11 research outputs found
Calculating Entanglement Eigenvalues for Non-Symmetric Quantum Pure States Based on the Jacobian Semidefinite Programming Relaxation Method
The geometric measure of entanglement is a widely used entanglement measure
for quantum pure states. The key problem of computation of the geometric
measure is to calculate the entanglement eigenvalue, which is equivalent to
computing the largest unitary eigenvalue of a corresponding complex tensor. In
this paper, we propose a Jacobian semidefinite programming relaxation method to
calculate the largest unitary eigenvalue of a complex tensor. For this, we
first introduce the Jacobian semidefinite programming relaxation method for a
polynomial optimization with equality constraint, and then convert the problem
of computing the largest unitary eigenvalue to a real equality constrained
polynomial optimization problem, which can be solved by the Jacobian
semidefinite programming relaxation method. Numerical examples are presented to
show the availability of this approach
An adaptive gradient method for computing generalized tensor eigenpairs
High order tensor arises more and more often in signal processing,data
analysis, higher-order statistics, as well as imaging sciences. In this paper,
an adaptive gradient (AG) method is presented for generalized tensor
eigenpairs. Global convergence and linear convergence rate are established
under some suitable conditions. Numerical results are reported to illustrate
the efficiency of the proposed method. Comparing with the GEAP method, an
adaptive shifted power method proposed by Tamara G. Kolda and Jackson R. Mayo
[SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1563-1581], the AG method is much
faster and could reach the largest eigenpair with a higher probability
Spectral projected gradient methods for generalized tensor eigenvalue complementarity problem
This paper looks at the tensor eigenvalue complementarity problem (TEiCP)
which arises from the stability analysis of finite dimensional mechanical
systems and is closely related to the optimality conditions for polynomial
optimization. We investigate two monotone ascent spectral projected gradient
(SPG) methods for TEiCP. We also present a shifted scaling-and-projection
algorithm (SPA), which is a great improvement of the original SPA method
proposed by Ling, He and Qi [Comput. Optim. Appl., DOI
10.1007/s10589-015-9767-z]. Numerical comparisons with some existed gradient
methods in the literature are reported to illustrate the efficiency of the
proposed methods.Comment: arXiv admin note: text overlap with arXiv:1601.0139
Iterative methods for computing U-eigenvalues of non-symmetric complex tensors with application in quantum entanglement
The purpose of this paper is to study the problem of computing unitary
eigenvalues (U-eigenvalues) of non-symmetric complex tensors. By means of
symmetric embedding of complex tensors, the relationship between U-eigenpairs
of a non-symmetric complex tensor and unitary symmetric eigenpairs
(US-eigenpairs) of its symmetric embedding tensor is established. An algorithm
(Algorithm \ref{algo:1}) is given to compute the U-eigenvalues of non-symmetric
complex tensors by means of symmetric embedding. Another algorithm, Algorithm
\ref{algo:2}, is proposed to directly compute the U-eigenvalues of
non-symmetric complex tensors, without the aid of symmetric embedding. Finally,
a tensor version of the well-known Gauss-Seidel method is developed. Efficiency
of these three algorithms are compared by means of various numerical examples.
These algorithms are applied to compute the geometric measure of entanglement
of quantum multipartite non-symmetric pure states.Comment: 19 pages; submitted for publication; comments are welcom
Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs
Finding the maximum eigenvalue of a symmetric tensor is an important topic in
tensor computation and numerical multilinear algebra. This paper is devoted to
a semi-definite program algorithm for computing the maximum -eigenvalue of a
class of tensors with sign structure called -tensors. The class of
-tensors extends the well-studied nonnegative tensors and essentially
nonnegative tensors, and covers some important tensors arising naturally from
spectral hypergraph theory. Our algorithm is based on a new structured
sums-of-squares (SOS) decomposition result for a nonnegative homogeneous
polynomial induced by a -tensor. This SOS decomposition enables us to show
that computing the maximum -eigenvalue of an even order symmetric -tensor
is equivalent to solving a semi-definite program, and hence can be accomplished
in polynomial time. Numerical examples are given to illustrate that the
proposed algorithm can be used to find maximum -eigenvalue of an even order
symmetric -tensor with dimension up to . We present two applications
for our proposed algorithm: we first provide a polynomial time algorithm for
computing the maximum -eigenvalues of large size Laplacian tensors of
hyper-stars and hyper-trees; second, we show that the proposed SOS algorithm
can be used to test the copositivity of a multivariate form associated with
symmetric extended -tensors, whose order may be even or odd. Numerical
experiments illustrate that our structured semi-definite program algorithm is
effective and promising
Computing Eigenvalues of Large Scale Hankel Tensors
Large scale tensors, including large scale Hankel tensors, have many
applications in science and engineering. In this paper, we propose an inexact
curvilinear search optimization method to compute Z- and H-eigenvalues of th
order dimensional Hankel tensors, where is large. Owing to the fast
Fourier transform, the computational cost of each iteration of the new method
is about . Using the Cayley transform, we obtain an
effective curvilinear search scheme. Then, we show that every limiting point of
iterates generated by the new algorithm is an eigen-pair of Hankel tensors.
