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REVIEW 3 major objections 6 minor 52 references

Under a linear latent causal model, observations from at least d heterogeneous environments identify the latent features and causal graph up to a surrounded-node equivalence, and the new algorithm CREATOR provably recovers them.

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2026-08-04 14:48 UTC pith:NL3CRAG3

load-bearing objection The noise assumption is genuinely relaxed and the algorithm is new, but Theorem 1 is only proved for p=d—the p≥d+1 case rests on an unjustified row-selection step. the 3 major comments →

arxiv 2509.22553 v2 pith:NL3CRAG3 submitted 2025-09-26 stat.ML cs.LG

Linear Causal Representation Learning by Topological Ordering, Pruning, and Disentanglement

Hao Chen , Lin Liu , Yu Guang Wang This is my paper
classification stat.ML cs.LG
keywords causal representation learninglinear structural causal modelidentifiabilitytopological orderingindependent component analysisheterogeneous environmentslatent causal graphDAG
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper tries to show that hidden causal variables can be recovered from observed data even when the data are only a linear mixture of those variables and come from ordinary heterogeneous environments, not from carefully designed interventions. The setting is a linear structural causal model over d latent features, with independent non-Gaussian noise and a linear mixing map to p measurements. The main theorem says that data from at least d environments identify both the latent features and the causal DAG up to permutation, scaling, and mixing among a node's 'surrounding' ancestors, provided each node's environment-specific coefficients vary enough. The proposed algorithm, CREATOR, achieves this by repeatedly finding a root-node noise component as a linear combination of measurements that is independent of its residual, projecting it out to reveal the next root, pruning edges with rank tests, and finally disentangling the recovered features. A sympathetic reader would care because this removes the need for interventional data and for restrictive assumptions that noise distributions match across environments, while still giving a formal identifiability and consistency guarantee.

Core claim

The central claim is an identifiability theorem: under a linear structural causal model over latent features, with a full-column-rank linear mixing map and independent non-Gaussian noise (at most one Gaussian component), the distribution of observed data from at least d environments uniquely determines the latent features and the causal DAG up to permutation, scale, and the 'surrounded-node' equivalence. The surrounded-node ambiguity means a recovered latent variable may be mixed with certain ancestors that share its children; those are the only unresolvable mixtures. The proof identifies each root node's exogenous noise as the unique linear functional of the observations that is independent

What carries the argument

The device that carries the argument is the root-detection characterization: for a candidate linear coefficient vector α, the derived variable α⊤x(k) is independent of the residual x(k)−E(x(k)|α⊤x(k)) in every environment if and only if it is proportional to a root-node exogenous noise component. This turns the search for latent features into a finite enumeration over rows of unmixing matrices produced by independent component analysis, evaluated by a kernel independence criterion. Iterative projection of each recovered root yields the topological ordering; the pruning subroutine compares ranks of environment-stacked regression coefficients to decide which candidate edges are real; and the d

Load-bearing premise

Every latent node's environment-specific coefficients—for the node itself plus its parents—must span a space of dimension one more than the number of its parents, so that the environment heterogeneity is strong enough for the pruning and disentanglement rank tests to discriminate parents from non-parents.

