Searcharxiv⌕ Search

arXiv subjects

Xu Wang

Publications and source records attributed to Xu Wang.

At least 469 records · Page 26Linked to original sources
arXiv2016

Entropy of a semigroup of maps from a set-valued view

In this paper, we introduce a new entropy-like invariant, named Hausdorff metric entropy, for finitely generated semigroups acting on compact metric spaces from a set-valued view and study its properties. We establish the relation between Hausdorff metric entropy and topological entropy of a semigroup defined by Biś. Some examples with positive or zero Hausdorff metric entropy are given. Moreover, some notions of chaos are also well generalized for finitely generated semigroups from a set-valued view.

math.DS↗
arXiv2015

Fast Landmark Subspace Clustering

Kernel methods obtain superb performance in terms of accuracy for various machine learning tasks since they can effectively extract nonlinear relations. However, their time complexity can be rather large especially for clustering tasks. In this paper we define a general class of kernels that can be easily approximated by randomization. These kernels appear in various applications, in particular, traditional spectral clustering, landmark-based spectral clustering and landmark-based subspace clustering. We show that for $n$ data points from $K$ clusters with $D$ landmarks, the randomization procedure results in an algorithm of complexity $O(KnD)$. Furthermore, we bound the error between the original clustering scheme and its randomization. To illustrate the power of this framework, we propose a new fast landmark subspace (FLS) clustering algorithm. Experiments over synthetic and real datasets demonstrate the superior performance of FLS in accelerating subspace clustering with marginal sacrifice of accuracy.

stat.ML↗
arXiv2015

Spectral Convergence Rate of Graph Laplacian

Laplacian Eigenvectors of the graph constructed from a data set are used in many spectral manifold learning algorithms such as diffusion maps and spectral clustering. Given a graph constructed from a random sample of a $d$-dimensional compact submanifold $M$ in $\mathbb{R}^D$, we establish the spectral convergence rate of the graph Laplacian. It implies the consistency of the spectral clustering algorithm via a standard perturbation argument. A simple numerical study indicates the necessity of a denoising step before applying spectral algorithms.

stat.ML↗
arXiv2015

Nonparametric Bayesian Regression on Manifolds via Brownian Motion

This paper proposes a novel framework for manifold-valued regression and establishes its consistency as well as its contraction rate. It assumes a predictor with values in the interval $[0,1]$ and response with values in a compact Riemannian manifold $M$. This setting is useful for applications such as modeling dynamic scenes or shape deformations, where the visual scene or the deformed objects can be modeled by a manifold. The proposed framework is nonparametric and uses the heat kernel (and its associated Brownian motion) on manifolds as an averaging procedure. It directly generalizes the use of the Gaussian kernel (as a natural model of additive noise) in vector-valued regression problems. In order to avoid explicit dependence on estimates of the heat kernel, we follow a Bayesian setting, where Brownian motion on $M$ induces a prior distribution on the space of continuous functions $C([0,1], M)$. For the case of discretized Brownian motion, we establish the consistency of the posterior distribution in terms of the $L_{q}$ distances for any $1 \leq q < \infty$. Most importantly, we establish contraction rate of order $O(n^{-1/4+ε})$ for any fixed $ε>0$, where $n$ is the number of observations. For the continuous Brownian motion we establish weak consistency.

math.ST↗
arXiv2014

Impact of Wavelength and Modulation Conversion on Transluscent Elastic Optical Networks Using MILP

Compared to legacy wavelength division multiplexing networks, elastic optical networks (EON) have added flexibility to network deployment and management. EONs can include previously available technology, such as signal regeneration and wavelength conversion, as well as new features such as finer-granularity spectrum assignment and modulation conversion. Yet each added feature adds to the cost of the network. In order to quantify the potential benefit of each technology, we present a link-based mixed-integer linear programming (MILP) formulation to solve the optimal resource allocation problem. We then propose a recursive model in order to either augment existing network deployments or speed up the resource allocation computation time for larger networks with higher traffic demand requirements than can be solved using an MILP. We show through simulation that systems equipped with signal regenerators or wavelength converters require a notably smaller total bandwidth, depending on the topology of the network. We also show that the suboptimal recursive solution speeds up the calculation and makes the running-time more predictable, compared to the optimal MILP.

cs.NI↗
arXiv2014

Riemannian Multi-Manifold Modeling

This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold $M$ into clusters lying around low-dimensional submanifolds of $M$. Important examples of $M$, for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the Grassmannian. The clustering problem with these examples of $M$ is already useful for numerous application domains such as action identification in video sequences, dynamic texture clustering, brain fiber segmentation in medical imaging, and clustering of deformed images. The proposed clustering algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. The intrinsic local geometry is encoded by local sparse coding and more importantly by directional information of local tangent spaces and geodesics. Theoretical guarantees are established for a simplified variant of the algorithm even when the clusters intersect. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical model as well as its superior performance over state-of-the-art techniques.

stat.ML↗
arXiv2014

Ohsawa-Takegoshi type theorem and extension of plurisubharmonic functions

We prove a Thullen type extension theorem of plurisubharmonic functions across a closed complete pluripolar set, which generalizes a theorem of Siu. Our approach depends on an Ohsawa-Takegoshi type extension theorem for a single point in a bounded complete Kähler domain, which is of independent interest.

