Linear Programming based Converses for Finite Blocklength Lossy Joint Source-Channel Coding

A linear programming (LP) based framework is presented for obtaining converses for finite blocklength lossy joint source-channel coding problems. The framework applies for any loss criterion, generalizes certain previously known converses, and also extends to multi-terminal settings. The finite blocklength problem is posed equivalently as a nonconvex optimization problem and using a lift-and-project-like method, a close but tractable LP relaxation of this problem is derived. Lower bounds on the original problem are obtained by the construction of feasible points for the dual of the LP relaxation. A particular application of this approach leads to the converse of Kostina and Verd{\'u} on the minimum excess distortion probability of a finite blocklength lossy joint source-channel code and to the converse of Wolfowitz for channel coding, showing thereby that our LP relaxation is asymptotically tight with increasing blocklengths for channel coding, lossless source coding and joint source-channel coding with the excess distortion probability as the loss criterion. The relaxation is also shown to be tight for all blocklengths for the minimization of the expected average bit-wise Hamming distortion of a binary uniform source over a binary symmetric memoryless channel. Using a duality based argument, a new general converse is derived for finite blocklength joint source-channel coding for a class of channels called the metric channels. We demonstrate how the optimization formulation and the lift-and-project method can be extended to networked settings.