nips nips2011 nips2011-156 knowledge-graph by maker-knowledge-mining

156 nips-2011-Learning to Learn with Compound HD Models

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Author: Antonio Torralba, Joshua B. Tenenbaum, Ruslan Salakhutdinov

Abstract: We introduce HD (or “Hierarchical-Deep”) models, a new compositional learning architecture that integrates deep learning models with structured hierarchical Bayesian models. Speciﬁcally we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a Deep Boltzmann Machine (DBM). This compound HDP-DBM model learns to learn novel concepts from very few training examples, by learning low-level generic features, high-level features that capture correlations among low-level features, and a category hierarchy for sharing priors over the high-level features that are typical of different kinds of concepts. We present efﬁcient learning and inference algorithms for the HDP-DBM model and show that it is able to learn new concepts from very few examples on CIFAR-100 object recognition, handwritten character recognition, and human motion capture datasets. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Abstract We introduce HD (or “Hierarchical-Deep”) models, a new compositional learning architecture that integrates deep learning models with structured hierarchical Bayesian models. [sent-6, score-0.368]

2 Speciﬁcally we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a Deep Boltzmann Machine (DBM). [sent-7, score-0.346]

3 We present efﬁcient learning and inference algorithms for the HDP-DBM model and show that it is able to learn new concepts from very few examples on CIFAR-100 object recognition, handwritten character recognition, and human motion capture datasets. [sent-9, score-0.316]

4 1 Introduction “Learning to learn”, or the ability to learn abstract representations that support transfer to novel but related tasks, lies at the core of many problems in computer vision, natural language processing, cognitive science, and machine learning. [sent-10, score-0.179]

5 For humans learners, however, just one or a few examples are often sufﬁcient to grasp a new category and make meaningful generalizations to novel instances [25, 16]. [sent-12, score-0.215]

6 The architecture we describe here takes a step towards this “one-shot learning” ability by learning several forms of abstract knowledge that support transfer of useful representations from previously learned concepts to novel ones. [sent-13, score-0.217]

7 We call our architectures compound HD models, where “HD” stands for “Hierarchical-Deep”, because they are derived by composing hierarchical nonparametric Bayesian models with deep networks, two inﬂuential approaches from the recent unsupervised learning literature with complementary strengths. [sent-14, score-0.481]

8 Recently introduced deep learning models, including Deep Belief Networks [5], Deep Boltzmann Machines [14], deep autoencoders [10], and others [12, 11], have been shown to learn useful distributed feature representations for many high-dimensional datasets. [sent-15, score-0.463]

9 The ability to automatically learn in multiple layers allows deep models to construct sophisticated domain-speciﬁc features without the need to rely on precise human-crafted input representations, increasingly important with the proliferation of data sets and application domains. [sent-16, score-0.385]

10 While the features learned by deep models can enable more rapid and accurate classiﬁcation learning, deep networks themselves are not well suited to one-shot learning of novel classes. [sent-17, score-0.615]

11 All units and parameters at all levels of the network are engaged in representing any given input and are adjusted together during learning. [sent-18, score-0.148]

12 This ability is the 1 hallmark of hierarchical Bayesian (HB) models, recently proposed in computer vision, statistics, and cognitive science [7, 25, 4, 13] for learning to learn from few examples. [sent-20, score-0.153]

13 Unlike deep networks, these HB models explicitly represent category hierarchies that admit sharing the appropriate abstract knowledge about the new class’s parameters via a prior abstracted from related classes. [sent-21, score-0.404]

14 HB approaches, however, have complementary weaknesses relative to deep networks. [sent-22, score-0.196]

15 GIST, SIFT features in computer vision, MFCC features in speech perception domains). [sent-25, score-0.214]

16 Moreover, many HB approaches often assume a ﬁxed hierarchy for sharing parameters [17, 3] instead of learning the hierarchy in an unsupervised fashion. [sent-27, score-0.256]

17 In this work we investigate compound HD (hierarchical-deep) architectures that integrate these deep models with structured hierarchical Bayesian models. [sent-28, score-0.386]

18 In particular, we show how we can learn a hierarchical Dirichlet process (HDP) prior over the activities of the top-level features in a Deep Boltzmann Machine (DBM), coming to represent both a layered hierarchy of increasingly abstract features, and a tree-structured hierarchy of classes. [sent-29, score-0.534]

19 Our model depends minimally on domain-speciﬁc representations and achieves state-of-the-art one-shot learning performance by unsupervised discovery of three components: (a) low-level features that abstract from the raw high-dimensional sensory input (e. [sent-30, score-0.254]

20 pixels, or 3D joint angles); (b) high-level part-like features that express the distinctive perceptual structure of a speciﬁc class, in terms of class-speciﬁc correlations over low-level features; and (c) a hierarchy of super-classes for sharing abstract knowledge among related classes. [sent-32, score-0.231]

