Related papers: Matrix Completion in Almost-Verification Time
In this paper we study the matrix completion problem: Suppose $X \in {\mathbb R}^{n_r \times n_c}$ is unknown except for a known upper bound $r$ on its rank. By measuring a small number $m \ll n_r n_c$ of elements of $X$, is it possible to…
We prove, using the subspace embedding guarantee in a black box way, that one can achieve the spectral norm guarantee for approximate matrix multiplication with a dimensionality-reducing map having $m = O(\tilde{r}/\varepsilon^2)$ rows.…
The problem of approximating a matrix by a low-rank one has been extensively studied. This problem assumes, however, that the whole matrix has a low-rank structure. This assumption is often false for real-world matrices. We consider the…
We consider the problem of matrix column subset selection, which selects a subset of columns from an input matrix such that the input can be well approximated by the span of the selected columns. Column subset selection has been applied to…
In many autonomous mapping tasks, the maps cannot be accurately constructed due to various reasons such as sparse, noisy, and partial sensor measurements. We propose a novel map prediction method built upon the recent success of Low-Rank…
In this paper, we introduce a novel and robust approach to Quantized Matrix Completion (QMC). First, we propose a rank minimization problem with constraints induced by quantization bounds. Next, we form an unconstrained optimization problem…
We propose an efficient matrix rank reduction method for non-negative matrices, whose time complexity is quadratic in the number of rows or columns of a matrix. Our key insight is to formulate rank reduction as a mean-field approximation by…
We present a novel method for matrix completion, specifically designed for matrices where one dimension is significantly larger than the other. Our Columns Selected Matrix Completion (CSMC) method combines Column Subset Selection with…
We study three fundamental problems of Linear Algebra, lying in the heart of various Machine Learning applications, namely: 1)"Low-rank Column-based Matrix Approximation". We are given a matrix A and a target rank k. The goal is to select a…
Randomized SVD has become an extremely successful approach for efficiently computing a low-rank approximation of matrices. In particular the paper by Halko, Martinsson, and Tropp (SIREV 2011) contains extensive analysis, and has made it a…
We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with…
Matrix completion constantly receives tremendous attention from many research fields. It is commonly applied for recommender systems such as movie ratings, computer vision such as image reconstruction or completion, multi-task learning such…
Matrix approximation is a common tool in machine learning for building accurate prediction models for recommendation systems, text mining, and computer vision. A prevalent assumption in constructing matrix approximations is that the…
Matrix rank minimization problems are gaining a plenty of recent attention in both mathematical and engineering fields. This class of problems, arising in various and across-discipline applications, is known to be NP-hard in general. In…
We study the low rank approximation problem of any given matrix $A$ over $\mathbb{R}^{n\times m}$ and $\mathbb{C}^{n\times m}$ in entry-wise $\ell_p$ loss, that is, finding a rank-$k$ matrix $X$ such that $\|A-X\|_p$ is minimized. Unlike…
This paper studies the matrix completion problem under arbitrary sampling schemes. We propose a new estimator incorporating both max-norm and nuclear-norm regularization, based on which we can conduct efficient low-rank matrix recovery…
We consider the problem of exact low-rank matrix completion from a geometric viewpoint: given a partially filled matrix M, we keep the positions of specified and unspecified entries fixed, and study how the minimal completion rank depends…
We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices…
We consider the problem of recovering a low-rank matrix from its clipped observations. Clipping is conceivable in many scientific areas that obstructs statistical analyses. On the other hand, matrix completion (MC) methods can recover a…
Matrix completion is a basic machine learning problem that has wide applications, especially in collaborative filtering and recommender systems. Simple non-convex optimization algorithms are popular and effective in practice. Despite recent…