Pith. sign in

REVIEW 2 major objections 5 minor 48 references

Identifiability and Estimation in High-Dimensional Nonparametric Latent Structure Models

T0 review · 2 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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

Pith's one-line read This paper proves a unified identifiability condition for high-dimensional nonparametric latent structure models: if the summed excess independence over some three-partition reaches $2m+2$, the mixture is identifiable, and the resulting…

desk verdict A genuine unification of identifiability conditions for latent structure models, with a repairable normalization error in the converse of Theorem 7; the forward theorem is novel and worth peer review. read the letter →

arxiv 2506.09165 v2 pith:BLYYYG5N submitted 2025-06-10 math.ST stat.TH

Yichen Lyu , Pengkun Yang This is my paper
classification math.STstat.TH MSC 62G0562G0762G20
keywords nonparametricestimationmultivariatemixturesidentifiabilityhighdimensionsKruskalrankHadamardproductminimaxrateslatentstructuremodels
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

Mixtures of product measures—data drawn from one of several latent groups, with conditional independence inside each group—are identifiable in high dimensions when the variables are sufficiently diverse. This paper establishes one condition, the summed per-variable "excess independence" over some three-partition of the coordinates reaching $2m+2$, that decides identifiability and generalizes the classical linear-independence, separability, and Bernoulli-mixture criteria. It also proves that the joint density can be estimated at minimax rates growing only polynomially in dimension, so the conditional-independence structure breaks the usual curse of dimensionality for Hölder-smooth densities. Under a separate incoherence condition, the component densities themselves can be recovered from a good estimate of the joint density with error of the same order, via a simultaneous-diagonalization algorithm. A reader should take away that dimension and diversity are resources for identification, not just obstacles.

What carries the argument

The load-bearing object is the Kruskal rank of a block of variables, computed through Gram matrices. For each coordinate $j$, form the $m\times m$ Gram matrix $A_j$ of the component densities $\{f_{kj}\}_{k=1}^m$; the Gram matrix of a block $S$ is then the Hadamard product $\circ_{j\in S} A_j$. Lemma 9 shows that the Kruskal rank of this Hadamard product is at least the sum of the individual Kruskal ranks minus $|S|+1$, so the quantity $\tau_\mu(S)=\min\{m,\sum_{j\in S}\mathrm{Ind}_\mu(j)-|S|+1\}$ lower-bounds the Kruskal rank of the block. Feeding these block ranks into a Hilbert-space extension of Kruskal's uniqueness theorem yields identifiability whenever three blocks have Kruskal ranks summing to at least $2m+2$. The same Gram-matrix and Hadamard-product structure, with Kruskal eigenvalues replacing ranks, drives the perturbation bounds in Theorem 12.

What would settle it

Exhaustively search small discrete product mixtures (e.g., Bernoulli, $m=3$, $d=5$) for two distinct parameter sets whose joint distributions coincide while their three-partition sum $\tau_\mu(S_1)+\tau_\mu(S_2)+\tau_\mu(S_3)$ equals $2m+2=8$; such a pair would directly contradict Theorem 7.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 7: if the coordinates can be partitioned into three groups $S_1,S_2,S_3$ with $\tau_\mu(S_1)+\tau_\mu(S_2)+\tau_\mu(S_3) \ge 2m+2$, where $\tau_\mu(S)$ is the total excess independence of the group—the sum over $j\in S$ of the $\ell$-independence rank of the $m$ component densities along variable $j$, minus $|S|$ plus one—then the mixture $\mu$ is identifiable, up to a global permutation of components. This extends the classical linear-independence condition and explains the previously observed $2m-1$ threshold for separable variables. The same machinery gives a companion minimax statement, Theorem 13: for a $q$-Hölder mixture density with $m$ components in dimension $d$, the Hellinger minimax risk is of order $n^{-q/(q+1)}$ times a polynomial in $m$ and $d$, so sample complexity scales polynomially in dimension. For component recovery, Theorem 12 shows that under a uniform incoherence assumption on the per-coordinate densities, the $L^2$ error of the recovered components and mixing weights is proportional to the $L^2$ error of the estimated joint density; Algorithm 1 implements this by simultaneous diagonalization.

Load-bearing premise

For the component-recovery and algorithmic guarantees, every coordinate must have its $m$ component densities uniformly non-parallel in $L^2$ (one incoherence parameter $\mu<1$ across all variables and all pairs), and every mixing weight must be bounded below by $\zeta>0$; if even one coordinate contains two nearly identical component densities, the stated error bounds blow up and the guarantees become empty.

