  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.
