediction error, we assume that the labels are smoothly varying with respect to the underlying graph. The simple smoothness assumption we use is that P i,j Wi,j|yi −yj| is small. Here || denotes absolute value, but the labels are binary so we can equivalently use squared difference. This smoothness assumption has been used by graph-based semi-supervised learning algorithms which compute ˆy using a labeled set L chosen uniformly at random from V [Blum and Chawla, 2001, Hanneke, 2006, Pelckmans et al., 2007, Bengio et al., 2006] and by online graph labeling methods that operate on an adversarially ordered stream of vertices [Pelckmans and Suykens, 2008, Brautbar, 2009, Herbster et al., 2008, 2005, Herbster, 2008] In this work we consider methods that make use of the smoothness assumption and structure of the graph in order to both select L as well as make predictions. Our hope is to achieve higher prediction accuracy as compared to random label selection and other methods for choosing L. We are particularly interested in batch offline methods which select L up front, receive yL and then predict ˆy. The single batch, offline label selection problem is important in many real-world applications because it is often the case that problem constraints make requesting more than one batch of labels very costly. For example, if requesting a label involves a time consuming, expensive experiment (potentially 1 involving human subjects), it may be significantly less costly to run a single batch of experiments in parallel as compared to running experiments in series. We give several methods which, under the assumption P i,j Wi,j|yi −yj| is small, guarantee the prediction error ||y −ˆy||2 will also be small. Some of the bounds provide interesting justifications for previously used methods, and we show improved performance over random label selection and baseline submodular maximization methods on several real world data sets. 2 General Worst Case Bound We first give a simple worst case bound on prediction error in terms of label smoothness using few assumptions about the method used to select labels or make predictions. In fact, the only assumption we make is that the predictions are consistent with the set of labeled points (i.e. ˆyL = yL). The bound motivates an interesting method for selecting labeled points and provides a new motivation for a standard prediction method Blum and Chawla [2001] when used with arbitrarily selected L. The bound also forms the basis of the other bounds we derive which make additional assumptions. Define the graph cut function Γ(A, B) ≜P i∈A,j∈B Wi,j. Let Ψ(L) ≜ min T ⊆(V \L)̸=0 Γ(T, (V \ T)) |T| Note this function is different from normalized cut (also called sparsest cut). In this function, the denominator is simply |T| while for normalized cut the denominator is min(|T|, |V \ T|). This difference is important: computing normalized cut is NP-hard, but we will show Ψ(L) can be computed in polynomial time. Ψ(L) measures how easily we can cut a large portion of the graph away from L. If Ψ(L) is small, then we can separate many nodes from L without cutting very many edges. We show that Ψ(L) where L is the set of labeled vertices measures to what extent prediction error can be high relative to label smoothness. This makes intuitive sense because if Ψ(L) is small than there is a large set of unlabeled nodes which are weakly connected to the remainder of the graph (including L). Theorem 1. For any ˆy consistent with a labeled set L ||y −ˆy||2 ≤ 1 2Ψ(L) X i,j Wi,j(|yi −yj|⊕|ˆyi −ˆyj|) ≤ 1 2Ψ(L)( X i,j Wi,j|yi −yj|+ X i,j Wi,j|ˆyi −ˆyj|) where ⊕is the XOR operator. Proof. Let I be the set of incorrectly classified points. First note that I ∩L = ∅(none of the labeled points are incorrectly classified). |I| = Γ(I, V \ I) |I| Γ(I, V \ I) ≤Γ(I, V \ I) Ψ(L) Note that for all of the edges (i, j) counted in Γ(I, V \ I), ˆyi = ˆyj implies yi ̸= yj and ˆyi ̸= ˆyj implies yi = yj. Then |I| ≤ 1 2Ψ(L) X i,j Wi,j(|yi −yj| ⊕|ˆyi −ˆyj|) The 1 2 term is introduced because the sum double counts edges. This bound is tight when the set of incorrectly classified points I is one of the sets minimizing minT ⊆(V \L)̸=0 Γ(T, (V \ T))/|T|. This bound provides an interesting justification for the algorithm in Blum and Chawla [2001] and related methods when used with arbitrarily selected labeled sets. The term involving the predicted labels, P i,j Wi,j|ˆyi −ˆyj|, is the objective function minimized under the constraint ˆyL = yL by the algorithm of Blum and Chawla [2001]. When this is used to compute ˆy, the bound simplifies. 2 ComputeCut(L) T ′ ←V \ L repeat T ←T ′ λ ←Γ(T,V \T ) |T | T ′ ←argmin A⊆(V \L) Γ(A, V \ A) −λ|A| until Γ(T ′, V \ T ′) −λ|T ′| = 0 return T MaximizeΨ(L, k) L ←∅ repeat T ←ComputeCut(L) i ←random vertex in T L ←L ∪{i} until |L| = k return L Figure 1: Left: Algorithm for computing Ψ(L). Right: Heuristic for maximizing Ψ(L). Lemma 1. If ˆy = argminˆy∈{0,1}n:ˆyL=yL X i,j Wi,j|ˆyi −ˆyj| for a labeled set L then ||y −ˆy||2 ≤ 1 Ψ(L) X i,j Wi,j|yi −yj| Proof. When we choose ˆy in this way P i,j Wi,j|ˆyi −ˆyj| ≤P i,j Wi,j|yi −yj| and the lemma follows from Theorem 1. Label propagation solves a version of this problem in which ˆy is real valued [Bengio et al., 2006]. The bound also motivates a simple label selection method. In particular, we would like to select a labeled L set that maximizes Ψ(L). We first describe how to compute Ψ(L) for a fixed L. Computing Ψ(L) is related to computing min T ⊆(V \L) Γ(T, V \ T) −λ|T| (1) with parameter λ > 0. The following result is paraphrased from Fujishige [2005] (pages 248-249). Theorem 2. λ′ = minT f(T ) g(T ) if and only if ∀λ ≤λ′ min T f(T) −λg(T) = 0 and ∀λ > λ′ min T f(T) −λg(T) < 0 We can compute Equation 1 for all λ via a parametric maxflow/mincut computation (it is known there are no more than n −1 distinct solutions). This gives a polynomial time algorithm for computing Ψ(L). Note this theorem is for unconstrained minimization of T, but restricting T ∩L = ∅ does not change the result: this constraint simply removes elements from the ground set. In practice, this constraint can be enforced by contracting the graph used in the flow computations or by giving certain edges infinite capacity. As an alternative to solving the parametric flow problem, we can find the desired λ value through an iterative method [Cunningham, 1985]. The left of Figure 1 shows this approach. The algorithm takes in a set L and computes argminT ⊆(V \L)̸=0 Γ(T, (V \T))/|T|. The correctness proof is simple. When the algorithm terminates, we know λ ≥λ′ = minT ⊆(V \L)̸=0 Γ(T, (V \ T))/|T| because we set λ to be Γ(T, (V \ T))/|T| for a particular T. By Theorem 2 and the termination condition, we also know λ ≤λ′ and can conclude λ = λ′ and the set T returned achieves this minimum. One can also show the algorithm terminates in at most |V | iterations [Cunningham, 1985]. Having shown how to compute Ψ(L), we now consider methods for maximizing it. Ψ is neither submodular nor supermodular. This seems to rule out straightforward set function optimization. In our experiments, we try a simple heuristic based on the following observation: for any L, if Ψ(L′) > Ψ(L) then it must be the case that L′ intersects one of the cuts minimizing minT ⊆(V \L)̸=∅Γ(T, (V \ 3 T))/|T|. In other words, in order to increase Ψ(L) we must necessarily include a point from the current cut. Our heuristic is then to simply add a random element from this cut to L. The right of Figure 1 shows this method. Several issues remain. First, although we have proposed a reasonable heuristic for maximizing Ψ(L), we do not have methods for maximizing it exactly or with guaranteed approximation. Aside from knowing the function is not submodular or supermodular, we also do not know the hardness of the problem. In the next section, we describe a lower bound on the Ψ function based on a notion of graph covering. This lower bound can be maximized approximately via a simple algorithm and has a well understood hardness of approximation. Second, we have found in experimenting with our heuristic for maximizing Ψ(L) that the function can be prone to imbalanced cuts; the computed cuts sometimes contain all or most of the unselected points V \ L and other times focus on small sets of outliers. We give a third bound on error which attempts to address some of this sensitivity. 3 Graph Covering Algorithm The method we consider in this section uses a notion of graph covering. We say a set L α-covers the graph if ∀i ∈V either i ∈L or P j∈L Wi,j ≥α. In other words, every node in the graph is either in L or connected with total weight at least α to nodes in L (or both). This is a simple real valued extension of dominating sets. A dominating set is a set L ⊆V such that ∀i ∈V either i ∈L or a neighbor of i is in L (or both). This notion of covering is related to the Ψ function discussed in the previous section. In particular, if a set L α-covers a graph than it is necessarily the case that Ψ(L) ≥α. The converse does not hold, however. In other words, α is a lower bound on Ψ(L). Then, α can replace Ψ(L) in the bound in the previous section for a looser upper bound on prediction error. Although the bound is looser, compared to maximizing Ψ(L) we better understand the complexity of computing an α-cover. Corollary 1. For any ˆy consistent with a labeled set L that is an α-cover ||y −ˆy||2 ≤1 2α X i,j Wi,j(|yi −yj| ⊕|ˆyi −ˆyj|) ≤1 2α( X i,j Wi,j|yi −yj| + X i,j Wi,j|ˆyi −ˆyj|) where ⊕is the XOR operator. Similar to Lemma 1, by making additional assumptions concerning the prediction method used we can derive a slightly simpler bound. In particular, for a labeled set L that is an α cover, we assume unlabeled nodes are labeled with the weighted majority vote of neighbors in L. In other words, set ˆyi = yi for i ∈L, and set ˆyi = y′ for i /∈L with y′ such that P j∈L:yj=y′ Wi,j ≥P j∈L:yj̸=y′ Wi,j. With this prediction method we get the following bound. Lemma 2. If L is an α-cover and V \ L is labeled according to majority vote ||y −ˆy||2 ≤1 α X i,j Wi,j|yi −yj|(1 −|ˆyi −ˆyj|) ≤1 α X i,j Wi,j|yi −yj| Proof. The right hand side follows immediately from the middle expression, so we focus on the first inequality. For every incorrectly labeled node, there is a set of nodes Li = {j ∈L : ˆyi = ˆyj} which satisfies yi ̸= yj∀j ∈Li, and P j∈Li Wi,j ≥α/2. We then have for every incorrectly labeled node a unique set of edges with total weight at least α/2 included inside the summation in the middle expression. In computing an α-cover, we want to solve. min L⊆V |L| : F(L) ≥α Where F(L) ≜min i∈V \L X j∈L Wi,j = F ′(L) ≜min i∈V X j∈L W ′ i,j where W ′ i,j = Wi,j for i ̸= j and W ′ i,i = ∞. F ′ is the minimum of a set of modular functions. F ′ is neither supermodular nor submodular. However, we can still compute an approximately minimal 4 α-cover using a trick introduced by Krause et al. [2008]. In particular, Krause et al. [2008] point out that min i∈V X j∈L W ′ i,j ≥α ⇔ X i 1 n min( X j∈L W ′ i,j, α) ≥α Also, min(P j∈L W ′ i,j, α) is submodular, and the sum of submodular functions is submodular. Then, we can replace F ′ with F ′ α(L) = X i 1 n min( X j∈L W ′ i,j, α) and solve min L⊆V |L| : F ′ α(L) ≥α This is a submodular set cover problem. The greedy algorithm has approximation guarantees for this problem for integer valued functions [Krause et al., 2008]. For binary weight graphs the approximation is O(log n). For real valued functions, it’s possible to round the function values to get an approximation guarantee. In practice, we apply the greedy algorithm directly. As previously mentioned, α-covers can be seen as real valued generalizations of dominating sets. In particular, an α-cover is a dominating set for binary weight graphs and α = 1. The hardness of approximation results for finding a minimum size dominating set then carry over to the more general α-cover problem. The next theorem shows that the α-cover problem is NP-hard and in fact the greedy algorithm for computing an α-cover is optimal up to constant factors for α = 1 and binary weight graphs. It is based on the well known connection between finding a minimum dominating set problem and finding a minimum set cover. Theorem 3. Finding the smallest dominating set L in a binary weight graph is NP-complete. Furthermore, if there is some ϵ > 0 such that a polynomial time algorithm approximates the smallest dominating set within (1 −ϵ) ln(n/2) then NP ⊂TIME(nO(log log n)). We have so far discussed computing a small α cover for a fixed α. If we instead have a fixed label budget and want to maximize α, we can do so by performing binary search over α. This is the approach used by Krause et al. [2008] and gives a bi-approximation. 4 Normalized Cut Algorithm In this section we consider an algorithm that clusters the data set and replaces the Ψ function with a normalized cut value. The normalized cut value for a set T ⊂V is Γ(T, V \ T) min(|T|, |V \ T|) In other words, normalized cut is the ratio between the cut value for T and minimum of the size of T and its complement. Computing the minimum normalized cut for a graph is NP-hard. Consider the following method: 1) partition the set of nodes V into clusters S1, S2, ...Sk, 2) for each cluster request sufficient labels to estimate the majority class with probability at least 1 −δ/k, and 3) label all nodes in each cluster with the majority label for that cluster. Here the probability 1−δ/k is with respect to the choice of the labeled nodes used to estimate the majority class for each cluster. Theorem 4. Let S1, S2, ...Sk be a partition of V , and assume we have estimates of the majority class of each Sl each of which are accurate with probability at least 1−δ/k. If ˆy labels every i ∈Sl according to the estimated majority label for Sl then with probability at least 1 −δ ||y −ˆy||2 ≤ X l 1 2φl X i,j∈Sl Wi,j|yi −yj| ≤1 2φ X i,j Wi,j|yi −yj| where φl = min T ⊂Sl Γ(T, Sl \ T) min(|T|, |Sl \ T|) and φ = min l φl 5 Proof. By the union bound, the estimated majority labels for all of the clusters are correct with probability at least 1 −δ. Let I be the set of incorrectly labeled nodes (errors). We consider the intersection of I with each of the clusters. Let Il ≜|I ∩Sl|. I = Sk l=1 Il Note that |Il| ≤|Sl \ Il| since we labeled cluster according to the majority label for the cluster. Then |I| = X l |Il| = X l Γ(Il, Sl \ Il) |Il| Γ(Il, Sl \ Il) = X l Γ(Il, Sl \ Il)min(|Il|, |Sl \ Il|) Γ(Il, Sl \ Il) ≤ X l Γ(Il, Sl \ Il) φl For any i, j, with i ∈Il and j ∈Sl \ Il, we must have yi ̸= yj. Also, for any i, j with yi ̸= yj and i, j ∈Sl, either i ∈Il or j ∈Il. In other words, there is a one-to-one correspondence between 1) edges i, j for which i, j ∈Sl and either i ∈Il or j ∈Il and 2) edges i, j for which i, j ∈Sl and yi ̸= yj. The desired result then follows. Note in practice we only label the unlabeled nodes in each cluster using the majority label estimates. Using the true labels for the labeled nodes only decreases error, so the theorem still holds. In this bound, φ is a measure of the density of the clusters. Computing φl for a particular cluster is NP-hard, but there are approximation algorithms. However, we are not aware of approximation algorithms for computing a partition such that φ is maximized. This is different from the standard normalized cut clustering problem; we do not care if clusters are strongly connected to each other only that each cluster is internally dense. In our experiments, we try several standard clustering algorithms and achieve good real world performance, but it remains an interesting open question to design a clustering algorithm for directly maximizing φ. An approach we have not yet tried is to use the error bound to choose between the results of different clustering algorithms. We now consider the problem of estimating the majority class for a cluster. If we uniformly sample labels from a cluster, standard results give that the probability of incorrectly estimating the majority decreases exponentially with the number of labels if the fraction of nodes in the minority class is bounded away from 1/2 by a constant. We now show that if the labels are sufficiently smooth and the cluster is sufficiently dense then the fraction of nodes in the minority class is small. Theorem 5. The fraction of nodes in the minority class of S is at most P i,j∈S Wi,j|yi −yj| φ|S| where φ = min T ⊂S Γ(T, S \ T) min(|T|, |S \ T|) Proof. Let S−be the set of nodes belonging to the minority class and S+ be the set of nodes belonging to the other class. Let f be the fraction of nodes in the minority class. f = |S−| |S| = |S−| |S| P i,j∈S Wi,j|yi −yj| min(|S+|, |S−|) min(|S+|, |S−|) P i,j∈S Wi,j|yi −yj| = P i,j∈S Wi,j|yi −yj| |S| min(|S+|, |S−|) Γ(S+, S−) ≤ P i,j∈S Wi,j|yi −yj| φ|S| If we have an estimate of the smoothness of the labels in a cluster, we can use this bound and an approximation of φ to determine the number of labels needed to estimate the majority class with high confidence. In our experiments, we simply request a single label per cluster. 6 Spectral k-Cut METIS Ψ Baseline Digit1/10 9.54 (4.42) 50.02 (1.04) 4.93 (4.05) 49.92 (3.18) 20.90 (15.67) Text/10 37.64 (8.64) 50.03 (0.3) 34.76 (6.05) 50.05 (0.06) 45.91 (7.96) BCI/10 50.13 (2.16) 50.16 (0.64) 49.68 (2.63) 50.32 (0.55) 50.12 (1.32) USPS/10 15.22 (6.22) 31.53 (23.65) 8.15 (5.51) 20.07 (2.70) 15.87 (4.82) g241c/10 39.63 (5.67) 50.03 (0.03) 29.18 (7.28) 50.29 (0.07) 47.26 (5.19) g241d/10 22.31 (7.06) 50.02 (0.23) 22.57 (7.26) 50.01 (0.09) 48.46 (3.39) Digit1/100 4.47 (1.35) 50.07 (1.46) 3.24 (0.76) 2.60 (0.83) 2.57 (0.67) Text/100 31.67 (2.41) 50.26 (2.73) 32.57 (1.88) 48.34 (0.67) 26.82 (3.88) BCI/100 47.37 (2.80) 50.14 (0.5) 45.35 (1.91) 48.17 (1.87) 47.48 (2.99) USPS/100 6.23 (1.49) 31.13 (26.31) 9.28 (1.38) 10.17 (0.39) 6.33 (2.46) g241c/100 44.31 (2.09) 50.02 (0.18) 37.47 (2.13) 52.48 (0.37) 42.86 (4.50) g241d/100 41.70 (2.44) 50.03 (0.18) 35.96 (1.99) 50.33 (0.21) 41.56 (4.34) Table 1: Error rate mean (standard deviation) for different data set, label count, method combinations. −3 −2 −1 0 1 2 3 4 5 6 7 −4 −2 0 2 4 6 8 −3 −2 −1 0 1 2 3 4 5 6 7 −4 −2 0 2 4 6 8 Figure 2: Left: Points selected by the Ψ function maximization method. Right: Points selected by the spectral clustering method. 5 Experiments We experimented with a method based on Lemma 1. We use the randomized method for maximizing Ψ and then predict with min-cuts [Blum and Chawla, 2001]. We also tried a method based on Theorem 4. We cluster the data then label each cluster according to a single randomly chosen point. We chose the number of clusters to be equal to the number of labeled points observing that if a cluster is split evenly amongst the two classes then we will have a high error rate regardless of how well we estimate the majority class. We tried three clustering algorithms: a spectral clustering method [Ng et al., 2001], the METIS package for graph partitioning [Karypis and Kumar, 1999], and a k-cut approximation algorithm [Saran and Vazirani, 1995, Gusfield, 1990]. As a baseline we use random label selection and prediction using the label propagation method of Bengio et al. [2006] with ϵ = 10−6 and µ = 10−6 and class mass normalization. We also experimented with a method motivated by the graph covering bound, but for lack of space we omit these results. We used six benchmark data sets [Chapelle et al., 2006]. We use graphs constructed with a Gaussian kernel with standard deviation chosen to be the average distance to the k1th nearest neighbor divided by 3 (a similar heuristic is used by Chapelle et al. [2006]). We then make this graph sparse by removing the edge between node i and j unless i is one of j’s k2 nearest neighbors or j is one of i’s k2 nearest neighbors. We use 10 and 100 labels. We set k1 and k2 for each data set and label count to be the parameters which give the lowest average error rate for label propagation averaging over 100 trials and choosing from the set {5, 10, 50, 100}. We tune the graph construction parameters to give low error for the baseline method to ensure any bias is in favor of the baseline as opposed to the new methods we propose. We then report average error over 1000 trials in the 10 label case and 100 trials in the 100 label case for each combination of data set and algorithm. Table 1 shows these results. We find that the Ψ function method does not perform well. We found on most of the data sets the cuts found by the method included all or almost all of V \L. In this case the points selected are essentially random. However, on the USPS data set and on some synthetic data sets we have tried, we have also observed the opposite behavior where the cuts are very small and seem to focus on small sets of outliers. Figure 2 shows an example of this. The k-cut method 7 also did not perform well. We’ve found this method has similar problems with outliers. We think these outlier sensitive methods are impractical for graphs constructed from real world data. The results for the spectral clustering and METIS clustering methods, however, are quite encouraging. These methods performed well matching or beating the baseline method on the 10 label trials and in some cases significantly improving performance. The METIS method seems particularly robust. On the 100 label trials, performance was not as good. In general, we expect label selection to help more when learning from very few labels. The choice in clustering method seems to be of great practical importance. The clustering methods which work best seem to be methods which minimize normalize cut like objectives. This is not surprising given the presence of the normalized cut term in Theorem 4, but it is an open problem to give a clustering method for directly minimizing the bound. We finally note that the numbers we report for our baseline method are in some cases significantly different than the published numbers [Chapelle et al., 2006]. This seems to be because of a variety of factors including differences in implementation as well as significant differences in experiment set up. We have also experimented with several heuristic modifications to our methods and compared our methods to simple greedy methods. One modification we tried is to use label propagation for prediction in conjunction with our label selection methods. We omit these results for lack of space. 6 Related Work Previous work has also used clustering, covering, and other graph properties to guide label selection on graphs. We are, however, the first to our knowledge to give bounds which relate prediction error to label smoothness for single batch label selection methods. Most previous work on label selection methods for learning on graphs has considered active (i.e. sequential) label selection [Zhu and Lafferty, 2003, Pucci et al., 2007, Zhao et al., 2008, Wang et al., 2007, Afshani et al., 2007]. Afshani et al. [2007] show in this setting O(c log(n/c)) where c = P i,j Wi,j|yi −yj| labels are sufficient and necessary to learn the labeling exactly under some balance assumptions. Without balance assumptions they show O(c log(1/ϵ)+c log(n/c)) labels are sufficient to achieve an ϵ error rate. In some cases, our bounds are better despite considering only non sequential label selection. Consider the case where c grows linearly with n so c/n = a for some constant a > 0. In this case, with the bound of Afshani et al. [2007] the number of labels required to achieve a fixed error rate ϵ also grows linearly with n. In comparison, our graph covering bound needs an α-cover with α = a/ϵ. For some graph topologies, the size of such a cover can grow sublinearly with n (for example if the graph contains large, dense clusters). Afshani et al. [2007] also use a kind of dominating set in their method, and it could be interesting to see if portions of their analysis could be adapted to the offline setting. Zhao et al. [2008] also use a clustering algorithm to select initial labels. Other work has given generalization error bounds in terms of label smoothness [Pelckmans et al., 2007, Hanneke, 2006, Blum et al., 2004] for transductive learning from randomly selected L. These bounds are PAC style which typically show that, roughly, the error rate decreases with O(P i,j Wi,j|yi −yj|/(b|L|)) where b is the minimum 2-cut of the graph. Depending on the graph structure, our bounds can be significantly better. For example, if a binary weight graph contains c cliques of size n/c then, we can find an α cover of size cα log(cα) giving an error rate of O(P i,j Wi,j|yi −yj|/(nα)). This is better if c log(cα) < n/b. A line of work has examined mistake bounds in terms of label smoothness for online learning on graphs [Pelckmans and Suykens, 2008, Brautbar, 2009, Herbster et al., 2008, 2005, Herbster, 2008]. These mistake bounds hold no matter how the sequence of vertices are chosen. Herbster [2008] also considers how cluster structure can improve mistake bounds in this setting and gives a mistake bound similar to our graph covering bound on prediction error. Herbster et al. [2005] discusses using an active learning method for the first several steps of an online algorithm. Our work differs from this previous work by considering prediction error bounds for offline learning as opposed to mistake bounds for online learning. The mistake bound setting is significantly different as the prediction method receives feedback after every prediction. Acknowledgments This material is based upon work supported by the National Science Foundation under grant IIS0535100. 8 References P. Afshani, E. Chiniforooshan, R. Dorrigiv, A. Farzan, M. Mirzazadeh, N. Simjour, and H. Zarrabi-Zadeh. On the complexity of finding an unknown cut via vertex queries. In COCOON, 2007. Y. Bengio, O. Delalleau, and N. Le Roux. Label propagation and quadratic criterion. In O. Chapelle, B. Sch¨olkopf, and A. Zien, editors, Semi-Supervised Learning. MIT Press, 2006. A. Blum and S. Chawla. Learning from labeled and unlabeled data using graph mincuts. In ICML, 2001. A. Blum, J. Lafferty, M. R. Rwebangira, and R. Reddy. Semi-supervised learning using randomized mincuts. In ICML, 2004. M. Brautbar. Online Learning a Labeling of a Graph. Mining and Learning with Graphs, 2009. O. Chapelle, B. Sch¨olkopf, and A. Zien. Semi-supervised learning. MIT press, 2006. W. Cunningham. Optimal attack and reinforcement of a network. Journal of the ACM, 1985. S. Fujishige. Submodular Functions and Optimization. Elsevier Science, 2005. D. Gusfield. Very simple methods for all pairs network flow analysis. SIAM Journal on Computing, 1990. S. Hanneke. An analysis of graph cut size for transductive learning. In ICML, 2006. M. Herbster. Exploiting Cluster-Structure to Predict the Labeling of a Graph. In ALT, 2008. M. Herbster, M. Pontil, and L. Wainer. Online learning over graphs. In ICML, 2005. M. Herbster, G. Lever, and M. Pontil. Online Prediction on Large Diameter Graphs. In NIPS, 2008. G. Karypis and V. Kumar. A fast and highly quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 1999. A. Krause, H. B. McMahan, C. Guestrin, and A. Gupta. Robust submodular observation selection. JMLR, 2008. A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: Analysis and an algorithm. In NIPS, 2001. K. Pelckmans and J. Suykens. An online algorithm for learning a labeling of a graph. In Mining and Learning with Graphs, 2008. K. Pelckmans, J. Shawe-Taylor, J. Suykens, and B. De Moor. Margin based transductive graph cuts using linear programming. 2007. A. Pucci, M. Gori, and M. Maggini. Semi-supervised active learning in graphical domains. In Mining and Learning With Graphs, 2007. H. Saran and V. V. Vazirani. Finding k cuts within twice the optimal. SIAM Journal on Computing, 1995. M. Wang, X. Hua, Y. Song, J. Tang, and L. Dai. Multi-Concept Multi-Modality Active Learning for Interactive Video Annotation. In International Conference on Semantic Computing, 2007. W. Zhao, J. Long, E. Zhu, and Y. Liu. A scalable algorithm for graph-based active learning. In Frontiers in Algorithmics, 2008. X. Zhu and J. Lafferty. Combining active learning and semi-supervised learning using gaussian fields and harmonic functions. In ICML, 2003. 9
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A Fast, Consistent Kernel Two-Sample Test Arthur Gretton Carnegie Mellon University MPI for Biological Cybernetics arthur.gretton@gmail.com Kenji Fukumizu Inst. of Statistical Mathematics Tokyo Japan fukumizu@ism.ac.jp Zaid Harchaoui Carnegie Mellon University Pittsburgh, PA, USA zaid.harchaoui@gmail.com Bharath K. Sriperumbudur Dept. of ECE, UCSD La Jolla, CA 92037 bharathsv@ucsd.edu Abstract A kernel embedding of probability distributions into reproducing kernel Hilbert spaces (RKHS) has recently been proposed, which allows the comparison of two probability measures P and Q based on the distance between their respective embeddings: for a sufficiently rich RKHS, this distance is zero if and only if P and Q coincide. In using this distance as a statistic for a test of whether two samples are from different distributions, a major difficulty arises in computing the significance threshold, since the empirical statistic has as its null distribution (where P = Q) an infinite weighted sum of χ2 random variables. Prior finite sample approximations to the null distribution include using bootstrap resampling, which yields a consistent estimate but is computationally costly; and fitting a parametric model with the low order moments of the test statistic, which can work well in practice but has no consistency or accuracy guarantees. The main result of the present work is a novel estimate of the null distribution, computed from the eigenspectrum of the Gram matrix on the aggregate sample from P and Q, and having lower computational cost than the bootstrap. A proof of consistency of this estimate is provided. The performance of the null distribution estimate is compared with the bootstrap and parametric approaches on an artificial example, high dimensional multivariate data, and text. 1 Introduction Learning algorithms based on kernel methods have enjoyed considerable success in a wide range of supervised learning tasks, such as regression and classification [25]. One reason for the popularity of these approaches is that they solve difficult non-parametric problems by representing the data points in high dimensional spaces of features, specifically reproducing kernel Hilbert spaces (RKHSs), in which linear algorithms can be brought to bear. While classical kernel methods have addressed the mapping of individual points to feature space, more recent developments [14, 29, 28] have focused on the embedding of probability distributions in RKHSs. When the embedding is injective, the RKHS is said to be characteristic [11, 29, 12], and the distance between feature mappings constitutes a metric on distributions. This distance is known as the maximum mean discrepancy (MMD). One well-defined application of the MMD is in testing whether two samples are drawn from two different distributions (i.e., a two-sample or homogeneity test). For instance, we might wish to find whether DNA microarrays obtained on the same tissue type by different labs are distributed identically, or whether differences in lab procedure are such that the data have dissimilar distributions (and cannot be aggregated) [8]. Other applications include schema matching in databases, where tests of distribution similarity can be used to determine which fields correspond [14], and speaker 1 verification, where MMD can be used to identify whether a speech sample corresponds to a person for whom previously recorded speech is available [18]. A major challenge when using the MMD in two-sample testing is in obtaining a significance threshold, which the MMD should exceed with small probability when the null hypothesis (that the samples share the same generating distribution) is satisfied. Following [14, Section 4], we define this threshold as an upper quantile of the asymptotic distribution of the MMD under the null hypothesis. Unfortunately this null distribution takes the form of an infinite weighted sum of χ2 random variables. Thus, obtaining a consistent finite sample estimate of this threshold — that is, an estimate that converges to the true threshold in the infinite sample limit — is a significant challenge. Three approaches have previously been applied: distribution-free large deviation bounds [14, Section 3], which are generally too loose for practical settings; fitting to the Pearson family of densities [14], a simple heuristic that performs well in practice, but has no guarantees of accuracy or consistency; and a bootstrap approach, which is guaranteed to be consistent, but has a high computational cost. The main contribution of the present study is a consistent finite sample estimate of the null distribution (not based on bootstrap), and a proof that this estimate converges to the true null distribution in the infinite sample limit. Briefly, the infinite sequence of weights that defines the null distribution is identical to the sequence of normalized eigenvalues obtained in kernel PCA [26, 27, 7]. Thus, we show that the null distribution defined using finite sample estimates of these eigenvalues converges to the population distribution, using only convergence results on certain statistics of the eigenvalues. In experiments, our new estimate of the test threshold has a smaller computational cost than that of resampling-based approaches such as the bootstrap, while providing performance as good as the alternatives for larger sample sizes. We begin our presentation in Section 2 by describing how probability distributions may be embedded in an RKHS. We also review the maximum mean discrepancy as our chosen distance measure on these embeddings, and recall the asymptotic behaviour of its finite sample estimate. In Section 3, we present both moment-based approximations to the null distribution of the MMD (which have no consistency guarantees); and our novel, consistent estimate of the null distribution, based on the spectrum of the kernel matrix over the aggregate sample. Our experiments in Section 4 compare the different approaches on an artificial dataset, and on high-dimensional microarray and neuroscience data. We also demonstrate the generality of a kernel-based approach by testing whether two samples of text are on the same topic, or on different topics. 2 Background In testing whether two samples are generated from the same distribution, we require both a measure of distance between probabilities, and a notion of whether this distance is statistically significant. For the former, we define an embedding of probability distributions in a reproducing kernel Hilbert space (RKHS), such that the distance between these embeddings is our test statistic. For the latter, we give an expression for the asymptotic distribution of this distance measure, from which a significance threshold may be obtained. Let F be an RKHS on the separable metric space X, with a continuous feature mapping φ(x) ∈F for each x ∈X. The inner product between feature mappings is given by the positive definite kernel function k(x, x′) := ⟨φ(x), φ(x′)⟩F. We assume in the following that the kernel k is bounded. Let P be the set of Borel probability measures on X. Following [4, 10, 14], we define the mapping to F of P ∈P as the expectation of φ(x) with respect to P, or µP : P → F P 7→ Z X φ(x)dP. The maximum mean discrepancy (MMD) [14, Lemma 7] is defined as the distance between two such mappings, MMD(P, Q) := ∥µP −µQ∥F = (Ex,x′(k(x, x′)) + Ey,y′k(y, y′) −2Ex,yk(x, y))1/2 , where x and x′ are independent random variables drawn according to P, y and y′ are independent and drawn according to Q, and x is independent of y. This quantity is a pseudo-metric on distributions: that is, it satisfies all the qualities of a metric besides MMD(P, Q) = 0 iff P = Q. For MMD 2 to be a metric, we require that the kernel be characteristic [11, 29, 12].1 This criterion is satisfied for many common kernels, such as the Gaussian kernel (both on compact domains and on Rd) and the B2l+1 spline kernel on Rd. We now consider two possible empirical estimates of the MMD, based on i.i.d. samples (x1, . . . , xm) from P and (y1, . . . , ym) from Q (we assume an equal number of samples for simplicity). An unbiased estimate of MMD is the one-sample U-statistic MMD2 u := 1 m(m −1) m X i̸=j h(zi, zj), (1) where zi := (xi, yi) and h(zi, zj) := k(xi, xj)+k(yi, yj)−k(xi, yj)−k(xj, yi). We also define the biased estimate MMD2 b by replacing the U-statistic in (1) with a V-statistic (the sum then includes terms i = j). Our goal is to determine whether P and Q differ, based on m samples from each. To this end, we require a measure of whether MMD2 u differs significantly from zero; or, if the biased statistic MMD2 b is used, whether this value is significantly greater than its expectation when P = Q. In other words we conduct a hypothesis test with null hypothesis H0 defined as P = Q, and alternative hypothesis H1 as P ̸= Q. We must therefore specify a threshold that the empirical MMD will exceed with small probability, when P = Q. For an asymptotic false alarm probability (Type I error) of α, an appropriate threshold is the 1 −α quantile of the asymptotic distribution of the empirical MMD assuming P = Q. According to [14, Theorem 8], this distribution takes the form mMMD2 u → D ∞ X l=1 λl(z2 l −2), (2) where → D denotes convergence in distribution, zl ∼N(0, 2) i.i.d., λi are the solutions to the eigenvalue equation Z X ˜k(xi, xj)ψl(xi)dP := λlψl(xj), (3) and ˜k(xi, xj) := k(xi, xj) −Exk(xi, x) −Exk(x, xi) + Ex,x′k(x, x′). Consistency in power of the resulting hypothesis test (that is, the convergence of its Type II error to zero for increasing m) is shown in [14]. The eigenvalue problem (3) has been studied extensively in the context of kernel PCA [26, 27, 7]: this connection will be used in obtaining a finite sample estimate of the null distribution in (2), and we summarize certain important results. Following [3, 10], we define the covariance operator C : F →F as ⟨f, Cf⟩F := var(f(x)) = Exf 2(x) −[Exf(x)]2 . (4) The eigenvalues λl of C are the solutions to the eigenvalue problem in (3) [19, Proposition 2]. Following e.g. [27, p.2511], empirical estimates of these eigenvalues are ˆλl = 1 mνl (5) where νl are the eigenvalues of the centered Gram matrix eK := HKH, Ki,j := k(xi, xj), and H = I −1 m11⊤is a centering matrix. Finally, by subtracting mMMD2 u from mMMD2 b, we observe that these differ by a quantity with expectation tr(C) = P∞ l=1 λl, and thus mMMD2 b → D ∞ X l=1 λlz2 l . 1Other interpretations of the MMD are also possible, for particular kernel choices. The most closely related is the L2 distance between probability density estimates [1], although this requires the kernel bandwidth to decrease with increasing sample size. See [1, 14] for more detail. Yet another interpretation is given in [32]. 3 3 Theory In the present section, we describe three approaches for approximating the null distribution of MMD. We first present the Pearson curve and Gamma-based approximations, which consist of parametrized families of distributions that we fit by matching the low order moments of the empirical MMD. Such approximations can be accurate in practice, although they remain heuristics with no consistency guarantees. Second, we describe a null distribution estimate based on substituting the empirical estimates (5) of the eigenvalues into (2). We prove that this estimate converges to its population counterpart in the large sample limit. 3.1 Moment-based null distribution estimates The Pearson curves and the Gamma approximation are both based on the low order moments of the empirical MMD. The second and third moments for MMD are obtained in [14]: E MMD2 u 2 = 2 m(m −1)Ez,z′ h2(z, z′) and (6) E MMD2 u 3 = 8(m −2) m2(m −1)2 Ez,z′ [h(z, z′)Ez′′ (h(z, z′′)h(z′, z′′))] + O(m−4). (7) Pearson curves take as arguments the variance, skewness and kurtosis As in [14], we replace the kurtosis with a lower bound due to [31], kurt MMD2 u ≥ skew MMD2 u 2 + 1. An alternative, more computationally efficient approach is to use a two-parameter Gamma approximation [20, p. 343, p. 359], mMMDb(Z) ∼xα−1e−x/β βαΓ(α) where α = (E(MMDb(Z)))2 var(MMDb(Z)) , β = mvar(MMDb(Z)) E(MMDb(Z)) , (8) and we use the biased statistic MMD2 b. Although the Gamma approximation is necessarily less accurate than the Pearson approach, it has a substantially lower computational cost (O(m2) for the Gamma approximation, as opposed to O(m3) for Pearson). Moreover, we will observe in our experiments that it performs remarkably well, at a substantial cost saving over the Pearson curves. 3.2 Null distribution estimates using Gram matrix spectrum In [14, Theorem 8], it was established that for large sample sizes, the null distribution of MMD approaches an infinite weighted sum of independent χ2 1 random variables, the weights being the population eigenvalues of the covariance operator C. Hence, an efficient and theoretically grounded way to calibrate the test is to compute the quantiles by replacing the population eigenvalues of C with their empirical counterparts, as computed from the Gram matrix (see also [18], where a similar strategy is proposed for the KFDA test with fixed regularization). The following result shows that this empirical estimate of the null distribution converges in distribution to its population counterpart. In other words, a test using the MMD statistic, with the threshold computed from quantiles of the null distribution estimate, is asymptotically consistent in level. Theorem 1 Let z1, . . . , zl, . . . be an infinite sequence of i.i.d. random variables, with z1 ∼N(0, 2). Assume P∞ l=1 λ1/2 l < ∞. Then, as m →∞ ∞ X l=1 ˆλl(z2 l −2) → D ∞ X l=1 λl(z2 l −2) . Furthermore, as m →∞ sup t P mMMD2 u > t −P ∞ X l=1 ˆλl(z2 l −2) > t ! →0 . 4 Proof (sketch) We begin with a proof of conditions under which the sum P∞ l=1 λl(z2 l −2) is finite w.p. 1. According to [16, Exercise 30, p. 358], we may use Kolmogorov’s inequality to determine that this sum converges a.s. if ∞ X l=1 Ez[λ2 l (z2 l −2)2] < ∞, from which it follows that the covariance operator must be Hilbert-Schmidt: this is guaranteed by the assumption P∞ l=1 λ1/2 l < ∞(see also [7]). We now proceed to the convergence result. Let C and bC be the covariance operator and its empirical estimator. Let λl and bλl (l = 1, 2, . . .) be the eigenvalues of C and bC, respectively, in descending order. We want to prove ∞ X p=1 (bλl −λl)Z2 l → 0 (9) in probability as n →∞, where Zp ∼N(0, 2) are i.i.d. random variables. The constant −2 in Z2 p −2 can be neglected as Tr[ bC] →Tr[C], where the proof is given in the online supplement. Thus X l (bλl −λl)Z2 l ≤ X l bλ1/2 l bλ1/2 l −λ1/2 l )Z2 l + X l bλ1/2 l −λ1/2 l )λ1/2 l Z2 l ≤ nX l bλlZ4 l o1/2nX l bλ1/2 l −λ1/2 l 2o1/2 + nX l λlZ4 l o1/2nX l bλ1/2 l −λ1/2 l 2o1/2 (Cauchy-Schwarz). (10) We now establish P l λlZ4 l and P l bλlZ4 l are of Op(1). The former follows from Chebyshev’s inequality. To prove the latter, we use that since ˆλi and Zi are independent, E X i ˆλiZ4 i = X i E[ˆλi]E[Z4 i ] = κE[tr( ˆC)], (11) where κ = E[Z4]. Since E[tr( ˆC)] is bounded when the kernel has bounded expectation, we again have the desired result by Chebyshev’s inequality. The proof is complete if we show X l bλ1/2 l −λ1/2 l )2 = op(1). (12) From bλ1/2 l −λ1/2 l 2 ≤ bλ1/2 l −λ1/2 l (bλ1/2 l + λ1/2 l ) = bλl −λl , (13) we have X l bλ1/2 l −λ1/2 l 2 ≤ X l |bλl −λl|. It is known as an extension of the Hoffmann-Wielandt inequality that X l bλl −λl ≤∥bC −C∥1, where ∥· ∥1 is the trace norm (see [23], also shown in [5, p. 490]). Using [18, Prop. 12], which gives ∥bC −C∥1 →0 in probability, the proof of the first statement is completed. The proof of the second statement follows immediately from the Polya theorem [21], as in [18]. 3.3 Discussion We now have several ways to calibrate the MMD test statistic, ranked in order of increasing computational cost: 1) the Gamma approximation, 2) the “empirical null distribution”: that is, the null distribution estimate using the empirical Gram matrix spectrum, and 3) the Pearson curves, and 5 the resampling procedures (subsampling or bootstrap with replacement). We include the final two approaches in the same cost category since even though the Pearson approach scales worse with m than the bootstrap (O(m3) vs O(m2)), the bootstrap has a higher cost for sample sizes less than about 103 due the requirement to repeatedly re-compute the test statistic. We also note that our result of large-sample consistency in level holds under a restrictive condition on the decay of the spectrum of the covariance operator, whereas the Gamma approximation calculations are straightforward and remain possible for any spectrum decay behaviour. The Gamma approximation remains a heuristic, however, and we give an example of a distribution and kernel for which it performs less accurately than the spectrum-based estimate in the upper tail, which is of most interest for testing purposes. 4 Experiments In this section, we compare the four approaches to obtaining the null distribution, both in terms of the approximation error computed with respect to simulations from the true null, and when used in homogeneity testing. Our approaches are denoted Gamma (the two-parameter Gamma approximation), Pears (the Pearson curves based on the first three moments, using a lower bound for the kurtosis), Spec (our new approximation to the null distribution, using the Gram matrix eigenspectrum), and Boot (the bootstrap approach). Artificial data: We first provide an example of a distribution P for which the heuristics Gamma and Pears have difficulty in approximating the null distribution, whereas Spec converges. We chose P to be a mixture of normals P = 0.5 ∗N(−1, 0.44) + 0.5 ∗N(+1, 0.44), and k as a Gaussian kernel with bandwidth ranging over σ = 2−4, 2−3, 2−2, 2−1, 20, 21, 22. The sample sizes were set to m = 5000, the total sample size hence being 10, 000, and the results were averaged over 50, 000 replications. The eigenvalues of the Gram matrix were estimated in this experiment using [13], which is slower but more accurate than standard Matlab routines. The true quantiles of the MMD null distribution, referred to as the oracle quantiles, were estimated by Monte Carlo simulations with 50, 000 runs. We report the empirical performance of Spec compared to the oracle in terms of ∆q = maxtr:q<r<1 |P(mMMD2 u > tr) −bPm(mMMD2 u > tr)|, where tq is such that P(mMMD2 u > tq) = q for q = 0.6, 0.7, 0.8, 0.9, and bPm is the Spec null distribution estimate obtained with m samples from each of P and Q. We also use this performance measure for the Gamma and Pears approximations. This focuses the performance comparison on the quantiles corresponding to the upper tail of the null distribution, while still addressing uniform accuracy over a range of thresholds so as to ensure reliable p-values. The results are shown in Figure 1, and demonstrate that for this combination of distribution and kernel, Spec performs almost uniformly better than both Gamma and Pears. We emphasize that the performance advantage of Spec is greatest when we restrict ourselves to higher quantiles, which are of most interest in testing. −4 −2 0 2 0.02 0.03 0.04 0.05 0.06 0.07 0.08 log2(σ) ∆0.6 ∆0.6 vs σ Gam Spec Pears −4 −2 0 2 0 0.02 0.04 0.06 0.08 log2(σ) ∆0.7 ∆0.7 vs σ Gam Spec Pears −4 −2 0 2 0 0.01 0.02 0.03 0.04 0.05 0.06 log2(σ) ∆0.8 ∆0.8 vs σ Gam Spec Pears −4 −2 0 2 0.01 0.02 0.03 0.04 0.05 log2(σ) ∆0.9 ∆0.9 vs σ Gam Spec Pears Figure 1: Evolution of ∆q for resp. the Gamma (Gam), Spectrum (Spec), and Pearson (Pears) approximations to the null distribution, as the Gaussian kernel bandwidth parameter varies. From left to right, plots of ∆q versus σ = 2−4, 2−3, . . . , 22 for q = 0.6, 0.7, 0.8, 0.9. Benchmark data: We next demonstrate the performance of the MMD tests on a number of multivariate datasets, taken from [14, Table 1]. We compared microarray data from normal and tumor tissues (Health status), microarray data from different subtypes of cancer (Subtype), and local field potential (LFP) electrode recordings from the Macaque primary visual cortex (V1) with and without spike events (Neural Data I and II, described in [24]). In all cases, we were provided with two samples having different statistical properties, where the detection of these differences was made difficult by the high data dimensionality (for the microarray data, density estimation is impossi6 ble given the small sample size and high data dimensionality, and a successful test cannot rely on accurate density estimates as an intermediate step). In computing the null distributions for both the Spec and Pears cases, we drew 500 samples from the associated null distribution estimates, and computed the test thresholds using the resulting empirical quantiles. For the Spec case, we computed the eigenspectrum on the gram matrix of the aggregate data from P and Q, retaining in all circumstances the maximum number 2m −1 of nonzero eigenvalues of the empirical Gram matrix. This is a conservative approach, given that the Gram matrix spectrum may decay rapidly [2, Appendix C], in which case it might be possible to safely discard the smallest eigenvalues. For the bootstrap approach Boot, we aggregated points from the two samples, then assigned these randomly without replacement to P and Q. In our experiments, we performed 500 such iterations, and used the resulting histogram of MMD values as our null distribution. We used a Gaussian kernel in all cases, with the bandwidth set to the median distance between points in the aggregation of samples from P and Q. We applied our tests to the benchmark data as follows: Given datasets A and B, we either drew one sample with replacement from A and the other from B (in which case a Type II error was made when the null hypothesis H0 was accepted); or we drew both samples with replacement from a single pool consisting of A and B combined (in which case a Type I error was made when H0 was rejected: this should happen a fraction 1 −α of the time). This procedure was repeated 1000 times to obtain average performance figures. We summarize our results in Table 1. Note that an extensive benchmark of the MMD Boot and Pears tests against other nonparametric approaches to two-sample testing is provided in [14]: these include the the Friedman-Rafsky generalisation of the Kolmogorov-Smirnov and Wald-Wolfowitz tests [9], the Biau-Gy¨orfitest [6], and the Hall-Tajvidi test [17]. See [14] for details. We observe that the kernel tests perform extremely well on these data: the Type I error is in the great majority of cases close to its design value of 1 −α, and the Type II error is very low (and often zero). The Spec test is occasionally slightly conservative, and has a lower Type I error than required: this is most pronounced in the Health Status dataset, for which the sample size m is low. The computational cost shows the expected trend, with Gamma being least costly, followed by Spec, Pears, and finally Boot (this trend is only visible for the larger m = 500 datasets). Note that for yet larger sample sizes, however, we expect the cost of Pears to exceed that of the remaining methods, due to its O(m3) cost requirement (vs O(m2) for the other approaches). Dataset Attribute Gamma Pears Spec Boot Neural Data I Type I/Type II 0.95 / 0.00 0.96 / 0.00 0.96 / 0.00 0.96 / 0.00 Time (sec) 0.06 3.92 2.79 5.79 Neural Data II Type I/Type II 0.96 / 0.00 0.96 / 0.00 0.97 / 0.00 0.96 / 0.00 Time (sec) 0.08 3.97 2.91 8.08 Health status Type I/Type II 0.96 / 0.00 0.96 / 0.00 0.98 / 0.00 0.95 / 0.00 Time (sec) 0.01 0.01 0.01 0.03 Subtype Type I/Type II 0.95 / 0.02 0.95 / 0.01 0.96 / 0.01 0.94 / 0.01 Time (sec) 0.05 0.05 0.05 0.07 Table 1: Benchmarks for the kernel two-sample tests on high dimensional multivariate data. Type I and Type II errors are provided, as are average run times. Sample size (dimension): Neural I 500 (63) ; Neural II 500 (100); Health Status 25 (12,600); Subtype 25 (2,118). Finally, we demonstrate the performance of the test on structured (text) data. Our data are taken from the Canadian Hansard corpus (http : //www.isi.edu/natural−language/download/hansard/). As in the earlier work on dependence testing presented in [15], debate transcripts on the three topics of agriculture, fisheries, and immigration were used. Transcripts were in English and French, however we confine ourselves to reporting results on the English data (the results on the French data were similar). Our goal was to distinguish samples on different topics, for instance P being drawn from transcripts on agriculture and Q from transcripts on immigration (in the null case, both samples were from the same topic). The data were processed following the same procedures as in [15]. We investigated two different kernels on text: the k-substring kernel of [22, 30] with k = 10, and a bag-of-words kernel. In both cases, we computed kernels between five-line extracts, ignoring lines shorter than five words long. Results are presented in Figure 2, and represent an average over all three combinations of 7 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Test performance,bow Sample size m Type II error Gamma Pears Spec Boot 10 20 30 40 50 0 0.2 0.4 0.6 0.8 1 Test performance,spec Sample size m Type II error 0 5 10 15 20 0 0.02 0.04 0.06 0.08 Eig. index Eig. amplitude Gram matrix spectrum, bow Figure 2: Canadian Hansard data. Left: Average Type II error over all of agriculture-fisheries, agricultureimmigration, and fisheries-immigration, for the bag-of-words kernel. Center: Average Type II error for the k-substring kernel. Right: Eigenspectrum of a centered Gram matrix obtained by drawing m = 10 points from each of P and Q, where P ̸= Q, for the bag-of-words kernel. different topic pairs: agriculture-fisheries, agriculture-immigration, and fisheries-immigration. For each topic pairing, results are averaged over 300 repetitions. We observe that in general, the MMD is very effective at distinguishing distributions of text fragments on different topics: for sample sizes above 30, all the test procedures are able to detect differences in distribution with zero Type II error, for both kernels. When the k-substring kernel is used, the Boot, Gamma, and Pears approximations can distinguish the distributions for sample sizes as low as 10: this indicates that a more sophisticated encoding of the text than provided by bag-of-words results in tests of greater sensitivity (consistent with the independence testing observations of [15]). We now investigate the fact that for sample sizes below m = 30 on the Hansard data, the Spec test has a much higher Type II error the alternatives. The k-substring and bag-of-words kernels are diagonally dominant: thus for small sample sizes, the empirical estimate of the kernel spectrum is effectively truncated at a point where the eigenvalues remain large, introducing a bias (Figure 2). This effect vanishes on the Hansard benchmark once the number of samples reaches 25-30. By contrast, for the Neural data using a Gaussian kernel, this small sample bias is not observed, and the Spec test has equivalent Type II performance to the other three tests (see Figure 1 in the online supplement). In this case, for sample sizes of interest (i.e., where there are sufficient samples to obtain a Type II error of less than 50%), the bias in the Spec test due to spectral truncation is negligible. We emphasize that the speed advantage of the Spec test becomes important only for larger sample sizes (and the consistency guarantee is only meaningful in this regime). 5 Conclusion We have presented a novel method for estimating the null distribution of the RKHS distance between probability distribution embeddings, for use in a nonparametric test of homogeneity. Unlike previous parametric heuristics based on moment matching, our new distribution estimate is consistent; moreover, it is computationally less costly than the bootstrap, which is the only alternative consistent approach. We have demonstrated in experiments that our method performs well on high dimensional multivariate data and text, as well as for distributions where the parametric heuristics show inaccuracies. We anticipate that our approach may also be generalized to kernel independence tests [15], and to homogeneity tests based on the kernel Fisher discriminant [18]. Acknowledgments: The ordering of the second through fourth authors is alphabetical. We thank Choon-Hui Teo for generating the Gram matrices for the text data, Malte Rasch for his assistance in the experimental evaluation, and Karsten Borgwardt for his assistance with the microarray data. A. G. was supported by grants DARPA IPTO FA8750-09-1-0141, ONR MURI N000140710747, and ARO MURI W911NF0810242. Z. H. was supported by grants from the Technical Support Working Group through funding from the Investigative Support and Forensics subgroup and NIMH 51435, and from Agence Nationale de la Recherche under contract ANR-06-BLAN-0078 KERNSIG. B. K. S. was supported by the MPI for Biological Cybernetics, NSF (grant DMS-MSPA 0625409), the Fair Isaac Corporation and the University of California MICRO program. References [1] N. Anderson, P. Hall, and D. Titterington. Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates. Journal of 8 Multivariate Analysis, 50:41–54, 1994. [2] F. R. Bach and M. I. Jordan. Kernel independent component analysis. J. Mach. Learn. Res., 3:1–48, 2002. [3] C. Baker. Joint measures and cross-covariance operators. Transactions of the American Mathematical Society, 186:273–289, 1973. [4] A. Berlinet and C. Thomas-Agnan. Reproducing Kernel Hilbert Spaces in Probability and Statistics. Springer-Verlag, Berlin, 2003. [5] Rajendra Bhatia and Ludwig Elsner. The Hoffman-Wielandt inequality in infinite dimensions. Proceedings of Indian Academy of Science (Mathematical Sciences), 104(3):483–494, 1994. [6] G. Biau and L. Gyorfi. On the asymptotic properties of a nonparametric l1-test statistic of homogeneity. IEEE Transactions on Information Theory, 51(11):3965–3973, 2005. [7] G. Blanchard, O. Bousquet, and L. Zwald. Statistical properties of kernel principal component analysis. Machine Learning, 66:259–294, 2007. [8] K. M. Borgwardt, A. Gretton, M. J. Rasch, H.-P. Kriegel, B. Sch¨olkopf, and A. J. Smola. Integrating structured biological data by kernel maximum mean discrepancy. Bioinformatics (ISMB), 22(14):e49– e57, 2006. [9] J. Friedman and L. Rafsky. Multivariate generalizations of the Wald-Wolfowitz and Smirnov two-sample tests. The Annals of Statistics, 7(4):697–717, 1979. [10] K. Fukumizu, F. R. Bach, and M. I. Jordan. Dimensionality reduction for supervised learning with reproducing kernel Hilbert spaces. J. Mach. Learn. Res., 5:73–99, 2004. [11] K. Fukumizu, A. Gretton, X. Sun, and B. Sch¨olkopf. Kernel measures of conditional dependence. In NIPS 20, pages 489–496, 2008. [12] K. Fukumizu, B. Sriperumbudur, A. Gretton, and B. Sch¨olkopf. Characteristic kernels on groups and semigroups. In NIPS 21, pages 473–480, 2009. [13] G. Golub and Q. Ye. An inverse free preconditioned krylov subspace method for symmetric generalized eigenvalue problems. SIAM Journal on Scientific Computing, 24:312–334, 2002. [14] A. Gretton, K. Borgwardt, M. Rasch, B. Sch¨olkopf, and A. Smola. A kernel method for the two-sampleproblem. In NIPS 19, pages 513–520, 2007. [15] A. Gretton, K. Fukumizu, C.-H. Teo, L. Song, B. Sch¨olkopf, and A. Smola. A kernel statistical test of independence. In NIPS 20, pages 585–592, 2008. [16] G. R. Grimmet and D. R. Stirzaker. Probability and Random Processes. Oxford University Press, Oxford, third edition, 2001. [17] P. Hall and N. Tajvidi. Permutation tests for equality of distributions in high-dimensional settings. Biometrika, 89(2):359–374, 2002. [18] Z. Harchaoui, F. Bach, and E. Moulines. Testing for homogeneity with kernel fisher discriminant analysis. In NIPS 20, pages 609–616. 2008. (long version: arXiv:0804.1026v1). [19] M. Hein and O. Bousquet. Kernels, associated structures, and generalizations. Technical Report 127, Max Planck Institute for Biological Cybernetics, 2004. [20] N. L. Johnson, S. Kotz, and N. Balakrishnan. Continuous Univariate Distributions. Volume 1 (Second Edition). John Wiley and Sons, 1994. [21] E. Lehmann and J. Romano. Testing Statistical Hypothesis (3rd ed.). Wiley, New York, 2005. [22] C. Leslie, E. Eskin, and W. S. Noble. The spectrum kernel: A string kernel for SVM protein classification. In Proceedings of the Pacific Symposium on Biocomputing, pages 564–575, 2002. [23] A. S. Markus. The eigen- and singular values of the sum and product of linear operators. Russian Mathematical Surveys, 19(4):93–123, 1964. [24] M. Rasch, A. Gretton, Y. Murayama, W. Maass, and N. K. Logothetis. Predicting spiking activity from local field potentials. Journal of Neurophysiology, 99:1461–1476, 2008. [25] B. Sch¨olkopf and A. Smola. Learning with Kernels. MIT Press, Cambridge, MA, 2002. [26] 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. [27] J. Shawe-Taylor, C. Williams, N. Cristianini, and J. Kandola. On the eigenspectrum of the Gram matrix and the generalisation error of kernel PCA. IEEE Trans. Inf. Theory, 51(7):2510–2522, 2005. [28] A. J. Smola, A. Gretton, L. Song, and B. Sch¨olkopf. A Hilbert space embedding for distributions. In ALT 18, pages 13–31, 2007. [29] B. Sriperumbudur, A. Gretton, K. Fukumizu, G. Lanckriet, and B. Sch¨olkopf. Injective hilbert space embeddings of probability measures. In COLT 21, pages 111–122, 2008. [30] C. H. Teo and S. V. N. Vishwanathan. Fast and space efficient string kernels using suffix arrays. In ICML, pages 929–936, 2006. [31] J. E. Wilkins. A note on skewness and kurtosis. Ann. Math. Stat., 15(3):333–335, 1944. [32] G. Zech and B. Aslan. A multivariate two-sample test based on the concept of minimum energy. In PHYSTAT, pages 97–100, 2003. 9
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Robust Nonparametric Regression with Metric-Space valued Output Matthias Hein Department of Computer Science, Saarland University Campus E1 1, 66123 Saarbr¨ucken, Germany hein@cs.uni-sb.de Abstract Motivated by recent developments in manifold-valued regression we propose a family of nonparametric kernel-smoothing estimators with metric-space valued output including several robust versions. Depending on the choice of the output space and the metric the estimator reduces to partially well-known procedures for multi-class classification, multivariate regression in Euclidean space, regression with manifold-valued output and even some cases of structured output learning. In this paper we focus on the case of regression with manifold-valued input and output. We show pointwise and Bayes consistency for all estimators in the family for the case of manifold-valued output and illustrate the robustness properties of the estimators with experiments. 1 Introduction In recent years there has been an increasing interest in learning with output which differs from the case of standard classification and regression. The need for such approaches arises in several applications which possess more structure than the standard scenarios can model. In structured output learning, see [1, 2, 3] and references therein, one generalizes multiclass classification to more general discrete output spaces, in particular incooperating structure of the joint input and output space. These methods have been successfully applied in areas like computational biology, natural language processing and information retrieval. On the other hand there has been a recent series of work which generalizes regression with multivariate output to the case where the output space is a Riemannian manifold, see [4, 5, 6, 7], with applications in signal processing, computer vision, computer graphics and robotics. One can also see this branch as structured output learning if one thinks of a Riemannian manifold as isometrically embedded in a Euclidean space. Then the restriction that the output has to lie on the manifold can be interpreted as constrained regression in Euclidean space, where the constraints couple several output features together. In this paper we propose a family of kernel estimators for regression with metric-space valued input and output motivated by estimators proposed in [6, 8] for manifold-valued regression. We discuss loss functions and the corresponding Bayesian decision theory for this general regression problem. Moreover, we show that this family of estimators has several well known estimators as special cases for certain choices of the output space and its metric. However, our main emphasis lies on the problem of regression with manifold-valued input and output which includes the multivariate Euclidean case. In particular, we show for all our proposed estimators their pointwise and Bayes consistency, that is in the limit as the sample size goes to infinity the estimated mapping converges to the Bayes optimal mapping. This includes estimators implementing several robust loss functions like the L1-loss, Huber loss or the ε-insensitive loss. This generality is possible since our proof considers directly the functional which is minimized instead of its minimizer as it is usually done in consistency proofs of the Nadaraya-Watson estimator. Finally, we conclude with a toy experiment illustrating the robustness properties and difference of the estimators. 1 2 Bayesian decision theory and loss functions for metric-space valued output We consider the structured output learning problem where the task is to learn a mapping φ : M →N between two metric spaces M and N, where dM denotes the metric of M and dN the metric of N. We assume that both metric spaces M and N are separable1. In general, we are in a statistical setting where the given input/output pairs (Xi, Yi) are i.i.d. samples from a probability measure P on M × N. In order to prove later on consistency of our metric-space valued estimator we first have to define the Bayes optimal mapping φ∗: M →N in the case where M and N are general metric spaces which depends on the employed loss function. In multivariate regression the most common loss function is, L(y, f(x)) = ∥y −f(x)∥2 2. However, it is well known that this loss is sensitive to outliers. In univariate regression one therefore uses the L1-loss or other robust loss functions like the Huber or εinsensitive loss. For the L1-loss the Bayes optimal function f ∗is given as f ∗(x) = Med[Y |X = x], where Med denotes the median of P(Y |X = x) which is a robust location measure. Several generalizations of the median for multivariate output have been proposed, see e.g. [9]. In this paper we refer to the minimizer of the loss function L(y, f(x)) = ∥y −f(x)∥Rn resp. L(y, f(x)) = dN(y, f(x)) as the (generalized) median, since this seems to be the only generalization of the univariate median which has a straightforward extension to metric spaces. In analogy to Euclidean case, we will therefore use loss functions penalizing the distance between predicted output and desired output: L(y, φ(x)) = Γ dN(y, φ(x)) , y ∈N, x ∈M, where Γ : R+ →R+. We will later on restrict Γ to a certain family of functions. The associated risk (or expected loss) is: RΓ(φ) = E[L(Y, φ(X))] and its Bayes optimal mapping φ∗ Γ : M →N can then be determined by φ∗ Γ := arg min φ:M→N, φ measurable RΓ(φ) = arg min φ:M→N, φ measurable E[Γ dN(Y, φ(X)) ] = arg min φ:M→N, φ measurable EX[EY |X[Γ dN(Y, φ(X)) | X]. (1) In the second step we used a result of [10] which states that a joint probability measure on the product of two separable metric spaces can always be factorized into a conditional probability measure and the marginal. In order that the risk is well-defined, we assume that there exists a measurable mapping φ : M →N so that E[Γ dN(Y, φ(X)) ] < ∞. This holds always once N has bounded diameter. Apart from the global risk RΓ(φ) we analyze for each x ∈M the pointwise risk R′ Γ(x, φ(x)), R′ Γ(x, φ(x)) = EY |X[Γ dN(Y, φ(X)) | X = x], which measures the loss suffered by predicting φ(x) for the input x ∈M. The total loss RΓ(φ) of the mapping φ is then RΓ(φ) = E[R′ Γ(X, φ(X))]. As in standard regression the factorization allows to find the Bayes optimal mapping φ∗pointwise, φ∗ Γ(x) = arg min p∈N R′ Γ(x, p) = arg min p∈N E[Γ dN(Y, p) | X = x] = arg min p∈N Z N Γ dN(y, p) dµx(y), where dµx is the conditional probability of Y conditioned on X = x. Later on we prove consistency for a set of kernel estimators each using a different loss function Γ from the following class of functions. Definition 1 A convex function Γ : R+ →R+ is said to be (α, s)-bounded if • Γ : R+ →R+ is continuously differentiable, monotonically increasing and Γ(0) = 0, • Γ(2x) ≤α Γ(x) for x ≥s and Γ(s) > 0 and Γ′(s) > 0. Several functions Γ corresponding to standard loss functions in regression are (α, s)-bounded: • Lp-type loss: Γ(x) = xγ for γ ≥1 is (2γ, 1)-bounded, • Huber-loss: Γ(x) = 2x2 ε for x ≤ε 2 and Γ(x) = 2x −ε 2 for x > ε 2 is (3, ε 2)-bounded. 1A metric space is separable if it contains a countable dense subset. 2 • ε-insensitive loss: Γ(x) = 0 for x ≤ε and Γ(x) = x −ε if x > ε is (3, 2ε)-bounded. While uniqueness of the minimizer of the pointwise loss functional R′ Γ(x, ·) cannot be guaranteed anymore in the case of metric space valued output, the following lemma shows that R′ Γ(x, ·) has reasonable properties (all longer proofs can be found in Section 7 or in the supplementary material). It generalizes a result provided in [11] for Γ(x) = x2 to all (α, s)-bounded losses. Lemma 1 Let N be a complete and separable metric space such that d(x, y) < ∞for all x, y ∈N and every closed and bounded set is compact. If Γ is (α, s)-bounded and R′ Γ(x, q) < ∞for some q ∈N, then • R′ Γ(x, p) < ∞for all p ∈N, • R′ Γ(x, ·) is continuous on N, • The set of minimizers Q∗= arg min q∈N R′ Γ(x, q) exists and is compact. It is interesting to have a look at one special loss, the case Γ(x) = x2. The minimizer of the pointwise risk, F(p) = arg min p∈N Z N d2 N(y, p) dµx(y), is called the Frech´et mean2 or Karcher mean in the case where N is a manifold. It is the generalization of a mean in Euclidean space to a general metric space. Unfortunately, it needs to be no longer unique as in the Euclidean case. A simple example is the sphere as the output space together with a uniform probability measure on it. In this case every point p on the sphere attains the same value F(p) and thus the global minimum is non-unique. We refer to [12, 13, 11] for more information under which conditions one can prove uniqueness of the global minimizer if N is a Riemannian manifold. The generalization of the median to Riemannian manifolds, that is Γ(x) = x, is discussed in [9, 4, 8]. For a discussion of the computation of the median in general metric spaces see [14]. 3 A family of kernel estimators with metric-space valued input and output In the following we provide the definition of the kernel estimator with metric-space valued output motivated by the two estimators proposed in [6, 8] for manifold-valued output. We use in the following the notation kh(x) = 1 hm k(x/h). Definition 2 Let (Xi, Yi)l i=1 be the sample with Xi ∈M and Yi ∈N. The metric-space-valued kernel estimator φl : M →N from metric space M to metric space N is defined for all x ∈M as φl(x) = arg min q∈N 1 l l X i=1 Γ dN(q, Yi) kh dM(x, Xi) , (2) where Γ : R+ →R+ is (α, s)-bounded and k : R+ →R+. If the data contains a large fraction of outliers one should use a robust loss function Γ, see Section 6. Usually the kernel function should be monotonically decreasing since the interpretation of kh dM(x, Xi) is to measure the similarity between x and Xi in M which should decrease as the distance increases. The computational complexity to determine φl(x) is quite high as for each test point one has to solve an optimization problem but comparable to structured output learning (see discussion below) where one maximizes for each test point the score function over the output space. For manifold-valued output we will describe in the next section a simple gradient-descent type optimization scheme in order to determine φl(x). It is interesting to see that several well-known nonparametric estimators for classification and regression can be seen as special cases of this estimator (or a slightly more general form) for different choices of the output space, its metric and the loss function. In particular, the approach shows a certain analogy of a generalization of regression into a continuous space (manifold-valued regression) and regression into a discrete space (structured output learning). 2In some cases the set of all local minimizers is denoted as the Frech´et mean set and the Frech´et mean is called unique if there exists only one global minimizer. 3 Multiclass classification: Let N = {1, . . . , K} where K denotes the number of classes K. If there is no special class-structure, then we use the discrete metric on N, dN(q, q′) = 1 if q ̸= q′ and 0 else leads for any Γ to the standard multiclass classification scheme using a majority vote. Costsensitive multiclass classification can be done by using dN(q, q′) to model the cost of misclassifying class q by class q′. Since general costs can generally not be modeled by a metric, it should be noted that the estimator can be modified using a similarity function, s : N × N →R, φl(x) = arg max q∈N 1 l l X i=1 s q, Yi kh dM(x, Xi) , (3) The consistency result below can be generalized to this case given that N has finite cardinality. Multivariate regression: Let N = Rn and M be a metric space. Then for Γ(x) = x2, one gets φl(x) = arg min q∈N 1 l l X i=1 ∥q −Yi∥2 kh dM(x, Xi) , which has the solution, φl(x) = 1 l P l i=1 kh dM(x,Xi) Yi 1 l P l i=1 kh dM(x,Xi) . This is the well-known Nadaraya-Watson estimator, see [15, 16], on a metric space. In [17] a related estimator is discussed when M is a closed Riemannian manifold and [18] discusses the Nadaraya-Watson estimator when M is a metric space. Manifold-valued regression: In [6] the estimator φl(x) has been proposed for the case where N is a Riemannian manifold and Γ(x) = x2, in particular with the emphasis on N being the manifold of shapes. The discussion of a robust median-type estimator, that is Γ(x) = x, has been done recently in [8]. While it has been shown in [7] that an approach using a global smoothness regularizer outperforms the estimator φl(x), it is a well working baseline with a simple implementation, see Section 4. Structured output: Structured output learning, see [1, 2, 3] and references therein, can be formulated using kernels k (x1, q1), (x2, q2) on the product M ×N of input and output space, which are supposed to measure jointly the similarity and thus can capture non-trivial dependencies between input and output. Using such kernels [1, 2, 3] learn a score function s : M × N →R, with Ψ(x) = arg max q∈N s(x, q). being the final prediction for x ∈M. The similarity to our estimator φl(x) in (2) becomes more obvious when we use that in the framework of [1] the learned score function can be written as Ψl(x) = arg max q∈N 1 l l X i=1 αi k (x, q), (Xi, Yi) , (4) where α ∈Rl is the learned coefficient vector. Apart from the coefficient vector α this has almost the form of the previously discussed estimator in Equation (3), using a joint similarity function on input and output space. Clearly, a structured output method where the coefficients α have been optimized, should perform better than αi = const. In cases where training time is prohibitive the estimator without α is an alternative, at least it provides a useful baseline for structured output learning. Moreover, if the joint kernel factorizes, k (x1, q1), (x2, q2) = kM(x1, x2) kN(q1, q2) on M and N, and kN(q, q) = const., then one can rewrite the problem in (4) as, Ψl(x) = arg min q∈N 1 l l X i=1 αi kM(x, Xi)d2 N(q, Yi), where dN is the induced (semi)-metric3 of kN. Apart from the learned coefficients this is basically equivalent to φl(x) in (2) for Γ(x) = x2. In the following we restrict ourselves to the case where M and N are Riemannian manifolds. In this case the optimization to obtain φl(x) can still be done very efficiently as the next section shows. 3The kernel kN induces a (semi)-metric dN on N via: d2 N(p, q) = kN(p, p) + kN(q, q) −2kN(p, q). 4 4 Implementation of the kernel estimator for manifold-valued output For fixed x ∈M, the functional F(q) for q ∈N which is optimized in the kernel estimator φl(x) can be rewritten with wi = kh(dM(x, Xi)) as, F(q) = l X i=1 wi Γ dN(q, Yi) . The covariant gradient of F(q) is given as, ∇F q = Pl i=1 wiΓ′ dN(p, Yi) vi, where vi ∈TqN is a tangent vector at q with ∥vi∥TqN = 1 given by the tangent vector at q of the minimizing4 geodesic from Yi to q (pointing “away” from Yi). Denoting by expq : TqN →N the exponential map at q, the simple gradient descent based optimization scheme can be written as • choose a random point q0 from N, • while stopping criteria not fulfilled, 1. compute gradient ∇F at qk 2. one has: qk+1 = expqk −α∇F|qk 3. determine stepsize α by Armijo rule [19]. As stopping criterion we use either the norm of the gradient or a threshold on the change of F. For the experiments in Section 6 we get convergence in 5 to 40 steps. 5 Consistency of the kernel estimator for manifold-valued input and output In this section we show the pointwise and Bayes consistency of the kernel estimator φl in the case where M and N are Riemannian manifolds. This case already subsumes several of the interesting applications discussed in [6, 8]. The proof of consistency of the general metric-space valued kernel estimator (for a restricted class of metric spaces including all Riemannian manifolds) requires high technical overload which is interesting in itself but which would make the paper hard accessible. The consistency of φl will be proven under the following assumptions: Assumptions (A1): 1. The loss Γ : R+ →R+ is (α, s)-bounded. 2. (Xi, Yi)l i=1 is an i.i.d. sample of P on M × N, 3. M and N are compact m-and n-dimensional manifolds, 4. The data-generating measure P on M × N is absolutely continuous with respect to the natural volume element, 5. The marginal density on M fulfills: p(x) ≥pmin, ∀x ∈M, 6. The density p(·, y) is continuous on M for all y ∈N, 7. The kernel fulfills: a 1s≤r1 ≤k(s) ≤b e−γ s2 and R Rm ∥x∥k(∥x∥) dx < ∞, Note, that existence of a density is not necessary for consistency. However, in order to keep the proofs simple, we restrict ourselves to this setting. In the following dV = √det g dx denotes the natural volume element of a Riemannian manifold with metric g, vol(S) and diam(N) are the volume and diameter of the set S. For the proof of our main theorem we need the following two propositions. The first one summarizes two results from [20]. Proposition 1 Let M be a compact m-dimensional Riemannian manifold. Then, there exists r0 > 0 and S1, S2 > 0 such that for all x ∈M the volume of the balls B(x, r) with radius r ≤r0 satisfies, S1 rm ≤vol B(x, r) ≤S2 rm. Moreover, the cardinality K of a δ-covering of M is upper bounded as, K ≤vol(N) S1 2 δ m . 4The set of points where there the minimizing geodesic is not unique, the so called cut locus, has measure zero and therefore plays no role in the optimization. 5 Moreover, we need a result about convolutions on manifolds. Proposition 2 Let the assumptions A1 hold, then if f is continuous we get for any x ∈M\∂M, lim h→0 Z M kh(dM(x, z))f(z) dV (z) = Cxf(x), where Cx = limh→0 R M kh(dM(x, z)) dV (z) > 0. If moreover f is Lipschitz continuous with Lipschitz constant L, then there exists a h0 > 0 such that for all h < h0(x), Z M kh(dM(x, z))f(z) dV (z) = Cx f(x) + O(h). The following main theorem proves the almost sure pointwise convergence of the manifold-valued kernel estimator for all (α, s)-bounded loss functions Γ. Theorem 1 Suppose the assumptions in A1 hold. Let φl(x) be the estimate of the kernel estimator for sample size l. If h →0 and lhm/ log l →∞, then for any x ∈M\∂M, lim l→∞|R′ Γ(x, φl(x)) −arg min q∈N R′ Γ(x, q)| = 0, almost surely. If additionally p(·, y) is Lipschitz-continuous for any y ∈N, then lim l→∞|R′ Γ(x, φl(x)) −arg min q∈N R′ Γ(x, q)| = O(h) + O p log l/(l hm) , almost surely. The optimal rate is given by h = O (log l/l) 1 2+m so that lim l→∞R′ Γ(x, φl(x)) −arg min q∈N R′ Γ(x, q) = O log l/l 1 2+m , almost surely. Note, that the condition l hm/ log l →∞for convergence is the same as for the Nadaraya-Watson estimator on a m-dimensional Euclidean space. This had to be expected as this condition still holds if one considers multivariate output, see [15, 16]. Thus, doing regression with manifold-valued output is not more “difficult” than standard regression with multivariate output. Next, we show Bayes consistency of the manifold-valued kernel estimator. Theorem 2 Let the assumptions A1 hold. If h →0 and lhm/ log l →∞, then lim l→∞RΓ(φl) −RΓ(φ∗) = 0, almost surely. Proof: We have, RΓ(φl) −RΓ(φ∗) ≤ E[|R′ Γ(X, φl(X)) −R′ Γ(X, φ∗(X))|]. Moreover, we have almost everywhere, liml→∞R′ Γ(x, φl(x)) = R′ Γ(x, φ∗(x)) almost surely. Since E[R′ Γ(X, φ(X))] < ∞and E[R′ Γ(X, φ∗(X))] < ∞, an extension of the dominated convergence theorem proven by Glick, see [21], provides the result. □ 6 Experiments We illustrate the differences of median and mean type estimator on a synthetic dataset with the task of estimating a curve on the sphere, that is M = [0, 1] and N = S1. The kernel used had the form, k |x −y|/h = 1 −|x −y|/h. The parameter h was found by 5-fold cross validation from the set [5, 10, 20, 40] ∗10−3. The results are summarized for different levels of outliers and different levels of van-Mises noise (note that the parameter k is inverse to the variance of the distribution) in Table 1. As expected the the L1-loss and the Huber loss as robust loss functions outperform the L2-loss in the presence of outliers, whereas the L2-loss outperforms the robust versions when no outliers are present. Note, that the Huber loss as a hybrid version between L1- and L2-loss is even slightly better than the L1-loss in the presence of outliers as well as in the outlier free case. Thus for a given dataset it makes sense not only to do cross-validation of the parameter h of the kernel function but also over different loss functions in order to adapt to possible outliers in the data. 6 Figure 1: Regression problem on the sphere with 1000 training points (black points). The blue points are the ground truth disturbed by van Mises noise with parameter k = 100 and 20% (outliers) with k = 3. The estimated curves are shown in green. Left: Result of L1-loss, mean error (ME) 0.256, mean squared error (MSE) 0.165. Middle: Result of L2-loss: ME = 0.265, MSE = 0.169. Right: Result of Huber loss with ε = 0.1: ME = 0.255, MSE = 0.165. In particular, the curves found using L1 and Huber loss are very close to the ground truth. Table 1: Mean squared error (unit 10−1) for regression on the sphere - for different noise levels k, number of labeled points, without and with outliers. Results are averaged over 10 runs. no outliers 20% outliers Number of samples 100 500 1000 100 500 1000 L1-Loss k = 100 0.63 ± 0.11 0.260 ± 0.027 0.219 ± 0.003 2.1 ± 0.2 1.57 ± 0.05 1.521 ± 0.015 Γ(x) = x k = 1000 0.43 ± 0.12 0.043 ± 0.005 0.030 ± 0.001 2.1 ± 0.5 1.45 ± 0.03 1.400 ± 0.008 L2-Loss k = 100 0.43 ± 0.10 0.230 ± 0.007 0.208 ± 0.001 2.0 ± 0.2 1.59 ± 0.02 1.549 ± 0.021 Γ(x) = x2 k = 1000 0.28 ± 0.16 0.032 ± 0.003 0.025 ± 0.001 2.0 ± 0.4 1.51 ± 0.03 1.447 ± 0.015 Huber-Loss k = 100 0.61 ± 0.11 0.257 ± 0.026 0.218 ± 0.003 2.1 ± 0.2 1.57 ± 0.05 1.520 ± 0.021 with ε = 0.1 k = 1000 0.42 ± 0.12 0.040 ± 0.005 0.028 ± 0.001 2.1 ± 0.5 1.44 ± 0.02 1.397 ± 0.008 7 Proofs Lemma 2 Let φ : R+ →R be convex, differentiable and monotonically increasing. Then min{φ′(x), φ′(y)}|y −x| ≤|φ(y) −φ(x)| ≤max{φ′(x), φ′(y)}|y −x|. Proof of Theorem 1 We define R′ Γ,l(x, q) = 1 l P l i=1 Γ(dN(q,Yi)) kh(dM(x,Xi)) E[kh(dM(x,X))] . Note that φl(x) = arg min q∈N R′ Γ,l(x, q) as we have only divided by a constant factor. We use the standard technique for the pointwise estimate, R′ Γ(x, φl(x)) −min q∈N R′ Γ(x, q) ≤R′ Γ(x, φl(x)) −R′ Γ,l(x, φl(x)) + R′ Γ,l(x, φl(x)) −min q∈N R′ Γ(x, q) ≤2 sup q∈N |R′ Γ,l(x, q) −R′ Γ(x, q)|. In order to bound the supremum, we will work on the event E, where we assume, 1 l P l i=1 kh(dM(x,Xi)) E[kh(dM(x,X))] −1 < 1 2, which holds with probability 1 −2 e−C l hm for some constant C. Moreover, we assume to have a δ-covering of N with centers Nδ = {qα}K α=1 where using Lemma 1 we have K ≤vol(N) S1 2 δ n . Thus for each q ∈N there exists qα ∈Nδ such that dN(q, qα) ≤δ. Introducing RE Γ (x, q) = E[Γ(dN(q,Y ))kh(dM(x,X))] E[kh(dM(x,X))] and using the decomposition, R′ Γ,l(x, q) −R′ Γ(x, q) =R′ Γ,l(x, q) −R′ Γ,l(x, qα) + R′ Γ,l(x, qα) −RE Γ (x, qα) + RE Γ (x, qα) −RE Γ (x, q) + RE Γ (x, q) −R′ Γ(x, q), we have to control four terms, R′ Γ,l(x, q)−R′ Γ,l(x, qα) = 1 l Pl i=1 Γ dN(q, Yi) −Γ dN(qα, Yi) kh(dM(x, Xi)) E[kh(dM(x, X))] ≤2 dN(q, qα) Γ′ diam(N) 1 l Pl i=1 kh(dM(x, Xi)) E[kh(dM(x, X))] ≤3 Γ′ diam(N) δ. 7 where we have used Lemma 2 and the fact that E holds. Then, there exists a constant C such that P max 1≤α≤K |R′ Γ,l(x, qα) −RE Γ (x, qα)| > ε ≤2vol(N) S1 2 δ n e−C l hmε2, which can be shown using Bernstein’s inequality for 1 l Pl i=1 Wi −E[Wi] where Wi = Γ(dN(qα,Yi))kh(dM(x,Xi)) E[kh(dM(x,X))] together with a union bound over the elements in the covering Nδ using |Wi| ≤b a Γ(diam(N)) hmS1rm 1 pmin , Var Wi ≤Γ(diam(N))2E[k2 h(dM(x, X))] (E[kh(dM(x, X))])2 ≤b a Γ(diam(N))2 hmS1rm 1 pmin , where we used Proposition 1 to lower bound vol(B(x, h r1)) for small enough h. Third, we get for the third term using again Lemma 2, |RE Γ (x, qα) −RE Γ (x, q)| ≤2Γ′(diam(N))dN(q, qα) ≤2Γ′(diam(N))δ. Last, we have to bound the approximation error RE Γ (x, q)−R′ Γ(x, q), Under the continuity assumption on the joint density p(x, y) we can use Proposition 2. For every x ∈M\∂M we get, lim h→0 Z M kh(dM(x, z))p(z, y)dV (z) = Cx p(x, y), lim h→0 Z M kh(dM(x, z))p(z)dV (z) = Cxp(x), where Cx > 0. Thus with fh = Z M kh(dM(x, z))p(z, y)dV (z), gh = Z M kh(dM(x, z))p(z)dV (z), we get for every x ∈M\∂M, lim h→0 fh gh −f g ≤lim h→0 |fh −f| gh + lim h→0 f |gh −g| g gh = 0, where we have used gh ≥aS1r1pmin > 0 and g = Cxp(x) > 0. Moreover, using results from the proof of Proposition 2 one can show fh < C for some constant C. Thus fh/gh < C for some constant and fh/gh →f/g as h →0. Using the dominated convergence theorem we thus get lim h→0 RE Γ (x, q) = lim h→0 E[Γ(dN(q, Y ))kh(dM(x, X))] E[kh(dM(x, X))] = Z N Γ dN(q, y) p(x, y) p(x) dy = R′ Γ(x, q). For the case where the joint density is Lipschitz continuous one gets using Proposition 2, RE Γ (x, q) = R′ Γ(x, q) + O(h). In total, there exist constants A, B, C, D1, D2, such that for sufficiently small h one has with probability 1 −AeB n log( 1 δ )−Clhmε2, sup q∈N |R′ Γ,l(x, q) −RE Γ (x, q)| ≤2D1δ + ε. With δ = l−s for some s > 0 one gets convergence if lhm log l → ∞together with limh→0 RE Γ (x, q) = R′ Γ(x, q). For the case where p(·, y) is Lipschitz continuous for all y ∈N we have RE Γ (x, q) = R′ Γ(x, q) + O(h) and can choose s large enough so that the bound from the approximation error dominates the one of the covering. Under the condition lhm log l →∞the probabilistic bound is summable in l which yields almost sure convergence by the Borel-CantelliLemma. The optimal rate in the Lipschitz continuous case is then determined by fixing h such that both terms of the bound are of the same order. □ Acknowledgments We thank Florian Steinke for helpful discussions about relations between generalized kernel estimators and structured output learning. This work has been partially supported by the Cluster of Excellence MMCI at Saarland University. 8 References [1] I. Tsochantaridis, T. Joachims, T. Hofmann, and Y. Altun. Large margin methods for structured and interdependent output variables. JMLR, 6:1453–1484, 2005. [2] J. Weston, G. BakIr, O. Bousquet, B. Sch¨olkopf, T. Mann, and W. S. Noble. Joint kernel maps. In Predicting Structured Data, pages 67–84. MIT Press, 2007. [3] E. Ricci, T. De Bie, and N. Cristianini. Magic moments for structured output prediction. JMLR, 9:2803–2846, 2008. [4] K.V. Mardia and P.E. Jupp. Directional statistics. Wiley New York, 2000. [5] Inam Ur Rahman, Iddo Drori, Victoria C. Stodden, David L. Donoho, and Peter Schroder. Multiscale representations for manifold-valued data. Multiscale Modeling and Simulation, 4(4):1201–1232, 2005. [6] B. C. Davis, P. T. Fletcher, E. Bullitt, and S. Joshi. Population shape regression from random design data. Computer Vision, 2007. ICCV 2007. IEEE 11th International Conference on, pages 1–7, 2007. [7] F. Steinke and M. Hein. Non-parametric regression between Riemannian manifolds. In Advances in Neural Information Processing Systems (NIPS) 21, pages 1561 – 1568, 2009. [8] P. T. Fletcher, S. Venkatasubramanian, and S. Joshi. The geometric median on Riemannian manifolds with application to robust atlas estimation. NeuroImage, 45:143 – 152, 2009. [9] C. G. Small. A survey of multidimensional medians. International Statistical Review, 58:263– 277, 1990. [10] D. Blackwell and M. Maitra. Factorization of probability measures and absolutely measurable sets. Proc. Amer. Math. Soc., 92(2):251–254, 1984. [11] R. Bhattacharya and V. Patrangenaru. Large sample theory of intrinsic and extrinsic sample means on manifolds I. Ann. Stat., 31(1):1–29, 2003. [12] H. Karcher. Riemannian center of mass and mollifier smoothing. Communications on Pure and Applied Mathematics, 30:509–541, 1977. [13] W. Kendall. Probability, convexity, and harmonic maps with small image. I. Uniqueness and fine existence. Proc. London Math. Soc., 61(2):371–406, 1990. [14] P. Indyk. Sublinear time algorithms for metric space problems. In Proceedings of the 31st Symposium on Theory of computing (STOC), pages 428 – 434, 1999. [15] L. Gy¨orfi, M. Kohler, A. Krzy˙zak, and H. Walk. A Distribution-Free Theory of Nonparametric Regression. Springer, New York, 2004. [16] W. Greblicki and M. Pawlak. Nonparametric System Identification. Cambridge University Press, Cambrige, 2008. [17] B. Pelletier. Nonparametric regression estimation on closed Riemannian manifolds. J. of Nonparametric Stat., 18:57–67, 2006. [18] S. Dabo-Niang and N. Rhomari. Estimation non parametrique de la regression avec variable explicative dans un espace metrique. C. R. Math. Acad. Sci. Paris, 1:75–80, 2003. [19] D. P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, Mass., 1999. [20] M. Hein. Uniform convergence of adaptive graph-based regularization. In G. Lugosi and H. Simon, editors, Proc. of the 19th Conf. on Learning Theory (COLT), pages 50–64, Berlin, 2006. Springer. [21] N. Glick. Consistency conditions for probability estimators and integrals of density estimators. Utilitas Math., 6:61–74, 1974. 9
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Kernels and learning curves for Gaussian process regression on random graphs Peter Sollich, Matthew J Urry King’s College London, Department of Mathematics London WC2R 2LS, U.K. {peter.sollich,matthew.urry}@kcl.ac.uk Camille Coti INRIA Saclay ˆIle de France, F-91893 Orsay, France Abstract We investigate how well Gaussian process regression can learn functions defined on graphs, using large regular random graphs as a paradigmatic example. Random-walk based kernels are shown to have some non-trivial properties: within the standard approximation of a locally tree-like graph structure, the kernel does not become constant, i.e. neighbouring function values do not become fully correlated, when the lengthscale σ of the kernel is made large. Instead the kernel attains a non-trivial limiting form, which we calculate. The fully correlated limit is reached only once loops become relevant, and we estimate where the crossover to this regime occurs. Our main subject are learning curves of Bayes error versus training set size. We show that these are qualitatively well predicted by a simple approximation using only the spectrum of a large tree as input, and generically scale with n/V , the number of training examples per vertex. We also explore how this behaviour changes for kernel lengthscales that are large enough for loops to become important. 1 Motivation and Outline Gaussian processes (GPs) have become a standard part of the machine learning toolbox [1]. Learning curves are a convenient way of characterizing their capabilities: they give the generalization error ϵ as a function of the number of training examples n, averaged over all datasets of size n under appropriate assumptions about the process generating the data. We focus here on the case of GP regression, where a real-valued output function f(x) is to be learned. The general behaviour of GP learning curves is then relatively well understood for the scenario where the inputs x come from a continuous space, typically Rn [2, 3, 4, 5, 6, 7, 8, 9, 10]. For large n, the learning curves then typically decay as a power law ϵ ∝n−α with an exponent α ≤1 that depends on the dimensionality n of the space as well as the smoothness properties of the function f(x) as encoded in the covariance function. But there are many interesting application domains that involve discrete input spaces, where x could be a string, an amino acid sequence (with f(x) some measure of secondary structure or biological function), a research paper (with f(x) related to impact), a web page (with f(x) giving a score used to rank pages), etc. In many such situations, similarity between different inputs – which will govern our prior beliefs about how closely related the corresponding function values are – can be represented by edges in a graph. One would then like to know how well GP regression can work in such problem domains; see also [11] for a related online regression algorithm. We study this 1 problem here theoretically by focussing on the paradigmatic example of random regular graphs, where every node has the same connectivity. Sec. 2 discusses the properties of random-walk inspired kernels [12] on such random graphs. These are analogous to the standard radial basis function kernels exp[−(x −x′)2/(2σ2)], but we find that they have surprising properties on large graphs. In particular, while loops in large random graphs are long and can be neglected for many purposes, by approximating the graph structure as locally tree-like, here this leads to a non-trivial limiting form of the kernel for σ →∞that is not constant. The fully correlated limit, where the kernel is constant, is obtained only because of the presence of loops, and we estimate when the crossover to this regime takes place. In Sec. 3 we move on to the learning curves themselves. A simple approximation based on the graph eigenvalues, using only the known spectrum of a large tree as input, works well qualitatively and predicts the exact asymptotics for large numbers of training examples. When the kernel lengthscale is not too large, below the crossover discussed in Sec. 2 for the covariance kernel, the learning curves depend on the number of examples per vertex. We also explore how this behaviour changes as the kernel lengthscale is made larger. Sec. 4 summarizes the results and discusses some open questions. 2 Kernels on graphs and trees We assume that we are trying to learn a function defined on the vertices of a graph. Vertices are labelled by i = 1 . . . V , instead of the generic input label x we used in the introduction, and the associated function values are denoted fi ∈R. By taking the prior P(f) over these functions f = (f1, . . . , fV ) as a (zero mean) Gaussian process we are saying that P(f) ∝exp(−1 2f TC−1f). The covariance function or kernel C is then, in our graph setting, just a positive definite V × V matrix. The graph structure is characterized by a V × V adjacency matrix, with Aij = 1 if nodes i and j are connected by an edge, and 0 otherwise. All links are assumed to be undirected, so that Aij = Aji, and there are no self-loops (Aii = 0). The degree of each node is then defined as di = PV j=1 Aij. The covariance kernels we discuss in this paper are the natural generalizations of the squaredexponential kernel in Euclidean space [12]. They can be expressed in terms of the normalized graph Laplacian, defined as L = 1 −D−1/2AD−1/2, where D is a diagonal matrix with entries d1, . . . , dV and 1 is the V × V identity matrix. An advantage of L over the unnormalized Laplacian D−A, which was used in the earlier paper [13], is that the eigenvalues of L (again a V ×V matrix) lie in the interval [0,2] (see e.g. [14]). From the graph Laplacian, the covariance kernels we consider here are constructed as follows. The p-step random walk kernel is (for a ≥2) C ∝(1 −a−1L)p = h 1 −a−1 1 + a−1D−1/2AD−1/2ip (1) while the diffusion kernel is given by C ∝exp −1 2σ2L ∝exp 1 2σ2D−1/2AD−1/2 (2) We will always normalize these so that (1/V ) P i Cii = 1, which corresponds to setting the average (over vertices) prior variance of the function to be learned to unity. To see the connection of the above kernels to random walks, assume we have a walker on the graph who at each time step selects randomly one of the neighbouring vertices and moves to it. The probability for a move from vertex j to i is then Aij/dj. The transition matrix after s steps follows as (AD−1)s: its ij-element gives the probability of being on vertex i, having started at j. We can now compare this with the p-step kernel by expanding the p-th power in (1): C ∝ p X s=0 ( p s )a−s(1−a−1)p−s(D−1/2AD−1/2)s =D−1/2 p X s=0 ( p s )a−s(1−a−1)p−s(AD−1)sD1/2 (3) Thus C is essentially a random walk transition matrix, averaged over the number of steps s with s ∼Binomial(p, 1/a) (4) 2 0 5 10 15 l 0 0.2 0.4 0.6 0.8 1 Cl,p p=1 p=2 p=3 p=4 p=5 p=10 p=20 p=50 p=100 p=200 p=500 p=infty a=2, d=3 1 10 100 1000 p/a 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 K1 ln V / ln(d-1) a=2, V=infty a=2, V=500 a=4, V=infty a=4, V=500 d=3 Figure 1: (Left) Random walk kernel Cℓ,p plotted vs distance ℓalong graph, for increasing number of steps p and a = 2, d = 3. Note the convergence to a limiting shape for large p that is not the naive fully correlated limit Cℓ,p→∞= 1. (Right) Numerical results for average covariance K1 between neighbouring nodes, averaged over neighbours and over randomly generated regular graphs. This shows that 1/a can be interpreted as the probability of actually taking a step at each of p “attempts”. To obtain the actual C the resulting averaged transition matrix is premultiplied by D−1/2 and postmultiplied by D1/2, which ensures that the kernel C is symmetric. For the diffusion kernel, one finds an analogous result but the number of random walk steps is now distributed as s ∼Poisson(σ2/2). This implies in particular that the diffusion kernel is the limit of the p-step kernel for p, a →∞at constant p/a = σ2/2. Accordingly, we discuss mainly the p-step kernel in this paper because results for the diffusion kernel can be retrieved as limiting cases. In the limit of a large number of steps s, the random walk on a graph will reach its stationary distribution p∞∝De where e = (1, . . . , 1). (This form of p∞can be verified by checking that it remains unchanged after multiplication with the transition matrix AD−1.) The s-step transition matrix for large s is then p∞eT = DeeT because we converge from any starting vertex to the stationary distribution. It follows that for large p or σ2 the covariance kernel becomes C ∝D1/2eeTD1/2, i.e. Cij ∝(didj)1/2. This is consistent with the interpretation of σ or (p/a)1/2 as a lengthscale over which the random walk can diffuse along the graph: once this lengthscale becomes large, the covariance kernel Cij is essentially independent of the distance (along the graph) between the vertices i and j, and the function f becomes fully correlated across the graph. (Explicitly f = vD1/2e under the prior, with v a single Gaussian random variable.) As we next show, however, the approach to this fully correlated limit as p or σ are increased is non-trivial. We focus in this paper on kernels on random regular graphs. This means we consider adjacency matrices A which are regular in the sense that they give for each vertex the same degree, di = d. A uniform probability distribution is then taken across all A that obey this constraint [15]. What will the above kernels look like on typical samples drawn from this distribution? Such random regular graphs will have long loops, of length of order ln(V ) or larger if V is large. Their local structure is then that of a regular tree of degree d, which suggests that it should be possible to calculate the kernel accurately within a tree approximation. In a regular tree all nodes are equivalent, so the kernel can only depend on the distance ℓbetween two nodes i and j. Denoting this kernel value Cℓ,p for a p-step random walk kernel, one has then Cℓ,p=0 = δℓ,0 and γp+1C0,p+1 = 1 −1 a C0,p + 1 a C1,p (5) γp+1Cℓ,p+1 = 1 ad Cℓ−1,p + 1 −1 a Cℓ,p + d−1 ad Cℓ+1,p for ℓ≥1 (6) where γp is chosen to achieve the desired normalization C0,p = 1 of the prior variance for every p. Fig. 1(left) shows results obtained by iterating this recursion numerically, for a regular graph (in the tree approximation) with degree d = 3, and a = 2. As expected the kernel becomes more longranged initially as p increases, but eventually it is seen to approach a non-trivial limiting form. This can be calculated as Cℓ,p→∞= [1 + ℓ(d −1)/d](d −1)−ℓ/2 (7) 3 and is also plotted in the figure, showing good agreement with the numerical iteration. There are (at least) two ways of obtaining the result (7). One is to take the limit σ →∞of the integral representation of the diffusion kernel on regular trees given in [16] (which is also quoted in [13] but with a typographical error that effectively removes the factor (d −1)−ℓ/2). Another route is to find the steady state of the recursion for Cℓ,p. This is easy to do but requires as input the unknown steady state value of γp. To determine this, one can map from Cℓ,p to the total random walk probability Sℓ,p in each “shell” of vertices at distance ℓfrom the starting vertex, changing variables to S0,p = C0,p and Sℓ,p = d(d −1)ℓ−1Cℓ,p (ℓ≥1). Omitting the factors γp, this results in a recursion for Sℓ,p that simply describes a biased random walk on ℓ= 0, 1, 2, . . ., with a probability of 1 −1/a of remaining at the current ℓ, probability 1/(ad) of moving to the left and probability (d −1)/(ad) of moving to the right. The point ℓ= 0 is a reflecting barrier where only moves to the right are allowed, with probability 1/a. The time evolution of this random walk starting from ℓ= 0 can now be analysed as in [17]. As expected from the balance of moves to the left and right, Sℓ,p for large p is peaked around the average position of the walk, ℓ= p(d −2)/(ad). For ℓsmaller than this Sℓ,p has a tail behaving as ∝(d −1)ℓ/2, and converting back to Cℓ,p gives the large-ℓscaling of Cℓ,p→∞∝(d −1)−ℓ/2; this in turn fixes the value of γp→∞and so eventually gives (7). The above analysis shows that for large p the random walk kernel, calculated in the absence of loops, does not approach the expected fully correlated limit; given that all vertices have the same degree, the latter would correspond to Cℓ,p→∞= 1. This implies, conversely, that the fully correlated limit is reached only because of the presence of loops in the graph. It is then interesting to ask at what point, as p is increased, the tree approximation for the kernel breaks down. To estimate this, we note that a regular tree of depth ℓhas V = 1 + d(d −1)ℓ−1 nodes. So a regular graph can be tree-like at most out to ℓ≈ln(V )/ ln(d −1). Comparing with the typical number of steps our random walk takes, which is p/a from (4), we then expect loop effects to appear in the covariance kernel when p/a ≈ln(V )/ ln(d −1) (8) To check this prediction, we measure the analogue of C1,p on randomly generated [15] regular graphs. Because of the presence of loops, the local kernel values are not all identical, so the appropriate estimate of what would be C1,p on a tree is K1 = Cij/ p CiiCjj for neighbouring nodes i and j. Averaging over all pairs of such neighbours, and then over a number of randomly generated graphs we find the results in Fig. 1(right). The results for K1 (symbols) accurately track the tree predictions (lines) for small p/a, and start to deviate just around the values of p/a expected from (8), as marked by the arrow. The deviations manifest themselves in larger values of K1, which eventually – now that p/a is large enough for the kernel to “notice” the loops - approach the fully correlated limit K1 = 1. 3 Learning curves We now turn to the analysis of learning curves for GP regression on random regular graphs. We assume that the target function f ∗is drawn from a GP prior with a p-step random walk covariance kernel C. Training examples are input-output pairs (iµ, f ∗ iµ + ξµ) where ξµ is i.i.d. Gaussian noise of variance σ2; the distribution of training inputs iµ is taken to be uniform across vertices. Inference from a data set D of n such examples µ = 1, . . . , n takes place using the prior defined by C and a Gaussian likelihood with noise variance σ2. We thus assume an inference model that is matched to the data generating process. This is obviously an over-simplification but is appropriate for the present first exploration of learning curves on random graphs. We emphasize that as n is increased we see more and more function values from the same graph, which is fixed by the problem domain; the graph does not grow. The generalization error ϵ is the squared difference between the estimated function ˆfi and the target f ∗ i , averaged across the (uniform) input distribution, the posterior distribution of f ∗given D, the distribution of datasets D, and finally – in our non-Euclidean setting – the random graph ensemble. Given the assumption of a matched inference model, this is just the average Bayes error, or the average posterior variance, which can be expressed explicitly as [1] ϵ(n) = V −1 X i Cii −k(i)TKk−1(i) D,graphs (9) 4 where the average is over data sets and over graphs, K is an n × n matrix with elements Kµµ′ = Ciµ,iµ′ + σ2δµµ′ and k(i) is a vector with entries kµ(i) = Ci,iµ. The resulting learning curve depends, in addition to n, on the graph structure as determined by V and d, and the kernel and noise level as specified by p, a and σ2. We fix d = 3 throughout to avoid having too many parameters to vary, although similar results are obtained for larger d. Exact prediction of learning curves by analytical calculation is very difficult due to the complicated way in which the random selection of training inputs enters the matrix K and vector k in (9). However, by first expressing these quantities in terms of kernel eigenvalues (see below) and then approximating the average over datasets, one can derive the approximation [3, 6] ϵ = g n ϵ + σ2 , g(h) = V X α=1 (λ−1 α + h)−1 (10) This equation for ϵ has to be solved self-consistently because ϵ also appears on the r.h.s. In the Euclidean case the resulting predictions approximate the true learning curves quite reliably. The derivation of (10) for inputs on a fixed graph is unchanged from [3], provided the kernel eigenvalues λα appearing in the function g(h) are defined appropriately, by the eigenfunction condition ⟨Cijφj⟩= λφi; the average here is over the input distribution, i.e. ⟨. . .⟩= V −1 P j . . . From the definition (1) of the p-step kernel, we see that then λα = κV −1(1 −λL α/a)p in terms of the corresponding eigenvalue of the graph Laplacian L. The constant κ has to be chosen to enforce our normalization convention P α λα = ⟨Cjj⟩= 1. Fortunately, for large V the spectrum of the Laplacian of a random regular graph can be approximated by that of the corresponding large regular tree, which has spectral density [14] ρ(λL) = q 4(d−1) d2 −(λL −1)2 2πdλL(2 −λL) (11) in the range λL ∈[λL −, λL +], λL ± = 1 + 2d−1(d −1)1/2, where the term under the square root is positive. (There are also two isolated eigenvalues λL = 0, 2 but these have weight 1/V each and so can be ignored for large V .) Rewriting (10) as ϵ = V −1 P α[(V λα)−1 + (n/V )(ϵ + σ2)−1]−1 and then replacing the average over kernel eigenvalues by an integral over the spectral density leads to the following prediction for the learning curve: ϵ = Z dλLρ(λL)[κ−1(1 −λL/a)−p + ν/(ϵ + σ2)]−1 (12) with κ determined from κ R dλLρ(λL)(1 −λL/a)p = 1. A general consequence of the form of this result is that the learning curve depends on n and V only through the ratio ν = n/V , i.e. the number of training examples per vertex. The approximation (12) also predicts that the learning curve will have two regimes, one for small ν where ϵ ≫σ2 and the generalization error will be essentially independent of σ2; and another for large ν where ϵ ≪σ2 so that ϵ can be neglected on the r.h.s. and one has a fully explicit expression for ϵ. We compare the above prediction in Fig. 2(left) to the results of numerical simulations of the learning curves, averaged over datasets and random regular graphs. The two regimes predicted by the approximation are clearly visible; the approximation works well inside each regime but less well in the crossover between the two. One striking observation is that the approximation seems to predict the asymptotic large-n behaviour exactly; this is distinct to the Euclidean case, where generally only the power-law of the n-dependence but not its prefactor come out accurately. To see why, we exploit that for large n (where ϵ ≪σ2) the approximation (9) effectively neglects fluctuations in the training input “density” of a randomly drawn set of training inputs [3, 6]. This is justified in the graph case for large ν = n/V , because the number of training inputs each vertex receives, Binomial(n, 1/V ), has negligible relative fluctuations away from its mean ν. In the Euclidean case there is no similar result, because all training inputs are different with probability one even for large n. Fig. 2(right) illustrates that for larger a the difference in the crossover region between the true (numerically simulated) learning curves and our approximation becomes larger. This is because the average number of steps p/a of the random walk kernel then decreases: we get closer to the limit of uncorrelated function values (a →∞, Cij = δij). In that limit and for low σ2 and large V the 5 0.1 1 10 ν = n / V 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 ε σ 2 = 0.1 σ 2 = 0.01 σ 2 = 0.001 σ 2 = 0.0001 σ 2 = 0 V=500 (filled) & 1000 (empty), d=3, a=2, p=10 0.1 1 10 ν = n / V 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 ε σ 2 = 0.1 σ 2 = 0.01 σ 2 = 0.001 σ 2 = 0.0001 σ 2 = 0 V=500, d=3, a=4, p=10 Figure 2: (Left) Learning curves for GP regression on random regular graphs with degree d = 3 and V = 500 (small filled circles) and V = 1000 (empty circles) vertices. Plotting generalization error versus ν = n/V superimposes the results for both values of V , as expected from the approximation (12). The lines are the quantitative predictions of this approximation. Noise level as shown, kernel parameters a = 2, p = 10. (Right) As on the left but with V = 500 only and for larger a = 4. 0.1 1 10 ν = n / V 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 ε σ 2 = 0.1 σ 2 = 0.01 σ 2 = 0.001 σ 2 = 0.0001 σ 2 = 0 V=500, d=3, a=2, p=20 1 10 100 1000 10000 n 10 -4 10 -3 10 -2 10 -1 10 0 ε simulation 1/(1+n/σ 2) theory (tree) theory (eigenv.) V=500, d=3, a=2, p=200, σ 2=0.1 Figure 3: (Left) Learning curves for GP regression on random regular graphs with degree d = 3 and V = 500, and kernel parameters a = 2, p = 20; noise level σ2 as shown. Circles: numerical simulations; lines: approximation (12). (Right) As on the left but for much larger p = 200 and for a single random graph, with σ2 = 0.1. Dotted line: naive estimate ϵ = 1/(1 + n/σ2). Dashed line: approximation (10) using the tree spectrum and the large p-limit, see (17). Solid line: (10) with numerically determined graph eigenvalues λL α as input. true learning curve is ϵ = exp(−ν), reflecting the probability of a training input set not containing a particular vertex, while the approximation can be shown to predict ϵ = max{1 −ν, 0}, i.e. a decay of the error to zero at ν = 1. Plotting these two curves (not displayed here) indeed shows the same “shape” of disagreement as in Fig. 2(right), with the approximation underestimating the true generalization error. Increasing p has the effect of making the kernel longer ranged, giving an effect opposite to that of increasing a. In line with this, larger values of p improve the accuracy of the approximation (12): see Fig. 3(left). One may ask about the shape of the learning curves for large number of training examples (per vertex) ν. The roughly straight lines on the right of the log-log plots discussed so far suggest that ϵ ∝1/ν in this regime. This is correct in the mathematical limit ν →∞because the graph kernel has a nonzero minimal eigenvalue λ−= κV −1(1−λL +/a)p: for ν ≫σ2/(V λ−), the square bracket 6 in (12) can then be approximated by ν/(ϵ+σ2) and one gets (because also ϵ ≪σ2 in the asymptotic regime) ϵ ≈σ2/ν. However, once p becomes reasonably large, V λ−can be shown – by analysing the scaling of κ, see Appendix – to be extremely (exponentially in p) small; for the parameter values in Fig. 3(left) it is around 4 × 10−30. The “terminal” asymptotic regime ϵ ≈σ2/ν is then essentially unreachable. A more detailed analysis of (12) for large p and large (but not exponentially large) ν, as sketched in the Appendix, yields ϵ ∝(cσ2/ν) ln3/2(ν/(cσ2)), c ∝p−3/2 (13) This shows that there are logarithmic corrections to the naive σ2/ν scaling that would apply in the true terminal regime. More intriguing is the scaling of the coefficient c with p, which implies that to reach a specified (low) generalization error one needs a number of training examples per vertex of order ν ∝cσ2 ∝p−3/2σ2. Even though the covariance kernel Cℓ,p – in the same tree approximation that also went into (12) – approaches a limiting form for large p as discussed in Sec. 2, generalization performance thus continues to improve with increasing p. The explanation for this must presumably be that Cℓ,p converges to the limit (7) only at fixed ℓ, while in the tail ℓ∝p, it continues to change. For finite graph sizes V we know of course that loops will eventually become important as p increases, around the crossover point estimated in (8). The approximation for the learning curve in (12) should then break down. The most naive estimate beyond this point would be to say that the kernel becomes nearly fully correlated, Cij ∝(didj)1/2 which in the regular case simplifies to Cij = 1. With only one function value to learn, and correspondingly only one nonzero kernel eigenvalue λα=1 = 1, one would predict ϵ = 1/(1 + n/σ2). Fig. 3(right) shows, however, that this significantly underestimates the actual generalization error, even though for this graph λα=1 = 0.994 is very close to unity so that the other eigenvalues sum to no more than 0.006. An almost perfect prediction is obtained, on the other hand, from the approximation (10) with the numerically calculated values of the Laplacian – and hence kernel – eigenvalues. The presence of the small kernel eigenvalues is again seen to cause logarithmic corrections to the naive ϵ ∝1/n scaling. Using the tree spectrum as an approximation and exploiting the large-p limit, one finds indeed (see Appendix, Eq. (17)) that ϵ ∝(c′σ2/n) ln3/2(n/c′σ2) where now n enters rather than ν = n/V , c′ being a constant dependent only on p and a: informally, the function to be learned only has a finite (rather than ∝V ) number of degrees of freedom. The approximation (17) in fact provides a qualitatively accurate description of the data Fig. 3(right), as the dashed line in the figure shows. We thus have the somewhat unusual situation that the tree spectrum is enough to give a good description of the learning curves even when loops are important, while (see Sec. 2) this is not so as far as the evaluation of the covariance kernel itself is concerned. 4 Summary and Outlook We have studied theoretically the generalization performance of GP regression on graphs, focussing on the paradigmatic case of random regular graphs where every vertex has the same degree d. Our initial concern was with the behaviour of p-step random walk kernels on such graphs. If these are calculated within the usual approximation of a locally tree-like structure, then they converge to a non-trivial limiting form (7) when p – or the corresponding lengthscale σ in the closely related diffusion kernel – becomes large. The limit of full correlation between all function values on the graph is only reached because of the presence of loops, and we have estimated in (8) the values of p around which the crossover to this loop-dominated regime occurs; numerical data for correlations of function values on neighbouring vertices support this result. In the second part of the paper we concentrated on the learning curves themselves. We assumed that inference is performed with the correct parameters describing the data generating process; the generalization error is then just the Bayes error. The approximation (12) gives a good qualitative description of the learning curve using only the known spectrum of a large regular tree as input. It predicts in particular that the key parameter that determines the generalization error is ν = n/V , the number of training examples per vertex. We demonstrated also that the approximation is in fact more useful than in the Euclidean case because it gives exact asymptotics for the limit ν ≫1. Quantitatively, we found that the learning curves decay as ϵ ∝σ2/ν with non-trivial logarithmic correction terms. Slower power laws ∝ν−α with α < 1, as in the Euclidean case, do not appear. 7 We attribute this to the fact that on a graph there is no analogue of the local roughness of a target function because there is a minimum distance (one step along the graph) between different input points. Finally we looked at the learning curves for larger p, where loops become important. These can still be predicted quite accurately by using the tree eigenvalue spectrum as an approximation, if one keeps track of the zero graph Laplacian eigenvalue which we were able to ignore previously; the approximation shows that the generalization error scales as σ2/n with again logarithmic corrections. In future work we plan to extend our analysis to graphs that are not regular, including ones from application domains as well as artificial ones with power-law tails in the distribution of degrees d, where qualitatively new effects are to be expected. It would also be desirable to improve the predictions for the learning curve in the crossover region ϵ ≈σ2, which should be achievable using iterative approaches based on belief propagation that have already been shown to give accurate approximations for graph eigenvalue spectra [18]. These tools could then be further extended to study e.g. the effects of model mismatch in GP regression on random graphs, and how these are mitigated by tuning appropriate hyperparameters. Appendix We sketch here how to derive (13) from (12) for large p. Eq. (12) writes ϵ = g(νV/(ϵ + σ2)) with g(h) = Z λL + λL − dλL ρ(λL)[κ−1(1 −λL/a)−p + hV −1]−1 (14) and κ determined from the condition g(0) = 1. (This g(h) is the tree spectrum approximation to the g(h) of (10).) Turning first to g(0), the factor (1 −λL/a)p decays quickly to zero as λL increases above λL −. One can then approximate this factor according to (1 −λL −/a)p[(a −λL)/(a −λL −)]p ≈ (1 −λL −/a)p exp[−(λL −λL −)p/(a −λL −)]. In the regime near λL −one can also approximate the spectral density (11) by its leading square-root increase, ρ(λL) = r(λL −λL −)1/2, with r = (d − 1)1/4d5/2/[π(d −2)2]. Switching then to a new integration variable y = (λL −λL −)p/(a −λL −) and extending the integration limit to ∞gives 1 = g(0) = κr(1 −λL −/a)p[p/(a −λL −)]−3/2 Z ∞ 0 dy√y e−y (15) and this fixes κ. Proceeding similarly for h > 0 gives g(h) = κr(1−λL −/a)p[p/(a−λL −)]−3/2F(hκV −1(1−λL −/a)p), F(z) = Z ∞ 0 dy√y (ey+z)−1 (16) Dividing by g(0) = 1 shows that simply g(h) = F(hV −1c−1)/F(0), where c = 1/[κ(1 − λL −/a)p] = rF(0)[p/(a −λL −)]−3/2 which scales as p−3/2. In the asymptotic regime ϵ ≪σ2 we then have ϵ = g(νV/σ2) = F(ν/(cσ2))/F(0) and the desired result (13) follows from the large-z behaviour of F(z) ≈z−1 ln3/2(z). One can proceed similarly for the regime where loops become important. Clearly the zero Laplacian eigenvalue with weight 1/V then has to be taken into account. If we assume that the remainder of the Laplacian spectrum can still be approximated by that of a tree [18], we get g(h) = (V + hκ)−1 + r(1 −λL −/a)p[p/(a −λL −)]−3/2F(hκV −1(1 −λL −/a)p) V −1 + r(1 −λL −/a)p[p/(a −λL −)]−3/2F(0) (17) The denominator here is κ−1 and the two terms are proportional respectively to the covariance kernel eigenvalue λ1, corresponding to λL 1 = 0 and the constant eigenfunction, and to 1−λ1. Dropping the first terms in the numerator and denominator of (17) by taking V →∞leads back to the previous analysis as it should. For a situation as in Fig. 3(right), on the other hand, where λ1 is close to unity, we have κ ≈V and so g(h) ≈(1 + h)−1 + rV (1 −λL −/a)p[p/(a −λL −)]−3/2F(h(1 −λL −/a)p) (18) The second term, coming from the small kernel eigenvalues, is the more slowly decaying because it corresponds to fine detail of the target function that needs many training examples to learn accurately. It will therefore dominate the asymptotic behaviour of the learning curve: ϵ = g(n/σ2) ∝ F(n/(c′σ2)) with c′ = (1 −λL −/a)−p independent of V . The large-n tail of the learning curve in Fig. 3(right) is consistent with this form. 8 References [1] C E Rasmussen and C K I Williams. Gaussian processes for regression. In D S Touretzky, M C Mozer, and M E Hasselmo, editors, Advances in Neural Information Processing Systems 8, pages 514–520, Cambridge, MA, 1996. MIT Press. [2] M Opper. Regression with Gaussian processes: Average case performance. In I K Kwok-Yee, M Wong, I King, and Dit-Yun Yeung, editors, Theoretical Aspects of Neural Computation: A Multidisciplinary Perspective, pages 17–23. Springer, 1997. [3] P Sollich. Learning curves for Gaussian processes. In M S Kearns, S A Solla, and D A Cohn, editors, Advances in Neural Information Processing Systems 11, pages 344–350, Cambridge, MA, 1999. MIT Press. [4] M Opper and F Vivarelli. General bounds on Bayes errors for regression with Gaussian processes. In M Kearns, S A Solla, and D Cohn, editors, Advances in Neural Information Processing Systems 11, pages 302–308, Cambridge, MA, 1999. MIT Press. [5] C K I Williams and F Vivarelli. Upper and lower bounds on the learning curve for Gaussian processes. Mach. Learn., 40(1):77–102, 2000. [6] D Malzahn and M Opper. Learning curves for Gaussian processes regression: A framework for good approximations. In T K Leen, T G Dietterich, and V Tresp, editors, Advances in Neural Information Processing Systems 13, pages 273–279, Cambridge, MA, 2001. MIT Press. [7] D Malzahn and M Opper. A variational approach to learning curves. In T G Dietterich, S Becker, and Z Ghahramani, editors, Advances in Neural Information Processing Systems 14, pages 463–469, Cambridge, MA, 2002. MIT Press. [8] P Sollich and A Halees. Learning curves for Gaussian process regression: approximations and bounds. Neural Comput., 14(6):1393–1428, 2002. [9] P Sollich. Gaussian process regression with mismatched models. In T G Dietterich, S Becker, and Z Ghahramani, editors, Advances in Neural Information Processing Systems 14, pages 519–526, Cambridge, MA, 2002. MIT Press. [10] P Sollich. Can Gaussian process regression be made robust against model mismatch? In Deterministic and Statistical Methods in Machine Learning, volume 3635 of Lecture Notes in Artificial Intelligence, pages 199–210. 2005. [11] M Herbster, M Pontil, and L Wainer. Online learning over graphs. In ICML ’05: Proceedings of the 22nd international conference on Machine learning, pages 305–312, New York, NY, USA, 2005. ACM. [12] A J Smola and R Kondor. Kernels and regularization on graphs. In M Warmuth and B Sch¨olkopf, editors, Proc. Conference on Learning Theory (COLT), Lect. Notes Comp. Sci., pages 144–158. Springer, Heidelberg, 2003. [13] R I Kondor and J D Lafferty. Diffusion kernels on graphs and other discrete input spaces. In ICML ’02: Proceedings of the Nineteenth International Conference on Machine Learning, pages 315–322, San Francisco, CA, USA, 2002. Morgan Kaufmann. [14] F R K Chung. Spectral graph theory. Number 92 in Regional Conference Series in Mathematics. Americal Mathematical Society, 1997. [15] A Steger and N C Wormald. Generating random regular graphs quickly. Combinator. Probab. Comput., 8(4):377–396, 1999. [16] F Chung and S-T Yau. Coverings, heat kernels and spanning trees. The Electronic Journal of Combinatorics, 6(1):R12, 1999. [17] C Monthus and C Texier. Random walk on the Bethe lattice and hyperbolic brownian motion. J. Phys. A, 29(10):2399–2409, 1996. [18] T Rogers, I Perez Castillo, R Kuehn, and K Takeda. Cavity approach to the spectral density of sparse symmetric random matrices. Phys. Rev. E, 78(3):031116, 2008. 9
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Fast subtree kernels on graphs Nino Shervashidze, Karsten M. Borgwardt Interdepartmental Bioinformatics Group Max Planck Institutes T¨ubingen, Germany {nino.shervashidze,karsten.borgwardt}@tuebingen.mpg.de Abstract In this article, we propose fast subtree kernels on graphs. On graphs with n nodes and m edges and maximum degree d, these kernels comparing subtrees of height h can be computed in O(mh), whereas the classic subtree kernel by Ramon & G¨artner scales as O(n24dh). Key to this efficiency is the observation that the Weisfeiler-Lehman test of isomorphism from graph theory elegantly computes a subtree kernel as a byproduct. Our fast subtree kernels can deal with labeled graphs, scale up easily to large graphs and outperform state-of-the-art graph kernels on several classification benchmark datasets in terms of accuracy and runtime. 1 Introduction Graph kernels have recently evolved into a branch of kernel machines that reaches deep into graph mining. Several different graph kernels have been defined in machine learning which can be categorized into three classes: graph kernels based on walks [5, 7] and paths [2], graph kernels based on limited-size subgraphs [6, 11], and graph kernels based on subtree patterns [9, 10]. While fast computation techniques have been developed for graph kernels based on walks [12] and on limited-size subgraphs [11], it is unclear how to compute subtree kernels efficiently. As a consequence, they have been applied to relatively small graphs representing chemical compounds [9] or handwritten digits [1], with approximately twenty nodes on average. But could one speed up subtree kernels to make them usable on graphs with hundreds of nodes, as they arise in protein structure models or in program flow graphs? It is a general limitation of graph kernels that they scale poorly to large, labeled graphs with more than 100 nodes. While the efficient kernel computation strategies from [11, 12] are able to compare unlabeled graphs efficiently, the efficient comparison of large, labeled graphs remains an unsolved challenge. Could one speed up subtree kernels to make them the kernel of choice for comparing large, labeled graphs? The goal of this article is to address both of the aforementioned questions, that is, to develop a fast subtree kernel that scales up to large, labeled graphs. The remainder of this article is structured as follows. In Section 2, we review the subtree kernel from the literature and its runtime complexity. In Section 3, we describe an alternative subtree kernel and its efficient computation based on the Weisfeiler-Lehman test of isomorphism. In Section 4, we compare these two subtree kernels to each other, as well as to a set of four other state-of-the-art graph kernels and report results on kernel computation runtime and classification accuracy on graph benchmark datasets. 1 2 The Ramon-G¨artner subtree kernel Terminology We define a graph G as a triplet (V, E, L), where V is the set of vertices, E the set of undirected edges, and L : V →Σ a function that assigns labels from an alphabet Σ to nodes in the graph1. The neighbourhood N(v) of a node v is the set of nodes to which v is connected by an edge, that is N(v) = {v′|(v, v′) ∈E}. For simplicity, we assume that every graph has n nodes, m edges, a maximum degree of d, and that there are N graphs in our given set of graphs. A walk is a sequence of nodes in a graph, in which consecutive nodes are connected by an edge. A path is a walk that consists of distinct nodes only. A (rooted) subtree is a subgraph of a graph, which has no cycles, but a designated root node. A subtree of G can thus be seen as a connected subset of distinct nodes of G with an underlying tree structure. The height of a subtree is the maximum distance between the root and any other node in the graph plus one. The notion of walk is extending the notion of path by allowing nodes to be equal. Similarly, the notion of subtrees can be extended to subtree patterns (also called ‘tree-walks’ [1]), which can have nodes that are equal. These repetitions of the same node are then treated as distinct nodes, such that the pattern is still a cycle-free tree. Note that all subtree kernels compare subtree patterns in two graphs, not (strict) subtrees. Let S(G) refer to the set of all subtree patterns in graph G. Definition The first subtree kernel on graphs was defined by [10]. It compares all pairs of nodes from graphs G = (V, E, L) and G′ = (V ′, E′, L′) by iteratively comparing their neighbourhoods: k(h) Ramon(G, G′) = X v∈V X v′∈V ′ kh(v, v′), (1) where kh(v, v′) = δ(L(v), L′(v′)), if h = 1 λrλs P R∈M(v,v′) Q (w,w′)∈R kh−1(w, w′), if h > 1 (2) and M(v, v′) = {R ⊆N(v) × N(v′)|(∀(u, u′), (w, w′) ∈R : u = w ⇔u′ = w′) ∧(∀(u, u′) ∈R : L(u) = L′(u′))}. (3) Intuitively, kRamon iteratively compares all matchings M(v, v′) between neighbours of two nodes v from G and v′ from G′. Complexity The runtime complexity of the subtree kernel for a pair of graphs is O(n2h4d), including a comparison of all pairs of nodes (n2), and a pairwise comparison of all matchings in their neighbourhoods in O(4d), which is repeated in h iterations. h is a multiplicative factor, not an exponent, as one can implement the subtree kernel recursively, starting with k1 and recursively computing kh from kh−1. For a dataset of N graphs, the resulting runtime complexity is then obviously in O(N 2n2h4d). Related work The subtree kernels in [9] and [1] refine the above definition for applications in chemoinformatics and hand-written digit recognition. Mah´e and Vert [9] define extensions of the classic subtree kernel that avoid tottering [8] and consider unbalanced subtrees. Both [9] and [1] propose to consider α-ary subtrees with at most α children per node. This restricts the set of matchings to matchings of up to α nodes, but the runtime complexity is still exponential in this parameter α, which both papers describe as feasible on small graphs (with approximately 20 nodes) with many distinct node labels. We present a subtree kernel that is efficient to compute on graphs with hundreds and thousands of nodes next. 1The extension of this definition and our results to graphs with edge labels is straightforward, but omitted for clarity of presentation. 2 3 Fast subtree kernels 3.1 The Weisfeiler-Lehman test of isomorphism Our algorithm for computing a fast subtree kernel builds upon the Weisfeiler-Lehman test of isomorphism [14], more specifically its 1-dimensional variant, also known as “naive vertex refinement”, which we describe in the following. Assume we are given two graphs G and G′ and we would like to test whether they are isomorphic. The 1-dimensional Weisfeiler-Lehman test proceeds in iterations, which we index by h and which comprise the following steps: Algorithm 1 One iteration of the 1-dimensional Weisfeiler-Lehman test of graph isomorphism 1: Multiset-label determination • For h = 1, set Mh(v) := l0(v) = L(v) for labeled graphs, and Mh(v) := l0(v) = | N(v)| for unlabeled graphs. • For h > 1, assign a multiset-label Mh(v) to each node v in G and G′ which consists of the multiset {lh−1(u)|u ∈N(v)}. 2: Sorting each multiset • Sort elements in Mh(v) in ascending order and concatenate them into a string sh(v). • Add lh−1(v) as a prefix to sh(v). 3: Sorting the set of multisets • Sort all of the strings sh(v) for all v from G and G′ in ascending order. 4: Label compression • Map each string sh(v) to a new compressed label, using a function f : Σ∗→Σ such that f(sh(v)) = f(sh(w)) if and only if sh(v) = sh(w). 5: Relabeling • Set lh(v) := f(sh(v)) for all nodes in G and G′. The sorting step 3 allows for a straightforward definition and implementation of f for the compression step 4: one keeps a counter variable for f that records the number of distinct strings that f has compressed before. f assigns the current value of this counter to a string if an identical string has been compressed before, but when one encounters a new string, one increments the counter by one and f assigns its value to the new string. The sorted order from step 3 guarantees that all identical strings are mapped to the same number, because they occur in a consecutive block. The Weisfeiler-Lehman algorithm terminates after step 5 of iteration h if {lh(v)|v ∈V } ̸= {lh(v′)|v′ ∈V ′}, that is, if the sets of newly created labels are not identical in G and G′. The graphs are then not isomorphic. If the sets are identical after n iterations, the algorithm stops without giving an answer. Complexity The runtime complexity of Weisfeiler-Lehman algorithm with h iterations is O(hm). Defining the multisets in step 1 for all nodes is an O(m) operation. Sorting each multiset is an O(m) operation for all nodes. This efficiency can be achieved by using Counting Sort, which is an instance of Bucket Sort, due to the limited range that the elements of the multiset are from. The elements of each multiset are a subset of {f(sh(v))|v ∈V }. For a fixed h, the cardinality of this set is upper-bounded by n, which means that we can sort all multisets in O(m) by the following procedure: We assign the elements of all multisets to their corresponding buckets, recording which multiset they came from. By reading through all buckets in ascending order, we can then extract the sorted multisets for all nodes in a graph. The runtime is O(m) as there are O(m) elements in the multisets of a graph in iteration h. Sorting the resulting strings is of time complexity O(m) via the Radix Sort. The label compression requires one pass over all strings and their characters, that is O(m). Hence all these steps result in a total runtime of O(hm) for h iterations. 3.2 The Weisfeiler-Lehman kernel on pairs of graphs Based on the Weisfeiler-Lehman algorithm, we define the following kernel function. 3 Definition 1 The Weisfeiler-Lehman kernel on two graphs G and G′ is defined as: k(h) W L(G, G′) = |{(si(v), si(v′))|f(si(v)) = f(si(v′)), i ∈{1, . . . , h}, v ∈V, v′ ∈V ′}|, (4) where f is injective and the sets {f(si(v))|v ∈V ∪V ′} and {f(sj(v))|v ∈V ∪V ′} are disjoint for all i ̸= j. That is, the Weisfeiler-Lehman kernel counts common multiset strings in two graphs. Theorem 2 The Weisfeiler-Lehman kernel is positive definite. Proof Intuitively, k(h) W L is a kernel because it counts matching subtree patterns of up to height h in two graphs. More formally, let us define a mapping φ that counts the occurrences of a particular label sequence s in G (generated in h iterations of Weisfeiler-Lehman). Let φ(h) s (G) denote the number of occurrences of s in G, and analogously φ(h) s (G′) for G′. Then k(h) s (G, G′) = φ(h) s (G)φ(h) s (G′) = = |{(si(v), si(v′))|si(v) = si(v′), i ∈{1, . . . , h}, v ∈V, v′ ∈V ′}|, (5) and if we sum over all s from Σ∗, we obtain k(h) W L(G, G′) = X s∈Σ∗ k(h) s (G, G′) = X s∈Σ∗ φ(h) s (G)φ(h) s (G′) = = |{(si(v), si(v′))|si(v) = si(v′), i ∈{1, . . . , h}, v ∈V, v′ ∈V ′}| = = |{(si(v), si(v′))|f(si(v)) = f(si(v′)), i ∈{1, . . . , h}, v ∈V, v′ ∈V ′}|, (6) where the last equality follows from the fact that f is injective. As f(s) ̸= s and hence each string s corresponds to exactly one subtree pattern t, k(h) W L defines a kernel with corresponding feature map φ(h) W L, such that φ(h) W L(G) = (φ(h) s (G))s∈Σ∗= (φ(h) t (G))t∈S(G). (7) Theorem 3 The Weisfeiler-Lehman kernel on a pair of graphs G and G′ can be computed in O(hm). Proof This follows directly from the definition of the Weisfeiler-Lehman kernel and the runtime complexity of the Weisfeiler-Lehman test, as described in Section 3.1. The number of matching multiset strings can be counted as part of step 3, as they occur consecutively in the sorted order. 3.3 The Weisfeiler-Lehman kernel on N graphs For computing the Weisfeiler-Lehman kernel on N graphs we propose the following algorithm which improves over the naive, N 2-fold application of the kernel from (4). We now process all N graphs simultaneously and conduct the steps given in the Algorithm 2 in each of h iterations on each graph G. The hash function g can be implemented efficiently: it again keeps a counter variable x which counts the number of distinct strings that g has mapped to compressed labels so far. If g is applied to a string that is different from all previous ones, then the string is mapped to x + 1, and x increments. As before, g is required to keep sets of compressed labels from different iterations disjoint. Theorem 4 For N graphs, the Weisfeiler-Lehman kernel on all pairs of these graphs can be computed in O(Nhm + N 2hn). Proof Naive application of the kernel from definition (4) for computing an N × N kernel matrix would require a runtime of O(N 2hm). One can improve upon this runtime complexity by computing φ(h) W L explicitly. This can be achieved by replacing the compression mapping f in the classic Weisfeiler-Lehman algorithm by a hash function g that is applied to all N graphs simultaneously. 4 Algorithm 2 One iteration of the Weisfeiler-Lehman kernel on N graphs 1: Multiset-label determination • Assign a multiset-label Mh(v) to each node v in G which consists of the multiset {lh−1(u)|u ∈N(v)}. 2: Sorting each multiset • Sort elements in Mh(v) in ascending order and concatenate them into a string sh(v). • Add lh−1(v) as a prefix to sh(v). 3: Label compression • Map each string sh(v) to a compressed label using a hash function g : Σ∗→Σ such that g(sh(v)) = g(sh(w)) if and only if sh(v) = sh(w). 4: Relabeling • Set lh(v) := g(sh(v)) for all nodes in G. This has the following effects on the runtime of Weisfeiler-Lehman: Step 1, the multiset-label determination, still requires O(Nm). Step 2, the sorting of the elements in each multiset can be done via a joint Bucket Sort (Counting Sort) of all strings, requiring O(Nn + Nm) time. The use of the hash function g renders the sorting of all strings unnecessary (Step 3 from Section 3.1), as identical strings will be mapped to the same (compressed) label anyway. Step 4 and Step 5 remain unchanged. The effort of computing φ(h) W L on all N graphs in h iterations is then O(Nhm), assuming that m > n. To get all pairwise kernel values we have to multiply all feature vectors, which requires a runtime of O(N 2hn), as each graph G has at most hn non-zero entries in φ(h) W L(G). 3.4 Link to the Ramon-G¨artner kernel The Weisfeiler-Lehman kernel can be defined in a recursive fashion which elucidates its relation to the Ramon-G¨artner kernel. Theorem 5 The kernel k(h) recursive defined as k(h) recursive(G, G′) = h X i=1 X v∈V X v′∈V ′ ki(v, v′), (8) where ki(v, v′) = δ(L(v), L′(v′)), if i = 1 ki−1(v, v′) maxR∈M(v,v′) Q (w,w′)∈R ki−1(w, w′), if i > 1 and M ̸= ∅ 0, if i > 1 and M = ∅ (9) and M(v, v′) = {R ⊆N(v) × N(v′)|(∀(u, u′), (w, w′) ∈R : u = w ⇔u′ = w′) ∧(∀(u, u′) ∈R : L(u) = L′(u′) ∧|R| = | N(v)| = | N(v′)|)} (10) is equivalent to the Weisfeiler-Lehman kernel k(h) W L. Proof We prove this theorem by induction over h. Induction initialisation: h = 1: k(1) W L = |{(s1(v), s1(v′))|f(s1(v)) = f(s1(v′)), v ∈V, v′ ∈V ′}| = (11) = X v∈V X v′∈V ′ δ(L(v), L′(v′)) = k(1) recursive. (12) The equality follows from the definition of M(v, v′). Induction step h →h + 1: Assume that k(h) W L = k(h) recursive. Then k(h+1) recursive = X v∈V X v′∈V ′ kh+1(v, v′) + h X i=1 X v∈V X v′∈V ′ ki(v, v′) = (13) = |{(sh+1(v), sh+1(v′))|f(sh+1(v)) = f(sh+1(v′)), v ∈V, v′ ∈V ′}| + k(h) W L = k(h+1) W L , (14) 5 10 1 10 2 10 3 10 !1 10 0 10 1 10 2 10 3 10 4 10 5 Number of graphs N Runtime in seconds 0 200 400 600 800 1000 0 50 100 150 200 250 300 350 400 Graph size n Runtime in seconds 2 3 4 5 6 7 8 0 5 10 15 20 Subtree height h Runtime in seconds 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 5 10 15 20 Graph density c Runtime in seconds pairwise global Figure 1: Runtime in seconds for kernel matrix computation on synthetic graphs using the pairwise (red, dashed) and the global (green) Weisfeiler-Lehman kernel (Default values: dataset size N = 10, graph size n = 100, subtree height h = 5, graph density c = 0.4). where the equality of (13) and (14) follows from the fact that kh+1(v, v′) = 1 if and only if the neigborhoods of v and v′ are identical, that is if f(sh+1(v)) = f(sh+1(v′)). Theorem 5 highlights the following differences between the Weisfeiler-Lehman and the RamonG¨artner kernel: In (8), Weisfeiler-Lehman considers all subtrees up to height h and the RamonG¨artner kernel the subtrees of exactly height h. In (9) and (10), the Weisfeiler-Lehman kernel checks whether the neighbourhoods of v and v′ match exactly, whereas the Ramon-G¨artner kernel considers all pairs of matching subsets of the neighbourhoods of v and v′ in (3). In our experiments, we next examine the empirical differences between these two kernels in terms of runtime and prediction accuracy on classification benchmark datasets. 4 Experiments 4.1 Runtime behaviour of Weisfeiler-Lehman kernel Methods We empirically compared the runtime behaviour of our two variants of the WeisfeilerLehman (WL) kernel. The first variant computes kernel values pairwise in O(N 2hm). The second variant computes the kernel values in O(Nhm + N 2hn) on the dataset simultaneously. We will refer to the former variant as the ‘pairwise’ WL, and the latter as ‘global’ WL. Experimental setup We assessed the behaviour on randomly generated graphs with respect to four parameters: dataset size N, graph size n, subtree height h and graph density c. The density of an undirected graph of n nodes without self-loops is defined as the number of its edges divided by n(n −1)/2, the maximal number of edges. We kept 3 out of 4 parameters fixed at their default values and varied the fourth parameter. The default values we used were 10 for N, 100 for n, 5 for h and 0.4 for the graph density c. In more detail, we varied N and n in range {10, 100, 1000}, h in {2, 4, 8} and c in {0.1, 0.2, . . . , 0.9}. For each individual experiment, we generated N graphs with n nodes, and inserted edges randomly until the number of edges reached ⌊cn(n −1)/2⌋. We then computed the pairwise and the global 6 WL kernel on these synthetic graphs. We report CPU runtimes in seconds in Figure 1, as measured in Matlab R2008a on an Apple MacPro with 3.0GHz Intel 8-Core with 16GB RAM. Results Empirically, we observe that the pairwise kernel scales quadratically with dataset size N. Interestingly, the global kernel scales linearly with N. The N 2 sparse vector multiplications that have to be performed for kernel computation with global WL do not dominate runtime here. This result on synthetic data indicates that the global WL kernel has attractive scalability properties for large datasets. When varying the number of nodes n per graph, we observe that the runtime of global WL scales linearly with n, and is much faster than the pairwise WL for large graphs. We observe the same picture for the height h of the subtree patterns. The runtime of both kernels grows linearly with h, but the global WL is more efficient in terms of runtime in seconds. Varying the graph density c, both methods show again a linearly increasing runtime, although the runtime of the global WL kernel is close to constant. The density c seems to be a graph property that affects the runtime of the pairwise kernel more severely than that of global WL. Across all different graph properties, the global WL kernel from Section 3.3 requires less runtime than the pairwise WL kernel from Section 3.2. Hence the global WL kernel is the variant of our Weisfeiler-Lehman kernel that we use in the following graph classification tasks. 4.2 Graph classification Datasets We employed the following datasets in our experiments: MUTAG, NCI1, NCI109, and D&D. MUTAG [3] is a dataset of 188 mutagenic aromatic and heteroaromatic nitro compounds labeled according to whether or not they have a mutagenic effect on the Gram-negative bacterium Salmonella typhimurium. We also conducted experiments on two balanced subsets of NCI1 and NCI109, which classify compounds based on whether or not they are active in an anti-cancer screen ([13] and http://pubchem.ncbi.nlm.nih.gov). D&D is a dataset of 1178 protein structures [4]. Each protein is represented by a graph, in which the nodes are amino acids and two nodes are connected by an edge if they are less than 6 Angstroms apart. The prediction task is to classify the protein structures into enzymes and non-enzymes. Experimental setup On these datasets, we compared our Weisfeiler-Lehman kernel to the RamonG¨artner kernel (λr = λs = 1), as well as to several state-of-the-art graph kernels for large graphs: the fast geometric random walk kernel from [12] that counts common labeled walks (with λ chosen from the set {10−2, 10−3, . . . , 10−6} by cross-validation on the training set), the graphlet kernel from [11] that counts common induced labeled connected subgraphs of size 3, and the shortest path kernel from [2] that counts pairs of labeled nodes with identical shortest path distance. We performed 10-fold cross-validation of C-Support Vector Machine Classification, using 9 folds for training and 1 for testing. All parameters of the SVM were optimised on the training dataset only. To exclude random effects of fold assignments, we repeated the whole experiment 10 times. We report average prediction accuracies and standard errors in Tables 1 and 2. We choose h for our Weisfeiler-Lehman kernel by cross-validation on the training dataset for h ∈ {1, . . . , 10}, which means that we computed 10 different WL kernel matrices in each experiment. We report the total runtime of this computation (not the average per kernel matrix). Results In terms of runtime the Weisfeiler-Lehman kernel can easily scale up even to graphs with thousands of nodes. On D&D, subtree-patterns of height up to 10 were computed in 11 minutes, while no other comparison method could handle this dataset in less than half an hour. The shortest path kernel is competitive to the WL kernel on smaller graphs (MUTAG, NCI1, NCI109), but on D&D its runtime degenerates to more than 23 hours. The Ramon and G¨artner kernel was computable on MUTAG in approximately 40 minutes, but for the large NCI datasets it only finished computation on a subsample of 100 graphs within two days. On D&D, it did not even finish on a subsample of 100 graphs within two days. The random walk kernel is competitive on MUTAG, but as the RamonG¨artner kernel, does not finish computation on the full NCI datasets and on D&D within two days. The graphlet kernel is faster than our WL kernel on MUTAG and the NCI datasets, and about a 7 Method/Dataset MUTAG NCI1 NCI109 D & D Weisfeiler-Lehman 82.05 (±0.36) 82.19 (± 0.18) 82.46 (±0.24) 79.78 (±0.36) Ramon & G¨artner 85.72 (±0.49) ———Graphlet count 75.61 (±0.49) 66.00 (±0.07) 66.59 (±0.08) 78.59 (±0.12) Random walk 80.72 (±0.38) ———Shortest path 87.28 (±0.55) 73.47 (±0.11) 73.07 (±0.11) 78.45 (±0.26) —-: did not finish in 2 days. Table 1: Prediction accuracy (± standard error) on graph classification benchmark datasets Dataset MUTAG NCI1 NCI109 D & D Maximum # nodes 28 111 111 5748 Average # nodes 17.93 29.87 29.68 284.32 # labels 7 37 54 89 Number of graphs 188 100 4110 100 4127 100 1178 Weisfeiler-Lehman 6” 5” 7’20” 5” 7’21” 58” 11’ Ramon & G¨artner 40’6” 25’9” 29 days∗ 26’40” 31 days∗ ——Graphlet count 3” 2” 1’27” 2” 1’27” 2’40” 30’21” Random walk 12” 58’30” 68 days∗ 2h 9’41” 153 days∗ ——Shortest path 2” 3” 4’38” 3” 4’39” 58’45” 23h 17’2” —-: did not finish in 2 days, * = extrapolated. Table 2: CPU runtime for kernel computation on graph classification benchmark datasets factor of 3 slower on D&D. However, this efficiency comes at a price, as the kernel based on size-3 graphlets turns out to lead to poor accuracy levels on three datasets. Using larger graphlets with 4 or 5 nodes that might have been more expressive led to infeasible runtime requirements in initial experiments (not shown here). On NCI1, NCI109 and D&D, the Weisfeiler-Lehman kernel reached the highest accuracy. On D&D the shortest path and graphlet kernels yielded similarly good results, while on NCI1 and NCI109 the Weisfeiler-Lehman kernel improves by more than 8% the best accuracy attained by other methods. On MUTAG, it reaches the third best accuracy among all methods considered. We could not assess the performance of the Ramon & G¨artner kernel and the random walk kernel on larger datasets, as their computation did not finish in 48 hours. The labeled size-3 graphlet kernel achieves low accuracy levels, except on D&D. To summarize, the WL kernel turns out to be competitive in terms of runtime on all smaller datasets, fastest on the large protein dataset, and its accuracy levels are highest on three out of four datasets. 5 Conclusions We have defined a fast subtree kernel on graphs that combines scalability with the ability to deal with node labels. It is competitive with state-of-the-art kernels on several classification benchmark datasets in terms of accuracy, even reaching the highest accuracy level on three out of four datasets, and outperforms them significantly in terms of runtime on large graphs, even the efficient computation schemes for random walk kernels [12] and graphlet kernels [11] that were recently defined. This new kernel opens the door to applications of graph kernels on large graphs in bioinformatics, for instance, protein function prediction via detailed graph models of protein structure on the amino acid level, or on gene networks for phenotype prediction. An exciting algorithmic question for further studies will be to consider kernels on graphs with continuous or high-dimensional node labels and their efficient computation. Acknowledgements The authors would like to thank Kurt Mehlhorn, Pascal Schweitzer, and Erik Jan van Leeuwen for fruitful discussions. 8 References [1] F. R. Bach. Graph kernels between point clouds. In ICML, pages 25–32, 2008. [2] K. M. Borgwardt and H.-P. Kriegel. Shortest-path kernels on graphs. In Proc. Intl. Conf. Data Mining, pages 74–81, 2005. [3] A. K. Debnath, R. L. Lopez de Compadre, G. Debnath, A. J. Shusterman, and C. Hansch. Structure-activity relationship of mutagenic aromatic and heteroaromatic nitro compounds. correlation with molecular orbital energies and hydrophobicity. J Med Chem, 34:786–797, 1991. [4] P. D. Dobson and A. J. Doig. Distinguishing enzyme structures from non-enzymes without alignments. J Mol Biol, 330(4):771–783, Jul 2003. [5] 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. [6] T. Horvath, T. G¨artner, and S. Wrobel. Cyclic pattern kernels for predictive graph mining. In Proceedings of the International Conference on Knowledge Discovery and Data Mining, 2004. [7] 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. [8] P. Mah´e, N. Ueda, T. Akutsu, J.-L. Perret, and J.-P. Vert. Extensions of marginalized graph kernels. In Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [9] P. Mah´e and J.-P. Vert. Graph kernels based on tree patterns for molecules. q-bio/0609024, September 2006. [10] J. Ramon and T. G¨artner. Expressivity versus efficiency of graph kernels. Technical report, First International Workshop on Mining Graphs, Trees and Sequences (held with ECML/PKDD’03), 2003. [11] N. Shervashidze, S.V.N. Vishwanathan, T. Petri, K. Mehlhorn, and K. M. Borgwardt. Efficient graphlet kernels for large graph comparison. In Artificial Intelligence and Statistics, 2009. [12] S. V. N. Vishwanathan, Karsten Borgwardt, and Nicol N. Schraudolph. Fast computation of graph kernels. In B. Sch¨olkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, Cambridge MA, 2007. MIT Press. [13] N. Wale and G. Karypis. Comparison of descriptor spaces for chemical compound retrieval and classification. In Proc. of ICDM, pages 678–689, Hong Kong, 2006. [14] B. Weisfeiler and A. A. Lehman. A reduction of a graph to a canonical form and an algebra arising during this reduction. Nauchno-Technicheskaya Informatsia, Ser. 2, 9, 1968. 9
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Thresholding Procedures for High Dimensional Variable Selection and Statistical Estimation Shuheng Zhou Seminar f¨ur Statistik ETH Z¨urich CH-8092, Switzerland Abstract Given n noisy samples with p dimensions, where n ≪p, we show that the multistep thresholding procedure can accurately estimate a sparse vector β ∈Rp in a linear model, under the restricted eigenvalue conditions (Bickel-Ritov-Tsybakov 09). Thus our conditions for model selection consistency are considerably weaker than what has been achieved in previous works. More importantly, this method allows very significant values of s, which is the number of non-zero elements in the true parameter. For example, it works for cases where the ordinary Lasso would have failed. Finally, we show that if X obeys a uniform uncertainty principle and if the true parameter is sufficiently sparse, the Gauss-Dantzig selector (Cand`esTao 07) achieves the ℓ2 loss within a logarithmic factor of the ideal mean square error one would achieve with an oracle which would supply perfect information about which coordinates are non-zero and which are above the noise level, while selecting a sufficiently sparse model. 1 Introduction In a typical high dimensional setting, the number of variables p is much larger than the number of observations n. This challenging setting appears in linear regression, signal recovery, covariance selection in graphical modeling, and sparse approximations. In this paper, we consider recovering β ∈Rp in the following linear model: Y = Xβ + ǫ, (1.1) where X is an n × p design matrix, Y is a vector of noisy observations and ǫ is the noise term. We assume throughout this paper that p ≥n (i.e. high-dimensional), ǫ ∼N(0, σ2In), and the columns of X are normalized to have ℓ2 norm √n. Given such a linear model, two key tasks are to identify the relevant set of variables and to estimate β with bounded ℓ2 loss. In particular, recovery of the sparsity pattern S = supp(β) := {j : βj ̸= 0}, also known as variable (model) selection, refers to the task of correctly identifying the support set (or a subset of “significant” coefficients in β) based on the noisy observations. Even in the noiseless case, recovering β (or its support) from (X, Y ) seems impossible when n ≪p. However, a line of recent research shows that it becomes possible when β is also sparse: when it has a relatively small number of nonzero coefficients and when the design matrix X is also sufficiently nice, which we elaborate below. One important stream of research, which we also adopt here, requires computational feasibility for the estimation methods, among which the Lasso and the Dantzig selector are both well studied and shown with provable nice statistical properties; see for example [11, 9, 19, 21, 5, 18, 12, 2]. For a chosen penalization parameter λn ≥0, regularized estimation with the ℓ1-norm penalty, also known 1 as the Lasso [16] or Basis Pursuit [6] refers to the following convex optimization problem bβ = arg min β 1 2n∥Y −Xβ∥2 2 + λn∥β∥1, (1.2) where the scaling factor 1/(2n) is chosen by convenience; The Dantzig selector [5] is defined as, (DS) arg min bβ∈Rp
bβ
1 subject to
1 nXT (Y −X bβ)
∞ ≤λn. (1.3) Our goal in this work is to recover S as accurately as possible: we wish to obtain bβ such that | supp(bβ) \ S| (and sometimes |S△supp(bβ)| also) is small, with high probability, while at the same time ∥bβ−β∥2 2 is bounded within logarithmic factor of the ideal mean square error one would achieve with an oracle which would supply perfect information about which coordinates are non-zero and which are above the noise level (hence achieving the oracle inequality as studied in [7, 5]); We deem the bound on ℓ2-loss as a natural criteria for evaluating a sparse model when it is not exactly S. Let s = |S|. Given T ⊆{1, . . . , p}, let us define XT as the n × |T| submatrix obtained by extracting columns of X indexed by T; similarly, let βT ∈R|T |, be a subvector of β ∈Rp confined to T. Formally, we study a Multi-step Procedure: First we obtain an initial estimator βinit using the Lasso as in (1.2) or the Dantzig selector as in (1.3), with λn = Θ(σ p 2 log p/n). 1. We then threshold the estimator βinit with t0, with the general goal such that, we get a set I1 with cardinality at most 2s; in general, we also have |I1 ∪S| ≤2s, where I1 = {j ∈{1, . . . , p} : βj,init ≥t0} for some t0 to be specified. Set I = I1. 2. We then feed (Y, XI) to either the Lasso estimator as in (1.2) or the ordinary least squares (OLS) estimator to obtain bβ, where we set bβI = (XT I XI)−1XT I Y and bβIc = 0. 3. We then possibly threshold bβI1 with t1 = 4λn p |I1| (to be specified), to obtain I2, repeat step 2 with I = I2 to obtain bβI and set all other coordinates to zero; return bβ. Our algorithm is constructive in that it does not rely on the unknown parameters s, βmin := minj∈S |βj| or those that characterize the incoherence conditions on X; instead, our choice of λn and thresholding parameters only depends on σ, n, and p. In our experiments, we apply only the first two steps, which we refer to as a two-step procedure; In particular, the Gauss-Dantzig selector is a two-step procedure with the Dantzig selector as βinit [5]. In theory, we apply the third step only when βmin is sufficiently large and when we wish to get a “sparser” model I. More definitions. For a matrix A, let Λmin(A) and Λmax(A) denote the smallest and the largest eigenvalues respectively. We refer to a vector υ ∈Rp with at most s non-zero entries, where s ≤p, as a s-sparse vector. Throughout this paper, we assume that n ≥2s and Λmin(2s) △= min υ̸=0;2s−sparse ∥Xυ∥2 2 /(n ∥υ∥2 2) > 0. (1.4) It is clear that n ≥2s is necessary, as any submatrix with more than n columns must be singular. In general, we also assume Λmax(s) △= maxυ̸=0;s−sparse ∥Xυ∥2 2 /(n ∥υ∥2 2) < ∞. As defined in [4], the s-restricted isometry constant δs of X is the smallest quantity such that (1 −δs) ∥υ∥2 2 ≤∥XT υ∥2 2 /n ≤(1 + δs) ∥υ∥2 2 , for all T ⊆{1, . . . , p} with |T| ≤s and coefficients sequences (υj)j∈T . It is clear that δs is non-decreasing in s and 1 −δs ≤Λmin(s) ≤Λmax(s) ≤1 + δs. Hence δ2s < 1 implies (1.4). Occasionally, we use βT ∈R|T |, where T ⊆{1, . . . , p}, to also represent its 0-extended version β′ ∈Rp such that β′ T c = 0 and β′ T = βT ; for example in (1.5) below. Oracle inequalities. The following idea has been explained in [5]; we hence describe it here only briefly. Note that due to different normalization of columns of X, our expressions are slightly 2 different from those in [5]. Consider the least square estimator bβI = (XT I XI)−1XT I Y , where |I| ≤s and consider the ideal least-squares estimator β⋄ β⋄= arg min I⊆{1,...,p}, |I|≤s E
β −bβI
2 2 , (1.5) which minimizes the expected mean squared error. It follows from [5] that for Λmax(s) < ∞, E ∥β −β⋄∥2 2 ≥min (1, 1/Λmax(s)) p X i=1 min(β2 i , σ2/n). (1.6) Now we check if for Λmax(s) < ∞, it holds with high probability that
bβ −β
2 2 = O(log p) p X i=1 min(β2 i , σ2/n), so that (1.7)
bβ −β
2 2 = O(log p) max(1, Λmax(s))E ∥β⋄−β∥2 2 in view of (1.6). (1.8) These bounds are meaningful since p X i=1 min(β2 i , σ2/n) = min I⊆{1,...,p} ∥β −βI∥2 2 + |I|σ2 n represents the ideal squared bias and variance. We elaborate on conditions on the design, under which we accomplish these goals using the multi-step procedures in the rest of this section. We now define a constant λσ,a,p for each a > 0, by which we bound the maximum correlation between the noise and covariates of X, which we only apply to X with column ℓ2 norm bounded by √n; Let Ta := ǫ :
XT ǫ n
∞ ≤λσ,a,p , where λσ,a,p = σ √ 1 + a r 2 log p n , hence (1.9) P (Ta) ≥1 −( p π log ppa)−1, for a ≥0; see [5]. (1.10) Variable selection. Our first result in Theorem 1.1 shows that consistent variable selection is possible under the Restricted Eigenvalue conditions, as formalized in [2]. Similar conditions have been used by [10] and [17]. Assumption 1.1 (Restricted Eigenvalue assumption RE(s, k0, X) [2]) For some integer 1 ≤ s ≤p and a positive number k0, the following holds: 1 K(s, k0, X) △= min J0⊆{1,...,p}, |J0|≤s min υ̸=0, ‚‚‚υJc 0 ‚‚‚ 1≤k0∥υJ0∥1 ∥Xυ∥2 √n ∥υJ0∥2 > 0. (1.11) If RE(s, k0, X) is satisfied with k0 ≥1, then the square submatrices of size ≤2s of XT X are necessarily positive definite (see [2]) and hence (1.4) must hold. We do not impose any extra constraint on s besides what is allowed in order for (1.11) to hold. Note that when s > n/2, it is impossible for the restricted eigenvalue assumption to hold as XI for any I such that |I| = 2s becomes singular in this case. Hence our algorithm is especially relevant if one would like to estimate a parameter β such that s is very close to n; See Section 4 for such examples. Let βmin := minj∈S |βj|. Theorem 1.1 (Variable selection under Assumption 1.1) Suppose that RE(s, k0, X) condition holds, where k0 = 1 for the DS and = 3 for the Lasso. Suppose λn ≥Bλσ,a,p for λσ,a,p as in (1.9), where B ≥1 for the DS and ≥2 for the Lasso. Let B2 = 1 BΛmin(2s). Let s ≥K4(s, k0, X) and βmin ≥4 √ 2 max(K(s, k0, X), 1)λn √s + max 4K2(s, k0, X), √ 2B2 λn √s. Then with probability at least P (Ta), the multi-step procedure returns bβ such that S ⊆I := supp(bβ), where |I \ S| < B2 2 16 and ∥bβ −β∥2 2 ≤ λ2 σ,a,p|I| Λ2 min(|I|) ≤2 log p(1 + a)sσ2(1 + B2 2/16) nΛ2 min(2s) , which satisfies (1.7) and (1.8) given that βmin ≥σ/√n and Pp i=1 min(β2 i , σ2/n) = sσ2/n. 3 Our analysis builds upon the rate of convergence bounds for βinit derived in [2]. The first implication of this work and also one of the motivations for analyzing the thresholding methods is: under Assumption 1.1, one can obtain consistent variable selection for very significant values of s, if only a few extra variables are allowed to be included in the estimator bβ. In our simulations, we recover the exact support set S with very high probability using a two-step procedure. Note that we did not optimize the lower bound on s as we focus on cases when the support of S is large. Thresholding that achieves the oracle inequalities. The natural question upon obtaining Theorem 1.1 is: is there a good thresholding rule that enables us to obtain a sufficiently sparse estimator bβ when some components of βS (and hence βmin) are well below σ/√n, which also satisfies the oracle inequality as in (1.7)? Before we answer this question, we define s0 as the smallest integer such that p X i=1 min(β2 i , λ2σ2) ≤s0λ2σ2, where λ = p 2 log p/n, (1.12) and the (s, s′)-restricted orthogonality constant [4] θs,s′ as the smallest quantity such that | ⟨XT c, XT ′c′ ⟩/n| ≤θs,s′ ∥c∥2 ∥c′∥2 (1.13) holds for all disjoint sets T, T ′ ⊆{1, . . . , p} of cardinality |T| ≤s and |T ′| < s′, where s + s′ ≤ p. Note that θ is non-decreasing in s, s′ and small values of θs,s′ indicates that disjoint subsets covariates in XT and XT ′ span nearly orthogonal subspaces. Theorem 1.2 says that under a uniform uncertainty principle (UUP), thresholding of an initial Dantzig selector βinit, at the level of Θ(σ p 2 log p/n) indeed identifies a sparse model I of cardinality at most 2s0 such that the ℓ2 2-loss for its corresponding least-squares estimator is indeed bounded within O(log p) of the ideal mean square error as in (1.5), when β is as sparse as required by the Dantzig selector to achieve such an oracle inequality [5]. This is accomplished without any knowledge of the significant coordinates of β and not being able to observe parameter values. Assumption 1.2 (A Uniform Uncertainly Principle) [5] For some integer 1 ≤s < n/3, assume δ2s + θs,2s < 1, which implies that λmin(2s) > θs,2s given that 1 −δ2s ≤Λmin(2s). Theorem 1.2 Choose τ, a > 0 and set λn = λp,τσ, where λp,τ := (√1 + a + τ −1) p 2 log p/n, in (1.3). Suppose β is s-sparse with δ2s + θs,2s < 1 −τ. Let threshold t0 be chosen from the range (C1λp,τσ, C4λp,τσ] for some constants C1, C4 to be defined. Then with probability at least 1−(√π log ppa)−1, the Gauss-Dantzig selector bβ selects a model I := supp(bβ) such that |I| ≤2s0, |I \ S| ≤s0 ≤s, and ∥bβ −β∥2 2 ≤2C2 3 log p σ2/n + p X i=1 min(β2 i , σ2/n) ! , (1.14) where C3 depends on a, τ, δ2s, θs,2s and C4; see (3.3). Our analysis builds upon [5]. Note that allowing t0 to be chosen from a range (as wide as one would like, with the cost of increasing the constant C3 in (1.14)), saves us from having to estimate C1, which indeed depends on δ2s and θs,2s. Assumption 1.2 implies that Assumption 1.1 holds for k0 = 1 with K(s, k0, X) = p Λmin(2s)/(Λmin(2s) −θs,2s) ≤ p Λmin(2s)/(1 −δ2s −θs,2s) (see [2]); It is an open question if we can derive the same result under Assumption 1.1. Previous work. Finally, we briefly review related work in multi-step procedures and the role of sparsity for high-dimensional statistical inference. Before this work, hard thresholding idea has been shown in [5] (via Gauss-Dantzig selector) as a method to correct the bias of the initial Dantzig selector. The empirical success of the Gauss-Dantzig selector in terms of improving the statistical accuracy is strongly evident in their experimental results. Our theoretical analysis on the oracle inequalities, which hold for the Gauss-Dantzig selector under a uniform uncertainty principle, is exactly inspired by their theoretical analysis of the initial Dantzig selector under the same conditions. For the Lasso, [12] has also shown in theoretical analysis that thresholding is effective in obtaining 4 a two-step estimator bβ that is consistent in its support with β; however, the choice of threshold level depends on the unknown value βmin (which needs to be sufficiently large) and s, and their theory does not directly yield (or imply) an algorithm for finding such parameters. Further, as pointed out by [2], a weakening of their condition is still sufficient for Assumption 1.1 to hold. The sparse recovery problem under arbitrary noise is also well studied, see [3, 15, 14]. Although as argued in [3, 14], the best accuracy under arbitrary noise has essentially been achieved in both work, their bounds are worse than that in [5] (hence the present paper) under the stochastic noise as discussed in the present paper; see more discussions in [5]. Moreover, greedy algorithms in [15, 14] require s to be part of their input, while the iterative algorithms in the present paper do not have such requirement, and hence adapt to the unknown level of sparsity s well. A more general framework on multi-step variable selection was studied by [20]. They control the probability of false positives at the price of false negatives, similar to what we aim for in the present paper. Unfortunately, their analysis is constrained to the case when s is a constant. Finally, under a restricted eigenvalue condition slightly stronger than Assumption 1.1, [22] requires s = O( p n/ log p) in order to achieve variable selection consistency using the adaptive Lasso [23] as the second step procedure. Organization of the paper. We prove Theorem 1.1 essentially in Section 2. A thresholding framework for the general setting is described in Section 3, which also sketches the proof of Theorem 1.2. Section 4 briefly discusses the relationship between linear sparsity and random design matrices. Section 5 includes simulation results showing that our two-step procedure is consistent with our theoretical analysis on variable selection. 2 Thresholding procedure when βmin is large We use a penalization parameter λn = Bλσ,a,p and assume βmin > Cλn √s for some constants B, C throughout this section; we first specify the thresholding parameters in this case. We then show in Theorem 2.1 that our algorithm works under any conditions so long as the rate of convergence of the initial estimator obeys the bounds in (2.2). Theorem 1.1 is a corollary of Theorem 2.1 under Assumption 1.1, given the rate of convergence bounds for βinit following derivations in [2]. The Iterative Procedure. We obtain an initial estimator βinit using the Lasso or the Dantzig selector. Let bS0 = {j : βj,init > 4λn}, and bβ(0) := βinit; Iterate through the following steps twice, for i = 0, 1: (a) Set ti = 4λn q |bSi|; (b) Threshold bβ(i) with ti to obtain I := bSi+1, where bSi+1 = j ∈bSi : bβ(i) j ≥4λn q |bSi| and compute bβ(i+1) I = (XT I XI)−1XT I Y. (2.1) Return the final set of variables in bS2 and output bβ such that bβbS2 = bβ(2) bS2 and bβj = 0, ∀j ∈bSc 2. Theorem 2.1 Let λn ≥Bλσ,a,p, where B ≥1 is a constant suitably chosen such that the initial estimator βinit satisfies on Ta, for υinit = βinit −β and some constants B0, B1, ∥υinit,S∥2 ≤ B0λn √s and ∥υinit,Sc∥1 ≤B1λns; (2.2) Suppose βmin ≥ max p B1, 2 2 √ 2 + max B0, √ 2B2 λn √s, (2.3) where B2 = 1/(BΛmin(2s)). Then for s ≥B2 1/16, it holds on Ta that |bSi| ≤2s, ∀i = 1, 2, and ∥bβ(i) −β∥2 ≤ λσ,a,p q |bSi|/Λmin(|bSi|) ≤λnB2 √ 2s, ∀i = 1, 2, (2.4) where bβ(i) are the OLS estimators based on I = bSi; Finally, the Iterative Procedure includes the correct set of variables in bS2 such that S ⊆bS2 ⊆bS1 and bS2 \ S := supp(bβ) \ S ≤ 1 16B2Λ2 min(|bS1|) ≤B2 2 16 . (2.5) 5 Remark 2.2 Without the knowledge of σ, one could use bσ ≥σ in λn; this will put a stronger requirement on βmin, but all conclusions of Theorem 2.1 hold. We also note that in order to obtain bS1 such that |bS1| ≤2s and bS1 ⊇S, we only need to threshold βinit at t0 = B1λn (see Section 3 and Lemma 3.2 for an example); instead of having to estimate B1, we use t0 = Θ(λn √s) to threshold. 3 A thresholding framework for the general setting In this section, we wish to derive a meaningful criteria for consistency in variable selection, when βmin is well below the noise level. Suppose that we are given an initial estimator βinit that achieves the rate of convergence bound as in (1.14), which adapts nearly ideally to the uncertainty in the support set S and the “significant” set. We show that although we cannot guarantee the presence of variables indexed by {j : |βj| < σ p 2 log p/n} to be included in the final set I (cf. (3.7)) due to their lack of strength, we wish to include the significant variables from S in I such that the OLS estimator based on I achieves this almost ideal rate of convergence as βinit does, even though some variables from S are missing in I. Here we pay a price for the missing variables in order to obtain a sparse model I. Toward this goal, we analyze the following algorithm under Assumption 1.2. The General Two-step Procedure: Assume δ2s + θs,2s < 1 −τ, where τ > 0; 1. First we obtain an initial estimator βmin using the Dantzig selector with λp,τ := (√1 + a+ τ −1) p 2 log p/n, where τ, a ≥0; we then threshold βinit with t0, chosen from the range (C1λp,τσ, C4λp,τσ], to obtain a set I of cardinality at most 2s, (we prove a stronger result in Lemma 3.2), where I := {j ∈{1, . . . , p} : βj,init > t0} , for C1 as defined in (3.3); (3.1) 2. In the second step, given a set I of cardinality at most 2s, we run the OLS regression to obtain obtained via (3.1), bβI = (XT I XI)−1XT I Y and set bβj = 0, ∀j ̸∈I. Theorem 2 in [5] has shown that the Dantzig selector achieves nearly the ideal level of MSE. Proposition 3.1 [5] Let Y = Xβ + ǫ, for ǫ being i.i.d. N(0, σ2) and ∥Xj∥2 2 = n. Choose τ, a > 0 and set λn = λp,τσ := (√1 + a + τ −1)σ p 2 log p/n in (1.3). Then if β is s-sparse with δ2s + θs,2s < 1−τ, the Dantzig selector obeys with probability at least 1−(√π log ppa)−1,
bβ −β
2 2 ≤ 2C2 2(√1 + a + τ −1)2 log p σ2/n + Pp i=1 min β2 i , σ2/n . From this point on we let δ := δ2s and θ := θs,2s; Analysis in [5] (Theorem 2) and the current paper yields the following constants, where C3 has not been optimized, C2 = 2C′ 0 + 1 + δ 1 −δ −θ where C′ 0 = C0 1 −δ −θ + θ(1 + δ) (1 −δ −θ)2 , (3.2) where C0 = 2 √ 2 1 + 1−δ2 1−δ−θ + (1 + 1/ √ 2) (1+δ)2 1−δ−θ; We now define C1 = C′ 0 + 1 + δ 1 −δ −θ and C2 3 = 3( √ 1 + a + τ −1)2((C′ 0 + C4)2 + 1) + 4(1 + a) Λ2 min(2s0). (3.3) We first set up the notation following that in [5]. We order the βj’s in decreasing order of magnitude |β1| ≥|β2|... ≥|βp|. (3.4) Recall that s0 is the smallest integer such that Pp i=1 min(β2 i , λ2σ2) ≤s0λ2σ2, where λ = p 2 log p/n. Thus by definition of s0, as essentially shown in [5], that 0 ≤s0 ≤s and s0λ2σ2 ≤ λ2σ2 + p X i=1 min(β2 i , λ2σ2) ≤2 log p σ2 n + p X i=1 min β2 i , σ2 n ! (3.5) and s0λ2σ2 ≥ s0+1 X j=1 min(β2 j , λ2σ2) ≥(s0 + 1) min(β2 s0+1, λ2σ2) for s < p, (3.6) 6 which implies that min(β2 s0+1, λ2σ2) < λ2σ2 and hence by (3.4), |βj| < λσ for all j > s0. (3.7) We now show in Lemma 3.2 that thresholding at the level of Cλσ at step 1 selects a set I of at most 2s0 variables, among which at most s0 are from Sc. Lemma 3.2 Choose τ > 0 such that δ2s + θs,2s < 1 −τ. Let βinit be the ℓ1-minimizer subject to the constraints, for λ := p 2 log p/n and λp,τ := (√1 + a + t−1) p 2 log p/n,
1 nXT (Y −Xβinit)
∞ ≤λp,τσ. (3.8) Given some constant C4 ≥C1, for C1 as in (3.3), choose a thresholding parameter t0 so that C4λp,τσ ≥t0 > C1λp,τσ; Set I = {j : |βj,init| > t0}. Then with probability at least P (Ta), as detailed in Proposition 3.1, we have for C′ 0 as in (3.2), |I| ≤ 2s0, and |I ∪S| ≤s + s0, and (3.9) ∥βD∥2 ≤ q (C′ 0 + C4)2 + 1λp,τσ√s0, where D := {1, . . . , p} \ I. (3.10) Next we show that even if we miss some columns of X in S, we can still hope to get the convergence rate as required in Theorem 1.2 so long as ∥βD∥2 is bounded and I is sufficiently sparse, for example, as bounded in Lemma 3.2. We first show in Lemma 3.3 a general result on rate of convergence of the OLS estimator based on a chosen model I, where a subset of relevant variables are missing. Lemma 3.3 (OLS estimator with missing variables) Let D := {1, . . . , p} \ I and SR = D ∩S such that I ∩SR = ∅. Suppose |I ∪SR| ≤2s. Then we have on Ta, for the least squares estimator based on I, bβI = (XT I XI)−1XT I Y , it holds that
bβI −β
2 2 ≤ θ|I|,|SR| ∥βD∥2 + λσ,a,p p |I| /Λmin(|I|) 2 + ∥βD∥2 2 . Now Theorem 1.2 is an immediate corollary of Lemma 3.2 and 3.3 in view of (3.5), given that |SR| < s, and |I| ≤2s0 and |I ∪SR| ≤|I ∪S| ≤s + s0 ≤2s as in Lemma 3.2 (3.9). Hence it is clear by (3.10) that we cannot cut too many “significant” variables; in particular, for those that are larger λσ√s0, we can cut at most a constant number of them. 4 Linear sparsity and random matrices A special case of design matrices that satisfy the Restricted Eigenvalue assumptions are the random design matrices. This is shown in a large body of work, for example [3, 4, 5, 1, 13], which shows that the uniform uncertainty principle (UUP) holds for “generic” or random design matrices for very significant values of s. For example, it is well known that for a random matrix with i.i.d. Gaussian variables (that is, Gaussian Ensemble, subject to normalizations of columns), and the Bernoulli and Subgaussian Ensembles [1, 13], the UUP holds for s = O(n/ log(p/n)); hence the thresholding procedure can recover a sparse model using nearly a constant number of measurements per nonzero component despite the stochastic noise, when n is a nonnegligible fraction of p. See [5] for other examples of random designs. In our simulations as shown in Section 5, exact recovery rate of the sparsity pattern is very high for a few types of random matrices using a two-step procedure, once the number of samples passes a certain threshold. For example, for an i.i.d. Gaussian Ensemble, the threshold for exact recovery is n = Θ(s log(p/n)), where Θ hides a very small constant, when βmin is sufficiently large; this shows a strong contrast with the ordinary Lasso, for which the probability of success in terms of exact recovery of the sparsity pattern tends to zero when n < 2s log(p −s) [19]. In an ongoing work, the author is exploring thresholding algorithms for a broader class of random designs that satisfy the Restricted Eigenvalue assumptions. 7 20 50 100 200 500 1000 0.0 0.2 0.4 0.6 0.8 1.0 (a) p = 256 n Prob. of success s = 8 Two−step s = 8 Lasso s = 64 Two−step s = 64 Lasso 0 100 200 300 400 500 0.0 0.2 0.4 0.6 0.8 1.0 (b) p = 512 n Prob. of success s=9 s=18 s=32 s=57 s=64 s=96 s=128 200 400 600 800 0.0 0.2 0.4 0.6 0.8 1.0 (c) p = 1024 n Prob. of success s=18 s=36 s=64 s=103 s=128 s=192 s=256 50 100 150 200 250 200 300 400 500 600 700 800 (d) p = 1024 Sample size vs. Sparsity s n Prob. of succ. 90% 80% Figure 1: (a) Compare the probability of success under s = 8 and 64 for p = 256. The two-step procedure requires much fewer samples than the ordinary Lasso. (b) (c) show the probability of success of the two-step procedure under different levels of sparsity when n increases for p = 512 and 1024 respectively; (d) The number of samples n increases almost linearly with s for p = 1024. 5 Illustrative experiments In our implementation, we choose to use the Lasso as the initial estimator. We show in Figure 1 that the two-step procedure indeed recovers a sparse model using a small number of samples per non-zero component in β when X is a Gaussian Ensemble. Similar behavior was also observed for the Bernoulli Ensemble in our simulations. We run under three cases of p = 256, 512, 1024; for each p, we increase the sparsity s by roughly equal steps from s = 0.2p/log 0.2p to p/4. For each tuple (p, s, n), we first generate a random Gaussian Ensemble of size n × p as X, where Xij ∼N(0, 1), which is then normalized to have column ℓ2-norm √n. For a given (p, s, n) and X, we repeat the following experiment 100 times: 1) Generate a vector β of length p: within β randomly choose s non-zero positions; for each position, we assign a value of 0.9 or −0.9 randomly. 2) Generate a vector ǫ of length p according to N(0, Ip), where Ip is the identity matrix. 3) Compute Y = Xβ + ǫ. Y and X are then fed to the two-step procedure to obtain bβ. 4) We then compare bβ with β; if all components match in signs, we count this experiment as a success. At the end of the 100 experiments, we compute the percentage of successful runs as the probability of success. We compare with the ordinary Lasso, for which we search over the full path of LARS [8] and always choose the bβ that best matches β in terms of support. Inside the two-step procedure, we always fix λn ≈0.69 p 2 log p/n and threshold βinit at t0 = ft q log p n √ bs, where bs = |bS0| for bS0 = {j : βj,init ≥0.5λn}, and ft is a constant chosen from the range of [1/6, 1/3]. Acknowledgments. This research was supported by the Swiss National Science Foundation (SNF) Grant 20PA21-120050/1. The author thanks Larry Wasserman, Sara van de Geer and Peter B¨uhlmann for helpful discussions, comments and their kind support throughout this work. 8 References [1] R. G. Baraniuk, M. Davenport, R. A. DeVore, and M. B. Wakin. A simple proof of the restricted isometry property for random matrices. Constructive Approximation, 28(3):253–263, 2008. [2] P. J. Bickel, Y. Ritov, and A. B. Tsybakov. Simultaneous analysis of Lasso and Dantzig selector. The Annals of Statistics, 37(4):1705–1732, 2009. [3] E. Cand`es, J. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications in Pure and Applied Mathematics, 59(8):1207–1223, August 2006. [4] E. Cand`es and T. Tao. Decoding by Linear Programming. IEEE Trans. Info. Theory, 51:4203–4215, 2005. [5] E. Cand`es and T. Tao. The Dantzig selector: statistical estimation when p is much larger than n. Annals of Statistics, 35(6):2313–2351, 2007. [6] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal on Scientific and Statistical Computing, 20:33–61, 1998. [7] D. L. Donoho and I. M. Johnstone. Ideal spatial adaptation by wavelet shrinkage. Biometrika, 81:425–455, 1994. [8] B. Efron, T. Hastie, I. Johnstone, and R. Tibshirani. Least angle regression. Annals of Statistics, 32(2):407– 499, 2004. [9] E. Greenshtein and Y. Ritov. Persistency in high dimensional linear predictor-selection and the virtue of over-parametrization. Bernoulli, 10:971–988, 2004. [10] V. Koltchinskii. Dantzig selector and sparsity oracle inequalities. Bernoulli, 15(3):799–828, 2009. [11] N. Meinshausen and P. B¨uhlmann. High dimensional graphs and variable selection with the Lasso. Annals of Statistics, 34(3):1436–1462, 2006. [12] N. Meinshausen and B. Yu. Lasso-type recovery of sparse representations for high-dimensional data. Annals of Statistics, 37(1):246–270, 2009. [13] S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann. Uniform uncertainty principle for bernoulli and subgaussian ensembles. Constructive Approximation, 28(3):277–289, 2008. [14] D. Needell and J. A. Tropp. CoSaMP: Iterative signal recovery from incomplete and inaccurate samples. Applied and Computational Harmonic Analysis, 26(3):301–321, 2008. [15] D. Needell and R. Vershynin. Signal recovery from incomplete and inaccurate measurements via regularized orthogonal matching pursuit. IEEE Journal of Selected Topics in Signal Processing, to appear, 2009. [16] R. Tibshirani. Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B, 58(1):267–288, 1996. [17] S. A. van de Geer. The deterministic Lasso. The JSM Proceedings, American Statistical Association, 2007. [18] S. A. van de Geer. High-dimensional generalized linear models and the Lasso. The Annals of Statistics, 36:614–645, 2008. [19] M. Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery using ℓ1-constrained quadratic programming. IEEE Trans. Inform. Theory, 2008. to appear, also posted as Technical Report 709, 2006, Department of Statistics, UC Berkeley. [20] L. Wasserman and K. Roeder. High dimensional variable selection. The Annals of Statistics, 37(5A):2178– 2201, 2009. [21] P. Zhao and B. Yu. On model selection consistency of Lasso. Journal of Machine Learning Research, 7:2541–2567, 2006. [22] S. Zhou, S. van de Geer, and P. B¨uhlmann. Adaptive Lasso for high dimensional regression and gaussian graphical modeling, 2009. arXiv:0903.2515. [23] H. Zou. The adaptive Lasso and its oracle properties. Journal of the American Statistical Association, 101:1418–1429, 2006. 9
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Beyond Categories: The Visual Memex Model for Reasoning About Object Relationships Tomasz Malisiewicz, Alexei A. Efros Robotics Institute Carnegie Mellon University {tmalisie,efros}@cs.cmu.edu Abstract The use of context is critical for scene understanding in computer vision, where the recognition of an object is driven by both local appearance and the object’s relationship to other elements of the scene (context). Most current approaches rely on modeling the relationships between object categories as a source of context. In this paper we seek to move beyond categories to provide a richer appearancebased model of context. We present an exemplar-based model of objects and their relationships, the Visual Memex, that encodes both local appearance and 2D spatial context between object instances. We evaluate our model on Torralba’s proposed Context Challenge against a baseline category-based system. Our experiments suggest that moving beyond categories for context modeling appears to be quite beneficial, and may be the critical missing ingredient in scene understanding systems. 1 Introduction Image understanding is one of the Holy Grail problems in computer vision. Understanding a scene arguably requires parsing the image into its constituent objects. In real scenes composed of many different objects, the spatial configuration of one object can facilitate recognition of related objects [1], and quite often ambiguities in recognition cannot be resolved without looking beyond the spatial extent of the object in question. Thus, algorithms which jointly recognize many objects at once by taking account of contextual relationships have been quite popular. While early systems relied on hand-coded rules for inter-object context (e.g. [2, 3]), more modern approaches typically perform inference in a probabilistic graphical model with respect to categories where object interactions are modeled as higher order potentials [4, 5, 6, 7, 8, 9, 10]. One important implicit assumption made by all such models is that interactions between object instances can be adequately modeled as relationships between human-defined object categories. In this paper we challenge this “category assumption” for object-object interactions and propose a novel category-free approach for modeling object relationships. We propose a new framework, the Visual Memex Model, for representing and reasoning about object identities and their contextual relationships in an exemplar-based, non-parametric way. We evaluate our model on Antonio Torralba’s proposed Context Challenge [11] against a baseline category-based system. 2 Motivation The use of categories (classes) to represent concepts (e.g. visual objects) is so prevalent in computer vision and machine learning that most researchers don’t give it a second thought. Faced with a new task, one simply carves up the solution space into classes (e.g. cars, people, buildings), assigns class labels to training examples and applies one of the many popular classifiers to arrive at a solution. 1 However, we believe that it is worthwhile to re-examine the basic assumption behind categorization, and especially its role in modeling relationships between objects. Theories of categorization date back to the ancient Greeks. Aristotle defined categories as discrete entities characterized by a set of properties shared by all their members [12]. His categories are mutually exclusive, and every member of a category is equal. This classical view is still the most widely accepted way of reasoning about categories and taxonomies in hard sciences. However, as pointed out by Wittgenstein, this is almost certainly not the way most of our everyday concepts work (e.g. what is the set of properties that define the concept “game” and nothing else? [13]). Empirical evidence for typicality (e.g. a robin is a more commonly cited example of “bird” than a chicken) and multiple category memberships (e.g. chicken is both “bird” and “food”) further complicate the Aristotelian view. The ground-breaking work of cognitive psychologist Eleanor Rosch [14] demonstrated that humans do not cut up the world into neat categories defined by shared properties, but instead use similarity as the basis of categorization. Her Prototype Theory postulates that an object’s class is determined by its similarity to (a set of) prototypes which define each category, allowing for varying degree of membership. Such Prototype models have been successfully used for object recognition [15, 16]. Going even further, Exemplar Theory [17, 18] rejects the need for explicit category representation, arguing instead that a concept can be implicitly formed via all its observed instances. This allows for a dynamic definition of categories based on data availability and task (e.g. an object can be a vehicle, a car, a Volvo, or Bob’s Volvo). A recent operationalization of the exemplar model in the visual domain can be found in [19]. But it might not be too productive to concentrate on the various categorization theories without considering the final aim – what do we need categories for? One argument is that categorization is a tool to facilitate knowledge transfer. E.g. having been attacked once by a tiger, it’s critically important to determine if a newly observed object belongs to the tiger category so as to utilize the information from the previous encounter. Note that here recognizing the explicit category is unimportant, as long as the two tigers could be associated with each other. Guided by this intuition and evidence from cognitive neuroscience, Bar [20] outlined the importance of analogies, associations, and prediction in the human brain. He argues that the goal of visual perception is not to recognize an object in the traditional sense of categorizing it (i.e. asking ’what is this?’), but instead linking the input with an analogous representation in memory (i.e. asking ’what is this like?’). Once a novel input is linked with analogous representations, associated representations are activated rapidly and predict the representations of what is most likely to occur next. These ideas regarding analogies, associations, and prediction are surprisingly similar to Vannevar Bush’s 1945 concept of the Memex [21] – which was seen decades later as pioneering hypertext and the World Wide Web. Concerned with the transmission and accessibility of scientific ideas, Bush faulted the “artificiality of systems of indexing” and proposed the Memory Extender (Memex), a physical device which would help find information based on association instead of strict categorical indexing. The associative links were to be entered manually by the user and could be of several different types. Chains of links would form into longer “associative trails” creating new narratives in the concept space. For Bush “the process of tying two items together is the important thing.” Inspired by these diverse ideas that are, nonetheless, all pointing in the same general direction, we have been motivated to try to evaluate them on a concrete problem, to see if they can offer benefits over the more traditional classification framework. One particular area where we feel these ideas might prove very useful is in modeling relationships between objects within an image. Therefore, in this paper we propose, in an homage to Bush, the Visual Memex Model, as a first step towards operationalizing the direct modeling of associations between visual objects, and compare it with more standard tools for the same task. 3 The Visual Memex Model Our starting point is Vannevar Bush’s observation that strict categorical indexing of concepts has severe limitations. Abandoning rigid object categories, we embrace Bush’s and Bar’s belief in the primary role of associations, but unlike Bush, we aim to discover these associations automatically from the data. At the core of our model is an exemplar-based representation of objects [18, 19]. The Visual Memex can then be thought of as a vast graph, with nodes representing all the object instances 2 Figure 1: The Visual Memex graph encodes object similarity (solid black edge) and spatial context (dotted red edge) between pairs of object exemplars. A spatial context feature is stored for each context edge. The Memex graph can be used to interpret a new image (left) by associating image segments with exemplars in the graph (orange edges) and propagating the information. Figure best viewed in color. in the dataset, and arcs representing the different types of associations between them (Figure 1). There are two types of arcs in our model, encoding two different relationships between objects: 1) visual similarity (e.g. this car looks like that car), and 2) contextual associations (e.g. this car is next to this building). Once the graph is built, it can be used to interpret a novel image (Figure 1, left) by first connecting segments within the image with similar stored exemplars, and then propagating contextual information between these exemplars through the graph. When an exemplar gets activated, visually similar exemplars as well as other contextually relevant objects get activated as well. This way, exemplarto-exemplar similarity in the Memex graph can serve as Bush’s “trails” to link concepts together in a non-parametric, query-dependent way, without the use of predefined categories. For example, in Figure 1, we should be able to infer that a car seen from the rear often co-occurs with an oblique building wall (but not a frontal wall) – something which category-based models would be hard-pressed to achieve. Formally, we define the Visual Memex Model as a graph G = (V, ES, EC, {D}, {f}) consisting of N object exemplar nodes V , similarity edges ES, context edges EC, N per-exemplar similarity functions {D}, and the spatial features {f} associated with each context edge. We now describe how to learn the similarity functions {D} from data to create the structure of the Visual Memex. 3.1 Similarity Edges We use the per-exemplar distance-function learning algorithm of Malisiewicz et al [19] to learn the object similarity edges. For each exemplar, the algorithm learns which other exemplars it is similar to as well as a distance function. A distance function is a linear combination of elementary distances used to measure similarity to the exemplar. We use the same 14 color, shape, texture, and location features as used in [19]. For the j-th exemplar, wj is the vector of 14 weights, bj is a scalar bias, and αj ∈{0, 1}|C| is a binary indicator vector which encodes which other exemplars the current exemplar is similar to. We solve [w∗ j, b∗ j, α∗ j] = arg minw,b,αfj(w, b, α), but since the exemplars’ optimization problems are independent we drop the j suffix for clarity. Let di be the vector of 14 Euclidean distances between the exemplar whose similarity we are learning (the focal exemplar) and the i-th exemplar. C is the set of exemplars that have the same label as the focal exemplar. Let L(x) = max(1 −x, 0)2 be the hinge-squared loss function. A different w, b, and α are learned per-exemplar by optimizing the following functional: f(w, b, α) = λ 2 ||w||2 + X i∈C αi L(−(wT di + b)) + X i/∈C L(wT di + b) −σ|| α ||2 (1) We minimize the above SVM-like objective function via an alternating optimization strategy as in [19]. The algorithm uses labels (see Section 3.3) during learning where the regularization term favors connecting to many similarly-labeled exemplars and the loss term favors separability in distance 3 associations ? car door road window window tree car tree window wheel wheel ? Category Estimation car person wheel road window fence building sidewalk door tree Figure 2: Torralba’s Context Challenge: “How far can you go without running a local object detector?” The task is to reason about the identity of the hidden object (denoted by a “?”) without local information. In our category-free Visual Memex model, object predictions are generated in the form of exemplar associations for the hidden object. In a category-based model, the category of the hidden object is directly estimated. space. We create a similarity edge between two exemplars if they are deemed similar by each others’ distance functions. We use a fixed λ = .00001 and σ = 100 for all exemplars. 3.2 Context Edges When two objects occur inside a single image, we encode their 2-D spatial relationship into a context feature vector f ∈ℜ10 (visualized as red dotted edges in Figure 1). The context feature vector encodes relative overlap, relative displacement, relative scale, and relative height of the bottom-most pixel between two exemplar regions in a single image. This feature captures the spatial relationship between two regions and does not take into account any appearance information – it is a generalization of the spatial features used in [8]. We measure the similarity between two context features using a Gaussian kernel: K(f, f ′) = e−α1|| f −f ′ ||2 with α1 = 1.0. 3.3 Building the Visual Memex We extract a large database of exemplar objects and their ground-truth segmentation masks from the LabelMe [22] dataset and learn the structure of the Visual Memex in an offline setting. We use objects from the 30 most frequently occurring categories in LabelMe. Similarity edges are created using the per-exemplar distance function learning framework of [19], and context edges are created each time two exemplars are observed in the same image. We have a total of N = 87, 802 exemplars in the Visual Memex, |ES| = 276, 782 similarity edges, and |EC| = 989, 106 context edges. 4 Evaluating on the Context Challenge The intuition that we would like to evaluate is that many useful regularities of the visual world are lost when dealing solely with categories (e.g. the side view of a building should associate more with a side view of a car than a frontal view of a car). The key motivation behind the Visual Memex is that context should depend on the appearance of an object and not just the category it belongs to. In order to test this hypothesis against the commonly held practice of abstracting away appearance into categories, we need a rich evaluation dataset as well as a meaningful evaluation task. We found that the Context Challenge [11] recently proposed by Antonio Torralba fits our needs perfectly. The evaluation task is inspired by the question: “How far can you go without running an object detector?” The goal is to recognize a single object in the image without peeking at pixels belonging to that object. Torralba presented an algorithm for predicting the category and scale of an object using only contextual information [23], but his notion of context is scene-centered (where the appearance of the entire image is used for prediction). Since the context we wish study in this paper is object-centered, we use an object-centered formulation of the Context Challenge. While it is not clear if the absolute performance numbers on the Context Challenge are very meaningful in themselves, we feel that it is an ideal task for studying object-centered context and the role of categorization assumptions in such models. 4 In our variant of the Context Challenge, the goal is to predict the category of a hidden object yi solely based on its spatial relationships to some provided objects – without using the pixels belonging to the hidden object at all. For our study, we use manually provided regions and category labels of K supporting objects inside a single image. We refer to the identities of the K supporting objects in the image as {y1, . . . , yK} (where y ∈{1, . . . , |C|}) and the set of K 2D spatial relationship features between each supporting object and the hidden object as {f i1, . . . , f iK}. 4.1 Inference in the Visual Memex Model In this section, we explain how to use the Visual Memex graph (automatically constructed from data) to perform inference for the Context Challenge hidden-object prediction task. Not making the “category assumption,” the model is defined with respect to exemplar associations for the hidden object. Inference in the model returns a compatibility score between every exemplar and the hidden object, and can be though of as returning an ordered list of exemplar associations. Due to the nature of exemplar associations as opposed to category assignments, a supporting object can be associated with multiple exemplars as opposed to a single category. We create soft exemplar associations between each of the supporting objects and the exemplars in the Visual Memex using the similarity functions {D} (see Section 3.1). {S1, . . . , SK} are the appearance features for the K supporting objects. Aa j is the affinity between exemplar a in the Visual Memex and the j-th supporting object and is created by evaluating Sj under a’s distance function Aa j = e−Da(Sj). Ψ(ei, ej, f ij) is the pairwise compatibility between exemplar ei and ej under the spatial feature f ij. Let Wab be the adjacency matrix representation of the similarity edges (Wuv = [(u, v) ∈ES]). Inference in the Visual Memex Model is done by optimizing the following conditional distribution which scores the assignment of an arbitrary exemplar ei to the hidden object based on contextual relations: p(ei|A1, . . . , AK, f i1, . . . , f iK) ∝ K Y j=1 N X a=1 Aa j Ψ(ei, ea, f ij) (2) log Ψ(ei, ej, f ij) = P (u,v)∈EC WiuWjvK(f ij, f uv) P (u,v)∈EC WiuWjv (3) The reason for the summation inside Equation 3 is that it aggregates contextual interactions from similar exemplars. By doing this, we effectively “densify” the contextual interactions in the Visual Memex. An interpretation of this densification procedure is that we are creating a kernel density estimator for an arbitrary pair of exemplars (ei, ej) via a weighted sum of kernels placed at context features in the data set {f uv} : (u, v) ∈EC where the weights WiuWjv measure visual similarity between pairs (ei, ej) and (eu, ev) . We experimented with using a single kernel, Ψ(ei, ej| f ij) = K(f ij, f ei,ej), and found that the integration of multiple features via the densification described above is a key ingredient for successful Visual Memex inference. Finally, after performing inference in the Visual Memex Model, we are left with a score for each exemplar. At this stage, as far as our model is concerned, the recognition has already been performed. However, since the task we are evaluated on is category-based, we combine the returned exemplars into a vote for categories using Luce’s Axiom of Choice [17] which averages the exemplar responses per-category. 4.2 CoLA-based Parametric Model We would like to evaluate the Visual Memex model against a more traditional, category-based framework with parametric inter-category relationships. One of the most recent and successful approaches is the CoLA model [8]. CoLA learns a set of parameters for each pair of categories which correspond to relative strengths of the four different top,above,below,inside spatial relationships. In the case of dealing with categories directly we consider a conditional distribution over the category of the hidden object yi that factors as a star graph with K leaves (with the hidden object being connected to 5 all the supporting objects). θ are model parameters, Ψ is a pairwise potential that measures the compatibility of two categories with a specified spatial relationship, and Z is a normalization constant such that the conditional distribution sums to 1. p(yi|y1, . . . , yK, f i1, . . . , f iK, θ) = 1 Z K Y j=1 Ψ(yi, yj, f ij, θ) (4) Following [8], we use a feature function h(f) that computes the affinity between feature f and a set of prototypical spatial relationships. We automatically find P prototypical spatial relationships by clustering all spatial feature vectors {f} in the training set via the popular Kmeans algorithm. Let h(f) ∈ℜP be the normalized vector of affinities to cluster centers {c1, . . . , cP }. θ is the set of all parameters in this model, with θ(yi, yj) ∈ℜP being the parameters associated with the pair of categories (yi, yj). log Ψ(yi, yj, f ij, θ) = [h(f ij)T ] θ(yi, yj) (5) hi(f) ∝ e−α|| f −ci||2 (6) We tried using the four prototypical relationships corresponding to above, below, inside, and outside as in [8], but found that using Kmeans with significantly larger number of prototypes P = 30 produced superior results. For learning θ, we found the maximum likelihood θ using gradient descent. The training objective function was optimized to mimic what happens during testing on the Context Challenge task. Since the distributions for the Context Challenge task are defined with respect to a single category variable (see Equation 4), we could compute the partition function directly and didn’t require any approximations as in [8] (which required training in a loopy graph). 4.3 Reduced KDE Memex Model Since the Visual Memex Model and the CoLA-inspired model make different assumptions with respect to objects (category-based vs. exemplar-based) and context (parametric vs. nonparametric), we feel it would also be useful to examine a hybrid model – dubbed the Reduced KDE Memex Model – which uses a nonparametric model of context but operates on object categories. The Reduced KDE Memex Model is created by collapsing all exemplars belonging to a single category into fullyconnected components which can be thought of as adding categories into the Visual Memex graph. Identities between individual exemplars are lost, and thus we lose the fine details of a spatial context. By forming categories, we can no longer say a particular spatial relationship is between a blue side view of a car and an oblique brick building, we can only say it is a relationship between a car and a building. Now that we are left with an unordered bag of spatial relationships {f} between two categories, we need a way to measure compatibility between a newly observed f and the stored relationships. We use the same form of the Context Challenge conditional distribution as in Equation 4. We use a Kernel Density Estimator(KDE) for every pair of categories, and the potential Ψ can be thought of as a matrix of such estimators. The use of nonparametric potentials in graphical models has been already explored in the domain of texture analysis [24]. δij is the Kronecker delta function. log Ψ(yi, yj, f ij) = P (u,v)∈EC δyiyuδyjyvK(f ij, f uv) P (u,v)∈EC δyiyuδyjyv (7) The Reduced Memex model, being category-based and nonparametric, aggregates the spatial relationships across many different pairs of exemplars from two categories. While we used a fixed kernel K which measures distance isotropically across the dimensions of f, the advantage of such a nonparametric approach is that with enough data the particularities of K do not matter. We also experimented with a Nearest Neighbor based model, but found the Kernel Density Estimation approach to be superior. 6 Visual Memex person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm KDE person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm CoLA person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm 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 Recall Precision Context Challenge Prediction Confidence Visual Memex KDE CoLA 0 0.2 0.4 0.6 0.8 1 Context Challenge Recognition Accuracy for 30 Categories person car tree window head building sky wall road sidewalk sign chair door mountain table floor streetlight lamp plant pole balcony wheel text grass column pane trash blind ground arm Visual Memex KDE CoLA Figure 3: a.) Context Challenge confusion matrices for the 3 methods: Visual Memex, KDE, and CoLA. b.) Recognition Precision versus Recall when thresholding output based on confidence. c) Side by side comparison of the 3 methods’ accuracies for 30 categories. 5 Results and Discussion For the Context Challenge evaluation, we use 200 randomly selected densely labeled images from LabelMe [22]. Our testset contains 3048 total objects from 30 different categories. For an image with K objects, we solve K Context Challenge problems with one hidden object and K-1 supporting objects. Qualitative results on this prediction task can be seen in Figure 4. We evaluate the performance of our Visual Memex model, the Reduced Memex KDE model, and the CoLA-inspired model with respect to categorization performance (confusion matrices can be seen in top left of Figure 3). The overall recognition accuracy of the Visual Memex Model, Reduced Memex Model, and CoLA are .527, .430, and .457 respectively. Note that the Visual Memex Model performs significantly better than the baselines. Taking a closer look at the per-category accuracies of the three methods (see bottom of Figure 3), we see that the CoLA-based method fails on many categories. The average per-category recognition accuracies of the three methods are: .534, .454, and .213. The Visual Memex Model still performs the best, but we see a significant drop in performance for the category-based CoLA method. CoLA is biased towards the popular categories, returning the most frequently occurring category (window) quite often. Overall, the Visual Memex Model achieves the best performance for 21 out of the 30 categories. In addition, we plot precision recall curves for each of the three methods to determine if high confidence returned by each model is correlated with high recognition rates (top right of Figure 3). The Visual Memex model has the most significant high-precision low-recall regime, suggesting that its confidence is a good measure of success. The relatively flat curve for the CoLA method is related to the problem of overcompensation for popular classes as mentioned above. The distributions returned by CoLA tend to degenerate to a single non-zero value (most often on one of the popular categories such as window). This is why the maximum probability returned by CoLA isn’t a good measure of confidence. We also demonstrate the power of the Visual Memex to predict appearance solely based on contextual interactions with other objects and their visual appearance. The middle row of Figure 4 demonstrates some of these associations. Note how in row 1, a plausible viewpoint is selected rather than just a random car. In row 3 we see that the appearance of snow on one mountain suggests that the other portion of the image also contains a snowy mountain. In summary, we presented a category-free Visual Memex Model and applied it to the task of contextual object recognition within the experimental framework of the Context Challenge. Our experiments confirm our intuition that moving beyond categories is beneficial for improved modeling of relationships between objects. Acknowledgements. This research was in part funded by NSF CAREER award IIS-0546547, NSF Graduate Research Fellowship, a Guggenheim Fellowship, as well as generous gift from Google. A. Efros thanks the WILLOW team at ENS, Paris for their hospitality. 7 table floor wall door floor table door wall sidewalk car road person Visual Memex Exemplar Predictions Input Image + Hidden Region Categorization Results Visual Memex KDE CoLA Visual Memex KDE CoLA Visual Memex KDE CoLA Visual Memex KDE CoLA Visual Memex KDE CoLA Visual Memex KDE CoLA Visual Memex KDE CoLA Figure 4: Qualitative Results on the Context Challenge. Exemplar predictions are from the Visual Memex model and categorization results are from the Visual Memex model, the KDE Model, and CoLA[8]. 8 References [1] Moshe Bar and Shimon Ullman. Spatial context in recognition. Perception, 25:343–352, 1996. 1 [2] A.R. Hanson and E.M. Riseman. Visions: A computer system for interpreting scenes. Computer Vision Systems, pages 303–333, 1978. 1 [3] T.M. Strat and M.A. Fischler. Context-based vision: Recognizing objects using information from both 2-d and 3-d imagery. PAMI, 13:1050–1065, 1991. 1 [4] Xuming He, Richard S. Zemel, and Miguel ´A. Carreira-Perpi˜n´an. Multiscale conditional random fields for image labeling. CVPR, pages 695–702, 2004. 1 [5] Sanjiv Kumar and Martial Hebert. A hierarchical field framework for unified context-based classification. ICCV, 2005. 1 [6] Jamie Shotton, John M. Winn, Carsten Rother, and Antonio Criminisi. Textonboost: Joint appearance, shape and context modeling for multi-class object recognition and segmentation. ECCV, 2006. 1 [7] Andrew Rabinovich, Anrea Vedaldi, Carolina Galleguillos, Eric Wiewora, and Serge Belongie. Objects in context. ICCV, 2007. 1 [8] Carolina Galleguillos, Andrew Rabinovich, and Serge Belongie. Object categorization using cooccurrence, location and appearance. ECCV, 2008. 1, 4, 5, 6, 8 [9] Devi Parikh, C. Lawrence Zitnick, and Tsuhan Chen. From appearance to context-based recognition: Dense labeling in small images. CVPR, 2008. 1 [10] Bryan C. Russell, Antonio Torralba, Ce Liu, Rob Fergus, and William T. Freeman. Object recognition by scene alignment. NIPS, 2007. 1 [11] Antonio Torralba. The context challenge. http://web.mit.edu/torralba/www/carsAndFacesInContext.html. 1, 4 [12] Aristotle. Categories. 2 [13] Ludwig Wittgenstein. Philosophical Investigations. Blackwell Publishing, 1953. 2 [14] Eleanor Rosch. Principles of categorization. Cognition and Categorization, pages 27–48, 1978. 2 [15] Shimon Edelman. Representation, similarity and the chorus of prototypes. Minds and Machines, 1995. 2 [16] Ariadna Quattoni, M. Collins, and Trevor Darrell. Transfer learning for image classification with sparse prototype representations. CVPR, 2008. 2 [17] D. L. Medin and M.M. Schaffer. Context theory of classification learning. Psychological Review, 85:207– 238, 1978. 2, 5 [18] Robert M. Nosofsky. Attention, similarity, and the identification-categorization relationship. Journal of Experimental Psychology: General, 115(1):39–57, 1986. 2 [19] Tomasz Malisiewicz and Alexei A. Efros. Recognition by association via learning per-exemplar distances. CVPR, 2008. 2, 3, 4 [20] Moshe Bar. The proactive brain: memory for predictions. Philosophical Transactions of the Royal Society B, 364:1235–1243, 2009. 2 [21] Vannevar Bush. As we may think. The Atlantic Monthly, 1945. 2 [22] Bryan Russell, Antonio Torralba, Kevin Murphy, , and William T. Freeman. Labelme: a database and web-based tool for image annotation. International Journal of Computer Vision, 77:157–173, 2008. 4, 7 [23] Antonio Torralba. Contextual priming for object detection. International Journal of Computer Vision, 53:169–191, 2003. 4 [24] Rupert Paget and I. D. Longstaff. Texture synthesis via a noncausal nonparametric multiscale markov random field. IEEE Transactions on Image Processing, 1998. 6 9
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Conditional Random Fields with High-Order Features for Sequence Labeling Nan Ye Wee Sun Lee Department of Computer Science National University of Singapore {yenan,leews}@comp.nus.edu.sg Hai Leong Chieu DSO National Laboratories chaileon@dso.org.sg Dan Wu Singapore MIT Alliance National University of Singapore dwu@nus.edu.sg Abstract Dependencies among neighbouring labels in a sequence is an important source of information for sequence labeling problems. However, only dependencies between adjacent labels are commonly exploited in practice because of the high computational complexity of typical inference algorithms when longer distance dependencies are taken into account. In this paper, we show that it is possible to design efficient inference algorithms for a conditional random field using features that depend on long consecutive label sequences (high-order features), as long as the number of distinct label sequences used in the features is small. This leads to efficient learning algorithms for these conditional random fields. We show experimentally that exploiting dependencies using high-order features can lead to substantial performance improvements for some problems and discuss conditions under which high-order features can be effective. 1 Introduction In a sequence labeling problem, we are given an input sequence x and need to label each component of x with its class to produce a label sequence y. Examples of sequence labeling problems include labeling words in sentences with its type in named-entity recognition problems [16], handwriting recognition problems [15], and deciding whether each DNA base in a DNA sequence is part of a gene in gene prediction problems [2]. Conditional random fields (CRF) [8] has been successfully applied in many sequence labeling problems. Its chief advantage lies in the fact that it models the conditional distribution P(y|x) rather than the joint distribution P(y, x). In addition, it can effectively encode arbitrary dependencies of y on x as the learning cost mainly depends on the parts of y involved in the dependencies. However, the use of high-order features, where a feature of order k is a feature that encodes the dependency between x and (k + 1) consecutive elements in y, can potentially lead to an exponential blowup in the computational complexity of inference. Hence, dependencies are usually assumed to exist only between adjacent components of y, giving rise to linear-chain CRFs which limits the order of the features to one. In this paper, we show that it is possible to learn and predict CRFs with high-order features efficiently under the following pattern sparsity assumption (which is often observed in real problems): the number of observed label sequences of length, say k, that the features depend on, is much smaller than nk, where n is the number of possible labels. We give an algorithm for computing the marginals and the CRF log likelihood gradient that runs in time polynomial in the number and length of the label sequences that the features depend on. The gradient can be used with quasi-newton methods to efficiently solve the convex log likelihood optimization problem [14]. We also provide an efficient decoding algorithm for finding the most probable label sequence in the presence of long label sequence features. This can be used with cutting plane methods to train max-margin solutions for sequence labeling problems in polynomial time [18]. We show experimentally that using high-order features can improve performance in sequence labeling problems. We show that in handwriting recognition, using even simple high-order indicator features improves performance over using linear-chain CRFs, and impressive performance improvement is observed when the maximum order of the indicator features is increased. We also use a synthetic data set to discuss the conditions under which higher order features can be helpful. We further show that higher order label features can sometimes be more stable under change of data distribution using a named entity data set. 2 Related Work Conditional random fields [8] are discriminately trained, undirected Markov models, which has been shown to perform well in various sequence labeling problems. Although a CRF can be used to capture arbitrary dependencies among components of x and y, in practice, this flexibility of the CRF is not fully exploited as inference in Markov models is NP-hard in general (see e.g. [1]), and can only be performed efficiently for special cases such as linear chains. As such, most applications involving CRFs are limited to some tractable Markov models. This observation also applies to other structured prediction methods such as structured support vector machines [15, 18]. A commonly used inference algorithm for CRF is the clique tree algorithm [5]. Defining a feature depending on k (not necessarily consecutive) labels will require forming a clique of size k, resulting in a clique-tree with tree-width greater or equal to k. Inference on such a clique tree will be exponential in k. For sequence models, a feature of order k can be incorporated into a k-order Markov chain, but the complexity of inference is again exponential in k. Under the pattern sparsity assumption, our algorithm achieves efficiency by maintaining only information related to a few occurred patterns, while previous algorithms maintain information about all (exponentially many) possible patterns. In the special case of a semi-Markov random fields, where high-order features depend on segments of identical labels, the complexity of inference is linear in the maximum length of the segments [13]. The semi-Markov assumption can be seen as defining a sparse feature representation: though the number of length k label patterns is exponential in k, the semi-Markov assumption effectively allows only n2 of them (n is the cardinality of the label set), as the features are defined on a sequence of identical labels that can only depend on the label of the preceding segment. Compared to this approach, our algorithm has the advantage of being able to efficiently handle high-order features having arbitrary label patterns. Long distance dependencies can also be captured using hierarchical models such as Hierarchical Hidden Markov Model (HHMM) [4] or Probabilistic Context Free Grammar (PCFG) [6]. The time complexity of inference in an HHMM is O(min{nl3, n2l}) [4, 10], where n is the number of states and l is the length of the sequence. Discriminative versions such as hierarchical CRF has also been studied [17]. Inference in PCFG and its discriminative version can also be efficiently done in O(ml3) where m is the number of productions in the grammar [6]. These methods are able to capture dependencies of arbitrary lengths, unlike k-order Markov chains. However, to do efficient learning with these methods, the hierarchical structure of the examples need to be provided. For example, if we use PCFG to do named entity recognition, we need to provide the parse trees for efficient learning; providing the named entity labels for each word is not sufficient. Hence, a training set that has not been labeled with hierarchical labels will need to be relabeled before it can be trained efficiently. Alternatively, methods that employ hidden variables can be used (e.g. to infer the hidden parse tree) but the optimization problem is no longer convex and local optima can sometimes be a problem. Using high-order features captures less expressive form of dependencies than these models but allows efficient learning without relabeling the training set with hierarchical labels. Similar work on using higher order features for CRFs was independently done in [11]. Their work apply to a larger class of CRFs, including those requiring exponential time for inference, and they did not identify subclasses for which inference is guaranteed to be efficient. 3 CRF with High-order Features Throughout the remainder of this paper, x, y, z (with or without decorations) respectively denote an observation sequence of length T, a label sequence of length T, and an arbitrary label sequence. The function | · | denotes the length of any sequence. The set of labels is Y = {1, . . . , n}. If z = (y1, . . . , yt), then zi:j denotes (yi, . . . , yj). When j < i, zi:j is the empty sequence (denoted by ϵ). Let the features being considered be f1, . . . , fm. Each feature fi is associated with a label sequence zi, called fi’s label pattern, and fi has the form fi(x, y, t) = gi(x, t), if yt−|zi|+1:t = zi 0, otherwise. We call fi a feature of order |zi|−1. Consider, for example, the problem of named entity recognition. The observations x = (x1, . . . , xT ) may be a word sequence; gi(x, t) may be an indicator function for whether xt is capitalized or may output a precomputed term weight if xt matches a particular word; and zi may be a sequence of two labels, such as (person, organization) for the named entity recognition task, giving a feature of order one. A CRF defines conditional probability distributions P(y|x) = Zx(y)/Zx, where Zx(y) = exp(Pm i=1 PT t=|zi| λifi(x, y, t)), and Zx = P y Zx(y). The normalization factor Zx is called the partition function. In this paper, we will use the notation P x:P red(x) f(x) to denote the summation of f(x) over all elements of x satisfying the predicate Pred(x). 3.1 Inference for High-order CRF In this section, we describe the algorithms for computing the partition function, the marginals and the most likely label sequence for high-order CRFs. We give rough polynomial time complexity bounds to give an idea of the effectiveness of the algorithms. These bounds are pessimistic compared to practical performance of the algorithms. It can also be verified that the algorithms for linear chain CRF [8] are special cases of our algorithms when only zero-th and first order features are considered. We show a work example illustrating the computations in the supplementary material. 3.1.1 Basic Notations As in the case of hidden Markov models (HMM) [12], our algorithm uses a forward and backward pass. First, we describe the equivalent of states used in the forward and backward computation. We shall work with three sets: the pattern set Z, the forward-state set P and the backward-state set S. The pattern set, Z, is the set of distinct label patterns used in the m features. For notational simplicity, assume Z = {z1, . . . , zM}. The forward-state set, P = {p1, . . . p|P|}, consists of distinct elements in Y ∪{zj 1:k}0≤k≤|zj|−1,1≤j≤M; that is, P consists of all labels and all proper prefixes (including ϵ) of label patterns, with duplicates removed. Similarly, S = {s1, . . . s|S|} consists of the labels and proper suffixes: distinct elements in Y ∪{zj 1:k}1≤k≤|zj|,1≤j≤M. The transitions between states are based on the prefix and suffix relationships defined below. Let z1 ≤p z2 denote that z1 is a prefix of z2 and let z1 ≤s z2 denote that z1 is a suffix of z2. We define the longest prefix and suffix relations with respect to the sets P and S as follows z1 ≤p S z2 if and only if (z1 ∈S) and (z1 ≤p z2) and (∀z ∈S, z ≤p z2 ⇒z ≤p z1) z1 ≤s P z2 if and only if (z1 ∈P) and (z1 ≤s z2) and (∀z ∈P, z ≤s z2 ⇒z ≤s z1). Finally, the subsequence relationship defined below are used when combining forward and backward variables to compute marginals. Let z ⊆z′ denote that z is a subsequence of z′, z ⊂z′ denote that z is a subsequence of z′ 2:|z′|−1. The addition of subscript j in ⊆j and ⊂j indicates that the condition z ≤s z′ 1:j is satisfied as well (that is, z ends at position j in z′). We shall give rough time bounds in terms of m (the total number of features), n (the number of labels), T (the length of the sequence), M (the number of distinct label patterns in Z), and the maximum order K = max{|z1| −1, . . . , |zM| −1}. 3.1.2 The Forward and Backward Variables We now define forward vector αx and backward vector βx. Suppose z ≤p y, then define y’s prefix score Zp x(z) = exp(Pm i=1 P|z| t=|zi| λifi(x, y, t)). Similarly, if z ≤s y, then define y’s suffix score Zs x(z) = exp(Pm i=1 PT t=T −|z|+|zi| λifi(x, y, t)). Zp x(z) and Zs x(z) only depend on z. Let αx(t, pi) = X z:|z|=t,pi≤s Pz Zp x(z) βx(t, si) = X z:|z|=T +1−t,si≤p Sz Zs x(z). The variable αx(t, pi) computes for x1:t the sum of the scores of all its label sequences z having pi as the longest suffix. Similarly, the variable βx(t, si) computes for xt:T the sum of scores of all its label sequence z having si as the longest prefix. Each vector αx(t, ·) is of dimension |P|, while βx(t, ·) has dimension |S|. We shall compute the αx and βx vectors with dynamic programming. Let Ψp x(t, p) = exp(P i:zi≤sp λigi(x, t)). For y with p ≤s y1:t, this function counts the contribution towards Zx(y) by all features fi with their label patterns ending at position t and being suffixes of p. Let piy be the concatenation of pi with a label y. The following proposition is immediate. Proposition 1 (a) For any z, there is a unique pi such that pi ≤s P z. (b) For any z, y, if pi ≤s P z and pk ≤s P piy, then pk ≤s P zy and Zp x(zy) = Ψp x(t, piy)Zp x(z). Proposition 1(a) means that we can induce partitions of label sequences using the forward states. and Proposition 1(b) shows how to make well-defined transition from one forward state at a time slice to another forward state at the next time slice. By definition, αx(0, ϵ) = 1, and αx(0, pi) = 0 for all pi ̸= ϵ. Using Proposition 1(b), the recurrence for αx is αx(t, pk) = X (pi,y):pk≤s Ppiy Ψp x(t, piy)αx(t −1, pi), for 1 ≤t ≤T. Similarly, for the backward vectors βx, let Ψs x(t, s) = exp(P i:zi≤ps λigi(x, t + |zi| −1)). By definition, βx(T + 1, ϵ) = 1, and βx(T + 1, si) = 0 for all si ̸= ϵ. The recurrence for βx is βx(t, sk) = X (si,y):sk≤p Sysi Ψs x(t, ysi)βx(t + 1, si), for 1 ≤t ≤T. Once αx or βx is computed, then using Proposition 1(a), Zx can be easily obtained: Zx = |P| X i=1 αx(T, pi) = |S| X i=1 βx(1, si). Time Complexity: We assume that each evaluation of the function gi(·, ·) can be performed in unit time for all i. All relevant values of Ψp x that are used can hence be computed in O(mn|P|T) (thus O(mnMKT)) time. In practice, this is pessimistic, and the computation can be done more quickly. For all following analyses, we assume that Ψp x has already been computed and stored in an array. Now all values of αx can be computed in Θ(n|P|T), thus O(nMKT) time. Similar bounds for Ψs x and βx hold. 3.1.3 Computing the Most Likely Label Sequence As in the case of HMM [12], Viterbi decoding (calculating the most likely label sequence) is obtained by replacing the sum operator in the forward backward algorithm with the max operator. Formally, let δx(t, pi) = maxz:|z|=t,pi≤s Pz Zp x(z). By definition, δx(0, ϵ) = 1, and δx(0, pi) = 0 for all pi ̸= ϵ, and using Proposition 1, we have δx(t, pk) = max (pi,y):pk≤s Ppiy Ψp x(t, piy)δx(t −1, pi), for 1 ≤t ≤T. We use Φx(t, pk) to record the pair (pi, y) chosen to obtain δx(t, pk), Φx(t, pk) = arg max(pi,y):pk≤s PpiyΨp x(t, piy)δx(t −1, pi). Let p∗ T = arg maxpi δx(T, pi), then the most likely path y∗= (y∗ 1, . . . , y∗ T ) has y∗ T as the last label in p∗ T , and the full sequence can be traced backwards using Φx(·, ·) as follows (p∗ t , y∗ t ) = Φx(t + 1, p∗ t+1), for 1 ≤t < T. Time Complexity: Either Ψp x or Ψs x can be used for decoding; hence decoding can be done in Θ(n min{|P|, |S|}T) time. 3.1.4 Computing the Marginals We need to compute marginals of label sequences and single variables, that is, compute P(yt−|z|:t = z|x) for z ∈Z ∪Y. Unlike in the traditional HMM, additional care need to be taken regarding features having label patterns that are super or sub sequences of z. We define Wx(t, z) = exp( X (i,j):zi⊂jz λigi(x, t −|z| + j)). This function computes the sum of all features that may activate strictly within z. If z1:|z|−1 ≤s pi and z2:|z| ≤p sj, define [pi, z, sj] as the sequence pi 1:|pi|−(|z|−1)zsj |z|−1:|sj|, and Ox(t, pi, sj, z) = exp( X (k,k′):z⊆zk,zk⊆k′[pi,z,sj] λkgk(x, t −|pi| + k′ −1)). Ox(t, pi, sj, z) counts the contribution of features with their label patterns properly containing z but within [pi, z, sj]. Proposition 2 Let z ∈Z ∪Y. For any y with yt−|z|+1:t = z, there exists unique pi, sj such that z1:|z|−1 ≤s pi, z2:|z| ≤p sj, pi ≤s P y1:t−1, and sj ≤p S yt−|z|+2:T . In addition, Zx(y) = 1 Wx(t,z)Zp x(t −1, y1:t−1)Zs x(T + 1 −(t −|z| + 2), yt−|z|+2:T )Ox(t, pi, sj, z). Multiplying by Ox counts features that are not counted in Zp xZs x while division by Wx removes features that are double-counted. By Proposition 2, we have P(yt−|z|+1:t = z|x) = P (i,j):z1:|z|−1≤spi,z2:|z|≤psj αx(t −1, pi)βx(t −|z| + 2, sj)Ox(t, pi, sj, z) ZxWx(t, z) . Time Complexity: Both Wx(t, z) and Ox(t, pi, sj, z) can be computed in O(|pi||sj|) = O(K2) time (with some precomputation). Thus a very pessimistic time bound for computing P(yt−|z|+1:t = z|x) is O(K2|P||S|) = O(M 2K4). 3.2 Training Given a training set T , the model parameters λi’s can be chosen by maximizing the regularized log-likelihood LT = log Π(x,y)∈T P(y|x) −Pm i=1 λ2 i 2σ2 reg , where σreg is a parameter that controls the degree of regularization. Note that LT is a concave function of λ1, . . . , λm, and its maximum is achieved when ∂LT ∂λi = ˜E(fi) −E(fi) −λk σ2reg = 0 where ˜E(fi) = P (x,y)∈T P|x| t=|zi| fi(x, y, t) is the empirical sum of the feature fi in the observed data, and E(fi) = P (x,y)∈T P |y′|=|x| P(y′|x) P|x| t=|zi| fi(x, y′, t) is the expected sum of fi. Given the gradient and value of LT , we use the L-BFGS optimization method [14] for maximizing the regularized log-likelihood. The function LT can now be computed because we have shown how to compute Zx, and computing the value of Zx(y) is straightforward, for all (x, y) ∈T . For the gradient, computing ˜E(fi) is straightforward, and E(fi) can be computed using marginals computed in previous section: E(fi) = X (x,y)∈T |x| X t=|zi| P(y′ t−|zi|+1:t = zi|x)gi(x, t). Time Complexity: Computing the gradient is clearly more time-consuming than LT , thus we shall just consider the time needed to compute the gradient. Let X = P (x,y)∈T |x|. We need to compute at most MX marginals, thus total time needed to compute all the marginals has O(M 3K4X) as an upper bound. Given the marginals, we can compute the gradient in O(mX) time. If the total number of gradient computations needed in maximization is I, then the total running time in training is bounded by O((M 3K4 + m)XI) (very pessimistic). 4 Experiments The practical feasibility of making use of high-order features based on our algorithm lies in the observation that the pattern sparsity assumption often holds. Our algorithm can be applied to take those high-order features into consideration; high-order features now form a component that one can play with in feature engineering. Now, the question is whether high-order features are practically significant. We first use a synthetic data set to explore conditions under which high-order features can be expected to help. We then use a handwritten character recognition problem to demonstrate that even incorporating simple highorder features can lead to impressive performance improvement on a naturally occurring dataset. Finally, we use a named entity data set to show that for some data sets, higher order label features may be more robust to changes in data distributions than observation features. 4.1 Synthetic Data Generated Using k-Order Markov Model We randomly generate an order k Markov model with n states s1, . . . , sn as follows. To increase pattern sparsity, we allow at most r randomly chosen possible next state given the previous k states. This limits the number of possible label sequences in each length k + 1 segment from nk+1 to nkr. The conditional probabilities of these r next states is generated by randomly selecting a vector from uniform distribution over [0, 1]r and normalizing them. Each state si generates an observation (a1, . . . , am) such that aj follows a Gaussian distribution with mean µij and standard deviation σ. Each µi,j is independently randomly generated from the uniform distribution over [0, 1]. In the experiments, we use values of n = 5, r = 2 and m = 3. The standard deviation, σ, has an important role in determining the characteristics of the data generated by this Markov model. If σ is very small as compared to most µij’s, then using the observations alone as features is likely to be good enough to obtain a good classifier of the states; the label correlations becomes less important for classification. However, if σ is large, then it is difficult to distinguish the states based on the observations alone and the label correlations, particularly those captured by higher order features are likely to be helpful. In short, the standard deviation, σ, is used to to control how much information the observations reveal about the states. We use the current, previous and next observations, rather than just the current observation as features, exploiting the conditional probability modeling strength of CRFs. For higher order features, we simply use all indicator features that appeared in the training data up to a maximum order. We considered the case k = 2 and k = 3, and varied σ and the maximum order. The training set and test set each contains 500 sequences of length 20; each sequence was initialized with a random sequence of length k and generated using the randomly generated order k Markov model. Training was done by maximizing the regularized log likelihood with regularization parameter σreg = 1 in all experiments in this paper. The experimental results are shown in Figure 1. Figure 1 shows that the high-order indicator features are useful in this case. In particular, we can see that it is beneficial to increase the order of the high-order features when the underlying model has longer distance correlations. As expected, increasing the order of the features beyond the order of the underlying model is not helpful. The results also suggests that in general, if the observations are closely coupled with the states (in the sense that different states correspond to very different observations), then feature engineering on the observations is generally enough to perform well, and Generated by 2nd-Order Markov Model 82 84 86 88 90 92 94 96 98 1 2 3 4 Maximum Order of Features Accuracy Sigma = 0.01 Sigma = 0.05 Sigma = 0.10 Generated by 3rd-Order Markov Model 79 81 83 85 87 89 91 93 95 1 2 3 4 Maximum Order of Features Accuracy Sigma = 0.01 Sigma = 0.05 Sigma = 0.10 Figure 1: Accuracy as a function of maximum order on the synthetic data set. Handwritten Character Recognition 74 76 78 80 82 84 86 88 1 2 3 4 5 Maximum Order of Features Accuracy Runtimes for Character Recognition Training 0 10 20 30 40 50 60 70 80 90 2 3 4 5 Maximum Order of Features Time (s) 0 500 1000 1500 2000 2500 3000 3500 Time (s) Per Iteration Time (Left Axis) Total Time (Right Axis) Figure 2: Accuracy (left) and running time (right) as a function of maximum order for the handwriting recognition data set. it is less important to use high-order features to capture label correlations. On the other hand, when such coupling is not clear, it becomes important to capture the label correlations, and high-order features can be useful. 4.2 Handwriting Recognition We used the handwriting recognition data set from [15], consisting of around 6100 handwritten words with an average length of around 8 characters. The data was originally collected by Kassel [7] from around 150 human subjects. The words were segmented into characters, and each character was converted into an image of 16 by 8 binary pixels. In this labeling problem, each xi is the image of a character, and each yi is a lower-case letter. The experimental setup is the same as that used in [15]: the data set was divided into 10 folds with each fold having approximately 600 training and 5500 test examples and the zero-th order features for a character are the pixel values. For higher order features, we again used all indicator features that appeared in the training data up to a maximum order. The average accuracy over the 10 folds are shown in Figure 2, where strong improvements are observed as the maximum order increases. Figure 2 also shows the total training time and the running time per iteration of the L-BFGS algorithm (which requires computation of the gradient and value of the function at each iteration). The running time appears to grow no more than linearly with the maximum order of the features for this data set. 4.3 Named Entity Recognition with Distribution Change The Named Entity Recognition (NER) problem asks for identification of named entities from texts. With carefully engineered observation features, there does not appear to be very much to be gained from using higher order features. However, in some situations, the training data does not come from the same distribution as the test data. In such cases, we hypothesize that higher order label features may be more stable than observation features and can sometimes offer performance gain. In our experiment, we used the Automatic Content Extraction (ACE) data [9], which is labeled with seven classes: Organization, Geo-political, Location, Facility, Vehicle, and Weapon. The ACE data comes from several genres and we use the following in our experiment: Broadcast conversation (BC), Newswire (NW), Weblog (WL) and Usenet (UN). Named Entity Recognition (Domain Adaptation) Average Improvement = 0.62 25 30 35 40 45 50 55 60 65 70 bc:nw bc:un bc:wl nw:bc nw:un nw:wl un:bc un:nw un:wl wl:bc wl:nw wl:un Training Domain : Test Domain F1 Score Linear Chain Second Order Figure 3: Named entity recognition results. We use all pairs of genres as training and test data. Scoring was done with the F1 score [16]. The features used are previous word, next word, current word, case patterns for these words, and all indicator label features of order up to k. The results for the case k = 1 and k = 2 are shown in Figure 3. Introducing second order indicator features shows improvement in 10 out of the 12 combinations and degrades performance in two of the combinations. However, the overall effect is small, with an average improvement of 0.62 in F1 score. 4.4 Discussion In our experiments, we used indicator features of all label patterns that appear in the training data. For real applications, if the pattern sparsity assumption is not satisfied, but certain patterns do not appear frequently enough and are not really important, then it is useful to see how we can select a subset of features with few distinct label patterns automatically. One possible approach would be to use boosting type methods [3] to sequentially select useful features. An alternate approach to feature selection is to use all possible features and maximize the margin of the solution instead. Generalization error bounds [15] show that it is possible to obtain good generalization with a relatively small training set size despite having a very large number of features if the margin is large. This indicates that feature selection may not be critical in some cases. Theoretically, it is also interesting to note that minimizing the regularized training cost when all possible high-order features of arbitrary length are used is computationally tractable. This is because the representer theorem [19] tells us that the optimum solution for minimizing quadratically regularized cost functions lies on the span of the training examples. Hence, even when we are learning with arbitrary sets of high-order features, we only need to use the features that appear in the training set to obtain the optimal solution. Given a training set of N sequences of length l, only O(l2N) long label sequences of all orders are observed. Using cutting plane techniques [18] the computational complexity of optimization is polynomial in inverse accuracy parameter, the training set size and maximum length of the sequences. It should also be possible to use kernels within the approach here. On the handwritten character problem, [15] reports substantial improvement in performance with the use of kernels. Use of kernels together with high-order features may lead to further improvements. However, we note that the advantage of the higher order features may become less substantial as the observations become more powerful in distinguishing the classes. Whether the use of higher order features together with kernels brings substantial improvement in performance is likely to be problem dependent. Similarly, observation features that are more distribution invariant such as comprehensive name lists can be used for the NER task we experimented with and may reduce the improvements offered by higher order features. 5 Conclusion The pattern sparsity assumption often holds in real applications, and we give efficient inference algorithms for CRF with high-order features when the pattern sparsity assumption is satisfied. This allows high-order features to be explored in feature engineering for real applications. We studied the conditions that are favourable for using high-order features using a synthetic data set, and demonstrated that using simple high-order features can lead to performance improvement on a handwriting recognition problem and a named entity recognition problem. Acknowledgements This work is supported by DSO grant R-252-000-390-592 and AcRF grant R-252-000-327-112. References [1] B. A. Cipra, “The Ising model is NP-complete,” SIAM News, vol. 33, no. 6, 2000. [2] A. Culotta, D. Kulp, and A. McCallum, “Gene prediction with conditional random fields,” University of Massachusetts, Amherst, Tech. Rep. UM-CS-2005-028, 2005. [3] T. G. Dietterich, A. Ashenfelter, and Y. Bulatov, “Training conditional random fields via gradient tree boosting,” in Proceedings of the Twenty-First International Conference on Machine Learning, 2004. [4] S. Fine, Y. Singer, and N. Tishby, “The hierarchical hidden markov model: Analysis and applications,” Machine Learning, vol. 32, no. 1, pp. 41–62, 1998. [5] C. Huang and A. Darwiche, “Inference in belief networks: A procedural guide,” International Journal of Approximate Reasoning, vol. 15, no. 3, pp. 225–263, 1996. [6] F. Jelinek, J. D. Lafferty, and R. L. Mercer, “Basic methods of probabilistic context free grammars,” in Speech Recognition and Understanding. Recent Advances, Trends, and Applications. Springer Verlag, 1992. [7] R. H. Kassel, “A comparison of approaches to on-line handwritten character recognition,” Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, USA, 1995. [8] J. Lafferty, A. McCallum, and F. Pereira, “Conditional random fields: Probabilistic models for segmenting and labeling sequence data,” in Proceedings of the Eighteenth International Conference on Machine Learning, 2001, pp. 282–289. [9] Linguistic Data Consortium, “ACE (Automatic Content Extraction) English Annotation Guidelines for Entities,” 2005. [10] K. P. Murphy and M. A. Paskin, “Linear-time inference in hierarchical HMMs,” in Advances in Neural Information Processing Systems 14, vol. 14, 2002. [11] X. Qian, X. Jiang, Q. Zhang, X. Huang, and L. Wu, “Sparse higher order conditional random fields for improved sequence labeling,” in ICML, 2009, p. 107. [12] L. R. Rabiner, A tutorial on hidden Markov models and selected applications in speech recognition. San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1990. [13] S. Sarawagi and W. W. Cohen, “Semi-Markov conditional random fields for information extraction,” in Advances in Neural Information Processing Systems 17. Cambridge, MA: MIT Press, 2005, pp. 1185–1192. [14] F. Sha and F. Pereira, “Shallow parsing with conditional random fields,” in Proceedings of the Twentieth International Conference on Machine Learning, 2003, pp. 282–289. [15] B. Taskar, C. Guestrin, and D. Koller, “Max-margin Markov networks,” in Advances in Neural Information Processing Systems 16. Cambridge, MA: MIT Press, 2004. [16] E. Tjong and F. D. Meulder, “Introduction to the CoNLL-2003 shared task: Languageindependent named entity recognition,” in Proceedings of Conference on Computational Natural Language Learning, 2003. [17] T. T. Tran, D. Phung, H. Bui, and S. Venkatesh, “Hierarchical semi-Markov conditional random fields for recursive sequential data,” in NIPS’08: Advances in Neural Information Processing Systems 20. Cambridge, MA: MIT Press, 2008, pp. 1657–1664. [18] I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun, “Support vector machine learning for interdependent and structured output spaces,” in Proceedings of the Twenty-First international conference on Machine learning, 2004, pp. 104–112. [19] G. Wahba, Spline models for observational data, ser. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1990, vol. 59.
