Pith. sign in

REVIEW 1 major objections 5 minor 295 references

Geometric Information Decomposition for Weighted Empirical Measures on the Sphere

T0 review · 1 major / 5 minor · reviewed 2026-08-02 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read For weighted data on a sphere, the paper claims the entropy deficit from uniformity can be decomposed into levelwise information gaps, with the first gap recovering vMF mean-direction information, the second isolating quadratic anisotropy,

desk verdict A solid, honestly scoped method paper: the GID machinery works as stated on interior moments, the boundary caveat is real but acknowledged, and it deserves a careful referee. read the letter →

arxiv 2607.02943 v3 pith:ARA7QIE6 submitted 2026-07-03 stat.ME

Kisung You , Boram Cho This is my paper · ORCID
classification stat.ME MSC 62H1162B1060D05
keywords geometricinformationdecompositionmaximumentropyvonMises-FisherFisher-BinghamsphericalharmonicsdirectionalstatisticsKLdivergenceweightedempiricalmeasures
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes that a weighted point cloud on a sphere should be summarized not by a single concentration number but by a sequence of information gaps, each measuring how much entropy is removed by adding one level of harmonic features. It establishes the key identity that the cumulative entropy deficit equals the sum of KL divergences between successive maximum-entropy fits, making this profile a true decomposition into first-order, second-order, and higher-order angular information. The first gap recovers the vMF mean-direction summary, the second gap isolates antipodal, axial, elliptical, and girdle structure, and later gaps reveal finer multimodality. This matters because vMF entropy alone can badly understate uncertainty when structure is antipodal or multimodal. The paper proves invariance, consistency, normal limits away from zero gaps, and chi-square-type null calibration, and shows experimentally that the gaps separate these geometric cases.

What carries the argument

The central object is the nested family of maximum-entropy (exponential-family) projections p_L that match the moments of the measure P on feature space V_L. The mechanism carrying the argument is the information-geometric Pythagorean identity for nested exponential families: the KL divergence between successive projections equals the entropy gap I_L. On the sphere, the feature spaces are the spans of spherical harmonics through degree L, with V_1 giving vMF, V_2 giving Fisher-Bingham-type quadratic structure, and higher levels giving finer angular detail. The estimable quantities are the plug-in gaps computed from weighted empirical moments, with inference relying on the moment CLT and seco

What would settle it

Generate a large sample from a symmetric three-mode distribution on the circle (e.g., equal von Mises components at 0, 2π/3, 4π/3) and compute the Fourier gap profile with accurate quadrature. The theory predicts I_3 is the dominant gap while I_1 and I_2 are near zero. If the observed I_2 is comparable to or larger than I_3 at large n, the claimed harmonic-level separation of structure would be refuted.

Watch

Extended reading notes

Core claim

The central claim is that for any weighted empirical measure on a compact manifold, nested maximum-entropy projections onto feature spaces V_0 ⊂ V_1 ⊂ ⋯ give I_L(P) = D_L(P) − D_{L−1}(P) = KL(p_L ν ∥ p_{L−1}ν), so the cumulative entropy deficit D_L is exactly the sum of levelwise gaps. On the sphere, level 1 is the vMF fit, level 2 adds traceless quadratic features to capture Fisher-Bingham-type anisotropy, and levels 3 and above use spherical harmonic exponential families. The profile I_1, I_2, I_3, … therefore separates mean-direction information from axial, girdle-like, and finer angular structure that a vMF fit misses. The paper also establishes that the decomposition is invariant under

Load-bearing premise

The main load-bearing assumption is that every relevant moment vector lies in the relative interior of its convex moment body; if a weighted sample's moments sit on the boundary (e.g., equal antipodal masses, point mass, or a great-circle/girdle support), the maximum-entropy projection is degenerate or undefined, and the main results apply only after mixture shrinkage that changes the target measure.

Editorial extensions

If this is right

  • When a vMF fit is adequate, I_1 dominates and the later gaps are near zero; when structure is antipodal or girdle-like, I_2 is the large gap; when structure is trimodal or tetrahedral, the signal first appears at I_3 or higher.
  • The profile is invariant under rotation and basis changes, so it describes the geometry of the distribution rather than its coordinate representation.
  • Under the null at level L, the standardized plug-in gap converges to a weighted sum of chi-square variables; with uniform weights and correct specification, 2n Î_L converges to χ²_q, giving a formal test that the new feature level adds no information.
  • With informative weights, a naive chi-square calibration fails and a sandwich-covariance quadratic form is needed to control Type I error, as the calibration experiments demonstrate.
  • The same construction applies to any compact manifold with nested feature spaces, making the decomposition a general tool for uncertainty assessment beyond the sphere.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Extension: because the identity holds for any compact manifold and any nested feature spaces, the same gap profile could be built on tori, projective spaces, or rotation groups, as long as interior moments and nested subspaces are available.
  • Extension: for high-dimensional embeddings where full harmonic fitting is impractical, the paper's monotonicity result implies that a structured subspace (low-rank, diagonal, or sketched quadratics) gives a lower bound on the full information gap, so the difference between structured and full gaps measures omitted information.
  • Extension: the profile offers a natural diagnostic for attention-weighted or importance-weighted representations: a dominant I_2 or I_3 indicates geometric structure that a single softmax-like direction cannot capture, which could be tested as a practical screening rule.
  • Extension: a concrete follow-up is to analyze the bias-variance tradeoff of the epsilon-mixture boundary remedy, since the paper leaves the shrinkage rate and its effect on asymptotic inference open.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper introduces geometric information decomposition (GID) for probability measures on the sphere, and more generally on compact Riemannian manifolds. The construction fits nested maximum-entropy exponential families to spherical harmonic and related feature spaces: level 1 recovers the von Mises-Fisher mean-direction information, level 2 captures Fisher-Bingham-type quadratic anisotropy (antipodal, axial, girdle structure), and higher levels describe finer harmonic structure. The central structural result is Theorem 1, which proves the monotonicity of entropy deficits and the KL-gap identity I_L(P)=D_L(P)-D_{L-1}(P)=KL(p_L^P nu || p_{L-1}^P nu), so that the cumulative deficit decomposes into the level-wise gaps. The paper establishes basis invariance, isometry invariance, consistency of plug-in estimators, delta-method asymptotic normality away from zero gaps, and a second-order null calibration with chi-square and weighted-chi-square limits. Simulation experiments on S^1 and S^2, including importance-weight calibration, and a query-weighted digit projection are reported. The key regularity condition is that all relevant moment vectors lie in the relative interior of the moment body.

Significance. If the stated scope is properly qualified, the paper is a solid contribution to directional statistics and information geometry. The mathematical core is sound: the proofs in Appendix C are standard and correct, the KL-gap identity follows from the nesting assumption rather than being assumed, and the delta-method and second-order null-calibration derivations check out. The experiments are carefully specified with seeds, Monte Carlo standard errors, quadrature sensitivity checks, and explicit statements about what is and is not being tested. The code is provided. The main caveat is that the GID profile is defined only when the relevant moment vectors are in the relative interior of the moment body; exact antipodal point masses, zero-thickness girdles, and other boundary configurations are outside the stated theory. This is a scope limitation rather than an internal inconsistency, and it is acknowledged in the paper, but it needs to be more prominent because the title and abstract claim a broader scope.

