Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel

TL;DR

This paper presents a method to construct multiscale wavelet frames on various domains using spectral filtering of reproducing kernels, bridging continuous and discrete frames with Monte Carlo estimation, and providing stability and convergence guarantees.

Contribution

It introduces a novel construction of wavelet frames on general domains and analyzes their stability and convergence in a sampling regime, extending classical wavelet theory to non-Euclidean structures.

Findings

Frames tend to their population counterparts with finite-sample rates.

Discrete frames serve as Monte Carlo estimates of continuous frames.

Stability of frames is proven regardless of initial training samples.

Abstract

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.