The multivariate L 1 -median and associated data depth
Open accessAbstract
This paper gives three related results: ( i ) a new, simple, fast, monotonically converging algorithm for deriving the L 1 -median of a data cloud in ℝ d , a problem that can be traced to Fermat and has fascinated applied mathematicians for over three centuries; ( ii ) a new general definition for depth functions, as functions of multivariate medians, so that different definitions of medians will, correspondingly, give rise to different dept functions; and ( iii ) a simple closed-form formula of the L 1 -depth function for a given data cloud in ℝ d .
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The geometric median attains the optimal 1/2 breakdown for any translation-equivariant estimator.
“The geometric median attains the optimal 1/2 breakdown for any translation-equivariant estimator (Lopuhaä and Rousseeuw, 1991;Small, 1990;Vardi and Zhang, 2000)”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Vardi and Zhang, 2000
Section 4, “The L1-depth” (page not supplied), explicitly states that the breakdown point of the L1-median is 1/2. Our copy does not state that 1/2 is the optimal bound for all translation-equivariant estimators, so it supports the median’s breakdown point but not that comparison. (LLM, conf=0.86)
Experimental — accuracy not yet measuredThe modified Weiszfeld iteration converges to the unique geometric median at a linear rate whenever the data are not collinear.
“Vardi and Zhang (2000) prove that the modified Weiszfeld iteration converges to the unique geometric median at a linear rate whenever the data are not collinear: ∥z (t) -ŷGM ∥ 2 ≤ ρ t ∥z (0) -ŷGM ∥ 2 for some ρ ∈ (0, 1).”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Vardi and Zhang, 2000
Section 2, “The L1-Median” (page unavailable), states that noncollinear data have a unique median. Section 5, “Proofs of 2.9–2.11” (page unavailable), establishes convergence of the modified iteration, but the available text does not establish a linear rate or the stated geometric error bound; it describes monotonic convergence instead. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe modified Weiszfeld iteration converges to the unique geometric median at a linear rate whenever the data are not collinear.
“Vardi and Zhang (2000) prove that the modified Weiszfeld iteration converges to the unique geometric median at a linear rate whenever the data are not collinear: ∥z (t) -ŷGM ∥ 2 ≤ ρ t ∥z (0) -ŷGM ∥ 2 for some ρ ∈ (0, 1).”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Vardi and Zhang (2000)
Section 2, “The L1-Median,” says noncollinear data have a unique median. The Introduction and Section 5 say the modified algorithm converges to it from any starting point, but the readable passages do not establish a linear rate or the stated bound; the relevant formulas are not decoded in our copy. (LLM, conf=0.85)
Experimental — accuracy not yet measuredThe geometric median objective has a unique minimizer when the data are not collinear.
“Suppose the data are not collinear: then for any line L ⊂ R d there exists at least one ŷi / ∈ L, so the corresponding summand is strictly convex along L. Hence F is strictly convex along every line, hence strictly convex on R d , and the minimizer is unique”
From RoPoLL: Robust Panel of LLM Judges (2026) · cites this paper as Vardi and Zhang, 2000
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Research works printed but not linked: 2
- [19]Liu, R. (1988) Proc. Natl. Acad. Sci. USA 85, 1732-1734.
- [24]Rousseeuw, P . J. & Leroy, A. M. (1987) Robust Regression and Outlier Detection (Wiley, New York).
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RoPoLL: Robust Panel of LLM Judges
Acharya, A., Pan, K. W., & Verkhovsky, B. (2026) · arXiv · doi:10.48550/arxiv.2606.30931
- Authors
- Vardi, Y., & Zhang, C.-H.
- Affiliations
- Rutgers, The State University of New Jersey (US)
- Venue
- Proceedings of the National Academy of Sciences
- Publisher
- National Academy of Sciences
- Published
- 2000-02-15
- Volume
- 97
- Issue
- 4
- Pages
- 1423-1426
- ISSN
- 0027-8424 (ISSN-L)
- 1091-6490
- PMC
- PMC26449
- OpenAlex
- W2040324983
- Cited by
- 430
- References
- 0
- Citations by year
- 2026: 17 · 2025: 12 · 2024: 22 · 2023: 27 · 2022: 26 · 2021: 27 · 2020: 31 · 2019: 22 · 2018: 42 · 2017: 33 · 2016: 24 · 2015: 20 · 2014: 16 · 2013: 12 · 2012: 13
- FWCI
- 5.82
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Proceedings of the National Academy of Sciences (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- Funders
- National Science Foundation
- National Security Agency
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Statistical Methods and Models
- Physical Sciences › Mathematics › Applied Mathematics › Point processes and geometric inequalities
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Applications
- Keywords
- data depth, depth functions, iterative methods, Fermat-Torricelli problem, multivariate median, closed-form expressions
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