Sparse model selection is ubiquitous from linear regression to graphical models where regularization paths, as a family of estimators upon the regularization parameter varying, are computed when the regularization parameter is unknown or decided data-adaptively. Traditional computational methods rely on solving a set of optimization problems where the regularization parameters are fixed on a grid that might be inefficient. In this paper, we introduce a simple iterative regularization path, which follows the dynamics of a sparse Mirror Descent algorithm or a generalization of Linearized Bregman Iterations with nonlinear loss. Its performance is competitive to \texttt{glmnet} with a further bias reduction. A path consistency theory is presented that under the Restricted Strong Convexity (RSC) and the Irrepresentable Condition (IRR), the path will first evolve in a subspace with no false positives and reach an estimator that is sign-consistent or of minimax optimal $\ell_2$ error rate. Early stopping regularization is required to prevent overfitting. Application examples are given in sparse logistic regression and Ising models for NIPS coauthorship.
This work proposes new inference methods for a regression coefficient of interest in a (heterogeneous) quantile regression model. We consider a high-dimensional model where the number of regressors potentially exceeds the sample size but a subset of them suffice to construct a reasonable approximation to the conditional quantile function. The proposed methods are (explicitly or implicitly) based on orthogonal score functions that protect against moderate model selection mistakes, which are often inevitable in the approximately sparse model considered in the present paper. We establish the unif…
We propose a sparse coefficient estimation and automated model selection procedure for autoregressive (AR) processes with heavy-tailed innovations based on penalized conditional maximum likelihood. Under mild moment conditions on the innovation processes, the penalized conditional maximum likelihood estimator (PCMLE) satisfies a strong consistency, $O_P(N^{-1/2})$ consistency, and the oracle properties, where N is the sample size. We have the freedom in choosing penalty functions based on the weak conditions on them. Two penalty functions, least absolute shrinkage and selection operator (LASSO…
In this paper, we consider estimating sparse inverse covariance of a Gaussian graphical model whose conditional independence is assumed to be partially known. Similarly as in [5], we formulate it as an $l_1$-norm penalized maximum likelihood estimation problem. Further, we propose an algorithm framework, and develop two first-order methods, that is, the adaptive spectral projected gradient (ASPG) method and the adaptive Nesterov's smooth (ANS) method, for solving this estimation problem. Finally, we compare the performance of these two methods on a set of randomly generated instances. Our comp…
In this paper we first study a smooth optimization approach for solving a class of nonsmooth strictly concave maximization problems whose objective functions admit smooth convex minimization reformulations. In particular, we apply Nesterov's smooth optimization technique [Y.E. Nesterov, Dokl. Akad. Nauk SSSR, 269 (1983), pp. 543--547; Y. E. Nesterov, Math. Programming, 103 (2005), pp. 127--152] to their dual counterparts that are smooth convex problems. It is shown that the resulting approach has ${\cal O}(1/{\sqrtε})$ iteration complexity for finding an $ε$-optimal solution to both primal and…
Arizona State University（亚利桑那州立大学）；The Biodesign Institute, Arizona State University（亚利桑那州立大学生物设计研究所）；University of Minnesota（明尼苏达大学）
Sparse feature selection has been demonstrated to be effective in handling high-dimensional data. While promising, most of the existing works use convex methods, which may be suboptimal in terms of the accuracy of feature selection and parameter estimation. In this paper, we expand a nonconvex paradigm to sparse group feature selection, which is motivated by applications that require identifying the underlying group structure and performing feature selection simultaneously. The main contributions of this article are twofold: (1) statistically, we introduce a nonconvex sparse group feature sele…
In this thesis we discuss machine learning methods performing automated variable selection for learning sparse predictive models. There are multiple reasons for promoting sparsity in the predictive models. By relying on a limited set of input variables the models naturally counteract the overfitting problem ubiquitous in learning from finite sets of training points. Sparse models are cheaper to use for predictions, they usually require lower computational resources and by relying on smaller sets of inputs can possibly reduce costs for data collection and storage. Sparse models can also contrib…
Xi’an Jiaotong University（西安交通大学）；State Key Laboratory of Management and Control for Complex Systems（复杂系统管理与控制国家重点实验室）
Given $m$ $d$-dimensional responsors and $n$ $d$-dimensional predictors, sparse regression finds at most $k$ predictors for each responsor for linear approximation, $1\leq k \leq d-1$. The key problem in sparse regression is subset selection, which usually suffers from high computational cost. Recent years, many improved approximate methods of subset selection have been published. However, less attention has been paid on the non-approximate method of subset selection, which is very necessary for many questions in data analysis. Here we consider sparse regression from the view of correlation, a…
