Pith. sign in

REVIEW 3 major objections 5 minor 53 references

Sparse Signal Recovery From Quadratic Systems with Full-Rank Matrices

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read The paper proves that $s$-sparse signals are uniquely recoverable from $m \geq 2s$ generic quadratic measurements, and that a two-stage Sparse Gauss-Newton algorithm refines an accurate initialization to quadratic convergence using only…

desk verdict The uniqueness results are clean and the refinement claim is promising, but the printed initialization has a real scaling bug and the key uniform concentration lemma is only sketched; worth a serious referee, conditional on fixes. read the letter →

arxiv 2507.07557 v1 pith:STRWFVBV submitted 2025-07-10 cs.IT math.IT

Jinming Wen , Yi Hu , Meng Huang This is my paper
classification cs.ITmath.IT MSC 65F1015A2994A12
keywords sparsequadraticsystemsGauss-NewtonmethodhardthresholdingpursuitspectralinitializationconvergencesamplecomplexitygenericuniquenessGaussianmeasurements
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks how many quadratic measurements are needed to recover an $s$-sparse signal, and whether a practical algorithm can meet that count. It proves that $m \geq 2s$ generic real measurements (and $m \geq 4s-2$ complex Hermitian ones) uniquely determine every $s$-sparse signal up to a global phase. For Gaussian measurements it proposes a two-stage Sparse Gauss-Newton (SGN) algorithm: a support-restricted spectral initialization lands within distance $\delta\|x\|$ of the truth using $O(s^2\log n)$ measurements, and a hard-thresholding Gauss-Newton refinement then converges quadratically using only $O(s\log n)$ measurements, with no resampling. If the analysis is right, the $O(s^2\log n)$ bottleneck for the refinement stage is broken and sparse quadratic recovery becomes practical at near-optimal sampling rates.

What carries the argument

The engine is the support-restricted Gauss-Newton step of equations (3.7)-(3.8), driven by a uniform concentration lemma for the empirical Jacobian $J(x) = \frac{1}{\sqrt{m}}[(A_1+A_1^{\top})x, \dots, (A_m+A_m^{\top})x]^{\top}$ of the quadratic map. Lemma 7.11 asserts that once $m \geq C(\delta)s\log n$, the sample Jacobian Gram matrix stays within $\delta\|p\|\|q\|$ of its expectation over every support of size at most $4s$ at once; Lemma 7.13 converts that into uniform bounds $(2-\delta)\|p\|^2 \leq \|J_S(p)^{\top}J_S(p)\| \leq (4+\delta)\|p\|^2$ and controls the cross-support term. These bounds make each hard-thresholded Gauss-Newton step (keep the $s$ largest entries, solve the restricted linearized least-squares problem) contract linearly while the support is still being recovered, and quadratically once the support is fixed, because the residual then factors as a product of the current error with itself. A diagonal-marginal thresholding step in Algorithm 1 supplies the required $\delta < 0.01$ starting point.

What would settle it

Run Algorithm 2 on Gaussian problems with $n=500$, $s=10$, $m = C s\log n$ for a constant $C$ at the level required by Lemma 7.11, starting from an initialization with $\mathrm{dist}(x_0,x)=0.005\|x\|$; if the quadratic contraction $\mathrm{dist}(x_{k+1},x) \leq \nu'\,\mathrm{dist}^2(x_k,x)$ has not appeared by iteration $c_1\log(\|x\|/x_{\min}) + c_2$, or if a direct Monte-Carlo evaluation of the uniform sup in (7.7) over randomly sampled supports of size $4s$ exceeds $\delta$ at $m = C s \log n$, then the hypothesis of Theorem 4.2 is violated and the central claim fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that sparse quadratic systems are both identifiable and computationally tractable at near-linear-in-sparsity sampling rates. The identifiability result, proved by dimension counting on projective varieties, is that a generic set of $m \geq 2s$ real matrices, or $m \geq 4s-2$ complex Hermitian matrices, makes the measurement map $x \mapsto (x^{\top} A_1 x, \dots, x^{\top} A_m x)$ injective on all $s$-sparse signals up to global phase. The algorithmic claim, Theorem 4.2, is that under i.i.d. Gaussian matrices, an initial estimate with $\mathrm{dist}(x_0, x) \leq \delta\|x\|$, $\delta < 0.01$, and $m \geq C' s \log n$ suffices for Algorithm 2 to satisfy $\mathrm{dist}(x_{k+1}, x) \leq \nu'\,\mathrm{dist}^2(x_k, x)$ for every $k \geq c_1 \log(\|x\|/x_{\min}) + c_2$ in the noise-free case, with a linear contraction phase before that; in the noisy case it attains the minimax error rate $O\big(\frac{\sigma}{\|x\|}\sqrt{\frac{s\log(en/s)}{m}}\big)$. The paper concludes that SGN is the first sparse full-rank quadratic recovery method whose refinement stage reaches quadratic convergence at near-optimal sample complexity without sample splitting.

