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PMODE: Theoretically Grounded and Modular Mixture Modeling

T0 review · 2 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read Mixture modeling as data partitioning: fit each subset with its own estimator, pick the partition by held-out loss, and get near-optimal rates — plus a multi-view model that runs at thousands of dimensions.

desk verdict A genuinely useful generalization of mixture-rate theory, with an MV-PMODE application whose rate claim is softer than it reads. read the letter →

arxiv 2508.21396 v1 pith:LWKCGRHF submitted 2025-08-29 cs.LG

Robert A. Vandermeulen This is my paper · ORCID
classification cs.LG MSC 62G0762G20
keywords mixturemodelingdensityestimationmulti-viewmodelsnaiveBayesmixturesconvergenceratesanomalydetectionkernelCIFAR-10
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

This paper tries to establish that the hard part of mixture modeling — jointly estimating component weights and densities — can be replaced by a divide-and-conquer procedure: partition the data, fit a separate density estimator to each part, and keep the partition whose induced mixture scores best on held-out data. The paper proves that, despite exponentially many candidate partitions, this 'partitioned mixture of density estimators' (PMODE) attains near-optimal rates whenever the component estimators do: eO(n^{-1/(4+max q_i)}) in squared L2 distance and eO(n^{-1/(2+max q_i)}) in KL divergence (eO suppresses log factors), with an extension of prior L1 guarantees to heterogeneous component families. As an application, MV-PMODE instantiates the recipe for multi-view (naive-Bayes) mixtures whose components are products of univariate kernel density estimates, and the paper reports that this shallow, permutation-invariant model scales to thousands of dimensions and holds its own against deep baselines on CIFAR-10 anomaly detection. A sympathetic reader would care because PMODE's objectives, unlike the comparison-based selection procedures of earlier provable methods, support practical local-search heuristics, so a theoretically grounded estimator also points to usable algorithms.

What carries the argument

The central object is the PMODE estimator: split the sample into estimation and validation sets, search assignments of estimation samples to k components, and select the assignment whose induced mixture minimizes an empirical loss on validation data — squared L2, whose cross-term is estimated by averaging the candidate density on validation points while the integral term is computed exactly, or negative log-likelihood, computed in log space. The load-bearing identity for the multi-view application is the product-density decomposition (Reiss 1989, Lemma 3.3.7; Tsybakov 2009, Section 2.3): the L1/KL error of a product of univariate densities is bounded by the sum of per-coordinate errors, turn

What would settle it

A concrete test would settle the scope. Build an anomaly-detection task from two pattern classes with identical per-pixel marginal histograms but different spatial arrangements (stripes versus a checkerboard from the same pixel palette) and measure MV-PMODE's AUROC. Being permutation-invariant, MV-PMODE must score near chance even though a convolutional model separates the classes trivially, showing the CIFAR-10 numbers rest on the multi-view assumption, not on data difficulty. On the theory side, simulate a known naive-Bayes mixture and check whether exhaustive L2/KL-PMODE's empirical error d

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Extended reading notes

Core claim

PMODE rests on a simple observation: mixture estimation is trivial if latent labels are known — group data by label, fit an estimator per group. The claim: this survives unknown labels — search over assignments, keep the mixture minimizing an empirical loss on held-out data. Component estimators of rate eO(n^{-1/q_i}) yield mixture rates eO(n^{-1/(4+max q_i)}) in squared L2 and eO(n^{-1/(2+max q_i)}) in KL. A union bound collapses the exponential search to the best component's rate: one candidate partition matches the true labeling. L2 and log-likelihood objectives allow local-search heuristics, and MV-PMODE instantiates the recipe with product-of-univariate-KDE components.

Load-bearing premise

The load-bearing premise is that the true density is exactly a multi-view mixture (Eq. 3): every component factorizes as a product of independent univariate densities. This is plainly false for CIFAR-10 images, where pixels are strongly spatially correlated, so the method's competitive anomaly-detection numbers depend on per-pixel marginal statistics carrying enough signal by themselves.

Editorial extensions

If this is right

  • The comparison-based selection with its factor-of-three slack — the obstacle that kept prior provable mixture estimators impractical — is bypassed; L2 and log-likelihood objectives admit incremental local moves, so hill climbing and parallel search become heuristics for the same estimator class the theory covers.
  • Multi-view density estimation becomes computationally feasible at thousands of dimensions and dataset sizes like CIFAR-10's; the paper states this is the first application of multi-view modeling at this scale.
  • The framework is modular: an improvement to any univariate estimator — robust, adaptive, or otherwise — transfers directly to the full high-dimensional model without new theory or reworking.
  • The theorems give concrete split-ratio defaults (s = n^{-4/(max q_i+4)} for L2, s = n^{-2/(max q_i+2)} for KL) that practitioners can use and that favor leaving more data for validation.
  • A permutation-invariant shallow density estimator matches deep, spatially aware baselines on several CIFAR-10 classes, indicating that per-pixel marginal statistics carry much of the anomaly signal on this benchmark.