We consider the two-group classification problem and propose a kernel classifier based on the optimal scoring framework. Unlike previous approaches, we provide theoretical guarantees on the expected risk consistency of the method. We also allow for feature selection by imposing structured sparsity using weighted kernels. We propose fully-automated methods for selection of all tuning parameters, and in particular adapt kernel shrinkage ideas for ridge parameter selection. Numerical studies demonstrate the superior classification performance of the proposed approach compared to existing nonparam…
University of Twente（特文特大学）；Eindhoven University of Technology（埃因霍温理工大学）；University of Jyväskylä（于韦斯屈莱大学）
Major complications arise from the recent increase in the amount of high-dimensional data, including high computational costs and memory requirements. Feature selection, which identifies the most relevant and informative attributes of a dataset, has been introduced as a solution to this problem. Most of the existing feature selection methods are computationally inefficient; inefficient algorithms lead to high energy consumption, which is not desirable for devices with limited computational and energy resources. In this paper, a novel and flexible method for unsupervised feature selection is pr…
Knot-based, sparse Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Problems can occur, however, when knot selection is done by optimizing the marginal likelihood. For example, the marginal likelihood surface is highly multimodal, which can cause suboptimal knot placement where some knots serve practically no function. This is especially a problem when many more knots are used than are necessary, resulting in extra computational cost for little to no gains in accuracy. We propose a one-at-a-time knot selection algorithm to select both …
Sparse, knot-based Gaussian processes have enjoyed considerable success as scalable approximations to full Gaussian processes. Certain sparse models can be derived through specific variational approximations to the true posterior, and knots can be selected to minimize the Kullback-Leibler divergence between the approximate and true posterior. While this has been a successful approach, simultaneous optimization of knots can be slow due to the number of parameters being optimized. Furthermore, there have been few proposed methods for selecting the number of knots, and no experimental results exi…
Variable selection and dimension reduction are two commonly adopted approaches for high-dimensional data analysis, but have traditionally been treated separately. Here we propose an integrated approach, called sparse gradient learning (SGL), for variable selection and dimension reduction via learning the gradients of the prediction function directly from samples. By imposing a sparsity constraint on the gradients, variable selection is achieved by selecting variables corresponding to non-zero partial derivatives, and effective dimensions are extracted based on the eigenvectors of the derived s…
Multi-label learning is usually used to mine the correlation between features and labels, and feature selection can retain as much information as possible through a small number of features. $\ell_{2,1}$ regularization method can get sparse coefficient matrix, but it can not solve multicollinearity problem effectively. The model proposed in this paper can obtain the most relevant few features by solving the joint constrained optimization problems of $\ell_{2,1}$ and $\ell_{F}$ regularization.In manifold regularization, we implement random walk strategy based on joint information matrix, and ge…
Sparse and geometry-aware generalisation of the mutual information for joint discriminative clustering and feature selection
Université Côte d’Azur（蔚蓝海岸大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Maasai Team（马赛团队）；CHU de Québec Research Centre（魁北克大学中心医院研究中心）；Laval University（拉瓦尔大学）
Factorization machines (FMs) are machine learning predictive models based on second-order feature interactions and FMs with sparse regularization are called sparse FMs. Such regularizations enable feature selection, which selects the most relevant features for accurate prediction, and therefore they can contribute to the improvement of the model accuracy and interpretability. However, because FMs use second-order feature interactions, the selection of features often causes the loss of many relevant feature interactions in the resultant models. In such cases, FMs with regularization specially d…
We present a new method for estimating multivariate, second-order stationary Gaussian Random Field (GRF) models based on the Sparse Precision matrix Selection (SPS) algorithm, proposed by Davanloo et al. (2015) for estimating scalar GRF models. Theoretical convergence rates for the estimated between-response covariance matrix and for the estimated parameters of the underlying spatial correlation function are established. Numerical tests using simulated and real datasets validate our theoretical findings. Data segmentation is used to handle large data sets.
