arXiv subjects

Matthias Katzfuss

Publications and source records attributed to Matthias Katzfuss.

At least 19 recordsLinked to original sources

arXiv2026

Generative multi-scale modeling via spatial autoregressive transport maps

Spatial fields in the Earth and environmental sciences are often available at multiple scales or resolutions. While coarse-scale data (e.g., from global circulation models) are often abundant, they lack the local detail provided by fine-scale data (e.g., from regional climate models), which are typically computationally expensive to generate. Statistical downscaling and multi-scale data fusion address this challenge by predicting high-resolution fields from low-resolution or related inputs. We propose a highly scalable Bayesian approach that can learn the joint non-Gaussian distribution and nonlinear dependence structure of nonstationary spatial fields across multiple scales from a small number of training samples. Our method employs scale-aware autoregressive Gaussian processes with suitably chosen regularization-inducing priors to model the conditional distribution of fine-scale fields given coarse-scale data. Exploiting conjugacy, the integrated likelihood is available in closed form, enabling efficient parameter optimization via stochastic gradient descent. Once trained, the method provides a closed-form characterization of the posterior distribution of fine-scale fields given coarse-scale inputs. In numerical comparisons, we demonstrate that our approach substantially outperforms existing methods and effectively characterizes and simulates fine-scale climate behavior based on output from coarse global circulation models.

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arXiv2026

Tensor Covariance Estimation via Kronecker-Structured Sparse Inverse Cholesky

High-dimensional multi-way (tensor) data pose significant challenges for covariance estimation due to the curse of dimensionality. We introduce a unified framework for scalable estimation of tensor covariances based on a Kronecker-structured sparse inverse Cholesky (KSIC) projection. Our approach is grounded in the geometry of information projection, defining the estimator as the moment-matching projection of a target distribution onto a manifold characterized by sparse, Kronecker-factored inverse Cholesky factors. By leveraging physical or data-driven nearest-neighbor sparsity, KSIC provides a geometry-aware representation that is both statistically interpretable and computationally efficient. Our framework integrates two estimation regimes: a nonparametric estimator that projects the empirical covariance directly onto the manifold, utilizing the KSIC structure to implicitly regularize rank-deficient data; and a parametric estimator that fits generative covariance models (e.g., Matérn) by maximizing the likelihood of their KSIC projections, formulated as a nested double forward Kullback-Leibler minimization. Theoretically, we establish the conditions for the existence of the KSIC projection and finite-sample concentration rates for the nonparametric regime, proving that the KSIC estimator gainfully exploits cross-mode information and is robust to data scarcity. Numerical experiments demonstrate that the proposed KSIC estimators achieve state-of-the-art accuracy and scalability, particularly in settings with high dimensionality and limited sample sizes. We apply KSIC to spatiotemporal temperature anomalies and functional MRI data, demonstrating its broad applicability across diverse multi-way data domains.

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arXiv2026

Data-Efficient Generative Modeling of Non-Gaussian Global Climate Fields via Scalable Composite Transformations

Quantifying uncertainty in climate-model output requires characterizing internal variability, often through large ensembles of physical climate-model runs. Since each additional ensemble member is computationally expensive, only limited numbers of replicated fields are typically available under a fixed model configuration and forcing scenario. We propose a data-efficient stochastic generator for the internal variability of global climate fields, specifically designed to overcome these sample-size constraints. The framework targets the distribution of climate-model output under fixed forcing and model physics. Inspired by copula modeling, our approach constructs a highly expressive joint distribution via a composite transformation to a multivariate standard normal space. We combine a nonparametric Bayesian transport map for spatial dependence modeling with flexible, spatially varying marginal models, essential for capturing non-Gaussian behavior and heavy-tailed extremes. These marginals are defined by a parametric model followed by a semi-parametric B-spline correction to capture complex distributional features. The marginal parameters are spatially smoothed using Gaussian-process priors with low-rank approximations, rendering the computational cost linear in the spatial dimension. When applied to global log-precipitation-rate fields under fixed forcing at more than 50,000 grid locations, our stochastic surrogate achieves high fidelity, accurately quantifying the climate distribution's spatial dependence and marginal characteristics, including the tails. Using only 10 training fields, it outperforms a state-of-the-art competitor trained on 80 fields, effectively octupling the computational budget for climate research. We provide a Python implementation at https://github.com/jobrachem/ppptm .

