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# The Sylvester Graphical Lasso (SyGlasso)

*Proceedings of the Twenty Third International Conference on Artificial Intelligence and Statistics*, PMLR 108:1943-1953, 2020.

#### Abstract

This paper introduces the Sylvester graphical lasso (SyGlasso) that captures multiway dependencies present in tensor-valued data. The model is based on the Sylvester equation that defines a generative model. The proposed model complements the tensor graphical lasso (Greenewald et al., 2019) that imposes a Kronecker sum model for the inverse covariance matrix, by providing an alternative Kronecker sum model that is generative and interpretable. A nodewise regression approach is adopted for estimating the conditional independence relationships among variables. The statistical convergence of the method is established, and empirical studies are provided to demonstrate the recovery of meaningful conditional dependency graphs. We apply the SyGlasso to an electroencephalography (EEG) study to compare the brain connectivity of alcoholic and nonalcoholic subjects. We demonstrate that our model can simultaneously estimate both the brain connectivity and its temporal dependencies.