Random channel coding and blind deconvolution

Asif, M. Salman; Mantzel, William; Romberg, Justin

doi:10.1109/allerton.2009.5394881

Cited by 24 publications

(26 citation statements)

References 14 publications

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“…These properties were first defined in [27] for the development of a probabilistic and RIPless theory of compressed sensing, and then used in [28] for the problem of blind spikes deconvolution. Other random subspace assumptions are also used in [29,30] for random channel coding and blind deconvolution. The transmitted signal in multi-user communication systems [24] can also be represented in a known low-dimensional random subspace in the case when each of the transmitters sends out a random signal for the sake of security, privacy, or spread spectrum communications.…”

Section: Theoretical Guarantee For Atomic Norm Denoisingmentioning

confidence: 99%

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Wakin

Tang

2020

IEEE Trans. Inform. Theory

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Non-stationary blind super-resolution is an extension of the traditional super-resolution problem, which deals with the problem of recovering fine details from coarse measurements. The non-stationary blind super-resolution problem appears in many applications including radar imaging, 3D single-molecule microscopy, computational photography, etc. There is a growing interest in solving non-stationary blind super-resolution task with convex methods due to their robustness to noise and strong theoretical guarantees. Motivated by the recent work on atomic norm minimization in blind inverse problems, we focus here on the signal denoising problem in non-stationary blind super-resolution. In particular, we use an atomic norm regularized least-squares problem to denoise a sum of complex exponentials with unknown waveform modulations. We quantify how the mean square error depends on the noise variance and the true signal parameters. Numerical experiments are also implemented to illustrate the theoretical result.Index terms-Atomic norm denoising, non-stationary blind deconvolution, line spectrum estimation, demodulation, mean square error.

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Section: Theoretical Guarantee For Atomic Norm Denoisingmentioning

confidence: 99%

Atomic Norm Denoising for Complex Exponentials With Unknown Waveform Modulations

Wakin

Tang

2020

IEEE Trans. Inform. Theory

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“…Subspace membership and sparsity have been used as priors in blind deconvolution for a long time. Previous works either use these priors without theoretical justification [5][6][7][8][9], or impose probabilistic models and show successful recovery with high probability [10,15,16,18]. The sufficient conditions for the identifiability in BD in our prequel paper [19] are (except for a special class of so-called sub-band structured signals or filters) suboptimal.…”

Section: Identifiability Resultsmentioning

confidence: 99%

“…In this paper, we focus on subspace or sparsity assumptions on the signal and the filter. These priors, which render BD better-posed by reducing the search space, were shown to be effective constraints or regularizers in various applications [5][6][7][8][9][10]. However, despite the success in practice, the theoretical results on uniqueness in BD with a subspace or sparsity constraint are limited.Recently, the "lifting" scheme -recasting bilinear or quadratic inverse problems, such as blind deconvolution and phase retrieval, as rank-1 matrix recovery from linear measurements -has attracted considerable attention [10,11].…”

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Identifiability and Stability in Blind Deconvolution Under Minimal Assumptions

Lee

Bresler

2017

IEEE Trans. Inform. Theory

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Blind deconvolution (BD) arises in many applications. Without assumptions on the signal and the filter, BD does not admit a unique solution. In practice, subspace or sparsity assumptions have shown the ability to reduce the search space and yield the unique solution. However, existing theoretical analysis on uniqueness in BD is rather limited. In an earlier paper, we provided the first algebraic sample complexities for BD that hold for almost all bases or frames.We showed that for BD of a pair of vectors in C n , with subspace constraints of dimensions m 1 and m 2 , respectively, a sample complexity of n ≥ m 1 m 2 is sufficient. This result is suboptimal, since the number of degrees of freedom is merely m 1 + m 2 − 1. We provided analogus results, with similar suboptimality, for BD with sparsity or mixed subspace and sparsity constraints. In this paper, taking advantage of the recent progress on the information-theoretic limits of unique low-rank matrix recovery, we finally bridge this gap, and derive an optimal sample complexity result for BD with generic bases or frames. We show that for BD of an arbitrary pair (resp. all pairs) of vectors in C n , with sparsity constraints of sparsity levels s 1 and s 2 , a sample complexity of n > s 1 + s 2 (resp. n > 2(s 1 + s 2 )) is sufficient. We also present analogous results for BD with subspace constraints or mixed constraints, with the subspace dimension replacing the sparsity level. Last but not least, in all the above scenarios, if the bases or frames follow a probabilistic distribution specified in the paper, the recovery is not only unique, but also stable against small perturbations in the measurements, under the same sample complexities.

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“…We employ lifting in the same spirit as [6], [7] but our goals are different. Firstly, we deal with general bilinear inverse problems which include the linear convolution model of [8], the circular convolution model of [7], [9] and compressed bilinear observation model of [10] as special cases. Secondly, we focus solely on identifiability (as opposed to recoverability by convex optimization [6], [7]) and thus require far milder assumptions on the distribution of the input signals.…”

Section: Introductionmentioning

confidence: 99%