Gaussian process regression is a widely-applied method for function approximation and uncertainty quantification. The technique has gained popularity recently in the machine learning community due to its robustness and interpretability. The mathematical methods we discuss in this paper are an extension of the Gaussian-process framework. We are proposing advanced kernel designs that only allow for functions with certain desirable characteristics to be elements of the reproducing kernel Hilbert space (RKHS) that underlies all kernel methods and serves as the sample space for Gaussian process regression. These desirable characteristics reflect the underlying physics; two obvious examples are symmetry and periodicity constraints. In addition, non-stationary kernel designs can be defined in the same framework to yield flexible multi-task Gaussian processes. We will show the impact of advanced kernel designs on Gaussian processes using several synthetic and two scientific data sets. The results show that including domain knowledge, communicated through advanced kernel designs, has a significant impact on the accuracy and relevance of the function approximation. Gaussian processes (GPs) [14] provide a powerful mathematical framework for function approximation from data. The associated technique is generally referred to as Gaussian process regression (GPR). GPs are flexible, robust, non-parametric and naturally include uncertainty quantification.

Principal Component Analysis (PCA) is a fundamental tool for data visualization, denoising, and dimensionality reduction. It is widely popular in Statistics, Machine Learning, Computer Vision, and related fields. However, PCA is well known to fall prey to the presence of outliers and often fails to detect the true underlying low-dimensional structure within the dataset. Recent supervised learning methods, following the Median of Means (MoM) philosophy, have shown great success in dealing with outlying observations without much compromise to their large sample theoretical properties. In this paper, we propose a PCA procedure based on the MoM principle. Called the Median of Means Principal Component Analysis (MoMPCA), the proposed method is not only computationally appealing but also achieves optimal convergence rates under minimal assumptions. In particular, we explore the non-asymptotic error bounds of the obtained solution via the aid of Vapnik-Chervonenkis theory and Rademacher complexity, while granting absolutely no assumption on the outlying observations. The efficacy of the proposal is also thoroughly showcased through simulations and real data applications.

Collecting and aggregating information from several probability measures or histograms is a fundamental task in machine learning. One of the popular solution methods for this task is to compute the barycenter of the probability measures under the Wasserstein metric. However, approximating the Wasserstein barycenter is numerically challenging because of the curse of dimensionality. This paper proposes the projection robust Wasserstein barycenter (PRWB) that mitigates the curse of dimensionality. This new model projects the probability measures onto a lower-dimensional subspace that maximizes the Wasserstein barycenter objective. The resulting problem is a max-min problem over the Stiefel manifold, which is numerically challenging in practice. Combining the iterative Bregman projection algorithm and Riemannian optimization, we propose two new algorithms for computing the PRWB. The complexity of arithmetic operations of the proposed algorithms for obtaining an $\epsilon$-stationary solution is analyzed. We incorporate the PRWB into a discrete distribution clustering algorithm, and the numerical results on real text datasets confirm that our PRWB model helps improve the clustering performance significantly.

As gradient-free stochastic optimization gains emerging attention for a wide range of applications recently, the demand for uncertainty quantification of parameters obtained from such approaches arises. In this paper, we investigate the problem of statistical inference for model parameters based on gradient-free stochastic optimization methods that use only function values rather than gradients. We first present central limit theorem results for Polyak-Ruppert-averaging type gradient-free estimators. The asymptotic distribution reflects the trade-off between the rate of convergence and function query complexity. We next construct valid confidence intervals for model parameters through the estimation of the covariance matrix in a fully online fashion. We further give a general gradient-free framework for covariance estimation and analyze the role of function query complexity in the convergence rate of the covariance estimator. This provides a one-pass computationally efficient procedure for simultaneously obtaining an estimator of model parameters and conducting statistical inference. Finally, we provide numerical experiments to verify our theoretical results and illustrate some extensions of our method for various machine learning and deep learning applications.

This paper describes a general-purpose extension of max-value entropy search, a popular approach for Bayesian Optimisation (BO). A novel approximation is proposed for the information gain -- an information-theoretic quantity central to solving a range of BO problems, including noisy, multi-fidelity and batch optimisations across both continuous and highly-structured discrete spaces. Previously, these problems have been tackled separately within information-theoretic BO, each requiring a different sophisticated approximation scheme, except for batch BO, for which no computationally-lightweight information-theoretic approach has previously been proposed. GIBBON (General-purpose Information-Based Bayesian OptimisatioN) provides a single principled framework suitable for all the above, out-performing existing approaches whilst incurring substantially lower computational overheads. In addition, GIBBON does not require the problem's search space to be Euclidean and so is the first high-performance yet computationally light-weight acquisition function that supports batch BO over general highly structured input spaces like molecular search and gene design. Moreover, our principled derivation of GIBBON yields a natural interpretation of a popular batch BO heuristic based on determinantal point processes.

