Related papers: Geometric Matrix Midranges
The main goal of this article is to present new types of inequalities refining and reversing inequalities of the harmonic mean of scalars and matrices. Furthermore, implementing the spectral decomposition of positive matrices, we present a…
The accurate computation of the covariance matrix of fitted model parameters is a somewhat neglected task in Statistics. Algorithms are given for computing accurate covariance matrices derived from computing the Hessian matrix by numerical…
We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite…
Deep Matrix Factorization (DMF) is an emerging approach to the problem of matrix completion. Recent works have established that gradient descent applied to a DMF model induces an implicit regularization on the rank of the recovered matrix.…
Geodesic metric spaces support a variety of averaging constructions for given finite sets. Computing such averages has generated extensive interest in diverse disciplines. Here we consider the inverse problem of recognizing computationally…
In an earlier work, the author together with Guo [Hermitian adjacency matrix of digraphs and mixed graphs, J. Graph Theory 85 (2017) 217-248] introduced the Hermitian adjacency matrix of directed (and partially directed) graphs. However, it…
We show how the numerical range of a matrix can be used to bound the optimal value of certain optimization problems over real tensor product vectors. Our bound is stronger than the trivial bounds based on eigenvalues, and can be computed…
In this paper, we review the problem of matrix completion and expose its intimate relations with algebraic geometry, combinatorics and graph theory. We present the first necessary and sufficient combinatorial conditions for matrices of…
We briefly introduce several problems: (1) a generalization of the convex fair partition conjecture, (2) on non-trivial invariants among polyhedrons that can be formed from the same set of face polygons, (3) two questions on assembling…
One of the main applications of semidefinite programming lies in linear systems and control theory. Many problems in this subject, certainly the textbook classics, have matrices as variables, and the formulas naturally contain…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic…
We revisit the matrix problems sparse null space and matrix sparsification, and show that they are equivalent. We then proceed to seek algorithms for these problems: We prove the hardness of approximation of these problems, and also give a…
We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on…
A common optimization problem is the minimization of a symmetric positive definite quadratic form $< x,Tx >$ under linear constrains. The solution to this problem may be given using the Moore-Penrose inverse matrix. In this work we extend…
It is known that semi-magic square matrices form a 2-graded algebra or superalgebra with the even and odd subspaces under centre-point reflection symmetry as the two components. We show that other symmetries which have been studied for…
We consider the problem of embedding the nodes of a hypergraph into Euclidean space under the assumption that the interactions arose through closeness to unknown hyperedge centres. In this way, we tackle the inverse problem associated with…
An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. Here, the genuinely quantum mechanical approach is considered in detail for (2x2) matrices. The method is based on the measurement of…
An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms…
The main goal of this paper is to discuss the recent advancements of operator means for accretive matrices in a more general setting. In particular, we present the general form governing the well established definition of geometric mean,…
High dimensional data analysis is known to be as a challenging problem. In this article, we give a theoretical analysis of high dimensional classification of Gaussian data which relies on a geometrical analysis of the error measure. It…