Reading between the lines

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

  • MV-PMODE's permutation invariance draws a sharp boundary: any anomaly signal living only in spatial layout (identical pixel marginals, different arrangements) is invisible to it. Constructing such paired classes and measuring the gap between MV-PMODE and a convolutional model would quantify how much of CIFAR-10's signal is marginal — a test the paper does not run.
  • The split-ratio formulas suggest a finite-sample bias-variance trade the paper leaves open: a larger estimation set feeds the component estimators but weakens selection. A data-driven or cross-validated rule for s could change practical performance materially and is directly testable.
  • The Diabetes experiments hint that hard partitions lose flexibility as the number of components grows; the paper floats overlapping partitions (components sharing data points) as a variant, which a reader could implement and compare against the current hard-partition version to see whether the decline is a local-minima artifact or a modeling ceiling.
  • Because each marginal is an independent univariate KDE, MV-PMODE composes with any univariate improvement — robust KDEs, adaptive bandwidths — without further theory, making it a convenient testbed for isolating which one-dimensional choices actually drive high-dimensional performance.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces PMODE (Partitioned Mixture Of Density Estimators), a framework for mixture density estimation in which the data are partitioned into subsets, a base estimator is applied to each subset, and the partition is selected by minimizing an empirical validation loss (squared L2 or KL divergence). Theoretical rates are proved for the exhaustive-search version of PMODE: Theorems 2.2 and 2.3 give eO(n^{-1/(4+q)}) L2 and eO(n^{-1/(2+q)}) KL rates when component estimators have rates eO(n^{-1/q}), generalizing a prior result of Ashtiani et al. to heterogeneous components and to direct loss evaluation. The paper then proposes MV-PMODE, an instantiation where each mixture component is a product of univariate Gaussian KDEs with Silverman bandwidths, and reports CIFAR-10 one-vs-rest anomaly detection results that are competitive with DSVDD and ADGAN. Supplementary experiments compare PMODE to EM for Gaussian mixtures on Diabetes and Iris data.

Significance. The theoretical results are a useful extension of the Ashtiani et al. line of work: they separate the choice of loss (L2 or KL) from the Scheffé-comparison machinery, and they explicitly allow heterogeneous component estimator families, which is a meaningful step toward a modular mixture-modeling toolbox. The MV-PMODE application is also notable: it demonstrates that a shallow, parallelizable, nonparametric estimator can be applied to thousands of dimensions and achieves reasonable anomaly detection on CIFAR-10, with the paper reporting full per-class variance and implementation details. If the MV-PMODE rate claim were properly supported, this would be a strong contribution. The empirical finding is interesting even without that support, but the paper's title and framing stake more on the theoretical grounding than the current proof supports.

major comments (2)
  1. [§3.2, Eq. (4), Theorem 2.3] The claim that 'Theorem 2.3 implies that MV-PMODE could plausibly achieve dimension-independent rates on the order of eO(n^{-1/5})' is not a valid consequence of Theorem 2.3. The theorem's hypotheses require each component estimator to satisfy D_KL(p_i || V_i) = eO(n^{-1/q_i}) and all densities in the range of V_i to be bounded below on [0,1]^d. The product of univariate Gaussian KDEs with Silverman bandwidth used in Eq. (4) has no established KL-rate guarantee (Silverman's rule is an MISE-oriented heuristic, not a KL-minimizing bandwidth), and the KDE is not uniformly bounded away from zero on compact support without modification. The paper's own caveat that the conditions 'could be enforced with mild modifications' and that the method 'may still inherit favorable convergence behavior' is not a proof. Because the dimension-independent rate is the main theoretical motivation for MV-PMODE
  2. [Appendix A, proofs of Theorems 2.1–2.3] The proofs contain a systematic omission of the mixture weights in the error-accumulation steps. In the proof of Theorem 2.1, the bound on ||p_{c*} - p||_1 is written as a sum of (|w_i - λ_{B_i}| + ||\hat p_{B_i} - p_i||_1), omitting the factor λ_{B_i} in the second term. The same omission occurs in Step 2 of Theorem 2.3 after applying Lemma A.2: the sum should be α_i D_KL(p_i || \hat p_{B_i}), not an unweighted sum. Without the weights, the displayed inequalities are not valid for components with very small weights, since the unweighted component error can be large while the weighted contribution to the mixture is negligible. The L2 proof in Eq. (5) does include λ_{B_i}, so the intended argument is clear, but the other two proofs must be corrected or supplemented with an argument that handles small weights (e.g., by ignoring negligible-mass components). Since these theorems are the cent
minor comments (5)
  1. [Abstract, throughout] The dataset name is rendered as 'CIF AR-10' in several places (abstract, Section 3.4, Appendix C); it should be 'CIFAR-10'.
  2. [Section 2.3/2.4] In Theorem 2.2 and Theorem 2.3, the split ratio s is given as n^{-4/(max_i q_i+4)} and n^{-2/(max_i q_i+2)}, respectively, but the proofs write 'Letting s = n^{-4/(q+4)}' and 's = n^{-2/(q+2)}' without first defining q. Define q = max_i q_i at the start of each proof.
  3. [Section 3.2, end] The phrase 'Theorem 2.3 implies that MV-PMODE could plausibly achieve...' is internally contradictory: a theorem cannot imply a plausible rate. Rephrase to separate the proven statement from the heuristic extrapolation.
  4. [Appendix D] The experimental section reports that 1200 samples are used for estimation and 3800 for validation, but does not connect this to the split ratio s or to the theory in Section 2. Adding one sentence on why this split was chosen would help the reader relate the experiment to the theoretical framework.
  5. [Section 2.5 and Appendix E] The paper emphasizes that PMODE is 'embarrassingly parallel,' but the corresponding implementation details (10 parallel perturbations, 50-attempt reduction) are given only in the appendix; a brief note in the main text would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PMODE rate theorems are derived from stated component-estimator assumptions, and the MV-PMODE eO(n^{-1/5}) remark is explicitly flagged by the paper as an extrapolation outside the theorem's hypotheses, not a derived claim.

