A Debiased LASSO Estimator for Design Matrices with Non-zero Mean Elements

The {LASSO} estimator has been popular in diverse fields such as statistics, bio-informatics and compressed sensing for estimation of sparse signals from under-sampled measurements. As the {LASSO} does not provide confidence intervals for the estimates of the individual elements, techniques such as the debiased {LASSO} have been developed that provide such confidence intervals and also mitigate the inherent bias in the {LASSO} estimates. However the theoretical guarantees underlying the debiased {LASSO} have been developed for a design or measurement matrix whose elements have zero mean. On the other hand, many commonly used measurement matrices in sparse regression applications in optical imaging or pooled biological testing involve non-negative, particularly binary, matrices whose elements have a non-zero mean. In this paper, we propose a debiased {LASSO} technique based on computing differences between different row subsets of the underlying design matrix iteratively which effectively converts the design matrix into one with zero-mean elements. We show analytically that this results in lower variances for the individual elements of the debiased {LASSO} estimate as compared to the standard debiased {LASSO}. We also provide numerical results to support our theory.