{"id":"a6ac7e6e-4028-4cad-8c1c-ce3b241a9424","arxiv_id":"2505.15351","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A workshop-based survey of open problems in phase retrieval, with a section proposing a unified framework that combines generative priors with conventional data fidelity.","lead":"This workshop report surveys open problems in phase retrieval, where one must reconstruct a signal from intensity measurements that hide the phase. It pairs each problem with practical motivation, and Section 5 adds a preliminary framework that blends generative models with classical reconstruction.","discovery_kind":"review","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 5's error bounds do not apply to phase retrieval as stated, because A(f)=|Af|^2 is neither injective (global phase) nor globally bi-Lipschitz (quadratic scaling).","rationale":"The reader correctly identifies the bi-Lipschitz injectivity assumption on A and G as the weakest point of Section 5. I agree that Section 5 is the part that makes the paper more than a roadmap and that its guarantees are not independently checkable here. However, the reader frames the issue as 'largely unverifiable' and 'no explicit Lipschitz constant is known.' The stronger, load-bearing defect is that the phase-retrieval map A(f)=|Af|^2 violates the assumption in a basic, non-parametric way: the global-phase symmetry breaks injectivity and the quadratic homogeneity breaks global Lipschitzness on C^n. This is an internal inconsistency with the paper's own phase-retrieval setup, not merely a gap between theory and practice. The survey as a whole remains a useful roadmap of open problems, and the numerical illustration in Section 5.2 is suggestive, so I would not reject the paper. The final verdict should remain CONDITIONAL: Section 5 needs either a corrected statement on the quotient/bounded domain or an explicit restriction of the claims. Since this does not change the reader's overall verdict, I mark verdict_should_be as UNCHANGED.","tokens_in":26603,"tokens_out":11117,"duration_ms":101964,"concrete_test":"Analytical check: take any nonzero f0 and ε=0 with y=|Af0|^2. Both f0 and −f0 are global minimizers of (11) and (13), but Lemma 5.2 would require ||−f0−f0||=2||f0||≤0, a contradiction. To settle the intended scope, re-derive Lemmas 5.2 and 5.5 with the phase-quotient distance inf_θ||f−e^{iθ}f0|| and with signals restricted to a bounded set; report the resulting constants α. If no such reformulation exists, Section 5's bounds are not phase-retrieval results as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The technical centerpiece, Lemma 5.5 (with Lemma 5.2), claims that the unified objective (13) interpolates between classical and generative phase retrieval with controlled error. The proof rests on the Section 5.1 blanket assumption that A and G are injective and bi-Lipschitz on all of C^n with constants α, β. For the phase-retrieval instance A(f)=|Af|^2, this assumption is not merely hard to verify; it is false. First, A is never injective: A(e^{iθ}f0)=A(f0) for every θ, so in the noiseless case ε=0 both f0 and e^{iθ}f0 are exact minimizers of (11) and (13). Lemma 5.2 then claims ||f_tilde−f0||≤0, contradicted by ||e^{iθ}f0−f0||=2||f0|| for θ=π. Second, A is not globally Lipschitz on C^n: for f=tg with t>1, ||A(tg)−A(g)||/||tg−g|| = (t+1)||A(g)||/||g||, which is unbounded as t→∞. Hence no finite α can satisfy the stated bi-Lipschitz inequality. Remark 5.1's appeal to injectivity does not address the global-phase symmetry or the unbounded domain. Unless the lemmas are restated in a quotient metric and on a bounded domain (or for a restricted class of bounded generative outputs), the headline interpolation result is not a theorem about phase retrieval as formulated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This survey, produced from the PRiMA 2024 Lorentz Center workshop, identifies and contextualizes six families of open problems in phase retrieval: intrinsic difficulty/complexity measures, physics-informed object and probe parameterization, generative-model regularization ('PtyGenography'), structured random measurements, one-bit quantized event-driven 4D STEM, and mathematical foundations of Wigner distribution deconvolution. Each section formulates questions and cites relevant literature. The paper also contains a self-contained technical core in Section 5: under an injectivity/bi-Lipschitz assumption on the