  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.
