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Sparse Approximate Solutions to Linear Systems
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Symmetric Tensors and Symmetric Tensor Rank
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IEEE Transactions on Visualization and Computer Graphics
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Communications of the ACM - The Status of the P versus NP Problem
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STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
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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.