How does missing data affect our ability to learn signal structures? It has been shown that learning signal structure in terms of principal components is dependent on the ratio of sample size and dimensionality and that a critical number of observations is needed before learning starts (Biehl and Mietzner, 1993). Here we generalize this analysis to include missing data. Probabilistic principal component analysis is regularly used for estimating signal structures in datasets with missing data. Our analytic result suggests that the effect of missing data is to effectively reduce signal-to-noise ratio rather than - as generally believed - to reduce sample size. The theory predicts a phase transition in the learning curves and this is indeed found both in simulation data and in real datasets.
Hong Kong University of Science and Technology（香港科技大学）；Zhejiang University（浙江大学）
The real-time crash likelihood prediction has been an important research topic. Various classifiers, such as support vector machine (SVM) and tree-based boosting algorithms, have been proposed in traffic safety studies. However, few research focuses on the missing data imputation in real-time crash likelihood prediction, although missing values are commonly observed due to breakdown of sensors or external interference. Besides, classifying imbalanced data is also a difficult problem in real-time crash likelihood prediction, since it is hard to distinguish crash-prone cases from non-crash cases…
SimulaMet（西穆拉大都会研究所）；Warsaw University of Technology（华沙理工大学）
Missing data is a commonly occurring problem in practice. Many imputation methods have been developed to fill in the missing entries. However, not all of them can scale to high-dimensional data, especially the multiple imputation techniques. Meanwhile, the data nowadays tends toward high-dimensional. Therefore, in this work, we propose Principal Component Analysis Imputation (PCAI), a simple but versatile framework based on Principal Component Analysis (PCA) to speed up the imputation process and alleviate memory issues of many available imputation techniques, without sacrificing the imputatio…
We analyze a practical algorithm for sparse PCA on incomplete and noisy data under a general non-random sampling scheme. The algorithm is based on a semidefinite relaxation of the $\ell_1$-regularized PCA problem. We provide theoretical justification that under certain conditions, we can recover the support of the sparse leading eigenvector with high probability by obtaining a unique solution. The conditions involve the spectral gap between the largest and second-largest eigenvalues of the true data matrix, the magnitude of the noise, and the structural properties of the observed entries. The …
University of Michigan（密歇根大学）；Carnegie Mellon University（卡内基梅隆大学）；Harvard University（哈佛大学）
For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many infer…
Lawrence Livermore National Laboratory（劳伦斯利弗莫尔国家实验室）；Sandia National Laboratories（桑迪亚国家实验室）
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 probabilistic interpretation of low-rank PCA, the data has a normal multivariate distribution and, therefore, normal marginal distributions for each column. If some marginals are continuous but not normal, the semiparametric copula-based principal component analysis (COCA) method is an alternative to PCA that combines a Gaussian copula with nonparametr…
Edmund Mach Foundation – Research and Innovation Center（埃德蒙·马赫基金会——研究与创新中心）
Linear principal component analysis (PCA) can be extended to a nonlinear PCA by using artificial neural networks. But the benefit of curved components requires a careful control of the model complexity. Moreover, standard techniques for model selection, including cross-validation and more generally the use of an independent test set, fail when applied to nonlinear PCA because of its inherent unsupervised characteristics. This paper presents a new approach for validating the complexity of nonlinear PCA models by using the error in missing data estimation as a criterion for model selection. It i…
Inference for Heteroskedastic PCA with Missing Data
Institute for Data, Systems, and Society, MIT（麻省理工学院数据、系统与社会研究所）；Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院）；Department of Operations Research and Financial Engineering, Princeton University（普林斯顿大学运筹学与金融工程系）
This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-à-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of co…
We study a practical algorithm for sparse principal component analysis (PCA) of incomplete and noisy data. Our algorithm is based on the semidefinite program (SDP) relaxation of the non-convex $l_1$-regularized PCA problem. We provide theoretical and experimental evidence that SDP enables us to exactly recover the true support of the sparse leading eigenvector of the unknown true matrix, despite only observing an incomplete (missing uniformly at random) and noisy version of it. We derive sufficient conditions for exact recovery, which involve matrix incoherence, the spectral gap between the la…
