Geometric modeling and regularization of algebraic problems

TL;DR

This paper introduces a geometric modeling approach that regularizes algebraic problems by ensuring solutions are Lipschitz continuous on complex analytic manifolds, transforming ill-posed problems into well-posed ones.

Contribution

It proposes a method to model algebraic problems with auxiliary equations satisfying specific conditions, enabling regularization and stability of solutions under data perturbations.

Findings

Solutions are Lipschitz continuous on the manifold.

Regularized problems are well-posed with unique solutions.

Empirical data solutions are stable within a tubular neighborhood.

Abstract

Discontinuity with respect to data perturbations is common in algebraic computation where solutions are often highly sensitive. Such problems can be modeled as solving systems of equations at given data parameters. By appending auxiliary equations, the models can be formulated to satisfy four easily verifiable conditions so that the data form complex analytic manifolds on which the solutions maintain their structures and the Lipschitz continuity. When such a problem is given with empirical data, solving the system becomes a least squares problem whose solution uniquely exists and enjoys Lipschitz continuity as long as the data point is in a tubular neighborhood of the manifold. As a result, the singular problem is regularized as a well-posed computational problem.