Publications
Preprints
Getter, Max; Trapasso, S. Ivan
The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds Preprint
2026.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, microlocal analysis, multiscale analysis, scattering transform
@workingpaper{GT_2026,
title = {The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds},
author = {Max Getter and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2607.21578, arXiv},
year = {2026},
date = {2026-07-24},
urldate = {2026-07-24},
abstract = {A central challenge in modern harmonic analysis is to quantify the balance between the approximation power of finely resolved multiscale representations and their robustness to nonlinear changes of coordinates, a problem arising naturally in signal processing and partial differential equations. Motivated by Mallat's pioneering results on the wavelet scattering transform, we identify a sharp resolution--robustness trade-off for scattering-type nonlinear multiscale representations built upon general wave-packet systems, showing that stability under small diffeomorphisms is governed by the geometry of the underlying frequency decomposition. In particular, for wave-packet systems with finer transverse resolution than wavelets, including curvelets and shearlets, we establish a geometric rigidity phenomenon: arbitrarily small, smooth, compactly supported deformations can move high-frequency mass across adjacent channels, leading to instability already at the first scattering layer. We complement this obstruction by identifying the sharp Sobolev threshold for deformation stability: below the critical regularity no Mallat-type estimate can hold, while at and above it stability is recovered by means of matched commutator bounds that allow deformations to be propagated through the frequency channels. Together, these results provide a systematic deformation-stability theory for Euclidean scattering transforms and yield the first stability estimates intrinsic to the scattering architecture beyond the classical wavelet setting.},
keywords = {deformation stability, harmonic analysis, machine learning, microlocal analysis, multiscale analysis, scattering transform},
pubstate = {published},
tppubtype = {workingpaper}
}
A central challenge in modern harmonic analysis is to quantify the balance between the approximation power of finely resolved multiscale representations and their robustness to nonlinear changes of coordinates, a problem arising naturally in signal processing and partial differential equations. Motivated by Mallat's pioneering results on the wavelet scattering transform, we identify a sharp resolution–robustness trade-off for scattering-type nonlinear multiscale representations built upon general wave-packet systems, showing that stability under small diffeomorphisms is governed by the geometry of the underlying frequency decomposition. In particular, for wave-packet systems with finer transverse resolution than wavelets, including curvelets and shearlets, we establish a geometric rigidity phenomenon: arbitrarily small, smooth, compactly supported deformations can move high-frequency mass across adjacent channels, leading to instability already at the first scattering layer. We complement this obstruction by identifying the sharp Sobolev threshold for deformation stability: below the critical regularity no Mallat-type estimate can hold, while at and above it stability is recovered by means of matched commutator bounds that allow deformations to be propagated through the frequency channels. Together, these results provide a systematic deformation-stability theory for Euclidean scattering transforms and yield the first stability estimates intrinsic to the scattering architecture beyond the classical wavelet setting.
Journal Articles
2.
