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.
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