Editorial extensions

If this is right

  • Identifiability of the mixture model (1) is decided by a single partition condition, unifying linear independence, the $2m-1$ separability threshold, conditional i.i.d. models, and Bernoulli mixtures.
  • Adding variables with even modest per-coordinate diversity provably helps identification, because the excess-independence sum grows with $d$; high dimensionality becomes a resource rather than only a curse.
  • The joint density admits minimax rates with only polynomial dimension dependence (of order $n^{-q/(q+1)}\cdot\mathrm{poly}(m,d)$ in Hellinger distance), so latent structure converts an exponential-rate problem into a nearly parametric one.
  • Under the $(\mu,\zeta)$-estimable condition, component densities and mixing proportions inherit the joint-density error linearly, making component estimation essentially as hard as joint density estimation.
  • Algorithm 1 recovers components from a plug-in density estimator using only incoherence rather than linear independence, with explicit error bounds in terms of $\epsilon$, $\zeta$, and $(1-\mu)^{-m}$.

Reading between the lines

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

  • The separation between the identifiability condition (Theorem 7) and the estimation condition (Assumption 11) suggests that joint density estimation remains achievable when components are nearly parallel, while component recovery becomes hard; this dichotomy could guide practical diagnostics.
  • The Hadamard-product Kruskal-rank lemma is likely transferable: any latent structure model whose blocks are conditionally independent should inherit a similar principle that excess independence accumulates across coordinates, for instance in topic models or nonparametric factor analysis.
  • The $(1-\mu)^{-O(m)}$ factors in Theorems 12 and 14 predict a sharp degradation as one coordinate's component densities approach parallel; this is testable by simulation and could motivate preprocessing that drops near-degenerate variables.
  • The minimax upper bound in Theorem 13 does not rely on incoherence, so the joint density can be estimated at the polynomial rate without separation; only the component-identification step pays the incoherence price.
Share X Bluesky LinkedIn Reddit HN

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 studies model (1), mixtures of product measures, and addresses identifiability and estimation in high-dimensional nonparametric latent structure models. It introduces the notion of ℓ-independence and proves Theorem 7: if a three-part partition of the variables satisfies the condition in (2), the mixture is identifiable, with a converse example showing that the condition is not necessary. It then develops a perturbation theory under a µ-incoherence assumption (Theorem 12), derives minimax rates for Hellinger and TV distances that depend only polynomially on dimension (Theorem 13), and proposes a simultaneous-diagonalization recovery algorithm (Algorithm 1 and Theorem 14). The positive identifiability argument and the estimation framework are developed in detail in the appendices.

Significance. If the main results hold, this is a substantial unifying contribution: linear independence, conditional i.i.d., and Bernoulli mixture identifiability all follow from a single condition, and the threshold 2m−1 emerges as a special case. The minimax rates show that the latent product structure avoids the curse of dimensionality, and the perturbation and algorithmic results work under incoherence rather than linear independence. The proof machinery, built on a Hilbert-space extension of Kruskal's theorem and a Hadamard-product Kruskal-rank lemma, is well matched to the problem and is mostly carefully executed. The paper also provides an explicit quantitative perturbation theorem and an operational recovery algorithm, which are valuable strengths.

