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Scattering Networks on Noncommutative Finite Groups

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper introduces a scattering transform on arbitrary finite groups, including noncommutative ones, and proves it is stable, energy preserving, equivariant, and increasingly translation-insensitive at deeper layers.

desk verdict A clean theoretical extension of scattering to arbitrary finite groups; the main theorems hold under an admissibility condition, and the only real weakness is an unverified external lemma plus a thin experimental section. read the letter →

arxiv 2505.20950 v1 pith:4JVGA5TP submitted 2025-05-27 math.NA cs.ITcs.LGcs.NAeess.SPmath.IT

Maria Teresa Arias , Davide Barbieri , Eugenio Hernández This is my paper · ORCID
classification math.NAcs.ITcs.LGcs.NAeess.SPmath.IT MSC 43A3042C4020C1568T07
keywords scatteringtransformfinitegroupsgroup-equivariantneuralnetworkswaveletsonnoncommutativeenergypreservationadmissibilityconditionequivariance
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper extends Mallat's scattering transform from Euclidean spaces to signals defined on any finite group, abelian or not. It constructs wavelets from group characters and builds a multi-layer cascade of convolutions and moduli that acts like a group-equivariant convolutional network. The main claim is that, under a Parseval-frame condition plus an admissibility condition on the low-pass filter, this transform controls signal energy, is nonexpansive and Lipschitz stable, preserves energy, is equivariant under left and right translations, and becomes approximately translation-invariant as depth grows. Concrete classification experiments on MNIST digits, audio barks versus meows, and functions on symmetric groups support the theory.

What carries the argument

The central object is the G-wavelet ψ_γ(x)=∑_{π∈Ĝ} d_π γ(π)χ_π(x), whose Fourier transform is the scalar matrix γ(π)Id_π; this makes convolution with ψ_γ a frequency-domain multiplication by the kernel γ. The Parseval-frame condition C(π)=∑_j |γ_j(π)|^2=1 makes the family of group translates a tight frame and yields the energy-splitting identity ‖U[p]f‖²=‖S[p]f‖²+∑_{j}‖U[p+j]f‖². The admissibility condition β=min_π |γ_0(π)|²>0 strengthens this into exponential decay of propagated energy and injectivity of the scattering map.

What would settle it

Choose a finite group and a kernel satisfying the Parseval condition with γ_0(π)=0 for every irreducible π, so φ=0; then for any nonzero f∈L²(G) the scattering transform outputs zero, while ‖f‖>0, directly contradicting energy preservation. For the relaxed theorem, construct a signed signal f where U[p]f becomes negative and check whether the claimed decay rate α(S)<1 still holds, which would test the necessity of the nonnegativity assumption.

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Extended reading notes

Core claim

For any finite group G, a G-wavelet is a class function built from the irreducible characters as ψ(x)=∑_{π∈Ĝ} d_π γ(π) χ_π(x), and the scattering transform is a cascade of modulus-of-convolution operators with such wavelets, followed by averaging with a low-pass filter φ=ψ_0. The paper proves that if the kernel satisfies the Calderón condition ∑_{j=0}^J |γ_j(π)|^2=1 for every irreducible π, then the transform is nonexpansive and Lipschitz stable; if additionally β=min_π |γ_0(π)|^2>0, then it is injective, preserves energy exactly, and its sensitivity to group translations decays exponentially with depth. These properties hold for arbitrary finite groups, so the construction provides a finite, provably stable representation for data with noncommutative symmetries.

Load-bearing premise

The main energy-preservation, injectivity, and approximate-invariance theorems all rely on admissibility: the low-pass filter's Fourier coefficient must be nonzero on every irreducible representation, and if that fails the scattering transform can lose all signal energy.

Editorial extensions

If this is right

  • Because nonexpansivity bounds total scattering energy by the input norm, truncating the transform at finite depth loses only a controlled amount of energy.
  • The Lipschitz stability bound means small perturbations of the input signal cause only proportionally small changes in the resulting representation.
  • Equivariance under left and right translations lets the same representation be used for signals whose labeling is insensitive to group action, while deeper layers become approximately invariant to translations.
  • Energy preservation and injectivity for admissible kernels mean the scattering coefficients retain all information about the original signal, making the transform a lossless feature extractor.
  • The classification experiments show the construction can be implemented for abelian groups, affine groups over finite fields, and symmetric groups, with accuracy gains over using raw signals.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The relaxed admissibility theorem in Section 5 borrows a Fourier-coefficient lower bound from an external result and applies only to nonnegative propagated signals; testing whether the same exponential decay holds for signed signals would show how far the relaxation genuinely extends.
  • Because the wavelet kernels are the only free parameters of the network, one could train those kernels end-to-end while keeping the proven stability and equivariance guarantees, a direction the paper does not explore.
  • For abelian groups the G-wavelets coincide with spectral graph wavelets on the Cayley graph, so the theory directly connects to graph scattering and could be used to design provably stable features on graphs that are Cayley graphs of finite groups.
  • The approximate-invariance result suggests a principled replacement for learned pooling layers in group-equivariant CNNs: deeper scattering layers give controlled translation insensitivity without explicit averaging or data augmentation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 6 minor