full rationale

Walked the derivation chain. Theorems 2.2 and 2.3 take per-component L2/KL rates as hypotheses and prove mixture rates via the same three-step structure: enumerate all labeled partitions, show one candidate (matching true labels) is close with the stated component rates plus a weight-error n1^{-1/2} term, then select by empirical risk minimization with a union bound. The component rates appear as inputs, not as conclusions; the final exponents (4+max q_i and 2+max q_i) are produced by balancing n1 = s n against component-error and uniform-deviation terms, so the claims are derived, not assumed. No fitted parameter is renamed as a prediction: the split ratio s is set as a function of n in the theorems, and in the experiments validation loss is used in the standard ERM way. The MV-PMODE passage in Section 3.2 explicitly says MV-PMODE 'does not fully satisfy the assumptions of Theorem 2.3' and calls the eO(n^{-1/5}) rate 'plausible' rather than proved. This is a correctness/omitted-support gap — the actual Silverman-rule product-KDE components do not have the theorem's KL-rate or lower-bound hypotheses — but it is not a circular reduction. Same-author citations (Vandermeulen & Ledent 2021; Vandermeulen 2023) and co-authored baselines (DSVDD, ADGAN) are used as background and comparison, not as proof of the PMODE theorems; no load-bearing self-citation chain was found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

PMODE introduces no new physical or mathematical entities. The free parameters listed are practical hyperparameters for the anomaly detection experiment, not part of the theoretical claims. The core theoretical results rest on standard concentration arguments plus the stated rate and boundedness assumptions for component estimators.

free parameters (3)
  • Number of components k = 20 (CIFAR-10 experiment)
    Chosen from a few simple preliminary tests, as stated in Appendix D; affects AUROC results.
  • Split ratio s = 1200 estimation / 3800 validation = 0.24
    The theory suggests s = n^{-2/(q+2)} (roughly 0.04 for q=3 at n=5000), but the experiment uses a much larger estimation fraction, not justified against the theory.
  • Hill-climbing perturbation rates = 5%, 2%, 1%, 0.1%
    Manually set for the discrete optimization heuristic; influences which local optimum is found.
assumptions (4)
  • domain assumption Component estimators Vi achieve eO(n^{-1/q_i}) rates in the relevant metric (L1, L2, or KL)
    Invoked in Theorems 2.1, 2.2, and 2.3; the practical KDEs achieve eO(n^{-1/3}) for Lipschitz densities.
  • domain assumption Densities in the component families and in the range of the estimators are uniformly bounded above (L2 theorem) or bounded above and below (KL theorem)
    Required for Hoeffding concentration on the empirical loss terms in Theorems 2.2 and 2.3.
  • domain assumption True data density has the multi-view form p(x) = sum_i w_i prod_j p_{i,j}(x_j)
    Equation (3) in Section 3.1; the entire MV-PMODE model is misspecified without it.
  • standard math Hoeffding's inequality and union bound arguments
    Used in all proof steps that control uniform deviations of empirical losses over exponentially many candidate partitions.

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Cite this review

Pith. "Pith review of PMODE: Theoretically Grounded and Modular Mixture Modeling." pith.science (2026). https://pith.science/paper/LWKCGRHF

@misc{pith2026250821396,
  author       = {Pith},
  title        = {Pith review of: PMODE: Theoretically Grounded and Modular Mixture Modeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LWKCGRHF}},
  note         = {Machine review of arXiv:2508.21396}
}
read the original abstract

We introduce PMODE (Partitioned Mixture Of Density Estimators), a general and modular framework for mixture modeling with both parametric and nonparametric components. PMODE builds mixtures by partitioning the data and fitting separate estimators to each subset. It attains near-optimal rates for this estimator class and remains valid even when the mixture components come from different distribution families. As an application, we develop MV-PMODE, which scales a previously theoretical approach to high-dimensional density estimation to settings with thousands of dimensions. Despite its simplicity, it performs competitively against deep baselines on CIFAR-10 anomaly detection.

Figures

Figures reproduced from arXiv: 2508.21396 by the authors.

