Enhancing interpretability of rule-based classifiers through feature graphs

Christel Sirocchi; Damiano Verda

REVIEW 4 major objections 7 minor 28 references

Enhancing interpretability of rule-based classifiers through feature graphs

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that the degree centrality of its projected feature graph is a valid model-agnostic feature importance metric for rule-based classifiers, performing comparably to permutation importance and with more stable rankings.

desk verdict A clearly specified graph-based framework for analyzing feature contributions in rule-based classifiers, with a novel weighted projection and degree-centrality importance score, but the validation is thinner than the claims warrant: synthetic results are visual only and the input relevance metric is adopted without independent scrutiny. read the letter →

arxiv 2506.13903 v1 pith:W7QJZCAK submitted 2025-06-16 cs.LG cs.AI

Christel Sirocchi , Damiano Verda This is my paper · ORCID

classification cs.LGcs.AI

keywords rule-basedsystemsfeatureimportancegraphsinterpretablemachinelearninggraphcentralityrulesetcomparisoninteractionsclinicaldecisionsupport

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

This paper tries to establish a unified way to read rule-based classifiers: turn the rule set into a graph whose nodes are input features, and let the connections record when two features contribute jointly to the same rules. The authors define feature importance as the total weight attached to a node (degree centrality) and claim this score is model-agnostic, computationally cheap, and more stable across data splits than permutation importance, Gini importance, or average Shapley values. On 15 public benchmarks the score matches permutation importance in top-k accuracy and is the most robust of the four. The same graph carries interaction information that a one-dimensional importance list cannot, and a Frobenius distance between graphs lets users compare rule sets over the same features. The practical payoff, illustrated on two clinical datasets, is the ability to see which pairs of clinical measurements are predictive only in combination.

What carries the argument

The central object is the projected feature graph: a weighted undirected graph on input features obtained from a tripartite network of features, rules, and class labels. Its adjacency matrix is computed by Equation (3) as a normalized product over rules, with each rule contributing a factor $1 - p_{ki} p_{kj} q_k$, where $p$ is a feature-to-rule relevance score and $q$ is a rule-to-class relevance score. Degree centrality of this adjacency matrix is the proposed importance measure, self-edges keep features that appear alone in rules, and the class-restricted projection gives class-specific feature graphs. The Frobenius distance between two normalized adjacency matrices is the proposed distance between rule sets.

What would settle it

On a synthetic rule set whose true pairwise contributions are known by construction, compare the centrality ranking from Equation (5) with the ranking obtained when Equation (8) is replaced by support, confidence, lift, or impurity gain; if the ranking materially changes with the injected score, the graph is measuring that score rather than the model's actual feature use.

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Extended reading notes

Core claim

The central claim is that projecting a tripartite features-rules-classes network onto the feature set produces a feature graph whose degree centrality is a valid feature importance metric, independent of the rule-induction algorithm. In that graph, an edge between two features is strong when the pair appears together with high relevance in at least one rule that matters, while self-edges preserve the contribution of features that appear alone in rules. The authors evaluate this on synthetic data with known independent and combined predictive structure, on two clinical datasets with four rule-induction schemes, and on 15 public benchmarks against permutation importance, Gini importance, and average Shapley values. They report that the centrality score performs comparably to permutation importance in top-5 and top-10 feature-selection accuracy and is the most stable ranking across cross-validation folds. The same construction restricted to rules with one class gives class-specific interaction graphs, and the Frobenius distance between adjacency matrices serves as a rule-set distance that does not require pairwise rule matching.

Load-bearing premise

The whole graph is built from two input scores, feature-to-rule relevance and rule-to-outcome relevance, adopted from an existing method without independent validation, so if those scores misrepresent true contributions the edge weights, centrality rankings, and graph distances all inherit the error.

Editorial extensions

If this is right

Rule sets from different induction schemes, such as decision trees, logic learning machines, association rules, and extracted neural-network rules, can be compared on a common feature-importance scale without re-training or pairwise rule matching.
Clinicians can see which measurements are predictive only in combination, so data collection can prioritize pairs like glucose with age or with body mass index rather than treating all risk factors independently.
Replacing a rule set in an expert system can be guided by graph distance, choosing the update whose feature-contribution graph is closest to the existing one and thereby preserving continuity of clinical explanations.
Class-specific projections expose interactions that aggregate importance hides, such as phase angle and impedance carrying the signal for carcinoma while a single spectral feature suffices for adipose tissue in the breast-tissue case.
The reported rank stability across decision-tree depths suggests that the centrality score can serve as a reliable feature-ranking tool when data splits are noisy, reducing the need for extensive hyperparameter tuning.

