Fast rates for empirical vector quantization

TL;DR

This paper proves that the expected distortion of empirically optimal vector quantizers converges at a rate of O(1/n), improving upon the previously known O(log n/n) rate for certain distributions.

Contribution

The authors establish a faster convergence rate of O(1/n) for empirical vector quantization under specific distribution conditions, expanding the theoretical understanding of quantizer performance.

Findings

Expected distortion decreases at rate O(1/n)

Applicable to well-polarized distributions with continuous densities

Improves upon previous O(log n/n) rate

Abstract

We consider the rate of convergence of the expected loss of empirically optimal vector quantizers. Earlier results show that the mean-squared expected distortion for any fixed distribution supported on a bounded set and satisfying some regularity conditions decreases at the rate O(log n/n). We prove that this rate is actually O(1/n). Although these conditions are hard to check, we show that well-polarized distributions with continuous densities supported on a bounded set are included in the scope of this result.