nips nips2012 nips2012-22 knowledge-graph by maker-knowledge-mining

22 nips-2012-A latent factor model for highly multi-relational data

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Author: Rodolphe Jenatton, Nicolas L. Roux, Antoine Bordes, Guillaume R. Obozinski

Abstract: Many data such as social networks, movie preferences or knowledge bases are multi-relational, in that they describe multiple relations between entities. While there is a large body of work focused on modeling these data, modeling these multiple types of relations jointly remains challenging. Further, existing approaches tend to breakdown when the number of these types grows. In this paper, we propose a method for modeling large multi-relational datasets, with possibly thousands of relations. Our model is based on a bilinear structure, which captures various orders of interaction of the data, and also shares sparse latent factors across different relations. We illustrate the performance of our approach on standard tensor-factorization datasets where we attain, or outperform, state-of-the-art results. Finally, a NLP application demonstrates our scalability and the ability of our model to learn efﬁcient and semantically meaningful verb representations. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 fr Abstract Many data such as social networks, movie preferences or knowledge bases are multi-relational, in that they describe multiple relations between entities. [sent-8, score-0.297]

2 While there is a large body of work focused on modeling these data, modeling these multiple types of relations jointly remains challenging. [sent-9, score-0.381]

3 Finally, a NLP application demonstrates our scalability and the ability of our model to learn efﬁcient and semantically meaningful verb representations. [sent-14, score-0.235]

4 1 Introduction Statistical Relational Learning (SRL) [7] aims at modeling data consisting of relations between entities. [sent-15, score-0.32]

5 Social networks, preference data from recommender systems, relational databases used for the semantic web or in bioinformatics, illustrate the diversity of applications in which such modeling has a potential impact. [sent-16, score-0.256]

6 Relational data typically involve different types of relations between entities or attributes. [sent-17, score-0.642]

7 These entities can be users in the case of social networks or recommender systems, words in the case of lexical knowledge bases, or genes and proteins in the case of bioinformatics ontologies, to name a few. [sent-18, score-0.389]

8 For binary relations, the data is naturally represented as a so called multi-relational graph consisting of nodes associated with entities and of different types of edges between nodes corresponding to the different types of relations. [sent-19, score-0.383]

9 Equivalently the data consists of a collection of triplets of the form (subject, relation, object), listing the actual relationships where we will call subject and object respectively the ﬁrst and second term of a binary relation. [sent-20, score-0.199]

10 Besides relational databases, SRL can also be used to model natural language semantics. [sent-23, score-0.244]

11 A standard way of representing the meaning of language is to identify entities and relations in texts or speech utterances and to organize them. [sent-24, score-0.691]

12 In this paper, we introduce a model for relational data and apply it to multi-relational graphs and to natural language. [sent-32, score-0.179]

13 In assigning high probabilities to valid relations and low probabilities to all the others, this model extracts meaningful representations of the various entities and relations in the data. [sent-33, score-0.97]

14 Besides, thanks to a sparse distributed representation of relation types, our model can handle data with a signiﬁcantly larger number of relation types than was considered so far in the literature (a crucial aspect for natural language data). [sent-37, score-0.325]

15 2 Related work A branch of relational learning, motivated by applications such as collaborative ﬁltering and link prediction in networks, models relations between entities as resulting from intrinsic latent attributes of these entities. [sent-39, score-0.878]

16 1 Work in what we will call relational learning from latent attributes (RLA) focused mostly on the problem of modeling a single relation type as opposed to trying to model simultaneously a collection of relations which can themselves be similar. [sent-40, score-0.682]

17 As reﬂected by several formalisms proposed for relational learning [7], it is the latter multi-relational learning problem which is needed to model efﬁciently large scale relational databases. [sent-41, score-0.358]

18 The fact that relations can be similar or related suggests that a superposition of independently learned models for each relation would be highly inefﬁcient especially since the relationships observed for each relation are extremely sparse. [sent-42, score-0.565]

19 A natural extension to learning multiple relations consists in stacking the matrices to be factorized and applying classical tensor factorization methods such as CANDECOMP / PARAFAC [25, 8]. [sent-44, score-0.477]

20 Another natural extension to learning several relations simultaneously can be to share the common embedding or the entities across relations via collective matrix factorization as proposed in RESCAL [15] and other related work [18, 23]. [sent-46, score-0.994]

21 To share parameters between relations, [9, 24, 14, 28] and [10] build models that cluster not only entities but relations as well. [sent-49, score-0.604]

22 As for SME [2], its modeling of relations by vectors allowed it to scale to several thousands of relations. [sent-52, score-0.32]

23 2 3 Relational data modeling We consider relational data consisting of triplets that encode the existence of relation between two entities that we will call the subject and the object. [sent-57, score-0.737]

24 Speciﬁcally, we consider a set of ns subjects {Si }i∈ 1;ns along with no objects {Ok }k∈ 1;no which are related by some of nr relations {Rj }j∈ 1;nr . [sent-58, score-0.496]

25 A triplet encodes that the relation Rj holds between the subject Si and the object Ok , which we will write Rj (Si , Ok ) = 1. [sent-59, score-0.258]

26 A typical example which we will discuss in greater detail is in natural language processing where a triplet (Si , Rj , Ok ) corresponds to the association of a subject and a direct object through a transitive verb. [sent-61, score-0.236]

27 The goal is to learn a model of the relations to reliably predict unseen triplets. [sent-62, score-0.297]

28 For instance, one might be interested in ﬁnding a likely relation Rj based only on the subject and object (Si , Ok ). [sent-63, score-0.19]

29 4 Model description In this work, we formulate the problem of learning a relation as a matrix factorization problem. [sent-64, score-0.167]

30 Following a rationale underlying several previous approaches [15, 24], we consider a model in which entities are embedded in Rp and relations are encoded as bilinear operators on the entities. [sent-65, score-0.627]

31 Each of the p-dimensional representations si , ok will have to be learned. [sent-75, score-0.583]

32 The relations are represented by a collection of matrices (Rj )1≤j≤nr , with Rj ∈ Rp×p , which together form a three-dimensional tensor. [sent-76, score-0.32]

33 Assuming ﬁrst that si (j) and ok are ﬁxed, our model is derived from a logistic model P[Rj (Si , Ok ) = 1] σ ηik , with (j) σ(t) 1/(1 + e−t ). [sent-78, score-0.562]

34 A natural form for ηik is a linear function of the tensor product si ⊗ ok (j) which we can write ηik = si , Rj ok where ·, · is the usual inner product in Rp . [sent-79, score-1.173]

35 If we think now of learning si , Rj and ok for all (i, j, k) simultaneously, this model learns together the matrices Rj and optimal embeddings si , ok of the entities so that the usual logistic regressions based on si ⊗ok predict well the probability of the observed relationships. [sent-80, score-1.625]

36 In NLP, we often refer to these as respectively unigram, bigram and trigram terms, a terminology which we will reuse in the rest of the paper. [sent-85, score-0.213]

37 We therefore design E(si , Rj , ok ) to account for these interactions of various orders, retaining only 2 terms involving Rj . [sent-86, score-0.435]

38 (j) In particular, introducing new parameters y, y , z, z ∈ Rp , we deﬁne ηik = E(si , Rj , ok ) as E(si , Rj , ok ) i y, Rj y + si , Rj z + z , Rj ok + si , Rj ok , k i (1) k where y, Rj y , s , Rj z + z , Rj o and s , Rj o are the uni-, bi- and trigram terms. [sent-87, score-1.928]

39 This parametrization is redundant in general given that E(si , Rj , ok ) is of the form (si + z), Rj (ok + z ) + bj ; but it is however useful in the context of a regularized model (see Section 5). [sent-88, score-0.373]

40 2 This is motivated by the fact that we are primarily interested in modelling the relations terms, and that it is not necessary to introduce all terms to fully parameterize the model. [sent-89, score-0.297]

41 2 Sharing parameters across relations through latent factors When learning a large number of relations, the number of observations for many relations can be quite small, leading to a risk of overﬁtting. [sent-91, score-0.642]

42 [24] addressed this issue with a nonparametric Bayesian model inducing clustering of both relations and entities. [sent-93, score-0.297]

43 SME [2] proposed to embed relations as vectors of Rp , like entities, to tackle problems with hundreds of relation types. [sent-94, score-0.408]

44 (2) r=1 The combined effect of (a) the sparsity of the decomposition and (b) the fact that d nr leads to sharing parameters across relations. [sent-96, score-0.129]

45 Further, constraining Θr to be the outer product ur vr also speeds up all computations relying on linear algebra. [sent-97, score-0.19]

46 (i ,j ,k )∈N (j) (i,j,k)∈P ηik − (j) (i,j,k)∈P∪N log(1 + exp(ηik )), with (j) ηik = E(si , Rj , ok ). [sent-101, score-0.373]