Summary. This paper constructs a scattering transform for functions on arbitrary finite groups, extending Mallat's Euclidean scattering to non-abelian groups. The authors define central G-wavelets as class functions whose Fourier multipliers are prescribed by kernels γ_j on the dual, and show (Theorem 2.6) that a Calderón-type condition is equivalent to the left translates forming a Parseval frame. They define a group scattering transform by cascading modulus-of-convolution propagators U[j] with a low-pass filter ϕ=ψ_0, and prove: the transform is non-expansive (Lemma 4.1), Lipschitz stable (Proposition 4.2), equivariant under left and right translations (Proposition 4.7), and, under the admissibility condition β = min_r |γ_0(r)|^2 > 0, it preserves energy and is injective (Theorem 4.6) and becomes approximately invariant to translations at exponential depth rate α^m (Theorem 4.8). A relaxation of admissibility is proposed in Section 5 using an external theorem of Kueh, Olson, Rockmore, and Tan [18], and numerical experiments on MNIST, meow/bark audio classification, and functions on symmetric groups are presented as illustrations.

Significance. If correct, the paper provides a finite, provably stable and equivariant scattering representation for signals on any finite group, a genuinely useful extension of the scattering framework to noncommutative group-structured data. The main theoretical contribution is valuable: the representation-theoretic derivation is largely self-contained, the frame equivalence in Theorem 2.6 is proven in full, the energy-splitting identity in Lemma 3.3 is clean, and the proofs of non-expansivity, stability, energy preservation, and approximate invariance are coherent and internally consistent. The paper also states an explicit parameter count and gives concrete design rules for Parseval frames via (27), which is practically useful. Its main weaknesses are that the relaxed-admissibility results in Section 5 rest entirely on an unverified and unproved quotation of an external theorem, and that the numerical section does not verify the admissibility hypotheses of the theorems for the kernels actually used; the experiments are best read as proof-of-concept illustrations rather than confirmations of the theoretical guarantees.

major comments (3)
  1. [Section 5, Theorems 5.1–5.2 and Remark 5.3] The entire relaxed-admissibility program is load-bearing on Theorem 5.1, quoted as "Theorem 5 in [18]". The paper neither proves this theorem nor reproduces its precise statement and normalization. Since Theorem 5.2 applies Theorem 5.1 to the nonnegative propagated signals U[p]f, any mismatch between the Fourier normalization used here (Plancherel in (12), with ̂f(π_r) = (1/|G|)∑_x f(x)π_r(x^{-1})) and the normalization used in [18] would change the constant deg(S)/|G| and, with it, β_γ(S), the decay rate α(S), and the energy preservation claimed in Remark 5.3. I request that the authors either prove the needed inequality, or state the exact version of [18, Theorem 5] with its constants and verify explicitly that the hypotheses and Fourier normalization agree with those of this paper.
  2. [Section 6.1 and Section 6.2] The numerical experiments do not verify the admissibility condition (36), or the relaxed condition β_γ(S)>0 of Theorem 5.2, for the kernels that are actually used. In Section 6.1 the kernels γ_j are normalized to satisfy (27) only; in Section 6.2 the kernel γ_0 is defined as the positive square root of 1 - (|γ_1|^2 + |γ_2|^2) without reporting min_k |γ_0(k)|^2 or checking the relaxed condition. Consequently, the experiments do not demonstrate the energy-preservation or exponential-invariance guarantees that are conditional on admissibility; they only illustrate the Parseval-frame properties and classification accuracy.
  3. [Section 6, Tables 1, 3, 6, 7, 9] The experimental section is not fully reproducible as reported: no code, data, or random seeds are provided, the test sets are small (e.g., Table 3 is based on only 56 test sounds; Tables 6 and 7 on 18 and 420 samples, respectively), and no variance or confidence intervals are reported for the accuracy numbers. This does not affect the mathematical claims, but it should be clearly labeled as indicative proof-of-concept experimentation rather than as a systematic benchmark.
minor comments (6)
  1. [Section 5, Theorem 5.2] The notation λ_m^J appears in the statement and proof of Theorem 5.2; it should be Λ_m^J, consistent with the rest of the paper.
  2. [Section 2.3] The author name "Givonval" should be "Gribonval" in both the text and reference [16].
  3. [Section 4.2, after Definition 4.3] The word "inyective" should be "injective".
  4. [Section 6.2] The phrase "Kaiser Fast transform" appears to refer to "Kaiser fast Fourier transform" or a Kaiser-window-based resampling; please use a standard name or define the term more precisely.
  5. [Equation (46) and Table 3] There are minor typos in the text around (46), e.g., "sucha as", and inconsistent decimal separators in Table 3 (41,61% vs 87.5%).
  6. [Section 6.2.4] The sentence "It may happen that the number of sounds is small" is imprecise; the 56-sample test set is indeed small, and stating the sample size explicitly would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the core theorems are self-contained derivations from stated admissibility and Parseval-frame hypotheses, and the one external bound cited in Section 5 is independent prior work.