21 It contains a set of visible units v ∈ {0, 1}D , and a sequence of layers of hidden units h1 ∈ {0, 1}F1 , h2 ∈ {0, 1}F2 ,. [sent-36, score-0.402]

22 There are connections only between hidden units in adjacent layers, as well as between visible and hidden units in the ﬁrst hidden layer. [sent-40, score-0.482]

23 Multinomial DBMs: To allow DBMs to express more information and introduce more structured hierarchical priors, we will use a conditional multinomial distribution to model activities of the toplevel units. [sent-50, score-0.304]

24 Speciﬁcally, we will use M softmax units, each with “1-of-K” encoding (so that each 1 For clarity, we use three hidden layers. [sent-51, score-0.192]

25 Middle: A different interpretation: M softmax units are replaced by a single multinomial unit which is sampled M times. [sent-54, score-0.365]

26 All M separate softmax units will share the same set of weights, connecting them to binary hidden units at the lower-level (Fig. [sent-57, score-0.488]

27 A key observation is that M separate copies of softmax units that all share the same set of weights can be viewed as a single multinomial unit that is samples M times [15, 19]. [sent-59, score-0.365]

28 A pleasing property of using softmax units is that the mathematics underlying the learning algorithm for binary-binary DBMs remains the same. [sent-60, score-0.278]

29 One way to express what has been learned is the conditional model P (v, h1 , h2 |h3 ) and a prior term P (h3 ). [sent-62, score-0.131]

30 a restricted Boltzmann machine), to model P (h3 ), we can place a hierarchical Dirichlet process prior over h3 , that will allow us to learn category hierarchies, and more importantly, useful representations of classes that contain few training examples. [sent-68, score-0.451]

31 l m (4) lm A Hierarchical Bayesian Topic Prior In a typical hierarchical topic model, we observe a set of N documents, each of which is modeled as a mixture over topics, that are shared among documents. [sent-71, score-0.181]

32 A topic t is a discrete distribution over K words with probability vector φt . [sent-73, score-0.123]

33 Each document n has its own distribution over topics given by probabilities θn . [sent-74, score-0.154]

34 In our compound HDP-DBM model, we will use a hierarchical topic model as a prior over the activities of the DBM’s top-level features. [sent-75, score-0.4]

35 Speciﬁcally, the term “document” will refer to the toplevel multinomial unit h3 , and M “words” in the document will represent the M samples, or active DBM’s top-level features, generated by this multinomial unit. [sent-76, score-0.262]

36 Words in each document are drawn by choosing a topic t with probability θnt , and then choosing a word w with probability φtw . [sent-77, score-0.147]

37 We will often refer to topics as our learned higher-level features, each of which deﬁnes a topic speciﬁc distribution over DBM’s h3 features. [sent-78, score-0.238]

38 Let h3 be the ith word in document n, and xin be its topic: in θn |π ∼ Dir(απ), φt |τ ∼ Dir(βτ ), xin |θn ∼ Mult(θn ), h3 |xin , φxin ∼ Mult(φxin ), in (5) where π is the global distribution over topics, τ is the global distribution over K words, and α and β are concentration parameters. [sent-79, score-0.492]

39 3 Let us further assume that we are presented with a ﬁxed two-level category hierarchy. [sent-80, score-0.123]

40 We b represent such partition by a vector zb of length N , each entry of which is zn ∈ {1, . [sent-84, score-0.212]

41 These partitions deﬁne a ﬁxed two-level tree hierarchy (see Fig. [sent-94, score-0.13]

42 The hierarchical topic model can be readily extended to modeling the above hierarchy. [sent-97, score-0.181]

43 For each document n that belong to the basic category c, we place a common Dirichlet prior over θn with (1) parameters πc . [sent-98, score-0.259]

44 , N xin |θn φt |τ ∼ Mult(θn ), ∼ Dir(βτ ), n (6) (1) n for each word i = 1, . [sent-107, score-0.22]

45 For a ﬁxed number of topics T , the above model represents a hierarchical extension of LDA. [sent-111, score-0.188]

46 Making use of topic index variables xin , we denote φ∗ = φxin (see Eq. [sent-115, score-0.315]

47 If, however, we are not given any level-1 or level-2 category labels, we need to infer the distribution over the possible category structures. [sent-127, score-0.246]

48 We place a nonparametric two-level nested Chinese Restaurant Prior (CRP) [2] over z, which deﬁnes a prior over tree structures and is ﬂexible enough to learn arbitrary hierarchies. [sent-128, score-0.274]

49 The main building block of the nested CRP is the Chinese restaurant process, a distribution on partition of integers. [sent-129, score-0.142]

50 Its assignment b s to the basic-level category zn , that is placed under a super-category zn , is again recursively drawn from Eq. [sent-137, score-0.343]

