Polynomial Regression on Riemannian Manifolds
Polynomial Regression on Riemannian Manifolds
Polynomial Regression on Riemannian Manifolds
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<strong>Riemannian</strong> <str<strong>on</strong>g>Polynomial</str<strong>on</strong>g>s<br />
At least three ways to define polynomial in R d<br />
Algebraic: γ(t) = c 0 + 1 1! c 1t + 1 2! c 2t 2 + · · · + 1 k! c kt k<br />
Variati<strong>on</strong>al: γ = argmin ϕ<br />
∫ T<br />
0 | ( d<br />
dt<br />
) k+1<br />
2<br />
ϕ(t)| 2 dt s.t. BC/ICs<br />
Differential:<br />
( d<br />
) k+1 (<br />
dt γ(t) = 0 s.t. initial c<strong>on</strong>diti<strong>on</strong>s d<br />
) i<br />
dt γ(0) = ci<br />
Covariant derivative: replace d dt of vectors with ∇ ˙γ<br />
k-order polynomial satisfies<br />
(∇ ˙γ ) k ˙γ = 0<br />
subject to initial c<strong>on</strong>diti<strong>on</strong>s γ(0), (∇ ˙γ ) i ˙γ(0), i = 0, . . . , k − 1<br />
Introduced via rolling maps by Jupp&Kent1987<br />
Studied by Leite (2008), in rolling map setting<br />
<str<strong>on</strong>g>Polynomial</str<strong>on</strong>g> <str<strong>on</strong>g>Regressi<strong>on</strong></str<strong>on</strong>g> <strong>on</strong> <strong>Riemannian</strong> <strong>Manifolds</strong> 5