major comments (1)
  1. [Section 3, Assumption 4 and Section B.3] The decomposition is undefined on boundary moment vectors. The definition of p_L^P requires m_L(P) in ri(M_L), and Appendix C.1 shows the natural-parameter map is a diffeomorphism only onto the relative interior. Thus, for a weighted empirical measure supported on an exact antipodal pair or a zero-thickness girdle, the level-2 moment lies on the boundary of M_2 and no finite Fisher-Bingham projection exists; Eq. (3) and the profile are simply not defined. This is not a purely formal edge case: the antipodal and girdle examples in the Introduction are the sharp limits of exactly these boundary configurations. The remedy in B.3, mixing with the uniform measure (1-epsilon) P_w + epsilon nu, forces interiority but changes the target; the gaps then describe the shrunken measure, not P_w. The reported experiments use smooth vMF blobs and quadrature grids and therefore do not exercise the bound
minor comments (5)
  1. [Section 6.1] The sentence 'At bR=0, bkappa=0 and the direction is unidentified' could be misread as a failure of the first-level projection. For the sphere, bR=0 is actually in the interior of the moment body and the uniform vMF distribution exists; only bR=1 is a true boundary case with no finite fit. Please clarify this distinction.
  2. [Section 8.3 and Table 3] The calibration experiment tests the quadratic surrogate eI2, not the exactly fitted gap bI2. The text explicitly and honestly notes that the finite-sample remainder bI2 - eI2 is not assessed, but the section title 'Null calibration on S^2' may still mislead readers. A sentence in the table caption or section heading clarifying that only the leading quadratic term is validated would help.
  3. [Appendix A.1 and Section 7] The theory assumes exact log-partitions and optimizers, while implementation uses fixed quadrature rules. The paper correctly lists the required o_p(a_n^-1) and o_p(a_n^-2) numerical-error rates and reports sensitivity checks. Still, a short main-text remark that the inferential guarantees are asymptotic in both sample size and quadrature error would make the idealization easier for readers to track.
  4. [Theorem 8] In the displayed equation for the weighted chi-square limit, the matrix S is the Schur complement previously defined, but the notation S^{-1/2} N0 Sigma_L N0^T S^{-1/2} could be misread as using the sample covariance. A parenthetical reminder that S is the Schur complement and Sigma_L is the moment covariance would improve readability.
  5. [Header] The arXiv header lists a future version date, '29 Jul 2026'. Please ensure the final version has the correct submission/revision dates.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained.

full rationale

The derivation chain is self-contained. Theorem 1's KL-gap identity is proved in Appendix C.2 from nesting of feature spaces and shared moment constraints, not assumed: because p_L^P and p_{L-1}^P share all V_{L-1} moments, the proof shows KL(p_L^P nu || p_{L-1}^P nu) = D_L(P)-D_{L-1}(P). The estimation results are plug-in delta-method and second-order quadratic-form results for the population functionals D_L and I_L; no fitted constant is relabeled as a prediction. Section 6's identifications of level 1 with vMF and level 2 with Fisher-Bingham-type structure are consequences of the chosen feature spaces, not circular outputs. The main caveat, Assumption 4's interior-moment restriction, is explicit: the paper states that boundary moments can force degenerate solutions, and Appendix B.3 discloses that the shrinkage remedy changes the target measure. This is a scope limitation rather than a circular step. No load-bearing self-citation appears: the Pythagorean identity is attributed to external sources (Csiszar 1975; Amari 2001), and the authors' code repository link is not part of the argument. Accordingly, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central derivation rests on standard exponential-family duality and the Pythagorean KL decomposition; these are legitimate axioms from prior literature. The main paper-specific assumptions are interior moments and exact integration, both stated explicitly. User choices (feature hierarchy, quadrature size, tolerances, residual cutoff) are degrees of freedom but not hidden fitted constants. No invented entities are postulated.

free parameters (3)
  • Feature hierarchy (V_0 ⊂ V_1 ⊂ ... ⊂ V_L) = User-chosen; experiments use Fourier levels 1-4 on S^1 and linear+quadratic on S^2
    The gaps are defined relative to the chosen feature spaces; changing levels or structured subspaces changes the profile. No data-driven selection rule is provided.
  • Quadrature/resolution and optimization tolerances = 4096-point grid (S^1); 30,000/50,000/80,000 Fibonacci points (S^2); BFGS dual-gradient tol 1e-6; moment-residual tol 1e-
    Theory assumes exact log-partitions and optimizers; finite rules are user-selected and their error rates for null inference are stated as requirements, not verified.
  • Residual cutoff K and weights a_l = K=8, a_l=1 in the main-text residual example
    The MMD/harmonic residual diagnostic in B.1 requires choosing K and weights; used only descriptively.
assumptions (6)
  • domain assumption The feature spaces are nested finite-dimensional subspaces of L^2_0(M, ν) (Section 3)
    All identities and monotonicity rely on nesting; structured or projected features must also be nested to remain exact GID.
  • domain assumption Relevant population and empirical moments lie in the relative interior of the moment body (Assumption 4, Section 3)
    Existence of a finite max-entropy projection fails for boundary moments; bR=0 and bR=1 are explicitly acknowledged in Section 6.1; Section B.3 shrinkage changes the target measure.
  • standard math Standard regular exponential-family duality and existence theory (Brown 1986; Csiszar & Matus 2003)
    Used in Appendix C.1 to construct p_L,m and identify ri(M_L) with the natural parameter space.
  • domain assumption Moment CLT for the relevant sampling design (Eq. 5; C.10-C.11)
    Consistency and normality require a law of large numbers/CLT; for i.i.d. deterministic weights a Lindeberg condition is imposed, and for self-normalized IS finite second moment of the influence function is required.
  • ad hoc to paper Exact log-partition evaluation and optimization (Appendix A.1, Numerical inference)
    The theory assumes exact integration; a fixed quadrature rule guarantees neither o_p(a_n^-1) nor o_p(a_n^-2) numerical error. This is an idealization the paper itself flags.
  • domain assumption Correct specification and uniform weights for the exact chi-square null (Theorem 8, final statement)
    The clean chi-square limit holds only under the reduced model with equal weights; otherwise the weighted/sandwich quadratic-form calibration (8) is needed.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Geometric Information Decomposition for Weighted Empirical Measures on the Sphere." pith.science (2026). https://pith.science/paper/ARA7QIE6

@misc{pith2026260702943,
  author       = {Pith},
  title        = {Pith review of: Geometric Information Decomposition for Weighted Empirical Measures on the Sphere},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARA7QIE6}},
  note         = {Machine review of arXiv:2607.02943}
}
read the original abstract

Weighted observations on the unit sphere arise in importance sampling, quadrature, and attention-weighted embeddings. Directional uncertainty is often summarized through a von Mises-Fisher (vMF) fit and its concentration or entropy. This summary uses only mean-direction information. It can miss antipodal, axial, girdle-like, or multimodal structure. We introduce geometric information decomposition (GID), which fits nested maximum-entropy projections to spherical features. Each gap measures the entropy reduction contributed by one feature level. The first gap is the fitted vMF distribution's KL divergence from uniformity. The second measures residual quadratic information, including Fisher-Bingham anisotropy. Later gaps describe finer angular structure. We establish invariance, consistency, alternative-regime asymptotic normality, and quadratic-form null calibration. Circular and spherical experiments include importance-weight calibration and a query-weighted digit projection. The results separate settings where vMF uncertainty is adequate from settings with higher-order structure.

Figures

Figures reproduced from arXiv: 2607.02943 by the authors.

