An Optimal Diffusion Approach to Quadratic Rate-Distortion Problems: New Solution and Approximation Methods

When compressing continuous data, some loss of information is inevitable, and this incurred a distortion upon reconstruction. The Rate–Distortion (RD) function characterizes the minimum achievable rate for a code whose decoding permits a specified amount of distortion. We exploit the connection between rate-distortion theory and entropic optimal transport to propose a novel stochastic-control formulation for the former, and use a classic result dating back to Schrodinger to show that the tradeoff between rate and mean squared error distortion is equivalent to a tradeoff between control energy and the differential entropy of the terminal state, whose probability law defines the reconstruction distribution. For a special class of sources, we show that the optimal control law and the corresponding trajectory in the space of probability measures are obtained by solving a backward heat equation. In more general settings, our approach yields a numerical method that estimates the RD function using diffusion processes with a constant diffusion coefficient. We demonstrate the effectiveness of our method through several examples.