Proposes FARM model combining latent factor and sparse regression.
problem Testing adequacy of latent factor and sparse regression models.
method Factor Augmented sparse linear Regression Model (FARM) with FabTest and ANOVA type tests.
result Model robustness and effectiveness validated through experiments.
Efficiently estimates sparse linear regression with heavy-tailed data and outliers.
problem Sparse estimation of linear regression coefficients with heavy-tailed covariates and noises, including outliers.
method Efficient computation of robust estimator with nearly optimal error bound.
result Nearly optimal error bound for robust sparse estimation.
Study finds exact limits for sparse regression with fewer observations than usual.
problem Understanding sparse linear regression with sublinear sparsity.
method Adaptive interpolation method and modified AMP algorithm.
result Exact asymptotic expressions for mutual information and MMSE in sublinear sparsity.
Optimal sketching bounds for sparse linear regression under various loss functions are established.
problem Sparse linear regression under different loss functions.
method Distribution over oblivious sketches for sparse ℓ2 norm regression and hinge-like loss functions. result Optimal sketching bounds with O(klog(d/k)/ε2) rows for sparse ℓ2 norm regression and O(μ2klog(μnd/ε)/ε2) rows for hinge-like loss functions. Efficiently estimates sparse linear regression with heavy-tailed and outlier-contaminated data.
problem Estimating sparse linear regression coefficients with heavy-tailed and outlier-contaminated data.
method Efficient computation of estimators with sharp error bounds.
result Sharp error bounds for efficient estimators.
The generalized linear model (GLM) plays a key role in regression analyses. In high-dimensional data, the sparse GLM has been used but it is not robust against outliers. Recently, the robust methods have been proposed for the specific example of the sparse GLM. Among them, we focus on the robust and sparse linear regre…
Study improves robustness and sparsity in linear regression with adversarial outliers and heavy-tailed noise.
problem Outliers and heavy-tailed noise in linear regression coefficients.
method Sharp concentration inequalities and generic chaining.
result Sharper error bounds under weaker assumptions.
New algorithm reduces sample complexity for sparse linear regression.
problem Sparse linear regression with correlated covariates and approximate dependencies.
method Polynomial-time algorithm that adapts the Lasso to tolerate approximate dependencies.
result Achieves near-optimal sample complexity for constant sparsity and ill-conditioned covariates.
Picasso is a new library for sparse learning problems in R and Python.
problem Sparse learning problems in high-dimensional data analysis.
method Unified framework of pathwise coordinate optimization with efficient active set selection strategies.
result picasso can efficiently handle large-scale problems.
OOMP selects features online for sparse linear regression.
problem Feature selection in high-dimensional sparse linear models.
method Online algorithm that alternates between feature selection and coefficient estimation.
result Theoretical guarantees and computational complexity analysis of OOMP.
Study shows inefficiency of sparse linear regression learning with fewer than Ω(k^2) samples.
problem Efficiency of sparse linear regression learning with minimal samples.
method Reduction to sparse PCA problems and lower bounds.
result Efficient algorithms for sparse linear regression require at least Ω(k^2) samples.
New insights into statistical and computational limits for mixed sparse linear regression.
problem Recovering two sparse signals from noisy linear measurements.
method Analysis of low-degree polynomials and a simple thresholding algorithm.
result Identification of a smooth information-computation tradeoff and order-optimality of the thresholding algorithm.
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 s…
PROBE algorithm efficiently solves sparse high-dimensional linear regression.
problem Sparse high-dimensional linear regression models with complex parameter spaces.
method Partitioned empirical Bayes ECM algorithm for computationally efficient MAP estimation.
result PROBE algorithm provides robust and efficient coordinate-wise optimization.
OKRidge solves sparse ridge regression problems for nonlinear systems.
problem Identifying sparse governing equations for nonlinear dynamical systems.
method OKRidge algorithm using saddle point formulation and ADMM-based approach with efficient proximal operators.
result OKRidge achieves provable optimality with significantly faster run times than Gurobi.
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 Matchi…
This paper uses quantum computing to solve sparse linear regression problems efficiently.
problem Sparse linear regression to identify important features from a large set of variables.
method Formulates the ℓ0 optimization problem as a QUBO problem and solves it using the D-Wave adiabatic quantum computer. result The QUBO solution matches the optimal solution for a wide range of sparsity penalty values across datasets.
Paper solves NP-hard sparse mixed linear regression problem with provable guarantees.
problem Sparse mixed linear regression on unlabeled data.
method Invex relaxation for intractable problem with theoretical guarantees.
result Exact recovery of data labels and close approximation of regression parameters.
Efficiently finds sparse solutions to max-plus equations for convex regression.
problem Finding sparse solutions to max-plus equations for convex multivariate regression.
method Polynomial-time algorithm for sparse approximate solutions.
result Optimal piecewise-linear fitting with minimum number of regions.
Study improves error bounds for sparse regression with heavy-tailed covariates.
problem Estimating sparse coefficients in linear regression with heavy-tailed covariates.
method Employed an ℓ1-penalized Huber regression method. result Error bound identical to Gaussian case for L-subexponential covariates. Paper proposes an online sparse linear regression method for streaming data.
problem Sparse regression for variable selection and prediction accuracy.
method Online sparse linear regression framework with memory efficiency and relaxed assumptions.
result The ℓ2-norm statistical error of the estimator diminishes to zero with optimal order. The paper analyzes sparse high-dimensional linear regression with random design and unknown error variance, providing adaptiveness and concentration rates.
problem Sparse high-dimensional linear regression with random design and unknown error variance.
method Analysis of posterior concentration rates, employing techniques to address model misspecification.
result Adaptiveness and concentration rates of the posterior for sparse high-dimensional linear regression.