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arXiv2026

Amortized Probabilistic Retrieval of Atmospheric CO2 from OCO-2 Spectra Using Deep Learning with Laplace Approximations and Normalizing Flows

Space-based monitoring of atmospheric carbon dioxide (CO2) is essential for constraining the global carbon budget. NASA's Orbiting Carbon Observatory-2 (OCO-2) estimates column-averaged dry-air mole fractions of CO2 (XCO2) using high-resolution spectra. However, current operational retrieval algorithms are computationally expensive and do not properly quantify uncertainties. We present a novel deep learning framework that addresses these challenges. Due to the difficulties of ground-truth data for real satellite observations, we develop and validate our approach using a high-fidelity simulation dataset. This dataset, created to support OCO-2 uncertainty quantification (UQ), incorporates realistic forward model errors. Our architecture encodes spectral bands using a multi-branch neural network and estimates posteriors of the full CO2 column or desired summaries thereof using two scalable UQ methods: Laplace approximations and normalizing flows. Our approach has five key advantages relative to operational "full-physics" solvers: (1) Amortization: Inference is orders of magnitude faster, enabling real-time processing of massive data streams; (2) Model error robustness: By training on simulations that explicitly include model discrepancies, our method accounts for systematic errors often neglected by standard inversions; (3) Point estimate accuracy: We achieve superior predictive accuracy compared to baseline methods; (4) Improved UQ: The probabilistic outputs yield better-calibrated uncertainty estimates; and (5) Non-Gaussian posteriors: When utilizing normalizing flows, our framework successfully models complex, asymmetric posterior distributions, overcoming the limitations of the Gaussian assumption. These results suggest that simulation-based deep learning is a viable path toward next-generation operational processing systems.

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arXiv2026

Scalable Derivative Gaussian Processes via Exact Gradient Reduction

Gradient observations can substantially improve Gaussian process (GP) surrogates, particularly in high-dimensional settings where function evaluations are expensive. However, exact inference with $n$ function values and $n$ full gradients in $d$ dimensions scales cubically in the joint state size, imposing an intractable $\mathcal{O}(n^3 d^3)$ computational bottleneck. We introduce TERA, a highly scalable derivative GP method based on target-specific exact gradient reduction. We prove that for stationary kernels, the gradient components orthogonal to the directions connecting the target and conditioning points are conditionally independent of the target function value; consequently, the exact conditional density is fully characterized by at most $m^2$ directional derivatives once a conditioning set of size $m$ is specified. By using these reduced, dimension-free conditionals as local factors in a Vecchia approximation, TERA effectively decouples $n$ and $d$ from the dense matrix inversion. This reduces the per-target evaluation cost to $\mathcal{O}(dm^2 + m^6)$ time and $\mathcal{O}(dm^2 + m^4)$ memory, leaving the underlying derivative GP model mathematically unchanged. Empirical evaluations demonstrate that TERA achieves state-of-the-art predictive accuracy while operating orders of magnitude faster than standard derivative GPs. Crucially, both computation time and peak GPU memory remain essentially flat with respect to $d$, enabling highly scalable inference in high-dimensional spaces.

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arXiv2026

Scalable generative modeling of non-Gaussian spatio-temporal fields via autoregressive Gaussian processes

Generative modeling of spatio-temporal fields is crucial for a variety of applications, including stochastic weather generators and climate-model surrogates. However, many such fields exhibit complex dependence structures that vary across space and time and are nonlinear, resulting in nonstationary and non-Gaussian joint distributions. Our approach represents the joint density of a spatio-temporal field as a product of univariate conditional distributions and models these conditionals using Gaussian processes within an autoregressive transport-map construction. This prior distribution provides regularization, making our method suitable for a small number of training samples. Data-dependent sparsity in the conditioning sets ensures scalability to high-dimensional distributions. We also propose a variant of the method designed to sample or predict forward in time from a given incomplete space-time trajectory. We demonstrate the accuracy and scalability of our approach on non-Gaussian climate-model output with tens of millions of data points.

