Adaptive strategies for solving parameterized systems using homotopy continuation

TL;DR

This paper explores adaptive methods in homotopy continuation for solving parameterized polynomial systems, focusing on affine patch selection, subsystem constraint management, and real solution path identification, with applications in computer vision.

Contribution

It introduces adaptive strategies for affine patch selection, subsystem constraint choice, and real solution path truncation in homotopy continuation methods for parameterized systems.

Findings

Improved numerical stability through adaptive subsystem selection.

Enhanced efficiency by heuristically truncating nonreal solution paths.

Validated approaches on computer vision problems.

Abstract

Three aspects of applying homotopy continuation, which is commonly used to solve parameterized systems of polynomial equations, are investigated. First, for parameterized systems which are homogeneous, we investigate options for performing computations on an adaptively chosen affine coordinate patch. Second, for parameterized systems which are overdetermined, we investigate options for adaptively selecting a well-constrained subsystem to restore numerical stability. Finally, since one is typically interested in only computing real solutions for parameterized problems which arise from applications, we investigate a scheme for heuristically identifying solution paths which appear to be ending at nonreal solutions and truncating them. We demonstrate these three aspects on two problems arising in computer vision.