Separation Capacity of Scattering Networks on Low-Dimensional Datasets

2026-07-08 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Theoretical Machine Learning · Depth: Expert, extended

Summary

The paper "Separation Capacity of Scattering Networks on Low-Dimensional Datasets" investigates scattering network architectures to maximize separation capacity on data with low intrinsic dimension, modeled as rectifiable sets. Focusing on networks with fixed monomial nonlinearity and no pooling, the research characterizes and bounds separation capacity based on dataset geometry. Two main design criteria are established: filters must meet data on sufficient frequencies, and matrices coupling the frame to data geometry should be well-conditioned. The study applies these bounds to sparse signals and polynomially parametrized rectifiable sets, providing upper and lower bounds for separation capacity, and deriving filter design recommendations for maximizing classification capabilities.

Key takeaway

For AI Scientists and Research Scientists designing scattering networks for classification tasks on low-dimensional data, prioritize filter design that ensures broad spectral overlap with the data and optimizes the conditioning of matrices linking the filter frame to the data's geometry. This approach, particularly for sparse signals or rectifiable sets, directly enhances the network's separation capacity, improving classification performance. Evaluate filter choices based on the restricted isometry constant for robust feature extraction.

Key insights

Scattering networks' separation capacity on low-dimensional data is maximized by specific filter design criteria.

Principles

• Filters must meet data on sufficiently many frequencies.
• Frame-to-data coupling matrices need to be well-conditioned.
• Separation capacity relates to data's tangential geometry.

Method

The paper characterizes separation capacity using Cover's theory, then applies bounds derived from approximate tangent spaces and second-moment matrices to scattering networks, analyzing sparse signals and rectifiable sets.

In practice

• Design filters for broad spectral support.
• Optimize filter matrices for low restricted isometry constant.
• Consider data's intrinsic dimension for network design.

Topics

• Scattering Networks
• Separation Capacity
• Low-Dimensional Data
• Rectifiable Sets
• Filter Design
• Geometric Measure Theory
• Sparse Signals

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.