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 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.
- Received 15 June 2019
DOI:https://doi.org/10.1103/PhysRevResearch.1.033159
Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.
Published by the American Physical Society