Fast Recovery and Approximation of Hidden Cauchy Structure

TL;DR

This paper introduces optimal complexity algorithms for identifying and approximating Cauchy matrices from data, including noisy cases, with proven bounds and numerical validation.

Contribution

It presents the first algorithms with optimal complexity for exact recovery and approximation of Cauchy matrices, including noisy data scenarios.

Findings

Algorithm for exact Cauchy matrix detection and recovery

Optimal complexity approximation algorithm with bounds

Numerical experiments confirming theoretical results

Abstract

We derive an algorithm of optimal complexity which determines whether a given matrix is a Cauchy matrix, and which exactly recovers the Cauchy points defining a Cauchy matrix from the matrix entries. Moreover, we study how to approximate a given matrix by a Cauchy matrix with a particular focus on the recovery of Cauchy points from noisy data. We derive an approximation algorithm of optimal complexity for this task, and prove approximation bounds. Numerical examples illustrate our theoretical results.