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.

/pdf/most-tensor-problems-are-np-hard-cqgnykhb6t.pdf

Most Tensor Problems Are NP-Hard

Whitfield Diffie and Martin Hellman startled the computer security world in 1975 with their paper “New Directions in Cryptography”, which introduced public-key cryptography [7]. There is now evidence that three cryptographers at the British Government Communications Headquarters (GCHQ), the British equivalent of the National Security Agency, may have predated this work as well as the discovery of the Rivest-Shamir-Adleman (RSA) public-key encryption algorithm [9]. It appears, however, that the British researchers did not appreciate the significance of their discovery. Since the mid 1970s, cryptography has become big business, a bestseller (Applied Cryptography by Bruce Schneier has sold over two hundred thousand copies), and an extremely active area of research at the intersection of mathematics and computer science. A plethora of new books in the area currently floods the bookstores. This review cannot discuss them all. Instead I will delineate the subareas of this burgeoning field and focus on some of the texts that I believe will be of more interest to Bulletin readers.

The Yamabe problem

Introduction to lie algebras and representation theory

Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

In this window, all groups are assumed finite. Here we collect a number of results that play a significant role in the book (further material of an elementary nature that we sometimes take for granted is easily available in textbooks such as [H], [R] and [A]).

Finite Group Theory

Prelude Part I. Pro-p Groups: 1. Profinite groups and pro-p groups 2. Powerful p-groups 3. Pro-p groups of finite rank 4. Uniformly powerful groups 5. Automorphism groups Interlude A. Fascicule de resultats: pro-p groups of finite rank Part II. Analytic Groups: 6. Normed algebras 7. The group algebra Interlude B. Linearity criteria 8. P-adic analytic groups Interlude C. Finitely generated groups, p-adic analytic groups and Poincare series 9. Lie theory Part III. Further Topics: 10. Pro-p groups of finite co-class 11. Dimension subgroup methods 12. Some graded algebras Interlude D. The Golod Shafarevic inequality Interlude E. Groups of sub-exponential growth 13. Analytic groups over pro-p rings.

Analytic Pro-P Groups

Subgroups of finite index in nilpotent groups

We prove that in every finitely generated profinite group, every subgroup of finite index is open; this implies that the topology on such groups is determined by the algebraic structure. This is deduced from the main result about finite groups: let w be a 'locally finite' group word and d ∈ N. Then there exists f = f(w, d) such that in every d-generator finite group G, every element of the verbal subgroup w(G) is equal to a product of f w-values. An analogous theorem is proved for commutators; this implies that in every finitely generated profinite group, each term of the lower central series is closed. The proofs rely on some properties of the finite simple groups, to be established in Part II.

http://annals.math.princeton.edu/wp-content/uploads/annals-v165-n1-p05.pdf

On finitely generated profinite groups, I: strong completeness and uniform bounds

Lie methods in the theory of pro-p groups, A. Shalev on the classification of prop-p groups and finite p-groups, C.R. Leedham-Green, S. McKay prop-p trees and applications, L. Ribes, P. Zalesskii just infinite branch groups, R. I. Grigorchuk on just infinite abstract and profinite groups, J.S. Wilson the Nottingham group, R. Camina on groups satisfying the Golod-Shafarevich condition, E. Zelmanov sub-group growth in prop-p groups, A. Mann zeta functions of groups, M. du Sautoy, D. Segal where the wild things are - ramification groups and the Nottingham group, M. du Sautoy, I. Fesenko p-adic Galois representations and prop-p Galois groups, N. Boston the cohomology of p-adic analytic groups, P. Symonds, T. Weigel. Appendix: further problems.

New Horizons in pro-p Groups

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