Whitney's Theorem, Triangular Sets and Probabilistic Descent on Manifolds

TL;DR

This paper explores probabilistic descent on manifolds defined by polynomial systems, utilizing Whitney's embedding theorem and numerical continuation methods to improve optimization in reduced-dimensional spaces.

Contribution

It introduces a novel approach combining Whitney's theorem with probabilistic descent on polynomial-defined manifolds, leveraging triangularization and embedding techniques.

Findings

Effective dimensionality reduction via Whitney's embedding theorem

Successful application of numerical continuation methods

Potential improvements in optimization on polynomial manifolds

Abstract

We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney's embedding theorem allows us to work in a reduced dimensional embedding space. A numerical continuation method applied to the reduced-dimensional embedded manifold is used to drive the procedure.