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.

翻译：内核平均嵌入是一个流行的工具,它代表着概率度量, 以其无限维平均值嵌入于复制的内核 Hilbert 空间。 当内核是特性时, 内核平均嵌入可以用来确定概率度量之间的距离, 称为最大平均值差异( MMD ) 。 内核平均嵌入和 MMD 的众所周知的优点是它们的计算成本低, 样本复杂性低。 但是, 内核平均嵌入的应用有限, 问题包括优化分布, 因为Hilbert 空间矢量与概率分布相对应的特性很难。 在此注释中, 我们提议利用内核总和方数值的参数参数来调整 Marteau- Ferey et al. [2020] 的正函数的分布, 以适应MMD 几何的分布。 首先, 我们显示, 当内核具有特性时, 内核总和方密度密度密度密度的分布是密度密度密度密度的。 然后, 我们提供算法来优化在定式的分布。