Load-bearing premise

The load-bearing assumption is a single uniform concentration estimate (Lemma 7.11): with only $m \geq C(\delta)s\log n$ Gaussian matrices, the derivative of the quadratic measurement map is uniformly well behaved on every support of size up to $4s$ at once; the appendix sketches the proof by analogy to a fixed-support lemma from [15], and if that uniform bound fails at this sampling rate, the $O(s\log n)$ refinement and quadratic convergence guarantees do not follow.

Editorial extensions

If this is right

  • Refinement of sparse quadratic signals becomes possible with $m = O(s\log n)$ measurements, so the remaining sampling bottleneck is the $O(s^2\log n)$ initialization stage rather than the iteration stage.
  • Because no resampling or sample splitting is needed, the same Gaussian measurements can drive both initialization and refinement, simplifying hardware and protocol design in computational imaging and cryo-electron microscopy.
  • The corollary iteration count $O(\log\log(1/\epsilon) + \log(\|x\|/x_{\min}))$ means an $\epsilon$-accurate sparse solution is reached in very few passes, matching the experiments where SGN needs roughly a tenth of the iterations of the comparison methods.
  • The noisy-case error bound $O\big(\frac{\sigma}{\|x\|}\sqrt{\frac{s\log(en/s)}{m}}\big)$ matches the minimax rate for sparse estimation, so the estimator is statistically optimal in the sparse regime.
  • The generic uniqueness theorems put the measurement floor at $2s$ (real) and $4s-2$ (complex), independent of ambient dimension $n$, so the $m \ll n$ regime is information-theoretically viable.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves the initialization at $O(s^2\log n)$; a natural next step, not pursued here, would be to redesign the support-restricted spectral step so the whole pipeline runs at $O(s\log n)$ and becomes sample-optimal end to end.
  • The same Jacobian concentration machinery should transfer to other structured priors, such as generative models or low-rank matrices, by replacing the sparsity support sets with the corresponding prior tangent sets; running SGN with a generative projection would test this directly.
  • The finite linear phase of length $c_1\log(\|x\|/x_{\min}) + c_2$ predicts that signals with very small nonzero entries spend more iterations in linear convergence; this is a testable prediction about how the phase transition shifts with $x_{\min}$.
  • One could empirically probe whether keeping $2s$ rather than $s$ entries in the hard-thresholding step accelerates support identification without destroying the quadratic phase; the current theory is restricted to exactly $s$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper studies recovery of an s-sparse signal x from m quadratic measurements y_i = x^T A_i x, where A_i are full-rank Gaussian matrices. It proves generic uniqueness with m >= 2s in the real case and m >= 4s - 2 in the complex case, and proposes a two-stage algorithm: a support-restricted spectral initialization (Algorithm 1) followed by a hard-thresholding Gauss-Newton refinement (Algorithm 2). The main theoretical claims are that the initialization reaches error delta0||x|| with m = O(s^2 log n) measurements, and that the refinement converges linearly and, in the noise-free case, quadratically after O(log(||x||/x_min)) iterations with m = O(s log n), without resampling. Numerical experiments compare the proposed SGN against Wirtinger Flow and Thresholded Wirtinger Flow.

