In this paper, we address the critical need for interpretable and uncertainty-aware machine learning models in the context of online learning for high-risk industries, particularly cyber-security. While deep learning and other complex models have demonstrated impressive predictive capabilities, their opacity and lack of uncertainty quantification present significant questions about their trustworthiness. We propose a novel pipeline for online supervised learning problems in cyber-security, that harnesses the inherent interpretability and uncertainty awareness of Additive Gaussian Processes (AGPs) models. Our approach aims to balance predictive performance with transparency while improving the scalability of AGPs, which represents their main drawback, potentially enabling security analysts to better validate threat detection, troubleshoot and reduce false positives, and generally make trustworthy, informed decisions. This work contributes to the growing field of interpretable AI by proposing a class of models that can be significantly beneficial for high-stake decision problems such as the ones typical of the cyber-security domain. The source code is available.
Gaussian Processes face two primary challenges: constructing models for large datasets and selecting the optimal model. This master's thesis tackles these challenges in the low-dimensional case. We examine recent convergence results to identify models with optimal convergence rates and pinpoint essential parameters. Utilizing this model, we propose a Samplet-based approach to efficiently construct and train the Gaussian Processes, reducing the cubic computational complexity to a log-linear scale. This method facilitates optimal regression while maintaining efficient performance.
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 processes. Traditional implementations of GPs involve stationary kernels (also termed covariance functions) that limit their flexibility, and exact methods for inference that prevent application to data sets with more than about ten thousand points. Modern approaches to address stationarity assumptions generally fail to accommodate large data sets, while all attempts to address scalability focus on approximating the Gaussian likelihood, which can involve subjectivity and lead to inaccuracies. In this work, we explicitly derive an alternative kernel that can discover and encode both sparsity and nonstationarity. We embed the kernel within a fully Bayesian GP model and leverage high-performance computing resources to enable the analysis of massive data sets. We demonstrate the favorable performance of our novel kernel relative to existing exact and approximate GP methods across a variety of synthetic data examples. Furthermore, we conduct space-time prediction based on more than one million measurements of daily maximum temperature and verify that our results outperform state-of-the-art methods in the Earth sciences. More broadly, having access to exact GPs that use ultra-scalable, sparsity-discovering, nonstationary kernels allows GP methods to truly compete with a wide variety of machine learning methods.
Gaussian processes (GPs) are a ubiquitous tool for geostatistical modeling with high levels of flexibility and interpretability, and the ability to make predictions at unseen spatial locations through a process called Kriging. Estimation of Kriging weights relies on the inversion of the process' covariance matrix, creating a computational bottleneck for large spatial datasets. In this paper, we propose an Amortized Bayesian Local Interpolation NetworK (A-BLINK) for fast covariance parameter estimation, which uses two pre-trained deep neural networks to learn a mapping from spatial location coordinates and covariance function parameters to Kriging weights and the spatial variance, respectively. The fast prediction time of these networks allows us to bypass the matrix inversion step, creating large computational speedups over competing methods in both frequentist and Bayesian settings, and also provides full posterior inference and predictions using Markov chain Monte Carlo sampling methods. We show significant increases in computational efficiency over comparable scalable GP methodology in an extensive simulation study with lower parameter estimation error. The efficacy of our approach is also demonstrated using a temperature dataset of US climate normals for 1991--2020 based on over 7,000 weather stations.
Rahimi and Recht (2007) introduced the idea of decomposing positive definite shift-invariant kernels by randomly sampling from their spectral distribution. This famous technique, known as Random Fourier Features (RFF), is in principle applicable to any such kernel whose spectral distribution can be identified and simulated. In practice, however, it is usually applied to the Gaussian kernel because of its simplicity, since its spectral distribution is also Gaussian. Clearly, simple spectral sampling formulas would be desirable for broader classes of kernels. In this paper, we show that the spectral distribution of positive definite isotropic kernels in $\mathbb{R}^{d}$ for all $d\geq1$ can be decomposed as a scale mixture of $\alpha$-stable random vectors, and we identify the mixing distribution as a function of the kernel. This constructive decomposition provides a simple and ready-to-use spectral sampling formula for many multivariate positive definite shift-invariant kernels, including exponential power kernels, generalized Mat\'ern kernels, generalized Cauchy kernels, as well as newly introduced kernels such as the Beta, Kummer, and Tricomi kernels. In particular, we retrieve the fact that the spectral distributions of these kernels are scale mixtures of the multivariate Gaussian distribution, along with an explicit mixing distribution formula. This result has broad applications for support vector machines, kernel ridge regression, Gaussian processes, and other kernel-based machine learning techniques for which the random Fourier features technique is applicable.
