Related papers: Recent Progress on Matrix Rigidity -- A Survey
A matroid is a machine capturing linearity of mathematical objects and producing combinatorial structures. Matroid structure arises everywhere since linearity is a ubiquitous concept. One natural way to obtain matroids is by considering…
In these lectures notes, we review our recent works addressing various problems of finding the nearest stable system to an unstable one. After the introduction, we provide some preliminary background, namely, defining Port-Hamiltonian…
This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning. Recently there has been some research activity in replacing…
We introduce the concept of strongly independent matrices over any field, and prove the existence of such matrices for certain fields and the non-existence for algebraically closed fields. Then we apply strongly independent matrices over…
Combinatorial rigidity theory seeks to describe the rigidity or flexibility of bar-joint frameworks in R^d in terms of the structure of the underlying graph G. The goal of this article is to broaden the foundations of combinatorial rigidity…
A matrix-compression algorithm is derived from a novel isogenic block decomposition for square matrices. The resulting compression and inflation operations possess strong functorial and spectral-permanence properties. The basic observation…
Matrices are exceptionally useful in various fields of study as they provide a convenient framework to organize and manipulate data in a structured manner. However, modern matrices can involve billions of elements, making their storage and…
We construct new, previously unknown parametric families of complex conference matrices and of complex Hadamard matrices of square orders and related them to complex equiangular tight frames.
In this article we survey recent progress in the algorithmic theory of matrix semigroups. The main objective in this area of study is to construct algorithms that decide various properties of finitely generated subsemigroups of an infinite…
We introduced a new coordinate-free approach to study the Cauchy-Riemann (CR) maps between the real hyperquadrics in the complex projective space. The central theme is based on a notion of orthogonality on the projective space induced by…
A precision matrix is the inverse of a covariance matrix. In this paper, we study the problem of estimating the precision matrix with a known graphical structure under high-dimensional settings. We propose a simple estimator of the…
Rigid, hard and soft problems and results in arithmetic geometry are presented. "Soft" and "hard" in our paper are limited to the framework of solutions of quadratic forms over rings of integers of local and global fields, the…
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a…
Considering matrices with missing entries, we study NP-hard matrix completion problems where the resulting completed matrix shall have limited (local) radius. In the pure radius version, this means that the goal is to fill in the entries…
An Hadamard matrix is a square matrix $H\in M_N(\pm1)$ whose rows and pairwise orthogonal. More generally, we can talk about the complex Hadamard matrices, which are the square matrices $H\in M_N(\mathbb C)$ whose entries are on the unit…
The aim of this note (as well as of the course itself) is to give a largely self-contained proof of two of the main results in the field of low-rank matrix recovery. This field aims for identification of low-rank matrices from only limited…
A rigidity theory is developed for bar-joint frameworks in linear matrix spaces endowed with a unitarily invariant norm. Analogues of Maxwell's counting criteria are obtained and minimally rigid matrix frameworks are shown to belong to the…
We discuss in details a modified variational matrix-product-state algorithm for periodic boundary conditions, based on a recent work by P. Pippan, S.R. White and H.G. Everts, Phys. Rev. B 81, 081103(R) (2010), which enables one to study…
A rigidity theory is developed for frameworks in a metric space with two types of distance constraints. Mixed sparsity graph characterisations are obtained for the infinitesimal and continuous rigidity of completely regular bar-joint…
In recent years, several algorithms, which approximate matrix decomposition, have been developed. These algorithms are based on metric conservation features for linear spaces of random projection types. We show that an i.i.d sub-Gaussian…