Related papers: Fooling-sets and rank in nonzero characteristic (e…

Consider a matrix $A$ of rank $n$ that approximates the $N\times N$ identity matrix with elementwise error at most $1/3$. We give a lower bound on the number of elements s.t. $|A_{i,j}|>\gamma$, for a certain threshold. Two corollaries are…

Let $H\_0, ..., H\_n$ be $m \times m$ matrices with entries in $\QQ$ and Hankel structure, i.e. constant skew diagonals. We consider the linear Hankel matrix $H(\vecx)=H\_0+\X\_1H\_1+...+\X\_nH\_n$ and the problem of computing sample points…

Factorization of matrices where the rank of the two factors diverges linearly with their sizes has many applications in diverse areas such as unsupervised representation learning, dictionary learning or sparse coding. We consider a setting…

We prove that if $\mathbb{F}$ is a field of positive odd characteristic $p,$ and $m,$ and $n$ are positive integers such that $m\geq2,$ and $n\leq p,$ every $n\times n$ nonderogatory matrix $A\in \mathbb{M}_n(\mathbb{F})$ which is sum of…

In this paper we consider the trace regression model. Assume that we observe a small set of entries or linear combinations of entries of an unknown matrix $A_0$ corrupted by noise. We propose a new rank penalized estimator of $A_0$. For…

We establish lower bounds on the rank of matrices in which all but the diagonal entries lie in a multiplicative group of small rank. Applying these bounds we show that the distance sets of finite pointsets in $\mathbb{R}^d$ generate high…

An $L$-matrix is a matrix whose off-diagonal entries belong to a set $L$, and whose diagonal is zero. Let $N(r,L)$ be the maximum size of a square $L$-matrix of rank at most $r$. Many applications of linear algebra in extremal combinatorics…

The girth of a matrix is the least number of linearly dependent columns, in contrast to the rank which is the largest number of linearly independent columns. This paper considers the construction of {\it high-girth} matrices, whose…

For any field k and any integers m,n with 0 <= 2m <= n+1, let W_n be the k-vector space of sequences (x_0,...,x_n), and let H_m be the subset of W_n consisting of the sequences that satisfy a degree-m linear recursion, that is, for which…

The notion of `stable rank' of a matrix is central to the analysis of randomized matrix algorithms, covariance estimation, deep neural networks, and recommender systems. We compare the properties of the stable rank and intrinsic dimension…

Randomized sampling has recently been demonstrated to be an efficient technique for computing approximate low-rank factorizations of matrices for which fast methods for computing matrix vector products are available. This paper describes an…

In this letter, we study the deterministic sampling patterns for the completion of low rank matrix, when corrupted with a sparse noise, also known as robust matrix completion. We extend the recent results on the deterministic sampling…

We develop several efficient algorithms for the classical \emph{Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input $n\times n$ matrix $A$, this…

An $m \times (n+1)$ multiplicity matrix is a matrix $M = ( \mu_{i,j} )$ with rows enumerated by $i \in \{ 1,\ 2, \ldots, m \}$ and columns enumerated by $j \in \{ 0,1,\ldots, n \}$ whose coordinates are nonnegative integers satisfying the…

This paper is concerned with the interplay between statistical asymmetry and spectral methods. Suppose we are interested in estimating a rank-1 and symmetric matrix $\mathbf{M}^{\star}\in \mathbb{R}^{n\times n}$, yet only a randomly…

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly-exponential in $r$. This…

We show several ways to round a real matrix to an integer one such that the rounding errors in all rows and columns as well as the whole matrix are less than one. This is a classical problem with applications in many fields, in particular,…

Let $k$ be an algebraically closed field and $A$ the polynomial algebra in $r$ variables with coefficients in $k$. In case the characteristic of $k$ is $2$, Carlsson conjectured that for any $DG$-$A$-module $M$ of dimension $N$ as a free…

In continuation to our recent work on noncommutative polynomial factorization, we consider the factorization problem for matrices of polynomials and show the following results. (1) Given as input a full rank $d\times d$ matrix $M$ whose…

This paper considers the problem of completing a matrix with many missing entries under the assumption that the columns of the matrix belong to a union of multiple low-rank subspaces. This generalizes the standard low-rank matrix completion…