Multiresolution Low-rank Tensor Formats
© 2020 Society for Industrial and Applied Mathematics. We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we...
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| Format: | Article |
| Language: | English |
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Society for Industrial & Applied Mathematics (SIAM)
2021
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| Online Access: | https://hdl.handle.net/1721.1/135371 |
| _version_ | 1826191908224892928 |
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| author | Mickelin, Oscar Karaman, Sertac |
| author_facet | Mickelin, Oscar Karaman, Sertac |
| author_sort | Mickelin, Oscar |
| collection | MIT |
| description | © 2020 Society for Industrial and Applied Mathematics. We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors. |
| first_indexed | 2024-09-23T09:03:10Z |
| format | Article |
| id | mit-1721.1/135371 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T09:03:10Z |
| publishDate | 2021 |
| publisher | Society for Industrial & Applied Mathematics (SIAM) |
| record_format | dspace |
| spelling | mit-1721.1/1353712021-10-28T04:49:08Z Multiresolution Low-rank Tensor Formats Mickelin, Oscar Karaman, Sertac © 2020 Society for Industrial and Applied Mathematics. We describe a simple, black-box compression format for tensors with a multiscale structure. By representing the tensor as a sum of compressed tensors defined on increasingly coarse grids, we capture low-rank structures on each grid-scale, and we show how this leads to an increase in compression for a fixed accuracy. We devise an alternating algorithm to represent a given tensor in the multiresolution format and prove local convergence guarantees. In two dimensions, we provide examples that show that this approach can beat the Eckart-Young theorem, and for dimensions higher than two, we achieve higher compression than the tensor-train format on six real-world datasets. We also provide results on the closedness and stability of the tensor format and discuss how to perform common linear algebra operations on the level of the compressed tensors. 2021-10-27T20:23:10Z 2021-10-27T20:23:10Z 2020 2021-04-30T17:39:35Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/135371 en 10.1137/19M1284579 SIAM Journal on Matrix Analysis and Applications Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. application/pdf Society for Industrial & Applied Mathematics (SIAM) SIAM |
| spellingShingle | Mickelin, Oscar Karaman, Sertac Multiresolution Low-rank Tensor Formats |
| title | Multiresolution Low-rank Tensor Formats |
| title_full | Multiresolution Low-rank Tensor Formats |
| title_fullStr | Multiresolution Low-rank Tensor Formats |
| title_full_unstemmed | Multiresolution Low-rank Tensor Formats |
| title_short | Multiresolution Low-rank Tensor Formats |
| title_sort | multiresolution low rank tensor formats |
| url | https://hdl.handle.net/1721.1/135371 |
| work_keys_str_mv | AT mickelinoscar multiresolutionlowranktensorformats AT karamansertac multiresolutionlowranktensorformats |