Signal representations with minimum &amp;#x2113;&lt;inf&gt;&amp;#x221E;&lt;/inf&gt;-norm

Studer, Christoph; Yin, Wotao; Baraniuk, Richard G.

doi:10.1109/allerton.2012.6483364

Cited by 22 publications

(22 citation statements)

References 21 publications

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“…We also compare to the recovery guarantee provided for PhaseMax using classical machine learning methods in [3]. 6 We see that PhaseMax requires the same sample complexity (number of required measurements) as compared to PhaseLift, TWF, and TAW, when used together with the truncated spectral initializer [19]. While the constants c 0 , c 1 , and c 2 in the recovery guarantees for all of the other methods are generally very large, our recovery guarantees contain no unspecified constants, explicitly depend on the approximation factor α, and are surprisingly sharp.…”

Section: A Comparison With Existing Recovery Guaranteesmentioning

confidence: 99%

PhaseMax: Convex Phase Retrieval via Basis Pursuit

Goldstein

Studer

2018

IEEE Trans. Inform. Theory

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252

162

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Abstract-We consider the recovery of a (real-or complexvalued) signal from magnitude-only measurements, known as phase retrieval. We formulate phase retrieval as a convex optimization problem, which we call PhaseMax. Unlike other convex methods that use semidefinite relaxation and lift the phase retrieval problem to a higher dimension, PhaseMax is a "non-lifting" relaxation that operates in the original signal dimension. We show that the dual problem to PhaseMax is Basis Pursuit, which implies that phase retrieval can be performed using algorithms initially designed for sparse signal recovery. We develop sharp lower bounds on the success probability of PhaseMax for a broad range of random measurement ensembles, and we analyze the impact of measurement noise on the solution accuracy. We use numerical results to demonstrate the accuracy of our recovery guarantees, and we showcase the efficacy and limits of PhaseMax in practice.

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Section: A Comparison With Existing Recovery Guaranteesmentioning

confidence: 99%

PhaseMax: Convex Phase Retrieval via Basis Pursuit

Goldstein

Studer

2018

IEEE Trans. Inform. Theory

Self Cite

252

162

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“…, 256. As it has been empirically shown in [29] for randomly subsampled DCT matrices, the SNR y is expected to be an increasing function of N/M for a given PAPR level of anti-sparsity. The rational is that the number of combinations of ± y 2 /N increases when N grows, leading to more chance of finding a better representation for a given PAPR, due to increased redundancy.…”

Section: A a Toy Examplementioning

confidence: 82%

“…Resorting to an uncertainty principle (UP), Lyubarskii et al have also introduced several examples of frames yielding computable Kashin's representations, such as random orthogonal matrices, random subsampled discrete Fourier transform (DFT) matrices, and random sub-Gaussian matrices [24]. The properties of the alternate optimization problem, which consists of minimizing the maximum magnitude of the representation coefficients for an upper-bounded 2 -reconstruction error, have been deeply investigated in [29], [30]. In these latest contributions, the optimal expansion is called the democratic representation and some bounds associated with archetypal matrices ensuring the UP are derived.…”

Section: Introductionmentioning

confidence: 99%

Bayesian Antisparse Coding

Elvira

Chainais

Dobigeon

2017

IEEE Trans. Signal Process.

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Sparse representations have proven their efficiency in solving a wide class of inverse problems encountered in signal and image processing. Conversely, enforcing the information to be spread uniformly over representation coefficients exhibits relevant properties in various applications such as robust encoding in digital communications. Anti-sparse regularization can be naturally expressed through an ∞-norm penalty. This paper derives a probabilistic formulation of such representations. A new probability distribution, referred to as the democratic prior, is first introduced. Its main properties as well as three random variate generators for this distribution are derived. Then this probability distribution is used as a prior to promote anti-sparsity in a Gaussian linear model, yielding a fully Bayesian formulation of anti-sparse coding. Two Markov chain Monte Carlo (MCMC) algorithms are proposed to generate samples according to the posterior distribution. The first one is a standard Gibbs sampler. The second one uses Metropolis-Hastings moves that exploit the proximity mapping of the log-posterior distribution. These samples are used to approximate maximum a posteriori and minimum mean square error estimators of both parameters and hyperparameters. Simulations on synthetic data illustrate the performances of the two proposed samplers, for both complete and over-complete dictionaries. All results are compared to the recent deterministic variational FITRA algorithm.

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“…The ℓ ∞ regularization has found applications in several fields [32,38,53]. Suppose that θ = 0, and define the saturation support of θ as I sat…”

Section: Anti-sparsitymentioning

confidence: 99%

Sharp oracle inequalities for low-complexity priors

Luu¹,

Fadili²,

Chesneau³

2018

Ann Inst Stat Math

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In this paper, we consider a high-dimensional statistical estimation problem in which the the number of parameters is comparable or larger than the sample size. We present a unified analysis of the performance guarantees of exponential weighted aggregation and penalized estimators with a general class of data losses and priors which encourage objects which conform to some notion of simplicity/complexity. More precisely, we show that these two estimators satisfy sharp oracle inequalities for prediction ensuring their good theoretical performances. We also highlight the differences between them. When the noise is random, we provide oracle inequalities in probability using concentration inequalities. These results are then applied to several instances including the Lasso, the group Lasso, their analysis-type counterparts, the ℓ ∞ and the nuclear norm penalties. All our estimators can be efficiently implemented using proximal splitting algorithms.

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Signal representations with minimum ℓ<inf>∞</inf>-norm

Cited by 22 publications

References 21 publications

PhaseMax: Convex Phase Retrieval via Basis Pursuit

PhaseMax: Convex Phase Retrieval via Basis Pursuit

Bayesian Antisparse Coding

Sharp oracle inequalities for low-complexity priors

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