measurement map and generative model, it states error bounds for the classical formulation (11), the generative formulation (12), and the unified objective (13), and reports a small MNIST-based numerical experiment supporting the unified approach.","tokens_in":26882,"tokens_out":7178,"duration_ms":66395,"significance":"The survey's breadth is its main value: it collects hard open questions from a diverse community, connects theory to concrete imaging applications, and provides an extensive reference list. Sections 3, 4, 6, 7, and 8 are appropriately hedged and useful as a field map. The Section 5 material, however, cannot currently serve as the advertised technical centerpiece: its assumptions are not satisfied by the phase retrieval map, and the numerical support is not reproducible. As a survey, the paper can be repaired by restricting the theorems to a setting where the assumptions hold or are explicitly conjectured, and by reporting numerical results with proper uncertainty quantification and code/data availability.","major_comments":[{"comment":"The blanket assumption that A and G are injective and bi-Lipschitz on all of C^n is false for the phase-retrieval map A(f)=|Af|^2. The map is not injective (A(e^{iθ}f)=A(f)), and it is not bi-Lipschitz: for g≠0 and t>1, ||A(tg)-A(g)||_2 / ||tg-g||_2 = (t+1) || |Ag|^2 ||_2 / ||g||_2, which is unbounded, while for t→0 the ratio tends to 0. Consequently, in the noiseless case ε=0, Lemma 5.2 would imply ||tilde f - f0||≤0, contradicting the global-phase twin e^{iθ}f0. The lemmas must be restated in a quotient metric (identifying global phase) and on a bounded domain, or explicitly restricted to a class of bounded outputs; otherwise the interpolation claim (13) is not a theorem about phase retrieval. Problem 5.6 acknowledges the absence of a global bi-Lipschitz property as an open question, but this does not repair the stated lemmas, since the assumption is not merely unverified: it is false.","section":"Section 5.1 (first paragraph; Lemmas 5.2 and 5.5)"},{"comment":"The numerical demonstration is a single small-scale experiment: n=64, k=30, ℓ=100 Bernoulli masks, one MNIST-derived PCA model, and no error bars, confidence intervals, or code. The text states that 'the combined method (13) achieves the best result for both low and high signal-to-noise ratio levels,' but the plotted differences may be within run-to-run variability. Please provide standard deviations or confidence intervals, release the code and data, or explicitly soften the claim to an illustrative observation. This is important because the methods involve random masks and random noise realizations.","section":"Section 5.2 (Figure 6 and surrounding text)"}],"minor_comments":[{"comment":"The keyword 'deconvilution' is a typo for 'deconvolution'.","section":"Abstract/keywords"},{"comment":"The phrase 'bandlimited [?,93]' contains an unresolved citation placeholder; the missing reference should be supplied.","section":"Section 8.2"},{"comment":"References [34] and [35] appear to be duplicate entries for the same tutorial by Dong et al.; they should be consolidated.","section":"References [34] and [35]"},{"comment":"The heading 'Vanderlugt correlation' is inconsistently capitalized; the text elsewhere uses 'VanderLugt'.","section":"Section 5.1.3"},{"comment":"In the first bullet, 'For the measurement setup when A= F DF, we observe...' should read 'when A=FDF,'; the sentence also does not indicate whether the reported phase transition is empirical or proven.","section":"Section 6.2"}],"recommendation":"major_revision","confidential_remarks":"The Section 5 lemmas are attributed to the authors' companion paper [11], and the present manuscript cannot be independently checked against that derivation. Since the stated assumptions fail for the phase retrieval map, the section needs a substantial rewrite before the paper can be accepted. I do not see this as a question of author intent; it is a technical gap that is fixable by a careful restatement of the assumptions and scope."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the punchline: this is a useful survey, worth having on the shelf, but the supposedly new technical result in Section 5 does not hold for phase retrieval as stated. The paper defines A(f)=|Af|^2 and then assumes A is injective and bi-Lipschitz. That can't be true: global phase gives A(e^{iθ}f)=A(f), and the quadratic scaling makes the Lipschitz bound blow up. The lemmas in Section 5 are not theorems about phase retrieval; they are theorems about a hypothetical map that satisfies stronger properties. The authors even hint at this in Problem 5.6, but they still present the lemmas without proof and defer to a companion paper.