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REVIEW 4 major objections 5 minor 34 references

Horizontal and Vertical Federated Causal Structure Learning via Higher-order Cumulants

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Under a linear non-Gaussian acyclic model, a variable is the causal source exactly when its cumulant-based tau statistic sums to zero across all other variables, which lets federated clients with disjoint variable sets recover the global…

desk verdict Under-specified cumulant theory and missing experiments sink this federated causal discovery draft, though the aggregation idea is worth a second look. read the letter →

arxiv 2507.06888 v1 pith:IXNOMAZY submitted 2025-07-09 cs.LG

Wei Chen , Wanyang Gu , Linjun Peng , Ruichu Cai , Zhifeng Hao , Kun Zhang This is my paper
classification cs.LG
keywords federatedcausaldiscoveryhigher-ordercumulantsverticallearninghorizontalLiNGAMsourceidentificationstrengthmatrixnon-Gaussianmodels
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes a federated causal discovery method, FedISHC, that works when clients hold different, possibly non-overlapping variable sets (vertical federation) as well as when they share variables (horizontal federation). It claims that under the linear non-Gaussian acyclic model (LiNGAM), the global source variable can be identified from aggregated third-order cumulants alone: a variable is a source exactly when its tau statistic summed against all other variables is zero. Because cumulants can be aggregated as sample-size-weighted averages of local cumulants, clients never need to send raw data or perform local causal discovery. The paper proves that eliminating each identified source can be done at the cumulant level, yielding the global causal order and the causal strength matrix B. A sympathetic reader would care because this removes the main obstacle to vertical federated causal discovery, missing variables causing spurious local edges, while also estimating edge strengths.

What carries the argument

The central object is the third-order cumulant tensor, and the operative criterion is the tau statistic, $\tau_{ij} = |C_3(x_i)C_{1,2}(x_i,x_j) - C_{2,1}(x_i,x_j)C_{1,2}(x_j,x_i)|$, which distinguishes a cause from its effect because the effect variable carries an extra noise term that cannot be canceled. Theorem 1 says a variable is a source if and only if its tau values summed against all other variables are zero, converting source identification into a sum test on aggregated cumulants. The update equations (5), (6), and (7) subtract the identified source's contribution from the global cumulants, so the recursion operates entirely at the cumulant level rather than on client data.

What would settle it

Generate a three-variable LiNGAM chain x1→x2→x3, split the variables so one client holds {x1,x2} and another holds {x2,x3}, give the two clients different nonzero means, have each client center its own data before computing local cumulants, aggregate by equation (2), and run the recursive source search; if the recovered order or causal strength matrix differs from the pooled-data result, the aggregation assumption is false.

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Extended reading notes

Core claim

The paper's central claim is Theorem 1: under LiNGAM, the source variable is exactly the one whose tau values summed against every other variable equal zero. This turns source search into a test on the global third-order cumulant tensor, which is assembled on the server from client-submitted local cumulants. Theorem 2 then shows that the influence of an identified source can be removed from the global cumulant tensor using only update equations (5), (6), and (7), so the recursive search continues without any client performing residual regression on raw data. Running this recursion produces the global causal order and, as a by-product, the causal strength matrix B. The paper therefore claims to solve both horizontal and vertical federated causal structure learning in one framework, estimating graph structure and edge strengths while exchanging only statistics.

Load-bearing premise

The load-bearing premise is that averaging the cumulants computed by each client is equivalent to computing the cumulant on the combined dataset; this equivalence is exact only when all clients use the same global mean, and the paper does not specify how that mean is established.

Editorial extensions

If this is right

  • Clients can contribute to global causal discovery without ever sharing raw observations; only local cumulant tensors and sample sizes leave the client.
  • Vertical splits with no common samples are handled, because cumulants depend only on the joint distribution of the variables involved, not on which other variables a client happens to record.
  • The recursive source elimination gives the complete causal ordering and the causal strength matrix B in one pass, so the output includes edge weights, not just a skeleton.
  • The framework covers both non-Gaussian linear data via third-order cumulants and Gaussian linear data via partial correlations, selected by checking whether third-order cumulants vanish.

