Robust Principal Component Analysis (RPCA) is a widely used method for recovering low-rank structure from data matrices corrupted by significant and sparse outliers. These corruptions may arise from occlusions, malicious tampering, or other causes for anomalies, and the joint identification of such corruptions with low-rank background is critical for process monitoring and diagnosis. However, existing RPCA methods and their extensions largely do not account for the underlying probabilistic distribution for the data matrices, which in many applications are known and can be highly non-Gaussian. We thus propose a new method called Robust Principal Component Analysis for Exponential Family distributions ($e^{\text{RPCA}}$), which can perform the desired decomposition into low-rank and sparse matrices when such a distribution falls within the exponential family. We present a novel alternating direction method of multiplier optimization algorithm for efficient $e^{\text{RPCA}}$ decomposition. The effectiveness of $e^{\text{RPCA}}$ is then demonstrated in two applications: the first for steel sheet defect detection, and the second for crime activity monitoring in the Atlanta metropolitan area.

Fair Principal Component Analysis (PCA) is a problem setting where we aim to perform PCA while making the resulting representation fair in that the projected distributions, conditional on the sensitive attributes, match one another. However, existing approaches to fair PCA have two main problems: theoretically, there has been no statistical foundation of fair PCA in terms of learnability; practically, limited memory prevents us from using existing approaches, as they explicitly rely on full access to the entire data. On the theoretical side, we rigorously formulate fair PCA using a new notion called \emph{probably approximately fair and optimal} (PAFO) learnability. On the practical side, motivated by recent advances in streaming algorithms for addressing memory limitation, we propose a new setting called \emph{fair streaming PCA} along with a memory-efficient algorithm, fair noisy power method (FNPM). We then provide its {\it statistical} guarantee in terms of PAFO-learnability, which is the first of its kind in fair PCA literature. Lastly, we verify the efficacy and memory efficiency of our algorithm on real-world datasets.

We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis (RPCA), that aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called Robust Tensor CUR Decompositions (RTCUR), for large-scale non-convex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.

Principal Component Analysis (PCA) is a popular tool in data analysis, especially when the data is high-dimensional. PCA aims to find subspaces, spanned by the so-called \textit{principal components}, that best explain the variance in the dataset. The deflation method is a popular meta-algorithm -- used to discover such subspaces -- that sequentially finds individual principal components, starting from the most important one and working its way towards the less important ones. However, due to its sequential nature, the numerical error introduced by not estimating principal components exactly -- e.g., due to numerical approximations through this process -- propagates, as deflation proceeds. To the best of our knowledge, this is the first work that mathematically characterizes the error propagation of the inexact deflation method, and this is the key contribution of this paper. We provide two main results: $i)$ when the sub-routine for finding the leading eigenvector is generic, and $ii)$ when power iteration is used as the sub-routine. In the latter case, the additional directional information from power iteration allows us to obtain a tighter error bound than the analysis of the sub-routine agnostic case. As an outcome, we provide explicit characterization on how the error progresses and affects subsequent principal component estimations for this fundamental problem.

Understanding feature representation for deep neural networks (DNNs) remains an open question within the general field of explainable AI. We use principal component analysis (PCA) to study the performance of a k-nearest neighbors classifier (k-NN), nearest class-centers classifier (NCC), and support vector machines on the learned layer-wise representations of a ResNet-18 trained on CIFAR-10. We show that in certain layers, as little as 20% of the intermediate feature-space variance is necessary for high-accuracy classification and that across all layers, the first ~100 PCs completely determine the performance of the k-NN and NCC classifiers. We relate our findings to neural collapse and provide partial evidence for the related phenomenon of intermediate neural collapse. Our preliminary work provides three distinct yet interpretable surrogate models for feature representation with an affine linear model the best performing. We also show that leveraging several surrogate models affords us a clever method to estimate where neural collapse may initially occur within the DNN.

