High dimensional errors-in-variables models with dependent measurements

Abstract

Suppose that we observe y R^f and X R^f m in the following errors-in-variables model: eqnarray* y & = & X_0 ^* + \\ X & = & X_0 + W eqnarray* where X_0 is a f m design matrix with independent subgaussian row vectors, R^f is a noise vector and W is a mean zero f m random noise matrix with independent subgaussian column vectors, independent of X_0 and . 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 ^* R^m from the model given a single observation matrix X and the response vector y. We establish consistency in estimating ^* and obtain the rates of convergence in the _q norm, where q = 1, 2 for the Lasso-type estimator, and for q [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.

Tasks

Reproductions