pith. sign in

arxiv: 2606.18074 · v2 · pith:37SVIT7Pnew · submitted 2026-06-16 · 📊 stat.ML · cs.LG· stat.ME

Tensor-based second-order causal discovery

Nathan Ouyang , Kexin Wang , Anna Seigal This is my paper

Pith reviewed 2026-06-26 22:28 UTC · model grok-4.3

classification 📊 stat.ML cs.LGstat.ME
keywords causal discoverytensor methodsinterventional dataDAG recoverysecond-order statisticslinear SEMlogarithmic interventionscovariance tensor
0
0 comments X

The pith

Tensor method recovers causal order and parameters from logarithmic interventions.

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

The paper presents TSCD, which builds a tensor from the covariance matrices of observational and interventional data. Under a linear structural equation model on a DAG with uncorrelated noise, the algorithm recovers both the graph structure and the functions on the edges. The key result is that the causal order and parameters remain identifiable even when the number of interventions grows only logarithmically with the number of variables. The approach also extends to nonlinear models, relies solely on second-order statistics, and experiments indicate robustness to noise along with scaling to hundreds of variables.

Core claim

TSCD takes as input a tensor formed from covariance matrices of observational and interventional data and outputs the DAG together with the functions on its edges; under the linear SEM assumption with uncorrelated noise, both the causal order and the parameters are identifiable from a number of interventions that is logarithmic in the number of variables.

What carries the argument

The tensor assembled from covariance matrices of observational and interventional data, which encodes second-order statistics to enable recovery of the DAG and edge parameters.

If this is right

  • The method requires only logarithmically many interventions for identifiability of order and parameters.
  • Second-order statistics suffice for recovery without assuming Gaussianity.
  • The algorithm extends to nonlinear models while preserving the use of covariance tensors.
  • Scalability to hundreds of variables follows from the computational efficiency of covariance matrices.

Where Pith is reading between the lines

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

  • Real-world data with approximately uncorrelated noise could allow causal discovery at substantially lower experimental cost than methods needing more interventions.
  • Hybrid algorithms might combine the tensor construction with other second-order techniques to handle mixed linear and nonlinear relations.
  • Testing the nonlinear variant on data with known ground-truth graphs would clarify how far the logarithmic intervention bound carries over beyond the linear case.

Load-bearing premise

Causal dependencies follow a linear structural equation model on a DAG and the noise variables are uncorrelated.

What would settle it

A dataset generated from a linear SEM on a DAG with uncorrelated noise where TSCD, given only a logarithmic number of interventions, returns an incorrect causal order.

Figures

Figures reproduced from arXiv: 2606.18074 by Anna Seigal, Kexin Wang, Nathan Ouyang.

Figure 1
Figure 1. Figure 1: Constructing the precision tensor for 10 variables and five contexts (one observational). Top row: Precision matrices for each context. Middle row: The matrices after rank reduction. Bottom row: The matrices are stacked to form the precision tensor. The boxed entries are the only ones that change in the rank reduction step; they correspond to the zeros of the intervention￾pattern matrix B ∈ {0, 1} 10×5 wit… view at source ↗
Figure 2
Figure 2. Figure 2: Intervention-asymmetric correlation test. If intervening on i pre￾serves correlation with j, then the dependence must flow from i to j, so i is an ancestor of j. If the correlation disappears, the observational dependence may instead come from j → i or from a latent common cause. 6.4. Root selection algorithm. We use pairwise intervention-asymmetric tests to se￾lect a root from a candidate set C. For each … view at source ↗
Figure 3
Figure 3. Figure 3: Performance comparison across different noise settings, with Gaussian ratios 0, 0.5, and 1. Ordering accuracy is measured by the number of parent-child errors, edges where the parent appears after the child in the recovered order. Our method recovers the correct causal order, while the baseline produces 13 parent-child errors. A representative example is in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scalability of TSCD. Left: F1 score versus number of nodes. Right: runtime versus number of nodes. Each point is averaged across 10 random sparse DAGs, each with sample size p 2 . The error bars are given by the standard deviation across trials [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Ground-truth DAG (left) and recovered graph (right) for a non￾linear SEM. The recovered structure matches the true causal order, with an edge error rate of 0.34. Here R, G, B denote the brightness of the red, green, and blue LEDs respectively on the main light source. The variables ˜I1 and V˜ 1 are the uncalibrated infrared and visible-light intensity measurements from the first light sensor, which is plac… view at source ↗
Figure 6
Figure 6. Figure 6: The distribution of best ranks for root nodes in projection score, as p (the number of nodes) varies, for two values of e (the edge probability), cf. Section B.1. All experiments were conducted with 1000 samples, half of the nodes given Gaussian noise and the rest given Student-t5 distribution noise. 102 103 104 10−1 Sample size per context Runtime (sec) Runtime (sec) 102 103 104 0 0.2 0.4 0.6 0.8 Sample s… view at source ↗
Figure 7
Figure 7. Figure 7: TSCD performance comparison with ablated versions across dif￾ferent noise settings, with Gaussian ratios 0, 0.5, and 1. Gaussian ratio de￾notes the ratio of nodes given Gaussian distributed noise, with the rest given Student-t5 distributions [PITH_FULL_IMAGE:figures/full_fig_p027_7.png] view at source ↗
read the original abstract