major comments (2)
  1. [Appendix A.2, proof of the converse in Theorem 7] The converse construction is not valid as written because the weights π_k = binom(2m−1,2k−2)/2^{2m−1} and \tildeπ_k = binom(2m−1,2k−1)/2^{2m−1} sum to 1/2, not to 1: the even and odd binomial coefficients of order 2m−1 each sum to 2^{2m−2}. Consequently, the measures µ0 and \tildeµ0 have total mass 1/2 and are not probability measures in the model family (1). The moment identity (16) therefore proves equality of two subprobability measures only, and the stated non-identifiable example is not established. The construction is likely repairable by changing the denominator to 2^{2m−2}, but the converse direction of Theorem 7 is not proven by the current text.
  2. [Section 3.1 and Appendix B, Eq. (23)] The bound λKru_m(A2∘⋯∘Am) ≥ ∏_{j=2}^m λKru_2(A_j)/(m−1)! used in (23) appears to be stronger than what Lemma 23 yields as stated. Iterating Lemma 23 with k1=2 gives denominators k1+k2, leading to a product of the form (m+1)!/6 rather than (m−1)!. Since this bound enters the constants and the ϵ threshold in Theorem 12, the explicit constants in the theorem should be re-derived, or Lemma 23 should be strengthened/proved in the form used. The qualitative perturbation claim is likely unaffected, but the proof as written does not justify the displayed constants.
minor comments (5)
  1. [Appendix A.2, converse proof] The definition of \tildeµ0 writes µ_k^{×2m−1} with d=2m−2; the exponent should be 2m−2.
  2. [Section 2, converse proof] The notation µ0 is overloaded: it denotes both the d=2m−2 non-identifiable example and the extra factor µ_0^{d−2m+2} in the d>2m−2 case. Distinct symbols would avoid confusion.
  3. [Appendix B, proof of Theorem 12, Step 3] The definition of A1 lists all columns as ⊗_{j=4}^{m+3} f_{2j}; the columns should depend on the component index k, e.g., ⊗_{j=4}^{m+3} f_{kj}.
  4. [Appendix A.2, proof of Lemma 9] In Case 1 of the proof, the term q_i^⊤ w_1 in the expression 0=Cw_j should read q_i^⊤ w_j; this is a typographical error in an otherwise clear argument.
  5. [Section 4.2, simulations] The caption text for Figure 1 and the surrounding discussion would be clearer if the exact error measure e were defined in the caption, since the figure is referenced before the definition is fully described.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the identifiability, minimax, and recovery results are proved from stated assumptions and external black-box theorems; no fitted input is renamed as a prediction.

full rationale

The paper's central claims are self-contained relative to their stated assumptions. The identifiability criterion in Theorem 7 is expressed through the rank-like quantity tau_mu(S), which is defined from the per-coordinate diversity index Ind_mu(j); it is not defined in terms of the target conclusion (identifiability). The forward proof uses the Hilbert-space Kruskal theorem of Vandermeulen and Saitenmacher (Lemma 20) as an external black box and proves the needed Hadamard-product Kruskal-rank bound (Lemma 9) internally. The converse uses classical moment theory, citing Wu and Yang only for the standard fact that 2m-2 moments do not identify an m-atomic distribution; this fact is external and does not assume Theorem 7. The minimax bounds in Theorem 13 follow from metric-entropy bounds (Lemma 31) built on one-dimensional Holder entropy estimates, and the lower and upper bounds are matched against standard d-dimensional rates; no parameter is fitted to the target risk. The perturbation theorem and Algorithm 1 give conditional guarantees: given a joint-density estimator with L2 error epsilon, they bound the component-recovery error. This is a stability statement, not a circular prediction. No ansatz is smuggled in via self-citation, and no load-bearing result rests solely on a self-citation. The only notable defect is a non-circular correctness concern in the converse construction of Theorem 7: as printed, the binomial weights pi_k and tilde pi_k sum to 1/2 rather than 1, so the constructed measure is not a probability measure; the moment identity (16) is independent of this normalization, so the proof is likely repairable, but this is a bug in the argument, not a circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's proofs rest on standard theorems from tensor analysis (Kruskal uniqueness, Hilbert-space tensor embeddings) and metric entropy for Holder classes, plus explicit modeling assumptions: product components, known m, incoherence, and bounded densities. There are no fitted free parameters and no invented physical entities.

assumptions (5)
  • standard math Kruskal's theorem characterizing uniqueness of CP decompositions of 3-way tensors
    Invoked via Lemma 20 (a Hilbert-space extension from VS22) to conclude permutation equivalence from the CP decompositions in (4).
  • standard math Properties of tensor products of Hilbert spaces and the unitary embedding in Lemma 16
    Used throughout proofs of Theorems 7 and 12 to move between product densities and tensor products.
  • standard math Known metric entropy bounds for the 1-dimensional Holder density class
    Used in Lemma 31 and Theorem 13 to derive covering and packing numbers of the mixture class.
  • domain assumption The model in (1) assumes component measures are product measures, m is known and fixed, and for estimation the densities are supported on [0,1] and are q-Holder smooth
    This defines the class under study; the identifiability theorem itself does not require density structure, but the minimax and algorithmic results do.
  • domain assumption Assumption 11: per-coordinate component densities are mu-incoherent in L2 with mu < 1, and mixing proportions are bounded below by zeta > 0
    Needed for the perturbation bounds in Theorem 12 and the algorithm correctness in Theorem 14; this is the strongest working assumption.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Identifiability and Estimation in High-Dimensional Nonparametric Latent Structure Models." pith.science (2026). https://pith.science/paper/BLYYYG5N

