Ψ-Resilience：基于一维拓扑信号的无模型特征重要性方法

# Model-Free Feature Importance from 1D Topological Signals
Source: [https://arxiv.org/html/2610.02299](https://arxiv.org/html/2610.02299)
Darian Onchis,Affiliation:West University of Timi oaraEmail:[darian\.onchis@e\-uvt\.ro](mailto:)Pedro Real JuradoAffiliation:University of SevillaEmail:[real@us\.es](mailto:)

###### Abstract

We introduceΨ\\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,Ψ\\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 thatΨ\\Psi\-Resilience is a stable explanation method that enables rigorous, distribution\-level auditing of feature importance without relying on a predictive model\.

## 1Introduction

Feature importance has become the standard tool for communicating and assessing the behavior of complex predictive models, but despite its frequent usage, there’s an ongoing confusion over what exactly is a “feature importance score” measuring\. Model\-based scores such as SHAP values\[[12](https://arxiv.org/html/2610.02299#bib.bib1)\]quantify how a particular fitted model changes its predictions when a feature is perturbed, while regulators and domain experts often ask a different question: How important is this feature within our data, regardless of any particular model we may use?

Most works formalise this distinction between model\-specific and intrinsic \(also called algorithm\-agnostic\) importance\[[22](https://arxiv.org/html/2610.02299#bib.bib7),[17](https://arxiv.org/html/2610.02299#bib.bib9)\]\. Model\-specific measures depend, as the name suggests, on the learning algorithm, but also on its hyperparameters, and even random initialisation, and can disagree dramatically across equally accurate models trained on the same data\[[17](https://arxiv.org/html/2610.02299#bib.bib9),[15](https://arxiv.org/html/2610.02299#bib.bib10)\]\. Intrinsic importance is defined as a functional of the joint distribution of\(X,Y\)\(X,Y\), typically via a population\-level contrast in predictive performance with versus without a feature, or feature subset\[[22](https://arxiv.org/html/2610.02299#bib.bib7),[21](https://arxiv.org/html/2610.02299#bib.bib8)\]\. This yields a property of the data\-generating process itself, not of any particular estimator\.

Recent work has made a lot of progress on intrinsic importance but there are still large gaps in this regard\. Many estimators are basically model\-mediated even when the target is conceptually model\-free: they require fitting flexible predictive models as nuisance regressors, with final importance estimates sensitive to modelling choices and optimisation instability\[[22](https://arxiv.org/html/2610.02299#bib.bib7)\]\. Another problem is that existing scores based on data only reveal little about the local structure of class separation along a feature, such as whether importance arises from a single large separation region or from many fragile and small pockets of discriminative signal\.

In this work, we proposeΨ\\Psi\-Resilience: a new approach to model\-free global feature importance driven by one\-dimensional topological signals\. We first construct a class\-disagreement landscape for each feature by estimating class\-conditional densities and taking their point\-wise absolute difference along the feature axis\. This gives us a one\-dimensional signal whose prominent peaks correspond to the regions in which the feature separates the classes locally\. We then build a cubical complex on the negative of this signal and compute zero\-dimensional persistence intervals\. The resilience functionalΨΔ\\Psi\_\{\\Delta\}is the main product of the process \- it assigns to each feature the aggregated “cost” of removing all topological features \(connected components\) that survive perturbations up to scaleΔ\\Delta\. Intuitively,ΨΔ\\Psi\_\{\\Delta\}quantifies how much robust class separation the feature supports at resolutionΔ\\Delta\. A general overview of our method can be seen in Figure[1](https://arxiv.org/html/2610.02299#S1.F1)\.

