Related papers: The rank of sparse random matrices

In an earlier paper, we discussed the probability that the determinant of a matrix undergoes the least change upon perturbation of one of its elements, provided that most or all of the elements of the matrix are chosen at random and that…

Based on the ranks of reduced density matrices, we derive necessary conditions for the separability of multiparticle arbitrary-dimensional mixed states, which are equivalent to sufficient conditions for entanglement. In a similar way we…

The low-complexity assumption in linear systems can often be expressed as rank deficiency in data matrices with generalized Hankel structure. This makes it possible to denoise the data by estimating the underlying structured low-rank…

When factorizing binary matrices, we often have to make a choice between using expensive combinatorial methods that retain the discrete nature of the data and using continuous methods that can be more efficient but destroy the discrete…

This paper provides the best bounds to date on the number of randomly sampled entries required to reconstruct an unknown low rank matrix. These results improve on prior work by Candes and Recht, Candes and Tao, and Keshavan, Montanari, and…

A central question in random matrix theory is universality. When an emergent phenomena is observed from a large collection of chosen random variables it is natural to ask if this behavior is specific to the chosen random variable or if the…

We investigate the methods that simultaneously enforce sparsity and low-rank structure in a matrix as often employed for sparse phase retrieval problems or phase calibration problems in compressive sensing. We propose a new approach for…

This paper studies inference in linear models with a high-dimensional parameter matrix that can be well-approximated by a ``spiked low-rank matrix.'' A spiked low-rank matrix has rank that grows slowly compared to its dimensions and nonzero…

The problem of identifying sparse solutions for the link structure and dynamics of an unknown linear, time-invariant network is posed as finding sparse solutions x to Ax=b. If the sensing matrix A satisfies a rank condition, this problem…

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…

1. A standard Gaussian random matrix has full rank with probability 1 and is well-conditioned with a probability quite close to 1 and converging to 1 fast as the matrix deviates from square shape and becomes more rectangular. 2. If we…

This paper considers sparse spiked covariance matrix models in the high-dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection. The optimal rate of…

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…

In structured system theory, a pattern matrix is a matrix with entries either fixed to zero or free to take arbitrary numbers. The (generic) rank of a pattern matrix is the rank of almost all its realizations. The resilience of various…

We show that the process matrix in the basis of tensor products of Pauli operators or SU(N) generators representing low rank and sparse dynamical maps will have only a few distinct entries which goes as $O(r^2)$ ($r$ is the rank).

We explain theoretically a curious empirical phenomenon: "Approximating a matrix by deterministically selecting a subset of its columns with the corresponding largest leverage scores results in a good low-rank matrix surrogate". To obtain…

In this paper, we prove the existence of capacity achieving linear codes with random binary sparse generating matrices. The results on the existence of capacity achieving linear codes in the literature are limited to the random binary codes…

Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and…

We consider the set of $n\times n$ matrices with rational entries having numerator and denominator of size at most $H$ and obtain upper and lower bounds on the number of such matrices of a given rank and then apply them to count such…

Complex systems are high-dimensional nonlinear dynamical systems with intricate interactions among their constituents. To make interpretable predictions about their large-scale behavior, it is typically assumed, without a clear statement,…