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Spectral partitioning of large and sparse 3‐tensors using low‐rank tensor approximation
Author(s) -
Eldén Lars,
Dehghan Maryam
Publication year - 2022
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2435
Subject(s) - tensor (intrinsic definition) , mathematics , rank (graph theory) , generalization , symmetric tensor , tensor contraction , cartesian tensor , adjacency list , tensor density , adjacency matrix , exact solutions in general relativity , combinatorics , tensor field , graph , mathematical analysis , pure mathematics
The problem of partitioning a large and sparse tensor is considered, where the tensor consists of a sequence of adjacency matrices. Theory is developed that is a generalization of spectral graph partitioning. A best rank‐ ( 2 , 2 , λ ) approximation is computed for λ = 1 , 2 , 3 , and the partitioning is computed from the orthogonal matrices and the core tensor of the approximation. It is shown that if the tensor has a certain reducibility structure, then the solution of the best approximation problem exhibits the reducibility structure of the tensor. Further, if the tensor is close to being reducible, then still the solution of the exhibits the structure of the tensor. Numerical examples with synthetic data corroborate the theoretical results. Experiments with tensors from applications show that the method can be used to extract relevant information from large, sparse, and noisy data.

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