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211 nips-2011-Penalty Decomposition Methods for Rank Minimization

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Author: Yong Zhang, Zhaosong Lu

Abstract: In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We ﬁrst show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this result, we then propose penalty decomposition methods for general rank minimization problems. The convergence results of the PD methods have been shown in the longer version of the paper [19]. Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. The computational results demonstrate that our methods generally outperform the existing methods in terms of solution quality and/or speed. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Penalty Decomposition Methods for Rank Minimization ∗ Zhaosong Lu † Yong Zhang ‡ Abstract In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. [sent-1, score-0.796]

2 We ﬁrst show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. [sent-2, score-0.276]

3 As a consequence, we establish that a class of rank minimization problems have closed form solutions. [sent-3, score-0.605]

4 Using this result, we then propose penalty decomposition methods for general rank minimization problems. [sent-4, score-0.762]

5 Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. [sent-6, score-0.52]

6 Recently, some image recovery and machine learning problems are formulated as (1) or (2) (see, for example, [27, 31]). [sent-13, score-0.17]

7 In addition, the problem of ﬁnding nearest low-rank correlation matrix is in the form of (1), which has important application in ﬁnance (see, for example, [4, 29, 36, 38, 25, 30, 12]). [sent-14, score-0.281]

8 , MAXCUT), one novel method is to ﬁrst solve the semideﬁnite programming (SDP) relaxation of (1) and then obtain an approximate solution of (1) by applying some heuristics to the solution of the SDP (see, for example, [11]). [sent-18, score-0.28]

9 In addition, the nuclear norm relaxation approach has been proposed for problems (1) or (2). [sent-20, score-0.374]

10 In their approach, a convex relaxation is applied to (1) or (2) by replacing the rank of X by the nuclear norm of X and numerous efﬁcient methods can then be applied to solve the resulting convex problems. [sent-30, score-0.751]

11 Additionally, for some application problems, the nuclear norm stays constant in feasible region. [sent-34, score-0.293]

12 For example, as for nearest low-rank correlation matrix problem (see Subsection 3. [sent-35, score-0.281]

13 2), any feasible point is a symmetric positive semideﬁnite matrix with all diagonal entries equal to one. [sent-36, score-0.133]

14 For those problems, nuclear norm relaxation approach is obviously inappropriate. [sent-37, score-0.322]

15 In this approach, problem (1) is cast into an NLP problem by replacing the constraint rank(X) ≤ r by X = U V where U ∈ m×r and V ∈ r×n , and then numerous optimization methods can be applied to solve the resulting NLP. [sent-39, score-0.155]

16 In this paper we consider general rank minimization problems (1) and (2). [sent-42, score-0.484]

17 We ﬁrst show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. [sent-43, score-0.276]

18 As a consequence, we establish that a class of rank minimization problems have closed form solutions. [sent-44, score-0.605]

19 Using this result, we then propose penalty decomposition methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. [sent-45, score-1.002]

20 Finally, we test the performance of our methods by applying them to matrix completion and nearest low-rank correlation matrix problems. [sent-47, score-0.52]

21 In Section 2, we ﬁrst establish some technical results on a class of rank minimization problems and then use them to develop the penalty decomposition methods for solving problems (1) and (2). [sent-52, score-1.037]

22 In Section 3, we conduct numerical experiments to test the performance of our penalty decomposition methods for solving matrix completion and nearest low-rank correlation matrix problems. [sent-53, score-1.005]

23 The Frobenius norm of a real matrix X is deﬁned as X F := Tr(XX T ) where Tr(·) denotes the trace of a matrix, and the nuclear norm of X, denoted by X ∗ , is deﬁned as the sum of all singular values of X. [sent-60, score-0.546]

24 For a real symmetric matrix X, λ(X) denotes the vector of all eigenvalues of X arranged in nondecreasing order and Λ(X) is the diagonal matrix whose ith diagonal entry is λi (X) for all i. [sent-63, score-0.244]

25 2 2 Penalty decomposition methods In this section, we ﬁrst establish some technical results on a class of rank minimization problems. [sent-69, score-0.697]

