Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity

Haocheng Zhang; Hao Zhang; Jihong Guan; Shuigeng Zhou; Yewei Xia; Yixin Ren

REVIEW 4 major objections 4 minor 63 references

Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper proposes an approximate kernel-based generalized score function with $\mathcal{O}(n)$ time and space complexity, enabling score-based causal discovery to scale linearly in sample size while preserving accuracy.

desk verdict Genuine speedup with a real but clearly-stated gap: the asymptotic consistency claim for the approximate score is unsupported. read the letter →

arxiv 2412.17717 v2 pith:IIL2UD5K submitted 2024-12-23 cs.LG stat.ML

Yixin Ren , Haocheng Zhang , Yewei Xia , Hao Zhang , Jihong Guan , Shuigeng Zhou This is my paper · ORCID

classification cs.LGstat.ML

keywords causaldiscoveryscore-basedlearningkernelmethodslow-rankapproximationincompleteCholeskydecompositiongreedyequivalencesearchconditionalindependencelarge-scale

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

Kernel-based score functions for causal discovery are attractive because they avoid assumptions about data distributions, but evaluating them costs $\mathcal{O}(n^3)$ time and $\mathcal{O}(n^2)$ memory. This paper shows that replacing the kernel matrix with a low-rank approximation, and regrouping every matrix product, inverse, and determinant inside the score into a so-called dumbbell form, cuts the cost to $\mathcal{O}(n)$ time and space for a fixed pivot budget. The resulting approximate score, CV-LR, gives accuracy comparable to the exact cross-validated likelihood score in experiments, while running orders of magnitude faster. The paper argues that when the approximation is good enough, the approximate score inherits local consistency, so the greedy equivalence search still converges to the correct Markov equivalence class asymptotically.

What carries the argument

The central object is the dumbbell-form matrix chain: a product of a wide outer factor of size $n \times m$, small inner factors of size $m \times m$, and a wide outer factor of size $m \times n$, possibly with a scalar-preconditioned identity. This form is closed under multiplication and, via the Woodbury matrix identity, under regularized inversion; traces are evaluated by the cyclic property and determinants by the Weinstein–Aronszajn identity, all at $\mathcal{O}(n m^2)$ or $\mathcal{O}(m^3)$ cost. It is the algebraic device that turns the composite kernel-matrix operations of the cross-validated likelihood score into a linear-in-$n$ computation.

What would settle it

Construct a synthetic dataset whose kernel matrix has effective rank growing past the pivot budget—for example, a nonlinear structural equation model with many independent influential covariates so that 100 pivots cannot keep the score within 0.5% relative error—then run greedy equivalence search with CV-LR and with exact CV on the same data. If the two searches return different Markov equivalence classes and CV-LR's class has a strictly worse normalized structural Hamming distance, the claim that the approximate score preserves the local score ordering would be falsified.

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Extended reading notes

Core claim

The central claim is that the cross-validated likelihood score of [15] can be computed with the kernel matrix replaced by a rank-$m$ factor $\Lambda\Lambda^\top$, using incomplete Cholesky decomposition for continuous data and an exact low-rank decomposition for discrete data with few states. All matrix products, regularized inverses, and log-determinants appearing in the score are reorganized into dumbbell-form matrix chains, which are closed under multiplication and under Woodbury-style inversion, yielding $\mathcal{O}(n m^2)$ time and $\mathcal{O}(n m)$ storage. With $m$ fixed (at most 100 in the experiments), this is linear in sample size. The paper further claims that, provided the low-rank score is a good enough approximation of the exact score, it satisfies the local consistency property needed for greedy equivalence search to return the true Markov equivalence class asymptotically, and it reports F1/SHD results on synthetic and real-world data matching or nearly matching the exact score.

Load-bearing premise

The load-bearing premise is that approximating the kernel with a small, fixed number of representative samples (at most 100) preserves the relative ordering of scores for neighboring candidate causal graphs, so the greedy search still converges to the correct answer; the paper asserts this 'good enough approximation' without a formal error bound.