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Abstraction and relational learning Charles Kemp & Alan Jern Department of Psychology Carnegie Mellon University {ckemp,ajern}@cmu.edu Abstract Most models of categorization learn categories defined by characteristic features but some categories are described more naturally in terms of relations. We present a generative model that helps to explain how relational categories are learned and used. Our model learns abstract schemata that specify the relational similarities shared by instances of a category, and our emphasis on abstraction departs from previous theoretical proposals that focus instead on comparison of concrete instances. Our first experiment suggests that abstraction can help to explain some of the findings that have previously been used to support comparison-based approaches. Our second experiment focuses on one-shot schema learning, a problem that raises challenges for comparison-based approaches but is handled naturally by our abstraction-based account. Categories such as family, sonnet, above, betray, and imitate differ in many respects but all of them depend critically on relational information. Members of a family are typically related by blood or marriage, and the lines that make up a sonnet must rhyme with each other according to a certain pattern. A pair of objects will demonstrate “aboveness” only if a certain spatial relationship is present, and an event will qualify as an instance of betrayal or imitation only if its participants relate to each other in certain ways. All of the cases just described are examples of relational categories. This paper develops a computational approach that helps to explain how simple relational categories are acquired. Our approach highlights the role of abstraction in relational learning. Given several instances of a relational category, it is often possible to infer an abstract representation that captures what the instances have in common. We refer to these abstract representations as schemata, although others may prefer to call them rules or theories. For example, a sonnet schema might specify the number of lines that a sonnet should include and the rhyming pattern that the lines should follow. Once a schema has been acquired it can support several kinds of inferences. A schema can be used to make predictions about hidden aspects of the examples already observed—if the final word in a sonnet is illegible, the rhyming pattern can help to predict the identity of this word. A schema can be used to decide whether new examples (e.g. new poems) qualify as members of the category. Finally, a schema can be used to generate novel examples of a category (e.g. novel sonnets). Most researchers would agree that abstraction plays some role in relational learning, but Gentner [1] and other psychologists have emphasized the role of comparison instead [2, 3]. Given one example of a sonnet and the task of deciding whether a second poem is also a sonnet, a comparison-based approach might attempt to establish an alignment or mapping between the two. Approaches that rely on comparison or mapping are especially prominent in the literature on analogical reasoning [4, 5], and many of these approaches can be viewed as accounts of relational categorization [6]. For example, the problem of deciding whether two systems are analogous can be formalized as the problem of deciding whether these systems are instances of the same relational category. Despite some notable exceptions [6, 7], most accounts of analogy focus on comparison rather than abstraction, and suggest that “analogy passes from one instance of a generalization to another without pausing for explicit induction of the generalization” (p 95) [8]. 1 0∀Q ∀x ∀y Q(x) < Q(y) ↔D1(x) < D1(y) Observation o Group g Schema s Figure 1: A hierarchical generative model for learning and using relational categories. The schema s at the top level is a logical sentence that specifies which groups are valid instances of the category. The group g at the second level is randomly sampled from the set of valid instances, and the observation o is a partially observed version of group g. Researchers that focus on comparison sometimes discuss abstraction, but typically suggest that abstractions emerge as a consequence of comparing two or more concrete instances of a category [3, 5, 9, 10]. This view, however, will not account for one-shot inferences, or inferences based on a single instance of a relational category. Consider a learner who is shown one instance of a sonnet then asked to create a second instance. Since only one instance is provided, it is hard to see how comparisons between instances could account for success on the task. A single instance, however, will sometimes provide enough information for a schema to be learned, and this schema should allow subsequent instances to be generated [11]. Here we develop a formal framework for exploring relational learning in general and one-shot schema learning in particular. Our framework relies on the hierarchical Bayesian approach, which provides a natural way to combine abstraction and probabilistic inference [12]. The hierarchical Bayesian approach supports representations at multiple levels of abstraction, and helps to explains how abstract representations (e.g. a sonnet schema) can be acquired given observations of concrete instances (e.g. individual sonnets). The schemata we consider are represented as sentences in a logical language, and our approach therefore builds on previous probabilistic methods for learning and using logical theories [13, 14]. Following previous authors, we propose that logical representations can help to capture the content of human knowledge, and that Bayesian inference helps to explain how these representations are acquired and how they support inductive inference. The following sections introduce our framework then evaluate it using two behavioral experiments. Our first experiment uses a standard classification task where participants are shown one example of a category then asked to decide which of two alternatives is more likely to belong to the same category. Tasks of this kind have previously been used to argue for the importance of comparison, but we suggest that these tasks can be handled by accounts that focus on abstraction. Our second experiment uses a less standard generation task [15, 16] where participants are shown a single example of a category then asked to generate additional examples. As predicted by our abstraction-based account, we find that people are able to learn relational categories on the basis of a single example. 1 A generative approach to relational learning Our examples so far have used real-world relational categories such as family and sonnet but we now turn to a very simple domain where relational categorization can be studied. Each element in the domain is a group of components that vary along a number of dimensions—in Figure 1, the components are figures that vary along the dimensions of size, color, and circle position. The groups can be organized into categories—one such category includes groups where every component is black. Although our domain is rather basic it allows some simple relational regularities to be explored. We can consider categories, for example, where all components in a group must be the same along some dimension, and categories where all components must be different along some dimension. We can also consider categories defined by relationships between dimensions—for example, the category that includes all groups where the size and color dimensions are correlated. Each category is associated with a schema, or an abstract representation that specifies which groups are valid instances of the category. Here we consider schemata that correspond to rules formulated 2 1 ∀x ∃x ff Di(x) ˘ =, ̸=, <, > ¯ vk 2 ∀x ∃x ff∀y x ̸= y → ∃y x ̸= y ∧ ff Di(x) ˘ =, ̸=, <, > ¯ Di(y) 3 ∀x Di(x) ˘ =, ̸= ¯ vk 8 < : ∧ ∨ ↔ 9 = ; Dj(x) ˘ =, ̸= ¯ vl 4 ∀x∀y x ̸= y → 0 @Di(x) ˘ =, ̸=, <, > ¯ Di(y) 8 < : ∧ ∨ ↔ 9 = ; Dj(x) ˘ =, ̸=, <, > ¯ Dj(y) 1 A 5 ∀Q ∃Q ff∀x ∃x ff∀y x ̸= y → ∃y x ̸= y ∧ ff Q(x) ˘ =, ̸=, <, > ¯ Q(y) 6 ∀Q Q ̸= Di → ∃Q Q ̸= Di ∧ ff ∀x∀y x ̸= y → 0 @Q(x) ˘ =, ̸=, <, > ¯ Q(y) 8 < : ∧ ∨ ↔ 9 = ; Di(x) ˘ =, ̸=, <, > ¯ Di(y) 1 A 7 ∀Q ∃Q ff∀R Q ̸= R → ∃R Q ̸= R ∧ ff ∀x∀y x ̸= y → 0 @Q(x) ˘ =, ̸=, <, > ¯ Q(y) 8 < : ∧ ∨ ↔ 9 = ; R(x) ˘ =, ̸=, <, > ¯ R(y) 1 A Table 1: Templates used to construct a hypothesis space of logical schemata. An instance of a given template can be created by choosing an element from each set enclosed in braces (some sets are laid out horizontally to save space), replacing each occurrence of Di or Dj with a dimension (e.g. D1) and replacing each occurrence of vk or vl with a value (e.g. 1). in a logical language. The language includes three binary connectives—and (∧), or (∨), and if and only if (↔). Four binary relations (=, ̸=, <, and >) are available for comparing values along dimensions. Universal quantification (∀x) and existential quantification (∃x) are both permitted, and the language includes quantification over objects (∀x) and dimensions (∀Q). For example, the schema in Figure 1 states that all dimensions are aligned. More precisely, if D1 is the dimension of size, the schema states that for all dimensions Q, a component x is smaller than a component y along dimension Q if and only if x is smaller in size than y. It follows that all three dimensions must increase or decrease together. To explain how rules in this logical language are learned we work with the hierarchical generative model in Figure 1. The representation at the top level is a schema s, and we assume that one or more groups g are generated from a distribution P(g|s). Following a standard approach to category learning [17, 18], we assume that g is uniformly sampled from all groups consistent with s: p(g|s) ∝ 1 g is consistent with s 0 otherwise (1) For all applications in this paper, we assume that the number of components in a group is known and fixed in advance. The bottom level of the hierarchy specifies observations o that are generated from a distribution P(o|g). In most cases we assume that g can be directly observed, and that P(o|g) = 1 if o = g and 0 otherwise. We also consider the setting shown in Figure 1 where o is generated by concealing a component of g chosen uniformly at random. Note that the observation o in Figure 1 includes only four of the components in group g, and is roughly analogous to our earlier example of a sonnet with an illegible final word. To convert Figure 1 into a fully-specified probabilistic model it remains to define a prior distribution P(s) over schemata. An appealing approach is to consider all of the infinitely many sentences in the logical language already mentioned, and to define a prior favoring schemata which correspond to simple (i.e. short) sentences. We approximate this approach by considering a large but finite space of sentences that includes all instances of the templates in Table 1 and all conjunctions of these instances. When instantiating one of these templates, each occurrence of Di or Dj should be replaced by one of the dimensions in the domain. For example, the schema in Figure 1 is a simplified instance of template 6 where Di is replaced by D1. Similarly, each instance of vk or vl should be replaced by a value along one of the dimensions. Our first experiment considers a problem where there are are three dimensions and three possible values along each dimension (i.e. vk = 1, 2, or 3). As a result there are 1568 distinct instances of the templates in Table 1 and roughly one million 3 conjunctions of these instances. Our second experiment uses three dimensions with five values along each dimension, which leads to 2768 template instances and roughly three million conjunctions of these instances. The templates in Table 1 capture most of the simple regularities that can be formulated in our logical language. Template 1 generates all rules that include quantification over a single object variable and no binary connectives. Template 3 is similar but includes a single binary connective. Templates 2 and 4 are similar to 1 and 3 respectively, but include two object variables (x and y) rather than one. Templates 5, 6 and 7 add quantification over dimensions to Templates 2 and 4. Although the templates in Table 1 capture a large class of regularities, several kinds of templates are not included. Since we do not assume that the dimensions are commensurable, values along different dimensions cannot be directly compared (∃x D1(x) = D2(x) is not permitted. For the same reason, comparisons to a dimension value must involve a concrete dimension (∀x D1(x) = 1 is permitted) rather than a dimension variable (∀Q ∀x Q(x) = 1 is not permitted). Finally, we exclude all schemata where quantification over objects precedes quantification over dimensions, and as a result there are some simple schemata that our implementation cannot learn (e.g. ∃x∀y∃Q Q(x) = Q(y)). The extension of each schema is a set of groups, and schemata with the same extension can be assigned to the same equivalence class. For example, ∀x D1(x) = v1 (an instance of template 1) and ∀x D1(x) = v1 ∧D1(x) = v1 (an instance of template 3) end up in the same equivalence class. Each equivalence class can be represented by the shortest sentence that it contains, and we define our prior P(s) over a set that includes a single representative for each equivalence class. The prior probability P(s) of each sentence is inversely proportional to its length: P(s) ∝λ|s|, where |s| is the length of schema s and λ is a constant between 0 and 1. For all applications in this paper we set λ = 0.8. The generative model in Figure 1 can be used for several purposes, including schema learning (inferring a schema s given one or more instances generated from the schema), classification (deciding whether group gnew belongs to a category given one or more instances of the category) and generation (generating a group gnew that belongs to the same category as one or more instances). Our first experiment explores all three of these problems. 2 Experiment 1: Relational classification Our first experiment is organized around a triad task where participants are shown one example of a category then asked to decide which of two choice examples is more likely to belong to the category. Triad tasks are regularly used by studies of relational categorization, and have been used to argue for the importance of comparison [1]. A comparison-based approach to this task, for instance, might compare the example object to each of the choice objects in order to decide which is the better match. Our first experiment is intended in part to explore whether a schema-learning approach can also account for inferences about triad tasks. Materials and Method. 18 adults participated for course credit and interacted with a custom-built computer interface. The stimuli were groups of figures that varied along three dimensions (color, size, and ball position, as in Figure 1). Each shape was displayed on a single card, and all groups in Experiment 1 included exactly three cards. The cards in Figure 1 show five different values along each dimension, but Experiment 1 used only three values along each dimension. The experiment included inferences about 10 triads. Participants were told that aliens from a certain planet “enjoy organizing cards into groups,” and that “any group of cards will probably be liked by some aliens and disliked by others.” The ten triad tasks were framed as questions about the preferences of 10 aliens. Participants were shown a group that Mr X likes (different names were used for the ten triads), then shown two choice groups and told that “Mr X likes one of these groups but not the other.” Participants were asked to select one of the choice groups, then asked to generate another 3-card group that Mr X would probably like. Cards could be added to the screen using an “Add Card” button, and there were three pairs of buttons that allowed each card to be increased or decreased along the three dimensions. Finally, participants were asked to explain in writing “what kind of groups Mr X likes.” The ten triads used are shown in Figure 2. Each group is represented as a 3 by 3 matrix where rows represent cards and columns show values along the three dimensions. Triad 1, for example, 4 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 0.5 1 311 321 331 311 331 331 112 212 312 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 0.5 1 311 232 123 211 232 123 331 122 213 0.5 1 312 312 312 313 312 312 133 133 133 0.5 1 111 212 313 211 312 113 311 322 333 0.5 1 211 222 233 211 232 223 311 122 333 0.5 1 331 323 333 231 323 333 321 332 313 4 8 12 16 20 24 0.5 1 0.5 1 311 113 313 311 113 323 132 332 233 0.5 1 311 212 113 111 312 213 331 322 313 0.5 1 321 222 123 321 122 223 331 212 133 0.5 1 231 222 213 231 322 213 231 132 333 (j) Some dimension has no repeats (g) All dimensions uniform (e) Two dimensions aligned (c) D2 and D3 aligned (a) D1 value always 3 (b) D2 uniform (d) D1 and D3 anti-aligned (f) Two dimensions anti-aligned (h) Some dimension uniform (i) All dimensions have no repeats Figure 2: Human responses and model predictions for the ten triads in Experiment 1. The plot at the left of each panel shows model predictions (white bars) and human preferences (black bars) for the two choice groups in each triad. The plots at the right of each panel summarize the groups created during the generation phase. The 23 elements along the x-axis correspond to the regularities listed in Table 2. 5 1 All dimensions aligned 13 One dimension has no repeats 2 Two dimensions aligned 14 D1 has no repeats 3 D1 and D2 aligned 15 D2 has no repeats 4 D1 and D3 aligned 16 D3 has no repeats 5 D2 and D3 aligned 17 All dimensions uniform 6 All dimensions aligned or anti-aligned 18 Two dimensions uniform 7 Two dimensions anti-aligned 19 One dimension uniform 8 D1 and D2 anti-aligned 20 D1 uniform 9 D1 and D3 anti-aligned 21 D2 uniform 10 D2 and D3 anti-aligned 22 D3 uniform 11 All dimensions have no repeats 23 D1 value is always 3 12 Two dimensions have no repeats Table 2: Regularities used to code responses to the generation tasks in Experiments 1 and 2 has an example group including three cards that each take value 3 along D1. The first choice group is consistent with this regularity but the second choice group is not. The cards in each group were arrayed vertically on screen, and were initially sorted as shown in Figure 2 (i.e. first by D3, then by D2 and then by D1). The cards could be dragged around on screen, and participants were invited to move them around in order to help them understand each group. The mapping between the three dimensions in each matrix and the three dimensions in the experiment (color, position, and size) was randomized across participants, and the order in which triads were presented was also randomized. Model predictions and results. Let ge be the example group presented in the triad task and g1 and g2 be the two choice groups. We use our model to compute the relative probability of two hypotheses: h1 which states that ge and g1 are generated from the same schema and that g2 is sampled randomly from all possible groups, and h2 which states that ge and g2 are generated from the same schema. We set P(h1) = P(h2) = 0.5, and compute posterior probabilities P(h1|ge, g1, g2) and P(h2|ge, g1, g2) by integrating over all schemata in the hypothesis space already described. Our model assumes that two groups are considered similar to the extent that they appear to have been generated by the same underlying schema, and is consistent with the generative approach to similarity described by Kemp et al. [19]. Model predictions for the ten triads are shown in Figure 2. In each case, the choice probabilities plotted (white bars) are the posterior probabilities of hypotheses h1 and h2. In nine out of ten cases the best choice according to the model is the most common human response. Responses to triads 2c and 2d support the idea that people are sensitive to relationships between dimensions (i.e. alignment and anti-alignment). Triads 2e and 2f are similar to triads studied by Kotovsky and Gentner [1], and we replicate their finding that people are sensitive to relationships between dimensions even when the dimensions involved vary from group to group. The one case where human responses diverge from model predictions is shown in Figure 2h. Note that the schema for this triad involves existential quantification over dimensions (some dimension is uniform), and according to our prior P(s) this kind of quantification is no more complex than other kinds of quantification. Future applications of our approach can explore the idea that existential quantification over dimensions (∃Q) is psychologically more complex than universal quantification over dimensions (∀Q) or existential quantification over cards (∃x), and can consider logical languages that incorporate this inductive bias. To model the generation phase of the experiment we computed the posterior distribution P(gnew|ge, g1, g2) = X s,h P(gnew|s)P(s|h, ge, g1, g2)P(h|ge, g1, g2) where P(h|ge, g1, g2) is the distribution used to model selections in the triad task. Since the space of possible groups is large, we visualize this distribution using a profile that shows the posterior probability assigned to groups consistent with the 23 regularities shown in Table 2. The white bar plots in Figure 2 show profiles predicted by the model, and the black plots immediately above show profiles computed over the groups generated by our 18 participants. In many of the 10 cases the model accurately predicts regularities in the groups generated by people. In case 2c, for example, the model correctly predicts that generated groups will tend to have no repeats along dimensions D2 and D3 (regularities 15 and 16) and that these two dimensions will be aligned (regularities 2 and 5). There are, however, some departures from the model’s predictions, and a notable example occurs in case 2d. Here the model detects the regularity that dimensions D1 and D3 are anti-aligned (regularity 9). Some groups generated by participants are consistent with 6 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 0.5 1 344 322 312 354 311 322 333 355 311 322 333 354 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 4 8 12 16 20 24 0.5 1 0.5 1 222 555 223 333 111 333 444 555 121 232 443 555 0.5 1 424 224 324 514 124 224 324 524 0.5 1 314 315 214 414 314 314 314 314 0.5 1 423 443 324 314 251 532 314 145 0.5 1 432 134 433 335 431 433 135 335 (e) All dimensions uniform (a) All dimensions aligned (c) D1 has no repeats, D2 and D3 uniform (b) D2 and D3 aligned (f) All dimensions have no repeats (d) D2 uniform Figure 3: Human responses and model predictions for the six cases in Experiment 2. In (a) and (b), the 4 cards used for the completion and generation phases are shown on either side of the dashed line (completion cards on the left). In the remaining cases, the same 4 cards were used for both phases. The plots at the right of each panel show model predictions (white bars) and human responses (black bars) for the generation task. In each case, the 23 elements along each x-axis correspond to the regularities listed in Table 2. The remaining plots show responses to the completion task. There are 125 possible responses, and the four responses shown always include the top two human responses and the top two model predictions. this regularity, but people also regularly generate groups where two dimensions are aligned rather than anti-aligned (regularity 2). This result may indicate that some participants are sensitive to relationships between dimensions but do not consider the difference between a positive relationship (alignment) and an inverse relationship (anti-alignment) especially important. Kotovsky and Gentner [1] suggest that comparison can explain how people respond to triad tasks, although they do not provide a computational model that can be compared with our approach. It is less clear how comparison might account for our generation data, and our next experiment considers a one-shot generation task that raises even greater challenges for a comparison-based approach. 3 Experiment 2: One-shot schema learning As described already, comparison involves constructing mappings between pairs of category instances. In some settings, however, learners make confident inferences given a single instance of a category [15, 20], and it is difficult to see how comparison could play a major role when only one instance is available. Models that rely on abstraction, however, can naturally account for one-shot relational learning, and we designed a second experiment to evaluate this aspect of our approach. 7 Several previous studies have explored one-shot relational learning. Holyoak and Thagard [21] developed a study of analogical reasoning using stories as stimuli and found little evidence of oneshot schema learning. Ahn et al. [11] demonstrated, however, that one-shot learning can be achieved with complex materials such as stories, and modeled this result using explanation-based learning. Here we use much simpler stimuli and explore a probabilistic approach to one-shot learning. Materials and Method. 18 adults participated for course credit. The same individuals completed Experiments 1 and 2, and Experiment 2 was always run before Experiment 1. The same computer interface was used in both experiments, and the only important difference was that the figures in Experiment 2 could now take five values along each dimension rather than three. The experiment included two phases. During the generation phase, participants saw a 4-card group that Mr X liked and were asked to generate two 5-card groups that Mr X would probably like. During the completion phase, participants were shown four members of a 5-card group and were asked to generate the missing card. The stimuli used in each phase are shown in Figure 3. In the first two cases, slightly different stimuli were used in the generation and completion phases, and in all remaining cases the same set of four cards was used in both cases. All participants responded to the six generation questions before answering the six completion questions. Model predictions and results. The generation phase is modeled as in Experiment 1, but now the posterior distribution P(gnew|ge) is computed after observing a single instance of a category. The human responses in Figure 3 (white bars) are consistent with the model in all cases, and confirm that a single example can provide sufficient evidence for learners to acquire a relational category. For example, the most common response in case 3a was the 5-card group shown in Figure 1—a group with all three dimensions aligned. To model the completion phase, let oe represent a partial observation of group ge. Our model infers which card is missing from ge by computing the posterior distribution P(ge|oe) ∝ P(oe|ge) P s P(ge|s)P(s), where P(oe|ge) captures the idea that oe is generated by randomly concealing one component of ge. The white bars in Figure 3 show model predictions, and in five out of six cases the best response according to the model is the same as the most common human response. In the remaining case (Figure 3d) the model generates a diffuse distribution over all cards with value 3 on dimension 2, and all human responses satisfy this regularity. 4 Conclusion We presented a generative model that helps to explain how relational categories are learned and used. Our approach captures relational regularities using a logical language, and helps to explain how schemata formulated in this language can be learned from observed data. Our approach differs in several respects from previous accounts of relational categorization [1, 5, 10, 22]. First, we focus on abstraction rather than comparison. Second, we consider tasks where participants must generate examples of categories [16] rather than simply classify existing examples. Finally, we provide a formal account that helps to explain how relational categories can be learned from a single instance. Our approach can be developed and extended in several ways. For simplicity, we implemented our model by working with a finite space of several million schemata, but future work can consider hypothesis spaces that assign non-zero probability to all regularities that can be formulated in the language we described. The specific logical language used here is only a starting point, and future work can aim to develop languages that provide a more faithful account of human inductive biases. Finally, we worked with a domain that provides one of the simplest ways to address core questions such as one-shot learning. Future applications of our general approach can consider domains that include more than three dimensions and a richer space of relational regularities. Relational learning and analogical reasoning are tightly linked, and hierarchical generative models provide a promising approach to both problems. We focused here on relational categorization, but future studies can explore whether probabilistic accounts of schema learning can help to explain the inductive inferences typically considered by studies of analogical reasoning. Although there are many models of analogical reasoning, there are few that pursue a principled probabilistic approach, and the hierarchical Bayesian approach may help to fill this gap in the literature. Acknowledgments We thank Maureen Satyshur for running the experiments. This work was supported in part by NSF grant CDI-0835797. 8 References [1] L. Kotovsky and D. Gentner. Comparison and categorization in the development of relational similarity. Child Development, 67:2797–2822, 1996. [2] D. Gentner and A. B. Markman. Structure mapping in analogy and similarity. American Psychologist, 52:45–56, 1997. [3] D. Gentner and J. Medina. Similarity and the development of rules. Cognition, 65:263–297, 1998. [4] B. Falkenhainer, K. D. Forbus, and D. Gentner. The structure-mapping engine: Algorithm and examples. Artificial Intelligence, 41:1–63, 1989. [5] J. E. Hummel and K. J. Holyoak. A symbolic-connectionist theory of relational inference and generalization. Psychological Review, 110:220–264, 2003. [6] M. Mitchell. Analogy-making as perception: a computer model. MIT Press, Cambridge, MA, 1993. [7] D. R. Hofstadter and the Fluid Analogies Research Group. Fluid concepts and creative analogies: computer models of the fundamental mechanisms of thought. 1995. [8] W. V. O. Quine and J. Ullian. The Web of Belief. Random House, New York, 1978. [9] J. Skorstad, D. Gentner, and D. Medin. Abstraction processes during concept learning: a structural view. In Proceedings of the 10th Annual Conference of the Cognitive Science Society, pages 419–425. 2009. [10] D. Gentner and J. Loewenstein. Relational language and relational thought. In E. Amsel and J. P. Byrnes, editors, Language, literacy and cognitive development: the development and consequences of symbolic communication, pages 87–120. 2002. [11] W. Ahn, W. F. Brewer, and R. J. Mooney. Schema acquisition from a single example. Journal of Experimental Psychology: Learning, Memory and Cognition, 18(2):391–412, 1992. [12] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin. Bayesian data analysis. Chapman & Hall, New York, 2nd edition, 2003. [13] C. Kemp, N. D. Goodman, and J. B. Tenenbaum. Learning and using relational theories. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 753–760. MIT Press, Cambridge, MA, 2008. [14] S. Kok and P. Domingos. Learning the structure of Markov logic networks. In Proceedings of the 22nd International Conference on Machine Learning, 2005. [15] J. Feldman. The structure of perceptual categories. Journal of Mathematical Psychology, 41: 145–170, 1997. [16] A. Jern and C. Kemp. Category generation. In Proceedings of the 31st Annual Conference of the Cognitive Science Society, pages 130–135. Cognitive Science Society, Austin, TX, 2009. [17] D. Conklin and I. H. Witten. Complexity-based induction. Machine Learning, 16(3):203–225, 1994. [18] J. B. Tenenbaum and T. L. Griffiths. Generalization, similarity, and Bayesian inference. Behavioral and Brain Sciences, 24:629–641, 2001. [19] C. Kemp, A. Bernstein, and J. B. Tenenbaum. A generative theory of similarity. In B. G. Bara, L. Barsalou, and M. Bucciarelli, editors, Proceedings of the 27th Annual Conference of the Cognitive Science Society, pages 1132–1137. Lawrence Erlbaum Associates, 2005. [20] C. Kemp, N. D. Goodman, and J. B. Tenenbaum. Theory acquisition and the language of thought. In Proceedings of the 30th Annual Conference of the Cognitive Science Society, pages 1606–1611. Cognitive Science Society, Austin, TX, 2008. [21] K. J. Holyoak and P. Thagard. Analogical mapping by constraint satisfaction. Cognitive Science, 13(3):295–355, 1989. [22] L. A. A. Doumas, J. E. Hummel, and C. M. Sandhofer. A theory of the discovery and predication of relational concepts. Psychological Review, 115(1):1–43, 2008. [23] M. L. Gick and K. J. Holyoak. Schema induction and analogical transfer. Cognitive Psychology, 15:1–38, 1983. 9
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Fast Graph Laplacian Regularized Kernel Learning via Semidefinite–Quadratic–Linear Programming Xiao-Ming Wu Dept. of IE The Chinese University of Hong Kong wxm007@ie.cuhk.edu.hk Anthony Man-Cho So Dept. of SE&EM The Chinese University of Hong Kong manchoso@se.cuhk.edu.hk Zhenguo Li Dept. of IE The Chinese University of Hong Kong zgli@ie.cuhk.edu.hk Shuo-Yen Robert Li Dept. of IE The Chinese University of Hong Kong bobli@ie.cuhk.edu.hk Abstract Kernel learning is a powerful framework for nonlinear data modeling. Using the kernel trick, a number of problems have been formulated as semidefinite programs (SDPs). These include Maximum Variance Unfolding (MVU) (Weinberger et al., 2004) in nonlinear dimensionality reduction, and Pairwise Constraint Propagation (PCP) (Li et al., 2008) in constrained clustering. Although in theory SDPs can be efficiently solved, the high computational complexity incurred in numerically processing the huge linear matrix inequality constraints has rendered the SDP approach unscalable. In this paper, we show that a large class of kernel learning problems can be reformulated as semidefinite-quadratic-linear programs (SQLPs), which only contain a simple positive semidefinite constraint, a second-order cone constraint and a number of linear constraints. These constraints are much easier to process numerically, and the gain in speedup over previous approaches is at least of the order m2.5, where m is the matrix dimension. Experimental results are also presented to show the superb computational efficiency of our approach. 1 Introduction Kernel methods provide a principled framework for nonlinear data modeling, where the inference in the input space can be transferred intactly to any feature space by simply treating the associated kernel as inner products, or more generally, as nonlinear mappings on the data (Sch¨olkopf & Smola, 2002). Some well-known kernel methods include support vector machines (SVMs) (Vapnik, 2000), kernel principal component analysis (kernel PCA) (Sch¨olkopf et al., 1998), and kernel k-means (Shawe-Taylor & Cristianini, 2004). Naturally, an important issue in kernel methods is kernel design. Indeed, the performance of a kernel method depends crucially on the kernel used, where different choices of kernels often lead to quite different results. Therefore, substantial efforts have been made to design appropriate kernels for the problems at hand. For instance, in (Chapelle & Vapnik, 2000), parametric kernel functions are proposed, where the focus is on model selection (Chapelle & Vapnik, 2000). The modeling capability of parametric kernels, however, is limited. A more natural idea is to learn specialized nonparametric kernels for specific problems. For instance, in cases where only inner products of the input data are involved, kernel learning is equivalent to the learning of a kernel matrix. This is the main focus of recent kernel methods. Currently, many different kernel learning frameworks have been proposed. These include spectral kernel learning (Li & Liu, 2009), multiple kernel learning (Lanckriet et al., 2004), and the Breg1 man divergence-based kernel learning (Kulis et al., 2009). Typically, a kernel learning framework consists of two main components: the problem formulation in terms of the kernel matrix, and an optimization procedure for finding the kernel matrix that has certain desirable properties. As seen in, e.g., the Maximum Variance Unfolding (MVU) method (Weinberger et al., 2004) for nonlinear dimensionality reduction (see (So, 2007) for related discussion) and Pairwise Constraint Propagation (PCP) (Li et al., 2008) for constrained clustering, a nice feature of such a framework is that the problem formulation often becomes straightforward. Thus, the major challenge in optimizationbased kernel learning lies in the second component, where the key is to find an efficient procedure to obtain a positive semidefinite kernel matrix that satisfies certain properties. Using the kernel trick, most kernel learning problems (Graepel, 2002; Weinberger et al., 2004; Globerson & Roweis, 2007; Song et al., 2008; Li et al., 2008) can naturally be formulated as semidefinite programs (SDPs). Although in theory SDPs can be efficiently solved, the high computational complexity has rendered the SDP approach unscalable. An effective and widely used heuristic for speedup is to perform low-rank kernel approximation and matrix factorization (Weinberger et al., 2005; Weinberger et al., 2007; Li et al., 2009). In this paper, we investigate the possibility of further speedup by studying a class of convex quadratic semidefinite programs (QSDPs). These QSDPs arise in many contexts, such as graph Laplacian regularized low-rank kernel learning in nonlinear dimensionality reduction (Sha & Saul, 2005; Weinberger et al., 2007; Globerson & Roweis, 2007; Song et al., 2008; Singer, 2008) and constrained clustering (Li et al., 2009). In the aforementioned papers, a QSDP is reformulated as an SDP with O(m2) variables and a linear matrix inequality of size O(m2) × O(m2). Such a reformulation is highly inefficient and unscalable, as it has an order of m9 time complexity (Ben-Tal & Nemirovski, 2001, Lecture 6). In this paper, we propose a novel reformulation that exploits the structure of the QSDP and leads to a semidefinite-quadratic-linear program (SQLP) that can be solved by the standard software SDPT3 (T¨ut¨unc¨u et al., 2003). Such a reformulation has the advantage that it only has one positive semidefinite constraint on a matrix of size m × m, one second-order cone constraint of size O(m2) and a number of linear constraints on O(m2) variables. As a result, our reformulation is much easier to process numerically. Moreover, a simple complexity analysis shows that the gain in speedup over previous approaches is at least of the order m2.5. Experimental results show that our formulation is indeed far more efficient than previous ones. The rest of the paper is organized as follows. We review related kernel learning problems in Section 2 and present our formulation in Section 3. Experiment results are reported in Section 4. Section 5 concludes the paper. 2 The Problems In this section, we briefly review some kernel learning problems that arise in dimensionality reduction and constrained clustering. They include MVU (Weinberger et al., 2004), Colored MVU (Song et al., 2008), (Singer, 2008), Pairwise Semidefinite Embedding (PSDE) (Globerson & Roweis, 2007), and PCP (Li et al., 2008). MVU maximizes the variance of the embedding while preserving local distances of the input data. Colored MVU generalizes MVU with side information such as class labels. PSDE derives an embedding that strictly respects known similarities, in the sense that objects known to be similar are always closer in the embedding than those known to be dissimilar. PCP is designed for constrained clustering, which embeds the data on the unit hypersphere such that two objects that are known to be from the same cluster are embedded to the same point, while two objects that are known to be from different clusters are embedded orthogonally. In particular, PCP seeks the smoothest mapping for such an embedding, thereby propagating pairwise constraints. Initially, each of the above problems is formulated as an SDP, whose kernel matrix K is of size n×n, where n denotes the number of objects. Since such an SDP is computationally expensive, one can try to reduce the problem size by using graph Laplacian regularization. In other words, one takes K = QY QT , where Q ∈Rn×m consists of the smoothest m eigenvectors of the graph Laplacian (m ≪n), and Y is of size m × m (Sha & Saul, 2005; Weinberger et al., 2007; Song et al., 2008; Globerson & Roweis, 2007; Singer, 2008; Li et al., 2009). The learning of K is then reduced to the learning of Y , leading to a quadratic semidefinite program (QSDP) that is similar to a standard quadratic program (QP), except that the feasible set of a QSDP resides in the positive semidefinite cone as well. The intuition behind this low-rank kernel approximation is that a kernel matrix of the 2 form K = QY QT actually, to some degree, preserves the proximity of objects in the feature space. Detailed justification can be found in the related work mentioned above. Next, we use MVU and PCP as representatives to demonstrate how the SDP formulations emerge from nonlinear dimensionality reduction and constrained clustering. 2.1 MVU The SDP of MVU (Weinberger et al., 2004) is as follows: max K tr(K) = I • K (1) s.t. n X i,j=1 kij = 0, (2) kii + kjj −2kij = d2 ij, ∀(i, j) ∈N, (3) K ⪰0, (4) where K = (kij) denotes the kernel matrix to be learned, I denotes the identity matrix, tr(·) denotes the trace of a square matrix, • denotes the element-wise dot product between matrices, dij denotes the Euclidean distance between the i-th and j-th objects, and N denotes the set of neighbor pairs, whose distances are to be preserved in the embedding. The constraint in (2) centers the embedding at the origin, thus removing the translation freedom. The constraints in (3) preserve local distances. The constraint K ⪰0 in (4) specifies that K must be symmetric and positive semidefinite, which is necessary since K is taken as the inner product matrix of the embedding. Note that given the constraint in (2), the variance of the embedding is characterized by V(K) = 1 2n P i,j(kii + kjj −2kij) = tr(K) (Weinberger et al., 2004) (see related discussion in (So, 2007), Chapter 4). Thus, the SDP in (1-4) maximizes the variance of the embedding while keeping local distances unchanged. After K is obtained, kernel PCA is applied to K to compute the low-dimensional embedding. 2.2 PCP The SDP of PCP (Li et al., 2008) is: min K ¯L • K (5) s.t. kii = 1, i = 1, 2, . . . , n, (6) kij = 1, ∀(i, j) ∈M, (7) kij = 0, ∀(i, j) ∈C, (8) K ⪰0, (9) where ¯L denotes the normalized graph Laplacian, M denotes the set of object pairs that are known to be from the same cluster, and C denotes those that are known to be from different clusters. The constraints in (6) map the objects to the unit hypersphere. The constraints in (7) map two objects that are known to be from the same cluster to the same vector. The constraints in (8) map two objects that are known to be from different clusters to vectors that are orthogonal. Let X = {xi}n i=1 be the data set, F be the feature space, and φ : X →F be the associated feature map of K. Then, the degree of smoothness of φ on the data graph can be captured by (Zhou et al., 2004): S(φ) = 1 2 n X i,j=1 wij °°°°° φ(xi) √dii −φ(xj) p djj °°°°° 2 F = ¯L • K, (10) where wij is the similarity of xi and xj, dii = Pn j=1 wij, and ∥· ∥F is the distance metric in F. The smaller the value S(φ), the smoother is the feature map φ. Thus, the SDP in (5-9) seeks the smoothest feature map that embeds the data on the unit hypersphere and at the same time respects the pairwise constraints. After K is solved, kernel k-means is then applied to K to obtain the clusters. 3 2.3 Low-Rank Approximation: from SDP to QSDP The SDPs in MVU and PCP are difficult to solve efficiently because their computational complexity scales at least cubically in the size of the matrix variable and the number of constraints (Borchers, 1999). A useful heuristic is to use low-rank kernel approximation, which is motivated by the observation that the degree of freedom in the data is often much smaller than the number of parameters in a fully nonparametric kernel matrix K. For instance, it may be equal to or slightly larger than the intrinsic dimension of the data manifold (for dimensionality reduction) or the number of clusters (for clustering). Another more specific observation is that it is often desirable to have nearby objects mapped to nearby points, as is done in MVU or PCP. Based on these observations, instead of learning a fully nonparametric K, one learns a K of the form K = QY QT , where Q and Y are of sizes n×m and m×m, respectively, where m ≪n. The matrix Q should be smooth in the sense that nearby objects in the input space are mapped to nearby points (the i-th row of Q is taken as a new representation of xi). Q is computed prior to the learning of K. In this way, the learning of a kernel matrix K is reduced to the learning of a much smaller Y , subject to the constraint that Y ⪰0. This idea is used in (Weinberger et al., 2007) and (Li et al., 2009) to speed up MVU and PCP, respectively, and is also adopted in Colored MVU (Song et al., 2008) and PSDE (Globerson & Roweis, 2007) for efficient computation. The choice of Q can be different for MVU and PCP. In (Weinberger et al., 2007), Q = (v2, . . . , vm+1), where {vi} are the eigenvectors of the graph Laplacian. In (Li et al., 2009), Q = (u1, . . . , um), where {ui} are the eigenvectors of the normalized graph Laplacian. Since such Q’s are obtained from graph Laplacians, we call the learning of K of the form K = QY QT the Graph Laplacian Regularized Kernel Learning problem, and denote the methods in (Weinberger et al., 2007) and (Li et al., 2009) by RegMVU and RegPCP, respectively. With K = QY QT , RegMVU and RegPCP become: RegMVU : max Y ⪰0 tr(Y ) −ν X (i,j)∈N ((QY QT )ii −2(QY QT )ij + (QY QT )jj −d2 ij)2, (11) RegPCP : min Y ⪰0 X (i,j,tij)∈S ((QY QT )ij −tij)2, (12) where ν > 0 is a regularization parameter and S = {(i, j, tij) | (i, j) ∈M ∪C, or i = j, tij = 1 if (i, j) ∈M or i = j, tij = 0 if (i, j) ∈C}. Both RegMVU and RegPCP can be succinctly rewritten in the unified form: min y yT Ay + bT y (13) s.t. Y ⪰0, (14) where y = vec(Y ) ∈Rm2 denotes the vector obtained by concatenating all the columns of Y , and A ⪰0 (Weinberger et al., 2007; Li et al., 2009). Note that this problem is convex since both the objective function and the feasible set are convex. Problem (13-14) is an instance of the so-called convex quadratic semidefinite program (QSDP), where the objective is a quadratic function in the matrix variable Y . Note that similar QSDPs arise in Colored MVU, PSDE, Conformal Eigenmaps (Sha & Saul, 2005), Locally Rigid Embedding (Singer, 2008), and Kernel Matrix Completion (Graepel, 2002). Before we present our approach for tackling the QSDP (13-14), let us briefly review existing approaches in the literature. 2.4 Previous Approach: from QSDP to SDP Currently, a typical approach for tackling a QSDP is to use the Schur complement (Boyd & Vandenberghe, 2004) to rewrite it as an SDP (Sha & Saul, 2005; Weinberger et al., 2007; Li et al., 2009; Song et al., 2008; Globerson & Roweis, 2007; Singer, 2008; Graepel, 2002), and then solve it using an SDP solver such as CSDP1 (Borchers, 1999) or SDPT32 (Toh et al., 2006). In this paper, we call 1https://projects.coin-or.org/Csdp/ 2http://www.math.nus.edu.sg/˜mattohkc/sdpt3.html 4 this approach the Schur Complement Based SDP (SCSDP) formulation. For the QSDP in (13-14), the equivalent SDP takes the form: min y,ν ν + bT y (15) s.t. Y ⪰0 and · Im2 A 1 2 y (A 1 2 y)T ν ¸ ⪰0, (16) where A 1 2 is the matrix square root of A, Im2 is the identity matrix of size m2 ×m2, and ν is a slack variable serving as an upper bound of yT Ay. The second semidefinite cone constraint is equivalent to (A 1 2 y)T (A 1 2 y) ≤ν by the Schur complement. Although the SDP in (15-16) has only m(m + 1)/2 + 1 variables, it has two semidefinite cone constraints, of sizes m×m and (m2+1)×(m2+1), respectively. Such an SDP not only scales poorly, but is also difficult to process numerically. Indeed, by considering Problem (15-16) as an SDP in the standard dual form, the number of iterations required by standard interior-point algorithms is of the order m, and the total number of arithmetic operations required is of the order m9 (Ben-Tal & Nemirovski, 2001, Lecture 6). In practice, it takes only a few seconds to solve the aforementioned SDP when m = 10, but can take more than 1 day when m = 40 (see Section 4 for details). Thus, it is not surprising that m is set to a very small value in the related methods—for example, m = 10 in (Weinberger et al., 2007) and m = 15 in (Li et al., 2009). However, as noted by the authors in (Weinberger et al., 2007), a larger m does lead to better performance. In (Li et al., 2009), the authors suggest that m should be larger than the number of clusters. Is this formulation from QSDP to SDP the best we can have? The answer is no. In the next section, we present a novel formulation that leads to a semidefinite-quadratic-linear program (SQLP), which is much more efficient and scalable than the one above. For instance, it takes about 15 seconds when m = 30, 2 minutes when m = 40, and 1 hour when m = 100, as reported in Section 4. 3 Our Formulation: from QSDP to SQLP In this section, we formulate the QSDP in (13-14) as an SQLP. Though our focus here is on the QSDP in (13-14), we should point out that our method applies to any convex QSDP. Recall that the size of A is m2 × m2. Let r be the rank of A. With Cholesky factorization, we can obtain an r ×m2 matrix B such that A = BT B, as A is symmetric positive semidefinite and of rank r (Golub & Loan, 1996). Now, let z = By. Then, the QSDP in (13-14) is equivalent to: min y,z,µ µ + bT y (17) s.t. z = By, (18) zT z ≤µ, (19) Y ⪰0. (20) Next, we show that the constraint in (19) is equivalent to a second-order cone constraint. Let Kn be the second-order cone of dimension n, i.e., Kn = {(x0; x) ∈Rn : x0 ≥∥x∥}, where ∥· ∥denotes the standard Euclidean norm. Let u = ( 1+µ 2 , 1−µ 2 , zT )T . Then, the following holds. Theorem 3.1. zT z ≤µ if and only if u ∈Kr+2. Proof. Note that u ∈Rr+2, since z ∈Rr. Also, note that µ = ( 1+µ 2 )2 −( 1−µ 2 )2. If zT z ≤µ, then ( 1+µ 2 )2 −( 1−µ 2 )2 = µ ≥zT z, which means that 1+µ 2 ≥∥( 1−µ 2 , zT )T ∥. In particular, we have u ∈Kr+2. Conversely, if u ∈Kr+2, then ( 1+µ 2 )2 ≥( 1−µ 2 )2 + zT z, thus implying zT z ≤µ. Let ei (where i = 1, 2, . . . , r + 2) be the i-th basis vector, and let C = (0r×2, Ir×r). Then, we have (e1 −e2)T u = µ, (e1 + e2)T u = 1, and z = Cu. Hence, by Theorem 3.1, the problem in (17-20) 5 swiss roll sample (n=2000) (a) (b) Figure 1: Swiss Roll. (a) The true manifold. (b) A set of 2000 points sampled from the manifold. is equivalent to: min y,u (e1 −e2)T u + bT y (21) s.t. (e1 + e2)T u = 1, (22) By −Cu = 0, (23) u ∈Kr+2, (24) Y ⪰0, (25) which is an instance of the SQLP problem (T¨ut¨unc¨u et al., 2003). Note that in this formulation, we have traded the semidefinite cone constraint of size (m2 + 1) × (m2 + 1) in (16) with one second-order cone constraint of size r + 2 and r + 1 linear constraints. As it turns out, such a formulation is much easier to process numerically and can be solved much more efficiently. Indeed, using standard interior-point algorithms, the number of iterations required is of the order √m (BenTal & Nemirovski, 2001, Lecture 6), and the total number of arithmetic operations required is of the order m6.5 (T¨ut¨unc¨u et al., 2003). This compares very favorably with the m9 arithmetic complexity of the SCSDP approach, and our experimental results indicate that the speedup in computation is quite substantial. Moreover, in contrast with the SCSDP formulation, which does not take advantage of the low rank structure of A, our formulation does take advantage of such a structure. 4 Experimental Results In this section, we perform several experiments to demonstrate the viability of our SQLP formulation and its superior computational performance. Since both the SQLP formulation and the previous SCSDP formulation can be solved by standard softwares to a satisfying gap tolerance, the focus in this comparison is not on the accuracy aspect but on the computational efficiency aspect. We set the relative gap tolerance for both algorithms to be 1e-08. We use SDPT3 (Toh et al., 2006; T¨ut¨unc¨u et al., 2003) to solve the SQLP. We follow (Weinberger et al., 2007; Li et al., 2009) and use CSDP 6.0.1 (Borchers, 1999) to solve the SCSDP. All experiments are conducted in Matlab 7.6.0(R2008a) on a PC with 2.5GHz CPU and 4GB RAM. Two benchmark databases, Swiss Roll3 and USPS4 are used in our experiments. Swiss Roll (Figure 1(a)) is a standard manifold model used for manifold learning and nonlinear dimensionality reduction. In the experiments, we use the data set shown in Figure 1(b), which consists of 2000 points sampled from the Swiss Roll manifold. USPS is a handwritten digits database widely used for clustering and classification. It contains images of handwritten digits from 0 to 9 of size 16×16, and has 7291 training examples and 2007 test examples. In the experiments, we use a subset of USPS with 2000 images, containing the first 200 examples of each digit from 0-9 in the training data. The feature to represent each image is a vector formed by concatenating all the columns of the image intensities. In the sequel, we shall refer to the two subsets used in the experiments simply as Swiss Roll and USPS. 3http://www.cs.toronto.edu/˜roweis/lle/code.html 4http://www-stat.stanford.edu/˜tibs/ElemStatLearn/ 6 Table 1: Computational Results on Swiss Roll (Time: second; # Iter: number of iterations) SCSDP SQLP m Time # Iter Time per Iter Time # Iter Time per Iter rank(A) 10 3.84 29 0.13 0.96 32 0.03 64 15 60.36 30 2.01 1.75 31 0.06 153 20 557.79 32 17.43 4.48 35 0.13 264 25 2821.76 34 82.99 7.84 37 0.21 403 30 13039.30 37 352.41 13.39 35 0.38 537 35 38559.50 33 1168.50 29.74 35 0.85 670 40 > 1 day — — 74.01 35 2.12 852 50 — — — 213.26 35 6.09 1152 60 — — — 467.90 35 13.37 1451 80 — — — 1729.65 39 44.35 2062 100 — — — 3988.31 36 110.79 2623 Table 2: Computational Results on USPS (Time: second; # Iter: number of iterations) SCSDP SQLP m Time # Iter Time per Iter Time # Iter Time per Iter rank(A) 10 2.84 21 0.14 0.47 16 0.03 100 15 42.96 22 1.95 1.26 17 0.07 225 20 461.38 27 17.09 3.35 17 0.20 400 25 2572.72 31 82.99 5.97 14 0.43 625 30 10576.01 30 352.53 15.72 19 0.83 900 35 35173.60 30 1172.50 44.53 17 2.62 1225 40 > 1 day — — 133.58 20 6.68 1600 50 — — — 362.24 16 22.64 2379 60 — — — 936.58 19 49.29 2938 80 — — — 1784.12 17 104.95 3112 100 — — — 2900.44 17 170.61 3111 The Swiss Roll (resp. USPS) is used to derive the QSDP in RegMVU (resp. RegPCP). For RegMVU, the 4NN graph is used, i.e., wij = 1 if xi is within the 4NN of xj or vice versa, and wij = 0 otherwise. We verified that the 4NN graph derived from our Swiss Roll data is connected. For RegPCP, we construct the graph following the approach suggested in (Li et al., 2009). Specifically, we have wij = exp(−d2 ij/(2σ2)) if xi is within 20NN of xj or vice versa, and wij = 0 otherwise. Here, σ is the averaged distance from each object to its 20-th nearest neighbor. For the pairwise constraints used in RegPCP, we randomly generate 20 similarity constraints for each class, and 20 dissimilarity constraints for every two classes, yielding a total of 1100 constraints. For each data set, m ranges over {10, 15, 20, 25, 30, 35, 40, 50, 60, 80, 100}. In summary, for each data set, 11 QSDPs are formed. Each QSDP gives rise to one SQLP and one SCSDP. Therefore, for each data set, 11 SQLPs and 11 SCSDPs are derived. 4.1 The Results The computational results of the programs are shown in Tables 1 and 2. For each program, we report the total computation time, the number of iterations needed to achieve the required tolerance, and the average time per iteration in solving the program. A dash (—) in the box indicates that the corresponding program takes too much time to solve. We choose to stop the program if it fails to converge within 1 day. This happens for the SCSDP with m = 40 on both data sets. ¿From the tables, we see that solving an SQLP is consistently much more faster than solving an SCSDP. To see the scalability, we plot the solution time (Time) against the problem size (m) in Figure 2. It can be seen that the solution time of the SCSDP grows much faster than that of the SQLP. This demonstrates the superiority of our proposed approach. 7 10 15 20 25 30 35 40 50 60 80 100 0.5 1 1.5 2 2.5 3 3.5 x 10 4 m Time (second) Swiss Roll SCSDP SQLP 10 15 20 25 30 35 40 50 60 80 100 0.5 1 1.5 2 2.5 3 3.5 x 10 4 m Time (second) USPS SCSDP SQLP (a) (b) Figure 2: Curves on computational cost: Solution Time vs. Problem Scale. We also note that the per-iteration computational costs of SCSDP and SQLP are drastically different. Indeed, for the same problem size m, it takes much less time per iteration for the SQLP than that for the SCSDP. This is not very surprising, as the SQLP formulation takes advantage of the low rank structure of the data matrix A. 5 Conclusions We have studied a class of convex optimization programs called convex Quadratic Semidefinite Programs (QSDPs), which arise naturally from graph Laplacian regularized kernel learning (Sha & Saul, 2005; Weinberger et al., 2007; Li et al., 2009; Song et al., 2008; Globerson & Roweis, 2007; Singer, 2008). A QSDP is similar to a QP, except that it is subject to a semidefinite cone constraint as well. To tackle the QSDP, one typically uses the Schur complement to rewrite it as an SDP (SCSDP), thus resulting in a large linear matrix inequality constraint. In this paper, we argue that this formulation is not computationally optimal and have proposed a novel formulation that leads to a semidefinite-quadratic-linear program (SQLP). Our formulation introduces one positive semidefinite constraint, one second-order cone constraint and a set of linear constraints. This should be contrasted with the two large semidefinite cone constraints in the SCSDP. Our complexity analysis and experimental results have shown that the proposed SQLP formulation scales far better than the SCSDP formulation. Acknowledgements The authors would like to thank Professor Kim-Chuan Toh for his valuable comments. This research work was supported in part by GRF grants CUHK 2150603, CUHK 414307 and CRF grant CUHK2/06C from the Research Grants Council of the Hong Kong SAR, China, as well as the NSFC-RGC joint research grant N CUHK411/07. References Ben-Tal, A., & Nemirovski, A. (2001). Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications. MPS–SIAM Series on Optimization. Philadelphia, Pennsylvania: Society for Industrial and Applied Mathematics. Borchers, B. (1999). CSDP, a C Library for Semidefinite Programming. Optimization Methods and Software, 11/12, 613–623. Boyd, S., & Vandenberghe, L. (2004). Convex Optimization. Cambridge: Cambridge University Press. Available online at http://www.stanford.edu/˜boyd/cvxbook/. Chapelle, O., & Vapnik, V. (2000). Model Selection for Support Vector Machines. In S. A. Solla, T. K. Leen and K.-R. M¨uller (Eds.), Advances in Neural Information Processing Systems 12: Proceedings of the 1999 Conference, 230–236. Cambridge, Massachusetts: The MIT Press. 8 Globerson, A., & Roweis, S. (2007). Visualizing Pairwise Similarity via Semidefinite Programming. Proceedings of the 11th International Conference on Artificial Intelligence and Statistics (pp. 139–146). Golub, G. H., & Loan, C. F. V. (1996). Matrix Computations. Baltimore, Maryland: The Johns Hopkins University Press. Third edition. Graepel, T. (2002). Kernel Matrix Completion by Semidefinite Programming. Proceedings of the 12th International Conference on Artificial Neural Networks (pp. 694–699). Springer–Verlag. Kulis, B., Sustik, M. A., & Dhillon, I. S. (2009). Low–Rank Kernel Learning with Bregman Matrix Divergences. The Journal of Machine Learning Research, 10, 341–376. Lanckriet, G. R. G., Cristianini, N., Bartlett, P., El Ghaoui, L., & Jordan, M. I. (2004). Learning the Kernel Matrix with Semidefinite Programming. The Journal of Machine Learning Research, 5, 27–72. Li, Z., & Liu, J. (2009). Constrained Clustering by Spectral Kernel Learning. To appear in the Proceedings of the 12th IEEE International Conference on Computer Vision. Li, Z., Liu, J., & Tang, X. (2008). Pairwise Constraint Propagation by Semidefinite Programming for Semi– Supervised Classification. Proceedings of the 25th International Conference on Machine Learning (pp. 576–583). Li, Z., Liu, J., & Tang, X. (2009). Constrained Clustering via Spectral Regularization. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition 2009 (pp. 421–428). Sch¨olkopf, B., & Smola, A. J. (2002). Learning with Kernels: Support Vector Machines, Regularization, Optimization, and Beyond. Cambridge, Massachusetts: The MIT Press. Sch¨olkopf, B., Smola, A. J., & M¨uller, K.-R. (1998). Nonlinear Component Analysis as a Kernel Eigenvalue Problem. Neural Computation, 10, 1299–1319. Sha, F., & Saul, L. K. (2005). Analysis and Extension of Spectral Methods for Nonlinear Dimensionality Reduction. Proceedings of the 22nd International Conference on Machine Learning (pp. 784–791). Shawe-Taylor, J., & Cristianini, N. (2004). Kernel Methods for Pattern Analysis. Cambridge: Cambridge University Press. Singer, A. (2008). A Remark on Global Positioning from Local Distances. Proceedings of the National Academy of Sciences, 105, 9507–9511. So, A. M.-C. (2007). A Semidefinite Programming Approach to the Graph Realization Problem: Theory, Applications and Extensions. Doctoral dissertation, Stanford University. Song, L., Smola, A., Borgwardt, K., & Gretton, A. (2008). Colored Maximum Variance Unfolding. In J. C. Platt, D. Koller, Y. Singer and S. Roweis (Eds.), Advances in Neural Information Processing Systems 20: Proceedings of the 2007 Conference, 1385–1392. Cambridge, Massachusetts: The MIT Press. Toh, K. C., T¨ut¨unc¨u, R. H., & Todd, M. J. (2006). On the Implementation and Usage of SDPT3 — A MATLAB Software Package for Semidefinite–Quadratic–Linear Programming, Version 4.0. User’s Guide. T¨ut¨unc¨u, R. H., Toh, K. C., & Todd, M. J. (2003). Solving Semidefinite–Quadratic–Linear Programs using SDPT3. Mathematical Programming, 95, 189–217. Vapnik, V. N. (2000). The Nature of Statistical Learning Theory. Statistics for Engineering and Information Science. New York: Springer–Verlag. Second edition. Weinberger, K. Q., Packer, B. D., & Saul, L. K. (2005). Nonlinear Dimensionality Reduction by Semidefinite Programming and Kernel Matrix Factorization. Proceedings of the 10th International Workshop on Artificial Intelligence and Statistics (pp. 381–388). Weinberger, K. Q., Sha, F., & Saul, L. K. (2004). Learning a Kernel Matrix for Nonlinear Dimensionality Reduction. Proceedings of the 21st International Conference on Machine Learning (pp. 85–92). Weinberger, K. Q., Sha, F., Zhu, Q., & Saul, L. K. (2007). Graph Laplacian Regularization for Large–Scale Semidefinite Programming. Advances in Neural Information Processing Systems 19: Proceedings of the 2006 Conference (pp. 1489–1496). Cambridge, Massachusetts: The MIT Press. Zhou, D., Bousquet, O., Lal, T. N., Weston, J., & Sch¨olkopf, B. (2004). Learning with Local and Global Consistency. Advances in Neural Information Processing Systems 16: Proceedings of the 2003 Conference (pp. 595–602). Cambridge, Massachusetts: The MIT Press. 9
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A Data-Driven Approach to Modeling Choice Vivek F. Farias Srikanth Jagabathula Devavrat Shah∗ Abstract We visit the following fundamental problem: For a ‘generic’ model of consumer choice (namely, distributions over preference lists) and a limited amount of data on how consumers actually make decisions (such as marginal preference information), how may one predict revenues from offering a particular assortment of choices? This problem is central to areas within operations research, marketing and econometrics. We present a framework to answer such questions and design a number of tractable algorithms (from a data and computational standpoint) for the same. 1 Introduction Consider a seller who must pick from a universe of N products, N, a subset M of products to offer to his customers. The ith product has price pi. Given a probabilistic model of how customers make choices, P(·|·), where P(i|M) is the probability that a potential customer purchases product i when faced with options M, the seller may solve (1) max M⊂N Ø i∈M piP(i|M). In addition to being a potentially non-trivial optimization problem, one faces a far more fundamental obstacle here: specifying the ‘choice’ model P(·|·) is a difficult task and it is unlikely that a seller will have sufficient data to estimate a generic such model. Thus, simply predicting expected revenues, R(M) = q i∈M piP(i|M), for a given offer set, M, is a difficult task. This problem, and variants thereof, are central in the fields of marketing, operations research and econometrics. With a few exceptions, the typical approach to dealing with the challenge of specifying a choice model with limited data has been to make parametric assumptions on the choice model that allow for its estimation from a limited amount of data. This approach has a natural deficiency – the implicit assumptions made in specifying a parametric model of choice may not hold. Indeed, for one of the most commonly used parametric models in modern practice (the multinomial logit), it is a simple task to come up with a list of deficiencies, ranging from serious economic fallacies presumed by the model ([5]), to a lack of statistical fit to observed data for real-world problems ([1, 8]). These issues have led to a proliferation of increasingly arcane parametric choice models. The present work considers the following question: given a limited amount of data on customer preferences and assuming only a ‘generic’ model of customer choice, what can one predict about expected revenues from a given set of products? We take as our ‘generic’ model of customer choice, the set of distributions over all possible customer preference lists (i.e. all possible permutations of N). We will subsequently see that essentially all extant models of customer choice can be viewed as a special case of this generic model. We view ‘data’ as some linear transformation of the distribution specifying the choice model, yielding marginal information. Again, we will see that this view is consistent with reality. ∗VF, DS are affiliated with ORC; VF with Sloan School of Management; SJ and DS with LIDS and Department of EECS at MIT. Emails: vfarias, jskanth, devavrat@mit.edu. The work was supported in part by NSF CAREER CNS 0546590. 1 Given these views, we first consider finding the ‘simplest’ choice model, consistent with the observed marginal data on customer preferences. Here we take as our notion of simple, a distribution over permutations of N with the sparsest support. We present two simple abstract properties that if satisfied by the ‘true’ choice model, allow us to solve the sparsest fit problem exactly via a simple combinatorial procedure (Theorem 2). In fact, the sparsest fit in this case coincides with the true model (Theorem 1). We present a generative family of choice models that illustrates when the two properties we posit may be expected to hold (see Theorem 3). More generally, when we may not anticipate the above abstract properties, we seek to find a ‘worst-case’ distribution consistent with the observed data in the sense that this distribution yields minimum expected revenues for a given offer set M while remaining consistent with the observed marginal data. This entails solving mathematical programs with as many variables as there are permutations (N!). In spite of this, we present a simple efficient procedure to solve this problem that is exact for certain interesting types of data and produces approximations (and computable error bounds) in general. Finally, we present a computational study illustrating the efficacy of our approach relative to a parametric technique on a real-world data set. Our main contribution is thus a novel approach to modeling customer choice given limited data. The approach we propose is complemented with efficient, implementable algorithms. These algorithms yield subroutines that make non-parametric revenue predictions for any given offer set (i.e. predict R(M) for any M) given limited data. Such subroutines could then be used in conjunction with generic set-function optimization heuristics to solve (1). Relevant Literature: There is a vast body of literature on the parametric modeling of customer choice; a seminal paper in this regard is [10]. See also [14] and references therein for an overview of the area with an emphasis on applications. There is a stream of research (eg. [6]) on estimating and optimizing (parametric) choice models when products possess measurable attributes that are the sole influencers of choice; we do not assume the availability of such attributes and thus do not consider this situation here. A non-parametric approach to choice modeling is considered by [12]; that work studies a somewhat distinct pricing problem, and assumes the availability of a specific type of rich observable data. Fitting a sparsest model to observable data has recently become of great interest in the area of compressive sensing in signal processing [3, 7], and in the design of sketches for streaming algorithms [2, 4]. This work focuses on deriving precise conditions on the support size of the true model, which, when satisfied, guarantee that the sparsest solution is indeed the true solution. However, these prior methods do not apply in the present context (see [9]); therefore, we take a distinct approach to the problem in this paper. 2 The Choice Model and Problem Formulations We consider a universe of N products, N = {0, 1, 2, . . . , N −1}. We assume that the 0th product in N corresponds to the ‘outside’ or ‘no-purchase’ option. A consumer is associated with a permutation σ of the elements of N; the customer prefers product i to product j iffσ(i) < σ(j). Given that the customer is faced with a set of alternatives M ⊂N, she chooses to purchase her single most preferred product among those in M. In particular, she purchases argmini∈M σ(i). Choice Model: We take as our model of customer choice a distribution, λ : SN →[0, 1], over all possible permutations (i.e. the set of all permutations SN). Define the set Sj(M) = {σ ∈SN : σ(j) < σ(i), ∀i ∈M, i ̸= j} as the set of all customer types that would result in a purchase of j when the offer set is M. Our choice model is thus P(j|M) = Ø σ∈Sj(M) λ(σ) ≜λj(M). This model subsumes a vast body of extant parametric choice models. Revenues: We associate every product in N with a retail price pj. Of course, p0 = 0. The expected revenues to a retailer from offering a set of products M to his customers under our choice model is thus given by R(M) = q j∈M pjλj(M). 2 Data: A seller will have limited data with which to estimate λ. We simply assume that the data observed by the seller is given by an m-dimensional ‘partial information’ vector y = Aλ, where A ∈{0, 1}m×N! makes precise the relationship between the observed data and the underlying choice model. For the purposes of illustration, we consider the following concrete examples of data vectors y: Ranking Data: This data represents the fraction of customers that rank a given product i as their rth choice. Here the partial information vector y is indexed by i, r with 0 ≤i, r ≤N. For each i, r, yri denotes the probability that product i is ranked at position r. The matrix A is thus in {0, 1}N2×N!. For a column of A corresponding to the permuation σ, A(σ), we will thus have A(σ)ri = 1 iffσ(i) = r. Comparison Data: This data represents the fraction of customers that prefer a given product i to a product j. The partial information vector y is indexed by i, j with 0 ≤i, j ≤ N; i ̸= j. For each i, j, yi,j denotes the probability that product i is preferred to product j. The matrix A is thus in {0, 1}N(N−1)×N!. A column of A, A(σ), will thus have A(σ)ij = 1 if and only if σ(i) < σ(j). Top Set Data: This data refers to a concatenation of the “Comparison Data” and information on the fraction of customers who have a given product i as their topmost choice for each i. Thus A⊤= [A⊤ 1 A⊤ 2 ] where A1 is simply the A matrix for comparison data, and A2 ∈{0, 1}N×N! has A2(σ)i = 1 iffσ(i) = 1. Many other types of data vectors consistent with the above view are possible; all we anticipate is that the dimension of the observed data m is substantially smaller than N!. We are now in a position to formulate the questions broached in the previous section precisely: “Simplest” Model: In finding the simplest choice model consistent with the observed data we attempt to solve: (2) minimize ∥λ∥0 subject to Aλ = y, 1⊤λ = 1, λ ≥0. Robust Approach: For a given offer set M, and data vector y, what are the minimal expected revenues we might expect from M consistent with the observed data? To answer this question, we attempt to solve : (3) minimize λ q j∈M pjλj(M) subject to Aλ = y, 1⊤λ = 1, λ ≥0. 3 Estimating Sparse Choice Models Here we consider finding the sparsest model consistent with the observed data (i.e. problem (2)). We face two questions: (a) Why is sparsity an interesting criterion? (b) Is there an efficient procedure to solve the program in (2)? We begin by identifying two simple conditions that define a class of choice models (i.e. a class of distributions λ). Assuming that the ‘true’ underlying model λ belongs to this class, we prove that the sparsest model (i.e the solution to (2)) is in fact this true model. This answers the first question. We then propose a simple procedure inspired by [9] that correctly solves the program in (2) assuming these conditions. It is difficult to expect the program in (2) to recover the true solution in general (see [9] for a justification). Nonetheless, we show that the conditions we impose are not overly restrictive: we prove that a “sufficiently” sparse model generated uniformly at random from the set of all possible choice models satisfies the two conditions with a high probability. Before we describe the conditions we impose on the true underlying distribution, we introduce some notation. Let λ denote the true underlying distribution, and let K denote the support size, ∥λ∥0. Let σ1, σ2, . . . , σK denote the permutations in the support, i.e, λ(σi) ̸= 0 for 1 ≤i ≤K, and λ(σ) = 0 for all σ ̸= σi, 1 ≤i ≤K. Recall that y is of dimension m and we index its elements by d. The two conditions we impose are as follows: Signature Condition: For every permutation σi in the support, there exists a d(i) ∈ {1, 2, . . . , m} such that A(σi)d(i) = 1 and A(σj)d(i) ̸= 0, for every j ̸= i and 1 ≤i, j ≤K. In other words, for each permutation σi in the support, yd(i) serves as its ‘signature’. 3 Linear Independence Condition: qK i=1 ciλ(σi) ̸= 0, for any ci ∈Z and |ci| ≤C, where Z denotes the set of integers and C is a sufficiently large number ≥K. This condition is satisfied with probability 1 if [λ1λ2 . . . λK]⊤is drawn uniformly from K-dim simplex. When the two conditions are satisfied, the sparsest solution is indeed the true solution as stated in the following theorem: Theorem 1. Suppose we are given y = Aλ and λ satisfies the “Signature” condition and the “Linear Independence” condition. Then, λ is the unique solution to the program in (2). The proof of Theorem 1 is given in the appendix. Next we describe the algorithm we propose for recovery. The algorithm takes y and A and as the input and outputs λi (denotes λ(σi)) and A(σi) for every permutation σi in the support. The algorithm assumes the observed values yd are sorted. Therefore, without loss of generality, assume that y1 < y2 < . . . < ym. Then, the algorithm is as follows: Algorithm: Initialization: λ0 = 0, k(0) = 0 and A(σi)d = 0, 1 ≤i ≤K and 1 ≤d ≤m. for d = 1 to m if yd = q i∈T λi for some T ⊆{1, . . . , k(d −1)} k(d) = k(d −1), A(σi)d = 1 ∀ i ∈T else k(d) = k(d −1) + 1, λk(d) = yd, A(σk(d))d = 1, end if end for Output K = k(m) and (λi, A(σi)), 1 ≤i ≤K. Now, we have the following theorem: Theorem 2. Suppose we are given y = Aλ and λ satisfies the “signature” and the “linear independence” conditions. Then, the above described algorithm recovers λ. Theorem 2 is proved in the appendix. The algorithm we have described either succeeds in finding a valid λ or else determines that the two properties are not satisfied. We now show that the conditions we have imposed do not restrict the class of plausible models severely. For this, we show that models drawn from the following generative model satisfy the conditions with high probability. Generative Model. Given K and an interval [a, b] on the positive real line, we generate a choice model λ as follows: choose K permutations, σ1, σ2, . . . , σK, uniformly at random with replacement, choose K numbers uniformly at random from the interval [a, b], normalize the numbers so that they sum to 1, and assign them to the permutations σi, 1 ≤i ≤K. For all other permutations σ ̸= σi, λ(σ) = 0. Note that, since we are choosing permutations in the support with replacement, there could be repetitions. However, for large N and K ≪N!, this happens with a vanishing probability. Depending on the observed data, we characterize values of sparsity K for which distributions generated by the above generative model can be recovered with a high probability. Specifically, we have the following theorem for the three forms of observed data mentioned in Section 2. The proof may be found in the appendix. Theorem 3. Suppose λ is a choice model of support size K drawn from the generative model. Then, λ satisfies the “signature” and “linear independence” conditions with probability 1 − o(1) as N →∞provided K = O(N) for ranking data, K = o(log N) for comparison data, and K = o( √ N) for the top set data. Of course, in general, the underlying choice model may not satisfy the two conditions we have posited or be exactly recoverable from the observed data. In order to deal with this more general scenario, we next propose an approach that implicitly identifies a ‘worst-case’ distribution consistent with the observed data. 4 4 Robust Revenue Estimates Consistent with Data In this section, we propose a general algorithm for the solution of program (3). This LP has N! variables and is clearly not amenable to direct solution; hence we consider its dual. In preparation for taking the dual, let Aj(M) ≜{A(σ) : σ ∈Sj(M)}, where, recall that, Sj(M) denotes the set of all permutations that result in the purchase of j ∈M when offered the assortment M. Since SN = ∪j∈MSj(M), we have implicitly specified a partition of the columns of the matrix A. Armed with this notation, the dual of (3) is: (4) maximize α,ν α⊤y + ν subject to max xj∈Aj(M) ! α⊤xj + ν " ≤pj, for each j ∈M. Our solution procedure will rely on an effective representation of the sets Aj(M). 4.1 A Canonical Representation of Aj(M) and its Application We assume that every set Sj(M) can be expressed as a disjoint union of Dj sets. We denote the dth such set by Sjd(M) and let Ajd(M) be the corresponding set of columns. Consider the convex hull of the set Ajd(M), conv{Ajd(M)} ≜¯ Ajd(M). ¯ Ajd(M) is by definition a polytope contained in the m-dimensional unit cube, [0, 1]m. In other words, (5) ¯ Ajd(M) = {xjd : Ajd 1 xjd ≥bjd 1 , Ajd 2 xjd = bjd 2 , Ajd 3 xjd ≤bjd 3 , xjd ≥0.} for appropriately defined Ajd · , bjd · . By a canonical representation of Aj(M), we will thus understand a partition of Sj(M) and a polyhedral representation of the columns corresponding to every set in the partition as given by (5). Ignoring the problem of actually obtaining this representation for now, we assume access to a canonical representation and present a simple program whose size is polynomial in the size of this representation that is equivalent to (3), (4). For simplicity of notation, we assume that each of the polytopes ¯ Ajd(M) is in standard form, i.e. ¯ Ajd(M) = {xjd : Ajd xjd = bjd, xjd ≥0.}. Now since an affine function is always optimized at the vertices of a polytope, we know: max xj∈Aj(M) ! α⊤xj + ν " = max d,xjd∈¯ Ajd(M) ! α⊤xjd + ν " . We have thus reduced (4) to a ‘robust’ LP. By strong duality we have: (6) max xjd∈¯ Ajd(M) ! α⊤xjd + ν " ≜ maximize xjd α⊤xjd + ν subject to Ajd xjd = bjd xjd ≥0. = minimize γjd bjd⊤γjd + ν subject to γjd⊤Ajd ≥α We have thus established the following useful equality: ; α, ν : max xj∈¯ Aj(M) ! α⊤xj + ν " ≤pj < = î α, ν : bjd⊤γjd + ν ≤pj, γjd⊤Ajd ≥α, d = 1, 2, . . . , Dj ï . It follows that solving (3) is equivalent to the following LP whose complexity is polynomial in the description of our canonical representation: (7) maximize α,ν α⊤y + ν subject to bjd⊤γjd + ν ≤pj for all j ∈M, d = 1, 2, . . . , Dj γjd⊤Ajd ≥α for all j ∈M, d = 1, 2, . . . , Dj. Our ability to solve (7) relies on our ability to produce an efficient canonical representation of Sj(M). In what follows, we first consider an example where such a representation is readily available, and then consider the general case. Canonical Representation for Ranking Data: Recall the definition of ranking data from Section 2. Consider partitioning Sj(M) into N sets wherein the dth set is given by 5 Sjd(M) = {σ ∈Sj(M): σ(j) = d}. It is not difficult to show that the set Ajd(M) is equal to the set of all vectors xjd in {0, 1}N satisfying: (8) N−1 q i=0 xjd ri = 1 for 0 ≤i ≤N −1 N−1 q r=0 xjd ri = 1 for 0 ≤r ≤N −1 xjd ri ∈{0, 1} for 0 ≤i, r ≤N −1. xjd dj = 1 xjd d′i = 0 for all i ∈M, i ̸= jand 0 ≤d′ < d. Our goal is, of course, to find a description for ¯ Ajd(M) of the type (5). Now consider replacing the third (integrality) constraint in (8) with simply the non-negativity constraint xjd ri ≥0. It is clear that the resulting polytope contains ¯ Ajd(M). In addition, one may show that the resulting polytope has integral vertices since it is simply a matching polytope with some variables forced to be integers, so that in fact the polytope is precisely ¯ Ajd(M), and we have our canonical representation. Further, notice that this representation yields an efficient algorithm to solve (3) via (7)! 4.2 Computing a Canonical Representation: Comparison Data Recall the definition of comparison data from Section 2. We use this data as an example to illustrate a general procedure for computing a canonical representation. Consider Sj(M). It is not difficult to see that the corresponding set of columns Aj(M) is equal to the set of vectors in {0, 1}(N−1)N satisfying the following constraints: (9) xj il ≥xj ik + xj kl −1 for all i, k, l ∈N, i ̸= k ̸= l xj ik + xj ki = 1 for all i, k ∈N, i ̸= k xj ji = 1 for all i ∈M, i ̸= j xj ik ∈{0, 1} for all i, k ∈N, i ̸= k Briefly, the second constraint follows since for any i, k, i ̸= k, either σ(i) > σ(k) or else σ(i) < σ(k). The first constraint enforces transitivity: σ(i) < σ(k) and σ(k) < σ(l) together imply σ(i) < σ(l). The third constraint enforces that all σ ∈Sj(M) must satisfy σ(j) < σ(i) for all i ∈M. Now consider the polytope obtained by relaxing the fourth (integrality) constraint to simply xj ik ≥0. Call this polytope ¯ Ao j(M). Of course, we must have ¯ Ao j(M) ⊇¯ Aj(M). Unlike the case of ranking data, however, ¯ Ao j(M) can in fact be shown to be non-integral, so that ¯ Ao j(M) ̸= ¯ Aj(M) in general. In this case we resort to the following procedure. [1.] Solve (7) using the representation of ¯ Ao j(M) in place of ¯ Aj(M). This yields a lower bound on (3) since ¯ Ao j(M) ⊃¯ Aj(M). Call the corresponding solution α(1), ν(1). [2.] Solve the optimization problem max α⊤ (1)xj subject to xj ∈¯ Ao j(M) for each j. If the optimal solution ˆxj is integral for each j, then stop; the solution computed in the first step is in fact optimal. [3.] Let ˆxj ik be a non-integral variable. Partition Sj(M) on this variable - i.e. define Sj1(M) = {σ : σ ∈Sj(M), σ(i) < σ(k)} and Sj2(M) = {σ : σ ∈Sj(M), σ(i) > σ(k)}. Define outer-approximations to ¯ Aj1(M) and ¯ Aj2(M) as the projection of ¯ Ao j(M) on xj ik = 1 and xj ik = 0 respectively. Go to step 1. The above procedure is finite, but the size of the LP we solve at each iteration doubles. Nonetheless, each iteration produces a lower bound to (3) whose quality is easily measured (for instance, by solving the maximization version of (3) using the same procedure), and this quality improves with each iteration. In our computational experiments with a related type of data, it sufficed to stop after a single iteration. 6 5 An Empirical Evaluation of the Approach We have presented simple sub-routines to estimate the revenues R(M) from a particular offer set M, given marginal preference data y. These sub-routines are effectively ‘non-parametric’ and can form the basis of a procedure that solves the revenue optimization problem posed in the introduction. Here we seek to contrast this approach with a commonly used parametric approach. We consider two types of observable data: ranking data and a ‘censored’ version of the comparison data which gives us for every pair of products i, j, ̸= 0, the fraction of customers that prefer i to j, and in addition prefer i to 0 (i.e. not buying). The latter type of data is quite realistic. The parametric recipe we consider is the following: One fits a Multinomial Logit (MNL) model to the observable data and picks an optimal offer set by evaluating R(M) = q j∈M pjP(j|M) assuming P(·|M) follows the estimated model. The MNL is a commonly used parametric model that associates with each product i in N a positive scalar wi; w0 = 1 by convention. The model assumes P(i|M) = wi/q j∈M wj. In place of making this parametric assumption, we could instead evaluate R(M) using the robust sub-routine developed in the previous section and pick M to maximize this conservative estimate. It is clear that if the MNL model is a poor fit to the true choice model, P, our robust approach is likely to outperform the parametric approach substantially. Instead, what we focus on here is what happens if the MNL model is a perfect fit to the true choice model. In this case, the parametric approach is the best possible. How sub-optimal is our non-parametric approach here? We consider an MNL model on N = 25 products. The model and prices were specified using customer utilities for Amazon.com’s highest selling DVDs (and their prices) during a 3-month period from 1 July 2005 to 30 September 2005 estimated by [13] 1. We generate synthetic observed data (of both the ranking type and the comparison type) according to this fitted MNL model. This represents a scenario where the fitted MNL is a perfect descriptor of reality. We conduct the following experiments: Quality of Revenue Predictions: For each type of observable data we compute our estimate of the minimum value that R(M) can take on, consistent with that data, by solving (3). We compare this with the value of R(M) predicted under the MNL model (which in this case, is exact). Figures 1(b) and 1(d) compare these two quantities for a set of randomly chosen subsets M of the 25 potential DVD’s assuming ranking data and the censored comparison data respectively. In both cases, our procedure produces excellent predictions of expected revenue without making the assumptions on P(·|·) inherent in the MNL model. Quality of Optimal Solutions to Revenue Maximization Problems: For each type of observable data, we compute optimal offer sets M of varying capacities assuming the fitted MNL model and an optimization procedure described in [13]. We then evaluate the revenue predictions for these optimal offer sets by solving (3). Figures 1(a) and 1(c) plot these estimates for the two types of observable data. The gap between the ‘MNL’ and the ‘MIN’ curves is thus an upper bound on the expected revenue loss if one used our nonparametric procedure to pick an optimal offer set M over the parametric procedure (which in this setting is optimal). Again, we see that the revenue loss is surprisingly small. 6 Conclusion and Potential Future Directions We have presented a general framework that allows us to answer questions related to how consumers choose among alternatives using limited observable data and without making additional parametric assumptions. The approaches we have proposed are feasible from a data availability standpoint as well as a computational standpoint and provide a much needed non-parametric ‘sub-routine’ for the revenue optimization problems described at the outset. This paper also opens up the potential for a stream of future work. 1The problem of optimizing over M is particularly relevant to Amazon.com given limited screen real-estate and cannibilization effects 7 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue (a) Ranking Data: Optimal M 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Assortment index Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue (b) Ranking Data: Random M 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Optimal MNL assortment (size) Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue (c) Comparison Data: Optimal M 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Assortment index Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue 0 5 10 15 20 25 0 2 4 6 8 10 12 14 Assortment index Expected Revenue (dollars) MNL Expected Revenue MIN Expected Revenue (d) Comparison Data: Random M References [1] K. Bartels, Y. Boztug, and M. M. Muller. Testing the multinomial logit model. Working Paper, 1999. [2] R. Berinde, AC Gilbert, P. Indyk, H. Karloff, and MJ Strauss. Combining geometry and combinatorics: A unified approach to sparse signal recovery. Preprint, 2008. [3] E.J. Candes, J.K. Romberg, and T. Tao. Stable signal recovery from incomplete and inaccurate measurements. Communications on Pure and Applied Mathematics, 59(8), 2006. [4] G. Cormode and S. Muthukrishnan. Combinatorial algorithms for compressed sensing. Lecture Notes in Computer Science, 4056:280, 2006. [5] G. Debreu. Review of r.d. luce, ‘individual choice behavior: A theoretical analysis’. American Economic Review, 50:186–188, 1960. [6] G. Dobson and S. Kalish. Positioning and pricing a product line. Marketing Science, 7(2):107–125, 1988. [7] DL Donoho. Compressed sensing. IEEE Transactions on Information Theory, 52(4): 1289–1306, 2006. [8] J. L. Horowitz. Semiparametric estimation of a work-trip mode choice model. Journal of Econometrics, 58:49–70, 1993. [9] S. Jagabathula and D. Shah. Inferring rankings under constrained sensing. In NIPS, pages 7–1, 2008. [10] D. McFadden. Econometric models for probabiistic choice among products. The Journal of Business, 53(3):S13–S29, 1980. [11] EH McKinney. Generalized birthday problem. American Mathematical Monthly, pages 385–387, 1966. [12] P. Rusmevichientong, B. Van Roy, and P. Glynn. A nonparametric approach to multiproduct pricing. Operations Research, 54(1), 2006. [13] P. Rusmevichientong, ZJ Shen, and D.B. Shmoys. Dynamic Assortment Optimization with a Multinomial Logit Choice Model and Capacity Constraint. Technical report, Working Paper, 2008. [14] Kalyan T. Talluri and Garrett J. van Ryzin. The Theory and Practice of Revenue Management. Springer Science+Business Media, 2004. 8
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Anomaly Detection with Score functions based on Nearest Neighbor Graphs Manqi Zhao ECE Dept. Boston University Boston, MA 02215 mqzhao@bu.edu Venkatesh Saligrama ECE Dept. Boston University Boston, MA, 02215 srv@bu.edu Abstract We propose a novel non-parametric adaptive anomaly detection algorithm for high dimensional data based on score functions derived from nearest neighbor graphs on n-point nominal data. Anomalies are declared whenever the score of a test sample falls below α, which is supposed to be the desired false alarm level. The resulting anomaly detector is shown to be asymptotically optimal in that it is uniformly most powerful for the specified false alarm level, α, for the case when the anomaly density is a mixture of the nominal and a known density. Our algorithm is computationally efficient, being linear in dimension and quadratic in data size. It does not require choosing complicated tuning parameters or function approximation classes and it can adapt to local structure such as local change in dimensionality. We demonstrate the algorithm on both artificial and real data sets in high dimensional feature spaces. 1 Introduction Anomaly detection involves detecting statistically significant deviations of test data from nominal distribution. In typical applications the nominal distribution is unknown and generally cannot be reliably estimated from nominal training data due to a combination of factors such as limited data size and high dimensionality. We propose an adaptive non-parametric method for anomaly detection based on score functions that maps data samples to the interval [0, 1]. Our score function is derived from a K-nearest neighbor graph (K-NNG) on n-point nominal data. Anomaly is declared whenever the score of a test sample falls below α (the desired false alarm error). The efficacy of our method rests upon its close connection to multivariate p-values. In statistical hypothesis testing, p-value is any transformation of the feature space to the interval [0, 1] that induces a uniform distribution on the nominal data. When test samples with p-values smaller than α are declared as anomalies, false alarm error is less than α. We develop a novel notion of p-values based on measures of level sets of likelihood ratio functions. Our notion provides a characterization of the optimal anomaly detector, in that, it is uniformly most powerful for a specified false alarm level for the case when the anomaly density is a mixture of the nominal and a known density. We show that our score function is asymptotically consistent, namely, it converges to our multivariate p-value as data length approaches infinity. Anomaly detection has been extensively studied. It is also referred to as novelty detection [1, 2], outlier detection [3], one-class classification [4, 5] and single-class classification [6] in the literature. Approaches to anomaly detection can be grouped into several categories. In parametric approaches [7] the nominal densities are assumed to come from a parameterized family and generalized likelihood ratio tests are used for detecting deviations from nominal. It is difficult to use parametric approaches when the distribution is unknown and data is limited. A K-nearest neighbor 1 (K-NN) anomaly detection approach is presented in [3, 8]. There an anomaly is declared whenever the distance to the K-th nearest neighbor of the test sample falls outside a threshold. In comparison our anomaly detector utilizes the global information available from the entire K-NN graph to detect deviations from the nominal. In addition it has provable optimality properties. Learning theoretic approaches attempt to find decision regions, based on nominal data, that separate nominal instances from their outliers. These include one-class SVM of Sch¨olkopf et. al. [9] where the basic idea is to map the training data into the kernel space and to separate them from the origin with maximum margin. Other algorithms along this line of research include support vector data description [10], linear programming approach [1], and single class minimax probability machine [11]. While these approaches provide impressive computationally efficient solutions on real data, it is generally difficult to precisely relate tuning parameter choices to desired false alarm probability. Scott and Nowak [12] derive decision regions based on minimum volume (MV) sets, which does provide Type I and Type II error control. They approximate (in appropriate function classes) level sets of the unknown nominal multivariate density from training samples. Related work by Hero [13] based on geometric entropic minimization (GEM) detects outliers by comparing test samples to the most concentrated subset of points in the training sample. This most concentrated set is the K-point minimum spanning tree(MST) for n-point nominal data and converges asymptotically to the minimum entropy set (which is also the MV set). Nevertheless, computing K-MST for n-point data is generally intractable. To overcome these computational limitations [13] proposes heuristic greedy algorithms based on leave-one out K-NN graph, which while inspired by K-MST algorithm is no longer provably optimal. Our approach is related to these latter techniques, namely, MV sets of [12] and GEM approach of [13]. We develop score functions on K-NNG which turn out to be the empirical estimates of the volume of the MV sets containing the test point. The volume, which is a real number, is a sufficient statistic for ensuring optimal guarantees. In this way we avoid explicit high-dimensional level set computation. Yet our algorithms lead to statistically optimal solutions with the ability to control false alarm and miss error probabilities. The main features of our anomaly detector are summarized. (1) Like [13] our algorithm scales linearly with dimension and quadratic with data size and can be applied to high dimensional feature spaces. (2) Like [12] our algorithm is provably optimal in that it is uniformly most powerful for the specified false alarm level, α, for the case that the anomaly density is a mixture of the nominal and any other density (not necessarily uniform). (3) We do not require assumptions of linearity, smoothness, continuity of the densities or the convexity of the level sets. Furthermore, our algorithm adapts to the inherent manifold structure or local dimensionality of the nominal density. (4) Like [13] and unlike other learning theoretic approaches such as [9, 12] we do not require choosing complex tuning parameters or function approximation classes. 2 Anomaly Detection Algorithm: Score functions based on K-NNG In this section we present our basic algorithm devoid of any statistical context. Statistical analysis appears in Section 3. Let S = {x1, x2, · · · , xn} be the nominal training set of size n belonging to the unit cube [0, 1]d. For notational convenience we use η and xn+1 interchangeably to denote a test point. Our task is to declare whether the test point is consistent with nominal data or deviates from the nominal data. If the test point is an anomaly it is assumed to come from a mixture of nominal distribution underlying the training data and another known density (see Section 3). Let d(x, y) be a distance function denoting the distance between any two points x, y ∈[0, 1]d. For simplicity we denote the distances by dij = d(xi, xj). In the simplest case we assume the distance function to be Euclidean. However, we also consider geodesic distances to exploit the underlying manifold structure. The geodesic distance is defined as the shortest distance on the manifold. The Geodesic Learning algorithm, a subroutine in Isomap [14, 15] can be used to efficiently and consistently estimate the geodesic distances. In addition by means of selective weighting of different coordinates note that the distance function could also account for pronounced changes in local dimensionality. This can be accomplished for instance through Mahalanobis distances or as a by product of local linear embedding [16]. However, we skip these details here and assume that a suitable distance metric is chosen. Once a distance function is defined our next step is to form a K nearest neighbor graph (K-NNG) or alternatively an ϵ neighbor graph (ϵ-NG). K-NNG is formed by connecting each xi to the K closest 2 points {xi1, · · · , xiK} in S −{xi}. We then sort the K nearest distances for each xi in increasing order di,i1 ≤· · · ≤di,iK and denote RS(xi) = di,iK, that is, the distance from xi to its K-th nearest neighbor. We construct ϵ-NG where xi and xj are connected if and only if dij ≤ϵ. In this case we define NS(xi) as the degree of point xi in the ϵ-NG. For the simple case when the anomalous density is an arbitrary mixture of nominal and uniform density1 we consider the following two score functions associated with the two graphs K-NNG and ϵ-NNG respectively. The score functions map the test data η to the interval [0, 1]. K-LPE: ˆpK(η) = 1 n n X i=1 I{RS(η)≤RS(xi)} (1) ϵ-LPE: ˆpϵ(η) = 1 n n X i=1 I{NS(η)≥NS(xi)} (2) where I{·} is the indicator function. Finally, given a pre-defined significance level α (e.g., 0.05), we declare η to be anomalous if ˆ pK(η), ˆpϵ(η) ≤α. We call this algorithm Localized p-value Estimation (LPE) algorithm. This choice is motivated by its close connection to multivariate p-values(see Section 3). The score function K-LPE (or ϵ-LPE) measures the relative concentration of point η compared to the training set. Section 3 establishes that the scores for nominally generated data is asymptotically uniformly distributed in [0, 1]. Scores for anomalous data are clustered around 0. Hence when scores below level α are declared as anomalous the false alarm error is smaller than α asymptotically (since the integral of a uniform distribution from 0 to α is α). Bivariate Gaussian mixture distribution −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 anomaly detection via K−LPE, n=200, K=6, α=0.05 −6 −4 −2 0 2 4 −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 level set at α=0.05 labeled as anomaly labeled as nominal 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 12 empirical distribution of the scoring function K−LPE value of K−LPE empirical density nominal data anomaly data α=0.05 Figure 1: Left: Level sets of the nominal bivariate Gaussian mixture distribution used to illustrate the KLPE algorithm. Middle: Results of K-LPE with K = 6 and Euclidean distance metric for m = 150 test points drawn from a equal mixture of 2D uniform and the (nominal) bivariate distributions. Scores for the test points are based on 200 nominal training samples. Scores falling below a threshold level 0.05 are declared as anomalies. The dotted contour corresponds to the exact bivariate Gaussian density level set at level α = 0.05. Right: The empirical distribution of the test point scores associated with the bivariate Gaussian appear to be uniform while scores for the test points drawn from 2D uniform distribution cluster around zero. Figure 1 illustrates the use of K-LPE algorithm for anomaly detection when the nominal data is a 2D Gaussian mixture. The middle panel of figure 1 shows the detection results based on K-LPE are consistent with the theoretical contour for significance level α = 0.05. The right panel of figure 1 shows the empirical distribution (derived from the kernel density estimation) of the score function K-LPE for the nominal (solid blue) and the anomaly (dashed red) data. We can see that the curve for the nominal data is approximately uniform in the interval [0, 1] and the curve for the anomaly data has a peak at 0. Therefore choosing the threshold α = 0.05 will approximately control the Type I error within 0.05 and minimize the Type II error. We also take note of the inherent robustness of our algorithm. As seen from the figure (right) small changes in α lead to small changes in actual false alarm and miss levels. 1When the mixing density is not uniform but, say f1, the score functions must be modified to ˆpK(η) = 1 n P n i=1 I ½ 1 RS (η)f1(η) ≤ 1 RS (xi)f1(xi) ¾ and ˆpϵ(η) = 1 n P n i=1 I ½ NS (η) f1(η) ≥NS (xi) f1(xi) ¾ for the two graphs K-NNG and ϵ-NNG respectively. 3 To summarize the above discussion, our LPE algorithm has three steps: (1) Inputs: Significance level α, distance metric (Euclidean, geodesic, weighted etc.). (2) Score computation: Construct K-NNG (or ϵ-NG) based on dij and compute the score function K-LPE from Equation 1 (or ϵ-LPE from Equation 2). (3) Make Decision: Declare η to be anomalous if and only if ˆpK(η) ≤α (or ˆpϵ(η) ≤α). Computational Complexity: To compute each pairwise distance requires O(d) operations; and O(n2d) operations for all the nodes in the training set. In the worst-case computing the K-NN graph (for small K) and the functions RS(·), NS(·) requires O(n2) operations over all the nodes in the training data. Finally, computing the score for each test data requires O(nd+n) operations(given that RS(·), NS(·) have already been computed). Remark: LPE is fundamentally different from non-parametric density estimation or level set estimation schemes (e.g., MV-set). These approaches involve explicit estimation of high dimensional quantities and thus hard to apply in high dimensional problems. By computing scores for each test sample we avoid high-dimensional computation. Furthermore, as we will see in the following section the scores are estimates of multivariate p-values. These turn out to be sufficient statistics for optimal anomaly detection. 3 Theory: Consistency of LPE A statistical framework for the anomaly detection problem is presented in this section. We establish that anomaly detection is equivalent to thresholding p-values for multivariate data. We will then show that the score functions developed in the previous section is an asymptotically consistent estimator of the p-values. Consequently, it will follow that the strategy of declaring an anomaly when a test sample has a low score is asymptotically optimal. Assume that the data belongs to the d-dimensional unit cube [0, 1]d and the nominal data is sampled from a multivariate density f0(x) supported on the d-dimensional unit cube [0, 1]d. Anomaly detection can be formulated as a composite hypothesis testing problem. Suppose test data, η comes from a mixture distribution, namely, f(η) = (1−π)f0(η)+πf1(η) where f1(η) is a mixing density supported on [0, 1]d. Anomaly detection involves testing the nominal hypotheses H0 : π = 0 versus the alternative (anomaly) H1 : π > 0. The goal is to maximize the detection power subject to false alarm level α, namely, P(declare H1 | H0) ≤α. Definition 1. Let P0 be the nominal probability measure and f1(·) be P0 measurable. Suppose the likelihood ratio f1(x)/f0(x) does not have non-zero flat spots on any open ball in [0, 1]d. Define the p-value of a data point η as p(η) = P0 µ x : f1(x) f0(x) ≥f1(η) f0(η) ¶ Note that the definition naturally accounts for singularities which may arise if the support of f0(·) is a lower dimensional manifold. In this case we encounter f1(η) > 0, f0(η) = 0 and the p-value p(η) = 0. Here anomaly is always declared(low score). The above formula can be thought of as a mapping of η →[0, 1]. Furthermore, the distribution of p(η) under H0 is uniform on [0, 1]. However, as noted in the introduction there are other such transformations. To build intuition about the above transformation and its utility consider the following example. When the mixing density is uniform, namely, f1(η) = U(η) where U(η) is uniform over [0, 1]d, note that Ωα = {η | p(η) ≥α} is a density level set at level α. It is well known (see [12]) that such a density level set is equivalent to a minimum volume set of level α. The minimum volume set at level α is known to be the uniformly most powerful decision region for testing H0 : π = 0 versus the alternative H1 : π > 0 (see [13, 12]). The generalization to arbitrary f1 is described next. Theorem 1. The uniformly most powerful test for testing H0 : π = 0 versus the alternative (anomaly) H1 : π > 0 at a prescribed level α of significance P(declare H1 | H0) ≤α is: φ(η) = ½ H1, p(η) ≤α H0, otherwise 4 Proof. We provide the main idea for the proof. First, measure theoretic arguments are used to establish p(X) as a random variable over [0, 1] under both nominal and anomalous distributions. Next when X d∼f0, i.e., distributed with nominal density it follows that the random variable p(X) d∼ U[0, 1]. When X d∼f = (1 −π)f0 + πf1 with π > 0 the random variable, p(X) d∼g where g(·) is a monotonically decreasing PDF supported on [0, 1]. Consequently, the uniformly most powerful test for a significance level α is to declare p-values smaller than α as anomalies. Next we derive the relationship between the p-values and our score function. By definition, RS(η) and RS(xi) are correlated because the neighborhood of η and xi might overlap. We modify our algorithm to simplify our analysis. We assume n is odd (say) and can be written as n = 2m + 1. We divide training set S into two parts: S = S1 ∩S2 = {x0, x1, · · · , xm} ∩{xm+1, · · · , x2m} We modify ϵ-LPE to ˆpϵ(η) = 1 m P xi∈S1 I{NS2(η)≥NS1(xi)} (or K-LPE to ˆpK(η) = 1 m P xi∈S1 I{RS2(η)≤RS1(xi)}). Now RS2(η) and RS1(xi) are independent. Furthermore, we assume f0(·) satisfies the following two smoothness conditions: 1. the Hessian matrix H(x) of f0(x) is always dominated by a matrix with largest eigenvalue λM, i.e., ∃M s.t. H(x) ⪯M ∀x and λmax(M) ≤λM 2. In the support of f0(·), its value is always lower bounded by some β > 0. We have the following theorem. Theorem 2. Consider the setup above with the training data {xi}n i=1 generated i.i.d. from f0(x). Let η ∈[0, 1]d be an arbitrary test sample. It follows that for a suitable choice K and under the above smoothness conditions, |ˆpK(η) −p(η)| n→∞ −→0 almost surely, ∀η ∈[0, 1]d For simplicity, we limit ourselves to the case when f1 is uniform. The proof of Theorem 2 consists of two steps: • We show that the expectation ES1 [ˆpϵ(η)] n→∞ −→p(η) (Lemma 3). This result is then extended to K-LPE (i.e. ES1 [ˆpK(η)] n→∞ −→p(η)) in Lemma 4. • Next we show that ˆpK(η) n→∞ −→ES1 [ˆpK(η)] via concentration inequality (Lemma 5). Lemma 3 (ϵ-LPE). By picking ϵ = m−3 5d q d 2πe, with probability at least 1 −e−βm1/15/2, lm(η) ≤ES1 [ˆpϵ(η)] ≤um(η) (3) where lm(η) = P0{x : (f0(η) −∆1) (1 −∆2) ≥(f0(x) + ∆1) (1 + ∆2)} −e−βm1/15/2 um(η) = P0{x : (f0(η) + ∆1) (1 + ∆2) ≥(f0(x) −∆1) (1 −∆2)} + e−βm1/15/2 ∆1 = λMm−6/5d/(2πe(d + 2)) and ∆2 = 2m−1/6. Proof. We only prove the lower bound since the upper bound follows along similar lines. By interchanging the expectation with the summation, ES1 [ˆpϵ(η)] = ES1 " 1 m X xi∈S1 I{NS2(η)≥NS1(xi)} # = 1 m X xi∈S1 ExiES1\xi h I{NS2(η)≥NS1(xi)} i = Ex1[PS1\x1(NS2(η) ≥NS1(x1))] 5 where the last inequality follows from the symmetric structure of {x0, x1, · · · , xm}. Clearly the objective of the proof is to show PS1\x1(NS2(η) ≥NS1(x1)) n→∞ −→I{f0(η)≥f0(x1)}. Skipping technical details, this can be accomplished in two steps. (1) Note that NS(x1) is a binomial random variable with success probability q(x1) := R Bϵ f0(x1 +t)dt. This relates PS1\x1(NS2(η) ≥ NS1(x1)) to I{q(η)≥q(x1)}. (2) We relate I{q(η)≥q(x1)} to I{f0(η)≥f0(x1)} based on the function smoothness condition. The details of these two steps are shown in the below. Note that NS1(x1) ∼Binom(m, q(x1)). By Chernoff bound of binomial distribution, we have PS1\x1(NS1(x1) −mq(x1) ≥δ) ≤e− δ2 2mq(x1) that is, NS1(x1) is concentrated around mq(x1). This implies, PS1\x1(NS2(η) ≥NS1(x1)) ≥I{NS2(η)≥mq(x1)+δx1} −e− δ2 x1 2mq(x1) (4) We choose δx1 = q(x1)mγ(γ will be specified later) and reformulate equation (4) as PS1\x1(NS2(η) ≥NS1(x1)) ≥I½ NS2 (η) mVol(Bϵ) ≥q(x1) Vol(Bϵ)(1+ 2 m1−γ ) ¾ −e−q(x1)m2γ−1 2 (5) Next, we relate q(x1)(or R Bϵ f0(x1 +t)dt) to f0(x1) via the Taylor’s expansion and the smoothness condition of f0, ¯¯¯¯¯ R Bϵ f0(x1 + t)dt Vol(Bϵ) −f0(x1) ¯¯¯¯¯ ≤λM 2 · 1 Vol(Bϵ) Z Bϵ ∥t∥2dt = λMϵ2 2d(d + 2) (6) and then equation (5) becomes PS1\x1(NS2(η) ≥NS1(x1)) ≥I½ NS2 (η) mVol(Bϵ) ≥ ³ f0(x1)+ λM ϵ2 2d(d+2) ´ (1+ 2 m1−γ ) ¾ −e−q(x1)m2α−1 2 By applying the same steps to NS2(η) as equation 4 (Chernoff bound) and equation 6 (Taylor’s explansion), we have with probability at least 1 −e−q(η)m2α−1 2 , Ex1[PS1\x1(NS2(η) ≥NS1(x1))] ≥Px1 ½µ f0(η)− λM ϵ2 2d(d+2) ¶³ 1− 2 m1−γ ´ ≥ µ f0(x1)+ λM ϵ2 2d(d+2) ¶³ 1+ 2 m1−γ ´¾ −e−q(x1)m2α−1 2 Finally, by choosing ϵ2 = m−6 5d · d 2πe and γ = 5/6, we prove the lemma. Lemma 4 (K-LPE). By picking K = ¡ 1 −2m−1/6¢ m2/5 (f0(η) −∆1), with probability at least 1 −e−βm1/15/2, lm(η) ≤ES1 [ˆpK(η)] ≤um(η) (7) Proof. The proof is very similar to the proof to Lemma 3 and we only give a brief outline here. Now the objective is to show PS1\x1(RS2(η) ≤RS1(x1)) n→∞ −→I{f0(η)≥f0(x1)}.The basic idea is to use the result of Lemma 3. To accomplish this, we note that {RS2(η) ≤RS1(x1)} contains the events {NS2(η) ≥K} ∩{NS1(x1) ≤K}, or equivalently {NS2(η) −q(η)m ≥K −q(η)m} ∩{NS1(x1) −q(x1)m ≤K −q(x1)m} (8) By the tail probability of Binomial distribution, the probability of the above two events converges to 1 exponentially fast if K −q(η)m < 0 and K −q(x1)m > 0. By using the same two-step bounding techniques developed in the proof to Lemma 3, these two inequalities are implied by K −m2/5 (f0(η) −∆1) < 0 and K −m2/5 (f0(x1) + ∆1) > 0 Therefore if we choose K = ¡ 1 −2m−1/6¢ m2/5 (f0(η) −∆1), we have with probability at least 1 −e−βm−1/15/2, PS1\x1(RS2(η) ≤RS1(x1)) ≥I{(f0(η)−∆1)(1−∆2)≥(f0(x1)+∆1)(1+∆2)} −e−βm−1/15/2 6 Remark: Lemma 3 and Lemma 4 were proved with specific choices for ϵ and K. However, they can be chosen in a range of values, but will lead to different lower and upper bounds. We will show in Section 4 via simulation that our LPE algorithm is generally robust to choice of parameter K. Lemma 5. Suppose K = cm2/5 and denote ˆpK(η) = 1 m P xi∈S1 I{RS2(η)≤RS1(xi)}. We have P0 (|ES1 [ˆpK(η)] −ˆpK(η)| > δ) ≤2e −2δ2m1/5 c2γ2 d where γd is a constant and is defined as the minimal number of cones centered at the origin of angle π/6 that cover Rd. Proof. We can not apply Law of Large Number in this case because I{RS2(η)≤RS1(xi)} are correlated. Instead, we need to use the more generalized concentration-of-measure inequality such as MacDiarmid’s inequality[17]. Denote F(x0, · · · , xm) = 1 m P xi∈S1 I{RS2(η)≤RS1(xi)}. From Corollary 11.1 in [18], sup x0,··· ,xm,x′ i |F(x0, · · · , xi, · · · , xm) −F(x0, · · · , x′ i, · · · , xn)| ≤Kγd/m (9) Then the lemma directly follows from applying McDiarmid’s inequality. Theorem 2 directly follows from the combination of Lemma 4 and Lemma 5 and a standard application of the first Borel-Cantelli lemma. We have used Euclidean distance in Theorem 2. When the support of f0 lies on a lower dimensional manifold (say d′ < d) adopting the geodesic metric leads to faster convergence. It turns out that d′ replaces d in the expression for ∆1 in Lemma 3. 4 Experiments First, to test the sensitivity of K-LPE to parameter changes, we run K-LPE on the benchmark dataset Banana [19] with K varying from 2 to 12. We randomly pick 109 points with “+1” label and regard them as the nominal training data. The test data comprises of 108 “+1” points and 183 “−1” points (ground truth) and the algorithm is supposed to predict “+1” data as nominal and “−1” data as anomalous. Scores computed for test set using Equation 1 is oblivious to true f1 density (“−1” labels). Euclidean distance metric is adopted for this experiment. To control false alarm at level α, points with score smaller than α are predicted as anomaly. Empirical false alarm and true positives are computed from ground truth. We vary α to obtain the empirical ROC curve. The above procedure is followed for the rest of the experiments in this section. As shown in 2(a), the LPE algorithm is insensitive to K. For comparison we plot the empirical ROC curve of the one-class SVM of [9]. For our OC-SVM implementation, for a fixed bandwidth, c, we obtain the empirical ROC curve by varying ν. We then vary the bandwidth, c, to obtain the best (in terms of AUC) ROC curve. The optimal bandwidth turns out to be c = 1.5. In LPE if we set α = 0.05 we get empirical FA = 0.06 and for α = 0.08, empirical FA = 0.09. For OC-SVM we are unaware of any natural way of picking c and ν to control FA rate based on training data. Next, we apply our K-LPE to the problem where the nominal and anomalous data are generated in the following way: f0 ∼1 2N µ· 8 0 ¸ , · 1 0 0 9 ¸¶ + 1 2N µ· −8 0 ¸ , · 1 0 0 9 ¸¶ , f1 ∼N µ 0, · 49 0 0 49 ¸¶ (10) We call ROC curve corresponding to the optimal Bayesian classifier as the Clairvoyant ROC (the red dashed curve in Figure 2(b)). The other two curves are averaged (over 15 trials) empirical ROC curves via LPE. Here we set K = 6 and n = 40 or n = 160. We see that for a relatively small training set of size 160 the average empirical ROC curve is very close to the clairvoyant ROC curve. Finally, we ran LPE on three real-world datasets: Wine, Ionosphere[20] and MNIST US Postal Service (USPS) database of handwritten digits. If there are more than 2 labels in the data set, we artificially regard points with one particular label as nominal and regard the points with other labels as anomalous. For example, for the USPS dataset, we regard instances of digit 0 as nominal and instances of digits 1, · · · , 9 as anomaly. The data points are normalized to be within [0, 1]d and we 7 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 false positives true positives banana data set ROC of LPE (K=2) ROC of LPE (K=4) ROC of LPE (K=6) ROC of LPE (K=8) ROC of LPE (K=10) ROC of LPE (K=12) ROC of one−class SVM (a) SVM vs. K-LPE for Banana Data 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 false positives true positives 2D Gaussian mixture ROC of LPE(n=40) ROC of LPE(n=160) Clairvoyant ROC (b) Clairvoyant vs. K-LPE Figure 2: (a) Empirical ROC curve of K-LPE on the banana dataset with K = 2, 4, 6, 8, 10, 12 (with n = 400) vs the empirical ROC curve of one class SVM developed in [9]; (b) Empirical ROC curves of K-LPE algorithm vs clairvoyant ROC curve (f0 is given by Equation 10) for K = 6 and for n = 40 or 160. use geodesic distance [14]. The ROC curves are shown in Figure 3. The feature dimension of Wine is 13 and we apply the ϵ-LPE algorithm with ϵ = 0.9 and n = 39. The test set is a mixture of 20 nominal points and 158 anomaly points. The feature dimension of Ionosphere is 34 and we apply the K-LPE algorithm with K = 9 and n = 175. The test set is a mixture of 50 nominal points and 126 anomaly points. The feature dimension of USPS is 256 and we apply the K-LPE algorithm with K = 9 and n = 400. The test set is a mixture of 367 nominal points and 33 anomaly points. In USPS, setting α = 0.5 induces empirical false-positive 6.1% and empirical false alarm rate 5.7% (In contrast FP = 7% and FA = 9% with ν = 5% for OC-SVM as reported in [9]). Practically we find that K-LPE is more preferable to ϵ-LPE and as a rule of thumb setting K ≈n2/5 is generally effective. 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 false positive true positive 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 false positive true positive 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 false positive true positive (a) Wine (b) Ionosphere (c) USPS Figure 3: ROC curves on real datasets via LPE; (a) Wine dataset with D = 13, n = 39, ϵ = 0.9; (b) Ionosphere dataset with D = 34, n = 175, K = 9; (c) USPS dataset with D = 256, n = 400, K = 9. 5 Conclusion In this paper, we proposed a novel non-parametric adaptive anomaly detection algorithm which leads to a computationally efficient solution with provable optimality guarantees. Our algorithm takes a K-nearest neighbor graph as an input and produces a score for each test point. Scores turn out to be empirical estimates of the volume of minimum volume level sets containing the test point. While minimum volume level sets provide an optimal characterization for anomaly detection, they are high dimensional quantities and generally difficult to reliably compute in high dimensional feature spaces. Nevertheless, a sufficient statistic for optimal tradeoff between false alarms and misses is the volume of the MV set itself, which is a real number. By computing score functions we avoid computing high dimensional quantities and still ensure optimal control of false alarms and misses. The computational cost of our algorithm scales linearly in dimension and quadratically in data size. 8 References [1] C. Campbell and K. P. Bennett, “A linear programming approach to novelty detection,” in Advances in Neural Information Processing Systems 13. MIT Press, 2001, pp. 395–401. [2] M. Markou and S. Singh, “Novelty detection: a review – part 1: statistical approaches,” Signal Processing, vol. 83, pp. 2481–2497, 2003. [3] R. Ramaswamy, R. Rastogi, and K. Shim, “Efficient algorithms for mining outliers from large data sets,” in Proceedings of the ACM SIGMOD Conference, 2000. [4] R. Vert and J. Vert, “Consistency and convergence rates of one-class svms and related algorithms,” Journal of Machine Learning Research, vol. 7, pp. 817–854, 2006. [5] D. Tax and K. R. M¨uller, “Feature extraction for one-class classification,” in Artificial neural networks and neural information processing, Istanbul, TURQUIE, 2003. [6] R. El-Yaniv and M. Nisenson, “Optimal singl-class classification strategies,” in Advances in Neural Information Processing Systems 19. MIT Press, 2007. [7] I. V. Nikiforov and M. Basseville, Detection of abrupt changes: theory and applications. Prentice-Hall, New Jersey, 1993. [8] K. Zhang, M. Hutter, and H. Jin, “A new local distance-based outlier detection approach for scattered real-world data,” March 2009, arXiv:0903.3257v1[cs.LG]. [9] B. Sch¨olkopf, J. C. Platt, J. Shawe-Taylor, A. J. Smola, and R. Williamson, “Estimating the support of a high-dimensional distribution,” Neural Computation, vol. 13, no. 7, pp. 1443–1471, 2001. [10] D. Tax, “One-class classification: Concept-learning in the absence of counter-examples,” Ph.D. dissertation, Delft University of Technology, June 2001. [11] G. R. G. Lanckriet, L. E. Ghaoui, and M. I. Jordan, “Robust novelty detection with single-class MPM,” in Neural Information Processing Systems Conference, vol. 18, 2005. [12] C. Scott and R. D. Nowak, “Learning minimum volume sets,” Journal of Machine Learning Research, vol. 7, pp. 665–704, 2006. [13] A. O. Hero, “Geometric entropy minimization(GEM) for anomaly detection and localization,” in Neural Information Processing Systems Conference, vol. 19, 2006. [14] J. B. Tenenbaum, V. de Silva, and J. C. Langford, “A global geometric framework fo nonlinear dimensionality reduction,” Science, vol. 290, pp. 2319–2323, 2000. [15] M. Bernstein, V. D. Silva, J. C. Langford, and J. B. Tenenbaum, “Graph approximations to geodesics on embedded manifolds,” 2000. [16] S. T. Roweis and L. K. Saul, “Nonlinear dimensionality reduction by local linear embedding,” Science, vol. 290, pp. 2323–2326, 2000. [17] C. McDiarmid, “On the method of bounded differences,” in Surveys in Combinatorics. Cambridge University Press, 1989, pp. 148–188. [18] L. Devroye, L. Gy¨orfi, and G. Lugosi, A Probabilistic Theory of Pattern Recognition. Springer Verlag New York, Inc., 1996. [19] “Benchmark repository.” [Online]. Available: http://ida.first.fhg.de/projects/bench/benchmarks.htm [20] A. Asuncion and D. J. Newman, “UCI machine learning repository,” 2007. [Online]. Available: http://www.ics.uci.edu/∼mlearn/MLRepository.html 9
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Submodularity Cuts and Applications Yoshinobu Kawahara∗ The Inst. of Scientific and Industrial Res. (ISIR), Osaka Univ., Japan kawahara@ar.sanken.osaka-u.ac.jp Kiyohito Nagano Dept. of Math. and Comp. Sci., Tokyo Inst. of Technology, Japan nagano@is.titech.ac.jp Koji Tsuda Comp. Bio. Research Center, AIST, Japan koji.tsuda@aist.go.jp Jeff A. Bilmes Dept. of Electrical Engineering, Univ. of Washington, USA bilmes@u.washington.edu Abstract Several key problems in machine learning, such as feature selection and active learning, can be formulated as submodular set function maximization. We present herein a novel algorithm for maximizing a submodular set function under a cardinality constraint — the algorithm is based on a cutting-plane method and is implemented as an iterative small-scale binary-integer linear programming procedure. It is well known that this problem is NP-hard, and the approximation factor achieved by the greedy algorithm is the theoretical limit for polynomial time. As for (non-polynomial time) exact algorithms that perform reasonably in practice, there has been very little in the literature although the problem is quite important for many applications. Our algorithm is guaranteed to find the exact solution finitely many iterations, and it converges fast in practice due to the efficiency of the cutting-plane mechanism. Moreover, we also provide a method that produces successively decreasing upper-bounds of the optimal solution, while our algorithm provides successively increasing lower-bounds. Thus, the accuracy of the current solution can be estimated at any point, and the algorithm can be stopped early once a desired degree of tolerance is met. We evaluate our algorithm on sensor placement and feature selection applications showing good performance. 1 Introduction In many fundamental problems in machine learning, such as feature selection and active learning, we try to select a subset of a finite set so that some utility of the subset is maximized. A number of such utility functions are known to be submodular, i.e., the set function f satisfies f(S) + f(T) ≥ f(S ∩T) + f(S ∪T) for all S, T ⊆V , where V is a finite set [2, 5]. This type of function can be regarded as a discrete counterpart of convex functions, and includes entropy, symmetric mutual information, information gain, graph cut functions, and so on. In recent years, treating machine learning problems as submodular set function maximization (usually under some constraint, such as limited cardinality) has been addressed in the community [10, 13, 22]. In this paper, we address submodular function maximization under a cardinality constraint: max S⊆V f(S) s.t. |S| ≤k, (1) where V = {1, 2, . . . , n} and k is a positive integer with k ≤n. Note that this formulation is considerably general and covers a broad range of problems. The main difficulty of this problem comes from a potentially exponentially large number of locally optimal solutions. In the field of ∗URL: http://www.ar.sanken.osaka-u.ac.jp/ kawahara/ 1 combinatorial optimization, it is well-known that submodular maximization is NP-hard and the approximation factor of (1 −1/e) (≈0.63) achieved by the greedy algorithm [19] is the theoretical limit of a polynomial-time algorithm for positive and nondecreasing submodular functions [3]. That is, in the worst case, any polynomial-time algorithm cannot give a solution whose function value is more than (1 −1/e) times larger than the optimal value unless P=NP. In recent years, it has been reported that greedy-based algorithms work well in several machine-learning problems [10, 1, 13, 22]. However, in some applications of machine learning, one seeks a solution closer to the optimum than what is guaranteed by this bound. In feature selection or sensor placement, for example, one may be willing to spend much more time in the selecting phase, since once selected, items are used many times or for a long duration. Unfortunately, there has been very little in the literature on finding exact but still practical solutions to submodular maximization [17, 14, 8]. To the best of our knowledge, the algorithm by Nemhauser and Wolsey [17] is the only way for exactly maximizing a general form of nondecreasing submodular functions (other than naive brute force). However, as stated below, this approach is inefficient even for moderate problem sizes. In this paper, we present a novel algorithm for maximizing a submodular set function under a cardinality constraint based on a cutting-plane method, which is implemented as an iterative small-scale binary-integer linear programming (BILP) procedure. To this end, we derive the submodularity cut, a cutting plane that cuts off the feasible sets on which the objective function values are guaranteed to be not better than current best one, and this is based on the submodularity of a function and its Lov´asz extension [15, 16]. This cut assures convergence to the optimum in finite iterations and allows the searching for better subsets in an efficient manner so that the algorithm can be applied to suitably-sized problems. The existing algorithm [17] is infeasible for such problems since, as originally presented, it has no criterion for improving the solution efficiently at each iteration (we compare these algorithms empirically in Sect. 5.1). Moreover, we present a new way to evaluate an upper bound of the optimal value with the help of the idea of Nemhauser and Wolsey [17]. This enables us to judge the accuracy of the current best solution and to calculate an ϵ-optimal solution for a predetermined ϵ > 0 (cf. Sect. 4). In our algorithm, one needs to iteratively solve smallscale BILP (and mixed integer programming (MIP) for the upper-bound) problems, which are also NP-hard. However, due to their small size, these can be solved using efficient modern software packages such as CPLEX. Note that BILP is a special case of MIP and more efficient to solve in general, and the presented algorithm can be applied to any submodular functions while the existing one needs the nondecreasing property.1 We evaluate the proposed algorithm on the applications of sensor placement and feature selection in text classification. The remainder of the paper is organized as follows: In Sect. 2, we present submodularity cuts and give a general description of the algorithm using this cutting plane. Then, we describe a specific procedure for performing the submodularity cut algorithm in Sect. 3 and the way of updating an upper bound for calculating an ϵ-optimal solution in Sect. 4. And finally, we give several empirical examples in Sect. 5, and conclude the paper in Sect. 6. 2 Submodularity Cuts and Cutting-Plane Algorithm We start with a subset S0 ⊆V of some ground set V with a reasonably good lower bound γ = f(S0) ≤max{f(S) : S ⊆V }. Using this information, we cut off the feasible sets on which the objective function values are guaranteed to be not better than f(S0). In this section, we address a method for solving the submodular maximization problem (1) based on this idea along the line of cutting-plane methods, as described by Tuy [23] (see also [6, 7]) and often successfully used in algorithms for solving mathematical programming problems [18, 11, 20]. 2.1 Lov´asz extension For dealing with the submodular maximization problem (1) in a way analogous to the continuous counterpart, i.e., convex maximization, we briefly describe an useful extension to submodular functions, called the Lov´asz extension [15, 16]. The relationship between the discrete and the continuous, described in this subsection, is summarized in Table 1. 1A submodular function is called nondecreasing if f(A) ≤f(B) for (A ⊆B). For example, an entropy function is nondecreasing but a cut function on nodes is not. 2 Table 1: Correspondence between continuous and discrete. (discrete) (continuous) f : 2V →R Eq. (2) =⇒ ˆf : Rn →R S ⊆V Eq. (3) ⇐⇒ IS ∈Rn f is submodular Thm. 1 ⇐⇒ ˆf is convex v H H* c* d1 d2 y2 y1 P H+ H Figure 1: Illustration of cutting plane H. For H∗ and c∗, see Section 3.2. Given any real vector p ∈Rn, we denote the m distinct elements of p by ˆp1 > ˆp2 > · · · > ˆpm. Then, the Lov´asz extension ˆf : Rn →R corresponding to a general set function f : 2V →R, which is not necessarily submodular, is defined as ˆf(p) = Pm−1 k=1 (ˆpk −ˆpk+1)f(Uk) + ˆpmf(Um), (2) where Uk = {i ∈V : pi ≥ˆpk}. From the definition, ˆf is a piecewise linear (i.e., polyhedral) function.2 In general, ˆf is not convex. However, the following relationship between the submodularity of f and the convexity of ˆf is given [15, 16]: Theorem 1 For a set function f : 2V →R and its Lov´asz extension ˆf : Rn →R, f is submodular if and only if ˆf is convex. Now, we define IS ∈{0, 1}n as IS = P i∈S ei, where ei is the i-th unit vector. Obviously, there is a one-to-one correspondence between IS and S. IS is called the characteristic vector of S.3 Then, the Lov´asz extension ˆf is a natural extension of f in the sense that it satisfies the following [15, 16]: ˆf(IS) = f(S) (S ⊆V ). (3) In what follows, we assume that f is submodular. Now we introduce a continuous relaxation of the problem (1) using the Lov´asz extension ˆf. A polytope P ⊆Rn is a bounded intersection of a finite set of half-spaces — that is, P is of the form P = {x ∈Rn : A⊤ j x ≤bj, j = 1, · · · , m}, where Aj is a real vector and bj is a real scalar. According to the correspondence between discrete and continuous functions described above, it is natural to replace the objective function f : 2V →R and the feasible region {S ⊆V : |S| ≤k} of the problem (1) by the Lov´asz extension ˆf : Rn →R and a polytope D0 ⊆Rn defined by D0 = {x ∈Rn : 0 ≤xi ≤1 (i = 1, · · · , n), Pn i=1xi ≤k}, respectively. The resulting problem is a convex maximization problem. For problem (1), we will use the analogy with the way of solving the continuous problem: max { ˆf(x) : x ∈D0}. The question is, can we solve it and how good is the solution? 2.2 Submodularity cuts Here, we derive what we call the submodularity cut, a cutting plane that cuts off the feasible sets with optimality guarantees using the submodularity of f, and with the help of the relationship between submodularity and convexity described in Thm. 1. Note that the algorithm using this cutting plane, described later, converges to an optimal solution in a finite number of iterations (cf. Thm. 5). The presented technique is essentially a discrete analog of concavity cut techniques for continuous concave minimization, which rests on the following property (see, e.g., [11]). Theorem 2 A convex function g : Rn →R attains its global maximum over a polytope P ⊂Rn at a vertex of P. 2For a submodular function, the Lov´asz extension (2) is known to be equal to ˆf(p) = sup{pT x : x ∈B(f)} (p ∈Rn), where B(f) = {x ∈Rn : x(S) ≤f(S) (∀S ⊂V ), x(V ) = f(V )} is the base polyhedron associated with f [15] and x(S) = P i∈S xi. 3For example in case of |V | = 6, the characteristic vector of S = {1, 3, 4} becomes IS = (1, 0, 1, 1, 0, 0). 3 First, we clarify the relation between discrete and continuous problems. Let P be a polytope with P ⊆D0. Denote by S(P) the subsets of V whose characteristic vectors are inside of P, i.e., IS′ ∈P for any S′ ∈S(P), and denote by V (P) the set consisting of all vertices of P. Note that any characteristic vector IS ∈P is a vertex of P. Also, there is a one-to-one correspondence between S(D0) and V (D0). Now clearly, we have max{f(S′) : S′ ∈S(P)} ≤max{ ˆf(x) : x ∈P}. (4) If we can find a subset ¯P where the function value of ˆf is always smaller than the currently-known largest value, any f( ¯S) for ¯S ∈S( ¯P) is also smaller than the value. Thus, the cutting plane for the problem max{ ˆf(x) : x ∈D0} can be applied to our problem (1) through the relationship (4). To derive the submodularity cut, we use the following definition: Definition 3 (γ-extension) Let g : Rn →R be a convex function, x ∈Rn, γ be a real number satisfying γ ≥g(x) and t > 0. Then, a point y ∈Rn defined by the following formula is called γ-extension of x in direction d ∈Rn \ {0} (with respect to g) where θ ∈R ∪{∞}: y = x + θd with θ = sup{t : g(x + td) ≤γ}. (5) We may have θ = ∞depending on g and d, but this is unproblematic in practice. The γ-extension of x ∈Rn can be defined with respect to the Lov´asz extension because it is a convex function. The submodular cut algorithm is an iterative procedure. At each iteration, the algorithm keeps a polytope P ⊆D0, the current best function value γ, and a set S∗⊆V satisfying f(S∗) = γ. We construct a submodular cut as follows. Let v ∈V (P) be a vertex of P such that v = IS for some S ∈S(P), and let K = K(v; d1, . . . , dn) be a convex polyhedral cone with vertex v generated by linearly independent vectors d1, . . . , dn, i. e., K = {v + t1d1 + · · · + tndn : tl ≥0}. For each i = 1, · · · , n, let yl = v + θldl be the γ-extension of v in direction dl with respect to ˆf. We choose the vectors d1, . . . , dn so that P ⊂K and θl > 0 (cf. Sect. 3.1). These directions are not necessarily chosen tightly on P (in fact, the directions described in Sect. 3.1 enclose P but also a set larger). Since the vectors dl are linearly independent, there exists a unique hyperplane H = H(y1, · · · , yn) that contains yl (l = 1, · · · , n), which we call a submodular cut. It is defined by (cf. Fig. 1) H = {x : eT Y −1x = 1 + eT Y −1v}. (6) where e = (1, · · · , 1)T ∈Rn and Y = ((y1 −v), · · · , (yn −v)). The hyperplane H generates two halfspaces H−= {x : eT Y −1x ≤1 + eT Y v} and H+ = {x : eT Y −1x ≥1 + eT Y v}. Obviously the point v is in the halfspace H−, and moreover, we have: Lemma 4 Let P ⊆D0 be a polytope, γ be the current best function value, v be a vertex of P such that v = IS for some S ∈S(P) and H−be the halfspace determined by the cutting plane, i.e., H−= {x : eT Y −1x ≤1 + eT Y v}, where Y = ((y1 −v), · · · , (yn −v)) and y1, . . . , yn are the γ-extensions of v in linearly independent directions d1, . . . , dn. Then, it holds that f(S′) ≤γ for all S′ ∈S(P ∩H−). Proof Since P ⊂K = K(IS; d1, · · · , dn), it follows that P ∩H−is contained in the simplex R = [IS, y1, · · · , yn]. Since the Lov´asz extension ˆf is convex and the maximum of a convex function over a compact convex set is attained at a vertex of the convex set (Thm. 2), the maximum of ˆf over R is attained at a vertex of R. Therefore, we have max{ ˆf(x) : x ∈P ∩H−} ≤max{f(x) : x ∈R} = max{ ˆf(v); ˆf(y1), · · · , ˆf(yn)} ≤γ. From Eq. (4), max{f(S′) : S′ ∈S(P ∩H−)} ≤max{ ˆf(x) : x ∈P ∩H−} ≤γ. The above lemma shows that we can cut off the feasible subsets S(P ∩H−) from S(P) without loss of any feasible set whose objective function value is better than γ. If S(P) = S(P ∩H−), then γ = max{f(S) : |S| ≤k} is achieved. A specific way to check whether S(P) = S(P ∩H−) will be given in Sect. 3.2. As v ∈S(P ∩H−) and v /∈S(P ∩H+), we have |S(P)| > |S(P ∩H+)|. (7) The submodular cut algorithm updates P ←P ∩H+ until the global optimality of γ is guaranteed. The general description is shown in Alg. 1 (also see Fig. 2). Furthermore, the finiteness of the algorithm is assured by the following theorem. 4 S0 S(P0) Sopt S(Popt) H 0 S1 S(P1) <continuous> <discrete> Lovasz extension Prop. 7 ... ... H1 P1=P0 ∩ H 0 Sopt-1 S(Popt-1) Hopt-1 Ѯ Ѯ Ѯ Ѯ + Popt=Popt-1 ∩ H opt-1 + Figure 2: Outline of the submodularity cuts algorithm. Algorithm 1 General description of the submodularity cuts algorithm. 1. Compute a subset S0 s.t. |S0| ≤k, and set a lower bound γ0 = f(S0). 2. Set P0 ←D0, stop ←false, i ←1 and S∗= S0. 3. while stop=false do 4. Construct with respect to Si−1, Pi−1 and γi−1 a submodularity cut Hi. 5. if S(Pi−1) = S(Pi−1 ∩Hi −) then 6. stop ←true (S∗is an optimal solution and γi−1 the optimal value). 7. else 8. Update γi (using Si and other available information) and set S∗s.t. f(S∗) = γi. 9. Compute Si ∈S(Pi), and set Pi ←Pi−1 ∩Hi + and i ←i + 1. 10. end if 11. end while Theorem 5 Alg. 1 gives an optimal solution to the problem (1) in a finite number of iterations. Proof In the beginning, |S(D0)| is finite. In view of (7), each iteration decreases |S(P)| by at least 1. So, the number of iterations is finite. 3 Implementation In this section, we describe a specific way to perform Alg. 1 using a binary-integer linear programming (BILP) solver. The pseudo-code of the resulting algorithm is shown in Alg. 2. 3.1 Construction of submodularity cuts Given a vertex of a polytope P ⊆D0, which is of the form IS, we describe how to compute linearly independent directions d1, · · · , dn for the construction of the submodularity cut at each iteration of the algorithm (Line 4 in Alg. 1). Note that the way described here is just one option and any other choice satisfying P ⊂K can be substituted. If |S| < k, then directions d1, . . . , dn can be chosen as −el (l ∈S) and el (l ∈V \ S). Now we focus on the case where |S| = k. Define a neighbor S(i,j) of S as S(i,j) := (S \ {i}) ∪{j} (i ∈S, j ∈V \ S). That is, the neighbor S(i,j) is given by replacing one of the elements of S with that of V \ S. Note that IS(i,j) −IS = ej −ei for any neighbor S(i,j) of S. Let S(i∗,j∗) be a neighbor that maximizes f(S(i,j)) among all neighbors of S. Since a subset S of size k has k × (n −k) neighbors S(i,j) (i ∈S, j ∈V \ S), this computation is O(nk). Suppose that S = {i1, . . . , ik} with i1 = i∗ and V \ S = {jk+1, . . . , jn} with jn = j∗. If f(S(i∗,j∗)) > γ, we update γ ←f(S(i∗,j∗)) and S∗←S(i∗,j∗). Thus, in either case it holds that γ ≥f(S(i∗,j∗)). As an example of the set of directions {d1, . . . , dn}, we choose dl = ( ej∗−eil if l ∈{1, . . . , k} ejl −ej∗ if l ∈{k + 1, . . . , n −1} −ej∗ if l = n. (8) It is easy to see that d1, . . . , dn are linearly independent. Moreover, we obtain the following lemma: Lemma 6 For the directions d1, . . . , dn defined in (8), a cone K(IS; d1, . . . , dn) = {IS + t1d1 + · · · + tndn : tl ≥0} contains the polytope D0 = {x ∈Rn : 0 ≤xl ≤1 (l = 1, · · · , n), Pn l=1xl ≤k}. The proof of this lemma is included in the supplementary material (Sect. A). The γ-extensions, i.e., θ’s, in these directions can be obtained in closed forms. The details of this are also included in the supplementary material (Sect. A). 5 Algorithm 2 Pseudo-code of the submodularity cuts algorithm using BILP. 1. Compute a subset S0 s.t. |S0| ≤k, and set a lower bound γ0 = f(S0). 2. Set P0 ←D0, stop ←false, i ←1 and S∗= S0. 3. while stop=false do 4. Construct with respect to Si−1, Pi−1 and γi−1 a submodularity cut H. 5. Solve the BILP problem (9) with respect to Aj and bj (j = 1, · · · , nk), and let the optimal solution and value Si and c∗, respectively. 6. if c∗≤1 + eT Y −1vi−1 then 7. stop ←true (S∗is an optimal solution and γi−1 the optimal value). 8. else 9. Update γi (using Si and other available information) and set S∗s.t. f(S∗) = γi. 10. Set Pi ←Pi−1 ∩H+ and i ←i + 1. 11. end if 12. end while 3.2 Stopping criterion and next starting point Next, we address the checking of optimality, i.e., whether S(P) = S(P ∩H−), and also finding the next starting subset Si (respectively, in Lines 5 and 9 in Alg. 1). Let eP ⊆Rn be the minimum polytope containing S(P). Geometrically, checking S(P) = S(P ∩H−) can be done by considering a parallel hyperplane H∗of H which is tangent to eP. If H = H∗or H∗is given by translating H towards v, then S(P) = S(P ∩H−). Numerically, such a translation corresponds to linear programming. Using Eq. (6), we obtain: Proposition 7 Let c∗be the optimal value of the binary integer program max x∈{0,1}n{eT Y −1x : Ajx ≥bj, j = 1, · · · , mk}. (9) Then S(P) ⊂H−if c∗≤1 + eT Y −1v. Note that, if c∗> 1+eT Y −1v, then the optimal solution x∗of Eq. (9) yields a subset of S(P \H−) which can be used as a starting subset of the next iteration (see Fig. 1). 4 Upper bound and ϵ-optimal solution Although our algorithm can find an exact solution in a finite number of iterations, the computational cost could be expensive for a high-dimensional case. Therefore, we present here an iterative update of an upper bound of the current solution, and thus a way to allow us to obtain an ϵ-optimal solution. To this end, we combine the idea of the algorithm by Nemhauser and Wolsey [17] with our cutting plane algorithm. Note that this hybrid approach is effective only when f is nondecreasing. If the submodular function f : 2V →R is nondecreasing, the submodular maximization problem (1) can be reformulated [17] as max η s.t. η ≤f(S) + P j∈V \Sρj(S)yj (S ⊆V ), P j∈V yj = k, yj ∈{0, 1} (j ∈V ) (10) where ρj(S) := f(S ∪{j}) −f(S). This formulation is a MIP with regard to one continuous and n binary variables, and has approximately 2n constraints. The first type of constraint corresponds to all feasible subsets S, and the number of inequalities is as large as 2n. This approach is therefore infeasible for certain problem sizes. Nemhauser and Wolsey [17] address this problem by adding the constraints one by on and calculating a reduced MIP problem iteratively. In the worse case, however, the number of iterations becomes equal to the case of when all constraints are added. The solution of a maximization problem with a subset of constraints is larger than the one with all constraints, so the good news is that this solution is guaranteed to improve (by monotonically decreasing down to the true solution) at each iteration. In our algorithm, by contrast, the best current solution increases monotonically to the true solution. Therefore, by adding the constraint corresponding to Si at each iteration of our algorithm and solving the reduced MIP above, we can evaluate an upper bound of the current solution. Thus, we can assure the optimality of a current solution, or obtain a desired ϵ-optimal solution using both the lower and upper bound. 6 b b 6XEPRGXODULW\b RSW 6XEPRGXODULW\b 6XEPRGXODULW\b 1HPKDXVHUb b:ROVH\ Dimensionality (n) Time (log-scale) [s] k = 5 b b 6XEPRGXODULW\b RSW 6XEPRGXODULW\b 6XEPRGXODULW\b 1HPKDXVHUb b:ROVH\ Dimensionality (n) Time (log-scale) [s] k = 8 Figure 3: Averaged computational time (log-scale) for computing exact and ϵ-optimal solutions by the submodularity cut algorithm and existing algorithm by Nemhauser and Wolsey. b b Upper bound Lower bound Function value of the solution by the greedy algorithm Time (log-scale) [s] Function value Figure 4: An example of computational time (log-scale) versus the calculated upper and lower bounds. 5 Experimental Evaluation We first empirically compare the proposed algorithm with the existing algorithm by Nemhauser and Wolsey [17] in Sect. 5.1, and then apply the algorithm to the real-world applications of sensor placement, and feature selection in text classification (Sect. 5.2 and 5.3, respectively). In the experiments, we used the solution by a greedy algorithm as initial subset S0. The experiments below were run on a 2.5GHz 64-bit workstation using Matlab and a Parallel CPLEX ver. 11.2 (8 threads) through a mex function. If θ = ∞in Eq. (5), we set θ = θ1, where θ1 is large (i.e. θ1 = 106). 5.1 Artificial example Here, we evaluate empirically and illustrate the submodularity cut algorithm (Alg. 2) with respect to (1) computational time for exact solutions compared with the existing algorithm and (2) how fast the algorithm can sandwich the true solution between the upper and lower bounds, using artificial datasets. The considered problem here is the K-location problem [17], i.e., the submodular maximization problem (1) with respect to the nondecreasing submodular function: f(S) = Pm i=1 maxj∈S cij, where C = cij is an m×n nonnegative matrix and V = {1, · · · , n}. We generated several matrices C of different size n (we fixed m = n+1), and solved the above problem with respect to k = 5, 8 for exact and ϵ optimal solutions, using the two algorithms. The graphs in Fig. 3 show the computational time (log-scale) for several n and k = 5, 8, where the results were averaged over randomly generated 3 matrices C. Note that, for example, the number of combination becomes more than two hundred millions for n = 45 and k = 8. As the figure shows, the required costs for Alg. 2 were less than the existing algorithm, especially in the case of high search spaces. This could be because the cuttingplane algorithm searches feasible subsets in an efficient manner by eliminating worse ones with the submodularity cuts. And Fig. 4 shows an example of the calculated upper and lower bounds vs. time (k = 5 and n = 45). The lower bound is updated rarely and converges to the optimal solution quickly while the upper bound decreases gradually. 5.2 Sensor placements Our first example with real data is the sensor placements problem, where we try to select sensor locations to minimize the variance of observations. The dataset we used here is temperature measurements at discretized finite locations V obtained using the NIMS sensor node deployed at a lake near the University of California, Merced [9, 12] (|V | = 86).4 As in [12], we evaluated the set of locations S ⊆V using the averaged variance reduction f(S) = V ar(∅) −V ar(S) = 1 n P sFs(S), where Fs(S) = σ2 s −σ2 s|S is the variance reduction and σ2 s|S denote the predictive variance at location s ∈V after observing locations S ⊆V . This function is monotone and submodular. The graphs in Fig. 5 show the computation time of our algorithm, and the accuracy improvement of our calculated solution over that of the greedy algorithm (%), respectively, for ϵ = 0.05, 0.1, 0.2. Both the computation time and improvement are large at around k = 5 compared with other choices of k. This is because the greedy solutions are good when k is either very small or large. 4The covariance matrix of the Gaussian process that models the measurements is available in Matlab Toolbox for Submodular Function Optimization (http://www.cs.caltech.edu/∼krausea/sfo/). 7 Cardinality ( k ) Time (log-scale) [s] b b Cardinality ( k ) Improvement [%] b b Figure 5: Computational time (left) and accuracy improvement over the greedy algorithm (right). Table 1: Selected words with [the values of information gain, classification precision]. k greedy submodularity cuts 5 (tonn,‘agricultur’,trade,pct,‘market’)[2.59,0.53]→(‘week’,tonn,trade,pct,‘washington’)[2.66,0.58] 10 ( . . .,week,oil,price,‘dollar’,‘offici’)[3.55,0.57]→( . . .,price,oil,‘bank’,‘produc’,‘blah’)[3.88,0.62] 5.3 Feature selection in text classification Our second real test case is feature selection in document classification using the Reuters-21578 dataset. We applied the greedy and submodularity cuts algorithms to the training set that includes 7,770 documents with 5,180 words (features) and 90 categories, where we used the information gain as a criterion [4]. Table 1 shows the selected words by the algorithms in the cases of k = 5, 10 (for the proposed algorithm ϵ = 0.003 in both cases) with the values of information gain and classification precision (tp/(tp + fp), tp; true positive, fp; false positive). For classification on the test set (3,019 documents with 5,180 words and 90 categories), we applied a Naive Bayes classifier with the selected features. The submodularity cuts algorithm selected several different words from that of the greedy algorithm. We can see that the words selected by our algorithm would have high predictive power even though the number of the chosen words is very small. 6 Conclusions In this paper, we presented a cutting-plane algorithm for submodular maximization problems, which can be implemented as an iterative binary-integer linear programming procedure. We derived a cutting plane procedure, called the submodularity cut, based on the submodularity of a set function through the Lov´asz extension, and showed this cut assures that the algorithm converges to the optimum in finite iterations. Moreover, we presented a way to evaluate an upper bound of the optimal value with the help of Nemhauser and Wolsey [17], which enables us to ensure the accuracy of the current best solution and to calculate an intended ϵ-optimal solution for a predetermined ϵ > 0. Our new algorithm computationally compared favorably against the existing algorithm on artificial datasets, and also showed improved performance on the real-world applications of sensor placements and feature selection in text classification. The submodular maximization problem treated in this paper covers broad range of applications in machine learning. In future works, we will develop frameworks with ϵ-optimality guarantees for more general problem settings such as knapsack constraints [21] and not nondecreasing submodular functions. This will be make the submodularity cuts framework applicable to a still wider variety of machine learning problems. Acknowledgments This research was supported in part by JSPS Global COE program “Computationism as a Foundation for the Sciences”, KAKENHI (20800019 and 21680025), the JFE 21st Century Foundation, and the Functional RNA Project of New Energy and Industrial Technology Development Organization (NEDO). Further support was received from a PASCAL2 grant, and by NSF grant IIS-0535100. Also, we are very grateful to the reviewers for helpful comments. 8 References [1] A. Das and D. Kempe. Algorithms for subset selection in linear regression. In R. E. Ladner and C. Dwork, editors, Proc. of the 40th Annual ACM Symp. on Theory of Computing (STOC 2008), pages 45–54, 2008. [2] J. Edmonds. Submodular functions, matroids, and certain polyhedra. In R. Guy, H. Hanani, N. Sauer, and J. Sh¨onheim, editors, Combinatorial Structures and Their Applications, pages 69–87. Gordon and Breach, 1970. [3] U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45:634–652, 1998. [4] G. Forman. An extensive empirical study of feature selection metrics for text classification. Journal of Machine Learning Research, 3:1289–1305, 2003. [5] S. Fujishige. Submodular Functions and Optimization. Elsevier, second edition, 2005. [6] F. Glover. Convexity cuts and cut search. Operations Research, 21:123–134, 1973. [7] F. Glover. Polyhedral convexity cuts and negative edge extension. Zeitschrift f¨ur Operations Research, 18:181–186, 1974. [8] B. Goldengorin. Maximization of submodular functions: Theory and enumeration algorithms. European Journal of Operational Research, 198(1):102–112, 2009. [9] T. C. Harmon, R. F. Ambrose, R. M. Gilbert, J. C. Fisher, M. Stealey, and W. J. Kaiser. High resolution river hydraulic and water quality characterization using rapidly deployable. Technical report, CENS,, 2006. [10] S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu. Batch mode active learning and its application to medical image classification. In Proc. of the 23rd int’l conf. on Machine learning (ICML 2006), pages 417–424, 2006. [11] R. Horst and H. Tuy. Global Optimization (Deterministic Approaches). Springer, 3 edition, 1996. [12] A. Krause, H. B. McMahan, C. Guestrin, and A. Gupta. Robust submodular observation selection. Journal of Machine Learning Research, 9:2761–2801, 2008. [13] A. Krause, A. Singh, and C. Guestrin. Near-optimal sensor placements in Gaussian processes: Theory, efficient algorithms and empirical studies. Journal of Machine Learning Research, 9:235–284, 2009. [14] H. Lee, G. L. Nemhauser, and Y. Wang. Maximizing a submodular function by integer programming: Polyhedral results for the quadratic case. European Journal of Operational Research, 94:154–166, 1996. [15] L. Lov´asz. Submodular functions and convexity. In A. Bachem, M. Gr¨otschel, and B. Korte, editors, Mathematical Programming – The State of the Art, pages 235–257. 1983. [16] K. Murota. Discrete Convex Analysis, volume 10 of Monographs on Discrete Math and Applications. Society for Industrial & Applied, 2000. [17] G. L. Nemhauser and L. A. Wolsey. Maximizing submodular set functions: formulations and analysis of algorithms. In P. Hansen, editor, Studies on Graphs and Discrete Programming, volume 11 of Annals of Discrete Mathematics. 1981. [18] G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. Wiley-Interscience, 1988. [19] G. L. Nemhauser, L. A. Wolsey, and M. L. Fisher. An analysis of approximations for maximizing for submodular set functions – I. Mathematical Programming, 14:265–294, 1978. [20] M. Porembski. Finitely convergent cutting planes for concave minimization. Journal of Global Optimization, 20(2):109–132, 2001. [21] M. Sviridenko. A note on maximizing a submodular set function subject to a knapsack constraint. Operations Research Letters, 32(1):41–43, 2004. [22] M. Thoma, H. Cheng, A. Gretton, J. Han, H. P. Kriegel, A. J. Smola, L. Song, P. S. Yu, X. Yan, and K. M. Borgwardt. Near-optimal supervised feature selection among frequent subgraphs. In Proc. of the 2009 SIAM Conference on Data Mining (SDM 2009), pages 1075–1086, 2009. [23] H. Tuy. Concave programming under linear constraints. Soviet Mathematics Doklady, 5:1437–1440, 1964. 9
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Bayesian Sparse Factor Models and DAGs Inference and Comparison Ricardo Henao DTU Informatics Technical University of Denmark 2800 Lyngby, Denmark Bioinformatics Centre University of Copenhagen 2200 Copenhagen, Denmark rhenao@binf.ku.dk Ole Winther DTU Informatics Technical University of Denmark 2800 Lyngby, Denmark Bioinformatics Centre University of Copenhagen 2200 Copenhagen, Denmark owi@imm.dtu.dk Abstract In this paper we present a novel approach to learn directed acyclic graphs (DAGs) and factor models within the same framework while also allowing for model comparison between them. For this purpose, we exploit the connection between factor models and DAGs to propose Bayesian hierarchies based on spike and slab priors to promote sparsity, heavy-tailed priors to ensure identifiability and predictive densities to perform the model comparison. We require identifiability to be able to produce variable orderings leading to valid DAGs and sparsity to learn the structures. The effectiveness of our approach is demonstrated through extensive experiments on artificial and biological data showing that our approach outperform a number of state of the art methods. 1 Introduction Sparse factor models have proven to be a very versatile tool for detailed modeling and interpretation of multivariate data, for example in the context of gene expression data analysis [1, 2]. A sparse factor model encodes the prior knowledge that the latent factors only affect a limited number of the observed variables. An alternative way of modeling the data is through linear regression between the measured quantities. This multiple regression model is a well-defined multivariate probabilistic model if the connectivity (non-zero weights) defines a directed acyclic graph (DAG). What usually is done in practice is to consider either factor or DAG models. Modeling the data with both types of models at the same time and then perform model comparison should provide additional insight as these models are complementary and often closely related. Unfortunately, existing off-the-shelf models are specified in such a way that makes direct comparison difficult. A more principled idea that can phrased in Bayesian terms is for example to find an equivalence between both models, then represent them using a common/comparable hierarchy, and finally use a marginal likelihood or a predictive density to select one of them. Although a formal connection between factor models and DAGs has been already established in [3], this paper makes important extensions such as explicitly modeling sparsity, stochastic search over the order of the variables and model comparison. Is well known that learning the structure of graphical models, in particular DAGs is a very difficult task because it turns out to be a combinatorial optimization problem known to be NP-hard [4]. A commonly used approach for structure learning is to split the problem into two stages using the fact that the space of variable orderings is far more smaller than the space of all possible structures, e.g. by first attempting to learn a suitable permutation of the variables and then the skeleton of the structure given the already found ordering or viceversa. Most of the work so far for continuous data assumes linearity and Gaussian variables hence they can only recover the DAG structure up 1 to Markov equivalence [5, 6, 7, 8], which means that some subset of links can be reversed without changing the likelihood [9]. To break the Markov equivalence usually experimental (interventional) data in addition to the observational (non-interventional) data is required [10]. In order to obtain identifiability from purely observational data, strong assumptions have to to be made [11, 3, 12]. In this work we follow the line of [3] by starting from a linear factor model and ensure identifiability by using non-normal heavy-tailed latent variables. As a byproduct we find a set of candidate orderings compatible with a linear DAG, i.e. a mixing matrix which is “close to” triangular. Finally, we may perform model comparison between the factor and DAG models inferred with fixed orderings taken from the candidate set. The rest of the paper is organized as follows. Sections 2 to 5 we motivate and describe the different ingredients in our method, in Section 6 we discuss existing work, in Section 7 experiments on both artificial and real data are presented, and Section 8 concludes with a discussion and perspectives for future work. 2 From DAGs to factor models We will assume that an ordered d-dimensional data vector Px can be represented as a directed acyclic graph with only observed nodes, where P is the usually unknown true permutation matrix. We will focus entirely on linear models such that the value of each variable is a linear weight combination of parent nodes plus a driving signal z x = P−1BPx + z , (1) where B is a strictly lower triangular square matrix. In this setting, each non-zero element of B corresponds to a link in the DAG. Solving for x we can rewrite the problem as x = P−1APz = P−1(I −B)−1Pz , (2) which corresponds to a noise-free linear factor model with the restriction that P−1AP must have a sparsity pattern that can be permuted to a triangular form since (I−B)−1 is triangular. This requirement alone is not enough to ensure identifiability (up to scaling and permutation of columns Pf)1. We further have to use prior knowledge about the distribution of the factors z. A necessary condition is that these must be a set of non-Gaussian independent variables [11]. For heavy-tailed data is it often sufficient in practice to use a model with heavier tails than Gaussian [13]. If the requirements for A and for the distribution of z are met, we can first estimate P−1AP and subsequently find P searching over the space of all possible orderings. Recently, [3] applied the fastICA algorithm to solve for the inverse mixing matrix P−1A−1P. To find a candidate solution for B, P is set such that B found from the direct relation equation (1), B = I −A−1 (according to magnitude-based criterion) is as close as possible to lower triangular. In the final step the Wald statistic is used for pruning B and the chi-square test is used for model selection. In our work we also exploit the relation between the factor models and linear DAGs. We apply a Bayesian approach to learning a sparse factor models and DAGs, and the stochastic search for P is performed as an integrated part of inference of the sparse factor model. The inference of factor model (including order) and DAG parameters are performed as two separate inferences such that the only input that comes from the first part is a set of candidate orders. 3 From factor models to DAGs Our first goal is to perform model inference in the families of factor and linear DAG models. We specify the joint distribution or probability of everything, e.g. for the factor model, as p(X, A, Z, Ψ, P, ·) = p(X|A, Z, P, ·)p(A|·)p(Z|·)p(Ψ|·)p(P|·)p(·) , where X = [x1, . . . , xN], Z = [z1, . . . zN], N is the number of observations and (·) indicates additional parameters in the hierarchical models. The prior over permutation p(P|·) will always be chosen to be uniform over the d! possible values. The actual sampling based inference for P is discussed in the next section and the standard Gibbs sampling components are provided in the supplementary material. Model comparison should ideally be performed using the marginal likelihood. This is more difficult to calculate with sampling than obtaining samples from the posterior so we use the predictive densities on a test set as a yardstick. 1These ambiguities are not affecting our ability to find correct permutation P of the rows. 2 Factor model Instead of using the noise-free factor model of equation (2) we allow for additive noise x = P−1 r APcz + ǫ, where ǫ is an additional Gaussian noise term with diagonal covariance matrix Ψ, i.e. uncorrelated noise, to account for independent measurement noise, Pr = P is the permutation matrix for the rows of A and Pc = PfPr another permutation for the columns with Pf accounting for the permutation freedom of the factors. We will not restrict the mixing matrix A to be triangular. Instead we infer Pr and Pc using a stochastic search based upon closeness to triangular as measured by a masked likelihood, see below. Now we can specify a hierarchy for the Bayesian model as follows X|Pr, A, Pc, Z, Ψ ∼N(X|P−1 r APcZ, Ψ) , Z ∼π(Z|·) , ψ−1 i |ss, sr ∼Gamma(ψ−1 i |ss, sr) , A ∼ρ(A|·) , (3) where ψi are elements of Ψ. For convenience, to exploit conjugate exponential families we are placing a gamma prior on the precision of ǫ with shape ss and rate sr. Given that the data is standardized, the selection of hyperparameters for ψi is not very critical as long as both “signal and noise” are supported. The prior should favor small values of ψi as well as providing support for ψi = 1 such that certain variables can be explained solely by noise (we set ss = 2 and sr = 0.05 in the experiments). For the factors we use a heavy-tailed prior π(Z|·) in the form of a Laplace distribution parameterized for convenience as a scale mixture of Gaussians [14] zjn|µ, λ ∼Laplace(zjn|µ, λ) = Z ∞ 0 N(zjn|µ, υ)Exponential(υjn|λ2)dυjn , (4) λ2|ℓs, ℓr ∼Gamma(λ2|ℓs, ℓr) , (5) where zjn is an element of Z, λ is the rate and υ has an exponential distribution acting as mixing density. Furthermore, we place a gamma distribution on λ2 to get conditionals for υ and λ2 in standard conjugate families. We let the components of Z have on average unit variance. This is achieved by setting ℓs/ℓr = 2 (we set ℓs = 4 and ℓr = 2). Alternatively one may use a t distribution—again as scale mixture of Gaussians—which can to interpolate between very heavy-tailed (power law) and very light tails, i.e. becoming Gaussian when degrees of freedom approaches infinity. However such flexibility comes at the price of being more difficult to select its hyperparameters, because the model could become unidentified for some settings. xin zjn υjn λ aij ψi rij ηij qij νj τij n = 1 : N j = 1 : d i = 1 : d Figure 1: Graphical model for Bayesian hierarchy in equation (3). The prior ρ(A|·) for the mixing matrix should be biased towards sparsity because we want to infer something close to a triangular matrix. Here we adopt a two-layer discrete spike and slab prior for the elements aij of A similar to the one in [2]. The first layer in the prior control the sparsity of each element aij individually, whereas the second layer impose a per-factor sparsity level to allow elements within the same factor to share information. The hierarchy can be written as aij|rij, ψi, τij ∼(1 −rij)δ(aij) + rijN(aij|0, ψiτij) , τ −1 ij |ts, tr ∼Gamma(τ −1 ij |ts, tr) , rij|ηij ∼Bernoulli(rij|ηij) , ηij|qij, αp, αm ∼(1 −qij)δ(ηij) + qijBeta(ηij|αpαm, αp(1 −αm)) , qij|νj ∼Bernoulli(qij|νj) , νj|βm, βp ∼Beta(νj|βpβm, βp(1 −βm)) , (6) where δ(·) is a Dirac δ-function. The prior above specify a point mass mixture over aij with mask rij. The expected probability of aij to be non-zero is ηij and is controlled through a beta hyperprior with mean αm and precision αp. Besides, each factor has a common sparsity rate νj that let the elements ηij to be exactly zero with probability 1−νj through a beta distribution with mean βm and 3 precision βp, turning the distribution of ηij bimodal over the unit interval. The magnitude of nonzero elements in A is specified through the slab distribution depending on τij. The parameters for τij should be specified in the same fashion as ψi but putting more probability mass around aij = 1, for instance ts = 4 and tr = 10. Note that we scale the variances with ψi since it makes the model easier to specify and tend to have better mixing properties [15]. The masking matrix rij with parameters ηij should be somewhat diffuse while favoring relatively large masking probabilities, e.g. αp = 10 and αm = 0.9. Additionally, qj and should favor very small values with low variance, this is for example βp = 1000 and βm = 0.005. The graphical model for the entire hierarchy in (3) omitting parameters is shown in Figure 1. DAG We make the following Bayesian specification of linear DAG model of equation (1) as X|Pr, B, X, · ∼π(X −P−1 r B|·) , B ∼ρ(B|·) , (7) where π and ρ are given by equations (4) and (6). The Bayesian specification for the DAG has a similar graphical model to the one in Figure 1 but without noise variances Ψ. The factor model needs only shared variance parameter λ for the Laplace distributed zjn because a change of scale in A is equivalent to change of variance in zjn. The DAG on the other hand, needs individual variance parameters because it has no scaling freedom. Given that we know that B is strictly lower triangular, it should be in general less sparse than A, thus we use a different setting for the sparsity prior, i.e. βp = 100 and βm = 0.01. 4 Sampling based inference For given permutation P, Gibbs sampling can be used for inference of the remaining parameters. Details of Gibbs sampler is given in the supplementary material and we will focus on the non-standard inference corresponding to the sampling over permutations. There are basically two approaches to find P, one is perform the inference for parameters and P jointly with B restricted to be triangular. The other is to let the factor model be unrestricted and search for P according to a criterion that does not affect parameter inference. Here we prefer the latter for two reasons. First, joint combinatorial and parameter inference in this model will probably have poor mixing with slow convergence. Second, we are also interested in comparing the factor model against the DAG for cases when we cannot really assume that the data is well approximated by a DAG. In our approach the proposal P⋆corresponds to picking two of the elements in the order vector by random and exchanging them. Other approaches such as restricting to pick two adjacent elements have been suggested as well [16, 7]. For the linear DAG model we are not performing joint inference of P and the model parameters. Rather we use a set of Ps found for the factor model to be good candidates for the DAG. The stochastic search for P = Pc goes as follows: we make inference for the unrestricted factor model, propose P⋆ r and P⋆ c independently according q(P⋆ r|Pr)q(P⋆ c|Pc) which is the uniform two variable random exchange. With this proposal and the flat prior over P, we use a MetropolisHastings acceptance probability simply as the ratio of likelihoods with A masked to have zeros above its diagonal (through masking matrix M) ξ→⋆= N(X|(P⋆ r)−1(M ⊙P⋆ rA(P⋆ c)−1)P⋆ c, Ψ) N(X|P−1 r (M ⊙PrAP−1 c )Pc, Ψ) , The procedure can be seen as a simple approach for generating hypotheses about good, close to triangular A, orderings in a model where the spike and slab prior provides bias towards sparsity. To learn DAGs we first perform inference on the factor model specified by the hierarchy in (3) to obtain a set of ordering candidates sorted according to their usage during sampling—after the burnin period. It is possible that the estimation of A might contain errors, e.g. a false zero entry on A allowing several orderings leading to several lower triangular versions of A, only one of those being actually correct. Thus, we propose not only to use the best candidate but a set of top candidates of size mtop = 10. Then we perform inference on the DAG model corresponding to the structure search hierarchy in (7), for each one of the permutation candidates being considered, P(1) r , . . . , P(mtop) r . Finally, we select the DAG model among candidates using the predictive distribution for the DAG when a test set is available or just the likelihood if not. 4 5 Predictive distributions and model comparison Given that our model produces both DAG and a factor model estimates at the same time, it could be interesting to estimate also whether one option is better than the other given the observed data, for example in exploratory analysis when the DAG assumption is just one reasonable option. In order to perform the model comparison, we use predictive densities p(X⋆|X, M) with M = {MFA, MDAG}, instead of marginal likelihoods because the latter is difficult and expensive to compute by sampling, requiring for example thermodynamic integration. With Gibbs sampling, we draw samples from the posterior distributions p(A, Ψ, λ|X, ·) and p(B, λ1, . . . , λm|X, ·). The average over the extensive variables associated with the test points p(Z⋆|·) is a bit more complicated because naively drawing samples from p(Z⋆|·) gives an estimator with high variance—for ψi ≪υjn. In the following we describe how to do it for each model, omitting the permutation matrices for clarity. Factor model We can compute the predictive distribution by taking the likelihood in equation (3) and marginalizing Z. Since the integral has no closed form we can approximate it using the Gaussian distribution from the scale mixture representation as p(X⋆|A, Ψ, ·) = Z p(X⋆|A, Z, Ψ)p(Z|·)dZ ≈ 1 rep Y n rep X r N(x⋆ n|0, A⊤UnA + Ψ) , where Un = diag(υ1n, . . . , υdn), the υjn are sampled from the prior and rep is the number of samples generated to approximate the intractable integral (rep = 500 in the experiments). Then we can average over p(A, Ψ, λ|X, ·) to obtain p(X⋆|X, MFA). DAG In this case the predictive distribution is rather easy because the marginal over Z in equation (4) is just a Laplace distribution with mean BX p(X⋆|B, ·) = Z p(X⋆|B, X, Z)p(Z|·)dZ = Y i,n Laplace(xij|[BX]in, λi) , where [BX]ij is the element indexed by the i-th row and n-th column of BX. In practice we compute the predictive densities for a particular X⋆during sampling and then select the model based on its ratio. Note that both predictive distributions depend directly on λ—the rate of Laplace distribution, making the estimates highly dependent on its value. This is why it is important to have the hyperprior on λ of equation (5) instead of just fixing its value. 6 Existing work Among the existing approaches to DAG learning, our work is most closely related to LiNGAM (Linear Non-Gaussian Acyclic Model for causal discovery) [3] with several important differences: Since LiNGAM relies on fastICA to learn the mixing is not inherently sparse, hence a pruning procedure based on Wald statistic and model fit second order information should be applied after obtaining an ordering for the variables. The order search in LiNGAM assumes that there is not estimation errors during fastICA model inference, then a single ordering candidate is produced. LiNGAM produces and select a final model among several candidates, but in contrast to our method such candidates are not different DAGs with different variable orderings but DAGs with different sparsity levels. The factor model inference in LiNGAM, namely fastICA is very efficient however their structure search involves repeated inversions of matrices of sizes d2 × d2 which can make it prohibitive for large problems. More explicitly, the computational complexity of LiNGAM is roughly O(Nfitd6) where Nfit is the number of model fit evaluations. In contrast, the complexity in our case is O(Nited2N) where Nite is the total number of samples including burn-in periods for both, factor model and DAG inferences. Finally, our model is more principled in the sense that all the approach is within the same Bayesian framework, as a result it can be extended to for example binary data or time series by selecting some suitable prior distributions. Much work on Bayesian models for DAG learning already exist. For example, the approach presented in [16] is a Gaussian Bayesian network and therefore suffers from lack of identifiability. Besides, order search is performed directly for the DAG model making necessary the use of longer 5 sampler runs with a number of computational tricks when the problem is large (d > 10), i.e. when exhaustive order enumeration is not an option. 7 Experiments We consider four sets of experiments in the following. The first two consist on extensive experiments using artificial data, the third addresses the model comparison scenario and the last one uses real data previously published in [17]. In every case we ran 2000 samples after a burn-in period of 4000 iterations and three independent chains for the factor model, and a single chain with 1000 samples and 2000 as burn-in for the DAG2. Hyperparameter settings are discussed in Section 3. LiNGAM suite We evaluate the performance of our model against LiNGAM3 using the artificial model generator presented in [3]. The generator produces both dense and sparse networks with different degree of sparsity, Z is generated from a non-Gaussian heavy-tailed distribution, X is generated using equation (1) and then randomly permuted to hide the correct order, P. For the experiment we have generated 1000 different dataset/models using d = {5, 10}, N = {200, 500, 1000, 2000} and the DAG was selected using the (training set) likelihood in equation (7). Results are summarized in Figure 2 using several performance measures. For the particular case of the area under the ROC curve (AUC), we use the conditional posterior of the masking matrix, i.e. p(R|X, ·) where R is a matrix with elements rij. AUC is an important measure because it quantifies how the model accounts for the uncertainty of presence or absence of links in the DAG. Such uncertainty assessment is not possible in LiNGAM where the probability of having a link is simply zero or one, however the AUC can be still computed. 200 500 1000 2000 0.76 0.78 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 True positive rate N (a) 200 500 1000 2000 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 True negative rate N (b) 200 500 1000 2000 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 d=5 Ours d=5 LINGAM d=10 Ours d=10 LINGAM AUC N (c) 200 500 1000 2000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Orderings error rate N (d) Figure 2: Performance measures for LiNGAM suite. Symbols are: square for 5 variables, star for 10 variables, solid line for sFA and dashed line for LiNGAM. (a) True positive rate. (b) True negative rate. (c) Frequency of AUC being greater than 0.9. (d) Number of estimated correct orderings. In terms of true negative rates, AUC and ordering error rate, our approach is significantly better than LiNGAM. The true positive rate results in Figure 2(a) show that LiNGAM outperform our approach only for N = 2000. However by comparing it to the true positive rate, it seems than LiNGAM prefer more dense models which could be an indication of overfitting. Looking to the ordering errors, our model is clearly superior. It is important to mention that being able to compute a probability for a link in the DAG to be zero, p(bij ̸= 0|X, ·), turns out to be very useful in practice, for example to reject links with high uncertainty or to rank them. To give an idea of running times on a regular two-core 2.5GHz machine, for d = 10 and N = 500: LiNGAM took in average 10 seconds and our method 170 seconds. However, when doubling the number of variables the times were 730 and 550 seconds for LiNGAM and our method respectively, which is in agreement with our complexity estimates. 2Source code available upon request (C with Matlab interface). 3Matlab package available at http://www.cs.helsinki.fi/group/neuroinf/lingam/. 6 Bayesian networks repository Next we want to compare some of the state of the art (Gaussian) approaches to DAG learning on 7 well known structures4, namely alarm, barley, carpo, hailfinder, insurance, mildew and water (d = 37, 48, 61, 56, 27, 35, 32 respectively). A single dataset of size 1000 per structure was generated using a similar procedure to the one used before. Apart from ours (sFA), we considered the following methods5: standard DAG search (DS), order-search (OS), sparse candidate pruning then DAG-search (DSC) [6], L1MB then DAG-search (DSL) [8], sparsecandidate pruning then order-search (OSC) [7]. Results are shown in Figure 3, including the number of reversed links found due to ordering errors. 0 0.1 0.2 False positive rate water mildew insurance hailfinder carpo barley alarm DS OS OSC DSC DSL sFA (a) 0 0.2 0.4 False negative rate water mildew insurance hailfinder carpo barley alarm (b) 0 0.2 0.4 0.6 0.8 AUC water mildew insurance hailfinder carpo barley alarm (c) 0 0.2 0.4 0.6 Reversed links water mildew insurance hailfinder carpo barley alarm (d) Figure 3: Performance measures for Bayesian networks repository experiments. In this case, our approach obtained slightly better results when looking at the false positive rate, Figure 3(a). The true negative rate is comparable to the other methods suggesting that our model in some cases is sparser than the others. AUC estimates are significantly better because we have continuous probabilities for links to be zero (in the other methods we had to use a binary value). From Figure 3(d), the number of reversed links in the other methods is quite high as expected due to lack of identifiability. Our model produced a small amount reversed links because it was not able to find any of the true orderings, but indeed something quite close. This results could be improved by running the sampler for a longer time or by considering more candidates. We also tried to run the other approaches with data generated from Gaussian distributions but the results were approximately equal to those shown in Figure 3. On the other hand, our approach performs similarly but the number of reversed links increases significantly since the model is no longer identified. The most important advantage of the (Gaussian) methods used in this experiment is their speed. In all cases they are considerably faster than sampling based methods. Their speed make them very suitable for large scale problems regardless of their identifiability issues. Model comparison For this experiment we have generated 1000 different datasets/models with d = 5 and N = {500, 1000} in a similar way to the first experiment but this time we selected the true model to be a factor model or a DAG uniformly. In order to generate a factor model we basically just need to be sure that A cannot be permuted to a triangular form. We kept 20% of the data to compute the predictive densities to then select between all estimated DAG candidates and the factor model. We found that for N = 500 our approach was able to select true DAGs 91.5% of the times and true factor models 89.2%, corresponding to an overall error of 9.6%, For N = 1000 the true DAG and true factor model rates increased to 98.5% and 94.6% respectively. This results demonstrate that our approach is very effective at selecting the true underlying structure in the data between the two proposed hypotheses. Protein-signaling network The dataset introduced in [17] consists on flow cytometry measurements of 11 phosphorylated proteins and phospholipids (Raf, Erk, p38, Jnk, Akt, Mek, PKA, PKC, PIP2, PIP3, PLCγ). Each observation is a vector of quantitative amounts measured from single cells, generated from a series of stimulatory cues and inhibitory interventions. The dataset contains both observational and experimental data. Here we are only using 1755 samples corresponding to 4http://compbio.cs.huji.ac.il/Repository/. 5Parameters: 10000 iterations, 5 candidates (SC, DSC), max fan-in of 5 (OS, OSC) and Or strategy and MDL penalty (DSL). 7 Raf Erk p38 Jnk Akt Mek PKA PKC PIP2 PIP3 PLCγ (a) Raf Erk p38 Jnk Akt Mek PKA PKC PIP2 PIP3 PLCγ (b) 0.02 0.44 0.15 0.46 0.23 0.93 0.23 0.37 0.77 0.93 0.24 Raf Erk p38 Jnk Akt Mek PKA PKC PIP2 PIP3 PLCγ (c) −4000 −3800 −3600 −3400 −3200 −3000 0 1 2 3 4 5 6 7 8 x 10 −3 Likelihood (d) 1 2 3 4 5 6 7 8 9 10 0.9 0.95 1 1 2 3 4 5 6 7 8 9 100.7 0.8 0.9 Orderings Ratio Accuracy (e) Figure 4: Result for protein-signaling network. (a) Textbook signaling network as reported in [17]. (b) Estimated structure using Bayesian networks [17]. (c) Estimated structure using our model. (e) Test likelihoods for the best ordering DAG (dashed) and the factor model (solid). (d) Likelihood ratios (solid) and structure errors (dashed) for all candidates considered by our method and their usage. The Bayesian network is not able to identify the direction of the links with only observational data. pure observational data and randomly selected 20% of the data to compute the predictive densities. Using the entire set will produce a richer model, however interventions are out of the scope of this paper. The textbook ground truth and results are presented in figure 4. From the 21 possible links in figure 4(a), the model from [17] was able to find 9, but also one falsely added link. In 4(b), a marginal likelihood equivalent prior is used and they therefore cannot make any inferences about directionality from observational data alone, see Figure 4(b). Our model in Figure 4(c) was able to find 10 true links, one falsely added link and only two reversed links (RL), one of them is PIP2 → PIP3 which according to the ground truth is bidirectional and the other one, PLCγ →PIP3 which was also found reversed using experimental data in [17]. Note from figure 4(e) that the predictive density ratios correlate quite well with the structural accuracy. The predictive densities for the best candidate (sixth in Figure 4(e)) is shown in Figure 4(d) and suggests that the factor model is a better option which makes sense considering that estimated DAG in figure 4(c) is a substructure of the ground truth. We also examined the estimated factor model and we found out that three factors could correspond to unmeasured proteins (PI3K, MKK and IP3), see Figure 2 and table 3 in [17]. We also tried the above methods. Results were very similar to our method in terms of true positives (≈9) and true negatives (≈32), however none of them were able to produce less than 6 reversed links that corresponds to approximately two-thirds of total true positives. 8 Discussion We have proposed a novel approach to perform inference and model comparison of sparse factor models and DAGs within the same framework. The key ingredients for both Bayesian models are spike and slab priors to promote sparsity, heavy-tailed priors to ensure identifiability and predictive densities to perform the comparison. A set of candidate orderings is produced by the factor model. Subsequently, a linear DAG is learned for each of the candidates. To the authors’ knowledge this is the first time that a method for comparing such a closely related linear models is proposed. This setting can be very beneficial in situations where the prior evidence suggests both DAG structure and/or unmeasured variables in the data. For example in the protein signaling network [17], the textbook ground truth suggests both DAG structure and a number of unmeasured proteins. The previous approach [17] only performed structure learning in DAGs but our results suggest that the data is better explained by the factor model. For further exploration of this data set, we obviously need to modify our approach to handle hybrid models, i.e. graphs with directed/undirected links and observed/latent nodes as well as being able to use experimental data. Our Bayesian hierarchical approach is very flexible. We are currently investigating extensions to other source distributions (non-parametric Dirichlet process, temporal Gaussian processes and discrete). 8 References [1] M. West. Bayesian factor regression models in the “large p, small n” paradigm. In J. Bernardo, M. Bayarri, J. Berger, A. Dawid, D. Heckerman, A. Smith, and M. West, editors, Bayesian Statistics 7, pages 723–732. Oxford University Press, 2003. [2] J. Lucas, C. Carvalho, Q. Wang, A. Bild, J. R. Nevins, and M. West. Bayesian Inference for Gene Expression and Proteomics, chapter Sparse Statistical Modeling in Gene Expression Genomics, pages 155–176. Cambridge University Press, 2006. [3] S. Shimizu, P. O. Hoyer, A. Hyv¨arinen, and A. Kerminen. A linear non-Gaussian acyclic model for causal discovery. Journal of Machine Learning Research, 7:2003–2030, October 2006. [4] D. M. Chickering. Learning Bayesian networks is NP-complete. In D. Fisher and H.-J. Lenz, editors, Learning from Data: AI and Statistics, pages 121–130. Springer-Verlag, 1996. [5] I. Tsamardinos, L. E. Brown, and C. F. Aliferis. The max-min hill-climbing Bayesian network structure learning algorithm. Machine Learning, 65(1):31–78, October 2006. [6] N. Friedman, I. Nachman, and D. Pe’er. Learning Bayesian network structure from massive datasets: The “sparse candidate” algorithm. In K. B. Laskey and H. Prade, editors, UAI, pages 206–215, 1999. [7] M. Teyssier and D. Koller. Ordering-based search: A simple and effective algorithm for learning Bayesian networks. In UAI, pages 548–549, 2005. [8] M. W. Schmidt, A. Niculescu-Mizil, and K. P. Murphy. Learning graphical model structure using L1regularization paths. In AAAI, pages 1278–1283, 2007. [9] D. Heckerman, D. Geiger, and D. M. Chickering. Learning Bayesian networks: The combination of knowledge and statistical data. Machine Learning, 20(3):197–243, January 1995. [10] J. Pearl. Causality: Models, Reasoning, and Inference. Cambridge University Press, March 2000. [11] P. Comon. Independent component analysis, a new concept? Signal Processing, 36(3):287–314, December 1994. [12] C. M. Carvalho, J. Chang, J. E. Lucas, J. R. Nevins, Q. Wang, and M. West. High-dimensional sparse factor modeling: Applications in gene expression genomics. Journal of the American Statistical Association, 103(484):1438–1456, December 2008. [13] A. Hyv¨arinen, J. Karhunen, and E. Oja. Independent Component Analysis. Wiley-Interscience, May 2001. [14] D. F. Andrews and C. L. Mallows. Scale mixtures of normal distributions. Journal of the Royal Statistical Society: Series B (Methodology), 36(1):99–102, 1974. [15] T. Park and G. Casella. The Bayesian lasso. Journal of the American Statistical Association, 103(482):681–686, June 2008. [16] N. Friedman and D. Koller. Being Bayesian about network structure: A Bayesian approach to structure discovery in Bayesian networks. Machine Learning, 50(1–2):95–125, January 2003. [17] K. Sachs, O. Perez, D. Pe’er, D. A. Lauffenburger, and G. P. Nolan. Causal protein-signaling networks derived from multiparameter single-cell data. Science, 308(5721):523–529, April 2005. 9
