High-dimensional classification has challenges mainly due to the singularity issue of sample covariance matrix. In this work, we propose a different approach get more reliable matrix by adjusting estimated eigenvalues. This procedure also brings us nonsingular as by-product. We improve optimization obtain linear classifier incorporating adjusted and shrinkage mean vector into original problem. ...

A bipartite graph is said to be symmetric if it has symmetry of reflecting two vertex sets. This paper investigates matching structure of symmetric bipartite graphs. We first apply the Dulmage-Mendelsohn decomposition to a symmetric bipartite graph. The resulting components, which are matching-covered, turn out to have symmetry. We then decompose a matching-covered bipartite graph via an ear de...

For statistical inferences that involve covariance matrices, it is desirable to obtain an accurate covariance matrix estimate with a well-structured eigen-system. We propose to estimate the covariance matrix through its matrix logarithm based on an approximate log-likelihood function. We develop a generalization of the Leonard and Hsu (1992) log-likelihood approximation that no longer requires ...

We construct a reduction which proves that the fooling set number and the determinantal rank of a Boolean matrix are NP-hard to compute. This note is devoted to the functions of determinantal rank and fooling set number, which are receiving attention in different applications, see [1, 3] and references therein. The purpose of this note is to give an NP-completeness proof for those functions, th...

The probability for two monic polynomials of a positive degree n with coefficients in the finite field Fq to be relatively prime turns out to be identical with the probability for an n × n Hankel matrix over Fq to be nonsingular. Motivated by this, we give an explicit map from pairs of coprime polynomials to nonsingular Hankel matrices that explains this connection. A basic tool used here is th...

Let q denote the finite field GF (q) and let b be a positive integer. MDS codes over the symbol alphabet b q are considered that are linear over q and have sparse (“low-density”) parity-check and generator matrices over q that are systematic over bq . Lower bounds are presented on the number of nonzero elements in any systematic parity-check or generator matrix of an q-linear MDS code over bq, ...

We prove that the determinantal complexity of a hypersurface of degree d > 2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the 3×3 permanent is 7. We also prove that for n > 3, there is no nonsingular hypersurface in Pn of degree d that has an expression as a dete...

Abstract We consider systems of $n$ diagonal equations in $k$th powers. Our main result shows that if the coefficient matrix such a system is sufficiently nonsingular, then partition regular and only it satisfies Rado’s columns condition. Furthermore, also admits constant solutions, we prove has nontrivial solutions over every set integers positive upper density.