This paper introduces AL$\ell_0$CORE, a new form of probabilistic non-negative tensor decomposition. AL$\ell_0$CORE is a Tucker decomposition where the number of non-zero elements (i.e., the $\ell_0$-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$\ell_0$CORE -- i.e., $allo$cated $\ell_0$-$co$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$\ell_0$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.

翻译：本文提出AL$\ell_0$CORE——一种新型概率非负张量分解方法。AL$\ell_0$CORE是一种Tucker分解，其中核心张量的非零元素数量（即$\ell_0$-范数）被约束为预设值$Q$，且该值远小于核心张量的大小。虽然用户指定总预算$Q$，但非零元素的位置和数值作为隐变量，在推理过程中被分配到核心张量中。因此，AL$\ell_0$CORE（即$allo$cated $\ell_0$-$co$nstrained $core$，分配式$\ell_0$约束核心）既享有CP分解的计算易处理性，又具备Tucker分解在定性上吸引人的隐结构。在一系列真实数据实验中，我们证明AL$\ell_0$CORE通常仅需完整核心的极小比例（例如约1%），即可达到与完整Tucker分解相同的结果，而计算成本仅相应降低至极小比例。