09:00 to 09:50 Registration 09:50 to 10:00 Welcome from David Abrahams (INI Director) 10:00 to 11:00 Victor Panaretos (EPFL - Ecole Polytechnique Fédérale de Lausanne)Procrustes Analysis of Covariance Operators and Optimal Transport of Gaussian Processes Covariance operators are fundamental in functional data analysis, providing the canonical means to analyse functional variation via the celebrated Karhunen-Loève expansion. These operators may themselves be subject to variation, for instance in contexts where multiple functional populations are to be compared. Statistical techniques to analyse such variation are intimately linked with the choice of metric on covariance operators, and the intrinsic infinite-dimensionality of these operators. We will describe the manifold-like geometry of the space of trace-class infinite-dimensional covariance operators and associated key statistical properties, under the recently proposed infinite-dimensional version of the Procrustes metric. In particular, we will identify this space with that of centred Gaussian processes equipped with the Wasserstein metric of optimal transportation. The identification allows us to provide a description of those aspects of the geometry that are important in terms of statistical inference, and establish key properties of the Fréchet mean of a random sample of covariances, as well as generative models that are canonical for such metrics. The latter will allow us to draw connections with the problem of registration of warped functional data. Based on joint work with V. Masarotto (EPFL) and Y. Zemel (Göttingen). INI 1 11:00 to 11:30 Morning Coffee 11:30 to 12:30 Jim Ramsay (McGill University)Dynamic Smoothing Meets Gravity authors: Jim Ramsay, Michelle Carey and Juan Li institutions: McGill University, University College Dublin, and McGill University Systems of differential equations are often used to model buffering processes that modulate a non-smooth high-energy input so as to produce an output that is smooth and that distributes the energy load over time and space. Handwriting is buffered in this way. We show that the smooth complex script that spells `"statistics" in Chinese can be represented as buffered version of a series of 46 equal-interval step inputs. The buffer consists of three undamped oscillating springs, one for each orthogonal coordinate. The periods of oscillation vary slightly over the three coordinate in a way that reflects the masses that are moved by muscle activations. Our analyses of data on juggling three balls and on lip motion during speech confirm that this model works for a wide variety of human motions. We use the term "dynamic smoothing" for the estimation of a structured input functional object along with the buffer characteristics. INI 1 12:30 to 13:30 Lunch @ Churchill College 13:30 to 14:30 Bodhisattva Sen (Columbia University)Adaptive Confidence Bands for Shape-Restricted Regression in Multi-dimension using Multiscale Tests Co-author: Pratyay Datta (Columbia University) We consider a multidimensional (continuous) Gaussian white noise regression model. We define suitable multiscale tests in this scenario. Utilizing these tests we construct confidence bands for the underlying regression function with guaranteed coverage probability, assuming that the underlying function is isotonic or convex. These confidence bands are shown to be asymptotically optimal in an appropriate sense. Computational issues will also be discussed. INI 1 14:30 to 15:30 Richard Samworth (University of Cambridge)Isotonic regression in general dimensions Co-authors: Qiyang Han (University of Washington), Tengyao Wang (University of Cambridge), Sabyasachi Chatterjee (University of Illinois)We study the least squares regression function estimator over the class of real-valued functions on $[0,1]^d$ that are increasing in each coordinate. For uniformly bounded signals and with a fixed, cubic lattice design, we establish that the estimator achieves the minimax rate of order $n^{−min\{2/(d+2),1/d\}}$ in the empirical $L_2$ loss, up to poly-logarithmic factors. Further, we prove a sharp oracle inequality, which reveals in particular that when the true regression function is piecewise constant on $k$ hyperrectangles, the least squares estimator enjoys a faster, adaptive rate of convergence of $(k/n)^{min(1,2/d)}$, again up to poly-logarithmic factors. Previous results are confined to the case $d\leq 2$. Finally, we establish corresponding bounds (which are new even in the case $d=2$) in the more challenging random design setting. There are two surprising features of these results: first, they demonstrate that it is possible for a global empirical risk minimisation procedure to be rate optimal up to poly-logarithmic factors even when the corresponding entropy integral for the function class diverges rapidly; second, they indicate that the adaptation rate for shape-constrained estimators can be strictly worse than the parametric rate. INI 1 15:30 to 16:00 Afternoon Tea 16:00 to 17:00 Rainer von Sachs (Université Catholique de Louvain)Intrinsic wavelet regression for curves and surfaces of Hermitian positive definite matrices Co-author: Joris Chau (ISBA, UC Louvain) In multivariate time series analysis, non-degenerate autocovariance and spectral density matrices are necessarily Hermitian and positive definite. An estimation methodology which preserves these properties is developed based on intrinsic wavelet transforms being applied to nonparametric wavelet regression for curves in the non-Euclidean space of Hermitian positive definite matrices. Via intrinsic average-interpolation in a Riemannian manifold equipped with a natural invariant Riemannian metric, we derive the wavelet coefficient decay and linear wavelet thresholding convergence rates of intrinsically smooth curves. Applying this more specifically to nonparametric spectral density estimation, an important property of the intrinsic linear or nonlinear wavelet spectral estimator under the invariant Riemannian metric is that it is independent of the choice of coordinate system of the time series, in contrast to most existing approaches. As a generalisation of this one-dimensional denoising of matrix-valued curves in the Riemannian manifold we also present higher-dimensional intrinsic wavelet transforms, applied in particular to time-varying spectral estimation of non-stationary multivariate time series, i.e. surfaces of Hermitian positive definite matrices.Related Linkshttps://cran.r-project.org/web/packages/pdSpecEst/index.html - R-package "pdSpecEst" (v1.2.1) on CRANhttps://jchau.shinyapps.io/pdSpecEst/ - Shiny-App "pdSpecEst" INI 1 17:00 to 18:00 Welcome Wine Reception & Poster Session