Tensors, or multidimensional arrays, are extensions to vectors and matrices, containing data arranged in a structured fashion. In this work, we explore the problem of deriving missing data from incomplete tensors, known as tensor completion. A common method in this field is Low-Rank Tensor Completion (LRTC), which conducts completion by assuming that the structure of the completed tensor should be low-rank. We innovatively design a framework which leverages inherent numerical properties of tensor elements, which we term as numerical priors, in a generalizable way. Most existing works only exploit priors regarding relations between tensor elements, such as smoothness, standardization, etc. However, our main insight is that numerical priors can carry essential information as well. For instance, if it is known that the tensor elements will be nonnegative integers (as is the case for most tensor completion tasks in Computer Vision), exploiting this information could help the algorithm converge to more accurate values (as we achieve with our SPTC algorithm). Based on our insights, we design the first generalizable framework that leverages numerical priors, which we name the Generalized CP Decomposition Tensor Completion (GCDTC) Framework. We also create the Smooth Poisson Tensor Completion (SPTC) algorithm as an instantiation of GCDTC in the specific task of nonnegative tensor completion, achieving state-of-the-art results and thus verifying our methods’ effectiveness.