Lower Bounds on the Rate-Distortion Function of Individual LDGM Codes

We consider lossy compression of a binary symmetric source by means of a low-density generator-matrix code. We derive two lower bounds on the rate distortion function which are valid for any low-density generator-matrix code with a given node degree distribution L(x) on the set of generators and for any encoding algorithm. These bounds show that, due to the sparseness of the code, the performance is strictly bounded away from the Shannon rate-distortion function. In this sense, our bounds represent a natural generalization of Gallager's bound on the maximum rate at which low-density parity-check codes can be used for reliable transmission. Our bounds are similar in spirit to the technique recently developed by Dimakis, Wainwright, and Ramchandran, but they apply to individual codes.