A Note on Optimizing Distributions using Kernel Mean Embeddings - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2021

A Note on Optimizing Distributions using Kernel Mean Embeddings


Kernel mean embeddings are a popular tool that consists in representing probability measures by their infinite-dimensional mean embeddings in a reproducing kernel Hilbert space. When the kernel is characteristic, mean embeddings can be used to define a distance between probability measures, known as the maximum mean discrepancy (MMD). A well-known advantage of mean embeddings and MMD is their low computational cost and low sample complexity. However, kernel mean embeddings have had limited applications to problems that consist in optimizing distributions, due to the difficulty of characterizing which Hilbert space vectors correspond to a probability distribution. In this note, we propose to leverage the kernel sums-of-squares parameterization of positive functions of Marteau-Ferey et al. [2020] to fit distributions in the MMD geometry. First, we show that when the kernel is characteristic, distributions with a kernel sum-of-squares density are dense. Then, we provide algorithms to optimize such distributions in the finite-sample setting, which we illustrate in a density fitting numerical experiment.
Fichier principal
Vignette du fichier
MMD_Balls_Technical_note (1).pdf (979.32 Ko) Télécharger le fichier
Origin Files produced by the author(s)

Dates and versions

hal-03454259 , version 1 (29-11-2021)



Boris Muzellec, Francis Bach, Alessandro Rudi. A Note on Optimizing Distributions using Kernel Mean Embeddings. 2021. ⟨hal-03454259⟩
24 View
396 Download



Gmail Mastodon Facebook X LinkedIn More