Recovery algorithms for high-dimensional rank one tensors

TL;DR

This paper introduces a deterministic algorithm for recovering high-dimensional rank one tensors from function values, effectively addressing the curse of dimensionality in certain parameter regimes.

Contribution

It provides a less costly deterministic recovery algorithm and characterizes the problem's tractability across different parameter ranges.

Findings

Deterministic algorithms enable uniform recovery of rank one tensors.

The algorithm's cost is lower than previous randomized methods for certain parameters.

The paper fully characterizes the problem's tractability based on the parameter M.

Abstract

We present deterministic algorithms for the uniform recovery of $d$-variate rank one tensors from function values. These tensors are given as product of $d$ univariate functions whose $r$th weak derivative is bounded by $M$. The recovery problem is known to suffer from the curse of dimensionality for $M\geq 2^r r!$. For smaller $M$, a randomized algorithm is known which breaks the curse. We construct a deterministic algorithm which is even less costly. In fact, we completely characterize the tractability of this problem by three different ranges of the parameter $M$.