What would settle it

Construct two linear structural causal models with the same number of latent variables and at least as many environments, both satisfying the paper's assumptions—non-Gaussian noise, full-rank linear mixing, and node-level non-degeneracy—whose observed distributions are identical but whose latent DAGs are not equivalent under ~sur; such a pair would refute Theorem 1.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • No interventional data are required: the identifiability guarantee relies only on heterogeneous observational environments, with at least as many environments as latent variables.
  • The relaxed distributional assumptions mean data from different labs, devices, or conditions can be pooled even when their noise levels and distributions differ, as long as each noise component remains non-Gaussian.
  • The recovered objects are honest: variables that cannot be separated from their 'surrounding' ancestors remain mixed, so downstream users know exactly which causal quantities are ambiguous and which are not.
  • With growing sample size, the probability that the algorithm returns the correct equivalence class converges to one, so the procedure is not only identifiable in principle but asymptotically consistent.
  • The LLM proof-of-concept suggests the same linear assumptions can expose interpretable causal structure (background, condition, ending) in hidden states of several large language models, giving a concrete path toward causal interpretability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to nonlinear additive-noise latent models: the root-detection step would need a different statistical criterion, since the appealing linear-form independence characterization no longer holds; the projection-and-prune skeleton might survive.
  • The paper leaves sample complexity open; bounding the finite-sample error of the kernel-independence search, especially under weak causal effects, is the obvious next analytical step before using CREATOR on small datasets.
  • For the LLM case study specifically, K=3 environments with a two-parent node sits at the edge of the node-level non-degeneracy condition; checking that condition on the actual hidden states would clarify whether the recovered DAG is guaranteed by the theorem or empirically lucky.
  • The surrounded-node ambiguity has a practical consequence: if a user wants direct causal effects rather than equivalence classes, they must bring additional assumptions or data to break the remaining mixtures before estimating effects.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper studies linear causal representation learning (CRL) from K heterogeneous environments. Latent features y^(k) follow a linear SCM with environment-specific weights W^(k) and non-Gaussian exogenous noises z^(k); observations are x^(k)=H y^(k) with H∈R^{p×d}, p≥d. Under a node-level non-degeneracy assumption (Assumption 2) and full-column-rank H (Assumption 3), Theorem 1 claims identifiability of y^(k) and the DAG G up to the equivalence relation ∼sur using at least d environments. The proposed algorithm CREATOR has three subroutines: sequential recovery of root noise components via ICA/HSIC (topological ordering), rank-based edge pruning, and feature disentanglement. Theorems 2–5 state guarantees for these subroutines; Theorem 6 claims pointwise consistency. Synthetic experiments compare CREATOR with LiNGCReL, and an LLM case study is used to illustrate the method.

Significance. If the identifiability claim is correct, the paper makes a useful contribution by relaxing distributional assumptions relative to Jin & Syrgkanis (2024): exogenous noise distributions may vary across environments and components need not have distinct distributions. The topological-ordering-plus-pruning strategy is well motivated, and the authors are transparent about adapting the Jin–Syrgkanis proof and about the scope of their theoretical claims. The paper also ships code and provides synthetic and real-data illustrations. However, the proof of the central theorem has a gap in the overcomplete (p>d) setting, and the algorithm's ICA-based recovery step is only justified for square mixtures; these issues need to be resolved before the stated guarantees can be accepted.

major comments (3)
  1. [Appendix B, proof of Theorem 1, p≥d+1 case] The final paragraph of the proof asserts that for p≥d+1 one can 'simply consider the first d dimension' and apply the p=d argument. This requires the d×d submatrix H_[d],· to be invertible, which is not implied by Assumption 3 (full column rank). Example: d=2, p=3, H=[[1,0],[0,0],[0,1]] has rank 2 but H_[1:2],· has rank 1. Full column rank only guarantees that some d-row submatrix is invertible. As written, the stated scope of Theorem 1 is not established for p≥d+1. The gap is likely repairable by selecting a full-rank d-row submatrix, but the proof must make that selection and justify it.
  2. [Section 3.1 and Appendix D, proof of Theorem 3 / Theorem 6] Subroutine 1 and the proof of Theorem 3 rely on 'classical results on ICA' (including Theorem 11 of Reyhani et al. 2012) to recover z^(k) from x^(k). Those results are for square invertible linear mixtures. The model allows p>d, and no overcomplete-ICA identifiability result is stated or proved under Assumptions 1–3. Consequently the algorithm's guarantees for p>d, including Theorems 3–6, are not justified as written. Either restrict the identifiability statements to p=d or add an explicit argument (for example, reducing to a full-rank d-dimensional coordinate subset and proving that ICA then applies).
  3. [Section 4.2, Assumption 2 in the LLM case study] The LLM case study has d=3 and K=3. For a node with two parents, Assumption 2 requires the three environment-specific vectors U_i^(k) to span R^3; for K=3 this is an exact full-rank condition on unknown matrices. The paper does not test or report whether Assumption 2 holds in this data. If it fails, Theorems 1–6 do not apply even though the algorithm may still run and output a DAG. Table 1 should be interpreted with this limitation, or a sensitivity/stability check should be added.
minor comments (6)
  1. [Section 2, Assumption 3] Typo: 'mixing matrix H∈R^{n×d}' should be H∈R^{p×d}.
  2. [Section 3.1, Eq. (2)] The definition of r_j^(k) uses proj⊥_i x_j − E(proj⊥_i x_j | α^T proj⊥_j x), but the optimization uses α^T proj⊥_i x. The notation should be aligned to avoid ambiguity about which projection is applied to α.
  3. [Section 4.1, 'Statistical Complexity' paragraph] The text says 'Theorem 3 in Appendix D.1 further establishes the point-wise consistency', but the theorem in Appendix D.1 is numbered Theorem 6. The cross-reference should be corrected.
  4. [Appendix C, Algorithm 2] The inner loop runs j∈{1,...,i}; for j=i the matrix C_{i,i} is empty and the rank condition is degenerate. The loop should be restricted to j<i (candidate parents).
  5. [Definition 3] The equivalence relation ∼sur is stated with a lower triangular matrix B and the condition 'for ∀j∈[d], i∉sur(j), B_{i,j}=0'. The indexing of rows and columns relative to the permutation π is not immediately clear; a brief clarification would help.
  6. [Appendix E.4] The dimensionality reduction description is hard to follow: 'multiply these them by a matrix i.i.d. drawn from standard Gaussian distribution with column number p=2' and then 'in turn multiplied by a random matrix ... with column number p=30'. Please state the dimensions of the random matrices explicitly.

