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Regularized Least-Squares

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Outline Why regularization? Truncated Singular Value Decomposition Damped least-squares Quadratic constraints

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Regularized Least-Squares Why regularization? We have seen that

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Regularized Least-Squares Why regularization? We have seen that But what happens if the system is almost dependent? –The solution becomes very sensitive to the data –Poor conditioning

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Regularized Least-Squares The 1-dimensional case The 1-dimensional normal equation

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Regularized Least-Squares The 1-dimensional case The 1-dimensional normal equation

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Regularized Least-Squares The 1-dimensional case The 1-dimensional normal equation

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Regularized Least-Squares Why regularization Contradiction between data and model

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Regularized Least-Squares A more interesting example: scattered data interpolation

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Regularized Least-Squares “True” curve

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Regularized Least-Squares Radial basis functions

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Regularized Least-Squares Radial basis functions

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Regularized Least-Squares Rbf are popular Modeling –J. C. Carr, R. K. Beatson, J. B. Cherrie, T. J. Mitchell,W. R. Fright, B. C. McCallum, and T. R. Evans. Reconstruction and representation of 3d objects with radial basis functions. In Proceedings of ACM SIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pages 67–76, August 2001. –G. Turk and J. F. O’Brien. Modelling with implicit surfaces that interpolate. ACM Transactions on Graphics, 21(4):855–873, October 2002. Animation –J. Noh and U. Neumann. Expression cloning. In Proceedings of ACMSIGGRAPH 2001, Computer Graphics Proceedings, Annual Conference Series, pages 277–288, August 2001. –F. Pighin, J. Hecker, D. Lischinski, R. Szeliski, and D. H. Salesin. Synthesizing realistic facial expressions from photographs. In Proceedings of SIGGRAPH 98, Computer Graphics Proceedings, Annual Conference Series, pages 75–84, July 1998.

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Regularized Least-Squares Radial basis functions At every point

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Regularized Least-Squares Radial basis functions At every point Solve the least-squares problem

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Regularized Least-Squares Radial basis functions At every point Solve the least-squares problem

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Regularized Least-Squares Rbf results

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Regularized Least-Squares p i 0 close to p i 1

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Regularized Least-Squares Radial basis functions At every point Solve the least-squares problem

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Regularized Least-Squares Radial basis functions At every point Solve the least-squares problem If p i 0 close to p i 1, A is near singular

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Regularized Least-Squares p i 0 close to p i 1

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Regularized Least-Squares p i 0 close to p i 1

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Regularized Least-Squares Rbf results with noise

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Regularized Least-Squares Rbf results with noise

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Regularized Least-Squares The Singular Value Decomposition Every matrix A ( nxm ) can be decomposed into: –where U is an nxn orthogonal matrix V is an mxm orthogonal matrix D is an nxm diagonal matrix

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Regularized Least-Squares The Singular Value Decomposition Every matrix A ( nxm ) can be decomposed into: –where U is an nxn orthogonal matrix V is an mxm orthogonal matrix D is an nxm diagonal matrix

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Regularized Least-Squares Geometric interpretation

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Regularized Least-Squares Solving with the SVD

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Regularized Least-Squares Solving with the SVD

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Regularized Least-Squares Solving with the SVD

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Regularized Least-Squares Solving with the SVD

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Regularized Least-Squares Solving with the SVD

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Regularized Least-Squares A is nearly rank defficient

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Regularized Least-Squares A is nearly rank defficient

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Regularized Least-Squares A is nearly rank defficient

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Regularized Least-Squares A is nearly rank defficient =>some of the are close to 0

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Regularized Least-Squares A is nearly rank defficient =>some of the are close to 0 =>some of the are close to

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Regularized Least-Squares A is nearly rank defficient =>some of the are close to 0 =>some of the are close to Problem with

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Regularized Least-Squares A is nearly rank defficient =>some of the are close to 0 =>some of the are close to Problem with Truncate the SVD

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Regularized Least-Squares p i 0 close to p i 1

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Regularized Least-Squares Rbf fit with truncated SVD

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Regularized Least-Squares Rbf results with noise

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Regularized Least-Squares Rbf fit with truncated SVD

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Regularized Least-Squares Choosing cutoff value k The first k such as

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning ?

