Truncated Linear Regression in High Dimensions

As in standard linear regression, in truncated linear regression, we are given access to observations (Ai, yi)i whose dependent variable equals yi Ai {\rm T} \cdot x * \etai, where x * is some fixed unknown vector of interest and \etai is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S \subset \mathbb{R}. The goal is to recover x * under some favorable conditions on the Ai's and the noise distribution. We prove that there exists a computationally and statistically efficient method for recovering k-sparse n-dimensional vectors x * from m truncated samples, which attains an optimal \ell2 reconstruction error of O(\sqrt{(k \log n)/m}). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, such as that which may arise from measurement saturation effects. Our result follows from a statistical and computational analysis of the Stochastic Gradient Descent (SGD) algorithm for solving a natural adaption of the LASSO optimization problem that accommodates truncation.