Multivariate data are typically represented by a rectangular matrix (table) in which the rows are the objects (cases) and the columns are the variables (measurements). When there are many variables one often reduces the dimension by principal component analysis (PCA), which in its basic form is not robust to outliers. Much research has focused on handling rowwise outliers, i.e. rows that deviate from the majority of the rows in the data (for instance, they might belong to a different population). In recent years also cellwise outliers are receiving attention. These are suspicious cells (entrie…
Daugavpils University（达加夫皮尔斯大学）；Data Science Institute（数据科学研究所）；Hasselt University（哈瑟尔特大学）；Amsterdam UMC（阿姆斯特丹大学医学中心）；Institute for Computing Applications（计算应用研究所）
Principal Component Analysis (PCA) is a dimension reduction technique. It produces inconsistent estimators when the dimensionality is moderate to high, which is often the problem in modern large-scale applications where algorithm scalability and model interpretability are difficult to achieve, not to mention the prevalence of missing values. While existing sparse PCA methods alleviate inconsistency, they are constrained to the Gaussian assumption of classical PCA and fail to address algorithm scalability issues. We generalize sparse PCA to the broad exponential family distributions under high-…
Inria（法国国家信息与自动化研究所）；University College London（伦敦大学学院）
Principal Component Analysis (PCA) is a popular method for dimension reduction and has attracted an unfailing interest for decades. More recently, kernel PCA (KPCA) has emerged as an extension of PCA but, despite its use in practice, a sound theoretical understanding of KPCA is missing. We contribute several lower and upper bounds on the efficiency of KPCA, involving the empirical eigenvalues of the kernel Gram matrix and new quantities involving a notion of variance. These bounds show how much information is captured by KPCA on average and contribute a better theoretical understanding of its …
TU Berlin（柏林工业大学）；Univ. Bordeaux（波尔多大学）；Bordeaux INP（波尔多国立理工学院）；CNRS（法国国家科学研究中心）
Despite the rapid development of computational hardware, the treatment of large and high dimensional data sets is still a challenging problem. This paper provides a twofold contribution to the topic. First, we propose a Gaussian Mixture Model in conjunction with a reduction of the dimensionality of the data in each component of the model by principal component analysis, called PCA-GMM. To learn the (low dimensional) parameters of the mixture model we propose an EM algorithm whose M-step requires the solution of constrained optimization problems. Fortunately, these constrained problems do not d…
PCA (Principal Component Analysis) and its variants areubiquitous techniques for matrix dimension reduction and reduced-dimensionlatent-factor extraction. One significant challenge in using PCA, is thechoice of the number of principal components. The information-theoreticMDL (Minimum Description Length) principle gives objective compression-based criteria for model selection, but it is difficult to analytically applyits modern definition - NML (Normalized Maximum Likelihood) - to theproblem of PCA. This work shows a general reduction of NML prob-lems to lower-dimension problems. Applying this …
Accurate predictions of pollutant concentrations at new locations are often of interest in air pollution studies on fine particulate matters (PM$_{2.5}$), in which data is usually not measured at all study locations. PM$_{2.5}$ is also a mixture of many different chemical components. Principal component analysis (PCA) can be incorporated to obtain lower-dimensional representative scores of such multi-pollutant data. Spatial prediction can then be used to estimate these scores at new locations. Recently developed predictive PCA modifies the traditional PCA algorithm to obtain scores with spatia…
This paper introduces a novel paradigm to impute missing data that combines a decision tree with an auto-associative neural network (AANN) based model and a principal component analysis-neural network (PCA-NN) based model. For each model, the decision tree is used to predict search bounds for a genetic algorithm that minimize an error function derived from the respective model. The models' ability to impute missing data is tested and compared using HIV sero-prevalance data. Results indicate an average increase in accuracy of 13% with the AANN based model's average accuracy increasing from 75.8…
We consider the problem of learning a mixture of Random Utility Models (RUMs). Despite the success of RUMs in various domains and the versatility of mixture RUMs to capture the heterogeneity in preferences, there has been only limited progress in learning a mixture of RUMs from partial data such as pairwise comparisons. In contrast, there have been significant advances in terms of learning a single component RUM using pairwise comparisons. In this paper, we aim to bridge this gap between mixture learning and single component learning of RUM by developing a `reduction' procedure. We propose to …
This work obtains novel finite sample guarantees for Principal Component Analysis (PCA). These hold even when the corrupting noise is non-isotropic, and a part (or all of it) is data-dependent. Because of the latter, in general, the noise and the true data are correlated. The results in this work are a significant improvement over those given in our earlier work where this "correlated-PCA" problem was first studied. In fact, in certain regimes, our results imply that the sample complexity required to achieve subspace recovery error that is a constant fraction of the noise level is near-optimal…
