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Tensor Train Rank-1 decomposition

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Tensor Train Rank-1 Singular Value and Symmetric Eigenvalue Decomposition for Matlab©/Octave©

The Tensor Train Rank-1 singular value decomposition (TTr1SVD) decomposes an arbitrary tensor A into a unique linear combination of orthonormal rank-1 terms. The Tensor Train Rank-1 symmetric eigenvalue decomposition (TTr1SED) does the same for a tensor that is symmetric in the last 2 modes of equal dimension. This allows for a very straightforward determination of a low-rank approximation as well as an easy quantification of the approximation error.

  1. Functions

  • [U,S,V,sigmas]=ttr1svd(A) or [U,S,V,sigmas]=ttr1sed(A)

Use this function to compute the TTr1SVD(/SED) decomposition.

  • Atilde=getAtilde(U,sigmas,V,sigmaI,n)

Use this function to compute a rank-R approximation Atilde of A from the TTr1SVD(/SED) decomposition.

  • demo.m

Small demo that illustrates the use of most functions in this pacakge.

  • [S,Q]=ttr12tucker(U,sigmas,V,n)

Converts a tensor A of size n in the TTr1SVD format to the Tucker (HOSVD) format. Usually results in a more sparse core S compared to traditional methods (e.g. Alternating Least Squares).

  • O=orthc(A) or O=orthc(A,tol)

This function computes all outer vector products that form tensors orthogonal to A.

  • ocv=verifyOrtc(A,O)

Use this function to verify that all tensors that can be formed with vectors in O are orthogonal to A.

  1. Reference

"A Constructive Algorithm for Decomposing a Tensor into a Finite Sum of Orthonormal Rank-1 Terms"

http://arxiv.org/abs/1407.1593

Authors: Kim Batselier, Haotian Liu, Ngai Wong

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Tensor Train Rank-1 decomposition

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LGPL-3.0, GPL-3.0 licenses found

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COPYING.LESSER
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