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arxiv: 2606.19148 · v1 · pith:RV2PWLYYnew · submitted 2026-06-17 · 📊 stat.CO

Fast Computation of Free-Support Wasserstein Medians

Kisung You , Mauro Giuffr\'e , Dennis Shung This is my paper

Pith reviewed 2026-06-26 18:19 UTC · model grok-4.3

classification 📊 stat.CO
keywords Wasserstein medianfree-supportoptimal transportmajorization-minimizationWeiszfeld algorithmbarycentric projectionrobust aggregation
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The pith

A direct fixed-weight relocation using exact OT projections computes free-support Wasserstein medians without nested barycenter solves.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a direct solver for free-support Wasserstein medians that updates each support atom by solving exact optimal transport problems to the input measures and then moving the atom to an inverse-distance-weighted average of the barycentric projections of those plans. This replaces the nested Weiszfeld scheme, which would require solving a full weighted barycenter problem at every outer step. For a smoothed version of the median objective the relocation step is shown to be the exact minimizer of a tight majorization-minimization surrogate, which immediately yields monotone descent, convex-hull invariance, a finite-time best-residual rate, and stationarity characterizations. Experiments indicate that the resulting medians match the quality of tightly solved nested baselines while using substantially fewer exact transport subproblems, and they remain less sensitive to outliers than Wasserstein barycenters.

Core claim

The relocation rule that sends each current support atom to the inverse-distance-weighted average of its barycentric projections obtained from exact optimal transport plans to the input measures is the exact minimizer of a tight majorization-minimization surrogate for the smoothed median objective; this surrogate property supplies monotone descent on exact transport subproblems together with convex-hull invariance, a finite-time best-residual convergence rate, residual-to-gradient control when the objective is differentiable, and fixed-point and stationarity characterizations.

What carries the argument

the inverse-distance-weighted average of barycentric projections obtained from exact optimal transport plans to the input measures

If this is right

  • Monotone descent holds on every exact transport subproblem.
  • The support of the median stays inside the convex hull of the input supports.
  • A finite-time best-residual convergence rate is obtained.
  • Under differentiability the residual controls the gradient norm.
  • Fixed points of the iteration are stationary for the smoothed objective.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The resolution-consistency result implies that the same fixed-weight rule can be applied at successively finer discretizations without re-tuning weights.
  • The stability analysis suggests the method remains useful for online aggregation when new measures arrive one at a time.
  • Because only exact transport subproblems are required, the approach can be paired with any fast exact OT solver without having to implement a nested barycenter routine.

Load-bearing premise

The fixed-weight approximation to the true median objective remains close enough to the variable-weight optimum that the computational savings still produce accurate medians.

What would settle it

A benchmark instance in which the direct solver's final objective value is shown to exceed the value attained by a nested Weiszfeld solver by more than a small tolerance when the same number of support points and the same smoothing parameter are used.

read the original abstract

The Wasserstein median is a robust alternative to the Wasserstein barycenter for averaging probability measures, but exact empirical computation can be expensive. A natural metric-space Weiszfeld scheme updates the current candidate by solving a weighted Wasserstein barycenter problem at each outer iteration, producing a nested optimization problem. We propose a direct fixed-weight free-support solver that avoids this inner barycenter loop. At each iteration, the method solves exact optimal transport (OT) subproblems from the current candidate to the input measures, computes barycentric projections of the selected plans, and relocates each support atom to an inverse-distance-weighted average of its projected destinations. For a smoothed median objective, we show that this relocation is the exact minimizer of a tight majorization--minimization surrogate. This yields monotone descent for exact transport subproblems, convex-hull invariance, a finite-time best-residual rate, residual-to-gradient control under differentiability, and fixed-point and stationarity characterizations. We also give smoothing, stability, and resolution-consistency results clarifying the fixed-weight approximation. In exact-OT benchmarks, the direct solver attains median objectives close to tightly solved nested Weiszfeld baselines while using substantially fewer exact transport subproblems. Additional contamination, posterior aggregation, and image-prototype experiments show that the direct solver produces median summaries comparable to nested computation and less sensitive to outlying distributions than Wasserstein barycenters.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a direct fixed-weight free-support solver for Wasserstein medians. At each iteration it solves exact OT subproblems from the current candidate to the input measures, computes barycentric projections, and relocates each support atom to an inverse-distance-weighted average of its projected destinations. For a smoothed median objective the relocation is shown to be the exact minimizer of a tight MM surrogate; this yields monotone descent (even with exact transport), convex-hull invariance, a finite-time best-residual rate, residual-to-gradient control, and fixed-point/stationarity characterizations. Separate smoothing, stability, and resolution-consistency results are given to justify the fixed-weight approximation. In exact-OT benchmarks the method attains objective values close to nested Weiszfeld baselines while using substantially fewer transport subproblems; additional experiments on contamination, posterior aggregation, and image prototypes show comparable summaries that are less sensitive to outliers than barycenters.