26 Then we propose penalty decomposition (PD) methods for solving problems (1) and (2) by using these technical results. [sent-70, score-0.507]

27 1 Technical results on special rank minimization In this subsection we ﬁrst show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. [sent-72, score-0.837]

28 As a consequence, we establish a result that a class of rank minimization problems have closed form solutions, which will be used to develop penalty decomposition methods in Subsection 2. [sent-73, score-0.935]

29 A norm · is a unitarily invariant norm on m×n if U XV = X for all U ∈ U m , V ∈ U n , X ∈ n×n . [sent-78, score-0.557]

30 More generally, a function F : m×n → is a unitarily invariant function if F (U XV ) = F (X) for all U ∈ U m , V ∈ U n , X ∈ m×n . [sent-79, score-0.385]

31 A set X ⊆ m×n is a unitarily invariant set if {U XV : U ∈ U m , V ∈ U n , X ∈ X } = X . [sent-80, score-0.385]

32 Similarly, a function F : S n → is a unitary similarity invariant function if F (U XU T ) = F (X) for all U ∈ U n , X ∈ S n . [sent-81, score-0.173]

33 A set X ⊆ S n is a unitary similarity invariant set if {U XU T : U ∈ U n , X ∈ X } = X . [sent-82, score-0.173]

34 The following result establishes that a class of matrix optimization problems over a subset of can be solved as lower dimensional vector optimization problems. [sent-83, score-0.276]

35 1 Let · be a unitarily invariant norm on m×n , and let F : m×n → be a unitarily invariant function. [sent-85, score-0.856]

36 Suppose that X ⊆ m×n is a unitarily invariant set. [sent-86, score-0.385]

37 Suppose that U Σ(A)V T is the singular value decomposition of A. [sent-88, score-0.236]

38 Then, X ∗ = U D(x∗ )V T is an optimal solution of the problem min F (X) + φ( X − A ) (3) s. [sent-89, score-0.137]

39 X ∈ X , ∗ q where x ∈ is an optimal solution of the problem min F (D(x)) + φ( D(x) − Σ(A) ) s. [sent-91, score-0.137]

40 1, we next state that a class of rank minimization problems on a subset of m×n can be solved as lower dimensional vector minimization problems. [sent-95, score-0.662]

41 Suppose that X ⊆ m×n is a unitarily invariant set, and U Σ(A)V T is the singular value decomposition of A. [sent-98, score-0.621]

42 Suppose that X ⊆ m×n is a unitarily invariant set, and U Σ(A)V T is the singular value decomposition of A. [sent-102, score-0.621]

43 When X is simple enough, problems (5) and (7) have closed form solutions. [sent-105, score-0.127]

44 In this case, it is not hard to observe that problems (6) and (8) have closed form solutions (see [20]). [sent-108, score-0.127]

45 3 that problems (5) and (7) also have closed form solutions. [sent-111, score-0.127]

46 The following results are heavily used in [6, 22, 34] for developing algorithms for solving the nuclear norm relaxation of matrix completion problems. [sent-112, score-0.615]

47 Suppose that U Σ(A)V T is the singular value decomposition of A. [sent-117, score-0.236]

48 Then, X ∗ = U D(x∗ )V T is an optimal solution of the problem 1 min ν X ∗ + X − A 2 , F 2 ∗ q where x ∈ is an optimal solution of the problem 1 min ν x 1 + x − σ(A) 2 . [sent-118, score-0.274]

49 Suppose that U Σ(A)V T is the singular value decomposition of A. [sent-121, score-0.236]

50 Clearly, the above results can be generalized to solve a class of matrix optimization problems over a subset of S n . [sent-123, score-0.24]

51 2 Penalty decomposition methods for solving (1) and (2) In this subsection, we consider the rank minimization problems (1) and (2). [sent-126, score-0.686]

52 In particular, we ﬁrst propose a penalty decomposition (PD) method for solving problem (1), and then extend it to solve problem (2) at end of this subsection. [sent-127, score-0.48]

53 Assumption 1 Problems (1) and (2) are feasible, and moreover, at least a feasible solution, denoted by X feas , is known. [sent-129, score-0.21]

