# Efficient Truncated Linear Regression with Unknown Noise Variance

arXiv: https://arxiv.org/abs/2208.12042 · HTML: https://mlsanity.com/paper/2208.12042/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 30 references that are in the map.
- First posted on arXiv 25 Aug 2022.

## Builds on (most cited first, 7 of 7)

- [An Introduction to Matrix Concentration Inequalities](https://mlsanity.com/paper/1501.01571/) (2015)
- [Efficient Statistics, in High Dimensions, from Truncated Samples](https://mlsanity.com/paper/1809.03986/) (2018)
- [Computationally and Statistically Efficient Truncated Regression](https://mlsanity.com/paper/2010.12000/) (2020)
- [Efficient Truncated Statistics with Unknown Truncation](https://mlsanity.com/paper/1908.01034/) (2019)
- [Efficient Parameter Estimation of Truncated Boolean Product Distributions](https://mlsanity.com/paper/2007.02392/) (2020)
- [Learning from Censored and Dependent Data: The case of Linear Dynamics](https://mlsanity.com/paper/2104.05087/) (2021)
- [Learning to Sample from Censored Markov Random Fields](https://mlsanity.com/paper/2101.06178/) (2021)

## Built on by (most cited first, 10 of 10)

- [Domain constraints improve risk prediction when outcome data is missing](https://mlsanity.com/paper/2312.03878/) (2023)
- [Efficient Statistics With Unknown Truncation, Polynomial Time Algorithms, Beyond Gaussians](https://mlsanity.com/paper/2410.01656/) (2024)
- [Statistical Query Lower Bounds for Learning Truncated Gaussians](https://mlsanity.com/paper/2403.02300/) (2024)
- [Transfer Learning Beyond Bounded Density Ratios](https://mlsanity.com/paper/2403.11963/) (2024)
- [Smoothed Analysis of Learning from Positive Samples](https://mlsanity.com/paper/2504.10428/) (2025)
- [Learning High-dimensional Gaussians from Censored Data](https://mlsanity.com/paper/2504.19446/) (2025)
- [Private Statistical Estimation via Truncation](https://mlsanity.com/paper/2505.12541/) (2025)
- [Linear Regression under Missing or Corrupted Coordinates](https://mlsanity.com/paper/2509.19242/) (2025)
- [High-Dimensional Gaussian Mean Estimation under Realizable Contamination](https://mlsanity.com/paper/2603.16798/) (2026)
- [Efficient Learning of Truncated Boolean Product Distributions: Influence to the Rescue](https://mlsanity.com/paper/2607.22889/) (2026)

## Abstract (arXiv)

Truncated linear regression is a classical challenge in Statistics, wherein a label, y = wT x + varepsilon, and its corresponding feature vector, x ∈ ℝk, are only observed if the label falls in some subset S subseteq ℝ; otherwise the existence of the pair (x, y) is hidden from observation. Linear regression with truncated observations has remained a challenge, in its general form, since the early works of~citettobin1958estimation,amemiya1973regression. When the distribution of the error is normal with known variance, recent work of~citetdaskalakis2019truncatedregression provides computationally and statistically efficient estimators of the linear model, w. In this paper, we provide the first computationally and statistically efficient estimators for truncated linear regression when the noise variance is unknown, estimating both the linear model and the variance of the noise. Our estimator is based on an efficient implementation of Projected Stochastic Gradient Descent on the negative log-likelihood of the truncated sample. Importantly, we show that the error of our estimates is asymptotically normal, and we use this to provide explicit confidence regions for our estimates.