Principal component analysis (PCA) is by far the most widespread tool for unsupervised learning with high-dimensional data sets. Its application is popularly studied for the purpose of exploratory data analysis and online process monitoring. Unfortunately, fine-tuning PCA models and particularly the number of components remains a challenging task. Today, this selection is often based on a combination of guiding principles, experience, and process understanding. Unlike the case of regression, where cross-validation of the prediction error is a widespread and trusted approach for model selection, there are no tools for PCA model selection which reach this level of acceptance. In this work, we address this challenge and evaluate the utility of the cross-validated ignorance score with both simulated and experimental data sets. Application of this method is based on the interpretation of PCA as a density model, as in probabilistic principal component analysis, and is shown to be a valuable tool to identify an optimal number of principal components.
The release of NBA player tracking data greatly enhances the granularity and dimensionality of basketball statistics used to evaluate and compare player performance. However, the high dimensionality of this new data source can be troublesome as it demands more computational resources and reduces the ability to easily interpret findings. Therefore, we must find a way to reduce the dimensionality of the data while retaining the ability to differentiate and compare player performance. In this paper, Principal Component Analysis (PCA) is used to identify four principal components that account for …
High dimensional data has introduced challenges that are difficult to address when attempting to implement classical approaches of statistical process control. This has made it a topic of interest for research due in recent years. However, in many cases, data sets have underlying structures, such as in advanced manufacturing systems. If extracted correctly, efficient methods for process control can be developed. This paper proposes a robust sparse dimensionality reduction approach for correlated high-dimensional process monitoring to address the aforementioned issues. The developed monitoring …
Principal Component Analysis (PCA) is a commonly used tool for dimension reduction in analyzing high dimensional data; Multilinear Principal Component Analysis (MPCA) has the potential to serve the similar function for analyzing tensor structure data. MPCA and other tensor decomposition methods have been proved effective to reduce the dimensions for both real data analyses and simulation studies (Ye, 2005; Lu, Plataniotis and Venetsanopoulos, 2008; Kolda and Bader, 2009; Li, Kim and Altman, 2010). In this paper, we investigate MPCA's statistical properties and provide explanations for its adva…
A particularly challenging context for dimensionality reduction is multivariate circular data, i.e., data supported on a torus. Such kind of data appears, e.g., in the analysis of various phenomena in ecology and astronomy, as well as in molecular structures. This paper introduces Scaled Torus Principal Component Analysis (ST-PCA), a novel approach to perform dimensionality reduction with toroidal data. ST-PCA finds a data-driven map from a torus to a sphere of the same dimension and a certain radius. The map is constructed with multidimensional scaling to minimize the discrepancy between pair…
George Washington University（乔治华盛顿大学）；National Institute of Standards and Technology（美国国家标准与技术研究院）
The movement of large quantities of data during the training of a Deep Neural Network presents immense challenges for machine learning workloads. To minimize this overhead, especially on the movement and calculation of gradient information, we introduce streaming batch principal component analysis as an update algorithm. Streaming batch principal component analysis uses stochastic power iterations to generate a stochastic k-rank approximation of the network gradient. We demonstrate that the low rank updates produced by streaming batch principal component analysis can effectively train convolut…
For very large datasets, random projections (RP) have become the tool of choice for dimensionality reduction. This is due to the computational complexity of principal component analysis. However, the recent development of randomized principal component analysis (RPCA) has opened up the possibility of obtaining approximate principal components on very large datasets. In this paper, we compare the performance of RPCA and RP in dimensionality reduction for supervised learning. In Experiment 1, study a malware classification task on a dataset with over 10 million samples, almost 100,000 features, …
