Conference Agenda

Overview and details of the sessions of this conference. Please select a date or location to show only sessions at that day or location. Please select a single session for detailed view (with abstracts and downloads if available).

 
Session Overview
Session
MS181, part 2: Integral and algebraic geometric methods in the study of Gaussian random fields
Time:
Friday, 12/Jul/2019:
10:00am - 12:00pm

Location: Unitobler, F007
30 seats, 59m^2

Presentations
10:00am - 12:00pm

Integral and algebraic geometric methods in the study of Gaussian random fields

Chair(s): David Ginsbourger (Idiap Research Institute and University of Bern, Switzerland), Jean-Marc Azaïs (Institut de Mathématiques de Toulouse)

Integral and algebraic geometry are at the heart of a number of contributions pertaining to the study of Gaussian random fields and related topics, not only from probabilistic and statistical viewpoints but also from the realm of interpolation and function approximation. This minisymposium will gather a team of junior researchers and established experts presenting original research results reflecting diverse challenges of geometrical and applied geometrical nature primarily involving Gaussian fields.

These encompass the study of geometrical and topological properties of sets implicitly defined by random fields such as zeros of random polynomials, excursion sets, as well as integral curves stemming for instance from filament estimation. Also, Gaussian field approximations dedicated to the estimation of excursion probabilities and more general geometric questions will be tackled, as well as algebraic methods in sparse grids for polynomial and Gaussian process interpolation.

 

(25 minutes for each presentation, including questions, followed by a 5-minute break; in case of x<4 talks, the first x slots are used unless indicated otherwise)

 

On some Karhunen-Loève expansions related to two-point homogeneous spaces

Jean-Renaud Pycke
Université d'Évry Val d'Essonne

Karhunen-Loève expansions provide a powerful tool in the study of Gaussian processes. However their statistical applications are not straightforward, because whereas the existence of such developments is assured by some general theorems, explicit computations remain cumbersome. These developments involve families of orthogonal functions and the theory of classical orthogonal polynomials provides such families. Some of them are connected to families of Riemannian manifolds enjoying remarkable symmetries. By using this interplay between geometry and special functions we derive families of explicit Karhunen-Loève expansions and some of the statistical tests that can be based upon them.

 

Geometry-driven finite-rank approximations of Gaussian random fields

Cédric Travelletti1, David Ginsbourger2, Dario Azzimonti3
1Idiap Research Institute and University of Bern, 2Idiap Research Institute and Unversity of Bern, 3Istituto "Dalle Molle" di Studi sull'Intelligenza Artificiale

We investigate new approaches to uncertainty quantification on target regions that are implicitly defined by a Gaussian random field, such as level and excursion sets of the field itself or derivatives thereof. The key idea is to appeal to finite rank approximations of the field with respect to linear functionals tailored so as to best capture geometric features of interest, contrasting with the L^2 optimality property of the celebrated Karhunen-Loève expansion. The inclusion of linear forms provides a natural link to Bayesian linear inverse problems, which we will exemplify through geophysical applications.

 

Algebraic methods in sparse grids for interpolation

Henry Wynn1, Hugo Maruri-Aguilar2
1London School of Economics, 2Queen Mary University of London

Sparse grids are specially construction for designs or sets of quadrature points used for polynomial interpolation and quadrature in solving differential equations with stochastic inputs. The grids are unions of tensor grids which in the so called nested case can be derived by permutation of levels from a reference grid which has a special hierarchical structure. It is shown how this structure gives rise to a monomial ideal and that the inclusion exclusion (IE) used to unravel the grid can be derived from the Hilbert series of the ideal and the coefficients use in the IE are the Betti number based on the minimal free resolution of the ideal. Remarkably, this IE structure carries over to the polynomial interpolators, not only for the reference grid but also the sparse grid itself. The considerable reduction in complexity achieved by using the algebraic method leads to the sparsity of the matrices used in the interpolation allowing the methods to be used in very high dimensions. The construction carries over to Gaussian Process interpolation when the covariance is of product type. Finally, the spacing of the grid need not be uniform but can be chosen to achieve optimal approximation.