Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel
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...
| Main Authors: | , , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
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Springer US
2021
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| Online Access: | https://hdl.handle.net/1721.1/136775 |
| _version_ | 1826194153574236160 |
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| author | De Vito, Ernesto Kereta, Zeljko Naumova, Valeriya Rosasco, Lorenzo Vigogna, Stefano |
| author2 | Center for Brains, Minds, and Machines |
| author_facet | Center for Brains, Minds, and Machines De Vito, Ernesto Kereta, Zeljko Naumova, Valeriya Rosasco, Lorenzo Vigogna, Stefano |
| author_sort | De Vito, Ernesto |
| collection | MIT |
| description | 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. |
| first_indexed | 2024-09-23T09:51:45Z |
| format | Article |
| id | mit-1721.1/136775 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:51:45Z |
| publishDate | 2021 |
| publisher | Springer US |
| record_format | dspace |
| spelling | mit-1721.1/1367752023-09-07T20:28:01Z Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel De Vito, Ernesto Kereta, Zeljko Naumova, Valeriya Rosasco, Lorenzo Vigogna, Stefano Center for Brains, Minds, and Machines 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. 2021-11-01T14:33:17Z 2021-11-01T14:33:17Z 2021-04-16 2021-04-18T03:49:23Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/136775 Journal of Fourier Analysis and Applications. 2021 Apr 16;27(2):37 PUBLISHER_CC en https://doi.org/10.1007/s00041-021-09835-0 Creative Commons Attribution https://creativecommons.org/licenses/by/4.0/ The Author(s) application/pdf Springer US Springer US |
| spellingShingle | De Vito, Ernesto Kereta, Zeljko Naumova, Valeriya Rosasco, Lorenzo Vigogna, Stefano Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title | Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title_full | Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title_fullStr | Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title_full_unstemmed | Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title_short | Construction and Monte Carlo Estimation of Wavelet Frames Generated by a Reproducing Kernel |
| title_sort | construction and monte carlo estimation of wavelet frames generated by a reproducing kernel |
| url | https://hdl.handle.net/1721.1/136775 |
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