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$\Psi$-Resilience: Model-Free Feature Importance from 1D Topological Signals
Fabian Galis, Darian Onchis, Pedro Real Jurado
TL;DR
Feature-importance scores can reflect a particular fitted model, while data-based approaches may not show the local structure of class separation. Ψ-Resilience derives scores from one-dimensional topological signals; it is competitive on synthetic ground-truth rankings and agrees positively with SHAP on real datasets.
Problem
Feature-importance scores may describe a fitted model rather than intrinsic properties of the data, and existing data-based scores reveal little about local class-separation structure.
Source: §1 · pp. 1–2Method
Ψ-Resilience forms one-dimensional class-disagreement landscapes from class-conditional densities and aggregates persistence features surviving perturbations up to a user-set scale ∆.
Source: §1 · p. 2Results
On synthetic data, Ψ∆ falls in the same performance category as SHAP, permutation importance, and Parr impact, with correlations between 0.748 and 0.818; on real data, its rankings positively correlate with SHAP, reaching ρ = 0.964 for Diabetes 130-US.
Source: §4.3 · p. 8Takeaways & Limitations
The method provides model-free importance scores with a visual audit trail through one-dimensional signals and persistence.
Source: §5 · p. 9Because Ψ∆ is univariate, it does not fully capture purely interaction-driven importance.
Source: §5 · p. 9
Abstract
from arXiv · showhide
We introduce $\Psi$-Resilience, a model-free feature importance method that derives explanations directly from the data itself via 1D topological signals. Our method constructs a class-disagreement landscape by estimating class-conditional densities and taking their pointwise absolute difference along the feature axis. Then, the 0-dimensional persistence of this 1D signal defines a resilience functional that aggregates only those topological features that survive perturbations up to a robustness scale which is set by the user. This gives us a context-robust importance score that is inherently auditable via the underlying 1D landscapes and their persistence. We evaluate our method on both synthetic and real datasets. On synthetic generators with specified ground-truth importance, $\Psi$-Resilience recovers the ranking of features with high fidelity, achieving Spearman rank correlations up to 0.8 and performing competitively with multiple feature importance methods, including SHAP and mutual information. On real datasets with no known ground truth, our technique agrees with these methods, with correlations up to 0.9. These results show that $\Psi$-Resilience is a stable explanation method that enables rigorous, distribution-level auditing of feature importance without relying on a predictive model.
1 Introduction
The introduction distinguishes data-intrinsic feature importance from model-specific scores and identifies a need for model-free measures that reveal local class separation. It presents Ψ-Resilience as a topological approach and reports competitive synthetic-benchmark results.
1 Introduction
- Model-specific scores can vary across equally accurate models, whereas intrinsic importance is a functional of the joint distribution of (X, Y), not a particular estimator.The distinction matters because feature-importance scores are widely used despite confusion about what they measure.Source: §1 · p. 1
- Existing data-based scores reveal little about whether separation along a feature comes from one large region or many small, fragile discriminative pockets.Some estimators also rely on flexible predictive models as nuisance regressors, making estimates sensitive to modeling choices and optimization instability.Source: §1 · pp. 1–2
- Ψ-Resilience estimates class-conditional densities, takes their pointwise absolute difference to form a 1D disagreement landscape, and aggregates persistence features surviving perturbations up to scale ∆.Its score is intended to quantify robust class separation at the chosen resolution, with a visual audit trail through the signal and persistence diagram.Source: §1 · p. 2, Figure 1 · p. 2
- On synthetic benchmarks with known ground-truth importance, Ψ-Resilience achieves competitive ranking alignment and offers stability and auditability beyond the compared importance methods.The comparison includes mutual information, F-scores, permutation importance, drop-column importance, SHAP [12], SAGE [5], and partial-dependence impact.Source: §1 · p. 2
2 Related Work
Related work spans model-specific explainers and data-intrinsic importance measures, but the latter commonly emphasize predictive-loss contrasts rather than the geometry of class separation along a feature. Ψ-Resilience instead uses inspectable one-dimensional disagreement signals and is positioned as complementary to other measures.
2 Related Work
- Model-specific explainers describe how a fitted model uses features, but equally accurate models can assign divergent importance scores.SAGE is defined using an abstract predictive loss but instantiated with a chosen model and imputation strategy [5]; variability across model-based measures has prompted caution in interpretation.Source: §2 · pp. 2–3
- Ψ∆ is presented as a model-independent function that can operate alongside other importance measures to capture robust separation information in the data.The paper explicitly distinguishes this goal from proposing another measure tailored to a particular model.Source: §2 · p. 3
- Intrinsic importance can be defined through oracle-predictiveness contrasts over feature sets [22, 21], and FIRST connects these parameters to global Sobol’ sensitivities [11].FIRST is described as a regression-focused ranking and selection procedure that can be approximated from noisy observational data [11].Source: §2 · p. 3
- Unlike loss-difference approaches, Ψ∆ uses one-dimensional class-disagreement signals to make spatial patterns of class separation inspectable.The related methods share a model-free target based on the joint distribution, but do not reveal those separation details.Source: §2 · p. 3
- Data-driven alternatives include nonparametric partial-dependence impact, correlation, and mutual information; Ψ-Resilience instead favors 1D class-decomposition geometry and adds robustness parameter ∆.The cited partial-dependence measures study regression-like response curves rather than class-decomposition geometry.Source: §2 · p. 3
3 Methodology
Ψ-Resilience assigns feature importance from one-dimensional signals of local class separation, using persistence to measure robust structure at a user-chosen scale. Class-conditional density estimates form the signals, and their persistent features are aggregated into feature-level scores.
