Compressed sensing with ℓ 0-norm: statistical physics analysis & algorithms for signal recovery
Noiseless compressive sensing is a protocol that enables undersampling and later recovery of a signal without loss of information. This compression is possible because the signal is usually sufficiently sparse in a given basis. Currently, the algorithm offering the best tradeoff between compression rate, robustness, and speed for compressive sensing is the LASSO (l(1)-norm bias) algorithm. However, many studies have pointed out the possibility that the implementation of l(p)-norms biases, with p smaller than one, could give better performance while sacrificing convexity. In this work, we focus specifically on the extreme case of the l(0)-based reconstruction, a task that is complicated by the discontinuity of the loss. In the first part of the paper, we describe via statistical physics methods, and in particular the replica method, how the solutions to this optimization problem are arranged in a clustered structure. We observe two distinct regimes: one at low compression rate where the signal can be recovered exactly, and one at high compression rate where the signal cannot be recovered accurately. In the second part, we present two message-passing algorithms based on our first results for the l(0)-norm optimization problem. The proposed algorithms are able to recover the signal at compression rates higher than the ones achieved by LASSO while being computationally efficient.
WOS:001031733100058
2023-01-01
979-8-3503-0149-6
New York
Information Theory Workshop
323
328
REVIEWED
Event name | Event place | Event date |
Saint-Malo, FRANCE | Apr 23-28, 2023 | |