Causal discovery seeks to uncover the causal dependencies among variables. For this purpose, we propose an algorithm called Tensor-based Second-order Causal Discovery (TSCD). Its input is a tensor obtained from the covariance matrices of observational and interventional data. Assuming the causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG), TSCD outputs the DAG and the functions on its edges, requiring only that the noise variables are uncorrelated. We also implement a version of the approach for nonlinear models. Our focus on second-order statistics (via the covariance matrices) is motivated by their statistical and computational efficiency relative to higher-order moments, their identifiability relative to first-order statistics, and that they work regardless of whether the variables are Gaussian. We show that TSCD has identifiable causal order and parameters from a number of interventions that is logarithmic in the number of variables. Experiments show that TSCD is robust to noise, competitive with existing methods, and scales to hundreds of variables.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proposes Tensor-based Second-order Causal Discovery (TSCD), which constructs a tensor from covariance matrices of observational and interventional data to recover both the causal order and the parameters (edge functions) of a linear SEM on a DAG, assuming only that noise terms are uncorrelated. It claims that the causal order and parameters are identifiable from a number of interventions logarithmic in the number of variables, provides a nonlinear implementation, and reports experiments indicating robustness to noise, competitiveness with existing methods, and scalability to hundreds of variables.

Significance. If the identifiability result is established, the work would be a meaningful contribution by showing that second-order statistics suffice for identifiability under standard linear SEM assumptions while requiring only logarithmically many interventions. This efficiency, combined with the reported scalability, could be useful for high-dimensional causal discovery where interventions are costly.

minor comments (2)
  1. The abstract asserts identifiability but supplies no theorem number, proof outline, or key equation; adding a pointer to the relevant section or theorem would improve readability without altering the technical content.
  2. In the experimental section, explicitly state how many interventions were used relative to the claimed logarithmic bound and whether the observed performance matches the theoretical scaling.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper presents TSCD as recovering the DAG and edge functions from a tensor of covariance matrices under linear SEM assumptions on a DAG with uncorrelated noise. The identifiability result for causal order and parameters (logarithmic interventions) is stated as following from the tensor construction and second-order statistics properties, without any quoted reduction of the target claim to a fitted parameter, self-definition, or load-bearing self-citation chain. The derivation is self-contained against the explicit assumptions; no steps match the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two standard domain assumptions in causal discovery; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption The causal dependencies follow a linear structural equation model on a directed acyclic graph (DAG)
    Stated explicitly as the modeling assumption for the main algorithm.
  • domain assumption The noise variables are uncorrelated
    Required for the method to output the DAG and edge functions.

pith-pipeline@v0.9.1-grok · 5697 in / 1138 out tokens · 40061 ms · 2026-06-26T22:28:57.126375+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

56 extracted references · 1 canonical work pages

  1. [1]

    Stat , volume=

    Learning directed acyclic graph models based on sparsest permutations , author=. Stat , volume=. 2018 , publisher=

    2018

  2. [2]

    SIAM review , volume=

    A rank--one reduction formula and its applications to matrix factorizations , author=. SIAM review , volume=. 1995 , publisher=

    1995

  3. [3]

    Biometrika , volume=

    Identifiability of Gaussian structural equation models with equal error variances , author=. Biometrika , volume=. 2014 , publisher=

    2014

  4. [4]

    Advances in neural information processing systems , volume=

    Nonlinear causal discovery with additive noise models , author=. Advances in neural information processing systems , volume=

  5. [5]

    Uncertainty in artificial intelligence , pages=

    Causal discovery with general non-linear relationships using non-linear ICA , author=. Uncertainty in artificial intelligence , pages=. 2020 , organization=

    2020

  6. [6]

    International conference on artificial intelligence and statistics , pages=

    Learning sparse nonparametric dags , author=. International conference on artificial intelligence and statistics , pages=. 2020 , organization=

    2020

  7. [7]

    Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence , pages=

    On the number of experiments sufficient and in the worst case necessary to identify all causal relations among N variables , author=. Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence , pages=

