DeepAI AI Chat
Log In Sign Up

A Tensor Rank Theory and Maximum Full Rank Subtensors

04/22/2020
by   Liqun Qi, et al.
South China Normal University
Tianjin University
The Hong Kong Polytechnic University
0

A matrix always has a full rank submatrix such that the rank of this matrix is equal to the rank of that submatrix. This property is one of the corner stones of the matrix rank theory. We call this property the max-full-rank-submatrix property. 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 max-full-rank-submatrix 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, the max-Tucker rank and the submax-Tucker rank, which are associated with the Tucker decomposition. We define a partial order among tensor rank functions and show that there exists a unique smallest tensor rank function. We introduce the full rank tensor concept, and define the max-full-rank-subtensor property. We show the max-Tucker tensor rank function and the smallest tensor rank function have this property. We define the closure for an arbitrary proper tensor rank function, and show that it is still a proper tensor rank function and has the max-full-rank-subtensor property. An application of the submax-Tucker rank is also presented.

READ FULL TEXT
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 Unified Theory for Tensor Ranks and its Application

In this paper, we present a unified theory for tensor ranks such that th...
04/13/2023

Solving Tensor Low Cycle Rank Approximation

Large language models have become ubiquitous in modern life, finding app...
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...
12/01/2021

Lévy copulas: a probabilistic point of view

There is a one-to-one correspondence between Lévy copulas and proper cop...
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 Tensor Rank Theory, Full-Rank Tensors and Base Subtensors

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