We live in an increasingly data-driven world in which mathematical models are crucial for uncovering properties of systems from measured data. Graphical models are commonly used for capturing the relationships between the parameters of a system using graphs. Graphical models have applications in many areas, such as social sciences, linguistics, neuroscience, and biology. Learning graphical models is of fundamental importance in machine learning and statistics, and is often challenged by the fact that only a small number of samples are available. Several algorithms (such as Graphical Lasso) have been proposed to address this problem. Despite the popularity of graphical lasso, 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 other techniques for learning graphical models. However, it is not clear how reliable these methods are and which method should be used for each particular application. To address these problems, we will design a novel framework for generating synthetic data based on stochastic electrical circuits, and use it as a platform to assess the performance of various techniques. We will show that our platform can be used to first find the best algorithm and then identify the best model by optimally adjusting the controllable parameters of the algorithm. We will illustrate our results on fMRI data and uncover new properties of brain networks.
Somayeh Sojoudi is an Assistant Project Scientist at the University of California, Berkeley. She received her PhD degree in Control & Dynamical Systems from California Institute of Technology in 2013. She was an Assistant Research Scientist at New York University School of Medicine from 2013 to 2015. She has worked on several interdisciplinary problems in optimization, control theory, machine learning, data analytics, and power systems. Somayeh Sojoudi is an associate editor for the IEEE Transactions on Smart Grid. She is a co-recipient of the 2015 INFORMS Optimization Society Prize for Young Researchers and a co-recipient of the 2016 INFORMS ENRE Energy Best Publication Award. She is a co-author of a best student paper award finalist for the 53rd IEEE Conference on Decision and Control 2014.