Figure 1
Figure 1. Examples of original and permuted images. The same permutation was used for both images. [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Diabetes test log-likelihood. Markers: means over 30 shuffles; error bars: standard deviation. E.2 Iris Gaussian Mixture Experiment: PMODE vs. EM This experiment followed the same procedure as the previous one, but was conducted on the Iris dataset. We used 120 samples for training and cross-validation, with the remaining 30 used for evaluation. The experiment was run for k ∈ 2, 3, 4, 5, with 30 independent runs for… view at source ↗
Figure 3
Figure 3. Iris test log-likelihood. Markers: means over 30 shuffles; error bars: standard deviation. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_3.png] view at source ↗

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Reference graph

Works this paper leans on

77 extracted references · 58 canonical work pages

  1. [1]

    Sidiropoulos

    Magda Amiridi, Nikos Kargas, and Nicholas D. Sidiropoulos. Low-rank characteristic tensor density estimation part I : Foundations. IEEE Transactions on Signal Processing, 70: 0 2654--2668, 2022 a . doi:10.1109/TSP.2022.3175608

    arXiv 2022

  2. [2]

    Sidiropoulos

    Magda Amiridi, Nikos Kargas, and Nicholas D. Sidiropoulos. Low-rank characteristic tensor density estimation part II : Compression and latent density estimation. Trans. Sig. Proc., 70: 0 2669–2680, January 2022 b . ISSN 1053-587X. doi:10.1109/TSP.2022.3158422. URL https://doi.org/10.1109/TSP.2022.3158422

    work page arXiv 2022

  3. [3]

    Kakade, and Matus Telgarsky

    Animashree Anandkumar, Rong Ge, Daniel Hsu, Sham M. Kakade, and Matus Telgarsky. Tensor decompositions for learning latent variable models. Journal of Machine Learning Research, 15 0 (80): 0 2773--2832, 2014. URL http://jmlr.org/papers/v15/anandkumar14b.html

    work page 2014

  4. [4]

    The more, the merrier: The blessing of dimensionality for learning large G aussian mixtures

    Joseph Anderson, Mikhail Belkin, Navin Goyal, Luis Rademacher, and James Voss. The more, the merrier: The blessing of dimensionality for learning large G aussian mixtures. In Proceedings of The 27th Conference on Learning Theory, pages 1135--1164, 2014

    work page 2014

  5. [5]

    Tight Bounds on the Hardness of Learning Simple Nonparametric Mixtures

    Bryon Aragam and Wai Ming Tai. A super-polynomial lower bound for learning nonparametric mixtures. CoRR, abs/2203.15150, 2022. URL https://doi.org/10.48550/arXiv.2203.15150

    work page Pith review arXiv doi:10.48550/arxiv.2203.15150 2022

  6. [6]

    Uniform consistency in nonparametric mixture models

    Bryon Aragam and Ruiyi Yang. Uniform consistency in nonparametric mixture models . The Annals of Statistics, 51 0 (1): 0 362 -- 390, 2023. doi:10.1214/22-AOS2255. URL https://doi.org/10.1214/22-AOS2255

    work page doi:10.1214/22-aos2255 2023

  7. [7]

    Xing, and Pradeep Ravikumar

    Bryon Aragam, Chen Dan, Eric P. Xing, and Pradeep Ravikumar. Identifiability of nonparametric mixture models and Bayes optimal clustering . The Annals of Statistics, 48 0 (4): 0 2277 -- 2302, 2020 a . doi:10.1214/19-AOS1887. URL https://doi.org/10.1214/19-AOS1887

    doi:10.1214/19-aos1887 2020

  8. [8]

    Xing, and Pradeep Ravikumar

    Bryon Aragam, Chen Dan, Eric P. Xing, and Pradeep Ravikumar. Identifiability of nonparametric mixture models and bayes optimal clustering. Ann. Statist., 48 0 (4): 0 2277--2302, 2020 b . ISSN 0090-5364. doi:10.1214/19-AOS1887. arXiv:1802.04397

    arXiv 2020

Show all 77 references
  1. [9]

    Learning topic models -- going beyond svd

    Sanjeev Arora, Rong Ge, and Ankur Moitra. Learning topic models -- going beyond svd. In Proceedings of the 2012 IEEE 53rd Annual Symposium on Foundations of Computer Science, FOCS '12, pages 1--10, Washington, DC, USA, 2012. IEEE Computer Society. ISBN 978-0-7695-4874-6. doi:1...

    2012 doi

  2. [10]

    A practical algorithm for topic modeling with provable guarantees

    Sanjeev Arora, Rong Ge, Yonatan Halpern, David Mimno, Ankur Moitra, David Sontag, Yichen Wu, and Michael Zhu. A practical algorithm for topic modeling with provable guarantees. In Sanjoy Dasgupta and David McAllester, editors, Proceedings of the 30th International Conference o...

    2013

  3. [11]

    Nearly tight sample complexity bounds for learning mixtures of gaussians via sample compression schemes

    Hassan Ashtiani, Shai Ben-David, Nicholas Harvey, Christopher Liaw, Abbas Mehrabian, and Yaniv Plan. Nearly tight sample complexity bounds for learning mixtures of gaussians via sample compression schemes. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, a...

    2018

  4. [12]

    Sample-efficient learning of mixtures

    Hassan Ashtiani, Shai Ben-David, and Abbas Mehrabian. Sample-efficient learning of mixtures. Proceedings of the AAAI Conference on Artificial Intelligence, 32 0 (1), Apr. 2018 b . doi:10.1609/aaai.v32i1.11627. URL https://ojs.aaai.org/index.php/AAAI/article/view/11627

    2018 doi

  5. [13]

    Hunter and

    Tatiana Benaglia, Didier Chauveau, and David R. Hunter and. An em-like algorithm for semi- and nonparametric estimation in multivariate mixtures. Journal of Computational and Graphical Statistics, 18 0 (2): 0 505--526, 2009. doi:10.1198/jcgs.2009.07175. URL https://doi.org/10....

