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Most tensor problems are NP-hard

Christopher Hillar, Lek-Heng Lim

TL;DR

The paper demonstrates that a broad class of multilinear tensor problems—including tensor eigenvalues, singular values, spectral norms, rank decompositions, and hyperdeterminants—are NP-hard, even when restricted to symmetric tensors. By constructing reductions from classical NP-hard problems like graph 3-colorability, the authors show that these problems remain hard in real, complex, and even certain finite-field settings, and they discuss related #P- and VNP-hardness results for eigenvector counting and hyperdeterminants. They also explore approximation limits, showing there is no PTAS for key tensor tasks like eigenvector approximation and spectral-norm evaluation unless P=NP, and they discuss the limits of computability, including undecidability results for certain bilinear and bivariate matrix-function problems. The work places tensor problems at a boundary between tractable linear/convex computations and intractable nonlinear/nonconvex problems, influencing both theoretical understanding and practical use of tensor methods. Overall, the article underscores the substantial computational obstacles in extending linear-algebraic tools to higher-order multilinear settings.

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.

Most tensor problems are NP-hard

TL;DR

The paper demonstrates that a broad class of multilinear tensor problems—including tensor eigenvalues, singular values, spectral norms, rank decompositions, and hyperdeterminants—are NP-hard, even when restricted to symmetric tensors. By constructing reductions from classical NP-hard problems like graph 3-colorability, the authors show that these problems remain hard in real, complex, and even certain finite-field settings, and they discuss related #P- and VNP-hardness results for eigenvector counting and hyperdeterminants. They also explore approximation limits, showing there is no PTAS for key tensor tasks like eigenvector approximation and spectral-norm evaluation unless P=NP, and they discuss the limits of computability, including undecidability results for certain bilinear and bivariate matrix-function problems. The work places tensor problems at a boundary between tractable linear/convex computations and intractable nonlinear/nonconvex problems, influencing both theoretical understanding and practical use of tensor methods. Overall, the article underscores the substantial computational obstacles in extending linear-algebraic tools to higher-order multilinear settings.

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.

Paper Structure

This paper contains 26 sections, 45 theorems, 93 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tensors
  3. Tensor Eigenvalue
  4. Computability and Complexity
  5. NP-hardness
  6. Approximation Schemes
  7. Tensor Singular Value, Hyperdeterminant, and Spectral Norm
  8. Tensor Rank
  9. Symmetric Tensors
  10. #P-hardness and VNP-hardness
  11. Intractable Matrix Problems
  12. Quantum Computers
  13. Finite Fields
  14. Quadratic feasibility is NP-hard
  15. Bilinear system is NP-hard
  16. ...and 11 more sections

Key Result

Theorem 1.3

Graph $3$-colorability is polynomially reducible to tensor $0$-eigenvalue over $\mathbb{R}$. Thus, deciding tensor eigenvalue over $\mathbb{R}$ is NP-hard.

Figures (2)

  • Figure 1: Simple graphs with six proper $3$-colorings (graph at the left) or none (graph at the right).
  • Figure 2: It is NP-hard to approximate a real eigenvector. Each colored circle above in the complex plane represents a pair of real numbers which are coordinates of a cube root of unity. If one could approximate an eigenvector of a rational tensor to within $\varepsilon = \frac{3}{4}$ in each real coordinate, then one would be able to properly color the vertices of a $3$-colorable graph $G$ (see Example \ref{['3OL_ex']}).

Theorems & Definitions (80)

  • Theorem 1.3
  • Example 1.4: Real tensor $0$-eigenvalue solves $3$-colorability
  • Theorem 1.5
  • Corollary 1.6
  • Theorem 1.7
  • Conjecture 1.9
  • Theorem 1.10
  • Theorem 1.11
  • Corollary 1.12
  • proof
  • ...and 70 more