Multichannel Sparse Blind Deconvolution on the Sphere

Li, Yanjun; Bresler, Yoram

doi:10.1109/icassp.2019.8683334

Cited by 6 publications

(42 citation statements)

References 64 publications

(133 reference statements)

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“…In particular, consider the interesting regime when θ = O(1) and κ = O(1), it is sufficient to set µ = O (log n) −1/6 n −3/4 , which leads to a sample size p = O(n 4.5 ) up to logarithmic factors. This significantly improves over the prior work of Li and Bresler [4], which requires a sample complexity of p = O(n 9 ) up to logarithmic factors. See further discussions in Section 5.…”

Section: Convergence Guarantees Of Mgdmentioning

confidence: 85%

“…As we shall see later, while this approach works well when C(g) is an orthogonal matrix, further care needs to be taken when it is a general invertible matrix in order to guarantee a benign optimization geometry. Following [4,10], we introduce the following pre-conditioned optimization problem:…”

Section: Nonconvex Optimization On the Spherementioning

confidence: 99%

“…Extension to the general case. To extend the geometry in Theorem 2 to the general case when C(g) is invertible, we adopt the trick in [4,10] and introduce the pre-conditioning matrix R:…”

Section: Geometry Of the Empirical Lossmentioning

confidence: 99%

“…When the filter is unknown, it leads to the so-called blind deconvolution problem. This problem is ill-posed without extra assumptions on the filter, since the number of unknowns is much larger than the number of observations [4,5,6,7,8]. Luckily, in many situations, one can make multiple observations sharing the same filter, but with diverse sparse inputs, either spatially or temporally, thanks to the advances of sensing technologies.…”

Section: Introductionmentioning

confidence: 99%

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Manifold Gradient Descent Solves Multi-Channel Sparse Blind Deconvolution Provably and Efficiently

Shi

Chi

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Multi-channel sparse blind deconvolution, or convolutional sparse coding, refers to the problem of learning an unknown filter by observing its circulant convolutions with multiple input signals that are sparse. This problem finds numerous applications in signal processing, computer vision, and inverse problems. However, it is challenging to learn the filter efficiently due to the bilinear structure of the observations with the respect to the unknown filter and inputs, leading to global ambiguities of identification. In this paper, we propose a novel approach based on nonconvex optimization over the sphere manifold by minimizing a smooth surrogate of the sparsity-promoting loss function. It is demonstrated that the manifold gradient descent with random initializations will provably recover the filter, up to scaling and shift ambiguity, as soon as the number of observations is sufficiently large under an appropriate random data model. Numerical experiments are provided to illustrate the performance of the proposed method with comparisons to existing methods.

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Section: Convergence Guarantees Of Mgdmentioning

confidence: 85%

Section: Nonconvex Optimization On the Spherementioning

confidence: 99%

“…Extension to the general case. To extend the geometry in Theorem 2 to the general case when C(g) is invertible, we adopt the trick in [4,10] and introduce the pre-conditioning matrix R:…”

Section: Geometry Of the Empirical Lossmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Manifold Gradient Descent Solves Multi-Channel Sparse Blind Deconvolution Provably and Efficiently

Shi

Chi

2020

ICASSP 2020 - 2020 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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“…On the other hand, we believe the probability tools of decoupling and measure concentration we developed here can form a solid foundation for studying other nonconvex problems under the random convolutional model. Those problems include blind calibration [80][81][82], sparse blind deconvolution [76,[83][84][85][86][87][88][89][90][91][92][93], and convolutional dictionary learning [16,[94][95][96][97]].…”

Section: Geometric Analysis and Global Resultmentioning

confidence: 99%

Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

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We study the convolutional phase retrieval problem, of recovering an unknown signal x ∈ C n from m measurements consisting of the magnitude of its cyclic convolution with a given kernel a ∈ C m . This model is motivated by applications such as channel estimation, optics, and underwater acoustic communication, where the signal of interest is acted on by a given channel/filter, and phase information is difficult or impossible to acquire. We show that when a is random and the number of observations m is sufficiently large, with high probability x can be efficiently recovered up to a global phase shift using a combination of spectral initialization and generalized gradient descent. The main challenge is coping with dependencies in the measurement operator. We overcome this challenge by using ideas from decoupling theory, suprema of chaos processes and the restricted isometry property of random circulant matrices, and recent analysis of alternating minimization methods. ). Most of the work has been conducted when QQ and YZ were with Department of Electrical Engineering and Data Science Institute at Columbia University. An extended abstract of the current work has appeared in NeurIPS'17 [1]. 1 The convolution can be written in matrix-vector form as a x = Caιn→mx, where Ca ∈ R m×m denotes the circulant matrix generated by a and ιn→m is a zero padding operator. In other words, a x = Ax with A ∈ C m×n being a matrix formed by the first n columns of Ca. arXiv:1712.00716v3 [stat.CO] 6 Oct 2019Structured random measurements. The study of structured random measurements in signal processing has quite a long history [44]. For compressed sensing [45], the work [46-48] studied random Fourier measurements, and later [13,14] proved similar results for partial random convolution measurements. However, the study of structured random measurements for phase retrieval is still quite limited. In particular, [49] and [50] studied t-designs and coded diffraction patterns (i.e., random masked Fourier measurements) using semidefinite programming. Recent work studied nonconvex optimization using coded diffraction patterns [34] and STFT measurements [51], both of which minimize a nonconvex objective similar to (1.5). These measurement models are motivated by different applications. For instance, coded diffraction is designed for imaging applications such as X-ray diffraction imaging, STFT can be applied to frequency resolved optical gating [52] and some speech processing tasks [53]. Both of the results show iterative contraction in a region that is at most O(1/ √ n)-close to the optimum. Unfortunately, for both results either the radius of the contraction region is not large enough for initialization to reach, or they require extra artificial technique such as resampling the data. In comparison, the contraction region we show for the random convolutional model is larger O(1/polylog(n)), which is achievable in the initialization stage via the spectral method. For a more detailed review of this subject, we refer the readers to Section 4 of [44]....

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Identification of Pulse Streams Of Unknown Shape From Time Encoding Machine Samples

Kalra

Bresler

Lee

2022

ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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We consider the problem of resolving overlapping pulses from noisy multi-snapshot measurements, which has been a problem central to various applications including medical imaging and array signal processing. ESPRIT algorithm has been used to estimate the pulse locations. However, existing theoretical analysis is restricted to ideal assumptions on signal and measurement models. We present a novel perturbation analysis that overcomes the previous theoretical limitation, which is derived without a stringent assumption on the signal model. Our unifying analysis applies to various sub-array designs of the ESPRIT algorithm. We demonstrate the usefulness of the perturbation analysis by specifying the result in two practical scenarios. In the first scenario, we quantify how the number of snapshots for stable recovery scales when the number of Fourier measurements per snapshot is sufficiently large. In the second scenario, we propose compressive blind array calibration by ESPRIT with random sub-arrays and provide the corresponding non-asymptotic error bound. Furthermore, we demonstrate that the empirical performance of ESPRIT corroborates the theoretical analysis through extensive numerical results.

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Multichannel Sparse Blind Deconvolution on the Sphere

Cited by 6 publications

References 64 publications

Manifold Gradient Descent Solves Multi-Channel Sparse Blind Deconvolution Provably and Efficiently

Manifold Gradient Descent Solves Multi-Channel Sparse Blind Deconvolution Provably and Efficiently

Convolutional Phase Retrieval via Gradient Descent

Identification of Pulse Streams Of Unknown Shape From Time Encoding Machine Samples

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