Reading between the lines

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

  • The same aggregation logic would plausibly extend to fourth-order cumulants, which would cover non-Gaussian noises with zero skewness; the paper only uses third-order statistics.
  • In practice, clients would need an explicit mean-alignment or double-centering protocol before computing local cumulants, because the aggregation formula in equation (2) is exact only under a common global mean.
  • The sign-test or permutation-test threshold for treating tau as zero is a free parameter whose influence on the recovered order is not analyzed; a practitioner would need to calibrate it against the available sample size.
  • Because the aggregation only requires knowing which clients share each variable pair, the method could be applied to hybrid federated settings with partially overlapping samples and features, not just the pure horizontal and vertical cases.
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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

4 major / 5 minor

Summary. The paper proposes FedISHC, a federated causal structure learning method intended to work in both horizontal and vertical settings with overlapping variable sets. Each client computes third-order cumulants of its local variables and sends them to a server; the server aggregates these into a global cumulant matrix, recursively identifies source variables using a zero-sum condition on products of third-order cumulants, removes the identified source by cumulant-level updates, and outputs the causal order and causal strength matrix. The paper states identifiability theorems for the non-Gaussian case, sketches an extension to linear Gaussian models, and its abstract claims superior performance on synthetic and real-world data.

Significance. If the theory were correct and the missing empirical support were supplied, the paper would address an important problem: federated causal discovery in vertical settings with overlapping variables, where constraint-based methods often fail because valid separating sets are unavailable to any single client. The idea of aggregating cumulant information instead of conditional-independence statistics is interesting, and the cumulant-level source-removal update is creative. However, the central identifiability theorem is false under the stated LiNGAM assumption, the aggregation formula is not justified, the appendix proofs are absent, and the experiments advertised in the abstract do not exist in the manuscript. As written, the contribution is not established.

major comments (4)
  1. [§3.2 Definition 2 and §4 Step 3, Theorem 1] Theorem 1 is not valid under the stated LiNGAM assumption. LiNGAM requires only non-Gaussian noise, not nonzero third-order cumulants. If the noise variables are symmetric non-Gaussian (e.g., Laplace or Uniform(-1,1)), then C3(e_i)=0 for every i, so C3(x_i), C1,2(x_i,x_j), and C2,1(x_i,x_j) all vanish for every pair and τ_ij=0 for all i,j. Consequently every variable satisfies the zero-sum condition of Theorem 1, the recursive source identification cannot separate sources from non-sources, and the influence coefficient α=C2,1(x_s,x_j)/C3(x_s) is 0/0. The paper nowhere states a nonzero-third-cumulant assumption, and the proof in Appendix A1 is absent. This is a population-level identifiability gap, not a finite-sample or federated-aggregation issue.
  2. [§4 Step 2, Eq. (2)] The claim that the global cumulant is a sample-size-weighted average of local cumulants is unsubstantiated. Cumulants are nonlinear functions of moments, so the cumulant of a pooled sample is not in general equal to the weighted average of local cumulants unless the local samples share the same mean and higher-order sufficient statistics are aggregated with appropriate corrections. In addition, the formula as written uses the total sample size across all clients in the denominator while restricting the sum to clients in I_ij; the denominator should be the sum of sample sizes over I_ij only, not over all clients. Because the entire federated protocol depends on Eq. (2), this requires a formal derivation or an explicit correction.
  3. [§1 and entire manuscript] The abstract states that the algorithm "demonstrates superior performance in experiments conducted on both synthetic data and real-world data," but the manuscript contains no experimental section, no datasets, no baselines, and no metrics. The empirical claim motivating the method is therefore completely unsupported.
  4. [Theorem 1, Theorem 2, and Eq. (8)] The proofs of the central identifiability theorem, the cumulant-update theorem, and the Gaussian extension are deferred to Appendices A1, A2, and A3, respectively, but none of these appendices is present in the manuscript. Without those proofs, the main theoretical claims are unverified; the text only provides an intuitive calculation for the two-variable direct-edge case.
minor comments (5)
  1. [§4 Step 3, Theorem 1 statement] The statement of Theorem 1 is garbled: "xs = X_{s∈O, j∈On{s}} τsj = 0" is not a well-formed proposition; it should state that x_s is the source if and only if Σ_{j∈O\{s}} τ_{sj}=0.
  2. [§3.1 Assumption 1] Assumption 1 is ambiguous: "there must exist at least one client among them" presumably means every pair of variables co-occurs in at least one client, but this is not stated clearly. This condition is essential for the global cumulant matrix to be complete.
  3. [§1, citation [13]] The statement that "higher-order cumulants rely solely on the joint distribution of the relevant variables and are not influenced by the absence of variables" cites reference [13], which is a survey on homomorphic encryption, not a source on cumulant properties; this citation appears to be incorrect.
  4. [§6] The claim that the approach has "unified all linear cases under a single framework" is an overstatement: in the linear Gaussian case the method can only recover Markov equivalence classes, not the full DAG orientation.
  5. [§5.2] The communication-cost analysis writes "O(d_k ×(2d_k−1)×N×1 + 1)"; the "+1" is informal and the analysis does not state whether the count is in numeric entries or bits, nor does it account for the cost of transmitting the local sample sizes beyond the one scalar.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: source identification and influence removal are parameter-free cumulant identities, not fits renamed as predictions.