@misc{pith2026250609165,
  author       = {Pith},
  title        = {Pith review of: Identifiability and Estimation in High-Dimensional Nonparametric Latent Structure Models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BLYYYG5N}},
  note         = {Machine review of arXiv:2506.09165}
}
read the original abstract

This paper studies the problems of identifiability and estimation in high-dimensional nonparametric latent structure models. We introduce an identifiability theorem that generalizes existing conditions, establishing a unified framework applicable to diverse statistical settings. Our results rigorously demonstrate how increased dimensionality, coupled with diversity in variables, inherently facilitates identifiability. For the estimation problem, we establish near-optimal minimax rate bounds for the high-dimensional nonparametric density estimation under latent structures with smooth marginals. Contrary to the conventional curse of dimensionality, our sample complexity scales only polynomially with the dimension. Additionally, we develop a perturbation theory for component recovery and propose a recovery procedure based on simultaneous diagonalization.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

48 extracted references · 44 canonical work pages

  1. [1]

    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(80):2773--2832, 2014

    work page 2014

  2. [2]

    Allman, Catherine Matias, and John A

    Elizabeth S. Allman, Catherine Matias, and John A. Rhodes. Identifiability of parameters in latent structure models with many observed variables. The Annals of Statistics , 37(6A), December 2009

    work page 2009

  3. [3]

    Tatiana Benaglia, Didier Chauveau, and David R. Hunter. An EM - Like Algorithm for Semi - and Nonparametric Estimation in Multivariate Mixtures . Journal of Computational and Graphical Statistics , 18(2):505--526, January 2009

    work page 2009

  4. [4]

    Tatiana Benaglia, Didier Chauveau, and David R. Hunter. Bandwidth Selection in an EM - Like Algorithm for Nonparametric Multivariate Mixtures . In Nonparametric Statistics and Mixture Models , pages 15--27, The Pennsylvania State University, USA, January 2011. WORLD SCIENTIFIC

    work page 2011

  5. [5]

    Smoothed analysis of tensor decompositions

    Aditya Bhaskara, Moses Charikar, Ankur Moitra, and Aravindan Vijayaraghavan. Smoothed analysis of tensor decompositions. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing , STOC '14, page 594–603, New York, NY, USA, 2014. Association for Computing Machinery

    work page 2014

  6. [6]

    Uniqueness of tensor decompositions with applications to polynomial identifiability

    Aditya Bhaskara, Moses Charikar, and Aravindan Vijayaraghavan. Uniqueness of tensor decompositions with applications to polynomial identifiability. In Maria Florina Balcan, Vitaly Feldman, and Csaba Szepesvári, editors, Proceedings of The 27th Conference on Learning Theory , volume 35 of Proceedings of Machine Learning Research , pages 742--778, Barcelona...

    work page 2014

  7. [7]

    Matrix Analysis

    Rajendra Bhatia. Matrix Analysis . Number 169 in Graduate Texts in Mathematics . Springer, New York, NY, 1997

    work page 1997

  8. [8]

    Approximation dans les espaces m \'e triques et th \'e orie de l'estimation

    Lucien Birg \'e . Approximation dans les espaces m \'e triques et th \'e orie de l'estimation. Zeitschrift f \"u r Wahrscheinlichkeitstheorie und verwandte Gebiete , 65:181--237, 1983

    work page 1983

Show all 48 references
  1. [9]

    Estimating multivariate latent-structure models

    Stéphane Bonhomme, Koen Jochmans, and Jean-Marc Robin. Estimating multivariate latent-structure models. The Annals of Statistics , 44(2), April 2016

    2016

  2. [10]

    Optimal Rate of Convergence for Finite Mixture Models

    Jiahua Chen. Optimal Rate of Convergence for Finite Mixture Models . The Annals of Statistics , 23(1), February 1995

    1995

  3. [11]

    Hunter, and Michael Levine

    Didier Chauveau, David R. Hunter, and Michael Levine. Semi-parametric estimation for conditional independence multivariate finite mixture models. Statistics Surveys , 9(none), January 2015

    2015

  4. [12]

    The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces

    Guoliang Chen and Yifeng Xue. The expression of the generalized inverse of the perturbed operator under Type I perturbation in Hilbert spaces. Linear Algebra and its Applications , 285(1-3):1--6, December 1998

    1998

  5. [13]

    Efficient Algorithms for Sparse Moment Problems without Separation

    Zhiyuan Fan and Jian Li. Efficient Algorithms for Sparse Moment Problems without Separation . 36th Annual Conference on Learning Theory , 2023