54 Given a penalty parameter > 0, the associated quadratic penalty function for (9) is deﬁned as Q (X, Y ) := f (X) + 2 X −Y 2 F. [sent-131, score-0.364]

55 (10) We now propose a PD method for solving problem (9) (or, equivalently, (1)) in which each penalty subproblem is approximately solved by a block coordinate descent (BCD) method. [sent-132, score-0.419]

56 Penalty decomposition method for (9) (asymmetric matrices): Let 0 > 0, σ > 1 be given. [sent-133, score-0.184]

57 Choose an arbitrary Y00 ∈ Y and a constant Υ ≥ max{f (X feas ), minX∈X Q 0 (X, Y00 )}. [sent-134, score-0.135]

58 1) Set l = 0 and apply the BCD method to ﬁnd an approximate solution (X k , Y k ) ∈ X × Y for the penalty subproblem min{Q k (X, Y ) : X ∈ X , Y ∈ Y} (11) by performing steps 1a)-1d): 4 k 1a) Solve Xl+1 ∈ Arg min Q k (X, Ylk ). [sent-136, score-0.444]

59 3) If min Q X∈X k+1 (X, Y k ) > Υ, set Y0k+1 := X feas . [sent-140, score-0.216]

60 (12) max(|Q k (Xlk , Ylk )|, 1) Moreover, we can terminate the outer iterations of the above method once k k max |Xij − Yij | ≤ O (13) ij for some O > 0. [sent-147, score-0.164]

61 To enhance the quality of approximate solutions, one may execute the BCD method multiple times starting from a suitable perturbation of the current approximate solution. [sent-149, score-0.149]

62 In detail, at the kth outer iteration, let (X k , Y k ) be a current approximate solution of (11) obtained by the BCD method, and let rk = rank(Y k ). [sent-150, score-0.155]

63 Before starting the (k + 1)th outer iteration, one can apply the BCD method again starting from Y0k ∈ Arg min{ Y − Y k F : rank(Y ) ≤ rk − 1} (namely, a rank-one perturbation of Y k ) and obtain a new approximate ˜ ˜ ˜ ˜ solution (X k , Y k ) of (11). [sent-152, score-0.234]

64 Otherwise, one can terminate the kth outer iteration and start the next outer iteration. [sent-154, score-0.195]

65 3, the subproblem in step 1b) can be reduced to the problem in form of (8), which has closed form solution when Ω is simple enough. [sent-156, score-0.22]

66 Under some suitable assumptions, we have established that any accumulation point of the sequence generated by our method when applied to problem (1) is a stationary point of a nonlinear reformulation of the problem. [sent-158, score-0.209]

67 (14) X,Y Given a penalty parameter > 0, the associated quadratic penalty function for (14) is deﬁned as P (X, Y ) := f (X) + ν rank(Y ) + 2 X −Y 2 F. [sent-161, score-0.364]

68 (15) Then we can easily adapt the PD method for solving (9) to solve (14) (or, equivalently, (2)) by setting the constant Υ ≥ max{f (X feas ) + ν rank(X feas ), minX∈X P 0 (X, Y00 )}. [sent-162, score-0.42]

69 2, the BCD subproblem in step 1b) when applied to minimize the penalty function (15) can be reduced to the problem in form of (6), which has closed form solution when Ω is simple enough. [sent-165, score-0.402]

70 To enhance the quality of approximate solutions, one may apply a similar strategy as described for the previous PD method by executing the BCD method multiple times starting from a suitable perturbation of the current approximate solution. [sent-168, score-0.185]

71 5 3 Numerical results In this section, we conduct numerical experiments to test the performance of our penalty decomposition (PD) methods proposed in Section 2 by applying them to solve matrix completion and nearest low-rank correlation matrix problems. [sent-173, score-1.011]

72 1 Matrix completion problem In this subsection, we apply our PD method proposed in Section 2 to the matrix completion problem, which has numerous applications in control and systems theory, image recovery and data mining (see, for example, [33, 24, 9, 16]). [sent-180, score-0.568]

