Principal component regression (PCR) is a widely used two-stage procedure: principal component analysis (PCA), followed by regression in which the selected principal components are regarded as new explanatory variables in the model. Note that PCA is based only on the explanatory variables, so the principal components are not selected using the information on the response variable. In this paper, we propose a one-stage procedure for PCR in the framework of generalized linear models. The basic loss function is based on a combination of the regression loss and PCA loss. An estimate of the regression parameter is obtained as the minimizer of the basic loss function with a sparse penalty. We call the proposed method sparse principal component regression for generalized linear models (SPCR-glm). Taking the two loss function into consideration simultaneously, SPCR-glm enables us to obtain sparse principal component loadings that are related to a response variable. However, a combination of loss functions may cause a parameter identification problem, but this potential problem is avoided by virtue of the sparse penalty. Thus, the sparse penalty plays two roles in this method. The parameter estimation procedure is proposed using various update algorithms with the coordinate descent algorithm. We apply SPCR-glm to two real datasets, doctor visits data and mouse consomic strain data. SPCR-glm provides more easily interpretable principal component (PC) scores and clearer classification on PC plots than the usual PCA.
We propose a principal components regression method based on maximizing a joint pseudo-likelihood for responses and predictors. Our method uses both responses and predictors to select linear combinations of the predictors relevant for the regression, thereby addressing an oft-cited deficiency of conventional principal components regression. The proposed estimator is shown to be consistent in a wide range of settings, including ones with non-normal and dependent observations; conditions on the first and second moments suffice if the number of predictors ($p$) is fixed and the number of observat…
Wisconsin Institutes for Discovery（威斯康星发现研究院）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
This paper proposes a fast and accurate method for sparse regression in the presence of missing data. The underlying statistical model encapsulates the low-dimensional structure of the incomplete data matrix and the sparsity of the regression coefficients, and the proposed algorithm jointly learns the low-dimensional structure of the data and a linear regressor with sparse coefficients. The proposed stochastic optimization method, Sparse Linear Regression with Missing Data (SLRM), performs an alternating minimization procedure and scales well with the problem size. Large deviation inequalities…
We perform a finite sample analysis of the detection levels for sparse principal components of a high-dimensional covariance matrix. Our minimax optimal test is based on a sparse eigenvalue statistic. Alas, computing this test is known to be NP-complete in general, and we describe a computationally efficient alternative test using convex relaxations. Our relaxation is also proved to detect sparse principal components at near optimal detection levels, and it performs well on simulated datasets. Moreover, using polynomial time reductions from theoretical computer science, we bring significant ev…
Regularization is an essential element of virtually all kernel methods for nonparametric regression problems. A critical factor in the effectiveness of a given kernel method is the type of regularization that is employed. This article compares and contrasts members from a general class of regularization techniques, which notably includes ridge regression and principal component regression. We derive an explicit finite-sample risk bound for regularization-based estimators that simultaneously accounts for (i) the structure of the ambient function space, (ii) the regularity of the true regression…
Quantile regression is studied in combination with a penalty which promotes structured (or group) sparsity. A mixed $\ell_{1,\infty}$-norm on the parameter vector is used to impose structured sparsity on the traditional quantile regression problem. An algorithm is derived to calculate the piece-wise linear solution path of the corresponding minimization problem. A Matlab implementation of the proposed algorithm is provided and some applications of the methods are also studied.
