High dimensional errors-in-variables models with dependent measurements

Abstract

Suppose that we observe $y\in {\mathbb{R}}^f$ and $X\in {\mathbb{R}}^{f\times m}$ in the following errors-in-variables model: \begin{eqnarray*} y & = & X_0 β^* + ε\\ X & = & X_0 + W \end{eqnarray*} where $X_0$ is a $f\times m$ design matrix with independent subgaussian row vectors, $\epsilon \in {\mathbb{R}}^f$ is a noise vector and $W$ is a mean zero $f\times m$ random noise matrix with independent subgaussian column vectors, independent of $X_0$ and $\epsilon$. This model is significantly different from those analyzed in the literature in the sense that we allow the measurement error for each covariate to be a dependent vector across its $f$ observations. Such error structures appear in the science literature when modeling the trial-to-trial fluctuations in response strength shared across a set of neurons. Under sparsity and restrictive eigenvalue type of conditions, we show that one is able to recover a sparse vector ${\beta }^{\ast }\in {\mathbb{R}}^m$ from the model given a single observation matrix $X$ and the response vector $y$. We establish consistency in estimating ${\beta }^{\ast }$ and obtain the rates of convergence in the ${\ell }_q$ norm, where $q=1,2$ for the Lasso-type estimator, and for $q\in [1,2]$ for a Dantzig-type conic programming estimator. We show error bounds which approach that of the regular Lasso and the Dantzig selector in case the errors in $W$ are tending to 0.

61 pages

References in corpus (5)

• High-dimensional regression with noisy and missing data: Provable guarantees with nonconvexity
• Convergence Properties of Kronecker Graphical Lasso Algorithms
• Adaptive Lasso for High Dimensional Regression and Gaussian Graphical Modeling
• Are a set of microarrays independent of each other?
• Measurement Error in Lasso: Impact and Correction

Cited by in corpus (2)

• Pivotal Estimation via Self-Normalization for High-Dimensional Linear Models with Error in Variables
• Errors-in-variables models with dependent measurements