11:00 AM

Tue

15

Apr

Basic rate-distortion theory consists of two theorems: 1. The rate-distortion theorem: let R(D) denote the infimum of rates needed to compress a source X to within a distortion D. In the rigorous framework of information theory, R(D) is an infinite dimensional optimization program. The rate-distortion theory states that a simplified expression for R(D) is a finite dimensional minimum mutual information expression RI(D). 2. Optimality of source-channel separation for communication with distortion First, Point 2 above, will be considered. The usual proof for Point 2 goes as follows: Let c be a channel. Let C denote the capacity, that is, the supremum of rates at which reliable communication is possible over c. In the framework of information theory, C is an infinite dimensional optimization program. The noisy channel coding theorem gives a finite dimensional simplified maximum mutual information expression CI for the capacity c. If it is known that a source X can be communicated to within a distortion D over c, the first step in proving the optimality of source-channel separation for communication with distortion is to prove that CI>RI(D). From this, it follows that C>R(D). Then, by use of a standard source-coding followed by channel-coding argument, the optimality of source-channel separation for communication with distortion follows. Note that this proof first proves CI RI(D). Thus, the proof goes through using simplified finite dimensional expressions for the infinite dimensional capacity C and the infinite-dimensional rate-distortion function R(D). We provide a proof where simplifications need to be carried out at a ``lower'' level. The exact nature of these ``lower'' level of simplifications will become clearer in the talks. If time permits, we will consider Point 1. Point 1. is a statement regarding a simplified finite-dimensional expression for R(D) and hence, the proof has to go through simplifications. We provide a proof where the simplification is carried out ``very late'' in the proof, as compared to in the existing proofs. We believe that these proofs shed insight into the workings of rate-distortion theory.