![Refer to caption](https://arxiv.org/html/2610.02299v1/overview.png)Figure 1:Overview ofΨ\\Psi\-Resilience\. Starting from class\-conditional densities \(a\), we construct a disagreement signal \(b\), compute its persistence diagram, and aggregate theΔ\\Delta\-resilient contributions intoΨΔ​\(j\)\\Psi\_\{\\Delta\}\(j\), our final score, for each feature \(c\)\.We testΨ\\Psi\-Resilience on synthetic benchmarks where ground\-truth feature importance is known and compare it to a wide range of intrinsic and model\-based global importance measures: mutual information, F\-scores, permutation importance, drop\-column variable importance\[[22](https://arxiv.org/html/2610.02299#bib.bib7)\], SHAP\[[12](https://arxiv.org/html/2610.02299#bib.bib1)\], SAGE\[[5](https://arxiv.org/html/2610.02299#bib.bib12)\], and nonparametric partial\-dependence impact\[[15](https://arxiv.org/html/2610.02299#bib.bib10)\]\. The comparison of our method with model\-based approaches is intended to move beyond the validation of whether a model behaves like other models with respect to feature importance, to the capturing of the real structure of data and to determine its intrinsic geometry, rather than just fitting to noise\. TheΨΔ\\Psi\_\{\\Delta\}score obtains competitive alignment with ground\-truth rankings while offering stability and auditability properties that are unlike these existing methods\. Our results show thatΨ\\Psi\-Resilience can serve as a “topology\-for\-auditors” layer that can work in tandem with model\-based explainability by exposing the robust separation structure inherent in the data itself\.

## 2Related Work

Much of the current research in explainable AI focuses onfeature importance for a particular predictive model\. Well\-known examples include SHAP\[[12](https://arxiv.org/html/2610.02299#bib.bib1)\], permutation importance \(originally developed in the context of random forests\), as well as “leave\-one\-feature\-out” and “drop\-column” importance scores\. The above methods detail how a certain modelf^\\hat\{f\}uses the features of the data, either in a local or global context\. Yet, as Williamson et al\. mentioned in the recent literature, these features do not tend to indicate the feature’s intrinsic predictive capability\. Two models might have the same accuracy, but assign widely divergent importance scores to features, given the same data\[[22](https://arxiv.org/html/2610.02299#bib.bib7),[17](https://arxiv.org/html/2610.02299#bib.bib9),[15](https://arxiv.org/html/2610.02299#bib.bib10)\]\.

Covert et al\. propose SAGE, a Shapley\-style global measure of importance that combines the marginal contributions of a given feature for predictive accuracy over all possible subsets of features\[[5](https://arxiv.org/html/2610.02299#bib.bib12)\]\. SAGE is algorithm\-agnostic because it is defined in terms of an abstract predictive loss function, but it’s instantiated with a chosen model and an imputation strategy, so the resulting scores are still model\-dependent\. Saarela and Jauhiainen conducted a comparison study demonstrating large variability across various algorithm/model\-based measures of importance \(such as SHAP, permutation importance, among others\) for classification tasks in various models, concluding that interpretations of results for predictive accuracy should be taken cautiously\[[17](https://arxiv.org/html/2610.02299#bib.bib9)\]\. The risk of over\-interpretation of algorithm/model\-based measures of importance has further been noted in recent biomedical research, urging for greater statistical verification of these claims\[[14](https://arxiv.org/html/2610.02299#bib.bib13),[20](https://arxiv.org/html/2610.02299#bib.bib14)\]\.

We emphasise in this regard that we are not proposing yet another importance measure that might work well on a particular model, but develop a purely model\-independent function that can operate together with the others\. The purpose ofΨΔ\\Psi\_\{\\Delta\}can be seen as capturing the robust separation information within the data\.

Regardingintrinsic and algorithm\-agnostic variable importance, Williamson et al\. defines this notion in more concrete terms by considering a contrast in oracle predictiveness between the set of all available features and a set where a particular setSSof features becomes unavailable\[[22](https://arxiv.org/html/2610.02299#bib.bib7),[21](https://arxiv.org/html/2610.02299#bib.bib8)\]\. This introduces a set of target quantities that depend solely upon the distribution of\(X,Y\)\(X,Y\)and the chosen loss function, as opposed to being sensitive to specifics of learning algorithms\. They provide semiparametric efficient estimates using flexible machine learning as nuisance regressors in low and high dimensions\.

Building upon this viewpoint,\[[11](https://arxiv.org/html/2610.02299#bib.bib11)\]establishes that the importance parameters in the Williamson et al\. method are equivalent to the global sensitivities calculated using the Sobol’ indices\. This relationship gives birth to FIRST, a factor importance ranking and selection procedure, specialized in regression, that can be directly approximated from noisy observational data, removing the necessity for the construction of a parametric surrogate\. The paper illustrates that the estimation of the total Sobol’ indices can in fact be both statistic\-justified and model\-agnostic and further supports the idea that the definition of importance should instead be based at the scale of the data\-generating mechanism\.