Significance. If the main claims hold, the paper would provide the first sparse full-rank quadratic recovery algorithm with a refinement stage at O(s log n) measurements and quadratic convergence, a clear improvement over the O(s^2 log n) bottleneck of prior work. The algebraic-geometry uniqueness results are clean and appear correct. The paper is generally clearly written and the appendix contains substantial proof detail. However, there is an internal inconsistency in the initialization scaling and a flawed noisy initialization claim; these issues need to be addressed before the results can be accepted as stated.

major comments (3)
  1. [Algorithm 1 Step 4 and Section 3.1] The printed scaling phi = (1/(2m) sum y_i^2)^{1/4} is inconsistent with the measurement model. For A_i with iid N(0,1) entries, E[y_i^2] = E[(x^T A_i x)^2] = ||x||^4, so phi converges to 2^{-1/4}||x||, not ||x||. Lemma 7.6 and the proof of Theorem 4.1 in Appendix 7.3 use phi = (1/m sum y_i^2)^{1/4}, so the algorithm as stated does not implement the analyzed estimator. Even with perfect directional alignment, dist(x0,x) >= (1 - 2^{-1/4})||x|| ≈ 0.159||x||, which violates the delta < 0.01 hypothesis of Theorem 4.2 and the 'any delta0' claim of Theorem 4.1. Please correct Algorithm 1 and Section 3.1 to use 1/m, or justify the 1/2 factor under a different measurement model.
  2. [Theorem 4.1 (noisy case)] The noisy initialization guarantee is false for arbitrary sub-exponential noise with large sigma. With y_i = x^T A_i x + epsilon_i, E[y_i^2] = ||x||^4 + E[epsilon_i^2], so the estimator phi = (1/m sum y_i^2)^{1/4} converges to (||x||^4 + E[epsilon^2])^{1/4}, which is not ||x|| when E[epsilon^2] is comparable to ||x||^4. Thus no amount of additional measurements can drive dist(x0,x) below |(1 + E[epsilon^2]/||x||^4)^{1/4} - 1| ||x||, contradicting the 'for any delta0' claim. The proof invokes Lemma 7.6, which is stated only for the noise-free case; the sentence in Lemma 7.10 that Lemma 7.6 'still remains valid' when noise variance is small does not cover the stated regime. Please either restrict Theorem 4.1 to bounded noise (e.g., sigma/||x||^2 below a constant depending on delta0) or modify the norm estimation to be noise-aware.
  3. [Lemma 7.11, Appendix 7.4] Lemma 7.11 is the probabilistic foundation for the O(s log n) sample complexity and the quadratic convergence in Theorem 4.2, yet its proof is only sketched as 'the proof is similar to that of [15, Lemma 12]'. The extension from a fixed subset S to a uniform bound over all |S| <= 4s with m >= C(delta) s log n involves a nontrivial union bound over (en/s)^{4s} supports; please provide the complete argument, including the net argument for both displays (7.7) and (7.8) and the precise constants and probability exponents.
minor comments (5)
  1. [Section 7.2, proof of Theorem 2.2] In the proof of Theorem 2.2, the expression dim_R(Y union C^n_I) appears to be a typo; the intended set is likely Z intersect C^n_J. Please fix this and ensure all symbols are defined.
  2. [Lemma 7.9] The assertion that the estimated support satisfies S^+ subset of S-hat subset of S is not guaranteed by the top-s selection when |S^+| < s, because off-support indices with large Y_jj can enter S-hat. The subsequent error bound only requires S^+ subset of S-hat, so the proof can be corrected by deleting the false inclusion, but the statement as written is inaccurate.
  3. [Equation (7.24) and surrounding text] The derivation for omega1 includes a spurious mu_k multiplying the noise term; the Gauss-Newton update (3.7)-(3.8) does not contain mu_k, and the sentence 'with the fact F(x) = 0' is inconsistent with the noisy case under consideration. Please clarify the derivation and state the noisy counterpart of F(x) explicitly.
  4. [Lemma 7.10, equation (7.5)] The denominator sigma_1(E[Y_{S-hat}]) should be ||x_{S-hat}||^2, and its lower bound should use the support approximation from Lemma 7.9; as written, replacing it with ||x||^2 ignores the possible loss of support mass. The constants can be adjusted, but the inequality as printed is not immediate.
  5. [Theorem 4.1 and Corollary 1] The displayed success probability 1 - 8n^{-5} - 4 exp(-c delta0^2 m) - 2 exp(-c' s) can be negative for small n; please state an assumption such as n >= C to keep the probability meaningful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the proof chain is self-contained and rests on external lemmas; only non-circular consistency issues are noted.

