162 research outputs found
Numerical Algorithms for Polynomial Optimisation Problems with Applications
In this thesis, we study tensor eigenvalue problems and polynomial optimization problems. In particular, we present a fast algorithm for computing the spectral radii of symmetric nonnegative tensors without requiring the partition of the tensors. We also propose some polynomial time approximation algorithms with new approximation bounds for nonnegative polynomial optimization problems over unit spheres. Furthermore, we develop an efficient and effective algorithm for the maximum clique problem
Finding the spectral radius of a nonnegative irreducible symmetric tensor via DC programming
The Perron-Frobenius theorem says that the spectral radius of an irreducible
nonnegative tensor is the unique positive eigenvalue corresponding to a
positive eigenvector. With this in mind, the purpose of this paper is to find
the spectral radius and its corresponding positive eigenvector of an
irreducible nonnegative symmetric tensor. By transferring the eigenvalue
problem into an equivalent problem of minimizing a concave function on a closed
convex set, which is typically a DC (difference of convex functions)
programming, we derive a simpler and cheaper iterative method. The proposed
method is well-defined. Furthermore, we show that both sequences of the
eigenvalue estimates and the eigenvector evaluations generated by the method
-linearly converge to the spectral radius and its corresponding eigenvector,
respectively. To accelerate the method, we introduce a line search technique.
The improved method retains the same convergence property as the original
version. Preliminary numerical results show that the improved method performs
quite well
Computing the extremal nonnegative solutions of the M-tensor equation with a nonnegative right side vector
We consider the tensor equation whose coefficient tensor is a nonsingular
M-tensor and whose right side vector is nonnegative. Such a tensor equation may
have a large number of nonnegative solutions. It is already known that the
tensor equation has a maximal nonnegative solution and a minimal nonnegative
solution (called extremal solutions collectively). However, the existing proofs
do not show how the extremal solutions can be computed. The existing numerical
methods can find one of the nonnegative solutions, without knowing whether the
computed solution is an extremal solution. In this paper, we present new proofs
for the existence of extremal solutions. Our proofs are much shorter than
existing ones and more importantly they give numerical methods that can compute
the extremal solutions. Linear convergence of these numerical methods is also
proved under mild assumptions. Some of our discussions also allow the
coefficient tensor to be a Z-tensor or allow the right side vector to have some
negative elements
Examples of Riemannian Manifolds with non-negative sectional curvature
An updated version with a few corrections.Comment: 32 page
Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs
Laplacian mixture models identify overlapping regions of influence in
unlabeled graph and network data in a scalable and computationally efficient
way, yielding useful low-dimensional representations. By combining Laplacian
eigenspace and finite mixture modeling methods, they provide probabilistic or
fuzzy dimensionality reductions or domain decompositions for a variety of input
data types, including mixture distributions, feature vectors, and graphs or
networks. Provable optimal recovery using the algorithm is analytically shown
for a nontrivial class of cluster graphs. Heuristic approximations for scalable
high-performance implementations are described and empirically tested.
Connections to PageRank and community detection in network analysis demonstrate
the wide applicability of this approach. The origins of fuzzy spectral methods,
beginning with generalized heat or diffusion equations in physics, are reviewed
and summarized. Comparisons to other dimensionality reduction and clustering
methods for challenging unsupervised machine learning problems are also
discussed.Comment: 13 figures, 35 reference
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