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Mixtures Closest to a Given Measure: A Semidefinite Programming Approach

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read A measure known only through finitely many moments can be optimally approximated by a parametric mixture, with a semidefinite hierarchy that converges — and sometimes outright recovers the mixture order and parameters.

desk verdict A useful asymptotic moment-SOS hierarchy for mixture approximation, but the advertised finite-convergence and recovery theorem is false as stated; the flatness condition on the mixing measure alone does not certify a true coupling. read the letter →

arxiv 2509.22879 v2 pith:5MSKZASF submitted 2025-09-26 math.OC cs.LG

Srećko {DJ}urašinović , Jean-Bernard Lasserre , Victor Magron This is my paper
classification math.OCcs.LG MSC 90C2290C2644A6062H3060E99
keywords mixturemodelssemidefiniteprogrammingmomentrelaxationsWassersteindistancetotalvariationflatextensionclusteringdeterminacy
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 studies the problem of approximating a probability measure, observed only through its moments up to some finite order, by a mixture of distributions from a parametric family such as Gaussians, Poissons, or exponentials. The approximation quality is measured by the squared 2-Wasserstein distance or the total variation distance, and the parameter set is not assumed finite; instead it is a compact semialgebraic set. The authors construct a hierarchy of semidefinite relaxations whose optimal values increase to the true optimal distance, and prove that when a rank-flatness condition holds on the optimal mixing pseudo-moment matrix, the convergence is finite, the mixing measure is K-atomic, and the K component parameters can be extracted by linear algebra. They also show that these estimates can initialize k-means or EM to reduce iteration counts and variance in clustering tasks.

What carries the argument

The load-bearing object is the pseudo-moment matrix M_d(ϕ) of the mixing measure ϕ, together with the flat-extension criterion rank M_d(ϕ*) = rank M_{d−d_min}(ϕ*) = K, which certifies that the truncated moment sequence comes from a K-atomic measure on the parameter set. The polynomial moment assumption — that every moment of the mixture is a polynomial in the parameters — converts mixture moments into linear functionals of ϕ and makes the problem a generalized moment problem amenable to a semidefinite hierarchy. For the total variation case, the additional device is the decomposition of the signed measure μ−ν into two positive measures bounded respectively by μ and ν; those domination constr

What would settle it

Construct a target measure μ that is itself a known K-component mixture, solve the SDP relaxation at order d, and check whether the flatness condition on M_d(ϕ*) triggers while the extracted K parameters yield a distance strictly larger than the SDP lower bound. That outcome would falsify the recovery claim. A more direct search: find a case where flatness holds on M_d(ϕ*) but M_d(λ*) (the coupling pseudo-moment matrix) admits no representing measure — then the claimed best approximation is not realized.

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Extended reading notes

Core claim

The central claim is that the infinite-dimensional problem of minimizing a distance between a target measure and a mixture from a parametric family becomes a tractable semidefinite program once the target is represented by its moments up to order 2d, provided the family's moments are polynomials in the mixture parameters. The mixture moments are then linear functionals of the mixing measure, making the whole problem an instance of the generalized moment problem. The paper proves that as the relaxation order d increases, the optimal value converges to the true distance; if the optimal pseudo-moment matrix of the mixing measure satisfies the flat-extension condition (its rank at order d equals

Load-bearing premise

The proofs of finite convergence rely on the flatness condition on the mixing measure alone (rank M_d(ϕ*) = rank M_{d−d_min}(ϕ*)) being enough to extend the entire truncated solution to a true coupling or to true dominating measures; no flatness on those auxiliary variables is established, so if that extension fails, the recovery guarantee collapses.

Editorial extensions

If this is right

  • When the flatness condition holds, the mixture order K and the component parameters can be read off the optimal solution without any a priori bound on the number of components.
  • The framework applies to any parametric family with polynomial moments and compact semialgebraic parameter sets, including Gaussian, Poisson, and exponential mixtures.
  • The recovered estimates can serve as high-quality initializations for k-means and EM, reducing both iteration counts and variance, as demonstrated on random Gaussian mixtures and MNIST.
  • For the total variation distance, asymptotic convergence holds even without flatness, and any accumulation point of the optimal mixing measures is an optimal (not necessarily atomic) mixture.