full rationale

No step in the derivation chain reduces to its own inputs. The algebraic-geometry uniqueness theorems (Theorems 2.1 and 2.2) are proved by dimension counting using the external [24, Cor. 11.13]. The initialization guarantee (Theorem 4.1) is derived from concentration inequalities ([40], [15, Lemma 4], in-paper Lemmas 7.7-7.8) and a Wedin-type sinTheta bound [19], none of which presuppose the conclusion. The refinement analysis (Theorem 4.2 and Lemmas 7.11-7.14) builds on external prior results ([15, Lemma 12], [35, Theorem 1], [19], [53], [9, Lemma A.7]); the cited [15] is by other authors, and the step-size interval and initialization radius are stated as universal hypotheses, not fitted to the target signal. The only self-authored citations ([6] and [25] by M. Huang) are not invoked in the proof chain. Two non-circular caveats are flagged: Lemma 7.11's proof is only sketched ('The proof is similar to that of [15, Lemma 12]', with (7.8) omitted), which is an external-dependency completeness risk rather than circularity; and Algorithm 1 Step 4 prints phi = (1/(2m) sum y_i^2)^{1/4} while the proof's Lemma 7.6 and Section 7.3 use phi = (1/m sum y_i^2)^{1/4}, an internal inconsistency that would preclude the delta<0.01 basin as printed. These are correctness issues outside the circularity definition.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claims rest on the Gaussian measurement model, exact sparsity with known s, sub-exponential noise bounds, and standard algebraic-geometry and concentration results. No new physical entity is introduced. One implementation parameter, the step size mu_k, must be chosen within an interval depending on the unknown ||x||; the paper does not specify a practical rule.

free parameters (1)
  • step size mu_k
    Theorem 4.2 requires mu_k within (mu_min/||x||^2, mu_max/||x||^2) with universal constants; no practical selection rule or experimental value is reported.
assumptions (4)
  • domain assumption The measurement matrices Ai are i.i.d. standard Gaussian, or complex Gaussian in the complex case.
    Theorems 4.1 and 4.2 assume Ai ~ N(0,1). The uniqueness theorems assume generic, not random, matrices.
  • domain assumption The signal x is exactly s-sparse and the sparsity level s is known.
    Algorithm 1 and Algorithm 2 use the hard thresholding operator with s; the guarantees hold for signals with ||x||_0 <= s.
  • domain assumption Noise is centered sub-exponential with max psi1 norm sigma.
    Theorem 4.1 and 4.2 assume this noise model.
  • standard math Standard results in algebraic geometry (dimension counts for projective varieties, Harris Cor. 11.13) and probability (Hoeffding, Bernstein, sub-exponential bounds) hold.
    Used in the proofs in Section 7.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Sparse Signal Recovery From Quadratic Systems with Full-Rank Matrices." pith.science (2026). https://pith.science/paper/STRWFVBV

@misc{pith2026250707557,
  author       = {Pith},
  title        = {Pith review of: Sparse Signal Recovery From Quadratic Systems with Full-Rank Matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/STRWFVBV}},
  note         = {Machine review of arXiv:2507.07557}
}
abstract

