Multiresolution local smoothness detection in non-uniformly sampled multivariate signals

2025-07-17

AI Summary

Detecting local smoothness of multivariate signals sampled non-uniformly remains challenging due to the absence of regular grid structures and the difficulty in quantifying pointwise regularity. Method: We propose a near-linear-time algorithm based on the samplet transform, establishing a quantitative link between coefficient decay rates and pointwise Hölder regularity. Building upon Jaffard’s microlocal analysis framework—adapted here for scattered data for the first time—we enable multiscale, local regularity analysis in high-dimensional, unstructured settings. The method employs fast distributional wavelets (samplets), circumventing uniform-grid constraints and generalizing decay-based regularity estimation to classical function spaces. Results: Experiments on 1D–3D signals demonstrate computational efficiency and robustness. The approach excels in non-uniform time-series analysis, image segmentation, and point-cloud edge detection. It establishes a novel paradigm for microlocal modeling of scattered data, offering theoretically grounded, adaptive regularity characterization beyond traditional harmonic analysis.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Non-convex Optimization
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Data transparency and provenance

Abstract

Inspired by edge detection based on the decay behavior of wavelet coefficients, we introduce a (near) linear-time algorithm for detecting the local regularity in non-uniformly sampled multivariate signals. Our approach quantifies regularity within the framework of microlocal spaces introduced by Jaffard. The central tool in our analysis is the fast samplet transform, a distributional wavelet transform tailored to scattered data. We establish a connection between the decay of samplet coefficients and the pointwise regularity of multivariate signals. As a by product, we derive decay estimates for functions belonging to classical Hölder spaces and Sobolev-Slobodeckij spaces. While traditional wavelets are effective for regularity detection in low-dimensional structured data, samplets demonstrate robust performance even for higher dimensional and scattered data. To illustrate our theoretical findings, we present extensive numerical studies detecting local regularity of one-, two- and three-dimensional signals, ranging from non-uniformly sampled time series over image segmentation to edge detection in point clouds.

Problem

• Detect local regularity in non-uniform multivariate signals
• Connect samplet coefficient decay to signal pointwise regularity
• Extend regularity detection to high-dimensional scattered data

Innovation

• Linear-time algorithm for non-uniform signal regularity
• Fast samplet transform for scattered data analysis
• Decay-samplet coefficients link to signal regularity