Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

Cai, Jian-Feng; Jiao, Yuling; Lu, Xiliang; You, Juntao

doi:10.1109/tsp.2022.3214091

Cited by 4 publications

(4 citation statements)

References 44 publications

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“…Therefore, the sample efficiency of many two-stage algorithms can be improved when combined with the proposed initialization methods. Examples of such algorithms are SPARTA [50], CoPRAM [25], thresholding/projected Wirtinger flow [14,42], SAM [10] and HTP [11], to just name a few. We use the two-stage HTP algorithm [11] as a typical example to illustrate this.…”

Section: Theoretical Guarantee and Sample Complexitymentioning

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%

“…which together with(10) implies S = S; otherwise, those indices j ∈ S satisfying |[EY] jj − [Y] jj | > γ 0 might be missed in S. Therefore, the gap γ 0 is the tolerance of the concentration error |[EY] jj − [Y] jj | for an accurate S and an accurate initial estimation. The larger gap γ 0 , the larger tolerance of |[EY] jj − [Y] jj |, and the fewer samples required.…”

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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: Theoretical Guarantee and Sample Complexitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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

Cai

You

2023

Inverse Problems

Self Cite

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“…Next, a series of breakthrough results [9,[13][14][15] provided provably valid algorithmic processes for special cases, where the measurement vectors are randomly derived from some certain multivariate probability distribution, such as the Gaussian distributions. Phase retrieval problems [9,13,14] require the number of observations m to exceed the problem dimension n. However, for the sparse phase retrieval problem, the true signal can be successfully recovered, even if the number of measurements m is less than the length of the signal n. In particular, a recent paper [16] utilized the random sampling technique to achieve the best empirical sampling complexity; in other words, it requires less measurements than state-of-the-art algorithms for sparse phase retrieval problems. For more on phase recovery, interested readers can refer to [17][18][19][20][21][22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

Heavy-Ball-Based Hard Thresholding Pursuit for Sparse Phase Retrieval Problems

Zhou

Sun

et al. 2023

Mathematics

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We introduce a novel iterative algorithm, termed the Heavy-Ball-Based Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HBHTP), to reconstruct a sparse signal from a small number of magnitude-only measurements. Our algorithm is obtained via a natural combination of the Hard Thresholding Pursuit for sparse phase retrieval (SPR-HTP) and the classical Heavy-Ball (HB) acceleration method. The robustness and convergence for the proposed algorithm were established with the help of the restricted isometry property. Furthermore, we prove that our algorithm can exactly recover a sparse signal with overwhelming probability in finite steps whenever the initialization is in the neighborhood of the underlying sparse signal, provided that the measurement is accurate. Extensive numerical tests show that SPR-HBHTP has a markedly improved recovery performance and runtime compared to existing alternatives, such as the Hard Thresholding Pursuit for sparse phase retrieval problem (SPR-HTP), the SPARse Truncated Amplitude Flow (SPARTA), and Compressive Phase Retrieval with Alternating Minimization (CoPRAM).

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Phase retrieval for radar waveform design

Pinilla,

Mishra,

Sadler

et al. 2024

Information and Inference: A Journal of the IMA

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The ability of a radar to discriminate in both range and Doppler velocity is completely characterized by the ambiguity function (AF) of its transmit waveform. Mathematically, it is obtained by correlating the waveform with its Doppler-shifted and delayed replicas. We consider the inverse problem of designing a radar transmit waveform that satisfies the specified AF magnitude. This process may be viewed as a signal reconstruction with some variation of phase retrieval methods. We provide a trust-region algorithm that minimizes a smoothed non-convex least-squares objective function to iteratively recover the underlying signal-of-interest for either time- or band-limited support. The method first approximates the signal using an iterative spectral algorithm and then refines the attained initialization based on a sequence of gradient iterations. Our theoretical analysis shows that unique signal reconstruction is possible using signal samples no more than thrice the number of signal frequencies or time samples. Numerical experiments demonstrate that our method recovers both time- and band-limited signals from sparsely and randomly sampled, noisy and noiseless AFs.

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Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

Cited by 4 publications

References 44 publications

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

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

Heavy-Ball-Based Hard Thresholding Pursuit for Sparse Phase Retrieval Problems

Phase retrieval for radar waveform design

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