Related papers: Generalized rank-constrained matrix approximations
The importance of accurate recommender systems has been widely recognized by academia and industry. However, the recommendation quality is still rather low. Recently, a linear sparse and low-rank representation of the user-item matrix has…
We provide new approximation guarantees for greedy low rank matrix estimation under standard assumptions of restricted strong convexity and smoothness. Our novel analysis also uncovers previously unknown connections between the low rank…
We consider the problem of approximating an affinely structured matrix, for example a Hankel matrix, by a low-rank matrix with the same structure. This problem occurs in system identification, signal processing and computer algebra, among…
The low-rank matrix completion problem can be solved by Riemannian optimization on a fixed-rank manifold. However, a drawback of the known approaches is that the rank parameter has to be fixed a priori. In this paper, we consider the…
In this paper we develop algorithms for approximating matrix multiplication with respect to the spectral norm. Let A\in{\RR^{n\times m}} and B\in\RR^{n \times p} be two matrices and \eps>0. We approximate the product A^\top B using two…
The application of generalized inverses is usually neglected in pure mathematical research. However, it is very effective for this paper. We expand the famous matrix rank theorem due to R. Penrose to operators between Banach paces.…
This paper extends the framework of randomised matrix multiplication to a coarser partition and proposes an algorithm as a complement to the classical algorithm, especially when the optimal probability distribution of the latter one is…
Given an input matrix polynomial whose coefficients are floating point numbers, we consider the problem of finding the nearest matrix polynomial which has rank at most a specified value. This generalizes the problem of finding a nearest…
The low-rank matrix approximation problem with respect to the entry-wise $\ell_{\infty}$-norm is the following: given a matrix $M$ and a factorization rank $r$, find a matrix $X$ whose rank is at most $r$ and that minimizes $\max_{i,j}…
The rank minimization problem is to find the lowest-rank matrix in a given set. Nuclear norm minimization has been proposed as an convex relaxation of rank minimization. Recht, Fazel, and Parrilo have shown that nuclear norm minimization…
We address the problem of finding the nearest graph Laplacian to a given matrix, with the distance measured using the Frobenius norm. Specifically, for the directed graph Laplacian, we propose two novel algorithms by reformulating the…
We study the problem of low-rank matrix completion for symmetric matrices. The minimum rank of a completion of a generic partially specified symmetric matrix depends only on the location of the specified entries, and not their values, if…
Motivated in part by a problem of combinatorial optimization and in part by analogies with quantum computations, we consider approximations of orthogonal matrices U by ``non-commutative convex combinations'' A of permutation matrices of the…
$\newcommand{\MatA}{\mathcal{M}}$ $\newcommand{\eps}{\varepsilon}$ $\newcommand{\NSize}{\mathsf{N}{}}$ $\newcommand{\MatB}{\mathcal{B}}$ $\newcommand{\Fnorm}[1]{\left\| {#1} \right\|_F}$ $\newcommand{\PrcOpt}[2]{\mu_{\mathrm{opt}}\pth{#1,…
Low rank approximation of matrices has been well studied in literature. Singular value decomposition, QR decomposition with column pivoting, rank revealing QR factorization (RRQR), Interpolative decomposition etc are classical deterministic…
Several recent randomized linear algebra algorithms rely upon fast dimension reduction methods. A popular choice is the Subsampled Randomized Hadamard Transform (SRHT). In this article, we address the efficacy, in the Frobenius and spectral…
The low-rank matrix approximation problems within a threshold are widely applied in information retrieval, image processing, background estimation of the video sequence problems and so on. This paper presents an adaptive randomized…
Low-rank approximation of a matrix by means of structured random sampling has been consistently efficient in its extensive empirical studies around the globe, but adequate formal support for this empirical phenomenon has been missing so…
We address the problem of minimizing a convex function over the space of large matrices with low rank. While this optimization problem is hard in general, we propose an efficient greedy algorithm and derive its formal approximation…
Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the…