These methods all share the common target of specifying importance as a model\-free function of the joint distribution\. They mainly focus on a loss difference perspective, without revealing the intricate details of the separation between classes along a feature dimension\. Instead,ΨΔ\\Psi\_\{\\Delta\}is derived from one\-dimensional class disagreement signals, making the spatial separation patterns fully inspectable\.

There are methods that seek to estimatefeature importance straight from the datawithout fitting a predictive model at all\. Parr et al\. consider feature impact to be a measure of the area under the ideal partial dependence curve and feature importance as a weighted distribution form of this curve\. Both of these measures can be expressed as a function of conditional expectations over a conditional joint distribution of\(X,Y\)\(X,Y\)and can each be approximated, nonparametrically\. Empirically, their values rank well in a comparative manner with other traditional model\-based feature selection algorithms\[[15](https://arxiv.org/html/2610.02299#bib.bib10)\]\. Oka et al\., as well as Takefuji, argue that correlation, mutual information, and data\-only conditions are valuable secondary tools for feature relevance\[[14](https://arxiv.org/html/2610.02299#bib.bib13),[20](https://arxiv.org/html/2610.02299#bib.bib14)\]\.Ψ\\Psi\-Resilience is closest in spirit to these data\-driven approaches, asΨΔ\\Psi\_\{\\Delta\}takes class\-conditional probabilities and signals of class disagreement from\(X,Y\)\(X,Y\), rather than requiring or relying on predictive models\. Our method differs from impact measures derived from partial dependence\[[15](https://arxiv.org/html/2610.02299#bib.bib10)\]through favoring geometry related to class decomposition in 1D rather than the study of regression\-like response curves\. In comparison to these methods, ours is accompanied by both importance scores and the robustness parameterΔ\\Delta\.

## 3Methodology

ΨΔ\\Psi\_\{\\Delta\}can be viewed as an intrinsically model\-free importance functional that operates on one\-dimensional density\-difference signals encoding local class separation, incorporates an explicit robustness scale via resilience homology, and yields importance scores that are by construction accompanied by interpretable 1D explanation signals\.

Let𝒟=\{\(xi,yi\)\}i=1n\\mathcal\{D\}=\\\{\(x\_\{i\},y\_\{i\}\)\\\}\_\{i=1\}^\{n\}be a supervised classification dataset withxi∈ℝdx\_\{i\}\\in\\mathbb\{R\}^\{d\}andyi∈\{1,…,K\}y\_\{i\}\\in\\\{1,\\dots,K\\\}\. We denote byxi,jx\_\{i,j\}the value of featurej∈\{1,…,d\}j\\in\\\{1,\\dots,d\\\}for sampleii, and by

𝒟c,j=\{xi,j:yi=c\},c∈\{1,…,K\},\\mathcal\{D\}\_\{c,j\}=\\\{x\_\{i,j\}:y\_\{i\}=c\\\},\\quad c\\in\\\{1,\\dots,K\\\},the class\-conditional sample for featurejj\.

Our goal is to assign each featurejja scalar importance scoreΨΔ​\(j\)\\Psi\_\{\\Delta\}\(j\)that depends only on the joint distribution of\(X,Y\)\(X,Y\), and which quantifies how much robust class\-separation structure this feature supports at a user\-chosen robustness scaleΔ\>0\\Delta\>0\.

### 3\.1Class\-Disagreement Signals via KDE

For each featurejj, we work with a one\-dimensional grid

tk=aj\+km−1\(bj−aj\),k=0,…,m−1,t\_\{k\}=a\_\{j\}\+\\frac\{k\}\{m\-1\}\(b\_\{j\}\-a\_\{j\}\),\\quad k=0,\\dots,m\-1,whereaj=mini⁡xi,ja\_\{j\}=\\min\_\{i\}x\_\{i,j\},bj=maxi⁡xi,jb\_\{j\}=\\max\_\{i\}x\_\{i,j\}, andm∈ℕm\\in\\mathbb\{N\}is the grid size\.

