Rank-one matrix completion is solved by the sum-of-squares relaxation of order two

Abstract

Abstract-This note studies the problem of nonsymmetric rank-one matrix completion. We show that in every instance where the problem has a unique solution, one can recover the original matrix through the second round of the sum-ofsquares/Lasserre hierarchy with minimization of the trace of the moments matrix. Our proof system is based on iteratively building a sum of N − 1 linearly independent squares, where N is the number of monomials of degree at most two, corresponding to the canonical basis (z α − z α 0 ) 2 . Those squares are constructed from the ideal I generated by the constraints and the monomials provided by the minimization of the trace. I. NON SYMMETRIC MATRIX COMPLETIONThis paper introduces a deterministic recovery result for non symmetric rank-1 matrix completion by using the Lasserre hierarchy of semidefinite programming relaxations. To our knowledge, the closest result in the current literature is [1] where the authors use the same hierarchy and certify recovery in the case of tensor decomposition. Our paper also shares its deterministic nature with [2] where the authors derive recovery from the spectral properties of a graph Laplacian. Finally, this paper can also be related to [3] in which the authors study noisy tensor completion and use the sixth round of the Lasserre hierarchy to derive probabilistic recovery guarantees.We will use M(r; m × n) to denote the set of matrices of rank r. This set is an algebraic determinantal variety that can be completely characterized through the vanishing of the (r + 1)-minors. This determinantal variety has dimension (m+n−r)r.The general nonsymmetric rank-1 matrix completion problem consists in recovering an unknown matrix X ∈ M(1; m× n) such that X = xy T , given a fixed subset of its entries [4], find X subject to rank(X) = 1As a slight abuse, we also speak of constraints X ij = A ij as belonging to Ω. In relation to problem (1), we introduce the mapping R Ω : R m×n → R |Ω| that corresponds to extracting the observed entries of the matrix. We let R 1 Ω denote the restriction of R Ω to matrices of rank-1, i.e R 1 Ω : M(1; m × n) → R |Ω| . Invertibility of this restriction R 1 Ω is a natural question. In other words, when can one uniquely recover the matrix X from the knowledge of R Ω (X) and the fact that X has rank 1 ?In particular, this paper considers the completion problem on M * (1, m×n), where M * (1, m×n) denotes the restriction of M(1; m × n) to matrices for which none of the entries are zero. The reason for this is that if a rank-1 matrix has a zero element, then the corresponding row or column will be zero, and it is easy to see that the completion problem will generically lack injectivity.Respectively denote by V 1 , V 2 the row and column indices of X. We consider the bipartite graph G(V 1 , V 2 , E) associated to problem (1), where the set of edges in the graph is defined by (i, j) ∈ E iff (i, j) ∈ Ω. The conditions for the recovery of the matrix X from the set Ω are related to the properties of this bipartite graph. In particular, we have the f...

Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

“…Along that line, [15,19] solves the symmetric rank one completion problem when the diagonal entries are given. The noiseless result of this paper was presented in the introductory note [13].…”

Stable rank one matrix completion is solved by two rounds of semidefinite programming relaxation

“…The proof system in this paper relies on spectral graph theory, and use the fact that the eigenvector of the exact solution X 0 is also an eigenvector of the data weighted graph Laplacian, to derive a bound on the recovery. Finally, the noiseless result of this paper was presented in the introductory note [19].…”

From Blind deconvolution to Blind Super-Resolution through convex programming

“…The probabilistic method reveals an elegant tool to derive recovery guarantees yet it does not make use of the particular structure of the problem, and as a consequence, is unable to explain the efficiency of nuclear norm minimization for that particular problem structure. Blind deconvolution however seems a natural candidate for a better understanding of the propagation of information in semidefinite relaxations such as discussed in [18], in the framework of matrix completion.…”