47 To properly normalize the terms appearing in (1) and (2), we carry out the minimization of the negative log-likelihood over a speciﬁc constraint set, namely  j  α 1 ≤ λ, Θr = ur · vr , minj − log(L), with z = z , O = S,  j k S,O,{α }, s , o , y, y , z, ur and vr in the ball w; w 2 ≤ 1 . [sent-102, score-0.405]

48 The regularization parameter λ ≥ 0 controls the sparsity of the relation representations in (2). [sent-104, score-0.158]

49 Given the fact that the model is conditional on a pair (si , ok ), only a single scale parameter, namely αj , is r j necessary in the product αr si , Θr ok , which motivates all the Euclidean unit ball constraints. [sent-106, score-0.909]

50 In practice, for each positive triplet, we sample a number of artiﬁcial negative triplets containing the same subject and object as our positive triplet but different verbs. [sent-118, score-0.246]

51 However, such a function is highly sensitive to the particular choice of negative verbs and using all the verbs as negative ones would be too costly. [sent-123, score-0.522]

52 Another more robust approach consists in using the likelihood function deﬁned above where we try to classify the positive verbs as a valid relationship and the negative ones as invalid relationships. [sent-124, score-0.261]

53 Finally, we observed that it was advantageous to down-weight the inﬂuence of the negative verbs to avoid swamping the inﬂuence of the positive ones. [sent-126, score-0.261]

54 Second, our model is also related to classical tensor factorization model such as PARAFAC which ap˜ proximate the tensor [Rk (Si , Oj )]i,j,k in the least-square sense by a low rank tensor H of the form d nr ×ns ×no . [sent-129, score-0.532]

55 The parameterization of all Rj as linear combir=1 αr ⊗ β r ⊗ γr for (αr , β r , γ r ) ∈ R nations of d rank one matrices is in fact equivalent to constraining the tensor R = {Rj }j∈ 1;nr to d be the low rank tensor R = r=1 αr ⊗ ur ⊗ vr . [sent-130, score-0.63]

56 As a consequence, the tensor of all trigram terms4 d can be written also as r=1 αr ⊗ β r ⊗ γ r with β r = S ur and γ r = O vr . [sent-131, score-0.401]

57 This shows that our model is a particular form of tensor factorization which reduces to PARAFAC (up to a change of loss function) when p is sufﬁciently large. [sent-132, score-0.157]

58 Finally, the approach considered in [2] seems a priori quite different from ours, in particular since relations are in that work embedded as vectors of Rp like the entities as opposed to matrices of Rp×p in our case. [sent-133, score-0.627]

59 In addition, no parameterization of the model [2] is able of handling both bigram and trigram interactions as we propose. [sent-135, score-0.301]

60 Australian tribes are renowned among anthropologists for the complex relational structure of their kinship systems. [sent-139, score-0.23]

61 This results in graph of 104 entities and 26 relation types, each of them depicting a different kinship term, such as Adiadya or Umbaidya. [sent-142, score-0.499]

62 This consists in a graph with 135 entities and 49 relation types. [sent-146, score-0.418]

63 The entities are high-level concepts like ’Disease or Syndrome’, ’Diagnostic Procedure’, or ’Mammal’. [sent-147, score-0.335]

64 The relations represent verbs depicting causal inﬂuence between concepts like ’affect’ or ’cause’. [sent-148, score-0.591]

65 ) with 56 binary relation types representing interactions among them like ’economic aid’, ’treaties’ or ’rel diplomacy’, and 111 features describing each country, which we treated as 111 additional entities interacting with the country through an additional ’has feature’ relation5 . [sent-151, score-0.543]

66 Interestingly, the trigram term from (1) is essential to obtain good performance on Kinships (with the trigram term removed, we obtain 0. [sent-195, score-0.22]

67 14 in LL), thus showing the need for modeling 3-way interactions in complex relational data. [sent-197, score-0.264]

68 Moreover, and as expected due to the low number of relations, the value of λ selected by cross-validation is quite large (λ = nr × d), and as consequence does not lead to sparsity in (2). [sent-198, score-0.129]

69 Results on this dataset also exhibits the beneﬁt of modeling relations with matrices instead of vectors as does SME [2]. [sent-199, score-0.343]

70 Zhu [28] recently reported results on Nations and Kinships evaluated in terms of area under the receiver-operating-characteristic curve instead of area under the precision-recall curve as we display in Table 1. [sent-200, score-0.136]

71 8 Learning semantic representations of verbs By providing an approach to model the relational structure of language, SRL can be of great use for learning natural language semantics. [sent-206, score-0.581]

72 This data was then ﬁltered to only select sentences for which the syntactic structure was (subject, verb, direct object) with each term of the triplet being a single word from the WordNet lexicon [13]. [sent-212, score-0.151]

73 The total number of relations in this dataset (i. [sent-214, score-0.297]

74 Moreover, to speed up the training, we gradually increased the number of sampled negative verbs (cf. [sent-259, score-0.261]

75 We ﬁrst consider a direct evaluation of our approach based on the test set of 250,000 instances by measuring how well we predict a relevant and meaningful verb given a pair (subject, direct object). [sent-264, score-0.263]

76 To this end, for each test relationship, we rank all verbs using our probability estimates given a pair (subject, direct object). [sent-265, score-0.304]

77 Table 2 displays our results with two kinds of metrics, namely, (1) the rank of the correct verb and (2) the fraction of test examples for which the correct verb is ranked in the top z% of the list. [sent-266, score-0.43]

78 In order to evaluate if some language semantics is captured by the representations, we also consider a less conservative approach where, instead of focusing on the correct verb only, we measure the minimum rank achieved over its set of synonyms obtained from WordNet. [sent-268, score-0.347]

79 The ﬁrst observation is that the task of verb prediction can be quite well addressed by a simple model based on 2-way interactions, as shown by the good median rank obtained by the bigram model. [sent-270, score-0.34]

80 On this data, we experienced that using bigram interactions in our energy function was essential to achieve good predictions. [sent-272, score-0.186]

81 1 · nr × d for which the coefﬁcients α of the representations (2) are sparse in the sense they are dominated by few large values (e. [sent-276, score-0.176]

82 8 1 Figure 1: Precision-recall curves for the task of lexical similarity classiﬁcation. [sent-302, score-0.131]

83 59 Table 3: Performance obtained on a task of lexical similarity classiﬁcation [27], where we compare our approach, SME [2], Collobert et al. [sent-315, score-0.131]

84 Our method learns latent representations for verbs and imposes them some structure via shared parameters, as shown in Section 4. [sent-319, score-0.354]

85 We consider the task of lexical similarity classiﬁcation described in [27] to evaluate this hypothesis. [sent-322, score-0.131]

86 Their dataset consists of 130 pairs of verbs labeled by humans with a score in {0, 1, 2, 3, 4}. [sent-323, score-0.236]

87 Higher scores means a stronger semantic similarity between the verbs composing the pair. [sent-324, score-0.339]

88 Based on the pairwise Euclidean distances10 between our learned verb representations Rj , we try to predict the class 4 (and also the “merged” classes {3, 4}) by using the assumption that the smallest the distance between Ri and Rj , the more likely the pair (i, j) should be labeled as 4. [sent-326, score-0.24]

89 We compare to representations learnt by [2] on the same training data, to word embeddings of [5] (which are considered as efﬁcient features in Natural Language Processing), and with three similarity measures provided by WordNet Similarity [19]. [sent-327, score-0.159]

90 Our method is capable of encoding meaningful semantic embeddings for verbs, even though it has been trained on noisy, automatically collected data and in spite of the fact that it was not our primary goal that distance in parameter space should satisfy any condition. [sent-331, score-0.13]

91 9 Conclusion Designing methods capable of handling large amounts of linked relations seems necessary to be able to model the wealth of relations underlying the semantics of any real-world problems. [sent-333, score-0.594]

92 We tackle this problem by using a shared representation of relations naturally suited to multi-relational data, in which entities have a unique representation shared between relation types, and where we propose that relation themselves decompose over latent “relational” factors. [sent-334, score-0.92]

93 This new approach ties or beats state-of-the art models on both standard relational learning problems and an NLP task. [sent-335, score-0.204]

94 The decomposition of relations over latent factors allows a signiﬁcant reduction of the number of parameters and is motivated both by computational and statistical reasons. [sent-336, score-0.345]

95 In particular, our approach is quite scalable both with respect to the number of relations and to the data samples. [sent-337, score-0.297]

96 Interestingly, though the presence of the trigram term was crucial in the tensor factorization problems, it played a marginal role in the NLP experiment, where most of the information was contained in the bigram and unigram terms. [sent-339, score-0.397]

97 Finally, we believe that exploring the similarities of the relations through an analysis of the latent factors could provide some insight on the structures shared between different relation types. [sent-340, score-0.479]

98 Learning systems of concepts with an inﬁnite relational model. [sent-407, score-0.207]

99 Learning distributed representations of concepts using linear relational embedding. [sent-466, score-0.254]

100 Dimensionality of nations project: Attributes of nations and behavior of nation dyads. [sent-485, score-0.202]

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tfidf for this paper:

