# On the compressibility of tensors

## TL;DR

This paper develops three methodologies to bound tensor compressibility based on algebraic structure, smoothness, and displacement structure, explaining why many tensors in applied mathematics are compressible.

## Contribution

It introduces three new bounds on tensor compressibility that help explain the prevalence of compressible tensors in applied mathematics.

## Key findings

- Solution tensor for Poisson equation can be approximated with O(n (log n)^2 (log(1/ε))^2) degrees of freedom.
- Constructive bounds enable spectral solution of Poisson equation with O(n (log n)^3 (log(1/ε))^3) complexity.
- Bound methods relate tensor structure to compressibility, aiding efficient tensor approximations.

## Abstract

Tensors are often compressed by expressing them in low rank tensor formats. In this paper, we develop three methodologies that bound the compressibility of a tensor: (1) Algebraic structure, (2) Smoothness, and (3) Displacement structure. For each methodology, we derive bounds on storage costs that partially explain the abundance of compressible tensors in applied mathematics. For example, we show that the solution tensor $\mathcal{X} \in \mathbb{C}^{n \times n \times n}$ of a discretized Poisson equation $-\nabla^2 u =1$ on $[-1,1]^3$ with zero Dirichlet conditions can be approximated to a relative accuracy of $0<\epsilon<1$ in the Frobenius norm by a tensor in tensor-train format with $\mathcal{O}(n (\log n)^2 (\log(1/\epsilon))^2)$ degrees of freedom. As this bound is constructive, we are also able to solve this equation spectrally with $\mathcal{O}(n (\log n)^3 (\log(1/\epsilon))^3)$ complexity.