Significance. If the MM-surrogate derivation and the listed descent/invariance/stationarity properties hold, the work supplies an efficient, theoretically grounded algorithm for free-support Wasserstein medians that avoids an inner barycenter loop. The explicit credit given to monotone descent for exact OT subproblems, convex-hull invariance, and the finite-time rate is a strength. The stability results, even if only qualitative, help bridge the smoothed analysis to the exact-OT setting that is actually benchmarked.

major comments (2)
  1. [stability and resolution-consistency results (referenced after the MM analysis)] The central claim that the direct solver produces medians comparable to nested Weiszfeld baselines with fewer OT calls rests on the fixed-weight approximation being sufficiently close to the true (non-smooth) median objective. The smoothing/stability/resolution-consistency results are invoked for this justification, yet they appear to supply only qualitative or asymptotic control rather than explicit quantitative bounds relating the surrogate gap to the smoothing parameter and the number of atoms; without such bounds the monotone descent on the surrogate does not automatically guarantee closeness of the attained objective to the true median in the exact-OT regime.
  2. [exact-OT benchmarks section] § on empirical benchmarks: the reported closeness of median objectives to the tightly solved nested baselines is presented without an accompanying error table or plot that quantifies the gap attributable to the fixed-weight approximation versus the gap due to early termination; this makes it difficult to isolate whether the computational saving is obtained at the cost of a measurable increase in objective value.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'finite-time best-residual rate' is not immediately clear; a brief parenthetical gloss on what 'best-residual' denotes would help readers.
  2. [smoothing parameter definition] Notation: the smoothing parameter is introduced as a free parameter; its dependence (or independence) on the number of atoms should be stated explicitly when the resolution-consistency result is presented.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will incorporate revisions to improve clarity and empirical presentation.

read point-by-point responses

  1. Referee: The central claim that the direct solver produces medians comparable to nested Weiszfeld baselines with fewer OT calls rests on the fixed-weight approximation being sufficiently close to the true (non-smooth) median objective. The smoothing/stability/resolution-consistency results are invoked for this justification, yet they appear to supply only qualitative or asymptotic control rather than explicit quantitative bounds relating the surrogate gap to the smoothing parameter and the number of atoms; without such bounds the monotone descent on the surrogate does not automatically guarantee closeness of the attained objective to the true median in the exact-OT regime.

    Authors: We agree that the smoothing, stability, and resolution-consistency results provide primarily qualitative and asymptotic justification rather than explicit quantitative bounds on the surrogate gap as a function of the smoothing parameter and atom count. The monotone descent property therefore does not yield a rigorous guarantee of closeness to the exact median objective. In the revision we will add an explicit remark acknowledging this limitation of the current analysis and discuss its implications for interpreting the exact-OT benchmarks. revision: yes

  2. Referee: § on empirical benchmarks: the reported closeness of median objectives to the tightly solved nested baselines is presented without an accompanying error table or plot that quantifies the gap attributable to the fixed-weight approximation versus the gap due to early termination; this makes it difficult to isolate whether the computational saving is obtained at the cost of a measurable increase in objective value.

