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Thank you to the following supporters of ICASSP 2007

TUT-13: Manifold Learning: A Geometric Perspective on Machine Learning

Monday Afternoon, April 16
14:00 - 17:00
Room 323A

Presented by

Prof. Partha Niyogi, University of Chicago

Abstract

Increasingly, we face a number of a number of data analysis, information processing, and pattern recognition problems in very high dimensional spaces. One approach to modeling such data is to proceed with the assumption that natural data is generated by physical systems with few degrees of freedom. A consequence of this assumption is that the data would then lie on (or near) some low dimensional manifold embedded in this high dimensional space.

Manifold learning refers to the class of statistical inference and machine learning problems that arise in this geometric setting. These include problems of (i) dimensionality reduction, data representation, and clustering (ii) regression and pattern classification (iii) semi-supervised and transductive learning (iv) density estimation (v) inverse problems, operator inversion, signal estimation, and so on.

Manifold methods include a number of non-linear approaches to data analysis that exploit the geometric properties of the manifold on which the data is presumed to lie. These include algorithms like LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps, and their variants. In this tutorial, we will provide an introduction to this exciting new area of manifold learning and consider applications in speech and image processing that are of interest to the ICASSP community.

1. Introduction and Motivation: Why it is reasonable to model data as lying on a low dimensional manifold. Examples from real life.
2. A Brief Introduction to Basic Differential Geometry: Manifolds, Charts, Tangent Spaces, Gradients, Exponential Map, Metric Tensor, etc.
3. The Laplace Beltrami Operator on a Riemannian Manifold: The Implications of this operator and its eigenfunctions for clustering, classification, and regression problems.
4. Dimensionality Reduction and Data Representation: Discussion of LLE, ISOMAP, Laplacian Eigenmaps, Diffusion Maps
5. Semi-supervised Learning: How to use the geometry of data from unlabeled examples to boost accuracy of classifier constructed from labeled examples.
6. Clustering: Spectral Methods for clustering and related problems.
7. Harmonic Analysis and Generalized Fourier Series: Implications for speech and signal processing.
8. The heat kernel on a Riemannian manifold and its many implications. The use of the heat kernel to construct a Reproducing Kernel Hilbert Space on the manifold. The connection to Support Vector Machines (SVMs) on the manifold.
9. Miscellaneous examples: Related computational geometry problems such as volume computation, homology computation, and so on.

Target Audience

The tutorial is aimed at researchers and practitioners interested in machine learning theory and its applications in a variety of domains involving speech, image, text, genomic, and other data. It will be at the level of beginning graduate students in engineering, physics, and the mathematical sciences. No knowledge of geometry will be assumed. Knowledge of calculus, linear algebra, and basic probability is necessary.

Speaker Biography

Partha Niyogi is Professor of Computer Science and Statistics at The University of Chicago. He obtained his B.Tech. from IIT Delhi, and S.M. and Ph. D. from MIT in Electrical Engineering and Computer Science. He was a Research Fellow at MIT, and a Member of Technical Staff at Bell Laboratories, Lucent Technologies before joining The University of Chicago. His research interests are in statistical inference, machine learning, speech and signal processing, computational linguistics, and artificial intelligence. He has written two books and many journal and conference papers on these subjects. For more information, see http://www.cs.uchicago.edu/~niyogi.