Semidefinite relaxations for semi-infinite polynomial programming

This paper studies how to solve semi-infinite polynomial programming (SIPP) problems by semidefinite relaxation method. We first introduce two SDP relaxation methods for solving polynomial optimization problems with finitely many constraints. Then we propose an exchange algorithm with SDP relaxations to solve SIPP problems with compact index set. At last, we extend the proposed method to SIPP problems with noncompact index set via homogenization. Numerical results show that the algorithm is efficient in practice.Comment: 23 pages, 4 figure

Lipschitz-Volume rigidity on limit spaces with Ricci curvature bounded from below

We prove a Lipschitz-Volume rigidity theorem for the non-collapsed Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below. This is a counterpart of the Lipschitz-Volume rigidity in Alexandrov geometry

Modular Properties of 3D Higher Spin Theory

In the three-dimensional sl(N) Chern-Simons higher-spin theory, we prove that the conical surplus and the black hole solution are related by the S-transformation of the modulus of the boundary torus. Then applying the modular group on a given conical surplus solution, we generate a 'SL(2,Z)' family of smooth constant solutions. We then show how these solutions are mapped into one another by coordinate transformations that act non-trivially on the homology of the boundary torus. After deriving a thermodynamics that applies to all the solutions in the 'SL(2,Z)' family, we compute their entropies and free energies, and determine how the latter transform under the modular transformations. Summing over all the modular images of the conical surplus, we write down a (tree-level) modular invariant partition function.Comment: 51 pages; v2: minor corrections and additions; v3: final version, to appear in JHE

Constructing Hierarchical Image-tags Bimodal Representations for Word Tags Alternative Choice

This paper describes our solution to the multi-modal learning challenge of ICML. This solution comprises constructing three-level representations in three consecutive stages and choosing correct tag words with a data-specific strategy. Firstly, we use typical methods to obtain level-1 representations. Each image is represented using MPEG-7 and gist descriptors with additional features released by the contest organizers. And the corresponding word tags are represented by bag-of-words model with a dictionary of 4000 words. Secondly, we learn the level-2 representations using two stacked RBMs for each modality. Thirdly, we propose a bimodal auto-encoder to learn the similarities/dissimilarities between the pairwise image-tags as level-3 representations. Finally, during the test phase, based on one observation of the dataset, we come up with a data-specific strategy to choose the correct tag words leading to a leap of an improved overall performance. Our final average accuracy on the private test set is 100%, which ranks the first place in this challenge.Comment: 6 pages, 1 figure, Presented at the Workshop on Representation Learning, ICML 201

Minimizing Rational Functions by Exact Jacobian SDP Relaxation Applicable to Finite Singularities

This paper considers the optimization problem of minimizing a rational function. We reformulate this problem as polynomial optimization by the technique of homogenization. These two problems are shown to be equivalent under some generic conditions. The exact Jacobian SDP relaxation method proposed by Nie is used to solve the resulting polynomial optimization. We also prove that the assumption of nonsingularity in Nie's method can be weakened as the finiteness of singularities. Some numerical examples are given to illustrate the efficiency of our method.Comment: 23 page

Two-Body Strong Decay of Z(3930) as the χc2(2P)\chi_{c2} (2P) State

The new particle Z(3930) found by the Belle and BaBar Collaborations through the $\gamma\gamma\rightarrow D\bar D$ process is identified to be the $\chi_{c2}(2P)$ state. Since the mass of this particle is above the $D\bar D^{(\ast)}$ threshold, the OZI-allowed two-body strong decays are the main decay modes. In this paper, these strong decay modes are studied with two methods. One is the instantaneous Bethe-Salpeter method within Mandelstam formalism. The other is the combination of the $^3P_0$ model and the former formalism. The total decay widths are 26.3 and 27.3 MeV for the methods with or without the $^3P_0$ vertex, respectively. The ratio of $\Gamma_{D\bar D}$ over $\Gamma_{D\bar D^\ast}$ which changes along with the mass of the initial meson is also presented.Comment: 11 pages, 3 figure