    Authors: We concur that an explicit quantification of the objective gaps would help readers separate the contribution of the fixed-weight approximation from early termination effects. We will revise the empirical section to include a table (or supplementary plot) reporting objective values for both solvers together with the number of OT subproblems used, thereby clarifying the observed trade-off. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent MM surrogate analysis

full rationale

The paper derives the fixed-weight relocation update as the exact minimizer of a majorization-minimization surrogate constructed for a smoothed median objective, then establishes monotone descent and related properties from that surrogate. This chain depends on external exact OT subproblems and standard MM arguments rather than reducing any claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation. Smoothing/stability results address the fixed-weight approximation separately without circular reduction. The central algorithmic claim therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach relies on standard optimal transport theory for existence of plans and introduces a smoothing parameter to enable the majorization-minimization analysis; no new entities are postulated.

free parameters (1)
  • smoothing parameter
    Chosen to create a differentiable smoothed median objective that admits an exact MM surrogate for the relocation step.
axioms (1)
  • standard math Optimal transport plans exist between the candidate and input measures and can be solved exactly
    The method requires exact OT subproblems at each iteration to compute the projections.

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Reference graph

Works this paper leans on

300 extracted references · 214 canonical work pages · 5 internal anchors

  1. [1]

    , editor =

    Danskin, John M. , editor =. The. doi:10.1007/978-3-642-46092-0 , urldate =

    work page doi:10.1007/978-3-642-46092-0

  2. [2]

    Mathematical Programming , author =

    Proximal alternating linearized minimization for nonconvex and nonsmooth problems , volume =. Mathematical Programming , author =. 2014 , pages =. doi:10.1007/s10107-013-0701-9 , language =

    work page doi:10.1007/s10107-013-0701-9 2014

  3. [3]

    Mathematical Programming , author =

    Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized. Mathematical Programming , author =. 2013 , pages =. doi:10.1007/s10107-011-0484-9 , language =

    work page doi:10.1007/s10107-011-0484-9 2013

  4. [4]

    Les équations aux dérivées partielles , author =

    Une propriété topologique des sous-ensembles analytiques réels , volume =. Les équations aux dérivées partielles , author =

  5. [5]

    USSR Computational Mathematics and Mathematical Physics , author =

    Gradient methods for the minimisation of functionals , volume =. USSR Computational Mathematics and Mathematical Physics , author =. 1963 , pages =. doi:10.1016/0041-5553(63)90382-3 , language =

    work page doi:10.1016/0041-5553(63)90382-3 1963

  6. [6]

    Mathematical Programming , author =

    A note on. Mathematical Programming , author =. 1973 , pages =. doi:10.1007/BF01584648 , language =

    work page doi:10.1007/bf01584648 1973

  7. [7]

    Learning multiple layers of features from tiny images , institution =

    Krizhevsky, Alex and Hinton, Geoffrey , year =. Learning multiple layers of features from tiny images , institution =

  8. [8]

    Topics in

    Villani, Cédric , year =. Topics in

  9. [9]

    Villani, Cédric , year =. Optimal

  10. [10]

    Probability measures on metric spaces of nonpositive curvature , volume =

    Sturm, Karl-Theodor , editor =. Probability measures on metric spaces of nonpositive curvature , volume =. Contemporary. 2003 , pages =. doi:10.1090/conm/338/06080 , language =

    work page doi:10.1090/conm/338/06080 2003

  11. [11]

    Computational optimal transport: With applications to data science.Foundations and Trends in Machine Learning, 11(5-6):355–607, 2019

    Computational. Foundations and Trends® in Machine Learning , author =. 2019 , pages =. doi:10.1561/2200000073 , language =

    work page doi:10.1561/2200000073 2019

  12. [12]

    Optimal transport mapping via input convex neural networks , volume =

    Makkuva, Ashok and Taghvaei, Amirhossein and Oh, Sewoong and Lee, Jason , editor =. Optimal transport mapping via input convex neural networks , volume =. Proceedings of the 37th international conference on machine learning , publisher =. 2020 , pages =

    2020

  13. [13]

