Monday, 13:45 - 14:10 h, Room: H 0112

 

Lek-Heng Lim
3-tensors as the boundary of tractability

Coauthor: Christopher Hillar

 

Abstract:
Why do problems in numerical computing become intractable as they transition from linear to nonlinear or convex to nonconvex? We shall argue that 3-tensor problems form the boundary separating the tractability of linear algebra and convex optimization from the intractability of polynomial algebra and nonconvex optimization - 3-tensor analogues of many efficiently computable matrix problems 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 spectral norm; determining the rank or a best rank-1 approximation to a 3-tensor. Additionally, some of these problems have no polynomial time approximation schemes, some are undecidable over Q, and at least one enumerative version is #P-complete. Restricting these problems to symmetric 3-tensors does not alleviate their NP-hardness.

 

Talk 2 of the invited session Mon.2.H 0112
"Structures, complexities, and eigenvalues of tensor forms and polynomial functions" [...]
Cluster 16
"Nonlinear programming" [...]

 

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