3 Methodology
- Ψ-Resilience scores how much robust class-separation structure each feature supports at a user-chosen robustness scale.The functional depends only on the joint distribution of (X, Y).Source: §3 · p. 3
3.1 Class-Disagreement Signals via KDE
- Class-conditional density estimates yield one-dimensional disagreement signals that encode local class separation along each feature.For two classes, the signal uses their density difference; for more than two, the method constructs pairwise signals.Source: §3.1 · p. 4
- A fixed kernel and bandwidth determine the class-conditional density estimates, with Scott’s rule used as the default bandwidth choice.The paper notes that any consistent bandwidth rule could be substituted.Source: §3.1 · p. 4
3.2 Resilience Homology on 1D Signals
- The method builds a cubical complex on the negated signal and computes 0-dimensional persistence from its filtration.Finite bars represent connected components born at one level and merged at another.Source: §3.2 · p. 4
- Bars at or below the robustness threshold contribute zero, while longer bars contribute linearly beyond it.The threshold may be specified in absolute units or derived from a relative robustness scale.Source: §3.2 · pp. 4–5
- Resilience contributions are aggregated across bars by sum, mean, or max, then across a feature’s disagreement signals.In the reported implementation, max is primarily used; multiclass pairwise scores are aggregated across class pairs.Source: §3.2 · p. 5
4 Experiments
The experiments test Ψ-Resilience on synthetic data with known feature importance and real datasets without ground truth, comparing its rankings with data-only and model-based methods. Synthetic results show competitive agreement with ground truth, while real-data rankings often agree with SHAP.
4.1 Synthetic Data Generation
- The synthetic generator creates multiclass data with known feature importance from correlated latent Gaussian variables mapped through a copula to user-specified marginals.Class signals can include nonlinear transforms and pairwise interactions; the target ranking is defined by the L2 norm of class-specific linear coefficients.Source: §4.1 · p. 6
- Experiments vary class complexity, correlation, noise, and marginal structure, and report that Ψ-Resilience is robust to these variations.The configurations include mixed marginals, nonlinear effects, and pairwise interactions.Source: §4.1 · p. 6
4.2 Experimental Setup
- Evaluation measured agreement between model-derived feature-importance scores and synthetic ground truth using Spearman rank correlation over features; real datasets demonstrate the approach and enable comparisons with other feature-importance methods where ground truth is unavailable.Source: §4 · p. 5, §4.3 · p. 7
4.3 Results
- Ψ-Resilience correlations with synthetic ground-truth importance ranged from 0.748 to 0.818, placing it alongside SHAP, permutation importance, and Parr impact.SHAP reached ρ = 0.819 for K = 3 and ρ = 0.811 for K = 5; results are averaged over five seeds with standard deviation below 10−2.Source: Table 1 · p. 7, §4.3 · p. 8
- On real datasets, Ψ-Resilience and SHAP rankings correlated from ρ = 0.645–0.811 on challenging datasets, reaching ρ = 0.964 for Diabetes 130-US and ρ = 0.904 for Covertype.These correlations compare rankings where ground-truth importance is unavailable.Source: Table 2 · p. 7, §4.3 · p. 8
5 Conclusion
Ψ-Resilience provides model-free global importance scores from robust one-dimensional class-separation signals, with an audit trail. Synthetic results are competitive with other methods, while real-data rankings are consistent with SHAP-based rankings.
5 Conclusion
- On synthetic benchmarks, Ψ-Resilience usually matches SHAP, permutation importance, and partial-dependence impact, outperforming or matching mutual information and ANOVA F-scores; real rankings strongly track SHAP.Real datasets have no known ground-truth importance.Source: §5 · p. 9
- Unlike loss-based intrinsic scores, Ψ-Resilience quantifies robust one-dimensional class separation at a specified resolution.Its scores are data-driven and accompanied by an audit trail via 1D signals and persistence.Source: §5 · p. 9
- Because it is univariate, Ψ-Resilience captures marginal separation along each feature and may miss importance driven purely by interactions.A multivariate extension would introduce major computational and statistical challenges.Source: §5 · p. 9
- With fixed-bandwidth kernel density estimation, disagreement signals and persistence diagrams are sensitive to bandwidth, sample size, and class imbalance.More systematic bandwidth selection and uncertainty quantification remain to be developed.Source: §5 · p. 9
- The evaluation focuses on classification; adapting the method to regression and structured outputs is not trivial.Source: §5 · p. 9