801 research outputs found
M-tensors and The Positive Definiteness of a Multivariate Form
We study M-tensors and various properties of M-tensors are given. Specially,
we show that the smallest real eigenvalue of M-tensor is positive corresponding
to a nonnegative eigenvector. We propose an algorithm to find the smallest
positive eigenvalue and then apply the property to study the positive
definiteness of a multivariate form
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
Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations
In this paper we study multivariate polynomial functions in complex variables
and the corresponding associated symmetric tensor representations. The focus is
on finding conditions under which such complex polynomials/tensors always take
real values. We introduce the notion of symmetric conjugate forms and general
conjugate forms, and present characteristic conditions for such complex
polynomials to be real-valued. As applications of our results, we discuss the
relation between nonnegative polynomials and sums of squares in the context of
complex polynomials. Moreover, new notions of eigenvalues/eigenvectors for
complex tensors are introduced, extending properties from the Hermitian
matrices. Finally, we discuss an important property for symmetric tensors,
which states that the largest absolute value of eigenvalue of a symmetric real
tensor is equal to its largest singular value; the result is known as Banach's
theorem. We show that a similar result holds in the complex case as well
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