Optimal Bregman quantization : existence and uniqueness of optimal quantizers revisited

TL;DR

This paper revisits the existence and uniqueness of optimal Bregman quantizers, providing new conditions under which these properties hold, especially for quadratic cases and certain distribution types.

Contribution

It establishes existence of optimal quantizers under weaker assumptions and proves a one-dimensional uniqueness theorem for specific distribution and divergence types.

Findings

Existence of optimal quantizers under lighter assumptions.

Uniqueness in one dimension for strongly unimodal distributions.

Results applicable to quadratic and certain convex divergence cases.

Abstract

In this paper we revisit the exsistence theorem for $L^r$-optimal quantization, $r\ge 2$, with respect to a Bregman divergence: we establish the existence of optimal quantizaers under lighter assumptions onthe strictly convex function which generates the divergence, espcially in the quadratic case ($r=2$). We then prove a uniqueness theorem ``\`a la Trushkin'' in one dimension for strongly unimodal distributions and divergences gerated by strictly convex functions whiose thire dervative is either stictly $\log$-convex or $\log$-concave.