A typical reconstruction limit for compressed sensing based on
L<sub>p</sub>-norm minimization

Kabashima, Yoshiyuki; Wadayama, Tadashi; Tanaka, Toshiyuki

doi:10.1088/1742-5468/2009/09/l09003

Cited by 178 publications

(306 citation statements)

References 29 publications

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“…During the last few months, several papers investigated compressed sensing problems using the replica method [RFG09, KWT09,GBS09]. In view of the discussion above, it is not surprising that these results can be recovered from the state evolution formalism put forward in [DMM09a].…”

Section: State Evolution and Replica Calculationsmentioning

confidence: 99%

The Noise-Sensitivity Phase Transition in Compressed Sensing

Donoho

Maleki

Montanari

2011

IEEE Trans. Inform. Theory

241

302

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Consider the noisy underdetermined system of linear equations: y = Ax 0 + z 0 , with n × N measurement matrix A, n < N , and Gaussian white noise z 0 ∼ N(0, σ 2 I). Both y and A are known, both x 0 and z 0 are unknown, and we seek an approximation to x 0 . When x 0 has few nonzeros, useful approximations are often obtained by 1 -penalized 2 minimization, in which the reconstructionx 1,λ solves min y − Ax 2 2 /2 + λ x 1 . Evaluate performance by mean-squared error (MSE = E||xConsider matrices A with iid Gaussian entries and a large-system limit in which n, N → ∞ with n/N → δ and k/n → ρ. Call the ratio MSE/σ 2 the noise sensitivity. We develop formal expressions for the MSE ofx 1,λ , and evaluate its worst-case formal noise sensitivity over all types of k-sparse signals. The phase space 0 ≤ δ, ρ ≤ 1 is partitioned by curve ρ = ρ MSE (δ) into two regions. Formal noise sensitivity is bounded throughout the region ρ < ρ MSE (δ) and is unbounded throughout the region ρ > ρ MSE (δ).The phase boundary ρ = ρ MSE (δ) is identical to the previously-known phase transition curve for equivalence of 1 − 0 minimization in the k-sparse noiseless case. Hence a single phase boundary describes the fundamental phase transitions both for the noiseless and noisy cases.Extensive computational experiments validate the predictions of this formalism, including the existence of game theoretical structures underlying it (saddlepoints in the payoff, least-favorable signals and maximin penalization).Underlying our formalism is an approximate message passing soft thresholding algorithm (AMP) introduced earlier by the authors. Other papers by the authors detail expressions for the formal MSE of AMP and its close connection to 1 -penalized reconstruction. Here we derive the minimax formal MSE of AMP and then read out results for 1 -penalized reconstruction.

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Section: State Evolution and Replica Calculationsmentioning

confidence: 99%

The Noise-Sensitivity Phase Transition in Compressed Sensing

Donoho

Maleki

Montanari

2011

IEEE Trans. Inform. Theory

241

302

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“…The problem actually becomes identical to CS with l p norm by assuming P (σ i ) ∝ exp(−β|σ i | p ) in the limit of β → ∞ instead of Eq. (11) as a prior [17].…”

Section: Formulationmentioning

confidence: 99%

“…We then compare performance with that of image restoration and error-correcting codes to clarify the central issue, viz., "universal feature" of the equivalence between thermal and quantum fluctuations in Bayesian MPM estimation. We should mention that the demodulating process in the CDMA system is quite similar to so-called compressed sensing (CS) [15][16][17]. CS and related techniques have been developed to solve various types of modern problems in engineering such as functional magnetic resonance imaging (f-MRI) and image processing.…”

Section: Introductionmentioning

confidence: 99%

Code-division multiple-access multiuser demodulator by using quantum fluctuations

et al. 2014

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We examine the average-case performance of a code-division multiple-access (CDMA) multiuser demodulator in which quantum fluctuations are utilized to demodulate the original message within the context of Bayesian inference. The quantum fluctuations are built into the system as a transverse field in the infinite-range Ising spin glass model. We evaluate the performance measurements by using statistical mechanics. We confirm that the CDMA multiuser modulator using quantum fluctuations achieve roughly the same performance as the conventional CDMA multiuser modulator through thermal fluctuations on average. We also find that the relationship between the quality of the original information retrieval and the amplitude of the transverse field is somehow a "universal feature" in typical probabilistic information processing, viz., in image restoration, error-correcting codes, and CDMA multiuser demodulation.

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“…The threshold δ = δ c (ρ) is called the DonohoTanner threshold. It should be noted that the same result can be derived in several different ways, including the approach on the basis of statistical mechanics of disordered systems [30], [31] and the study on the basis of a modified version of the null space characterization [32]- [34]. For further extensions, see e.g., [35]- [42].…”

Section: Random Matrix Ensemblesmentioning

confidence: 81%

A User's Guide to Compressed Sensing for Communications Systems

Hayashi

Nagahara

Tanaka

2013

IEICE Trans. Commun.

205

110

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SUMMARYThis survey provides a brief introduction to compressed sensing as well as several major algorithms to solve it and its various applications to communications systems. We firstly review linear simultaneous equations as ill-posed inverse problems, since the idea of compressed sensing could be best understood in the context of the linear equations. Then, we consider the problem of compressed sensing as an underdetermined linear system with a prior information that the true solution is sparse, and explain the sparse signal recovery based on 1 optimization, which plays the central role in compressed sensing, with some intuitive explanations on the optimization problem. Moreover, we introduce some important properties of the sensing matrix in order to establish the guarantee of the exact recovery of sparse signals from the underdetermined system. After summarizing several major algorithms to obtain a sparse solution focusing on the 1 optimization and the greedy approaches, we introduce applications of compressed sensing to communications systems, such as wireless channel estimation, wireless sensor network, network tomography, cognitive radio, array signal processing, multiple access scheme, and networked control.

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A typical reconstruction limit for compressed sensing based on L_p-norm minimization

Cited by 178 publications

References 29 publications

The Noise-Sensitivity Phase Transition in Compressed Sensing

The Noise-Sensitivity Phase Transition in Compressed Sensing

Code-division multiple-access multiuser demodulator by using quantum fluctuations

A User's Guide to Compressed Sensing for Communications Systems

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A typical reconstruction limit for compressed sensing based on Lp-norm minimization

Cited by 178 publications

References 29 publications

The Noise-Sensitivity Phase Transition in Compressed Sensing

The Noise-Sensitivity Phase Transition in Compressed Sensing

Code-division multiple-access multiuser demodulator by using quantum fluctuations

A User's Guide to Compressed Sensing for Communications Systems

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Resources

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A typical reconstruction limit for compressed sensing based on L_p-norm minimization