Analysis via Orthonormal Systems in Reproducing Kernel Hilbert $C^*$-Modules and Applications

Kernel Mean Embeddings of Von Neumann-Algebra-Valued Measures

“…•, • : M × M → A that satisfies the following four conditions for u, v, w ∈ M and c, d ∈ A: 1. u, vc + wd = u, v c + u, w d, 2. v, u = u, v * , 3. u, u ≥ 0, and 4. u, u = 0 implies u = 0. The A-valued inner product induces the notion of orthonormal, which is important for solving various minimization problems [15]. A set of vectors {p 1 , .…”

“…PCA is a statistical procedure to find a low dimensional subspace that preserves information of samples, which has been applied to, for example, visualization and anomaly detection [23,27,15]. We introduce a PCA for A-valued measures in terms of the proposed KME in RKHM.…”

“…In this paper, we apply theories of von Neumann-algebra-valued measures (more generally, vectorvalued measures) to define KME of von Neumann-algebra valued measures, and generalize the KME in RKHS to reproducing kernel Hilbert modules (RKHMs), which enables us to embed von Neumannalgebra-valued measures into RKHMs. RKHM is a generalization of RKHS [20,18,41,15], and von Neumann-algebra is a special class of bounded linear operators on a Hilbert space. An important example of von Neumann-algebras is the space of matrices C m×m , where a C m×m -valued measure can describe m 2 variables simultaneously; thus, it can be employed to describe relations of variable pairs in the distributions.…”