Principal component analysis (PCA) for binary data, known as logistic PCA, has become a popular alternative to dimensionality reduction of binary data. It is motivated as an extension of ordinary PCA by means of a matrix factorization, akin to the singular value decomposition, that maximizes the Bernoulli log-likelihood. We propose a new formulation of logistic PCA which extends Pearson's formulation of a low dimensional data representation with minimum error to binary data. Our formulation does not require a matrix factorization, as previous methods do, but instead looks for projections of th…
This paper introduces a new unsupervised method for dimensionality reduction via regression (DRR). The algorithm belongs to the family of invertible transforms that generalize Principal Component Analysis (PCA) by using curvilinear instead of linear features. DRR identifies the nonlinear features through multivariate regression to ensure the reduction in redundancy between he PCA coefficients, the reduction of the variance of the scores, and the reduction in the reconstruction error. More importantly, unlike other nonlinear dimensionality reduction methods, the invertibility, volume-preservati…
Mitsubishi Electric Research Laboratories（三菱电机研究实验室）
We propose a new concept of rateless auto-encoders (RL-AEs) that enable a flexible latent dimensionality, which can be seamlessly adjusted for varying distortion and dimensionality requirements. In the proposed RL-AEs, instead of a deterministic bottleneck architecture, we use an over-complete representation that is stochastically regularized with weighted dropouts, in a manner analogous to sparse AE (SAE). Unlike SAEs, our RL-AEs employ monotonically increasing dropout rates across the latent representation nodes such that the latent variables become sorted by importance like in principal com…
Dimension reduction for high-dimensional compositional data plays an important role in many fields, where the principal component analysis of the basis covariance matrix is of scientific interest. In practice, however, the basis variables are latent and rarely observed, and standard techniques of principal component analysis are inadequate for compositional data because of the simplex constraint. To address the challenging problem, we relate the principal subspace of the centered log-ratio compositional covariance to that of the basis covariance, and prove that the latter is approximately iden…
Stochastic principal component analysis (SPCA) has become a popular dimensionality reduction strategy for large, high-dimensional datasets. We derive a simplified algorithm, called Lazy SPCA, which has reduced computational complexity and is better suited for large-scale distributed computation. We prove that SPCA and Lazy SPCA find the same approximations to the principal subspace, and that the pairwise distances between samples in the lower-dimensional space is invariant to whether SPCA is executed lazily or not. Empirical studies find downstream predictive performance to be identical for bo…
Though there is a growing body of literature on fairness for supervised learning, the problem of incorporating fairness into unsupervised learning has been less well-studied. This paper studies fairness in the context of principal component analysis (PCA). We first present a definition of fairness for dimensionality reduction, and our definition can be interpreted as saying that a reduction is fair if information about a protected class (e.g., race or gender) cannot be inferred from the dimensionality-reduced data points. Next, we develop convex optimization formulations that can improve the f…
Variable-Projection Sparse Functional Principal Component Analysis: Interpretable Functional Dimensionality Reduction with Applications to Raman Spectral Data
Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya（马来亚大学理学院数学科学研究所）；Centre of Research for Statistical Modelling and Methodology, Faculty of Science, Universiti Malaya（马来亚大学理学院统计建模与方法研究中心）；School of Mathematics and Statistics, University of Glasgow（格拉斯哥大学数学与统计学院）；Univ. Lille, CNRS, UMR 8524 – Laboratoire Paul Painlevé, Inria-Datavers（里尔大学、法国国家科学研究中心UMR 8524保罗·潘勒瓦实验室、Inria-Datavers）
In dimensionality reduction problems, the adopted technique may produce disparities between the representation errors of different groups. For instance, in the projected space, a specific class can be better represented in comparison with another one. In some situations, this unfair result may introduce ethical concerns. Aiming at overcoming this inconvenience, a fairness measure can be considered when performing dimensionality reduction through Principal Component Analysis. However, a solution that increases fairness tends to increase the overall re-construction error. In this context, this p…