In signal processing and data recovery, reconstructing a signal from quadratic measurements poses a significant challenge, particularly in high-dimensional settings where measurements $m$ is far less than the signal dimension $n$ (i.e., $m \ll n$). This paper addresses this problem by exploiting signal sparsity. Using tools from algebraic geometry, we derive theoretical recovery guarantees for sparse quadratic systems, showing that $m\ge 2s$ (real case) and $m\ge 4s-2$ (complex case) generic measurements suffice to uniquely recover all $s$-sparse signals. Under a Gaussian measurement model, we propose a novel two-stage Sparse Gauss-Newton (SGN) algorithm. The first stage employs a support-restricted spectral initialization, yielding an accurate initial estimate with $m=O(s^2\log{n})$ measurements. The second stage refines this estimate via an iterative hard-thresholding Gauss-Newton method, achieving quadratic convergence to the true signal within finitely many iterations when $m\ge O(s\log{n})$. Compared to existing second-order methods, our algorithm achieves near-optimal sampling complexity for the refinement stage without requiring resampling. Numerical experiments indicate that SGN significantly outperforms state-of-the-art algorithms in both accuracy and computational efficiency. In particular, (1) when sparsity level $s$ is high, compared with existing algorithms, SGN can achieve the same success rate with fewer measurements. (2) SGN converges with only about $1/10$ iterations of the best existing algorithm and reach lower relative error.

Figures

Figures reproduced from arXiv: 2507.07557 by the authors.

Figure 1
Figure 1. Initialization experiments: Average relative error be￾tween x 0 and x for n = 500, s = 5 and m/n varying within the range [0.1, 1]. Corollary 1. Let {Ai} m i=1 be i.i.d. standard Gaussian random matrices with Ai ∼ N n×n(0, 1), and x ∈ R n be any fixed signal with kxk0 ≤ s. Assume that the initial guess x 0 is generated by Algorithm 1. There exists some positive constants C, C1, C2, C3, ǫ ∈ (0, 1), µ, µ such that if … view at source ↗
Figure 2
Figure 2. The phase transitions of WF, TWF, and proposed algorithm. Experiment 1: Comparative evaluation of initialization methods. In this experiment, we conduct a systematic comparison between our proposed initialization method and the Thresholded Spectral Initialization (TSI) approach from [15]. We fix the signal dimension n = 500 and sparsity level s = 5. The measurement ratio m/n is varied from 0.1 to 1 with step size be… view at source ↗
Figure 3
Figure 3. Convergence behavior: Plot of relative errors versus number of iterations for the proposed algorithm (SGN) and TWF. 6 8 10 12 14 16 18 20 22 Sparsity 100 102 104 106 108 110 112 114 Iterations required TWF (a) m = ⌊10s log n⌋ 6 8 10 12 14 16 18 20 22 Sparsity 4 6 8 10 12 14 Iterations required SGN (b) m = ⌊10s log n⌋ [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The average number of iterations required when suc￾cessful recovery is achieved. the same success rate. It means that our proposed algorithm outperforms WF and TWF in terms of sampling complexity, demonstrating a substantial improvement. Experiment 3 : Convergence beha…
Figure 5
Figure 5. Figure 5: Robustness of SGN to additional Gaussian noise Experiment 4: Robustness to noise. In this experiment, we investigate the robustness of the proposed SGN algo￾rithm to additive Gaussian noise. The experiment result is shown in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

53 extracted references · 52 canonical work pages

  1. [1]

    Akccakaya and V

    M. Akccakaya and V. Tarokh , Sparse signal recovery from a mixture of linear and magnitude-only measurements, IEEE Signal Process. Lett., 22 (2015), pp. 1220–1223

    work page 2015

  2. [2]

    Basu and Y

    S. Basu and Y. Bresler , Uniqueness of tomography with unknown view angles , IEEE Trans. Inform. Theory, 9 (2000), pp. 1094–1106

    work page 2000

  3. [3]

    Bendory and D

    T. Bendory and D. Edidin , The sample complexity of sparse multireference alignment a nd single-particle cryo-electron microscopy, SIAM J. Math. Data Sci., 6 (2024), pp. 254–282

    work page 2024

  4. [4]

    Bendory, Y

    T. Bendory, Y. Khoo, J. Kileel, O. Mickelin, and A. Singer , Autocorrelation analy- sis for cryo-em with sparsity constraints: improved sample complexity and projection-based algorithms, Proc. Natl. Acad. Sci., 120 (2023), p. e2216507120

    work page 2023

  5. [5]