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A Sparse Non-Parametric Approach for Single Channel Separation of Known Sounds Paris Smaragdis Adobe Systems Inc. paris@adobe.com Madhusudana Shashanka Mars Inc. shashanka@alum.bu.edu Bhiksha Raj Carnegie Mellon University bhiksha@cs.cmu.edu Abstract In this paper we present an algorithm for separating mixed sounds from a monophonic recording. Our approach makes use of training data which allows us to learn representations of the types of sounds that compose the mixture. In contrast to popular methods that attempt to extract compact generalizable models for each sound from training data, we employ the training data itself as a representation of the sources in the mixture. We show that mixtures of known sounds can be described as sparse combinations of the training data itself, and in doing so produce significantly better separation results as compared to similar systems based on compact statistical models. Keywords: Example-Based Representation, Signal Separation, Sparse Models. 1 Introduction This paper deals with the problem of single-channel signal separation – separating out signals from individual sources in a mixed recording. As of recently, a popular statistical approach has been to obtain compact characterizations of individual sources and employ them to identify and extract their counterpart components from mixture signals. Statistical characterizations may include codebooks [1], Gaussian mixture densities [2], HMMs [3], independent components [4, 5], sparse dictionaries [6], non-negative decompositions [7–9] and latent variable models [10,11]. All of these methods attempt to derive a generalizable model that captures the salient characteristics of each source. Separation is achieved by abstracting components from the mixed signal that conform to the statistical characterizations of the individual sources. The key here is the specific statistical model employed – the more effectively it captures the specific characteristics of the signal sources, the better the separation that may be achieved. In this paper we argue that, given any sufficiently large collection of data from a source, the best possible characterization of any data is, quite simply, the data themselves. This has been the basis of several example-based characterizations of a data source, such as nearest-neighbor, K-nearest neighbor, Parzen-window based models of source distributions etc. Here, we use the same idea to develop a monaural source-separation algorithm that directly uses samples from the training data to represent the sources in a mixture. Using this approach we sidestep the need for a model training step, and we can rely on a very flexible reconstruction process, especially as compared with previously used statistical models. Identifying the proper samples from the training data that best approximate a sample of the mixture is of course a hard combinatorial problem, which can be computationally demanding. We therefore formulate this as a sparse approximation problem and proceed to solve it with an efficient algorithm. We additionally show that this approach results in 1 source estimates which are guaranteed to lie on the source manifold, as opposed to trainedbasis approaches which can produce arbitrary outputs that will not necessarily be plausible source estimates. Experimental evaluations show that this approach results in separated signals that exhibit significantly higher performance metrics as compared to conceptually similar techniques which are based on various types of combinations of generalizable bases representing the sources. 2 Proposed Method In this section we cover the underlying statistical model we will use, introduce some of the complications that one might encounter when using it and finally we propose an algorithm that resolves these issues. 2.1 The Basic Model Given a magnitude spectrogram of a single source, each spectral frame is modeled as a histogram of repeated draws from a multinomial distribution over the frequency bins. At a given time frame t, consider a random process characterized by the probability Pt(f) of drawing frequency f in a given draw. The distribution Pt(f) is unknown but what one can observe instead is the result of multiple draws from the process, that is the observed spectral vector. The model assumes that Pt(f) is comprised of bases indexed by a latent variable z. The latent factors are represented by P(f|z). The probability of picking the z-th distribution in the t-th time frame can be represented by Pt(z). We use this model to learn the source-specific bases given by Pt(f|z) as done in [10,11]. At this point this model is conceptually very similar to the non-negative factorization models in [8,9]. Now let the matrix VF ×T of entries vft represent the magnitude spectrogram of the mixture sound and vt represent time frame t (the t-th column vector of matrix V). Each mixture spectral frame is again modeled as a histogram of repeated draws, from the multinomial distributions corresponding to every source. The model for each mixture frame includes an additional latent variable s representing each source, and is given by Pt(f) = X s Pt(s) X z∈{zs} Ps(f|z)Pt(z|s), (1) where Pt(f) is the probability of observing frequency f in time frame t in the mixture spectrogram, Ps(f|z) is the probability of frequency f in the z-th learned basis vector from source s, Pt(z|s) is the probability of observing the z-th basis vector of source s at time t, {zs} represents the set of values the latent variable z can take for source s, and Pt(s) is the probability of observing source s at time t. We can assume that for each source in the mixture we have an already trained model in the form of basis vectors Ps(f|z). These bases will represent a dictionary of spectra that best describe each source. Armed with this knowledge we can decompose a new mixture of these known sources in terms of the contributions of the dictionaries for each source. To do so we can use the EM algorithm to estimate Pt(z|s) and Pt(s): Pt(s, z|f) = Pt(s)Pt(z|s)Ps(f|z) P s Pt(s) P z∈{zs} Ps(f|z)Pt(z|s) (2) Pt(z|s) = P f vftPt(s, z|f) P f,z vftPt(s, z|f) (3) Pt(s) = P f vft P z∈{zs} Pt(s, z|f) P f vft P s P z∈{zs} Pt(s, z|f) (4) The reconstruction of the contribution of source s in the mixture can then be computed as ˆv(s) ft = Pt(s) P z∈{zs} Ps(f|z)Pt(z|s) P s Pt(s) P z∈{zs} Ps(f|z)Pt(z|s)vft 2 Source A Source B Mixture Convex Hull A Convex Hull B Simplex Figure 1: Illustration of the basic model. The triangles denote the position of basis functions for two source classes. The square is an instance of a mixture of the two sources. The mixture point is not within the convex hull which covers either source, but it is within the convex hull defined by all the bases combined. These reconstructions will approximate the magnitude spectrogram of each source in the mixture. Once we obtain these reconstructions we can use them to modulate the original phase spectrogram of the mixture and obtain the time-series representation of the sources. Let us now pursue a brief pictorial understanding of this algorithm, which will help us introduce the concepts in the next section. Each basis vector and the mixture input will lie in a F −1 dimensional simplex (due to the fact that these quantities are normalized to sum to unity). Each source’s basis set will define a convex hull within which any point can be approximated using these bases. Assuming that the training data is accurate, all potential inputs from that source should lie in that area. The union of all the source bases will define a larger space in which a mixture input will be inside. Any mixture point can then be approximated as a weighted sum of multiple bases from both sources. For visualization of these concepts for F = 3, see figure 1. 2.2 Using Training Data Directly as a Dictionary In this paper, we would like to explain the mixture frame from the training spectral frames instead of using a smaller set of learned bases. There are two rationales behind this decision. The first is that the resulting large dictionary provides a better description of the sources, as opposed to the less expressive learned-basis models. As we show later on, this holds even for learned-basis models with dictionaries as large as the proposed method’s. The secondary rationale behind this operation is based on the observation that the points defined by the convex hull of a source’s model, do not necessarily all fall on that source’s manifold. To visualize this problem consider the plots in figure 2. In both of these plots the sources exhibit a clear structure. In the left plot both sources appear in a circular pattern, and in the right plot in a spiral form. As shown in [12], learning a set of bases that explains these sources results in defining a convex hull that surrounds the training data. Under this model potential source estimates can now lie anywhere inside these hulls. Using trainedbasis models, if we decompose the mixture points in these figures we obtain two source estimates which do not lie in the same manifold as the original sources. Although the input was adequately approximated, there is no guarantee that the extracted sources are indeed appropriate outcomes for their sound class. In order to address this problem and to also provide a richer dictionary for the source reconstructions, we will make direct use of the training data in order to explain the mixture, and bypass the basis representation as an abstraction. To do so we will use each frame of the spectrograms of the training sequences as the bases Ps(f|z). More specifically, let W(s) F ×T (s) be the training spectrogram from source s and let w(s) t represent the time frame t from the spectrogram. In this case, the latent variable z for source s takes T (s) values, and the z-th basis function will be given by the (normalized) z-th column vector of W(s). 3 Source A Source B Mixture Convex Hull A Convex Hull B Estimate for A Estimate for B Approximation of mixture Source A Source B Mixture Convex Hull A Convex Hull B Estimate for A Estimate for B Approximation of mixture Figure 2: Two examples where the separation process using trained bases provides poor source estimates. In both plots the training data for each source are denoted by △and ▽, and the mixture sample by □. The learned bases of each source are the vertices of the two dashed convex hulls that enclose each class. The source estimates and the approximation of the mixture are denoted by ×, + and ⃝. In the left case the two sources lie on two overlapping circular areas, the source estimates however lie outside these areas. On the right, the two sources form two intertwined spirals. The recovered sources lie very closely on the competing source’s area, thereby providing a highly inappropriate decomposition. Although the mixture was well approximated in both cases, the estimated sources were poor representations of their classes. With the above model we would ideally want to use one dictionary element per source at any point in time. Doing so will ensure that the outputs would lie on the source manifold, and also offset any issues of potential overcompleteness. One way to ensure this is to perform a reconstruction such that we only use one element of each source at any time, much akin to a nearest-neighbor model, albeit in an additive setting. This kind of search can be computationally very demanding so we instead treat this as a sparse approximation problem. The intuition is that at any given point in time, the mixture frame is explained by very few active elements from the training data. In other words, we need the mixture weight distributions and the speaker priors to be sparse at every time instant. We use the concept of entropic prior introduced in [13] to enforce sparsity. Given a probability distribution θ, entropic prior is defined as Pe(θ) = e−H(θ) (5) where H(θ) = −P i θi log θi is the entropy of the distribution. A sparse representation, by definition, has few “active” elements which means that the representation has low entropy. Hence, imposing this prior during maximum a posteriori estimation is a way to minimize entropy during estimation which will result in a sparse θ distribution. We would like to minimize the entropies of both the speaker dependent mixture weight distributions (given by Pt(z|s)) and the source priors (given by Pt(s)) at every frame. In other words, we want to minimize H(z|s) and H(s) at every time frame. However, we know from information theory that H(z, s) = H(z|s) + H(s). Thus, reducing the entropy of the joint distribution Pt(z, s) is equivalent to reducing the conditional entropy of the source dependent mixture weights and the entropy of the source priors. Since the dictionary is already known and is given by the normalized spectral frames from source training spectrograms, the parameter to be estimated is given by Pt(z, s). The model, written in terms of this parameter, is given by Pt(f) = X s X z∈{zs} Ps(f|z)Pt(z, s). where we have modified equation (1) by representing Pt(s)Pt(z|s) as Pt(z, s). We use the Expectation-Maximization algorithm to derive the update equations. Let all parameters to be estimated be represented by Λ. We impose an entropic prior distribution on Pt(z, s) 4 Source A Source B Mixture Estimate for A Estimate for B Approximation of mixture Source A Source B Mixture Estimate for A Estimate for B Approximation of mixture Figure 3: Using a sparse reconstruction on the data in figure 2. Note how in contrast to that figure the source estimates are now identified as training data points, and are thus plausible solutions. The approximation of the mixture is the nearest point of the line connecting the two source estimates, to the actual mixture input. Note that the proper solution is the one that results in such a line that is as close as possible to the mixture point, and not one that is defined by two training points close to the mixture. given by log P(Λ) = β X t X s X z∈{zs} Pt(z, s) log Pt(z, s), where β is a parameter indicating the extent of sparsity desired. The E-step is given by Pt(z, s|f) = Pt(z, s)Ps(f|z) P s P z∈{zs} Pt(z, s)Ps(f|z) and the M-step by ωt Pt(z, s) + β + β log Pt(z, s) + λt = 0 (6) where we have let ω represent P f vftPt(s, z|f) and λt is the Lagrange multiplier. The above M-step equation is a system of simultaneous transcendental equations for Pt(z, s). Brand [13] proposes a method to solve such problems using the Lambert W function [14]. It can be shown that Pt(z, s) can be estimated as ˆPt(z, s) = −ω/β W(−ωe1+λt/β/β). (7) Equations (6),(7) form a set of fixed point iterations that typically converge in 2-5 iterations [13]. Once Pt(z, s) is estimated, the reconstruction of source s can be computed as ˆv(s) ft = P z∈{zs} Ps(f|z)Pt(z, s) P s P z∈{zs} Ps(f|z)Pt(z, s)vft Now let us consider how this problem resolves the issues presented in figure 2. In figure 3 we show the results obtained using this approach on the same data. The sparsity parameter β as set to 0.1. In both plots we see that the source reconstructions lie on a training point, thereby being a plausible source estimate. The approximation of the mixture is not as exact as before, since now it has to lie on the line connecting the two active source elements. This is not however an issue of concern since in practice the approximation is always good enough, and the guarantee of a plausible source estimate is more valuable than the exact approximation of the mixture. Alternative means to strive towards similar results would be to make use of priors such as in [15, 16]. In these approaches the priors are imposed on the mixture weights and thus are not as effective for this particular task since they still suffer from the symptoms of learned-basis models. This was verified through cursory simulations, which also revealed an additional computational complexity penalty against such models. 5 −5 0 5 Input source 1 −5 0 5 10 Amplitude Input source 2 Training samples index Training sample weights 1 Time frame index 5 10 15 20 25 5 10 15 20 25 Training sample weights 2 Time frame index 5 10 15 20 25 5 10 15 20 25 Figure 4: An oracle case where we fit training data from two speakers, on the mixture of that data. The top plots show the input waveforms, and the bottom plots shows the estimated weights multiplied with the source priors. As expected the weights exhibit two diagonal traces which imply that the algorithm we used has fit the data appropriately. 3 Experimental Results In this section we present the results of experiments done with real speech data. All of these experiments we performed on data from the TIMIT speech database on 0dB male/female mixtures. The sources were sampled as 16 kHz, we used 64 ms windows for the spectrogram computation, and an overlap of 32 ms. Before the FFT computation, the input was tapered using a square-root Hann window. The training data was around 25 sec worth of speech for each speaker, and the testing mixture was about 3 sec long. We evaluated the separation performance using the metrics provided in [17]. These metrics include the Signal to Interference Ratio (SIR), the Signal to Distortion Ratio (SDR), and the Signal to Artifacts Ratio (SAR). The first is a measure of how well we suppress the interfering speaker, whereas the other two provide us with a sense of how much the extracted source is corrupted due to the separation process. All of these are measured in dB and the higher they are the better the performance is deemed to be. In the following sections we first present some “oracle tests” that validate that indeed this algorithm is performing as expected, and we then proceed to more realistic testing. Finally, we show the performance impact of pruning the training data in order to speed up computation time. 3.1 Oracle tests In order to verify that this approach works we go through a few oracle experiments. In these tests we include the actual solutions as training data and we make sure that the answers are exactly what we would expect to find. The first experiment we perform is on a mixture for which the training data includes its isolated constituent sentences. In this experiment we would expect to see two dictionary components active at each point in time, one from each speaker’s dictionary, and both of these progressing through the component index linearly through time. As shown in figure 4, we observe exactly that behavior. This test provides a sanity check which verifies that given an answer this algorithm can properly identify it. A more comprehensive oracle test is shown in figure 5. In this experiment, the training data were again the same as the testing data. We averaged the results from 10 runs using different combinations of speakers, varying sparsity parameters and number of bases. The sparsity parameter β was checked for various values from 0 to 0.8, and we used trained-basis models with 5, 10, 20, 40, 80, 160 and 320 bases, as well as the proposed scenario where all the training data is used as a dictionary. The primary observation from this experiment is that the more bases we use the better the results get. We also see that increasing the sparsity parameter we see a modest improvement in most cases. 6 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 5 10 Bases Signal to Distortion Ratio β 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 10 20 Bases Signal to Interference Ratio β 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 5 10 Bases Signal to Artifacts Ratio β Figure 5: Average separation performance metrics for oracle cases, as dependent on the choice of different number of elements in the speaker’s dictionary, and different choices of the entropic prior parameter β. The left plot shows the SDR, the middle plot the SIR, and the right plot the SAR, all in dB. The basis row labeled as “Train” is the case where we use all the training data as a basis set. 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 5 10 Bases Signal to Distortion Ratio β 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 10 20 Bases Signal to Interference Ratio β 0 0.01 0.1 0.2 0.3 0.5 0.8 5 10 20 40 80 160320Train 0 5 10 Bases Signal to Artifacts Ratio β Figure 6: Average separation performance metrics for real-world cases, as dependent on the choice of different number of elements in the speaker’s dictionary, and different choices of the entropic prior parameter β. The left plot shows the SDR, the middle plot the SIR, and the right plot the SAR, all in dB. Sparsely using all of the training data clearly outperforms low-rank models by a significant margin on all metrics. 3.2 Results on Realistic Situations Let us now consider the more realistic case where the mixture data is different from the training set. In the following simulation we repeat the previous experiment, but in this case there are no common elements between the training and testing data. The input mixture has to be reconstructed using approximate samples. The results are now very different in nature. We do not obtain such high numbers in performance as in the oracle case, but we also see a stronger trend in favor of sparsity and the use of all the training data as a dictionary. The results are shown in figure 6. We can clearly see that in all metrics using all the training data significantly outperforms trained-basis models. More importantly, we see that this is not because we have a larger dictionary. For trained-bases we see a performance peak at around 80 bases, but then we observe a deterioration in performance as we use a larger dictionary. Using the actual training data results in a significant boost though. Due to the high dimensionality of the data the effect of sparsity is a little more subtle, but we still see a helpful boost especially for the SIR which is the most important of the performance measures. We see some decrease in the SAR, which is expected since the reconstructions are made using elements that look like the remaining data, and are not made to approximate the actual input mixture. This does not mean that the extracted sources are distorted and of poor quality, but rather that they don’t match the original inputs exactly. The use of sparsity ensures that the output is a plausible speech signal devoid of artifacts like distortion and musical noise. The effects of sparsity alone in the proposed case are shown separately in figure 7. 7 0 0.01 0.1 0.2 0.3 0.5 0.8 0 5 10 15 Sparsity parameter β dB SDR SIR SAR Figure 7: A slice of the results in figure 6 in which we only show the case where we use all the training data as adictionary. The horizontal axis represents various values for the sparsity parameter β. 0% 20% 40% 60% 70% 80% 90% 95% 0 5 10 15 Percentage of discarded training frames dB SDR SIR SAR Figure 8: Effect of discarding low energy training frames. The horizontal axis denotes the percentage of training frames that have been discarded. These are averaged results using a sparsity parameter β = 0.1. The unfortunate side effect of the proposed method is that we need to use a dictionary which can be substantially larger than otherwise. In order to address this concern we show that the size of the training data can be easily pruned down to a size comparable to trainedbasis models and still outperform them. Since sound signals, especially speech, tend to have a considerable amount of short-term pauses and regions of silence, we can use an energy threshold to in order to select the loudest frames of the training spectrogram as bases. In figure 8 we show how the separation performance metrics are influenced as we increasingly remove bases which lie under various energy percentiles. It is clear that even after discarding up to at least 70% of the lowest energy training frames the performance is still approximately the same. After that we see some degradation since we start discarding significant parts of the training data. Regardless this scheme outperforms trained-basis models of equivalent size. For the 80% percentile case, a trained-basis model of the same size dictionary results in roughly half the values in all performance metrics, a very significant handicap for the same amount of computational and memory requirements. The experiments in this paper were all conducted in MATLAB on an average modern desktop machine. Overall computations for a single mixture took roughly 4 sec when not using the sparsity prior, 14 sec when using the sparsity prior (primarily due to slow computation of Lambert’s function), and dropped down to 5 sec when using the 30% highest energy frames from the training data. 4 Conclusion In this paper we present a new approach to solving the monophonic source separation problem. The contributions of this paper lies primarily in the choice of using all the training data as opposed to a trained-basis model. In order to do so we present a sparse learning algorithm which can efficiently solve this problem, and also guarantees that the returned source estimates are plausible given the training data. We provide experiments that show how this approach is influenced by the use of varying sparsity constraints and training data selection. Finally we demonstrate how this approach can generate significantly superior results as compared to trained-basis methods. 8 References [1] S. T. Roweis, One microphone source separation, in Advances in Neural Information Processing Systems, 2001. [2] Reddy, A.M. and B. Raj. Soft Mask Methods for Single-Channel Speaker Separation, in IEEE Transactions of Audio, Speech, and Language Processing, Volume: 15, Issue: 6, Aug 2007. [3] T. Kristjansson, J. Hershey, P. Olsen, S. Rennie, and R. Gopinath, Super-human multitalker speech recognition: The IBM 2006 speech separation challenge system, in International Conference on Spoken Language Processing (INTERSPEECH), 2006, pp. 97–100, Kluwer Academic Publishers, ch. 20, pp. 295304. [4] Casey, M.A., and A. Westner. Separation of mixed audio sources by independent subspace analysis, in Proceedings of the International Conference of Computer Music, 2000. [5] Jang, G.-J., T.-W. Lee. A Maximum Likelihood Approach to Single-channel Source Separation, in Journal of Machine Learning Research 4 (2003) pp. 1365–1392. [6] Pearlmutter, B., M. Zibulevsky, Blind Source Separation by Sparse Decomposition in a Signal Dictionary, in Neural Computation 13, pp. 863–882. 2001. [7] L. Benaroya, L. M. Donagh, F. Bimbot, and R. Gribonval, Non negative sparse representation for wiener based source separation with a single sensor, in Acoustics, Speech, and Signal Processing, IEEE International Conference on, 2003, pp. 613–616. [8] M. N. Schmidt and R. K. Olsson, Single-channel speech separation using sparse nonnegative matrix factorization, in International Conference on Spoken Language Processing (INTERSPEECH), 2006. [9] T. Virtanen, Sound source separation using sparse coding with temporal continuity objective, in International Computer Music Conference, ICMC, 2003. [10] Smaragdis, P. Raj, B. and Shashanka, M.V. 2007. Supervised and Semi-Supervised Separation of Sounds from Single-Channel Mixtures. In proceedings of ICA 2007. London, UK. September 2007. [11] Raj, B.; Smaragdis, P. 2005. Latent Variable Decomposition of Spectrograms for single channel speaker separation. In Proceedings of the IEEE Workshop on Applications of Signal Processing to Audio and Acoustics, New Paltz, NY, October, 2005. [12] Shashanka, M.V., B. Raj, P. Smaragdis, 2007. Sparse Overcomplete Latent Variable Decoposition of Counts Data. In Neural Information Processing Systems (NIPS), Vancouver, BC, Canada. December 2007. [13] Brand, M.E. Pattern Discovery via Entropy Minimization. In Uncertainty 99, AISTATS99,1999. [14] Corless, R.M., G.H. Gonnet, D.E.G. Hare, D.J. Jeffrey, and D.E. Knuth. On the Lambert W Function. Advances in Computational Mathematics,1996. [15] Bouguila N. and D. Ziou. Using unsupervised learning of a finite Dirichlet mixture model to improve pattern recognition applications, Pattern Recognition Letters, Volume 26, Issue 12, September 2005. [16] Hinneburg, A., Gabriel, H.-H. and Gohr, A. Bayesian Folding-In with Dirichlet Kernels for PLSI, in Seventh IEEE International Conference on Data Mining, Oct. 2007 [17] F´evotte, C., R. Gribonval and E. Vincent. 2005. BSS EVAL Toolbox User Guide, IRISA Technical Report 1706, Rennes, France, April 2005. 9
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Directed Regression Yi-hao Kao Stanford University Stanford, CA 94305 yihaokao@stanford.edu Benjamin Van Roy Stanford University Stanford, CA 94305 bvr@stanford.edu Xiang Yan Stanford University Stanford, CA 94305 xyan@stanford.edu Abstract When used to guide decisions, linear regression analysis typically involves estimation of regression coefficients via ordinary least squares and their subsequent use to make decisions. When there are multiple response variables and features do not perfectly capture their relationships, it is beneficial to account for the decision objective when computing regression coefficients. Empirical optimization does so but sacrifices performance when features are well-chosen or training data are insufficient. We propose directed regression, an efficient algorithm that combines merits of ordinary least squares and empirical optimization. We demonstrate through a computational study that directed regression can generate significant performance gains over either alternative. We also develop a theory that motivates the algorithm. 1 Introduction When used to guide decision-making, linear regression analysis typically treats estimation of regression coefficients separately from their use to make decisions. In particular, estimation is carried out via ordinary least squares (OLS) without consideration of the decision objective. The regression coefficients are then used to optimize decisions. When there are multiple response variables and features do not perfectly capture their relationships, it is beneficial to account for the decision objective when computing regression coefficients. Imperfections in feature selection are common since it is difficult to identify the right features and the number of features is typically restricted in order to avoid over-fitting. Empirical optimization (EO) is an alternative to OLS which selects coefficients that minimize empirical loss in the training data. Though it accounts for the decision objective when computing regression coefficients, EO sacrifices performance when features are well-chosen or training data is insufficient. In this paper, we propose a new algorithm – directed regression (DR) – which is a hybrid between OLS and EO. DR selects coefficients that are a convex combination of those that would be selected by OLS and those by EO. The weights of OLS and EO coefficients are optimized via crossvalidation. We study DR for the case of decision problems with quadratic objective functions. The algorithm takes as input a training set of data pairs, each consisting of feature vectors and response variables, together with a quadratic loss function that depends on decision variables and response variables. Regression coefficients are computed for subsequent use in decision-making. Each future decision depends on newly sampled feature vectors and is made prior to observing response variables with the goal of minimizing expected loss. We present computational results demonstrating that DR can substantially outperform both OLS and EO. These results are for synthetic problems with regression models that include subsets of relevant 1 features. In some cases, OLS and EO deliver comparable performance while DR reduces expected loss by about 20%. In none of the cases considered does either OLS or EO outperform DR. We also develop a theory that motivates DR. This theory is based on a model in which selected features do not perfectly capture relationships among response variables. We prove that, for this model, the optimal vector of coefficients is a convex combination of those that would be generated by OLS and EO. 2 Linear Regression for Decision-Making Suppose we are given a set of training data pairs O = {(x(1), y(1)), · · · , (x(N), y(N))}. Each nth data pair is comprised of feature vectors x(n) 1 , . . . , x(n) K ∈ℜM and a vector y(n) ∈ℜM of response variables. We would like to compute regression coefficients r ∈ℜK so that given a data pair (x, y), the linear combination P k rkxk of feature vectors estimates the expectation of y conditioned on x. We restrict attention to cases where M > 1, with special interest in problems where M is large, because it is in such situations that DR offers the largest performance gains. We consider a setting where the regression model is used to guide future decisions. In particular, after computing regression coefficients, each time we observe feature vectors x1, . . . , xK we will have to select a decision u ∈ℜL before observing the response vector y. The choice incurs a loss ℓ(u, y) = u⊤G1u + u⊤G2y, where the matrices G1 ∈ℜL×L and G2 ∈ℜL×M are known, and the former is positive definite and symmetric. We aim to minimize expected loss, assuming that the conditional expectation of y given x is PK k=1 rkxk. As such, given x and r, we select a decision ur(x) = argmin u ℓ Ã u, K X k=1 rkxk ! = −1 2G−1 1 G2 K X k=1 rkxk. The question is how best to compute the regression coefficients r for this purpose. To motivate the setting we have described, we offer a hypothetical application. Example 1. Consider an Internet banner ad campaign that targets M classes of customers. An average revenue of ym is received per customer of class m that the campaign reaches. This quantity is random and influenced by K observable factors x1m, . . . , xKm. These factors may be correlated across customers classes; for example, they could capture customer preferences as they relate to ad content or how current economic conditions affect customers. For each mth class, the cost of reaching the umth customer increases with um because ads are first targeted at customers that can be reached at lower cost. This cost is quadratic, so that we pay γmu2 m to reach um customers, where γm is a known constant. The application we have described fits our general problem context. It is natural to predict the response vector y using a linear combination P k rkxk of factors with the regression coefficients rk computed based on past observations O = {(x(1), y(1)), · · · , (x(N), y(N))}. The goal is to maximize expected revenue less advertising costs. This gives rise to a loss function that is quadratic in u and y: ℓ(u, y) = M X m=1 (γmu2 m −umym). One might ask why not construct M separate linear regression models, one for each response variable, each with a separate set of K coefficients. The reason is that this gives rise to MK coefficients; when M is large and data is limited, this could lead to over-fitting. Models of the sort we consider, where regression coefficients are shared across multiple response variables, are sometimes referred to as general linear models and have seen a wide range of applications [7, 8]. It is well-known that the quality of results is highly sensitive to the choice of features, even more so than for models involving a single response variable [7]. 2 3 Algorithms Ordinary least squares (OLS) is a conventional approach to computing regression coefficients. This would produce a coefficient vector rOLS = argmin r∈ℜK N X n=1 °°°°°y(n) − K X k=1 rkx(n) k °°°°° 2 . (1) Note that OLS does not take the decision objective into account when computing regression coefficients. Empirical optimization (EO), as studied for example in [2, 6], offers an alternative that does so. This approach minimizes empirical loss on the training data: rEO = argmin r∈ℜK N X n=1 ℓ(ur(x(n)), y(n)). (2) Note that EO does not explicitly aim to estimate the conditional expectation of the response vector. Instead it focusses on decision loss that would be incurred with the training data. Both rOLS and rEO can be computed efficiently by minimizing convex quadratic functions. As we will see in our computational and theoretical analyses, OLS and EO can be viewed as two extremes, each offering room for improvement. In this paper, we propose an alternative algorithm – directed regression (DR) – which produces a convex combination rDR = (1 −λ)rOLS + λrEO of coefficients computed by OLS and EO. The term directed is chosen to indicate that DR is influenced by the decision objective though, unlike EO, it does not simply minimize empirical loss. The parameter λ ∈[0, 1] is computed via cross-validation, with an objective of minimizing average loss on validation data. Average loss is a convex quadratic function of λ, and therefore can be easily minimized over λ ∈[0, 1]. DR is designed to generate decisions that are more robust to imperfections in feature selection than OLS. As such, DR addresses issues similar to those that have motivated work in data-driven robust optimization, as surveyed in [3]. Our focus on making good decisions despite modeling inaccuracies also complements recent work that studies how models deployed in practice can generate effective decisions despite their failure to pass basic statistical tests [4]. 4 Computational Results In this section, we present results from applying OLS, EO, and DR to synthetic data. To generate a data set, we first sample parameters of a generative model as follows: 1. Sample P matrices C1, . . . , CP ∈ℜM×Q, with each entry from each matrix drawn independently from N(0, 1). 2. Sample a vector ˜r ∈ℜP from N(0, I). 3. Sample Ga ∈ℜL×L and Gb ∈ℜL×M, with each entry of each matrix drawn from N(0, 1). Let G1 = G⊤ a Ga and G2 = G⊤ a Gb. Given generative model parameters C1, . . . , CP and ˜r, we sample each training data pair (x(n), y(n)) as follows: 1. Sample a vector φ(n) ∈ℜQ from N(0, I) and a vector w(n) ∈ℜM from N(0, σ2 wI). 2. Let y(n) = PP i=1 ˜riCiφ(n) + w(n). 3. For each k = 1, 2, · · · , K, let x(n) k = Ckφ(n). The vector φ(n) can be viewed as a sample from an underlying information space. The matrices C1, . . . , CP extract feature vectors from φ(n). Note that, though response variables depend on P feature vectors, only K ≤P are used in the regression model. Given generative model parameters and a coefficient vector r ∈ℜK, it is easy to evaluate the expected loss ℓ(r) = Ex,y[ℓ(ur(x), y)]. It is also easy to evaluate the minimal expected loss ℓ∗= 3 10 20 30 40 50 0 2000 4000 6000 8000 10000 N Excess Loss OLS EO DR (a) 45 50 55 60 0 1000 2000 3000 4000 5000 6000 K Excess Loss OLS EO DR (b) Figure 1: (a) Excess losses delivered by OLS, EO, and DR, for different numbers N of training samples. (b) Excess losses delivered by OLS, EO, and DR, using different numbers K of the 60 features. minr Ex,y[ℓ(ur(x), y)]. We will assess each algorithm in terms of the excess loss ℓ(r)−ℓ∗delivered by the coefficient vector r that the algorithm computes. Excess loss is nonnegative, and this allows us to make comparisons in percentage terms. We carried out two sets of experiments to compare the performance of OLS, EO, and DR. In the first set, we let M = 15, L = 15, P = 60, Q = 20, σw = 5, and K = 50. For each N ∈ {10, 15, 20, 30, 50}, we ran 100 trials, each with an independently sampled generative model and training data set. In each trial, each algorithm computes a coefficient vector given the training data and loss function. With DR, λ is selected via leave-one-out cross-validation when N ≤20, and via 5-fold cross-validation when N > 20. Figure 1(a) plots excess losses averaged over trials. Note that the excess loss incurred by DR is never larger than that of OLS or EO. Further, when N = 20, the excess loss of OLS and EO are both around 20% larger than that of DR. For small N, OLS is as effective as DR, while, EO becomes as effective as DR as N grows large. In the second set of experiments, we use the same parameter values as in the first set, except we fix N = 20 and consider use of K ∈{45, 50, 55, 58, 60} feature vectors. Again, we ran 100 trials for each K, applying the three algorithms as in the first set of experiments. Figure 1(b) plots excess losses averaged over trials. Note that when K = 55, DR delivers excess loss around 20% less than EO and OLS. When K = P = 60, there are no missing features and OLS matches the performance of DR. Figure 2 plots the values of λ selected by cross-validation, each averaged over the 100 trials, as a function of N and K. As the number of training samples N grows, so does λ, indicating that DR is weighted more heavily toward EO. As the number of feature vectors K grows, λ diminishes, indicating that DR is weighted more heavily toward OLS. 5 Theoretical Analysis In this section, we formulate a generative model for the training data and future observations. For this model, optimal coefficients are convex combinations of rOLS and rEO. As such, our model and analysis motivate the use of DR. 5.1 Model In this section, we describe a generative model that samples the training data set, as well as “missing features,” and a representative future observation. We then formulate an optimization problem where the objective is to minimize expected loss on the future observation conditioned on the training data and missing features. It may seem strange to condition on missing features since in practice they are unavailable when computing regression coefficients. However, we will later establish that optimal 4 10 20 30 40 50 0 0.2 0.4 0.6 0.8 N λ (a) 45 50 55 60 0 0.2 0.4 0.6 0.8 K λ (b) Figure 2: (a) The average values of selected λ, for different numbers N of training samples. (b) The average values of selected λ, using different numbers K of the 60 features. coefficients are convex combinations of rOLS and rEO, each of which can be computed without observing missing features. Since directed regression searches over these convex combinations, it should approximate what would be generated by a hypothetical algorithm that observes missing features. We will assume that each feature, whether observed or missing, is a linear function of an “information vector” drawn from ℜQ. Specifically, the N training data samples depend on information vectors φ(1), . . . , φ(N) ∈ℜQ. A linear function mapping an information vector to a feature vector can be represented by a matrix in ℜM×Q, and to describe our generative model, it is useful to define an inner product for such matrices. In particular, we define the inner product between matrices A and B by ⟨A, B⟩= 1 N N X n=1 (Aφ(n))⊤(Bφ(n)). Our generative model takes several parameters as input. First, there are the number of samples N, the number of response variables M, and the number of feature vectors K. Second, a parameter µQ specifies the expected dimension of the information vector. Finally, there are standard deviations σr, σϵ, and σw, of observed feature coefficients, missing feature coefficients, and noise, respectively. Given parameters N, M, K, µQ, σr, σϵ, and σw, the generative model produces data as follows: 1. Sample Q from the geometric distribution with mean µQ. 2. Sample φ(1), . . . , φ(N) ∈ℜQ from N(0, IQ). 3. Sample C1, . . . , CK and D1, · · · , DJ ∈ℜM×Q with each entry i.i.d. from N(0, 1), where K + J = MQ. 4. Apply the Gram-Schmidt algorithm with respect to the inner product defined above to generate an orthonormal basis ˜C1, . . . , ˜CK, ˜D1, . . . , ˜DJ from the sequence C1, . . . , CK, D1, . . . , DJ. 5. Sample r∗∈ℜK from N(0, σ2 rIK) and r⊥∗∈ℜJ from N(0, σ2 ϵ IJ). 6. For n = 1, · · · , N, sample w(n) ∈ℜM from N(0, σ2 wIM), and let x(n) = h C1φ(n) · · · CKφ(n)i , (3) z(n) = h ˜D1φ(n) · · · ˜DJφ(n)i , (4) y(n) = K X k=1 r∗ kx(n) k + J X j=1 r⊥∗ j z(n) j + w(n). (5) 5 7. Sample ˜φ uniformly from {φ(1), · · · , φ(N)} and ˜w ∈ℜM from N(0, σ2 wIM). Generate ˜x, ˜z, and ˜y by the same functions in (3), (4), and (5). The samples z(1), . . . , z(N), ˜z represent missing features. The Gram-Schmidt procedure ensures two properties. First, since ⟨Ck, ˜Dj⟩= 0, missing features are uncorrelated with observed features. If this were not the case, observed features would provide information about missing features. Second, since ˜D1, . . . , ˜DJ are orthonormal, the distribution of missing features is invariant to rotations in the J-dimensional subspace from which they are drawn. In other words, all directions in that space are equally likely. We define an augmented training set O = {(x(1), z(1), y(1)), · · · , (x(N), z(N), y(N))} and consider selecting regression coefficients ˆr ∈ℜK that solve min r∈ℜK E[ℓ(ur(˜x), ˜y)|O]. Note that the probability distribution here is implicitly defined by our generative model, and as such, ˆr may depend on N, M, K , µQ, σr, σϵ, σw, and O. 5.2 Optimal Solutions Our primary interest is in cases where prior knowledge about the coefficients r∗is weak and does not significantly influence ˆr. As such, we will from here on restrict attention to the case where σr is asymptotically large. Hence, ˆr will no longer depend on σr. It is helpful to consider two special cases. One is where σϵ = 0 and the other is where σϵ is asymptotically large. We will refer to ˆr in these extreme cases as ˆr0 and ˆr∞. The following theorem establishes that these extremes are delivered by OLS and EO. Theorem 1. For all N, M, K, µQ, σw, and O, ˆr0 = argmin r∈ℜK N X n=1 °°°°°y(n) − K X k=1 rkx(n) k °°°°° 2 and ˆr∞= argmin r∈ℜK N X n=1 ℓ(ur(x(n)), y(n)). Note that σϵ represents the degree of bias in a regression model that assumes there are no missing features. Hence, the above theorem indicates that OLS is optimal when there is no bias while EO is optimal as the bias becomes asymptotically large. It is also worth noting that the coefficient vectors ˆr0 and ˆr∞can be computed without observing the missing features, though ˆr is defined by an expectation that is conditioned on their realizations. Further, computation of ˆr0 and ˆr∞does not require knowledge of Q or σw. Our next theorem establishes that the coefficient vector ˆr is always a convex combination of ˆr0 and ˆr∞. Theorem 2. For all N, M, K, µQ, σw, σϵ, and O, ˆr = (1 −λ)ˆr0 + λˆr∞, where λ = 1 1+ σ2w Nσ2ϵ . Our two theorems together imply that, with an appropriately selected λ ∈[0, 1], (1 −λ)rOLS + λrEO = ˆr. This suggests that directed regression, which optimizes λ via cross-validation to generate a coefficient vector rDR = (1 −λ)rOLS + λrEO, should approximate ˆr well without observing the missing features or requiring knowledge of Q, σϵ, or σw. 6 5.3 Interpretation To develop intuition for our results, we consider an idealized situation where the coefficients r∗and r⊥∗are provided to us by an oracle. Then the optimal coefficient vector would be rO = argmin r∈ℜK E[ℓ(ur(˜x), ˜y)|O, r∗, r⊥∗]. It can be shown that rOLS is a biased estimator of rO, while rEO is an unbiased one. However, the variance of rOLS is smaller than that of rEO. The optimal tradeoff is indeed captured by the value of λ provided in Theorem 2. In particular, as the number of training samples N increases, variance diminishes and λ approaches 1, placing increasing weight on EO. On the other hand, as the number of observed features K increases, model bias decreases and λ approaches 0, placing increasing weight on OLS. Our experimental results demonstrate that the value of λ selected by cross-validation exhibits the same behavior. 6 Extensions Though we only treated linear models and quadratic objective functions, our work suggests that there can be significant gains in broader problem settings from a tighter coupling between machine learning and decision-making. In particular, machine learning algorithms should factor decision objectives into the learning process. It will be interesting to explore how to do this with other classes of models and objectives. One might argue that feature mis-specification is not a critical issue in light of effective methods for subset selection. In particular, rather than selecting a few features and facing the consequences of model bias, one might select an enormous set of features and apply a method like the lasso [10] to identify a small subset. Our view is that even this enormous set will result in model biases that might be ameliorated by generalizations of DR. There is also the concern that data requirements grow with the size of the large feature set, albeit slowly. Understanding how to synthesize DR with subset selection methods is an interesting direction for future research. Another issue that should be explored is the effectiveness of cross-validation in optimizing λ. In particular, it would be helpful to understand how the estimate relates to the ideal value of λ identified by Theorem 2. More general work on the selection of convex combinations of models (e.g., [1, 5]) may lend insights to our setting. Let us close by mentioning that the ideas behind DR ought to play a role in reinforcement learning (RL) as presented in [9]. RL algorithms learn from experience to predict a sum of future rewards as a function of a state, typically by fitting a linear combination of features of the state. This socalled approximate value function is then used to guide sequential decision-making. The problem we addressed in this paper can be viewed as a single-period version of RL, in the sense that each decision incurs an immediate cost but bears no further consequences. It would be interesting to extend our idea to the multi-period case. Acknowledgments We thank James Robins for helpful comments and suggestions. The first author is supported by a Stanford Graduate Fellowship. This research was supported in part by the National Science Foundation through grant CMMI-0653876. Appendix Proof of Theorem 1. For each n, let x(n) = h x(n) 1 · · · x(n) K i , z(n) = h z(n) 1 · · · z(n) J i . Let X = £ x(1)⊤ · · · x(N)⊤¤⊤, Z = £ z(1)⊤ · · · z(N)⊤¤⊤, Y = £ y(1)⊤ · · · y(N)⊤¤⊤, ¯r = E[r∗|O], ¯r⊥= E[r⊥∗|O]. For any matrix V , let V † denote (V ⊤V )−1V ⊤. Recall that ⟨Ck, ˜Dj⟩= 0, ∀k, j implies that each column of X is orthogonal to 7 each column of Z. Because r∗, r⊥∗, O are jointly Gaussian, as σr →∞, we have · ¯r ¯r⊥ ¸ = argmin (r,r⊥) 1 2σ2w N X n=1 °°°°°° y(n) − K X k=1 rkx(n) k − J X j=1 r⊥ j z(n) j °°°°°° 2 + 1 2σ2ϵ J X j=1 r⊥2 j = argmin (r,r⊥) °°°° · 1 σw Y 0 ¸ − · 1 σw X 1 σw Z 0 1 σϵ IJ ¸ · r r⊥ ¸°°°° 2 = " (X⊤X)−1X⊤Y (Z⊤Z + σ2 w σ2ϵ I)−1Z⊤Y # . Let a(n) = G −1 2 1 G2x(n), b(n) = G −1 2 1 G2z(n), A = £ a(1)⊤ · · · a(N)⊤¤⊤, B = £ b(1)⊤ · · · b(N)⊤¤⊤. We have ˆr = argmin r E[ℓ(ur(˜x), ˜y)|O] = argmin r 1 N N X n=1 E˜y[ℓ(ur(˜x), ˜y)|˜x = x(n), O] = argmin r N X n=1 ur(x(n))⊤G1ur(x(n)) + ur(x(n))⊤G2 E[˜y|˜x = x(n), O] = argmin r N X n=1 1 4r⊤a(n)⊤a(n)r −1 2r⊤a(n)⊤(a(n)¯r + b(n)¯r⊥) = ¯r + A†B¯r⊥= X†Y + A†B(Z⊤Z + σ2 w σ2ϵ I)−1Z⊤Y. (6) Taking σϵ →0 and σϵ →∞yields ˆr0 = X†Y , (7) ˆr∞= X†Y + A†BZ†Y . (8) The first part of the theorem then follows because ˆr0 = X†Y = argmin r ∥Y −Xr∥2 = argmin r N X n=1 °°°°°y(n) − K X k=1 rkx(n) k °°°°° 2 . We now prove the second part. Note that argmin r N X n=1 ℓ(ur(x(n)), y(n)) = argmin r N X n=1 ur(x(n))⊤G1ur(x(n)) + ur(x(n))⊤G2y(n) = argmin r r⊤A⊤Ar −2r⊤ N X n=1 h(n)⊤y(n) = (A⊤A)−1H⊤Y, where h(n) = G⊤ 2 G−1 1 G2x(n) and H = £ h(1)⊤ · · · h(N)⊤¤⊤. Each kth column of H hk = G⊤ 2 G−1 1 G2Ckφ(1) ... G⊤ 2 G−1 1 G2Ckφ(N) is in span{col X, col Z} because G⊤ 2 G−1 1 G2Ck ∈ℜM×Q = span{C1, · · · , CK, ˜D1, · · · , ˜DJ}. Since the residual Y ′ = Y −XX†Y −ZZ†Y upon projecting Y onto span {col X, col Z} is orthogonal to the subspace, we have h⊤ k Y ′ = 0, ∀k and hence H⊤Y ′ = 0. This implies H⊤Y = H⊤XX†Y + H⊤ZZ†Y . Further, since a(n)⊤a(n) = h(n)⊤x(n), a(n)⊤b(n) = h(n)⊤z(n), ∀n, we have ˆr∞ = X†Y + A†BZ†Y = (A⊤A)−1 ¡ A⊤AX†Y + A⊤BZ†Y ¢ = (A⊤A)−1 ¡ H⊤XX†Y + H⊤ZZ†Y ¢ = (A⊤A)−1H⊤Y. Proof of Theorem 2. Because ⟨˜Di, ˜Dj⟩= 1{i = j}, we have Z⊤Z = NI. Plugging this into (6) and comparing the resultant expression with (7) and (8) yield the desired result. 8 References [1] J.-Y. Audibert. Aggregated estimators and empirical complexity for least square regression. Annales de l’Institut Henri Poincare Probability and Statistics, 40(6):685–736, 2004. [2] P. L. Bartlett and S. Mendelson. Empirical minimization. Probability Theory and Related Fields, 135(3):311–334, 2006. [3] D. Bertsimas and A. Thiele. Robust and data-driven optimization: Modern decision-making under uncertainty. In Tutorials on Operations Research. INFORMS, 2006. [4] O. Besbes, R. Philips, and A. Zeevi. Testing the validity of a demand model: An operations perspective. 2007. [5] F. Bunea, A. B. Tsybakov, and M. H. Wegkamp. Aggregation for Gaussian regression. The Annals of Statistics, 35(4):1674–1697, 2007. [6] D. Haussler. Decision theoretic generalizations of the PAC model for neural net and other learning applications. Information and Computation, 100:78–150, 1992. [7] K. Kim and N. Timm. Univariate and Multivariate General Linear Models: Theory and Applications with SAS. Chapman & Hall/CRC, 2006. [8] K. E. Muller and P. W. Stewart. Linear Model Theory: Univariate, Multivariate, and Mixed Models. Wiley, 2006. [9] R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, 1998. [10] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of Royal Statistical Society, 1996. 9
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Data-driven calibration of linear estimators with minimal penalties Sylvain Arlot ∗ CNRS ; Willow Project-Team Laboratoire d’Informatique de l’Ecole Normale Superieure (CNRS/ENS/INRIA UMR 8548) 23, avenue d’Italie, F-75013 Paris, France sylvain.arlot@ens.fr Francis Bach † INRIA ; Willow Project-Team Laboratoire d’Informatique de l’Ecole Normale Superieure (CNRS/ENS/INRIA UMR 8548) 23, avenue d’Italie, F-75013 Paris, France francis.bach@ens.fr Abstract This paper tackles the problem of selecting among several linear estimators in non-parametric regression; this includes model selection for linear regression, the choice of a regularization parameter in kernel ridge regression or spline smoothing, and the choice of a kernel in multiple kernel learning. We propose a new algorithm which first estimates consistently the variance of the noise, based upon the concept of minimal penalty which was previously introduced in the context of model selection. Then, plugging our variance estimate in Mallows’ CL penalty is proved to lead to an algorithm satisfying an oracle inequality. Simulation experiments with kernel ridge regression and multiple kernel learning show that the proposed algorithm often improves significantly existing calibration procedures such as 10-fold cross-validation or generalized cross-validation. 1 Introduction Kernel-based methods are now well-established tools for supervised learning, allowing to perform various tasks, such as regression or binary classification, with linear and non-linear predictors [1, 2]. A central issue common to all regularization frameworks is the choice of the regularization parameter: while most practitioners use cross-validation procedures to select such a parameter, data-driven procedures not based on cross-validation are rarely used. The choice of the kernel, a seemingly unrelated issue, is also important for good predictive performance: several techniques exist, either based on cross-validation, Gaussian processes or multiple kernel learning [3, 4, 5]. In this paper, we consider least-squares regression and cast these two problems as the problem of selecting among several linear estimators, where the goal is to choose an estimator with a quadratic risk which is as small as possible. This problem includes for instance model selection for linear regression, the choice of a regularization parameter in kernel ridge regression or spline smoothing, and the choice of a kernel in multiple kernel learning (see Section 2). The main contribution of the paper is to extend the notion of minimal penalty [6, 7] to all discrete classes of linear operators, and to use it for defining a fully data-driven selection algorithm satisfying a non-asymptotic oracle inequality. Our new theoretical results presented in Section 4 extend similar results which were limited to unregularized least-squares regression (i.e., projection operators). Finally, in Section 5, we show that our algorithm improves the performances of classical selection procedures, such as GCV [8] and 10-fold cross-validation, for kernel ridge regression or multiple kernel learning, for moderate values of the sample size. ∗http://www.di.ens.fr/∼arlot/ †http://www.di.ens.fr/∼fbach/ 1 2 Linear estimators In this section, we define the problem we aim to solve and give several examples of linear estimators. 2.1 Framework and notation Let us assume that one observes Yi = f(xi) + εi ∈R for i = 1 . . . n , where ε1, . . . , εn are i.i.d. centered random variables with E[ε2 i ] = σ2 unknown, f is an unknown measurable function X 7→R and x1, . . . , xn ∈X are deterministic design points. No assumption is made on the set X . The goal is to reconstruct the signal F = (f(xi))1≤i≤n ∈Rn , with some estimator bF ∈Rn , depending only on (x1, Y1), . . . , (xn, Yn) , and having a small quadratic risk n−1∥bF −F∥2 2 , where ∀t ∈Rn , we denote by ∥t∥2 the ℓ2-norm of t , defined as ∥t∥2 2 := Pn i=1 t2 i . In this paper, we focus on linear estimators bF that can be written as a linear function of Y = (Y1, . . . , Yn) ∈Rn , that is, bF = AY , for some (deterministic) n × n matrix A . Here and in the rest of the paper, vectors such as Y or F are assumed to be column-vectors. We present in Section 2.2 several important families of estimators of this form. The matrix A may depend on x1, . . . , xn (which are known and deterministic), but not on Y , and may be parameterized by certain quantities—usually regularization parameter or kernel combination weights. 2.2 Examples of linear estimators In this paper, our theoretical results apply to matrices A which are symmetric positive semi-definite, such as the ones defined below. Ordinary least-squares regression / model selection. If we consider linear predictors from a design matrix X ∈Rn×p , then bF = AY with A = X(X⊤X)−1X⊤, which is a projection matrix (i.e., A⊤A = A); bF = AY is often called a projection estimator. In the variable selection setting, one wants to select a subset J ⊂{1, . . . , p} , and matrices A are parameterized by J . Kernel ridge regression / spline smoothing. We assume that a positive definite kernel k : X × X →R is given, and we are looking for a function f : X →R in the associated reproducing kernel Hilbert space (RKHS) F , with norm ∥· ∥F . If K denotes the n × n kernel matrix, defined by Kab = k(xa, xb) , then the ridge regression estimator—a.k.a. spline smoothing estimator for spline kernels [9]—is obtained by minimizing with respect to f ∈F [2]: 1 n n X i=1 (Yi −f(xi))2 + λ∥f∥2 F . The unique solution is equal to bf = Pn i=1 αik(·, xi) , where α = (K +nλI)−1Y . This leads to the smoothing matrix Aλ = K(K + nλIn)−1 , parameterized by the regularization parameter λ ∈R+ . Multiple kernel learning / Group Lasso / Lasso. We now assume that we have p different kernels kj , feature spaces Fj and feature maps Φj : X →Fj , j = 1, . . . , p . The group Lasso [10] and multiple kernel learning [11, 5] frameworks consider the following objective function J(f1, . . . , fp)= 1 n n X i=1 yi−Pp j=1⟨fj, Φj(xi)⟩ 2+2λ p X j=1 ∥fj∥Fj = L(f1, . . . , fp)+2λ p X j=1 ∥fj∥Fj . Note that when Φj(x) is simply the j-th coordinate of x ∈Rp , we get back the penalization by the ℓ1-norm and thus the regular Lasso [12]. Using a1/2 = minb⩾0 1 2{ a b + b} , we obtain a variational formulation of the sum of norms 2 Pp j=1 ∥fj∥= minη∈Rp + Pp j=1 n ∥fj∥2 ηj + ηj o . Thus, minimizing J(f1, . . . , fp) with respect to (f1, . . . , fp) is equivalent to minimizing with respect to η ∈Rp + (see [5] for more details): min f1,...,fp L(f1, . . . , fp) + λ p X j=1 ∥fj∥2 ηj + λ p X j=1 ηj = 1 ny⊤ Pp j=1 ηjKj + nλIn −1y + λ p X j=1 ηj , 2 where In is the n × n identity matrix. Moreover, given η , this leads to a smoothing matrix of the form Aη,λ = (Pp j=1 ηjKj)(Pp j=1 ηjKj + nλIn)−1 , (1) parameterized by the regularization parameter λ ∈R+ and the kernel combinations in Rp +—note that it depends only on λ−1η , which can be grouped in a single parameter in Rp + . Thus, the Lasso/group lasso can be seen as particular (convex) ways of optimizing over η . In this paper, we propose a non-convex alternative with better statistical properties (oracle inequality in Theorem 1). Note that in our setting, finding the solution of the problem is hard in general since the optimization is not convex. However, while the model selection problem is by nature combinatorial, our optimization problems for multiple kernels are all differentiable and are thus amenable to gradient descent procedures—which only find local optima. Non symmetric linear estimators. Other linear estimators are commonly used, such as nearestneighbor regression or the Nadaraya-Watson estimator [13]; those however lead to non symmetric matrices A , and are not entirely covered by our theoretical results. 3 Linear estimator selection In this section, we first describe the statistical framework of linear estimator selection and introduce the notion of minimal penalty. 3.1 Unbiased risk estimation heuristics Usually, several estimators of the form bF = AY can be used. The problem that we consider in this paper is then to select one of them, that is, to choose a matrix A . Let us assume that a family of matrices (Aλ)λ∈Λ is given (examples are shown in Section 2.2), hence a family of estimators ( bFλ)λ∈Λ can be used, with bFλ := AλY . The goal is to choose from data some bλ ∈Λ , so that the quadratic risk of bFbλ is as small as possible. The best choice would be the oracle: λ⋆∈arg min λ∈Λ n n−1∥bFλ −F∥2 2 o , which cannot be used since it depends on the unknown signal F . Therefore, the goal is to define a data-driven bλ satisfying an oracle inequality n−1∥bFbλ −F∥2 2 ≤Cn inf λ∈Λ n n−1∥bFλ −F∥2 2 o + Rn , (2) with large probability, where the leading constant Cn should be close to 1 (at least for large n) and the remainder term Rn should be negligible compared to the risk of the oracle. Many classical selection methods are built upon the “unbiased risk estimation” heuristics: If bλ minimizes a criterion crit(λ) such that ∀λ ∈Λ, E [crit(λ)] ≈E h n−1∥bFλ −F∥2 2 i , then bλ satisfies an oracle inequality such as in Eq. (2) with large probability. For instance, crossvalidation [14, 15] and generalized cross-validation (GCV) [8] are built upon this heuristics. One way of implementing this heuristics is penalization, which consists in minimizing the sum of the empirical risk and a penalty term, i.e., using a criterion of the form: crit(λ) = n−1∥bFλ −Y ∥2 2 + pen(λ) . The unbiased risk estimation heuristics, also called Mallows’ heuristics, then leads to the ideal (deterministic) penalty penid(λ) := E h n−1∥bFλ −F∥2 2 i −E h n−1∥bFλ −Y ∥2 2 i . 3 When bFλ = AλY , we have: ∥bFλ −F∥2 2 = ∥(Aλ −In)F∥2 2 + ∥Aλε∥2 2 + 2 ⟨Aλε, (Aλ −In)F⟩, (3) ∥bFλ −Y ∥2 2 = ∥bFλ −F∥2 2 + ∥ε∥2 2 −2 ⟨ε, Aλε⟩+ 2 ⟨ε, (In −Aλ)F⟩, (4) where ε = Y −F ∈Rn and ∀t, u ∈Rn , ⟨t, u⟩= Pn i=1 tiui . Since ε is centered with covariance matrix σ2In , Eq. (3) and Eq. (4) imply that penid(λ) = 2σ2 tr(Aλ) n , (5) up to the term −E[n−1∥ε∥2 2]= −σ2 , which can be dropped off since it does not vary with λ . Note that df(λ) = tr(Aλ) is called the effective dimensionality or degrees of freedom [16], so that the ideal penalty in Eq. (5) is proportional to the dimensionality associated with the matrix Aλ— for projection matrices, we get back the dimension of the subspace, which is classical in model selection. The expression of the ideal penalty in Eq. (5) led to several selection procedures, in particular Mallows’ CL (called Cp in the case of projection estimators) [17], where σ2 is replaced by some estimator c σ2 . The estimator of σ2 usually used with CL is based upon the value of the empirical risk at some λ0 with df(λ0) large; it has the drawback of overestimating the risk, in a way which depends on λ0 and F [18]. GCV, which implicitly estimates σ2 , has the drawback of overfitting if the family (Aλ)λ∈Λ contains a matrix too close to In [19]; GCV also overestimates the risk even more than CL for most Aλ (see (7.9) and Table 4 in [18]). In this paper, we define an estimator of σ2 directly related to the selection task which does not have similar drawbacks. Our estimator relies on the concept of minimal penalty, introduced by Birg´e and Massart [6] and further studied in [7]. 3.2 Minimal and optimal penalties We deduce from Eq. (3) the bias-variance decomposition of the risk: E h n−1∥bFλ −F∥2 2 i = n−1 ∥(Aλ −In)F∥2 2 + tr(A⊤ λ Aλ)σ2 n = bias + variance , (6) and from Eq. (4) the expectation of the empirical risk: E h n−1∥bFλ −Y ∥2 2 −∥ε∥2 2 i = n−1 ∥(Aλ −In)F∥2 2 − 2 tr(Aλ) −tr(A⊤ λ Aλ) σ2 n . (7) Note that the variance term in Eq. (6) is not proportional to the effective dimensionality df(λ) = tr(Aλ) but to tr(A⊤ λ Aλ) . Although several papers argue these terms are of the same order (for instance, they are equal when Aλ is a projection matrix), this may not hold in general. If Aλ is symmetric with a spectrum Sp(Aλ) ⊂[0, 1] , as in all the examples of Section 2.2, we only have 0 ≤tr(A⊤ λ Aλ) ≤tr(Aλ) ≤2 tr(Aλ) −tr(A⊤ λ Aλ) ≤2 tr(Aλ) . (8) In order to give a first intuitive interpretation of Eq. (6) and Eq. (7), let us consider the kernel ridge regression example and assume that the risk and the empirical risk behave as their expectations in Eq. (6) and Eq. (7); see also Fig. 1. Completely rigorous arguments based upon concentration inequalities are developed in [20] and summarized in Section 4, leading to the same conclusion as the present informal reasoning. First, as proved in [20], the bias n−1 ∥(Aλ −In)F∥2 2 is a decreasing function of the dimensionality df(λ) = tr(Aλ) , and the variance tr(A⊤ λ Aλ)σ2n−1 is an increasing function of df(λ) , as well as 2 tr(Aλ) −tr(A⊤ λ Aλ) . Therefore, Eq. (6) shows that the optimal λ realizes the best trade-off between bias (which decreases with df(λ)) and variance (which increases with df(λ)), which is a classical fact in model selection. Second, the expectation of the empirical risk in Eq. (7) can be decomposed into the bias and a negative variance term which is the opposite of penmin(λ) := n−1 2 tr(Aλ) −tr(A⊤ λ Aλ) σ2 . (9) 4 0 200 400 600 800 −0.5 0 0.5 degrees of freedom ( tr A ) generalization errors σ2trA2 − 2σ2trA σ2trA bias variance ∼ σ2tr A2 generalization error ∼ bias + σ2 tr A2 empirical error−σ2 ∼ bias+σ2trA2−2σ2 tr A Figure 1: Bias-variance decomposition of the generalization error, and minimal/optimal penalties. As suggested by the notation penmin , we will show it is a minimal penalty in the following sense. If ∀C ≥0, bλmin(C) ∈arg min λ∈Λ n n−1∥bFλ −Y ∥2 2 + C penmin(λ) o , then, up to concentration inequalities that are detailed in Section 4.2, bλmin(C) behaves like a minimizer of gC(λ) = E h n−1∥bFλ −Y ∥2 2 + C penmin(λ) i −n−1σ2 = n−1 ∥(Aλ −In)F∥2 2+(C−1) penmin(λ) . Therefore, two main cases can be distinguished: • if C < 1 , then gC(λ) decreases with df(λ) so that df(bλmin(C)) is huge: bλmin(C) overfits. • if C > 1 , then gC(λ) increases with df(λ) when df(λ) is large enough, so that df(bλmin(C)) is much smaller than when C < 1 . As a conclusion, penmin(λ) is the minimal amount of penalization needed so that a minimizer bλ of a penalized criterion is not clearly overfitting. Following an idea first proposed in [6] and further analyzed or used in several other papers such as [21, 7, 22], we now propose to use that penmin(λ) is a minimal penalty for estimating σ2 and plug this estimator into Eq. (5). This leads to the algorithm described in Section 4.1. Note that the minimal penalty given by Eq. (9) is new; it generalizes previous results [6, 7] where penmin(Aλ) = n−1 tr(Aλ)σ2 because all Aλ were assumed to be projection matrices, i.e., A⊤ λ Aλ = Aλ . Furthermore, our results generalize the slope heuristics penid ≈2 penmin (only valid for projection estimators [6, 7]) to general linear estimators for which penid / penmin ∈(1, 2] . 4 Main results In this section, we first describe our algorithm and then present our theoretical results. 4.1 Algorithm The following algorithm first computes an estimator of bC of σ2 using the minimal penalty in Eq. (9), then considers the ideal penalty in Eq. (5) for selecting λ . Input: Λ a finite set with Card(Λ) ≤Knα for some K, α ≥0 , and matrices Aλ . • ∀C > 0 , compute bλ0(C) ∈arg minλ∈Λ{∥bFλ −Y ∥2 2 + C 2 tr(Aλ) −tr(A⊤ λ Aλ) } . • Find bC such that df(bλ0( bC)) ∈ n3/4, n/10 . • Select bλ ∈arg minλ∈Λ{∥bFλ −Y ∥2 2 + 2 bC tr(Aλ)} . In the steps 1 and 2 of the above algorithm, in practice, a grid in log-scale is used, and our theoretical results from the next section suggest to use a step-size of order n−1/4 . Note that it may not be 5 possible in all cases to find a C such that df(bλ0(C)) ∈[n3/4, n/10] ; therefore, our condition in step 2, could be relaxed to finding a bC such that for all C > bC + δ , df(bλ0(C)) < n3/4 and for all C < bC −δ , df(bλ0(C)) > n/10 , with δ = n−1/4+ξ , where ξ > 0 is a small constant. Alternatively, using the same grid in log-scale, we can select bC with maximal jump between successive values of df(bλ0(C))—note that our theoretical result then does not entirely hold, as we show the presence of a jump around σ2 , but do not show the absence of similar jumps elsewhere. 4.2 Oracle inequality Theorem 1 Let bC and bλ be defined as in the algorithm of Section 4.1, with Card(Λ) ≤Knα for some K, α ≥0 . Assume that ∀λ ∈Λ , Aλ is symmetric with Sp(Aλ) ⊂[0, 1] , that εi are i.i.d. Gaussian with variance σ2 > 0 , and that ∃λ1, λ2 ∈Λ with df(λ1) ≥n 2 , df(λ2) ≤√n, and ∀i ∈{1, 2} , n−1 ∥(Aλi −In)F∥2 2 ≤σ2 r ln(n) n . (A1−2) Then, a numerical constant Ca and an event of probability at least 1 −8Kn−2 exist on which, for every n ≥Ca , 1 −91(α + 2) r ln(n) n ! σ2 ≤bC ≤ 1 + 44(α + 2) p ln(n) n1/4 ! σ2 . (10) Furthermore, if ∃κ ≥1, ∀λ ∈Λ, n−1 tr(Aλ)σ2 ≤κE h n−1∥bFλ −F∥2 2 i , (A3) then, a constant Cb depending only on κ exists such that for every n ≥Cb , on the same event, n−1∥bFbλ −F∥2 2 ≤ 1 + 40κ ln(n) inf λ∈Λ n n−1∥bFλ −F∥2 2 o + 36(κ + α + 2) ln(n)σ2 n . (11) Theorem 1 is proved in [20]. The proof mainly follows from the informal arguments developed in Section 3.2, completed with the following two concentration inequalities: If ξ ∈Rn is a standard Gaussian random vector, α ∈Rn and M is a real-valued n × n matrix, then for every x ≥0 , P |⟨α, ξ⟩| ≤ √ 2x ∥α∥2 ≥1 −2e−x (12) P ∀θ > 0, ∥Mξ∥2 2 −tr(M ⊤M) ≤θ tr(M ⊤M) + 2(1 + θ−1) ∥M∥2 x ≥1 −2e−x , (13) where ∥M∥is the operator norm of M . A proof of Eq. (12) and (13) can be found in [20]. 4.3 Discussion of the assumptions of Theorem 1 Gaussian noise. When ε is sub-Gaussian, Eq. (12) and Eq. (13) can be proved for ξ = σ−1ε at the price of additional technicalities, which implies that Theorem 1 is still valid. Symmetry. The assumption that matrices Aλ must be symmetric can certainly be relaxed, since it is only used for deriving from Eq. (13) a concentration inequality for ⟨Aλξ, ξ⟩. Note that Sp(Aλ) ⊂ [0, 1] barely is an assumption since it means that Aλ actually shrinks Y . Assumptions (A1−2). (A1−2) holds if maxλ∈Λ {df(λ)} ≥n/2 and the bias is smaller than c df(λ)−d for some c, d > 0 , a quite classical assumption in the context of model selection. Besides, (A1−2) is much less restrictive and can even be relaxed, see [20]. Assumption (A3). The upper bound (A3) on tr(Aλ) is certainly the strongest assumption of Theorem 1, but it is only needed for Eq. (11). According to Eq. (6), (A3) holds with κ = 1 when Aλ is a projection matrix since tr(A⊤ λ Aλ) = tr(Aλ) . In the kernel ridge regression framework, (A3) holds as soon as the eigenvalues of the kernel matrix K decrease like j−α—see [20]. In general, (A3) means that bFλ should not have a risk smaller than the parametric convergence rate associated with a model of dimension df(λ) = tr(Aλ) . When (A3) does not hold, selecting among estimators whose risks are below the parametric rate is a rather difficult problem and it may not be possible to attain the risk of the oracle in general. 6 −2 0 2 0 100 200 300 400 log(C/σ2) selected degrees of freedom minimal penalty optimal penalty / 2 −2 0 2 0 50 100 150 200 log(C/σ2) selected degrees of freedom optimal/2 minimal (discrete) minimal (continuous) Figure 2: Selected degrees of freedom vs. penalty strength log(C/σ2) : note that when penalizing by the minimal penalty, there is a strong jump at C = σ2 , while when using half the optimal penalty, this is not the case. Left: single kernel case, Right: multiple kernel case. Nevertheless, an oracle inequality can still be proved without (A3), at the price of enlarging bC slightly and adding a small fraction of σ2n−1 tr(Aλ) in the right-hand side of Eq. (11), see [20]. Enlarging bC is necessary in general: If tr(A⊤ λ Aλ) ≪tr(Aλ) for most λ ∈Λ , the minimal penalty is very close to 2σ2n−1 tr(Aλ) , so that according to Eq. (10), overfitting is likely as soon as bC underestimates σ2 , even by a very small amount. 4.4 Main consequences of Theorem 1 and comparison with previous results Consistent estimation of σ2 . The first part of Theorem 1 shows that bC is a consistent estimator of σ2 in a general framework and under mild assumptions. Compared to classical estimators of σ2 , such as the one usually used with Mallows’ CL, bC does not depend on the choice of some model assumed to have almost no bias, which can lead to overestimating σ2 by an unknown amount [18]. Oracle inequality. Our algorithm satisfies an oracle inequality with high probability, as shown by Eq. (11): The risk of the selected estimator bFbλ is close to the risk of the oracle, up to a remainder term which is negligible when the dimensionality df(λ⋆) grows with n faster than ln(n) , a typical situation when the bias is never equal to zero, for instance in kernel ridge regression. Several oracle inequalities have been proved in the statistical literature for Mallows’ CL with a consistent estimator of σ2 , for instance in [23]. Nevertheless, except for the model selection problem (see [6] and references therein), all previous results were asymptotic, meaning that n is implicitly assumed to be larged compared to each parameter of the problem. This assumption can be problematic for several learning problems, for instance in multiple kernel learning when the number p of kernels may grow with n . On the contrary, Eq. (11) is non-asymptotic, meaning that it holds for every fixed n as soon as the assumptions explicitly made in Theorem 1 are satisfied. Comparison with other procedures. According to Theorem 1 and previous theoretical results [23, 19], CL, GCV, cross-validation and our algorithm satisfy similar oracle inequalities in various frameworks. This should not lead to the conclusion that these procedures are completely equivalent. Indeed, second-order terms can be large for a given n , while they are hidden in asymptotic results and not tightly estimated by non-asymptotic results. As showed by the simulations in Section 5, our algorithm yields statistical performances as good as existing methods, and often quite better. Furthermore, our algorithm never overfits too much because df(bλ) is by construction smaller than the effective dimensionality of bλ0( bC) at which the jump occurs. This is a quite interesting property compared for instance to GCV, which is likely to overfit if it is not corrected because GCV minimizes a criterion proportional to the empirical risk. 5 Simulations Throughout this section, we consider exponential kernels on Rd , k(x, y)=Qd i=1 e−|xi−yi| , with the x’s sampled i.i.d. from a standard multivariate Gaussian. The functions f are then selected randomly as Pm i=1 αik(·, zi) , where both α and z are i.i.d. standard Gaussian (i.e., f belongs to the RKHS). 7 4 5 6 7 0.5 1 1.5 2 2.5 3 log(n) mean( error / errororacle ) 10−fold CV GCV min. penalty 3.5 4 4.5 5 5.5 1 1.5 2 2.5 3 3.5 log(n) mean( error / errorMallows ) MKL+CV GCV kernel sum min. penalty Figure 3: Comparison of various smoothing parameter selection (minikernel, GCV, 10-fold cross validation) for various values of numbers of observations, averaged over 20 replications. Left: single kernel, right: multiple kernels. Jump. In Figure 2 (left), we consider data xi ∈R6 , n = 1000, and study the size of the jump in Figure 2 for kernel ridge regression. With half the optimal penalty (which is used in traditional variable selection for linear regression), we do not get any jump, while with the minimal penalty we always do. In Figure 2 (right), we plot the same curves for the multiple kernel learning problem with two kernels on two different 4-dimensional variables, with similar results. In addition, we show two ways of optimizing over λ ∈Λ = R2 + , by discrete optimization with n different kernel matrices—a situation covered by Theorem 1—or with continuous optimization with respect to η in Eq. (1), by gradient descent—a situation not covered by Theorem 1. Comparison of estimator selection methods. In Figure 3, we plot model selection results for 20 replications of data (d = 4, n = 500), comparing GCV [8], our minimal penalty algorithm, and cross-validation methods. In the left part (single kernel), we compare to the oracle (which can be computed because we can enumerate Λ), and use for cross-validation all possible values of λ . In the right part (multiple kernel), we compare to the performance of Mallows’ CL when σ2 is known (i.e., penalty in Eq. (5)), and since we cannot enumerate all λ’s, we use the solution obtained by MKL with CV [5]. We also compare to using our minimal penalty algorithm with the sum of kernels. 6 Conclusion A new light on the slope heuristics. Theorem 1 generalizes some results first proved in [6] where all Aλ are assumed to be projection matrices, a framework where assumption (A3) is automatically satisfied. To this extent, Birg´e and Massart’s slope heuristics has been modified in a way that sheds a new light on the “magical” factor 2 between the minimal and the optimal penalty, as proved in [6, 7]. Indeed, Theorem 1 shows that for general linear estimators, penid(λ) penmin(λ) = 2 tr(Aλ) 2 tr(Aλ) −tr(A⊤ λ Aλ) , (14) which can take any value in (1, 2] in general; this ratio is only equal to 2 when tr(Aλ) ≈tr(A⊤ λ Aλ) , hence mostly when Aλ is a projection matrix. Future directions. In the case of projection estimators, the slope heuristics still holds when the design is random and data are heteroscedastic [7]; we would like to know whether Eq. (14) is still valid for heteroscedastic data with general linear estimators. In addition, the good empirical performances of elbow heuristics based algorithms (i.e., based on the sharp variation of a certain quantity around good hyperparameter values) suggest that Theorem 1 can be generalized to many learning frameworks (and potentially to non-linear estimators), probably with small modifications in the algorithm, but always relying on the concept of minimal penalty. Another interesting open problem would be to extend the results of Section 4, where Card(Λ) ≤ Knα is assumed, to continuous sets Λ such as the ones appearing naturally in kernel ridge regression and multiple kernel learning. We conjecture that Theorem 1 is valid without modification for a “small” continuous Λ , such as in kernel ridge regression where taking a grid of size n in log-scale is almost equivalent to taking Λ = R+ . On the contrary, in applications such as the Lasso with p ≫n variables, the natural set Λ cannot be well covered by a grid of cardinality nα with α small, and our minimal penalty algorithm and Theorem 1 certainly have to be modified. 8 References [1] J. Shawe-Taylor and N. Cristianini. Kernel Methods for Pattern Analysis. Cambridge University Press, 2004. [2] B. Sch¨olkopf and A. J. Smola. Learning with Kernels. MIT Press, 2001. [3] O. Chapelle and V. Vapnik. Model selection for support vector machines. In Advances in Neural Information Processing Systems (NIPS), 1999. [4] C. E. Rasmussen and C. Williams. Gaussian Processes for Machine Learning. MIT Press, 2006. [5] F. Bach. Consistency of the group Lasso and multiple kernel learning. Journal of Machine Learning Research, 9:1179–1225, 2008. [6] L. Birg´e and P. Massart. Minimal penalties for Gaussian model selection. Probab. Theory Related Fields, 138(1-2):33–73, 2007. [7] S. Arlot and P. Massart. Data-driven calibration of penalties for least-squares regression. J. Mach. Learn. Res., 10:245–279, 2009. [8] P. Craven and G. Wahba. Smoothing noisy data with spline functions. Estimating the correct degree of smoothing by the method of generalized cross-validation. Numer. Math., 31(4):377– 403, 1978/79. [9] G. Wahba. Spline Models for Observational Data. SIAM, 1990. [10] M. Yuan and Y. Lin. Model selection and estimation in regression with grouped variables. Journal of The Royal Statistical Society Series B, 68(1):49–67, 2006. [11] G. R. G. Lanckriet, N. Cristianini, P. Bartlett, L. El Ghaoui, and M. I. Jordan. Learning the kernel matrix with semidefinite programming. J. Mach. Learn. Res., 5:27–72, 2003/04. [12] R. Tibshirani. Regression shrinkage and selection via the Lasso. Journal of The Royal Statistical Society Series B, 58(1):267–288, 1996. [13] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. SpringerVerlag, 2001. [14] D. M. Allen. The relationship between variable selection and data augmentation and a method for prediction. Technometrics, 16:125–127, 1974. [15] M. Stone. Cross-validatory choice and assessment of statistical predictions. J. Roy. Statist. Soc. Ser. B, 36:111–147, 1974. [16] T. Zhang. Learning bounds for kernel regression using effective data dimensionality. Neural Comput., 17(9):2077–2098, 2005. [17] C. L. Mallows. Some comments on Cp. Technometrics, 15:661–675, 1973. [18] B. Efron. How biased is the apparent error rate of a prediction rule? J. Amer. Statist. Assoc., 81(394):461–470, 1986. [19] Y. Cao and Y. Golubev. On oracle inequalities related to smoothing splines. Math. Methods Statist., 15(4):398–414 (2007), 2006. [20] S. Arlot and F. Bach. Data-driven calibration of linear estimators with minimal penalties, September 2009. Long version. arXiv:0909.1884v1. [21] ´E. Lebarbier. Detecting multiple change-points in the mean of a gaussian process by model selection. Signal Proces., 85:717–736, 2005. [22] C. Maugis and B. Michel. Slope heuristics for variable selection and clustering via gaussian mixtures. Technical Report 6550, INRIA, 2008. [23] K.-C. Li. Asymptotic optimality for Cp, CL, cross-validation and generalized cross-validation: discrete index set. Ann. Statist., 15(3):958–975, 1987. 9
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fMRI-Based Inter-Subject Cortical Alignment Using Functional Connectivity Bryan R. Conroy1 Benjamin D. Singer2 James V. Haxby3∗ Peter J. Ramadge1 1 Department of Electrical Engineering, 2 Neuroscience Institute, Princeton University 3 Department of Psychology, Dartmouth College Abstract The inter-subject alignment of functional MRI (fMRI) data is important for improving the statistical power of fMRI group analyses. In contrast to existing anatomically-based methods, we propose a novel multi-subject algorithm that derives a functional correspondence by aligning spatial patterns of functional connectivity across a set of subjects. We test our method on fMRI data collected during a movie viewing experiment. By cross-validating the results of our algorithm, we show that the correspondence successfully generalizes to a secondary movie dataset not used to derive the alignment. 1 Introduction Functional MRI (fMRI) studies of human neuroanatomical organization commonly analyze fMRI data across a population of subjects. The effective use of this data requires deriving a spatial correspondence across the set of subjects, i.e., the data must be aligned, or registered, into a common coordinate space. Current inter-subject registration techniques derive this correspondence by aligning anatomically-defined features, e.g. major sulci and gyri, across subjects, either in the volume or on extracted cortical surfaces. Talairach normalization [1], for example, derives a piecewise affine transformation by matching a set of major anatomical landmarks in the brain volume. More advanced techniques match a denser set of anatomical features, such as cortical curvature [2], and derive nonlinear transformations between a reference space and each subject’s cortical surface. It is known, however, that an accurate inter-subject functional correspondence cannot be derived using only anatomical features, since the size, shape and anatomical location of functional loci vary across subjects [3], [4]. Because of this deficiency in current alignment methods, it is common practice to spatially smooth each subject’s functional data prior to a population based analysis. However, this incurs the penalty of blurring the functional data within and across distinct cortical regions. Thus, the functional alignment of multi-subject fMRI data remains an important problem. We propose to register functional loci directly by using anatomical and functional data to learn an inter-subject cortical correspondence. This approach was first explored in [5], where subject cortices were registered by maximizing the inter-subject correlation of the functional response elicited by a common stimulus (a movie viewing). In essence, the correspondence was selected to maximize the correlation of the fMRI time series between subjects. This relies on the functional response being time-locked with the experimental stimulus. Large regions of visual and auditory cortex stimulated by a movie viewing do indeed show consistent inter-subject synchrony [6]. However, other areas in the intrinsic [7] or default [8] system fail to exhibit significant correlations across repeated stimulus trials. The technique of [5] is hence not expected to improve alignment in these intrinsic regions. In contrast to [5], we propose to achieve inter-subject alignment by aligning intra-subject patterns of cortical functional connectivity. By functional connectivity, we mean within-subject similarity of ∗This work was funded by a grant from the National Institute of Mental Health (5R01MH075706-02) 1 the temporal response of remote regions of cortex [9]. This can be estimated from fMRI data, for example, by correlating the functional time series between pairs of cortical nodes within a subject. This yields a dense set of functional features for each subject from which we learn an inter-subject correspondence. Unlike other functional connectivity work (see e.g. [10]), we define connectivity between pairs of cortical nodes rather than with respect to anatomical regions of interest. Our approach is inspired by studies showing that the patterns of functional connectivity in the intrinsic network are consistent across subjects [7], [11]. This suggests that our method has the potential to learn an inter-subject functional correspondence within both extrinsic and intrinsic cortical networks. In summary, we formulate a multi-subject cortical alignment algorithm that minimizes the difference between functional connectivity vectors of corresponding cortical nodes across subjects. We do so by learning a dense-deformation field on the cortex of each subject, suitably regularized to preserve cortical topology [2]. Our key contributions are: a) the novel alignment objective, b) a principled algorithm for accomplishing the alignment, and c) experimental verification on fMRI data. The paper is organized as follows. In §2 we formulate the multi-subject alignment problem, followed by a detailed exposition of the algorithm in §3 and §4. Finally, we exhibit results of the algorithm applied to multi-subject fMRI data in §5 and draw conclusions in §6. 2 Formulation of the Multi-Subject Alignment Problem For each subject we are given volumetric anatomical MRI data and fMRI data. The anatomical data is used to extract a two-dimensional surface model of cortex. This greatly facilitates cortical based analysis and subsequent visualization [12], [13], [14]. Cortex is segmented, then each cortical hemisphere is inflated to obtain a smooth surface, which is projected to the sphere, S2, represented by a discrete spherical mesh Ms = {pk ∈S2; 1 ≤k ≤Nv/2}. The two cortical hemispheres are hence modeled by the disjoint union S = S2 ⊎S2, represented by the corresponding disjoint union of mesh points M = Ms ⊎Ms. Anatomical cortical features, such as cortical curvature, are functions Da : S →RNa sampled on M. Thus, our analysis is restricted to cortex only. The fMRI volumeric data is first aligned with the anatomical scan, then mapped onto S. This assigns each mesh node pk ∈M a “volumetric cortical voxel” vk ∈R3, with associated functional time series fk ∈RNt. The functional time series data is then a function Df : S →RNt sampled on M. As indicated in the introduction, we do not directly register the fMRI time series but instead register the functional connectivity derived from the time series. Let σ(f1, f2) denote a similarity measure on pairs of time series f1, f2 ∈RNt. A useful example is empirical correlation: σ(f1, f2) = corr(f1, f2); another possibility is an estimate of the mutual information between the pairwise entries of f1, f2. Define the functional connectivity of the fMRI data under σ as the map C(pi, pj) = σ(Df(pi), Df(pj)), i.e., the similarity of the functional times series at the pairs of cortical nodes. Functional connections both within and across cortical hemispheres are considered. Functional connectivity can be conceptualized as the adjacency matrix of an edge-weighted graph on all cortical nodes. The edge between nodes pi, pj is weighted by the pairwise similarity measure σ(fi, fj) codifying the functional similarity of pi and pj. In the case of correlation, C is the correlation matrix of the time series data. For typical values of Nv (≈72, 000), the functional connectivity data structure is huge. Hence we need efficient mechanisms for working with C. We are given the data discussed above for Ns subjects. Subject k’s training data is specified by samples of the functions Da,k : Sk →RNa, Df,k : Sj →RNt, and the derived functional connectivity Ck, all sampled on the mesh Mk, k = 1, . . . , Ns. Our objective is to learn a relation consisting of Ns-tuples of corresponding points across the set of cortices. To do so, we could select a node from M1 for subject 1 and learn the corresponding points on the cortices of the remaining Ns −1 subjects through smooth and invertible mappings gk : S1 →Sk, k = 2, . . . , Ns. However, this arbitrarily and undesirably gives special status to one subject. Instead, we introduce a reference model Sref = S2 ⊎S2 with mesh Mref. For each node p ∈Mref on Sref, we seek to learn the Ns-tuple of corresponding points (g1(p), g2(p), . . . , gNs(p)), parameterized by gk : Sref →Sk, k = 1, . . . , Ns. In general terms, we can now summarize our task as follows: use the functional connectivity data Ck, in conjunction with the anatomical data Da,k, k = 1, . . . , Ns, to estimate warping functions {gk : k = 1, . . . , Ns}, subject to specified regularity conditions, that bring some specified balance of anatomy and functional connectivity into alignment across subjects. That said, for the remainder 2 of the paper we restrict attention to aligning only functional connectivity across subjects. There is no doubt that anatomy must be an integral part of a full solution; but that aspect is not new, and is already well understood. Restricting attention to the alignment of functional connectivity will allow us to concentrate on the most novel and important aspects of our approach. To proceed, assume a reference connectivity Cref, such that for each subject k = 1, . . . , Ns, Ck(gk(pi), gk(pj)) = Cref(pi, pj) + ϵk(pi, pj), pi, pj ∈Mref (1) where Ck(gk(pi), gk(pj)) = σ(Df,k(gk(pi)), Df,k(gk(pj))), and ϵk is zero-mean random noise. Since gk(p) may not be a mesh point, computation of Df,k(gk(p)) requires interpolation of the time series using mesh nodes in a neighborhood of gk(p). This will be important as we proceed. Given (1), we estimate g by maximizing a regularized log likelihood: ˆg = arg max g=(g1,··· ,gNs) log P(C1, . . . , CNs|g) −λ P k Reg(gk) (2) where Reg(gk) constrains each warping function gk to be smooth and invertible. Here, we will focus on the log likelihood term and delay the discussion of regularization to §3. Optimization of (2) is complicated by the fact that Cref is a latent variable, so it must be estimated along with g. We use Expectation-Maximization to iteratively alternate between computing an expectation of Cref (E-step), and a maximum likelihood estimate of g given both the observed and estimated unobserved data (M-step) [15]. In the E-step, the expectation of Cref, Cref, conditioned on the current estimate of g, bg, is computed by averaging the connectivity across subjects: Cref(pi, pj) = 1/Ns PNs k=1 Ck(bgk(pi), bgk(pj)), pi, pj ∈Sref (3) In the M-step, the estimate bg is refined to maximize the likelihood of the full data: bg = arg max g=(g1,··· ,gNs) log P(Cref, C1, C2, · · · , CNs|g) (4a) = arg min g=(g1,··· ,gNs) PNs k=1 P pi,pj∈Sref(Cref(pi, pj) −Ck(gk(pi), gk(pj)))2 (4b) where we have assumed that the noise in (1) is i.i.d. Gaussian. Because (4b) decouples, we can optimize over each subject’s warp separately, i.e., these optimizations can be done in parallel: bgk = arg min gk P pi,pj∈Sref(Cref(pi, pj) −Ck(gk(pi), gk(pj)))2 (5) However, an interesting alternative is to perform these sequentially with an E-step after each that updates the reference estimate Cref. This also allows some other interesting adaptations. We note: (Cref(pi, pj) −Ck(gk(pi), gk(pj)))2 ∝ (Ck(pi, pj) −Ck(gk(pi), gk(pj)))2 (6a) where Ck(pi, pj) = 1 (Ns−1) P n̸=k Cn(bgn(pi), bgn(pj)), pi, pj ∈Mref, (7) is the leave-one-out template for subject k, which is indepedendent of gk. Thus, we replace (5) by: bgk = arg min gk P pi,pj∈Sref(Ck(pi, pj) −Ck(gk(pi), gk(pj)))2 (8) From (5) and (8) we observe that the multi-subject alignment problem reduces to a sequence of pairwise registrations, each of which registers one subject to an average of connectivity matrices. If we use (5), each round of pairwise registrations can be done in parallel and the results used to update the average template. The difficulty is the computational update of Cref. Alternatively, using (8) we do the pairwise registrations sequentially and compute a new leave-one-out template after each registration. This is the approach we pursue. An algorithm for solving the pairwise registration is derived in the next section and we examine the computation of leave-one-out templates in §4. 3 Pairwise Cortical Alignment We now develop an algorithm for aligning one subject, with connectivity CF , to a reference, with connectivity CR, with CF , CR ∈RNv×Nv. For concreteness, from this point forward we let σ(f1, f2) = corr(f1, f2) and assume that the time series have zero mean and unit norm. 3 A function g: MR →SF maps a reference mesh point pi ∈MR to g(pi) ∈SF . By interpolating the floating subject’s times series at the points g(pi) ∈SF we obtain the associated warped functional connectivity: ˜CF = [σ(f F g(pi), f F g(pj))]. We seek bg that best matches ˜CF to CR in the sense: bg = arg min g ∥˜CF −CR∥2 f + λReg(g) (9) Here ∥· ∥f is the matrix Frobenius norm and the regularization term Reg(g) serves as a prior over the space of allowable mappings. In the following steps, we examine how to efficiently solve (9). Step 1: Parameterizing the dependence of ˜CF on the warp. We first develop the dependence of the matrix ˜CF on the warping function g. This requires specifying how the time series at the warped points g(pi) ∈SF is interpolated using the time series data {f F i ∈RNt, i = 1, . . . , Nv} at the mesh points {pF i ∈MF , i = 1, . . . , Nv}. Here, we employ linear interpolation with a spherical kernel Φ: f F (p) = PNv i=1 f F i Φ(p, pi), p ∈SF . The kernel should be matched to the following specific objectives: (a) The kernel should be monomodal. Since the gradient of the registration objective depends on the derivative of the interpolation kernel, this will reduce the likelihood of the algorithm converging to a local minimum; (b) The support of the kernel should be finite. This will limit interpolation complexity. However, as the size of the support decreases, so will the capture range of the algorithm. At the initial stages of the algorithm, the kernel should have a broad extent, due to higher initial uncertainty, and become increasingly more localized as the algorithm converges. Thus, (c) The support of the kernel should be easily adjustable. With these considerations in mind, we select Φ(p, pi) to be a spherical radial basis function Φi : S2 →R centered at pi ∈S2 and taking the form: Φi(p) = ϕ(d(p, pi)), p ∈S2, where ϕ : [0, π] → R and d(p, pi) is the spherical geodesic distance between p and pi [16]. Then Φi(p) is monomodal with a maximum at pi, it depends only on the distance between p and pi and is radially symmetric. In detail, we employ the particular spherical radial basis function: Φi(p) = ϕ(d(p, pi)) = (1 −(2/r) sin(d(p, pi)/2))4 +((8/r) sin(d(p, pi)/2) + 1) (10) where r is a fixed parameter, and (a)+ = a1{a ≥0}. Φi(p) has two continuous derivatives and its support is {p ∈S2 : d(p, pi) < 2 sin−1(r/2)}. Note that the support can be easily adjusted through the parameter r. So the kernel has all of our desired properties. We can now make the dependence of ˜CF on g more explicit. Let TF = [f F 1 , f F 2 , · · · , f F Nv]. Then eTF = f F (g(p1)) f F (g(p2)) · · · f F (g(pNv)) = TF A where A = [Φi(g(pj))] is the Nv × Nv matrix of interpolation coefficients dependent on g and the interpolation kernel. Next, noting that CF = T T F TF , we use A to write the post-warp correlation matrix as: eCF = DAT CF AD (11) where D = diag(d1, d2, · · · , dNv) serves to normalize the updated data to unit norm: dj = ∥f F (g(pj))∥−1. Finally, we use ˜A = AD to write: ∥˜CF −CR∥2 f = ∥˜AT CF ˜A −CR∥2 f (12) Here, (12) encodes the dependence of the registration objective on g through the matrix ˜A. It is also important to note that since the interpolation kernel is locally supported, ˜A is a sparse matrix. Step 2: Efficient Representation/Computation of the Registration Objective. We now consider the Nv × Nv matrices CF and CR. At a spatial resolution of 2 mm, the spherical model of human cortex can yield Nv ≈72, 000 total mesh points. In this situation, direct computation with CF and CR is prohibitive. Hence we need an efficient way to represent and compute the objective (12). For fMRI data it is reasonable to assume that Nt ≪Nv. Hence, since the data has been centered, the rank of CF = T T F TF and of CR = T T R TR is at most Nt −1. For simplicity, we make the reasonable assumption that rank(TF ) = rank(TR) = d. Then CF and CR can be efficiently represented by compact d-dimensional SVDs CF = VF ΣF V T F and CR = VRΣRV T R . Moveover, these can be computed directly from SVDs of the data matrices: TF = UTF ΣTF V T TF and TR = UTRΣTRV T TR. In detail: VF = VTF , VR = VTR, ΣF = ΣT TF ΣTF , and ΣR = ΣT TRΣTR. 4 The above representation avoids computing CF and CR, but we must also show that it enables efficient evaluation of (12). To this end, introduce the following linear transformation: B = W T F ˜AWR (13) where WF = VF V ⊥ F , WR = VR V ⊥ R , are orthogonal with the Nv −d columns of V ⊥ F and V ⊥ R forming orthonormal bases for range(VF )⊥and range(VR)⊥, respectively. Write B as: B = B1 B2 B3 B4 (14) with B1 ∈Rd×d, B2 ∈Rd×Nv, B3 ∈R(Nv−d)×d and B4 ∈R(Nv−d)×(Nv−d). Substituting (13) and (14) into (12) and simplifying yields: ∥˜CF −CR∥2 f = ∥BT 1 ΣF B1 −ΣR∥2 f + 2∥BT 1 ΣF B2∥2 f + ∥BT 2 ΣF B2∥2 f (15) with B1 = V T F ˜AVR and B2 = V T F ˜AV ⊥ R (16) The d × d matrix B1 is readily computed since VF , VR are of manageable size. Computation of the d×Nv matrix B2 depends on V ⊥ R . This has ON columns spanning the Nv −d dimensional subspace null(CR). Since there is residual freedom in the choice of V ⊥ R and B2 is large, its selection merits closer examination. Now (16) can be viewed as a projection of the rows of V T F ˜A onto the columns of VR and V ⊥ R . The columns of ˜AVF −VRBT 1 lie in null(CR) and BT 2 = (V ⊥ R )T ( ˜AVF −VRBT 1 ). Hence a QR-factorization QR = ˜AT VF −VRBT 1 yields d ON vectors in null(CR). Choosing these as the first d columns of V ⊥ R , yields B2 = [R , 0], i.e., B2 is very sparse. In summary, we have derived the following efficient means of evaluating the objective. By onetime preprocessing of the time series data we obtain ΣF , ΣR and VF , VR. Then given a warp g, we compute: the interpolation matrix ˜A, B1 = V T F ˜AVR, and finally B2 via QR factorization of ˜AT VF −VRBT 1 . Then we evaluate (15). Step 3: The Transformation Space and Regularization. We now examine the specification of g in greater detail. We allow each mesh point to move freely (locally) in two directions. The use of such nonlinear warp models for inter-subject cortical alignment has been validated over, for example, rigid-body transformations [17]. To specify g, we first need to set up a coordinate system on the sphere. Let U = {(φ, θ); 0 < φ < π, 0 < θ < 2π}. Then the sphere can be parameterized by x: U →R3 with x(φ, θ) = (sin φ cos θ, sin φ sin θ, cos φ). Here, φ is a zenith angle measured against one of the principal axes, and θ is an azimuthal angle measured in one of the projective planes (i.e., xy-plane, xz-plane, or yz-plane). Note that x omits a semicircle of S2; so at least two such parameterizations are required to cover the entire sphere [18]. Consider pi ∈S2 parameterized by x(φ, θ) such that pi = x(φi, θi). Then the warp field at pi is: g(pi) = x(φi + ∆φi, θi + ∆θi) = x(˜φi, ˜θi) (17) for displacements ∆φi and ∆θi. The warp g is thus parameterized by: {˜φi, ˜θi, i = 1, . . . , Nv}. The warp g must be regularized to avoid undesired topological distortions (e.g. folding and excessive expansion) and to avoid over-fitting the data. This is achieved by adding a regularization term to the objective that penalizes such distortions. There are several ways this can be done. Here we follow [14] and regularize g by penalizing both metric and areal distortion. The metric distortion term penalizes warps that disrupt local distances between neighboring mesh nodes. This has the effect of limiting the expansion/contraction of cortex. The areal distortion term seeks to preserve a consistent orientation of the surface. Given a triangularization of the spherical mesh, each triangle is given an oriented normal vector that initially points radially outward from the sphere. Constraining the oriented area of all triangles to be positive prevents folds in the surface [14]. Step 4: Optimization of the objective. We optimize (3) over g by gradient descent. Denote the objective by S(g), let eaij = aijdj be the (i, j)-th entry of ˜A = AD and a(p) = [Φ1(p) Φ2(p) · · · ΦNv(p)]T . From the parameterization of the warp (17), we see that eaij = 5 Algorithm 1 Pairwise algorithm 1: Given: SVD of floating dataset ΣF , VF and reference dataset ΣR, VR 2: Given: Initial warp estimate g(0) 3: Given: Sequence r1 > r2 > · · · > rM of spatial resolutions 4: for m = 1 to M do 5: Set the kernel Φi in (10), with r = rm 6: Smooth the reference to resolution rm 7: Solve for ˆg in (9) by gradient descent with initial condition g(m−1) 8: Set g(m) = ˆg 9: end for 10: Output result: g(M) Algorithm 2 Multi-subject algorithm 1: Given: SVD of datasets, {Σk, Vk}Ns k=1 2: Initialize g(0) k to identity, k = 1, . . . , Ns 3: for t = 1 to T do 4: for k = 1 to Ns do 5: Construct Ck as explained in §4 6: Align Ck to Ck by Algorithm 1 with initial condition g(t−1) k 7: Set g(t) k to the output of the alignment 8: Use g(t) k to update Σk, Vk 9: end for 10: end for 11: Output result: g = {g(T ) 1 , . . . , g(T ) Ns ) Figure 1: The registration algorithms. Φi(x(eφj, eθj))∥TF a(x(eφj, eθj))∥−1 depends only on the warp parameters of the jth mesh node, eφj and eθj. Then, by the chain rule, the partial derivative of S(g) with respect to eφj is given by: ∂S(g) ∂e φj = PNv i=1 ∂∥˜ CF −CR∥2 f ∂eaij ∂eaij ∂e φj + λ ∂Reg(g) ∂e φj (18) A similar expression is obtained for the partial derivative with respect to eθj. Since the interpolation kernel is supported locally, the summation in (18) is taken over a small number of terms. A full expression for ∂S/∂eφj is given in the supplemental, and that of ∂Reg(g)/∂eφj in [14]. To help avoid local minima we take a multi-resolution optimization approach [19]. The registration is run on a sequence of spatial resolutions r1 > r2 > · · · > rM, with rM given by the original resolution of the data. The result at resolution rm is used to initialize the alignment at resolution rm+1. The alignment for rm is performed by matching the kernel parameter r in (10) to rm. Note that the reference dataset is also spatially smoothed at each rm by the transformation in (11), with A = [a(p1) a(p2) · · · a(pNv)]. The pairwise algorithm is summarized as Algorithm 1 in Figure 1. 4 Multi-Subject Alignment: Computing Leave-one-out Templates We now return to the multi-subject alignment problem, which is summarized as Algorithm 2 in Figure 1. It only remains to discuss efficient computation of the leave-one-out-template (7). Since Ck is an average of Ns −1 positive semi-definite matrices each of rank d, the rank d of Ck is bounded as follows d ≤d ≤(Ns −1)d. Assume that eCn, the connectivity matrix of subject n after warp gn (see (11)), has an efficient d ≪Nv dimensional SVD representation eCn = eVneΣn eV T n . To compute the SVD for Ck, we exploit the sequential nature of the multi-subject alignment algorithm by refining the SVD of the leave-one-out template for subject k−1, Ck−1 = V k−1Σk−1V T k−1, computed in the previous iteration. This is achieved by expressing Ck in terms of Ck−1: Ck = Ck−1 + 1 Ns−1( eCk−1 −eCk) (19) and computing matrix decompositions for the singular vectors of eCk−1 and eCk in terms of V k−1: eVk−1 = V k−1Pk−1 + Qk−1Rk−1 (20a) eVk = V k−1Pk (20b) where Pj = V T k−1 eVj ∈Rd×d, for j = k −1, k, projects the columns of eVj onto the columns of V k−1. The second term of (20a), Qk−1Rk−1, is the QR-decomposition of the residual components 6 of eVk−1 after projection onto range(V k−1). Since Ck−1 is an average of positive semi-definite matrices that includes eCk, we are sure that range(eVk) ⊆range(V k−1), (supplementary material). Using the matrix decompositions (20a) and (20b), Ck in (19) above can be expressed as: Ck = V k−1 Qk−1 G V k−1 Qk−1 T (21) where G is the symmetric (d + d) × (d + d) matrix: G = Σk−1 0 0 0 + 1 Ns −1( " Pk−1eΣk−1P T k−1 Pk−1eΣk−1RT k−1 Rk−1eΣk−1P T k−1 Rk−1eΣk−1RT k−1 # − Pk eΣkP T k 0 0 0 ) (22) We now compute the SVD of G = VGΣGV T G . Then, using (21), we obtain the SVD for Ck as: V k = V k−1 Qk−1 VG and Σk = ΣG (23) For a moderate number of subjects, (d + d) ≤Nsd ≪Nv, this approach is more efficient than a brute-force O(N 3 v ) SVD. Additionally, it works directly on the singular values eΣk and vectors eVk of each warped connectivity matrix eCk, alleviating the need to store large Nv × Nv matrices. 5 Experimental Results We tested the algorithm using fMRI data collected from 10 subjects viewing a movie split into 2 sessions separated by a short break. The data was preprocessed following [5]. For each subject, a structural scan was acquired before each session, from which the cortical surface model was derived (§2) and then anatomically aligned to a template using FreeSurfer (Fischl, http://surfer.nmr.mgh.harvard.edu). Similar to [5], we find that anatomical alignment based on cortical curvature serves as a superior starting point for functional alignment over Talairach alignment. First, functional connectivity was found for each subject and session: Ck,i, k = 1, . . . , Ns, i = 1, 2. These were then aligned within subjects, Ck,1 ↔Ck,2, and across subjects, Ck,1 ↔Cj,2, using Algorithm 1. Since the data starts in anatomical correspondence, we expect small warp displacements within subject and larger ones across subjects. The mean intra-subject warp displacement was 0.72 mm (σ = 0.48), with 77% of the mesh nodes warped less than 1 mm and fewer than 1.5% warped by more than the data spatial resolution (2 mm). In contrast, the mean inter-subject warp displacement was 1.46 mm (σ = 0.92 mm), with 22% of nodes warped more than 2 mm. See Figures 2(a)-(b). In a separate analysis, each subject was aligned to its leave-one-out template on each session using Algorithm 1, yielding a set of warps gk,i(pj), k = 1, . . . , Ns, i = 1, 2, j = 1, . . . , Nv. To evaluate the consistency of the correspondence derived from different sessions, we compared the warps gk,1 to gk,2 for each subject k. Here, we only consider nodes that are warped by at least the data resolution. This analysis provides a measure of the sensitivity to noise present in the fMRI data. At node pj, we compute the angle 0 ≤θ ≤π between the warp tangent vectors of gk,1(pj) and gk,2(pj). This measures the consistency of the direction of the warp across sessions: smaller values of θ suggest a greater warp coherence across sessions. Figure 2(c) shows a histogram of θ averaged across the cortical nodes of all 10 subjects. The tight distribution centered near θ = 0 suggests significant consistency in the warp direction across sessions. In particular, 93% of the density for θ lies inside π/2, 81% inside π/4, and 58% inside π/8. As a secondary comparison, we compute a normalized consistency measure WNC(pj) = d(gk,1(pj), gk,2(pj))/(d(gk,1(pj), pj) + d(gk,2(pj), pj)), where d(·, ·) is spherical geodesic distance. The measure takes variability in both warp angle and magnitude into account; it is bounded between 0 and 1, and WNC(pj) = 0 only if gk,1(pj) = gk,2(pj). A histogram for WNC is given in 2(d); WNC exhibits a peak at 0.15, with a mean of 0.28 (σ = 0.22). Finally, Algorithm 2 was applied to the first session fMRI data to learn a set of warps g = (g1, . . . , gNs) for 10 subjects. The alignment required approximately 10 hours on a Intel 3.8GHz Nehalem quad-core processor with 12GB RAM. To evaluate the alignment, we apply the warps to the held out second session fMRI data, where subjects viewed a different segment of the movie. This warping yields data {f k gk(pi)} for each subject k, with interpolation performed in the original volume to avoid artificial smoothing. The cross-validated inter-subject correlation ISC(pi) is the mean 7 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Warp Distance (mm) Frequency (a) 0 1 2 3 4 5 6 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Warp Distance (mm) Frequency (b) 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 Angle between tangent vectors (radians/!) Frequency (c) 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1 Normalized consistency Frequency (d) Figure 2: Consistency Histograms. (a) Intra-subject warp distances; (b) Inter-subject warp distances; (c) Angle between warp vectors across sessions; (d) Across-session normalized warp consistency measure WNC. (a) Lateral View (b) Medial View (c) Ventral View (d) Lateral View (e) Medial View (f) Ventral View Figure 3: Map of ISC on right cortical hemisphere, alignment: anatomical (top), functional (bottom). correlation of each subject’s functional time series with the mean time series of the other subjects: ISC(pi) = (1/Ns) PNs k=1 corr(f k gk(pi), P n̸=k f n gn(pi)), pi ∈Mref (24) We also compute the mean inter-subject correlation, ISC = (1/Nv) PNv i=1 ISC(pi). We compare the cross-validated ISC map with the ISC map of the second session movie viewing computed under anatomical correspondence. Mean ISC improved by 18%, from 0.072 to 0.085. In addition, the number of significant inter-subject correlations (ISC(pi) > 0.1, P < 0.01) increased by 22.9%, from 19, 362 to 23, 789. Figure 3 shows the ISC maps computed under anatomical alignment and functional alignment on the inflated right cortical hemisphere. As expected, the areas of improvement in inter-subject correlation are consistent with the extrinsic regions of cortex [6]. 6 Conclusion We have proposed a novel cortical registration algorithm that produces a functional correspondence across a set of subjects. The algorithm uses the fMRI data directly to align the spatial patterns of functional response elicited by a movie viewing. Despite the high-dimensionality of the data under consideration, the algorithm is efficient in both space and time complexity. By comparing the inter-subject alignments derived from different fMRI experimental sessions, we show that the correspondence is consistent and robust to noise and variability in the fMRI temporal response. We also cross-validate the correspondence on independent test data that was not used to derive the alignment. On the test data, the algorithm produces a consistent increase in intersubject correlation of fMRI time series, suggesting that functional alignment of extrinsic regions of cortex that are directly driven by the movie viewing experiment, such as visual and auditory areas, is improved considerably. Further testing is warranted to evaluate improvement in intrinsic areas of cortex whose response is not temporally synchronized with the experimental stimulus. 8 References [1] J. Talairach and P. Tournoux. Co-planar Stereotaxic Atlas of the Human Brain. Thieme Publishing Group, 1988. [2] B. Fischl, R.B.H. Tootell, and A.M. Dale. High-resolution intersubject averaging and a coordinate system for the cortical surface. Human Brain Mapping, 8:272–284, 1999. [3] J.D.G. Watson, R. Myers, R.S.F. Frackowiak, J.V. Hajnal, R.P. Woods, J.C. Mazziotta, S. Shipp, and S. Zeki. Area v5 of the human brain: evidence from a combined study using positron emission tomography and magnetic resonance imaging. Cerebral Cortex, 3:79–94, 1993. [4] J. Rademacher, V.S. Caviness, H. Steinmetz, and A.M. Galaburda. Topographical variation of the human primary cortices: implications for neuroimaging, brain mapping and neurobiology. Cerebral Cortex, 3:313–329, 1995. [5] M.R. Sabuncu, B.D. Singer, B. Conroy, R.E. Bryan, P.J. Ramadge, and J.V. Haxby. Functionbased inter-subject alignment of human cortical anatomy. Cerebral Cortex Advance Access published on May 6, 2009, DOI 10.1093/cercor/bhp085. [6] U. Hasson, Y. Nir, G. Fuhrmann, and R. Malach. Intersubject synchronization of cortical activity during natural vision. Science, 303:1634–1640, 2004. [7] Y. Golland, S. Bentin, H. Gelbard, Y. Benjamini, R. Heller, Y. Nir, U. Hasson, and R. Malach. Extrinsic and intrinsic systems in the posterior cortex of the human brain revealed during natural sensory stimulation. Cerebral Cortex, 17:766–777, 2007. [8] M.E. Raichle, A.M. MacLeod, A.Z. Snyder, W.J. Powers, D.A. Gusnard, and G.L. Shulman. A default mode of brain function. PNAS, 98:676–682, 2001. [9] K.J. Friston. Functional and effective connectivity in neuroimaging. Human Brain Mapping, 2:56–78, 1994. [10] Michael D. Greicius, Ben Krasnow, Allan L. Reiss, and Vinod Menon. Functional connectivity in the resting brain: A network analysis of the default mode hypothesis. PNAS, 100:253–258, 2003. [11] J.L. Vincent, A.Z. Snyder, M.D. Fox, B.J. Shannon, J.R. Andrews, M.E. Raichle, and R.L. Buckner. Coherent spontaneous activity identifies a hippocampal-parietal memory network. J. Neurophysiol, 96:3517–3531, 2006. [12] D.C. Van Essen, H.A. Drury, J. Dickson, J. Harwell, D. Hanlon, and C.H. Anderson. An integrated software suite for surface-based analyses of cerebral cortex. J. Am. Med. Inform. Assoc., 8:443–459, 2001. [13] A.M. Dale, B. Fischl, and M.I. Sereno. Cortical surface-based analysis. i. segmentation and surface reconstruction. NeuroImage, 9:179–194, 1999. [14] B. Fischl, M.I. Sereno, and A.M. Dale. Cortical surface-based analysis. ii. inflation, flattening, and a surface-based coordinate system. NeuroImage, 9:195–207, 1999. [15] G.J. McLachlan and T. Krishnan. The EM Algorithm and Extensions. Wiley, 1997. [16] G.E. Fasshauer and L.L. Schumaker. Scattered data fitting on the sphere. Proceedings of the international conference on mathematical methods for curves and surfaces II, pages 117–166, 1998. [17] B.A. Ardekani, A.H. Bachman, S.C. Strother, Y. Fujibayashi, and Y. Yonekura. Impact of inter-subject image registration on group analysis of fmri data. International Congress Series, 1265:49–59, 2004. [18] M. Do Carmo. Differential Geometry of Curves and Surfaces. Prentice Hall, 1976. [19] R. Bajcsy and S. Kovacic. Multiresolution elastic matching. Computer Vision, Graphics, and Image Processing, 46:1–21, 1989. 9
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Unsupervised feature learning for audio classification using convolutional deep belief networks Honglak Lee Yan Largman Peter Pham Andrew Y. Ng Computer Science Department Stanford University Stanford, CA 94305 Abstract In recent years, deep learning approaches have gained significant interest as a way of building hierarchical representations from unlabeled data. However, to our knowledge, these deep learning approaches have not been extensively studied for auditory data. In this paper, we apply convolutional deep belief networks to audio data and empirically evaluate them on various audio classification tasks. In the case of speech data, we show that the learned features correspond to phones/phonemes. In addition, our feature representations learned from unlabeled audio data show very good performance for multiple audio classification tasks. We hope that this paper will inspire more research on deep learning approaches applied to a wide range of audio recognition tasks. 1 Introduction Understanding how to recognize complex, high-dimensional audio data is one of the greatest challenges of our time. Previous work [1, 2] revealed that learning a sparse representation of auditory signals leads to filters that closely correspond to those of neurons in early audio processing in mammals. For example, when sparse coding models are applied to natural sounds or speech, the learned representations (basis vectors) showed a striking resemblance to the cochlear filters in the auditory cortex. In related work, Grosse et al. [3] proposed an efficient sparse coding algorithm for auditory signals and demonstrated its usefulness in audio classification tasks. However, the proposed methods have been applied to learn relatively shallow, one-layer representations. Learning more complex, higher-level representation is still a non-trivial, challenging problem. Recently, many promising approaches have been proposed to learn the processing steps of the “second stage and beyond” [4, 5, 6, 7, 8]. These “deep learning” algorithms try to learn simple features in the lower layers and more complex features in the higher layers. However, to the best of our knowledge, these “deep learning” approaches have not been extensively applied to auditory data. The deep belief network [4] is a generative probabilistic model composed of one visible (observed) layer and many hidden layers. Each hidden layer unit learns a statistical relationship between the units in the lower layer; the higher layer representations tend to become more complex. The deep belief network can be efficiently trained using greedy layerwise training, in which the hidden layers are trained one at a time in a bottom-up fashion [4]. Recently, convolutional deep belief networks [9] have been developed to scale up the algorithm to high-dimensional data. Similar to deep belief networks, convolutional deep belief networks can be trained in a greedy, bottom-up fashion. By applying these networks to images, Lee et al. (2009) showed good performance in several visual recognition tasks [9]. In this paper, we will apply convolutional deep belief networks to unlabeled auditory data (such as speech and music) and evaluate the learned feature representations on several audio classification tasks. In the case of speech data, we show that the learned features correspond to phones/phonemes. In addition, our feature representations outperform other baseline features (spectrogram and MFCC) 1 for multiple audio classification tasks. In particular, our method compares favorably with other stateof-the-art algorithms for the speaker identification task. For the phone classification task, MFCC features can be augmented with our features to improve accuracy. We also show for certain tasks that the second-layer features produce higher accuracy than the first-layer features, which justifies the use of deep learning approaches for audio classification. Finally, we show that our features give better performance in comparison to other baseline features for music classification tasks. In our experiments, the learned features often performed much better than other baseline features when there was only a small number of labeled training examples. To the best of our knowledge, we are the first to apply deep learning algorithms to a range of audio classification tasks. We hope that this paper will inspire more research on deep learning approaches applied to audio recognition tasks. 2 Algorithms 2.1 Convolutional deep belief networks We first briefly review convolutional restricted Boltzmann machines (CRBMs) [9, 10, 11] as building blocks for convolutional deep belief networks (CDBNs). We will follow the formulation of [9] and adapt it to a one dimensional setting. For the purpose of this explanation, we assume that all inputs to the algorithm are single-channel time-series data with nV frames (an nV dimensional vector); however, the formulation can be straightforwardly extended to the case of multiple channels. The CRBM is an extension of the “regular” RBM [4] to a convolutional setting, in which the weights between the hidden units and the visible units are shared among all locations in the hidden layer. The CRBM consists of two layers: an input (visible) layer V and a hidden layer H. The hidden units are binary-valued, and the visible units are binary-valued or real-valued. Consider the input layer consisting of an nV dimensional array of binary units. To construct the hidden layer, consider K nW -dimensional filter weights W K (also referred to as “bases” throughout this paper). The hidden layer consists of K “groups” of nH-dimensional arrays (where nH ≜ nV −nW + 1) with units in group k sharing the weights W k. There is also a shared bias bk for each group and a shared bias c for the visible units. The energy function can then be defined as: E(v, h) = − K X k=1 nH X j=1 nW X r=1 hk j W k r vj+r−1 − K X k=1 bk nH X j=1 hk j −c nV X i=1 vi. (1) Similarly, the energy function of CRBM with real-valued visible units can be defined as: E(v, h) = 1 2 nV X i v2 i − K X k=1 nH X j=1 nW X r=1 hk j W k r vj+r−1 − K X k=1 bk nH X j=1 hk j −c nV X i=1 vi. (2) The joint and conditional probability distributions are defined as follows: P(v, h) = 1 Z exp(−E(v, h)) (3) P(hk j = 1|v) = sigmoid(( ˜W k ∗v v)j + bk) (4) P(vi = 1|h) = sigmoid( X k (W k ∗f hk)i + c) (for binary visible units) (5) P(vi|h) = Normal( X k (W k ∗f hk)i + c, 1) (for real visible units), (6) where ∗v is a “valid” convolution, ∗f is a “full” convolution,1 and ˜W k j ≜W k nW −j+1. Since all units in one layer are conditionally independent given the other layer, inference in the network can be efficiently performed using block Gibbs sampling. Lee et al. [9] further developed a convolutional RBM with “probabilistic max-pooling,” where the maxima over small neighborhoods of hidden units are computed in a probabilistically sound way. (See [9] for more details.) In this paper, we use CRBMs with probabilistic max-pooling as building blocks for convolutional deep belief networks. 1Given an m-dimensional vector and an n-dimensional kernel (where m > n), valid convolution gives a (m −n + 1)-dimensional vector, and full convolution gives a (m + n −1)-dimensional vector. 2 For training the convolutional RBMs, computing the exact gradient for the log-likelihood term is intractable. However, contrastive divergence [12] can be used to approximate the gradient effectively. Since a typical CRBM is highly overcomplete, a sparsity penalty term is added to the log-likelihood objective [8, 9]. More specifically, the training objective can be written as minimizeW,b,c Llikelihood(W, b, c) + Lsparsity(W, b, c), (7) where Llikelihood is a negative log-likelihood that measures how well the CRBM approximates the input data distribution, and Lsparsity is a penalty term that constrains the hidden units to having sparse average activations. This sparsity regularization can be viewed as limiting the “capacity” of the network, and it often results in more easily interpretable feature representations. Once the parameters for all the layers are trained, we stack the CRBMs to form a convolutional deep belief network. For inference, we use feed-forward approximation. 2.2 Application to audio data For the application of CDBNs to audio data, we first convert time-domain signals into spectrograms. However, the dimensionality of the spectrograms is large (e.g., 160 channels). We apply PCA whitening to the spectrograms and create lower dimensional representations. Thus, the data we feed into the CDBN consists of nc channels of one-dimensional vectors of length nV , where nc is the number of PCA components in our representation. Similarly, the first-layer bases are comprised of nc channels of one-dimensional filters of length nW . 3 Unsupervised feature learning 3.1 Training on unlabeled TIMIT data We trained the first and second-layer CDBN representations using a large, unlabeled speech dataset. First, we extracted the spectrogram from each utterance of the TIMIT training data [13]. The spectrogram had a 20 ms window size with 10 ms overlaps. The spectrogram was further processed using PCA whitening (with 80 components) to reduce the dimensionality. We then trained 300 first-layer bases with a filter length (nW ) of 6 and a max-pooling ratio (local neighborhood size) of 3. We further trained 300 second-layer bases using the max-pooled first-layer activations as input, again with a filter length of 6 and a max-pooling ratio of 3. 3.2 Visualization In this section, we illustrate what the network “learns” through visualization. We visualize the firstlayer bases by multiplying the inverse of the PCA whitening on each first-layer basis (Figure 1). Each second-layer basis is visualized as a weighted linear combination of the first-layer bases. high freq. low freq. Figure 1: Visualization of randomly selected first-layer CDBN bases trained on the TIMIT data. Each column represents a “temporal receptive field” of a first-layer basis in the spectrogram space. The frequency channels are ordered from the lowest frequency (bottom) to the highest frequency (top). All figures in the paper are best viewed in color. 3.2.1 Phonemes and the CDBN features In Figure 2, we show how our bases relate to phonemes by comparing visualizations of each phoneme with the bases that are most activated by that phoneme. For each phoneme, we show five spectrograms of sound clips of that phoneme (top five columns in each phoneme group), and the five first-layer bases with the highest average activations on the given phoneme (bottom five columns in each phoneme group). Many of the first-layer bases closely match the shapes of phonemes. There are prominent horizontal bands in the lower frequencies of the firstlayer bases that respond most to vowels (for example, “ah” and “oy”). The bases that respond most 3 Example phones ("ah") Example phones ("oy") Example phones ("el") Example phones ("s") First layer bases First layer bases First layer bases First layer bases Figure 2: Visualization of the four different phonemes and their corresponding first-layer CDBN bases. For each phoneme: (top) the spectrograms of the five randomly selected phones; (bottom) five first-layer bases with the highest average activations on the given phoneme. to fricatives (for example, “s”) typically take the form of widely distributed areas of energy in the high frequencies of the spectrogram. Both of these patterns reflect the structure of the corresponding phoneme spectrograms. Closer inspection of the bases provides slight evidence that the first-layer bases also capture more fine-grained details. For example, the first and third “oy” bases reflect the upward-slanting pattern in the phoneme spectrograms. The top “el” bases mirror the intensity patterns of the corresponding phoneme spectrograms: a high intensity region appears in the lowest frequencies, and another region of lesser intensity appears a bit higher up. 3.2.2 Speaker gender information and the CDBN features In Figure 3, we show an analysis of two-layer CDBN feature representations with respect to the gender classification task (Section 4.2). Note that the network was trained on unlabeled data; therefore, no information about speaker gender was given during training. Example phones (female) First layer bases ("female") Second layer bases ("female") Example phones (male) First layer bases ("male") Second layer bases ("male") Figure 3: (Left) five spectrogram samples of “ae” phoneme from female (top)/male (bottom) speakers. (Middle) Visualization of the five first-layer bases that most differentially activate for female/male speakers. (Right) Visualization of the five second-layer bases that most differentially activate for female/male speakers. For comparison with the CDBN features, randomly selected spectrograms of female (top left five columns) and male (bottom left five columns) pronunciations of the “ae” phoneme from the TIMIT dataset are shown. Spectrograms for the female pronunciations are qualitatively distinguishable by a finer horizontal banding pattern in low frequencies, whereas male pronunciations have more blurred 4 patterns. This gender difference in the vowel pronunciation patterns is typical across the TIMIT data. Only the bases that are most biased to activate on either male or female speech are shown. The bases that are most active on female speech encode the horizontal band pattern that is prominent in the spectrograms of female pronunciations. On the other hand, the male-biased bases have more blurred patterns, which again visually matches the corresponding spectrograms. 4 Application to speech recognition tasks In this section, we demonstrate that the CDBN feature representations learned from the unlabeled speech corpus can be useful for multiple speech recognition tasks, such as speaker identification, gender classification, and phone classification. In most of our experiments, we followed the selftaught learning framework [14]. The motivation for self-taught learning comes from situations where we are given only a small amount of labeled data and a large amount of unlabeled data;2 therefore, one of our main interests was to evaluate the different feature representations given a small number of labeled training examples (as often assumed in self-taught learning or semi-supervised learning settings). More specifically, we trained the CDBN on unlabeled TIMIT data (as described in Section 3.1); then we used the CDBN features for classification on labeled training/test data3 that were randomly selected from the TIMIT corpus.4 4.1 Speaker identification We evaluated the usefulness of the learned CDBN representations for the speaker identification task. The subset of the TIMIT corpus that we used for speaker identification has 168 speakers and 10 utterances (sentences) per speaker, resulting in a total of 1680 utterances. We performed 168-way classification on this set. For each number of utterances per speaker, we randomly selected training utterances and testing utterances and measured the classification accuracy; we report the results averaged over 10 random trials.5 To construct training and test data for the classification task, we extracted a spectrogram from each utterance in the TIMIT corpus. We denote this spectrogram representation as “RAW” features. We computed the first and second-layer CDBN features using the spectrogram as input. We also computed MFCC features, widely-used standard features for generic speech recognition tasks. As a result, we obtained spectrogram/MFCC/CDBN representations for each utterance with multiple (typically, several hundred) frames. In our experiments, we used simple summary statistics (for each channel) such as average, max, or standard deviation over all the frames. We evaluated the features using standard supervised classifiers, such as SVM, GDA, and KNN. The choices of summary statistics and hyperparameters for the classifiers were done using crossvalidation. We report the average classification accuracy (over 10 random trials) with a varying number of training examples. Table 1 shows the average classification accuracy for each feature representation. The results show that the first and second CDBN representations both outperform baseline features (RAW and MFCC). The numbers compare MFCC and CDBN features with as many of the same factors (such as preprocessing and classification algorithms) as possible. Further, to make a fair comparison between CDBN features and MFCC, we used the best performing implementation6 among several standard implementations for MFCC. Our results suggest that without special preprocessing or postprocess2In self-taught learning, the labeled data and unlabeled data don’t need to share the same labels or the same generative distributions. 3There are two disjoint TIMIT data sets. We drew unlabeled data from the larger of the two for unsupervised feature learning, and we drew labeled data from the other data set to create our training and test set for the classification tasks. 4In the case of phone classification, we followed the standard protocol (e.g., [15]) rather than self-taught learning framework to evaluate our algorithm in comparison to other methods. 5Details: There were some exceptions to this; for the case of eight training utterances, we followed Reynolds (1995) [16]; more specifically, we used eight training utterances (2 sa sentences, 3 si sentences and first 3 sx sentences); the two testing utterances were the remaining 2 sx sentences. We used cross validation for selecting hyperparameters for classification, except for the case of 1 utterance per speaker, where we used a randomly selected validation sentence per speaker. 6We used Dan Ellis’ implementation available at: http://labrosa.ee.columbia.edu/matlab/ rastamat. 5 Table 1: Test classification accuracy for speaker identification using summary statistics #training utterances per speaker RAW MFCC CDBN L1 CDBN L2 CDBN L1+L2 1 46.7% 54.4% 74.5% 62.8% 72.8% 2 43.5% 69.9% 76.7% 66.2% 76.7% 3 67.9% 76.5% 91.3% 84.3% 91.8% 5 80.6% 82.6% 93.7% 89.6% 93.8% 8 90.4% 92.0% 97.9% 95.2% 97.0% Table 2: Test classification accuracy for speaker identification using all frames #training utterances per speaker MFCC ([16]’s method) CDBN MFCC ([16]) + CDBN 1 40.2% 90.0% 90.7% 2 87.9% 97.9% 98.7% 3 95.9% 98.7% 99.2% 5 99.2% 99.2% 99.6% 8 99.7% 99.7% 100.0% ing (besides the summary statistics which were needed to reduce the number of features), the CDBN features outperform MFCC features, especially in a setting with a very limited number of labeled examples. We further experimented to determine if the CDBN features can achieve competitive performance in comparison to other more sophisticated, state-of-the-art methods. For each feature representation, we used the classifier that achieved the highest performance. More specifically, for the MFCC features we replicated Reynolds (1995)’s method,7 and for the CDBN features we used a SVM based ensemble method.8 As shown in Table 2, the CDBN features consistently outperformed MFCC features when the number of training examples was small. We also combined both methods by taking a linear combination of the two classifier outputs (before taking the final classification prediction from each algorithm).9 The resulting combined classifier performed the best, achieving 100% accuracy for the case of 8 training utterances per speaker. 4.2 Speaker gender classification We also evaluated the same CDBN features which were learned for the speaker identification task on the gender classification task. We report the classification accuracy for various quantities of training examples (utterances) per gender. For each number of training examples, we randomly sampled training examples and 200 testing examples; we report the test classification accuracy averaged over 20 trials. As shown in Table 3, both the first and second CDBN features outperformed the baseline features, especially when the number of training examples were small. The second-layer CDBN features consistently performed better than the first-layer CDBN features. This suggests that the second-layer representation learned more invariant features that are relevant for speaker gender classification, justifying the use of “deep” architectures. 4.3 Phone classification Finally, we evaluated our learned representation on phone classification tasks. For this experiment, we treated each phone segment as an individual example and computed the spectrogram (RAW) and MFCC features for each phone segment. Similarly, we computed the first-layer CDBN representations. Following the standard protocol [15], we report the 39 way phone classification accuracy on the test data (TIMIT core test set) for various numbers of training sentences. For each number of training examples, we report the average classification accuracy over 5 random trials. The summary 7Details: In [16], MFCC features (with multiple frames) were computed for each utterance; then a Gaussian mixture model was trained for each speaker (treating each individual MFCC frame as a input example to the GMM. For the a given test utterance, the prediction was made by determining the GMM model that had the highest test log-likelihood. 8In detail, we treated each single-frame CDBN features as an individual example. Then, we trained a multiclass linear SVM for these individual frames. For testing, we computed SVM prediction score for each speaker, and then aggregated predictions from all the frames. Overall, the highest scoring speaker was selected for the prediction. 9The constant for the linear combination was fixed across all the numbers of training utterances, and it was selected using cross validation. 6 Table 3: Test accuracy for gender classification problem #training utterances per gender RAW MFCC CDBN L1 CDBN L2 CDBN L1+L2 1 68.4% 58.5% 78.5% 85.8% 83.6% 2 76.7% 78.7% 86.0% 92.5% 92.3% 3 79.5% 84.1% 88.9% 94.2% 94.2% 5 84.4% 86.9% 93.1% 95.8% 95.6% 7 89.2% 89.0% 94.2% 96.6% 96.5% 10 91.3% 89.8% 94.7% 96.7% 96.6% Table 4: Test accuracy for phone classification problem #training utterances RAW MFCC MFCC ([15]’s method) CDBN L1 MFCC+CDBN L1 ([15]) 100 36.9% 58.3% 66.6% 53.7% 67.2% 200 37.8% 61.5% 70.3% 56.7% 71.0% 500 38.7% 64.9% 74.1% 59.7% 75.1% 1000 39.0% 67.2% 76.3% 61.6% 77.1% 2000 39.2% 69.2% 78.4% 63.1% 79.2% 3696 39.4% 70.8% 79.6% 64.4% 80.3% results are shown in Table 4. In this experiment, the first-layer CDBN features performed better than spectrogram features, but they did not outperform the MFCC features. However, by combining MFCC features and CDBN features, we could achieve about 0.7% accuracy improvement consistently over all the numbers of training utterances. In the realm of phone classification, in which significant research effort is often needed to achieve even improvements well under a percent, this is a significant improvement. [17, 18, 19, 20] This suggests that the first-layer CDBN features learned somewhat informative features for phone classification tasks in an unsupervised way. In contrast to the gender classification task, the secondlayer CDBN features did not offer much improvement over the first-layer CDBN features. This result is not unexpected considering the fact that the time-scale of most phonemes roughly corresponds to the time-scale of the first-layer CDBN features. 5 Application to music classification tasks In this section, we assess the applicability of CDBN features to various music classification tasks. Table 5: Test accuracy for 5-way music genre classification Train examples RAW MFCC CDBN L1 CDBN L2 CDBN L1+L2 1 51.6% 54.0% 66.1% 62.5% 64.3% 2 57.0% 62.1% 69.7% 67.9% 69.5% 3 59.7% 65.3% 70.0% 66.7% 69.5% 5 65.8% 68.3% 73.1% 69.2% 72.7% 5.1 Music genre classification For the task of music genre classification, we trained the first and second-layer CDBN representations on an unlabeled collection of music data.10 First, we computed the spectrogram (20 ms window size with 10 ms overlaps) representation for individual songs. The spectrogram was PCA-whitened and then fed into the CDBN as input data. We trained 300 first-layer bases with a filter length of 10 and a max-pooling ratio of 3. In addition, we trained 300 second-layer bases with a filter length of 10 and a max-pooling ratio of 3. We evaluated the learned CDBN representation for 5-way genre classification tasks. The training and test songs for the classification tasks were randomly sampled from 5 genres (classical, electric, jazz, pop, and rock) and did not overlap with the unlabeled data. We randomly sampled 3-second segments from each song and treated each segment as an individual training or testing example. We report the classification accuracy for various numbers of training examples. For each number of training examples, we averaged over 20 random trials. The results are shown in Table 5. In this task, the first-layer CDBN features performed the best overall. 10Available from http://ismir2004.ismir.net/ISMIR_Contest.html. 7 5.2 Music artist classification Furthermore, we evaluated whether the CDBN features are useful in identifying individual artists.11 Following the same procedure as in Section 5.1, we trained the first and second-layer CDBN representations from an unlabeled collection of classical music data. Some representative bases are shown in Figure 4. Then we evaluated the learned CDBN representation for 4-way artist identification tasks. The disjoint sets of training and test songs for the classification tasks were randomly sampled from the songs of four artists. The unlabeled data and the labeled data did not include the same artists. We randomly sampled 3-second segments from each song and treated each segment as an individual example. We report the classification accuracy for various quantities of training examples. For each number of training examples, we averaged over 20 random trials. The results are shown in Table 6. The results show that both the first and second-layer CDBN features performed better than the baseline features, and that either using the second-layer features only or combining the first and the second-layer features yielded the best results. This suggests that the second-layer CDBN representation might have captured somewhat useful, higher-level features than the first-layer CDBN representation. high freq. low freq. Figure 4: Visualization of randomly selected first-layer CDBN bases trained on classical music data. Table 6: Test accuracy for 4-way artist identification Train examples RAW MFCC CDBN L1 CDBN L2 CDBN L1+L2 1 56.0% 63.7% 67.6% 67.7% 69.2% 2 69.4% 66.1% 76.1% 74.2% 76.3% 3 73.9% 67.9% 78.0% 75.8% 78.7% 5 79.4% 71.6% 80.9% 81.9% 81.4% 6 Discussion and conclusion Modern speech datasets are much larger than the TIMIT dataset. While the challenge of larger datasets often lies in considering harder tasks, our objective in using the TIMIT data was to restrict the amount of labeled data our algorithm had to learn from. It remains an interesting problem to apply deep learning to larger datasets and more challenging tasks. In this paper, we applied convolutional deep belief networks to audio data and evaluated on various audio classification tasks. By leveraging a large amount of unlabeled data, our learned features often equaled or surpassed MFCC features, which are hand-tailored to audio data. Furthermore, even when our features did not outperform MFCC, we could achieve higher classification accuracy by combining both. Also, our results show that a single CDBN feature representation can achieve high performance on multiple audio recognition tasks. We hope that our approach will inspire more research on automatically learning deep feature hierarchies for audio data. Acknowledgment We thank Yoshua Bengio, Dan Jurafsky, Yun-Hsuan Sung, Pedro Moreno, Roger Grosse for helpful discussions. We also thank anonymous reviewers for their constructive comments. This work was supported in part by the National Science Foundation under grant EFRI-0835878, and in part by the Office of Naval Research under MURI N000140710747. 11In our experiments, we found that artist identification task was more difficult than the speaker identification task because the local sound patterns can be highly variable even for the same artist. 8 References [1] E. C. Smith and M. S. Lewicki. Efficient auditory coding. Nature, 439:978–982, 2006. [2] B. A. Olshausen and D. J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381:607–609, 1996. [3] R. Grosse, R. Raina, H. Kwong, and A.Y. Ng. Shift-invariant sparse coding for audio classification. In UAI, 2007. [4] G. E. Hinton, S. Osindero, and Y.-W. Teh. A fast learning algorithm for deep belief nets. Neural Computation, 18(7):1527–1554, 2006. [5] M. Ranzato, C. Poultney, S. Chopra, and Y. LeCun. Efficient learning of sparse representations with an energy-based model. In NIPS, 2006. [6] Y. Bengio, P. Lamblin, D. Popovici, and H. Larochelle. Greedy layer-wise training of deep networks. In NIPS, 2006. [7] H. Larochelle, D. Erhan, A. Courville, J. Bergstra, and Y. Bengio. An empirical evaluation of deep architectures on problems with many factors of variation. In ICML, 2007. [8] H. Lee, C. Ekanadham, and A. Y. Ng. Sparse deep belief network model for visual area V2. In NIPS, 2008. [9] H. Lee, R. Grosse, R. Ranganath, and A. Y. Ng. Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. In ICML, 2009. [10] G. Desjardins and Y. Bengio. Empirical evaluation of convolutional RBMs for vision. Technical report, 2008. [11] M. Norouzi, M. Ranjbar, and G. Mori. Stacks of convolutional restricted boltzmann machines for shift-invariant feature learning. In CVPR, 2009. [12] G. E. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14:1771–1800, 2002. [13] W. Fisher, G. Doddington, and K. Goudie-Marshall. The darpa speech recognition research database: Specifications and status. In DARPA Speech Recognition Workshop, 1986. [14] R. Raina, A. Battle, H. Lee, B. Packer, and A. Y. Ng. Self-taught learning: Transfer learning from unlabeled data. In ICML, 2007. [15] P. Clarkson and P. J. Moreno. On the use of support vector machines for phonetic classification. In ICASSP99, pages 585–588, 1999. [16] D. A. Reynolds. Speaker identification and verification using gaussian mixture speaker models. Speech Commun., 17(1-2):91–108, 1995. [17] F. Sha and L. K. Saul. Large margin gaussian mixture modeling for phonetic classication and recognition. In ICASSP’06, 2006. [18] Y.-H. Sung, C. Boulis, C. Manning, and D. Jurafsky. Regularization, adaptation, and nonindependent features improve hidden conditional random fields for phone classification. In IEEE ASRU, 2007. [19] S. Petrov, A. Pauls, and D. Klein. Learning structured models for phone recognition. In EMNLP-CoNLL, 2007. [20] D. Yu, L. Deng, and A. Acero. Hidden conditional random field with distribution constraints for phone classification. In Interspeech, 2009. 9
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Accelerating Bayesian Structural Inference for Non-Decomposable Gaussian Graphical Models Baback Moghaddam Jet Propulsion Laboratory California Institute of Technology baback@jpl.nasa.gov Benjamin M. Marlin Department of Computer Science University of British Columbia bmarlin@cs.ubc.ca Mohammad Emtiyaz Khan Department of Computer Science University of British Columbia emtiyaz@cs.ubc.ca Kevin P. Murphy Department of Computer Science University of British Columbia murphyk@cs.ubc.ca Abstract We make several contributions in accelerating approximate Bayesian structural inference for non-decomposable GGMs. Our first contribution is to show how to efficiently compute a BIC or Laplace approximation to the marginal likelihood of non-decomposable graphs using convex methods for precision matrix estimation. This optimization technique can be used as a fast scoring function inside standard Stochastic Local Search (SLS) for generating posterior samples. Our second contribution is a novel framework for efficiently generating large sets of high-quality graph topologies without performing local search. This graph proposal method, which we call “Neighborhood Fusion” (NF), samples candidate Markov blankets at each node using sparse regression techniques. Our third contribution is a hybrid method combining the complementary strengths of NF and SLS. Experimental results in structural recovery and prediction tasks demonstrate that NF and hybrid NF/SLS out-perform state-of-the-art local search methods, on both synthetic and real-world datasets, when realistic computational limits are imposed. 1 Introduction There are two main reasons to learn the structure of graphical models: knowledge discovery (to interpret the learned topology) and density estimation (to compute log-likelihoods and make predictions). The main difficulty in graphical model structure learning is that the hypothesis space is extremely large, containing up to 2d(d−1)/2 graphs on d nodes. When the sample size n is small, there can be significant uncertainty with respect to the graph structure. It is therefore advantageous to adopt a Bayesian approach and maintain an approximate posterior over graphs instead of using a single “best” graph, especially since Bayesian model averaging (BMA) can improve predictions. There has been much work on Bayesian inference for directed acyclic graphical model (DAG) structure, mostly based on Markov chain Monte Carlo (MCMC) or stochastic local search (SLS) [22, 19, 16, 14]. MCMC and SLS methods for DAGs exploit the important fact that the marginal likelihood of a DAG, or an approximation such as the Bayesian Information Criterion (BIC) score, can be computed very efficiently under standard assumptions including independent conjugate priors, and complete data. An equally important property in the DAG setting is that the score can be quickly updated when small local changes are made to the graph. This conveniently allows one to move rapidly through the very large graph space of DAGs. However, for knowledge discovery, a DAG may be an unsuitable representation for several reasons. First, it does not allow directed cycles, which may be an unnatural restriction in certain domains. Second, DAGs can only be identified up to Markov equivalence in the general case. In contrast, undirected graphs (UGs) avoid these issues and may be a more natural representation for some problems. Also, for UGs there are fast methods available for identifying the local connectivity at each node (the node’s Markov blanket). We note that while the UG and DAG representations have different properties and enable different inference and structure learning algorithms, the distinction between UGs and DAGs from a density estimation perspective may be less important [12]. Most prior work on Bayesian inference for Gaussian Graphical Models (GGMs) has focused on the special case of decomposable graphs (e.g., [17, 2, 29]). The popularity of decomposable GGMs is mostly due to the fact that one can compute the marginal likelihood in closed form using similar assumptions to the DAG case. In addition, one can update the marginal likelihood in constant time after single-edge moves in graph space [17]. However, the space of decomposable graphs is much smaller than the space of general undirected graphs. For example, the number of decomposable graphs on d nodes for d = 2, . . . , 8 is 2, 8, 61, 822, 18154, 617675, 30888596 [1, p.158]. If we divide the number of decomposable graphs by the number of general undirected graphs, we get the “volume” ratios: 1, 1, 0.95, 0.80, 0.55, 0.29, 0.12. This means that decomposability significantly limits the subclass of UGs available for modeling purposes, even for small d. Several authors have studied Bayesian inference for GGM structure in the general case using approximations to the marginal likelihood based on Monte Carlo methods (e.g., [8, 31, 20, 3]). However, these methods cannot scale to large graphs because of the high computational cost of Monte Carlo approximation. In this paper, we propose several techniques to help accelerate approximate Bayesian structural inference for non-decomposable GGMs. In Section 2, we show how to efficiently compute BIC and Laplace approximations to the marginal likelihood p(D|G) by using recent convex optimization methods for estimating the precision matrix of a GGM. In Section 3, we present a novel framework for generating large sets of high-quality graphs which we call “Neighborhood Fusion” (NF). This framework is quite general in scope and can use any Markov blanket finding method to devise a set of probability distributions (proposal densities) over the local topology at each node. It then specifies rules for “fusing” these local densities (via sampling) into an approximate posterior over whole graphs p(G|D). In Section 4, we combine the complementary strengths of NF and existing SLS methods to obtain even higher quality posterior distributions in certain cases. In Section 5, we present an empirical evaluation of both knowledge discovery and predictive performance of our methods. For knowledge discovery, we measure structural recovery in terms of accuracy of finding true edges in synthetic GGMs (with known structure). For predictive performance, we evaluate test set log-likelihood as well as missing-data imputation on real data (with unknown structure). We show that the proposed NF and hybrid NF/SLS methods for general graphs outperform current approaches to GGM learning for both decomposable and general (non-decomposable) graphs. Throughout this paper we will view the marginal likelihood p(D|G) as the key to structural inference and as being equivalent to the graph posterior p(G|D) by adopting a flat structural prior p(G) w.l.o.g. 2 Marginal Likelihood for General Graphs In this section we will review the G-Wishart distribution and discuss approximations to the marginal likelihood of a non-decomposable GGM under the G-Wishart prior. Unlike the decomposable case, here the marginal likelihood can not be found in closed form. Our main contribution is the insight that recently proposed convex optimization methods for precision matrix estimation can be used to efficiently find the mode of a G-Wishart distribution, which in turn allows for more efficient computation of BIC and Laplace modal approximations to the marginal likelihood. We begin with some notation. We define n to be the number of data cases and d to be the number of data dimensions. We denote the ith data case by xi and a complete data set D with the n × d matrix X, with the corresponding scatter matrix S = XT X (we assume centered data). We use G to denote an undirected graph, or more precisely its adjacency matrix. Graph edges are denoted by unordered pairs (i, j) and the edge (i, j) is in the graph G if Gij = 1. The space of all positive definite matrices having the same zero-pattern as G is denoted by S++ G . The covariance matrix is denoted by Σ and its inverse or the precision matrix by Ω= Σ−1. We also define ⟨A, B⟩= Trace(AB). The Gaussian likelihood p(D|Ω) is expressed in terms of the data scatter matrix S in Equation 1. We denote the prior distribution over precision matrices given a graph G by p(Ω|G). The standard measure of model quality in the Bayesian model selection setting is the marginal likelihood p(D|G) which is obtained by integrating p(D|Ω)p(Ω|G) over the space S++ G as shown in Equation 2. p(D|Ω) = n Y i=1 N(xi| 0, Ω−1) ∝|Ω|n/2 exp(−1 2⟨Ω, S⟩) (1) p(D|G) = Z S++ G p(D|Ω) p(Ω|G) dΩ (2) The G-Wishart density in Equation 3 is the Diaconis-Ylvisaker conjugate form [10] for the GGM likelihood as shown in [27]. The indicator function I[Ω∈S++ G ] in Equation 3 restricts the density’s support to S++ G . The G-Wishart generalizes the hyper inverse Wishart (HIW) distribution to general non-decomposable graphs. The G-Wishart normalization constant Z is shown in Equation 4. W(Ω|G, δ0, S0) = I[Ω∈S++ G ] Z(G, δ0, S0) |Ω|(δ0−2)/2 exp(−1 2⟨Ω, S0⟩) (3) Z(G, δ0, S0) = Z S++ G |Ω|(δ0−2)/2 exp(−1 2⟨Ω, S0⟩) dΩ (4) p(D|G) = Z S++ G p(D|Ω) W(Ω|G, δ0, S0) dΩ ∝ Z(G, δn, Sn) Z(G, δ0, S0) (5) Because of the conjugate prior in Equation 3, the Ωposterior has a similar form W(Ω|G, δn, Sn) where δn = δ0 + n is the posterior degrees of freedom and the posterior scatter matrix Sn = S + S0. The resulting marginal likelihood is then the ratio of the two normalizing terms shown in Equation 5 (which we refer to as Zn and Z0 for short). The main drawback of the G-Wishart for general graphs, compared to the HIW for decomposable graphs, is that one cannot compute the normalization terms Zn and Z0 in closed form. As a result, Bayesian model selection for non-decomposable GGMs relies on approximating the marginal likelihood p(D|G). The existing literature focuses on Monte Carlo and Laplace approximations. One strategy that makes use of Monte Carlo estimates of both Zn and Z0 is given by [3]. However, the computation time required to find accurate estimates can be extremely high [20] (see Section 6). An effective approximation strategy based on using a Laplace approximation to Zn and a Monte Carlo approximation to Z0 is given in [21]. This requires finding the mode of the G-Wishart, with which a closed-form expression for the Hessian is derived [21]. We consider a simpler method which applies the Laplace approximation to both Zn and Z0 for greater speed, which we call full-Laplace. Nevertheless, computing the Hessian determinant has a computational complexity of O(E3), where E is the number of edges in G. Since E = O(d2) in the worst-case scenario, computing a full Hessian determinant becomes infeasible for large d in all but the sparsest of graphs. Due to the high computational cost of Monte Carlo and Laplace approximation in high dimensions, we consider two alternative marginal likelihood approximations that are significantly more efficient. The first alternative is to approximate Zn and Z0 by Laplace computations in which the Hessian matrix is replaced by its diagonal (by setting off-diagonal elements to zero). We refer to this method as the diagonal-Laplace score. The other alternative is the Bayesian Information Criterion (BIC) score shown in Equation 6, which is another large-sample Laplace approximation BIC(G) = log p(D|ˆΩG) −1 2 dof(G) log n , dof(G) = d + X i<j Gij (6) where, by analogy to [34], we define the GGM’s degrees-of-freedom (dof) to be the number of free parameters in the precision matrix. For BIC we use the G-Wishart posterior mode ˆΩG as the plug-in estimate, since the MLE is undefined for n < d. But we use a vague and proper prior (δ0 = 3). Therefore, all three approximations will require finding the mode of a G-Wishart (for the posterior and/or the prior). In [21] an Iterative Proportional Scaling (IPS) algorithm [30] is proposed to find the G-Wishart mode. However, IPS requires finding the maximal cliques of the graph, which is an NP-hard problem. We will now derive a much more efficient G-Wishart mode-finder using convex optimization techniques. We apply this method to find ˆΩG when computing BIC scores, as well as the prior and posterior G-Wishart modes when computing Laplace approximations to Z0 and Zn. Observe that we can express the mode of any G-Wishart distribution with the optimization problem in Equation 7, where the density is parameterized by graph G, degree δ and the scatter matrix S. ˆΩG = arg max Ω∈S++ G log W(Ω|G, δ, S) = arg min Ω∈S++ G −log |Ω| + Ω, S δ −2 (7) This “COVSEL” type problem [9] is equivalent to finding the maximum likelihood precision matrix of a GGM with known structure G, and is a convex optimization problem. Several new methods for solving this precision estimation problem have been recently proposed, and unlike IPS they do not require computing the clique structure of the underlying graph. Hastie et al. [18] present one such method which consists of iteratively solving a series of least square problems on the free elements of the precision matrix, which has O(Pd i (P j̸=i Gij)3) complexity per iteration [18, p.634]. The G-Wishart mode in Equation 7 can also be found more directly with a gradient-based optimizer such as L-BFGS [6], by using the implementation convention that the objective function is ∞for a non-positive definite matrix. This technique has been used previously by Duchi et al. for the more difficult problem of ℓ1 penalized precision matrix estimation [13]. The gradient of the objective function is simply set to (−Ω−1 + S) ⊙G, where ⊙indicates element-wise multiplication. The elements of the precision matrix corresponding to absent edges in G are fixed to zero, and we optimize over the remaining elements. The complexity per iteration is O(d3). In practice, initializing the above optimization with the output of few iterations of the block coordinate descent method of [18] (Glasso with known G) is quite effective, as it requires fewer subsequent L-BFGS steps. In Section 5 we explore the speed vs. accuracy trade-off of the various marginal likelihood approximation schemes discussed above; comparing full-Laplace, diagonal-Laplace and the BIC score functions to the marginal likelihood values obtained with the Monte Carlo method of [3]. 3 Neighborhood Fusion In this section we describe a novel framework we call “Neighborhood Fusion” (NF) for generating an approximate posterior distribution p(G|D) over general graphs. An important advantage of working with general graphs, instead of decomposable graphs, is that we can leverage simple and stable methods for quickly exploring Markov blankets. One popular method for structural recovery is Glasso which imposes an l1 penalty on Ω[4, 15, 32]. Finding the corresponding graph takes O(d3) time per iteration for each setting of the regularization parameter λ. However, the choice of the λ parameter is critical, and in practice we often find that no setting of this parameter leads to good recovery. A related approach, proposed in [23], uses l1-regularized linear regression or Lasso to identify the Markov blanket (MB) of each node. These Markov blankets are then combined using intersection or union (AND/OR) to give the global graph G. These methods essentially produce a single “best” graph, but our main interest is in approximating the full posterior p(G|D). Our NF framework uses a Markov blanket finding method to derive a set of probability distributions over the local topology at each node, and specifies a rule for combining these into an approximate posterior over graphs. The detailed steps of the generic NF algorithm are: 1. Regress each node i on all others to find neighborhoods of all cardinalities k = 0 : d −1 using a sparse regression method. Denote the set of Markov blankets for node i by Ni 2. Compute the linear regression scores s(b) for each Markov blanket b in Ni, and define pi(b) = exp(s(b))/(P b′∈Ni exp(s(b′))) as the node’s Markov blanket proposal density 3. Independently sample a Markov blanket for each node i from its proposal density pi(b), and then combine all d sampled Markov blankets to assemble a single graph G 4. Find G’s precision matrix using Equation 7 and compute the graph score as in Section 2 5. Repeat sampling step 3 and 4 to produce a large ensemble of posterior-weighted graphs The design choices in the NF framework are the choice of a sparse linear regression method (and its score function), the choice of a method for combining Markov blankets, and the choice of a graph score function (for marginal likelihood). In all the results that follow we use the linear regression BIC score induced by regressing node i on Ni, and generate whole graphs by intersecting the Markov blankets using the AND operator. This essentially constitutes sampling from the “ANDcensored” pseudo marginal likelihood and is therefore likely to produce good candidate MBs that can be fused into high-quality graphs. Note that the uncertainty modeled by the MB proposal density is critical, as it promotes efficient exploration of model space to generate a large variety of highscoring models. Indeed, the best NF-sampled graphs typically have higher scores than the pseudo “MAP” graph obtained by simply intersecting the best MBs [23], due to the inherent noise in the linear regression BIC scores and the possibility of over-fitting. Moreover, our MB proposals can be “flattened” with a temperature parameter to trade-off exploration vs. fidelity of the sampled graphs, though we generally find it unnecessary to go to such extremes and use a default temperature of one. We next consider two further specialized instances of the NF framework using different sparse linear regression methods. The first method uses the full Lasso/LARS regularization path and is called L1MB (“L1 Markov Blanket”) which we adapted from the DAG-learning method of [28]. NF based on these l1-derived MBs we call NF-L1MB (or NF-L1 for short). In light of recent theoretical results on the superiority of greedy forward/backward search over Lasso [33] we also use the l0-based method of [24] which we call L0MB (“L0 Markov blanket”). And NF based on L0MB we will call NF-L0MB (or NF-L0 for short). Our experimental results show that the improvement of the l0-based greedy search of [24] over Lasso/LARS translates directly to obtaining improved MB proposals with NF-L0MB compared to NF-L1MB. Similar forward/backward greedy variable selection techniques were put to good use in the “compositional network” DAG-to-UG method of [11], however not for deriving proposal distributions for parents/MBs as we do here for NF. Our overall computational scheme is quite fast by design: finding MB proposals is at most O(d4) with L1MB/L0MB (although L0MB has a smaller constant for both the forward and backward passes). Thereafter, we sample full graphs in O(d2) time (since we are sampling a discrete p.m.f. for d MB candidates at each node) and computing a G-Wishart mode ˆΩG is just O(d3) per iteration. 4 Stochastic Local Search Stochastic Local Search (SLS) can also be viewed as a mechanism for generating an approximate posterior distribution over graphs. Like MCMC methods, SLS explores high probability regions of graph space, but unlike MCMC it computes approximate model probabilities directly for each graph it visits. This is sensible for large discrete hypothesis spaces like the space of UGs since the chance of visiting the same graph multiple times is extremely small. We note that SLS represents an orthogonal and complementary approach to structural inference relative to the NF framework presented in Section 3. In this section we discuss SLS for both decomposable and general (non-decomposable) GGMs. Specifically, we describe new initialization and edge-marginal updating methods for nondecomposable GGMs, and also introduce a highly effective hybrid NF/SLS method. SLS with decomposable graphs has the advantage that its natural scoring function, the marginal likelihood, can be computed exactly under the conjugate Hyper Inverse Wishart prior. The marginal likelihood can also be updated efficiently when local changes are made to the underlying graph. A state-of-the-art SLS method for decomposable GGMs is given in [29], which can be used with an arbitrary score function over the space of general graphs. Here we consider SLS for general graphs using the Laplace score described in Section 2. In the SLS in [29], at iteration t, an edge (i, j) from Gt is chosen at random and flipped with probability qij. If the resulting graph is admissible and has not been visited before, this graph becomes Gt+1, and we evaluate its score. In the general case, every new graph generated is admissible. In the decomposable case, only decomposable graphs are admissible. We should note that unlike exhaustive search methods, this method avoids evaluating the score of all O(d2) neighboring graphs at each iteration, and instead picks one at random. There are two key modifications used in [29] which help this method work well in practice. First, the marginal edge probabilities qij are updated online, so edges that have proved useful in the past are more likely to be proposed in the future. Second, on each iteration the algorithm chooses to perform a resampling step with probability pr or a global move with probability pg. In a resampling step we set Gt+1 to Gv, where v ≤t, with probability proportional to the score (or exponentiated score) of Gv. In a global move we sample a completely new graph (based on the edge marginals qij) for Gt+1. We note that a similar idea of using edge-marginals to propose moves in DAG space was suggested in [14]. In this paper, we set pr = 0.02 and pg = 0 (i.e., we do not use global moves). 6 8 10 12 14 16 0 0.1 0.2 0.3 0.4 0.5 0.6 Dimension (d) KL (bits) BIC Laplace Diag Laplace (a) Posterior Error Dimension (d) Time (sec) 6 8 10 12 14 16 10−3 10−2 10−1 1 BIC Laplace Diag Laplace MC (b) Score CPU Time 1000 2000 3000 4000 5000 9150 9200 9250 9300 9350 9400 9450 9500 9550 DLS−T GLS−T GLS−NF NF−L0 (5k) NF−L1 (5k) NF−L0 (100) C−L Tree Iteration Score (c) MF Score Trace Figure 1: Score trade-offs: (a) average KL error of posterior approximations and (b) the average time to score a single graph as a function of data dimensionality. (c) Results on the MF dataset: scores for various methods. We now propose a new initialization and updating scheme for non-decomposable SLS based on a set of k initial graphs G1 0, ..., Gk 0 (with positive weights w1 0, ..., wk 0 defined by normalized scores) obtained from our NF graph-sampling framework. Our approach views qij as a Beta random variable with prior parameters αij = Pk l=1 wl 0Gl 0,ij and βij = Pk l=1 wl 0(1 −Gl 0,ij). We update this distribution online using p(qij|G1:t) = Beta(αij +tf t ij , βij +t(1−f t ij)), where f t ij = Pt l=1 Gl ij p(D|Gl) Pt l=1 p(D|Gl) . We then flip an edge with probability E[qij] = (αij + tf t ij)/(αij + βij + t). SLS’s main drawback is that, if started from the empty graph as in [29], it will necessarily take at least E steps to find the highest scoring graph, where E is the number of true edges. This means that it will likely require a very large number of iterations even in moderately large and dense graphs. An improved initialization strategy is to start the search from the optimal tree, which can be found in O(d2) time using the Chow-Liu algorithm [7]. An even better initialization strategy, for nondecomposable graphs, is to “seed” SLS with a batch of NF-sampled graphs for G1 0, ..., Gk 0 and then start the search by executing a resampling step. In this way, a limited number of SLS steps can effectively explore the space around these initial high-quality graphs. We refer to this new method, where NF is used to both initialize the edge-marginals and seed the graph history, as hybrid NF/SLS. 5 Experiments We begin our experimental analysis by first assessing the speed vs. accuracy trade-off of the different marginal likelihood approximations in Section 2. For this evaluation we use the Monte Carlo method of [3] as a proxy for the ground truth marginal likelihood. For data dimensions d = 6, ..., 16, we sample 100 random, sparse precision matrices with an average edge density of 0.5. For each sampled precision matrix Ωwe generate 10d observations from the corresponding GGM. Using each approximation method, we score all d(d −1)/2 neighbors of G obtained from G by single edge flips. We then compute a posterior distribution over this set of graphs by normalizing the scores (or exponentiated scores as appropriate). We then compute the Kullback-Leibler (KL) divergence from the Monte Carlo based posterior to each approximate posterior. We also record the time required to score each graph. The scoring methods we use are BIC, full-Laplace and diagonal-Laplace for Zn and Z0. We use a G-Wishart prior with parameters δ0 = 3 and S0 = I. In Figure 1(a) we show the average error of these posterior approximations as a function of data dimensionality d, as measured by KL divergence. In Figure 1(b) we show the average time required to score a single graph as a function of graph size d. As expected, full-Laplace is the most accurate and most costly of the approximations next to Monte Carlo. Interestingly, diagonal-Laplace appears to be significantly more accurate than BIC (for this test) and is in fact only twice as costly. Moreover, diagonalLaplace is already more than 20 times faster than Monte Carlo and full-Laplace at d = 16. On the basis of the speed vs. accuracy trade-off seen in Figure 1(a) and Figure 1(b), we will report only the diagonal-Laplace score in the remainder of our experiments. We next evaluate the NF-L1MB and NF-L0MB methods described in Section 3 (note that we will use the short labels NF-L1 and NF-L0 in the Figures), and SLS for decomposable and general graphs 9100 9150 9200 9250 9300 9350 9400 9450 9500 9550 DLS−T GLS−T GLS−NF NF−L0 NF−L1 (a) Diagonal-Laplace Score 4500 4550 4600 4650 4700 DLS−T GLS−T GLS−NF NF−L0 NF−L1 (b) Test log-likelihood 242 244 246 248 250 252 254 256 DLS−T GLS−T GLS−NF NF−L0 NF−L1 (c) Imputed log-likelihood Figure 2: Mutual Fund results: box plots of the (a) scores, (b) test set log-likelihoods and (c) test set imputation log-likelihoods (averaged over all possible missing 3-nodes). The BMA performance is indicated with a circle. initialized from the optimal tree as described in Section 4 (denoted as DLS-T and GLS-T, respectively), and a L0MB-based hybrid NF/SLS method as described in Section 4 (denoted as GLS-NF). We sample 5000 graphs for each of the NF methods and run each of the SLS methods for 5000 steps, also producing 5000 graphs. The hybrid NF/SLS method is initialized with a sample of 100 NF graphs, and then run for 5000 steps. We compute the score for each set of graphs (diagonal-Laplace for non-decomposable and exact marginal likelihood for decomposables). We extract the 100 best graphs by score, and produce an approximation to p(G|D) by normalizing the exponentiated scores. We report results for individual graphs in the best 100, but our main focus is on performance statistics under Bayesian model averaging (BMA) with approximate scores of each method. In the following experiments we use a G-Wishart prior degree δ0 = 3 (the smallest integer yielding a proper prior) and unless otherwise noted, a default prior scatter matrix of S0 = mean(diag(cov(X))) · Id. We examine the two main inferential tasks of prediction and knowledge discovery. We first measure the predictive ability of each method by computing both test set log-likelihoods and test set imputation log-likelihoods. For this task we use the “Mutual Funds” (MF) dataset used by [29] for SLS with decomposable GGMs, with d = 59, which they split into 60 months of training data and 26 months of test data. But due to the resulting critical sampling (n ≈d), here we use a more stable S0 = ρ · Diag(XT X) with ρ = 0.055 (a Ledoit-Wolf shrinkage). In Figure 1(c) we show a trace plot of scores for the SLS methods and best scores for the NF and tree methods. Box plots of diagonalLaplace scores for each method on the MF data are shown in Figure 2(a). The corresponding test set log-likelihoods are shown in Figure 2(b). For the imputation experiment, we impute “missing” triplets of variables given the values of the remaining variables. We compute the log-likelihood of this predictive (imputed) distribution by averaging it over all 59-choose-3 = 32509 possible missing patterns and all 26 test cases. The imputation log-likelihoods are shown in Figure 2(c). We can see that NF-L0MB out-performs NF-L1MB on both predictive tasks (full and missing). Interestingly, on this small data set SLS for general graphs (GLS-T) performs rather well. But our hybrid NF-L0MB “seeding” approach for SLS (GLS-NF) has the best overall BMA performance. In the second set of tasks, we evaluate the structural recovery of each method by measuring the true positive and false positive rates for edge inclusion w.r.t. a ground-truth GGM. The synthetic data sets contain d = 100 nodes, E = 300 edges and n/d ratios of 5.0 (Synth-1) and 0.5 (Synth-2). Synth-1 is thus generously oversampled while Synth-2 is undersampled. Both synthetic GGMs were generated by moralizing a random DAG. Figures 3(a) and 3(b) show plots of TPR vs. FPR for edge recovery. The rates for individual graphs are shown as small grey symbols while the BMA rate is shown with a large bold colored symbol. The results show that NF-L0MB and GLS-NF (based on seeding GLS with 100 NF-L0MB graphs) are the best methods on both data sets. We also see that NF-L0MB dominates NF-L1MB, while the hybrid GLS-NF dominates both GLS-T and DLS-T. For the d = 59 MF dataset in Figure 1(c), NF-sampling 5000 graphs and doing the G-Wishart modefits and diagonal-Laplace scoring, takes a total of 13 mins, and likewise 30 mins for the synthetic d = 100 dataset in Figure 3. Generating and scoring 5000 graphs with non-decomposable SLS takes 37 mins on the MF dataset and 59 mins on the synthetic one. Decomposable SLS takes 31 mins on MF and 43 mins on the synthetic. All times quoted are for Matlab code running on a 3.16 GHz PC. 0 0.2 0.4 0.6 0.8 1 TPR FPR 10−4 10−3 10−2 10−1 1 DLS−T GLS−T GLS−NF NF−L0 NF−L1 (a) Synth-1: n/d = 5.0 (d = 100) 0.3 0.35 0.4 0.45 0.5 0.55 TPR FPR 10−2 10−1 1 DLS−T GLS−T GLS−NF NF−L0 NF−L1 (b) Synth-2: n/d = 0.5 (d = 100) Figure 3: True Positive vs. False Positive rates for (a) Synth-1 and (b) Synth-2 datasets for each recovery method. The top 100 graphs are shown with a grey symbol and the bold colored symbol is the BMA graph. 6 Discussion We offer a practical framework for fast inference in non-decomposable GGMs providing reasonable accuracy for marginal likelihoods. While Monte Carlo methods are the “gold standard” (modulo the usual convergence issues) they are exorbitantly costly for even moderately large d. For example, scoring all the neighbors of a 150-node graph via SLS required over 40 days of computation in [20]. A similar size task would take less than 40 mins with our diagonal-Laplace approximation method. As pointed out by [21] there may not always be sufficient concentration for a Laplace approximation to Z0 to be very accurate, which is why they use MC for this quantity. We chose Laplace for both Zn and Z0 solely for speed (to avoid MC altogether) and found good agreement between full-Laplace and BIC for much larger graphs than in Figure 1(a). Our Laplace scores also roughly matched the MC values for the Fisher Iris data in [3], selecting essentially the same top-ranked 16 graphs (see Figure 5 in [3]). Using a diagonal instead of a full Hessian was yet another compromise for speed. An issue that should be explored further is the sensitivity of these approximations to different priors. We experimentally validated NF on nearly 104 synthetic cases ranging in size from d = 10, ..., 500, with various edge densities and n/d ratios, with consistently good results, typified by the two test cases shown in Figure 3. Note that the sub-par performance of NF-L1 is not a failing of NF but due to l1-based MBs, and superiority of l0-based F/B greedy search is not without precedent [25, 24, 33]. We note that NF can be partially justified as a pseudo marginal likelihood (PML), but whereas most authors rely only on its maximizer [23] we exploit the full (pseudo) density. Without the AND filter, NF-drawn MBs are sampled from a set of “consistent” full-conditionals in the sense of Besag [5], and their max-BIC MBs are collectively the PML mode (note that here we mean the node regression BIC, not graph BIC). Enforcing AND is a necessary domain truncation for a valid UG which alters the mode. This symmetrized “pseudo-MAP” G is often an average-scoring one compared to the best and worst found by NF, which motivates BMA and justifies NF. We can also view NF as an overdispersed proposal density; its weighted graphs a rough proxy for p(G|D). This approximation may be biased but our results show it is quite useful for prediction and imputation (and seeding SLS with high-quality graphs). Finally, while use of BIC/Laplace for hypothesis testing is often criticized, it can still be useful for estimation [26], and nowhere in our framework are these scores being used to select a single “best” model (whether it be a MB or a G) due to our reliance on sampling and BMA. Acknowledgments We like to thank the reviewers for their helpful and encouraging feedback. BMM was supported by the Killam Trusts at UBC and KPM would like to thank NSERC and CIFAR. This work was in part carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. References [1] H. Armstrong. Bayesian Estimation of Decomposable GGMs. PhD thesis, UNSW, 2005. [2] H. Armstrong, C. Carter, K. Wong, and R. Kohn. Bayesian covariance matrix estimation using a mixture of decomposable graphical models. Statistics and Computing, 2008. [3] A. Atay-Kayis and H. Massam. A Monte Carlo method for computing the marginal likelihood in nondecomposable Gaussian graphical models. Biometrika, 92, 2005. [4] O. Banerjee, L. El Ghaoui, A. d’Aspremont, and G. Natsoulis. Convex optimization techniques for fitting sparse Gaussian graphical models. In Intl. Conf. on Machine Learning, 2006. [5] J. Besag. Efficiency of pseudo-likelihood estimation for simple Gaussian fields. Biometrika, 1977. [6] R. Byrd, P. Lu, J. Nocedal, and C. Zhu. A limited memory algorithm for bound constrained optimization. SIAM J. of Scientific & Statistical Computing, 16(5), 1995. [7] C. Chow and C. Liu. Approximating discrete probability distributions with dependence trees. IEEE Trans. on Info. Theory, 14, 1968. [8] P. Dellaportas, P. Giudici, and G. Roberts. Bayesian inference for nondecomposable graphical Gaussian models. Sankhya, Ser. A, 65, 2003. [9] A. Dempster. Covariance selection. Biometrics, 28(1), 1972. [10] P. Diaconis and D. Ylvisaker. Conjugate priors for exponential families. Annals of statistics, 7(2), 1979. [11] D. Dobra, C. Hans, B. Jones, J. Nevins, G. Yao, and M. West. Sparse graphical models for exploring gene expression data. J. Multivariate analysis, 90, 2004. [12] J. Domke, A. Karapurkar, and Y. Aloimonos. Who killed the directed model? In CVPR, 2008. [13] J. Duchi, S. Gould, and D. Koller. Projected subgradients for learning sparse Gaussians. In UAI, 2008. [14] D. Eaton and K. Murphy. Bayesian structure learning using DP and MCMC. In UAI, 2007. [15] J. Friedman, T. Hastie, and R. Tibshirani. Sparse inverse covariance estimation in Glasso. Biostats, 2007. [16] N. Friedman and D. Koller. Being Bayesian about network structure: A Bayesian approach to structure discovery in Bayesian networks. Machine Learning, 50, 2003. [17] P. Giudici and P. Green. Decomposable graphical Gaussian model determination. Biometrika, 1999. [18] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning. Springer, 2009. [19] D. Heckerman, D. Geiger, and M. Chickering. Learning Bayesian networks: the combination of knowledge and statistical data. Machine Learning, 20(3), 1995. [20] B. Jones, C. Carvalho, A. Dobra, C. Hans, C. Carter, and M. West. Experiments in stochastic computation for high-dimensional graphical models. Statistical Science, 20, 2005. [21] A. Lenkoski and A. Dobra. Bayesian structural learning and estimation in Gaussian graphical models. Technical Report 545, Department of Statistics, University of Washington, 2008. [22] D. Madigan and A. Raftery. Model selection and accounting for model uncertainty in graphical models using Occam’s window. J. of the Am. Stat. Assoc., 89, 1994. [23] N. Meinshausen and P. Buhlmann. High dimensional graphs and variable selection with the Lasso. The Annals of Statistics, 2006. [24] B. Moghaddam, A. Gruber, Y. Weiss, and S. Avidan. Sparse regression as a sparse eigenvalue problem. In Information Theory & Applications Workshop (ITA’08), 2008. [25] B. Moghaddam, Y. Weiss, and S. Avidan. Spectral bounds for sparse PCA: Exact & greedy algorithms. In NIPS, 2006. [26] A. Raftery. Bayesian model selection in social research. Sociological Methodology, 25, 1995. [27] A. Roverato. Hyper inverse Wishart distribution for non-decomposable graphs and its application to Bayesian inference for Gaussian graphical models. Scand. J. Statistics, 29, 2002. [28] M. Schmidt, A Niculescu-Mizil, and K Murphy. Learning graphical model structure using l1 regularization paths. In AAAI, 2007. [29] J. Scott and C. Carvalho. Feature-inclusion stochastic search for Gaussian graphical models. J. of Computational and Graphical Statistics, 17(4), 2008. [30] T. Speed and H. Kiiveri. Gaussian Markov distributions over finite graphs. Annals of Statistics, 1986. [31] F. Wong, C. Carter, and R. Kohn. Efficient estimation of covariance selection models. Biometrika, 2003. [32] M. Yuan and Yi Lin. Model selection and estimation in the GGM. Biometrika, 94(1), 2007. [33] T. Zhang. Adaptive forward-backward greedy algorithm for sparse learning. In NIPS, 2008. [34] H. Zou, T. Hastie, and R. Tibshirani. On the ”degrees of freedom” of Lasso. Annals of Statistics, 2007.