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arXiv2026

Linear-Cost Vecchia Approximation of Multivariate Normal Probabilities

Multivariate normal (MVN) probabilities arise in myriad applications, but they are analytically intractable and need to be evaluated via Monte-Carlo-based numerical integration. For the state-of-the-art minimax exponential tilting (MET) method, we show that the complexity of each of its components can be greatly reduced through an integrand parameterization that utilizes the sparse inverse Cholesky factor produced by the Vecchia approximation, whose approximation error is often negligible relative to the Monte-Carlo error. Based on this idea, we derive algorithms that can estimate MVN probabilities and sample from truncated MVN distributions in linear time (and that are easily parallelizable) at the same convergence or acceptance rate as MET, whose complexity is cubic in the dimension of the MVN probability. We showcase the advantages of our methods relative to existing approaches using several simulated examples. We also analyze a groundwater-contamination dataset with over twenty thousand censored measurements to demonstrate the scalability of our method for partially censored Gaussian-process models.

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arXiv2025

Fast Gaussian Process Approximations for Autocorrelated Data

This paper is concerned with the problem of how to speed up computation for Gaussian process models trained on autocorrelated data. The Gaussian process model is a powerful tool commonly used in nonlinear regression applications. Standard regression modeling assumes random samples and an independently, identically distributed noise. Various fast approximations that speed up Gaussian process regression work under this standard setting. But for autocorrelated data, failing to account for autocorrelation leads to a phenomenon known as temporal overfitting that deteriorates model performance on new test instances. To handle autocorrelated data, existing fast Gaussian process approximations have to be modified; one such approach is to segment the originally correlated data points into blocks in which the blocked data are de-correlated. This work explains how to make some of the existing Gaussian process approximations work with blocked data. Numerical experiments across diverse application datasets demonstrate that the proposed approaches can remarkably accelerate computation for Gaussian process regression on autocorrelated data without compromising model prediction performance.

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arXiv2025

Sparse inverse Cholesky factorization of dense kernel matrices by greedy conditional selection

Dense kernel matrices resulting from pairwise evaluations of a kernel function arise naturally in machine learning and statistics. Previous work in constructing sparse approximate inverse Cholesky factors of such matrices by minimizing Kullback-Leibler divergence recovers the Vecchia approximation for Gaussian processes. These methods rely only on the geometry of the evaluation points to construct the sparsity pattern. In this work, we instead construct the sparsity pattern by leveraging a greedy selection algorithm that maximizes mutual information with target points, conditional on all points previously selected. For selecting $k$ points out of $N$, the naive time complexity is $\mathcal{O}(N k^4)$, but by maintaining a partial Cholesky factor we reduce this to $\mathcal{O}(N k^2)$. Furthermore, for multiple ($m$) targets we achieve a time complexity of $\mathcal{O}(N k^2 + N m^2 + m^3)$, which is maintained in the setting of aggregated Cholesky factorization where a selected point need not condition every target. We apply the selection algorithm to image classification and recovery of sparse Cholesky factors. By minimizing Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization, Gaussian process regression, and preconditioning with the conjugate gradient, improving over $k$-nearest neighbors selection.

