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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.

Randomized numerical linear algebra - RandNLA, for short - concerns the use of randomization as a resource to develop improved algorithms for large-scale linear algebra computations. The origins of contemporary RandNLA lay in theoretical computer science, where it blossomed from a simple idea: randomization provides an avenue for computing approximate solutions to linear algebra problems more efficiently than deterministic algorithms. This idea proved fruitful in the development of scalable algorithms for machine learning and statistical data analysis applications. However, RandNLA's true potential only came into focus upon integration with the fields of numerical analysis and "classical" numerical linear algebra. Through the efforts of many individuals, randomized algorithms have been developed that provide full control over the accuracy of their solutions and that can be every bit as reliable as algorithms that might be found in libraries such as LAPACK. Recent years have even seen the incorporation of certain RandNLA methods into MATLAB, the NAG Library, NVIDIA's cuSOLVER, and SciKit-Learn. For all its success, we believe that RandNLA has yet to realize its full potential. In particular, we believe the scientific community stands to benefit significantly from suitably defined "RandBLAS" and "RandLAPACK" libraries, to serve as standards conceptually analogous to BLAS and LAPACK. This 200-page monograph represents a step toward defining such standards. In it, we cover topics spanning basic sketching, least squares and optimization, low-rank approximation, full matrix decompositions, leverage score sampling, and sketching data with tensor product structures (among others). Much of the provided pseudo-code has been tested via publicly available MATLAB and Python implementations., Comment: v1: this is the first arXiv release of LAPACK Working Note 299. v2: complete rewrite of the subsection on trace estimation, among other changes. See frontmatter page ii (pdf page 5) for revision history

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

The statistical methods used to analyze medical data are becoming increasingly complex. Novel statistical methods increasingly rely on simulation studies to assess their validity. Such assessments typically appear in statistical or computational journals, and the methodology is later introduced to the medical community through tutorials. This can be problematic if applied researchers use the methodologies in settings that have not been evaluated. In this paper, we explore a case study of one such method that has become popular in the analysis of coronavirus disease 2019 (COVID-19) data. The integrated nested Laplace approximations (INLA), as implemented in the R-INLA package, approximates the marginal posterior distributions of target parameters that would have been obtained from a fully Bayesian analysis. We seek to answer an important question: Does existing research on the accuracy of INLA's approximations support how researchers are currently using it to analyze COVID-19 data? We identify three limitations to work assessing INLA's accuracy: 1) inconsistent definitions of accuracy, 2) a lack of studies validating how researchers are actually using INLA, and 3) a lack of research into the reproducibility of INLA's output. We explore the practical impact of each limitation with simulation studies based on models and data used in COVID-19 research. Our results suggest existing methods of assessing the accuracy of the INLA technique may not support how COVID-19 researchers are using it. Guided in part by our results, we offer a proposed set of minimum guidelines for researchers using statistical methodologies primarily validated through simulation studies.

Topological data analysis (TDA) allows us to explore the topological features of a dataset. Among topological features, lower dimensional ones have recently drawn the attention of practitioners in mathematics and statistics due to their potential to aid the discovery of low dimensional structure in a data set. However, lower dimensional features are usually challenging to detect from a probabilistic perspective. In this paper, lower dimensional topological features occurring as zero-density regions of density functions are introduced and thoroughly investigated. Specifically, we consider sequences of coverings for the support of a density function in which the coverings are comprised of balls with shrinking radii. We show that, when these coverings satisfy certain sufficient conditions as the sample size goes to infinity, we can detect lower dimensional, zero-density regions with increasingly higher probability while guarding against false detection. We supplement the theoretical developments with the discussion of simulated experiments that elucidate the behavior of the methodology for different choices of the tuning parameters that govern the construction of the covering sequences and characterize the asymptotic results., Comment: 28 pages