Homotopy-theoretic least squares regression
Abstract
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 \v{C}ech-theoretic complex where the -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 \v{C}ech-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.
Cite
@article{arxiv.2603.05870,
title = {Homotopy-theoretic least squares regression},
author = {Cheyne Glass},
journal= {arXiv preprint arXiv:2603.05870},
year = {2026}
}