We propose practical deep Gaussian process models on Riemannian manifolds, similar in spirit to residual neural networks. With manifold-to-manifold hidden layers and an arbitrary last layer, they can model manifold- and scalar-valued functions, as well as vector fields. We target data inherently supported on manifolds, which is too complex for shallow Gaussian processes thereon. For example, while the latter perform well on high-altitude wind data, they struggle with the more intricate, nonstationary patterns at low altitudes. Our models significantly improve performance in these settings, enhancing prediction quality and uncertainty calibration, and remain robust to overfitting, reverting to shallow models when additional complexity is unneeded. We further showcase our models on Bayesian optimisation problems on manifolds, using stylised examples motivated by robotics, and obtain substantial improvements in later stages of the optimisation process. Finally, we show our models to have potential for speeding up inference for non-manifold data, when, and if, it can be mapped to a proxy manifold well enough.
Model selection in Gaussian processes scales prohibitively with the size of the training dataset, both in time and memory. While many approximations exist, all incur inevitable approximation error. Recent work accounts for this error in the form of computational uncertainty, which enables -- at the cost of quadratic complexity -- an explicit tradeoff between computation and precision. Here we extend this development to model selection, which requires significant enhancements to the existing approach, including linear-time scaling in the size of the dataset. We propose a novel training loss for hyperparameter optimization and demonstrate empirically that the resulting method can outperform SGPR, CGGP and SVGP, state-of-the-art methods for GP model selection, on medium to large-scale datasets. Our experiments show that model selection for computation-aware GPs trained on 1.8 million data points can be done within a few hours on a single GPU. As a result of this work, Gaussian processes can be trained on large-scale datasets without significantly compromising their ability to quantify uncertainty -- a fundamental prerequisite for optimal decision-making.
We present MetaMetrics-MT, an innovative metric designed to evaluate machine translation (MT) tasks by aligning closely with human preferences through Bayesian optimization with Gaussian Processes. MetaMetrics-MT enhances existing MT metrics by optimizing their correlation with human judgments. Our experiments on the WMT24 metric shared task dataset demonstrate that MetaMetrics-MT outperforms all existing baselines, setting a new benchmark for state-of-the-art performance in the reference-based setting. Furthermore, it achieves comparable results to leading metrics in the reference-free setting, offering greater efficiency.
Gaussian processes (GPs) are non-parametric probabilistic regression models that are popular due to their flexibility, data efficiency, and well-calibrated uncertainty estimates. However, standard GP models assume homoskedastic Gaussian noise, while many real-world applications are subject to non-Gaussian corruptions. Variants of GPs that are more robust to alternative noise models have been proposed, and entail significant trade-offs between accuracy and robustness, and between computational requirements and theoretical guarantees. In this work, we propose and study a GP model that achieves robustness against sparse outliers by inferring data-point-specific noise levels with a sequential selection procedure maximizing the log marginal likelihood that we refer to as relevance pursuit. We show, surprisingly, that the model can be parameterized such that the associated log marginal likelihood is strongly concave in the data-point-specific noise variances, a property rarely found in either robust regression objectives or GP marginal likelihoods. This in turn implies the weak submodularity of the corresponding subset selection problem, and thereby proves approximation guarantees for the proposed algorithm. We compare the model's performance relative to other approaches on diverse regression and Bayesian optimization tasks, including the challenging but common setting of sparse corruptions of the labels within or close to the function range.
Descriptions of the Galactic Center using Fermi gamma-ray data have so far modeled the Galactic Center Excess (GCE) as a template with fixed spatial morphology or as a linear combination of such templates. Although these templates are informed by various physical expectations, the morphology of the excess is a priori unknown. For the first time, we describe the GCE using a flexible, non-parametric machine learning model -- the Gaussian process (GP). We assess our model's performance on synthetic data, demonstrating that the model can recover the templates used to generate the data. We then fit the \Fermi data with our model in a single energy bin from 2-20 GeV (leaving a spectral GP analysis of the GCE for future work) using a variety of template models of diffuse gamma-ray emission to quantify our fits' systematic uncertainties associated with diffuse emission modeling. We interpret our best-fit GP in terms of GCE templates consisting of an NFW squared template and a bulge component to determine which bulge models can best describe the fitted GP and to what extent the best-fit GP is described better by an NFW squared template versus a bulge template. The best-fit GP contains morphological features that are typically not associated with traditional GCE studies. These include a localized bright source at around $(\ell,b) = (20^{\circ}, 0^{\circ})$ and a diagonal arm extending Northwest from the Galactic Center. In spite of these novel features, the fitted GP is explained best by a template-based model consisting of the bulge presented in Coleman et al. (2020) and a squared NFW component. Our results suggest that the physical interpretation of the GCE in terms of stellar bulge and NFW-like components is highly sensitive to the assumed morphologies, background models, and the region of the sky used for inference.
