Entropic graphs: Applications Alfred O. Hero Dept. EECS, Dept BME, Dept. Statistics University of Michigan - Ann Arbor 1.Dimension reduction and pattern matching 2.Entropic graphs for manifold learning 3.Simulation studies 4.Applications to face and digit databases

1.Dimension Reduction and Pattern Matching 128x128 images of faces Different poses, illuminations, facial expressions The set of all face images evolve on a lower dimensional imbedded manifold in R^(16384)

Face Manifold

Classification on Face Manifold

Manifold Learning: What is it good for? Interpreting high dimensional data Discovery and exploitation of lower dimensional structure Deducing non-linear dependencies between populations Improving detection and classification performance Improving image compression performance

Background on Manifold Learning 1.Manifold intrinsic dimension estimation 1.Local KLE, Fukunaga, Olsen (1971) 2.Nearest neighbor algorithm, Pettis, Bailey, Jain, Dubes (1971) 3.Fractal measures, Camastra and Vinciarelli (2002) 4.Packing numbers, Kegl (2002) 2.Manifold Reconstruction 1.Isomap-MDS, Tenenbaum, de Silva, Langford (2000) 2.Locally Linear Embeddings (LLE), Roweiss, Saul (2000) 3.Laplacian eigenmaps (LE), Belkin, Niyogi (2002) 4.Hessian eigenmaps (HE), Grimes, Donoho (2003) 3.Characterization of sampling distributions on manifolds 1.Statistics of directional data, Watson (1956), Mardia (1972) 2.Data compression on 3D surfaces, Kolarov, Lynch (1997) 3.Statistics of shape, Kendall (1984), Kent, Mardia (2001)

A statistical sample Sampling distribution 2dim manifold Domain Sampling Embedding Sampling on a Domain Manifold Observed sample

Learning 3D Manifolds Ref: Tenenbaum&etal (2000) Sampling density fy = Uniform on manifold N=400N=800 Ref: Roweiss&etal (2000) Swiss RollS-Curve

Sampled S-curve What is shortest path between points A and B along manifold? A B Geodesic from A to B is shortest path Euclidean Path is poor approximation

Geodesic Graph Path Approximation A B Dykstra’s shortest path approximates geodesic k-NNG skeleton k=4

ISOMAP (PCA) Reconstruction Compute k-NN skeleton on observed sample Run Dykstra’s shortest path algorithm between all pairs of vertices of k-NN Generate Geodesic pairwise distance matrix approximation Perform MDS on Reconstruct sample in manifold domain

ISOMAP Convergence When domain mapping is an isometry, domain is open and convex, and true domain dimension d is known (de Silva&etal:2001): How to estimate d? How to estimate attributes of sampling density?

How to Estimate d? Landmark-ISOMAP residual curve For Abilene Netflow data set

2. Entropic Graphs in D-dimensional Euclidean space Euclidean MST with edge power weighting gamma: pairwise distance matrix over edge length matrix of spanning trees over Euclidean k-NNG with edge power weighting gamma: When obtain Geodesic MST

Example: Uniform Planar Sample

Example: MST on Planar Sample

Example: k-NNG on Planar Sample

Convergence of Euclidean MST Beardwood, Halton, Hammersley Theorem:

GMST Convergence Theorem Ref: Costa&Hero:TSP2003

k-NNG Convergence Theorem

Shrinkwrap Interpretation n=400 n=800 Dimension = “Shrinkage rate” as vary number of resampled points on M

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

3. Simulation Studies: Swiss Roll GMSTkNN K=4 n=400, f=Uniform on manifold

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

loglogLinear Fit to GMST Length

GMST Dimension and Entropy Estimates From LS fit find: Intrinsic dimension estimate Alpha-entropy estimate ( ) –Ground truth:

MST/kNN Comparisons n=800n=400 n=800n=400 MST kNN

Entropic Graphs on S2 Sphere in 3D n=500, f=Uniform on manifold GMSTkNN

k-NNG on Sphere S4 in 5D Histogram of resampled d-estimates of k-NNG N=1000 points uniformly distributed on S4 (sphere) in 5D k=7 for all algorithms kNN resampled 5 times Length regressed on 10 or 20 samples at end of mean length sequence 30 experiments performed ISOMAP always estimates d=5 Table of relative frequencies of correct d estimate n

Estimated entropy (n = 600) True Entropy kNN/GMST Comparisons

GMST4-NN kNN/GMST Comparisons for Uniform Hyperplane

Improve Performance by Bootstrap Resampling Main idea: Averaging of weak learners –Using fewer (N) samples per MST estimate, generate large number (M) of weak estimates of d and H –Reduce bias by averaging these estimates (M>>1,N=1) –Better than optimizing estimate of MST length (M=1,N>>1) Illustration of bootstrap resampling method: A,B: N=1 vs C: M=1

Table of relative frequencies of correct d estimate using the GMST, with (N = 1) and without (M = 1) bias correction. kNN/GMST Comparisons for Uniform Hyperplane

4. Application: ISOMAP Face Database Synthesized 3D face surface Computer generated images representing 700 different angles and illuminations Subsampled to 64 x 64 resolution (D=4096) Disagreement over intrinsic dimensionality –d=3 (Tenenbaum) vs d=4 (Kegl) Resampling Histogram of d hat Mean GMST Length Function d=3 H=21.1 bits Mean kNNG (k=7) length d=4 H=21.8 bits

Application: Yale Face Database Description of 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

Samples from Face database B

GMST for 3 Face Folios

Real valued intrinsic dimension estimates using 3-NN graph for face 1. Real valued intrinsic dimension estimates using 3-NN graph for face 2. Dimension Estimator Histograms for Face database B

Remarks on Yale Facebase B GMST LS estimation parameters –Local Geodesic approximation used to generate pairwise distance matrix –Estimates based on 25 resamplings over 18 largest folio sizes To represent any folio we might hope to attain –factor > 600 reduction in degrees of freedom (dim) –only 1/10 bit per pixel for compression –a practical parameterization/encoder?

Sample: MNIST Handwritten Digits Application: MNIST Digit Database

Estimated intrinsic dimension Histogram of intrinsic dimension estimates: GMST (left) and 5-NN (right) (M = 1, N = 10, Q = 15). MNIST Digit Database

ISOMAP (k = 6) residual variance plot. The digits database contains nonlinear transformations, such as width distortions of each digit, that are not adequately modeled by ISOMAP! MNIST Digit Database

Conclusions Entropic graphs give accurate global and consistent estimators of dimension and entropy Manifold learning and model reduction –LLE, LE, HE estimate d by finding local linear representation of manifold –Entropic graph estimates d from global resampling –Initialization of ISOMAP… with entropic graph estimator Computational considerations –GMST, kNN with pairwise distance matrix: O(E log E) –GMST with greedy neighborhood search: O(d n log n) –kNN with kdb tree partitioning: O(d n log n)

References A. O. Hero, B. Ma, O. Michel and J. D. Gorman, “Application of entropic graphs,” IEEE Signal Processing Magazine, Sept H. Neemuchwala, A.O. Hero and P. Carson, “Entropic graphs for image registration,” to appear in European Journal of Signal Processing, J. Costa and A. O. Hero, “Manifold learning with geodesic minimal spanning trees,” to appear in IEEE T- SP (Special Issue on Machine Learning), A. O. Hero, J. Costa and B. Ma, "Convergence rates of minimal graphs with random vertices," submitted to IEEE T-IT, March 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