Circularity Check

0 steps flagged

No significant circularity: the identifiability theorems are derived from stated assumptions plus external results (ICA, Darmois-Skitovitch, HSIC), and the algorithm is not fitted to its claimed outputs. Non-circular proof gaps exist, notably the p>=d+1 extension of Theorem 1.

full rationale

I traced the paper's derivation chain. Theorem 1 is proved from Assumptions 1-3 using classical ICA results, Darmois-Skitovitch, and an explicitly disclosed adaptation of the proof of Theorem 1 in Jin & Syrgkanis (2024); Assumptions 2-3 are also adopted from that external work, not from the authors' own prior results. Theorems 2-5 are proved from the same model and assumptions via linear algebra, rank arguments, and independence criteria; Theorem 6 uses consistency of FastICA and HSIC. No target quantity (the latent features or the causal DAG) is fitted into the proofs, and the algorithm's subroutines are deterministic functions of the data once ICA, HSIC, regression, and rank estimation are fixed. The evaluation metric LocR2 is defined in terms of the same ~sur equivalence class that the identifiability claim uses, but this is not circular: it measures exactly the claimed recovery, and CREATOR is not optimized against LocR2. There are no load-bearing self-citations; the cited prior works are external and independent. I also flag, as non-circular weaknesses: the p>=d+1 generalization of Theorem 1 in Appendix B asserts one may 'simply consider the first d dimension' x_[d] = H_[d],. y, which requires H_[d],. to be invertible, while Assumption 3 (full column rank) only guarantees that some d-row submatrix is invertible; this is a proof gap, not a circular reduction. The LLM case study does not verify Assumption 2, and the main text's pointer to 'Theorem 3 in Appendix D.1' appears to be a typo for Theorem 6. These issues affect correctness and rigor but do not make the derivation equivalent to its inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 7 axioms · 0 invented entities

The central theoretical result is a theorem proved from explicit assumptions and standard external results; there are no fitted constants in the identifiability claims and no new postulated entities. The two implementation choices listed as free parameters affect finite-sample behavior but not the population-level theory.

