Statistical physics-based reconstruction in compressed sensing

By Florent Krzakala, M. M'ezard, François Sausset, Yifan Sun, Lenka Zdeborov'a. 2011

318 citations recorded by the source corpus.

Compressed sensing is triggering a major evolution in signal acquisition. It consists in sampling a sparse signal at low rate and later using computational power for its exact reconstruction, so that only the necessary information is measured. Currently used reconstruction techniques are, however, limited to acquisition rates larger than the true density of the signal. We design a new procedure which is able to reconstruct exactly the signal with a number of measurements that approaches the theoretical limit in the limit of large systems. It is based on the joint use of three essential ingredients: a probabilistic approach to signal reconstruction, a message-passing algorithm adapted from belief propagation, and a careful design of the measurement matrix inspired from the theory of crystal nucleation. The performance of this new algorithm is analyzed by statistical physics methods. The obtained improvement is confirmed by numerical studies of several cases.

Subfields: Artificial Intelligence, Computational Mechanics, Structural Biology

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Citations in the corpus

The Effect of Gradient Noise on the Energy Landscape of Deep Networks (2015, background): “…of spin glasses to combinatorial optimization, random k-SAT, LDPC codes etc. have become popular (Mezard and Montanari, 2009) and an understanding of the energy landscape and thermodynamics of such systems has led to a number of efficient algorithms (Krzakala et al., 2011; Braunstein et al., 2005).”
Generalized approximate message passing for estimation with random linear mixing (2010, methodology): “Such approximate methods originated in CDMA multiuser detection problems in [1]– [5], and, more recently, have attracted considerable interest in compressed sensing [6]–[9], [20], [21].”
Optimal errors and phase transitions in high-dimensional generalized linear models (2017, background): “The phase diagram of this case is well known (67, 68).”
Probabilistic reconstruction in compressed sensing: algorithms, phase diagrams, and threshold achieving matrices (2012, methodology): “We introduced seeded measurement matrices for CS in [1], and showed there, both numerically and using the density evolution, that with such matrices it is possible to achieve the information theoretic optimal measurement rates.”
Spatially coupled ensembles universally achieve capacity under belief propagation (2012, background): “ompressed sensing [88]– [91]. In [88], Kudekar and Pﬁster use sparse measurement matrices with sub-optimal veriﬁcation decoding and show that spatial coupling boosts thresholds of sparse recovery. In [90], [91], Krzakala, Me´zard, Sausset, Sun, and Zdeborova as well as Donoho, Javanmard, and Montana”
Statistical physics of inference: thresholds and algorithms (2015, methodology): “[132] where spatially coupled measurement matrices were designed that allowed AMP to reconstruct the signal in compressed sensing anytime the measurement rate α is larger than the density of non-zero elements in the signal ρ, i.”