Reading between the lines

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

A testable extension is to feed the edge weights, not just node centrality, into feature selection: strong edges between a predictive feature and a spurious one might flag correlations that survive single-feature rankings but not leave-one-out evaluation.
Because the projection multiplies contributions, a single highly relevant rule containing a rare feature pair can dominate an edge; comparing multiplicative aggregation with additive aggregation on the same synthetic data would reveal whether that emphasis is a strength or a distortion.
The framework's stated agnosticism to the input relevance scores suggests it could be used as a general contribution-to-graph wrapper around any local explanation method, although the paper only demonstrates it with scores adopted from logic learning machines.
The robustness result hints at a practical rule of thumb: when data splits are noisy, graph centrality may be a safer basis than permutation importance for deciding which patient information to collect, a claim that could be tested directly on the same benchmarks with different training seeds.

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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

The paper is a solid, workmanlike contribution to interpretability for rule-based models. What is actually new: the weighted projection in Eq. (3), which uses a complement-product form to aggregate joint feature relevance across rules, plus class-specific feature graphs, a Frobenius-norm distance between normalized adjacency matrices, and a degree-centrality feature importance score. These are sensible extensions of known bipartite-projection ideas, and the paper does a good job of specifying the framework formally and making the code public. The benchmark evaluation on 15 datasets, with four rule-based methods and two detailed clinical case studies, is a real strength. The clinical findings (glucose–age and glucose–BMI interactions on Pima, class-specific interactions on breast tissue) are plausible and well connected to the medical literature.

The soft spots are real but not fatal. The stress-test concern is the most important: every downstream object—adjacency, centrality, distance—inherits the feature and rule relevance scores from Eqs. (7)–(8), which are adopted from the LLM literature (one author is involved) and never independently validated within the paper. The synthetic evaluation is only qualitative heatmaps; it does not quantitatively show that the graph recovers known feature structure. The benchmark performance is measured indirectly, by training DTs on top-k features, which can wash out the contribution of the importance metric itself. Table 1 also has no error bars, and the robustness column is a single Spearman value per metric/dataset, so "superior robustness" is asserted on thin evidence. The saturation behavior of Eq. (3)—where the product quickly approaches one when many rules mention the same pair—is a minor concern but worth a sentence of analysis. These issues are fixable: the paper already mentions alternative relevance metrics in the code, and the method is agnostic to P and q, so a sensitivity analysis (e.g., using support, confidence, lift, or impurity gain) would substantially strengthen the claims.

That said, the central framework is sound and the paper is honest about its scope. It is internally consistent, and the input-relevance dependency is a feature of any interpretability pipeline, not a unique flaw. With added error bars, a quantitative synthetic check, and a sensitivity analysis of P and q, the claims would hold up. I think this deserves a serious referee, not a desk rejection. I would read it myself if I were working on rule-based interpretability, though I am unlikely to cite it directly in my own near-term work.

Referee Report

4 major / 7 minor

Summary. The paper proposes a graph-based framework for estimating feature contributions in rule-based classifiers. It constructs a tripartite graph among features, rules, and classes, projects it onto a feature-level weighted graph via Eq. (3), and defines a feature importance score as degree centrality (Eq. 5) and a distance metric between rule sets as the Frobenius norm of the adjacency difference (Eq. 4). The graph adjacency is built from feature relevance and rule relevance scores adopted from the Logic Learning Machine literature (Eqs. 7-8). The authors evaluate the approach on synthetic data, two clinical datasets, and 15 public benchmarks, comparing the importance score against permutation importance, Gini importance, and SHAP in terms of top-k classification accuracy and rank robustness.

