We introduce a randomized procedure that, given an m×n matrix A and a positive integer k, approximates A with a matrix Z of rank k. The algorithm relies on applying a structured l×m random matrix R to each column of A, where l is an integer near to, but greater than, k. The structure of R allows us to apply it to an arbitrary m× 1 vector at a cost proportional to m log(l); the resulting procedu...

Let Ω be an open convex domain of C. We study constants K such that Ω is K-spectral or complete K-spectral for each continuous linear Hilbert space operator with numerical range included in Ω. Several approaches are discussed.

An n×n correlation matrix has k factor structure if its off-diagonal agrees with that of a rank k matrix. Such correlation matrices arise, for example, in factor models of collateralized debt obligations (CDOs) and multivariate time series. We analyze the properties of these matrices and, in particular, obtain an explicit formula for the rank in the one factor case. Our main focus is on the nea...

In this paper, we consider the supervised learning task which consists in predicting the normalized rank of a numerical variable. We introduce a novel probabilistic approach to estimate the posterior distribution of the target rank conditionally to the predictors. We turn this learning task into a model selection problem. For that, we define a 2D partitioning family obtained by discretizing num...

We address the problem of computing a low-rank estimate Y of the solution X of the Lyapunov equation AX + XA′ + Q = 0 without computing the matrix X itself. This problem has applications in both the reduced-order modeling and the control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternating direction implicit ...

The canonical polyadic decomposition (CPD) of a low-rank tensor plays major role in data analysis and signal processing by allowing for unique recovery underlying factors. However, it is well known that the CPD approximation problem ill-posed. That is, may fail to have best rank $R$ when $R>1$. This article gives deterministic bounds existence approximations over ${\mathbb{K}}={\mathbb{R}}$ or ...