\n\nWhat is genuinely good: the survey is a well-organized map of the gap between theory and practice. Sections 3, 4, 6, 7, 8 give a fair picture of where things stand—object complexity measures, parameterized models in crystallography, structured random masks, one-bit quantization in 4D STEM, Wigner deconvolution. The open problems are concrete and some are not widely known outside narrow communities. The bibliography is broad, and the authors are honest about what is open.\n\nThe soft spots are real, but they are mostly concentrated in Section 5. Besides the false assumptions, the proofs are in a companion paper, so this version cannot be independently checked. The numerical experiment is a single masked-Fourier MNIST setup with 8x8 images, no error bars, no code, and one regularization schedule. That is fine for an illustrative Figure 6, but it does not support the quantitative claims. The rest of the survey does not have this problem.\n\nThe citation pattern looks fine. Self-citing the companion paper is appropriate; the survey explicitly attributes Section 5 to it and the companion provides the derivations. The circularity burden is low.\n\nWho is this for? Someone looking for a concise briefing on open phase retrieval problems with pointers to the literature will get value. I would not cite the Section 5 bounds, but I might cite the survey for the open problems. A serious editor should send it to peer review, but with the expectation that Section 5 either be rewritten with a quotient metric and bounded-domain assumptions (or a different map), or be cut back to a literature review with the error bounds removed. As is, a referee should not let the paper stand as-is.\n\nRecommendation: accept for peer review, but require fixing or removing Section 5's technical claims.","headline":"Useful workshop survey of phase retrieval open problems; the Section 5 technical claims fail as stated because A(f)=|Af|^2 is neither injective nor globally bi-Lipschitz.","tokens_in":27508,"tokens_out":4042,"would_cite":false,"duration_ms":33744,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["94A12","42A38","94A08"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single variational objective with a generative prior interpolates between classical phase retrieval and generative-only reconstruction, and its reconstruction error is bounded by a linear trade-off between generative bias and…","keywords":["phase retrieval","ptychography","inverse problem complexity","generative priors","random matrices","quantization","Wigner distribution deconvolution"],"falsifier":"Construct an out-of-distribution signal $f_0$ whose distance to the generative model's best approximation, $\\|G(z_0)-f_0\\|$, is known; choose a measurement map whose Lipschitz constant $\\alpha$ can be numerically estimated; solve the unified objective over a grid of $\\lambda$; and check whether $\\|f-f_0\\|$ stays below $\\lambda\\alpha\\|G(z_0)-f_0\\| + 2\\alpha\\|\\varepsilon\\|$ at every $\\lambda$. An observed error above the predicted line at any $\\lambda$ would refute the paper's Lemma 5.5.","tokens_in":26373,"feed_emoji":"🔬","tokens_out":19448,"duration_ms":147295,"temperature":0.7,"pith_summary":"Phase retrieval is the task of recovering a signal from intensity-only measurements, and it is central to X-ray crystallography, electron microscopy, and ptychography. This survey argues that the main open problems of the field—measuring object difficulty, physics-informed parametrization, structured random measurements, one-bit quantization, and Wigner distribution deconvolution—are what keep theory from matching practice. Its technical centerpiece is a claim about generative priors: a single objective that fits both the phaseless measurements and a generative model interpolates between classical reconstruction and generative-only reconstruction, with reconstruction error bounded by a linear trade-off between model bias and noise amplification. If this bound holds in realistic settings, it gives practitioners a principled way to decide how much to trust a learned prior.","feed_headline":"One objective unifies classical and generative phase retrieval","feed_subtitle":"A new