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arxiv: 1907.10773 · v1 · pith:O2TITOPPnew · submitted 2019-07-24 · 🧮 math.NA · cs.NA

Inverting Spectrogram Measurements via Aliased Wigner Distribution Deconvolution and Angular Synchronization

Michael Perlmutter , Sami Merhi , Aditya Viswanathan , Mark Iwen This is my paper

Pith reviewed 2026-05-24 16:25 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords phase retrievalSTFT magnitudespectrogram inversionWigner distributionangular synchronizationcoded diffraction patterns
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The pith

A two-step method recovers discrete bandlimited signals from fewer than d STFT magnitude measurements using aliased Wigner deconvolution and angular synchronization.

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

The paper proposes recovering a signal from subsampled short-time Fourier transform magnitude measurements by first using aliased Wigner distribution deconvolution to recover part of the rank-one outer product matrix, then applying angular synchronization to find the Fourier transform of the signal. This allows reconstruction via Fourier inversion. The approach yields two new phase retrieval algorithms that are efficient and perform well numerically. It proves guarantees for recovery of bandlimited signals from fewer measurements and introduces a new class of deterministic coded diffraction patterns that enable noise-robust recovery.

Core claim

The central claim is that discrete bandlimited signals in C^d can be recovered from fewer than d STFT magnitude measurements using a two-step process of aliased Wigner distribution deconvolution followed by angular synchronization, and that this method also supports a new class of deterministic coded diffraction pattern measurements for efficient and noise robust recovery.

What carries the argument

The aliased Wigner distribution deconvolution step that recovers a portion of the rank-one matrix x-hat x-hat^*, followed by angular synchronization to recover x-hat.

If this is right

  • Recovery is guaranteed for discrete bandlimited signals from subsampled STFT magnitudes.
  • New deterministic coded diffraction patterns allow efficient and noise robust recovery.
  • The resulting algorithms are efficient and compare favorably to standard phase retrieval methods numerically.
  • Two theorems establish the recovery guarantees and the new class of measurements.

Where Pith is reading between the lines

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

  • The two-step structure may apply to other magnitude-based recovery problems where partial rank-one information can be extracted.
  • The angular synchronization step could be swapped for alternative phase estimation techniques in related settings.
  • The method's performance on signals that are approximately bandlimited rather than exactly so remains open for testing.

Load-bearing premise

The signals are discrete and bandlimited, and the subsampled STFT measurements have sufficient structure for the deconvolution step to recover a usable portion of the rank-one matrix.

What would settle it

A counterexample discrete bandlimited signal that cannot be recovered from the subsampled STFT magnitudes using the aliased Wigner deconvolution plus angular synchronization procedure.

Figures

Figures reproduced from arXiv: 1907.10773 by Aditya Viswanathan, Mark Iwen, Michael Perlmutter, Sami Merhi.

Figure 5.1
Figure 5.1. Figure 5.1: Selection of HIO+ER iteration parameters1 We performed experiments with both Algorithm 1, as presented in Section 4, and also with a modified version which uses a post-processing procedure to obtain improved accuracy. The modified algorithm replaces Steps (5) and (7) of Algorithm 1 (referred to as Diag. Mag. Est. and Norm. Ang. Sync. in [PITH_FULL_IMAGE:figures/full_fig_p020_5_1.png] view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Evaluating the performance of Algorithm 1 for various parameter choices [PITH_FULL_IMAGE:figures/full_fig_p020_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Evaluating the robustness and efficiency of Algorithm 1 (and Theorem 3) as in (5.1), with ρ = 8. We observe that for larger values of L, performance improved by about 10dB. We also note that, in practice, a suitable value of L can be chosen depending on whether the proposed method is used as a reconstruction procedure or as an initializer for another algorithm. In [PITH_FULL_IMAGE:figures/full_fig_p021_… view at source ↗
Figure 5
Figure 5. Figure 5: b plots the corresponding execution time for the various algorithms as a function of the problem [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Evaluating the performance of the Lemma 11 based approach As can be seen in [PITH_FULL_IMAGE:figures/full_fig_p022_5_4.png] view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Evaluating the robustness and efficiency of the Lemma 11 based approach Algorithm 2. One possible solution is to utilize the Tikhonov regularized solution A = (W∗W +σ 2 I) −1W∗V (see [27] for example), where the regularization parameter σ 2 is chosen using a procedure such as the L-curve method [26]. However, empirical simulations suggest that this procedure is not sufficiently robust to achieve reconstr… view at source ↗
Figure 5
Figure 5. Figure 5: presents empirical evaluation of the noise robustness and computational efficiency of Algorithm [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Empirical validation of Theorem 2 (Algorithm 2) and the Modified Iterated Tikhonov Method of Algorithm 3 [PITH_FULL_IMAGE:figures/full_fig_p024_5_6.png] view at source ↗
read the original abstract

We propose a two-step approach for reconstructing a signal ${\bf x}\in\mathbb{C}^d$ from subsampled short-time Fourier transform magnitude (spectogram) measurements: First, we use an aliased Wigner distribution deconvolution approach to solve for a portion of the rank-one matrix ${\bf \widehat{{\bf x}}}{\bf \widehat{{\bf x}}}^{*}.$ Second, we use angular syncrhonization to solve for ${\bf \widehat{{\bf x}}}$ (and then for ${\bf x}$ by Fourier inversion). Using this method, we produce two new efficient phase retrieval algorithms that perform well numerically in comparison to standard approaches and also prove two theorems, one which guarantees the recovery of discrete, bandlimited signals ${\bf x}\in\mathbb{C}^{d}$ from fewer than $d$ STFT magnitude measurements and another which establishes a new class of deterministic coded diffraction pattern measurements which are guaranteed to allow efficient and noise robust recovery.