Estimating for each classc∈\{1,…,K\}c\\in\\\{1,\\dots,K\\\}the class\-conditional density of featurejjusing a univariate kernel density estimator,

p^j,c​\(t\)=1\|𝒟c,j\|​hj​∑xi,j∈𝒟c,jK⁡\(t−xi,jhj\),\\hat\{p\}\_\{j,c\}\(t\)=\\frac\{1\}\{\|\\mathcal\{D\}\_\{c,j\}\|\\,h\_\{j\}\}\\sum\_\{x\_\{i,j\}\\in\\mathcal\{D\}\_\{c,j\}\}K\\\!\\left\(\\frac\{t\-x\_\{i,j\}\}\{h\_\{j\}\}\\right\),\(1\)whereKKis a fixed kernel \(Gaussian in our implementation\) andhj\>0h\_\{j\}\>0is a bandwidth parameter\. We use Scott’s rulehj=σ^jn−1/5h\_\{j\}=\\widehat\{\\sigma\}\_\{j\}\\,n^\{\-1/5\}, whereσ^j\\widehat\{\\sigma\}\_\{j\}is the empirical standard deviation ofxi,jx\_\{i,j\}, but any consistent rule could be substituted\.

WhenK=2K=2, let the two classes bec0c\_\{0\}andc1c\_\{1\}\. The disagreement signal for featurejjis

fj​\(t\)=\|p^j,c1​\(t\)−p^j,c0​\(t\)\|,f\_\{j\}\(t\)=\\bigl\|\\hat\{p\}\_\{j,c\_\{1\}\}\(t\)\-\\hat\{p\}\_\{j,c\_\{0\}\}\(t\)\\bigr\|,\(2\)and we denote by𝐟j=\(fj​\(t0\),…,fj​\(tm−1\)\)\\mathbf\{f\}\_\{j\}=\(f\_\{j\}\(t\_\{0\}\),\\dots,f\_\{j\}\(t\_\{m\-1\}\)\)its discrete vector representation\.

WhenK\>2K\>2, we define pairwise signals\.

For each unordered pair\(c1,c2\)\(c\_\{1\},c\_\{2\}\)withc1<c2c\_\{1\}<c\_\{2\}, we compute

fj\(c1,c2\)​\(t\)=\|p^j,c1​\(t\)−p^j,c2​\(t\)\|\.f\_\{j\}^\{\(c\_\{1\},c\_\{2\}\)\}\(t\)=\\bigl\|\\hat\{p\}\_\{j,c\_\{1\}\}\(t\)\-\\hat\{p\}\_\{j,c\_\{2\}\}\(t\)\\bigr\|\.\(3\)
In the end, we obtain one or more one\-dimensional scalar fields on the grid\{tk\}\\\{t\_\{k\}\\\}, which we treat as topological signals encoding local class separation along featurejj\.

### 3\.2Resilience Homology on 1D Signals

Given a discrete one\-dimensional signal𝐟=\(f⁡\(t0\),…,f⁡\(tm−1\)\)\\mathbf\{f\}=\(f\(t\_\{0\}\),\\dots,f\(t\_\{m\-1\}\)\), we construct a cubical complex𝒞\\mathcal\{C\}on thesemmgrid points and them−1m\-1intervals\[tk,tk\+1\]\[t\_\{k\},t\_\{k\+1\}\]between them\. We then consider the scalar field−f\-fon𝒞\\mathcal\{C\}, assigning each vertex the value−f⁡\(tk\)\-f\(t\_\{k\}\)and extending to edges via the standard upper\-star filtration\[[7](https://arxiv.org/html/2610.02299#bib.bib15)\]\.

We then form the sublevel\-set filtration of−f\-f,

𝒞α=\{σ∈𝒞:\(−f\)​\(x\)≤α​for all​x∈σ\},α∈ℝ,\\mathcal\{C\}^\{\\alpha\}=\\\{\\sigma\\in\\mathcal\{C\}:\(\-f\)\(x\)\\leq\\alpha\\text\{ for all \}x\\in\\sigma\\\},\\quad\\alpha\\in\\mathbb\{R\},\(4\)and compute the00\-dimensional persistent homology of this filtration, obtaining a multiset of birth–death pairs

Dgm0​\(f\)=\{\(bℓ,dℓ\)\}ℓ=1L,bℓ<dℓ,\\mathrm\{Dgm\}\_\{0\}\(f\)=\\\{\(b\_\{\\ell\},d\_\{\\ell\}\)\\\}\_\{\\ell=1\}^\{L\},\\quad b\_\{\\ell\}<d\_\{\\ell\},\(5\)whereLLis the number of finite bars\. Each bar corresponds to a connected component of the sublevel sets that is born at levelbℓb\_\{\\ell\}and merges at leveldℓd\_\{\\ell\}\.