Nicola, Fabio; Trapasso, S. Ivan
Generalized moduli of continuity under irregular or random deformations via multiscale analysis Journal Article
In: Inf. Inference, vol. 14, iss. 2, no. iaaf006, 2025.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, multiscale analysis
@article{NT_IMAIAI_25,
title = {Generalized moduli of continuity under irregular or random deformations via multiscale analysis},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2104.11977, arXiv
https://academic.oup.com/imaiai/article/doi/10.1093/imaiai/iaaf006/8108050?utm_source=authortollfreelink&utm_campaign=imaiai&utm_medium=email&guestAccessKey=87ddedf1-38c9-401d-b53d-cf41144903cf, PDF},
doi = {10.1093/imaiai/iaaf006},
year = {2025},
date = {2025-04-08},
urldate = {2025-04-08},
journal = {Inf. Inference},
volume = {14},
number = {iaaf006},
issue = {2},
abstract = {Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields (tauin L^infty(mathbb{R}^d;mathbb{R}^d)), to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces (U_s) at scale (s), stability in (L^2) holds in the regime (|tau|_{L^infty}/sll 1) - essentially as an effect of the uncertainty principle. Instability occurs when (|tau|_{L^infty}/sgg 1), and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space (B^{d/2}_{2,1}) tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when (tau(x)) is modeled as a random field (not bounded, in general) with identically distributed variables (|tau(x)|), (xinmathbb{R}^d).},
keywords = {deformation stability, harmonic analysis, machine learning, multiscale analysis},
pubstate = {published},
tppubtype = {article}
}
Motivated by the problem of robustness to deformations of the input for deep convolutional neural networks, we identify signal classes which are inherently stable to irregular deformations induced by distortion fields \(\tau\in L^\infty(\mathbb{R}^d;\mathbb{R}^d)\), to be characterized in terms of a generalized modulus of continuity associated with the deformation operator. Resorting to ideas of harmonic and multiscale analysis, we prove that for signals in multiresolution approximation spaces \(U_s\) at scale \(s\), stability in \(L^2\) holds in the regime \(\|\tau\|_{L^\infty}/s\ll 1\) – essentially as an effect of the uncertainty principle. Instability occurs when \(\|\tau\|_{L^\infty}/s\gg 1\), and we provide a sharp upper bound for the asymptotic growth rate. The stability results are then extended to signals in the Besov space \(B^{d/2}_{2,1}\) tailored to the given multiresolution approximation. We also consider the case of more general time-frequency deformations. Finally, we provide stochastic versions of the aforementioned results, namely we study the issue of stability in mean when \(\tau(x)\) is modeled as a random field (not bounded, in general) with identically distributed variables \(|\tau(x)|\), \(x\in\mathbb{R}^d\).
1.
Nicola, Fabio; Trapasso, S. Ivan
Stability of the scattering transform for deformations with minimal regularity Journal Article
In: J. Math. Pures Appl., vol. 180, pp. 122–150, 2023.
Abstract | Links | BibTeX | Tags: deformation stability, harmonic analysis, machine learning, multiscale analysis, scattering transform
@article{NT_JMPA_23,
title = {Stability of the scattering transform for deformations with minimal regularity},
author = {Fabio Nicola and S. Ivan Trapasso},
url = {https://arxiv.org/abs/2205.11142, arXiv
https://www.sciencedirect.com/science/article/pii/S0021782423001496/pdfft?md5=b95043909dc30fb3845bb0cac8a65b05&pid=1-s2.0-S0021782423001496-main.pdf, PDF},
doi = {10.1016/j.matpur.2023.10.008},
year = {2023},
date = {2023-12-31},
urldate = {2023-12-31},
journal = {J. Math. Pures Appl.},
volume = {180},
pages = {122–150},
publisher = {Elsevier Masson},
abstract = {Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small (C^2) diffeomorphisms - a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale (C^alpha), (alpha >0). We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class (C^{alpha}), (alpha>1), whereas instability phenomena can occur at lower regularity levels modelled by (C^alpha), (0le alpha <1). While the behaviour at the threshold given by Lipschitz (or even (C^1)) regularity remains beyond reach, we are able to prove a stability bound in that case, up to (varepsilon) losses.},
keywords = {deformation stability, harmonic analysis, machine learning, multiscale analysis, scattering transform},
pubstate = {published},
tppubtype = {article}
}
Within the mathematical analysis of deep convolutional neural networks, the wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of multiscale analysis can be combined with a cascade of modulus nonlinearities to build a nonexpansive, translation invariant signal representation with provable geometric stability properties, namely Lipschitz continuity to the action of small \(C^2\) diffeomorphisms – a remarkable result for both theoretical and practical purposes, inherently depending on the choice of the filters and their arrangement into a hierarchical architecture. In this note, we further investigate the intimate relationship between the scattering structure and the regularity of the deformation in the Hölder regularity scale \(C^\alpha\), \(\alpha >0\). We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class \(C^{\alpha}\), \(\alpha>1\), whereas instability phenomena can occur at lower regularity levels modelled by \(C^\alpha\), \(0\le \alpha <1\). While the behaviour at the threshold given by Lipschitz (or even \(C^1\)) regularity remains beyond reach, we are able to prove a stability bound in that case, up to \(\varepsilon\) losses.