Eigenvectors of Z-tensors associated with least H-eigenvalue with application to hypergraphs

Unlike an irreducible $Z$-matrices, a weakly irreducible $Z$-tensor $\mathcal{A}$ can have more than one eigenvector associated with the least H-eigenvalue. We show that there are finitely many eigenvectors of $\mathcal{A}$ associated with the least H-eigenvalue. If $\mathcal{A}$ is further combinatorial symmetric, the number of such eigenvectors can be obtained explicitly by the Smith normal form of the incidence matrix of $\mathcal{A}$. When applying to a connected uniform hypergraph $G$, we prove that the number of Laplacian eigenvectors of $G$ associated with the zero eigenvalue is equal to the the number of adjacency eigenvectors of $G$ associated with the spectral radius, which is also equal to the number of signless Laplacian eigenvectors of $G$ associated with the zero eigenvalue if zero is an signless Laplacian eigenvalue

Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than O(1/ϵ)O(1/\epsilon)

In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is $O(1/\epsilon)$ without any assumption on the strong convexity. In this work, we will show that the proposed HOPS achieved a lower iteration complexity of $\widetilde O(1/\epsilon^{1-\theta})$\footnote{$\widetilde O()$ suppresses a logarithmic factor.} with $\theta\in(0,1]$ capturing the local sharpness of the objective function around the optimal solutions. To the best of our knowledge, this is the lowest iteration complexity achieved so far for the considered non-smooth optimization problems without strong convexity assumption. The HOPS algorithm employs Nesterov's smoothing technique and Nesterov's accelerated gradient method and runs in stages, which gradually decreases the smoothing parameter in a stage-wise manner until it yields a sufficiently good approximation of the original function. We show that HOPS enjoys a linear convergence for many well-known non-smooth problems (e.g., empirical risk minimization with a piece-wise linear loss function and $\ell_1$ norm regularizer, finding a point in a polyhedron, cone programming, etc). Experimental results verify the effectiveness of HOPS in comparison with Nesterov's smoothing algorithm and the primal-dual style of first-order methods.Comment: This is a long version of the paper accepted by NIPS 201

Learning Discriminators as Energy Networks in Adversarial Learning

We propose a novel framework for structured prediction via adversarial learning. Existing adversarial learning methods involve two separate networks, i.e., the structured prediction models and the discriminative models, in the training. The information captured by discriminative models complements that in the structured prediction models, but few existing researches have studied on utilizing such information to improve structured prediction models at the inference stage. In this work, we propose to refine the predictions of structured prediction models by effectively integrating discriminative models into the prediction. Discriminative models are treated as energy-based models. Similar to the adversarial learning, discriminative models are trained to estimate scores which measure the quality of predicted outputs, while structured prediction models are trained to predict contrastive outputs with maximal energy scores. In this way, the gradient vanishing problem is ameliorated, and thus we are able to perform inference by following the ascent gradient directions of discriminative models to refine structured prediction models. The proposed method is able to handle a range of tasks, e.g., multi-label classification and image segmentation. Empirical results on these two tasks validate the effectiveness of our learning method

Half-arc-transitive graphs of prime-cube order of small valencies

A graph is called {\em half-arc-transitive} if its full automorphism group acts transitively on vertices and edges, but not on arcs. It is well known that for any prime $p$ there is no tetravalent half-arc-transitive graph of order $p$ or $p^2$. Xu~[Half-transitive graphs of prime-cube order, J. Algebraic Combin. 1 (1992) 275-282] classified half-arc-transitive graphs of order $p^3$ and valency $4$. In this paper we classify half-arc-transitive graphs of order $p^3$ and valency $6$ or $8$. In particular, the first known infinite family of half-arc-transitive Cayley graphs on non-metacyclic $p$-groups is constructed.Comment: 13 page

Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

A graph $\Gamma$ is a bi-Cayley graph over a group $G$ if $G$ is a semiregular group of automorphisms of $\Gamma$ having two orbits. Let $G$ be a non-abelian metacyclic $p$-group for an odd prime $p$, and let $\Gamma$ be a connected bipartite bi-Cayley graph over the group $G$. In this paper, we prove that $G$ is normal in the full automorphism group ${\rm Aut}(\Gamma)$ of $\Gamma$ when $G$ is a Sylow $p$-subgroup of ${\rm Aut}(\Gamma)$. As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $2p$. Furthermore, it is shown that there are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the group $G$ of valency less than $p$.Comment: 20 pages, 1 figur

A Combinatorial Method for Computing Characteristic Polynomials of Starlike Hypergraphs

By using the Poisson formula for resultants and the variants of chip-firing game on graphs, we provide a combinatorial method for computing a class of of resultants, i.e. the characteristic polynomials of the adjacency tensors of starlike hypergraphs including hyperpaths and hyperstars,which are given recursively and explicitly

The layered compound CaClFeP is an Arsenic-free high TcT_c iron-pnictide

We first analyze why the iron pnictides with high $T_c$ superconductivity so far are As-based, by the Hund's rule correlation picture, then examine the P-based and Sb-based cases, respectively. Consequently, we propose that CaClFeP with ZrCuSiAs-type structure is an As-free high $T_c$ iron-pnictide. The subsequent density functional theory calculations show that the ground state of CaClFeP is of a collinearly antiferromagnetic order on Fe moments with structural distortion, resulting from the interplay between the strong nearest and next-nearest neighbor antiferromagnetic superexchange interactions bridged by P atoms, similar as the As-based pnictides. The other P-based pnictides are either nonmagnetic or magnetic but with weak exchange interactions. The Sb-based pnictides unlikely show high $T_c$ superconductivity because of the existence of robust ferromagnetic order.Comment: 4 pages, 5 figures, and 1 tabl

A Note On the Rank of the Optimal Matrix in Symmetric Toeplitz Matrix Completion Problem

We consider the symmetric Toeplitz matrix completion problem, whose matrix under consideration possesses specific row and column structures. This problem, which has wide application in diverse areas, is well-known to be computationally NP-hard. This note provides an upper bound on the objective of minimizing the rank of the symmetric Toeplitz matrix in the completion problem based on the theorems from the trigonometric moment problem and semi-infinite problem. We prove that this upper bound is less than twice the number of linear constraints of the Toeplitz matrix completion problem. Compared with previous work in the literature, ours is one of the first efforts to investigate the bound of the objective value of the Toeplitz matrix completion problem

DSNet: Deep and Shallow Feature Learning for Efficient Visual Tracking

In recent years, Discriminative Correlation Filter (DCF) based tracking methods have achieved great success in visual tracking. However, the multi-resolution convolutional feature maps trained from other tasks like image classification, cannot be naturally used in the conventional DCF formulation. Furthermore, these high-dimensional feature maps significantly increase the tracking complexity and thus limit the tracking speed. In this paper, we present a deep and shallow feature learning network, namely DSNet, to learn the multi-level same-resolution compressed (MSC) features for efficient online tracking, in an end-to-end offline manner. Specifically, the proposed DSNet compresses multi-level convolutional features to uniform spatial resolution features. The learned MSC features effectively encode both appearance and semantic information of objects in the same-resolution feature maps, thus enabling an elegant combination of the MSC features with any DCF-based methods. Additionally, a channel reliability measurement (CRM) method is presented to further refine the learned MSC features. We demonstrate the effectiveness of the MSC features learned from the proposed DSNet on two DCF tracking frameworks: the basic DCF framework and the continuous convolution operator framework. Extensive experiments show that the learned MSC features have the appealing advantage of allowing the equipped DCF-based tracking methods to perform favorably against the state-of-the-art methods while running at high frame rates.Comment: To appear at ACCV 2018. 14 pages, 8 figure

Quantum speed limit time of a qubit system with non-Hermitian detuning

We investigated the quantum speed limit time of a qubit system with non-Hermitian detuning. Our results show that, with respect to two distinguishable states of the non-Hermitian system, the evolutionary time does not have a nonzero lower bound. And the quantum evolution of the system can be effectively accelerated by adjusting the non-Hermitian detuning parameter, as well as the quantum speed limit time can be arbitrarily small even be zero