Deep Gaussian Processes (DGPs) leverage a compositional structure to model non-stationary processes. DGPs typically rely on local inducing point approximations across intermediate GP layers. Recent advances in DGP inference have shown that incorporating global Fourier features from the Reproducing Kernel Hilbert Space (RKHS) can enhance the DGPs’ capability to capture complex non-stationary patterns. This paper extends the use of these features to compositional GPs involving linear transformations. In particular, we introduce Ordinary Differential Equation(ODE)–based RKHS Fourier features that allow for adaptive amplitude and phase modulation through convolution operations. This convolutional formulation relates our work to recently proposed deep latent force models, a multi-layer structure designed for modelling nonlinear dynamical systems. By embedding these adjustable RKHS Fourier features within a doubly stochastic variational inference framework, our model exhibits improved predictive performance across various regression tasks.

Recent advances in general-purpose foundation models have stimulated the development of large biological sequence models. While natural language shows symbolic granularity (characters, words, sentences), biological sequences exhibit hierarchical granularity whose levels (nucleotides, amino acids, protein domains, genes) further encode biologically functional information. In this paper, we investigate the integration of crossgranularity knowledge from models through a case study of BiGCARP, a Pfam domain-level model for biosynthetic gene clusters, and ESM, an amino acid-level protein language model. Using representation analysis tools and a set of probe tasks, we first explain why a straightforward cross-model embedding initialization fails to improve downstream performance in BiGCARP, and show that deeper-layer embeddings capture a more contextual and faithful representation of the model's learned knowledge. Furthermore, we demonstrate that representations at different granularities encode complementary biological knowledge, and that combining them yields measurable performance gains in intermediate-level prediction tasks. Our findings highlight crossgranularity integration as a promising strategy for improving both the performance and interpretability of biological foundation models. Our code is available at https://github.com/Nugkta/cgrep.

Gaussian Process (GP) Variational Autoencoders (VAEs) extend standard VAEs by replacing the fully factorised Gaussian prior with a GP prior, thereby capturing richer correlations among latent variables. However, performing exact GP inference in large-scale GPVAEs is computationally prohibitive, often forcing existing approaches to rely on restrictive kernel assumptions or large sets of inducing points. In this work, we propose a neighbour-driven approximation strategy that exploits local adjacencies in the latent space to achieve scalable GPVAE inference. By confining computations to the nearest neighbours of each data point, our method preserves essential latent dependencies, allowing more flexible kernel choices and mitigating the need for numerous inducing points. Through extensive experiments on tasks including representation learning, data imputation, and conditional generation, we demonstrate that our approach outperforms other GPVAE variants in both predictive performance and computational efficiency.

Modelling the behaviour of highly nonlinear dynamical systems with robust uncertainty quantification is a challenging task which typically requires approaches specifically designed to address the problem at hand. We introduce a domain-agnostic model to address this issue termed the deep latent force model (DLFM), a deep Gaussian process with physics-informed kernels at each layer, derived from ordinary differential equations using the framework of process convolutions. Two distinct formulations of the DLFM are presented which utilise weight-space and variational inducing points-based Gaussian process approximations, both of which are amenable to doubly stochastic variational inference. We present empirical evidence of the capability of the DLFM to capture the dynamics present in highly nonlinear real-world multi-output time series data. Additionally, we find that the DLFM is capable of achieving comparable performance to a range of non-physics-informed probabilistic models on benchmark univariate regression tasks. We also empirically assess the negative impact of the inducing points framework on the extrapolation capabilities of LFM-based models.

