Related papers: The full rank condition for sparse random matrices
We observe a $N\times M$ matrix of independent, identically distributed Gaussian random variables which are centered except for elements of some submatrix of size $n\times m$ where the mean is larger than some $a>0$. The submatrix is sparse…
Square matrices appear in many machine learning problems and models. Optimization over a large square matrix is expensive in memory and in time. Therefore an economic approximation is needed. Conventional approximation approaches factorize…
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 compute analytically the probability distribution and moments of the sum and product of the non-zero eigenvalues and singular values of random matrices with (i) non-negative entries, (ii) fixed rank, and (iii) prescribed sums of the…
We consider the problem of robust matrix completion, which aims to recover a low rank matrix $L_*$ and a sparse matrix $S_*$ from incomplete observations of their sum $M=L_*+S_*\in\mathbb{R}^{m\times n}$. Algorithmically, the robust matrix…
We consider a square random matrix made by i.i.d. rows with any distribution and prove that, for any given dimension, the probability for the least singular value to be in [0; $\epsilon$) is at least of order $\epsilon$. This allows us to…
Suppose $\mathbb{F}$ is a field and let $\mathbf{a} := (a_1, a_2, \dotsc)$ be a sequence of non-zero elements in $\mathbb{F}$. For $\mathbf{a}_n := (a_1, \dotsc, a_n)$, we consider the family $\mathcal{M}_n(\mathbf{a})$ of $n \times n$…
A positive definite matrix is called logarithmically sparse if its matrix logarithm has many zero entries. Such matrices play a significant role in high-dimensional statistics and semidefinite optimization. In this paper, logarithmically…
We consider the related tasks of matrix completion and matrix approximation from missing data and propose adaptive sampling procedures for both problems. We show that adaptive sampling allows one to eliminate standard incoherence…
A matrix is incomplete when some of its entries are missing. A Robinson incomplete symmetric matrix is an incomplete symmetric matrix whose non-missing entries do not decrease along rows and columns when moving toward the diagonal. A…
In this paper, we have obtained a necessary and suffcient condition for the series.
The concepts of differentiation and integration for matrices are known. As far as each matrix is differentiable, it is not clear a priori whether a given matrix is integrable or not. Recently some progress was obtained for diagonalizable…
The problem of finding a rank-one solution to a system of linear matrix equations arises from many practical applications. Given a system of linear matrix equations, however, such a low-rank solution does not always exist. In this paper, we…
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually…
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…
A matrix completion problem is to recover the missing entries in a partially observed matrix. Most of the existing matrix completion methods assume a low rank structure of the underlying complete matrix. In this paper, we introduce an…
In this paper, we analyze the fundamental conditions for low-rank tensor completion given the separation or tensor-train (TT) rank, i.e., ranks of unfoldings. We exploit the algebraic structure of the TT decomposition to obtain the…
In this paper, we introduce and study the notion of phase rank. Given a matrix filled with phases, i.e., with complex entries of modulus $1$, its phase rank is the smallest possible rank for a complex matrix with the same phase, but…
The number of non-negative integer matrices with given row and column sums appears in a variety of problems in mathematics and statistics but no closed-form expression for it is known, so we rely on approximations of various kinds. Here we…
Random sampling has become a critical tool in solving massive matrix problems. For linear regression, a small, manageable set of data rows can be randomly selected to approximate a tall, skinny data matrix, improving processing time…