Homotopy-theoretic least squares regression
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910045851090944 |
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| author | Glass, Cheyne |
| author_facet | Glass, Cheyne |
| contents | A presheaf of complexes is constructed on a category of weighted finite subsets of a fixed Euclidean space. To each object, a Koszul complex is assigned which resolves the coordinate ring of least squares solutions on that data set for a choice of particular model (ie ``y=mx+b''). In order to obtain a total Čech-theoretic complex where the $0$-cocycles resemble locally defined least squares solutions gluing together up to homotopy, the coefficient rings for the Koszul complexes over each subset are linearized near a least squares solution. While these new linearized complexes do not immediately assemble into a presheaf, additional change-of-coordinates maps restore functoriality. Evaluating this new presheaf of complexes on a cover, its total-degree-0-cocycles of this Čech-Koszul bicomplex reveals (higher) homotopies between the discrepancies of least squares solutions on (higher) overlaps. A toy example with 5 data points is worked out in full elementary detail. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_05870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Homotopy-theoretic least squares regression Glass, Cheyne Algebraic Topology Algebraic Geometry 55N30, 18G35, 62J05 A presheaf of complexes is constructed on a category of weighted finite subsets of a fixed Euclidean space. To each object, a Koszul complex is assigned which resolves the coordinate ring of least squares solutions on that data set for a choice of particular model (ie ``y=mx+b''). In order to obtain a total Čech-theoretic complex where the $0$-cocycles resemble locally defined least squares solutions gluing together up to homotopy, the coefficient rings for the Koszul complexes over each subset are linearized near a least squares solution. While these new linearized complexes do not immediately assemble into a presheaf, additional change-of-coordinates maps restore functoriality. Evaluating this new presheaf of complexes on a cover, its total-degree-0-cocycles of this Čech-Koszul bicomplex reveals (higher) homotopies between the discrepancies of least squares solutions on (higher) overlaps. A toy example with 5 data points is worked out in full elementary detail. |
| title | Homotopy-theoretic least squares regression |
| topic | Algebraic Topology Algebraic Geometry 55N30, 18G35, 62J05 |
| url | https://arxiv.org/abs/2603.05870 |