Polynomial Regression on Riemannian Manifolds
Polynomial Regression on Riemannian Manifolds
Polynomial Regression on Riemannian Manifolds
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Lagrange multiplier (adjoint) vector fields λ i al<strong>on</strong>g γ:<br />
E ∗ (γ, {v i }, {λ i }) =<br />
N∑<br />
g i (γ(t i )) +<br />
i=1<br />
∑k−1<br />
+<br />
i=1<br />
∫ T<br />
0<br />
∫ T<br />
0<br />
〈λ 0 , ˙γ − v 1 〉dt<br />
〈λ i , ∇ ˙γ v i − v i+1 〉dt +<br />
Euler-Lagrange for {λ i } gives forward system.<br />
Vector field integrati<strong>on</strong> by parts:<br />
∫ T<br />
0<br />
〈λ i , ∇ ˙γ v i 〉dt = [〈λ i , v i 〉] T 0 −<br />
∫ T<br />
0<br />
∫ T<br />
0<br />
〈∇ ˙γ λ i , v i 〉dt<br />
〈λ k , ∇ ˙γ v k 〉dt<br />
<str<strong>on</strong>g>Polynomial</str<strong>on</strong>g> <str<strong>on</strong>g>Regressi<strong>on</strong></str<strong>on</strong>g> <strong>on</strong> <strong>Riemannian</strong> <strong>Manifolds</strong> 9