  This paper is concerned with a rate-distortion theory for sequences of i.i.d.
random variables with general distribution supported on general sets including
manifolds and fractal sets. Manifold structures are prevalent in data science,
e.g., in compressed sensing, machine learning, image processing, and
handwritten digit recognition. Fractal sets find application in image
compression and in modeling of Ethernet traffic. We derive a lower bound on the
(single-letter) rate-distortion function that applies to random variables X of
general distribution and for continuous X reduces to the classical Shannon
lower bound. Moreover, our lower bound is explicit up to a parameter obtained
by solving a convex optimization problem in a nonnegative real variable. The
only requirement for the bound to apply is the existence of a sigma-finite
reference measure for X satisfying a certain subregularity condition. This
condition is very general and prevents the reference measure from being highly
concentrated on balls of small radii. To illustrate the wide applicability of
our result, we evaluate the lower bound for a random variable distributed
uniformly on a manifold, namely, the unit circle, and a random variable
distributed uniformly on a self-similar set, namely, the middle third Cantor
set.
