2020/06/17 by Ernesto De Vito, De Vito, Ernesto, Željko Kereta +7
Engineering · #42C15 #42C40 #46E22 #47A52 #65T60 #68T05 #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Machine Learning (stat.ML) #Photoacoustic and Ultrasonic Imaging
paper · pdf · doi:10.48550/arxiv.2006.09870
openalex publication_date 2020/06/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a construction of multiscale tight frames on general domains.\nThe frame elements are obtained by spectral filtering of the integral operator\nassociated with a reproducing kernel. Our construction extends classical\nwavelets as well as generalized wavelets on both continuous and discrete\nnon-Euclidean structures such as Riemannian manifolds and weighted graphs.\nMoreover, it allows to study the relation between continuous and discrete\nframes in a random sampling regime, where discrete frames can be seen as Monte\nCarlo estimates of the continuous ones. Pairing spectral regularization with\nlearning theory, we show that a sample frame tends to its population\ncounterpart, and derive explicit finite-sample rates on spaces of Sobolev and\nBesov regularity. Our results prove the stability of frames constructed on\nempirical data, in the sense that all stochastic discretizations have the same\nunderlying limit regardless of the set of initial training samples.\n