Principal components computed via PCA (principal component analysis) are traditionally used to reduce dimensionality in genomic data or to correct for population stratification. In this paper, we explore the penalized eigenvalue problem (PEP) which reformulates the computation of the first eigenvector as an optimization problem and adds an L1 penalty constraint. The contribution of our article is threefold. First, we extend PEP by applying Nesterov smoothing to the original LASSO-type L1 penalty. This allows one to compute analytical gradients which enable faster and more efficient minimization of the objective function associated with the optimization problem. Second, we demonstrate how higher order eigenvectors can be calculated with PEP using established results from singular value decomposition (SVD). Third, using data from the 1000 Genome Project dataset, we empirically demonstrate that our proposed smoothed PEP allows one to increase numerical stability and obtain meaningful eigenvectors. We further investigate the utility of the penalized eigenvector approach over traditional PCA.

Principal component analysis (PCA) is a widely employed statistical tool used primarily for dimensionality reduction. However, it is known to be adversely affected by the presence of outlying observations in the sample, which is quite common. Robust PCA methods using M-estimators have theoretical benefits, but their robustness drop substantially for high dimensional data. On the other end of the spectrum, robust PCA algorithms solving principal component pursuit or similar optimization problems have high breakdown, but lack theoretical richness and demand high computational power compared to the M-estimators. We introduce a novel robust PCA estimator based on the minimum density power divergence estimator. This combines the theoretical strength of the M-estimators and the minimum divergence estimators with a high breakdown guarantee regardless of data dimension. We present a computationally efficient algorithm for this estimate. Our theoretical findings are supported by extensive simulations and comparisons with existing robust PCA methods. We also showcase the proposed algorithm's applicability on two benchmark datasets and a credit card transactions dataset for fraud detection.

In this paper, we develop new methods for analyzing high-dimensional tensor datasets. A tensor factor model describes a high-dimensional dataset as a sum of a low-rank component and an idiosyncratic noise, generalizing traditional factor models for panel data. We propose an estimation algorithm, called tensor principal component analysis (TPCA), which generalizes the traditional PCA applicable to panel data. The algorithm involves unfolding the tensor into a sequence of matrices along different dimensions and applying PCA to the unfolded matrices. We provide theoretical results on the consistency and asymptotic distribution for the TPCA estimator of loadings and factors. We also introduce a novel test for the number of factors in a tensor factor model. The TPCA and the test feature good performance in Monte Carlo experiments and are applied to sorted portfolios.

Anomaly detection (AD) in images, identifying significant deviations from normality, is a critical issue in computer vision. This paper introduces a novel approach to dimensionality reduction for AD using pre-trained convolutional neural network (CNN) that incorporate EfficientNet models. We investigate the importance of component selection and propose two types of tree search approaches, both employing a greedy strategy, for optimal eigencomponent selection. Our study conducts three main experiments to evaluate the effectiveness of our approach. The first experiment explores the influence of test set performance on component choice, the second experiment examines the performance when we train on one anomaly type and evaluate on all other types, and the third experiment investigates the impact of using a minimum number of images for training and selecting them based on anomaly types. Our approach aims to find the optimal subset of components that deliver the highest performance score, instead of focusing solely on the proportion of variance explained by each component and also understand the components behaviour in different settings. Our results indicate that the proposed method surpasses both Principal Component Analysis (PCA) and Negated Principal Component Analysis (NPCA) in terms of detection accuracy, even when using fewer components. Thus, our approach provides a promising alternative to conventional dimensionality reduction techniques in AD, and holds potential to enhance the efficiency and effectiveness of AD systems.

Motivated by the increasing availability of data of functional nature, we develop a general probabilistic and statistical framework for extremes of regularly varying random elements $X$ in $L^2[0,1]$. We place ourselves in a Peaks-Over-Threshold framework where a functional extreme is defined as an observation $X$ whose $L^2$-norm $\|X\|$ is comparatively large. Our goal is to propose a dimension reduction framework resulting into finite dimensional projections for such extreme observations. Our contribution is double. First, we investigate the notion of Regular Variation for random quantities valued in a general separable Hilbert space, for which we propose a novel concrete characterization involving solely stochastic convergence of real-valued random variables. Second, we propose a notion of functional Principal Component Analysis (PCA) accounting for the principal `directions' of functional extremes. We investigate the statistical properties of the empirical covariance operator of the angular component of extreme functions, by upper-bounding the Hilbert-Schmidt norm of the estimation error for finite sample sizes. Numerical experiments with simulated and real data illustrate this work.