Classical Gaussian processes and Kriging models are commonly based on stationary kernels, whereby correlations between observations depend exclusively on the relative distance between scattered data. While this assumption ensures analytical tractability, it limits the ability of Gaussian processes to represent heterogeneous correlation structures. In this work, we investigate variably scaled kernels as an effective tool for constructing non-stationary Gaussian processes by explicitly modifying the correlation structure of the data. Through a scaling function, variably scaled kernels alter the correlations between data and enable the modeling of targets exhibiting abrupt changes or discontinuities. We analyse the resulting predictive uncertainty via the variably scaled kernel power function and clarify the relationship between variably scaled kernels-based constructions and classical non-stationary kernels. Numerical experiments demonstrate that variably scaled kernels-based Gaussian processes yield improved reconstruction accuracy and provide uncertainty estimates that reflect the underlying structure of the data

Mean-field variational inference (VI) iscomputationally scalable, but its highlydemanding independence requirement hinders it from being applied to wider scenarios. Although many VI methods that take correlation into account have been proposed, these methods generally are not scalable enough to capture the correlation among data instances, which often arises in applications involving graphs or explicit constraints among instances.

However, the Achilles' heel of all these approaches is that

In Partially Observable Markov Decision Processes (POMDPs), maintaining and updating belief distributions over possible underlying states provides a principled way to summarize action-observation history for effective decision-making under uncertainty. As environments grow more realistic, belief distributions develop complexity that standard mathematical models cannot accurately capture, creating a fundamental challenge in maintaining representational accuracy. Despite advances in deep learning and probabilistic modeling, existing POMDP belief approximation methods fail to accurately represent complex uncertainty structures such as high-dimensional, multi-modal belief distributions, resulting in estimation errors that lead to suboptimal agent behaviors. To address this challenge, we present ESCORT (Efficient Stein-variational and sliced Consistency-Optimized Representation for Temporal beliefs), a particle-based framework for capturing complex, multi-modal distributions in high-dimensional belief spaces. ESCORT extends SVGD with two key innovations: correlation-aware projections that model dependencies between state dimensions, and temporal consistency constraints that stabilize updates while preserving correlation structures. This approach retains SVGD's attractive-repulsive particle dynamics while enabling accurate modeling of intricate correlation patterns. Unlike particle filters prone to degeneracy or parametric methods with fixed representational capacity, ESCORT dynamically adapts to belief landscape complexity without resampling or restrictive distributional assumptions. We demonstrate ESCORT's effectiveness through extensive evaluations on both POMDP domains and synthetic multi-modal distributions of varying dimensionality, where it consistently outperforms state-of-the-art methods in terms of belief approximation accuracy and downstream decision quality.

The increasing need for data privacy and the demand for robust machine learning models have fueled the development of synthetic data generation techniques. However, current methods often succeed in replicating simple summary statistics but fail to preserve both the pairwise and higher-order correlation structure of the data that define the complex, multi-variable interactions inherent in real-world systems. This limitation can lead to synthetic data that is superficially realistic but fails when used for sophisticated modeling tasks. In this white paper, we introduce Generative Correlation Manifolds (GCM), a computationally efficient method for generating synthetic data. The technique uses Cholesky decomposition of a target correlation matrix to produce datasets that, by mathematical proof, preserve the entire correlation structure -- from simple pairwise relationships to higher-order interactions -- of the source dataset. We argue that this method provides a new approach to synthetic data generation with potential applications in privacy-preserving data sharing, robust model training, and simulation.

This paper focuses on reinforcement learning (RL) with clustered data, which is commonly encountered in healthcare applications. We propose a generalized fitted Q-iteration (FQI) algorithm that incorporates generalized estimating equations into policy learning to handle the intra-cluster correlations. Theoretically, we demonstrate (i) the optimalities of our Q-function and policy estimators when the correlation structure is correctly specified, and (ii) their consistencies when the structure is mis-specified. Empirically, through simulations and analyses of a mobile health dataset, we find the proposed generalized FQI achieves, on average, a half reduction in regret compared to the standard FQI.