    Scalable

    Fan, Amirhossein, Jiaojiao, Taghvaei and Chen, Yongxin , editor =. Scalable. Proceedings of the 38th international conference on machine learning , publisher =. 2021 , pages =

    2021

  14. [14]

    Manole, Tudor and Balakrishnan, Sivaraman and Niles-Weed, Jonathan and Wasserman, Larry , year =. Plugin. doi:10.48550/ARXIV.2107.12364 , urldate =

    work page doi:10.48550/arxiv.2107.12364

  15. [15]

    Management Science , author =

    On the. Management Science , author =. 1958 , pages =. doi:10.1287/mnsc.5.1.1 , language =

    work page doi:10.1287/mnsc.5.1.1 1958

  16. [16]

    Uscidda, Théo and Cuturi, Marco , month = feb, year =. The

  17. [17]

    The Annals of Statistics , author =

    Minimax estimation of smooth optimal transport maps , volume =. The Annals of Statistics , author =. doi:10.1214/20-AOS1997 , number =

    work page doi:10.1214/20-aos1997

  18. [18]

    Communications on Pure and Applied Mathematics , author =

    Polar factorization and monotone rearrangement of vector‐valued functions , volume =. Communications on Pure and Applied Mathematics , author =. 1991 , pages =. doi:10.1002/cpa.3160440402 , language =

    work page doi:10.1002/cpa.3160440402 1991

  19. [19]

    doi:10.21227/KATB-ZV89 , abstract =

    Brunner, Clemens and Leeb, Robert and Müller-Putz, Gernot , year =. doi:10.21227/KATB-ZV89 , abstract =

    work page doi:10.21227/katb-zv89

  20. [20]

    Fan, Jiaojiao and Liu, Shu and Ma, Shaojun and Zhou, Haomin and Chen, Yongxin , month = nov, year =. Neural

  21. [21]

    IMA Journal of Numerical Analysis , author =

    Quantitative stability and error estimates for optimal transport plans , volume =. IMA Journal of Numerical Analysis , author =. 2021 , pages =. doi:10.1093/imanum/draa045 , abstract =

    work page doi:10.1093/imanum/draa045 2021

  22. [22]

    SIAM Journal on Mathematical Analysis , author =

    Minimizing. SIAM Journal on Mathematical Analysis , author =. 2003 , pages =. doi:10.1137/S0036141002410927 , language =

    work page doi:10.1137/s0036141002410927 2003

  23. [23]

    Mémoire sur la théorie des déblais et des remblais , publisher =

    Monge, Gaspard , year =. Mémoire sur la théorie des déblais et des remblais , publisher =

  24. [24]

    arXiv:2103.04621 [math] , author =

    Geodesic of the. arXiv:2103.04621 [math] , author =. 2021 , keywords =

    arXiv 2021

  25. [25]

    International Journal of Computer Vision , author =

    A. International Journal of Computer Vision , author =. 2006 , pages =. doi:10.1007/s11263-005-3222-z , language =

    work page doi:10.1007/s11263-005-3222-z 2006

  26. [26]

    Journal of Mathematical Imaging and Vision , author =

    Intrinsic. Journal of Mathematical Imaging and Vision , author =. 2006 , pages =. doi:10.1007/s10851-006-6228-4 , language =

    work page doi:10.1007/s10851-006-6228-4 2006

  27. [27]

    Mapping estimation for discrete optimal transport , volume =

    Perrot, Michaël and Courty, Nicolas and Flamary, Rémi and Habrard, Amaury , editor =. Mapping estimation for discrete optimal transport , volume =. Advances in

  28. [28]

    Joint distribution optimal transportation for domain adaptation , volume =

    Courty, Nicolas and Flamary, Rémi and Habrard, Amaury and Rakotomamonjy, Alain , editor =. Joint distribution optimal transportation for domain adaptation , volume =. Advances in

  29. [29]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , author =

    Optimal. IEEE Transactions on Pattern Analysis and Machine Intelligence , author =. 2017 , pages =. doi:10.1109/TPAMI.2016.2615921 , number =

    work page doi:10.1109/tpami.2016.2615921 2017

  30. [30]

