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= Airm T · x^* + ηi, where x^* is some fixed unknown vector of interest and ηi is independent noise; except we are only given an observation if its dependent variable yi lies in some "truncation set" S subset 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 ℓ2 reconstruction error of O(√(k log n)/m). As a corollary, our guarantees imply a computationally efficient and information-theoretically optimal algorithm for compressed sensing with truncation, 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 adaptation of the LASSO optimization problem that accommodates truncation. This generalizes the works of both: (1) [Daskalakis et al. 2018], where no regularization is needed due to the low-dimensionality of the data, and (2) [Wainright 2009], where the objective function is simple due to the absence of truncation. In order to deal with both truncation and high-dimensionality at the same time, we develop new techniques that not only generalize the existing ones but we believe are of independent interest.