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Homotopy-theoretic least squares regression

Cheyne Glass

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 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.

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 Čech-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 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.
Paper Structure (3 sections, 5 theorems, 22 equations)

This paper contains 3 sections, 5 theorems, 22 equations.

Table of Contents

  1. The least-squares Koszul presheaf
  2. Homotopy theoretic least squares solutions
  3. A toy example

Key Result

Lemma 1.2

If $f(x, a) = \phi(x) \cdot a$ is linear in $\Lambda$ then $\eta^k= \nu^k(\omega) + N^k(\omega) \cdot a$ where $\nu^k$ and $N^{k, \bullet}$ are the scalar and vector-valued functions on $\omega$ respectively given by

Theorems & Definitions (12)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Theorem 1.3
  • proof
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • ...and 2 more