  The success of the compressed sensing paradigm has shown that a substantial
reduction in sampling and storage complexity can be achieved in certain linear
and non-adaptive estimation problems. It is therefore an advisable strategy for
noncoherent information retrieval in, for example, sporadic blind and
semi-blind communication and sampling problems. But, the conventional model is
not practical here since the compressible signals have to be estimated from
samples taken solely on the output of an un-calibrated system which is unknown
during measurement but often compressible. Conventionally, one has either to
operate at suboptimal sampling rates or the recovery performance substantially
suffers from the dominance of model mismatch. In this work we discuss such type
of estimation problems and we focus on bilinear inverse problems. We link this
problem to the recovery of low-rank and sparse matrices and establish stable
low-dimensional embeddings of the uncalibrated receive signals whereby
addressing also efficient communication-oriented methods like universal random
demodulation. Exemplary, we investigate in more detail sparse convolutions
serving as a basic communication channel model. In using some recent results
from additive combinatorics we show that such type of signals can be
efficiently low-rate sampled by semi-blind methods. Finally, we present a
further application of these results in the field of phase retrieval from
intensity Fourier measurements.
