Related papers: On tight bounds for the Lasso

We consider the problem of Bayesian optimization of a one-dimensional Brownian motion in which the $T$ adaptively chosen observations are corrupted by Gaussian noise. We show that as the smallest possible expected cumulative regret and the…

In high dimensional settings, sparse structures are crucial for efficiency, both in term of memory, computation and performance. It is customary to consider $\ell_1$ penalty to enforce sparsity in such scenarios. Sparsity enforcing methods,…

We show that two polynomial time methods, a Lasso estimator with adaptively chosen tuning parameter and a Slope estimator, adaptively achieve the exact minimax prediction and $\ell_2$ estimation rate $(s/n)\log (p/s)$ in high-dimensional…

We consider the problems of confidence estimation and hypothesis testing on a parameter of signal observed in Gaussian white noise. For these problems we point out lower bounds of asymptotic efficiency in the zone of moderate deviation…

We consider the problem of detecting a small subset of defective items from a large set via non-adaptive "random pooling" group tests. We consider both the case when the measurements are noiseless, and the case when the measurements are…

The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified…

Volatility forecasts are key inputs in financial analysis. While lasso based forecasts have shown to perform well in many applications, their use to obtain volatility forecasts has not yet received much attention in the literature. Lasso…

We consider a linear regression model, with the parameter of interest a specified linear combination of the regression parameter vector. We suppose that, as a first step, a data-based model selection (e.g. by preliminary hypothesis tests or…

This paper presents uniform-in-time finite-sample bounds for regularized linear regression with vector-valued outputs and conditionally zero-mean subgaussian noise. By revisiting classical self-normalized martingale arguments, we obtain…

We study the fundamental problem of learning the parameters of a high-dimensional Gaussian in the presence of noise -- where an $\varepsilon$-fraction of our samples were chosen by an adversary. We give robust estimators that achieve…

This paper studies the statistical properties of the group Lasso estimator for high dimensional sparse quantile regression models where the number of explanatory variables (or the number of groups of explanatory variables) is possibly much…

The $\ell_1$-penalized method, or the Lasso, has emerged as an important tool for the analysis of large data sets. Many important results have been obtained for the Lasso in linear regression which have led to a deeper understanding of…

In this paper, we introduce a new probability distribution, the Lasso distribution. We derive several fundamental properties of the distribution, including closed-form expressions for its moments and moment-generating function.…

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…

The high-dimensional linear model $y = X \beta^0 + \epsilon$ is considered and the focus is put on the problem of recovering the support $S^0$ of the sparse vector $\beta^0.$ We introduce Lasso-Zero, a new $\ell_1$-based estimator whose…

The estimation of parameters in a linear model is considered under the hypothesis that the noise, with finite second order statistics, can be represented in a given deterministic basis by random coefficients. An extended underdetermined…

Variance estimation in the linear model when $p > n$ is a difficult problem. Standard least squares estimation techniques do not apply. Several variance estimators have been proposed in the literature, all with accompanying asymptotic…

Model selection in penalized regression critically depends on an accurate assessment of model complexity, commonly quantified through the effective degrees of freedom. While the Lasso admits a simple and unbiased characterization, given by…

A lower bound on the minimum mean-squared error (MSE) in a Bayesian estimation problem is proposed in this paper. This bound utilizes a well-known connection to the deterministic estimation setting. Using the prior distribution, the bias…

This paper describes a flexible framework for generalized low-rank tensor estimation problems that includes many important instances arising from applications in computational imaging, genomics, and network analysis. The proposed estimator…