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Learning transport operators for image manifolds Benjamin J. Culpepper Department of EECS Computer Science Division University of California, Berkeley Berkeley, CA 94720 bjc@cs.berkeley.edu Bruno A. Olshausen Helen Wills Neuroscience Institute & School of Optometry University of California, Berkeley Berkeley, CA 94720 baolshausen@berkeley.edu Abstract We describe an unsupervised manifold learning algorithm that represents a surface through a compact description of operators that traverse it. The operators are based on matrix exponentials, which are the solution to a system of first-order linear differential equations. The matrix exponents are represented by a basis that is adapted to the statistics of the data so that the infinitesimal generator for a trajectory along the underlying manifold can be produced by linearly composing a few elements. The method is applied to recover topological structure from low dimensional synthetic data, and to model local structure in how natural images change over time and scale. 1 Introduction It is well known that natural images occupy a small fraction of the space of all possible images. Moreover, as images change over time in response to observer motion or changes in the environment they trace out particular trajectories along manifolds in this space. It is reasonable to expect that perceptual systems have evolved ways to efficiently model these manifolds, and thus mathematical models that capture their structure in operators that transport along them may be of use for understanding perceptual systems, as well as for engineering artificial vision systems. In this paper, we derive methods for learning these transport operators from data. Rather than simply learning a mapping of individual data points to a low-dimensional space, we seek a compact representation of the entire manifold via the operators that traverse it. We investigate a direct application of the Lie approach to invariance [1] utilizing a matrix exponential generative model for transforming images. This is in contrast to previous methods that rely mainly upon a firstorder Taylor series approximation of the matrix exponential [2,3], and bilinear models, in which the transformation variables interact multiplicatively with the input [4,5,6]. It is also distinct from the class of methods that learn embeddings of manifolds from point cloud data [7,8,9,10]. The spirit of this work is similar to [11], which also uses a spectral decomposition to make learning tractable in extremely high dimensional Lie groups, such as those over images. We share the goal of [12] of learning a model of the manifold which can then be generalized to new data. Here we show how a particular class of transport operators for moving along manifolds may be learned from data. The model is first applied to synthetic datasets to demonstrate interesting cases where it can recover topology, and that for more difficult cases it neatly approximates the local structure. Subsequently, we apply it to time-varying natural images and extrapolate along inferred trajectories to demonstrate super-resolution and temporal filling-in of missing video frames. 1 2 Problem formulation Let us consider an image of the visual world at time t as a point x ∈RN, where the elements of x correspond to image pixels. We describe the evolution of x as ˙x = A x , (1) where the matrix A is a linear operator capturing some action in the environment that transforms the image. Such an action belongs to a family that occupies a subspace of RN×N given by A = M X m=1 Ψm cm (2) for some M ≤N 2 (usually M << N 2), with Ψm ∈RN×N. The amount of a particular action from the dictionary Ψm that occurs is controlled by the corresponding cm. At t = 0, a vision system takes an image x0, and then makes repeated observations at intervals ∆t. Given x0, the solution to (1) traces out a continuously differentiable manifold of images given by xt = exp(At) x0, which we observe periodically. Our goal is to learn an appropriate set of bases, Ψ, that allow for a compact description of this set of transformations by training on many pairs of related observations. This generative model for transformed images has a number of attractive properties. First, it factors apart the time-varying image into an invariant part (the initial image, x0) and variant part (the transformation, parameterized by the coefficient vector c), thus making explicit the underlying causes. Second, the learned exponential operators are quite powerful in terms of modeling capacity, compared to their linear counterparts. Lastly, the partial derivatives of the objective function have a simple form that may be computed efficiently. 3 Algorithm The model parameters are learned by maximizing the log-likelihood of the model. Consider two ‘close’ states of the system in isolation. Let x0 be our initial condition, and x1 be a second observation. These points are related through an exponentiated matrix that itself is composed of a few basis elements, plus zero-mean white i.i.d. Gaussian noise, n: x1 = T(c) x0 + n (3) T(c) = exp( X m Ψm cm) . (4) We assume a factorial sparse prior over the transform variables c of the form P(cm) ∝ exp(−ζ |cm|). The negative log of the posterior probability of the data under the model is given by E = 1 2||x1 −T(c) x0||2 2 + γ 2 X m ||Ψm||2 F + ζ||c||1 , (5) where || · ||F is the Frobenius norm, which acts to regularize the dictionary element lengths. The 1-norm encourages sparsity. Given two data points, the solution of the c variables which relate them through Ψ is found by a fast minimization of E with respect to c. Learning of the basis Ψ proceeds by gradient descent with respect to E. (Note that this constitutes a variational approximation to the log-likelihood, similar to [13].) The Ψ variables are initialized randomly, and adjusted according to ∆Ψ = −η ∂E ∂Ψ, using the solution, c, for a pair of observations x0, x1. Figure 1 outlines the steps of the algorithm. The partial derivatives of E w.r.t. c and Ψ can be cast in a simple form using the spectral decomposition of A, given by P α λαuαvT α, with right eigenvectors uα, left eigenvectors vα, and eigenvalues λα [14]. Let U = [u1u2...uN], V = [v1v2...vN] and D be a diagonal matrix of the eigenvalues λα. Then ∂exp(A)ij ∂Akl = X αβ FαβUiαVkαUlβVjβ , (6) 2 1 choose M ≤N 2 2 initialize Ψ 3 while stopping criteria is not met, 4 pick x0, x1 5 initialize c to zeros 6 c ←arg minc E 7 ∆Ψ = −η ∂E ∂Ψ 8 sort Ψm by ||Ψm||F 9 M ←max m s.t. ||Ψm||F > ϵ Figure 1: Pseudo-code for the learning algorithm. Steps 1-2 initialize. A typical stopping criteria in step 3 is that the reconstruction error or sparsity on some held-out data falls below a threshold. Steps 4-6 compute an E-step on some pair of data points. Step 7 computes a ‘partial’ M-step. Steps 8-9 shrink the subspace spanned by the dictionary if one or more of the elements have shrunk sufficiently in norm. where the matrix F is given by: Fαβ = ( exp(λβ)−exp(λα) λβ−λα if λβ ̸= λα exp(λα) otherwise (7) Application of the chain rule and a re-arrangement of terms yields simplified forms for the partials of E w.r.t. c and Ψ. After computing two intermediate terms P and Q, P = UT(x1x0 T + x0x0 TTT)V (8) Qkl = X αβ VkαUlβFαβPαβ , (9) the two partial derivatives for inference and learning are: ∂E ∂cm = X kl Qkl Ψklm + ζ sgn(cm) (10) ∂E ∂Ψklm = Qkl cm + γ Ψklm . (11) The order of complexity for both derivatives is determined by the computation of Q, which requires an eigen-decomposition and a few matrix multiplications, giving O(N p) with 2 < p < 3. 4 Experiments on point sets We first test the model by applying it to simple datasets where the solutions are known: learning the topology of a sphere and a torus. Second, we apply the model to learn the manifold of timevarying responses to a natural movie from complex oriented filters. These demonstrations illustrate the algorithm’s capability for learning significant non-linear structure. We have also applied the model to the Klein bottle. Though closely related to the torus, it is an example of a low-dimensional surface whose topology can not be captured by a first-order Lie operator, though our model is able to interpolate between points on the surface using a piecewise approximation (see the supplementary material accompanying this paper for further discussion of this point). Related pairs of points on a torus are generated by choosing two angles θ0, φ0 uniformly at random from [0, 2π]; two related angles θ1, φ1 are produced by sampling from two von Mises distributions with means θ0 and φ0, and concentration κ = 5 using the circular statistics toolbox of [15]. For the sphere, we generate the first pair of angles using the normal-deviate method, to avoid concentration of samples near the poles. Though parameterized by two angles, the coordinates of points on these surfaces are 3- and 4-dimensional; pairs of points xt for t = 0, 1 on the unit sphere are given by xt = (sin θt cos φt, sin θt sin φt, cos θt), and points on a torus by xt = (cos θt, sin θt, cos φt, sin φt). 3 −1 0 1 −1 0 1 −1 −0.5 0 0.5 1 −1 0 1 −1 0 1 −1 −0.5 0 0.5 1 −1 0 1 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 (a) (b) Figure 2: Orbits of learned sphere operators. (a) Three Ψm basis elements applied to points at the six poles of the sphere, (1, 0, 0), (0, 1, 0), (0, 0, 1), (−1, 0, 0), (0, −1, 0), and (0, 0, −1). The orbits are generated by setting x0 to a pole, then plotting xt = exp(Ψm t) x0 for t = [−100, 100]. (b) When superimposed on top of each other, the three sets of orbits clearly define the surface of a sphere. −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x1 x2 x3 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x1 x2 x4 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x2 x3 x4 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x1 x2 x3 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x1 x2 x4 −1 0 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 x2 x3 x4 Figure 3: Orbits of learned torus operators. Each row shows three projections of a Ψm basis element applied to a point on the surface of the torus. The orbits shown are generated by setting x0 = (0, 1, 0, 1) then plotting xt = exp(Ψm t) x0 for t = [−1000, 1000] in projections constructed from each triplet of the four coordinates. In each plot, two coordinates always obey a circular relationship, while the third varies more freely. 4 Figure 4: Learning transformations of oriented filter pairs across time. The orbits of three three complex filter outputs in response to a natural movie. The blue points denote the complex output for each frame in the movie sequence and are linked to their neighbors via the blue line. The points circled in red were observed by the model, and the red curve shows an extrapolation along the estimated trajectory. For the sphere, N = 3, thus setting M = 9 gives the model the freedom to generate the full space of A operators. The Ψ are initialized to mean-zero white Gaussian noise with variance 0.01, and 10, 000 learning updates are computed by generating a pair of related points, minimizing E w.r.t. c, then updating Ψ according to ∆Ψ = −η ∂E ∂Ψ. In all of the point set experiments, γ = 0.0001 and ζ = 0.01. For cases where topology can be recovered, the solution is robust to the settings of γ and ζ – changing either variable by an order of magnitude does not change the solution, though it may increase the number of learning steps required to get to it. In cases where the topology can not be recovered, the influence on the solution of the settings of γ and ζ is more subtle, as their relative values effectively trade-off the importance of data reconstruction and the sparsity of the vector c. We adjust η during learning as follows: when ∆Ψ causes E to decrease, we multiply η by 1.01; otherwise, we multiply η by 0.99. When the model has more parameters than it needs to fully capture the topology of the sphere this fact is evident from the solution it learns: six of the dictionary elements Ψm drop out (they have norm less than 10−6), since the F-norm ‘weight decay’ term kills off dictionary elements that are used rarely. Figure 2 shows orbits produced by applying each of the remaining Ψm operators to points on the sphere. Similar experiments are successful for the torus; Figure 3 shows trajectories of the operators learned for the torus. As an intermediate step towards modeling time varying natural images, we investigate the model’s ability to learn the response surface for a single complex oriented filter to a moving image. A complex pyramid is built from each frame in a movie, and pairs of filter responses 1 to 4 frames apart are observed by the model. Four 2x2 basis functions are learned in the manner described above. Figure 4 shows three representative examples that illustrate how well the model is able to extrapolate from the solution estimated using the learned basis Ψ, and complex responses from the same filter within a 4 frame time interval. In most cases, this trajectory follows the data closely for several frames. 5 Experiments on movies In the image domain, our model has potential applications in temporal interpolation/filling in of video, super-resolution, compression, and geodesic distance estimation. We apply the model to moving natural images and investigate the first three applications; the third will be the subject of future work. Here we report on the ability of the model to learn transformations across time, as well as across scales of the Laplacian pyramid. Our data is many short grayscale video sequences of Africa from the BBC. 5.1 Time We apply the model to natural movies by presenting it with patches of adjacent frame pairs. Using an analytically generated infinitesimal shift operator, we first run a series of experiments to determine the effect of local minima on the recovery of a known displacement through the minimization of E 5 Figure 5: Shift operator learned from synthetically transformed natural images. The operator Ψ1, displayed as an array of weights that, for each output pixel, shows the strength of its connection to each input pixel. Each of the 15x15 arrays represents one output pixel’s connections. Because of the 1/f 2 falloff in the power spectrum of natural images, synthetic images with a wildly different distribution of spatial frequency content, such as uncorrelated noise, will not be properly shifted by this operator. t = 0.00 t = 0.25 t = 0.50 t = 0.75 t = 1.00 t = 0.00 t = 0.25 t = 0.50 t = 0.75 t = 1.00 exponential linear 1st order Taylor approximation Figure 6: Interpolating between shifted images to temporally fill in missing video frames. Two images x0 and x1 are generated by convolving an image of a diagonal line and a shifted diagonal line by a 3x3 Gaussian kernel with σ = 0.8, and the operator A is inferred. The top row shows the sequence of images xt = exp(At) x0 for t = 0.25, 0.50, 0.75, 1.00. The middle row shows linear interpolation between x0 and x1. The bottom row shows the sequence of images xt = (I + At) x0, that is, the first-order Taylor expansion of the matrix exponential, which performs poorly for shifts greater than one pixel. 6 w.r.t. c. When initialized to zero, the c vector often converges to the wrong displacement, but this problem can be avoided with high probability using a coarse-to-fine technique [16,17]. Doing so requires a slight alteration to our inference algorithm: now we must solve a sequence of optimization problems on frame pairs convolved with a Gaussian kernel whose variance is progressively decreased. At each step in the sequence, both frames are convolved by the kernel before a patch is selected. For the first step, the c variables are initialized to zero; for subsequent steps they are initialized to the solution of the previous step. For our analytical shifting operator, two blurring filters – first a 5x5 kernel with variance 10, then a 3x3 kernel with variance 5 – reliably gives a proper initialization for the final minimization that runs on the unaltered data. For control purposes, the video for this experiment comes from a camera fly over; thus, most of the motion in the scene is due to camera motion. We apply the model pairs of 11x11 patches, selected from random locations in the video, but discarding patches near the horizon where there is little or no motion. We initialize M = 16; after learning, the basis function with the longest norm has the structure of a shift operator in the primary direction of motion taking place in the video sequence. Using these 16 operators, we run inference on 1,000 randomly selected pairs of patches from a second video, not used during learning, and measure the quality of the reconstruction as the trajectory is used to predict into the future. At 5 frames into the future, our model is able to maintain an average SNR of 7, compared to SNR 5 when a first-order Taylor approximation is used in place of the matrix exponential; for comparison, the average SNR for the identity transformation model on this data is 1. Since the primary form of motion going on in these small patches is translation, we also train a single operator using artificially translated natural data to make clear that the model can learn this case completely. For this last experiment we take a 360x360 pixel frame of our natural movie, and continuously translate the entire frame in the Fourier domain by a displacement chosen uniformly at random from [0, 3] pixels. We then randomly select a 15x15 region on the interior of the pair of frames and use the two 225 pixel vectors as our x0 and x1. We modify the objective function to be E = 1 2||W [x1 −T(c) x0]||2 2 + γ 2 X m ||Ψm||2 F + ζ||c||1 , (12) where W is a binary windowing function that selects the central 9x9 region from a 15x15 patch; thus, the residual errors that come from new content translating into the patch are ignored. After learning, the basis function Ψ1 (shown in figure 5) is capable of translating natural images up to 3 pixels while maintaining an average SNR of 16 in the 9x9 center region. Figure 6 shows how this operator is able to correctly interpolate between two measurements of a shifted image in order to temporally up-sample a movie. 5.2 Scale The model can also learn to transform between successive scales in the Laplacian pyramid built from a single frame of a video sequence. Figure 7 depicts the system transforming an image patch from scale 2 to 3 of a 256x256 pixel image. We initialize M = 100, but many basis elements shrink during learning; we use only the 16 Ψm with non-negligible norm to encode a scale change. The basis Ψ is initialized to mean-zero white Gaussian noise with variance 0.01; the same inference and learning procedures as described for the point sets are then run on pairs x0, x1 selected at random from the corpus of image sequences in the following way. First, we choose a random frame from a random sequence, then up-sample and blur scale 3 of its Laplacian pyramid. Second, we select an 8x8 patch from scale 2 (x0) of the corresponding up-blurred image patch (x1). Were it not for the highly structured manifold on which natural images live, the proposition of finding an operator that maps a blurred, subsampled image to its high-resolution original state would seem untenable. However, our results show that in many cases, a reduced representation of such two-way mappings can be found, even for small patches. 6 Discussion and conclusions We have shown that it is possible to learn low-dimensional parameterizations of operators that transport along non-linear manifolds formed by natural images, both across time and scale. Our focus thus far has been primarily on understanding the model and how to properly optimize its parameters, 7 (a) (b) (c) (d) Scale 3 Estimated Scale 2 Actual Scale 2 Error Figure 7: Learning transformations across scale. (a) Scale 3 of the Laplacian pyramid for a natural scene we wish to code, by describing how it transforms across scale, in terms of our learned dictionary. (b) The estimated scale 2, computed by transforming 8x8 regions of the up-sampled and blurred scale 3. The estimated scale 2 has SNR 9.60; (c) shows the actual scale 2 and (d) shows the errors made by our estimation. For reconstruction we use only 16 dictionary elements. as little work has previously been done on learning such high dimensional Lie groups. A promising direction for future work is to explore higher-order models capable of capturing non-commutative operators, such as xt = exp(Ψ1 c1) exp(Ψ2 c2) · · · exp(ΨK cK) x0 , (13) as this formulation may be more parsimonious for factoring apart transformations which are prevalent in natural movies, such as combinations of translation and rotation. Early attempts to model the manifold structure of images train on densely sampled point clouds and find an embedding into a small number of coordinates along the manifold. However such an approach does not actually constitute a model, since there is no function for mapping arbitrary points, or moving along the manifold. One must always refer back to original data points on which the model was trained – i.e., it works as a lookup table rather than being an abstraction of the data. Here, by learning operators that transport along the manifold we have been able to learn a compact description of its structure. This model-based representation can be leveraged to compute geodesics using a numerical approximation to the arc length integral: S = Z 1 0 ||A exp(A t)||2 2 dt = lim T →∞ T X t=1 || exp(A t T ) x0 −exp(A t −1 T ) x0||2 2 , (14) where T is the number of segments chosen to use in the piecewise linear approximation of the curve, and each term in the summation gives the length of a segment. We believe that this aspect of our model will be of use in difficult classification problems, such as face identification, where Euclidean distances measured in pixel-space give poor results. Previous attempts to learn Lie group operators have focused on linear approximations. Here we show that utilizing the full Lie operator/matrix exponential in learning, while computationally intensive, is tractable, even in the extremely high dimensional cases required by models of natural movies. Our spectral decomposition is the key component that enables this, and, in combination with careful mitigation of local minima in the objective function using a coarse-to-fine technique, gives us the power to factor out large transformations from data. One shortcoming of the approach described here is that transformations are modeled in the original pixel domain. Potentially these transformations may be described more economically by working in a feature space, such as a sparse decomposition of the image. This is a direction of ongoing work. Acknowledgments The authors gratefully acknowledge many useful discussions with Jascha Sohl-Dickstein, Jimmy Wang, Kilian Koepsell, Charles Cadieu, and Amir Khosrowshahi, and the insightful comments from our anonymous reviewers. 8 References [1] VanGool, L., Moons, T., Pauwels, E. & Oosterlinck, A. (1995) Vision and Lie’s approach to invariance. Image and Vision Computing, 13(4): 259-277. [2] Miao, X. & Rao, R.P.N. (2007) Learning the Lie groups of visual invariance. Neural Computation, 19(10): 2665-2693. [3] Rao, R.P.N & Ruderman D.L. (1999) Learning Lie Groups for Invariant Visual Perception. Advances in Neural Information Processing Systems, 11:810-816. Cambridge, MA: MIT Press. [4] Grimes, D.B., & Rao, R.P.N. (2002). A Bilinear Model for Sparse Coding. Advances in Neural Information Processing Systems, 15. Cambridge, MA: MIT Press. [5] Olshausen, B.A., Cadieu, C., Culpepper, B.J. & Warland, D. (2007) Bilinear Models of Natural Images. SPIE Proceedings vol. 6492: Human Vision Electronic Imaging XII (B.E. Rogowitz, T.N. Pappas, S.J. Daly, Eds.), Jan 28 - Feb 1, 2007, San Jose, California. [6] Tenenbaum, J. B. & Freeman, W. T. (2000) Separating style and content with bilinear models. Neural Computation, 12(6):1247-1283. [7] Roweis, S. & Saul, L. (2000) Nonlinear dimensionality reduction by locally linear embedding. Science, 290(5500): 2323-2326. [8] Weinberger, K. Q. & Saul, L. K. (2004) Unsupervised learning of image manifolds by semidefinite programming. Computer Vision and Pattern Recognition. [9] Tenenbaum, J. B., de Silva, V. & Langford, J. C. (2000) A Global Geometric Framework for Nonlinear Dimensionality Reduction. Science, 22 December 2000: 2319-2323. [10] Belkin, M., & Niyogi, P. (2002). Laplacian eigenmaps and spectral techniques for embedding and clustering. Advances in Neural Information Processing Systems, 14. Cambridge, MA: MIT Press. [11] Wang, C. M., Sohl-Dickstein, J., & Olshausen, B. A. (2009) Unsupervised Learning of Lie Group Operators from Natural Movies. Redwood Center for Theoretical Neuroscience, Technical Report; RCTR 01-09. [12] Dollar, P., Rabaud, V., & Belongie, S (2007) Non-isometric Manifold Learning: Analysis and an Algorithm. Int. Conf. on Machine Learning , 241-248. [13] Olshausen, B.A. & Field, D.J. (1997) Sparse Coding with an Overcomplete Basis Set: A Strategy Employed by V1? Vision Research, 37: 3311-3325. [14] Ortiz, M., Radovitzky, R.A. & Repetto, E.A (2001) The computation of the exponential and logarithmic mappings and their first and second linearizations. International Journal For Numerical Methods In Engineering. 52: 1431-1441. [15] Berens, P. & Velasco, M. J. (2009) The circular statistics toolbox for Matlab. MPI Technical Report, 184. [16] Anandan, P. (1989) A computational framework and an algorithm for the measurement of visual motion. Int. J. Comput. Vision, 2(3): 283-310. [17] Glazer, F. (1987) Hierarchical Motion Detection. 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Manifold Embeddings for Model-Based Reinforcement Learning under Partial Observability Keith Bush School of Computer Science McGill University Montreal, Canada kbush@cs.mcgill.ca Joelle Pineau School of Computer Science McGill University Montreal, Canada jpineau@cs.mcgill.ca Abstract Interesting real-world datasets often exhibit nonlinear, noisy, continuous-valued states that are unexplorable, are poorly described by first principles, and are only partially observable. If partial observability can be overcome, these constraints suggest the use of model-based reinforcement learning. We experiment with manifold embeddings to reconstruct the observable state-space in the context of offline, model-based reinforcement learning. We demonstrate that the embedding of a system can change as a result of learning, and we argue that the best performing embeddings well-represent the dynamics of both the uncontrolled and adaptively controlled system. We apply this approach to learn a neurostimulation policy that suppresses epileptic seizures on animal brain slices. 1 Introduction The accessibility of large quantities of off-line discrete-time dynamic data—state-action sequences drawn from real-world domains—represents an untapped opportunity for widespread adoption of reinforcement learning. By real-world we imply domains that are characterized by continuous state, noise, and partial observability. Barriers to making use of this data include: 1) goals (rewards) are not well-defined, 2) exploration is expensive (or not permissible), and 3) the data does not preserve the Markov property. If we assume that the reward function is part of the problem description, then to learn from this data we must ensure the Markov property is preserved before we approximate the optimal policy with respect to the reward function in a model-free or model-based way. For many domains, particularly those governed by differential equations, we may leverage the inductive bias of locality during function approximation to satisfy the Markov property. When applied to model-free reinforcement learning, function approximation typically assumes that the value function maps nearby states to similar expectations of future reward. As part of model-based reinforcement learning, function approximation additionally assumes that similar actions map to nearby future states from nearby current states [10]. Impressive performance and scalability of local modelbased approaches [1, 2] and global model-free approaches [6, 17] have been achieved by exploiting the locality of dynamics in fully observable state-space representations of challenging real-world problems. In partially observable systems, however, locality is not preserved without additional context. First principle models offer some guidance in defining local dynamics, but the existence of known first principles cannot always be assumed. Rather, we desire a general framework for reconstructing state-spaces of partially observable systems which guarantees the preservation of locality. Nonlinear dynamic analysis has long used manifold embeddings to reconstruct locally Euclidean state-spaces of unforced, partially observable systems [24, 18] and has identified ways of finding these embeddings non-parametrically [7, 12]. Dynamicists have also used embeddings as generative models of partially observable unforced systems [16] by numerically integrating over the resultant embedding. 1 Recent advances have extended the theory of manifold embeddings to encompass deterministically and stochastically forced systems [21, 22]. A natural next step is to apply these latest theoretical tools to reconstruct and control partially observable forced systems. We do this by first identifying an appropriate embedding for the system of interest and then leveraging the resultant locality to perform reinforcement learning in a modelbased way. We believe it may be more practical to address reinforcement learning under partial observability in a model-based way because it facilitates reasoning about domain knowledge and off-line validation of the embedding parameters. The primary contribution of this paper is to formally combine and empirically evaluate these existing, but not well-known, methods by incorporating them in off-line, model-based reinforcement learning of two domains. First, we study the use of embeddings to learn control policies in a partially observable variant of the well-known Mountain Car domain. Second, we demonstrate the embedding-driven, model-based technique to learn an effective and efficient neurostimulation policy for the treatment of epilepsy. The neurostimulation example is important because it resides among the hardest classes of learning domain—a continuous-valued state-space that is nonlinear, partially observable, prohibitively expensive to explore, noisy, and governed by dynamics that are currently not well-described by mathematical models drawn from first principles. 2 Methods In this section we combine reinforcement learning, partial observability, and manifold embeddings into a single mathematical formalism. We then describe non-parametric means of identifying the manifold embedding of a system and how the resultant embedding may be used as a local model. 2.1 Reinforcement Learning Reinforcement learning (RL) is a class of problems in which an agent learns an optimal solution to a multi-step decision task by interacting with its environment [23]. Many RL algorithms exist, but we will focus on the Q-learning algorithm. Consider an environment (i.e. forced system) having a state vector, s ∈RM, which evolves according to a nonlinear differential equation but is discretized in time and integrated numerically according to the map, f. Consider an agent that interacts with the environment by selecting action, a, according to a policy function, π. Consider also that there exists a reward function, g, which informs the agent of the scalar goodness of taking an action with respect to the goal of some multi-step decision task. Thus, for each time, t, a(t) = π(s(t)), (1) s(t + 1) = f(s(t), a(t)), and (2) r(t + 1) = g(s(t), a(t)). (3) RL is the process of learning the optimal policy function, π∗, that maximizes the expected sum of future rewards, termed the optimal action-value function or Q-function, Q∗, such that, Q∗(s(t), a(t)) = r(t + 1) + γ max a Q∗(s(t + 1), a), (4) where γ is the discount factor on [0, 1). Equation 4 assumes that Q∗is known. Without a priori knowledge of Q∗an approximation, Q, must be constructed iteratively. Assume the current Qfunction estimate, Q, of the optimal, Q∗, contains error, δ, δ(t) = r(t + 1) + γ max a Q (s(t + 1), a) −Q (s(t), a(t)) , where δ(t) is termed the temporal difference error or TD-error. The TD-error can be used to improve the approximation of Q by Q (s(t), a(t)) = Q (s(t), a(t)) + αδ(t), (5) where α is the learning rate. By selecting action a that maximizes the current estimate of Q, Qlearning specifies that over many applications of Equation 5, Q approaches Q∗. 2 2.2 Manifold Embeddings for Reinforcement Learning Under Partial Observability Q-learning relies on complete state observability to identify the optimal policy. Nonlinear dynamic systems theory provides a means of reconstructing complete state observability from incomplete state via the method of delayed embeddings, formalized by Takens’ Theorem [24]. Here we present the key points of Takens’ Theorem utilizing the notation of Huke [8] in a deterministically forced system. Assume s is an M-dimensional, real-valued, bounded vector space and a is a real-valued action input to the environment. Assuming that the state update f and the policy π are deterministic functions, Equation 1 may be substituted into Equation 2 to compose a new function, φ, s(t + 1) = f (s(t), π(s(t))) , = φ(s(t)), (6) which specifies the discrete time evolution of the agent acting on the environment. If φ is a smooth map φ : RM →RM and this system is observed via function, y, such that ˜s(t) = y(s(t)), (7) where y : RM →R, then if φ is invertible, φ−1 exists, and φ, φ−1, and y are continuously differentiable we may apply Takens’ Theorem [24] to reconstruct the complete state-space of the observed system. Thus, for each ˜s(t), we can construct a vector sE(t), sE(t) = [˜s(t), ˜s(t −1), ..., ˜s(t −(E −1))], E > 2M, (8) such that sE lies on a subset of RE which is an embedding of s. Because embeddings preserve the connectivity of the original vector-space, in the context of RL the mapping ψ, sE(t + 1) = ψ(sE(t)), (9) may be substituted for f (Eqn. 6) and vectors sE(t) may be substituted for corresponding vectors s(t) in Equations 1–5 without loss of generality. 2.3 Non-parametric Identification of Manifold Embeddings Takens’ Theorem does not define how to compute the embedding dimension of arbitrary sequences of observations, nor does it provide a test to determine if the theorem is applicable. In general. the intrinsic dimension, M, of a system is unknown. Finding high-quality embedding parameters of challenging domains, such as chaotic and noise-corrupted nonlinear signals, occupy much of the fields of subspace identification and nonlinear dynamic analysis. Numerous methods of note exist, drawn from both disciplines. We employ a spectral approach [7]. This method, premised by the singular value decomposition (SVD), is non-parametric, computationally efficient, and robust to additive noise—all of which are useful in practical application. As will be seen in succeeding sections, this method finds embeddings which are both accurate in theoretical tests and useful in practice. We summarize the spectral parameter selection algorithm as follows. Given a sequence of state observations ˜s of length ˜S, we choose a sufficiently large fixed embedding dimension, ˆE. Sufficiently large refers to a cardinality of dimension which is certain to be greater than twice the dimension in which the actual state-space resides. For each embedding window size, ˆTmin ∈{ ˆE, ..., ˜S}, we: 1) define a matrix S ˆ E having row vectors, s ˆ E(t), t ∈{ ˆTmin, ..., ˜S}, constructed according to the rule, s ˆ E(t) = [˜s(t), ˜s(t −τ), ..., ˜s(t −( ˆE −1)τ)], (10) where τ = ˆTmin/( ˆE −1), 2) compute the SVD of the matrix S ˆ E, and 3) record the vector of singular values, σ( ˆTmin). Embedding parameters of ˜s are found by analysis of the second singular values, σ2( ˆTmin), ˆTmin ∈{ ˆE, ..., ˜S}. The ˆTmin value of the first local maxima of this sequence is the approximate embedding window, Tmin, of ˜s. The approximate embedding dimension, E, is the number of non-trivial singular values of σ(Tmin) where we define non-trivial as a value greater than the long-term trend of σ ˆ E with respect to ˆTmin. Embedding ˜s according to Equation 10 via parameters E and Tmin yields the matrix SE of row vectors, sE(t), t ∈{Tmin, ..., ˜S}. 3 2.4 Generative Local Models from Embeddings The preservation of locality and dynamics afforded by the embedding allows an approximation of the underlying dynamic system. To model this space we assume that the derivative of the Voronoi region surrounding each embedded point is well-approximated by the derivative at the point itself, a nearest-neighbors derivative [16]. Using this, we simulate trajectories as iterative numerical integration of the local state and gradient. We define the model and integration process formally. Consider a dataset D as a set of temporally aligned sequences of state observations ˜s(t), action observations a(t), and reward observations r(t), t ∈{1, ..., ˜S}. Applying the spectral embedding method to D yields a sequence of vectors sE(t) in RE indexed by t ∈{Tmin, ..., ˜S}. A local model M of D is the set of 3-tuples, m(t) = {sE(t), a(t), r(t)}, t ∈{Tmin, ..., ˜S}, as well as operations on these tuples, A(m(t)) ≡a(t), S(m(t)) ≡sE(t), Z(m(t)) ≡z(t) where z(t) = [s(t), a(t)], and U(M, a) ≡Ma where Ma is the subset of tuples in M containing action a. Consider a state vector x(i) in RE indexed by simulation time, i. To numerically integrate this state we define the gradient according to our definition of locality, namely the nearest neighbor. This step is defined differently for models having discrete and continuous actions. The model’s nearest neighbor of x(i) when taking action a(i) is defined in the case of a discrete set of actions, A, according to Equation 11 and in the continuous case it is defined by Equation 12, m(tx(i)) = argmin m(t)∈U(M,a(i)) ∥S(m(t)) −x(i)∥, a ∈A, (11) m(tx(i)) = argmin m(t)∈M ∥Z(m(t)) −[x(i), ωa(i)] ∥, a ∈R. (12) where ω is a scaling parameter on the action space. The model gradient and numerical integration are defined, respectively, as, ∇x(i) = S(m(tx(i) + 1)) −S(m(tx(i))) and (13) x(i + 1) = x(i) + ∆i ¡ ∇x(i) + η ¢ , (14) where η is a vector of noise and ∆i is the integration step-size. Applying Equations 11–14 iteratively simulates a trajectory of the underlying system, termed a surrogate trajectory. Surrogate trajectories are initialized from state x(0). Equation 14 assumes that dataset D contains noise. This noise biases the derivative estimate in RE, via the embedding rule (Eqn. 10). In practice, a small amount of additive noise facilitates generalization. 2.5 Summary of Approach Our approach is to combine the practices of dynamic analysis and RL to construct useful policies in partially observable, real-world domains via off-line learning. Our meta-level approach is divided into two phases: the modeling phase and the learning phase. We perform the modeling phase in steps: 1) record a partially observable system (and its rewards) under the control of a random policy or some other policy or set of policies that include observations of high reward value; 2) identify good candidate parameters for the embedding via the spectral embedding method; and 3) construct the embedding vectors and define the local model of the system. During the learning phase, we identify the optimal policy on the local model with respect to the rewards, R(m(t)) ≡r(t), via batch Q-learning. In this work we consider strictly local function approximation of the model and Q-function, thus, we define the Q-function as a set of values, Q, indexed by the model elements, Q(m), m ∈M. For a state vector x(i) in RE at simulation time i, and an associated action, a(i), the reward and Q-value of this state can be indexed by either Equation 11 or 12, depending on whether the action is discrete or continuous. Note, our technique does not preclude the use of non-local function approximation, but here we assume a sufficient density of data exists to reconstruct the embedded state-space with minimal bias. 3 Case Study: Mountain Car The Mountain Car problem is a second-order, nonlinear dynamic system with low-dimensional, continuous-valued state and action spaces. This domain is perhaps the most studied continuousvalued RL domain in the literature, but, surprisingly, there is little study of the problem in the case where the velocity component of state is unobserved. While not a real-world domain as imagined in the introduction, Mountain Car provides a familiar benchmark to evaluate our approach. 4 50000 50000 0 100000 100000 150000 2.5 150000 200000 5.0 200000 −1.0 7.5 10.0 −0.5 0.0 0.5 1000 1000 100 100 0 −1.0 5 10 −0.5 Training Samples 15 0.0 Path−to−goal Length Path−to−goal Length 20 0.5 x(t) Tmin (sec) x(t) Tmin (sec) τ ) x(t− τ ) x(t− Singular Values Singular Values −1.0 −0.5 0.0 0.5 0 2.5 5.0 −1.0 7.5 10.0 −0.5 0.0 0.5 0 5 10 Learned Policy (d) 15 (a) Embedding Performance, E=2 (b) Random Policy Embedding Performance, E=3 (c) Training Samples σ5 σ2 σ1 σ3 σ1 σ2 σ5 0.20 0.70 1.20 1.70 2.20 Tmin Best Max Random 0.20 0.70 1.20 1.70 2.20 Tmin Best Max Random Figure 1: Learning experiments on Mountain Car under partial observability. (a) Embedding spectrum and accompanying trajectory (E = 3, Tmin = 0.70 sec.) under random policy. (b) Learning performance as a function of embedding parameters and quantity of training data. (c) Embedding spectrum and accompanying trajectory (E = 3, Tmin = 0.70 sec.) for the learned policy. We use the Mountain Car dynamics and boundaries of Sutton and Barto [23]. We fix the initial state for all experiments (and resets) to be the lowest point of the mountain domain with zero velocity, which requires the longest path-to-goal in the optimal policy. Only the position element of the state is observable. During the modeling phase, we record this domain under a random control policy for 10,000 time-steps (∆t = 0.05 seconds), where the action is changed every ∆t = 0.20 seconds. We then compute the spectral embedding of the observations (Tmin = [0.20, 9.95] sec., ∆Tmin = 0.25 sec., and ˆE = 5). The resulting spectrum is presented in Figure 1(a). We conclude that the embedding of Mountain Car under the random policy requires dimension E = 3 with a maximum embedding window of Tmin = 1.70 seconds. To evaluate learning phase outcomes with respect to modeling phase outcomes, we perform an experiment where we model the randomly collected observations using embedding parameters drawn from the product of the sets Tmin = {0.20, 0.70, 1.20, 1.70, 2.20} seconds and E = {2, 3}. While we fix the size of the local model to 10,000 elements we vary the total amount of training samples observed from 10,000 to 200,000 at intervals of 10,000. We use batch Q-learning to identify the optimal policy in a model-based way—in Equation 5 the transition between state-action pair and the resulting state-reward pair is drawn from the model (η = 0.001). After learning converges, we execute the learned policy on the real system for 10,000 time-steps, recording the mean path-to-goal length over all goals reached. Each configuration is executed 30 times. We summarize the results of these experiments by log-scale plots, Figures 1(b) and (c), for embeddings of dimension two and three, respectively. We compare learning performance against three measures: the maximum performing policy achievable given the dynamics of the system (path-togoal = 63 steps), the best (99th percentile) learned policy for each quantity of training data for each embedding dimension, and the random policy. Learned performance is plotted as linear regression fits of the data. Policy performance results of Figures 1(b) and (c) may be summarized by the following observations. Performance positively relates to the quantity of off-line training data for all embedding parameters. Except for the configuration (E = 2, Tmin = 0.20), influence of Tmin on learning performance relative to E is small. Learning performance of 3-dimensional embeddings dominate 5 all but the shortest 2-dimensional embeddings. These observations indicate that the parameters of the embedding ultimately determine the effectiveness of RL under partial observability. This is not surprising. What is surprising is that the best performing parameter configurations are linked to dynamic characteristics of the system under both a random policy and the learned policy. To support this claim we collected 1,000 sample observations of the best policy (E = 3, Tmin = 0.70 sec., Ntrain = 200, 000) during control of the real Mountain Car domain (path-to-goal = 79 steps). We computed and plotted the embedding spectrum and first two dimensions of the embedding in Figure 1(d). We compare these results to similar plots for the random policy in Figure 1(a). We observe that the spectrum of the learned system has shifted such that the optimal embedding parameters require a shorter embedding window, Tmin = 0.70–1.20 sec. and a lower embedding dimension E = 2 (i.e., σ3 peaks at Tmin = 0.70–1.20 and σ3 falls below the trend of σ5 at this window length). We confirm this by observing the embedding directly, Figure 1(d). Unlike the random policy, which includes both an unstable spiral fixed point and limit cycle structure and requires a 3-dimensional embedding to preserve locality, the learned policy exhibits a 2-dimensional unstable spiral fixed point. Thus, the fixed-point structure (embedding structure) of the combined policy-environment system changes during learning. To reinforce this claim, we consider the difference between a 2-dimensional and 3-dimensional embedding. An agent may learn to project into a 2-dimensional plane of the 3-dimensional space, thus decreasing its embedding dimension if the training data supports a 2-dimensional policy. We believe it is no accident that (E = 3, Tmin = 0.70) is the best performing configuration across all quantities of training data. This configuration can represent both 3-dimensional and 2-dimensional policies, depending on the amount of training data available. It can also select between 2-dimensional embeddings having window sizes of Tmin = {0.35, 0.70} sec., depending on whether the second or third dimension is projected out. One resulting parameter configuration (E = 2, Tmin = 0.35) is near the optimal 2-dimensional configuration of Figure 1(b). 4 Case Study: Neurostimulation Treatment of Epilepsy Epilepsy is a common neurological disorder which manifests itself, electrophysiologically, in the form of intermittent seizures—intense, synchronized firing of neural populations. Researchers now recognize seizures as artifacts of abnormal neural dynamics and rely heavily on the nonlinear dynamic systems analysis and control literature to understand and treat seizures [4]. Promising techniques have emerged from this union. For example, fixed frequency electrical stimulation of slices of the rat hippocampus under artificially induced epilepsy have been demonstrated to suppress the frequency, duration, or amplitude of seizures [9, 5]. Next generation epilepsy treatments, derived from machine learning, promise maximal seizure suppression via minimal electrical stimulation by adapting control policies to patients’ unique neural dynamics. Barriers to constructing these treatments arise from a lack of first principles understanding of epilepsy. Without first principles, neuroscientists have only vague notions of what effective neurostimulation treatments should look like. Even if effective policies could be envisioned, exploration of the vast space of policy parameters is impractical without computational models. Our specific control problem is defined as follows. Given labeled field potential recordings of brain slices under fixed-frequency electrical stimulation policies of 0.5, 1.0, and 2.0 Hz, as well as unstimulated control data, similar to the time-series depicted in Figure 2(a), we desire to learn a stimulation policy that suppresses seizures of a real, previously unseen, brain slice with an effective mean frequency (number of stimulations divided by the time the policy is active) of less than 1.0 Hz (1.0 Hz is currently known to be the most robust suppression policy for the brain slice model we use [9, 5]). As a further complication, on-line exploration is extremely expensive because the brain slices are experimentally viable for periods of less than 2 hours. Again, we approach this problem as separate modeling and learning phases. We first compute the embedding spectrum of our dataset assuming ˆE = 15, presented in Figure 2(b). Using our knowledge of the interaction between embedding parameters and learning we select the embedding dimension E = 3 and embedding window Tmin = 1.05 seconds. Note, the strong maxima of σ2 at Tmin = 110 seconds is the result of periodicity of seizures in our small training dataset. Periodicity of spontaneous seizure formation, however, varies substantially between slices. We select a shorter embedding window and rely on integration of the local model to unmask long-term dynamics. 6 Control 1 Hz 0 50 100 150 0.5 Hz σ3 σ2 σ1 σ2 σ3 (a) Example Field Potentials (b) Neurostimulation Embedding Spectrum −0.6 −0.4 −0.2 0.6 0.0 0.2 0.4 −0.6 −0.4 −0.2 0.6 0.0 0.2 0.4 −0.4 −0.2 0.6 0.0 0.2 0.4 −0.6 −3 0 −1 −2 (c) Neurostimulation Model 2 Hz 200 sec 1 mV Singular Values 0 50 100 150 200 250 0.0 0.5 1.0 1.5 2.0 0 10 20 30 40 50 60 Singular Values * * T T min min (s) (s) 2nd Principal Component 3rd Principal Component 1st Principal Component 2nd Principal Component Figure 2: Graphical summary of the modeling phase of our adaptive neurostimulation study. (a) Sample observations from the fixed-frequency stimulation dataset. Seizures are labeled with horizontal lines. (b) The embedding spectrum of the fixed-frequency stimulation dataset. The large maximum of σ2 at approximately 100 sec. is an artifact of the periodicity of seizures in the dataset. *Detail of the embedding spectrum for Tmin = [0.05, 2.0] depicting a maximum of σ2 at the timescale of individual stimulation events. (c) The resultant neurostimulation model constructed from embedding the dataset with parameters (E = 3, Tmin = 1.05 sec.). Note, the model has been desampled 5× in the plot. In this complex domain we apply the spectral method differently than described in Section 2. Rather than building the model directly from the embedding (E = 3, Tmin = 1.05), we perform a change of basis on the embedding ( ˆE = 15, Tmin = 1.05), using the first three columns of the right singular vectors, analogous to projecting onto the principal components. This embedding is plotted in Figure 2(c). Also, unlike the previous case study, we convert stimulation events in the training data from discrete frequencies to a continuous scale of time-elapsed-since-stimulation. This allows us to combine all of the data into a single state-action space and then simulate any arbitrary frequency. Based on exhaustive closed-loop simulations of fixed-frequency suppression efficacy across a spectrum of [0.001, 2.0] Hz, we constrain the model’s action set to discrete frequencies a = {2.0, 0.25} Hz in the hopes of easing the learning problem. We then perform batch Q-learning over the model (∆t = 0.05, ω = 0.1, and η = 0.00001), using discount factor γ = 0.9. We structure the reward function to penalize each electrical stimulation by −1 and each visited seizure state by −20. Without stimulation, seizure states comprise 25.6% of simulation states. Under a 1.0 Hz fixedfrequency policy, stimulation events comprise 5.0% and seizures comprise 6.8% of the simulation states. The policy learned by the agent also reduces the percent of seizure states to 5.2% of simulation states while stimulating only 3.1% of the time (effective frequency equals 0.62 Hz). In simulation, therefore, the learned policy achieves the goal. We then deployed the learned policy on real brain slices to test on-line seizure suppression performance. The policy was tested over four trials on two unique brain slices extracted from the same animal. The effective frequencies of these four trials were {0.65, 0.64, 0.66, 0.65} Hz. In all trials seizures were effectively suppressed after a short transient period, during which the policy and slice achieved equilibrium. (Note: seizures occurring at the onset of stimulation are common artifacts of neurostimulation). Figure 3 displays two of these trials spaced over four sequential phases: (a) a control (no stimulation) phase used to determine baseline seizure activity, (b) a learned policy trial lasting 1,860 seconds, (c) a recovery phase to ensure slice viability after stimulation and to recompute baseline seizure activity, and (d) a learned policy trial lasting 2,130 seconds. 7 (a) Control Phase 60 sec 2 mV (d) Policy Phase 2 (b) Policy Phase 1 (c) Recovery Phase Stimulations * Figure 3: Field potential trace of a real seizure suppression experiment using a policy learned from simulation. Seizures are labeled as horizontal lines above the traces. Stimulation events are marked by vertical bars below the traces. (a) A control phase used to determine baseline seizure activity. (b) The initial application of the learned policy. (c) A recovery phase to ensure slice viability after stimulation and recompute baseline seizure activity. (d) The second application of the learned policy. *10 minutes of trace are omitted while the algorithm was reset. 5 Discussion and Related Work The RL community has long studied low-dimensional representations to capture complex domains. Approaches for efficient function approximation, basis function construction, and discovery of embeddings has been the topic of significant investigations [3, 11, 20, 15, 13]. Most of this work has been limited to the fully observable (MDP) case and has not been extended to partially observable environments. The question of state space representation in partially observable domains was tackled under the POMDP framework [14] and recently in the PSR framework [19]. These methods address a similar problem but have been limited primarily to discrete action and observation spaces. The PSR framework was extended to continuous (nonlinear) domains [25]. This method is significantly different from our work, both in terms of the class of representations it considers and in the criteria used to select the appropriate representation. Furthermore, it has not yet been applied to real-world domains. An empirical comparison with our approach is left for future consideration. The contribution of our work is to integrate embeddings with model-based RL to solve real-world problems. We do this by leveraging locality preserving qualities of embeddings to construct dynamic models of the system to be controlled. While not improving the quality of off-line learning that is possible, these models permit embedding validation and reasoning over the domain, either to constrain the learning problem or to anticipate the effects of the learned policy on the dynamics of the controlled system. To demonstrate our approach, we applied it to learn a neurostimulation treatment of epilepsy, a challenging real-world domain. We showed that the policy learned off-line from an embedding-based, local model can be successfully transferred on-line. This is a promising step toward widespread application of RL in real-world domains. Looking to the future, we anticipate the ability to adjust the embedding a priori using a non-parametric policy gradient approach over the local model. An empirical investigation into the benefits of this extension are also left for future consideration. Acknowledgments The authors thank Dr. Gabriella Panuccio and Dr. Massimo Avoli of the Montreal Neurological Institute for generating the time-series described in Section 4. The authors also thank Arthur Guez, Robert Vincent, Jordan Frank, and Mahdi Milani Fard for valuable comments and suggestions. The authors gratefully acknowledge financial support by the Natural Sciences and Engineering Research Council of Canada and the Canadian Institutes of Health Research. 8 References [1] Christopher G. Atkeson, Andrew W. Moore, and Stefan Schaal. Locally weighted learning for control. Artificial Intelligence Review, 11:75–113, 1997. [2] Christopher G. Atkeson and Jun Morimoto. Nonparametric representation of policies and value functions: A trajectory-based approach. In Advances in Neural Information Processing, 2003. [3] M. Bowling, A. Ghodsi, and D. Wilkinson. Action respecting embedding. In Proceedings of ICML, 2005. [4] F. Lopes da Silva, W. Blanes, S. Kalitzin, J. Parra, P. Suffczynski, and D. Velis. Dynamical diseases of brain systems: Different routes to epileptic seizures. IEEE Transactions on Biomedical Engineering, 50(5):540–548, 2003. [5] G. D’Arcangelo, G. Panuccio, V. Tancredi, and M. Avoli. Repetitive low-frequency stimulation reduces epileptiform synchronization in limbic neuronal networks. Neurobiology of Disease, 19:119–128, 2005. [6] Damien Ernst, Pierre Guerts, and Louis Wehenkel. Tree-based batch mode reinforcement learning. Journal of Machine Learning Research, 6:503–556, 2005. [7] A. Galka. Topics in Nonlinear Time Series Analysis: with implications for EEG Analysis. World Scientific, 2000. [8] J.P. Huke. Embedding nonlinear dynamical systems: A guide to Takens’ Theorem. Technical report, Manchester Institute for Mathematical Sciences, University of Manchester, March, 2006. [9] K. Jerger and S. Schiff. Periodic pacing and in vitro epileptic focus. Journal of Neurophysiology, 73(2):876–879, 1995. [10] Nicholas K. Jong and Peter Stone. Model-based function approximation in reinforcement learning. In Proceedings of AAMAS, 2007. [11] P.W. Keller, S. Mannor, and D. Precup. Automatic basis function construction for approximate dynamic programming and reinforcement learning. In Proceedings of ICML, 2006. [12] M. Kennel and H. Abarbanel. False neighbors and false strands: A reliable minimum embedding dimension algorithm. Physical Review E, 66:026209, 2002. [13] S. Mahadevan and M. Maggioni. Proto-value functions: A Laplacian framework for learning representation and control in Markov decision processes. Journal of Machine Learning Research, 8:2169–2231, 2007. [14] A. K. McCallum. Reinforcement Learning with Selective Perception and Hidden State. PhD thesis, University of Rochester, 1996. [15] R. Munos and A. Moore. Variable resolution discretization in optimal control. Machine Learning, 49:291– 323, 2002. [16] U. Parlitz and C. Merkwirth. Prediction of spatiotemporal time series based on reconstructed local states. Physical Review Letters, 84(9):1890–1893, 2000. [17] Jan Peters, Sethu Vijayakumar, and Stefan Schaal. Natural actor-critic. In Proceedings of ECML, 2005. [18] Tim Sauer, James A. Yorke, and Martin Casdagli. Embedology. Journal of Statistical Physics, 65:3/4:579–616, 1991. [19] S. Singh, M. L. Littman, N. K. Jong, D. Pardoe, and P. Stone. Learning predictive state representations. In Proceedings of ICML, 2003. [20] W. Smart. Explicit manifold representations for value-functions in reinforcement learning. In Proceedings of ISAIM, 2004. [21] J. Stark. Delay embeddings for forced systems. I. Deterministic forcing. Journal of Nonlinear Science, 9:255–332, 1999. [22] J. Stark, D.S. Broomhead, M.E. Davies, and J. Huke. Delay embeddings for forced systems. II. Stochastic forcing. Journal of Nonlinear Science, 13:519–577, 2003. [23] R. Sutton and A. Barto. Reinforcement learning: An introduction. The MIT Press, Cambridge, MA, 1998. [24] F. Takens. Detecting strange attractors in turbulence. In D. A. Rand & L. S. Young, editor, Dynamical Systems and Turbulence, volume 898, pages 366–381. Warwick, 1980. [25] D. Wingate and S. Singh. On discovery and learning of models with predictive state representations of state for agents with continuous actions and observations. In Proceedings of AAMAS, 2007. 9
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Ensemble Nystr¨om Method Sanjiv Kumar Google Research New York, NY sanjivk@google.com Mehryar Mohri Courant Institute and Google Research New York, NY mohri@cs.nyu.edu Ameet Talwalkar Courant Institute of Mathematical Sciences New York, NY ameet@cs.nyu.edu Abstract A crucial technique for scaling kernel methods to very large data sets reaching or exceeding millions of instances is based on low-rank approximation of kernel matrices. We introduce a new family of algorithms based on mixtures of Nystr¨om approximations, ensemble Nystr¨om algorithms, that yield more accurate low-rank approximations than the standard Nystr¨om method. We give a detailed study of variants of these algorithms based on simple averaging, an exponential weight method, or regression-based methods. We also present a theoretical analysis of these algorithms, including novel error bounds guaranteeing a better convergence rate than the standard Nystr¨om method. Finally, we report results of extensive experiments with several data sets containing up to 1M points demonstrating the significant improvement over the standard Nystr¨om approximation. 1 Introduction Modern learning problems in computer vision, natural language processing, computational biology, and other areas are often based on large data sets of tens of thousands to millions of training instances. But, several standard learning algorithms such as support vector machines (SVMs) [2, 4], kernel ridge regression (KRR) [14], kernel principal component analysis (KPCA) [15], manifold learning [13], or other kernel-based algorithms do not scale to such orders of magnitude. Even the storage of the kernel matrix is an issue at this scale since it is often not sparse and the number of entries is extremely large. One solution to deal with such large data sets is to use an approximation of the kernel matrix. As shown by [18], later by [6, 17, 19], low-rank approximations of the kernel matrix using the Nystr¨om method can provide an effective technique for tackling large-scale scale data sets with no significant decrease in performance. This paper deals with very large-scale applications where the sample size can reach millions of instances. This motivates our search for further improved low-rank approximations that can scale to such orders of magnitude and generate accurate approximations. We show that a new family of algorithms based on mixtures of Nystr¨om approximations, ensemble Nystr¨om algorithms, yields more accurate low-rank approximations than the standard Nystr¨om method. Moreover, these ensemble algorithms naturally fit distributed computing environment where their computational cost is roughly the same as that of the standard Nystr¨om method. This issue is of great practical significance given the prevalence of distributed computing frameworks to handle large-scale learning problems. The remainder of this paper is organized as follows. Section 2 gives an overview of the Nystr¨om low-rank approximation method and describes our ensemble Nystr¨om algorithms. We describe several variants of these algorithms, including one based on simple averaging of p Nystr¨om solutions, 1 an exponential weight method, and a regression method which consists of estimating the mixture parameters of the ensemble using a few columns sampled from the matrix. In Section 3, we present a theoretical analysis of ensemble Nystr¨om algorithms, namely bounds on the reconstruction error for both the Frobenius norm and the spectral norm. These novel generalization bounds guarantee a better convergence rate for these algorithms in comparison to the standard Nystr¨om method. Section 4 reports the results of extensive experiments with these algorithms on several data sets containing up to 1M points, comparing different variants of our ensemble Nystr¨om algorithms and demonstrating the performance improvements gained over the standard Nystr¨om method. 2 Algorithm We first give a brief overview of the Nystr¨om low-rank approximation method, introduce the notation used in the following sections, and then describe our ensemble Nystr¨om algorithms. 2.1 Standard Nystr¨om method We adopt a notation similar to that of [5, 9] and other previous work. The Nystr¨om approximation of a symmetric positive semidefinite (SPSD) matrix K is based on a sample of m ≪n columns of K [5, 18]. Let C denote the n×m matrix formed by these columns and W the m×m matrix consisting of the intersection of these m columns with the corresponding m rows of K. The columns and rows of K can be rearranged based on this sampling so that K and C be written as follows: K = W K⊤ 21 K21 K22 and C = W K21 . (1) Note that W is also SPSD since K is SPSD. For a uniform sampling of the columns, the Nystr¨om method generates a rank-k approximation eK of K for k≤m defined by: eK = CW+ k C⊤≈K, (2) where Wk is the best k-rank approximation of W for the Frobenius norm, that is Wk = argminrank(V)=k ∥W −V∥F and W+ k denotes the pseudo-inverse of Wk [7]. W+ k can be derived from the singular value decomposition (SVD) of W, W = UΣU⊤, where U is orthonormal and Σ = diag(σ1, . . . , σm) is a real diagonal matrix with σ1 ≥· · ·≥σm ≥0. For k ≤rank(W), it is given by W+ k = Pk i=1 σ−1 i UiUi⊤, where Ui denotes the ith column of U. Since the running time complexity of SVD is O(m3) and O(nmk) is required for multiplication with C, the total complexity of the Nystr¨om approximation computation is O(m3+nmk). 2.2 Ensemble Nystr¨om algorithm The main idea behind our ensemble Nystr¨om algorithm is to treat each approximation generated by the Nystr¨om method for a sample of m columns as an expert and to combine p ≥1 such experts to derive an improved hypothesis, typically more accurate than any of the original experts. The learning set-up is defined as follows. We assume a fixed kernel function K : X ×X →R that can be used to generate the entries of a kernel matrix K. The learner receives a sample S of mp columns randomly selected from matrix K uniformly without replacement. S is decomposed into p subsamples S1,. . ., Sp. Each subsample Sr, r ∈[1, p], contains m columns and is used to define a rank-k Nystr¨om approximation eKr. Dropping the rank subscript k in favor of the sample index r, eKr can be written as eKr = CrW+ r C⊤ r , where Cr and Wr denote the matrices formed from the columns of Sr and W+ r is the pseudo-inverse of the rank-k approximation of Wr. The learner further receives a sample V of s columns used to determine the weight µr ∈R attributed to each expert eKr. Thus, the general form of the approximation of K generated by the ensemble Nystr¨om algorithm is eKens = p X r=1 µr eKr. (3) The mixture weights µr can be defined in many ways. The most straightforward choice consists of assigning equal weight to each expert, µr = 1/p, r ∈[1, p]. This choice does not require the additional sample V , but it ignores the relative quality of each Nystr¨om approximation. Nevertheless, 2 this simple uniform method already generates a solution superior to any one of the approximations eKr used in the combination, as we shall see in the experimental section. Another method, the exponential weight method, consists of measuring the reconstruction error ˆǫr of each expert eKr over the validation sample V and defining the mixture weight as µr =exp(−ηˆǫr)/Z, where η > 0 is a parameter of the algorithm and Z a normalization factor ensuring that the vector µ = (µ1, . . . , µp) belongs to the simplex ∆of Rp: ∆= {µ ∈Rp : µ ≥0 ∧Pp r=1 µr = 1}. The choice of the mixture weights here is similar to those used in the weighted-majority algorithm [11]. Let KV denote the matrix formed by using the samples from V as its columns and let eKV r denote the submatrix of eKr containing the columns corresponding to the columns in V . The reconstruction error ˆǫr =∥eKV r −KV ∥can be directly computed from these matrices. A more general class of methods consists of using the sample V to train the mixture weights µr to optimize a regression objective function such as the following: min µ λ∥µ∥2 2 + ∥ p X r=1 µr eKV r −KV ∥2 F , (4) where KV denotes the matrix formed by the columns of the samples S and V and λ > 0. This can be viewed as a ridge regression objective function and admits a closed form solution. We will refer to this method as the ridge regression method. The total complexity of the ensemble Nystr¨om algorithm is O(pm3 +pmkn+Cµ), where Cµ is the cost of computing the mixture weights, µ, used to combine the p Nystr¨om approximations. In general, the cubic term dominates the complexity since the mixture weights can be computed in constant time for the uniform method, in O(psn) for the exponential weight method, or in O(p3+ pms) for the ridge regression method. Furthermore, although the ensemble Nystr¨om algorithm requires p times more space and CPU cycles than the standard Nystr¨om method, these additional requirements are quite reasonable in practice. The space requirement is still manageable for even large-scale applications given that p is typically O(1) and m is usually a very small percentage of n (see Section 4 for further details). In terms of CPU requirements, we note that our algorithm can be easily parallelized, as all p experts can be computed simultaneously. Thus, with a cluster of p machines, the running time complexity of this algorithm is nearly equal to that of the standard Nystr¨om algorithm with m samples. 3 Theoretical analysis We now present a theoretical analysis of the ensemble Nystr¨om method for which we use as tools some results previously shown by [5] and [9]. As in [9], we shall use the following generalization of McDiarmid’s concentration bound to sampling without replacement [3]. Theorem 1. Let Z1, . . . , Zm be a sequence of random variables sampled uniformly without replacement from a fixed set of m + u elements Z, and let φ: Zm →R be a symmetric function such that for all i ∈[1, m] and for all z1, . . . , zm ∈Z and z′ 1, . . . , z′ m ∈Z, |φ(z1, . . . , zm) − φ(z1, . . . , zi−1, z′ i, zi+1, . . . , zm)|≤c. Then, for all ǫ>0, the following inequality holds: Pr φ −E[φ] ≥ǫ ≤exp −2ǫ2 α(m,u)c2 , (5) where α(m, u) = mu m+u−1/2 1 1−1/(2 max{m,u}). We define the selection matrix corresponding to a sample of m columns as the matrix S ∈Rn×m defined by Sii =1 if the ith column of K is among those sampled, Sij =0 otherwise. Thus, C=KS is the matrix formed by the columns sampled. Since K is SPSD, there exists X ∈RN×n such that K = X⊤X. We shall denote by Kmax the maximum diagonal entry of K, Kmax = maxi Kii, and by dK max the distance maxij p Kii + Kjj −2Kij. 3.1 Error bounds for the standard Nystr¨om method The following theorem gives an upper bound on the norm-2 error of the Nystr¨om approximation of the form ∥K−eK∥2/∥K∥2 ≤∥K−Kk∥2/∥K∥2 +O(1/√m) and an upper bound on the Frobenius 3 error of the Nystr¨om approximation of the form ∥K −eK∥F /∥K∥F ≤∥K −Kk∥F /∥K∥F + O(1/m 1 4 ). Note that these bounds are similar to the bounds in Theorem 3 in [9], though in this work we give new results for the spectral norm and present a tighter Lipschitz condition (9), the latter of which is needed to derive tighter bounds in Section 3.2. Theorem 2. Let eK denote the rank-k Nystr¨om approximation of K based on m columns sampled uniformly at random without replacement from K, and Kk the best rank-k approximation of K. Then, with probability at least 1 −δ, the following inequalities hold for any sample of size m: ∥K −eK∥2 ≤∥K −Kk∥2 + 2n √mKmax h 1 + q n−m n−1/2 1 β(m,n) log 1 δ dK max/K 1 2max i ∥K −eK∥F ≤∥K −Kk∥F + 64k m 1 4 nKmax h 1 + q n−m n−1/2 1 β(m,n) log 1 δ dK max/K 1 2max i 1 2 , where β(m, n) = 1− 1 2 max{m,n−m}. Proof. To bound the norm-2 error of the Nystr¨om method in the scenario of sampling without replacement, we start with the following general inequality given by [5][proof of Lemma 4]: ∥K −eK∥2 ≤∥K −Kk∥2 + 2∥XX⊤−ZZ⊤∥2, (6) where Z = p n m XS. We then apply the McDiarmid-type inequality of Theorem 1 to φ(S) = ∥XX⊤−ZZ⊤∥2. Let S′ be a sampling matrix selecting the same columns as S except for one, and let Z′ denote p n m XS′. Let z and z′ denote the only differing columns of Z and Z′, then |φ(S′) −φ(S)| ≤∥z′z′⊤−zz⊤∥2 = ∥(z′ −z)z′⊤+ z(z′ −z)⊤∥2 (7) ≤2∥z′ −z∥2 max{∥z∥2, ∥z′∥2}. (8) Columns of Z are those of X scaled by p n/m. The norm of the difference of two columns of X can be viewed as the norm of the difference of two feature vectors associated to K and thus can be bounded by dK. Similarly, the norm of a single column of X is bounded by K 1 2max. This leads to the following inequality: |φ(S′) −φ(S)| ≤2n m dK maxK 1 2max. (9) The expectation of φ can be bounded as follows: E[Φ] = E[∥XX⊤−ZZ⊤∥2] ≤E[∥XX⊤−ZZ⊤∥F ] ≤ n √mKmax, (10) where the last inequality follows Corollary 2 of [9]. The inequalities (9) and (10) combined with Theorem 1 give a bound on ∥XX⊤−ZZ⊤∥2 and yield the statement of the theorem. The following general inequality holds for the Frobenius error of the Nystr¨om method [5]: ∥K −eK∥2 F ≤∥K −Kk∥2 F + √ 64k ∥XX⊤−ZZ⊤∥2 F nKmax ii . (11) Bounding the term ∥XX⊤−ZZ⊤∥2 F as in the norm-2 case and using the concentration bound of Theorem 1 yields the result of the theorem. 3.2 Error bounds for the ensemble Nystr¨om method The following error bounds hold for ensemble Nystr¨om methods based on a convex combination of Nystr¨om approximations. Theorem 3. Let S be a sample of pm columns drawn uniformly at random without replacement from K, decomposed into p subsamples of size m, S1, . . . , Sp. For r ∈[1, p], let eKr denote the rank-k Nystr¨om approximation of K based on the sample Sr, and let Kk denote the best rank-k approximation of K. Then, with probability at least 1 −δ, the following inequalities hold for any sample S of size pm and for any µ in the simplex ∆and eKens = Pp r=1 µr eKr: ∥K −eKens∥2 ≤∥K −Kk∥2 + 2n √mKmax h 1 + µmaxp 1 2 q n−pm n−1/2 1 β(pm,n) log 1 δ dK max/K 1 2max i ∥K −eKens∥F ≤∥K −Kk∥F + 64k m 1 4 nKmax h 1 + µmaxp 1 2 q n−pm n−1/2 1 β(pm,n) log 1 δ dK max/K 1 2max i 1 2 , where β(pm, n) = 1− 1 2 max{pm,n−pm} and µmax = maxp r=1 µr. 4 Proof. For r ∈[1, p], let Zr = p n/m XSr, where Sr denotes the selection matrix corresponding to the sample Sr. By definition of eKens and the upper bound on ∥K −eKr∥2 already used in the proof of theorem 2, the following holds: ∥K −eKens∥2 =
p X r=1 µr(K −eKr)
2 ≤ p X r=1 µr∥K −eKr∥2 (12) ≤ p X r=1 µr ∥K −Kk∥2 + 2∥XX⊤−ZrZ⊤ r ∥2 (13) = ∥K −Kk∥2 + 2 p X r=1 µr∥XX⊤−ZrZ⊤ r ∥2. (14) We apply Theorem 1 to φ(S) = Pp r=1 µr∥XX⊤−ZrZ⊤ r ∥2. Let S′ be a sample differing from S by only one column. Observe that changing one column of the full sample S changes only one subsample Sr and thus only one term µr∥XX⊤−ZrZ⊤ r ∥2. Thus, in view of the bound (9) on the change to ∥XX⊤−ZrZ⊤ r ∥2, the following holds: |φ(S′) −φ(S)| ≤2n m µmaxdK maxK 1 2max, (15) The expectation of Φ can be straightforwardly bounded by E[Φ(S)] = Pp r=1 µr E[∥XX⊤− ZrZ⊤ r ∥2] ≤Pp r=1 µr n √mKmax = n √mKmax using the bound (10) for a single expert. Plugging in this upper bound and the Lipschitz bound (15) in Theorem 1 yields our norm-2 bound for the ensemble Nystr¨om method. For the Frobenius error bound, using the convexity of the Frobenius norm square ∥·∥2 F and the general inequality (11), we can write ∥K −eKens∥2 F =
p X r=1 µr(K −eKr)
2 F ≤ p X r=1 µr∥K −eKr∥2 F (16) ≤ p X r=1 µr h ∥K −Kk∥2 F + √ 64k ∥XX⊤−ZrZ⊤ r ∥F nKmax ii i . (17) = ∥K −Kk∥2 F + √ 64k p X r=1 µr∥XX⊤−ZrZ⊤ r ∥F nKmax ii . (18) The result follows by the application of Theorem 1 to ψ(S)=Pp r=1 µr∥XX⊤−ZrZ⊤ r ∥F in a way similar to the norm-2 case. The bounds of Theorem 3 are similar in form to those of Theorem 2. However, the bounds for the ensemble Nystr¨om are tighter than those for any Nystr¨om expert based on a single sample of size m even for a uniform weighting. In particular, for µ=1/p, the last term of the ensemble bound for norm-2 is smaller by a factor larger than µmaxp 1 2 = 1/√p. 4 Experiments In this section, we present experimental results that illustrate the performance of the ensemble Nystr¨om method. We work with the datasets listed in Table 1. In Section 4.1, we compare the performance of various methods for calculating the mixture weights (µr). In Section 4.2, we show the effectiveness of our technique on large-scale datasets. Throughout our experiments, we measure the accuracy of a low-rank approximation eK by calculating the relative error in Frobenius and spectral norms, that is, if we let ξ = {2, F}, then we calculate the following quantity: % error = ∥K −eK∥ξ ∥K∥ξ × 100. (19) 5 Dataset Type of data # Points (n) # Features (d) Kernel PIE-2.7K [16] face images 2731 2304 linear MNIST [10] digit images 4000 784 linear ESS [8] proteins 4728 16 RBF AB-S [1] abalones 4177 8 RBF DEXT [1] bag of words 2000 20000 linear SIFT-1M [12] Image features 1M 128 RBF Table 1: A summary of the datasets used in the experiments. 4.1 Ensemble Nystr¨om with various mixture weights In this set of experiments, we show results for our ensemble Nystr¨om method using different techniques to choose the mixture weights as discussed in Section 2.2. We first experimented with the first five datasets shown in Table 1. For each dataset, we fixed the reduced rank to k=50, and set the number of sampled columns to m = 3% n.1 Furthermore, for the exponential and the ridge regression variants, we sampled an additional set of s = 20 columns and used an additional 20 columns (s′) as a hold-out set for selecting the optimal values of η and λ. The number of approximations, p, was varied from 2 to 30. As a baseline, we also measured the minimal and mean percent error across the p Nystr¨om approximations used to construct eKens. For the Frobenius norm, we also calculated the performance when using the optimal µ, that is, we used least-square regression to find the best possible choice of combination weights for a fixed set of p approximations by setting s=n. The results of these experiments are presented in Figure 1 for the Frobenius norm and in Figure 2 for the spectral norm. These results clearly show that the ensemble Nystr¨om performance is significantly better than any of the individual Nystr¨om approximations. Furthermore, the ridge regression technique is the best of the proposed techniques and generates nearly the optimal solution in terms of the percent error in Frobenius norm. We also observed that when s is increased to approximately 5% to 10% of n, linear regression without any regularization performs about as well as ridge regression for both the Frobenius and spectral norm. Figure 3 shows this comparison between linear regression and ridge regression for varying values of s using a fixed number of experts (p = 10). Finally we note that the ensemble Nystr¨om method tends to converge very quickly, and the most significant gain in performance occurs as p increases from 2 to 10. 4.2 Large-scale experiments Next, we present an empirical study of the effectiveness of the ensemble Nystr¨om method on the SIFT-1M dataset in Table 1 containing 1 million data points. As is common practice with large-scale datasets, we worked on a cluster of several machines for this dataset. We present results comparing the performance of the ensemble Nystr¨om method, using both uniform and ridge regression mixture weights, with that of the best and mean performance across the p Nystr¨om approximations used to construct eKens. We also make comparisons with a recently proposed k-means based sampling technique for the Nystr¨om method [19]. Although the k-means technique is quite effective at generating informative columns by exploiting the data distribution, the cost of performing k-means becomes expensive for even moderately sized datasets, making it difficult to use in large-scale settings. Nevertheless, in this work, we include the k-means method in our comparison, and we present results for various subsamples of the SIFT-1M dataset, with n ranging from 5K to 1M. To fairly compare these techniques, we performed ‘fixed-time’ experiments. To do this, we first searched for an appropriate m such that the percent error for the ensemble Nystr¨om method with ridge weights was approximately 10%, and measured the time required by the cluster to construct this approximation. We then alloted an equal amount of time (within 1 second) for the other techniques, and measured the quality of the resulting approximations. For these experiments, we set k=50 and p=10, based on the results from the previous section. Furthermore, in order to speed up computation on this large dataset, we decreased the size of the validation and hold-out sets to s=2 and s′ =2, respectively. 1Similar results (not reported here) were observed for other values of k and m as well. 6 0 5 10 15 20 25 30 3 3.5 4 4.5 Number of base learners (p) Percent Error (Frobenius) Ensemble Method − PIE−2.7K mean b.l. best b.l. uni exp ridge optimal 0 5 10 15 20 25 30 10 11 12 13 14 15 16 Number of base learners (p) Percent Error (Frobenius) Ensemble Method − MNIST mean b.l. best b.l. uni exp ridge optimal 0 5 10 15 20 25 30 0.4 0.45 0.5 0.55 0.6 0.65 Number of base learners (p) Percent Error (Frobenius) Ensemble Method − ESS mean b.l. best b.l. uni exp ridge optimal 0 5 10 15 20 25 30 24 26 28 30 32 34 36 38 40 Number of base learners (p) Percent Error (Frobenius) Ensemble Method − AB−S mean b.l. best b.l. uni exp ridge optimal 0 5 10 15 20 25 30 52 54 56 58 60 62 64 66 68 70 Number of base learners (p) Percent Error (Frobenius) Ensemble Method − DEXT mean b.l. best b.l. uni exp ridge optimal Figure 1: Percent error in Frobenius norm for ensemble Nystr¨om method using uniform (‘uni’), exponential (‘exp’), ridge (‘ridge’) and optimal (‘optimal’) mixture weights as well as the best (‘best b.l.’) and mean (‘mean b.l.’) performance of the p base learners used to create the ensemble approximation. 0 5 10 15 20 25 30 0.8 1 1.2 1.4 1.6 1.8 2 Number of base learners (p) Percent Error (Spectral) Ensemble Method − PIE−2.7K mean b.l. best b.l. uni exp ridge 0 5 10 15 20 25 30 2 3 4 5 6 7 8 9 10 Number of base learners (p) Percent Error (Spectral) Ensemble Method − MNIST mean b.l. best b.l. uni exp ridge 0 5 10 15 20 25 30 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 Number of base learners (p) Percent Error (Spectral) Ensemble Method − ESS mean b.l. best b.l. uni exp ridge 0 5 10 15 20 25 30 10 15 20 25 30 35 40 45 Number of base learners (p) Percent Error (Spectral) Ensemble Method − AB−S mean b.l. best b.l. uni exp ridge 0 5 10 15 20 25 30 10 15 20 25 30 35 40 45 Number of base learners (p) Percent Error (Spectral) Ensemble Method − DEXT mean b.l. best b.l. uni exp ridge Figure 2: Percent error in spectral norm for ensemble Nystr¨om method using various mixture weights as well as the best and mean performance of the p approximations used to create the ensemble approximation. Legend entries are the same as in Figure 1. The results of this experiment, presented in Figure 4, clearly show that the ensemble Nystr¨om method is the most effective technique given a fixed amount of time. Furthermore, even with the small values of s and s′, ensemble Nystr¨om with ridge-regression weighting outperforms the uniform ensemble Nystr¨om method. We also observe that due to the high computational cost of k-means for large datasets, the k-means approximation does not perform well in this ‘fixed-time’ experiment. It generates an approximation that is worse than the mean standard Nystr¨om approximation and its performance increasingly deteriorates as n approaches 1M. Finally, we note that al7 5 10 15 20 25 3.35 3.4 3.45 3.5 Relative size of validation set Percent Error (Frobenius) Effect of Ridge − PIE−2.7K no−ridge ridge optimal 5 10 15 20 25 10.495 10.5 10.505 10.51 10.515 10.52 10.525 Relative size of validation set Percent Error (Frobenius) Effect of Ridge − MNIST no−ridge ridge optimal 0 5 10 15 20 25 0.445 0.45 0.455 Relative size of validation set Percent Error (Frobenius) Effect of Ridge − ESS no−ridge ridge optimal 0 5 10 15 20 25 26 26.5 27 27.5 28 28.5 Relative size of validation set Percent Error (Frobenius) Effect of Ridge − AB−S no−ridge ridge optimal 5 10 15 20 25 54.5 55 55.5 56 Relative size of validation set Percent Error (Frobenius) Effect of Ridge − DEXT no−ridge ridge optimal Figure 3: Comparison of percent error in Frobenius norm for the ensemble Nystr¨om method with p= 10 experts with weights derived from linear regression (‘no-ridge’) and ridge regression (‘ridge’). The dotted line indicates the optimal combination. The relative size of the validation set equals s/n×100%. 10 4 10 5 10 6 9 10 11 12 13 14 15 16 17 Size of dataset (n) Percent Error (Frobenius) Large Scale Ensemble Study mean b.l. best b.l. uni ridge kmeans Figure 4: Large-scale performance comparison with SIFT-1M dataset. Given fixed computational time, ensemble Nystr¨om with ridge weights tends to outperform other techniques. though the space requirements are 10 times greater for ensemble Nystr¨om in comparison to standard Nystr¨om (since p = 10 in this experiment), the space constraints are nonetheless quite reasonable. For instance, when working with the full 1M points, the ensemble Nystr¨om method with ridge regression weights only required approximately 1% of the columns of K to achieve a percent error of 10%. 5 Conclusion We presented a novel family of algorithms, ensemble Nystr¨om algorithms, for accurate low-rank approximations in large-scale applications. The consistent and significant performance improvement across a number of different data sets, along with the fact that these algorithms can be easily parallelized, suggests that these algorithms can benefit a variety of applications where kernel methods are used. Interestingly, the algorithmic solution we have proposed for scaling these kernel learning algorithms to larger scales is itself derived from the machine learning idea of ensemble methods. We also gave the first theoretical analysis of these methods. We expect that finer error bounds and theoretical guarantees will further guide the design of the ensemble algorithms and help us gain a better insight about the convergence properties of our algorithms. 8 References [1] A. Asuncion and D. Newman. UCI machine learning repository, 2007. [2] B. E. Boser, I. Guyon, and V. N. Vapnik. A training algorithm for optimal margin classifiers. In COLT, volume 5, pages 144–152, 1992. [3] C. Cortes, M. Mohri, D. Pechyony, and A. Rastogi. Stability of transductive regression algorithms. In ICML, 2008. [4] C. Cortes and V. N. Vapnik. Support-Vector Networks. Machine Learning, 20(3):273–297, 1995. [5] P. Drineas and M. W. Mahoney. On the Nystr¨om method for approximating a Gram matrix for improved kernel-based learning. JMLR, 6:2153–2175, 2005. [6] C. Fowlkes, S. Belongie, F. Chung, and J. Malik. Spectral grouping using the Nystr¨om method. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(2), 2004. [7] G. Golub and C. V. Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, 2nd edition, 1983. [8] A. Gustafson, E. Snitkin, S. Parker, C. DeLisi, and S. Kasif. Towards the identification of essential genes using targeted genome sequencing and comparative analysis. BMC:Genomics, 7:265, 2006. [9] S. Kumar, M. Mohri, and A. Talwalkar. Sampling techniques for the Nystr¨om method. In AISTATS, pages 304–311, 2009. [10] Y. LeCun and C. Cortes. The MNIST database of handwritten digits, 2009. [11] N. Littlestone and M. K. Warmuth. The weighted majority algorithm. Information and Computation, 108(2):212261, 1994. [12] D. G. Lowe. Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60:91–110, 2004. [13] J. C. Platt. Fast embedding of sparse similarity graphs. In NIPS, 2004. [14] C. Saunders, A. Gammerman, and V. Vovk. Ridge Regression Learning Algorithm in Dual Variables. In Proceedings of the ICML ’98, pages 515–521, 1998. [15] B. Sch¨olkopf, A. Smola, and K.-R. M¨uller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10(5):1299–1319, 1998. [16] T. Sim, S. Baker, and M. Bsat. The CMU PIE database. In Conference on Automatic Face and Gesture Recognition, 2002. [17] A. Talwalkar, S. Kumar, and H. Rowley. Large-scale manifold learning. In CVPR, 2008. [18] C. K. I. Williams and M. Seeger. Using the Nystr¨om method to speed up kernel machines. In NIPS, pages 682–688, 2000. [19] K. Zhang, I. Tsang, and J. Kwok. Improved Nystr¨om low-rank approximation and error analysis. In ICML, pages 273–297, 2008. 9
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Manifold Regularization for SIR with Rate Root-n Convergence Wei Bian School of Computer Engineering Nanyang Technological University Singapore, 639798 weibian@pmail.ntu.edu.sg Dacheng Tao School of Computer Engineering Nanyang Technological University Singapore, 639798 dctao@ntu.edu.sg Abstract In this paper, we study the manifold regularization for the Sliced Inverse Regression (SIR). The manifold regularization improves the standard SIR in two aspects: 1) it encodes the local geometry for SIR and 2) it enables SIR to deal with transductive and semi-supervised learning problems. We prove that the proposed graph Laplacian based regularization is convergent at rate root-n. The projection directions of the regularized SIR are optimized by using a conjugate gradient method on the Grassmann manifold. Experimental results support our theory. 1 Introduction Sliced inverse regression (SIR) [7] was proposed for sufficient dimension reduction. In a regression setting, with the predictors X and the response Y, the sufficient dimension reduction (SDR) subspace B is defined by the conditional independency Y⊥X| BTX. Under the assumption that the distribution of X is elliptic symmetric [7], it has been proved that the SDR subsapce B is related to the inverse regression curve E(X|Y). It can be estimated at least partially by a generalized eigendecomposition between the covariance matrix of the predictors Cov(X) and the covariance matrix of the inverse regression curve Cov(E(X|Y)). When Y is a continuous random variable, it is discretized by slicing its range into several slices so as to estimate E(X|Y) empirically. This procedure reflects the name of SIR. For practical applications, the elliptic symmetric assumption on P(X) in SIR cannot be fully satisfied, because many real datasets are embedded on manifolds [1]. Therefore, SIR cannot select an efficient subspace for predicting the response Y because the local geometry of the predictors X is ignored. Additionally, SIR only utilizes labeled (given response) data (predictors). Thus, it is valuable to extend SIR to deal with transductive and semi-supervised learning problems by considering unlabelled samples. We solve the above two problems of SIR by using the manifold regularization [2], which has been developed to incorporate the local geometry in learning classification or regression functions. In this paper, we utilize it to preserve the local geometry of predictors in learning the SDR subspace B. In addition, it helps SIR to solve transductive/semi-supervised learning problems because the regularization encodes the marginal distribution of the unlabelled predictors. Different regularizations for SIR have been well studied, e.g., the non-singular regularization [14], the ridge regularization [9], and the sparse regularization [8]. However, all existing regularizations do not encode the local geometry of the predictors. Although the localized sliced inverse regression [12] considers the local geometry, it is heuristic and does not follow up the regularization framework. The rest of the paper is organized as following. Section 2 presents the manifold regularization for SIR. Section 3 proves the convergence of the new manifold regularization. We discuss the optimiza1 tion algorithm of the regularized SIR by using the conjugate gradient method on the Grassmann manifold in Section 4. Section 5 presents the experimental results on synthetic and real datasets. Section 6 concludes this paper. 2 Manifold Regularization for SIR In the rest of the paper, we use terminologies in regression and deem classification as regression with the category response. Upper case letters X ∈Rp and Y ∈R are respectively the predictors and the response, and lower case letters x and y are corresponding realizations. Given a sample set, containing nl labeled samples {xi, yi}nl i=1 and nu unlabeled samples {xi}n=nl+nu i=nl+1 , we seek an optimal k-dimensional subspace spanned by B = [β1, ..., βk] such that the response Y is predictable with the projected predictors BT X. We also use matrix X = [x1, x2, ..., xn] to denote all predictors in the sample set. 2.1 Sliced Inverse Regression Suppose the response Y is predictable with a sufficient k-dimensional projection of the original predictors X. We can consider the following regression model [7]. Y = f ¡ βT 1 X, βT 2 X, ..., βT k X, ε ¢ (1) where β’s are linear independent projection vectors and ε is the independent noise. Given a set of samples {xi, yi}nl i=1, SIR estimates the projection subspace B = [β1, ..., βk] via following steps: discretize Y by slicing its range into H slices; calculate the sample frequency fh of Y falling into the h-th slice and the sample estimation of the conditional mean ¯Xh = E(X|Y = h); estimate the mean ¯X and covariance matrix Σ of predictors X; calculate the matrix Γ = P h fh ¡¯Xh −¯X ¢ ¡¯Xh −¯X ¢T ; and B is finally obtained by using the generalized eigen-decomposition Σβ = λΓβ. It can be proved that the generalized eigen-decomposition is equivalent to the following optimization, max B trace ³¡ BT ΣB ¢−1 BT ΓB ´ . (2) We refer to (2) as the objective function of SIR and thus we can impose with the manifold regularization on (2). Remark 2.1 Another way to get the objective (2) is based on the least square formulation for SIR proposed in [3], min B L (B, C) = H X h=1 fh ¡¯Xh −¯X −ΣBCh ¢T Σ−1 ¡¯Xh −¯X −ΣBCh ¢ (3) where C = [C1, C2, ..., Ch] are auxiliary variables. Eliminate Ch by setting the partial derivative ∂L/∂Ch = 0, and then (2) can be obtained directly. Additionally, (2) shows that SIR could have a similar objective as linear discriminant analysis, although they are obtained from different understandings of discriminative dimension reduction. 2.2 Manifold Regularization for SIR Each dimension reduction projection β can be deemed as a linear function or a mapping g(x) = βT x. We expect to preserve the local geometry of the distribution of the predictors X while doing mapping g(x). Suppose the predictors X are embedded on a manifold M, this can be achieved by penalizing the gradient ∇Mg along the manifold M. Because we are dealing with random variables with the distribution P(X), the following formulation can be applied, R = Z X∈M ∥∇Mg∥2 dP (X). (4) The above formulation is different from the original manifold regularization [2]on the point that the function g(x) is a dimension reduction mapping here while it is a classification or regression 2 function in [2]. Usually, both the manifold and the marginal distribution of X are unknown. It has been well studied in manifold learning, however, that the regularization (4) can be approximated by using the associated graph Laplacian of labeled and unlabeled {xi}n=nl+nu i=1 . Construct an adjacent graph for {xi}n=n1+nu i=1 , where the pairwise edge weight (W)ij = φ (∥xi −xj∥) is defined by the kernel function φ (·), e.g., the heat kernel φ (d) = exp ¡ −d2¢ , and then the associated graph Laplacian is L = D −W, where D is a diagonal matrix given by Dii = P j Wij. Thus, the regularization in (4) can be approximated by R = gT Lg, where g = [βT x1, ..., βT xn]. Furthermore, because there are k independent projections B = [β1, ..., βk] , we take the summation of k regularizations R = k X i=1 gT i Lgi = trace ¡ GT LG ¢ (5) where G = [g1, ..., gk]. In manifold learning, it is suggested to use the normalized graph Laplacian D−1/2LD−1/2 to replace L, or to use an equivalent constraint GT DG = I, to get a better performance [1], and the solution obtained by the normalized graph Laplacian is consistent with weaker conditions than the unnormalized one [13]. In the proposed regularized SIR, we normalize the regularization (5) as R = trace ³¡ GT DG ¢−1 GT LG ´ , which is equivalent to the constraint GT DG = I. This normalization makes R invariant to scalar and rotation transformations of the projections B = [β1, ..., βk], which is preferred for dimension reduction problems. By adding the regularization R = trace ³¡ GT DG ¢−1 GT LG ´ to SIR (2), and substituting G = XT B, we get the regularized SIR max B SIRr (B) = trace ³¡ BT ΣB ¢−1 BT ΓB ´ −ηtrace ³¡ BT SB ¢−1 BT QB ´ (6) where Q = 1/n (n −1) XLXT , S = 1/n (n −1) XDXT , and η is the positive weighting factor. 3 Convergence of the Regularization Different from the existing regularizations [8,9,14] for SIR, which are constructed as deterministic terms, the manifold regularization in (6) is a random term that involves two data dependent variables (matrices) Q and S. Therefore, it is necessary to discuss the convergence property of the proposed manifold regularization. It has been well proved that both Σ and Γ converge at rate root-n [7,11,15]. Therefore, the convergence rate of the objective (6) depends on whether the regularization term converges at rate root-n. Below, we prove that both the sample based estimations Q = 1/n (n −1) XLXT and S = 1/n (n −1) XDXT converge to deterministic matrices at rate root-n. Note that the convergence of a special case where the graph Laplacian is built by the kernel function φ (d) = 1 (d < ε) was proved in [6]. Our proof scheme, however, is quite other than that used in [6]. Additionally, we target a general choice of kernel φ (·) and also prove the root-n convergence rate which has not been obtained before. Although samples {xi}n=nl+nu i=1 are independent, the dependency of L and D on samples makes Q and S cannot be expanded as a summation of independent items. Therefore, it is difficult to apply the law of large numbers and the central limit theorem to prove the convergence and obtain the corresponding convergence rate. However, we can prove them by constructing the converged limitation and show that the variance of the sample based estimation with respect to the constructed limitation decades at rate root-n. Throughout the results obtained in this Section, we assume the following conditions hold. Conditions 3.1 For kernel function φ (d) , it satisfies φ (0) = 1 and |φ (d)| ⩽1. For the distribution of predictors P(X), the fourth order moment exists, i.e.,E ³°°° ¡ vec(xxT ) ¢ ¡ vec(xxT ) ¢T °°° ´ < ∞, where vec() vectorizes a matrix into a column vector. 3 We start by splitting Q into two parts T1 and T2, Q = 1 n (n −1)XLXT = 1 n (n −1) n X i=1 (Dii −Wii) xixT i − 1 n (n −1) n X i̸=j WijxixT j = T1 −T2. (7) Substituting the function φ (·) into (7), we have T1 = 1 n(n−1) nP i=1 à nP j=1 φ (∥xi −xj∥) −φ (0) ! xixT i = 1 n nP i=1 à 1 n−1 nP j̸=i φ (∥xi −xj∥) ! xixT i T2 = 1 n(n−1) nP i̸=j WijxixT j = 1 n nP i xi à 1 n−1 nP j̸=i φ (∥xi −xj∥) xT j ! . (8) Under the condition 3.1, the next two lemmas show the convergence of T1 and T2, respectively. Lemma 3.1 Let the conditional expectation ϕ (x) = E (φ (∥z −x∥) |x), wherez and x are independent and both are sampled from P(X). The E ¡ ϕ (x) xxT ¢ exists, and T1 in (8) converges almost surely at rate n−1/2, i.e., T1 a.s = E ¡ ϕ (x) xxT ¢ + O ³ n−1/2´ . (9) Lemma 3.2 Let the conditional expectation η (x) = E (φ (∥z −x∥) z |x), where z and x are independent and both are sampled from P(X). The E ³ xη (x)T ´ exists, and T2 in (8) converges almost surely at rate n−1/2, i.e., T2 a.s = E ³ xη (x)T ´ + O ³ n−1/2´ . (10) The proofs of above two lemmas are given in Section 6. Based on Lemmas 1 and 2, we have the following two theorems for the convergence of Q and S. Theorem 3.1 Given the Conditions 3.1, the sample based estimation Q converges almost surely to a deterministic matrix E (Q) = E ¡ ϕ (x) xxT ¢ −E ³ xη (x)T ´ at rate n−1/2, i.e., Q a.s = E (Q) + O ¡ n−1/2¢ . Proof. Because Q = T1 −T2, the theorem is an immediate result from Lemmas 3.1 and 3.2. Theorem 3.2 Given the Conditions 3.1, the sample based estimation S converges almost surely to a deterministic matrix E ¡ ϕ (x) xxT ¢ at rate n−1/2, i.e., S a.s = E ¡ ϕ (x) xxT ¢ + O ¡ n−1/2¢ . Proof. Dii = P j Wij = nP j=1 φ (∥xi −xj∥), so S = 1 n(n−1) nP i=1 DiixixT i = 1 n(n−1) nP i=1 à nP j=1 φ (∥xi −xj∥) ! xixT i = 1 n(n−1) nP i=1 à P j̸=i φ (∥xi −xj∥) + φ (0) ! xixT i = T1 + 1 n(n−1) nP i=1 xixT i . Because 1 (n−1) nP i=1 xixT i is an unbiased estimation of Cov(X), we have 1 n(n−1) nP i=1 xixT i a.s. = O ¡ n−1¢ . Therefore, according to Lemma 3.1, we have S = T1 + O ¡ n−1¢ a.s. = E ¡ ϕ (x) xxT ¢ + O ¡ n−1/2¢ . Note that here E (S) ̸= E ¡ ϕ (x) xxT ¢ , but equality can be asymptotically achieved when n →∞. 4 Optimization on the Grassmann Manifold The optimization of the regularized SIR (6) is much more difficult than that of the standard SIR (2), which can be solved by the generalized eigen-decomposition. In this section, we present a conjugate 4 gradient method on the Grassmann manifold to solve (6), based on the fact it is invariant to scalar and rotation transformations of the projection B. By exploiting the geometry of the Grassmann manifold, the conjugate gradient algorithm converges faster than the gradient scheme in the Euclidean space. Given a constrained optimization problem min F (A) subject to A ∈Rp×k and AT A = I, if the problem further satisfies F (A) = F (AO) for an arbitrary orthonormal matrix O, then it is called an optimization problem defined on the Grassmann manifold Gpk. By the following theorem, we can transform (6) into its equivalent form (11) which is defined on the Grassmann manifold. Theorem 4.1 Suppose that Σ is nonsingular and given the eigen-decomposition Σ−1/2SΣ−1/2 = U ˜ΛU T , problem (6) is equivalent to min AT A=I F (A) = −trace ³ AT ˜ΓA ´ + ηtrace µ³ AT ˜ΛA ´−1 AT ˜QA ¶ (11) where ˜Γ = U T Σ−1/2ΓΣ−1/2U and ˜Q = U T Σ−1/2QΣ−1/2U. Given the optimal solution A of (11), the optimal solution of (6) is given by B = Σ−1/2UA . Proof. Substituting B = Σ−1/2UA into (6), we have SIRr (A) = trace ³¡ AT A ¢−1 AT ˜ΓA ´ − ηtrace µ³ AT ˜ΛA ´−1 AT ˜QA ¶ . Given a nonsingular Σ, B = Σ−1/2UA is an invertible variable transform. Thus, we know that if A maximizes SIRr (A) then B maximizes SIRr (B). Because SIRr (A) is invariant to scalar and rotation transformations, a constraint AT A = I can be added to (6). We then get (11). This completes the proof. To implement the conjugate gradient method on the Grassmann manifold, the gradient of F (A) in (11) is required. According to [4], the gradient GA of F (A) on the manifold is defined by GA = ΠAFA where FA is the gradient of F (A) in the Euclidian space and ΠA = I −AAT is the projection onto the tangent space at A of the manifold. In case of F (A) in (11), it is given by, GA = ¡ I −AAT ¢ ˜ΓA −η µ I −˜ΛA ³ AT ˜ΛA ´−1 AT ¶ ˆQA ³ AT ˜ΛA ´−1 . (12) Next, we present the conjugate gradient method on the Grassmann manifold [4] to solve (11). The algorithm is given by the following three steps: • 1-D searching along the geodesic: given the current position Ak , the gradient Gk and the searching direction Hk , the 1-D searching along the geodesic is given by min t F (A (t)) s.t. A (t) = F ¡ AkV cos (Σt) V T + U sin (Σt) V T ¢ (13) where UΣV T is the compact SVD of Hk. Record the minimum solution tk = tmin, and Ak+1 = A (tk) as the starting position for next searching. • Transporting gradient and search direction: parallel transport Gk and Hk from Ak to Ak+1 by using τGk = Gk −(AkV sin Σtk + U (I −cos Σtk)) U T Gk (14) τHk = (−AkV sin Σtk + U cos Σtk) ΣV T (15) • Calculating the conjugate direction: given the gradient Gk+1 at Ak+1, the conjugate searching direction is Hk+1 = −Gk+1 + trace ³ (Gk+1 −τGk)T Gk+1 ´ /trace ¡ GT k Gk ¢ τHk. (16) Initialize A0 by a random guess (subject to AT 0 A0 = I ) and let H0 = −G0, and then repeat the above three steps iteratively to minimize F (A) until convergence, i.e., |F (Ak+1) −F (Ak)| < ε0. Note that, the same as the conjugate gradient method in the Euclidian space, the searching direction Hk has to be resetting as Hk = −Gk with a period of p (n −p), i.e., the dimension of the searching space. 5 5 Experiments In this section, we evaluate the proposed regularized SIR on two real datasets. We show the results of the standard SIR and the localized SIR on the same experiments for reference. 5.1 USPS Test The USPS dataset contains 9,298 handwriting characters of digits 0 to 9. The entire USPS database is divided into two parts, a training set is with 7,291 samples and a test set is with 2,007 samples [5]. In our experiment, dimension reduction is first implemented and then the nearest neighbor rule is used for classification. By using the 1/3 of the data in training set as labeled data and the rest 2/3 as unlabeled data, we conduct supervised and semisupervised dimension reduction by the following five methods: supervised training of standard SIR, the manifold regularized SIR, and the localized SIR, and semi-supervised training of the manifold regularized SIR and the localized SIR. Performances are evaluated on the independent testing set. Table 1 summarizes the experimental results. It shows that both the regularized SIR and the localized SIR [12] can achieve superior performance to the standard SIR, and the manifold regularized SIR performs better than the localized SIR in both the supervised and the semi-supervised training. Experimental results reflect that the manifold regularized SIR is effective on exploiting the local geometry of a dataset. Table 1: Experimental results on the USPS dataset: SIR; the manifold regularized SIR (RSIR); the localized SIR (LSIR); semi-supervised training of the manifold regularized SIR (sRSIR); semisupervised training of the localized SIR (sLSIR). Dimensionality 7 9 11 13 15 17 19 21 SIR 0.8635 0.8794 — — — — — — RSIR 0.8575 0.8809 0.8859 0.8889 0.9028 0.9108 0.9148 0.9193 sRSIR 0.8685 0.8864 0.8934 0.8909 0.9053 0.9128 0.9208 0.9193 LSIR 0.8301 0.8421 0.8535 0.8724 0.8789 0.8949 0.8989 0.9003 sLSIR 0.8526 0.8675 0.8795 0.8826 0.8914 0.8954 0.9038 0.9063 5.2 Transductive Visualization In Coil-20 database [10], each object has 72 images taken from different view angles. All images are cropped into 128×128 pixel arrays with 256 gray levels. We then reduce the size to 32×32, and used the first 10 objects for 2-D visualization, with randomly labeled 6 out of 72 images. Figure 1 shows the visualization results obtained by SIR, the proposed regularized SIR and the localized SIR [12]. The figure shows that by exploiting the unlabeled data via the manifold regularization for dimension reduction, the performance for data visualization can be significantly improved. The localized SIR performs better than SIR, but not as good as the regularized SIR. -4000 -3000 -2000 -1000 0 1000 2000 3000 4000 0 1000 2000 3000 4000 5000 6000 -2000 -1500 -1000 -500 0 500 1000 1500 -1500 -1000 -500 0 500 1000 -1500 -1000 -500 0 500 1000 1500 2000 -1500 -1000 -500 0 500 1000 Figure 1: Visualization of the first 10 objects in Coil-20 database: from left to right, by the standard SIR, the manifold regularized SIR, and the localized SIR. 6 6 Proofs of Lemmas Proof of Lemma 3.1 Because the kernel function φ (·) is bounded by |φ (d)| ⩽1, we have |ϕ (x)| = |E (φ (∥z −x∥) |x)| ⩽1, which implies that E ¡ ϕ (x) xxT ¢ exists. Then, to prove T1 a.s = E ¡ ϕ (x) xxT ¢ + O ¡ n−1/2¢ , it is sufficient to show that E (T1) = E ¡ ϕ (x) xxT ¢ and Cov (vec (T1)) = E ³ (vec (T1)) (vec (T1))T ´ −(vec (E (T1))) (vec (E (T1)))T = O ¡ n−1¢ . First, because xi and xj are independent when i ̸= j, it follows, E (T1) = E 1 n n X i=1 1 n −1 n X j̸=i,j=1 φ (∥xi −xj∥) xixT i = 1 n n X i=1 E xixT i 1 n −1 n X j̸=i,j=1 E (φ (∥xi −xj∥) |xi ) = 1 n n X i=1 E ¡ xixT i ϕ (xi) ¢ = E ¡ ϕ (x) xxT ¢ . (17) Next, we show E ³ (vec (T1)) (vec (T1))T ´ is a summation of two terms, of which one is (vec(E (T1))) (vec(E (T1)))T and the other is O ¡ n−1¢ . E ³ (vec (T1)) (vec (T1))T ´ = 1 n2 (n −1)2 E vec n X i̸=j φ(∥xi −xj∥)xixT i vec n X i̸=j φ (∥xi −xj∥) xixT i T = 1 n2 (n −1)2 n X i̸=j n X i′̸=j′ E ³ vec ¡ φ (∥xi −xj∥) xixT i ¢ ¡ vec ¡ φ (∥xi′ −xj′∥) xi′xT i′ ¢¢T ´ = 1 n2 (n −1)2 X i,j,i′,j′distinct E (Φi,j,i′,j′) + 1 n2 (n −1)2 X else E (Φi,j,i′,j′), (18) where Φi,j,i′,j′ = vec ¡ φ (∥xi −xj∥) xixT i ¢ ¡ vec ¡ φ (∥xi′ −xj′∥) xi′xT i′ ¢¢T . When i, j, i′, j′ are distinct, xi,xj,xi′, and xj′ are independent, we have E (Φi,j,i′,j′) = E ³¡ vec ¡ φ (∥xi −xj∥) xixT i ¢¢ ¡ vec ¡ φ (∥xi′ −xj′∥) xi′xT i′ ¢¢T ´ = ¡ vec ¡ E ¡ ϕ (x) xxT ¢¢¢ ¡ vec ¡ E ¡ ϕ (x) xxT ¢¢¢T = (vec (E (T1))) (vec (E (T1)))T . (19) Therefore, the first term in E ³ vec (T1) (vec (T1))T ´ is 1 n2(n−1)2 P i,j,i′,j′ distinct E (Φi,j,i′,j′) = n(n−1)(n−2)(n−3) n2(n−1)2 (vec (E (T1))) (vec (E (T1)))T = (vec (E (T1))) (vec (E (T1)))T + O ¡ n−1¢ . (20) For the second term in E ³ (vec (T1)) (vec (T1))T ´ , E (Φi,j,i′,j′) is bounded by a constant (matrix) M under the Conditions 3.1, and thus we have ¯¯¯¯¯ 1 n2 (n −1)2 X else E (Φi,j,i′,j′) ¯¯¯¯¯ ⩽ 1 n2 (n −1)2 X else M = n(n −1) (4n −6) n2 (n −1)2 M = O ¡ n−1¢ . (21) 7 Combining the above two results, we have Cov (vec (T1)) = E(vec (T1)) (vec (T1))T −(vec (E (T1))) (vec (E (T1)))T = O ¡ n−1¢ (22) Proof of Lemma 3.2 Similar to the proof of Lemma 3.1, E ³ xη (x)T ´ exists. Then, it is sufficient to show that E (T2) = E ³ xη (x)T ´ and Cov (vec (T2)) = O ¡ n−1¢ . First, we have E (T2) = E 1 n n X i=1 xi 1 n −1 n X j̸=i,j=1 φ (∥xi −xj∥) xT j = 1 n n X i=1 E xi 1 n −1 n X j̸=i,j=1 E ¡ φ (∥xi −xj∥) xT j |xi ¢ = 1 n n X i=1 E (xiη (xi)) = E (xη (x)) . (23) Next, we split E ³ (vec (T2)) (vec (T2))T ´ into two terms E ³ (vec (T2)) (vec (T2))T ´ = 1 n2 (n −1)2 E vec n X i̸=j φ (∥xi −xj∥) xixT j vec n X i̸=j φ (∥xi −xj∥) xixT j T = 1 n2 (n −1)2 n X i̸=j n X i′̸=j′ E ³¡ vec ¡ φ (∥xi −xj∥) xixT j ¢¢ ¡ vec ¡ φ (∥xi′ −xj′∥) xi′xT j′ ¢¢T ´ = 1 n2 (n −1)2 X i,j,i′,j′distinct E (Ψi,j,i′,j′) + 1 n2 (n −1)2 X else E (Ψi,j,i′,j′) (24) where Ψi,j,i′,j′ = vec ¡ φ (∥xi −xj∥) xixT j ¢ ¡ vec ¡ φ (∥xi′ −xj′∥) xi′xT j′ ¢¢T . Following the same method used in the proof of Lemma 3.1, we have 1 n2 (n −1)2 X else E (Ψi,j,i′,j′) = (vec (E (T2))) (vec (E (T2)))T + O ¡ n−1¢ (25) and ¯¯¯¯¯ 1 n2 (n −1)2 X else E (Ψi,j,i′,j′) ¯¯¯¯¯ ⩽O ¡ n−1¢ . (26) Therefore, we have Cov (vec (T2)) = O ¡ n−1¢ . 7 Conclusion We have studied the manifold regularization for Sliced Inverse Regression (SIR). The regularized SIR extended the original SIR in many ways, i.e., it utilizes the local geometry that is ignored originally and enables SIR to deal with the tranductive/semisupervised learning problems. We also discussed the statistical properties of the proposed regularization, that under mild conditions, the manifold regularization converges at rate root-n. To solve the regularized SIR problem, we present a conjugate gradient method conducted on the Grassmann manifold. Experiments on real datasets validate the effectiveness of the regularized SIR. Acknowledgments This project was supported by the Nanyang Technological University Nanyang SUG Grant (under project number M58020010). 8 References [1] Belkin, M. & Niyogi, P. (2003) Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15(6): 1373-1396. [2] Belkin, M., Niyogi, P. & Sindhwani, V. (2006) Manifold regularization: Ageometric framework for learning from labeled and unlabeled examples. Journal of Machine Learning Research, 1: 1-48. [3] Cook, R.D.(2004) Testing predictor contributions in sufficient dimension reduction. Annals of Statistics, 32: 1061-1092. [4] Edelman, A., Arias, T.A., & Smith, S.T. (1998) The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20(2):303-353. [5] Hastie, T., Buja, A., & Tibshirani, R. (1995) Penalized discriminant analysis. Annals of Statistics, 2: 73-102. [6] He, X., Deng, C., & Min, W. (2005) Statistical and computational analysis of locality preserving projection. In 22th International Conference on Machine Learning (ICML). [7] Li, K. (1991) Sliced inverse regression for dimension reduction (with discussion). J. Amer. Statist. Assoc., 86:316-342. [8] Li, L. (2007). Sparse sufficient dimension reduction. Biometrika 94(3): 603-613. [9] Li, L., & YIN, X. (2008). Sliced inverse regression with regularizations. Biometrics 64: 124-131. [10] Nene, S.A., Nayar, S.K., & Murase, H. (1996) Columbia object image library: COIL-20. Technical Report No. CUCS-006-96, Dept. of Computer Science, Columbia University. [11] Saracco, J. (1997). An asymptotic theory for sliced inverse regression. Comm. Statist. Theory Methods 26: 2141-2171. [12] Wu, Q., Mukherjee, S., & Liang, F. (2008) Localized sliced inverse regression. Advances in neural information processing systems 20, Cambridge, MA: MIT Press. [13] von Luxburg, U., Bousquet, O., & Belkin, M. (2005) Limits of spectral clustering. In L. K. Saul, Y. Weiss and L. Bottou (Eds.), Advances in neural information processing systems 17, Cambridge, MA: MIT Press. [14] Zhong, W., Zeng, P., Ma, P., Liu, J. S., & Zhu, Y. (2005) RSIR: Regularized sliced inverse regression for motif discovery. Bioinformatics 21: 4169-4175. [15] Zhu, L.X., & NG, K.W. (1995) Asymptotics of sliced inverse regression. 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STDP enables spiking neurons to detect hidden causes of their inputs Bernhard Nessler, Michael Pfeiffer, and Wolfgang Maass Institute for Theoretical Computer Science, Graz University of Technology A-8010 Graz, Austria {nessler,pfeiffer,maass}@igi.tugraz.at Abstract The principles by which spiking neurons contribute to the astounding computational power of generic cortical microcircuits, and how spike-timing-dependent plasticity (STDP) of synaptic weights could generate and maintain this computational function, are unknown. We show here that STDP, in conjunction with a stochastic soft winner-take-all (WTA) circuit, induces spiking neurons to generate through their synaptic weights implicit internal models for subclasses (or “causes”) of the high-dimensional spike patterns of hundreds of pre-synaptic neurons. Hence these neurons will fire after learning whenever the current input best matches their internal model. The resulting computational function of soft WTA circuits, a common network motif of cortical microcircuits, could therefore be a drastic dimensionality reduction of information streams, together with the autonomous creation of internal models for the probability distributions of their input patterns. We show that the autonomous generation and maintenance of this computational function can be explained on the basis of rigorous mathematical principles. In particular, we show that STDP is able to approximate a stochastic online Expectation-Maximization (EM) algorithm for modeling the input data. A corresponding result is shown for Hebbian learning in artificial neural networks. 1 Introduction It is well-known that synapses change their synaptic efficacy (“weight”) w in dependence of the difference tpost −tpre of the firing times of the post- and presynaptic neuron according to variations of a generic STDP rule (see [1] for a recent review). However, the computational benefit of this learning rule is largely unknown [2, 3]. It has also been observed that local WTA-circuits form a common network-motif in cortical microcircuits [4]. However, it is not clear how this network-motif contributes to the computational power and adaptive capabilities of laminar cortical microcircuits, out of which the cortex is composed. Finally, it has been conjectured for quite some while, on the basis of theoretical considerations, that the discovery and representation of hidden causes of their high-dimensional afferent spike inputs is a generic computational operation of cortical networks of neurons [5]. One reason for this belief is that the underlying mathematical framework, ExpectationMaximization (EM), arguably provides the most powerful approach to unsupervised learning that we know of. But one has so far not been able to combine these three potential pieces (STDP, WTAcircuits, EM) of the puzzle into a theory that could help us to unravel the organization of computation and learning in cortical networks of neurons. We show in this extended abstract that STDP in WTA-circuits approximates EM for discovering hidden causes of large numbers of input spike trains. We first demonstrate this in section 2 in an application to a standard benchmark dataset for the discovery of hidden causes. In section 3 we show that the functioning of this demonstration can be explained on the basis of EM for simpler non-spiking approximations to the spiking network considered in section 2. 1 2 Discovery of hidden causes for a benchmark dataset We applied the network architecture shown in Fig. 1A to handwritten digits from the MNIST dataset [6].1 This dataset consists of 70, 000 28 × 28-pixel images of handwritten digits2, from which we picked the subset of 20, 868 images containing only the digits 0, 3 and 4. Training examples were randomly sampled from this subset with a uniform distribution of digit classes. 0 0 tpost − tpre ∆ wki c · e−wki -1 σ -1 Simple STDP curve Complex STDP curve A B Figure 1: A) Architecture for learning with STDP in a WTA-network of spiking neurons. B) Learning curve for the two STDP rules that were used (with σ = 10ms). The synaptic weight wki is changed in dependence of the firing times tpre of the presynaptic neuron yi and tpost of the postsynaptic neuron zk. If zk fires at time t without a firing of yi in the interval [t −σ, t + 2σ], wki is reduced by 1. The resulting weight change is in any case multiplied with the current learning rate η, which was chosen in the simulations according to the variance tracking rule7. Pixel values xj were encoded through population coding by binary variables yi (spikes were produced for each variable yi by a Poisson process with a rate of 40 Hz for yi = 1, and 0 Hz for yi = 0, at a simulation time step of 1ms, see Fig. 2A). Every training example x was presented for 50ms. Every neuron yi was connected to all K = 10 output neurons z1, . . . , z10. A Poisson process caused firing of one of the neurons zk on average every 5ms (see [8] for a more realistic firing mechanism). The WTA-mechanism ensured that only one of the output neurons could fire at any time step. The winning neuron at time step t was chosen from the soft-max distribution p(zk fires at time t|y) = euk(t) PK l=1 eul(t) , (1) where uk(t) = Pn i=1 wki˜yi(t) + wk0 represents the current membrane potential of neuron zk (with ˜yi(t) = 1 if yi fired within the time interval [t −10ms, t], else ˜yi(t) = 0).3 STDP with the learning curves shown in Fig. 1B was applied to all synapses wki for an input consisting of a continuous sequence of spike encodings of handwritten digits, each presented for 50ms (see 1A similar network of spiking neurons had been applied successfully in [7] to learn with STDP the classification of symbolic (i.e., not handwritten) characters. Possibly our theoretical analysis could also be used to explain their simulation result. 2Pixels were binarized to black/white. All pixels that were black in less than 5% of the training examples were removed, leaving m = 429 external variables xj, that were encoded by n = 858 spiking neurons yi. Our approach works just as well for external variables xj that assume any finite number of values, provided that they are presented to the network through population coding with one variable yi for every possible value of xj. In fact, the approach appears to work also for the commonly considered population coding of continuous external variables. 3This amounts to a representation of the EPSP caused by a firing of neuron yi by a step function, which facilitates the theoretical analysis in section 3. Learning with the spiking network works just as well for biologically realistic EPSP forms. 2 Time [ms] Input Neurons Input Spike Trains 0 50 100 150 100 200 300 400 500 600 700 800 Time [ms] Output Neurons Output before Learning 0 50 100 150 1 2 3 4 5 6 7 8 9 10 Time [ms] Output Neurons Output after Learning 0 50 100 150 1 2 3 4 5 6 7 8 9 10 A B C Figure 2: Unsupervised classification learning and sparsification of firing of output neurons after training. For testing we presented three examples from an independent test set of handwritten digits 0, 3, 4 from the MNIST dataset, and compared the firing of the output-neurons before and after learning. A) Representation of the three handwritten digits 0, 3, 4 for 50ms each by 858 spiking neurons yi. B) Response of the output neurons before training. C) Response of the output neurons after STDP (according to Fig. 1B) was applied to their weights wki for a continuous sequence of spike encodings of 4000 randomly drawn examples of handwritten digits 0, 3, 4, each represented for 50ms (like in panel A). The three output neurons z4, z9, z6 that respond have generated internal models for the three shown handwritten digits according to Fig. 3C. Fig. 2A).4 The learning rate η was chosen locally according to the variance tracking rule7. Fig. 2C shows that for subsequent representations of new handwritten samples of the same digits only one neuron responds during each of the 50ms while a handwritten digit is shown. The implicit internal models which the output neurons z1, . . . , z10 had created in their weights after applying STDP are made explicit in Fig. 3B and C. Since there were more output neurons than digits, several output neurons created internal models for different ways of writing the same digit. When after applying STDP to 2000 random examples of handwritten digits 0 and 3 also examples of handwritten digit 4 were included in the next 2000 examples, the internal models of the 10 output neurons reorganized autonomously, to include now also two internal models for different ways of writing the digit 4. The adaptation of the spiking network to the examples shown so far is measured in Fig. 3A by the normalized conditional entropy H(L|Z)/H(L, Z), where L denotes the correct classification of each handwritten digit y, and Z is the random variable which denotes the cluster assignment with p(Z = k|y) = p(zk = 1|y), the firing probabilities at the presentation of digit y, see (1). Since after training by STDP each of the output neurons fire preferentially for one digit, we can measure the emergent classification capability of the network. The resulting weight-settings achieve a classification error of 2.19% on the digits 0 and 3 after 2000 training steps and 3.68% on all three digits after 4000 training steps on independent test sets of 10,000 new samples each. 3 Underlying theoretical principles We show in this section that one can analyze the learning dynamics of the spiking network considered in the preceding section (with the simple STDP curve of Fig. 1B with the help of Hebbian learning (using rule (12)) in a corresponding non-spiking neural network Nw. Nw is a stochastic artificial neural network with the architecture shown in Fig. 1A, and with a parameter vector w consisting of thresholds wk0 (k = 1, . . . , K) for the K output units z1, . . . , zK and weights wki for the connection from the ith input node yi (i = 1, . . . , n) to the kth output unit zk. We assume that this network receives at each discrete time step a binary input vector y ∈{0, 1}n and outputs a binary vector z ∈{0, 1}K with PK k=1 zk = 1, where the k such that zk = 1 is drawn from the distribution 4Whereas the weights in the theoretical analysis of section 3 will approximate logs of probabilities (see (6)), one can easily make all weights non-negative by restricting the range of these log-probabilities to [−5, 0], and then adding a constant 5 to all weight values. This transformation gives rise to the factor c = e5 in Fig. 1B. 3 0 500 1000 1500 2000 2500 3000 3500 4000 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 Training Examples Conditional Entropy Spiking Network (simple STDP curve) Spiking Network (complex STDP curve) Non−spiking Network (no missing attributes) Non−spiking Network (35% missing attributes) A B C Figure 3: Analysis of the learning progress of the spiking network for the MNIST dataset. A) Normalized conditional entropy (see text) for the spiking network with the two variants of STDP learning rules illustrated in Fig. 1B (red solid and blue dashed lines), as well as two non-spiking approximations of the network with learning rule (12) that are analyzed in section 3. According to this analysis the non-spiking network with 35% missing attributes (dash-dotted line) is expected to have a very similar learning behavior to the spiking network. 2000 random examples of handwritten digits 0 and 3 were presented (for 50ms each) to the spiking network as the first 2000 examples. Then for the next 2000 examples also samples of handwritten digit 4 were included. B) The implicit internal models created by the neurons after 2000 training examples are made explicit by drawing for each pixel the difference wki −wk(i+1) of the weights for input yi and yi+1 that encode the two possible values (black/white) of the variable xj that encodes this pixel value. One can clearly see that neurons created separate internal models for different ways of writing the two digits 0 and 3. C) Re-organized internal models after 2000 further training examples that included digit 4. Two output neurons had created internal models for the newly introduced digit 4. over {1, . . . , K} defined by p(zk = 1|y, w) = euk K P l=1 eul with uk = n X i=1 wki yi + wk0 . (2) We consider the case where there are arbitrary discrete external variables x1, . . . , xm, each ranging over {1, . . . , M} (we had M = 2 in section 2), and assume that these are encoded through binary variables y1, . . . , yn for n = m · M with Pn i=1 yi = m according to the rule y(j−1)·M+r = 1 ⇐⇒ xj = r , for j = 1, . . . , m and r = 1, . . . , M. (3) In other words: the group Gj of variables y(j−1)·M+1, . . . , y(j−1)·M+M provides a population coding for the discrete variable xj. We now consider a class of probability distributions that is particularly relevant for our analysis: mixtures of multinomial distributions [9], a generalization of mixtures of Bernoulli distributions (see section 9.3.3 of [10]). This is a standard model for latent class analysis [11] in the case of discrete variables. Mixtures of multinomial distributions are arbitrary mixtures of K distributions p1(x), . . . , pK(x) that factorize, i.e., pk(x) = m Y j=1 pkj(xj) for arbitrary distributions pkj(xj) over the range {1, . . . , M} of possible values for xj. In other words: there exists some distribution over hidden binary variables zk with PK k=1 zk = 1, where the k with zk = 1 is usually referred to as a hidden “cause” in the generation of x, such that p(x) = K X k=1 p(zk = 1) · pk(x). (4) 4 We first observe that any such distribution p(x) can be represented with some suitable weight vector w by the neural network Nw, after recoding of the multinomial variables xj by binary variables yi as defined before: p(y|w) = K X k=1 eu∗ k with u∗ k := n X i=1 w∗ ki yi + w∗ k0 , (5) for w∗ ki := log p(yi = 1|zk = 1) and w∗ k0 := log p(zk = 1) . (6) In addition, Nw defines for any weight vector w whose components are normalized, i.e. K X k=1 ewk0 = 1 and X i∈Gj ewki = 1 , for j = 1, . . . , m; k = 1, . . . , K, (7) a mixture of multinomials of the type (4). The problem of learning a generative model for some arbitrarily given input distribution p∗(x) (or p∗(y) after recoding according to (3)), by the neural network Nw is to find a weight vector w such that p(y|w) defined by (5) models p∗(y) as accurately as possible. As usual, we quantify this goal by demanding that Ep∗[log p(y|w)] (8) is maximized. Note that the architecture Nw is very useful from a functional point of view, because if (7) holds, then the weighted sum uk at its unit zk has according to (2) the value log p(zk = 1|y, w), and the stochastic WTA rule of Nw picks the “winner” k with zk = 1 from this internally generated model p(zk = 1|y, w) for the actual distribution p∗(zk = 1|y) of hidden causes. We will not enforce the normalization (7) explicitly during the subsequently considered learning process, but rather use a learning rule (12) that turns out to automatically approximate such normalization in the limit. Expectation Maximization (EM) is the standard method for maximizing Ep∗[log p(y|w)]. We will show that the simple STDP-rule of Fig. 1B for the spiking network of section 2 can be viewed as an approximation to an online version of this EM method. We will first consider in section 3.1 the standard EM-approach, and show that the Hebbian learning rule (12) provides a stochastic approximation to the maximization step. 3.1 Reduction to EM The standard method for maximizing the expected log-likelihood Ep∗[log p(y|w)] with a distribution p of the form p(y|w) = P z p(y, z|w) with hidden variables z, is to observe that Ep∗[log p(y|w)] can be written for arbitrary distributions q(z|y) in the form Ep∗[log p(y|w)] = L(q, w) + Ep∗[KL(q(z|y)||p(z|y, w))] (9) L(q, w) = Ep∗ "X z q(z|y) log p(y, z|w) q(z|y) # , (10) where KL(.) denotes the Kullback-Leibler divergence. In the E-step one sets q(z|y) = p(z|y, wold) for the current parameter values w = wold, thereby achieving Ep∗[KL(q(z|y)||p(z|y, wold))] = 0. In the M-step one replaces wold by new parameters w that maximize L(q, w) for this distribution q(z|y). One can easily show that this is achieved by setting w∗ ki = log p∗(yi = 1|zk = 1), and w∗ k0 = log p∗(zk = 1), (11) with values for the variables zk generated by q(z|y) = p(z|y, wold), while the values for the variables y are generated by the external distribution p∗. Note that this distribution of z is exactly the distribution (2) of the output of the neural network Nw for inputs y generated by p∗.5 In the following section we will show that this M-step can be approximated by applying iteratively a simple Hebbian learning rule to the weights w of the neural network Nw. 5Hence one can extend p∗(y) for each fixed w to a joint distribution p∗(y, z), where the z are generated for each y by Nw. 5 3.2 A Hebbian learning rule for the M-step We show here that the target weight values (11) are the only equilibrium points of the following Hebbian learning rule: ∆wki= (η (e−wki −1), if yi=1 and zk=1 −η, if yi=0 and zk=1 0, if zk = 0, ∆wk0= η (e−wk0 −1), if zk=1 −η, if zk=0 (12) It is obvious (using for the second equivalence the fact that yi is a binary variable) that E[∆wki] = 0 ⇔ p∗(yi=1|zk=1)η(e−wki −1) −p∗(yi=0|zk=1)η = 0 ⇔ p∗(yi=1|zk=1)(e−wki −1) + p∗(yi=1|zk=1) −1 = 0 ⇔ p∗(yi=1|zk=1)e−wki = 1 ⇔ wki = log p∗(yi=1|zk=1) . (13) Analogously one can show that E[∆wk0] = 0 ⇔wk0 = log p∗(zk=1). With similar elementary calculations one can show that E[∆wki] has for any w a value that moves wki in the direction of w∗ ki (in fact, exponentially fast). One can actually show that one single step of (12) is a linear approximation of the ideal incremental update of wki = log aki Nk , with aki and Nk representing the values of the corresponding sufficient statistics, as log aki+1 Nk+1 = wki + log(1 + ηe−wki) −log(1 + η) for η = 1 Nk . This also reveals the role of the learning rate η as the reciprocal of the equivalent sample size6. In order to guarantee the stochastic convergence (see [12]) of the learning rule one has to use a decaying learning rate η(t) such that P∞ t=1 η(t) = ∞and P∞ t=1(η(t))2 = 0.7 The learning rule (12) is similar to a rule that had been introduced in [13] in the context of supervised learning and reinforcement learning. That rule had satisfied an equilibrium condition similar to (13). But to the best of our knowledge, such type of rule has so far not been considered in the context of unsupervised learning. One can easily see the correspondence between the update of wki in (12) and in the simple STDP rule of Fig. 1B. In fact, if each time where neuron zk fires in the spiking network, each presynaptic neuron yi that currently has a high firing rate has fired within the last σ = 10ms before the firing of zk, the two learning rules become equivalent. However since the latter condition could only be achieved with biologically unrealistic high firing rates, we need to consider in section 3.4 the case for the non-spiking network where some attributes are missing (i.e., yi = 0 for all i ∈Gj; for some group Gj that encodes an external variable xj via population coding). We first show that the Hebbian learning rule (12) is also meaningful in the case of online learning of Nw, which better matches the online learning process for the spiking network. 3.3 Stochastic online EM The preceding arguments justify an application of learning rule (12) for a number of steps within each M-step of a batch EM approach for maximizing E∗ p[log p(y|w)]. We now show that it is also meaningful to apply the same rule (12) in an online stochastic EM approach (similarly as in [14]), where at each combined EM-step only one example y is generated by p∗, and the learning rule (12) 6The equilibrium condition (13) only sets a necessary constraint for the the quotient of the two directions of the update in (12). The actual formulation of (12) is motivated by the goal of updating a sufficient statistics. 7In our experiments we used an adaptation of the variance tracking heuristic from [13]. If we assume that the consecutive values of the weights represent independent samples of their true stochastic distribution at the current learning rate, then this observed distribution is the log of a beta-distribution of the above mentioned parameters of the sufficient statistics. Analytically this distribution has the first and second moments E[wki] ≈ log aki Ni and E[w2 ki] ≈E[wki]2 + 1 aki + 1 Ni , leading to the estimate ηnew ki = 1 Ni = E[w2 ki]−E[wki]2 e−E[wki]+1 . The empirical estimates of these first two moments can be gathered online by exponentially decaying averages using the same learning rate ηki. 6 is applied just once (for zk resulting from p(z|y, w) for the current weights w, or simpler: for the zk that is output by Nw for the current input y). Our strategy for showing that a single application of learning rule (12) is expected to provide progress in an online EM-setting is the following. We consider the Lagrangian F for maximizing Ep∗[log p(y|w)] under the constraints (7), and show that an application of rule (12) is expected to increase the value of F. We set F(w, λ) = Ep∗[log p(y|w)] −λ0 1 − K X k=1 ewk0 ! − K X k=1 m X j=1 λkj 1 − X i∈Gj ewki . (14) According to (5) one can write p(y|w) = PK k=1 euk for uk = PK i=1 wki yi + wk0. Hence one arrives at the following conditions for the Lagrange multipliers λ: K X k=1 ∂F ∂wk0 = K X k=1 Ep∗[ euk PK l=1 eul ] −λ0ewk0 ! = 0 (15) X i∈Gj ∂F ∂wki = X i∈Gj Ep∗[yi euk PK l=1 eul ] −λkjewki ! = 0, (16) which yield λ0 = 1 and λkj = Ep∗[ euk PK l=1 eul ]. Plugging these values for λ into ∇wF · E∗ p[∆w] with ∆w defined by (12) shows that this vector product is always positive. Hence even a single application of learning rule (12) to a single new example y, drawn according to p∗, is expected to increase Ep∗[log p(y|w)] under the constraints (7). 3.4 Impact of missing attributes We had shown at the end of 3.2 that learning in the spiking network corresponds to learning in the non-spiking network Nw with missing attributes. A profound analysis of the correct handling of missing attribute values in EM can be found in [15]. Their analysis implies that the correct learning action is then not to change the weights wki for i ∈Gj. However the STDP rule of Fig. 1B, as well as (12), reduce also these weights by η if zk fires. This yields a modification of the equilibrium analysis (13): E[∆wki] = 0 ⇔(1 −r) p∗(yi=1|zk=1)η(e−wki −1) −p∗(yi=0|zk=1)η −rη = 0 ⇔ wki = log p∗(yi=1|zk=1) + log(1 −r) , (17) where r is the probability that i belongs to a group Gj where the value of xj is missing. Since this probability r is independent of the neuron zk and also independent of the current value of the external variable xi, this offset of log(1 −r) is expected to be the same for all weights. It can easily be verified, that such an offset does not change the resulting probabilities of the competition in the E-step according to (2). 3.5 Relationship between the spiking and the non-spiking network As indicated at the end of section 3.2, the learning process for the spiking network from section 2 with the simple STDP curve from Fig. 1B (and external variables xj encoded by input spike trains from neurons yi) is equivalent to a somewhat modified learning process of the non-spiking network Nw with the Hebbian learning rule (12) and external variables xj encoded by binary variables yi. Each firing of a neuron zk at some time t corresponds to a discrete time step in Nw with an application of the Hebbian learning rule (12). Each neuron yi that had fired during the time interval [t −10ms, t] contributes a value ˜yi(t) = 1 to the membrane potential uk(t) of the neuron zk at time t, and a value ˜yi(0) = 0 if it did not fire during [t −10ms, t]. Hence the weight updates at time t according to the simple STDP curve are exactly equal to those of (12) in the non-spiking network. However (12) will in general be applied to a corresponding input y where it may occur that for some 7 j ∈{1, . . . , m} one has yi = 0 for all i ∈Gj (since none of the neurons yi with i ∈Gj fired in the spiking network during [t −10ms, t]). Hence we arrive at an application of (12) to an input y with missing attributes, as discussed in section 3.4. Since several neurons zk are likely to fire during the presentation of an external input x (each handwritten digit was presented for 50ms in section 2; but a much shorter presentation time of 10ms also works quite well), this external input x gives in general rise to several applications of the STDP rule. This corresponds to several applications of rule (12) to the same input (but with different choices of missing attributes) in the non-spiking network. In the experiments in section 2, every example in the non-spiking network with missing attributes was therefore presented for 10 steps, such that the average number of learning steps is the same as in the spiking case. The learning process of the spiking network corresponds to a slight variation of the stochastic online EM algorithm that is implemented through (12) according to the analysis of section 3.3. 4 Discussion The model for discovering hidden causes of inputs that is proposed in this extended abstract presents an interesting shortcut for implementing and learning generative models for input data in networks of neurons. Rather than building and adapting an explicit model for re-generating internally the distribution of input data, our approach creates an implicit model of the input distribution (see Fig. 3B) that is encoded in the weights of neurons in a simple WTA-circuit. One might call it a Vapnik-style [16] approach towards generative modeling, since it focuses directly on the task to represent the most likely hidden causes of the inputs through neuronal firing. As the theoretical analysis via nonspiking networks in section 3 has shown, this approach also offers a new perspective for generating self-adapting networks on the basis of traditional artificial neural networks. One just needs to add the stochastic and non-feedforward parts required for implementing stochastic WTA circuits to a 1-layer feedforward network, and apply the Hebbian learning rule (12) to the feedforward weights. One interesting aspect of the “implicit generative learning” approach that we consider in this extended abstract is that it retains important advantages of the generative learning approach, faster learning and better generalization [17], while retaining the algorithmic simplicity of the discriminative learning approach. Our approach also provides a new method for analyzing details of STDP learning rules. The simulation results of section 2 show that a simplified STDP rule that can be understood clearly from the perspective of stochastic online EM with a suitable Hebbian learning rule, provides good performance in discovering hidden causes for a standard benchmark dataset. A more complex STDP rule, whose learning curve better matches experimentally recorded average changes of synaptic weights, provides almost the same performance. For a comparison of the STDP curves in Fig. 1B with experimentally observed STDP curves one should keep in mind, that most experimental data on STDP curves are for very low firing rates. The STDP curve of Fig. 7C in [18] for a firing rate of 20Hz has, similarly as the STDP curves in Fig. 1B of this extended abstract, no pronounced negative dip, and instead an almost constant negative part. In our upcoming paper [8] we will provide full proofs for the results announced in this extended abstract, as well as further applications and extensions of the learning result. We will also demonstrate, that the learning rules that we have proposed are robust to noise, and that they are matched quite well by experimental data. Acknowledgments We would like to thank the anonymous reviewer for a hint in the notational formalism. Written under partial support by the Austrian Science Fund FWF, project # P17229-N04, project # S9102-N04, and project # FP6-015879 (FACETS) as well as # FP7-216593 (SECO) of the European Union. 8 References [1] Y. Dan and M. Poo. Spike timing-dependent plasticity of neural circuits. Neuron, 44:23–30, 2004. [2] L. F. Abbott and S. B. Nelson. Synaptic plasticity: taming the beast. Nature Neuroscience, 3:1178–1183, 2000. [3] A. Morrison, A. Aertsen, and M. Diesmann. Spike-timing-dependent plasticity in balanced random networks. Neural Computation, 19:1437–1467, 2007. [4] R. J. Douglas and K. A. Martin. Neuronal circuits of the neocortex. Annu Rev Neurosci, 27:419–451, 2004. [5] G. E. Hinton and Z. Ghahramani. Generative models for discovering sparse distributed representations. Philos Trans R Soc Lond B Biol Sci., 352(1358):1177–1190, 1997. [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, 1998. [7] A. Gupta and L. N. Long. Character recognition using spiking neural networks. IJCNN, pages 53–58, 2007. [8] B. Nessler, M. Pfeiffer, and W. Maass. Spike-timing dependent plasticity performs stochastic expectation maximization to reveal the hidden causes of complex spike inputs. (in preparation). [9] M. Meil˘a and D. Heckerman. An experimental comparison of model-based clustering methods. Machine Learning, 42(1):9–29, 2001. [10] C. M. Bishop. Pattern Recognition and Machine Learning. Springer, New York, 2006. [11] G. McLachlan and D. Peel. Finite mixture models. Wiley, 2000. [12] J.H. Kushner and G.G. Yin. Stochastic approximation algorithms and applications. Springer, 1997. [13] B. Nessler, M. Pfeiffer, and W. Maass. Hebbian learning of bayes optimal decisions. In Advances in Neural Information Processing Systems 21, pages 1169–1176. MIT Press, 2009. [14] M. Sato. Fast learning of on-line EM algorithm. Rapport Technique, ATR Human Information Processing Research Laboratories, 1999. [15] Z. Ghahramani and M.I. Jordan. Mixture models for learning from incomplete data. Computational Learning Theory and Natural Learning Systems, 4:67–85, 1997. [16] V. Vapnik. Universal learning technology: Support vector machines. NEC Journal of Advanced Technology, 2:137–144, 2005. [17] A. Y. Ng and M. I. Jordan. On discriminative vs. generative classifiers: A comparison of logistic regression and naive Bayes. Advances in Neural Information Processing Systems (NIPS), 14:841–848, 2002. [18] P. J. Sj¨ostr¨om, G. G. Turrigiano, and S. B. Nelson. Rate, timing, and cooperativity jointly determine cortical synaptic plasticity. Neuron, 32:1149–1164, 2001. 9
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Locality-Sensitive Binary Codes from Shift-Invariant Kernels Maxim Raginsky Duke University Durham, NC 27708 m.raginsky@duke.edu Svetlana Lazebnik UNC Chapel Hill Chapel Hill, NC 27599 lazebnik@cs.unc.edu Abstract This paper addresses the problem of designing binary codes for high-dimensional data such that vectors that are similar in the original space map to similar binary strings. We introduce a simple distribution-free encoding scheme based on random projections, such that the expected Hamming distance between the binary codes of two vectors is related to the value of a shift-invariant kernel (e.g., a Gaussian kernel) between the vectors. We present a full theoretical analysis of the convergenceproperties of the proposed scheme, and report favorable experimental performance as compared to a recent state-of-the-art method, spectral hashing. 1 Introduction Recently, there has been a lot of interest in the problem of designing compact binary codes for reducing storage requirements and accelerating search and retrieval in large collections of highdimensional vector data [11, 13, 15]. A desirable property of such coding schemes is that they should map similar data points to similar binary strings, i.e., strings with a low Hamming distance. Hamming distances can be computed very efficiently in hardware, resulting in very fast retrieval of strings similar to a given query, even for brute-force search in a database consisting of millions of data points [11, 13]. Moreover, if code strings can be effectively used as hash keys, then similarity searches can be carried out in sublinear time. In some existing schemes, e.g. [11, 13], the notion of similarity between data points comes from supervisory information, e.g., two documents are similar if they focus on the same topic or two images are similar if they contain the same objects. The binary encoder is then trained to reproduce this “semantic” similarity measure. In this paper, we are more interested in unsupervised schemes, where the similarity is given by Euclidean distance or by a kernel defined on the original feature space. Weiss et al. [15] have recently proposed a spectral hashing approach motivated by the idea that a good encoding scheme should minimize the sum of Hamming distances between pairs of code strings weighted by the value of a Gaussian kernel between the corresponding feature vectors. With appropriate heuristic simplifications, this objective can be shown to yield a very efficient encoding rule, where each bit of the code is given by the sign of a sine function applied to a one-dimensional projection of the feature vector. Spectral hashing shows promising experimental results, but its behavior is not easy to characterize theoretically. In particular, it is not clear whether the Hamming distance between spectral hashing code strings converges to any function of the Euclidean distance or the kernel value between the original vectors as the number of bits in the code increases. In this paper, we propose a coding method that is similar to spectral hashing computationally, but is derived from completely different considerations, is amenable to full theoretical analysis, and shows better practical behavior as a function of code size. We start with a low-dimensional mapping of the original data that is guaranteed to preserve the value of a shift-invariant kernel (specifically, the random Fourier features of Rahimi and Recht [8]), and convert this mapping to a binary one with similar guarantees. In particular, we show that the normalized Hamming distance (i.e., Hamming distance divided by the number of bits in the code) between any two embedded points sharply concentrates around a well-defined continuous function of the kernel value. This leads to a Johnson– Lindenstrauss type result [4] which says that a set of any N points in a Euclidean feature space can be embedded in a binary cube of dimension O(log N) in a similarity-preserving way: with high probability, the binary encodings of any two points that are similar (as measured by the kernel) are nearly identical, while those of any two points that are dissimilar differ in a constant fraction of their bits. Using entropy bounds from the theory of empirical processes, we also prove a stronger result of this type that holds for any compact domain of RD, provided the number of bits is proportional to the intrinsic dimension of the domain. Our scheme is completely distribution-free with respect to the data: its structure depends only on the underlying kernel. In this, it is similar to locality sensitive hashing (LSH) [1], which is a family of methods for deriving low-dimensional discrete representations of the data for sublinear near-neighbor search. However, our scheme differs from LSH in that we obtain both upper and lower bounds on the normalized Hamming distance between any two embedded points, while in LSH the goal is only to preserve nearest neighbors (see [6] for further discussion of the distinction between LSH and more general similarity-preserving embeddings). To the best of our knowledge, our scheme is among the first random projection methods for constructing a similarity-preserving embedding into a binary cube. In addition to presenting a thorough theoretical analysis, we have evaluated our approach on both synthetic and real data (images from the LabelMe database [10] represented by high-dimensional GIST descriptors [7]) and compared its performance to that of spectral hashing. Despite the simplicity and distribution-free nature of our scheme, we have been able to obtain very encouraging experimental results. 2 Binary codes for shift-invariant kernels Consider a Mercer kernel K(·, ·) on RD that satisfies the following for all points x, y ∈RD: (K1) It is translation-invariant (or shift-invariant), i.e., K(x, y) = K(x −y). (K2) It is normalized, i.e., K(x −y) ≤1 and K(x −x) ≡K(0) = 1. (K3) For any real number α ≥1, K(αx −αy) ≤K(x −y). The Gaussian kernel K(x, y) = exp(−γ∥x −y∥2/2) or the Laplacian kernel K(x, y) = exp(−γ∥x −y∥1) are two well-known examples. We would like to construct an embedding F n of RD into the binary cube {0, 1}n such that for any pair x, y the normalized Hamming distance 1 ndH(F n(x), F n(y)) △= 1 n n X i=1 1{Fi(x)̸=Fi(y)} between F n(x) = (F1(x), . . . , Fn(x)) and F n(y) = (F1(y), . . . , Fn(y)) behaves like h1(K(x −y)) ≤1 ndH(F n(x), F n(y)) ≤h2(K(x −y)) where h1, h2 : [0, 1] →R+ are continuous decreasing functions, and h1(1) = h2(1) = 0 and h1(0) = h2(0) = c > 0. In other words, we would like to map D-dimensional real vectors into n-bit binary strings in a locality-sensitive manner, where the notion of locality is induced by the kernel K. We will achieve this goal by drawing F n appropriately at random. Random Fourier features. Recently, Rahimi and Recht [8] gave a scheme that takes a Mercer kernel satisfying (K1) and (K2) and produces a random mapping Φn : RD →Rn, such that, with high probability, the inner product of any two transformed points approximates the kernel: Φn(x)·Φn(y) ≈K(x−y) for all x, y. Their scheme exploits Bochner’s theorem [9], a fundamental result in harmonic analysis which says that any such K is a Fourier transform of a uniquely defined probability measure PK on RD. They define the random Fourier features (RFF) via Φω,b(x) △= √ 2 cos(ω · x + b), (1) where ω ∼PK and b ∼Unif[0, 2π]. For example, for the Gaussian kernel K(s) = e−γ∥s∥2/2, we take ω ∼Normal(0, γID×D). With these features, we have E[Φω,b(x)Φω,b(y)] = K(x −y). The scheme of [8] is as follows: draw an i.i.d. sample ((ω1, b1), . . . , (ωn, bn)), where each ωi ∼PK and bi ∼Unif[0, 2π], and define a mapping Φn : RD →Rn via Φn(x) △= 1 √n Φω1,b1(x), . . . , Φωn,bn(x) for x ∈X. Then E[Φn(x) · Φn(y)] = K(x −y) for all x, y. From random Fourier features to random binary codes. We will compose the RFFs with random binary quantizers. Draw a random threshold t ∼Unif[−1, 1] and define the quantizer Qt : [−1, 1] →{−1, +1} via Qt(u) △= sgn(u + t), where we let sgn(u) = −1 if u < 0 and sgn(u) = +1 if u ≥0. We note the following basic fact (we omit the easy proof): Lemma 2.1 For any u, v ∈[−1, 1], Pt {Qt(u) ̸= Qt(v)} = |u −v|/2. Now, given a kernel K, we define a random map Ft,ω,b : RD →{0, 1} through Ft,ω,b(x) △= 1 2 [1 + Qt (cos(ω · x + b))] , (2) where t ∼Unif[−1, 1], ω ∼PK, and b ∼Unif[0, 2π] are independent of one another. From now on, we will often omit the subscripts t, ω, b and just write F for the sake of brevity. We have: Lemma 2.2 E 1{F (x)̸=F (y)} = hK(x −y) △= 8 π2 ∞ X m=0 1 −K(mx −my) 4m2 −1 , ∀x, y (3) Proof: Using Lemma 2.1, we can show E 1{F (x)̸=F (y)} = 1 2 Eω,b | cos(ω ·x+b)−cos(ω ·y +b)|. Using trigonometric identities and the independence of ω and b, we can express this expectation as Eb,ω |cos(ω · x + b) −cos(ω · y + b)| = 4 π Eω sin ω · (x −y) 2 . We now make use of the Fourier series representation of the full rectified sine wave g(τ) = | sin(τ)|: g(τ) = 2 π + 4 π ∞ X m=1 1 1 −4m2 cos(mτ) = 4 π ∞ X m=1 1 −cos(2mτ) 4m2 −1 . Using this together with the fact that Eω cos(ω · s) = K(s) for any s ∈RD [8], we obtain (3). ■ Lemma 2.2 shows that the probability that F(x) ̸= F(y) is a well-defined continuous function of x−y. The infinite series in (3) can, of course, be computed numerically to any desired precision. In addition, we have the following upper and lower bounds solely in terms of the kernel value K(x−y): Lemma 2.3 Define the functions h1(u) △= 4 π2 (1 −u) and h2(u) △= min 1 2 √ 1 −u, 4 π2 (1 −2u/3) , where u ∈[0, 1]. Note that h1(0) = h2(0) = 4/π2 ≈0.405 and that h1(1) = h2(1) = 0. Then h1(K(x −y)) ≤hK(x −y) ≤h2(K(x −y)) for all x, y. Proof: Let ∆ △= cos(ω · x + b) −cos(ω · y + b). Then E |∆| = E √ ∆2 ≤ √ E ∆2 (the last step uses concavity of the square root). Using the properties of the RFF, E ∆2 = (1/2) E[(Φω,b(x) − Φω,b(y))2] = 1 −K(x −y). Therefore, E 1{F (x)̸=F (y)} = (1/2) E |∆| ≤(1/2) p 1 −K(x −y). We also have E 1{F (x)̸=F (y)} = 4 π2 −8 π2 ∞ X m=1 K(mx −my) 4m2 −1 ≤4 π2 −8 3π2 K(x−y) = 4 π2 1−2K(x−y)/3 . This proves the upper bound in the lemma. On the other hand, since K satisfies (K3), hK(x −y) ≥ 1 −K(x −y) · 8 π2 ∞ X m=1 1 4m2 −1 = 4 π2 1 −K(x −y) , because the mth term of the series in (3) is not smaller than 1 −K(x −y) /(4m2 −1). ■ Fig. 1 shows a comparison of the kernel approximation properties of the RFFs [8] with our scheme for the Gaussian kernel. (a) (b) (c) Figure 1: (a) Approximating the Gaussian kernel by random features (green) and random signs (red). (b) Relationship of normalized Hamming distance between random signs to functions of the kernel. The scatter plots in (a) and (b) are obtained from a synthetic set of 500 uniformly distributed 2D points with n = 5000. (c) Bounds for normalized Hamming distance in Lemmas 2.2 and 2.3 vs. the Euclidean distance. Now we concatenate several mappings of the form Ft,ω,b to construct an embedding of X into the binary cube {0, 1}n. Specifically, we draw n i.i.d. triples (t1, ω1, b1), . . . , (tn, ωn, bn) and define F n(x) △= F1(x), . . . , Fn(y) , where Fi(x) ≡Fti,ωi,bi(x), i = 1, . . . , n As we will show next, this construction ensures that, for any two points x and y, the fraction of the bits where the binary strings F n(x) and F n(y) disagree sharply concentrates around hK(x −y), provided n is large enough. Using the results proved above, we conclude that, for any two points x and y that are “similar,” i.e., K(x −y) ∼1, most of the bits of F n(x) and F n(y) will agree, whereas for any two points x and y that are “dissimilar,” i.e., K(x −y) ∼0, F n(x) and F n(y) will disagree in about 40% or more of their bits. Analysis of performance. We first prove a Johnson–Lindenstrauss type result which says that, for any finite subset of RD, the normalized Hamming distance respects the similarities between points. It should be pointed out that the analogy with Johnson–Lindenstrauss is only qualitative: our embedding is highly nonlinear, in contrast to random linear projections used there [4], and the resulting distortion of the neighborhood structure, although controllable, does not amount to a mere rescaling by constants. Theorem 2.4 Fix ǫ, δ ∈(0, 1). For any finite data set D = {x1, . . . , xN} ⊂RD, F n is such that hK(xj −xk) −δ ≤1 ndH(F n(xj), F n(xk)) ≤hK(xj −xk) + δ (4) h1(K(xj −xk)) −δ ≤1 ndH(F n(xj), F n(xk)) ≤h2(K(xj −xk)) + δ (5) for all j, k with probability ≥1 −N 2e−2nδ2. Moreover, the events (4) and (5) will hold with probability ≥1 −ǫ if n ≥(1/2δ2) log(N 2/ǫ). Thus, any N-point subset of RD can be embedded, with high probability, into the binary cube of dimension O(log N) in a similarity-preserving way. The proof (omitted) is by a standard argument using Hoeffding’s inequality and the union bound, as well as the bounds of Lemma 2.3. We also prove a much stronger result: any compact subset X ⊂ RD can be embedded into a binary cube whose dimension depends only on the intrinsic dimension and the diameter of X and on the second moment of PK, such that the normalized Hamming distance behaves in a similarity-preserving way for all pairs of points in X simultaneously. We make use of the following [5]: Definition 2.5 The Assouad dimension of X ⊂RD, denoted by dX, is the smallest integer k, such that, for any ball B ⊂RD, the set B ∩X can be covered by 2k balls of half the radius of B. The Assouad dimension is a widely used measure of the intrinsic dimension [2, 6, 3]. For example, if X is an ℓp ball in RD, then dX = O(D); if X is a d-dimensional hyperplane in RD, then dX = O(d) [2]. Moreover, if X is a d-dimensional Riemannian submanifold of RD with a suitably bounded curvature, then dX = O(d) [3]. We now have the following result: Theorem 2.6 Suppose that the kernel K is such that LK △= p Eω∼PK ∥ω∥2 < +∞. Then there exists a constant C > 0 independent of D and K, such that the following holds. Fix any ǫ, δ > 0. If n ≥max CLKdX diam X δ2 , 2 δ2 log 2 ǫ , then, with probability at least 1 −ǫ, the mapping F n is such that, for every pair x, y ∈X, hK(x −y) −δ ≤1 ndH(F n(x), F n(y)) ≤hK(x −y) + δ (6) Proof: For every pair x, y ∈X, let Ax,y be the set of all θ ≡(t, ω, b), such that Ft,ω,b(x) ̸= Ft,ω,b(y), and let A = {Ax,y : x, y ∈X}. Then we can write 1 ndH(F n(x), F n(y)) = 1 n n X i=1 1{θi∈Ax,y}. For any sequence θn = (θ1, . . . , θn), define the uniform deviation ∆(θn) △= sup x,y∈X 1 n n X i=1 1{θi∈Ax,y} −E 1{Ft,ω,b(x)̸=Ft,ω,b(y)} . For every 1 ≤i ≤n and an arbitrary θ′ i, let θn (i) denote θn with the ith component replaced by θ′ i. Then |∆(θn) −∆(θn (i))| ≤1/n for any i and any θ′ i. Hence, by McDiarmid’s inequality, P {|∆(θn) −Eθn ∆(θn)| > β} ≤2e−2nβ2, ∀β > 0. (7) Now we need to bound Eθn ∆(θn). Using a standard symmetrization technique [14], we can write Eθn ∆(θn) ≤2R(A) △= 2 Eθn,σn " sup x,y∈X 1 n n X i=1 σi1{θi∈Ax,y} # , (8) where σn = (σ1, . . . , σn) is an i.i.d. Rademacher sequence, P{σi = −1} = P(σi = +1} = 1/2. The quantity R(A) can be bounded by the Dudley entropy integral [14] R(A) ≤C0 √n Z ∞ 0 q log N(ǫ, A, ∥· ∥L2(µ))dǫ, (9) where C0 > 0 is a universal constant, and N(ǫ, A, ∥· ∥L2(µ)) is the ǫ-covering number of the function class {θ 7→1{θ∈A} : A ∈A} with respect to the L2(µ) norm, where µ is the distribution of θ ≡(t, ω, b). We will bound these covering numbers by the covering numbers of X with respect to the Euclidean norm on RD. It can be shown that, for any four points x, x′, y, y′ ∈X,
1Ax,y −1Ax′,y′
2 L2(µ) = Z 1{θ∈Ax,y} −1{θ∈Ax′,y′} 2dµ(θ) ≤µ(Bx△Bx′) + µ(By△By′), where △denotes symmetric difference of sets, and Bx △= {(t, ω, b) : Qt(cos(ω · x + b)) = +1} (details omitted for lack of space). Now, 2µ (Bx△Bx′) = 2 Eω,b h Pt Qt(cos(ω · x + b)) ̸= Qt(cos(ω · y + b)) i = Eω,b |cos(ω · x + b) −cos(ω · x′ + b)| ≤Eω |ω · (x −x′)| ≤LK∥x −x′∥. Then µ (Bx△Bx′) + µ (By△By′) ≤ LK 2 (∥x −x′∥+ ∥y −y′∥). This implies that N(ǫ, A, ∥· ∥L2(µ)) ≤N(ǫ2/LK, X, ∥·∥)2, where N(δ, X, ∥·∥) are the covering numbers of X w.r.t. the Euclidean norm ∥·∥. By definition of the Assouad dimension, N(δ, X, ∥·∥) ≤(2 diam X/δ)dX , so N(ǫ, A, ∥· ∥L2(µ)) ≤ 2LK diam X ǫ2 2dX . We can now estimate the integral in (9) by R(A) ≤C1 r LKdX diam X n , (10) for some constant C1 > 0. From (10) and (8), we obtain Eθn ∆(θn) ≤C2 q LKdX diam X n , where C2 = 2C1. Using this and (7) with β = δ/2, we obtain (6) with C = 16C2 2. ■ For example, with the Gaussian kernel K(s) = e−γ∥s∥2/2 on RD, we have LK = √Dγ. The kernel bandwidth γ is often chosen as γ ∝1/[D(diam X)2] (see, e.g., [12, Sec. 7.8]); with this setting, the number of bits needed to guarantee the bound (6) is n = Ω((dX /δ2) log(1/ǫ)). It is possible, in principle, to construct a dimension-reducing embedding of X into a binary cube, provided the number of bits in the embedding is larger than the intrinsic dimension of X. Our method Spectral hashing (a) (b) (c) (d) (e) (f) Figure 2: Synthetic results. First row: scatter plots of normalized Hamming distance vs. Euclidean distance for our method (a) and spectral hashing (b) with code size 32 bits. Green indicates pairs of data points that are considered true “neighbors” for the purpose of retrieval. Second row: scatter plots for our method (c) and spectral hashing (d) with code size 512 bits. Third row: recall-precision plots for our method (e) and spectral hashing (f) for code sizes from 8 to 512 bits (best viewed in color). 3 Empirical Evaluation In this section, we present the results of our scheme with a Gaussian kernel, and compare our performance to spectral hashing [15].1 Spectral hashing is a recently introduced, state-of-the-art approach that has been reported to obtain better results than several other well-known methods, including LSH [1] and restricted Boltzmann machines [11]. Unlike our method, spectral hashing chooses code parameters in a deterministic, data-dependent way, motivated by results on convergence of 1We use the code made available by the authors of [15] at http://www.cs.huji.ac.il/˜yweiss/SpectralHashing/. Our method Spectral hashing Figure 3: Recall-precision curves for the LabelMe database for our method (left) and for spectral hashing (right). Best viewed in color. eigenvectors of graph Laplacians to Laplacian eigenfunctions on manifolds. Though spectral hashing is derived from completely different considerations than our method, its encoding scheme is similar to ours in terms of basic computation. Namely, each bit of a spectral hashing code is given by sgn(cos(k ω · x)), where ω is a principal direction of the data (instead of a randomly sampled direction, as in our method) and k is a weight that is deterministically chosen according to the analytical form of certain kinds of Laplacian eigenfunctions. The structural similarity between spectral hashing and our method makes comparison between them appropriate. To demonstrate the basic behavior of our method, we first report results for two-dimensional synthetic data using a protocol similar to [15] (we have also conducted tests on higher-dimensional synthetic data, with very similar results). We sample 10,000 “database” and 1,000 “query” points from a uniform distribution defined on a 2d rectangle with aspect ratio 0.5. To distinguish true positives from false positives for evaluating retrieval performance, we select a “nominal” neighborhood radius so that each query point on average has 50 neighbors in the database. Next, we rescale the data so that this radius is 1, and set the bandwidth of the kernel to γ = 1. Fig. 2 (a,c) shows scatter plots of normalized Hamming distance vs. Euclidean distance for each query point paired with each database point for 32-bit and 512-bit codes. As more bits are added to our code, the variance of the scatter plots decreases, and the points cluster tighter around the theoretically expected curve (Eq. (3), Fig. 1). The scatter plots for spectral hashing are shown in Fig. 2 (b,d). As the number of bits in the spectral hashing code is increased, normalized Hamming distance does not appear to converge to any clear function of the Euclidean distance. Because the derivation of spectral hashing in [15] includes several heuristic steps, the behavior of the resulting scheme appears to be difficult to analyze, and shows some undesirable effects as the code size increases. Figure 2 (e,f) compares recall-precision curves for both methods using a range of code sizes. Since the normalized Hamming distance for our method converges to a monotonic function of the Euclidean distance, its performance keeps improving as a function of code size. On the other hand, spectral hashing starts out with promising performance for very short codes (up to 32 bits), but then deteriorates for higher numbers of bits. Next, we present retrieval results for 14,871 images taken from the LabelMe database [10]. The images are represented by 320-dimensional GIST descriptors [7], which have proven to be effective at capturing perceptual similarity between scenes. For this experiment, we randomly select 1,000 images to serve as queries, and the rest make up the “database.” As with the synthetic experiments, a nominal threshold of the average distance to the 50th nearest neighbor is used to determine whether a database point returned for a given query is considered a true positive. Figure 3 shows precisionrecall curves for code sizes ranging from 16 bits to 1024 bits. As in the synthetic experiments, spectral hashing appears to have an advantage over our method for extremely small code sizes, up to about 32 bits. However, this low bit regime may not be very useful in practice, since below 32 bits, neither method achieves performance levels that would be satisfactory for real-world applications. For larger code sizes, our method begins to dominate. For example, with a 128-bit code (which is equivalent to just two double-precision floating point numbers), our scheme achieves 0.8 precision Euclidean neighbors 32 bit code 512 bit code Precision: 0.81 Precision: 1.00 Precision: 0.38 Precision: 0.96 Figure 4: Examples of retrieval for two query images on the LabelMe database. The left column shows top 48 neighbors for each query according to Euclidean distance (the query image is in the top left of the collage). The middle (resp. right) column shows nearest neighbors according to normalized Hamming distance with a 32-bit (resp. 512-bit) code. The precision of retrieval is evaluated as the proportion of top Hamming neighbors that are also Euclidean neighbors within the “nominal” radius. Incorrectly retrieved images in the middle and right columns are shown with a red border. Best viewed in color. at 0.2 recall, whereas spectral hashing only achieves about 0.5 precision at the same recall. Moreover, the performance of spectral hashing actually begins to decrease for code sizes above 256 bits. Finally, Figure 4 shows retrieval results for our method on a couple of representative query images. In addition to being completely distribution-free and exhibiting more desirable behavior as a function of code size, our scheme has one more practical advantage. Unlike spectral hashing, we retain the kernel bandwidth γ as a “free parameter,” which gives us flexibility in terms of adapting to target neighborhood size, or setting a target Hamming distance for neighbors at a given Euclidean distance. This can be especially useful for making sure that a significant fraction of neighbors for each query are mapped to strings whose Hamming distance from the query is no greater than 2. This is a necesary condition for being able to use binary codes for hashing as opposed to brute-force search (although, as demonstrated in [11, 13], even brute-force search with binary codes can already be quite fast). To ensure high recall within a low Hamming radius, we can progressively increase the kernel bandwidth γ as the code size increases, thus counteracting the increase in unnormalized Hamming distance that inevitably accompanies larger code sizes. Preliminary results (omitted for lack of space) show that this strategy can indeed increase recall for low Hamming radius while sacrificing some precision. In the future, we will evaluate this tradeoff more extensively, and test our method on datasets consisting of millions of data points. At present, our promising initial results, combined with our comprehensive theoretical analysis, convincingly demonstrate the potential usefulness of our scheme for large-scale indexing and search applications. Acknowledgments This work was supported by NSF CAREER Award No. IIS 0845629. References [1] A. Andoni and P. Indyk. Near-optimal hashing algorithms for approximate nearest neighbor in high dimensions. Commun. ACM, 51(1):117–122, 2008. [2] K. Clarkson. Nearest-neighbor searching and metric space dimensions. In Nearest-Neighbor Methods for Learning and Vision: Theory and Practice, pages 15–59. MIT Press, 2006. [3] S. Dasgupta and Y. Freund. Random projection trees and low dimensional manifolds. In STOC, 2008. [4] S. Dasgupta and A. Gupta. An elementary proof of a theorem of Johnson and Lindenstrauss. Random Struct. Alg., 22(1):60–65, 2003. [5] J. Heinonen. Lectures on Analysis on Metric Spaces. Springer, New York, 2001. [6] P. Indyk and A. Naor. Nearest-neighbor-preserving embeddings. ACM Trans. Algorithms, 3(3):Art. 31, 2007. [7] A. Oliva and A. Torralba. Modeling the shape of the scene: a holistic representation of the spatial envelope. Int. J. Computer Vision, 42(3):145–175, 2001. [8] A. Rahimi and B. Recht. Random features for large-scale kernel machines. In NIPS, 2007. [9] M. Reed and B. Simon. Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. Academic Press, 1975. [10] B. Russell, A. Torralba, K. Murphy, and W. T. Freeman. LabelMe: a database and web-based tool for image annotation. Int. J. Computer Vision, 77:157–173, 2008. [11] R. Salakhutdinov and G. Hinton. Semantic hashing. In SIGIR Workshop on Inf. Retrieval and App. of Graphical Models, 2007. [12] B. Sch¨olkopf and A. J. Smola. Learning With Kernels. MIT Press, 2002. [13] A. Torralba, R. Fergus, and Y. Weiss. Small codes and large databases for recognition. In CVPR, 2008. [14] A. W. van der Vaart and J. A. Wellner. Weak Convergence and Empirical Processes. Springer, 1996. [15] Y. Weiss, A. Torralba, and R. Fergus. Spectral hashing. In NIPS, 2008.
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Nonparametric Greedy Algorithms for the Sparse Learning Problem Han Liu and Xi Chen School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 Abstract This paper studies the forward greedy strategy in sparse nonparametric regression. For additive models, we propose an algorithm called additive forward regression; for general multivariate models, we propose an algorithm called generalized forward regression. Both algorithms simultaneously conduct estimation and variable selection in nonparametric settings for the high dimensional sparse learning problem. Our main emphasis is empirical: on both simulated and real data, these two simple greedy methods can clearly outperform several state-ofthe-art competitors, including LASSO, a nonparametric version of LASSO called the sparse additive model (SpAM) and a recently proposed adaptive parametric forward-backward algorithm called Foba. We also provide some theoretical justifications of specific versions of the additive forward regression. 1 Introduction The linear model is a mainstay of statistical inference. At present, there are two major approaches to fit sparse linear models: convex regularization and greedy pursuit. The convex regularization approach regularizes the model by adding a sparsity constraint, leading to methods like LASSO [19, 7] or the Dantzig selector [6]. The greedy pursuit approach regularizes the model by iteratively selecting the current optimal approximation according to some criteria, leading to methods like the matching pursuit [14] or orthogonal matching pursuit (OMP) [20]. Substantial progress has been made recently on applying the convex regularization idea to fit sparse additive models. For splines, Lin and Zhang [12] propose a method called COSSO, which uses the sum of reproducing kernel Hilbert space norms as a sparsity inducing penalty, and can simultaneously conduct estimation and variable selection; Ravikumar et al. [17, 16] develop a method called SpAM. The population version of SpAM can be viewed as a least squares problem penalized by the sum of L2(P)-norms; Meier et al. [15] develop a similar method using a different sparsitysmoothness penalty, which guarantees the solution to be a spline. All these methods can be viewed as different nonparametric variants of LASSO. They have similar drawbacks: (i) it is hard to extend them to handle general multivariate regression where the mean functions are no longer additive; (ii) due to the large bias induced by the regularization penalty, the model estimation is suboptimal. One way to avoid this is to resort to two-stage procedures as in [13], but the method becomes less robust due to the inclusion of an extra tuning parameter in the first stage. In contrast to the convex regularization methods, the greedy pursuit approaches do not suffer from such problems. Instead of trying to formulate the whole learning task to be a global convex optimization, the greedy pursuit approaches adopt iterative algorithms with a local view. During each iteration, only a small number of variables are actually involved in the model fitting so that the whole inference only involves low dimensional models. Thus they naturally extend to the general multivariate regression and do not induce large estimation bias, which makes them especially suitable for high dimensional nonparametric inference. However, the greedy pursuit approaches do not attract as 1 much attention as the convex regularization approaches in the nonparametric literature. For additive models, the only work we know of are the sparse boosting [4] and multivariate adaptive regression splines (MARS) [9]. These methods mainly target on additive models or lower-order functional ANOVA models, but without much theoretical analysis. For general multivariate regression, the only available method we are aware of is rodeo [11]. However, rodeo requires the total number of variables to be no larger than a double-logarithmic of the data sample size, and does not explicitly identify relevant variables. In this paper, we propose two new greedy algorithms for sparse nonparametric learning in high dimensions. By extending the idea of the orthogonal matching pursuit to nonparametric settings, the main contributions of our work include: (i) we formulate two greedy nonparametric algorithms: additive forward regression (AFR) for sparse additive models and generalized forward regression (GFR) for general multivariate regression models. Both of them can simultaneously conduct estimation and variable selection in high dimensions. (ii) We present theoretical results for AFR using specific smoothers. (iii) We report thorough numerical results on both simulated and real-world datasets to demonstrate the superior performance of these two methods over the state-of-the-art competitors, including LASSO, SpAM, and an adaptive parametric forward-backward algorithm called Foba [22]. The rest of this paper is organized as follows: in the next section we review the basic problem formulation and notations. In Section 3 we present the AFR algorithm, in section 4, we present the GFR algorithm. Some theoretical results are given in Section 5. In Section 6 we present numerical results on both simulated and real datasets, followed by a concluding section at the end. 2 Sparse Nonparametric Learning in High Dimensions We begin by introducing some notations. Assuming n data points (Xi, Y i) n i=1 are observed from a high dimensional regression model Y i = m(Xi) + ϵi, ϵi ∼N(0, σ2) i = 1, . . . , n, (1) where Xi = (Xi 1, . . . , Xi p)T ∈Rp is a p-dimensional design point, m : Rp →R is an unknown smooth mean function. Here we assume m lies in a p-dimensional second order Sobolev ball with finite radius. In the sequel, we denote the response vector (Y 1, . . . , Y n)T by Y and the vector (X1 j , . . . , Xn j )T by Xj for 1 ≤j ≤p. We assume m is functional sparse, i.e. there exists an index set S ⊂{1, . . . , p}, such that (General) m(x) = m(xS), (2) where |S| = r ≪p and xS denotes the sub-vector of x with elements indexed by S. Sometimes, the function m can be assumed to have more structures to obtain a better estimation result. The most popular one is additivity assumption [10]. In this case, m decomposes into the sum of r univariate functions {mj}j∈S: (Additive) m(x) = α + X j∈S mj(xj), (3) where each component function mj is assumed to lie in a second order Sobolev ball with finite radius so that each element in the space is smooth enough. For the sake of identifiability, we also assume Emj(Xj) = 0 for j = 1, . . . , p, where the expectation is taken with respect to the marginal distribution of Xj. Given the models in (2) or (3), we have two tasks: function estimation and variable selection. For the first task, we try to find an estimate bm, such that ∥bm −m∥→0 as n goes to infinity, where ∥· ∥ is some function norm. For the second task, we try to find an estimate bS, which is an index set of variables, such that P bS = S →1 as n goes to infinity. 3 Additive Forward Regression In this section, we assume the true model is additive, i.e. m(x) = α + P j∈S mj(xj). In general, if the true index set for the relevant variables is known, the backfitting algorithm can be directly 2 applied to estimate bm [10]. It is essentially a Gauss-Seidel iteration for solving a set of normal equations in a function space. In particular, we denote the estimates on the jth variable Xj to be bmj ≡( bmj(X1 j ), . . . , bmj(Xn j ))T ∈Rn. Then bmj can be estimated by regressing the partial residual vector Rj = Y −α −P k̸=j bmk on the variable Xj. This can be calculated by bmj = SjRj, where Sj : Rn →Rn is a smoothing matrix, which only depends on X1, . . . , Xn but not on Y . Once bmj is updated, the algorithm holds it fixed and repeats this process by cycling through each variable until convergence. Under mild conditions on the smoothing matrices S1, . . . , Sp, the backfitting algorithm is a first order algorithm that guarantees to converge [5] and achieves the minimax rate of convergence as if only estimating a univariate function. However, for sparse learning problems, since the true index set is unknown, the backfitting algorithm no longer works due to the uncontrolled estimation variance. By extending the idea of the orthogonal matching pursuit to sparse additive models, we design a forward greedy algorithm called the additive forward regression (AFR), which only involves a few variables in each iteration. Under this framework, we only need to conduct the backfitting algorithm on a small set of variables. Thus the variance can be well controlled. The algorithm is described in Figure 1, where we use ⟨·, ·⟩n to denote the inner product of two vectors. Input: (Xi, Y i) n i=1 and η > 0 let A(0) = ∅, α = Pn i=1 Y i/n and the residual R(0) = Y −α for k = 1, 2, 3, . . . for each j ̸∈A(k−1), estimate bmj by smoothing: bmj = SjR(k−1) let j(k) = arg max j̸∈A(k−1)|⟨bmj, R(k−1)⟩n| let A(k) = A(k−1) ∪j(k) estimate M(k) = {mj : j ∈A(k)} by the backfitting algorithm compute the residual R(k) = Y −α −P mj∈M(k) mj(Xj) if (∥R(k−1)∥2 2 −∥R(k)∥2 2)/n ≤η k = k −1 break end if end for Output: selected variables A(k) and estimated component functions M(k) = {mj : j ∈A(k)} Figure 1: THE ADDITIVE FORWARD REGRESSION ALGORITHM The algorithm uses an active set A to index the variables included in the model during each iteration and then performs a full optimization over all “active” variables via the backfitting algorithm. The main advantage of this algorithm is that during each iteration, the model inference is conducted in low dimensions and thus avoids the curse of dimensionality. The stopping criterion is controlled by a predefined parameter η which is equivalent to the regularization tuning parameter in convex regularization methods. Other stopping criteria, such as the maximum number of steps, can also be adopted. In practice, we always recommend to use data-dependent technique, such as crossvalidation, to automatically tune this parameter. Moreover, the smoothing matrix Sj can be fairly general, e.g. univariate local linear smoothers as described below, kernel smoothers or spline smoothers [21], etc. 4 Generalized Forward Regression This section only assume m(x) to be functional sparse, i.e. m(x) = m(xS), without restricting the model to be additive. In this case, to find a good estimate bm becomes more challenging. To estimate the general multivariate mean function m(x), one of the most popular methods is the local linear regression: given an evaluation point x = (x1, . . . , xp)T , the estimate bm(x) is the 3 Input: (Xi, Y i) n i=1 and η > 0 let A(0) = ∅, α = Pn i=1 Y i/n and δ(0) = Pn i=1(Y i −α)2/n for k = 1, 2, 3, . . . let j(k) = arg min j̸∈A(k−1) Pn i=1 Y i −S(A(k−1) ∪j)XiY 2 /n let A(k) = A(k−1) ∪j(k) let δ(k) = Pn i=1 Y i −S(A(k))XiY 2 /n if (δ(k−1) −δ(k)) ≤η k = k −1 break end if end for Output: selected variables A(k) and local linear estimates (S(A(k))X1Y, . . . , S(A(k))XnY ) Figure 2: THE GENERALIZED FORWARD REGRESSION ALGORITHM solution bαx to the following locally kernel weighted least squares problem: min αx,βx n X i=1 Y i −αx −βT x (Xi −x) 2 p Y j=1 Khj(Xi j −xj), (4) where K(·) is a one dimensional kernel function and the kernel weight function in (4) is taken as a product kernel with the diagonal bandwidth matrix H1/2 = diag{h1, . . . , hp}. Such a problem can be re-casted as a standard weighted least squares regression. Therefore a closed-form solution to the the local linear estimate can be explicitly given by bαx = eT 1 (XT x WxXx)−1XT x WxY = SxY, where e1 = (1, 0, . . . , 0)T is the first canonical vector in Rp+1 and Wx = diag p Y j=1 Khj(X1 j −xj), . . . , p Y j=1 Khj(Xn j −xj) , Xx = 1 (X1 −x)T ... ... 1 (Xn −x)T . Here, Sx is the local linear smoothing matrix. Note that if we constrain βx = 0, then the local linear estimate reduces to the kernel estimate. The pointwise rate of convergence of such an estimate has been characterized in [8]: | bm(x) −m(x)|2 = OP (n−4/(4+p)), which is extremely slow when p > 10. To handle the large p case, we again extend the idea of the orthogonal matching pursuit to this setting. For an index subset A ⊂{1, . . . , p} and the evaluation point x, the local linear smoother restricted on A is denoted as S(A) and S(A)x = eT 1 X(A)T x W(A)xX(A)x −1 X(A)T x W(A)x, where W(A)x is a diagonal matrix whose diagonal entries are the product of univariate kernels over the set A and X(A)x is a submatrix of Xx that only contains the columns indexed by A. Given these definitions, the generalized forward regression (GFR) algorithm is described in Figure 2. Similar to AFR, GFR also uses an active set A to index the variables included in the model. Such mechanism allows all the statistical inference to be conducted only in low-dimensional spaces. The GFR algorithm using the multivariate local linear smoother can be computationally heavy for very high dimensional problems. However, GFR is a generic framework and can be equipped with arbitrary multivariate smoothers, e.g. kernel/Nearest Neighbor/spline smoothers. These smoothers lead to much better scalability. The only reason we use the local linear smoother as an illustrative example in this paper is due to its popularity and potential advantage on correcting the boundary bias. 4 5 Theoretical Properties In this section, we provide the theoretical properties of the additive forward regression estimates using the spline smoother. Due to the asymptotic equivalence of the spline smoother and the local linear smoother [18], we deduce that these results should also hold for the local linear smoother. Our main result in Theorem 1 says when using the spline smoother with certain truncation rate to implement AFR algorithm, the resulting estimator is consistent with a certain rate. When the underlying true component functions do not go to zeroes too fast, we also achieve variable selection consistency. Our analysis relies heavily on [3]. A similar analysis has also been reported in the technical report version of [16]. Theorem 1. Assuming there exists some ξ > 0 which can be arbitrarily large, such that p = O(nξ). For ∀j ∈{1, . . . , p}, we assume mj lies in a second-order Sobolev ball with finite radius, and m = α + Pp j=1 mj. For the additive forward regression algorithm using the spline smoother with a truncation rate at n1/4, after (n/log n)1/2 steps, we obtain that ∥m −bm∥2 = OP r log n n ! . (5) Furthermore, if we also assume minj∈S ∥mj∥= Ω log n n 1/4 , then P bS = S →1 as n goes to infinity. Here, bS is the index set for nonzero component functions in bm. The rate for ∥bm −m∥2 obtained from Theorem 1 is only O(n−1/2), which is slower than the minimax rate O(n−4/5). This is mainly an artifact of our analysis instead of a drawback of the additive forward regression algorithm. In fact, if we perform a basis expansion for each component function to first cast the problem to be a finite dimensional linear model with group structure, under some more stringent smallest eigenvalue conditions on the augmented design as in [23], we can show that AFR using spline smoothers can actually achieves the minimax rate O(n−4/5) up to a logarithmic factor. A detailed treatment will be reported in a follow up paper. Sketch of Proof: We first describe an algorithm called group orthogonal greedy algorithm (GOGA), which solves a noiseless function approximation problem in a direct-sum Hilbert space. AFR can then be viewed as an empirical realization of such an “ideal” algorithm. GOGA is a group extension of the orthogonal greedy algorithm (OGA) in [3]. For j = 1, . . . , p, let Hj be a Hilbert space of continuous functions with a Hamel basis Dj. Then for a function m in the direct-sum Hilbert space H = H1 + H2 + . . . + Hp, we want to approximate m using the union of many truncated bases D = D′ 1 ∪. . . ∪D′ p, where for all j, D′ j ⊂Dj. We equip an inner product ⟨·, ·⟩on H: ∀f, g ∈H, ⟨f, g⟩= R f(X)g(X)dPX where PX is the marginal distribution for X. Let ∥· ∥be the norm induced by the inner product ⟨·, ·⟩on H. GOGA begins by setting m(0) = 0, and then recursively defines the approximant m(k) based on m(k−1) and its residual r(k−1) ≡m −m(k−1). More specifically: we proceed as the following: define f (k) j to be the projection of r(k−1) onto the truncated basis D′ j, i.e. f (k) j = arg ming∈D′ j ∥r(k−1) −g∥2. We calculate j(k) as j(k) = arg maxj |⟨r(k−1), f (k) j ⟩|. m(k) can then be calculated by projecting m onto the additive function space generated by A(k) = D′ j(1) + · · · + D′ j(k): bm(k) = arg min g∈span(A(k)) ∥m −g∥2. AFR using regression splines is exactly GOGA when there is no noise. For noisy samples, we replace the unknown function m by its n-dimensional output vector Y , and replace the inner product ⟨·, ·⟩by ⟨·, ·⟩n, which is defined as ⟨f, g⟩n = 1 n Pn i=1 f(Xi)g(Xi). The projection of the current residual vector onto each dictionary D′ j is replaced by the corresponding nonparametric smoothers. Considering any function m ∈H, we proceed in the same way as in [3], but replacing the OGA arguments in their analysis by those of GOGA. The desired results of the theorem follow from a simple argument on bounding the random random covering number of spline spaces. 5 6 Experimental Results In this section, we present numerical results for AFR and GFR applied to both synthetic and real data. The main conclusion is that, in many cases, their performance on both function estimation and variable selection can clearly outperform those of LASSO, Foba, and SpAM. For all the reported experiments, we use local linear smoothers to implement AFR and GFR. The results for other smoothers, such as smoothing splines, are similar. Note that different bandwidth parameters will have big effects on the performances of local linear smoothers. Our experiments simply use the plug-in bandwidths according to [8] and set the bandwidth for each variable to be the same. For AFR, the bandwidth h is set to be 1.06n−1/5 and for GFR, the bandwidth is varying over each iteration such that h = 1.06n−1/(4+|A|), where |A| is the size of the current active set. For an estimate bm, the estimation performance for the synthetic data is measured by the mean square error (MSE), which is defined as MSE( bm) = 1 n Pn i=1 m(Xi) −bm(Xi) 2 . For the real data, since we do not know the true function m(x), we approximate the mean squared error using 5-fold crossvalidation scores. 6.1 The Synthetic Data For the synthetic data experiments, we consider the compound symmetry covariance structure of the design matrix X ∈Rn×p with n = 400 and p = 20. Each dimension Xj is generated according to Xj = Wj + tU 1 + t , j = 1, . . . , p, where W1, . . . , Wp and U are i.i.d. sampled from Uniform(0,1). Therefore the correlation between Xj and Xk is t2/(1 + t2) for j ̸= k. We assume the true regression functions have r = 4 relevant variables: Y = m(X) + ϵ = m(X1, . . . , X4) + ϵ. (6) To evaluate the variable selection performance of different methods, we generate 50 designs and 50 trials for each design. For each trial, we run the greedy forward algorithm r steps. If all the relevant variables are included in, the variable selection task for this trial is said to be successful. We report the mean and standard deviation of the success rate in variable selection for various correlation between covariates by varying the values of t. We adopt some synthetic examples as in [12] and define the following four functions: g1(x) = x, g2(x) = (2x−1)2, g3(x) = sin(2πx)/(2−sin(2πx)), and g4(x) = 0.1 sin(2πx)+0.2 cos(2πx)+ 0.3 sin2(2πx) + 0.4 cos3(2πx) + 0.5 sin3(2πx). The following four regression models are studied. The first model is linear; the second is additive; the third and forth are more complicated nonlinear models with at least two way interactions: (Model1) : Y i = 2Xi 1 + 3Xi 2 + 4Xi 3 + 5Xi 4 + 2N(0, 1), with t = 1 ; (Model2) : Y i = 5g1(Xi 1) + 3g2(Xi 2) + 4g3(Xi 3) + 6g4(Xi 4) + 4N(0, 1), with t = 1 ; (Model3) : Y i = exp(2Xi 1Xi 2 + Xi 3) + 2Xi 4 + N(0, 1), with t = 0.5 ; (Model4) : Y i = X4 j=1 gj(Xi j) + g1(Xi 3Xi 4) + g2((Xi 1 + Xi 3)/2) + g3(Xi 1Xi 2) + N(0, 1) with t = 0.5. Compared with LASSO, Foba, and SpAM, the estimation performance using MSE as evaluation criterion is presented in Figure 3. And Table 1 shows the rate of success for variable selection of these models with different correlations controlled by t. From Figure 3, we see that AFR and GFR methods provide very good estimates for the underlying true regression functions as compared to others. Firstly, LASSO and SpAM perform very poorly when the selected model is very sparse. This is because they are convex regularization based approaches: to obtain a very sparse model, they induce very large estimation bias. On the other hand, the greedy pursuit based methods like Foba, AFR and GFR do not suffer from such a problem. Secondly, when the true model is linear, all methods perform similarly. For the nonlinear true regression 6 Model 1 Model 2 Model 3 Model 4 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 sparsity MSE GFR AFR SpAM Foba LASSO 1 2 3 4 5 6 7 8 9 10 4 5 6 7 8 9 10 11 12 13 14 sparsity MSE GFR AFR SpAM Foba LASSO 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 2 2.5 3 3.5 sparsity MSE GFR AFR SpAM Foba LASSO 1 2 3 4 5 6 7 8 9 10 0.2 0.25 0.3 0.35 0.4 0.45 0.5 sparsity MSE GFR AFR SpAM Foba LASSO Figure 3: Performance of the different algorithms on synthetic data: MSE versus sparsity level function, AFR, GFR and SpAM outperform LASSO and Foba. It is expectable since LASSO and Foba are based on linear assumptions. Furthermore, we notice that when the true model is additive (Model 2) or nearly additive (Model 4), AFR performs the best. However, for the non-additive general multivariate regression function (Model 3), GFR performs the best. For all examples, when more and more irrelevant variables are included in the model, SpAM has a better generalization performance due to the regularization effect. Table 1: Comparison of variable selection Model 1 LASSO(sd) Foba SpAM AFR GFR t = 0 1.000 (0.0000) 1.000 (0.0000) 0.999 (0.0028) 0.999 (0.0039) 0.990 (0.0229) t = 1 0.879 (0.0667) 0.882 (0.0557) 0.683 (0.1805) 0.879 (0.0525) 0.839 (0.0707) t = 2 0.559 (0.0913) 0.553 (0.0777) 0.190 (0.1815) 0.564 (0.0739) 0.515 (0.0869) Model 2 LASSO(sd) Foba SpAM AFR GFR t = 0 0.062 (0.0711) 0.069 (0.0774) 0.842 (0.1128) 0.998 (0.0055) 0.769 (0.1751) t = 1 0.056 (0.0551) 0.060 (0.0550) 0.118 (0.0872) 0.819 (0.1293) 0.199 (0.2102) t = 2 0.004 (0.0106) 0.029 (0.0548) 0.008 (0.0056) 0.260 (0.1439) 0.021 (0.0364) Model 3 LASSO(sd) Foba SpAM AFR GFR t = 0 0.997 (0.0080) 0.999 (0.0039) 0.980 (0.1400) 1.000 (0.0000) 1.000 (0.0000) t = 1 0.818 (0.1137) 0.802 (0.1006) 0.934 (0.1799) 1.000 (0.0000) 0.995 (0.0103) t = 2 0.522 (0.1520) 0.391 (0.1577) 0.395 (0.3107) 0.902 (0.1009) 0.845 (0.1623) Model 4 LASSO(sd) Foba SpAM AFR GFR t = 0 0.043 (0.0482) 0.043 (0.0437) 0.553 (0.1864) 0.732 (0.1234) 0.967 (0.0365) t = 0.5 0.083 (0.0823) 0.049 (0.0511) 0.157 (0.1232) 0.126 (0.0688) 0.708 (0.1453) t = 1 0.048 (0.0456) 0.085 (0.0690) 0.095 (0.0754) 0.192 (0.0679) 0.171 (0.1067) The variable selection performances of different methods in Table 1 are very similar to their estimation performances. We observe that, when correlation parameter t becomes larger, the performances of all methods decrease. But SpAM is most sensitive to the correlation increase. In all models, the performance of SpAM can decrease more than 70% for the larger t; in contrast, AFR and GFR are more robust to the increased correlation between different covariates. Another interesting observation is on model 4. From the previous discussion, on this model, AFR achieves a better estimation performance. However, when comparing the variable selection performance, GFR is the best. This suggests that for nonparametric inference, the goals of estimation consistency and variable selection consistency might not be always coherent. Some tradeoffs might be needed to balance them. 6.2 The real data In this subsection, we compare five methods on three real datasets: Boston Housing, AutoMPG, and Ionosphere data set 1. Boston Housing contains 556 data points, with 13 features; AutoMPG 392 data points (we delete those with missing values), with 7 features and Ionosphere 351 data points, with 34 features and the binary output. We treat Ionosphere as a regression problem although the 1Available from UCI Machine Learning Database Repository: http:archive.ics.uci.edu/ml. 7 response is binary. We run 10 times 5-fold cross validation on each dataset and plot the mean and standard deviation of MSE versus different sparsity levels in Figure 4. Boston Housing AutoMPG Ionosphere 1 2 3 4 5 6 7 8 9 10 11 12 13 10 20 30 40 50 60 70 80 90 sparsity CV Error GFR AFR SpAM Foba LASSO 1 2 3 4 5 6 7 0 10 20 30 40 50 60 70 sparsity CV Error GFR AFR SpAM Foba LASSO 1 5 10 15 20 25 30 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 sparsity CV Error GFR AFR SpAM Foba LASSO Figure 4: Performance of the different algorithms on real datasets: CV error versus sparsity level From Figure 4, since all the error bars are tiny, we deem all the results significant. On the Boston Housing and AutoMPG datasets, the generalization performances of AFR and GFR are clearly better than LASSO, Foba, and SpAM. For all these datasets, if we prefer very sparse models, the performance of the greedy methods are much better than the convex regularization methods due to the much less bias being induced. On the Ionosphere data, we only need to run GFR up to 15 selected variables, since the generalization performance with 15 variables is already worse than the null model due to the curse of dimensionality. Both AFR and GFR on this dataset achieve the best performances when there are no more than 10 variables included; while SpAM achieves the best CV score with 25 variables. However, this is not to say that the true model is not sparse. The main reason that SpAM can achieve good generalization performance when many variables included is due to its regularization effect. We think the true model should be sparse but not additive. Similar trend among different methods has also appeared in Model 4 of previous synthetic datasets. 7 Conclusions and Discussions We presented two new greedy algorithms for nonparametric regression with either additive mean functions or general multivariate regression functions. Both methods utilize the iterative forward stepwise strategy, which guarantees the model inference is always conducted in low dimensions in each iteration. These algorithms are very easy to implement and have good empirical performance on both simulated and real datasets. One thing worthy to note is: people sometimes criticize the forward greedy algorithms since they can never have the chance to correct the errors made in the early steps. This is especially true for high dimensional linear models, which motivates the outcome of some adaptive forward-backward procedures such as Foba [22]. We addressed a similar question: Whether a forward-backward procedure also helps in the nonparametric settings? AFR and GFR can be trivially extended to be forward-backward procedures using the same way as in [22]. We conducted a comparative study to see whether the backward steps help or not. However, the backward step happens very rarely and the empirical performance is almost the same as the purely forward algorithm. This is very different from the linear model cases, where the backward step can be crucial. In summary, in the nonparametric settings, the backward ingredients will cost much more computational efforts with very tiny performance improvement. We will investigate more on this phenomenon in the near future. A very recent research strand is to learn nonlinear models by the multiple kernel learning machinery [1, 2], another future work is to compare our methods with the multiple kernel learning approach from both theoretical and computational perspectives. Acknowledgements We thank John Lafferty, Larry Wasserman, Pradeep Ravikumar, and Jamie Carbonell for very helpful discussions on this work. This research was supported in part by NSF grant CCF-0625879 and a Google Fellowship to Han Liu. 8 References [1] Francis Bach. Consistency of the group lasso and multiple kernel learning. Journal of Machine Learning Research, 8:1179–1225, 2008. [2] Francis Bach. Exploring large feature spaces with hierarchical multiple kernel learning. In Advances in Neural Information Processing Systems 21. MIT Press, 2008. [3] Andrew R. Barron, Albert Cohen, Wolfgang Dahmen, and Ronald A. DeVore. Approximation and learning by greedy algorithms. The Annals of Statistics, 36:64–94, 2008. [4] Peter B¨uhlmann and Bin Yu. Sparse boosting. Journal of Machine Learning Research, 7:1001– 1024, 2006. [5] Andreas Buja, Trevor Hastie, and Robert Tibshirani. Linear smoothers and additive models. The Annals of Statistics, 17:453–510, 1989. [6] Emmanuel Candes and Terence Tao. The dantzig selector: statistical estimation when p is much larger than n. The Annals of Statistics, 35:2313–2351, 2007. [7] Scott Shaobing Chen, David L. Donoho, and Michael A. Saunders. Atomic decomposition by basis pursuit. SIAM Journal on Scientific and Statistical Computing, 20:33–61, 1998. [8] Jianqing Fan and Ir`ene Gijbels. Local polynomial modelling and its applications. Chapman and Hall, 1996. [9] Jerome H. Friedman. Multivariate adaptive regression splines. The Annals of Statistics, 19:1– 67, 1991. [10] Trevor Hastie and Robert Tibshirani. Generalized additive models. Chapman & Hall Ltd., 1999. [11] John Lafferty and Larry Wasserman. Rodeo: Sparse, greedy nonparametric regression. The Annals of Statistics, 36(1):28–63, 2008. [12] Yi Lin and Hao Helen Zhang. Component selection and smoothing in multivariate nonparametric regression. The Annals of Statistics., 34(5):2272–2297, 2006. [13] Han Liu and Jian Zhang. On the estimation consistency of the group lasso and its applications. Proceedings of the Twelfth International Conference on Artificial Intelligence and Statistics, 2009. [14] S. Mallat and Z. Zhang. Matching pursuit with time-frequency dictionaries. IEEE Transactions on Signal Processing, 41:3397–3415, 1993. [15] Lukas Meier, Sara van de Geer, and Peter B¨uhlmann. High-dimensional additive modelling. The Annals of Statistics (to appear), 2009. [16] Pradeep Ravikumar, John Lafferty, Han Liu, and Larry Wasserman. Sparse additive models. Journal of the Royal Statistical Society, Series B, Methodological, 2009. To appear. [17] Pradeep Ravikumar, Han Liu, John Lafferty, and Larry Wasserman. Spam: Sparse additive models. In Advances in Neural Information Processing Systems 20, 2007. [18] B. W. Silverman. Spline smoothing: The equivalent variable kernel method. The Annals of Statistics, 12:898–916, 1984. [19] Robert Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, Methodological, 58:267–288, 1996. [20] Joel A. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theory, 50(10):2231–2241, October 2004. [21] Grace Wahba. Spline models for observational data. SIAM [Society for Industrial and Applied Mathematics], 1990. [22] Tong Zhang. Adaptive forward-backward greedy algorithm for learning sparse representations. Technical report, Rutgers University, 2008. [23] Tong Zhang. On the consistency of feature selection usinggreedy least squares regression. Journal of Machine Learning Research, 10:555–568, 2009. 9
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Regularized Distance Metric Learning: Theory and Algorithm Rong Jin1 Shijun Wang2 Yang Zhou1 1Dept. of Computer Science & Engineering, Michigan State University, East Lansing, MI 48824 2Radiology and Imaging Sciences, National Institutes of Health, Bethesda, MD 20892 rongjin@cse.msu.edu wangshi@cc.nih.gov zhouyang@msu.edu Abstract In this paper, we examine the generalization error of regularized distance metric learning. We show that with appropriate constraints, the generalization error of regularized distance metric learning could be independent from the dimensionality, making it suitable for handling high dimensional data. In addition, we present an efficient online learning algorithm for regularized distance metric learning. Our empirical studies with data classification and face recognition show that the proposed algorithm is (i) effective for distance metric learning when compared to the state-of-the-art methods, and (ii) efficient and robust for high dimensional data. 1 Introduction Distance metric learning is a fundamental problem in machine learning and pattern recognition. It is critical to many real-world applications, such as information retrieval, classification, and clustering [6, 7]. Numerous algorithms have been proposed and examined for distance metric learning. They are usually classified into two categories: unsupervised metric learning and supervised metric learning. Unsupervised distance metric learning, or sometimes referred to as manifold learning, aims to learn a underlying low-dimensional manifold where the distance between most pairs of data points are preserved. Example algorithms in this category include ISOMAP [13] and Local Linear Embedding (LLE) [8]. Supervised metric learning attempts to learn distance metrics from side information such as labeled instances and pairwise constraints. It searches for the optimal distance metric that (a) keeps data points of the same classes close, and (b) keeps data points from different classes far apart. Example algorithms in this category include [17, 10, 15, 5, 14, 19, 4, 12, 16]. In this work, we focus on supervised distance metric learning. Although a large number of studies were devoted to supervised distance metric learning (see the survey in [18] and references therein), few studies address the generalization error of distance metric learning. In this paper, we examine the generalization error for regularized distance metric learning. Following the idea of stability analysis [1], we show that with appropriate constraints, the generalization error of regularized distance metric learning is independent from the dimensionality of data, making it suitable for handling high dimensional data. In addition, we present an online learning algorithm for regularized distance metric learning, and show its regret bound. Note that although online metric learning was studied in [9], our approach is advantageous in that (a) it is computationally more efficient in handling the constraint of SDP cone, and (b) it has a proved regret bound while [9] only shows a mistake bound for the datasets that can be separated by a Mahalanobis distance. To verify the efficacy and efficiency of the proposed algorithm for regularized distance metric learning, we conduct experiments with data classification and face recognition. Our empirical results show that the proposed online algorithm is (1) effective for metric learning compared to the state-of-the-art methods, and (2) robust and efficient for high dimensional data. 1 2 Regularized Distance Metric Learning Let D = {zi = (xi, yi), i = 1, . . . , n} denote the labeled examples, where xk = (x1 k, . . . , xd k) ∈Rd is a vector of d dimension and yi ∈{1, 2, . . ., m} is class label. In our study, we assume that the norm of any example is upper bounded by R, i.e., supx |x|2 ≤R. Let A ∈Sd×d + be the distance metric to be learned, where the distance between two data points x and x′ is calculated as |x −x′|2 A = (x −x′)⊤A(x −x′). Following the idea of maximum margin classifiers, we have the following framework for regularized distance metric learning: min A 1 2|A|2 F + 2C n(n −1) X i<j g yi,j 1 −|xi −xj|2 A : A ⪰0, tr(A) ≤η(d) (1) where • yi,j is derived from class labels yi and yj, i.e., yi,j = 1 if yi = yj and −1 otherwise. • g(z) is the loss function. It outputs a small value when z is a large positive value, and a large value when z is large negative. We assume g(z) to be convex and Lipschitz continuous with Lipschitz constant L. • |A|2 F is the regularizer that measures the complexity of the distance metric A. • tr(A) ≤η(d) is introduced to ensure a bounded domain for A. As will be revealed later, this constraint will become active only when the constraint constant η(d) is sublinear in d, i.e., η ∼O(dp) with p < 1. We will also show how this constraint could affect the generalization error of distance metric learning. 3 Generalization Error Let AD be the distance metric learned by the algorithm in (1) from the training examples D. Let ID(A) denote the empirical loss , i.e., ID(A) = 2 n(n −1) X i<j g yi,j 1 −|xi −xj|2 A (2) For the convenience of presentation, we also write g yi,j(1 −|xi −xj|2 A) = V (A, zi, zj) to highlight its dependence on A and two examples zi and zj. We denote by I(A) the loss of A over the true distribution, i.e., I(A) = E(zi,zj)[V (A, zi, zj)] (3) Given the empirical loss ID(A) and the loss over the true distribution I(A), we define the estimation error as DD = I(AD) −ID(AD) (4) In order to show the behavior of estimation error, we follow the analysis based on the stability of the algorithm [1]. The uniform stability of an algorithm determines the stability of the algorithm when one of the training examples is replaced with another. More specifically, an algorithm A has uniform stability β if ∀(D, z), ∀i, sup u,v |V (AD, u, v) −V (ADz,i, u, v)| ≤β (5) where Dz,i stands for the new training set that is obtained by replacing zj ∈D with a new example z. We further define β = κ/n as the uniform stability β behaves like O(1/n). The advantage of using stability analysis for the generalization error of regularized distance metric learning. This is because the example pair (zi, zj) used for training distance metrics are not I.I.D. although zi is, making it difficult to directly utilize the results from statistical learning theory. In the analysis below, we first show how to derive the generalization error bound for regularized distance metric learning given the uniform stability β (or κ). We then derive the uniform stability constant for the regularized distance metric learning framework in (1). 2 3.1 Generalization Error Bound for Given Uniform Stability Analysis in this section follows closely [1], and we therefore omit the detailed proofs. Our analysis utilizes the McDiarmid inequality that is stated as follows. Theorem 1. (McDiarmid Inequality) Given random variables {vi}l i=1, v′ i, and a function F : vl → R satisfying sup v1,...,vl,v′ i ˛˛˛F(v1, . . . , vl) −F(v1, . . . , vi−1, v ′ i, vi+1, . . . , vl) ˛˛˛ ≤ci, the following statement holds Pr (|F(v1, . . . , vl) −E(F(v1, . . . , vl))| > ǫ) ≤2 exp − 2ǫ Pl i=1 c2 i ! To use the McDiarmid inequality, we first compute E(DD). Lemma 1. Given a distance metric learning algorithm A has uniform stability κ/n, we have the following inequality for E(DD) E(DD) ≤2κ n (6) where n is the number of training examples in D. The result in the following lemma shows that the condition in McDiarmid inequality holds. Lemma 2. Let D be a collection of n randomly selected training examples, and Di,z be the collection of examples that replaces zi in D with example z. We have |DD −DDi,z| bounded as follows |DD −DDi,z| ≤2κ + 8Lη(d) + 2g0 n (7) where g0 = supz,z′ |V (0, z, z′)| measures the largest loss when distance metric A is 0. Combining the results in Lemma 1 and 2, we can now derive the the bound for the generalization error by using the McDiarmid inequality. Theorem 2. Let D denote a collection of n randomly selected training examples, and AD be the distance metric learned by the algorithm in (1) whose uniform stability is κ/n. With probability 1 −δ, we have the following bound for I(AD) I(AD) −ID(AD) ≤2κ n + (2κ + 4Lη(d) + 2g0) r ln(2/δ) 2n (8) 3.2 Generalization Error for Regularized Distance Metric Learning First, we show that the superium of tr(AD) is O(d1/2), which verifies that η(d) should behave sublinear in d. This is summarized by the following proposition. Proposition 1. The trace constraint in (1) will be activated only when η(d) ≤ p 2dg0C (9) where g0 = supz,z′ |V (0, z, z′)|. Proof. It follows directly from [tr(AD)/d]2 ≤|AD|2 F ≤2C sup z,z′ |V (0, z, z′)| ≤Cg0. To bound the uniform stability, we need the following proposition Proposition 2. For any two distance metrics A and A′, we have the following inequality hold for any examples zu and zv |V (A, zu, zv) −V (A′, zu, zv)| ≤4LR2|A −A′|F (10) 3 The above proposition follows directly from the fact that (a) V (A, z, z′) is Lipschitz continuous and (b) |x|2 ≤R for any example x. The following lemma bounds |AD −AD′|F . Lemma 3. Let D denote a collection of n randomly selected training examples, and by z = (x, y) a randomly selected example. Let AD be the distance metric learned by the algorithm in (1). We have |AD −ADi,z|F ≤8CLR2 n (11) The proof of the above lemma can be found in Appendix A. By putting the results in Lemma 3 and Proposition 2, we have the following theorem for the stability of the Frobenius norm based regularizer. Theorem 3. The uniform stability for the algorithm in (1) using the Frobenius norm regularizer, denoted by β, is bounded as follows β = κ n ≤32CL2R4 n (12) where κ = 32CL2R4 Combing Theorem 3 and 2, we have the following theorem for the generalization error of distance metric learning algorithm in (1) using the Frobenius norm regularizer Theorem 4. Let D be a collection of n randomly selected examples, and AD be the distance metric learned by the algorithm in (1) with h(A) = |A|2 F . With probability 1 −δ, we have the following bound for the true loss function I(AD) where AD is learned from (1) using the Frobenius norm regularizer I(AD) −ID(AD) ≤32CL2R4 n + 32CL2R4 + 4Ls(d) + 2g0 r ln(2/δ) 2n (13) where s(d) = min √2dg0C, η(d) . Remark The most important feature in the estimation error is that it converges in the order of O(s(d)/√n). By choosing η(d) to have a low dependence of d (i.e., η(d) ∼dp with p ≪1), the proposed framework for regularized distance metric learning will be robust to the high dimensional data. In the extreme case, by setting η(d) to be a constant, the estimation error will be independent from the dimensionality of data. 4 Algorithm In this section, we discuss an efficient algorithm for solving (1). We assume a hinge loss for g(z), i.e., g(z) = max(0, b −z), where b is the classification margin. To design an online learning algorithm for regularized distance metric learning, we follow the theory of gradient based online learning [2] by defining potential function Φ(A) = |A|2 F /2. Algorithm 1 shows the online learning algorithm. The theorem below shows the regret bound for the online learning algorithm in Figure 1. Theorem 5. Let the online learning algorithm 1 run with learning rate λ > 0 on a sequence (xt, x′ t), yt, t = 1, . . . , n. Assume |x|2 ≤R for all the training examples. Then, for all distance metric M ∈Sd×d + , we have bLn ≤ 1 1 −8R4λ/b Ln(M) + 1 2λ|M|2 F where Łn(M) = n X t=1 max 0, b −yt(1 −|xt −x′ t|2 M) , bLn = n X t=1 max 0, b −yt(1 −|xt −x′ t|2 At−1) 4 Algorithm 1 Online Learning Algorithm for Regularized Distance Metric Learning 1: INPUT: predefined learning rate λ 2: Initialize A0 = 0 3: for t = 1, . . . , T do 4: Receive a pair of training examples {(x1 t, y1 t ), (x2 t , y2 t )} 5: Compute the class label yt: yt = +1 if y1 t = y2 t , and yt = −1 otherwise. 6: if the training pair (x1 t , x2 t), yt is classified correctly, i.e., yt 1 −|x1 t −x2 t|2 At−1 > 0 then 7: At = At−1. 8: else 9: At = πS+(At−1 −λyt(xt −x′ t)(xt −x′ t)⊤), where πS+(M) projects matrix M into the SDP cone. 10: end if 11: end for The proof of this theorem can be found in Appendix B. Note that the above online learning algorithm require computing πS+(M), i.e., projecting matrix M onto the SDP cone, which is expensive for high dimensional data. To address this challenge, first notice that M ′ = πS+(M) is equivalent to the optimization problem M ′ = arg minM′⪰0 |M ′ −M|F. We thus approximate At = πS+(At−1 − λyt(xt −x′ t)(xt −x′ t)⊤) with At = At−1 −λtyt(xt −x′ t)(xt −x′ t)⊤where λt is computed as follows λt = arg min λt |λt −λ| : λt ∈[0, λ], At−1 −λtyt(xt −x′ t)(xt −x′ t)⊤⪰0 (14) The following theorem shows the solution to the above optimization problem. Theorem 6. The optimal solution λt to the problem in (14) is expressed as λt = λ yt = −1 min λ, [(xt −x′ t)⊤A−1 t−1(xt −x′ t)]−1 yt = +1 Proof of this theorem can be found in the supplementary materials. Finally, the quantity (xt − x′ t)A−1 t−1(xt −x′ t) can be computed by solving the following optimization problem max u 2u⊤(xt −x′ t) −u⊤Au whose optimal value can be computed efficiently using the conjugate gradient method [11]. Note that compared to the online metric learning algorithm in [9], the proposed online learning algorithm for metric learning is advantageous in that (i) it is computationally more efficient by avoiding projecting a matrix into a SDP cone, and (ii) it has a provable regret bound while [9] only presents the mistake bound for the separable datasets. 5 Experiments We conducted an extensive study to verify both the efficiency and the efficacy of the proposed algorithms for metric learning. For the convenience of discussion, we refer to the propoesd online distance metric learning algorithm as online-reg. To examine the efficacy of the learned distance metric, we employed the k Nearest Neighbor (k-NN) classifier. Our hypothesis is that the better the distance metric is, the higher the classification accuracy of k-NN will be. We set k = 3 for k-NN for all the experiments according to our experience. We compare our algorithm to the following six state-of-the-art algorithms for distance metric learning as baselines: (1) Euclidean distance metric; (2) Mahalanobis distance metric, which is computed as the inverse of covariance matrix of training samples, i.e., (Pn i=1 xixi)−1; (3) Xing’s algorithm proposed in [17]; (4) LMNN, a distance metric learning algorithm based on the large margin nearest neighbor classifier [15]; (5) ITML, an Information-theoretic metric learning based on [4]; and (6) Relevance Component Analysis (RCA) [10]. We set the maximum number of iterations for Xing’s method to be 10, 000. The number of target neighbors in LMNN and parameter γ in ITML 5 Table 1: Classification error (%) of a k-NN (k = 3) classifier on the ten UCI data sets using seven different metrics. Standard deviation is included. Dataset Eclidean Mahala Xing LMNN ITML RCA Online-reg 1 19.5 ± 2.2 18.8 ± 2.5 29.3 ± 17.2 13.8 ± 2.5 8.6 ± 1.7 17.4 ± 1.5 13.2 ± 2.2 2 39.9 ± 2.3 6.7 ± 0.6 40.1 ± 2.6 3.6 ± 1.1 40.0 ± 2.3 3.8 ± 0.4 3.7 ± 1.2 3 36.0 ± 2.0 42.1 ± 4.0 43.5 ± 12.5 33.1 ± 0.6 39.8 ± 3.3 41.6 ± 0.7 37.3 ± 4.1 4 4.0 ± 1.7 10.4 ± 2.7 3.1 ± 2.0 3.9 ± 1.6 3.2 ± 1.6 2.9 ± 1.5 3.2 ± 1.3 5 30.6 ± 1.9 29.1 ± 2.1 30.6 ± 1.9 29.6 ± 1.8 28.8 ± 2.1 28.6 ± 2.3 27.7 ± 1.3 6 25.4 ± 4.2 18.4 ± 3.4 23.3 ± 3.4 15.2 ± 3.1 17.1 ± 4.1 13.9 ± 2.2 12.9 ± 2.2 7 31.9 ± 2.8 10.0 ± 2.8 24.6 ± 7.5 4.5 ± 2.4 28.7 ± 3.7 1.8 ± 1.5 1.8 ± 1.1 8 18.9 ± 0.5 37.3 ± 0.5 16.1 ± 0.6 18.4 ± 0.4 23.3 ± 1.3 30.6 ± 0.7 19.8 ± 0.6 9 2.0 ± 0.4 6.1 ± 0.5 12.4 ± 0.8 1.6 ± 0.3 2.5 ± 0.4 2.8 ± 0.4 2.9 ± 0.4 Table 2: p-values of the Wilcoxon signed-rank test of the 7 methods on the 9 datasets. Methods Eclidean Mahala Xing LMNN ITML RCA Online-reg Euclidean 1.000 0.734 0.641 0.004 0.496 0.301 0.129 Mahala 0.734 1.000 0.301 0.008 0.570 0.004 0.004 Xing 0.641 0.301 1.000 0.027 0.359 0.074 0.027 LMNN 0.004 0.008 0.027 1.000 0.129 0.496 0.734 ITML 0.496 0.570 0.359 0.129 1.000 0.820 0.164 RCA 0.301 0.004 0.074 0.496 0.820 1.000 0.074 Online-reg 0.129 0.004 0.027 0.734 0.164 0.074 1.000 were tuned by cross validation over the range from 10−4 to 104. All the algorithms are implemented and run using Matlab. All the experiment are run on a AMD Processor 2.8G machine, with 8GMB RAM and Linux operation system. 5.1 Experiment (I): Comparison to State-of-the-art Algorithms We conducted experiments of data classification over the following nine datasets from UCI repository: (1) balance-scale, with 3 classes, 4 features, and 625 instances; (2) breast-cancer, with 2 classes, 10 features, and 683 instance; (3) glass, with 6 classes, 9 features, and 214 instances; (4) iris, with 3 classes, 4 features, and 150 instances; (5) pima, with 2 classes, 8 features, and 768 instances; (6) segmentation, with 7 classes, 19 features, and 210 instances; (7)wine, with 3 classes, 13 features, and 178 instances; (8) waveform, with 3 classes, 21 features, and 5000 instances; (9) optdigits, with 10 classes, 64 features, 3823 instances. For all the datasets, we randomly select 50% samples for training, and use the remaining samples for testing. Table 1 shows the classification errors of all the metric learning methods over 9 datasets averaged over 10 runs, together with the standard deviation. We observe that the proposed metric learning algorithm deliver performance that comparable to the state-of-the-art methods. In particular, for almost all datasets, the classification accuracy of the proposed algorithm is close to that of LMNN, which has yielded overall the best performance among six baseline algorithms. This is consistent with the results of the other studies, which show LMNN is among the most effective algorithms for distance metric learning. To further verify if the proposed method performs statistically better than the baseline methods, we conduct statistical test by using Wilcoxon signed-rank test [3]. The Wilcoxon signed-rank test is a non-parametric statistical hypothesis test for the comparisons of two related samples. It is known to be safer than the Student’s t-test because it does not assume normal distributions. From table 2, we find that the regularized distance metric learning improves the classification accuracy significantly compared to Mahalanobis distance, Xing’s method and RCA at significant level 0.1. It performs slightly better than ITML and is comparable to LMNN. 6 0.1 0.12 0.14 0.16 0.18 0.2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Image resize ratio Classification accuracy att−face Euclidean Mahalanobis LMNN ITML RCA Online_reg 0.1 0.12 0.14 0.16 0.18 0.2 0 1000 2000 3000 4000 5000 6000 7000 Image resize ratio Running time (seconds) att−face LMNN ITML RCA Online_reg (a) (b) Figure 1: (a) Face recognition accuracy of kNN and (b) running time of LMNN, ITML, RCA and online reg algorithms on the “att-face” dataset with varying image sizes. 5.2 Experiment (II): Results for High Dimensional Data To evaluate the dependence of the regularized metric learning algorithms on data dimensions, we tested it by the task of face recognition. The AT&T face database 1 is used in our study. It consists of grey images of faces from 40 distinct subjects, with ten pictures for each subject. For every subject, the images were taken at different times, with varied the lighting condition and different facial expressions (open/closed-eyes, smiling/not-smiling) and facial details (glasses/no-glasses). The original size of each image is 112 × 92 pixels, with 256 grey levels per pixel. To examine the sensitivity to data dimensionality, we vary the data dimension (i.e., the size of images) by compressing the original images into size different sizes with the image aspect ratio preserved. The image compression is achieved by bicubic interpolation (the output pixel value is a weighted average of pixels in the nearest 4-by-4 neighborhood). For each subject, we randomly spit its face images into training set and test set with ratio 4 : 6. A distance metric is learned from the collection of training face images, and is used by the kNN classifier (k = 3) to predict the subject ID of the test images. We conduct each experiment 10 times, and report the classification accuracy by averaging over 40 subjects and 10 runs. Figure 1 (a) shows the average classification accuracy of the kNN classifier using different distance metric learning algorithms. The running times of different metric learning algorithms for the same dataset is shown in Figure 1 (b). Note that we exclude Xing’s method in comparison because its extremely long computational time. We observed that with increasing image size (dimensions), the regularized distance metric learning algorithm yields stable performance, indicating that the it is resilient to high dimensional data. In contrast, for almost all the baseline methods except ITML, their performance varied significantly as the size of the input image changed. Although ITML yields stable performance with respect to different size of images, its high computational cost (Figure 1), arising from solving a Bregman optimization problem in each iteration, makes it unsuitable for high-dimensional data. 6 Conclusion In this paper, we analyze the generalization error of regularized distance metric learning. We show that with appropriate constraint, the regularized distance metric learning could be robust to high dimensional data. We also present efficient learning algorithms for solving the related optimization problems. Empirical studies with face recognition and data classification show the proposed approach is (i) robust and efficient for high dimensional data, and (ii) comparable to the state-of-theart approaches for distance learning. In the future, we plan to investigate different regularizers and their effect for distance metric learning. 1http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html 7 ACKNOWLEDGEMENTS The work was supported in part by the National Science Foundation (IIS-0643494) and the U. S. Army Research Laboratory and the U. S. Army Research Office (W911NF-09-1-0421). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF and ARO. Appendix A: Proof of Lemma 3 Proof. We introduce the Bregmen divergence for the proof of this lemma. Given a convex function of matrix ϕ(X), the Bregmen divergence between two matrices A and B is computed as follows: dϕ(A, B) = ϕ(B) −ϕ(A) −tr ∇ϕ(A)⊤(B −A) We define convex function N(X) and VD(X) as follows: N(X) = ∥X∥2 F, VD(X) = 2 n(n −1) X i<j V (X, zi, zj) and furthermore convex function TD(X) = N(X) + CVD(X). We thus have dN(AD, ADi,z) + dN(ADi,z, AD) ≤dTD(AD, ADi,z) + dTDi,z (ADi,z, AD) = C n(n −1) X j̸=i [V (ADi,z, zi, zj) −V (ADi,z, z, zj) + V (AD, z, zj) −V (AD, zi, zj)] ≤ 8CLR2 n |AD −ADi,z|F The first inequality follows from the fact that both N(X) and VD(X) are convex in X. The second step holds because matrix AD and ADi,z minimize the objective function TD(X) and TDi,z(X), respectively, and therefore (ADi,z −AD)⊤∇TD(AD) ≥0, (AD −ADi,z)⊤∇TDi,z(ADi,z) ≥0 Since dN(A, B) = ∥A −B∥2 F , we therefore have |AD −ADi,z|2 F ≤8CLR2 n |AD −ADi,z|F , which leads to the result in the lemma. Appendix B: Proof of Theorem 7 Proof. We denote by A′ t = At−1 −λy(xt −x′ t)(xt −x′ t)⊤and At = πS+(A′ t). Following Theorem 11.1 and Theorem 11.4 [2], we have bLn −Ln(M) ≤1 λDΦ∗(M, A0) + 1 λ n X t=1 DΦ∗(At−1, A′ t) where DΦ∗(A, B) = 1 2|A −B|2 F , Φ(A) = Φ∗(A) = 1 2|A|2 F Using the relation A′ t = At−1 −λy(xt −x′ t)(xt −x′ t)⊤and A0 = 0, we have bLn −Ln(M) ≤1 2λ|M|2 F + 1 2λ n X t=1 I h yt(1 −|xt −x′ t|2 At−1) < 0 i |xt −x′ t|4 By assuming |x|2 ≤R for any training example, we have |xt −x′ t|4 2 ≤16R4. Since n X t=1 I h yt(1 −|xt −x′ t|2 At−1) < 0 i |xt −x′ t|4 ≤ n X t=1 max(0, b −yt(1 −|xt −x′ t|2 At−1))16R4 b = 16R4 b bLn we thus have the result in the theorem 8 References [1] Bousquet, Olivier, and Andr´e Elisseeff. Stability and generalization. Journal of Machine Learning Research, 2:499–526, March 2002. [2] Nicolo Cesa-Bianchi and Gabor Lugosi. Prediction, Learning, and Games. Cambridge University Press, New York, NY, USA, 2006. [3] G.W. Corder and D.I. Foreman. Nonparametric Statistics for Non-Statisticians: A Step-by-Step Approach. New Jersey: Wiley, 2009. [4] J.V. Davis, B. Kulis, P. Jain, S. Sra, and I.S. Dhillon. Information-theoretic metric learning. In Proceedings of the 24th international conference on Machine Learning, 2007. [5] A. Globerson and S. Roweis. Metric learning by collapsing classes. In Advances in Neural Information Processing Systems, 2005. [6] Steven C.H. Hoi, Wei Liu, and Shih-Fu Chang. Semi-supervised distance metric learning for collaborative image retrieval. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2008. [7] Steven C.H. Hoi, Wei Liu, Michael R. Lyu, and Wei-Ying Ma. Learning distance metrics with contextual constraints for image retrieval. In Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2006. [8] L. K. Saul and S. T. Roweis. Think globally, fit locally: Unsupervised learning of low dimensional manifolds. Journal of Machine Learning Research, 4, 2003. [9] Shai Shalev-Shwartz, Yoram Singer, and Andrew Y. Ng. Online and batch learning of pseudometrics. In Proceedings of the twenty-first international conference on Machine learning, pages 94–101, 2004. [10] N. Shental, T. Hertz, D. Weinshall, and M. Pavel. Adjustment learning and relevant component analysis. In Proceedings of the Seventh European Conference on Computer Vision, volume 4, pages 776–792, 2002. [11] Jonathan R Shewchuk. An introduction to the conjugate gradient method without the agonizing pain. Technical report, Carnegie Mellon University, Pittsburgh, PA, USA, 1994. [12] Luo Si, Rong Jin, Steven C. H. Hoi, and Michael R. Lyu. Collaborative image retrieval via regularized metric learning. In ACM Multimedia Systems Journal (MMSJ), 2006. [13] J.B. Tenenbaum, V. de Silva, and J. C. Langford. A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2000. [14] I.W. Tsang, P.M. Cheung, and J.T. Kwok. Kernel relevant component analysis for distance metric learning. In IEEE International Joint Conference on Neural Networks (IJCNN), 2005. [15] K. Weinberger, J. Blitzer, and L. Saul. Distance metric learning for large margin nearest neighbor classification. In Advances in Neural Information Processing Systems, 2005. [16] Lei Wu, Steven C.H. Hoi, Rong Jin, Jianke Zhu, and Nenghai Yu. Distance metric learning from uncertain side information with application to automated photo tagging. In Proceedings of ACM International Conference on Multimedia (MM), 2009. [17] E.P. Xing, A.Y. Ng, M.I. Jordan, and S. Russell. Distance metric learning, with application to clustering with side-information. In Advances in Neural Information Processing Systems, 2002. [18] L. Yang and R. Jin. Distance metric learning: A comprehensive survey. Michigan State University, Tech. Rep., 2006. [19] L. Yang, R. Jin, R. Sukthankar, and Y. Liu. An efficient algorithm for local distance metric learning. In the Proceedings of the Twenty-First National Conference on Artificial Intelligence Proceedings (AAAI), 2006. 9