51 The proposed model allows for both: a nonparametric prior over potentially unbounded number of global topics, or higher-level features, as well as a nonparametric prior that allow learning an arbitrary tree taxonomy. [sent-140, score-0.26]

52 Sampling HDP parameters: Given category assignment vectors z, and the states of the top-level DBM features h3 , we use posterior representation sampler of [20]. [sent-143, score-0.298]

53 In particular, the HDP sampler (1) (2) (3) maintains the stick-breaking weights {θ}N , and {πc , πs , πg }; and topic indicator variables x n=1 (parameters φ can be integrated out). [sent-144, score-0.133]

54 The sampler alternatives between: (a) sampling cluster indices xin using Gibbs updates in the Chinese restaurant franchise (CRF) representation of the HDP; (b) sampling the weights at all three levels conditioned on x using the usual posterior of a DP2 . [sent-145, score-0.371]

55 Sampling category assignments z: Given current instantiation of the stick-breaking weights, using a deﬁning property of a DP, for each input n, we have: (1) (1) (1) (θ1,n , . [sent-146, score-0.123]

56 8), the posterior over the category assignment can be calculated as follows: p(zn |θn , z−n , π (1) ) ∝ p(θn |π (1) , zn )p(zn |z−n ), (10) where z−n denotes variables z for all observations other than n. [sent-153, score-0.263]

57 Sampling DBM’s hidden units: Given the states of the DBM’s top-level multinomial unit h3 , conditional samples from P (h1 , h2 |h3 , vn ) can be obtained by running a Gibbs sampler that alternates n n n between sampling the states of h1 independently given h2 , and vice versa. [sent-155, score-0.222]

58 Conditioned on topic n n assignments xin and h2 , the states of the multinomial unit h3 for each input n are sampled using n n Gibbs conditionals: P (h3 |h2 , h3 , xn ) ∝ P (h2 |h3 )P (h3 |xin ), (11) in n −in n n in where the ﬁrst term is given by the product of logistic functions (see Eq. [sent-156, score-0.402]

59 2, followed by running the full Gibbs sampler to get approximate samples from the posterior over the category assignments. [sent-163, score-0.191]

60 In practice, for faster inference, we ﬁx learned topics φt and approximate the marginal likelihood that h3 belongs to category zt by assuming that t (1) document speciﬁc DP can be well approximated by the class-speciﬁc DP3 Gt ≈ Gzt (see Fig. [sent-164, score-0.362]

62 8) allows us to efﬁciently infer approximate posterior over category assignments4 . [sent-166, score-0.153]

63 1st DBM features layer 2nd layer HDP high-level features 2. [sent-171, score-0.462]

64 forest Figure 2: A random subset of the 1st , 2nd layer DBM features, Figure 3: A typical partition of the 100 and higher-level class-sensitive HDP features/topics. [sent-184, score-0.153]

65 basic-level categories 5 Experiments We present experimental results on the CIFAR-100 [8], handwritten character [9], and human motion capture recognition datasets. [sent-185, score-0.199]

66 For all datasets, we ﬁrst pretrain a DBM model in unsupervised fashion on raw sensory input (e. [sent-186, score-0.15]

67 Across all datasets, we also assume that the basic-level category labels are given, but no super-category labels are available. [sent-191, score-0.123]

68 The training set includes many examples of familiar categories but only a few examples of a novel class. [sent-192, score-0.225]

69 1 CIFAR-100 dataset The CIFAR-100 image dataset [8] contains 50,000 training and 10,000 test images of 100 object categories (100 per class), with 32 × 32 × 3 RGB pixels. [sent-201, score-0.243]

70 The ﬁrst DBM layer contained 10,000 binary hidden units, and the second layer contained M=1000 softmax units, each deﬁning a distribution over 10, 000 second layer features6 . [sent-205, score-0.564]

71 2 displays a random subset of the 1st and 2nd layer DBM features, as well as higher-level classsensitive features, or topics, learned by the HDP model. [sent-208, score-0.202]

72 To visualize a particular higher-level feature, we ﬁrst sample M words from a ﬁxed topic φt , followed by sampling RGB pixel values from the conditional DBM model. [sent-209, score-0.158]

73 While DBM features capture mostly low-level structure, including edges and corners, the HDP features tend to capture higher-level structure, including contours, shapes, color components, and surface boundaries. [sent-210, score-0.288]

74 More importantly, features at all levels of the hierarchy evolve without incorporating any image-speciﬁc priors. [sent-211, score-0.201]

75 We trained the model on 99 classes containing 500 training images each, but only one training example of a “pear” class. [sent-215, score-0.182]

76 Hence the novel category can inherit 5 The dataset contains random images of natural scenes downloaded from the web We also experimented with a 3-layer DBM model, as well as various softmax parameters: M = 500 and M = 2000. [sent-218, score-0.381]