Figure 1
Figure 1. Geometry-to-information fingerprint on S 2 . The top row shows representative point clouds under an orthographic projection and the bottom row reports the first three information gaps from the harmonic maximum-entropy hierarchy. vMF structure is first-order, antipodal and girdle structure are second-order, and the tetrahedral example first appears at a higher harmonic level. level L. The increment IL(P) = DL(P) − DL… view at source ↗
Figure 2
Figure 2. Empirical CDFs of null-calibration p-values for the three settings in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Trimodal circle example. The empirical ker [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Numerical validation of the local interpretations on [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

295 extracted references · 11 linked inside Pith

  1. [1]

    Causality:

    Pearl, Judea , year =. Causality:

  2. [2]

    and Ivanov, Sergej and Ivanov, Sergei V

    Burago, Dmitrij Jurʹevič and Burago, Jurij D. and Ivanov, Sergej and Ivanov, Sergei V. , year =. A

  3. [3]

    Methods of

    Amari, Shun'ichi and Nagaoka, Hiroshi and Amari, Shun'ichi and Amari, Shun'ichi , translator =. Methods of

  4. [4]

    Lange, Richard D and Kwok, Devin and Matelsky, Jordan Kyle and Wang, Xinyue and Rolnick, David and Kording, Konrad , editor =. Deep. Proceedings of 2nd. 2023 , pages =

    2023

  5. [5]

    Nature Communications , author =

    Single-cell analysis reveals inflammatory interactions driving macular degeneration , volume =. Nature Communications , author =. 2023 , pages =. doi:10.1038/s41467-023-37025-7 , language =

    doi:10.1038/s41467-023-37025-7 2023

  6. [6]

    The Annals of Statistics , author =

    Data-driven. The Annals of Statistics , author =. doi:10.1214/009053607000000541 , number =

    doi:10.1214/009053607000000541

  7. [7]

    , month = jan, year =

    Brown, Lawrence D. , month = jan, year =. Fundamentals of. doi:10.1214/lnms/1215466757 , urldate =

    arXiv

  8. [8]

    Transactions of the American Mathematical Society , author =

    Strong laws for \. Transactions of the American Mathematical Society , author =. 1996 , pages =. doi:10.1090/S0002-9947-96-01681-9 , language =

    doi:10.1090/s0002-9947-96-01681-9 1996

Show all 295 references
  1. [9]

    Journal of the American Statistical Association , author =

    Asymptotic. Journal of the American Statistical Association , author =. 1987 , pages =. doi:10.1080/01621459.1987.10478472 , language =

    1987

  2. [10]

    Journal of Machine Learning Research , author =

    Scikit-learn:. Journal of Machine Learning Research , author =. 2011 , pages =

    2011

  3. [11]

    Biometrika , author =

    Saddlepoint approximations for the. Biometrika , author =. 2005 , pages =. doi:10.1093/biomet/92.2.465 , language =

    2005 doi

  4. [12]

    Computational Statistics , author =

    On maximum likelihood estimation of the concentration parameter of von. Computational Statistics , author =. 2014 , pages =. doi:10.1007/s00180-013-0471-0 , language =

    2014 doi

  5. [13]

    IEEE Transactions on Information Theory , author =

    Information projections revisited , volume =. IEEE Transactions on Information Theory , author =. 2003 , pages =. doi:10.1109/TIT.2003.810633 , language =

    2003

  6. [15]

    IEEE Transactions on Information Theory , author =

    Information geometry on hierarchy of probability distributions , volume =. IEEE Transactions on Information Theory , author =. 2001 , pages =. doi:10.1109/18.930911 , number =

    2001 doi

  7. [16]

    Nature Reviews Neuroscience , author =

    Neural tuning and representational geometry , volume =. Nature Reviews Neuroscience , author =. 2021 , pages =. doi:10.1038/s41583-021-00502-3 , language =

    2021 doi

  8. [17]

    New insights and perspectives on the natural gradient method , url =

    Martens, James , month = sep, year =. New insights and perspectives on the natural gradient method , url =

  9. [18]

    International Statistical Review , author =

    Directions. International Statistical Review , author =. 2022 , pages =. doi:10.1111/insr.12481 , abstract =

    2022 doi

  10. [19]

    doi:10.1007/978-981-19-1044-9 , language =

    Directional. doi:10.1007/978-981-19-1044-9 , language =

    doi

  11. [20]

    On the spherical

    You, Kisung and Shung, Dennis , year =. On the spherical. doi:10.48550/ARXIV.2208.11929 , abstract =

    doi arXiv

  12. [21]

    Statistics and Computing , author =

    Learning over von. Statistics and Computing , author =. 2026 , pages =. doi:10.1007/s11222-026-10844-6 , language =

    2026 doi

  13. [22]

    Diethe, Tom , month = feb, year =. A. doi:10.48550/arXiv.1502.07104 , abstract =

    doi arXiv

  14. [23]

    doi:10.18637/jss.v058.i10 , language =

    Journal of Statistical Software , author =. doi:10.18637/jss.v058.i10 , language =

    doi

  15. [24]

    Communications in Statistics - Theory and Methods , author =

    A solution to the multivariate behrens-fisher problem , volume =. Communications in Statistics - Theory and Methods , author =. 1986 , pages =. doi:10.1080/03610928608829342 , language =

    1986 doi

  16. [25]

    Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , author =

    Dispersion on a. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences , author =. 1953 , pages =. doi:10.1098/rspa.1953.0064 , language =

    1953

  17. [26]

    Annals of Eugenics , author =

    The. Annals of Eugenics , author =. 1936 , pages =. doi:10.1111/j.1469-1809.1936.tb02137.x , language =

    1936

  18. [27]

    Translational Psychiatry , author =

    Brain network dysfunctions in addiction: a meta-analysis of resting-state functional connectivity , volume =. Translational Psychiatry , author =. 2022 , pages =. doi:10.1038/s41398-022-01792-6 , number =

    2022 doi

  19. [28]

    SIAM Journal on Matrix Analysis and Applications , author =

    Permutation-. SIAM Journal on Matrix Analysis and Applications , author =. 2024 , pages =. doi:10.1137/22M1538144 , language =

    2024 doi

  20. [29]

    NeuroImage , author =

    Resting state functional connectivity in addiction:. NeuroImage , author =. 2012 , pages =. doi:10.1016/j.neuroimage.2012.01.117 , language =

    2012 doi

  21. [30]

    Frontiers in Human Neuroscience , author =

    Sex. Frontiers in Human Neuroscience , author =. 2019 , pages =. doi:10.3389/fnhum.2019.00104 , urldate =

    2019

  22. [31]

    Trends in Cognitive Sciences , author =

    Functional connectomics from resting-state. Trends in Cognitive Sciences , author =. 2013 , pages =. doi:10.1016/j.tics.2013.09.016 , language =

    2013 doi

  23. [32]

    Cerebral Cortex , author =

    Linked. Cerebral Cortex , author =. 2015 , pages =. doi:10.1093/cercor/bhu036 , language =

    2015 doi

  24. [33]

    ETS Research Bulletin Series , author =

    Estimating causal effects of treatments in randomized and nonrandomized studies , volume =. ETS Research Bulletin Series , author =. doi:10.1002/j.2333-8504.1972.tb00631.x , abstract =

    1972

  25. [34]

    Biometrika , author =

    The. Biometrika , author =. 1983 , pages =. doi:10.1093/biomet/70.1.41 , language =

    1983 doi

  26. [35]

    Journal of the American Statistical Association , author =

    Estimation of. Journal of the American Statistical Association , author =. 1994 , pages =. doi:10.1080/01621459.1994.10476818 , language =

    1994

  27. [36]

    Biometrika , author =

    Quasi-oracle estimation of heterogeneous treatment effects , volume =. Biometrika , author =. 2021 , pages =. doi:10.1093/biomet/asaa076 , abstract =

    2021 doi

  28. [37]

    The Annals of Applied Statistics , author =

    Propensity score weighting for causal inference with multiple treatments , volume =. The Annals of Applied Statistics , author =. doi:10.1214/19-AOAS1282 , number =

    doi

  29. [38]

    Journal of the American Statistical Association , author =

    Balancing. Journal of the American Statistical Association , author =. 2018 , pages =. doi:10.1080/01621459.2016.1260466 , language =

    2018

  30. [39]

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

    Causal inference on distribution functions , volume =. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2023 , pages =. doi:10.1093/jrsssb/qkad008 , language =