    Blumensath and M

    T. Blumensath and M. E. Davies , Iterative thresholding for sparse approximations , J. Fourier Anal. Appl., 14 (2008), pp. 629–654

    work page 2008

  6. [6]

    J.-F. Cai, M. Huang, D. Li, and Y. W ang , Solving phase retrieval with random initial guess is nearly as good as by spectral initialization , Appl. Comput. Harmon. Anal., 58 (2022), pp. 60–84

    work page 2022

  7. [7]

    J.-F. Cai, H. Liu, and Y. W ang , Fast rank-one alternating minimization algorithm for phas e retrieval, J. Sci. Comput., 79 (2019), pp. 128–147

    work page 2019

  8. [8]

    J.-F. Cai, Y. Long, R. Wen, and J. Ying , A fast and provable algorithm for sparse phase retrieval, in Proc. Int. Conf. Learn. Represent., (2023)

    work page 2023

Show all 53 references
  1. [9]

    T. T. Cai, X. Li, and Z. Ma , Optimal rates of convergence for noisy sparse phase retriev al via thresholded wirtinger flow , Ann. Stat., 44 (2016), pp. 2221–2251

    2016

  2. [10]

    T. T. CAI and A. ZHANG , ROP: Matrix recovery via rank-one projections , Ann. Stat., 43 (2015), pp. 102–138

    2015

  3. [11]

    E. J. Cand `es and X. Li , Solving quadratic equations via phaselift when there are ab out as many equations as unknowns , Found. Comput. Math., 14 (2014), pp. 1017–1026

    2014

  4. [12]

    E. J. Candes, X. Li, and M. Soltanolkotabi , Phase retrieval via wirtinger flow: Theory and algorithms , IEEE Trans. Inf. Theory, 61 (2015), pp. 1985–2007

    2015

  5. [13]

    E. J. Candes, T. Strohmer, and V. Voroninski , Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming , Commun. Pure Appl. Math., 66 (2013), pp. 1241–1274

    2013

  6. [14]

    Chen and M

    J. Chen and M. K. Ng , Error bound of empirical l2 risk minimization for noisy standard and generalized phase retrieval problems , arXiv preprint arXiv:2205.13827, (2022)

    2022 arXiv

  7. [15]

    J. Chen, M. K. Ng, and Z. Liu , Solving quadratic systems with full-rank matrices using sparse or generative priors , IEEE Trans. Signal Process., (2025)

    2025

  8. [16]

    Chen and E

    Y. Chen and E. J. Cand `es, Solving random quadratic systems of equations is nearly as e asy as solving linear systems , Commun. Pure Appl. Math., 70 (2017), pp. 822–883

    2017

  9. [17]

    Y. Chen, Y. Chi, J. F an, and C. Ma , Gradient descent with random initialization: Fast global convergence for nonconvex phase retrieval , Math. Program., 176 (2019), pp. 5–37

    2019

  10. [18]

    L. Dai, X. Lu, and J. You , Grahtp: A provable newton-like algorithm for sparse phase retrieval, arXiv preprint arXiv:2410.04034, (2024)

    2024 arXiv

  11. [19]

    F. M. Dopico , A note on sin Θ theorems for singular subspace variations , BIT, 40 (2000), pp. 395–403

    2000

  12. [20]

    J. C. Duchi and F. Ruan , Solving (most) of a set of quadratic equalities: Composite o pti- mization for robust phase retrieval , Inf. Inference, 8 (2019), pp. 471–529

    2019

  13. [21]

    Foucart, Hard thresholding pursuit: an algorithm for compressive se nsing, SIAM J

    S. Foucart, Hard thresholding pursuit: an algorithm for compressive se nsing, SIAM J. Nu- mer. Anal., 49 (2011), pp. 2543–2563

    2011

  14. [22]

    Gao and Z

    B. Gao and Z. Xu , Phaseless recovery using the Gauss–Newton method , IEEE Trans. Signal Process., 65 (2017), pp. 5885–5896

