The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, drawback of the known approaches is that rank parameter has to fixed priori. In this paper, we consider set bounded-rank matrices. We propose rank-adaptive method, which consists optimization, increase step and reduction step. explore its performance applied problem. Numerical expe...

A rank-adaptive integrator for the dynamical low-rank approximation of matrix and tensor differential equations is presented. The fixed-rank recently proposed by two authors extended to allow an adaptive choice rank, using subspaces that are generated itself. first updates evolving bases then does a Galerkin step in subspace both new old bases, which followed rank truncation given tolerance. It...

Various applications in signal processing and machine learning give rise to highly structured spectral optimization problems characterized by low-rank solutions. Two important examples that motivate this work are optimization problems from phase retrieval and from blind deconvolution, which are designed to yield rank-1 solutions. An algorithm is described based on solving a certain constrained ...

We study the frequent problem of approximating a target matrix with a matrix of lower rank. We provide a simple and efficient (EM) algorithm for solving weighted low rank approximation problems, which, unlike simple matrix factorization problems, do not admit a closed form solution in general. We analyze, in addition, the nature of locally optimal solutions that arise in this context, demonstra...

We consider in this paper the multivariate regression problem, when the target regression matrix A is close to a low rank matrix. Our primary interest is in on the practical case where the variance of the noise is unknown. Our main contribution is to propose in this setting a criterion to select among a family of low rank estimators and prove a non-asymptotic oracle inequality for the resulting...

Approximating a tensor by another of lower rank is in general an ill posed problem. Yet, this kind of approximation is mandatory in the presence of measurement errors or noise. We show how tools recently developed in compressed sensing can be used to solve this problem. More precisely, a minimal angle between the columns of loading matrices allows to restore both existence and uniqueness of the...

Abstract. This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected ...

Many problems in computer vision and recommender systems involve low-rank matrices. In this work, we study the problem of finding the maximum entry of a stochastic low-rank matrix from sequential observations. At each step, a learning agent chooses pairs of row and column arms, and receives the noisy product of their latent values as a reward. The main challenge is that the latent values are un...