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arXiv2025

Vecchia Gaussian Process Ensembles on Internal Representations of Deep Neural Networks

For regression tasks, standard Gaussian processes (GPs) provide natural uncertainty quantification (UQ), while deep neural networks (DNNs) excel at representation learning. Deterministic UQ methods for neural networks have successfully combined the two and require only a single pass through the neural network. However, current methods necessitate changes to network training to address feature collapse, where unique inputs map to identical feature vectors. We propose an alternative solution, the deep Vecchia ensemble (DVE), which allows deterministic UQ to work in the presence of feature collapse, negating the need for network retraining. DVE comprises an ensemble of GPs built on hidden-layer outputs of a DNN, achieving scalability via Vecchia approximations that leverage nearest-neighbor conditional independence. DVE is compatible with pretrained networks and incurs low computational overhead. We demonstrate DVE's utility on several datasets and carry out experiments to understand the inner workings of the proposed method.

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arXiv2025

Learning non-Gaussian spatial distributions via Bayesian transport maps with parametric shrinkage

Many applications, including climate-model analysis and stochastic weather generators, require learning or emulating the distribution of a high-dimensional and non-Gaussian spatial field based on relatively few training samples. To address this challenge, a recently proposed Bayesian transport map (BTM) approach consists of a triangular transport map with nonparametric Gaussian-process (GP) components, which is trained to transform the distribution of interest distribution to a Gaussian reference distribution. To improve the performance of this existing BTM, we propose to shrink the map components toward a ``base'' parametric Gaussian family combined with a Vecchia approximation for scalability. The resulting ShrinkTM approach is more accurate than the existing BTM, especially for small numbers of training samples. It can even outperform the ``base'' family when trained on a single sample of the spatial field. We demonstrate the advantage of ShrinkTM though numerical experiments on simulated data and on climate-model output.

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arXiv2025

Probabilistic Skip Connections for Deterministic Uncertainty Quantification in Deep Neural Networks

Deterministic uncertainty quantification (UQ) in deep learning aims to estimate uncertainty with a single pass through a network by leveraging outputs from the network's feature extractor. Existing methods require that the feature extractor be both sensitive and smooth, ensuring meaningful input changes produce meaningful changes in feature vectors. Smoothness enables generalization, while sensitivity prevents feature collapse, where distinct inputs are mapped to identical feature vectors. To meet these requirements, current deterministic methods often retrain networks with spectral normalization. Instead of modifying training, we propose using measures of neural collapse to identify an existing intermediate layer that is both sensitive and smooth. We then fit a probabilistic model to the feature vector of this intermediate layer, which we call a probabilistic skip connection (PSC). Through empirical analysis, we explore the impact of spectral normalization on neural collapse and demonstrate that PSCs can effectively disentangle aleatoric and epistemic uncertainty. Additionally, we show that PSCs achieve uncertainty quantification and out-of-distribution (OOD) detection performance that matches or exceeds existing single-pass methods requiring training modifications. By retrofitting existing models, PSCs enable high-quality UQ and OOD capabilities without retraining.

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arXiv2024

Scalable Sampling of Truncated Multivariate Normals Using Sequential Nearest-Neighbor Approximation

We propose a linear-complexity method for sampling from truncated multivariate normal (TMVN) distributions with high fidelity by applying nearest-neighbor approximations to a product-of-conditionals decomposition of the TMVN density. To make the sequential sampling based on the decomposition feasible, we introduce a novel method that avoids the intractable high-dimensional TMVN distribution by sampling sequentially from $m$-dimensional TMVN distributions, where $m$ is a tuning parameter controlling the fidelity. This allows us to overcome the existing methods' crucial problem of rapidly decreasing acceptance rates for increasing dimension. Throughout our experiments with up to tens of thousands of dimensions, we can produce high-fidelity samples with $m$ in the dozens, achieving superior scalability compared to existing state-of-the-art methods. We study a tetrachloroethylene concentration dataset that has $3{,}971$ observed responses and $20{,}730$ undetected responses, together modeled as a partially censored Gaussian process, where our method enables posterior inference for the censored responses through sampling a $20{,}730$-dimensional TMVN distribution.

