A homogeneous geometry of low-rank tensors

TL;DR

This paper investigates the geometric structure of low-rank tensor sets, establishing conditions for smooth homogeneous manifold structures and deriving efficient Riemannian metrics for these tensors.

Contribution

It introduces conditions under which low-rank tensor sets form smooth homogeneous manifolds and develops Riemannian metrics with complete geodesics for these structures.

Findings

Low-rank tensor sets are smooth homogeneous manifolds under certain conditions.

Riemannian metrics with complete geodesics are derived for these tensor manifolds.

The geometric framework facilitates efficient tensor computations.

Abstract

We consider sets of fixed CP, multilinear, and TT rank tensors, and derive conditions for when (the smooth parts of) these sets are smooth homogeneous manifolds. For CP and TT ranks, the conditions are essentially that the rank is sufficiently low. These homogeneous structures are then used to derive Riemannian metrics whose geodesics are both complete and efficient to compute.