    2023 doi

  31. [40]

    Geodesic

    Kurisu, Daisuke and Zhou, Yidong and Otsu, Taisuke and Müller, Hans-Georg , year =. Geodesic. doi:10.48550/ARXIV.2406.19604 , abstract =

    doi arXiv

  32. [41]

    Proceedings of the National Academy of Sciences , author =

    Sex beyond the genitalia:. Proceedings of the National Academy of Sciences , author =. 2015 , pages =. doi:10.1073/pnas.1509654112 , abstract =

    2015 doi

  33. [42]

    and Robins, James M

    Hernan, Miguel A. and Robins, James M. , year =. Causal

  34. [43]

    The Econometrics Journal , author =

    Double/debiased machine learning for treatment and structural parameters , volume =. The Econometrics Journal , author =. 2018 , pages =. doi:10.1111/ectj.12097 , language =

    2018 doi

  35. [44]

    The Annals of Probability , author =

    Central limit theorems and bootstrap in high dimensions , volume =. The Annals of Probability , author =. doi:10.1214/16-AOP1113 , number =

    doi

  36. [45]

    The Annals of Statistics , author =

    Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors , volume =. The Annals of Statistics , author =. doi:10.1214/13-AOS1161 , number =

    doi

  37. [46]

    Trends in Cognitive Sciences , author =

    Large-scale brain systems in. Trends in Cognitive Sciences , author =. 2012 , pages =. doi:10.1016/j.tics.2011.11.007 , language =

    2012 doi

  38. [47]

    Biological Psychiatry: Cognitive Neuroscience and Neuroimaging , author =

    Intrinsic. Biological Psychiatry: Cognitive Neuroscience and Neuroimaging , author =. 2016 , pages =. doi:10.1016/j.bpsc.2016.03.004 , language =

    2016 doi

  39. [48]

    Riemannian

    Collas, Antoine and Ju, Ce and Salvy, Nicolas and Thirion, Bertrand , year =. Riemannian. doi:10.48550/ARXIV.2505.18193 , abstract =

    doi

  40. [49]

    Biometrics , author =

    Doubly. Biometrics , author =. 2005 , pages =. doi:10.1111/j.1541-0420.2005.00377.x , language =

    2005

  41. [50]

    The Annals of Statistics , author =

    Generalized random forests , volume =. The Annals of Statistics , author =. doi:10.1214/18-AOS1709 , number =

    doi

  42. [51]

    Magnetic Resonance in Medicine , author =

    Log‐. Magnetic Resonance in Medicine , author =. 2006 , pages =. doi:10.1002/mrm.20965 , abstract =

    2006 doi

  43. [52]

    1998 , keywords =

    Efficient and. 1998 , keywords =

    1998

  44. [53]

    Chwialkowski, Kacper and Strathmann, Heiko and Gretton, Arthur , editor =. A. Proceedings of. 2016 , pages =

    2016

  45. [54]

    Physical Review , author =

    On the. Physical Review , author =. 1930 , pages =. doi:10.1103/PhysRev.36.823 , language =

    1930 doi

  46. [55]

    Orthogonal polynomials , isbn =

    Szegő, Gábor , year =. Orthogonal polynomials , isbn =

  47. [56]

    Journal of Multivariate Analysis , author =

    A new test for multivariate normality , volume =. Journal of Multivariate Analysis , author =. 2005 , pages =. doi:10.1016/j.jmva.2003.12.002 , language =

    2005 doi

  48. [57]

    Rayner, J. C. W. and Thas, O. and Best, D. J. , month = apr, year =. Smooth. doi:10.1002/9780470824443 , language =

    doi

  49. [58]

    The Annals of Statistics , author =

    \. The Annals of Statistics , author =. doi:10.1214/aos/1176350374 , number =

  50. [59]

    Scandinavian Actuarial Journal , author =

    Smooth test for goodness of fit , volume =. Scandinavian Actuarial Journal , author =. 1937 , pages =. doi:10.1080/03461238.1937.10404821 , language =

    1937

  51. [60]

    Journal für die reine und angewandte Mathematik , author =

    Ueber die. Journal für die reine und angewandte Mathematik , author =. 1866 , pages =

  52. [61]

    Biometrika , author =

    Measures of multivariate skewness and kurtosis with applications , volume =. Biometrika , author =. 1970 , pages =. doi:10.1093/biomet/57.3.519 , language =

    1970 doi

  53. [62]

    Liu, Qiang and Lee, Jason and Jordan, Michael , editor =. A. Proceedings of. 2016 , pages =

    2016

  54. [63]

    Gaussian

    Janson, Svante , month = jun, year =. Gaussian. doi:10.1017/CBO9780511526169 , abstract =

    doi

  55. [64]

    IEEE Transactions on Neural Networks , author =

    Iterative. IEEE Transactions on Neural Networks , author =. 2011 , pages =. doi:10.1109/TNN.2011.2106511 , number =

    2011

  56. [65]

    Journal of Machine Learning Research , author =

    Composite. Journal of Machine Learning Research , author =. 2025 , pages =

    2025

  57. [66]

    Bernoulli , author =

    A one-sample test for normality with kernel methods , volume =. Bernoulli , author =. doi:10.3150/18-BEJ1037 , number =

    doi

  58. [67]

    Communications in Statistics - Theory and Methods , author =

    A class of invariant consistent tests for multivariate normality , volume =. Communications in Statistics - Theory and Methods , author =. 1990 , pages =. doi:10.1080/03610929008830400 , language =

    1990 doi

  59. [68]

    Journal of Machine Learning Research , author =

    Spectral. Journal of Machine Learning Research , author =. 2024 , pages =

    2024

  60. [69]

    Biometrika , author =

    A test for normality based on the empirical characteristic function , volume =. Biometrika , author =. 1983 , pages =. doi:10.1093/biomet/70.3.723 , language =

    1983 doi

  61. [70]

    Gaussianization , volume =

    Chen, Scott and Gopinath, Ramesh , editor =. Gaussianization , volume =. Advances in

  62. [71]

    Russian Mathematical Surveys , author =

    Ornstein–. Russian Mathematical Surveys , author =. 2018 , pages =. doi:10.1070/RM9812 , number =

    2018 doi

  63. [72]

    Metrika , author =

    A consistent test for multivariate normality based on the empirical characteristic function , volume =. Metrika , author =. 1988 , pages =. doi:10.1007/BF02613322 , language =

    1988 doi

  64. [73]

    Journal of Machine Learning Research , author =

    On the. Journal of Machine Learning Research , author =. 2021 , pages =

    2021

  65. [74]

    Transactions of the American Mathematical Society , author =

    Theory of. Transactions of the American Mathematical Society , author =. 1950 , pages =. doi:10.1090/S0002-9947-1950-0051437-7 , language =

    1950 doi

  66. [75]

    The Annals of Probability , author =

    Limit. The Annals of Probability , author =. doi:10.1214/aop/1176989128 , number =

  67. [77]

    Rosenberg, Steven , month = jan, year =. The. doi:10.1017/CBO9780511623783 , urldate =

    doi

  68. [78]

    Smola, Alex and Gretton, Arthur and Song, Le and Schölkopf, Bernhard , editor =. A. Algorithmic. 2007 , pages =. doi:10.1007/978-3-540-75225-7_5 , urldate =

    2007 doi

  69. [79]

    Signal Processing , author =

    Effective sample size for importance sampling based on discrepancy measures , volume =. Signal Processing , author =. 2017 , pages =. doi:10.1016/j.sigpro.2016.08.025 , language =

    2017 doi

  70. [80]

    Ecology , author =

    Measuring diversity: the importance of species similarity , volume =. Ecology , author =. 2012 , pages =. doi:10.1890/10-2402.1 , language =

    2012 doi

  71. [81]

