Related papers: Differential equations for real-structured (and un…
In this paper, we present the separability criteria to identify non-$k$-separability and genuine multipartite entanglement in mixed multipartite states using elements of density matrices. Our criteria can detect the non-$k$-separability of…
This work considers the problem of estimating the distance between two covariance matrices directly from the data. Particularly, we are interested in the family of distances that can be expressed as sums of traces of functions that are…
Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This…
The Euclidean distance geometry problem arises in a wide variety of applications, from determining molecular conformations in computational chemistry to localization in sensor networks. When the distance information is incomplete, the…
Similarity search based on a distance function in metric spaces is a fundamental problem for many applications. Queries for similar objects lead to the well-known machine learning task of nearest-neighbours identification. Many data…
Computationally efficient solutions for pseudometrics quantifying deviation from topological conjugacy between dynamical systems are presented. Deviation from conjugacy is quantified in a Pareto optimal sense that accounts for spectral…
In quantum mechanics, the Schrieffer--Wolff (SW) transformation (also called quasi-degenerate perturbation theory) is known as an approximative method to reduce the dimension of the Hamiltonian. We present a geometric interpretation of the…
We propose a new approach to computing global minimizers of singular value functions in two real variables. Specifically, we present new algorithms to compute the Kreiss constant of a matrix and the distance to uncontrollability of a linear…
For given k distinct complex conjugate pairs, l distinct real numbers, and a given graph G on 2k+l vertices with a matching of size at least k, we will show that there is a real matrix whose eigenvalues are the given numbers and its graph…
Differential privacy is a widely studied notion of privacy for various models of computation. Technically, it is based on measuring differences between probability distributions. We study $\epsilon,\delta$-differential privacy in the…
Many methods in differentially private model training rely on computing the similarity between a query point (such as public or synthetic data) and private data. We abstract out this common subroutine and study the following fundamental…
We prove that the distance from an $n\times n$ complex matrix $M$ to the set of nilpotents is at least $\frac{1}{2}\sec\frac{\pi}{n+2}$ if there is a nonzero projection $P$ such that $PMP=M$ and $M^*M\geq P$. In the particular case where…
We review some basic results of convex analysis and geometry in $\mathbb{R}^n$ in the context of formulating a differential equation to track the distance between an observer flying outside a convex set $K$ and $K$ itself.
Distance weighted discrimination (DWD) is a linear discrimination method that is particularly well-suited for classification tasks with high-dimensional data. The DWD coefficients minimize an intuitive objective function, which can solved…
The aim of this paper is to find an optimal matching between manifold-valued curves, and thereby adequately compare their shapes, seen as equivalent classes with respect to the action of reparameterization. Using a canonical decomposition…
The paper introduces a special case of the Euclidean distance matrix completion problem (edmcp) of interest in statistical data analysis where only the minimal spanning tree distances are given and the matrix completion must preserve the…
This paper addresses the task of estimating a covariance matrix under a patternless sparsity assumption. In contrast to existing approaches based on thresholding or shrinkage penalties, we propose a likelihood-based method that regularizes…
Performing analog computations with metastructures is an emerging wave-based paradigm for solving mathematical problems. For such devices, one major challenge is their reconfigurability, especially without the need for a priori mathematical…
A differential geometric approach to singular perturbation theory is presented. It is shown that singular perturbation problems such as multiple-scale and boundary layer problems can be treated more easily on a differential geometric basis.…
We introduce a new geometric framework for the set of symmetric positive-definite (SPD) matrices, aimed to characterize deformations of SPD matrices by individual scaling of eigenvalues and rotation of eigenvectors of the SPD matrices. To…