77 6 6 Learning with 3 examples Shared HDP high-level features Shape Color HDP−DBM DBM SVM 0. [sent-220, score-0.153]

78 1 0 0 200 400 600 800 1000 1200 1400 Sorted Class Index Figure 4: Left: Training examples along with eight most probable topics φt , ordered by hand. [sent-238, score-0.148]

79 CIFAR Dataset Model Tuned HDP-DBM HDP-DBM Flat HDP-DBM DBM DBN SVM 1-NN GIST Handwritten Characters Motion Capture Number of examples Number of examples Number of examples 1 3 5 10 50 1 3 5 10 1 3 5 10 50 0. [sent-241, score-0.138]

80 Table 1 also shows that ﬁne-tuning parameters of all layers jointly as well as learning super-category hierarchy signiﬁcantly improves model performance. [sent-356, score-0.138]

81 2 Handwritten Characters The handwritten characters dataset [9] can be viewed as the “transpose” of MNIST. [sent-362, score-0.209]

82 Instead of containing 60,000 images of 10 digit classes, the dataset contains 30,000 images of 1500 characters (20 examples each) with 28 × 28 pixels. [sent-363, score-0.301]

83 These characters are from 50 alphabets from around the world, including Bengali, Cyrillic, Arabic, Sanskrit, Tagalog (see Fig. [sent-364, score-0.125]

84 We split the dataset into 15,000 training and 15,000 test images (10 examples of each class). [sent-366, score-0.175]

85 Similar to the CIFAR dataset, we pretrain a two-layer DBM model, with the ﬁrst layer containing 1000 hidden units, and the second layer containing M=100 softmax units, each deﬁning a distribution over 1000 second layer features. [sent-367, score-0.6]

86 2 displays a random subset of training images, along with the 1st and 2nd layer DBM features, as well as higher-level class-sensitive HDP features. [sent-369, score-0.208]

87 The HDP features tend to capture higher-level parts, many of which resemble pen “strokes”. [sent-370, score-0.144]

88 The HDP-DBM model signiﬁcantly outperforms other methods, particularly when learning characters with few training examples. [sent-373, score-0.172]

89 Learned Super-Classes (by row) Learning with 3 examples Sampled Novel Characters Training Examples Conditional Samples Figure 6: Left: Learned super-classes along with examples of novel characters, generated by the model for the same super-class. [sent-376, score-0.138]

90 Right: Three training examples along with 8 conditional samples. [sent-377, score-0.128]

91 3 Motion capture We next applied our model to human motion capture data consisting of sequences of 3D joint angles plus body orientation and translation [18]. [sent-382, score-0.159]

92 For the two-layer DBM model, the ﬁrst layer contained 500 hidden units, with the second layer containing M =50 softmax units, each deﬁning a distribution over 500 second layer features. [sent-387, score-0.564]

93 The difference is particularly large in the regime when we observe only a handful number of training examples of a novel walking style. [sent-390, score-0.182]

94 6 Conclusions We developed a compositional architecture that learns an HDP prior over the activities of top-level features of the DBM model. [sent-391, score-0.308]

95 The resulting compound HDP-DBM model is able to learn low-level features from raw sensory input, high-level features, as well as a category hierarchy for parameter sharing. [sent-392, score-0.542]

96 Indeed, any other deep learning module, including Deep Belief Networks, sparse auto-encoders, or any other hierarchical Bayesian model can be adapted. [sent-395, score-0.282]

97 The nested chinese restaurant process and bayesian nonparametric inference of topic hierarchies. [sent-409, score-0.339]

98 Modeling human transfer learning with the hierarchical dirichlet process. [sent-415, score-0.2]

99 Convolutional deep belief networks for scalable unsupervised learning of hierarchical representations. [sent-460, score-0.387]

100 80 million tiny images: a large dataset for nonparametric object and scene recognition. [sent-554, score-0.163]

