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 $\mathbb{R}^n$.Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608

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 $n^{m-1}\sin\frac{\pi}{n}$, and an upper bound of its $E$-spectral radius is $n^{\frac{m}2}\sin\frac{\pi}{n}$. 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 $\frac{\pi}{\sqrt6}$

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

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.Comment: arXiv admin note: text overlap with arXiv:1405.1288 by other author

End-to-End Quantum-like Language Models with Application to Question Answering

Language Modeling (LM) is a fundamental research topic ina range of areas. Recently, inspired by quantum theory, a novel Quantum Language Model (QLM) has been proposed for Information Retrieval (IR). In this paper, we aim to broaden the theoretical and practical basis of QLM. We develop a Neural Network based Quantum-like Language Model (NNQLM) and apply it to Question Answering. Specifically, based on word embeddings, we design a new density matrix, which represents a sentence (e.g., a question or an answer) and encodes a mixture of semantic subspaces. Such a density matrix, together with a joint representation of the question and the answer, can be integrated into neural network architectures (e.g., 2-dimensional convolutional neural networks). Experiments on the TREC-QA and WIKIQA datasets have verified the effectiveness of our proposed models