On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution

Zhang, Yuqian; Lau, Yenson; Kuo, Han-Wen; Cheung, Sky C.; Pasupathy, Abhay; Wright, John

doi:10.1109/cvpr.2017.466

Cited by 47 publications

(27 citation statements)

References 23 publications

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“…Cx 2 . Therefore, combining the results in (C. 15) and (C.16), for any δ ≥ 0, whenever m ≥ C 4 δ −1 C x 2 n log m, choosing t = C 5 δ, we have…”

Section: C1 Concentration Of Y (G)mentioning

confidence: 63%

“…Instead, we control the bulk effect of phase errors uniformly in a neighborhood around the ground truth. This requires us to develop new decoupling and concentration tools for controlling nonlinear phase functions of circulant random matrices, which could be potentially useful for analyzing other random circulant convolution problems, such as sparse blind deconvolution [15] and convolutional dictionary learning [16].…”

Section: Introductionmentioning

confidence: 99%

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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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“…Cx 2 . Therefore, combining the results in (C. 15) and (C.16), for any δ ≥ 0, whenever m ≥ C 4 δ −1 C x 2 n log m, choosing t = C 5 δ, we have…”

Section: C1 Concentration Of Y (G)mentioning

confidence: 63%

Section: Introductionmentioning

confidence: 99%

Convolutional Phase Retrieval via Gradient Descent

Zhang

Eldar

et al. 2020

IEEE Trans. Inform. Theory

Self Cite

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“…This is a non-smooth optimization problem on the product manifold of a sphere and R m . Some related background and the corresponding algorithms can be found in [112].…”

Section: Deep Learningmentioning

confidence: 99%

A Brief Introduction to Manifold Optimization

Liu

Wen

et al. 2020

J. Oper. Res. Soc. China

148

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Manifold optimization is ubiquitous in computational and applied mathematics, statistics, engineering, machine learning, physics, chemistry and etc. One of the main challenges usually is the non-convexity of the manifold constraints. By utilizing the geometry of manifold, a large class of constrained optimization problems can be viewed as unconstrained optimization problems on manifold. From this perspective, intrinsic structures, optimality conditions and numerical algorithms for manifold optimization are investigated. Some recent progress on the theoretical results of manifold optimization are also presented.

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“…On the other hand, some methods employed different blur kernel priors [12,[31][32][33], either encouraging the estimated the blur kernel to be sparse or discouraging the delta kernel. In addition, Zhang et al [21] imposed a unit Frobenius norm constraint on the blur kernel. However, an explicit solution of the latent image, which was very critical to their method, was often not easy to achieve.…”

Section: Related Workmentioning

confidence: 99%

Blind Deconvolution with Scale Ambiguity

Fan

Wang

et al. 2020

Applied Sciences

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Recent years have witnessed significant advances in single image deblurring due to the increasing popularity of electronic imaging equipment. Most existing blind image deblurring algorithms focus on designing distinctive image priors for blur kernel estimation, which usually play regularization roles in deconvolution formulation. However, little research effort has been devoted to the relative scale ambiguity between the latent image and the blur kernel. The well-known L 1 normalization constraint, i.e., fixing the sum of all the kernel weights to be one, is commonly selected to remove this ambiguity. In contrast to this arbitrary choice, we in this paper introduce the L p -norm normalization constraint on the blur kernel associated with a hyper-Laplacian prior. We show that the employed hyper-Laplacian regularizer can be transformed into a joint regularized prior based on a scale factor. We quantitatively show that the proper choice of p makes the joint prior sufficient to favor the sharp solutions over the trivial solutions (the blurred input and the delta kernel). This facilitates the kernel estimation within the conventional maximum a posterior (MAP) framework. We carry out numerical experiments on several synthesized datasets and find that the proposed method with p = 2 generates the highest average kernel similarity, the highest average PSNR and the lowest average error ratio. Based on these numerical results, we set p = 2 in our experiments. The evaluation on some real blurred images demonstrate that the results by the proposed methods are visually better than the state-of-the-art deblurring methods.

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On the Global Geometry of Sphere-Constrained Sparse Blind Deconvolution

Cited by 47 publications

References 23 publications

Convolutional Phase Retrieval via Gradient Descent

Convolutional Phase Retrieval via Gradient Descent

A Brief Introduction to Manifold Optimization

Blind Deconvolution with Scale Ambiguity

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