Most tensor problems are NP-hard

Christopher Hillar; Lek-Heng Lim

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arXiv · 0911.1393

Most tensor problems are NP-hard

Christopher Hillar·Lek-Heng Lim

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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.

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Christopher Hillar, Lek-Heng Lim. 2013-07-01. Most tensor problems are NP-hard. https://arxiv.org/abs/0911.1393

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