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Matrix Completion with Deterministic Pattern: a Geometric Perspective Alexander Shapiro, Yao Xie, Rui Zhang

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Matrix Completion with Deterministic Pattern: a Geometric Perspective Alexander Shapiro, Yao Xie, Rui Zhang 1 Matrix completion with deterministic pattern: A geometric perspective Alexander Shapiro, Yao Xie, Rui Zhang Abstract—We consider the matrix completion problem with problem: the uniqueness conditions for minimum rank matrix a deterministic pattern of observed entries. In this setting, we recovery with random linear measurements of the true matrix; aim to answer the question: under what condition there will here the linear measurements correspond to inner product of be (at least locally) unique solution to the matrix completion problem, i.e., the underlying true matrix is identifiable. We a measurement mask matrix with the true matrix, and hence, answer the question from a certain point of view and outline the observations are different from that in matrix completion). a geometric perspective. We give an algebraically verifiable With a deterministic pattern of observed entries, a complete sufficient condition, which we call the well-posedness condition, characterization of the identifiable matrix for matrix com- for the local uniqueness of MRMC solutions. We argue that this pletion remains an important yet open question: under what condition is necessary for local stability of MRMC solutions, and we show that the condition is generic using the characteristic conditions for the pattern, there will be (at least locally) unique rank. We also argue that the low-rank approximation approaches solution? Recent work [8] provides insights into this problem are more stable than MRMC and further propose a sequential by studying the so-called completable problems and estab- statistical testing procedure to determine the “true” rank from lishing conditions ensuring the existence of at most finitely observed entries. Finally, we provide numerical examples aimed many rank-r matrices that agree with all its observed entries. at verifying validity of the presented theory. A related work [9] studied this problem when there is a sparse noise that corrupts the entries. The rank estimation problem I. INTRODUCTION has been discussed in [10], [11], and related tensor completion Matrix completion (e.g., [1]–[3]) is a fundamental problem problem in [12]: the goal in these works are different though; in signal processing and machine learning, which studies the they aim to find upper and lower bound for the true rank, recovery of a low-rank matrix from an observation of a subset whereas our rank selection test in SectionIV determines the of its entries. It has attracted a lot attention from researchers most plausible rank from a statistical point of view. and practitioners and there are various motivating real-world In this paper, we aim to answer the question from a applications including recommender systems and the Netflix somewhat different point of view and to give a geometric challenge (see a recent overview in [4]). A popular approach perspective. In particular, we consider the solution of the for matrix completion is to find a matrix of minimal rank sat- Minimum Rank Matrix Completion (MRMC) formulation, isfying the observation constraints. Due to the non-convexity which leads to a non-convex optimization problem. We address of the rank function, popular approaches are convex relaxation the following questions: (i) Given observed entries arranged (see, e.g., [5]) and nuclear norm minimization. There is a rich according to a (deterministic) pattern, by solving the MRMC literature, both in establishing performance bounds, developing problem, what is the minimum achievable rank? (ii) Under efficient algorithms and providing performance guarantees. what conditions, there will be a unique matrix that is a solution Recently there has also been new various results for non- to the MRMC problem? We give a sufficient condition (which convex formulations of matrix completion problem (see, e.g., we call the well-posedness condition) for the local uniqueness of MRMC solutions, and illustrate how such condition can be arXiv:1802.00047v4 [cs.LG] 29 Aug 2018 [6]). Existing conditions ensuring recovery of the minimal rank verified. We also show that such well-posedness condition is matrix are usually formulated in terms of missing-at-random generic using the concept of characteristic rank. In addition, entries and under an assumption of the so-called bounded- we also consider the convex relaxation and nuclear norm coherence (see a survey for other approaches in [4]; we do not minimization formulations. Based on our theoretical results, we argue that given m aim to give a complete overview of the vast literature). These ∗ observations of an n1 × n2 matrix, if the minimal rank r is results