Symmetric Tensor Eigen-Rank-One Iterative Decomposition (STEROID) and the Permuted QR algorithm for symmetric tensors (PQRST)

The Symmetric Tensor Eigen-Rank-One Iterative Decomposition (STEROID) decomposes an arbitrary symmetric tensor A into a real linear combination of unit-norm symmetric rank-1 terms. Sparse symmetric tensors are also supported. Z-eigenpairs of A can be computed using a STEROID-based implementation of the Permuted QR algorithm for symmetric tensors (PQRST).

1. Functions

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• [V,d,lambdas,e,tail]=steroid(A) or [V,d,lambdas,e,tail]=steroid(A,method)

Use this function to compute the STEROID of a symmetric tensor A. The least-squares problem W*d=vec(A) is solved according to the given 'method' input argument.

• [V,d,lambdas,e,tail]=steroids(polyA,R)

Use this function to compute the STEROID of a sparse symmetric tensor A, where A is represented as a homogeneous polynomial. This method can be highly advantageous in terms of required memory and time complexity for very sparse tensors of high dimension. Instead of the full eigenvalue decomposition, only R terms corresponding with the highest eigenvalues are computed. The least-squares problem Wd=vec(A) is always solved using the W^TW method.

• [lambda,V,err,itr]=pqrst(A,tol,kmax,shifting,R)

STEROID-based implementation of the Permuted QR algorithm for Symmetric Tensors (PQRST), computes possible Z-eigenpairs (lambda,V) of A.

• A=randsymten(d,n)

Creates a random symmetric tensor of order d and dimension n.

• e=symcheck(A)

Checks the symmetry of A for all permutations of its indices.

• B=embed(A)

Embeds a symmetric d-way tensor A with odd d into a symmetric d+1-way tensor B such that B(:,:,...,:,1)=A.

• [lindex index]=exp2ind(mons)

Converts homogeneous monomial exponents into a linear and tensor index of a d-way symmetric tensor of dimension n.

• [U,V,D,e]=svdsteroid(A,method)

Combination of an SVD and STEROID for a tensor A that is symmetric in all but the first mode.

• demo.m

Small demo that illustrates the use of steroid.m

2. Reference

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"Symmetric Tensor Decomposition by an Iterative Eigendecomposition Algorithm"

http://arxiv.org/abs/1409.4926

• A QR Algorithm for Symmetric Tensors

http://arxiv.org/abs/1411.1926

Authors: Kim Batselier, Ngai Wong