    2023

  6. [14]

    Servedio

    Jon Feldman, Ryan O'Donnell, and Rocco A. Servedio. Learning Mixtures of Product Distributions over Discrete Domains . SIAM Journal on Computing , 37(5):1536--1564, January 2008

    2008

  7. [15]

    Kaashoek

    Israel Gohberg, Seymour Goldberg, and Marinus A. Kaashoek. Singular Values of Compact Operators , pages 96--108. Birkh \"a user Basel, Basel, 1990

    1990

  8. [16]

    Identification of Mixtures of Discrete Product Distributions in Near - Optimal Sample and Time Complexity

    Spencer L Gordon, Erik Jahn, Bijan Mazaheri, Yuval Rabani, and Leonard J Schulman. Identification of Mixtures of Discrete Product Distributions in Near - Optimal Sample and Time Complexity . 37th Annual Conference on Learning Theory , 2024

    2024

  9. [17]

    Source identification for mixtures of product distributions

    Spencer Gordon, Bijan H Mazaheri, Yuval Rabani, and Leonard Schulman. Source identification for mixtures of product distributions. In Mikhail Belkin and Samory Kpotufe, editors, Proceedings of Thirty Fourth Conference on Learning Theory , volume 134 of Proceedings of Machine L...

    2021

  10. [18]

    Schulman, and Yuval Rabani

    Spencer Gordon, Bijan Mazaheri, Leonard J. Schulman, and Yuval Rabani. The Sparse Hausdorff Moment Problem , with Application to Topic Models , September 2020. arXiv:2007.08101 [cs, stat]

    2020 arXiv

  11. [19]

    Gordon and Leonard J

    Spencer L. Gordon and Leonard J. Schulman. Hadamard Extensions and the Identification of Mixtures of Product Distributions . IEEE Transactions on Information Theory , 68(6):4085--4089, June 2022

    2022

  12. [20]

    van der Vaart

    Subhashis Ghosal and Aad W. van der Vaart. Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. The Annals of Statistics , 29(5), October 2001

    2001

  13. [21]

    Horn and Charles R

    Roger A. Horn and Charles R. Johnson. Matrix analysis . Cambridge University Press, Cambridge ; New York, 2nd ed edition, 2012

    2012

  14. [22]

    Strong identifiability and optimal minimax rates for finite mixture estimation

    Philippe Heinrich and Jonas Kahn. Strong identifiability and optimal minimax rates for finite mixture estimation. The Annals of Statistics , 46(6A), December 2018

    2018

  15. [23]

    Horn and Zai Yang

    Roger A. Horn and Zai Yang. Rank of a Hadamard product. Linear Algebra and its Applications , 591:87--98, April 2020

    2020

  16. [24]

    Nonparametric estimation of component distributions in a multivariate mixture

    Peter Hall and Xiao-Hua Zhou. Nonparametric estimation of component distributions in a multivariate mixture. The Annals of Statistics , 31(1), February 2003

    2003

  17. [25]

    On the use of Bernoulli mixture models for text classiÿcation

    A Juan and E Vidal. On the use of Bernoulli mixture models for text classiÿcation. Pattern Recognition , 2002

    2002

  18. [26]

    Juan and E

    A. Juan and E. Vidal. Bernoulli mixture models for binary images. In Proceedings of the 17th International Conference on Pattern Recognition , 2004. ICPR 2004. , pages 367--370 Vol.3, Cambridge, UK, 2004. IEEE

    2004

  19. [27]

    Kolda and Brett W

    Tamara G. Kolda and Brett W. Bader. Tensor decompositions and applications. SIAM Review , 51(3):455--500, 2009

    2009

  20. [28]

    Fundamentals of the theory of operator algebras

    Richard V Kadison and John R Ringrose. Fundamentals of the theory of operator algebras. Volume I: Elementary Theory . Academic press New York, 1983

    1983

  21. [29]

    Joseph B. Kruskal. Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and its Applications , 18(2):95--138, 1977

    1977

  22. [30]

    Non-parametric identification and estimation of the number of components in multivariate mixtures

    Hiroyuki Kasahara and Katsumi Shimotsu. Non-parametric identification and estimation of the number of components in multivariate mixtures. Journal of the Royal Statistical Society: Series B (Statistical Methodology) , 76(1):97--111, January 2014

    2014

  23. [31]