Principal Component Analysis (PCA) is the workhorse tool for dimensionality reduction in this era of big data. While often overlooked, the purpose of PCA is not only to reduce data dimensionality, but also to yield features that are uncorrelated. Furthermore, the ever-increasing volume of data in the modern world often requires storage of data samples across multiple machines, which precludes the use of centralized PCA algorithms. This paper focuses on the dual objective of PCA, namely, dimensionality reduction and decorrelation of features, but in a distributed setting. This requires estimati…
University of Konstanz（康斯坦茨大学）；University of Stuttgart（斯图加特大学）
We present a technique to perform dimensionality reduction on data that is subject to uncertainty. Our method is a generalization of traditional principal component analysis (PCA) to multivariate probability distributions. In comparison to non-linear methods, linear dimensionality reduction techniques have the advantage that the characteristics of such probability distributions remain intact after projection. We derive a representation of the PCA sample covariance matrix that respects potential uncertainty in each of the inputs, building the mathematical foundation of our new method: uncertain…
Principal Component Analysis (PCA) is a very successful dimensionality reduction technique, widely used in predictive modeling. A key factor in its widespread use in this domain is the fact that the projection of a dataset onto its first $K$ principal components minimizes the sum of squared errors between the original data and the projected data over all possible rank $K$ projections. Thus, PCA provides optimal low-rank representations of data for least-squares linear regression under standard modeling assumptions. On the other hand, when the loss function for a prediction problem is not the l…
Principal Component Analysis for Multivariate Extremes
Colorado State University（科罗拉多州立大学）；Université Paris Cité, Université Paris Saclay, ENS Paris Saclay, CNRS, SSA, INSERM, Centre Borelli（巴黎西岱大学、巴黎萨克雷大学、巴黎高等师范学校、法国国家科学研究中心、社会卫生与老龄化研究所、博尔内利中心）
The University of Tokyo（东京大学）；Technical University of Darmstadt（达姆施塔特工业大学）；RIKEN（日本理化学研究所）
Direct contextual policy search methods learn to improve policy parameters and simultaneously generalize these parameters to different context or task variables. However, learning from high-dimensional context variables, such as camera images, is still a prominent problem in many real-world tasks. A naive application of unsupervised dimensionality reduction methods to the context variables, such as principal component analysis, is insufficient as task-relevant input may be ignored. In this paper, we propose a contextual policy search method in the model-based relative entropy stochastic search…
We derive the asymptotic distributions of the spiked eigenvalues and eigenvectors under a generalized and unified asymptotic regime, which takes into account the spike magnitude of leading eigenvalues, sample size, and dimensionality. This new regime allows high dimensionality and diverging eigenvalue spikes and provides new insights into the roles the leading eigenvalues, sample size, and dimensionality play in principal component analysis. The results are proven by a technical device, which swaps the role of rows and columns and converts the high-dimensional problems into low-dimensional one…
Motivation: Although principal component analysis is frequently applied to reduce the dimensionality of matrix data, the method is sensitive to noise and bias and has difficulty with comparability and interpretation. These issues are addressed by improving the fidelity to the study design. Principal axes and the components for variables are found through the arrangement of the training data set, and the centers of data are found according to the design. By using both the axes and the center, components for an observation that belong to various studies can be separately estimated. Both of the c…
A principal component analysis based on the generalized Gini correlation index is proposed (Gini PCA). The Gini PCA generalizes the standard PCA based on the variance. It is shown, in the Gaussian case, that the standard PCA is equivalent to the Gini PCA. It is also proven that the dimensionality reduction based on the generalized Gini correlation matrix, that relies on city-block distances, is robust to outliers. Monte Carlo simulations and an application on cars data (with outliers) show the robustness of the Gini PCA and provide different interpretations of the results compared with the var…
College of Information and Communications University of South Carolina（南卡罗来纳大学信息与通信学院）；Arnold School of Public Health at the University of South Carolina（南卡罗来纳大学阿诺德公共卫生学院）
Large textual corpora are often represented by the document-term frequency matrix whose elements are the frequency of terms; however, this matrix has two problems: sparsity and high dimensionality. Four dimension reduction strategies are used to address these problems. Of the four strategies, unsupervised feature transformation (UFT) is a popular and efficient strategy to map the terms to a new basis in the document-term frequency matrix. Although several UFT-based methods have been developed, fuzzy clustering has not been considered for dimensionality reduction. This research explores fuzzy c…