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Time-Varying Dynamic Bayesian Networks Le Song, Mladen Kolar and Eric P. Xing School of Computer Science, Carnegie Mellon University {lesong, mkolar, epxing}@cs.cmu.edu Abstract Directed graphical models such as Bayesian networks are a favored formalism for modeling the dependency structures in complex multivariate systems such as those encountered in biology and neural science. When a system is undergoing dynamic transformation, temporally rewiring networks are needed for capturing the dynamic causal influences between covariates. In this paper, we propose time-varying dynamic Bayesian networks (TV-DBN) for modeling the structurally varying directed dependency structures underlying non-stationary biological/neural time series. This is a challenging problem due the non-stationarity and sample scarcity of time series data. We present a kernel reweighted ℓ1-regularized auto-regressive procedure for this problem which enjoys nice properties such as computational efficiency and provable asymptotic consistency. To our knowledge, this is the first practical and statistically sound method for structure learning of TVDBNs. We applied TV-DBNs to time series measurements during yeast cell cycle and brain response to visual stimuli. In both cases, TV-DBNs reveal interesting dynamics underlying the respective biological systems. 1 Introduction Analysis of biological networks has led to numerous advances in understanding the organizational principles and functional properties of various biological systems, such as gene regulatory systems [1] and central nervous systems [2]. However, most such results are based on static networks, that is, networks with invariant topology over a given set of biological entities. A major challenge in systems biology is to understand and model, quantitatively, the dynamic topological and functional properties of biological networks. We refer to these time or condition specific biological circuitries as time-varying networks or structural non-stationary networks, which are ubiquitous in biological systems. For example (i) over the course of a cell cycle, there may exist multiple biological “themes” that determine functions of each gene and their regulatory relations, and these “themes” are dynamic and stochastic. As a result, the molecular networks at each time point are context-dependent and can undergo systematic rewiring rather than being invariant over time [3]. (ii) The emergence of a unified cognitive moment relies on the coordination of scattered mosaics of functionally specialized brain regions. Neural assemblies, distributed local networks of neurons transiently linked by dynamic connections, enable the emergence of coherent behaviour and cognition [4]. A key technical hurdle preventing us from an in-depth investigation of the mechanisms that drive temporal biological processes is the unavailability of serial snapshots of time-varying networks underlying biological processes. Current technology does not allow for experimentally determining a series of time specific networks for a realistic dynamic biological system. Usually, only time series measurements of the activities of the nodes can be made, such as microarray, EEG or fMRI. Our goal is to recover the latent time-varying networks underlying biological processes, with temporal resolution up to every single time point based on time series measurements of the nodal states. Recently, there has been a surge of interests along this direction [5, 6, 7, 8, 9, 10]. However, most existing approaches are computationally expensive, making large scale genome-wide reverse engineering nearly infeasible. Furthermore, these methods also lack formal statistical characterization of 1 the estimation procedure. For instance, non-stationary dynamic Bayesian networks are introduced in [9], where the structures are learned via MCMC sampling; such approach is not likely to scale up to more than 1000 nodes and without a regularization term it is also prone to overfitting when the dimension of the data is high but the number of observations is small. More recent efforts have focused on efficient kernel reweighted or total-variation penalized sparse structure recovery methods for undirected time-varying networks [10, 11, 12], which possess both attractive computational schemes and rigorous statistical consistency results. However, what has not been addressed so far is how to recover directed time-varying networks. Our current paper advances in this direction. More specifically, we propose time-varying dynamic Bayesian networks (TV-DBN) for modeling the directed time-evolving network structures underlying non-stationary biological time series. To make this problem statistically tractable, we rely on the assumption that the underlying network structures are sparse and vary smoothly across time. We propose a kernel reweighted ℓ1-regularized auto-regressive approach for learning this sequence of networks. Our approach has the following attractive properties: (i) The aggregation of observations from adjacent time points by kernel reweighting greatly alleviates the statistical problem of sample scarcity when the networks can change at each time point whereas only one or a few time series replicates are available. (ii) The problem of structural estimation for a TV-DBN decomposes into a collection of simpler and atomic structural learning problems. We can choose from a battery of highly scalable ℓ1-regularized least-square solvers for learning each structure. (iii) We can formally characterize the conditions under which our estimation procedure is structurally consistent: as time series are sampled in increasing resolution, our algorithm can recover the true structure of the underlying TV-DBN with high probability. It is worth emphasizing that our approach is very different from earlier approaches, such as the structure learning algorithms for dynamic Bayesian networks [13], which learn time-homogeneous dynamic systems with fixed node dependencies, or approaches which start from an a priori static network and then trace time-dependent activities [3]. The Achilles’ heel of this latter approach is that edges that are transient over a short period of time may be missed by the summary static network in the first place. Furthermore, our approach is also different from change point based algorithms [14, 8] which first segment time series and then fit an invariant structure to each segment. These approaches can only recover piece-wise stationary models rather than constantly varying networks. In our experiments, we demonstrate the advantange of TV-DBNs using synthetic networks. We also apply TV-DBNs to real world datasets: a gene expression dataset measured during yeast cell cycle; and an EEG dataset recorded during a motor imagination task. In both cases, TV-DBNs reveal interesting time-varying causal structures of the underlying biological systems. 2 Preliminary We concern ourselves with stochastic processes in time or space domains, such as the dynamic control of gene expression during cell cycle, or the sequential activation of brain areas during cognitive decision making, of which the state of a variable at one time point is determined by the states of a set of variables at previous time points. Models describing a stochastic temporal processes can be naturally represented as dynamic Bayesian networks (DBN) [15]. Taking the transcriptional regulation of gene expression as an example, let Xt := (Xt 1, . . . , Xt p)⊤∈Rp be a vector representing the expression levels of p genes at time t, a stochastic dynamic process can be modeled by a “first-order Markovian transition model” p(Xt|Xt−1), which defines the probabilistic distribution of gene expressions at time t given those at time t −1. Under this assumption, the likelihood of the observed expression levels of these genes over a time series of T steps can be expressed as: p(X1, . . . , XT ) = p(X1) YT t=2 p(Xt|Xt−1) = p(X1) YT t=2 Yp i=1 p(Xt i|Xt−1 πi ), (1) where we assume that the topology of the networks is specified by a set of regulatory relations Xt−1 πi := {Xt−1 j : Xt−1 j regulates Xt i}, and hence the transition model p(Xt|Xt−1) factors over individual genes, i.e., Q i p(Xt i|Xt−1 πi ). Each p(Xt i|Xt−1 πi ) can be viewed as a regulatory gate function that takes multiple covariates (regulators) and produce a single response. A simple form of the transition model p(Xt|Xt−1) in a DBN is a linear dynamics model: Xt = A · Xt−1 + ϵ, ϵ ∼N(0, σ2I), (2) where A ∈Rp×p is a matrix of coefficients relating the expressions at time t −1 to those of the next time point, and ϵ is a vector of isotropic zero mean Gaussian noise with variance σ2. In this 2 case, the gate function that defines the conditional distribution p(Xt i|Xt−1 πi ) can be expressed as a univariate Gaussian, i.e., p(Xt i|Xt−1 πi ) = N(Xt i; Ai·Xt−1, σ2), where Ai· denotes the ith row of the matrix A. This model is also known as an auto-regressive model. The major reason for favoring DBNs over standard Bayesian networks (BN) or undirected graphical models is its enhanced semantic interpretability. An edge in a BN does not necessarily imply causality due to the Markov equivalence of different edge configurations in the network [16]. In DBNs (of the type defined above), all directed edges only point from time t −1 to t, which bear a natural causal implication and are more likely to suggest regulatory relations. The auto-regressive model in (2) also offers an elegant formal framework for consistent estimation of the structures of DBNs; we can read off the edges between variables in Xt−1 and Xt by simply identifying the nonzero entries in the transition matrix A. For example, the non-zero entries of Ai· represent the set of regulator Xπi that directly lead to a response on Xi. Contrary to the name of dynamic Bayesian networks may suggest, DBNs are time-invariant models and the underlying network structures do not change over time. That is, the dependencies between variables in Xt−1 and Xt are fixed, and both p(Xt|Xt−1) and A are invariant over time. The term “dynamic” only means that the DBN can model dynamical systems. In the sequel, we will present a new formalism where the structures of DBNs are time-varying rather than invariant. 3 A New Formalism: Time-Varying Dynamic Bayesian Networks We will focus on recovering the directed time-varying network structure (or the locations of nonzero entries in A) rather than the exact edge values. This is related to the structure estimation problems studied in [11, 12], but in our case for auto-regressive models (and hence directed networks). Structure estimation results in parse models for easy interpretation, but it is statistically more challenging than the value estimation problem. This is also different from estimating a nonstationary model in the conventional sense, where one interests in recovering the exact values of the varying coefficients [17, 18]. To make this distinction clear, we use the following 3 examples: B1 = 0 1 0 0 0 1 0 0 0 ! , B2 = 0 0.1 0 0 0 3 0 0 0 ! , B3 = 0 1 0.1 0 0 1.1 0 0.1 0 ! . (3) Matrices B1 and B2 encode the same graph structure, since the locations of their non-zero entries are exactly same. Although B1 is closer to B3 than B2 in terms of matrix values (eg. measured in Frobenius norm), they encodes very different graph strucutres. Formally, let graph Gt = (V, Et) represents the conditional independence relations between the components of random vectors Xt−1 and Xt. The vertex set V is a common set of variables underlying X1:T , i.e., each node in V corresponds to a sequence of variables X1:T i . The edge set Et ⊆V × V contains directed edges from components of Xt−1 to those of Xt; an edge (i, j) ̸∈Et if and only if Xt i is conditionally independent of Xt−1 j given the rest of the variables in the model. Due to the time-varying nature of the networks, the transition model pt(Xt|Xt−1) in (1) becomes time dependent. In the case of the auto-regressive DBN in (2), its time-varying extension becomes: Xt = At · Xt−1 + ϵ, ϵ ∼N(0, σ2I), (4) and our goal is to estimate the non-zero entries in the sequence of time dependent transition matrices {At} (t = 1 . . . T). The directed edges Et := Et(At) in network Gt associated with each At can be recovered via Et = (i, j) ∈V × V | i ̸= j, At ij ̸= 0 . 4 Estimating Time-Varying DBN Note that if we follow the naive assumption that each temporal snapshot is a completely different network, the task of jointly estimating {At} by maximizing the log-likelihood would be statistically impossible because the estimator would suffer from extremely high variance due to sample scarcity. Therefore, we make a statistically tractable yet realistic assumption that the underlying network structures are sparse and vary smoothly across time; and hence temporally adjacent networks are likely to share common edges than temporally distal networks. 3 Overall, we have designed a procedure that decomposes the problem of estimating the time-varying networks along two orthogonal axes. The first axis is along the time, where we estimate the network for each time point separately by reweighting the observations accordingly; and the second axis is along the set of genes, where we estimate the neighborhood for each gene separately and then join these neighborhoods to form the overall network. One benefit of such decomposition is that the estimation problem is reduced to a set of atomic optimizations, one for each node i (i = 1 . . . |V|) at each time point t∗(t∗= 1 . . . T): ˆ At∗ i· = argmin At∗ i· ∈R1×n 1 T XT t=1 wt∗(t)(xt i −At∗ i· xt−1)2 + λ
At∗ i·
1 , (5) where λ is a parameter for the ℓ1-regularization term, which controls the number of non-zero entries in the estimated ˆ At∗ i· , and hence the sparsity of the networks; wt∗(t) is the weighting of an observation from time t when we estimate the network at time t∗. More specifically, it is defined as wt∗(t) = Kh(t−t∗) PT t=1 Kh(t−t∗), where Kh(·) = K( · h) is a symmetric nonnegative kernel function and h is the kernel bandwidth. We use a Gaussian RBF kernel, Kh(t) = exp(−t2 h ), in our later experiments. Note that multiple measurements at the same time point are considered as i.i.d. observations and can be trivially handled by assigning them the same weights. The objective defined in (5) is essentially a weighted regression problem. The square loss function is due to the fact that we are fitting a linear model with uncorrelated Gaussian noise. Two other key components of the objective are: (i) a kernel reweighting scheme for aggregating observations across time; and (ii) an ℓ1-regularization for sparse structure estimation. The first component originates from our assumption that the structural changes of the network vary smoothly across time. This assumption allows us to borrow information across time by reweighting the observations from different time points and then treating them as if they were i.i.d. observations. Intuitively, the weighting should place more emphasis on observations at or near time point t∗with weights becoming smaller as observations move further away from time point t∗. The second component is to promote sparse structure and avoid model overfitting. This is also consistent with the biological observation that networks underlying biological processes are parsimonious in structure. For example, a transcription factor only controls a small fraction of target genes at particular time point or under a specific condition [19]. It is well-known that ℓ1-regularized least square linear regression, has a parsimonious property, and exhibits model-selection consistency (i.e., recovers the set of true non-zero regression coefficients asymptotically) in noisy settings even when p ≫T [20]. Note that our procedure can also be easily extended to learn the structure of auto-regressive models of higher order D: Xt = PD d=1 At(d) · Xt−d + ϵ, ϵ ∼N(0, σ2I). The change we need to make is to incorporate the higher order auto-regressive coefficients in the square loss function, i.e., (xt i −PD d=1 At∗ i· (d)xt−d)2, and penalize the ℓ1-norms of these At∗ i· (d) correspondingly. 5 Optimization Estimating time-varying networks using the decomposition scheme above requires solving a collection of optimization problems in (5). In a genome-wide reverse engineering task, there can be tens of thousands of genes and hundreds of time points, so one can easily have a million optimization problems. Therefore, it is essential to use an efficient algorithm for solving the atomic optimization problem in (5), which can be trivially parallelized for each genes at each time point. Instead of solving the form of the optimization problem in (5), we will push the weighting wt∗(t) into the square loss function by scaling the covariates and response variables by p wt∗(t), i.e. ˜xt i ← p wt∗(t)xt i and ˜xt−1 ← p wt∗(t)xt−1. After this transformation, the optimization problem becomes a standard ℓ1-regularized least-square problem which can be solved via a battery of highly scalable and specialized solvers, such as the shooting algorithm [21]. The shooting algorithm is a simple, straightforward and fast algorithm that iteratively solves a system of nonlinear equations related to the optimality condition of problem (5): 2 T PT t=1(At∗ i· ˜xt−1 −˜xt i)˜xt−1 j = −λ sign(At∗ ij ) (∀j = 1 . . . p). At each iteration of the shooting algorithm, one entries of Ai· is updated by holding all other entries fixed. Overall, our procedure for estimating time-varying networks is summarized in Algorithm 1, which uses the shooting algorithm as the key building block (step 4 Algorithm 1: Procedure for Estimating Time-Varying DBN Input: Time series {x1, . . . , xT }, regularization parameter λ and kernel parameter h. Output: Time-varying networks {A1, . . . , AT }. begin 1 Introduce variable A0 and randomly initialize it 2 for i = 1 . . . p do 3 for t∗= 1 . . . T do 4 Initialize: At∗ i· ←At∗−1 i· 5 Scale time series: ˜xt i ← p wt∗(t)xt i, ˜xt−1 ← p wt∗(t)xt−1 (t = 1 . . . T) 6 while At∗ i· not converges do 7 for j = 1 . . . p do 8 Compute: Sj ←2 T PT t=1(P k̸=j At∗ ik˜xt−1 k −˜xt i)˜xt−1 j , bj = 2 T PT t=1 ˜xt−1 j ˜xt−1 j 9 Update: At∗ ij ←(sign(Sj −λ)λ −Sj)/bj, if |Sj| > λ, otherwise 0 10 end 11 7-10). In step 5, we uses a warm start for each atomic optimization problem: since the networks vary smoothly across time, we can use At∗−1 i· as a good initialization for At∗ i· for further speedup. 6 Statistical Properties In this section, we study the statistical consistency of the estimation procedure in Section 4. Our analysis is different from the consistency results presented by [11] on recovering time-varying undirected graphical models. Their analysis deals with Frobenius norm consistency which is a weaker result than the structural consistency we pursue here. Our structural consistency result for TV-DBNs estimation procedure follows the proof strategy of [20]; however, the analysis is complicated by two major factors. First, times series observations are very often non-i.i.d.— current observations may depend on past history. Second, we are modeling non-stationary processes, where we need to deal with the additional bias term that arises due to locally stationary approximation to non-stationarity. In the following, we state our assumptions and theorem, but leave the detailed proof of this theorem for a full version of the paper (a sketch of the proof can be found in the appendix). Theorem 1 Assume that the conditions below hold: 1. Elements of At are smooth functions with bounded second derivatives, i.e. there exists a constant L > 0 s.t. | ∂ ∂tAt ij| < L and | ∂2 ∂t2 At ij| < L. 2. The minimum absolute value of non-zero elements of At is bounded away from zero at observation points, and this bound tends to zero as we observe more and more samples, i.e., amin := mint∈{1/T,2/T,...,1} mini∈[p],j∈St i |At ij| > 0. 3. Let Σt = E[Xt(Xt)T ] = [σij(t)]p i,j=1 and let St i denote the set of non-zero elements of the i-th row of the matrix At, i.e. St i = {j ∈[p] : At ij ̸= 0}. Assume that there exist a constant d ∈(0, 1] s.t. maxj∈St i,k̸=j |σjk(t)| ≤d s, ∀i ∈[p], t ∈[0, 1], where s is an upper bound on the number of non-zero elements, i.e. s = maxt∈[0,1] maxi∈[p] |St i|. 4. The kernel K(·) : R 7→R is a symmetric function and has bounded support on [0, 1]. There exists a constant MK s.t. maxx∈R |K(x)| ≤MK and maxx∈R K(x)2 ≤MK. Let the regularization parameter scale as λ = O( p (log p)/Th), the minimum absolute non-zero entry amin of At∗be sufficiently large (amin ≥2λ). If h = O(T 1/3) and s log p T h = o(1) then P[supp( ˆ At∗) = supp(At∗)] →1, T →∞, ∀t∗∈[0, 1]. (6) 5 7 Experiments To the best of our knowledge, this is the first practical method for structure learning of non-stationary DBNs. Thus we mainly compare with static DBN structure learning methods. The goal is to demonstrate the advantage of TV-DBNs for modeling time-varying structures of non-stationary processes which are ignored by traditional approaches. We conducted 3 experiments using synthetic data, gene expression data and EEG signals. In these experiments, TV-DBNs either better recover the underlying networks, or provide better explanatory power for the underlying biological processes. Synthetic Data In this experiment, we generate synthetic time series using a first order autoregressive models with smoothly varying model structures. More specifically, we first generate 8 different anchor transition matrices At1 . . . At8, each of which corresponds to an Erd¨os-R´enyi random graph of node size p = 50 and average indegree of 2 (we have also experimented with p = 75 and 100 which provides similar results). We then evenly space these 8 anchor matrices, and interpolate a suitable number of intermediate matrices to match the number of observations T. Due to the interpolation, the average indegree of each node is around 4. With the sequence of {At}(t = 1 . . . T), we simulate the time series according to equation (4) with noise variance σ2 = 1. We then study the behavior of TV-DBNs and static DBNs [22] in recovering the underlying varying networks as we increase the number of observations T. We also compare with a piecewise constant DBN that estimate a static network for each segment obtained from change point detection [14]. For the TV-DBN, we choose the bandwidth parameter h of the Gaussian kernel according to the spacing between two adjacent anchor matrices (T/7) such that exp(−T 2 49h) = exp(−1). For all methods, we choose the regularization parameter such that the resulting networks has an average indegree of 4. We evaluate the performance using an F1 score, which is the harmonic mean of precision and recall scores in retrieving the true time-varying network edges. Figure 1: F1 score of estimating timevarying networks for different methods. We can see that estimating a static DBN or a piecewise constant DBN does not provide a good estimation of the network structures (Figure 1). In contrast, the TV-DBN leads to a significantly higher F1 score, and its performance also benefit quickly from increasing the number of observations. Note that these results are not surprising since time-varying networs simply fit better with the data generating process. As time-varying networks occur often in biological systems, we expect TV-DBNs will be useful for studying biological systems. Yeast Gene Regulatory Networks. In this experiment, we will reverse engineer the time varying gene regulatory networks from time series of gene expression measured across two yeast cell cycles. A yeast cell cycle is divided into four stages: S phase for DNA synthesis, M phase for mitosis, and G1 and G2 phase separating S and M phases. We use two time series (alpha30 and alpha38) from [23] which are technical replicates of each other with a sampling interval of 5 minutes and a total of 25 time points across two yeast cell cycles. We consider a set of 3626 genes which are common to both arrays. We choose the bandwidth parameter h such that the weighting decay to exp(−1) for half of a cell cycle, i.e. exp(−62/h) = exp(−1). We choose the regularization parameter such that the sparsity of the networks are around 0.01. During the cell cycle of yeasts, there exist multiple underlying “themes” that determine the functionalities of each gene and their relationships to each other, and such themes are dynamical and stochastic. As a result, the gene regulatory networks at each time point are context-dependent and can undergo systematic rewiring, rather than being invariant over time. A summary of the estimated time-varying networks are visualized in Figure 2. We group genes according to 50 ontology groups. We can see that the most active groups of genes are related to background processes such as cytoskeleton organization, enzyme regulator activity, ribosome activity. We can also spot transient interactions, for instance, between genes related to site of polarized growth and nucleolus (time point 18), and between genes related to ribosome and cellular homeostasis (time point 24). Note that, although gene expressions are measured across two cell cycles, the values do not necessarily exhibit periodic behavior. In fact, only a small fraction of yeast genes (less than 20%) has been reported to exhibit cycling behavior [23]. 6 (a) t1 (b) t2 (c) t4 (d) t6 (e) t8 (f) t10 (g) t12 (h) t14 (i) t16 (j) t18 (k) t20 (l) t22 (m) t24 Figure 2: Interactions between gene ontological groups. The weight of an edge between two ontological groups is the total number of connection between genes in the two groups. We thresholded the edge weight such that only the dominant interactions are displayed. Table 1: The number of enriched unique gene sets discovered by the static and time-varying networks respectively. Here we are interested in recall score: the time-varying networks better models the biological system. DBN TV-DBN TF 7 23 Knockout 7 26 Ontology 13 77 Next we study genes sets that are related to specific stage of cell cycle where we expect to see periodic behavior. In particular, we obtain gene sets known to be related to G1, S and S/G2 stage respectively.1 We use interactivity, which is the total number of edges a group of genes is connected to, to describe the activity of each group of genes. Since the regulatory networks are directed, we can examine both indegree and outdegree separately for each gene sets. In Figure 3(a)(b)(c), the interactivities of these genes indeed exhibit periodic behavior which corresponds well with their supposed functions in cell cycles. We also plot the histogram of indegree and outdegree (averaged across time) for the time-varying networks in Figure 3(d). We find that the outdegrees approximately follow a scale free distribution with largest outdegree reaching 90. This corresponds well with the biological observation that there are a few genes (regulators) that regulate a lot of other genes. The indegree distribution is very different from that of the outdegree, and it exhibits a clear peak between 5 and 6. This also corresponds well with biological observations that most genes are controlled only by a few regulators. To further assess the modeling power of the time-varying networks and its advantage over static network, we perform gene set enrichment studies. More specifically, we use three types of information to define the gene sets: transcription factor binding targets (TF), gene knockout signatures (Knockout), and gene ontology (Ontology) groups [24]. We partition the genes in the time varying networks at each time point into 50 groups using spectral clustering, and then test whether these groups are enriched with genes from any predefined gene sets. We use a max-statistic and a 99% confidence level for the test [25]. Table 1 indicates that time-varying networks are able to discover more functional groups as defined by the genes sets than static networks as commonly used in biological literature. In the appendix, we also visualize the time spans of these active functional groups. It can be seen that many of them are dynamic and transient, and not captured by a static network. Brain Response to Visual Stimuli. In this experiment, we will explore the interactions between brain regions in response to visual stimuli using TV-DBNs. We use the EEG dataset from [26] where five healthy subjects (labeled ‘aa’, ‘al’, ‘av’, ‘aw’ and ‘ay’ respectively) were required to imagine body part movement based on visual cues in order to generate EEG changes. We focus our 1We obtain gene sets from http://genome-www.stanford.edu/cellcycle/data/rawdata/KnowGenes.doc. 7 (a) (b) (c) (d) Figure 3: (a) Genes specific to G1 phase are being regulated periodically; we can see that the average indegree of these genes increases during G1 stage and starts to decline right after the G1 phase. (b) S phase specific genes periodically regulate other genes; we can see that the average outdegree of these genes peaks at the end of S phase and starts to decline right after S phase. (c) The interactivity of S/G2 specific genes also show nice correspondence with their functional roles; we can see that the average outdegree increases till G2 phase and then starts to decline. (d) Indegree and outdegree distribution averaged over 24 time points. SB t = 1.0s t = 1.5s t = 2.0s t = 2.5s al av Figure 4: Temporal progression of brain interactions for subject ‘al’ and BCI “illiterate” ‘av’. The plot for the other 3 subjects can be found in the appendix. The dots correspond to EEG electrode positions in 10-5 system. analysis on trials related to right hand imagination, and signals in the window [1.0, 2.5] second after the visual cue is presented. We bandpass filter data at 8–12 Hz to obtain EEG alpha activity. We further normalize each EEG channel to zero mean and unit variance, and estimate the time-varying networks for all 5 subject using exactly the same regularization parameter and kernel bandwidth (h s.t. exp(−(0.5)2/h) = exp(−1)). We tried a range of different regularization parameters, but obtained qualitatively similar results to Figure 4. What is particularly interesting in this dataset is that subject ‘av’ is called BCI “illiterate”; he/she is unable to generate clear EEG changes during motor imagination. The estimated time-varying networks reveal that the brain interactions of subject ‘av’ is particularly weak and the brain connectivity actually decreases as the experiment proceeds. In contrast, all other four subjects show an increased brain interaction as they engage in active imagination. Particularly, these increased interactions occur between visual and motor cortex. This dynamic coherence between visual and motor cortex corresponds nicely to the fact that subjects are consciously transforming visual stimuli into motor imaginations which involves the motor cortex. It seems that subject ‘av’ fails to perform such integration due to the disruption of brain interactions. 8 Conclusion In this paper, we propose time-varying dynamic Bayesian networks (TV-DBN) for modeling the varying network structures underlying non-stationary biological time series. We have designed a simple and scalable kernel reweighted structural learning algorithm to make the learning possible. Given the rapid advances in data collection technologies for biological systems, we expect that complex, high-dimensional, and feature rich data from complex dynamic biological processes, such as cancer progression, immune responses, and developmental processes, will continue to grow. Thus, we believe our new method is a timely contribution that can narrow the gap between imminent methodological needs and the available data and offer deeper understanding of the mechanisms and processes underlying biological networks. Acknowledgments LS is supported by a Ray and Stephenie Lane Research Fellowship. EPX is supported by grant ONR N000140910758, NSF DBI-0640543, NSF DBI-0546594, NSF IIS-0713379 and an Alfred P. Sloan Research Fellowship. We also thank Grace Tzu-Wei Huang for helpful discussions. 8 References [1] A. L. Barabasi and Z. N. Oltvai. Network biology: Understanding the cell’s functional organization. Nature Reviews Genetics, 5(2):101–113, 2004. [2] Francisco Varela, Jean-Philippe Lachaux, Eugenio Rodriguez, and Jacques Martinerie. The brainweb: Phase synchronization and large-scale integration. Nature Reviews Neuroscience, 2:229–239, 2001. [3] N. Luscombe, M. Babu, H. Yu, M. Snyder, S. Teichmann, and M. Gerstein. Genomic analysis of regulatory network dynamics reveals large topological changes. Nature, 431:308–312, 2004. [4] Eugenio Rodriguez, Nathalie George, Jean-Philippe Lachaux, Jacques Martinerie, Bernard Renault, and Francisco J. Varela1. Perception’s shadow: long-distance synchronization of human brain activity. Nature, 397(6718):430–433, 1999. [5] M. Talih and N. Hengartner. Structural learning with time-varying components: Tracking the crosssection of financial time series. J. Royal Stat. Soc. B, 67(3):321C341, 2005. [6] S. Hanneke and E. P. Xing. Discrete temporal models of social networks. In Workshop on Statistical Network Analysis, ICML06, 2006. [7] F. Guo, S. Hanneke, W. Fu, and E. P. Xing. Recovering temporally rewiring networks: A model-based approach. In International Conference in Machine Learning, 2007. [8] X. Xuan and K. Murphy. Modeling changing dependency structure in multivariate time series. In International Conference in Machine Learning, 2007. [9] J. Robinson and A. Hartemink. Non-stationary dynamic bayesian networks. In Neural Information Processing Systems, 2008. [10] Amr Ahmed and Eric P. Xing. Tesla: Recovering time-varying networks of dependencies in social and biological studies. Proceeding of the National Academy of Sciences, in press, 2009. [11] S. Zhou, J. Lafferty, and L. Wasserman. Time varying undirected graphs. In Computational Learning Theory, 2008. [12] L. Song, M. Kolar, and E. Xing. Keller: Estimating time-evolving interactions between genes. In Bioinformatics (ISMB), 2009. [13] N. Friedman, M. Linial, I. Nachman, and D. Peter. Using bayesian networks to analyze expression data. Journal of Computational Biology, 7:601–620, 2000. [14] N. Dobingeon, J. Tourneret, and M. Davy. Joint segmentation of piecewise constant autoregressive processes by using a hierarchical model and a bayesian sampling approach. IEEE Transactions on Signal Processing, 55(4):1251–1263, 2007. [15] K. Kanazawa, D. Koller, and S. Russell. Stochastic simulation algorithms for dynamic probabilistic networks. Uncertainty in AI, 1995. [16] L. Getoor, N. Friedman, D. Koller, and B. Taskar. Learning probabilistic models with link uncertainty. Journal of Machine Learning Research, 2002. [17] R. Dahlhaus. Fitting time series models to nonstationary processes. Ann. Statist, (25):1–37, 1997. [18] C. Andrieu, M. Davy, and A. Doucet. Efficient particle filtering for jump markov systems: Application to time-varying autoregressions. IEEE Transactions on Signal Processing, 51(7):1762–1770, 2003. [19] E. H. Davidson. Genomic Regulatory Systems. Academic Press, 2001. [20] Florentina Bunea. Honest variable selection in linear and logistic regression models via ℓ1 and ℓ1 + ℓ2 penalization. Electronic Journal of Statistics, 2:1153, 2008. [21] W. Fu. Penalized regressions: the bridge versus the lasso. Journal of Computational and Graphical Statistics, 7(3):397–416, 1998. [22] M. Schmidt, A. Niculescu-Mizil, and K Murphy. Learning graphical model structure using l1regularization paths. In AAAI, 2007. [23] Tata Pramila, Wei Wu, Shawna Miles, William Noble, and Linda Breeden. The forkhead transcription factor hcm1 regulates chromosome segregation genes and fills the s-phase gap in the transcriptional circuitry of the cell cycle. Gene and Development, 20:2266–2278, 2006. [24] Jun Zhu, Bin Zhang, Erin Smith, Becky Drees, Rachel Brem, Leonid Kruglyak, Roger Bumgarner, and Eric E Schadt. Integrating large-scale functional genomic data to dissect the complexity of yeast regulatory networks. Nature Genetics, 40:854–861, 2008. [25] T. Nichols and A. Holmes. Nonparametric permutation tests for functional neuroimaging: a primer with examples. Human Brain Mapping, 15:1–25, 2001. [26] G. Dornhege, B. Blankertz, G. Curio, and K.R. M¨uller. Boosting bit rates in non-invasive eeg single-trial classifications by feature combination and multi-class paradigms. IEEE Trans. Biomed. Eng., 51:993– 1002, 2004. 9
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A General Projection Property for Distribution Families Yao-Liang Yu Yuxi Li Dale Schuurmans Csaba Szepesv´ari Department of Computing Science University of Alberta Edmonton, AB, T6G 2E8 Canada {yaoliang,yuxi,dale,szepesva}@cs.ualberta.ca Abstract Surjectivity of linear projections between distribution families with fixed mean and covariance (regardless of dimension) is re-derived by a new proof. We further extend this property to distribution families that respect additional constraints, such as symmetry, unimodality and log-concavity. By combining our results with classic univariate inequalities, we provide new worst-case analyses for natural risk criteria arising in classification, optimization, portfolio selection and Markov decision processes. 1 Introduction In real applications, the model of the problem at hand inevitably embodies some form of uncertainty: the parameters of the model are usually (roughly) estimated from data, which themselves can be uncertain due to various kinds of noises. For example, in finance, the return of a financial product can seldom be known exactly beforehand. Despite this uncertainty, one still usually has to take action in the underlying application. However, due to uncertainty, any attempt to behave “optimally” in the world must take into account plausible alternative models. Focusing on problems where uncertain data/parameters are treated as random variables and the model consists of a joint distribution over these variables, we initially assume prior knowledge that the first and second moments of the underlying distribution are known, but the distribution is otherwise arbitrary. A parametric approach to handling uncertainty in such a setting would be to fit a specific parametric model to the known moments and then apply stochastic programming techniques to solve for an optimal decision. For example, fitting a Gaussian model to the constraints would be a popular choice. However, such a parametric strategy can be too bold, hard to justify, and might incur significant loss if the fitting distribution does not match the true underlying distribution very well. A conservative, but more robust approach would be to take a decision that was “protected” in the worst-case sense; that is, behaves optimally assuming that nature has the freedom to choose an adverse distribution. Such a minimax formulation has been studied in several fields [1; 2; 3; 4; 5; 6] and is also the focus of this paper. Although Bayesian optimal decision theory is a rightfully wellestablished approach for decision making under uncertainty, minimax has proved to be a useful alternative in many domains, such as finance, where it is difficult to formulate appropriate priors over models. In these fields, minimax formulation combined with stochastic programming [7] have been extensively studied and successfully applied. We make a contribution to minimax probability theory and apply the results to problems arising in four different areas. Specifically, we generalize a classic result on the linear projection property of distribution families: we show that any linear projection between distribution families with fixed mean and covariance, regardless of their dimensions, is surjective. That is, given any matrix X and any random vector r with mean XT µ and covariance XT ΣX, one can always find another random vector R with mean µ and covariance Σ such that XT R = r (almost surely). Our proof imposes no conditions on the deterministic matrix X, hence extends the classic projection result in [6], which assumes X is a vector. We furthermore extend this surjective property to some restricted distribution 1 families, which allows additional prior information to be incorporated and hence less conservative solutions to be obtained. In particular, we prove that surjectivity of linear projections remains to hold for distribution families that are additionally symmetric, log-concave, or symmetric linear unimodal. In each case, our proof strategy allows one to construct the worst-case distribution(s). An immediate application of these results is to reduce the worst-case analysis of multivariate expectations to the univariate (or reduced multivariate) ones, which have been long studied and produced many fruitful results. In this direction, we conduct worst-case analyses of some common restricted distribution families. We illustrate our results on problems that incorporate a classic worst case value-at-risk constraint: minimax probability classification [2]; chance constrained linear programming (CCLP) [3]; portfolio selection [4]; and Markov decision processes (MDPs) with reward uncertainty [8]. Although some of the results we obtain have been established in the respective fields [2; 3; 4], we unify them through a much simpler proof strategy. Additionally, we provide extensions to other constrained distribution families, which makes the minimax formulation less conservative in each case. These results are then extended to the more recent conditional value-at-risk constraint, and new bounds are proved, including a new bound on the survival function for symmetric unimodal distributions. 2 A General Projection Property First we establish a generalized linear projection property for distribution families. The key application will be to reduce worst-case multivariate stochastic programming problems to lower dimensional equivalents; see Corollary 1. Popescu [6] has proved the special case of reduction to one dimension, however we provide a simpler proof that can be more easily extended to other distribution families1. Let (µ, Σ) denote the family of distributions sharing common mean µ and covariance Σ, and let µX = XT µ and ΣX = XT ΣX. Below we denote random variables by boldface letters, and use I to denote the identity matrix. We use † to denote the pseudo-inverse. Theorem 1 (General Projection Property (GPP)) For all µ, Σ ≽0, and X ∈Rm×d, the projection XT R = r from m-variate distributions R ∼(µ, Σ) to d-variate distributions r ∼(µX, ΣX) is surjective and many-to-one. That is, every r ∼(µX, ΣX) can be obtained from some R ∼(µ, Σ) via XT R = r (almost surely). Proof: The proof is constructive. Given a r ∼(µX, ΣX), we can construct a pre-image R by letting R = ΣXΣ† Xr+(Im−ΣXΣ† XXT )M, where M ∼(µ, Σ) is independent of r, for example, one can choose M as a Gaussian random vector. It is easy to verify that R ∼(µ, Σ) and XT R = ΣXΣ† Xr+ (Im −ΣXΣ† X)XT M = r. The last equality holds since (Im −ΣXΣ† X)r = (Im −ΣXΣ† X)XT M (the two random vectors on both sides have the same mean and zero covariance). Note that since M can be chosen arbitrarily in (µ, Σ), the projections are always many-to-one. ■ Although this establishes the general result, we extend it to distribution families under additional constraints below. That is, one often has additional prior information about the underlying distribution, such as symmetry, unimodality, and/or support. In such cases, if a general linear projection property can still be shown to hold, the additional assumptions can be used to make the minimax approach less conservative in a simple, direct manner. We thus consider a number of additionally restricted distribution families. Definition 1 A random vector X is called (centrally) symmetric about µ, if for all vectors x, Pr(X ≥µ + x) = Pr(X ≤µ −x). A univariate random variable is called unimodal about a if its cumulative distribution function (c.d.f.) is convex on (−∞, a] and concave on [a, ∞). A random vector X is called log-concave if its c.d.f. is log-concave. A random m-vector X is called linear unimodal about 0m if for all a ∈Rm, aT X is (univariate) unimodal about 0. Let (µ, Σ)S denote the family of distributions in (µ, Σ) that are additionally symmetric about µ, and similarly, let (µ, Σ)L denote the family of distributions that are additionally log-concave, and 1In preparing the final version of this paper, we noticed that a very recent work [9] proved the one dimensional case by a similar technique as ours. 2 let (µ, Σ)SU denote the family of distributions that are additionally symmetric and linear unimodal about µ. For each of these restricted families, we require the following properties to establish our next main result. Lemma 1 (a) If random vector X is symmetric about 0, then AX + µ is symmetric about µ. (b) If X, Y are independent and both symmetric about 0, Z = X + Y is also symmetric about 0. Although once misbelieved, it is now clear that the convolution of two (univariate) unimodal distributions need not be unimodal. However, for symmetric, unimodal distributions we have Lemma 2 ([10] Theorem 1.6) If two independent random variables x and y are both symmetric and unimodal about 0, then z = x + y is also unimodal about 0. There are several non-equivalent extensions of unimodality to multivariate random variables. We consider two specific (multivariate) unimodalities in this paper: log-concave and linear unimodal.2 Lemma 3 ([10] Lemma 2.1, Theorem 2.4, Theorem 2.18) 1. Linearity: If random m-vector X is log-concave, aT X is also log-concave for all a ∈Rm. 2. Cartesian Product: If X and Y are log-concave, then Z = [ X Y ] is also log-concave . 3. Convolution: If X and Y are independent and log-concave, then Z = X+Y is also log-concave. Given the above properties, we can now extend Theorem 1 to (µ, Σ)S, (µ, Σ)L and (µ, Σ)SU. Theorem 2 (GPP for Symmetric, Log-concave, and Symmetric Linear Unimodal Distributions) For all µ, Σ ≽0 and X ∈Rm×d, the projection XT R = r from m-variate R ∼(µ, Σ)S to d-variate r ∼(µX, ΣX)S is surjective and many-to-one. The same is true for (µ, Σ)L and (µ, Σ)SU.3 Proof: The proofs follow the same basic outline as Theorem 1 except that in the first step we now choose N ∼(0m, Im)S or (0m, Im)L or (0m, Im)SU. Then, respectively, symmetry of the constructed R follows from Lemma 1; log-concavity of R follows from Lemma 3; and linear unimodality of R follows from the definition and Lemma 2. The maps remain many-to-one. ■ An immediate application of the general projection property is to reduce worst-case analyses of multivariate expectations to the univariate case. Note that in the following corollary, the optimal distribution of R can be easily constructed from the optimal distribution of r. Corollary 1 For any matrix X and any function g(·) (including in particular when X is a vector) sup R∼(µ,Σ) E[g(XT R)] = sup r∼(XT µ,XT ΣX) E[g(r)]. (1) The equality continues to hold if we restrict (µ, Σ) to (µ, Σ)S, (µ, Σ)L, or (µ, Σ)SU respectively. Proof: It is obvious that the right hand side is an upper bound on the left hand side, since for every R ∼(µ, Σ) there exists an r ∼(XT µ, XT ΣX) given by r = XT R. Similarly for (µ, Σ)S, (µ, Σ)L, and (µ, Σ)SU. However, given Theorems 1 and 2, one can then establish the converse.4 ■ 3 Application to Worst-case Value-at-risk We now apply these projection properties to analyze the worst case value-at-risk (VaR) —a useful risk criterion in many application areas. Consider the following constraint on a distribution R Pr(−xT R ≤α) ≥1 −ϵ, (2) 2A sufficient but not necessary condition for log-concavity is having log-concave densities. This can be used to verify log-concavity of normal and uniform distributions. In the univariate case, log-concave distributions are called strongly unimodal, which is only a proper subset of univariate unimodal distributions [10]. 3If X is a vector we can also extend this theorem to other multivariate unimodalities such as symmetric star/block/convex unimodal. 4The closure of (µ, Σ), (µ, Σ)S, (µ, Σ)L, and (µ, Σ)SU under linear projection is critical for Corollary 1 to hold. Corollary 1 fails for other kinds of multivariate unimodalities, such as symmetric star/block/convex unimodal. It also fails for (µ, Σ)+, a distribution family whose support is contained in the nonnegative orthant. This is not surprising since determining whether the set (µ, Σ)+ is empty is already NP-hard [11]. 3 for given x, α and ϵ ∈(0, 1). In this case, the infimum over α such that (2) is satisfied is referred to as the ϵ-VaR of R. Within certain restricted distribution families, such as Q-radially symmetric distributions, (2) can be (equivalently) transformed to a deterministic second order cone constraint (depending on the range of ϵ) [3]. Unfortunately, determining whether (2) can be satisfied for given x, α and ϵ ∈(0, 1) is NP-hard in general [8]. Suppose however that one knew the distribution of R belonged to a certain family, such as (µ, Σ).5 Given such knowledge, it is natural to consider whether (2) can be satisfied in a worst case sense. That is, consider [ inf R∼(µ,Σ) Pr(−xT R ≤α) ] ≥1 −ϵ. (3) Here the infimum of α values satisfying (3) is referred to as the worst-case ϵ-VaR. If we have additional information about the underlying distribution, such as symmetry or unimodality, the worstcase ϵ-VaR can be reduced. Importantly, using the results of the previous section, we can easily determine the worst-case ϵ-VaR for various distribution families. These can also be used to provide a tractable bound on the ϵ-VaR even when the distribution is known. Proposition 1 For alternative distribution families, the worst-case ϵ-VaR constraint (3) is given by: if R ∼(µ, Σ) then α ≥−µx + √ 1 −ϵ ϵ σx, (4) if R ∼(µ, Σ)S then { α ≥−µx + √ 1 2ϵσx, if ϵ ∈(0, 1 2) α ≥−µx, if ϵ ∈[ 1 2, 1) (5) if R ∼(µ, Σ)SU then { α ≥−µx + 2 3 √ 1 2ϵσx, if ϵ ∈(0, 1 2) α ≥−µx, if ϵ ∈[ 1 2, 1) (6) if R ∼N(µ, Σ) then α ≥−µx + Φ−1(1 −ϵ)σx, (7) where µx = xT µ, σx = √ xT Σx and Φ(·) is the c.d.f. of the standard normal distribution N(0, 1). It turns out some results of Proposition 1 are known. In fact, the first bound (4) has been extensively studied. However, given the results of the previous section, we can now provide a much simpler proof.6 (This simplicity will also allow us to achieve some useful new bounds in Section 4 below.) Proof: From Corollary 1 it follows that inf R∼(µ,Σ) Pr(−xT R ≤α) = inf r∼(−µx,σ2x) Pr(r ≤α) = 1 − sup r∼(−µx,σ2 x) Pr(r > α). (8) Given that the problem is reduced to the univariate case, we simply exploit classical inequalities: if x ∼(µ, σ2) then Pr(x > t) ≤ σ2 σ2 + (µ −t)2 , (9) if x ∼(µ, σ2)S then Pr(x > t) ≤1 2 min(1, σ2 (µ −t)2 ), (10) if x ∼(µ, σ2)SU then Pr(x > t) ≤1 2 min(1, 4 9 σ2 (µ −t)2 ), (11) for t ≥µ.7 Now to prove (4), simply plug (8) into (3) and notice that an application of (9) leads to α ≥−µx and 1 − σ2 x σ2x + (−µx −α)2 ≥1 −ϵ. (4) then follows by simple rearrangement. The same procedure can be used to prove (5), (6), (7). ■ 5We will return to the question of when such moment information is also subject to uncertainty in Section 5. 6[2] and [3] provide a proof of (4) based on the multivariate Chebyshev inequality in [12]; [4] proves (4) from dual optimality; and the proof in [6] utilizes two point support property of the general constraint (3). 7(9) is known as the (one-sided) Chebyshev inequality. Two-sided version of (11) is known as the Gauss inequality. These classical bounds are tight. Proofs can be found in [13], for example. 4 Figure 1: Comparison of the coefficients in front of σx for different distribution families in Proposition 1 (left) and Proposition 2 (right). Only the range ϵ ∈(0, 1 2) is depicted. Proposition 1 clearly illustrates the benefit of prior knowledge. Figure 1 compares the coefficients on σx among the different worst case VaR for different distribution families. The large gap between coefficients for general and symmetric (linear) unimodal distributions demonstrates how additional constraints can generate much less conservative solutions while still ensuring robustness. Beyond simplifying existing proofs, Proposition 1 can be used to extend some of the uses of the VaR criterion in different application areas. Minimax probability classification [2]: Lanckriet et al. [2] first studied the value-at-risk constraint in binary classification. In this scenario, one is given labeled data from two different sources and seeks a robust separating hyperplane. From the data, the distribution families (µ1, Σ1) and (µ2, Σ2) can be estimated. Then a robust hyperplane can be recovered by minimizing the worst-case error min x̸=0,α,ϵ ϵ s.t. [ inf R1∼(µ1,Σ1) Pr(xT R1 ≤α) ] ≥1 −ϵ and [ inf R2∼(µ2,Σ2) Pr(xT R2 ≥α) ] ≥1 −ϵ, (12) where x is the normal vector of the hyperplane, α is the offset and ϵ controls the error probability. Note that the results in [2] follow from using the bound (4). However, interesting additional facts arise when considering alternative distribution families. For example, consider symmetric distributions. In this case, suppose we knew in advance that the optimal ϵ lay in [ 1 2, 1), meaning that no hyperplane predicts better than random guessing. Then the constraints in (12) become linear, covariance information becomes useless in determining the optimal hyperplane, and the optimization concentrates solely on separating the means of two classes. Although such a result might seem surprising, it is a direct consequence of symmetry: the worst-case distributions are forced to put probability mass arbitrarily far away on both sides of the mean, thereby eliminating any information brought by covariance. When the optimal ϵ lies in (0, 1 2), however, covariance information becomes meaningful, since the worst-case distributions can no longer put probability mass arbitrarily far away on both sides of the mean (owing to the existence of a hyperplane that predicts labels better than random guessing). In this case, the optimization problems involving (µ, Σ)S and (µ, Σ)SU are equivalent to that for (µ, Σ) except that the maximum error probability ϵ becomes smaller, which is to be expected since more information about the marginal distributions should make one more confident to predict the labels of future data. Chance Constrained Linear Programming (CCLP) [3]: Consider a linear program minx aT x s.t. rT x ≥0. If the coefficient r is uncertain, it is clear that solving the linear program merely using the expected value of r could result in a solution x that was sub-optimal or even infeasible. Calafiore and El Ghaoui studied this problem in [3], and imposed the inequality constraint with high probability, leading to the the so-called chance constrained linear program (CCLP): min x aT x s.t. [ inf R∼(µ,Σ) Pr(−xT R ≤0) ] ≥1 −ϵ. (13) 5 In this case, α is simply 0 and ϵ is given by the user. Depending on the value of ϵ, the chance constraint can be equivalently transformed into a second order cone constraint or a linear constraint. The work in [3] concentrates on the general and symmetric distribution families. In the latter case, [3] uses the first part of inequality (5) as a sufficient condition for guaranteeing robust solutions. Note however that from Corollary 1 and Proposition 1 one can now see that (5) is also a necessary condition. Although the symmetric linear unimodal case is not discussed in [3], from Proposition 1 again one can see that incorporating bound (6) in (13) yields a looser constraint than does (5), hence the feasible region will be enlarged and the optimum value of the CCLP potentially reduced, corresponding to the intuition that increased prior knowledge leads to more optimized results. Portfolio Selection [4]: In portfolio selection, let R represent the (uncertain) returns of a suite of financial assets, and x the weighting one would like to put on the various assets. Here α > −xT R represents an upper bound on the loss one might suffer with weighting x. The goal is to minimize an upper bound on the loss that holds with high probability,8 say 1 −ϵ, specified by the user min x,α α s.t. [ inf R∼(µ,Σ) Pr(−xT R ≤α) ] ≥1 −ϵ. (14) This criterion has been studied by El Ghaoui et al. [4] in the worst case setting. Previous work has not addressed the case when additional symmetry or linear unimodal information is available. However, comparing the minimal value of α in Proposition 1, we see that such additional information, such as symmetry or unimodality, indeed decreases our potential loss, as shown clearly in Figure 1. This makes sense, since the more one knows about uncertain returns the less risk one should have to bear. Note also that when incorporating additional information, the optimal portfolio, represented by x, is changed as well but remains mean-variance efficient when ϵ ∈(0, 1 2). Uncertain MDPs with reward uncertainty: The standard planning problem in Markov decision processes (MDPs) is to find a policy such that maximizes the expected total discounted return. This nonlinear optimization problem can be efficiently solved by dynamic programming, provided that the model parameters (transition kernel and reward function) are exactly known. Unfortunately, this is rarely the case in practice. Delage and Mannor [8] extend this problem to the uncertain case by employing the value-at-risk type constraint (2) and assuming the unknown reward model and transition kernel are drawn from a known distribution (Gaussian and Dirichlet respectively).Unfortunately, [8] also proves that the constraint (2) is generally NP-hard to satisfy unless one assumes some very restricted form of distribution, such as Gaussian. Alternatively, note that one can use the worst case value-at-risk formulation (3) to obtain a tractable approximation to (2) min x,α α s.t. [ inf R∼(µ,Σ) Pr(−xT R ≤α) ] ≥1 −ϵ, (15) where R is the reward function (unknown but assumed to belong to (µ, Σ)) and x represents a discounted-stationary state-action visitation distribution (which can be used to recover an optimal behavior policy). Although this worst case formulation (15) might appear to be conservative compared to working with a known distribution on R and using (2), when additional information about the distribution is available, such as symmetry or unimodality, (15) can be brought very close to using a Gaussian distribution, as shown in Figure 1. Thus, given reasonable constraints, the minimax approach does not have to be overly conservative, while providing robustness and tractability. 4 Application to Worst-case Conditional Value-at-risk Finally, we investigate the more refined conditional value-at-risk (CVaR) criterion that bounds the conditional expectation of losses beyond the value-at-risk (VaR). This criterion has been of growing prominence in many areas recently. Consider the following quantity defined as the mean of a tail distribution: ˆf = E [ −xT R Pr(−xT R ≤α∗) ≥1−ϵ ] where α∗= arg min α α s.t. Pr(−xT R ≤α) ≥1−ϵ. (16) Here, α∗is the value-at-risk and ˆf is the conditional value-at-risk of R. It is well-known that the CVaR, ˆf, is always an upper bound on the VaR, α∗. Although it might appear that dealing with 8Note that seeking to minimize the loss surely leads to a meaningless outcome. For example, if ϵ = 0, the optimization problem trivially says that the loss of any portfolio will be no larger than ∞. 6 the CVaR criterion entails greater complexity than the VaR, since VaR is directly involved in the definition of CVaR, it turns out that CVaR can be more directly expressed as ˆf = min α α + 1 ϵ E [ (−xT R −α)+ ] , (17) where (x)+ = max(0, x) [14]. Unlike the VaR constraint (2), (17) is always (jointly) convex in x and α. Thus if R were discrete, ˆf could be easily computed by a linear program [14; 5]. However, the expectation in (17) involves a high dimensional integral in general, whose analytical solution is not always available, thus ˆf is still hard to compute in practice. Although one potential remedy might be to use Monte Carlo techniques to approximate the expectation, we instead take a robust approach: As before, suppose one knew the distribution of R belonged to a certain family, such as (µ, Σ). Given such knowledge, it is natural to consider the worst-case CVaR f = sup R∼(µ,Σ) min α α + 1 ϵ E [ (−xT R −α)+ ] = min α sup R∼(µ,Σ) α + 1 ϵ E [ (−xT R −α)+ ] , (18) where the interchangeability of the min and sup operators follows from the classic minimax theorem [15]. Importantly, as in the previous section, we can determine the worst-case CVaR for various distribution families. If one has additional information about the underlying distribution, such as symmetry or unimodality, the worst-case CVaR can be reduced. These can be used to provide a tractable bound on the CVaR even when the distribution is known. Proposition 2 For alternative distribution families, the worst-case CVaR is given by: if R ∼(µ, Σ) then α = −µx + (2ϵ −1) 2 √ ϵ(1 −ϵ) σx, f = −µx + √ 1 −ϵ ϵ σx, (19) if R ∼(µ, Σ)S then { α = −µx + 1 √ 8ϵσx, f = −µx + 1 √ 2ϵσx if ϵ ∈(0, 1 2] α = −µx − 1 √ 8(1−ϵ)σx, f = −µx + √1−ϵ √ 2ϵ σx if ϵ ∈[ 1 2, 1) (20) if R ∼(µ, Σ)SU then α = −µx + 1 3√ϵσx, f = −µx + 2 3√ϵσx if ϵ ∈(0, 1 3] α = −µx + √ 3(1 −2ϵ)σx, f = −µx + √ 3(1 −ϵ)σx if ϵ ∈[ 1 3, 2 3] α = −µx − 1 3√1−ϵσx, f = −µx + 2√1−ϵ 3ϵ σx, if ϵ ∈[ 2 3, 1) (21) if R ∼N(µ, Σ) then f = −µx + e−(Φ−1(1−ϵ))2 2 √ 2πϵ σx, (22) where µx = xT µ, σx = √ xT Σx and Φ(·) is the c.d.f. of a standard normal distribution N(0, 1). The results of Proposition 2 are a novel contribution of this paper, with the exception of (22), which is a standard result in stochastic programming [7]. Proof: We know from Corollary 1 that sup R∼(µ,Σ) E [ (−xT R −α)+ ] = sup r∼(−µx,σ2x) E [ (r −α)+ ] , (23) which reduces the problem to the univariate case. To proceed, we will need to make use of the univariate results given in Proposition 3 below. Assuming Proposition 3 for now, we show how to prove (19): In this case, substitute (23) into (18) and apply (24) from Proposition 3 below to obtain f = min α α + 1 2ϵ [ (−µx −α) + √ σ2x + (−µx −α)2 ] . This is a convex univariate optimization problem in α. Taking the derivative with respect to α and setting to zero gives α = −µx + (2ϵ−1) 2√ ϵ(1−ϵ)σx. Substituting back we obtain f = −µx + √ 1−ϵ ϵ σx. A similar strategy can be used to prove (20), (21), and (22). ■ As with Proposition 1, Proposition 2 illustrates the benefit of prior knowledge. Figure 1 (right) compares the coefficients on σx among different worst-case CVaR quantities for different families. Comparing VaR and CVaR in Figure 1 shows that unimodality has less impact on improving CVaR. A key component of Proposition 2 is its reliance on the following important univariate results. The following proposition gives tight bounds of the expected survival function for the various families. 7 Proposition 3 For alternative distribution families, the expected univariate survival functions are: sup x∼(µ,σ2) E [ (x −t)+ ] = 1 2 [ (µ −t) + √ σ2 + (µ −t)2 ] , (24) sup x∼(µ,σ2)S E [ (x −t)+ ] = σ−t+µ 2 , if µ −σ 2 ≤t ≤µ + σ 2 σ2 8(t−µ), if t > µ + σ 2 −σ2+8(t−µ)2 8(t−µ) , if t < µ −σ 2 (25) sup x∼(µ,σ2)SU E [ (x −t)+ ] = ( √ 3σ−t+µ)2 4 √ 3σ , if µ − σ √ 3 ≤t ≤µ + σ √ 3 σ2 9(t−µ), if t > µ + σ √ 3 −σ2+9(t−µ)2 9(t−µ) , if t < µ − σ √ 3 (26) Here (26) is a further novel contribution of this paper. Proofs of (24) and (25) can be found in [1]. Interestingly, to the best of our knowledge, the worst-case CVaR criterion has not yet been applied to any of the four problems mentioned in the previous section9. Given the space constraints, we can only discuss the direct application of worst-case CVaR to the portfolio selection problem. We note that CVaR has been recently applied to ν-SVM learning in [16]. Implications for Portfolio Selection: By comparing Propositions 1 and 2, the first interesting conclusion one can reach about portfolio selection is that, without considering any additional information, the worst-case CVaR criterion yields the same optimal portfolio weighting x as the worst-case VaR criterion (recall that VaR minimizes α in Proposition 1 by adjusting x while CVaR minimizes f by adjusting x in Proposition 2). However, the worst-case distributions for the two approaches are not the same, which can be seen from the relation (16) between VaR and CVaR and observing that α in (4) is not the same as in (19). Next, when additional symmetry information is taken into account and ϵ ∈(0, 1 2), CVaR and VaR again select the same portfolio but under different worst-case distributions. When unimodality is added, the CVaR criterion finally begins to select different portfolios than VaR. 5 Concluding Remarks We have provided a simpler yet broader proof of the general linear projection property for distribution families with given mean and covariance. The proof strategy can be easily extended to more restricted distribution families. A direct implication of our results is that worst-case analyses of multivariate expectations can often be reduced to those of univariate ones. By combining this trick with classic univariate inequalities, we were able to provide worst-case analyses of two widely adopted constraints (based on value-at-risk criteria). Our analysis recovers some existing results in a simpler way while also provides new insights on incorporating additional information. Above, we assumed the first and second moments of the underlying distribution were precisely known, which of course is questionable in practice. Fortunately, there are standard techniques for handling such additional uncertainty. One strategy, proposed in [2], is to construct a (bounded and convex) uncertainty set U over (µ, Σ), and then applying a similar minimax formulation but with respect to (µ, Σ) ∈U. As shown in [2], appropriately chosen uncertainty sets amount to adding straightforward regularizations to the original problem. A second approach is simply to lower one’s confidence of the constraints and rely on the fact that the moment estimates are close to their true values within some additional confidence bound [17]. That is, instead of enforcing the constraint (3) or (18) surely, one can instead plug-in the estimated moments and argue that constraints will be satisfied within some diminished probability. For an application of this strategy in CCLP, see [3]. Acknowledgement We gratefully acknowledge support from the Alberta Ingenuity Centre for Machine Learning, the Alberta Ingenuity Fund, iCORE and NSERC. Csaba Szepesv`ari is on leave from MTA SZTAKI, Bp. Hungary. 9Except the very recent work of [9] on portfolio selection. 8 References [1] R. Jagannathan. “Minimax procedure for a class of linear programs under uncertainty”. Operations Research, vol. 25(1):pp. 173–177, 1977. [2] Gert R.G. Lanckriet, Laurent El Ghaoui, Chiranjib Bhattacharyya and Michael I. Jordan. “A robust minimax approach to classification”. Journal of Machine Learning Research, vol. 03:pp. 555–582, 2002. [3] G.C.Calafiore and Laurent El Ghaoui. “On distributionally robust chance-constrained linear programs”. Journal of Optimization Theory and Applications, vol. 130(1):pp. 1–22, 2006. [4] Laurent El Ghaoui, Maksim Oks and Francois Oustry. “Worst-case value-at-risk and robust portfolio optimization: a conic programming approach”. Operations Research, vol. 51(4):pp. 542–556, 2003. [5] Shu-Shang Zhu and Masao Fukushima. “Worst-case conditional value-at-risk with application to robust portfolio management”. Operations Research, vol. 57(5):pp. 1155–1168, 2009. [6] Ioana Popescu. “Robust mean-covariance solutions for stochastic optimization”. Operations Research, vol. 55(1):pp. 98–112, 2007. [7] Andr´as Pr´ekopa. Stochastic Programming. Springer, 1995. [8] Erick Delage and Shie Mannor. “Percentile optimization for Markov decision processes with parameter uncertainty”. Operations Research, to appear 2009. [9] Li Chen, Simai He and Shuzhong Zhang. “Tight Bounds for Some Risk Measures, with Applications to Robust Portfolio Selection”. Tech. rep., Department of Systerms Engineering and Engineering Management, The Chinese University of Hongkong, 2009. [10] Sudhakar Dharmadhikari and Kumar Joag-Dev. Unimodality, Convexity, and Applications. Academic Press, 1988. [11] Dimitris Bertsimas and Ioana Popescu. “Optimal inequalities in probability theory a convex optimization approach”. SIAM Journal on Optimization, vol. 15(3):pp. 780–804, 2005. [12] Albert W. Marshall and Ingram Olkin. “Multivariate Chebyshev inequalities”. Annals of Mathematical Statistics, vol. 31(4):pp. 1001–1014, 1960. [13] Ioana Popescu. “A semidefinite programming approach to optimal moment bounds for convex classes of distributions”. Mathematics of Operations Research, vol. 30(3):pp. 632–657, 2005. [14] R. Tyrrell Rockafellar and Stanislav Uryasev. “Optimization of conditional value-at-risk”. Journal of Risk, vol. 2(3):pp. 493–517, 2000. [15] Ky Fan. “Minimax Theorems”. Proceedings of the National Academy of Sciences, vol. 39(1):pp. 42–47, 1953. [16] Akiko Takeda and Masashi Sugiyama. “ν-support vector machine as conditional value-at-risk minimization”. In Proceedings of the 25th International Conference on Machine Learning, pp. 1056–1063. 2008. [17] John Shawe-Taylor and Nello Cristianini. “Estimating the moments of a random vector with applications”. In Proceedings of GRETSI 2003 Conference, pp. 47–52. 2003. 9
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Region-based Segmentation and Object Detection Stephen Gould1 Tianshi Gao1 Daphne Koller2 1 Department of Electrical Engineering, Stanford University 2 Department of Computer Science, Stanford University {sgould,tianshig,koller}@cs.stanford.edu Abstract Object detection and multi-class image segmentation are two closely related tasks that can be greatly improved when solved jointly by feeding information from one task to the other [10, 11]. However, current state-of-the-art models use a separate representation for each task making joint inference clumsy and leaving the classification of many parts of the scene ambiguous. In this work, we propose a hierarchical region-based approach to joint object detection and image segmentation. Our approach simultaneously reasons about pixels, regions and objects in a coherent probabilistic model. Pixel appearance features allow us to perform well on classifying amorphous background classes, while the explicit representation of regions facilitate the computation of more sophisticated features necessary for object detection. Importantly, our model gives a single unified description of the scene—we explain every pixel in the image and enforce global consistency between all random variables in our model. We run experiments on the challenging Street Scene dataset [2] and show significant improvement over state-of-the-art results for object detection accuracy. 1 Introduction Object detection is one of the great challenges of computer vision, having received continuous attention since the birth of the field. The most common modern approaches scan the image for candidate objects and score each one. This is typified by the sliding-window object detection approach [22, 20, 4], but is also true of most other detection schemes (such as centroid-based methods [13] or boundary edge methods [5]). The most successful approaches combine cues from inside the object boundary (local features) with cues from outside the object (contextual cues), e.g., [9, 20, 6]. Recent works are adopting a more holistic approach by combining the output of multiple vision tasks [10, 11] and are reminiscent of some of the earliest work in computer vision [1]. However, these recent works use a different representation for each subtask, forcing information sharing to be done through awkward feature mappings. Another difficulty with these approaches is that the subtask representations can be inconsistent. For example, a bounding-box based object detector includes many pixels within each candidate detection window that are not part of the object itself. Furthermore, multiple overlapping candidate detections contain many pixels in common. How these pixels should be treated is ambiguous in such approaches. A model that uniquely identifies each pixel is not only more elegant, but is also more likely to produce reliable results since it encodes a bias of the true world (i.e., a visible pixel belongs to only one object). In this work, we propose a more integrated region-based approach that combines multi-class image segmentation with object detection. Specifically, we propose a hierarchical model that reasons simultaneously about pixels, regions and objects in the image, rather than scanning arbitrary windows. At the region level we label pixels as belonging to one of a number of background classes (currently sky, tree, road, grass, water, building, mountain) or a single foreground class. The foreground class is then further classified, at the object level, into one of our known object classes (currently car and pedestrian) or unknown. 1 Our model builds on the scene decomposition model of Gould et al. [7] which aims to decompose an image into coherent regions by dynamically moving pixel between regions and evaluating these moves relative to a global energy objective. These bottom-up pixel moves result in regions with coherent appearance. Unfortunately, complex objects such as people or cars are composed of several dissimilar regions which will not be combined by this bottom-up approach. Our new hierarchical approach facilitates both bottom-up and top-down reasoning about the scene. For example, we can propose an entire object comprised of multiple regions and evaluate this joint move against our global objective. Thus, our hierarchical model enjoys the best of two worlds: Like multi-class image segmentation, our model uniquely explains every pixel in the image and groups these into semantically coherent regions. Like object detection, our model uses sophisticated shape and appearance features computed over candidate object locations with precise boundaries. Furthermore, our joint model over regions and objects allows context to be encoded through direct semantic relationships (e.g., “car” is usually found on “road”). 2 Background and Related Work Our method inherits features from the sliding-window object detector works, such as Torralba et al. [19] and Dalal and Triggs [4], and the multi-class image segmentation work of Shotton et al. [16]. We further incorporate into our model many novel ideas for improving object detection via scene context. The innovative works that inspire ours include predicting camera viewpoint for estimating the real world size of object candidates [12], relating “things” (objects) to nearby “stuff” (regions) [9], co-occurrence of object classes [15], and general scene “gist” [18]. Recent works go beyond simple appearance-based context and show that holistic scene understanding (both geometric [11] and more general [10]) can significantly improve performance by combining related tasks. These works use the output of one task (e.g., object detection) to provide features for other related tasks (e.g., depth perception). While they are appealing in their simplicity, current models are not tightly coupled and may result in incoherent outputs (e.g., the pixels in a bounding box identified as “car” by the object detector, may be labeled as “sky” by an image segmentation task). In our method, all tasks use the same region-based representation which forces consistency between variables. Intuitively this leads to more robust predictions. The decomposition of a scene into regions to provide the basis for vision tasks exists in some scene parsing works. Notably, Tu et al. [21] describe an approach for identifying regions in the scene. Their approach has only be shown to be effective on text and faces, leaving much of the image unexplained. Sudderth et al. [17] relate scenes, objects and parts in a single hierarchical framework, but do not provide an exact segmentation of the image. Gould et al. [7] provides a complete description of the scene using dynamically evolving decompositions that explain every pixel (both semantically and geometrically). However, the method cannot distinguish between between foreground objects and often leaves them segmented into multiple dissimilar pieces. Our work builds on this approach with the aim of classifying objects. Other works attempt to integrate tasks such as object detection and multi-class image segmentation into a single CRF model. However, these models either use a different representation for object and non-object regions [23] or rely on a pixel-level representation [16]. The former does not enforce label consistency between object bounding boxes and the underlying pixels while the latter does not distinguish between adjacent objects of the same class. Recent work by Gu et al. [8] also use regions for object detection instead of the traditional slidingwindow approach. However, unlike our method, they use a single over-segmentation of the image and make the strong assumption that each segment represents a (probabilistically) recognizable object part. Our method, on the other hand, assembles objects (and background regions) using segments from multiple different over-segmentations. The multiple over-segmentations avoids errors made by any one segmentation. Furthermore, we incorporate background regions which allows us to eliminate large portions of the image thereby reducing the number of component regions that need to be considered for each object. Liu et al. [14] use a non-parametric approach to image labeling by warping a given image onto a large set of labeled images and then combining the results. This is a very effective approach since it scales easily to a large number of classes. However, the method does not attempt to understand the scene semantics. In particular, their method is unable to break the scene into separate objects (e.g., a row of cars will be parsed as a single region) and cannot capture combinations of classes not present in the training set. As a result, the approach performs poorly on most foreground object classes. 2 3 Region-based Model for Object Detection We now present an overview of our joint object detection and scene segmentation model. This model combines scene structure and semantics in a coherent energy function. 3.1 Energy Function Our model builds on the work of Gould et al. [7] which aims to decompose a scene into a number (K) of semantically consistent regions. In that work, each pixel p in the image I belongs to exactly one region, identified by its region-correspondence variable Rp ∈{1, . . . , K}. The r-th region is then simply the set of pixels Pr whose region-correspondence variable equals r, i.e., Pr = {p : Rp = r}. In our notation we will always use p and q to denote pixels, r and s to denote regions, and o to denote objects. Double indices indicate pairwise terms between adjacent entities (e.g., pq or rs). Regions, while visually coherent, may not encompass entire objects. Indeed, in the work of Gould et al. [7] foreground objects tended to be over-segmented into multiple regions. We address this deficiency by allowing an object to be composed of many regions (rather than trying to force dissimilar regions to merge). The object to which a region belongs is denoted by its object-correspondence variable Or ∈{∅, 1, . . . , N}. Some regions, such as background, do not belong to any object which we denote by Or = ∅. Like regions, the set of pixels that comprise the o-th object is denoted by Po = S r:Or=o Pr. Currently, we do not allow a single region or object to be composed of multiple disconnected components. Random variables are associated with the various entities (pixels, regions and objects) in our model. Each pixel has a local appearance feature vector αp ∈Rn (see [7]). Each region has an appearance variable Ar that summarizes the appearance of the region as a whole, a semantic class label Sr (such as “road” or “foreground object”), and an object-correspondence variable Or. Each object, in turn, has an associated object class label Co (such as “car” or “pedestrian”). The final component in our model is the horizon which captures global geometry information. We assume that the image was taken by a camera with horizontal axis parallel to the ground and model the horizon vhz ∈[0, 1] as the normalized row in the image corresponding to its location. We quantize vhz into the same number of rows as the image. We combine the variables in our model into a single coherent energy function that captures the structure and semantics of the scene. The energy function includes terms for modeling the location of the horizon, region label preferences, region boundary quality, object labels, and contextual relationships between objects and regions. These terms are described in detail below. The combined energy function E(R, S, O, C, vhz | I, θ) has the form: E = ψhz(vhz) + X r ψreg r (Sr, vhz) + X r,s ψbdry rs + X o ψobj o (Co, vhz) + X o,r ψctxt or (Co, Sr) (1) where for notational clarity the subscripts on the factors indicate that they are functions of the pixels (appearance and shape) belonging to the regions, i.e., ψreg r is also a function of Pr, etc. It is assumed that all terms are conditioned on the observed image I and model parameters θ. The summation over context terms includes all ordered pairs of adjacent objects and regions, while the summation over boundary terms is over unordered pairs of regions. An illustration of the variables in the energy function is shown in Figure 1. The first three energy terms are adapted from the model of [7]. We briefly review them here: Horizon term. The ψhz term captures the a priori location of the horizon in the scene and, in our model, is implemented as a log-gaussian ψhz(vhz) = −log N(vhz; µ, σ2) with parameters µ and σ learned from labeled training images. Knowing the location of the horizon allows us to compute the world height of an object in the scene. Using the derivation from Hoiem et al. [12], it can be shown that the height yk of an object (or region) in the scene can be approximated as yk ≈h vt−vb vhz−vb where h is the height of the camera origin above the ground, and vt and vb are the row of the top-most and bottom-most pixels in the object/region, respectively. In our current work, we assume that all images were taken from the same height above the ground, allowing us to use vt−vb vhz−vb as a feature in our region and object terms. Region term. The region term ψreg in our energy function captures the preference for a region to be assigned different semantic labels (currently sky, tree, road, grass, water, building, mountain, foreground). For convenience we include the vhz variable in this term to provide rough geometry information. If a region is associated with an object, then we constrain the assignment of its class label to foreground (e.g., a “sky” region cannot be part of a “car” object). 3 Procedure SceneInference Generate over-segmentation dictionary Ω Initialize Rp using any of the over-segmentations Repeat until convergence Phase 1: Propose a pixel move {Rp : p ∈ω} ←r Update region and boundary features Run inference over regions S and vhz Phase 2: Propose a pixel {Rp} ←r or region move {Or} ←o Update region, boundary and object features Run inference over regions and objects (S, C) and vhz Compute total energy E If (E < Emin) then Accept move and set Emin = E Else reject move Figure 1: Illustration of the entities in our model (left) and inference algorithm (right). See text for details. More formally, let Nr be the number of pixels in region r, i.e., Nr = P p 1{Rp = r}, and let φr : Pr, vhz, I 7→Rn denote the features for the r-th region. The region term is then ψreg r (Sr, vhz) = ∞ if Or ̸= ∅and Sr ̸= foreground −ηregNr log σ (Sr | φr; θreg) otherwise (2) where σ(·) is the multi-class logit σ(y | x; θ) = exp{θT y x} P y′ exp n θT y′x o and ηreg is the relative weight of the region term versus the other terms in the model. Boundary term. The term ψbdry penalizes two adjacent regions with similar appearance or lack of boundary contrast. This helps to merge coherent pixels into a single region. We combine two metrics in this term: the first captures region similarity as a whole, the second captures contrast along the common boundary between the regions. Specifically, let d (x, y; S) = p (x −y)T S−1(x −y) denote the Mahalanobis distance between vectors x and y, and Ers be the set of pixels along the boundary. Then the boundary term is ψbdry rs = ηbdry A · |Ers| · e−1 2 d(Ar,As;ΣA)2 + ηbdry α X (p,q)∈Ers e−1 2 d(αp,αq;Σα)2 (3) where the ΣA and Σα are the image-specific pixel appearance covariance matrix computed over all pixels and neighboring pixels, respectively. In our experiments we restrict ΣA to be diagonal and set Σα = βI with β = E ∥αp −αq∥2 as in Shotton et al. [16]. The parameters ηbdry A and ηbdry α encode the trade-off between the region similarity and boundary contrast terms and weight them against the other terms in the energy function (Equation 1). Note that the boundary term does not include semantic class or object information. The term purely captures segmentation coherence in terms of appearance. Object term. Going beyond the model in [7], we include object terms ψobj in our energy function that score the likelihood of a group of regions being assigned a given object label. We currently classify objects as either car, pedestrian or unknown. The unknown class includes objects like trash cans, street signs, telegraph poles, traffic cones, bicycles, etc. Like the region term, the object term is defined by a logistic function that maps object features φo : Po, vhz, I 7→Rn to probability of each object class. However, since our region layer already identifies foreground regions, we would like our energy to improve only when we recognize known object classes. We therefore bias the object term to give zero contribution to the energy for the class unknown.1 Formally we have ψobj n (Co, vhz) = −ηobjNo log σ Co | φo; θobj −log σ unknown | φo; θobj (4) where No is the number of pixels belonging to the object. Context term. Intuitively, contextual information which relates objects to their local background can improve object detection. For example, Heitz and Koller [9] showed that detection rates improve by relating “things” (objects) to “stuff” (background). Our model has a very natural way of 1This results in the technical condition of allowing Or to take the value ∅for unknown foreground regions without affecting the energy. 4 encoding such relationships through pairwise energy terms between objects Co and regions Sr. We do not encode contextual relationships between region classes (i.e., Sr and Ss) since these rarely help.2 Contextual relationships between foreground objects (i.e., Co and Cm) may be beneficial (e.g., people found on bicycles), but are not considered in this work. Formally, the context term is ψctxt or (Co, Sr) = −ηctxt log σ Co × Sr | φor; θctxt (5) where φor : (Po, Pr, I) 7→Rn is a pairwise feature vector for object o and region r, σ(·) is the multi-class logit, and ηctxt weights the strength of the context term relative to other terms in the energy function. Since the pairwise context term is between objects and (background) regions it grows linearly with the number of object classes. This has a distinct advantage over approaches which include a pairwise term between all classes resulting in quadratic growth. 3.2 Object Detectors Performing well at object detection requires more than simple region appearance features. Indeed, the power of state-of-the-art object detectors is their ability to model localized appearance and general shape characteristics of an object class. Thus, in addition to raw appearance features, we append to our object feature vector φo features derived from such object detection models. We discuss two methods for adapting state-of-the-art object detector technologies for this purpose. In the first approach, we treat the object detector as a black-box that returns a score per (rectangular) candidate window. However, recall that an object in our model is defined by a contiguous set of pixels Po, not a rectangular window. In the black-box approach, we naively place a bounding box (at the correct aspect ratio) around these pixels and classify the entire contents of the box. To make classification more robust we search candidate windows in a small neighborhood (defined over scale and position) around this bounding box, and take as our feature the output of highest scoring window. In our experiments we test this approach using the HOG detector of Dalal and Triggs [4] which learns a linear SVM classifier over feature vectors constructed by computing histograms of gradient orientations in fixed-size overlapping cells within the candidate window. Note that in the above black-box approach many of the pixels within the bounding box are not actually part of the object (consider, for example, an L-shaped region). A better approach is to mask out all pixels not belonging to the object. In our implementation, we use a soft mask that attenuates the intensity of pixels outside the object based on their distance to the object boundary (see Figure 2). This has the dual advantage of preventing hard edge artifacts and being less sensitive to segmentation errors. The masked window is used at both training and test time. In our experiments we test this more integrated approach using the patch-based features of Torralba et al. [19, 20]. Here features are extracted by matching small rectangular patches at various locations within the masked window and combining these weak responses using boosting. Object appearance and shape are captured by operating on both the original (intensity) image and the edge-filtered image. For both approaches, we append the score (for each object) from the object detection classifiers— linear SVM or boosted decision trees—to the object feature vector φo. (a) full window (b) hard region mask (c) hard window (d) soft region mask (e) soft window Figure 2: Illustration of soft mask for proposed object regions. An important parameter for sliding-window detectors is the base scale at which features are extracted. Scale-invariance is achieved by successively down-sampling the image. Below the basescale, feature matching becomes inaccurate, so most detectors will only find objects above some minimum size. Clearly there exists a trade-off between the desire to detect small objects, feature quality, and computational cost. To reduce the computational burden of running our model on high-resolution images while still being able to identify small objects, we employ a multi-scale approach. Here we run our scene decomposition algorithm on a low-resolution (320 × 240) version of the scene, but extract features from the original high-resolution version. That is, when we extract object-detector features we map the object pixels Po onto the original image and extract our features at the higher resolution. 2The most informative region-to-region relationship is that sky tends to be above ground (road, grass, or water). This information is already captured by including the horizon in our region term. 5 4 Inference and Learning We now describe how we perform inference and learn the parameters of our energy function. 4.1 Inference We use a modified version of the hill-climbing inference algorithm described in Gould et al. [7], which uses multiple over-segmentations to propose large moves in the energy space. An overview of this procedure is shown in the right of Figure 1. We initialize the scene by segmenting the image using an off-the-shelf unsupervised segmentation algorithm (in our experiments we use meanshift [3]). We then run inference using a two-phased approach. In the first phase, we want to build up a good set of initial regions before trying to classify them as objects. Thus we remove the object variables O and C from the model and artificially increase the boundary term weights (ηbdry α and ηbdry A ) to promote merging. In this phase, the algorithm behaves exactly as in [7] by iteratively proposing re-assignments of pixels to regions (variables R) and recomputes the optimal assignment to the remaining variables (S and vhz). If the overall energy for the new configuration is lower, the move is accepted, otherwise the previous configuration is restored and the algorithm proposes a different move. The algorithm proceeds until no further reduction in energy can be found after exhausting all proposal moves from a pre-defined set (see Section 4.2). In the second phase, we anneal the boundary term weights and introduce object variables over all foreground regions. We then iteratively propose merges and splits of objects (variables O) as well as high-level proposals (see Section 4.2 below) of new regions generated from sliding-window object candidates (affecting both R and O). After a move is proposed, we recompute the optimal assignment to the remaining variables (S, C and vhz). Again, this process repeats until the energy cannot be reduced by any of the proposal moves. Since only part of the scene is changing during any iteration we only need to recompute the features and energy terms for the regions affected by a move. However, inference is still slow given the sophisticated features that need to be computed and the large number of moves considered. To improve running time, we leave the context terms ψctxt out of the model until the last iteration through the proposal moves. This allows us to maximize each region term independently during each proposal step—we use an iterated conditional modes (ICM) update to optimize vhz after the region labels have been inferred. After introducing the context term, we use max-product belief propagation to infer the optimal joint assignment to S and C. Using this approach we can process an image in under five minutes. 4.2 Proposal Moves We now describe the set of pixel and region proposal moves considered by our algorithm. These moves are relative to the current best scene decomposition and are designed to take large steps in the energy space to avoid local minima. As discussed above, each move is accepted if it results in a lower overall energy after inferring the optimal assignment for the remaining variables. The main set of pixel moves are described in [7] but briefly repeated here for completeness. The most basic move is to merge two adjacent regions. More sophisticated moves involve local re-assignment of pixels to neighboring regions. These moves are proposed from a pre-computed dictionary of image segments Ω. The dictionary is generated by varying the parameters of an unsupervised over-segmentation algorithm (in our case mean-shift [3]) and adding each segment ω to the dictionary. During inference, these segments are used to propose a re-assignment of all pixels in the segment to a neighboring region or creation of new region. These bottom-up proposal moves work well for background classes, but tend to result in over-segmented foreground classes which have heterogeneous appearance, for example, one would not expect the wheels and body of a car to be grouped together by a bottom-up approach. An analogous set of moves can be used for merging two adjacent objects or assigning regions to objects. However, if an object is decomposed into multiple regions, this bottom-up approach is problematic as multiple such moves may be required to produce a complete object. When performed independently, these moves are unlikely to improve the energy. We get around this difficulty by introducing a new set of powerful top-down proposal moves based on object detection candidates. Here we use pre-computed candidates from a sliding-window detector to propose new foreground regions with corresponding object variable. Instead of proposing the entire bounding-box from the detector, we propose the set of intersecting segments (from our segmentation dictionary Ω) that are fully contained within the bounding-box in a single move. 6 EXPERIMENT CARS PED. Patch baseline 0.40 0.15 HOG baseline 0.35 0.37 Patch RB (w/o cntxt) 0.55 0.22 Patch RB (full model) 0.56 0.21 HOG RB (w/o cntxt) 0.58 0.35 HOG RB (full model) 0.57 0.35 Figure 3: PR curves for car (left) and pedestrian (right) detection on the Street Scene dataset [2]. The table shows 11-pt average precision for variants of the baseline sliding-window and our region-based (RB) approach. 4.3 Learning We learn the parameters of our model from labeled training data in a piecewise fashion. First, the individual terms are learned using the maximum-likelihood objective for the subset of variables within each term. The relative weights (ηreg, ηobj, etc.) between the terms are learned through crossvalidation on a subset of the training data. Boosted pixel appearance features (see [7]) and object detectors are learned separately and their output provided as input features to the combined model. For both the base object detectors and the parameters of the region and object terms, we use a closed-loop learning technique where we first learn an initial set of parameters from training data. We then run inference on our training set and record mistakes made by the algorithm (false-positives for object detection and incorrect moves for the full algorithm). We augment the training data with these mistakes and re-train. This process gives a significant improvement to the final results. 5 Experiments We conduct experiments on the challenging Street Scene dataset [2]. This is a dataset consisting of 3547 high-resolution images of urban environments. We rescaled the images to 320 × 240 before running our algorithm. The dataset comes with hand-annotated region labels and object boundaries. However, the annotations use rough overlapping polygons, so we used Amazon’s Mechanical Turk to improve the labeling of the background classes only. We kept the original object polygons to be consistent with other results on this dataset. We divided the dataset into five folds—the first fold (710 images) was used for testing and the remaining four used for training. The multi-class image segmentation component of our model achieves an overall pixel-level accuracy of 84.2% across the eight semantic classes compared to 83.0% for the pixel-based baseline method described in [7]. More interesting was our object detection performance. The test set contained 1183 cars and 293 pedestrians with average size of 86 × 48 and 22 × 49 pixels, respectively. Many objects are occluded making this a very difficult dataset. Since our algorithm produces MAP estimation for the scene we cannot simply generate a precision-recall curve by varying the object classifier threshold as is usual for reporting object detection results. Instead we take the max-marginals for each Cn variable at convergence of our algorithm and sweep over thresholds for each object separately to generate a curve. An attractive aspect of this approach is that our method does not have overlapping candidates and hence does not require arbitrary post-processing such as non-maximal suppression of sliding-window detections. Our results are shown in Figure 3. We also include a comparison to two baseline sliding-window approaches. Our method significantly improves over the baselines for car detection. For pedestrian detection, our method shows comparable performance to the HOG baseline which has been specifically engineered for this task. Notice that our method does not achieve 100% recall (even at low precision) due to the curves being generated from the MAP assignment in which pixels have already been grouped into regions. Unlike the baselines, this forces only one candidate object per region. However, by trading-off the strength (and hence operating point) of the energy terms in our model we can increase the maximum recall for a given object class (e.g., by increasing the weight of the object term by a factor of 30 we were able to increase pedestrian recall from 0.556 to 0.673). Removing the pairwise context term does not have a significant affect on our results. This is due to the encoding of semantic context through the region term and the fact that all images were of urban scenes. However, we believe that on a dataset with more varied backgrounds (e.g., rural scenes) context would play a more important role. We show some example output from our algorithm in Figure 4. The first row shows the original image (left) together with annotated regions and objects (middle-left), regions (middle-right) and predicted horizon (right). Notice how multiple regions get grouped together into a single object. The remaining rows show a selection of results (image and annotated output) from our method. 7 Figure 4: Qualitative results from our experiments. Top row shows original image, annotated regions and objects, region boundaries, and predicted horizon. Other examples show original image (left) and overlay colored by semantic class and detected objects (right). 6 Discussion In this paper we have presented a hierarchical model for joint object detection and image segmentation. Our novel approach overcomes many of the problems associated with trying to combine related vision tasks. Importantly, our method explains every pixel in the image and enforces consistency between random variables from different tasks. Furthermore, our model is encapsulated in a modular energy function which can be easily analyzed and improved as new computer vision technologies become available. One of the difficulties in our model is learning the trade-off between energy terms—too strong a boundary penalty and all regions will be merged together, while too weak a penalty and the scene will be split into too many segments. We found that a closed-loop learning regime where mistakes from running inference on the training set are used to increase the diversity of training examples made a big difference to performance. Our work suggests a number of interesting directions for future work. First, our greedy inference procedure can be replaced with a more sophisticated approach that makes more global steps. More importantly, our region-based model has the potential for providing holistic unified understanding of an entire scene. This has the benefit of eliminating many of the implausible hypotheses that plague current computer vision algorithms. Furthermore, by clearly delineating what is recognized, our framework directly present hypotheses for objects that are currently unknown providing the potential for increasing our library of characterized objects using a combination of supervised and unsupervised techniques. Acknowledgments. This work was supported by the NSF under grant IIS 0917151, MURI contract N000140710747, and The Boeing Company. We thank Pawan Kumar and Ben Packer for helpful discussions. 8 References [1] H.G. Barrow and J.M. Tenenbaum. Computational vision. IEEE, 1981. [2] S. Bileschi and L. Wolf. A unified system for object detection, texture recognition, and context analysis based on the standard model feature set. In BMVC, 2005. [3] D. Comaniciu and P. Meer. Mean shift: A robust approach toward feature space analysis. PAMI, 2002. [4] N. Dalal and B. Triggs. Histograms of oriented gradients for human detection. In CVPR, 2005. [5] V. Ferrari, L. Fevrier, F. Jurie, and C. Schmid. Groups of adjacent contour segments for object detection. PAMI, 2008. [6] M. Fink and P. Perona. Mutual boosting for contextual inference. In NIPS, 2003. [7] Stephen Gould, Rick Fulton, and Daphne Koller. Decompsing a scene into geometric and semantically consistent regions. In ICCV, 2009. [8] C. Gu, J. J. Lim, P. Arbelaez, and J. Malik. Recognition using regions. In CVPR, 2009. [9] G. Heitz and D. Koller. Learning spatial context: Using stuff to find things. In ECCV, 2008. [10] G. Heitz, S. Gould, A. Saxena, and D. Koller. Cascaded classification models: Combining models for holistic scene understanding. In NIPS, 2008. [11] D. Hoiem, A. A. Efros, and M. Hebert. Closing the loop on scene interpretation. CVPR, 2008. [12] D. Hoiem, A. A. Efros, and M. Hebert. Putting objects in perspective. IJCV, 2008. [13] B. Leibe, A. Leonardis, and B. Schiele. Combined object categorization and segmentation with an implicit shape model. In ECCV, 2004. [14] C. Liu, J. Yuen, and A. Torralba. Nonparametric scene parsing: Label transfer via dense scene alignment. In CVPR, 2009. [15] A. Rabinovich, A. Vedaldi, C. Galleguillos, E. Wiewiora, and S. Belongie. Objects in context. In ICCV, 2007. [16] J. Shotton, J. Winn, C. Rother, and A. Criminisi. TextonBoost: Joint appearance, shape and context modeling for multi-class object recognition and segmentation. In ECCV, 2006. [17] E. Sudderth, A. Torralba, W. Freeman, and A. Willsky. Describing visual scenes using transformed objects and parts. In IJCV, 2007. [18] A. Torralba, K. P. Murphy, W. T. Freeman, and M. A. Rubin. Context-based vision system for place and object recognition, 2003. [19] A. Torralba, K. Murphy, and W. Freeman. Sharing features: efficient boosting procedures for multiclass object detection. In CVPR, 2004. [20] A. Torralba, K. Murphy, and W. Freeman. Contextual models for object detection using boosted random fields. In NIPS, 2004. [21] Z. Tu, X. Chen, A. L. Yuille, and S.-C. Zhu. Image parsing: Unifying segmentation, detection, and recognition. In ICCV, 2003. [22] P. Viola and M. J. Jones. Robust real-time face detection. IJCV, 2004. [23] C. Wojek and B. Schiele. A dynamic conditional random field model for joint labeling of object and scene classes. In ECCV, 2008. 9
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Hierarchical Mixture of Classification Experts Uncovers Interactions between Brain Regions Bangpeng Yao1 Dirk B. Walther2 Diane M. Beck2,3∗ Li Fei-Fei1∗ 1Computer Science Department, Stanford University, Stanford, CA 94305 2Beckman Institute, University of Illinois at Urbana-Champaign, Urbana, IL 61801 3Psychology Department, University of Illinois at Urbana-Champaign, Champaign, IL 61820 {bangpeng,feifeili}@cs.stanford.edu {walther,dmbeck}@illinois.edu Abstract The human brain can be described as containing a number of functional regions. These regions, as well as the connections between them, play a key role in information processing in the brain. However, most existing multi-voxel pattern analysis approaches either treat multiple regions as one large uniform region or several independent regions, ignoring the connections between them. In this paper we propose to model such connections in an Hidden Conditional Random Field (HCRF) framework, where the classifier of one region of interest (ROI) makes predictions based on not only its voxels but also the predictions from ROIs that it connects to. Furthermore, we propose a structural learning method in the HCRF framework to automatically uncover the connections between ROIs. We illustrate this approach with fMRI data acquired while human subjects viewed images of different natural scene categories and show that our model can improve the top-level (the classifier combining information from all ROIs) and ROI-level prediction accuracy, as well as uncover some meaningful connections between ROIs. 1 Introduction In recent years, machine learning approaches for analyzing fMRI data have become increasingly popular [15, 24, 18, 16]. In these multi-voxel pattern analysis (MVPA) approaches, patterns of voxels are associated with particular stimuli, leading to verifiable predictions about independent test data. Voxels are extracted from previously known regions of interest (ROIs) [15, 31], selected from the brain by some statistical criterion [24], or defined by a sliding window (“searchlight”) positioned at each location in the brain in turn [20]. All of these methods, however, ignore the highly interconnected nature of the brain. Neuroanatomical evidence from macaque monkeys [10] indicates that brain regions involved in visual processing are indeed highly interconnected. Since research on human subjects is largely limited to non-invasive procedures, considerably less is known about interactions between visual areas in the human brain. Here we demonstrate a method of learning the interactions between regions from fMRI data acquired while human subjects view images of natural scenes. Determining the category of a natural scene (e.g. classifying a scene as a beach, or a forest) is important for many human activities such as navigation or object perception [30]. Despite the large variety of images within and across categories, humans are very good at categorizing natural scenes [27, 9]. In our recent study of natural scene categorization in humans with functional magnetic resonance imaging (fMRI), we discovered that information about natural scene categories is represented in patterns of activity in the parahippocampal place area (PPA), the retrosplenial cortex (RSC), the lateral occipital complex (LOC), and the primary visual cortex (V1) [31]. We demonstrated that this information can be read out from fMRI activity with a linear support vector machine (SVM) classifier. ∗Diane M. Beck and Li Fei-Fei contributed equally to this work. 1 Given the highly interconnected nature of the brain, however, it is unlikely that these regions encode natural scene categories independently of each other. As previous ROI-based MVPA methods studies, in [31] we built predictors for each ROI independently, ignoring their interactions. The method in [31] neither explores connections among the ROIs nor uses the connections to build a classifier on top of all ROIs. In this work, we propose a method for simultaneously learning the voxel patterns associated with natural scene categories in several ROIs and their interactions in a Hidden Conditional Random Field (HCRF) [28] framework. In our model, the classifier of each ROI makes predictions based on not only its voxels, but also the prediction results of the ROIs that it connects to. Using the same fMRI data set, we also explore a mutual information based method to discover functional connectivity [5]. Our current model differs from [5], however, by applying a generative model to concurrently estimate the structure of connectivity as well as maximize the end behavioral task (in this case, a scene classification task). Furthermore, we propose a structural learning method to automatically uncover the structure of the interactions between ROIs for natural scene categorization, i.e. to decide which ROIs should be and which ones should not be connected. Unlike existing models for functional connectivity, which mostly rely on the correlation of time courses of voxels [23], our approach makes use of the patterns of activity in ROIs as well as the category labels of the images presented to the subjects. Built in the hierarchical framework of HCRF, our structural learning method utilizes information in the voxel values at the bottom layer of the network as well as categorical labels at the top layer. In our method, the connections between each pair of ROIs are evaluated for their potential to improve prediction accuracy, and only those that show improvement will be added to the final structural map. In the remaining part of this paper, we first elaborate on our model and structural learning approach in Section 2. We discuss related work on MVPA and connectivity analysis in Section 3. Finally, we present experimental results in Section 4 and conclude the paper in Section 5. 2 Modeling Interactions of Brain Regions: a HCRF Representation The brain is highly interconnected, and the nature of the connections determines to a large extent how information is processed in the brain. We model the connections of brain regions in a Hidden Conditional Random Field (HCRF) framework for the task of natural scene categorization and propose a structural learning method to uncover the pattern of connectivity. In the first part of this section we assume that the structural connections between brain regions are already known. We will discuss in Section 2.2 how these connections are automatically learned. 2.1 Integrating Information across Brain Regions Suppose we are given a set of regions of interest (ROIs) and connections between these regions (see the intermediate layer of Fig.1). Existing ROI-based MVPA approaches build a classifier for each ROI independently [15, 24, 18, 16, 31], neglecting the connections between ROIs. It is our objective here to explore the structure of the connections between ROIs to improve prediction accuracy for decoding viewed scene category from fMRI data. In order to achieve these goals, we propose a Hidden Conditional Random Field (HCRF) model (Fig.1) to allow each ROI to be influenced by the ROIs that it connects to and build a top-level classifier which makes use of information in all ROIs. In this framework, the classifier for one ROI makes prediction based on the voxels in this region as well as the results of the classifiers of its connected ROIs, thereby improving the accuracy of each ROI. In the absence of evidence about the directionality of connections, we assume them to be symmetric, i.e., to allow the information between two ROIs to go in both directions to the same extent. On the technical side, using an undirected model avoids the difficulties of defining a coherent generative process for graph structures in directed models, thereby giving us more flexibility in representing complex patterns [29]. Our model starts with independently trained classifiers for each ROI as in [31] (the bottom layer of Fig.1). Consider an fMRI data set whose individual brain acquisitions are associated with one of 𝐶class labels. For an acquisition sample 𝑖, the decision values of the 𝐶independent classifiers are represented as 𝒳𝑖= {X𝑖 1, ⋅⋅⋅, X𝑖 𝑀}, where 𝑀is the number of ROIs. X𝑖 𝑚= {𝑥𝑖 𝑚,1, ⋅⋅⋅, 𝑥𝑖 𝑚,𝐶} are the decision values for the 𝑚-th ROI, where 𝑥𝑖 𝑚,𝑐is the probability that region 𝑚assigns sample 𝑖to the 𝑐-th class, irrespective of the information in any other ROI. 2 Y1 Y2 Y3 Y4 1 2 3 4 Z Top-layer Intermediate-layer Bottom-layer Type-I Type-II Type-III Potentials Potentials Potentials Figure 1: Illustration of the HCRF model for modeling connections between ROIs. Four ROIs, placed figuratively on a schematic brain, are shown here for illustration of the model. Superscripts indexing different samples are omitted in the figure. 𝑍 is the category label predicted from all ROIs. 𝑌𝑚, the hidden variable of the model, is the prediction result of the classifier of ROI 𝑚. X𝑚is the output of an independently trained classifier for ROI 𝑚. Section 2.1 gives details about the three types of connections. In the figure thicker lines represent stronger connections, thinner lines weaker connections. The weights of all connections and connectivity pattern of the type-II potentials are estimated by the model. Given X𝑖 𝑚as input, the classifier for ROI 𝑚can directly predict sample 𝑖as belonging to the 𝑐∗-th class if 𝑥𝑖 𝑚,𝑐∗is the largest component of X𝑖 𝑚. However, this method ignores the dependencies between ROIs. To remedy this, our model allows collaborative error-correction over the ROIs by using the given structure of connections (the intermediate layer of Fig.1). Denoting the prediction results of the ROI classifiers as 𝒴= {𝑌1, ⋅⋅⋅, 𝑌𝑀}, where 𝑌𝑚∈{1, ⋅⋅⋅, 𝐶} is the classifier output for ROI 𝑚, our model allows for the predictions 𝑌𝑚and 𝑌𝑙to interact if ROIs 𝑚and 𝑙are connected in the given structure (the intermediate layer in Fig.1). Based on the ROI-level prediction results 𝒴, our model outputs the category label of sample 𝑖: 𝑍𝑖∈ {1, ⋅⋅⋅, 𝐶} (the top layer of Fig.1). Furthermore, because we cannot directly observe the prediction of each ROI when acquiring the fMRI data, we treat 𝒴as hidden variables. The underlying graphical model is shown in Fig.1. To estimate the overall classification probability given the observed voxel values, we marginalize over all possible values of 𝒴. The HCRF model is therefore defined as 𝑝(𝑍𝑖∣𝒳𝑖; 𝜽) = ∑ 𝒴 𝑝(𝑍𝑖, 𝒴∣𝒳𝑖; 𝜽) = ∑ 𝒴exp(Ψ(𝑍𝑖, 𝒴, 𝒳𝑖; 𝜽)) ∑ 𝑍 ∑ 𝒴exp(Ψ(𝑍, 𝒴, 𝒳𝑖; 𝜽)) (1) where 𝜽are the parameters of the model, and Ψ(𝑍, 𝒴, 𝒳; 𝜽) is a potential function parameterized by 𝜽. We define the potential function Ψ(𝑍, 𝒴, 𝒳; 𝜽) as the weighted sum of edge potential functions defined on every edge 𝑒(2-clique) of the model: Ψ(𝑍, 𝒴, 𝒳; 𝜽) = ∑ 𝑒 𝜃𝑒𝜓𝑒(𝑍, 𝒴, 𝒳) (2) As shown in Fig.1, there are three types of potentials which describe different edges in the model: Type-I Potential 𝑒= (X𝑚, 𝑌𝑚). Such edges model the distribution of class labels of different ROIs conditioned on the observations X𝑚. The edge connects an X𝑚node and a 𝑌𝑚node where 𝑚= 1, ⋅⋅⋅, 𝑀. The edge potential function is defined by: 𝜓𝑒(𝑍, 𝒴, 𝒳) = 𝑓𝑦𝑥(𝑌𝑚, X𝑚) = 𝑥𝑚,𝑌𝑚 (3) where 𝑥𝑚,𝑌𝑚is the 𝑌𝑚-th component of the vector X𝑚. A large weight for (X𝑚, 𝑌𝑚) implies that the independent classifier trained on voxels of ROI 𝑚is effective in giving correct predictions. Type-II Potential 𝑒= (𝑌𝑚, 𝑌𝑙). Such edges model the dependencies between the ROIs. Note that not all pairs of ROIs are connected. The edge potential function is defined by: 𝜓𝑒(𝑍, 𝒴, 𝒳) = 𝑓𝑦𝑦(𝑌𝑚, 𝑌𝑙) = { 𝛼, 𝑌𝑚= 𝑌𝑙 0, 𝑌𝑚∕= 𝑌𝑙 (4) where 𝛼> 0. If two ROIs are connected, they tend to make similar predictions. A large weight for (𝑌𝑚, 𝑌𝑙) means the connection between 𝑌𝑚and 𝑌𝑙is strong. Type-III Potential 𝑒= (𝑍, 𝑌𝑚). Such edges define a joint distribution over the class label and the prediction result of each ROI. The edge connects a 𝑌𝑚node and the 𝑍node where 𝑚= 1, ⋅⋅⋅, 𝑀. 3 The edge potential function is defined by: 𝜓𝑒(𝑍, 𝒴, 𝒳) = 𝑓𝑦𝑧(𝑌𝑚, 𝑍) = { 𝛽, 𝑌𝑚= 𝑍 0, 𝑌𝑚∕= 𝑍 (5) where 𝛽> 0. A large weight for (𝑍, 𝑌𝑚) means ROI 𝑚has a big contribution to the top-level prediction of the brain. Allowing connected ROIs to interact with each other makes our model significantly different from existing MVPA methods [15, 24, 18, 16], and can improve the prediction accuracy of each ROI. Intuitively, if the values of all components in X𝑖 𝑚are similar, then ROI 𝑚is likely to have incorrect predictions if its classifier merely relies on X𝑖 𝑚. In such situations it is possible for the classifier for one ROI to make better predictions if it can use the information in its connected ROIs. 2.2 Learning the Structural Connections of the Hidden Layer in HCRF Model We have described a method that models the connections between ROIs to build a classification predictor on top of all ROIs. However, for many tasks (e.g. scene categorization), one critical scientific goal is to uncover which ROIs are functionally connected for that task. Automatic learning of the structures of graphical models is a difficult problem in machine learning. To illustrate the difficulty, let us assume that we have 4 ROIs and that we want to explore all possible models of connectivity between them. There are 6 possible connections between the ROIs, so in order to investigate whether all possible combinations of connections are present, we need to evaluate 26 = 64 different models. For 5 ROIs we have 10 potential connections, leading to 210 = 1024. In general, given 𝑀ROIs, there are 2𝑀(𝑀−1)/2 possible combinations of connections. In situations with many ROIs, evaluating all possible structures quickly becomes impractical because of the computational constraints. Approximate approaches to learn the structures of directed graphs use the generative process in the model [21, 19, 32]. For undirected graphs, it is usually assumed that the structures are pre-defined [29]. Some incremental approaches [26, 22] were proposed for random fields construction. However the computational complexity of these approaches is still high. In our model shown in Fig.1, the potentials represented by solid lines are fixed (type-I and type-III). That is to say, each ROI always makes predictions based on the information in its voxels, and the response at the top level is always influenced by the prediction results of all ROIs. That leaves the dependencies between ROIs (type-II edges, the dashed line in Fig.1) to be learned. Therefore, our structural learning starts from a graphical model containing only type-I and type-III potentials, without any interactions between ROIs. Based on this initial model, we evaluate each type-II potential respectively to decide if it should be added to the model. As we have described in Section 1, connections among ROIs play a key role in information processing. Executing a specific task (e.g., scene categorization) activates certain ROIs as well as rely on connections between some of them. Inspired by this fact, we evaluate whether two ROIs, say ROIs 𝑚and 𝑙, should be connected by comparing two models with and without an edge between 𝑌𝑚and Y1 Y2 Y3 Y4 1 2 3 4 Z Training accuracy: cP Training accuracy: nP 2 4 Connect and Y Y If and only if c n P P > Y1 Y2 Y3 Y4 1 2 3 4 Z Figure 2: An illustration for evaluating if ROIs 2 and 4 should be connected. All other ROIs are omitted. We compare the performance of two modes with (left) and without (right) interactions between ROIs 2 and 4. 4 Input: 𝑀ROIs and their feature vectors 𝒳= {X1, ⋅⋅⋅, X𝑀}. A HCRF model 𝒢with nodes 𝑍, 𝑌1, ⋅⋅⋅, 𝑌𝑀, X1, ⋅⋅⋅, X𝑀, and edges (𝑌1, X1), ⋅⋅⋅,(𝑌𝑀, X𝑀), (𝑍, 𝑌1), ⋅⋅⋅, (𝑍, 𝑌𝑀). foreach pair of ROIs 𝑚and 𝑙do Train an HCRF model with nodes 𝑍, 𝑌𝑚, 𝑌𝑙, 𝒳𝑚, 𝒳𝑙, and edges (𝑌𝑚, X𝑚), (𝑌𝑙, X𝑙), (𝑍, 𝑌𝑚), (𝑍, 𝑌𝑙), (𝑌𝑚, 𝑌𝑙). Obtain training accuracy 𝑃𝑐; Train an HCRF model with nodes 𝑍, 𝑌𝑚, 𝑌𝑙, 𝒳𝑚, 𝒳𝑙, and edges (𝑌𝑚, X𝑚), (𝑌𝑙, X𝑙), (𝑍, 𝑌𝑚), (𝑍, 𝑌𝑙). Obtain training accuracy 𝑃𝑛; if 𝑃𝑐> 𝑃𝑛then Add edge (𝑌𝑚, 𝑌𝑙) to the input model 𝒢; Output: The updated model 𝒢. Algorithm 1: The algorithm for uncovering structural connections between ROIs in the HCRF model. 𝑌𝑙. If allowing interactions between ROIs 𝑚and 𝑙helps to improve top-level recognition performance, thus more closely approximating human performance, then 𝑚and 𝑙should be connected. Furthermore, we ignore information in all other ROIs when evaluating the connection between ROIs 𝑚and 𝑙(Fig.2). So the model will only contain 5 nodes: 𝑍, 𝑌𝑚, 𝑌𝑙, X𝑚, and X𝑙. Although some useful information might be lost compared to evaluating all possible combinations of connections, approximating the algorithm in this way can enable the evaluation of many possible connections in a reasonable amount of time, making this algorithm much more practical. The structural learning algorithm is shown in Algorithm 1, and an illustration of evaluating the connection between ROI 2 and 4 is in Fig.2. 2.3 Model Learning and Inference Learning In the step of structural learning, we need to estimate model parameters to compare the models with or without a type-II connection (see Fig.2 for an illustration). Once we have determined which ROIs should interact, i.e. which type-II potentials should be set, we would like to find out the strength of these connections as well as type-I and III potentials. Here the parameters 𝜽= {𝜃𝑒}𝑒are learned by maximizing the conditional log-likelihood of class label 𝑍on training data 𝒳: 𝜽∗= arg max 𝜽 𝐿(𝜽) = arg max 𝜽 ∑ 𝑖 log 𝑝(𝑍𝑖∣𝒳𝑖; 𝜽) = arg max 𝜽 ∑ 𝑖 log ∑ 𝒴exp(Ψ(𝑍𝑖, 𝒴, 𝒳𝑖; 𝜽)) ∑ 𝑍 ∑ 𝒴exp(Ψ(𝑍, 𝒴, 𝒳𝑖; 𝜽)) (6) The objective function is not concave due to the hidden variables 𝒴. Although finding the global optimum is difficult, we can still find a local optimum by iteratively updating the values of 𝜽using the gradient descent method. To be specific, we first set 𝜽to be initial values 𝜽(0), and for each iteration we adopt the following formula to update 𝜽(𝑛) to 𝜽(𝑛+1): 𝜽(𝑛+1) = 𝜽(𝑛) − G(𝜽(𝑛))⊤G(𝜽(𝑛)) G(𝜽(𝑛))⊤H(𝜽(𝑛))G(𝜽(𝑛))G(𝜽(𝑛)) (7) where G(𝜽) and H(𝜽) are the gradient vector and Hessian matrix of 𝐿(𝜽) respectively. This iterative updating continues until reaching a maximum number of iterations or ∥G(𝜽)∥is smaller than a threshold. When the number of ROIs is large, marginalizing over all possible values of 𝒴is time-consuming. In such situations we can use Gibbs sampling to compute the gradient vector and Hessian matrix of 𝐿(𝜽). In the case of natural scene categorization, evidence from neuroscience studies have postulated that 7 regions are likely to play critical roles in this task [31]. We therefore consider 7 ROIs in our experiment, allowing us to marginalize over all possible values of Y. Inference Given the model parameters 𝜽∗and a sample 𝒳, the top-level prediction result is 𝑍∗= arg max 𝑍 𝑝(𝑍∣𝒳; 𝜽∗) (8) After 𝑍∗is obtained, we can get the prediction results corresponding to each ROI by 𝒴∗= arg max 𝒴 𝑝(𝑍∗, 𝒴∣𝒳; 𝜽∗) (9) 5 3 Related Work In this paper, we model the dependencies between ROIs in an HCRF framework, which improves the ROI-level as well as the top-level decoding accuracy by allowing ROIs to exchange information. Other approaches to inferring connections between brain regions from fMRI data can be broadly separated into effective connectivity and functional connectivity [11]. Models for effective connectivity, such as Granger causality mapping [14] and dynamic causal modeling [13], model directed connections between brain regions. These approaches were developed to account for biological temporal dependencies, which is not the case in this work. Functional connectivity refers to undirected connections, which can be either model-driven or data-driven [23]. Model-driven methods usually test a prior hypothesis by correlating the time courses of a seed voxel and a target voxel [12]. Datadriven methods, such as Independent Component Analysis [8], are typically used to identify spatial modes of coherent activity in the brain at rest. None of these methods, however, has the ability to use the specific relation between the patterns of voxel activations inside ROIs and the ground truth of the experimental condition. The structural learning method proposed in this paper offers an entirely new way to assess the interactions between brain regions based on the exchange of information between ROIs so that the accuracy of decoding experimental conditions from the data is improved. Furthermore in contrast with the conventional model comparison approaches of trying to optimize the evidence of each model [2], our method relates the connectivity structure to observed brain activities as well as the classes of stimuli that elicited the activities. Therefore the model proposed here provides a novel and natural way to model the implicit dependencies between different ROIs. 4 Experimental Evaluation 4.1 Data Set and Experimental Design In order to evaluate the proposed method we re-analyze the fMRI data set from our work in [31]. In this experiment, 5 subjects were presented with color images of 6 scene categories: beaches, buildings, forests, highways, industry, and mountains. Photographs were chosen to capture the high variability within each scene category. Images were presented in blocks of 10 images of the same category lasting for 16 seconds (8 brain acquisitions). Each subject performed 12 runs, with each run containing one block for each of the six categories. Please refer to [31] for more details. We use 7 ROIs that are likely to play critical roles for natural scene categorization. They were determined in separate localizer scans: V1, left/right LOC, left/right PPA, left/right RSC. The data for two subjects were excluded, because not all of the ROIs could be found in the localizer scans for these subjects. For the analysis we use two nested cross validations over the 12 runs for each subject. In the outer loop we cross-validate on each subject to test the performance of the proposed method. For each subject, 11 runs out of 12 are selected as training samples and the remaining run is used as the testing set. For each subject this procedure is repeated 12 times, in turn leaving each run out for testing once. Average accuracy of the 36 experiments across all subjects is used to evaluate the performance of the model. In the inner loop, we use 10 of the 11 training runs to train an SVM classifier for each ROI and each subject, and the remaining run to learn the connections between ROIs and train the HCRF model by using outputs of the SVM classifiers. We repeat this procedure 11 times, giving us 11 models. Results of the 11 models on the test data in the inner loop are combined using bagging [4]. We empirically set both 𝛼in Equ.(4) and 𝛽in Equ.(5) to 0.5. 4.2 Scene Classification Results and Analysis In order to comprehensively evaluate the performance of the proposed structural learning and modeling approach, we consider different settings of the intermediate layer of our HCRF model. While always keeping all type-I and type-III potentials connected, we consider five different dependencies between the ROIs as shown in Fig.3. The setting in Fig.3(e) possesses all properties of our method: the connections between ROIs are determined by structural learning, and the weights of the connections are obtained by estimating model parameters in Equ.(6). In order to estimate the effectiveness of our structural learning method, we compare this setting with the situations where no connections exists between any of the ROIs (Fig.3(a)), and all ROIs are fully connected (Fig.3(b,c)). In each connectivity situation, we either use the same (Fig.3(b,d)) or different (Fig.3(c,e)) weights for type-II 6 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 Y1 Y2 Y3 Y4 Figure 3: Various settings of the intermediate layer of our model. Dashed lines represent type-II potentials. In each setting we keep all type-I and III potentials connected. For simplicity, we omit the visualizations of type-I and III potentials here. Different line widths represent different potential weights. (a) No connection exists between any pair of ROIs. (b,c) The ROIs are fully connected. (d,e) The connections between ROIs are obtained by structural learning. (b,d) All type-II potentials have equal weights. (c,e) The weights of different type-II potentials can be different. Note that (e) is the full model in this paper. Table 1: Recognition accuracy for predicting natural scene categories with different methods (chance is 1/6). “Overall classification” means the accuracy for predicting the categories by the top-level node in Fig.1. We carry out experiments on the HCRF models with different settings of the type-II potentials, as shown in Fig.3. Note that we always learn the weights of type-I and type-III potentials. We also list classification results of the SVM classifiers independently trained on each ROI as the baseline. The bolded numbers indicate superior performance compared to all other settings for each ROI. ∗𝑝< 0.01; ∗∗𝑝< 0.005. Method SVM Fig.3(a) Fig.3(b) Fig.3(c) Fig.3(d) Fig.3(e) Overall classification N/A 31%∗ 29%∗ 33%∗∗ 34%∗∗ 36%∗∗ V1 21% 22% 25% 24% 27% 28%∗ left LOC 22% 23% 27% 29%∗ 31%∗ 32%∗∗ right LOC 25% 24% 27% 30%∗ 29%∗ 33%∗∗ ROI left PPA 27% 27% 26% 28%∗ 31%∗ 31%∗ right PPA 26% 28%∗ 28%∗ 31%∗ 31%∗ 32%∗∗ left RSC 30% 30%∗ 30%∗ 32%∗ 33%∗∗ 35%∗∗ right RSC 26%∗ 27% 29%∗ 30%∗ 30%∗ 32%∗∗ potentials. Note that the type-II potentials of the models in Fig.3(b,d) are also obtaining by learning. Classification accuracy of the five different HCRF models, along with individual SVM classification accuracy for each ROI, is shown in Tbl.1. Note that the model with no type-II potentials (Fig.3(a)) is different from independent SVM classifiers because of the type-I potentials. From Table 1 it becomes clear that learning both the structure of the connections and their strengths leads to more improvement in decoding accuracy than either one of these alone. The overall, toplevel classification rate increases from 31% for the variant of the model without any connections (Fig.3(a)) to 36% for the variant with the structure of the model as well as the connection strengths learned (Fig.3(e)). We see similar improvements for the individual ROIs: 4-5% for PPA and RSC, 6% for V1, and 9% for LOC. The fact that decoding from LOC benefits most from interacting with other ROIs is interesting and significant. We will discuss this finding in more detail below. 4.3 Structural Learning Results and Analysis Having established that our full HCRF model outperforms other comparison models in the recognition task, we now investigate how our model can shed light on learning connectivity between brain regions. In the nested cross validation procedure, 12×11=132 structural maps are learned for each subject. Tbl.2 reports for each subject which connections are present in what fraction of these structural maps. A connection is regarded as a strong connection for a subject if it presents in at least half of the models learned for this subject. In Tbl.2 we use larger font size to denote the connections which are strong on more subjects. Connections that are strong for all subjects are marked in bold. We see that both LOC and PPA show strong interactions between the contralateral counterparts, which makes sense for integrating information across the visual hemifields. We also observe strong interactions between PPA and RSC across hemispheres, which underscores the importance of acrosshemifield integration of visual information. We see a similar effect in the interactions between LOC and PPA: strong contralateral interactions. Left LOC also interacts strongly with right RSC. 7 Table 2: Statistics of structural connections. For each subject we have 132 learned structural maps (12-fold cross-validation, each one has 11 models). This table shows the percentage of the times that the corresponding connection is learned in the 132 experiments. Larger font size denotes connections that are strong on more subjects. Connections that are strong on all subjects are marked in bold. Connection Sbj.1 Sbj.2 Sbj.3 Connection Sbj.1 Sbj.2 Sbj.3 V1-leftLOC 0.67 0.25 0.33 rightLOC-leftPPA 0.58 0.58 0.66 V1-rightLOC 0.50 0.29 0.54 rightLOC-rightPPA 0.36 0.58 0.89 V1-leftPPA 0.44 0.29 0.36 rightLOC-leftRSC 0.63 0.38 0.31 V1-rightPPA 0.38 0.33 0.69 rightLOC-rightRSC 0.36 0.30 0.87 V1-leftRSC 0.29 0.30 0.23 leftPPA-rightPPA 0.99 0.56 0.78 V1-rightRSC 0.36 0.29 0.59 leftPPA-leftRSC 0.97 0.34 0.46 leftLOC-rightLOC 0.66 0.88 0.71 leftPPA-rightRSC 0.61 0.53 0.40 leftLOC-leftPPA 0.46 0.64 0.76 rightPPA-leftRSC 0.67 0.74 0.51 leftLOC-rightPPA 0.75 0.96 0.65 rightPPA-rightRSC 0.93 0.74 0.41 leftLOC-leftRSC 0.41 0.78 0.61 leftRSC-rightRSC 0.65 0.20 0.45 leftLOC-rightRSC 0.75 0.83 0.76 The strong interactions between PPA and RSC are not surprising, since both are typically associated with the processing of natural scenes [25], albeit with slightly different roles [7]. The interactions between LOC and PPA are somewhat more surprising, since LOC is usually associated with the processing of isolated objects. Together with the strong improvement of decoding accuracy for natural scene categories from LOC when it is allowed to interact with other ROIs (see above), this suggests a role for LOC in scene categorization. It is conceivable that the detection of typical objects (e.g., a car) helps with determining the scene category (e.g., highway), as has been shown in [17, 6]. On the other hand, it is also possible that information flows the other way, that scene-specific information in PPA and RSC feeds into LOC to bias object detection based on the scene category (see [3, 1]), and that the classifier decodes this bias signal in LOC. Fig.4 shows the connections which are strong on at least two subjects. right LOC V1 left LOC right PPA left PPA left right RSC RSC Figure 4: Schematic illustration of the connections between the seven ROIs obtained by our structural learning method. Activated regions for the seven ROIs are marked in red. The connections shown in this figure are strong on at least two of the three subjects. Connections that are strong for all three subjects (marked with bold in Table 2) are marked with thicker lines in this figure. 5 Conclusion In this paper we modeled the interactions between brain regions in an HCRF framework. We also presented a structural learning method to automatically uncover the connections between ROIs. Experimental results showed that our approach can improve the top-level as well as ROI-level prediction accuracy, as well as uncover some meaningful connections between ROIs. One direction for future work is to use an exploratory “searchlight” approach [20] to automatically discover ROIs, and apply our structural learning and modeling method to those ROIs. Acknowledgements This work is funded by National Institutes of Health Grant 1 R01 EY019429 (to L.F.-F., D.M.B., D.B.W.), a Beckman Postdoctoral Fellowship (to D.B.W.), a Microsoft Research New Faculty Fellowship (to L.F.-F.), and the Frank Moss Gift Fund (to L.F-F.). The authors would like to thank Barry Chai, Linjie Luo, and Hao Su for helpful comments and discussions. 8 References [1] M. Bar. Visual objects in context. Nature Rev Neurosci, 5(8):617–629, 2004. [2] D. Barber and C. M. Bishop. Bayesian model comparison by monte carlo chaining. In NIPS, 1997. [3] I. Biederman. Perceiving real-world scenes. Science, 177(4043):77–80, 1972. [4] L. Breiman. Bagging predictors. Mach Learn, 24:123–140, 1996. [5] B. Chai†, D. B. Walther†, D. M. Beck∗, and L. Fei-Fei∗. Exploring functional connectivities of the human brain using multivariate information analysis. In NIPS, 2009. (†,∗indicates equal contribution). [6] J. L. Davenport and M. C. Potter. Scene consistency in object and background perception. Psychol Sci, 15(8):559–564, 2004. [7] R. A. Epstein and J. S. Higgins. Differential parahippocampal and retrosplenial involvement in three types of scene recognition. Cereb Cortex, 17:1680–1693, 2007. [8] F. Esposito, E. Formisano, E. Seifritz, R. Geobel, R. Morrone, G. Tedeschi, and F. D. Salle. Spatial independent component analysis of functional MRI time-series: To what extent do results depend on the algorithm used. Hum Brain Mapp, 16:146–157, 2002. [9] L. Fei-Fei, A. Iyer, C. Koch, and P. Perona. What do we perceive in a glance of a real-world scene? J Vision, 7(1):1–29, 2007. [10] D. J. Felleman and D. C. van Essen. Distributed hierarchical processing in the primate cerebral cortex. Cereb Cortex, 1:1–47, 1991. [11] K. J. Friston. Functional and effective connectivity in neuroimaging: a synthesis. Hum Brain Mapp, 2:56–78, 1995. [12] K. J. Friston, C. Frith, F. P. Liddle, and R. Frackowiak. Functional connectivity: The principal-component analysis of large (PET) data sets. J Cerebr Blood F Met, 13:5–14, 1993. [13] K. J. Friston, L. Harrison, and W. Penny. Dynamic causal modeling. NeuroImage, 19:1273–1302, 2003. [14] R. Goebel, A. Roebroeck, D.-S. Kim, and E. Formisano. Investigating directed cortical interactions in time-resolved fMRI data using vector autoregressive modeling and granger causality mapping. Magn Reson Imaging, 21:1251–1261, 2003. [15] J. V. Haxby, M. I. Gobbini, M. L. Furey, A. Ishai, J. Schouten, and P. Pietrini. Distributed and overlapping representations of faces and objects in ventral temporal cortex. Science, 293(5539):2425–2430, 2001. [16] J.-D. Haynes and G. Rees. Predicting the orientation of invisible stimuli from activity in human primary visual cortex. Nat Neurosci, 8:686–691, 2005. [17] A. Hollingworth and J. M. Henderson. Accurate visual memory for previously attended objects in natural scenes. J Exp Psychol Human, 28:113–136, 2002. [18] Y. Kamitani and F. Tong. Decoding the visual and subjective contents of the human brain. Nat Neurosci, 8:679–685, 2005. [19] C. Kemp and J. B. Tenenbaum. The discovery of structural form. P Natl Acad Sci USA, 105(31):10687– 10692, 2008. [20] N. Kriegeskorte, R. Goebel, and P. Bandettini. Information-based functional brain mapping. P Natl Acad Sci USA, 103(10):3863–3868, 2006. [21] W. Lam and F. Bacchus. Learning Bayesian belief networks: An approach based on the mdl principle. Comput Intell, 10(4):269–293, 1994. [22] S. Lee, V. Ganapahthi, and D. Koller. Efficient structure learning of markov networks using 𝑙1regularization. In NIPS, 2006. [23] K. Li, L. Guo, J. Nie, G. Li, and T. Liu. Review of methods for functional brain connectivity detection using fmri. Comput Med Imag Grap, 33:131–139, 2009. [24] D. Neill, A. Moore, F. Pereira, and T. Mitchell. Detecting significant multidimensional spatial clusters. In NIPS, 2004. [25] K. O’Craven and N. Kanwisher. Mental imagery of faces and places activates corresponding stimulusspecific brain regions. J Cognitive Neurosci, 12:1013–1023, 2000. [26] S. D. Pietra, V. D. Pietra, and J. Lafferty. Inducing features of random fields. IEEE T Pattern Anal, 19(4):380–393, 1997. [27] M. C. Potter. Short-term conceptual memory for pictures. J Exp Psychol - Hum L, 2(5):509–522, 1976. [28] A. Quattoni, S. Wang, L.-P. Morency, M. Collins, and T. Darrell. Hidden conditional random fields. IEEE T Pattern Anal, 29(10):1848–1852, 2007. [29] B. Taskar, P. Abbeel, and D. Koller. Discriminative probabilistic models for relational data. In UAI, 2002. [30] B. Tversky and K. Hemenway. Categories of scenes. Cognitive Psychol, 15:121–149, 1983. [31] D. B. Walther, E. Caddigan, L. Fei-Fei∗, and D. M. Beck∗. Natural scene categories revealed in distributed patterns of activity in the human brain. J Neurosci, 29(34):10573–10581, 2009. (∗indicates equal contribution). [32] M. L. Wong, W. Lam, and K. S. Leung. Using evoluntionary programming and minimum description length principle for data mining of bayesian networks. IEEE T Pattern Anal, 21(2):174–178, 1999. 9
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Unsupervised Detection of Regions of Interest Using Iterative Link Analysis Gunhee Kim School of Computer Science Carnegie Mellon University gunhee@cs.cmu.edu Antonio Torralba Computer Science and Artificial Intelligence Laboratory Massachusetts Institute of Technology torralba@csail.mit.edu Abstract This paper proposes a fast and scalable alternating optimization technique to detect regions of interest (ROIs) in cluttered Web images without labels. The proposed approach discovers highly probable regions of object instances by iteratively repeating the following two functions: (1) choose the exemplar set (i.e. a small number of highly ranked reference ROIs) across the dataset and (2) refine the ROIs of each image with respect to the exemplar set. These two subproblems are formulated as ranking in two different similarity networks of ROI hypotheses by link analysis. The experiments with the PASCAL 06 dataset show that our unsupervised localization performance is better than one of state-of-the-art techniques and comparable to supervised methods. Also, we test the scalability of our approach with five objects in Flickr dataset consisting of more than 200K images. 1 Introduction This paper proposes an unsupervised approach to the detection of regions of interest (ROIs) from a Web-sized dataset (Fig.1). We define the regions of interest as highly probable rectangular regions of object instances in the images. The extraction of ROIs is extremely helpful for recognition and Web user interfaces. For example, [3, 5] showed comparative studies in which ROI detection is useful to learn more accurate models, which leads to nontrivial improvement of classification and localization performance. In the recognition of indoor scenes [17], the local regions that contain objects may have special meaning to characterize the scene description. Also, many Web applications allow a user to attach notes on user-specified regions in a cluttered image (e.g. Flickr Notes). Our algorithm can make this cumbersome annotation easier by suggesting the regions a user may be interested in. Our solution to the problem of unsupervised ROI detection is inspired by an alternating optimization. Alternating optimization is one of widely used heuristics where optimization over two sets of variables is not straightforward, but optimization with respect to one while keeping the other fixed is much easier and solvable. This approach has been successful in a wide range of areas such as K-means, Expectation-Maximization, and Iterative Closest Point algorithms [2]. Figure 1: Detection of regions of interest (ROIs). Given a Web-sized dataset, our algorithm detects bounding box-shaped ROIs that are statistically significant across the dataset in an unsupervised manner. The yellow boxes are groundtruth labels, and the red and blue ones are ROIs detected by the proposed method. 1 The unsupervised ROI detection can be though of as a chicken-and-egg problem between (1) finding exemplars of objects in the dataset and (2) localizing object instances in each image. If classrepresentative exemplars are given, the detection of objects in images is solvable (i.e. a conventional detection or localization problem). Conversely, if object instances are clearly annotated beforehand, the exemplars can be easily obtained (i.e. a conventional modeling or ranking problem). Given an image set, first we assume that each image itself is the best ROI (i.e. the most confident object region). Then a small number of highly ranked ones among the selected ROIs are chosen as exemplars (called hub seeking), which serve as references to refine the ROIs of each image (called ROI refinement). We repeat these two updates until convergence. The two steps are formulated as ranking in two different similarity networks of ROI hypotheses by link analysis. The hub seeking corresponds to finding a central and diverse hub set in a network of the selected ROIs (i.e. interimage level). The ROI refinement is the ranking in a bipartite graph between the hub sets and all possible ROI hypotheses of each image (i.e. intra-image level). Our work is closely related to topics on ROI detection [3, 5, 17, 14], unsupervised localization [9, 24, 21, 18, 1, 12], and online image collection [13, 19, 6]. The ROI detection and unsupervised localization share a similar goal of detecting the regions of objects in cluttered images. However, most previous work has been successful for standard datasets with thousands of images. On the other hand, our goal is to propose a simple and fast method that can take advantage of enormous amounts of Web data. The main objective of online image collection is to collect relevant images from highly noisy data queried by keywords from the Web. Its main limitation is that much of the previous work requires additional assumptions such as a small number of seed images in the beginning [13], texts and HTML tags associated with images [19], and user-labeled images [6]. On the other hand, no additional meta-data are required in our approach. Recently, link analysis techniques on visual similarity networks were successfully exploited in computer vision problems [12, 15, 11, 16]. [15] applied the random walk with restart technique to the auto-captioning task. However, their work is a supervised method requiring annotated caption words for the segmented regions in training images. [12] is similar to ours in that the unsupervised classification and localization are the main objectives. However, their method suffers from a scalability issue, and thus their experiments were performed using only 600 images. [11] successfully applied the PageRank technique to a large-scale image search, but unlike ours their approach is evaluated with quite clean images and sub-image level localization is not dealt with. Likewise, [16] also exploited the matching graph of a large-scale image set, but the localization was not discussed. The main advantages of our approach are summarized as follows. First, the proposed method is extremely simple and fast, with compelling performance. Our approach shows superior results over a state-of-the-art unsupervised localization method [18] for the PASCAL 06 dataset. We proposed a simple heuristic for scalability to make the computation time linear with the data size without severe performance drop. For example, the localization of 200K images took only 4.5 hours with naive matlab implementation on a single PC equipped with Intel Xeon 2.83 GHz CPU (once image segmentation and feature extraction were done). Second, our approach is dynamic thanks to the evolving network representation. At every iteration, new ROI hypothesis are added and trivial ones are removed from the network while reusing a large portion of previously computed information. Third, unlike most previous work, our approach requires neither human annotation, meta-data, nor initial seed images. Finally, we evaluate our approach with a challenging Flickr dataset of up to 200K images. Although some work [22] in image retrieval uses millions of images, this work has a different goal from ours. The objective of image retrieval is to quickly index and search the nearest images to a given query. On the other hand, our goal is to localize objects in every single image of a dataset without supervision. 2 ROI Candidates and Description The input to our algorithm is a set of images I = {I1, I2, ..., I|I|}. The first task is to define a set of ROI hypotheses from the image set R = {R1, R2, ..., R|I|}. Ideally, the set of ROI hypotheses Ra = {ra1, ..., ram} of an image Ia enumerates all plausible bounding boxes, and at least one of them is supposed to be a good object annotation. Fig.2 shows the procedure of ROI hypothesis generation. Given an image, 15 segments are extracted by Normalized cuts [20]. The minimum rectangle to enclose each segment is defined as initial ROI hypotheses. Since the over-segmentation 2 Figure 2: An example of ROI extraction and description. From left to right: (a) An input image. (b) 15 segments. (c) 43 ROI hypotheses. (d) Distribution of visual words. (e) Edge gradients. is unavoidable in most cases, the combinations of the initial hypotheses are also considered. We first compute pairwise minimum paths between the initial hypotheses using the Dijkstra algorithm. Then the bounding boxes to enclose those minimum paths are added to the ROI hypothesis set. Finally, a largely overlapped pair of ROIs is merged if rai∩raj rai∪raj > 0.8. Note that the hypothesis set always includes the image itself as the largest candidate, and the average set size is about 50. Each ROI hypothesis is represented by two types of descriptors, which are spatial pyramids of visual words [17] and HOG [3]. As usual, the visual words are generated by vector quantization to randomly selected SIFT descriptors. K-means is applied to form a dictionary of 200 visual words. A visual word is assigned to each pixel of an image by finding nearest cluster center in the dictionary, and then binned using a two-level spatial pyramid. The oriented gradients are computed by Canny edge detection and Sobel mask. Then the HOG descriptor is discretized into 20 orientation bins in the range of [0◦,180◦] by following [3]. The pyramid level is up to three. The similarity measure between a pair of ROIs is cosine similarity, which is simply calculated by dot product of two L2 normalized histograms. Here both descriptors are equally weighted. 3 The Algorithm 3.1 Similarity Networks and Link Analysis Techniques All inferences in our approach are based on the link analysis of k-nearest neighbor similarity network between ROI hypotheses. The similarity network is a weighted graph G = (V, E, W), where V is the set of vertices that are ROI hypotheses. E and W are edge and weight sets discovered by the similarity measure in the previous section. Each vertex is only connected to its k-nearest neighbors with k = a·log |V| [23], where a is a constant set to 10. It results in a sparse network, which is more advantageous in terms of computational speed and accuracy. It guarantees that the complexity of network analysis is O(|V| log |V|) at worst. Finally, the network is row normalized so that the edge weight from note i and j indicates the probability of a random surfer jumping from i to j. The link analysis technique we use is PageRank [4, 10]. Given a similarity matrix G, it computes the same length of PageRank vector p, which assigns a ranked score to each vertex of the network. Intuitively, the PageRank scores of the network of ROI hypotheses are indices of the goodness of hypotheses. 3.2 Overview of the Algorithm Algorithm 1 summarizes the proposed algorithm. The main input is the set of ROI hypotheses R generated by the method of section 2. The output is the set of selected ROIs S∗(⊂R). In each image, usually one or two, and rarely more than three, of the most promising ROIs are chosen. The basic idea of our approach is to jointly optimize the ROI selection of each image and the examplar detection among the selected ROIs. Examplars correspond to hubs in our network representation. We begin with images themselves as an initial set of ROI selection S(0) (Step 1). Even though this initialization is quite poor, highly ranked hubs among the ROIs are likely to be much more reliable. They are detected by the function Hub seeking (Step 3). Then, the hub sets are exploited to refine the ROIs of each images by the function Hub seeking (Step 5). In turn, those refined ROIs are likely to lead to a better hub set at the next iteration. The alternating iterations of those two functions are expected to lead convergence for not only the best ROI selection of each image but also the most representative ROIs of the data set. An example of evolution of ROI selection is shown in Fig.4.(c). Although our algorithm forces to select at least one ROI for each image, the PageRank vector by Hub seeking can indicate the confidence of each ROI, which can be used to filter out wrongly selected ROIs. Conceptually, both functions share a similar ranking problem to 3 Figure 3: Examples of hub images. The pictures illustrate highest-ranked images in 10,000 randomly selected images from five objects of our Flickr dataset and all {train+val}images from two objects of the PASCAL06. select a small subset of highly ranked nodes from the input networks of ROI hypotheses. They will be discussed in the following subsections in detail. Inherently, a good initialization is essential for alternating optimization. Our key assumption is as follows: Provided that the similarity network includes a sufficiently large number of images, the hub images are likely to be good references. This is based on the finding of our previous work [12]: If each visual entity votes for others that are similar to itself, this democratic voting can reveal the dominant statistics of the image set. Although the images in a dataset are highly variable, the more repetitive visual information may get more similarity votes, which can be easily and quickly discovered as hubs by link analysis. Fig.3 supports this argument in our dataset. It illustrates topranked images of our dataset in which the objects are clearly shown in the center with significant size. Obviously, they are excellent initialization candidates. Since we deal with discrete patches from unordered natural images on the Web, it is extremely difficult to analytically understand several important behaviors of our algorithm such as convexity, convergence, sensitivity to initial guess, and quality of our solution. One widely used assumption in the optimization with image patches is linearity with small incremental displacement (e.g. AAM [8]). However, it is not the case in our problem and causes severe computation increase. These issues may be open challenges for the optimization of large-scale image analysis. Algorithm 1 The Algorithm Input: ROI hypothesis R associated with image set I. Output: The set of selected ROIs S∗(⊂R), where S∗= S(T ) when converged at T. 1: S(0) ←largest ROI hypothesis in each image. while S(t−1) ̸= S(t) do 2: Generate k-NN similarity network G(t) of S(t). 3: H(t) ←Hub seeking(G(t)), where the hub set H(t) ⊂S(t) for all Ia ∈I unless ROI selection of Ia is not changed for several consecutive times do 4: s(t) a ←ROI refinement(H(t), Ra), where s(t) a : ROI selection of Ia, Ra: ROI hypotheses of Ia. 5: S(t) ←S(t) ∪s(t) a \s(t−1) a . end for end while Algorithm 2 Hub seeking function Input: (1) Network G(t). (2) Window size: d. Output: (1) Hub set H(t). 1: Compute PageRank vector p of G(t). for all vertex v ∈G(t) do 2: Find the neighbor set of v Nv = {u| max reachable probability from v to u > d}. 3: Find local maxima node of v m(v) = arg maxu p(Nv) where u ∈Nv. 4: H(t) ←v if v = m(v). end for Algorithm 3 ROI refinement function Input: (1) Hub set H(t). (2) Ra, ROI hypotheses of Ia Output: (1) The selected ROIs s(t) a (⊂Ra). 1: Generate k-NN self-similarity matrix Wi of Ra and k-NN similarity matrix Wo between Ra and H(t) . Both of them are row-normalized. 2: Generate augmented bipartite graph W = αWi (1 −α)Wo WT o 0 3: Compute PageRank vector p of W. 4: s∗ a = arg maxraj p(raj) where raj ∈Ra. 3.3 Hub Seeking with Centrality and Diversity The goal of this step is to detect a hub set H(t) from S(t) by analyzing the network G(t). The main criteria are centrality and diversity. In other words, the selected hub set should be not only highly ranked but also diverse enough not to lose various aspects of an object. To meet this requirement, we design the hub seeking inspired by Mean Shift [7]. Given feature points, the algorithm creates a fixed-radius window at each point. Then each window iteratively moves into the direction of 4 Figure 4: (a) An example of a bipartite graph between the hub set and ROI hypotheses of an image. The similarity between hubs and hypotheses is captured by Wo and the affinity between the hypotheses by Wi. The hub set is sorted by PageRank values from left and right. The values of leftmost and rightmost are 0.0081 and 0.0024, respectively. They successfully capture various aspects related to the car object. (b) The effect of the augmented bipartite graph. The left image is with α = 0 and the right with α = 0.1. The ranking of hypotheses is represented by jet colormap from red (high) to blue (low). In the left, the weights from the red box to the blue one are (0.052, 0.050, 0.049, 0.049, 0.049); in the right, (0.060, 0.060, 0.059, 0.059, 0.057). (c) An example of ROI evolution. At T = 0, the selected ROI is an image itself but is converged to the real object after T = 5. the maximum increase in the local density function until it reaches a local maximum. Those local maxima become the modes, and the data points that converge to the same maxima are clustered. The proposed algorithm 2 works in the same manner. For each vertex, we define the search window in the form of maximum reachable probability d (Step 2). The window covers the vertices whose maximum reachable probability is larger than d. For example, given d = 0.1, wij = 0.6, wjk = 0.2, the probability of vertices i to k is 0.6 × 0.2 = 0.12 > d. Then k is considered inside the search window of i. For the density function, we use the PageRank vector because it is proportional to the vertex degree if the graph is symmetric and connected [25]. In Step 3, we compute the vector m that assigns the local maximum vertex within the window of each vertex in G(t). If v = m(v), the v is a local maximum, and it is added to H(t). Additionally, we can easily perform the clustering from m. For each node, the search window keeps moving the maximum direction indicated by m until it reaches the local maximum. Then the nodes that converge to the same maxima can be clustered. 3.4 ROI Refinement Formally, this step is to define a nonparametric function for each image fa : Ra →R+ (positive real number) with respect to the hub set H(t). Then the hypothesis with maximum ranked value is chosen as the best ROI. In order to solve this problem, we first construct an augmented bipartite graph W between the hub set H(t) and all possible ROIs Ra as shown in Step 2 of Algorithm 3 (see Fig.4(a)). For better understanding, let us first consider a pure bipartite graph with α = 0. Then the matrix W represents the similarity voting between the ROI candidates and the hub set. If the PageRank vector p of W is computed, then p(Ra) summarizes the relative importance of each ROI hypothesis with respect to the H(t), which is exactly what we require. Rather than a pure bipartite graph (α = 0), we augment it by nonzero α. Fig.4.(b) explains the effects of α. The left image shows the result of α = 0. Even though the red hypothesis is the maximum, several hypotheses near the dark gray car have significant values. With nonzero α = 0.1, those hypotheses are allowed to augment each other, so the maximum ROI is changed to a hypothesis on the car. In terms of link analysis, if a random surfer visits nodes of ROI hypotheses (Ra), it jumps to other hypotheses with probability α or other hubs with 1 −α. Since the nearby hypotheses share large portions of rectangles, they have higher similarity, which results in more votes for nearby hypotheses. 3.5 Scalability Setting The bottleneck of our approach is the Step 3 of Algorithm 1. The network generation requires quadratic computation of cosine similarity of S(t). In order to bound the computational complexity, we limit the maximum number of images to be considered each run of Algorithm 1 by constant number N. N should be small enough not to suffer from computational burden. Simultaneously, it should be large enough to successfully detect the meaningful statistics from an extremely variable 5 dataset. (In experiments, N is set to 10,000.) If the dataset size |I| > N, we randomly sample N images from I and construct initial consideration set Ic ⊂I. Algorithm 1 is applied to the image set Ic to obtain S∗ c . Then we generate new Ic by sampling unvisited images from I. In order to reuse the result of S∗ c for the next iteration, we sample x% of N from previous S∗ c based on the PageRank values of the network G∗of S∗ c . In other words, the highly ranked (i.e. highly confident) ROIs in the previous step are reused to expedite the convergence of next iteration. We iterate the above strategy until all images are examined. This simple heuristic allows our technique to analyze an extremely large dataset in a linear time without significant performance drop. 4 Results We evaluate our approach with two different experiments, (1) performance tests with PASCAL VOC 20061 and (2) scalability tests with Flickr images. The PASCAL dataset provides groundtruth labels, so our approach is quantitatively evaluated and compared with other approaches. Using Flickr dataset, we examine the scalability of our method in a real-world problem. The images are collected by a query that consists of one object word and one context word. We downloaded images of the objects {butterfly+insect(69,990), classic+car(265,731), motorcycle+bike(106,590), sunflower(165,235), giraffe+zoo(53,620)}. The numbers in parentheses are dataset sizes. 4.1 Performance Tests The input of our algorithm consists of unlabeled images, which may include a single object (called as weakly supervised) or multiple objects (called unsupervised). For unsupervised cases, we perform not only localization but also classification according to object types. The PASCAL 06 dataset is so challenging to use that only very rare previous work has used it for unsupervised localization. For comparison, we ran publicly available code of one of the state-of-the-art techniques proposed by Russell et al2 [18] in the identical setting. The PASCAL dataset consists of {train+val+test}. However, our approach requires only images as an input, and thus all of the {train+val+test} images are used without discrimination between them. Note that our task is an image annotation not a learning problem that requires training and testing steps. The performance is measured by following the protocol of PASCAL evaluation: (1) The performance is evaluated from only the {test} set. In practice, there is very little performance difference between analysis of all {train+val+test} and {test} only. (2) The detection is considered correct if the overlap between the prediction and ground truth exceeds 50%. Weakly supervised localization. Fig.5 shows the detection performance as Precision-Recall (PR) curves. For [18], we iterate experiments by changing the number of topics from two to six, and the best results are reported. For clear comparison between our results and [18], we select only the best bounding box in each image. We also present the best result of each object in VOC06 competition. Strictly speaking, it is not a valid comparison because the experimental setups of VOC06 competition and ours are totally different. However, we illustrate them as references to show how closely our approach can reach the best supervised methods in VOC 06 for the localization. Although the performance varies according to objects, our approach significantly outperformed [18] except in cow. Promisingly, the performances of our approach for bicycle and motorbike are comparable, and those for bus, cat, and dog objects are superior to the bests of the supervised methods in VOC06. Unsupervised classification and localization. Here we evaluate how well our approach works for unsupervised classification and localization tasks (i.e. images of multiple objects without any annotation are given). Since both our method and [18] aim at sub-image level classification and detection, we first find out the most confident region of each image, and run the clustering by LDA in [18] and spectral clustering [20] in our method. The evaluation of classification follows the rule of VOC06 by the ROC curves as shown in Fig.6. We also show the best of the VOC06 submissions for supervised classification as a reference. As shown in Fig.6.(a)−(c), our method and [18] present similar ROC performance. In other words, both methods are quite good at ranking for classification. However, the classification rates of our method are better by about 10% for both 3-object and 4object cases. (Ours: 69.08%; [18]: 59.05% for {bicycle, car, dog}. Ours: 59.51%; [18]: 50.99% 1The dataset is available at http://www.pascal-network.org/challenges/VOC/ 2The code is available at http://www.di.ens.fr/∼russell/projects/mult seg discovery/index.html 6 Figure 5: Results of weakly supervised localization. PR curves for the {test} sets of all objects in the PASCAL 06 dataset. (Ours: blue; [18]: red; the best of VOC06: green). Note that our localization and that of [18] are unsupervised, but the VOC06 localization is supervised. (X-axis: recall; Y-axis: precision). Figure 6: Results of unsupervised classification and localization. (a)−(c) ROC curves for {test} set of {bicycle, car, dog}. (Ours: blue; [18]: red; the best of VOC06: green). The AUCs of ours, [18], and the best of VOC06 are bicycle:(0.892, 0.869, 0.948), car:(0.968, 0.965, 0.977), and dog:(0.932, 0.954, 0.876), respectively. (X-axis: false positive rates, Y-axis: true positive rates). (d)−(f) PR curves for unsupervised localization of ours (blue) and [18] (magenta). For comparison, we also represent the results of our weakly supervised localization (red) and the best of VOC 06 (green). (X-axis: recall, Y-axis: precision). for {bicycle, car, dog, sheep}.) We also show the unsupervised localization performance as PRcurves in Fig.6.(d)−(f). For comparison, we also represent the results of our weakly supervised experiments and the bests of VOC 06 for corresponding objects. The nontrivial performance drop is observed due to the classification errors and distraction by other objects in the dataset. 4.2 Scalability Tests It is an open question how to evaluate the results of a large number of Web-downloaded images that have no ground-truth. For a quantitative evaluation, we manually annotated 0.5% randomly selected images of datasets, and they are used as limited but approximate indices of performance measures. According to the data sizes used in experiments, we randomly pick x% from the annotated set and (100 −x)% from the non-annotated set. The x is {20, 10, 5, 1, 0.5, 0.5} for the dataset size of {500, 5K, 10K, 50K, 100K, 200K}. Weakly supervised localization. One interesting question we address here is how performances and computation times vary as a function of data sizes. The experiments are repeated ten times for each dataset size, and the median (i.e. fifth-best) performance scores are reported. Similarly to previous tests, we select only the best ROI per image. As shown in Fig.7, the performances of 500 images fluctuate, but the results of the dataset size above 5K are stable. As the dataset size increases, a small performance improvement is observed. Since the maximum number of images at each running of the algorithm is bounded by N(= 10, 000), the computation times are linear to the number of images, and the performances of the data size above N are similar each other. Perturbation tests. Here we test the goodness of selected ROIs from a different view: robustness of ROI detection against random network formation. For example, given an image Ia, we can generate 100 sets of 200 randomly selected images that include Ia. If the ROI selection for Ia is repetitive across 100 different sets, we can say the ROI estimator for Ia is confident. This procedure is similar to bootstrapping or cross-validation. 7 Figure 7: Weakly supervised localization. (a) PR curves for five objects of our Flickr dataset by varying dataset sizes from 500 to 200K. (b) The log-log plot between the number of images and computation times for the car object. The slope of each range is {1.23, 2.05, 0.95, 1.05, 1.28} from left to right. Figure 8: Examples of perturbation tests. The histograms summarize how many times each ROI is selected in 100 random sets. The frequencies of particular ROIs are represented by the thickness of bounding boxes and the jet colormap from red (high) to blue (low). From left to right, the entropies of the distributions are {0.2419, 1.6846, 2.4331}, respectively. (X-axis: ROI hypotheses; Y-axis: Frequency). Fig.8 shows some examples of the perturbation tests. The histogram indicates how many times each ROI hypothesis is selected among 100 random sets. From left to right, one can see the increase of the difficulty of ROI detection. A peak is observed in an obvious image, but the distribution is wider in a challenging image. The entropy of the distribution can be an index of the measure of difficulty or the confidence of the estimator of the image. More localization examples. Fig.9 shows more examples of localization in our approach. The third row illustrates some typical examples of failure. Frequently co-occurred objects can be detected instead such as flowers in butterfly images, insects on sunflowers, other animals in the zoo, and persons everywhere. Also, sometimes small multiple instances are detected by one ROI or a part of an object is discovered (e.g. a giraffe face rather than the whole body). 5 Discussion We proposed an alternating optimization approach for scalable unsupervised ROI detection by analyzing the statistics of similarity links between ROI hypotheses. Both tests with PASCAL 06 and Flickr datasets showed that our approach is not only comparable to other unsupervised and supervised techniques but also applicable to real images on the Web. Acknowledgement. Funding for this research was provided by NSF Career award (IIS 0747120). Figure 9: More examples of object localization. The first and second rows represent successful detection, and the third row illustrates some typical failures. The yellow boxes are groundtruth labels, and the red and blue ones are ROIs detected by the proposed method. 8 References [1] N. Ahuja and S. Todorovic. Learning the taxonomy and models of categories present in arbitrary images. In ICCV, 2007. [2] P. J. Besl and N. D. McKay. A method for registration of 3-d shapes. IEEE Trans. on Pattern Analysis and Machine Intelligence, 14(2):239–256, 1992. [3] A. Bosch, A. Zisserman, and X. Munoz. Image classification using random forests and ferns. In ICCV, 2007. [4] S. Brin and L. Page. The anatomy of a large-scale hypertextual web search engine. In WWW, 1998. [5] O. Chum and A. Zisserman. An exemplar model for learning object classes. In CVPR, 2007. [6] B. Collins, J. Deng, K. Li, and L. Fei-Fei. Towards scalable dataset construction: An active learning approach. In ECCV, 2008. [7] D. Comaniciu and P. Meer. Mean shift: A robutst approach toward feature space analysis. IEEE Trans. on Pattern Analysis and Machine Intelligence, 24(5):603–619, 2002. [8] T. F. Cootes, G. J. Edwards, and C. J. Taylor. Active appearance models. IEEE Trans. on Pattern Analysis and Machine Intelligence, 23(6):681–685, 2001. [9] R. Fergus, L. Fei-Fei, P. Perona, and A. Zisserman. Learning object categories from google’s image search. In ICCV, pages 1816–1823, Oct. 2005. [10] G. Jeh and J. Widom. Scaling personalized web search. In WWW, 2003. [11] Y. Jing and S. Baluja. Visualrank, pagerank for google image search. IEEE Trans. on Pattern Analysis and Machine Intelligence, 30(11):1–31, 2008. [12] G. Kim, C. Faloutsos, and M. Hebert. Unsupervised modeling of object categories using link analysis techniques. In CVPR, 2008. [13] L.-J. Li, G. Wang, and L. Fei-Fei. Optimol: automatic object picture collection via incremental model learning. In CVPR, 2007. [14] T. Liu, J. Sun, N.-N. Zheng, X. Tang, and H.-Y. Shum. Learning to detect a salient object. In CVPR, 2007. [15] J.-Y. Pan, H.-J. Yang, C. Faloutsos, and P. Duygulu. Automatic multimedia cross-modal correlation discovery. In SIGKDD, 2004. [16] J. Philbin and A. Zisserman. Object mining using a matching graph on very large image collections. In ICVGIP, 2008. [17] A. Quattoni and A. Torralba. Recognizing indoor scenes. In CVPR, 2009. [18] B. C. Russell, A. A. Efros, J. Sivic, W. T. Freeman, and A. Zisserman. Using multiple segmentations to discover objects and their extent in image collections. In CVPR, 2006. [19] F. Schroff, A. Criminisi, and A. Zisserman. Harvesting image databases from the web. In ICCV, 2007. [20] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Trans. on Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000. [21] J. Sivic, B. C. Russell, A. A. Efros, A. Zisserman, and W. T. Freeman. Discovering objects and their location in images image features. In ICCV, 2005. [22] A. Torralba, R. Fergus, and W. T. Freeman. 80 million tiny images: a large dataset for non-parametric object and scene recognition. IEEE Trans. on Pattern Analysis and Machine Intelligence, 30(11):1958– 1970, 2008. [23] U. von Luxburg. A tutorial on spectral clustering. Statistics and Computing, 17(4):395–416, 2007. [24] J. Winn and N. Jojic. Locus: Learning object classes with unsupervised segmentation. In ICCV, 2005. [25] D. Zhou, J. Weston, A. Gretton, O. Bousquet, and B. Sch¨olkopf. Ranking on data manifolds. In NIPS, 2004. 9
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Fast Learning from Non-i.i.d. Observations Ingo Steinwart Information Sciences Group CCS-3 Los Alamos National Laboratory Los Alamos, NM 87545, USA ingo@lanl.gov Andreas Christmann University of Bayreuth Department of Mathematics D-95440 Bayreuth Andreas.Christmann@uni-bayreuth.de Abstract We prove an oracle inequality for generic regularized empirical risk minimization algorithms learning from α-mixing processes. To illustrate this oracle inequality, we use it to derive learning rates for some learning methods including least squares SVMs. Since the proof of the oracle inequality uses recent localization ideas developed for independent and identically distributed (i.i.d.) processes, it turns out that these learning rates are close to the optimal rates known in the i.i.d. case. 1 Introduction In the past, most articles investigating statistical properties of learning algorithms assumed that the observed data was generated in an i.i.d. fashion. However, in many applications this assumption cannot be strictly justified since the sample points are intrinsically temporal and thus often weakly dependent. Typical examples for this phenomenon are applications where observations come from (suitably pre-processed) time series, i.e., for example, financial predictions, signal processing, system observation and diagnosis, and speech or text recognition. A set of natural and widely accepted notions for describing such weak dependencies1 are mixing concepts such as α-, β-, and φ-mixing, since a) they offer a generalization to i.i.d. processes that is satisfied by various types of stochastic processes including Markov chains and many time series models, and b) they quantify the dependence in a conceptionally simple way that is accessible to various types of analysis. Because of these features, the machine learning community is currently in the process of appreciating and accepting these notions as the increasing number of articles in this direction shows. Probably the first work in this direction goes back to Yu [20], whose techniques for β-mixing processes inspired subsequent work such as [18, 10, 11], while the analysis of specific learning algorithms probably started with [9, 5, 8]. More recently, [7] established consistency of regularized boosting algorithms learning from β-mixing processes, while [15] established consistency of support vector machines (SVMs) learning from α-mixing processes, which constitute the largest class of mixing processes. For the latter, [21] established generalization bounds for empirical risk minimization (ERM) and [19, 17] analyzed least squares support vector machines (LS-SVMs). In this work, we establish a general oracle inequality for generic regularized learning algorithms and α-mixing observations by combining a Bernstein inequality for such processes [9] with localization ideas for i.i.d. processes pioneered by [6] and refined by e.g. [1]. To illustrate this oracle inequality, we then use it to show learning rates for some algorithms including ERM over finite sets and LSSVMs. In the ERM case our results match those in the i.i.d. case if one replaces the number of observations with the “effective number of observations”, while, for LS-SVMs, our rates are at least quite close to the recently obtained optimal rates [16] for i.i.d. observations. However, the latter difference is not surprising, when considering the fact that [16] used heavy machinery from 1For example, [4] write on page 71: “. . . it is a common practice to assume a certain mild asymptotic independence (such as α-mixing) as a precondition in the context of . . . nonlinear times series.” 1 empirical process theory such as Talagrand’s inequality and localized Rademacher averages, while our results only use a light-weight argument based on Bernstein’s inequality. 2 Definitions, Results, and Examples Let X be a measurable space and Y ⊂R be closed. Furthermore, let (Ω, A, µ) be a probability space and Z := (Zi)i≥1 be a stochastic process such that Zi : Ω→X ×Y for all i ≥1. For n ≥1, we further write Dn := ((X1, Y1), . . . , (Xn, Yn)) := (Z1, . . . , Zn) for a training set of length n that is distributed according to the first n components of Z. Throughout this work, we assume that Z is stationary, i.e., the (X × Y )n-valued random variables (Zi1, . . . , Zin) and (Zi1+i, . . . , Zin+i) have the same distribution for all n, i, i1, . . . , in ≥1. We further write P for the distribution of one (and thus all) Zi, i.e., for all measurable A ⊂X × Y , we have P(A) = µ {ω ∈Ω: Zi(ω) ∈A} . (1) To learn from stationary processes whose components are not independent, [15] suggests that it is necessary to replace the independence assumption by a notion that still guarantees certain concentration inequalities. We will focus on α-mixing, which is based on the α-mixing coefficients α(Z, µ, n) := sup n µ(A ∩B) −µ(A)µ(B) : i ≥1, A ∈Ai 1 and B ∈A∞ i+n o , n ≥1, where Ai 1 and A∞ i+n are the σ-algebras generated by (Z1, . . . , Zi) and (Zi+n, Zi+n+1, . . . ), respectively. Throughout this work, we assume that the process Z is geometrically α-mixing, that is α(Z, µ, n) ≤c exp(−bnγ) , n ≥1, (2) for some constants b > 0, c ≥0, and γ > 0. Of course, i.i.d. processes satisfy (2) for c = 0 and all b, γ > 0. Moreover, several time series models such as ARMA and GARCH, which are often used to describe, e.g. financial data, satisfy (2) under natural conditions [4, Chapter 2.6.1], and the same is true for many Markov chains including some dynamical systems perturbed by dynamic noise, see e.g. [18, Chapter 3.5]. An extensive and thorough account on mixing concepts including stronger mixing notions such as β- and φ-mixing is provided by [3]. Let us now describe the learning algorithms we are interested in. To this end, we assume that we have a hypothesis set F consisting of bounded measurable functions f : X →R that is pre-compact with respect to the supremum norm ∥· ∥∞, i.e., for all ε > 0, the covering numbers N(F, ∥· ∥∞, ε) := inf n ≥1 : ∃f1, . . . , fn ∈F such that F ⊂ n [ i=1 B(fi, ε) are finite, where B(fi, ε) := {f ∈ℓ∞(X) : ∥f −fi∥∞≤ε} denotes the ε-ball with center fi in the space ℓ∞(X) of bounded functions f : X →R. Moreover, we assume that we have a regularizer, that is, a function Υ : F →[0, ∞). Following [13, Definition 2.22], we further say that a function L : X × Y × R →[0, ∞) is a loss that can be clipped at some M > 0, if L is measurable and L(x, y, ¯t) ≤L(x, y, t) , (x, y, t) ∈X × Y × R, (3) where ¯t denotes the clipped value of t at ±M, that is ¯t := t if t ∈[−M, M], ¯t := −M if t < −M, and ¯t := M if t > M. Various often used loss functions can be clipped. For example, if Y := {−1, 1} and L is a convex, margin-based loss represented by ϕ : R →[0, ∞), that is L(y, t) = ϕ(yt) for all y ∈Y and t ∈R, then L can be clipped, if and only if ϕ has a global minimum, see [13, Lemma 2.23]. In particular, the hinge loss, the least squares loss for classification, and the squared hinge loss can be clipped, but the logistic loss for classification and the AdaBoost loss cannot be clipped. On the other hand, [12] established a simple technique, which is similar to inserting a small amount of noise into the labeling process, to construct a clippable modification of an arbitrary convex, margin-based loss. Moreover, if Y := [−M, M] and L is a convex, distance-based loss represented by some ψ : R →[0, ∞), that is L(y, t) = ψ(y −t) for all y ∈Y and t ∈R, then L can be clipped whenever ψ(0) = 0, see again [13, Lemma 2.23]. In particular, the least squares loss and the pinball loss used for quantile regression can be clipped, if the space of labels Y is bounded. Given a loss function L and an f : X →R, we often use the notation L ◦f for the function (x, y) 7→L(x, y, f(x)). Moreover, the L-risk is defined by RL,P (f) := Z X×Y L(x, y, f(x)) dP(x, y) , 2 and the minimal L-risk is R∗ L,P := inf{RL,P (f) | f : X →R}. In addition, a function f ∗ L,P satisfying RL,P (f ∗ L,P ) = R∗ L,P is called a Bayes decision function. Finally, we denote empirical risks based on Dn by RL,Dn(f), that is, for a realization of Dn(ω) of the training set Dn we have RL,Dn(ω)(f) = 1 n n X i=1 L Xi(ω), Yi(ω), f(Xi(ω)) . Given a regularizer Υ : F →[0, ∞), a clippable loss, and an accuracy δ ≥0, we consider learning methods that, for all n ≥1, produce a decision function fDn,Υ ∈F satisfying Υ(fDn,Υ) + RL,Dn( ¯fDn,Υ) ≤inf f∈F Υ(f) + RL,Dn(f) + δ . (4) Note that methods such as SVMs (see below) that minimize the right-hand side of (4) exactly, satisfy (4), because of (3). The following theorem, which is our main result, establishes an oracle inequality for methods (4), when the training data is generated by Z. Theorem 2.1 Let L : X × Y × R →[0, ∞) be a loss that can be clipped at M > 0 and that satisfies L(x, y, 0) ≤1, L(x, y, t) ≤B, and L(x, y, t) −L(x, y, t′) ≤|t −t′| (5) for all (x, y) ∈X × Y and t, t′ ∈[−M, M], where B > 0 is some constant. Moreover, let Z := (Zi)i≥1 be an X × Y -valued process that satisfies (2), and P be defined by (1). Assume that there exist a Bayes decision function f ∗ L,P and constants ϑ ∈[0, 1] and V ≥B2−ϑ such that EP L ◦¯f −L ◦f ∗ L,P 2 ≤V · EP (L ◦¯f −L ◦f ∗ L,P ) ϑ , f ∈F, (6) where F is a hypothesis set and L ◦f denotes the function (x, y) 7→L(x, y, f(x)). Finally, let Υ : F →[0, ∞) be a regularizer, f0 ∈F be a fixed function and B0 ≥B be a constant such that ∥L ◦f0∥∞≤B0. Then, for all fixed ε > 0, δ ≥0, τ > 0, and n ≥max{b/8, 22+5/γb−1/γ}, every learning method defined by (4) satisfies with probability µ not less than 1 −3Ce−τ: Υ(fDn,Υ) + RL,P ( ¯fDn,Υ) −R∗ L,P < 3 Υ(f0) + RL,P (f0) −R∗ L,P + 4B0cBτ nα + 4ε + 2δ + 36cσV (τ + ln N(F, ∥· ∥∞, ε)) nα 1/(2−ϑ) , where α := γ γ+1, C := 1 + 4e−2c, cσ := ( 82+γ b )1/(1+γ), and cB := cσ/3. Before we illustrate this theorem by a few examples, let us briefly discuss the variance bound (6). For example, if Y = [−M, M] and L is the least squares loss, then it is well-known that (6) is satisfied for V := 16M 2 and ϑ = 1, see e.g. [13, Example 7.3]. Moreover, under some assumptions on the distribution P, [14] established a variance bound of the form (6) for the so-called pinball loss used for quantile regression. In addition, for the hinge loss, (6) is satisfied for ϑ := q/(q + 1), if Tsybakov’s noise assumption holds for q, see [13, Theorem 8.24]. Finally, based on [2], [12] established a variance bound with ϑ = 1 for the earlier mentioned clippable modifications of strictly convex, twice continuously differentiable margin-based loss functions. One might wonder, why the constant B0 is necessary in Theorem 2.1, since appearently it only adds further complexity. However, a closer look reveals that the constant B only bounds functions of the form L ◦¯f, while B0 bounds the function L ◦f0 for an unclipped f0 ∈F. Since we do not assume that all f ∈F satisfy ¯f = f, we conclude that in general B0 is necessary. We refer to Examples 2.4 and 2.5 for situations, where B0 is significantly larger than B. Let us now consider a few examples of learning methods to which Theorem 2.1 applies. The first one is empirical risk minimization over a finite set. Example 2.2 Let the hypothesis set F be finite and Υ(f) = 0 for all f ∈F. Moreover, assume that ∥f∥∞≤M for all f ∈F. Then, for accuracy δ := 0, the learning method described by (4) is ERM, and Theorem 2.1 provides, by some simple estimates, the oracle inequality RL,P (fDn,Υ) −R∗ L,P < 3 inf f∈F RL,P (f) −R∗ L,P + 36cσV (τ + ln |F|) nα 1/(2−ϑ) + 4BcBτ nα . 3 Besides constants, this oracle inequality is an exact analogue to the standard oracle inequality for ERM learning from i.i.d. processes, [13, Theorem 7.2]. ◁ Before we present another example, let us first reformulate Theorem 2.1 for the case that the involved covering numbers have a certain polynomial behavior. Corollary 2.3 Consider the situation of Theorem 2.1 and additionally assume that there exist constants a > 0 and p ∈(0, 1] such that ln N(F, ∥· ∥∞, ε) ≤a ε−2p , ε > 0. Then there is cp,ϑ > 0 only depending on p and ϑ such that the inequality of Theorem 2.1 reduces to Υ(fDn,Υ) + RL,P ( ¯fDn,Υ) −R∗ L,P < 3 Υ(f0) + RL,P (f0) −R∗ L,P + cp,ϑ cσV a nα 1/(2+2p−ϑ) + 36cσV τ nα 1/(2−ϑ) + 4B0cBτ nα + 2δ . For the learning rates considered in the following examples, the exact value of cp,ϑ is of no importance. However, a careful numerical analysis shows that cp,ϑ ≤40 for all p ∈(0, 1] and ϑ ∈[0, 1]. Corollary 2.3 can be applied to various methods including e.g. SVMs with the hinge loss or the pinball loss, and regularized boosting algorithms. For the latter, we refer to e.g. [2] for some learning rates in the i.i.d. case and to [7] for a consistency result in the case of geometrically β-mixing observations. Unfortunately, a detailed exposition of the learning rates resulting from Corollary 2.3 for all these algorithms is clearly out of scope this paper, and hence we will only discuss learning rates for LS-SVMs. However, the only reason we picked LS-SVMs is that they are one of the few methods for which both rates for learning from α-mixing processes and optimal rates in the i.i.d. case are known. By considering LS-SVMs we can thus assess the sharpness of our results. Let us begin by briefly recalling LS-SVMs. To this end, let X be a compact metric space and k be a continuous kernel on X with reproducing kernel Hilbert space (RKHS) H. Given a regularization parameter λ > 0 and the least squares loss L(y, t) := (y −t)2, the LS-SVM finds the unique solution fDn,λ = arg min f∈H λ∥f∥2 H + RL,Dn(f) . To describe the approximation properties of H, we further need the approximation error function A(λ) := inf f∈H λ∥f∥2 H + RL,P (f) −R∗ L,P , λ > 0 . Example 2.4 (Rates for least squares SVMs) Let X be a compact metric space, Y = [−1, 1], and Z and P as above. Furthermore, let L be the least squares loss and H be the RKHS of a continuous kernel k over X. Assume that the closed unit ball BH of H satisfies ln N(BH, ∥· ∥∞, ε) ≤a ε−2p , ε > 0, (7) where a > 0 and p ∈(0, 1] are some constants. In addition, assume that the approximation error function satisfies A(λ) ≤cλβ for some c > 0, β ∈(0, 1], and all λ > 0. We define ρ := min n β, β β + 2pβ + p o . Then Corollary 2.3 applied to F := λ−1/2BH shows that the LS-SVM using λn := n−αρ/β learns with rate n−αρ. Let us compare this rate with other recent results: [17] establishes the learning rate n−2β β+3 , whenever (2) is satisfied for some α. At first glance, this rate looks stronger, since it is independent of α. However, a closer look shows that it depends on the confidence level 1 −3Ce−τ by a factor of eτ rather than by the factor of τ appearing in our analysis, and hence these rates are not comparable. Moreover, in the case α = 1, our rates are still faster whenever p ∈(0, 1/3], which is e.g. satisfied 4 for sufficiently smooth kernels, see e.g. [13, Theorem 6.26]. Moreover, [19] has recently established the rate n− αβ 2p+1 , (8) which is faster than ours, if and only if β > 1+p 1+2p. In particular, for highly smooth kernels such as the Gaussian RBF kernels, where p can be chosen arbitrarily close to 0, their rate is never faster. Moreover, [19] requires knowing α, which, as we will briefly discuss in Remark 2.6, is not the case for our rates. In this regard, it is interesting to note that their iterative proof procedure, see [13, Chapter 7.1] for a generic description of this technique, can also be applied to our oracle inequality. The resulting rate is essentially n−α min{β,β/(β+pβ+p)}, which is always faster than (8). Due to space constraints and the fact that these rates require knowing α and β, we skip a detailed exposition. Finally, both [19] and [17] only consider LS-SVMs, while Theorem 2.1 applies to various learning methods. ◁ Example 2.5 (Almost optimal rates for least squares SVMs) Consider the situation of Example 2.4, and additionally assume that there exists a constant Cp > 0 such that ∥f∥∞≤Cp ∥f∥p H∥f∥1−p L2(PX) , f ∈H. (9) As in [16], we can then bound B0 ≤λ(β−1)p, and hence the SVM using λn := n− α β+2pβ+p learns with rate n− αβ β+2pβ+p , compared to the optimal rate n− β β+p in the i.i.d. case, see [16]. In particular, if H = W m(X) is a Sobolev space over X ⊂Rd with smoothness m > d/2, and the marginal distribution PX is absolutely continuous with respect to the uniform distribution, where corresponding density is bounded away from 0 and ∞, then (7) and (9) are satisfied for p := d 2m. Moreover, the assumption on the approximation error function is satisfied for β := s/m, whenever f ∗ L,P ∈W s(X) and s ∈(d/2, m]. Consequently, the resulting learning rate is n− 2sα 2s+d+2ds/m , which in the i.i.d. case, where α = 1, is worse than the optimal rate n− 2s 2s+d by the term 2ds/m. Note that this difference can be made arbitrarily small by picking a sufficiently large m. Unfortunately, we do not know, whether the extra term 2ds/m is an artifact of our proof techniques, which are relatively light-weighted compared to the heavy machinery used in the i.i.d. case. Similarly, we do not know, whether the used Bernstein inequality for α-mixing processes, see Theorem 3.1, is optimal, but it is the best inequality we could find in the literature. However, if there is, or will be, a better version of this inequality, our oracle inequalities can be easily improved, since our techniques only require a generic form of Bernstein’s inequality. ◁ Remark 2.6 In the examples above, the rates were achieved by picking particular regularization sequences that depend on both α and β, which in turn, are almost never known in practice. Fortunately, there exists an easy way to achieve the above rates without such knowledge. Indeed, let us assume we pick a polynomially growing n−1/p-net Λn of (0, 1], split the training sample Dn into two (almost) equally sized and consecutive parts D(1) n and D(2) n , compute fD(1) n ,λ for all λ ∈Λn, and pick a λ∗∈Λn whose fD(1) n ,λ∗minimizes the RL,D(2) n -risk over Λn. Then combining Example 2.2 with the oracle inequality of Corollary 2.3 for LS-SVMs shows that the learning rates of the Examples 2.4 and 2.5 are also achieved by this training-validation approach. Although the proof is a straightforward modification of [13, Theorem 7.24], it is out of the page limit of this paper. ◁ 3 Proofs In the following, ⌊t⌋denotes the largest integer n satisfying n ≤t, and similarly, ⌈t⌉denotes the smallest integer n satisfying n ≥t. The key result we need to prove the oracle inequality of Theorem 2.1 is the following Bernstein type inequality for geometrically α-mixing processes, which was established in [9, Theorem 4.3]: 5 Theorem 3.1 Let Z := (Zi)i≥1 be an X × Y -valued stochastic process that satisfies (2) and P be defined by (1). Furthermore, let h : X × Y →R be a bounded measurable function for which there exist constants B > 0 and σ ≥0 such that EP h = 0, EP h2 ≤σ2, and ∥h∥∞≤B. For n ≥1 we define n(γ) := $ n 8n b 1 γ+1 −1% . Then, for all n ≥1 and all ε > 0, we have µ ω ∈Ω: 1 n n X i=1 h(Zi(ω)) ≥ε ≤ 1 + 4e−2c e−3ε2n(γ) 6σ2+2εB . (10) Before we prove Theorem 2.1, we need to slightly modify (10). To this end, we first observe that ⌈t⌉≤2t for all t ≥1 and ⌊t⌋≥t/2 for all t ≥2. From this it is easy to conclude that, for all n satisfying n ≥n0 := max{b/8, 22+5/γb−1/γ}, we have n(γ) ≥2−2γ+5 γ+1 b 1 γ+1 nα , where α := γ γ+1. For C := 1 + 4e−2c, cσ := ( 82+γ b )1/(1+γ), and cB := cσ/3, we thus obtain µ ω ∈Ω: 1 n n X i=1 h(Zi(ω)) ≥ε ≤Ce−τ , n ≥n0, where τ := ε2nα cσσ2+εcBB . Simple transformations and estimations then yield µ ω ∈Ω: 1 n n X i=1 h(Zi(ω)) ≥ r τcσσ2 nα + cBBτ nα ≤Ce−τ (11) for all n ≥max{b/8, 22+5/γb−1/γ} and τ > 0. In the following, we will use only this inequality. In addition, we will need the following simple and well-known lemma: Lemma 3.2 For q ∈(1, ∞), define q′ ∈(1, ∞) by 1/q + 1/q′ = 1. Then, for all a, b ≥0, we have (qa)2/q(q′b)2/q′ ≤(a + b)2 and ab ≤aq/q + bq′/q′. Proof of Theorem 2.1: For f : X →R we define hf := L ◦f −L ◦f ∗ L,P . By the definition of fDn,Υ, we then have Υ(fDn,Υ) + EDnh ¯ fDn,Υ ≤Υ(f0) + EDnhf0 + δ, and consequently we obtain Υ(fDn,Υ) + RL,P ( ¯fDn,Υ) −R∗ L,P = Υ(fDn,Υ) + EP h ¯ fDn,Υ ≤ Υ(f0) + EDnhf0 −EDnh ¯ fDn,Υ + EP h ¯ fDn,Υ + δ = (Υ(f0) + EP hf0) + (EDnhf0 −EP hf0) + (EP h ¯ fDn,Υ −EDnh ¯ fDn,Υ) + δ . (12) Let us first bound the term EDnhf0 −EP hf0. To this end, we further split this difference into EDnhf0 −EP hf0 = EDn(hf0 −h ¯ f0) −EP (hf0 −h ¯ f0) + EDnh ¯ f0 −EP h ¯ f0 . (13) Now L ◦f0 −L ◦¯f0 ≥0 implies hf0 −h ¯ f0 = L ◦f0 −L ◦¯f0 ∈[0, B0], and hence we obtain EP (hf0 −h ¯ f0) −EP (hf0 −h ¯ f0) 2 ≤EP (hf0 −h ¯ f0)2 ≤B0 EP (hf0 −h ¯ f0) . Inequality (11) applied to h := (hf0 −h ¯ f0) −EP (hf0 −h ¯ f0) thus shows that EDn(hf0 −h ¯ f0) −EP (hf0 −h ¯ f0) < r τcσB0 EP (hf0 −h ¯ f0) nα + cBB0τ nα holds with probability µ not less than 1 −Ce−τ. Moreover, using √ ab ≤a 2 + b 2, we find q n−ατcσB0 EP (hf0 −h ¯ f0) ≤EP (hf0 −h ¯ f0) + n−αcσB0τ/4 , 6 and consequently we have with probability µ not less than 1 −Ce−τ that EDn(hf0 −h ¯ f0) −EP (hf0 −h ¯ f0) < EP (hf0 −h ¯ f0) + 7cBB0τ 4nα . (14) In order to bound the remaining term in (13), that is EDnh ¯ f0 −EP h ¯ f0, we first observe that (5) implies ∥h ¯ f0∥∞≤B, and hence we have ∥h ¯ f0 −EP h ¯ f0∥∞≤2B. Moreover, (6) yields EP (h ¯ f0 −EP h ¯ f0)2 ≤EP h2 ¯ f0 ≤V (EP h ¯ f0)ϑ . In addition, if ϑ ∈(0, 1], Lemma 3.2 implies for q := 2 2−ϑ, q′ := 2 ϑ, a := (n−αcσ2−ϑϑϑV τ)1/2, and b := (2ϑ−1EP h ¯ f0)ϑ/2, that s cσV τ(EP h ¯ f0)ϑ nα ≤ 1 −ϑ 2 cσ2−ϑϑϑV τ nα 1 2−ϑ + EP h ¯ f0 ≤ cσV τ nα 1 2−ϑ + EP h ¯ f0. Since EP h ¯ f0 ≥0, this inequality also holds for ϑ = 0, and hence (11) shows that we have EDnh ¯ f0 −EP h ¯ f0 < EP h ¯ f0 + cσV τ nα 1 2−ϑ + 2cBBτ nα (15) with probability µ not less than 1 −Ce−τ. By combining this estimate with (14) and (13), we now obtain that with probability µ not less than 1 −2Ce−τ we have EDnh ¯ f0 −EP h ¯ f0 < EP h ¯ f0 + cσV τ nα 1 2−ϑ + 2cBBτ nα + 7cBB0τ 4nα , (16) i.e., we have established a bound on the second term in (12). Let us now fix a minimal ε-net C of F, that is, an ε-net of cardinality |C| = N(F, ∥· ∥∞, ε). Let us first consider the case nα < 3cB(τ + ln |C|). Combining (16) with (12) and using B ≤B0, B2−ϑ ≤V , 3cB ≤cσ, 2 ≤41/(2−ϑ), and EP h ¯ fDn,Υ −EDnh ¯ fDn,Υ ≤2B, we then find Υ(fDn,Υ) + RL,P (fDn,Υ) −R∗ L,P ≤ Υ(f0) + 2EP hf0 + cσV τ nα 1 2−ϑ + 2cBBτ nα + 7cBB0τ 4nα + (EP h ¯ fDn,Υ −EDnh ¯ fDn,Υ) + δ ≤ Υ(f0) + 2EP hf0 + cσV (τ + ln |C|) nα 1 2−ϑ + 4cBB0τ nα + 2B cσ(τ + ln |C|) nα 1 2−ϑ + δ ≤ 3Υ(f0) + 3EP hf0 + 36cσV (τ + ln |C|) nα 1 2−ϑ + 4cBB0τ nα + δ with probability µ not less than 1−2e−τ. It thus remains to consider the case nα ≥3cB(τ +ln |C|). To establish a non-trivial bound on the term EP h ¯ fD −EDnh ¯ fD in (12), we define functions gf,r := EP h ¯ f −h ¯ f EP h ¯ f + r , f ∈F , where r > 0 is a real number to be fixed later. For f ∈F, we then have ∥gf,r∥∞≤2Br−1, and for ϑ > 0, q := 2 2−ϑ, q′ := 2 ϑ, a := r, and b := EP h ¯ f ̸= 0, the first inequality of Lemma 3.2 yields EP g2 f,r ≤ EP h2 ¯ f (EP h ¯ f + r)2 ≤ (2 −ϑ)2−ϑϑϑ EP h2 ¯ f 4r2−ϑ(EP h ¯ f )ϑ ≤V rϑ−2 . (17) Moreover, for ϑ ∈(0, 1] and EP h ¯ f = 0, we have EP h2 ¯ f = 0 by the variance bound (6), which in turn implies EP g2 f,r ≤V rϑ−2. Finally, it is not hard to see that EP g2 f,r ≤V rϑ−2 also holds for ϑ = 0. Now, (11) together with a simple union bound yields µ Dn ∈(X × Y )n : sup f∈C EDngf,r < r cσV τ nαr2−ϑ + 2cBBτ nαr ≥1 −C |C| e−τ , 7 and consequently we see that, with probability µ not less than 1 −C |C| e−τ, we have EP h ¯ f −EDnh ¯ f < EP h ¯ f + r r cσV τ nαr2−ϑ + 2cBBτ nαr (18) for all f ∈C. Since fDn,Υ ∈F, there now exists an fDn ∈C with ∥fDn,Υ −fDn∥∞≤ε. By the assumed Lipschitz continuity of L the latter implies h ¯ fDn(x, y) −h ¯ fDn,Υ(x, y) ≤ ¯fDn(x) −¯fDn,Υ(x) ≤ fDn(x) −fDn,Υ(x) ≤ε for all (x, y) ∈X × Y . Combining this with (18), we obtain EP h ¯ fDn,Υ −EDnh ¯ fDn,Υ < EP h ¯ f + ε + r r cσV (τ + ln |C|) nαr2−ϑ + 2cBB(τ + ln |C|) nαr + 2ε with probability µ not less than 1 −C e−τ. By combining this estimate with (12) and (16), we then obtain that Υ(fDn,Υ) + EP h ¯ fDn,Υ < Υ(f0) + 2EP hf0 + cσV τ nα 1 2−ϑ + 2cBBτ nα + 7cBB0τ 4nα + 2ε + δ + EP h ¯ fDn,Υ + ε + r r cσV (τ + ln |C|) nαr2−ϑ + 2cBB(τ + ln |C|) nαr (19) holds with probability µ not less than 1 −3Ce−τ. Consequently, it remains to bound the various terms. To this end, we first observe that for r := 36cσV (τ + ln |C|) nα 1/(2−ϑ) , we obtain, since 6 ≤361/(2−ϑ), cσV τ nα 1 2−ϑ ≤r 6 and r cσV (τ + ln |C|) nαr2−ϑ ≤1 6 . In addition, V ≥B2−ϑ, cσ ≥3cB, 6 ≤361/(2−ϑ), and nα ≥3cB(τ + ln |C|) imply 2cBB(τ + ln |C|) rnα = 6 9 · 3cB(τ + ln |C|) nα · B r ≤ 6 9 · 3cB(τ + ln |C|) nα 1 2−ϑ · V 1 2−ϑ r ≤ 1 9 · 36cσV (τ + ln |C|) nαr2−ϑ 1 2−ϑ = 1 9 . Using these estimates together with 1/6 + 1/9 ≤1/3 in (19), we see that Υ(fDn,Υ) + EP h ¯ fDn,Υ < Υ(f0) + 2EP hf0 + r 3 + 7cBB0τ 4nα + EP h ¯ fDn,Υ + ε + r 3 + 2ε + δ holds with probability µ not less than 1 −3Ce−τ. Consequently, we have Υ(fDn,Υ)+EP h ¯ fDn,Υ < 3Υ(f0)+3EP hf0 + 36cσV (τ + ln |C| nα 1/(2−ϑ) + 4cBB0τ nα +4ε+2δ , i.e. we have shown the assertion. Proof of Corollary 2.3: The result follows from minimizing the right-hand side of the oracle inequality of Theorem 2.1 with respect to ε. References [1] P. L. Bartlett, O. Bousquet, and S. Mendelson. Local Rademacher complexities. Ann. Statist., 33:1497–1537, 2005. [2] G. Blanchard, G. Lugosi, and N. Vayatis. On the rate of convergence of regularized boosting classifiers. J. Mach. Learn. Res., 4:861–894, 2003. 8 [3] R. C. Bradley. Introduction to Strong Mixing Conditions. Vol. 1-3. Kendrick Press, Heber City, UT, 2007. [4] J. Fan and Q. Yao. Nonlinear Time Series. Springer, New York, 2003. [5] A. Irle. On consistency in nonparametric estimation under mixing conditions. J. Multivariate Anal., 60:123–147, 1997. [6] W. S. Lee, P. L. Bartlett, and R. C. Williamson. The importance of convexity in learning with squared loss. IEEE Trans. Inform. Theory, 44:1974–1980, 1998. [7] A. Lozano, S. Kulkarni, and R. Schapire. Convergence and consistency of regularized boosting algorithms with stationary β-mixing observations. In Y. Weiss, B. Sch¨olkopf, and J. Platt, editors, Advances in Neural Information Processing Systems 18, pages 819–826. MIT Press, Cambridge, MA, 2006. [8] R. Meir. Nonparametric time series prediction through adaptive model selection. Mach. Learn., 39:5–34, 2000. [9] D. S. Modha and E. Masry. Minimum complexity regression estimation with weakly dependent observations. IEEE Trans. Inform. Theory, 42:2133–2145, 1996. [10] M. Mohri and A. Rostamizadeh. Stability bounds for non-i.i.d. processes. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1025–1032. MIT Press, Cambridge, MA, 2008. [11] M. Mohri and A. Rostamizadeh. Rademacher complexity bounds for non-i.i.d. processes. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Advances in Neural Information Processing Systems 21, pages 1097–1104. 2009. [12] I. Steinwart. Two oracle inequalities for regularized boosting classiers. Statistics and Its Interface, 2:271284, 2009. [13] I. Steinwart and A. Christmann. Support Vector Machines. Springer, New York, 2008. [14] I. Steinwart and A. Christmann. Estimating conditional quantiles with the help of the pinball loss. Bernoulli, accepted with minor revision. [15] I. Steinwart, D. Hush, and C. Scovel. Learning from dependent observations. J. Multivariate Anal., 100:175–194, 2009. [16] I. Steinwart, D. Hush, and C. Scovel. Optimal rates for regularized least squares regression. In S. Dasgupta and A. Klivans, editors, Proceedings of the 22nd Annual Conference on Learning Theory, pages 79–93. 2009. [17] H. Sun and Q. Wu. Regularized least square regression with dependent samples. Adv. Comput. Math., to appear. [18] M. Vidyasagar. A Theory of Learning and Generalization: With Applications to Neural Networks and Control Systems. Springer, London, 2nd edition, 2003. [19] Y.-L. Xu and D.-R. Chen. Learning rates of regularized regression for exponentially strongly mixing sequence. J. Statist. Plann. Inference, 138:2180–2189, 2008. [20] B. Yu. Rates of convergence for empirical processes of stationary mixing sequences. Ann. Probab., 22:94–116, 1994. [21] B. Zou and L. Li. The performance bounds of learning machines based on exponentially strongly mixing sequences. Comput. Math. Appl., 53:1050–1058, 2007. 9