The majority of finite mixture models suffer from not allowing asymmetric tail dependencies within components and not capturing non-elliptical clusters in clustering applications. Since vine copulas are very flexible in capturing these types of dependencies, we propose a novel vine copula mixture model for continuous data. We discuss the model selection and parameter estimation problems and further formulate a new model-based clustering algorithm. The use of vine copulas in clustering allows for a range of shapes and dependency structures for the clusters. Our simulation experiments illustrate a significant gain in clustering accuracy when notably asymmetric tail dependencies or/and non-Gaussian margins within the components exist. The analysis of real data sets accompanies the proposed method. We show that the model-based clustering algorithm with vine copula mixture models outperforms the other model-based clustering techniques, especially for the non-Gaussian multivariate data.

Data acquisition forms the primary step in all empirical research. The availability of data directly impacts the quality and extent of conclusions and insights. In particular, larger and more detailed datasets provide convincing answers even to complex research questions. The main problem is that 'large and detailed' usually implies 'costly and difficult', especially when the data medium is paper and books. Human operators and manual transcription have been the traditional approach for collecting historical data. We instead advocate the use of modern machine learning techniques to automate the digitisation process. We give an overview of the potential for applying machine digitisation for data collection through two illustrative applications. The first demonstrates that unsupervised layout classification applied to raw scans of nurse journals can be used to construct a treatment indicator. Moreover, it allows an assessment of assignment compliance. The second application uses attention-based neural networks for handwritten text recognition in order to transcribe age and birth and death dates from a large collection of Danish death certificates. We describe each step in the digitisation pipeline and provide implementation insights.

Motivated by the recent successes of neural networks that have the ability to fit the data perfectly and generalize well, we study the noiseless model in the fundamental least-squares setup. We assume that an optimum predictor fits perfectly inputs and outputs $\langle \theta_* , \phi(X) \rangle = Y$, where $\phi(X)$ stands for a possibly infinite dimensional non-linear feature map. To solve this problem, we consider the estimator given by the last iterate of stochastic gradient descent (SGD) with constant step-size. In this context, our contribution is two fold: (i) from a (stochastic) optimization perspective, we exhibit an archetypal problem where we can show explicitly the convergence of SGD final iterate for a non-strongly convex problem with constant step-size whereas usual results use some form of average and (ii) from a statistical perspective, we give explicit non-asymptotic convergence rates in the over-parameterized setting and leverage a fine-grained parameterization of the problem to exhibit polynomial rates that can be faster than $O(1/T)$. The link with reproducing kernel Hilbert spaces is established.

We consider the problem of sparse normal means estimation in a distributed setting with communication constraints. We assume there are $M$ machines, each holding a $d$-dimensional observation of a $K$-sparse vector $\mu$ corrupted by additive Gaussian noise. A central fusion machine is connected to the $M$ machines in a star topology, and its goal is to estimate the vector $\mu$ with a low communication budget. Previous works have shown that to achieve the centralized minimax rate for the $\ell_2$ risk, the total communication must be high - at least linear in the dimension $d$. This phenomenon occurs, however, at very weak signals. We show that once the signal-to-noise ratio (SNR) is slightly higher, the support of $\mu$ can be correctly recovered with much less communication. Specifically, we present two algorithms for the distributed sparse normal means problem, and prove that above a certain SNR threshold, with high probability, they recover the correct support with total communication that is sublinear in the dimension $d$. Furthermore, the communication decreases exponentially as a function of signal strength. If in addition $KM\ll d$, then with an additional round of sublinear communication, our algorithms achieve the centralized rate for the $\ell_2$ risk. Finally, we present simulations that illustrate the performance of our algorithms in different parameter regimes.

Factorization-based gradient descent is a scalable and efficient algorithm for solving low-rank matrix completion. Recent progress in structured non-convex optimization has offered global convergence guarantees for gradient descent under certain statistical assumptions on the low-rank matrix and the sampling set. However, while the theory suggests gradient descent enjoys fast linear convergence to a global solution of the problem, the universal nature of the bounding technique prevents it from obtaining an accurate estimate of the rate of convergence. In this paper, we perform a local analysis of the exact linear convergence rate of gradient descent for factorization-based matrix completion for symmetric matrices. Without any additional assumptions on the underlying model, we identify the deterministic condition for local convergence of gradient descent, which only depends on the solution matrix and the sampling set. More crucially, our analysis provides a closed-form expression of the asymptotic rate of convergence that matches exactly with the linear convergence observed in practice. To the best of our knowledge, our result is the first one that offers the exact rate of convergence of gradient descent for matrix factorization in Euclidean space for matrix completion.