Principal component analysis (PCA) is a standard dimensionality reduction technique used in various research and applied fields. From an algorithmic point of view, classical PCA can be formulated in terms of operations on a multivariate Gaussian likelihood. As a consequence of the implied Gaussian formulation, the principal components are not robust to outliers. In this paper, we propose a modified formulation, based on the use of a multivariate Cauchy likelihood instead of the Gaussian likelihood, which has the effect of robustifying the principal components. We present an algorithm to comput…
Graph Signal Processing (GSP) is a promising framework to analyze multi-dimensional neuroimaging datasets, while taking into account both the spatial and functional dependencies between brain signals. In the present work, we apply dimensionality reduction techniques based on graph representations of the brain to decode brain activity from real and simulated fMRI datasets. We introduce seven graphs obtained from a) geometric structure and/or b) functional connectivity between brain areas at rest, and compare them when performing dimension reduction for classification. We show that mixed graphs …
Principal component analysis is an important pattern recognition and dimensionality reduction tool in many applications. Principal components are computed as eigenvectors of a maximum likelihood covariance $\widehatΣ$ that approximates a population covariance $Σ$, and these eigenvectors are often used to extract structural information about the variables (or attributes) of the studied population. Since PCA is based on the eigendecomposition of the proxy covariance $\widehatΣ$ rather than the ground-truth $Σ$, it is important to understand the approximation error in each individual eigenvector …
This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the exi…
High-resolution spectroscopic surveys of the Milky Way have entered the Big Data regime and have opened avenues for solving outstanding questions in Galactic archaeology. However, exploiting their full potential is limited by complex systematics, whose characterization has not received much attention in modern spectroscopic analyses. In this work, we present a novel method to disentangle the component of spectral data space intrinsic to the stars from that due to systematics. Using functional principal component analysis on a sample of $18,933$ giant spectra from APOGEE, we find that the intri…
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 of the data. However, PCA has two major weaknesses. Firstly, it only considers linear correlations among variables (features), and secondly it is not suitable for categorical data. We resolve these issues by proposing Maximally Correlated Principal Component Analysis (MCPCA). MCPCA computes transformations of variables whose covariance matrix has the largest …
Principal component analysis (PCA) is widely used for feature extraction and dimensionality reduction, with documented merits in diverse tasks involving high-dimensional data. Standard PCA copes with one dataset at a time, but it is challenged when it comes to analyzing multiple datasets jointly. In certain data science settings however, one is often interested in extracting the most discriminative information from one dataset of particular interest (a.k.a. target data) relative to the other(s) (a.k.a. background data). To this end, this paper puts forth a novel approach, termed discriminative…
Principal Component Analysis (PCA) is a popular tool for dimensionality reduction and feature extraction in data analysis. There is a probabilistic version of PCA, known as Probabilistic PCA (PPCA). However, standard PCA and PPCA are not robust, as they are sensitive to outliers. To alleviate this problem, this paper introduces the Self-Paced Learning mechanism into PPCA, and proposes a novel method called Self-Paced Probabilistic Principal Component Analysis (SP-PPCA). Furthermore, we design the corresponding optimization algorithm based on the alternative search strategy and the expectation-…
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Technion（以色列理工学院）
We consider the dimensionality-reduction problem (finding a subspace approximation of observed data) for contaminated data in the high dimensional regime, where the number of observations is of the same magnitude as the number of variables of each observation, and the data set contains some (arbitrarily) corrupted observations. We propose a High-dimensional Robust Principal Component Analysis (HR-PCA) algorithm that is tractable, robust to contaminated points, and easily kernelizable. The resulting subspace has a bounded deviation from the desired one, achieves maximal robustness -- a breakdow…
