Rates of Convergence for Sparse Variational Gaussian Process Regression

Excellent variational approximations to Gaussian process posteriors have been developed which avoid the O(N³) scaling with dataset size N. They reduce the computational cost to O(NM²), with M≪N being the number of inducing variables, which summarise the process. While the computational cost seems to be linear in N, the true complexity of the algorithm depends on how M must increase to ensure a certain quality of approximation. We address this by characterising the behavior of an upper bound on the KL divergence to the posterior. We show that with high probability the KL divergence can be made arbitrarily small by growing M more slowly than N. A particular case of interest is that for regression with normally distributed inputs in D-dimensions with the popular Squared Exponential kernel, M = O(log^DN) is sufficient. Our results show that as datasets grow, Gaussian process posteriors can truly be approximated cheaply, and provide a concrete rule for how to increase M in continual learning scenarios

Open problems in artificial life

This article lists fourteen open problems in artificial life, each of which is a grand challenge requiring a major advance on a fundamental issue for its solution. Each problem is briefly explained, and, where deemed helpful, some promising paths to its solution are indicated

Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees

Gaussian processes are frequently deployed as part of larger machine learning and decision-making systems, for instance in geospatial modeling, Bayesian optimization, or in latent Gaussian models. Within a system, the Gaussian process model needs to perform in a stable and reliable manner to ensure it interacts correctly with other parts of the system. In this work, we study the numerical stability of scalable sparse approximations based on inducing points. To do so, we first review numerical stability, and illustrate typical situations in which Gaussian process models can be unstable. Building on stability theory originally developed in the interpolation literature, we derive sufficient and in certain cases necessary conditions on the inducing points for the computations performed to be numerically stable. For low-dimensional tasks such as geospatial modeling, we propose an automated method for computing inducing points satisfying these conditions. This is done via a modification of the cover tree data structure, which is of independent interest. We additionally propose an alternative sparse approximation for regression with a Gaussian likelihood which trades off a small amount of performance to further improve stability. We provide illustrative examples showing the relationship between stability of calculations and predictive performance of inducing point methods on spatial tasks

Design innovation for the 1990's

Statement of responsibility on title-page reads: Richard K. Lester, Michael J. Driscoll, Michael W. Golay, David D. Lanning, Lawrence M. Lidsky, Norman C. Rasmussen and Neil E. Todreas"September 1983."Includes bibliographical reference

The Origin of the Hot Gas in the Galactic Halo: Confronting Models with XMM-Newton Observations

We compare the predictions of three physical models for the origin of the hot halo gas with the observed halo X-ray emission, derived from 26 high-latitude XMM-Newton observations of the soft X-ray background between l=120\degr and l=240\degr. These observations were chosen from a much larger set of observations as they are expected to be the least contaminated by solar wind charge exchange emission. We characterize the halo emission in the XMM-Newton band with a single-temperature plasma model. We find that the observed halo temperature is fairly constant across the sky (~1.8e6-2.3e6 K), whereas the halo emission measure varies by an order of magnitude (~0.0005-0.006 cm^-6 pc). When we compare our observations with the model predictions, we find that most of the hot gas observed with XMM-Newton does not reside in isolated extraplanar supernova remnants -- this model predicts emission an order of magnitude too faint. A model of a supernova-driven interstellar medium, including the flow of hot gas from the disk into the halo in a galactic fountain, gives good agreement with the observed 0.4-2.0 keV surface brightness. This model overpredicts the halo X-ray temperature by a factor of ~2, but there are a several possible explanations for this discrepancy. We therefore conclude that a major (possibly dominant) contributor to the halo X-ray emission observed with XMM-Newton is a fountain of hot gas driven into the halo by disk supernovae. However, we cannot rule out the possibility that the extended hot halo of accreted material predicted by disk galaxy formation models also contributes to the emission.Comment: 20 pages, 14 figures. New version accepted for publication in ApJ. Changes include new section discussing systematic errors (Section 3.2), improved method for characterizing our model spectra (4.2.2), changes to discussion of other observations (5.1). Note that we can no longer rule out possibility that extended hot halo of accreted material contributes to observed halo emission (see 5.2.1

Semaglutide and cardiovascular outcomes by baseline HbA1c in diabetes: the SUSTAIN 6 and PIONEER 6 trials.

No abstract available

Type-Ia Supernova-driven Galactic Bulge Wind

Stellar feedback in galactic bulges plays an essential role in shaping the evolution of galaxies. To quantify this role and facilitate comparisons with X-ray observations, we conduct 3D hydrodynamical simulations with the adaptive mesh refinement code, FLASH, to investigate the physical properties of hot gas inside a galactic bulge, similar to that of our Galaxy or M31. We assume that the dynamical and thermal properties of the hot gas are dominated by mechanical energy input from SNe, primarily Type Ia, and mass injection from evolved stars as well as iron enrichment from SNe. We study the bulge-wide outflow as well as the SN heating on scales down to ~4 pc. An embedding scheme that is devised to plant individual SNR seeds, allows to examine, for the first time, the effect of sporadic SNe on the density, temperature, and iron ejecta distribution of the hot gas as well as the resultant X-ray morphology and spectrum. We find that the SNe produce a bulge wind with highly filamentary density structures and patchy ejecta. Compared with a 1D spherical wind model, the non-uniformity of simulated gas density, temperature, and metallicity substantially alters the spectral shape and increases the diffuse X-ray luminosity. The differential emission measure as a function of temperature of the simulated gas exhibits a log-normal distribution, with a peak value much lower than that of the corresponding 1D model. The bulk of the X-ray emission comes from the relatively low temperature and low abundance gas shells associated with SN blastwaves. SN ejecta are not well mixed with the ambient medium, at least in the bulge region. These results, at least partly, account for the apparent lack of evidence for iron enrichment in the soft X-ray-emitting gas in galactic bulges and intermediate-mass elliptical galaxies.[...]Comment: 37 pages, 19 figures, submitted to MNRAS; comments are welcom