# Quantum Mean Embedding of Probability Distributions

## TL;DR

This paper introduces a quantum approach to represent probability distributions explicitly as quantum states, potentially accelerating kernel mean embedding methods used in machine learning.

## Contribution

It proposes a novel quantum representation of probability distributions that allows explicit mean embedding computation, unlike classical implicit methods.

## Key findings

- Quantum representation enables explicit mean embedding calculation.
- Potential for speedup in kernel-based machine learning methods.
- Discussion of theoretical and experimental challenges for implementation.

## 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 maximum mean discrepancy (MMD). 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. 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.