Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures
This work provides a theoretical extension for analyzing complex multivariate and quantum data, but it is incremental as it builds on existing kernel mean embedding frameworks.
The authors generalized kernel mean embeddings to von Neumann-algebra-valued measures using reproducing kernel Hilbert modules, enabling probabilistic analyses with higher-order variable interactions and applications in quantum mechanics, with empirical validation on synthetic and real-world data.
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.