This work presents an omics-driven modeling pipeline that integrates machine-learning tools to facilitate the dynamic modeling of multiscale biological systems. Random forests and permutation feature importance are proposed to mine omics datasets, guiding feature selection and dimensionality reduction for dynamic modeling. Continuous and differentiable machine-learning functions can be trained to link the reduced omics feature set to key components of the dynamic model, resulting in a hybrid model. As proof of concept, we apply this framework to a high-dimensional proteomics dataset of $\textit{Saccharomyces cerevisiae}$. After identifying key intracellular proteins that correlate with cell growth, targeted dynamic experiments are designed, and key model parameters are captured as functions of the selected proteins using Gaussian processes. This approach captures the dynamic behavior of yeast strains under varying proteome profiles while estimating the uncertainty in the hybrid model's predictions. The outlined modeling framework is adaptable to other scenarios, such as integrating additional layers of omics data for more advanced multiscale biological systems, or employing alternative machine-learning methods to handle larger datasets. Overall, this study outlines a strategy for leveraging omics data to inform multiscale dynamic modeling in systems biology and bioprocess engineering.
Bayesian inference offers a robust framework for updating prior beliefs based on new data using Bayes' theorem, but exact inference is often computationally infeasible, necessitating approximate methods. Though widely used, these methods struggle to estimate marginal likelihoods accurately, particularly due to the rigid functional structures of deterministic models like Gaussian processes and the limitations of small sample sizes in stochastic models like the ensemble Kalman method. In this work, we introduce BI-EqNO, an equivariant neural operator framework for generalized approximate Bayesian inference, designed to enhance both deterministic and stochastic approaches. BI-EqNO transforms priors into posteriors conditioned on observation data through data-driven training. The framework is flexible, supporting diverse prior and posterior representations with arbitrary discretizations and varying numbers of observations. Crucially, BI-EqNO's architecture ensures (1) permutation equivariance between prior and posterior representations, and (2) permutation invariance with respect to observational data. We demonstrate BI-EqNO's utility through two examples: (1) as a generalized Gaussian process (gGP) for regression, and (2) as an ensemble neural filter (EnNF) for sequential data assimilation. Results show that gGP outperforms traditional Gaussian processes by offering a more flexible representation of covariance functions. Additionally, EnNF not only outperforms the ensemble Kalman filter in small-ensemble settings but also has the potential to function as a "super" ensemble filter, capable of representing and integrating multiple ensemble filters for enhanced assimilation performance. This study highlights BI-EqNO's versatility and effectiveness, improving Bayesian inference through data-driven training while reducing computational costs across various applications.
In computational physics, machine learning has now emerged as a powerful complementary tool to explore efficiently candidate designs in engineering studies. Outputs in such supervised problems are signals defined on meshes, and a natural question is the extension of general scalar output regression models to such complex outputs. Changes between input geometries in terms of both size and adjacency structure in particular make this transition non-trivial. In this work, we propose an innovative strategy for Gaussian process regression where inputs are large and sparse graphs with continuous node attributes and outputs are signals defined on the nodes of the associated inputs. The methodology relies on the combination of regularized optimal transport, dimension reduction techniques, and the use of Gaussian processes indexed by graphs. In addition to enabling signal prediction, the main point of our proposal is to come with confidence intervals on node values, which is crucial for uncertainty quantification and active learning. Numerical experiments highlight the efficiency of the method to solve real problems in fluid dynamics and solid mechanics.
Accurate uncertainty quantification for causal effects is essential for robust decision making in complex systems, but remains challenging in non-parametric settings. One promising framework represents conditional distributions in a reproducing kernel Hilbert space and places Gaussian process priors on them to infer posteriors on causal effects, but requires restrictive nuclear dominant kernels and approximations that lead to unreliable uncertainty estimates. In this work, we introduce a method, IMPspec, that addresses these limitations via a spectral representation of the Hilbert space. We show that posteriors in this model can be obtained explicitly, by extending a result in Hilbert space regression theory. We also learn the spectral representation to optimise posterior calibration. Our method achieves state-of-the-art performance in uncertainty quantification and causal Bayesian optimisation across simulations and a healthcare application.
The computational cost for inference and prediction of statistical models based on Gaussian processes with Mat\'ern covariance functions scales cubicly with the number of observations, limiting their applicability to large data sets. The cost can be reduced in certain special cases, but there are currently no generally applicable exact methods with linear cost. Several approximate methods have been introduced to reduce the cost, but most of these lack theoretical guarantees for the accuracy. We consider Gaussian processes on bounded intervals with Mat\'ern covariance functions and for the first time develop a generally applicable method with linear cost and with a covariance error that decreases exponentially fast in the order $m$ of the proposed approximation. The method is based on an optimal rational approximation of the spectral density and results in an approximation that can be represented as a sum of $m$ independent Gaussian Markov processes, which facilitates easy usage in general software for statistical inference, enabling its efficient implementation in general statistical inference software packages. Besides the theoretical justifications, we demonstrate the accuracy empirically through carefully designed simulation studies which show that the method outperforms all state-of-the-art alternatives in terms of accuracy for a fixed computational cost in statistical tasks such as Gaussian process regression.
