Abstract: This paper introduces AL$\ell_0$CORE, a new form of probabilistic non-negative tensor decomposition.AL$\ell_0$CORE is a Tucker decomposition that constrains the number of non-zero elements (i.e., the $\ell_0$-norm) of the core tensor to be at most $Q$.While the user dictates the total budget $Q$, the locations and values of the non-zero elements are latent variables allocated across the core tensor during inference.AL$\ell_0$CORE---i.e., allocated $\ell_0$-constrained core---thus enjoys both the computational tractability of canonical polyadic (CP) decomposition and the qualitatively appealing latent structure of Tucker.In a suite of real-data experiments, we demonstrate that AL$\ell_0$CORE typically requires only tiny fractions (e.g., 1\%) of the core to achieve the same results as Tucker at a correspondingly small fraction of the cost.