  In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$
describes how much a data source can be compressed (in bit-rate) at any given
level of fidelity (distortion). Obtaining $R(D)$ for a given data source
establishes the fundamental performance limit for all compression algorithms.
We propose a new method to estimate $R(D)$ from the perspective of optimal
transport. Unlike the classic Blahut--Arimoto algorithm which fixes the support
of the reproduction distribution in advance, our Wasserstein gradient descent
algorithm learns the support of the optimal reproduction distribution by moving
particles. We prove its local convergence and analyze the sample complexity of
our R-D estimator based on a connection to entropic optimal transport.
Experimentally, we obtain comparable or tighter bounds than state-of-the-art
neural network methods on low-rate sources while requiring considerably less
tuning and computation effort. We also highlight a connection to
maximum-likelihood deconvolution and introduce a new class of sources that can
be used as test cases with known solutions to the R-D problem.