Institut de Mathématiques de Bourgogne（勃艮第数学研究所）；Université de Bourgogne Franche Comté（勃艮第-弗朗什孔泰大学）；DePaul University（德保罗大学）
In the current context of data explosion, online techniques that do not require storing all data in memory are indispensable to routinely perform tasks like principal component analysis (PCA). Recursive algorithms that update the PCA with each new observation have been studied in various fields of research and found wide applications in industrial monitoring, computer vision, astronomy, and latent semantic indexing, among others. This work provides guidance for selecting an online PCA algorithm in practice. We present the main approaches to online PCA, namely, perturbation techniques, incremen…
Most high-dimensional matrix recovery problems are studied under the assumption that the target matrix has certain intrinsic structures. For image data related matrix recovery problems, approximate low-rankness and smoothness are the two most commonly imposed structures. For approximately low-rank matrix recovery, the robust principal component analysis (PCA) is well-studied and proved to be effective. For smooth matrix problem, 2d fused Lasso and other total variation based approaches have played a fundamental role. Although both low-rankness and smoothness are key assumptions for image data …
Principal Component Analysis (PCA) is a method for estimating a subspace given noisy samples. It is useful in a variety of problems ranging from dimensionality reduction to anomaly detection and the visualization of high dimensional data. PCA performs well in the presence of moderate noise and even with missing data, but is also sensitive to outliers. PCA is also known to have a phase transition when noise is independent and identically distributed; recovery of the subspace sharply declines at a threshold noise variance. Effective use of PCA requires a rigorous understanding of these behaviors…
Principal components analysis (PCA) is a well-known technique for approximating a tabular data set by a low rank matrix. Here, we extend the idea of PCA to handle arbitrary data sets consisting of numerical, Boolean, categorical, ordinal, and other data types. This framework encompasses many well known techniques in data analysis, such as nonnegative matrix factorization, matrix completion, sparse and robust PCA, $k$-means, $k$-SVD, and maximum margin matrix factorization. The method handles heterogeneous data sets, and leads to coherent schemes for compressing, denoising, and imputing missing…
The article proposes and theoretically analyses a \emph{computationally efficient} multi-task learning (MTL) extension of popular principal component analysis (PCA)-based supervised learning schemes \cite{barshan2011supervised,bair2006prediction}. The analysis reveals that (i) by default learning may dramatically fail by suffering from \emph{negative transfer}, but that (ii) simple counter-measures on data labels avert negative transfer and necessarily result in improved performances. Supporting experiments on synthetic and real data benchmarks show that the proposed method achieves comparable…
KAIST（韩国科学技术院）；UC Berkeley IEOR（加州大学伯克利分校工业工程与运筹学系）；Citadel（城堡投资）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper defines fair principal component analysis (PCA) as minimizing the maximum mean discrepancy (MMD) between dimensionality-reduced conditional distributions of different protected classes. The incorporation of MMD naturally leads to an exact and tractable mathematical formulation of fairness with good statistical properties. We formulate the problem of fair PCA subject to MMD constraints as a non-convex optimization over the Stiefel manifold and solve it using the Riemannian Exact Penalty Method with Smoothing (REPMS; Liu and Boumal, 2019). Importantly, we provide local optimality guar…
In this paper, we study the problem of sparse Principal Component Analysis (PCA) in the high-dimensional setting with missing observations. Our goal is to estimate the first principal component when we only have access to partial observations. Existing estimation techniques are usually derived for fully observed data sets and require a prior knowledge of the sparsity of the first principal component in order to achieve good statistical guarantees. Our contributions is threefold. First, we establish the first information-theoretic lower bound for the sparse PCA problem with missing observations…
In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the $k$-dimensional subspace which can explain the most variation of the $d$-dimensional data points that come into memory sequentially. In order to deal with large $d$ and large $N$ (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the informat…
University of Wisconsin Madison（威斯康星大学麦迪逊分校）；Amazon Lab126（亚马逊Lab126）