    2009 arXiv

  6. [14]

    Latent dirichlet allocation

    David Blei, Andrew Ng, and Michael Jordan. Latent dirichlet allocation. In T. Dietterich, S. Becker, and Z. Ghahramani, editors, Advances in Neural Information Processing Systems, volume 14. MIT Press, 2001

    2001

  7. [15]

    Estimating multivariate latent-structure models

    St \'e phane Bonhomme, Koen Jochmans, and Jean-Marc Robin. Estimating multivariate latent-structure models . The Annals of Statistics, 44 0 (2): 0 540 -- 563, 2016. doi:10.1214/15-AOS1376. URL https://doi.org/10.1214/15-AOS1376

    2016 doi

  8. [16]

    The optimal approximation factor in density estimation

    Olivier Bousquet, Daniel Kane, and Shay Moran. The optimal approximation factor in density estimation. In Alina Beygelzimer and Daniel Hsu, editors, Proceedings of the Thirty-Second Conference on Learning Theory, volume 99 of Proceedings of Machine Learning Research, pages 318...

    2019

  9. [17]

    Bruni and G

    C. Bruni and G. Koch. Identifiability of continuous mixtures of unknown G aussian distributions. Ann. Probab., 13 0 (4): 0 1341--1357, 11 1985. doi:10.1214/aop/1176992817. URL http://dx.doi.org/10.1214/aop/1176992817

    1985

  10. [18]

    Generalized multi-view model: Adaptive density estimation under low-rank constraints

    Julien Chhor , Olga Klopp , and Alexandre Tsybakov . Generalized multi-view model: Adaptive density estimation under low-rank constraints . arXiv e-prints, art. arXiv:2404.17209, April 2024. doi:10.48550/arXiv.2404.17209

    2024 doi arXiv

  11. [19]

    The sample complexity of semi-supervised learning with nonparametric mixture models

    Chen Dan, Liu Leqi, Bryon Aragam, Pradeep K Ravikumar, and Eric P Xing. The sample complexity of semi-supervised learning with nonparametric mixture models. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Inform...

    2018

  12. [20]

    Image anomaly detection with generative adversarial networks

    Lucas Deecke, Robert Vandermeulen, Lukas Ruff, Stephan Mandt, and Marius Kloft. Image anomaly detection with generative adversarial networks. In Michele Berlingerio, Francesco Bonchi, Thomas G \"a rtner, Neil Hurley, and Georgiana Ifrim, editors, Machine Learning and Knowledge...

    2019

  13. [21]

    A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the em algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39 0 (1): 0 1--22, 1977. doi:https://doi.org/10.1111/j.2517-6161.1977.tb01600.x. URL https://rss.onl...

    1977

  14. [22]

    Devroye and L

    L. Devroye and L. Gyorfi. Nonparametric Density Estimation: The L^1 View. Wiley Interscience Series in Discrete Mathematics. Wiley, 1985. ISBN 9780471816461

    1985

  15. [23]

    Devroye and G

    L. Devroye and G. Lugosi. Combinatorial Methods in Density Estimation. Springer, New York, 2001

    2001

  16. [24]

    An application of classical invariant theory to identifiability in nonparametric mixtures

    Ryan Elmore, Peter Hall, and Amnon Neeman. An application of classical invariant theory to identifiability in nonparametric mixtures. In Annales de l'institut Fourier, volume 55, pages 1--28, 2005

    2005

  17. [25]

    Deep anomaly detection using geometric transformations

    Izhak Golan and Ran El-Yaniv. Deep anomaly detection using geometric transformations. In S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett, editors, Advances in Neural Information Processing Systems, volume 31. Curran Associates, Inc., 2018. URL...

    2018

  18. [26]

    A distance measure between gmms based on the unscented transform and its application to speaker recognition

    Jacob Goldberger and Hagai Aronowitz. A distance measure between gmms based on the unscented transform and its application to speaker recognition. In Interspeech 2005, pages 1985--1988, 2005. doi:10.21437/Interspeech.2005-624

    2005 doi

  19. [27]

    Nonparametric estimation of component distributions in a multivariate mixture

    Peter Hall and Xiao-Hua Zhou. Nonparametric estimation of component distributions in a multivariate mixture. Annals of Statistics, pages 201--224, 2003

    2003

  20. [28]

    Nonparametric inference in multivariate mixtures

    Peter Hall, Amnon Neeman, Reza Pakyari, and Ryan Elmore. Nonparametric inference in multivariate mixtures. Biometrika, 92 0 (3): 0 667--678, 2005

    2005

  21. [29]

    Harris, K

    Charles R. Harris, K. Jarrod Millman, St \' e fan J. van der Walt, Ralf Gommers, Pauli Virtanen, David Cournapeau, Eric Wieser, Julian Taylor, Sebastian Berg, Nathaniel J. Smith, Robert Kern, Matti Picus, Stephan Hoyer, Marten H. van Kerkwijk, Matthew Brett, Allan Haldane, Jai...

