Most Tensor Problems Are NP-Hard

• Published in: Journal of the ACM Vol. 60; no. 6; pp. 1 - 39
• Main Authors: Hillar, Christopher J., Lim, Lek-Heng
• Format: Journal Article
• Language: English
• Published: New York, NY Association for Computing Machinery 01.11.2013
• Subjects: Algorithmics. Computability. Computer arithmetics; Algorithms; Applied sciences; Approximation; Computer science; control theory; systems; Eigenvalues; Exact sciences and technology; Linear algebra; Linear equations; Mathematical problems; Numerical analysis; Studies; Theoretical computing; Singularity; Bilinear system; symmetric tensors; hyperdeterminants; tensor spectral norm; Theory; Polynomial approximation; Undecidability; Modeling; Numerical multilinear algebra; Tensor calculus; Eigenvalue problem; VNP-hardness; Polynomial matrix; Numerical linear algebra; tensor eigenvalue; nonnegative definite tensors; Computational complexity; tensor rank; Polynomial time; NP-hardness; P-hardness; Algorithms; bivariate matrix polynomials; NP hard problem; tensor singular value; Best approximation; Feasibility; Matrix polynomial; system of multilinear equations; polynomial time approximation schemes
• ISSN: 0004-5411; 1557-735X
• DOI: 10.1145/2512329

We prove that multilinear (tensor) analogues of many efficiently computable problems in numerical linear algebra are NP-hard. Our list 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 is NP-, #P-, and VNP-hard.