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Abstract: We investigate the representational power of sum-product networks (computation networks analogous to neural networks, but whose individual units compute either products or weighted sums), through a theoretical analysis that compares deep (multiple hidden layers) vs. shallow (one hidden layer) architectures. We prove there exist families of functions that can be represented much more efﬁciently with a deep network than with a shallow one, i.e. with substantially fewer hidden units. Such results were not available until now, and contribute to motivate recent research involving learning of deep sum-product networks, and more generally motivate research in Deep Learning. 1 Introduction and prior work Many learning algorithms are based on searching a family of functions so as to identify one member of said family which minimizes a training criterion. The choice of this family of functions and how members of that family are parameterized can be a crucial one. 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It is not obvious that circuit complexity results (that typically consider only boolean or at least discrete nodes) are directly applicable in the context of typical machine learning algorithms such as neural networks (that compute continuous representations of their input). Orponen [23] surveys theoretical results in computational complexity that are relevant to learning algorithms. For instance, H˚ stad and Goldmann [12] extended some results to the case of networks of linear threshold units a with positivity constraints on the weights. Bengio et al. [5, 7] investigate, respectively, complexity issues in networks of Gaussian radial basis functions and decision trees, showing intrinsic limitations of these architectures e.g. on tasks similar to the parity problem. Utgoff and Stracuzzi [35] informally discuss the advantages of depth in boolean circuit in the context of learning architectures. Bengio [3] suggests that some polynomials could be represented more efﬁciently by deep sumproduct networks, but without providing any formal statement or proofs. This work partly addresses this void by demonstrating families of circuits for which a deep architecture can be exponentially more efﬁcient than a shallow one in the context of real-valued polynomials. Note that we do not address in this paper the problem of learning these parameters: even if an efﬁcient deep representation exists for the function we seek to approximate, in general there is no 2 guarantee for standard optimization algorithms to easily converge to this representation. This paper focuses on the representational power of deep sum-product circuits compared to shallow ones, and studies it by considering particular families of target functions (to be represented by the learner). We ﬁrst formally deﬁne sum-product networks. We consider two families of functions represented by deep sum-product networks (families F and G). For each family, we establish a lower bound on the minimal number of hidden units a depth-2 sum-product network would require to represent a function of this family, showing it is much less efﬁcient than the deep representation. 2 Sum-product networks Deﬁnition 1. A sum-product network is a network composed of units that either compute the product of their inputs or a weighted sum of their inputs (where weights are strictly positive). Here, we restrict our deﬁnition of the generic term “sum-product network” to networks whose summation units have positive incoming weights1 , while others are called “negative-weight” networks. Deﬁnition 2. A “negative-weight“ sum-product network may contain summation units whose weights are non-positive (i.e. less than or equal to zero). Finally, we formally deﬁne what we mean by deep vs. shallow networks in the rest of the paper. Deﬁnition 3. A “shallow“ sum-product network contains a single hidden layer (i.e. a total of three layers when counting the input and output layers, and a depth equal to two). Deﬁnition 4. A “deep“ sum-product network contains more than one hidden layer (i.e. a total of at least four layers, and a depth at least three). The family F 3 3.1 Deﬁnition The ﬁrst family of functions we study, denoted by F, is made of functions built from deep sumproduct networks that alternate layers of product and sum units with two inputs each (details are provided below). The basic idea we use here is that composing layers (i.e. using a deep architecture) is equivalent to using a factorized representation of the polynomial function computed by the network. Such a factorized representation can be exponentially more compact than its expansion as a sum of products (which can be associated to a shallow network with product units in its hidden layer and a sum unit as output). This is what we formally show in what follows. + ℓ2 = λ11ℓ1 + µ11ℓ1 = x1x2 + x3x4 = f (x1, x2, x3, x4) 2 1 1 λ11 = 1 µ11 = 1 × ℓ1 = x1x2 1 x1 x2 × ℓ1 = x3x4 2 x3 x4 Figure 1: Sum-product network computing the function f ∈ F such that i = λ11 = µ11 = 1. Let n = 4i , with i a positive integer value. Denote by ℓ0 the input layer containing scalar variables {x1 , . . . , xn }, such that ℓ0 = xj for 1 ≤ j ≤ n. Now deﬁne f ∈ F as any function computed by a j sum-product network (deep for i ≥ 2) composed of alternating product and sum layers: • ℓ2k+1 = ℓ2k · ℓ2k for 0 ≤ k ≤ i − 1 and 1 ≤ j ≤ 22(i−k)−1 2j−1 2j j • ℓ2k = λjk ℓ2k−1 + µjk ℓ2k−1 for 1 ≤ k ≤ i and 1 ≤ j ≤ 22(i−k) j 2j 2j−1 where the weights λjk and µjk of the summation units are strictly positive. The output of the network is given by f (x1 , . . . , xn ) = ℓ2i ∈ R, the unique unit in the last layer. 