Resampling methods for spatial regression models under a class of stochastic designs

In this paper we consider the problem of bootstrapping a class of spatial regression models when the sampling sites are generated by a (possibly nonuniform) stochastic design and are irregularly spaced. It is shown that the natural extension of the existing block bootstrap methods for grid spatial data does not work for irregularly spaced spatial data under nonuniform stochastic designs. A variant of the blocking mechanism is proposed. It is shown that the proposed block bootstrap method provides a valid approximation to the distribution of a class of M-estimators of the spatial regression parameters. Finite sample properties of the method are investigated through a moderately large simulation study and a real data example is given to illustrate the methodology.Comment: Published at http://dx.doi.org/10.1214/009053606000000551 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Exact heat kernel on a hypersphere and its applications in kernel SVM

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of using heat diffusion on a hypersphere to measure similarity has been previously proposed, demonstrating promising results based on a heuristic heat kernel obtained from the zeroth order parametrix expansion; however, how well this heuristic kernel agrees with the exact hyperspherical heat kernel remains unknown. This paper presents a higher order parametrix expansion of the heat kernel on a unit hypersphere and discusses several problems associated with this expansion method. We then compare the heuristic kernel with an exact form of the heat kernel expressed in terms of a uniformly and absolutely convergent series in high-dimensional angular momentum eigenmodes. Being a natural measure of similarity between sample points dwelling on a hypersphere, the exact kernel often shows superior performance in kernel SVM classifications applied to text mining, tumor somatic mutation imputation, and stock market analysis

Open string instantons and relative stable morphisms

We show how topological open string theory amplitudes can be computed by using relative stable morphisms in the algebraic category. We achieve our goal by explicitly working through an example which has been previously considered by Ooguri and Vafa from the point of view of physics. By using the method of virtual localization, we successfully reproduce their results for multiple covers of a holomorphic disc, whose boundary lies in a Lagrangian submanifold of a Calabi-Yau 3-fold, by Riemann surfaces with arbitrary genera and number of boundary components. In particular we show that in the case we consider there are no open string instantons with more than one boundary component ending on the Lagrangian submanifold.Comment: This is the version published by Geometry & Topology Monographs on 22 April 200

A Low-order Model of Water Vapor, Clouds, and Thermal Emission for Tidally Locked Terrestrial Planets

In the spirit of minimal modeling of complex systems, we develop an idealized two-column model to investigate the climate of tidally locked terrestrial planets with Earth-like atmospheres in the habitable zone of M-dwarf stars. The model is able to approximate the fundamental features of the climate obtained from three-dimensional (3D) atmospheric general circulation model (GCM) simulations. One important reason for the two-column model's success is that it reproduces the high cloud albedo of the GCM simulations, which reduces the planet's temperature and delays the onset of a runaway greenhouse state. The two-column model also clearly illustrates a secondary mechanism for determining the climate: the nightside acts as a ``radiator fin'' through which infrared energy can be lost to space easily. This radiator fin is maintained by a temperature inversion and dry air on the nightside, and plays a similar role to the subtropics on modern Earth. Since 1D radiative-convective models cannot capture the effects of the cloud albedo and radiator fin, they are systematically biased towards a narrower habitable zone. We also show that cloud parameters are most important for determining the day--night thermal emission contrast in the two-column model, which decreases and eventually reverses as the stellar flux increases. This reversal is important because it could be detected by future extrasolar planet characterization missions, which would suggest that the planet has Earth-like water clouds and is potentially habitable.Comment: The Astrophysical Journal (in press), 14 pages, 11 figures, 1 tabl

Entanglement and spin-squeezing in a network of distant optical lattice clocks

We propose an approach for collective enhancement of precision for remotely located optical lattice clocks and a way of generation of the Einstein-Podolsky-Rosen state of remote clocks. Close to Heisenberg scaling of the clock precision with the number of clocks M can be achieved even for an optical channel connecting clocks with substantial losses. This scenario utilizes a collective quantum nondemolition measurement on clocks with parallel Bloch vectors for enhanced measurement precision. We provide an optimal network solution for distant clocks as well as for clocks positioned in close proximity of each other. In the second scenario, we employ collective dissipation to drive two clocks with oppositely oriented Bloch vectors into a steady state entanglement. The corresponding EPR entanglement provides enhanced time sharing beyond the projection noise limit between the two quantum synchronized clocks protected from eavesdropping, as well as allows better characterization of systematic effects

Complexity Results for MCMC derived from Quantitative Bounds

This paper considers how to obtain MCMC quantitative convergence bounds which can be translated into tight complexity bounds in high-dimensional settings. We propose a modified drift-and-minorization approach, which establishes a generalized drift condition defined in a subset of the state space. The subset is called the ``large set'' and is chosen to rule out some ``bad'' states which have poor drift property when the dimension gets large. Using the ``large set'' together with a ``centered'' drift function, a quantitative bound can be obtained which can be translated into a tight complexity bound. As a demonstration, we analyze a certain realistic Gibbs sampler algorithm and obtain a complexity upper bound for the mixing time, which shows that the number of iterations required for the Gibbs sampler to converge is constant under certain conditions on the observed data and the initial state. It is our hope that this modified drift-and-minorization approach can be employed in many other specific examples to obtain complexity bounds for high-dimensional Markov chains.Comment: 42 page

Of McKay Correspondence, Non-linear Sigma-model and Conformal Field Theory

The ubiquitous ADE classification has induced many proposals of often mysterious correspondences both in mathematics and physics. The mathematics side includes quiver theory and the McKay Correspondence which relates finite group representation theory to Lie algebras as well as crepant resolutions of Gorenstein singularities. On the physics side, we have the graph-theoretic classification of the modular invariants of WZW models, as well as the relation between the string theory nonlinear $\sigma$-models and Landau-Ginzburg orbifolds. We here propose a unification scheme which naturally incorporates all these correspondences of the ADE type in two complex dimensions. An intricate web of inter-relations is constructed, providing a possible guideline to establish new directions of research or alternate pathways to the standing problems in higher dimensions.Comment: 35 pages, 4 figures; minor corrections, comments on toric geometry and references adde