Construction and Monte Carlo estimation of wavelet frames generated by a reproducing kernel
This work provides a stable framework for wavelet analysis on complex domains, which is incremental as it builds on existing spectral and learning theory approaches.
The authors introduced a method for constructing multiscale tight frames on general domains using spectral filtering of reproducing kernel integral operators, extending classical wavelets to non-Euclidean structures like Riemannian manifolds and graphs. They proved that discrete frames from random sampling converge to continuous ones with explicit finite-sample rates in Sobolev and Besov spaces, ensuring stability across different training samples.
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.