Abstract
In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Z-eigenvalues of a real symmetric tensor with even order. We first establish that the maximum Z-eigenvalue function is a continuous and convex function on the symmetric tensor space and so provide formulas of the convex conjugate function and ε-subdifferential of the maximum Z-eigenvalue function. Consequently, for an mth-order N-dimensional tensor A, we show that the normalized eigenspace associated with maximum Z-eigenvalue function is ρth-order Hölder stable at A with ρ=1m(3m-3)n-1-1. As a by-product, we also establish that the maximum Z-eigenvalue function is always at least ρth-order semismooth at A. As an application, we introduce the characteristic tensor of a hypergraph and show that the maximum Z-eigenvalue function of the associated characteristic tensor provides a natural link for the combinatorial structure and the analytic structure of the underlying hypergraph. Finally, we establish a variational formula for the second largest Z-eigenvalue for the characteristic tensor of a hypergraph and use it to provide lower bounds for the bipartition width of a hypergraph. Some numerical examples are also provided to show how one can compute the largest/second-largest Z-eigenvalue of a medium size tensor, using polynomial optimization techniques and our variational formula.
Original language | English |
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Pages (from-to) | 1001-1029 |
Number of pages | 29 |
Journal | Numerical Linear Algebra with Applications |
Volume | 20 |
Issue number | 6 |
DOIs | |
Publication status | Published - 1 Dec 2013 |
Keywords
- Characteristic tensor
- Maximum Z-eigenvalue
- Polynomial optimization
- Semismoothness
- Spectral graph theory
- Symmetric tensor
ASJC Scopus subject areas
- Algebra and Number Theory
- Applied Mathematics