Polynomial Regression on Riemannian Manifolds

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