A key challenge in the practical application of Gaussian processes (GPs) is selecting a proper covariance function. The process convolutions construction of GPs allows some additional flexibility, but still requires choosing a proper smoothing kernel, which is non-trivial. Previous approaches have built covariance functions by using GP priors over the smoothing kernel, and by extension the covariance, as a way to bypass the need to specify it in advance. However, these models have been limited in several ways: they are restricted to single dimensional inputs, e.g. time; they only allow modelling of single outputs and they do not scale to large datasets since inference is not straightforward. In this paper, we introduce a nonparametric process convolution formulation for GPs that alleviates these weaknesses. We achieve this using a functional sampling approach based on Matheron’s rule to perform fast sampling using interdomain inducing variables. We test the performance of our model on benchmarks for single output, multi-output and large-scale GP regression, and find that our approach can provide improvements over standard GP models, particularly for larger datasets.

Diffusion MRI (dMRI) is a widely used imaging modality, but requires long scanning times to acquire high resolution datasets. By leveraging the unique geometry present within this domain, we present a novel approach to dMRI angular super-resolution that extends upon the parametric continuous convolution (PCConv) framework. We introduce several additions to the operation including a Fourier feature mapping, 'global' co-ordinates, and domain specific context. Using this framework, we build a fully parametric continuous convolution network (PCCNN) and compare against existing models. We demonstrate the PCCNN performs competitively while using significantly fewer parameters. Moreover, we show that this formulation generalises well to clinically relevant downstream analyses such as fixel-based analysis, and neurite orientation dispersion and density imaging.

Gaussian processes (GPs) can provide a principled approach to uncertainty quantification with easy-to-interpret kernel hyperparameters, such as the lengthscale, which controls the correlation distance of function values.However, selecting an appropriate kernel can be challenging. Deep GPs avoid manual kernel engineering by successively parameterizing kernels with GP layers, allowing them to learn low-dimensional embeddings of the inputs that explain the output data. Following the architecture of deep neural networks, the most common deep GPs warp the input space layer-by-layer but lose all the interpretability of shallow GPs. An alternative construction is to successively parameterize the lengthscale of a kernel, improving the interpretability but ultimately giving away the notion of learning lower-dimensional embeddings. Unfortunately, both methods are susceptible to particular pathologies which may hinder fitting and limit their interpretability. This work proposes a novel synthesis of both previous approaches: {Thin and Deep GP} (TDGP). Each TDGP layer defines locally linear transformations of the original input data maintaining the concept of latent embeddings while also retaining the interpretation of lengthscales of a kernel. Moreover, unlike the prior solutions, TDGP induces non-pathological manifolds that admit learning lower-dimensional representations. We show with theoretical and experimental results that i) TDGP is, unlike previous models, tailored to specifically discover lower-dimensional manifolds in the input data, ii) TDGP behaves well when increasing the number of layers, and iii) TDGP performs well in standard benchmark datasets.

Abstract Multi-output Gaussian processes (MOGPs) can help to improve predictive performance for some output variables, by leveraging the correlation with other output variables. In this paper, our main motivation is to use multiple-output Gaussian processes to exploit correlations between outputs where each output is a multi-class classification problem. MOGPs have been mostly used for multi-output regression. There are some existing works that use MOGPs for other types of outputs, e.g., multi-output binary classification. However, MOGPs for multi-class classification has been less studied. The reason is twofold: 1) when using a softmax function, it is not clear how to scale it beyond the case of a few outputs; 2) most common type of data in multi-class classification problems consists of image data, and MOGPs are not specifically designed to image data. We thus propose a new MOGPs model called Multi-output Gaussian Processes with Augment & Reduce (MOGPs-AR) that can deal with large scale classification and downsized image input data. Large scale classification is achieved by subsampling both training data sets and classes in each output whereas downsized image input data is handled by incorporating a convolutional kernel into the new model. We show empirically that our proposed model outperforms single-output Gaussian processes in terms of different performance metrics and multi-output Gaussian processes in terms of scalability, both in synthetic and in real classification problems. We include an example with the Ommiglot dataset where we showcase the properties of our model.

