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arxiv: 2506.12201 · v2 · pith:CAGNODBOnew · submitted 2025-06-13 · 💻 cs.IT · eess.SP· math.IT· math.ST· stat.TH

Functional Multi-Reference Alignment via Deconvolution

Omar Al-Ghattas , Anna Little , Daniel Sanz-Alonso , Mikhail Sweeney This is my paper

Pith reviewed 2026-05-21 23:55 UTC · model grok-4.3

classification 💻 cs.IT eess.SPmath.ITmath.STstat.TH
keywords multi-reference alignmentdeconvolutionKotlarski's formulafunctional datasecond-order statisticssignal estimationFourier transform
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The pith

The multi-reference alignment problem reduces to deconvolution, where the signal is recovered from second-order statistics via Kotlarski's formula.

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

This paper shows that estimating a signal from its shifted and noisy copies can be reframed as a deconvolution task. The central step is to recover the signal directly from the second-order statistics of the observations by applying Kotlarski's identification result. The authors extend the formula to general dimensions and develop estimators that remain valid when the signal's Fourier transform vanishes at isolated frequencies. This connection supplies a new route to algorithms that avoid explicit shift estimation.

Core claim

In the functional multi-reference alignment problem the signal is estimated from shifted, noisy observations by treating the problem as deconvolution and recovering the signal from second-order statistics via Kotlarski's formula. The formula is extended to general dimension, and the estimation procedure is studied for signals whose Fourier transforms vanish at certain points.

What carries the argument

Kotlarski's formula, which identifies the characteristic function of the signal from the joint distribution of summed independent copies, applied here to the covariance structure of the MRA observations.

If this is right

  • MRA algorithms can be built from existing deconvolution estimators that use only second moments.
  • The procedure works in arbitrary dimensions without change of formulation.
  • Recovery remains possible for signals whose Fourier transforms have isolated zeros.

Where Pith is reading between the lines

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

  • Rates and minimax results already known for deconvolution could be imported to give concrete error bounds for functional MRA.
  • The same second-order approach might apply to other alignment tasks such as image registration under random translations.
  • Higher-order moment versions of Kotlarski's formula could yield alternative estimators with different robustness properties.

Load-bearing premise

Kotlarski's identification result applies directly to the functional MRA observations under the paper's noise and shift models.

What would settle it

A simulation in which the signal reconstructed from empirical second-order statistics via the extended Kotlarski formula matches the true signal within the expected statistical error as the number of observations grows.

Figures

Figures reproduced from arXiv: 2506.12201 by Anna Little, Daniel Sanz-Alonso, Mikhail Sweeney, Omar Al-Ghattas.

Figure 1
Figure 1. Figure 1: Performance of our algorithms in four illustrative examples. True signal (thick blue), [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Error decay with varying sample size for fixed [PITH_FULL_IMAGE:figures/full_fig_p019_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Recovery error as a function of noise parameters. [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Error decay with varying sample size for fixed [PITH_FULL_IMAGE:figures/full_fig_p040_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Recovery error as a function of noise intensity [PITH_FULL_IMAGE:figures/full_fig_p041_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: In thick blue: true signals in space and frequency. In thin red, their recoveries [PITH_FULL_IMAGE:figures/full_fig_p042_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Error decay with varying sample size for fixed [PITH_FULL_IMAGE:figures/full_fig_p043_7.png] view at source ↗
read the original abstract

This paper studies the multi-reference alignment (MRA) problem of estimating a signal function from shifted, noisy observations. Our functional formulation reveals a new connection between MRA and deconvolution: the signal can be estimated from second-order statistics via Kotlarski's formula, an important identification result in deconvolution with replicated measurements. To design our MRA algorithms, we extend Kotlarski's formula to general dimension and study the estimation of signals with vanishing Fourier transform, thus also contributing to the deconvolution literature. We validate our deconvolution approach to MRA through both theory and numerical experiments.

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 / 3 minor

Summary. The paper studies the functional multi-reference alignment (MRA) problem of estimating a signal from shifted noisy observations. It establishes a connection to deconvolution by showing that the signal can be recovered from second-order statistics using Kotlarski's formula, derives a multivariate extension of this identification result, provides explicit conditions (including controlled vanishing of the Fourier transform) under which recovery is possible up to global shift, and validates the approach through theoretical proofs and numerical experiments.

Significance. If the central claims hold, the work is significant for bridging the MRA and deconvolution literatures with a new identification result from second-order statistics. The explicit derivation of the multivariate extension, the matching of noise and shift models to the replicated-measurement structure required by Kotlarski's formula, and the supply of proofs for the recoverability conditions (including vanishing Fourier transforms) are clear strengths that advance both fields and support practical algorithm design.

minor comments (3)
  1. §3 (multivariate extension): the statement of the extended Kotlarski formula would benefit from an explicit list of the minimal assumptions on the characteristic function of the noise that are carried over from the univariate case, to make the applicability to the functional MRA model immediately verifiable.
  2. Numerical experiments section: the description of how the vanishing Fourier transform condition is enforced in the simulated signals and how error is measured relative to the global shift ambiguity could be expanded for reproducibility.
  3. Notation: the use of the same symbol for the signal and its Fourier transform in different sections occasionally leads to ambiguity; a consistent hat notation or separate symbols would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript, which correctly identifies the core contribution: extending Kotlarski's deconvolution formula to the functional MRA setting with multivariate signals and controlled vanishing of the Fourier transform. We appreciate the recommendation for minor revision and the recognition that the work bridges the MRA and deconvolution literatures.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper applies Kotlarski's formula—an established external identification result from the deconvolution literature—to the functional MRA model and derives a multivariate extension with explicit conditions on vanishing Fourier transforms. The model assumptions (noise and shift structure) are matched to the replicated-measurement requirements of the formula, with proofs supplied in the relevant sections. This constitutes independent mathematical support rather than a reduction to fitted parameters or self-citation chains within the paper. The central claim is therefore self-contained against external benchmarks and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of Kotlarski's formula to functional observations with unknown shifts and additive noise; no free parameters or new entities are mentioned in the abstract.

axioms (1)
  • domain assumption Kotlarski's identification result holds for the functional multi-reference alignment model with replicated measurements.
    The paper invokes this result to connect second-order statistics to signal recovery.

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