Reading between the lines

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

  • The same moment-relaxation logic could be applied to other distances that admit moment-based lower bounds, such as energy distances or maximum mean discrepancy, as long as the distance can be certified by polynomial constraints.
  • The flatness condition used here is only checked on the mixing measure; testing whether joint flatness on the coupling (W2 case) or on the TV components (ψ+*, ψ−*) is necessary would clarify the actual robustness of the extraction step under numerical perturbation.
  • The paper's univariate projection experiments suggest that a systematic study of random projections could turn this method into a practical dimension-agnostic preprocessing routine for large-scale clustering.
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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

3 major / 4 minor

Summary. The manuscript develops semidefinite programming hierarchies for the following problem: given a target probability measure μ known only through moments up to order 2d, find a mixture ν_φ from a parametric family (Gaussian, Poisson, exponential, etc.) over a compact basic semi-algebraic parameter set that minimizes a regularized squared 2-Wasserstein or total variation distance. The main theoretical claims are (i) asymptotic convergence of the relaxation values to the true optimum, and (ii) finite convergence and recovery of the mixture order and parameters whenever the optimal pseudo-moment matrix M_d(φ*) satisfies the flatness condition rank M_d(φ*) = rank M_{d-dmin}(φ*). The paper also presents an algorithm for extracting mixture parameters and numerical experiments on synthetic GMMs and MNIST, with an application to initializing k-means and EM.

Significance. The problem is timely, and the general approach—reformulating mixture approximation as a generalized moment problem and applying moment-SOS relaxations—is potentially valuable. The TV reformulation via the Hahn-Jordan decomposition is a nice device, and the paper ships reproducible code with experiments. The asymptotic convergence arguments follow standard moment-SOS patterns, and the Carleman-type bounds for Gaussian, Poisson, and exponential mixtures are useful ingredients. However, the central advertised finite-convergence/recovery theorem is not correct as stated, and the numerical claims that rely on it are not supported.

major comments (3)
  1. [Theorem 3.2 / Appendix A.3] The flatness condition (13) on M_d(φ*) alone does not certify finite convergence or optimality. Counterexample: n=1, μ=(δ_0+δ_1)/2, family N(m,σ^2) with S_θ=[0,1]×[0.05,1], R≡1, ε>0 small, d=1. Let φ*=δ_(0.5,0.5). Then rank M_1(φ*)=rank M_0(φ*)=1, and (12) is feasible with λ_{1,0}=λ_{0,1}=λ_{2,0}=λ_{0,2}=λ_{1,1}=1/2, giving objective 0. Yet W2^2(μ,N(0.5,0.5^2))=0.5, while 1/2 N(0,0.05^2)+1/2 N(1,0.05^2) has W2^2≤0.0025. Thus (13) holds but φ* is not optimal for (11) and K=1 is not the optimal mixture order. The proof's appeal to Theorem A.1 is inapplicable: that theorem concerns accumulation points as d→∞; a flat extension at fixed d need not be such a point, and no flatness is imposed on λ*. This invalidates the finite-convergence claim.
  2. [Theorem 3.3 / Appendix A.4] The same gap occurs in the TV case. The rank condition is only on M_d(φ*); no flatness or extension condition is stated for M_d(ψ_+*) or M_d(ψ_-*). The proof says 'proceed as in proof of Theorem 3.2' and inherits the defect. Moreover, (17c)-(17d) are truncated moment inequalities; even if φ* is K-atomic, the pseudo-moment pair (ψ_+*,ψ_-*) need not arise from measures satisfying ψ_+*≤μ and ψ_-*≤ν_φ*. The d=1 example above can be adapted to show K-atomic recovery can fail.
  3. [Lemma A.1 / Theorems A.1-A.2] Lemma A.1 is stated for any mixture family satisfying Assumption 2.2, but its proof treats only Gaussian, Poisson, and exponential families; the final sentence asserts other families 'can also be shown' without proof. Theorems A.1 and A.2 invoke Lemma A.1 to obtain moment determinacy of ν_φ* and of the y-marginal of λ*. As a result, the asymptotic convergence theorems are not established at the stated level of generality. Please restrict the claims to the named families or provide a general proof.
minor comments (4)
  1. [Eq. (18)] The summation on the right-hand side should run over s(p,d), not s(n,p).
  2. [Algorithm 2] Line 1 has an extra closing parenthesis in M_d(φ*(d))); line 7 repeats 'global global'.
  3. [§3.2] 'Hanh-Jordan' should be 'Hahn-Jordan'.
  4. [Figure 1 caption] The value τ^{W2}_{4,0,R}=0 is the truncated relaxation value, not the true W2 distance. The sentence claiming the recovered mixtures are 'best' in the W2/TV sense relies on the invalid finite-convergence theorem and should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the SDP hierarchies are derived from first principles; self-citations are not load-bearing. A finite-convergence proof gap is flagged but is a correctness issue, not circularity.

