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

• URL: http://arxiv.org/abs/2603.16712v1
• Date: Tue, 17 Mar 2026 16:02:41 GMT
• Title: High-dimensional estimation with missing data: Statistical and computational limits
• Abstract summary: We consider computationally-efficient estimation of population parameters when observations are subject to missing data.<n>For mean estimation in $ell$ norm, we show that in order to obtain error at most $$, for any constant contamination $in (0, 1)$, (roughly) $n gtrsim d e1/2$ samples are necessary.<n>We turn to linear regression with missing observations and show that such a gap does not persist.
• Score: 13.480462575790334
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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