Sparse linear regression, which entails finding a sparse solution to an underdetermined system of linear equations, can formally be expressed as an $l_0$-constrained least-squares problem. The Orthogonal Least-Squares (OLS) algorithm sequentially selects the features (i.e., columns of the coefficient matrix) to greedily find an approximate sparse solution. In this paper, a generalization of Orthogonal Least-Squares which relies on a recursive relation between the components of the optimal solution to select L features at each step and solve the resulting overdetermined system of equations is proposed. Simulation results demonstrate that the generalized OLS algorithm is computationally efficient and achieves performance superior to that of existing greedy algorithms broadly used in the literature.
We study a new class of codes for lossy compression with the squared-error distortion criterion, designed using the statistical framework of high-dimensional linear regression. Codewords are linear combinations of subsets of columns of a design matrix. Called a Sparse Superposition or Sparse Regression codebook, this structure is motivated by an analogous construction proposed recently by Barron and Joseph for communication over an AWGN channel. For i.i.d Gaussian sources and minimum-distance encoding, we show that such a code can attain the Shannon rate-distortion function with the optimal er…
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…
Machine learning and statistics typically focus on building models that capture the vast majority of the data, possibly ignoring a small subset of data as "noise" or "outliers." By contrast, here we consider the problem of jointly identifying a significant (but perhaps small) segment of a population in which there is a highly sparse linear regression fit, together with the coefficients for the linear fit. We contend that such tasks are of interest both because the models themselves may be able to achieve better predictions in such special cases, but also because they may aid our understanding …
Here we propose an algorithm, named generalized orthogonal components regression (GOCRE), to explore the relationship between a categorical outcome and a set of massive variables. A set of orthogonal components are sequentially constructed to account for the variation of the categorical outcome, and together build up a generalized linear model (GLM). This algorithm can be considered as an extension of the partial least squares (PLS) for GLMs, but overcomes several issues of existing extensions based on iteratively reweighted least squares (IRLS). First, existing extensions construct a differen…
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.
Significant attention has been given to minimizing a penalized least squares criterion for estimating sparse solutions to large linear systems of equations. The penalty is responsible for inducing sparsity and the natural choice is the so-called $l_0$ norm. In this paper we develop a Momentumized Iterative Shrinkage Thresholding (MIST) algorithm for minimizing the resulting non-convex criterion and prove its convergence to a local minimizer. Simulations on large data sets show superior performance of the proposed method to other methods.
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…
Renmin University of China（中国人民大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）；MinJiang University（闽江学院）
High-dimensional linear regression model is the most popular statistical model for high-dimensional data, but it is quite a challenging task to achieve a sparse set of regression coefficients. In this paper, we propose a simple heuristic algorithm to construct sparse high-dimensional linear regression models, which is adapted from the shortest solution-guided decimation algorithm and is referred to as ASSD. This algorithm constructs the support of regression coefficients under the guidance of the least-squares solution of the recursively decimated linear equations, and it applies an early-stop…
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 regress…
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…
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…
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.
The dramatic growth of big datasets presents a new challenge to data storage and analysis. Data reduction, or subsampling, that extracts useful information from datasets is a crucial step in big data analysis. We propose an orthogonal subsampling (OSS) approach for big data with a focus on linear regression models. The approach is inspired by the fact that an orthogonal array of two levels provides the best experimental design for linear regression models in the sense that it minimizes the average variance of the estimated parameters and provides the best predictions. The merits of OSS are thr…
We consider estimation in a sparse additive regression model with the design points on a regular lattice. We establish the minimax convergence rates over Sobolev classes and propose a Fourier-based rate-optimal estimator which is adaptive to the unknown sparsity and smoothness of the response function. The estimator is derived within Bayesian formalism but can be naturally viewed as a penalized maximum likelihood estimator with the complexity penalties on the number of nonzero univariate additive components of the response and on the numbers of the nonzero coefficients of their Fourer expansio…
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…
Penalized regression estimators are a popular tool for the analysis of sparse and high-dimensional data sets. However, penalized regression estimators defined using an unbounded loss function can be very sensitive to the presence of outlying observations, especially high leverage outliers. Moreover, it can be particularly challenging to detect outliers in high-dimensional data sets. Thus, robust estimators for sparse and high-dimensional linear regression models are in need. In this paper, we study the robust and asymptotic properties of MM-Bridge and adaptive MM-Bridge estimators: $\ell_q$-pe…
