Abstract

The i.i.d. assumption in machine learning is endemic, but often flawed. Complex data sets exhibit partial correlations between both instances and features. A model specifying both types of correlation can have a number of parameters that scales quadratically with the number of features and data points. We introduce the bigraphical lasso, an estimator for precision matrices of matrix-normals based on the Cartesian product of graphs. A prominent product in spectral graph theory, this structure has appealing properties for regression, enhanced sparsity and interpretability. To deal with the parameter explosion we introduce `1 penalties and fit the model through a flip-flop algorithm that results in a linear number of lasso regressions. We demonstrate the performance of our approach with simulations and an example from the COIL image data set.