1,346 research outputs found
The extremal spectral radii of -uniform supertrees
In this paper, we study some extremal problems of three kinds of spectral
radii of -uniform hypergraphs (the adjacency spectral radius, the signless
Laplacian spectral radius and the incidence -spectral radius).
We call a connected and acyclic -uniform hypergraph a supertree. We
introduce the operation of "moving edges" for hypergraphs, together with the
two special cases of this operation: the edge-releasing operation and the total
grafting operation. By studying the perturbation of these kinds of spectral
radii of hypergraphs under these operations, we prove that for all these three
kinds of spectral radii, the hyperstar attains uniquely the
maximum spectral radius among all -uniform supertrees on vertices. We
also determine the unique -uniform supertree on vertices with the second
largest spectral radius (for these three kinds of spectral radii). We also
prove that for all these three kinds of spectral radii, the loose path
attains uniquely the minimum spectral radius among all
-th power hypertrees of vertices. Some bounds on the incidence
-spectral radius are given. The relation between the incidence -spectral
radius and the spectral radius of the matrix product of the incidence matrix
and its transpose is discussed
A Tensor Analogy of Yuan's Theorem of the Alternative and Polynomial Optimization with Sign structure
Yuan's theorem of the alternative is an important theoretical tool in
optimization, which provides a checkable certificate for the infeasibility of a
strict inequality system involving two homogeneous quadratic functions. In this
paper, we provide a tractable extension of Yuan's theorem of the alternative to
the symmetric tensor setting. As an application, we establish that the optimal
value of a class of nonconvex polynomial optimization problems with suitable
sign structure (or more explicitly, with essentially non-positive coefficients)
can be computed by a related convex conic programming problem, and the optimal
solution of these nonconvex polynomial optimization problems can be recovered
from the corresponding solution of the convex conic programming problem.
Moreover, we obtain that this class of nonconvex polynomial optimization
problems enjoy exact sum-of-squares relaxation, and so, can be solved via a
single semidefinite programming problem.Comment: acceted by Journal of Optimization Theory and its application, UNSW
preprint, 22 page
SOS-Hankel Tensors: Theory and Application
Hankel tensors arise from signal processing and some other applications. SOS
(sum-of-squares) tensors are positive semi-definite symmetric tensors, but not
vice versa. The problem for determining an even order symmetric tensor is an
SOS tensor or not is equivalent to solving a semi-infinite linear programming
problem, which can be done in polynomial time. On the other hand, the problem
for determining an even order symmetric tensor is positive semi-definite or not
is NP-hard. In this paper, we study SOS-Hankel tensors. Currently, there are
two known positive semi-definite Hankel tensor classes: even order complete
Hankel tensors and even order strong Hankel tensors. We show complete Hankel
tensors are strong Hankel tensors, and even order strong Hankel tensors are
SOS-Hankel tensors. We give several examples of positive semi-definite Hankel
tensors, which are not strong Hankel tensors. However, all of them are still
SOS-Hankel tensors. Does there exist a positive semi-definite non-SOS-Hankel
tensor? The answer to this question remains open. If the answer to this
question is no, then the problem for determining an even order Hankel tensor is
positive semi-definite or not is solvable in polynomial-time. An application of
SOS-Hankel tensors to the positive semi-definite tensor completion problem is
discussed. We present an ADMM algorithm for solving this problem. Some
preliminary numerical results on this algorithm are reported
Investigating into segmentation methods for diagnosis of respiratory diseases using adventitious respiratory sounds
Respiratory condition has received a great amount of attention nowadays since respiratory diseases recently become the globally leading causes of death. Traditionally, stethoscope is applied in early diagnosis but it requires clinician with extensive training experience to provide accurate diagnosis. Accordingly, a subjective and fast diagnosing solution of respiratory diseases is highly demanded. Adventitious respiratory sounds (ARSs), such as crackle, are mainly concerned during diagnosis since they are indication of various respiratory diseases. Therefore, the characteristics of crackle are informative and valuable regarding to develop a computerised approach for pathology-based diagnosis. In this work, we propose a framework combining random forest classifier and Empirical Mode Decomposition (EMD) method focusing on a multi-classification task of identifying subjects in 6 respiratory conditions (healthy, bronchiectasis, bronchiolitis, COPD, pneumonia and URTI). Specifically, 14 combinations of respiratory sound segments were compared and we found segmentation plays an important role in classifying different respiratory conditions. The classifier with best performance (accuracy = 0.88, precision = 0.91, recall = 0.87, specificity = 0.91, F1-score = 0.81) was trained with features extracted from the combination of early inspiratory phase and entire inspiratory phase. To our best knowledge, we are the first to address the challenging multi-classification problem
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