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Evaluating multi-class learning strategies in a hierarchical framework for object detection Sanja Fidler Marko Boben Aleˇs Leonardis Faculty of Computer and Information Science University of Ljubljana, Slovenia {sanja.fidler, marko.boben, ales.leonardis}@fri.uni-lj.si Abstract Multi-class object learning and detection is a challenging problem due to the large number of object classes and their high visual variability. Specialized detectors usually excel in performance, while joint representations optimize sharing and reduce inference time — but are complex to train. Conveniently, sequential class learning cuts down training time by transferring existing knowledge to novel classes, but cannot fully exploit the shareability of features among object classes and might depend on ordering of classes during learning. In hierarchical frameworks these issues have been little explored. In this paper, we provide a rigorous experimental analysis of various multiple object class learning strategies within a generative hierarchical framework. Specifically, we propose, evaluate and compare three important types of multi-class learning: 1.) independent training of individual categories, 2.) joint training of classes, and 3.) sequential learning of classes. We explore and compare their computational behavior (space and time) and detection performance as a function of the number of learned object classes on several recognition datasets. We show that sequential training achieves the best trade-off between inference and training times at a comparable detection performance and could thus be used to learn the classes on a larger scale. 1 Introduction Object class detection has been one of the mainstream research areas in computer vision. In recent years we have seen a significant trend towards larger recognition datasets with an increasing number of object classes [1]. This necessitates representing, learning and detecting multiple object classes, which is a challenging problem due to the large number and the high visual variability of objects. To learn and represent multiple object classes there have mainly been two strategies: the detectors for each class have either been trained in isolation, or trained on all classes simultaneously. Both exert certain advantages and disadvantages. Training independently allows us to apply complex probabilistic models that use a significant amount of class specific features and allows us to tune the parameters for each class separately. For object class detection, these approaches had notable success [2]. However, representing multiple classes in this way, means stacking together specific class representations. This, on the one hand, implies that each novel class can be added in constant time, however, the representation grows clearly linearly with the number of classes and is thus also linear in inference. On the other hand, joint representations enlarge sublinearly by virtue of sharing the features among several object classes [3, 4]. This means sharing common computations and increasing the speed of the joint detector. Training, however, is usually quadratic in the number of classes. Furthermore, adding just one more class forces us to re-train the representation altogether. Receiving somewhat less attention, the strategy to learn the classes sequentially (but not independently) potentially enjoys the traits of both learning types [4, 5, 6]. By learning one class after 1 another, we can transfer the knowledge acquired so far to novel classes and thus likely achieve both, sublinearity in inference and cut down training time. In order to scale to a higher number of object classes, learning them sequentially lends itself as the best choice. In literature, the approaches have mainly used one of these three learning strategies in isolation. To the best of our knowledge, little research has been done on analyzing and comparing them with respect to one another. This is important because it allows us to point to losses and gains of each particular learning setting, which could focus further research and improve the performance. This is exactly what this paper is set to do — we present a hierarchical framework within which all of the aforementioned learning strategies can be unbiasedly evaluated and put into perspective. Prominent work on these issues has been done in the domain of flat representations [4, 3], where each class is modeled as an immediate aggregate of local features. However, there is an increasing literature consensus, that hierarchies provide a more suitable form of multi-class representation [7, 8, 9, 10, 11, 12]. Hierarchies not only share complex object parts among similar classes, but can re-use features at several levels of granularity also for dissimilar objects. In this paper, we provide a rigorous experimental evaluation of several important multi-class learning strategies for object detection within a generative hierarchical framework. We make use of the hierarchical learning approach by [13]. Here we propose and evaluate three types of multi-class learning: 1.) independent training of individual categories, 2.) joint training, 3.) sequential training of classes. Several issues were evaluated on multiple object classes: 1.) growth of representation, 2.) training and 3.) inference time, 4.) degree of feature sharing and re-use at each level of the hierarchy, 5.) influence of class ordering in sequential learning, and 6.) detection performance, all as a function of the number of classes learned. We show that sequential training achieves the best trade-off between inference and training times at a comparable detection performance and could thus be used to learn the classes on a larger scale. Related work. Prior work on multi-class learning in generative hierarchies either learns separate hierarchies for each class [14, 15, 16, 10, 17], trains jointly [7, 18, 9, 19, 20, 11], whereas work on sequential learning of classes has been particularly scarce [6, 13]. However, to the best of our knowledge, no work has dealt with, evaluated and compared multiple important learning concepts under one hierarchical framework. 2 The hierarchical model and inference The hierarchical model. We use the hierarchical model of [13, 21], which we summarize here. Objects are represented with a recursive compositional shape vocabulary which is learned from images. The vocabulary contains a set of shape models or compositions at each layer. Each shape model in the hierarchy is modeled as a conjunction of a small number of parts (shapes from the previous layer). Each part is spatially constrained on the parent shape model via a spatial relation which is modeled with a two-dimensional Gaussian distribution. The number and the type of parts can differ across the shape models and is learned from the data without supervision. At the lowest layer, the vocabulary consists of a small number of short oriented contour fragments, while the vocabulary at the top-most layer contains models that code the shapes of the whole objects. For training, we need a positive and a validation set of class images, while the structure of the representation is learned in an unsupervised way (no labels on object parts or smaller subparts need to be given). The hierarchical vocabulary V = (V, E) is represented with a directed graph, where multiple edges between two vertices are allowed. The vertices V of the graph represent the shape models and the edges E represent the composition relations between them. The graph V has a hierarchical structure, where the set of vertices V is partitioned into subsets V 1, . . . , V O, each containing the shapes at a particular layer. The vertices {v1 i }6 i=1 at the lowest layer V 1 represent 6 oriented contour fragments. The vertices at the top-most layer V O, referred to as the object layer represent the whole shapes of the objects. Each object class C is assigned a subset of vertices V O C ⊆V O that code the object layer shapes of that particular class. We denote the set of edges between the vertex layers V ℓand V ℓ−1 with Eℓ. Each edge eℓ Ri = vℓ Rvℓ−1 i in Eℓis associated with the Gaussian parameters θℓ Ri := θ(eℓ Ri) = (µℓ Ri, Σℓ Ri) of the spatial relation between the parent shape vℓ R and its part vℓ−1 i . We will use θℓ R = (θℓ Ri)i to denote the vector of all the parameters of a shape model vℓ R. The pair Vℓ:= (V ℓ, Eℓ) will be referred to as the vocabulary at layer ℓ. 2 Inference. We infer object class instances in a query image I in the following way. We follow the contour extraction of [13], which finds local maxima in oriented Gabor energy. This gives us the contour fragments F and their positions X. In the process of inference we build a (directed acyclic) inference graph G = (Z, Q). The vertices Z are partitioned into vertex layers 1 to O (object layer), Z = Z1∪· · ·∪ZO, and similarly also the edges, Q = Q1∪· · ·∪QO. Each vertex zℓ= (vℓ, xℓ) ∈Zℓ represents a hypothesis that a particular shape vℓ∈V ℓfrom the vocabulary is present at location xℓ. The edges in Qℓconnect each parent hypothesis zℓ R to all of its part hypotheses zℓ−1 i . The edges in the bottom layer Q1 connect the hypotheses in the first layer Z1 with the observations. With S(z) we denote the subgraph of G that contains the vertices and edges of all descendants of vertex z. Since our definition of each vocabulary shape model assumes that its parts are conditionally independent, we can calculate the likelihood of the part hypotheses zℓ−1 i = (vℓ−1 i , xℓ−1 i ) under a parent hypothesis zℓ R = (vℓ R, xℓ R) by taking a product over the individual likelihoods of the parts: p(vℓ−1, xℓ−1 | vℓ R, xℓ R, θℓ R) = Y eℓ Ri=vℓ Rvℓ−1 i p(xℓ−1 i | xℓ R, vℓ−1 i , vℓ R, θℓ Ri) (1) The term pRi := p(xℓ−1 i | xℓ R, vℓ−1 i , vℓ R, θℓ Ri) stands for the spatial constraint imposed by a vocabulary edge eℓ Ri between a parent hypothesis zℓ R and its part hypothesis zℓ−1 i . It is modeled by a normal distribution, pRi = N(xℓ−1 i −xℓ R | θℓ Ri), where θℓ Ri = (µℓ Ri, Σℓ Ri). If the likelihood in (1) is above a threshold, we add edges between zℓ R and its most likely part hypotheses. The log-likelihood of the observations under a hypothesis zℓ R is then calculated recursively over the subgraph S(zℓ R): log p(F, X, z1:ℓ−1 | zℓ R; V) = X zR′zi′∈E(S(zℓ R)) log pR′i′ + X z1 i′∈V (S(zℓ R)) log p(F, X | z1 i′), (2) where E(S(zℓ R)) and V (S(zℓ R)) denote the edges and vertices of the subgraph S(zℓ R), respectively. The last term is the likelihood of the Gabor features under a particular contour fragment hypothesis. 3 Multi-class learning strategies We first define the objective function for multi-class learning and show how different learning strategies can be used with it in the following subsections. Our goal is to find a hierarchical vocabulary V that well represents the distribution p(I | C) ≈p(F, X | C; V) at minimal complexity of the representation (C denotes the class variable). Specifically, we seek for a vocabulary V = ∪ℓVℓthat optimizes the function f over the data D = {(Fn, Xn, Cn)}N n=1 (N training images): V∗= arg max V f(V), where f(V) = L(D | V) −λ · T(V) (3) The first term in (3) represents the log-likelihood: L(D | V) = N X n=1 log p(Fn, Xn | C; V) = N X n=1 log X z p(Fn, Xn, z | C; V), (4) while T(V) penalizes the complexity of the model [21] and λ controls the amount of penalization. Several approximations are made to learn the vocabulary; namely, the vocabulary is learned layer by layer (in a bottom-up way) by finding frequent spatial layouts of parts from the previous layer [13] and then using f to select a minimal set of models at each layer that still produce a good wholeobject shape representation at the final, object layer [21]. The top layer models are validated on a set of validation images and those yielding a high rate of false-positives are removed from V. We next show how different training strategies are performed to learn a joint multi-class vocabulary. 3.1 Independent training of individual classes In independent training, a class specific vocabulary Vc is learned using the training images of each particular class C = c. We learn Vc by maximizing f over the data D = {(Fn, Xn, C = c)}. For the negative images in the validation step, we randomly sample images from other classes. The joint multi-class representation V is then obtained by stacking the class specific vocabularies Vc together, V ℓ= ∪cV ℓ c (the edges E are added accordingly). Note that V 1 c is the only layer common to all classes (6 oriented contour fragments), thus V 1 = V 1 c . 3 3.2 Joint training of classes In joint training, the learning phase has two steps. In the first step, the training data D for all the classes is presented to the algorithm simultaneously, and is treated as unlabeled. The spatial parameters θ of the models at each layer are then inferred from images of all classes, and will code “average” spatial part dispositions. The joint statistics also influences the structure of the models by preferring those that are most repeatable over the classes. This way, the jointly learned vocabulary V will be the best trade-off between the likelihood L and the complexity T over all the classes in the dataset. However, the final, top-level likelihood for each particular class could be low because the more discriminative class-specific information has been lost. Thus, we employ a second step which revisits each class separately. Here, we use the joint vocabulary V and add new models vℓ R to each layer ℓif they further increase the score f for each particular class. This procedure is similar to that used in sequential training and will be explained in more detail in the following subsection. Object layer VO is consequently learned and added to V for each class. We validate the object models after all classes have been trained. A similarity measure is used to compare every two classes based on the degree of feature sharing between them. In validation, we choose the negative images by sampling the images of the classes according to the distribution defined by the similarity measure. This way, we discard the models that poorly discriminate between the similar classes. 3.3 Sequential training of classes When training the classes sequentially, we train on each class separately, however, our aim is to 1.) maximize the re-use of compositions learned for the previous classes, and 2.) add those missing (class-specific) compositions that are needed to represent class k sufficiently well. Let V1:k−1 denote the vocabulary learned for classes 1 to k −1. To learn a novel class k, for each layer ℓwe seek a new set of shape models that maximizes f over the data D = {(Fn, Xn, C = k)} conditionally on the already learned vocabulary Vℓ 1:k−1. This is done by treating the hypotheses inferred with respect to Vℓ 1:k−1 as fixed, which gives us a starting value of the score function f. Each new model vℓ R is then evaluated and selected conditionally on this value, i.e such that the difference f(Vℓ 1:k−1 ∪ vℓ R) −f(Vℓ 1:k−1) is maximized. Since according to the definition in (4) the likelihood L increases the most when the hypotheses have largely disjoint supports, we can greatly speed up the learning process: the models need to be learned only with respect to those (F, X) in an image that have a low likelihood under the vocabulary Vℓ 1:k−1, which can be determined prior to training. 4 Experimental results We have evaluated the hierarchical multi-class learning strategies on several object classes. Specifically, we used: UIUC multi-scale cars [22], GRAZ [4] cows and persons, Weizmann multi-scale horses (adapted by Shotton et al. [23]), all five classes from the ETH dataset [24], and all ten classes from TUD shape2 [25]. Basic information is given in Table 1. A 6-layer vocabulary was learned. 1 The bounding box information was used during training. When evaluating detection performance, a detection will be counted as correct, if the predicted bounding box coincides with groundtruth more than 50%. On the ETH dataset alone, this threshold is lowered to 0.3 to enable a fair comparison with the related work [24]. The performance will be given either with recall at equal error rate (EER), positive detection rate at low FPPI, or as classif.by-detection (on TUD shape2), depending on the type of results reported on that dataset thus-far. To evaluate the shareability of compositions between the classes, we will use the following measure: deg share(ℓ) = 1 |V ℓ| X vℓ R∈V ℓ (# of classes that use vℓ R) −1 # of all classes −1 , defined for each layer ℓseparately. By “vℓ R used by class C” it is meant that there is a path of edges connecting any of the class specific shapes V O C and vℓ R. To give some intuition behind the measure: 1The number of layers depends on the objects’ size in the training images (it is logarithmic with the number of non-overlapping contour fragments in an image). To enable a consistent evaluation of feature sharing, etc, we have scaled the training images in a way which produced the whole-shape models at layer 6 for each class. 4 deg share = 0 if no shape from layer ℓis shared (each class uses its own set of shapes), and it is 1 if each shape is used by all the classes. Beside the mean (which defines deg share), the plots will also show the standard deviation. In sequential training, we can additionally evaluate the degree of re-use when learning each novel class. Higher re-use means lower training time and a more compact representation. We expect a tendency of higher re-use as the number k of classes grows, thus we define it with respect to the number of learned classes: deg transfer(k, ℓ) = # of vℓ R ∈V ℓ 1:k−1 used by ck # of all vℓ R ∈V ℓ 1:k used by ck (5) Evaluation was performed by progressively increasing the number of object classes (from 2 to 10). The individual training will be denoted by I, joint by J, and sequential by S. Table 2 relates the detection performances of I to those of the related work. On the left side, we report detection accuracy at low FPPI rate for the ETH dataset, averaged over 5 random splits of training/test images as in [24]. On the right side, recall at EER is given for a number of classes. Two classes. We performed evaluation on two visually very similar classes (cow, horse), and two dissimilar classes (person, car). Table 3 gives information on 1.) size (the number of compositions at each layer), 2.) training and 3.) inference times, 4.) recall at EER. In sequential training, both possible orders were used (denoted with S1 and S2) to see whether different learning orders (of classes) affect the performance. The first two rows show the results for each class individually, while the last row contains information with respect to the conjoined representations. Already for two classes, the cumulative training time is slightly lower for S than I, while both being much smaller than that of J. Degree of sharing. The hierarchies learned in I, J, and S on cows and horses, and J for car-person are shown in Fig. 2 in a respective order from left to right. The red nodes depict cow/car and blue horse/person compositions. The green nodes depict the shared compositions. We can observe a slightly lower number of shareable nodes for S compared to J, yet still the lower layers for cowhorse are almost completely re-used. Even for the visually dissimilar classes (car-person) sharing is present at lower layers. Numerically, the degrees of sharing and transfer are plotted in Fig. 1. Detection rate. The recall values for each class are reported in Table 3. Interestingly, “knowing” horses improved the performance for cows. For car-person, individual training produced the best result, while training person before car turned out to be a better strategy for S. Fig. 1 shows the detection rates for cows and horses on the joint test set (the strongest class hypothesis is evaluated), which allows for a much higher false-positive rate. We evaluate it with F-measure (to account for FP). A higher performance for all joint representations over the independent one can be observed. This is due to the high degree of sharing in J and S, which puts similar hypotheses in perspective and thus discriminates between them better. Five classes. The results for ETH-5 are reported in Table 4. We used half of the images for training, and the other half for testing. The split was random, but the same for I, J, and S. We also test whether different orders in S affect performance (we report an average over 3 random S runs). Ordering does slightly affect performance, which means we may try to find an optimal order of classes in training. We can also observe that the number of compositions at each layer is higher for S as for J (both being much smaller than I), but this only slightly showed in inference times. Ten classes. The results on TUD-10 are presented in Table 5. A few examples of the learned shapes for S are shown in Fig. 3. Due to the high training complexity of J, we have only ran J for 2, 5 and 10 classes. We report classif.-by-detection (the strongest class hypothesis in an image must overlap with groundtruth more than 50%). To demonstrate the strength of our representation, we have also ran (linear) SVM on top of hypotheses from Layers 1−3, and compared the performances. Already here, Layer 3 + SVM outperforms prior work [25] by 10%. Fig. 4-(11.) shows classification as a number of learned classes. Our approach consistently outperforms SVM, which is likely due to the high scale- and rotation- variability of images with which our approach copes well. Fig. 4 shows: inference time, cumulative training time, degree of sharing (for the final 10-class repr.), transfer, and classification rates as a function of the number of learned classes. Vocabulary size. The top row in Fig 4 shows representation size for I, J and S as a function of learned classes. With respect to worst case (I), both J and S have a highly sublinear growth. Moreover, in layers 2 and 3, where the burden on inference is the highest (the highest number of 5 inferred hypotheses), an almost constant tendency can be seen. We also compare the curves with those reported for a flat approach by Opelt et al. [4] in Fig 4-(5). We plot the number of models at Layer 5 which are approximately of the same granularity as the learned boundary parts in [4]. Both, J and S hierarchical learning types show a significantly better logarithmic tendency as in [4]. Fig 4-(6) shows the size of the hierarchy file stored on disk. It is worth emphasizing that the hierarchy subsuming 10 classes uses only 0.5Mb on disk and could fit on an average mobile device. 50 classes. To increase the scale of the experiments we show the performance of sequential training on 50 classes from LabelMe [1]. The results are presented in Fig. 5. For I in the inference time plot we used the inference time for the first class linearly extrapolated with the number of classes. We can observe that S achieves much lower inference times than I, although it is clear that for a higher number of classes more research is needed to cut down the inference times to a practical value. I J S1 S2 0.7 0.75 0.8 0.85 0.9 0.95 1 Detection rate: JOINT cow+horse dataset learning type F−measure 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of sharing / re−use: cow−horse layer degree of sharing Joint training Sequential 1: cow + horse Sequential 2: horse + car 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of sharing / re−use: car−person layer degree of sharing Joint training Sequential 1: car + person Sequential 2: person + car Figure 1: From left to right: 1.) detection rate (F measure) on the joint cow-horse test set. 2.) degree of sharing for cow-horse, 3.) car-person vocabularies, 4.) an example detection of a person and horse. Figure 2: Learned 2-class vocabularies for different learning types (the nodes depict the compositions vℓ R, the links represent the edges eℓ Ri between them — the parameters θℓare not shown). From left to right: cow-horse hierarchy for 1.) I, 2.) J, 3.) S1, and 4.) car-person J. Green nodes denoted shared compositions. 5 Conclusions and discussion We evaluated three types of multi-class learning strategies in a hierarchical compositional framework, namely 1.) independent, 2). joint, and 3.) sequential training. A comparison was made through several important computational aspects as well as by detection performance. We conclude that: 1.) Both joint and sequential training strategies exert sublinear growth in vocabulary size (more evidently so in the lower layers) and, consequently, sublinear inference time. This is due to a high degree of sharing and transfer within the resulting vocabularies. The hierarchy obtained by sequential training grows somewhat faster, but not significantly so. 2.) Training time was expectedly worst for joint training, while training time even reduced with each additional class during sequential training. 3.) Different training orders of classes did perform somewhat differently — this means we might try to find an “optimal” order of learning. 4.) Training independently has mostly yielded the best detection rates, but the discrepancy with the other two strategies was low. For similar classes (cow-horse), sequential learning even improved the detection performance, and was in most cases above the joint’s performance. By training sequentially, we can learn class specific features (yet still have a high degree of sharing) which boost performance. Most importantly, sequential training has achieved the best trade-off between detection performance, re-usability, inference and training time. The observed computational properties of all the strategies in general, and sequential learning in particular, go well beyond the reported behavior of flat approaches [4]. This makes sequential learning of compositional hierarchies suitable for representing the classes on a larger scale. Acknowledgments This research has been supported in part by the following funds: EU FP7-215843 project POETICON, EU FP7-215181 project CogX, Research program Computer Vision P2-0214 and Project J2-2221 (ARRS). 6 layer 1 layer 2 layer 3 layer 4 layer 5 layer 6 hammer pliers saucepan scissors Figure 3: A few examples from the learned hierarchical shape vocabulary for S on TUD-10. Each shape in the hierarchy is a composition of shapes from the layer below. Only the mean of each shape is shown. method size on disk classf. rate Stark et al.[25] / 44% Level 1 + SVM 206 Kb 32% Level 2 + SVM 3, 913 Kb 44% train. time infer. time size of representation Level 3 + SVM 34, 508 Kb 54% L2 L3 L4 L5 Independent 1, 249 Kb 71% 207 min 12.2 sec 74 96 159 181 Joint 408 Kb 69% 752 min 2.0 sec 14 23 39 59 Sequential 490 Kb 71% 151 min 2.4 sec 9 21 49 76 Table 5: Results on the TUD-10. Classification obtained as classification-by-detection. 1 2 3 4 5 6 7 8 9 10 0 10 20 30 40 50 60 70 number of classes size of representation Growth of representation independent joint sequential Layer 2 1 2 3 4 5 6 7 8 9 10 0 20 40 60 80 100 number of classes size of representation Growth of representation independent joint sequential Layer 3 1 2 3 4 5 6 7 8 9 10 0 50 100 150 number of classes size of representation Growth of representation independent joint sequential Layer 4 1 2 3 4 5 6 7 8 9 10 0 50 100 150 number of classes size of representation Growth of representation independent joint sequential Layer 5 1 2 3 4 5 6 7 8 9 10 0 100 200 300 400 500 number of classes size of representation Growth of representation Opelt et al. independent joint sequential 2 4 6 8 10 0 2 4 6 8 10 12 number of classes inference time (sec) Inference time per image independent joint sequential 2 4 6 8 10 0 200 400 600 800 1000 1200 1400 number of classes cumulative size on disk (Kb) Size on disk independent joint sequential Layer 2 + SVM Layer 3 + SVM 2 4 6 8 10 0 50 100 150 200 number of classes cumulative training time (min) Cumulative training time independent sequential 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of sharing: TUD layer degree of sharing Joint training Sequential: alphabetical order 1 2 3 4 5 6 7 8 9 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Degree of transfer per layer number of learnt classes degree of transfer Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 2 4 6 8 40 50 60 70 80 90 100 number of classes Classification rate (%) Classification rate sequential SVM on Layer 2 SVM on Layer 3 Figure 4: Results on TUD-10. Top: (1-4) repr. size as a function of the number of learned classes. Middle: 5.) repr. size compared to [4], 6.) size of hierarchy on disk, 7.) avg. inference time per image, 8.) cumulative train. time. Bottom: degree of 9.) sharing and 10.) transfer, 11.) classif. rates, 12.) example detection of cup. 10 20 30 40 50 0 50 100 150 200 250 300 number of classes number of compositions Size of representation (# classes) Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 10 20 30 40 50 0 50 100 150 200 250 300 350 number of classes inference time per image (sec) Inference time per image (# classes) independent sequential 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 Degree of transfer per layer number of learnt classes degree of transfer Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Figure 5: Results on 50 object classes from LabelMe [1]. From left to right: Size of representation (number of compositions per layer), inference times, and deg transfer, all as a function of the number of learned classes. 7 Table 1: Dataset information. dataset ETH shape TUD shape1 (train) + TUD shape2 (test) Graz UIUC Weizm. class apple bottle giraffe mug swan cup fork hammer knife mug pan pliers pot saucepan scissors cow person car horse # train 19 21 30 24 15 20 20 20 20 20 20 20 20 20 20 20 19 40 20 # test 21 27 57 24 17 10 10 10 10 10 10 10 10 10 10 65 19 108 228 Table 2: Comparison of detection rates with related work. Left: Average detection-rate (in %) at 0.4 FPPI for the related work, while we also report the actual FPPI, for the ETH shape database. Right: Recall at EER for various classes. The approaches that use more than just contour information (and are thus not directly comparable to ours) are shaded gray. applelogo bottle giraffe mug swan average [24] 83.2(1.7) 83.2(7.5) 58.6(14.6) 83.6(8.6) 75.4(13.4) 76.8 [26] 89.9(4.5) 76.8(6.1) 90.5(5.4) 82.7(5.1) 84.0(8.4) 84.8 ind. train. 87.3(2.6) 86.2(2.8) 83.3(4.3) 84.6(2.3) 78.2(5.4) 83.7 0.32 FPPI 0.36 FPPI 0.21 FPPI 0.27 FPPI 0.26 FPPI cow horse car scale person Related work 100. [4] 89.0 [23] 90.6 [27] 52.6 [4] 100. [7] 93.0 [28] 93.5 [29] 52.4 [23] 98.5 [23] / 92.1 [13] 50.0 [28] ind. train. 98.5 94.3 93.5 60.4 Table 3: Results for different learning types on the cow-horse, and car-person classes. class size of representation (number of compositions per layer) train. time (min) infer. time (sec) rec. at EER (%) I J S1 (1 + 2) S2 (2 + 1) I J S1 S2 I J S1 S2 I J S1 S2 L2 L3 L4 L5 L2 L3 L4 L5 L2 L3 L4 L5 L2 L3 L4 L5 cow (1) 17 17 23 25 17 25 27 25 17 17 23 25 14 17 20 20 25 / 25 19 1.9 2.0 2.1 2.0 96.9 96.9 96.9 98.5 horse (2) 12 12 18 24 17 26 26 27 18 18 24 21 12 12 18 24 20 / 17 20 2.3 2.6 2.7 2.7 94.3 93.4 93.4 94.3 cow+hrs. 29 29 41 49 17 26 30 36 18 21 33 40 14 19 29 38 45 65 42 39 4.3 2.5 2.6 2.5 95.6 95.6 95.6 96.4 car (1) 6 10 13 16 7 16 20 20 6 10 13 16 11 19 18 18 35 / 35 33 3.4 5.2 5.3 5.4 93.5 91.7 93.5 92.4 person (2) 9 16 19 21 7 12 14 22 11 12 15 23 9 16 19 21 17 / 15 17 2.3 2.6 2.8 3.0 60.4 58.3 56.3 60.4 car+prsn. 15 26 32 37 7 18 25 42 12 19 27 39 11 25 31 38 52 85 50 50 6.3 4.8 4.9 5.0 77.0 75.0 74.9 76.4 Table 4: Results for different learning types on the 5−class ETH shape dataset. class size of representation trn.time(min) infr.time(sec) det. rate (%) FPPI I J S, mean (std) - over 3 runs I J S I J S I J S I J S L2 L3 L4 L5 L2 L3 L4 L5 L2 L3 L4 L5 applelogo 11 30 27 28 14 15 21 28 10(0.6) 25(1.7) 27(12.7) 23(7.5) 23 / 23 3.6 11.1 12.1 88.6 86.4 86.4 0.34 0.27 0.28 bottle 7 11 22 22 13 16 21 22 9(0.6) 22(8.1) 28(2) 22(3.6) 25 / 21 3.4 11.1 12.1 85.5 80.0 80.0 0.4 0.34 0.32 giraffe 5 13 22 37 15 19 26 26 10(0.6) 28(5.9) 35(1.7) 30(1.2) 31 / 26 3.2 11.1 12.1 82.4 81.3 84.6 0.19 0.16 0.18 mug 9 16 25 23 16 19 25 34 10(0.6) 25(7.9) 30(4.7) 29(4.9) 31 / 18 3.6 11.1 12.1 84.9 83.3 83.3 0.31 0.22 0.22 swan 11 18 29 26 16 20 26 27 10(0) 23(6.4) 30(4.0) 27(1.5) 17 / 12 2.8 11.1 12.1 75.8 69.7 72.7 0.28 0.22 0.21 all 43 88 125 136 16 22 32 55 11(0.6) 33(2.5) 61(9.5) 79(13.7) 127 235 100 16.6 11.1 12.1 83.4 80.1 81.4 0.30 0.24 0.24 8 References [1] Russell, B., Torralba, A., Murphy, K., and Freeman, W. T. (2008) Labelme: a database and web-based tool for image annotation. IJCV, 77, 157–173. [2] Leibe, B., Leonardis, A., and Schiele, B. (2008) Robust object detection with interleaved categorization and segmentation. IJCV, 77, 259–289. [3] Torralba, A., Murphy, K. P., and Freeman, W. T. (2007) Sharing visual features for multiclass and multiview object detection. IEEE PAMI, 29, 854–869. [4] Opelt, A., Pinz, A., and Zisserman, A. (2008) Learning an alphabet of shape and appearance for multiclass object detection. IJCV, 80, 16–44. [5] Fei-Fei, L., Fergus, R., and Perona, P. (2004) Learning generative visual models from few training examples: an incremental bayesian approach tested on 101 object categories. IEEE CVPR’04 Workshop on Generative-Model Based Vision. [6] Krempp, S., Geman, D., and Amit, Y. (2002) Sequential learning of reusable parts for object detection. Tech. rep. [7] Todorovic, S. and Ahuja, N. (2007) Unsupervised category modeling, recognition, and segmentation in images. IEEE PAMI. [8] Zhu, S. and Mumford, D. (2006) A stochastic grammar of images. Found. and Trends in Comp. Graphics and Vision, 2, 259–362. [9] Ranzato, M. A., Huang, F.-J., Boureau, Y.-L., and LeCun, Y. (2007) Unsupervised learning of invariant feature hierarchies with applications to object recognition. CVPR. [10] Ullman, S. and Epshtein, B. (2006) Visual Classification by a Hierarchy of Extended Features.. Towards Category-Level Object Recognition, Springer-Verlag. [11] Sivic, J., Russell, B. C., Zisserman, A., Freeman, W. T., and Efros, A. A. (2008) Unsupervised discovery of visual object class hierarchies. CVPR. [12] Bart, I., Porteous, E., Perona, P., and Wellings, M. (2008) Unsupervised learning of visual taxonomies. CVPR. [13] Fidler, S. and Leonardis, A. (2007) Towards scalable representations of visual categories: Learning a hierarchy of parts. CVPR. [14] Scalzo, F. and Piater, J. H. (2005) Statistical learning of visual feature hierarchies. W. on Learning, CVPR. [15] Zhu, L., Lin, C., Huang, H., Chen, Y., and Yuille, A. (2008) Unsupervised structure learning: Hierarchical recursive composition, suspicious coincidence and competitive exclusion. ECCV, vol. 2, pp. 759–773. [16] Fleuret, F. and Geman, D. (2001) Coarse-to-fine face detection. IJCV, 41, 85–107. [17] Schwartz, J. and Felzenszwalb, P. (2007) Hierarchical matching of deformable shapes. CVPR. [18] Ommer, B. and Buhmann, J. M. (2007) Learning the compositional nature of visual objects. CVPR. [19] Serre, T., Wolf, L., Bileschi, S., Riesenhuber, M., and Poggio, T. (2007) Object recognition with cortexlike mechanisms. IEEE PAMI, 29, 411–426. [20] Sudderth, E., Torralba, A., Freeman, W. T., and Willsky, A. (2008) Describing visual scenes using transformed objects and parts. IJCV, pp. 291–330. [21] Fidler, S., Boben, M., and Leonardis, A. (2009) Optimization framework for learning a hierarchical shape vocabulary for object class detection. BMVC. [22] Agarwal, S., Awan, A., and Roth, D. (2004) Learning to detect objects in images via a sparse, part-based representation. IEEE PAMI, 26, 1475–1490. [23] Shotton, J., Blake, A., and Cipolla, R. (2008) Multi-scale categorical object recognition using contour fragments. PAMI, 30, 1270–1281. [24] Ferrari, V., Fevrier, L., Jurie, F., and Schmid, C. (2007) Accurate object detection with deformable shape models learnt from images. CVPR. [25] Stark, M. and Schiele, B. (2007) How good are local features for classes of geometric objects? ICCV. [26] Fritz, M. and Schiele, B. (2008) Decomposition, discovery and detection of visual categories using topic models. CVPR. [27] Mutch, J. and Lowe, D. G. (2006) Multiclass object recognition with sparse, localized features. CVPR, pp. 11–18. [28] Shotton, J., Blake, A., and Cipolla, R. (2008) Efficiently combining contour and texture cues for object recognition. BMVC. [29] Ahuja, N. and Todorovic, S. (2008) Connected segmentation tree – a joint representation of region layout and hierarchy. 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Optimal Scoring for Unsupervised Learning Zhihua Zhang and Guang Dai College of Computer Science & Technology Zhejiang University Hangzhou, Zhejiang, 310027 China Abstract We are often interested in casting classification and clustering problems as a regression framework, because it is feasible to achieve some statistical properties in this framework by imposing some penalty criteria. In this paper we illustrate optimal scoring, which was originally proposed for performing the Fisher linear discriminant analysis by regression, in the application of unsupervised learning. In particular, we devise a novel clustering algorithm that we call optimal discriminant clustering. We associate our algorithm with the existing unsupervised learning algorithms such as spectral clustering, discriminative clustering and sparse principal component analysis. Experimental results on a collection of benchmark datasets validate the effectiveness of the optimal discriminant clustering algorithm. 1 Introduction The Fisher linear discriminant analysis (LDA) is a classical method that considers dimensionality reduction and classification jointly. LDA estimates a low-dimensional discriminative space defined by linear transformations through maximizing the ratio of between-class scatter to within-class scatter. It is well known that LDA is equivalent to a least mean squared error procedure in the binary classification problem [4]. It is of great interest to obtain a similar relationship in multi-class problems. A significant literature has emerged to address this issue [6, 8, 12, 14]. This provides another approach to performing LDA by regression, in which penalty criteria are tractably introduced to achieve some statistical properties such as regularized LDA [5] and sparse discriminant analysis [2]. It is also desirable to explore unsupervised learning problems in a regression framework. Recently, Zou et al. [17] reformulated principal component analysis (PCA) as a regression problem and then devised a sparse PCA by imposing the lasso (the elastic net) penalty [10, 16] on the regression vector. In this paper we consider unsupervised learning problems by optimal scoring, which was originally proposed to perform LDA by regression [6]. In particular, we devise a novel unsupervised framework by using the optimal scoring and the ridge penalty. This framework can be used for dimensionality reduction and clustering simultaneously. We are mainly concerned with the application in clustering. In particular, we propose a clustering algorithm that we called optimal discriminant clustering (ODC). Moreover, we establish a connection of our clustering algorithm with discriminative clustering algorithms [3, 13] and spectral clustering algorithms [7, 15]. This implies that we can cast these clustering algorithms as regression-type problems. In turn, this facilitates the introduction of penalty terms such as the lasso and elastic net so that we have sparse unsupervised learning algorithms. Throughout this paper, Im denotes the m×m identity matrix, 1m the m×1 vector of ones, 0 the zero vector or matrix with appropriate size, and Hm = Im −1 m1m1′ m the m×m centering matrix. For an m×1 vector a = (a1, . . . , am)′, diag(a) represents the m×m diagonal matrix with a1, . . . , am as its diagonal entries. For an m×m matrix A = [aij], we let A+ be the Moore-Penrose inverse of A, tr(A) be the trace of A, rk(A) be the rank of A and ∥A∥F = p tr(A′A) be the Frobenius norm of A. 1 2 Problem Formulation We are concerned with a multi-class classification problem. Given a set of n p-dimensional data points, {x1, . . . , xn} ∈X ⊂Rp, we assume that the xi are grouped into c disjoint classes and that each xi belongs to one class. Let V = {1, 2, . . . , n} denote the index set of the data points xi and partition V into c disjoint subsets Vj; i.e., Vi ∩Vj = ∅for i ̸= j and ∪c j=1Vj = V , where the cardinality of Vj is nj so that Pc j=1 nj = n. We also make use of a matrix representation for the problem in question. In particular, we let X = [x1, . . . , xn]′ be an n×p data matrix, and E = [eij] be an n×c indicator matrix with eij = 1 if input xi is in class j and eij = 0 otherwise. Let Π = diag(n1, . . . , nc), Π 1 2 = diag(√n1, . . . , √nc), π = (n1, . . . , nc)′ and √π = (√n1, . . . , √nc)′. It follows that 1′ nE = 1′ cΠ = π′, E1c = 1n, 1′ cπ = n, E′E = Π and Π−1π = 1c. 2.1 Scoring Matrices Hastie et al. [6] defined a scoring matrix for the c-class classification problem. That is, it is such a c×(c−1) matrix Θ ∈Rc×(c−1) that Θ′(E′E)Θ = Θ′ΠΘ = Ic−1. The jth row of Θ defines a scoring or scaling for the jth class. Here we refine this definition as: Definition 1 Given a c-class classification problem with the cardinality of the jth class being nj, a c×(c−1) matrix Θ is referred to as the class scoring matrix if it satisfies Θ′ΠΘ = Ic−1 and π′Θ = 0. It follows from this definition that ΘΘ′ = Π−1−1 n1c1′ c. In the literature [15], the authors presented a specific example for Θ = (θ1, . . . , θc−1)′. That is, θ′ 1 = ³ √n−n1 √nn1 , − √n1 √ n(n−n1)1′ c−1 ´ and θ′ l = µ 0 ∗1′ l−1, qPc j=l+1 nj q nl Pc j=l nj , √nl qPc j=l nj Pc j=l+1 nj 1′ c−l ¶ for l = 2, . . . , c−1. Especially, when c = 2, Θ = ( √n2 √nn1 , − √n1 √nn2 )′ is a 2-dimensional vector. Let Y = EΘ (n×(c−1)). We then have Y′Y = Ic−1 and 1′ nY = 0. To address an unsupervised clustering problem with c classes, we relax the setting of Y = EΘ and give the following definition. Definition 2 An n×(c−1) matrix Y is referred to as the sample scoring matrix if it satisfies Y′Y = Ic−1 and 1′ nY = 0. Note that c does not necessarily represent the number of classes in this definition. For example, we view c−1 as the dimension of a reduced dimensional space in the dimensionality reduction problem. 2.2 Optimal Scoring for LDA To devise a classifier for the c-class classification problem, we consider a penalized optimal scoring model, which is defined by min Θ, W n f(Θ, W) ≜1 2∥EΘ −HnXW∥2 F + σ2 2 tr(W′W) o (1) under the constraints Θ′ΠΘ = Ic−1 and π′Θ = 0 where Θ ∈Rc×(c−1) and W ∈Rp×(c−1). Compared with the setting in [6], we add the constraint π′Θ = 0. The reason is due to 1′ nHnXW = 0. We thus impose 1′ nEΘ = π′Θ = 0 for consistency. Denote R = Π−1 2 E′HnX(X′HnX + σ2Ip)−1X′HnEΠ−1 2 . Since Rπ 1 2 = 0, there exists a c×(c−1) orthogonal matrix ∆, the columns of which are the eigenvectors of R. That is, ∆satisfies ∆′∆= Ic−1 and ∆′π 1 2 = 0. 2 Theorem 1 A minimizer of Problem (1) is ˆΘ = Π−1 2 ∆and ˆ W = (X′HnX+σ2Ip)−1X′HnE ˆΘ. Here [∆, 1 √nπ 1 2 ] is the c×c matrix of the orthonormal eigenvectors of R. Since for an arbitrary class scoring matrix Θ, its rank is c−1, we have Θ = ˆΘΥ where Υ is some (c−1)×(c−1) orthonormal matrix. Moreover, it follows from ΘΘ′ = Π−1 −1 n1c1′ c that the between-class scatter matrix is given by Σb = X′HnEΘΘ′E′HnX = X′HnE ˆΘ ˆΘ ′E′HnX. Accordingly, we can also write the generalized eigenproblem for the penalized LDA as X′HnE ˆΘ ˆΘ ′E′HnXA = (X′HnX + σ2Ip)AΛ, because the total scatter matrix Σ is Σ = X′HnX. We now obtain ˆ W ˆΘ ′E′HnXA = AΛ. It is well known that ˆ W ˆΘ ′E′HnX and ˆΘ ′E′HnX ˆ W have the same nonzero eigenvalues. Moreover, ˆΘ ′E′HnXA is the eigenvector matrix of ˆΘ ′E′HnX ˆ W. We thus establish the relationship between A in the penalized LDA and W in the penalized optimal scoring model (1). 3 Optimal Scoring for Unsupervised Learning In this section we extend the notion of optimal scoring to unsupervised learning problems, leading to a new framework for dimensionality reduction and clustering analysis simultaneously. 3.1 Framework In particular, we relax EΘ in (1) as a sample scoring matrix Y and define the following penalized model: min Y, W n f(Y, W) ≜1 2∥Y −HnXW∥2 F + σ2 2 tr(W′W) o (2) under the constraints 1′ nY = 0 and Y′Y = Ic−1. The following theorem provides a solution for this problem. Theorem 2 A minimizer of Problem (2) is ˆY and ˆ W = (X′HnX + σ2Ip)−1X′Hn ˆY, where ˆY is the n×(c−1) orthogonal matrix of the top eigenvectors of HnX(X′HnX + σ2Ip)−1X′Hn. The proof is given in Appendix A. Note that all the eigenvalues of HnX(X′HnX + σ2Ip)−1X′Hn are between 0 and 1. Especially, when σ2 = 0, the eigenvalues are either 1 or 0. In this case, if rk(HnX) ≥c−1, f( ˆY, ˆ W) achieves its minimum 0, otherwise the minimum value is c−1−rk(HnX) 2 . With the estimates of Y and W, we can develop an unsupervised learning procedure. It is clear that W can be treated as a non-orthogonal projection matrix and HnXW is then the low-dimensional configuration of X. Using this treatment, we obtain a new alternative to the regression formulation of PCA by Zou et al. [17]. In this paper, however, we concentrate on the application of the framework in clustering analysis. 3.2 Optimal Discriminant Clustering Our clustering procedure is given in Algorithm 1. We refer to this procedure as optimal discriminant clustering due to its relationship with LDA, which is shown by the connection between (1) and (2). Assume that ˜X = [˜x1, . . . , ˜xn]′ (n×r) is a feature matrix corresponding to the data matrix X. In this case, we have S = Hn ˜X( ˜X′Hn ˜X + σ2Ir)−1 ˜X′Hn = C(C + σ2In)−1, 3 where C = Hn ˜X ˜X′Hn is the n×n centered kernel matrix. This implies that we can obtain ˆY without the explicit use of the feature matrix ˜X. Moreover, we can compute Z by Z = Hn ˜X( ˜X′Hn ˜X + σ2Ir)−1 ˜X′HnY = SY. We are thus able to devise this clustering algorithm by using the reproducing kernel k(·, ·) : X×X →R such that K(xi, xj) = ˜x′ i˜xj and K = ˜X ˜X′. Algorithm 1 Optimal Discriminant Clustering Algorithm 1: procedure ODC(HnX, c, σ2) 2: Estimate ˆY and ˆ W according to Theorem 2; 3: Calculate Z = [z1, . . . , zn]′ = HnX ˆ W; 4: Perform K-means on the zi; 5: Return the partition of the zi as the partition of the xi. 6: end procedure 3.3 Related Work We now explore the connection of the optimal discriminant clustering with the discriminative clustering algorithm [3] and spectral clustering [7]. Recall that ˆY is the matrix of the c−1 top eigenvectors of C(C + σ2In)−1. Consider that if λ ̸= 0 is an eigenvalue of C with associated eigenvector u, then λ/(λ + σ2) (̸= 0) is an eigenvalue of C(C + σ2In)−1 with associated eigenvector u. Moreover, λ/(λ + σ2) is increasing as λ increases. This implies that ˆY is also the matrix of the c−1 top eigenvectors of C. As we know, the spectral clustering applies a rounding scheme such as K-means directly on ˆY. We thus have a relationship between the spectral clustering and optimal discriminant clustering. We study the relationship between the discriminative clustering algorithm and the spectral clustering algorithm. Let M be a linear transformation from the r-dimensional ˜X to an s-dimensional transformed feature space F, namely F = ˜XM, where M is an r×s matrix of rank s (s < r). The corresponding scatter matrices in the F-space are thus given by M′ΣM and M′ΣbM. The discriminative clustering algorithm [3, 13] in the reproducing kernel Hilbert space (RKHS) tries to solve the problem of argmax E, M f(E, M) ≜tr((M′(Σ+σ2Ir)M)−1M′ΣbM) = tr ¡ (M′( ˜X′Hn ˜X+σ2Ir)M)−1M′ ˜X′HnE ¡ E′E ¢−1E′Hn ˜XM ¢ Applying the discussion in [15] to Hn ˜XM(M′( ˜X′Hn ˜X+σ2Ir)M)−1M′ ˜X′Hn, we have the following relaxation problem max Y∈Rn×(c−1),M∈Rr×s tr(Y′Hn ˜XM(M′( ˜X′Hn ˜X+σ2Ir)M)−1M′ ˜X′HnY), s.t. Y′Y = Ic−1 and Y′1n = 0. (3) Express M = ˜X′HnB + N where N satisfies N′ ˜X′Hn = 0 (i.e., N ∈span{ ˜X′Hn}⊥) and B is some n×s matrix. Under the condition of either σ2 = 0 or N = 0 (i.e., M ∈span{ ˜X′Hn}), we can obtain that Hn ˜XM(M′( ˜X′Hn ˜X+σ2Ir)M)−1M′ ˜X′Hn = CB(B′(CC + σ2C)B)−1B′C. Again consider that if λ ̸= 0 is an eigenvalue of C with associated eigenvector u, then λ/(λ+σ2) ̸= 0 is an eigenvalue of C(CC + σ2C)+C with associated eigenvector u. Moreover, λ/(λ + σ2) is increasing in λ. We now directly obtain the following theorem from Theorem 3.1 in [13]. Theorem 3 Let Y∗and M∗be the solution of Problem (3). Then 4 Table 1: Summary of the benchmark datasets, where c is the number of classes, p is the dimension of the input vector, and n is the number of samples in the dataset. Types Dataset c p n Face ORL 40 1024 400 Yale 15 1024 165 PIE 68 1024 6800 Gene SRBCT 4 2308 63 UCI Iris 4 4 150 Yeast 10 8 1484 Image segmentation 7 19 2100 Statlog landsat satellite 7 36 2000 (i) If σ2 = 0, Y∗is the solution of the following problem argmaxY∈Rn×(c−1) tr(Y′CC+Y), s.t. Y′Y = Ic−1 and Y′1n = 0. (ii) If M ∈span{ ˜X′Hn}, Y∗is the solution of the following problem: argmaxY ∈Rn×(c−1) tr(Y′CY), s.t. Y′Y = Ic−1 and Y′1n = 0. Theorem 3 shows that discriminative clustering is essentially equivalent to spectral clustering. This further leads us to a relationship between the discriminative clustering and optimal discriminant clustering from the relationship between the spectral clustering and optimal discriminant clustering. In summary, we are able to unify the discriminative clustering as well as spectral clustering into the optimal scoring framework in (2). 4 Experimental Study To evaluate the performance of our optimal discriminant clustering (ODC) algorithm, we conducted experimental comparisons with other related clustering algorithms on several real-world datasets. In particular, the comparison was implemented on three face datasets, the “SRBCT” gene dataset, and four UCI datasets. Further details of these datasets are summarized in Table 1. To effectively evaluate the performance, we employed two typical measurements: the Normalized Mutual Information (NMI) and the Clustering Error (CE). It should be mentioned that for NMI, the larger this value, the better the performance. For CE, the smaller the value, the better the performance. More details and the corresponding implementations for both can be found in [11]. In the experiments, we compared our ODC with four different clustering algorithms, i.e., the conventional K-means [1], normalized cut (NC) [9], DisCluster [3] and DisKmeans [13]. It is worth noting that two discriminative clustering algorithms: DisCluster [3] and DisKmeans [13], are very closely related to our ODC, because they are derived from the discriminant analysis criteria in essence (also see the analysis in Section 3.3). In addition, the implementation code for NC is available at http://www.cis.upenn.edu/∼jshi/software/. For the sake of simplicity, the parameter σ2 in ODC is sought from the range σ2 ∈ {10−3, 10−2.5, 10−2, 10−1.5, 10−1, 10−0.5, 100, 100.5, 101, 101.5, 102, 102.5, 103}. Similarly, the parameters in other clustering algorithms compared here are also searched in a wide range. For simplicity, we just reported the best results of clustering algorithms with respect to different parameters on each dataset. Table 2 summaries the NMI and CE on all datasets. According to the NMI values in Table 2, our ODC outperforms other clustering algorithms on five datasets: ORL, SRBCT, iris, yeast and image segmentation. According to the CE values in Table 2, it is obvious that the performance of our ODC is best in comparison with other algorithms on all the datasets, and NC and DisKmeans algorithms can achieve the almost same performance with ODC on the SRBCT and iris datasets respectively. Also, it is seen that the DisCluster algorithm has dramatically different performance based on the NMI and CE. The main reason is that the final solution in DisCluster is very sensitive to the initial variables and numerical computation. 5 −6 −4 −2 0 2 4 6 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.86 0.88 2log(σ) NMI K−means ODC (a) −6 −4 −2 0 2 4 6 0.5 0.52 0.54 0.56 0.58 0.6 0.62 0.64 2log(σ) NMI K−means ODC (b) −6 −4 −2 0 2 4 6 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 2log(σ) NMI K−means ODC (c) −6 −4 −2 0 2 4 6 0.2 0.25 0.3 0.35 0.4 2log(σ) NMI K−means ODC (d) −6 −4 −2 0 2 4 6 0.6 0.62 0.64 0.66 0.68 0.7 0.72 0.74 2log(σ) NMI K−means ODC (e) −6 −4 −2 0 2 4 6 0.255 0.26 0.265 0.27 0.275 0.28 0.285 0.29 0.295 0.3 0.305 2log(σ) NMI K−means ODC (f) −6 −4 −2 0 2 4 6 0.54 0.55 0.56 0.57 0.58 0.59 0.6 2log(σ) NMI K−means ODC (g) −6 −4 −2 0 2 4 6 0.57 0.58 0.59 0.6 0.61 0.62 0.63 2log(σ) NMI K−means ODC (h) Figure 1: The NMI versus the parameter σ tuning in ODC on all datasets, where the NMI of Kmeans is used as the baseline: (a) ORL; (b) Yale; (c) PIE; (d) SRBCT; (e) iris; (f) yeast; (g) image segmentation; (h) statlog landsat satellite. In order to reveal the effect of the parameter σ on ODC, Figures 1 and 2 depict the NMI and CE results of ODC with respect to different parameters σ on all datasets. Similar to [11, 13], we used the results of K-means as a baseline. From Figures 1 and 2, we can see that similar to the conventional clustering algorithms (including the compared algorithms), the parameter σ has a significant impact on the performance of ODC, especially when the evaluation results are measured by NMI. In contrast to the result in Figure 1, the effect of the parameter σ becomes less pronounced in Figure 2. Table 2: Clustering results: the Normalized Mutual Information (NMI) and the Clustering Error (CE) (%) of all clustering algorithms are calculated on different datasets. Measure Dataset K-means NC DisCluster DisKmeans ODC NMI ORL 0.7971 0.8015 0.7978 0.8531 0.8567 Yale 0.6237 0.6203 0.5974 0.5641 0.5766 PIE 0.1140 0.2232 0.1940 0.3360 0.3035 SRBCT 0.2509 0.3722 0.3216 0.2683 0.3966 Iris 0.6595 0.6876 0.7248 0.7353 0.7353 Yeast 0.2968 0.2915 0.2993 0.3020 0.3041 Image segmentation 0.5830 0.5500 0.5700 0.5934 0.5942 Statlog landsat satellite 0.6126 0.6316 0.6152 0.6009 0.6166 CE (%) ORL 38.25 34.50 38.75 29.00 28.50 Yale 45.45 46.06 45.45 45.45 44.84 PIE 79.82 79.82 77.35 66.23 65.52 SRBCT 55.55 47.61 50.79 53.96 47.61 Iris 16.66 15.33 12.66 11.33 11.33 Yeast 59.43 59.90 59.43 57.07 56.73 Image segmentation 45.14 49.47 45.95 41.66 40.23 Statlog landsat satellite 32.30 32.65 32.25 31.20 30.50 5 Concluding Remarks In this paper we have proposed a regression framework to deal with unsupervised dimensionality reduction and clustering simultaneously. The framework is based on the optimal scoring and ridge penalty. In particular, we have developed a new clustering algorithm which is called optimal discriminant clustering (ODC). ODC can efficiently identify the optimal solution and it has an underlying relationship with the discriminative clustering and spectral clustering. 6 −6 −4 −2 0 2 4 6 28 30 32 34 36 38 40 2log(σ) CE (%) K−means ODC (a) −6 −4 −2 0 2 4 6 44.8 44.9 45 45.1 45.2 45.3 45.4 45.5 45.6 CE (%) 2log(σ) K−means ODC (b) −6 −4 −2 0 2 4 6 64 66 68 70 72 74 76 78 80 82 2log(σ) CE (%) K−means ODC (c) −6 −4 −2 0 2 4 6 47 48 49 50 51 52 53 54 55 56 2log(σ) CE (%) K−means ODC (d) −6 −4 −2 0 2 4 6 11 12 13 14 15 16 17 2log(σ) CE (%) K−means ODC (e) −6 −4 −2 0 2 4 6 56.5 57 57.5 58 58.5 59 59.5 2log(σ) CE (%) K−means ODC (f) −6 −4 −2 0 2 4 6 40 41 42 43 44 45 46 2log(σ) CE (%) K−means ODC (g) −6 −4 −2 0 2 4 6 30.5 31 31.5 32 32.5 2log(σ) CE (%) K−means ODC (h) Figure 2: The CE (%) versus the parameter σ tuning in ODC on all datasets, where the CE (%) of K-means is used as the baseline: (a) ORL; (b) Yale; (c) PIE; (d) SRBCT; (e) iris; (f) yeast; (g) image segmentation; (h) statlog landsat satellite. This framework allows us for developing a sparse unsupervised learning algorithm; that is, we alternatively consider the following optimization problem: min Y, W f(Y, W) = 1 2∥Y −HnXW∥2 F + λ1 2 tr(W′W) + λ2∥W∥1 under the constraints 1′ nY = 0 and Y′Y = Ic−1. We will study this further. Acknowledgement This work has been supported in part by program for Changjiang Scholars and Innovative Research Team in University (IRT0652, PCSIRT), China. A Proof of Theorem 2 For simplicity, we replace HnX by X and let q = c−1 in the following derivation. Consider the Lagrange function: L(Y, W, B, b) = 1 2tr(Y′Y) −tr(Y′XW) + 1 2tr(W′(X′X+σ2Ip)W) −1 2tr(B(Y′Y−Iq)) −tr(b′Y′1n), where B is a q×q symmetric matrix of Lagrange multipliers and b is a q×1 vector of Lagrange multipliers. By direct differentiation, it can be shown that ∂L ∂Y = Y −XW −YB −1nb′, ∂L ∂W = (X′X + σ2Ip)W −X′Y. Letting ∂L ∂Y = 0, we have Y −XW −YB −1nb′ = 0. Pre-multiplying both sides of the above equation by 1′ n, we obtain b = 0. Thus, it follows from ∂L ∂Y = 0 and ∂L ∂W = 0 that ½ Y −XW −YB = 0, W = (X′X + σ2Ip)−1X′Y. 7 Substituting the second equation into the first equation, we further have (In −X(X′X + σ2Ip)−1X′)Y = YB. Now we take the spectral decomposition of B as B = UBΛBU′ B where UB is a q×q orthonormal matrix and ΛB is a q×q diagonal matrix. We thus have (In −X(X′X + σ2Ip)−1X′)YUB = YUBΛB. This shows that the diagonal entries of ΛB and the columns of YUB are the eigenvalues and the associated eigenvectors of In −X(X′X + σ2Ip)−1X′. We consider the case that n ≥p. Let the SVD of X be X = UΓV′ where U (n×p) and V (p×p) are orthogonal, and Γ = diag(γ1, . . . , γp) (p×p) is a diagonal matrix with γ1 ≥γ2 ≥· · · ≥γp ≥0. We then have X(X′X + σ2Ip)−1X′ = UΛU′, where Λ = diag(λ1, . . . , λp) with λi = γ2 i /(γ2 i + σ2). There exists such an n×(n−p) orthogonal matrix U3 that its last column is 1 √n1n and [U, U3] is an n×n orthonormal matrix. That is, U3 is the eigenvector matrix of X(X′X + σ2Ip)−1X′ corresponding to the eigenvalue 0. Let U1 be the n×q matrix of the first q columns of [U, U3]. We now define ˆY = U1, ˆ W = (X′X + σ2Ip)−1X′U1, UB = Iq and ΛB = diag(1 −λ1, . . . , 1 − λq) where λi = 0 whenever i > p. It is easily seen that such a ˆY satisfies ˆY′ ˆY = Iq and ˆY′1n = 0 due to U′ 1U′ 1 = Iq and X′1n = 0. Moreover, we have f( ˆY, ˆ W) = q 2 −1 2 q X i=1 λi = q 2 −1 2 q X i=1 γ2 i γ2 i + σ2 where γi = 0 whenever i > p. Note that all the eigenvalues of X(X′X + σ2Ip)−1X′ are between 0 and 1. Especially, when σ2 = 0, the eigenvalues are either 1 or 0. In this case, if rk(X) ≥q, f( ˆY, ˆ W) achieves its minimum 0, otherwise the minimum value is q−rk(X) 2 . To verify that ( ˆY, ˆ W) is a minimizer of problem (2), we consider the Hessian matrix of L with respect to (Y, W). Let vec(Y′) = (y11, . . . , y1q, y21, . . . , ynq)′ and vec(W′) = (w11, . . . , w1q, w21, . . . , wpq)′. The Hessian matrix is then given by H(Y, W) = " ∂2L ∂vec(Y′)∂vec(Y′)′ ∂2L ∂vec(Y′)∂vec(W′)′ ∂2L ∂vec(W′)∂vec(Y′)′ ∂2L ∂vec(W′)∂vec(W′)′ # = · (Iq−B)⊗In −Iq⊗X −Iq⊗X′ Iq⊗(X′X + σ2Ip) ¸ . Let C′ = [C′ 1, C′ 2], where C1 and C2 are n×q and p×q, be an arbitrary nonzero (n+p)×q matrix such that C′ 1[1n, ˆY] = 0, which is equivalent to C′ 11n = 0 and C′ 1U1 = 0. If rk(X) ≤q, we have C′ 1X = 0. Hence, vec(C′)′H( ˆY, ˆ W)vec(C′) = tr(C′ 1C1(Iq −B)) −2tr(C′ 1XC2) + tr(C′ 2(X′X + σ2Ip)C2) = tr(C′ 1C1(Iq −B)) + tr(C′ 2(X′X + σ2Ip)C2) ≥0. This implies that ( ˆY, ˆ W) is a minimizer of problem (2). In the case that rk(X) = m > q, we have p > q. Thus we can partition U and V into U = [U1, U2] and V = [V1, V2] where V1 and V2 are p×q and p×(p−q). Thus, vec(C′)′H( ˆY, ˆ W)vec(C′) = tr(C′ 1C1(Iq −B)) −2tr(C′ 1XC2) + tr(C′ 2(X′X + σ2Ip)C2) ≥ tr(C′ 1U2Λ2U′ 2C1)−2tr(C′ 1U2Γ2V′ 2C2)+tr(C′ 2V2D2V′ 2C2) +tr(C′ 1U3U′ 3C1Λ1) + tr(C′ 2V1D1V′ 1C2) = tr £ (Λ1/2 2 U′ 2C1 −D1/2 2 V′ 2C2)′(Λ1/2 2 U′ 2C1 −D1/2 2 V′ 2C2) ¤ +tr(C′ 1U3U′ 3C1Λ1) + tr(C′ 2V1D1V′ 1C2) ≥0. Here Λ1 = diag(λ1, . . . , λq), Λ2 = diag(λq+1, . . . , λp), Γ1 = diag(γ1, . . . , γq), Γ2 = diag(γq+1, . . . , γp), D1 = Γ2 1 + σ2Iq and D2 = Γ2 2 + σ2Ip−q, so we have Γ2 = D1/2 2 Λ1/2 2 . Moreover, we use the fact that tr(C′ 1U2U′ 2C1Λ1) ≥tr(C′ 1U2Λ2U′ 2C1) because λiIq −Λ2 for i = 1, . . . , q are positive semidefinite. If n < p, we also make the SVD of X as X = UΓV′. But, right now, U is n×n, V is n×p, and Λ is n×n. Using this SVD, we have the same result as the case of n ≥p. 8 References [1] C. M. Bishop. Pattern Recognition and Machine Learning. Springer, first edition, 2007. [2] L. Clemmensen, T. Hastie, and B. Erbøll. Sparse discriminant analysis. Technical report, June 2008. [3] F. De la Torre and T. Kanade. Discriminative cluster analysis. In The 23rd International Conference on Machine Learning, 2006. [4] R. O. Duda, P. E. Hart, and D. G. Stork. Pattern Classification. John Wiley and Sons, New York, second edition, 2001. [5] T. Hastie, A. Buja, and R. Tibshirani. Penalized discriminant analysis. The Annals of Statistics, 23(1):73–102, 1995. [6] T. Hastie, R. Tibshirani, and A. Buja. Flexible discriminant analysis by optimal scoring. Journal of the American Statistical Association, 89(428):1255–1270, 1994. [7] A. Y. Ng, M. I. Jordan, and Y. Weiss. On spectral clustering: analysis and an algorithm. In Advances in Neural Information Processing Systems 14, volume 14, 2002. [8] C. H. Park and H. Park. A relationship between linear discriminant analysis and the generalized minimum squared error solution. SIAM Journal on Matrix Analysis and Applications, 27(2):474–492, 2005. [9] J. Shi and J. Malik. Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8):888–905, 2000. [10] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58:267–288, 1996. [11] M. Wu and B. Sch¨olkopf. A local learning approach for clustering. In Advances in Neural Information Processing Systems 19, 2007. [12] J. Ye. Least squares linear discriminant analysis. In The Twenty-Fourth International Conference on Machine Learning, 2007. [13] J. Ye, Z. Zhao, and M. Wu. Discriminative k-means for clustering. In Advances in Neural Information Processing Systems 20, 2008. [14] Z. Zhang, G. Dai, and M. I. Jordan. A flexible and efficient algorithm for regularized Fisher discriminant analysis. In The European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD), 2009. [15] Z. Zhang and M. I. Jordan. Multiway spectral clustering: A margin-based perspective. Statistical Science, 23(3):383–403, 2008. [16] H. Zou and T. Hastie. Regularization and variable selection via the elastic net. Journal of the Royal Statistical Society, Series B, 67:301–320, 2005. [17] H. Zou, T. Hastie, and R. Tibshirani. Sparse principal component analysis. Journal of Computational and Graphical Statistics, 15:265–286, 2006. 9
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Matrix Completion from Noisy Entries Raghunandan H. Keshavan∗, Andrea Montanari∗†, and Sewoong Oh∗ Abstract Given a matrix M of low-rank, we consider the problem of reconstructing it from noisy observations of a small, random subset of its entries. The problem arises in a variety of applications, from collaborative filtering (the ‘Netflix problem’) to structure-from-motion and positioning. We study a low complexity algorithm introduced in [1], based on a combination of spectral techniques and manifold optimization, that we call here OPTSPACE. We prove performance guarantees that are order-optimal in a number of circumstances. 1 Introduction Spectral techniques are an authentic workhorse in machine learning, statistics, numerical analysis, and signal processing. Given a matrix M, its largest singular values –and the associated singular vectors– ‘explain’ the most significant correlations in the underlying data source. A low-rank approximation of M can further be used for low-complexity implementations of a number of linear algebra algorithms [2]. In many practical circumstances we have access only to a sparse subset of the entries of an m × n matrix M. It has recently been discovered that, if the matrix M has rank r, and unless it is too ‘structured’, a small random subset of its entries allow to reconstruct it exactly. This result was first proved by Cand´es and Recht [3] by analyzing a convex relaxation indroduced by Fazel [4]. A tighter analysis of the same convex relaxation was carried out in [5]. A number of iterative schemes to solve the convex optimization problem appeared soon thereafter [6, 7, 8] (also see [9] for a generalization). In an alternative line of work, the authors of [1] attacked the same problem using a combination of spectral techniques and manifold optimization: we will refer to their algorithm as OPTSPACE. OPTSPACE is intrinsically of low complexity, the most complex operation being computing r singular values and the corresponding singular vectors of a sparse m × n matrix. The performance guarantees proved in [1] are comparable with the information theoretic lower bound: roughly nr max{r, log n} random entries are needed to reconstruct M exactly (here we assume m of order n). A related approach was also developed in [10], although without performance guarantees for matrix completion. The above results crucially rely on the assumption that M is exactly a rank r matrix. For many applications of interest, this assumption is unrealistic and it is therefore important to investigate their robustness. Can the above approaches be generalized when the underlying data is ‘well approximated’ by a rank r matrix? This question was addressed in [11] within the convex relaxation approach of [3]. The present paper proves a similar robustness result for OPTSPACE. Remarkably the guarantees we obtain are order-optimal in a variety of circumstances, and improve over the analogus results of [11]. ∗Department of Electrical Engineering, Stanford University †Departments of Statistics, Stanford University 1 1.1 Model definition Let M be an m × n matrix of rank r, that is M = UΣV T . (1) where U has dimensions m × r, V has dimensions n × r, and Σ is a diagonal r × r matrix. We assume that each entry of M is perturbed, thus producing an ‘approximately’ low-rank matrix N, with Nij = Mij + Zij , (2) where the matrix Z will be assumed to be ‘small’ in an appropriate sense. Out of the m × n entries of N, a subset E ⊆[m] × [n] is revealed. We let N E be the m × n matrix that contains the revealed entries of N, and is filled with 0’s in the other positions N E ij = Nij if (i, j) ∈E , 0 otherwise. (3) The set E will be uniformly random given its size |E|. 1.2 Algorithm For the reader’s convenience, we recall the algorithm introduced in [1], which we will analyze here. The basic idea is to minimize the cost function F(X, Y ), defined by F(X, Y ) ≡ min S∈Rr×r F(X, Y, S) , (4) F(X, Y, S) ≡ 1 2 X (i,j)∈E (Nij −(XSY T )ij)2 . (5) Here X ∈Rn×r, Y ∈Rm×r are orthogonal matrices, normalized by XT X = m1, Y T Y = n1. Minimizing F(X, Y ) is an a priori difficult task, since F is a non-convex function. The key insight is that the singular value decomposition (SVD) of N E provides an excellent initial guess, and that the minimum can be found with high probability by standard gradient descent after this initialization. Two caveats must be added to this decription: (1) In general the matrix N E must be ‘trimmed’ to eliminate over-represented rows and columns; (2) For technical reasons, we consider a slightly modified cost function to be denoted by eF(X, Y ). OPTSPACE( matrix N E ) 1: Trim N E, and let e N E be the output; 2: Compute the rank-r projection of e N E, Tr( e N E) = X0S0Y T 0 ; 3: Minimize eF(X, Y ) through gradient descent, with initial condition (X0, Y0). We may note here that the rank of the matrix M, if not known, can be reliably estimated from e N E. We refer to the journal version of this paper for further details. The various steps of the above algorithm are defined as follows. Trimming. We say that a row is ‘over-represented’ if it contains more than 2|E|/m revealed entries (i.e. more than twice the average number of revealed entries). Analogously, a column is overrepresented if it contains more than 2|E|/n revealed entries. The trimmed matrix e N E is obtained from N E by setting to 0 over-represented rows and columns. f M E and eZE are defined similarly. Hence, e N E = f M E + eZE. Rank-r projection. Let e N E = min(m,n) X i=1 σixiyT i , (6) 2 be the singular value decomposition of e N E, with singular vectors σ1 ≥σ2 ≥. . . . We then define Tr( e N E) = mn |E| r X i=1 σixiyT i . (7) Apart from an overall normalization, Tr( e N E) is the best rank-r approximation to e N E in Frobenius norm. Minimization. The modified cost function eF is defined as eF(X, Y ) = F(X, Y ) + ρ G(X, Y ) (8) ≡ F(X, Y ) + ρ m X i=1 G1 ||X(i)||2 3µ0r + ρ n X j=1 G1 ||Y (j)||2 3µ0r , (9) where X(i) denotes the i-th row of X, and Y (j) the j-th row of Y . See Section 1.3 below for the definition of µ0. The function G1 : R+ →R is such that G1(z) = 0 if z ≤1 and G1(z) = e(z−1)2 −1 otherwise. Further, we can choose ρ = Θ(nǫ). Let us stress that the regularization term is mainly introduced for our proof technique to work (and a broad family of functions G1 would work as well). In numerical experiments we did not find any performance loss in setting ρ = 0. One important feature of OPTSPACE is that F(X, Y ) and eF(X, Y ) are regarded as functions of the r-dimensional subspaces of Rm and Rn generated (respectively) by the columns of X and Y . This interpretation is justified by the fact that F(X, Y ) = F(XA, Y B) for any two orthogonal matrices A, B ∈Rr×r (the same property holds for eF). The set of r dimensional subspaces of Rm is a differentiable Riemannian manifold G(m, r) (the Grassman manifold). The gradient descent algorithm is applied to the function eF : M(m, n) ≡G(m, r) × G(n, r) →R. For further details on optimization by gradient descent on matrix manifolds we refer to [12, 13]. 1.3 Main results Our first result shows that, in great generality, the rank-r projection of e N E provides a reasonable approximation of M. Throughout this paper, without loss of generality, we assume α ≡m/n ≥1. Theorem 1.1. Let N = M + Z, where M has rank r and |Mij| ≤Mmax for all (i, j) ∈[m] × [n], and assume that the subset of revealed entries E ⊆[m] × [n] is uniformly random with size |E|. Then there exists numerical constants C and C′ such that 1 √mn||M −Tr( e N E)||F ≤CMmax nrα3/2 |E| 1/2 + C′ n√rα |E| || eZE||2 , (10) with probability larger than 1 −1/n3. Projection onto rank-r matrices through SVD is pretty standard (although trimming is crucial for achieving the above guarantee). The key point here is that a much better approximation is obtained by minimizing the cost eF(X, Y ) (step 3 in the pseudocode above), provided M satisfies an appropriate incoherence condition. Let M = UΣV T be a low rank matrix, and assume, without loss of generality, U T U = m1 and V T V = n1. We say that M is (µ0, µ1)-incoherent if the following conditions hold. A1. For all i ∈[m], j ∈[n] we have, Pr k=1 U 2 ik ≤µ0r, Pr k=1 V 2 ik ≤µ0r. A2. There exists µ1 such that | Pr k=1 Uik(Σk/Σ1)Vjk| ≤µ1r1/2. Theorem 1.2. Let N = M + Z, where M is a (µ0, µ1)-incoherent matrix of rank r, and assume that the subset of revealed entries E ⊆[m] × [n] is uniformly random with size |E|. Further, let Σmin = Σ1 ≤· · · ≤Σr = Σmax with Σmax/Σmin ≡κ. Let c M be the output of OPTSPACE on input N E. Then there exists numerical constants C and C′ such that if |E| ≥ Cn√ακ2 max µ0r√α log n ; µ2 0r2ακ4 ; µ2 1r2ακ4 , (11) 3 then, with probability at least 1 −1/n3, 1 √mn ||c M −M||F ≤C′ κ2 n√αr |E| ||ZE||2 . (12) provided that the right-hand side is smaller than Σmin. Apart from capturing the effect of additive noise, these two theorems improve over the work of [1] even in the noiseless case. Indeed they provide quantitative bounds in finite dimensions, while the results of [1] were only asymptotic. 1.4 Noise models In order to make sense of the above results, it is convenient to consider a couple of simple models for the noise matrix Z: Independent entries model. We assume that Z’s entries are independent random variables, with zero mean E{Zij} = 0 and sub-gaussian tails. The latter means that P{|Zij| ≥x} ≤2 e−x2 2σ2 , (13) for some bounded constant σ2. Worst case model. In this model Z is arbitrary, but we have an uniform bound on the size of its entries: |Zij| ≤Zmax. The basic parameter entering our main results is the operator norm of eZE, which is bounded as follows. Theorem 1.3. If Z is a random matrix drawn according to the independent entries model, then there is a constant C such that, || eZE||2 ≤Cσ √α|E| log |E| n 1/2 , (14) with probability at least 1 −1/n3. If Z is a matrix from the worst case model, then || eZE||2 ≤2|E| n√α Zmax , (15) for any realization of E. Note that for |E| = Ω(n log n) , no row or column is over-represented with high probability. It follows that in the regime of |E| for which the conditions of Theorem 1.2 are satisfied, we have ZE = eZE. Then, among the other things, this result implies that for the independent entries model the right-hand side of our error estimate, Eq. (12), is with high probability smaller than Σmin, if |E| ≥Crα3/2n log n κ4(σ/Σmin)2. For the worst case model, the same statement is true if Zmax ≤ Σmin/C√rκ2. Due to space constraints, the proof of Theorem 1.3 will be given in the journal version of this paper. 1.5 Comparison with related work Let us begin by mentioning that a statement analogous to our preliminary Theorem 1.1 was proved in [14]. Our result however applies to any number of revealed entries, while the one of [14] requires |E| ≥(8 log n)4n (which for n ≤5 · 108 is larger than n2). As for Theorem 1.2, we will mainly compare our algorithm with the convex relaxation approach recently analyzed in [11]. Our basic setting is indeed the same, while the algorithms are rather different. 4 0 0.2 0.4 0.6 0.8 1 0 100 200 300 400 500 600 Convex Relaxation Lower Bound rank-r projection OptSpace : 1 iteration 2 iterations 3 iterations 10 iterations |E|/n RMSE Figure 1: Root mean square error achieved by OPTSPACE for reconstructing a random rank-2 matrix, as a function of the number of observed entries |E|, and of the number of line minimizations. The performance of nuclear norm minimization and an information theory lower bound are also shown. Figure 1 compares the average root mean square error for the two algorithms as a function of |E|. Here M is a random rank r = 2 matrix of dimension m = n = 600, generated by letting M = eU eV T with eUij, eVij i.i.d. N(0, 20/√n). The noise is distributed according to the independent entries model with Zij ∼N(0, 1). This example is taken from [11] Figure 2, from which we took the data for the convex relaxation approach, as well as the information theory lower bound. After one iteration, OPTSPACE has a smaller root mean square error than [11], and in about 10 iterations it becomes indistiguishable from the information theory lower bound. Next let us compare our main result with the performance guarantee in [11], Theorem 7. Let us stress that we require some bound on the condition number κ, while the analysis of [11, 5] requires a stronger incoherence assumption. As far as the error bound is concerned, [11] proved 1 √mn ||c M −M||F ≤7 r n |E| ||ZE||F + 2 n√α ||ZE||F . (16) (The constant in front of the first term is in fact slightly smaller than 7 in [11], but in any case larger than 4 √ 2). Theorem 1.2 improves over this result in several respects: (1) We do not have the second term on the right hand side of (16), that actually increases with the number of observed entries; (2) Our error decreases as n/|E| rather than (n/|E|)1/2; (3) The noise enters Theorem 1.2 through the operator norm ||ZE||2 instead of its Frobenius norm ||ZE||F ≥||ZE||2. For E uniformly random, one expects ||ZE||F to be roughly of order ||ZE||2 √n. For instance, within the intependent entries model with bounded variance σ, ||ZE||F = Θ( p |E|) while ||ZE||2 is of order p |E|/n (up to logarithmic terms). 2 Some notations The matrix M to be reconstructed takes the form (1) where U ∈Rm×r, V ∈Rn×r. We write U = [u1, u2, . . . , ur] and V = [v1, v2, . . . , vr] for the columns of the two factors, with ||ui|| = √m, ||vi|| = √n, and uT i uj = 0, vT i vj = 0 for i ̸= j (there is no loss of generality in this, since normalizations can be absorbed by redefining Σ). 5 We shall write Σ = diag(Σ1, . . . , Σr) with Σ1 ≥Σ2 ≥· · · ≥Σr > 0. The maximum and minimum singular values will also be denoted by Σmax = Σ1 and Σmin = Σr. Further, the maximum size of an entry of M is Mmax ≡maxij |Mij|. Probability is taken with respect to the uniformly random subset E ⊆[m]×[n] given |E| and (eventually) the noise matrix Z. Define ǫ ≡|E|/√mn. In the case when m = n, ǫ corresponds to the average number of revealed entries per row or column. Then it is convenient to work with a model in which each entry is revealed independently with probability ǫ/√mn. Since, with high probability |E| ∈[ǫ√α n −A√n log n, ǫ√α n + A√n log n], any guarantee on the algorithm performances that holds within one model, holds within the other model as well if we allow for a vanishing shift in ǫ. We will use C, C′ etc. to denote universal numerical constants. Given a vector x ∈Rn, ||x|| will denote its Euclidean norm. For a matrix X ∈Rn×n′, ||X||F is its Frobenius norm, and ||X||2 its operator norm (i.e. ||X||2 = supu̸=0 ||Xu||/||u||). The standard scalar product between vectors or matrices will sometimes be indicated by ⟨x, y⟩or ⟨X, Y ⟩, respectively. Finally, we use the standard combinatorics notation [N] = {1, 2, . . . , N} to denote the set of first N integers. 3 Proof of Theorem 1.1 As explained in the introduction, the crucial idea is to consider the singular value decomposition of the trimmed matrix e N E instead of the original matrix N E. Apart from a trivial rescaling, these singular values are close to the ones of the original matrix M. Lemma 3.1. There exists a numerical constant C such that, with probability greater than 1−1/n3, σq ǫ −Σq ≤CMmax rα ǫ + 1 ǫ || eZE||2 , (17) where it is understood that Σq = 0 for q > r. Proof. For any matrix A, let σq(A) denote the qth singular value of A. Then, σq(A+B) ≤σq(A)+ σ1(B), whence σq ǫ −Σq ≤ σq(f M E)/ǫ −Σq + σ1( eZE)/ǫ ≤ CMmax rα ǫ + 1 ǫ || eZE||2 , where the second inequality follows from the following Lemma as shown in [1]. Lemma 3.2 (Keshavan, Montanari, Oh, 2009 [1]). There exists a numerical constant C such that, with probability larger than 1 −1/n3, 1 √mn M − √mn ǫ f M E 2 ≤CMmax rα ǫ . (18) We will now prove Theorem 1.1. Proof. (Theorem 1.1) For any matrix A of rank at most 2r, ||A||F ≤ √ 2r||A||2, whence 1 √mn||M −Tr( e N E)||F ≤ √ 2r √mn M − √mn ǫ e N E − X i≥r+1 σixiyT i 2 ≤ √ 2r √mn M − √mn ǫ f M E 2 + √mn ǫ || eZE||2 + √mn ǫ σr+1 ≤ 2CMmax p 2αr/ǫ + (2 √ 2r/ǫ) || eZE||2 ≤ C′Mmax nrα3/2 |E| 1/2 + 2 √ 2 n√rα |E| || eZE||2 . This proves our claim. 6 4 Proof of Theorem 1.2 Recall that the cost function is defined over the Riemannian manifold M(m, n) ≡G(m, r)×G(n, r). The proof of Theorem 1.2 consists in controlling the behavior of F in a neighborhood of u = (U, V ) (the point corresponding to the matrix M to be reconstructed). Throughout the proof we let K(µ) be the set of matrix couples (X, Y ) ∈Rm×r × Rn×r such that ||X(i)||2 ≤µr, ||Y (j)||2 ≤µr for all i, j 4.1 Preliminary remarks and definitions Given x1 = (X1, Y1) and x2 = (X2, Y2) ∈M(m, n), two points on this manifold, their distance is defined as d(x1, x2) = p d(X1, X2)2 + d(Y1, Y2)2, where, letting (cos θ1, . . . , cos θr) be the singular values of XT 1 X2/m, d(X1, X2) = ||θ||2 . (19) Given S achieving the minimum in Eq. (4), it is also convenient to introduce the notations d−(x, u) ≡ q Σ2 mind(x, u)2 + ||S −Σ||2 F , (20) d+(x, u) ≡ q Σ2maxd(x, u)2 + ||S −Σ||2 F . (21) 4.2 Auxiliary lemmas and proof of Theorem 1.2 The proof is based on the following two lemmas that generalize and sharpen analogous bounds in [1] (for proofs we refer to the journal version of this paper). Lemma 4.1. There exists numerical constants C0, C1, C2 such that the following happens. Assume ǫ ≥C0µ0r√α max{ log n ; µ0r√α(Σmin/Σmax)4 } and δ ≤Σmin/(C0Σmax). Then, F(x) −F(u) ≥ C1nǫ√α d−(x, u)2 −C1n√rα||ZE||2d+(x, u) , (22) F(x) −F(u) ≤ C2nǫ√α Σ2 max d(x, u)2 + C2n√rα||ZE||2d+(x, u) , (23) for all x ∈M(m, n) ∩K(4µ0) such that d(x, u) ≤δ, with probability at least 1 −1/n4. Here S ∈Rr×r is the matrix realizing the minimum in Eq. (4). Corollary 4.2. There exist a constant C such that, under the hypotheses of Lemma 4.1 ||S −Σ||F ≤CΣmaxd(x, u) + C √r ǫ ||ZE||2 . (24) Further, for an appropriate choice of the constants in Lemma 4.1, we have σmax(S) ≤2Σmax + C √r ǫ ||ZE||2 , (25) σmin(S) ≥1 2Σmin −C √r ǫ ||ZE||2 . (26) Lemma 4.3. There exists numerical constants C0, C1, C2 such that the following happens. Assume ǫ ≥C0µ0r√α (Σmax/Σmin)2 max{ log n ; µ0r√α(Σmax/Σmin)4 } and δ ≤Σmin/(C0Σmax). Then, ||grad eF(x)||2 ≥C1 nǫ2 Σ4 min d(x, u) −C2 √rΣmax ǫΣmin ||ZE||2 Σmin 2 + , (27) for all x ∈M(m, n) ∩K(4µ0) such that d(x, u) ≤δ, with probability at least 1 −1/n4. (Here [a]+ ≡max(a, 0).) We can now turn to the proof of our main theorem. 7 Proof. (Theorem 1.2). Let δ = Σmin/C0Σmax with C0 large enough so that the hypotheses of Lemmas 4.1 and 4.3 are verified. Call {xk}k≥0 the sequence of pairs (Xk, Yk) ∈M(m, n) generated by gradient descent. By assumption, the following is true with a large enough constant C: ||ZE||2 ≤ ǫ C√r Σmin Σmax 2 Σmin . (28) Further, by using Corollary 4.2 in Eqs. (22) and (23) we get F(x) −F(u) ≥ C1nǫ√αΣ2 min d(x, u)2 −δ2 0,− , (29) F(x) −F(u) ≤ C2nǫ√αΣ2 max d(x, u)2 + δ2 0,+ , (30) where δ0,−≡C √rΣmax ǫΣmin ||ZE||2 Σmin , δ0,+ ≡C √rΣmax ǫΣmin ||ZE||2 Σmax . (31) By Eq. (28), we can assume δ0,+ ≤δ0,−≤δ/10. For ǫ ≥Cαµ2 1r2(Σmax/Σmin)4 as per our assumptions, using Eq. (28) in Theorem 1.1, together with the bound d(u, x0) ≤||M −X0SY T 0 ||F /n√αΣmin, we get d(u, x0) ≤δ 10 . (32) We make the following claims : 1. xk ∈K(4µ0) for all k. Indeed without loss of generality we can assume x0 ∈K(3µ0) (because otherwise we can rescale those lines of X0, Y0 that violate the constraint). Therefore eF(x0) = F(x0) ≤ 4C2nǫ√αΣ2 maxδ2/100. On the other hand eF(x) ≥ρ(e1/9 −1) for x ̸∈K(4µ0). Since eF(xk) is a non-increasing sequence, the thesis follows provided we take ρ ≥ C2nǫ√αΣ2 min. 2. d(xk, u) ≤δ/10 for all k. Assuming ǫ ≥Cαµ2 1r2(Σmax/Σmin)6, we have d(x0, u)2 ≤(Σ2 min/C′Σ2 max)(δ/10)2. Also assuming Eq. (28) with large enough C we can show the following. For all xk such that d(xk, u) ∈[δ/10, δ], we have eF(x) ≥F(x) ≥F(x0). This contradicts the monotonicity of eF(x), and thus proves the claim. Since the cost function is twice differentiable, and because of the above, the sequence {xk} converges to Ω= x ∈K(4µ0) ∩M(m, n) : d(x, u) ≤δ , grad eF(x) = 0 . (33) By Lemma 4.3 for any x ∈Ω, d(x, u) ≤C √rΣmax ǫΣmin ||ZE||2 Σmin (34) which implies the thesis using Corollary 4.2. Acknowledgements This work was partially supported by a Terman fellowship, the NSF CAREER award CCF-0743978 and the NSF grant DMS-0806211. 8 References [1] R. H. Keshavan, A. Montanari, and S. Oh. Matrix completion from a few entries. arXiv:0901.3150, January 2009. [2] A. Frieze, R. Kannan, and S. Vempala. Fast monte-carlo algorithms for finding low-rank approximations. J. ACM, 51(6):1025–1041, 2004. [3] E. J. Cand`es and B. Recht. Exact matrix completion via convex optimization. arxiv:0805.4471, 2008. [4] M. Fazel. Matrix Rank Minimization with Applications. PhD thesis, Stanford University, 2002. [5] E. J. Cand`es and T. Tao. The power of convex relaxation: Near-optimal matrix completion. arXiv:0903.1476, 2009. [6] J-F Cai, E. J. Cand`es, and Z. Shen. A singular value thresholding algorithm for matrix completion. arXiv:0810.3286, 2008. [7] S. Ma, D. Goldfarb, and L. Chen. Fixed point and Bregman iterative methods for matrix rank minimization. arXiv:0905.1643, 2009. [8] K. Toh and S. Yun. An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. http://www.math.nus.edu.sg/∼matys, 2009. [9] J. Wright, A. Ganesh, S. Rao, and Y. Ma. Robust principal component analysis: Exact recovery of corrupted low-rank matrices. arXiv:0905.0233, 2009. [10] K. Lee and Y. Bresler. Admira: Atomic decomposition for minimum rank approximation. arXiv:0905.0044, 2009. [11] E. J. Cand`es and Y. Plan. Matrix completion with noise. arXiv:0903.3131, 2009. [12] A. Edelman, T. A. Arias, and S. T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matr. Anal. Appl., 20:303–353, 1999. [13] P.-A. Absil, R. Mahony, and R. Sepulchrer. Optimization Algorithms on Matrix Manifolds. Princeton University Press, 2008. [14] D. Achlioptas and F. McSherry. Fast computation of low-rank matrix approximations. J. ACM, 54(2):9, 2007. 9
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Rethinking LDA: Why Priors Matter Hanna M. Wallach David Mimno Andrew McCallum Department of Computer Science University of Massachusetts Amherst Amherst, MA 01003 {wallach,mimno,mccallum}@cs.umass.edu Abstract Implementations of topic models typically use symmetric Dirichlet priors with fixed concentration parameters, with the implicit assumption that such “smoothing parameters” have little practical effect. In this paper, we explore several classes of structured priors for topic models. We find that an asymmetric Dirichlet prior over the document–topic distributions has substantial advantages over a symmetric prior, while an asymmetric prior over the topic–word distributions provides no real benefit. Approximation of this prior structure through simple, efficient hyperparameter optimization steps is sufficient to achieve these performance gains. The prior structure we advocate substantially increases the robustness of topic models to variations in the number of topics and to the highly skewed word frequency distributions common in natural language. Since this prior structure can be implemented using efficient algorithms that add negligible cost beyond standard inference techniques, we recommend it as a new standard for topic modeling. 1 Introduction Topic models such as latent Dirichlet allocation (LDA) [3] have been recognized as useful tools for analyzing large, unstructured collections of documents. There is a significant body of work applying LDA to an wide variety of tasks including analysis of news articles [14], study of the history of scientific ideas [2, 9], topic-based search interfaces1 and navigation tools for digital libraries [12]. In practice, users of topic models are typically faced with two immediate problems: First, extremely common words tend to dominate all topics. Second, there is relatively little guidance available on how to set T, the number of topics, or studies regarding the effects of using a suboptimal setting for T. Standard practice is to remove “stop words” before modeling using a manually constructed, corpus-specific stop word list and to optimize T by either analyzing probabilities of held-out documents or resorting to a more complicated nonparametric model. Additionally, there has been relatively little work in the machine learning literature on the structure of the prior distributions used in LDA: most researchers simply use symmetric Dirichlet priors with heuristically set concentration parameters. Asuncion et al. [1] recently advocated inferring the concentration parameters of these symmetric Dirichlets from data, but to date there has been no rigorous scientific study of the priors used in LDA—from the choice of prior (symmetric versus asymmetric Dirichlets) to the treatment of hyperparameters (optimize versus integrate out)—and the effects of these modeling choices on the probability of held-out documents and, more importantly, the quality of inferred topics. In this paper, we demonstrate that practical implementation issues (handling stop words, setting the number of topics) and theoretical issues involving the structure of Dirichlet priors are intimately related. We start by exploring the effects of classes of hierarchically structured Dirichlet priors over the document–topic distributions and topic–word distributions in LDA. Using MCMC simulations, we find that using an asymmetric, hierarchical Dirichlet prior over the document–topic distributions and 1http://rexa.info/ 1 a symmetric Dirichlet prior over the topic–word distributions results in significantly better model performance, measured both in terms of the probability of held-out documents and in the quality of inferred topics. Although this hierarchical Bayesian treatment of LDA produces good results, it is computationally intensive. We therefore demonstrate that optimizing the hyperparameters of asymmetric, nonhierarchical Dirichlets as part of an iterative inference algorithm results in similar performance to the full Bayesian model while adding negligible computational cost beyond standard inference techniques. Finally, we show that using optimized Dirichlet hyperparameters results in dramatically improved consistency in topic usage as T is increased. By decreasing the sensitivity of the model to the number of topics, hyperparameter optimization results in robust, data-driven models with substantially less model complexity and computational cost than nonparametric models. Since the priors we advocate (an asymmetric Dirichlet over the document–topic distributions and a symmetric Dirichlet over the topic–word distributions) have significant modeling benefits and can be implemented using highly efficient algorithms, we recommend them as a new standard for LDA. 2 Latent Dirichlet Allocation LDA is a generative topic model for documents W = {w(1), w(2), . . . , w(D)}. A “topic” t is a discrete distribution over words with probability vector φt. A Dirichlet prior is placed over Φ = {φ1, . . . φT }. In almost all previous work on LDA, this prior is assumed to be symmetric (i.e., the base measure is fixed to a uniform distribution over words) with concentration parameter β: P(Φ) = Q t Dir (φt; βu) = Q t Γ(β) Q w Γ( β W ) Q w φ β W −1 w|t δ P w φw|t −1 . (1) Each document, indexed by d, has a document-specific distribution over topics θd. The prior over Θ = {θ1, . . . θD} is also assumed to be a symmetric Dirichlet, this time with concentration parameter α. The tokens in every document w(d) = {w(d) n }Nd n=1 are associated with corresponding topic assignments z(d) = {z(d) n }Nd n=1, drawn i.i.d. from the document-specific distribution over topics, while the tokens are drawn i.i.d. from the topics’ distributions over words Φ = {φ1, . . . , φT }: P(z(d) | θd) = Q n θz(d) n |d and P(w(d) | z(d), Φ) = Q n φw(d) n |z(d) n . (2) Dirichlet–multinomial conjugacy allows Θ and Φ to be marginalized out. For real-world data, documents W are observed, while the corresponding topic assignments Z are unobserved. Variational methods [3, 16] and MCMC methods [7] are both effective at inferring the latent topic assignments Z. Asuncion et al. [1] demonstrated that the choice of inference method has negligible effect on the probability of held-out documents or inferred topics. We use MCMC methods throughout this paper—specifically Gibbs sampling [5]—since the internal structure of hierarchical Dirichlet priors are typically inferred using a Gibbs sampling algorithm, which can be easily interleaved with Gibbs updates for Z given W. The latter is accomplished by sequentially resampling each topic assignment z(d) n from its conditional posterior given W, αu, βu and Z\d,n (the current topic assignments for all tokens other than the token at position n in document d): P(z(d) n | W, Z\d,n, αu, βu) ∝P(w(d) n | z(d) n , W\d,n, Z\d,n, βu) P(z(d) n | Z\d,n, αu) ∝ N \d,n w(d) n |z(d) n + β W N \d,n z(d) n + β N \d,n z(d) n |d + α T Nd −1 + α , (3) where sub- or super-script “\d, n” denotes a quantity excluding data from position n in document d. 3 Priors for LDA The previous section outlined LDA as it is most commonly used—namely with symmetric Dirichlet priors over Θ and Φ with fixed concentration parameters α and β, respectively. The simplest way to vary this choice of prior for either Θ or Φ is to infer the relevant concentration parameter from data, either by computing a MAP estimate [1] or by using an MCMC algorithm such as slice sampling [13]. A broad Gamma distribution is an appropriate choice of prior for both α and β. 2 u α φt θd zn wn β u T D N (a) u α φt θd zn wn β u n β′ T D N (b) t|d t′|d t|d t|d γ1 =t γ2 =t′ γ3 =t m (c) u α m α′ φt θd zn wn β u T D N (d) u α m α′ φt θd zn wn β u n β′ T D N (e) t|d t′|d t|d t|d γ1 =t γ2 =t′ γ3 =t t′|d′ t′|d′ t′|d′ t′|d′ γ1 =t′ γ2 =t′ γ1 =t γ2 =t′ γ3 =t′ u (f) Figure 1: (a)-(e): LDA with (a) symmetric Dirichlet priors over Θ and Φ, (b) a symmetric Dirichlet prior over Θ and an asymmetric Dirichlet prior over Φ, (d) an asymmetric Dirichlet prior over Θ and a symmetric Dirichlet prior over Φ, (e) asymmetric Dirichlet priors over Θ and Φ. (c) Generating {z(d) n }4 n=1 = (t, t′, t, t) from the asymmetric, predictive distribution for document d; (f) generating {z(d) n }4 n=1 = (t, t′, t, t) and {z(d′) n }4 n=1 = (t′, t′, t′, t′) from the asymmetric, hierarchical predictive distributions for documents d and d′, respectively. Alternatively, the uniform base measures in the Dirichlet priors over Θ and Φ can be replaced with nonuniform base measures m and n, respectively. Throughout this section we use the prior over Θ as a running example, however the same construction and arguments also apply to the prior over Φ. In section 3.1, we describe the effects on the document-specific conditional posterior distributions, or predictive distributions, of replacing u with a fixed asymmetric (i.e., nonuniform) base measure m. In section 3.2, we then treat m as unknown, and take a fully Bayesian approach, giving m a Dirichlet prior (with a uniform base measure and concentration parameter α′) and integrating it out. 3.1 Asymmetric Dirichlet Priors If Θ is given an asymmetric Dirichlet prior with concentration parameter α and an known (nonuniform) base measure m, the predictive probability of topic t occurring in document d given Z is P(z(d) Nd+1 =t | Z, αm) = Z dθd P(t | θd) P(θd | Z, αm) = Nt|d + αmt Nd + α . (4) If topic t does not occur in z(d), then Nt|d will be zero, and the probability of generating z(d) Nd+1 =t will be mt. In other words, under an asymmetric prior, Nt|d is smoothed with a topic-specific quantity αmt. Consequently, different topics can be a priori more or less probable in all documents. One way of describing the process of generating from (4) is to say that generating a topic assignment z(d) n is equivalent to setting the value of z(d) n to the the value of some document-specific draw from m. While this interpretation provides little benefit in the case of fixed m, it is useful for describing the effects of marginalizing over m on the predictive distributions (see section 3.2). Figure 1c depicts the process of drawing {z(d) n }4 n=1 using this interpretation. When drawing z(d) 1 , there are no existing document-specific draws from m, so a new draw γ1 must be generated, and z(d) 1 assigned the value of this draw (t in figure 1c). Next, z(d) 2 is drawn by either selecting γ1, with probability proportional to the number of topic assignments that have been previously “matched” to γ1, or a new draw from m, with probability proportional to α. In figure 1c, a new draw is selected, so γ2 is drawn from m and z(d) 2 assigned its value, in this case t′. The next topic assignment is drawn in the same way: existing draws γ1 and γ2 are selected with probabilities proportional to the numbers of topic assignments to which they have previously been matched, while with probability proportional to α, z(d) 3 is matched to a new draw from m. In figure 1c, γ1 is selected and z(d) 3 is assigned the value of γ1. In general, the probability of a new topic assignment being assigned the value of an existing document-specific draw γi from m is proportional to N (i) d , the number of topic assignments 3 previously matched to γi. The predictive probability of topic t in document d is therefore P(z(d) Nd+1 =t | Z, αm) = PI i=1 N (i) d δ (γi −t) + αmt Nd + α , (5) where I is the current number of draws from m for document d. Since every topic assignment is matched to a draw from m, PI i=1 N (i) d δ (γi −t) = Nt|d. Consequently, (4) and (5) are equivalent. 3.2 Integrating out m In practice, the base measure m is not fixed a priori and must therefore be treated as an unknown quantity. We take a fully Bayesian approach, and give m a symmetric Dirichlet prior with concentration parameter α′ (as shown in figures 1d and 1e). This prior over m induces a hierarchical Dirichlet prior over Θ. Furthermore, Dirichlet–multinomial conjugacy then allows m to be integrated out. Giving m a symmetric Dirichlet prior and integrating it out has the effect of replacing m in (5) with a “global” P´olya conditional distribution, shared by the document-specific predictive distributions. Figure 1f depicts the process of drawing eight topic assignments—four for document d and four for document d′. As before, when a topic assignment is drawn from the predictive distribution for document d, it is assigned the value of an existing (document-specific) internal draw γi with probability proportional to the number of topic assignments previously matched to that draw, and to the value of a new draw γi′ with probability proportional to α. However, since m has been integrated out, the new draw must be obtained from the “global” distribution. At this level, γi′ treated as if it were a topic assignment, and assigned the value of an existing global draw γj with probability proportional to the number of document-level draws previously matched to γj, and to a new global draw, from u, with probability proportional to α′. Since the internal draws at the document level are treated as topic assignments the global level, there is a path from every topic assignment to u, via the internal draws. The predictive probability of topic t in document d given Z is now P(z(d) Nd+1 =t | Z, α, α′u) = Z dm P(z(d) Nd+1 =t | Z, αm) P(m | Z, α′u) = Nt|d + α ˆNt + α′ T P t ˆNt + α′ Nd + α , (6) where I and J are the current numbers of document-level and global internal draws, respectively, Nt|d = PI i=1 N (i) d δ (γi −t) as before and ˆNt = PJ j=1 N (j) δ (γj −t). The quantity N (j) is the total number of document-level internal draws matched to global internal draw γj. Since some topic assignments will be matched to existing document-level draws, P d δ (Nt|d > 0) ≤ˆNt ≤Nt, where P d δ (Nt|d > 0) is the number of unique documents in Z in which topic t occurs. An important property of (6) is that if concentration parameter α′ is large relative to P t ˆNt, then counts ˆNt and P t ˆNt are effectively ignored. In other words, as α′ →∞the hierarchical, asymmetric Dirichlet prior approaches a symmetric Dirichlet prior with concentration parameter α. For any given Z for real-world documents W, the internal draws and the paths from Z to u are unknown. Only the value of each topic assignment is known, and hence Nt|d for each topic t and document d. In order to compute the conditional posterior distribution for each topic assignment (needed to resample Z) it is necessary to infer ˆNt for each topic t. These values can be inferred by Gibbs sampling the paths from Z to u [4, 15]. Resampling the paths from Z to u can be interleaved with resampling Z itself. Removing z(d) n = t from the model prior to resampling its value consists of decrementing Nt|d and removing its current path to u. Similarly, adding a newly sampled value z(d) n =t′ into the model consists of incrementing Nt′|d and sampling a new path from z(d) n to u. 4 Comparing Priors for LDA To investigate the effects of the priors over Θ and Φ, we compared the four combinations of symmetric and asymmetric Dirichlets shown in figure 1: symmetric priors over both Θ and Φ (denoted 4 0 1000 3000 5000 −760000 −720000 −680000 50 topics Iteration Log Probability (a) α Frequency 3.5 4.5 5.5 0 50 150 α' Frequency 0 50 100 150 0 50 150 β Frequency 60 70 80 90 0 200 400 log β' Frequency 10 30 50 70 0 40 80 140 (b) -6.90 -6.85 -6.80 -6.75 -6.70 -6.65 Patent abstracts patents[, 1] -9.28 -9.27 -9.26 -9.25 -9.24 -9.23 -9.22 NYT nyt[, 1] -8.33 -8.32 -8.31 -8.30 -8.29 -8.28 20News (c) Figure 2: (a) log P(W, Z | Ω) (patent abstracts) for SS, SA, AS and AS, computed every 20 iterations and averaged over 5 Gibbs sampling runs. AS (red) and AA (black) perform similarly and converge to higher values of log P(W, Z | Ω) than SS (blue) and SA (green). (b) Histograms of 4000 (iterations 1000-5000) concentration parameter values for AA (patent abstracts). Note the log scale for β′: the prior over Φ approaches a symmetric Dirichlet, making AA equivalent to AS. (c) log P(W, Z | Ω) for all three data sets at T = 50. AS is consistently better than SS. SA is poor (not shown). AA is capable of matching AS, but does not always. Data set D ¯ Nd N W Stop Patent abstracts 1016 101.87 103499 6068 yes 20 Newsgroups 540 148.17 80012 14492 no NYT articles 1768 270.06 477465 41961 no Table 1: Data set statistics. D is the number of documents, ¯ Nd is the mean document length, N is the number of tokens, W is the vocabulary size. “Stop” indicates whether stop words were present (yes) or not (no). SS), a symmetric prior over Θ and an asymmetric prior over Φ (denoted SA), an asymmetric prior over Θ and a symmetric prior over Φ (denoted AS), and asymmetric priors over both Θ and Φ (denoted AA). Each combination was used to model three collections of documents: patent abstracts about carbon nanotechnology, New York Times articles, and 20 Newsgroups postings. Due to the computationally intensive nature of the fully Bayesian inference procedure, only a subset of each collection was used (see table 1). In order to stress each combination of priors with respect to skewed distributions over word frequencies, stop words were not removed from the patent abstracts. The four models (SS, SA, AS, AA) were implemented in Java, with integrated-out base measures, where appropriate. Each model was run with T ∈{25, 50, 75, 100} for five runs of 5000 Gibbs sampling iterations, using different random initializations. The concentration parameters for each model (denoted by Ω) were given broad Gamma priors and inferred using slice sampling [13]. During inference, log P(W, Z | Ω) was recorded every twenty iterations. These values, averaged over the five runs for T = 50, are shown in figure 2a. (Results for other values of T are similar.) There are two distinct patterns: models with an asymmetric prior over Θ (AS and AA; red and black, respectively) perform very similarly, while models with a symmetric prior over Θ (SS and SA; blue and green, respectively) also perform similarly, with significantly worse performance than AS and AA. Results for all three data sets are summarized in figure 2c, with the log probability divided by the number of tokens in the collection. SA performs extremely poorly on NYT and 20 Newsgroups, and is not therefore shown. AS consistently achieves better likelihood than SS. The fully asymmetric model, AA, is inconsistent, matching AS on the patents and 20 Newsgroups but doing poorly on NYT. This is most likely due to the fact that although AA can match AS, it has many more degrees of freedom and therefore a much larger space of possibilities to explore. We also calculated the probability of held-out documents using the “left-to-right” evaluation method described by Wallach et al. [17]. These results are shown in figure 3a, and exhibit a similar pattern to the results in figure 2a—the best-performing models are those with an asymmetric priors over Θ. We can gain intuition about the similarity between AS and AA by examining the values of the sampled concentration parameters. As explained in section 3.2, as α′ or β′ grows large relative to P t ˆNt or P w ˆNw, an asymmetric Dirichlet prior approaches a symmetric Dirichlet with concentration parameter α or β. Histograms of 4000 concentration parameter values (from iterations 1000-4000) from the five Gibbs runs of AA with T = 50 are shown in figure 2b. The values for α, α′ and β 5 -6.18 -6.16 -6.14 -6.12 -6.10 Held-out probability Topics Nats / token 25 50 75 100 (a) 0.080 a field emission an electron the 0.080 a the carbon and gas to an 0.080 the of a to and about at 0.080 of a surface the with in contact 0.080 the a and to is of liquid 0.895 the a of to and is in 0.187 carbon nanotubes nanotube catalyst 0.043 sub is c or and n sup 0.061 fullerene compound fullerenes 0.044 material particles coating inorganic 0.042 a field the emission and carbon is 0.042 the carbon catalyst a nanotubes 0.042 a the of substrate to material on 0.042 carbon single wall the nanotubes 0.042 the a probe tip and of to 1.300 the a of to and is in 0.257 and are of for in as such 0.135 a carbon material as structure nanotube 0.065 diameter swnt about nm than fiber swnts 0.029 compositions polymers polymer contain Asymmetric α Symmetric α Asymmetric β Symmetric β (b) Figure 3: (a) Log probability of held-out documents (patent abstracts). These results mirror those in figure 2a. AS (red) and AA (black) again perform similarly, while SS (blue) and SA (green) are also similar, but exhibit much worse performance. (b) αmt values and the most probable words for topics obtained with T = 50. For each model, topics were ranked according to usage and the topics at ranks 1, 5, 10, 20 and 30 are shown. AS and AA are robust to skewed word frequency distributions and tend to sequester stop words in their own topics. are all relatively small, while the values for β′ are extremely large, with a median around exp 30. In other words, given the values of β′, the prior over Φ is effectively a symmetric prior over Φ with concentration parameter β. These results demonstrate that even when the model can use an asymmetric prior over Φ, a symmetric prior gives better performance. We therefore advocate using model AS. It is worth noting the robustness of AS to stop words. Unlike SS and SA, AS effectively sequesters stop words in a small number of more frequently used topics. The remaining topics are relatively unaffected by stop words. Creating corpus-specific stop word lists is seen as an unpleasant but necessary chore in topic modeling. Also, for many specialized corpora, once standard stop words have been removed, there are still other words that occur with very high probability, such as “model,” “data,” and “results” in machine learning literature, but are not technically stop words. If LDA cannot handle such words in an appropriate fashion then they must be treated as stop words and removed, despite the fact that they play meaningful semantic roles. The robustness of AS to stop words has implications for HMM-LDA [8] which models stop words using a hidden Markov model and “content” words using LDA, at considerable computational cost. AS achieves the same robustness to stop words much more efficiently. Although there is empirical evidence that topic models that use asymmetric Dirichlet priors with optimized hyperparameters, such as Pachinko allocation [10] and Wallach’s topic-based language model [18], are robust to the presence of extremely common words, these studies did not establish whether the robustness was a function of a more complicated model structure or if careful consideration of hyperparameters alone was sufficient. We demonstrate that AS is capable of learning meaningful topics even with no stop word removal. For efficiency, we do not necessarily advocate doing away with stop word lists entirely, but we argue that using an asymmetric prior over Θ allows practitioners to use a standard, conservative list of determiners, prepositions and conjunctions that is applicable to any document collection in a given language, rather than hand-curated corpus-specific lists that risk removing common but meaningful terms. 5 Efficiency: Optimizing rather than Integrating Out Inference in the full Bayesian formulation of AS is expensive because of the additional complexity in sampling the paths from Z to u and maintaining hierarchical data structures. It is possible to retain the theoretical and practical advantages of using AS without sacrificing the advantages of simple, efficient models by directly optimizing m, rather than integrating it out. The concentration parameters α and β may also be optimized (along with m for α and by itself for β). In this section, we therefore compare the fully Bayesian version of AS with optimized AS, using SS as a baseline. Wallach [19] compared several methods for jointly the maximum likelihood concentration parameter and asymmetric base measure for a Dirichlet–multinomial model. We use the most efficient of these methods. The advantage of optimizing m is considerable: although it is likely that further optimizations would reduce the difference, 5000 Gibbs sampling iterations (including sampling α, 6 Patents NYT 20 NG ASO -6.65 ± 0.04 -9.24 ± 0.01 -8.27 ± 0.01 AS -6.62 ± 0.03 -9.23 ± 0.01 -8.28 ± 0.01 SS -6.91 ± 0.01 -9.26 ± 0.01 -8.31 ± 0.01 25 50 75 100 ASO -6.18 -6.12 -6.12 -6.08 AS -6.15 -6.13 -6.11 -6.10 SS -6.18 -6.18 -6.16 -6.13 Table 2: log P(W, Z | Ω) / N for T = 50 (left) and log P(Wtest | W, Z, Ω) / N test for varying values of T (right) for the patent abstracts. AS and ASO (optimized hyperparameters) consistently outperform SS except for ASO with T = 25. Differences between AS and ASO are inconsistent and within standard deviations. ASO AS SS ASO 4.37 ± 0.08 4.34 ± 0.09 5.43 ± 0.05 AS — 4.18 ± 0.09 5.39 ± 0.06 SS — — 5.93 ± 0.03 ASO AS SS ASO 3.36 ± 0.03 3.43 ± 0.05 3.50 ± 0.07 AS — 3.36 ± 0.02 3.56 ± 0.07 SS — — 3.49 ± 0.04 Table 3: Average VI distances between multiple runs of each model with T = 50 on (left) patent abstracts and (right) 20 newsgroups. ASO partitions are approximately as similar to AS partitions as they are to other ASO partitions. ASO and AS partitions are both are further from SS partitions, which tend to be more dispersed. α′ and β) for the patent abstracts using fully Bayesian AS with T = 25 took over four hours, while 5000 Gibbs sampling iterations (including hyperparameter optimization) took under 30 minutes. In order to establish that optimizing m is a good approximation to integrating it out, we computed log P(W, Z | Ω) and the log probability of held-out documents for fully Bayesian AS, optimized AS (denoted ASO) and as a baseline SS (see table 2). AS and ASO consistently outperformed SS, except for ASO when T = 25. Since twenty-five is a very small number of topics, this is not a cause for concern. Differences between AS and ASO are inconsistent and within standard deviations. From a point of view of log probabilities, ASO therefore provides a good approximation to AS. We can also compare topic assignments. Any set of topic assignments can be thought of as partition of the corresponding tokens into T topics. In order to measure similarity between two sets of topic assignments Z and Z′ for W, we can compute the distance between these partitions using variation of information (VI) [11, 6] (see suppl. mat. for a definition of VI for topic models). VI has several attractive properties: it is a proper distance metric, it is invariant to permutations of the topic labels, and it can be computed in O (N + TT ′) time, i.e., time that is linear in the number of tokens and the product of the numbers of topics in Z and Z′. For each model (AS, ASO and SS), we calculated the average VI distance between all 10 unique pairs of topic assignments from the 5 Gibbs runs for that model, giving a measure of within-model consistency. We also calculated the between-model VI distance for each pair of models, averaged over all 25 unique pairs of topic assignments for that pair. Table 3 indicates that ASO partitions are approximately as similar to AS partitions as they are to other ASO partitions. ASO and AS partitions are both further away from SS partitions, which tend to be more dispersed. These results confirm that ASO is indeed a good approximation to AS. 6 Effect on Selecting the Number of Topics Selecting the number of topics T is one of the most problematic modeling choices in finite topic modeling. Not only is there no clear method for choosing T (other than evaluating the probability of held-out data for various values of T), but degree to which LDA is robust to a poor setting of T is not well-understood. Although nonparametric models provide an alternative, they lose the substantial computational efficiency advantages of finite models. We explore whether the combination of priors advocated in the previous sections (model AS) can improve the stability of LDA to different values of T, while retaining the static memory management and simple inference algorithms of finite models. Ideally, if LDA has sufficient topics to model W well, the assignments of tokens to topics should be relatively invariant to an increase in T—i.e., the additional topics should be seldom used. For example, if ten topics is sufficient to accurately model the data, then increasing the number of topics to twenty shouldn’t significantly affect inferred topic assignments. If this is the case, then using large T should not have a significant impact on either Z or the speed of inference, especially as recentlyintroduced sparse sampling methods allow models with large T to be trained efficiently [20]. Figure 4a shows the average VI distance between topic assignments (for the patent abstracts) inferred by models with T = 25 and models with T ∈{50, 75, 100}. AS and AA, the bottom two lines, are 7 4.0 4.5 5.0 5.5 6.0 6.5 Clustering distance from T=25 Topics Variation of Information 50 75 100 (a) 50 topics 75 topics 100 topics AS prior 0.0 0.2 0.4 0.6 0.8 1.0 50 topics 75 topics 100 topics SS prior 0.0 0.2 0.4 0.6 0.8 1.0 (b) Figure 4: (a) Topic consistency measured by average VI distance from models with T = 25. As T increases, AS (red) and AA (black) produce Zs that stay significantly closer to those obtained with T = 25 than SA (green) and SS (blue). (b) Assignments of tokens (patent abstracts) allocated to the largest topic in a 25 topic model, as T increases. For AS, the topic is relatively intact, even at T = 100: 80% of tokens assigned to the topic at T = 25 are assigned to seven topics. For SS, the topic has been subdivided across many more topics. much more stable (smaller average VI distances) than SS and SA at 50 topics and remain so as T increases: even at 100 topics, AS has a smaller VI distance to a 25 topic model than SS at 50 topics. Figure 4b provides intuition for this difference: for AS, the tokens assigned to the largest topic at T = 25 remain within a small number of topics as T is increased, while for SS, topic usage is more uniform and increasing T causes the tokens to be divided among many more topics. These results suggest that for AS, new topics effectively “nibble away” at existing topics, rather than splitting them more uniformly. We therefore argue that the risk of using too many topics is lower than the risk of using too few, and that practitioners should be comfortable using larger values of T. 7 Discussion The previous sections demonstrated that AS results in the best performance over AA, SA and SS, measured in several ways. However, it is worth examining why this combination of priors results in superior performance. The primary assumption underlying topic modeling is that a topic should capture semantically-related word co-occurrences. Topics must also be distinct in order to convey information: knowing only a few co-occurring words should be sufficient to resolve semantic ambiguities. A priori, we therefore do not expect that a particular topic’s distribution over words will be like that of any other topic. An asymmetric prior over Φ is therefore a bad idea: the base measure will reflect corpus-wide word usage statistics, and a priori, all topics will exhibit those statistics too. A symmetric prior over Φ only makes a prior statement (determined by the concentration parameter β) about whether topics will have more sparse or more uniform distributions over words, so the topics are free to be as distinct and specialized as is necessary. However, it is still necessary to account for power-law word usage. A natural way of doing this is to expect that certain groups of words will occur more frequently than others in every document in a given corpus. For example, the words “model,” “data,” and “algorithm” are likely to appear in every paper published in a machine learning conference. These assumptions lead naturally to the combination of priors that we have empirically identified as superior: an asymmetric Dirichlet prior over Θ that serves to share commonalities across documents and a symmetric Dirichlet prior over Φ that serves to avoid conflicts between topics. Since these priors can be implemented using efficient algorithms that add negligible cost beyond standard inference techniques, we recommend them as a new standard for LDA. 8 Acknowledgments This work was supported in part by the Center for Intelligent Information Retrieval, in part by CIA, NSA and NSF under NSF grant number IIS-0326249, and in part by subcontract number B582467 from Lawrence Livermore National Security, LLC under prime contract number DEAC52-07NA27344 from DOE/NNSA. Any opinions, findings and conclusions or recommendations expressed in this material are the authors’ and do not necessarily reflect those of the sponsor. 8 References [1] A. Asuncion, M. Welling, P. Smyth, and Y. W. Teh. On smoothing and inference for topic models. In Proceedings of the 25th Conference on Uncertainty in Artificial Intelligence, 2009. [2] D. Blei and J. Lafferty. A correlated topic model of Science. Annals of Applied Statistics, 1(1):17–35, 2007. [3] D. M. Blei, A. Y. Ng, and M. I. Jordan. Latent Dirichlet allocation. Journal of Machine Learning Research, 3:993–1022, January 2003. [4] P. J. Cowans. Probabilistic Document Modelling. PhD thesis, University of Cambridge, 2006. [5] S. Geman and D. Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transaction on Pattern Analysis and Machine Intelligence 6, pages 721–741, 1984. [6] S. Goldwater and T. L. Griffiths. A fully Bayesian approach to unsupervised part-of-speech tagging. In Association for Computational Linguistics, 2007. [7] T. L. Griffiths and M. Steyvers. Finding scientific topics. Proceedings of the National Academy of Sciences, 101(suppl. 1):5228–5235, 2004. [8] T. L. Griffiths, M. Steyvers, D. M. Blei, and J. B. Tenenbaum. Integrating topics and syntax. In L. K. Saul, Y. Weiss, and L. Bottou, editors, Advances in Neural Information Processing Systems 17, pages 536–544. The MIT Press, 2005. [9] D. Hall, D. Jurafsky, and C. D. Manning. Studying the history of ideas using topic models. In Proceedings of EMNLP 2008, pages 363–371. [10] W. Li and A. McCallum. Mixtures of hierarchical topics with pachinko allocation. In Proceedings of the 24th International Conference on Machine learning, pages 633–640, 2007. [11] M. Meil˘a. Comparing clusterings by the variation of information. In Conference on Learning Theory, 2003. [12] D. Mimno and A. McCallum. Organizing the OCA: Learning faceted subjects from a library of digital books. In Proceedings of the 7th ACM/IEEE joint conference on Digital libraries, pages 376–385, Vancouver, BC, Canada, 2007. [13] R. M. Neal. Slice sampling. Annals of Statistics, 31:705–767, 2003. [14] D. Newman, C. Chemudugunta, P. Smyth, and M. Steyvers. Analyzing entities and topics in news articles using statistical topic models. In Intelligence and Security Informatics, Lecture Notes in Computer Science. 2006. [15] Y. W. Teh, M. I. Jordan, M. J. Beal, and D. M. Blei. Hierarchical Dirichlet processes. Journal of the American Statistical Association, 101:1566–1581, 2006. [16] Y. W. Teh, D. Newman, and M. Welling. A collapsed variational Bayesian inference algorithm for latent Dirichlet allocation. In Advances in Neural Information Processing Systems 18, 2006. [17] H. Wallach, I. Murray, R. Salakhutdinov, and D. Mimno. Evaluation methods for topic models. In Proceedings of the 26th Interational Conference on Machine Learning, 2009. [18] H. M. Wallach. Topic modeling: Beyond bag-of-words. In Proceedings of the 23rd International Conference on Machine Learning, pages 977–984, Pittsburgh, Pennsylvania, 2006. [19] H. M. Wallach. Structured Topic Models for Language. Ph.D. thesis, University of Cambridge, 2008. [20] L. Yao, D. Mimno, and A. McCallum. Efficient methods for topic model inference on streaming document collections. In Proceedings of KDD 2009, 2009. 9
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A Generalized Natural Actor-Critic Algorithm Tetsuro Morimura†, Eiji Uchibe‡, Junichiro Yoshimoto‡, Kenji Doya‡ †: IBM Research – Tokyo, Kanagawa, Japan ‡: Okinawa Institute of Science and Technology, Okinawa, Japan tetsuro@jp.ibm.com, {uchibe,jun-y,doya}@oist.jp Abstract Policy gradient Reinforcement Learning (RL) algorithms have received substantial attention, seeking stochastic policies that maximize the average (or discounted cumulative) reward. In addition, extensions based on the concept of the Natural Gradient (NG) show promising learning efficiency because these regard metrics for the task. Though there are two candidate metrics, Kakade’s Fisher Information Matrix (FIM) for the policy (action) distribution and Morimura’s FIM for the stateaction joint distribution, but all RL algorithms with NG have followed Kakade’s approach. In this paper, we describe a generalized Natural Gradient (gNG) that linearly interpolates the two FIMs and propose an efficient implementation for the gNG learning based on a theory of the estimating function, the generalized Natural Actor-Critic (gNAC) algorithm. The gNAC algorithm involves a near optimal auxiliary function to reduce the variance of the gNG estimates. Interestingly, the gNAC can be regarded as a natural extension of the current state-of-the-art NAC algorithm [1], as long as the interpolating parameter is appropriately selected. Numerical experiments showed that the proposed gNAC algorithm can estimate gNG efficiently and outperformed the NAC algorithm. 1 Introduction Policy Gradient Reinforcement Learning (PGRL) attempts to find a policy that maximizes the average (or time-discounted) reward, based on gradient ascent in the policy parameter space [2, 3, 4]. Since it is possible to handle the parameters controlling the randomness of the policy, the PGRL, rather than the value-based RL, can find the appropriate stochastic policy and has succeeded in several practical applications [5, 6, 7]. However, depending on the tasks, PGRL methods often require an excessively large number of learning time-steps to construct a good stochastic policy, due to the learning plateau where the optimization process falls into a stagnant state, as was observed for a very simple Markov Decision Process (MDP) with only two states [8]. In this paper, we propose a new PGRL algorithm, a generalized Natural Actor-Critic (gNAC) algorithm, based on the natural gradient [9]. Because “natural gradient” learning is the steepest gradient method in a Riemannian space and the direction of the natural gradient is defined on that metric, it is an important issue how to design the Riemannian metric. In the framework of PGRL, the stochastic policies are represented as parametric probability distributions. Thus the Fisher Information Matrices (FIMs) with respect to the policy parameter induce appropriate Riemannian metrics. Kakade [8] used an average FIM for the policy over the states and proposed a natural policy gradient (NPG) learning. Kakade’s FIM has been widely adopted and various algorithms for the NPG learning have been developed by many researchers [1, 10, 11]. These are based on the actor-critic framework, called the natural actor-critic (NAC) [1]. Recently, the concept of “Natural State-action Gradient” (NSG) learning has been proposed in [12], which shows potential to reduce the learning time spent by being better at avoiding the learning plateaus than the NPG. This natural gradient is on the FIM of the state-action joint distribution as the Riemannian metric for RL, which is directly associated with the average rewards as the objective function. Morimura et al. [12] showed that the metric of the NSG corresponds with 1 the changes in the stationary state-action joint distribution. In contrast, the metric of the NPG takes into account only changes in the action distribution and ignores changes in the state distribution, which also depends on the policy in general. They also showed experimental results with exact gradients where the NSG learning outperformed NPG learning, especially with large numbers of states in the MDP. However, no algorithm for estimating the NSG has been proposed, probably because the estimation for the derivative of log stationary state distribution was difficult [13]. Therefore, the development of a tractable algorithm for NSG would be of great importance, and this is the one of the primary goals of this paper. Meanwhile, it would be very difficult to select an appropriate FIM because it would be dependent on the given task. Accordingly, we created a linear interpolation of both of the FIMs as a generalized Natural Gradient (gNG) and derived an efficient approach to estimate the gNG by applying the theory of the estimating function for stochastic models [14] in Section 3. In Section 4, we derive a gNAC algorithm with an instrumental variable, where a policy parameter is updated by a gNG estimate that is a solution of the estimating function derived in Section 3, and show that the gNAC can be regarded as a natural extension of the current state-of-the-art NAC algorithm [1]. To validate the performance of the proposed algorithm, numerical experiments are shown in Section 5, where the proposed algorithm can estimate the gNG efficiently and outperformed the NAC algorithm [1]. 2 Background of Policy Gradient and Natural Gradient for RL We briefly review the policy gradient and natural gradient learning as gradient ascent methods for RL and also present the motivation of the gNAC approach. 2.1 Policy Gradient Reinforcement Learning PGRL is modeled on a discrete-time Markov Decision Process (MDP) [15, 16]. It is defined by the quintuplet (S, A, p, r, π), where S ∋s and A ∋a are finite sets of states and actions, respectively. Also, p : S × A × S →[0, 1] is a state transition probability function of a state s, an action a, and the following state s+1, i.e.1, p(s+1|s, a) ≜Pr(s+1|s, a). R : S × A × S →R is a bounded reward function of s, a, and s+1, which defines an immediate reward r = R(s, a, s+1) observed by a learning agent at each time step. The action probability function π : A × S × Rd →[0, 1] uses a, s, and a policy parameter θ ∈Rd to define the decision-making rule of the learning agent, which is also called a policy, i.e., π(a|s; θ) ≜Pr(a|s, θ). The policy is normally parameterized by users and is controlled by tuning θ. Here, we make two assumptions in the MDP. Assumption 1 The policy is always differentiable with respect to θ and is non-redundant for the task, i.e., the statistics F a(θ) ∈Rd×d (defined in Section 2.2) are always bounded and non-singular. Assumption 2 The Markov chain M(θ)≜{S, A, p, π, θ} is always ergodic (irreducible and aperiodic). Under Assumption 2, there exists a unique stationary state distribution dθ(s) ≜Pr(s|M(θ)), which is equal to the limiting distribution and independent of the initial state, dθ(s′) = limt→∞Pr(S+t = s′|S = s, M(θ)), ∀s ∈ S. This distribution satisfies the balance equation: dθ(s+1) = P s∈S P a∈A p(s+1|s, a)π(a|s;θ)dθ(s). The goal of PGRL is to find the policy parameter θ∗that maximizes the average of the immediate rewards, the average reward, η(θ) ≜Eθ[r] = X s∈S X a∈A X s+1∈S dθ(s)π(a|s;θ)p(s+1|s, a)R(s, a, s+1), (1) where Eθ[a] denotes the expectation of a on the Markov chain M(θ). The derivative of the average reward for (1) with respect to the policy parameter, ∇θη(θ) ≜[∂η(θ)/∂θ1, ..., ∂η(θ)/∂θd]⊤, which is referred to as the Policy Gradient (PG), is ∇θη(θ) = Eθ [ r∇θln{dθ(s)π(a|s;θ)}] . 1Although to be precise it should be Pr(S+1 =s+1|St =s, A=a) for the random variables S+1, S, and A, we write Pr(s+1|s, a) for simplicity. The same simplification is applied to the other distributions. 2 Therefore, the average reward η(θ) will be increased by updating the policy parameter as θ := θ + α∇θη(θ), where := denotes the right-to-left substitution and α is a sufficiently small learning rate. This framework is called the PGRL [4]. It is noted that the ordinary PGRL methods omit the differences in sensitivities and the correlations between the elements of θ, as defined by the probability distributions of the MDP, while most probability distributions expressed in the MDP have some form of a manifold structure instead of a Euclidean structure. Accordingly, the updating direction of the policy parameter by the ordinary gradient method will be different from the steepest directions on these manifolds. Therefore, the optimization process sometimes falls into a stagnant state, commonly called a plateau [8, 12]. 2.2 Natural Gradients for PGRL To avoid the plateau problem, the concept of the natural gradient was proposed by Amari [9], which is a gradient method on a Riemannian space. The parameter space being a Riemannian space implies that the parameter θ ∈Rd is on the manifold with the Riemannian metric G(θ) ∈Rd×d (a semipositive definite matrix), instead of being on a Euclidean manifold of an arbitrarily parameterized policy, and the squared length of a small incremental vector ∆θ connecting θ to θ + ∆θ is given by ∥∆θ∥2 G = ∆θ⊤G(θ)∆θ, where ⊤denotes the transpose. Under the constraint ∥∆θ∥2 G = ε2 for a sufficiently small constant ε, the steepest ascent direction of the function η(θ) on the manifold G(θ) is given by e∇G(θ)η(θ) = G(θ)−1∇θη(θ), which is called the natural gradient (NG). Accordingly, to (locally) maximize η(θ), θ is incrementally updated with θ := θ + α e∇G(θ)η(θ). The direction of the NG is defined using a Riemannian metric. Thus, an appropriate choice of the Riemannian metric for the task is required. With RL, two kinds of Fisher Information Matrices (FIMs) F (θ) have been proposed as the Riemannian metric matrices G(θ):2 (I) Kakade [8] focuses only on the changes in the policy (action) distributions and proposes defining the metric matrix with the notation ∇θaθbθ ≜(∇θaθ)bθ, as F a(θ) ≜Eθ ∇θlnπ(a|s;θ)∇θlnπ(a|s;θ)⊤ = Eθ [Fa(θ, s)] , (2) where Fa(θ, s)≜Eθ ∇θlnπ(a|s;θ)∇θlnπ(a|s;θ)⊤|s is the FIM of the policy at a state s. The NG on this FIM, e∇F a(θ)η(θ) = F a(θ)−1∇θη(θ), is called the Natural Policy Gradient (NPG). (II) Considering that the average reward η(θ) in (1) is affected not only by the policy distributions π(a|s;θ) but also by the stationary state distribution dθ(s), Moimura et al. [12] proposed the use of the FIM of the state-action joint distribution for RL, Fs,a(θ) ≜Eθ h ∇θ ln {dθ(s)π(a|s;θ)} ∇θ ln {dθ(s)π(a|s;θ)}⊤i = Fs(θ) + F a(θ), (3) where Fs(θ) ≜P s∈S dθ(s)∇θln dθ(s)∇θln dθ(s)⊤is the FIM of dθ(s). The NG on this FIM, e∇Fs,a(θ)η(θ) = Fs,a(θ)−1∇θη(θ), is called the Natural State-action Gradient (NSG). Some algorithms for the NPG learning, such as NAC [1] and NTD [10, 11], can be successfully implemented using modifications of the actor-critic frameworks based on the LSTDQ(λ) [18] and TD(λ) [16]. In contrast, no tractable algorithm for the NSG learning has been proposed to date. However, it has been suggested that the NSG learning it better than the NPG learning due to the three differences [12]: (a) The NSG learning appropriately benefits from the concepts of Amari’s NG learning, since the metric Fs,a(θ) necessarily and sufficiently accounts for the probability distribution that the average reward depends on. (b) Fs,a(θ) is an analogy to the Hessian matrix of the average reward. (c) Numerical experiments show a strong tendency to avoid entrapment in a learning plateau3, especially with large numbers of states. Therefore, the development of a tractable algorithm for NSG is important, and this is one of the goals of our work. 2The reason for using F (θ) as G(θ) is because the FIM Fx(θ) is a unique metric matrix of the secondorder Taylor expansion of the Kullback-Leibler divergence Pr(x|θ+∆θ) from Pr(x|θ) [17]. 3Although there were numerical experiments involving the NSG in [12], they computed the NSG analytically with the state transition probabilities and the reward function, which is typically unknown in RL. 3 On the other hand, it was proven that the metric of NPG learning, F a(θ), accounts for the infinite time-steps joint distribution in the Markov chain M(θ) [19, 1], while the metric of NSG learning, Fs,a(θ) accounts only for the single time-step distribution, which is the stationary state-action joint distribution dθ(s)π(a|s;θ). Accordingly, the mixing time of M(θ) might be drastically changed with NSG learning compared to NPG learning, since the mixing time depends on the multiple (not necessarily infinite) time-steps rather than the single time-step, i.e., while various policies can lead to the same stationary state distribution, Markov chains associated with these policies have different mixing times. A larger mixing time makes it difficult for the learning agent to explore the environment and to estimate the gradient with finite samples. The ranking of the performances of the NPG and NSG learning will be dependent on the RL task properties. Thus, we consider a mixture of NPG and NSG as a generalized NG (gNG) and propose the approach of ’generalized Natural Actor-Critic’ (gNAC), in which the policy parameter of an actor is updated by an estimate of the gNG of a critic. 3 Generalized Natural Gradient for RL First we explain the definition and properties of the generalized Natural Gradient (gNG). Then we introduce the estimating functions to build up a foundation for an efficient estimation of the gNG. 3.1 Definition of gNG for RL In order to define an interpolation between NPG and NSG with a parameter ι ∈[0, 1], we consider a linear interpolation from the FIM of (2) for the NPG to the FIM of (3) for the NSG, written as ˜Fs,a(θ, ι) ≜ιFs(θ) + F a(θ). (4) Then the natural gradient of the interpolated FIM is e∇˜ Fs,a(θ,ι)η(θ) = ˜Fs,a(θ, ι)−1∇θη(θ), (5) which we call the “generalized natural gradient” for RL with the interpolating parameter ι, gNG(ι). Obviously, gNG(ι=0) and gNG(ι=1) are equivalent to the NPG and the NSG, respectively. When ι is equal to 1/t, this FIM ˜Fs,a(θ, ι) is equivalent to the FIM of the t time-steps joint distribution from the stationary state distribution dθ(s) on M(θ) [12]. Thus, this interpolation controlled by ι can be interpreted as a continuous interpolation with respect to the time-steps of the joint distribution, so that ι : 1 →0 is inversely proportional to t : 1 →∞. The term ’generalized’ of gNG(ι) reflects the generalization as the time steps on the joint distribution that the NG follows. 3.2 Estimating Function of gNG(ι) We provide a general view of the estimation of the gNG(ι) using the theory of the estimating function, which provides well-established results for parameter estimation [14]. Such a function g ∈Rd for an estimator ω ∈Rd (and a variable x) is called an estimating function when it satisfies these conditions for all θ: Eθ [g(x, ω∗)] = 0 (6) det | Eθ[∇ωg(x, ω)] | ̸= 0, ∀ω (7) Eθ g(x, ω)⊤g(x, ω) < ∞, ∀ω (8) where ω∗and det | · | denote the exact solution of this estimation and the determinant, respectively. Proposition 1 The d-dimensional (random) function g′ ι,θ(s, a; ω) ≜∇θln{dθ(s)π(a|s;θ)} r −∇θln{dθ(s)ιπ(a|s;θ)}⊤ω (9) is an estimating function for gNG(ι), such that the unique solution of Eθ[g′ ι,θ(s, a; ω)] = 0 with respect to ω is equal to the gNG(ι). Proof: From (1) and (4), the equation 0 = Eθ[g′ ι,θ(s, a; ω∗)] = ∇θη(θ) −˜Fs,a(θ, ι)ω∗ holds. Thus, ω∗is equal to the gNG(ι) from (5). The remaining conditions from (7) and (8), which the estimating function must satisfy, also obviously hold (under Assumption 1). □ 4 In order to estimate gNG(ι) by using the estimating function (9) with finite T samples on M(θ), the simultaneous equation 1 T T −1 X t=0 g′ ι,θ(st, at; bω) = 0 is solved with respect to ω. The solution bω, which is also called the M-estimator [20], is an unbiased estimate of gNG(ι), so that bω = ω∗holds in the limit as T →∞. Note that the conduct of solving the estimating function of (9) is equivalent to the linear regression with the instrumental variable ∇θln{dθ(s)π(a|s;θ)} where the regressand, the regressor, and the model parameter (estimator) are r (or R(s, a)), ∇θln{dθ(s)ιπ(a|s;θ)}, and ω, respectively [21], so that the regression residuals ‘r −∇θln{dθ(s)ιπ(a|s;θ)}⊤ω’ are not correlated with the instrumental variables ∇θln{dθ(s)π(a|s;θ)}. 3.3 Auxiliary Function of Estimating Function Although we made a simple algorithm implementing the gNAC approach with the M-estimator of the estimating function in (9), the performance of the estimation of gNG(ι) may be unacceptable for real RL applications, since the variance of the estimates of gNG(ι) tends to become too large. For that reason, we extend the estimating function using (9) by embedding an auxiliary function to create space for improvement in (9). Lemma 1 The d-dimensional (random) function is an estimating function for gNG(ι), gι,θ(s, a; ω) ≜∇θln{dθ(s)π(a|s;θ)} r −∇θln{dθ(s)ιπ(a|s;θ)}⊤ω −ρ(s, s+1) , (10) where ρ(s, s+1) is called the auxiliary function for (9): ρ(s, s+1) ≜c + b(s) −b(s+1). (11) The c and b(s) are an arbitrary bounded constant and an arbitrary bounded function of the state. respectively. Proof: See supplementary material. Let Gθ denote the class of such functions gθ with various auxiliary functions ρ. An optimal auxiliary function, which leads to minimizing the variance of the gNG estimate bω, is defined by the optimality criterion of the estimating functions [22]. An estimating function g∗ ι,θ is optimal in Gι,θ if det | Σg∗ ι,θ | ≤det | Σgι,θ | where Σgθ ≜Eθ gι,θ(s, a; ω∗)gι,θ(s, a; ω∗)⊤ . Lemma 2 Let us approximate (or assume) r ≈Eθ[R(s, a, s+1)|s, a] ≜R(s, a), (12) ρ(s, s+1) ≈Eθ[ρ(s, s+1)|s, a] ≜ρ(s, a). (13) If the policy is non-degenerate for the task (so the dimension of θ, d, is equal to P|S| i=1(|Ai| −1), where |S| and |Ai| are the numbers of states and the available actions at state si, respectively) and ω∗denotes the gNG(ι), then the ‘near’ optimal auxiliary function ρ∗in the ‘near’ optimal estimating function g∗ ι,θ(s, a; ω) satisfies4 R(s, a) = ∇θln{dθ(s)ιπ(a|s;θ)}⊤ω∗+ Eθ[ρ∗(s, s+1)|s, a]. (14) Proof Sketch: The covariance matrix for the criterion of the auxiliary function ρ is approximated as Σgθ ≈Eθ ∇θln{dθ(s)π(a|s;θ)}∇θln{dθ(s)π(a|s;θ)}⊤(R(s, a) −∇θln{dθ(s)ιπ(a|s;θ)}⊤ω∗−ρ(s, a))2 ≜ˆΣgθ. (15) The function ρ(s, a) usually has |S| degrees of freedom over all of the (s, a) couplets with the ergodicity of M(θ), because “b(s)−b(s+1)” in ρ has (|S|−1) degrees of freedom over all of the (s, s+1) 4The ‘near’ of the near estimating function comes from the approximations of (12) and (13), which implicitly assume that the sum of the (co)variances, Eθ[(r −R(s, a))2 + (ρ(s, a, s+1) −ρ(s, a))2 −2(r − R(s, a))(ρ(s, a, s+1)−ρ(s, a))|s, a], are not large. This assumption seems to hold in many RL tasks. 5 couplets. The value of ∇θln{dθ(s)ιπ(a|s;θ)}⊤ω has P|S| i=1(|Ai| −1) degrees of freedom. R(s, a) has at most P|S| i=1 |Ai| degrees of freedom. Therefore, there exist ρ∗and ∇θln{dθ(s)ιπ(a|s;θ)}⊤ω⋆ that satisfy (14). Remembering that ∇θln{dθ(s)ιπ(a|s;θ)}⊤ω∗is the approximator of R(s, a) (or r) and ω∗is independent of the choice of ρ due to Lemma 1, we know that ω⋆= ω∗holds. Therefore, if the estimating function has an auxiliary function ρ∗satisfying (14), the criterion of the optimality for ρ is minimized for det | ˆΣg∗ θ | = 0 due to (15). □ From Lemma 2, the near optimal auxiliary function ρ∗can be regarded as minimizing the mean squared residuals to zero between R(s, a) and the estimator bRρ(s, a; ω) ≜ ∇θln{dθ(s)ιπ(a|s;θ)}⊤ω + ρ(s, s+1). Thus, the meaning of this near optimality of g∗ ι,θ(s, a; bω) is interpreted as a near minimization of the Euclidean distance between r and its approximator bRρ∗(s, a; bω), so that ρ∗works to reduce the distance of the regressand r and the subspace of the regressor ∇θln{dθ(s)ιπ(a|s;θ)} of the M-estimator bω. In particular, R(s, a) is almost in this subspace at the point bω = ω∗. Lemma 2 leads directly to Corollary 1. Corollary 1 Let b∗ ι=0(s) and c∗ ι=0 be the functions in the near optimal auxiliary function ρ∗(s, s+1) at ι = 0, then b∗ ι=0(s) and c∗ ι=0 are equal to the (un-discounted) state value function [23] and the average reward, respectively. Proof: For all s, ω, and θ, the following equation holds, Eθ ∇θln{dθ(s)ι=0π(a|s;θ)}⊤ω | s = ω⊤X a∈A ∇θπ(a|s;θ) = 0. Therefore, the following equation, which is the same as the definition of the value function b∗ ι=0(s) with the average reward c∗ ι=0 as the solution of the Poisson equation [23], can be derived from (14): b∗ ι=0(s) + c∗ ι=0 = Eθ[r + b∗ ι=0(s+1) | s], ∀s. □ 4 A Generalized NAC Algorithm We now propose a useful instrumental variable for the gNG(ι) estimation and then derive a gNAC algorithm along with an algorithm for ∇θln dθ(s) estimation. 4.1 Bias from Estimation of ∇θln dθ(s) For computation of the M-estimator of gι,θ(s, a; ω) as the gNG(ι) estimate on M(θ), the computations of both of the derivatives, ∇θlnπ(a|s;θ) and ∇θln dθ(s), are required. While we can easily compute ∇θlnπ(a|s;θ) since we have parameterized the policy, we cannot compute the Logarithm stationary State distribution Derivative (LSD) ∇θln dθ(s) analytically unless the state transition probabilities and the reward function are known. Thus, we use the LSD estimate from the algorithm, LSLSD [13]. These LSD estimates b∇θln dθ(s) usually have some estimation errors with finite samples, while the estimates are unbiased, so that b∇θln dθ(s) = ∇θln dθ(s) + ϵ(s), where ϵ(s) is an d-dimensional random variable satisfying E{ϵ(s)|s} = 0. In such cases, the estimate of gNG(ι) from the estimating function (9) or (10) would be biased, because the first condition of (6) for gι,θ is violated unless Eθ[ϵ(s)ϵ(s)⊤] = 0. Thus, in Section 4.2, we consider a refinement of the instrumental variable as the part ∇θln{dθ(s)π(a|s;θ)} in the estimating function (10), since the instrumental variable can be replaced with any function I that satisfies their conditions5 for any s, θ, and ω [22] and makes the solution ω∗become the gNG(ι): Eθ h I(r −{ b∇θln dθ(s)ι + ∇θlnπ(a|s;θ)}⊤ω∗−ρ(s, s+1) i = 0, (16) det | Eθ[I { b∇θln dθ(s)ι + ∇θlnπ(a|s;θ)}⊤] | ̸= 0, (17) Eθ h (r −{ b∇θln dθ(s)ι + ∇θlnπ(a|s;θ)}⊤ω −ρ(s, s+1))2I⊤I i < ∞. (18) 4.2 Instrumental variables of near optimal estimating function for gNG(ι) We use a linear function to introduce the auxiliary function (defined in (11)), ρ(s, s+1; ν) ≜( ˜φ(s) −[φ(s+1)⊤0]⊤)⊤ν, 5These correspond to the conditions for the estimating function, (6), (7), and (8). 6 A Generalized Natural Actor-Critic Algorithm with LSLSD(λ) Given: A policy π(a|s;θ) with an adjustable θ and a feature vector function of the state, φ(s). Initialize: θ, β ∈[0, 1], α, λ ∈[0, 1),. Set: A := 0; B := 0; C := 0; D := 0; E := 0; x := 0; y := 0. For t = 0 to T −1 do Critic: Compute the gNG(ι) estimate bωι A := βA + ∇θlnπ(at|st;θ)∇θlnπ(at|st;θ)⊤; B := βB + ∇θlnπ(at|st;θ) ˜ψ(st, st+1)⊤; C := βC + ˜φ(st)∇θlnπ(at|st;θ)⊤; D := βD + ˜φ(st)φ(st)⊤; E := βE + ˜φ(st) ˜ψ(st, st+1)⊤; x := βx + rt∇θlnπ(at|st;θ); y := βy + rt ˜φ(st); Ω:= “LSLSD(λ) algorithm” [13] bωι := {A + ι ˜ C⊤Ω−BE−1(C + ιDΩ)}−1 (x −BE−1y) Actor: Update θ by the gNG(ι)estimate bω θ := θ + αbωι; End Return: the policy π(a|s;θ). ∗˜ C is the sub-matrix of C getting off the lowest row. where ν ∈R|S|+1 and φ(s) ∈R|S| are the model parameter and the regressor (feature vector function) of the state s, respectively, and ˜φ(s) ≜[φ(s)⊤, 1]⊤. We assume that the set of φ(s) is linearly independent. Accordingly, the whole model parameter of the estimating function is now [ω⊤, ν⊤]⊤≜ϖ. We propose the following instrumental variable I⋆(s, a) ≜[∇θlnπ(a|s;θ), ˜φ(s)]⊤. (19) Because this instrumental variable I⋆has the desirable property as shown in Theorem 1, the estimating function g⋆ ι,θ(s, a; ϖ) with I⋆is a useful function, even if the LSD is estimated. Theorem 1 To estimate gNG(ι), let I⋆(s, a) be used for the estimating function as g⋆ ι,θ(s, a; ϖ) = I⋆(s, a) r −(b∇θln dθ(s)ι + ∇θlnπ(a|s;θ))⊤ω −ρ(s, s+1; ν) , (20) and ω∗and ν∗be the solutions, so ω∗is equal to the gNG(ι), ω∗= e∇˜ Fs,a(θ,ι)η(θ), and the auxiliary function with ν∗is the near optimal auxiliary function provided in Lemma 2, ρ(s, s+1; ν∗) = ρ∗(s, s+1), even if the LSD estimates include (zero mean) random noises. Proof Sketch: (i) The condition (18) for the instrumental variable is satisfied due to Assumption 1. (ii) Considering Eθ[b∇θln dθ(s)∇θlnπ(a|s;θ)⊤] = 0 and Assumption 1, the condition (17) det | Eθ[∇ϖg⋆ ι,θ] |̸=0, is satisfied. This guarantees that the solution ϖ∗≜[ω∗⊤, ν∗⊤]⊤of (20) that satisfies Eθ[g⋆ ι,θ]=0 is unique. (iii) Assuming that Theorem 1 is true so that “ω∗= e∇˜ Fs,a(θ,ι)η(θ)” and “ρ(s, s+1; ν∗)=ρ∗(s, s+1)” hold, then E[g⋆ ι,θ(s, a; ϖ⊤)|s, a] becomes I⋆(s, a) r −R(s, a) from (14) and its expectation over M(θ) becomes equal to 0. This means that (20) also satisfies the condition (16). From (i), (ii), and (iii), this theorem is proven. □ The optimal instrumental variable I∗(s, a) with respect to the variance minimization is derived straightforwardly with the results of [21, 24]. However, since I∗is usually to be estimated, we do not adress I∗here. Note that the proposed I⋆(s, a) of (19) can be computed analytically. 4.3 A Generalized Natural Actor-Critic Algorithm with LSLSD We can straightforwardly derive a generalized Natural Actor-Critic algorithm, gNAC(ι), by solving the estimating function g⋆ ι,θ(s, a; ϖ) in (20), using the LSD estimate b∇θln dθ(s) ≜Ω⊤φ(s). However, since ν in the mode parameter is not required in updating the policy parameter θ, to reduce the computational cost, we compute only ω by using the results of the block matrices, The above algorithm table shows an instance of the gNAC(ι) algorithm with LSLSD(λ) [13] with the forgetting parameter β for the statistics, the learning rate of the policy α, and the the definitions ψ(st−1, st) ≜[φ(st−1) −φ(st)] and ˜ψ(st−1, st) ≜[ψ(st−1, st)⊤, 1]⊤. Note that the LSD estimate is not used at all in the proposed gNAC(ι = 0). In addition, note that gNAC(ι = 0) is equivalent to a non-episodic NAC algorithm modified to optimize the average reward, instead of the discounted cumulative reward [1]. This interpretation is consistent with the results of Corollary 1. 7 (A) (B) 0 200 400 600 800 1000 0 0.5 1 1.5 2 Time Step Angle [radian] gNG(1) estimate with ρ gNG(0.5) estimate with ρ gNG(0.25) estimate with ρ gNG(1) estimate without ρ gNG(0.5) estimate without ρ gNG(0.25) estimate without ρ 0 1 2 3 4 5 x 10 5 0 0.2 0.4 0.6 0.8 1 Time Step Average Reward gNAC(1) gNAC(0.5) gNAC(0.25) NAC = gNAC(0) AC Figure 1: Averages and standard deviations over 50 independent episodes: (A) The angles between the true gNG(ι) and estimates with and without the auxiliary function ρ(s, s+1, ν) on the 5-states MDP, (B) The learning performances (average rewards) for the various (N)PGRL algorithms with the auxiliary functions on the 30-states MDP. 5 Numerical Experiment We studied the results of the proposed gNAC algorithm with the various ι = {0, 0.25, 0.5, 1} and randomly synthesized MDPs with |S| = {5, 30} states and |A| = 2 actions. Starting with the performance baseline of the existing PG methods, we used Konda’s actor-critic algorithm [23]. This algorithm uses the baseline function in which the state value estimates are estimated by LSTD(0) [25], while the original version did not use any baseline function. Note that gNAC(ι = 0) can be regarded as the NAC proposed by [1], which serves as the baseline for the current state-of-the-art PGRL algorithm. We initialized the setting of the MDP in each episode so the set of the actions was always |A|={l, m}. The state transition probability function was set by using the Dirichlet distribution Dir(α∈R2) and the uniform distribution U(a; b) generating an integer from 1 to a other than b: we first initialized it such that p(s′|s, a):=0, ∀(s′, s, a) and then with q(s, a)∼Dir(α=[.3, .3]) and s\b ∼U(|S|; b),p(s+1|s, l) := q1(s, l) p(x\s+1|s, l) := q2(s, l) , p(s|s, m) := q1(s, m) p(x\s|s, m) := q2(s, m) , where s′ = 1 and s′ = |S|+1 are the identical states. The the reward function R(s, a, s+1) was set temporarily with the Gaussian distribution N(µ=0, σ2 =1), normalized so that maxθ η(θ)=1 and minθ η(θ)=−1; R(s, a, s+1):=2(R(s, a, s+1) −minθ η(θ))/(maxθ η(θ) −minθ η(θ)) −1. The policy is represented by the sigmoidal function: π(l|s; θ) = 1/(1 + exp(−θ⊤φ(s))). Each ith element of the initial policy parameter θ0 ∈R|S| and the feature vector of the jth state, φ(sj) ∈R|S|, were drawn from N(0, 1) and N(δij, 0.5), respectively, where δij is the Kronecker delta. Figure 1 (A) shows the angles between the true gNG(ι) and the gNG(ι) estimates with and without the auxiliary function ρ(s, s+1, ν) at α := 0 (fixed policy), β := 1, λ := 0. The estimation without the auxiliary function was implemented by solving the estimating function of (9). We can confirm that the estimate using g⋆ ι,θ(s, a; ϖ) in (20) that implements the near-optimal estimating function became a much more efficient estimator than without the auxiliary function. Figure 1 (B) shows the comparison results in terms of the learning performances, where the learning rates for the gNACs and Konda’s actor-critic were set as α := 3 × 10−4 and αKonda := 60α. The other hyper parameters β := 1−α and λ := 0 were the same in each of the algorithms. We thus confirmed that our gNAC(ι>0) algorithm outperformed the current state-of-the-art NAC algorithm (gNAC(ι=0)). 6 Summary In this paper, we proposed a generalized NG (gNG) learning algorithm that combines two Fisher information matrices for RL. The theory of the estimating function provided insight to prove some important theoretical results from which our proposed gNAC algorithm was derived. Numerical experiments showed that the gNAC algorithm can estimate gNGs efficiently and that it can outperform a current state-of-the-art NAC algorithm. In order to utilize the auxiliary function of the estimating function for the gNG, we defined an auxiliary function on the criterion of the near optimality of the estimating function, by minimizing the distance between the immediate reward as the regressand and the subspace of the regressors of the gNG at the solution of the gNG. However, it may be possible to use different criterion, such as the optimality on the Fisher information matrix metric instead of the Euclidean metric. Also, an analysis of the properties of gNG itself will be necessary to more deeply understand the properties and efficacy of our proposed gNAC algorithm. 8 References [1] J. Peters, S. Vijayakumar, and S. Schaal. Natural actor-critic. In European Conference on Machine Learning, 2005. [2] V. Gullapalli. A stochastic reinforcement learning algorithm for learning real-valued functions. Neural Networks, 3(6):671–692, 1990. [3] R. J. Williams. Simple statistical gradient-following algorithms for connectionist reinforcement learning. Machine Learning, 8:229–256, 1992. [4] J. Baxter and P. Bartlett. Infinite-horizon policy-gradient estimation. Journal of Artificial Intelligence Research, 15:319–350, 2001. [5] R. Tedrake, T.W. T. W. Zhang, and H. S. Seung. Stochastic policy gradient reinforcement learning on a simple 3D biped. In IEEE International Conference on Intelligent Robots and Systems, 2004. [6] J. Peters and S. Schaal. Policy gradient methods for robotics. In IEEE International Conference on Intelligent Robots and Systems, 2006. [7] S. Richter, D. Aberdeen, and J. Yu. Natural actor-critic for road traffic optimisation. In Advances in Neural Information Processing Systems. MIT Press, 2007. [8] S. Kakade. A natural policy gradient. In Advances in Neural Information Processing Systems, volume 14. MIT Press, 2002. [9] S. Amari. Natural gradient works efficiently in learning. Neural Computation, 10(2):251–276, 1998. [10] T. Morimura, E. Uchibe, and K. Doya. Utilizing natural gradient in temporal difference reinforcement learning with eligibility traces. In International Symposium on Information Geometry and its Applications, pages 256–263, 2005. [11] S. Bhatnagar, R. Sutton, M. Ghavamzadeh, and M. Lee. Incremental natural actor-critic algorithms. In Advances in Neural Information Processing Systems, pages 105–112. MIT Press, 2008. [12] T. Morimura, E. Uchibe, J. Yoshimoto, and K. Doya. A new natural policy gradient by stationary distribution metric. In European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases, 2008. [13] T. Morimura, E. Uchibe, J. Yoshimoto, J. Peters, and K. Doya. Derivatives of logarithmic stationary distributions for policy gradient reinforcement learning. Neural Computation. (in press). [14] V. Godambe. Estimating function. Oxford Science, 1991. [15] D. P. Bertsekas. Dynamic Programming and Optimal Control, Volumes 1 and 2. Athena Scientific, 1995. [16] R. S. Sutton and A. G. Barto. Reinforcement Learning. MIT Press, 1998. [17] S. Amari and H. Nagaoka. Method of Information Geometry. Oxford University Press, 2000. [18] M. G. Lagoudakis and R. Parr. Least-squares policy iteration. Journal of Machine Learning Research, 4:1107–1149, 2003. [19] D. Bagnell and J. Schneider. Covariant policy search. In Proceedings of the International Joint Conference on Artificial Intelligence, July 2003. [20] S. Amari and M. Kawanabe. Information geometry of estimating functions in semi-parametric statistical models. Bernoulli, 3(1), 1997. [21] A. C. Singh and R. P. Rao. Optimal instrumental variable estimation for linear models with stochastic regressors using estimating functions. In Symposium on Estimating Functions, pages 177–192, 1996. [22] B. Chandrasekhar and B. K. Kale. Unbiased statistical estimating functions in presence of nuisance parameters. Journal of Statistical Planning and. Inference, 9:45–54, 1984. [23] V. S. Konda and J. N. Tsitsiklis. On actor-critic algorithms. SIAM Journal on Control and Optimization, 42(4):1143–1166, 2003. [24] T. Ueno, M. Kawanabe, T. Mori, S. Maeda, and S. Ishii. A semiparametric statistical approach to modelfree policy evaluation. In International Conference on Machine Learning, pages 857–864, 2008. [25] J. A. Boyan. Technical update: Least-squares temporal difference learning. Machine Learning, 49(23):233–246, 2002. 9
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Efficient Moments-based Permutation Tests Chunxiao Zhou Huixia Judy Wang Dept. of Electrical and Computer Eng. Dept. of Statistics University of Illinois at Urbana-Champaign North Carolina State University Champaign, IL 61820 Raleigh, NC 27695 czhou4@gmail.com wang@stat.ncsu.edu Yongmei Michelle Wang Depts. of Statistics, Psychology, and Bioengineering University of Illinois at Urbana-Champaign Champaign, IL 61820 ymw@illinois.edu Abstract In this paper, we develop an efficient moments-based permutation test approach to improve the test’s computational efficiency by approximating the permutation distribution of the test statistic with Pearson distribution series. This approach involves the calculation of the first four moments of the permutation distribution. We propose a novel recursive method to derive these moments theoretically and analytically without any permutation. Experimental results using different test statistics are demonstrated using simulated data and real data. The proposed strategy takes advantage of nonparametric permutation tests and parametric Pearson distribution approximation to achieve both accuracy and efficiency. 1 Introduction Permutation tests are flexible nonparametric alternatives to parametric tests in small samples, or when the distribution of a test statistic is unknown or mathematically intractable. In permutation tests, except exchangeability, no other statistical assumptions are required. The p-values can be obtained by using the permutation distribution. Permutation tests are appealing in many biomedical studies, which often have limited observations with unknown distribution. They have been used successfully in structural MR image analysis [1, 2, 3], in functional MR image analysis [4], and in 3D face analysis [5]. There are three common approaches to construct the permutation distribution [6, 7, 8]: (1) exact permutation enumerating all possible arrangements; (2) approximate permutation based on random sampling from all possible permutations; (3) approximate permutation using the analytical moments of the exact permutation distribution under the null hypothesis. The main disadvantage of the exact permutation is the computational cost, due to the factorial increase in the number of permutations with the increasing number of subjects. The second technique often gives inflated type I errors caused by random sampling. When a large number of repeated tests are needed, the random permutation strategy is also computationally expensive to achieve satisfactory accuracy. Regarding the third approach, the exact permutation distribution may not have moments or moments with tractability. In most applications, it is not the existence but the derivation of moments that limits the third approach. To the best of our knowledge, there is no systematic and efficient way to derive the moments of the permutation distribution. Recently, Zhou [3] proposed a solution by converting the permutation of data to that of the statistic coefficients that are symmetric to the permutation. Since the test statistic coefficients usually have simple presentations, it is easier to track the permutation of the test statistic coefficients than that of data. However, this method requires the derivation of the permutation for each specific test statistic, which is not accessible to practical users. In this paper, we propose a novel strategy by employing a general theoretical method to derive the moments of the permutation distribution of any weighted v-statistics, for both univariate and multivariate data. We note that any moments of the permutation distribution for weighted v-statistics [9] can be considered as a summation of the product of data function term and index function term over a high dimensional index set and all possible permutations. Our key idea is to divide the whole index set into several permutation equivalent (see Definition 2) index subsets such that the summation of the data/index function term over all permutations is invariant within each subset and can be calculated without conducting any permutation. Then we can obtain the moments by summing up several subtotals. The proposed method can be extended to equivalent weighted v-statistics by replacing them with monotonic weighted v-statistics. This is due to the fact that only the order of test statistics of all permutations matters for obtaining the p-values, so that the monotonic weighted v-statistics shares the same p-value with the original test statistic. Given the first four moments, the permutation distribution can be well fitted by Pearson distribution series. The p-values are then obtained without conducting any real permutation. For multiple comparison of two-group difference, given the sample size n1 = 21 and n2 = 21, the number of tests m = 2,000, we need to conduct m×(n1+ n2)!/n1!/n2! ≈ 1.1×1015 permutations for the exact permutation test. Even for 20,000 random permutations per test, we still need m×20,000 ≈ 4×107 permutations. Alternatively, our moments-based permutation method using Pearson distribution approximation only involves the calculation of the first four analytically-derived moments of exact permutation distributions to achieve high accuracy (see section 3). Instead of calculating test statistics in factorial scale with exact permutation, our moments-based permutation only requires computation of polynomial order. For example, the computational cost for univariate mean difference test statistic and modified multivariate Hotelling's T2 test statistics [8] are O(n) and O(n3), respectively, where n = n1+ n2. 2 Methodology In this section, we shall mainly discuss how to calculate the moments of the permutation distribution for weighted v-statistics. For other test statistics, a possible solution is to replace them with their equivalent weighted v-statistics by monotonic transforms. The detailed discussion about equivalent test statistics can be found in [7, 8, 10]. 2.1 Computational challenge Let us first look at a toy example. Suppose we have a two-group univariate data 1 1 1 2 1 1 ( , , , , , ) n n n n x x x x x + + = L L , where the first n1 elements are in group A and the rest, n2 ,are in group B. For comparison of the two groups, the hypothesis is typically constructed as: 0 : A B H µ µ = vs. : a A B H µ µ ≠ , where , A B µ µ are the population means of the groups A and B, respectively. Define 1 1 1 / n A i i x x n = = ∑ and 1 2 1 / n B i i n x x n = + = ∑ as the sample means of two groups, where n=n1+n2. We choose the univariate group mean difference as the test statistic, i.e., 1 ( ) ( ) n A B i i T x x x w i x = = − = ∑ , where the index function 1 ( ) 1/ w i n = , if 1 {1, , } i n ∈ L and 2 ( ) 1/ w i n = − , if 1 { 1, , } i n n ∈ + L . Then the total number of all possible permutations of {1, , } n L is n!. To calculate the fourth moment of the permutation distribution, 1 2 3 4 1 2 3 4 4 4 ( ) 1 2 3 4 ( ) ( ) ( ) ( ) 1 1 1 1 1 1 1 ( ( ))= ( ( ) ) = ( ) ( ) ( ) ( ) ! ! n n n n n n n i i i i i S i S i i i i E T x w i x w i w i w i w i x x x x n n π π π π π π π π ∈ = ∈ = = = = ∑ ∑ ∑ ∑∑∑∑ , where π is the permutation operator and the symmetric group Sn [11] includes all distinct permutations. The above example shows that the moment calculation can be considered as a summation over all possible permutations and a large index set. It is noticeable that the computational challenge here is to go through the factorial level permutations and polynomial level indices. 2.2 Partition the index set In this paper, we assume that the test statistic T can be expressed as a weighted v-statistic of degree d [9], that is, 1 1 1 1 1 ( ) ( , , ) ( , , ) d d n n d i i i i T x w i i h x x = = = ∑ ∑ L L L , where 1 2 ( , , , )T n x x x x = L is a data with n observations, and w is a symmetric index function. h is a symmetric data function, i.e., invariant under permutation of 1 ( , , ) d i i L . Though the symmetry property is not required for our method, it helps reduce the computational cost. Here, each observation kx can be either univariate or multivariate. In the above toy example, d=1 and h is the identity function. Therefore, the r-th moment of the test statistic from the permutated data is: 1 1 2 1 ( ) ( ) , , , ( ( )) ( ( , , ) ( , , )) d d r r d i i i i i E T x E w i i h x x π π π π = ∑ L L L ( ) ( ) 1 (1) (1) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 1 , , , , , [ { ( , , ) ( , , )}] . k k d d r r d r r k k d i i k k i i i i E w i i h x x π π π = = = ∑ ∏ ∏ L L L L L Then we can exchange the summation order of permutations and that of indices, ( ) ( ) 1 (1) (1) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 1 , , , , , ( ( )) {( ( , , )) ( ( , , ))}. k k d d r r d r r r k k d i i k k i i i i E T x w i i E h x x π π π π = = = ∑ ∏ ∏ L L L L L Thus any moment of permutation distribution can be considered as a summation of the product of data function term and index function term over a high dimensional index set and all possible permutations. Since all possible permutations map any index value between 1 and n to all possible index values from 1 to n with equal probability, ( ) ( ) 1 ( ) ( ) 1 ( ( , , )) k k d r i i k E h x x π π π =∏ L , the summation of data function over all permutations is only related to the equal/unequal relationship among indices. It is natural to divide the whole index set (1) (1) 1 , , , 1 { , , } {( ) d r i i d U i i = = L L L ( ) ( ) 1 , , , ( )} r r d i i L into the union of disjoint index subsets, in which ( ) ( ) 1 ( ) ( ) 1 ( ( , , )) k k d r i i k E h x x π π π =∏ L is invariant. Definition 1. Since h is a symmetric function, two index elements 1 ( , , ) d i i L and 1 ( , , ) d j j L are said to be equivalent if they are the same up to the order. For example, for d = 3, (1, 4, 5) = (1,5,4) = (4,1,5) = (4,5,1) = (5,1,4) = (5,4,1). Definition 2. Two indices (1) (1) ( ) ( ) 1 1 , , , , , , {( ) ( )} r r d d i i i i L L L and (1) (1) ( ) ( ) 1 1 , , , , , , {( ) ( )} r r d d j j j j L L L are said to be permutation equivalent/≡ if there exists a permutation n S π ∈ such that (1) (1) ( ) ( ) (1) (1) ( ) ( ) 1 1 1 1 , , , , , , , , , , , , {( ( ) ( )) ( ( ) ( ))} {( ) ( )} r r r r d d d d i i i i j j j j π π π π = L L L L L L . Here "=" means they have same index elements by Definition 1. For example, for d = 2, n = 4, r = 2, {(1, 2), (2, 3)} ≡ {(2, 4), (1, 4)} since we can apply π: 1→1, 2→4, 3→2, 4→3, such that {( π(1), π(2)), (π(2), π(3))} = {(1, 4), (4, 2)}= {(2, 4), (1, 4)}. As a result, the whole index set for d = 2, r = 2, can be divided into seven permutation equivalent subsets, [{(1, 1), (1, 1)}], [{(1, 1), (1, 2)}], [{(1, 1), (2, 2)}], [{(1, 2), (1, 2)}], [{(1, 1), (2, 3)}], [{(1, 2), (1, 3)}], [{(1, 2), (3, 4)}], where [ ] denotes the equivalence class. Note that the number of the permutation equivalent subsets is only related to the order of weighted v-test statistic d and the order of moment r , but not related to the data size n, and it is small for the first several moments calculation (small r) with low order test statistics (small d). Using the permutation equivalent relationship defined in Definition 2, the whole index set U can be partitioned into several permutation equivalent index subsets. Then we can calculate the r-th moment by summing up subtotals of all index subsets. This procedure can be done without any real permutations based on Proposition 1 and Proposition 2 below. Proposition 1. We claim that the data function sum ( ) ( ) 1 ( ) ( ) 1 ( ( , , )) k k d r i i k E h x x π π π =∏ L is invariant within each equivalent index subset, and ( ) ( ) 1 (1) (1) ( ) ( ) (1) (1) ( ) ( ) 1 1 1 1 ( ) ( ) 1 1 {( , , ), ,( , , )} [{( , , ), ,( , , )}] ( ) ( ) (1) (1) ( ) ( ) 1 1 1 ( , , ) ( ( , , )) , ([{( , , ), ,( , , )}]) k k d r r r r d d d d k k d r j j r k j j j j i i i i i i r r k d d h x x E h x x card i i i i π π π = ∈ = ∑ ∏ = ∏ L L L L L L L L L L L where (1) (1) ( ) ( ) 1 1 ([{( , , ), ,( , , )}]) r r d d card i i i i L L L is the number of indices falling into the permutation equivalent index subset (1) (1) ( ) ( ) 1 1 [{( , , ), ,( , , )}] r r d d i i i i L L L . Proof sketch: Since all indices in the same permutation equivalent subset are equivalent with respect to the symmetric group Sn, ( ) ( ) ( ) ( ) 1 1 ( ) ( ) ( ) ( ) 1 1 1 ( ( , , )) ( , , ) ! k k k k d d n r r i i i i S k k E h x x h x x n π π π π π π∈ = = = = ∑ ∏ ∏ L L ( ) ( ) 1 (1) (1) ( ) ( ) (1) (1) ( ) ( ) 1 1 1 1 1 {( , , ), ,( , , )} [{( , ), ,( , )}] (1) (1) ( ) ( ) 1 1 ( ( , , ) !) ([{( , , ), ,( , , )}]) , ! k k d r r r r d d d d r j j k j j j j i i i i r r d d h x x n card i i i i n = ∈ ∑ ∏ = L L L L L L L L L L ( ) ( ) 1 (1) (1) ( ) ( ) (1) (1) ( ) ( ) 1 1 1 1 1 {( , , ), ,( , , )} [{( , , ), ,( , , )}] (1) (1) ( ) ( ) 1 1 ( ( , , )) . ([{( , , ), ,( , , )}]) k k d r r r r d d d d r j j k j j j j i i i i r r d d h x x card i i i i = ∈ ∑ ∏ = L L L L L L L L L L Proposition 2. Thus we can obtain the r-th moment by summing up the production of the data partition sum wλ and the index partition sum hλ over all permutation equivalent subsets, i.e., [ ] ( ( )) r U E T x w h π λ λ λ∈ = ∑ , where (1) (1) ( ) ( ) 1 1 [{( , , ), ,( , , )}] r r d d i i i i λ = L L L is any permutation equivalent subset of the whole index set U. [U] denotes the set of all distinct permutation equivalent classes of U. The data partition sum is ( ) ( ) 1 (1) (1) ( ) ( ) 1 1 1 {( , , ), ,( , , )} ( ( , , )) , ( ) k k d r r d d r j j k j j j j h x x h card λ λ λ = ∈ ∑ ∏ = L L L L and the index partition sum is ( ) ( ) 1 (1) (1) ( ) ( ) 1 1 1 {( , , ), ,( , , )} ( ( , , )) . k k d r r d d r j j k j j j j w w x x λ λ = ∈ = ∑ ∏ L L L L Proof sketch: With Proposition 1, ( ) ( ) 1 ( ) ( ) 1 ( ( , , )) k k d r i i k E h x x π π π =∏ L is invariant within each equivalent index subset, therefore, ( ) ( ) 1 (1) (1) 1 ( ) ( ) 1 ( ) ( ) 1 ( ) ( ) 1 1 , , , , , ( ( )) {( ( , , )) ( ( , , ))} k k d d r r d r r r k k d i i k k i i i i E T x w i i E h x x π π π π = = = = ∑ ∏ ∏ L L L L L ( ) ( ) 1 (1) (1) ( ) ( ) 1 1 ( ) ( ) 1 ( ) ( ) [ ] 1 1 {( , , ), ,( , , )} {( ( , , )) ( ( , , ))} k k d r r d d r r k k d j j U k k j j j j w j j E h x x π π π λ λ ∈ = = ∈ = = ∑ ∑ ∏ ∏ L L L L L (1) (1) ( ) ( ) 1 1 ( ) ( ) 1 [ ] [ ] 1 {( , , ), ,( , , )} { ( , , ) } . r r d d r k k d U U k j j j j w j j h w h λ λ λ λ λ λ ∈ ∈ = ∈ = = ∑ ∑ ∑ ∏ L L L L Since both data partition sum wλ and the index partition sum hλ can be calculated by summation over all distinct indices within each permutation equivalent index subset, no any real permutation is needed for computing the moments. 2.3 Recursive calculation Direct calculation of the data partition sum and index partition sum leads to traversing throughout the whole index set. So the computational cost is O(ndr). In the following, we shall discuss how to reduce the cost by a recursive calculation algorithm. Definition 3. Let (1) (1) ( ) ( ) 1 1 [{( , , ), ,( , , )}] r r d d i i i i λ = L L L and (1) (1) ( ) ( ) 1 1 , , , , , , [( ) ( )] r r d d j j j j ν = L L L . λ and ν are two different permutation equivalent subsets of the whole index set U. We say that the partition order of ν is less than that of λ , i.e., ν λ ≺ , if λ can be converted to ν by merging two or more index elements. For instance, [(1,1),(2,3)] [(1,2),(3,4)] , ν λ = = ≺ since by merging 1 and 2, λ is converted to [{(1, 1), (3, 4)}] = [{(1, 1), (2, 3)}]. [{(1, 1), (3, 4)}] and [{(1, 1), (2, 3)}] are the same permutation equivalent index subsets because we can apply the permutation ¹: 1→1, 2→4, 3→3, 4→2 to [{(1, 1), (3, 4)}]. Note that the merging operation may not be unique, for example, ν can also be converted to λ by merging 3 and 4. To clarify the concept of partition order, we list the order of all partitions when d=2 and r=2 in figure 1. The partition order of a permutation equivalent subset ν is said to be lower than that of another permutation equivalent subset λ if there is a directed path from λ to ν . ( ) ( ) { } [ 1, 1 , 1,1 ] ( ) ( ) { } [ 1, 1 , 2, 2 ] ( ) ( ) { } [ 1, 2 , 1, 2 ] ( ) ( ) { } [ 1, 1 , 1, 2 ] ( ) ( ) { } [ 1, 1 , 2, 3 ] ( ) ( ) { } [ 1, 2 , 1, 3 ] ( ) ( ) { } [ 1, 2 , 3, 4 ] Figure 1: Order of all permutation equivalent subsets when d = 2 and r = 2. The difficulty for computing data partition sum and index partition sum comes from two constraints; equal constraint and unequal constraint. For example, in the permutation equivalent subset [{(1, 1), (2, 2)}], the equal constraint is that the first and the second index number are equal and the third and fourth index are also equal. On the other hand, the unequal constraint requires that the first two index numbers are different from those of the last two. Due to the difficulties mentioned, we solve this problem by first relaxing the unequal constraint and then applying the principle of inclusion and exclusion. Thus, the calculation of a partition sum can be separated into two parts: the relaxed partition sum without unequal constraint, and lower order partition sums. For example, [(1,1), (2,2)] [(1,1), (2,2)]* [(1,1), (1,1)] ( ( , ) ( , )) i j w w i i w j j w w λ λ λ = = = ≠ = = − = ∑ 2 2 , ( ( , ) ( , )) ( ( , ) ( , )) ( ( , )) ( , ) i j i j i i w i i w j j w i i w j j w i i w i i = = − = − ∑ ∑ ∑ ∑ , as the relaxed index partition sum 2 [(1,1), (2,2)]* , ( ( , ) ( , )) ( ( , )) i j i w w i i w j j w i i λ= = = ∑ ∑ . Proposition 3. The index partition sum wλ can be calculated by subtracting all lower order partition sums from the corresponding relaxed index partition sum * wλ , i.e., * #( ) #( ) #( ) w w w λ λ ν ν λ λ λ ν ν = − → ∑ ≺ , where #( ) λ is the number of distinct order-sensitive permutation equivalent subsets. For example, there are 2!2!2!/2!/2!=2 order-sensitive index partition types for [(1,1),(2, 3)] λ = . They are [(1, 1), (2, 3)] and [(2, 3), (1, 1)]. Note that [(1, 1), (2, 3)] and [(1, 1), (3, 2)] are the same type. #( ) λ ν → is the number of different ways of merging a higher order permutation equivalent subset λ to a low order permutation equivalent subset ν . The calculation of the data index partition sum is similar. Therefore, the computational cost mainly depends on the calculation of relaxed partition sum and the lowest order partition sum. Since the computational cost of the lowest order term is O(n), we mainly discuss the calculation of relaxed partition sums in the following paragraphs. To reduce the computational cost, we develop a greedy graph search algorithm. For demonstration, we use the following example. * [(1,1),(1,2),(1,2),(1,3),(2,3),(1,4)] , , , #( ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i j k l w w i i w i j w i j w i k w j k w i l λ λ = = ∑ . The permutation equivalent index subset is represented by an undirected graph. Every node denotes an index number. We connect two different nodes if these two corresponding index numbers are in the same index element, i.e., in the same small bracket. In figure 2, the number 2 on the edge ij denotes that the pair (i, j) is used twice. The self-connected node is also allowed. We assume there is no isolated subgraph in the following discussion. If any isolated subgraph exists, we only need to repeat the same procedure for all isolated subgraphs. Now we shall discuss the steps to compute the * [(1,1),(1,2),(1,2),(1,3),(2,3),(1,4)] wλ= . Firstly, we get rid of the weights of edges and self-connections, i.e., , , , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i j k l w i i w i j w i j w i k w j k w i l ∑ = , , , ( , ) ( , ) ( , ) ( , ) i j k l a i j w i k w j k w i l ∑ , as ( , ) ( , ) ( , ) ( , ) a i j w i i w i j w i j = . Then we search a node with the lowest degree and do summation for all indices connected with respect to the chosen node, i.e., , , , , , ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) ( , ) i j k l i j k a i j w i k w j k w i l b i j w i k w j k = ∑ ∑ , as ( , ) ( , ) ( , ) l b i j a i j w i l = ∑ . The chosen nodes and connected edges are deleted after the above computation. We repeat the same step until a symmetric graph occurs. Since every node in the symmetric graph has the same degree, we randomly choose any node; for example, k for summation, then , , , ( , ) ( , ) ( , ) ( , ) ( , ) i j k i j b i j w i k w j k b i j c i j = ∑ ∑ , as ( , ) ( , ) ( , ) k c i j w i k w j k = ∑ . Finally, we clear the whole graph and obtain the relaxed index partition sum. i j k l i k i j k l 2 i j j Figure 2: Greedy Search Algorithm for computing The most computational-expensive case is the complete graph in which every pair of nodes is connected. Hence, the computational cost of * cλ is determined by the subtotal that has the largest symmetric subgraph in its graph representation. For example, the most expensive relaxed index partition sum for d=2 and r=3 is ( , ) ( , ) ( , ) w i j w i k w j k , which is a triangle in the graph representation. Proposition 4 For d>=2, let ( 1) / 2 ( 1) / 2 ( 1) / 2 m m r d d m m − ≤ − < + , where r is the order of moment and m is an integer. For a d-th order test statistic, the computational cost of the partition sum for the r-th moment is bounded by O(nm). When d = 1 the computational complexity of the partition sum is O(n). Specifically, the computational cost of the 3rd and 4th moments for a second order test statistic is O(n3). The computational cost for the 1st and 2nd moments is O(n2). 2.4 Fitting The Pearson distribution series (Pearson I ~ VII) is a family of probability distributions that are more general than the normal distribution [12]. It covers all distributions in the (β1, β2) plane including normal, beta, gamma, log-normal, and etc., where distribution shape parameters β1, β2 are the square of standardized skewness and kurtosis measurements, respectively. Given the first four moments, the Pearson distribution series can be utilized to approximate the permutation distribution of the test statistic without conducting real permutation. 3 Experimental results To evaluate the accuracy and efficiency of our moments-based permutation tests, we generate simulated data and conduct permutation tests for both linear and quadratic test statistics. We consider six simulated cases in the first experiment for testing the difference between two groups, A and B. We use mean difference statistics here. For group A, n1 observations are generated independently from Normal(0,1) in Cases 1-2, from Gamma(3,3) in Cases 3-4, and from Beta(0.8, 0.8) in Cases 5-6. For group B, n2 independent observations are generated from Normal(1, 0.5) in Cases 1-2, from Gamma (3,2) in Cases 3-4, and from Beta(0.1, 0.1) in Cases 5-6. The design is balanced in Cases 1, 3, and 5 with n1 = n2 = 10, and unbalanced in Cases 2, 4, and 6 with n1 = 6, n2 = 18. Table 1 illustrates the high accuracy of our moments-based permutation technique. Furthermore, comparing with exact permutation or random 10,000 permutations, the moments-based permutation tests reduce more than 99.8% of the computation cost, and this efficiency gain increases with sample size. Table 1 shows the computation time and p-values of three permutation methods from one simulation. In order to demonstrate the robustness of our method, we repeated the simulation for 10 times in each case, and calculated the mean and variance of the absolute biases of p-values of both moments-based permutation and random permutation, treating the pvalues of exact permutation as gold standard. In most cases, our moments-based permutation is less biased and more stable than random permutation (Table 2), which demonstrates the robustness and accuracy of our method. Table 1: Comparison of computation costs and p-values of three permutation methods: Momentsbased permutation (MP), random permutation (RP), and exact permutation (EP). The t_MP, t_RP, and t_EP denote the computation time (in seconds), and p_MP, p_RP, and p_EP are the p-values of the three permutation methods. C a s e 1 C a s e 2 C a s e 3 C a s e 4 C a s e 5 C a s e 6 t _ M P 6.79e-4 5.37e-4 5.54e-4 5.16e-4 5.79e-4 6.53e-4 t _ R P 5.07e-1 5.15e-1 5.06e-1 1.30e-1 2.78e-1 5.99e-1 t _ E P 3.99e-0 1.21e-0 3.71e-0 1.21e-0 3.71e-0 1.22e-0 p_MP 1.19e-1 2.45e-2 1.34e-1 1.19e-1 3.58e-2 5.07e-5 p _ R P 1.21e-1 2.56e-2 1.36e-1 1.20e-1 3.53e-2 5.09e-2 p _ E P 1.19e-1 2.39e-2 1.34e-1 1.15e-1 3.55e-2 5.11e-2 We consider three simulated cases in the second experiment for testing the difference among three groups D, E, and F. We use modified F statistics [7] here. For group D, n1 observations are generated independently from Normal(0,1) in Case 7, from Gamma(3,2) in Case 8, and from Beta(0.8, 0.8) in Case 9. For group E, n2 independent observations are generated from Normal(0,1) in Case 7, from Gamma(3,2) in Case 8, and from Beta(0.8, 0.8) in Case 9. For group F, n3 independent observations are generated from Normal(0.1,1) in Case 7, from Gamma(3,1) in Case 8, and from Beta(0.1, 0.1) in Case 9.The design is unbalanced with n1 = 6, n2 = 8, and n3 =12. Since the exact permutation is too expensive here, we consider the p-values of 200,000 random permutations (EP) as gold standard. Our methods are more than one hundred times faster than 2,000 random permutation (RP) and also more accurate and robust (Table 3). We applied the method to the MRI hippocampi belonging to 2 groups, with 21 subjects in group A and 15 in group B. The surface shapes of different objects are represented by the same number of location vectors (with each location vector consisting of the spatial x, y, and z coordinates of the corresponding vertex) for our subsequent statistical shape analysis. There is no shape difference at a location if the corresponding location vector has an equal mean between two groups. Evaluation of the hypothesis test using our moments-based permutation with the modified Hotelling’s T2 test statistics [8] is shown in Fig. 3(a) and 3(b). It can be seen that the Pearson distribution approximation leads to ignorable discrepancy with the raw p-value map from real permutation. The false positive error control results are shown in Fig. 3(c). Table 2: Robustness and accuracy comparison of moments-based permutation and random permutation across 10 simulations, considering the p-values of exact permutation as gold standard. Mean_ABias_MP and VAR_MP are the mean of the absolute biases and the variance of the biases of p-values of moments-based permutation; Mean_ABias_RP and VAR_RP are the mean of the absolute biases and the variance of the biases of p-values of random permutation. Mean difference statistic is used. Case1 Case2 Case3 Case4 Case5 Case6 Mean_ABias_MP 1.62e-4 3.04e-4 6.36e-4 8.41e-4 1.30e-3 3.50e-3 Mean_ABias_RP 7.54e-4 3.39e-4 9.59e-4 8.39e-4 1.30e-3 2.00e-3 VAR_MP 6.42e-8 2.74e-7 1.54e-6 1.90e-6 3.76e-6 2.77e-5 VAR_RP 7.85e-7 1.86e-7 1.69e-6 3.03e-6 4.24e-5 1.88e-5 Table 3: Computation cost, robustness, and accuracy comparison of moments-based permutation and random permutation across 10 simulations. Modified F statistic is used. C a s e 7 C a s e 8 C a s e 9 C a s e 7 C a s e 8 C a s e 9 t_MP 1.03e-3 1.42e-3 1.64e-3 Mean_ABias_MP 9.23e-4 2.37e-4 2.11e-3 t_RP 1.51e-1 1.48e-1 1.38e-1 Mean_ABias_RP 3.94e-3 2.79e-3 3.42e-3 t_EP 1.76e+1 1.86e+1 2.37e+1 VAR_MP 1.10e-6 8.74e-8 1.23e-5 VAR_RP 2.27e-5 1.48e-5 1.85e-5 (a) (b) (c) (d) =0.0 p-value>0.05 =0.05 (e) Figure 3. (a) and (b): Comparison of techniques in raw p-value measurement at 0.05 α = (without correction), through real permutation ((a); number of permutations = 10,000) and using the present moments-based permutation (b). (c) p-map after BH’s FDR correction of (b). (e) Facial differences between Asian male and white male. Locations in red on the 3D surface denote significant face shape differences (significance level α = 0.01 with false discovery rate control). We also applied our method to the 3D face comparison between Asian males and white males. We choose 10 Asian males and 10 white males out of the USF face database to calculate their differences with the modified Hotelling’s T2 test statistics. Each face surface is represented by 4,000 voxels. All surfaces are well aligned. Results from our algorithm in Fig. 3(e) show that significant differences occur at eye edge, nose, lip corners, and cheeks. They are consistent with anthropology findings and suggest the discriminant surface regions for ethnic group recognition. 4 Conclusion We present and develop novel moments-based permutation tests where the permutation distributions are accurately approximated through Pearson distributions for considerably reduced computation cost. Comparing with regular random permutation, the proposed method considerably reduces computation cost without loss of accuracy. General and analytical formulations for the moments of permutation distribution are derived for weighted v-test statistics. The proposed strategy takes advantage of nonparametric permutation tests and parametric Pearson distribution approximation to achieve both accuracy/flexibility and efficiency. References [1] Nichols, T. E., and A. P. Holmes (2001), Nonparametric permutation tests for functional neuroimaging: A primer with examples, Human Brain Mapping, 15, 1-25. [2] Zhou, C., D. C. Park, M. Styner, and Y. M. Wang (2007), ROI constrained statistical surface morphometry, IEEE International Symposium on Biomedical Imaging, Washington, D. C., 1212-1215. [3] Zhou, C., and Y. M. Wang (2008), Hybrid permutation test with application to surface shape analysis, Statistica Sinica, 18, 1553-1568. [4] Pantazis, D., R. M. Leahy, T. E. Nichols, and M. Styner (2004), Statistical surfacebased morphometry using a non-parametric approach, IEEE International Symposium on Biomedical Imaging, 2, 1283-1286. [5] Zhou, C., Y. Hu, Y. Fu., H. Wang, Y. M. Wang, and T. S. Huang (2008), 3D face analysis for distinct features using statistical randomization, IEEE International Conference on Acoustics, Speech, and Signal Processing, Las Vegas, Nevada, 981984. [6] Hubert, L. (1987), Assignment Methods in Combinatorial Data Analysis, Marcel Dekker, New York. [7] Mielke, P. W., and K. J. Berry (2001), Permutation Methods: A Distance Function Approach, Springer, New York. [8] Good, P. (2005), Permutation, Parametric and Bootstrap Tests of Hypotheses, 3rd ed., Springer, New York. [9] Serfling, R. J. (1980), Approximation Theorems of Mathematical Statistics, Wiley, New York. [10] Edgington, E., and P. Onghena (2007), Randomization Tests, 4th ed., Chapman & Hall, London. [11] Nicholson, W. K. (2006), Introduction to Abstract Algebra, 3rd ed., Wiley, New York. [12] Hahn, G. J., and S. S. Shapiro (1967), Statistical Models in Engineering, John Wiley and Sons, Chichester, England.
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Maximum likelihood trajectories for continuous-time Markov chains Theodore J. Perkins Ottawa Hospital Research Institute Ottawa, Ontario, Canada tperkins@ohri.ca Abstract Continuous-time Markov chains are used to model systems in which transitions between states as well as the time the system spends in each state are random. Many computational problems related to such chains have been solved, including determining state distributions as a function of time, parameter estimation, and control. However, the problem of inferring most likely trajectories, where a trajectory is a sequence of states as well as the amount of time spent in each state, appears unsolved. We study three versions of this problem: (i) an initial value problem, in which an initial state is given and we seek the most likely trajectory until a given final time, (ii) a boundary value problem, in which initial and final states and times are given, and we seek the most likely trajectory connecting them, and (iii) trajectory inference under partial observability, analogous to finding maximum likelihood trajectories for hidden Markov models. We show that maximum likelihood trajectories are not always well-defined, and describe a polynomial time test for well-definedness. When well-definedness holds, we show that each of the three problems can be solved in polynomial time, and we develop efficient dynamic programming algorithms for doing so. 1 Introduction A continuous-time Markov chain (CTMC) is a model of a dynamical system which, upon entering some state, remains in that state for a random real-valued amount of time (called the dwell time or occupancy time) and then transitions randomly to a new state. CTMCs are used in a wide variety of domains. In stochastic chemical kinetics, states may correspond to the conformation of a molecule such as a protein, peptide or nucleic acid polymer, and transitions correspond to conformational changes (e.g., [1]). Or, the state may correspond to the numbers of different types of molecules in an interacting system, and transitions are the result of chemical reactions between molecules [2]. In phylogenetics, the states may correspond to the genomes of different organisms, and transitions to the evolutionary events (mutations) that separate those organisms [3]. Other application domains include queueing theory, process control and manufacturing, quality control, formal verification, and robot nagivation. Many computational problems associated with CTMCs have been solved, often by generalizing methods developed for discrete-time Markov chains (DTMCs). For example, stationary distributions for CTMCs can be computed in a manner very similar to that for DTMCs [4]. Estimating the parameters of a CTMC from fully observed data involves estimating state transition probabilities, just as for DTMCs, but adds estimation of the state dwell time distributions. Estimating parameters from partially observed data can be done by a generalization of the well-known Baum-Welch algorithm for parameter estimation for hidden Markov models [5] or by Bayesian methods [6, 7]. When the state of a CTMC is observed periodically through time, but some transitions between observation times may go unseen, the parameter estimation problem can also be solved through embedding 1 techniques [8]. In scenarios such as manufacturing or robot navigation, one may assume that the state transitions or dwell times are under at least partial control. When control choices are made once for each state entered, dynamic programming and related methods can be used to develop optimal control strategies [9]. When control choices are made continuously in time, methods for hybrid system control are more appropriate [10]. Another fundamental and well-studied problem for CTMCs is to compute, given an initial state and time, the state distribution or most likely state at a later time. These problems are readily solved for DTMCs by dynamic programming [11], but for the CTMCs, solutions have a somewhat different flavor. One approach is based on the forward Chapman-Kolmogorov equations [4], called the Master equation in the stochastic chemical kinetics literature [12]. These specify a system of ordinary differential equations the describe how the probabilities of being in each state change over time. Solving the equations, sometimes analytically but more often numerically, yields the entire state distribution as a function of time. Alternatively, one can uniformize the CTMC, which produces a DTMC along with a probability distribution for a number of transitions to perform. The process obtained by choosing the number of transitions, and then producing a trajectory with that many transitions from the DTMC, has the same state distribution as the original CTMC. This representation allows particularly efficient computation of the state distribution if that distribution is only required at one or a smaller number of different times. Finally, especially in the chemical kinetics community, stochastic simulation algorithms are popular [13]. These approaches act by simply simulating trajectories from the CTMC to produce empirical, numerical estimates of state distributions or other features of the dynamics. Despite the extensive work on a variety of problems related to to CTMCs, to the best of our knowledge, the problem of finding most likely trajectories has not been addressed. With this paper, we attempt to fill that gap. We propose dynamic programming solutions to three variants of the problem: (i) an initial value problem, where a starting state and final time are given, and we seek the most likely sequence of states and dwell times occurring up until the final time, (ii) a boundary value problem, where initial and final states and times are given, and we seek the most likely intervening trajectory, and (iii) a problem involving partial observability, where we have a sequence of “observations” that may not give full state information, and we want to infer the most likely trajectory that the system followed in producing the observations. 2 Definitions A CTMC is defined by four things: (i) a finite state set S, (ii) initial state probabilities, Ps for s ∈S, (iii) state transition probabilities Pss′ for s, s′ ∈S, and (iv) state dwell time parameters λs for each s ∈S. Let St ∈S denote the state of the system at time t ∈[0, +∞). The rules for the evolution of the system are that it starts in state S0, which is chosen according to the distribution Ps. At any time t, when the system is in state St = s, the system stays in state s for a random amount of time that is exponentially distributed with parameter λs. When the system finally leaves state s, the next state of the system is s′ ̸= s with probability Pss′. A trajectory of the CTMC is a sequence of states along with the dwell times in all but the last state U = (s0, t0, s1, t1, . . . , sk−1, tk−1, sk). The meaning of this trajectory is that the system started in state s0, where it stayed for time t0, then transitioned to state s1, where it stayed for time t1, and so on. Eventually, the system reaches state sk, where it remains. Let Ut = (s0, t0, s1, t1, . . . , skt−1, tkt−1, skt) be a random variable describing the trajectory of the system up until time t. In particular, this means that there are kt state transitions up until time t (where kt is itself a random variable), the system enters state skt sometime at or before time t, and remains in state skt until sometime after time t. Given the initial state, S0, and a time t, the likelihood of a particular trajectory U is l(Ut = U|S0) = ( 0 if s0 ̸= S0 or Pk−1 i=0 ti > t Πk−1 i=0 λsie−λsitiPsisi+1 e−λsk(t−P i ti) otherwise (1) When P i ti > t, the likelihood is zero, because it means that the specified transitions have not completed by time t. Otherwise, the terms inside the first parentheses account for the likelihood of the dwell times and the state transitions in the sequence, and the term inside the second parentheses 2 accounts for the probability that the dwell time in the final state does not complete before time t. With this notation, the initial value problem we study is easily stated as arg max U l(Ut = U|S0 = s) , (2) where s ∈S and t > 0 are both given. The boundary value problem we study is arg max U l(Ut = U|S0 = s, St = s′). (3) Here, the given s and s′ are any states in S, possibly the same state, and t > 0 is also given. A hidden continuous-time Markov chain (HCTMC) adds an observation model to the CTMC. In particular, we assume a finite set of possible observations O. When the system is observed and it is in state s ∈S, the observer sees observation o ∈O with probability Pso. Let O = (o1, τ1, o2, τ2, . . . , om, τm) denote a sequence of observations and the times at which they are made. We assume that the observation times are fixed, being chosen ahead of time, and depend in no way on the evolution of the chain itself. Given a trajectory of the system U = (s0, t0, s1, t1, . . . , tk−1, sk), let U(t) denote the state of the system at time t implied by that sequence. Then, the probability of an observation sequence O given the trajectory U can be written as P(O|Uτm = U) = Πm i=1PU(τi)oi (4) The final problem we study in this paper is that of finding the most likely trajectory given an observation sequence: arg max U l(Uτm = U|O) ∝arg max U P(O|Uτm = U)l(Uτm = U) (5) 3 Solving the initial and boundary value problems In this section we develop solutions to problems (2) and (3). The first step in this development is to show that we can analytically optimize the dwell times if we are given the state sequence. This is covered in the next subsection. Following that, we develop a dynamic program to find optimal state sequences, assuming that the dwell times are set to their optimal values relative to the state sequence. 3.1 Maximum likelihood dwell times Consider a particular trajectory U = (s0, t0, s1, t1, . . . , sk−1, tk−1, sk). Given S0 and a time t, the likelihood of that particular trajectory, l(Ut = U|S0) is given above by Equation (1). Let us assume that S0 = s0, as we have no need to consider U starting from the wrong state, and let us maximize l(Ut = U|S0) with respect to the dwell times. To be concise, let Ttk = {(t0, t1, . . . , tk−1) : ti ≥ 0 for all 0 ≤i < k and P i ti ≤t}. This is the set of all feasible dwell times for the states up until state sk. Then we can write the desired optimization as arg max(t0,...,tk−1)∈Ttk Πk−1 i=0 λsie−λsitiPsisi+1 e−λsk (t−Σiti) . (6) It is more convenient to maximize the logarithm, which gives us arg max(t0,...,tk−1)∈Ttk k−1 X i=0 log λsi −λsiti + log Psisi+1 ! −λsk (t −Σjtj) (7) Dropping the terms that do not depend on any of the ti and rearranging, we find the equivalent problem arg max(t0,...,tk−1)∈Ttk k−1 X i=0 (λsk −λsi)ti (8) The solution can be obtained by inspection. If λsk ≤λsi for all 0 ≤i < k, then we must have all ti = 0. That is, the system transitions instantaneously through the states s0, s1, ..., sk−1 and then 3 dwells in state sk for (at least) time t.1 Otherwise, let j be such that λsj is minimal for 0 ≤j < k. Then an optimal solution has tj = t, and all other ti = 0. Intuitively, this says that if state sj has the largest expected dwell time (corresponding to the smallest λ parameter), then the most likely setting of dwell times is obtained by assuming all of the time t is spent in state sj, and all other transitions happen instantaneously. This is not unintuitive, although it is dissatisfying in the sense that the most likely set of dwell times are not typical in some sense. For example, none are near their expected value. Moreoever, the basic character of the solution—that all the time t goes into waiting at the slowest state—is independent of t. Nevertheless, being able to solve explicitly for the most likely dwell times for a given state sequence makes it much easier to find the most likely Ut. So, let us press onwards. 3.2 Dynamic programming for the most likely state sequence Substituting back our solution for the ti into Equation (1), and continuing our assumption that s0 = S0, we obtain max (t0,...,tk−1)∈Ttk l(Ut = U|S0) = Πk−1 i=0 λsiPsisi+1 e−λsk t if λsk ≤λsi for all 0 ≤i < k Πk−1 i=0 λsiPsisi+1 e−(mink−1 i=0 λsi)t otherwise = Πk−1 i=0 λsiPsisi+1 e−(mink i=0 λsi)t (9) This leads to a dynamic program for finding the state sequence that maximizes the likelihood. As is typical, we build maximum likelihood paths of increasing length by finding the best ways of extending shorter paths. The main difference with a more typical scenario is that to score an extension we need to know not just the score and final state of the shorter path, but also the smallest dwell time parameter along that path. Define a (k, s, λ)-trajectory to be one that includes k ∈{0, 1, 2, . . .} state transitions, ends at state sk = s, and for which the smallest dwell time parameter of any state along the trajectory is λ. Then define Fk(s, λ) to be the maximum achievable l(Ut = U|S0), where we restrict attention to U that are (k, s, λ)-trajectories. We initialize the dynamic program as: F0(S0, λS0) = e−tλS0 F0(s, λ) = 0 for all (s, λ) ̸= (S0, λS0) To compute Fk(s, λ) for larger k, we first observe that Fk(s, λ) is undefined if λ > λs. This is because there are no (k, s, λ)-trajectories if λ > λs. The fact that a trajectory ends at state s implies that the minimum dwell time parameter along the trajectory can be no greater than λs. So, we only compute Fk(s, λ) for λ ≤λs. To determine Fk+1(s, λ), we must consider two cases. If λ < λs, then the best (k + 1, s, λ)trajectory must come from some (k, s′, λ)-trajectory. That is, the length k trajectory must already have a dwell time parameter of λ along it. The state s′ can be any state other than s. If λ = λs, then the best (k + 1, s, λ)-trajectory may be an extension of any (k, s′, λ′)-trajectory with λ′ ≥λ and s ̸= s′. To be more concise, define G(s, λ) = {λ} if λ < λs {λs′ : λs′ ≥λ} if λ = λs (10) We then compute F for increasing k as: Fk+1(s, λ) = max s′̸=s,λ′∈G(s,λ) Fk(s′, λ′)λs′Ps′se−t(λ−λ′) The first term on the right hand side accounts for the likelihood of the best (k, s′, λ′)-trajectory. The next two terms account for the dwell in s′ and the transition probability to s. The final term accounts for any difference between the smallest dwell time parameters along the k and k + 1 transition trajectories. 1If the reader is not comfortable with a dwell time exactly equal to zero, one may instead take ti = 0 as a shorthand for an infinitesimal but positive dwell time. Alternatively, the optimization problem can be modified to explicitly require ti > 0. However, this does nothing to change the fundamental nature of the solution, while resulting in a significantly more laborious exposition. 4 a b x z y λy=1/10 λx=1 λz=1 λa Pxz=1/3 Pxy=2/3 Pyz=Pza=Pab=Pba=1 λb Figure 1: A continuous-time Markov chain used as a demonstration domain. The five circles correspond to states, and the arrows to transitions between states. States are also labeled with their dwell time parameters. Because the set of possible states, S, is finite, so is the set of possible dwell time parameters, λs for s ∈S. The size of the table Fk for each k is thus at most |S|2. If we limit k to some maximum value K, then the total size of all the tables is at most K|S|2, and the total computational effort O(K|S|3). To solve the initial value problem (2), we scan over all values of k, s and λ to find the maximum value of Fk(s, λ). Such a value implies that the most likely state sequence ends at state s after k state transitions. We can use a traceback to reconstitute the full sequence of states, and the result of the previous section to obtain the most likely dwell times. To solve the boundary value problem (3), we do the same, except that we only scan over values of k and λ, looking for the maximum value of Fk(St, λ). 3.3 Examples In this section, we use the toy chain depicted in Figure 1 to demonstrate the algorithm of the previous section, and to highlight some properties of maximum likelihood trajectories. First, suppose that we know the system is in state x at time zero and in state z at time t. There are two different paths, (x, z) and (x, y, z), that lead from x to z. If we ignore the issue of dwell times and consider only the transition probabilities, then the path (x, y, z) seems more probable. Its probability is PxyPyz = 2 3 · 1 = 2 3, whereas the direct path (x, z) simply has probability Pxz = 1 3. However, if we consider the dwell times as well, the story can change. For example, suppose that t = 1. Note that λy = 1 10, so that the expected dwell time in state y is 10. If the chain enters state y, the chance of it leaving y before time t = 1 is quite small. If we run the dynamic programming algorithm of the previous section to find the most likely trajectory, it finds (s0 = x, t0 = 0, s1 = z) to be most likely, with a score of 0.1226. Along the way, it computes the likelihood of the most likely path going through y, which is (s0 = x, t0 = 0, s1 = y, t1 = t, s2 = x). It prefers to place all the dwell time t in state y, because that state is most likely to have a long dwell time. However, the total score of this trajectory is still only 0.0603, making the direct path the more likely one. On the other hand, if t = 2, then the path through y becomes more likely by a score of 0.0546 to 0.0451. If t = 10, then the path through y still has a likelihood of 0.0245, whereas the direct path has a likelihood below 2 × 10−5, because it is highly unlikely to remain in x and/or z for so long. Next, suppose that we know S0 = a and that we are interested in knowing the most likely trajectory out until time t, regardless of the final state of that trajectory. For simplicity, suppose also that λa = λb. There is only one possible state sequence containing k transitions for each k = 0, 1, 2, . . ., and the likelihood of any such sequence turns out to be independent of the dwell times (assuming the dwell times total no more than time t): (Πk−1 i=0 λe−λti)e−λ(t−Σiti) = e−λtλk (11) If λ < 1, this implies the optimal trajectory has the system remaining at state a. However, if λ = 1 then all trajectories of all lengths have the same likelihood. If λ > 1, then there are trajectories of arbitrarily large likelihood, but no maximum likelihood trajectory. Intuitively, because the likelihood of a dwell time can be greater than one, the likelihood of a trajectory can be increased by including short dwells in states with high dwell parameters λ. In general, if a continuous-time Markov chain has a cycle of states (s0, s1, . . . , sk = s0), such that Πk−1 i=0 Psisi+1λsi > 1, then maximum likelihood trajectories do not exist. Rather, a sequence of 5 s0 s1 s2 s3 o1 o2 o3 o4 time Figure 2: Abstract example of a continuous-time trajectory of a chain, along with observations taken at fixed time intervals. trajectories with ever-increasing likelihood can be found starting from any state from which the cycle is reachable. One should, thus, always check the chain for this property before seeking maximum likelihood trajectories. This can be easily done in polynomial time. For example, one can label the edges of the transition graph with the weights log Pss′λs for the edge from s to s′, and then check the graph for the existence of a positive-weight cycle—a well-known polynomial-time computation. 4 Solving the partially observable problem We now turn to problem (12), where we are given an observation sequence O = (o1, τ1, o2, τ2, . . . , om, τm) and want to find the most likely trajectory U. For simplicity, we assume that τ1 = 0. The following can be straightforwardly generalized to allow the first observation to take place sometime after the trajectory begins. Similarly, we restrict attention to trajectories U = (s0, t0, s1, t0, . . . , tk−1, sk) where P i tk ≤τm, so that we do not concern ourselves with extrapolating the trajectory beyond the final observation time. The conditional likelihood of such a trajectory can be written as l(Uτm = U|O) ∝ P(O|Uτm = U)l(Uτm = U) (12) = Πm i=1PU(τi)oi Ps0 Πk−1 i=0 λsie−λsitiPsisi+1 e−λsk (t−Σiti) (13) The term in the first parentheses is P(O|Uτm = U), and the term in the second parentheses is l(Uτm = U). The only differences between the second parentheses and Equation (1) is that we now include the probability of starting in state s0, and we have implicitly assumed that P i tk ≤τm, as mentioned above. This form, however, is not convenient for optimizing U. To do this, we need to rewrite l(Uτm = U) in a way that separates the likelihood into events happening in each interval of time between observations. 4.1 Decomposing trajectory likelihood by observation intervals For simplicity, let us further restrict attention to trajectories U that do not include a transition into a state si precisely at any observation time τj. We do not have space here to show that this restriction does not affect the value of the optimization problem; this will be addressed in the full paper. The likelihood of the trajectory can be written in terms of the events in each observation interval. For example, consider the trajectory and observations depicted in Figure 2. In the first interval, the system starts in state s0 and transitions to s1, where it stays until time τ2. The likelihood of this happening is Ps0λs0e−λs0t0Ps0s1e−λs1(τ2−t0). In the second observation interval, the system never leaves state s1. The probability of this happening is e−λs1(τ3−τ2). Finally, in the third interval, the system continues in state s1 before transitioning to state s2 and then s3, where it remains until the final observation. The likelihood of this happening is λs1e−λs1(t0+t1−τ3)Ps1s2λs2e−λs2t2ps2s3e−λs3(τ4−t0−t1−t2). If we multiply these together, we obtain the full likelihood of the trajectory, Ps0(Π2 i=0λsie−λsiti)e−λs3(τ4−Σjtj). In general, let Ui = (si0, ti0, si1, ti1, . . . , siki) denote the sequence of states and dwell times of trajectory U during the time interval [τi, τi+1). The first dwell time ti0, if any, is measured with respect to the start of the time interval. The component of the likelihood of the whole trajectory U attributable to the ith time interval is nothing other than l(Uτi+1−τi = Ui|S0 = si0). Thus, the likelihood of the whole trajectory can be written as l(Uτm = U) = Ps0Πm−1 i=1 l(Uτi+1−τi = Ui|S0 = si0) (14) 6 4.2 Dynamic programming for the optimal trajectory Combining Equations (12) and (14), we find l(Uτm = U|O) ∝PU(0)PU(0)o1Πm−1 i=1 l(Uτi+1−τi = Ui|S0 = U(τi))PU(τi+1)oi+1 (15) The first two terms account for the probability of the initial state and the probability of the first observation given the initial state. The terms inside the product account for the likelihood of the ith interval of the trajectory, and the probability of the (i + 1)st observation, given the state at the end of the ith interval of the trajectory. One immediate implication of this rewriting of the conditional likelihood is the following. At times τi and τi+1, the system is in states U(τi) and U(τi+1). If U is to maximize the conditional likelihood, it had better be that the fragment of the trajectory between those two times, Ui, is a maximum likelihood trajectory from state U(τi) to state U(τi+1) in time τi+1 −τi. If it is not, then an alternative, higher likelihood trajectory fragment could be swapped into U, resulting in a higher conditional likelihood. Let us define Ht(s, s′) = max U′ l(Ut = U′|S0 = s, St = s′) (16) to be the maximum achievable likelihood by any trajectory from state s to state s′ in time t. Then a necessary condition for U to maximize the conditional likelihood is l(Uτi+1−τi = Ui|S0 = U(τi)) = Hτi+1−τi(U(τi), U(τi+1)) . (17) Moreover, to find an optimal U, we can simply assume that the above condition holds, and concern ourselves only with finding the best endpoints for the each time interval, U(τi) and U(τi+1). (Of course, the endpoint of one interval must be the same as the initial point of the next interval.) Specifically, define Ji(s) to be the likelihood of the most likely trajectory covering the time interval [τ1, τi], accounting for the first i observations, and ending at state s. The we can compute J as follows. To initialize, we set J1(s) = PsPso1 . (18) Then, for i = 1, 2, . . . , m −1, Ji+1(s) = max s′ Ji(s′)Hτi+1−τi(s′, s)Psoi+1 . (19) We can then reconstruct the most likely trajectory by finding s that maximizes Jm(s) and tracing back to the beginning. This algorithm is identical to the Viterbi algorithm for finding most likely state sequences for hidden Markov models, with the exception that the state transition probabilities in the Viterbi algorithm are replaced by the Hτi+1−τi(s′, s) terms above—which can, of course, be computed based on the results of the previous section. 4.3 Examples To demonstrate this algorithm, let us return to the CTMC depicted in Figure 1. We assume that λa = λb = 1, that the system always starts in state x, and that when we observe the system, we get a real-valued Gaussian observation with standard deviation 1 and means 0, 10, 3, 100 and 100 for states x, y, z, a and b respectively.2 The left side of Figure 3 shows three sample sequences of 20 observations. The right side of the figure shows the most likely trajectories inferred under different assumptions. First, if we assume the time interval between observations is t = 1, and we consider observations OA, then the most likely trajectory has the system in state x up through the 10th observation, after which it instantly transitions to state z and remains there. This makes sense, as the lower observations at the start of the series are more likely in state x. If we consider instead observations OB, which has a high observation at time t = 11, the procedure infers that the system was in state y at that time. Moreover, it predicts that the system switches into y immediately after the 10th observation, and says there until just before the 12th observation, taking advantage of the fact that longer dwell times are more likely in state y than in the other states. If we consider observations OC, which have a spike at t = 5, the transit to state y is moved earlier, and state z is used to explain observations at t = 6 onward, even though the first few are relatively unlikely in that state. If we 2Although our derivations above assume the observation set O is finite, the same approach goes through if O is continuous and individual observations have likelihoods instead of probabilities. 7 x z OA, t=20 y x z OA, t=2 y x z OC, t=1 y x z OB, t=1 y x z OA, t=1 observation number 1 5 10 15 20 1 5 10 15 20 −2 0 2 4 6 8 10 observation number observation OA OB OC Figure 3: Left: three length-20 observation sequences, OA, OB, and OC. All three are the same at most points, but the 11th observation of OB is 10, and the 5th observation of OC is 10. Right: most likely trajectories inferred by our algorithm, assuming the underlying CTMC is the one given in Figure 1, with parameters given in the text. return to observations OA, but we assume that the time interval between observations is t = 2, then the most likely trajectory is different than it is for t = 1. Although the same states are used to explain the observations, the most likely trajectory has the system transitioning from x to y immediately after the 10th observation and dwelling there until just before the 11th observation, where the state becomes z. This is because, as explained previously, this is the more likely trajectory from x to z given t = 2. If we assume the time interval between observations is t = 20, then a wider range of observations during the trajectory are attributed to state y. Intuitively, this is because, although the observations are somewhat unlikely under state y, it is extremely unlikely for the system to dwell for so long in state z as to account for all of the observations from the 11th onward. 5 Discussion We have provided correct, efficient algorithms for inferring most likely trajectories of CTMCs given either initial or initial and final states of the chain, or given noisy/partial observations of the chain. Given the enormous practical import of the analogous problems for discrete-time chains, we are hopeful that our methods will prove useful additions to the toolkit of methods available for analyzing continuous-time chains. An alternative, existing approach to the problems we have addressed here is to discretize time, producing a DTMC which is then analyzed by standard methods [14]. A problem with this approach, however, is that if the time step is taken too large, the discretized chain can collapse a whole set of transition sequences of the CTMC into a single “pseudotransition”, obscuring the real behavior of the system in continuous time. If the time step is taken to be sufficiently small, then the DTMC should produce substantially the same solutions as our approach. However, the time complexity of the calculations increases as the time step shrinks, which can be a problem if we are interested in long time intervals and/or there are states with very short expected dwell times, necessitating very small time steps. A related problem on which we are working is to find the most probable state sequence of a continuous-time chain under similar informational assumptions. By this, we mean that the dwell times, rather than being optimized, are marginalized out, so that we are left with only the sequence of states and not the particular times they occurred. In many applications, this state sequence may be of greater interest than the dwell times—especially since, as we have shown, maximum likelihood dwell times are often infinitessimal and hence non-representative of typical system behavior. Morever, this version of the problem has the advantage of always being well-defined. Because state sequences have probabilities rather than likelihoods, a most probable state sequence will always exist. Acknowledgments Funding for this work was provided in part by the National Sciences and Engineering Research Council of Canada and by the Ottawa Hospital Research Institute. 8 References [1] FG Ball and JA Rice. Stochastic models for ion channels: introduction and bibliography. Mathematical biosciences, 112(2):189, 1992. [2] D.J. Wilkinson. Stochastic modelling for systems biology. Chapman & Hall/CRC, 2006. [3] M. Holder and P.O. Lewis. Phylogeny estimation: traditional and Bayesian approaches. Nature Reviews Genetics, 4(4):275–284, 2003. [4] H.M. Taylor and S. Karlin. An introduction to stochastic modeling. Academic Press, 1998. [5] D.R. Fredkin and J.A. Rice. Maximum likelihood estimation and identification directly from single-channel recordings. Proceedings: Biological Sciences, pages 125–132, 1992. [6] R. Rosales, J.A. Stark, W.J. Fitzgerald, and S.B. Hladky. Bayesian restoration of ion channel records using hidden Markov models. Biophysical Journal, 80(3):1088–1103, 2001. [7] M.A. Suchard, R.E. Weiss, and J.S. Sinsheimer. Bayesian selection of continuous-time Markov chain evolutionary models. Molecular Biology and Evolution, 18(6):1001–1013, 2001. [8] DT Crommelin and E. Vanden-Eijnden. Fitting timeseries by continuous-time Markov chains: A quadratic programming approach. Journal of Computational Physics, 217(2):782–805, 2006. [9] M. L. Puterman. Markov Decision Processes: Discrete Stochastic Dynamic Programming. John Wiley and Sons, New York, 1994. [10] S. Hedlund and A. Rantzer. Optimal control of hybrid systems. In Decision and Control, 1999. Proceedings of the 38th IEEE Conference on, volume 4, 1999. [11] D.P. Bertsekas. Dynamic programming and optimal control. Athena Scientific Belmont, Mass, 1995. [12] NG Van Kampen. Stochastic processes in physics and chemistry. North-Holland, 2007. [13] D. T. Gillespie. Exact stochastic simulation of coupled chemical reactions. Journal of Physical Chemistry, 81:2340–2361, 1977. [14] A. Hordijk, D.L. Iglehart, and R. Schassberger. Discrete time methods for simulating continuous time Markov chains. Advances in Applied Probability, pages 772–788, 1976. 9
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Lower bounds on minimax rates for nonparametric regression with additive sparsity and smoothness Garvesh Raskutti1, Martin J. Wainwright1,2, Bin Yu1,2 1UC Berkeley Department of Statistics 2UC Berkeley Department of Electrical Engineering and Computer Science Abstract We study minimax rates for estimating high-dimensional nonparametric regression models with sparse additive structure and smoothness constraints. More precisely, our goal is to estimate a function f ∗: Rp →R that has an additive decomposition of the form f ∗(X1, . . . , Xp) = P j∈S h∗ j(Xj), where each component function h∗ j lies in some class H of “smooth” functions, and S ⊂{1, . . . , p} is an unknown subset with cardinality s = |S|. Given n i.i.d. observations of f ∗(X) corrupted with additive white Gaussian noise where the covariate vectors (X1, X2, X3, ..., Xp) are drawn with i.i.d. components from some distribution P, we determine lower bounds on the minimax rate for estimating the regression function with respect to squared-L2(P) error. Our main result is a lower bound on the minimax rate that scales as max s log(p/s) n , s ǫ2 n(H) . The first term reflects the sample size required for performing subset selection, and is independent of the function class H. The second term s ǫ2 n(H) is an s-dimensional estimation term corresponding to the sample size required for estimating a sum of s univariate functions, each chosen from the function class H. It depends linearly on the sparsity index s but is independent of the global dimension p. As a special case, if H corresponds to functions that are m-times differentiable (an mth-order Sobolev space), then the s-dimensional estimation term takes the form sǫ2 n(H) ≍s n−2m/(2m+1). Either of the two terms may be dominant in different regimes, depending on the relation between the sparsity and smoothness of the additive decomposition. 1 Introduction Many problems in modern science and engineering involve high-dimensional data, by which we mean that the ambient dimension p in which the data lies is of the same order or larger than the sample size n. A simple example is parametric linear regression under high-dimensional scaling, in which the goal is to estimate a regression vector β∗∈Rp based on n samples. In the absence of additional structure, it is impossible to obtain consistent estimators unless the ratio p/n converges to zero which precludes the regime p ≫n. In many applications, it is natural to impose sparsity conditions, such as requiring that β∗have at most s non-zero parameters for some s ≪p. The method of ℓ1-regularized least squares, also known as the Lasso algorithm [14], has been shown to have a number of attractive theoretical properties for such high-dimensional sparse models (e.g., [1, 19, 10]). Of course, the assumption of a parametric linear model may be too restrictive for some applications. Accordingly, a natural extension is the non-parametric regression model y = f ∗(x1, . . . , xp)+w, where w ∼N(0, σ2) is additive observation noise. Unfortunately, this general non-parametric model is known to suffer severely from the “curse of dimensionality”, in that for most natural function classes, the sample size n required to achieve a given estimation accuracy grows exponentially in the dimension. This challenge motivates the use of additive non-parametric models (see the book [6] and references therein), in which the function f ∗is decomposed additively as a sum f ∗(x1, x2, ..., xp) = Pp j=1 h∗ j(xj) of univariate functions h∗ j. A natural sub-class of these 1 models are the sparse additive models, studied by Ravikumar et. al [12], in which f ∗(x1, x2, ..., xp) = X j∈S h∗ j(xj), (1) where S ⊂{1, 2, . . . , p} is some unknown subset of cardinality |S| = s. A line of past work has proposed and analyzed computationally efficient algorithms for estimating regression functions of this form. Just as ℓ1-based relaxations such as the Lasso have desirable properties for sparse parametric models, similar ℓ1-based approaches have proven to be successful. Ravikumar et al. [12] propose a back-fitting algorithm to recover the component functions hj and prove consistency in both subset recovery and consistency in empirical L2(Pn) norm. Meier et al. [9] propose a method that involves a sparsity-smoothness penalty term, and also demonstrate consistency in L2(P) norm. In the special case that H is a reproducing kernel Hilbert space (RKHS), Koltchinskii and Yuan [7] analyze a least-squares estimator based on imposing an ℓ1 −ℓH-penalty. The analysis in these paper demonstrates that under certain conditions on the covariates, such regularized procedures can yield estimators that are consistent in the L2(P)-norm even when n ≪p. Of complementary interest to the rates achievable by practical methods are the fundamental limits of the estimating sparse additive models, meaning lower bounds that apply to any algorithm. Although such lower bounds are well-known under classical scaling (where p remains fixed independent of n), to the best of our knowledge, lower bounds for minimax rates on sparse additive models have not been determined. In this paper, our main result is to establish a lower bound on the minimax rate in L2(P) norm that scales as max s log(p/s) n , sǫ2 n(H) . The first term s log(p/s) n is a subset selection term, independent of the univariate function space H in which the additive components lie, that reflects the difficulty of finding the subset S. The second term sǫ2 n(H) in an s-dimensional estimation term, which depends on the low dimension s but not the ambient dimension p, and reflects the difficulty of estimating the sum of s univariate functions, each drawn from function class H. Either the subset selection or s-dimensional estimation term dominates, depending on the relative sizes of n, p, and s as well as H. Importantly, our analysis applies both in the low-dimensional setting (n ≫p) and the highdimensional setting (p ≫n) provided that n, p and s are going to ∞. Our analysis is based on informationtheoretic techniques centered around the use of metric entropy, mutual information and Fano’s inequality in order to obtain lower bounds. Such techniques are standard in the analysis of non-parametric procedures under classical scaling [5, 2, 17], and have also been used more recently to develop lower bounds for high-dimensional inference problems [16, 11]. The remainder of the paper is organized as follows. In the next section, the results are stated including appropriate preliminary concepts, notation and assumptions. In Section 3, we state the main results, and provide some comparisons to the rates achieved by existing algorithms. In Section 4, we provide an overview of the proof. We discuss and summarize the main consequences in Section 5. 2 Background and problem formulation In this paper, we consider a non-parametric regression model with random design, meaning that we make n observations of the form y(i) = f ∗(X(i)) + w(i), for i = 1, 2, . . . , n. (2) Here the random vectors X(i) ∈Rp are the covariates, and have elements X(i) j drawn i.i.d. from some underlying distribution P. We assume that the noise variables w(i) ∼N(0, σ2) are drawn independently, and independent of all X(i)’s. Given a base class H of univariate functions with norm ∥· ∥H, consider the class of functions f : Rp →R that have an additive decomposition: F : = f : Rp →R | f(x1, x2, ..., xp) = p X j=1 hj(xj), and ∥hj∥H ≤1 ∀j = 1, . . . , p . Given some integer s ∈{1, . . . , p}, we define the function class F0(s), which is a union of p s s-dimensional subspaces of F, given by F0(s) : = f ∈F | p X j=1 I(hj ̸= 0) ≤s . (3) The minimax rate of estimation over F0(s) is defined by the quantity min b f maxf ∗∈F0(s) E∥bf −f ∗∥2 L2(P), where the expectation is taken over the noise w, and randomness in the sampling, and bf ranges over all (measurable) 2 functions of the observations {(y(i), X(i))}n i=1. The goal of this paper is to determine lower bounds on this minimax rate. 2.1 Inner products and norms Given a univariate function hj ∈H, we define the usual L2(P) inner product ⟨hj, h′ j⟩L2(P) : = Z R hj(x)h′ j(x) dP(x). (With a slight abuse of notation, we use P to refer to the measure over Rp as well as the induced marginal measure in each direction defined over R). Without loss of generality (re-centering the functions as needed), we may assume E[hj(X)] = Z R hj(x) dP(x) = 0, for all hj ∈H. As a consequence, we have E[f(X1, . . . , Xp)] = 0 for all functions f ∈F0(s). Given our assumption that the covariate vector X = (X1, . . . , Xp) has independent components, the L2(P) inner product on F has the additive decomposition ⟨f, f ′⟩L2(P) = Pp j=1 ⟨hj, h′ j⟩L2(P). (Note that if independence were not assumed the L2(P) inner product over F would involve cross-terms.) 2.2 Kullback-Leibler divergence Since we are using information theoretic techniques, we will be using the Kullback-Leibler (KL) divergence as a measure of “distance” between distributions. For a given pair of functions f and ef, consider the n-dimensional vectors f(X) = f(X(1)), f(X(2)), . . . , f(X(n)) T and ef(X) = ef(X(1)), ef(X(2)), . . . , ef(X(n)) T . Since Y |f(X) ∼N(f(X), σ2In×n) and Y | ef(X) ∼N( ef(X), σ2In×n), D(Y |f(X) ∥Y | ef(X)) = 1 2σ2 ∥f(X) −ef(X)∥2 2. (4) We also use the notation D(f ∥ef) to mean the average K-L divergence between the distributions of Y induced by the functions f and ef respectively. Therefore we have the relation D(f ∥ef) = EX D(Y |f(X) ∥Y | ef(X)) = n 2σ2 ∥f −ef∥2 L2(P). (5) This relation between average K-L divergence and squared L2(P) distance plays an important role in our proof. 2.3 Metric entropy for function classes In this section, we define the notion of metric entropy, which provides a way in which to measure the relative sizes of different function classes with respect to some metric ρ. More specifically, central to our results is the metric entropy of F0(s) with respect to the L2(P) norm. Definition 1 (Covering and packing numbers). Consider a metric space consisting of a set S and a metric ρ : S × S →R+. (a) An ǫ-covering of S in the metric ρ is a collection {f 1, . . . , f N} ⊂S such that for all f ∈S, there exists some i ∈{1, . . . , N} with ρ(f, f i) ≤ǫ. The ǫ-covering number Nρ(ǫ) is the cardinality of the smallest ǫ-covering. (b) An ǫ-packing of S in the metric ρ is a collection {f 1, . . . , f M} ⊂S such that ρ(f i, f j) ≥ǫ for all i ̸= j. The ǫ-packing number Mρ(ǫ) is the cardinality of the largest ǫ-packing. The covering and packing entropy (denoted by log Nρ(ǫ) and log Mρ(ǫ) respectively) are simply the logarithms of the covering and packing numbers, respectively. It can be shown that for any convex set, the quantities log Nρ(ǫ) and log Mρ(ǫ) are of the same order (within constant factors independent of ǫ). 3 In this paper, we are interested in packing (and covering) subsets of the function class F0(s) in the L2(P) metric, and so drop the subscript ρ from here onwards. En route to characterizing the metric entropy of F0(s), we need to understand the metric entropy of the unit balls of our univariate function class H—namely, the sets BH(1) : = {h ∈H | ∥h∥H ≤1}. The metric entropy (both covering and packing entropy) for many classes of functions are known. We provide some concrete examples here: (i) Consider the class H = {hβ : R →R | hβ(x) = βx} of all univariate linear functions with the norm ∥hβ∥H = |β|. Then it is known [15] that the metric entropy of BH(1) scales as log M(ǫ; H) ∼log(1/ǫ). (ii) Consider the class H = {h : [0, 1] →[0, 1] | |h(x) −h(y)| ≤|x −y|} of all 1-Lipschitz functions on [0, 1] with the norm ∥h∥H = supx∈[0,1] |h(x)|. In this case, it is known [15] that the metric entropy scales as log M H(ǫ; H) ∼1/ǫ. Compared to the previous example of linear models, note that the metric entropy grows much faster as ǫ →0, indicating that the class of Lipschitz functions is much richer. (iii) Consider the class of Sobolev spaces W m for m ≥1, consisting of all functions that have m derivatives, and the mth derivative is bounded in L2(P) norm. In this case, it is known that log M(ǫ; H) ∼ǫ−1 m (e.g., [3]). Clearly, increasing the smoothness constraint m leads to smaller classes. Such Sobolev spaces are a particular class of functions whose packing/covering entropy grows at a rate polynomial in 1 ǫ . In our analysis, we require that the metric entropy of BH(1) satisfy the following technical condition: Assumption 1. Using log M(ǫ; H) to denote the packing entropy of the unit ball BH(1) in the L2(P)-norm, assume that there exists some α ∈(0, 1) such that lim ǫ→0 log M(αǫ; H) log M(ǫ; H) > 1. The condition is required to ensure that log M(cǫ)/ log M(ǫ) can be made arbitrarily small or large uniformly over small ǫ by changing c, so that a bound due to Yang and Barron [17] can be applied. It is satisfied for most non-parametric classes, including (for instance) the Lipschitz and Sobolev classes defined in Examples (ii) and (iii) above. It may fail to hold for certain parametric classes, such as the set of linear functions considered in Example (i); however, we can use an alternative technique to derive bounds for the parametric case (see Corollary 2). 3 Main result and some consequences In this section, we state our main result and then develop some of its consequences. We begin with a theorem that covers the function class F0(s) in which the univariate function classes H have metric entropy that satisfies Assumption 1. We state a corollary for the special cases of univariate classes H with metric entropy growing polynomial in (1/ǫ), and also a corollary for the special case of sparse linear regression. Consider the observation model (2) where the covariate vectors have i.i.d. elements Xj ∼P, and the regression function f ∗∈F0(s). Suppose that the univariate function class H that underlies F0(s) satisfies Assumption 1. Under these conditions, we have the following result: Theorem 1. Given n i.i.d. samples from the sparse additive model (2), the minimax risk in squared-L2(P) norm is lower bounded as min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥max σ2s log(p/s) 32n , s 16ǫ2 n(H) , (6) where, for a fixed constant c, the quantity ǫn(H) = ǫn > 0 is largest positive number satisfying the inequality nǫ2 n 2σ2 ≤log M c ǫn . (7) For the case where H has an entropy that is growing to ∞at a polynomial rate as ǫ →0—say log M(ǫ; H) = Θ(ǫ−1/m) for some m > 1 2, we can compute the rate for the s-dimensional estimation term explicitly. 4 Corollary 1. For the sparse additive model (2) with univariate function space H such that such that log M(ǫ; H) = Θ(ǫ−1/m), we have min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥max σ2s log(p/s) 32n , C s σ2 n 2m 2m+1 , (8) for some C > 0. 3.1 Some consequences In this section, we discuss some consequences of our results. Effect of smoothness: Focusing on Corollary 1, for spaces with m bounded derivatives (i.e., functions in the Sobolev space W m), the minimax rate is n− 2m 2m+1 (for details, see e.g. Stone [13]). Clearly, faster rates are obtained for larger smoothness indices m, and as m →∞, the rate approaches the parametric rate of n−1. Since we are estimating over an s-dimensional space (under the assumption of independence), we are effectively estimating s univariate functions, each lying within the function space H. Therefore the uni-dimensional rate is multiplied by s. Smoothness versus sparsity: It is worth noting that depending on the relative scalings of s, n and p and the metric entropy of H, it is possible for either the subset selection term or s-dimensional estimation term to dominate the lower bound. In general, if log(p/s) n = o(ǫ2 n(H)), the s-dimensional estimation term dominates, and vice versa (at the boundary, either term determines the minimax rate). In the case of a univariate function class H with polynomial entropy as in Corollary 1, it can be seen that for n = o((log(p/s))2m+1), the s-dimensional estimation term dominates while for n = Ω((log(p/s))2m+1), the subset selection term dominates. Rates for linear models: Using an alternative proof technique (not the one used in this paper), it is possible [11] to derive the exact minimax rate for estimation in the sparse linear regression model, in which we observe y(i) = X j∈S βjX(i) j + w(i), for i = 1, 2, ..., n. (9) Note that this is a special case of the general model (2) in which H corresponds to the class of univariate linear functions (see Example (i)). Corollary 2. For sparse linear regression model (9), the the minimax rate scales as max s log(p/s) n , s n . In this case, we see clearly the subset selection term dominates for p →∞, meaning the subset selection problem is always “harder” (in a statistical sense) than the s-dimensional estimation problem. As shown by Bickel et al. [1], the rate achieved by ℓ1-regularized methods is s log p n under suitable conditions on the covariates X. Upper bounds: To show that the lower bounds are tight, upper bounds that are matching need to be derived. Upper bounds (matching up to constant factors) can be derived via a classical information-theoretic approach (e.g., [5, 2]), which involves constructing an estimator based on a covering set and bounding the covering entropy of F0(s). While this estimation approach does not lead to an implementable algorithm, it is a simple theoretical device to demonstrate that lower bounds are tight. We turn our focus on implementable algorithms in the next point. Comparison to existing bounds: We now provide a brief comparison of the minimax lower bounds with upper bounds on rates achieved by existing implementable algorithms provided by past work [12, 7, 9]. Ravikumar et al. [12] propose a back-fitting algorithm to minimize the least-squares objective with a sparsity constraint on the the function f. The rates derived in Koltchinskii and Yuan [7] do not match the lower bounds derived in Theorem 1. Further, it is difficult to directly compare the rates in Ravikumar et al. [12] and Meier et al. [9] with our minimax lower bounds since their analysis does not explicitly track the sparsity index s. We are currently in the process of conducting a thorough comparison with the above-mentioned ℓ1-based methods. 4 Proof outline In this section, we provide an outline of the proof of Theorem 1; due to space constraints, we defer some of the technical details to the full-length version. The proof is based on a combination of information-theoretic 5 techniques and the concepts of packing and covering entropy, as defined previously in Section 2.3. First, we provide a high-level overview of the proof. The basic idea is to carefully choose two subsets T1 and T2 of the function class F0(s) and lower bound the minimax rates over these two subsets. In Section 4.1, application of the generalized Fano method—a technique based on Fano’s inequality—to the set T1 defined in equation (10) yields a lower bound on the subset selection term. In Section 4.2, we apply an alternative method for obtaining lower bounds over a second set T2 defined in equation (11) that captures the difficulty of estimating the sum of s univariate functions.. The second technique also exploits Fano’s inequality but uses a more refined upper bound on the mutual information developed by Yang and Barron [17]. Before procedding, we first note that for any T ⊂F0(s), we have min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥min b f max f ∗∈T E∥bf −f ∗∥2 L2(P). Moreover, for any subsets T1, T2 ⊂F0(s), we have min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥max min b f max f ∗∈T1 E∥bf −f ∗∥2 L2(P), min b f max f ∗∈T2 E∥bf −f ∗∥2 L2(P) , since the bound holds for each of the two terms. We apply this lower bound using the subsets T1 and T2 defined in equations (10) and (11). 4.1 Bounding the complexity of subset selection For part of the proof, we use the generalized Fano’s method [4], which we state below without proof. Given some parameter space, we let d be a metric on it. Lemma 1. (Generalized Fano Method) For a given integer r ≥2, consider a collection Mr = {P1, . . . , Pr} of r probability distributions such that d(θ(Pi), θ(Pj)) ≥αr for all i ̸= j, and the pairwise KL divergence satisfies D(Pi ∥Pj) ≤βr for all i, j = 1, . . . , r. Then the minimax risk over the family is lower bounded as max j Ejd(θ(Pj), bθ) ≥αr 2 1 −βr + log 2 log r . The proof of Lemma 1 involves applying Fano’s inequality over the discrete set of parameters θ ∈Θ indexed by the set of distributions Mr. Now we construct the set T1 which creates the set of probability distributions Mr. Let g be an arbitrary function in H such that ∥g∥L2(P) = σ 4 q log (p/s) n . The set T1 is defined as T1 : = f : f(X1, X2, ..., Xp) = p X j=1 cjg(Xj), cj ∈{−1, 0, 1} | ∥c∥0 = s . (10) T1 may be viewed as a hypercube of F0(s) and will lead to the lower bound for the ’subset selection’ term. This hypercube construction is often used to prove lower bounds (see Yu [18]). Next, we require a further reduction of the set T1 to a set A (defined in Lemma 2) to ensure that elements of A are well-separated in L2(P) norm. The construction of A is as follows: Lemma 2. There exists a subset A ⊂T1 such that: (i) log |A| ≥1 2s log(p/s), (ii) ∥f −f ′∥2 L2(P) ≥σ2s log(p/s) 16n ∀f, f ′ ∈A, and (iii) D(f ∥f ′) ≤1 8s log(p/s) ∀f, f ′ ∈A. The proof involves using a combinatoric argument to construct the set A. For an argument on how the set is constructed, see K¨uhn [8]. For s log p s ≥8 log 2, applying the Generalized Fano Method (Lemma 1) together with Lemma 2 yields the bound min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥min b f max f ∗∈A E∥bf −f ∗∥2 L2(P) ≥σ2s log(p/s) 32n . This completes the proof for the subset selection term ( s log(p/s) n ) in Theorem 1. 6 4.2 Bounding the complexity of s-dimensional estimation Next we derive a bound for the s-dimensional estimation term by determining a lower bound over T2. Let S be an arbitrary subset of s integers in {1, 2, .., p}, and define the set FS as T2 : = FS : = f ∈F : f(X) = X j∈S hj(Xj) . (11) Clearly FS ⊂F0(s) meaning that min b f max f ∗∈F0(s) E∥bf −f ∗∥2 L2(P) ≥min b f max f ∗∈FS E∥bf −f ∗∥2 L2(P). We use a technique used in Yang and Barron [17] to lower bound the minimax rate over FS. The idea is to construct a maximal δn-packing set for FS and a minimal ǫn-covering set for FS, and then to apply Fano’s inequality to a carefully chosen mixture distribution involving the covering and packing sets (see the full-length version for details). Following these steps yields the following result: Lemma 3. min b f max f ∗∈FS E∥bf −f ∗∥2 L2(P) ≥δ2 n 4 1 −log N(ǫn; FS) + nǫ2 n/2σ2 + log 2 log M(δn; FS) . Now we have a bound with expressions involving the covering and packing entropies of the s-dimensional space FS. The following Lemma allows bounds on log M(ǫ; FS) and log N(ǫ; FS) in terms of the unidimensional packing and covering entropies respectively: Lemma 4. Let H be function space with a packing entropy log M(ǫ; H) that satisfies Assumption 1. Then we have the bounds log M(ǫ; FS) ≥s log M(ǫ/√s; H), and log N(ǫ; FS) ≤s log N(ǫ/√s; H). The proof involves constructing ǫ √s- packing set and covering sets in each of the s dimensions and displaying that these are ǫ-packing and coverings sets in FS (respectively). Combining Lemmas 3 and 4 leads to the inequality min b f max f ∗∈FS E∥bf −f ∗∥2 L2(P) ≥δ2 n 4 1 −s log N(ǫn/√s; H) + nǫ2 n/2σ2 + log 2 s log M(δn/√s; H) . (12) Now we choose ǫn and δn to meet the following constraints: n 2σ2 ǫ2 n ≤ s log N( ǫn √s; H), and (13a) 4 log N( ǫn √s; H) ≤ log M( δn √s; H). (13b) Combining Assumption 1 with the well-known relations log M(2ǫ; H) ≤log N(2ǫ; H) ≤log M(ǫ; H), we conclude that in order to satisfy inequalities (13a) and (13b), it is sufficient to choose ǫn = cδn for a constant c, and then require that s log M( cδn √s ; H) ≥nδ2 n 2σ2 . Furthermore, if we define δn/√s = eδn, then this inequality can be re-expressed as log M(c eδn) ≥nf δn 2 2σ2 . For n 2σ2 ǫ2 n ≥log 2, using inequalities (13a) and (13b) together with equation (12) yields the desired rate min b f max f ∗∈FS E∥bf −f ∗∥2 L2(P) ≥s eδn 2 16 , thereby completing the proof. 5 Discussion In this paper, we have derived lower bounds for the minimax risk in squared L2(P) error for estimating sparse additive models based on the sum of univariate functions from a function class H. The rates show that the estimation problem effectively decomposes into a subset selection problem and an s-dimensional estimation 7 problem, and the “harder” of the two problems (in a statistical sense) determines the rate of convergence. More concretely, we demonstrated that the subset selection term scales as s log(p/s) n , depending linearly on the number of components s and only logarithmically in the ambient dimension p. This subset selection term is independent of the univariate function space H. On the other hand, the s-dimensional estimation term depends on the “richness” of the univariate function class, measured by its metric entropy; it scales linearly with s and is independent of p. Ongoing work suggests that our lower bounds are tight in many cases, meaning that the rates derived in Theorem 1 are minimax optimal for many function classes. There are a number of ways in which the work can be extended. One implicit and strong assumption in our analysis was that the covariates Xj, j = 1, 2, ..., p are independent. It would be interesting to investigate the case when the random variables are endowed with some correlation structure. One would expect the rates to change, particularly if many of the variables are collinear. It would also be interesting to develop a more complete understanding of whether computationally efficient algorithms [7, 12, 9] based on regularization achieve the lower bounds on the minimax rate derived in this paper. References [1] P. Bickel, Y. Ritov, and A. Tsybakov. Simultaneous analysis of the Lasso and Dantzig selector. Annals of Statistics, 2009. To appear. [2] L. Birg´e. Approximation dans les espaces metriques et theorie de l’estimation. Z. Wahrsch. verw. Gebiete, 65:181–327, 1983. [3] M. S. Birman and M. Z. Solomjak. Piecewise-polynomial approximations of functions of the classes W α p . Math. USSR-Sbornik, 2(3):295–317, 1967. [4] T. S. Han and S. Verdu. Generalizing the Fano inequality. IEEE Transactions on Information Theory, 40:1247–1251, 1994. [5] R. Z. Has’minskii. A lower bound on the risks of nonparametric estimates of densities in the uniform metric. Theory Prob. Appl., 23:794–798, 1978. [6] T. Hastie and R. Tibshirani. Generalized Additive Models. Chapman and Hall Ltd, Boca Raton, 1999. [7] V. Koltchinskii and M. Yuan. Sparse recovery in large ensembles of kernel machines. In Proceedings of COLT, 2008. [8] T. K¨uhn. A lower estimate for entropy numbers. Journal of Approximation Theory, 110:120–124, 2001. [9] L. Meier, S. van de Geer, and P. Buhlmann. High-dimensional additive modeling. Annals of Statistics, To appear. [10] N. Meinshausen and B.Yu. Lasso-type recovery of sparse representations for high-dimensional data. Annals of Statistics, 37(1):246–270, 2009. [11] G. Raskutti, M. J. Wainwright, and B. Yu. Minimax rates of estimation for high-dimensional linear regression over ℓq-balls. Technical Report arXiv:0910.2042, UC Berkeley, Department of Statistics, 2009. [12] P. Ravikumar, H. Liu, J. Lafferty, and L. Wasserman. Sparse additive models. Journal of the Royal Statistical Society, To appear. [13] C. J. Stone. Optimal global rates of convergence for nonparametric regression. Annals of Statistics, 10:1040–1053, 1982. [14] R. Tibshirani. Regression shrinkage and selection via the lasso. Journal of the Royal Statistical Society, Series B, 58(1):267–288, 1996. [15] S. van de Geer. Empirical Processes in M-Estimation. Cambridge University Press, 2000. [16] M. J. Wainwright. Information-theoretic bounds for sparsity recovery in the high-dimensional and noisy setting. IEEE Trans. Info. Theory, December 2009. Presented at International Symposium on Information Theory, June 2007. [17] Y. Yang and A. Barron. Information-theoretic determination of minimax rates of convergence. Annals of Statistics, 27(5):1564–1599, 1999. [18] B. Yu. Assouad, Fano and Le Cam. Research Papers in Probability and Statistics: Festschrift in Honor of Lucien Le Cam, pages 423–435, 1996. [19] C. H. Zhang and J. Huang. The sparsity and bias of the lasso selection in high-dimensional linear regression. Annals of Statistics, 36:1567–1594, 2006. 8
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Sufficient Conditions for Agnostic Active Learnable Liwei Wang Key Laboratory of Machine Perception, MOE, School of Electronics Engineering and Computer Science, Peking University, wanglw@cis.pku.edu.cn Abstract We study pool-based active learning in the presence of noise, i.e. the agnostic setting. Previous works have shown that the effectiveness of agnostic active learning depends on the learning problem and the hypothesis space. Although there are many cases on which active learning is very useful, it is also easy to construct examples that no active learning algorithm can have advantage. In this paper, we propose intuitively reasonable sufficient conditions under which agnostic active learning algorithm is strictly superior to passive supervised learning. We show that under some noise condition, if the Bayesian classification boundary and the underlying distribution are smooth to a finite order, active learning achieves polynomial improvement in the label complexity; if the boundary and the distribution are infinitely smooth, the improvement is exponential. 1 Introduction Active learning addresses the problem that the algorithm is given a pool of unlabeled data drawn i.i.d. from some underlying distribution. The algorithm can then pay for the label of any example in the pool. The goal is to learn an accurate classifier by requesting as few labels as possible. This is in contrast with the standard passive supervised learning, where the labeled examples are chosen randomly. The simplest example that demonstrates the potential of active learning is to learn the optimal threshold on an interval. If there exists a perfect threshold separating the two classes (i.e. there is no noise), then binary search only needs O(ln 1 ϵ ) labels to learn an ϵ-accurate classifier, while passive learning requires O( 1 ϵ ) labels. Another encouraging example is to learn a homogeneous linear separator for data uniformly distributed on the unit sphere of Rd. In this case active learning can still give exponential savings in the label complexity [Das05]. However, there are also very simple problems that active learning does not help at all. Suppose the instances are uniformly distributed on [0, 1], and the positive class could be any interval on [0, 1]. Any active learning algorithms needs O( 1 ϵ ) label requests to learn an ϵ-accurate classifier [Han07]. There is no improvement over passive learning. All above are noise-free (realizable) problems. Of more interest and more realistic is the agnostic setting, where the class labels can be noisy so that the best classifier in the hypothesis space has a non-zero error ν. For agnostic active learning, there is no active learning algorithm that can always reduce label requests due to a lower bound Ω( ν2 ϵ2 ) for the label complexity [Kaa06]. It is known that whether active learning helps or not depends on the distribution of the instance-label pairs and the hypothesis space. Thus a natural question would be that under what conditions is active learning guaranteed to require fewer labels than passive learning. 1 In this paper we propose intuitively reasonable sufficient conditions under which active learning achieves lower label complexity than that of passive learning. Specifically, we focus on the A2 algorithm [BAL06] which works in the agnostic setting. Earlier work has discovered that the label complexity of A2 can be upper bounded by a parameter of the hypothesis space and the data distribution called disagreement coefficient [Han07]. This parameter often characterizes the intrinsic difficulty of the learning problem. By an analysis of the disagreement coefficient we show that, under some noise condition, if the Bayesian classification boundary and the underlying distribution are smooth to a finite order, then A2 gives polynomial savings in the label complexity; if the boundary and the distribution are infinitely smooth, A2 gives exponential savings. 1.1 Related Works Our work is closely related to [CN07], in which the authors proved sample complexity bounds for problems with smooth classification boundary under Tsybakov’s noise condition [Tsy04]. They also assumed that the distribution of the instances is bounded from above and below. The main difference to our work is that their analysis is for the membership-query setting [Ang88], in which the learning algorithm can choose any point in the instance space and ask for its label; while the pool-based model analyzed here assumes the algorithm can only request labels of the instances it observes. Another related work is due to Friedman [Fri09]. He introduced a different notion of smoothness and showed that this guarantees exponential improvement for active learning. But his work focused on the realizable case and does not apply to the agnostic setting studied here. Soon after A2, Dasgupta, Hsu and Monteleoni [DHM07] proposed an elegant agnostic active learning algorithm. It reduces active learning to a series of supervised learning problems. If the hypothesis space has a finite VC dimension, it has a better label complexity than A2. However, this algorithm relies on the normalized uniform convergence bound for the VC class. It is not known whether it holds for more general hypothesis space such as the smooth boundary class analyzed in this paper. (For recent advances on this topic, see [GKW03].) It is left as an open problem whether our results apply to this algorithm by refined analysis of the normalized bounds. 2 Preliminaries Let X be an instance space, D a distribution over X × {−1, 1}. Let H be the hypothesis space, a set of classifiers from X to {±1}. Denote DX the marginal of D over X. In our active learning model, the algorithm has access to a pool of unlabeled examples from DX . For any unlabeled point x, the algorithm can ask for its label y, which is generated from the conditional distribution at x. The error of a hypothesis h according to D is erD(h) = Pr(x,y)∼D(h(x) ̸= y). The empirical error on a finite sample S is erS(h) = 1 |S| P (x,y)∈S I[h(x) ̸= y], where I is the indicator function. We use h∗denote the best classifier in H. That is, h∗= arg minh∈H erD(h). Let ν = erD(h∗). Our goal is to learn a ˆh ∈H with error rate at most ν + ϵ, where ϵ is a predefined parameter. A2 is the first rigorous agnostic active learning algorithm. A description of the algorithm is given in Fig.1. It was shown that A2 is never much worse than passive learning in terms of the label complexity. The key observation that A2 can be superior to passive learning is that, since our goal is to choose an ˆh such that erD(ˆh) ≤erD(h∗) + ϵ, we only need to compare the errors of hypotheses. Therefore we can just request labels of those x on which the hypotheses under consideration have disagreement. To do this, the algorithm keeps track of two spaces. One is the current version space Vi, consisting of hypotheses that with statistical confidence are not too bad compared to h∗. To achieve such a statistical guarantee, the algorithm must be provided with a uniform convergence bound over the hypothesis space. That is, with probability at least 1 −δ over the draw of sample S according to D, LB(S, h, δ) ≤erD(h) ≤UB(S, h, δ), hold simultaneously for all h ∈H, where the lower bound LB(S, h, δ) and upper bound UB(S, h, δ) can be computed from the empirical error erS(h). The other space is the region of disagreement DIS(Vi), which is the set of all x ∈X for which there are hypotheses in Vi that disagree on x. Formally, for any V ⊂H, DIS(V ) = {x ∈X : ∃h, h′ ∈V, h(x) ̸= h′(x)}. 2 Input: concept space H, accuracy parameter ϵ ∈(0, 1), confidence parameter δ ∈(0, 1); Output: classifier ˆh ∈H; Let ˆn = 2(2 log2 λ ϵ + ln 1 δ ) log2 2 ϵ (λ depends on H and the problem, see Theorem 5) ; Let δ′ = δ/ˆn ; V0 ←H, S0 ←∅, i ←0, j1 ←0, k ←1 ; while ∆(Vi)(minh∈Vi UB(Si, h, δ′) −minh∈Vi LB(Si, h, δ′)) > ϵ do Vi+1 ←{h ∈Vi : LB(Si, h, δ′) ≤minh′∈Vi UB(Si, h′, δ′)}; i ←i+1; if ∆(Vi) < 1 2∆(Vjk) then k ←k + 1; jk ←i; end S′ i ←Rejection sample 2i−jk samples x from D satisfying x ∈DIS(Vi); Si ←{(x, y = label(x)) : x ∈S′ i}; end Return ˆh= argminh∈ViUB(Si, h, δ′). Algorithm 1: The A2 algorithm (this is the version in [Han07]) The volume of DIS(V ) is denoted by ∆(V ) = PrX∼DX (X ∈DIS(V )). Requesting labels of the instances from DIS(Vi) allows A2 require fewer labels than passive learning. Hence the key issue is how fast ∆(Vi) reduces. This process, and in turn the label complexity of A2, are nicely characterized by the disagreement coefficient θ introduced in [Han07]. Definition 1 Let ρ(·, ·) be the pseudo-metric on a hypothesis space H induced by DX . That is, for h, h′ ∈H, ρ(h, h′) = PrX∼DX (h(X) ̸= h′(X)). Let B(h, r) = {h′ ∈H: ρ(h, h′) ≤r}. The disagreement coefficient θ(ϵ) is θ(ϵ) = sup r≥ϵ PrX∼DX (X ∈DIS(B(h∗, r))) r , (1) where h∗= arg minh∈H erD(h). Note that θ depends on H and D, and 1 ≤θ(ϵ) ≤1 ϵ . 3 Main Results As mentioned earlier, whether active learning helps or not depends on the distribution and the hypothesis space. There are simple examples such as learning intervals for which active learning has no advantage. However, these negative examples are more or less “artificial”. It is important to understand whether problems with practical interest are actively learnable or not. In this section we provide intuitively reasonable conditions under which the A2 algorithm is strictly superior to passive learning. Our main results (Theorem 11 and Theorem 12) show that if the learning problem has a smooth Bayes classification boundary, and the distribution DX has a density bounded by a smooth function, then under some noise condition A2 saves label requests. It is a polynomial improvement for finite smoothness, and exponential for infinite smoothness. In Section 3.1 we formally define the smoothness and introduce the hypothesis space, which contains smooth classifiers. We show a uniform convergence bound of order O(n−1/2) for this hypothesis space. This bound determines UB(S, h, δ) and LB(S, h, δ) in A2. Section 3.2 is the main technical part, where we give upper bounds for the disagreement coefficient of smooth problems. In Section 3.3 we show that under some noise condition, there is a sharper bound for the label complexity in terms of the disagreement coefficient. These lead to our main results. 3 3.1 Smoothness Let f be a function defined on Ω⊂Rd. For any vector k = (k1, · · · , kd) of d nonnegative integers, let |k| = Pd i=1 ki. Define the K-norm as ∥f∥K := max |k|≤K−1sup x∈Ω |Dkf(x)| + max |k|=K−1 sup x,x′∈Ω Dkf(x) −Dkf(x′) ∥x −x′∥ , (2) where Dk = ∂|k| ∂k1x1 · · · ∂kdxd , is the differential operator. Definition 2 (Finite Smooth Functions) A function f is said to be Kth order smooth with respect to a constant C, if ∥f∥K ≤C. The set of Kth order smooth functions is defined as F K C := {f : ∥f∥K ≤C}. (3) Thus Kth order smooth functions have uniformly bounded partial derivatives up to order K −1, and the K −1th order partial derivatives are Lipschitz. Definition 3 (Infinitely Smooth Functions) A function f is said to be infinitely smooth with respect to a constant C, if ∥f∥K ≤C for all nonnegative integers K. The set of infinitely smooth functions is denoted by F ∞ C . With the definitions of smoothness, we introduce the hypothesis space we use in the A2 algorithm. Definition 4 (Hypotheses with Smooth Boundaries) A set of hypotheses HK C defined on [0, 1]d+1 is said to have Kth order smooth boundaries, if for every h ∈HK C , the classification boundary is a Kth order smooth function on [0, 1]d. To be precise, let x = (x1, x2, . . . , xd+1) ∈[0, 1]d+1. The classification boundary is the graph of function xd+1 = f(x1, . . . , xd), where f ∈F K C . Similarly, a hypothesis space H∞ C is said to have infinitely smooth boundaries, if for every h ∈H∞ C the classification boundary is the graph an infinitely smooth function on [0, 1]d. Previous results on the label complexity of A2 assumes the hypothesis space has finite VC dimension. The goal is to ensure a O(n−1/2) uniform convergence bound so that UB(S, h, δ) − LB(S, h, δ) = O(n−1/2). The hypothesis space HK C and H∞ C do not have finite VC dimensions. Compared with the VC class, HK C and H∞ C are exponentially larger in terms of the covering numbers [vdVW96]. But uniform convergence bound still holds for HK C and H∞ C under a broad class of distributions. The following theorem is a consequence of some known results in empirical processes. Theorem 5 For any distribution D over [0, 1]d+1 × {−1, 1}, whose marginal distribution DX on [0, 1]d+1 has a density upper bounded by a constant M, and any 0 < δ ≤δ0 (δ0 is a constant), with probability at least 1 −δ over the draw of the training set S of n examples, |erD(h) −erS(h)| ≤λ s log 1 δ n , (4) holds simultaneously for all h ∈HK C provided K > d (or K = ∞). Here λ is a constant depending only on d, K, C and M. Proof It can be seen, from Corollary 2.7.3 in [vdVW96] that the bracketing numbers N[ ] of HK C satisfies log N[ ](ϵ, HK C , L2(DX )) = O(( 1 ϵ ) 2d K ). Since K > d, then there exist constants c1, c2 such that PD à sup h∈HK C |er(h) −erS(h)| ≥t ! ≤c1 exp µ −nt2 c2 ¶ for all nt2 ≥t0, where t0 is some constant (see Theorem 5.11 and Lemma 5.10 of [vdG00]). Let δ = c1 exp ³ −nt2 c2 ´ , the theorem follows. 4 Now we can determine UB(S, h, δ) and LB(S, h, δ) for A2 by simply letting UB(S, h, δ) = erS(h) + λ q ln 1 δ n and LB(S, h, δ) = erS(h) −λ q ln 1 δ n , where S is of size n. 3.2 Disagreement Coefficient The disagreement coefficient θ plays an important role for the label complexity of active learning algorithms. In fact previous negative examples for which active learning does not work are all the results of large θ. For instance the interval learning problem, θ(ϵ) = 1 ϵ , which leads to the same label complexity as passive learning. In the following two theorems we show that the disagreement coefficient θ(ϵ) for smooth problems is small. Theorem 6 Let the hypothesis space be HK C . If the distribution DX has a density p(x1, . . . , xd+1) such that there exists a Kth order smooth function g(x1, . . . , xd+1) and two constants 0 < α ≤ β such that αg(x1, . . . , xd+1) ≤p(x1, . . . , xd+1) ≤βg(x1, . . . , xd+1) for all (x1, . . . , xd+1) ∈ [0, 1]d+1, then θ(ϵ) = O ³¡ 1 ϵ ¢ d K+d ´ . Theorem 7 Let the hypothesis space be H∞ C . If the distribution DX has a density p(x1, . . . , xd+1) such that there exist an infinitely smooth function g(x1, . . . , xd) and two constants 0 < α ≤β such that αg(x1, . . . , xd) ≤p(x1, . . . , xd+1) ≤βg(x1, . . . , xd) for all (x1, . . . , xd+1) ∈[0, 1]d+1, then θ(ϵ) = O(logd( 1 ϵ )). The key points in the theorems are: the classification boundaries are smooth; and the density is bounded from above and below by constants times a smooth function. These two conditions include a large class of learning problems. Note that the density itself is not necessarily smooth. We just require the density does not change too rapidly. The intuition behind the two theorems above is as follows. Let fh∗(x) and fh(x) be the classification boundaries of h∗and h, and suppose ρ(h, h∗) is small, where ρ(h, h∗) = Prx∼DX (h(x) ̸= h∗(x)) is the pseudo metric. If the classification boundaries and the density are all smooth, then the two boundaries have to be close to each other everywhere. That is, |fh(x) −ff ∗(x)| is small uniformly for all x. Hence only the points close to the classification boundary of h∗can be in DIS(B(h∗, ϵ)), which leads to a small disagreement coefficient. The proofs of Theorem 6 and Theorem 7 rely on the following two lemmas. Lemma 8 Let Φ be a function defined on [0, 1]d and R [0,1]d |Φ(x)|dx ≤r. If there exists a Kth order smooth function ˜Φ and 0 < α ≤β such that α|˜Φ(x)| ≤|Φ(x)| ≤β|˜Φ(x)| for all x ∈[0, 1]d, then ∥Φ∥∞= O(r K K+d ) = O(r · ( 1 r) d K+d ), where ∥Φ∥∞= supx∈[0,1]d |Φ(x)|. Lemma 9 Let Φ be a function defined on [0, 1]d and R [0,1]d |Φ(x)|dx ≤r. If there exists an infinitely smooth function ˜Φ and 0 < α ≤β such that α|˜Φ(x)| ≤|Φ(x)| ≤β|˜Φ(x)| for all x ∈[0, 1]d, then ∥Φ∥∞= O(r · logd( 1 r)) We will briefly describe the ideas of the proofs of these two lemmas in the Appendix. The formal proofs are given in the supplementary file. Proof of Theorem 6 First of all, since we focus on binary classification, DIS(B(h∗, r)) can be written equivalently as DIS(B(h∗, r)) = {x ∈X, ∃h ∈B(h∗, r), s.t. h(x) ̸= h∗(x)}. Consider any h ∈B(h∗, r). Let fh, fh∗∈F K C be the corresponding classification boundaries of h and h∗respectively. If r is sufficiently small, we must have ρ(h, h∗) = Pr X∼DX(h(X) ̸= h∗(X)) = Z [0,1]d dx1 . . . dxd ¯¯¯¯¯ Z fh(x1,...,xd) fh∗(x1,...,xd) p(x1, . . . , xd+1)dxd+1 ¯¯¯¯¯ . 5 Denote Φh(x1, . . . , xd) = Z fh(x1,...,xd) fh∗(x1,...,xd) p(x1, . . . , xd+1)dxd+1. We assert that there is a Kth order smooth function ˜Φh(x1, . . . , xd) and two constants 0 < u ≤v such that u|˜Φh| ≤|Φh| ≤v|˜Φh|. To see this, remember that fh and fh∗are Kth order smooth functions; and the density p is upper and lower bounded by constants times a Kth order smooth function g(x1, . . . , xd+1); and note that ˜Φh(x1, . . . , xd) = R fh(x1,...,xd) fh∗(x1,...,xd) g(x1, . . . , xd+1)dxd+1 is a Kth order smooth function. The latter is easy to check by taking derivatives. By Lemma 8, we have ∥Φh∥∞= O(r · ( 1 r) d K+d ), since R |Φh| = ρ(h, h∗) ≤r. Because this holds for all h ∈B(h∗, r), we have suph∈B(h∗,r) ∥Φh∥∞= O(r · ( 1 r) d K+d ). Now consider the region of disagreement of B(h∗, r). Clearly DIS(B(h∗, r)) = ∪h∈B(h∗,r){x : h(x) ̸= h∗(x)}. Hence Pr X∼DX (x ∈DIS(B(h∗, r))) = Pr X∼DX ¡ x ∈∪h∈B(h∗,r){x : h(x) ̸= h∗(x)} ¢ ≤ 2 sup h∈B(h∗,r) Z [0,1]d ∥Φh∥∞dx1 . . . dxd = O à r · µ1 r ¶ d K+d ! . The theorem follows by the definition of θ(ϵ). Theorem 7 can be proved similarly by using Lemma 9. 3.3 Label Complexity It was shown in [Han07] that the label complexity of A2 is O µ θ2 µν2 ϵ2 + 1 ¶ polylog µ1 ϵ ¶ ln 1 δ ¶ , (5) where ν = minh∈H erD(h). When ϵ ≥ν, our previous results on the disagreement coefficient already imply polynomial or exponential improvements for A2. However, when ϵ < ν, the label complexity becomes O( 1 ϵ2 ), the same as passive learning whatever θ is. In fact, without any assumption on the noise, the O( 1 ϵ2 ) result is inevitable due to the Ω( ν2 ϵ2 ) lower bound of agnostic active learning [Kaa06]. Recently, there has been considerable interest in how noise affects the learning rate. A remarkable notion is due to Tsybakov [Tsy04], which was first introduced for passive learning. Let η(x) = P(Y = 1|X = x). Tsybakov’s noise condition assumes that for some c > 0, 0 < α ≤∞ Pr X∼DX(|η(X) −1/2| ≤t) ≤ct−α, (6) for all 0 < t ≤t0, where t0 is some constant. (6) implies a connection between the pseudo distance ρ(h, h∗) and the excess risk erD(h) −erD(h∗): ρ(h, h∗) ≤c′ (erD(h) −erD(h∗))1/κ , (7) where h∗is the Bayes classifier, c′ is some finite constant. Here κ = 1+α α ≥1 is called the noise exponent. κ = 1 is the optimal case, where the problem has bounded noise; κ > 1 correspond to unbounded noise. Castro and Nowak [CN07] noticed that Tsybakov’s noise condition is also important in active learning. They proved label complexity bounds in terms of κ for the membership-query setting. A notable fact is that ˜O(( 1 ϵ ) 2κ−2 κ ) (κ > 1) is both an upper and a lower bound for membership-query in the minimax sense. It is important to point out that the lower bound automatically applies to pool-based model, since pool makes weaker assumptions than membership-query. Hence for large κ, active learning has very limited improvement over passive learning whatever other factors are. Recently, Hanneke [Han09] obtained similar label complexity for pool-based model. He showed the labels requested by A2 is O(θ2 ln 1 ϵ ln 1 δ ) for the bounded noise case, i.e. κ = 1. Here we slightly 6 generalize Hanneke’s result to unbounded noise by introducing the following noise condition. We assume there exist c1, c2 > 0 and T0 > 0 such that Pr X∼DX(|η(X) −1/2| ≤1 T ) ≤c1e−c2T , (8) for all T ≥T0. It is not difficult to show that (8) implies ρ(h, h∗) = O µ (er(h) −er(h∗)) ln 1 (er(h) −er(h∗)) ¶ . (9) This condition assumes unbounded noise. Under this noise condition, A2 has a better label complexity. Theorem 10 Assume that the learning problem satisfies the noise condition (8) and DX has a density upper bounded by a constant M. For any hypothesis space H that has a O(n−1/2) uniform convergence bound, if the Bayes classifier h∗is in H, then with probability at least 1−δ, A2 outputs ˆh ∈H with erD(ˆh) ≤erD(h∗) + ϵ, and the number of labels requested by the algorithm is at most O(θ2(ϵ) · ln 1 δ · polylog( 1 ϵ )). Proof As the proof of [Han07], one can show that with probability 1 −δ we never remove h∗from Vi, and for any h, h′ ∈Vi we must have ∆(Vi)(eri(h) −eri(h′)) = erD(h) −erD(h′), where eri(h) is the error rate of h conditioned on DIS(Vi). These guarantees erD(ˆh) ≤erD(h∗) + ϵ. If ∆(Vi) ≤2ϵθ(ϵ), due to the O(n−1/2) uniform convergence bound, O(θ2(ϵ) ln 1 δ ) labels suffices to make ∆(Vi)(UB(Si, h, δ′) −LB(Si, h, δ′)) ≤ϵ for all h ∈DIS(Vi) and the algorithm stops. Hence we next consider ∆(Vi) > 2ϵθ(ϵ). Note that there are at most O(ln 1 ϵ ) times ∆(Vi) < 1 2∆(Vjk) occurs. So below we bound the number of labels needed to make ∆(Vi) < 1 2∆(Vjk) occurs. By the definition of θ(ϵ), if ρ(h, h∗) ≤ ∆(Vjk ) 2θ(ϵ) for all h ∈Vi, then ∆(Vi) < 1 2∆(Vjk). Let γ(h) = erD(h) −erD(h∗). By the noise assumption (9) we have that if γ(h) ln 1 γ(h) ≤c∆(Vjk) 2θ(ϵ) , (10) then ∆(Vi) < 1 2∆(Vjk). Here and below, c is appropriate constant but may be different from line to line. Note that (10) holds if γ(h) ≤c ∆(Vjk ) θ(ϵ) ln θ(ϵ) ∆(Vjk ) , and in turn if γ(h) ≤c ∆(Vjk ) θ(ϵ) ln 1 ϵ since ∆(Vjk) ≥∆(Vi) > 2ϵθ(ϵ). But to have the last inequality, the algorithm only needs to label O(θ2(ϵ) ln2 1 ϵ ln 1 δ ) instances from DIS(Vi). So the total number of labels requested by A2 is O(θ2(ϵ) ln 1 δ ln3 1 ϵ ) Now we give our main label complexity bounds for agnostic active learning. Theorem 11 Let the instance space be [0, 1]d+1. Let the Hypothesis space be HK C , where K > d. Assume that the Bayes classifier h∗of the learning problem is in HK C ; the noise condition (8) holds; and DX has a density bounded by a Kth order smooth function as in Theorem 6. Then the A2 algorithm outputs ˆh with error rate erD(ˆh) ≤erD(h∗) + ϵ and the number of labels requested is at most ˜O ³¡ 1 ϵ ¢ 2d K+d ln 1 δ ´ , where in ˜O we hide the polylog ¡ 1 ϵ ¢ term. Proof Note that the density DX is upper bounded by a smooth function implies that it is also upper bounded by a constant M. Combining Theorem 5, 6 and 10 the theorem follows. Combining Theorem 5, 7 and 10 we can show the following theorem. Theorem 12 Let the instance space be [0, 1]d+1. Let the Hypothesis space be H∞ C . Assume that the Bayes classifier h∗of the learning problem is in H∞ C ; the noise condition (8) holds; and DX has a density bounded by an infinitely smooth function as in Theorem 7. Then the A2 algorithm outputs ˆh with error rate erD(ˆh) ≤erD(h∗) + ϵ and the number of labels requested is at most O ¡ polylog ¡ 1 ϵ ¢ ln 1 δ ¢ . 7 4 Conclusion We show that if the Bayesian classification boundary is smooth and the distribution is bounded by a smooth function, then under some noise condition active learning achieves polynomial or exponential improvement in the label complexity than passive supervised learning according to whether the smoothness is of finite order or infinite. Although we assume that the classification boundary is the graph of a function, our results can be generalized to the case that the boundaries are a finite number of functions. To be precise, consider N functions f1(x) ≤· · · ≤fN(x), for all x ∈[0, 1]d. Let f0(x) ≡0, fN+1(x) ≡1. The positive (or negative) set defined by these functions is {(x, xd+1) : f2i(x) ≤x ≤f2i+1(x), i = 0, 1, . . . , N 2 }. Our theorems still hold in this case. In addition, by techniques in [Dud99] (page 259), our results may generalize to problems which have intrinsic smooth boundaries (not only graphs of functions). Appendix In this appendix we describe very briefly the ideas to prove Lemma 8 and Lemma 9. The formal proofs can be found in the supplementary file. Ideas to Prove Lemma 8 First consider the d = 1 case. Note that if f ∈F K C , then |f (K−1)(x) − f (K−1)(x′)| ≤C|x −x′| for all x, x′ ∈[0, 1]. It is not difficult to see that we only need to show for any f such that |f (K−1)(x)−f (K−1)(x′)| ≤C|x−x′|, if R 1 0 |f(x)|dx = r, then ∥f∥∞= O(r K K+1 ). To show this, note that in order that ∥f∥∞achieves the maximum while R |f| = r, the derivatives of f must be as large as possible. Indeed, it can be shown that (one of) the optimal f is of the form f(x) = C K!|x −ξ|K 0 ≤x ≤ξ, 0 ξ < x ≤1. (11) That is, |f (K−1)(x) −f (K−1)(x′)| = C|x −x′| (i.e. the K −1 order derivatives reaches the upper bound of the Lipschitz constant.) for all x, x′ ∈[0, ξ], where ξ is determined by R 1 0 f(x)dx = r. It is then easy to check that ∥f∥∞= O(r K K+1 ). For the general d > 1 case, we relax the constraint. Note that all K −1th order partial derivatives are Lipschitz implies that all K −1th order directional derivatives are Lipschitz too. Under the latter constraint, (one of) the optimal f has the form f(x) = C K!|∥x∥−ξ|K 0 ≤∥x∥≤ξ, 0 ξ < ∥x∥. where ξ is determined by R [0,1]d |f(x)|dx = r. This implies ∥f∥∞= O(r K K+d ). Ideas to Prove Lemma 9 Similar to the proof of Lemma 8, we only need to show that for any f ∈F ∞ C , if R [0,1]d |f(x)|dx = r, then ∥f∥∞= O(r · logd( 1 r)). Since f is infinitely smooth, we can choose K large and depending on r. For the d = 1 case, let K + 1 = log 1 r log log 1 r . We know that the optimal f is of the form of Eq.(11). (Actually this choice of K is approximately the largest K such that Eq.(11) is still the optimal form. If K is larger than this, ξ will be out of [0, 1].) Since R 1 0 |f(x)| = r, we have ξK+1 = (K+1)! C . Now, ∥f∥∞= C K!ξK. Note that ( 1 r)K+1 = ( 1 r) log log 1 r log 1 r = log 1 r. By Stirling’s formula we can show ∥f∥∞= O(r · log 1 r). For the d > 1 case, let K +d = log 1 r log log 1 r . By similar arguments we can show ∥f∥∞= O(r·logd 1 r). Acknowledgement This work was supported by NSFC(60775005). 8 References [Ang88] D. Angluin. Queries and concept learning. Machine Learning, 2:319–342, 1988. [BAL06] M.-F. Balcan, A.Beygelzimer, and J. Langford. Agnostic active learning. In 23th International Conference on Machine Learning, 2006. [CN07] R. Castro and R. Nowak. Minimax bounds for active learning. In 20th Annual Conference on Learning Theory, 2007. [Das05] S. Dasgupta. Coarse sample complexity bounds for active learning. In Advances in Neural Information Processing Systems, 2005. [DHM07] S. Dasgupta, D. Hsu, and C. Monteleoni. A general agnostic active learning algorithm. In Advances in Neural Information Processing Systems, 2007. [Dud99] R.M. Dudley. Uniform Central Limit Theorems. Cambridge University Press, 1999. [Fri09] E. Friedman. Active learning for smooth problems. In 22th Annual Conference on Learning Theory, 2009. [GKW03] V.E. Gine, V.I. Koltchinskii, and J. Wellner. Ratio limit theorems for empirical processes. Stochastic Inequalities and Applications, 56:249–278, 2003. [Han07] S. Hanneke. A bound on the label complexity of agnostic active learning. In 24th International Conference on Machine Learning, 2007. [Han09] S. Hanneke. Adaptive rates of convergence in active learning. In 22th Annual Conference on Learning Theory, 2009. [Kaa06] M. Kaariainen. Active learning in the non-realizable case. In 17th International Conference on Algorithmic Learning Theory, 2006. [Tsy04] A. Tsybakov. Optimal aggregation of classifiers in statistical learning. The Annals of Statistics, 32:135–166, 2004. [vdG00] S. van de Geer. Applications of Empirical Process Theory. Cambridge University Press, 2000. [vdVW96] A. van der Vaart and J. Wellner. Weak Convergence and Empirical Processes with Application to Statistics. Springer Verlag, 1996. 9
