Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be unfeasible. We propose a new class of methods to overcome these challenges in the common case of sparse constraints, where one has a large number of constraints and each only involves a few elements. Our methods rely on a basis transformation into blocks of constrained versus non-constrained subspaces, and we show that the methods greatly outperform existing alternatives in terms of computational cost. By combining the proposed methods with the stochastic partial differential equation approach for Gaussian random fields, we also show how to formulate Gaussian process regression with linear constraints in a GMRF setting to reduce computational cost. This is illustrated in two applications with simulated data.

Almost all scientific data have uncertainties originating from different sources. Gaussian process regression (GPR) models are a natural way to model data with Gaussian-distributed uncertainties. GPR also has the benefit of reducing I/O bandwidth and storage requirements for large scientific simulations. However, the reconstruction from the GPR models suffers from high computation complexity. To make the situation worse, classic approaches for visualizing the data uncertainties, like probabilistic marching cubes, are also computationally very expensive, especially for data of high resolutions. In this paper, we accelerate the level-crossing probability calculation efficiency on GPR models by subdividing the data spatially into a hierarchical data structure and only reconstructing values adaptively in the regions that have a non-zero probability. For each region, leveraging the known GPR kernel and the saved data observations, we propose a novel approach to efficiently calculate an upper bound for the level-crossing probability inside the region and use this upper bound to make the subdivision and reconstruction decisions. We demonstrate that our value occurrence probability estimation is accurate with a low computation cost by experiments that calculate the level-crossing probability fields on different datasets.

This paper introduces a new probabilistic framework for supervised learning in neural systems. It is designed to model complex, uncertain systems whose random outputs are strongly non-Gaussian given deterministic inputs. The architecture itself is a random object stochastically generated by a latent anisotropic Gaussian random field defined on a compact, boundaryless, multiply-connected manifold. The goal is to establish a novel conceptual and mathematical framework in which neural architectures are realizations of a geometry-aware, field-driven generative process. Both the neural topology and synaptic weights emerge jointly from a latent random field. A reduced-order parameterization governs the spatial intensity of an inhomogeneous Poisson process on the manifold, from which neuron locations are sampled. Input and output neurons are identified via extremal evaluations of the latent field, while connectivity is established through geodesic proximity and local field affinity. Synaptic weights are conditionally sampled from the field realization, inducing stochastic output responses even for deterministic inputs. To ensure scalability, the architecture is sparsified via percentile-based diffusion masking, yielding geometry-aware sparse connectivity without ad hoc structural assumptions. Supervised learning is formulated as inference on the generative hyperparameters of the latent field, using a negative log-likelihood loss estimated through Monte Carlo sampling from single-observation-per-input datasets. The paper initiates a mathematical analysis of the model, establishing foundational properties such as well-posedness, measurability, and a preliminary analysis of the expressive variability of the induced stochastic mappings, which support its internal coherence and lay the groundwork for a broader theory of geometry-driven stochastic learning.

Probabilistic Graphical Models (PGMs) encode conditional dependencies among random variables using a graph -nodes for variables, links for dependencies- and factorize the joint distribution into lower-dimensional components. This makes PGMs well-suited for analyzing complex systems and supporting decision-making. Recent advances in topological signal processing highlight the importance of variables defined on topological spaces in several application domains. In such cases, the underlying topology shapes statistical relationships, limiting the expressiveness of canonical PGMs. To overcome this limitation, we introduce Colored Markov Random Fields (CMRFs), which model both conditional and marginal dependencies among Gaussian edge variables on topological spaces, with a theoretical foundation in Hodge theory. CMRFs extend classical Gaussian Markov Random Fields by including link coloring: connectivity encodes conditional independence, while color encodes marginal independence. We quantify the benefits of CMRFs through a distributed estimation case study over a physical network, comparing it with baselines with different levels of topological prior.

Collecting large training datasets, annotated with high-quality labels, is costly and time-consuming. This paper proposes a novel framework for training deep convolutional neural networks from noisy labeled datasets that can be obtained cheaply. The problem is formulated using an undirected graphical model that represents the relationship between noisy and clean labels, trained in a semi-supervised setting.

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Learning algorithms must consider complex relationships between variables to provide useful predictions in many structured prediction problems.

In this work, we analyze robust alternatives.

In the present work, we address (i) by extending ACV to CV schemes with dependence structure between the folds.