full rationale

The paper's central chain is: aggregate local third-order cumulants (Eq. 2), compute tau by Eq. (3), identify the source by Theorem 1, remove its influence by Eqs. (5)-(7), and read causal strengths as alpha = C_{2,1}(x_s,x_j)/C_3(x_s). Each step is a population-level cumulant identity derived from LiNGAM, not a fit to the quantity being predicted. The tau criterion for a two-variable chain is explicitly derived in Eq. (4), and the update equations are stated as parameter-free cancellations in Theorem 2. The self-citations [14,15] provide motivation but are not the load-bearing derivation; the paper states Theorem 1's proof is in Appendix A1, not in those citations. Several verifiability gaps are real but are not circularity: the appendices A1-A3 are absent from the arXiv v1, the aggregation identity in Eq. (2) lacks explicit mean-centering conditions, and Theorem 1 is false as stated for symmetric non-Gaussian noises with zero third-order cumulants (e.g., zero-mean Laplace or uniform errors), since all tau values vanish and every variable satisfies the source condition. That is an identifiability/assumption gap, not a circular reduction. No step reduces to its own input by construction, so the circularity score is 0.

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

The central claim rests on the LiNGAM model, cumulant decomposability, and the existence of a client covering each variable pair. The proofs are deferred to missing appendices, and the method implicitly requires nonzero third-order cumulants for all noise variables.

assumptions (5)
  • domain assumption Linear non-Gaussian acyclic model (LiNGAM): X=BX+E with independent non-Gaussian noise components.
    Stated in Definition 2 and used by Theorem 1 and Theorem 2 to relate cumulants to causal coefficients.
  • domain assumption The global cumulant can be obtained as a weighted average of local cumulants (Eq. 2).
    This sample-wise decomposability is asserted in Section 4 Step 2, but is only valid for central moments with respect to the global mean, which the paper does not establish.
  • domain assumption For any pair of variables, at least one client contains both.
    Assumption 1 in Section 3.1; required so every joint cumulant term can be computed by some client.
  • domain assumption Causal strength coefficients b_ij are identical across all clients.
    Stated in Definition 2; needed for a single global causal matrix to exist.
  • domain assumption All noise variables have nonzero third-order cumulants.
    Not stated, but necessary for tau_j != 0 for non-source variables; symmetric non-Gaussian noises would break source identification.

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Cite this review

Pith. "Pith review of Horizontal and Vertical Federated Causal Structure Learning via Higher-order Cumulants." pith.science (2026). https://pith.science/paper/IXNOMAZY

@misc{pith2026250706888,
  author       = {Pith},
  title        = {Pith review of: Horizontal and Vertical Federated Causal Structure Learning via Higher-order Cumulants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IXNOMAZY}},
  note         = {Machine review of arXiv:2507.06888}
}
read the original abstract

Federated causal discovery aims to uncover the causal relationships between entities while protecting data privacy, which has significant importance and numerous applications in real-world scenarios. Existing federated causal structure learning methods primarily focus on horizontal federated settings. However, in practical situations, different clients may not necessarily contain data on the same variables. In a single client, the incomplete set of variables can easily lead to spurious causal relationships, thereby affecting the information transmitted to other clients. To address this issue, we comprehensively consider causal structure learning methods under both horizontal and vertical federated settings. We provide the identification theories and methods for learning causal structure in the horizontal and vertical federal setting via higher-order cumulants. Specifically, we first aggregate higher-order cumulant information from all participating clients to construct global cumulant estimates. These global estimates are then used for recursive source identification, ultimately yielding a global causal strength matrix. Our approach not only enables the reconstruction of causal graphs but also facilitates the estimation of causal strength coefficients. Our algorithm demonstrates superior performance in experiments conducted on both synthetic data and real-world data.