Abstract Protecting ML classifiers from adversarial examples is crucial. We propose that the main threat is an attacker perturbing a confidently classified input to produce a confident misclassification. We consider in this paper the $$L_0$$ L 0 attack in which a small number of inputs can be perturbed by the attacker at test-time. To quantify the risk of this form of attack we have devised a formal guarantee in the form of an adversarial bound (AB) for a binary, Gaussian process classifier using the EQ kernel. This bound holds for the entire input domain, bounding the potential of any future adversarial attack to cause a confident misclassification. We explore how to extend to other kernels and investigate how to maximise the bound by altering the classifier (for example by using sparse approximations). We test the bound using a variety of datasets and show that it produces relevant and practical bounds for many of them.

The labeling process within a supervised learning task is usually carried out by an expert, which provides the ground truth (gold standard) for each sample. However, in many real-world applications, we typically have access to annotations provided by crowds holding different and unknown expertise levels. Learning from crowds (LFC) intends to configure machine learning paradigms in the presence of multilabelers, residing on two key assumptions: the labeler's performance does not depend on the input space, and independence among the annotators is imposed. Here, we propose the correlated chained Gaussian processes from the multiple annotators (CCGPMA) approach, which models each annotator's performance as a function of the input space and exploits the correlations among experts. Experimental results associated with classification and regression tasks show that our CCGPMA performs better modeling of the labelers' behavior, indicating that it consistently outperforms other state-of-the-art LFC approaches.

Linear systems occur throughout engineering and the sciences, most notably as differential equations. In many cases the forcing function for the system is unknown, and interest lies in using noisy observations of the system to infer the forcing, as well as other unknown parameters. In differential equations, the forcing function is an unknown function of the independent variables (typically time and space), and can be modelled as a Gaussian process (GP). In this paper we show how the adjoint of a linear system can be used to efficiently infer forcing functions modelled as GPs, after using a truncated basis expansion of the GP kernel. We show how exact conjugate Bayesian inference for the truncated GP can be achieved, in many cases with substantially lower computation than would be required using MCMC methods. We demonstrate the approach on systems of both ordinary and partial differential equations, and show that the basis expansion approach approximates well the true forcing with a modest number of basis vectors. Finally, we show how to infer point estimates for the non-linear model parameters, such as the kernel length-scales, using Bayesian optimisation.

Modelling the mobility of people in a city depends on counting data with inherent problems of overdispersion. Such dispersion issues are caused by massive amounts of data with zero values. Though traditional machine learning models have been used to overcome said problems, they lack the ability to appropriately model the spatio-temporal correlations in data. To improve the modelling of such spatio-temporal correlations, in this work we propose to model the citizens mobility, for the Chinese city of Guangzhou, by means of a Zero-inflated Poisson likelihood in conjunction with Gaussian process priors generated from convolution processes. We follow the idea of chaining the likelihood’s parameters to latent functions drawn from Gaussian process priors; this way allowing a higher flexibility to model heteroscedasticity. Additionally, we derive a stochastic variational inference framework that allow us to use two types of convolution process models in the context of large datasets: correlated chained Gaussian processes with a convolution processes model, and correlated chained Gaussian processes with variational inducing kernels. We present quantitative and qualitative results comparing the performance between Negative Binomial and Zero-inflated Poisson likelihoods, both in combination with three types of Gaussian process priors: a linear model of coregionalisation, and our two proposed methods based on a convolution processes model and variational inducing kernels.

Effectively modeling phenomena present in highly nonlinear dynamical systems whilst also accurately quantifying uncertainty is a challenging task, which often requires problem-specific techniques. We present a novel, domain-agnostic approach to tackling this problem, using compositions of physics-informed random features, derived from ordinary differential equations. The architecture of our model leverages recent advances in approximate inference for deep Gaussian processes, such as layer-wise weight-space approximations which allow us to incorporate random Fourier features, and stochastic variational inference for approximate Bayesian inference. We provide evidence that our model is capable of capturing highly nonlinear behaviour in real-world multivariate time series data. In addition, we find that our approach achieves comparable performance to a number of other probabilistic models on benchmark regression tasks.