free parameters (2)
  • RBF kernel bandwidth for HSIC
    Subroutine 1 uses HSIC with RBF kernel to test independence; the kernel bandwidth is a tuning choice that affects finite-sample selection of alpha but is not analyzed or used in the identifiability theorems.
  • Rank threshold in pruning
    Subroutine 2 compares ranks of matrices; in finite samples a threshold for counting singular values is needed, but the paper does not specify how it is chosen. This affects practical performance but not the population-level theorem.
axioms (7)
  • domain assumption Exogenous noise z^(k) has independent components and at most one is Gaussian (Assumption 1, Section 2).
    Needed for Darmois-Skitovitch and ICA identifiability; central to root detection in subroutine 1.
  • domain assumption Node-level non-degeneracy: for each node i, dim span{U_i^(k) : k in [K]} = |pa(i)|+1 (Assumption 2, Section 2).
    Used in Theorem 1 proof to force permutation consistency, in Theorem 2 to conclude alpha corresponds to a root node, and in Theorem 4 for pruning.
  • domain assumption Mixing matrix H has full column rank with d <= p and latent dimension d known (Assumption 3, Section 2).
    Ensures observations span the latent space and ICA can recover the d sources.
  • domain assumption Linear SCM y = W^T y + Omega z with DAG G, and x = H y across environments (Model (1), Section 2).
    This is the problem setup; the paper only considers linear models and linear mixing.
  • standard math Darmois-Skitovitch theorem (Darmois 1953; Skitovitch 1953).
    Used in the proof of Theorem 2 to characterize when a linear form of independent non-Gaussian variables is independent of a residual.
  • standard math ICA identifiability and consistency of unmixing matrix estimation (Comon 1994; Hyvarinen and Oja 2000; Reyhani et al. 2012).
    Subroutine 1 enumerates candidate alpha from ICA rows; Theorem 3 and Lemma 3 use consistency of ICA.
  • standard math HSIC is a valid independence criterion with asymptotic consistency (Gretton et al. 2005).
    Used in optimization (2) and in the convergence proof of Lemma 3.

pith-pipeline@v1.3.0-alltime-deepseek · 24801 in / 15392 out tokens · 127130 ms · 2026-08-04T14:48:40.902434+00:00 · methodology

0 comments
read the original abstract

Causal representation learning (CRL) has garnered increasing interest from the causal inference and artificial intelligence communities due to its potential to disentangle complex data-generating mechanism into causally interpretable latent features by leveraging the heterogeneity of modern datasets. In this paper, we further contribute to the CRL literature, by focusing on the stylized linear structural causal model over latent features and assuming a linear mixing function that maps latent features to the observed data or measurements. Existing linear CRL methods often rely on stringent assumptions, such as access to single-node interventional data or restrictive distributional constraints on latent features and/or exogenous measurement noise. However, these prerequisites can be easy to violate in practice. In this work, we propose a novel linear CRL algorithm that, unlike existing methods, operates under weaker assumptions on environment heterogeneity and data-generating distributions while still recovering latent causal features up to an equivalence class. We further validate our new algorithm via synthetic experiments and an interpretability analysis of large language models, demonstrating both its superiority over competing methods in finite samples and its potential in integrating causality into understanding artificial intelligence. The source code is available at https://github.com/utulie/code_for_linear_crl_paper_creator.

Figures

Figures reproduced from arXiv: 2509.22553 by Hao Chen, Lin Liu, Yu Guang Wang.

Figure 1
Figure 1. Figure 1: An illustration of subroutine 1. Dashed nodes and edges are eliminated. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: LocR2 and SHD metric for different data generation setup. Figure 2a and 2c compare the performance of latent feature and causal DAG identification in setting (1). Figure 2b and 2d present performance in setting (2). 9 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: LocR2 and SHD metric for different data generation setup. Figures 2a and 2c compare the performance of latent feature and causal DAG identification in setting (1). Figures 2b and 2d compare the performance in setting (2). 21 [PITH_FULL_IMAGE:figures/full_fig_p021_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The impact of topological ordering inference on the performance of [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: LocR2 and SHD metric for noise variables with Gaussian and non-Gaussian with 𝐾 = 𝑑 and 𝐾 = 2𝑑. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_5.png] view at source ↗

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Reference graph

Works this paper leans on

52 extracted references · 11 linked inside Pith

  1. [1]

    Phi-3 technical report: A highly capable language model locally on your phone

    Marah Abdin, Jyoti Aneja, Hany Awadalla, Ahmed Awadallah, Ammar Ahmad Awan, Nguyen Bach, Amit Bahree, Arash Bakhtiari, Jianmin Bao, Harkirat Behl, et al. Phi-3 technical report: A highly capable language model locally on your phone. arXiv preprint arXiv:2404.14219, 2024