1 The corresponding (shallow) network for i = 1 and additive weights set to one is shown in Figure 1 1 This condition is required by some of the proofs presented here. 3 (this architecture is also the basic building block of bigger networks for i > 1). Note that both the input size n = 4i and the network’s depth 2i increase with parameter i. 3.2 Theoretical results The main result of this section is presented below in Corollary 1, providing a lower bound on the minimum number of hidden units required by a shallow sum-product network to represent a function f ∈ F. The high-level proof sketch consists in the following steps: (1) Count the number of unique products found in the polynomial representation of f (Lemma 1 and Proposition 1). (2) Show that the only possible architecture for a shallow sum-product network to compute f is to have a hidden layer made of product units, with a sum unit as output (Lemmas 2 to 5). (3) Conclude that the number of hidden units must be at least the number of unique products computed in step 3.2 (Lemma 6 and Corollary 1). Lemma 1. Any element ℓk can be written as a (positively) weighted sum of products of input varij ables, such that each input variable xt is used in exactly one unit of ℓk . Moreover, the number mk of products found in the sum computed by ℓk does not depend on j and obeys the following recurrence j rule for k ≥ 0: if k + 1 is odd, then mk+1 = m2 , otherwise mk+1 = 2mk . k Proof. We prove the lemma by induction on k. It is obviously true for k = 0 since ℓ0 = xj . j Assuming this is true for some k ≥ 0, we consider two cases: k+1 k • If k + 1 is odd, then ℓj = ℓk 2j−1 · ℓ2j . By the inductive hypothesis, it is the product of two (positively) weighted sums of products of input variables, and no input variable can k appear in both ℓk 2j−1 and ℓ2j , so the result is also a (positively) weighted sum of products k of input variables. Additionally, if the number of products in ℓk 2j−1 and ℓ2j is mk , then 2 mk+1 = mk , since all products involved in the multiplication of the two units are different (since they use disjoint subsets of input variables), and the sums have positive weights. Finally, by the induction assumption, an input variable appears in exactly one unit of ℓk . This unit is an input to a single unit of ℓk+1 , that will thus be the only unit of ℓk+1 where this input variable appears. k • If k + 1 is even, then ℓk+1 = λjk ℓk 2j−1 + µjk ℓ2j . Again, from the induction assumption, it j must be a (positively) weighted sum of products of input variables, but with mk+1 = 2mk such products. As in the previous case, an input variable will appear in the single unit of ℓk+1 that has as input the single unit of ℓk in which this variable must appear. 2i Proposition 1. The number of products in the sum computed in the output unit l1 of a network √ n−1 . computing a function in F is m2i = 2 Proof. We ﬁrst prove by induction on k ≥ 1 that for odd k, mk = 22 k 22 1+1 2 2 k+1 2 −2 , and for even k, . This is obviously true for k = 1 since 2 = 2 = 1, and all units in ℓ1 are mk = 2 single products of the form xr xs . Assuming this is true for some k ≥ 1, then: −1 0 −2 • if k + 1 is odd, then from Lemma 1 and the induction assumption, we have: mk+1 = m2 = k 2 k 22 2 −1 k +1 = 22 2 • if k + 1 is even, then instead we have: mk+1 = 2mk = 2 · 22 k+1 2 −2 −2 = 22 = 22 (k+1)+1 2 (k+1) 2 −2 −1 which shows the desired result for k + 1, and thus concludes the induction proof. Applying this result with k = 2i (which is even) yields 2i m2i = 22 2 −1 √ =2 4 22i −1 √ =2 n−1 . 2i Lemma 2. The products computed in the output unit l1 can be split in two groups, one with products containing only variables x1 , . . . , x n and one containing only variables x n +1 , . . . , xn . 2 2 Proof. This is obvious since the last unit is a “sum“ unit that adds two terms whose inputs are these two groups of variables (see e.g. Fig. 1). 2i Lemma 3. The products computed in the output unit l1 involve more than one input variable. k Proof. It is straightforward to show by induction on k ≥ 1 that the products computed by lj all involve more than one input variable, thus it is true in particular for the output layer (k = 2i). Lemma 4. Any shallow sum-product network computing f ∈ F must have a “sum” unit as output. Proof. By contradiction, suppose the output unit of such a shallow sum-product network is multiplicative. This unit must have more than one input, because in the case that it has only one input, the output would be either a (weighted) sum of input variables (which would violate Lemma 3), or a single product of input variables (which would violate Proposition 1), depending on the type (sum or product) of the single input hidden unit. Thus the last unit must compute a product of two or more hidden units. It can be re-written as a product of two factors, where each factor corresponds to either one hidden unit, or a product of multiple hidden units (it does not matter here which speciﬁc factorization is chosen among all possible ones). Regardless of the type (sum or product) of the hidden units involved, those two factors can thus be written as weighted sums of products of variables xt (with positive weights, and input variables potentially raised to powers above one). From Lemma 1, both x1 and xn must be present in the ﬁnal output, and thus they must appear in at least one of these two factors. Without loss of generality, assume x1 appears in the ﬁrst factor. Variables x n +1 , . . . , xn then cannot be present in the second