Georgia Institute of Technology College of Sciences

Constrained low rank approximation is a general framework for data analysis, which usually has the advantage of being simple, fast, scalable and domain general. One of the most known constrained low rank approximation method is nonnegative matrix factorization (NMF). This research studies the design and implementation of several variants of NMF for text, graph and hybrid data analytics. It will address challenges including solving new data analytics problems and improving the scalability of existing NMF algorithms. There are two major types of matrix representation of data: feature-data matrix and similarity matrix. Previous work showed successful application of standard NMF for feature-data matrix to areas such as text mining and image analysis, and Symmetric NMF (SymNMF) for similarity matrix to areas such as graph clustering and community detection. In this work, a divide-and-conquer strategy is applied to both methods to improve their time complexity from cubic growth with respect to the reduced low rank to linear growth, resulting in DC-NMF and HierSymNMF2 method. Extensive experiments on large scale real world data shows improved performance of these two methods.Furthermore, in this work NMF and SymNMF are combined into one formulation called JointNMF, to analyze hybrid data that contains both text content and connection structure information. Typical hybrid data where JointNMF can be applied includes paper/patent data where there are citation connections among content and email data where the sender/receipts relation is represented by a hypergraph and the email content is associated with hypergraph edges. An additional capability of the JointNMF is prediction of unknown network information which is illustrated using several real world problems such as citation recommendations of papers and activity/leader detection in organizations.The dissertation also includes brief discussions of relationship among different variants of NMF.

In this talk I will present an information theoretic approach to stochastic optimal control and inference that has advantages over classical methodologies and theories for decision making under uncertainty. The main idea is that there are certain connections between optimality principles in control and information theoretic inequalities in statistical physics that allow us to solve hard decision making problems in robotics, autonomous systems and beyond. There are essentially two different points of view of the same "thing" and these two different points of view overlap for a fairly general class of dynamical systems that undergo stochastic effects. I will also present a holistic view of autonomy that collapses planning, perception and control into one computational engine, and ask questions such as how organization and structure relates to computation and performance. The last part of my talk includes computational frameworks for uncertainty representation and suggests ways to incorporate these representations within decision making and control.

This is an intoductory talk for the currently using methods for certifying roots for system of equations. First we discuss about alpha-theory which was constructed by Smale and Shub, and explain how this theory could be modified in order to apply in actual problems. In this step, we point out that alpha theory is still restricted only into polynomial systems and polynomial-exponential systems. After that as a remedy for this problem, we will introduce an interval arithmetic, and the Krawczyk method. We will end the talk with a discussion about how these current methods could be used in more general setting.

For a graph G, a set of subtrees of G are edge-independent with root r ∈ V(G) if, for every vertex v ∈ V(G), the paths between v and r in each tree are edge-disjoint. A set of k such trees represent a set of redundant broadcasts from r which can withstand k-1 edge failures. It is easy to see that k-edge-connectivity is a necessary condition for the existence of a set of k edge-independent spanning trees for all possible roots. Itai and Rodeh have conjectured that this condition is also sufficient. This had previously been proven for k=2, 3. We prove the case k=4 using a decomposition of the graph similar to an ear decomposition. Joint work with Robin Thomas.