arXiv subjects

Yisheng Song

Publications and source records attributed to Yisheng Song.

46 records · Page 3Linked to original sources

arXiv2016

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.

math.OC↗

arXiv2015

Eigenvalues and structured properties of P-tensors

We define two new constants associated with real eigenvalues of a P-tensor. With the help of these two constants, in the case of P-tensors, we establish upper bounds of two important quantities, whose positivity is a necessary and sufficient condition for a general tensor to be a P-tensor.

math.OC↗

arXiv2015

Tensor Complementarity Problem and Semi-positive Tensors

The tensor complementarity problem $(\q, \mathcal{A})$ is to $$\mbox{ find } \x \in \mathbb{R}^n\mbox{ such that }\x \geq \0, \q + \mathcal{A}\x^{m-1} \geq \0, \mbox{ and }\x^\top (\q + \mathcal{A}\x^{m-1}) = 0.$$ We prove that a real tensor $\mathcal{A}$ is a (strictly) semi-positive tensor if and only if the tensor complementarity problem $(\q, \mathcal{A})$ has a unique solution for $\q>\0$ ($\q\geq\0$), and a symmetric real tensor is a (strictly) semi-positive tensor if and only if it is (strictly) copositive. That is, for a strictly copositive symmetric tensor $\mathcal{A}$, the tensor complementarity problem $(\q, \mathcal{A})$ has a solution for all $\q \in \mathbb{R}^n$.

math.OC↗

arXiv2014

Properties of Some Classes of Structured Tensors

In this paper, we extend some classes of structured matrices to higher order tensors. We discuss their relationships with positive semi-definite tensors and some other structured tensors. We show that every principal sub-tensor of such a structured tensor is still a structured tensor in the same class, with a lower dimension. The potential links of such structured tensors with optimization, nonlinear equations, nonlinear complementarity problems, variational inequalities and the nonnegative tensor theory are also discussed.

math.SP↗

arXiv2014

An Even Order Symmetric B Tensor is Positive Definite

It is easily checkable if a given tensor is a B tensor, or a B$_0$ tensor or not. In this paper, we show that a symmetric B tensor can always be decomposed to the sum of a strictly diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors, and a symmetric B$_0$ tensor can always be decomposed to the sum of a diagonally dominated symmetric M tensor and several positive multiples of partially all one tensors. When the order is even, this implies that the corresponding B tensor is positive definite, and the corresponding B$_0$ tensor is positive semi-definite. This gives a checkable sufficient condition for positive definite and semi-definite tensors. This approach is different from the approach in the literature for proving a symmetric B matrix is positive definite, as that matrix approach cannot be extended to the tensor case.

math.SP↗

arXiv2014

Strong convergence for the modified Mann's iteration of $λ$-strict pseudocontraction

In this paper, for an $λ$-strict pseudocontraction $T$, we prove strong convergence of the modified Mann's iteration defined by $$x_{n+1}=β_{n}u+γ_nx_n+(1-β_{n}-γ_n)[α_{n}Tx_n+(1-α_{n})x_n],$$ where $\{α_{n}\}$, $ \{β_{n}\}$ and $\{γ_n\}$ in $(0,1)$ satisfy: (i) $0 \leq α_{n}\leq \fracλ{K^2}$ with $\liminf\limits_{n\to\infty}α_n(λ-K^2α_n)> 0$; (ii) $\lim\limits_{n\to\infty}β_n= 0$ and $\sum\limits_{n=1}^\inftyβ_n=\infty$; (iii) $\limsup\limits_{n\to\infty}γ_n<1$.Our results unify and improve some existing results.

math.FA↗

arXiv2014

Infinite and finite dimensional Hilbert tensors

For an $m$-order $n-$dimensional Hilbert tensor (hypermatrix) $\mathcal{H}_n=(\mathcal{H}_{i_1i_2\cdots i_m})$, $$\mathcal{H}_{i_1i_2\cdots i_m}=\frac1{i_1+i_2+\cdots+i_m-m+1},\ i_1,\cdots, i_m=1,2,\cdots,n$$ its spectral radius is not larger than $n^{m-1}\sin\fracπ{n}$, and an upper bound of its $E$-spectral radius is $n^{\frac{m}2}\sin\fracπ{n}$. Moreover, its spectral radius is strictly increasing and its $E$-spectral radius is nondecreasing with respect to the dimension $n$. When the order is even, both infinite and finite dimensional Hilbert tensors are positive definite. We also show that the $m$-order infinite dimensional Hilbert tensor (hypermatrix) $\mathcal{H}_\infty=(\mathcal{H}_{i_1i_2\cdots i_m})$ defines a bounded and positively $(m-1)$-homogeneous operator from $l^1$ into $l^p$ ($1<p<\infty$), and the norm of corresponding positively homogeneous operator is smaller than or equal to $\fracπ{\sqrt6}$.

math.SP↗

arXiv2013

The necessary and sufficient conditions of copositive tensors

In this paper, it is proved that (strict) copositivity of a symmetric tensor $\mathcal{A}$ is equivalent to the fact that every principal sub-tensor of $\mathcal{A}$ has no a (non-positive) negative $H^{++}$-eigenvalue. The necessary and sufficient conditions are also given in terms of the $Z^{++}$-eigenvalue of the principal sub-tensor of the given tensor. This presents a method of testing (strict) copositivity of a symmetric tensor by means of the lower dimensional tensors. Also the equivalent definition of strictly copositive tensors is given on entire space $\mathbb{R}^n$.

math.OC↗

arXiv2013

Eigenvalue analysis of constrained minimization problem for homogeneous polynomial

In this paper, the concepts of Pareto $H$-eigenvalue and Pareto $Z$-eigenvalue are introduced for studying constrained minimization problem and the necessary and sufficient conditions of such eigenvalues are given. It is proved that a symmetric tensor has at least one Pareto $H$-eigenvalue (Pareto $Z$-eigenvalue). Furthermore, the minimum Pareto $H$-eigenvalue (or Pareto $Z$-eigenvalue) of a symmetric tensor is exactly equal to the minimum value of constrained minimization problem of homogeneous polynomial deduced by such a tensor, which gives an alternative methods for solving the minimum value of constrained minimization problem. In particular, a symmetric tensor $\mathcal{A}$ is copositive if and only if every Pareto $H$-eigenvalue ($Z-$eigenvalue) of $\mathcal{A}$ is non-negative.

math.OC↗

arXiv2012

Geometric measure of entanglement of multipartite mixed states

The geometric measure of entanglement of a pure state, defined by its distance to the set of pure separable states, is extended to multipartite mixed states. We characterize the nearest disentangled mixed state to a given mixed state with respect to this measure by a system of equations. The entanglement eigenvalue for a mixed state is introduced. For a given mixed state, we show that its nearest disentangled mixed state is associated with its entanglement eigenvalue.

quant-ph↗