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arxiv: 2606.23713 · v1 · pith:2J6FGYUFnew · submitted 2026-06-16 · 📡 eess.SP · stat.ME

Unifying Adaptive Fourier and M\"obius-Based Models for Efficient and Interpretable Biomedical Signal Decomposition

Christian Canedo , Roc\'io Carratal\'a-S\'aez , Cristina Rueda This is my paper

Pith reviewed 2026-06-26 23:12 UTC · model grok-4.3

classification 📡 eess.SP stat.ME
keywords Adaptive Fourier DecompositionFrequency-Modulated Möbius modelsignal decompositionECGEEGmathematical equivalencebiomedical signalsmulti-channel analysis
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The pith

Finite-order AFD and FMM decompositions of oscillatory signals are mathematically equivalent.

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

The paper establishes that Adaptive Fourier Decomposition using Takenaka-Malmquist expansions and the Frequency-Modulated Möbius model produce identical results when limited to the same finite number of components. This holds because both frameworks reduce to the same optimization problem under independent Gaussian noise assumptions. The connection lets computationally fast AFD methods, including FFT implementations, deliver the morphologically labeled parameters that FMM provides. The equivalence extends directly to multi-channel recordings typical in ECG and EEG data. Examples in the paper compare approximation quality on EEG traces and component labeling on ECG beats.

Core claim

We prove that finite-order AFD and FMM decompositions are mathematically equivalent. Under mild regularity assumptions, we further show that their associated estimation procedures solve the same underlying optimization problem when FMM is formulated with independent Gaussian noise. The results are extended to multi-channel signals, which are central in multilead bioelectric recordings.

What carries the argument

The shared optimization problem solved by finite Takenaka-Malmquist expansions in AFD and parametric Möbius transforms in FMM.

If this is right

  • Fast AFD approximations including FFT-based versions can be substituted for FMM fitting while retaining morphological parameter labels.
  • Components obtained from either method become directly interpretable in physiological terms via the FMM parametrization.
  • The unified procedure applies without change to multi-channel signals in multilead recordings.
  • Approximation quality can be tracked as the number of components grows using either computational route.

Where Pith is reading between the lines

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

  • Hybrid algorithms could alternate between AFD speed for initial fitting and FMM labels for final reporting in clinical pipelines.
  • The equivalence suggests testing whether the same identity holds for other parametric families built on Möbius transforms.
  • Real-time monitoring devices might adopt the faster route while preserving the interpretability clinicians expect from FMM.

Load-bearing premise

FMM estimation is formulated with independent Gaussian noise.

What would settle it

A direct numerical comparison on the same multi-channel recording where the fitted parameters or residual errors from the two estimation procedures differ under an independent Gaussian noise model.

Figures

Figures reproduced from arXiv: 2606.23713 by Christian Canedo, Cristina Rueda, Roc\'io Carratal\'a-S\'aez.

Figure 1
Figure 1. Figure 1: Illustration of FMM decomposition and AFD for [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Results of the Experiment 1. The original EEG signal corresponds to Patient 316 from the ICARE Database [48] (Hour 26, Channel Fp1-F7). Panel (a) shows the cumulative R2 measure when each component k is fitted over the residuals of the previous k − 1 components, which remain fixed. Panel (b) shows the fitted signal with 15 components to the given signal. 6.1. Experimental Comparison of Optimization Methods… view at source ↗
Figure 3
Figure 3. Figure 3: Results of the Experiment 2. Examples 1-4 from [46]: Each subplot rep￾resents the true and the estimated a values using Cyclic AFD and FMM backfitting al￾gorithms. The dashed line represents the unit circle (∂D), which is the bound for the a parameters. where µˆK denotes the predicted signal of any model with K components. The core AFD algorithm extracts variability slightly faster than the FMM algorithm, … view at source ↗
Figure 4
Figure 4. Figure 4: ECG signal analysis for heartbeats corresponding to patients 13022 (panel (a)) [PITH_FULL_IMAGE:figures/full_fig_p021_4.png] view at source ↗
read the original abstract