Editorial extensions

If this is right

Score-based causal discovery with kernel scores becomes feasible for datasets with tens of thousands of samples, where the original $\mathcal{O}(n^3)$ computation would be impractical.
Datasets with simple structure—discrete variables with few distinct values, or empty conditioning sets—receive the largest speedups, since the adaptive low-rank algorithm terminates early.
Because the approximate score is designed to preserve local consistency, greedy equivalence search retains its asymptotic correctness guarantee whenever the approximation is good enough.
The method extends to mixed continuous/discrete and multi-dimensional variables, where the exact kernel score also applies but was previously too costly.
Memory use drops from $\mathcal{O}(n^2)$ to $\mathcal{O}(n m)$, making kernel-based scores usable in storage-limited settings.

Reading between the lines

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

The speedup is only linear for a fixed pivot budget; if the kernel's effective rank grows with sample size or conditioning-set size, the method would need more pivots and the $\mathcal{O}(n)$ claim would degrade accordingly.
The same dumbbell-form algebra could be applied to other kernel-based scores, such as the marginal-likelihood score in [15, 43], potentially yielding a similar speedup.
A formal error bound connecting approximation rank to score differences would turn the 'good enough approximation' heuristic into a verifiable condition, and could guide adaptive pivot selection per variable.
The 0.5% relative-error threshold used to choose $m=100$ is a calibration on score values; a more direct test would measure the ordering of local scores on neighboring DAGs as the paper's own consistency argument requires.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: the paper earns its place on the algorithmic side. Ren et al. show that the cross-validated kernel score from Huang et al. can be computed in O(n m^2) time and O(n m) memory by never forming the n x n kernel matrices; the dumbbell-form algebra is the actual contribution, and it is worked out in enough detail that I believe the complexity claim. The exact low-rank decomposition for discrete variables is a real plus, and the experiments deliver what they promise: speedups of 100x-10,000x with accuracy close to the exact CV score, on synthetic and real data. That is worth having.

The soft spot is exactly where the reader's report puts it: Section 6. The paper asserts that "with a good enough approximation" the approximate score inherits local consistency, but it never defines "good enough" and never bounds the effect of the low-rank error on score differences between neighboring DAGs. That matters because GES compares those differences, and the 0.5% relative error reported in Appendix B.1 on the score value does not imply sign preservation. Moreover m=100 is fixed while n grows, with no spectral decay assumption to keep the error from growing. So the asymptotic recovery guarantee is unsupported. It is an honest statement—they don't hide it—but it is still a load-bearing gap. It is fixable in one of two ways: prove the error bound, or reframe Section 6 as an empirical finding and drop the consistency claim.

Minor issues: no error bars in the accuracy plots despite averaging over 20 runs; the abstract's O(n) is loose since the body says O(n m^2), which is only O(n) for fixed m; and the real-data experiments cap at n=2000 and 20-node graphs, so the extrapolation to "large" datasets is fair but not demonstrated.

I'd send this to review. The algorithmic result is solid, the experiments are useful, and the theoretical gap is clear enough that a referee could ask for a specific fix. A desk reject would be a mistake. My own use: I would cite it if I were working on scalable kernel-based causal discovery, and I'd bring it up in a reading group to discuss how much of a consistency guarantee you actually need for a practical speedup.

Referee Report

4 major / 4 minor

Summary. The paper proposes a low-rank approximation of the cross-validated kernel-based generalized score function of Huang et al. (2018), with the stated goal of reducing the cost of evaluating each local score from O(n^3) time and O(n^2) storage to O(n m^2) time and O(n m) storage, where m is the number of Nyström/ICL pivots. Section 5 introduces a 'dumbbell form' for matrix chains and uses Woodbury and Weinstein–Aronszajn identities to rewrite the score's inverse, determinant, and trace terms so that all computations are on m-by-m matrices plus rank-m factors. Section 6 argues that the approximate score inherits local consistency from the exact CV score when the approximation is 'good enough,' and therefore GES recovers the correct Markov equivalence class asymptotically. Experiments on synthetic data (continuous, mixed, multi-dimensional, up to n=1000 and 7 variables) and on the SACHS and CHILD networks (up to n=2000, 11–20 nodes) report large speedups over the exact CV score with comparable F1/SHD accuracy.

