Separation Capacity of Scattering Networks on Low-Dimensional Datasets

2026-07-07

AI Summary

This work investigates the design of pooling-free scattering networks employing fixed monomial nonlinearities to maximize separability for data with low intrinsic dimensionality. By integrating frame theory, geometric measure theory, and moment analysis, the study provides the first geometric characterization of a scattering network’s separation capacity, establishing theoretical bounds for feature extractors operating on low-dimensional rectifiable data. The core contribution consists of two practical design principles: the network’s filters must span a sufficiently broad frequency range, and the frame formed by these filters—when coupled with the data’s geometric structure through a coupling matrix—must exhibit a well-conditioned condition number. These criteria jointly ensure significantly enhanced separation performance, offering concrete guidance for the construction of effective scattering architectures tailored to geometrically structured low-dimensional data.

Abstract

We 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.

Problem

• separation capacity
• scattering networks
• low-dimensional datasets
• intrinsic dimension
• feature extractors

Innovation

• scattering networks
• separation capacity
• low-dimensional data
• frame design
• geometric characterization