factor, since otherwise one product in the output 2 would contain both x1 and one of these variables (this product cannot cancel out since weights must be positive), violating Lemma 2. But with a similar reasoning, since as a result xn must appear in the ﬁrst factor, variables x1 , . . . , x n cannot be present in the second factor either. Consequently, no 2 input variable can be present in the second factor, leading to the desired contradiction. Lemma 5. Any shallow sum-product network computing f ∈ F must have only multiplicative units in its hidden layer. Proof. By contradiction, suppose there exists a “sum“ unit in the hidden layer, written s = t∈S αt xt with S the set of input indices appearing in this sum, and αt > 0 for all t ∈ S. Since according to Lemma 4 the output unit must also be a sum (and have positive weights according to Deﬁnition 1), then the ﬁnal output will also contain terms of the form βt xt for t ∈ S, with βt > 0. This violates Lemma 3, establishing the contradiction. Lemma 6. Any shallow negative-weight sum-product network (see Deﬁnition 2) computing f ∈ F √ must have at least 2 n−1 hidden units, if its output unit is a sum and its hidden units are products. Proof. Such a network computes a weighted sum of its hidden units, where each hidden unit is a γ product of input variables, i.e. its output can be written as Σj wj Πt xt jt with wj ∈ R and γjt ∈ {0, 1}. In order to compute a function in F, this shallow network thus needs a number of hidden units at least equal to the number of unique products in that function. From Proposition 1, this √ number is equal to 2 n−1 . √ Corollary 1. Any shallow sum-product network computing f ∈ F must have at least 2 units. n−1 hidden Proof. This is a direct corollary of Lemmas 4 (showing the output unit is a sum), 5 (showing that hidden units are products), and 6 (showing the desired result for any shallow network with this speciﬁc structure – regardless of the sign of weights). 5 3.3 Discussion Corollary 1 above shows that in order to compute some function in F with n inputs, the number of √ √ units in a shallow network has to be at least 2 n−1 , (i.e. grows exponentially in n). On another hand, the total number of units in the deep (for i > 1) network computing the same function, as described in Section 3.1, is equal to 1 + 2 + 4 + 8 + . . . + 22i−1 (since all units are binary), which is √ also equal to 22i − 1 = n − 1 (i.e. grows only quadratically in n). It shows that some deep sumproduct network with n inputs and depth O(log n) can represent with O(n) units what would √ require O(2 n ) units for a depth-2 network. Lemma 6 also shows a similar result regardless of the sign of the weights in the summation units of the depth-2 network, but assumes a speciﬁc architecture for this network (products in the hidden layer with a sum as output). 4 The family G In this section we present similar results with a different family of functions, denoted by G. Compared to F, one important difference of deep sum-product networks built to deﬁne functions in G is that they can vary their input size independently of their depth. Their analysis thus provides additional insight when comparing the representational efﬁciency of deep vs. shallow sum-product networks in the case of a ﬁxed dataset. 4.1 Deﬁnition Networks in family G also alternate sum and product layers, but their units have as inputs all units from the previous layer except one. More formally, deﬁne the family G = ∪n≥2,i≥0 Gin of functions represented by sum-product networks, where the sub-family Gin is made of all sum-product networks with n input variables and 2i + 2 layers (including the input layer ℓ0 ), such that: 1. ℓ1 contains summation units; further layers alternate multiplicative and summation units. 2. Summation units have positive weights. 3. All layers are of size n, except the last layer ℓ2i+1 that contains a single sum unit that sums all units in the previous layer ℓ2i . k−1 4. In each layer ℓk for 1 ≤ k ≤ 2i, each unit ℓk takes as inputs {ℓm |m = j}. j An example of a network belonging to G1,3 (i.e. with three layers and three input variables) is shown in Figure 2. ℓ3 = x2 + x2 + x2 + 3(x1x2 + x1x3 + x2x3) = g(x1, x2, x3) 3 2 1 1 + ℓ2 = x2 + x1x2 × 1 1 +x1x3 + x2x3 ℓ1 = x2 + x3 1 × ℓ2 = . . . 2 × ℓ2 = x2 + x1x2 3 3 +x1x3 + x2x3 + + ℓ1 = x1 + x3 2 + ℓ1 = x1 + x2 3 x1 x2 x3 Figure 2: Sum-product network computing a function of G1,3 (summation units’ weights are all 1’s). 4.2 Theoretical results The main result is stated in Proposition 3 below, establishing a lower bound on the number of hidden units of a shallow sum-product network computing g ∈ G. The proof sketch is as follows: 1. We show that the polynomial expansion of g must contain a large set of products (Proposition 2 and Corollary 2). 2. We use both the number of products in that set as well as their degree to establish the desired lower bound (Proposition 3). 