{"title":"Stability of the scattering transform for deformations with minimal regularity","authors":"Fabio Nicola, S. Ivan Trapasso","doi":"10.1016/j.matpur.2023.10.008","DOIUrl":null,"url":null,"abstract":"<div><p>The wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of harmonic and multiscale analysis can be ingeniously exploited to build a signal representation with provable geometric stability properties, such as the Lipschitz continuity to the action of small <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> 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 <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>, <span><math><mi>α</mi><mo>></mo><mn>0</mn></math></span>. We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>, <span><math><mi>α</mi><mo>></mo><mn>1</mn></math></span>, whereas instability phenomena can occur at lower regularity levels modeled by <span><math><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi></mrow></msup></math></span>, <span><math><mn>0</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>1</mn></math></span>. While the analysis at the threshold given by Lipschitz (or even <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>) regularity remains beyond reach, we are able to prove a stability bound in that case, up to <em>ε</em> losses.</p></div>","PeriodicalId":51071,"journal":{"name":"Journal de Mathematiques Pures et Appliquees","volume":"180 ","pages":"Pages 122-150"},"PeriodicalIF":2.1000,"publicationDate":"2023-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0021782423001496/pdfft?md5=b95043909dc30fb3845bb0cac8a65b05&pid=1-s2.0-S0021782423001496-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal de Mathematiques Pures et Appliquees","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0021782423001496","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The wavelet scattering transform introduced by Stéphane Mallat is a unique example of how the ideas of harmonic and multiscale analysis can be ingeniously exploited to build a signal representation with provable geometric stability properties, such as the Lipschitz continuity to the action of small 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 , . We are able to precisely identify the stability threshold, proving that stability is still achievable for deformations of class , , whereas instability phenomena can occur at lower regularity levels modeled by , . While the analysis at the threshold given by Lipschitz (or even ) regularity remains beyond reach, we are able to prove a stability bound in that case, up to ε losses.
期刊介绍:
Published from 1836 by the leading French mathematicians, the Journal des Mathématiques Pures et Appliquées is the second oldest international mathematical journal in the world. It was founded by Joseph Liouville and published continuously by leading French Mathematicians - among the latest: Jean Leray, Jacques-Louis Lions, Paul Malliavin and presently Pierre-Louis Lions.