On a Relation Between the Rate-Distortion Function and Optimal Transport

We discuss a relationship between rate-distortion and optimal transport (OT) theory, even though they seem to be unrelated at first glance. In particular, we show that a function defined via an extremal entropic OT distance is equivalent to the rate-distortion function. We numerically verify this result as well as previous results that connect the Monge and Kantorovich problems to optimal scalar quantization. Thus, we unify solving scalar quantization and rate-distortion functions in an alternative fashion by using their respective optimal transport solvers. The asymptotic limit on the minimum number of bits required to represent X with average distortion at most D is given by the rate-distortion function (Cover & Thomas (2006)), defined as R(D):= inf I(X; Y). (1) Any rate-distortion pair (R, D) satisfying R > R(D) is achievable by some lossy source code, and no code can achieve a rate-distortion less than R(D).