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We introduce an efficient and robust auto-tuning framework for hyperparameter selection in dimension reduction (DR) algorithms, focusing on large-scale datasets and arbitrary performance metrics. By leveraging Bayesian optimization (BO) with a surrogate model, our approach enables efficient hyperparameter selection with multi-objective trade-offs and allows us to perform data-driven sensitivity analysis. By incorporating normalization and subsampling, the proposed framework demonstrates versatility and efficiency, as shown in applications to visualization techniques such as t-SNE and UMAP. We evaluate our results on various synthetic and real-world datasets using multiple quality metrics, providing a robust and efficient solution for hyperparameter selection in DR algorithms., 20 pages, 16 figures

In this paper we develop the randomized Sharded Bayesian Additive Regression Trees (SBT) model. We introduce a randomization auxiliary variable and a sharding tree to decide partitioning of data, and fit each partition component to a sub-model using Bayesian Additive Regression Tree (BART). By observing that the optimal design of a sharding tree can determine optimal sharding for sub-models on a product space, we introduce an intersection tree structure to completely specify both the sharding and modeling using only tree structures. In addition to experiments, we also derive the theoretical optimal weights for minimizing posterior contractions and prove the worst-case complexity of SBT., 46 pages, 10 figures (Appendix included)

Supervised dimension reduction (SDR) has been a topic of growing interest in data science, as it enables the reduction of high-dimensional covariates while preserving the functional relation with certain response variables of interest. However, existing SDR methods are not suitable for analyzing datasets collected from case-control studies. In this setting, the goal is to learn and exploit the low-dimensional structure unique to or enriched by the case group, also known as the foreground group. While some unsupervised techniques such as the contrastive latent variable model and its variants have been developed for this purpose, they fail to preserve the functional relationship between the dimension-reduced covariates and the response variable. In this paper, we propose a supervised dimension reduction method called contrastive inverse regression (CIR) specifically designed for the contrastive setting. CIR introduces an optimization problem defined on the Stiefel manifold with a non-standard loss function. We prove the convergence of CIR to a local optimum using a gradient descent-based algorithm, and our numerical study empirically demonstrates the improved performance over competing methods for high-dimensional data.

The modeling and uncertainty quantification of closed curves is an important problem in the field of shape analysis, and can have significant ramifications for subsequent statistical tasks. Many of these tasks involve collections of closed curves, which often exhibit structural similarities at multiple levels. Modeling multiple closed curves in a way that efficiently incorporates such between-curve dependence remains a challenging problem. In this work, we propose and investigate a multiple-output (a.k.a. multi-output), multi-dimensional Gaussian process modeling framework. We illustrate the proposed methodological advances, and demonstrate the utility of meaningful uncertainty quantification, on several curve and shape-related tasks. This model-based approach not only addresses the problem of inference on closed curves (and their shapes) with kernel constructions, but also opens doors to nonparametric modeling of multi-level dependence for functional objects in general., 66 pages, 20 figures

We systematically describe the problem of simultaneous surrogate modeling of mixed variables (i.e., continuous, integer and categorical variables) in the Bayesian optimization (BO) context. We provide a unified hybrid model using both Monte-Carlo tree search (MCTS) and Gaussian processes (GP) that encompasses and generalizes multiple state-of-the-art mixed BO surrogates. Based on the architecture, we propose applying a new dynamic model selection criterion among novel candidate families of covariance kernels, including non-stationary kernels and associated families. Different benchmark problems are studied and presented to support the superiority of our model, along with results highlighting the effectiveness of our method compared to most state-of-the-art mixed-variable methods in BO., 56 pages, 22 Figures

Visualizing very large matrices involves many formidable problems. Various popular solutions to these problems involve sampling, clustering, projection, or feature selection to reduce the size and complexity of the original task. An important aspect of these methods is how to preserve relative distances between points in the higher-dimensional space after reducing rows and columns to fit in a lower dimensional space. This aspect is important because conclusions based on faulty visual reasoning can be harmful. Judging dissimilar points as similar or similar points as dissimilar on the basis of a visualization can lead to false conclusions. To ameliorate this bias and to make visualizations of very large datasets feasible, we introduce two new algorithms that respectively select a subset of rows and columns of a rectangular matrix. This selection is designed to preserve relative distances as closely as possible. We compare our matrix sketch to more traditional alternatives on a variety of artificial and real datasets., Comment: 38 pages, 13 figures

Modern datasets often exhibit high dimensionality, yet the data reside in low-dimensional manifolds that can reveal underlying geometric structures critical for data analysis. A prime example of such a dataset is a collection of cell cycle measurements, where the inherently cyclical nature of the process can be represented as a circle or sphere. Motivated by the need to analyze these types of datasets, we propose a nonlinear dimension reduction method, Spherical Rotation Component Analysis (SRCA), that incorporates geometric information to better approximate low-dimensional manifolds. SRCA is a versatile method designed to work in both high-dimensional and small sample size settings. By employing spheres or ellipsoids, SRCA provides a low-rank spherical representation of the data with general theoretic guarantees, effectively retaining the geometric structure of the dataset during dimensionality reduction. A comprehensive simulation study, along with a successful application to human cell cycle data, further highlights the advantages of SRCA compared to state-of-the-art alternatives, demonstrating its superior performance in approximating the manifold while preserving inherent geometric structures., Comment: 60 pages