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Isolating the hard core of phaseless inference: the Phase selection formulation

T0 review · 3 major / 6 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read Splitting phase retrieval into sign selection plus regression lets standard heuristics reach the Bayes-optimal sample efficiency.

desk verdict A clever reformulation of phase retrieval with a genuine large-deviation analysis, but the headline 'approaching Bayes-optimal' claim is softer than the abstract suggests. read the letter →

arxiv 2502.04282 v1 pith:ZI4ANRO3 submitted 2025-02-06 cond-mat.dis-nn

Davide Straziota , Luca Saglietti This is my paper · ORCID
classification cond-mat.dis-nn
keywords phaseretrievalselectionreplicatheorylargedeviationssimulatedannealingapproximatemessagepassingLangevindynamicsBayes-optimalthreshold
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

In noiseless real-valued phase retrieval, the signal is hidden behind the absolute values of linear measurements; the genuinely hard part is the missing signs. The paper introduces 'Phase selection,' a two-level formulation in which a combinatorial loop chooses the signs and an inner ridge regression recovers the signal in closed form. A replica large-deviation analysis maps the free-energy landscape and shows two branches—one uninformed, one aligned with the signal—that merge at a first-order transition whose location depends on dataset size and on an L2 penalty. The paper's central proposal is to anneal temperature and regularization simultaneously, keeping their product fixed, so that early regularization steers the search toward informative signs and late weak regularization removes the bias. Numerical experiments with simulated annealing, approximate message passing, and Langevin dynamics all approach the Bayes-optimal threshold $\alpha_{BO}=1.13$, which would make the hard core of phase retrieval accessible to standard solvers.

What carries the argument

The load-bearing object is the Phase selection loss $H_{A,y}(x,S)=\|S\odot y-Ax\|_2^2+\frac{\lambda}{2}\|x\|_2^2$, with binary sign variables $S\in\{-1,1\}^M$ and signal $x\in\mathbb{R}^N$. For fixed signs the inner problem is ridge regression with the closed-form estimator $\hat{x}(S)=(A^\top A+\lambda I)^{-1}A^\top(S\odot y)$; the outer problem is a combinatorial optimization over signs. The analytical machinery is a replica large-deviation computation of the free entropy $\Phi(\mu,\phi)$, whose Legendre transform gives the complexity $\Sigma(O,e)$ of sign configurations at overlap $O$ and energy $e$, plus a stability analysis of the replica-symmetric saddle point. The proposed annealing schedule keeps $\mu\lambda$ constant while $\mu\to\infty$, so early strong regularization guides the search toward informative signs and late weak regularization removes the bias.

What would settle it

Run the simulated-annealing solver with the fixed-$\mu\lambda$ schedule on instances of size $N=3200$ at a dataset size strictly between $\alpha_{BO}=1.13$ and the RS merging prediction $\alpha\approx1.7$, and measure the success fraction over many instances as $N$ grows; if it tends to zero instead of one, simultaneous annealing does not reach the Bayes-optimal efficiency. Independently, a one-step replica-symmetry-breaking calculation of the free entropy at the same $(\mu,\lambda)$ values would show whether the informed and uninformed branches still merge at a finite $\alpha$.

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Extended reading notes

Core claim

The central claim is that noiseless real-valued phase retrieval should be solved by selecting the signs first and regressing second, and that this decomposition changes the algorithmic picture even though it leaves the information-theoretic threshold untouched. For each sign assignment the inner problem is a convex ridge regression with the closed-form estimator $\hat{x}(S)=(A^\top A+\lambda I)^{-1}A^\top(S\odot y)$, so the whole difficulty resides in the binary outer optimization over $S\in\{-1,1\}^M$. The replica computation of the free entropy $\Phi(\mu,\phi)$ reveals, at low temperature near the Bayes-optimal threshold, two coexisting branches: an uninformed branch connected to random initialization and an informed branch connected to the signal, with a first-order transition at which they merge. The paper shows that an $L_2$ regularization anticipates this merging to smaller dataset sizes, and that a simultaneous annealing of the inverse temperature $\mu$ and the regularization $\lambda$ (holding $\mu\lambda$ fixed) removes the bias that a fixed $\lambda$ would leave. The three solvers tested—simulated annealing, approximate message passing, and Langevin dynamics on relaxed signs—are reported to approach the Bayes-optimal sample efficiency $\alpha_{BO}=1.13$ as the annealing is slowed, in the sense that their finite-size success curves move toward that threshold.