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Abstract: How can neural networks learn to represent information optimally? We answer this question by deriving spiking dynamics and learning dynamics directly from a measure of network performance. We ﬁnd that a network of integrate-and-ﬁre neurons undergoing Hebbian plasticity can learn an optimal spike-based representation for a linear decoder. The learning rule acts to minimise the membrane potential magnitude, which can be interpreted as a representation error after learning. In this way, learning reduces the representation error and drives the network into a robust, balanced regime. The network becomes balanced because small representation errors correspond to small membrane potentials, which in turn results from a balance of excitation and inhibition. The representation is robust because neurons become self-correcting, only spiking if the representation error exceeds a threshold. 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Furthermore, the learning rule that we derive reproduces the main features of STDP (property (d) above). In this way, a network can learn to represent information optimally, with synaptic, neural and network dynamics consistent with those observed experimentally. 1 Derivation of the learning rule for a single neuron We begin by deriving a learning rule for a single neuron with an autapse (a self-connection) (Fig. 1A). Our approach is to derive synaptic dynamics for the autapse and spiking dynamics for the neuron such that the neuron learns to optimally represent a time-varying input signal. We will derive a learning rule for networks of neurons later, after we have developed the fundamental concepts for the single neuron case. Our ﬁrst step is to derive optimal spiking dynamics for the neuron, so that we have a target for our learning rule. We do this by making two simple assumptions [11]. First, we assume that the neuron can provide an estimate or read-out x(t) of a time-dependent signal x(t) by ﬁltering its spike train ˆ o(t) as follows: ˙ x(t) = −ˆ(t) + Γo(t), ˆ x (1) where Γ is a ﬁxed read-out weight, which we will refer to as the neuron’s “output kernel” and the spike train can be written as o(t) = i δ(t − ti ), where {ti } are the spike times. Next, we assume that the neuron only produces a spike if that spike improves the read-out, where we measure the read-out performance through a simple squared-error loss function: 2 L(t) = x(t) − x(t) . ˆ (2) With these two assumptions, we can now derive optimal spiking dynamics. First, we observe that if the neuron produces an additional spike at time t, the read-out increases by Γ, and the loss function becomes L(t|spike) = (x(t) − (x(t) + Γ))2 . This allows us to restate our spiking rule as follows: ˆ the neuron should only produce a spike if L(t|no spike) > L(t|spike), or (x(t) − x(t))2 > (x(t) − ˆ (x(t) + Γ))2 . Now, squaring both sides of this inequality, deﬁning V (t) ≡ Γ(x(t) − x(t)) and ˆ ˆ deﬁning T ≡ Γ2 /2 we ﬁnd that the neuron should only spike if: V (t) > T. (3) We interpret V (t) to be the membrane potential of the neuron, and we interpret T as the spike threshold. This interpretation allows us to understand the membrane potential functionally: the voltage is proportional to a prediction error—the difference between the read-out x(t) and the actual ˆ signal x(t). A spike is an error reduction mechanism—the neuron only spikes if the error exceeds the spike threshold. This is a greedy minimisation, in that the neuron ﬁres a spike whenever that action decreases L(t) without considering the future impact of that spike. Importantly, the neuron does not require direct access to the loss function L(t). 2 To determine the membrane potential dynamics, we take the derivative of the voltage, which gives ˙ ˙ us V = Γ(x − x). (Here, and in the following, we will drop the time index for notational brevity.) ˙ ˆ ˙ Now, using Eqn. (1) we obtain V = Γx − Γ(−x + Γo) = −Γ(x − x) + Γ(x + x) − Γ2 o, so that: ˙ ˆ ˆ ˙ ˙ V = −V + Γc − Γ2 o, (4) where c = x + x is the neural input. This corresponds exactly to the dynamics of a leaky integrate˙ and-ﬁre neuron with an inhibitory autapse1 of strength Γ2 , and a feedforward connection strength Γ. The dynamics and connectivity guarantee that a neuron spikes at just the right times to optimise the loss function (Fig. 1B). In addition, it is especially robust to noise of different forms, because of its error-correcting nature. If x is constant in time, the voltage will rise up to the threshold T at which point a spike is ﬁred, adding a delta function to the spike train o at time t, thereby producing a read-out x that is closer to x and causing an instantaneous drop in the voltage through the autapse, ˆ by an amount Γ2 = 2T , effectively resetting the voltage to V = −T . We now have a target for learning—we know the connection strength that a neuron must have at the end of learning if it is to represent information optimally, for a linear read-out. We can use this target to derive synaptic dynamics that can learn an optimal representation from experience. Speciﬁcally, we consider an integrate-and-ﬁre neuron with some arbitrary autapse strength ω. The dynamics of this neuron are given by ˙ V = −V + Γc − ωo. (5) This neuron will not produce the correct spike train for representing x through a linear read-out (Eqn. (1)) unless ω = Γ2 . Our goal is to derive a dynamical equation for the synapse ω so that the spike train becomes optimal. We do this by quantifying the loss that we are incurring by using the suboptimal strength, and then deriving a learning rule that minimises this loss with respect to ω. The loss function underlying the spiking dynamics determined by Eqn. (5) can be found by reversing the previous membrane potential analysis. First, we integrate the differential equation for V , assuming that ω changes on time scales much slower than the membrane potential. We obtain the following (formal) solution: V = Γx − ω¯, o (6) ˙ where o is determined by o = −¯ + o. The solution to this latter equation is o = h ∗ o, a convolution ¯ ¯ o ¯ of the spike train with the exponential kernel h(τ ) = θ(τ ) exp(−τ ). As such, it is analogous to the instantaneous ﬁring rate of the neuron. Now, using Eqn. (6), and rewriting the read-out as x = Γ¯, we obtain the loss incurred by the ˆ o sub-optimal neuron, L = (x − x)2 = ˆ 1 V 2 + 2(ω − Γ2 )¯ + (ω − Γ2 )2 o2 . o ¯ Γ2 (7) We observe that the last two terms of Eqn. (7) will vanish whenever ω = Γ2 , i.e., when the optimal reset has been found. We can therefore simplify the problem by deﬁning an alternative loss function, 1 2 V , (8) 2 which has the same minimum as the original loss (V = 0 or x = x, compare Eqn. (2)), but yields a ˆ simpler learning algorithm. We can now calculate how changes to ω affect LV : LV = ∂LV ∂V ∂o ¯ =V = −V o − V ω ¯ . (9) ∂ω ∂ω ∂ω We can ignore the last term in this equation (as we will show below). Finally, using simple gradient descent, we obtain a simple Hebbian-like synaptic plasticity rule: τω = − ˙ ∂LV = V o, ¯ ∂ω (10) where τ is the learning time constant. 1 This contribution of the autapse can also be interpreted as the reset of an integrate-and-ﬁre neuron. Later, when we generalise to networks of neurons, we shall employ this interpretation. 3 This synaptic learning rule is capable of learning the synaptic weight ω that minimises the difference between x and x (Fig. 1B). During learning, the synaptic weight changes in proportion to the postˆ synaptic voltage V and the pre-synaptic ﬁring rate o (Fig. 1C). As such, this is a Hebbian learning ¯ rule. Of course, in this single neuron case, the pre-synaptic neuron and post-synaptic neuron are the same neuron. The synaptic weight gradually approaches its optimal value Γ2 . However, it never completely stabilises, because learning never stops as long as neurons are spiking. Instead, the synapse oscillates closely about the optimal value (Fig. 1D). This is also a “greedy” learning rule, similar to the spiking rule, in that it seeks to minimise the error at each instant in time, without regard for the future impact of those changes. To demonstrate that the second term in Eqn. (5) can be neglected we note that the equations for V , o, and ω deﬁne a system ¯ of coupled differential equations that can be solved analytically by integrating between spikes. This results in a simple recurrence relation for changes in ω from the ith to the (i + 1)th spike, ωi+1 = ωi + ωi (ωi − 2T ) . τ (T − Γc − ωi ) (11) This iterative equation has a single stable ﬁxed point at ω = 2T = Γ2 , proving that the neuron’s autaptic weight or reset will approach the optimal solution. 