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As industrial systems become more complex and monitoring sensors for everything from surveillance to our health become more ubiquitous, multivariate time series prediction is taking an important place in the smooth-running of our society. A…

Principal component analysis (PCA) is arguably the most popular tool in multivariate exploratory data analysis. In this paper, we consider the question of how to handle heterogeneous variables that include continuous, binary, and ordinal.…

In the era of big data, reducing data dimensionality is critical in many areas of science. Widely used Principal Component Analysis (PCA) addresses this problem by computing a low dimensional data embedding that maximally explain variance…

We propose a spectral clustering method based on local principal components analysis (PCA). After performing local PCA in selected neighborhoods, the algorithm builds a nearest neighbor graph weighted according to a discrepancy between the…

Principal component analysis (PCA) is a statistical technique commonly used in multivariate data analysis. However, PCA can be difficult to interpret and explain since the principal components (PCs) are linear combinations of the original…

We tackle the challenges of modeling high-dimensional data sets, particularly those with latent low-dimensional structures hidden within complex, non-linear, and noisy relationships. Our approach enables a seamless integration of concepts…

Principal component analysis (PCA) is perhaps the most widely used method for data dimensionality reduction. A key question in PCA is deciding how many factors to retain. This manuscript describes a new approach to automatically selecting…

Principal Component Analysis (PCA) is a highly useful topic within an introductory Linear Algebra course, especially since it can be used to incorporate a number of applied projects. This method represents an essential application and…

Persistent homology (PH) is one of the most popular tools in topological data analysis (TDA), while graph theory has had a significant impact on data science. Our earlier work introduced the persistent spectral graph (PSG) theory as a…

Sparse principal component analysis (PCA), an important variant of PCA, attempts to find sparse loading vectors when conducting dimension reduction. This paper considers the nonsmooth Riemannian optimization problem associated with the…

In the past decade, sparse principal component analysis has emerged as an archetypal problem for illustrating statistical-computational tradeoffs. This trend has largely been driven by a line of research aiming to characterize the…

Neural operator learning has emerged as a powerful approach for solving partial differential equations (PDEs) in a data-driven manner. However, applying principal component analysis (PCA) to high-dimensional solution fields incurs…

Genetic mutations frequently disrupt protein structure, stability, and solubility, acting as primary drivers for a wide spectrum of diseases. Despite the critical importance of these molecular alterations, existing computational models…

The effectiveness of Spatio-temporal Graph Neural Networks (STGNNs) in time-series applications is often limited by their dependence on fixed, hand-crafted input graph structures. Motivated by insights from the Topological Data Analysis…

Single-cell RNA-sequencing (scRNA-seq) has become a routinely used technique to quantify the gene expression profile of thousands of single cells simultaneously. Analysis of scRNA-seq data plays an important role in the study of cell states…

Deep neural networks (DNNs) often produce overconfident out-of-distribution predictions, motivating Bayesian uncertainty quantification. The Linearized Laplace Approximation (LLA) achieves this by linearizing the DNN and applying Laplace…

Principal Component Analysis (PCA) is a classical method for reducing the dimensionality of data by projecting them onto a subspace that captures most of their variation. Effective use of PCA in modern applications requires understanding…

Deep learning models hold state of the art performance in many fields, yet their design is still based on heuristics or grid search methods that often result in overparametrized networks. This work proposes a method to analyze a trained…

Protein representation learning (PRL) is crucial for understanding structure-function relationships, yet current sequence- and graph-based methods fail to capture the hierarchical organization inherent in protein structures. We introduce…

Dimensionality reduction, cluster analysis, and sparse representation are basic components in machine learning. However, their relationships have not yet been fully investigated. In this paper, we find that the spectral graph theory…