Load-bearing premise

The analysis assumes that a single symmetric description of the disorder captures the relevant free-energy branches; if that description breaks down, the predicted transition and the annealing schedule that depends on it could give the wrong threshold.

Editorial extensions

If this is right

  • If the central claim holds, generic solvers can solve noiseless real-valued phase retrieval with roughly $1.13$ measurements per signal dimension, close to the information-theoretic limit.
  • The L2 regularization's role is demystified: it biases the optimization toward correcting large-magnitude sign errors, which anticipates the branch merging in the free-energy landscape.
  • The simultaneous-annealing protocol (fixed $\mu\lambda$) provides a concrete schedule for removing the regularization bias while keeping its early benefit.
  • The Phase selection decomposition preserves the original problem's information-theoretic threshold, so it is a reformulation, not a relaxation that changes what is recoverable.

Reading between the lines

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

  • Editorial extension: the same decomposition should apply to other problems where a discrete selection is coupled to continuous estimation, such as quantized compressed sensing or spike-and-slab regression; the paper suggests this direction but does not test it.
  • Editorial extension: a sharper test of the central claim is to fix $\alpha\in(1.13,1.7)$ and check whether the success probability at fixed computational budget converges to 1 as $N\to\infty$; the presented curves are consistent with this but do not prove it.
  • Editorial extension: if replica-symmetry-breaking corrections are the reason the RS threshold $\alpha\approx1.7$ disagrees with simulated annealing, a one-step RSB large-deviation computation would predict a different merging point, and the simultaneous-annealing schedule might be improvable by adapting the $\mu\lambda$ product to that corrected phase diagram.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces a two-level formulation of real-valued phase retrieval, called Phase selection, in which the missing signs of the measurements are treated as explicit binary variables and the signal is recovered by a nested convex linear regression. The authors perform a replica-theoretic large-deviation analysis of the sign-configuration complexity and free-energy landscape under a replica-symmetric ansatz, showing that an L2 regularization can anticipate the merging of the informed and uninformed branches. They then propose a simultaneous annealing of inverse temperature and regularization and report finite-size success rates for three heuristics (Simulated Annealing, Approximate Message Passing, and Langevin dynamics), claiming that these approach the Bayes-optimal sample efficiency alpha_BO = 1.13.

Significance. If the headline claim is established, the paper would provide a physically motivated decomposition of phase retrieval into a hard combinatorial sign-selection subproblem and a convex regression subproblem, and it would show that standard heuristics with a carefully designed annealing schedule can reach near-information-theoretic sample efficiency. The large-deviation computation and the stability analysis are valuable contributions, and the paper is honest in reporting that the replica-symmetric ansatz is not globally stable. However, the central numerical claim is currently supported only by finite-size success-probability curves without error bars, confidence intervals, or scaling analysis, so the asymptotic statement in the abstract is not yet justified.

major comments (3)
  1. [Sec. IV A-C, Figs. 6-8] The central claim of approaching alpha_BO = 1.13 rests on success-probability curves for 100 instances per point and system sizes up to N=800 (SA, Langevin) or N=3200 (AMP), but the curves are plotted without error bars and the half-success thresholds are read off by eye. No scaling analysis in N or in the annealing rate epsilon is provided: there is no fit of the threshold alpha_h(N, epsilon) to a limit, no extrapolation as epsilon -> 0, and no statement about how finite-size effects behave. Please add binomial confidence intervals, describe the threshold extraction procedure, and provide a scaling or extrapolation analysis, or explicitly soften the asymptotic claim in the abstract.
  2. [Sec. III C 2 and Appendix A 2] The theoretically motivated annealing schedule is derived from a replica-symmetric analysis, but the paper itself states (Appendix A 2 a) that the RS solutions are not globally stable and (Sec. IV) that the RS prediction of branch merging at alpha = 1.7 disagrees with the unregularized SA recovery threshold, attributing the discrepancy to replica symmetry breaking. Since the RSB effects are not computed quantitatively, the free-energy branch structure that motivates the simultaneous annealing is not established beyond the RS approximation, and the only direct support for the headline claim is the finite-size numerics. A quantitative 1RSB analysis of the thresholds, or an explicit argument that the qualitative branch picture survives RSB, is needed.
  3. [Abstract and Sec. V] The abstract's claim that the algorithms 'are shown to approach the Bayes-optimal sample efficiency' is stronger than the paper's own hedging in the body, e.g., 'the heuristic seems to approach the alpha_BO threshold in the limit of vanishing annealing rate' (Sec. IV B) and 'seems to approach' in the caption of Fig. 6. Without the missing scaling analysis, the strong asymptotic claim is not supported. Please either provide the scaling evidence or revise the claim to state that the finite-size thresholds decrease with N and slower annealing and extrapolate to a value near alpha_BO.
minor comments (6)
  1. [Fig. 3 caption] The word 'istogram' should be 'histogram'.
  2. [Fig. 6 caption] 'sistem size' should be 'system size'.
  3. [Sec. IV C] The sentence 'The continuous optimization of theta an be performed' should read 'can be performed'.
  4. [Algorithm 1] The loop condition 'while t <= T' is unclear because the pseudocode never increments t; please state explicitly how T relates to the number of Monte Carlo steps and annealing updates.
  5. [Figs. 6-8] The definition of a successful run as 'm ~ 1 and MSE_y = 0' needs explicit numerical thresholds, since exact equality will never be attained in floating point.
  6. [Fig. 8 caption] The caption calls the solver 'GD-based' while the text describes Langevin dynamics; please unify the terminology.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular reduction found: the Bayes-optimal threshold is imported from external phase-retrieval literature, and the numerical approach-to-threshold claim is measured, not fitted.