Without the assumption of a second-order sufficient condition, we analyze the
linear convergence rate of iterate sequence by the Kurdyka-{\L}ojasiewicz
property. Finally, numerical experiments for Hankel tensors, whose dimension
may up to one million, are reported to show the efficiency of the proposed
curvilinear search method
All Real Eigenvalues of Symmetric Tensors
This paper studies how to compute all real eigenvalues of a symmetric tensor.
As is well known, the largest or smallest eigenvalue can be found by solving a
polynomial optimization problem, while the other middle eigenvalues can not. We
propose a new approach for computing all real eigenvalues sequentially, from
the largest to the smallest. It uses Jacobian SDP relaxations in polynomial
optimization. We show that each eigenvalue can be computed by solving a finite
hierarchy of semidefinite relaxations. Numerical experiments are presented to
show how to do this
Successive Rank-One Approximations for Nearly Orthogonally Decomposable Symmetric Tensors
Many idealized problems in signal processing, machine learning and statistics
can be reduced to the problem of finding the symmetric canonical decomposition
of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing
inspiration from the matrix case, the successive rank-one approximations (SROA)
scheme has been proposed and shown to yield this tensor decomposition exactly,
and a plethora of numerical methods have thus been developed for the tensor
rank-one approximation problem. In practice, however, the inevitable errors
(say) from estimation, computation, and modeling necessitate that the input
tensor can only be assumed to be a nearly SOD tensor---i.e., a symmetric tensor
slightly perturbed from the underlying SOD tensor. This article shows that even
in the presence of perturbation, SROA can still robustly recover the symmetric
canonical decomposition of the underlying tensor. It is shown that when the
perturbation error is small enough, the approximation errors do not accumulate
with the iteration number. Numerical results are presented to support the
theoretical findings
Newton correction methods for computing real eigenpairs of symmetric tensors
Real eigenpairs of symmetric tensors play an important role in multiple
applications. In this paper we propose and analyze a fast iterative
Newton-based method to compute real eigenpairs of symmetric tensors. We derive
sufficient conditions for a real eigenpair to be a stable fixed point for our
method, and prove that given a sufficiently close initial guess, the
convergence rate is quadratic. Empirically, our method converges to a
significantly larger number of eigenpairs compared to previously proposed
iterative methods, and with enough random initializations typically finds all
real eigenpairs. In particular, for a generic symmetric tensor, the sufficient
conditions for local convergence of our Newton-based method hold simultaneously
for all its real eigenpairs
Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph
The spectral theory of higher-order symmetric tensors is an important tool to
reveal some important properties of a hypergraph via its adjacency tensor,
Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these
tensors, we propose an efficient approach to calculate products of these
tensors and any vectors. Using the state-of-the-art L-BFGS approach, we develop
a first-order optimization algorithm for computing H- and Z-eigenvalues of
these large scale sparse tensors (CEST). With the aid of the
Kurdyka-{\L}ojasiewicz property, we prove that the sequence of iterates
generated by CEST converges to an eigenvector of the tensor.When CEST is
started from multiple randomly initial points, the resulting best eigenvalue
could touch the extreme eigenvalue with a high probability. Finally, numerical
experiments on small hypergraphs show that CEST is efficient and promising.
Moreover, CEST is capable of computing eigenvalues of tensors corresponding to
a hypergraph with millions of vertices