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We study a family of sparse estimators defined as minimizers of some empirical Lipschitz loss function -- which include the hinge loss, the logistic loss and the quantile regression loss -- with a convex, sparse or group-sparse…
We consider the problem of learning a coefficient vector x_0\in R^N from noisy linear observation y=Ax_0+w \in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x'. In…
We study the total least squares (TLS) problem that generalizes least squares regression by allowing measurement errors in both dependent and independent variables. TLS is widely used in applied fields including computer vision, system…
The effects of several nonlinear regularization techniques are discussed in the framework of 3D seismic tomography. Traditional, linear, $\ell_2$ penalties are compared to so-called sparsity promoting $\ell_1$ and $\ell_0$ penalties, and a…
A recent technique of randomized smoothing has shown that the worst-case (adversarial) $\ell_2$-robustness can be transformed into the average-case Gaussian-robustness by "smoothing" a classifier, i.e., by considering the averaged…
The main result of this article is that we obtain an elementwise error bound for the Fused Lasso estimator for any general convex loss function $\rho$. We then focus on the special cases when either $\rho$ is the square loss function (for…
We consider an $\ell_2$-regularized non-convex optimization problem for recovering signals from their noisy phaseless observations. We design and study the performance of a message passing algorithm that aims to solve this optimization…
We propose two novel approaches to the recovery of an (approximately) sparse signal from noisy linear measurements in the case that the signal is a priori known to be non-negative and obey given linear equality constraints, such as simplex…
Most detection algorithms in spatial modulation (SM) are formulated as linear regression via the regularized least-squares (RLS) method. In this method, the transmit signal is estimated by minimizing the residual sum of squares penalized…
Regularized linear regression under the $\ell_1$ penalty, such as the Lasso, has been shown to be effective in variable selection and sparse modeling. The sampling distribution of an $\ell_1$-penalized estimator $\hat{\beta}$ is hard to…
Suppose that $\mathbf{y}=\lvert A\mathbf{x_0}\rvert+\eta$ where $\mathbf{x_0} \in \mathbb{R}^d$ is the target signal and $\eta\in \mathbb{R}^m$ is a noise vector. The aim of phase retrieval is to estimate $\mathbf{x_0}$ from $\mathbf{y}$. A…
We consider the compressed sensing problem, where the object $x_0 \in \bR^N$ is to be recovered from incomplete measurements $y = Ax_0 + z$; here the sensing matrix $A$ is an $n \times N$ random matrix with iid Gaussian entries and $n < N$.…
This paper studies well-posedness and parameter sensitivity of the Square Root LASSO (SR-LASSO), an optimization model for recovering sparse solutions to linear inverse problems in finite dimension. An advantage of the SR-LASSO (e.g., over…
We consider the problem of recovering a $k$-sparse signal ${\mbox{$\beta$}}_0\in\mathbb{R}^p$ from noisy observations $\bf y={\bf X}\mbox{$\beta$}_0+{\bf w}\in\mathbb{R}^n$. One of the most popular approaches is the $l_1$-regularized least…
We provide a geometric approach to the lasso. We study the tangency of the level sets of the least square objective function with the polyhedral boundary sets $B(t)$ of the parameters in $\mathbb R^p$ with the $\ell_1$ norm equal to $t$.…
We study a class of fused lasso problems where the estimated parameters in a sequence are regressed toward their respective observed values (fidelity loss), with $\ell_1$ norm penalty (regularization loss) on the differences between…
The Chebyshev or $\ell_{\infty}$ estimator is an unconventional alternative to the ordinary least squares in solving linear regressions. It is defined as the minimizer of the $\ell_{\infty}$ objective function \begin{align*}…
We study the classical problem of predicting an outcome variable, $Y$, using a linear combination of a $d$-dimensional covariate vector, $\mathbf{X}$. We are interested in linear predictors whose coefficients solve: % \begin{align*}…
We study the effect of norm based regularization on the size of coresets for regression problems. Specifically, given a matrix $ \mathbf{A} \in {\mathbb{R}}^{n \times d}$ with $n\gg d$ and a vector $\mathbf{b} \in \mathbb{R} ^ n $ and…
In this paper, we study the issue of estimating a structured signal $x_0 \in \mathbb{R}^n$ from non-linear and noisy Gaussian observations. Supposing that $x_0$ is contained in a certain convex subset $K \subset \mathbb{R}^n$, we prove that…