Nonanticipative Rate Distortion Function and Relations to Filtering Theory

TL;DR

This paper explores the connection between nonanticipative rate distortion functions and filtering theory, establishing existence, deriving solutions, and illustrating with an example in the context of stationary processes.

Contribution

It introduces a realizability constraint to relate RDF to filtering, proves the existence of extremum solutions, and derives closed-form solutions for stationary processes.

Findings

Existence of extremum solutions via weak* convergence.

Closed-form extremum reconstruction distribution for stationary processes.

Illustrative example demonstrating the theoretical concepts.

Abstract

The relation between nonanticipative Rate Distortion Function (RDF) and filtering theory is discussed on abstract spaces. The relation is established by imposing a realizability constraint on the reconstruction conditional distribution of the classical RDF. Existence of the extremum solution of the nonanticipative RDF is shown using weak$^*$-convergence on appropriate topology. The extremum reconstruction conditional distribution is derived in closed form, for the case of stationary processes. The realization of the reconstruction conditional distribution which achieves the infimum of the nonanticipative RDF is described. Finally, an example is presented to illustrate the concepts.