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

2 major / 2 minor

Summary. The paper proposes a two-step phase retrieval method for discrete signals from subsampled STFT magnitude measurements. The first step applies aliased Wigner distribution deconvolution to recover a portion of the rank-one matrix formed by the Fourier transform of the signal. The second step uses angular synchronization on the recovered entries to determine the phases (up to global phase), followed by Fourier inversion to obtain the original signal. Theoretical contributions include a guarantee of unique recovery for bandlimited signals in C^d from fewer than d STFT magnitude measurements and a new class of deterministic coded diffraction pattern measurements that permit efficient, noise-robust recovery. Numerical experiments compare the resulting algorithms favorably to standard phase retrieval methods.

Significance. If the connectivity and uniqueness conditions for angular synchronization are rigorously established, the work would provide deterministic sub-d measurement guarantees and practical algorithms for structured phase retrieval, extending beyond random measurement ensembles. The combination of Wigner deconvolution with angular synchronization offers a concrete algorithmic pathway with potential for applications in signal processing where measurement budgets are limited.

major comments (2)
  1. [Abstract (two-step approach paragraph) and recovery theorem for <d STFT measurements] Abstract (paragraph describing the two-step approach) and the theorem guaranteeing recovery from fewer than d STFT magnitude measurements: the uniqueness claim for angular synchronization requires that the support of entries recovered by aliased Wigner deconvolution induces a graph with sufficient connectivity (or satisfies the relevant rank conditions) to determine phases uniquely up to global phase. The manuscript does not appear to verify that the chosen subsampling pattern and bandlimit together guarantee this connectivity property rather than merely a nonempty recovered support; this is load-bearing for the central recovery theorem.
  2. [Theorem on deterministic coded diffraction patterns] Theorem establishing the new class of deterministic coded diffraction pattern measurements: the noise-robust recovery claim requires explicit stability bounds or perturbation analysis showing how noise in the magnitude measurements propagates through the deconvolution and synchronization steps. Without such analysis, it is unclear whether the efficiency and robustness hold with constants independent of dimension or only under additional assumptions on the noise level.
minor comments (2)
  1. [Abstract] Abstract contains a typo: 'syncrhonization' should be 'synchronization'.
  2. [Introduction / notation section] Notation for the rank-one matrix is introduced as M = x̂ x̂* but the subsequent text uses boldface inconsistently; a single consistent notation should be adopted throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments on the manuscript. Below we provide point-by-point responses to the major comments, indicating the revisions we intend to make.

read point-by-point responses

  1. Referee: Abstract (paragraph describing the two-step approach) and the theorem guaranteeing recovery from fewer than d STFT magnitude measurements: the uniqueness claim for angular synchronization requires that the support of entries recovered by aliased Wigner deconvolution induces a graph with sufficient connectivity (or satisfies the relevant rank conditions) to determine phases uniquely up to global phase. The manuscript does not appear to verify that the chosen subsampling pattern and bandlimit together guarantee this connectivity property rather than merely a nonempty recovered support; this is load-bearing for the central recovery theorem.

    Authors: We appreciate the referee pointing out the need for explicit verification of the connectivity condition. The proof of the central recovery theorem (Theorem 3.1) does establish that the subsampling pattern combined with the bandlimit assumption yields a connected graph on the recovered entries, specifically by showing that the aliased Wigner deconvolution recovers a set of entries whose induced graph contains a spanning tree. This argument is currently embedded within the main proof rather than isolated as a preliminary lemma. In the revised manuscript we will extract this argument into a dedicated lemma placed immediately before the statement of the theorem, thereby making the connectivity guarantee fully explicit and self-contained. revision: yes

  2. Referee: Theorem establishing the new class of deterministic coded diffraction pattern measurements: the noise-robust recovery claim requires explicit stability bounds or perturbation analysis showing how noise in the magnitude measurements propagates through the deconvolution and synchronization steps. Without such analysis, it is unclear whether the efficiency and robustness hold with constants independent of dimension or only under additional assumptions on the noise level.

    Authors: We agree that the manuscript currently asserts noise robustness at a high level without supplying quantitative perturbation bounds. While the numerical section demonstrates practical stability, a formal analysis of error propagation through the two steps is absent. In the revision we will add a new subsection (or appendix) that derives first-order perturbation bounds for the aliased Wigner deconvolution step under additive bounded noise and then invokes existing stability results for angular synchronization to obtain an overall recovery guarantee. The resulting constants will be shown to depend only on the measurement parameters and the bandlimit, remaining independent of the ambient dimension d. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The abstract describes a two-step procedure (aliased Wigner deconvolution followed by angular synchronization) and states that two theorems are proved guaranteeing recovery under stated conditions on bandlimited signals and measurement structure. No quoted equations or steps reduce a claimed prediction or guarantee to a fitted parameter, self-definition, or unverified self-citation chain. The connectivity requirement for angular synchronization is presented as part of the theorem hypotheses rather than smuggled in by construction. This matches the default expectation of an independent derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides insufficient detail to enumerate free parameters, axioms, or invented entities; no explicit fitting, background lemmas, or new postulated objects are named.

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Reference graph

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