error bound lets users tune how much to trust a learned prior against measurement noise.","key_machinery":"The load-bearing object is the unified variational objective $\\min_{z,f} \\|A(f)-y\\|^2 + \\lambda^2\\|G(z)-f\\|^2$, which makes $\\lambda$ a continuous dial between pure data fidelity and pure generative reconstruction; under the bi-Lipschitz assumptions, Lemma 5.5 converts this objective into the linear error bound. A second load-bearing identity, used in the Wigner deconvolution section, is $I = W_f \\ast K_w$: ptychographic intensities are a convolution of the object's Wigner distribution with the window's, so a Fourier-domain division separates object from window when the window's ambiguity function has no zeros.","core_discovery":"On the paper's own terms, classical phase retrieval and generative-model-based reconstruction are two endpoints of one optimization problem. Given a measurement map $A$, a generative model $G$, and noisy phaseless data $y$, the estimator obtained from minimizing $\\|A(f)-y\\|^2 + \\lambda^2\\|G(z)-f\\|^2$ interpolates between the classical estimator at $\\lambda=0$ and the generative-only estimator as $\\lambda\\to\\infty$. Under the assumption that $A$ and $G$ are injective and bi-Lipschitz, Lemma 5.5 bounds the reconstruction error by $\\lambda\\alpha\\|G(z_0)-f_0\\| + 2\\alpha\\|\\varepsilon\\|$, a linear trade-off between the bias introduced by the generative model and the amplification of measurement noise. The survey also contends that the open problems it lists are the essential bottlenecks separating such guarantees from practical imaging.","pith_inferences":["The linear form of the bound suggests a calibration rule the paper leaves implicit: choose λ proportional to the Lipschitz constant times the ratio of noise to model bias, so the two error terms balance; this could be tested from residuals alone.","If the bi-Lipschitz assumption can be relaxed to local versions, the same interpolation objective should apply to other nonlinear inverse problems with learned priors, such as deblurring and limited-angle tomography.","The subspace completion approach for missing Wigner-deconvolution coefficients could extend to arbitrary missing patterns by viewing the diagonals of $ff^*$ as a low-rank matrix completion problem; the paper analyzes only the one-missing-coefficient case.","The weak-versus-full recovery distinction offers a detector-design criterion: for low-dose event-driven STEM, the oversampling ratio should be compared with the algorithmic full-recovery threshold rather than the information-theoretic one."],"forward_implications":["The unified objective reduces to classical phase retrieval at $\\lambda=0$ and to generative-only reconstruction as $\\lambda\\to\\infty$, and in the paper's masked-Fourier experiments it gives the lowest relative error for both in-distribution and out-of-distribution signals.","A Fourier-domain correlation procedure can single out sparse imperfections that a generative model cannot express, demonstrated on a letter with a diacritic mark, without an explicit model of the defect.","For Gaussian random measurement matrices, weak recovery begins at oversampling ratio 1 and full recovery at 2, with an algorithmic full-recovery threshold near 2.03 for approximate message passing; structured two-layer lens–diffuser systems show a phase transition near 2.","In the low-dose regime, Poisson phase retrieval can be reinterpreted as one-bit quantization with bit-flip noise, and a hybrid method achieves weak recovery with four measurements per unknown for Gaussian sensing, the starting point for event-driven scanning transmission electron microscopy.","Wigner distribution deconvolution recovers an object from ptychographic intensities in $O(n\\delta\\log n)$ operations when all scan positions are used and the window's ambiguity function has no zeros; symmetric windows introduce zeros and require a subspace completion step whose guarantees are incomplete."],"supporting_citations":[{"why":"Supplies the error-bound lemmas and the unified objective that Section 5's central claim rests on.","marker":"[11]"},{"why":"Establishes the bi-Lipschitz property of injective phase retrieval maps that the bounds assume.","marker":"[12]"},{"why":"Reports the numerical observation that generative reconstruction wins at low signal-to-noise ratio and classical reconstruction at high ratio, motivating interpolation.","marker":"[99]"},{"why":"Provides