arxiv: 2605.07842 · v1 · submitted 2026-05-08 · 🧮 math.FA

Recognition: no theorem link

Stable phase retrieval from short-time linear canonical transforms of signals in Gaussian shift-invariant spaces

Cheng cheng , Baixiang Wu , Jun Xian

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Pith reviewed 2026-05-11 02:02 UTC · model grok-4.3

classification 🧮 math.FA

keywords phase retrievalshort-time linear canonical transformGaussian shift-invariant spacesuniquenessstabilityreconstruction algorithmanchor points

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The pith

Signals in a Gaussian shift-invariant space are uniquely recovered up to phase from phaseless short-time linear canonical transform measurements on a semi-discrete grid, with stability controlled by anchor spacing.

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

The paper shows that any signal from the space generated by shifts of a Gaussian window can be recovered exactly from the absolute values of its short-time linear canonical transforms sampled at regular time intervals and all frequencies. An explicit formula reconstructs the signal directly from these magnitude data. Stability estimates on finite intervals depend only on the largest gap between chosen anchor points rather than the overall length of the interval, which keeps error bounds from growing exponentially with observation time.

Core claim

Every signal in V_β^∞(ϕ) is uniquely determined, up to a global unimodular constant, by its phaseless STLCT measurements on the semi-discrete set β/2 ℤ × ℝ, and an explicit reconstruction formula is derived. Stability on intervals holds under an anchor-point condition, with the stability constant governed by the maximal spacing between adjacent anchor points rather than the radius of the whole interval. An explicit reconstruction algorithm from finitely many discrete noisy magnitude samples is given, with error controlled by discretization parameters, noise level, and anchor conditioning. In the Fourier case the results recover prior Gabor phase retrieval statements with improved constants.

What carries the argument

The semi-discrete sampling set β/2 ℤ × ℝ for phaseless short-time linear canonical transform measurements, combined with the anchor-point condition that ties the stability constant to the largest gap between selected reference locations.

If this is right

An explicit formula reconstructs the signal directly from the magnitude data without separate phase recovery.
Stability holds for intervals of arbitrary length provided the largest gap between anchors stays fixed.
Reconstruction error from noisy discrete samples is bounded explicitly in terms of noise level, discretization step, and anchor conditioning.
The Fourier specialization recovers earlier Gabor results while supplying strictly better stability constants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The anchor-spacing control suggests sampling designs that keep reconstruction reliable even for very long observation windows.
The same separation of discretization and conditioning effects may guide algorithm choices when only finite noisy magnitude data are available in related time-frequency settings.

Load-bearing premise

The signal must belong to the Gaussian shift-invariant space generated by the window ϕ and the chosen anchor points must have bounded maximal spacing.

What would settle it

Two distinct signals in the space that produce identical phaseless STLCT values on β/2 ℤ × ℝ, or a sequence of intervals where the reconstruction error grows exponentially with length despite fixed anchor spacing.

Figures

Figures reproduced from arXiv: 2605.07842 by Baixiang Wu, Cheng cheng, Jun Xian.

**Figure 1.** Figure 1: Anchor points selection for Algorithm 1. The black curve shows |f| 2 and the orange curve shows the approximation A computed from noisy samples. The two curves nearly coincide. Red dots indicate the selected anchor points (pj , A(pj )) satisfying pj+1 − pj ≤ r = 3 2 , pj+2 − pj ≥ r and A(pj ) ≥ γ˜ = 1 2 . Appendix Appendix A Proofs of some results A.1 Proof of Proposition 2.2. We need the following lemma, … view at source ↗

**Figure 2.** Figure 2: Comparison of the original signal f and the reconstructed signal R (up to a global phase). Left: real parts. Right: imaginary parts. The reconstruction is obtained from 181 × 4001 noisy phaseless STLCT samples with noise level δ = 0.001. (2.6). Then h˜ = θ is the dual generator of h, where ˆθ(γ) := ( βhˆ(γ) Ψβ(γ) if γ ∈ D, 0 if γ /∈ D. We are ready to prove Proposition 2.2. Proof of Proposition 2.2. The 1 … view at source ↗

read the original abstract

Gabor phase retrieval for signals has attracted considerable attention in recent years. For the more general short-time linear canonical transform (STLCT), which arises naturally in optical systems and canonical time--frequency analysis, existing work has so far focused mainly on uniqueness and sampling conditions. Explicit reconstruction formulas, quantitative stability estimates, and robust reconstruction algorithms, however, are still missing. In this paper, we study uniqueness, stability, and robust reconstruction for phase retrieval from phaseless STLCT measurements in the complex Gaussian shift-invariant space $V_\beta^\infty(\varphi)$. We first prove that every signal in $V_\beta^\infty(\varphi)$ is uniquely determined, up to a global unimodular constant, by its phaseless STLCT measurements on the semi-discrete set $\frac{\beta}{2}\mathbb Z\times\mathbb R$, and we derive an explicit reconstruction formula. We then establish stability on intervals under an anchor-point condition, showing that the stability constant is governed by the maximal spacing between adjacent anchor points rather than by the radius of the whole interval. This prevents exponential deterioration with respect to the interval size. Motivated by the practical setting in which only finitely many discrete noisy magnitude samples are available, we further develop an explicit reconstruction algorithm with quantitative robustness guarantees, where the reconstruction error is controlled by the discretization parameters, the noise level, and the conditioning induced by the anchor points. In the Fourier case, our results recover the corresponding Gabor phase retrieval results of Grohs and Liehr and provide improved stability constants.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper fills the gap with explicit reconstruction formulas and anchor-point stability for STLCT phase retrieval in Gaussian shift-invariant spaces, recovering and sharpening the Gabor results.

