Homotopy-theoretic least squares regression

TL;DR

This paper develops a homotopy-theoretic framework for analyzing least squares solutions across data subsets, using presheaves of complexes and cech cohomology to understand solution gluing and higher homotopies.

Contribution

It introduces a novel homotopy-theoretic approach to least squares regression using presheaves of complexes and cech cohomology, providing new insights into solution consistency.

Findings

Constructed a presheaf of complexes on weighted finite subsets.

Revealed higher homotopies between least squares solutions on overlaps.

Worked out a detailed toy example with 5 data points.

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 $0$-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 \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.