    Machine Learning , author =

    Wasserstein discriminant analysis , volume =. Machine Learning , author =. 2018 , pages =. doi:10.1007/s10994-018-5717-1 , language =

    work page doi:10.1007/s10994-018-5717-1 2018

  31. [31]

    Bonet, Clément and Malézieux, Benoît and Rakotomamonjy, Alain and Drumetz, Lucas and Moreau, Thomas and Kowalski, Matthieu and Courty, Nicolas , month = mar, year =. Sliced-

  32. [32]

    2018 , pages =

    Journal of Neural Engineering , author =. 2018 , pages =. doi:10.1088/1741-2552/aadea0 , number =

    work page doi:10.1088/1741-2552/aadea0 2018

  33. [33]

    Frontiers in Neuroscience , author =

    Review of the. Frontiers in Neuroscience , author =. doi:10.3389/fnins.2012.00055 , urldate =

    work page doi:10.3389/fnins.2012.00055 2012

  34. [34]

    Mathematical Programming , author =

    An alternative to. Mathematical Programming , author =. 2020 , pages =. doi:10.1007/s10107-019-01381-4 , language =

    work page doi:10.1007/s10107-019-01381-4 2020

  35. [35]

    2019 , keywords =

    Applied directional statistics , isbn =. 2019 , keywords =

    2019

  36. [36]

    Proceedings of the American Mathematical Society , author =

    Positive definite matrices and the. Proceedings of the American Mathematical Society , author =. 2015 , pages =. doi:10.1090/proc/12953 , language =

    work page doi:10.1090/proc/12953 2015

  37. [37]

    Computational Statistics , author =

    A short note on parameter approximation for von. Computational Statistics , author =. 2012 , pages =. doi:10.1007/s00180-011-0232-x , language =

    work page doi:10.1007/s00180-011-0232-x 2012

  38. [38]

    A new metric on the manifold of kernel matrices with application to matrix geometric means , volume =

    Sra, Suvrit , editor =. A new metric on the manifold of kernel matrices with application to matrix geometric means , volume =. Advances in neural information processing systems , publisher =

  39. [39]

    Efficient similarity search for covariance matrices via the

    Cherian, Anoop and Sra, Suvrit and Banerjee, Arindam and Papanikolopoulos, Nikolaos , month = nov, year =. Efficient similarity search for covariance matrices via the. 2011. doi:10.1109/ICCV.2011.6126523 , urldate =

    work page doi:10.1109/iccv.2011.6126523 2011

  40. [40]

    Journal of Machine Learning Research , author =

    Clustering on the unit hypersphere using von mises-fisher distributions , volume =. Journal of Machine Learning Research , author =. 2005 , pages =

    2005

  41. [41]

    First-order

    Zhang, Hongyi and Sra, Suvrit , editor =. First-order. 29th. 2016 , pages =

    2016

  42. [42]

    SIAM Journal on Mathematical Analysis , author =

    Linearized. SIAM Journal on Mathematical Analysis , author =. 2024 , pages =. doi:10.1137/23M1564535 , language =

    work page doi:10.1137/23m1564535 2024

  43. [43]

    Santambrogio, Filippo , year =. Optimal. doi:10.1007/978-3-319-20828-2 , language =

    work page doi:10.1007/978-3-319-20828-2

  44. [44]

    Communications on Pure and Applied Mathematics , author =

    Riemannian. Communications on Pure and Applied Mathematics , author =. 1977 , pages =. doi:10.1002/cpa.3160300502 , language =

    work page doi:10.1002/cpa.3160300502 1977

  45. [45]

    Van Den Bosch, Jasper Jf and Golan, Tal and Peters, Benjamin and Taylor, JohnMark and Shahbazi, Mahdiyar and Lin, Baihan and Charest, Ian and Diedrichsen, Jörn and Kriegeskorte, Nikolaus and Mur, Marieke and Schütt, Heiko H , month = sep, year =. A. doi:10.7554/eLife.107828.1 , abstract =

    work page doi:10.7554/elife.107828.1

  46. [46]

