Freiburg 2019 – scientific programme

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FM: Fall Meeting

FM 55: Quantum & Information Science: Neural Networks, Machine Learning, and Artificial Intelligence II

FM 55.1: Invited Talk

Wednesday, September 25, 2019, 14:00–14:30, 1098

Quantum Mean Embedding of Probability Distributions — •Jonas M. Kübler, Krikamol Muandet, and Bernhard Schölkopf — Max Planck Institute for Intelligent Systems, Tübingen, Germany

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 and prove that the representation is unique if the corresponding kernel function is c₀-universal. This is a new method for encoding classical data in a quantum state and 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.