full rationale

The paper's central claims—non-expansivity (Lemma 4.1), stability (Proposition 4.2), energy preservation and injectivity under admissibility (Theorem 4.6 and Section 4.2), equivariance (Proposition 4.7), and approximate invariance (Theorem 4.8)—are derived directly from the Parseval-frame condition (27), the admissibility condition βγ > 0 in Definition 4.3, and standard finite-group Fourier analysis. Admissibility is a stated hypothesis, not an output, and the proofs do not assume the conclusions. The Section 5 relaxation invokes Theorem 5.1 from Kueh–Olson–Rockmore–Tan ([18]), which is an external, non-overlapping prior result; its correctness is a matter of verification, not circularity, and the paper does not present it as its own contribution. The numerical sections fit kernels to training labels, but the reported accuracies are evaluated on held-out test sets, and the theoretical results do not depend on those fitted values. There are no load-bearing self-citations, no definitions that secretly encode the target result, and no fitted parameter renamed as a prediction. Any concern about the unverified external bound in Theorem 5.2 is a correctness risk, not a circularity risk.

Assumptions & free parameters 4 free parameters · 2 assumptions · 0 invented entities

The central theorems rest on standard representation theory plus one cited external theorem in Section 5. No new physical or external entities are postulated. The numerical examples introduce kernel parameters fitted to training data, but those do not affect the theoretical claims and are flagged as empirical degrees of freedom.

free parameters (4)
  • Audio scattering kernel gamma0, gamma1, gamma2 on Aff(Fp) = Computed from class-average coefficients CB(k) and CM(k) over 20 training sounds
    The wavelet filters are derived from labelled training data as supervised feature design. This is a real empirical degree of freedom and the main reason the audio result should be seen as a demonstration, not a benchmark.
  • Symmetric-group kernels gamma_j(r) = |<d_j, chi_r>| / sqrt(J) and gamma0(r) = Computed from training distance functions for S3 and S5
    The scattering filters depend on the training distance functions and their class labels. The perfect S5 test accuracy may reflect the data-adaptive kernel design rather than a general method.
  • S6 kernels gamma_j(r) = |<f_j, chi_r>| = Fourier magnitudes of three random training functions on S6
    The filters are chosen from the training functions, and the reported accuracy is 73.33 percent, so the gains are limited and dataset-specific.
  • MNIST filter families, scales, and normalization = Mexican hat sigma=2, Shannon, Daubechies db2, J=1,5,8, normalized to satisfy the Parseval condition
    These are manually chosen hyperparameters. The normalization enforces equation (27) but is not a fit to the class labels.
assumptions (2)
  • standard math Standard finite-group Fourier analysis: Schur orthogonality, Plancherel theorem, and decomposition of the left regular representation into irreducibles.
    Used throughout Section 2 to derive the frame equivalence in Theorem 2.6 and the Fourier multiplier form of the wavelets in equation (15).
  • domain assumption Theorem 5 of Kueh, Olson, Rockmore and Tan [18]: a lower bound on Fourier coefficients of nonnegative functions over compact groups.
    The relaxed admissibility decay in Theorem 5.2 relies on this external theorem, applied to U[p]f, which is nonnegative by construction. The paper does not prove the bound, so its correctness is inherited from [18].