    2020

  22. [30]

    Deep anomaly detection with outlier exposure

    Dan Hendrycks, Mantas Mazeika, and Thomas Dietterich. Deep anomaly detection with outlier exposure. In International Conference on Learning Representations, 2019 a . URL https://openreview.net/forum?id=HyxCxhRcY7

    2019

  23. [31]

    Using self-supervised learning can improve model robustness and uncertainty

    Dan Hendrycks, Mantas Mazeika, Saurav Kadavath, and Dawn Song. Using self-supervised learning can improve model robustness and uncertainty. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d Alch\' e -Buc, E. Fox, and R. Garnett, editors, Advances in Neural Information Process...

    2019

  24. [32]

    Convergence rates of parameter estimation for some weakly identifiable finite mixtures

    Nhat Ho and XuanLong Nguyen. Convergence rates of parameter estimation for some weakly identifiable finite mixtures . The Annals of Statistics, 44 0 (6): 0 2726 -- 2755, 2016. doi:10.1214/16-AOS1444. URL https://doi.org/10.1214/16-AOS1444

    2016 doi

  25. [33]

    Dasvdd: Deep autoencoding support vector data descriptor for anomaly detection

    Hadi Hojjati and Narges Armanfard. Dasvdd: Deep autoencoding support vector data descriptor for anomaly detection. IEEE Trans. on Knowl. and Data Eng., 36 0 (8): 0 3739–3750, August 2024. ISSN 1041-4347. doi:10.1109/TKDE.2023.3328882. URL https://doi.org/10.1109/TKDE.2023.3328882

    2024

  26. [34]

    Nonlinear ica using auxiliary variables and generalized contrastive learning

    Aapo Hyvarinen, Hiroaki Sasaki, and Richard Turner. Nonlinear ica using auxiliary variables and generalized contrastive learning. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Stati...

    2019

  27. [35]

    Sidiropoulos

    Nikos Kargas and Nicholas D. Sidiropoulos. Learning mixtures of smooth product distributions: Identifiability and algorithm. In Kamalika Chaudhuri and Masashi Sugiyama, editors, Proceedings of the Twenty-Second International Conference on Artificial Intelligence and Statistics...

    2019

  28. [36]

    Variational autoencoders and nonlinear ica: A unifying framework

    Ilyes Khemakhem, Diederik Kingma, Ricardo Monti, and Aapo Hyvarinen. Variational autoencoders and nonlinear ica: A unifying framework. In Silvia Chiappa and Roberto Calandra, editors, Proceedings of the Twenty Third International Conference on Artificial Intelligence and Stati...

    2020

  29. [37]

    JooSeuk Kim and Clayton D. Scott. Robust kernel density estimation. Journal of Machine Learning Research, 13 0 (82): 0 2529--2565, 2012. URL http://jmlr.org/papers/v13/kim12b.html

    2012

  30. [38]

    Identifiability of deep generative models without auxiliary information

    Bohdan Kivva, Goutham Rajendran, Pradeep Kumar Ravikumar, and Bryon Aragam. Identifiability of deep generative models without auxiliary information. In Alice H. Oh, Alekh Agarwal, Danielle Belgrave, and Kyunghyun Cho, editors, Advances in Neural Information Processing Systems,...

    2022

  31. [39]

    Estimation of the number of components of nonparametric multivariate finite mixture models

    Caleb Kwon and Eric Mbakop. Estimation of the number of components of nonparametric multivariate finite mixture models . The Annals of Statistics, 49 0 (4): 0 2178 -- 2205, 2021. doi:10.1214/20-AOS2032. URL https://doi.org/10.1214/20-AOS2032

    2021 doi

  32. [40]

    The em algorithm gives sample-optimality for learning mixtures of well-separated gaussians

    Jeongyeol Kwon and Constantine Caramanis. The em algorithm gives sample-optimality for learning mixtures of well-separated gaussians. In Jacob Abernethy and Shivani Agarwal, editors, Proceedings of Thirty Third Conference on Learning Theory, volume 125 of Proceedings of Machin...

    2020

  33. [41]

    Numba: A llvm-based python jit compiler

    Siu Kwan Lam, Antoine Pitrou, and Stanley Seibert. Numba: A llvm-based python jit compiler. In Proceedings of the Second Workshop on the LLVM Compiler Infrastructure in HPC, pages 1--6, 2015

    2015

  34. [42]

    Vandermeulen, Billy Joe Franks, Klaus Robert Muller, and Marius Kloft

    Philipp Liznerski, Lukas Ruff, Robert A. Vandermeulen, Billy Joe Franks, Klaus Robert Muller, and Marius Kloft. Exposing outlier exposure: What can be learned from few, one, and zero outlier images. Transactions on Machine Learning Research, 2022. ISSN 2835-8856. URL https://o...

    2022

  35. [43]

    Density estimation in linear time

    Satyaki Mahalanabis and Daniel Stefankovic . Density estimation in linear time . arXiv e-prints, art. arXiv:0712.2869, December 2007. doi:10.48550/arXiv.0712.2869

    2007 doi arXiv

  36. [44]

    Minimax Density Estimation for Growing Dimension

    Daniel McDonald. Minimax Density Estimation for Growing Dimension . In Aarti Singh and Jerry Zhu, editors, Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, volume 54 of Proceedings of Machine Learning Research, pages 194--203. PMLR, 2...