    Pith/arXiv arXiv 2024

  2. [2]

    GPT -4 technical report

    Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. GPT -4 technical report. arXiv preprint arXiv:2303.08774, 2023

    Pith/arXiv arXiv 2023

  3. [3]

    Interventional causal representation learning

    Kartik Ahuja, Divyat Mahajan, Yixin Wang, and Yoshua Bengio. Interventional causal representation learning. In International Conference on Machine Learning, pp.\ 372--407. PMLR, 2023

    2023

  4. [4]

    Physics of language models: Part 1, learning hierarchical language structures

    Zeyuan Allen-Zhu and Yuanzhi Li. Physics of language models: Part 1, learning hierarchical language structures. arXiv preprint arXiv:2305.13673, 2023

    Pith/arXiv arXiv 2023

  5. [5]

    A latent variable model approach to PMI -based word embeddings

    Sanjeev Arora, Yuanzhi Li, Yingyu Liang, Tengyu Ma, and Andrej Risteski. A latent variable model approach to PMI -based word embeddings. Transactions of the Association for Computational Linguistics, 4: 0 385--399, 2016

    2016

  6. [6]

    GPT-Neo: Large Scale Autoregressive Language Modeling with Mesh-Tensorflow , March 2021

    Sid Black, Leo Gao, Phil Wang, Connor Leahy, and Stella Biderman. GPT-Neo: Large Scale Autoregressive Language Modeling with Mesh-Tensorflow , March 2021. URL https://doi.org/10.5281/zenodo.5297715

    doi:10.5281/zenodo.5297715 2021

  7. [7]

    Sparks of artificial general intelligence: Early experiments with GPT -4

    S \'e bastien Bubeck, Varun Chandrasekaran, Ronen Eldan, Johannes Gehrke, Eric Horvitz, Ece Kamar, Peter Lee, Yin Tat Lee, Yuanzhi Li, Scott Lundberg, Harsha Nori, Hamid Palangi, Marco Tulio Ribeiro, and Yi Zhang. Sparks of artificial general intelligence: Early experiments with GPT -4. arXiv preprint arXiv:2303.12712, 2023

    Pith/arXiv arXiv 2023

  8. [8]

    Learning linear causal representations from interventions under general nonlinear mixing

    Simon Buchholz, Goutham Rajendran, Elan Rosenfeld, Bryon Aragam, Bernhard Sch \"o lkopf, and Pradeep Ravikumar. Learning linear causal representations from interventions under general nonlinear mixing. In Proceedings of the 37th International Conference on Neural Information Processing Systems, pp.\ 45419--45462, 2023

    2023

  9. [9]

    Invariance, causality and robustness

    Peter B \"u hlmann. Invariance, causality and robustness. Statistical Science, 35 0 (3): 0 404--426, 2020

    2020

  10. [10]

    Causal inference of general treatment effects using neural networks with a diverging number of confounders

    Xiaohong Chen, Ying Liu, Shujie Ma, and Zheng Zhang. Causal inference of general treatment effects using neural networks with a diverging number of confounders. Journal of Econometrics, 238 0 (1): 0 105555, 2024

    2024

  11. [11]

    Independent component analysis, a new concept? Signal Processing, 36 0 (3): 0 287--314, 1994

    Pierre Comon. Independent component analysis, a new concept? Signal Processing, 36 0 (3): 0 287--314, 1994

    1994

  12. [12]

    Analyse g \'e n \'e rale des liaisons stochastiques: etude particuli \`e re de l'analyse factorielle lin \'e aire

    George Darmois. Analyse g \'e n \'e rale des liaisons stochastiques: etude particuli \`e re de l'analyse factorielle lin \'e aire. Revue de l'Institut international de statistique, pp.\ 2--8, 1953

    1953

  13. [13]

    Identifying the consequences of dynamic treatment strategies: A decision theoretic overview

    A Philip Dawid and Vanessa Didelez. Identifying the consequences of dynamic treatment strategies: A decision theoretic overview. Statistics Surveys, 4: 0 184--231, 2010