Significance. If the central claims hold, the paper makes an important practical contribution: kernel-based generalized score causal discovery, previously limited to a few thousand samples by O(n^3) cost, becomes feasible for larger n while retaining the flexibility of nonparametric scores. The algebraic transformation in Section 5 is a genuine technical contribution, and the authors ship code, which supports reproducibility. The speedups reported (100x–10000x in favorable settings) are plausible and valuable. However, the asymptotic consistency guarantee for GES with the approximate score is asserted rather than proved, and the empirical evidence would be stronger with variance reporting. The contribution is therefore best assessed as an algorithmic acceleration with heuristic justification, rather than a fully supported theoretical claim about causal discovery.

major comments (4)

[Section 6, after Definition 6.1] The sentence 'With a good enough approximation, S_LR also naturally satisfies the local consistency' is the load-bearing link between the fast score and the causal discovery guarantee, but no result in the paper defines 'good enough.' Local consistency requires the approximate score to preserve the sign of S_CV(G') - S_CV(G) for every edge insertion and deletion as n tends to infinity. The relative-error check in Appendix B.1 (score values within 0.5%) does not imply sign preservation, because score differences between neighboring DAGs can be orders of magnitude smaller than the scores themselves. Moreover, the pivot-count m is fixed at 100 while n grows, and no spectral-decay assumption is stated to guarantee that the low-rank approximation error remains below the true score gap. Please provide a formal error bound linking the approximation error to the score differences, or at least an empirical study of sign agreement between S_CV and S_LR for all GES moves.
[Section 5, complexity statement] The paper claims O(n) time and space complexity, but the detailed statement is O(n m^2) time and O(n m) storage with m = max(m_x, m_z). This is linear in n only if m is treated as a constant. No argument is given that a fixed m (e.g., m=100) keeps the approximation error below the relevant threshold as n grows for general continuous data and nonempty conditioning sets. Since the experiment in Section 7.3 shows speedup for n up to 4000, the claim 'linear computational complexity' should be qualified as 'linear for fixed rank m,' and the dependence of the required m on n and on the conditioning set should be discussed or empirically characterized.
[Section 7.4, Figures 2–4 and Section 7.5, Figure 5] The accuracy plots and runtime plots do not show error bars, confidence intervals, or any measure of variability, despite the experiments being repeated 20 times (and 10 times in the appendices). The central empirical claim is that CV-LR achieves accuracy 'comparable' to CV; without variance or paired comparison statistics, the small differences visible in Figures 2–5 cannot be distinguished from noise. Please report standard deviations or confidence intervals, and for the runtime claims in Figure 1, report the spread over repeated runs.
[Section 7.2 and Appendix B.1] The choice of m=100 is justified by requiring the relative approximation error to be below 0.5% on a small set of configurations (one node, one or six conditioning variables, two data types). This is a data-dependent tuning decision, and no sensitivity analysis is given for other data sizes, graph densities, or conditioning set sizes. Because m and the ICL precision eta are the only free parameters of the method, the paper should provide guidance on how they interact with the sample size and the data distribution, and should report the actual number of pivots selected by the adaptive algorithm in the experiments, rather than only the upper bound m=100.

minor comments (4)

[Eq. (8)] In the first line of Eq. (8), the term '- n2 0 / 2 log(2π)' appears to be a typographical corruption of '- n_0 / 2 log(2π)'; please correct the typesetting.
[Section 4, Algorithm 1] Algorithm 1's stopping rule 'if sum_j=i..n d_j < eta' is described as reaching precision, but the output guarantee in the caption states '||ΛΛ^T - K_X|| <= eta'; the relationship between this diagonal residual criterion and the spectral or Frobenius norm bound should be stated precisely.
[Section 5, Eq. (18)] The notation after Eq. (18), with terms involving '2I - DF' and 'G', is dense and would benefit from a short derivation or a reference to the supplementary material; the typo 'Woddbury' in Section 5 should also be corrected to 'Woodbury'.
[Section 7.1] The sentence explaining why BIC, BDeu, and SC are restricted ('SC score... is unsuitable for multi-dimensional data') is helpful, but the text does not specify the kernel width choice for the proposed method's continuous kernels beyond 'twice the median distance'; please give the exact formula.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the complexity derivation is self-contained algebra, and the causal-consistency step is an unproven assertion rather than a reduction to inputs.

