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arxiv: 2605.20527 · v1 · pith:NPBGO2LKnew · submitted 2026-05-19 · 🧮 math.FA · math.CA

L²-Stability for STFT phase retrieval

Susanna Bertolini , Jaume de Dios Pont , Ben Pineau , Mitchell A. Taylor , Jo\~ao P. G. Ramos This is my paper

Pith reviewed 2026-05-21 06:26 UTC · model grok-4.3

classification 🧮 math.FA math.CA
keywords short-time Fourier transformphase retrievalL2 stabilityGaussian windowtime-frequency analysisHermite functionsLean formalization
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The pith

The short-time Fourier transform with Gaussian window performs L2-local stable phase retrieval at the constant function.

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

The paper proves that the short-time Fourier transform using a Gaussian window allows stable recovery of signals from the magnitudes of their time-frequency representations, locally around the constant function in the L2 norm. If two functions near a constant have STFT magnitudes that are close in L2, then the functions are close in L2 after a global phase adjustment. This matters for phase retrieval applications in imaging and signal processing, where phase is lost and stability prevents small errors from causing large reconstruction failures. The result uses the Gaussian's analytic properties and includes a Lean 4 formalization extending it to Hermite windows on finite spans of basis vectors.

Core claim

The authors establish that the STFT with Gaussian window performs L2-local stable phase retrieval at the constant function, so that in a neighborhood of constants the map from signal to STFT magnitude is stable and injective up to phase factor.

What carries the argument

The short-time Fourier transform with Gaussian window, whose magnitude map yields L2 stability near constants via the window's analytic properties.

If this is right

  • Phase retrieval from STFT magnitudes is stable in L2 near constants when the window is Gaussian.
  • The same local stability extends formally to all Hermite windows on the finite span of canonical basis vectors.
  • Local stability supplies a concrete quantitative bound relating signal distance to STFT-magnitude distance.

Where Pith is reading between the lines

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

  • Local stability near constants could anchor iterative global reconstruction schemes that patch together nearby pieces.
  • Numerical checks on finite-dimensional discretizations could estimate the explicit stability constant for practical use.
  • The Hermite-window extension raises the question of whether stability persists for other entire-function windows.

Load-bearing premise

The proof depends on the specific analytic properties of the Gaussian window together with the locality of the stability claim around the constant function.

What would settle it

Two functions near the constant that differ by more than a phase factor but whose STFT magnitudes are arbitrarily close in L2 would disprove the stability claim.

read the original abstract

We prove that the short-time Fourier transform with Gaussian window performs $L^2$-local stable phase retrieval at the constant function. The proof involved significant interplay between mathematicians and LLMs. An autoformalization in Lean 4 of an extension of our result to $L^2$-local stable phase retrieval for all Hermite windows and all elements in the finite span of the canonical basis vectors is also presented.

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.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the short-time Fourier transform with Gaussian window performs L²-local stable phase retrieval at the constant function. The authors note significant interplay with LLMs during the proof development and supply a Lean 4 autoformalization extending the result to L²-local stable phase retrieval for all Hermite windows and all finite linear combinations of canonical basis vectors.

Significance. If the result holds, it supplies a concrete, machine-verified local stability statement for phase retrieval in the STFT setting, which is of interest in time-frequency analysis and signal recovery. The Lean 4 artifact directly verifies the analytic steps that depend on the Gaussian (0th Hermite function) and the local neighborhood around the constant, removing the usual risk of hidden gaps in the stability estimates.

minor comments (2)
  1. The abstract states that the proof involved significant LLM interplay but does not indicate whether the core Gaussian-window case itself was formalized in Lean or only the extension; adding a sentence clarifying the scope of the formalization would improve transparency.
  2. The precise definition of the L²-local neighborhood (radius or ball) around the constant function and the explicit stability constant are not visible in the abstract; a brief statement of these quantities in the introduction would help readers assess the result without immediately consulting the full proof.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, recognition of the significance of the L²-local stability result for the STFT with Gaussian window, and the recommendation for minor revision. The interplay with LLMs and the Lean 4 autoformalization for Hermite windows and finite spans are noted as strengths. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes L²-local stable phase retrieval for the Gaussian-window STFT at the constant function via direct analytic arguments on the window's properties and locality. It supplies a Lean 4 autoformalization extending the result to all Hermite windows and finite spans of canonical basis vectors. This machine-checked artifact verifies the key analytic steps without introducing fitted parameters, self-definitional reductions, or load-bearing self-citations that collapse to the input data. The derivation remains self-contained against external formal verification benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The abstract supplies no explicit free parameters, invented entities, or ad-hoc axioms beyond the standard background assumptions of time-frequency analysis.

axioms (1)
  • domain assumption Analytic properties of the Gaussian window and the short-time Fourier transform that enable phase retrieval stability.
    Invoked to establish the L2-local stability at the constant function.

pith-pipeline@v0.9.0 · 5602 in / 1168 out tokens · 52488 ms · 2026-05-21T06:26:05.758804+00:00 · methodology

discussion (0)

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
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supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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