73 Recently, numerous methods were proposed to solve the nuclear norm relaxation or the variant of (16) (see, for example, [18, 6, 22, 8, 13, 14, 21, 23, 32, 17, 37, 35]). [sent-184, score-0.439]

74 It is not hard to see that problem (16) is a special case of the general rank minimization problem (2) with f (X) ≡ 0, ν = 1, Ω = m×n , and X = {X ∈ m×n : Xij = Mij , (i, j) ∈ Θ}. [sent-185, score-0.432]

75 Next we conduct numerical experiments to test the performance of our PD method for solving matrix completion problem (16) on real data. [sent-189, score-0.43]

76 In our experiment, we aim to test the performance of our PD method for solving a grayscale image inpainting problem [2]. [sent-190, score-0.278]

77 For an image inpainting problem, our goal is to ﬁll the missing pixel values of the image at given pixel locations. [sent-192, score-0.301]

78 As shown in [33, 24], this problem can be solved as a matrix completion problem if the image is of low-rank. [sent-194, score-0.384]

79 In our test, the original 512 × 512 grayscale image is shown in Figure 1(a). [sent-195, score-0.123]

80 To obtain the data for problem (16), we ﬁrst apply the singular value decomposition to the original image and truncate the resulting decomposition to get an image of rank 40 shown in Figure 1(e). [sent-196, score-0.87]

81 We now apply our PD method to solve problem (16) with the data given in Figures 1(b), 1(c) and 1(d), and the resulting recovered images are presented in Figures 1(f), 1(g) and 1(h), respectively. [sent-199, score-0.149]

82 2 Nearest low-rank correlation matrix problem In this subsection, we apply our PD method proposed in Section 2 to ﬁnd the nearest low-rank correlation matrix, which has important applications in ﬁnance (see, for example, [4, 29, 36, 38, 30]). [sent-206, score-0.42]

83 diag(X) = e, rank(X) ≤ r, X 0 n for some correlation matrix C ∈ S+ and some integer r ∈ [1, n], where diag(X) denotes the vector consisting of the diagonal entries of X and e is the all-ones vector. [sent-209, score-0.193]

84 6 (a) original image (b) 50% masked original (c) 50% masked rank 40 (d) 6. [sent-211, score-0.541]

85 Next we conduct numerical experiments to test the performance of our method for solving (17) on three classes of benchmark testing problems. [sent-216, score-0.191]

86 Then we apply our PD method and the method named as Major developed in [25] to solve problem (17) on the instances generated above. [sent-226, score-0.132]

87 4 Concluding remarks In this paper we proposed penalty decomposition (PD) methods for general rank minimization problems in which each subproblem is solved by a block coordinate descend method. [sent-238, score-1.002]

88 In the longer version of the paper [20], we have showed that under some suitable assumptions any accumulation point of the sequence generated by our method when applied to the rank constrained minimization problem is a stationary point of a nonlinear reformulation of the problem. [sent-239, score-0.684]

89 A singular value thresholding algorithm for matrix come pletion. [sent-341, score-0.178]

90 A rank minimization heuristic with application to minimum order system approximation. [sent-366, score-0.432]

91 A sequential semismooth Newton method for the nearest low-rank correlation matrix problem. [sent-403, score-0.317]

92 Interior-point method for nuclear norm approximation with application to system identiﬁcation. [sent-408, score-0.286]

93 An implementable proximal point algorithmic framework for nuclear norm minimization. [sent-418, score-0.25]

94 Fixed point and Bregman iterative methods for matrix rank minimization. [sent-450, score-0.402]

95 A sequential factorization method for recovering shape and motion from image streams. [sent-464, score-0.16]

96 Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. [sent-487, score-0.34]

97 On the simultaneous calibration of multifactor lognormal interest rate models to Black volatilities and to the correlation matrix. [sent-492, score-0.142]

98 An accelerated proximal gradient algorithm for nuclear norm regularized least squares problems. [sent-520, score-0.25]

99 Shape and motion from image streams under orthography: a factorization method. [sent-528, score-0.124]

100 Solving a low-rank factorization model for matrix completion by a nonlinear successive over-relaxation algorithm. [sent-543, score-0.276]

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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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