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We study the problem of estimating, in the sense of optimal transport metrics, a measure which is assumed supported on a manifold embedded in a Hilbert space. By establishing a precise connection between optimal transport metrics, optimal…
In Bayesian statistics, a continuity property of the posterior distribution with respect to the observable variable is crucial as it expresses well-posedness, i.e., stability with respect to errors in the measurement of data. Essentially,…
In supervised learning with distributional inputs in the two-stage sampling setup, relevant to applications like learning-based medical screening or causal learning, the inputs (which are probability distributions) are not accessible in the…
In this paper, we characterize Probabilistic Principal Component Analysis in Hilbert spaces and demonstrate how the optimal solution admits a representation in dual space. This allows us to develop a generative framework for kernel methods.…
We consider linear approximation based on function evaluations in reproducing kernel Hilbert spaces of certain analytic weighted power series kernels and stationary kernels on the interval $[-1,1]$. Both classes contain the popular Gaussian…
We propose a novel class of Gaussian processes (GPs) whose spectra have compact support, meaning that their sample trajectories are almost-surely band limited. As a complement to the growing literature on spectral design of covariance…
This paper studies the probabilistic function approximation problem over reproducing kernel Hilbert spaces. We show the existence and uniqueness of the optimizer under mild assumptions. Furthermore, we generalize the celebrated representer…
The family of Mat\'ern kernels are often used in spatial statistics, function approximation and Gaussian process methods in machine learning. One reason for their popularity is the presence of a smoothness parameter that controls, for…
Gaussian processes are important models in the field of probabilistic numerics. We present a procedure for optimizing Mat\'ern kernel temporal Gaussian processes with respect to the kernel covariance function's hyperparameters. It is based…
The optimal mass transport problem gives a geometric framework for optimal allocation, and has recently gained significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be…
The Gaussian process (GP) is a widely used probabilistic machine learning method with implicit uncertainty characterization for stochastic function approximation, stochastic modeling, and analyzing real-world measurements of nonlinear…
We present a novel variation of online kernel machines in which we exploit a consensus based optimization mechanism to guide the evolution of decision functions drawn from a reproducing kernel Hilbert space, which efficiently models the…
Flexible Bayesian models are typically constructed using limits of large parametric models with a multitude of parameters that are often uninterpretable. In this article, we offer a novel alternative by constructing an exponentially tilted…
Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately,…
The Sinkhorn algorithm is a numerical method for the solution of optimal transport problems. Here, I give a brief survey of this algorithm, with a strong emphasis on its geometric origin: it is natural to view it as a discretization, by…
Optimal transport (OT) has become a widely used tool in the machine learning field to measure the discrepancy between probability distributions. For instance, OT is a popular loss function that quantifies the discrepancy between an…
Kernel based methods provide a way to reconstruct potentially high-dimensional functions from meshfree samples, i.e., sampling points and corresponding target values. A crucial ingredient for this to be successful is the distribution of the…
We develop an exact and scalable algorithm for one-dimensional Gaussian process regression with Mat\'ern correlations whose smoothness parameter $\nu$ is a half-integer. The proposed algorithm only requires $\mathcal{O}(\nu^3 n)$ operations…
Gaussian processes (GPs) are Bayesian non-parametric models popular in a variety of applications due to their accuracy and native uncertainty quantification (UQ). Tuning GP hyperparameters is critical to ensure the validity of prediction…
In modern data analysis, nonparametric measures of discrepancies between random variables are particularly important. The subject is well-studied in the frequentist literature, while the development in the Bayesian setting is limited where…