full rationale

The derivation is self-contained: (7) defines the target value directly from W2/TV plus a user-chosen regularizer, with no fitted parameter later renamed as a prediction; (12) and (17) are obtained from the infinite-dimensional moment formulations (11) and (16) by standard truncation and localizing constraints, and their duality and asymptotic convergence are proved in the appendix rather than imported by citation. The flatness condition (13) is a genuine Curto–Fialkow sufficient condition for a representing atomic measure, not an algebraic restatement of the desired conclusion. The paper builds on Lasserre 2024a,b, but those are independently developed published hierarchies, and the present arguments reproduce the needed steps, so the self-citations are not load-bearing. I therefore find no circularity. For completeness, I flag a non-circular correctness gap: in the proofs of Theorems 3.2 and 3.3 (Appendices A.3–A.4), flatness of M_d(phi*) alone is used to assert that phi* solves the infinite-dimensional problem via Theorem A.1, but a fixed-order flat extension need not be an accumulation point of the hierarchy as d→∞, and no flatness/extension condition is imposed on λ*(d) or ψ±*(d); moreover Lemma A.1 is proven only for Gaussian, Poisson, and exponential families while the theorems are stated for arbitrary polynomial-moment families. This is an unsupported inference, not an equivalence-by-construction, so it does not affect the circularity score.

Assumptions & free parameters 4 free parameters · 9 assumptions · 0 invented entities

The central claim rests on standard moment-SOS theorems plus two domain assumptions (moment determinacy and polynomial moments), and the finite-convergence arguments additionally depend on unproved or partially proved facts: Slater's condition, the flat-extension-to-recovery step, and the general Carleman lemma. No new physical or conceptual entities are introduced; the regularization terms are user inputs, not invented objects.

free parameters (4)
  • Regularization polynomial R = sum of squared parameter monomials, for example theta -> sum_{gamma != 0} theta^{2gamma}; user selected
    Added to the objective in (7), (11), and (16) to promote low-rank and sparse solutions; the hierarchy value and extracted K depend on it.
  • Regularization strength epsilon = 10^-3 in Figures 2 and 3; 10^-5 and 0.1 in Table 1; 0 in Figure 1
    Controls the trade-off between distance and sparsity. Table 1 shows the estimated mixture order shifts with epsilon, so it is not a fixed constant of the method.
  • Rank and eigenvalue tolerance tol = 10^-2 in Figures 2 and 3; 10^-6 in Table 1
    Used in criterion (18) to estimate the mixture order K and to decide when the flatness condition is numerically satisfied.
  • Relaxation order d = 4 in all numerical experiments; d >= d_min in theory
    Determines the number of moments matched and the SDP size. Convergence and rank detection depend on increasing d.
assumptions (9)
  • domain assumption Assumption 2.1: the target measure mu is moment-determinate (multivariate Carleman condition).
    Required in Theorems A.1 and A.2 to ensure truncated pseudo-moment sequences admit representing measures; not verified in experiments.
  • domain assumption Assumption 2.2: every moment of the parametric family is polynomial in theta, p_alpha in R_{|alpha|}[theta].
    Makes the infinite dimensional problem a moment-SOS generalized moment problem; holds for Gaussian, Poisson, and exponential families but not all parametric families.
  • domain assumption S_theta is compact and basic semi-algebraic, Eq. (4).
    Used to make the quadratic module Archimedean and to bound moment sequences in the compactness and duality arguments.
  • standard math Putinar Positivstellensatz and Archimedeanity of the quadratic module Q(r).
    Invoked in Theorem A.1 Part II to pass from PSD localizing matrices to representing measures for phi*.
  • standard math Curto-Fialkow flat extension theorem.
    Used in proofs of Theorems 3.2 and 3.3 to extract a K-atomic mixing measure from a flat truncated moment matrix.
  • standard math Multivariate Carleman criterion (Nussbaum) for measure determinacy.
    Used repeatedly to convert moment sequences with PSD matrices and growth conditions into actual measures.
  • standard math Hahn-Jordan decomposition of signed measures.
    Underlies the equivalent infinite linear program (15) for total variation distance.
  • ad hoc to paper Slater's condition or strict feasibility holds for the SDP pairs (12),(35) and (17),(43).
    The paper states 'one can prove that Slater's condition holds' and asserts no duality gap without explicit proof; this underpins the equality between primal and dual values used in convergence arguments.
  • ad hoc to paper Lemma A.1: mixture families satisfy the Carleman-type growth bound (23) for all p_alpha.
    The lemma is stated generally but proven only for Gaussian, Poisson, and exponential families; the general polynomial-moment case needs an additional growth condition not stated in Assumption 2.2.

