Scalable and consistent embedding of probability measures into Hilbert spaces via measure quantization

Abstract: This paper is focused on statistical learning from data that come as probability measures. In this setting, popular approaches consist in embedding such data into a Hilbert space with either Linearized Optimal Transport or Kernel Mean Embedding. However, the cost of computing such embeddings prohibits their direct use in large-scale settings. We study two methods based on measure quantization for approximating input probability measures with discrete measures of small-support size. The first one is based on optimal quantization of each input measure, while the second one relies on mean-measure quantization. We study the consistency of such approximations, and its implication for scalable embeddings of probability measures into a Hilbert space at a low computational cost. We finally illustrate our findings with various numerical experiments.