We study sparse principal components analysis in high dimensions, where $p$ (the number of variables) can be much larger than $n$ (the number of observations), and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix. We introduce two complementary notions of $\ell_q$ subspace sparsity: row sparsity and column sparsity. We prove nonasymptotic lower and upper bounds on the minimax subspace estimation error for $0\leq q\leq1$. The bounds are optimal for row sparse subspaces and nearly optimal for column sparse subspaces, they ap…
In this study, we propose a function-on-function linear quantile regression model that allows for more than one functional predictor to establish a more flexible and robust approach. The proposed model is first transformed into a finite-dimensional space via the functional principal component analysis paradigm in the estimation phase. It is then approximated using the estimated functional principal component functions, and the estimated parameter of the quantile regression model is constructed based on the principal component scores. In addition, we propose a Bayesian information criterion to …
This paper studies sparse linear regression analysis with outliers in the responses. A parameter vector for modeling outliers is added to the standard linear regression model and then the sparse estimation problem for both coefficients and outliers is considered. The $\ell_{1}$ penalty is imposed for the coefficients, while various penalties including redescending type penalties are for the outliers. To solve the sparse estimation problem, we introduce an optimization algorithm. Under some conditions, we show the algorithmic and statistical convergence property for the coefficients obtained by…
Scaled sparse linear regression jointly estimates the regression coefficients and noise level in a linear model. It chooses an equilibrium with a sparse regression method by iteratively estimating the noise level via the mean residual square and scaling the penalty in proportion to the estimated noise level. The iterative algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty levels above a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a convex minimization of a penalized joint loss function for the regres…
Regression with sparse inputs is a common theme for large scale models. Optimizing the underlying linear algebra for sparse inputs allows such models to be estimated faster. At the same time, centering the inputs has benefits in improving the interpretation and convergence of the model. However, centering the data naturally makes sparse data become dense, limiting opportunities for optimization. We propose an efficient strategy that estimates centered regression while taking advantage of sparse structure in data, improving computational performance and decreasing the memory footprint of the es…
We consider the related problems of estimating the $l_2$-norm and the squared $l_2$-norm in sparse linear regression with unknown variance, as well as the problem of testing the hypothesis that the regression parameter is null under sparse alternatives with $l_2$ separation. We establish the minimax optimal rates of estimation (respectively, testing) in these three problems.
We consider the problem of testing the hypothesis that the parameter of linear regression model is 0 against an s-sparse alternative separated from 0 in the l2-distance. We show that, in Gaussian linear regression model with p < n, where p is the dimension of the parameter and n is the sample size, the non-asymptotic minimax rate of testing has the form sqrt((s/n) log(1 + sqrt(p)/s )). We also show that this is the minimax rate of estimation of the l2-norm of the regression parameter.
Under a standard assumption in complexity theory (NP not in P/poly), we demonstrate a gap between the minimax prediction risk for sparse linear regression that can be achieved by polynomial-time algorithms, and that achieved by optimal algorithms. In particular, when the design matrix is ill-conditioned, the minimax prediction loss achievable by polynomial-time algorithms can be substantially greater than that of an optimal algorithm. This result is the first known gap between polynomial and optimal algorithms for sparse linear regression, and does not depend on conjectures in average-case com…
Principal component regression (PCR) is a two-stage procedure that selects some principal components and then constructs a regression model regarding them as new explanatory variables. Note that the principal components are obtained from only explanatory variables and not considered with the response variable. To address this problem, we propose the sparse principal component regression (SPCR) that is a one-stage procedure for PCR. SPCR enables us to adaptively obtain sparse principal component loadings that are related to the response variable and select the number of principal components sim…
We propose a new method for supervised learning, especially suited to wide data where the number of features is much greater than the number of observations. The method combines the lasso ($\ell_1$) sparsity penalty with a quadratic penalty that shrinks the coefficient vector toward the leading principal components of the feature matrix. We call the proposed method the "principal components lasso" ("pcLasso"). The method can be especially powerful if the features are pre-assigned to groups (such as cell-pathways, assays or protein interaction networks). In that case, pcLasso shrinks each group…
Principal component regression (PCR) is a two-stage procedure: the first stage performs principal component analysis (PCA) and the second stage constructs a regression model whose explanatory variables are replaced by principal components obtained by the first stage. Since PCA is performed by using only explanatory variables, the principal components have no information about the response variable. To address the problem, we propose a one-stage procedure for PCR in terms of singular value decomposition approach. Our approach is based upon two loss functions, a regression loss and a PCA loss, w…
We propose a new two stage algorithm LING for large scale regression problems. LING has the same risk as the well known Ridge Regression under the fixed design setting and can be computed much faster. Our experiments have shown that LING performs well in terms of both prediction accuracy and computational efficiency compared with other large scale regression algorithms like Gradient Descent, Stochastic Gradient Descent and Principal Component Regression on both simulated and real datasets.