2 Learning in a homogeneous network We now generalise our learning rule derivation to a network of N identical, homogeneously connected neurons. This generalisation is reasonably straightforward because many characteristics of the single neuron case are shared by a network of identical neurons. We will return to the more general case of heterogeneously connected neurons in the next section. We begin by deriving optimal spiking dynamics, as in the single neuron case. This provides a target for learning, which we can then use to derive synaptic dynamics. As before, we want our network to produce spikes that optimally represent a variable x for a linear read-out. We assume that the read-out x is provided by summing and ﬁltering the spike trains of all the neurons in the network: ˆ ˙ x = −ˆ + Γo, ˆ x (12) 2 where the row vector Γ = (Γ, . . . , Γ) contains the read-out weights of the neurons and the column vector o = (o1 , . . . , oN ) their spike trains. Here, we have used identical read-out weights for each neuron, because this indirectly leads to homogeneous connectivity, as we will demonstrate. Next, we assume that a neuron only spikes if that spike reduces a loss-function. This spiking rule is similar to the single neuron spiking rule except that this time there is some ambiguity about which neuron should spike to represent a signal. Indeed, there are many different spike patterns that provide exactly the same estimate x. For example, one neuron could ﬁre regularly at a high rate (exactly like ˆ our previous single neuron example) while all others are silent. To avoid this ﬁring rate ambiguity, we use a modiﬁed loss function, that selects amongst all equivalent solutions, those with the smallest neural ﬁring rates. We do this by adding a ‘metabolic cost’ term to our loss function, so that high ﬁring rates are penalised: ¯ L = (x − x)2 + µ o 2 , ˆ (13) where µ is a small positive constant that controls the cost-accuracy trade-off, akin to a regularisation parameter. Each neuron in the optimal network will seek to reduce this loss function by ﬁring a spike. Speciﬁcally, the ith neuron will spike whenever L(no spike in i) > L(spike in i). This leads to the following spiking rule for the ith neuron: Vi > Ti (14) where Vi ≡ Γ(x − x) − µoi and Ti ≡ Γ2 /2 + µ/2. We can naturally interpret Vi as the membrane ˆ potential of the ith neuron and Ti as the spiking threshold of that neuron. As before, we can now derive membrane potential dynamics: ˙ V = −V + ΓT c − (ΓT Γ + µI)o, 2 (15) The read-out weights must scale as Γ ∼ 1/N so that ﬁring rates are not unrealistically small in large networks. We can see this by calculating the average ﬁring rate N oi /N ≈ x/(ΓN ) ∼ O(N/N ) ∼ O(1). i=1 ¯ 4 where I is the identity matrix and ΓT Γ + µI is the network connectivity. We can interpret the selfconnection terms {Γ2 +µ} as voltage resets that decrease the voltage of any neuron that spikes. This optimal network is equivalent to a network of identical integrate-and-ﬁre neurons with homogeneous inhibitory connectivity. The network has some interesting dynamical properties. The voltages of all the neurons are largely synchronous, all increasing to the spiking threshold at about the same time3 (Fig. 1F). Nonetheless, neural spiking is asynchronous. The ﬁrst neuron to spike will reset itself by Γ2 + µ, and it will inhibit all the other neurons in the network by Γ2 . This mechanism prevents neurons from spik- x 3 The ﬁrst neuron to spike will be random if there is some membrane potential noise. V (A) (B) x x ˆ x 10 1 0.1 0 50 100 150 200 250 300 350 400 0 50 100 150 200 250 300 350 400 1 D 0.5 V V 0 ˆ x V ˆ x (C) 1 0 1 2 0 0.625 25 25.625 (D) start of learning 1 V 50 200.625 400 400.625 1 2.4 O 1.78 ω 1.77 25 neuron$ 0 1 2 !me$ 3 4 25 1 5 V 400.625 !me$ (F) 25 1 2.35 1.05 1.049 400 25.625 !me$ (E) neuron$ 100.625 200 end of learning 1.4 1.35 ω 100 !me$ 1 V 1 O 50.625 0 1 2 !me$ 3 4 5 V !me$ !me$ Figure 1: Learning in a single neuron and a homogeneous network. (A) A single neuron represents an input signal x by producing an output x. (B) During learning, the single neuron output x (solid red ˆ ˆ line, top panel) converges towards the input x (blue). Similarly, for a homogeneous network the output x (dashed red line, top panel) converges towards x. Connectivity also converges towards optimal ˆ connectivity in both the single neuron case (solid black line, middle panel) and the homogeneous net2 2 work case (dashed black line, middle panel), as quantiﬁed by D = maxi,j ( Ωij − Ωopt / Ωopt ) ij ij at each point in time. Consequently, the membrane potential reset (bottom panel) converges towards the optimal reset (green line, bottom panel). Spikes are indicated by blue vertical marks, and are produced when the membrane potential reaches threshold (bottom panel). Here, we have rescaled time, as indicated, for clarity. (C) Our learning rule dictates that the autapse ω in our single neuron (bottom panel) changes in proportion to the membrane potential (top panel) and the ﬁring rate (middle panel). (D) At the end of learning, the reset ω ﬂuctuates weakly about the optimal value. (E) For a homogeneous network, neurons spike regularly at the start of learning, as shown in this raster plot. Membrane potentials of different neurons are weakly correlated. (F) At the end of learning, spiking is very irregular and membrane potentials become more synchronous. 5 ing synchronously. The population as a whole acts similarly to the single neuron in our previous example. Each neuron ﬁres regularly, even if a different neuron ﬁres in every integration cycle. The design of this optimal network requires the tuning of N (N − 1) synaptic parameters. How can an arbitrary network of integrate-and-ﬁre neurons learn this optimum? As before, we address this question by using the optimal network as a target for learning. We start with an arbitrarily connected network of integrate-and-ﬁre neurons: ˙ V = −V + ΓT c − Ωo, (16) where Ω is a matrix of connectivity weights, which includes the resets of the individual neurons. Assuming that learning occurs on a slow time scale, we can rewrite this equation as V = ΓT x − Ω¯ . o (17) Now, repeating the arguments from the single neuron derivation, we modify the loss function to obtain an online learning rule. Speciﬁcally, we set LV = V 2 /2, and calculate the gradient: ∂LV = ∂Ωij Vk k ∂Vk =− ∂Ωij Vk δki oj − ¯ k Vk Ωkl kl ∂ ol ¯ . ∂Ωij (18) We can simplify this equation considerably by observing that the contribution of the second summation is largely averaged out under a wide variety of realistic conditions4 . Therefore, it can be neglected, and we obtain the following local learning rule: ∂LV ˙ = V i oj . ¯ τ Ωij = − ∂Ωij (19) This is a Hebbian plasticity rule, whereby connectivity changes in proportion to the presynaptic ﬁring rate oj and post-synaptic membrane potential Vi . We assume that the neural thresholds are set ¯ to a constant T and that the neural resets are set to their optimal values −T . In the previous section we demonstrated that these resets can be obtained by a Hebbian plasticity rule (Eqn. (10)). This learning rule minimises the difference between the read-out and the signal, by approaching the optimal recurrent connection strengths for the network (Fig. 1B). As in the single neuron case, learning does not stop, so the connection strengths ﬂuctuate close to their optimal value. During learning, network activity becomes progressively more asynchronous as it progresses towards optimal connectivity (Fig. 1E, F). 3 Learning in the general case Now that we have developed the fundamental concepts underlying our learning rule, we can derive a learning rule for the more general case of a network of N arbitrarily connected leaky integrateand-ﬁre neurons. Our goal is to understand how such networks can learn to optimally represent a ˙ J-dimensional signal x = (x1 , . . . , xJ ), using the read-out equation x = −x + Γo. We consider a network with the following membrane potential dynamics: ˙ V = −V + ΓT c − Ωo, (20) where c is a J-dimensional input. We assume that this input is related to the signal according to ˙ c = x + x. This assumption can be relaxed by treating the input as the control for an arbitrary linear dynamical system, in which case the signal represented by the network is the output of such a computation [11]. However, this further generalisation is beyond the scope of this work. As before, we need to identify the optimal recurrent connectivity so that we have a target for learning. Most generally, the optimal recurrent connectivity is Ωopt ≡ ΓT Γ + µI. The output kernels of the individual neurons, Γi , are given by the rows of Γ, and their spiking thresholds by Ti ≡ Γi 2 /2 + 4 From the deﬁnition of the membrane potential we can see that Vk ∼ O(1/N ) because Γ ∼ 1/N . Therefore, the size of the ﬁrst term in Eqn. (18) is k Vk δki oj = Vi oj ∼ O(1/N ). Therefore, the second term can ¯ ¯ be ignored if kl Vk Ωkl ∂ ol /∂Ωij ¯ O(1/N ). This happens if Ωkl O(1/N 2 ) as at the start of learning. It also happens towards the end of learning if the terms {Ωkl ∂ ol /∂Ωij } are weakly correlated with zero mean, ¯ or if the membrane potentials {Vi } are weakly correlated with zero mean. 6 µ/2. With these connections and thresholds, we ﬁnd that a network of integrate-and-ﬁre neurons ˆ ¯ will produce spike trains in such a way that the loss function L = x − x 2 + µ o 2 is minimised, ˆ where