The AL${\mathrm{\ell }}_0$CORE Tensor Decomposition for Sparse Count Data

This paper introduces AL${\mathrm{\ell }}_0$CORE, a new form of probabilistic non-negative tensor decomposition. AL${\mathrm{\ell }}_0$CORE is a Tucker decomposition where the number of non-zero elements (i.e., the ${\mathrm{\ell }}_0$-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. AL${\mathrm{\ell }}_0$CORE – i.e., $allo$cated ${\mathrm{\ell }}_0$-$co$nstrained $core$– 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 AL${\mathrm{\ell }}_0$CORE 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.