    2010

  14. [14]

    On random graphs

    Paul Erd o s and Alfr \'e d R \'e nyi. On random graphs. I . Publications Mathematicae, 6: 0 290--297, 1959

    1959

  15. [15]

    Causal disentanglement for single-cell representations and controllable counterfactual generation

    Yicheng Gao, Kejing Dong, Caihua Shan, Dongsheng Li, and Qi Liu. Causal disentanglement for single-cell representations and controllable counterfactual generation. Nature Communications, 16 0 (1): 0 6775, 2025

    2025

  16. [16]

    Measuring statistical dependence with hilbert-schmidt norms

    Arthur Gretton, Olivier Bousquet, Alex Smola, and Bernhard Sch \"o lkopf. Measuring statistical dependence with hilbert-schmidt norms. In Sanjay Jain, Hans Ulrich Simon, and Etsuji Tomita (eds.), Algorithmic Learning Theory, pp.\ 63--77, Berlin, Heidelberg, 2005. Springer Berlin Heidelberg. ISBN 978-3-540-31696-1

    2005

  17. [17]

    Independent component analysis: Algorithms and applications

    Aapo Hyv \"a rinen and Erkki Oja. Independent component analysis: Algorithms and applications. Neural Networks, 13 0 (4-5): 0 411--430, 2000

    2000

  18. [18]

    Learning causal representations from general environments: Identifiability and intrinsic ambiguity

    Jikai Jin and Vasilis Syrgkanis. Learning causal representations from general environments: Identifiability and intrinsic ambiguity. In Proceedings of the 38th International Conference on Neural Information Processing Systems, 2024

    2024

  19. [19]

    What is causal about causal models and representations? arXiv preprint arXiv:2501.19335, 2025

    Frederik Hytting J rgensen, Luigi Gresele, and Sebastian Weichwald. What is causal about causal models and representations? arXiv preprint arXiv:2501.19335, 2025

    Pith/arXiv arXiv 2025

  20. [20]

    Causal reasoning and large language models: Opening a new frontier for causality

    Emre K c man, Robert Ness, Amit Sharma, and Chenhao Tan. Causal reasoning and large language models: Opening a new frontier for causality. arXiv preprint arXiv:2305.00050, 2023

    Pith/arXiv arXiv 2023

  21. [21]

    Optimality of the J ohnson- L indenstrauss lemma

    Kasper Green Larsen and Jelani Nelson. Optimality of the J ohnson- L indenstrauss lemma. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pp.\ 633--638. IEEE, 2017

    2017

  22. [22]

    Factor analysis as a statistical method

    David N Lawley and Adam E Maxwell. Factor analysis as a statistical method. Journal of the Royal Statistical Society Series D: The Statistician, 12 0 (3): 0 209--229, 1962

    1962

  23. [23]

    Causal estimation of memorisation profiles

    Pietro Lesci, Clara Meister, Thomas Hofmann, Andreas Vlachos, and Tiago Pimentel. Causal estimation of memorisation profiles. In Proceedings of the 62nd Annual Meeting of the Association for Computational Linguistics (ACL), volume 1, pp.\ 15616--15635, 2024

    2024

  24. [24]

    Deep S eek-v3 technical report

    Aixin Liu, Bei Feng, Bing Xue, Bingxuan Wang, Bochao Wu, Chengda Lu, Chenggang Zhao, Chengqi Deng, Chenyu Zhang, Chong Ruan, et al. Deep S eek-v3 technical report. arXiv preprint arXiv:2412.19437, 2024

    Pith/arXiv arXiv 2024

  25. [25]

    The Llama 3 family of models

    AI @ Meta Llama Team. The Llama 3 family of models. https://github.com/meta-llama/PurpleLlama/blob/main/Llama-Guard3/1B/MODEL_CARD.md, 2024

    2024

  26. [26]

    Fourth moments and independent component analysis

    Jari Miettinen, Sara Taskinen, Klaus Nordhausen, and Hannu Oja. Fourth moments and independent component analysis. Statistical Science, 30 0 (3): 0 372--390, 2015

    2015

  27. [27]

