Polynomial Regression on Riemannian Manifolds

Landmark Space
Space L of N points in R d . Geodesic equations:
d
dt x i =
N∑
γ(|x i − x j | 2 )α j
j=1
d
dt α i = −2
N∑
(x i − x j )γ ′ (|x i − x j |) 2 αi T α j
j=1
Usually use Gaussian kernel
γ(r) = e −r/(2σ2 )
x ∈ L and α ∈ T ∗ x L is a covector (momentum)
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 26