    Entropy , author =

    Approximation of. Entropy , author =. 2019 , pages =. doi:10.3390/e21010043 , language =

    2019 doi

  72. [82]

    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , author =

    The heat equation on manifolds as a gradient flow in the. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques , author =. doi:10.1214/08-AIHP306 , number =

    doi

  73. [83]

    Computer Graphics Forum , author =

    A. Computer Graphics Forum , author =. 2009 , pages =. doi:10.1111/j.1467-8659.2009.01515.x , language =

    2009

  74. [84]

    TEST , author =

    Recent advances in directional statistics , volume =. TEST , author =. 2021 , pages =. doi:10.1007/s11749-021-00759-x , language =

    2021 doi

  75. [85]

    Statistics & Probability Letters , author =

    Kernel density estimation on. Statistics & Probability Letters , author =. 2005 , pages =. doi:10.1016/j.spl.2005.04.004 , language =

    2005 doi

  76. [86]

    Entropy , author =

    On. Entropy , author =. 2026 , pages =. doi:10.3390/e28030282 , abstract =

    2026 doi

  77. [87]

    Applied and Computational Harmonic Analysis , author =

    Small time asymptotics of the entropy of the heat kernel on a. Applied and Computational Harmonic Analysis , author =. 2024 , pages =. doi:10.1016/j.acha.2024.101642 , language =

    2024

  78. [88]

    Entropy , author =

    k-. Entropy , author =. 2011 , pages =. doi:10.3390/e13030650 , abstract =

    2011 doi

  79. [89]

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

    The. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 1982 , pages =. doi:10.1111/j.2517-6161.1982.tb01189.x , abstract =

    1982

  80. [90]

    Physical Review , author =

    Information. Physical Review , author =. 1957 , pages =. doi:10.1103/PhysRev.106.620 , language =

    1957 doi

  81. [91]

    Biometrika , author =

    Kernel density estimation with spherical data , volume =. Biometrika , author =. 1987 , pages =. doi:10.1093/biomet/74.4.751 , language =

    1987 doi

  82. [92]

    Grigoryan, Alexander , year =. Heat

  83. [93]

    An overview of uniformity tests on the hypersphere , copyright =

    García-Portugués, Eduardo and Verdebout, Thomas , year =. An overview of uniformity tests on the hypersphere , copyright =. doi:10.48550/ARXIV.1804.00286 , abstract =

    doi arXiv

  84. [94]

    , month = feb, year =

    Dupuis, Paul and Ellis, Richard S. , month = feb, year =. A. doi:10.1002/9781118165904 , language =

    doi

  85. [95]

    Communications on Pure and Applied Mathematics , author =

    Asymptotic evaluation of certain markov process expectations for large time,. Communications on Pure and Applied Mathematics , author =. 1975 , pages =. doi:10.1002/cpa.3160280102 , language =

    1975 doi

  86. [96]

    The Annals of Probability , author =

    \. The Annals of Probability , author =. doi:10.1214/aop/1176996454 , number =

  87. [97]

    and Welling, Max , year =

    Cohen, Taco S. and Welling, Max , year =. Harmonic. Proceedings of the 32nd

  88. [99]

    Intrinsic effective sample size for manifold-valued

    You, Kisung , year =. Intrinsic effective sample size for manifold-valued. doi:10.48550/ARXIV.2605.03266 , abstract =

    doi arXiv

  89. [100]

    Colourful

    Bárány, Imre and Onn, Shmuel , month = aug, year =. Colourful. Mathematics of Operations Research , publisher =

  90. [101]

    Riemannian

    do Carmo, Manfredo Perdigão , year =. Riemannian

  91. [102]

    Bayesian Analysis , author =

    Rank-. Bayesian Analysis , author =. doi:10.1214/20-BA1221 , number =

    doi

  92. [103]

    Proceedings of the London Mathematical Society , author =

    Probability,. Proceedings of the London Mathematical Society , author =. 1990 , pages =. doi:10.1112/plms/s3-61.2.371 , language =

    1990 doi

  93. [104]

    Biometrika , author =

    Fréchet analysis of variance for random objects , volume =. Biometrika , author =. 2019 , pages =. doi:10.1093/biomet/asz052 , abstract =

    2019 doi

  94. [105]

    The Annals of Statistics , author =

    Fréchet change-point detection , volume =. The Annals of Statistics , author =. doi:10.1214/19-AOS1930 , number =

    doi

  95. [106]

    and Border, Kim C

    Aliprantis, Charalambos D. and Border, Kim C. , year =. Infinite. doi:10.1007/3-540-29587-9 , language =

    doi

  96. [107]

    Bayesian Analysis , author =

    Nested. Bayesian Analysis , author =. doi:10.1214/24-BA1453 , number =

    doi

  97. [108]

    Bayesian Analysis , author =

    R∗:. Bayesian Analysis , author =. doi:10.1214/20-BA1252 , number =

    doi

  98. [109]

    Bayesian Analysis , author =

    A. Bayesian Analysis , author =. doi:10.1214/14-BA888 , number =

    doi

  99. [110]

    2016 , pages =

    Geochemistry, Geophysics, Geosystems , author =. 2016 , pages =. doi:10.1002/2016GC006307 , abstract =

    2016 doi

  100. [111]

    Information Systems , author =

    H-. Information Systems , author =. 2004 , pages =. doi:10.1016/j.is.2003.10.002 , language =

    2004 doi

  101. [112]

    IEEE Transactions on Medical Imaging , author =

    A. IEEE Transactions on Medical Imaging , author =. 2004 , pages =. doi:10.1109/TMI.2004.831218 , language =

    2004

  102. [113]

    Self-tuning spectral clustering , booktitle =

    Zelnik-manor, Lihi and Perona, Pietro , editor =. Self-tuning spectral clustering , booktitle =. 2004 , pages =

    2004

  103. [114]

    and Girotra, A

    Costa, J.A. and Girotra, A. and Hero, A.O. , year =. Estimating. doi:10.1109/SSP.2005.1628631 , urldate =

    2005

  104. [115]

    and Thomas, Joy A

    Cover, Thomas M. and Thomas, Joy A. , month = sep, year =. Elements of. doi:10.1002/047174882X , language =

    doi

  105. [116]

    and Paganini, Fernando , year =

    Dullerud, Geir E. and Paganini, Fernando , year =. A course in robust control theory: a convex approach , isbn =

  106. [117]

    Acta Numerica , author =

    Random matrix theory , volume =. Acta Numerica , author =. 2005 , pages =. doi:10.1017/S0962492904000236 , abstract =

    2005 doi

  107. [118]

    Measuring

    Gretton, Arthur and Bousquet, Olivier and Smola, Alex and Schölkopf, Bernhard , editor =. Measuring. Algorithmic. 2005 , pages =. doi:10.1007/11564089_7 , urldate =

    2005 doi

  108. [119]

    Laplacian score for feature selection , booktitle =

    He, Xiaofei and Cai, Deng and Niyogi, Partha , year =. Laplacian score for feature selection , booktitle =

  109. [120]

    Hilbertian metrics and positive definite kernels on probability measures , booktitle =

    Hein, Matthias and Bousquet, Olivier , year =. Hilbertian metrics and positive definite kernels on probability measures , booktitle =

  110. [121]

    and Principe, Jose C

    Jenssen, Robert and Erdogmus, Deniz and Hild, Kenneth E. and Principe, Jose C. and Eltoft, Torbjørn , editor =. Optimizing the. Energy. 2005 , pages =. doi:10.1007/11585978_3 , urldate =

    2005 doi

  111. [122]

    N-distances and their applications , isbn =

    Klebanov, Lev Borisovič , year =. N-distances and their applications , isbn =

  112. [123]