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Pith. "Pith review of Mixtures Closest to a Given Measure: A Semidefinite Programming Approach." pith.science (2026). https://pith.science/paper/5MSKZASF

@misc{pith2026250922879,
  author       = {Pith},
  title        = {Pith review of: Mixtures Closest to a Given Measure: A Semidefinite Programming Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5MSKZASF}},
  note         = {Machine review of arXiv:2509.22879}
}
read the original abstract

Mixture models, such as Gaussian mixture models, are widely used in machine learning to represent complex data distributions. A key challenge, especially in high-dimensional settings, is to determine the mixture order and estimate the mixture parameters. We study the problem of approximating a target measure, available only through finitely many of its moments, by a mixture of distributions from a parametric family (e.g., Gaussian, exponential, Poisson), with approximation quality measured by the 2-Wasserstein or the total variation distance. Unlike many existing approaches, the parameter set is not assumed to be finite; it is modeled as a compact basic semi-algebraic set. We introduce a hierarchy of semidefinite relaxations with asymptotic convergence to the desired optimal value. In addition, when a certain rank condition is satisfied, the convergence is even finite and recovery of an optimal mixing measure is obtained. We also present an application to clustering, where our framework serves either as a stand-alone method or as a preprocessing step that yields both the number of clusters and strong initial parameter estimates, thereby accelerating convergence of standard (local) clustering algorithms.

Figures

Figures reproduced from arXiv: 2509.22879 by the authors.

Figure 1
Figure 1. Fitting GMMs to a sample from dµ(x) = 1[0,1](x) dx. Blue and red densities are obtained by computing τ W2 4,0,R and τ TV 4,0,R, respectively, with Sm,σ =  (m, σ) | m ≥ m2 , σ ≥ σ 2 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Impact of Algorithm 1-based initialization on the number of iterations required by [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Effect of Algorithm 1-based initialization (with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

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Works this paper leans on

62 extracted references · 15 canonical work pages

  1. [1]

    A new look at the statistical model identification

    Hirotugu Akaike. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19 0 (6): 0 716--723, 1974. doi:10.1109/TAC.1974.1100705

    arXiv 1974

  2. [2]

    Algebraic identifiability of gaussian mixtures

    Carlos Améndola, Kristian Ranestad, and Bernd Sturmfels. Algebraic identifiability of gaussian mixtures. International Mathematics Research Notices, 2018 0 (21): 0 6556--6580, 2018. doi:10.1093/imrn/rnx126

    work page doi:10.1093/imrn/rnx126 2018

  3. [3]

    The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm

    Erling D Andersen and Knud D Andersen. The MOSEK interior point optimizer for linear programming: an implementation of the homogeneous algorithm . In High performance optimization, pp.\ 197--232. Springer, 2000

    2000

  4. [4]

    Antoniak

    Charles E. Antoniak. Mixtures of Dirichlet Processes with Applications to Bayesian Nonparametric Problems . The Annals of Statistics, 2 0 (6): 0 1152 -- 1174, 1974. doi:10.1214/aos/1176342871. URL https://doi.org/10.1214/aos/1176342871

    arXiv 1974

  5. [5]

    Polynomial time and private learning of unbounded gaussian mixture models, 2023

    Jamil Arbas, Hassan Ashtiani, and Christopher Liaw. Polynomial time and private learning of unbounded gaussian mixture models, 2023. URL https://arxiv.org/abs/2303.04288

    arXiv 2023

  6. [6]

    k-means++: the advantages of careful seeding

    David Arthur and Sergei Vassilvitskii. k-means++: the advantages of careful seeding. In Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA '07, pp.\ 1027–1035, USA, 2007. Society for Industrial and Applied Mathematics. ISBN 9780898716245