In this paper, we consider a new variant for principal component analysis (PCA), aiming to capture the grouping and/or sparse structures of factor loadings simultaneously. To achieve these goals, we employ a non-convex truncated regularization with naturally adjustable sparsity and grouping effects, and propose the Feature Grouping and Sparse Principal Component Analysis (FGSPCA). The proposed FGSPCA method encourages the factor loadings with similar values to collapse into disjoint homogeneous groups for feature grouping or into a special zero-valued group for feature selection, which in turn…
We study sparse principal component analysis for high dimensional vector autoregressive time series under a doubly asymptotic framework, which allows the dimension $d$ to scale with the series length $T$. We treat the transition matrix of time series as a nuisance parameter and directly apply sparse principal component analysis on multivariate time series as if the data are independent. We provide explicit non-asymptotic rates of convergence for leading eigenvector estimation and extend this result to principal subspace estimation. Our analysis illustrates that the spectral norm of the transit…
In high-dimensional data, many sparse regression methods have been proposed. However, they may not be robust against outliers. Recently, the use of density power weight has been studied for robust parameter estimation and the corresponding divergences have been discussed. One of such divergences is the $γ$-divergence and the robust estimator using the $γ$-divergence is known for having a strong robustness. In this paper, we consider the robust and sparse regression based on $γ$-divergence. We extend the $γ$-divergence to the regression problem and show that it has a strong robustness under hea…
When the design matrix has orthonormal columns, "soft thresholding" the ordinary least squares (OLS) solution produces the Lasso solution [Tibshirani, 1996]. If one uses the Puffer preconditioned Lasso [Jia and Rohe, 2012], then this result generalizes from orthonormal designs to full rank designs (Theorem 1). Theorem 2 refines the Puffer preconditioner to make the Lasso select the same model as removing the elements of the OLS solution with the largest p-values. Using a generalized Puffer preconditioner, Theorem 3 relates ridge regression to the preconditioned Lasso; this result is for the hi…
We introduce a new method for sparse principal component analysis, based on the aggregation of eigenvector information from carefully-selected axis-aligned random projections of the sample covariance matrix. Unlike most alternative approaches, our algorithm is non-iterative, so is not vulnerable to a bad choice of initialisation. We provide theoretical guarantees under which our principal subspace estimator can attain the minimax optimal rate of convergence in polynomial time. In addition, our theory provides a more refined understanding of the statistical and computational trade-off in the pr…
Principal Components Regression (PCR) is a traditional tool for dimension reduction in linear regression that has been both criticized and defended. One concern about PCR is that obtaining the leading principal components tends to be computationally demanding for large data sets. While random projections do not possess the optimality properties of the leading principal subspace, they are computationally appealing and hence have become increasingly popular in recent years. In this paper, we present an analysis showing that for random projections satisfying a Johnson-Lindenstrauss embedding prop…
We present an extension of sparse PCA, or sparse dictionary learning, where the sparsity patterns of all dictionary elements are structured and constrained to belong to a prespecified set of shapes. This \emph{structured sparse PCA} is based on a structured regularization recently introduced by [1]. While classical sparse priors only deal with \textit{cardinality}, the regularization we use encodes higher-order information about the data. We propose an efficient and simple optimization procedure to solve this problem. Experiments with two practical tasks, face recognition and the study of the …
The computation of the sparse principal component of a matrix is equivalent to the identification of its principal submatrix with the largest maximum eigenvalue. Finding this optimal submatrix is what renders the problem ${\mathcal{NP}}$-hard. In this work, we prove that, if the matrix is positive semidefinite and its rank is constant, then its sparse principal component is polynomially computable. Our proof utilizes the auxiliary unit vector technique that has been recently developed to identify problems that are polynomially solvable. Moreover, we use this technique to design an algorithm wh…
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…
Modern technologies are generating ever-increasing amounts of data. Making use of these data requires methods that are both statistically sound and computationally efficient. Typically, the statistical and computational aspects are treated separately. In this paper, we propose an approach to entangle these two aspects in the context of regularized estimation. Applying our approach to sparse and group-sparse regression, we show that it can improve on standard pipelines both statistically and computationally.
