The project aims at developing new methods for dynamical system modeling, based on convex optimization formulations and recent algorithms for large-scale non-smooth optimization. A first component of the project addresses the estimation and topology selection of graphical models of time series. A graphical model provides a graph representation of relations between random variables, for example, conditional dependence. These relations can be translated into sparsity constraints on the parameters of the model. A fundamental challenge in the estimation of a graphical model is the selection of a sparse graph topology from observed data. The project aims at developing methods for sparse topology selection via non-smooth convex regularizations. The main application that motivates this work is connectivity analysis from functional magnetic resonance imaging time series. A second part is concerned with new methods for system identification based on convex algorithms for low-rank approximation of structured matrices. This work requires the formulation of system identification problems as constrained rank optimization problems and the development of large-scale algorithms for convex relaxations of the rank optimization problems.
The project combines techniques from optimization, system theory, and machine learning to address fundamental problems in the modeling of dynamical systems. Graphical models, an important topic in machine learning, are not widely studied in system identification. Conversely, system identification can provide tools for modeling dynamical aspects in machine learning problems. The use of convex formulations and fast first-order algorithms will enable an efficient solution of large instances in practical applications.
Software implementations of the algorithms developed in the project will be made freely available. The outcomes will be integrated in the graduate optimization sequence in the Electrical Engineering Department at UCLA, in particular an advanced course on large-scale optimization. Research opportunities will be offered to students via individual study courses and summer internships.