Rate-Distortion Dimension of Stochastic Processes

The rate-distortion dimension (RDD) of an analog stationary process is studied as a measure of complexity that captures the amount of information contained in the process. It is shown that the RDD of a process, defined as the asymptotic ratio of its rate-distortion function R(D) to log 1/D as distortion D approaches zero, is equal to its information dimension (ID). This generalizes an earlier result by Kawabata and Dembo and provides an operational approach to evaluate the ID of a process, which previously was shown to be closely related to the effective dimension of the underlying process and also to the fundamental limits of compressed sensing. This is illustrated through the computation of the RDD of a piecewise constant process.