Thanks to their versatility, ease of deployment and high-performance, surrogate models have become staple tools in the arsenal of uncertainty quantification (UQ). From local interpolants to global spectral decompositions, surrogates are characterised by their ability to efficiently emulate complex computational models based on a small set of model runs used for training. An inherent limitation of many surrogate models is their susceptibility to the curse of dimensionality, which traditionally limits their applicability to a maximum of $\mathcal{O}(10^2)$ input dimensions. We present a novel ap…
The merit of projecting data onto linear subspaces is well known from, e.g., dimension reduction. One key aspect of subspace projections, the maximum preservation of variance (principal component analysis), has been thoroughly researched and the effect of random linear projections on measures such as intrinsic dimensionality still is an ongoing effort. In this paper, we investigate the less explored depths of linear projections onto explicit subspaces of varying dimensionality and the expectations of variance that ensue. The result is a new family of bounds for Euclidean distances and inner pr…
Graph Neural Networks (GNNs) have become a popular approach for various applications, ranging from social network analysis to modeling chemical properties of molecules. While GNNs often show remarkable performance on public datasets, they can struggle to learn long-range dependencies in the data due to over-smoothing and over-squashing tendencies. To alleviate this challenge, we propose PCAPass, a method which combines Principal Component Analysis (PCA) and message passing for generating node embeddings in an unsupervised manner and leverages gradient boosted decision trees for classification …
Institute of Medical Genetics, University Medical Centre Ljubljana（卢布尔雅那大学医学中心医学遗传学研究所）
Class prediction is an important application of microarray gene expression data analysis. The high-dimensionality of microarray data, where number of genes (variables) is very large compared to the number of samples (obser- vations), makes the application of many prediction techniques (e.g., logistic regression, discriminant analysis) difficult. An efficient way to solve this prob- lem is by using dimension reduction statistical techniques. Increasingly used in psychology-related applications, Rasch model (RM) provides an appealing framework for handling high-dimensional microarray data. In th…
As a widely used method in machine learning, principal component analysis (PCA) shows excellent properties for dimensionality reduction. It is a serious problem that PCA is sensitive to outliers, which has been improved by numerous Robust PCA (RPCA) versions. However, the existing state-of-the-art RPCA approaches cannot easily remove or tolerate outliers by a non-iterative manner. To tackle this issue, this paper proposes Angular Embedding (AE) to formulate a straightforward RPCA approach based on angular density, which is improved for large scale or high-dimensional data. Furthermore, a trimm…
Molecular simulations produce very high-dimensional data-sets with millions of data points. As analysis methods are often unable to cope with so many dimensions, it is common to use dimensionality reduction and clustering methods to reach a reduced representation of the data. Yet these methods often fail to capture the most important features necessary for the construction of a Markov model. Here we demonstrate the results of various dimensionality reduction methods on two simulation data-sets, one of protein folding and another of protein-ligand binding. The methods tested include a k-means c…
Huazhong University of Science and Technology（华中科技大学）；University of Technology Sydney（悉尼科技大学）
Dimensionality reduction is an important operation in information visualization, feature extraction, clustering, regression, and classification, especially for processing noisy high dimensional data. However, most existing approaches preserve either the global or the local structure of the data, but not both. Approaches that preserve only the global data structure, such as principal component analysis (PCA), are usually sensitive to outliers. Approaches that preserve only the local data structure, such as locality preserving projections, are usually unsupervised (and hence cannot use label inf…
Principal Component Analysis is a novel way of of dimensionality reduction. This problem essentially boils down to finding the top k eigen vectors of the data covariance matrix. A considerable amount of literature is found on algorithms meant to do so such as an online method be Warmuth and Kuzmin, Matrix Stochastic Gradient by Arora, Oja's method and many others. In this paper we see some of these stochastic approaches to the PCA optimization problem and comment on their convergence and runtime to obtain an epsilon sub-optimal solution. We revisit convex relaxation based methods for stochasti…
