Work

Linear Regression under Missing or Corrupted Coordinates

Year: 2025

Type: preprint

Abstract: We study multivariate linear regression under Gaussian covariates in two settings, where data may be erased or corrupted by an adversary under a coordinate-wise budget. In the incomplete data setting, an adversary may inspect the dataset and delete entries in up to an $η$-fraction of samples per coordinate; a strong form of the Missing Not At Random model. In the corrupted data setting, the adversary instead replaces values arbitrarily, and the corruption locations are unknown to the learner. Despite substantial work on missing data, linear regression under such adversarial missingness remains poorly understood, even information-theoretically. Unlike the clean setting, where estimation error vanishes with more samples, here the optimal error remains a positive function of the problem parameters. Our main contribution is to characterize this error up to constant factors across essentially the entire parameter range. Specifically, we establish novel information-theoretic lower bounds on the achievable error that match the error of (computationally efficient) algorithms. A key implication is that, perhaps surprisingly, the optimal error in the missing data setting matches that in the corruption setting-so knowing the corruption locations offers no general advantage.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2509.19242

Authors: Diakonikolas, Ilias, Diakonikolas, Jelena, Kane, Daniel M., Lee, Jasper C. H., Pittas, Thanasis

Institutions: University of Wisconsin–Madison, University of California San Diego, University of California, Davis

Language: English

Topic: Advanced Statistical Methods and Models

Subfield: Statistics and Probability

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, U.S. Department of Defense, Office of Naval Research

Awards: CAREER: Optimization and Learning with Changing Distributions, Collaborative Research: AF: Medium: Algorithmic High-Dimensional Robust Statistics, G3683668120, G4720003262, G4805816520, Collaborative Research: AF: Medium: Algorithmic High-Dimensional Robust Statistics, TRIPODS: Institute for Foundations of Data Science, G6098521345, G6671297155, CAREER: Structure and Analysis of Low Degree Polynomials, AitF: Collaborative Research: Fast, Accurate, and Practical: Adaptive Sublinear Algorithms for Scalable Visualization, G8876996369

OpenAlex ID: W4416255293

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