full rationale

Walking the claimed derivation chain, the free-energy and large-deviation analysis starts from the two-level Phase selection loss and is solved with a stated replica-symmetric ansatz and standard rescalings; it does not insert the target threshold alpha_BO as an input. The value alpha_BO = 1.13 is not derived from any fit in this paper; it is taken from the phase-retrieval literature, and the paper explicitly says the Phase selection threshold is inherited because the new formulation only adds suboptimal local minima. The central numerical claim is based on measured success fractions over freshly generated instances with fixed annealing schedules; the thresholds in Figs. 6-8 are read from those curves rather than tuned to land on 1.13. The acknowledged discrepancy between the RS branch-merging prediction (alpha = 1.7) and the simulated-annealing recovery threshold (alpha > 1.7), attributed to replica symmetry breaking, is a falsifiable prediction that partially fails, which is evidence of non-circularity rather than circularity. Some citations to prior work by one of the authors ([5], [24], [33]) appear in methodological or stability discussions, but the core claim does not reduce to those citations: local stability is independently checked in Appendix A2, and alpha_BO is an external benchmark. The lack of error bars and scaling analysis in the finite-size success-rate curves weakens the asymptotic 'approach alpha_BO' claim as numerical evidence, but that is an extrapolation and statistical-support concern, not an equivalence-by-construction concern under the stated circularity rules.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new physical entities. The main theoretical burden is carried by the RS ansatz, the replica trick, and the Gaussian assumptions, all of which are standard in the statistical physics of disordered systems but not fully justified. The hand-chosen algorithm hyperparameters affect the numerical demonstration but are not fitted to the Bayes-optimal threshold.

free parameters (4)
  • initial regularization lambda_0 = 1.0 for SA and AMP, 0.75 for Langevin
    Hand-chosen starting point of the annealing schedule; performance depends on it but it is not fitted to the target result.
  • initial inverse temperature mu_0 = 1.0 for SA and Langevin, 1000 for AMP
    Chosen by hand; AMP requires a large initial inverse temperature.
  • annealing rate epsilon = 0.1 and 0.01
    Controls the multiplicative temperature schedule; slower rates improve success probability, so the choice affects the extrapolation to alpha_BO.
  • Monte Carlo steps per temperature T_A = 100
    Schedule hyperparameter in the simulated annealing solver.
assumptions (6)
  • domain assumption Replica Symmetric ansatz captures the relevant free-energy branches
    The RS ansatz in Eq. (17) is used for all free-entropy computations; Appendix A.2 shows the RS solution is not globally stable, so the branch structure near the claimed transition may be altered by RSB.
  • standard math Replica trick analytic continuation s -> 0 and beta -> 0 is valid
    The quenched free entropy is obtained via two replica tricks and analytic continuation in Eqs. (13)-(14), a standard but non-rigorous physics technique.
  • domain assumption i.i.d. Gaussian measurement matrix and Gaussian signal prior
    Assumed in Sec. II A; the replica computation and AMP equations depend on these distributions.
  • domain assumption Large deviation principle for the complexity Sigma(O,e)
    The analysis assumes exponentially many sign configurations obey a large deviation principle with rate function Sigma, introduced in Sec. III A without proof.
  • domain assumption Proportional scaling regime with alpha > alpha_IT = 1 and noiseless measurements
    The theoretical and numerical study is restricted to M/N = O(1) in the noiseless case, as stated in Sec. II A.
  • domain assumption Flat prior for the estimated signal x in the internal regression
    In Sec. III B 1, P(x_ac) is taken flat while the true signal has a Gaussian prior; this asymmetry is part of the MAP setting.