    2017

  37. [45]

    Do deep generative models know what they don't know? In International Conference on Learning Representations, 2019

    Eric Nalisnick, Akihiro Matsukawa, Yee Whye Teh, Dilan Gorur, and Balaji Lakshminarayanan. Do deep generative models know what they don't know? In International Conference on Learning Representations, 2019. URL https://openreview.net/forum?id=H1xwNhCcYm

    2019

  38. [46]

    On w -mixtures: Finite convex combinations of prescribed component distributions

    Frank Nielsen and Richard Nock . On w -mixtures: Finite convex combinations of prescribed component distributions . arXiv e-prints, art. arXiv:1708.00568, August 2017. doi:10.48550/arXiv.1708.00568

    2017 doi arXiv

  39. [47]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in P ython. Journal of Machine Learnin...

    2011

  40. [48]

    Pedregosa, G

    F. Pedregosa, G. Varoquaux, A. Gramfort, V. Michel, B. Thirion, O. Grisel, M. Blondel, P. Prettenhofer, R. Weiss, V. Dubourg, J. Vanderplas, A. Passos, D. Cournapeau, M. Brucher, M. Perrot, and E. Duchesnay. Scikit-learn: Machine learning in P ython. Journal of Machine Learnin...

    2011

  41. [49]

    R.D. Reiss. Approximate distributions of order statistics: with applications to nonparametric statistics. Springer series in statistics. Springer, 1989. ISBN 9783540968511. URL https://books.google.de/books?id=DxzvAAAAMAAJ

    1989

  42. [50]

    Consistent estimation of identifiable nonparametric mixture models from grouped observations

    Alexander Ritchie, Robert A Vandermeulen, and Clayton Scott. Consistent estimation of identifiable nonparametric mixture models from grouped observations. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Syst...

    2020

  43. [51]

    Deep one-class classification

    Lukas Ruff, Robert Vandermeulen, Nico Goernitz, Lucas Deecke, Shoaib Ahmed Siddiqui, Alexander Binder, Emmanuel M \"u ller, and Marius Kloft. Deep one-class classification. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on Machine ...

    2018

  44. [52]

    Kauffmann, Robert A

    Lukas Ruff, Jacob R. Kauffmann, Robert A. Vandermeulen, Gr \' e goire Montavon, Wojciech Samek, Marius Kloft, Thomas G. Dietterich, and Klaus - Robert M \" u ller. A unifying review of deep and shallow anomaly detection. Proc. IEEE , 109 0 (5): 0 756--795, 2021. doi:10.1109/JP...

    2021

  45. [53]

    A Useful Convergence Theorem for Probability Distributions

    Henry Scheffe. A Useful Convergence Theorem for Probability Distributions . The Annals of Mathematical Statistics, 18 0 (3): 0 434 -- 438, 1947. doi:10.1214/aoms/1177730390. URL https://doi.org/10.1214/aoms/1177730390

    1947

  46. [54]

    Waldstein, Ursula Schmidt-Erfurth, and Georg Langs

    Thomas Schlegl, Philipp Seeb \"o ck, Sebastian M. Waldstein, Ursula Schmidt-Erfurth, and Georg Langs. Unsupervised anomaly detection with generative adversarial networks to guide marker discovery. In Marc Niethammer, Martin Styner, Stephen Aylward, Hongtu Zhu, Ipek Oguz, Pew-T...

    2017

  47. [55]

    Waldstein, Georg Langs, and Ursula Schmidt-Erfurth

    Thomas Schlegl, Philipp Seeböck, Sebastian M. Waldstein, Georg Langs, and Ursula Schmidt-Erfurth. f-anogan: Fast unsupervised anomaly detection with generative adversarial networks. Medical Image Analysis, 54: 0 30--44, 2019. ISSN 1361-8415. doi:https://doi.org/10.1016/j.media...

    2019 doi

  48. [56]

    B. W. Silverman. Density Estimation for Statistics and Data Analysis. Chapman & Hall/CRC monographs on statistics. Chapman & Hall, London, 1986

    1986

  49. [57]

    Robust low rank kernel embeddings of multivariate distributions

    Le Song and Bo Dai. Robust low rank kernel embeddings of multivariate distributions. In C.J. Burges, L. Bottou, M. Welling, Z. Ghahramani, and K.Q. Weinberger, editors, Advances in Neural Information Processing Systems, volume 26. Curran Associates, Inc., 2013. URL https://pro...

    2013

  50. [58]

    Nonparametric estimation of multi-view latent variable models

    Le Song, Animashree Anandkumar, Bo Dai, and Bo Xie. Nonparametric estimation of multi-view latent variable models. In Eric P. Xing and Tony Jebara, editors, Proceedings of the 31st International Conference on Machine Learning, volume 32 of Proceedings of Machine Learning Resea...

    2014

  51. [60]

    Charles J. Stone. Optimal Global Rates of Convergence for Nonparametric Regression . The Annals of Statistics, 10 0 (4): 0 1040 -- 1053, 1982. doi:10.1214/aos/1176345969. URL https://doi.org/10.1214/aos/1176345969

    1982

  52. [61]

    Csi: Novelty detection via contrastive learning on distributionally shifted instances

    Jihoon Tack, Sangwoo Mo, Jongheon Jeong, and Jinwoo Shin. Csi: Novelty detection via contrastive learning on distributionally shifted instances. In H. Larochelle, M. Ranzato, R. Hadsell, M.F. Balcan, and H. Lin, editors, Advances in Neural Information Processing Systems, volum...