We present a framework for transfer learning based on modular variational Gaussian processes (GP). We develop a module-based method that having a dictionary of well fitted GPs, each model being characterised by its hyperparameters, pseudo-inputs and their corresponding posterior densities, one could build ensemble GP models without revisiting any data. Our method avoids undesired data centralisation, reduces rising computational costs and allows the transfer of learned uncertainty metrics after training. We exploit the augmentation of high-dimensional integral operators based on the Kullback-Leibler divergence between stochastic processes to introduce an efficient lower bound under all the sparse variational GPs, with different complexity and even likelihood distribution. The method is also valid for multi-output GPs, learning correlations a posteriori between independent modules. Extensive results illustrate the usability of our framework in large-scale and multi-task experiments, also compared with the exact inference methods in the literature.

This paper introduces a method for the nonparametric Bayesian learning of nonlinear operators, through the use of the Volterra series with kernels represented using Gaussian processes (GPs), which we term the nonparametric Volterra kernels model (NVKM). When the input function to the operator is unobserved and has a GP prior, the NVKM constitutes a powerful method for both single and multiple output regression, and can be viewed as a nonlinear and nonparametric latent force model. When the input function is observed, the NVKM can be used to perform Bayesian system identification. We use recent advances in efficient sampling of explicit functions from GPs to map process realisations through the Volterra series without resorting to numerical integration, allowing scalability through doubly stochastic variational inference, and avoiding the need for Gaussian approximations of the output processes. We demonstrate the performance of the model for both multiple output regression and system identification using standard benchmarks.

A continuing challenge for machine learning is providing methods to perform computation on data while ensuring the data remains private. In this paper we build on the provable privacy guarantees of differential privacy which has been combined with Gaussian processes through the previously published cloaking method, an approach that tackles the problem of providing privacy for the outputs of a training set. In this paper we solve several shortcomings of this method, starting with the problem of predictions in regions with low data density. We experiment with the use of inducing points to provide a sparse approximation and show that these can provide robust differential privacy in outlier areas and at higher dimensions. We then look at classification, and modify the Laplace approximation approach to provide differentially private predictions. We then combine this with the sparse approximation and demonstrate the capability to perform classification in high dimensions. We finally explore the issue of hyperparameter selection and develop a method for their private selection. This paper and associated libraries provide a robust toolkit for combining differential privacy and Gaussian processes in a practical manner.

The regulatory process of Drosophila is thoroughly studied for understanding a great variety of biological principles. While pattern-forming gene networks are analysed in the transcription step, post-transcriptional events (e.g. translation, protein processing) play an important role in establishing protein expression patterns and levels. Since the post-transcriptional regulation of Drosophila depends on spatiotemporal interactions between mRNAs and gap proteins, proper physically-inspired stochastic models are required to study the link between both quantities. Previous research attempts have shown that using Gaussian processes (GPs) and differential equations lead to promising predictions when analysing regulatory networks. Here we aim at further investigating two types of physically-inspired GP models based on a reaction-diffusion equation where the main difference lies in where the prior is placed. While one of them has been studied previously using protein data only, the other is novel and yields a simple approach requiring only the differentiation of kernel functions. In contrast to other stochastic frameworks, discretising the spatial space is not required here. Both GP models are tested under different conditions depending on the availability of gap gene mRNA expression data. Finally, their performances are assessed on a high-resolution dataset describing the blastoderm stage of the early embryo of Drosophila melanogaster

Latent force models are systems whereby there is a mechanistic model describing the dynamics of the system state, with some unknown forcing term that is approximated with a Gaussian process. If such dynamics are non-linear, it can be difficult to estimate the posterior state and forcing term jointly, particularly when there are system parameters that also need estimating. This paper uses black-box variational inference to jointly estimate the posterior, designing a multivariate extension to local inverse autoregressive flows as a flexible approximator of the system. We compare estimates on systems where the posterior is known, demonstrating the effectiveness of the approximation, and apply to problems with non-linear dynamics, multi-output systems and models with non-Gaussian likelihoods.

