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.

翻译：量化问题旨在通过有限离散测度来寻找$\mathbb{R}^d$上概率测度的最佳逼近。Wasserstein距离是衡量逼近质量的典型选择。本文研究了熵正则化量化问题的性质与鲁棒性，该问题是对标准量化问题的松弛。所提出的逼近技术自然地采用softmin函数，该函数从理论和实践角度均以其鲁棒性著称。此外，我们利用熵正则化Wasserstein距离评估软量化问题逼近的质量，并通过随机梯度方法实现最优解。所提方法中的控制参数可调整优化问题的难度水平，在处理特别具有挑战性的关键问题时展现出显著优势。本文还通过多种案例实证展示了该方法的性能。