Locality regularized reconstruction: structured sparsity and Delaunay triangulations

• Published in: Sampling theory, signal processing, and data analysis Vol. 23; no. 2
• Main Authors: Mueller, Marshall, Murphy, James M., Tasissa, Abiy
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025
• Subjects: Abstract Harmonic Analysis; Machine Learning; Mathematics; Mathematics and Statistics; Original Article; Signal,Image and Speech Processing; Computational geometry; 65F22; 90C25; Delaunay triangulation; 52C35; 68U05; 94A12; Regularization; 42C15; Optimization; Sparse signal processing
• DOI: 10.1007/s43670-025-00104-5
• ISSN: 2730-5716

Linear representation learning is widely studied due to its conceptual simplicity and empirical utility in tasks such as compression, classification, and feature extraction. Given a set of points and a vector , the goal is to find coefficients so that , subject to some desired structure on . In this work we seek that forms a local reconstruction of by solving a regularized least squares regression problem. We obtain local solutions through a locality function that promotes the use of columns of that are close to when used as a regularization term. We prove that, for all levels of regularization and under a mild condition that the columns of have a unique Delaunay triangulation, the optimal coefficients’ number of non-zero entries is upper bounded by , thereby providing local sparse solutions when . Under the same condition we also show that for any contained in the convex hull of there exists a regime of regularization parameter such that the optimal coefficients are supported on the vertices of the Delaunay simplex containing . This provides an interpretation of the sparsity as having structure obtained implicitly from the Delaunay triangulation of . We demonstrate that our locality regularized problem can be solved in comparable time to other methods that identify the containing Delaunay simplex.