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The Ordered Residual Kernel for Robust Motion Subspace Clustering Tat-Jun Chin, Hanzi Wang and David Suter School of Computer Science The University of Adelaide, South Australia {tjchin, hwang, dsuter}@cs.adelaide.edu.au Abstract We present a novel and highly effective approach for multi-body motion segmentation. Drawing inspiration from robust statistical model fitting, we estimate putative subspace hypotheses from the data. However, instead of ranking them we encapsulate the hypotheses in a novel Mercer kernel which elicits the potential of two point trajectories to have emerged from the same subspace. The kernel permits the application of well-established statistical learning methods for effective outlier rejection, automatic recovery of the number of motions and accurate segmentation of the point trajectories. The method operates well under severe outliers arising from spurious trajectories or mistracks. Detailed experiments on a recent benchmark dataset (Hopkins 155) show that our method is superior to other stateof-the-art approaches in terms of recovering the number of motions, segmentation accuracy, robustness against gross outliers and computational efficiency. 1 Introduction1 Multi-body motion segmentation concerns the separation of motions arising from multiple moving objects in a video sequence. The input data is usually a set of points on the surface of the objects which are tracked throughout the video sequence. Motion segmentation can serve as a useful preprocessing step for many computer vision applications. In recent years the case of rigid (i.e. nonarticulated) objects for which the motions could be semi-dependent on each other has received much attention [18, 14, 19, 21, 22, 17]. Under this domain the affine projection model is usually adopted. Such a model implies that the point trajectories from a particular motion lie on a linear subspace of at most four, and trajectories from different motions lie on distinct subspaces. Thus multi-body motion segmentation is reduced to the problem of subspace segmentation or clustering. To realize practical algorithms, motion segmentation approaches should possess four desirable attributes: (1) Accuracy in classifying the point trajectories to the motions they respectively belong to. This is crucial for success in the subsequent vision applications, e.g. object recognition, 3D reconstruction. (2) Robustness against inlier noise (e.g. slight localization error) and gross outliers (e.g. mistracks, spurious trajectories), since getting imperfect data is almost always unavoidable in practical circumstances. (3) Ability to automatically deduce the number of motions in the data. This is pivotal to accomplish fully automated vision applications. (4) Computational efficiency. This is integral for the processing of video sequences which are usually large amounts of data. Recent work on multi-body motion segmentation can roughly be divided into algebraic or factorization methods [3, 19, 20], statistical methods [17, 7, 14, 6, 10] and clustering methods [22, 21, 5]. Notable approaches include Generalized PCA (GPCA) [19, 20], an algebraic method based on the idea that one can fit a union of m subspaces with a set of polynomials of degree m. Statistical methods often employ concepts such random hypothesis generation [4, 17], Expectation-Maximization [14, 6] 1This work was supported by the Australian Research Council (ARC) under the project DP0878801. 1 and geometric model selection [7, 8]. Clustering based methods [22, 21, 5] are also gaining attention due to their effectiveness. They usually include a dimensionality reduction step (e.g. manifold learning [5]) followed by a clustering of the point trajectories (e.g. via spectral clustering in [21]). A recent benchmark [18] indicated that Local Subspace Affinity (LSA) [21] gave the best performance in terms of classification accuracy, although their result was subsequently surpassed by [5, 10]. However, we argue that most of the previous approaches do not simultaneously fulfil the qualities desirable of motion segmentation algorithms. Most notably, although some of the approaches have the means to estimate the number of motions, they are generally unreliable in this respect and require manual input of this parameter. In fact this prior knowledge was given to all the methods compared in [18]2. Secondly, most of the methods (e.g. [19, 5]) do not explicitly deal with outliers. They will almost always breakdown when given corrupted data. These deficiencies reduce the usefulness of available motion segmentation algorithms in practical circumstances. In this paper we attempt to bridge the gap between experimental performance and practical usability. Our previous work [2] indicates that robust multi-structure model fitting can be achieved effectively with statistical learning. Here we extend this concept to motion subspace clustering. Drawing inspiration from robust statistical model fitting [4], we estimate random hypotheses of motion subspaces in the data. However, instead of ranking these hypotheses we encapsulate them in a novel Mercer kernel. The kernel can function reliably despite overwhelming sampling imbalance, and it permits the application of non-linear dimensionality reduction techniques to effectively identify and reject outlying trajectories. This is then followed by Kernel PCA [11] to maximize the separation between groups and spectral clustering [13] to recover the number of motions and clustering. Experiments on the Hopkins 155 benchmark dataset [18] show that our method is superior to other approaches in terms of the qualities described above, including computational efficiency. 1.1 Brief review of affine model multi-body motion segmentation Let {tfp ∈R2}f=1,...,F p=1,...,P be the set of 2D coordinates of P trajectories tracked across F frames. In multi-body motion segmentation the tfp’s correspond to points on the surface of rigid objects which are moving. The goal is to separate the trajectories into groups corresponding to the motion they belong to. In other words, if we arrange the coordinates in the following data matrix T = t11 · · · t1P ... ... ... tF 1 . . . tF P ∈R2F ×P , (1) the goal is to find the permutation Γ ∈RP ×P such that the columns of T · Γ are arranged according to the respective motions they belong to. It turns out that under affine projection [1, 16] trajectories from the same motion lie on a distinct subspace in R2F , and each of these motion subspaces is of dimensions 2, 3 or 4. Thus motion segmentation can be accomplished via clustering subspaces in R2F . See [1, 16] for more details. Realistically actual motion sequences might contain trajectories which do not correspond to valid objects or motions. These trajectories behave as outliers in the data and, if not taken into account, can be seriously detrimental to subspace clustering algorithms. 2 The Ordered Residual Kernel (ORK) First, we take a statistical model fitting point of view to motion segmentation. Let {xi}i=1,...,N be the set of N samples on which we want to perform model fitting. We randomly draw p-subsets from the data and use it to fit a hypothesis of the model, where p is the number of parameters that define the model. In motion segmentation, the xi’s are the columns of matrix T, and p = 4 since the model is a four-dimensional subspace3. Assume that M of such random hypotheses are drawn. For each data point xi compute its absolute residual set ri = {ri 1, . . . , ri M} as measured to the M hypotheses. For motion segmentation, the residual is the orthogonal distance to a hypothesis 2As confirmed through private contact with the authors of [18]. 3Ideally we should also consider degenerate motions with subspace dimensions 2 or 3, but previous work [18] using RANSAC [4] and our results suggest this is not a pressing issue for the Hopkins 155 dataset. 2 subspace. We sort the elements in ri to obtain the sorted residual set ˜ri = {ri λi 1, . . . , ri λi M }, where the permutation {λi 1, . . . , λi M} is obtained such that ri λi 1 ≤· · · ≤ri λi M . Define the following ˜θi := {λi 1, . . . , λi M} (2) as the sorted hypothesis set of point xi, i.e. ˜θi depicts the order in which xi becomes the inlier of the M hypotheses as a fictitious inlier threshold is increased from 0 to ∞. We define the Ordered Residual Kernel (ORK) between two data points as k˜r(xi1, xi2) := 1 Z M/h X t=1 zt · kt ∩(˜θi1, ˜θi2), (3) where zt = 1 t are the harmonic series and Z = PM/h t=1 zt is the (M/h)-th harmonic number. Without lost of generality assume that M is wholly divisible by h. Step size h is used to obtain the Difference of Intersection Kernel (DOIK) kt ∩(˜θi1, ˜θi2) := 1 h(|˜θ1:αt i1 ∩˜θ1:αt i2 | −|˜θ1:αt−1 i1 ∩˜θ1:αt−1 i2 |) (4) where αt = t · h and αt−1 = (t −1) · h. Symbol ˜θa:b i indicates the set formed by the a-th to the b-th elements of ˜θi. Since the contents of the sorted hypotheses set are merely permutations of {1 . . . M}, i.e. there are no repeating elements, 0 ≤k˜r(xi1, xi2) ≤1. (5) Note that k˜r is independent of the type of model to be fitted, thus it is applicable to generic statistical model fitting problems. However, we concentrate on motion subspaces in this paper. Let τ be a fictitious inlier threshold. The kernel k˜r captures the intuition that, if τ is low, two points arising from the same subspace will have high normalized intersection since they share many common hypotheses which correspond to that subspace. If τ is high, implausible hypotheses fitted on outliers start to dominate and decrease the normalized intersection. Step size h allows us to quantify the rate of change of intersection if τ is increased from 0 to ∞, and since zt is decreasing, k˜r will evaluate to a high value for two points from the same subspace. In contrast, k˜r is always low for points not from the same subspace or that are outliers. Proof of satisfying Mercer’s condition. Let D be a fixed domain, and P(D) be the power set of D, i.e. the set of all subsets of D. Let S ⊆P(D), and p, q ∈S. If µ is a measure on D, then k∩(p, q) = µ(p ∩q), (6) called the intersection kernel, is provably a valid Mercer kernel [12]. The DOIK can be rewritten as kt ∩(˜θi1, ˜θi2) = 1 h(|˜θ(αt−1+1):αt i1 ∩˜θ(αt−1+1):αt i2 | +|˜θ1:(αt−1) i1 ∩˜θ(αt−1+1):αt i2 | + |˜θ(αt−1+1):αt i1 ∩˜θ1:(αt−1) i2 |). (7) If we let D = {1 . . .M} be the set of all possible hypothesis indices and µ be uniform on D, each term in Eq. (7) is simply an intersection kernel multiplied by |D|/h. Since multiplying a kernel with a positive constant and adding two kernels respectively produce valid Mercer kernels [12], the DOIK and ORK are also valid Mercer kernels.• Parameter h in k˜r depends on the number of random hypotheses M, i.e. step size h can be set as a ratio of M. The value of M can be determined based on the size of the p-subset and the size of the data N (e.g. [23, 15]), and thus h is not contingent on knowledge of the true inlier noise scale or threshold. Moreover, our experiments in Sec. 4 show that segmentation performance is relatively insensitive to the settings of h and M. 2.1 Performance under sampling imbalance Methods based on random sampling (e.g. RANSAC [4]) are usually affected by unbalanced datasets. The probability of simultaneously retrieving p inliers from a particular structure is tiny if points 3 from that structure represent only a small minority in the data. In an unbalanced dataset the “pure” p-subsets in the M randomly drawn samples will be dominated by points from the majority structure in the data. This is a pronounced problem in motion sequences, since there is usually a background “object” whose point trajectories form a large majority in the data. In fact, for motion sequences from the Hopkins 155 dataset [18] with typically about 300 points per sequence, M has to be raised to about 20,000 before a pure p-subset from the non-background objects is sampled. However, ORK can function reliably despite serious sampling imbalance. This is because points from the same subspace are roughly equi-distance to the sampled hypotheses in their vicinity, even though these hypotheses might not pass through that subspace. Moreover, since zt in Eq. (3) is decreasing only residuals/hypotheses in the vicinity of a point are heavily weighted in the intersection. Fig. 1(a) illustrates this condition. Results in Sec. 4 show that ORK excelled even with M = 1, 000. (a) Data in R2F . (b) Data in RKHS Fk˜r. Figure 1: (a) ORK under sampling imbalance. (b) Data in RKHS induced by ORK. 3 Multi-Body Motion Segmentation using ORK In this section, we describe how ORK is used for multi-body motion segmentation. 3.1 Outlier rejection via non-linear dimensionality reduction Denote by Fk˜r the Reproducing Kernel Hilbert Space (RKHS) induced by k˜r. Let matrix A = [φ(x1) . . . φ(xN)] contain the input data after it is mapped to Fk˜r. The kernel matrix K = AT A is computed using the kernel function k˜r as Kp,q = ⟨φ(xp), φ(xq)⟩= k˜r(xp, xq), p, q ∈{1 . . . N}. (8) Since k˜r is a valid Mercer kernel, K is guaranteed to be positive semi-definite [12]. Let K = Q∆QT be the eigenvalue decomposition (EVD) of K. Then the rank-n Kernel Singular Value Decomposition (Kernel SVD) [12] of A is An = [AQn(∆n)−1 2 ][(∆n) 1 2 ][(Qn)T ] ≡UnΣn(Vn)T . (9) Via the Matlab notation, Qn = Q:,1:n and ∆n = ∆1:n,1:n. The left singular vectors Un is an orthonormal basis for the n-dimensional principal subspace of the whole dataset in Fk˜r. Projecting the data onto the principal subspace yields B = [AQn(∆n)−1 2 ]T A = (∆n) 1 2 (Qn)T , (10) where B = [b1 . . . bN] ∈Rn×N is the reduced dimension version of A. Directions of the principal subspace are dominated by inlier points, since k˜r evaluates to a high value generally for them, but always to a low value for gross outliers. Moreover the kernel ensures that points from the same subspace are mapped to the same cluster and vice versa. Fig. 1(b) illustrates this condition. Fig. 2(a)(left) shows the first frame of sequence “Cars10” from the Hopkins 155 dataset [18] with 100 false trajectories of Brownian motion added to the original data (297 points). The corresponing RKHS norm histogram for n = 3 is displayed in Fig. 2(b). The existence of two distinct modes, 4 (a) (left) Before and (right) after outlier removal. Blue dots are inliers while red dots are added outliers. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 5 10 15 Vector norm in principal subspace Bin count Inlier mode Outlier mode (b) Actual norm histogram of “cars10”. Figure 2: Demonstration of outlier rejection on sequence “cars10” from Hopkins 155. corresponding respectively to inliers and outliers, is evident. We exploit this observation for outlier rejection by discarding data with low norms in the principal subspace. The cut-off threshold ψ can be determined by analyzing the shape of the distribution. For instance we can fit a 1D Gaussian Mixture Model (GMM) with two components and set ψ as the point of equal Mahalanobis distance between the two components. However, our experimentation shows that an effective threshold can be obtained by simply setting ψ as the average value of all the norms, i.e. ψ = 1 N N X i=1 ∥bi∥. (11) This method was applied uniformly on all the sequences in our experiments in Sec. 4. Fig. 2(a)(right) shows an actual result of the method on Fig. 2(a)(left). 3.2 Recovering the number of motions and subspace clustering After outlier rejection, we further take advantage of the mapping induced by ORK for recovering the number of motions and subspace clustering. On the remaining data, we perform Kernel PCA [11] to seek the principal components which maximize the variance of the data in the RKHS, as Fig. 1(b) illustrates. Let {yi}i=1,...,N ′ be the N ′-point subset of the input data that remains after outlier removal, where N ′ < N. Denote by C = [φ(y1) . . . φ(yN ′)] the data matrix after mapping the data to Fk˜r, and by symbol ˜C the result of adjusting C with the empirical mean of {φ(y1), . . . , φ(yN ′)}. The centered kernel matrix ˜K′ = ˜CT ˜C [11] can be obtained as ˜K′ = νT K′ν, ν = [IN ′ −1 N ′ 1N ′,N ′], (12) where K′ = CT C is the uncentered kernel matrix, Is and 1s,s are respectively the s × s identity matrix and a matrix of ones. If ˜K′ = RΩRT is the EVD of ˜K′, then we obtain first-m kernel principal components Pm of C as the first-m left singular vectors of ˜C , i.e. Pm = ˜CRm(Ωm)−1 2 , (13) where Rm = R:,1:m and Ω1:m,1:m; see Eq. (9). Projecting the data on the principal components yields D = [d1 . . . dN ′] = (Ωm) 1 2 (Rm)T , (14) where D ∈Rm×N ′. The affine subspace span(Pm) maximizes the spread of the centered data in the RKHS, and the projection D offers an effective representation for clustering. Fig. 3(a) shows the Kernel PCA projection results for m = 3 on the sequence in Fig. 2(a). The number of clusters in D is recovered via spectral clustering. More specifically we apply the Normalized Cut (Ncut) [13] algorithm. A fully connected graph is first derived from the data, where its weighted adjacency matrix W ∈RN ′×N ′ is obtained as Wp,q = exp(−∥dp −dq∥2/2δ2), (15) and δ is taken as the average nearest neighbour distance in the Euclidean sense among the vectors in D. The Laplacian matrix [13] is then derived from W and eigendecomposed. Under Ncut, 5 0.06 0.08 0.1 −0.1 −0.05 0 0.05 0.1 0.15 −0.15 −0.1 −0.05 0 0.05 0.1 (a) Kernel PCA and Ncut results. (b) W matrix. (c) Final result for “cars10”. Figure 3: Actual results on the motion sequence in Fig. 2(a)(left). the number of clusters is revealed as the number of eigenvalues of the Laplacian that are zero or numerically insignificant. With this knowledge, a subsequent k-means step is then performed to cluster the points. Fig. 3(b) shows W for the input data in Fig. 2(a)(left) after outlier removal. It can be seen that strong affinity exists between points from the same cluster, thus allowing accurate clustering. Figs. 3(a) and 3(c) illustrate the final clustering result for the data in Fig. 2(a)(left). There are several reasons why spectral clustering under our framework is more successful than previous methods. Firstly, we perform an effective outlier rejection step that removes bad trajectories that can potentially mislead the clustering. Secondly, the mapping induced by ORK deliberately separates the trajectories based on their cluster membership. Finally, we perform Kernel PCA to maximize the variance of the data. Effectively this also improves the separation of clusters, thus facilitating an accurate recovery of the number of clusters and also the subsequent segmentation. This distinguishes our work from previous clustering based methods [21, 5] which tend to operate without maximizing the between-class scatter. Results in Sec. 4 validate our claims. 4 Results Henceforth we indicate the proposed method as “ORK”. We leverage on a recently published benchmark on affine model motion segmentation [18] as a basis of comparison. The benchmark was evaluated on the Hopkins 155 dataset4 which contains 155 sequences with tracked point trajectories. A total of 120 sequences have two motions while 35 have three motions. The sequences contain degenerate and non-degenerate motions, independent and partially dependent motions, articulated motions, nonrigid motions etc. In terms of video content three categories exist: Checkerboard sequences, traffic sequences (moving cars, trucks) and articulated motions (moving faces, people). 4.1 Details on benchmarking Four major algorithms were compared in [18]: Generalized PCA (GPCA) [19], Local Subspace Affinity (LSA) [21], Multi-Stage Learning (MSL) [14] and RANSAC [17]. Here we extend the benchmark with newly reported results from Locally Linear Manifold Clustering (LLMC) [5] and Agglomerative Lossy Compression (ALC) [10, 9]. We also compare our method against Kanatani and Matsunaga’s [8] algorithm (henceforth, the “KM” method) in estimating the number of independent motions in the video sequences. Note that KM per se does not perform motion segmentation. For the sake of objective comparisons we use only implementations available publicly5. Following [18], motion segmentation performance is evaluated in terms of the labelling error of the point trajectories, where each point in a sequence has a ground truth label, i.e. classification error = number of mislabeled points total number of points . (16) Unlike [18], we also emphasize on the ability of the methods in recovering the number of motions. However, although the methods compared in [18] (except RANSAC) theoretically have the means to 4Available at http://www.vision.jhu.edu/data/hopkins155/. 5For MSL and KM, see http://www.suri.cs.okayama-u.ac.jp/e-program-separate.html/. For GPCA, LSA and RANSAC, refer to the url for the Hopkins 155 dataset. 6 do so, their estimation of the number of motions is generally unrealiable and the benchmark results in [18] were obtained by revealing the actual number of motions to the algorithms. A similar initialization exists in [5, 10] where the results were obtained by giving LLMC and ALC this knowledge a priori (for LLMC, this was given at least to the variant LLMC 4m during dimensionality reduction [5], where m is the true number of motions). In the following subsections, where variants exist for the compared algorithms we use results from the best performing variant. In the following the number of random hypotheses M and step size h for ORK are fixed at 1000 and 300 respectively, and unlike the others, ORK is not given knowledge of the number of motions. 4.2 Data without gross outliers We apply ORK on the Hopkins 155 dataset. Since ORK uses random sampling we repeat it 100 times for each sequence and average the results. Table 1 depicts the obtained classification error among those from previously proposed methods. ORK (column 9) gives comparable results to the other methods for sequences with 2 motions (mean = 7.83%, median = 0.41%). For sequences with 3 motions, ORK (mean = 12.62%, median = 4.75%) outperforms GPCA and RANSAC, but is slightly less accurate than the others. However, bear in mind that unlike the other methods ORK is not given prior knowledge of the true number of motions and has to estimate this independently. Column 1 2 3 4 5 6 8 9 10 Method REF GPCA LSA MSL RANSAC LLMC ALC ORK ORK∗ Sequences with 2 motions Mean 2.03 4.59 3.45 4.14 5.56 3.62 3.03 7.83 1.27 Median 0.00 0.38 0.59 0.00 1.18 0.00 0.00 0.41 0.00 Sequences with 3 motions Mean 5.08 28.66 9.73 8.23 22.94 8.85 6.26 12.62 2.09 Median 2.40 28.26 2.33 1.76 22.03 3.19 1.02 4.75 0.05 Table 1: Classification error (%) on Hopkins 155 sequences. REF represents the reference/control method which operates based on knowledge of ground truth segmentation. Refer to [18] for details. We also separately investigate the accuracy of ORK in estimating the number of motions, and compare it against KM [8] which was proposed for this purpose. Note that such an experiment was not attempted in [18] since approaches compared therein generally do not perform reliably in estimating the number of motions. The results in Table 2 (columns 1–2) show that for sequences with two motions, KM (80.83%) outperforms ORK (67.37%) by ≈15 percentage points. However, for sequences with three motions, ORK (49.66%) vastly outperforms KM (14.29%) by more than doubling the percentage points of accuracy. The overall accuracy of KM (65.81%) is slightly better than ORK (63.37%), but this is mostly because sequences with two motions form the majority in the dataset (120 out of 155). This leads us to conclude that ORK is actually the superior method here. Dataset Hopkins 155 Hopkins 155 + Outliers Column 1 2 3 4 Method KM ORK KM ORK 2 motions 80.83% 67.37% 00.00% 47.58% 3 motions 14.29% 49.66% 100.00% 50.00% Overall 65.81% 63.37% 22.58% 48.13% Table 2: Accuracy in determining the number of motions in a sequence. Note that in the experiment with outliers (columns 3–4), KM returns a constant number of 3 motions for all sequences. We re-evaluate the performance of ORK by considering only results on sequences where the number of motions is estimated correctly by ORK (there are about 98 ≡63.37% of such cases). The results are tabulated under ORK∗(column 10) in Table 1. It can be seen that when ORK estimates the number of motions correctly, it is significantly more accurate than the other methods. Finally, we compare the speed of the methods in Table 3. ORK was implemented and run in Matlab on a Dual Core Pentium 3.00GHz machine with 4GB of main memory (this is much less powerful 7 than the 8 Core Xeon 3.66GHz with 32GB memory used in [18] for the other methods in Table 3). The results show that ORK is comparable to LSA, much faster than MSL and ALC, but slower than GPCA and RANSAC. Timing results of LLMC are not available in the literature. Method GPCA LSA MSL RANSAC ALC ORK 2 motions 324ms 7.584s 11h 4m 175ms 10m 32s 4.249s 3 motions 738ms 15.956s 1d 23h 258ms 10m 32s 8.479s Table 3: Average computation time on Hopkins 155 sequences. 4.3 Data with gross outliers We next examine the ability of the proposed method in dealing with gross outliers in motion data. For each sequence in Hopkins 155, we add 100 gross outliers by creating trajectories corresponding to mistracks or spuriously occuring points. These are created by randomly initializing 100 locations in the first frame and allowing them to drift throughout the sequence according to Brownian motion. The corrupted sequences are then subjected to the algorithms for motion segmentation. Since only ORK is capable of rejecting outliers, the classification error of Eq. (16) is evaluated on the inlier points only. The results in Table 4 illustrate that ORK (column 4) is the most accurate method by a large margin. Despite being given the true number of motions a priori, GPCA, LSA and RANSAC are unable to provide satisfactory segmentation results. Column 1 2 3 4 5 Method GPCA LSA RANSAC ORK ORK∗ Sequences with 2 motions Mean 28.66 24.25 30.64 16.50 1.62 Median 30.96 26.51 32.36 10.54 0.00 Sequences with 3 motions Mean 40.61 30.94 42.24 19.99 2.68 Median 41.30 27.68 43.43 8.49 0.09 Table 4: Classification error (%) on Hopkins 155 sequences with 100 gross outliers per sequence. In terms of estimating the number of motions, as shown in column 4 in Table 2 the overall accuracy of ORK is reduced to 48.13%. This is contributed mainly by the deterioration in accuracy on sequences with two motions (47.58%), although the accuracy on sequences with three motions are maintained (50.00%). This is not a surprising result, since sequences with three motions generally have more (inlying) point trajectories than sequences with two motions, thus the outlier rates for sequences with three motions are lower (recall that a fixed number of 100 false trajectories are added). On the other hand, the KM method (column 3) is completely overwhelmed by the outliers— for all the sequences with outliers it returned a constant “3” as the number of motions. We again re-evaluate ORK by considering results from sequences (now with gross outliers) where the number of motions is correctly estimated (there are about 75 ≡48.13% of such cases). The results tabulated under ORK∗(column 5) in Table 4 show that the proposed method can accurately segment the point trajectories without being influenced by the gross outliers. 5 Conclusions In this paper we propose a novel and highly effective approach for multi-body motion segmentation. Our idea is based on encapsulating random hypotheses in a novel Mercel kernel and statistical learning. We evaluated our method on the Hopkins 155 dataset with results showing that the idea is superior other state-of-the-art approaches. It is by far the most accurate in terms of estimating the number of motions, and it excels in segmentation accuracy despite lacking prior knowledge of the number of motions. The proposed idea is also highly robust towards outliers in the input data. Acknowledgements. We are grateful to the authors of [18] especially Ren´e Vidal for discussions and insights which have been immensely helpful. 8 References [1] T. Boult and L. Brown. Factorization-based segmentation of motions. In IEEE Workshop on Motion Understanding, 1991. [2] T.-J. Chin, H. Wang, and D. Suter. Robust fitting of multiple structures: The statistical learning approach. In ICCV, 2009. [3] J. Costeira and T. Kanade. A multibody factorization method for independently moving objects. IJCV, 29(3):159–179, 1998. [4] M. A. Fischler and R. C. Bolles. Random sample concensus: A paradigm for model fitting with applications to image analysis and automated cartography. Comm. of the ACM, 24:381–395, 1981. [5] A. Goh and R. Vidal. Segmenting motions of different types by unsupervised manifold clustering. In CVPR, 2007. [6] A. Gruber and Y. Weiss. Multibody factorization with uncertainty and missing data using the EM algorithm. In CVPR, 2004. [7] K. Kanatani. Motion segmentation by subspace separation and model selection. In ICCV, 2001. [8] K. Kanatani and C. Matsunaga. Estimating the number of independent motions for multibody segmentation. In ACCV, 2002. [9] Y. Ma, H. Derksen, W. Hong, and J. Wright. Segmentation of multivariate mixed data via lossy coding and compression. TPAMI, 29(9):1546–1562, 2007. [10] S. Rao, R. Tron, Y. Ma, and R. Vidal. Motion segmentation via robust subspace separation in the presence of outlying, incomplete, or corrupted trajectories. In CVPR, 2008. [11] B. Sch¨olkopf, A. Smola, and K. R. M¨uller. Nonlinear component analysis as a kernel eigenvalue problem. Neural Computation, 10:1299–1319, 1998. [12] J. Shawe-Taylor and N. Cristianini. Kernel methods for pattern analysis. Cambridge University Press, 2004. [13] J. Shi and J. Malik. Normalized cuts and image segmentation. TPAMI, 22(8):888–905, 2000. [14] Y. Sugaya and K. Kanatani. Geometric structure of degeneracy for multi-body motion segmentation. In Workshop on Statistical Methods in Video Processing, 2004. [15] R. Toldo and A. Fusiello. Robust multiple structures estimation with J-Linkage. In ECCV, 2008. [16] C. Tomasi and T. Kanade. Shape and motion from image streams under orthography. IJCV, 9(2):137–154, 1992. [17] P. Torr. Geometric motion segmentation and model selection. Phil. Trans. Royal Society of London, 356(1740):1321–1340, 1998. [18] R. Tron and R. Vidal. A benchmark for the comparison of 3-D motion segmentation algorithms. In CVPR, 2007. [19] R. Vidal and R. Hartley. Motion segmentation with missing data by PowerFactorization and Generalized PCA. In CVPR, 2004. [20] R. Vidal, Y. Ma, and S. Sastry. Generalized Principal Component Analysis (GPCA). TPAMI, 27(12):1–15, 2005. [21] J. Yan and M. Pollefeys. A general framework for motion segmentation: independent, articulated, rigid, non-rigid, degenerate and non-degenerate. In ECCV, 2006. [22] L. Zelnik-Manor and M. Irani. Degeneracies, dependencies and their implications on multibody and multi-sequence factorization. In CVPR, 2003. [23] W. Zhang and J. Koseck´a. Nonparametric estimation of multiple structures with outliers. In Dynamical Vision, ICCV 2005 and ECCV 2006 Workshops, 2006. 9
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Learning to Rank by Optimizing NDCG Measure Hamed Valizadegan Rong Jin Computer Science and Engineering Michigan State University East Lansing, MI 48824 {valizade,rongjin}@cse.msu.edu Ruofei Zhang Jianchang Mao Advertising Sciences, Yahoo! Labs 4401 Great America Parkway, Santa Clara, CA 95054 {rzhang,jmao}@yahoo-inc.com Abstract Learning to rank is a relatively new field of study, aiming to learn a ranking function from a set of training data with relevancy labels. The ranking algorithms are often evaluated using information retrieval measures, such as Normalized Discounted Cumulative Gain (NDCG) [1] and Mean Average Precision (MAP) [2]. Until recently, most learning to rank algorithms were not using a loss function related to the above mentioned evaluation measures. The main difficulty in direct optimization of these measures is that they depend on the ranks of documents, not the numerical values output by the ranking function. We propose a probabilistic framework that addresses this challenge by optimizing the expectation of NDCG over all the possible permutations of documents. A relaxation strategy is used to approximate the average of NDCG over the space of permutation, and a bound optimization approach is proposed to make the computation efficient. Extensive experiments show that the proposed algorithm outperforms state-of-the-art ranking algorithms on several benchmark data sets. 1 Introduction Learning to rank has attracted the focus of many machine learning researchers in the last decade because of its growing application in the areas like information retrieval (IR) and recommender systems. In the simplest form, the so-called pointwise approaches, ranking can be treated as classification or regression by learning the numeric rank value of documents as an absolute quantity [3, 4]. The second group of algorithms, the pairwise approaches, considers the pair of documents as independent variables and learns a classification (regression) model to correctly order the training pairs [5, 6, 7, 8, 9, 10, 11]. The main problem with these approaches is that their loss functions are related to individual documents while most evaluation metrics of information retrieval measure the ranking quality for individual queries, not documents. This mismatch has motivated the so called listwise approaches for information ranking, which treats each ranking list of documents for a query as a training instance [2, 12, 13, 14, 15, 16, 17]. Unlike the pointwise or pairwise approaches, the listwise approaches aim to optimize the evaluation metrics such as NDCG and MAP. The main difficulty in optimizing these evaluation metrics is that they are dependent on the rank position of documents induced by the ranking function, not the numerical values output by the ranking function. In the past studies, this problem was addressed either by the convex surrogate of the IR metrics or by heuristic optimization methods such as genetic algorithm. In this work, we address this challenge by a probabilistic framework that optimizes the expectation of NDCG over all the possible permutation of documents. To handle the computational difficulty, we present a relaxation strategy that approximates the expectation of NDCG in the space of permutation, and a bound optimization algorithm [18] for efficient optimization. Our experiment with several benchmark data sets shows that our method performs better than several state-of-the-art ranking techniques. 1 The rest of this paper is organized as follows. The related work is presented in Section 2. The proposed framework and optimization strategy is presented in Section 3. We report our experimental study in Section 4 and conclude this work in Section 5. 2 Related Work We focus on reviewing the listwise approaches that are closely related to the theme of this work. The listwise approaches can be classified into two categories. The first group of approaches directly optimizes the IR evaluation metrics. Most IR evaluation metrics, however, depend on the sorted order of documents, and are non-convex in the target ranking function. To avoid the computational difficulty, these approaches either approximate the metrics with some convex functions or deploy methods (e.g., genetic algorithm [19]) for non-convex optimization. In [13], the authors introduced LambdaRank that addresses the difficulty in optimizing IR metrics by defining a virtual gradient on each document after the sorting. While [13] provided a simple test to determine if there exists an implicit cost function for the virtual gradient, theoretical justification for the relation between the implicit cost function and the IR evaluation metric is incomplete. This may partially explain why LambdaRank performs very poor when compared to MCRank [3], a simple adjustment of classification for ranking (a pointwise approach). The authors of MCRank paper even claimed that a boosting model for regression produces better results than LambdaRank. Volkovs and Zemel [17] proposed optimizing the expectation of IR measures to overcome the sorting problem, similar to the approach taken in this paper. However they use monte carlo sampling to address the intractable task of computing the expectation in the permutation space which could be a bad approximation for the queries with large number of documents. AdaRank [20] uses boosting to optimize NDCG, similar to our optimization strategy. However they deploy heuristics to embed the IR evaluation metrics in computing the weights of queries and the importance of weak rankers; i.e. it uses NDCG value of each query in the current iteration as the weight for that query in constructing the weak ranker (the documents of each query have similar weight). This is unlike our approach that the contribution of each single document to the final NDCG score is considered. Moreover, unlike our method, the convergence of AdaRank is conditional and not guaranteed. Sun et al. [21] reduced the ranking, as measured by NDCG, to pairwise classification and applied alternating optimization strategy to address the sorting problem by fixing the rank position in getting the derivative. SVMMAP [2] relaxes the MAP metric by incorporating it into the constrains of SVM. Since SVM-MAP is designed to optimize MAP, it only considers the binary relevancy and cannot be applied to the data sets that have more than two levels of relevance judgements. The second group of listwise algorithms defines a listwise loss function as an indirect way to optimize the IR evaluation metrics. RankCosine [12] uses cosine similarity between the ranking list and the ground truth as a query level loss function. ListNet [14] adopts the KL divergence for loss function by defining a probabilistic distribution in the space of permutation for learning to rank. FRank [9] uses a new loss function called fidelity loss on the probability framework introduced in ListNet. ListMLE [15] employs the likelihood loss as the surrogate for the IR evaluation metrics. The main problem with this group of approaches is that the connection between the listwise loss function and the targeted IR evaluation metric is unclear, and therefore optimizing the listwise loss function may not necessarily result in the optimization of the IR metrics. 3 Optimizing NDCG Measure 3.1 Notation Assume that we have a collection of n queries for training, denoted by Q = {q1, . . . , qn}. For each query qk, we have a collection of mk documents Dk = {dk i , i = 1, . . . , mk}, whose relevance to qk is given by a vector rk = (rk 1, . . . , rk mk) ∈Zmk. We denote by F(d, q) the ranking function that takes a document-query pair (d, q) and outputs a real number score, and by jk i the rank of document dk i within the collection Dk for query qk. The NDCG value for ranking function F(d, q) is then computed as following: L(Q, F) = 1 n n X k=1 1 Zk mk X i=1 2rk i −1 log(1 + jk i ) (1) where Zk is the normalization factor [1]. NDCG is usually truncated at a particular rank level (e.g. the first 10 retrieved documents) to emphasize the importance of the first retrieved documents. 2 3.2 A Probabilistic Framework One of the main challenges faced by optimizing the NDCG metric defined in Equation (1) is that the dependence of document ranks (i.e., jk i ) on the ranking function F(d, q) is not explicitly expressed, which makes it computationally challenging. To address this problem, we consider the expectation of L(Q, F) over all the possible rankings induced by the ranking function F(d, q), i.e., ¯L(Q, F) = 1 n n X k=1 1 Zk mk X i=1 * 2rk i −1 log(1 + jk i ) + F = 1 n n X k=1 1 Zk mk X i=1 X πk∈Smk Pr(πk|F, qk) 2rk i −1 log(1 + πk(i)) (2) where Smk stands for the group of permutations of mk documents, and πk is an instance of permutation (or ranking). Notation πk(i) stands for the rank position of the ith document by πk. To this end, we first utilize the result in the following lemma to approximate the expectation of 1/ log(1+πk(i)) by the expectation of πk(i). Lemma 1. For any distribution Pr(π|F, q), the inequality ¯L(Q, F) ≥¯H(Q, F) holds where ¯H(Q, F) = 1 n n X k=1 1 Zk mk X i=1 2rk i −1 log(1 + ⟨πk(i)⟩F ) (3) Proof. The proof follows from the fact that (a) 1/x is a convex function when x > 0 and therefore ⟨1/ log(1+x)⟩≥1/⟨log(1+x)⟩; (b) log(1+x) is a concave function, and therefore ⟨log(1+x)⟩≤ log(1 + ⟨x⟩). Combining these two factors together, we have the result stated in the lemma. Given ¯H(Q, F) provides a lower bound for ¯L(Q, F), in order to maximize ¯L(Q, F), we could alternatively maximize ¯H(Q, F), which is substantially simpler than ¯L(Q, F). In the next step of simplification, we rewrite πk(i) as πk(i) = 1 + mk X j=1 I(πk(i) > πk(j)) (4) where I(x) outputs 1 when x is true and zero otherwise. Hence, ⟨πk(i)⟩is written as ⟨πk(i)⟩= 1 + mk X j=1 ⟨I(πk(i) > πk(j))⟩= 1 + mk X j=1 Pr(πk(i) > πk(j)) (5) As a result, to optimize ¯H(Q, F), we only need to define Pr(πk(i) > πk(j)), i.e., the marginal distribution for document dk j to be ranked before document dk i . In the next section, we will discuss how to define a probability model for Pr(πk|F, qk), and derive pairwise ranking probability Pr(πk(i) > πk(j)) from distribution Pr(πk|F, qk). 3.3 Objective Function We model Pr(πk|F, qk) as follows Pr(πk|F, qk) = 1 Z(F, qk) exp mk X i=1 X j:πk(j)>πk(i) (F(dk i , qk) −F(dk j , qk)) = 1 Z(F, qk) exp à mk X i=1 (mk −2πk(i) + 1)F(dk i , qk) ! (6) where Z(F, qk) is the partition function that ensures the sum of probability is one. Equation (6) models each pair (dk i , dk j ) of the ranking list πk by the factor exp(F(dk i , qk) −F(dk j , qk)) if dk i is ranked before dk j (i.e., πk(dk i ) < πk(dk j )) and vice versa. This modeling choice is consistent with the idea of ranking the documents with largest scores first; intuitively, the more documents in a permutation are in the decreasing order of score, the bigger the probability of the permutation is. Using Equation (6) for Pr(πk|F, qk), we have ¯H(Q, F) expressed in terms of ranking function F. By maximizing ¯H(Q, F) over F, we could find the optimal solution for ranking function F. As indicated by Equation (5), we only need to compute the marginal distribution Pr(πk(i) > πk(j)). To approximate Pr(πk(i) > πk(j)), we divide the group of permutation Smk into two sets: 3 Gk a(i, j) = {πk|πk(i) > πk(j)} and Gk b(i, j) = {πk|πk(i) < πk(j)}. Notice that there is a one-to-one mapping between these two sets; namely for any ranking πk ∈Gk a(i, j), we could create a corresponding ranking πk ∈Gk b(i, j) by switching the rankings of document dk i and dk j and vice versa. The following lemma allows us to bound the marginal distribution Pr(πk(i) > πk(j)). Lemma 2. If F(dk i , qk) > F(dk j , qk), we have Pr(πk(i) > πk(j)) ≤ 1 1 + exp £ 2(F(dk i , qk) −F(dk j , qk)) ¤ (7) Proof. 1 = X πk∈Gka(i,j) Pr(πk|F, qk) + X πk∈Gk b (i,j) Pr(πk|F, qk) = X πk∈Gka(i,j) Pr(πk|F, qk) ³ 1 + exp £ 2(πk(i) −πk(j))(F(dk i , qk) −F(dk j , qk)) ¤ ´ ≥ X πk∈Gk a(i,j) ³ Pr(πk|F, qk) ¡ 1 + exp £ 2(F(dk i , qk) −F(dk j , qk)) ¤¢ ´ = ¡ 1 + exp £ 2(F(dk i , qk) −F(dk j , qk)) ¤¢ Pr ¡ πk(i) > πk(j) ¢ We used the definition of Pr(πk|F, qk) in Equation (6) to find Gk b(i, j) as the dual of Gk a(i, j) in the first step of the proof. The inequality in the proof is because πk(i) −πk(j) ≥1 and the last step is because Pr(πk|F, qk) is the only term dependent on π. This lemma indicates that we could approximate Pr(πk(i) > πk(j)) by a simple logistic model. The idea of using logistic model for Pr(πk(i) > πk(j)) is not new in learning to rank [7, 9]; however it has been taken for granted and no justification has been provided in using it for learning to rank. Using the logistic model approximation introduced in Lemma 2, we now have ⟨πk(i)⟩written as ⟨πk(i)⟩ ≈ 1 + mk X j=1 1 1 + exp £ 2(F(dk i , qk) −F(dk j , qk)) ¤ (8) To simplify our notation, we define F k i = 2F(dk i , qk), and rewrite the above expression as ⟨πk(i)⟩= 1 + mk X j=1 Pr(πk(i) > πk(j)) ≈1 + mk X j=1 1 1 + exp(F k i −F k j ) Using the above approximation for ⟨πk(i)⟩, we have ¯H in Equation (3) written as ¯H(Q, F) ≈1 n n X k=1 1 Zk mk X i=1 2rk i −1 log(2 + Ak i ) (9) where Ak i = mk X j=1 I(j ̸= i) 1 + exp(F k i −F k j ) (10) We define the following proposition to further simplify the objective function: Proposition 1. 1 log(2 + Ak i ) ≥ 1 log(2) − Ak i 2 [log(2)]2 The proof is due to the Taylor expansion of convex function 1/log(2 + x), x > −1 around x = 0 noting that Ak i > 0 (the proof of convexity of 1/log(1 + x) is given in Lemma 1). By plugging the result of this proposition to the objective function in Equation (9), the new objective is to minimize the following quantity: ¯ M(Q, F) ≈1 n n X k=1 1 Zk mk X i=1 (2rk i −1)Ak i (11) The objective function in Equation (11) is explicitly related to F via term Ak i . In the next section, we aim to derive an algorithm that learns an effective ranking function by efficiently minimizing ¯ M. It is also important to note that although ¯ M is no longer a rigorous lower bound for the original objective function ¯L, our empirical study shows that this approximation is very effective in identifying the appropriate ranking function from the training data. 4 3.4 Algorithm To minimize ¯ M(Q, F) in Equation (11), we employ the bound optimization strategy [18] that iteratively updates the solution for F. Let F k i denote the value obtained so far for document dk i . To improve NDCG, following the idea of Adaboost, we restrict the new ranking value for document dk i , denoted by ˜F k i , is updated as to the following form: ˜F k i = F k i + αf k i (12) where α > 0 is the combination weight and f k i = f(dk i , qk) ∈{0, 1} is a binary value. Note that in the above, we assume the ranking function F(d, q) is updated iteratively with an addition of binary classification function f(d, q), which leads to efficient computation as well as effective exploitation of the existing algorithms for data classification. . To construct a lower bound for ¯ M(Q, F), we first handle the expression [1 + exp(F k i −F k j )]−1, summarized by the following proposition. Proposition 2. 1 1 + exp( ˜F k i −˜F k j ) ≤ 1 1 + exp(F k i −F k j ) + γk i,j £ exp(α(f k j −f k i )) −1 ¤ (13) where γk i,j = exp(F k i −F k j ) ¡ 1 + exp(F k i −F k j ) ¢2 (14) The proof of this proposition can be found in Appendix A. This proposition separates the term related to F k i from that related to αf k i in Equation (11), and shows how the new weak ranker (i.e., the binary classification function f(d, q)) will affect the current ranking function F(d, q). Using the above proposition, we can derive the upper bound for M (Theorem 1) as well as a closed form solution for α given the solution for F (Theorem 2). Theorem 1. Given the solution for binary classifier f d i , the optimal α that minimizes the objective function in Equation (11) is α = 1 2 log Pn k=1 Pmk i,j=1 2rk i −1 Zk θk i,jI(f k j < f k i ) Pn k=1 Pmk i,j=1 2rk i −1 Zk θk i,jI(f k j > f k i ) (15) where θk i,j = γk i,jI(j ̸= i). Theorem 2. ¯ M(Q, ˜F) ≤¯ M(Q, F) + γ(α) + exp(3α) −1 3 n X k=1 mk X i=1 f k i mk X j=1 2rk i −2rk j Zk θk i,j where γ(α) is only a function of α with γ(0) = 0. The proofs of these theorems are provided in Appendix B and Appendix C respectively. Note that the bound provided by Theorem 2 is tight because by setting α = 0, the inequality reduces to equality resulting ¯ M(Q, ˜F) = ¯ M(Q, F). The importance of this theorem is that the optimal solution for f k i can be found without knowing the solution for α. Algorithm 1 1 summarizes the procedure in minimizing the objective function in Equation (11). First, it computes θk ij for every pair of documents of query k. Then, it computes wk i , a weight for each document which can be positive or negative. A positive weight wk i indicates that the ranking position of dk i induced by the current ranking function F is less than its true rank position, while a negative weight wk i shows that ranking position of dk i induced by the current F is greater than its true rank position. Therefore, the sign of weight wk i provides a clear guidance for how to construct the next weak ranker, the binary classifier in our case; that is, the documents with a positive wk i should be labeled as +1 by the binary classifier and those with negative wk i should be labeled as −1. The magnitude of wk i shows how much the corresponding document is misplaced in the ranking. In other words, it shows the importance of correcting the ranking position of document dk i in terms of improving the value of NDCG. This leads to maximizing η given in Equation (17) which can be considered as some sort of classification accuracy. We use sampling strategy in order to maximize η because most binary classifiers do not support the weighted training set; that is, we first sample the documents according to |wk i | and then construct a binary classifier with the sampled documents. It can be shown that the proposed algorithm reduces the objective function M exponentially (the proof is removed due to the lack of space). 1Notice that we use F(dk i ) instead of F(dk i , qk) to simplify the notation in the algorithm. 5 Algorithm 1 NDCG Boost: A Boosting Algorithm for Maximizing NDCG 1: Initialize F(dk i ) = 0 for all documents 2: repeat 3: Compute θk i,j = γk i,jI(j ̸= i) for all document pairs of each query. γk i,j is given in Eq. (14). 3: Compute the weight for each document as wk i = mk X j=1 2rk i −2rk j Zk θk i,j (16) 3: Assign each document the following class label yk i = sign(wk i ). 4: Train a classifier f(x) : Rd →{0, 1} that maximizes the following quantity η = n X k=1 mk X i=1 |wk i |f(dk i )yk i (17) 5: Predict fi for all documents in {Dk, i = 1, . . . , n} 6: Compute the combination weight α as provided in Equation (15). 7: Update the ranking function as F k i ←F k i + αf k i . 8: until reach the maximum number of iterations 4 Experiments To study the performance of NDCG Boost we use the latest version (version 3.0) of LETOR package provided by Microsoft Research Asia [22]. LETOR Package includes several benchmark data data, baselines and evaluation tools for research on learning to rank. 4.1 Letor Data Sets There are seven data sets provided in the LETOR package: OHSUMED, Top Distillation 2003 (TD2003), Top Distillation 2004 (TD2004), Homepage Finding 2003 (HP2003), Homepage Finding 2003 (HP2003), Named Page Finding 2003 (NP2003) and Named Page Finding 2004 (NP2004) 2. There are 106 queries in the OSHUMED data sets with a number of documents for each query. The relevancy of each document in OHSUMED data set is scored 0 (irrelevant), 1 (possibly) or 2 (definitely). The total number of query-document relevancy judgments provided in OHSUMED data set is 16140 and there are 45 features for each query-document pair. For TD2003, TD2004, HP2003, HP2004 and NP2003, there are 50, 75, 75, 75 and 150 queries, respectively, with about 1000 retrieved documents for each query. This amounts to a total number of 49171, 74170, 74409, 73834 and 147606 query-document pairs for TD2003, TD2004, HP2003, HP2004 and NP2003 respectively. For these data sets, there are 63 features extracted for each query-document pair and a binary relevancy judgment for each pair is provided. For every data sets in LETOR, five partitions are provided to conduct the five-fold cross validation, each includes training, test and validation sets. The results of a number of state-of-the-art learning to rank algorithms are also provided in the LETOR package. Since these baselines include some of the most well-known learning to rank algorithms from each category (pointwise, pairwise and listwise), we use them to study the performance of NDCG Boost. Here is the list of these baselines (the details can be found in the LETOR web page): Regression: This is a simple linear regression which is a basic pointwise approach and can be considered as a reference point. RankSVM: RankSVM is a pairwise approach using Support Vector Machine [5]. FRank: FRank is a pairwise approach. It uses similar probability model to RankNet [7] for the relative rank position of two documents, with a novel loss function called Fidelity loss function [9]. TSai et al [9] showed that FRank performs much better than RankNet. ListNet: ListNet is a listwise learning to rank algorithm [14]. It uses cross-entropy loss as its listwise loss function. AdaRank NDCG: This is a listwise boosting algorithm that incorporates NDCG in computing the samples and combination weights [20]. 2The experiment result for the last data set is not reported due to the lack of space. 6 1 2 3 4 5 6 7 8 9 10 0.4 0.42 0.44 0.46 0.48 0.5 0.52 0.54 @ n NDCG OHSUMED dataset Regression FRank ListNet RankSVM AdaRankNDCG SVM_MAP NDCG_\BOOST (a) OHSUMED 1 2 3 4 5 6 7 8 9 10 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4 0.42 @n NDCG TD2003 dataset (b) TD2003 1 2 3 4 5 6 7 8 9 10 0.25 0.3 0.35 0.4 0.45 0.5 @n NDCG TD2004 dataset (c) TD2004 1 2 3 4 5 6 7 8 9 10 0.65 0.7 0.75 0.8 0.85 @n NDCG HP2003 dataset (d) HP2003 1 2 3 4 5 6 7 8 9 10 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 @n NDCG HP2004 dataset (e) HP2004 1 2 3 4 5 6 7 8 9 10 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 @n NDCG NP2003 dataset (f) NP2003 Figure 1: The experimental results in terms of NDCG for Letor 3.0 data sets SVM MAP: SVM MAP is a support vector machine with MAP measure used in the constraints. It is a listwise approach [2]. While the validation set is used in finding the best set of parameters in the baselines in LETOR, it is not being used for NDCG Boost in our experiments. For NDCG Boost, we set the maximum number of iteration to 100 and use decision stump as the weak ranker. Figure 1 provides the the average results of five folds for different learning to rank algorithms in terms of NDCG @ each of the first 10 truncation level on the LETOR data sets 3. Notice that the performance of algorithms in comparison varies from one data set to another; however NDCG Boost performs almost always the best. We would like to point out a few statistics; On OHSUMED data set, NDCG Boost performs 0.50 at NDCG@3, a 4% increase in performance, compared to FRANK, the second best algorithm. On TD2003 data set, this value for NDCG Boost is 0.375 that shows a 10% increase, compared with RankSVM (0.34), the second best method. On HP2004 data set, NDCG Boost performs 0.80 at NDCG@3, compared to 0.75 of SVM MAP, the second best method, which indicates a 6% increase. Moreover, among all the methods in comparison, NDCG Boost appears to be the most stable method across all the data sets. For example, FRank, which performs well in OHSUMED and TD2004 data sets, yields a poor performance on TD2003, HP2003 and HP 2004. Similarly, AdaRank NDCG achieves a decent performance on OHSUMED data set, but fails to deliver accurate ranking results on TD2003, HP2003 and NP2003. In fact, both AdaRank NDCG and FRank perform even worse than the simple Regression approach on TD2003, which further indicates their instability. As another example, ListNet and RankSVM, which perform well on TD2003 are not competitive to NDCG boost on OHSUMED and TD2004 data sets. 5 Conclusion Listwise approach is a relatively new approach to learning to rank. It aims to use a query-level loss function to optimize a given IR measure. The difficulty in optimizing IR measure lies in the inherited sort function in the measure. We address this challenge by a probabilistic framework that optimizes the expectation of NDCG over all the possible permutations of documents. We present a relaxation strategy to effectively approximate the expectation of NDCG, and a bound optimization strategy for efficient optimization. Our experiments on benchmark data sets shows that our method is superior to the state-of-the-art learning to rank algorithms in terms of performance and stability. 3NDCG is commonly measured at the first few retrieved documents to emphasize their importance. 7 6 Acknowledgements The work was supported in part by the Yahoo! Labs4 and National Institute of Health (1R01GM079688-01). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of Yahoo! and NIH. A Proof of Proposition 2 1 1 + exp( ˜F k i −˜F k j ) = 1 1 + exp(F k i −F k j + α(f k i −f k j )) = 1 1 + exp(F k i −F k j ) ³ 1 1 + exp(F k i −F k j ) + exp(F k i −F k j ) 1 + exp(F k i −F k j ) exp(α(f k i −f k j ) ´−1 ≤ 1 1 + exp(F k i −F k j ) ³ 1 − exp(F k i −F k j ) 1 + exp(F k i −F k j ) + exp(F k i −F k j ) 1 + exp(F k i −F k j ) exp(α(f k j −f k i ) ´ = 1 1 + exp(F k i −F k j ) + γk i,j £ exp(α(f k j −f k i ) −1 ¤ The first step is a simple manipulations of the terms and the second step is due to the convexity of inverse function on R+. B Proof of Theorem 1 In order to obtain the result of the Theorem 1, we first plug Equation (13) in Equation (11). This leads to minimizing Pn k=1 Pmk i,j=1 2rk i −1 Zk θk i,j £ exp(α(f k j −f k i )) ¤ , the term related to α . Since f k i takes binary values 0 and 1, we have the following: n X k=1 mk X i,j=1 2rk i −1 Zk θk i,j exp(α(f k j −f k i )) = n X k=1 mk X i,j=1 2rk i −1 Zk θk i,j ³ exp(α)I(f k j > f k i ) + exp(−α)I(f k j < f k i ) ´ Getting the partial derivative of this term respect to α and having it equal to zero results the theorem. C Proof of Theorem 2 First, we provide the following proposition to handle exp(α(f k j −f k i )). Proposition 3. If x, y ∈[0, 1], we have exp(α(x −y)) ≤exp(3α) −1 3 (x −y) + exp(3α) + exp(−3α) + 1 3 (18) Proof. Due to the convexity of exp function, we have: exp(α(x −y)) = exp(3αx −y + 1 3 + 0 × 1 −x + y 3 + 1 3 × −3α) ≤x −y + 1 3 exp(3α) + 1 −x + y 3 + 1 3 exp(−3α) Using the result in the above proposition, we can bound the last term in Equation (13) as follows: θk i,j £ exp(α(f k j −f k i ) −1 ¤ ≤θk i,j ³exp(3α) −1 3 (f k j −f k i ) + exp(3α) + exp(−3α) −2 3 ´ (19) Using the result in Equation (19) and (13), we have ¯ M(Q, ˜F) in Equation (11) bounded as ¯ M(Q, ˜F) ≤¯ M(Q, F) + γ(α) + exp(3α) −1 3 n X k=1 mk X i=1 2rk i −1 Zk mk X j=1 θk i,j(f k i −f k j ) = ¯ M(Q, F) + γ(α) + exp(3α) −1 3 n X k=1 mk X i=1 f k i mk X j=1 2rk i −2rk j Zk θk i,j 4The first author has been supported as a part-time intern in Yahoo! 8 References [1] Kalervo J¨arvelin and Jaana Kek¨al¨ainen. Ir evaluation methods for retrieving highly relevant documents. In SIGIR 2000: Proceedings of the 23th annual international ACM SIGIR conference on Research and development in information retrieval, pages 41–48, 2000. [2] Yisong Yue, Thomas Finley, Filip Radlinski, and Thorsten Joachims. A support vector method for optimizing average precision. In SIGIR 2007: Proceedings of the 30th annual Int. ACM SIGIR Conf. on Research and development in information retrieval, pages 271–278, 2007. [3] Ping Li, Christopher Burges, and Qiang Wu. Mcrank: Learning to rank using multiple classification and gradient boosting. In Neural Information Processing System 2007. [4] Ramesh Nallapati. Discriminative models for information retrieval. In SIGIR ’04: Proceedings of the 27th annual international ACM SIGIR conference on Research and development in information retrieval, pages 64–71, New York, NY, USA, 2004. ACM. [5] Ralf Herbrich, Thore Graepel, and Klaus Obermayer. Support vector learning for ordinal regression. In Int. Conf. on Artificial Neural Networks 1999, pages 97–102, 1999. [6] Yoav Freund, Raj Iyer, Robert E. Schapire, and Yoram Singer. An efficient boosting algorithm for combining preferences. Journal of Machine Learning Research, 4:933–969, 2003. [7] Chris Burges, Tal Shaked, Erin Renshaw, Ari Lazier, Matt Deeds, Nicole Hamilton, and Greg Hullender. Learning to rank using gradient descent. In International Conference on Machine Learning 2005, 2005. [8] Yunbo Cao, Jun Xu, Tie-Yan Liu, Hang Li, Yalou Huang, and Hsiao-Wuen Hon. Adapting ranking svm to document retrieval. In SIGIR 2006: Proceedings of the 29th annual international ACM SIGIR conference on Research and development in information retrieval, pages 186–193, 2006. [9] Ming Feng Tsai, Tie yan Liu, Tao Qin, Hsin hsi Chen, and Wei ying Ma. Frank: A ranking method with fidelity loss. In SIGIR 2007: Proceedings of the 30th annual international ACM SIGIR conference on Research and development in information retrieval, 2007. [10] Rong Jin, Hamed Valizadegan, and Hang Li. Ranking refinement and its application to information retrieval. In WWW ’08: Proc. of the 17th int. conference on World Wide Web. [11] Steven C.H. Hoi and Rong Jin. Semi-supervised ensemble ranking. In Proceedings of Association for the Advancement of Artificial Intelligence (AAAI2008). [12] Tao Qin, Tie yan Liu, Ming feng Tsai, Xu dong Zhang, and Hang Li. Learning to search web pages with query-level loss functions. Technical report, 2006. [13] Christopher J. C. Burges, Robert Ragno, and Quoc V. Le. Learning to rank with nonsmooth cost functions. In Neural Information Processing System 2006, 2006. [14] Zhe Cao and Tie yan Liu. Learning to rank: From pairwise approach to listwise approach. In International Conference on Machine Learning 2007, pages 129–136, 2007. [15] Fen Xia, Tie-Yan Liu, Jue Wang, Wensheng Zhang, and Hang Li. Listwise approach to learning to rank: theory and algorithm. In Int. Conf. on Machine Learning 2008, pages 1192–1199, 2008. [16] Michael Taylor, John Guiver, Stephen Robertson, and Tom Minka. Softrank: optimizing nonsmooth rank metrics. [17] Maksims N. Volkovs and Richard S. Zemel. Boltzrank: learning to maximize expected ranking gain. In ICML ’09: Proceedings of the 26th Annual International Conference on Machine Learning, pages 1089–1096, New York, NY, USA, 2009. ACM. [18] Ruslan Salakhutdinov, Sam Roweis, and Zoubin Ghahramani. On the convergence of bound optimization algorithms. In Proc. 19th Conf. in Uncertainty in Artificial Intelligence (UAI 03). [19] Jen-Yuan Yeh, Yung-Yi Lin, Hao-Ren Ke, and Wei-Pang Yang. Learning to rank for information retrieval using genetic programming. In SIGIR 2007 workshop: Learning to Rank for Information Retrieval. [20] Jun Xu and Hang Li. Adarank: a boosting algorithm for information retrieval. In SIGIR ’07: Proceedings of the 30th annual international ACM SIGIR conference on Research and development in information retrieval, pages 391–398, 2007. [21] Zhengya Sun, Tao Qin, Qing Tao, and Jue Wang. Robust sparse rank learning for non-smooth ranking measures. In SIGIR ’09: Proceedings of the 32nd international ACM SIGIR conference on Research and development in information retrieval, pages 259–266, New York, NY, USA, 2009. ACM. [22] Tie-Yan Liu, Tao Qin, Jun Xu, Wenying Xiong, and Hang Li. Letor: Benchmark dataset for research on learning to rank for information retrieval. 9
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Slow Learners are Fast John Langford, Alexander J. Smola, Martin Zinkevich Machine Learning, Yahoo! Labs and Australian National University 4401 Great America Pky, Santa Clara, 95051 CA {jl, maz, smola}@yahoo-inc.com Abstract Online learning algorithms have impressive convergence properties when it comes to risk minimization and convex games on very large problems. However, they are inherently sequential in their design which prevents them from taking advantage of modern multi-core architectures. In this paper we prove that online learning with delayed updates converges well, thereby facilitating parallel online learning. 1 Introduction Online learning has become the paradigm of choice for tackling very large scale estimation problems. The convergence properties are well understood and have been analyzed in a number of different frameworks such as by means of asymptotics [12], game theory [8], or stochastic programming [13]. Moreover, learning-theory guarantees show that O(1) passes over a dataset suffice to obtain optimal estimates [3, 2]. This suggests that online algorithms are an excellent tool for learning. This view, however, is slightly deceptive for several reasons: current online algorithms process one instance at a time. That is, they receive the instance, make some prediction, incur a loss, and update an associated parameter. In other words, the algorithms are entirely sequential in their nature. While this is acceptable in single-core processors, it is highly undesirable given that the number of processing elements available to an algorithm is growing exponentially (e.g. modern desktop machines have up to 8 cores, graphics cards up to 1024 cores). It is therefore very wasteful if only one of these cores is actually used for estimation. A second problem arises from the fact that network and disk I/O have not been able to keep up with the increase in processor speed. A typical network interface has a throughput of 100MB/s and disk arrays have comparable parameters. This means that current algorithms reach their limit at problems of size 1TB whenever the algorithm is I/O bound (this amounts to a training time of 3 hours), or even smaller problems whenever the model parametrization makes the algorithm CPU bound. Finally, distributed and cloud computing are unsuitable for today’s online learning algorithms. This creates a pressing need to design algorithms which break the sequential bottleneck. We propose two variants. To our knowledge, this is the first paper which provides theoretical guarantees combined with empirical evidence for such an algorithm. Previous work, e.g. by [6] proved rather inconclusive in terms of theoretical and empirical guarantees. In a nutshell, we propose the following two variants: several processing cores perform stochastic gradient descent independently of each other while sharing a common parameter vector which is updated asynchronously. This allows us to accelerate computationally intensive problems whenever gradient computations are relatively expensive. A second variant assumes that we have linear function classes where parts of the function can be computed independently on several cores. Subsequently the results are combined and the combination is then used for a descent step. A common feature of both algorithms is that the update occurs with some delay: in the first case other cores may have updated the parameter vector in the meantime, in the second case, other cores may have already computed parts of the function for the subsequent examples before an update. 1 2 Algorithm 2.1 Platforms We begin with an overview of three platforms which are available for parallelization of algorithms. They differ in their structural parameters, such as synchronization ability, latency, and bandwidth and consequently they are better suited to different styles of algorithms. The description is not comprehensive by any means. For instance, there exist numerous variants of communication paradigms for distributed and cloud computing. Shared Memory Architectures: The commercially available 4-16 core CPUs on servers and desktop computers fall into this category. They are general purpose processors which operate on a joint memory space where each of the processors can execute arbitrary pieces of code independently of other processors. Synchronization is easy via shared memory/interrupts/locks. A second example are graphics cards. There the number of processing elements is vastly higher (1024 on high-end consumer graphics cards), although they tend to be bundled into groups of 8 cores (also referred to as multiprocessing elements), each of which can execute a given piece of code in a data-parallel fashion. An issue is that explicit synchronization between multiprocessing elements is difficult — it requires computing kernels on the processing elements to complete. Clusters: To increase I/O bandwidth one can combine several computers in a cluster using MPI or PVM as the underlying communications mechanism. A clear limit here is bandwidth constraints and latency for inter-computer communication. On Gigabit Ethernet the TCP/IP latency can be in the order of 100µs, the equivalent of 105 clock cycles on a processor and network bandwidth tends to be a factor 100 slower than memory bandwdith. Grid Computing: Computational paradigms such as MapReduce [4] and Hadoop are well suited for the parallelization of batch-style algorithms [17]. In comparison to cluster configurations communication and latency are further constrained. For instance, often individual processing elements are unable to communicate directly with other elements with disk / network storage being the only mechanism of inter-process data transfer. Moreover, the latency is significantly increased. We consider only the first two platforms since latency plays a critical role in the analysis of the class of algorithms we propose. While we do not exclude the possibility of devising parallel online algorithms suited to grid computing, we believe that the family of algorithm proposed in this paper is unsuitable and a significantly different synchronization paradigm would be needed. 2.2 Delayed Stochastic Gradient Descent Many learning problems can be written as convex minimization problems. It is our goal to find some parameter vector x (which is drawn from some Banach space X with associated norm ∥·∥) such that the sum over convex functions fi : X →R takes on the smallest value possible. For instance, (penalized) maximum likelihood estimation in exponential families with fully observed data falls into this category, so do Support Vector Machines and their structured variants. This also applies to distributed games with a communications constraint within a team. At the outset we make no special assumptions on the order or form of the functions fi. In particular, an adversary may choose to order or generate them in response to our previous choices of x. In other cases, the functions fi may be drawn from some distribution (e.g. whenever we deal with induced losses). It is our goal to find a sequence of xi such that the cumulative loss P i fi(xi) is minimized. With some abuse of notation we identify the average empirical and expected loss both by f ∗. This is possible, simply by redefining p(f) to be the uniform distribution over F. Denote by f ∗(x) := 1 |F| X i fi(x) or f ∗(x) := Ef∼p(f)[f(x)] (1) and correspondingly x∗:= argmin x∈X f ∗(x) (2) the average risk. We assume that x∗exists (convexity does not guarantee a bounded minimizer) and that it satisfies ∥x∗∥≤R (this is always achievable, simply by intersecting X with the unit-ball of radius R). We propose the following algorithm: 2 Algorithm 1 Delayed Stochastic Gradient Descent Input: Feasible space X ⊆Rn, annealing schedule ηt and delay τ ∈N Initialization: set x1 . . . , xτ = 0 and compute corresponding gt = ∇ft(xt). for t = τ + 1 to T + τ do Obtain ft and incur loss ft(xt) Compute gt := ∇ft(xt) Update xt+1 = argminx∈X ∥x −(xt −ηtgt−τ)∥(Gradient Step and Projection) end for loss gradient data source x Figure 1: Data parallel stochastic gradient descent with shared parameter vector. Observations are partitioned on a per-instance basis among n processing units. Each of them computes its own gradient gt = ∂xft(xt). Since each computer is updating x in a round-robin fashion, it takes a delay of τ = n −1 between gradient computation and when the gradients are applied to x. In this paper the annealing schedule will be either ηt = σ (t−τ) or ηt = σ √t−τ . Often, X = Rn. If we set τ = 0, algorithm 1 becomes an entirely standard stochastic gradient descent algorithm. The only difference with delayed stochastic gradient descent is that we do not update the parameter vector xt with the current gradient gt but rather with a delayed gradient gt−τ that we computed τ steps previously. We extend this to bounds which are dependent on strong convexity [1, 7] to obtain adaptive algorithms which can take advantage of well-behaved optimization problems in practice. An extension to Bregman divergences is possible. See [11] for details. 2.3 Templates Asynchronous Optimization: Assume that we have n processors which can process data independently of each other, e.g. in a multicore platform, a graphics card, or a cluster of workstations. Moreover, assume that computing the gradient of ft(x) is at least n times as expensive as it is to update x (read, add, write). This occurs, for instance, in the case of conditional random fields [15, 18], in planning [14], and in ranking [19]. The rationale for delayed updates can be seen in the following setting: assume that we have n cores performing stochastic gradient descent on different instances ft while sharing one common parameter vector x. If we allow each core in a round-robin fashion to update x one at a time then there will be a delay of τ = n −1 between when we see ft and when we get to update xt+τ. The delay arises since updates by different cores cannot happen simultaneously. This setting is preferable whenever computation of ft itself is time consuming. Note that there is no need for explicit thread-level synchronization between individual cores. All we need is a read / write-locking mechanism for x or alternatively, atomic updates on the parameter vector. On a multi-computer cluster we can use a similar mechanism simply by having one server act as a state-keeper which retains an up-to-date copy of x while the loss-gradient computation clients can retrieve at any time a copy of x and send gradient update messages to the state keeper. Pipelined Optimization: The key impediment in the previous template is that it required significant amounts of bandwidth solely for the purpose of synchronizing the state vector. This can be addressed by parallelizing computing the function value fi(x) explicitly rather than attempting to compute several instances of fi(x) simultaneously. Such situations occur, e.g. when fi(x) = g(⟨φ(zi), x⟩) for high-dimensional φ(zi). If we decompose the data zi (or its features) over n nodes we can compute partial function values and also all partial updates locally. The only communication required is to combine partial values and to compute gradients with respect to ⟨φ(zi), x⟩. This causes delay since the second stage is processing results of the first stage while the latter has already moved on to processing ft+1 or further. While the architecture is quite different, the effects are identical: the parameter vector x is updated with some delay τ. Note that here τ can be much smaller than the number of processors and mainly depends on the latency of the communication channel. Also note that in this configuration the memory access for x is entirely local. 3 Randomization: Order of observations matters for delayed updates: imagine that an adversary, aware of the delay τ bundles each of the τ most similar instances ft together. In this case we will incur a loss that can be τ times as large as in the non-delayed case and require a learning rate which is τ times smaller. The reason being that only after seeing τ instances of ft will we be able to respond to the data. Such highly correlated settings do occur in practice: for instance, e-mails or search keywords have significant temporal correlation (holidays, political events, time of day) and cannot be treated as iid data. Randomization of the order of data can be used to alleviate the problem. 3 Lipschitz Continuous Losses Due to space constraints we only state the results and omit the proofs. A more detailed analysis can be found in [11]. We begin with a simple game theoretic analysis that only requires ft to be convex and where the subdifferentials are bounded ∥∇ft(x)∥≤L by some L > 0. Denote by x∗the minimizer of f ∗(x). It is our goal to bound the regret R associated with a sequence X = {x1, . . . , xT } of parameters. If all terms are convex we obtain R[X] := T X t=1 ft(xt) −ft(x∗) ≤ T X t=1 ⟨∇ft(xt), xt −x∗⟩= T X t=1 ⟨gt, xt −x∗⟩. (3) Next define a potential function measuring the distance between xt and x∗. In the more general analysis this will become a Bregman divergence. We define D(x∥x′) := 1 2 ∥x −x′∥2. At the heart of our regret bounds is the following which bounds the instantaneous risk at a given time [16]: Lemma 1 For all x∗and for all t > τ, if X = Rn, the following expansion holds: ⟨xt−τ −x∗, gt−τ⟩= 1 2ηt ∥gt−τ∥2 + D(x∗∥xt) −D(x∗∥xt+1) ηt + min(τ,t−(τ+1)) X j=1 ηt−j ⟨gt−τ−j, gt−τ⟩ Note that the decomposition of Lemma 1 is very similar to standard regret decomposition bounds, such as [21]. The key difference is that we now have an additional term characterizing the correlation between successive gradients which needs to be bounded. In the worst case all we can do is bound ⟨gt−τ−j, gt−τ⟩≤L2, whenever the gradients are highly correlated, which yields the following: Theorem 2 Suppose all the cost functions are Lipschitz continuous with a constant L and maxx,x′∈X D(x∥x′) ≤F 2. Given ηt = σ √t−τ for some constant σ > 0, the regret of the delayed update algorithm is bounded by R[X] ≤σL2√ T + F 2 √ T σ + L2 στ 2 2 + 2L2στ √ T (4) and consequently for σ2 = F 2 2τL2 and T ≥τ 2 we obtain the bound R[X] ≤4FL √ τT (5) In other words the algorithm converges at rate O( √ τT). This is similar to what we would expect in the worst case: an adversary may reorder instances such as to maximally slow down progress. In this case a parallel algorithm is no faster than a sequential code. This result may appear overly pessimistic but the following example shows that such worst-case scaling behavior is to be expected: Lemma 3 Assume that an optimal online algorithm with regard to a convex game achieves regret R[m] after seeing m instances. Then any algorithm which may only use information that is at least τ instances old has a worst case regret bound of τR[m/τ]. Our construction works by designing a sequence of functions fi where for a fixed n ∈N all fnτ+j are identical (for j ∈{1, . . . , n}). That is, we send identical functions to the algorithm while it has no chance of responding to them. Hence, even an algorithm knowing that we will see τ identical instances in a row but being disallowed to respond to them for τ instances will do no better than one which sees every instance once but is allowed to respond instantly. 4 The useful consequence of Theorem 2 is that we are guaranteed to converge at all even if we encounter delay (the latter is not trivial — after all, we could end up with an oscillating parameter vector for overly aggressive learning rates). While such extreme cases hardly occur in practice, we need to make stronger assumptions in terms of correlation of ft and the degree of smoothness in ft to obtain tighter bounds. We conclude this section by studying a particularly convenient case: the setting when the functions fi are strongly convex satisfying f(x∗) ≥fi(x) + ⟨x∗−x, ∂xf(x)⟩+ hi 2 ∥x −x∗∥2 (6) Here we can get rid of the D(x∗∥x1) dependency in the loss bound. Theorem 4 Suppose that the functions fi are strongly convex with parameter λ > 0. Moreover, choose the learning rate ηt = 1 λ(t−τ) for t > τ and ηt = 0 for t ≤τ. Then under the assumptions of Theorem 2 we have the following bound: R[X] ≤λτF 2 + 1 2 + τ L2 λ (1 + τ + log T) (7) The key difference is that now we need to take the additional contribution of the gradient correlations into account. As before, we pay a linear price in the delay τ. 4 Decorrelating Gradients To improve our bounds beyond the most pessimistic case we need to assume that the adversary is not acting in the most hostile fashion possible. In the following we study the opposite case — namely that the adversary is drawing the functions fi iid from an arbitrary (but fixed) distribution. The key reason for this requirement is that we need to control the value of ⟨gt, gt′⟩for adjacent gradients. The flavor of the bounds we use will be in terms of the expected regret rather than an actual regret. Conversions from expected to realized regret are standard. See e.g. [13, Lemma 2] for an example of this technique. For this purpose we need to take expectations of sums of copies of the bound of Lemma 1. Note that this is feasible since expectations are linear and whenever products between more than one term occur, they can be seen as products which are conditionally independent given past parameters, such as ⟨gt, gt′⟩for |t −t′| ≤τ (in this case no information about gt can be used to infer gt′ or vice versa, given that we already know all the history up to time min(t, t′) −1. A key quantity in our analysis are bounds on the correlation between subsequent instances. In some cases we will only be able to obtain bounds on the expected regret rather than the actual regret. For the reasons pointed out in Lemma 3 this is an in-principle limitation of the setting. Our first strategy is to assume that ft arises from a scalar function of a linear function class. This leads to bounds which, while still bearing a linear penalty in τ, make do with considerably improved constants. The second strategy makes stringent smoothness assumptions on ft, namely it assumes that the gradients themselves are Lipschitz continuous. This will lead to guarantees for which the delay becomes increasingly irrelevant as the algorithm progresses. 4.1 Covariance bounds for linear function classes Many functions ft(x) depend on x only via an inner product. They can be expressed as ft(x) = l(yt, ⟨zt, x⟩) and hence gt(x) = ∇ft(x) = zt∂⟨zt,x⟩l(yt, ⟨zt, x⟩) (8) Now assume that ∂⟨zt,x⟩l(yt, ⟨zt, x⟩) ≤Λ for all x and all t. This holds, e.g. in the case of logistic regression, the soft-margin hinge loss, novelty detection. In all three cases we have Λ = 1. Robust loss functions such as Huber’s regression score [9] also satisfy (8), although with a different constant (the latter depends on the level of robustness). For such problems it is possible to bound the correlation between subsequent gradients via the following lemma: Lemma 5 Denote by (y, z), (y′, z′) ∼Pr(y, z) random variables which are drawn independently of x, x′ ∈X. In this case Ey,z,y′,z′ [⟨∂xl(y, ⟨z, x⟩), ∂xl(y′, ⟨z′, x′⟩)⟩] ≤Λ2
Ez,z′ z′z⊤
Frob =: L2α (9) 5 Here we defined α to be the scaling factor which quantifies by how much gradients are correlated. This yields a tighter version of Theorem 2. Corollary 6 Given ηt = σ √t−τ and the conditions of Lemma 5 the regret of the delayed update algorithm is bounded by R[X] ≤σL2√ T + F 2 √ T σ + L2αστ 2 2 + 2L2αστ √ T (10) Hence for σ2 = F 2 2ταL2 (assuming that τα ≥1) and T ≥τ 2 we obtain R[X] ≤4FL √ ατT. 4.2 Bounds for smooth gradients The key to improving the rate rather than the constant with regard to which the bounds depend on τ is to impose further smoothness constraints on ft. The rationale is quite simple: we want to ensure that small changes in x do not lead to large changes in the gradient. This is precisely what we need in order to show that a small delay (which amounts to small changes in x) will not impact the update that is carried out to a significant amount. More specifically we assume that the gradient of f is a Lipschitz-continuous function. That is, ∥∇ft(x) −∇ft(x′)∥≤H ∥x −x′∥. (11) Such a constraint effectively rules out piecewise linear loss functions, such as the hinge loss, structured estimation, or the novelty detection loss. Nonetheless, since this discontinuity only occurs on a set of measure 0 delayed stochastic gradient descent still works very well on them in practice. Theorem 7 In addition to the conditions of Theorem 2 assume that the functions fi are i.i.d., H ≥ L 4F √τ and that H also upper-bounds the change in the gradients as in Equation 11. Moreover, assume that we choose a learning rate ηt = σ √t−τ with σ = F L . In this case the risk is bounded by E[R[X]] ≤ 28.3F 2H + 2 3FL + 4 3F 2H log T τ 2 + 8 3FL √ T. (12) Note that the convergence bound which is O(τ 2 log T + √ T) is governed by two different regimes. Initially, a delay of τ can be quite harmful since subsequent gradients are highly correlated. At a later stage when optimization becomes increasingly an averaging process a delay of τ in the updates proves to be essentially harmless. The key difference to bounds of Theorem 2 is that now the rate of convergence has improved dramatically and is essentially as good as in sequential online learning. Note that H does not influence the asymptotic convergence properties but it significantly affects the initial convergence properties. This is exactly what one would expect: initially while we are far away from the solution x∗parallelism does not help much in providing us with guidance to move towards x∗. However, after a number of steps online learning effectively becomes an averaging process for variance reduction around x∗since the stepsize is sufficiently small. In this case averaging becomes the dominant force, hence parallelization does not degrade convergence further. Such a setting is desirable — after all, we want to have good convergence for extremely large amounts of data. 4.3 Bounds for smooth gradients with strong convexity We conclude this section with the tightest of all bounds — the setting where the losses are all strongly convex and smooth. This occurs, for instance, for logistic regression with ℓ2 regularization. Such a requirement implies that the objective function f ∗(x) is sandwiched between two quadratic functions, hence it is not too surprising that we should be able to obtain rates comparable with what is possible in the minimization of quadratic functions. Also note that the ratio between upper and lower quadratic bound loosely corresponds to the condition number of a quadratic function — the ratio between the largest and smallest eigenvalue of the matrix involved in the optimization problem. 6 -12 -10 -8 -6 -4 -2 0 2 0 10 20 30 40 50 60 70 80 90 100 Log_2 Error Thousands of Iterations Performance on TREC Data no delay delay of 10 delay of 100 delay of 1000 -6 -5 -4 -3 -2 -1 0 1 2 0 10 20 30 40 50 60 70 80 90 100 Log_2 Error Thousands of Iterations Performance on Real Data no delay delay of 10 delay of 100 delay of 1000 Figure 2: Experiments with simulated delay on the TREC dataset (left) and on a propietary dataset (right). In both cases a delay of 10 has no effect on the convergence whatsoever and even a delay of 100 is still quite acceptable. 0 50 100 150 200 250 300 350 400 450 1 2 3 4 5 6 7 Percent Speedup Threads Performance on TREC Data Figure 3: Time performance on a subset of the TREC dataset which fits into memory, using the quadratic representation. There was either one thread (a serial implementation) or 3 or more threads (master and 2 or more slaves). Theorem 8 Under the assumptions of Theorem 4, in particular, assuming that all functions fi are i.i.d and strongly convex with constant λ and corresponding learning rate ηt = 1 λ(t−τ) and provided that Equation 11 holds we have the following bound on the expected regret: E [R[X]] ≤10 9 λτF 2 + 1 2 + τ L2 λ [1 + τ + log(3τ + (Hτ/λ))] + L2 2λ[1 + log T] + π2τ 2HL2 6λ2 . (13) As before, this improves the rate of the bound. Instead of a dependency of the form O(τ log T) we now have the dependency O(τ log τ + log T). This is particularly desirable for large T. We are now within a small factor of what a fully sequential algorithm can achieve. In fact, we could make the constant arbitrary small for large enough T. 5 Experiments In our experiments we focused on pipelined optimization. In particular, we used two different training sets that were based on e-mails: the TREC dataset [5], consisting of 75,419 e-mail messages, and a proprietary (significantly harder) dataset of which we took 100,000 e-mails. These e-mails were tokenized by whitespace. The problem there is one of binary classification where we minimized a ’Huberized’ soft-margin loss function ft(x) = l(yt ⟨zt, x⟩) where l(χ) = 1 2 −χ if χ ≤0 1 2(χ −1)2 if χ ∈[0, 1] 0 otherwise (14) Here yt ∈{±1} denote the labels of the binary classification problem, and l is the smoothed quadratic soft-margin loss of [10]. We used two feature representations: a linear one which amounted to a simple bag of words representation, and a quadratic one which amounted to generating a bag of word pairs (consecutive or not). 7 To deal with high-dimensional feature spaces we used hashing [20]. In particular, for the TREC dataset we used 218 feature bins and for the proprietary dataset we used 224 bins. Note that hashing comes with performance guarantees which state that the canonical distortion due to hashing is sufficiently small for the dimensionality we picked. We tried to address the following issues: 1. The obvious question is a systematic one: how much of a convergence penalty do we incur in practice due to delay. This experiment checks the goodness of our bounds. We checked convergence for a system where the delay is given by τ ∈{0, 10, 100, 1000}. 2. Secondly, we checked on an actual parallel implementation whether the algorithm scales well. Unlike the previous check includes issues such as memory contention, thread synchronization, and general feasibility of a delayed updating architecture. Implementation The code was written in Java, although several of the fundamentals were based upon VW [10], that is, hashing and the choice of loss function. We added regularization using lazy updates of the parameter vector (i.e. we rescale the updates and occasionally rescale the parameter). This is akin to Leon Bottou’s SGD code. For robustness, we used ηt = 1 √ t. All timed experiments were run on a single, 8 core machine with 32 GB of memory. In general, at least 6 of the cores were free at any given time. In order to achieve advantages of parallelization, we divide the feature space {1 . . . n} into roughly equal pieces, and assign a slave thread to each piece. Each slave is given both the weights for its pieces, as well as the corresponding pieces of the examples. The master is given the label of each example. We compute the dot product separately on each piece, and then send these results to a master. The master adds the pieces together, calculates the update, and then sends that back to the slaves. Then, the slaves update their weight vectors in proportion to the magnitude of the central classifier. What makes this work quickly is that there are multiple examples in flight through this dataflow simultaneously. Note that between the time when a dot product is calculated for an example and when the results have been transcribed, the weight vector has been updated with several other earlier examples and the dot products have been calculated from several later examples. As a safeguard we limited the maximum delay to 100 examples. In this case the compute slave would simply wait for the pipeline to clear. The first experiment that we ran was a simulation where we artificially added a delay between the update and the product (Figure 2a). We ran this experiment using linear features, and observed that the performance did not noticeably degrade with a delay of 10 examples, did not significantly degrade with a delay of 100, but with a delay of 1000, the performance became much worse. The second experiment that we ran was with a proprietary dataset (Figure 2b). In this case, the delays hurt less; we conjecture that this was because the information gained from each example was smaller. In fact, even a delay of 1000 does not result in particularly bad performance. Since even the sequential version already handled 150,000 examples per second we tested parallelization only for quadratic features where throughput would be in the order of 1000 examples per second. Here parallelization dramatically improved performance — see Figure 3. To control for disk access we loaded a subset of the data into memory and carried out the algorithm on it. Summary and Discussion The type of updates we presented is a rather natural one. However, intuitively, having a delay of τ is like having a learning rate that is τ times larger. In this paper, we have shown theoretically how independence between examples can make the actual effect much smaller. The experimental results showed three important aspects: first of all, small simulated delayed updates do not hurt much, and in harder problems they hurt less; secondly, in practice it is hard to speed up “easy” problems with a small amount of computation, such as e-mails with linear features; finally, when examples are larger or harder, the speedups can be quite dramatic. 8 References [1] Peter L. Bartlett, Elad Hazan, and Alexander Rakhlin. Adaptive online gradient descent. In J. C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, Cambridge, MA, 2008. MIT Press. [2] L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In J. C. Platt, D. Koller, Y. Singer, and S.T. Roweis, editors, NIPS. MIT Press, 2007. [3] L´eon Bottou and Yann LeCun. Large scale online learning. In S. Thrun, L. Saul, and B. Sch¨olkopf, editors, Advances in Neural Information Processing Systems 16, pages 217– 224, Cambridge, MA, 2004. MIT Press. [4] C.T. Chu, S.K. Kim, Y. A. Lin, Y. Y. Yu, G. Bradski, A. Ng, and K. Olukotun. Map-reduce for machine learning on multicore. In B. Sch¨olkopf, J. Platt, and T. Hofmann, editors, Advances in Neural Information Processing Systems 19, 2007. [5] G. Cormack. TREC 2007 spam track overview. In The Sixteenth Text REtrieval Conference (TREC 2007) Proceedings, 2007. [6] O. Delalleau and Y. Bengio. Parallel stochastic gradient descent, 2007. CIAR Summer School, Toronto. [7] C.B. Do, Q.V. Le, and C.-S. Foo. Proximal regularization for online and batch learning. In A.P. Danyluk, L. Bottou, and M. L. Littman, editors, Proceedings of the 26th Annual International Conference on Machine Learning, ICML 2009, Montreal, Quebec, Canada, June 14-18, 2009, volume 382, page 33. ACM, 2009. [8] Elad Hazan, Amit Agarwal, and Satyen Kale. Logarithmic regret algorithms for online convex optimization. Machine Learning, 69(2-3):169–192, 2007. [9] P. J. Huber. Robust Statistics. John Wiley and Sons, New York, 1981. [10] J. Langford, L. Li, and A. Strehl. Vowpal wabbit online learning project, 2007. http://hunch.net/?p=309. [11] J. Langford, A.J. Smola, and M. Zinkevich. Slow learners are fast. arXiv:0911.0491. [12] N. Murata, S. Yoshizawa, and S. Amari. Network information criterion — determining the number of hidden units for artificial neural network models. IEEE Transactions on Neural Networks, 5:865–872, 1994. [13] Y. Nesterov and J.-P. Vial. Confidence level solutions for stochastic programming. Technical Report 2000/13, Universit´e Catholique de Louvain - Center for Operations Research and Economics, 2000. [14] N. Ratliff, J. Bagnell, and M. Zinkevich. Maximum margin planning. In International Conference on Machine Learning, July 2006. [15] N. Ratliff, J. Bagnell, and M. Zinkevich. (online) subgradient methods for structured prediction. In Eleventh International Conference on Artificial Intelligence and Statistics (AIStats), March 2007. [16] Shai Shalev-Shwartz, Yoram Singer, and Nathan Srebro. Pegasos: Primal estimated subgradient solver for SVM. In Proc. Intl. Conf. Machine Learning, 2007. [17] Choon Hui Teo, S. V. N. Vishwanthan, Alex J. Smola, and Quoc V. Le. Bundle methods for regularized risk minimization. J. Mach. Learn. Res., 2009. Submitted in February 2009. [18] S. V. N. Vishwanathan, Nicol N. Schraudolph, Mark Schmidt, and Kevin Murphy. Accelerated training conditional random fields with stochastic gradient methods. In Proc. Intl. Conf. Machine Learning, pages 969–976, New York, NY, USA, 2006. ACM Press. [19] M. Weimer, A. Karatzoglou, Q. Le, and A. Smola. Cofirank - maximum margin matrix factorization for collaborative ranking. In J.C. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20. MIT Press, Cambridge, MA, 2008. [20] K. Weinberger, A. Dasgupta, J. Attenberg, J. Langford, and A.J. Smola. Feature hashing for large scale multitask learning. In L. Bottou and M. Littman, editors, International Conference on Machine Learning, 2009. [21] M. Zinkevich. Online convex programming and generalised infinitesimal gradient ascent. In Proc. Intl. Conf. Machine Learning, pages 928–936, 2003. 9
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Tracking Dynamic Sources of Malicious Activity at Internet-Scale Shobha Venkataraman∗, Avrim Blum†, Dawn Song⋄, Subhabrata Sen∗, Oliver Spatscheck∗ ∗AT&T Labs – Research {shvenk,sen,spatsch}@research.att.com †Carnegie Mellon University avrim@cs.cmu.edu ⋄University of California, Berkeley dawnsong@cs.berkeley.edu Abstract We formulate and address the problem of discovering dynamic malicious regions on the Internet. We model this problem as one of adaptively pruning a known decision tree, but with additional challenges: (1) severe space requirements, since the underlying decision tree has over 4 billion leaves, and (2) a changing target function, since malicious activity on the Internet is dynamic. We present a novel algorithm that addresses this problem, by putting together a number of different “experts” algorithms and online paging algorithms. We prove guarantees on our algorithm’s performance as a function of the best possible pruning of a similar size, and our experiments show that our algorithm achieves high accuracy on large real-world data sets, with significant improvements over existing approaches. 1 Introduction It is widely acknowledged that identifying the regions that originate malicious traffic on the Internet is vital to network security and management, e.g., in throttling attack traffic for fast mitigation, isolating infected sub-networks, and predicting future attacks [6,18,19,24,26]. In this paper, we show how this problem can be modeled as a version of a question studied by Helmbold and Schapire [11] of adaptively learning a good pruning of a known decision tree, but with a number of additional challenges and difficulties. These include a changing target function and severe space requirements due to the enormity of the underlying IP address-space tree. We develop new algorithms able to address these difficulties that combine the underlying approach of [11] with the sleeping experts framework of [4, 10] and the online paging problem of [20]. We show how to deal with a number of practical issues that arise and demonstrate empirically on real-world datasets that this method substantially improves over existing approaches of /24 prefixes and network-aware clusters [6,19,24] in correctly identifying malicious traffic. Our experiments on data sets of 126 million IP addresses demonstrate that our algorithm is able to achieve a clustering that is both highly accurate and meaningful. 1.1 Background Multiple measurement studies have indicated that malicious traffic tends to cluster in a way that aligns with the structure of the IP address space, and that this is true for many different kinds of malicious traffic – spam, scanning, botnets, and phishing [6, 18, 19, 24]. Such clustered behaviour can be easily explained: most malicious traffic originates from hosts in poorly-managed networks, and networks are typically assigned contiguous blocks of the IP address space. Thus, it is natural that malicious traffic is clustered in parts of the IP address space that belong to poorly-managed networks. From a machine learning perspective, the problem of identifying regions of malicious activity can be viewed as one of finding a good pruning of a known decision tree – the IP address space may be naturally interpreted as a binary tree (see Fig.1(a)), and the goal is to learn a pruning of this tree that is not too large and has low error in classifying IP addresses as malicious or non-malicious. The structure of the IP address space suggests that there may well be a pruning with only a modest number of leaves that can classify most of the traffic accurately. Thus, identifying regions of malicious activity from an online stream of labeled data is much like the problem considered by Helmbold and Schapire [11] of adaptively learning a good pruning of a known decision tree. However, there are a 1 number of real-world challenges, both conceptual and practical, that must be addressed in order to make this successful. One major challenge in our application comes from the scale of the data and size of a complete decision tree over the IP address space. A full decision tree over the IPv4 address space would have 232 leaves, and over the IPv6 address space (which is slowly being rolled out), 2128 leaves. With such large decision trees, it is critical to have algorithms that do not build the complete tree, but instead operate in space comparable to the size of a good pruning. These space constraints are also important because of the volume of traffic that may need to be analyzed – ISPs often collect terabytes of data daily and an algorithm that needs to store all its data in memory simultaneously would be infeasible. A second challenge comes from the fact that the regions of malicious activity may shift longitudinally over time [25]. This may happen for many reasons, e.g., administrators may eventually discover and clean up already infected bots, and attackers may target new vulnerabilities and attack new hosts elsewhere. Such dynamic behaviour is a primary reason why individual IP addresses tend to be such poor indicators of future malicious traffic [15,26]. Thus, we cannot assume that the data comes from a fixed distribution over the IP address space; the algorithm needs to adapt to dynamic nature of the malicious activity, and track these changes accurately and quickly. That is, we must consider not only an online sequence of examples but also a changing target function. While there have been a number of measurement studies [6,18,19,24] that have examined the origin of malicious traffic from IP address blocks that are kept fixed apriori, none of these have focused on developing online algorithms that find the best predictive IP address tree. Our challenge is to develop an efficient high-accuracy online algorithm that handles the severe space constraints inherent in this problem and accounts for the dynamically changing nature of malicious behavior. We show that we can indeed do this, both proving theoretical guarantees on adaptive regret and demonstrating successful performance on real-world data. 1.2 Contributions In this paper, we formulate and address the problem of discovering and tracking malicious regions of the IP address space from an online stream of data. We present an algorithm that adaptively prunes the IP address tree in a way that maintains at most m leaves and performs nearly as well as the optimum adaptive pruning of the IP address tree with a comparable size. Intuitively, we achieve the required adaptivity and the space constraints by combining several “experts” algorithms together with a tree-based version of paging. Our theoretical results prove that our algorithm can predict nearly as well as the best adaptive decision tree with k leaves when using O(k log k) leaves. Our experimental results demonstrate that our algorithm identifies malicious regions of the IP address space accurately, with orders of magnitude improvement over previous approaches. Our experiments focus on classifying spammers and legitimate senders on two mail data sets, one with 126 million messages collected over 38 days from the mail servers of a tier-1 ISP, and a second with 28 million messages collected over 6 months from an enterprise mail server. Our experiments also highlight the importance of allowing the IP address tree to be dynamic, and the resulting view of the IP address space that we get is both compelling and meaningful. 2 Definitions and Preliminaries We now present some basic definitions as well as our formal problem statement. The IP address hierarchy can be naturally interpreted as a full binary tree, as in Fig. 1: the leaves of the tree correspond to individual IP addresses, and the non-leaf nodes correspond to the remaining IP prefixes. Let P denote the set of all IP prefixes, and I denote the set of all IP addresses. We also use term clusters to denote the IP prefixes. We define an IPTree TP to be a pruning of the full IP address tree: a tree whose nodes are IP prefixes P ∈P, and whose leaves are each associated with a label, i.e., malicious or non-malicious. An IPtree can thus be interpreted as a classification function for the IP addresses I: an IP address i gets the label associated with its longest matching prefix in P. Fig. 1 shows an example of an IPtree. We define the size of an IPtree to be the number of leaves it has. For example, in Fig. 1(a), the size of the IPtree is 6. As described in Sec. 1, we focus on online learning in this paper. A typical point of comparison used in the online learning model is the error of the optimal offline fixed algorithm. In this case, the optimal offline fixed algorithm is the IPtree of a given size k i.e., the tree of size k that makes 2 0.0.0.0/0 0.0.0.0/1 128.0.0.0/1 0.0.0.0/2 192.0.0.0/2 + + + + 160.0.0.0/3 128.0.0.0/4 152.0.0.0/4 (a) An example IPTree (b) A real IPTree (Color coding explained in Sec. 5) Figure 1: IPTrees: example and real. Recall that an IP address is interpreted as a 32-bit string, read from left to right. This defines a path on the binary tree, going left for 0 and right for 1. An IP prefix is denoted by IP/n, where n indicates the number of bits relevant to the prefix. the fewest mistakes on the entire sequence. However, if the true underlying IPtree may change over time, a better point of comparison would allow the offline tree to also change over time. To make such a comparison meaningful, the offline tree must pay an additional penalty each time it changes (otherwise the offline tree would not be a meaningful point of comparison – it could change for each IP address in the sequence, and thus make no mistakes). We therefore limit the kinds of changes the offline tree can make, and compare the performance of our algorithm to every IPtree with k leaves, as a function of the errors it makes and the changes it makes. We define an adaptive IPtree of size k to be an adaptive tree that can (a) grow nodes over time so long as it never has more than k leaves, (b) change the labels of its leaf nodes, and (c) occasionally reconfigure itself completely. Our goal is to develop an online algorithm T such that for any sequence of IP addresses, (1) for every adaptive tree T ′ of size k, the number of mistakes made by T is bounded by a (small) function of the mistakes and the changes of types (a), (b), and (c) made by T ′, and (2) T uses no more than ˜O(k) space. In the next section, we describe an algorithm meeting these requirements. 3 Algorithms and Analysis In this section, we describe our main algorithm TrackIPTree, and present theoretical guarantees on its performance. At a high-level, our approach keeps a number of experts in each prefix of the IPtree, and combines their predictions to classify every IP address. The inherent structure in the IPtree allows us to decompose the problem into a number of expert problems, and provide lower memory bounds and better guarantees than earlier approaches. We begin with an overview. Define the path-nodes of an IP address to be the set of all prefixes of i in T , and denote this set by Pi,T . To predict the label of an IP i, the algorithm looks up all the pathnodes in Pi,T , considers their predictions, and combines these predictions to produce a final label for i. To update the tree, the algorithm rewards the path-nodes that predicted correctly, penalizes the incorrect ones, and modifies the tree structure if necessary. To fill out this overview, there are four technical questions that we need to address: (1) Of all the path-nodes in Pi,T , how do we learn the ones that are the most important? (2) How do we learn the correct label to predict at a particular path-node in Pi,T (i.e., positive or negative)? (3) How do we grow the IPtree appropriately, ensuring that it grows primarily the prefixes needed to improve the classification accuracy? (4) How do we ensure that the size of the IPtree stays bounded by m? We address these questions by treating them as separate subproblems, and we show how they fit together to become the complete algorithm in Figure 3.1. 3.1 Subproblems of TrackIPTree We now describe our algorithm in detail. Since our algorithm decomposes naturally into the four subproblems mentioned above, we focus on each subproblem separately to simplify the presentation. We use the following notation in our descriptions: Recall from Sec. 2 that m is the maximum number of leaves allowed to our algorithm, k is the size of the optimal offline tree, and Pi,T denotes the set of path-nodes, i.e., the prefixes of IP i in the current IPtree T . Relative Importance of the Path Nodes First, we consider the problem of deciding which of the prefix nodes in the path Pi,T is most important. We formulate this as a sleeping experts problem [4, 10]. We set an expert in each node, and call them the path-node experts, and for an IP i, we consider the set of path-node experts in Pi,T to be the “awake” experts, and the rest to be “asleep”. The 3 x1 x0 x2 x3 x4 x5 x6 y+ y(a) Sleeping Experts: Relative Importance (b) Shifting Experts: Determining of Path-Nodes Node Labels Figure 2: Decomposing the TrackIPTree Algorithm sleeping experts algorithm makes predictions using the awake experts, and intuitively, has the goal of predicting nearly as well as the best awake expert on the instance i 1. In our context, the best awake expert on the IP i corresponds to the prefix of i in the optimal IPtree, which remains sleeping until the IPtree grows that prefix. Fig. 2(a) illustrates the sleeping experts framework in our context: the shaded nodes are “awake” and the rest are “asleep”. Specifically, let xt denote the weight of the path-node expert at node t, and let Si,T = P t∈Pi,T xt. To predict on IP address i, the algorithm chooses the expert at node t with probability xt/Si,T . To update, the algorithm penalizes all incorrect experts in Pi,T , reducing their weight xt to γxt. (e.g., γ = 0.8). It then renormalizes the weights of all the experts in Pi,T so that their sum Si,T does not change. (In our proof, we use a slightly different version of the sleeping experts algorithm [4]). Deciding Labels of Individual Nodes Next, we need to decide whether the path-node expert at a node n should predict positive or negative. We use a different experts algorithm to address this subproblem – the shifting experts algorithm [12]. Specifically, we allow each node n to have two additional experts – a positive expert, which always predicts positive, and a negative expert, which always predicts negative. We call these experts node-label experts. Let yn,+ and yn,−denote the weights of the positive and negative node-label experts respectively, with yn,−+yn,+ = 1. The algorithm operates as follows: to predict, the node predicts positive with probability yn,+ and negative with probability yn,−. To update, when the node receives a label, it increases the weight of the correct node-label expert by ǫ, and decreases the weight of the incorrect node-label expert by ǫ (upto a maximum of 1 and a minimum of 0). Note that this algorithm naturally adapts when a leaf of the optimal IPtree switches labels – the relevant node in our IPtree will slowly shift weights from the incorrect node-label expert to the correct one, making an expected 1 ǫ mistakes in the process. Fig. 2(b) illustrates the shifting experts setting on an IPtree: each node has two experts, a positive and a negative. Fig. 3 shows how it fits in with the sleeping experts algorithm. Building Tree Structure We next address the subproblem of building the appropriate structure for the IPtree. The intuition here is: when a node in the IPtree makes many mistakes, then either that node has a subtree in the optimal IPtree that separates the positive and negative instances, or the optimal IPtree must also make the same mistakes. Since TrackIPTree cannot distinguish between these two situations, it simply splits any node that makes sufficient mistakes. In particular, TrackIPTree starts with only the root node, and tracks the number of mistakes made at every node. Every time a leaf makes 1 ǫ mistakes, TrackIPTree splits that leaf into its children, and instantiates and initializes the relevant path-node experts and node-label experts of the children. In effect, it is as if the path-node experts of the children had been asleep till this point, but will now be “awake” for the appropriate IP addresses. TrackIPTree waits for 1 ǫ mistakes at each node before growing it, so that there is a little resilence with noisy data – otherwise, it would split a node every time the optimal tree made a mistake, and the IPtree would grow very quickly. Note also that it naturally incorporates the optimal IPtree growing a leaf; our tree will grow the appropriate nodes when that leaf has made 1 ǫ mistakes. Bounding Size of IPtree Since TrackIPTree splits any node after it makes 1 ǫ mistakes, it is likely that the IPtree it builds is split much farther than the optimal IPtree – TrackIPTree does not know when to stop growing a subtree, and it splits even if the same mistakes are made by the optimal IPtree. While this excessive splitting does not impact the predictions of the path-node experts or the node-label experts significantly, we still need to ensure that the IPtree built by our algorithm does not become too large. 1We leave the exact statement of the guarantee to the proof in [23] 4 TRACKIPTREE Input: tree size m, learning rate ǫ, penalty factor γ Initialize: Set T := root InitializeNode(root) Prediction Rule: Given IP i //Select a node-label expert for n ∈Pi,T flip coin of bias yn,+ if heads, predict[n] := + else predict[n] := − //Select a path-node expert rval := predict[n] with weight xn/ P t∈P xt Return rval Update Rule: Given IP i, label r //Update node-label experts for n ∈Pi,T for label z ∈{+, −} if z = r, yn,z := yn,z + ǫ else yn,z := yn,z −ǫ Update Rule (Contd.): //Update path-node experts s := P n∈Pt,T xn for n ∈Pi,T if predict[n] ̸= r, penalize xn := γxn mistakes[xn] + + if mistakes[xn] > 1/ǫ and n is leaf, GrowTree(n) Renormalize xn := xn s P j∈Pi,T xj sub INITIALIZENODE Input: node t xt := 1; yt,+ := yt,−:= 0.5 mistakes[t] := 0 sub GROWTREE Input: leaf l if size(T) ≥m Select nodes N to discard with paging algorithm Split leaf l into children lc, rc. InitializeNode(lc), InitializeNode(rc) Figure 3: The Complete TrackIPTree Algorithm We do this by framing it as a paging problem [20]: consider each node in the IPtree to be a page, and the maximum allowed nodes in the IPtree to be the size of the cache. The offline IPtree, which has k leaves, needs a cache of size 2k. The IPtree built by our algorithm may have at most m leaves (and thus, 2m nodes, since it is a binary tree), and so the size of its cache is 2m and the offline cache is 2k. We may then select nodes to be discarded as if they were pages in the cache once the IPtree grows beyond 2m nodes; so, for example, we may choose the least recently used nodes in the IPtree, with LRU as the paging algorithm. Our analysis shows that setting m = O( k ǫ2 log k ǫ ) suffices, when TrackIPTree uses FLUSH-WHEN-FULL (FWF) as its paging algorithm – this is a simple paging algorithm that discards all the pages in the cache when the cache is full, and restarts with an empty cache. We use FWF here for a clean analysis, and especially since in simple paging models, many algorithms achieve no better guarantees [20]. For our experiments, we implement LRU, and our results show that this approach, while perhaps not sophisticated, still maintains an accurate predictive IPtree. 3.2 Analysis In this section, we present theoretical guarantees on TrackIPTree’s performance. We show our algorithm performs nearly as well as best adaptive k-IPtree, bounding the number of mistakes made by our algorithm as a function of the number of mistakes, number of labels changes and number of complete reconfigurations of the optimal such tree in hindsight. Theorem 3.1 Fix k. Set the maximum number of leaves allowed to the TrackIPTree algorithm m to be 10k ǫ2 log k ǫ . Let T be an adaptive k-IPtree. Let ∆T,z denote the number of times T changes labels on the its leaves over the sequence z, and RT,z denote the number of times times T has completely reconfigured itself over z. The algorithm TrackIPTreeensures that on any sequence of instances z, for each T , the number of mistakes made by TrackIPTree is at most (1 + 3ǫ)MT,z + ( 1 ǫ + 3)∆T,z + 10k ǫ3 log k ǫ (RT,z + 1) with probability at least 1 − 1 k k 2ǫ2 . In other words, if there is an offline adaptive k-IPtree, that makes few changes and few mistakes on the input sequence of IP addresses, then TrackIPTree will also make only a small number of mistakes. Due to space constraints, we present the proof in the technical report [23]. 4 Evaluation Setup We now describe our evaluation set-up: data, practical changes to the algorithm, and baseline schemes that compare against. While there are many issues that go into converting the algorithm in Sec. 3 for practical use, we describe here those most important to our experiments, and defer the rest to the technical report [23]. 5 Data We focus on IP addresses derived from mail data, since spammers represent a significant fraction of the malicious activity and compromised hosts on the Internet [6], and labels are relatively easy to obtain from spam-filtering run by the mail servers. For our evaluation, we consider labels from the mail servers’ spam-filtering to be ground truth. Any errors in the spam-filtering will influence the tree that we construct and our experimental results are limited by this assumption. One data set consists of log extracts collected at the mail servers of a tier-1 ISP with 1 million active mailboxes. The extracts contain the IP addresses of the mail servers that send mail to the ISP, the number of messages they sent, and the fraction of those messages that are classified as spam, aggregated over 10 minute intervals. The mail server’s spam-filtering software consists of a combination of hand-crafted rules, DNS blacklists, and Brightmail [1], and we take their results as labels for our experiments. The log extracts were collected over 38 days from December 2008 to January 2009, and contain 126 million IP addresses, of which 105 million are spam and 21 million are legitimate. The second data set consists of log extracts from the enterprise mail server of a large corporation with 1300 active mailboxes. These extracts also contain the IP addresses of mail servers that attempted to send mail, along with the number of messages they sent and the fraction of these messages that were classified spam by SpamAssassin [2], aggregated over 10 minute intervals. The extracts contain 28 million IP addresses, of which around 1.2 million are legitimate and the rest are spammers. Note that in both cases, our data only contains aggregate information about the IP addresses of the mail servers sending mail to the ISP and enterprise mail servers, and so we do not have the ability to map any information back to individual users of the ISP or enterprise mail servers. TrackIPTree For the experimental results, we use LRU as the paging algorithm when nodes need to be discarded from the IPtree (Sec. 3.1). In our implementation, we set TrackIPTree to discard 1% of m, the maximum leaves allowed, every time it needs to expire nodes. The learning rate ǫ is set to 0.05 and the penalty factor γ for sleeping experts is set to 0.1 respectively. Our results are not affected if these parameters are changed by a factor of 2-3. While we have presented an online learning algorithm, in practice, it will often need to predict on data without receiving labels of the instances right away. Therefore, we study TrackIPTree’s accuracy on the following day’s data, i.e., to compute prediction accuracy of day i, TrackIPTree is allowed to update until day i−1. We choose intervals of a day’s length to allow the tree’s predictions to be updated at least every day. Apriori Fixed Clusters We compare TrackIPTree to two sets of apriori fixed clusters: (1) networkaware clusters, which are a set of unique prefixes derived from BGP routing table snapshots [17], and (2) /24 prefixes. We choose these clusters as a baseline, as they have been the basis of measurement studies discussed earlier (Sec. 1), prior work in IP-based classification [19, 24], and are even used by popular DNS blacklists [3]. We use the fixed clusters to predict the label of an IP in the usual manner: we simply assign an IP the label of its longest matching prefix among the clusters.Of course, we first need to assign these clusters their own labels. To ensure that they classify as well as possible, we assign them the optimal labeling over the data they need to classify; we do this by allowing them to make multiple passes over the data. That is, for each day, we assign labels so that the fixed clusters maximize their accuracy on spam for a given required accuracy on legitimate mail 2. It is clear that this experimental set-up is favourable to the apriori fixed clusters. We do not directly compare against the algorithm in [11], as it requires every unique IP address in the data set to be instantiated in the tree. In our experiments (e.g., with the ISP logs), this means that it requires over 90 million leaves in the tree. We instead focus on practical prior approaches with more cluster sizes in our experiments. 5 Results We report three sets of experimental results regarding the prediction accuracy of TrackIPTree using the experimental set-up of Section 4. While we do not provide an extensive evaluation of our algorithm’s computational efficiency, we note that our (unoptimized) implementation of TrackIPTree takes under a minute to learn over a million IP addresses, on a 2.4GHz Sparc64-VI core. 2For space reasons, we defer the details of how we assign this labeling to the technical report [23] 6 0 0.5 1 0.2 0.4 0.6 0.8 1 Coverage on Legit IPs Accuracy on Spam IPs TrackIPTree Network−Aware /24 Prefixes (a) Expt 1: ISP logs 0 0.5 1 0.2 0.4 0.6 0.8 1 Coverage on Legit IPs Accuracy on Spam IPs TrackIPTree Network−Aware /24 Prefixes (b) Expt 1: Enterprise logs 0 0.5 1 0.85 0.9 0.95 1 Coverage on Legit IPs Accuracy on Spam IPs 200k 100k 50k 20k (c) Expt 2: ISP logs 0 0.5 1 0.92 0.94 0.96 0.98 1 Coverage on Legit IPs Accuracy on Spam IPs 50k 10k 5k 1k (d) Expt 2: Enterprise logs 10 20 30 0.05 0.1 0.15 0.2 0.25 Time in days Error on Legit IPs Dynamic Static: 5 Days Static: 10 Days (e) Expt 3: Legitimate IPs 10 20 30 0.05 0.1 0.15 Time in days Error on Spam IPs Dynamic Static: 5 Days Static: 10 Days (f) Expt 3: Spam IPs Figure 4: Results for Experiments 1, 2, and 3 Our results compare the fraction of spamming IPs that the clusters classify correctly, subject to the constraint that they classify at least x% legitimate mail IPs correctly (we term this to be the coverage of the legitimate IPs required). Thus, we effectively plot the true positive rate against the true negative rate. (This is just the ROC curve with the x-axis reversed, since we plot the true positive against the true negative, instead of plotting the true positive against the false positive.) Experiment 1: Comparisons with Apriori Fixed Clusters Our first set of experiments compares the performance of our algorithm with network-aware clusters and /24 IP prefixes. Figs. 4(a) & 4(b) illustrate the accuracy tradeoff of the three sets of clusters on the two data sets. Clearly, the accuracy of TrackIPTree is a tremendous improvement on both sets of apriori fixed clusters – for any choice of coverage on legitimate IPs, the accuracy of spam IPs by TrackIPTree is far higher than the apriori fixed clusters, even by as much as a factor of 2.5. In particular, note that when the coverage required on legitimate IPs is 95%, TrackIPTree achieves 95% accuracy in classifying spam on both data sets, compared to the 35 −45% achieved by the other clusters. In addition, TrackIPTree gains this classification accuracy using a far smaller tree. Table 1 shows the median number of leaves instantiated by the tree at the end of each day. (To be fair to the fixed clusters, we only instantiate the prefixes required to classify the day’s data, rather than all possible prefixes in the clustering scheme.) Table 1 shows that the tree produced by TrackIPTree is a factor of 2.5-17 smaller with the ISP logs, and a factor of 20-100 smaller with the enterprise logs. These numbers highlight that the apriori fixed clusters are perhaps too coarse to classify accurately in parts of the IP address space, and also are insufficiently aggregated in other parts of the address space. Experiment 2: Changing the Maximum Leaves Allowed Next, we explore the effect of changing m, the maximum number of leaves allowed to TrackIPTree. Fig. 4(c) & 4(d) show the accuracycoverage tradeoff for TrackIPTree when m ranges between 20,000-200,000 leaves for the ISP logs, and 1,000-50,000 leaves for the enterprise logs. Clearly, in both cases, the predictive accuracy increases with m only until m is “sufficiently large” – once m is large enough to capture all the distinct subtrees in the underlying optimal IPtree, the predictive accuracy will not increase. While the actual values of m are specific to our data sets, the results highlight the importance of having a space-efficient and flexible algorithm – both 10,000 and 100,000 are very modest sizes compared to the number of possible apriori fixed clusters, or the size of the IPv4 address space, and this suggests that the underlying decision tree required is indeed of a modest size. Experiment 3: Does a Dynamic Tree Help? In this experiment, we demonstrate empirically that our algorithm’s dynamic aspects do indeed significantly enhance its accuracy over static clustering schemes. The static clustering that we compare to is a tree generated by our algorithm, but one that learns over the first z days, and then stays unchanged. For ease of reference, we call such a tree a z-static tree; in our experiments, we set z = 5 and z = 10. We compare these trees by examining separately the errors incurred on legitimate and spam IPs. 7 ISP Enterprise TrackIPTree 99942 9963 /24 Prefixes 1732441 1426445 Network-aware 260132 223025 wt Implication Colour ≥0.2 Strongly Legit Dark Green [0, 0.2) Weakly Legit Light Green (−0.2, 0) Weakly Malicious Blue ≤−0.2 Strongly Malicious White Table 1: Sizes of Clustering Schemes Table 2: Colour coding for IPtree in Fig 1(b) Fig. 4(e) & 4(f) compare the errors of the z-static trees and the dynamic tree on legitimate and spam IPs respectively, using the ISP logs. Clearly, both z-static trees degrade in accuracy over time, and they do so on both legitimate and spam IPs. On the other hand, the accuracy of the dynamic tree does not degrade over this period. Further, the in error grows with time; after 28 days, the 10-static tree has almost a factor of 2 higher error on both spam IPs and legitimate IPs. Discussion and Implications Our experiments demonstrate that our algorithm is able to achieve high accuracy in predicting legitimate and spam IPs, e.g., it can predict 95% of the spam IPs correctly, when misclassifying only 5% of the legitimate IPs. However, it does not classify the IPs perfectly. This is unsurprising – achieving zero classification error in these applications is practically infeasible, given IP address dynamics [25]. Nevertheless, our IPtree still provides insight into the malicious activity on the Internet. As an example, we examine a high-level view of the Internet obtained from our tree, and its implications. Fig. 1(b) visualizes an IPtree on the ISP logs with 50,000 leaves. It is laid out so that the root prefix is near the center, and the prefixes grow their children outwards. The nodes are coloured depending on their weights, as shown in Table 2: for node t, define wt = P j∈Q xj(yj,+ −yj,−), where Q is the set of prefixes of node t (including node t itself. Thus, the blue central nodes are the large prefixes (e.g., /8 prefixes), and the classification they output is slightly malicious; this means that an IP address without a longer matching prefix in the tree is typically classified to be malicious. This suggests, for example, that an unseen IP address is typically classified as a spammer by our IPtree, which is consistent with the observations of network administrators. A second observation we can make is that the tree has many short branches as well as long branches, suggesting that some IP prefixes are grown to much greater depth than others. This might happen, for instance, if active IP addresses for this application are not distributed uniformly in the address space (and so all prefixes do not need to be grown at uniform rates), which is also what we might expect to see based on prior work [16]. Of course, these observations are only examples; a complete analysis of our IPtree’s implications is part of our future work. Nevertheless, these observations suggest that our tree does indeed capture an appropriate picture of the malicious activity on the Internet. 6 Other Related Work In the networking and databases literature, there has been much interest in designing streaming algorithms to identify IP prefixes with significant network traffic [7, 9, 27], but these algorithms do not explore how to predict malicious activity. Previous IP-based approaches to reduce spam traffic [22, 24], as mentioned earlier, have also explored individual IP addresses, which are not particularly useful since they are so dynamic [15, 19, 25]. Zhang et al [26] also examine how to predict whether known malicious IP addresses may appear at a given network, by analyzing the co-occurence of all known malicious IP addresses at a number of different networks. More closely related is [21], who present algorithms to extract prefix-based filtering rules for IP addresses that may be used in offline settings. There has also been work on computing decision trees over streaming data [8,13], but this work assumes that data comes from a fixed distribution. 7 Conclusion We have addressed the problem of discovering dynamic malicious regions on the Internet. We model this problem as one of adaptively pruning a known decision tree, but with the additional challenges coming from real-world settings – severe space requirements and a changing target function. We developed new algorithms to address this problem, by combining “experts” algorithms and online paging algorithms. We showed guarantees on our algorithm’s performance as a function of the best possible pruning of a similar size, and our experimental results on real-world datasets are orders of magnitude better than current approaches. Acknowledgements We are grateful to Alan Glasser and Gang Yao for their help with the data analysis efforts. 8 References [1] Brightmail. http://www.brightmail.com. [2] SpamAssassin. http://www.spamassassin.apache.org. [3] SpamHaus. http://www.spamhaus.net. [4] BLUM, A., AND MANSOUR, Y. From external to internal regret. In In Proceedings of 18th Annual Conference on Computational Learning Theory (COLT 2005) (2005). [5] CESA-BIANCHI, N., FREUND, Y., HAUSSLER, D., HELMBOLD, D. P., SCHAPIRE, R. E., AND WARMUTH, M. K. How to use expert advice. J. ACM 44, 3 (1997), 427–485. [6] COLLINS, M. P., SHIMEALL, T. J., FABER, S., NAIES, J., WEAVER, R., AND SHON, M. D. Using uncleanliness to predict future botnet addresses. In Proceedings of the Internet Measurement Conference (2007). [7] CORMODE, G., KORN, F., MUTHUKRISHNAN, S., AND SRIVASTAVA, D. Diamond in the rough: Finding hierarchical heavy hitters in multi-dimensional data. In SIGMOD ’04: Proceedings of the 2004 ACM SIGMOD international conference on Management of data (2004). [8] DOMINGOS, P., AND HULTEN, G. Mining high-speed data streams. In Proceedings of ACM SIGKDD (2000), pp. 71–80. [9] ESTAN, C., SAVAGE, S., AND VARGHESE, G. Automatically inferring patterns of resource consumption in network traffic. In Proceedings of SIGCOMM’03 (2003). [10] FREUND, Y., SCHAPIRE, R. E., SINGER, Y., AND WARMUTH, M. K. Using and combining predictors that specialize. In Proceedings of the Twenty-Ninth Annual Symposium on the Theory of Computing (STOC) (1997), pp. 334–343. [11] HELMBOLD, D. P., AND SCHAPIRE, R. E. Predicting nearly as well as the best pruning of a decision tree. Machine Learning 27, 1 (1997), 51–68. [12] HERBSTER, M., AND WARMUTH, M. Tracking the best expert. Machine Learning 32, 2 (August 1998). [13] JIN, R., AND AGARWAL, G. Efficient and effective decision tree construction on streaming data. In Proceedings of ACM SIGKDD (2003). [14] JUNG, J., KRISHNAMURTHY, B., AND RABINOVICH, M. Flash crowds and denial of service attacks: Characterization and implications for cdns and websites. In Proceedings of the International World Wide Web Conference (May 2002). [15] JUNG, J., AND SIT, E. An empirical study of spam traffic and the use of DNS black lists. In Proceedings of Internet Measurement Conference (IMC) (2004). [16] KOHLER, E., LI, J., PAXSON, V., AND SHENKER, S. Observed structure of addresses in IP traffic. IEEE/ACM Transactions in Networking 14, 6 (2006). [17] KRISHNAMURTHY, B., AND WANG, J. On network-aware clustering of web clients. In Proceedings of ACM SIGCOMM (2000). [18] MAO, Z. M., SEKAR, V., SPATSCHECK, O., VAN DER MERWE, J., AND VASUDEVAN, R. Analyzing large ddos attacks using multiple data sources. In ACM SIGCOMM Workshop on Large Scale Attack Defense (2006). [19] RAMACHANDRAN, A., AND FEAMSTER, N. Understanding the network-level behavior of spammers. In Proceedings of ACM SIGCOMM (2006). [20] SLEATOR, D. D., AND TARJAN, R. E. Amortized efficiency of list update and paging rules. In Communications of the ACM (1985), vol. 28, pp. 202–208. [21] SOLDO, F., MARKOPOULO, A., AND ARGYRAKI, K. Optimal filtering of source address prefixes: Models and algorithms. In Proceedings of IEEE Infocom 2009 (2009). [22] TWINING, D., WILLIAMSON, M. M., MOWBRAY, M., AND RAHMOUNI, M. Email prioritization: Reducing delays on legitimate mail caused by junk mail. In USENIX Annual Technical Conference (2004). [23] VENKATARAMAN, S., BLUM, A., SONG, D., SEN, S., AND SPATSCHECK, O. Tracking dynamic sources of malicious activity at internet-scale. Tech. Rep. TD-7NZS8K, AT&T Labs, 2009. [24] VENKATARAMAN, S., SEN, S., SPATSCHECK, O., HAFFNER, P., AND SONG, D. Exploiting network structure for proactive spam mitigation. In Proceedings of Usenix Security’07 (2007). [25] XIE, Y., YU, F., ACHAN, K., GILLUM, E., , GOLDSZMIDT, M., AND WOBBER, T. How dynamic are IP addresses? In Proceedings of ACM SIGCOMM (2007). [26] ZHANG, J., PORRAS, P., AND ULRICH, J. Highly predictive blacklists. In Proceedings of Usenix Security’08 (2008). [27] ZHANG, Y., SINGH, S., SEN, S., DUFFIELD, N., AND LUND, C. Online identification of hierarchical heavy hitters: algorithms, evaluation, and applications. In IMC ’04: Proceedings of the 4th ACM SIGCOMM conference on Internet measurement (New York, NY, USA, 2004), ACM, pp. 101–114. 9
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Graph Zeta Function in the Bethe Free Energy and Loopy Belief Propagation Yusuke Watanabe The Institute of Statistical Mathematics 10-3 Midori-cho, Tachikawa Tokyo 190-8562, Japan watay@ism.ac.jp Kenji Fukumizu The Institute of Statistical Mathematics 10-3 Midori-cho, Tachikawa Tokyo 190-8562, Japan fukumizu@ism.ac.jp Abstract We propose a new approach to the analysis of Loopy Belief Propagation (LBP) by establishing a formula that connects the Hessian of the Bethe free energy with the edge zeta function. The formula has a number of theoretical implications on LBP. It is applied to give a sufficient condition that the Hessian of the Bethe free energy is positive definite, which shows non-convexity for graphs with multiple cycles. The formula clarifies the relation between the local stability of a fixed point of LBP and local minima of the Bethe free energy. We also propose a new approach to the uniqueness of LBP fixed point, and show various conditions of uniqueness. 1 Introduction Pearl’s belief propagation [1] provides an efficient method for exact computation in the inference with probabilistic models associated to trees. As an extension to general graphs allowing cycles, Loopy Belief Propagation (LBP) algorithm [2] has been proposed, showing successful performance in various problems such as computer vision and error correcting codes. One of the interesting theoretical aspects of LBP is its connection with the Bethe free energy [3]. It is known, for example, the fixed points of LBP correspond to the stationary points of the Bethe free energy. Nonetheless, many of the properties of LBP such as exactness, convergence and stability are still unclear, and further theoretical understanding is needed. This paper theoretically analyzes LBP by establishing a formula asserting that the determinant of the Hessian of the Bethe free energy equals the reciprocal of the edge zeta function up to a positive factor. This formula derives a variety of results on the properties of LBP such as stability and uniqueness, since the zeta function has a direct link with the dynamics of LBP as we show. The first application of the formula is the condition for the positive definiteness of the Hessian of the Bethe free energy. The Bethe free energy is not necessarily convex, which causes unfavorable behaviors of LBP such as oscillation and multiple fixed points. Thus, clarifying the region where the Hessian is positive definite is an importance problem. Unlike the previous approaches which consider the global structure of the Bethe free energy such as [4, 5], we focus the local structure. Namely, we provide a simple sufficient condition that determines the positive definite region: if all the correlation coefficients of the pseudomarginals are smaller than a value given by a characteristic of the graph, the Hessian is positive definite. Additionally, we show that the Hessian always has a negative eigenvalue around the boundary of the domain if the graph has at least two cycles. Second, we clarify a relation between the local stability of a LBP fixed point and the local structure of the Bethe free energy. Such a relation is not necessarily obvious, since LBP is not the gradient descent of the Bethe free energy. In this line of studies, Heskes [6] shows that a locally stable fixed point of LBP is a local minimum of the Bethe free energy. It is thus interesting to ask which local 1 minima of the Bethe free energy are stable or unstable fixed points of LBP. We answer this question by elucidating the conditions of the local stability of LBP and the positive definiteness of the Bethe free energy in terms of the eigenvalues of a matrix, which appears in the graph zeta function. Finally, we discuss the uniqueness of LBP fixed point by developing a differential topological result on the Bethe free energy. The result shows that the determinant of the Hessian at the fixed points, which appears in the formula of zeta function, must satisfy a strong constraint. As a consequence, in addition to the known result on the one-cycle case, we show that the LBP fixed point is unique for any unattractive connected graph with two cycles without restricting the strength of interactions. 2 Loopy belief propagation algorithm and the Bethe free energy Throughout this paper, G = (V, E) is a connected undirected graph with V the vertices and E the undirected edges. The cardinality of V and E are denoted by N and M respectively. In this article we focus on binary variables, i.e., xi ∈{±1}. Suppose that the probability distribution over the set of variables x = (xi)i∈V is given by the following factorization form with respect to G: p(x) = 1 Z ∏ ij∈E ψij(xi, xj) ∏ i∈V ψi(xi), (1) where Z is a normalization constant and ψij and ψi are positive functions given by ψij(xi, xj) = exp(Jijxixj) and ψi(xi) = exp(hixi) without loss of generality. In various applications, the computation of marginal distributions pi(xi) := ∑ x\{xi} p(x) and pij(xi, xj) := ∑ x\{xixj} p(x) is required though the exact computation is intractable for large graphs. If the graph is a tree, they are efficiently computed by Pearl’s belief propagation algorithm [1]. Even if the graph has cycles, it is empirically known that the direct application of this algorithm, called Loopy Belief Propagation (LBP), often gives good approximation. LBP is a message passing algorithm. For each directed edge, a message vector µi→j(xj) is assigned and initialized arbitrarily. The update rule of messages is given by µnew i→j(xj) ∝ ∑ xi ψji(xj, xi)ψi(xi) ∏ k∈Ni\j µk→i(xi), (2) where Ni is the neighborhood of i ∈V . The order of edges in the update is arbitrary. In this paper we consider parallel update, that is, all edges are updated simultaneously. If the messages converge to a fixed point {µ∞ i→j}, the approximations of pi(xi) and pij(xi, xj) are calculated by the beliefs, bi(xi) ∝ψi(xi) ∏ k∈Ni µ∞ k→i(xi), bij(xi, xj) ∝ψij(xi, xj)ψi(xi)ψj(xj) ∏ k∈Ni\j µ∞ k→i(xi) ∏ k∈Nj\i µ∞ k→j(xj), (3) with normalization ∑ xi bi(xi) = 1 and ∑ xi,xj bij(xi, xj) = 1. From (2) and (3), the constraints bij(xi, xj) > 0 and ∑ xj bij(xi, xj) = bi(xi) are automatically satisfied. We introduce the Bethe free energy as a tractable approximation of the Gibbs free energy. The exact distribution (1) is characterized by a variational problem p(x) = argminˆp FGibbs(ˆp), where the minimum is taken over all probability distributions on (xi)i∈V and FGibbs(ˆp) is the Gibbs free energy defined by FGibbs(ˆp) = KL(ˆp||p) −log Z. Here KL(ˆp||p) = ∫ ˆp log(ˆp/p) is the KullbackLeibler divergence from ˆp to p. Note that FGibbs(ˆp) is a convex function of ˆp. In the Bethe approximation, we confine the above minimization to the distribution of the form b(x) ∝∏ ij∈E bij(xi, xj) ∏ i∈V bi(xi)1−di , where di := |Ni| is the degree and the constraints bij(xi, xj) > 0, ∑ xi,xj bij(xi, xj) = 1 and ∑ xj bij(xi, xj) = bi(xi) are satisfied. A set {bi(xi), bij(xi, xj)} satisfying these constraints is called pseudomarginals. For computational tractability, we modify the Gibbs free energy to the objective function called Bethe free energy: F(b) := − ∑ ij∈E ∑ xixj bij(xi, xj) log ψij(xi, xj) − ∑ i∈V ∑ xi bi(xi) log ψi(xi) + ∑ ij∈E ∑ xixj bij(xi, xj) log bij(xi, xj) + ∑ i∈V (1 −di) ∑ xi bi(xi) log bi(xi). (4) 2 The domain of the objective function F is the set of pseudomarginals. The function F does not necessarily have a unique minimum. The outcome of this modified variational problem is the same as that of LBP [3]. To put it more precisely, There is a one-to-one correspondence between the set of stationary points of the Bethe free energy and the set of fixed points of LBP. It is more convenient if we work with minimal parameters, mean mi = Ebi[xi] and correlation χij = Ebij[xixj]. Then we have an effective parametrization of pseudomarginals: bij(xi, xj) = 1 4(1 + mixi + mjxj + χijxixj), bi(xi) = 1 2(1 + mi). (5) The Bethe free energy (4) is rewritten as F({mi, χij}) = − ∑ ij∈E Jijχij − ∑ i∈V himi + ∑ ij∈E ∑ xixj η (1+mixi +mjxj + χijxixj 4 ) + ∑ i∈V (1 −di) ∑ xi η (1 +mixi 2 ) , (6) where η(x) := x log x. The domain of F is written as L(G) := { {mi, χij} ∈RN+M|1 + mixi + mjxj + χijxixj > 0 for all ij ∈E and xi, xj = ±1 } . The Hessian of F, which consists of the second derivatives with respect to {mi, χij}, is a square matrix of size N + M and denoted by ∇2F. This is considered to be a matrix-valued function on L(G). Note that, from (6), ∇2F does not depend on Jij and hi. 3 Zeta function and Hessian of Bethe free energy 3.1 Zeta function and Ihara’s formula For each undirected edge of G, we make a pair of oppositely directed edges, which form a set of directed edges ⃗E. Thus | ⃗E| = 2M. For each directed edge e ∈⃗E, o(e) ∈V is the origin of e and t(e) ∈V is the terminus of e. For e ∈⃗E, the inverse edge is denoted by ¯e, and the corresponding undirected edge by [e] = [¯e] ∈E. A closed geodesic in G is a sequence (e1, . . . , ek) of directed edges such that t(ei) = o(ei+1) and ei ̸= ¯ei+1 for i ∈Z/kZ. Two closed geodesics are said to be equivalent if one is obtained by cyclic permutation of the other. An equivalent class of closed geodesics is called a prime cycle if it is not a repeated concatenation of a shorter closed geodesic. Let P be the set of prime cycles of G. For given weights u = (ue)e∈⃗E, the edge zeta function [7, 8] is defined by ζG(u) := ∏ p∈P (1 −g(p))−1, g(p) := ue1 · · · uek for p = (e1, . . . , ek), where ue ∈C is assumed to be sufficiently small for convergence. This is an analogue of the Riemann zeta function which is represented by the product over all the prime numbers. Example 1. If G is a tree, which has no prime cycles, ζG(u) = 1. For 1-cycle graph CN of length N, the prime cycles are (e1, e2, . . . , eN) and (¯eN, ¯eN−1, . . . , ¯e1), and thus ζCN (u) = (1 − ∏N l=1 uel)−1(1 −∏N l=1 u¯el)−1. Except for these two types of graphs, the number of prime cycles is infinite. It is known that the edge zeta function has the following simple determinant formula, which gives analytical continuation to the whole C2M. Let C( ⃗E) be the set of functions on the directed edges. We define a matrix on C( ⃗E), which is determined by the graph G, by Me,e′ := {1 if e ̸= ¯e′ and o(e) = t(e′), 0 otherwise. (7) Theorem 1 ([8], Theorem 3). ζG(u) = det(I −UM)−1, (8) where U is a diagonal matrix defined by Ue,e′ := ueδe,e′. 3 We need to show another determinant formula of the edge zeta function, which is used in the proof of theorem 3. We leave the proof of theorem 2 to the supplementary material. Theorem 2 (Multivariable version of Ihara’s formula). Let C(V ) be the set of functions on V . We define two linear operators on C(V ) by ( ˆDf)(i) := ( ∑ e∈⃗ E t(e)=i ueu¯e 1 −ueu¯e ) f(i), ( ˆ Af)(i) := ∑ e∈⃗ E t(e)=i ue 1 −ueu¯e f(o(e)), where f ∈C(V ). (9) Then we have ( ζG(u)−1 = ) det(I −UM) = det(I + ˆD −ˆ A) ∏ [e]∈E (1 −ueu¯e). (10) If we set ue = u for all e ∈⃗E , the edge zeta function is called the Ihara zeta function [9] and denoted by ζG(u). In this single variable case, theorem 2 is reduced to Ihara’s formula [10]: ζG(u)−1 = det(I −uM) = (1 −u2)M det(I + u2 1 −u2 D − u 1 −u2 A), (11) where D is the degree matrix and A is the adjacency matrix defined by (Df)(i) := dif(i), (Af)(i) := ∑ e∈⃗E,t(e)=i f(o(e)), f ∈C(V ). 3.2 Main formula Theorem 3 (Main Formula). The following equality holds at any point of L(G): ( ζG(u)−1= ) det(I −UM) = det(∇2F) ∏ ij∈E ∏ xi,xj=±1 bij(xi, xj) ∏ i∈V ∏ xi=±1 bi(xi)1−di 22N+4M, (12) where bij and bi are given by (5) and ui→j := χij −mimj 1 −m2 j . (13) Proof. (The detail of the computation is given in the supplementary material.) From (6), it is easy to see that the (E,E)-block of the Hessian is a diagonal matrix given by ∂2F ∂χij∂χkl = δij,kl 1 4 ( 1 1+mi +mj +χij + 1 1−mi +mj −χij + 1 1+mi −mj −χij + 1 1−mi −mj +χij ) . Using this diagonal block, we erase (V,E)-block and (E,V)-block of the Hessian. In other words, we choose a square matrix X such that det X = 1 and XT (∇2F)X = [ Y 0 0 ( ∂2F ∂χij∂χkl ) ] . After the computation given in the supplementary material, we see that (Y )i,j = 1 1−m2 i + ∑ k∈Ni (χik−mimk)2 (1−m2 i )(1−m2 i −m2 k+2mimkχik−χ2 ik) if i = j, −Ai,j χij−mimj 1−m2 i −m2 j+2mimjχij−χ2 ij otherwise. (14) From uj→i = χij−mimj 1−m2 i , it is easy to check that IN + ˆD −ˆ A = Y W, where ˆ A and ˆD is defined in (9) and W is a diagonal matrix defined by Wi,j := δi,j(1 −m2 i ). Therefore, det(I −UM) = det(Y ) ∏ i∈V (1 −m2 i ) ∏ [e]∈E (1 −ueu¯e) = R.H.S. of (12) For the left equality, theorem 2 is used. Theorem 3 shows that the determinant of the Hessian of the Bethe free energy is essentially equal to det(I−UM), the reciprocal of the edge zeta function. Since the matrix UM has a direct connection with LBP as seen in section 5, the above formula derives many consequences shown in the rest of the paper. 4 4 Application to positive definiteness conditions The convexity of the Bethe free energy is an important issue, as it guarantees uniqueness of the fixed point. Pakzad et al [11] and Heskes [5] derive sufficient conditions of convexity and show that the Bethe free energy is convex for trees and graphs with one cycle. In this section, instead of such global structure, we shall focus the local structure of the Bethe free energy as an application of the main formula. For given square matrix X, Spec(X) ⊂C denotes the set of eigenvalues (spectra), and ρ(X) the spectral radius of a matrix X, i.e., the maximum of the modulus of the eigenvalues. Theorem 4. Let M be the matrix given by (7). For given {mi, χij} ∈L(G), U is defined by (13). Then, Spec(UM) ⊂C \ R≥1 =⇒ ∇2F is a positive definite matrix at {mi, χij}. Proof. We define mi(t) := mi and χij(t) := tχij + (1 −t)mimj. Then {mi(t), χij(t)} ∈L(G) and {mi(1), χij(1)} = {mi, χij}. For t ∈[0, 1], we define U(t) and ∇2F(t) in the same way by {mi(t), χij(t)}. We see that U(t) = tU. Since Spec(UM) ⊂C\R≥1, we have det(I −tUM) ̸= 0 ∀t ∈[0, 1]. From theorem 3, det(∇2F(t)) ̸= 0 holds on this interval. Using (14) and χij(0) = mi(0)mj(0), we can check that ∇2F(0) is positive definite. Since the eigenvalues of ∇2F(t) are real and continuous with respect t, the eigenvalues of ∇2F(1) must be positive reals. We define the symmetrization of ui→j and uj→i by βi→j = βj→i := χij −mimj {(1 −m2 i )(1 −m2 j)}1/2 = Covbij[xi, xj] {Varbi[xi]Varbj[xj]}1/2 . (15) Thus, ui→juj→i = βi→jβj→i. Since βi→j = βj→i, we sometimes abbreviate βi→j as βij. From the final expression, we see that |βij| < 1. Define diagonal matrices Z and B by (Z)e,e′ := δe,e′(1 −m2 t(e))1/2 and (B)e,e′ := δe,e′βe respectively. Then we have ZUMZ−1 = BM, because (ZUMZ−1)e,e′ = (1 −m2 t(e))1/2ue(M)e,e′(1 −m2 o(e))−1/2 = βe(M)e,e′. Therefore Spec(UM) = Spec(BM). The following corollary gives a more explicit condition of the region where the Hessian is positive definite in terms of the correlation coefficients of the pseudomarginals. Corollary 1. Let α be the Perron Frobenius eigenvalue of M and define Lα−1(G) := {{mi, χij} ∈ L(G)||βe| < α−1 for all e ∈⃗E}. Then, the Hessian ∇2F is positive definite on Lα−1(G). Proof. Since |βe| < α−1, we have ρ(BM) < ρ(α−1M) = 1 ([12] Theorem 8.1.18). Therefore Spec(BM) ∩R≥1 = ϕ. As is seen from (11), α−1 is the distance from the origin to the nearest pole of Ihara’s zeta ζG(u). From example 1, we see that ζG(u) = 1 for a tree G and ζCN (u) = (1 −uN)−2 for a 1-cycle graph CN. Therefore α−1 is ∞and 1 respectively. In these cases, Lα−1(G) = L(G) and F is a strictly convex function on L(G), because |βe| < 1 always holds. This reproduces the results shown in [11]. In general, using theorem 8.1.22 of [12], we have mini∈V di −1 ≤α ≤maxi∈V di −1. Theorem 3 is also useful to show non-convexity. Corollary 2. Let {mi(t) := 0, χij(t) := t} ∈L(G) for t < 1. Then we have lim t→1 det(∇2F(t))(1 −t)M+N−1 = −2−M−N+1(M −N)κ(G), (16) where κ(G) is the number of spanning trees in G. In particular, F is never convex on L(G) for any connected graph with at least two linearly independent cycles, i.e. M −N ≥1. Proof. The equation (16) is obtained by Hashimoto’s theorem [13], which gives the u →1 limit of the Ihara zeta function. (See supplementary material for the detail.) If M −N ≥1, the right hand side of (16) is negative. As approaches to {mi = 0, χij = 1} ∈L(G), the determinant of the Hessian diverges to −∞. Therefore the Hessian is not positive definite near the point. Summarizing the results in this section, we conclude that F is convex on L(G) if and only if G is a tree or a graph with one cycle. To the best of our knowledge, this is the first proof of this fact. 5 5 Application to stability analysis In this section we discuss the local stability of LBP and the local structure of the Bethe free energy around a LBP fixed point. Heskes [6] shows that a locally stable fixed point of sufficiently damped LBP is a local minima of the Bethe free energy. The converse is not necessarily true in general, and we will elucidate the gap between these two properties. First, we regard the LBP update as a dynamical system. Since the model is binary, each message µi→j(xj) is parametrized by one parameter, say ηi→j. The state of LBP algorithm is expressed by η = (ηe)e∈⃗E ∈C( ⃗E), and the update rule (2) is identified with a transform T on C( ⃗E), ηnew = T(η). Then, the set of fixed points of LBP is {η∞∈C( ⃗E)|T(η∞) = η∞}. A fixed point η∞is called locally stable if LBP starting with a point sufficiently close to η∞converges to η∞. The local stability is determined by the linearizion T ′ around the fixed point. As is discussed in [14], η∞is locally stable if and only if Spec(T ′(η∞)) ⊂{λ ∈C||λ| < 1}. To suppress oscillatory behaviors of LBP, damping of update Tϵ := (1 −ϵ)T + ϵI is sometimes useful, where 0 ≤ϵ < 1 is a damping strength and I is the identity. A fixed point is locally stable with some damping if and only if Spec(T ′(η∞)) ⊂{λ ∈C|Reλ < 1}. There are many representations of the linearization (derivative) of LBP update (see [14, 15]), we choose a good coordinate following Furtlehner et al [16]. In section 4 of [16], they transform messages as µi→j →µi→j/µ∞ i→j and functions as ψij →bij/(bibj) and ψi →bi, where µ∞ i→j is the message of the fixed point. This changes only the representations of messages and functions, and does not affect LBP essentially. This transformation causes T ′(η∞) →P T ′(η∞)P −1 with an invertible matrix P. Using this transformation, we see that the following fact holds. (See supplementary material for the detail.) Theorem 5 ([16], Proposition 4.5). Let ui→j be given by (3), (5) and (13) at a LBP fixed point η∞. The derivative T ′(η∞) is similar to UM, i.e. UM = P T ′(η∞)P −1 with an invertible matrix P. Since det(I −T ′(η∞)) = det(I −UM), the formula in theorem 3 implies a direct link between the linearization T ′(η∞) and the local structure of the Bethe free energy. From theorem 4, we have that a fixed point of LBP is a local minimum of the Bethe free energy if Spec(T ′(η∞)) ⊂C\R≥1. It is now clear that the condition for positive definiteness, local stability of damped LBP and local stability of undamped LBP are given in terms of the set of eigenvalues, C \ R≥1, {λ ∈C|Reλ < 1} and {λ ∈C||λ| < 1} respectively. A locally stable fixed point of sufficiently damped LBP is a local minimum of the Bethe free energy, because {λ ∈C|Reλ < 1} is included in C \ R≥1. This reproduces Heskes’s result [6]. Moreover, we see the gap between the locally stable fixed points with some damping and the local minima of the Bethe free energy: if Spec(T ′(η∞)) is included in C \ R≥1 but not in {λ ∈C|Reλ < 1}, the fixed point is a local minimum of the Bethe free energy though it is not a locally stable fixed point of LBP with any damping. It is interesting to ask under which condition a local minimum of the Bethe free energy is a stable fixed point of (damped) LBP. While we do not know a complete answer, for an attractive model, which is defined by Jij ≥0, the following theorem implies that if a stable fixed point becomes unstable by changing Jij and hi, the corresponding local minimum also disappears. Theorem 6. Let us consider continuously parametrized attractive models {ψij(t), ψi(t)}, e.g. t is a temperature: ψij(t) = exp(t−1Jijxixj) and ψi(t) = exp(t−1hixi). For given t, run LBP algorithm and find a (stable) fixed point. If we continuously change t and see the LBP fixed point becomes unstable across t = t0, then the corresponding local minimum of the Bethe free energy becomes a saddle point across t = t0. Proof. From (3), we see bij(xi, xj) ∝exp(Jijxixj + θixi + θjxj) for some θi and θj. From Jij ≥0, we have Covbij[xi, xj] = χij −mimj ≥0, and thus ui→j ≥0. When the LBP fixed point becomes unstable, the Perron Frobenius eigenvalue of UM goes over 1, which means det(I −UM) crosses 0. From theorem 3 we see that det(∇2F) becomes positive to negative at t = t0. Theorem 6 extends theorem 2 of [14], which discusses only the case of vanishing local fields hi = 0 and the trivial fixed point (i.e. mi = 0). 6 6 Application to uniqueness of LBP fixed point The uniqueness of LBP fixed point is a concern of many studies, because the property guarantees that LBP finds the global minimum of the Bethe free energy if it converges. The major approaches to the uniqueness is to consider equivalent minimax problem [5], contraction property of LBP dynamics [17, 18], and to use the theory of Gibbs measure [19]. We will propose a different, differential topological approach to this problem. In our approach, in combination with theorem 3, the following theorem is the basic apparatus. Theorem 7. If det ∇2F(q) ̸= 0 for all q ∈(∇F)−1(0) then ∑ q:∇F (q)=0 sgn ( det ∇2F(q) ) = 1, where sgn(x) := {1 if x > 0, −1 if x < 0. We call each summand, which is +1 or −1, the index of F at q. Note that the set (∇F)−1(0), which is the stationary points of the Bethe free energy, coincides with the fixed points of LBP. The above theorem asserts that the sum of indexes of all the fixed points must be one. As a consequence, the number of the fixed points of LBP is always odd. Note also that the index is a local quantity, while the assertion expresses the global structure of the function F. For the proof of theorem 7, we prepare two lemmas. The proof of lemma 1 is shown in the supplementary material. Lemma 2 is a standard result in differential topology, and we refer [20] theorem 13.1.2 and comments in p.104 for the proof. Lemma 1. If a sequence {qn} ⊂L(G) converges to a point q∗∈∂L(G), then ∥∇F(qn)∥→∞, where ∂L(G) is the boundary of L(G) ⊂RN+M. Lemma 2. Let M1 and M2 be compact, connected and orientable manifolds with boundaries. Assume that the dimensions of M1 and M2 are the same. Let f : M1 →M2 be a smooth map satisfying f(∂M1) ⊂∂M2. For a regular value of p ∈M2, i.e. det(∇f(q)) ̸= 0 for all q ∈f −1(p), we define the degree of the map f by deg f := ∑ q∈f −1(p) sgn(det ∇f(q)). Then deg f does not depend on the choice of a regular value p ∈M2. Sketch of proof. Define a map Φ : L(G) →RN+M by Φ := ∇F + (h J ) . Note that Φ does not depend on h and J as seen from (6). Then it is enough to prove ∑ q∈Φ−1(( h J)) sgn(det ∇Φ(q)) = ∑ q∈Φ−1(0) sgn(det ∇Φ(q)), (17) because Φ−1(0) has a unique element {mi = 0, χij = 0}, at which ∇2F is positive definite, and the right hand side of (17) is equal to one. Define a sequence of manifolds {Cn} by Cn := {q ∈ L(G)| ∑ ij∈E ∑ xi,xj−log bij ≤n}, which increasingly converges to L(G). Take K > 0 and ϵ > 0 to satisfy K −ϵ > ∥ (h J ) ∥. From lemma 1, for sufficiently large n0, we have Φ−1(0), Φ−1(h J ) ⊂Cn0 and Φ(∂Cn0) ∩B0(K) = ϕ, where B0(K) is the closed ball of radius K at the origin. Let Πϵ : RN+M →B0(K) be a smooth map that is the identity on B0(K −ϵ), monotonically increasing on ∥x∥, and Πϵ(x) = K ∥x∥x for ∥x∥≥K. We obtain a map ˜Φ := Πϵ ◦Φ : Cn0 →B0(K) such that ˜Φ(∂Cn0) ⊂∂B0(K). Applying lemma 2 yields (17). If we can guarantee that the index of every fixed point is +1 in advance of running LBP, we conclude that fixed point of LBP is unique. We have the following a priori information for β. Lemma 3. Let βij be given by (15) at any fixed point of LBP. Then |βij| ≤tanh(|Jij|) and sgn(βij) = sgn(Jij) hold. Proof. From (3), we see that bij(xi, xj) ∝exp(Jijxixj + θixi + θjxj) for some θi and θj. With (15) and straightforward computation, we obtain βij = sinh(2Jij)(cosh(2θi) + cosh(2Jij))−1/2(cosh(2θj)+cosh(2Jij))−1/2. The bound is attained when θi = 0 and θj = 0. From theorem 7 and lemma 3, we can immediately obtain the uniqueness condition in [18], though the stronger contractive property is proved under the same condition in [18]. 7 Figure 1: Graph of Example 2. Figure 2: Graph ˆG. Figure 3: Two other types. Corollary 3 ([18]). If ρ(J M) < 1, then the fixed point of LBP is unique, where J is a diagonal matrix defined by Je,e′ = tanh(|Je|)δe,e′. Proof. Since |βij| ≤tanh(|Jij|), we have ρ(BM) ≤ρ(J M) < 1. ([12] Theorem 8.1.18.) Then det(I −BM) = det(I −UM) > 0 implies that the index of any LBP fixed point must be +1. In the proof of the above corollary, we only used the bound of modulus. In the following case of corollary 4, we can utilize the information of signs. To state the corollary, we need a terminology. The interactions {Jij, hi} and {J′ ij, h′ i} are said to be equivalent if there exists (si) ∈{±1}V such that J′ ij = Jijsisj and h′ i = hisi. Since an equivalent model is obtained by gauge transformation xi →xisi, the uniqueness property of LBP for equivalent models is unchanged. Corollary 4. If the number of linearly independent cycle of G is two (i.e. M −N + 1 = 2), and the interaction is not equivalent to attractive model, then the LBP fixed point is unique. The proof is shown in the supplementary material. We give an example to illustrate the outline. Example 2. Let V := {1, 2, 3, 4} and E := {12, 13, 14, 23, 34}. The interactions are given by arbitrary {hi} and {−J12, J13, J14, J23, J34} with Jij ≥0. See figure 1. It is enough to check that det(I −BM) > 0 for arbitrary 0 ≤β13, β23, β14, β34 < 1 and −1 < β12 ≤0. Since the prime cycles of G bijectively correspond to those of ˆG (in figure 2), we have det(I−BM) = det(I−ˆB ˆ M), where ˆβe1 = β12β23, ˆβe2 = β13, and ˆβe3 = β34. We see that det(I −ˆB ˆ M) = (1 −ˆβe1 ˆβe2 − ˆβe1 ˆβe3 −ˆβe2 ˆβe3 −2ˆβe1 ˆβe2 ˆβe3)(1 −ˆβe1 ˆβe2 −ˆβe1 ˆβe3 −ˆβe2 ˆβe3 + 2ˆβe1 ˆβe2 ˆβe3) > 0. In other cases, we can reduce to the graph ˆG or the graphs in figure 3 similarly (see the supplementary material). For attractive models, the fixed point of the LBP is not necessarily unique. For graphs with multiple cycles, all the existing results on uniqueness make assumptions that upperbound |Jij| essentially. In contrast, corollary 4 applies to arbitrary strength of interactions if the graph has two cycles and the interactions are not attractive. It is noteworthy that, from corollary 2, the Bethe free energy is non-convex in the situation of corollary 4, while the fixed point is unique. 7 Concluding remarks For binary pairwise models, we show the connection between the edge zeta function and the Bethe free energy in theorem 3, in the proof of which the multi-variable version of Ihara’s formula (theorem 2) is essential. After the initial submission of this paper, we found that theorem 3 is extended to a more general class of models including multinomial models and Gaussian models represented by arbitrary factor graphs. We will discuss the extended formula and its applications in a future paper. Some recent researches on LBP have suggested the importance of zeta function. In the context of the LDPC code, which is an important application of LBP, Koetter et al [21, 22] show the connection between pseudo-codewords and the edge zeta function. On the LBP for the Gaussian graphical model, Johnson et al [23] give zeta-like product formula of the partition function. While these are not directly related to our work, pursuing covered connections is an interesting future research topic. Acknowledgements This work was supported in part by Grant-in-Aid for JSPS Fellows 20-993 and Grant-in-Aid for Scientific Research (C) 19500249. 8 References [1] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers, San Mateo, CA, 1988. [2] K. Murphy, Y. Weiss, and M.I. Jordan. Loopy belief propagation for approximate inference: An empirical study. Proc. of Uncertainty in AI, 15:467–475, 1999. [3] J.S. Yedidia, W.T. Freeman, and Y. Weiss. Generalized belief propagation. Adv. in Neural Information Processing Systems, 13:689–95, 2001. [4] Y. Weiss. Correctness of Local Probability Propagation in Graphical Models with Loops. Neural Computation, 12(1):1–41, 2000. [5] T. Heskes. On the uniqueness of loopy belief propagation fixed points. Neural Computation, 16(11):2379–2413, 2004. [6] T. Heskes. Stable fixed points of loopy belief propagation are minima of the Bethe free energy. Adv. in Neural Information Processing Systems, 15, pages 343–350, 2002. [7] K. Hashimoto. Zeta functions of finite graphs and representations of p-adic groups. Automorphic forms and geometry of arithmetic varieties, 15:211–280, 1989. [8] H.M. Stark and A.A. Terras. Zeta functions of finite graphs and coverings. Advances in Mathematics, 121(1):124–165, 1996. [9] Y. Ihara. On discrete subgroups of the two by two projective linear group over p-adic fields. Journal of the Mathematical Society of Japan, 18(3):219–235, 1966. [10] H. Bass. The Ihara-Selberg zeta function of a tree lattice. Internat. J. Math, 3(6):717–797, 1992. [11] P. Pakzad and V. Anantharam. Belief propagation and statistical physics. Conference on Information Sciences and Systems, (225), 2002. [12] R.A. Horn and C.R. Johnson. Matrix analysis. Cambridge University Press, 1990. [13] K. Hashimoto. On zeta and L-functions of finite graphs. Internat. J. Math, 1(4):381–396, 1990. [14] JM Mooij and HJ Kappen. On the properties of the Bethe approximation and loopy belief propagation on binary networks. Journal of Statistical Mechanics: Theory and Experiment, (11):P11012, 2005. [15] S. Ikeda, T. Tanaka, and S. Amari. Information geometry of turbo and low-density parity-check codes. IEEE Transactions on Information Theory, 50(6):1097–1114, 2004. [16] C. Furtlehner, J.M. Lasgouttes, and A. De La Fortelle. Belief propagation and Bethe approximation for traffic prediction. INRIA RR-6144, Arxiv preprint physics/0703159, 2007. [17] A.T. Ihler, JW Fisher, and A.S. Willsky. Loopy belief propagation: Convergence and effects of message errors. Journal of Machine Learning Research, 6(1):905–936, 2006. [18] J. M. Mooij and H. J. Kappen. Sufficient Conditions for Convergence of the Sum-Product Algorithm. IEEE Transactions on Information Theory, 53(12):4422–4437, 2007. [19] S. Tatikonda and M.I. Jordan. Loopy belief propagation and Gibbs measures. Uncertainty in AI, 18:493–500, 2002. [20] B.A. Dubrovin, A.T. Fomenko, S.P. Novikov, and Burns R.G. Modern Geometry: Methods and Applications: Part 2: the Geometry and Topology of Manifolds . Springer-Verlag, 1985. [21] R. Koetter, W.C.W. Li, PO Vontobel, and JL Walker. Pseudo-codewords of cycle codes via zeta functions. IEEE Information Theory Workshop, pages 6–12, 2004. [22] R. Koetter, W.C.W. Li, P.O. Vontobel, and J.L. Walker. Characterizations of pseudo-codewords of (low-density) parity-check codes. Advances in Mathematics, 213(1):205–229, 2007. [23] J.K. Johnson, V.Y. Chernyak, and M. Chertkov. Orbit-Product Representation and Correction of Gaussian Belief Propagation. Proceedings of the 26th International Conference on Machine Learning, (543), 2009. 9
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Augmenting Feature-driven fMRI Analyses: Semi-supervised Learning and Resting State Activity Matthew B. Blaschko Visual Geometry Group Department of Engineering Science University of Oxford blaschko@robots.ox.ac.uk Jacquelyn A. Shelton Max Planck Institute for Biological Cybernetics Fakult¨at f¨ur Informations- und Kognitionswissenschaften Universit¨at T¨ubingen jshelton@tuebingen.mpg.de Andreas Bartels Max Planck Institute for Biological Cybernetics Centre for Integrative Neuroscience, Universit¨at T¨ubingen abartels@tuebingen.mpg.de Abstract Resting state activity is brain activation that arises in the absence of any task, and is usually measured in awake subjects during prolonged fMRI scanning sessions where the only instruction given is to close the eyes and do nothing. It has been recognized in recent years that resting state activity is implicated in a wide variety of brain function. While certain networks of brain areas have different levels of activation at rest and during a task, there is nevertheless significant similarity between activations in the two cases. This suggests that recordings of resting state activity can be used as a source of unlabeled data to augment discriminative regression techniques in a semi-supervised setting. We evaluate this setting empirically yielding three main results: (i) regression tends to be improved by the use of Laplacian regularization even when no additional unlabeled data are available, (ii) resting state data seem to have a similar marginal distribution to that recorded during the execution of a visual processing task implying largely similar types of activation, and (iii) this source of information can be broadly exploited to improve the robustness of empirical inference in fMRI studies, an inherently data poor domain. 1 Introduction In this work we study the use of resting state activity for the semi-supervised analysis of human fMRI studies. We wish to use resting state activity as an additional source of unlabeled data in a semi-supervised regression setting. We analyze the weights of a trained regressor to infer brain regions that are implicated in visual processing tasks. As the recording of human fMRI data is constrained by limits on the time a subject can safely remain in a scanner, and by the high demand for high-resolution scanning facilities, it is important to fully utilize available data. One source of such additional data is resting state activity, the brain activation that arises in the absence of any task. This data has been the subject of many studies in recent years, and has the important advantage of not being biased by a specific task. We show in this work that the marginal distribution of resting state activity is suitable to improve regression performance when employed for semi-supervised learning. In neuroscience there has been a recent surge of interest in analyzing brain activity in more natural, complex settings, e.g. with volunteers viewing movies, in order to gain insight in brain processes and connectivity underlying more natural processing. The problem has been approached from dif1 ferent routes: linear regression was used to identify brain areas correlating with particular labels in the movie [2], the perceived content was inferred based on brain activity [23], data-driven methods were used to subdivide the brain into units with distinct response profiles [1], and correlation across subjects was used to infer stimulus-driven brain processes at different timescales [24]. Several pattern recognition techniques have previously been applied to fMRI data of brains, including support vector machines and Fisher linear discriminant analysis [26, 27, 29]. In [22], kernel canonical correlation analysis (KCCA) was applied to fMRI data from human subjects. We have recently applied a semi-supervised extension of KCCA to human fMRI data [32] where the unlabeled data source was given by the subjects viewing a movie for which the labels were not known. In this work, we explore the more realistic setting in which unlabeled data are available as a side product of other fMRI studies. This enables the more efficient use of available data, and obviates the necessity to waste scanner time and human labeling effort in order to produce sufficiently large data sets to achieve satisfactory results. In Section 2 we discuss the generation and significance of resting state activity. We then discuss the statistical assumptions implicit in semi-supervised learning in Section 3. We present the experimental setup for data acquisition in Section 4, and discuss the semi-supervised regression model in Section 5. In Section 6, we show empirically that resting state activity is an effective source of unlabeled data for semi-supervised learning. 2 Resting State Activity Resting state activity has attracted the attention of neuroscientists now for over a decade [8, 20]. It is defined as brain activation that arises in the absence of any task, and it is usually measured in awake subjects during prolonged fMRI scanning sessions, where no other instructions are given than to close the eyes and to do nothing. The basic idea is that spontaneous fluctuations of neural activity in the brain may reveal some fundamental characteristics of brain function. This may include functional aspects, but also structural ones. For example, certain networks of areas have been shown to be more active at rest than during the execution of a task, leading to the hypothesis that these areas may be involved in maintaining the default state of the brain, performing mental house-keeping functions, such as monitoring own bodily states or the self [9, 30, 31], or being involved in intrinsic as opposed to extrinsic (i.e. stimulusdriven) tasks [17]. Additionally, spontaneous fluctuations of brain activity in particular brain regions have been shown to be directly correlated with metabolic activity and also with behavioural task performance, thus providing evidence that these fluctuations do not merely reflect artefacts of vascular perfusion, heart rate or breathing [9, 7]. Instead, evidence suggests that spontaneous activity changes reflect to some extent neural activity that may account for trial-to-trial variability of human behaviour [14, 28]. Resting state activity however also has structural implications, in that temporal correlations between spatially separate regions (functional connectivity) may be indicative of underlying neural communication between them, which in turn may be mediated by anatomical connections. Several studies have shown that homologue regions of the hemispheres (e.g. left and right motor cortex, Wernickes regions, etc) have high temporal correlations during rest [8, 3]. Also networks known to be anatomically connected, such as those belonging to the language network (Brocas area, Wernickes area, Geschwinds territory) within a given hemisphere show strong correlations during resting state, indicating that spontaneous activity (or activity driven by mental imagery, etc) in one region may affect others that are directly connected with it [21, 3]. Some recent studies also attempt to reveal that resting state connectivity directly relates to structural activity as revealed using diffusion tensor imaging [33, 19]. Finally, alterations in resting state activity patterns have recently been shown to be diagnostic for clinical conditions such as neuropsychiatric disorders [18], and have been shown to alter with increasing age [34]. However, the analysis of resting state activity poses a challenge, as it is not stimulus-driven, and is therefore difficult to analyze or to reveal using hypothesis-driven analysis methods. One common approach has been to reveal functional networks and their connectivity by measuring the temporal correlations of a seed region with the remaining voxels of the brain [8, 21, 3, 33, 19]. Another approach has been to apply data-driven spatio-temporal clustering methods such as independent component analysis (ICA) to reveal distinct functional areas or networks at rest [35, 1]. The over2 whelming evidence of these studies shows that groups of voxels, but also widespread networks of cortical areas that are co-engaged during task performances are also consistently co-activated during rest [35, 1, 12]. We provide an alternative, computationally driven approach to assess whether and to which extent externally driven functional networks coincide with spontaneous fluctuations during rest. We stimulated volunteers using natural movies, and measured resting state activity during the same session in separate runs that each lasted 20 min. Prior work has shown that natural movie viewing leads not only to wide-spread cortical activation, but also to a higher functional separation of distinct networks and areas compared to that obtained using traditional stimulation with controlled stimuli [23, 1, 17]. This is most likely so because distinct cortical regions responded each to distinct features occurring in the movie, thus revealing the functional division of labor in cortex [2, 4, 23]. In subsequent sections we show that semi-supervised learning algorithms improve when resting state data are added to aide feature-regression of movie-viewing data. This improvement indicates that a similar cortical structure underlies resting state data as that underlying movie-viewing data. These results thus fall in line with prior work demonstrating consistency of resting state networks across subjects [35, 12], and reveal that feature-driven activity during natural viewing induces a similar functional clustering as that occurring during rest. Importantly however, this approach may also be of other methodological interest, in that data obtained at rest may actually be used to augment the performance of feature-driven regression of stimulus-driven data. 3 Semi-supervised Learning Semi-supervised learning makes use of a combination of labeled and unlabeled training points in order to better learn a mapping from an input space, X (in this case voxels recorded from fMRI), to an output space, Y (variables recording viewing conditions). Discriminative models typically attempt to infer a mapping f : X →Y based on properties the conditional distribution p(y|x). In order to incorporate training data in X for which no correspondence is known to Y, additional assumptions must be made about the properties of the joint distribution over X ×Y. This often gives semi-supervised learning more of a generative flavor in that we assume some properties of the joint distribution in order to better make use of the marginal distribution p(x) [11]. There are several closely related assumptions employed in the development of semi-supervised learning algorithms, but we focus here on the manifold assumption [6]. We assume that our high dimensional data lie on a low dimensional manifold, and that changes in p(y|x) vary slowly as measured by distances within the manifold. The additional unlabeled data in X allow us to better model the manifold on which the data lie. In the case of fMRI acquired data, we expect that brain activity follow certain common patterns of activation. Furthermore, transitions between these patterns of activation will not be discontinuous. We can therefore be fairly confident in the assumption that the manifold assumption holds in principle. Of crucial importance, however, is that the distribution of the unlabeled samples not result in a degenerate marginal distribution with respect to the discriminative task at hand, that is to say that p(y|x) be slowly varying as measured by distances measured within the manifold estimated from labeled and unlabeled samples from X. Theoretical accounts of semi-supervised learning frequently assume that all samples from X be drawn i.i.d. In practice, in a data poor domain, we may have to resort to a source of unlabeled data that is derived by a (slightly) different process than that of the labeled samples. As resting state data is a representative byproduct of the experimental design of fMRI studies, we explore the empirical performance of its employment as a source of unlabeled data. This gives us vital insight into whether the distribution of brain states is sufficiently similar to that of subjects who are performing a visual processing task, and suggests a general and powerful improvement to the design of fMRI studies by making use of this ready source of unlabeled information. 4 Data Acquisition A Siemens 3TTIM scanner was used to acquire the fMRI data of 5 human volunteers and consisted of 350 time slices of 3-dimensional fMRI brain volumes. Time-slices were separated by 3.2 s (TR), 3 each with a spatial resolution of 46 slices (2.6 mm width, 0.4 mm gap) with 64x64 pixels of 3x3 mm, resulting in a spatial resolution of 3x3x3 mm. Each subject watched 2 movies of 18.5 min length, wherein one movie had labels indicating the continuous content of the movie (i.e. degree of visual contrast, or the degree to which a face was present, etc.) and the other remained unlabeled. The subjects additionally were recorded during a resting state of the same length of time. The imaging data were pre-processed using standard procedures using the Statistical Parametric Mapping (SPM) toolbox before analysis [15]. Included was a slice-time correction to compensate for acquisition delays between slices, a spatial realignment to correct for small head-movements, a spatial normalization to the SPM standard brain space (near MNI), and spatial smoothing using a Gaussian filter of 12 mm full width at half maximum (FWHM). Subsequently, images were skull-and-eye stripped and the mean of each time-slice was set to the same value (global scaling). A temporal high-pass filter with a cut-off of 512 s was applied, as well as a low-pass filter with the temporal properties of the hemodynamic response function (hrf), in order to reduce temporal acquisition noise. For the movie with corresponding labels, the label time-series were obtained using two separate methods. First by using computer frame-by-frame analysis of the movie [4], and second using subjective ratings averaged across an independent set of five human observers [1]. The computerderived labels indicated luminance change over time (temporal contrast), visual motion energy (i.e. the fraction of temporal contrast that can be explained by motion in the movie). The human-derived labels indicated the intensity of subjectively experienced color, and the degree to which faces and human bodies were present in the movie. In prior studies, each of these labels had been shown to correlate with brain activity in particular and distinct sets of areas specialized to process the particular label in question [1, 4]. 5 Regression Model We have applied a semi-supervised Laplacian regularized ridge regression framework to learn our discriminant function. We assume multivariate data xi ∈X with associated labels yi ∈R, for i = 1, . . . , n, although the setting is directly extensible to arbitrary input and output domains [10]. Ridge regression is classically formulated as argminw X i (yi −⟨xi, w⟩)2 + λ∥w∥2, (1) where x and y are assumed to be zero mean [25]. This is equivalent to maximizing (Tikhonov regularized) correlation between y and the projection of x onto w [16]. In order to extend this to the semi-supervised setting [11], we assume the manifold assumption and employ Laplacian regularization [37, 38, 5, 36, 6]. We assume that we have px additional unlabeled training samples and use the variable mx = n+px for notational convenience. We denote the design matrix of labeled data as X and that of labeled and unlabeled data ˆX. We can now write our Laplacian regularized objective function as argminw(y −Xw)T (y −Xw) + λ∥w∥2 + γ m2x wT ˆXT L ˆXw (2) where L is an empirical graph Laplacian [6]. The two regularization parameters, λ and γ, are set using a model selection step. We have employed a variant of the model selection used in [32], which employs a grid search to maximize the difference in objective functions between a randomized permutation of the correspondences between x and y and the unpermuted data. We have used a symmetric normalized graph Laplacian where the weights are given by a Gaussian function with the bandwidth set to the median distance between training data points L = I −D−1 2 SD−1 2 , (3) where S is a similarity matrix and D is a diagonal matrix whose entries are the row sums of S. We have primarily chosen this regression model for its simplicity. Provided the manifold assumption holds for our source of data, and that the conditional distribution, p(y|x), is slowly varying as measured by the manifold estimated from both labeled and unlabeled data, we can expect that semi-supervised Laplacian regularization will improve results across a range of loss functions and output spaces. 4 Table 1: Mean holdout correlations for motion in the five subjects across all experiments. For a description of the experiments, see Section 5. In all cases, semi-supervision from resting state activity (Exp C) improves over regression using only fully labeled data (Exp A). Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Exp A −0.008 ± 0.12 −0.08 ± 0.07 −0.08 ± 0.04 −0.06 ± 0.07 −0.08 ± 0.08 Exp B −0.02 ± 0.17 −0.03 ± 0.10 0.01 ± 0.09 −0.02 ± 0.04 −0.03 ± 0.08 Exp C 0.12 ± 0.06 0.10 ± 0.10 0.17 ± 0.14 0.012 ± 0.09 0.06 ± 0.12 Exp D 0.09 ± 0.09 0.10 ± 0.14 0.15 ± 0.15 0.04 ± 0.04 0.02 ± 0.11 Exp E 0.11 ± 0.10 0.11 ± 0.15 0.12 ± 0.09 0.11 ± 0.08 0.16 ± 0.15 Table 2: Mean holdout correlations for human body in the five subjects across all experiments. For a description of the experiments, see Section 5. In all cases, semi-supervision from resting state activity (Exp C) improves over regression using only fully labeled data (Exp A). Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Exp A 0.13 ± 0.17 −0.003 ± 0.12 0.09 ± 0.11 0.06 ± 0.14 0.12 ± 0.17 Exp B 0.16 ± 0.16 0.16 ± 0.22 0.28 ± 0.15 0.16 ± 0.20 0.21 ± 0.16 Exp C 0.36 ± 0.17 0.29 ± 0.16 0.42 ± 0.15 0.30 ± 0.12 0.40 ± 0.06 Exp D 0.34 ± 0.09 0.30 ± 0.14 0.38 ± 0.25 0.25 ± 0.11 0.35 ± 0.11 Exp E 0.35 ± 0.22 0.37 ± 0.17 0.45 ± 0.08 0.33 ± 0.14 0.43 ± 0.05 As our data consist of (i) recordings from a completely labeled movie, (ii) recordings from resting state activity, and (iii) recordings from an unlabeled movie, we are able to employ several variants of semi-supervision in the above framework: • A: In this variant, we employ only fully supervised data and use the regression given by Equation (1). • B: We also use only fully supervised data in this variant, but we employ Laplacian regularization in addition to Tikhonov regularization (Equation (2)). • C: We introduce semi-supervision from resting state activity. • D: In this variant, semi-supervision comes from the unlabeled movie. This allows us to evaluate the effects of semi-supervision from data that are designed to be drawn from the same distribution as our labeled data. • E: Finally, we combine the unlabeled data from both resting state activity and from the unlabeled movie. 6 Experimental Results In order to evaluate the performance of the regression model with different semi-supervised variants, we have performed five fold cross validation. For each fold, we measure the correlation between the projected data and its associated labels. We have performed these experiments across five different subjects with three different output variables. Table 1 shows the test correlations for all subjects and experiments for the motion output variable, while Table 2 shows results for the human body variable, and Table 3 for the language variable. Wilcoxon signed-rank tests have shown significant Table 3: Mean holdout correlations for language in the five subjects across all experiments. For a description of the experiments, see Section 5. In all cases, semi-supervision from resting state activity (Exp C) improves over regression using only fully labeled data (Exp A). Sub 1 Sub 2 Sub 3 Sub 4 Sub 5 Exp A 0.10 ± 0.13 0.10 ± 0.10 0.11 ± 0.14 −0.03 ± 0.17 −0.03 ± 0.11 Exp B 0.15 ± 0.17 −0.05 ± 0.09 0.06 ± 0.23 0.14 ± 0.18 0.03 ± 0.14 Exp C 0.35 ± 0.10 0.15 ± 0.11 0.42 ± 0.03 0.07 ± 0.17 0.10 ± 0.13 Exp D 0.27 ± 0.17 0.29 ± 0.14 0.34 ± 0.20 0.08 ± 0.11 −0.03 ± 0.11 Exp E 0.34 ± 0.17 0.22 ± 0.15 0.30 ± 0.18 0.24 ± 0.15 0.07 ± 0.19 5 (a) Regression without Laplacian regularization. (b) Laplacian regularized solution. (c) Semi-supervised Laplacian regularized solution using resting state data. Figure 1: Illustration of weight maps obtained for the visual motion feature in experiments A, B, and D. Transverse slices are shown through a single subjects T1-weighted structural image with superimposed weight-maps, colored in red for positive weights (left column), and colored in blue for negative weights (right column). The positive weight maps (left column) reveal the motion processing area V5/MT+, as well as posterior in the midline a part of peripheral early visual area V1 (not labelled). The negative weight maps reveal a reduction of BOLD signal in the occipital poles (the foveal representation of early visual areas V1-V3). Both results are in agreement with the findings reported in a prior study[4]. (a) Regression without Laplacian regularization. (b) Laplacian regularized solution. (c) Semi-supervised Laplacian regularized solution using resting state data. Figure 2: Illustration of weight maps for the human body feature. Weight maps (in red) are show on transverse (left) and sagittal (right) brain sections of a single subject. Activity involves the objectresponsive lateral occipital cortex (LOC) extending dorsally into region responsive to human bodies, dubbed extrastriate body area (EBA) [13]. The weights in all experiments are very strong for this feature (see colorbar), and nearly no difference in the extent of activation is visible across experiments. improvement between ridge regression and semi-supervised Laplacian regularization with confidence > 95% for all variables. We also provide a qualitative evaluation of the results in the form of a map of the significant weights onto slices shown through single subjects. Figure 1 shows the weights for the motion variable, Figure 2 for the human body variable, and Figure 3 for the language variable. 7 Discussion One can observe several trends in Tables 1-3. First, we notice that the results for experiment A are not satisfactory. Correlations appear to be non-existent or low, and show high variation across subjects. We conclude that the labeled training data alone are not sufficient to learn a reliable regressor for these learning problems. The results in experiment B are mixed. For some subjects and variables performance improved, but it is not consistent. We expect that this indicates non-linearity in the data, but that the labeled data alone are not sufficient to accurately estimate the manifold. We see consistent improvement in experiment C over experiment A. This supports the primary hypothesis of this work – that the marginal distribution of resting state activity in combination of that from the visual 6 (a) Regression without Laplacian regularization. (b) Laplacian regularized solution. (c) Semi-supervised Laplacian regularized solution using resting state data. Figure 3: Illustration of weight maps obtained for the language feature across the different experiments. Weight maps (in red) are superimposed on sagittal, coronal and transverse sections of a single subjects brain. The activation associated to this feature involved the superior temporal sulcus (STS), extending anteriorly to include parts of Wernickes speech processing area, and posterior and ventrally (increasing with experiments A, B and D) object-responsive region LOC, involved in analyzing facial features (in accord with the findings in [2]). processing task allows us to robustly estimate a manifold structure that improves regression performance. The results for experiment C and D are similar, with neither data source dominating the other. As the unlabeled data for experiment D were generated specifically to match the distribution of the labeled data, we conclude that resting state activity gives a similar increase in performance to semi-supervised learning with i.i.d. data. Finally, the setup in experiment E – in which we use both sources of semi-supervised data – performs similarly on average to that in experiments C and D. We conclude that the two sources of unlabeled data may not hold complimentary data, indicating that a wholesale replacement of one source by another is an effective strategy. The feature-weight maps shown in Figures 1-3 were all in accord with established findings in neuroscience, in that distinct features such as visual motion, the perception of human bodies or of language correlated with activation of distinct brain regions, such as V5+/MT+, the lateral occipital complex (LOC) and the extrastriate body area (EBA), as well as regions of the STS and Wernickes area, respectively. These findings have now been established in studies using controlled stimuli, as well as those showing movie-clips to volunteers [13, 2, 4, 23]. Here we asked whether using semi-supervised learning methods can improve a feature-driven analysis when adding data obtained in the resting state. The motivation for this stems from prior studies that suggest a functionally relevant involvement of cortical regions during rest. Data-driven analyses of resting state activity reveals a similar functional architecture that can also be observed during stimulus-driven activity, and which can be reproducibly found across subjects [12, 35]. In addition, also the functional connectivity between distinct regions appears to be physiologically plausible at rest [21, 8, 20], and in fact is very similar to the functional connectivity observed during viewing of movies [3]. Taken together, these findings would suggest that resting state activity may in theory be able to augment in a non-biased way datasets obtained in a functional setting. At the same time, if resting state data were indeed found to augment results of feature-driven analyses, this would form an important finding, as it would directly indicate that resting state activity indeed is similar in its nature to that induced by stimulus-driven settings. Our findings indeed appear to show such an effect, as is illustrated in Figures 1-3. For example, the activation of visual motion responsive cortex V5+/MT+ clearly increased in experiments A-C. Note that this was not only reflected in the positive weights, but also in the negative ones; in complete consistency with the findings reported in [4] even the negative involvement of foveal visual representations with increase of visual motion 7 became amplified with the addition of resting state data. Similar findings concerned the cortical regions involved in the perception of language. However, this augmenting effect was not observed in all subjects for all features Figure 2 for example shows a subject in whom the human body feature obtained very high weights already in the most basic analysis, and no augmentation was apparent in the weight maps for the more complex analyses, perhaps reflecting a saturation effect. Since the resting state is not well-defined, it may also be that particular internal states, sleepiness, etc. would not guarantee augmenting in all datasets. All in all however our results show that adding resting state data can indeed augment findings obtained in stimulus-inducing settings. This method may therefore be useful for the increasing number of imaging centres acquiring resting state data for completely different purposes, which may then be used to augment functional data, entirely free of cost in terms of scan time. An even more promising prospect however is that also the baseline or rest condition within stimulus-driven sessions may be used to augment the results obtained in the stimulus conditions. This may be especially valuable, since almost all imaging sessions contain baseline conditions, that are often not used for further analysis, but take up considerable amount of scan time. Apart from the above, application-orientated considerations, our findings also provide new evidence that brain-states during rest which are difficult to characterize indeed resemble those during exposure to complex, natural stimulation. Our approach is therefore an extension of prior attempts to characterize the complex, rich, yet difficult to characterize brain activation during the absence of externally driven stimulation. 8 Conclusions In this work, we have proposed the use of resting state data as a source for the unlabeled component of semi-supervised learning for fMRI studies. Experimental results show that one of the primary assumptions of semi-supervised learning, the manifold assumption, holds well for this data, and that the marginal distribution of unlabeled resting state data is observed to augment that of labeled data to consistently improve regression performance. Semi-supervised Laplacian regularization is a widely applicable regularization technique that can be added to many kinds of machine learning algorithms. As we have shown that the basic assumptions of semi-supervised learning hold for this kind of data, we expect that this approach would work on these other discriminant/regression methods as well, including kernel logistic regression, support vector machines, and kernel canonical correlation analysis. As data acquisition and the manual labeling of stimulus data are expensive components of brain imaging, the benefits of exploiting additional unlabeled data are clear. Resting state data is a promising source as there are no task specific biases introduced. In future work we intend to further study the properties of the distribution of resting state activity. We also intend to pursue cross subject studies. If brain activity is consistent across subjects for the specific task measured by a study, a large cross subject sample of resting state data may be employed to improve results. Acknowledgments The first author is supported by the Royal Academy of Engineering through a Newton International Fellowship. The second author is supported by an ACM-W scholarship. References [1] A. Bartels and S. Zeki. The chronoarchitecture of the human brain–natural viewing conditions reveal a time-based anatomy of the brain. NeuroImage, 22(1):419 – 433, 2004. [2] A. Bartels and S. Zeki. Functional brain mapping during free viewing of natural scenes. Human Brain Mapping, 21(2):75–85, 02/01/ 2004. [3] A. Bartels and S. Zeki. Brain dynamics during natural viewing conditions–a new guide for mapping connectivity in vivo. NeuroImage, 24(2):339 – 349, 2005. [4] A. Bartels, S. Zeki, and N. K. Logothetis. Natural vision reveals regional specialization to local motion and to contrast-invariant, global flow in the human brain. Cereb. Cortex, pages bhm107+, July 2007. [5] M. Belkin and P. Niyogi. Semi-supervised learning on riemannian manifolds. Machine Learning, 56(13):209–239, 2004. [6] M. Belkin, P. Niyogi, and V. Sindhwani. Manifold Regularization: A Geometric Framework for Learning from Labeled and Unlabeled Examples. JMLR, 7:2399–2434, 2006. 8 [7] M. Bianciardi, M. Fukunaga, P. van Gelderen, S. G. Horovitz, J. A. de Zwart, and J. H. Duyn. Modulation of spontaneous fMRI activity in human visual cortex by behavioral state. NeuroImage, 45(1):160–168, 2009. [8] B. Biswal, Z. F. Yetkin, V. M. Haughton, and J. S. Hyde. Functional connectivity in the motor cortex of resting human brain using echo-planar mri. Magnetic Resonance in Medicine, 34(4):537–541, 1995. [9] B. B. Biswal, J. Van Kylen, and J. S. Hyde. Simultaneous assessment of flow and bold signals in restingstate functional connectivity maps. NMR Biomed, 10(4-5):165–170, 1997. [10] M. B. Blaschko, C. H. Lampert, and A. Gretton. Semi-supervised laplacian regularization of kernel canonical correlation analysis. In ECML PKDD ’08: Proceedings of the 2008 European Conference on Machine Learning and Knowledge Discovery in Databases I, pages 133–145. Springer-Verlag, 2008. [11] O. Chapelle, B. Sch¨olkopf, and A. Zien, editors. Semi-Supervised Learning. MIT Press, Cambridge, MA, 2006. [12] J. S. S. Damoiseaux, S. A. R. B. A. Rombouts, F. Barkhof, P. Scheltens, C. J. J. Stam, S. M. M. Smith, and C. F. F. Beckmann. Consistent resting-state networks across healthy subjects. Proc Natl Acad Sci U S A, August 2006. [13] P. E. Downing, Y. Jiang, M. Shuman, and N. Kanwisher. A Cortical Area Selective for Visual Processing of the Human Body. Science, 293(5539):2470–2473, 2001. [14] M. D. Fox, A. Z. Snyder, J. M. Zacks, and M. E. Raichle. Coherent spontaneous activity accounts for trial-to-trial variability in human evoked brain responses. Nature Neuroscience, 9(1):23–25, December 2005. [15] K. Friston, J. Ashburner, S. Kiebel, T. Nichols, and W. Penny, editors. Statistical Parametric Mapping: The Analysis of Functional Brain Images. Academic Press, 2007. [16] T. V. Gestel, J. A. K. Suykens, J. D. Brabanter, B. D. Moor, and J. Vandewalle. Kernel canonical correlation analysis and least squares support vector machines. In ICANN ’01: Proceedings of the International Conference on Artificial Neural Networks, pages 384–389, London, UK, 2001. Springer-Verlag. [17] Y. Golland, S. Bentin, H. Gelbard, Y. Benjamini, R. Heller, Y. Nir, U. Hasson, and R. Malach. Extrinsic and Intrinsic Systems in the Posterior Cortex of the Human Brain Revealed during Natural Sensory Stimulation. Cereb. Cortex, 17(4):766–777, 2007. [18] M. Greicius. Resting-state functional connectivity in neuropsychiatric disorders. Current opinion in neurology, 24(4):424–430, August 2008. [19] M. D. Greicius, K. Supekar, V. Menon, and R. F. Dougherty. Resting-State Functional Connectivity Reflects Structural Connectivity in the Default Mode Network. Cereb. Cortex, 19(1):72–78, 2009. [20] R. Gur, L. Mozley, P. Mozley, S. Resnick, J. Karp, A. Alavi, S. Arnold, and R. Gur. Sex differences in regional cerebral glucose metabolism during a resting state. Science, 267(5197):528–531, 1995. [21] M. Hampson, B. S. Peterson, P. Skudlarski, J. C. Gatenby, and J. C. Gore. Detection of functional connectivity using temporal correlations in mr images. Hum Brain Mapp, 15(4):247–262, April 2002. [22] D. R. Hardoon, J. Mour˜ao-Miranda, M. Brammer, and J. Shawe-Taylor. Unsupervised Analysis of fMRI Data Using Kernel Canonical Correlation. NeuroImage, 37(4):1250–1259, 2007. [23] U. Hasson, Y. Nir, I. Levy, G. Fuhrmann, and R. Malach. Intersubject Synchronization of Cortical Activity During Natural Vision. Science, 303(5664):1634–1640, 2004. [24] U. Hasson, E. Yang, I. Vallines, D. J. Heeger, and N. Rubin. A hierarchy of temporal receptive windows in human cortex. J. Neurosci., 28(10):2539–2550, March 2008. [25] T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, 2001. [26] J. V. Haxby, M. I. Gobbini, M. L. Furey, A. Ishai, J. L. Schouten, and P. Pietrini. Distributed and Overlapping Representations of Faces and Objects in Ventral Temporal Cortex. Science, 293(5539):2425–2430, 2001. [27] J.-D. Haynes and G. Rees. Decoding mental states from brain activity in humans. Nature Reviews Neuroscience, 7(7):523–534, July 2006. [28] C. Kelly, L. Uddin, B. Biswal, F. Castellanos, and M. Milham. Competition between functional brain networks mediates behavioral variability. NeuroImage, 39(1):527–537, January 2008. [29] S.-P. Ku, A. Gretton, J. Macke, and N. K. Logothetis. Comparison of pattern recognition methods in classifying high-resolution bold signals obtained at high magnetic field in monkeys. Magnetic Resonance Imaging, 26(7):1007 – 1014, 2008. [30] M. E. Raichle, A. M. MacLeod, A. Z. Snyder, W. J. Powers, D. A. Gusnard, and G. L. Shulman. A default mode of brain function. Proc Natl Acad Sci U S A, 98(2):676–682, January 2001. [31] M. E. E. Raichle and A. Z. Z. Snyder. A default mode of brain function: A brief history of an evolving idea. Neuroimage, 37(4):1083–1090, October 2007. [32] J. Shelton, M. Blaschko, and A. Bartels. Semi-supervised subspace analysis of human functional magnetic resonance imaging data. Technical Report 185, Max Planck Institute for Biological Cybernetics, 2009. [33] P. Skudlarski, K. Jagannathan, V. D. Calhoun, M. Hampson, B. A. Skudlarska, and G. Pearlson. Measuring brain connectivity: Diffusion tensor imaging validates resting state temporal correlations. NeuroImage, 43(3):554 – 561, 2008. [34] M. C. Stevens, G. D. Pearlson, and V. D. Calhoun. Changes in the interaction of resting-state neural networks from adolescence to adulthood. Human Brain Mapping, 8(30):2356–2366, 2009. [35] V. G. van de Ven, E. Formisano, D. Prvulovic, C. H. Roeder, and D. E. Linden. Functional connectivity as revealed by spatial independent component analysis of fmri measurements during rest. Hum Brain Mapp, 22(3):165–178, July 2004. [36] D. Zhou, O. Bousquet, T. N. Lal, J. Weston, and B. Sch¨olkopf. Learning with local and global consistency. In Advances in Neural Information Processing Systems 16, 2004. [37] X. Zhu and Z. Ghahramani. Learning from labeled and unlabeled data with label propagation. Technical Report CMU-CALD-02-107, Carnegie Mellon University, 2002. [38] X. Zhu, Z. Ghahramani, and J. Lafferty. Semi-supervised learning using gaussian fields and harmonic functions. In International Conference on Machine Learning, pages 912–919, 2003. 9
