Polynomial Regression on Riemannian Manifolds

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Variation with respect to the curve γ: Let {γ s : s ∈ (−ɛ, ɛ)} a smooth family of curves, with: γ 0 = γ W (t) := d ds γ s(t)| s=0 Extend v i , λ i away from curve via parallel transport: Then ∫ T ∇ W v i = 0 ∇ W λ i = 0 0 〈δ γ E ∗ (γ, {v i }, {λ i }), W 〉dt = d ds E∗ (γ s , {v i }, {λ i })| s=0 <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 11
For any smooth family of curves γ s (t), we have [ d ds γ s(t), d dt γ s(t)] = [W, ˙γ s ] = 0 so We also need the Leibniz rule ∇ W ˙γ = ∇ ˙γ W. d ds 〈X, Y 〉| s=0 = 〈∇ W X, Y 〉 + 〈X, ∇ W Y 〉, And the Riemann curvature tensor R(X, Y )Z = ∇ X ∇ Y Z − ∇ Y ∇ X Z − ∇ [X,Y ] Z ∇ W ∇ ˙γ Z = ∇ ˙γ ∇ W Z + R(W, ˙γ)Z <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 12
Page 1 and 2: Polynomial <strong
Page 3 and 4: Parametric Regression</stro
Page 5 and 6: Riemannian Polynomial</stro
Page 15 and 16: Rolling maps Leite 2008 Unroll curv
Page 19 and 20: Lagrange multiplier (adjoint) vecto
Page 21: Rewrite using integration by parts
Page 25 and 26: Now do the same with another term T
Page 27 and 28: Combine all terms to get adjoint eq
Page 31 and 32: Kendall Shape Space Space of N land
Page 33 and 34: Covariant derivative in shape space
Page 35 and 36: First fundamental form is (O’Neil
Page 37 and 38: Corpus Collosum Aging (www.oasis-br
Page 39 and 40: Landmark Space Space L of N points
Page 41 and 42: Have simple formula for cometric g
Page 43 and 44: For curvature, need Christoffel sym
Page 45 and 46: Landmark Regression</strong
Page 47: Thank You!

Polynomial Regression on Riemannian Manifolds

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