Tensor network ranks

TL;DR

This paper introduces a generalized concept of tensor network ranks based on undirected graphs, showing that many high-rank objects can have low $G$-rank, which broadens the applicability of low-rank approximations.

Contribution

It proposes a new framework for tensor ranks using graph-based notions, unifying and extending classical rank concepts and tensor network states.

Findings

$G$-rank can be exponentially lower than classical rank.

Many tensors have low $G$-rank despite high classical rank.

The framework applies to functions, matrices, and tensors across disciplines.

Abstract

In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we will see are all the same object in this context) has \emph{low rank} may be ill-justified. There are many natural instances where the object in question has high rank with respect to the classical notions of rank: matrix rank, tensor rank, multilinear rank --- the latter two being the most straightforward generalizations of the former. To remedy this, we show that one may vastly expand these classical notions of ranks: Given any undirected graph $G$, there is a notion of $G$-rank associated with $G$, which provides us with as many different kinds of ranks as there are undirected graphs. In particular, the popular tensor network states in physics (e.g., \textsc{mps}, \textsc{ttns}, \textsc{peps}) may be regarded as functions of a specific $G$-rank for various choices of $G$. Among other things, we will see that a function, matrix, or tensor may have very high matrix, tensor, or multilinear rank and yet very low $G$-rank for some $G$. In fact the difference is in the orders of magnitudes and the gaps between $G$-ranks and these classical ranks are arbitrarily large for some important objects in computer science, mathematics, and physics. Furthermore, we show that there is a $G$ such that almost every tensor has $G$-rank exponentially lower than its rank or the dimension of its ambient space.