Journal of the ACM· 2013Q1
Most Tensor Problems Are NP-Hard
- 1,146citations
- Q1SCImago
- 2013year
Short summary
Many fundamental problems involving tensors, the multidimensional generalization of matrices, are proven to be NP-hard, meaning they are computationally intractable for large inputs.
AI-generated from the title and abstract; the full text is not read.
Key points
- Multilinear (tensor) analogues of many numerical linear algebra problems are NP-hard.
- Problems proven hard include eigenvalue/singular value determination and approximation for 3-tensors.
- Restricting problems to symmetric tensors does not reduce their NP-hardness.
- Computing the combinatorial hyperdeterminant is NP-, #P-, and VNP-hard.
AI-generated from the title and abstract; the full text is not read.
Abstract
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.
The authors' abstract, as published at the source. Journal of the ACM, 2013 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Computational Mathematics
Computational MathematicsMathematics