CR eigenvalue estimate and Kohn-Rossi cohomology

Let $X$ be a compact connected CR manifold with a transversal CR $S^1$-action of dimension $2n-1$, which is only assumed to be weakly pseudoconvex. Let $\Box_b$ be the $\overline{\partial}_b$-Laplacian. Eigenvalue estimate of $\Box_b$ is a fundamental issue both in CR geometry and analysis. In this paper, we are able to obtain a sharp estimate of the number of eigenvalues smaller than or equal to $\lambda$ of $\Box_b$ acting on the $m$-th Fourier components of smooth $(n-1,q)$-forms on $X$, where $m\in \mathbb{Z}_+$ and $q=0,1,\cdots, n-1$. Here the sharp means the growth order with respect to $m$ is sharp. In particular, when $\lambda=0$, we obtain the asymptotic estimate of the growth for $m$-th Fourier components $H^{n-1,q}_{b,m}(X)$ of $H^{n-1,q}_b(X)$ as $m \rightarrow +\infty$. Furthermore, we establish a Serre type duality theorem for Fourier components of Kohn-Rossi cohomology which is of independent interest. As a byproduct, the asymptotic growth of the dimensions of the Fourier components $H^{0,q}_{b,-m}(X)$ for $m\in \mathbb{Z}_+$ is established. Compared with previous results in this field, the estimate for $\lambda=0$ already improves very much the corresponding estimate of Hsiao and Li . We also give appilcations of our main results, including Morse type inequalities, asymptotic Riemann-Roch type theorem, Grauert-Riemenscheider type criterion, and an orbifold version of our main results which answers an open problem.Comment: 39 pages, submitted on January 17, 2018. Comments welcome! arXiv admin note: text overlap with arXiv:1506.06459, arXiv:1502.02365 by other author

Learning Structured Inference Neural Networks with Label Relations

Images of scenes have various objects as well as abundant attributes, and diverse levels of visual categorization are possible. A natural image could be assigned with fine-grained labels that describe major components, coarse-grained labels that depict high level abstraction or a set of labels that reveal attributes. Such categorization at different concept layers can be modeled with label graphs encoding label information. In this paper, we exploit this rich information with a state-of-art deep learning framework, and propose a generic structured model that leverages diverse label relations to improve image classification performance. Our approach employs a novel stacked label prediction neural network, capturing both inter-level and intra-level label semantics. We evaluate our method on benchmark image datasets, and empirical results illustrate the efficacy of our model.Comment: Conference on Computer Vision and Pattern Recognition(CVPR) 201