Figures

Figures reproduced from arXiv: 2507.06888 by the authors.

Figure 1
Figure 1. Overall framework of FedISHC. First, the client sends its sample size and local cumulant [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The left figure shows that xi , xj , and xk form a common structure, while the right figure indicates that xi , xj , and xk are independent of each other. Step 5: The algorithm outputs the global causal order, the updated global cumulant matrix, and the causal strength matrix B. Throughout the algorithm’s execution, we compute the influence coefficient α = C2,1(xs, xj )/C3(xs) for the source variable xs and each oth… view at source ↗

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Reference graph

Works this paper leans on

34 extracted references · 29 canonical work pages

  1. [1]

    Causal mechanisms in the social sciences.Annual review of sociology, 36(1):49–67, 2010

    Peter Hedstr¨ om and Petri Ylikoski. Causal mechanisms in the social sciences.Annual review of sociology, 36(1):49–67, 2010. 7

    work page 2010

  2. [2]

    Academic Press, 2014

    Clark Glymour, Richard Scheines, and Peter Spirtes.Discovering causal structure: Artificial intelligence, philosophy of science, and statistical modeling. Academic Press, 2014

    work page 2014

  3. [3]

    Prob- abilistic computational causal discovery for systems biology.Uncertainty in biology: a computa- tional modeling approach, pages 33–73, 2016

    Vincenzo Lagani, Sofia Triantafillou, Gordon Ball, Jesper Tegn´ er, and Ioannis Tsamardinos. Prob- abilistic computational causal discovery for systems biology.Uncertainty in biology: a computa- tional modeling approach, pages 33–73, 2016

    work page 2016

  4. [4]

    Causal discovery analysis: A promising tool in advancing precision medicine for eating disorders.International Journal of Eating Disorders, 56(11):2012–2021, 2023

    Lisa M Anderson, Kelvin O Lim, Erich Kummerfeld, Ross D Crosby, Scott J Crow, Scott G Engel, Lauren Forrest, Stephen A Wonderlich, and Carol B Peterson. Causal discovery analysis: A promising tool in advancing precision medicine for eating disorders.International Journal of Eating Disorders, 56(11):2012–2021, 2023

    work page 2012

  5. [5]

    FedDAG: Federated DAG Structure Learning

    Erdun Gao, Junjia Chen, Li Shen, Tongliang Liu, Mingming Gong, and Howard Bondell. Feddag: Federated dag structure learning.arXiv preprint arXiv:2112.03555, 2021

    work page Pith review arXiv 2021

  6. [6]

    Federated causal discovery with missing data in a multicentric study on endometrial cancer.Available at SSRN 4998504, 2024

    Zanga Alessio, Alice Bernasconi, Peter JF Lucas, JMA Pijnenborg, Casper Reijnen, Marco Scutari, and Anthony C Constantinou. Federated causal discovery with missing data in a multicentric study on endometrial cancer.Available at SSRN 4998504, 2024

    work page 2024

  7. [7]

    Towards practical federated causal structure learning

    Zhaoyu Wang, Pingchuan Ma, and Shuai Wang. Towards practical federated causal structure learning. InJoint European Conference on Machine Learning and Knowledge Discovery in Databases, pages 351–367. Springer, 2023

    work page 2023

  8. [8]

    Federated Causal Discovery from Heterogeneous Data

    Loka Li, Ignavier Ng, Gongxu Luo, Biwei Huang, Guangyi Chen, Tongliang Liu, Bin Gu, and Kun Zhang. Federated causal discovery from heterogeneous data.arXiv preprint arXiv:2402.13241, 2024

    work page Pith review arXiv 2024

Show all 34 references
  1. [9]

    Dianwen Ng, Xiang Lan, Melissa Min-Szu Yao, Wing P Chan, and Mengling Feng. Federated learning: a collaborative effort to achieve better medical imaging models for individual sites that have small labelled datasets.Quantitative Imaging in Medicine and Surgery, 11(2):852, 2021