    2017

  15. [23]

    Gross, Y.-K

    D. Gross, Y.-K. Liu, S. T. Flammia, S. Becker, and J. Eisert , Quantum state tomography via compressed sensing , Phys. Rev. Lett., 105 (2010), p. 150401. SPARSE SIGNAL RECOVERY FROM QUADRATIC SYSTEMS 29

    2010

  16. [24]

    Harris , Algebraic Geometry: A First Course , Springer Science & Business Media, 133 (2013)

    J. Harris , Algebraic Geometry: A First Course , Springer Science & Business Media, 133 (2013)

    2013

  17. [25]

    Huang and Y

    M. Huang and Y. W ang , Linear convergence of randomized kaczmarz method for solvi ng complex-valued phaseless equations , SIAM J. Imaging Sci., 15 (2022), pp. 989–1016

    2022

  18. [26]

    Huang, S

    S. Huang, S. Gupta, and I. Dokmani ´c, Solving complex quadratic equations with full-rank random gaussian matrices , in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., (2 019), pp. 5596–5600

  19. [27]

    Huang, S

    S. Huang, S. Gupta, and I. Dokmani ´c, Solving complex quadratic systems with full-rank random matrices, IEEE Trans. Signal Process., 68 (2020), pp. 4782–4796

    2020

  20. [28]

    Jagatap and C

    G. Jagatap and C. Hegde , Sample-efficient algorithms for recovering structured sign als from magnitude-only measurements, IEEE Trans. Inf. Theory, 65 (2019), pp. 4434–4456

    2019

  21. [29]

    Jaganathan, Y

    K. Jaganathan, Y. C. Eldar, and B. Hassibi , Phase retrieval: An overview of recent developments, Optical Compressive Imaging, (2016), pp. 279–312

    2016

  22. [30]

    C. Ma, K. W ang, Y. Chi, and Y. Chen , Implicit regularization in nonconvex statistical estimation: Gradient descent converges linearly for phase retrieval and matrix completion , Found. Comput. Math., 20 (2020), pp. 451–632

    2020

  23. [31]

    Netrapalli, P

    P. Netrapalli, P. Jain, and S. Sanghavi , Phase retrieval using alternating minimization , IEEE Trans. Signal Process., 63 (2015), pp. 4814–4826

    2015

  24. [32]

    Rigollet and A

    P. Rigollet and A. Tsybakov , Exponential screening and optimal rates of sparse estima- tion, Ann. Statist., 39 (2011), pp. 731–771

    2011

  25. [33]

    Shechtman, Y

    Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. S egev, Phase retrieval with application to optical imaging: a contempor ary overview , IEEE Signal Process. Mag., 32 (2015), pp. 87–109

    2015

  26. [34]

    Shechtman, Y

    Y. Shechtman, Y. C. Eldar, A. Szameit, and M. Segev , Sparsity based sub-wavelength imaging with partially incoherent light via quadratic comp ressed sensing , Opt. Express, 19 (2011), pp. 14807–14822

    2011

  27. [35]

    Shen and P

    J. Shen and P. Li , A tight bound of hard thresholding , J. Mach. Learn. Res., 18 (2018), pp. 1–42

    2018

  28. [36]

    Soltanolkotabi, Structured signal recovery from quadratic measurements: B reaking sam- ple complexity barriers via nonconvex optimization , IEEE Trans

    M. Soltanolkotabi, Structured signal recovery from quadratic measurements: B reaking sam- ple complexity barriers via nonconvex optimization , IEEE Trans. Inf. Theory, 65 (2019), pp. 2374–2400

    2019

  29. [37]

    J. Sun, Q. Qu, and J. Wright , A geometric analysis of phase retrieval , Found. Comput. Math., 18 (2018), pp. 1131–1198

    2018

  30. [38]

    Szameit, Y

    A. Szameit, Y. Shechtman, E. Osherovich, E. Bullkich, P. Sidor enko, H. Dana, S. Steiner, E. B. Kley, S. Gazit, T. Cohen-Hyams, et al. , Sparsity-based single-shot subwavelength coherent diffractive imaging , Nat. Mater., 11 (2012), pp. 455–459

