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Abstract:
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We apply a recently proposed theory of computational sufficiency for expo-fam estimators with invariant generators (Vu, 2018) to large classes of sparse multi-dimensional matrix-variate estimators. These classes include a convex relaxation of sparse SVD, the bigraphical lasso, and tensor graphical lasso. This provides both computational and methodological insights. On the methodological front, We show that these estimators share an exact reduction by generalized single linkage thresholding operators. For example, one consequence is that these procedures share a common set of knots in their regularization paths. On the computational front, our results generalize the exact thresholding phenomenon for the graphical lasso (Witten et al. 2012, Mazumder & Hastie, 2012). We show how to efficiently reduce the feasible set for the optimization problem to enable faster algorithms, and demonstrate manyfold reduction in the runtime by employing these insights.
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