    2021

  2. [10]

    Federated recommendation systems

    Liu Yang, Ben Tan, Vincent W Zheng, Kai Chen, and Qiang Yang. Federated recommendation systems. InFederated Learning: Privacy and Incentive, pages 225–239. Springer, 2020

    2020

  3. [11]

    Federated graph learning–a position paper.arXiv preprint arXiv:2105.11099, 2021

    Huanding Zhang, Tao Shen, Fei Wu, Mingyang Yin, Hongxia Yang, and Chao Wu. Federated graph learning–a position paper.arXiv preprint arXiv:2105.11099, 2021

    2021 arXiv

  4. [12]

    Hybrid federated learning for feature & sample heterogeneity: Algorithms and implementation.Transactions on Machine Learning Research, 2024

    Xinwei Zhang, Wotao Yin, Mingyi Hong, and Tianyi Chen. Hybrid federated learning for feature & sample heterogeneity: Algorithms and implementation.Transactions on Machine Learning Research, 2024

    2024

  5. [13]

    A survey on homomorphic encryption schemes: Theory and implementation.ACM Computing Surveys (Csur), 51(4):1–35, 2018

    Abbas Acar, Hidayet Aksu, A Selcuk Uluagac, and Mauro Conti. A survey on homomorphic encryption schemes: Theory and implementation.ACM Computing Surveys (Csur), 51(4):1–35, 2018

    2018

  6. [14]

    Causal discovery with latent confounders based on higher-order cumulants

    Ruichu Cai, Zhiyi Huang, Wei Chen, Zhifeng Hao, and Kun Zhang. Causal discovery with latent confounders based on higher-order cumulants. InInternational conference on machine learning, pages 3380–3407. PMLR, 2023

    2023

  7. [15]

    Identification of causal structure with latent variables based on higher order cumulants

    Wei Chen, Zhiyi Huang, Ruichu Cai, Zhifeng Hao, and Kun Zhang. Identification of causal structure with latent variables based on higher order cumulants. InProceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 20353–20361, 2024

    2024

  8. [16]

    Towards federated bayesian network structure learning with con- tinuous optimization

    Ignavier Ng and Kun Zhang. Towards federated bayesian network structure learning with con- tinuous optimization. InInternational Conference on Artificial Intelligence and Statistics, pages 8095–8111. PMLR, 2022

    2022

  9. [17]

    Federated causality learning with explainable adaptive optimization

    Dezhi Yang, Xintong He, Jun Wang, Guoxian Yu, Carlotta Domeniconi, and Jinglin Zhang. Federated causality learning with explainable adaptive optimization. InProceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 16308–16315, 2024

    2024

  10. [18]

    Federated learning of generalized linear causal networks.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024

    Qiaoling Ye, Arash A Amini, and Qing Zhou. Federated learning of generalized linear causal networks.IEEE Transactions on Pattern Analysis and Machine Intelligence, 2024. 8

    2024

  11. [19]

    Nothing but regrets—privacy-preserving federated causal discovery

    Osman Mian, David Kaltenpoth, Michael Kamp, and Jilles Vreeken. Nothing but regrets—privacy-preserving federated causal discovery. InInternational Conference on Artificial Intelligence and Statistics, pages 8263–8278. PMLR, 2023

    2023

  12. [20]

    Towards privacy-aware causal structure learning in federated setting.IEEE Transactions on Big Data, 9(6):1525–1535, 2023

    Jianli Huang, Xianjie Guo, Kui Yu, Fuyuan Cao, and Jiye Liang. Towards privacy-aware causal structure learning in federated setting.IEEE Transactions on Big Data, 9(6):1525–1535, 2023

    2023

  13. [21]

    Enhancing causal discovery in federated settings with limited local samples

    Xianjie Guo, Liping Yi, Xiaohu Wu, Kui Yu, and Gang Wang. Enhancing causal discovery in federated settings with limited local samples. InInternational Workshop on Trustworthy Federated Learning, pages 164–179. Springer, 2024

    2024

  14. [22]

    Fedcsl: A scalable and accurate approach to feder- ated causal structure learning

    Xianjie Guo, Kui Yu, Lin Liu, and Jiuyong Li. Fedcsl: A scalable and accurate approach to feder- ated causal structure learning. InProceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 12235–12243, 2024

    2024

  15. [23]

    Sample quality heterogeneity-aware federated causal discovery through adaptive variable space selection

    Xianjie Guo, Kui Yu, Hao Wang, Lizhen Cui, Han Yu, and Xiaoxiao Li. Sample quality heterogeneity-aware federated causal discovery through adaptive variable space selection. In Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence, pages 4071...