We define the base resilience of barℓ\\ellas half its persistence:

rℓ=dℓ−bℓ2\.r\_\{\\ell\}=\\frac\{d\_\{\\ell\}\-b\_\{\\ell\}\}\{2\}\.\(6\)rℓr\_\{\\ell\}approximates the minimalL∞L\_\{\\infty\}perturbation offfrequired to eliminate that topological feature in the ideal continuous setting\.

LetΔabs≥0\\Delta\_\{\\mathrm\{abs\}\}\\geq 0be a robustness scale in the same units asff\. We define theΔ\\Delta\-resilient contribution of barℓ\\ellas

rℓ\(Δabs\)=max⁡\{rℓ−Δabs,0\}\.r\_\{\\ell\}^\{\(\\Delta\_\{\\mathrm\{abs\}\}\)\}=\\max\\\{r\_\{\\ell\}\-\\Delta\_\{\\mathrm\{abs\}\},0\\\}\.\(7\)Bars withrℓ≤Δabsr\_\{\\ell\}\\leq\\Delta\_\{\\mathrm\{abs\}\}are considered too fragile to be relevant at scaleΔabs\\Delta\_\{\\mathrm\{abs\}\}and contribute zero; longer bars contribute linearly beyond this threshold\.

Then, we aggregate theΔ\\Delta\-resilient contributions across bars using a chosen aggregation operator𝖠𝗀𝗀bar∈\{sum,mean,max\}\\mathsf\{Agg\}\_\{\\mathrm\{bar\}\}\\in\\\{\\mathrm\{sum\},\\mathrm\{mean\},\\mathrm\{max\}\\\}:

ΨΔ​\(f\)=𝖠𝗀𝗀bar​\(\{rℓ\(Δabs\)\}ℓ=1L\),\\Psi\_\{\\Delta\}\(f\)=\\mathsf\{Agg\}\_\{\\mathrm\{bar\}\}\\bigl\(\\\{r\_\{\\ell\}^\{\(\\Delta\_\{\\mathrm\{abs\}\}\)\}\\\}\_\{\\ell=1\}^\{L\}\\bigr\),\(8\)whereΔabs\\Delta\_\{\\mathrm\{abs\}\}is either a user\-specified absolute threshold or, as in our implementation, derived from a relative thresholdΔ\>0\\Delta\>0via

Δabs=Δ⋅\(maxk⁡f⁡\(tk\)−mink⁡f⁡\(tk\)\),\\Delta\_\{\\mathrm\{abs\}\}=\\Delta\\cdot\\bigl\(\\max\_\{k\}f\(t\_\{k\}\)\-\\min\_\{k\}f\(t\_\{k\}\)\\bigr\),\(9\)when a relative robustness scale is desired\.

In our experiments we primarily use𝖠𝗀𝗀bar=max\\mathsf\{Agg\}\_\{\\mathrm\{bar\}\}=\\mathrm\{max\}, so thatΨΔ​\(f\)\\Psi\_\{\\Delta\}\(f\)captures the resilience of the single most robust separation mode in the signal\.

Finally, the feature\-level importance scoreΨΔ​\(j\)\\Psi\_\{\\Delta\}\(j\)is defined by aggregatingΨΔ​\(f\)\\Psi\_\{\\Delta\}\(f\)over one or more disagreement signals associated with featurejj\.

WhenK=2K=2, there is a single disagreement signalfjf\_\{j\}given by \([2](https://arxiv.org/html/2610.02299#S3.E2)\), and we setΨΔ​\(j\)=ΨΔ​\(fj\)\\Psi\_\{\\Delta\}\(j\)=\\Psi\_\{\\Delta\}\(f\_\{j\}\)\.

WhenK\>2K\>2and we operate in pairwise mode, we have one disagreement signalfj\(c1,c2\)f\_\{j\}^\{\(c\_\{1\},c\_\{2\}\)\}per unordered class pair\(c1,c2\)\(c\_\{1\},c\_\{2\}\)\. We computeΨΔ​\(fj\(c1,c2\)\)\\Psi\_\{\\Delta\}\(f\_\{j\}^\{\(c\_\{1\},c\_\{2\}\)\}\)for each pair and aggregate the score across all pairs\.