    Journal of Machine Learning Research , author =

    Probabilistic non-linear principal component analysis with gaussian process latent variable models , volume =. Journal of Machine Learning Research , author =. 2005 , pages =

    2005

  113. [124]

    , year =

    Lehmann, Erich Leo and Romano, Joseph P. , year =. Testing statistical hypotheses , isbn =

  114. [125]

    SIAM Journal on Matrix Analysis and Applications , author =

    A. SIAM Journal on Matrix Analysis and Applications , author =. 2005 , pages =. doi:10.1137/S0895479803436937 , language =

    2005 doi

  115. [126]

    Communications in Statistics - Simulation and Computation , author =

    A. Communications in Statistics - Simulation and Computation , author =. 2005 , pages =. doi:10.1080/03610910500308495 , language =

    2005 doi

  116. [127]

    The Annals of Statistics , author =

    Markov. The Annals of Statistics , author =. doi:10.1214/aos/1176325750 , number =

  117. [128]

    Annual Review of Statistics and Its Application , author =

    Convergence. Annual Review of Statistics and Its Application , author =. 2020 , pages =. doi:10.1146/annurev-statistics-031219-041300 , abstract =

    2020 doi

  118. [129]

    Journal of Computational and Graphical Statistics , author =

    General. Journal of Computational and Graphical Statistics , author =. 1998 , pages =. doi:10.1080/10618600.1998.10474787 , language =

    1998

  119. [130]

    The Annals of Statistics , author =

    Large sample theory of intrinsic and extrinsic sample means on manifolds—. The Annals of Statistics , author =. doi:10.1214/009053605000000093 , number =

    doi

  120. [131]

    and Casella, George , year =

    Robert, Christian P. and Casella, George , year =. Monte. doi:10.1007/978-1-4757-3071-5 , urldate =

    doi

  121. [132]

    and Tweedie, Richard L

    Meyn, Sean P. and Tweedie, Richard L. , year =. Markov. doi:10.1007/978-1-4471-3267-7 , urldate =

    doi

  122. [133]

    Entropy , author =

    Information-. Entropy , author =. 2014 , pages =. doi:10.3390/e16063074 , abstract =

    2014 doi

  123. [134]

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

    Riemann. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2011 , pages =. doi:10.1111/j.1467-9868.2010.00765.x , abstract =

    2011

  124. [135]

    Journal of the American Statistical Association , author =

    Markov. Journal of the American Statistical Association , author =. 1996 , pages =. doi:10.1080/01621459.1996.10476956 , language =

    1996

  125. [136]

    Pattern Recognition , author =

    Multiple instance learning with bag dissimilarities , volume =. Pattern Recognition , author =. 2015 , pages =. doi:10.1016/j.patcog.2014.07.022 , language =

    2015 doi

  126. [137]

    Learning with kernels: support vector machines, regularization, optimization, and beyond , isbn =

    Schölkopf, Bernhard and Smola, Alexander Johannes , year =. Learning with kernels: support vector machines, regularization, optimization, and beyond , isbn =

  127. [138]

    IEEE Transactions on Neural Networks , author =

    A. IEEE Transactions on Neural Networks , author =. 2002 , pages =. doi:10.1109/72.991427 , number =

    2002 doi

  128. [139]

    Machine Learning , author =

    Support-vector networks , volume =. Machine Learning , author =. 1995 , pages =. doi:10.1007/BF00994018 , language =

    1995 doi

  129. [140]

    and Guyon, Isabelle M

    Boser, Bernhard E. and Guyon, Isabelle M. and Vapnik, Vladimir N. , month = jul, year =. A. Proceedings of the. doi:10.1145/130385.130401 , language =

  130. [141]

    Colourful linear programming , volume =

    Bárány, Imre and Onn, Shmuel , editor =. Colourful linear programming , volume =. Integer. 1996 , pages =. doi:10.1007/3-540-61310-2_1 , urldate =

    1996 doi

  131. [142]

    Israel Journal of Mathematics , author =

    Tverberg’s theorem via number fields , volume =. Israel Journal of Mathematics , author =. 1992 , pages =. doi:10.1007/BF02808223 , language =

    1992 doi

  132. [143]

    Journal of the London Mathematical Society , author =

    A. Journal of the London Mathematical Society , author =. 1966 , pages =. doi:10.1112/jlms/s1-41.1.123 , language =

    1966 doi

  133. [144]

    Probability in

    Ledoux, Michel and Talagrand, Michel , year =. Probability in. doi:10.1007/978-3-642-20212-4 , urldate =

    doi

  134. [146]

    Algorithmic

    Vovk, Vladimir and Gammerman, Alexander and Shafer, Glenn , year =. Algorithmic. doi:10.1007/978-3-031-06649-8 , language =

    doi

  135. [147]

    Journal of Machine Learning Research , author =

    On the. Journal of Machine Learning Research , author =. 2007 , pages =

    2007

  136. [148]

    Statistical

    Zhang, Tong , year =. Statistical. The Annals of Statistics , publisher =

  137. [149]

    Journal of the American Statistical Association , author =

    Multicategory. Journal of the American Statistical Association , author =. 2004 , pages =. doi:10.1198/016214504000000098 , language =

    2004 doi

  138. [150]

    Foundations of

    Mohri, Mehryar and Rostamizadeh, Afshin and Talwalkar, Ameet , year =. Foundations of

  139. [151]

    and Mendelson, Shahar , editor =

    Bartlett, Peter L. and Mendelson, Shahar , editor =. Rademacher and. Computational. 2001 , pages =. doi:10.1007/3-540-44581-1_15 , urldate =

    2001 doi

  140. [152]

    Sathiya and Sundararajan, S

    Hsieh, Cho-Jui and Chang, Kai-Wei and Lin, Chih-Jen and Keerthi, S. Sathiya and Sundararajan, S. , year =. A dual coordinate descent method for large-scale linear. Proceedings of the 25th international conference on. doi:10.1145/1390156.1390208 , language =

  141. [153]

    Mathematical Programming , author =

    Finding the nearest point in. Mathematical Programming , author =. 1976 , pages =. doi:10.1007/BF01580381 , language =

    1976 doi

  142. [154]

    and Freund, R

    Osuna, E. and Freund, R. and Girosi, F. , year =. An improved training algorithm for support vector machines , isbn =. Neural. doi:10.1109/NNSP.1997.622408 , urldate =

    1997

  143. [155]

    Selective classification for deep neural networks , isbn =

    Geifman, Yonatan and El-Yaniv, Ran , year =. Selective classification for deep neural networks , isbn =. Proceedings of the 31st

  144. [156]

    IEEE Transactions on Information Theory , author =

    On optimum recognition error and reject tradeoff , volume =. IEEE Transactions on Information Theory , author =. 1970 , pages =. doi:10.1109/TIT.1970.1054406 , number =

    1970

  145. [157]

    Steinwart, Ingo and Christmann, Andreas , year =. Support. doi:10.1007/978-0-387-77242-4 , language =

    doi

  146. [158]

    Shawe-Taylor, John and Cristianini, Nello , month = jun, year =. Kernel. doi:10.1017/CBO9780511809682 , urldate =

    doi

  147. [159]

    Sathiya , editor =

    Duan, Kai-Bo and Keerthi, S. Sathiya , editor =. Which. Multiple. 2005 , pages =. doi:10.1007/11494683_28 , urldate =

    2005 doi

  148. [160]

    Rifkin, Ryan and Klautau, Aldebaro , month = dec, year =. In. Journal of Machine Learning Research , publisher =

  149. [161]

    Weston, Jason and Watkins, Christopher , month = jan, year =. Support

  150. [162]

    , year =

    Vapnik, Vladimir N. , year =. The. doi:10.1007/978-1-4757-3264-1 , urldate =

    doi

  151. [163]

    Using the

    Williams, Christopher and Seeger, Matthias , editor =. Using the. Advances in

  152. [164]