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning Inverse skinning –Let be a set of pairs of geometry and skeleton configurations

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning Inverse skinning –Let be a set of pairs of geometry and skeleton configurations

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning Inverse skinning –Let be a set of pairs of geometry and skeleton configurations

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Regularized Least-Squares Example: inverse skinning “Skinning Mesh Animations”, James and Twigg, siggraph Skinning Inverse skinning –Let be a set of pairs of geometry and skeleton configurations

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Regularized Least-Squares “Skinning Mesh Animations”, James and Twigg, siggraph

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Regularized Least-Squares Problem with the TSVD We have to compute the SVD of A, and O() process: impractical for large marices Little control over regularization

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Regularized Least-Squares Damped least-squares Replace by where is a scalar and L is a matrix

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Regularized Least-Squares Damped least-squares Replace by where is a scalar and L is a matrix The solution verifies

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Regularized Least-Squares Examples of L DiagonalDifferential Limit scaleEnforce smoothness

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Regularized Least-Squares Rbf results with noise

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Regularized Least-Squares

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Example: “Least-Squares Meshes”, Sorkin and Cohen-Or, siggaph Reconstruct a mesh given –Control points –Connectivity (planar mesh)

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Regularized Least-Squares Example: “Least-Squares Meshes”, Sorkin and Cohen-Or, siggaph Reconstruct a mesh given –Control points –Connectivity (planar mesh) Smooth reconstruction

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Regularized Least-Squares Example: “Least-Squares Meshes”, Sorkin and Cohen-Or, siggaph Reconstruct a mesh given –Control points –Connectivity (planar mesh) Smooth reconstruction In matrix form

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Regularized Least-Squares Reconstruction Minimize reconstruction error where

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Regularized Least-Squares “Least-Squares Meshes”, Sorkine and Cohen-Or, siggraph

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Regularized Least-Squares Quadratic constraints Solve or

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Regularized Least-Squares Quadratic constraints Solve or

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Regularized Least-Squares Example

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Regularized Least-Squares Example

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Regularized Least-Squares Example

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Regularized Least-Squares Discussion If, there is no solution (since there is no x for which )

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Regularized Least-Squares Discussion If, there is no solution (since there is no x for which ) If, the solution exists and is unique

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Regularized Least-Squares Discussion If, there is no solution (since there is no x for which ) If, the solution exists and is unique –Either the solution of is in the feasible set

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Regularized Least-Squares Discussion If, there is no solution (since there is no x for which ) If, the solution exists and is unique –Either the solution of is in the feasible set –or the solution is at the boundary Solve

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Regularized Least-Squares Discussion Solve where is a Lagrange multiplier

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Regularized Least-Squares Conclusion TSVD really useful if you need an SVD

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Regularized Least-Squares Conclusion TSVD really useful if you need an SVD Regularization constrains the solution: –Value, differential operator, other properties –Soft (damping) or hard constraint (quadratic) –Linear or non-linear

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Regularized Least-Squares Conclusion TSVD really useful if you need an SVD Regularization constrains the solution: –Value, differential operator, other properties –Soft (damping) or hard constraint (quadratic) –Linear or non-linear Danger of over-damping or constraining

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Regularized Least-Squares Example: inverse kinematic Problem: solve for joint angles given end-effector positions

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Regularized Least-Squares Example: inverse kinematic Problem: solve for joint angles given end-effector positions ?

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Regularized Least-Squares Example: inverse kinematic Problem: solve for joint angles given end-effector positions

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Regularized Least-Squares Example: inverse kinematic Problem: solve for joint angles given end-effector positions

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