Volterra and polynomial regression models play a major role in nonlinear system identification and inference tasks. Exciting applications ranging from neuroscience to genome-wide association analysis build on these models with the additional requirement of parsimony. This requirement has high interpretative value, but unfortunately cannot be met by least-squares based or kernel regression methods. To this end, compressed sampling (CS) approaches, already successful in linear regression settings, can offer a viable alternative. The viability of CS for sparse Volterra and polynomial models is th…
Sharif University of Technology（谢里夫理工大学）；Carnegie Mellon University（卡内基梅隆大学）
We propose two nonlinear regression methods, named Adversarial Orthogonal Regression (AdOR) for additive noise models and Adversarial Orthogonal Structural Equation Model (AdOSE) for the general case of structural equation models. Both methods try to make the residual of regression independent from regressors while putting no assumption on noise distribution. In both methods, two adversarial networks are trained simultaneously where a regression network outputs predictions and a loss network that estimates mutual information (in AdOR) and KL-divergence (in AdOSE). These methods can be formulat…
We consider the Orthogonal Least-Squares (OLS) algorithm for the recovery of a $m$-dimensional $k$-sparse signal from a low number of noisy linear measurements. The Exact Recovery Condition (ERC) in bounded noisy scenario is established for OLS under certain condition on nonzero elements of the signal. The new result also improves the existing guarantees for Orthogonal Matching Pursuit (OMP) algorithm. In addition, This framework is employed to provide probabilistic guarantees for the case that the coefficient matrix is drawn at random according to Gaussian or Bernoulli distribution where we e…
In the multiple linear regression setting, we propose a general framework, termed weighted orthogonal components regression (WOCR), which encompasses many known methods as special cases, including ridge regression and principal components regression. WOCR makes use of the monotonicity inherent in orthogonal components to parameterize the weight function. The formulation allows for efficient determination of tuning parameters and hence is computationally advantageous. Moreover, WOCR offers insights for deriving new better variants. Specifically, we advocate weighting components based on their c…
In this paper, we address the challenging problem of selecting tuning parameters for high-dimensional sparse regression. We propose a simple and computationally efficient method, called path thresholding (PaTh), that transforms any tuning parameter-dependent sparse regression algorithm into an asymptotically tuning-free sparse regression algorithm. More specifically, we prove that, as the problem size becomes large (in the number of variables and in the number of observations), PaTh performs accurate sparse regression, under appropriate conditions, without specifying a tuning parameter. In fin…
Hashing is one of the most popular methods for similarity search because of its speed and efficiency. Dense binary hashing is prevalent in the literature. Recently, insect olfaction was shown to be structurally and functionally analogous to sparse hashing [6]. Here, we prove that this biological mechanism is the solution to a well-posed optimization problem. Furthermore, we show that orthogonality increases the accuracy of sparse hashing. Next, we present a novel method, Procrustean Orthogonal Sparse Hashing (POSH), that unifies these findings, learning an orthogonal transform from training da…
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 …
University of Reading（雷丁大学）；University of Southampton（南安普顿大学）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；CSIRO Mathematics and Information Sciences（联邦科学与工业研究组织数学与信息科学研究所）；Charles Sturt University（查尔斯特大学）
A l1-norm penalized orthogonal forward regression (l1-POFR) algorithm is proposed based on the concept of leaveone- out mean square error (LOOMSE). Firstly, a new l1-norm penalized cost function is defined in the constructed orthogonal space, and each orthogonal basis is associated with an individually tunable regularization parameter. Secondly, due to orthogonal computation, the LOOMSE can be analytically computed without actually splitting the data set, and moreover a closed form of the optimal regularization parameter in terms of minimal LOOMSE is derived. Thirdly, a lower bound for regular…
For the problem of multi-class linear classification and feature selection, we propose approximate message passing approaches to sparse multinomial logistic regression (MLR). First, we propose two algorithms based on the Hybrid Generalized Approximate Message Passing (HyGAMP) framework: one finds the maximum a posteriori (MAP) linear classifier and the other finds an approximation of the test-error-rate minimizing linear classifier. Then we design computationally simplified variants of these two algorithms. Next, we detail methods to tune the hyperparameters of their assumed statistical models…
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…
We consider the high-dimensional sparse linear regression problem of accurately estimating a sparse vector using a small number of linear measurements that are contaminated by noise. It is well known that the standard cadre of computationally tractable sparse regression algorithms---such as the Lasso, Orthogonal Matching Pursuit (OMP), and their extensions---perform poorly when the measurement matrix contains highly correlated columns. To address this shortcoming, we develop a simple greedy algorithm, called SWAP, that iteratively swaps variables until convergence. SWAP is surprisingly effecti…
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…
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.
