The rate-distortion dimension of sets and measures

Data compression of independent samples drawn from a fractal set is considered. The asymptotic ratio of rate to magnitude log distortion characterizes the effective dimension occupied by the underlying distribution. This quantity is shown to be identical to Renyi's (1959) information dimension. For self-similar fractal sets this dimension is distribution dependent-in sharp contrast with the behavior of absolutely continuous measures. The rate-distortion dimension of a set is defined as the maximal rate-distortion dimension for distributions supported on this set. Kolmogorov's metric dimension is an upper bound on the rate-distortion dimension, while the Hausdorff dimension is a lower bound. Examples of sets for which the rate-distortion dimension differs from these bounds are provided.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>