A Tensor Rank Theory and Maximum Full Rank Subtensors

TL;DR

This paper develops a theoretical framework for tensor ranks, proposing axioms, defining proper tensor rank functions, and extending the max-full-rank-submatrix property from matrices to tensors, with applications to tensor decompositions.

Contribution

It introduces axioms for tensor rank functions, identifies proper tensor ranks, and extends the max-full-rank-submatrix property to tensors, establishing a foundational theory.

Findings

Proper tensor rank functions exist, including max-Tucker and submax-Tucker ranks.

The max-full-rank-subtensor property extends to certain tensor ranks.

A minimal tensor rank function with this property is identified.

Abstract

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.