    Linguistic regularities in continuous space word representations

    Tomas Mikolov, Wen-tau Yih, and Geoffrey Zweig. Linguistic regularities in continuous space word representations. In Proceedings of the 2013 Conference of the North A merican Chapter of the Association for Computational Linguistics: Human Language Technologies , pp.\ 746--751, 2013

    2013

  28. [28]

    Determining the number of factors from empirical distribution of eigenvalues

    Alexei Onatski. Determining the number of factors from empirical distribution of eigenvalues. The Review of Economics and Statistics, 92 0 (4): 0 1004--1016, 2010

    2010

  29. [29]

    The linear representation hypothesis and the geometry of large language models

    Kiho Park, Yo Joong Choe, and Victor Veitch. The linear representation hypothesis and the geometry of large language models. In International Conference on Machine Learning, pp.\ 39643--39666. PMLR, 2024

    2024

  30. [30]

    Automatic differentiation in PyTorch

    Adam Paszke, Sam Gross, Soumith Chintala, Gregory Chanan, Edward Yang, Zachary DeVito, Zeming Lin, Alban Desmaison, Luca Antiga, and Adam Lerer. Automatic differentiation in PyTorch . In NIPS 2017 Workshop Autodiff, 2017

    2017

  31. [31]

    Elements of causal inference: foundations and learning algorithms

    Jonas Peters, Dominik Janzing, and Bernhard Sch \"o lkopf. Elements of causal inference: foundations and learning algorithms. The MIT Press, 2017

    2017

  32. [32]

    From causal to concept-based representation learning

    Goutham Rajendran, Simon Buchholz, Bryon Aragam, Bernhard Sch \"o lkopf, and Pradeep Ravikumar. From causal to concept-based representation learning. In Proceedings of the 38th International Conference on Neural Information Processing Systems, pp.\ 101250--101296, 2024

    2024

  33. [33]

    Consistency and asymptotic normality of FastICA and bootstrap FastICA

    Nima Reyhani, Jarkko Ylipaavalniemi, Ricardo Vig \'a rio, and Erkki Oja. Consistency and asymptotic normality of FastICA and bootstrap FastICA . Signal Processing, 92 0 (8): 0 1767--1778, 2012

    2012

  34. [34]

    a us Kleindessner, Chris Russell, Dominik Janzing, Bernhard Sch \

    Paul Rolland, Volkan Cevher, Matth \"a us Kleindessner, Chris Russell, Dominik Janzing, Bernhard Sch \"o lkopf, and Francesco Locatello. Score matching enables causal discovery of nonlinear additive noise models. In International Conference on Machine Learning, pp.\ 18741--18753. PMLR, 2022

    2022

  35. [35]

    BLOOM : A 176b-parameter open-access multilingual language model

    Teven Le Scao, Angela Fan, Christopher Akiki, Ellie Pavlick, Suzana Ili \'c , Daniel Hesslow, Roman Castagn \'e , Alexandra Sasha Luccioni, Fran c ois Yvon, et al. BLOOM : A 176b-parameter open-access multilingual language model. arXiv preprint arXiv:2211.05100, 2022

    Pith/arXiv arXiv 2022

  36. [36]

    Toward causal representation learning

    Bernhard Sch \"o lkopf, Francesco Locatello, Stefan Bauer, Nan Rosemary Ke, Nal Kalchbrenner, Anirudh Goyal, and Yoshua Bengio. Toward causal representation learning. Proceedings of the IEEE, 109 0 (5): 0 612--634, 2021

    2021

  37. [37]

    On a property of the normal distribution

    Viktor P Skitovitch. On a property of the normal distribution. Doklady Akademii Nauk SSSR, 89: 0 217--219, 1953

    1953

  38. [38]

    Causal temporal representation learning with nonstationary sparse transition

    Xiangchen Song, Zijian Li, Guangyi Chen, Yujia Zheng, Yewen Fan, Xinshuai Dong, and Kun Zhang. Causal temporal representation learning with nonstationary sparse transition. In Proceedings of the 38th International Conference on Neural Information Processing Systems, pp.\ 77098--77131, 2024

    2024

  39. [39]