  8. [8]

    Journal of Symbolic Computation , volume=

    Lower bounds on the rank and symmetric rank of real tensors , author=. Journal of Symbolic Computation , volume=. 2023 , publisher=

    2023

  9. [9]

    SIAM journal on Matrix Analysis and Applications , volume=

    A link between the canonical decomposition in multilinear algebra and simultaneous matrix diagonalization , author=. SIAM journal on Matrix Analysis and Applications , volume=. 2006 , publisher=

    2006

  10. [10]

    Meyer , title =

    Carl D. Meyer , title =. The American Mathematical Monthly , volume =. 2018 , publisher =. doi:10.1080/00029890.2017.1389199 , URL =

    work page doi:10.1080/00029890.2017.1389199 2018

  11. [11]

    2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) , pages=

    Computing linear sections of varieties: quantum entanglement, tensor decompositions and beyond , author=. 2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS) , pages=. 2023 , organization=

    2023

  12. [12]

    arXiv preprint arXiv:2510.18627 , year=

    Multi-subspace power method for decomposing all tensors , author=. arXiv preprint arXiv:2510.18627 , year=

    Pith/arXiv arXiv

  13. [13]

    Perturb-Seq: dissecting molecular circuits with scalable single-cell

    Dixit, Atray and Parnas, Oren and Li, Biyu and Chen, Jenny and Fulco, Charles P and Jerby-Arnon, Livnat and Marjanovic, Nemanja D and Dionne, Danielle and Burks, Tyler and Raychowdhury, Raktima and others , journal=. Perturb-Seq: dissecting molecular circuits with scalable single-cell. 2016 , publisher=

    2016

  14. [14]

    2000 , publisher=

    Causation, prediction, and search , author=. 2000 , publisher=

    2000

  15. [15]

    and Hyvärinen, Aapo and Kerminen, Antti , journal=

    Shimizu, Shohei and Hoyer, Patrik O. and Hyvärinen, Aapo and Kerminen, Antti , journal=. A Linear Non-

  16. [16]

    2017 , publisher=

    Elements of causal inference: foundations and learning algorithms , author=. 2017 , publisher=

    2017

  17. [17]

    2011 , publisher=

    Tensors: geometry and applications: geometry and applications , author=. 2011 , publisher=

    2011

  18. [18]

    The annals of mathematical statistics , volume=

    The method of path coefficients , author=. The annals of mathematical statistics , volume=. 1934 , publisher=

    1934

  19. [19]

    SIAM review , volume=

    Tensor decompositions and applications , author=. SIAM review , volume=. 2009 , publisher=

    2009

  20. [20]

    IEEE journal of selected topics in signal processing , volume=

    Structured data fusion , author=. IEEE journal of selected topics in signal processing , volume=. 2015 , publisher=

    2015

  21. [21]

    SIAM Journal on Matrix Analysis and Applications , volume=

    On generic identifiability of 3-tensors of small rank , author=. SIAM Journal on Matrix Analysis and Applications , volume=. 2012 , publisher=

    2012

  22. [22]

    The Annals of Statistics , volume=

    Global identifiability of linear structural equation models , author=. The Annals of Statistics , volume=

  23. [23]

    International conference on machine learning , pages=

    Linear causal disentanglement via interventions , author=. International conference on machine learning , pages=. 2023 , organization=

    2023

  24. [24]

    and Bollen, Kenneth , journal=

    Shimizu, Shohei and Inazumi, Takanori and Sogawa, Yasuhiro and Hyvärinen, Aapo and Kawahara, Yoshinobu and Washio, Takashi and Hoyer, Patrik O. and Bollen, Kenneth , journal=. DirectLiNGAM: A Direct Method for Learning a Linear Non-

  25. [25]

    Journal of Machine Learning Research , volume=

    Optimal Structure Identification with Greedy Search , author=. Journal of Machine Learning Research , volume=

  26. [26]

    arXiv e-prints , pages=

    Identifiable multi-view causal discovery without non-gaussianity , author=. arXiv e-prints , pages=

  27. [27]

    Characterization and Greedy Learning of Interventional

    Hauser, Alain and Bühlmann, Peter , journal=. Characterization and Greedy Learning of Interventional

  28. [28]

    The Annals of Statistics , pages=

    Half-trek criterion for generic identifiability of linear structural equation models , author=. The Annals of Statistics , pages=. 2012 , publisher=

    2012

  29. [29]

    Conference on Uncertainty in Artificial Intelligence , pages=

    Permutation-based causal structure learning with unknown intervention targets , author=. Conference on Uncertainty in Artificial Intelligence , pages=. 2020 , organization=

