A Gaussian Process Regression Model for Distribution Inputs

@article{Bachoc2018AGP,
  title={A Gaussian Process Regression Model for Distribution Inputs},
  author={F. Bachoc and F. Gamboa and Jean-Michel Loubes and N. Venet},
  journal={IEEE Transactions on Information Theory},
  year={2018},
  volume={64},
  pages={6620-6637}
}
Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provide a family of positive definite kernels built using transportation based distances. We provide a probabilistic understanding of these kernels and characterize the corresponding… Expand
Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding
In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has beenExpand
Bayesian regression and classification using Gaussian process priors indexed by probability density functions
TLDR
This paper introduces the notion of Gaussian processes indexed by probability density functions for extending the Matern family of covariance functions and shows how a Bayesian inference with a Gaussian process prior (covariance parameters estimation and prediction) can be put into action on the space of probabilitydensity functions. Expand
Distribution regression model with a Reproducing Kernel Hilbert Space approach
Abstract In this paper, we introduce a new distribution regression model for probability distributions. This model is based on a Reproducing Kernel Hilbert Space (RKHS) regression framework, whereExpand
Gaussian Processes indexed on the symmetric group: prediction and learning
TLDR
This paper proposes and study an harmonic analysis of the covariance operators that enables to consider Gaussian processes models and forecasting issues and is motivated by statistical ranking problems. Expand
Learning a Gaussian Process Model on the Riemannian Manifold of Non-decreasing Distribution Functions
TLDR
This work introduces a novel framework to learn a Gaussian process model on the space of Strictly Non-decreasing Distribution Functions (SNDF), and defines a Riemannian structure of the SNDF space and learns a GP model indexed by SNDF. Expand
Wasserstein Regression these are either Nadaraya – Watson type estimators that suffer from a severe curse of dimensionality
The analysis of samples of random objects that do not lie in a vector space is gaining increasing attention in statistics. An important class of such object data is univariate probability measuresExpand
Gaussian field on the symmetric group: Prediction and learning
In the framework of the supervised learning of a real function defined on an abstract space X , the so called Kriging method stands on a real Gaussian field defined on X . The Euclidean case is wellExpand
Deep Kernels with Probabilistic Embeddings for Small-Data Learning
Gaussian Processes (GPs) are known to provide accurate predictions and uncertainty estimates even with small amounts of labeled data by capturing similarity between data points through their kernelExpand
Deep Probabilistic Kernels for Sample-Efficient Learning
TLDR
Deep Probabilistic kernels are proposed which use a probabilistic neural network to map high-dimensional data to a probability distribution in a low dimensional subspace, and leverage the rich work on kernels between distributions to capture the similarity between these distributions. Expand
Statistical Aspects of Wasserstein Distances.
TLDR
A snapshot of the main concepts involved in Wasserstein distances and optimal transportation is provided, and a succinct overview of some of their many statistical aspects are provided. Expand
...
1
2
3
4
...

References

SHOWING 1-10 OF 68 REFERENCES
Sliced Wasserstein Kernels for Probability Distributions
  • S. Kolouri, Yang Zou, G. Rohde
  • Computer Science, Mathematics
  • 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR)
  • 2016
TLDR
This work provides a new perspective on the application of optimal transport flavored distances through kernel methods in machine learning tasks and provides a family of provably positive definite kernels based on the Sliced Wasserstein distance. Expand
Two-stage sampled learning theory on distributions
TLDR
This paper provides theoretical guarantees for a remarkably simple algorithmic alternative to solve the distribution regression problem: embed the distributions to a reproducing kernel Hilbert space, and learn a ridge regressor from the embeddings to the outputs. Expand
Nonparametric Divergence Estimation with Applications to Machine Learning on Distributions
TLDR
Estimation algorithms are presented, how to apply them for machine learning tasks on distributions are described, and empirical results on synthetic data, real word images, and astronomical data sets are shown. Expand
Kernel Mean Embedding of Distributions: A Review and Beyonds
TLDR
A comprehensive review of existing work and recent advances in the Hilbert space embedding of distributions, and to discuss the most challenging issues and open problems that could lead to new research directions. Expand
Gaussian Processes for Machine Learning (Adaptive Computation and Machine Learning)
TLDR
The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics, and includes detailed algorithms for supervised-learning problem for both regression and classification. Expand
Asymptotic properties of a maximum likelihood estimator with data from a Gaussian process
We consider an estimation problem with observations from a Gaussian process. The problem arises from a stochastic process modeling of computer experiments proposed recently by Sacks, Schiller, andExpand
Approximation, metric entropy and small ball estimates for Gaussian measures
A precise link proved by Kuelbs and Li relates the small ball behavior of a Gaussian measure μ on a Banach space E with the metric entropy behavior of K μ , the unit ball of the reproducing kernelExpand
Gromov-Wasserstein Averaging of Kernel and Distance Matrices
TLDR
This paper presents a new technique for computing the barycenter of a set of distance or kernel matrices, which define the interrelationships between points sampled from individual domains, and provides a fast iterative algorithm for the resulting nonconvex optimization problem. Expand
Distribution-Free Distribution Regression
TLDR
This paper develops theory and methods for distribution-free versions of distribution regression and proves that when the eective dimension is small enough (as measured by the doubling dimension), then the excess prediction risk converges to zero with a polynomial rate. Expand
Estimating structured correlation matrices in smooth Gaussian random field models
This article considers the estimation of structured correlation matrices in infinitely differentiable Gaussian random field models. The problem is essentially motivated by the stochastic modeling ofExpand
...
1
2
3
4
5
...