## 4Experiments

For reproducibility, all experiments were implemented in Python, utilizing scikit\-learn for basic modeling\[[16](https://arxiv.org/html/2610.02299#bib.bib5)\], the SHAP library\[[12](https://arxiv.org/html/2610.02299#bib.bib1)\]for attributions, and GUDHI\[[6](https://arxiv.org/html/2610.02299#bib.bib2)\]for computing cubical complexes\. The experiments were run in a cloud configuration hosted by Google Colab\[[8](https://arxiv.org/html/2610.02299#bib.bib6)\]\(T4 GPU\)\. In terms of hyperparameters, unless otherwise stated, we set the grid size tom=300m=300, the robustness parameter toΔ=0\.03\\Delta=0\.03\(interpreted via \([9](https://arxiv.org/html/2610.02299#S3.E9)\)\),𝖠𝗀𝗀bar=max\\mathsf\{Agg\}\_\{\\mathrm\{bar\}\}=\\mathrm\{max\}, and𝖠𝗀𝗀pair=sum\\mathsf\{Agg\}\_\{\\mathrm\{pair\}\}=\\mathrm\{sum\}\. Given fixed\(X,Y\)\(X,Y\)and hyperparameters, the procedure is deterministic\. Any randomness in our experiments arised only from the synthetic data generator and from external baselines, not fromΨ\\Psi\-Resilience itself\.

We evaluated the method on both real and synthetic datasets:

1. 1\.Wisconsin Breast Cancer, a modestly\-sized \(569 instances with 30 features\) medical dataset for binary classification\[[23](https://arxiv.org/html/2610.02299#bib.bib4)\];
2. 2\.Diabetes 130\-US, a medical dataset with 101k instances and 47 features\[[19](https://arxiv.org/html/2610.02299#bib.bib20)\];
3. 3\.Covertype, a classification of pixels into 7 forest cover types based on 54 attributes, containing 58k instances\[[2](https://arxiv.org/html/2610.02299#bib.bib21)\];
4. 4\.Spambasefor binary spam classification, consisting of 4601 x 57 numeric word\-frequency features\[[10](https://arxiv.org/html/2610.02299#bib.bib16)\];
5. 5\.Jannis, a dataset of considerable size \(83\.7k instances with 55 features\) built for a machine learning competition\[[9](https://arxiv.org/html/2610.02299#bib.bib17)\];
6. 6\.PhishingWebsites, consisting of 11k instances with 68 features\[[13](https://arxiv.org/html/2610.02299#bib.bib18)\];
7. 7\.HIGGS, a physics dataset with 1 million instances of 29 features for distinguishing between a signal process which produces Higgs bosons and a background process which does not\[[1](https://arxiv.org/html/2610.02299#bib.bib19)\];
8. 8\.Asynthetic datasetwhere true feature importance is known by design\. The generation of this dataset is described in the subsequent section\.

To evaluate scalability, We included both small\-scale and large\-scale datasets, such as the 1 million instance HIGGS dataset\. The synthetic dataset enables the validation of our scores against the ground truth, while the real datasets demonstrates the approach on real\-world data where ground truth is unavailable, but where comparisons with other feature importance methods can be made\.

### 4\.1Synthetic Data Generation

We built a synthetic generator to obtain multiclass datasets where there is a known ground\-truth importance along the feature dimensions\. For each of these setups, we drawX∈ℝn×dX\\in\\mathbb\{R\}^\{n\\times d\}from a correlated latent GaussianZ∼𝒩⁡\(0,Σ\)Z\\sim\\mathcal\{N\}\(0,\\Sigma\), whereΣ\\Sigmais either Toeplitz, equicorrelated, or a random symmetric positive definite matrix\. A Gaussian copula step mapsZZto uniform scores and then to user\-specified marginals \(e\.g\., Gaussian, lognormal, uniform, Laplace, exponential\), yielding the final design matrixXX\.