This paper presents a new variable selection approach integrated with Gaussian process (GP) regression. We consider a sparse projection of input variables and a general stationary covariance model that depends on the Euclidean distance between the projected features. The sparse projection matrix is considered as an unknown parameter. We propose a forward stagewise approach with embedded gradient descent steps to co-optimize the parameter with other covariance parameters based on the maximization of a non-convex marginal likelihood function with a concave sparsity penalty, and some convergence …
ControlBurn is a Python package to construct feature-sparse tree ensembles that support nonlinear feature selection and interpretable machine learning. The algorithms in this package first build large tree ensembles that prioritize basis functions with few features and then select a feature-sparse subset of these basis functions using a weighted lasso optimization criterion. The package includes visualizations to analyze the features selected by the ensemble and their impact on predictions. Hence ControlBurn offers the accuracy and flexibility of tree-ensemble models and the interpretability o…
Air Force Engineering University（空军工程大学）；Northwestern Polytechnical University（西北工业大学）
In the field of data mining, how to deal with high-dimensional data is an inevitable problem. Unsupervised feature selection has attracted more and more attention because it does not rely on labels. The performance of spectral-based unsupervised methods depends on the quality of constructed similarity matrix, which is used to depict the intrinsic structure of data. However, real-world data contain a large number of noise samples and features, making the similarity matrix constructed by original data cannot be completely reliable. Worse still, the size of similarity matrix expands rapidly as th…
Harbin Institute of Technology（哈尔滨工业大学）；Qufu Normal University（曲阜师范大学）；Anhui University（安徽大学）
Principal Component Analysis (PCA) has been used to study the pathogenesis of diseases. To enhance the interpretability of classical PCA, various improved PCA methods have been proposed to date. Among these, a typical method is the so-called sparse PCA, which focuses on seeking sparse loadings. However, the performance of these methods is still far from satisfactory due to their limitation of using unsupervised learning methods; moreover, the class ambiguity within the sample is high. To overcome this problem, this study developed a new PCA method, which is named the Supervised Discriminative …
Sparse PCA with False Discovery Rate Controlled Variable Selection
Technische Universität Darmstadt（达姆施塔特工业大学）；Conservatoire National des Arts et Métiers（法国国立工艺学院）；The Hong Kong University of Science and Technology（香港科技大学）
We study the problem of selecting a subset of k random variables from a large set, in order to obtain the best linear prediction of another variable of interest. This problem can be viewed in the context of both feature selection and sparse approximation. We analyze the performance of widely used greedy heuristics, using insights from the maximization of submodular functions and spectral analysis. We introduce the submodularity ratio as a key quantity to help understand why greedy algorithms perform well even when the variables are highly correlated. Using our techniques, we obtain the stronge…
Sparse Gaussian processes and various extensions thereof are enabled through inducing points, that simultaneously bottleneck the predictive capacity and act as the main contributor towards model complexity. However, the number of inducing points is generally not associated with uncertainty which prevents us from applying the apparatus of Bayesian reasoning for identifying an appropriate trade-off. In this work we place a point process prior on the inducing points and approximate the associated posterior through stochastic variational inference. By letting the prior encourage a moderate number …
Université de Rouen（鲁昂大学）；INSA de Rouen（鲁昂国立应用科学学院）
To answer the existence of optimal swimmer learning/teaching strategies, this work introduces a two-level clustering in order to analyze temporal dynamics of motor learning in breaststroke swimming. Each level have been performed through Sparse Fisher-EM, a unsupervised framework which can be applied efficiently on large and correlated datasets. The induced sparsity selects key points of the coordination phase without any prior knowledge.
Feature selection is an important and challenging task in high dimensional clustering. For example, in genomics, there may only be a small number of genes that are differentially expressed, which are informative to the overall clustering structure. Existing feature selection methods, such as Sparse K-means, rarely tackle the problem of accounting features that can only separate a subset of clusters. In genomics, it is highly likely that a gene can only define one subtype against all the other subtypes or distinguish a pair of subtypes but not others. In this paper, we propose a K-means based c…
Sparse Bayesian Deep Functional Learning with Structured Region Selection
School of Statistics and Data Science, Shanghai University of Finance and Economics（上海金融学院统计与数据科学学院）；School of Statistics（统计学院）；Data Science, Shanghai University of Finance（金融大学数据科学学院）；Department of Biostatistics, Yale School of Public Health（耶鲁大学公共卫生学院生物统计学系）
Gaussian graphical models are of great interest in statistical learning. Because the conditional independencies between different nodes correspond to zero entries in the inverse covariance matrix of the Gaussian distribution, one can learn the structure of the graph by estimating a sparse inverse covariance matrix from sample data, by solving a convex maximum likelihood problem with an $\ell_1$-regularization term. In this paper, we propose a first-order method based on an alternating linearization technique that exploits the problem's special structure; in particular, the subproblems solved i…