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-…
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 fractio…
Flexible Bayesian approach for generalized linear models, especially for sparse logistic regression.
problem Sparse logistic regression challenges in machine learning.
method Empirical Bayes approach with mean-field variational inference, tuning-free and scalable.
result Superior predictive performance in sparse logistic regression compared to existing methods.
New research shows how preconditioning can solve sparse linear regression problems efficiently.
problem Efficiently solving sparse linear regression problems without restrictive conditions.
method Preconditioned Lasso approach to solve sparse linear regression problems.
result Preconditioning can solve a large class of sparse linear regression problems nearly optimally.
Sparse linear regression is hard to solve efficiently, even with k-sparse solutions.
problem Sparse linear regression problem with k-sparse solutions.
method Fine-grained complexity and hardness assumptions.
result No better-than-brute-force algorithms exist for sparse linear regression.
High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). O…
New model leads to optimal test loss in sparse linear regression.
problem Sparse linear regression with low test loss despite interpolating training data.
method Developed a new parametrization of the model that combines benefits of ℓ1 and ℓ2 norms.
result Training via gradient descent leads to an interpolator with near-optimal test loss.
c-lasso is a Python tool for robust and sparse regression with linear constraints.
problem Sparse and robust linear regression with linear constraints.
method Estimates coefficients and scale under linear constraints using perspective M-estimators.
result Provides estimators for various loss functions with linear constraints.
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 s…
This paper studies robust regression in the settings of Huber's ε-contamination models. We consider estimators that are maximizers of multivariate regression depth functions. These estimators are shown to achieve minimax rates in the settings of ε-contamination models for various regression problems including nonpa…
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 …
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 iterat…
Folded concave penalization methods have been shown to enjoy the strong oracle property for high-dimensional sparse estimation. However, a folded concave penalization problem usually has multiple local solutions and the oracle property is established only for one of the unknown local solutions. A challenging fundamenta…
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 a…
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…
Sparse linear regression, which entails finding a sparse solution to an underdetermined system of linear equations, can formally be expressed as an l0-constrained least-squares problem. The Orthogonal Least-Squares (OLS) algorithm sequentially selects the features (i.e., columns of the coefficient matrix) to greedil…
The paper analyzes methods for sparse Bayesian regression in nonlinear system identification.
problem Learning sparse models in Bayesian regression with nonlinear applications.
method Two classes of methods: regularization and thresholding based, built on automatic relevance determination (ARD).
result Analytical demonstration of favorable performance with sparse solutions in linear problems.
Recent research has studied the role of sparsity in high dimensional regression and signal reconstruction, establishing theoretical limits for recovering sparse models from sparse data. This line of work shows that ℓ1-regularized least squares regression can accurately estimate a sparse linear model from n nois…
Lower bounds show OLS outperforms basis pursuit in overparameterized linear regression.
problem Excess risk of sparse interpolating procedures in overparameterized linear regression.
method Proved lower bounds on excess risk for OLS and basis pursuit.
result Excess risk of basis pursuit can converge at an exponentially slower rate than OLS.
Study on recovering supports of multiple sparse vectors from mixed linear measurements.
problem Recovering supports of multiple sparse vectors from a mixture of linear measurements.
method Developed algorithms to identify the support of all component vectors using polynomial and quasi-polynomial number of measurements.
result Polynomial and quasi-polynomial number of measurements sufficient for recovering the supports of all component vectors.
Efficiently performs robust and sparse kernel regression.
problem Robust and sparse kernel regression.
method Sign gradient descent and early stopping.
result Sign gradient descent achieves robust and sparse kernel regression efficiently.
The Lasso is suboptimal in sparse linear regression due to design matrix constraints.
problem The suboptimality of the Lasso estimator in sparse linear regression.
method Characterization of optimal estimation rates and analysis of the Lasso estimator.
result The Lasso is provably minimax rate-suboptimal when the minimum singular value is small.
SLR tackles sparse linear regression problems, showing hardness for efficient algorithms.
problem Sparse linear regression with noisy data and k-sparse solutions.
method Reduction from lattice problems to SLR instances, showing hardness.
result Hardness of SLR instances, even for isotropic Gaussian design matrices.
There has been a growing interest in using non-parametric regression methods like Gaussian Process (GP) regression for system identification. GP regression does traditionally have three important downsides: (1) it is computationally intensive, (2) it cannot efficiently implement newly obtained measurements online, and …
A new image interpolation model using sparse representation and nonlocal linear regression.
problem Image interpolation without blurring and noise.
method Sparse representation, nonlocal self-similarity, nonlocal linear regression, adaptive sub-dictionary learning, weighted encoding.
result Our method outperforms state-of-the-art methods in quantitative measures and visual quality.
Optimal multitask learning method for sparse heterogeneous datasets.
problem Efficiently learning from multiple related datasets with sparse task-specific differences.
method MOLAR estimator, combining weighted median and shrinkage.
result Improves estimation error dependence on data dimension compared to task-wise least squares.