    2007

  7. [7]

    Multimodal word distributions

    Ben Athiwaratkun and Andrew Wilson. Multimodal word distributions. In Proceedings of the 55th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pp.\ 1645--1656, Vancouver, Canada, July 2017. Association for Computational Linguistics. doi:10.18653/v1/P17-1151. URL https://aclanthology.org/P17-1151/

    work page doi:10.18653/v1/p17-1151 2017

  8. [8]

    Kane, Pravesh K

    Ainesh Bakshi, Ilias Diakonikolas, Huan Jia, Daniel M. Kane, Pravesh K. Kothari, and Santosh S. Vempala. Robustly learning mixtures of \( k \) arbitrary gaussians. In STOC '22—Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pp.\ 1234--1247. Association for Computing Machinery, 2022

    2022

Show all 62 references
  1. [9]

    Xin Chen and Anderson Y. Zhang. Achieving optimal clustering in gaussian mixture models with anisotropic covariance structures. In Proceedings of the 38th Annual Conference on Neural Information Processing Systems (NeurIPS), 2024

    2024

  2. [10]

    Aggregating continuous word embeddings for information retrieval

    St \'e phane Clinchant and Florent Perronnin. Aggregating continuous word embeddings for information retrieval. In Proceedings of the Workshop on Continuous Vector Space Models and their Compositionality, pp.\ 100--109, Sofia, Bulgaria, August 2013. Association for Computation...

    2013

  3. [11]

    Curto and Lawrence A

    Ra\'ul E. Curto and Lawrence A. Fialkow. Truncated k-moment problems in several variables. Journal of Operator Theory, 54 0 (1): 0 189--226, 2005. URL http://www.jstor.org/stable/24715679

    2005

  4. [12]

    Learning mixtures of gaussians

    Sanjoy Dasgupta. Learning mixtures of gaussians. In Proceedings of the 40th Annual Symposium on Foundations of Computer Science, FOCS '99, pp.\ 634, USA, 1999. IEEE Computer Society. ISBN 0769504094

    1999

  5. [13]

    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--38, 1977

    1977

  6. [14]

    Likelihood ratio tests and singularities

    Mathias Drton. Likelihood ratio tests and singularities. Annals of Statistics, 37 0 (2): 0 979--1012, 2009. doi:10.1214/07-AOS583

    2009 doi

  7. [15]

    Model-based clustering, discriminant analysis, and density estimation

    Chris Fraley and Adrian E Raftery. Model-based clustering, discriminant analysis, and density estimation. Journal of the American Statistical Association, 97 0 (458): 0 611--631, 2002. doi:10.1198/016214502760047131. URL https://doi.org/10.1198/016214502760047131

    2002 doi

  8. [16]

    Rong Ge, Qingqing Huang, and Sham M. Kakade. Learning mixtures of gaussians in high dimensions. In Proceedings of the Forty-Seventh Annual ACM Symposium on Theory of Computing, STOC '15, pp.\ 761–770, New York, NY, USA, 2015. Association for Computing Machinery. ISBN 978145033...

    2015

  9. [17]

    J. K. Ghosh and P. K. Sen. On the asymptotic performance of the log likelihood ratio statistic for the mixture model and related results. In Proceedings of the Berkeley Conference in Honor of J. Neyman and J. Kiefer, volume 2, pp.\ 789--806. Wadsworth, Monterey, CA, 1985

    1985

  10. [18]

    Alastair R. Hall. Generalized Method of Moments. Advanced Texts in Econometrics. Oxford University Press, Oxford, 2005

    2005

  11. [19]

    Large sample properties of generalized method of moments estimators

    Lars Peter Hansen. Large sample properties of generalized method of moments estimators. Econometrica, 50 0 (4): 0 1029--1054, 1982. doi:10.2307/1912775

    1982 doi

  12. [20]

    Strong identifiability and optimal minimax rates for finite mixture estimation

    Peter Heinrich and Jonas Kahn. Strong identifiability and optimal minimax rates for finite mixture estimation. Annals of Statistics, 46 0 (6A): 0 2844--2870, 2018. doi:10.1214/17-AOS1647

    2018 doi

  13. [21]

    Detecting Global Optimality and Extracting Solutions in GloptiPoly, pp.\ 293--310