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Pith. "Pith review of Isolating the hard core of phaseless inference: the Phase selection formulation." pith.science (2026). https://pith.science/paper/ZI4ANRO3

@misc{pith2026250204282,
  author       = {Pith},
  title        = {Pith review of: Isolating the hard core of phaseless inference: the Phase selection formulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZI4ANRO3}},
  note         = {Machine review of arXiv:2502.04282}
}
abstract

Real-valued Phase retrieval is a non-convex continuous inference problem, where a high-dimensional signal is to be reconstructed from a dataset of signless linear measurements. Focusing on the noiseless case, we aim to disentangle the two distinct sub-tasks entailed in the Phase retrieval problem: the hard combinatorial problem of retrieving the missing signs of the measurements, and the nested convex problem of regressing the input-output observations to recover the hidden signal. To this end, we introduce and analytically characterize a two-level formulation of the problem, called ``Phase selection''. Within the Replica Theory framework, we perform a large deviation analysis to characterize the minimum mean squared error achievable with different guesses for the hidden signs. Moreover, we study the free-energy landscape of the problem when both levels are optimized simultaneously, as a function of the dataset size. At low temperatures, in proximity to the Bayes-optimal threshold -- previously derived in the context of Phase retrieval -- we detect the coexistence of two free-energy branches, one connected to the random initialization condition and a second to the signal. We derive the phase diagram for a first-order transition after which the two branches merge. Interestingly, introducing an $L_2$ regularization in the regression sub-task can anticipate the transition to lower dataset sizes, at the cost of a bias in the signal reconstructions which can be removed by annealing the regularization intensity. Finally, we study the inference performance of three meta-heuristics in the context of Phase selection: Simulated Annealing, Approximate Message Passing, and Langevin Dynamics on the continuous relaxation of the sign variables. With simultaneous annealing of the temperature and the $L_2$ regularization, they are shown to approach the Bayes-optimal sample efficiency.

Figures

Figures reproduced from arXiv: 2502.04282 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. We present the istogram of stabilities for [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. In the [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The SA-based solver is employed for solving 100 Phase Selection instances, for [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The AMP-based solver is employed for on 100 Phase Selection instances, for [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The GD-based solver is employed for solving 100 Phase Selection instances, for [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png]

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Forward citations

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Reference graph

Works this paper leans on

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    Probabilistic formulation of Phase Selection It is possible to reformulate Phase Retrieval problem from a Bayesian perspective: the estimate x is drawn from a posterior measure p(x|A, y) where the missing phases do not appear explicitly. Using the marginal probability theorem, we can explicit the dependency on S, rewriting the posterior probability for x ...

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    We recover the full large deviation of the complexity at fixed values of O, allowing us to study the impact of a strategic selection of the signs compared to random sampling

    λ = 0, fixed sign overlap O We start by exploring the unregularized case, at an intermediate value α = 1.7. We recover the full large deviation of the complexity at fixed values of O, allowing us to study the impact of a strategic selection of the signs compared to random sampling. O is fixed implicitly, by finding the conjugated value of the chemical pot...

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    We thus set ϕ = 0 in the following, allowing the overlap O to vary with µ

    λ = 0, ϕ = 0 (unconstrained O) In an optimization scenario, the correct phases are unknown, making it unfeasible to bias toward a correct sign selection with the potential ϕ. We thus set ϕ = 0 in the following, allowing the overlap O to vary with µ. To assess the recoverability of the hidden signal, one can study the free entropy Φ as a function of the in...

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    Analysis of the stabilities, impact of µ and λ We have shown how annealing the inverse temperatureµ may be sufficient to recover the hidden signal x0, provided α is large enough. We now turn to understanding exactly what happens to the signs’ selection as µ increases, and whether switching on the regularization λ can help to achieve a lower recovery thres...

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    Problem generator In order to deal with numerical investigation a Phase Selection instances generator has been implemented, it follows in Alg.4. Algorithm 4 Phase Selection instances generator Input: Structure: N, α,act, ∆0, P0(·), Vx0 , ρ Initialization: M = αN, Mtst = max{20...

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.