Polynomial Regression on Riemannian Manifolds

For first term, T 1 = ∫ T
0 〈λ 0, γ˙
s 〉dt
d
ds T 1(γ s )| s=0 = d ds
=
=
∫ T
0
∫ T
0
∫ T
0
〈λ 0 , γ˙
s 〉dt| s=0
〈∇ W λ 0 , γ˙
s 〉 + 〈λ 0 , ∇ W γ˙
s 〉dt| s=0
〈0, γ˙
s 〉 + 〈λ 0 , ∇ ˙γ W 〉dt| s=0
= [〈λ 0 , W 〉] T 0 −
∫ T
Variation of this term with respect to γ is
0
δ γ(t) T 1 = −∇ ˙γ λ 0
δ γ(T ) T 1 = 0 = λ 0 (T )
δ γ(0) T 1 = −λ 0 (0)
〈∇ ˙γ λ 0 , W 〉dt
<strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 13