    2024

  16. [24]

    Integrating locally learned causal structures with overlapping variables.Advances in Neural Information Processing Systems, 21, 2008

    David Danks, Clark Glymour, and Robert Tillman. Integrating locally learned causal structures with overlapping variables.Advances in Neural Information Processing Systems, 21, 2008

    2008

  17. [25]

    Constraint-based causal discovery from multiple in- terventions over overlapping variable sets.The Journal of Machine Learning Research, 16(1):2147– 2205, 2015

    Sofia Triantafillou and Ioannis Tsamardinos. Constraint-based causal discovery from multiple in- terventions over overlapping variable sets.The Journal of Machine Learning Research, 16(1):2147– 2205, 2015

    2015

  18. [26]

    Causal discovery from multiple data sets with non-identical variable sets

    Biwei Huang, Kun Zhang, Mingming Gong, and Clark Glymour. Causal discovery from multiple data sets with non-identical variable sets. InProceedings of the AAAI conference on artificial intelligence, volume 34, pages 10153–10161, 2020

    2020

  19. [27]

    Causal discovery from unknown interventional datasets over overlapping variable sets.IEEE Transactions on Knowledge and Data Engineering, 2024

    Fuyuan Cao, Yunxia Wang, Kui Yu, and Jiye Liang. Causal discovery from unknown interventional datasets over overlapping variable sets.IEEE Transactions on Knowledge and Data Engineering, 2024

    2024

  20. [28]

    A novel data enhancement approach to dag learning with small data samples.Applied Intelligence, 53(22):27589–27607, 2023

    Xiaoling Huang, Xianjie Guo, Yuling Li, and Kui Yu. A novel data enhancement approach to dag learning with small data samples.Applied Intelligence, 53(22):27589–27607, 2023

    2023

  21. [29]

    Multi-target markov boundary discovery: Theory, algorithm, and application.IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4964–4980, 2022

    Xingyu Wu, Bingbing Jiang, Yan Zhong, and Huanhuan Chen. Multi-target markov boundary discovery: Theory, algorithm, and application.IEEE Transactions on Pattern Analysis and Machine Intelligence, 45(4):4964–4980, 2022

    2022

  22. [30]

    SIAM, 2001

    David R Brillinger.Time series: data analysis and theory. SIAM, 2001

    2001

  23. [31]

    Independent component analysis, adaptive and learning systems for signal processing, communications, and control.vol, 1:11–14, 2002

    Aapo Hyv¨ arinen, Juha Karhunen, and Erkki Oja. Independent component analysis, adaptive and learning systems for signal processing, communications, and control.vol, 1:11–14, 2002

    2002

  24. [32]

    A linear non-gaussian acyclic model for causal discovery.Journal of Machine Learning Research, 7(10), 2006

    Shohei Shimizu, Patrik O Hoyer, Aapo Hyv¨ arinen, Antti Kerminen, and Michael Jordan. A linear non-gaussian acyclic model for causal discovery.Journal of Machine Learning Research, 7(10), 2006

    2006

  25. [33]

    Directlingam: A direct method for learning a linear non-gaussian structural equation model.Journal of Machine Learning Research-JMLR, 12(Apr):1225–1248, 2011

    Shohei Shimizu, Takanori Inazumi, Yasuhiro Sogawa, Aapo Hyvarinen, Yoshinobu Kawahara, Takashi Washio, Patrik O Hoyer, Kenneth Bollen, and Patrik Hoyer. Directlingam: A direct method for learning a linear non-gaussian structural equation model.Journal of Machine Learning Resea...

    2011

  26. [34]

    Secure multiparty computation.Communications of the ACM, 64(1):86–96, 2020

    Yehuda Lindell. Secure multiparty computation.Communications of the ACM, 64(1):86–96, 2020. 9

    2020

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