math.CV↗
arXiv2014

Holomorphic maps with large images

We show that each pseudoconvex domain $Ω\subset {\mathbb C}^n$ admits a holomorphic map $F$ to ${\mathbb C}^m$ with $|F|\le C_1 e^{C_2 \hatδ^{-6}}$, where $\hatδ$ is the minimum of the boundary distance and $(1+|z|^2)^{-1/2}$, such that every boundary point is a Casorati-Weierstrass point of $F$. Based on this fact, we introduce a new anti-hyperbolic concept --- universal dominability. We also show that for each $α>6$ and each pseudoconvex domain $Ω\subset {\mathbb C}^n$, there is a holomorphic function $f$ on $Ω$ with $|f|\le C_αe^{C_α' \hatδ^{-α}}$, such that every boundary point is a Picard point of $F$. Applications to the construction of holomorphic maps of a given domain onto some ${\mathbb C}^m$ are given.

math.CV↗
arXiv2014

Variation of Bergman kernels of pseudoconvex domains

We shall give a variational formula of the full Bergman kernels associated to a family of smoothly bounded strongly pseudoconvex domains. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.

math.CV↗
arXiv2013

Variation of the Bergman kernels under deformations of complex structures

Inspired by Berndtsson's work on the subharmonicity property of the Bergman kernel, we give a local variation formula of the full Bergman kernels associated to deformations of complex manifolds. In compact case, it follows from the reproducing property of the Bergman kernel and the curvature formula of the 0-th direct image sheaf. In general, following Schumacher's idea, we use the Lie derivative to compute the variation. An equivalent criterion for the triviality of holomorphic motions of planar domains in terms of the Bergman kernel is given as an application.

math.CV↗
arXiv2013

Covering stability of Bergman kernels on Kähler hyperbolic manifolds

This paper is a sequel to \cite{Xu}. In this paper, an estimation of the Bergman Kernel of Kähler hyperbolic manifold is given by the $L^2$ estimate and the Bochner formula. As an application, an effective criterion of the very ampleness of the canonical line bundle of Kähler hyperbolic manifold is given, which is a generalization of Yeung's result.

math.CV↗
arXiv2013

Big Deformation in 17C

Reaction- and interaction cross sections of 17C on a carbon target have been reanalyzed by use of the modified Glauber model. The analysis with a deformed Woods-Saxon density/potential suggests a big deformation structure for 17C. The existence of a tail in the density distribution supports its possibility of being a one-neutron halo structure. Under a deformed core plus single-particle assumption, analysis shows a dominant d-wave of the valence neutron in 17C. Key words Cross section, Glauber model, Density distribution, Halo, Deformation.

nucl-th↗
arXiv2013

Revisiting Timed Specification Theory II : Realisability

In this paper we present an assume-guarantee specification theory (aka interface theory from [14]) for modular synthesis and verification of real-time systems with critical timing constraints. It is a further step of our earlier work [10] which achieved an elegant algebraic specification theory for real-time systems endowed with the capability to freeze time. In this paper we relinquish such (unrealisable) capability and target more realistic systems without the ability to stop time. Our theory, in a combined process-algebraic and reactive-synthesis style, provides the operations of parallel composition for system integration, logical conjunction/disjunction for viewpoint fusion and independent development, and quotient for incremental synthesis. We show that a substitutive refinement preorder, which is a coarsening of the pre-congruence in [10], constitutes the weakest pre-congruence preserving freedom of incompatibility errors. The coarsening requires a shift in the focus of our theory to a more game-theoretical treatment, where the coarsening constitutes a reactive synthesis game named normalisation and is efficiently implementable by a novel local bot-backpropagation algorithm. Previously, timed concurrent games have been studied in [1,14,13], where one of the key concern is the removal of time-blocking strategies by applying blame assignment [13]. Our timed games also have the issue of time-blocking strategies, which may arise through the composition of specifications. However, due to our distinctively different formulation of timed games, we have discovered another elegant solution to the problem without blame assignment. Our solution utilises a second reactive synthesis game called realisation, which is dual to normalisation and implementable by the dual local top-backpropagation algorithm.

cs.LO↗
arXiv2012

Revisiting Timed Specification Theories: A Linear-Time Perspective

We consider the setting of component-based design for real-time systems with critical timing constraints. Based on our earlier work, we propose a compositional specification theory for timed automata with I/O distinction, which supports substitutive refinement. Our theory provides the operations of parallel composition for composing components at run-time, logical conjunction/disjunction for independent development, and quotient for incremental synthesis. The key novelty of our timed theory lies in a weakest congruence preserving safety as well as bounded liveness properties. We show that the congruence can be characterised by two linear-time semantics, timed-traces and timed-strategies, the latter of which is derived from a game-based interpretation of timed interaction.

cs.SE↗
arXiv2012

Classical Theory of High Field Atomic Ionization Using Elliptical Polarization

Important information about strong-field atomic or molecular ionization can be missed when using linearly polarized laser fields. The field strength at which an electron was ionized, or the time during a pulse of the ionization event are examples of such missing information. In treating single, double, and triple ionization events we show that information of this kind is made readily available by use of elliptical polarization.

physics.atom-ph↗