Sparse subspace clustering (SSC) using greedy-based neighbor selection, such as matching pursuit (MP) and orthogonal matching pursuit (OMP), has been known as a popular computationally-efficient alternative to the conventional L1-minimization based methods. Under deterministic bounded noise corruption, in this paper we derive coherence-based sufficient conditions guaranteeing correct neighbor identification using MP/OMP. Our analyses exploit the maximum/minimum inner product between two noisy data points subject to a known upper bound on the noise level. The obtained sufficient condition clear…
EPFL（洛桑联邦理工学院）；MIT（麻省理工学院）；University of Texas at Austin（得克萨斯大学奥斯汀分校）；IIT Kharagpur（印度理工学院卡拉格普尔分校）
Group-based sparsity models are proven instrumental in linear regression problems for recovering signals from much fewer measurements than standard compressive sensing. The main promise of these models is the recovery of "interpretable" signals through the identification of their constituent groups. In this paper, we establish a combinatorial framework for group-model selection problems and highlight the underlying tractability issues. In particular, we show that the group-model selection problem is equivalent to the well-known NP-hard weighted maximum coverage problem (WMC). Leveraging a grap…
In this paper, we propose methods for functional predictor selection and the estimation of smooth functional coefficients simultaneously in a scalar-on-function regression problem under high-dimensional multivariate functional data setting. In particular, we develop two methods for functional group-sparse regression under a generic Hilbert space of infinite dimension. We show the convergence of algorithms and the consistency of the estimation and the selection (oracle property) under infinite-dimensional Hilbert spaces. Simulation studies show the effectiveness of the methods in both the selec…
We propose a K-sparse exhaustive search (ES-K) method and a K-sparse approximate exhaustive search method (AES-K) for selecting variables in linear regression. With these methods, K-sparse combinations of variables are tested exhaustively assuming that the optimal combination of explanatory variables is K-sparse. By collecting the results of exhaustively computing ES-K, various approximate methods for selecting sparse variables can be summarized as density of states. With this density of states, we can compare different methods for selecting sparse variables such as relaxation and sampling. Fo…
Sparse versions of principal component analysis (PCA) have imposed themselves as simple, yet powerful ways of selecting relevant features of high-dimensional data in an unsupervised manner. However, when several sparse principal components are computed, the interpretation of the selected variables is difficult since each axis has its own sparsity pattern and has to be interpreted separately. To overcome this drawback, we propose a Bayesian procedure called globally sparse probabilistic PCA (GSPPCA) that allows to obtain several sparse components with the same sparsity pattern. This allows the …
We propose Robust Lasso-Zero, an extension of the Lasso-Zero methodology, initially introduced for sparse linear models, to the sparse corruptions problem. We give theoretical guarantees on the sign recovery of the parameters for a slightly simplified version of the estimator, called Thresholded Justice Pursuit. The use of Robust Lasso-Zero is showcased for variable selection with missing values in the covariates. In addition to not requiring the specification of a model for the covariates, nor estimating their covariance matrix or the noise variance, the method has the great advantage of hand…
We study a seemingly unexpected and relatively less understood overfitting aspect of a fundamental tool in sparse linear modeling - best subset selection, which minimizes the residual sum of squares subject to a constraint on the number of nonzero coefficients. While the best subset selection procedure is often perceived as the "gold standard" in sparse learning when the signal to noise ratio (SNR) is high, its predictive performance deteriorates when the SNR is low. In particular, it is outperformed by continuous shrinkage methods, such as ridge regression and the Lasso. We investigate the be…
Interpretable machine learning has demonstrated impressive performance while preserving explainability. In particular, neural additive models (NAM) offer the interpretability to the black-box deep learning and achieve state-of-the-art accuracy among the large family of generalized additive models. In order to empower NAM with feature selection and improve the generalization, we propose the sparse neural additive models (SNAM) that employ the group sparsity regularization (e.g. Group LASSO), where each feature is learned by a sub-network whose trainable parameters are clustered as a group. We s…
Max Planck Institute for Molecular Genetics（马克斯·普朗克分子遗传学研究所）
We present a Bayesian method for feature selection in the presence of grouping information with sparsity on the between- and within group level. Instead of using a stochastic algorithm for parameter inference, we employ expectation propagation, which is a deterministic and fast algorithm. Available methods for feature selection in the presence of grouping information have a number of short-comings: on one hand, lasso methods, while being fast, underestimate the regression coefficients and do not make good use of the grouping information, and on the other hand, Bayesian approaches, while accura…