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Learning in Markov Random Fields using Tempered Transitions Ruslan Salakhutdinov Brain and Cognitive Sciences and CSAIL Massachusetts Institute of Technology rsalakhu@mit.edu Abstract Markov random fields (MRF’s), or undirected graphical models, provide a powerful framework for modeling complex dependencies among random variables. Maximum likelihood learning in MRF’s is hard due to the presence of the global normalizing constant. In this paper we consider a class of stochastic approximation algorithms of the Robbins-Monro type that use Markov chain Monte Carlo to do approximate maximum likelihood learning. We show that using MCMC operators based on tempered transitions enables the stochastic approximation algorithm to better explore highly multimodal distributions, which considerably improves parameter estimates in large, densely-connected MRF’s. Our results on MNIST and NORB datasets demonstrate that we can successfully learn good generative models of high-dimensional, richly structured data that perform well on digit and object recognition tasks. 1 Introduction Markov random fields (MRF’s) provide a powerful tool for representing dependency structure between random variables. They have been successfully used in various application domains, including machine learning, computer vision, and statistical physics. The major limitation of MRF’s is the need to compute the partition function, whose role is to normalize the joint distribution over the set of random variables. Maximum likelihood learning in MRF’s is often very difficult because of the hard inference problem induced by the partition function. When modeling high-dimensional, richly structured data, the inference problem becomes much more difficult because the distribution we need to infer is likely to be highly multimodal [17]. Multimodality is common in real-world distributions, such as the distribution of natural images, in which an exponentially large number of possible image configurations have extremely low probability, but there are many very different images that occur with similar probabilities. To date, there has been very little work addressing the problem of efficient learning in large, denselyconnected MRF’s that contain millions of parameters. While there exists a substantial literature on developing approximate learning algorithms for arbitrary MRF’s, many of these algorithms are unlikely to work well when dealing with high-dimensional inputs. Methods that are based on replacing the likelihood term with some tractable approximations, such as pseudo-likelihood [1] or mixtures of random spanning trees [11], perform very poorly for densely-connected MRF’s with strong dependency structures [3]. When using variational methods, such as loopy BP [18] and TRBP [16], learning often gets trapped in poor local optima [5, 13]. MCMC-based algorithms, including MCMC maximum likelihood estimators [3, 20] and Contrastive Divergence [4], typically suffer from high variance (or strong bias) in their estimates, and can sometimes be painfully slow. The main problem here is the inability of Markov chains to efficiently explore distributions with many isolated modes. 1 In this paper we concentrate on the class of stochastic approximation algorithms of the RobbinsMonro type that use MCMC to estimate the model’s expected sufficient statistics, needed for maximum likelihood learning. We first show that using this class of algorithms allows us to make very rapid progress towards finding a fairly good set of parameters, even for models containing millions of parameters. Second, we show that using MCMC operators based on tempered transitions [9] enables the stochastic algorithm to better explore highly multimodal distributions, which considerably improves parameter estimates, particularly in large, densely-connected MRF’s. Our results on the MNIST and NORB datasets demonstrate that the stochastic approximation algorithm together with tempered transitions can be successfully used to model high-dimensional real-world distributions. 2 Maximum Likelihood Learning in MRF’s Let x ∈X K be a random vector on K variables, where each xi takes on values in some discrete alphabet. Let φ(x) denote a D-dimensional vector of sufficient statistics, and let θ ∈RD be a vector of canonical parameters. The exponential family associated with sufficient statistics φ consists of the following parameterized set of probability distributions: p(x; θ) = p∗(x) Z(θ) = 1 Z(θ) exp (θ⊤φ(x)), Z(θ) = X x exp (θ⊤φ(x)), (1) where p∗(·) denotes the unnormalized probability distribution and Z(θ) is the partition function. For example, consider the following binary pairwise MRF. Given a graph G = (V, E) with vertices V and edges E, the probability distribution over a binary random vector x ∈{0, 1}K is given by: p(x; θ) = 1 Z(θ) exp θ⊤φ(x) = 1 Z(θ) exp X (i,j)∈E θijxixj + X i∈V θixi . (2) The derivative of the log-likelihood for an observation x0 with respect to parameter vector θ can be obtained from Eq. 1: ∂log p(x0; θ) ∂θ = φ(x0) −Ep(x;θ)[φ(x)], (3) where EP [·] denotes an expectation with respect to distribution P. Except for simple models such as the tree structured graphs exact maximum likelihood learning is intractable, because exact computation of the expectation Ep(x;θ)[·] takes time that is exponential in the treewidth of the graph1. One approach is to learn model parameters by maximizing the pseudo-likelihood (PL) [1], which replaces the likelihood with a tractable product of conditional probabilities: PPL(x0; θ) = K Y k=1 p(xk|x0,−k; θ), (4) where x0,−k denotes an observation vector x0 with xk omitted. Pseudo-likelihood provides good estimates for weak dependence, when p(xk|x−k) ≈p(xk), or when it well approximates the true likelihood function. For MRF’s with strong dependence structure, it is unlikely to work well. Another approach, called the MCMC maximum likelihood estimator (MCMC-MLE) [3], has been shown to sometimes provide considerably better results than PL [3, 20]. The key idea is to use importance sampling to approximate the model’s partition function. Consider running a Markov chain to obtain samples x(1), x(2), ..., x(n) from some fixed proposal distribution p(x; ψ)2. These samples can be used to approximate the log-likelihood ratio for an observation x0: L(θ) = log p(x0; θ) p(x0; ψ) = (θ −ψ)⊤φ(x0) −log Z(θ) Z(ψ) (5) ≈ (θ −ψ)⊤φ(x0) −log 1 n n X i=1 e(θ−ψ)⊤φ(x(i)) = Ln(θ), (6) 1For many interesting models considered in this paper exact computation of Ep(x;θ)[·] takes time that is exponential in the dimensionality of x. 2We will also assume that p(x; ψ) ̸= 0 whenever p(x; θ) ̸= 0, ∀θ. 2 Algorithm 1 Stochastic Approximation Procedure. 1: Given an observation x0. Randomly initialize θ1 and M sample particles {x1,1, ...., x1,M}. 2: for t = 1 : T (number of iterations) do 3: for m = 1 : M (number of parallel Markov chains) do 4: Sample xt+1,m given xt,m using transition operator Tθt(xt+1,m ←xt,m). 5: end for 6: Update: θt+1 = θt + αt h φ(x0) − 1 M PM m=1 φ(xt+1,m) i . 7: Decrease αt. 8: end for where we used the approximation: Z(θ) Z(ψ) = P x e(θ−ψ)⊤φ(x)p(x; ψ) ≈1 n Pn i=1 e(θ−ψ)⊤φ(x(i)). Provided our Markov chain is ergodic, it can be shown that Ln(θ) −→L(θ) for all θ. It can further be shown that, under the “usual” regularity conditions, if ˆθn maximizes Ln(θ) and θ∗maximizes L(θ), then ˆθn a.s. −−→θ∗. This implies that as the number of samples n, drawn from our proposal distributions, goes to infinity, MCMC-MLE will converge to the true maximum likelihood estimator. While this estimator provides nice asymptotic convergence guarantees, it performs very poorly in practice, particularly when the parameter vector θ is high-dimensional. In high-dimensional spaces, the variance of an estimator Ln(θ) will be very large, or possibly infinite, unless the proposal distribution p(x; ψ) is a near-perfect approximation to p(x; θ). While there have been some attempts to improve MCMC-MLE by considering a mixture of proposal distributions [20], they do not fix the problem when learning MRF’s with millions of parameters. 3 Stochastic Approximation Procedure (SAP) We now consider a stochastic approximation procedure that uses MCMC to estimate the model’s expected sufficient statistics. SAP belongs to the general class of well-studied stochastic approximation algorithms of the Robbins-Monro type [19, 12]. The algorithm itself dates back to 1988 [19], but only recently it has been shown to work surprisingly well when training large MRF’s, including restricted Boltzmann machines [15] and deep Boltzmann machines [14, 13]. The idea behind learning a parameter vector θ using SAP is straightforward. Let x0 be our observation. Then the state and the parameters are updated sequentially: θt+1 = θt + αt φ(x0) −φ(xt+1) , where xt+1 ∼Tθt(xt+1 ←xt). (7) Given xt, we sample a new state xt+1 using the transition operator Tθt(xt+1 ←xt) that leaves p(·; θt) invariant. A new parameter θt+1 is then obtained by replacing the intractable expectation Ep(x;θt)[φ(x)] with φ(xt+1). In practice, we typically maintain a set of M sample points Xt = {xt,1, ...., xt,M}, which we will often refer to as sample particles. In this case, the intractable model’s expectation is replaced by the sample average 1/M PM m=1 φ(xt+1,m). The procedure is summarized in Algorithm 1. One important property of this algorithm is that just like MCMC-MLE, it can be shown to asymptotically converge to the maximum likelihood estimator θ∗.3 In particular, for fully visible discrete MRF’s, if one uses a Gibbs transition operator and the learning rate is set to αt = 1 (t+1)U , where U is a positive constant, such that U > 2KC0C1, then θt a.s. −−→θ∗(see Theorem 4.1 of [19]). Here K is the dimensionality of x, C0 = max{||φ(x0) −φ(x)||; x ∈X K} is the largest magnitude of the gradient, and C1 is the maximum variation of φ when one changes the values of a single component only: C1 = max{||φ(x) −φ(y)||; x, y ∈X K, k ∈{1, ..., K}, y−k = x−k}. The proof of convergence relies on the following simple decomposition. First, let S(θ) denote the true gradient of the log-likelihood function: S(θ) = ∂log p(x0;θ) ∂θ = φ(x0) −Ep(x;θ)[φ(x)]. The parameter update rule then takes the following form: θt+1 = θt + αt φ(x0) −φ(xt+1) = θt + αtS(θt) + αt Ep(x;θ)[φ(x)] −φ(xt+1) = θt + αtS(θt) + αtǫt. (8) 3One necessary condition for almost sure convergence requires the learning rate to decrease with time, so that P∞ t=0 αt = ∞and P∞ t=0 α2 t < ∞. 3 Algorithm 2 Tempered Transitions Run. 1: Initialize β0 < β1 < ... < βS = 1. Given a current state xS. 2: for s = S −1 : 0 (Forward pass) do 3: Sample xs given xs+1 using Ts(xs ←xs+1). 4: end for 5: Set ˜x0 = x0. 6: for s = 0 : S −1 (Backward pass) do 7: Sample ˜xs+1 given ˜xs using eTs(˜xs+1 ←˜xs). 8: end for 9: Accept a new state ˜xS with probability: min h 1, QS s=1 p∗(xs)βs−1−βs p∗(˜xs)βs−βs−1 i . The first term (rhs. of Eq. 8) is the discretization of the ordinary differential equation ˙θ = S(θ). The algorithm is therefore a perturbation of this discretization with the noise term ǫt. The proof proceeds by showing that the noise term is not too large. Intuitively, as the learning rate becomes sufficiently small compared to the mixing rate of the Markov chain, the chain will stay close to the stationary distribution, even if it is only run for a few MCMC steps per parameter update. This, in turn, will ensure that the noise term ǫt goes to zero. When looking at the behavior of this algorithm in practice, we find that initially it makes very rapid progress towards finding a sensible region in the parameter space. However, as the algorithm begins to capture the multimodality of the data distribution, the Markov chain tends to mix poorly, producing highly correlated samples for successive parameter updates. This often leads to poor parameter estimates, especially when modeling complex, high-dimensional distributions. The main problem here is the inability of the Markov chain to efficiently explore a distribution with many isolated modes. However, the transition operators Tθt(xt+1 ←xt) used in the stochastic approximation algorithm do not necessarily need to be simple Gibbs or Metropolis-Hastings updates to guarantee almost sure convergence. Instead, we propose to use MCMC operators based on tempered transitions [9] that can more efficiently explore highly multimodal distributions. In addition, implementing tempered transitions requires very little extra work beyond the implementation of the Gibbs sampler. 3.1 Tempered Transitions Suppose that our goal is to sample from p(x; θ). We first define a sequence of intermediate probability distributions: p0, ..., pS, with pS = p(x; θ) and p0 being more spread out and easier to sample from than pS. Constructing a suitable sequence of intermediate probability distributions will in general depend on the problem. One general way to define this sequence is: ps(x) ∝ p∗(x; θ)βs, (9) with “inverse temperatures” β0 < β1 < ... < βS = 1 chosen by the user. For each s = 1, .., S−1 we define a transition operator Ts(x′ ←x) that leaves ps invariant. In our implementation Ts(x′ ←x) is the Gibbs sampling operator. We also need to define a reverse transition operator eTs(x←x′) that satisfies the following reversibility condition for all x and x′: ps(x)Ts(x′ ←x) = eTs(x←x′)ps(x′). (10) If Ts is reversible, then eTs is the same as Ts. Many commonly used transition operators, such as Metropolis–Hastings, are reversible. Non-reversible operators are usually composed of several reversible sub-transitions applied in sequence Ts = Q1...QK, such as the single component updates in a Gibbs sampler. The reverse operator can be simply constructed from the same sub-transitions, but applied in the reverse order eTs = QK...Q1. Given the current state x of the Markov chain, tempered transitions apply a sequence of transition operators TS−1 . . . T0 eT0 . . . eTS−1 that systematically “move” the sample particle x from the original complex distribution to the easily sampled distribution, and then back to the original distribution. A new candidate state ˜x is accepted or rejected based on ratios of probabilities of intermediate states. Since p0 is less concentrated than pS, the sample particle will have a chance to move around the state space more easily, and we may hope that the probability distribution of the resulting candidate 4 0 10e2 10e3 10e4 −600 −500 −400 −300 −200 −100 Number of Gibbs Updates (log−scale) Log−probability Stochastic Approximation Exact Maximum Likelihood 50 60 70 80 90 100 −160 −159 −158 −157 −156 −155 −154 Number of Gibbs Updates (× 1000 ) Log−probability Stochastic Approximation Maximum Likelihood Tempered Transitions 0 10e2 10e3 10e4 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Number of Gibbs Updates (log−scale) % correctly classified Stochastic Approximation 100 120 140 160 180 200 0.97 0.972 0.974 0.976 0.978 0.98 0.982 Number of Gibbs Updates (× 1000 ) % correctly classified Stochastic Approximation Tempered Transitions x h x h label Restricted Boltzmann Machine Semi-restricted Boltzmann Machine Figure 1: Experimental results on MNIST dataset. Top: Toy RBM with 10 hidden units. The x-axis show the number of Gibbs updates and the y-axis displays the training log-probability in nats. Bottom: Classification performance of the semi-restricted Boltzmann machines with 500 hidden units on the full MNIST datasets. state will be much broader than the mode in which the current start state resides. The procedure is shown in Algorithm 2. Note that there is no need to compute the normalizing constants of any intermediate distributions. Tempered transitions can make major changes to the current state, which allows the Markov chain to produce less correlated samples between successive parameter updates. This can greatly improve the accuracy of the estimator, but is also more computationally expensive. We therefore propose to alternate between applying a more expensive tempered transitions operator and the standard Gibbs updates. We call this algorithm Trans-SAP. 4 Experimental Results In our experiments we used the MNIST and NORB datasets. To speed-up learning, we subdivided datasets into minibatches, each containing 100 training cases, and updated the parameters after each minibatch. The number of sample particles used for estimating the model’s expected sufficient statistics was also set to 100. For the stochastic approximation algorithm, we always apply a single Gibbs update to the sample particles. In all experiments, the learning rates were set by quickly running a few preliminary experiments and picking the learning rates that worked best on the validation set. We also use natural logarithms, providing values in nats. 4.1 MNIST The MNIST digit dataset contains 60,000 training and 10,000 test images of ten handwritten digits (0 to 9), with 28×28 pixels. The dataset was binarized: each pixel value was stochastically set to 1 with probability proportional to its pixel intensity. From the training data, a random sample of 10,000 images was set aside for validation. In our first experiment we trained a small restricted Boltzmann machine (RBM). An RBM is a particular type of Markov random field that has a two-layer architecture, in which the visible binary stochastic units x are connected to hidden binary stochastic units h, as shown in Fig. 1. The probability that the model assigns to a visible vector x is: P(x; θ) = 1 Z(θ) X h exp X i,j θijxihj + X i θixi + X j θjhj . (11) 5 Samples before Tempered Transitions Samples after Tempered Transitions Model Samples Figure 2: Left: Sample particles produced by the stochastic approximation algorithm after 100,000 parameter updates. Middle: Sample particles after applying a tempered transitions run. Right: Samples generated from the current model by randomly initializing all binary states and running the Gibbs sampler for 500,000 steps. After applying tempered transitions, sample particles look more like the samples generated from the current model. The images shown are the probabilities of the visible units given the binary states of the hidden units. The model had 10 hidden units. This allowed us to calculate the exact value of the partition function simply by summing out the 784 visible units for each configuration of the hiddens. For the stochastic approximation procedure, the total number of parameter updates was 100,000, so the learning took about 25.6 minutes on a Pentium 4 3.00GHz machine. The learning rate was kept fixed at 0.01 for the first 10,000 parameter updates, and was then annealed as 10/(1000+t). For comparison, we also trained the same model using exact maximum likelihood with exactly the same learning schedule. Perhaps surprisingly, SAP makes very rapid progress towards the maximum likelihood solution, even though the model contains 8634 free parameters. The top panel of Fig. 1 further shows that combining regular Gibbs updates with tempered transitions provides a more accurate estimator. We applied tempered transitions only during the last 50,000 Gibbs steps, alternating between 200 Gibbs updates and a single tempered transitions run that used 50 β’s spaced uniformly from 1 to 0.9. The acceptance rate for the tempered transitions was about 0.8. To be fair, we compared different algorithms based on the total number of Gibbs steps. For SAP, parameters were updated after each Gibbs step (see Algorithm 1), whereas for Trans-SAP, parameters were updated after each Gibbs update but not during the tempered transitions run4. Hence Trans-SAP took slightly less computer time compared to the plain SAP. Pseudo-likelihood and MCMC maximum likelihood estimators perform quite poorly, even for this small toy problem. In our second experiment, we trained a larger semi-restricted Boltzmann machine that contained 705,622 parameters. In contrast to RBM’s, the visible units in this model form a fully connected pairwise binary MRF (see Fig. 1, bottom left panel). The model had 500 hidden units and was trained to model the joint probability distribution over the digit images and labels. The total number of Gibbs updates was set to 200,000, so the learning took about 19.5 hours. The learning rate was kept fixed at 0.05 for the first 50,000 parameter updates, and was then decreased as 100/(2000 + t). The bottom panel of Fig. 1 shows classification performance on the full MNIST test set. As expected, SAP makes very rapid progress towards finding a good setting of the parameter values. Using tempered transitions further improves classification performance. As in our previous experiment, tempered transitions were only applied during the last 100,000 Gibbs updates, alternating between 1000 Gibbs updates and a single tempered transitions run that used 500 β’s spaced uniformly from 1 to 0.9. The acceptance rate was about 0.7. After learning was complete, in addition to classification performance, we also estimated the log-probability that both models assigned to the test data. To estimate the models’ partition functions, we used Annealed Importance Sampling [10, 13] – a technique that is very similar to tempered transitions. The plain stochastic approximation algorithm achieved an average test log-probability of -87.12 per image, whereas Trans-SAP achieved a considerably better average test log-probability of -85.91. 4This reduced the total number of parameter updates from 100, 000 to 50, 000 + 50, 000 ∗2/3 = 83, 333. 6 Training Samples Model trained with Tempered Transitions Model trained without Tempered Transitions Figure 3: Results on the NORB dataset. Left: Random samples from the training set. Samples generated from the two RBM models, trained using SAP with (Middle) and without (Right) tempered transitions. Samples were generated by running the Gibbs sampler for 100,000 steps. To get an intuitive picture of how tempered transitions operate, we looked at the sample particles before and after applying a tempered transitions run. Figure 2 shows sample particles after 100,000 parameter updates. Observe that the particles look like the real handwritten digits. However, a run of tempered transitions reveals that the current model is very unbalanced, with more probability mass placed on images of four. To further test whether the “refreshed” particles were representative of the current model, we generated samples from the current model by randomly initializing binary states of the visible and hidden units, and running the Gibbs sampler for 500,000 steps. Clearly, the refreshed particles look more like the samples generated from the true model. This in turn allows Trans-SAP to better estimate the model’s expected sufficient statistics, which greatly facilitates learning a better generative model. 4.2 NORB Results on MNIST show that the stochastic approximation algorithm works well on the relatively simple task of handwritten digit recognition. In this section we present results on a considerably more difficult dataset. NORB [6] contains images of 50 different 3D toy objects with 10 objects in each of five generic classes: planes, cars, trucks, animals, and humans. The training set contains 24,300 stereo image pairs of 25 objects, whereas the test set contains 24,300 stereo pairs of the remaining, different 25 objects. The goal is to classify each object into its generic class. From the training data, 4,300 cases were set aside for validation. Each image has 96×96 pixels with integer greyscale values in the range [0,255]. We further reduced the dimensionality of each image from 9216 down to 4488 by using larger pixels around the edges of the image5. We also augmented the training data with additional unlabeled data by applying simple pixel translations, creating a total of 1,166,400 training instances. To deal with raw pixel data, we followed the approach of [8] by first learning a Gaussian-binary RBM with 4000 hidden units, and then treating the the activities of its hidden layer as “preprocessed” data. The model was trained using contrastive divergence learning for 500 epochs. The learned low-level RBM effectively acts as a preprocessor that transforms greyscale images into 4000-dimensional binary vectors, which we use as the input for training our models. We proceeded to training an RBM with 4000 hidden units using binary representations learned by the preprocessor module6. The RBM, containing over 16 million parameters, was trained in a completely unsupervised way. The total number of Gibbs updates was set to 400,000. The learning rate was kept fixed at 0.01 for the first 100,000 parameter updates, and was then annealed as 100/(1000 + t). Similar to the previous experiments, tempered transitions were applied during the last 200,000 Gibbs updates, alternating between 1000 Gibbs updates and a single tempered transitions run that used 1000 β’s spaced uniformly from 1 to 0.9. 5The dimensionality of each training vector, representing a stereo pair, was 2×4488 = 8976. 6The resulting model is effectively a Deep Belief Network with two hidden layers. 7 Figure 3 shows samples generated from two models, trained using stochastic approximation with and without tempered transitions. Both models were able to learn a lot of regularities in this highdimensional, highly-structured data, including various object classes, different viewpoints and lighting conditions. The plain stochastic approximation algorithm produced a very unbalanced model with a large fraction of the model’s probability mass placed on images of humans. Using tempered transitions allowed us to learn a better and more balanced generative model, including the lighting effects. Indeed, the plain SAP achieved a test log-probability of -611.08 per image, whereas Trans-SAP achieved a test log-probability of -598.58. We also tested the classification performance of both models simply by fitting a logistic regression model to the labeled data (using only the 24300 labeled training examples without any translations) using the top-level hidden activities as inputs. The model trained by SAP achieved an error rate of 8.7%, whereas the model trained using Trans-SAP reduced the error rate down to 8.4%. This is compared to 11.6% achieved by SVM’s, 22.5% achieved by logistic regression applied directly in the pixel space, and 18.4% achieved by K-nearest neighbors [6]. 5 Conclusions We have presented a class of stochastic approximation algorithms of the Robbins-Monro type that can be used to efficiently learn parameters in large densely-connected MRF’s. Using MCMC operators based on tempered transitions allows the stochastic approximation algorithm to better explore highly multimodal distributions, which in turn allows us to learn good generative models of handwritten digits and 3D objects in a reasonable amount of computer time. In this paper we have concentrated only on using tempered transition operators. There exist a variety of other methods for sampling from distributions with many isolated modes, including simulated tempering [7] and parallel tempering [3], all of which can be incorporated into SAP. In particular, the concurrent work of [2] employs parallel tempering techniques to imrpove mixing in RBM’s. There are, however, several advantages of using tempered transitions over other existing methods. First, tempered transitions do not require specifying any extra variables, such as the approximate values of normalizing constants of intermediate distributions, which are needed for the simulated tempering method. Second, tempered transitions have modest memory requirements, unlike, for example, parallel tempering, since the acceptance rule can be computed on the fly as the intermediate states are generated. Finally, the implementation of tempered transitions requires almost no extra work beyond implementing the Gibbs sampler, and can be easily integrated into existing code. Acknowledgments We thank Vinod Nair for sharing his code for blurring and translating NORB images. This research was supported by NSERC. References [1] J. Besag. Efficiency of pseudolikelihood estimation for simple Gaussian fields. Biometrica, 64:616–618, 1977. [2] G. Desjardins, A. Courville, Y. Bengio, P. Vincent, and O. Delalleau. Tempered Markov chain Monte Carlo for training of restricted Boltzmann machines. Technical Report 1345, University of Montreal, 2009. [3] C. Geyer. Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics, pages 156–163, 1991. [4] G. Hinton. Training products of experts by minimizing contrastive divergence. Neural Computation, 14(8):1711–1800, 2002. [5] A. Kulesza and F. Pereira. Structured learning with approximate inference. In NIPS, 2007. [6] Y. LeCun, F. J. Huang, and L. Bottou. Learning methods for generic object recognition with invariance to pose and lighting. In CVPR (2), pages 97–104, 2004. [7] E. Marinari and G. Parisi. Simulated tempering: A new Monte Carlo scheme. Europhysics Letters, 19:451–458, 1992. 8 [8] V. Nair and G. Hinton. Implicit mixtures of restricted Boltzmann machines. In Advances in Neural Information Processing Systems, volume 21, 2009. [9] R. Neal. Sampling from multimodal distributions using tempered transitions. Statistics and Computing, 6:353–366, 1996. [10] R. Neal. Annealed importance sampling. Statistics and Computing, 11:125–139, 2001. [11] P. Pletscher, C. Ong, and J. Buhmann. Spanning tree approximations for conditional random fields. In Proceedings of the International Conference on Artificial Intelligence and Statistics, volume 5, 2009. [12] H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Stat., 22:400–407, 1951. [13] R. Salakhutdinov. Learning and evaluating Boltzmann machines. Technical Report UTML TR 2008-002, Department of Computer Science, University of Toronto, 2008. [14] R. Salakhutdinov and G. Hinton. Deep Boltzmann machines. In Proceedings of the International Conference on Artificial Intelligence and Statistics, volume 5, pages 448–455, 2009. [15] T. Tieleman. Training restricted Boltzmann machines using approximations to the likelihood gradient. In Machine Learning, Proceedings of the Twenty-first International Conference (ICML 2008). ACM, 2008. [16] M. Wainwright, T. Jaakkola, and A. Willsky. Tree-reweighted belief propagation algorithms and approximate ML estimation by pseudo-moment matching. In AI and Statistics, volume 9, 2003. [17] M. Welling and C. Sutton. Learning in Markov random fields with Contrastive Free Energies. In Proceedings of the International Conference on Artificial Intelligence and Statistics, volume 10, 2005. [18] J. S. Yedidia, W. T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Transactions on Information Theory, 51(7):2282–2312, 2005. [19] L. Younes. Estimation and annealing for Gibbsian fields. Ann. Inst. Henri Poincar´e (B), 24(2):269–294, 1988. [20] S. Zhu and X. Liu. Learning in Gibbsian fields: How accurate and how fast can it be? In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR00), pages 2–9. IEEE, 2000. 9
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Statistical Consistency of Top-k Ranking Fen Xia Institute of Automation Chinese Academy of Sciences fen.xia@ia.ac.cn Tie-Yan Liu Microsoft Research Asia tyliu@microsoft.com Hang Li Microsoft Research Asia hanglig@microsoft.com Abstract This paper is concerned with the consistency analysis on listwise ranking methods. Among various ranking methods, the listwise methods have competitive performances on benchmark datasets and are regarded as one of the state-of-the-art approaches. Most listwise ranking methods manage to optimize ranking on the whole list (permutation) of objects, however, in practical applications such as information retrieval, correct ranking at the top k positions is much more important. This paper aims to analyze whether existing listwise ranking methods are statistically consistent in the top-k setting. For this purpose, we define a top-k ranking framework, where the true loss (and thus the risks) are defined on the basis of top-k subgroup of permutations. This framework can include the permutationlevel ranking framework proposed in previous work as a special case. Based on the new framework, we derive sufficient conditions for a listwise ranking method to be consistent with the top-k true loss, and show an effective way of modifying the surrogate loss functions in existing methods to satisfy these conditions. Experimental results show that after the modifications, the methods can work significantly better than their original versions. 1 Introduction Ranking is the central problem in many applications including information retrieval (IR). In recent years, machine learning technologies have been successfully applied to ranking, and many learning to rank methods have been proposed, including the pointwise [12] [9] [6], pairwise [8] [7] [2], and listwise methods [13] [3] [16]. Empirical results on benchmark datasets have demonstrated that the listwise ranking methods have very competitive ranking performances [10]. To explain the high ranking performances of the listwise ranking methods, a theoretical framework was proposed in [16]. In the framework, existing listwise ranking methods are interpreted as making use of different surrogate loss functions of the permutation-level 0-1 loss. Theoretical analysis shows that these surrogate loss functions are all statistically consistent in the sense that minimization of the conditional expectation of them will lead to obtaining the Bayes ranker, i.e., the optimal ranked list of the objects. Here we point out that there is a gap between the analysis in [16] and many real ranking problems, where the correct ranking of the entire permutation is not needed. For example, in IR, users usually care much more about the top ranking results and thus only correct ranking at the top positions is important. In this new situation, it is no longer clear whether existing listwise ranking methods are still statistically consistent. The motivation of this work is to perform formal study on the issue. For this purpose, we propose a new ranking framework, in which the “true loss” is defined on the top-k subgroup of permutations instead of on the entire permutation. The new true loss only measures errors occurring at the top k positions of a ranked list, therefore we refer to it as the top-k true loss (Note that when k equals the length of the ranked list, the top-k true loss will become exactly 1 the permutation-level 0-1 loss). We prove a new theorem which gives sufficient conditions for a surrogate loss function to be consistent with the top-k true loss. We also investigate the change of the conditions with respect to different k’s. Our analysis shows that, as k decreases, to guarantee the consistency of a surrogate loss function, the requirement on the probability space becomes weaker while the requirement on the surrogate loss function itself becomes stronger. As a result, a surrogate loss function that is consistent with the permutation-level 0-1 loss might not be consistent with the top-k true loss any more. Therefore, the surrogate loss functions in existing listwise ranking methods, which have been proved to be consistent with the permutation-level 0-1 loss, are not theoretically guaranteed to have good performances in the top-k setting. Modifications to these surrogate loss functions are needed to further make them consistent with the top-k true loss. We show how to make such modifications, and empirically verify that such modifications can lead to significant performance improvement. This validates the correctness of our theoretical analysis. 2 Permutation-level ranking framework We review the permutation-level ranking framework proposed in [16]. Let X be the input space whose elements are groups of objects to be ranked, Y be the output space whose elements are permutations of objects, and PXY be an unknown but fixed joint probability distribution of X and Y . Let h ∈H : X →Y be a ranking function. Let x ∈X and y ∈Y , and let y(i) be the index of the object that is ranked at position i in y. The task of learning to rank is to learn a function that can minimize the expected risk R(h), defined as, R(h) = Z X×Y l(h(x), y)dP(x, y), (1) where l(h(x), y) is the true loss such that l(h(x), y) = 1, if h(x) ̸= y 0, if h(x) = y. (2) The above true loss indicates that if the permutation of the predicted result is exactly the same as the permutation in the ground truth, then the loss is zero; otherwise the loss is one. For ease of reference, we call it permutation-level 0-1 loss. The optimal ranking function which can minimize the expected true risk R(h∗) = inf R(h) is referred to as the permutation-level Bayes ranker. h∗(x) = arg max y∈Y P(y|x). (3) In practice, for efficiency consideration, the ranking function is usually defined as h(x) = sort(g(x1), . . . , g(xn)), where g(·) denotes the scoring function, and sort(·) denotes the sorting function. Since the risk is non-continuous and non-differentiable with respect to the scoring function g, a continuous and differentiable surrogate loss function φ(g(x), y) is usually used as an approximation of the true loss. In this way, the expected risk becomes Rφ(g) = Z X×Y φ(g(x), y)dP(x, y), (4) where g(x) = (g(x1), . . . , g(xn)) is a vector-valued function induced by g. It has been shown in [16] that many existing listwise ranking methods fall into the above framework, with different surrogate loss functions used. Furthermore, their surrogate loss functions are statistically consistent under certain conditions with respect to the permutation-level 0-1 loss. However, as shown in the next section, the permutation-level 0-1 loss is not suitable to describe the ranking problem in many real applications. 3 Top-k ranking framework We next describe the real ranking problem, and then propose the top-k ranking framework. 2 3.1 Top-k ranking problem In real ranking applications like IR, people pay more attention to the top-ranked objects. Therefore the correct ranking on the top positions is critically important. For example, modern web search engines only return top 1, 000 results and 10 results in each page. According to a user study1, 62% of search engine users only click on the results within the first page, and 90% of users click on the results within the first three pages. It means that two ranked lists of documents will likely provide the same experience to the users (and thus suffer the same loss), if they have the same ranking results for the top positions. This, however, cannot be reflected in the permutation-level 0-1 loss in Eq.(2). This characteristic of ranking problems has also been explored in earlier studies in different settings [4, 5, 14]. We refer to it as the top-k ranking problem. 3.2 Top-k true loss To better describe the top-k ranking problem, we propose defining the true loss based on the top k positions in a ranked list, referred to as the top-k true loss. lk(h(x), y) = 0, if ˆy(i) = y(i) ∀i ∈{1, . . . , k}, where ˆy = h(x), 1, otherwise . (5) The actual value of k is determined by application. When k equals the length of the entire ranked list, the top-k true loss will become exactly the permutation-level 0-1 loss. In this regard, the top-k true loss is more general than the permutation-level 0-1 loss. With Eq.(5), the expected risk becomes Rk(h) = Z X×Y lk(h(x), y)dP(x, y). (6) It can be proved that the optimal ranking function with respect to the top-k true loss (i.e., the top-k Bayes ranker) is any permutation in the top-k subgroup having the highest probability2, i.e., h∗ k(x) ∈arg maxGk(j1,j2,...,jk)∈GkP(Gk(j1, j2, ..., jk)|x), (7) where Gk(j1, j2, ..., jk) = {y ∈Y |y(t) = jt, ∀t = 1, 2, . . . k} denotes a top-k subgroup in which all the permutations have the same top-k true loss; Gk denotes the collection of all top-k subgroups. With the above setting, we will analyze the consistency of the surrogate loss functions in existing ranking methods with the top-k true loss in the next section. 4 Theoretical analysis In this section, we first give the sufficient conditions of consistency for the top-k ranking problem. Next, we show how these conditions change with respect to k. Last, we discuss whether the surrogate loss functions in existing methods are consistent, and how to make them consistent if not. 4.1 Statistical consistency We investigate what kinds of surrogate loss functions φ(g(x), y) are statistically consistent with the top-k true loss. For this purpose, we study whether the ranking function that minimizes the conditional expectation of the surrogate loss function defined as follows coincides with the top-k Bayes ranker as defined in Eq.(7). Q(P(y|x), g(x)) = X y∈Y P(y|x)φ(g(x), y). (8) 1iProspect Search Engine User Behavior Study, April 2006, http://www.iprospect.com/ 2Note that the probability of a top-k subgroup is defined as the sum of the probabilities of the permutations in the subgroup (cf., Definitions 6 and 7 in [3]). 3 According to [1], the above condition is the weakest condition to guarantee that optimizing a surrogate loss function will lead to obtaining a model achieving the Bayes risk (in our case, the top-k Bayes ranker), when the training sample size approaches infinity. We denote Q(P(y|x), g(x)) as Q(p, g), g(x) as g and P(y|x) as py. Hence, Q(p, g) is the loss of g at x with respect to the conditional probability distribution py. The key idea is to decompose the sorting of g into pairwise relationship between scores of objects. To this end, we denote Yi,j as a permutation set in which each permutation ranks object i before object j, i.e., Yi,j ≜{y ∈Y : y−1(i) < y−1(j)} (here y−1(j) denotes the position of object j in permutation y), and introduce the following definitions. Definition 1. ΛGk is the a top-k subgroup probability space, such that ΛGk ≜{p ∈R|Gk| : P Gk(j1,j2,...,jk)∈Gk pGk(j1,j2,...,jk) = 1, pGk(j1,j2,...,jk) ≥0}. Definition 2. A top-k subgroup probability space ΛGk is order preserving with respect to objects i and j, if ∀y ∈Yi,j and Gk(y(1), y(2), ..., y(k)) ̸= Gk(σ−1 i,j y(1), σ−1 i,j y(2), ..., σ−1 i,j y(k)), we have pGk(y(1),y(2),...,y(k)) > pGk(σ−1 i,j y(1),σ−1 i,j y(2),...,σ−1 i,j y(k)). Here σ−1 i,j y denotes the permutation in which the positions of objects i and j are exchanged while those of the other objects remain the same as in y. Definition 3. A surrogate loss function φ is top-k subgroup order sensitive on a set Ω⊂Rn, if φ is a non-negative differentiable function and the following three conditions hold for ∀objects i and j: (1) φ(g, y) = φ(σ−1 i,j g, σ−1 i,j y); (2)Assume gi < gj, ∀y ∈Yi,j. If Gk(y(1), y(2), ..., y(k)) ̸= Gk(σ−1 i,j y(1), σ−1 i,j y(2), ..., σ−1 i,j y(k)), then φ(g, y) ≥φ(g, σ−1 i,j y) and for at least one y, the strict inequality holds; otherwise, φ(g, y) = φ(g, σ−1 i,j y). (3) Assume gi = gj. ∃y ∈Yi,j with Gk(y(1), y(2), ..., y(k)) ̸= Gk(σ−1 i,j y(1), σ−1 i,j y(2), ..., σ−1 i,j y(k)) satisfying ∂φ(g,σ−1 i,j y) ∂gi > ∂φ(g,y) ∂gi . The order preserving property of a top-k subgroup probability space (see Definition 2) indicates that if the top-k subgroup probability on a permutation y ∈Yi,j is larger than that on permutation σ−1 i,j y, then the relation holds for any other permutation y′ in Yi,j and and the corresponding σ−1 i,j y′ provided that the top-k subgroup of the former is different from that of the latter. The order sensitive property of a surrogate loss function (see Definition 3) indicates that (i) φ(g, y) exhibits a symmetry in the sense that simultaneously exchanging the positions of objects i and j in the ground truth and their scores in the predicted score list will not make the surrogate loss change. (ii) When a permutation is transformed to another permutation by exchanging the positions of two objects of it, if the two permutations do not belong to the same top-k subgroup, the loss on the permutation that ranks the two objects in the decreasing order of their scores will not be greater than the loss on its counterpart. (iii) There exists a permutation, for which the speed of change in loss with respect to the score of an object will become faster if exchanging its position with another object with the same score but ranked lower. A top-k subgroup order sensitive surrogate loss function has several nice properties as shown below. Proposition 4. Let φ(g, y) be a top-k subgroup order sensitive loss function. ∀y, ∀π ∈ Gk(y(1), y(2), . . . , y(k)), we have φ(g, π) = φ(g, y). Proposition 5. Let φ(g, y) be a top-k subgroup order sensitive surrogate loss function. ∀objects i and j with gi = gj, ∀y ∈Yi,j, if Gk(y(1), y(2), ..., y(k)) ̸= Gk(σ−1 i,j y(1), σ−1 i,j y(2), ..., σ−1 i,j y(k)), then ∂φ(g,σ−1 i,j y) ∂gi ≥∂φ(g,y) ∂gi . Otherwise, ∂φ(g,σ−1 i,j y) ∂gi = ∂φ(g,y) ∂gi . Proposition 4 shows that all permutations in the same top-k subgroup share the same loss φ(g, y) and thus share the same partial difference with respect to the score of a given object. Proposition 5 indicates that the partial difference of φ(g, y) also has a similar property to φ(g, y) (see the second condition in Definition 3). Due to space restriction, we omit the proofs (see [15] for more details). Based on the above definitions and propositions, we give the main theorem (Theorem 6), which states the sufficient conditions for a surrogate loss function to be consistent with the top-k true loss. Theorem 6. Let φ be a top-k subgroup order sensitive loss function on Ω⊂Rn . For ∀n objects, if its top-k subgroup probability space is order preserving with respect to n −1 object pairs {(ji, ji+1)}k i=1 and {(jk+si, jk+i : 0 ≤si < i)}n−k i=2 , then the loss φ(g, y) is consistent with the top-k true loss as defined in Eq.(5). 4 The proof of the main theorem is mostly based on Theorem 7, which specifies the score relation between two objects for the minimizer of Q(p, g). Due to space restriction, we only give Theorem 7 and its detailed proof. For the detailed proof of Theorem 6, please refer to [15]. Theorem 7. Let φ(g, y) be a top-k subgroup order sensitive loss function. ∀i and j, if the topk subgroup probability space is order preserving with respect to them, and g is a vector which minimizes Q(p, g) in Eq.(8), then gi > gj. Proof. Without loss of generality, we assume i = 1, j = 2, g′ 1 = g2, g′ 2 = g1, and g′ k = gk(k > 2). First, we prove g1 ≥g2 by contradiction. Assume g1 < g2, we have Q(p, g′) −Q(p, g) = X y∈Y (pσ−1 1,2y −py)φ(g, y) = X y∈Y1,2 (pσ−1 1,2y −py)(φ(g, y) −φ(g, σ−1 1,2y)). The first equation is based on the fact g′ = σ−1 1,2g, and the second equation is based on the fact σ−1 1,2σ−1 1,2y = y. After some algebra, by using Proposition 4, we have, Q(p, g′) −Q(p, g) = X Gk(y)∈{Gk:Gk(y)̸=Gk(σ−1 1,2y)}:y∈Y1,2 (pGk(σ−1 1,2y) −pGk(y))(φ(g, y) −φ(g, σ−1 1,2y)), where Gk(y) denotes the subgroup that y belongs to. Since g1 < g2, we have φ(g, y) ≥φ(g, σ−1 1,2y). Meanwhile, pGk(σ−1 1,2y) < pGk(y) due to the order preserving of the top-k subgroup probability space. Thus each component in the sum is non-positive and at least one of them is negative, which means Q(p, g′) < Q(p, g). This is a contradiction to the optimality of g. Therefore, we must have g1 ≥g2. Second, we prove g1 ̸= g2, again by contradiction. Assume g1 = g2. By setting the derivative of Q(p, g) with respect to g1 and g2 to zero and compare them3, we have, X y∈Y1,2 (py −pσ−1 1,2y)(∂φ(g, y) ∂g1 −∂φ(g, σ−1 1,2y) ∂g1 ) = 0. After some algebra, we obtain, X Gk(y)∈{Gk:Gk(y)̸=Gk(σ−1 1,2y)}:y∈Y1,2 (pGk(y) −pGk(σ−1 1,2y))(∂φ(g, y) ∂g1 −∂φ(g, σ−1 1,2y) ∂g1 ) = 0. According to Proposition 5, we have ∂φ(g,y) ∂g1 ≤ ∂φ(g,σ−1 1,2y) ∂g1 . Meanwhile, pGk(σ−1 1,2y) < pGk(y) due to the order preserving of the top-k subgroup probability space. Thus, the above equation cannot hold since at least one of components in the sum is negative according to Definition 3. 4.2 Consistency with respect to k We discuss the change of the consistency conditions with respect to various k values. First, we have the following proposition for the top-k subgroup probability space. Proposition 8. If the top-k subgroup probability space is order preserving with respect to object i and j, the top-(k −1) subgroup probability space is also order preserving with respect to i and j. The proposition can be proved by decomposing a top-(k −1) subgroup into the sum of top-k subgroups. One can find the detailed proof in [15]. Here we give an example to illustrate the basic idea. Suppose there are three objects {1, 2, 3} to be ranked. If the top-2 subgroup probability space is order preserving with respect to objects 1 and 2, then we have pG2(1,2) > pG2(2,1), pG2(1,3) > pG2(2,3) and pG2(3,1) > pG2(3,2). On the other hand, for top-1, we have pG1(1) > pG1(2). Note that pG1(1) = pG2(1,2) + pG2(1,3) and pG1(2) = pG2(2,1) + pG2(2,3). Thus, it is easy to verify that Proposition 8 holds for this case while the opposite does not. Second, we obtain the following proposition for the surrogate loss function φ. 3By trivial modifications, one can handle the case that g1 or g2 is infinite (cf. [17]). 5 Proposition 9. If the surrogate loss function φ is top-k subgroup order sensitive on a set Ω⊂Rn, then it is also top-(k + 1) subgroup order sensitive on the same set. Again, one can refer to [15] for the detailed proof of the proposition, and here we only provide an example. Let us consider the same setting in the previous example. Assume that g1 < g2. If φ is top-1 subgroup order sensitive, then we have φ(g, (1, 2, 3)) ≥φ(g, (2, 1, 3)), φ(g, (1, 3, 2)) ≥φ(g, (2, 3, 1)), and φ(g, (3, 1, 2)) = φ(g, (3, 2, 1)). From Proposition 4, we know that the two inequalities are strict. On the other hand, if φ is top-2 subgroup order sensitive, the following inequalities hold with at least one of them being strict: φ(g, (1, 2, 3)) ≥φ(g, (2, 1, 3)), φ(g, (1, 3, 2)) ≥φ(g, (2, 3, 1)), and φ(g, (3, 1, 2)) ≥φ(g, (3, 2, 1)). Therefore top-1 subgroup order sensitive is a special case of top-2 subgroup order sensitive. According to the above propositions, we can come to the following conclusions. • For the consistency with the top-k true loss, when k becomes smaller, the requirement on the probability space becomes weaker but the requirement on the surrogate loss function becomes stronger. Since we never know the real property of the (unknown) probability space, it is more likely the requirement on the probability space for the consistency with the top-k true loss can be satisfied than that for the top-l (l > k) true loss. Specifically, it is risky to assume the requirement for the permutation-level 0-1 loss to hold. • If we fix the true loss to be top-k and the probability space to be top-k subgroup order preserving, the surrogate loss function should be at most top-l (l ≤k) subgroup order sensitive in order to meet the consistency conditions. It is not guaranteed that a top-l (l > k) subgroup order sensitive surrogate loss function can be consistent with the top-k true loss. For example, a top-1 subgroup order sensitive surrogate loss function may be consistent with any top-k true loss, but a permutation-level order sensitive surrogate loss function may not be consistent with any top-k true loss, if k is smaller than the length of the list. For ease of understanding the above discussions, let us see an example shown in the following proposition (the proof of this proposition can be found in [15]). It basically says that given a probability space that is top-1 subgroup order preserving, a top-3 subgroup order sensitive surrogate loss function may not be consistent with the top-1 true loss. Proposition 10. Suppose there are three objects to be ranked. φ is a top-3 subgroup order sensitive loss function and the strict inequality φ(g, (3, 1, 2)) < φ(g, (3, 2, 1)) holds when g1 > g2. The probabilities of permutations are p123 = p1, p132 = 0, p213 = p2, p231 = 0, p312 = 0, p321 = p2 respectively, where p1 > p2. Then φ is not consistent with the top-1 true loss. The above discussions imply that although the surrogate loss functions in existing listwise ranking methods are consistent with the permutation-level 0-1 loss (under a rigid condition), they may not be consistent with the top-k true loss (under a mild condition). Therefore, it is necessary to modify these surrogate loss functions. We will make discussions on this in the next subsection. 4.3 Consistent surrogate loss functions In [16], the surrogate loss functions in ListNet, RankCosine, and ListMLE have been proved to be permutation-level order sensitive. According to the discussion in the previous subsection, however, they may not be top-k subgroup order sensitive, and therefore not consistent with the top-k true loss. Even for the consistency with the permutation-level 0-1 loss, in order to guarantee these surrogate loss functions to be consistent, the requirement on the probability space may be too strong in some real scenarios. To tackle the challenge, it is desirable to modify these surrogate loss functions to make them top-k subgroup order sensitive. Actually this is doable, and the modifications to the aforementioned surrogate loss functions are given as follows. 4.3.1 Likelihood loss The likelihood loss is the loss function used in ListMLE [16], which is defined as below, φ(g(x), y) = −log P(y|x; g), where P(y|x; g) = n Y i=1 exp(g(xy(i))) Pn t=i exp(g(xy(t))). (9) 6 We propose replacing the permutation probability with the top-k subgroup probability (which is also defined with the Luce model [11]) in the above definition: P(y|x; g) = k Y i=1 exp(g(xy(i))) Pn t=i exp(g(xy(t))). (10) It can be proved that the modified loss is top-k subgroup order sensitive (see [15]). 4.3.2 Cosine loss The cosine loss is the loss function used in RankCosine [13], which is defined as follows, φ(g(x), y) = 1 2(1 − ψy(x)T g(x) ∥ψy(x)∥∥g(x)∥), (11) where the score vector of the ground truth is produced by a mapping function ψy(·) : Rd →R, which retains the order in a permutation, i.e., ψy(xy(1)) > · · · > ψy(xy(n)). We propose changing the mapping function as follows. Let the mapping function retain the order for the top k positions of the ground truth permutation and assigns to all the remaining positions a small value (which is smaller than the score of any object ranked at the top-k positions), i.e., ψy(xy(1)) > · · · > ψy(xy(k)) > ψy(xy(k+1)) = · · · = ψy(xy(n)) = ϵ. It can be proved that after the modification, the cosine loss becomes top-k subgroup order sensitive (see [15]). 4.3.3 Cross entropy loss The cross entropy loss is the loss function used in ListNet [3], defined as follows, φ(g(x), y) = D(P(π|x; ψy)||P(π|x; g)), (12) where ψ is a mapping function whose definition is similar to that in RankCosine, and P(π|x; ψy) and P(π|x; g) are the permutation probabilities in the Luce model. We propose using a mapping function to modify the cross entropy loss in a similar way as in the case of the cosine loss4 It can be proved that such a modification can make the surrogate loss function top-k subgroup order sensitive (see [15]). 5 Experimental results In order to validate the theoretical analysis in this work, we conducted some empirical study. Specifically, we used OHSUMED, TD2003, and TD2004 in the LETOR benchmark dataset [10] to perform some experiments. As evaluation measure, we adopted Normalized Discounted Cumulative Gain (N) at positions 1, 3, and 10, and Precision (P) at positions 1, 3, and 10.5 It is obvious that these measures are top-k related and are suitable to evaluate the ranking performance in top-k ranking problems. We chose ListMLE as example method since the likelihood loss has nice properties such as convexity, soundness, and linear computational complexity [16]. We refer to the new method that we obtained by applying the modifications mentioned in Section 4.3 as top-k ListMLE. We tried different values of k (i.e., k=1, 3, 10, and the exact length of the ranked list). Obviously the last case corresponds to the original likelihood loss in ListMLE. Since the training data in LETOR is given in the form of multi-level ratings, we adopted the methods proposed in [16] to produce the ground truth ranked list. We then used stochastic gradient descent as the algorithm for optimization of the likelihood loss. As for the ranking model, we chose linear Neural Network, since the model has been widely used [3, 13, 16]. 4Note that in [3], a top-k cross entropy loss was also proposed, by using the top-k Luce model. However, it can be verified that the so-defined top-k cross entropy loss is still permutation-level order sensitive, but not top-k subgroup order sensitive. In other words, the proposed modification here is still needed. 5On datasets with only two ratings such as TD2003 and TD2004, N@1 equals P@1. 7 The experimental results are summarized in Tables 1-3. Methods N@1 N@3 N@10 P@1 P@3 P@10 ListMLE 0.548 0.473 0.446 0.642 0.582 0.495 Top-1 ListMLE 0.529 0.482 0.447 0.652 0.595 0.499 Top-3 ListMLE 0.535 0.484 0.445 0.671 0.608 0.504 Top-10 ListMLE 0.558 0.473 0.444 0.672 0.601 0.509 Table 1: Ranking accuracies on OHSUMED Methods N/P@1 N@3 N@10 P@3 P@10 ListMLE 0.24 0.253 0.261 0.22 0.146 Top-1 ListMLE 0.4 0.329 0.314 0.3 0.176 Top-3 ListMLE 0.44 0.382 0.343 0.34 0.204 Top-10 ListMLE 0.5 0.410 0.378 0.38 0.22 Table 2: Ranking accuracies on TD2003 Methods N/P@1 N@3 N@10 P@3 P@10 ListMLE 0.4 0.351 0.356 0.284 0.188 Top-1 ListMLE 0.52 0.469 0.451 0.413 0.248 Top-3 ListMLE 0.506 0.456 0.458 0.417 0.261 Top-10 ListMLE 0.52 0.469 0.472 0.413 0.269 Table 3: Ranking accuracies on TD2004 Methods N@1 N@3 N@10 P@1 P@3 P@10 RankBoost 0.497 0.472 0.435 0.604 0.586 0.495 Ranking SVM 0.495 0.464 0.441 0.633 0.592 0.507 ListNet 0.523 0.477 0.448 0.642 0.602 0.509 RankCosine 0.523 0.475 0.437 0.642 0.589 0.493 Top-10 ListMLE 0.558 0.473 0.444 0.672 0.601 0.509 Table 4: Ranking accuracies on OHSUMED From the tables, we can see that with the modifications the ranking accuracies of ListMLE can be significantly boosted, in terms of all measures, on both TD2003 and TD2004. This clearly validates our theoretical analysis. On OHSUMED, all the loss functions achieve comparable performances. The possible explanation is that the probability space in OHSUMED is well formed such that it is order preserving for many different k values. Next, we take Top-10 ListMLE as an example to make comparison with some other baseline methods such as Ranking SVM [8], RankBoost [7], ListNet [3], and RankCosine [13]. The results are listed in Tables 4-6. We can see from the tables, Top-10 ListMLE achieves the best performance among all the methods on the TD2003 and TD2004 datasets in terms of almost all measures. On the OHSUMED dataset, it also performs fairly well as compared to the other methods. Especially for N@1 and P@1, it significantly outperforms all the other methods on all the datasets. Methods N/P@1 N@3 N@10 P@3 P@10 RankBoost 0.26 0.270 0.285 0.24 0.178 Ranking SVM 0.42 0.378 0.341 0.34 0.206 ListNet 0.46 0.408 0.374 0.36 0.222 RankCosine 0.36 0.346 0.322 0.3 0.182 Top-10 ListMLE 0.5 0.410 0.378 0.38 0.22 Table 5: Ranking accuracies on TD2003 Methods N/P@1 N@3 N@10 P@3 P@10 RankBoost 0.48 0.463 0.471 0.404 0.253 Ranking SVM 0.44 0.409 0.420 0.351 0.225 ListNet 0.439 0.437 0.457 0.399 0.257 RankCosine 0.439 0.397 0.405 0.328 0.209 Top-10 ListMLE 0.52 0.469 0.472 0.413 0.269 Table 6: Ranking accuracies on TD2004 From the above experimental results, we can come to the conclusion that for real ranking applications like IR (where top-k evaluation measures are widely used), it is better to use the top-k true loss than the permutation-level 0-1 loss, and is better to use the modified surrogate loss functions than the original surrogate loss functions. 6 Conclusion In this paper we have proposed a top-k ranking framework, which can better describe real ranking applications like information retrieval. In the framework, the true loss is defined on the top-k subgroup of permutations. We have derived the sufficient conditions for a surrogate loss function to be statistically consistent with the top-k true loss. We have also discussed how to modify the loss functions in existing listwise ranking methods to make them consistent with the top-k true loss. Our experiments have shown that with the proposed modifications, algorithms like ListMLE can significantly outperform their original version, and also many other ranking methods. As future work, we plan to investigate the following issues. (1) we will empirically study the modified ListNet and RankCosine, to see whether their performances can also be significantly boosted in the top-k setting. (2) We will also study the consistency of the pointwise and pairwise loss functions with the top-k true loss. 8 References [1] P. L. Bartlett, M. I. Jordan, and J. D. McAuliffe. Convexity, classification, and risk bounds. Journal of the American Statistical Association, 101:138–156, 2006. [2] C. Burges, T. Shaked, E. Renshaw, A. Lazier, M. Deeds, N. Hamilton, and G. Hullender. Learning to rank using gradient descent. In Proc. of ICML’05, pages 89–96, 2005. [3] Z. Cao, T. Qin, T. Y. Liu, M. F. Tsai, and H. Li. Learning to rank: From pairwise approach to listwise approach. In Proc. of ICML’07, pages 129–136, 2007. [4] S. Clemencon and N. Vayatis. Ranking the best instances. Journal of Machine Learning Research, 8:2671–2699, 2007. [5] D. Cossock and T. Zhang. Subset ranking using regression. In Proc. of COLT, pages 605–619, 2006. [6] D. Cossock and T. Zhang. Statistical analysis of bayes optimal subset ranking. Information Theory, 54:5140–5154, 2008. [7] Y. Freund, R. Iyer, R. E. Schapire, and Y. Singer. An efficient boosting algorithm for combining preferences. In Proc. of ICML’98, pages 170–178, 1998. [8] R. Herbrich, T. Graepel, and K. Obermayer. Support vector vector learning for ordinal regression. In Proc. of ICANN’99, pages 97–102, 1999. [9] P. Li, C. Burges, and Q. Wu. Mcrank: Learning to rank using multiple classification and gradient boosting. In Advances in Neural Information Processing Systems 20(NIPS 07), pages 897–904, Cambridge, MA, 2008. MIT Press. [10] T. Y. Liu, T. Qin, J. Xu, W. Y. Xiong, and H. Li. Letor: Benchmark dataset for research on learning to rank for information retrieval. In LR4IR 2007, in conjunction with SIGIR 2007, 2007. [11] J. I. Marden, editor. Analyzing and Modeling Rank Data. Chapman and Hall, London, 1995. [12] R. Nallapati. Discriminative models for information retrieval. In Proc. of SIGIR’04, pages 64–71, 2004. [13] T. Qin, X.-D. Zhang, M.-F. Tsai, D.-S. Wang, T.-Y. Liu, and H. Li. Query-level loss functions for information retrieval. Information processing and management, 44:838–855, 2008. [14] C. Rudin. Ranking with a p-norm push. In Proc. of COLT, pages 589–604, 2006. [15] F. Xia, T. Y. Liu, and H. Li. Top-k consistency of learning to rank methods. Technical report, Microsoft Research, MSR-TR-2009-139, 2009. [16] F. Xia, T. Y. Liu, J. Wang, W. S. Zhang, and H. Li. Listwise approach to learning to rank theory and algorithm. In Proc. of ICML’08, pages 1192–1199, 2008. [17] T. Zhang. Statistical analysis of some multi-category large margin classification methods. Journal of Machine Learning Research, 5:1225–1251, 2004. 9
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Speaker Comparison with Inner Product Discriminant Functions W. M. Campbell MIT Lincoln Laboratory Lexington, MA 02420 wcampbell@ll.mit.edu Z. N. Karam DSPG, MIT RLE, Cambridge MA MIT Lincoln Laboratory, Lexington, MA zahi@mit.edu D. E. Sturim MIT Lincoln Laboratory Lexington, MA 02420 sturim@ll.mit.edu Abstract Speaker comparison, the process of finding the speaker similarity between two speech signals, occupies a central role in a variety of applications—speaker verification, clustering, and identification. Speaker comparison can be placed in a geometric framework by casting the problem as a model comparison process. For a given speech signal, feature vectors are produced and used to adapt a Gaussian mixture model (GMM). Speaker comparison can then be viewed as the process of compensating and finding metrics on the space of adapted models. We propose a framework, inner product discriminant functions (IPDFs), which extends many common techniques for speaker comparison—support vector machines, joint factor analysis, and linear scoring. The framework uses inner products between the parameter vectors of GMM models motivated by several statistical methods. Compensation of nuisances is performed via linear transforms on GMM parameter vectors. Using the IPDF framework, we show that many current techniques are simple variations of each other. We demonstrate, on a 2006 NIST speaker recognition evaluation task, new scoring methods using IPDFs which produce excellent error rates and require significantly less computation than current techniques. 1 Introduction Comparing speakers in speech signals is a common operation in many applications including forensic speaker recognition, speaker clustering, and speaker verification. Recent popular approaches to text-independent comparison include Gaussian mixture models (GMMs) [1], support vector machines [2, 3], and combinations of these techniques. When comparing two speech utterances, these approaches are used in a train and test methodology. One utterance is used to produce a model which is then scored against the other utterance. The resulting comparison score is then used to cluster, verify or identify the speaker. Comparing speech utterances with kernel functions has been a common theme in the speaker recognition SVM literature [2, 3, 4]. The resulting framework has an intuitive geometric structure. Variable length sequences of feature vectors are mapped to a large dimensional SVM expansion vector. These vectors are “smoothed” to eliminate nuisances [2]. Then, a kernel function is applied to the ∗This work was sponsored by the Federal Bureau of Investigation under Air Force Contract FA8721-05C-0002. Opinions, interpretations, conclusions, and recommendations are those of the authors and are not necessarily endorsed by the United States Government. 1 two vectors. The kernel function is an inner product which induces a metric on the set of vectors, so comparison is analogous to finding the distances between SVM expansion vectors. A recent trend in the speaker recognition literature has been to move towards a more linear geometric view for non-SVM systems. Compensation via linear subspaces and supervectors of mean parameters of GMMs is presented in joint factor analysis [5]. Also, comparison of utterances via linear scoring is presented in [6]. These approaches have introduced many new ideas and perform well in speaker comparison tasks. An unrealized effort in speaker recognition is to bridge the gap between SVMs and some of the new proposed GMM methods. One difficulty is that most SVM kernel functions in speaker comparison satisfy the Mercer condition. This restricts the scope of investigation of potential comparison strategies for two speaker utterances. Therefore, in this paper, we introduce the idea of inner product discriminant functions (IPDFs). IPDFs are based upon the same basic operations as SVM kernel functions with some relaxation in structure. First, we map input utterances to vectors of fixed dimension. Second, we compensate the input feature vectors. Typically, this compensation takes the form of a linear transform. Third, we compare two compensated vectors with an inner product. The resulting comparison function is then used in an application specific way. The focus of our initial investigations of the IPDF structure are the following. First, we show that many of the common techniques such as factor analysis, nuisance projection, and various types of scoring can be placed in the framework. Second, we systematically describe the various inner product and compensation techniques used in the literature. Third, we propose new inner products and compensation. Finally, we explore the space of possible combinations of techniques and demonstrate several novel methods that are computationally efficient and produce excellent error rates. The outline of the paper is as follows. In Section 2, we describe the general setup for speaker comparison using GMMs. In Section 3, we introduce the IPDF framework. Section 4 explores inner products for the IPDF framework. Section 5 looks at methods for compensating for variability. In Section 6, we perform experiments on the NIST 2006 speaker recognition evaluation and explore different combinations of IPDF comparisons and compensations. 2 Speaker Comparison A standard distribution used for text-independent speaker recognition is the Gaussian mixture model [1], g(x) = N X i=1 λiN(x|mi, Σi). (1) Feature vectors are typically cepstral coefficients with associated smoothed first- and second-order derivatives. We map a sequence of feature vectors, xNx 1 , from a speaker to a GMM by adapting a GMM universal background model (UBM). Here, we use the shorthand xNx 1 to denote the sequence, x1, · · · , xNx. For the purpose of this paper, we will assume only the mixture weights, λi, and means, mi, in (1) are adapted. Adaptation of the means is performed with standard relevance MAP [1]. We estimate the mixture weights using the standard ML estimate. The adaptation yields new parameters which we stack into a parameter vector, ax, where ax = λt x mt x t (2) = λx,1 · · · λx,N mt x,1 · · · mt x,N t . (3) In speaker comparison, the problem is to compare two sequences of feature vectors, e.g., xNx 1 and yNy 1 . To compare these two sequences, we adapt a GMM UBM to produce two sets of parameter vectors, ax and ay, as in (2). The goal of our speaker comparison process can now be recast as a function that compares the two parameter vectors, C(ax, ay), and produces a value that reflects the similarity of the speakers. Initial work in this area was performed using kernels from support vector machines [4, 7, 2], but we expand the scope to other types of discriminant functions. 2 3 Inner Product Discriminant Functions The basic framework we propose for speaker comparison functions is composed of two parts— compensation and comparison. For compensation, the parameter vectors generated by adaptation in (2) can be transformed to remove nuisances or projected onto a speaker subspace. The second part of our framework is comparison. For the comparison of parameter vectors, we will consider natural distances that result in inner products between parameter vectors. We propose the following inner product discriminant function (IPDF) framework for exploring speaker comparison, C(ax, ay) = (Lxax)tD2(Lyay) (4) where Lx, Ly are linear transforms and potentially dependent on λx and/or λy. The matrix D is positive definite, usually diagonal, and possibly dependent on λx and/or λy. Note, we also consider simple combinations of IPDFs to be in our framework—e.g., positively-weighted sums of IPDFs. Several questions from this framework are: 1) what inner product gives the best speaker comparison performance, 2) what compensation strategy works best, 3) what tradeoffs can be made between accuracy and computational cost, and 4) how do the compensation and the inner product interact. We explore theoretical and experimental answers to these questions in the following sections. 4 Inner Products for IPDFs In general, an inner product of the parameters should be based on a distance arising from a statistical comparison. We derive three straightforward methods in this section. We also relate some other methods, without being exhaustive, that fall in this framework that have been described in detail in the literature. 4.1 Approximate KL Comparison (CKL) A straightforward strategy for comparing the GMM parameter vectors is to use an approximate form of the KL divergence applied to the induced GMM models. This strategy was used in [2] successfully with an approximation based on the log-sum inequality; i.e., for the GMMs, gx and gy, with parameters ax and ay, D(gx∥gy) ≤ N X i=1 λx,iD (N(·; mx,i, Σi)∥N(·; my,i, Σi)) . (5) Here, D(·∥·) is the KL divergence, and Σi is from the UBM. By symmetrizing (5) and substituting in the KL divergence between two Gaussian distributions, we obtain a distance, ds, which upper bounds the symmetric KL divergence, ds(ax, ay) = Ds(λx∥λy) + N X i=1 (0.5λx,i + 0.5λy,i)(mx,i −my,i)tΣ−1 i (mx,i −my,i). (6) We focus on the second term in (6) for this paper, but note that the first term could also be converted to a comparison function on the mixture weights. Using polarization on the second term, we obtain the inner product CKL(ax, ay) = N X i=1 (0.5λx,i + 0.5λy,i)mt x,iΣ−1 i my,i. (7) Note that (7) can also be expressed more compactly as CKL(ax, ay) = mt x ((0.5λx + 0.5λy) ⊗In) Σ−1my (8) where Σ is the block matrix with the Σi on the diagonal, n is the feature vector dimension, and ⊗ is the Kronecker product. Note that the non-symmetric form of the KL distance in (5) would result in the average mixture weights in (8) being replaced by λx. Also, note that shifting the means by the UBM will not affect the distance in (6), so we can replace means in (8) by the UBM centered means. 3 4.2 GLDS kernel (CGLDS) An alternate inner product approach is to use generalized linear discriminants and the corresponding kernel [4]. The overall structure of this GLDS kernel is as follows. A per feature vector expansion function is defined as b(xi) = [b1(xi) · · · bm(xi)]t . (9) The mapping between an input sequence, xNx 1 is then defined as xNx 1 7→bx = 1 Nx Nx X i=1 b(xi). (10) The corresponding kernel between two sequences is then KGLDS(xNx 1 , yNy 1 ) = bt xΓ−1by (11) where Γ = 1 Nz Nz X i=1 b(zi)b(zi)t, (12) and zNz 1 is a large set of feature vectors which is representative of the speaker population. In the context of a GMM UBM, we can define an expansion as follows b(xi) = p(1|xi)(xi −m1)t · · · p(N|xi)(xi −mN)tt (13) where p(j|xi) is the posterior probability of mixture component j given xi, and mj is from a UBM. Using (13) in (10), we see that bx = (λx ⊗In)(mx −m) and by = (λy ⊗In)(my −m) (14) where m is the stacked means of the UBM. Thus, the GLDS kernel inner product is CGLDS(ax, ay) = (mx −m)t(λx ⊗In)Γ−1(λy ⊗In)(my −m). (15) Note that Γ in (12) is almost the UBM covariance matrix, but is not quite the same because of a squaring of the p(j|zi) in the diagonal. As is commonly assumed, we will consider a diagonal approximation of Γ, see [4]. 4.3 Gaussian-Distributed Vectors A common assumption in the factor analysis literature [5] is that the parameter vector mx as x varies has a Gaussian distribution. If we assume a single covariance for the entire space, then the resulting likelihood ratio test between two Gaussian distributions results in a linear discriminant [8]. More formally, suppose that we have a distribution with mean mx and we are trying to distinguish from a distribution with the UBM mean m, then the discriminant function is [8], h(x) = (mx −m)tΥ−1(x −m) + cx (16) where cx is a constant that depends on mx, and Υ is the covariance in the parameter vector space. We will assume that the comparison function can be normalized (e.g., by Z-norm [1]), so that cx can be dropped. We now apply the discriminant function to another mean vector, my, and obtain the following comparison function CG(ax, ay) = (mx −m)tΥ−1(my −m). (17) 4.4 Other Methods Several other methods are possible for comparing the parameter vectors that arise either from ad hoc methods or from work in the literature. We describe a few of these in this section. Geometric Mean Comparison (CGM). A simple symmetric function that is similar to the KL (8) and GLDS (15) comparison functions is arrived at by replacing the arithmetic mean in CKL by a geometric mean. The resulting kernel is CGM(ax, ay) = (mx −m)t(λ1/2 x ⊗In)Σ−1(λ1/2 y ⊗In)(my −m) (18) 4 where Σ is the block diagonal UBM covariances. Fisher Kernel (CF ). The Fisher kernel specialized to the UBM case has several forms [3]. The main variations are the choice of covariance in the inner product and the choice of normalization of the gradient term. We took the best performing configuration for this paper—we normalize the gradient by the number of frames which results in a mixture weight scaling of the gradient. We also use a diagonal data-trained covariance term. The resulting comparison function is CF (ax, ay) = (λx ⊗In)Σ−1(mx −m) t Φ−1 (λy ⊗In)Σ−1(my −m) (19) where Φ is a diagonal matrix acting as a variance normalizer. Linearized Q-function (CQ). Another form of inner product may be derived from the linear Qscoring shown in [6]. In this case, the scoring is given as (mtrain −m)tΣ−1(F −Nm) where N and F are the zeroth and first order sufficient statistics of a test utterance, m is the UBM means, mtrain is the mean of a training model, and Σ is the block diagonal UBM covariances. A close approximation of this function can be made by using a small relevance factor in MAP adaptation of the means to obtain the following comparison function CQ(ax, ay) = (mx −m)tΣ−1(λy ⊗In)(my −m). (20) Note that if we symmetrize CQ, this gives us CKL; this analysis ignores for a moment that in [6], compensation is also asymmetric. KL Kernel (KKL). By assuming the mixture weights are constant and equal to the UBM mixture in the comparison function CKL (7), we obtain the KL kernel, KKL(mx, my) = mt x (λ ⊗In) Σ−1my (21) where λ are the UBM mixture weights. This kernel has been used extensively in SVM speaker recognition [2]. An analysis of the different inner products in the preceding sections shows that many of the methods presented in the literature have a similar form, but are interestingly derived with quite disparate techniques. Our goal in the experimental section is to understand how these comparison function perform and how they interact with compensation. 5 Compensation in IPDFs Our next task is to explore compensation methods for IPDFs. Our focus will be on subspace-based methods. With these methods, the fundamental assumption is that either speakers and/or nuisances are confined to a small subspace in the parameter vector space. The problem is to use this knowledge to produce a higher signal (speaker) to noise (nuisance) representation of the speaker. Standard notation is to use U to represent the nuisance subspace and to have V represent the speaker subspace. Our goal in this section is to recast many of the methods in the literature in a standard framework with oblique and orthogonal projections. To make a cohesive presentation, we introduce some notation. We define an orthogonal projection with respect to a metric, PU,D, where D and U are full rank matrices as PU,D = U(U tD2U)−1U tD2 (22) where DU is a linearly independent set, and the metric is ∥x −y∥D = ∥Dx −Dy∥2. The process of projection, e.g. y = PU,Db, is equivalent to solving the least-squares problem, ˆx = argminx ∥Ux −b∥D and letting y = U ˆx. For convenience, we also define the projection onto the orthogonal complement of U, U ⊥, as QU,D = PU⊥,D = I −PU,D. Note that we can regularize the projection PU,D by adding a diagonal term to the inverse in (22); the resulting operation remains linear but is no longer a projection. We also define the oblique projection onto V with null space U + (U + V )⊥and metric induced by D. Let QR be the (skinny) QR decomposition of the matrix [UV ] in the D norm (i.e., QtD2Q = I), and QV be the columns corresponding to V in the matrix Q. Then, the oblique (non-orthogonal) projection onto V is OV,U,D = V (Qt V D2V )−1Qt V D2. (23) The use of projections in our development will add geometric understanding to the process of compensation. 5 5.1 Nuisance Attribute Projection (NAP) A framework for eliminating nuisances in the parameter vector based on projection was shown in [2]. The basic idea is to assume that nuisances are confined to a small subspace and can be removed via an orthogonal projection, mx 7→QU,Dmx. One justification for using subspaces comes from the perspective that channel classification can be performed with inner products along one-dimensional subspaces. Therefore, the projection removes channel specific directions from the parameter space. The NAP projection uses the metric induced by a kernel in an SVM. For the GMM context, the standard kernel used is the approximate KL comparison (8) [2]. We note that since D is known a priori to speaker comparison, we can orthonormalize the matrix DU and apply the projection as a matrix multiply. The resulting projection has D = λ1/2 ⊗In Σ−1/2. 5.2 Factor Analysis and Joint Factor Analysis The joint factor analysis (JFA) model assumes that the mean parameter vector can be expressed as ms,sess = m + Ux + V y (24) where ms,sess is the speaker- and session-dependent mean parameter vector, U and V are matrices with small rank, and m is typically the UBM. Note that for this section, we will use the standard variables for factor analysis, x and y, even though they conflict with our earlier development. The goal of joint factor analysis is to find solutions to the latent variables x and y given training data. In (24), the matrix U represents a nuisance subspace, and V represents a speaker subspace. Existing work on this approach for speaker recognition uses both maximum likelihood (ML) estimates and MAP estimates of x and y [9, 5]. In the latter case, a Gaussian prior with zero mean and diagonal covariance for x and y is assumed. For our work, we focus on the ML estimates [9] of x and y in (24), since we did not observe substantially different performance from MAP estimates in our experiments. Another form of modeling that we will consider is factor analysis (FA). In this case, the term V y is replaced by a constant vector representing the true speaker model, ms; the goal is then to estimate x. Typically, as a simplification, ms is assumed to be zero when calculating sufficient statistics for estimation of x [10]. The solution to both JFA and FA can be unified. For the JFA problem, if we stack the matrices [UV ], then the problem reverts to the FA problem. Therefore, we initially study the FA problem. Note that we also restrict our work to only one EM iteration of the estimation of the factors, since this strategy works well in practice. The standard ML solution to FA [9] for one EM iteration can be written as U tΣ−1(N ⊗In)U x = U tΣ−1 [F −(N ⊗In)m] (25) where F is the vector of first order sufficient statistics, and N is the diagonal matrix of zeroth order statistics (expected counts). The sufficient statistics are obtained from the UBM applied to an input set of feature vectors. We first let Nt = PN i=1 Ni and multiply both sides of (25) by 1/Nt. Now use relevance MAP with a small relevance factor and F and N to obtain ms; i.e., both ms −m and F −(N ⊗In)m will be nearly zero in the entries corresponding to small Ni. We obtain U tΣ−1(λs ⊗In)U x = U tΣ−1 (λs ⊗In) [ms −m] (26) where λs is the speaker dependent mixture weights. We note that (26) are the normal equations for the least-squares problem, ˆx = argminx ∥Ux −(ms −m)∥D where D is given below. This solution is not unexpected since ML estimates commonly lead to least-squares problems with GMM distributed data [11]. Once the solution to (26) is obtained, the resulting Ux is subtracted from an estimate of the speaker mean, ms to obtain the compensated mean. If we assume that ms is obtained by a relevance map adaptation from the statistics F and N with a small relevance factor, then the FA process is well approximated by ms 7→QU,Dms (27) where D = λ1/2 s ⊗In Σ−1/2. (28) 6 JFA becomes an extension of the FA process we have demonstrated. One first projects onto the stacked UV space. Then another projection is performed to eliminate the U component of variability. This can be expressed as a single oblique projection; i.e., the JFA process is ms 7→OV,U,IP[UV ],Dms = OV,U,Dms. (29) 5.3 Comments and Analysis Several comments should be made on compensation schemes and their use in speaker comparison. First, although NAP and ML FA (27) were derived in substantially different ways, they are essentially the same operation, an orthogonal projection. The main difference is in the choice of metrics under which they were originally proposed. For NAP, the metric depends on the UBM only, and for FA it is utterance and UBM dependent. A second observation is that the JFA oblique projection onto V has substantially different properties than a standard orthogonal projection. When JFA is used in speaker recognition [5, 6], typically JFA is performed in training, but the test utterance is compensated only with FA. In our notation, applying JFA with linear scoring [6] gives CQ(OV,U,D1m1, QU,D2m2) (30) where m1 and m2 are the mean parameter vectors estimated from the training and testing utterances, respectively; also, D1 = (λ1/2 1 ⊗In)Σ−1/2 and D2 = (λ1/2 2 ⊗In)Σ−1/2. Our goal in the experiments section is to disentangle and understand some of the properties of scoring methods such as (30). What is significant in this process—mismatched train/test compensation, data-dependent metrics, or asymmetric scoring? A final note is that training the subspaces for the various projections optimally is not a process that is completely understood. One difficulty is that the metric used for the inner product may not correspond to the metric for compensation. As a baseline, we used the same subspace for all comparison functions. The subspace was obtained with an ML style procedurefor training subspaces similar to [11] but specialized to the factor analysis problem as in [5]. 6 Speaker Comparison Experiments Experiments were performed on the NIST 2006 speaker recognition evaluation (SRE) data set. Enrollment/verification methodology and the evaluation criterion, equal error rate (EER) and minDCF, were based on the NIST SRE evaluation plan [12]. The main focus of our efforts was the one conversation enroll, one conversation verification task for telephone recorded speech. T-Norm models and Z-Norm [13] speech utterances were drawn from the NIST 2004 SRE corpus. Results were obtained for both the English only task (Eng) and for all trials (All) which includes speakers that enroll/verify in different languages. Feature extraction was performed using HTK [14] with 20 MFCC coefficients, deltas, and acceleration coefficients for a total of 60 features. A GMM UBM with 512 mixture components was trained using data from NIST SRE 2004 and from Switchboard corpora. The dimension of the nuisance subspace, U, was fixed at 100; the dimension of the speaker space, V , was fixed at 300. Results are in Table 1. In the table, we use the following notation, DUBM = λ1/2 ⊗In Σ−1/2, D1 = λ1/2 1 ⊗In Σ−1/2, D2 = λ1/2 2 ⊗In Σ−1/2 (31) where λ are the UBM mixture weights, λ1 are the mixture weights estimated from the enrollment utterance, and λ2 are the mixture weights estimated from the verification utterance. We also use the notation DL, DG, and DF to denote the parameters of the metric for the GLDS, Gaussian, and Fisher comparison functions from Sections 4.2, 4.3, and 4.4, respectively. An analysis of the results in Table 1 shows several trends. First, the performance of the best IPDF configurations is as good or better than the state of the art SVM and JFA implementations. Second, the compensation method that dominates good performance is an orthogonal complement of the nuisance subspace, QU,D. Combining a nuisance projection with an oblique projection is fine, but 7 Table 1: A comparison of baseline systems and different IPDF implementations Comparison Enroll Verify EER minDCF EER minDCF Function Comp. Comp. All (%) All (×100) Eng (%) Eng (×100) Baseline SVM QU,DUBM QU,DUBM 3.82 1.82 2.62 1.17 Baseline JFA, CQ OV,U,D1 QU,D2 3.07 1.57 2.11 1.23 CKL OV,U,D1 QU,D2 3.21 1.70 2.32 1.32 CKL OV,U,D1 OV,U,D2 8.73 5.06 8.06 4.45 CKL QU,D1 QU,D2 2.93 1.55 1.89 0.93 CKL QU,DUBM QU,DUBM 3.03 1.55 1.92 0.95 CKL I −OU,V,D1 I −OU,V,D2 7.10 3.60 6.49 3.13 CGM QU,D1 QU,D2 2.90 1.59 1.73 0.98 CGM QU,DUBM QU,DUBM 3.01 1.66 1.89 1.05 CGM QU,DUBM I 3.95 1.93 2.76 1.26 KKL QU,DUBM QU,DUBM 4.95 2.46 3.73 1.75 KKL QU,D1 QU,D2 5.52 2.85 4.43 2.15 CGLDS QU,DL QU,DL 3.60 1.93 2.27 1.23 CG QU,DG QU,DG 5.07 2.52 3.89 1.87 CF QU,DF QU,DF 3.56 1.89 2.22 1.12 Table 2: Summary of some IPDF performances and computation time normalized to a baseline system. Compute time includes compensation and inner product only. Comparison Enroll Verify EER minDCF Compute Function Comp. Comp. Eng (%) Eng (×100) time CQ OV,U,D1 QU,D2 2.11 1.23 1.00 CGM QU,D1 QU,D2 1.73 0.98 0.17 CGM QU,DUBM QU,DUBM 1.89 1.05 0.08 CGM QU,DUBM I 2.76 1.26 0.04 using only oblique projections onto V gives high error rates. A third observation is that comparison functions whose metrics incorporate λ1 and λ2 perform significantly better than ones with fixed λ from the UBM. In terms of best performance, CKL, CQ, and CGM perform similarly. For example, the 95% confidence interval for 2.90% EER is [2.6, 3.3]%. We also observe that a nuisance projection with fixed DUBM gives similar performance to a projection involving a “variable” metric, Di. This property is fortuitous since a fixed projection can be precomputed and stored and involves significantly reduced computation. Table 2 shows a comparison of error rates and compute times normalized by a baseline system. For the table, we used precomputed data as much as possible to minimize compute times. We see that with an order of magnitude reduction in computation and a significantly simpler implementation, we can achieve the same error rate. 7 Conclusions and future work We proposed a new framework for speaker comparison, IPDFs, and showed that several recent systems in the speaker recognition literature can be placed in this framework. We demonstrated that using mixture weights in the inner product is the key component to achieve significant reductions in error rates over a baseline SVM system. We also showed that elimination of the nuisance subspace via an orthogonal projection is a computationally simple and effective method of compensation. Most effective methods of compensation in the literature (NAP, FA, JFA) are straightforward variations of this idea. By exploring different IPDFs using these insights, we showed that computation can be reduced substantially over baseline systems with similar accuracy to the best performing systems. Future work includes understanding the performance of IPDFs for different tasks, incorporating them into an SVM system, and hyperparameter training. 8 References [1] Douglas A. Reynolds, T. F. Quatieri, and R. Dunn, “Speaker verification using adapted Gaussian mixture models,” Digital Signal Processing, vol. 10, no. 1-3, pp. 19–41, 2000. [2] W. M. Campbell, D. E. Sturim, D. A. Reynolds, and A. Solomonoff, “SVM based speaker verification using a GMM supervector kernel and NAP variability compensation,” in Proc. ICASSP, 2006, pp. I97– I100. [3] C. Longworth and M. J. F. Gales, “Derivative and parametric kernels for speaker verification,” in Proc. Interspeech, 2007, pp. 310–313. [4] W. M. Campbell, “Generalized linear discriminant sequence kernels for speaker recognition,” in Proc. ICASSP, 2002, pp. 161–164. [5] P. Kenny, P. Ouellet, N. Dehak, V. Gupta, and P. Dumouchel, “A study of inter-speaker variability in speaker verification,” IEEE Transactions on Audio, Speech and Language Processing, 2008. [6] Ondrej Glembek, Lukas Burget, Najim Dehak, Niko Brummer, and Patrick Kenny, “Comparison of scoring methods used in speaker recognition with joint factor analysis,” in Proc. ICASSP, 2009. [7] Pedro J. Moreno, Purdy P. Ho, and Nuno Vasconcelos, “A Kullback-Leibler divergence based kernel for SVM classification in multimedia applications,” in Adv. in Neural Inf. Proc. Systems 16, S. Thrun, L. Saul, and B. Schölkopf, Eds. MIT Press, Cambridge, MA, 2004. [8] Keinosuke Fukunaga, Introduction to Statistical Pattern Recognition, Academic Press, 1990. [9] Simon Lucey and Tsuhan Chen, “Improved speaker verification through probabilistic subspace adaptation,” in Proc. Interspeech, 2003, pp. 2021–2024. [10] Robbie Vogt, Brendan Baker, and Sridha Sriharan, “Modelling session variability in text-independent speaker verification,” in Proc. Interspeech, 2005, pp. 3117–3120. [11] Mark J. F. Gales, “Cluster adaptive training of hidden markov models,” IEEE Trans. Speech and Audio Processing, vol. 8, no. 4, pp. 417–428, 2000. [12] M. A. Przybocki, A. F. Martin, and A. N. Le, “NIST speaker recognition evaluations utilizing the Mixer corpora—2004,2005,2006,” IEEE Trans. on Speech, Audio, Lang., vol. 15, no. 7, pp. 1951–1959, 2007. [13] Roland Auckenthaler, Michael Carey, and Harvey Lloyd-Thomas, “Score normalization for textindependent speaker verification systems,” Digital Signal Processing, vol. 10, pp. 42–54, 2000. [14] J. Odell, D. Ollason, P. Woodland, S. Young, and J. Jansen, The HTK Book for HTK V2.0, Cambridge University Press, Cambridge, UK, 1995. 9
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Asymptotic Analysis of MAP Estimation via the Replica Method and Compressed Sensing∗ Sundeep Rangan Qualcomm Technologies Bedminster, NJ srangan@qualcomm.com Alyson K. Fletcher University of California, Berkeley Berkeley, CA alyson@eecs.berkeley.edu Vivek K Goyal Mass. Inst. of Tech. Cambridge, MA vgoyal@mit.edu Abstract The replica method is a non-rigorous but widely-accepted technique from statistical physics used in the asymptotic analysis of large, random, nonlinear problems. This paper applies the replica method to non-Gaussian maximum a posteriori (MAP) estimation. It is shown that with random linear measurements and Gaussian noise, the asymptotic behavior of the MAP estimate of an n-dimensional vector “decouples” as n scalar MAP estimators. The result is a counterpart to Guo and Verd´u’s replica analysis of minimum mean-squared error estimation. The replica MAP analysis can be readily applied to many estimators used in compressed sensing, including basis pursuit, lasso, linear estimation with thresholding, and zero norm-regularized estimation. In the case of lasso estimation the scalar estimator reduces to a soft-thresholding operator, and for zero normregularized estimation it reduces to a hard-threshold. Among other benefits, the replica method provides a computationally-tractable method for exactly computing various performance metrics including mean-squared error and sparsity pattern recovery probability. 1 Introduction Estimating a vector x ∈Rn from measurements of the form y = Φx + w, (1) where Φ ∈Rm×n represents a known measurement matrix and w ∈Rm represents measurement errors or noise, is a generic problem that arises in a range of circumstances. One of the most basic estimators for x is the maximum a posteriori (MAP) estimate ˆxmap(y) = arg max x∈Rn px|y(x|y), (2) which is defined assuming some prior on x. For most priors, the MAP estimate is nonlinear and its behavior is not easily characterizable. Even if the priors for x and w are separable, the analysis of the MAP estimate may be difficult since the matrix Φ couples the n unknown components of x with the m measurements in the vector y. The primary contribution of this paper—an abridged version of [1]—is to show that with certain large random Φ and Gaussian w, there is an asymptotic decoupling of (1) into n scalar MAP estimation problems. Each equivalent scalar problem has an appropriate scalar prior and Gaussian noise with an effective noise level. The analysis yields the asymptotic joint distribution of each component xj of x and its corresponding estimate ˆxj in the MAP estimate vector ˆxmap(y). From the joint distribution, various further computations can be made, such as the mean-squared error (MSE) of the MAP estimate or the error probability of a hypothesis test computed from the MAP estimate. ∗This work was supported in part by a University of California President’s Postdoctoral Fellowship and National Science Foundation CAREER Award 0643836. 1 Replica Method. Our analysis is based on a powerful but non-rigorous technique from statistical physics known as the replica method. The replica method was originally developed by Edwards and Anderson [2] to study the statistical mechanics of spin glasses. Although not fully rigorous from the perspective of probability theory, the technique was able to provide explicit solutions for a range of complex problems where many other methods had previously failed [3]. The replica method was first applied to the study of nonlinear MAP estimation problems by Tanaka [4] and M¨uller [5]. These papers studied the behavior of the MAP estimator of a vector x with i.i.d. binary components observed through linear measurements of the form (1) with a large random Φ and Gaussian w. The results were then generalized in a remarkable paper by Guo and Verd´u [6] to vectors x with arbitrary distributions. Guo and Verd´u’s result was also able to incorporate a large class of minimum postulated MSE estimators, where the estimator may assume a prior that is different from the actual prior. The main result in this paper is the corresponding MAP statement to Guo and Verd´u’s result. In fact, our result is derived from Guo and Verd´u’s by taking appropriate limits with large deviations arguments. The non-rigorous aspect of the replica method involves a set of assumptions that include a selfaveraging property, the validity of a “replica trick,” and the ability to exchange certain limits. Some progress has been made in formally proving these assumptions; a survey of this work can be found in [7]. Also, some of the predictions of the replica method have been validated rigorously by other means [8]. To emphasize our dependence on these unproven assumptions, we will refer to Guo and Verd´u’s result as the Replica MMSE Claim. Our main result, which depends on Guo and Verd´u’s analysis, will be called the Replica MAP Claim. Applications to Compressed Sensing. As an application of our main result, we will develop a few analyses of estimation problems that arise in compressed sensing [9–11]. In compressed sensing, one estimates a sparse vector x from random linear measurements. Generically, optimal estimation of x with a sparse prior is NP-hard [12]. Thus, most attention has focused on greedy heuristics such as matching pursuit and convex relaxations such as basis pursuit [13] or lasso [14]. While successful in practice, these algorithms are difficult to analyze precisely. Recent compressed sensing research has provided scaling laws on numbers of measurements that guarantee good performance of these methods [15–17]. However, these scaling laws are in general conservative. There are, of course, notable exceptions including [18] and [19] which provide matching necessary and sufficient conditions for recovery of strictly sparse vectors with basis pursuit and lasso. However, even these results only consider exact recovery and are limited to measurements that are noise-free or measurements with a signal-to-noise ratio (SNR) that scales to infinity. Many common sparse estimators can be seen as MAP estimators with certain postulated priors. Most importantly, lasso and basis pursuit are MAP estimators assuming a Laplacian prior. Other commonly-used sparse estimation algorithms, including linear estimation with and without thresholding and zero norm-regularized estimators, can also be seen as MAP-based estimators. For these algorithms, the replica method provides—under the assumption of the replica hypotheses—not just bounds, but the exact asymptotic behavior. This in turns permits exact expressions for various performance metrics such as MSE or fraction of support recovery. The expressions apply for arbitrary ratios k/n, n/m and SNR. 2 Estimation Problem and Assumptions Consider the estimation of a random vector x ∈Rn from linear measurements of the form y = Φx + w = AS1/2x + w, (3) where y ∈Rm is a vector of observations, Φ = AS1/2, A ∈Rm×n is a measurement matrix, S is a diagonal matrix of positive scale factors, S = diag (s1, . . . , sn) , sj > 0, (4) and w ∈Rm is zero-mean, white Gaussian noise. We consider a sequence of such problems indexed by n, with n →∞. For each n, the problem is to determine an estimate bx of x from the observations y knowing the measurement matrix A and scale factor matrix S. 2 The components xj of x are modeled as zero mean and i.i.d. with some prior probability distribution p0(xj). The per-component variance of the Gaussian noise is E|wj|2 = σ2 0. We use the subscript “0” on the prior and noise level to differentiate these quantities from certain “postulated” values to be defined later. In (3), we have factored Φ = AS1/2 so that even with the i.i.d. assumption on xjs above and an i.i.d. assumption on entries of A, the model can capture variations in powers of the components of x that are known a priori at the estimator. Variations in the power of x that are not known to the estimator should be captured in the distribution of x. We summarize the situation and make additional assumptions to specify the problem precisely as follows: (a) The number of measurements m = m(n) is a deterministic quantity that varies with n and satisfies lim n→∞n/m(n) = β for some β ≥0. (The dependence of m on n is usually omitted for brevity.) (b) The components xj of x are i.i.d. with probability distribution p0(xj). (c) The noise w is Gaussian with w ∼N(0, σ2 0Im). (d) The components of the matrix A are i.i.d. zero mean with variance 1/m. (e) The scale factors sj are i.i.d. and satisfy sj > 0 almost surely. (f) The scale factor matrix S, measurement matrix A, vector x and noise w are independent. 3 Review of the Replica MMSE Claim We begin by reviewing the Replica MMSE Claim of Guo and Verd´u [6]. Suppose one is given a “postulated” prior distribution ppost and a postulated noise level σ2 post that may be different from the true values p0 and σ2 0. We define the minimum postulated MSE (MPMSE) estimate of x as ˆxmpmse(y) = E x | y ; ppost, σ2 post = Z xpx|y(x | y ; ppost, σ2 post) dx, where px|y(x | y ; q, σ2) is the conditional distribution of x given y under the x distribution and noise variance specified as parameters after the semicolon: px|y(x | y ; q, σ2) = C−1 exp −1 2σ2 ∥y −AS1/2x∥2 q(x), q(x) = n Y j=1 q(xj), (5) where C is a normalization constant. The Replica MMSE Claim describes the asymptotic behavior of the postulated MMSE estimator via an equivalent scalar estimator. Let q(x) be a probability distribution defined on some set X ⊆R. Given µ > 0, let px|z(x | z ; q, µ) be the conditional distribution px|z(x | z ; q, µ) = Z x∈X φ(z −x ; µ)q(x) dx −1 φ(z −x ; µ)q(x) (6) where φ(·) is the Gaussian distribution φ(v ; µ) = 1 √2πµe−|v|2/(2µ). (7) The distribution px|z(x|z ; q, µ) is the conditional distribution of the scalar random variable x ∼ q(x) from an observation of the form z = x + √µv, (8) where v ∼N(0, 1). Using this distribution, we can define the scalar conditional MMSE estimate, ˆxmmse scalar(z ; q, µ) = Z x∈X xpx|z(x|z ; µ) dx. (9) 3 Also, given two distributions, p0(x) and p1(x), and two noise levels, µ0 > 0 and µ1 > 0, define mse(p1, p0, µ1, µ0, z) = Z x∈X |x −ˆxmmse scalar(z ; p1, µ1)|2px|z(x | z ; p0, µ0) dx, (10) which is the mean-squared error in estimating the scalar x from the variable z in (8) when x has a true distribution x ∼p0(x) and the noise level is µ = µ0, but the estimator assumes a distribution x ∼p1(x) and noise level µ = µ1. Replica MMSE Claim [6]. Consider the estimation problem in Section 2. Let ˆxmpmse(y) be the MPMSE estimator based on a postulated prior ppost and postulated noise level σ2 post. For each n, let j = j(n) be some deterministic component index with j(n) ∈{1, . . . , n}. Then there exist effective noise levels σ2 effand σ2 p−effsuch that: (a) As n →∞, the random vectors (xj, sj, ˆxmpmse j ) converge in distribution to the random vector (x, s, ˆx) where x, s, and v are independent with x ∼p0(x), s ∼pS(s), v ∼N(0, 1), and ˆx = ˆxmmse scalar(z ; ppost, µp), z = x + √µv. (11) where µ = σ2 eff/s and µp = σ2 p−eff/s. (b) The effective noise levels satisfy the equations σ2 eff = σ2 0 + βE [s mse(ppost, p0, µp, µ, z)] (12a) σ2 p−eff = σ2 post + βE [s mse(ppost, ppost, µp, µp, z)] , (12b) where the expectations are taken over s ∼pS(s) and z generated by (11). The Replica MMSE Claim asserts that the asymptotic behavior of the joint estimation of the ndimensional vector x can be described by n equivalent scalar estimators. In the scalar estimation problem, a component x ∼p0(x) is corrupted by additive Gaussian noise yielding a noisy measurement z. The additive noise variance is µ = σ2 eff/s, which is the effective noise divided by the scale factor s. The estimate of that component is then described by the (generally nonlinear) scalar estimator ˆx(z ; ppost, µp). The effective noise levels σ2 effand σ2 p−effare described by the solutions to fixed-point equations (12). Note that σ2 effand σ2 p−effappear implicitly on the left- and right-hand sides of these equations via the terms µ and µp. When there are multiple solutions to these equations, the true solution is the minimizer of a certain Gibbs’ function [6]. 4 Replica MAP Claim We now turn to MAP estimation. Let X ⊆R be some (measurable) set and consider an estimator of the form ˆxmap(y) = arg min x∈X n 1 2γ ∥y −AS1/2x∥2 2 + n X j=1 f(xj), (13) where γ > 0 is an algorithm parameter and f : X →R is some scalar-valued, non-negative cost function. We will assume that the objective function in (13) has a unique essential minimizer for almost all y. The estimator (13) can be interpreted as a MAP estimator. Specifically, for any u > 0, it can be verified that ˆxmap(y) is the MAP estimate ˆxmap(y) = arg max x∈X n px|y(x | y ; pu, σ2 u), where pu(x) and σ2 u are the prior and noise level pu(x) = Z x∈X n exp(−uf(x))dx −1 exp(−uf(x)), σ2 u = γ/u, (14) 4 where f(x) = P j f(xj). To analyze this MAP estimator, we consider a sequence of MMSE estimators bxu(y) = E x | y ; pu, σ2 u . (15) The proof of the Replica MAP Claim below (see [1]) uses a standard large deviations argument to show that lim u→∞bxu(y) = ˆxmap(y) for all y. Under the assumption that the behaviors of the MMSE estimators are described by the Replica MMSE Claim, we can then extrapolate the behavior of the MAP estimator. To state the claim, define the scalar MAP estimator ˆxmap scalar(z ; λ) = arg min x∈X F(x, z, λ), F(x, z, λ) = 1 2λ|z −x|2 + f(x). (16) where, again, we assume that (16) has a unique essential minimizer for almost all λ and almost all z. We also assume that the limit σ2(z, λ) = lim x→ˆx |x −ˆx|2 2(F(x, z, λ) −F(ˆx, z, λ)), (17) exists where ˆx = ˆxmap scalar(z; λ). We make the following additional assumptions: Assumption 1 Consider the MAP estimator (13) applied to the estimation problem in Section 2. Assume: (a) For all u > 0 sufficiently large, assume the postulated prior pu and noise level σ2 u satisfy the Replica MMSE Claim. Also, assume that for the corresponding effective noise levels, σ2 eff(u) and σ2 p−eff(u), the following limits exists: σ2 eff,map = lim u→∞σ2 eff(u), γp = lim u→∞uσ2 p−eff(u). (b) Suppose for each n, ˆxu j (n) is the MMSE estimate of the component xj for some index j ∈{1, . . . , n} based on the postulated prior pu and noise level σ2 u. Then, assume that the following limits can be interchanged: lim u→∞lim n→∞ˆxu j (n) = lim n→∞lim u→∞ˆxu j (n), where the limits are in distribution. (c) Assume that f(x) is non-negative and satisfies f(x)/ log |x| →∞as |x| →∞. Item (a) is stated to reiterate that we are assuming the Replica MMSE Claim is valid. See [1, Sect. IV] for additional discussion of technical assumptions. Replica MAP Claim [1]. Consider the estimation problem in Section 2. Let ˆxmap(y) be the MAP estimator (13) defined for some f(x) and γ > 0 satisfying Assumption 1. For each n, let j = j(n) be some deterministic component index with j(n) ∈{1, . . . , n}. Then: (a) As n →∞, the random vectors (xj, sj, ˆxmap j ) converge in distribution to the random vector (x, s, ˆx) where x, s, and v are independent with x ∼p0(x), s ∼pS(s), v ∼N(0, 1), and ˆx = ˆxmap scalar(z, λp), z = x + √µv, (18) where µ = σ2 eff,map/s and λp = γp/s. (b) The limiting effective noise levels σ2 eff,map and γp satisfy the equations σ2 eff,map = σ2 0 + βE s|x −ˆx|2 (19a) γp = γ + βE sσ2(z, λp) , (19b) where the expectations are taken over x ∼p0(x), s ∼pS(s), and v ∼N(0, 1), with ˆx and z defined in (18). 5 Analogously to the Replica MMSE Claim, the Replica MAP Claim asserts that asymptotic behavior of the MAP estimate of any single component of x is described by a simple equivalent scalar estimator. In the equivalent scalar model, the component of the true vector x is corrupted by Gaussian noise and the estimate of that component is given by a scalar MAP estimate of the component from the noise-corrupted version. 5 Analysis of Compressed Sensing Our results thus far hold for any separable distribution for x and under mild conditions on the cost function f. The role of f is to determine the estimator. In this section, we first consider choices of f that yield MAP estimators relevant to compressed sensing. We then additionally impose a sparse prior for x for numerical evaluations of asymptotic performance. Lasso Estimation. We first consider the lasso or basis pursuit estimate [13,14] given by ˆxlasso(y) = arg min x∈Rn 1 2γ ∥y −AS1/2x∥2 2 + ∥x∥1, (20) where γ > 0 is an algorithm parameter. This estimator is identical to the MAP estimator (13) with the cost function f(x) = |x|. With this cost function, the scalar MAP estimator in (16) is given by ˆxmap scalar(z ; λ) = T soft λ (z), (21) where T soft λ (z) is the soft thresholding operator T soft λ (z) = ( z −λ, if z > λ; 0, if |z| ≤λ; z + λ, if z < −λ. (22) The Replica MAP Claim now states that there exists effective noise levels σ2 eff,map and γp such that for any component index j, the random vector (xj, sj, ˆxj) converges in distribution to the vector (x, s, ˆx) where x ∼p0(x), s ∼pS(s), and ˆx is given by ˆx = T soft λp (z), z = x + √µv, (23) where v ∼N(0, 1), λp = γp/s, and µ = σ2 eff,map/s. Hence, the asymptotic behavior of lasso has a remarkably simple description: the asymptotic distribution of the lasso estimate ˆxj of the component xj is identical to xj being corrupted by Gaussian noise and then soft-thresholded to yield the estimate ˆxj. To calculate the effective noise levels, one can perform a simple calculation to show that σ2(z, λ) in (17) is given by σ2(z, λ) = λ, if |z| > λ; 0, if |z| ≤λ. (24) Hence, E sσ2(z, λp) = E [sλp Pr(|z| > λp)] = γp Pr(|z| > γp/s) (25) where we have use the fact that λp = γp/s. Substituting (21) and (25) into (19), we obtain the fixed-point equations σ2 eff,map = σ2 0 + βE h s|x −T soft λp (z)|2i (26a) γp = γ + βγp Pr(|z| > γp/s), (26b) where the expectations are taken with respect to x ∼p0(x), s ∼pS(s), and z in (23). Again, while these fixed-point equations do not have a closed-form solution, they can be relatively easily solved numerically given distributions of x and s. 6 Zero Norm-Regularized Estimation. Lasso can be regarded as a convex relaxation of zero normregularized estimation ˆxzero(y) = arg min x∈Rn 1 2γ ∥y −AS1/2x∥2 2 + ∥x∥0, (27) where ∥x∥0 is the number of nonzero components of x. For certain strictly sparse priors, zero norm-regularized estimation may provide better performance than lasso. While computing the zero norm-regularized estimate is generally very difficult, we can use the replica analysis to provide a simple characterization of its performance. This analysis can provide a bound on the performance achievable by practical algorithms. The zero norm-regularized estimator is identical to the MAP estimator (13) with the cost function f(x) = 0, if x = 0; 1, if x ̸= 0. (28) Technically, this cost function does not satisfy the conditions of the Replica MAP Claim. To avoid this problem, we can consider an approximation of (28), fδ,M(x) = 0, if |x| < δ; 1, if |x| ∈[δ, M], which is defined on the set X = {x : |x| ≤M}. We can then take the limits δ →0 and M →∞. To simplify the presentation, we will just apply the Replica MAP Claim with f(x) in (28) and omit the details in taking the appropriate limits. With f(x) given by (28), the scalar MAP estimator in (16) is given by ˆxmap scalar(z ; λ) = T hard t (z), t = √ 2λ, (29) where T hard t is the hard thresholding operator, T hard t (z) = z, if |z| > t; 0, if |z| ≤t. (30) Now, similar to the case of lasso estimation, the Replica MAP Claim states there exists effective noise levels σ2 eff,map and γp such that for any component index j, the random vector (xj, sj, ˆxj) converges in distribution to the vector (x, s, ˆx) where x ∼p0(x), s ∼pS(s), and ˆx is given by ˆx = T hard t (z), z = x + √µv, (31) where v ∼N(0, 1), λp = γp/s, µ = σ2 eff,map/s, and t = p 2λp = q 2γp/s. (32) Thus, the zero norm-regularized estimation of a vector x is equivalent to n scalar components corrupted by some effective noise level σ2 eff,map and hard-thresholded based on a effective noise level γp. The fixed-point equations for the effective noise levels σ2 eff,map and γp can be computed similarly to the case of lasso. Specifically, one can verify that (24) and (25) are both satisfied for the hard thresholding operator as well. Substituting (25) and (29) into (19), we obtain the fixed-point equations σ2 eff,map = σ2 0 + βE s|x −T hard t (z)|2 , (33a) γp = γ + βγp Pr(|z| > t), (33b) where the expectations are taken with respect to x ∼p0(x), s ∼pS(s), z in (31), and t given by (32). These fixed-point equations can be solved numerically. 7 0.5 1 1.5 2 2.5 3 −18 −16 −14 −12 −10 −8 −6 −4 −2 0 Measurement ratio β = n/m Median squared error (dB) Linear (replica) Linear (sim.) Lasso (replica) Lasso (sim.) Zero norm−reg Optimal MMSE Figure 1: MSE performance prediction with the Replica MAP Claim. Plotted is the median normalized SE for various sparse recovery algorithms: linear MMSE estimation, lasso, zero normregularized estimation, and optimal MMSE estimation. Solid lines show the asymptotic predicted MSE from the Replica MAP Claim. For the linear and lasso estimators, the circles and triangles show the actual median SE over 1000 Monte Carlo simulations. Numerical Simulation. To validate the predictive power of the Replica MAP Claim for finite dimensions, we performed numerical simulations where the components of x are a zero-mean Bernoulli–Gaussian process. Specifically, xj ∼ N(0, 1), with prob. 0.1; 0, with prob. 0.9. We took the vector x to have n = 100 i.i.d. components, and we used ten values of m to vary β = n/m from 0.5 to 3. For each problem size, we simulated the lasso and linear MMSE estimators over 1000 independent instances with noise levels chosen such that the SNR with perfect side information is 10 dB. Each set of trials is represented by its median squared error in Fig. 1. The simulated performance is matched very closely by the asymptotic values predicted by the replica analysis. (Analysis of the linear MMSE estimator using the Replica MAP Claim is detailed in [1]; the Replica MMSE Claim is also applicable to this estimator.) In addition, the replica analysis can be applied to zero norm-regularized and optimal MMSE estimators that are computationally infeasible for large problems. These results are also shown in Fig. 1, illustrating the potential of the replica method to quantify the precise performance losses of practical algorithms. Additional numerical simulations in [1] illustrate convergenceto the replica MAP limit, applicability to discrete distributions for x, effects of power variations in the components, and accurate prediction of the probability of sparsity pattern recovery. 6 Conclusions We have shown that the behavior of vector MAP estimators with large random measurement matrices and Gaussian noise asymptotically matches that of a set of decoupled scalar estimation problems. We believe that this equivalence to a simple scalar model will open up numerous doors for analysis, particularly in problems of interest in compressed sensing. One can use the model to dramatically improve upon existing performance analyses for sparsity pattern recovery and MSE. Also, the technique is sufficiently general to study effects of dynamic range. 8 References [1] S. Rangan, A. K. Fletcher, and V. K. Goyal. Asymptotic analysis of MAP estimation via the replica method and applications to compressed sensing. arXiv:0906.3234v1 [cs.IT]., June 2009. [2] S. F. Edwards and P. W. Anderson. Theory of spin glasses. J. Phys. F: Metal Physics, 5:965– 974, 1975. [3] H. Nishimori. Statistical physics of spin glasses and information processing: An introduction. International Series of Monographs on Physics. Oxford Univ. Press, Oxford, UK, 2001. [4] T. Tanaka. A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors. IEEE Trans. Inform. Theory, 48(11):2888–2910, November 2002. [5] R. R. M¨uller. Channel capacity and minimum probability of error in large dual antenna array systems with binary modulation. IEEE Trans. Signal Process., 51(11):2821–2828, November 2003. [6] D. Guo and S. Verd´u. Randomly spread CDMA: Asymptotics via statistical physics. IEEE Trans. Inform. Theory, 51(6):1983–2010, June 2005. [7] M. Talagrand. Spin Glasses: A Challenge for Mathematicians. Springer, New York, 2003. [8] A. Montanari and D. Tse. Analysis of belief propagation for non-linear problems: The example of CDMA (or: How to prove Tanaka’s formula). arXiv:cs/0602028v1 [cs.IT]., February 2006. [9] E. J. Cand`es, J. Romberg, and T. Tao. Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory, 52(2):489– 509, February 2006. [10] D. L. Donoho. Compressed sensing. IEEE Trans. Inform. Theory, 52(4):1289–1306, April 2006. [11] E. J. Cand`es and T. Tao. Near-optimal signal recovery from random projections: Universal encoding strategies? IEEE Trans. Inform. Theory, 52(12):5406–5425, December 2006. [12] B. K. Natarajan. Sparse approximate solutions to linear systems. SIAM J. Computing, 24(2):227–234, April 1995. [13] S. S. Chen, D. L. Donoho, and M. A. Saunders. Atomic decomposition by basis pursuit. SIAM J. Sci. Comp., 20(1):33–61, 1999. [14] R. Tibshirani. Regression shrinkage and selection via the lasso. J. Royal Stat. Soc., Ser. B, 58(1):267–288, 1996. [15] D. L. Donoho, M. Elad, and V. N. Temlyakov. Stable recovery of sparse overcomplete representations in the presence of noise. IEEE Trans. Inform. Theory, 52(1):6–18, January 2006. [16] J. A. Tropp. Greed is good: Algorithmic results for sparse approximation. IEEE Trans. Inform. Theory, 50(10):2231–2242, October 2004. [17] J. A. Tropp. Just relax: Convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory, 52(3):1030–1051, March 2006. [18] M. J. Wainwright. Sharp thresholds for high-dimensional and noisy sparsity recovery using ℓ1-constrained quadratic programming (lasso). IEEE Trans. Inform. Theory, 55(5):2183–2202, May 2009. [19] D. L. Donoho and J. Tanner. Counting faces of randomly-projected polytopes when the projection radically lowers dimension. J. Amer. Math. Soc., 22(1):1–53, January 2009. 9
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Statistical Models of Linear and Non–linear Contextual Interactions in Early Visual Processing Ruben Coen–Cagli AECOM Bronx, NY 10461 rcagli@aecom.yu.edu Peter Dayan GCNU, UCL 17 Queen Square, LONDON dayan@gatsby.ucl.ac.uk Odelia Schwartz AECOM Bronx, NY 10461 oschwart@aecom.yu.edu Abstract A central hypothesis about early visual processing is that it represents inputs in a coordinate system matched to the statistics of natural scenes. Simple versions of this lead to Gabor–like receptive fields and divisive gain modulation from local surrounds; these have led to influential neural and psychological models of visual processing. However, these accounts are based on an incomplete view of the visual context surrounding each point. Here, we consider an approximate model of linear and non–linear correlations between the responses of spatially distributed Gaborlike receptive fields, which, when trained on an ensemble of natural scenes, unifies a range of spatial context effects. The full model accounts for neural surround data in primary visual cortex (V1), provides a statistical foundation for perceptual phenomena associated with Li’s (2002) hypothesis that V1 builds a saliency map, and fits data on the tilt illusion. 1 Introduction That visual input at a given point is greatly influenced by its spatial context is manifest in a host of neural and perceptual effects (see, e.g., [1, 2]). For instance, stimuli surrounding the so-called classical receptive field (RF) lead to striking nonlinearities in the responses of visual neurons [3, 4]; spatial context results in intriguing perceptual illusions, such as the misjudgment of a center stimulus attribute in the presence of a surrounding stimulus [5–7]; it also plays a critical role in determining the salience of points in visual space, for instance controlling pop-out, contour integration, texture segmentation [8–10] and more generally locations where statistical homogeneity of the input breaks down [1]. Contextual effects are widespread across sensory systems, neural areas, and stimulus attributes — making them an attractive target for computational modeling. There are various mechanistic treatments of extra-classical RF effects (e.g.,[11–13]) and contour integration [14], and V1’s suggested role in computing salience has been realized in a large-scale dynamical model [1, 15]. There are also normative approaches to salience (e.g., [16–19]) with links to V1. However, these have not substantially encompassed neurophysiological data or indeed made connections with the perceptual literature on contour integration and the tilt illusion. Our aim is to build a principled model based on scene statistics that can ultimately account for, and therefore unify, the whole set of contextual effects above. Much seminal work has been done in the last two decades on learning linear filters from first principles from the statistics of natural images (see e.g. [20]). However, contextual effects emerge from the interactions among multiple filters; therefore here we address the much less well studied issue of the learned, statistical, basis of the coordination of the group of filters — the scene–dependent, linear and non–linear interactions among them. We focus on recent advances in models of scene statistics, using a Gaussian Scale Mixture generative model (GSM; [21–23]) that captures the joint dependencies (e.g., [24–31]) between the activations of Gabor–like filters to natural scenes. The 1 GSM captures the dependencies via two components, (i) covariance in underlying Gaussian variables, which accounts for linear correlations in the activations of filters; and (ii) a shared mixer variable, which accounts for the non–linear correlations in the magnitudes of the filter activations. As yet, the GSM has not been applied to the wide range of contextual phenomena discussed above. This is partly because linear correlations, which appear important to capture phenomena such as contour integration, have largely been ignored outside image processing (e.g., [23]). In addition, although the mixer variable of the GSM is closely related to bottom-up models of divisive normalization in cortex [32, 33], the assignment problem of grouping filters that share a common mixer for a given scene has yet to receive a computationally and neurobiologically realistic solution. Recent work has shown that incorporating a simple, predetermined, solution to the assignment problem in a GSM could capture the tilt illusion [34]. Nevertheless, the approach has not been studied in a more realistic model with Gabor-like filters, and learning assignments from natural scenes. Further, the implications of assignment for cortical V1 data and salience have not been explored. In this paper we extend the GSM model to learn both assignments and linear covariance (section 2). We then apply the model to contextual neural V1 data, noting its link to the tilt illusion (section 3); and then to perceptual salience examples (section 4). In the discussion (section 5), we also describe the relationship between our GSM model and other recent scene statistics approaches (e.g., [31, 35]). 2 Methods A recent focus in natural image statistics has been the joint conditional histograms of the activations of pairs of oriented linear filters (throughout the paper, filters come from the first level of a steerable pyramid with 4 orientations [36]). When filter pairs are proximal in space, these histograms have a characteristic bowtie shape: the variance of one filter depends on the magnitude of activation of the other. It has been shown [22] that this form of dependency can be captured by a class of generative model known as Gaussian Scale Mixture (GSM), which assumes that the linear filter activations x = vg are random variables defined as the product of two other random variables, a multivariate Gaussian g, and a (positive) scalar v which scales the variance of all the Gaussian components. Here, we address two additional properties of natural scenes. First, in addition to the variance dependency, filters which are close enough in space and feature space are linearly dependent, as shown by the tilt of the bowtie in fig. 1b. In order for the GSM to capture this effect, the multivariate Gaussian must be endowed with a non-diagonal covariance matrix. This matrix can be approximated by the sample covariance matrix of the filter activations or learned directly [23]; here, we learn it by maximizing the likelihood of the observed data. The second issue is that filter dependencies differ across image patches, implying that there is no fixed relationship between mixers and filters [28]. The general issue of learning multiple pools of filters, each assigned to a different mixer on an patch–dependent basis, has been addressed in recent work [30], but using a computationally and biologically impracticable scheme [37] which allowed for arbitrary pooling. We consider an approximation to the assignment problem, by allowing a group of surround filters to either share or not the same mixer with a target filter. While this is clearly an oversimplified model of natural images, here we aimed for a reasonable balance between the complexity of the model, and biological plausibility of the computations involved. 2.1 The generative model The basic repeating unit of our simplified model involves center and surround groups of filters: we use nc to denote the number of center filters, and xc their activations; similarly, we use ns and xs for the surround; finally, we define ncs .= nc+ns and x .= (x1 c, . . . xnc c , x1 s, . . . xns s )⊤. We consider a single assignment choice as to whether the center group’s mixer variable vc is (case ξ1), or is not (case ξ2) shared with the surround, which in the latter case would have its own mixer variable vs. Thus, there are 2 configurations, or competing models, which are themselves combined (i.e., a mixture of GSM’s, see also [35]). The graphical models of the two configurations are shown in Fig. 1a. We show this from the perspective of the center group, since in the implementation we will be reporting model neuron responses in the center location given the contextual surround. 2 Defining Gaussian components as g .= (g1 c, . . . gnc c , g1 s, . . . gns s )⊤, and assuming the mixers are independent and the pools are independent given the mixers, the mixture distribution is: p(x) = p(ξ1)p(x | ξ1) + p(ξ2)p(x | ξ2) (1) p(x | ξ1) = R dvc p(vc)p(x | vc, ξ1) (2) p(x | ξ2) = R dvc p(vc)p(xc | vc, ξ2) R dvs p(vs)p(xs | vs, ξ2) (3) We assume a Rayleigh prior distribution on the mixers, and covariance matrix Σcs for the Gaussian components for ξ1, and Σc and Σs for center and surround, respectively, for ξ2. The integrals in eqs. (2,3) can then be solved analytically: p(x | ξ1) = det(Σ−1 cs ) 1 2 (2π) ncs 2 B(1 −ncs 2 ; λcs) λ ( ncs 2 −1) cs (4) p(x | ξ2) = det(Σ−1 c ) 1 2 (2π) nc 2 B(1 −nc 2 ; λc) λ ( nc 2 −1) c det(Σ−1 s ) 1 2 (2π) ns 2 B(1 −ns 2 ; λs) λ ( ns 2 −1) s (5) where B is the modified Bessel function of the second kind, and λcs = p x⊤Σ−1 cs x. 2.2 Learning The parameters to be estimated are the covariance matrices (Σcs, Σc, Σs) and the prior probability (k) that center and surround share the same pool; we use a Generalized Expectation Maximization algorithm, specifically Multi Cycle EM [38], where a full EM cycle is divided into three subcycles, each involving a full E-step and a partial M-step performed only on one covariance matrix. E-step: In the E-step we compute an estimate, Q, of the posterior distribution over the assignment variable, given the filter activations and the previous estimates of the parameters, namely kold and Θold .= n Σc, Σs, Σcs o . This is obtained via Bayes rule: Q(ξ1) = p(ξ1 | x, Θold) ∝kold p(x | ξ1, Θold) (6) Q(ξ2) = p(ξ2 | x, Θold) ∝(1 −kold) p(x | ξ2, Θold) (7) M-step: In the M-step we increase the complete–data Log Likelihood, namely: f = Q(ξ1) log [k p(x | ξ1, Θ)] + Q(ξ2) log [(1 −k)p(x | ξ2, Θ)] (8) Solving ∂f/∂k = 0, we obtain k⋆.= arg maxk [f] = Q(ξ1). The other terms cannot be solved analytically, and a numerical procedure must be adopted to maximize f w.r.t. the covariance matrices. This requires an explicit form for the gradient: ∂f ∂Σ−1 cs = Q(ξ1) Σcs 2 − 1 2λcs B(−ncs 2 ; λcs) B(1 −ncs 2 ; λcs)xx⊤ (9) Similar expressions hold for the other partial derivatives. In practice, we add the constraint that the covariances of the surround filters are spatially symmetric. 2.3 Inference: patch–by–patch assignment and model neural unit Upon convergence of EM, the covariance matrices and prior k over the assignment are found. Then, for a new image patch, the probability p(ξ1 | x) that the surround shares a common mixer with the center is inferred. The output of the center group is taken to be the estimate (for the present, we consider just the mean) of the Gaussian component E [gc | x], which we take to be our model neural unit response. To estimate the normalized response of the center filter, we need to compute the following expected value under the full model: E [gc | x] = Z dgc gc p(gc | x) = p(ξ1 | x)E [gc | x , ξ1] + p(ξ2 | x)E [gc | xc , ξ2] (10) the r.h.s., obtained by a straightforward calculation applying Bayes rule and the conditional independence of xs from xc, gc given ξ2, is the sum of the expected value of gc in the two configurations, 3 Figure 1: (a) Graphical model for the two components of the mixture of GSMs, where the center filter is (ξ1; left) or is not (ξ2; right) normalized by the surround filters; (b) joint conditional histogram of two linear filters activations, showing the typical bowtie shape due to the variance dependency, as well as a tilt due to linear dependencies between the two filters; (c) marginal distribution of linear activations in black, estimated Gaussian component in blue, and ideal Gaussian in red. The estimated distribution is closer to a Gaussian than that of the original filter. weighted by their posterior probabilities. The explicit form for the estimate of the i-th component (corresponding in the implementation to a given orientation and phase) of gc under ξ1 is: E gi c | x , ξ1 = sign(xi c) p |xic| s |xic| λcs B( 1 2 −ncs 2 ; λcs) B(1 −ncs 2 ; λcs) (11) and a similar expression holds under ξ2, replacing the subscript cs by c. Note that in either configuration, the mixer variable’s effect on this is a form of divisive normalization or gain control, through λ (including for stability, as in [30], an additive constant set to 1 for the λ values; we omit the formulæ to save space). Under ξ1, but not ξ2, this division is influenced by the surround1. Note also that, due to the presence of the inverse covariance matrix in λcs = p x⊤Σ−1 cs x, the gain control signal is reduced when there is strong covariance, which in turn enhances the neural unit response. 3 Cortical neurophysiology simulations To simulate neurophysiological experiments, we consider the following filter configuration: 3 × 3 spatial positions separated by 6 pixels, 2 phases (quadrature pair), and one orientation (vertical), plus 3 additional orientations in the central position to allow for cross–orientation gain control. We first learn the parameters of the model for 25000 patches from an ensemble of 5 standard scenes (Einstein, Goldhill, and so on). We take as our model neuron the absolute value of the complex activation composed by the non–linear responses (eq. (11)) of two phases of the central vertical filter. We characterize the neuron’s basic properties with a procedure that is common in physiology experiments focusing on contextual non–linear modulation. First, we measure the so-called Area Summation curve, namely the response to gratings that are optimal in orientation and spatial frequency, as a function of size. Cat and monkey experiments have shown striking non–linearities, with the peak response at low contrasts being for significantly larger diameters than at high contrasts (Figure 2a). We obtain the same behavior in the model (Figure 2b; see also [33]). This behavior is due to the assignment: for small grating sizes, center and surround have a higher posterior probability at high contrast than at low contrast, and therefore the surround exerts stronger gain control. In a reduced model with no assignment, we obtain a much weaker effect (Figure 2c). We then assess the modulatory effect of a surround grating on a fixed, optimally–oriented central grating, as a function of their relative orientations (Figure 3a). As is common, we determine the spatial extent of the center and surround stimuli based on the area summation curves (see [4]). The model simulations (Figure 3b), as in the data, exhibit the most reduced responses when the center and surround have similar orientation (but note the ”blip” when they are exactly equal in Figure 3a;b, which arises in the model from the covariance of the Gaussian; see also [31]). In addition, 1To ensure that the mixer follows the same distribution under ξ1 and ξ2, after training with natural images we rescale vc — and therefore gc — so that their values span the same range in both configurations; since the assignment is made at a higher level in the hierarchy, such a rescaling is equivalent to downstream normalization processes that make the estimates of gc comparable under ξ1 and ξ2. 4 Figure 2: Area summation curves show the normalized firing rate of a neuron in response to optimal gratings of increasing size. (a) a V1 neuron, after [4]; (b) the model neuron described in Sec. 3; (c) a reduced model assuming that the surround filters are always in the gain pool of the center filter. Figure 3: Orientation tuning of the surround. (a) and (b): normalized firing rate in response to a stimulus composed by an optimal central grating surrounded by an annular grating of varying orientation, for (a) a V1 neuron, after [3]; and (b) the model neuron described in Sec. 3 (c) Probability that the surround normalizes the center as a function of the relative orientation of the annular grating. as the orientation difference between center and surround grows, the response increases and then decreases, an effect that arises from the assignments. In the model simulations, we find that the strength of this behavior depends on contrast, being larger at low contrasts, an effect of which there are hints in the neurophysiology experimental data (using contrasts of 0.2 to 0.4) but which has yet to be systematically explored. Figure 3c shows the posterior assignment probability for the same two contrasts as in figure 3b, as a function of the surround orientation. These remain close to 1 at all orientations at high contrast, but fall off more rapidly at low contrast. Note that a previous GSM population model assumed (but did not learn) this form of fall off of the posterior weights of of figure 3c, and showed that it is a basis for explaining the so-called direct and indirect biases in the tilt illusion; i.e., repulsion and attraction in the perception of a center stimulus orientation in the presence of a surround stimulus [34]. Figure 4 compares the GSM model of [34] designed with parameters matched to perceptual data, to the result of our learned model. The qualitative shape (although not the quantitative strength) of the effects are similar. 4 Salience popout and contour integration simulations To address perceptual salience effects, we need a population model of oriented units. We consider one distinct group of filters – arranged as for the model neuron in Sec. 3 – for each of four orientations (0, 45, 90, 135 deg, sampling more coarsely than [1]). We compute the non–linear response of each model neuron as in Sec. 3, and take the maximum across the four orientations as the population output, as in standard population decoding. This is performed at each pixel of the input image, and the result is interpreted as a saliency map. We first consider the popout of a target that differs from a background of distractors by a single feature (eg. [8]), in our case orientation. Input image and output saliency map (the brighter, the more salient) are shown in Fig. 5. As in [1], the target pops out since it is less suppressed by its own, orthogonally-oriented neighbors than the surround bars are by their parallel ones; here, this emerges straight from normative inference. [8] quantified relative target saliency above detection threshold 5 Figure 4: The tilt illusion. (a) Comparison of the learned GSM model (black, solid line with filled squares), with the GSM model in [34] (blue, solid line; parameters set to account for the illusion data of [39]), and the model in [34] with parameters modified to match the learned model (blue, dashed line). The response of each neuron in the population is plotted as a function of the difference between the surround stimulus orientation and the preferred center stimulus orientation. We assume all oriented neurons have identical properties to the learned vertical neuron (i.e., ignoring the oblique effect). The model of [34] includes idealized tuning curves. The learned model is as in the previous section, but with filters of narrower orientation tuning (because of denser sampling of 16 orientations in the pyramid), which results in an earlier point on the x axis of maximal response. Model simulations are normalized to a maximum of 1. (b) Simulations of the tilt illusion using the model in [34], based on parameters matched to the learned model (dashed line) versus parameters matched to the data of [39] (solid line). as a function of the difference in orientation between target and distractors using luminance (Fig. 5b). Fig. 5c plots saliency from the model; it exhibits non–linear saturation for large orientation contrast, an effect that not all saliency models capture (see [17] for discussion). The shape of the saturation is different for neural (Fig. 2a-b) versus perceptual (Fig. 5b-c) data, in both experiment and model; for the latter, this arises from differences in stimuli (gratings versus bars, how the center and surround extents were determined). The second class of saliency effects involves collinear facilitation. One example is the so called border effect, shown in figure 6a – one side of the border, whose individual bars are collinear, is more salient than the other (e.g. [1], but see also [40]). The middle and right plots in figure 6a depict the saliency map for the full model and a reduced model that uses a diagonal covariance matrix. Notice that the reduced model also shows an enhancement of the collinear side of the border vs the parallel, due to the partial overlap of the linear receptive fields; but, as explained in Sec. 2.3, the higher covariance between collinear filters in the full model, strengthens the effect. To quantify the difference, we report also the ratio between the salience values on the collinear and parallel sides of the border, after subtracting the saliency value of the homogeneous regions: the lower value for the reduced model (1.28; versus 1.74 for the full model) shows that the full model enhances the collinear relative to the parallel side. The ratio for the full model increases if we rescale the off– diagonal terms of the covariance matrix relative to the diagonal (2.1 for a scaling factor of 1.5; 2.73 for a factor of 2). Rescaling would come from more reasonably dense spatial sampling. Fig. 6b provides another, stronger example of the collinear facilitation. 5 Discussion We have extended a standard GSM generative model of scene statistics to encompass contextual effects. We modeled the covariance between the Gaussian components associated with neighboring locations, and suggested a simple, approximate, process for choosing whether or not to pool such locations under the same mixer. Using parameters learned from natural scenes, we showed that this model provides a promising account of neurophysiological data on area summation and centersurround orientation contrast, and perceptual data on the saliency of image elements. This form of model has previously been applied to the tilt illusion [34], but had just assumed the assignments of figure 3c, in order to account for the indirect tilt illusion. Here, this emerged from first principles. This model therefore unifies a wealth of data and ideas about contextual visual processing. To our 6 Figure 5: (a) An example of the stimulus and saliency map computed by the model. (b) Perceptual data reproduced after [8], and (c) model output, of the saliency of the central bar as a function of the orientation contrast between center and surround. Figure 6: (a) Border effect: the collinear side of the border is more salient than the parallel one; the center plot is the saliency map for the full model, right plot is for a reduced model with diagonal covariance matrix. (b) Another example of collinear facilitation: the center row of bars is more salient, relative to the background, when the bars are collinear (left) rather than when they are parallel (right). In both (a) and (b), Col/Par is the ratio between the salience values on the collinear and parallel sides of the border, after subtracting the saliency value of the homogeneous regions. knowledge, there have only been few previous attempts of this sort; one notable example is the extensive salience work of [17]; here we go further in terms of simulating neural non–linearities, and making connections with the contour integration and illusion literature: phenomena that have previously been addressed only individually, if at all. Our model is closely related to a number of suggestions in the literature. Previous bottom-up models of divisive normalization, which were the original inspiration for the application by [22] of the GSM, can account for some neural non–linearities by learning divisive weights instead of assignments (e.g., [33]). However they do not incorporate linear correlations, and they fix the divisive weights a priori rather than on a image–by–image basis such as in our model. Non-parametric statistical alternatives to divisive normalization, e.g. non-linear ICA [41], have also been proposed, but have been applied only to the orientation masking nonlinearity, therefore not addressing spatial context. There are also various top-down models based on related principles. Compared with previous GSM modelling [30], we have built a more computationally straightforward, and neurobiologically credible, approximate assignment mechanism. Other recent generative statistical models that capture the statistical dependencies of the filters in slightly different ways (notably [31, 35]), might also be able to encompass the data we have presented here. However, [35] has been applied only to the image processing domain, and the model of [31] has not been tied to the perceptual phenomena we have considered, nor to contrast data. There are also quantitative differences between the models, including issues of soft versus hard assignment (see discussion in [30]); the assumption about the link to data (here we adopted the mean of the Gaussian component of the GSM which incorporates 7 an explicit gain control, in contrast to the approach in [31]); and the richness of assignment versus approximation in the various models (here we have purposely taken an approximate version of a full assignment model). There are also many models devoted to saliency. We showed that our assignment process, and the normalization that results, is a good match for (and thus a normative justification of) at least some of the results that [1, 15] captured in a dynamical realization of the V1 saliency hypothesis. However, our model achieves suppression in regions of statistical homogeneity divisively rather than subtractively. The covariance between the Gaussian components captures some aspects of the long range excitatory effects in that model, which permit contour integration. However, some of the collinear facilitation arises just from receptive field overlap; and the structure of the covariance in natural scenes seems rather impoverished compared with that implied by the association field [42], and merits further examination with higher order statistics (see also [10, 26]). Note also that dynamical models have not previously been applied to the same range of data (such as the tilt illusion). Open theoretical issues include quantifying carefully the effect of the rather coarse assignment approximation, as well as the differences between the learned model and the idealized population model of the tilt illusion [34]. Other important issues include characterizing the nature and effect of uncertainty in the distributions of g and v rather than just the mean. This is critical to characterize psychophysical results on contrast detection in the face of noise and also orientation acuity, and also raises the issues aired by [31] as to how neural responses convey uncertainties. Open experimental issues include a range of other contextual effects as to salience, contour integration, and even perceptual crowding. Contextual effects are equally present at multiple levels of neural processing. An important future generalization would be to higher neural areas, and to mid and high level vision (which themselves exhibit gain-control related phenomena, see e.g. [43]). More generally, context is pervasive in time as well as space. The parallels are underexplored, and so pressing. Acknowledgements This work was funded by the Alfred P. Sloan Foundation (OS); and The Gatsby Charitable Foundation, the BBSRC, the EPSRC and the Wellcome Trust (PD). We are very grateful to Adam Kohn, Joshua Solomon, Adam Sanborn, and Li Zhaoping for discussion. References [1] Z. Li. A saliency map in primary visual cortex. Trends Cogn Sci, 6(1):9–16, 2002. [2] P. Series, J. Lorenceau, and Y. Fr´egnac. The ”silent” surround of v1 receptive fields: theory and experiments. J Physiol Paris, 97(4-6):453–474, 2003. [3] H. E. Jones, K. L. Grieve, W. Wang, and A. M. Sillito. Surround suppression in primate v1. J Neurophysiol, 86(4):2011–2028, 2001. [4] J. R. Cavanaugh, W. Bair, and J. A. Movshon. Selectivity and spatial distribution of signals from the receptive field surround in macaque v1 neurons. J Neurophysiol, 88(5):2547–2556, 2002. [5] J. J. Gibson and M. Radner. Adaptation, after-effect, and contrast in the perception of tilted lines. Journal of Experimental Psychology, 20:553–569, 1937. [6] C. W. Clifford, P. Wenderoth, and B. Spehar. A functional angle on some after-effects in cortical vision. Proc R Soc Lond B Biol Sci, 1454:1705–1710, 2000. [7] J. A. Solomon and M. J. Morgan. Stochastic re-calibration: contextual effects on perceived tilt. Proc Biol Sci, 273(1601):2681–2686, 2006. [8] H.C. Nothdurft. The conspicuousness of orientation and motion contrast. Spatial Vision, 7(4):341–363, 1993. [9] D. J. Field, A. Hayes, and R. F. Hess. Contour integration by the human visual system: evidence for a local ”association field”. Vision Res, 33(2):173–193, 1993. [10] W. S. Geisler, J. S. Perry, B. J. Super, and D. P. Gallogly. Edge co-occurrence in natural images predicts contour grouping performance. Vision Res, 41(6):711–724, 2001. [11] L. Schwabe, K. Obermayer, A. Angelucci, and P. C. Bressloff. The role of feedback in shaping the extraclassical receptive field of cortical neurons: a recurrent network model. J Neurosci, 26(36):9117–9129, 2006. [12] J. Wielaard and P. Sajda. Extraclassical receptive field phenomena and short-range connectivity in v1. Cereb Cortex, 16(11):1531–1545, 2006. [13] T. J. Sullivan and V. R. de Sa. A model of surround suppression through cortical feedback. Neural Netw, 19(5):564–572, 2006. 8 [14] T.N. Mundhenk and L. Itti. Computational modeling and exploration of contour integration for visual saliency. Biological Cybernetics, 93(3):188–212, 2005. [15] Z. Li. Visual segmentation by contextual influences via intracortical interactions in primary visual cortex. Network: Computation in Neural Systems, 10(2):187–212, 1999. [16] L. Itti and C. Koch. A saliency-based search mechanism for overt and covert shifts of visual attention. Vision Research, 40(10-12):1489–1506, 2000. [17] D. Gao, V. Mahadevan, and N. Vasconselos. On the plausibility of the discriminant center-surround hypothesis for visual saliency. Journal of Vision, 8(7)(13):1–18, 2008. [18] L. Zhang, M.H. Tong, T. Marks, H. Shan, and G.W. Cottrell. Sun: A bayesian framework for saliency using natural statistics. Journal of Vision, 8(7)(32):1–20, 2008. [19] N.D.B. Bruce and J.K. Tsotsos. Saliency, attention, and visual search: An information theoretic approach. Journal of Vision, 9(3)(5):1–24, 2009. [20] A. Hyv¨arinen, J. Hurri, and P.O. Hoyer. Natural Image Statistics. Springer, 2009. [21] D. Andrews and C. Mallows. Scale mixtures of normal distributions. J. Royal Stat. Soc., 36:99–102, 1974. [22] M. J. Wainwright, E. P. Simoncelli, and A. S. Willsky. Random cascades on wavelet trees and their use in modeling and analyzing natural imagery. Applied and Computational Harmonic Analysis, 11(1):89–123, 2001. [23] J. Portilla, V. Strela, M. Wainwright, and E. P. Simoncelli. Image denoising using a scale mixture of Gaussians in the wavelet domain. IEEE Trans Image Processing, 12(11):1338–1351, 2003. [24] C. Zetzsche, B. Wegmann, and E. Barth. Nonlinear aspects of primary vision: Entropy reduction beyond decorrelation. In Int’l Symposium, Society for Information Display, volume XXIV, pages 933–936, 1993. [25] E. P. Simoncelli. Statistical models for images: Compression, restoration and synthesis. In Proc 31st Asilomar Conf on Signals, Systems and Computers, pages 673–678, Pacific Grove, CA, 1997. IEEE Computer Society. [26] P. Hoyer and A. Hyv¨arinen. A multi-layer sparse coding network learns contour coding from natural images. Vision Research, 42(12):1593–1605, 2002. [27] A Hyv¨arinen, J. Hurri, and J. Vayrynen. Bubbles: a unifying framework for low-level statistical properties of natural image sequences. Journal of the Optical Society of America A, 20:1237–1252, 2003. [28] Y. Karklin and M. S. Lewicki. A hierarchical Bayesian model for learning nonlinear statistical regularities in nonstationary natural signals. Neural Computation, 17:397–423, 2005. [29] S. Osindero, M. Welling, and G. E. Hinton. Topographic product models applied to natural scene statistics. Neural Computation, 18(2):381–414, 2006. [30] O. Schwartz, T. J. Sejnowski, and P. Dayan. Soft mixer assignment in a hierarchical generative model of natural scene statistics. Neural Comput, 18(11):2680–2718, 2006. [31] Y. Karklin and M.S. Lewicki. Emergence of complex cell properties by learning to generalize in natural scenes. Nature, 457(7225):83–86, 2009. [32] D. J. Heeger. Normalization of cell responses in cat striate cortex. Visual Neuroscience, 9:181–198, 1992. [33] O. Schwartz and E. P. Simoncelli. Natural signal statistics and sensory gain control. Nature Neuroscience, 4(8):819–825, 2001. [34] O. Schwartz, T.J. Sejnowski, and P. Dayan. Perceptual organization in the tilt illusion. Journal of Vision, 9(4)(19):1–20, 2009. [35] J.A. Guerrero-Colon, E.P. Simoncelli, and J. Portilla. Image denoising using mixtures of gaussian scale mixtures. In Proc 15th IEEE Int’l Conf on Image Proc, pages 565–568, 2008. [36] E. P. Simoncelli, W. T. Freeman, E. H. Adelson, and D. J. Heeger. Shiftable multi-scale transforms. IEEE Trans Information Theory, 38(2):587–607, 1992. [37] C. K. I. Williams and N. J. Adams. Dynamic trees. In M. S. Kearns, S. A. Solla, and D. A. Cohn, editors, Adv. Neural Information Processing Systems, volume 11, pages 634–640, Cambridge, MA, 1999. MIT Press. [38] X.L. Meng and D.B. Rubin. Maximum likelihood estimation via the ecm algorithm: A general framework. Biometrika, 80(2):267–278, 1993. [39] E. Goddard, C.W.G. Clifford, and S.G. Solomon. Centre-surround effects on perceived orientation in complex images. Vision Research, 48:1374–1382, 2008. [40] A.V. Popple and Z. Li. Testing a v1 model: perceptual biases and saliency effects. Journal of Vision, 1,3:148, 2001. [41] J. Malo and J. Guti´errez. V1 non–linear properties emerge from local–to–global non–linear ica. Network: Comp. Neur. Syst., 17(1):85–102, 2006. [42] D. J. Field. Relations between the statistics of natural images and the response properties of cortical cells. J. Opt. Soc. Am. A, 4(12):2379–2394, 1987. [43] Q. Li and Z. Wang. General–purpose reduced–reference image quality assessment based on perceptually and statistically motivated image representation. In Proc 15th IEEE Int’l Conf on Image Proc, pages 1192–1195, 2008. 9
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A joint maximum-entropy model for binary neural population patterns and continuous signals Sebastian Gerwinn Philipp Berens Matthias Bethge MPI for Biological Cybernetics and University of T¨ubingen Computational Vision and Neuroscience Spemannstrasse 41, 72076 T¨ubingen, Germany {firstname.surname}@tuebingen.mpg.de Abstract Second-order maximum-entropy models have recently gained much interest for describing the statistics of binary spike trains. Here, we extend this approach to take continuous stimuli into account as well. By constraining the joint secondorder statistics, we obtain a joint Gaussian-Boltzmann distribution of continuous stimuli and binary neural firing patterns, for which we also compute marginal and conditional distributions. This model has the same computational complexity as pure binary models and fitting it to data is a convex problem. We show that the model can be seen as an extension to the classical spike-triggered average/covariance analysis and can be used as a non-linear method for extracting features which a neural population is sensitive to. Further, by calculating the posterior distribution of stimuli given an observed neural response, the model can be used to decode stimuli and yields a natural spike-train metric. Therefore, extending the framework of maximum-entropy models to continuous variables allows us to gain novel insights into the relationship between the firing patterns of neural ensembles and the stimuli they are processing. 1 Introduction Recent technical advances in systems neuroscience allow us to monitor the activity of increasingly large neural ensembles simultaneously (e.g. [5, 21]). To understand how such ensembles process sensory information and perform the complex computations underlying successful behavior requires not only collecting massive amounts of data, but also the use of suitable statistical models for data analysis. What degree of precision should be incorportated into such a model involves a tradeoff between the question of interest and mathematical tractability: Complex multi-compartmental models [8] allow inference concerning the underlying biophysical processes, but their applicability to neural populations is limited. The generalized linear model [15] on the other hand is tractable even for large ensembles and provides a phenomenological description of the data. Recently, several groups have used binary maximum entropy models incorporating pairwise correlations to model neural activity in large populations of neurons on short time scales [19, 22, 7, 25]. These models have two important features: (1) Since they only require measuring the mean activity of individual neurons and correlations in pairs of neurons, they can be estimated from moderate amounts of data. (2) They seem to capture the essential structure of neural population activity at these timescales even in networks of up to a hundred neurons [21]. Although the generality of these findings have been subject to debate [3, 18], pairwise maximum-entropy and related models [12] are an important tool for the description of neural population activity [23, 17]. 1 To find features to which a neuron is sensitive spike-triggered average and spike-triggered covariance are commonly used techniques [20, 16]. They correspond to fitting a Gaussian distribution to the spike-triggered ensemble. If one has access to multi-neuron recordings, a straightforward extension of this approach is to fit a different Gaussian distribution to each binary population pattern. In statistics, the corresponding model is known as the location model [14, 10, 9]. To estimate this model, one has to observe sufficient amounts of data for each population pattern. As the number of possible binary patterns grows exponentially with the number of neurons, it is desirable to include regularization constraints in order to make parameter estimation tractable. Here, we extend the framework of pairwise maximum entropy modeling to a joint model for binary and continuous variables. This allows us to analyze the functional connection structure in a neural population at the same time as its relationship with further continuous signals of interest. In particular, this approach makes it possible to include a stimulus as a continuous variable into the framework of maximum-entropy modeling. In this way, we can study the stimulus dependence of binary neural population activity in a regularized framework in a rigorous way. In particular, we can use it to extract non-linear features in the stimulus that a population of neurons is sensitive to, while taking the binary nature of spike trains into account. We discuss the relationship of the obtained features with classical approaches such as spike-triggered average (STA) and spike-triggered covariance (STC). In addition, we show how the model can be used to perform spike-by-spike decoding and yields a natural spike-train metric [24, 2]. We start with a derivation of the model and a discussion of its features. 2 Model In this section we derive the maximum-entropy model for joint continuous and binary data with second-order constraints and describe its basic properties. We write continuous variables x and binary variables b. Having observed the joint mean µ and joint covariance C, we want to find a distribution pME which achieves the maximal entropy under all distributions with these observed moments. Since we model continuous and binary variables jointly, we define entropy to be a mixed discrete entropy and differential entropy: H[p] = − X b Z p(x, b) log p(x, b)dx Formally, we require pME to satisfy the following constraints: E[x] = µx E[b] = µb E[xx⊤] = Cxx + µxµ⊤ x E[bb⊤] = Cbb + µbµ⊤ b (1) E[xb⊤] = Cxb + µxµ⊤ b E[bx⊤] = Cbx + µbµ⊤ x = Cxb + µxµ⊤ b where the expectations are taken over pME. Cxx, Cxb and Cbb are blocks in the observed covariance matrix corresponding to the respective subsets of variables. This problem can be solved analytically using the Lagrange formalism, which leads to a maximum entropy distribution of Boltzmann type: pME(x, b|Λ, λ) = 1 Z(Λ, λ) exp (Q (x, b|Λ, λ)) Q(x, b|Λ, λ) = 1 2 x b ⊤ Λ x b + λ⊤ x b (2) Z(Λ, λ) = X b Z exp (Q (x, b|Λ, λ)) dx, where Λ and λ are chosen such that the resulting distribution fulfills the constraints in equation (1), as we discuss below. Before we compute marginal and conditional distributions in this model, we explore its basic properties. First, we note that the joint distribution can be factorized in the following way: pME(x, b|Λ, λ) = pME(x|b, Λ, λ)pME(b|Λ, λ) (3) 2 The conditional density pME(x|b, Λ, λ) is a Normal distribution, given by: pME(x|b, Λ, λ) ∝exp 1 2x⊤Λxxx + x⊤(λx + Λxbb) (4) ∝N x|µx|b, Σ ,with µx|b = Σ (λx + Λxbb) , Σ = (−Λxx)−1 Here, Λxx, Λxb, Λbx, λx are the blocks in Λ which correspond to x and b, respectively. While the mean of this Normal distribution dependent on b, the covariance matrix is independent of the specific binary state. The marginal probability pME(b|Λ, λ) is given by: Z(Λ, λ)pME(b|Λ, λ) = exp 1 2b⊤Λbbb + b⊤λb Z exp 1 2x⊤Λxxx + x⊤(λx + Λxbb) dx = (2π) n 2 |−Λxx|−1 2 exp 1 2b⊤ Λbb + Λ⊤ xb (−Λxx)−1 Λxb b (5) +b⊤ λb + Λ⊤ xb (−Λxx)−1 λx + 1 2λ⊤ x (−Λxx)−1 λx To evaluate the maximum entropy distribution, we need to compute the partition function, which follows from the previous equation by summing over b: Z(Λ, λ) = (2π) n 2 |−Λxx|−1 2 X b exp 1 2b⊤ Λbb + Λ⊤ xb (−Λxx)−1 Λxb b +b⊤ λb + Λ⊤ xb (−Λxx)−1 λx + 1 2λ⊤ x (−Λxx)−1 λx (6) Next, we compute the marginal distribution with respect to x. From equation (5) and (4), we find that pME(x|Λ, λ) is a mixture of Gaussians, where each Gaussian of equation (4) is weighted by the corresponding pME(b|Λ, λ). While all mixture components have the same covariance, the different weighting terms affect each component’s influence on the marginal covariance of x. Finally, we also compute the conditional density pME(b|x, Λ, λ), which is given by: pME(b|x, Λ, λ) = 1 Z′ exp 1 2b⊤Λbbb + b⊤(λb + Λbxx) Z′ = X b exp 1 2b⊤Λbbb + b⊤(λb + Λbxx) (7) Note, that the distribution of the binary variables given the continuous variables is again of Boltzmann type. Parameter fitting To find suitable parameters for given data, we employ a maximum likelihood approach [1, 11], where we find the optimal parameters via gradient descent: l(Λ, λ) = log p({x(n), b(n)}N n=1|Λ, λ) = X n Q(x(n), b(n)|Λ, λ) −N log Z (Λ, λ) (8) ⇒∇Λl = N * x b x b ⊤+ data − * x b x b ⊤+ pME ∇λl = N " x b data − x b pME # To calculate the moments over the model distribution pME we make use of the above factorization: xx⊤ = xx⊤|b b = (−Λxx)−1 + D µx|bµ⊤ x|b E b D xb⊤E = D µx|bb⊤E b = bx⊤ ⊤, ⟨x⟩= D µx|b E b (9) 3 A 4 3 2 1 0 1 2 3 4 0.00 0.04 0.08 0.12 B 4 3 2 1 0 1 2 3 4 0.00 0.04 0.08 0.12 C 4 3 2 1 0 1 2 3 4 0.00 0.04 0.08 0.12 FIGURE 1: Illustration of different parameter settings. A:independent binary and continuous variables, B: correlations (0.4) between variables, C: changing mean of the binary variables (here: 0.7) corresponds to changing weightings of the Gaussians, correlations are 0.4. Blue lines indicate p(x|b = 1) and green ones p(x|b = 0). Hence, the only average we actually need to evaluate numerically is the one over the binary variables. Unfortunately, we cannot directly set the parameters for the continuous part, as they depend on the ones for the binary part. However, since the above equations can be evaluated analytically, the difficult part is finding the parameters for the binary variables. In particular, if the number of binary variables is large, calculating the partition function can become infeasible. To some extent, this can be remedied by the use of specialized Monte-Carlo algorithms [4]. 2.1 Example In order to gain intuition into the properties of the model, we illustrate it in a simple one-dimensional case. From equation (4) for the conditional mean of the continuous variables, we expect the distance between the conditional means µx|b to increase with increasing correlation between continuous and binary variables increases. We see that this is indeed the case: While the conditional Gaussians p(x|b = 1) and p(x|b = 0) are identical if x and b are uncorrelated (figure 1A), a correlation between x and b shifts them away from the unconditional mean (figure 1B). Also, the weight assigned to each of the two Gaussians can be changed. While in figures 1A and 1B b has a symmetric mean of 0.5, a non-symmetric mean leads to an asymmetry in the weighting of each Gaussian illustrated in figure 1C. 2.2 Comparison with other models for the joint modeling of binary and continous data There are two models in the literature which model the joint distribution of continuous and binary variables, which we will list in the following and compare them to the model derived in this paper. Location model The location model (LM) [14, 10, 9] also uses the same factorization as above p(x, b) = p(x|b)p(b). However, the distribution for the binary variables p(b) is not of Boltzmann type but a general multinomial distribution and therefore has more degrees of freedom. The conditional distribution p(x|b) is assumed to be Gaussian with moments (µb, Σb), which can both depend on the conditional state b. Thus to fit the LM usually requires much more data to estimate the moments for every possible binary state. The location model can also be seen as a maximum entropy model in the sense, that it is the distribution with maximal entropy under all distribution with the conditional moments. As fitting this model in its general form is prone to overfitting, various ad hoc constraints have been proposed; see [9] for details. Partially dichotomized Gaussian model Another simple possibility to obtain a joint distribution of continuous and binary variables is to take multivariate (latent) Gaussian distribution for all variables and then dichotomize those components which should represent the binary variables. Thus, a binary variable bi is set to 1 if the underlying Gaussian variables is greater than 0 and it is set to 0 if the Gaussian variable is smaller than 0. This model is known as the partially dichotomized Gaussian (PDG) [6]. Importantly the marginal distribution over the continuous variables is always Gaussian and not a mixture as in our model. The reason for this is that all marginals of a Gaussian distribution are again Gaussian. 4 A B C D E FIGURE 2: Illustration of the binary encoding with box-type tuning curves. A: shows the marginal distribution over stimuli. The true underlying stimulus distribution is a uniform distribution over the interval ( 0.5,0.5) and is plotted in shaded gray. The mixture of Gaussian approximation of the MaxEnt model is plotted in black. Each neuron has a tuning-curve, consisting of a superposition of box-functions. B shows the tuning-curve of the first neuron. This is equivalent to the conditional distribution, when conditioning on the first bit, which indicates if the stimulus is in the right part of the interval. The tuning-curve is a superposition of 5 box-functions. The true tuning curve is plotted in shaded gray whereas the MaxEnt approximation is plotted in black. C shows the tuning curve of neuron with index 2. D: Covariance between continuous and binary variables as a function of the index of the binary variables. This is the same as the STA for each neuron (see also equation (10)). E shows the conditional distribution, when conditioning on both variables (0,2) to be one. This corresponds to the product of the tuning-curves. 3 Applications 3.1 Spike triggering and feature extraction Spike triggering is a common technique in order to find features which a single neuron is sensitive to. The presented model can be seen as an extension in the following sense. Suppose that we have observed samples (xn,bn) from a population responding to a stimulus. The spike triggered average (STA) for a neuron i is then defined as STAi = n xnbn i n bn i = E[xbi]ri, (10) where ri = P n bn i N = p(bi = 1) is the firing rate of the i-th neuron or fraction of ones within the sample. Note, that the moment E[xbi] is one of the constraints we require for the maximum entropy model and therefore the STA is included in the model. In addition, the model has also similarities to spike-triggered covariance (STC) [20, 16]. STC denotes the distribution or, more precisely, the covariance of the stimuli that evoked a spiking response. Usually, this covariance is then compared to the total covariance over the entire stimulus distribution. In the joint maximum-entropy model, we have access to a similar distribution, namely the conditional distribution p(x| bi = 1), which is a compact description of the spike-triggered distribution. Note that p(x| bi = 1) can be highly non-Gaussian as all neurons j = i are marginalized out – this is why the current model is an extension to spike triggering. Additionally, we can also trigger or con5 A B C FIGURE 3: Illustration of a spike-by-spike decoding scheme. The MaxEnt model was fit to data from two deterministic integrate-and-fire models. The MaxEnt model can then be used for decoding spikes generated by the two independent deterministic models. The two green arrows correspond the weights of a two-pixel receptive field for each of the two neurons. The 2 dimensional stimulus was drawn from two independent Gamma distributions. The resulting spike-trains were discretized in 5 time-bins, each 200 ms long. A spike-train to a particular stimulus (x cross) is decoded. In A) the marginal distribution of the continuous variables is shown. In B) the posterior, when conditioning on the first temporal half of the response to that stimulus is shown. Finally in C) the conditional distribution, when conditioning on the full observed binary pattern is plotted. dition not on a single neuron but on any response pattern BS of a sub-population S. The resulting p(x| BS) with BS = { b : bi = Bi i S} is then also a mixture of Gaussians with 2n components, where n is the number of unspecified neurons j / S. As illustrated above (see figure 1B), correlations between neurons and stimuli lead to a separation of the individual Gaussians. Hence, stimulus correlations of other neurons j = i in the distribution p(x,bj =i| bi = 1) would have the same effect on the spike-triggered distribution of neuron i. Correlations within this distribution also imply, that there are correlations between neuron j and neuron i. Thus, stimulus as well as noise correlations cause deviations of the conditional p(x| BS) from a single Gaussian. Therefore, the full conditional distribution p(x| BS) in general contain more information about the features which trigger this sub-population to evoke the specified response pattern, than the conditional mean, i.e. the STA. We demonstrate the capabilities of this approach by considering the following encoding. As stimulus, we consider one continuous real valued variable that is drawn uniformly from the interval [ 0.5,0.5]. It is mapped to a binary population response in the following way. Each neuron i has a square-wave tuning function: bi(x) = (sin (2 (i + 1)x)) , where is the Heaviside function. In this way, the response of a neuron is set to 1 if its tuningfunction is positive and 0 otherwise. The first (index 0) neuron distinguishes the left and the right part of the entire interval. The (i + 1)st neuron distinguishes subsequently left from right in the subintervals of the ith neuron. That is, the response of the second neuron is always 1, if the stimulus is in the right part of the intervals [ 0.5,0] and [0,0.5]. These tuning curves can also be thought of as a mapping into a non-linear feature space in which the neuron acts linear again. Although the datageneration process is not contained in our model class we were able to extract the tuning curves as shown in figure 2. Note, that for this example neither the STA nor STC analysis alone would provide any insight into the feature selectivity of the neurons, in particular for the neurons which have multi-modal tuning curves (the ones with higher indexes in the above example). However, the tuning curves could be reconstructed with any kind of density estimation, given the STA. 3.2 Spike-by-Spike decoding Since we have a simple expression for the conditional distribution p(x| b, , ) (see equation (4)), we can use the model to analyze the decoding performance of a neural population. To illustrate this, we sampled spike trains from two leaky integrate-and-fire neurons for 1 second and discretized the resulting spike trains into 5 bins of 200 ms length each. Each trial, we used a constant two dimensional stimulus, which was drawn from two independent Gamma distributions with shape 6 A B FIGURE 4: Illustration of the conditional probability p(b| x) for the example in figure 3. In 4A, for every binary pattern the corresponding probability is plotted for the given stimulus from figure 3, where the brightness of each square indicates its probability. For the given stimulus the actual response pattern used for figure 3 is marked with a circle. Each pattern b is split into two halves by the contributions of the two neurons (32 possible patterns for each neuron) and response patterns of the first neuron are shown on the x-axis, while response patterns of the second neuron on the y-axis. In 4B we plotted for each pattern b its probability under the two conditional distributions p(b| x ) and p(b| x ) against each other with x = (0.85,0.72) and x = (1.5,1.5). parameter = 3 and scale parameter = 0.3. For each LIF neuron, this two dimensional stimulus was then projected onto the one-dimensional subspace spanned by its receptive field and used as input current. Hence, there are 10 binary variables, 5 for each spike-train of the neurons and 2 continuous variables for the stimulus to be modeled. We draw 5· 106 samples, calculated the second order moments of the joint stimulus and response vectors and fitted our maximum entropy model to these moments. The obtained distribution is shown in figure 3. In 3A, we show the marginal distribution of the stimuli, which is a mixture of 210 Gaussians. The receptive fields of the two neurons are indicated by green arrows. To illustrate the decoding process, we sampled a stimulus and corresponding response r, from which we try to reconstruct the stimulus. In 3B, we show the conditional distribution when conditioning on the first half of the response. Finally in 3C, the complete posterior is shown when conditioned on the full response. From a-c, the posterior is more and more concentrated around the true stimulus. Although there is no neural noise in the encoding model, the reconstruction is not perfect. This is due to the regularization properties of the maximum entropy approach. 3.3 Stimulus dependence of firing patterns While previous studies on the structure of neuronal firing patterns in the retina have compared how well second-order maximum entropy models fit the empirically observed distributions under different stimulation conditions [19, 22], the stimulus has never been explicitly taken into account into the model. In the proposed framework, we have access to p(b| x), so we can explicitly study how the pattern distribution of a neural population depends on the stimulus. We illustrate this by continuing the example of figure 3. First, we show how the individual firing probabilities depend on x (figure 4A). Note, that although the encoding process for the previous example was noiseless, that is, for every given stimulus there is only one response pattern, the conditional distribution p(b| x) is not a delta-function, but dispersed around the expected response. This is due to the second order approximation to the encoding model. Further, as it turns out, that a spike in the next bin after a spike is very unlikely under the model, which captures the property of the leaky integrator. Also, we compare how p(b| x) changes for different values of x. This is illustrated in figure 4B. 3.4 Spike train metric Oftentimes, it is desirable to measure distances between spike trains [24]. One problem, however, is that not every spike might be of equal importance. That is, if a spike train differs only in one spike, it might nevertheless represent a completely different stimulus. Therefore, Ahmadian [2] suggested to measure the distance between spike trains as the difference of stimuli when reconstructed based on 7 the one or the other spike train seems. If the population is noisy, we want to measure the difference of reconstructed stimuli on average. To this end, we need access to the posterior distribution, when conditioning on a particular spike train or binary pattern. Using the maximum entropy model, we can define the following spike-metric: d(b1, b2) = DKL pME(x|b1)||pME(x|b2) = 1 2 µx|b1 −µx|b2 ⊤ Λxx µx|b1 −µx|b2 (11) Here, DKL denotes the Kullback-Leibler divergence between the posterior densities. Equation 11 is symmetric in b, however, in order to get a symmetric expression for other types of posterior distributions, the Jensen-Shannon divergence might be used instead. As an example we consider the induced metrics for the encoding model of figure 2. The metric induced by the square-wave tuning functions of section 3.1 is relatively simple. When conditioning on a particular population response, the conditional distribution p(x|b) is always a Gaussian with approximately the width of the smallest wavelength. Flipping a neuron’s response within this pattern corresponds to shifting the conditional distribution. Suppose we have observed a population response consisting of only ones. This results in a Gaussian posterior distribution with mean in the middle of the rightmost interval (0.5 − 1 1024, 0.5). Now flipping the response of the “low-frequency” neuron, that is the one shown in figure 2B, shifts the mean of the posterior to the middle of the sub-interval (− 1 1024, 0). Whereas flipping the “high-frequency” neuron, the one which indicates left or right within the smallest possible sub-interval, corresponds to shifting the mean just by the amount of this smallest interval to the left. Flipping the response of single neurons within this population can result in posterior distribution which look quite different in terms of the Kullback-Leibler divergence. In particular, there is an ordering in terms of the frequency of the neurons with respect to the proposed metric. 4 Conclusion We have presented a maximum-entropy model based on the joint second order statistics of continuous valued variables and binary neural responses. This allows us to extend the maximum-entropy approach [19] for analyzing neural data to incorporate other variables of interest such as continuous valued stimuli. Alternatively, additional neurophysiological signals such as local field potentials [13] can be taken into account to study their relation with the joint firing patterns of local neural ensembles. We have demonstrated four applications of this approach: (1) It allows us to extract the features a (sub-)population of neurons is sensitive to, (2) we can use it for spike-by-spike decoding, (3) we can assess the impact of stimuli on the distribution of population patterns and (4) it yields a natural spike-train metric. We have shown that the joint maximum-entropy model can be learned in a convex fashion, although high-dimensional binary patterns might require the use of efficient sampling techniques. Because of the maximum-entropy approach the resulting distribution is well regularized and does not require any ad-hoc restrictions or regularity assumptions as have been proposed for related models [9]. Analogous to a Boltzmann machine with hidden variables, it is possible to further add hidden binary nodes to the model. This allows us to take higher-order correlations into account as well, although we stay essentially in the second-order framework. Fortunately, the learning scheme for fitting the modified model to observed data remains almost unchanged: The only difference is that the moments have to be averaged over the non-observed binary variables as well. In this way, the model can also be used as a clustering algorithm if we marginalize over all binary variables. The resulting mixture of Gaussian model will consist of 2N components, where N is the number of hidden binary variables. Unfortunately, convexity cannot be guaranteed if the model contains hidden nodes. In a similar fashion, we could also add hidden continuous variables, for example to model unobserved common inputs. In contrast to hidden binary nodes, this does not lead to an increased model complexity: averaging over hidden continuous variables corresponds to integrating out each Gaussian within the mixture, which results in another Gaussian. Also the restriction that all covariance matrices in the mixture need to be the same still holds, because each Gaussian is integrated in the same way. Acknowledgments We would like to thank J. Macke and J. Cotton for discussions and feedback on the manuscript. This work is supported by the German Ministry of Education, Science, Research and Technology through the Bernstein award to MB (BMBF; FKZ: 01GQ0601), the Werner-Reichardt Centre for Integrative Neuroscience T¨ubingen, and the Max Planck Society. 8 References [1] D.H. Ackley, G.E. Hinton, and T.J. Sejnowski. A learning algorithm for boltzmann machines. Cognitive Science, 9:147–169, 1985. [2] Y. Ahmadian, J. Pillow, J. Shlens, E. Simoncelli, E.J. Chichilinsky, and L. Paninski. A decoder-based spike train metric for analyzing the neural code in the retina. In Frontiers in Systems Neuroscience. Conference Abstract: Computational and systems neuroscience, 2009. [3] M. Bethge and P. Berens. Near-Maximum entropy models for binary neural representations of natural images. In Proceedings of the Twenty-First Annual Conference on Neural Information Processing Systems, volume 20, pages 97–104, Cambridge, MA, 2008. MIT Press. [4] Tamara Broderick, Miroslav Dudik, Gasper Tkacik, Robert E Schapire, and William Bialek. Faster solutions of the inverse pairwise ising problem. arXiv, q-bio.QM:0712.2437, Dec 2007. [5] G. Buzsaki. Large-scale recording of neuronal ensembles. Nature Neuroscience, 7(5):446–451, 2004. [6] D. R. Cox and Nanny Wermuth. Likelihood factorizations for mixed discrete and continuous variables. Scandinavian Journal of Statistics, 26(2):209–220, June 1999. [7] A. Tang et al. A maximum entropy model applied to spatial and temporal correlations from cortical networks in vitro. J. Neurosci., 28(2):505–518, 2008. [8] Q.J.M. Huys, M.B. Ahrens, and L. Paninski. Efficient estimation of detailed single-neuron models. Journal of neurophysiology, 96(2):872, 2006. [9] W. Krzanowski. The location model for mixtures of categorical and continuous variables. Journal of Classification, 10(1):25–49, 1993. [10] S. L. Lauritzen and N. Wermuth. Graphical models for associations between variables, some of which are qualitative and some quantitative. The Annals of Statistics, 17(1):31–57, March 1989. [11] D.J.C. MacKay. Information theory, inference and learning algorithms. Cambridge U. Press, 2003. [12] J.H. Macke, P. Berens, A.S. Ecker, A.S. Tolias, and M. Bethge. Generating spike trains with specified correlation coefficients. Neural Computation, 21(2):1–27, 2009. [13] Marcelo A. Montemurro, Malte J. Rasch, Yusuke Murayama, Nikos K. Logothetis, and Stefano Panzeri. Phase-of-Firing coding of natural visual stimuli in primary visual cortex. Current Biology, Vol 18:375– 380, March 2008. [14] I. Olkin and R. F. Tate. Multivariate correlation models with mixed discrete and continuous variables. The Annals of Mathematical Statistics, 32(2):448–465, June 1961. [15] J.W. Pillow, J. Shlens, L. Paninski, A. Sher, A.M. Litke, EJ Chichilnisky, and E.P. Simoncelli. Spatiotemporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207):995– 999, 2008. [16] J.W. Pillow and E.P. Simoncelli. Dimensionality reduction in neural models: an information-theoretic generalization of spike-triggered average and covariance analysis. Journal of Vision, 6(4):414–428, 2006. [17] Y. Roudi, E. Aurell, and J.A. Hertz. Statistical physics of pairwise probability models. Frontiers in Computational Neuroscience, 2009. [18] Yasser Roudi, Sheila Nirenberg, and Peter E. Latham. Pairwise maximum entropy models for studying large biological systems: When they can work and when they can’t. PLoS Comput Biol, 5(5), 2009. [19] Elad Schneidman, Michael J. Berry, Ronen Segev, and William Bialek. Weak pairwise correlations imply strongly correlated network states in a neural population. Nature, 440(7087):1007–1012, April 2006. [20] O. Schwartz, EJ Chichilnisky, and E.P. Simoncelli. Characterizing neural gain control using spiketriggered covariance. In Advances in Neural Information Processing Systems 14: Proceedings of the 2002 [sic] Conference, page 269. MIT Press, 2002. [21] J. Shlens, G. D. Field, J. L. Gauthier, M. Greschner, A. Sher, A. M. Litke, and E. J. Chichilnisky. The structure of Large-Scale synchronized firing in primate retina. Journal of Neuroscience, 29(15):5022, 2009. [22] Jonathon Shlens, Greg D. Field, Jeffrey L. Gauthier, Matthew I. Grivich, Dumitru Petrusca, Alexander Sher, Alan M. Litke, and E. J. Chichilnisky. The structure of Multi-Neuron firing patterns in primate retina. J. Neurosci., 26(32):8254–8266, August 2006. [23] Jonathon Shlens, Fred Rieke, and E. J. Chichilnisky. Synchronized firing in the retina. Current Opinion in Neurobiology, 18(4):396–402, August 2008. [24] J.D. Victor and K.P. Purpura. Metric-space analysis of spike trains: theory, algorithms and application. Network: computation in neural systems, 8(2):127–164, 1997. [25] Shan Yu, Debin Huang, Wolf Singer, and Danko Nikolic. A small world of neuronal synchrony. Cereb. Cortex, 18(12):2891–2901, April 2008. 9
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Hierarchical Modeling of Local Image Features through Lp-Nested Symmetric Distributions Fabian Sinz Max Planck Institute for Biological Cybernetics Spemannstraße 41 72076 T¨ubingen, Germany fabee@tuebingen.mpg.de Eero P. Simoncelli Center for Neural Science, and Courant Institute of Mathematical Sciences, New York University New York, NY 10003 eero.simoncelli@nyu.edu Matthias Bethge Max Planck Institute for Biological Cybernetics Spemannstraße 41 72076 T¨ubingen, Germany mbethge@tuebingen.mpg.de Abstract We introduce a new family of distributions, called Lp-nested symmetric distributions, whose densities are expressed in terms of a hierarchical cascade of Lpnorms. This class generalizes the family of spherically and Lp-spherically symmetric distributions which have recently been successfully used for natural image modeling. Similar to those distributions it allows for a nonlinear mechanism to reduce the dependencies between its variables. With suitable choices of the parameters and norms, this family includes the Independent Subspace Analysis (ISA) model as a special case, which has been proposed as a means of deriving filters that mimic complex cells found in mammalian primary visual cortex. Lp-nested distributions are relatively easy to estimate and allow us to explore the variety of models between ISA and the Lp-spherically symmetric models. By fitting the generalized Lp-nested model to 8 × 8 image patches, we show that the subspaces obtained from ISA are in fact more dependent than the individual filter coefficients within a subspace. When first applying contrast gain control as preprocessing, however, there are no dependencies left that could be exploited by ISA. This suggests that complex cell modeling can only be useful for redundancy reduction in larger image patches. 1 Introduction Finding a precise statistical characterization of natural images is an endeavor that has concerned research for more than fifty years now and is still an open problem. A thorough understanding of natural image statistics is desirable from an engineering as well as a biological point of view. It forms the basis not only for the design of more advanced image processing algorithms and compression schemes, but also for a better comprehension of the operations performed by the early visual 1 system and how they relate to the properties of the natural stimuli that are driving it. From both perspectives, redundancy reducing algorithms such as Principal Component Analysis (PCA), Independent Component Analysis (ICA), Independent Subspace Analysis (ISA) and Radial Factorization [11; 21] have received considerable interest since they yield image representations that are favorable for compression and image processing and at the same time resemble properties of the early visual system. In particular, ICA and ISA yield localized, oriented bandpass filters which are reminiscent of receptive fields of simple and complex cells in primary visual cortex [4; 16; 10]. Together with the Redundancy Reduction Hypothesis by Barlow and Attneave [3; 1], those observations have given rise to the idea that these filters represent an important aspect of natural images which is exploited by the early visual system. Several result, however, show that the density model of ICA is too restricted to provide a good model for natural images patches. Firstly, several authors have demonstrated that filter responses of ICA filters on natural images are not statistically independent [20; 23; 6]. Secondly, after whitening, the optimum of ICA in terms of statistical independence is very shallow or, in other words, all whitening filters yield almost the same redundancy reduction [5; 2]. A possible explanation for that finding is that, after whitening, densities of local image features are approximately spherical [24; 23; 12; 6]. This implies that those densities cannot be made independent by ICA because (i) all whitening filters differ only by an orthogonal transformation, (ii) spherical densities are invariant under orthogonal transformations, and (iii) the only spherical and factorial distribution is the Gaussian. Once local image features become more distant from each other, the contour lines of the density deviates from spherical and become more star-shaped. In order to capture this star-shaped contour lines one can use the more general Lp-spherically symmetric distributions which are characterized by densities of the form χ(y) = g( y p) with y p = ( | yi| p)1/ p and p > 0 [9; 10; 21]. p=0.8 p=2 p=1.5 p=0.8 Figure 1: Scatter plots and marginal histograms of neighboring (left) and distant (right) symmetric whitening filters which are shown at the top. The dashed Contours indicate the unit sphere for the optimal p of the best fitting non-factorial (dashed line) and factorial (solid line) Lp-spherically symmetric distribution, respectively. While close filters exhibit p = 2 (spherically symmetric distribution), the value of p decreases for more distant filters. As illustrated in Figure 1, the relationship between local bandpass filter responses undergoes a gradual transition from L2-spherical for nearby to star-shaped (Lp-spherical with p < 2) for more distant features [12; 21]. Ultimately, we would expect extremely distant features to become independent, having a factorial density with p 0.8. When using a single Lp-spherically symmetric model for the joint distribution of nearby and more distant features, a single value of p can only represent a compromise for the whole variety of iso-probability contours. This raises the question whether a combination of local spherical models, as opposed to a single Lp-spherical model, yields a better characterization of the statistics of natural image patches. Possible ways to join several local models are Independent Subspace Analysis (ISA) [10], which uses a factorial combination of locally Lpspherical densities, or Markov Random Fields (MRFs) [18; 13]. Since MRFs have the drawback of being implicit density models and computationally very expensive for inference, we will focus on ISA and our model. In principle, ISA could choose its subspaces such that nearby features are grouped into a joint subspace which can then be well described by a spherical symmetric model (p = 2) while more distant pixels, living in different subspaces, are assumed to be independent. In fact, previous studies have found ISA to perform better than ICA for image patches as small as 8× 8 and to yield an optimal p 2 for the local density models [10]. On the other hand, the ISA model assumes a binary partition into either a Lp-spherical or a factorial distribution which does not seem to be fully justified considering the gradual transition described above. 2 Here, we propose a new family of hierarchical models by replacing the Lp-norms in the Lp-spherical models by Lp-nested functions, which consist of a cascade of nested Lp-norms and therefore allow for different values of p for different groups of filters. While this family includes the Lp-spherical family and ISA models, it also includes densities that avoid the hard partition into either factorial or Lp-spherical. At the same time, parameter estimation for these models can still be similarly efficient and robust as for Lp-spherically symmetric models. We find that this family (i) fits the data significantly better than ISA and (ii) generates interesting filters which are grouped in a sensible way within the hierarchy. We also find that, although the difference in performance between Lp-spherical and Lp-nested models is significant, it is small on 8 × 8 patches, suggesting that within this limited spatial range, the iso-probability contours of the joint density can still be reasonably approximated by a single Lp-norm. Preliminary results on 16 × 16 patches exhibit a more pronounced difference between the Lp-nested and the Lp-spherically symmetric distribution, suggesting that the change in p becomes more important for modelling densities over a larger spatial range. 2 Models Lp-Nested Symmetric Distributions Consider the function f(y) = n1 X i=1 |yi|p1 ! p∅ p1 + ... + n X i=n1+...+nℓ−1+1 |yi|pℓ p∅ pℓ 1 p∅ (1) =
(∥y1:n1∥p1, ..., ∥yn−nℓ+1:n∥pℓ)⊤
p∅ . We call this type of functions Lp-nested and the resulting class of distributions Lp-nested symmetric. Lp-nested symmetric distributions are a special case of the ν-spherical distributions which have a density characterized by the form ρ(y) = g(ν(y)) where ν : Rn →R is a positively homogeneous function of degree one, i.e. it fulfills ν(ay) = aν(y) for any a ∈R+ and y ∈Rn [7]. Lpnested functions are obviously positively homogeneous. Of course, Lp-nested functions of Lpnested functions are again Lp-nested. Therefore, an Lp-nested function f in its general form can be visualized by a tree in which each inner node corresponds to an Lp-norm while the leaves stand for the coefficients of the vector y. Because of the positive homogeneity it is possible to normalize a vector y with respect to ν and obtain a coordinate respresentation x = r · u where r = ν(y) and u = y/ν(y). This implies that the random variable Y has the stochastic representation Y .= RU with independent U and R [7] which makes it a generalization of the Gaussian Scale Mixture model [23]. It can be shown that for a given ν, U always has the same distribution while the distribution ϱ(r) of R determines the specific ρ(y) [7]. For a general ν, it is difficult to determine the distribution of U since the partition function involves the surface area of the ν-unit sphere which is not analytically tractable in most cases. Here, we show that Lp-nested functions allow for an analytical expression of the partition function. Therefore, the corresponding distributions constitute a flexible yet tractable subclass of ν-spherical distributions. In the remaining paper we adopt the following notational convention: We use multi-indices to index single nodes of the tree. This means that I = ∅denotes the root node, I = (∅, i) = i denotes its ith child, I = (i, j) the jth child of i and so on. The function values at individual inner nodes I are denoted by fI, the vector of function values of the children of an inner node I by fI,1:ℓI = (fI,1, ..., fI,ℓI)⊤. By definition, parents and children are related via fI = ∥fI,1:ℓI∥pI. The number of children of a particular node I is denoted by ℓI. Lp-nested symmetric distributions are a very general class of densities. For instance, since every Lpnorm ∥· ∥p is an Lp-nested function, Lp-nested distributions includes the family of Lp-spherically symmetric distributions including (for p = 2) the family of spherically symmetric distributions. When e.g. setting f = ∥· ∥2 or f = (∥· ∥p 2)1/p, and choosing the radial distribution ϱ appropriately, one can recover the Gaussian ρ(y) = Z−1 exp −∥y∥2 2 or the generalized spherical Gaussian ρ(y) = Z−1 exp (−∥y∥p 2), respectively. On the other hand, when choosing the Lp-nested function f as in equation (1) and ϱ to be the radial distribution of a p-generalized Normal distribution ϱ(r) = 3 Z−1rn−1 exp (−rp∅/s) [8; 22], the inner nodes f1:ℓ∅become independent and we can recover an ISA model. Note, however, that not all ISA models are also Lp-nested since Lp-nested symmetry requires the radial distribution to be that of a p-generalized Normal. In general, for a given radial distribution ϱ on the Lp-nested radius f(y), an Lp-nested symmetric distribution has the form ρ(y) = 1 Sf(f(y)) · ϱ(f(y)) = 1 Sf(1) · f n−1(y) · ϱ(f(y)) (2) where Sf(f(y)) = Sf(1)·f n−1(y) is the surface area of the Lp-nested sphere with the radius f(y). This means that the partition function of a general Lp-nested symmetric distribution is the partition function of the radial distribution normalized by the surface area of the Lp-nested sphere with radius f(y). For a given f and a radius f∅= f(y) this surface area is given by the equation Sf(f∅) = fn−1 ∅ 2n Y I∈I 1 pℓI−1 I ℓI−1 Y k=1 B "Pk i=1 nI,k pI , nI,k+1 pI # = fn−1 ∅ 2n Y I∈I QℓI k=1 Γ h nI,k pI i pℓI−1 I Γ h nI pI i where I denotes the set of all multi-indices of inner nodes, nI the number of leaves of the subtree under I and B [a, b] the beta function. Therefore, if the partition function of the radial distribution can be computed easily, so can the partition function of the multivariate Lp-nested distribution. Since the only part of equation (2) that includes free parameters is the radial distribution ϱ, maximum likelihood estimation of those parameters ϑ can be carried out on the univariate distribution ϱ only, because argmaxϑ log ρ(y|ϑ) (2) = argmaxϑ (−log Sf(f(y)) + log ϱ(f(y)|ϑ)) = argmaxϑ log ϱ(f(y)|ϑ). This means that parameter estimation can be done efficiently and robustly on the values of the Lpnested function. Since, for a given f, an Lp-nested distribution is fully specified by a radial distribution, changing the radial distribution also changes the Lp-nested distribution. This suggests an image decomposition constructed from a cascade of nonlinear, gain-control-like mappings reducing the dependence between the filter coefficients. Similar to Radial Gaussianization or Lp-Radial Factorization algorithms [12; 21], the radial distribution ϱ∅of the root node is mapped into the radial distribution of a p-generalized Normal via histogram equalization, thereby making its children exponential power distributed and statistically independent [22]. This procedure is then repeated recursively for each of the children until the leaves of the tree are reached. Below, we estimate the multi-information (MI) between the filters or subtrees at different levels of the hierarchy. In order to do that robustly, we need to know the joint distribution of their values. In particular, we are interested in the joint distribution of the children fI,1:ℓI of a node I (e.g. layer 2 in Figure 2). Just from the form of an Lp-nested function one might guess that those children are Lp-spherically symmetric distributed. However, this is not the case. For example, the children f1:ℓ∅ of the root node (assuming that none of them is a leaf) follow the distribution ρ(f1:ℓ∅) = ϱ∅(∥f1:ℓ∅∥p∅) S∥·∥p∅(∥f1:ℓ∅∥p∅) ℓ∅ Y i=1 fni−1 i . (3) This implies that f1:ℓ∅can be represented as a product of two independent random variables u = f1:ℓ∅/∥f1:ℓ∅∥p∅∈Rℓ∅ + and r = ∥f1:ℓ∅∥p∅∈R+ with r ∼ϱ∅and up∅ 1 , ..., up∅ ℓ∅ ∼ Dir n1/p∅, ..., nℓ∅/p∅ following a Dirichlet distribution (see Additional Material). We call this distribution a Dirichlet Scale Mixture (DSM). A similar form can be shown for the joint distribution of leaves and inner nodes (summarizing the whole subtree below them). Unfortunately, only the children f1:ℓ∅of the root node are really DSM distributed. We were not able to analytically calculate the marginal distribution of an arbitrary node’s children fI,1:ℓI, but we suspect it to have a similar form. For that reason we fit DSMs to those children fI,1:ℓ∅in the experiments below and use the estimated model to assess the dependencies between them. We also use it for measuring the dependencies between the subspaces of ISA. 4 Fitting DSMs via maximum likelihood can be carried out similarly to estimating Lp-nested distributions: Since the radial variables u and r are independent, the Dirichlet and the radial distribution can be estimated on the normalized data points {ui}m i=1 and their respective norms {ri}m i=1 independently. Lp-Spherically Symmetric Distributions and Independent Subspace Analysis The family of Lp-spherically symmetric distributions are a special case of Lp-nested distributions for which f(y) = ∥y∥p [9]. We use the ISA model by [10] where the filter responses y are modelled by a factorial combination of Lp-spherically symmetric distributions sitting on each subspace ρ(y) = K Y k=1 ρk(∥yIk∥pk). 3 Experiments Given an image patch x, all models used in this paper define densities over filter responses y = Wx of linear filters. This means, that all models have the form ρ(y) = | det W|·ρ(Wx). The (n−1)×n matrix W has the form W = QSP where P ∈R(n−1)×n has mutually orthogonal rows and projects onto the orthogonal complement of the DC-filter (filter with equal coefficients), S ∈R(n−1)×(n−1) is a whitening matrix and Q ∈SOn−1 is an orthogonal matrix determining the final filter shapes of W. When we speak of optimizing the filters according to a model, we mean optimizing Q over SOn−1. The reason for projecting out the DC component is, that it can behave quite differently depending on the dataset. Therefore, it is usually removed and modelled separately. Since the DC component is the same for all models and would only add a constant offset to the measures we use in our experiments, we ignore it in the experiments below. Data We use ten pairs of independently sampled training and test sets of 8 × 8 (16 × 16) patches from the van Hateren dataset, each containing 100, 000 (500, 000) examples. Hyv¨arinen and K¨oster [10] report that ISA already finds several subspaces for 8 × 8 image patches. We perform all experiments with two different types of preprocessing: either we only whiten the data (WO-data), or we whiten it and apply an additional contrast gain control step (CGC-data), for which we use the radial factorization method described in [12; 21] with p = 2 in the symmetric whitening basis. We use the same whitening procedure as in [21; 6]: Each dataset is centered on the mean over examples and dimensions and rescaled such that whitening becomes volume conserving. Similarly, we use the same orthogonal matrix to project out the DC-component of each patch (matrix P above). On the remaining n−1 dimensions, we perform symmetric whitening (SYM) with S = C−1 2 where C denotes the covariance matrix of the DC-corrected data C = cov [PX]. Evaluation Measures We use the Average Log Loss per component (ALL) for assessing the quality of the different models, which we estimate by taking the empirical average over a large ensemble of test points ALL = − 1 n−1 ⟨log ρ(y)⟩Y ≈− 1 m(n−1) Pm i=1 log ρ(yi). The ALL equals the entropy if the model distribution equals the true distribution and is larger otherwise. For the CGC-data, we adjust the ALL by the log-determinant of the CGC transformation [11]. In contrast to [10] this allows us to quantitively compare models across the two different types of preprocessing (WO and CGC), which was not possible in [10]. In order to measure the dependence between different random variables, we use the multiinformation per component (MI) 1 n−1 Pd i=1 H[Yi] −H[Y ] which is the difference between the sum of the marginal entropies and the joint entropy. The MI is a positive quantity which is zero if and only if the joint distribution is factorial. We estimate the marginal entropies by a jackknifed MLE entropy estimator [17] (corrected for the log of the bin width in order to estimate the differential entropy) where we adjust the bin width of the histograms suggested by Scott [19]. Instead of the joint entropy, we use the ALL of an appropriate model distribution. Since the ALL is theoretically always larger than the true joint entropy (ignoring estimation errors) using the ALL instead of the joint entropy should underestimate the true MI, which is still sufficient for our purpose. Parameter Estimation For all models (ISA, DSM, Lp-spherical and Lp-nested), we estimate the parameters ϑ for the radial distribution as described above in Section 2. For a given filter matrix 5 W the values of the exponents p are estimated by minimizing the ALL at the ML estimates ˆϑ over p = (p1, ..., pq)⊤. For the Lp-nested distributions, we use the Nelder-Mead [15] method for the optimization over p = (p1, ..., pq)⊤and for the Lp-spherically symmetric distributions we use Golden Search over the single p. For the ISA model, we carry out a Golden Search over p for each subspace independently. For the Lp-spherical and the single models on the ISA subspaces, we use a search range of p ∈[0.1, 2.1] on p. For estimating the Dirichlet Scale Mixtures, we use the fastfit package by Tom Minka to estimate the parameters of the Dirichlet distribution. The radial distribution is estimated independently as described above. When fitting the filters W to the different models (ISA, Lp-spherical and Lp-nested), we use a gradient ascent on the log-likelihood over the orthogonal group by alternating between optimizing the parameters p and ϑ and optimizing for W. For the gradient ascent, we compute the standard Euclidean gradient with respect to W ∈R(n−1)×(n−1) and project it back onto the tangent space of SOn−1. Using the gradient ∇W obtained in that manner, we perform a line search with respect to t using the backprojections of W + t · ∇W onto SOn−1. This method is a simplified version of the one proposed by [14]. Experiments with Independent Subspace Analysis and Lp-Spherically Symmetric Distributions We optimized filters for ISA models with K = 2, 4, 8, 16 subspaces embracing 32, 16, 8, 4 components (one subspace always had one dimension less due to the removal of the DC component), and for an Lp-spherically symmetric model. When optimizing for W we use a radial Γ-distribution for the Lp-spherically symmetric models and a radial χp distribution (∥yIk∥pk pk is Γ-distributed) for the models on the single single subspaces of ISA, which is closer to the one used by [10]. After optimization, we make a final optimization for p and ϑ using a mixture of log normal distributions (log N) with K = 6 mixture components on the radial distribution(s). Lp-Nested Symmetric Distributions As for the Lp-spherically symmetric models, we use a radial Γ-distribution for the optimization of W and a mixture of log N distributions for the final fit. We use two different kind of tree structures for our experiments with Lp-nested symmetric distributions. In the deep tree (DT) structure we first group 2×2 blocks of four neighboring SYM filters. Afterwards, we group those blocks again in a quadtree manner until we reached the root node (see Figure 2A). The second tree structure (PNDk) was motivated by ISA. Here, we simply group the filter within each subspace and joined them at the root node afterwards (see Figure 2B). In order to speed up parameter estimation, each layer of the tree shared the same value of p. Multi-Information Measurements For the ISA models, we estimated the MI between the filter responses within each subspace and between the Lp-radii ∥yIk∥pk, 1 ≤k ≤K. In the former case we used the ALL of an Lp-spherically symmetric distribution with especially optimized p and ϑ, in the latter a DSM with optimized radial and Dirichlet distribution as a surrogate for the joint entropy. For the Lp-nested distribution, we estimate the MI between the children fI,1:ℓI of all inner nodes I. In case the children are leaves, we use the ALL of an Lp-spherically symmetric distribution as surrogate for the joint entropy, in case the children are inner nodes themselves, we use the ALL of an DSM. The red arrows in Figure 2A exemplarily depict the entities between which the MI was estimated. 4 Results and Discussion Figure (2) shows the optimized filters for the DT and the PND16 tree structure (we included the filters optimized on the first of ten datasets for all tree structures in the Additional Material). For both tree structures, the filters on the lowest level are grouped according to spatial frequency and orientation, whereas the variation in orientation is larger for the PND16 tree structure and some filters are unoriented. The next layer of inner nodes, which is only present in the DT tree structure, roughly joins spatial location, although each of those inner nodes has one child whose leaves are global filters. When looking at the various values of p at the inner nodes, we can observe that nodes which are higher up in the tree usually exhibit a smaller value of p. Surprisingly, as can be seen in Figure 3 B and C, a smaller value of p does not correspond to a larger independence between the subtrees, which are even more correlated because almost every subtree contains global filters. The small value of p is caused by the fact that the DSM (the distribution of the subtree values) has to account for this correlation which it can only do by decreasing the value of p (see Figure 3 and the DSM in 6 A B Layer 1 Layer 2 Layer 3 p1=0.77071 p2=0.8438 p3=2.276 p1=0.8413 p2=1.693 Figure 2: Examples for the tree structures of Lp-nested distributions used in the experiments: (A) shows the DT structure with the corresponding optimized values. The red arrows display examples of groups of filters or inner nodes, respectively, for which we estimated the MI. (B) shows the PND16 tree structure with the corresponding values of p at the inner nodes and the optimized filters. the Additional Material). Note that this finding is exactly opposite to the assumptions in the ISA model which can usually not generate such a behavior (Figure 3A) as it models the two subtrees to be independent. This is likely to be one reason for the higher ALL of the ISA models (see Table 1). 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 ||y 1:32 ||p 1 sampled ||y 32:63 ||p 2 sampled 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 ||y 1:32 ||p 1 ||y 32:63 ||p 2 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 f1 f2 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 f1 sampled f2 sampled A B C D Figure 3: Independence of subspaces for WO-data not justfied: (A) Subspace radii sampled from ISA2, (B) subspace radii of natural image patches in the ISA2 basis, (C) subtree values of the PND2 in the PND2 basis, and (D) samples from the PND2 model. While the ISA2 model spreads out the radii almost over the whole positive quadrant due to the independence assumption the samples from the Lp-nested subtrees are more concentrated around the diagonal like the true data. The Lp-nested model can achieve this behavior since (i) it does not assume a radial distribution that leads to independent radii on the subtrees and (ii) the subtree values f1 and f2 are DSM[n1/p∅, n2/p∅, ] distributed. By changing the value of p∅, the DSM model can put more mass towards the diagonal, which produces the ”beam-like” behavior shown in the plot. Table 1 shows the ALL and the MI measurements for all models. Except for the ISA models on WO-data, all performances are similar, whereas the Lp-nested models usually achieve the lowest ALL independent of the particular tree structure used. For the WO-data, the Lp-spherical and the ISA2 model come close to the performance of the Lp-nested models. For the other ISA models on WO-data the ALL gets worse with increasing number of subspaces (bold font number in Table 1). This reflects the effect described above: Contrary to the assumptions of the ISA model, the responses of the different subspaces become in fact more correlated than the single filter responses. This can also be seen in the MI measurements discussed below. When looking at the ALL for CGC data, on the other hand, ISA suddenly becomes competitive. This importance of CGC for ISA has already been noted in [10]. The small differences between all the models in the CGC case shows that the contour change of the joint density for 8×8 patches is too small to allow for a large advantage of the Lp-nested model, because contrast gain control (CGC) 7 directly corresponds to modeling the distribution with an Lp-spherically symmetric distribution [21]. Preliminary results on 16 × 16 data (1.39 ± 0.003 for the Lp-nested and 1.45 ± 0.003 for the Lpspherical model on WO-data), however, show a more pronounced improvement with for the Lpnested model, indicating that a single p does not suffice anymore to capture all dependencies when going to larger patch sizes. When looking at the MI measurements between the filters/subtrees at different levels of the hierarchy in the Lp-nested, Lp-spherically symmetric and ISA models, we can observe that for the WO-data, the MI actually increases when going from lower to higher layers. This means that the MI between the direct filter responses (layer 3 for DT and layer 2 for all others) is in fact lower than the MI between the subspace radii or the inner nodes of the Lp-nested tree (layer 1-2 for DT, layer 1 for all others). The highest MI is achieved between the children of the root node for the DT tree structure (DT layer 1). As explained above this observation contradicts the assumptions of the ISA model and probably causes it worse performance on the WO-data. For the CGC-data, on the other hand, the MI has been substantially decreased by CGC over all levels of the hierarchy. Furthermore, the single filter responses inside a particular subspace or subtree are now more dependent than the subtrees or subspaces themselves. This suggests that the competitive performance of ISA is not due to the model but only due to the fact that CGC made the data already independent. In order to double check this result, we fitted an ICA model to the CGC-data [21] and found an ALL of 1.41 ± 0.004 which is very close to the performance of ISA and the Lp-nested distributions (which would not be the case for WO-data [21]). Taken together, the ALL and the MI measurements suggest that ISA is not the best way to join multiple local models into a single joint model. The basic assumption of the ISA model for natural images is that filter coefficients can either be dependent within a subspace or must be independent between different subspaces. However, the increasing ALL for an increasing number of subspaces and the fact that the MI between subspaces is actually higher than within the subspaces, demonstrates that this hard partition is not justified when the data is only whitened. Family Lp-nested Model Deep Tree PND2 PND4 PND8 PND16 ALL 1.39 ± 0.004 1.39 ± 0.004 1.39 ± 0.004 1.40 ± 0.004 1.39 ± 0.004 ALL CGC 1.39 ± 0.005 1.40 ± 0.004 1.40 ± 0.005 1.40 ± 0.004 1.39 ± 0.004 MI Layer 1 0.84 ± 0.019 0.48 ± 0.008 0.7 ± 0.002 0.75 ± 0.003 0.61 ± 0.0036 MI Layer 1 CGC 0.0 ± 0.004 0.10 ± 0.002 0.02 ± 0.003 0.0 ± 0.009 0.0 ± 0.01 MI Layer 2 0.42 ± 0.021 0.35 ± 0.017 0.33 ± 0.017 0.28 ± 0.019 0.25 ± 0.025 MI Layer 2 CGC 0.002 ± 0.005 0.01 ± 0.0008 0.01 ± 0.004 0.01 ± 0.006 0.02 ± 0.008 MI Layer 3 0.28 ± 0.036 MI Layer 3 GCG 0.04 ± 0.005 Family Lp-spherical ISA Model ISA2 ISA4 ISA8 ISA16 ALL 1.41 ± 0.004 1.40 ± 0.005 1.43 ± 0.006 1.46 ± 0.006 1.55 ± 0.006 ALL CGC 1.41 ± 0.004 1.41 ± 0.008 1.39 ± 0.007 1.40 ± 0.005 1.41 ± 0.007 MI Layer 1 0.34 ± 0.004 0.47 ± 0.01 0.69 ± 0.012 0.7 ± 0.018 0.63 ± 0.0039 MI Layer 1 CGC 0.00 ± 0.005 0.00 ± 0.09 0.00 ± 0.06 0.00 ± 0.04 0.00 ± 0.02 MI Layer 2 0.36 ± 0.017 0.33 ± 0.019 0.31 ± 0.032 0.24 ± 0.024 MI Layer 2 CGC 0.004 ± 0.003 0.03 ± 0.012 0.02 ± 0.018 0.0006 ± 0.013 Table 1: ALL and MI for all models: The upper part shows the results for the Lp-nested models, the lower part show the results for the Lp-spherical and the ISA models. The ALL for the Lp-nested models is almost equal for all tree structures and a bit lower compared to the Lp-spherical and the ISA models. For the whitened only data, the ALL increases significantly with the number of subspaces (bold font). For the CGC data, most models perform similarly well. When looking at the MI, we can see that higher layers for whitened only data are in fact more dependent than lower ones. For CGC data, the MI has dropped substantially over all layers due to CGC. In that case, the lower layers are more independent. In summary, our results show that Lp-nested symmetric distributions yield a good performance on natural image patches, although the advantage over Lp-spherically symmetric distributions is fairly small, suggesting that the distribution within these small patches (8 × 8) is captured reasonably well by a single Lp-norm. Furthermore, our results demonstrate that—at least for 8 × 8 patches—the assumptions of ISA are too rigid for WO-data and are trivially fulfilled for the CGC-data, since CGC already removed most of the dependencies. We are currently working to extend this study to larger patches, which we expect will reveal a more significant advantage for Lp-nested models. 8 References [1] F. Attneave. Informational aspects of visual perception. Psychological Review, 61:183–193, 1954. [2] R. Baddeley. Searching for filters with “interesting” output distributions: an uninteresting direction to explore? Network: Computation in Neural Systems, 7(2):409–421, 1996. [3] H. B. Barlow. Sensory mechanisms, the reduction of redundancy, and intelligence. 1959. [4] Anthony J. Bell and Terrence J. Sejnowski. An Information-Maximization approach to blind separation and blind deconvolution. Neural Computation, 7(6):1129–1159, November 1995. [5] Matthias Bethge. Factorial coding of natural images: how effective are linear models in removing higherorder dependencies? Journal of the Optical Society of America A, 23(6):1253–1268, June 2006. [6] Jan Eichhorn, Fabian Sinz, and Matthias Bethge. Natural image coding in v1: How much use is orientation selectivity? PLoS Comput Biol, 5(4):e1000336, April 2009. [7] Carmen Fernandez, Jacek Osiewalski, and Mark F. J. Steel. Modeling and inference with ν-spherical distributions. Journal of the American Statistical Association, 90(432):1331–1340, Dec 1995. [8] Irwin R. Goodman and Samuel Kotz. Multivariate θ-generalized normal distributions. Journal of Multivariate Analysis, 3(2):204–219, Jun 1973. [9] A. K. Gupta and D. Song. lp-norm spherical distribution. Journal of Statistical Planning and Inference, 60:241–260, 1997. [10] A. Hyvarinen and U. Koster. Complex cell pooling and the statistics of natural images. Network: Computation in Neural Systems, 18(2):81–100, 2007. [11] S Lyu and E P Simoncelli. Nonlinear extraction of ’independent components’ of natural images using radial Gaussianization. Neural Computation, 21(6):1485–1519, June 2009. [12] S Lyu and E P Simoncelli. Reducing statistical dependencies in natural signals using radial Gaussianization. In D. Koller, D. Schuurmans, Y. Bengio, and L. Bottou, editors, Adv. Neural Information Processing Systems 21, volume 21, pages 1009–1016, Cambridge, MA, May 2009. MIT Press. [13] Siwei Lyu and E.P. Simoncelli. Modeling multiscale subbands of photographic images with fields of gaussian scale mixtures. Pattern Analysis and Machine Intelligence, IEEE Transactions on, 31(4):693– 706, 2009. [14] J. H. Manton. Optimization algorithms exploiting unitary constraints. IEEE Transactions on Signal Processing, 50:635 – 650, 2002. [15] J. A. Nelder and R. Mead. A simplex method for function minimization. The Computer Journal, 7(4):308– 313, Jan 1965. [16] Bruno A. Olshausen and David J. Field. Emergence of simple-cell receptive field properties by learning a sparse code for natural images. Nature, 381(6583):607–609, June 1996. [17] Liam Paninski. Estimation of entropy and mutual information. Neural Computation, 15(6):1191–1253, Jun 2003. [18] S. Roth and M.J. Black. Fields of experts: a framework for learning image priors. In Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, volume 2, pages 860–867 vol. 2, 2005. [19] David W. Scott. On optimal and data-based histograms. Biometrika, 66(3):605–610, Dec 1979. [20] E.P. Simoncelli. Statistical models for images: compression, restoration and synthesis. In Signals, Systems & Computers, 1997. Conference Record of the Thirty-First Asilomar Conference on, volume 1, pages 673–678 vol.1, 1997. [21] F. Sinz and M. Bethge. The conjoint effect of divisive normalization and orientation selectivity on redundancy reduction. In Neural Information Processing Systems 2008, 2009. [22] F. H. Sinz, S. Gerwinn, and M. Bethge. Characterization of the p-generalized normal distribution. Journal of Multivariate Analysis, 100(5):817–820, 05 2009. [23] M. J. Wainwright and E. P. Simoncelli. Scale mixtures of gaussians and the statistics of natural images. In Advances in neural information processing systems, volume 12, pages 855–861, 2000. [24] Christoph Zetzsche, Gerhard Krieger, and Bernhard Wegmann. The atoms of vision: Cartesian or polar? Journal of the Optical Society of America A, 16(7):1554–1565, Jul 1999. 9
|
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|
196
|
3,699
|
Optimizing Multi-class Spatio-Spectral Filters via Bayes Error Estimation for EEG Classification Wenming Zheng Research Center for Learning Science Southeast University Nanjing, Jiangsu 210096, P.R. China wenming zheng@seu.edu.cn Zhouchen Lin Microsoft Research Asia Beijing 100190, P.R. China zhoulin@microsoft.com Abstract The method of common spatio-spectral patterns (CSSPs) is an extension of common spatial patterns (CSPs) by utilizing the technique of delay embedding to alleviate the adverse effects of noises and artifacts on the electroencephalogram (EEG) classification. Although the CSSPs method has shown to be more powerful than the CSPs method in the EEG classification, this method is only suitable for two-class EEG classification problems. In this paper, we generalize the two-class CSSPs method to multi-class cases. To this end, we first develop a novel theory of multi-class Bayes error estimation and then present the multi-class CSSPs (MCSSPs) method based on this Bayes error theoretical framework. By minimizing the estimated closed-form Bayes error, we obtain the optimal spatio-spectral filters of MCSSPs. To demonstrate the effectiveness of the proposed method, we conduct extensive experiments on the BCI competition 2005 data set. The experimental results show that our method significantly outperforms the previous multi-class CSPs (MCSPs) methods in the EEG classification. 1 Introduction The development of non-invasive brain computer interface (BCI) using the electroencephalogram (EEG) signal has become a very hot research topic in the BCI community [1]. During the last several years, a large number of signal processing and machine learning methods have been proposed for EEG classification [6]. It is challenging to extract the discriminant features from the EEG signal for EEG classification. This is because in most cases the EEG data are centered at zero and thus many traditional discriminant feature extraction methods, e.g., Fisher’s linear discriminant analysis (FLDA) [7], cannot be successfully used. Among the various EEG feature extraction methods, the common spatial patterns (CSPs) method [2] is one of the most popular. Given two classes of EEG signal, the basic idea of CSPs is to find some projection directions such that the projections of the EEG signal onto these directions will maximize the variance of one class and simultaneously minimize the variance of the other class. Although CSPs have achieved great success in EEG classification, this method only utilizes the spatial information of the EEG signal. To utilize both the spatial and the temporal information of the EEG signal for classification, Lemm et al. [3] proposed a new EEG feature extraction method, called common spatio-spectral patterns (CSSPs), which extended the CSPs method by concatenating the original EEG data and a time-delayed one to form a longer vector sample, and then performed EEG feature extraction, which is similar to the CSPs method, from these padded samples. The experiments in [3] showed that the CSSPs method outperforms the CSPs method. A multi-class extension of the two-class CSPs method (MCSPs) was proposed by Dornhege et al. [4] who adopted a joint approximate diagonalization (JAD) technique to find the optimal spatial filters. Grosse-Wentrup and Buss [5] recently pointed out that the MCSPs method has two major 1 drawbacks. The first drawback is that this method lacks solid theoretical foundation with respect to its classification error. The second one is that the selection of the optimal spatial filters of MCSPs is based on heuristics. To overcome these drawbacks, they proposed a method based on mutual information to select the optimal spatial filters from the original MCSPs result. Nevertheless, it should be noted that both the MCSPs methods are based on the JAD technique, where a closed-form solution is unavailable, making the theoretical analysis difficult. In this paper, we generalize the two-class CSSPs method to multi-class cases, hereafter called the MCSSPs method. However, we do not adopt the same JAD technique used in the MCSPs method to derive our MCSSPs method. Instead, we derive our MCSSPs method directly based on the Bayes error estimation, and thus provide a solid theoretic foundation. To this end, we first develop a novel theory of multi-class Bayes error estimation, which has a closed-form solution to find the optimal discriminant vectors. Based on this new theoretic framework, we propose our MCSSPs method for EEG feature extraction and recognition. 2 Brief Review of CSPs and CSSPs Let Xt i = {xt i,j ∈IRd|j = 1, · · · , mi,t} (t = 1, · · · , ni; i = 1, · · · , c) denote the EEG data set from the tth trial of the ith class, where d, c, ni, and mi,t denote the number of channels (i.e., recording electrodes), the number of classes, the number of trials of the ith class, and the number of samples (i.e., recording points) in the tth trial of the ith class, respectively. Assume that the EEG data conditioned on each class follows a Gaussian distribution with a zero mean, i.e., pi(x) = N(0, Σi) (i = 1, · · · , c)1. Then the main task of EEG feature extraction is to find a linear transformation W ∈IRd×k (k < d), such that for finite training data using the projected vectors yt i,j = WT xt i,j to classify the EEG signal may lead to better classification accuracy than using xt i,j. 2.1 The CSPs Method For the two-class EEG classification problem, the basic idea of CSPs is to find a transformation matrix W that simultaneously diagonalizes both class covariance matrices Σ1 and Σ2 [2], i.e., WT ΣiW = Λi, (i = 1, 2), (1) where Λi = diag{λi,1, · · · , λi,d} (i = 1, 2) are diagonal matrices. The spatial filters can be chosen as the columns of W associated with the maximal or minimal ratio of λ1,j λ2,j (j = 1, · · · , d). Parra et al. [6] proved that the CSPs method can be formulated as the following optimization problem: ω = arg max ω max ½ωT Σ1ω ωT Σ2ω , ωT Σ2ω ωT Σ1ω ¾ , (2) and this optimization problem boils down to solving the following generalized eigenvalue decomposition problem: Σ1ω = λΣ2ω. (3) Let ω1, · · · , ωd and λ1, · · · , λd be the eigenvectors and the corresponding eigenvalues of equation (3), then the spatial filters ωi1, · · · , ωik can be chosen from the eigenvectors ω1, · · · , ωd associated with the largest and the smallest eigenvalues. Then W = [ωi1, · · · , ωik] and the projection of Xt i with W can be expressed as: Yt i = WT Xt i. (4) 2.2 The CSSPs Method The CSSPs method is an extension of CSPs by concatenating the original EEG data and a timedelayed one to form a longer vector sample, and then performing EEG feature extraction, which is similar to the CSPs method, from these padded samples. More specifically, let δτ denote the time-delay operator with the delayed time τ, i.e., δτ(xt i,j) = xt i,j−τ. (5) 1This model is often assumed in the literature, e.g., [5]. 2 Then, equation (4) can be re-written as the following: ˆYt i = WT (0)Xt i + WT (τ)δτ(Xt i), (6) where W(0) and W(τ) are the transformation matrices on the EEG data Xt and δτ(Xt), respectively. To express the above equation in a similar form as CSPs, we define ˆXt i = µ Xt i δτ(Xt i) ¶ . (7) In this way, solving the CSSPs problem boils down to solving a similar generalized eigenvalue problem as defined in equation (3), if we use the new class covariance matrices ˆΣ1 and ˆΣ2 to replace the original class covariance matrices Σ1 and Σ2, where ˆΣi = ˜Σi trace( ˜Σi) , and ˜Σi = X t ˆXt i( ˆXt i)T . (8) 3 MCSSPs Based on Multi-class Bayes Error Estimation In this section, we extend the CSSPs method to the multi-class case. To begin with, we develop a novel theory of multi-class Bayes error estimation. Then we present our MCSSPs method based on this Bayes error framework. 3.1 Multi-class Bayes Error Estimation It is well known that the Bayes error regarding classes i and j can be expressed as [7]: ε = Z min(Pipi(x), Pjpj(x))dx, (9) where Pi and pi(x) are the apriori probability and the probability density function of the ith class, respectively. Let εij = R p PiPjpi(x)pj(x)dx. By applying the following inequality: min(a, b) ≤ √ ab, ∀a, b ≥0, (10) and the assumption pi(x) = N(0, Σi), we obtain the following upper bound of the Bayes error: ε ≤εij = p PiPj exp à −1 2ln | ¯Σij| p |Σi||Σj| ! = p PiPj à | ¯Σij| p |Σi||Σj| !−1 2 , (11) where ¯Σij = 1 2(Σi + Σj). The expression in exp(·) is the simplified Bhattacharyya distance [7]. If we project the samples to 1D by a vector ω, then the upper bound εij becomes: εij = p PiPj à ωT ¯Σijω p (ωT Σiω)(ωT Σjω) !−1 2 . (12) Define u = ωT ¯Σijω and v = ωT ∆Σijω, where ∆Σij = 1 2(Σi −Σj). Then εij can be written as εij = p PiPj µ u √ u2 −v2 ¶−1 2 = p PiPj µ 1 − ³v u ´2¶ 1 4 ≤ p PiPj µ 1 −1 4 ³v u ´2¶ . (13) For the c classes problem, the upper bound of the Bayes error in the reduced feature space can be estimated as ε ≤Pc−1 i=1 Pc j=i+1 εij [8]. Then, from equation (13), we obtain that ε ≤ c−1 X i=1 c X j=i+1 εij ≤ c−1 X i=1 c X j=i+1 p PiPj à 1 −1 4 µωT ∆Σijω ωT ¯Σijω ¶2! = c−1 X i=1 c X j=i+1 p PiPj −1 8 c X i=1 c X j=1 p PiPj µωT (∆Σij)ω ωT ¯Σijω ¶2 . (14) 3 Recursively applying the following inequality ¡ a b ¢2 + ¡ c d ¢2 ≥ ³ a+c b+d ´2 , ∀a, c ≥0; b, d > 0 to the error bound in equation (14), we have ε ≤ c−1 X i=1 c X j=i+1 p PiPj −1 8 ÃPc i=1 Pc j=1(PiPj) 5 4 |ωT ∆Σijω| Pc i=1 Pc j=1 PiPjωT ¯Σijω !2 . (15) Let ¯Σ = Pc i=1 PiΣi be the global covariance matrix. Then we have c X i=1 c X j=1 PiPj ¯Σij = 1 2 c X i=1 c X j=1 PiPj(Σi + Σj) = ¯Σ. (16) Combining equations (15) and (16), we have ε ≤ c−1 X i=1 c X j=i+1 p PiPj −1 8 ÃPc i=1 Pc j=1(PiPj) 5 4 |ωT ∆Σijω| ωT ¯Σω !2 . (17) Assume that the prior probabilities of the classes are the same, i.e., Pi = Pj = P, which holds for most EEG experiments. Then equation (17) becomes ε ≤ c−1 X i=1 c X j=i+1 P −1 8 à P 5 2 Pc i=1 Pc j=1 |ωT (Σi −Σj)ω| 2ωT ¯Σω !2 . (18) On the other hand, from ¯Σ = Pc i=1 PiΣi = Pc i=1 PΣi, we obtain that P c X i=1 |ωT (Σi −Σj)ω| ≥ ¯¯¯¯¯¯ c X j=1 PωT (Σi −Σj)ω ¯¯¯¯¯¯ = |ωT (Σi −¯Σ)ω|. (19) Combining equations (19) and (18), we obtain that ε ≤ c−1 X i=1 c X j=i+1 P −1 8 à P 3 2 Pc i=1 |ωT (Σi −¯Σ)ω| 2ωT ¯Σω !2 . (20) 3.2 MCSSPs Based on Multi-class Bayes Error Estimation Let ˆΣi (k = 1, · · · , c) denote the new class covariance matrices computed via equation (8). Then to minimize the Bayes error, we should minimize its upper bound, which boils down to maximizing the following discriminant criterion J(ω) = Pc i=1 |ωT ( ˆΣi −ˆ¯Σ)ω| ωT ˆ¯Σω . (21) where ˆ¯Σ is the global covariance matrix. Based on this criterion, we define the k optimal spatial filters of MCSSPs as follows: ω1 = arg max ω Pc i=1 |ωT ( ˆΣi −ˆ¯Σ)ω| ωT ˆ¯Σω , · · · ωk = arg max ωT ˆ¯Σωj = 0, j = 1, · · · , k −1 Pc i=1 |ωT ( ˆΣi −ˆ¯Σ)ω| ωT ˆ¯Σω . (22) Let ˆˆΣi = ˆ¯Σ −1 2 ˆΣi ˆ¯Σ −1 2 (i = 1, · · · , c) and α = ˆ¯Σ 1 2 ω. Then solving the optimization problem of equation (22) is equivalent to solving the following optimization problem α1 = arg max α Pc i=1 |αT ( ˆˆΣi −I)α| αT α , · · · αk = arg max αT Uk−1 = 0 Pc i=1 |αT ( ˆˆΣi −I)α| αT α , (23) 4 where Uk−1 = [α1, · · · , αk−1] and I is the identity matrix. Suppose that si ∈{+1, −1} denotes the positive or negative sign of αT ( ˆˆΣi −I)α. Then |αT ( ˆΣi −I)α| = αT si( ˆΣi −I)α. (24) So equation (23) can be expressed as α1 = arg max α αT Pc i=1 si( ˆˆΣi −I)α αT α , · · · αk = arg max αT Uk−1 = 0 αT Pc i=1 si( ˆˆΣi −I)α αT α . (25) Let T(s) = Pc i=1 si( ˆˆΣi −I), where s = [s1, s2, · · · , sc]T and si ∈{+1, −1}. Then the first vector α1 defined in equation (25) is the principal eigenvector associated with the largest eigenvalue of the matrix T(s). Suppose that we have obtained the first k vectors α1, · · · , αk. To solve the (k + 1)-th vector αk+1, we introduce Theorems 1 and 2 below. The similar proofs of both theorems can be found in [9]. Theorem 1. Let QkRk be the QR decomposition of Uk. Then αk+1 defined in (25) is the principal eigenvector corresponding to the largest eigenvalue of the following matrix (Id −QkQT k )T(s)(Id −QkQT k ). Theorem 2. Suppose that QkRk is the QR decomposition of Uk. Let Uk+1 = (Uk αk+1), q = αk+1 −Qk(QT k αk+1), and Qk+1 = ³ Qk q ∥q∥ ´ . Then Qk+1 µ Rk QT k αk+1 0 ∥q∥ ¶ is the QR decomposition of Uk+1. The above two theorems are crucial to design our fast algorithm for solving MCSSPs: Theorem 1 makes it possible to use the power method to solve MCSSPs, while Theorem 2 makes it possible to update Qk+1 from Qk by adding a single column. Moreover, it is notable that Id −QkQT k = k Y i=1 (Id −qiqT i ) = (Id −Qk−1QT k−1)(Id −qkqT k ), (26) where qi is the i-th column of Qk. Equation (26) makes it possible to update the matrix (Id − QkQT k )T(s)(Id −QkQT k ) from (Id −Qk−1QT k−1)T(s)(Id −Qk−1QT k−1) by the rank-one update technique. Let S = {s|s ∈{+1, −1}c} denote the parameter vector set, whose cardinality is 2c. Then we have that max ∥α∥=1 c X i=1 |αT ( ˆˆΣi −I)α| = max s∈S max ∥α∥=1 αT T(s)α. (27) If c is not too large, a full search on S similar to that proposed in [9] is affordable. We present the pseudo-code of our MCSSPs method using the full search on S in Algorithm 1. However, if c is a bit large, we may adopt a similar approach as proposed in [10], which is based on a greedy search, to find the suboptimal solution. The pseudo-code based on the greedy search is given in Algorithm 2. 4 EEG Feature Extraction Based on the MCSSPs Let Xt i be the EEG sample points from the tth trial under the ith condition (i.e., the ith class). Let ωj be the jth optimal spatial filter of the MCSSPs method. Construct the new data ˆXt i = µ Xt i δτ(Xt i) ¶ , and let ˆpt i,j = ωT j ˆXt i (28) 5 Algorithm 1: The MCSSPs Algorithm Based on the Full Search Strategy Input: • Input data matrix X and the class label vector l. Initialization: 1. Compute the average covariance matrices ˆΣi (i = 1, · · · , c) and ˆ¯Σ; 2. Perform SVD of ˆ¯Σ: ˆ¯Σ = UΛUT , compute ˆ¯Σ −1 2 = UΛ−1 2 UT and ˆ¯Σ −1 = UΛ−1UT ; 3. Compute ˆˆΣi = ˆ¯Σ −1 2 ˆΣi ˆ¯Σ −1 2 and ∆ˆˆΣi = ˆˆΣi −I (i = 1, · · · , c); 4. Enumerate all the elements of S and denote them by S = {s1, s2, · · · , s2c}; For i = 1, 2, · · · , k, Do 1. For j=1 to 2c • Compute T(si); • Solve the principal eigenvector of T(si)α(j) = λ(j)α(j) via the power iteration method; 2. Select the eigenvector α with the largest eigenvalue maxj=1,···,2c{λ(j)}; 3. If i = 1, then qi ←α, qi ←qi/∥qi∥, and Q1 ←qi; else qi ←α −Qi−1(QT i−1α), qi ←qi/∥qi∥, and Qi ←(Qi−1 qi); 4. Compute ∆ˆˆΣp ←∆ˆˆΣp−(∆ˆˆΣpqi)qT i −qi(qT i ∆ˆˆΣp)+qi(qT i ∆ˆˆΣpqi)qT i (p = 1, · · · , c); Output: ωi = ˆ¯Σ −1 2 αi, i = 1, · · · , k. Algorithm 2: The MCSSPs Algorithm Based on the Greedy Search Strategy Input: • Input data matrix X and the class label vector l. Initialization: 1. Compute the average covariance matrices ˆΣi (i = 1, · · · , c) and ˆ¯Σ; 2. Perform SVD of ˆ¯Σ: ˆ¯Σ = UΛUT , compute ˆ¯Σ −1 2 = UΛ−1 2 UT and ˆ¯Σ −1 = UΛ−1UT ; 3. Compute ˆˆΣi = ˆ¯Σ −1 2 ˆΣi ˆ¯Σ −1 2 and ∆ˆˆΣi = ˆˆΣi −I (i = 1, · · · , c); For i = 1, 2, · · · , k, Do 1. Set s ←(1, · · · , 1)T , s1 ←−s, and compute T(s); 2. Solve the principal eigenvector of T(s)α = λα associated with the largest absolute eigenvalue |λ| via the power iteration method. Set λ0 ←|λ|; While s ̸= s1, Do (a) Set s1 ←s; (b) For j = 1, 2, · · · , c, Do • Set sj ←−sj, where sj denotes the jth element of s. Compute T(s); • Solve the principal eigenvector of T(s)α = λα associated with the largest absolute eigenvalue |λ| via the power iteration method, and set λ1 ←|λ|; • If λ1 ≤λ0, then sj ←−sj, else λ0 ←λ1; (c) Compute T(s) and solve the principal eigenvector αi of T(s)αi = λαi associated with the largest absolute eigenvalue |λ| via the power iteration method; 3. If i = 1, then qi ←αi, qi ←qi/∥qi∥, and Q1 ←qi; else qi ←αi −Qi−1(QT i−1αi), qi ←qi/∥qi∥, and Qi ←(Qi−1 qi); 4. Compute ∆ˆˆΣp ←∆ˆˆΣp−(∆ˆˆΣpqi)qT i −qi(qT i ∆ˆˆΣp)+qi(qT i ∆ˆˆΣpqi)qT i (p = 1, · · · , c); Output: ωi = ˆ¯Σ −1 2 αi, i = 1, · · · , k. 6 be the projections of the EEG data ˆXt onto the projection vector ωj. Then the covariance of the elements in the projections ˆpt i,j can be expressed as vt i,j = var(ωT j ˆXt i) = ωT j ˆΣt iωj. (29) where ˆΣt i denotes the covariance matrix of the EEG data in the tth trial of the ith class. For all the k spatio-spectral filters ω1, · · · , ωk, we obtain the k features vt i,j (j = 1, · · · , k) from the tth trial of EEG data. Now let vt i = [vt i,1, · · · , vt i,k]T be the feature vector associated with the tth trial of the ith class. Similar to the method used in [2], the following log-transformation form is used as the final feature vector of the EEG signal: f t i = log à vt i P k vt i,k ! , (30) where the log function is applied to each element of the vector independently. The logtransformation serves to approximate the normal distribution of the data [2]. For the given unknown EEG data Z, we use the same procedures to extract the corresponding features, i.e., we first construct the new data ˆZ = µ Z δτ(Z) ¶ , and then adopt the above method to extract the corresponding discriminant feature vector f z, where f z = log µ vz P k vz k ¶ , vz = [vz 1, · · · , vz k]T , and vz j = ωT j ˆΣzωj, (31) in which ˆΣz denotes the covariance matrix of ˆZ. After obtaining the discriminant feature vectors f t i (i = 1, · · · , c; t = 1, 2 · · · , ni) and f z, we can classify the unknown EEG data into one of the c classes by using a classifier, e.g., the K-nearest neighbor (K-NN) classifier [7]. 5 Experiments To test the performance of our MCSSPs method, we use the real world EEG data set to conduct experiments. The data set used here is from “BCI competition 2005” - data set IIIa [11]. This data set consists of recordings from three subjects (k3b, k6b, and l1b), which performed four different motor imagery tasks (left/right hand, one foot, or tongue) according to a cue. During the experiments, the EEG signal is recorded in 60 channels, using the left mastoid as reference and the right mastoid as ground. The EEG was sampled at 250 Hz and was filtered between 1 and 50 Hz with the notch filter on. Each trial lasted for 7 s, with the motor imagery performed during the last 4 s of each trial. For subjects k6b and l1b, a total of 60 trials per condition were recorded. For subject k3b, a total of 90 trials per condition were recorded. Similar to the method in [5], we discard the four trials of subject k6b with missing data. For each trial of the EEG raw data, we only use part of the sample points, i.e., from No.1001 to No.1750, as the experiment data since they carry most of the information in the EEG signal. Consequently, each trial contains 750 data points. We adopt the two-fold cross validation strategy to perform the experiment, i.e., for all the trials of each condition per subject, we divide them into two groups. Each group is used as training data and testing data once. We conduct five rounds of experiments in total, with different divisions of the training and testing data sets, to obtain ten recognition rates, which are averaged as the final recognition rate. For comparison, we also conduct the same experiment using both MCSPs methods proposed by [4] and [5], respectively. To better identify the effect of using different EEG filters, a simple classifier, K-NN classifier with the Euclidean distance and 7 nearest neighbors, is used for final classification. Table 1 shows the average classification rates (%) versus the standard deviations (%) of the three methods2, while figure 1 shows the average recognition rates of our MCSSPs method with different choices of the delayed time τ. From table 1, we can see that the MCSSPs method achieves much better classification performance than the MCSPs methods. 2The results using the MCSPs method proposed in [5] are inferior to those reported in [5] because we did not pre-filter the EEG signals with a Butterworth filter and did not use the logistic regression classifiers for classification either, as we are more interested in comparing the effect of different EEG filters. 7 Table 1: Comparison of the classification rates (%) versus standard deviations (%) between MCSPs and MCSSPs. Subject MCSPs [4] MCSPs [5] MCSSPs/Bayes k3b 46.17 (6.15) 84.89 (2.74) 85.83 (2.23) k6b 33.54 (4.27) 50.09 (2.59) 56.28 (3.87) l1b 35.17 (3.92) 62.08 (3.99) 68.58 (6.16) 1 2 3 4 5 6 7 8 9 10 50 55 60 65 70 75 80 85 90 τ Classification rates (%) k3b k6b l1b Figure 1: The classification rates (%) of our MCSSPs method with different choices of τ. 6 Conclusions In this paper, we extended the two-class CSSPs method to the multi-class cases via the Bayes error estimation. We first proposed a novel theory on multi-class Bayes error estimation, which has a closed-form solution to find the optimal discriminant vectors for feature extraction. Then we applied the multi-class Bayes error estimation theory to generalize the two-class CSSPs method to multiclass cases. The experiments on the data set IIIa from BCI competition 2005 have shown that our MCSSPs method is superior to the MCSPS methods. With more elaborate treatments, e.g., preprocessing the EEG signal and adopting a more advanced classifier, even higher classification rates are possible. These will be reported in our forthcoming papers. Acknowledgment This work was partly supported by National Natural Science Foundation of China under Grants 60503023 and 60872160. References [1] B. Blankertz, G. Curio, & K.-R. M¨uller (2002) Classifying single trial EEG: towards brain computer interfacing. In: T.G. Dietterich, S. Bechker, Z. Ghaharamani (Eds.), Advances in Neural Information Processing Systems14, pp.157-164. Cambridge, MA:MIT Press. [2] H. Ramoser, J. Mueller-Gerking, & G. Pfurtscheller (2000) Optimal spatial filtering of single trial EEG during imaged hand movement. IEEE Transactions on Rehabilitation Engineering. 8(4):441-446. 8 [3] S. Lemm, B. Blanketz, G. Curio, & K.-R. M¨uller (2005) Spatio-spectral filters for improved classification of single trial EEG. IEEE Transactions on Biomedical Engineering. 52(9):1541-1548. [4] G. Dornhege, B. Blankertz, G. Curio, & K.-R. M¨uller (2004) Boosting bit rates in noninvasive EEG singletrial classifications by feature combination and multiclass paradigms. IEEE Transactions on Biomedical Engineering. 51(6):993-1002. [5] M. Grosse-Wentrup, & M. Buss (2008) Multiclass Common Spatial Patterns and Information Theoretic Feature Extraction. IEEE Transactions on Biomedical Engineering. 55:1991-2000. [6] L. C. Parra, C. D. Spence, A. D. Gerson, & P. Sajda (2005) Recipes for linear analysis of EEG. Neuroimage, 28:326-341. [7] K. Fukunaga (1990) Introduction to Statistical Pattern Recognition (Second Edition). New York: Academic Press. [8] J.T. Chu & J.C. Chuen (1967) Error Probability in Decision Functions for Character Recognition. Journal of the Association for Computing Machinery. 14(2):273-280. [9] W. Zheng (2009) Heteroscedastic Feature Extraction for Texture Classification. IEEE Signal Processing Letters, 16(9):766-769. [10] W. Zheng, H. Tang, Z. Lin, & T.S. Huang (2009) A Novel Approach to Expression Recognition from Non-frontal Face Images. Proceedings of 2009 IEEE International Conference on Computer Vision (ICCV2009), pp.1901-1908. [11] G. Blankertz, K.R. Mueller, D. Krusienski, G. Schalk, J.R. Wolpaw, A. Schloegl, G. Pfurtscheller, J. R. Millan, M. Schroeder, & N. Birbaumer (2006) The BCI competition III: Validating alternative approaches to actual BCI problems. IEEE Transactions on Rehabilitation Engineering 14:153-159. 9
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