MMSE‐based version of OMP for recovery of discrete‐valued sparse signals

Sparrer, Susanne; Fischer, Robert F. H.

doi:10.1049/el.2015.0924

Cited by 22 publications

(9 citation statements)

References 9 publications

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“…uniform distribution on the (complex) power shell. We decode the inner code using the OMP [24] and its MMSE-based extension [32]. As the OMP is a sequential algorithm, the number of output codewords equals the number of steps.…”

Section: Numerical Resultsmentioning

confidence: 99%

Coded Compressed Sensing with List Recoverable Codes for the Unsourced Random Access

Andreev¹,

Rybin²,

Frolov³

2022

Preprint

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We consider a coded compressed sensing approach for the unsourced random access and replace the outer tree code proposed by Amalladinne et al. with the list recoverable code capable of correcting t errors. A finite-length random coding bound for such codes is derived. The numerical experiments in the single antenna quasi-static Rayleigh fading MAC show that transition to list recoverable codes correcting t errors improves the performance of coded compressed sensing scheme by 7-10 dB compared to the tree code-based scheme. We propose two practical constructions of outer codes. The first is a modification of the tree code. It utilizes the same code structure, and a key difference is a decoder capable of correcting up to t errors. The second is based on the Reed-Solomon codes and Guruswami-Sudan list decoding algorithm. The first scheme provides an energy efficiency very close to the random coding bound when the decoding complexity is unbounded. But for the practical parameters, the second scheme is better and improves the performance of a tree code-based scheme when the number of active users is less than 200.

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Section: Numerical Resultsmentioning

confidence: 99%

Coded Compressed Sensing with List Recoverable Codes for the Unsourced Random Access

Andreev¹,

Rybin²,

Frolov³

2022

Preprint

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“…Hence,x is the unique optimal solution of (P 1 ([ℓ, u] Z )) if and only ifx + andx − form the unique minimizer of (7). The claims now follow from Theorem 34 applied to the reformulation (7) of (P 1 ([ℓ, u] Z )).…”

Section: Which Reveals That a Violates The (Standard Continuous) Nsp(mentioning

confidence: 90%

“…There are only few articles in the literature that deal with this case. For instance, Sparrer and Fischer [7] present a heuristic approach based on orthogonal matching pursuit. A further heuristic was proposed by Flinth and Kutyniok [8] based on a combination of projection and orthogonal matching pursuit ideas.…”

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confidence: 99%

Sparse recovery with integrality constraints

Lange

Pfetsch

Seib

et al. 2020

Discrete Applied Mathematics

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In this paper, we investigate conditions for the unique recoverability of sparse integer-valued signals from few linear measurements. Both the objective of minimizing the number of nonzero components, the so-called ℓ 0 -norm, as well as its popular substitute, the ℓ 1 -norm, are covered. Furthermore, integer constraints and possible bounds on the variables are investigated. Our results show that the additional prior knowledge of signal integrality allows for recovering more signals than what can be guaranteed by the established recovery conditions from (continuous) compressed sensing. Moreover, even though the considered problems are NP-hard in general (even with an ℓ 1 -objective), we investigate testing the ℓ 0 -recovery conditions via some numerical experiments; it turns out that the corresponding problems are quite hard to solve in practice. However, medium-sized instances of ℓ 0 -and ℓ 1 -minimization with binary variables can be solved exactly within reasonable time. Index TermsSparse recovery, compressed sensing, integrality constraints, nullspace conditions I. INTRODUCTIONT HE recovery of sparse signals has received a tremendous interest in recent years. The basic setting without noise is as follows: Under the prior knowledge that a measurement vector b ∈ R m \ {0} is generated by a sparse signal x ∈ R n via Ax = b, where A ∈ R m×n with rank(A) = m < n is the sensing matrix, the question is whether x can be uniquely recovered, given A and b. Thus, one approach is to find the sparsest x that explains the measurements, i.e., one minimizes x 0 := |{i ∈ {1, . . . , n} : x i = 0}| under the constraint Ax = b. However, this problem is NP-hard, see Garey and Johnson [1]. The crucial idea in this context (see, e.g., Chen et al. [2]) is to replace x 0 by the ℓ 1 -norm x 1 := |x 1 | + · · · + |x n |, which results in a convex problem that can even be cast as a linear program (LP) and is therefore tractable. The literature contains an abundance of conditions under which minimizers of x 1 subject to Ax = b are unique and equal to the sparsest solution; at this place, we refer to the book by Foucart and Rauhut [3] for more information and an overview of selected specialized algorithms to solve the ℓ 1 -minimization problem.The key point for the mentioned series of striking results is the prior knowledge that b can be sparsely represented or approximated. A natural question is whether further knowledge about the structure of the representations x can lead to stronger results about the recoverability. In general terms, the two problems from above can be written aswhere X ⊆ R n is a constraint set representing further restrictions on the representations. The "classical" results in the literature refer to the case X = R n . One main example in which X = R n is the case in which x has to be nonnegative, i.e., X = R n + , see, for instance, Donoho and Tanner [4], Bruckstein et al. [5] and Khajehnejad et al. [6].In this paper, we investigate the case in which x is required to be integer, i.e., X ⊆ Z n . This is motivated by ...

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“…A recent study verified that these problems, as well as those with unbounded integer signals, are NP-hard [44]. Algorithms for the unconstrained integer SMV problem thus apply greedy heuristics such as OMP-based approaches [45,46].…”

Section: Previous Workmentioning

confidence: 99%

Extreme Compressed Sensing of Poisson Rates from Multiple Measurements

Kota,

LeJeune,

Drezek

et al. 2021

Preprint

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Compressed sensing (CS) is a signal processing technique that enables the efficient recovery of a sparse high-dimensional signal from low-dimensional measurements. In the multiple measurement vector (MMV) framework, a set of signals with the same support must be recovered from their corresponding measurements. Here, we present the first exploration of the MMV problem where signals are independently drawn from a sparse, multivariate Poisson distribution. We are primarily motivated by a suite of biosensing applications of microfluidics where analytes (such as whole cells or biomarkers) are captured in small volume partitions according to a Poisson distribution. We recover the sparse parameter vector of Poisson rates through maximum likelihood estimation with our novel Sparse Poisson Recovery (SPoRe) algorithm. SPoRe uses batch stochastic gradient ascent enabled by Monte Carlo approximations of otherwise intractable gradients. By uniquely leveraging the Poisson structure, SPoRe substantially outperforms a comprehensive set of existing and custom baseline CS algorithms. Notably, SPoRe can exhibit high performance even with one-dimensional measurements and high noise levels. This resource efficiency is not only unprecedented in the field of CS but is also particularly potent for applications in microfluidics in which the number of resolvable measurements per partition is often severely limited. We prove the identifiability property of the Poisson model under such lax conditions, analytically develop insights into system performance, and confirm these insights in simulated experiments. Our findings encourage a new approach to biosensing and are generalizable to other applications featuring spatial and temporal Poisson signals.

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MMSE‐based version of OMP for recovery of discrete‐valued sparse signals

Cited by 22 publications

References 9 publications

Coded Compressed Sensing with List Recoverable Codes for the Unsourced Random Access

Coded Compressed Sensing with List Recoverable Codes for the Unsourced Random Access

Sparse recovery with integrality constraints

Extreme Compressed Sensing of Poisson Rates from Multiple Measurements

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