This paper studies the problem of learning the correlation structure of a set of intervention functions defined on the directed acyclic graph (DAG) of a causal model. This is useful when we are interested in jointly learning the causal effects of interventions on different subsets of variables in a DAG, which is common in field such as healthcare or operations research. We propose the first multi-task causal Gaussian process (GP) model, which we call DAG-GP, that allows for information sharing across continuous interventions and across experiments on different variables. DAG-GP accommodates different assumptions in terms of data availability and captures the correlation between functions lying in input spaces of different dimensionality via a well-defined integral operator. We give theoretical results detailing when and how the DAG-GP model can be formulated depending on the DAG. We test both the quality of its predictions and its calibrated uncertainties. Compared to single-task models, DAG-GP achieves the best fitting performance in a variety of real and synthetic settings. In addition, it helps to select optimal interventions faster than competing approaches when used within sequential decision making frameworks, like active learning or Bayesian optimization.

The paper introduces a non-linear version of the process convolution formalism for building covariance functions for multi-output Gaussian processes. The non-linearity is introduced via Volterra series, one series per each output. We provide closed-form expressions for the mean function and the covariance function of the approximated Gaussian process at the output of the Volterra series. The mean function and covariance function for the joint Gaussian process are derived using formulae for the product moments of Gaussian variables. We compare the performance of the non-linear model against the classical process convolution approach in one synthetic dataset and two real datasets.

Gaussian process (GP) audio source separation is a time- domain approach that circumvents the inherent phase approx- imation issue of spectrogram based methods. Furthermore, through its kernel, GPs elegantly incorporate prior knowl- edge about the sources into the separation model. Despite these compelling advantages, the computational complexity of GP inference scales cubically with the number of audio samples. As a result, source separation GP models have been restricted to the analysis of short audio frames. We intro- duce an efficient application of GPs to time-domain audio source separation, without compromising performance. For this purpose, we used GP regression, together with spectral mixture kernels, and variational sparse GPs. We compared our method with LD-PSDTF (positive semi-definite tensor factorization), KL-NMF (Kullback-Leibler non-negative ma- trix factorization), and IS-NMF (Itakura-Saito NMF). Results show that the proposed method outperforms these techniques.

Aggregated data is commonplace in areas such as epidemiology and demography. For example, census data for a population is usually given as averages defined over time periods or spatial resolutions (cities, regions or countries). In this paper, we present a novel multi-task learning model based on Gaussian processes for joint learning of variables that have been aggregated at different input scales. Our model represents each task as the linear combination of the realizations of latent processes that are integrated at a different scale per task. We are then able to compute the cross-covariance between the different tasks either analytically or numerically. We also allow each task to have a potentially different likelihood model and provide a variational lower bound that can be optimised in a stochastic fashion making our model suitable for larger datasets. We show examples of the model in a synthetic example, a fertility dataset and an air pollution prediction application.

To survive environmental conditions, cells transcribe their response activities into encoded mRNA sequences in order to produce certain amounts of protein concentrations. The external conditions are mapped into the cell through the activation of special proteins called transcription factors (TFs). Due to the difficult task to measure experimentally TF behaviors, and the challenges to capture their quick-time dynamics, different types of models based on differential equations have been proposed. However, those approaches usually incur in costly procedures, and they present problems to describe sudden changes in TF regulators. In this paper, we present a switched dynamical latent force model for reverse-engineering transcriptional regulation in gene expression data which allows the exact inference over latent TF activities driving some observed gene expressions through a linear differential equation. To deal with discontinuities in the dynamics, we introduce an approach that switches between different TF activities and different dynamical systems. This creates a versatile representation of transcription networks that can capture discrete changes and non-linearities. We evaluate our model on both simulated data and real data (e.g., microaerobic shift in E. coli, yeast respiration), concluding that our framework allows for the fitting of the expression data while being able to infer continuous-time TF profiles.

