Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures

TL;DR

This paper extends kernel mean embeddings to von Neumann-algebra-valued measures within reproducing kernel Hilbert modules, enabling advanced probabilistic analysis and machine learning applications in quantum mechanics.

Contribution

It introduces a generalized KME framework for von Neumann-algebra-valued measures, preserving key properties like injectivity and universality in the new setting.

Findings

The generalized KME maintains injectivity and universality.

Empirical results demonstrate effectiveness on synthetic and real data.

Framework enables analysis of higher-order interactions in multivariate and quantum data.

Abstract

Kernel mean embedding (KME) is a powerful tool to analyze probability measures for data, where the measures are conventionally embedded into a reproducing kernel Hilbert space (RKHS). In this paper, we generalize KME to that of von Neumann-algebra-valued measures into reproducing kernel Hilbert modules (RKHMs), which provides an inner product and distance between von Neumann-algebra-valued measures. Von Neumann-algebra-valued measures can, for example, encode relations between arbitrary pairs of variables in a multivariate distribution or positive operator-valued measures for quantum mechanics. Thus, this allows us to perform probabilistic analyses explicitly reflected with higher-order interactions among variables, and provides a way of applying machine learning frameworks to problems in quantum mechanics. We also show that the injectivity of the existing KME and the universality of RKHS are generalized to RKHM, which confirms many useful features of the existing KME remain in our generalized KME. And, we investigate the empirical performance of our methods using synthetic and real-world data.