High-dimensional estimation with missing data: Statistical and computational limits

研究问题: 研究在观测存在缺失、且缺失机制为可实现污染模型（任意未知的 MNAR）时，如何对总体参数进行计算可行估计，并刻画统计极限与计算极限之间的差距。
方法与思路: 摘要称在真实数据为高斯、ε 比例观测受任意未知 MNAR 机制影响的设定下，分别考察均值估计（ℓ2 范数）、协方差估计（相对算子范数）与线性回归，比较信息论下界与多项式时间算法的样本复杂度。
材料陈述的贡献: 作者提出均值与协方差估计中存在统计—计算差距：信息论最优需约 n≳de^{1/ρ²} 样本但算法计算不可行，而高效算法需约 n≳d^{1/ρ²}，并给出基于平方和的多项式时间算法近乎达到该下界；线性回归中该差距不持续，最小化简单强凸经验风险即可多项式时间近乎达到信息论下界。
阅读关注点: 应核对缺失污染模型与 MNAR 的具体假设、下界所针对的算法族范围、平方和算法的阶数与运行时间、以及线性回归结论成立的条件与常数依赖。
阅读边界: 摘要未说明具体定理形式、证明技术细节与数值实验；且本文为 arXiv 预印本，未经同行评审，不能据此声称已被期刊接收或结论已被验证。

DeepSeek Flash · 2026-10-13

原文摘要

We consider computationally-efficient estimation of population parameters when observations are subject to missing data. In particular, we consider estimation under the realizable contamination model of missing data in which an $ε$ fraction of the observations are subject to an arbitrary (and unknown) missing not at random (MNAR) mechanism. When the true data is Gaussian, we provide evidence towards statistical-computational gaps in several problems. For mean estimation in $\ell_2$ norm, we show that in order to obtain error at most $ρ$ , for any constant contamination $ε\in (0, 1)$ , (roughly) $n \gtrsim d e^{1/ρ^2}$ samples are necessary and that there is a computationally-inefficient algorithm which achieves this error. On the other hand, we show that any computationally-efficient method within certain popular families of algorithms requires a much larger sample complexity of (roughly) $n \gtrsim d^{1/ρ^2}$ and that there exists a polynomial time algorithm based on sum-of-squares which (nearly) achieves this lower bound. For covariance estimation in relative operator norm, we show that a parallel development holds. Finally, we turn to linear regression with missing observations and show that such a gap does not persist. Indeed, in this setting we show that minimizing a simple, strongly convex empirical risk nearly achieves the information-theoretic lower bound in polynomial time.We consider computationally-efficient estimation of population parameters when observations are subject to missing data. In particular, we consider estimation under the realizable contamination model of missing data in which an $ε$ fraction of the observations are subject to an arbitrary (and unknown) missing not at random (MNAR) mechanism. When the true data is Gaussian, we provide evidence towards statistical-computational gaps in several problems. For mean estimation in $\ell_2$ norm, we show that in order to obtain error at most $ρ$, for any constant contamination $ε\in (0, 1)$, (roughly) $n \gtrsim d e^{1/ρ^2}$ samples are necessary and that there is a computationally-inefficient algorithm which achieves this error. On the other hand, we show that any computationally-efficient method within certain popular families of algorithms requires a much larger sample complexity of (roughly) $n \gtrsim d^{1/ρ^2}$ and that there exists a polynomial time algorithm based on sum-of-squares which (nearly) achieves this lower bound. For covariance estimation in relative operator norm, we show that a parallel development holds. Finally, we turn to linear regression with missing observations and show that such a gap does not persist. Indeed, in this setting we show that minimizing a simple, strongly convex empirical risk nearly achieves the information-theoretic lower bound in polynomial time.

摘要来源：arXiv

笔记与分组

分类依据与纠错

High-dimensional estimation with missing data: Statistical and computational limits

作者的期刊发表依据

代码、数据与补充材料

高维缺失数据估计：统计与计算极限

原文摘要

同方向的基础文献

同方向的基础文献