Related papers: A Computational Framework for Multivariate Convex …
In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric…
We consider a Bayesian framework for estimating a high-dimensional sparse precision matrix, in which adaptive shrinkage and sparsity are induced by a mixture of Laplace priors. Besides discussing our formulation from the Bayesian…
Minimax lower bounds are pessimistic in nature: for any given estimator, minimax lower bounds yield the existence of a worst-case target vector $\beta^*_{worst}$ for which the prediction error of the given estimator is bounded from below.…
In constrained stochastic optimization, one naturally expects that imposing a stricter feasible set does not increase the statistical risk of an estimator defined by projection onto that set. In this paper, we show that this intuition can…
Beside the minimization of the prediction error, two of the most desirable properties of a regression scheme are stability and interpretability. Driven by these principles, we propose continuous-domain formulations for one-dimensional…
For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models…
In this paper, we analyse the recovery properties of nonconvex regularized $M$-estimators, under the assumption that the true parameter is of soft sparsity. In the statistical aspect, we establish the recovery bound for any stationary point…
We consider the problem of estimating an unknown $n_1 \times n_2$ matrix $\mathbf{\theta^*}$ from noisy observations under the constraint that $\mathbf{\theta}^*$ is nondecreasing in both rows and columns. We consider the least squares…
We develop and analyze a set of new sequential simulation-optimization algorithms for large-scale multi-dimensional discrete optimization via simulation problems with a convexity structure. The "large-scale" notion refers to that the…
Sparse additive modeling is a class of effective methods for performing high-dimensional nonparametric regression. In this work we show how shape constraints such as convexity/concavity and their extensions, can be integrated into additive…
We propose a unified framework for estimating low-rank matrices through nonconvex optimization based on gradient descent algorithm. Our framework is quite general and can be applied to both noisy and noiseless observations. In the general…
We analyse the convergence of an approximate, fully inexact, ADMM algorithm under additive, deterministic and probabilistic error models. We consider the generalized ADMM scheme that is derived from generalized Lagrangian penalty with…
We introduce a novel scheme for choosing the regularization parameter in high-dimensional linear regression with Lasso. This scheme, inspired by Lepski's method for bandwidth selection in non-parametric regression, is equipped with both…
This paper introduces a smoothed proximal Lagrangian method for minimizing a nonconvex smooth function over a convex domain with additional explicit convex nonlinear constraints. Two key features are 1) the proposed method is single-looped,…
Consider the problem of minimizing the expected value of a (possibly nonconvex) cost function parameterized by a random (vector) variable, when the expectation cannot be computed accurately (e.g., because the statistics of the random…
We introduce the localized Lasso, which is suited for learning models that are both interpretable and have a high predictive power in problems with high dimensionality $d$ and small sample size $n$. More specifically, we consider a function…
Binary tomography is concerned with the recovery of binary images from a few of their projections (i.e., sums of the pixel values along various directions). To reconstruct an image from noisy projection data, one can pose it as a…
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…
We prove that the convex least squares estimator (LSE) attains a $n^{-1/2}$ pointwise rate of convergence in any region where the truth is linear. In addition, the asymptotic distribution can be characterized by a modified invelope process.…
We show both adaptive and non-adaptive minimax rates of convergence for a family of weighted Laplacian-Eigenmap based nonparametric regression methods, when the true regression function belongs to a Sobolev space and the sampling density is…