Truncated Linear Regression in High Dimensions

Abstract: As in standard linear regression, in truncated linear regression, we are given access to observations $(A_i, y_i)_i$ whose dependent variable equals $$y_i= A_i^{\rm T} \cdot x^* + \eta_i$$, where $x<sup>*$ 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" $S \subset \mathbb{R}$. The goal is to recover $x<sup>*$ 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 $x<sup>*$ from $m$ truncated samples, which attains an optimal $\ell_2$ 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, 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.