Modern physics simulation often involves multiple functions of interests, and traditional numerical approaches are known to be complex and computationally costly. While machine learning-based surrogate models can offer significant cost reductions, most focus on a single task, such as forward prediction, and typically lack uncertainty quantification -- an essential component in many applications. To overcome these limitations, we propose Arbitrarily-Conditioned Multi-Functional Diffusion (ACMFD), a versatile probabilistic surrogate model for multi-physics emulation. ACMFD can perform a wide range of tasks within a single framework, including forward prediction, various inverse problems, and simulating data for entire systems or subsets of quantities conditioned on others. Specifically, we extend the standard Denoising Diffusion Probabilistic Model (DDPM) for multi-functional generation by modeling noise as Gaussian processes (GP). We then introduce an innovative denoising loss. The training involves randomly sampling the conditioned part and fitting the corresponding predicted noise to zero, enabling ACMFD to flexibly generate function values conditioned on any other functions or quantities. To enable efficient training and sampling, and to flexibly handle irregularly sampled data, we use GPs to interpolate function samples onto a grid, inducing a Kronecker product structure for efficient computation. We demonstrate the advantages of ACMFD across several fundamental multi-physics systems.
We present a novel Bayesian tomography approach for Coherence Imaging Spectroscopy (CIS) that simultaneously reconstructs ion temperature and velocity distributions in plasmas. Utilizing nonlinear Gaussian Process Tomography (GPT) with the Laplace approximation, we model prior distributions of log-emissivity, temperature, and velocity as Gaussian processes. This framework rigorously incorporates nonlinear effects and temperature dependencies often neglected in conventional CIS tomography, enabling robust reconstruction even in the region of high temperature and velocity. By applying a log-Gaussian process, we also address issues like velocity divergence in low-emissivity regions. Validated with phantom simulations and experimental data from the RT-1 device, our method reveals detailed spatial structures of ion temperature and toroidal ion flow characteristic of magnetospheric plasma. This work significantly broadens the scope of CIS tomography, offering a robust tool for plasma diagnostics and facilitating integration with complementary measurement techniques.
A key task in AutoML is to model learning curves of machine learning models jointly as a function of model hyper-parameters and training progression. While Gaussian processes (GPs) are suitable for this task, na\"ive GPs require $\mathcal{O}(n^3m^3)$ time and $\mathcal{O}(n^2 m^2)$ space for $n$ hyper-parameter configurations and $\mathcal{O}(m)$ learning curve observations per hyper-parameter. Efficient inference via Kronecker structure is typically incompatible with early-stopping due to missing learning curve values. We impose $\textit{latent Kronecker structure}$ to leverage efficient product kernels while handling missing values. In particular, we interpret the joint covariance matrix of observed values as the projection of a latent Kronecker product. Combined with iterative linear solvers and structured matrix-vector multiplication, our method only requires $\mathcal{O}(n^3 + m^3)$ time and $\mathcal{O}(n^2 + m^2)$ space. We show that our GP model can match the performance of a Transformer on a learning curve prediction task.
The problem of classifying graphs is ubiquitous in machine learning. While it is standard to apply graph neural networks or graph kernel methods, Gaussian processes can be employed by transforming spatial features from the graph domain into spectral features in the Euclidean domain, and using them as the input points of classical kernels. However, this approach currently only takes into account features on vertices, whereas some graph datasets also support features on edges. In this work, we present a Gaussian process-based classification algorithm that can leverage one or both vertex and edges features. Furthermore, we take advantage of the Hodge decomposition to better capture the intricate richness of vertex and edge features, which can be beneficial on diverse tasks.
The performance of tracking algorithms strongly depends on the chosen model assumptions regarding the target dynamics. If there is a strong mismatch between the chosen model and the true object motion, the track quality may be poor or the track is easily lost. Still, the true dynamics might not be known a priori or it is too complex to be expressed in a tractable mathematical formulation. This paper provides a comparative study between three different methods that use machine learning to describe the underlying object motion based on training data. The first method builds on Gaussian Processes (GPs) for predicting the object motion, the second learns the parameters of an Interacting Multiple Model (IMM) filter and the third uses a Long Short-Term Memory (LSTM) network as a motion model. All methods are compared against an Extended Kalman Filter (EKF) with an analytic motion model as a benchmark and their respective strengths are highlighted in one simulated and two real-world scenarios.