[Paper Review] Numerical Nonlinear Algebra

Daniel J. Bates, Paul Breiding+4 more|arXiv (Cornell University)|Feb 16, 2023

Polynomial and algebraic computation4 citations

TL;DR

This paper presents numerical nonlinear algebra as a computational framework for solving systems of polynomial equations using numerical homotopy continuation, leveraging algebraic geometry and combinatorial structures to efficiently compute complex solutions. It demonstrates the method's effectiveness through certification of solutions and applications in computer vision, including minimal problems like the 5pt and 3pt camera calibration with degrees 20 and 8, respectively.

ABSTRACT

Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent growth of numerical nonlinear algebra continues to be informed by algebraic geometry and its applications. These include new approaches to solving, algorithms for studying positive-dimensional varieties, certification, and a range of applications both within mathematics and from other disciplines. With new implementations, numerical nonlinear algebra is now a fundamental computational tool for algebraic geometry and its applications. We survey some of these innovations and some recent applications.

Motivation & Objective

• To develop and systematize numerical methods for solving systems of polynomial equations using numerical analysis and algebraic geometry.
• To enable efficient computation of complex solutions to polynomial systems through homotopy continuation, particularly for high-degree minimal problems.
• To provide certification techniques for numerical solutions, ensuring correctness and reliability in practical applications.
• To demonstrate the utility of numerical nonlinear algebra in real-world domains such as computer vision and geometric optimization.
• To bridge symbolic and numerical computation by advancing tools that handle positive-dimensional varieties and overdetermined systems.

Proposed method

• Utilizes polynomial homotopy continuation to track solution paths from a known start system to the target system, solving F(x) = 0 via a homotopy H(x,t) interpolating between F and G.
• Employs predictor-corrector schemes to numerically solve the Davidenko differential equation governing the path tracking of solutions.
• Applies the polyhedral homotopy algorithm to exploit combinatorial structures in polynomial systems, improving efficiency in solving sparse systems.
• Uses endgames to handle singular solutions at t=0, ensuring robustness for non-regular zeros.
• Implements certification methods to validate approximate numerical solutions, ensuring correctness under perturbations.
• Combines homotopy continuation with machine learning in some cases, e.g., using neural networks to guide path tracking and avoid computing all solutions.

Experimental results

Research questions

• RQ1 How can numerical homotopy continuation be systematically applied to solve systems of polynomial equations arising in algebraic geometry and applications?
• RQ2 What combinatorial and geometric structures in polynomial systems can be exploited to improve the efficiency of numerical solution methods?
• RQ3 How can approximate numerical solutions to polynomial systems be rigorously certified as correct?
• RQ4 What are the practical limits and performance trade-offs of numerical nonlinear algebra in high-degree minimal problems such as the 5pt problem?
• RQ5 In what ways can numerical nonlinear algebra outperform symbolic methods in solving complex geometric problems, especially when degrees exceed practical symbolic computation thresholds?

Key findings

• The 5pt problem in computer vision has a branched cover of degree 20, and numerical homotopy continuation can compute all 20 complex solutions efficiently.
• The 3pt calibrated camera problem has a degree of 8, and numerical methods can compute all complex solutions, with only a subset having positive depths in practice.
• For problems of high degree such as 216 and 312 involving point and line correspondences, specialized solvers like MINUS can compute all complex solutions using optimized homotopy continuation.
• A hybrid approach using neural networks to guide path tracking reduces computation by focusing only on relevant real solutions, achieving performance comparable to state-of-the-art solvers.
• Certification techniques allow for rigorous validation of numerical solutions, ensuring reliability even when solutions are approximate.
• The framework enables the solution of overdetermined systems by solving relaxations and verifying solutions via the original formulation, enhancing robustness in geometric optimization.