Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel
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Author(s)
De Vito, Ernesto
Kereta, Zeljko
Naumova, Valeriya
Rosasco, Lorenzo
Vigogna, Stefano
Date Issued
April 16, 2021
Publisher
Springer US
Citation
Journal of Fourier Analysis and Applications. 2021 Apr 16;27(2):37
Version
Final published version
Abstract
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.
MIT Department
Center for Brains, Minds, and Machines
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Creative Commons Attribution
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DOI of Published Version
https://doi.org/10.1007/s00041-021-09835-0
