Yuanzhe Xi

Jiaying Lu, Yongchen Qian, Shifan Zhao et al.

Previous research has demonstrated the advantages of integrating data from multiple sources over traditional unimodal data, leading to the emergence of numerous novel multimodal applications. We propose a multimodal classification benchmark MuG with eight datasets that allows researchers to evaluate and improve their models. These datasets are collected from four various genres of games that cover tabular, textual, and visual modalities. We conduct multi-aspect data analysis to provide insights into the benchmark, including label balance ratios, percentages of missing features, distributions of data within each modality, and the correlations between labels and input modalities. We further present experimental results obtained by several state-of-the-art unimodal classifiers and multimodal classifiers, which demonstrate the challenging and multimodal-dependent properties of the benchmark. MuG is released at https://github.com/lujiaying/MUG-Bench with the data, tutorials, and implemented baselines.

Ru Huang, Kai Chang, Huan He et al.

We propose a data-driven and machine-learning-based approach to compute non-Galerkin coarse-grid operators in algebraic multigrid (AMG) methods, addressing the well-known issue of increasing operator complexity. Guided by the AMG theory on spectrally equivalent coarse-grid operators, we have developed novel ML algorithms that utilize neural networks (NNs) combined with smooth test vectors from multigrid eigenvalue problems. The proposed method demonstrates promise in reducing the complexity of coarse-grid operators while maintaining overall AMG convergence for solving parametric partial differential equation (PDE) problems. Numerical experiments on anisotropic rotated Laplacian and linear elasticity problems are provided to showcase the performance and compare with existing methods for computing non-Galerkin coarse-grid operators.

Difeng Cai, Edmond Chow, Yuanzhe Xi

In this paper, we present a comprehensive analysis of the posterior covariance field in Gaussian processes, with applications to the posterior covariance matrix. The analysis is based on the Gaussian prior covariance but the approach also applies to other covariance kernels. Our geometric analysis reveals how the Gaussian kernel's bandwidth parameter and the spatial distribution of the observations influence the posterior covariance as well as the corresponding covariance matrix, enabling straightforward identification of areas with high or low covariance in magnitude. Drawing inspiration from the a posteriori error estimation techniques in adaptive finite element methods, we also propose several estimators to efficiently measure the absolute posterior covariance field, which can be used for efficient covariance matrix approximation and preconditioning. We conduct a wide range of experiments to illustrate our theoretical findings and their practical applications.

Shifan Zhao, Jiaying Lu, Ji Yang et al.

Gaussian Process Regression (GPR) is widely used in statistics and machine learning for prediction tasks requiring uncertainty measures. Its efficacy depends on the appropriate specification of the mean function, covariance kernel function, and associated hyperparameters. Severe misspecifications can lead to inaccurate results and problematic consequences, especially in safety-critical applications. However, a systematic approach to handle these misspecifications is lacking in the literature. In this work, we propose a general framework to address these issues. Firstly, we introduce a flexible two-stage GPR framework that separates mean prediction and uncertainty quantification (UQ) to prevent mean misspecification, which can introduce bias into the model. Secondly, kernel function misspecification is addressed through a novel automatic kernel search algorithm, supported by theoretical analysis, that selects the optimal kernel from a candidate set. Additionally, we propose a subsampling-based warm-start strategy for hyperparameter initialization to improve efficiency and avoid hyperparameter misspecification. With much lower computational cost, our subsampling-based strategy can yield competitive or better performance than training exclusively on the full dataset. Combining all these components, we recommend two GPR methods-exact and scalable-designed to match available computational resources and specific UQ requirements. Extensive evaluation on real-world datasets, including UCI benchmarks and a safety-critical medical case study, demonstrates the robustness and precision of our methods.