6 We will also need the following lemma, which states that when n − 1 items each belong to n − 1 sets among a total of n sets, then we can associate to each item one of the sets it belongs to without using the same set for different items. Lemma 7. Let S1 , . . . , Sn be n sets (n ≥ 2) containing elements of {P1 , . . . , Pn−1 }, such that for any q, r, |{r|Pq ∈ Sr }| ≥ n − 1 (i.e. each element Pq belongs to at least n − 1 sets). Then there exist r1 , . . . , rn−1 different indices such that Pq ∈ Srq for 1 ≤ q ≤ n − 1. Proof. Omitted due to lack of space (very easy to prove by construction). Proposition 2. For any 0 ≤ j ≤ i, and any product of variables P = Πn xαt such that αt ∈ N and t=1 t j 2j whose computed value, when expanded as a weighted t αt = (n − 1) , there exists a unit in ℓ sum of products, contains P among these products. Proof. We prove this proposition by induction on j. First, for j = 0, this is obvious since any P of this form must be made of a single input variable xt , that appears in ℓ0 = xt . t Suppose now the proposition is true for some j < i. Consider a product P = Πn xαt such that t=1 t αt ∈ N and t αt = (n − 1)j+1 . P can be factored in n − 1 sub-products of degree (n − 1)j , β i.e. written P = P1 . . . Pn−1 with Pq = Πn xt qt , βqt ∈ N and t βqt = (n − 1)j for all q. By t=1 the induction hypothesis, each Pq can be found in at least one unit ℓ2j . As a result, by property 4 kq (in the deﬁnition of family G), each Pq will also appear in the additive layer ℓ2j+1 , in at least n − 1 different units (the only sum unit that may not contain Pq is the one that does not have ℓ2j as input). kq By Lemma 7, we can thus ﬁnd a set of units ℓ2j+1 such that for any 1 ≤ q ≤ n − 1, the product rq Pq appears in ℓ2j+1 , with indices rq being different from each other. Let 1 ≤ s ≤ n be such that rq 2(j+1) s = rq for all q. Then, from property 4 of family G, the multiplicative unit ℓs computes the n−1 2j+1 product Πq=1 ℓrq , and as a result, when expanded as a sum of products, it contains in particular P1 . . . Pn−1 = P . The proposition is thus true for j + 1, and by induction, is true for all j ≤ i. Corollary 2. The output gin of a sum-product network in Gin , when expanded as a sum of products, contains all products of variables of the form Πn xαt such that αt ∈ N and t αt = (n − 1)i . t=1 t Proof. Applying Proposition 2 with j = i, we obtain that all products of this form can be found in the multiplicative units of ℓ2i . Since the output unit ℓ2i+1 computes a sum of these multiplicative 1 units (weighted with positive weights), those products are also present in the output. Proposition 3. A shallow negative-weight sum-product network computing gin ∈ Gin must have at least (n − 1)i hidden units. Proof. First suppose the output unit of the shallow network is a sum. Then it may be able to compute gin , assuming we allow multiplicative units in the hidden layer in the hidden layer to use powers of their inputs in the product they compute (which we allow here for the proof to be more generic). However, it will require at least as many of these units as the number of unique products that can be found in the expansion of gin . In particular, from Corollary 2, it will require at least the number n of unique tuples of the form (α1 , . . . , αn ) such that αt ∈ N and t=1 αt = (n − 1)i . Denoting ni dni = (n − 1)i , this number is known to be equal to n+dni −1 , and it is easy to verify it is higher d than (or equal to) dni for any n ≥ 2 and i ≥ 0. Now suppose the output unit is multiplicative. Then there can be no multiplicative hidden unit, otherwise it would mean one could factor some input variable xt in the computed function output: this is not possible since by Corollary 2, for any variable xt there exist products in the output function that do not involve xt . So all hidden units must be additive, and since the computed function contains products of degree dni , there must be at least dni such hidden units. 7 4.3 Discussion Proposition 3 shows that in order to compute the same function as gin ∈ Gin , the number of units in the shallow network has to grow exponentially in i, i.e. in the network’s depth (while the deep network’s size grows linearly in i). The shallow network also needs to grow polynomially in the number of input variables n (with a degree equal to i), while the deep network grows only linearly in n. It means that some deep sum-product network with n inputs and depth O(i) can represent with O(ni) units what would require O((n − 1)i ) units for a depth-2 network. Note that in the similar results found for family F, the depth-2 network computing the same function as a function in F had to be constrained to either have a speciﬁc combination of sum and hidden units (in Lemma 6) or to have non-negative weights (in Corollary 1). On the contrary, the result presented here for family G holds without requiring any of these assumptions. 5 Conclusion We compared a deep sum-product network and a shallow sum-product network representing the same function, taken from two families of functions F and G. For both families, we have shown that the number of units in the shallow network has to grow exponentially, compared to a linear growth in the deep network, so as to represent the same functions. The deep version thus offers a much more compact representation of the same functions. This work focuses on two speciﬁc families of functions: ﬁnding more general parameterization of functions leading to similar results would be an interesting topic for future research. Another open question is whether it is possible to represent such functions only approximately (e.g. up to an error bound ǫ) with a much smaller shallow network. Results by Braverman [8] on boolean circuits suggest that similar results as those presented in this paper may still hold, but this topic has yet to be formally investigated in the context of sum-product networks. A related problem is also to look into functions deﬁned only on discrete input variables: our proofs do not trivially extend to this situation because we cannot assume anymore that two polynomials yielding the same output values must have the same expansion coefﬁcients (since the number of input combinations becomes ﬁnite). 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