Significance. If the proposed importance metric is shown to be reliable, the framework would provide a unified, model-agnostic way to visualize feature interactions and compare rule sets, which is of practical value in high-stakes domains such as healthcare. The paper makes the implementation publicly available and implements alternative relevance metrics, both of which are positive features. However, the central claims are conditional on the validity of the input relevance scores and on the projection in Eq. (3) not distorting the contribution structure; the current evidence is not yet sufficient to support the strong claims of competitive performance and superior robustness.

major comments (4)

[Section 3, Eq. (3)] The complement-product form in Eq. (3) can saturate: when even a few rules have moderately large p_ki * p_kj * q_k, the product over k of (1 - p_ki p_kj q_k) approaches zero, so A_ij tends to 1 and becomes insensitive to the magnitude of the relevance. As a result, the degree centrality in Eq. (5) increasingly reflects the number of rules in which a pair of features co-occurs rather than the strength of their joint contribution. This directly affects the validity of the proposed importance metric. Please provide a diagnostic (e.g., the distribution of A_ij values) and consider an alternative aggregation such as a sum or max that preserves magnitude information.
[Sections 4.1 and 5.1] The synthetic evaluation is entirely visual: no quantitative measure is reported for how well the graph, or the underlying relevance scores of Eqs. (7)-(8), recover the known ground-truth feature structure. Because the entire framework inherits the properties of these input scores, the evaluation should include a quantitative recovery test (e.g., Spearman correlation between the computed centrality and the ground-truth importance, with standard errors). The paper already states that alternative relevance metrics (support, confidence, lift, impurity gain) are implemented; comparing the results under these alternatives would substantially strengthen the claim that the framework is robust to the choice of input metric.
[Section 5.3, Table 1] The accuracy values are reported as point estimates without any measure of variance (standard deviation or confidence interval). Given that the robustness claim is central, the average Spearman correlations should also be reported with their variability across folds. Without error bars, it is impossible to assess whether the differences between methods are meaningful. Please report means and standard deviations across CV folds (or repeated runs) for both accuracy and rank robustness.
[Section 4.2] The paper states that the fidelity of the extracted rule sets was evaluated with accuracy and F1, and that the number of rules was set to 15, but these fidelity results are never reported. Since the clinical insights in Section 5.2 are derived from surrogate rule sets, the reader needs to see how faithful these surrogates are to the original black-box models.

minor comments (7)

[Abstract and throughout] The abstract and several places in the text contain spacing/formatting errors, e.g., 'discerncrucialfeatures' and 'uncovercrucial'; please fix the PDF compilation and proofread the text.
[Table 1] In Table 1, 'Hill V alley' should be 'Hill Valley'.
[Section 5.2] The centrality values and the pair-wise distance matrix are described qualitatively; please include the numerical values in a table so that the claims can be verified.
[Section 3] The graph is described as undirected, but this is only valid if the adjacency matrix is symmetric; please state explicitly that A' is symmetric given non-negative P and q.
[Section 5.3] The term 'model-agnostic' is used for the importance metric, but the benchmark evaluation only uses decision trees; please clarify the scope of the experiments.
[Eq. (6)] The rule relevance I_nabla = covering * (1 - error) is one of several possible choices; consider reporting how the main results change if lift or confidence is used instead.
[References] Reference [6] is a water-distribution application of LLM; for the relevance metrics defined in Section 3, please also cite the original LLM source where these metrics are introduced.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the graph construction is a compositional transformation of explicitly stated relevance metrics, and the central claims are independently benchmarked.

full rationale

The derivation chain is compositional rather than circular. The tripartite graph edges are defined in Eq. (3) as a product of feature relevance values and rule relevance values, the adjacency matrix is normalized, feature importance in Eq. (5) is a row sum of that normalized adjacency matrix, and graph distance in Eq. (4) is a Frobenius norm of adjacency differences. Each object is a new function of the preceding quantities, not a formal restatement of its inputs, and no parameter is fitted to the quantity it is later said to predict. The only self-citation signal is that the relevance metrics in Eqs. (7)-(8) are attributed to the LLM literature [6], whose author list includes the second author. This is source attribution rather than circular reduction: the formulas are stated in full, the pipeline is evaluated on 15 public benchmarks against permutation importance, Gini importance, and SHAP, and alternative relevance metrics are noted as implemented in code and deferred to future work. The concern that an unvalidated input metric may bias the graph is a validity or robustness limitation, not a circularity, because the centrality claim depends on the input but is not by construction equal to it. There is no fitted input renamed as a prediction, no imported uniqueness theorem, and no renaming of a known result presented as a derivation. Therefore the paper is self-contained against external benchmarks and exhibits no significant circularity.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The method introduces no fitted parameters and no new physical entities. It relies on four load-bearing domain assumptions: the validity of the input relevance metrics, the fidelity of the multiplicative aggregation, the appropriateness of degree centrality as importance, and the meaningfulness of the Frobenius distance.