the read-out is given by x = Γ¯ . We can show this by prescribing a greedy5 spike rule: o a spike is ﬁred by neuron i whenever L(no spike in i) > L(spike in i) [11]. The resulting spike generation rule is Vi > Ti , (21) ˆ where Vi ≡ ΓT (x − x) − µ¯i is interpreted as the membrane potential. o i 5 Despite being greedy, this spiking rule can generate ﬁring rates that are practically identical to the optimal solutions: we checked this numerically in a large ensemble of networks with randomly chosen kernels. (A) x1 … x … 1 1 (B) xJJ x 10 L 10 T T 10 4 6 8 1 Viii V D ˆˆ ˆˆ x11 xJJ x x F 0.5 0 0.4 … … 0.2 0 0 2000 4000 !me (C) x V V 1 x 10 x 3 ˆ x 8 0 x 10 1 2 3 !me 4 5 4 0 1 4 0 1 8 V (F) Ρ(Δt) E-‐I input 0.4 ˆ x 0 3 0 1 x 10 1.3 0.95 x 10 ˆ x 4 V (E) 1 x 0 end of learning 50 neuron neuron 50 !me 2 0 ˆ x 0 0.5 ISI Δt 1 2 !me 4 5 4 1.5 1.32 3 2 0.1 Ρ(Δt) x E-‐I input (D) start of learning 0 2 !me 0 0 0.5 ISI Δt 1 Figure 2: Learning in a heterogeneous network. (A) A network of neurons represents an input ˆ signal x by producing an output x. (B) During learning, the loss L decreases (top panel). The difference between the connection strengths and the optimal strengths also decreases (middle panel), as 2 2 quantiﬁed by the mean difference (solid line), given by D = Ω − Ωopt / Ωopt and the maxi2 2 mum difference (dashed line), given by maxi,j ( Ωij − Ωopt / Ωopt ). The mean population ﬁring ij ij rate (solid line, bottom panel) also converges towards the optimal ﬁring rate (dashed line, bottom panel). (C, E) Before learning, a raster plot of population spiking shows that neurons produce bursts ˆ of spikes (upper panel). The network output x (red line, middle panel) fails to represent x (blue line, middle panel). The excitatory input (red, bottom left panel) and inhibitory input (green, bottom left panel) to a randomly selected neuron is not tightly balanced. Furthermore, a histogram of interspike intervals shows that spiking activity is not Poisson, as indicated by the red line that represents a best-ﬁt exponential distribution. (D, F) At the end of learning, spiking activity is irregular and ˆ Poisson-like, excitatory and inhibitory input is tightly balanced and x matches x. 7 How can we learn this optimal connection matrix? As before, we can derive a learning rule by minimising the cost function LV = V 2 /2. This leads to a Hebbian learning rule with the same form as before: ˙ τ Ωij = Vi oj . ¯ (22) Again, we assume that the neural resets are given by −Ti . Furthermore, in order for this learning rule to work, we must assume that the network input explores all possible directions in the J-dimensional input space (since the kernels Γi can point in any of these directions). The learning performance does not critically depend on how the input variable space is sampled as long as the exploration is extensive. In our simulations, we randomly sample the input c from a Gaussian white noise distribution at every time step for the entire duration of the learning. We ﬁnd that this learning rule decreases the loss function L, thereby approaching optimal network connectivity and producing optimal ﬁring rates for our linear decoder (Fig. 2B). In this example, we have chosen connectivity that is initially much too weak at the start of learning. Consequently, the initial network behaviour is similar to a collection of unconnected single neurons that ignore each other. Spike trains are not Poisson-like, ﬁring rates are excessively large, excitatory and inhibitory ˆ input is unbalanced and the decoded variable x is highly unreliable (Fig. 2C, E). As a result of learning, the network becomes tightly balanced and the spike trains become asynchronous, irregular and Poisson-like with much lower rates (Fig. 2D, F). However, despite this apparent variability, the population representation is extremely precise, only limited by the the metabolic cost and the discrete nature of a spike. This learnt representation is far more precise than a rate code with independent Poisson spike trains [11]. In particular, shufﬂing the spike trains in response to identical inputs drastically degrades this precision. 4 Conclusions and Discussion In population coding, large trial-to-trial spike train variability is usually interpreted as noise [2]. We show here that a deterministic network of leaky integrate-and-ﬁre neurons with a simple Hebbian plasticity rule can self-organise into a regime where information is represented far more precisely than in noisy rate codes, while appearing to have noisy Poisson-like spiking dynamics. Our learning rule (Eqn. (22)) has the basic properties of STDP. Speciﬁcally, a presynaptic spike occurring immediately before a post-synaptic spike will potentiate a synapse, because membrane potentials are positive immediately before a postsynaptic spike. Furthermore, a presynaptic spike occurring immediately after a post-synaptic spike will depress a synapse, because membrane potentials are always negative immediately after a postsynaptic spike. This is similar in spirit to the STDP rule proposed in [12], but different to classical STDP, which depends on post-synaptic spike times [9]. This learning rule can also be understood as a mechanism for generating a tight balance between excitatory and inhibitory input. We can see this by observing that membrane potentials after learning can be interpreted as representation errors (projected onto the read-out kernels). Therefore, learning acts to minimise the magnitude of membrane potentials. Excitatory and inhibitory input must be balanced if membrane potentials are small, so we can equate balance with optimal information representation. Previous work has shown that the balanced regime produces (quasi-)chaotic network dynamics, thereby accounting for much observed cortical spike train variability [13, 14, 4]. Moreover, the STDP rule has been known to produce a balanced regime [16, 17]. Additionally, recent theoretical studies have suggested that the balanced regime plays an integral role in network computation [15, 13]. In this work, we have connected these mechanisms and functions, to conclude that learning this balance is equivalent to the development of an optimal spike-based population code, and that this learning can be achieved using a simple Hebbian learning rule. Acknowledgements We are grateful for generous funding from the Emmy-Noether grant of the Deutsche Forschungsgemeinschaft (CKM) and the Chaire d’excellence of the Agence National de la Recherche (CKM, DB), as well as a James Mcdonnell Foundation Award (SD) and EU grants BACS FP6-IST-027140, BIND MECT-CT-20095-024831, and ERC FP7-PREDSPIKE (SD). 8 References [1] Tolhurst D, Movshon J, and Dean A (1982) The statistical reliability of signals in single neurons in cat and monkey visual cortex. Vision Res 23: 775–785. [2] Shadlen MN, Newsome WT (1998) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J Neurosci 18(10): 3870–3896. [3] Zohary E, Newsome WT (1994) Correlated neuronal discharge rate and its implication for psychophysical performance. Nature 370: 140–143. [4] Renart A, de la Rocha J, Bartho P, Hollender L, Parga N, Reyes A, & Harris, KD (2010) The asynchronous state in cortical circuits. Science 327, 587–590. [5] Ecker AS, Berens P, Keliris GA, Bethge M, Logothetis NK, Tolias AS (2010) Decorrelated neuronal ﬁring in cortical microcircuits. Science 327: 584–587. [6] Okun M, Lampl I (2008) Instantaneous correlation of excitation and inhibition during ongoing and sensory-evoked activities. Nat Neurosci 11, 535–537. [7] Shu Y, Hasenstaub A, McCormick DA (2003) Turning on and off recurrent balanced cortical activity. Nature 423, 288–293. [8] Gentet LJ, Avermann M, Matyas F, Staiger JF, Petersen CCH (2010) Membrane potential dynamics of GABAergic neurons in the barrel cortex of behaving mice. Neuron 65: 422–435. [9] Caporale N, Dan Y (2008) Spike-timing-dependent plasticity: a Hebbian learning rule. Annu Rev Neurosci 31: 25–46. [10] Boerlin M, Deneve S (2011) Spike-based population coding and working memory. PLoS Comput Biol 7, e1001080. [11] Boerlin M, Machens CK, Deneve S (2012) Predictive coding of dynamic variables in balanced spiking networks. under review. [12] Clopath C, B¨ sing L, Vasilaki E, Gerstner W (2010) Connectivity reﬂects coding: a model of u voltage-based STDP with homeostasis. Nat Neurosci 13(3): 344–352. [13] van Vreeswijk C, Sompolinsky H (1998) Chaotic balanced state in a model of cortical circuits. Neural Comput 10(6): 1321–1371. [14] Brunel N (2000) Dynamics of sparsely connected networks of excitatory and inhibitory neurons. J Comput Neurosci 8, 183–208. [15] Vogels TP, Rajan K, Abbott LF (2005) Neural network dynamics. Annu Rev Neurosci 28: 357–376. [16] Vogels TP, Sprekeler H, Zenke F, Clopath C, Gerstner W. (2011) Inhibitory plasticity balances excitation and inhibition in sensory pathways and memory networks. Science 334(6062):1569– 73. [17] Song S, Miller KD, Abbott LF (2000) Competitive Hebbian learning through spike-timingdependent synaptic plasticity. Nat Neurosci 3(9): 919–926. 9

