Learning Gaussian DAG Models without Condition Number Bounds

Gaussian Graphical Models (GGMs) have wide-ranging applications in machine learning and the natural and social sciences. In most of the settings in which they are applied, the number of observed samples is much smaller than the dimension and they are assumed to be sparse. While there are a variety of algorithms (e.g. Graphical Lasso, CLIME) that provably recover the graph structure with a logarithmic number of samples, to do so they require various assumptions on the well-conditioning of the precision matrix which preclude long-range dependencies, present in many settings of interest. Here we give the first fixed polynomial-time algorithms for learning attractive GGMs and walk-summable GGMs with a logarithmic number of samples and without any such assumptions. In particular, our algorithms can tolerate strong dependencies among the variables. Our result for structure recovery in walksummable GGMs is derived from a more general result for efficient sparse linear regression in walk-summable models without any norm dependencies. We complement our results with experiments showing that many existing algorithms fail even in some simple settings where there are long dependency chains. Our algorithms do not.