full rationale

I find no circular step that can be exhibited by reduction. The approximate score S_LR is defined by substituting low-rank approximations of centered kernel matrices into the exact cross-validated likelihood score of Huang et al. (2018) and then algebraically rewriting the expression using Woodbury and Weinstein-Aronszajn identities (Section 5). The O(n m^2) time and O(n m) storage claims follow from the dimensions of the dumbbell-form matrix chains and are internal algebraic results, not fitted to any evaluation target. Local consistency of the exact CV score is imported from an external result ([15], Lemma 3), not from the present authors. The statement in Section 6 that "with a good enough approximation, S_LR(G_h,D) also naturally satisfies the local consistency" is an unproven assumption: the paper provides no error bound linking low-rank approximation quality to sign preservation of score differences S(G') - S(G). This is a rigor gap, and the 0.5% relative-error check in Appendix B.1 does not close it, but it is not circularity because the assertion is not defined in terms of its own conclusion and no parameter is fitted to the causal-discovery targets. The choice m = 100 is selected by score fidelity (relative error below 0.5%) on separate settings and then used in experiments against ground-truth graphs; that is standard hyperparameter selection, not a fitted-input-called-prediction pattern. Self-citations [26, 27, 47, 51] are related-work references and do not carry the derivation. Consequently, no step reduces the claimed result to its own inputs, and the honest finding is no significant circularity.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central fast-score derivation rests on standard RKHS and matrix algebra (Woodbury, Weinstein-Aronszajn), the causal sufficiency assumptions of the GES framework, and two unproved practical assumptions: the low-rank approximation with m pivots is accurate enough, and that accuracy suffices to preserve the ordering of local scores for GES consistency. The free parameters are the rank cap m and the ICL precision eta, both hand-set.

free parameters (2)

m (maximal rank / number of pivots) = 100
Set to 100 in experiments after checking that the relative score error stays below 0.5% (Section 7.2). The complexity is O(n m^2), so the claimed O(n) depends on m being a fixed constant.
eta (precision for incomplete Cholesky) = 1e-6
Precision threshold in Algorithm 1; controls when column sampling stops.

assumptions (6)

domain assumption The kernels are characteristic, bounded, and the RKHS is appropriately embedded in L2.
Needed to equate conditional independence with the zero of the partial cross-covariance operator in Section 2.
domain assumption The data is causally sufficient: no unobserved common causes and no feedback (DAG assumption).
Stated in Section 6 before applying GES.
domain assumption The CV-likelihood score is decomposable as a sum over variables of local scores.
Used in Eq. (31) to sum local scores; this property is inherited from the generalized score framework.
ad hoc to paper A sufficiently accurate low-rank approximation preserves the ordering of local scores, yielding local consistency of S_LR.
Asserted in Section 6 with 'naturally satisfies' but no formal bound or proof is given; this is the main theoretical gap.
standard math Woodbury matrix identity and Weinstein-Aronszajn determinant identity.
Used in Lemma 5.3 and Section 5 for the inverse and determinant of dumbbell-form matrix chains.
domain assumption The incomplete Cholesky decomposition produces a controlled low-rank approximation.
Algorithm 1 assumes the greedy pivot selection gives a good low-rank approximation; error bounds from the cited literature are not stated.

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Cite this review

Pith. "Pith review of Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity." pith.science (2026). https://pith.science/paper/IIL2UD5K

@misc{pith2026241217717,
  author       = {Pith},
  title        = {Pith review of: Fast Causal Discovery by Approximate Kernel-based Generalized Score Functions with Linear Computational Complexity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IIL2UD5K}},
  note         = {Machine review of arXiv:2412.17717}
}

abstract

Score-based causal discovery methods can effectively identify causal relationships by evaluating candidate graphs and selecting the one with the highest score. One popular class of scores is kernel-based generalized score functions, which can adapt to a wide range of scenarios and work well in practice because they circumvent assumptions about causal mechanisms and data distributions. Despite these advantages, kernel-based generalized score functions pose serious computational challenges in time and space, with a time complexity of $\mathcal{O}(n^3)$ and a memory complexity of $\mathcal{O}(n^2)$, where $n$ is the sample size. In this paper, we propose an approximate kernel-based generalized score function with $\mathcal{O}(n)$ time and space complexities by using low-rank technique and designing a set of rules to handle the complex composite matrix operations required to calculate the score, as well as developing sampling algorithms for different data types to benefit the handling of diverse data types efficiently. Our extensive causal discovery experiments on both synthetic and real-world data demonstrate that compared to the state-of-the-art method, our method can not only significantly reduce computational costs, but also achieve comparable accuracy, especially for large datasets.

Figures

Figures reproduced from arXiv: 2412.17717 by the authors.

**Figure 3.** The F1/SHD score of recovered causal graphs with [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 5.** Left two: F1 score results on the two discrete net [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

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