Polynomial Regression on Riemannian Manifolds

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N∑ ((DΓ(u, v))w)i = (wi − wj) T (ui − uj)γ ′ (|xi − xj| 2 )(K −1 v)j j=1 N∑ + 2 (xi − xj) T (ui − uj)(xi − xj) T (wi − wj)γ ′′ (|xi − xj| 2 )(K −1 v)j j=1 N∑ + (xi − xj) T (ui − uj)γ ′ (|xi − xj| 2 )(( d dɛ K−1 )v)j j=1 N∑ + (wi − wj) T (vi − vj)γ ′ (|xi − xj| 2 )(K −1 u)j j=1 N∑ + 2 (xi − xj) T (vi − vj)(xi − xj) T (wi − wj)γ ′′ (|xi − xj| 2 )(K −1 u)j j=1 N∑ + (xi − xj) T (vi − vj)γ ′ (|xi − xj| 2 )(( d dɛ K−1 )u)j j=1 N∑ N∑ − 2 (xi − xj) T (wi − wj)γ ′ (|xi − xj| 2 ) (xj − xk)γ ′ (|xj − xk| 2 )((K −1 u) T k (K−1 v)j + (K −1 u) T j (K −1 v)k) j=1 k=1 N∑ N∑ − γ(|xi − xj| 2 ) (wj − wk)γ ′ (|xj − xk| 2 )((K −1 u) T k (K−1 v)j + (K −1 u) T j (K −1 v)k) j=1 k=1 N∑ N∑ − 2 γ(|xi − xj| 2 ) (xj − xk)(xj − xk) T (wj − wk)γ ′′ (|xj − xk| 2 )((K −1 u) T k (K−1 v)j + (K −1 u) T j (K −1 v)k) j=1 k=1 N∑ N∑ − γ(|xi − xj| 2 ) (xj − xk)γ ′ (|xj − xk| 2 ) j=1 k=1 × (( d dɛ K−1 u) T k (K−1 v)j + (K −1 u) T k ( d dɛ K−1 v)j + ( d dɛ K−1 u) T j (K −1 v)k + (K −1 u) T j ( d dɛ K−1 v)k) ( ( d ) dɛ K−1 )v = −(K −1 d i dɛ KK−1 v)i ∑ N = −2(K −1 (xk − xj) T (wk − wj)γ ′ (|xk − xj| 2 )(K −1 v)j j=1 <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 30
Landmark <strong>Regression</strong> Results Same Bookstein rat data. Procrustes alignment, no scaling. 0.3 0.2 0.1 0 −0.1 −0.2 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 0.36 0.34 0.32 0.34 0.32 0.3 0.3 0.28 0.28 0.26 0.26 0.24 0.24 0.2 0.25 0.3 0.35 0.4 0.45 0.22 −0.2 −0.18 −0.16 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 0 R 2 = 0.92 geodesic, 0.94 quadratic <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 31
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 and 22: Rewrite using integration by parts
Page 23 and 24: For any smooth family of curves γ
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: For curvature, need Christoffel sym
Page 47: Thank You!

Polynomial Regression on Riemannian Manifolds

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