We present an efficient stochastic algorithm (RSG+) for canonical correlation analysis (CCA) using a reparametrization of the projection matrices. We show how this reparametrization (into structured matrices), simple in hindsight, directly presents an opportunity to repurpose/adjust mature techniques for numerical optimization on Riemannian manifolds. Our developments nicely complement existing methods for this problem which either require $O(d^3)$ time complexity per iteration with $O(\frac{1}{\sqrt{t}})$ convergence rate (where $d$ is the dimensionality) or only extract the top $1$ component…
Principal component analysis (PCA) is a well-known linear dimension-reduction method that has been widely used in data analysis and modeling. It is an unsupervised learning technique that identifies a suitable linear subspace for the input variable that contains maximal variation and preserves as much information as possible. PCA has also been used in prediction models where the original, high-dimensional space of predictors is reduced to a smaller, more manageable, set before conducting regression analysis. However, this approach does not incorporate information in the response during the dim…
The P300 event-related potential (ERP), evoked in scalp-recorded electroencephalography (EEG) by external stimuli, has proven to be a reliable response for controlling a BCI. The P300 component of an event related potential is thus widely used in brain-computer interfaces to translate the subjects' intent by mere thoughts into commands to control artificial devices. The main challenge in the classification of P300 trials in electroencephalographic (EEG) data is the low signal-to-noise ratio (SNR) of the P300 response. To overcome the low SNR of individual trials, it is common practice to avera…
City University（城市大学）；East West University（东西大学）；Bangladesh University of Engineering and Technology（孟加拉工程技术大学）
Worldwide research shows that millions of lives lost per year because of heart disease. The healthcare sector produces massive volumes of data on heart disease that are sadly not used to locate secret knowledge for successful decision making. One of the most important aspects at this moment is detecting heart disease at an early stage. Researchers have applied distinct techniques to the UCI Machine Learning heart disease dataset. Many researchers have tried to apply some complex techniques to this dataset, where detailed studies are still missing. In this paper, Principal Component Analysis (P…
We study the problem of high-dimensional Principal Component Analysis (PCA) with missing observations. In simple, homogeneous missingness settings with a noise level of constant order, we show that an existing inverse-probability weighted (IPW) estimator of the leading principal components can (nearly) attain the minimax optimal rate of convergence. However, deeper investigation reveals both that, particularly in more realistic settings where the missingness mechanism is heterogeneous, the empirical performance of the IPW estimator can be unsatisfactory, and moreover that, in the noiseless cas…
Suppose we observe data of the form $Y_i = D_i (S_i + \varepsilon_i) \in \mathbb{R}^p$ or $Y_i = D_i S_i + \varepsilon_i \in \mathbb{R}^p$, $i=1,\ldots,n$, where $D_i \in \mathbb{R}^{p\times p}$ are known diagonal matrices, $\varepsilon_i$ are noise, and we wish to perform principal component analysis (PCA) on the unobserved signals $S_i \in \mathbb{R}^p$. The first model arises in missing data problems, where the $D_i$ are binary. The second model captures noisy deconvolution problems, where the $D_i$ are the Fourier transforms of the convolution kernels. It is often reasonable to assume the …
In this paper, we study the application of sparse principal component analysis (PCA) to clustering and feature selection problems. Sparse PCA seeks sparse factors, or linear combinations of the data variables, explaining a maximum amount of variance in the data while having only a limited number of nonzero coefficients. PCA is often used as a simple clustering technique and sparse factors allow us here to interpret the clusters in terms of a reduced set of variables. We begin with a brief introduction and motivation on sparse PCA and detail our implementation of the algorithm in d'Aspremont et…
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…
Traditional principal component analysis (PCA) is well known in high-dimensional data analysis, but it requires to express data by a matrix with observations to be continuous. To overcome the limitations, a new method called flexible PCA (FPCA) for exponential family distributions is proposed. The goal is to ensure that it can be implemented to arbitrary shaped region for either count or continuous observations. The methodology of FPCA is developed under the framework of generalized linear models. It provides statistical models for FPCA not limited to matrix expressions of the data. A maximum …
We evaluate the following Machine Learning techniques for Green Energy (Wind, Solar) Prediction: Bayesian Inference, Neural Networks, Support Vector Machines, Clustering techniques (PCA). Our objective is to predict green energy using weather forecasts, predict deviations from forecast green energy, find correlation amongst different weather parameters and green energy availability, recover lost or missing energy (/ weather) data. We use historical weather data and weather forecasts for the same.
