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Regularized Least-Squares

Regularized Least-Squares. Outline. Why regularization? Truncated Singular Value Decomposition Damped least-squares Quadratic constraints. Why regularization?. We have seen that. Why regularization?. We have seen that But what happens if the system is almost dependent?

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Regularized Least-Squares

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  1. Regularized Least-Squares Regularized Least-Squares

  2. Outline • Why regularization? • Truncated Singular Value Decomposition • Damped least-squares • Quadratic constraints Regularized Least-Squares

  3. Why regularization? • We have seen that Regularized Least-Squares

  4. 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 Regularized Least-Squares

  5. The 1-dimensional case • The 1-dimensional normal equation Regularized Least-Squares

  6. The 1-dimensional case • The 1-dimensional normal equation Regularized Least-Squares

  7. The 1-dimensional case • The 1-dimensional normal equation Regularized Least-Squares

  8. Why regularization • Contradiction between data and model Regularized Least-Squares

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

  10. “True” curve Regularized Least-Squares

  11. Radial basis functions Regularized Least-Squares

  12. Radial basis functions Regularized Least-Squares

  13. 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. Regularized Least-Squares

  14. Radial basis functions • At every point Regularized Least-Squares

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

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

  17. Rbf results Regularized Least-Squares

  18. pi0 close topi1 Regularized Least-Squares

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

  20. Radial basis functions • At every point • Solve the least-squares problem • If pi0 close topi1, A is near singular Regularized Least-Squares

  21. pi0 close topi1 Regularized Least-Squares

  22. pi0 close topi1 Regularized Least-Squares

  23. Rbf results with noise Regularized Least-Squares

  24. Rbf results with noise Regularized Least-Squares

  25. 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 Regularized Least-Squares

  26. 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 Regularized Least-Squares

  27. Geometric interpretation Regularized Least-Squares

  28. Solving with the SVD Regularized Least-Squares

  29. Solving with the SVD Regularized Least-Squares

  30. Solving with the SVD Regularized Least-Squares

  31. Solving with the SVD Regularized Least-Squares

  32. Solving with the SVD Regularized Least-Squares

  33. A is nearly rank defficient Regularized Least-Squares

  34. A is nearly rank defficient Regularized Least-Squares

  35. A is nearly rank defficient Regularized Least-Squares

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

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

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

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

  40. pi0 close topi1 Regularized Least-Squares

  41. Rbf fit with truncated SVD Regularized Least-Squares

  42. Rbf results with noise Regularized Least-Squares

  43. Rbf fit with truncated SVD Regularized Least-Squares

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

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

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

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

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

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

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

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