    Rahimi, Ali and Recht, Benjamin , editor =. Random. Advances in

  153. [165]

    Sequential

    Platt, John , month = apr, year =. Sequential

  154. [166]

    Mathematical Programming , author =

    Pegasos: primal estimated sub-gradient solver for. Mathematical Programming , author =. 2011 , pages =. doi:10.1007/s10107-010-0420-4 , language =

    2011 doi

  155. [167]

    Revisiting

    Jaggi, Martin , editor =. Revisiting. Proceedings of the 30th. 2013 , pages =

    2013

  156. [168]

    Naval Research Logistics Quarterly , author =

    An algorithm for quadratic programming , volume =. Naval Research Logistics Quarterly , author =. 1956 , pages =. doi:10.1002/nav.3800030109 , language =

    1956 doi

  157. [169]

    2011 , pages =

    ACM Transactions on Intelligent Systems and Technology , author =. 2011 , pages =. doi:10.1145/1961189.1961199 , abstract =

    2011

  158. [170]

    Journal of the American Statistical Association , author =

    Least. Journal of the American Statistical Association , author =. 2019 , pages =. doi:10.1080/01621459.2017.1395341 , language =

    2019

  159. [171]

    Canadian Journal of Statistics , author =

    Classification with reject option , volume =. Canadian Journal of Statistics , author =. 2006 , pages =. doi:10.1002/cjs.5550340410 , abstract =

    2006 doi

  160. [172]

    Machine Learning , author =

    Support. Machine Learning , author =. 2004 , pages =. doi:10.1023/B:MACH.0000008084.60811.49 , language =

    2004

  161. [173]

    Neural Computation , author =

    Estimating the. Neural Computation , author =. 2001 , pages =. doi:10.1162/089976601750264965 , abstract =

    2001 doi

  162. [174]

    Journal of Machine Learning Research , author =

    Learning. Journal of Machine Learning Research , author =. 2005 , pages =

    2005

  163. [175]

    Neural Computation , author =

    On. Neural Computation , author =. 2005 , pages =. doi:10.1162/0899766052530802 , abstract =

    2005 doi

  164. [176]

    Journal of Artificial Intelligence Research , author =

    Solving. Journal of Artificial Intelligence Research , author =. 1995 , pages =. doi:10.1613/jair.105 , abstract =

    1995 doi

  165. [177]

    and Schapire, Robert E

    Allwein, Erin L. and Schapire, Robert E. and Singer, Yoram , month = sep, year =. Reducing. Journal of Machine Learning Research , publisher =. doi:10.1162/15324430152733133 , abstract =

    doi

  166. [178]

    Journal of Computational and Graphical Statistics , author =

    Reinforced. Journal of Computational and Graphical Statistics , author =. 2016 , pages =. doi:10.1080/10618600.2015.1043010 , language =

    2016

  167. [179]

    Equivalence of

    Pouliot, Guillaume , editor =. Equivalence of. Proceedings of the 35th. 2018 , pages =

    2018

  168. [180]

    Multiclass

    Mroueh, Youssef and Poggio, Tomaso and Rosasco, Lorenzo and Slotine, Jean-jeacques , editor =. Multiclass. Advances in. 2012 , keywords =

    2012

  169. [181]

    Journal of Artificial Intelligence Research , author =

    A. Journal of Artificial Intelligence Research , author =. 2007 , pages =. doi:10.1613/jair.2251 , abstract =

    2007 doi

  170. [182]

    Crammer, Koby and Singer, Yoram , month = mar, year =. On the. Journal of Machine Learning Research , publisher =

  171. [183]

    Journal of Machine Learning Research , author =

    Classification with a. Journal of Machine Learning Research , author =. 2008 , pages =

    2008

  172. [184]

    , editor =

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

    doi

  173. [185]

    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 =

    2014 doi

  174. [186]

    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 =

    2013 doi

  175. [187]

    Les équations aux dérivées partielles , author =

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

  176. [188]

    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 =

    1963 doi

  177. [189]

    Mathematical Programming , author =

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

    1973 doi

  178. [190]

    Learning multiple layers of features from tiny images , institution =

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

  179. [191]

    Topics in

    Villani, Cédric , year =. Topics in

  180. [192]

    Villani, Cédric , year =. Optimal

  181. [193]

    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 =

    2003 doi

  182. [194]

    Foundations and Trends® in Machine Learning , author =

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

    2019 doi

  183. [195]

    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

  184. [196]

    Scalable

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

    2021

  185. [197]

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

    doi arXiv

  186. [198]

    Management Science , author =

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

    1958 doi

  187. [199]

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

  188. [200]

    The Annals of Statistics , author =

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

    doi

  189. [201]

    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 =

    1991 doi

  190. [202]

    doi:10.21227/KATB-ZV89 , abstract =

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

    doi

  191. [203]

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

  192. [204]

    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 =

    2021 doi

  193. [205]

    SIAM Journal on Mathematical Analysis , author =

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

    2003 doi

  194. [206]

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

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

  195. [207]

    arXiv:2103.04621 [math] , author =

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

    2021

  196. [208]

    International Journal of Computer Vision , author =

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

    2006 doi

  197. [209]

    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 =

    2006 doi

  198. [210]

    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

  199. [211]

    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

  200. [212]

    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 =

    2017

  201. [213]

    Machine Learning , author =

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

    2018 doi

  202. [214]

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

  203. [215]

    2018 , pages =

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

    2018 doi

  204. [216]

    Frontiers in Neuroscience , author =

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

    2012 arXiv

  205. [217]

    Mathematical Programming , author =

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

    2020 doi

  206. [218]

    2019 , keywords =

    Applied directional statistics , isbn =. 2019 , keywords =

    2019

  207. [219]

    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 =

    2015 doi

  208. [220]

    Computational Statistics , author =

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

    2012 doi

  209. [221]

    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 =

  210. [222]

    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 =

    2011

  211. [223]

    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

  212. [224]

    First-order

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

    2016

  213. [225]

    SIAM Journal on Mathematical Analysis , author =

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

    2024 doi

  214. [226]

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

    doi

  215. [227]

    Communications on Pure and Applied Mathematics , author =

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

    1977 doi

  216. [228]

    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 ...

    doi

  217. [229]

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

    doi arXiv

  218. [230]

    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 =

    2026 doi

  219. [231]

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

    doi arXiv

  220. [232]

    Michigan Mathematical Journal , author =

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

    1984

  221. [233]

    , year =

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

  222. [234]

    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 =

    2018 doi

  223. [235]

    and Jones, M.C

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

    doi

  224. [236]

    The Annals of Statistics , author =

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

  225. [237]

    The Annals of Statistics , author =

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

  226. [238]

    Journal of Computational and Graphical Statistics , author =

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

    2010 arXiv

  227. [239]

    Journal of Classification , author =

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

    2003 doi

  228. [240]

    The Annals of Statistics , author =

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

    doi

  229. [241]

    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 =

    1959 doi

  230. [242]

    and Kumar, Abhishek and Ermon, Stefano and Poole, Ben , year =

    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 =

    2011 doi arXiv

  231. [243]

    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

  232. [244]

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

    2015

  233. [245]

    Cerebral Cortex , author =

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

    2018 doi

  234. [246]

    , month = feb, year =

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

    doi

  235. [247]

    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 =

    1981

  236. [248]

    Technometrics , author =

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

    2001 doi

  237. [249]

    , month = mar, year =

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

    doi

  238. [250]

    Journal d'Analyse Mathématique , author =

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

    1951 doi

  239. [251]

    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 =

    doi

  240. [253]

    The Annals of Statistics , author =

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

    doi

  241. [254]

    Bernoulli , author =

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

    doi

  242. [255]

    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 =

    1997

  243. [256]

