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
Cubic spline satisfies (Noakes1989, Leite, Machado,…)
∫ T
γ = argmin ϕ 0 |∇ ˙ϕ ˙ϕ(t)| 2 dt
Euler-Lagrange equation: (∇ ˙γ ) 3 ˙γ = R( ˙γ, ∇ ˙γ ˙γ) ˙γ
Shape splines (Trouve-Vialard)
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 5