Polynomial Regression on Riemannian Manifolds

For curvature, need Christoffel symbols and their derivatives:
N∑
(Γ(u, v)) i = − (x i − x j) T (v i − v j)γ ′ (|x i − x j| 2 )(K −1 u) j
j=1
N∑
− (x i − x j) T (u i − u j)γ ′ (|x i − x j| 2 )(K −1 v) j
j=1
N∑
N∑
+ γ(|x i − x j| 2 ) (x j − x k )γ ′ (|x j − x k | 2 )((K −1 u) T k (K −1 v) j + (K −1
j=1
k=1
Take derivative with respect to x, and combine using
R l ijk = Γl ki,j − Γl ji,k + Γl jmΓ m ki − Γl km Γm ji
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 29