    Didier Henrion and Jean-Bernard Lasserre. Detecting Global Optimality and Extracting Solutions in GloptiPoly, pp.\ 293--310. Springer Berlin Heidelberg, Berlin, Heidelberg, 2005. ISBN 978-3-540-31594-0. doi:10.1007/10997703_15. URL https://doi.org/10.1007/10997703_15

    2005 doi

  14. [22]

    Matrix manifold optimization for gaussian mixtures

    Reshad Hosseini and Suvrit Sra. Matrix manifold optimization for gaussian mixtures. In Advances in Neural Information Processing Systems, volume 28. Curran Associates, Inc., 2015. URL https://proceedings.neurips.cc/paper_files/paper/2015/file/dbe272bab69f8e13f14b405e038deb64-Paper.pdf

    2015

  15. [23]

    Bootstrapping log likelihood and eic, an extension of aic

    Masami Ishiguro, Yasunori Sakamoto, and Genshiro Kitagawa. Bootstrapping log likelihood and eic, an extension of aic. Annals of the Institute of Statistical Mathematics, 49: 0 411--434, 1997. doi:10.1023/A:1003183708705

    1997 doi

  16. [24]

    Finite mixtures of multivariate poisson distributions with application

    Dimitris Karlis and Loukia Meligkotsidou. Finite mixtures of multivariate poisson distributions with application. Journal of Statistical Planning and Inference, 137 0 (6): 0 1942--1960, 2007. ISSN 0378-3758. doi:https://doi.org/10.1016/j.jspi.2006.07.001. URL https://www.scien...

    1942 doi

  17. [25]

    Choosing initial values for the em algorithm for finite mixtures

    Dimitris Karlis and Evdokia Xekalaki. Choosing initial values for the em algorithm for finite mixtures. Computational Statistics & Data Analysis, 41 0 (3--4): 0 577--590, 2003. doi:10.1016/S0167-9473(02)00177-9

    2003 doi

  18. [26]

    Rousseeuw

    Leonard Kaufman and Peter J. Rousseeuw. Finding Groups in Data: An Introduction to Cluster Analysis. John Wiley & Sons, New York, 1990

    1990

  19. [27]

    Tensor decomposition for learning gaussian mixtures from moments

    Rima Khouja, Pierre-Alexandre Mattei, and Bernard Mourrain. Tensor decomposition for learning gaussian mixtures from moments. Journal of Symbolic Computation, 113: 0 193--210, 2022. ISSN 0747-7171. doi:https://doi.org/10.1016/j.jsc.2022.04.002. URL https://www.sciencedirect.co...

    2022 doi

  20. [28]

    Kiefer and J

    J. Kiefer and J. Wolfowitz. Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Annals of Mathematical Statistics, 27 0 (4): 0 887--906, 1956. doi:10.1214/aoms/1177728066

    1956

  21. [29]

    Minimizer extraction in polynomial optimization is robust

    Igor Klep, Janez Povh, and Jurij Vol c i c . Minimizer extraction in polynomial optimization is robust. SIAM Journal on Optimization, 28 0 (4): 0 3177--3207, 2018. doi:10.1137/17M1152061. URL https://doi.org/10.1137/17M1152061

    2018 doi

  22. [30]

    Kothari, Prasad Raghavendra Manohar, and B

    Pravesh K. Kothari, Prasad Raghavendra Manohar, and B. Haeupler Zhang. Polynomial-time sum-of-squares can robustly estimate mean and covariance of gaussians optimally. In Proceedings of The 33rd International Conference on Algorithmic Learning Theory, volume 167 of Proceedings...

    2022

  23. [31]

    Lasserre

    Jean B. Lasserre. Gaussian mixtures closest to a given measure via optimal transport. Comptes Rendus. Mathématique, 362: 0 1455--1473, 2024 a . doi:10.5802/crmath.657. URL https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.657/

    2024 doi

  24. [32]

    Moments, Positive Polynomials and Their Applications

    Jean Bernard Lasserre. Moments, Positive Polynomials and Their Applications. IMPERIAL COLLEGE PRESS, 2009. doi:10.1142/p665. URL https://www.worldscientific.com/doi/abs/10.1142/p665

    2009 doi

  25. [33]

    An Introduction to Polynomial and Semi-Algebraic Optimization

    Jean Bernard Lasserre. An Introduction to Polynomial and Semi-Algebraic Optimization. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2015

    2015

  26. [34]

