Sparse signal recovery from phaseless measurements via hard thresholding pursuit

Cai, Jian-Feng; Li, Jingzhi; Lu, Xiliang; You, Juntao

doi:10.1016/j.acha.2021.10.002

Cited by 9 publications

(22 citation statements)

References 38 publications

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“…Actually, the sample complexity of this approach is governed by the gap between diagonals [EY] jj for j ∈ S and j ∈ S c as in the following (11) finally leads to the sample complexity m = Ω(s 2 log n) for spectral initialization [25].…”

Section: Spectral Initialization For Sparse Phase Retrievalmentioning

confidence: 99%

“…Those nonconvex methods are usually divided into two stages, namely, the initialization stage and the local refinement stage. Provided an initial guess that is sufficiently close to the underlying signal, non-convex algorithms including SPARTA [50], CoPRAM [25], thresholding/projected Wirtinger flow [14,42], stochastic alternating minimization (SAM) [10] and hard thresholding pursuit (HTP) [11] are guaranteed to converge at least linearly to the ground truth with the near-optimal sample complexity Ω(s log n). Moreover, algorithms like SAM and HTP are guaranteed to give an exact recovery of the underlying signal in only a few iterations, implying that the local refinement stage can be efficiently implemented.…”

Section: Introductionmentioning

confidence: 99%

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Provable sample-efficient sparse phase retrieval initialized by truncated power method

Cai

You

2023

Inverse Problems

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We study the sparse phase retrieval problem, recovering an $s$-sparse length-$n$ signal from $m$ magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies for this problem. Despite non-convexity, many two-stage algorithms provably converge to the underlying solution linearly when appropriately initialized. However, in terms of sample complexity, the bottleneck of those algorithms with Gaussian random measurements often comes from the initialization stage. Although the refinement stage usually needs only $m=\Omega(s\log n)$ measurements, the widely used spectral initialization in the initialization stage requires $m=\Omega(s^2\log n)$ measurements to produce a desired initial guess, which causes the total sample complexity order-wisely more than necessary. To reduce the number of measurements, we propose a truncated power method to replace the spectral initialization for non-convex sparse phase retrieval algorithms. We prove that $m=\Omega(\bar{s} s\log n)$ measurements, where $\bar{s}$ is the stable sparsity of the underlying signal, are sufficient to produce a desired initial guess. When the underlying signal contains only very few significant components, the sample complexity of the proposed algorithm is $m=\Omega(s\log n)$ and optimal. Numerical experiments illustrate that the proposed method is more sample-efficient than state-of-the-art algorithms.

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Section: Spectral Initialization For Sparse Phase Retrievalmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Provable sample-efficient sparse phase retrieval initialized by truncated power method

Cai

You

2023

Inverse Problems

Self Cite

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“…Let consider a signal f that is sparse, x has only K non-zero entries. f = ψ x, where ψ is some transform domain [20], [21]. The measurement vector can be obtained as 𝑦 = 𝜙𝑓 = 𝜙𝜓𝑥 = 𝐴𝑥 .…”

Section: D((imentioning

confidence: 99%

Comparison of different sparse dictionaries for compressive sampling

Devendrappa¹,

Karthik²,

A.³

et al. 2022

Bulletin EEI

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Compressive sampling/compressed sensing (CS) is building on the observation that most of the signals in nature are sparse or compressible concerning some transform domain. And by converse, the same can be reconstructed with high accuracy by making use of far fewer samples than what is required by violating Shannon-Nyquist theorem. Some of the transform techniques like discrete cosine transform, fast fourier transforms discrete wavelet transform, discrete fourier transforms. In this paper, novel CS techniques like FFTCoSAMP, DCTCoSaMP, and DWTCoSaMP are introduced and compared on different sparse transforms for CS in magnetic resonance (MR) images based on sparse signal sequences/dictionaries by means of transform techniques with respect to objective quality assessment algorithms like PSNR, SSIM and RMSE, where CoSaMP stands for compressive sampling matching pursuit. DWTCoSaMP is giving the PSNR values of 37.16 (DB4), 38.12 (Coif3), 38.5 (Sym8), for DCTCoSaMP and FFTCoSaMP, it’s 36.33 and 36.01 respectively. For DWTCoSaMP, SSIM value is 0.81, and for DCTCoSaMP and FFTCoSaMP, it’s 0.73 and 0.7 respectively. And finally, for DWTCoSaMP, RMSE value is 0.66, and for DCTCoSaMP and FFTCoSaMP, it’s 0.53 and 0.41 respectively. DWTCoSaMP reveals the best than rest of the methods and traditional CS techniques by the detailed comparison and analysis.

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“…Sparse phase retrieval: Various algorithms have also been devised for sparse phase retrieval [25]- [27]. In particular, there exist various non-convex optimization based algorithms, including thresholded/projected WF [24], [28], sparse truncated amplitude flow [29], compressive phase retrieval with alternating minimization [30], and sparse phase retrieval by hard iterative pursuit [31]. All of these approaches are analyzed under the assumption of using a sensing matrix with i.i.d.…”

Section: A Related Workmentioning

confidence: 99%

“…Without loss of generality, we assume that |S| ≥ m 2 (otherwise, we have |T | ≥ m 2 and we can derive an upper bound for q + x 2 instead of for q − x 2 ). Expanding the squares in (31), we obtain…”

Section: B Proof Of Theoremmentioning

confidence: 99%

Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors

Liu¹,

Ghosh²,

Scarlett³

2021

Preprint

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Compressive phase retrieval is a popular variant of the standard compressive sensing problem, in which the measurements only contain magnitude information. In this paper, motivated by recent advances in deep generative models, we provide recovery guarantees with order-optimal sample complexity bounds for phase retrieval with generative priors. We first show that when using i.i.d. Gaussian measurements and an L-Lipschitz continuous generative model with bounded k-dimensional inputs, roughly O(k log L) samples suffice to guarantee that the signal is close to any vector that minimizes an amplitude-based empirical loss function. Attaining this sample complexity with a practical algorithm remains a difficult challenge, and a popular spectral initialization method has been observed to pose a major bottleneck. To partially address this, we further show that roughly O(k log L) samples ensure sufficient closeness between the signal and any globally optimal solution to an optimization problem designed for spectral initialization (though finding such a solution may still be challenging). We adapt this result to sparse phase retrieval, and show that O(s log n) samples are sufficient for a similar guarantee when the underlying signal is s-sparse and n-dimensional, matching an information-theoretic lower bound. While our guarantees do not directly correspond to a practical algorithm, we propose a practical spectral initialization method motivated by our findings, and experimentally observe significant performance gains over various existing spectral initialization methods of sparse phase retrieval.

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Sparse signal recovery from phaseless measurements via hard thresholding pursuit

Cited by 9 publications

References 38 publications

Provable sample-efficient sparse phase retrieval initialized by truncated power method

Provable sample-efficient sparse phase retrieval initialized by truncated power method

Comparison of different sparse dictionaries for compressive sampling

Towards Sample-Optimal Compressive Phase Retrieval with Sparse and Generative Priors

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