Manifold learning and pattern matching with entropic graphs

Alfred O. Hero Dept. EECS, Dept Biomed. Eng., Dept. Statistics

University of Michigan - Ann Arbor hero@eecs.umich.edu

http://www.eecs.umich.edu/~hero

Multimodality Face Matching

Clustering Gene Microarray Data

Cy5/Cy3 hybridization profiles

Image Registration

Vehicle Classification

• 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)

Courtesy of Center for Imaging Science, JHU

HMMVT62Truck

Image Manifold

What is manifold learning 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

Random Sampling on a Manifold

Classifying on a Manifold

Class A Class B

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)– Statistics of shape, Kendall (1984), Kent, Mardia (2001)– Data compression on 3D surfaces, Kolarov, Lynch (1997)

Assumption:

is a conformal mappingA statistical sample

Sampling distribution

2D manifold

Sampling

Embedding

Sampling on a Domain Manifold

Alpha-Entropy and Divergence• Alpha-entropy

• Alpha-divergence

• Other alpha-dissimilarity measures– Alpha-Jensen difference– Alpha geometric-arithmetic (GA) divergence

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

• pairwise distance matrix of complete graph

• When the matrix is constructed from geodesic distances between points on , e.g. using ISOMAP, we obtain the Geodesic MST

A Planar Sample and its Euclidean MST

Convergence of Euclidean MST

Beardwood, Halton, Hammersley Theorem:

Key Result for GMST

Ref: Costa&Hero:TSP2003

Special Cases

• Isometric embedding (ISOMAP)

• Conformal embedding (C-ISOMAP)

Remarks

• Result holds for many other combinatorial optimization algorithms (Costa&Hero:2003)– K-NNG– Steiner trees– Minimal matchings– Traveling Salesman Tours

• a.s. convergence rates (Hero&etal:2002)• For isometric embeddings Jacobian does not

have to be estimated for dimension estimation

Joint Estimation Algorithm

• Assume large-n log-affine 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

Random Samples on a Swiss Roll

• Ref: Grimes and Donoho (2003)

Bootstrap Estimates of GMST Length

785 790 795 800805

806

807

808

809

810

811

812

813

814

815

n

E[L

n]

Segment n=786:799 of MST sequence (=1,m=10) for unif sampled Swiss Roll

loglogLinear Fit to GMST Length

6.665 6.67 6.675 6.68 6.6856.692

6.694

6.696

6.698

6.7

6.702

6.704Segment of logMST sequence (=1,m=10) for unif sampled Swiss Roll

log(n)

log

(E[L

n])

y = 0.53*x + 3.2

log(E[Ln])

LS fit

Dimension and Entropy Estimates

• From LS fit find:

• Intrinsic dimension estimate

• Alpha-entropy estimate (nats)

Dimension Estimation Comparisons

Practical Application

• 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 face folio

GMST for 3 Face Folios

GMST for 3 Face Folios

Yale Face Database Results

• GMST LS estimation parameters– ISOMAP used to generate pairwise distance matrix– LS based on 25 resamplings over 26 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?

Ref: Costa&Hero 2003

Conclusions

• 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– Standard techniques (LLE, MDS, LE, HE) rely on local

linear fits – Entropic graph methods fit the manifold globally– Computational complexity is only n log n

Advantages of Geodesic Entropic Graph Methods

Summary of Algorithm

• Run ISOMAP or C-ISOMAP algorithm to generate pairwise distance matrix on intrinsic domain of manifold

• Build geodesic entropic graph from pairwise distance matrix– MST: consistent estimator of manifold dimension and

process alpha-entropy– K-NNG: consistent estimator of information divergence

between labeled vectors• Use bootstrap resampling and LS fitting to extract

rate of convergence (intrinsic dimension) and convergence factor (entropy) of entropic graph

Swiss Roll Example

Uniform Samples on 3D Imbedding of Swiss Roll

Geodesic Minimal Spanning Tree

GMST over Uniform Samples on Swiss Roll

Geodesic MST on Imbedded Mixture

GMST on Gaussian Samples on Swiss Roll

Classifying on a Manifold

Class A Class B