    The Annals of Statistics , author =

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

    doi

  244. [257]

    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 =

    2000 doi

  245. [258]

    The Annals of Statistics , author =

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

  246. [259]

    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 =

    2015 doi

  247. [260]

    The Annals of Mathematical Statistics , author =

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

    1962

  248. [261]

    Journal of Computational and Graphical Statistics , author =

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

    1993

  249. [262]

    The Annals of Statistics , author =

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

  250. [263]

    , year =

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

  251. [264]

    The Annals of Statistics , author =

    On a. The Annals of Statistics , author =. doi:10.1214/aos/1176346412 , number =

  252. [265]

    Journal of Applied Statistics , author =

    Scale-space theory: a basic tool for analyzing structures at different scales , volume =. Journal of Applied Statistics , author =. 1994 , pages =. doi:10.1080/757582976 , language =

    1994 doi

  253. [266]

    Biological Cybernetics , author =

    The structure of images , volume =. Biological Cybernetics , author =. 1984 , pages =. doi:10.1007/BF00336961 , language =

    1984 doi

  254. [267]

    The Annals of Mathematical Statistics , author =

    Consistency of the. The Annals of Mathematical Statistics , author =. 1956 , pages =. doi:10.1214/aoms/1177728066 , language =

    1956

  255. [268]

    SIAM Journal on Mathematical Analysis , author =

    The. SIAM Journal on Mathematical Analysis , author =. 1998 , pages =. doi:10.1137/S0036141096303359 , language =

    1998 doi

  256. [269]

    Denoising

    Ho, Jonathan and Jain, Ajay and Abbeel, Pieter , editor =. Denoising. Advances in. 2020 , pages =

    2020

  257. [270]

    , year =

    Hartigan, John A. , year =. Clustering algorithms , isbn =

  258. [271]

    Hall, Peter and York, Matthew , year =. On the. Statistica Sinica , publisher =

  259. [272]

    The Annals of Statistics , author =

    Confidence bands in density estimation , volume =. The Annals of Statistics , author =. doi:10.1214/09-AOS738 , number =

    doi

  260. [273]

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

    Non-. Journal of the Royal Statistical Society Series B: Statistical Methodology , author =. 2016 , pages =. doi:10.1111/rssb.12111 , abstract =

    2016 doi

  261. [274]

    The Annals of Statistics , author =

    Nonparametric ridge estimation , volume =. The Annals of Statistics , author =. doi:10.1214/14-AOS1218 , number =

    doi

  262. [275]

    IEEE Transactions on Information Theory , author =

    The estimation of the gradient of a density function, with applications in pattern recognition , volume =. IEEE Transactions on Information Theory , author =. 1975 , pages =. doi:10.1109/TIT.1975.1055330 , number =

    1975

  263. [276]

    Journal of the American Statistical Association , author =

    Model-. Journal of the American Statistical Association , author =. 2002 , pages =. doi:10.1198/016214502760047131 , language =

    2002 doi

  264. [277]

    Biometrika , author =

    Mode testing via the excess mass estimate , volume =. Biometrika , author =. 2001 , pages =. doi:10.1093/biomet/88.2.499 , language =

    2001 doi

  265. [278]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , author =

    Unsupervised. IEEE Transactions on Pattern Analysis and Machine Intelligence , author =. 2002 , pages =. doi:10.1109/34.990138 , number =

    2002 doi

  266. [279]

    The Annals of Statistics , author =

    A. The Annals of Statistics , author =. doi:10.1214/aos/1176342360 , number =

  267. [280]

    The Annals of Statistics , author =

    Confidence sets for persistence diagrams , volume =. The Annals of Statistics , author =. doi:10.1214/14-AOS1252 , number =

    doi

  268. [281]

    Journal of the American Statistical Association , author =

    Bayesian. Journal of the American Statistical Association , author =. 1995 , pages =. doi:10.1080/01621459.1995.10476550 , language =

    1995

  269. [282]

    Eldridge, Justin and Belkin, Mikhail and Wang, Yusu , editor =. Beyond. Proceedings of. 2015 , pages =

    2015

  270. [283]

    , month = may, year =

    Efron, Bradley and Tibshirani, R.J. , month = may, year =. An. doi:10.1201/9780429246593 , language =

    doi

  271. [284]

    The Annals of Statistics , author =

    Bootstrap. The Annals of Statistics , author =. doi:10.1214/aos/1176344552 , number =

  272. [285]

    , year =

    Edelsbrunner, Herbert and Harer, J. , year =. Computational

  273. [286]

    Computational Statistics & Data Analysis , author =

    Feature significance for multivariate kernel density estimation , volume =. Computational Statistics & Data Analysis , author =. 2008 , pages =. doi:10.1016/j.csda.2008.02.035 , language =

    2008 doi

  274. [287]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , author =

    Mean shift: a robust approach toward feature space analysis , volume =. IEEE Transactions on Pattern Analysis and Machine Intelligence , author =. 2002 , pages =. doi:10.1109/34.1000236 , number =

    2002 doi

  275. [288]

    IEEE Transactions on Pattern Analysis and Machine Intelligence , author =

    Mean shift, mode seeking, and clustering , volume =. IEEE Transactions on Pattern Analysis and Machine Intelligence , author =. 1995 , pages =. doi:10.1109/34.400568 , number =

    1995 doi

  276. [289]

    The Annals of Statistics , author =

    Asymptotic theory for density ridges , volume =. The Annals of Statistics , author =. doi:10.1214/15-AOS1329 , number =

    doi

  277. [290]

    Journal of Machine Learning Research , author =

    Robust. Journal of Machine Learning Research , author =. 2018 , pages =

    2018

  278. [291]

    Foundations of Computational Mathematics , author =

    Geometric. Foundations of Computational Mathematics , author =. 2011 , pages =. doi:10.1007/s10208-011-9098-0 , language =

    2011 doi

  279. [292]

    The Annals of Statistics , author =

    Scale space view of curve estimation , volume =. The Annals of Statistics , author =. doi:10.1214/aos/1016218224 , number =

  280. [293]

    1999 , pages =

    Journal of the American Statistical Association , author =. 1999 , pages =. doi:10.1080/01621459.1999.10474186 , language =

    1999

  281. [294]

    IEEE Transactions on Information Theory , author =

    Consistent. IEEE Transactions on Information Theory , author =. 2014 , pages =. doi:10.1109/TIT.2014.2361055 , number =

    2014

  282. [295]

    Rates of convergence for the cluster tree , booktitle =

    Chaudhuri, Kamalika and Dasgupta, Sanjoy , year =. Rates of convergence for the cluster tree , booktitle =

  283. [296]

    Electronic Journal of Statistics , author =

    Data-driven density derivative estimation, with applications to nonparametric clustering and bump hunting , volume =. Electronic Journal of Statistics , author =. doi:10.1214/13-EJS781 , number =

    doi

  284. [297]

    and Williams, Christopher K

    Carreira-Perpiñán, Miguel Á. and Williams, Christopher K. I. , editor =. On the. Scale. 2003 , pages =. doi:10.1007/3-540-44935-3_44 , language =

    2003 doi

  285. [298]

    Bulletin of the American Mathematical Society , author =

    Topology and data , volume =. Bulletin of the American Mathematical Society , author =. 2009 , pages =. doi:10.1090/S0273-0979-09-01249-X , language =

    2009 doi

  286. [299]

    ACM Transactions on Knowledge Discovery from Data , author =

    Hierarchical. ACM Transactions on Knowledge Discovery from Data , author =. 2015 , pages =. doi:10.1145/2733381 , abstract =

    2015 doi

  287. [300]

    The Annals of Statistics , author =

    Kernel density estimation via diffusion , volume =. The Annals of Statistics , author =. doi:10.1214/10-AOS799 , number =

    doi

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.