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 …
We devise a one-shot approach to distributed sparse regression in the high-dimensional setting. The key idea is to average "debiased" or "desparsified" lasso estimators. We show the approach converges at the same rate as the lasso as long as the dataset is not split across too many machines. We also extend the approach to generalized linear models.
Kernel regression is a popular non-parametric fitting technique. It aims at learning a function which estimates the targets for test inputs as precise as possible. Generally, the function value for a test input is estimated by a weighted average of the surrounding training examples. The weights are typically computed by a distance-based kernel function and they strongly depend on the distances between examples. In this paper, we first review the latest developments of sparse metric learning and kernel regression. Then a novel kernel regression method involving sparse metric learning, which is …
The Graduate University for Advanced Studies, SOKENDAI（综合研究大学院大学）
Robust and sparse estimation of linear regression coefficients is investigated. The situation addressed by the present paper is that covariates and noises are sampled from heavy-tailed distributions, and the covariates and noises are contaminated by malicious outliers. Our estimator can be computed efficiently. Further, the error bound of the estimator is nearly optimal.
Effective features can improve the performance of a model, which can thus help us understand the characteristics and underlying structure of complex data. Previous feature selection methods usually cannot keep more local structure information. To address the defects previously mentioned, we propose a novel supervised orthogonal least square regression model with feature weighting for feature selection. The optimization problem of the objection function can be solved by employing generalized power iteration (GPI) and augmented Lagrangian multiplier (ALM) methods. Experimental results show that …
Carnegie Mellon University（卡内基梅隆大学）；University of Chicago（芝加哥大学）
We study the problem of multivariate regression where the data are naturally grouped, and a regression matrix is to be estimated for each group. We propose an approach in which a dictionary of low rank parameter matrices is estimated across groups, and a sparse linear combination of the dictionary elements is estimated to form a model within each group. We refer to the method as conditional sparse coding since it is a coding procedure for the response vectors Y conditioned on the covariate vectors X. This approach captures the shared information across the groups while adapting to the structur…
In presence of sparse noise we propose kernel regression for predicting output vectors which are smooth over a given graph. Sparse noise models the training outputs being corrupted either with missing samples or large perturbations. The presence of sparse noise is handled using appropriate use of $\ell_1$-norm along-with use of $\ell_2$-norm in a convex cost function. For optimization of the cost function, we propose an iteratively reweighted least-squares (IRLS) approach that is suitable for kernel substitution or kernel trick due to availability of a closed form solution. Simulations using r…
Weill Cornell Medicine（威尔康奈尔医学院）；University of Connecticut（康涅狄格大学）；Michigan State Universtiy（密歇根州立大学）
We propose a sparse and low-rank tensor regression model to relate a univariate outcome to a feature tensor, in which each unit-rank tensor from the CP decomposition of the coefficient tensor is assumed to be sparse. This structure is both parsimonious and highly interpretable, as it implies that the outcome is related to the features through a few distinct pathways, each of which may only involve subsets of feature dimensions. We take a divide-and-conquer strategy to simplify the task into a set of sparse unit-rank tensor regression problems. To make the computation efficient and scalable, fo…
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）
Linear regression models contaminated by Gaussian noise (inlier) and possibly unbounded sparse outliers are common in many signal processing applications. Sparse recovery inspired robust regression (SRIRR) techniques are shown to deliver high quality estimation performance in such regression models. Unfortunately, most SRIRR techniques assume \textit{a priori} knowledge of noise statistics like inlier noise variance or outlier statistics like number of outliers. Both inlier and outlier noise statistics are rarely known \textit{a priori} and this limits the efficient operation of many SRIRR alg…
We propose a penalized orthogonal-components regression (POCRE) for large p small n data. Orthogonal components are sequentially constructed to maximize, upon standardization, their correlation to the response residuals. A new penalization framework, implemented via empirical Bayes thresholding, is presented to effectively identify sparse predictors of each component. POCRE is computationally efficient owing to its sequential construction of leading sparse principal components. In addition, such construction offers other properties such as grouping highly correlated predictors and allowing for…
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…
Information Security Lab, Institute of Big Data Technologies,\ Graduate School, Peking University