    2020

  53. [62]

    Identifiability of Finite Mixtures

    Henry Teicher. Identifiability of Finite Mixtures . The Annals of Mathematical Statistics, 34 0 (4): 0 1265 -- 1269, 1963. doi:10.1214/aoms/1177703862. URL https://doi.org/10.1214/aoms/1177703862

    1963

  54. [63]

    Identifiability of mixtures of product measures

    Henry Teicher. Identifiability of mixtures of product measures. The Annals of Mathematical Statistics, 38 0 (4): 0 1300--1302, 1967

    1967

  55. [64]

    Tsybakov

    A.B. Tsybakov. Introduction to nonparametric estimation. Springer Series in Statistics, New York, page 214, 2009. cited By 1

    2009

  56. [65]

    Sample complexity using infinite multiview models

    Robert A Vandermeulen. Sample complexity using infinite multiview models. arXiv preprint arXiv:2302.04292, 2023

    2023 arXiv

  57. [66]

    Beyond smoothness: Incorporating low-rank analysis into nonparametric density estimation

    Robert A Vandermeulen and Antoine Ledent. Beyond smoothness: Incorporating low-rank analysis into nonparametric density estimation. Advances in Neural Information Processing Systems, 34: 0 12180--12193, 2021

    2021

  58. [67]

    Vandermeulen and René Saitenmacher

    Robert A. Vandermeulen and René Saitenmacher. Generalized identifiability bounds for mixture models with grouped samples. IEEE Transactions on Information Theory, 70 0 (4): 0 2746--2758, 2024. doi:10.1109/TIT.2024.3367433

    2024

  59. [68]

    An operator theoretic approach to nonparametric mixture models

    Robert A Vandermeulen, Clayton D Scott, et al. An operator theoretic approach to nonparametric mixture models. Annals of Statistics, 47 0 (5): 0 2704--2733, 2019

    2019

  60. [69]

    Vandermeulen, Wai Ming Tai, and Bryon Aragam

    Robert A. Vandermeulen, Wai Ming Tai, and Bryon Aragam. Breaking the curse of dimensionality in structured density estimation. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024. URL https://openreview.net/forum?id=dWwin2uGYE

    2024

  61. [70]

    Dimension-independent rates for structured neural density estimation

    Robert A Vandermeulen, Wai Ming Tai, and Bryon Aragam. Dimension-independent rates for structured neural density estimation. In Forty-second International Conference on Machine Learning, 2025. URL https://openreview.net/forum?id=o7mxIi8jRv

    2025

  62. [71]

    Vankadara, Sebastian Bordt, Ulrike von Luxburg, and Debarghya Ghoshdastidar

    Leena C. Vankadara, Sebastian Bordt, Ulrike von Luxburg, and Debarghya Ghoshdastidar. Recovery guarantees for kernel-based clustering under non-parametric mixture models. In Arindam Banerjee and Kenji Fukumizu, editors, Proceedings of The 24th International Conference on Artif...

    2021

  63. [72]

    Mclachlan

    Cinzia Viroli and Geoffrey J. Mclachlan. Deep gaussian mixture models. Statistics and Computing, 29 0 (1): 0 43–51, January 2019. ISSN 0960-3174. doi:10.1007/s11222-017-9793-z. URL https://doi.org/10.1007/s11222-017-9793-z

    2019 doi

  64. [73]

    Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St \'e fan J

    Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser, Jonathan Bright, St \'e fan J. van der Walt , Matthew Brett, Joshua Wilson, K. Jarrod Millman, Nikolay Mayorov, Andrew R. J. Nels...

    2020

  65. [74]

    An equal-size hard EM algorithm for diverse dialogue generation

    Yuqiao Wen, Yongchang Hao, Yanshuai Cao, and Lili Mou. An equal-size hard EM algorithm for diverse dialogue generation. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=k5PEHHY4spM

    2023

  66. [75]

    On the identifiability of finite mixtures

    Sidney J Yakowitz and John D Spragins. On the identifiability of finite mixtures. The Annals of Mathematical Statistics, pages 209--214, 1968

    1968

  67. [76]

    Yatracos

    Yannis G. Yatracos. Rates of Convergence of Minimum Distance Estimators and Kolmogorov's Entropy . The Annals of Statistics, 13 0 (2): 0 768 -- 774, 1985. doi:10.1214/aos/1176349553. URL https://doi.org/10.1214/aos/1176349553

    1985

  68. [77]

    FastFlow: Unsupervised Anomaly Detection and Localization via 2D Normalizing Flows

    Jiawei Yu , Ye Zheng , Xiang Wang , Wei Li , Yushuang Wu , Rui Zhao , and Liwei Wu . FastFlow: Unsupervised Anomaly Detection and Localization via 2D Normalizing Flows . arXiv e-prints, art. arXiv:2111.07677, November 2021. doi:10.48550/arXiv.2111.07677

    2021 doi arXiv

  69. [78]

    P-kdgan: progressive knowledge distillation with gans for one-class novelty detection

    Zhiwei Zhang, Shifeng Chen, and Lei Sun. P-kdgan: progressive knowledge distillation with gans for one-class novelty detection. In Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence, IJCAI'20, 2021. ISBN 9780999241165

    2021

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.