    Levine, D

    M. Levine, D. R. Hunter, and D. Chauveau. Maximum smoothed likelihood for multivariate mixtures. Biometrika , 98(2):403--416, June 2011

    2011

  24. [32]

    S. E. Leurgans, R. T. Ross, and R. B. Abel. A Decomposition for Three - Way Arrays . SIAM Journal on Matrix Analysis and Applications , 14(4):1064--1083, October 1993

    1993

  25. [33]

    Schulman, and Chaitanya Swamy

    Jian Li, Yuval Rabani, Leonard J. Schulman, and Chaitanya Swamy. Learning Arbitrary Statistical Mixtures of Discrete Distributions , April 2015. arXiv:1504.02526 [cs]

    2015 arXiv

  26. [34]

    A nonparametric estimation method for the multivariate mixture models

    Nan Lu and Lihong Wang. A nonparametric estimation method for the multivariate mixture models. Journal of Statistical Computation and Simulation , 92(17):3727--3742, November 2022

    2022

  27. [35]

    Information Theory: From Coding to Learning

    Yury Polyanskiy and Yihong Wu. Information Theory: From Coding to Learning . Cambridge University Press, 2025

    2025

  28. [36]

    Methods of modern mathematical physics: Functional analysis , volume 1

    Michael Reed and Barry Simon. Methods of modern mathematical physics: Functional analysis , volume 1. Gulf Professional Publishing, 1980

    1980

  29. [37]

    Schulman, and Chaitanya Swamy

    Yuval Rabani, Leonard J. Schulman, and Chaitanya Swamy. Learning mixtures of arbitrary distributions over large discrete domains. In Proceedings of the 5th conference on Innovations in theoretical computer science , pages 207--224, Princeton New Jersey USA, January 2014. ACM

    2014

  30. [38]

    Sidiropoulos and Rasmus Bro

    Nicholas D. Sidiropoulos and Rasmus Bro. On the uniqueness of multilinear decomposition of n-way arrays. Journal of Chemometrics , 14(3):229--239, 2000

    2000

  31. [39]

    Matrix perturbation theory

    Gilbert W Stewart and Ji-guang Sun. Matrix perturbation theory. Academic Press , 1990

    1990

  32. [40]

    Identifiability of Mixtures of Product Measures

    Henry Teicher. Identifiability of Mixtures of Product Measures . The Annals of Mathematical Statistics , 38(4):1300--1302, August 1967. Publisher: Institute of Mathematical Statistics

    1967

  33. [41]

    On the Identifiability of Finite Mixtures of Finite Product Measures , July 2018

    Behrooz Tahmasebi, Seyed Abolfazl Motahari, and Mohammad Ali Maddah-Ali. On the Identifiability of Finite Mixtures of Finite Product Measures , July 2018. arXiv:1807.05444 [math, stat]

    2018 arXiv

  34. [42]

    Tsybakov

    Alexandre B. Tsybakov. Introduction to nonparametric estimation . Springer series in statistics. Springer, New York, english ed. edition, 2009

    2009

  35. [43]

    Vandermeulen and Clayton D

    Robert A. Vandermeulen and Clayton D. Scott. An operator theoretic approach to nonparametric mixture models. The Annals of Statistics , 47(5):2704--2733, October 2019. Publisher: Institute of Mathematical Statistics

    2019

  36. [44]

    Vandermeulen and René Saitenmacher

    Robert A. Vandermeulen and René Saitenmacher. Generalized Identifiability Bounds for Mixture Models with Grouped Samples , July 2022. arXiv:2207.11164 [cs, math, stat]

    2022 arXiv

  37. [45]

    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. The Annals of Statistics , 48(4), August 2020

    2020

  38. [46]

    Rates of convergence of minimum distance estimators and kolmogorov's entropy

    Yannis G Yatracos. Rates of convergence of minimum distance estimators and kolmogorov's entropy. The Annals of Statistics , 13(2):768--774, 1985

    1985

  39. [47]

    Information-theoretic determination of minimax rates of convergence

    Yuhong Yang and Andrew Barron. Information-theoretic determination of minimax rates of convergence. The Annals of Statistics , 27(5):1564--1599, October 1999. Publisher: Institute of Mathematical Statistics

    1999

  40. [48]

    Nonparametric Estimation of Multivariate Mixtures

    Chaowen Zheng and Yichao Wu. Nonparametric Estimation of Multivariate Mixtures . Journal of the American Statistical Association , 115(531):1456--1471, July 2020

    2020

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.