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

Qiao Wang

TL;DR

This work develops a heat-evolution framework to convexify even-degree polynomials for global minimization and analyzes the evolution of minimizers through a scale-space lens grounded in Yuille–Poggio fingerprint theory. By applying the heat equation with Gaussian filtering, the authors obtain a tractable, multiscale description of critical points and establish a necessary-and-sufficient criterion for when backward trajectory methods reach the global minimizer. For quartic polynomials, the paper provides explicit fingerprint structures, exact interrelations among FP1, FP2, and FP3, and a confinement zone, together with a simple Newton-like scheme that globally minimizes quartics without convexification. The results extend to higher-degree polynomials, revealing that global convergence can fail and introducing the concepts of confinement and escape zones to characterize when the convexified minimizer can be inverted to recover the original global minimizer. Overall, the work clarifies when heat-evolved convexification yields the global minimizer and offers detailed quartic-case prescriptions plus numerical pathways for higher-degree scenarios, with potential extensions to multivariate cases.

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.

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

TL;DR

This work develops a heat-evolution framework to convexify even-degree polynomials for global minimization and analyzes the evolution of minimizers through a scale-space lens grounded in Yuille–Poggio fingerprint theory. By applying the heat equation with Gaussian filtering, the authors obtain a tractable, multiscale description of critical points and establish a necessary-and-sufficient criterion for when backward trajectory methods reach the global minimizer. For quartic polynomials, the paper provides explicit fingerprint structures, exact interrelations among FP1, FP2, and FP3, and a confinement zone, together with a simple Newton-like scheme that globally minimizes quartics without convexification. The results extend to higher-degree polynomials, revealing that global convergence can fail and introducing the concepts of confinement and escape zones to characterize when the convexified minimizer can be inverted to recover the original global minimizer. Overall, the work clarifies when heat-evolved convexification yields the global minimizer and offers detailed quartic-case prescriptions plus numerical pathways for higher-degree scenarios, with potential extensions to multivariate cases.

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.
Paper Structure (14 sections, 16 theorems, 87 equations, 2 figures, 1 table)

This paper contains 14 sections, 16 theorems, 87 equations, 2 figures, 1 table.

Table of Contents

  1. Background and Motivations
  2. Heat evolution and convexification of polynomials
  3. Heat evolution of $p(x)$
  4. Comparison principle
  5. Global minimizer and scale space fingerprint
  6. Fingerprints of scale space
  7. Confinement zone and escape zone
  8. Case study of quartic polynomials
  9. The structure of fingerprints
  10. The structure of fingerprint $\mathcal{FP}_1$
  11. The structure of $\mathcal{FP}_2$ and $\mathcal{FP}_3$
  12. The intersection between fingerprints
  13. Confinement zone
  14. Differential equation of critical points across scale

Key Result

theorem 1

For each even degree monic polynomial $p(x)$, there exists an specified $T^*=T^*(p)$ such that the heat convolution $p(x,t)$ is convex w.r.t $x$ at any $t>T^*$.

Figures (2)

  • Figure 1: The illustration of Yuille-Poggio's flow as well as $\mathcal{FP}_2$ and $\mathcal{FP}_3$.
  • Figure 2: Illustration of Fingerprints and Trajectories in Example \ref{['example-14']}. To observe the change of trajectories with coefficient $c$ in the polynomial, we vary it in $c\in [-2,2]$. (a): $\mathcal{FP}_i,\ (i=1,2,3)$, $t_u$ and $t^*$. Notice that $\mathcal{FP}_1$ corresponds to $c=0.01$. (b): $\mathcal{FP}_2,\ \mathcal{FP}_3$ and trajectories of $c=-0.05$. (c): critical trajectories when $c=-0.051$, which are symmetric about $x=-\frac{a}{4}$. (d): $\mathcal{FP}_2,\ \mathcal{FP}_3$ and trajectories of $c=-0.2$. (e): trajectories by varying $c$. (f): more trajectories by varying $c$.

Theorems & Definitions (37)

  • remark thmcounterremark
  • theorem 1
  • proof
  • theorem 2: Comparison principle
  • proof
  • definition thmcounterdefinition
  • remark thmcounterremark
  • definition thmcounterdefinition
  • theorem 3
  • lemma thmcounterlemma
  • ...and 27 more