Manifold Learning Techniques for Signal and Visual Processing Lecture series by Radu HORAUD, INRIA Grenoble Rhone-Alpes Spring (March-May) 2013

This 3D shape (left) is represented by an undirected weighted graph whose vertices correspond to 3D points and whose edges correspond to the local topology of the shape. Each graph vertex is projected onto the first non-null eigenvector of the graph's Laplacian matrix. This eigenvector (shown as a straight line) may be viewed as the intrinsic principal direction of the shape, invariant to shape deformations such as articulated motion.

Lecture #1: Introduction to linear and non-linear dimensionality reduction.

Brief overview of spectral and graph-based methods, such as principal component analysis (PCA), multi dimensional scaling (MDS), ISOMAP, LLE, Laplacian embedding, etc.

Further readings: L. Saul et al. Spectral Methods for Dimensionality Reduction. O. Chapelle, B. Schoelkopf, and A. Zien (eds.), Semisupervised Learning, pages 293-308. MIT Press: Cambridge, MA.

Lecture #2: Properties of symmetric matrices.

Eigenvalues and eigenvectors. Practical computation for dense ans sparse matrices. Covariance matrix. Gram matrix.

Lecture #3: Graphs, graph matrices and spectral embeddings of graphs.

What is the matrix of a graph? Properties of graph matrices. Spectral graph theory. Undirected weighted graphs and Markov chains. Graph distances. Graph construction.

Lecture #4: Laplacian embedding.

Spectral clustering. Semi-supervised spectral clustering. Links with other methods: Normalized cuts and random walks on graphs. Image and shape segmentation

Lecture #5: An introduction to kernel methods

Properties of kernels. Kernel PCA.

Lecture #6: Graph kernels and the heat kernel.

Lecture #7: Wavelets on graphs

Lecture #8: The Gaussian distribution, Gaussian mixtures, and the EM algorithm.

The univariate and multivariate Gaussian distributions. The Gaussian mixture model. The expectaction-maximization algorithm for Gaussian mixtures. The curse of dimensionality.

Lecture #9: Probabilistic PCA.

EM algorithm for PPCA. Bayesian PCA and model selection. Factor analysis.

Lecture #10: Gaussian processes

A short introduction to Gaussian processes for regression, lecture entirely based on Bishop's book.