On Characteristic Rank for Matrix and Tensor Completion [Lecture Notes]

In this lecture note, we discuss a fundamental concept, referred to as the characteristic rank, that suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing. We illustrate this framework through two examples-matrix and three-way tensor completion problems-and consider basic properties, including the identifiability of matrices and tensors, given partial observations. We consider cases without observation noise to illustrate the principle.