14 research outputs found
On spectral hypergraph theory of the adjacency tensor
We study both and -eigenvalues of the adjacency tensor of a uniform
multi-hypergraph and give conditions for which the largest positive or
-eigenvalue corresponds to a strictly positive eigenvector. We also
investigate when the -spectrum of the adjacency tensor is symmetric
The Eigenvectors of the Zero Laplacian and Signless Laplacian Eigenvalues of a Uniform Hypergraph
In this paper, we show that the eigenvectors of the zero Laplacian and
signless Lapacian eigenvalues of a -uniform hypergraph are closely related
to some configured components of that hypergraph. We show that the components
of an eigenvector of the zero Laplacian or signless Lapacian eigenvalue have
the same modulus. Moreover, under a {\em canonical} regularization, the phases
of the components of these eigenvectors only can take some uniformly
distributed values in \{\{exp}(\frac{2j\pi}{k})\;|\;j\in [k]\}. These
eigenvectors are divided into H-eigenvectors and N-eigenvectors. Eigenvectors
with minimal support is called {\em minimal}. The minimal canonical
H-eigenvectors characterize the even (odd)-bipartite connected components of
the hypergraph and vice versa, and the minimal canonical N-eigenvectors
characterize some multi-partite connected components of the hypergraph and vice
versa.Comment: 22 pages, 3 figure
Nonnegative polynomial optimization over unit spheres and convex programming relaxations
We consider approximation algorithms for nonnegative polynomial optimization over unit spheres. Such optimization models have wide applications, e.g., in signal and image processing, high order statistics, and computer vision. Since polynomial functions are nonconvex, the problems under consideration are all NP-hard. In this paper, based on convex polynomial optimization relaxations, we propose polynomial-time approximation algorithms with new approximation bounds. Numerical results are reported to show the effectiveness of the proposed approximation algorithms