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Topological data analysis (TDA) is an emerging mathematical concept for characterizing shapes in complex data. In TDA, persistence diagrams are widely recognized as a useful descriptor of data, and can distinguish robust and noisy…

Gaussian processes are rich distributions over functions, with generalization properties determined by a kernel function. When used for long-range extrapolation, predictions are particularly sensitive to the choice of kernel parameters. It…

Bayesian Optimization is an iterative method, tailored to optimizing expensive black box objective functions. Surrogate models like Gaussian Processes, which are the gold standard in Bayesian Optimization, can be inefficient for inputs with…

In nonparametric regression, it is common for the inputs to fall in a restricted subset of Euclidean space. Typical kernel-based methods that do not take into account the intrinsic geometry of the domain across which observations are…

We propose a new scalable framework for spatio-temporal data fusion with multi-fidelity Gaussian processes (MFGPs) that enables fully likelihood-based inference for both stationary and non-stationary fidelity integration. The framework is…

Fisher's linear discriminant analysis is a classical method for classification, yet it is limited to capturing linear features only. Kernel discriminant analysis as an extension is known to successfully alleviate the limitation through a…

Learning generative models for graph-structured data is challenging because graphs are discrete, combinatorial, and the underlying data distribution is invariant to the ordering of nodes. However, most of the existing generative models for…

Sequential learning paradigms pose challenges for gradient-based deep learning due to difficulties incorporating new data and retaining prior knowledge. While Gaussian processes elegantly tackle these problems, they struggle with…

Analysis of geospatial data has traditionally been model-based, with a mean model, customarily specified as a linear regression on the covariates, and a covariance model, encoding the spatial dependence. We relax the strong assumption of…

Gaussian process regression (GPR) has been a well-known machine learning method for various applications such as uncertainty quantifications (UQ). However, GPR is inherently a data-driven method, which requires sufficiently large dataset.…

The Gaussian Process with a deep kernel is an extension of the classic GP regression model and this extended model usually constructs a new kernel function by deploying deep learning techniques like long short-term memory networks. A…

With view to applications in stochastic analysis and geometry, we introduce a new correspondence for positive definite kernels (p.d.) $K$ and their associated reproducing kernel Hilbert spaces. With this we establish two kinds of…

Gaussian processes (GPs) are used to make medical and scientific decisions, including in cardiac care and monitoring of atmospheric carbon dioxide levels. Notably, the choice of GP kernel is often somewhat arbitrary. In particular,…

A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive…

We introduce a family of multilayer graph kernels and establish new links between graph convolutional neural networks and kernel methods. Our approach generalizes convolutional kernel networks to graph-structured data, by representing…

Gaussian processes (GPs) are Bayesian nonparametric generative models that provide interpretability of hyperparameters, admit closed-form expressions for training and inference, and are able to accurately represent uncertainty. To model…

While (message-passing) graph neural networks have clear limitations in approximating permutation-equivariant functions over graphs or general relational data, more expressive, higher-order graph neural networks do not scale to large…

Accurately representing the sound field with the high spatial resolution is critical for immersive and interactive sound field reproduction technology. To minimize experimental effort, data-driven methods have been proposed to estimate…

Deep generative models (DGMs) have recently demonstrated remarkable success in capturing complex probability distributions over graphs. Although their excellent performance is attributed to powerful and scalable deep neural networks, it is,…

Gaussian process regression is a powerful method for predicting states based on given data. It has been successfully applied for probabilistic predictions of structural systems to quantify, for example, the crack growth in mechanical…