the information-theoretic weak and full recovery thresholds for random phase retrieval that Section 6 extends.","marker":"[82]"},{"why":"Introduces the one-bit quantization reformulation of low-dose Poisson phase retrieval and the hybrid method used as a baseline.","marker":"[70]"},{"why":"Introduces Wigner distribution deconvolution and the convolution identity used throughout Section 8.","marker":"[96]"},{"why":"Derives the discrete Wigner-deconvolution relation and the near-linear-time recovery procedure that Section 8 builds on.","marker":"[93]"},{"why":"Proposes lost subspace completion for the case where Fourier coefficients of the window ambiguity function are missing.","marker":"[40]"}],"fun_headline_variants":["Classical and generative phase retrieval now one objective","One loss function spans classical to generative phase retrieval","Survey unveils a single optimization for phase retrieval variants","Phase retrieval unified: classical and generative as two endpoints","New framing: phase retrieval is a tunable trade-off"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bound assumes the measurement process and the generative model both stretch distances by known, bounded factors; for real learned networks and realistic phase retrieval measurements these factors are essentially unverifiable, and the paper itself notes that explicit constants are unknown for most systems.","fun_headline_variants_meta":{"raw":{"variants":["Classical and generative phase retrieval now one objective","One loss function spans classical to generative phase retrieval","Survey unveils a single optimization for phase retrieval variants","Phase retrieval unified: classical and generative as two endpoints","New framing: phase retrieval is a tunable trade-off"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00012,"raw_usage":{"total_tokens":1019,"prompt_tokens":808,"completion_tokens":211,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":424,"completion_tokens_details":{"reasoning_tokens":136}},"tokens_in":424,"tokens_out":211,"duration_ms":2827,"temperature":1.0,"reasoning_tokens":136,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:19:02.401300+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Construct an out-of-distribution signal $f_0$ whose distance to the generative model's best approximation, $\\|G(z_0)-f_0\\|$, is known; choose a measurement map whose Lipschitz constant $\\alpha$ can be numerically estimated; solve the unified objective over a grid of $\\lambda$; and check whether $\\|f-f_0\\|$ stays below $\\lambda\\alpha\\|G(z_0)-f_0\\| + 2\\alpha\\|\\varepsilon\\|$ at every $\\lambda$. An observed error above the predicted line at any $\\lambda$ would refute the paper's Lemma 5.5.","supporting_citations":[{"cited_title":"PtyGenography: using generative models for regularization of the phase retrieval problem","cited_arxiv_id":"2502.01338","evidence_quote":"Supplies the error-bound lemmas and the unified objective that Section 5's central claim rests on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports the numerical observation that generative reconstruction wins at low signal-to-noise ratio and classical reconstruction at high ratio, motivating interpolation."},{"cited_title":"Phase retrieval in high dimensions: Statistical and computational phase transitions","cited_arxiv_id":null,"evidence_quote":"Provides the information-theoretic weak and full recovery thresholds for random phase retrieval that Section 6 extends."},{"cited_title":"A one-bit quantization approach for low-dose poisson phase retrieval","cited_arxiv_id":null,"evidence_quote":"Introduces the one-bit quantization reformulation of low-dose Poisson phase retrieval and the hybrid method used as a baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces Wigner distribution deconvolution and the convolution identity used throughout Section 8."},{"cited_title":"Inverting spectrogram measurements via aliased Wigner distribution de- convolution and angular synchronization","cited_arxiv_id":null,"evidence_quote":"Derives the discrete Wigner-deconvolution relation and the near-linear-time recovery procedure that Section 8 builds on."},{"cited_title":"Well- conditioned ptychographic imaging via lost subspace completion","cited_arxiv_id":null,"evidence_quote":"Proposes lost subspace completion for the case where Fourier coefficients of the window ambiguity function are missing."}],"review_version":1}