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
Geodesic (k = 1) has both forms
γ = argmin ϕ
∫ T
0 | ˙ϕ(t)|2 dt
∇ ˙γ ˙γ = 0 s.t. initial conditions γ(0), ˙γ(0)
Well-studied (Fletcher, Younes, Trouve, …)
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 5