assumptions (4)

domain assumption Feature relevance I_box and rule relevance I_nabla are valid measures of contribution.
These scores (Eqs. 7-8) are the only inputs to the graph, adopted from LLM literature [6], one author overlapping; if they misrepresent contribution, the graph inherits the error.
domain assumption The noisy-OR-like aggregation in Eq. (3) faithfully represents feature interactions.
A_ij = 1 - product(1 - p_ki p_kj q_k) treats rules as independent evidence and can saturate; no justification or sensitivity analysis is given.
domain assumption Degree centrality is a sensible feature importance.
Importance is defined as the row sum of the normalized adjacency matrix; this assumes total incident edge weight reflects predictive contribution, which is plausible but not proven.
domain assumption Frobenius norm of normalized adjacency matrices is a meaningful distance between rule sets.
Eq. (4) assumes matrix norm distance captures semantic differences in feature contribution; no comparison with existing rule-set distance methods is provided.

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Cite this review

Pith. "Pith review of Enhancing interpretability of rule-based classifiers through feature graphs." pith.science (2026). https://pith.science/paper/W7QJZCAK

@misc{pith2026250613903,
  author       = {Pith},
  title        = {Pith review of: Enhancing interpretability of rule-based classifiers through feature graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7QJZCAK}},
  note         = {Machine review of arXiv:2506.13903}
}

read the original abstract

In domains where transparency and trustworthiness are crucial, such as healthcare, rule-based systems are widely used and often preferred over black-box models for decision support systems due to their inherent interpretability. However, as rule-based models grow complex, discerning crucial features, understanding their interactions, and comparing feature contributions across different rule sets becomes challenging. To address this, we propose a comprehensive framework for estimating feature contributions in rule-based systems, introducing a graph-based feature visualisation strategy, a novel feature importance metric agnostic to rule-based predictors, and a distance metric for comparing rule sets based on feature contributions. By experimenting on two clinical datasets and four rule-based methods (decision trees, logic learning machines, association rules, and neural networks with rule extraction), we showcase our method's capability to uncover novel insights on the combined predictive value of clinical features, both at the dataset and class-specific levels. These insights can aid in identifying new risk factors, signature genes, and potential biomarkers, and determining the subset of patient information that should be prioritised to enhance diagnostic accuracy. Comparative analysis of the proposed feature importance score with state-of-the-art methods on 15 public benchmarks demonstrates competitive performance and superior robustness. The method implementation is available on GitHub: https://github.com/ChristelSirocchi/rule-graph.

Figures

Figures reproduced from arXiv: 2506.13903 by the authors.

**Figure 2.** Evaluation on the Pima Indians Diabetes dataset, comprising 8 features: [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Evaluation on the Breast Cancer dataset, comprising 9 breast tissue char [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Bioinformatics26(10), 1340–1347 (2010)

Altmann, A., Toloşi, L., Sander, O., Lengauer, T.: Permutation importance: a corrected feature importance measure. Bioinformatics26(10), 1340–1347 (2010)

work page 2010

[2] [2]

IEEE journal of biomedical and health in- formatics19(5), 1557–1566 (2015)

Banaee, H., Loutfi, A.: Data-driven rule mining and representation of temporal patterns in physiological sensor data. IEEE journal of biomedical and health in- formatics19(5), 1557–1566 (2015)

work page 2015

[3] [3]

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Chaves, R., Ramírez, J., Górriz, J., Puntonet, C.G., Initiative, A.D.N., et al.: Association rule-based feature selection method for alzheimer’s disease diagnosis. Expert Systems with Applications39(14), 11766–11774 (2012)

work page 2012

[4] [4]

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