TR 11-12:30 in 2-338
Instructor: Santosh Vempala (vempala@mit.edu)
The spectrum of a matrix (or a graph) captures many interesting properties in surprising ways. In this course, we will study spectral methods (i.e., methods based on eigenvalues and eigenvectors) in several contexts: computing a least-squares fit and its generalizations (find a set of low-dimensional subspaces that minimize the sum of squared distances to a given point set), clustering and partitioning, characterization of topological properties (a graph is planar iff the nullspace of a certain class of adjacency matrices has rank at most 3), the regularity lemma, learning mixtures of distributions, finding large cliques etc.. We will try to find and highlight the important common elements. We will pay special attention to the role of spectral projections (the representation in the space of the top $k$ singular vectors or in the nullspace). On the way, we will study the following fact (stated informally here) and its algorithmic consequences: any matrix has a "small" submatrix from which a "good" approximation to the spectral representation can be computed.
Students will present a paper and work on open problem related
to the topic of the course (possibly in groups of two).