# Fall 2017¶

We meet 5:30–7:00 in Evans room 1011.

## Thursday, September 28¶

One of the greatest advantages of representing data with graphs is access to generic algorithms for analytic tasks, such as clustering. In this talk I will describe some popular graph clustering algorithms, and explain why they are well-motivated from a theoretical perspective.

## Thursday, October 19¶

**Spectral Sparsification of Graphs**(

`slides`

)Many important properties of an undirected graph manifest themselves spectrally in the eigenvalues or quadratic forms of matrices related to the graph. For instance, the connectivity structure, electrical properties, and random walk behavior of a graph are determined by its Laplacian matrix. A spectral sparsifier of a graph G is a sparse graph H on the same set of vertices such that the Laplacians of H and G are close, so that H captures the spectral behavior of G while being much cheaper to store and perform computations on. We survey a line of work showing that spectral sparsifiers with constant degree exist for every graph and can be computed efficiently.

## Thursday, November 30¶

**Data-Driven Methods for Learning Sparse Graphical Models**

Learning models from data has a significant impact on many disciplines, including computer vision, medical imaging, social networks, neuroscience and signal processing. In the network inference problem, one may model the relationships between the network components through an underlying inverse covariance matrix. Learning this graphical model is often challenged by the fact that only a small number of samples are available. Despite the popularity of graphical lasso for solving this problem, there is not much known about the properties of this statistical method as an optimization algorithm. In this talk, we will develop new notions of sign-consistent matrices and inverse-consistent matrices to obtain key properties of graphical lasso. In particular, we will prove that although the complexity of solving graphical lasso is high, the sparsity pattern of its solution has a simple formula if a sparse graphical model is sought. Besides graphical lasso, there are several techniques for learning graphical models. We will design an optimization-based mathematical framework to study the performance of various techniques. We will illustrate our results in different case studies.