    Cheng, Zixiong and Liu, Hang , year =. Robust. doi:10.48550/ARXIV.2603.07563 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2603.07563

  47. [47]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    Huber means on. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2026 , pages =. doi:10.1093/jrsssb/qkaf054 , language =

    work page doi:10.1093/jrsssb/qkaf054 2026

  48. [48]

    You, Kisung , year =. Finite. doi:10.48550/ARXIV.2604.24895 , urldate =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2604.24895

  49. [49]

    Michigan Mathematical Journal , author =

    A class of. Michigan Mathematical Journal , author =. 1984 , note =. doi:10.1307/mmj/1029003026 , number =

    work page doi:10.1307/mmj/1029003026 1984

  50. [50]

    , year =

    Witkin, Andrew P. , year =. Scale-space filtering , booktitle =

  51. [51]

    Annual Review of Statistics and Its Application , author =

    Topological. Annual Review of Statistics and Its Application , author =. 2018 , pages =. doi:10.1146/annurev-statistics-031017-100045 , abstract =

    work page doi:10.1146/annurev-statistics-031017-100045 2018

  52. [52]

    and Jones, M.C

    Wand, M.P. and Jones, M.C. , month = dec, year =. Kernel. doi:10.1201/b14876 , language =

    work page doi:10.1201/b14876

  53. [53]

    The Annals of Statistics , author =

    On nonparametric estimation of density level sets , volume =. The Annals of Statistics , author =. doi:10.1214/aos/1069362732 , number =

    work page doi:10.1214/aos/1069362732

  54. [54]

    The Annals of Statistics , author =

    Variable. The Annals of Statistics , author =. doi:10.1214/aos/1176348768 , number =

    work page doi:10.1214/aos/1176348768

  55. [55]

    Journal of Computational and Graphical Statistics , author =

    A. Journal of Computational and Graphical Statistics , author =. 2010 , pages =. doi:10.1198/jcgs.2009.07049 , language =

    work page doi:10.1198/jcgs.2009.07049 2010

  56. [56]

    Journal of Classification , author =

    Estimating the. Journal of Classification , author =. 2003 , pages =. doi:10.1007/s00357-003-0004-6 , number =

    work page doi:10.1007/s00357-003-0004-6 2003

  57. [57]

    The Annals of Statistics , author =

    Fully adaptive density-based clustering , volume =. The Annals of Statistics , author =. doi:10.1214/15-AOS1331 , number =

    work page doi:10.1214/15-aos1331

  58. [58]

    Information and Control , author =

    Some inequalities satisfied by the quantities of information of. Information and Control , author =. 1959 , pages =. doi:10.1016/S0019-9958(59)90348-1 , language =

    work page doi:10.1016/s0019-9958(59)90348-1 1959

  59. [59]

    Score-Based Generative Modeling through Stochastic Differential Equations

    Song, Yang and Sohl-Dickstein, Jascha and Kingma, Diederik P. and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , year =. Score-. doi:10.48550/ARXIV.2011.13456 , abstract =

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2011.13456 2011

  60. [60]

    Generative modeling by estimating gradients of the data distribution , abstract =

    Song, Yang and Ermon, Stefano , year =. Generative modeling by estimating gradients of the data distribution , abstract =. Proceedings of the 33rd

  61. [61]

    Sohl-Dickstein, Jascha and Weiss, Eric and Maheswaranathan, Niru and Ganguli, Surya , editor =. Deep. Proceedings of the 32nd. 2015 , pages =

    2015

  62. [62]

    Cerebral Cortex , author =

    Local-. Cerebral Cortex , author =. 2018 , pages =. doi:10.1093/cercor/bhx179 , language =

    work page doi:10.1093/cercor/bhx179 2018

  63. [63]

