Robust Mean Estimation in High Dimensions: An Outlier-Fraction Agnostic and Efficient Algorithm

Aditya Deshmukh, Jing Liu, Venugopal V. Veeravalli

Research output: Contribution to journalArticlepeer-review


The problem of robust mean estimation in high dimensions is studied, in which a certain fraction (less than half) of the datapoints can be arbitrarily corrupted. Motivated by compressive sensing, the robust mean estimation problem is formulated as the minimization of the ℓ0-‘norm’ of an outlier indicator vector, under a second moment constraint on the datapoints. The ℓ0-‘norm’ is then relaxed to the ℓp-norm (0 < p ≤ 1) in the objective, and it is shown that the global minima for each of these objectives are order-optimal and have optimal breakdown point for the robust mean estimation problem. Furthermore, a computationally tractable iterative ℓp-minimization and hard thresholding algorithm is proposed that outputs an order-optimal robust estimate of the population mean. The proposed algorithm (with breakdown point ≈ 0.3) does not require prior knowledge of the fraction of outliers, in contrast with most existing algorithms, and for p = 1 it has near-linear time complexity. Both synthetic and real data experiments demonstrate that the proposed algorithm outperforms state-of-the-art robust mean estimation methods.

Original languageEnglish (US)
Pages (from-to)4675-4690
Number of pages16
JournalIEEE Transactions on Information Theory
Issue number7
StatePublished - Jul 1 2023
Externally publishedYes


  • Robust inference
  • data contamination
  • high-dimensional statistics
  • linear time-complexity algorithms

ASJC Scopus subject areas

  • Information Systems
  • Library and Information Sciences
  • Computer Science Applications


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