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Non-Isometric Manifold Learning Analysis and an Algorithm

Non-Isometric Manifold Learning Analysis and an Algorithm. Piotr Doll á r, Vincent Rabaud, Serge Belongie. University of California, San Diego. I. Motivation. Extend manifold learning to applications other than embedding

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Non-Isometric Manifold Learning Analysis and an Algorithm

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  1. Non-Isometric Manifold Learning Analysis and an Algorithm Piotr Dollár, Vincent Rabaud, Serge Belongie University of California, San Diego

  2. I. Motivation • Extend manifold learning to applications other than embedding • Establish notion of test error and generalization for manifold learning

  3. Linear Manifolds (subspaces) Typical operations: • project onto subspace • distance to subspace • distance between points • generalize to unseen regions

  4. Non-Linear Manifolds Desired operations: • project onto manifold • distance to manifold • geodesic distance • generalize to unseen regions

  5. II. Locally Smooth Manifold Learning (LSML) Represent manifold by its tangent space Non-local Manifold Tangent Learning [Bengio et al. NIPS05] Learning to Traverse Image Manifolds [Dollar et al. NIPS06]

  6. Learning the tangent space Data on d dim. manifold in D dim. space

  7. Learning the tangent space Learn function from point to tangent basis

  8. Loss function

  9. Optimization procedure

  10. III. Analyzing Manifold Learning Methods • Need evaluation methodology to: • objectively compare methods • control model complexity • extend to non-isometric manifolds

  11. Evaluation metric Applicable for manifolds that can be densely sampled

  12. Finite Sample Performance Performance: LSML > ISOMAP > MVU >> LLE Applicability: LSML >> LLE > MVU > ISOMAP

  13. Model Complexity All methods have at least one parameter: k Bias-Variance tradeoff

  14. LSML test error

  15. LSML test error

  16. IV. Using the Tangent Space • projection • manifold de-noising • geodesic distance • generalization

  17. Projection x' is the projection of x onto a manifold if it satisfies: gradient descent is performed after initializing the projection x' to some point on the manifold: tangents at projection matrix

  18. Manifold de-noising Goal: recover points on manifold ( ) from points corrupted by noise ( ) - original - de-noised

  19. Geodesic distance Shortest path: On manifold: gradient descent

  20. Geodesic distance  embedding of sparse and structured data can apply to non-isometric manifolds 

  21. Generalization Reconstruction within the support of training data Generalization beyond support of training data 

  22. Thank you!

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