Permutation-preserving Functions and Neural Vecchia Covariance Kernels

We present a theoretically grounded Gaussian process framework that leverages neural feature maps to construct expressive kernels. We show that the learned feature map can be interpreted as an optimal low-rank approximation to a Gram matrix derived from an implied RKHS, from which we establish consistency of the GP posterior. We further analyse the spectral properties of the induced kernels and introduce product feature-map kernels to address oversmoothing. This simple yet powerful approach enables fast, scalable, and accurate exact GP inference with minimal upfront work. The flexibility of kernel design supports seamless application to both regression and classification tasks across diverse data modalities, including tabular inputs and structured domains such as images. On benchmark datasets, this approach surpasses pre-existing methods in terms of accuracy and training and prediction efficiency.

Gaussian processes (GPs) provide a principled Bayesian framework for uncertainty estimation, but their computational complexity severely limits scalability to large datasets. We propose SIKA-GP, which accelerates GP inference using sparse inducing kernel approximations based on a dyadic ordered template basis, incurring only ${O}(\log M)$ complexity dependence on the number of inducing points. Our approach constructs compact and expressive kernel representations from sparsely activated bases, enabling efficient tensorized GPU computation and seamless integration with modern large-scale models. SIKA-GP can be naturally embedded into Bayesian neural networks (BNNs) with sparse activations, yielding significant speedups in both training and inference without sacrificing predictive performance. The method naturally extends to deep feature learning, addressing the scalability challenges introduced by deep architectures and high-dimensional feature representations. Empirical results on vision and transformer-based language benchmarks demonstrate that our approach consistently delivers fast and accurate GP models, providing a principled path toward scalable kernel learning.

This feature article provides an overview of the theoretical foundations for coVariance neural networks (VNNs), i.e., graph neural networks (GNNs) operating on covariance matrices as graphs. Covariance matrices are ubiquitous across domains, and hence, the deployment of GNNs often leverages graphs of pairwise statistical dependencies. Existing theoretical contributions on GNNs consider abstract graph representations and cannot accommodate the data-driven nuances associated with covariance matrices. This tutorial brings into focus various novel theoretical insights via mathematical analyses of VNNs that have broad signal processing implications, including: (i) a conceptual equivalence between VNNs and principal component analysis (PCA)-based information processing; (ii) refined stability bounds on predictive outcomes in the presence of finite sample-induced covariance matrix perturbations; and (iii) refined characterization of transferability of VNNs across multiscale datasets. The theoretical insights discussed herein provide the underlying principles and justification towards adopting VNNs over workhorse PCA-based learning pipelines, in applications where covariance matrices are useful descriptors of data structure. We also convey how impact of these foundational advances permeates to \textit{principled} designs and applications of learning methods across broad domains where covariance matrices emerge. Notably, we elucidate the conceptual insights facilitated by VNNs to the specific task of characterizing brain age gap for neurodegenerative conditions using neuroimaging datasets, a timely problem in computational neuroscience. Broader impacts to other application domains are discussed as well.