6 research outputs found
Almost Lossless Analog Compression without Phase Information
We propose an information-theoretic framework for phase retrieval.
Specifically, we consider the problem of recovering an unknown n-dimensional
vector x up to an overall sign factor from m=Rn phaseless measurements with
compression rate R and derive a general achievability bound for R.
Surprisingly, it turns out that this bound on the compression rate is the same
as the one for almost lossless analog compression obtained by Wu and Verd\'u
(2010): Phaseless linear measurements are as good as linear measurements with
full phase information in the sense that ignoring the sign of m measurements
only leaves us with an ambiguity with respect to an overall sign factor of x
Homomorphic Sensing of Subspace Arrangements
Homomorphic sensing is a recent algebraic-geometric framework that studies
the unique recovery of points in a linear subspace from their images under a
given collection of linear maps. It has been successful in interpreting such a
recovery in the case of permutations composed by coordinate projections, an
important instance in applications known as unlabeled sensing, which models
data that are out of order and have missing values. In this paper, we provide
tighter and simpler conditions that guarantee the unique recovery for the
single-subspace case, extend the result to the case of a subspace arrangement,
and show that the unique recovery in a single subspace is locally stable under
noise. We specialize our results to several examples of homomorphic sensing
such as real phase retrieval and unlabeled sensing. In so doing, in a unified
way, we obtain conditions that guarantee the unique recovery for those
examples, typically known via diverse techniques in the literature, as well as
novel conditions for sparse and unsigned versions of unlabeled sensing.
Similarly, our noise result also implies that the unique recovery in unlabeled
sensing is locally stable.Comment: 18 page