# A Unified Theory for Tensor Ranks and its Application

In this paper, we present a unified theory for tensor ranks such that they are natural extension of matrix ranks. We present some axioms for tensor rank functions. The CP rank, the max-Tucker rank and the submax-Tucker rank are tensor rank functions. The CP rank is subadditive but not proper. The max-Tucker rank naturally arises from the Tucker decomposition. It is proper and subadditive, but not strongly proper. The submax-Tucker rank is also associated with the Tucker decomposition, but is a new tensor rank function. It is strongly proper but not subadditive. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. The CP rank, and the max-Tucker rank are not the smallest tensor rank function. We define the closure of a strongly proper tensor rank function, and show that it is also a strongly proper tensor rank function. A strongly proper tensor rank function is closed if it is equal to its closure. We show that the smallest tensor rank function is strongly proper and closed. Our theoretic analysis indicates that the submax-Tucker rank is a good choice for low rank tensor approximation and tensor completion. An application of the submax-Tucker rank is presented.

• 31 publications
• 12 publications
• 11 publications
04/22/2020

### A Tensor Rank Theory and Maximum Full Rank Subtensors

A matrix always has a full rank submatrix such that the rank of this mat...
04/22/2020

### A Tensor Rank Theory, Full Rank Tensors and The Sub-Full-Rank Property

A matrix always has a full rank submatrix such that the rank of this mat...
04/22/2020

### A Tensor Rank Theory and The Sub-Full-Rank Property

One fundamental property in matrix theory is that the rank of a matrix i...
10/16/2022

### Finding the smallest or largest element of a tensor from its low-rank factors

We consider the problem of finding the smallest or largest entry of a te...
04/22/2020

### A Tensor Rank Theory, Full-Rank Tensors and Base Subtensors

A matrix always has a full-rank submatrix such that the rank of this mat...
10/23/2019

### Deterministic tensor completion with hypergraph expanders

We provide a novel analysis of low rank tensor completion based on hyper...
03/31/2021

### Low-CP-rank Tensor Completion via Practical Regularization

Dimension reduction techniques are often used when the high-dimensional ...