Tree-Adaptive Multiscale Kernel Lasso in Samplet Coordinates

TL;DR

This paper introduces a multiscale kernel approximation framework using samplets for large scattered data, enabling adaptive site selection and sparse solutions with efficient computations.

Contribution

It presents a novel samplet-based multiscale kernel approximation method with adaptive data site selection and efficient sparse regularized solutions.

Findings

Achieves accurate reconstructions with sparser representations.

Demonstrates computational efficiency in 2D and 3D experiments.

Effectively handles multi-kernel models with varying lengthscales.

Abstract

We develop a novel framework for sparse multiscale kernel approximation of large scattered data problems based on a samplet representation. Samplets form a multiresolution analysis of localized discrete signed measures and enable quasi-sparse representations of kernel matrices associated to asymptotically smooth kernels as well as smoothness detection of scattered data. Building on the latter, we introduce an adaptive data site selection strategy based on the localization of the native reproducing kernel Hilbert space norm in the samplet expansion coefficients. The selection results in a small set of representative data sites, significantly reducing the effective problem size. On the corresponding reduced kernel subspace, we solve an $\ell^1$-regularized least-squares problem using a trust-region semismooth Newton method in a normal-map formulation, stabilized by an online low-rank SVD on the active set to handle the notorious ill-conditioning of kernel matrices. Numerical experiments in two and three dimensions, including multi-kernel models with varying lengthscales, demonstrate that the proposed approach achieves accurate reconstructions with considerably sparser representations and good computational efficiency.