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Article overview
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Yuille-Poggio's Flow and Global Minimizer of polynomials through convexification by Heat Evolution | | Qiao Wang
; | | Date: |
1 Jan 2023 | | Abstract: | 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. | | Source: | arXiv, 2301.00326 | | Services: | Forum | Review | PDF | Favorites | |
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