A higher-order generalized singular value decomposition for rank-deficient matrices
The higher-order generalized singular value decomposition (HO-GSVD) is a matrix factorization technique that extends the GSVD to <i>N</i> ≥ 2 data matrices and can be used to identify common subspaces that are shared across multiple large-scale datasets with different row dimensions. The...
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| Format: | Journal article |
| Language: | English |
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Society for Industrial and Applied Mathematics
2023
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| _version_ | 1797111145720446976 |
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| author | Kempf, I Goulart, PJ Duncan, SR |
| author_facet | Kempf, I Goulart, PJ Duncan, SR |
| author_sort | Kempf, I |
| collection | OXFORD |
| description | The higher-order generalized singular value decomposition (HO-GSVD) is a matrix factorization technique that extends the GSVD to <i>N</i> ≥ 2 data matrices and can be used to identify common subspaces that are shared across multiple large-scale datasets with different row dimensions. The standard HO-GSVD factors <i>N</i> matrices <i>A</i><sub><i>i</i></sub> ∈ ℝ<sup><i>m</i><sub><i>i</i></sub>×<i>n</i></sup> as <i>A</i><sub><i>i</i></sub> = <i>U<sub>i</sub></i>∑<sub><i>i</i></sub><i>V</i><sup>T</sup> but requires that each of the matrices <i>A</i><sub><i>i</i></sub> has full column rank. We propose a modification of the HO-GSVD that extends its applicability to rank-deficient data matrices <i>A</i><sub><i>i</i></sub>. If the matrix of stacked <i>A</i><sub><i>i</i></sub> has full rank, we show that the properties of the original HO-GSVD extend to our approach. We extend the notion of common subspaces to isolated subspaces, which identify features that are unique to one <i>A</i><sub><i>i</i></sub>. We also extend our results to the higher-order cosine-sine decomposition (HO-CSD), which is closely related to the HO-GSVD. Our extension of the standard HO-GSVD allows its application to matrices with <i>m<sub>i</sub></i> < <i>n</i> or rank (<i>A</i><sub><i>i</i></sub>) < <i>n</i>, such as those encountered in bioinformatics, neuroscience, control theory, and classification problems. |
| first_indexed | 2024-03-07T08:04:39Z |
| format | Journal article |
| id | oxford-uuid:218bce04-a2f5-4b43-af60-ba3eab79d896 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-07T08:04:39Z |
| publishDate | 2023 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:218bce04-a2f5-4b43-af60-ba3eab79d8962023-10-30T12:03:50ZA higher-order generalized singular value decomposition for rank-deficient matricesJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:218bce04-a2f5-4b43-af60-ba3eab79d896EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2023Kempf, IGoulart, PJDuncan, SRThe higher-order generalized singular value decomposition (HO-GSVD) is a matrix factorization technique that extends the GSVD to <i>N</i> ≥ 2 data matrices and can be used to identify common subspaces that are shared across multiple large-scale datasets with different row dimensions. The standard HO-GSVD factors <i>N</i> matrices <i>A</i><sub><i>i</i></sub> ∈ ℝ<sup><i>m</i><sub><i>i</i></sub>×<i>n</i></sup> as <i>A</i><sub><i>i</i></sub> = <i>U<sub>i</sub></i>∑<sub><i>i</i></sub><i>V</i><sup>T</sup> but requires that each of the matrices <i>A</i><sub><i>i</i></sub> has full column rank. We propose a modification of the HO-GSVD that extends its applicability to rank-deficient data matrices <i>A</i><sub><i>i</i></sub>. If the matrix of stacked <i>A</i><sub><i>i</i></sub> has full rank, we show that the properties of the original HO-GSVD extend to our approach. We extend the notion of common subspaces to isolated subspaces, which identify features that are unique to one <i>A</i><sub><i>i</i></sub>. We also extend our results to the higher-order cosine-sine decomposition (HO-CSD), which is closely related to the HO-GSVD. Our extension of the standard HO-GSVD allows its application to matrices with <i>m<sub>i</sub></i> < <i>n</i> or rank (<i>A</i><sub><i>i</i></sub>) < <i>n</i>, such as those encountered in bioinformatics, neuroscience, control theory, and classification problems. |
| spellingShingle | Kempf, I Goulart, PJ Duncan, SR A higher-order generalized singular value decomposition for rank-deficient matrices |
| title | A higher-order generalized singular value decomposition for rank-deficient matrices |
| title_full | A higher-order generalized singular value decomposition for rank-deficient matrices |
| title_fullStr | A higher-order generalized singular value decomposition for rank-deficient matrices |
| title_full_unstemmed | A higher-order generalized singular value decomposition for rank-deficient matrices |
| title_short | A higher-order generalized singular value decomposition for rank-deficient matrices |
| title_sort | higher order generalized singular value decomposition for rank deficient matrices |
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