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Manifold Learning Using Geodesic Entropic Graphs

Manifold Learning Using Geodesic Entropic Graphs. Research supported in part by: ARO-DARPA MURI DAAD19-02-1-0262. Alfred O. Hero and Jose Costa Dept. EECS, Dept Biomed. Eng., Dept. Statistics University of Michigan - Ann Arbor hero@eecs.umich.edu http://www.eecs.umich.edu/~hero.

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Manifold Learning Using Geodesic Entropic Graphs

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  1. Manifold Learning Using Geodesic Entropic Graphs Research supported in part by: ARO-DARPA MURI DAAD19-02-1-0262 Alfred O. Hero and Jose Costa Dept. EECS, Dept Biomed. Eng., Dept. Statistics University of Michigan - Ann Arbor hero@eecs.umich.edu http://www.eecs.umich.edu/~hero • Manifold Learning and Dimension Reduction • Entropic Graphs • Examples

  2. 1.Dimension Reduction and Pattern Matching • 128x128 images of three vehicles over 1 deg increments of 360 deg azimuth at 0 deg elevation • The 3(360)=1080 images evolve on a lower dimensional imbedded manifold in R^(16384) HMMV Truck T62 Courtesy of Center for Imaging Science, JHU

  3. Land VehicleImage Manifold Quantities Of Interest Embediing (extrinsic) Dimension: D Manifold (intrinsic) Dimension: d Entropy:

  4. Sampling on a Domain Manifold Assumption: is a conformal mapping 2dim manifold Embedding Sampling distribution Sampling A statistical sample

  5. Background on Manifold Learning • Manifold intrinsic dimension estimation • Local KLE, Fukunaga, Olsen (1971) • Nearest neighbor algorithm, Pettis, Bailey, Jain, Dubes (1971) • Fractal measures, Camastra and Vinciarelli (2002) • Packing numbers, Kegl (2002) • Manifold Reconstruction • Isomap-MDS, Tenenbaum, de Silva, Langford (2000) • Locally Linear Embeddings (LLE), Roweiss, Saul (2000) • Laplacian eigenmaps (LE), Belkin, Niyogi (2002) • Hessian eigenmaps (HE), Grimes, Donoho (2003) • Characterization of sampling distributions on manifolds • Statistics of directional data, Watson (1956), Mardia (1972) • Data compression on 3D surfaces, Kolarov, Lynch (1997) • Statistics of shape, Kendall (1984), Kent, Mardia (2001)

  6. 2. Entropic GraphsA Planar Sample and its Euclidean MST

  7. MST and Geodesic MST • For a set of points in D-dimensional Euclidean space, the Euclidean MST with edge power weighting gamma is defined as • edge lengths of a spanning tree over • When pairwise distances are geodesic distances on obtain Geodesic MST • For dense samplings GMST length = MST length

  8. Convergence of Euclidean MST Beardwood, Halton, Hammersley Theorem:

  9. Convergence Theorem for GMST Ref: Costa&Hero:TSP2003

  10. Special Cases • Isometric embedding ( distance preserving) • Conformal embedding ( angle preserving)

  11. Joint Estimation Algorithm • Convergence theorem suggests log-linear model • Use bootstrap resampling to estimate mean MST length and apply LS to jointly estimate slope and intercept from sequence • Extract d and H from slope and intercept

  12. 3. ExamplesRandom Samples on the Swiss Roll • Ref: Tenenbaum&etal (2000)

  13. Bootstrap Estimates of GMST Length Bootstrap SE bar (83% CI)

  14. loglogLinear Fit to GMST Length

  15. Dimension and Entropy Estimates • From LS fit find: • Intrinsic dimension estimate • Alpha-entropy estimate ( ) • Ground truth:

  16. Dimension Estimation Comparisons

  17. Application to Faces • Yale face database 2 • Photographic folios of many people’s faces • Each face folio contains images at 585 different illumination/pose conditions • Subsampled to 64 by 64 pixels (4096 extrinsic dimensions) • Objective: determine intrinsic dimension and entropy of a typical face folio

  18. GMST for 3 Face Folios Ref: Costa&Hero 2003

  19. Conclusions Advantages of Geodesic Entropic Graph Methods • Characterizing high dimension sampling distributions • Standard techniques (histogram, density estimation) fail due to curse of dimensionality • Entropic graphs can be used to construct consistent estimators of entropy and information divergence • Robustification to outliers via pruning • Manifold learning and model reduction • LLE, LE, HE estimate d by finding local linear representation of manifold • Entropic graph estimates d from global resampling • Computational complexity of MST is only n log n

  20. References • A. O. Hero, B. Ma, O. Michel and J. D. Gorman, “Application of entropic graphs,” IEEE Signal Processing Magazine, Sept 2002. • H. Neemuchwala, A.O. Hero and P. Carson, “Entropic graphs for image registration,” to appear in European Journal of Signal Processing, 2003. • J. Costa and A. O. Hero, “Manifold learning with geodesic minimal spanning trees,” accepted in IEEE T-SP (Special Issue on Machine Learning), 2004. • A. O. Hero, J. Costa and B. Ma, "Convergence rates of minimal graphs with random vertices," submitted to IEEE T-IT, March 2001. • J. Costa, A. O. Hero and C. Vignat, "On solutions to multivariate maximum alpha-entropy Problems", in Energy Minimization Methods in Computer Vision and Pattern Recognition (EMM-CVPR), Eds. M. Figueiredo, R. Rangagaran, J. Zerubia, Springer-Verlag, 2003

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