We propose the Factor Augmented sparse linear Regression Model (FARM) that not only encompasses both the latent factor regression and sparse linear regression as special cases but also bridges dimension reduction and sparse regression together. We provide theoretical guarantees for the estimation of our model under the existence of sub-Gaussian and heavy-tailed noises (with bounded (1+x)-th moment, for all x>0), respectively. In addition, the existing works on supervised learning often assume the latent factor regression or the sparse linear regression is the true underlying model without just…
Regularized variants of Principal Components Analysis, especially Sparse PCA and Functional PCA, are among the most useful tools for the analysis of complex high-dimensional data. Many examples of massive data, have both sparse and functional (smooth) aspects and may benefit from a regularization scheme that can capture both forms of structure. For example, in neuro-imaging data, the brain's response to a stimulus may be restricted to a discrete region of activation (spatial sparsity), while exhibiting a smooth response within that region. We propose a unified approach to regularized PCA which…
Case Western Reserve University（凯斯西储大学）；Florida International University（佛罗里达国际大学）；Kent State University（肯特州立大学）；National Institute on Aging, National Institute of Health（美国国立卫生研究院国家老龄化研究所）；Fordham University（福特汉姆大学）
This paper presents a sublinear classical algorithm for principal component regression. The algorithm uses quantum-inspired linear algebra, an idea developed by Tang. Using this technique, her algorithm for recommendation systems achieved runtime only polynomially slower than its quantum counterpart. Her work was quickly adapted to solve many other problems in sublinear time complexity. In this work, we developed an algorithm for principal component regression that runs in time polylogarithmic to the number of data points, an exponential speed up over the state-of-the-art algorithm, under the …
Forward regression is a statistical model selection and estimation procedure which inductively selects covariates that add predictive power into a working statistical regression model. Once a model is selected, unknown regression parameters are estimated by least squares. This paper analyzes forward regression in high-dimensional sparse linear models. Probabilistic bounds for prediction error norm and number of selected covariates are proved. The analysis in this paper gives sharp rates and does not require beta-min or irrepresentability conditions.
Stanford University（斯坦福大学）；MIT（麻省理工学院）；Microsoft Research, New England（新英格兰微软研究院）
We show how to efficiently project a vector onto the top principal components of a matrix, without explicitly computing these components. Specifically, we introduce an iterative algorithm that provably computes the projection using few calls to any black-box routine for ridge regression. By avoiding explicit principal component analysis (PCA), our algorithm is the first with no runtime dependence on the number of top principal components. We show that it can be used to give a fast iterative method for the popular principal component regression problem, giving the first major runtime improvemen…
This work aims to give non-asymptotic results for estimating the first principal component of a multivariate random process. We first define the covariance function and the covariance operator in the multivariate case. We then define a projection operator. This operator can be seen as a reconstruction step from the raw data in the functional data analysis context. Next, we show that the eigenelements can be expressed as the solution to an optimization problem, and we introduce the LASSO variant of this optimization problem and the associated plugin estimator. Finally, we assess the estimator's…
School of Mathematics and Information Sciences, Yantai University（烟台大学数学与信息科学学院）；Department of Mathematics, Shandong Normal University（山东师范大学数学学院）
The implementation of conventional sparse principal component analysis (SPCA) on high-dimensional data sets has become a time consuming work. In this paper, a series of subspace projections are constructed efficiently by using Household QR factorization. With the aid of these subspace projections, a fast deflation method, called SPCA-SP, is developed for SPCA. This method keeps a good tradeoff between various criteria, including sparsity, orthogonality, explained variance, balance of sparsity, and computational cost. Comparative experiments on the benchmark data sets confirm the effectiveness …
UC Berkeley（加州大学伯克利分校）；University of Washington（华盛顿大学）
Sparse principal component analysis (SPCA) has emerged as a powerful technique for modern data analysis, providing improved interpretation of low-rank structures by identifying localized spatial structures in the data and disambiguating between distinct time scales. We demonstrate a robust and scalable SPCA algorithm by formulating it as a value-function optimization problem. This viewpoint leads to a flexible and computationally efficient algorithm. Further, we can leverage randomized methods from linear algebra to extend the approach to the large-scale (big data) setting. Our proposed innova…
We study least squares linear regression over $N$ uncorrelated Gaussian features that are selected in order of decreasing variance. When the number of selected features $p$ is at most the sample size $n$, the estimator under consideration coincides with the principal component regression estimator; when $p>n$, the estimator is the least $\ell_2$ norm solution over the selected features. We give an average-case analysis of the out-of-sample prediction error as $p,n,N \to \infty$ with $p/N \to α$ and $n/N \to β$, for some constants $α\in [0,1]$ and $β\in (0,1)$. In this average-case setting, the…
École Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）
Data analyses based on linear methods constitute the simplest, most robust, and transparent approaches to the automatic processing of large amounts of data for building supervised or unsupervised machine learning models. Principal covariates regression (PCovR) is an underappreciated method that interpolates between principal component analysis and linear regression, and can be used to conveniently reveal structure-property relations in terms of simple-to-interpret, low-dimensional maps. Here we provide a pedagogic overview of these data analysis schemes, including the use of the kernel trick t…
In this paper, we consider the classic measurement error regression scenario in which our independent, or design, variables are observed with several sources of additive noise. We will show that our motivating example's replicated measurements on both the design and dependent variables may be leveraged to enhance a sparse regression algorithm. Specifically, we estimate the variance and use it to scale our design variables. We demonstrate the efficacy of scaling from several points of view and validate it empirically with a biomass characterization data set using two of the most widely used spa…
School of Electrical Engineering, Korea Advanced Institute of Science and Technology（韩国科学技术院电气工程学院）
We consider the classical sparse regression problem of recovering a sparse signal $x_0$ given a measurement vector $y = Φx_0+w$. We propose a tree search algorithm driven by the deep neural network for sparse regression (TSN). TSN improves the signal reconstruction performance of the deep neural network designed for sparse regression by performing a tree search with pruning. It is observed in both noiseless and noisy cases, TSN recovers synthetic and real signals with lower complexity than a conventional tree search and is superior to existing algorithms by a large margin for various types of …
Multivariate regression model is a natural generalization of the classical univari- ate regression model for fitting multiple responses. In this paper, we propose a high- dimensional multivariate conditional regression model for constructing sparse estimates of the multivariate regression coefficient matrix that accounts for the dependency struc- ture among the multiple responses. The proposed method decomposes the multivariate regression problem into a series of penalized conditional log-likelihood of each response conditioned on the covariates and other responses. It allows simultaneous esti…
We propose and analyze a novel framework for learning sparse representations, based on two statistical techniques: kernel smoothing and marginal regression. The proposed approach provides a flexible framework for incorporating feature similarity or temporal information present in data sets, via non-parametric kernel smoothing. We provide generalization bounds for dictionary learning using smooth sparse coding and show how the sample complexity depends on the L1 norm of kernel function used. Furthermore, we propose using marginal regression for obtaining sparse codes, which significantly improv…
Sparse principal component analysis (sparse PCA) aims at finding a sparse basis to improve the interpretability over the dense basis of PCA, meanwhile the sparse basis should cover the data subspace as much as possible. In contrast to most of existing work which deal with the problem by adding some sparsity penalties on various objectives of PCA, in this paper, we propose a new method SPCArt, whose motivation is to find a rotation matrix and a sparse basis such that the sparse basis approximates the basis of PCA after the rotation. The algorithm of SPCArt consists of three alternating steps: r…
MIT Biological Engineering（麻省理工学院生物工程系）；MIT CSAIL（麻省理工学院计算机科学与人工智能实验室）
We introduce Graph-Sparse Logistic Regression, a new algorithm for classification for the case in which the support should be sparse but connected on a graph. We val- idate this algorithm against synthetic data and benchmark it against L1-regularized Logistic Regression. We then explore our technique in the bioinformatics context of proteomics data on the interactome graph. We make all our experimental code public and provide GSLR as an open source package.
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-…
Computer Science and Engineering, Arizona State University, USA（亚利桑那州立大学计算机科学与工程学院，美国）；SAS Institute Inc., USA（美国SAS研究所）
The l1-regularized logistic regression (or sparse logistic regression) is a widely used method for simultaneous classification and feature selection. Although many recent efforts have been devoted to its efficient implementation, its application to high dimensional data still poses significant challenges. In this paper, we present a fast and effective sparse logistic regression screening rule (Slores) to identify the 0 components in the solution vector, which may lead to a substantial reduction in the number of features to be entered to the optimization. An appealing feature of Slores is that …
In this paper, we propose a cone projected power iteration algorithm to recover the first principal eigenvector from a noisy positive semidefinite matrix. When the true principal eigenvector is assumed to belong to a convex cone, the proposed algorithm is fast and has a tractable error. Specifically, the method achieves polynomial time complexity for certain convex cones equipped with fast projection such as the monotone cone. It attains a small error when the noisy matrix has a small cone-restricted operator norm. We supplement the above results with a minimax lower bound of the error under t…