    2020

  30. [30]

    , booktitle=

    Zheng, Xun and Aragam, Bryon and Ravikumar, Pradeep and Xing, Eric P. , booktitle=

  31. [31]

    Beware of the simulated

    Reisach, Alexander and Seiler, Christof and Weichwald, Sebastian , journal=. Beware of the simulated

  32. [32]

    Journal of the Royal Statistical Society: Series B , volume=

    Causal Inference using Invariant Prediction: Identification and Confidence Intervals , author=. Journal of the Royal Statistical Society: Series B , volume=

  33. [33]

    Linear algebra and its applications , volume=

    Three-way arrays: rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics , author=. Linear algebra and its applications , volume=. 1977 , publisher=

    1977

  34. [34]

    Biometrika , pages=

    The probable error of a mean , author=. Biometrika , pages=. 1908 , publisher=

    1908

  35. [35]

    Biometrika , volume=

    Frequency distribution of the values of the correlation coefficient in samples from an indefinitely large population , author=. Biometrika , volume=. 1915 , publisher=

    1915

  36. [36]

    Friedman, Nir and Linial, Michal and Nachman, Iftach and Pe'er, Dana , booktitle=. Using

  37. [37]

    Science , volume=

    Causal protein-signaling networks derived from multiparameter single-cell data , author=. Science , volume=. 2005 , publisher=

    2005

  38. [38]

    Nature reviews neuroscience , volume=

    Causal mapping of human brain function , author=. Nature reviews neuroscience , volume=. 2022 , publisher=

    2022

  39. [39]

    Journal of Economic Literature , volume=

    Potential outcome and directed acyclic graph approaches to causality: Relevance for empirical practice in economics , author=. Journal of Economic Literature , volume=. 2020 , publisher=

    2020

  40. [40]

    Epidemiology , volume=

    Causal diagrams for epidemiologic research , author=. Epidemiology , volume=. 1999 , publisher=

    1999

  41. [41]

    Proceedings of the IEEE , volume=

    Toward causal representation learning , author=. Proceedings of the IEEE , volume=. 2021 , publisher=

    2021

  42. [42]

    arXiv preprint arXiv:2601.15239 , year=

    Multi-context principal component analysis , author=. arXiv preprint arXiv:2601.15239 , year=

    arXiv

  43. [43]

    Proceedings of the National Academy of Sciences , volume=

    Methods for causal inference from gene perturbation experiments and validation , author=. Proceedings of the National Academy of Sciences , volume=. 2016 , publisher=

    2016

  44. [44]

    Causality: Objectives and assessment , pages=

    Structure learning in causal cyclic networks , author=. Causality: Objectives and assessment , pages=. 2010 , organization=

    2010

  45. [45]

    Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence , pages=

    Equivalence and synthesis of causal models , author=. Proceedings of the Sixth Annual Conference on Uncertainty in Artificial Intelligence , pages=

  46. [46]

    Advances in neural information processing systems , volume=

    Permutation-based causal inference algorithms with interventions , author=. Advances in neural information processing systems , volume=

  47. [47]

    Biometrika , volume=

    Ancestor regression in linear structural equation models , author=. Biometrika , volume=. 2023 , publisher=

    2023

  48. [48]

    Advances in neural information processing systems , volume=

    BACKSHIFT: Learning causal cyclic graphs from unknown shift interventions , author=. Advances in neural information processing systems , volume=

  49. [49]

    2009 , publisher=

    Causality , author=. 2009 , publisher=

    2009

  50. [50]

    Probabilistic soft interventions in conditional

    Markowetz, Florian and Grossmann, Steffen and Spang, Rainer , booktitle=. Probabilistic soft interventions in conditional. 2005 , organization=

    2005

  51. [51]

    Causal chambers as a real-world physical testbed for

    Gamella, Juan L and Peters, Jonas and B. Causal chambers as a real-world physical testbed for. Nature Machine Intelligence , volume=. 2025 , publisher=

    2025

  52. [52]

    Structural Equation Modeling , volume=

    A new identification condition for recursive models with correlated errors , author=. Structural Equation Modeling , volume=. 2002 , publisher=

    2002

  53. [53]

    2023 , publisher=

    Algebraic statistics , author=. 2023 , publisher=

    2023

  54. [54]

    1934 , publisher=

    Lectures on matrices , author=. 1934 , publisher=

    1934

  55. [55]

    Numerical Algorithms , pages=

    Subspace power method for symmetric tensor decomposition , author=. Numerical Algorithms , pages=. 2025 , publisher=

    2025

  56. [56]

    Journal of Algebra , year=

    A real generalized trisecant trichotomy , author=. Journal of Algebra , year=