Quantifying uncertainties on excursion sets under a Gaussian random field prior

We focus on the problem of estimating and quantifying uncertainties on the excursion set of a function under a limited evaluation budget. We adopt a Bayesian approach where the objective function is assumed to be a realization of a Gaussian random field. In this setting, the posterior distribution on the objective function gives rise to a posterior distribution on excursion sets. Several approaches exist to summarize the distribution of such sets based on random closed set theory. While the recently proposed Vorob'ev approach exploits analytical formulae, further notions of variability require Monte Carlo estimators relying on Gaussian random field conditional simulations. In the present work we propose a method to choose Monte Carlo simulation points and obtain quasi-realizations of the conditional field at fine designs through affine predictors. The points are chosen optimally in the sense that they minimize the posterior expected distance in measure between the excursion set and its reconstruction. The proposed method reduces the computational costs due to Monte Carlo simulations and enables the computation of quasi-realizations on fine designs in large dimensions. We apply this reconstruction approach to obtain realizations of an excursion set on a fine grid which allow us to give a new measure of uncertainty based on the distance transform of the excursion set. Finally we present a safety engineering test case where the simulation method is employed to compute a Monte Carlo estimate of a contour line

Knowledge-based systems and geological survey

This personal and pragmatic review of the philosophy underpinning methods of geological surveying suggests that important influences of information technology have yet to make their impact. Early approaches took existing systems as metaphors, retaining the separation of maps, map explanations and information archives, organised around map sheets of fixed boundaries, scale and content. But system design should look ahead: a computer-based knowledge system for the same purpose can be built around hierarchies of spatial objects and their relationships, with maps as one means of visualisation, and information types linked as hypermedia and integrated in mark-up languages. The system framework and ontology, derived from the general geoscience model, could support consistent representation of the underlying concepts and maintain reference information on object classes and their behaviour. Models of processes and historical configurations could clarify the reasoning at any level of object detail and introduce new concepts such as complex systems. The up-to-date interpretation might centre on spatial models, constructed with explicit geological reasoning and evaluation of uncertainties. Assuming (at a future time) full computer support, the field survey results could be collected in real time as a multimedia stream, hyperlinked to and interacting with the other parts of the system as appropriate. Throughout, the knowledge is seen as human knowledge, with interactive computer support for recording and storing the information and processing it by such means as interpolating, correlating, browsing, selecting, retrieving, manipulating, calculating, analysing, generalising, filtering, visualising and delivering the results. Responsibilities may have to be reconsidered for various aspects of the system, such as: field surveying; spatial models and interpretation; geological processes, past configurations and reasoning; standard setting, system framework and ontology maintenance; training; storage, preservation, and dissemination of digital records

Combinatorial optimization over two random point sets

We analyze combinatorial optimization problems over a pair of random point sets of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set, or the connected bipartite r-regular graph of minimal length. As the cardinal of the sets goes to infinity, we investigate the convergence of such bipartite functionals.Comment: 34 page

Texture Segregation By Visual Cortex: Perceptual Grouping, Attention, and Learning

A neural model is proposed of how laminar interactions in the visual cortex may learn and recognize object texture and form boundaries. The model brings together five interacting processes: region-based texture classification, contour-based boundary grouping, surface filling-in, spatial attention, and object attention. The model shows how form boundaries can determine regions in which surface filling-in occurs; how surface filling-in interacts with spatial attention to generate a form-fitting distribution of spatial attention, or attentional shroud; how the strongest shroud can inhibit weaker shrouds; and how the winning shroud regulates learning of texture categories, and thus the allocation of object attention. The model can discriminate abutted textures with blurred boundaries and is sensitive to texture boundary attributes like discontinuities in orientation and texture flow curvature as well as to relative orientations of texture elements. The model quantitatively fits a large set of human psychophysical data on orientation-based textures. Object boundar output of the model is compared to computer vision algorithms using a set of human segmented photographic images. The model classifies textures and suppresses noise using a multiple scale oriented filterbank and a distributed Adaptive Resonance Theory (dART) classifier. The matched signal between the bottom-up texture inputs and top-down learned texture categories is utilized by oriented competitive and cooperative grouping processes to generate texture boundaries that control surface filling-in and spatial attention. Topdown modulatory attentional feedback from boundary and surface representations to early filtering stages results in enhanced texture boundaries and more efficient learning of texture within attended surface regions. Surface-based attention also provides a self-supervising training signal for learning new textures. Importance of the surface-based attentional feedback in texture learning and classification is tested using a set of textured images from the Brodatz micro-texture album. Benchmark studies vary from 95.1% to 98.6% with attention, and from 90.6% to 93.2% without attention.Air Force Office of Scientific Research (F49620-01-1-0397, F49620-01-1-0423); National Science Foundation (SBE-0354378); Office of Naval Research (N00014-01-1-0624

The three-dimensional random field Ising magnet: interfaces, scaling, and the nature of states

The nature of the zero temperature ordering transition in the 3D Gaussian random field Ising magnet is studied numerically, aided by scaling analyses. In the ferromagnetic phase the scaling of the roughness of the domain walls, $w\sim L^\zeta$, is consistent with the theoretical prediction $\zeta = 2/3$. As the randomness is increased through the transition, the probability distribution of the interfacial tension of domain walls scales as for a single second order transition. At the critical point, the fractal dimensions of domain walls and the fractal dimension of the outer surface of spin clusters are investigated: there are at least two distinct physically important fractal dimensions. These dimensions are argued to be related to combinations of the energy scaling exponent, $\theta$, which determines the violation of hyperscaling, the correlation length exponent $\nu$, and the magnetization exponent $\beta$. The value $\beta = 0.017\pm 0.005$ is derived from the magnetization: this estimate is supported by the study of the spin cluster size distribution at criticality. The variation of configurations in the interior of a sample with boundary conditions is consistent with the hypothesis that there is a single transition separating the disordered phase with one ground state from the ordered phase with two ground states. The array of results are shown to be consistent with a scaling picture and a geometric description of the influence of boundary conditions on the spins. The details of the algorithm used and its implementation are also described.Comment: 32 pp., 2 columns, 32 figure