4 0.65505022 184 nips-2012-Learning Probability Measures with respect to Optimal Transport Metrics

Author: Guillermo Canas, Lorenzo Rosasco

Abstract: We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal quantization, and learning theory, we derive new probabilistic bounds for the performance of a classic algorithm in unsupervised learning (k-means), when used to produce a probability measure derived from the data. In the course of the analysis, we arrive at new lower bounds, as well as probabilistic upper bounds on the convergence rate of empirical to population measures, which, unlike existing bounds, are applicable to a wide class of measures. 1 Introduction and Motivation In this paper we study the problem of learning from random samples a probability distribution supported on a manifold, when the learning error is measured using transportation metrics. The problem of learning a probability distribution is classic in statistics, and is typically analyzed for distributions in X = Rd that have a density with respect to the Lebesgue measure, with total variation, and L2 among the common distances used to measure closeness of two densities (see for instance [10, 32] and references therein.) The setting in which the data distribution is supported on a low dimensional manifold embedded in a high dimensional space has only been considered more recently. In particular, kernel density estimators on manifolds have been described in [36], and their pointwise consistency, as well as convergence rates, have been studied in [25, 23, 18]. A discussion on several topics related to statistics on a Riemannian manifold can be found in [26]. Interestingly, the problem of approximating measures with respect to transportation distances has deep connections with the ﬁelds of optimal quantization [14, 16], optimal transport [35] and, as we point out in this work, with unsupervised learning (see Sec. 4.) In fact, as described in the sequel, some of the most widely-used algorithms for unsupervised learning, such as k-means (but also others such as PCA and k-ﬂats), can be shown to be performing exactly the task of estimating the data-generating measure in the sense of the 2-Wasserstein distance. This close relation between learning theory, and optimal transport and quantization seems novel and of interest in its own right. Indeed, in this work, techniques from the above three ﬁelds are used to derive the new probabilistic bounds described below. Our technical contribution can be summarized as follows: (a) we prove uniform lower bounds for the distance between a measure and estimates based on discrete sets (such as the empirical measure or measures derived from algorithms such as kmeans); (b) we provide new probabilistic bounds for the rate of convergence of empirical to population measures which, unlike existing probabilistic bounds, hold for a very large class of measures; 1 (c) we provide probabilistic bounds for the rate of convergence of measures derived from k-means to the data measure. The structure of the paper is described at the end of Section 2, where we discuss the exact formulation of the problem as well as related previous works. 2 Setup and Previous work Consider the problem of learning a probability measure ρ supported on a space M, from an i.i.d. sample Xn = (x1 , . . . , xn ) ∼ ρn of size n. We assume M to be a compact, smooth d-dimensional manifold of bounded curvature, with C 1 metric and volume measure λM , embedded in the unit ball of a separable Hilbert space X with inner product ·, · , induced norm · , and distance d (for d instance M = B2 (1) the unit ball in X = Rd .) Following [35, p. 94], let Pp (M) denote the Wasserstein space of order 1 ≤ p < ∞: Pp (M) := x p dρ(x) < ∞ ρ ∈ P (M) : M of probability measures P (M) supported on M, with ﬁnite p-th moment. The p-Wasserstein distance 1/p Wp (ρ, µ) = inf [E X − Y p ] : Law(X) = ρ, Law(Y ) = µ (1) X,Y where the random variables X and Y are distributed according to ρ and µ respectively, is the optimal expected cost of transporting points generated from ρ to those generated from µ, and is guaranteed to be ﬁnite in Pp (M) [35, p. 95]. The space Pp (M) with the Wp metric is itself a complete separable metric space [35]. We consider here the problem of learning probability measures ρ ∈ P2 (M), where the performance is measured by the distance W2 . There are many possible choices of distances between probability measures [13]. Among them, Wp metrizes weak convergence (see [35] theorem 6.9), that is, in Pp (M), a sequence (µi )i∈N of measures converges weakly to µ iff Wp (µi , µ) → 0 and their p-th order moments converge to that of µ. There are other distances, such as the L´ vy-Prokhorov, or the weak-* distance, that also metrize e weak convergence. However, as pointed out by Villani in his excellent monograph [35, p. 98], 1. “Wasserstein distances are rather strong, [...]a deﬁnite advantage over the weak-* distance”. 2. “It is not so difﬁcult to combine information on convergence in Wasserstein distance with some smoothness bound, in order to get convergence in stronger distances.” Wasserstein distances have been used to study the mixing and convergence of Markov chains [22], as well as concentration of measure phenomena [20]. To this list we would add the important fact that existing and widely-used algorithms for unsupervised learning can be easily extended (see Sec. 4) to compute a measure ρ that minimizes the distance W2 (ˆn , ρ ) to the empirical measure ρ n ρn := ˆ 1 δx , n i=1 i a fact that will allow us to prove, in Sec. 5, bounds on the convergence of a measure induced by k-means to the population measure ρ. The most useful versions of Wasserstein distance are p = 1, 2, with p = 1 being the weaker of the two (by H¨ lder’s inequality, p ≤ q ⇒ Wp ≤ Wq .) In particular, “results in W2 distance are usually o stronger, and more difﬁcult to establish than results in W1 distance” [35, p. 95]. A discussion of p = ∞ would take us out of topic, since its behavior is markedly different. 2.1 Closeness of Empirical and Population Measures By the strong law of large numbers, the empirical measure converges almost surely to the population measure: ρn → ρ in the sense of the weak topology [34]. Since weak convergence and convergence ˆ in Wp plus convergence of p-th moments are equivalent in Pp (M), this means that, in the Wp sense, the empirical measure ρn converges to ρ, as n → ∞. A fundamental question is therefore how fast ˆ the rate of convergence of ρn → ρ is. ˆ 2 2.1.1 Convergence in expectation The rate of convergence of ρn → ρ in expectation has been widely studied in the past, resultˆ ing in upper bounds of order EW2 (ρ, ρn ) = O(n−1/(d+2) ) [19, 8], and lower bounds of order ˆ EW2 (ρ, ρn ) = Ω(n−1/d ) [29] (both assuming that the absolutely continuous part of ρ is ρA = 0, ˆ with possibly better rates otherwise). More recently, an upper bound of order EWp (ρ, ρn ) = O(n−1/d ) has been proposed [2] by proving ˆ a bound for the Optimal Bipartite Matching (OBM) problem [1], and relating this problem to the expected distance EWp (ρ, ρn ). In particular, given two independent samples Xn , Yn , the OBM ˆ problem is that of ﬁnding a permutation σ that minimizes the matching cost n−1 xi −yσ(i) p [24, p ˆ ˆ ˆ 30]. It is not hard to show that the optimal matching cost is Wp (ˆXn , ρYn ) , where ρXn , ρYn are ρ the empirical measures associated to Xn , Yn . By Jensen’s inequality, the triangle inequality, and (a + b)p ≤ 2p−1 (ap + bp ), it holds EWp (ρ, ρn )p ≤ EWp (ˆXn , ρYn )p ≤ 2p−1 EWp (ρ, ρn )p , ˆ ρ ˆ ˆ and therefore a bound of order O(n−p/d ) for the OBM problem [2] implies a bound EWp (ρ, ρn ) = ˆ O(n−1/d ). The matching lower bound is only known for a special case: ρA constant over a bounded set of non-null measure [2] (e.g. ρA uniform.) Similar results, with matching lower bounds are found for W1 in [11]. 