We present a new variable selection method based on model-based gradient boosting and randomly permuted variables. Model-based boosting is a tool to fit a statistical model while performing variable selection at the same time. A drawback of the fitting lies in the need of multiple model fits on slightly altered data (e.g. cross-validation or bootstrap) to find the optimal number of boosting iterations and prevent overfitting. In our proposed approach, we augment the data set with randomly permuted versions of the true variables, so called shadow variables, and stop the step-wise fitting as soo…
A Scalable Gradient-Based Optimization Framework for Sparse Minimum-Variance Portfolio Selection
School of Mathematics and Statistics（数学与统计学学院）；University of New South Wales（新南威尔士大学）；School of Mathematical and Physical Sciences（数学与物理科学学院）；University of Technology Sydney（技术大学悉尼分校）；Bayes Business School（贝叶斯商学院）；City St George’s, University of London（伦敦大学城圣乔治学院）；Faculty of Science and Engineering（科学与工程学院）；Macquarie University（麦考瑞大学）
Recently, sparsity-based algorithms are proposed for super-resolution spectrum estimation. However, to achieve adequately high resolution in real-world signal analysis, the dictionary atoms have to be close to each other in frequency, thereby resulting in a coherent design. The popular convex compressed sensing methods break down in presence of high coherence and large noise. We propose a new regularization approach to handle model collinearity and obtain parsimonious frequency selection simultaneously. It takes advantage of the pairing structure of sine and cosine atoms in the frequency dicti…
From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identification
AI Institute in Dynamic Systems, Department of Mechanical Engineering, University of Washington（动态系统人工智能研究所，机械工程系，华盛顿大学）；Department of Mechanical Engineering, University of Hawai‘i at Mānoa（机械工程系，夏威夷大学马诺阿分校）；Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista “Júlio de Mesquita Filho”（数学与计算系，科技学院，保罗斯州立大学“朱利奥·梅斯奎塔·弗洛雷斯”分校）
In this paper we propose a computationally efficient algorithm for on-line variable selection in multivariate regression problems involving high dimensional data streams. The algorithm recursively extracts all the latent factors of a partial least squares solution and selects the most important variables for each factor. This is achieved by means of only one sparse singular value decomposition which can be efficiently updated on-line and in an adaptive fashion. Simulation results based on artificial data streams demonstrate that the algorithm is able to select important variables in dynamic se…
Sparse polynomial chaos expansions (PCE) are an efficient and widely used surrogate modeling method in uncertainty quantification for engineering problems with computationally expensive models. To make use of the available information in the most efficient way, several approaches for so-called basis-adaptive sparse PCE have been proposed to determine the set of polynomial regressors ("basis") for PCE adaptively. The goal of this paper is to help practitioners identify the most suitable methods for constructing a surrogate PCE for their model. We describe three state-of-the-art basis-adaptive a…
Optimizing Sparse Generalized Singular Vectors for Feature Selection in Proximal Support Vector Machines with Application to Breast and Ovarian Cancer Detection
Tufts University（塔夫茨大学）；Colorado State University（科罗拉多州立大学）
This article considers the problem of multi-group classification in the setting where the number of variables $p$ is larger than the number of observations $n$. Several methods have been proposed in the literature that address this problem, however their variable selection performance is either unknown or suboptimal to the results known in the two-group case. In this work we provide sharp conditions for the consistent recovery of relevant variables in the multi-group case using the discriminant analysis proposal of Gaynanova et al., 2014. We achieve the rates of convergence that attain the opt…
We propose a method for recovering the structure of a sparse undirected graphical model when very few samples are available. The method decides about the presence or absence of bonds between pairs of variable by considering one pair at a time and using a closed form formula, analytically derived by calculating the posterior probability for every possible model explaining a two body system using Jeffreys prior. The approach does not rely on the optimisation of any cost functions and consequently is much faster than existing algorithms. Despite this time and computational advantage, numerical re…
DeepMind（深度思考）；Princeton University（普林斯顿大学）；University of Alberta（阿尔伯塔大学）
This paper provides a statistical analysis of high-dimensional batch Reinforcement Learning (RL) using sparse linear function approximation. When there is a large number of candidate features, our result sheds light on the fact that sparsity-aware methods can make batch RL more sample efficient. We first consider the off-policy policy evaluation problem. To evaluate a new target policy, we analyze a Lasso fitted Q-evaluation method and establish a finite-sample error bound that has no polynomial dependence on the ambient dimension. To reduce the Lasso bias, we further propose a post model-sele…
We consider the bridge linear regression modeling, which can produce a sparse or non-sparse model. A crucial point in the model building process is the selection of adjusted parameters including a regularization parameter and a tuning parameter in bridge regression models. The choice of the adjusted parameters can be viewed as a model selection and evaluation problem. We propose a model selection criterion for evaluating bridge regression models in terms of Bayesian approach. This selection criterion enables us to select the adjusted parameters objectively. We investigate the effectiveness of …
