Heat Kernel for Open Manifolds

In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a differential forms heat kernel on open manifolds of bounded geometry was proven. In that paper, it was shown that the heat kernel obeyed certain properties, one of which was a relationship between the derivative of heat kernel of different degrees. We will give a proof of this condition for complete manifolds with Ricci curvature bounded below, and then use it to give an integral representation of the heat kernel of degree $k$

Regularization Paths for Generalized Linear Models via Coordinate Descent

We develop fast algorithms for estimation of generalized linear models with convex penalties. The models include linear regression, two-class logistic regression, and multi- nomial regression problems while the penalties include âÂÂ_1 (the lasso), âÂÂ_2 (ridge regression) and mixtures of the two (the elastic net). The algorithms use cyclical coordinate descent, computed along a regularization path. The methods can handle large problems and can also deal efficiently with sparse features. In comparative timings we find that the new algorithms are considerably faster than competing methods.

The Volatility Trend of Protosolar and Terrestrial Elemental Abundances

We present new estimates of protosolar elemental abundances based on an improved combination of solar photospheric abundances and CI chondritic abundances. These new estimates indicate CI chondrites and solar abundances are consistent for 60 elements. We compare our new protosolar abundances with our recent estimates of bulk Earth composition (normalized to aluminium), thereby quantifying the devolatilization in going from the solar nebula to the formation of the Earth. The quantification yields a linear trend $\log(f) = \alpha\log(T_C) + \beta$, where $f$ is the Earth-to-Sun abundance ratio and $T_C$ is the 50$\%$ condensation temperature of elements. The best fit coefficients are: $\alpha = 3.676\pm 0.142$ and $\beta = -11.556\pm 0.436$. The quantification of these parameters constrains models of devolatilization processes. For example, the coefficients $\alpha$ and $\beta$ determine a critical devolatilization temperature for the Earth $T_{\mathrm{D}}(\mathrm{E}) = 1391 \pm 15$ K. The terrestrial abundances of elements with $T_{C} < T_{\mathrm{D}}(\mathrm{E})$ are depleted compared with solar abundances, whereas the terrestrial abundances of elements with $T_{C} > T_{\mathrm{D}}(\mathrm{E})$ are indistinguishable from solar abundances. The terrestrial abundance of Hg ($T_C$ = 252 K) appears anomalously high under the assumption that solar and CI chondrite Hg abundances are identical. To resolve this anomaly, we propose that CI chondrites have been depleted in Hg relative to the Sun by a factor of $13\pm7$. We use the best-fit volatility trend to derive the fractional distribution of carbon and oxygen between volatile and refractory components ($f_\mathrm{vol}$, $f_\mathrm{ref}$). We find ($0.91\pm 0.08$, $0.09 \pm 0.08$) for carbon and ($0.80 \pm 0.04$, $0.20 \pm 0.04$) for oxygen.Comment: Accepted for publication in Icarus. 28 pages, 12 figures, 5 tables. Compared to v1, the results and conclusion are the same, while discussion of results and implications is expanded considerabl

Envisat - taking the measure of North Atlantic storms

Envisat carries a number of sensors able to provide quantitative information on raining clouds: AATSR delivers information on cloud microphysics (particle size, temperature etc.), MWR-2 gives columnar totals for liquid and vapour forms of water, and RA-2 yields rain rate and wind speed. This paper examines the complementarity of these sensors, with a focussed study on significant rain events in the N. Atlantic, covering both coherent large storms and fronts with smaller scale structure. The difference in liquid water estimates from the infra-red and passive systems appears to be related to the temperature and sizes of drops being detected

Practical bounds on the error of Bayesian posterior approximations: A nonasymptotic approach

Bayesian inference typically requires the computation of an approximation to the posterior distribution. An important requirement for an approximate Bayesian inference algorithm is to output high-accuracy posterior mean and uncertainty estimates. Classical Monte Carlo methods, particularly Markov Chain Monte Carlo, remain the gold standard for approximate Bayesian inference because they have a robust finite-sample theory and reliable convergence diagnostics. However, alternative methods, which are more scalable or apply to problems where Markov Chain Monte Carlo cannot be used, lack the same finite-data approximation theory and tools for evaluating their accuracy. In this work, we develop a flexible new approach to bounding the error of mean and uncertainty estimates of scalable inference algorithms. Our strategy is to control the estimation errors in terms of Wasserstein distance, then bound the Wasserstein distance via a generalized notion of Fisher distance. Unlike computing the Wasserstein distance, which requires access to the normalized posterior distribution, the Fisher distance is tractable to compute because it requires access only to the gradient of the log posterior density. We demonstrate the usefulness of our Fisher distance approach by deriving bounds on the Wasserstein error of the Laplace approximation and Hilbert coresets. We anticipate that our approach will be applicable to many other approximate inference methods such as the integrated Laplace approximation, variational inference, and approximate Bayesian computationComment: 22 pages, 2 figure

Truncated Random Measures

Completely random measures (CRMs) and their normalizations are a rich source of Bayesian nonparametric priors. Examples include the beta, gamma, and Dirichlet processes. In this paper we detail two major classes of sequential CRM representations---series representations and superposition representations---within which we organize both novel and existing sequential representations that can be used for simulation and posterior inference. These two classes and their constituent representations subsume existing ones that have previously been developed in an ad hoc manner for specific processes. Since a complete infinite-dimensional CRM cannot be used explicitly for computation, sequential representations are often truncated for tractability. We provide truncation error analyses for each type of sequential representation, as well as their normalized versions, thereby generalizing and improving upon existing truncation error bounds in the literature. We analyze the computational complexity of the sequential representations, which in conjunction with our error bounds allows us to directly compare representations and discuss their relative efficiency. We include numerous applications of our theoretical results to commonly-used (normalized) CRMs, demonstrating that our results enable a straightforward representation and analysis of CRMs that has not previously been available in a Bayesian nonparametric context.Comment: To appear in Bernoulli; 58 pages, 3 figure