The problem of tensor completion is important to many areas such as computer vision, data analysis, signal processing, etc. Previously, a category of methods known as low-rank tensor completion has been proposed and developed, involving the enforcement of low-rank structures on completed tensors. While such methods have been constantly improved, none have previously considered exploiting the numerical properties of tensor elements. This work attempts to construct a new methodological framework called GCDTC (Generalized CP Decomposition Tensor Completion) based on these properties. In this newly introduced framework, the CP Decomposition is reformulated as a Maximum Likelihood Estimate (MLE) problem, and generalized via the introduction of differing loss functions. The generalized decomposition is subsequently applied to low-rank tensor completion. Such loss functions can also be easily adjusted to consider additional factors in completion, such as smoothness, standardization, etc. An example of nonnegative integer tensor decomposition via the Poisson CP Decomposition is given to demonstrate the new methodology's potentials. Through experimentation with real-life data, it is confirmed that this method could produce results superior to current state-of-the-art methodologies. It is expected that the proposed notion would inspire a new set of tensor completion methods based on the generalization of decompositions, thus contributing to related fields.

翻译：张量补全问题在计算机视觉、数据分析、信号处理等诸多领域具有重要意义。此前，一类被称为低秩张量补全的方法已被提出并发展，其核心在于对补全后的张量施加低秩约束。尽管这类方法不断得到改进，但此前尚未有研究考虑利用张量元素的数值特性。本文尝试基于这些特性构建一个名为GCDTC（广义CP分解张量补全）的新方法框架。在该新框架中，CP分解被重新表述为最大似然估计问题，并通过引入不同的损失函数实现广义化。随后，这种广义分解被应用于低秩张量补全。此类损失函数还可轻松调整以考虑补全中的额外因素，如平滑性、标准化等。本文以泊松CP分解为例展示了非负整数张量分解的实例，以论证新方法的潜力。通过真实数据实验，证实该方法能够产生优于当前最优方法的结果。预期所提出的概念将激发一系列基于分解广义化的新型张量补全方法，从而为相关领域做出贡献。