Yuille-Poggio's Flow and Global Minimizer of Polynomials through Convexification by Heat Evolution

TL;DR

This paper explores a convexification approach for polynomial minimization using heat evolution and fingerprint theory, establishing conditions for success and introducing a simplified Newton method for quartic polynomials.

Contribution

It introduces the Yuille-Poggio flow and a comprehensive interpretation of fingerprint theory, providing necessary and sufficient conditions for the success of convexified minimization algorithms.

Findings

Conditions for successful convexification are established.

A simplified Newton method guarantees global minimization of quartic polynomials.

The approach unifies heat evolution, fingerprint theory, and polynomial minimization.

Abstract

This study examines the convexification version of the backward differential flow algorithm for the global minimization of polynomials, introduced by O. Arikan \textit{et al} in \cite{ABK}. It investigates why this approach might fail with high-degree polynomials yet succeeds with quartic polynomials. We employ the heat evolution method for convexification combined with Gaussian filtering, which acts as a cumulative form of Steklov's regularization. In this context, we apply the fingerprint theory from computer vision. Originally developed by A.L. Yuille and T. Poggio in the 1980s for computer vision, the fingerprint theory, particularly the fingerprint trajectory equation, is used to illustrate the scaling (temporal) evolution of minimizers. In the case of general polynomials, our research has led to the creation of the Yuille-Poggio flow and a broader interpretation of the fingerprint concepts, in particular we establish the condition both sufficient and necessary for the convexified backward differential flow algorithms to successfully achieve global minimization. For quartic polynomials, our analysis not only reflects the results of O. Arikan et al. \cite{ABK} but also presents a significantly simpler version of Newton's method that can always globally minimize quartic polynomials without convexification.