A major challenge for machine learning is increasing the availability of data while respecting the privacy of individuals. Here we combine the provable privacy guarantees of the differential privacy framework with the flexibility of Gaussian processes (GPs). We propose a method using GPs to provide differentially private (DP) regression. We then improve this method by crafting the DP noise covariance structure to efficiently protect the training data, while minimising the scale of the added noise. We find that this cloaking method achieves the greatest accuracy, while still providing privacy guarantees, and offers practical DP for regression over multi-dimensional inputs. Together these methods provide a starter toolkit for combining differential privacy and GPs.

We present a novel extension of multi-output Gaussian processes for handling heterogeneous outputs. We assume that each output has its own likelihood function and use a vector-valued Gaussian process prior to jointly model the parameters in all likelihoods as latent functions. Our multi-output Gaussian process uses a covariance function with a linear model of coregionalisation form. Assuming conditional independence across the underlying latent functions together with an inducing variable framework, we are able to obtain tractable variational bounds amenable to stochastic variational inference. We illustrate the performance of the model on synthetic data and two real datasets: a human behavioral study and a demographic high-dimensional dataset.

Often in machine learning, data are collected as a combination of multiple conditions, e.g., the voice recordings of multiple persons, each labeled with an ID. How could we build a model that captures the latent information related to these conditions and generalize to a new one with few data? We present a new model called Latent Variable Multiple Output Gaussian Processes (LVMOGP) that allows to jointly model multiple conditions for regression and generalize to a new condition with a few data points at test time. LVMOGP infers the posteriors of Gaussian processes together with a latent space representing the information about different conditions. We derive an efficient variational inference method for LVMOGP for which the computational complexity is as low as sparse Gaussian processes. We show that LVMOGP significantly outperforms related Gaussian process methods on various tasks with both synthetic and real data.

In this paper, magnetic resonance image similarity metrics based on generative model induced spaces are introduced. Particularly, three generative-based similarities are proposed. Metrics are tested in an atlas selection task for multi-atlas-based image segmentation of basal ganglia structure, and compared with the mean square metric, as it is assessed on the high dimensional image domain. Attained results show that our proposal provides a suitable atlas selection and improves the segmentation of the structures of interest.

Purely data-driven approaches for machine learning present difficulties when data are scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data-driven modeling with a physical model of the system. We show how different, physically inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology, and geostatistics.

Latent force models encode the interaction between multiple related dynamical systems in the form of a kernel or covariance function. Each variable to be modeled is represented as the output of a differential equation and each differential equation is driven by a weighted sum of latent functions with uncertainty given by a Gaussian process prior. In this paper we consider employing the latent force model framework for the problem of determining robot motor primitives. To deal with discontinuities in the dynamical systems or the latent driving force we introduce an extension of the basic latent force model, that switches between different latent functions and potentially different dynamical systems. This creates a versatile representation for robot movements that can capture discrete changes and non-linearities in the dynamics. We give illustrative examples on both synthetic data and for striking movements recorded using a Barrett WAM robot as haptic input device. Our inspiration is robot motor primitives, but we expect our model to have wide application for dynamical systems including models for human motion capture data and systems biology.

We present a sparse approximation approach for dependent output Gaussian processes (GP). Employing a latent function framework, we apply the convolution process formalism to establish dependencies between output variables, where each latent function is represented as a GP. Based on these latent functions, we establish an approximation scheme using a conditional independence assumption between the output processes, leading to an approximation of the full covariance which is determined by the locations at which the latent functions are evaluated. We show results of the proposed methodology for synthetic data and real world applications on pollution prediction and a sensor network.