Sobolev Algorithm for Local Smoothness Analysis (SALSA) via Sharp Direct and Inverse Statements

2025/12/19 by Sara Avesani, Leevan Ling, Avesani, Sara +5
Computer Science · Engineering · Mathematics · #FOS: Mathematics #Medical Image Segmentation Techniques #Numerical Analysis (math.NA) #Numerical methods in engineering #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2512.17377

openalex publication_date 2025/12/19 · openalex created_date 2025/12/23 · openalex updated_date 2026/07/28

Abstract

We extend sharp direct and inverse approximation statements for kernel-based methods for finitely smooth kernels, i.e. those whose native spaces are norm-equivalent to Sobolev spaces. In particular, our inverse results are now formulated for a broad class of approximation schemes beyond interpolation, extending existing theory. Building on these results, we propose a novel Sobolev Algorithm for Local Smoothness Analysis (SALSA) for detecting local smoothness properties of target data, including their degree of smoothness and non-smoothness. The method is rigorously grounded based on the sharp direct and inverse statements. Numerical experiments in various settings highlight the effectiveness of the proposed algorithm.

Citations

• Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds
• Multiresolution local smoothness detection in non-uniformly sampled multivariate signals
• On the Convergence of Irregular Sampling in Reproducing Kernel Hilbert Spaces
• Sharp inverse statements for kernel interpolation
• Kernel Interpolation with Sparse Grids
• Bi-Directional Cascade Network for Perceptual Edge Detection
• Unsupervised Learning of Edges
• Optimal Convergence Rates Results for Linear Inverse Problems in Hilbert Spaces
• An inverse theorem for compact Lipschitz regions in R^d using\n localized kernel bases
• DeepEdge: A Multi-Scale Bifurcated Deep Network for Top-Down Contour\n Detection
• A Computational Approach to Edge Detection

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• On a regularization-correction approach for the approximation of piecewise smooth functions