Sparse Subspace Clustering (SSC) is one of the most popular methods for clustering data points into their underlying subspaces. However, SSC may suffer from heavy computational burden. Orthogonal Matching Pursuit applied on SSC accelerates the computation but the trade-off is the loss of clustering accuracy. In this paper, we propose a noise-robust algorithm, Restricted Connection Orthogonal Matching Pursuit for Sparse Subspace Clustering (RCOMP-SSC), to improve the clustering accuracy and maintain the low computational time by restricting the number of connections of each data point during th…
In this work we propose to fit a sparse logistic regression model by a weakly convex regularized nonconvex optimization problem. The idea is based on the finding that a weakly convex function as an approximation of the $\ell_0$ pseudo norm is able to better induce sparsity than the commonly used $\ell_1$ norm. For a class of weakly convex sparsity inducing functions, we prove the nonconvexity of the corresponding sparse logistic regression problem, and study its local optimality conditions and the choice of the regularization parameter to exclude trivial solutions. Despite the nonconvexity, a …
We provide a novel -- and to the best of our knowledge, the first -- algorithm for high dimensional sparse regression with constant fraction of corruptions in explanatory and/or response variables. Our algorithm recovers the true sparse parameters with sub-linear sample complexity, in the presence of a constant fraction of arbitrary corruptions. Our main contribution is a robust variant of Iterative Hard Thresholding. Using this, we provide accurate estimators: when the covariance matrix in sparse regression is identity, our error guarantee is near information-theoretically optimal. We then de…
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.
The main contribution of the paper is a new approach to subspace clustering that is significantly more computationally efficient and scalable than existing state-of-the-art methods. The central idea is to modify the regression technique in sparse subspace clustering (SSC) by replacing the $\ell_1$ minimization with a generalization called Ordered Weighted $\ell_1$ (OWL) minimization which performs simultaneous regression and clustering of correlated variables. Using random geometric graph theory, we prove that OWL regression selects more points within each subspace, resulting in better cluster…
This paper proposes a new algorithm for multiple sparse regression in high dimensions, where the task is to estimate the support and values of several (typically related) sparse vectors from a few noisy linear measurements. Our algorithm is a "forward-backward" greedy procedure that -- uniquely -- operates on two distinct classes of objects. In particular, we organize our target sparse vectors as a matrix; our algorithm involves iterative addition and removal of both (a) individual elements, and (b) entire rows (corresponding to shared features), of the matrix. Analytically, we establish that …
Division of Biostatistics, University of California, Berkeley（加州大学伯克利分校生物统计学系）
Tensor regression models, such as CP regression and Tucker regression, have many successful applications in neuroimaging analysis where the covariates are of ultrahigh dimensionality and possess complex spatial structures. The high-dimensional covariate arrays, also known as tensors, can be approximated by low-rank structures and fit into the generalized linear models. The resulting tensor regression achieves a significant reduction in dimensionality while remaining efficient in estimation and prediction. Brain functional connectivity is an essential measure of brain activity and has shown sig…
University of Sheffield（谢菲尔德大学）；University of Oxford（牛津大学）
This paper introduces a new sparse spatio-temporal structured Gaussian process regression framework for online and offline Bayesian inference. This is the first framework that gives a time-evolving representation of the interdependencies between the components of the sparse signal of interest. A hierarchical Gaussian process describes such structure and the interdependencies are represented via the covariance matrices of the prior distributions. The inference is based on the expectation propagation method and the theoretical derivation of the posterior distribution is provided in the paper. Th…
We describe a new library named picasso, which implements a unified framework of pathwise coordinate optimization for a variety of sparse learning problems (e.g., sparse linear regression, sparse logistic regression, sparse Poisson regression and scaled sparse linear regression) combined with efficient active set selection strategies. Besides, the library allows users to choose different sparsity-inducing regularizers, including the convex $\ell_1$, nonconvex MCP and SCAD regularizers. The library is coded in C++ and has user-friendly R and Python wrappers. Numerical experiments demonstrate th…
Operations Research Center, Massachusetts Institute of Technology（麻省理工学院运筹学中心）
We present a novel method for exact hierarchical sparse polynomial regression. Our regressor is that degree $r$ polynomial which depends on at most $k$ inputs, counting at most $\ell$ monomial terms, which minimizes the sum of the squares of its prediction errors. The previous hierarchical sparse specification aligns well with modern big data settings where many inputs are not relevant for prediction purposes and the functional complexity of the regressor needs to be controlled as to avoid overfitting. We present a two-step approach to this hierarchical sparse regression problem. First, we dis…
