Monte Carlo wavelets: a randomized approach to frame discretization

Kereta, Željko; Vigogna, Stefano; Naumova, Valeriya; Rosasco, Lorenzo; Vito, Ernesto De

doi:10.1109/sampta45681.2019.9030825

Cited by 1 publication

(1 citation statement)

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“…On the other hand, a representation built on empirical samples poses an additional problem of stability, accounted for by how well it generalizes to future data. In this paper, expanding upon the ideas outlined in [35], we introduce a data-driven construction of wavelet frames on non-Euclidean domains, and provide stability results in high probability.…”

Section: Introductionmentioning

confidence: 99%

Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel

Vito

Kereta

Naumova

et al. 2021

J Fourier Anal Appl

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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.

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Section: Introductionmentioning

confidence: 99%