arXiv subjects

Malena Español

Publications and source records attributed to Malena Español.

3 recordsLinked to original sources

arXiv2026

Structure-Informed Bounds on the Kronecker Rank of Block-Structured Matrices

We derive theoretical bounds on the Kronecker rank of block-structured matrices that possess both inner and outer structure. Building on the matrix-to-tensor and tensor-to-matrix framework of Kilmer and Saibaba ($\textit{SIAM J. Matrix Anal. Appl.}$, 2022), we show that the Kronecker rank of a matrix $\mathbf{A}$ equals the dimension of the span of its distinct blocks (the $\textit{inner blockspan}$), and equals as well the dimension of the corresponding span for a permuted matrix $\mathbf{B}$ whose inner and outer structures are interchanged. We give two proofs of this equality: one via a direct dimension argument, and one via an explicit isomorphism between the outer blockspan of $\mathbf{A}$ and the column space of the second-mode unfolding of the associated tensor. These results yield nested containments that translate structural information, such as Toeplitz, Hankel, banded, or sparse patterns, into computable upper bounds on the Kronecker rank. We also establish a duality between element-level sparsity in $\mathbf{A}$ and block-level sparsity in $\mathbf{B}$. Numerical experiments confirm the theory across several classes of structured matrices and, for two sparse matrices drawn from the SuiteSparse collection, provide a structural explanation for observed singular value decay.

math.NA↗

arXiv2024

Parameter Selection by GCV and a $χ^2$ test within Iterative Methods for $\ell_1$-regularized Inverse Problems

$\ell_1$ regularization is used to preserve edges or enforce sparsity in a solution to an inverse problem. We investigate the Split Bregman and the Majorization-Minimization iterative methods that turn this non-smooth minimization problem into a sequence of steps that include solving an $\ell_2$-regularized minimization problem. We consider selecting the regularization parameter in the inner generalized Tikhonov regularization problems that occur at each iteration in these $\ell_1$ iterative methods. The generalized cross validation and $χ^2$ degrees of freedom methods are extended to these inner problems. In particular, for the $χ^2$ method this includes extending the $χ^2$ result for problems in which the regularization operator has more rows than columns, and showing how to use the $A-$weighted generalized inverse to estimate prior information at each inner iteration. Numerical experiments for image deblurring problems demonstrate that it is more effective to select the regularization parameter automatically within the iterative schemes than to keep it fixed for all iterations. Moreover, an appropriate regularization parameter can be estimated in the early iterations and used fixed to convergence.

math.NA↗

arXiv2017

A Discrete-to-Continuum Model of Weakly Interacting Incommensurate Chains

In this paper we use a formal discrete-to-continuum procedure to derive a continuum variational model for two chains of atoms with slightly incommensurate lattices. The chains represent a cross-section of a three-dimensional system consisting of a graphene sheet suspended over a substrate. The continuum model recovers both qualitatively and quantitatively the behavior observed in the corresponding discrete model. The numerical solutions for both models demonstrate the presence of large commensurate regions separated by localized incommensurate domain walls.

cond-mat.mtrl-sci↗