Oscillatory biomedical signals such as electrocardiograms (ECG) and electroencephalograms (EEG) call for decompositions that are both computationally efficient and interpretable. This paper establishes a formal connection between two finite-order frameworks that have largely evolved independently: Adaptive Fourier Decomposition (AFD), based on orthonormal Takenaka-Malmquist expansions, and the Frequency-Modulated Mobius (FMM) model, a parametric decomposition built on Mobius transforms with morphologically meaningful parameters. We prove that finite-order AFD and FMM decompositions are mathematically equivalent. Under mild regularity assumptions, we further show that their associated estimation procedures solve the same underlying optimization problem when FMM is formulated with independent Gaussian noise. The results are extended to multi-channel signals, which are central in multilead bioelectric recordings. Practically, the equivalence clarifies how fast AFD approximations, including FFT-based implementations, relate to FMM-style parametrization and component interpretability. We illustrate these implications with an EEG example evaluating approximation behavior as the number of components increases, and with an ECG use case comparing five-component decompositions on representative beats, contrasting unlabeled AFD components with physiologically identified FMM components. Overall, the proposed equivalence provides a principled basis to leverage the computational advantages of AFD alongside the interpretability of FMM in biomedical signal analysis.

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 manuscript proves that finite-order Adaptive Fourier Decomposition (AFD) based on Takenaka-Malmquist orthonormal expansions is mathematically equivalent to the Frequency-Modulated Möbius (FMM) parametric decomposition. Under mild regularity assumptions, the associated estimation procedures solve the same optimization problem when the FMM model is formulated with independent Gaussian noise. The result is extended to multi-channel signals and illustrated on EEG approximation behavior and ECG five-component decompositions contrasting unlabeled AFD components with physiologically labeled FMM components.

Significance. If the equivalence holds, the work unifies two independently developed frameworks, enabling use of fast AFD approximations (including FFT-based implementations) together with the morphological interpretability of FMM parameters. This is directly relevant to multilead bioelectric recordings. The manuscript ships an explicit mathematical equivalence proof conditioned on the stated noise model, which is a clear strength.

minor comments (3)
  1. [Abstract and §3] The abstract refers to 'mild regularity assumptions' without listing them; the main text should state these assumptions explicitly (e.g., in the theorem statement) to allow readers to verify applicability.
  2. [§2] Notation for the Takenaka-Malmquist basis functions and the Möbius parameters should be aligned in a single table or definition block to ease comparison between the two representations.
  3. [§5] The EEG and ECG figures would benefit from explicit captions stating the number of components, the optimization criterion used, and whether the displayed components are ordered by energy or by physiological label.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their thorough and positive review of our manuscript. We are pleased that the equivalence result between finite-order AFD and FMM decompositions, along with its implications for multi-channel biomedical signals, was recognized as a strength. The recommendation for acceptance is appreciated.

Circularity Check

0 steps flagged

No significant circularity; direct mathematical equivalence proof

full rationale

The paper claims to prove mathematical equivalence between finite-order AFD (Takenaka-Malmquist) and FMM decompositions, plus equivalence of their estimation procedures under independent Gaussian noise. This is a self-contained mathematical argument conditioned on explicit assumptions, with no indication of self-definitional constructs, fitted parameters renamed as predictions, or load-bearing self-citations. The scoping in the abstract avoids overclaiming, and the result does not reduce to re-expressing inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; therefore the ledger is populated at the level of explicit statements in the abstract. No free parameters or invented entities are named. The key domain assumption is the independent Gaussian noise model required for the optimization equivalence.

axioms (2)
  • domain assumption Mild regularity assumptions on the signals
    Invoked in the abstract to extend the equivalence from the decompositions themselves to the estimation procedures.
  • domain assumption FMM formulated with independent Gaussian noise
    Explicitly required for the claim that the two estimation procedures solve the same optimization problem.

pith-pipeline@v0.9.1-grok · 5780 in / 1395 out tokens · 31238 ms · 2026-06-26T23:12:57.386188+00:00 · methodology

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Reference graph

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