Related papers: Nearly optimal minimax estimator for high-dimensio…
Robust statistical estimators offer resilience against outliers but are often computationally challenging, particularly in high-dimensional sparse settings. Modern optimization techniques are utilized for robust sparse association…
We prove an L2 recovery bound for a family of sparse estimators defined as minimizers of some empirical loss functions -- which include hinge loss and logistic loss. More precisely, we achieve an upper-bound for coefficients estimation…
We consider the problem of designing minimax estimators for estimating the parameters of a probability distribution. Unlike classical approaches such as the MLE and minimum distance estimators, we consider an algorithmic approach for…
When computing bounds, spatial branch-and-bound algorithms often linearly outer approximate convex relaxations for non-convex expressions in order to capitalize on the efficiency and robustness of linear programming solvers. Considering…
This paper considers solving the unconstrained $\ell_q$-norm ($0\leq q<1$) regularized least squares ($\ell_q$-LS) problem for recovering sparse signals in compressive sensing. We propose two highly efficient first-order algorithms via…
Although the standard formulations of prediction problems involve fully-observed and noiseless data drawn in an i.i.d. manner, many applications involve noisy and/or missing data, possibly involving dependence, as well. We study these…
We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = \langle X,w^* \rangle + \epsilon$ (with $X \in \mathbb{R}^d$ and $\epsilon$ independent), in…
In this paper, we discuss the statistical properties of the $\ell_q$ optimization methods $(0<q\leq 1)$, including the $\ell_q$ minimization method and the $\ell_q$ regularization method, for estimating a sparse parameter from noisy…
This paper proposes a theory for $\ell_1$-norm penalized high-dimensional $M$-estimators, with nonconvex risk and unrestricted domain. Under high-level conditions, the estimators are shown to attain the rate of convergence…
Estimation of linear functionals from observed data is an important task in many subjects. Juditsky & Nemirovski [The Annals of Statistics 37.5A (2009): 2278-2300] propose a framework for non-parametric estimation of linear functionals in a…
We propose dimension reduction methods for sparse, high-dimensional multivariate response regression models. Both the number of responses and that of the predictors may exceed the sample size. Sometimes viewed as complementary, predictor…
In high-dimensional linear regression, the goal pursued here is to estimate an unknown regression function using linear combinations of a suitable set of covariates. One of the key assumptions for the success of any statistical procedure in…
We develop a constructive approach to estimating sparse, high-dimensional linear regression models. The approach is a computational algorithm motivated from the KKT conditions for the $\ell_0$-penalized least squares solutions. It generates…
The Lasso is an attractive technique for regularization and variable selection for high-dimensional data, where the number of predictor variables $p_n$ is potentially much larger than the number of samples $n$. However, it was recently…
We consider parameter estimation under sparse linear regression -- an extensively studied problem in high-dimensional statistics and compressed sensing. While the minimax framework has been one of the most fundamental approaches for…
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…
We consider a class of $\ell_0$-regularized linear-quadratic (LQ) optimal control problems. This class of problems is obtained by augmenting a penalizing sparsity measure to the cost objective of the standard linear-quadratic regulator…
In this paper, a multi-parameterized proximal point algorithm combining with a relaxation step is developed for solving convex minimization problem subject to linear constraints. We show its global convergence and sublinear convergence rate…
The $\ell_0$-constrained empirical risk minimization ($\ell_0$-ERM) is a promising tool for high-dimensional statistical estimation. The existing analysis of $\ell_0$-ERM estimator is mostly on parameter estimation and support recovery…
In this chapter, we discuss recent work on learning sparse approximations to high-dimensional functions on data, where the target functions may be scalar-, vector- or even Hilbert space-valued. Our main objective is to study how the…