# Learning High-dimensional Gaussians from Censored Data

arXiv: https://arxiv.org/abs/2504.19446 · HTML: https://mlsanity.com/paper/2504.19446/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 98 references that are in the map.
- First posted on arXiv 28 Apr 2025.

## Builds on (most cited first, 30 of 36)

- [Robust Estimators in High Dimensions without the Computational Intractability](https://mlsanity.com/paper/1604.06443/) (2016)
- [Agnostic Estimation of Mean and Covariance](https://mlsanity.com/paper/1604.06968/) (2016)
- [Recent Advances in Algorithmic High-Dimensional Robust Statistics](https://mlsanity.com/paper/1911.05911/) (2019)
- [Learning from Untrusted Data](https://mlsanity.com/paper/1611.02315/) (2016)
- [HoloClean: Holistic Data Repairs with Probabilistic Inference](https://mlsanity.com/paper/1702.00820/) (2017)
- [Sampling from a log-concave distribution with Projected Langevin Monte Carlo](https://mlsanity.com/paper/1507.02564/) (2015)
- [Robust and Heavy-Tailed Mean Estimation Made Simple, via Regret Minimization](https://mlsanity.com/paper/2007.15839/) (2020)
- [Robustly Learning a Gaussian: Getting Optimal Error, Efficiently](https://mlsanity.com/paper/1704.03866/) (2017)
- [De-Biased Machine Learning of Global and Local Parameters Using Regularized Riesz Representers](https://mlsanity.com/paper/1802.08667/) (2018)
- [Learning Mixtures of Gaussians in High Dimensions](https://mlsanity.com/paper/1503.00424/) (2015)
- [Efficient Statistics, in High Dimensions, from Truncated Samples](https://mlsanity.com/paper/1809.03986/) (2018)
- [Robustly Learning any Clusterable Mixture of Gaussians](https://mlsanity.com/paper/2005.06417/) (2020)
- [Full Law Identification In Graphical Models Of Missing Data: Completeness Results](https://mlsanity.com/paper/2004.04872/) (2020)
- [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)
- [List Decodable Mean Estimation in Nearly Linear Time](https://mlsanity.com/paper/2005.09796/) (2020)
- [Identification In Missing Data Models Represented By Directed Acyclic Graphs](https://mlsanity.com/paper/1907.00241/) (2019)
- [High-Dimensional Robust Mean Estimation via Gradient Descent](https://mlsanity.com/paper/2005.01378/) (2020)
- [Robust Learning of Mixtures of Gaussians](https://mlsanity.com/paper/2007.05912/) (2020)
- [Efficient Parameter Estimation of Truncated Boolean Product Distributions](https://mlsanity.com/paper/2007.02392/) (2020)
- [Truncated Linear Regression in High Dimensions](https://mlsanity.com/paper/2007.14539/) (2020)
- [What to Expect of Classifiers? Reasoning about Logistic Regression with Missing Features](https://mlsanity.com/paper/1903.01620/) (2019)
- [A Fast Spectral Algorithm for Mean Estimation with Sub-Gaussian Rates](https://mlsanity.com/paper/1908.04468/) (2019)
- [Semiparametric Inference for Non-monotone Missing-Not-at-Random Data: the No Self-Censoring Model](https://mlsanity.com/paper/1909.01848/) (2019)
- [Learning Distributions Generated by One-Layer ReLU Networks](https://mlsanity.com/paper/1909.01812/) (2019)
- [Efficient Truncated Linear Regression with Unknown Noise Variance](https://mlsanity.com/paper/2208.12042/) (2022)
- [Learning from Censored and Dependent Data: The case of Linear Dynamics](https://mlsanity.com/paper/2104.05087/) (2021)
- [A Simple and General Debiased Machine Learning Theorem with Finite Sample Guarantees](https://mlsanity.com/paper/2105.15197/) (2021)
- [Testing Convex Truncation](https://mlsanity.com/paper/2305.03146/) (2023)
- [What Makes A Good Fisherman? Linear Regression under Self-Selection Bias](https://mlsanity.com/paper/2205.03246/) (2022)

## Built on by (most cited first, 2 of 2)

- [High-Dimensional Statistics: Reflections on Progress and Open Problems](https://mlsanity.com/paper/2605.05076/) (2026)
- [Linear Regression under Missing or Corrupted Coordinates](https://mlsanity.com/paper/2509.19242/) (2025)

## Abstract (arXiv)

We provide efficient algorithms for the problem of distribution learning from high-dimensional Gaussian data where in each sample, some of the variable values are missing. We suppose that the variables are missing not at random (MNAR). The missingness model, denoted by S(y), is the function that maps any point y in Rd to the subsets of its coordinates that are seen. In this work, we assume that it is known. We study the following two settings: (i) Self-censoring: An observation x is generated by first sampling the true value y from a d-dimensional Gaussian N(μ*, Sigma*) with unknown μ* and Sigma*. For each coordinate i, there exists a set Si subseteq Rd such that xi = yi if and only if yi in Si. Otherwise, xi is missing and takes a generic value (e.g., "?"). We design an algorithm that learns N(μ*, Sigma*) up to total variation (TV) distance epsilon, using poly(d, 1/ε) samples, assuming only that each pair of coordinates is observed with sufficiently high probability. (ii) Linear thresholding: An observation x is generated by first sampling y from a d-dimensional Gaussian N(μ*, Sigma) with unknown μ* and known Sigma, and then applying the missingness model S where S(y) = i in [d] : viT y <= bi for some v1, ..., vd in Rd and b1, ..., bd in R. We design an efficient mean estimation algorithm, assuming that none of the possible missingness patterns is very rare conditioned on the values of the observed coordinates and that any small subset of coordinates is observed with sufficiently high probability.