Truncated Linear Regression in High Dimensions

• URL: http://arxiv.org/abs/2007.14539v1
• Date: Wed, 29 Jul 2020 00:31:34 GMT
• Title: Truncated Linear Regression in High Dimensions
• Authors: Constantinos Daskalakis, Dhruv Rohatgi, Manolis Zampetakis
• Abstract summary: In truncated linear regression, $(A_i, y_i)_i$ whose dependent variable equals $y_i= A_irm T cdot x* + eta_i$ is some fixed unknown vector of interest. 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.
• Score: 26.41623833920794
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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 $x^*$ 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 $x^*$ 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 $\ell_2$ 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.

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