Truncated Linear Regression in High Dimensions

As in standard linear regression, in truncated linear regression, we are given access to observations (A(i), y(i))(i) whose dependent variable equals y(i) = A(t)(T). x* + eta(i), where x* is some fixed unknown vector of interest and eta(i) is independent noise; except we are only given an observation if its dependent variable y(i) lies in some "truncation set" S subset of R. The goal is to recover x* under some favorable conditions on the A(i)'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 (2) pound reconstruction error of O(J(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. [9], where no regularization is needed due to the low-dimensionality of the data, and (2) Wainright [27], 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.