Yuille-Poggio's Flow and Global Minimizer of polynomials through convexification by Heat Evolution

In this paper, we investigate the possibility of the backward-differential-flow-like algorithm which starts from the minimum of convexification version of the polynomial. We apply the heat evolution convexification approach through Gaussian filtering, which is actually an accumulation version of Steklov's regularization. We generalize the fingerprint theory which was proposed in the theory of computer vision by A.L. Yuille and T. Poggio in 1980s, in particular their fingerprint trajectory equation, to characterize the evolution of minimizers across the scale. On the other hand, we propose the "seesaw" polynomials $p(x|s)$ and we find a seesaw differential equation $\frac{\partial p(x|s)}{\,ds}=-\frac{1}{p''(x)}$ to characterize the evolution of global minimizer $x^*(s)$ of $p(x|s)$ while varying $s$. Essentially, both the fingerprints $\mathcal{FP}_2$ and $\mathcal{FP}_3$ of $p(x)$, consisting of the zeros of $\frac{\partial^2 p(x,t)}{\partial x^2}$ and $\frac{\partial^3 p(x,t)}{\partial x^3}$, respectively, are independent of seesaw coefficient $s$, upon which we define the Confinement Zone and Escape Zone. Meanwhile, varying $s$ will monotonically condition the location of global minimizer of $p(x|s)$, and all these location form the Attainable Zone. Based on these concepts, we prove that the global minimizer $x^*$ of $p(x)$ can be inversely evolved from the global minimizer of its convexification polynomial $p(x,t_0)$ if and only if $x^*$ is included in the Escape Zone. In particular, we give detailed analysis for quartic and six degree polynomials.