[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-84053-en":3,"doc-seo-84053-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},84053,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Separation Capacity of Scattering Networks on Low-Dimensional Datasets","Separation capacity is studied for scattering network architectures acting on data with low intrinsic dimension modeled as rectifiable sets. The work characterizes and bounds the separation capacity of general feature extractors using the geometry of the underlying dataset, expressed through ranks on approximate tangent spaces and through a second-moment matrix coupling global geometry to the extractor. Specialization to pooling-free scattering networks yields filter-design criteria: sufficient frequency coverage and well-conditioned coupling matrices, with sparse-signal cases reducing to restricted isometry minimization.","Separation Capacity of Scattering Networks on Low-Dimensional  \nDatasets  \narXiv :2607 .06048v1 [ stat .ML] 7 Jul 2026  \nKonstantin H¨aberle ETH Zurich [haeberlk@ethz. ch](haeberlk@ethz. ch)  \nHelmut B¨olcskei ETH Zurich [hboelcskei@ethz. ch](hboelcskei@ethz. ch)  \nAbstract  \nWe aim to identify scattering network architectures that maximize the separation capacity on data with low intrinsic dimension. The networks we consider employ a fixed monomial nonlinearity and no pooling, so that the only design variable is the frame generated by the network filters. For data modeled as rectifiable sets, we first characterize and bound the separation capacity of general feature extractors in terms of the geometry of the dataset. We then particularize to scattering networks and obtain two design criteria: (i) the filters should meet the data on sufficiently many frequencies, and (ii) the matrices coupling the frame to the geometry of the data should be well-conditioned.  \nKeywords: Learning theory, pattern classification, feature extraction, scattering networks, convolutional neural networks, geometric measure theory.  \n1 Introduction  \nA common intuition for the success of deep learning models in classification and regression tasks is that real-world data, although embedded in high-dimensional spaces, often exhibit an intrinsic low-dimensional structure [1, 2] . In this paper, we model this intrinsic low-dimensional structure using the framework of geometric measure theory [3, 4] . Specifically, we assume that data lie on a rectifiable set, i.e., a set that, up to Hausdorff measure zero, is covered by a countable union of Lipschitz images of subsets of a Euclidean space. This class contains not only smooth submanifolds but also countable unions thereof, such as the unions of linear subspaces that constitute sparse-signal models.  \nThis paper studies the classification capabilities of scattering networks [5, 6, 7] on rectifiable sets, as measured by Cover’s separation capacity [8, 9] . Scattering networks are multilayered, neural-network-type architectures in which each layer computes convolutions with frame-generating filters [10], followed by pointwise nonlinearities and, in general, pooling operators. We particularize to the pooling-free case with a fixed monomial nonlinearity, so that the design freedom lies entirely in the choice of the frame. The central question addressed in this paper is how the geometry of the data should enter this choice. Our aim is to characterize the filters that maximize the separation capacity for a given rectifiable dataset.  \nTwo main contributions are reported. First, we relate the separation capacity of general feature extractors to the geometry of the underlying rectifiable set. Notably, the separation capacity is bounded below by the rank of the differential on the approximate tangent spaces and bounded above by the rank of a second-moment matrix, which couples the feature extractor to the global geometry of the dataset. Second, by applying these bounds to scattering networks, we establish filter-design criteria. The upper bounds prescribe that the spectral supports of the filters, intersected with the spectrum of the dataset, should not be contained in a coset of a proper subgroup. The lower bounds, in turn, suggest choosing the filters so that, for each Lipschitz parametrization of the underlying rectifiable set, an associated filter-dependent matrix  \n2 K. H¨aberle and H. B¨olcskei  \nis as well-conditioned as possible. For sparse signals, the criterion reduces to minimizing the restricted isometry constant of a single matrix.  \nThe paper is organized as follows. Section 2 develops the connection between the separation capacity of general feature extractors on rectifiable sets and the geometry of those sets. In Section 3, we apply these results to scattering networks to establish filter-design criteria, first for sparse signals and then for more general rectifiable sets. Appendix A reviews the restr","cbCailgdWn7KECi3","https://ap.wps.com/l/cbCailgdWn7KECi3","pdf",397676,1,19,"English","en",105,"# Introduction\n# Separation capacity computations on rectifiable sets\n## s-Separation capacity definition\n## Subspace characterization (Lemma 2.2)","[{\"question\":\"What determines the separation capacity for low-dimensional datasets in this work?\",\"answer\":\"Separation capacity is tied to the dataset’s rectifiable geometry, quantified via ranks on approximate tangent spaces and a second-moment matrix that couples feature extraction to the global dataset structure.\"},{\"question\":\"Why does the pooling-free scattering network model restrict design freedom?\",\"answer\":\"With fixed monomial nonlinearity and no pooling, the only adjustable component is the frame generated by the network filters, so separation capacity optimization becomes a filter-design problem.\"},{\"question\":\"What are the two main filter-design criteria derived for scattering networks?\",\"answer\":\"Filters should (i) cover sufficiently many frequencies present in the data, and (ii) produce coupling matrices to the data geometry that are well-conditioned, improving lower and upper bound behavior.\"}]",1784192256,48,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"separation-capacity-of-scattering-networks-on-low-dimensional-datasets","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/separation-capacity-of-scattering-networks-on-low-dimensional-datasets/84053/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-21","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What determines the separation capacity for low-dimensional datasets in this work?","Question",{"text":75,"@type":76},"Separation capacity is tied to the dataset’s rectifiable geometry, quantified via ranks on approximate tangent spaces and a second-moment matrix that couples feature extraction to the global dataset structure.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Why does the pooling-free scattering network model restrict design freedom?",{"text":80,"@type":76},"With fixed monomial nonlinearity and no pooling, the only adjustable component is the frame generated by the network filters, so separation capacity optimization becomes a filter-design problem.",{"name":82,"@type":73,"acceptedAnswer":83},"What are the two main filter-design criteria derived for scattering networks?",{"text":84,"@type":76},"Filters should (i) cover sufficiently many frequencies present in the data, and (ii) produce coupling matrices to the data geometry that are well-conditioned, improving lower and 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