Quantum mean embedding of probability distributions

Abstract

The kernel mean embedding of probability distributions is commonly used in machine learning as an injective mapping from distributions to functions in an infinite-dimensional Hilbert space. It allows us, for example, to define a distance measure between probability distributions, called the maximum mean discrepancy. In this work, we propose to represent probability distributions in a pure quantum state of a system that is described by an infinite-dimensional Hilbert space and prove that the representation is unique if the corresponding kernel function is c_{0} universal. This enables us to work with an explicit representation of the mean embedding, whereas classically one can only work implicitly with an infinite-dimensional Hilbert space through the use of the kernel trick. We show how this explicit representation can speed up methods that rely on inner products of mean embeddings and discuss the theoretical and experimental challenges that need to be solved in order to achieve these speedups.