Most tensor problems are NP-hard

TL;DR

This paper establishes that many fundamental tensor problems, including eigenvalue determination and rank approximation, are NP-hard, highlighting the computational difficulty of nonlinear tensor computations even in symmetric cases.

Contribution

It proves NP-hardness for a wide range of tensor problems, extending the understanding of computational complexity in multilinear algebra.

Findings

Determining feasibility of bilinear systems is NP-hard.

Deciding tensor eigenvalues and spectral norms is NP-hard.

Computing tensor rank and best rank-1 approximation is NP-hard.

Abstract

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list here includes: determining the feasibility of a system of bilinear equations, deciding whether a 3-tensor possesses a given eigenvalue, singular value, or spectral norm; approximating an eigenvalue, eigenvector, singular vector, or the spectral norm; and determining the rank or best rank-1 approximation of a 3-tensor. Furthermore, we show that restricting these problems to symmetric tensors does not alleviate their NP-hardness. We also explain how deciding nonnegative definiteness of a symmetric 4-tensor is NP-hard and how computing the combinatorial hyperdeterminant of a 4-tensor is NP-, #P-, and VNP-hard. We shall argue that our results provide another view of the boundary separating the computational tractability of linear/convex problems from the intractability of nonlinear/nonconvex ones.