223 research outputs found
The necessary and sufficient conditions of copositive tensors
In this paper, it is proved that (strict) copositivity of a symmetric tensor
is equivalent to the fact that every principal sub-tensor of
has no a (non-positive) negative -eigenvalue. The
necessary and sufficient conditions are also given in terms of the
-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 .Comment: 13 pages. arXiv admin note: text overlap with arXiv:1302.608
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 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 , the tensor
complementarity problem (\q, \mathcal{A}) has a solution for all \q \in
\mathbb{R}^n
Infinite and finite dimensional Hilbert tensors
For an -order dimensional Hilbert tensor (hypermatrix)
, its
spectral radius is not larger than , and an upper
bound of its -spectral radius is . Moreover,
its spectral radius is strictly increasing and its -spectral radius is
nondecreasing with respect to the dimension . When the order is even, both
infinite and finite dimensional Hilbert tensors are positive definite. We also
show that the -order infinite dimensional Hilbert tensor (hypermatrix)
defines a bounded and
positively -homogeneous operator from into (),
and the norm of corresponding positively homogeneous operator is smaller than
or equal to
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
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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
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