are typically aimed at establishing the recovery with 2 1=2 a high probability. In addition, there has been much work on less than R(n1; n2; m) := (n1+n2)=2−[(n1+n2) =4−m] , low-rank matrix recovery (see, e.g.,[7], which studies a related then the corresponding solution is unstable in the sense that an arbitrary small perturbation of the observed values can Alexander Shapiro (e-mail: [email protected]), Yao make this rank unattainable. On the other hand if r∗ > Xie (e-mail: [email protected]) and Rui Zhang (e-mail: R(n1; n2; m), then almost surely the solution is not (even [email protected]) are with the H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, locally) unique (cf., [13]). This indicates that except on rare GA. occasions, the MRMC problem cannot have both properties Research of Alexander Shapiro was partly supported by NSF grant 1633196 of possessing unique and stable solutions. Consequently, what and DARPA EQUiPS program, grant SNL 014150709. Research of Yao Xie was partially supported by NSF grants CCF-1442635, CMMI-1538746, an makes sense is to try to solve the minimum rank problem NSF CAREER Award CCF-1650913, and a S.F. Express award. approximately and hence to consider low-rank approximation 2 approaches (such as an approach mentioned in [4], [14]) as a A. Definitions better alternative to the MRMC formulation. Lt us introduce some necessary definitions. Denote by M We also propose a sequential statistical testing procedure the n1 × n2 matrix with the specified entries Mij, (i; j) 2 Ω, c to determine the ‘true’ rank from noisy observed entries. and all other entries equal zero. Consider Ω := f1; :::; n1g × Such statistical approach can be useful for many existing f1; :::; n2g n Ω, the complement of the index set Ω, and define low-rank matrix completion algorithms, which require a pre- n1×n2 c specification of the matrix rank, such as the alternating min- VΩ := Y 2 R : Yij = 0; (i; j) 2 Ω : imization approach to solving the non-convex problem by This linear space represents the set of matrices that are filled representing the low-rank matrix as a product of two low-rank with zeros at the locations of the unobserved entries. Similarly matrix factors (see, e.g., [4], [15], [16]). define The paper is organized as follows. In the next section, we n1×n2 introduce the considered setting and some basic definitions. VΩc := Y 2 R : Yij = 0; (i; j) 2 Ω : In SectionII we present the problem set-up, including the By PΩ we denote the projection onto the space VΩ, i.e., MRMC, LRMA, and convex relaxation formulations. Section [PΩ(Y )]ij = Yij for (i; j) 2 Ω and [PΩ(Y )]ij = 0 for III contains the main theoretical results. A statistical test of c (i; j) 2 Ω . By this construction, fM + X : X 2 VΩc g rank is presented in SectionIV. In SectionV we present is the affine space of all matrices that satisfy the observation numerical results related to the developed theory. Finally constraints. Note that M 2 VΩ and the dimension of the linear SectionVI concludes the paper. All proofs are transferred to space VΩ is dim(VΩ) = m, while dim(VΩc ) = n1n2 − m. the Appendix. We say that a property holds for almost every (a.e.) Mij, We use conventional notations. For a 2 R we denote by or almost surely, if the set of matrices Y 2 VΩ for which dae the least integer that is greater than or equal to a. By this property does not hold has Lebesgue measure zero in the A ⊗ B we denote the Kronecker product of matrices (vectors) space VΩ. A and B, and by vec(A) column vector obtained by stacking columns of matrix A. We use the following matrix identity for B. Minimum Rank Matrix Completion (MRMC) matrices A; B; C of appropriate order Since the true rank is unknown, a natural approach is to find the minimum rank matrix that is consistent with vec(ABC) = (C> ⊗ A)vec(B): (1) the observations. This goal can be written as the following By Sp we denote the linear space of p×p symmetric matrices optimization problem referred to as the Minimum Rank Matrix and by writing X 0 we mean that matrix X 2 Sp is positive Completion (MRMC), semidefinite. By σi(Y ) we denote the i-th largest singular n ×n min rank(Y ) subject to Yij = Mij; (i; j) 2 Ω: (2) 1 2 n ×n value of matrix Y 2 R . By Ip we denote the identity Y 2R 1 2 matrix of dimension p. In general, the rank minimization problem is non-convex and NP-hard to solve. However, this problem is fundamental to various efficient heuristics derived from here. Largely, there II. MATRIX COMPLETION AND PROBLEM SET-UP are two categories of approximation heuristics: (i) approximate the rank function with some surrogate function such as the Consider the problem of recovering an n1 × n2 data matrix nuclear norm function, (ii) or solve a sequence of rank- of low rank when observing a small number m of its entries, constrained problems such as the matrix factorization based which are denoted as Mij, (i; j) 2 Ω.
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