Quasioptimality of maximum-volume cross interpolation of tensors

TL;DR

This paper proves that maximum-volume cross interpolation in tensor train format achieves near-optimal accuracy with a controlled error factor, and introduces greedy algorithms with numerical validation.

Contribution

It establishes quasioptimality of maximum-volume cross interpolation for tensors and develops greedy algorithms with theoretical and numerical support.

Findings

Maximum-volume sets yield quasioptimal interpolation accuracy

Greedy algorithms effectively implement the interpolation method

Numerical experiments confirm the efficiency and accuracy of the proposed algorithms

Abstract

We consider a cross interpolation of high-dimensional arrays in the tensor train format. We prove that the maximum-volume choice of the interpolation sets provides the quasioptimal interpolation accuracy, that differs from the best possible accuracy by the factor which does not grow exponentially with dimension. For nested interpolation sets we prove the interpolation property and propose greedy cross interpolation algorithms. We justify the theoretical results and test the speed and accuracy of the proposed algorithm with convincing numerical experiments.