One step RSB scheme for the rate distortion function

Abstract

We apply statistical mechanics to an inverse problem of linear mapping to investigate the physics of the irreversible compression. We use the replica symmetry breaking (RSB) technique with a toy model to demonstrate the Shannon's result. The rate distortion function, which is widely known as the theoretical limit of the compression with a fidelity criterion, is derived using the Parisi one step RSB scheme. The bound can not be achieved in the sparsely-connected systems, where suboptimal solutions dominate the capacity.