Soft Quantization using Entropic Regularization

TL;DR

This paper introduces a robust soft quantization method using entropy regularization and Wasserstein distance, with a stochastic gradient approach and adjustable difficulty, demonstrating strong empirical performance.

Contribution

It proposes a novel entropy-regularized soft quantization technique with a stochastic gradient solution and adjustable parameters, enhancing robustness and applicability.

Findings

Robustness of the entropy-regularized quantization demonstrated.

Effective stochastic gradient optimization implemented.

Performance shown across various experimental setups.

Abstract

The quantization problem aims to find the best possible approximation of probability measures on ${\mathbb{R}}^d$ using finite, discrete measures. The Wasserstein distance is a typical choice to measure the quality of the approximation. This contribution investigates the properties and robustness of the entropy-regularized quantization problem, which relaxes the standard quantization problem. The proposed approximation technique naturally adopts the softmin function, which is well known for its robustness in terms of theoretical and practicability standpoints. Moreover, we use the entropy-regularized Wasserstein distance to evaluate the quality of the soft quantization problem's approximation, and we implement a stochastic gradient approach to achieve the optimal solutions. The control parameter in our proposed method allows for the adjustment of the optimization problem's difficulty level, providing significant advantages when dealing with exceptionally challenging problems of interest. As well, this contribution empirically illustrates the performance of the method in various expositions.