2.1.2 Convergence in probability Results for convergence in probability, one of the main results of this work, appear to be considerably harder to obtain. One fruitful avenue of analysis has been the use of so-called transportation, or Talagrand inequalities Tp , which can be used to prove concentration inequalities on Wp [20]. In particular, we say that ρ satisﬁes a Tp (C) inequality with C > 0 iff Wp (ρ, µ)2 ≤ CH(µ|ρ), ∀µ ∈ Pp (M), where H(·|·) is the relative entropy [20]. As shown in [6, 5], it is possible to obtain probabilistic upper bounds on Wp (ρ, ρn ), with p = 1, 2, if ρ is known to satisfy a Tp inequality ˆ of the same order, thereby reducing the problem of bounding Wp (ρ, ρn ) to that of obtaining a Tp ˆ inequality. Note that, by Jensen’s inequality, and as expected from the behavior of Wp , the inequality T2 is stronger than T1 [20]. While it has been shown that ρ satisﬁes a T1 inequality iff it has a ﬁnite square-exponential moment 2 (E[eα x ] ﬁnite for some α > 0) [4, 7], no such general conditions have been found for T2 . As an example, consider that, if M is compact with diameter D then, by theorem 6.15 of [35], and the celebrated Csisz´ r-Kullback-Pinsker inequality [27], for all ρ, µ ∈ Pp (M), it is a Wp (ρ, µ)2p ≤ (2D)2p ρ − µ where · does not. TV 2 TV ≤ 22p−1 D2p H(µ|ρ), is the total variation norm. Clearly, this implies a Tp=1 inequality, but for p ≥ 2 it The T2 inequality has been shown by Talagrand to be satisﬁed by the Gaussian distribution [31], and then slightly more generally by strictly log-concave measures (see [20, p. 123], and [3].) However, as noted in [6], “contrary to the T1 case, there is no hope to obtain T2 inequalities from just integrability or decay estimates.” Structure of this paper. In this work we obtain bounds in probability (learning rates) for the problem of learning a probability measure in the sense of W2 . We begin by establishing (lower) bounds for the convergence of empirical to population measures, which serve to set up the problem and introduce the connection between quantization and measure learning (sec. 3.) We then describe how existing unsupervised learning algorithms that compute a set (k-means, k-ﬂats, PCA,. . . ) can be easily extended to produce a measure (sec. 4.) Due to its simplicity and widespread use, we focus here on k-means. Since the two measure estimates that we consider are the empirical measure, and the measure induced by k-means, we next set out to prove upper bounds on their convergence to the data-generating measure (sec. 5.) We arrive at these bounds by means of intermediate measures, which are related to the problem of optimal quantization. The bounds apply in a very broad setting (unlike existing bounds based on transportation inequalities, they are not restricted to log-concave measures [20, 3].) 3 3 Learning probability measures, optimal transport and quantization We address the problem of learning a probability measure ρ when the only observations we have at our disposal are n i.i.d. samples Xn = (x1 , . . . , xn ). We begin by establishing some notation and useful intermediate results. Given a closed set S ⊆ X , let {Vq : q ∈ S} be a Borel Voronoi partition of X composed of sets Vq closest to each q ∈ S, that is, such that each Vq ⊆ {x ∈ X : x − q = minr∈S x − r } is measurable (see for instance [15].) Consider the projection function πS : X → S mapping each x ∈ Vq to q. By virtue of {Vq }q∈S being a Borel Voronoi partition, the map πS is measurable [15], and it is d (x, πS (x)) = minq∈S x − q for all x ∈ X . For any ρ ∈ Pp (M), let πS ρ be the pushforward, or image measure of ρ under the mapping πS , −1 which is deﬁned to be (πS ρ)(A) := ρ(πS (A)) for all Borel measurable sets A. From its deﬁnition, it is clear that πS ρ is supported on S. We now establish a connection between the expected distance to a set S, and the distance between ρ and the set’s induced pushforward measure. Notice that, for discrete sets S, the expected Lp distance to S is exactly the expected quantization error Ep,ρ (S) := Ex∼ρ d(x, S)p = Ex∼ρ x − πS (x) p incurred when encoding points x drawn from ρ by their closest point πS (x) in S [14]. This close connection between optimal quantization and Wasserstein distance has been pointed out in the past in the statistics [28], optimal quantization [14, p. 33], and approximation theory [16] literatures. The following two lemmas are key tools in the reminder of the paper. The ﬁrst highlights the close link between quantization and optimal transport. Lemma 3.1. For closed S ⊆ X , ρ ∈ Pp (M), 1 ≤ p < ∞, it holds Ex∼ρ d(x, S)p = Wp (ρ, πS ρ)p . Note that the key element in the above lemma is that the two measures in the expression Wp (ρ, πS ρ) must match. When there is a mismatch, the distance can only increase. That is, Wp (ρ, πS µ) ≥ Wp (ρ, πS ρ) for all µ ∈ Pp (M). In fact, the following lemma shows that, among all the measures with support in S, πS ρ is closest to ρ. Lemma 3.2. For closed S ⊆ X , and all µ ∈ Pp (M) with supp(µ) ⊆ S, 1 ≤ p < ∞, it holds Wp (ρ, µ) ≥ Wp (ρ, πS ρ). When combined, lemmas 3.1 and 3.2 indicate that the behavior of the measure learning problem is limited by the performance of the optimal quantization problem. For instance, Wp (ρ, ρn ) can only ˆ be, in the best-case, as low as the optimal quantization cost with codebook of size n. The following section makes this claim precise. 3.1 Lower bounds Consider the situation depicted in ﬁg. 1, in which a sample X4 = {x1 , x2 , x3 , x4 } is drawn from a distribution ρ which we assume here to be absolutely continuous on its support. As shown, the projection map πX4 sends points x to their closest point in X4 . The resulting Voronoi decomposition of supp(ρ) is drawn in shades of blue. By lemma 5.2 of [9], the pairwise intersections of Voronoi regions have null ambient measure, and since ρ is absolutely continuous, the pushforward measure 4 can be written in this case as πX4 ρ = j=1 ρ(Vxj )δxj , where Vxj is the Voronoi region of xj . Note that, even for ﬁnite sets S, this particular decomposition is not always possible if the {Vq }q∈S form a Borel Voronoi tiling, instead of a Borel Voronoi partition. If, for instance, ρ has an atom falling on two Voronoi regions in a tiling, then both regions would count the atom as theirs, and double-counting would imply q ρ(Vq ) > 1. The technicalities required to correctly deﬁne a Borel Voronoi partition are such that, in general, it is simpler to write πS ρ, even though (if S is discrete) this measure can clearly be written as a sum of deltas with appropriate masses. By lemma 3.1, the distance Wp (ρ, πX4 ρ)p is the (expected) quantization cost of ρ when using X4 as codebook. Clearly, this cost can never be lower than the optimal quantization cost of size 4. This reasoning leads to the following lower bound between empirical and population measures. 4 Theorem 3.3. For ρ ∈ Pp (M) with absolutely continuous part ρA = 0, and 1 ≤ p < ∞, it holds Wp (ρ, ρn ) = Ω(n−1/d ) uniformly over ρn , where the constants depend on d and ρA only. ˆ ˆ Proof: Let Vn,p (ρ) := inf S⊂M,|S|=n Ex∼ρ d(x, S)p be the optimal quantization cost of ρ of order p with n centers. Since ρA = 0, and since ρ has a ﬁnite (p + δ)-th order moment, for some δ > 0 (since it is supported on the unit ball), then it is Vn,p (ρ) = Θ(n−p/d ), with constants depending on d and ρA (see [14, p. 78] and [16].) Since supp(ˆn ) = Xn , it follows that ρ Wp (ρ, ρn )p ˆ ≥ lemma 3.2 Wp (ρ, πXn ρ)p = lemma 3.1 Ex∼ρ d(x, Xn )p ≥ Vn,p (ρ) = Θ(n−p/d ) Note that the bound of theorem 3.3 holds for ρn derived from any sample Xn , and is therefore ˆ stronger than the existing lower bounds on the convergence rates of EWp (ρ, ρn ) → 0. In particular, ˆ it trivially induces the known lower bound Ω(n−1/d ) on the rate of convergence in expectation. 4 Unsupervised learning algorithms for learning a probability measure As described in [21], several of the most widely used unsupervised learning algorithms can be ˆ interpreted to take as input a sample Xn and output a set Sk , where k is typically a free parameter of the algorithm, such as the number of means in k-means1 , the dimension of afﬁne spaces in PCA, n ˆ etc. Performance is measured by the empirical quantity n−1 i=1 d(xi , Sk )2 , which is minimized among all sets in some class (e.g. sets of size k, afﬁne spaces of dimension k,. . . ) This formulation is general enough to encompass k-means and PCA, but also k-ﬂats, non-negative matrix factorization, and sparse coding (see [21] and references therein.) Using the discussion of Sec. 3, we can establish a clear connection between unsupervised learning and the problem of learning probability measures with respect to W2 . Consider as a running example the k-means problem, though the argument is general. Given an input Xn , the k-means problem is ˆ ˆ to ﬁnd a set |Sk | = k minimizing its average distance from points in Xn . By associating to Sk the pushforward measure πSk ρn , we ﬁnd that ˆ ˆ 1 n n ˆ ˆ d(xi , Sk )2 = Ex∼ρn d(x, Sk )2 ˆ i=1 = lemma 3.1 W2 (ˆn , πSk ρn )2 . ρ ˆ ˆ (2) Since k-means minimizes equation 2, it also ﬁnds the measure that is closest to ρn , among those ˆ with support of size k. This connection between k-means and W2 measure approximation was, to the best of the authors’ knowledge, ﬁrst suggested by Pollard [28] though, as mentioned earlier, the argument carries over to many other unsupervised learning algorithms. Unsupervised measure learning algorithms. We brieﬂy clarify the steps involved in using an existing unsupervised learning algorithm for probability measure learning. Let Uk be a parametrized algorithm (e.g. k-means) that takes a sample Xn and outputs a set Uk (Xn ). The measure learning algorithm Ak : Mn → Pp (M) corresponding to Uk is deﬁned as follows: ˆ 1. Ak takes a sample Xn and outputs the measure πSk ρn , supported on Sk = Uk (Xn ); ˆ ˆ 2. since ρn is discrete, then so must πSk ρn be, and thus Ak (Xn ) = ˆ ˆ ˆ 1 n n ˆ i=1 δπSk (xi ) ; 3. in practice, we can simply store an n-vector