# 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 matrix is equal to the rank of that submatrix. We call this property the sub-full-rank property. This property is one of the corner stones of the matrix rank theory. Tensor ranks play a crucial role in low rank tensor approximation, tensor completion and tensor recovery. However, their theory is still not matured yet. Can we set an axiom system for tensor ranks? Can we extend the full rank concept and the sub-full-rank property to tensors? We explore these in this paper. We first propose some axioms for tensor rank functions. Then we introduce proper tensor rank functions. The CP rank is a tensor rank function, but is not proper. There are two proper tensor rank functions associated with the Tucker decomposition, the max-Tucker rank and the submax-Tucker rank. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the concept of full rank tensors, and show that the max-Tucker tensor rank function and the smallest tensor rank function have the sub-full-rank property. We define the closure for a proper tensor rank function, and show that the closure of any proper tensor rank function is a proper tensor rank function with the sub-full-rank property. An application of the submax-Tucker rank is also presented.

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 Base Subtensors

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...
04/22/2020

### A Unified Theory for Tensor Ranks and its Application

In this paper, we present a unified theory for tensor ranks such that th...
06/23/2023

### An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank

An approximate projection onto the tangent cone to the variety of third-...
05/04/2020

### A Solution for Large Scale Nonlinear Regression with High Rank and Degree at Constant Memory Complexity via Latent Tensor Reconstruction

This paper proposes a novel method for learning highly nonlinear, multiv...
06/11/2020

### Tensor-Based Modulation for Unsourced Massive Random Access

We introduce a modulation for unsourced massive random access whereby th...