Convergence and asymptotic freeness of missing data matrices

TL;DR

This paper investigates the convergence and asymptotic freeness of missing data matrices formed by Hadamard products of deterministic and random matrices, establishing conditions under which classical free probability results extend to these incomplete data models.

Contribution

It provides necessary and sufficient conditions on deterministic matrices for the convergence and freeness of missing data matrices of iid, elliptic, and covariance types.

Findings

Conditions for convergence of missing data matrices.

Extension of free probability results to incomplete data models.

Characterization of asymptotic freeness in missing data scenarios.

Abstract

We consider a random matrix of the form $D_n \odot X_n$ (known as a variance profile matrix), where $\odot$ denotes the Hadamard product of the two matrices, $D_n$ is a deterministic matrix, and $X_n$ is a random matrix. We call $D_n\odot X_n$ as a missing data matrix of $X_n$ when the entries of $D_n$ are either $0$ or $1$. This framework is commonly used in various applied fields, such as biology, neuroscience, and network data analysis. We study the convergence and asymptotic freeness of missing data matrices of iid, elliptic, and covariance random matrices. Specifically, it is known that independent iid, elliptic, and covariance matrices converge to freely independent circular, elliptic, and Mar\v{c}enko-Pastur variables, respectively. In this article, we provide the necessary and sufficient conditions on deterministic matrices $D_n$ for which these results hold true for independent missing data matrices of these three types of random matrices.