Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions

TL;DR

This paper proves that for high-dimensional antisymmetric tensor functions, the minimal number of terms needed in tensor product approximations grows exponentially, highlighting fundamental limitations in using low-rank TPFs for such problems.

Contribution

It establishes a rigorous exponential lower bound on the complexity of antisymmetric tensor product functions, including neural network parameterized ones, in high dimensions.

Findings

Minimum number of terms grows exponentially with dimension for antisymmetric TPFs.

Low-rank TPFs are fundamentally unsuitable for high-dimensional antisymmetric problems.

The proof links antisymmetric TPFs to antisymmetric tensors and their Canonical Polyadic rank.

Abstract

Tensor product function (TPF) approximations have been widely adopted in solving high-dimensional problems, such as partial differential equations and eigenvalue problems, achieving desirable accuracy with computational overhead that scales linearly with problem dimensions. However, recent studies have underscored the extraordinarily high computational cost of TPFs on quantum many-body problems, even for systems with as few as three particles. A key distinction in these problems is the antisymmetry requirement on the unknown functions. In the present work, we rigorously establish that the minimum number of involved terms for a class of TPFs to be exactly antisymmetric increases exponentially fast with the problem dimension. This class encompasses both traditionally discretized TPFs and the recent ones parameterized by neural networks. Our proof exploits the link between the antisymmetric TPFs in this class and the corresponding antisymmetric tensors and focuses on the Canonical Polyadic rank of the latter. As a result, our findings reveal that low-rank TPFs are fundamentally unsuitable for high-dimensional problems where antisymmetry is essential.