    , month = feb, year =

    Silverman, B.W. , month = feb, year =. Density. doi:10.1201/9781315140919 , language =

    work page doi:10.1201/9781315140919

  64. [64]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    Using. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 1981 , pages =. doi:10.1111/j.2517-6161.1981.tb01155.x , abstract =

    work page doi:10.1111/j.2517-6161.1981.tb01155.x 1981

  65. [65]

    Technometrics , author =

    From. Technometrics , author =. 2001 , pages =. doi:10.1198/004017001316975916 , language =

    work page doi:10.1198/004017001316975916 2001

  66. [66]

    , month = mar, year =

    Scott, David W. , month = mar, year =. Multivariate. doi:10.1002/9781118575574 , language =

    work page doi:10.1002/9781118575574

  67. [67]

    Journal d'Analyse Mathématique , author =

    On. Journal d'Analyse Mathématique , author =. 1951 , pages =. doi:10.1007/BF02790092 , language =

    work page doi:10.1007/bf02790092 1951

  68. [68]

    The Annals of Statistics , author =

    Asymptotics and optimal bandwidth selection for highest density region estimation , volume =. The Annals of Statistics , author =. doi:10.1214/09-AOS766 , number =

    work page doi:10.1214/09-aos766

  69. [69]

    Probability of indecomposability of a random mapping function.Ann

    Remarks on. The Annals of Mathematical Statistics , author =. 1956 , pages =. doi:10.1214/aoms/1177728190 , language =

    work page doi:10.1214/aoms/1177728190 1956

  70. [70]

    The Annals of Statistics , author =

    Generalized density clustering , volume =. The Annals of Statistics , author =. doi:10.1214/10-AOS797 , number =

    work page doi:10.1214/10-aos797

  71. [71]

    Bernoulli , author =

    Optimal rates for plug-in estimators of density level sets , volume =. Bernoulli , author =. doi:10.3150/09-BEJ184 , number =

    work page doi:10.3150/09-bej184

  72. [72]

    Journal of the Royal Statistical Society Series B: Statistical Methodology , author =

    On. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 1997 , pages =. doi:10.1111/1467-9868.00095 , abstract =

    work page doi:10.1111/1467-9868.00095 1997

  73. [73]

    The Annals of Statistics , author =

    The topography of multivariate normal mixtures , volume =. The Annals of Statistics , author =. doi:10.1214/009053605000000417 , number =

    work page doi:10.1214/009053605000000417

  74. [74]

    Computational Statistics & Data Analysis , author =

    Alternating kernel and mixture density estimates , volume =. Computational Statistics & Data Analysis , author =. 2000 , pages =. doi:10.1016/S0167-9473(00)00003-7 , language =

    work page doi:10.1016/s0167-9473(00)00003-7 2000

  75. [75]

    The Annals of Statistics , author =

    Measuring. The Annals of Statistics , author =. doi:10.1214/aos/1176324626 , number =

    work page doi:10.1214/aos/1176324626

  76. [76]

    and Wang, Bei and Zheng, Yan , editor =

    Phillips, Jeff M. and Wang, Bei and Zheng, Yan , editor =. Geometric. 2015 , pages =. doi:10.4230/LIPICS.SOCG.2015.857 , booktitle =

    work page doi:10.4230/lipics.socg.2015.857 2015

  77. [77]

    The Annals of Mathematical Statistics , author =

    On. The Annals of Mathematical Statistics , author =. 1962 , pages =. doi:10.1214/aoms/1177704472 , language =

    work page doi:10.1214/aoms/1177704472 1962

  78. [78]

    Journal of Computational and Graphical Statistics , author =

    The. Journal of Computational and Graphical Statistics , author =. 1993 , pages =. doi:10.1080/10618600.1993.10474599 , language =

    work page doi:10.1080/10618600.1993.10474599 1993

  79. [79]

    The Annals of Statistics , author =

    Nonparametric testing of the existence of modes , volume =. The Annals of Statistics , author =. doi:10.1214/aos/1031594735 , number =

    work page doi:10.1214/aos/1031594735

  80. [80]

    , year =

    Lindsay, Bruce G. , year =. Mixture models: theory, geometry, and applications , isbn =

Showing first 80 references.