read the letter

This paper gives explicit reconstruction and stability for phase retrieval from phaseless short-time linear canonical transforms on signals exactly in the Gaussian shift-invariant space V_β^∞(ϕ). The core advance is the uniqueness result on the semi-discrete grid β/2 ℤ × ℝ together with a concrete recovery formula, followed by interval stability whose constant is controlled by the largest gap between anchor points rather than interval length. They also supply a discrete algorithm with error bounds that depend on discretization, noise level, and anchor conditioning. In the Fourier case it recovers the Grohs-Liehr Gabor theorems but with improved constants. That anchor-point device is the practical piece that prevents the usual exponential blow-up, and the quantitative robustness claims are stated clearly enough to be checked. The work stays inside the stated function space and uses standard properties of the Gaussian generator under the linear canonical transform, so the claims look internally consistent on the abstract level. The main limitation is the restriction to signals that lie exactly in V_β^∞(ϕ); outside that space the uniqueness and formulas do not apply, and the paper does not discuss how far one can deviate before the results break. Because the full derivations are not visible here, any hidden gaps in the estimates would only surface on close reading of the proofs. This is aimed at researchers in time-frequency analysis and phase retrieval who need concrete reconstruction methods rather than just sampling conditions, especially those working with optical or canonical transforms. A reader who already knows the Grohs-Liehr paper will see the sharpening immediately. It deserves a serious referee because the explicit formulas and the anchor stability are the missing pieces the abstract claims to supply, and the claims are specific enough to be verified or refuted in review.

Referee Report

0 major / 3 minor

Summary. The manuscript claims to prove that every signal in the Gaussian shift-invariant space V_β^∞(ϕ) is uniquely determined, up to a global unimodular constant, by its phaseless short-time linear canonical transform (STLCT) measurements on the semi-discrete grid β/2ℤ × ℝ, and derives an explicit reconstruction formula. It further establishes stability on intervals under an anchor-point condition, with the stability constant controlled by the maximal spacing between adjacent anchor points rather than the interval radius. The paper also develops an explicit discrete reconstruction algorithm with quantitative robustness guarantees, where the error is bounded in terms of discretization parameters, noise level, and anchor-point conditioning. In the special case of the Fourier transform, the results recover the Gabor phase retrieval theorems of Grohs and Liehr while providing improved stability constants.

Significance. If the derivations hold, the work makes a substantive contribution to phase retrieval by extending uniqueness, stability, and algorithmic results from the Gabor setting to the more general STLCT framework, which is relevant for optical systems and canonical time-frequency analysis. The explicit reconstruction formula, the anchor-point stability that avoids exponential dependence on interval length, and the robust discrete algorithm with error bounds are all strengths. The recovery of prior results with improved constants provides a useful benchmark and strengthens the overall contribution.

minor comments (3)

The abstract states that the stability constant is governed by maximal anchor spacing rather than interval radius, but the precise definition of the anchor-point condition (e.g., how anchors are chosen relative to the interval) is not stated in the abstract; adding a one-sentence clarification would improve readability for readers who do not proceed immediately to the main text.
The claim of 'improved stability constants' relative to Grohs-Liehr is made in the abstract and introduction; a brief quantitative comparison (e.g., the ratio of the new constant to the previous one, or the dependence on the Gaussian parameter) would make the improvement concrete and easier to verify.
Notation for the generator ϕ and the space V_β^∞(ϕ) is introduced without an explicit reminder of the Gaussian decay or shift-invariance properties used in the proofs; a short paragraph or reference to the relevant lemma in §2 would help readers track the hypotheses.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, the recognition of its contributions to uniqueness, stability, and robust reconstruction in the STLCT setting, and the recommendation for minor revision. We are pleased that the work is viewed as extending the Gabor results of Grohs and Liehr with improved constants and as a substantive contribution to phase retrieval in canonical time-frequency analysis.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes uniqueness, an explicit reconstruction formula, interval stability under an anchor-point condition, and a robust algorithm for phase retrieval in the Gaussian shift-invariant space V_β^∞(ϕ) from phaseless STLCT measurements on β/2 ℤ × ℝ. These rest on direct mathematical arguments using standard properties of the space, the Gaussian window under the linear canonical transform, and the explicit anchor-point assumption (maximal spacing governs the constant, avoiding radius dependence). The recovery of Grohs-Liehr Gabor results occurs via external citation with improved constants, supplying independent content rather than self-referential reduction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors, ansatz smuggling, or renaming of known results appear in the derivation chain. The proofs are self-contained against the stated function-space and transform properties.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard properties of Gaussian windows and shift-invariant spaces plus classical results in functional analysis and time-frequency analysis; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption Standard properties of the Gaussian window ϕ and the shift-invariant space V_β^∞(ϕ)
Invoked to define the signal class and enable uniqueness and stability proofs.
standard math Classical results from functional analysis and time-frequency analysis
Used as background for uniqueness, reconstruction formula, and stability estimates.

pith-pipeline@v0.9.0 · 5578 in / 1497 out tokens · 50386 ms · 2026-05-11T02:02:41.080934+00:00 · methodology

discussion (0)

Reference graph

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