Consensus-based sparse signal reconstruction algorithm for wireless sensor           networks

Bao, Peng; Zhao, Zhi; Han, Guangjie; Shen, Jian

doi:10.1177/1550147716666290

“…More specifically, we say that the unknown vector β * is k-sparse (exactly k-sparse) if it has at most k non-zero coordinates (exactly k-nonzero coordinates). The sparsity is a very useful assumption in applications, for example in compressed sensing [15], [20] , biomedical imaging [10], [34] and sensor networks [40], [39], but also in theory [20]. For our purposes we assume that the value of k is known for all the results.…”

Section: Introductionmentioning

confidence: 99%

Sparse high-dimensional linear regression. Estimating squared error and a phase transition

Gamarnik

¹

,

Zadik

²

2022

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We consider a sparse high dimensional regression model where the goal is to recover a ksparse unknown vector β * from n noisy linear observations of the form Y = Xβ * + W ∈ R n where X ∈ R n×p has iid N (0, 1) entries and W ∈ R n has iid N (0, σ 2 ) entries. Under certain assumptions on the parameters, an intriguing assymptotic gap appears between the minimum value of n, call it n * , for which the recovery is information theoretically possible, and the minimum value of n, call it n alg , for which an efficient algorithm is known to provably recover β * . In [26] it was conjectured that the gap is not artificial, in the sense that for sample sizes n ∈ [n * , n alg ] the problem is algorithmically hard.We support this conjecture in two ways. Firstly, we show that the optimal solution of the LASSO provably fails to ℓ 2 -stably recover the unknown vector β * when n ∈ [n * , cn alg ], for some sufficiently small constant c > 0. Secondly, we establish that n alg , up to a multiplicative constant factor, is a phase transition point for the appearance of a certain Overlap Gap Property (OGP) over the space of k-sparse vectors. The presence of such an Overlap Gap Property phase transition, which originates in statistical physics, is known to provide evidence of an algorithmic hardness. Finally we show that if n > Cn alg for some large enough constant C > 0, a very simple algorithm based on a local search improvement rule is able both to ℓ 2 -stably recover the unknown vector β * and to infer correctly its support, adding it to the list of provably successful algorithms for the high dimensional linear regression problem.

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“…Since then they have played a crucial part in computer science and the success related to distributed computing [7]. The same algorithm has been researched and is being used in different fields including robotics [9], autonomous vehicles [10], wireless sensors [11] and unmanned aerial vehicles [12].…”

Section: Literature Reviewmentioning

confidence: 99%

Consensus Based Distributed Control in Micro-Grid Clusters

Fuad¹

1

0

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Sparse High-Dimensional Linear Regression. Algorithmic Barriers and a Local Search Algorithm

Gamarnik,

Zadik

2017

Preprint

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We consider a sparse high dimensional regression model where the goal is to recover a ksparse unknown vector β * from n noisy linear observations of the form Y = Xβ * + W ∈ R n where X ∈ R n×p has iid N (0, 1) entries and W ∈ R n has iid N (0, σ 2 ) entries. Under certain assumptions on the parameters, an intriguing assymptotic gap appears between the minimum value of n, call it n * , for which the recovery is information theoretically possible, and the minimum value of n, call it n alg , for which an efficient algorithm is known to provably recover β * . In [26] it was conjectured that the gap is not artificial, in the sense that for sample sizes n ∈ [n * , n alg ] the problem is algorithmically hard.We support this conjecture in two ways. Firstly, we show that the optimal solution of the LASSO provably fails to ℓ 2 -stably recover the unknown vector β * when n ∈ [n * , cn alg ], for some sufficiently small constant c > 0. Secondly, we establish that n alg , up to a multiplicative constant factor, is a phase transition point for the appearance of a certain Overlap Gap Property (OGP) over the space of k-sparse vectors. The presence of such an Overlap Gap Property phase transition, which originates in statistical physics, is known to provide evidence of an algorithmic hardness. Finally we show that if n > Cn alg for some large enough constant C > 0, a very simple algorithm based on a local search improvement rule is able both to ℓ 2 -stably recover the unknown vector β * and to infer correctly its support, adding it to the list of provably successful algorithms for the high dimensional linear regression problem.

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Consensus-based sparse signal reconstruction algorithm for wireless sensor networks

Cited by 3 publications

References 37 publications

Sparse high-dimensional linear regression. Estimating squared error and a phase transition

Sparse high-dimensional linear regression. Estimating squared error and a phase transition

Consensus Based Distributed Control in Micro-Grid Clusters

Sparse High-Dimensional Linear Regression. Algorithmic Barriers and a Local Search Algorithm

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