    Linear causal disentanglement via interventions

    Chandler Squires, Anna Seigal, Salil S Bhate, and Caroline Uhler. Linear causal disentanglement via interventions. In International Conference on Machine Learning, pp.\ 32540--32560. PMLR, 2023

    2023

  40. [40]

    Causal representation learning from multimodal biomedical observations

    Yuewen Sun, Lingjing Kong, Guangyi Chen, Loka Li, Gongxu Luo, Zijian Li, Yixuan Zhang, Yujia Zheng, Mengyue Yang, Petar Stojanov, Eran Segal, Eric P Xing, and Kun Zhang. Causal representation learning from multimodal biomedical observations. In The Thirteenth International Conference on Learning Representations, 2025

    2025

  41. [41]

    Score-based causal representation learning from interventions: Nonparametric identifiability

    Burak Var c , Emre Acart \"u rk, Karthikeyan Shanmugam, and Ali Tajer. Score-based causal representation learning from interventions: Nonparametric identifiability. In Causal Representation Learning Workshop at NeurIPS 2023, 2023

    2023

  42. [42]

    General identifiability and achievability for causal representation learning

    Burak Var c , Emre Acart \"u rk, Karthikeyan Shanmugam, and Ali Tajer. General identifiability and achievability for causal representation learning. In International Conference on Artificial Intelligence and Statistics, pp.\ 2314--2322. PMLR, 2024 a

    2024

  43. [43]

    Linear causal representation learning from unknown multi-node interventions

    Burak Var c , Emre Acart \"u rk, Karthikeyan Shanmugam, and Ali Tajer. Linear causal representation learning from unknown multi-node interventions. arXiv preprint arXiv:2406.05937, 2024 b

    Pith/arXiv arXiv 2024

  44. [44]

    Score-based causal representation learning: Linear and general transformations

    Burak Var c , Emre Acart \"u rk, Karthikeyan Shanmugam, and Ali Tajer. Score-based causal representation learning: Linear and general transformations. arXiv preprint arXiv:2402.00849, 2024 c

    Pith/arXiv arXiv 2024

  45. [45]

    Dimension reduction via adaptive slicing

    Tao Wang. Dimension reduction via adaptive slicing. Statistica Sinica, 32 0 (1): 0 499--516, 2022

    2022

  46. [46]

    Identifiability guarantees for causal disentanglement from purely observational data

    Ryan Welch, Jiaqi Zhang, and Caroline Uhler. Identifiability guarantees for causal disentanglement from purely observational data. In Proceedings of the 38th International Conference on Neural Information Processing Systems, pp.\ 102796--102821, 2024

    2024

  47. [47]

    Sketching as a tool for numerical linear algebra

    David P Woodruff. Sketching as a tool for numerical linear algebra. Foundations and Trends in Theoretical Computer Science , 10 0 (1--2): 0 1--157, 2014

    2014

  48. [48]

    Stability

    Bin Yu. Stability. Bernoulli, 19 0 (4): 0 1484--1500, 2013

    2013

  49. [49]

    Identifiability guarantees for causal disentanglement from soft interventions

    Jiaqi Zhang, Kristjan Greenewald, Chandler Squires, Akash Srivastava, Karthikeyan Shanmugam, and Caroline Uhler. Identifiability guarantees for causal disentanglement from soft interventions. In Proceedings of the 37th International Conference on Neural Information Processing Systems, pp.\ 50254--50292, 2023

    2023

  50. [50]

    Causal representation learning from multiple distributions: A general setting

    Kun Zhang, Shaoan Xie, Ignavier Ng, and Yujia Zheng. Causal representation learning from multiple distributions: A general setting. In International Conference on Machine Learning, pp.\ 60057--60075. PMLR, 2024 a

    2024

  51. [51]

    TinyLlama : An open-source small language model

    Peiyuan Zhang, Guangtao Zeng, Tianduo Wang, and Wei Lu. TinyLlama : An open-source small language model. arXiv preprint arXiv:2401.02385, 2024 b

    Pith/arXiv arXiv 2024

  52. [52]

    write newline

    " write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION format.date year duplicate empty "emp...