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arXiv2024

Asymptotic properties of Vecchia approximation for Gaussian processes

Vecchia approximation has been widely used to accurately scale Gaussian-process (GP) inference to large datasets, by expressing the joint density as a product of conditional densities with small conditioning sets. We study fixed-domain asymptotic properties of Vecchia-based GP inference for a large class of covariance functions (including Matérn covariances) with boundary conditioning. In this setting, we establish that consistency and asymptotic normality of maximum exact-likelihood estimators imply those of maximum Vecchia-likelihood estimators, and that exact GP prediction can be approximated accurately by Vecchia GP prediction, given that the size of conditioning sets grows polylogarithmically with the data size. Hence, Vecchia-based inference with quasilinear complexity is asymptotically equivalent to exact GP inference with cubic complexity. This also provides a general new result on the screening effect. Our findings are illustrated by numerical experiments, which also show that Vecchia approximation can be more accurate than alternative approaches such as covariance tapering and reduced-rank approximations.

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arXiv2023

Bayesian nonparametric generative modeling of large multivariate non-Gaussian spatial fields

Multivariate spatial fields are of interest in many applications, including climate model emulation. Not only can the marginal spatial fields be subject to nonstationarity, but the dependence structure among the marginal fields and between the fields might also differ substantially. Extending a recently proposed Bayesian approach to describe the distribution of a nonstationary univariate spatial field using a triangular transport map, we cast the inference problem for a multivariate spatial field for a small number of replicates into a series of independent Gaussian process (GP) regression tasks with Gaussian errors. Due to the potential nonlinearity in the conditional means, the joint distribution modeled can be non-Gaussian. The resulting nonparametric Bayesian methodology scales well to high-dimensional spatial fields. It is especially useful when only a few training samples are available, because it employs regularization priors and quantifies uncertainty. Inference is conducted in an empirical Bayes setting by a highly scalable stochastic gradient approach. The implementation benefits from mini-batching and could be accelerated with parallel computing. We illustrate the extended transport-map model by studying hydrological variables from non-Gaussian climate-model output.

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arXiv2023

Variational sparse inverse Cholesky approximation for latent Gaussian processes via double Kullback-Leibler minimization

To achieve scalable and accurate inference for latent Gaussian processes, we propose a variational approximation based on a family of Gaussian distributions whose covariance matrices have sparse inverse Cholesky (SIC) factors. We combine this variational approximation of the posterior with a similar and efficient SIC-restricted Kullback-Leibler-optimal approximation of the prior. We then focus on a particular SIC ordering and nearest-neighbor-based sparsity pattern resulting in highly accurate prior and posterior approximations. For this setting, our variational approximation can be computed via stochastic gradient descent in polylogarithmic time per iteration. We provide numerical comparisons showing that the proposed double-Kullback-Leibler-optimal Gaussian-process approximation (DKLGP) can sometimes be vastly more accurate for stationary kernels than alternative approaches such as inducing-point and mean-field approximations at similar computational complexity.

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arXiv2023

Correlation-based sparse inverse Cholesky factorization for fast Gaussian-process inference

Gaussian processes are widely used as priors for unknown functions in statistics and machine learning. To achieve computationally feasible inference for large datasets, a popular approach is the Vecchia approximation, which is an ordered conditional approximation of the data vector that implies a sparse Cholesky factor of the precision matrix. The ordering and sparsity pattern are typically determined based on Euclidean distance of the inputs or locations corresponding to the data points. Here, we propose instead to use a correlation-based distance metric, which implicitly applies the Vecchia approximation in a suitable transformed input space. The correlation-based algorithm can be carried out in quasilinear time in the size of the dataset, and so it can be applied even for iterative inference on unknown parameters in the correlation structure. The correlation-based approach has two advantages for complex settings: It can result in more accurate approximations, and it offers a simple, automatic strategy that can be applied to any covariance, even when Euclidean distance is not applicable. We demonstrate these advantages in several settings, including anisotropic, nonstationary, multivariate, and spatio-temporal processes. We also illustrate our method on multivariate spatio-temporal temperature fields produced by a regional climate model.

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arXiv2023

Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields

A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and uncertainty quantification of the map estimation, while still resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional distributions due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for flexible marginals. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output.

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