On ANOVA decompositions of kernels and Gaussian random field paths

The FANOVA (or "Sobol'-Hoeffding") decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, a practical limitation is that computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on random field models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. In the present work, we focus on FANOVA decompositions of Gaussian random field sample paths, and we notably introduce an associated kernel decomposition (into 2^{2d} terms) called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of Gaussian random field sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging

Invariances of random fields paths, with applications in Gaussian Process Regression

We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Lo\`eve isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promising results and perspectives in Gaussian process regression

Task-Adaptive Robot Learning from Demonstration with Gaussian Process Models under Replication

Learning from Demonstration (LfD) is a paradigm that allows robots to learn complex manipulation tasks that can not be easily scripted, but can be demonstrated by a human teacher. One of the challenges of LfD is to enable robots to acquire skills that can be adapted to different scenarios. In this paper, we propose to achieve this by exploiting the variations in the demonstrations to retrieve an adaptive and robust policy, using Gaussian Process (GP) models. Adaptability is enhanced by incorporating task parameters into the model, which encode different specifications within the same task. With our formulation, these parameters can be either real, integer, or categorical. Furthermore, we propose a GP design that exploits the structure of replications, i.e., repeated demonstrations with identical conditions within data. Our method significantly reduces the computational cost of model fitting in complex tasks, where replications are essential to obtain a robust model. We illustrate our approach through several experiments on a handwritten letter demonstration dataset.Comment: 8 pages, 9 figure

Recent advances in directional statistics

Mainstream statistical methodology is generally applicable to data observed in Euclidean space. There are, however, numerous contexts of considerable scientific interest in which the natural supports for the data under consideration are Riemannian manifolds like the unit circle, torus, sphere and their extensions. Typically, such data can be represented using one or more directions, and directional statistics is the branch of statistics that deals with their analysis. In this paper we provide a review of the many recent developments in the field since the publication of Mardia and Jupp (1999), still the most comprehensive text on directional statistics. Many of those developments have been stimulated by interesting applications in fields as diverse as astronomy, medicine, genetics, neurology, aeronautics, acoustics, image analysis, text mining, environmetrics, and machine learning. We begin by considering developments for the exploratory analysis of directional data before progressing to distributional models, general approaches to inference, hypothesis testing, regression, nonparametric curve estimation, methods for dimension reduction, classification and clustering, and the modelling of time series, spatial and spatio-temporal data. An overview of currently available software for analysing directional data is also provided, and potential future developments discussed.Comment: 61 page

Kernels for sequentially ordered data

We present a novel framework for learning with sequential data of any kind, such as multivariate time series, strings, or sequences of graphs. The main result is a ”sequentialization” that transforms any kernel on a given domain into a kernel for sequences in that domain. This procedure preserves properties such as positive definiteness, the associated kernel feature map is an ordered variant of sample (cross-)moments, and this sequentialized kernel is consistent in the sense that it converges to a kernel for paths if sequences converge to paths (by discretization). Further, classical kernels for sequences arise as special cases of this method. We use dynamic programming and low-rank techniques for tensors to provide efficient algorithms to compute this sequentialized kernel

High-Dimensional Non-Linear Variable Selection through Hierarchical Kernel Learning

We consider the problem of high-dimensional non-linear variable selection for supervised learning. Our approach is based on performing linear selection among exponentially many appropriately defined positive definite kernels that characterize non-linear interactions between the original variables. To select efficiently from these many kernels, we use the natural hierarchical structure of the problem to extend the multiple kernel learning framework to kernels that can be embedded in a directed acyclic graph; we show that it is then possible to perform kernel selection through a graph-adapted sparsity-inducing norm, in polynomial time in the number of selected kernels. Moreover, we study the consistency of variable selection in high-dimensional settings, showing that under certain assumptions, our regularization framework allows a number of irrelevant variables which is exponential in the number of observations. Our simulations on synthetic datasets and datasets from the UCI repository show state-of-the-art predictive performance for non-linear regression problems

A survey on high-dimensional Gaussian process modeling with application to Bayesian optimization

International audienceBayesian Optimization, the application of Bayesian function approximation to finding optima of expensive functions, has exploded in popularity in recent years. In particular, much attention has been paid to improving its efficiency on problems with many parameters to optimize. This attention has trickled down to the workhorse of high dimensional BO, high dimensional Gaussian process regression, which is also of independent interest. The great flexibility that the Gaussian process prior implies is a boon when modeling complicated, low dimensional surfaces but simply says too little when dimension grows too large. A variety of structural model assumptions have been tested to tame high dimensions, from variable selection and additive decomposition to low dimensional embeddings and beyond. Most of these approaches in turn require modifications of the acquisition function optimization strategy as well. Here we review the defining structural model assumptions and discuss the benefits and drawbacks of these approaches in practice