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We consider the problem of learning a linear subspace from data corrupted by outliers. Classical approaches are typically designed for the case in which the subspace dimension is small relative to the ambient dimension. Our approach works…
High dimensional data and systems with many degrees of freedom are often characterized by covariance matrices. In this paper, we consider the problem of simultaneously estimating the dimension of the principal (dominant) subspace of these…
The two primary approaches for high-dimensional regression problems are sparse methods (e.g., best subset selection, which uses the L0-norm in the penalty) and ensemble methods (e.g., random forests). Although sparse methods typically yield…
The coresets approach, also called subsampling or subset selection, aims to select a subsample as a surrogate for the observed sample and has found extensive applications in large-scale data analysis. Existing coresets methods construct the…
Principal Component Analysis (PCA) is a fundamental data preprocessing tool in the world of machine learning. While PCA is often thought of as a dimensionality reduction method, the purpose of PCA is actually two-fold: dimension reduction…
We study the problem of choosing the best subset of p features in linear regression given n observations. This problem naturally contains two objective functions including minimizing the amount of bias and minimizing the number of…
A significant hurdle for analyzing large sample data is the lack of effective statistical computing and inference methods. An emerging powerful approach for analyzing large sample data is subsampling, by which one takes a random subsample…
A method is introduced to perform simultaneous sparse dimension reduction on two blocks of variables. Beyond dimension reduction, it also yields an estimator for multivariate regression with the capability to intrinsically deselect…
In current research, machine and deep learning solutions for the classification of temporal data are shifting from single-channel datasets (univariate) to problems with multiple channels of information (multivariate). The majority of these…
The statistical analysis of tree structured data is a new topic in statistics with wide application areas. Some Principal Component Analysis (PCA) ideas were previously developed for binary tree spaces. In this study, we extend these ideas…
An analysis of high-dimensional data can offer a detailed description of a system but is often challenged by the curse of dimensionality. General dimensionality reduction techniques can alleviate such difficulty by extracting a few…
Sufficient dimension reduction [J. Amer. Statist. Assoc. 86 (1991) 316-342] has long been a prominent issue in multivariate nonparametric regression analysis. To uncover the central dimension reduction space, we propose in this paper an…
Linear-parametric optimization, where multiple objectives are combined into a single objective using linear combinations with parameters as coefficients, has numerous links to other fields in optimization and a wide range of application…
Principal Component Analysis (PCA) is a powerful and popular dimensionality reduction technique. However, due to its linear nature, it often fails to capture the complex underlying structure of real-world data. While Kernel PCA (kPCA)…
This paper is about how to partition decision variables while decomposing a large-scale optimization problem for the best performance of distributed solution methods. Solving a large-scale optimization problem sequen- tially can be…
This work proposes a method for solving linear stochastic optimal control (SOC) problems using sum of squares and semidefinite programming. Previous work had used polynomial optimization to approximate the value function, requiring a high…
In this paper, we study the estimation of partially linear models for spatial data distributed over complex domains. We use bivariate splines over triangulations to represent the nonparametric component on an irregular two-dimensional…
We propose a new sparse regression method called the component lasso, based on a simple idea. The method uses the connected-components structure of the sample covariance matrix to split the problem into smaller ones. It then solves the…
Dimensionality reduction is a first step of many machine learning pipelines. Two popular approaches are principal component analysis, which projects onto a small number of well chosen but non-interpretable directions, and feature selection,…
This article presents two novel adaptive-sparse polynomial dimensional decomposition (PDD) methods for solving high-dimensional uncertainty quantification problems in computational science and engineering. The methods entail global…