We introduce the convolutional spectral kernel (CSK), a novel family of interpretable and non-stationary kernels derived from the convolution of two imaginary radial basis functions. We propose the input-frequency spectrogram as a novel tool to analyze nonparametric kernels as well as the kernels of deep Gaussian processes (DGPs). Observing through the lens of the spectrogram, we shed light on the interpretability of deep models, along with useful insights for effective inference. We also present scalable variational and stochastic Hamiltonian Monte Carlo inference to learn rich, yet interpretable frequency patterns from data using DGPs constructed via covariance functions. Empirically we show on simulated and real-world datasets that CSK extracts meaningful non-stationary periodicities.

Spectral mixture kernels have been proposed as general-purpose, flexible kernels for learning and discovering more complicated patterns in the data. Spectral mixture kernels have recently been generalized into non-stationary kernels by replacing the mixture weights, frequency means and variances by input-dependent functions. These functions have also been modelled as Gaussian processes on their own. In this paper we propose modelling the hyperparameter functions with neural networks, and provide an experimental comparison between the stationary spectral mixture and the two non-stationary spectral mixtures. Scalable Gaussian process inference is implemented within the sparse variational framework for all the kernels considered. We show that the neural variant of the kernel is able to achieve the best performance, among alternatives, on several benchmark datasets.

We propose a novel deep learning paradigm of differential flows that learn a stochastic differential equation transformations of inputs prior to a standard classification or regression function. The key property of differential Gaussian processes is the warping of inputs through infinitely deep, but infinitesimal, differential fields, that generalise discrete layers into a dynamical system. We demonstrate state-of-the-art results that exceed the performance of deep Gaussian processes and neural networks.

The expressive power of Gaussian processes depends heavily on the choice of kernel. In this work we propose the novel harmonizable mixture kernel (HMK), a family of expressive, interpretable, non-stationary kernels derived from mixture models on the generalized spectral representation. As a theoretically sound treatment of non-stationary kernels, HMK supports harmonizable covariances, a wide subset of kernels including all stationary and many non-stationary covariances. We also propose variational Fourier features, an inter-domain sparse GP inference framework that offers a representative set of 'inducing frequencies'. We show that harmonizable mixture kernels interpolate between local patterns, and that variational Fourier features offers a robust kernel learning framework for the new kernel family.

We propose deep convolutional Gaussian processes, a deep Gaussian process architecture with convolutional structure. The model is a principled Bayesian framework for detecting hierarchical combinations of local features for image classification. We demonstrate greatly improved image classification performance compared to current Gaussian process approaches on the MNIST and CIFAR-10 datasets. In particular, we improve CIFAR-10 accuracy by over 10 percentage points.

We introduce a novel paradigm for learning non-parametric drift and diffusion functions for stochastic differential equation (SDE). The proposed model learns to simulate path distributions that match observations with non-uniform time increments and arbitrary sparseness, which is in contrast with gradient matching that does not optimize simulated responses. We formulate sensitivity equations for learning and demonstrate that our general stochastic distribution optimisation leads to robust and efficient learning of SDE systems.

Metabolic flux balance analyses are a standard tool in analysing metabolic reaction rates compatible with measurements, steady-state and the metabolic reaction network stoichiometry. Flux analysis methods commonly place unrealistic assumptions on fluxes due to the convenience of formulating the problem as a linear programming model, and most methods ignore the notable uncertainty in flux estimates. We introduce a novel paradigm of Bayesian metabolic flux analysis that models the reactions of the whole genome-scale cellular system in probabilistic terms, and can infer the full flux vector distribution of genome-scale metabolic systems based on exchange and intracellular (e.g. 13C) flux measurements, steady-state assumptions, and target function assumptions. The Bayesian model couples all fluxes jointly together in a simple truncated multivariate posterior distribution, which reveals informative flux couplings. Our model is a plug-in replacement to conventional metabolic balance methods, such as flux balance analysis (FBA). Our experiments indicate that we can characterise the genome-scale flux covariances, reveal flux couplings, and determine more intracellular unobserved fluxes in C. acetobutylicum from 13C data than flux variability analysis. The COBRA compatible software is available at github.com/markusheinonen/bamfa

Proteins are commonly used by biochemical industry for numerous processes. Refining these proteins' properties via mutations causes stability effects as well. Accurate computational method to predict how mutations affect protein stability are necessary to facilitate efficient protein design. However, accuracy of predictive models is ultimately constrained by the limited availability of experimental data. We have developed mGPfusion, a novel Gaussian process (GP) method for predicting protein's stability changes upon single and multiple mutations. This method complements the limited experimental data with large amounts of molecular simulation data. We introduce a Bayesian data fusion model that re-calibrates the experimental and in silico data sources and then learns a predictive GP model from the combined data. Our protein-specific model requires experimental data only regarding the protein of interest and performs well even with few experimental measurements. The mGPfusion models proteins by contact maps and infers the stability effects caused by mutations with a mixture of graph kernels. Our results show that mGPfusion outperforms state-of-the-art methods in predicting protein stability on a dataset of 15 different proteins and that incorporating molecular simulation data improves the model learning and prediction accuracy.

Zero-inflated datasets, which have an excess of zero outputs, are commonly encountered in problems such as climate or rare event modelling. Conventional machine learning approaches tend to overestimate the non-zeros leading to poor performance. We propose a novel model family of zero-inflated Gaussian processes (ZiGP) for such zero-inflated datasets, produced by sparse kernels through learning a latent probit Gaussian process that can zero out kernel rows and columns whenever the signal is absent. The ZiGPs are particularly useful for making the powerful Gaussian process networks more interpretable. We introduce sparse GP networks where variable-order latent modelling is achieved through sparse mixing signals. We derive the non-trivial stochastic variational inference tractably for scalable learning of the sparse kernels in both models. The novel output-sparse approach improves both prediction of zero-inflated data and interpretability of latent mixing models.

In conventional ODE modelling coefficients of an equation driving the system state forward in time are estimated. However, for many complex systems it is practically impossible to determine the equations or interactions governing the underlying dynamics. In these settings, parametric ODE model cannot be formulated. Here, we overcome this issue by introducing a novel paradigm of nonparametric ODE modelling that can learn the underlying dynamics of arbitrary continuous-time systems without prior knowledge. We propose to learn non-linear, unknown differential functions from state observations using Gaussian process vector fields within the exact ODE formalism. We demonstrate the model's capabilities to infer dynamics from sparse data and to simulate the system forward into future.