Duality and geodesics for probabilistic frames

Abstract

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their properties. In particular, we formulate a theory of transport duals for probabilistic frames and prove certain properties of this class. We also investigate paths of probabilistic frames, identifying conditions under which geodesic paths between two such measures are themselves probabilistic frames. In the discrete case this is related to ranks of convex combinations of matrices, while in the continuous case this is related to the continuity of the optimal transport plan.

Harmonic analysis of network systems via kernels and their boundary realizations

“…In summary, condition (38) has its roots in probability and in the study of Gaussian processes. For the following result we cite the papers [7,11,9,10,12,17,35,30,31,32,37,36,38,58,43,44,45,57,46,65,68,67,72,73,78], and especially [62].…”

“…Application of positive definite kernels to harmonic analysis of fractals: [21,14,22,69,60,28,29,3,56]. Connections to stochastic analysis and diffusion: [62,34,11,31,32,36,59,78,80]. And use of kernel-tools in neural network models, data analysis, and in learning theory: [20,24,35,63,64].…”

Frames and subspaces

“…The investigation of probabilistic frames is still at its initial stage. For example, in [52] the authors introduced the notion of transport duals and used the setting of the Wasserstein metric to investigate the properties of such probabilistic frames. In particular, this setting offers the flexibility to find (non-discrete) probabilistic frames which are duals to a given probabilistic frame.…”