Time integration of rank-constrained Tucker tensors

TL;DR

This paper introduces a new discrete time integrator for rank-constrained Tucker tensors, extending matrix methods to tensors, with proven properties and demonstrated effectiveness in quantum dynamics and tensor optimization.

Contribution

It extends the projector-splitting integrator from matrices to Tucker tensors, providing a robust, exact reconstruction method for time-dependent tensors of fixed rank.

Findings

The integrator inherits properties from the matrix case.

It reconstructs Tucker tensors exactly.

It is robust to small singular values.

Abstract

Dynamical low-rank approximation in the Tucker tensor format of given large time-dependent tensors and of tensor differential equations is the subject of this paper. In particular, a discrete time integration method for rank-constrained Tucker tensors is presented and analyzed. It extends the known projector-splitting integrator for dynamical low-rank approximation of matrices to Tucker tensors and is shown to inherit the same favorable properties. The integrator is based on iteratively applying the matrix projector-splitting integrator to tensor unfoldings but with inexact solution in a substep. It has the property that it reconstructs time-dependent Tucker tensors of the given rank exactly. The integrator is also shown to be robust to the presence of small singular values in the tensor unfoldings. Numerical examples with problems from quantum dynamics and tensor optimization methods illustrate our theoretical results.