Computationally-efficient initialisation of GPs: The generalised variogram method

We present a computationally-efficient strategy to find the hyperparameters of a Gaussian process (GP) avoiding the computation of the likelihood function. The found hyperparameters can then be used directly for regression or passed as initial conditions to maximum-likelihood (ML) training. Motivated by the fact that training a GP via ML is equivalent (on average) to minimising the KL-divergence between the true and learnt model, we set to explore different metrics/divergences among GPs that are computationally inexpensive and provide estimates close to those of ML. In particular, we identify the GP hyperparameters by matching the empirical covariance to a parametric candidate, proposing and studying various measures of discrepancy. Our proposal extends the Variogram method developed by the geostatistics literature and thus is referred to as the Generalised Variogram method (GVM). In addition to the theoretical presentation of GVM, we provide experimental validation in terms of accuracy, consistency with ML and computational complexity for different kernels using synthetic and real-world data.

Streaming computation of optimal weak transport barycenters

We introduce the weak barycenter of a family of probability distributions, based on the recently developed notion of optimal weak transport of measures arXiv:1412.7480(v4). We provide a theoretical analysis of the weak barycenter and its relationship to the classic Wasserstein barycenter, and discuss its meaning in the light of convex ordering between probability measures. In particular, we argue that, rather than averaging the information of the input distributions as done by the usual optimal transport barycenters, weak barycenters contain geometric information shared across all input distributions, which can be interpreted as a latent random variable affecting all the measures. We also provide iterative algorithms to compute a weak barycenter for either finite or infinite families of arbitrary measures (with finite moments of order 2), which are particularly well suited for the streaming setting, i.e., when measures arrive sequentially. In particular, our streaming computation of weak barycenters does not require to smooth empirical measures or to define a common grid for them, as some of the previous approaches to Wasserstin barycenters do. The concept of weak barycenter and our computation approaches are illustrated on synthetic examples, validated on 2D real-world data and compared to the classical Wasserstein barycenters.

The Wasserstein-Fourier Distance for Stationary Time Series

We introduce a novel framework for analysing stationary time series based on optimal transport distances and spectral embeddings. First, we represent time series by their power spectral density (PSD), which summarises the signal energy spread across the Fourier spectrum. Second, we endow the space of PSDs with the Wasserstein distance, which capitalises its unique ability to preserve the geometric information of a set of distributions. These two steps enable us to define the Wasserstein-Fourier (WF) distance, which allows us to compare stationary time series even when they differ in sampling rate, length, magnitude and phase. We analyse the features of WF by blending the properties of the Wasserstein distance and those of the Fourier transform. The proposed WF distance is then used in three sets of key time series applications considering real-world datasets: (i) interpolation of time series leading to data augmentation, (ii) dimensionality reduction via non-linear PCA, and (iii) parametric and non-parametric classification tasks. Our conceptual and experimental findings validate the general concept of using divergences of distributions, especially the Wasserstein distance, to analyse time series through comparing their spectral representations.