21 research outputs found
Rank Minimization over Finite Fields: Fundamental Limits and Coding-Theoretic Interpretations
This paper establishes information-theoretic limits in estimating a finite
field low-rank matrix given random linear measurements of it. These linear
measurements are obtained by taking inner products of the low-rank matrix with
random sensing matrices. Necessary and sufficient conditions on the number of
measurements required are provided. It is shown that these conditions are sharp
and the minimum-rank decoder is asymptotically optimal. The reliability
function of this decoder is also derived by appealing to de Caen's lower bound
on the probability of a union. The sufficient condition also holds when the
sensing matrices are sparse - a scenario that may be amenable to efficient
decoding. More precisely, it is shown that if the n\times n-sensing matrices
contain, on average, \Omega(nlog n) entries, the number of measurements
required is the same as that when the sensing matrices are dense and contain
entries drawn uniformly at random from the field. Analogies are drawn between
the above results and rank-metric codes in the coding theory literature. In
fact, we are also strongly motivated by understanding when minimum rank
distance decoding of random rank-metric codes succeeds. To this end, we derive
distance properties of equiprobable and sparse rank-metric codes. These
distance properties provide a precise geometric interpretation of the fact that
the sparse ensemble requires as few measurements as the dense one. Finally, we
provide a non-exhaustive procedure to search for the unknown low-rank matrix.Comment: Accepted to the IEEE Transactions on Information Theory; Presented at
IEEE International Symposium on Information Theory (ISIT) 201
Recovery of binary sparse signals from compressed linear measurements via polynomial optimization
The recovery of signals with finite-valued components from few linear
measurements is a problem with widespread applications and interesting
mathematical characteristics. In the compressed sensing framework, tailored
methods have been recently proposed to deal with the case of finite-valued
sparse signals. In this work, we focus on binary sparse signals and we propose
a novel formulation, based on polynomial optimization. This approach is
analyzed and compared to the state-of-the-art binary compressed sensing
methods
Advances in the recovery of binary sparse signals
Recently, the recovery of binary sparse signals from compressed linear systems has received attention due to its several applications. In this contribution, we review the latest results in this framework, that are based on a suitable non-convex polynomial formulation of the
problem. Moreover, we propose novel theoretical results. Then, we show numerical results that highlight the enhancement obtained through the non-convex approach with respect to the state-of-the-art methods