On characteristic rank for matrix and tensor completion

TL;DR

This paper introduces the concept of characteristic rank as a unifying framework to analyze the fundamental properties of low-dimensional models like matrices and tensors in completion problems, focusing on noiseless scenarios.

Contribution

It proposes the characteristic rank framework for understanding identifiability in matrix and tensor completion, providing a new perspective on these problems.

Findings

Characteristic rank characterizes basic properties of low-dimensional models.

Framework applies to noiseless matrix and tensor completion.

Provides insights into identifiability of models from partial observations.

Abstract

In this lecture note, we discuss a fundamental concept, referred to as the {\it characteristic rank}, which suggests a general framework for characterizing the basic properties of various low-dimensional models used in signal processing. Below, we illustrate this framework using two examples: matrix and three-way tensor completion problems, and consider basic properties include identifiability of a matrix or tensor, given partial observations. In this note, we consider cases without observation noise to illustrate the principle.