  We propose an information-theoretic framework for analog signal separation.
Specifically, we consider the problem of recovering two analog signals, modeled
as general random vectors, from the noiseless sum of linear measurements of the
signals. Our framework is inspired by the groundbreaking work of Wu and Verd\'u
(2010) on analog compression and encompasses, inter alia, inpainting,
declipping, super-resolution, the recovery of signals corrupted by impulse
noise, and the separation of (e.g., audio or video) signals into two distinct
components. The main results we report are general achievability bounds for the
compression rate, i.e., the number of measurements relative to the dimension of
the ambient space the signals live in, under either measurability or H\"older
continuity imposed on the separator. Furthermore, we find a matching converse
for sources of mixed discrete-continuous distribution. For measurable
separators our proofs are based on a new probabilistic uncertainty relation
which shows that the intersection of generic subspaces with general sets of
sufficiently small Minkowski dimension is empty. H\"older continuous separators
are dealt with by introducing the concept of regularized probabilistic
uncertainty relations. The probabilistic uncertainty relations we develop are
inspired by embedding results in dynamical systems theory due to Sauer et al.
(1991) and---conceptually---parallel classical Donoho-Stark and Elad-Bruckstein
uncertainty principles at the heart of compressed sensing theory.
Operationally, the new uncertainty relations take the theory of sparse signal
separation beyond traditional sparsity---as measured in terms of the number of
non-zero entries---to the more general notion of low description complexity as
quantified by Minkowski dimension. Finally, our approach also allows to
significantly strengthen key results in Wu and Verd\'u (2010).
