This paper introduces ALL0CORE, a new form of probabilistic non-negative tensor decomposition. ALL0CORE is a Tucker decomposition where the number of non-zero elements (i.e., the L0-norm) of the core tensor is constrained to a preset value Q much smaller than the size of the core. While the user dictates the total budget Q, the locations and values of the non-zero elements are latent variables and allocated across the core tensor during inference. ALL0CORE -- i.e., allocated L0-constrained core -- thus enjoys both the computational tractability of CP decomposition and the qualitatively appealing latent structure of Tucker. In a suite of real-data experiments, we demonstrate that ALL0CORE typically requires only tiny fractions (e.g.,~1%) of the full core to achieve the same results as full Tucker decomposition at only a correspondingly tiny fraction of the cost.

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