The rate of increase of mean values of functions in weighted Hardy spaces

Let $0<p<\infty$ and $0\leq q<\infty$. For each $f$ in the weighted Hardy space $H_{p, q}$,\ we show that $d\|f_r\|_{p,q}^p/dr$ grows at most like $o(1/1- r)$ as $r\rightarrow 1$.Comment: 5 pages, accepted for publication in Acta Mathematica Vietnamic

High-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamics

This paper develops the high-order accurate entropy stable finite difference schemes for one- and two-dimensional special relativistic hydrodynamic equations. The schemes are built on the entropy conservative flux and the weighted essentially non-oscillatory (WENO) technique as well as explicit Runge-Kutta time discretization. The key is to technically construct the affordable entropy conservative flux of the semi-discrete second-order accurate entropy conservative schemes satisfying the semi-discrete entropy equality for the found convex entropy pair. As soon as the entropy conservative flux is derived, the dissipation term can be added to give the semi-discrete entropy stable schemes satisfying the semi-discrete entropy inequality with the given convex entropy function. The WENO reconstruction for the scaled entropy variables and the high-order explicit Runge-Kutta time discretization are implemented to obtain the fully-discrete high-order schemes. Several numerical tests are conducted to validate the accuracy and the ability to capture discontinuities of our entropy stable schemes.Comment: 26 pages, 14 figure

Order boundedness of weighted composition operators on weighted Dirichlet spaces and derivative Hardy spaces

In this paper, we completely characterize the order boundedness of weighted composition operators between different weighted Dirichlet spaces and different derivative Hardy spaces.Comment: 9 page

Group Sparse Additive Models

We consider the problem of sparse variable selection in nonparametric additive models, with the prior knowledge of the structure among the covariates to encourage those variables within a group to be selected jointly. Previous works either study the group sparsity in the parametric setting (e.g., group lasso), or address the problem in the non-parametric setting without exploiting the structural information (e.g., sparse additive models). In this paper, we present a new method, called group sparse additive models (GroupSpAM), which can handle group sparsity in additive models. We generalize the l1/l2 norm to Hilbert spaces as the sparsity-inducing penalty in GroupSpAM. Moreover, we derive a novel thresholding condition for identifying the functional sparsity at the group level, and propose an efficient block coordinate descent algorithm for constructing the estimate. We demonstrate by simulation that GroupSpAM substantially outperforms the competing methods in terms of support recovery and prediction accuracy in additive models, and also conduct a comparative experiment on a real breast cancer dataset.Comment: ICML201

Multi-Label Annotation Aggregation in Crowdsourcing

As a means of human-based computation, crowdsourcing has been widely used to annotate large-scale unlabeled datasets. One of the obvious challenges is how to aggregate these possibly noisy labels provided by a set of heterogeneous annotators. Another challenge stems from the difficulty in evaluating the annotator reliability without even knowing the ground truth, which can be used to build incentive mechanisms in crowdsourcing platforms. When each instance is associated with many possible labels simultaneously, the problem becomes even harder because of its combinatorial nature. In this paper, we present new flexible Bayesian models and efficient inference algorithms for multi-label annotation aggregation by taking both annotator reliability and label dependency into account. Extensive experiments on real-world datasets confirm that the proposed methods outperform other competitive alternatives, and the model can recover the type of the annotators with high accuracy. Besides, we empirically find that the mixture of multiple independent Bernoulli distribution is able to accurately capture label dependency in this unsupervised multi-label annotation aggregation scenario.Comment: 15 pages, 7 figure

Graph Clustering with Density-Cut

How can we find a good graph clustering of a real-world network, that allows insight into its underlying structure and also potential functions? In this paper, we introduce a new graph clustering algorithm Dcut from a density point of view. The basic idea is to envision the graph clustering as a density-cut problem, such that the vertices in the same cluster are densely connected and the vertices between clusters are sparsely connected. To identify meaningful clusters (communities) in a graph, a density-connected tree is first constructed in a local fashion. Owing to the density-connected tree, Dcut allows partitioning a graph into multiple densely tight-knit clusters directly. We demonstrate that our method has several attractive benefits: (a) Dcut provides an intuitive criterion to evaluate the goodness of a graph clustering in a more natural and precise way; (b) Built upon the density-connected tree, Dcut allows identifying the meaningful graph clusters of densely connected vertices efficiently; (c) The density-connected tree provides a connectivity map of vertices in a graph from a local density perspective. We systematically evaluate our new clustering approach on synthetic as well as real data to demonstrate its good performance

Convex-constrained Sparse Additive Modeling and Its Extensions

Sparse additive modeling is a class of effective methods for performing high-dimensional nonparametric regression. In this work we show how shape constraints such as convexity/concavity and their extensions, can be integrated into additive models. The proposed sparse difference of convex additive models (SDCAM) can estimate most continuous functions without any a priori smoothness assumption. Motivated by a characterization of difference of convex functions, our method incorporates a natural regularization functional to avoid overfitting and to reduce model complexity. Computationally, we develop an efficient backfitting algorithm with linear per-iteration complexity. Experiments on both synthetic and real data verify that our method is competitive against state-of-the-art sparse additive models, with improved performance in most scenarios.Comment: 17 pages, 2 figure

Norm-attaining integral operators on analytic function spaces

Any bounded analytic function $g$ induces a bounded integral operator $S_g$ on the Bloch space, the Dirichlet space and $BMOA$ respectively. $S_g$ attains its norm on the Bloch space and $BMOA$ for any $g$, but does not attain its norm on the Dirichlet space for non-constant $g$. Some results are also obtained for $S_g$ on the little Bloch space, and for another integral operator $T_g$ from the Dirichlet space to the Bergman space.Comment: 9 page

Analytic version of critical QQ spaces and their properties

In this paper, we establish an analytic version of critical spaces $Q_{\alpha}^{\beta}(\mathbb{R}^{n})$ on unit disc $\mathbb{D}$, denoted by $Q^{\beta}_{p}(\mathbb{D})$. Further we prove a relation between $Q^{\beta}_{p}(\mathbb{D})$ and Morrey spaces. By the boundedness of two integral operators, we give the multiplier spaces of $Q^{\beta}_{p}(\mathbb{D})$

Strict singularity of Volterra type operators on Hardy spaces

In this paper, we first characterize the boundedness and compactness of Volterra type operator $S_gf(z) = \int_0^z f'(\zeta)g(\zeta)d\zeta, \ z \in \mathbb{D},$ defined on Hardy spaces $H^p, \, 0< p <\infty$. The spectrum of $S_g$ is also obtained. Then we prove that $S_g$ fixes an isomorphic copy of $\ell^p$ and an isomorphic copy of $\ell^2$ if the operator $S_g$ is not compact on $H^p (1\leq p<\infty)$. In particular, this implies that the strict singularity of the operator $S_g$ coincides with the compactness of the operator $S_g$ on $H^p$. At last, we post an open question for further study.Comment: 9 page