Polynomial Regression on Riemannian Manifolds

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<strong>Polynomial</strong> <strong>Regression</strong> 0.04 0.035 0.03 0.025 Dependent Variable 0.02 0.015 0.01 0.005 0 −0.005 −0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Independent Variable <strong>Polynomial</strong>s provide a more flexible framework for parametric regression on Riemannian manifolds <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 4
Riemannian <strong>Polynomial</strong>s At least three ways to define polynomial in R d Algebraic: γ(t) = c 0 + 1 1! c 1t + 1 2! c 2t 2 + · · · + 1 k! c kt k <strong>Polynomial</strong> <strong>Regression</strong> on Riemannian Manifolds 5
Page 1 and 2: Polynomial <strong
Page 3: Parametric Regression</stro
Page 7 and 8: 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 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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