Johns Hopkins University（约翰斯·霍普金斯大学）；Flatiron Institute（熨斗研究所）；New York University（纽约大学）
Overparameterization in deep learning is powerful: Very large models fit the training data perfectly and yet often generalize well. This realization brought back the study of linear models for regression, including ordinary least squares (OLS), which, like deep learning, shows a "double-descent" behavior: (1) The risk (expected out-of-sample prediction error) can grow arbitrarily when the number of parameters $p$ approaches the number of samples $n$, and (2) the risk decreases with $p$ for $p>n$, sometimes achieving a lower value than the lowest risk for $p<n$. The divergence of the risk for O…
Highly robust factored principal component analysis for matrix-valued outlier accommodation and explainable detection via matrix minimum covariance determinant
Manifold learning is used for dimensionality reduction, with the goal of finding a projection subspace to increase and decrease the inter- and intraclass variances, respectively. However, a bottleneck for subspace learning methods often arises from the high dimensionality of datasets. In this paper, a hierarchical approach is proposed to scale subspace learning methods, with the goal of improving classification in large datasets by a range of 3% to 10%. Different combinations of methods are studied. We assess the proposed method on five publicly available large datasets, for different eigen-va…
VinAI Research（VinAI 研究院）；Hong Kong Polytechnic University（香港理工大学）
Principal component analysis is a simple yet useful dimensionality reduction technique in modern machine learning pipelines. In consequential domains such as college admission, healthcare and credit approval, it is imperative to take into account emerging criteria such as the fairness and the robustness of the learned projection. In this paper, we propose a distributionally robust optimization problem for principal component analysis which internalizes a fairness criterion in the objective function. The learned projection thus balances the trade-off between the total reconstruction error and t…
Sparse principal component analysis (PCA) is an important technique for dimensionality reduction of high-dimensional data. However, most existing sparse PCA algorithms are based on non-convex optimization, which provide little guarantee on the global convergence. Sparse PCA algorithms based on a convex formulation, for example the Fantope projection and selection (FPS), overcome this difficulty, but are computationally expensive. In this work we study sparse PCA based on the convex FPS formulation, and propose a new algorithm that is computationally efficient and applicable to large and high-d…
Robust principal component analysis (RPCA) can recover low-rank matrices when they are corrupted by sparse noises. In practice, many matrices are, however, of high-rank and hence cannot be recovered by RPCA. We propose a novel method called robust kernel principal component analysis (RKPCA) to decompose a partially corrupted matrix as a sparse matrix plus a high or full-rank matrix with low latent dimensionality. RKPCA can be applied to many problems such as noise removal and subspace clustering and is still the only unsupervised nonlinear method robust to sparse noises. Our theoretical analys…
School of Computer Science and Engineering, The Hebrew University of Jerusalem（耶路撒冷希伯来大学计算机科学与工程学院）；Google Research（谷歌研究院）
We consider Fair Principal Component Analysis (FPCA) and search for a low dimensional subspace that spans multiple target vectors in a fair manner. FPCA is defined as a non-concave maximization of the worst projected target norm within a given set. The problem arises in filter design in signal processing, and when incorporating fairness into dimensionality reduction schemes. The state of the art approach to FPCA is via semidefinite relaxation and involves a polynomial yet computationally expensive optimization. To allow scalability, we propose to address FPCA using naive sub-gradient descent. …
ECE Dept. and Digital Tech. Center, U. of Minnesota（明尼苏达大学电子与计算机工程系及数字技术中心）
Principal component analysis (PCA) has well-documented merits for data extraction and dimensionality reduction. PCA deals with a single dataset at a time, and it is challenged when it comes to analyzing multiple datasets. Yet in certain setups, one wishes to extract the most significant information of one dataset relative to other datasets. Specifically, the interest may be on identifying, namely extracting features that are specific to a single target dataset but not the others. This paper develops a novel approach for such so-termed discriminative data analysis, and establishes its optimalit…