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Pith. "Pith review of Scattering Networks on Noncommutative Finite Groups." pith.science (2026). https://pith.science/paper/4JVGA5TP

@misc{pith2026250520950,
  author       = {Pith},
  title        = {Pith review of: Scattering Networks on Noncommutative Finite Groups},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4JVGA5TP}},
  note         = {Machine review of arXiv:2505.20950}
}
read the original abstract

Scattering Networks were initially designed to elucidate the behavior of early layers in Convolutional Neural Networks (CNNs) over Euclidean spaces and are grounded in wavelets. In this work, we introduce a scattering transform on an arbitrary finite group (not necessarily abelian) within the context of group-equivariant convolutional neural networks (G-CNNs). We present wavelets on finite groups and analyze their similarity to classical wavelets. We demonstrate that, under certain conditions in the wavelet coefficients, the scattering transform is non-expansive, stable under deformations, preserves energy, equivariant with respect to left and right group translations, and, as depth increases, the scattering coefficients are less sensitive to group translations of the signal, all desirable properties of convolutional neural networks. Furthermore, we provide examples illustrating the application of the scattering transform to classify data with domains involving abelian and nonabelian groups.

Figures

Figures reproduced from arXiv: 2505.20950 by the authors.

Figure 1
Figure 1. The structure of the scattering transform of a signal [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Representation of the Cayley graph of (Z ∗ p , ∗) The adjacency matrix of Cp−1 is   0 1 0 . . . 0 1 1 0 1 . . . 0 0 0 1 0 . . . 0 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . 0 1 1 0 0 . . . 1 0   It is a circulant matrix and their eigenvectors are (χ i (x) : x ∈ Z ∗ p ). The wavelets associated to a kernel γz : {0, 1, p − 2} → C, z ∈ Ω are: ψγz (x m) = Xp−2 n=0 γz(n)ω nm. (26) 3 Construction … view at source ↗
Figure 3
Figure 3. G-Scattering transform 11 [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (14 more)
Figure 4
Figure 4. Figure 4: Example of data 18 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: Characters of Z/28Z × Z/28Z We apply a G-Scattering Transform with 1, 2, and 3 layers and filter scales J = 1, J = 5, and J = 8. We fix a mexican hat wavelet with σ = 2.0 (see [PITH_FULL_IMAGE:figures/full_fig_p019_5.png]
Figure 6
Figure 6. Figure 6: Mexican hat wavelet 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Shannon Wavelets [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: Daubechies wavelet db2 rotated by θ (top row) and dilated by 0.5 and rotated by θ (bottom row). We normalize the functions {ϕ, e ψfj : j ∈ 1, . . . , J} in order to satisfy the unitary conditions (27). To do this, write ϕe(n, m) = X (a,b)∈Z/28×Z/28Z γe0(a, b)χ(a,b)(n, …
Figure 9
Figure 9. Figure 9: First level scattering associated to Shannon wavelets [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: First level scattering associated to Daubechies [PITH_FULL_IMAGE:figures/full_fig_p021_10.png]
Figure 11
Figure 11. Figure 11: The original length of an example signal is 6 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: First row: data labeled as ‘Meow‘, second row: data labeled and as ’Bark‘. Signal’s amplitudes [PITH_FULL_IMAGE:figures/full_fig_p023_12.png]
Figure 13
Figure 13. Figure 13: Spectrograms of the audio exhibit in Figure 12 created with scipy.signal. [PITH_FULL_IMAGE:figures/full_fig_p023_13.png]
Figure 14
Figure 14. Figure 14: Real (top) and imaginary (bottom) parts of Morlet wavelet with [PITH_FULL_IMAGE:figures/full_fig_p025_14.png]
Figure 15
Figure 15. Figure 15: Color maps of 10 · log10 |Wpf(a, b)| for the preprocesed sounds of [PITH_FULL_IMAGE:figures/full_fig_p026_15.png]
Figure 16
Figure 16. Figure 16: Color maps of 10 · log10 |Wpf(a, b)| of the sound denoted ‘Bark 1’ in [PITH_FULL_IMAGE:figures/full_fig_p026_16.png]
Figure 17
Figure 17. Figure 17: values of CB(k) (red dots) and CM(k) (blue dots), 0 ≤ k ≤ p − 1, for p = 31. Lemma 6.1. For any f ∈ S, any p prime, and any character χ k , 0 ≤ k ≤ p − 1, of Aff(Fp), |⟨Wpf , χk ⟩| ≤ 1 √ 2 1 (2πB) 1/4 , where B is the decay of the Morlet wavelet (see (49)). Consequent…

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