Related papers: Typical ranks in symmetric matrix completion
Matrix completion algorithms recover a low rank matrix from a small fraction of the entries, each entry contaminated with additive errors. In practice, the singular vectors and singular values of the low rank matrix play a pivotal role for…
Matrix completion is the problem of recovering a low rank matrix by observing a small fraction of its entries. A series of recent works [KOM12,JNS13,HW14] have proposed fast non-convex optimization based iterative algorithms to solve this…
A tropical matrix is a matrix defined over the max-plus semiring. For such matrices, there exist several non-coinciding notions of rank: the row rank, the column rank, the Schein/Barvinok rank, the Kapranov rank, or the tropical rank, among…
The square root rank of a nonnegative matrix $A$ is the minimum rank of a matrix $B$ such that $A=B \circ B$, where $\circ$ denotes entrywise product. We show that the square root rank of the slack matrix of the correlation polytope is…
In this paper, we propose an efficient and scalable low rank matrix completion algorithm. The key idea is to extend orthogonal matching pursuit method from the vector case to the matrix case. We further propose an economic version of our…
The almost-principal rank characteristic sequence (apr-sequence) of an $n\times n$ symmetric matrix is introduced, which is defined to be the string $a_1 a_2 \cdots a_{n-1}$, where $a_k$ is either $\tt A$, $\tt S$, or $\tt N$, according as…
Hermitian and unitary matrices are two representatives of the class of normal matrices whose full eigenvalue decomposition can be stably computed in quadratic computing com plexity. Recently, fast and reliable eigensolvers dealing with low…
An n \times n matrix A has a normal defect of k if there exists an (n+k) \times (n+k) normal matrix A_{ext} with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4 \times 4…
Multicriteria Decision Making problems are important both for individuals and groups. Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. We focus on decision problems where the…
Matrix completion aims to reconstruct a data matrix based on observations of a small number of its entries. Usually in matrix completion a single matrix is considered, which can be, for example, a rating matrix in recommendation system.…
This paper studies the problem of deterministic rank-one matrix completion. It is known that the simplest semidefinite programming relaxation, involving minimization of the nuclear norm, does not in general return the solution for this…
In this paper, we consider matrix completion from non-uniformly sampled entries including fully observed and partially observed columns. Specifically, we assume that a small number of columns are randomly selected and fully observed, and…
We initiate a study of the zero-nonzero patterns of n by n alternating sign matrices. We characterize the row (column) sum vectors of these patterns and determine their minimum term rank. In the case of connected alternating sign matrices,…
The condition number of solutions to full rank linear least-squares problem are shown to be given by an optimization problem that involves nuclear norms of rank 2 matrices. The condition number is with respect to the least-squares…
Matrix factorization models have been extensively studied as a valuable test-bed for understanding the implicit biases of overparameterized models. Although both low nuclear norm and low rank regularization have been studied for these…
We consider a generalization of low-rank matrix completion to the case where the data belongs to an algebraic variety, i.e. each data point is a solution to a system of polynomial equations. In this case the original matrix is possibly…
The problem of completing a large matrix with lots of missing entries has received widespread attention in the last couple of decades. Two popular approaches to the matrix completion problem are based on singular value thresholding and…
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…
Random linear systems over the Galois Field modulo 2 have an interest in connection with problems ranging from computational optimization to complex networks. They are often approached using random matrices with Poisson-distributed or…
We give a new framework for solving the fundamental problem of low-rank matrix completion, i.e., approximating a rank-$r$ matrix $\mathbf{M} \in \mathbb{R}^{m \times n}$ (where $m \ge n$) from random observations. First, we provide an…