πSk (x1 ), . . . , πSk (xn ) , from which Ak (Xn ) ˆ ˆ can be reconstructed by placing atoms of mass 1/n at each point. In the case that Uk is the k-means algorithm, only k points and k masses need to be stored. Note that any algorithm A that attempts to output a measure A (Xn ) close to ρn can be cast in the ˆ above framework. Indeed, if S is the support of A (Xn ) then, by lemma 3.2, πS ρn is the measure ˆ closest to ρn with support in S . This effectively reduces the problem of learning a measure to that of ˆ 1 In a slight abuse of notation, we refer to the k-means algorithm here as an ideal algorithm that solves the k-means problem, even though in practice an approximation algorithm may be used. 5 ﬁnding a set, and is akin to how the fact that every optimal quantizer is a nearest-neighbor quantizer (see [15], [12, p. 350], and [14, p. 37–38]) reduces the problem of ﬁnding an optimal quantizer to that of ﬁnding an optimal quantizing set. Clearly, the minimum of equation 2 over sets of size k (the output of k-means) is monotonically ˆ ˆ non-increasing with k. In particular, since Sn = Xn and πSn ρn = ρn , it is Ex∼ρn d(x, Sn )2 = ˆ ˆ ˆ ˆ 2 W2 (ˆn , πSn ρn ) = 0. That is, we can always make the learned measure arbitrarily close to ρn ρ ˆ ˆ ˆ by increasing k. However, as pointed out in Sec. 2, the problem of measure learning is concerned with minimizing the 2-Wasserstein distance W2 (ρ, πSk ρn ) to the data-generating measure. The ˆ ˆ actual performance of k-means is thus not necessarily guaranteed to behave in the same way as the empirical one, and the question of characterizing its behavior as a function of k and n naturally arises. ˆ Finally, we note that, while it is Ex∼ρn d(x, Sk )2 = W2 (ˆn , πSk ρn )2 (the empirical performances ρ ˆ ˆ ˆ are the same in the optimal quantization, and measure learning problem formulations), the actual performances satisfy ˆ Ex∼ρ d(x, Sk )2 = W2 (ρ, π ˆ ρ)2 ≤ W2 (ρ, π ˆ ρn )2 , 1 ≤ k ≤ n. ˆ lemma 3.1 Sk lemma 3.2 Sk Consequently, with the identiﬁcation between sets S and measures πS ρn , the measure learning ˆ problem is, in general, harder than the set-approximation problem (for example, if M = Rd and ρ is absolutely continuous over a set of non-null volume, it is not hard to show that the inequality is ˆ almost surely strict: Ex∼ρ d(x, Sk )2 < W2 (ρ, πSk ρn )2 for 1 < k < n.) ˆ ˆ In the remainder, we characterize the performance of k-means on the measure learning problem, for varying k, n. Although other unsupervised learning algorithms could have been chosen as basis for our analysis, k-means is one of the oldest and most widely used, and the one for which the deep connection between optimal quantization and measure approximation is most clearly manifested. Note that, by setting k = n, our analysis includes the problem of characterizing the behavior of the distance W2 (ρ, ρn ) between empirical and population measures which, as indicated in Sec. 2.1, ˆ is a fundamental question in statistics (i.e. the speed of convergence of empirical to population measures.) 5 Learning rates In order to analyze the performance of k-means as a measure learning algorithm, and the convergence of empirical to population measures, we propose the decomposition shown in ﬁg. 2. The diagram includes all the measures considered in the paper, and shows the two decompositions used to prove upper bounds. The upper arrow (green), illustrates the decomposition used to bound the distance W2 (ρ, ρn ). This decomposition uses the measures πSk ρ and πSk ρn as intermediates to arrive ˆ ˆ at ρn , where Sk is a k-point optimal quantizer of ρ, that is, a set Sk minimizing Ex∼ρ d(x, S)2 over ˆ all sets of size |S| = k. The lower arrow (blue) corresponds to the decomposition of W2 (ρ, πSk ρn ) ˆ ˆ (the performance of k-means), whereas the labelled black arrows correspond to individual terms in the bounds. We begin with the (slightly) simpler of the two results. 5.1 Convergence rates for the empirical to population measures Let Sk be the optimal k-point quantizer of ρ of order two [14, p. 31]. By the triangle inequality and the identity (a + b + c)2 ≤ 3(a2 + b2 + c2 ), it follows that W2 (ρ, ρn )2 ≤ 3 W2 (ρ, πSk ρ)2 + W2 (πSk ρ, πSk ρn )2 + W2 (πSk ρn , ρn )2 . ˆ ˆ ˆ ˆ (3) This is the decomposition depicted in the upper arrow of ﬁg. 2. By lemma 3.1, the ﬁrst term in the sum of equation 3 is the optimal k-point quantization error of ρ over a d-manifold M which, using recent techniques from [16] (see also [17, p. 491]), is shown in the proof of theorem 5.1 (part a) to be of order Θ(k −2/d ). The remaining terms, b) and c), are slightly more technical and are bounded in the proof of theorem 5.1. Since equation 3 holds for all 1 ≤ k ≤ n, the best bound on W2 (ρ, ρn ) can be obtained by optimizˆ ing the right-hand side over all possible values of k, resulting in the following probabilistic bound for the rate of convergence of the empirical to population measures. 6 x2 x W2 (ρ, ρn ) ˆ supp ρ x1 π{x1 ,x2 ,x3 ,x4 } ρ a) x3 πSk ρ b) πSk ρn ˆ c) d) ρn ˆ πSk ρn ˆ ˆ W2 (ρ, πSk ρn ) ˆ ˆ x4 Figure 1: A sample {x1 , x2 , x3 , x4 } is drawn from a distribution ρ with support in supp ρ. The projection map π{x1 ,x2 ,x3 ,x4 } sends points x to their closest one in the sample. The induced Voronoi tiling is shown in shades of blue. Figure 2: The measures considered in this paper are linked by arrows for which upper bounds for their distance are derived. Bounds for the quantities of interest W2 (ρ, ρn )2 , and W2 (ρ, πSk ρn )2 , ˆ ˆ ˆ are decomposed by following the top and bottom colored arrows. Theorem 5.1. Given ρ ∈ Pp (M) with absolutely continuous part ρA = 0, sufﬁciently large n, and τ > 0, it holds W2 (ρ, ρn ) ≤ C · m(ρA ) · n−1/(2d+4) · τ, ˆ where m(ρA ) := 5.2 M 2 with probability 1 − e−τ . ρA (x)d/(d+2) dλM (x), and C depends only on d. Learning rates of k-means The key element in the proof of theorem 5.1 is that the distance between population and empirical measures can be bounded by choosing an intermediate optimal quantizing measure of an appropriate size k. In the analysis, the best bounds are obtained for k smaller than n. If the output of k-means is close to an optimal quantizer (for instance if sufﬁcient data is available), then we would similarly expect that the best bounds for k-means correspond to a choice of k < n. The decomposition of the bottom (blue) arrow in ﬁgure 2 leads to the following bound in probability. Theorem 5.2. Given ρ ∈ Pp (M) with absolutely continuous part ρA = 0, and τ > 0, then for all sufﬁciently large n, and letting k = C · m(ρA ) · nd/(2d+4) , it holds W2 (ρ, πSk ρn ) ≤ C · m(ρA ) · n−1/(2d+4) · τ, ˆ ˆ where m(ρA ) := M 2 with probability 1 − e−τ . ρA (x)d/(d+2) dλM (x), and C depends only on d. Note that the upper bounds in theorem 5.1 and 5.2 are exactly the same. Although this may appear ˆ surprising, it stems from the following fact. Since S = Sk is a minimizer of W2 (πS ρn , ρn )2 , the ˆ ˆ bound d) of ﬁgure 2 satisﬁes: W2 (πSk ρn , ρn )2 ≤ W2 (πSk ρn , ρn )2 ˆ ˆ ˆ ˆ ˆ and therefore (by the deﬁnition of c), the term d) is of the same order as c). It follows then that adding term d) to the bound only affects the constants, but otherwise leaves it unchanged. Since d) is the term that takes the output measure of k-means to the empirical measure, this implies that the rate of convergence of k-means (for suitably chosen k) cannot be worse than that of ρn → ρ. ˆ Conversely, bounds for ρn → ρ are obtained from best rates of convergence of optimal quantizers, ˆ whose convergence to ρ cannot be slower than that of k-means (since the quantizers that k-means produces are suboptimal.) 7 Since the bounds obtained for the convergence of ρn → ρ are the same as those for k-means with ˆ k of order k = Θ(nd/(2d+4) ), this suggests that estimates of ρ that are as accurate as those derived from an n point-mass measure ρn can be derived from k point-mass measures with k ˆ n. Finally, we note that the introduced bounds are currently limited by the statistical bound sup |W2 (πS ρn , ρn )2 − W2 (πS ρ, ρ)2 | ˆ ˆ |S|=k = sup |Ex∼ρn d(x, S)2 − Ex∼ρ d(x, S)2 | ˆ lemma 3.1 |S|=k (4) (see for instance [21]), for which non-matching lower bounds are known. This means that, if better upper bounds can be obtained for equation 4, then both bounds in theorems 5.1 and 5.2 would automatically improve (would become closer to the lower bound.) References [1] M. Ajtai, J. Komls, and G. Tusndy. On optimal matchings. Combinatorica, 4:259–264, 1984. [2] Franck Barthe and Charles Bordenave. Combinatorial optimization over two random point sets. Technical Report arXiv:1103.2734, Mar 2011. [3] Gordon Blower. The Gaussian isoperimetric inequality and transportation. Positivity, 7:203–224, 2003. [4] S. G. Bobkov and F. G¨ tze. Exponential integrability and transportation cost related to logarithmic o Sobolev inequalities. Journal of Functional Analysis, 163(1):1–28, April 1999. 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