    A hierarchy of convex relaxations for the total variation distance

    Jean-Bernard Lasserre. A hierarchy of convex relaxations for the total variation distance. arXiv preprint arXiv:2401.01086, 2024 b . URL https://arxiv.org/abs/2401.01086

    2024

  27. [35]

    Estimating gaussian mixtures using sparse polynomial moment systems

    Julia Lindberg, Carlos Am\' e ndola, and Jose Israel Rodriguez. Estimating gaussian mixtures using sparse polynomial moment systems. SIAM Journal on Mathematics of Data Science, 7 0 (1): 0 224--252, 2025. doi:10.1137/23M1610082. URL https://doi.org/10.1137/23M1610082

    2025 doi

  28. [36]

    Bruce G. Lindsay. Mixture Models: Theory, Geometry and Applications, volume 5 of NSF-CBMS Regional Conference Series in Probability and Statistics. Institute of Mathematical Statistics, Hayward, CA, 1995. Available via JSTOR

    1995

  29. [37]

    Clustering mixtures with almost optimal separation in polynomial time

    Allen Liu and Jerry Li. Clustering mixtures with almost optimal separation in polynomial time. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pp.\ 1248–1261, New York, NY, USA, 2022. Association for Computing Machinery. ISBN 978145039...

    2022

  30. [38]

    Likelihood ratio tests of the number of components in a normal mixture with unequal variances

    Yin Lo. Likelihood ratio tests of the number of components in a normal mixture with unequal variances. Statistics & Probability Letters, 71 0 (2): 0 225--235, 2005. doi:10.1016/j.spl.2004.12.015

    2005 doi

  31. [39]

    TSSOS: a Julia library to exploit sparsity for large-scale polynomial optimization

    Victor Magron and Jie Wang. TSSOS: a Julia library to exploit sparsity for large-scale polynomial optimization . ArXiv, abs/2103.00915, 2021. URL https://api.semanticscholar.org/CorpusID:232076082

    2021 arXiv

  32. [40]

    McLachlan and David Peel

    Geoffrey J. McLachlan and David Peel. Finite Mixture Models. Wiley Series in Probability and Statistics. John Wiley & Sons, New York, 2000. ISBN 978-0-471-00626-2

    2000

  33. [41]

    McLachlan and Suren Rathnayake

    Geoffrey J. McLachlan and Suren Rathnayake. On the number of components in a gaussian mixture model. WIREs Data Mining and Knowledge Discovery, 4 0 (5): 0 341--355, 2014. doi:https://doi.org/10.1002/widm.1135. URL https://wires.onlinelibrary.wiley.com/doi/abs/10.1002/widm.1135

    2014 doi

  34. [42]

    Moitra and G

    A. Moitra and G. Valiant. Settling the polynomial learnability of mixtures of gaussians. In IEEE 51st Annual Symposium on Foundations of Computer Science, pp.\ 93--102. IEEE, 2010

    2010

  35. [43]

    A. E. Nussbaum. Quasi-analytic vectors. Arkiv f\" o r Matematik , 6: 0 179--191, 1965

    1965

  36. [44]

    Pereira, Joe Kileel, and Tamara G

    João M. Pereira, Joe Kileel, and Tamara G. Kolda. Tensor moments of gaussian mixture models: Theory and applications, 2022. URL https://arxiv.org/abs/2202.06930

    2022 arXiv

  37. [45]

    A study of gaussian mixture models of color and texture features for image classification and segmentation

    Haim Permuter, Joseph Francos, and Ian Jermyn. A study of gaussian mixture models of color and texture features for image classification and segmentation. Pattern Recognition, 39 0 (4): 0 695--706, 2006. ISSN 0031-3203. doi:https://doi.org/10.1016/j.patcog.2005.10.028. URL htt...

    2006 doi

  38. [46]

    Positive polynomials on compact semi-algebraic sets

    Mihai Putinar. Positive polynomials on compact semi-algebraic sets. Indiana University Mathematics Journal, 42 0 (3): 0 969--984, 1993. ISSN 00222518, 19435258. URL http://www.jstor.org/stable/24897130

    1993

  39. [47]

    R. A. Redner and H. F. Walker. Mixture densities, maximum likelihood and the em algorithm. SIAM Review, 26 0 (2): 0 195--239, 1984. doi:10.1137/1026034

    1984 doi

  40. [48]

    Reynolds, Thomas F

    Douglas A. Reynolds, Thomas F. Quatieri, and Robert B. Dunn. Speaker verification using adapted gaussian mixture models. Digital Signal Processing, 10 0 (1): 0 19--41, 2000. ISSN 1051-2004. doi:https://doi.org/10.1006/dspr.1999.0361. URL https://www.sciencedirect.com/science/a...