This work presents a new robust PCA method for foreground-background separation on freely moving camera video with possible dense and sparse corruptions. Our proposed method registers the frames of the corrupted video and then encodes the varying perspective arising from camera motion as missing data in a global model. This formulation allows our algorithm to produce a panoramic background component that automatically stitches together corrupted data from partially overlapping frames to reconstruct the full field of view. We model the registered video as the sum of a low-rank component that ca…
In health-pollution cohort studies, accurate predictions of pollutant concentrations at new locations are needed, since the locations of fixed monitoring sites and study participants are often spatially misaligned. For multi-pollution data, principal component analysis (PCA) is often incorporated to obtain low-rank (LR) structure of the data prior to spatial prediction. Recently developed predictive PCA modifies the traditional algorithm to improve the overall predictive performance by leveraging both LR and spatial structures within the data. However, predictive PCA requires complete data or …
School of Applied Sciences (FCA), University of Campinas (UNICAMP)（坎皮纳斯大学（UNICAMP）应用科学学院（FCA））
Principal component analysis (PCA), a ubiquitous dimensionality reduction technique in signal processing, searches for a projection matrix that minimizes the mean squared error between the reduced dataset and the original one. Since classical PCA is not tailored to address concerns related to fairness, its application to actual problems may lead to disparity in the reconstruction errors of different groups (e.g., men and women, whites and blacks, etc.), with potentially harmful consequences such as the introduction of bias towards sensitive groups. Although several fair versions of PCA have be…
Multivariate binary data is becoming abundant in current biological research. Logistic principal component analysis (PCA) is one of the commonly used tools to explore the relationships inside a multivariate binary data set by exploiting the underlying low rank structure. We re-expressed the logistic PCA model based on the latent variable interpretation of the generalized linear model on binary data. The multivariate binary data set is assumed to be the sign observation of an unobserved quantitative data set, on which a low rank structure is assumed to exist. However, the standard logistic PCA …
University of California San Diego（加州大学圣迭戈分校）；Université de Sherbrooke（舍布鲁克大学）
We present a system for localizing sound sources in a room with several ad-hoc microphone arrays. Each circular array performs direction of arrival (DOA) estimation independently using commercial software. The DOAs are fed to a fusion center, concatenated, and used to perform the localization based on two proposed methods, which require only few labeled source locations (anchor points) for training. The first proposed method is based on principal component analysis (PCA) of the observed DOA and does not require any knowledge of anchor points. The array cluster can then perform localization on …
Artificial intelligence is revolutionizing our lives at an ever increasing pace. At the heart of this revolution is the recent advancements in deep neural networks (DNN), learning to perform sophisticated, high-level tasks. However, training DNNs requires massive amounts of data and is very computationally intensive. Gaining analytical understanding of the solutions found by DNNs can help us devise more efficient training algorithms, replacing the commonly used mthod of stochastic gradient descent (SGD). We analyze the dynamics of SGD and show that, indeed, direct computation of the solutions …
We consider multi-class classification problems for high dimensional data. Following the idea of reduced-rank linear discriminant analysis (LDA), we introduce a new dimension reduction tool with a flavor of supervised principal component analysis (PCA). The proposed method is computationally efficient and can incorporate the correlation structure among the features. Besides the theoretical insights, we show that our method is a competitive classification tool by simulated and real data examples.