    2012

  31. [39]

    Y. S. Tan and R. Vershynin , Phase retrieval via randomized kaczmarz: theoretical guar an- tees, Inf. Inference, 8 (2019), pp. 97–123

    2019

  32. [40]

    Vershynin , High-dimensional probability: An introduction with appli cations in data sci- ence, Camb

    R. Vershynin , High-dimensional probability: An introduction with appli cations in data sci- ence, Camb. Ser. Stat. Probab. Math. 47, Cambridge University Pr ess, Cambridge, UK, (2018)

    2018

  33. [41]

    W aldspurger, Phase retrieval with random gaussian sensing vectors by alt ernating pro- jections, IEEE Trans

    I. W aldspurger, Phase retrieval with random gaussian sensing vectors by alt ernating pro- jections, IEEE Trans. Inf. Theory, 64 (2018), pp. 3301–3312

    2018

  34. [42]

    W aldspurger, A

    I. W aldspurger, A. d’Aspremont, and S. Mallat , Phase recovery, maxcut and complex semidefinite programming, Math. Program., 149 (2015), pp. 47–81

    2015

  35. [43]

    W ang, G

    G. W ang, G. B. Giannakis, and Y. C. Eldar , Solving systems of random quadratic equations via truncated amplitude flow , IEEE Trans. Inf. Theory, 64 (2018), pp. 773–794

    2018

  36. [44]

    W ang, L

    G. W ang, L. Zhang, G. B. Giannakis, M. Akccakaya, and J. Chen , Sparse phase retrieval via truncated amplitude flow , IEEE Trans. Signal Process., 66 (2017), pp. 479–491

    2017

  37. [45]

    W ang and Z

    Y. W ang and Z. Xu , Phase retrieval for sparse signals , Appl. Comput. Harmon. Anal., 37 (2014), pp. 531–544

    2014

  38. [46]

    W ang and Z

    Y. W ang and Z. Xu , Generalized phase retrieval: measurement number, matrix r ecovery and beyond, Appl. Comput. Harmon. Anal., 47 (2019), pp. 423–446

    2019

  39. [47]

    M. Xu, D. Dong, and J. W ang , Subspace phase retrieval, IEEE Trans. Inf. Theory, 70 (2024)

    2024

  40. [48]

    Ye and C

    F. Ye and C. H. Zhang , Rate minimaxity of the Lasso and Dantzig selector for the ℓ1 loss in ℓr balls, J. Mach. Learn. Res., 11 (2010), pp. 3519–3540

    2010

  41. [49]

    X. Yuan, P. Li, and T. Zhang , Exact recovery of hard thresholding pursuit , Adv. Neural Inf. Process. Syst., 29 (2016). 30 JINMING WEN, YI HU, AND MENG HUANG

    2016

  42. [50]

    Zehni, S

    M. Zehni, S. Huang, I. Dokmani ´c, and Z. Zhao , 3D unknown view tomography via rotation invariants, in Proc. IEEE Int. Conf. Acoust. Speech Signal Process., (2 020), pp. 1449–1453

  43. [51]

    Zhang, Y

    H. Zhang, Y. Chi, and Y. Liang , Provable non-convex phase retrieval with outliers: Median truncatedwirtinger flow, in Proc. Int. Conf. Mach. Learn., (2016), pp. 1022–1031

    2016

  44. [52]

    Zhang, Y

    H. Zhang, Y. Liang, and Y. Chi , A nonconvex approach for phase retrieval: Reshaped wirtinger flow and incremental algorithms , J. Mach. Learn. Res., 18 (2017), pp. 5164–5198

    2017

  45. [53]

    Horn and C

    R. Horn and C. Johnson , Topics in matrix analysis , Cambridge university press, (1994). College of Information Science and Technology, Jinan Univer sity, Guangzhou, China Email address : jinming.wen@mail.mcgill.ca College of Information Science and Technology, Jinan Univer si...

    1994

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.