  41. [49]

    Lasserre

    Cordian Riener, Thorsten Theobald, Lina Jansson Andrén, and Jean B. Lasserre. Exploiting symmetries in sdp-relaxations for polynomial optimization. Mathematics of Operations Research, 38 0 (1): 0 122–141, February 2013. ISSN 1526-5471. doi:10.1287/moor.1120.0558. URL http://dx...

    2013

  42. [50]

    Estimating the dimension of a model

    Gideon Schwarz. Estimating the dimension of a model. Annals of Statistics, 6 0 (2): 0 461--464, 1978. doi:10.1214/aos/1176344136

    1978

  43. [51]

    A cautionary note on likelihood ratio tests in mixture models

    Wolfgang Seidel, Karl Mosler, and Matthias Alker. A cautionary note on likelihood ratio tests in mixture models. Annals of the Institute of Statistical Mathematics, 52: 0 481--487, 2000. doi:10.1023/A:1004181512763

    doi

  44. [52]

    Identifying the number of components in gaussian mixture models using numerical algebraic geometry

    Sara Shirinkam, Adel Alaeddini, and Elizabeth Gross. Identifying the number of components in gaussian mixture models using numerical algebraic geometry. Journal of Algebra and Its Applications, 19 0 (11): 0 2050204, 2020. doi:10.1142/S0219498820502047. URL https://doi.org/10.1...

    2020 doi

  45. [53]

    Chris Stauffer and W.E.L. Grimson. Adaptive background mixture models for real-time tracking. In CVPR, volume 2, pp.\ 252 Vol. 2, 02 1999. ISBN 0-7695-0149-4. doi:10.1109/CVPR.1999.784637

    1999

  46. [54]

    Estimating the number of clusters in a data set via the gap statistic

    Robert Tibshirani, Guenther Walther, and Trevor Hastie. Estimating the number of clusters in a data set via the gap statistic. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 63 0 (2): 0 411--423, 2001. doi:10.1111/1467-9868.00293

    2001

  47. [55]

    D. M. Titterington, A. F. M. Smith, and U. E. Makov. Statistical Analysis of Finite Mixture Distributions. Wiley, Chichester, 1985

    1985

  48. [56]

    Optimal estimation of gaussian mixtures via denoised method of moments

    Yihong Wu and Pengkun Yang. Optimal estimation of gaussian mixtures via denoised method of moments. Annals of Statistics, 48 0 (4): 0 1981--2007, 2020. doi:10.1214/19-AOS1873

    1981 doi

  49. [57]

    Learning gaussian mixtures using the wasserstein-fisher-rao gradient flow

    Yuling Yan, Kaizheng Wang, and Philippe Rigollet. Learning gaussian mixtures using the wasserstein-fisher-rao gradient flow. Arxiv 2301.01766, 2023. URL https://arxiv.org/abs/2301.01766

    2023 arXiv

  50. [58]

    Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity

    Guoshen Yu, Guillermo Sapiro, and Stéphane Mallat. Solving inverse problems with piecewise linear estimators: From gaussian mixture models to structured sparsity. IEEE Transactions on Image Processing, 21 0 (5): 0 2481--2499, 2012. doi:10.1109/TIP.2011.2176743

    2012

  51. [59]

    A mixture of mixture models for a classification problem: The unity measure error

    Marco Zio, Ugo Guarnera, and Roberto Rocci. A mixture of mixture models for a classification problem: The unity measure error. Computational Statistics & Data Analysis , 51: 0 2573--2585, 02 2007. doi:10.1016/j.csda.2006.01.001

    2007 doi

  52. [60]

    @esa (Ref

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  53. [61]

    \@lbibitem[] @bibitem@first@sw\@secondoftwo \@lbibitem[#1]#2 \@extra@b@citeb \@ifundefined br@#2\@extra@b@citeb \@namedef br@#2 \@nameuse br@#2\@extra@b@citeb \@ifundefined b@#2\@extra@b@citeb @num @parse #2 @tmp #1 NAT@b@open@#2 NAT@b@shut@#2 \@ifnum @merge>\@ne @bibitem@firs...

  54. [62]

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.