Hierarchical Low-rank Structure of Parameterized Distributions

TL;DR

This paper demonstrates that various one-parameter distribution families, including binomial, Poisson, and chi-squared, exhibit a hierarchical low-rank structure in their matrix forms, supported by theoretical proofs and numerical validation.

Contribution

It reveals a hierarchical low-rank structure in the matrix forms of several common one-parameter distributions, providing new insights into their mathematical properties.

Findings

Distribution matrices have hierarchical low-rank structure

The proof uses a uniform relative bound of a divergence function

Numerical results confirm the theoretical analysis

Abstract

This note shows that the matrix forms of several one-parameter distribution families satisfy a hierarchical low-rank structure. Such families of distributions include binomial, Poisson, and $\chi^2$ distributions. The proof is based on a uniform relative bound of a related divergence function. Numerical results are provided to confirm the theoretical findings.