We selectssfeatures as informative and endow each classk∈\{1,…,K\}k\\in\\\{1,\\dots,K\\\}with its own linear coefficientswk,jw\_\{k,j\}on this subset, withwk,j∼Unif⁡\(a,b\)⋅\{±1\}w\_\{k,j\}\\sim\\mathrm\{Unif\}\(a,b\)\\cdot\\\{\\pm 1\\\}\. The class\-specific linear signals aresk=X​wks\_\{k\}=Xw\_\{k\}, optionally augmented by nonlinear transforms \(square,sin\\sin,tanh\\tanh,ReLU,abs\) applied to a fraction of the informative features and a small number of pairwise interaction terms \(centered products\)\. To control difficulty, each signalsks\_\{k\}is rescaled to a target signal\-to\-noise ratio \(SNR\) and perturbed by Gaussian noise,

ℓi,k=snr⋅si,kSD⁡\(s⋅,k\)\+εi,k,εi,k∼𝒩⁡\(0,1\),\\ell\_\{i,k\}=\\mathrm\{snr\}\\cdot\\frac\{s\_\{i,k\}\}\{\\mathrm\{SD\}\(s\_\{\\cdot,k\}\)\}\+\\varepsilon\_\{i,k\},\\qquad\\varepsilon\_\{i,k\}\\sim\\mathcal\{N\}\(0,1\),and class\-specific interceptsbkb\_\{k\}are calibrated so that the empirical class frequencies approximately match a desired probability vectorπ\\piunder the softmax modelp⁡\(yi=k∣xi\)=softmaxk​\(ℓi\+b\)p\(y\_\{i\}=k\\mid x\_\{i\}\)=\\mathrm\{softmax\}\_\{k\}\(\\ell\_\{i\}\+b\)\. Labels are sampled asyi∼Categorical\(p\(yi=⋅∣xi\)\)y\_\{i\}\\sim\\mathrm\{Categorical\}\(p\(y\_\{i\}=\\cdot\\mid x\_\{i\}\)\)\.

The global importance of featurejjin ground truth is defined as theℓ2\\ell\_\{2\}\-norm of its class\-specific linear coefficients,‖w⋅,j‖2=\(∑k=1Kwk,j2\)1/2\\\|w\_\{\\cdot,j\}\\\|\_\{2\}=\\bigl\(\\sum\_\{k=1\}^\{K\}w\_\{k,j\}^\{2\}\\bigr\)^\{1/2\}; the nonlinear weights are used only to enhance realism and to add difficulty levels, not to define the target ranking\. The irrelevant features have the same correlation structure as the informative ones, so spurious associations and masking effects arise naturally\.

In all our experiments involving this data, we used multiple configurations:

1. 1\.Default:n=1000n=1000,d=30d=30,K=3K=3,s=10s=10informative features,SNR=1\.5\\mathrm\{SNR\}=1\.5, Toeplitz or random SPD covariance, Gaussian marginals, no nonlinearity\.
2. 2\.Varying SNR and classes:K∈\{3,5\}K\\in\\\{3,5\\\},SNR∈\{0\.8,1\.5,2\.0\}\\mathrm\{SNR\}\\in\\\{0\.8,1\.5,2\.0\\\}, Toeplitz covariance withρ∈\{0\.2,0\.6\}\\rho\\in\\\{0\.2,0\.6\\\}\.
3. 3\.Non\-Gaussian \+ nonlinear: mixed marginals \(Gaussian, lognormal, Laplace, uniform, exponential\), nonlinear effects on 50% of informative features with a few pairwise interactions\.

These settings allowed the performance ofΨ\\Psi\-Resilience and baseline approaches to be effectively evaluated for varying levels of class complexity, correlation, noise, and marginal structure\. The experiments show that the method is robust and resistant against such variations\.

### 4\.2Experimental Setup

We now describe the experimental process used to compareΨ\\Psi\-Resilience to various feature importance methods\. All methods and generators were reimplemented in Python to the best of our knowledge according to their respective articles using standard libraries\.

On each synthetic dataset configuration, we computed global feature importance scores for all features using the following methods:

- •Ψ\\Psi\-Resilience \(ΨΔ\\Psi\_\{\\Delta\}\)\.Our method, with a uniform grid ofm=300m=300evaluation points per feature and withΔ=0\.03\\Delta=0\.03\(taken as a fraction of the signal range\)\.
- •Mutual information \(MI\)\.The univariate mutual informationI⁡\(Xj,Y\)I\(X\_\{j\};Y\)estimated by the standard nonparametric estimator for classification implemented in scikit\-learn, which has been proposed as a model\-independent importance measure in a variety of contexts\[[14](https://arxiv.org/html/2610.02299#bib.bib13),[20](https://arxiv.org/html/2610.02299#bib.bib14)\]\.
- •ANOVA F\-score\.The one\-way F\-statisticFjF\_\{j\}of featurejjversus the class labelYY\. Although based on a parametric mean\-shift model, this is just a standard data\-only baseline for filtering and ranking features\.
- •SHAP\.Global SHAP importance was computed from an XGBoost classifier\[[4](https://arxiv.org/html/2610.02299#bib.bib3)\]trained on\(X,Y\)\(X,Y\)for a maximum depth of44\. We used the TreeSHAP implementation\[[12](https://arxiv.org/html/2610.02299#bib.bib1)\]to obtain the SHAP values for each sample and class and aggregated them by taking the mean absolute value across all samples and classes, yielding a single score per feature\.
- •Permutation importance\.We computed Breiman\-style permutation importance on the same trained XGBoost model by repeatedly permuting each featureXjX\_\{j\}and the negative log\-likelihood’s reduction \(i\.e\., the increase in multiclass log\-loss\)\[[3](https://arxiv.org/html/2610.02299#bib.bib22)\]\. The score for featurejjis the mean change in score over 20 permutations\.
- •Drop\-column VI\.Following the framework for algorithm\-agnostic variable importance\[[22](https://arxiv.org/html/2610.02299#bib.bib7)\], we calculated, for each featurejj, the difference in empirical risk between the full model trained on all features and a reduced model trained onXXwith columnjjremoved\. Risk was measured as multiclass log\-loss on the same training data; the resulting importance isVIdrop​\(j\)=R^−j−R^full\\mathrm\{VI\}\_\{\\mathrm\{drop\}\}\(j\)=\\hat\{R\}\_\{\-j\}\-\\hat\{R\}\_\{\\mathrm\{full\}\}\.
- •Parr partial\-dependence impact\.We implemented a model\-based approximation to the nonparametric feature impact of Parr et al\.\[[15](https://arxiv.org/html/2610.02299#bib.bib10)\]\. For each featurejj, we computed a model\-based partial dependence profile by overwritingX:,jX\_\{:,j\}with grid values and averaging the predicted probability of the true class over samples\. The impact score is the mean absolute deviation of this profile from its baseline value \(the average predicted probability before intervention\)\. This yields a scalar impact measure per feature\.

The methods mentioned above may be classified into purely data\-based, model\-free scores \(ΨΔ\\Psi\_\{\\Delta\}, MI, F\-score\) and model\-based, algorithm\-agnostic scores \(SHAP, Permutation, Drop\-column, Parr impact, and SAGE\)\. As far as publicly documented implementations of FIRST go, they are focused on regression and binary classification, and we couldn’t integrate this model for the multiclass tasks of our pipeline\.

### 4\.3Results

Table 1:Spearman correlationρ\\rhowith ground\-truth feature importance on synthetic benchmarks for different numbers of classesK\>2K\>2\. Results use default generator hyperparameters and are averaged over 5 random seeds, with a standard deviation less than10−210^\{\-2\}\.
Table 2:Spearman correlationρ\\rhoofΨ\\Psi\-Resilience with the SHAP scores on the real\-world benchmark datasets\. The values are deterministic, per\-dataset\.

For every dataset\-method configuration, we computed a vector of feature importance scores\. We measured the agreement between the models and the ground truth using Spearman rank correlation\[[18](https://arxiv.org/html/2610.02299#bib.bib23)\]calculated over the features\. The value of this coefficient doesn’t change under monotone transformations of the scores and highlights the consistency of the rankings\.

To complement the correlation comparisons, we also measured the runtime for each method and dataset configuration\. For each, we measured the elapsed time from the start of the method’s computation to its completion using high\-resolution timers\. The time differences across all methods remained consistent and proportional relative to each other and the scale of the data\. As a reference, in a representative six\-class synthetic setting with default generation parameters,Ψ\\Psi\-Resilience completed the process in 0\.502 seconds on our hardware configuration\. In comparison, other model\-based methods, such as SHAP \(0\.956 seconds for XGBoost plus 0\.785 seconds\), Permutation Importance \(0\.956 seconds for XGBoost plus 5\.784 seconds\), Drop\-column \(33\.3 seconds\), and PD\-based impact \(6\.324 seconds\), experienced high processing time due to the complexity of the underlying models\.

Our experimental outcome for synthetic data \(Table[2](https://arxiv.org/html/2610.02299#S4.T2)\) and real\-world data \(Table[2](https://arxiv.org/html/2610.02299#S4.T2)\) verifies our main points\.Ψ\\Psi\-Resilience is accurate as a global feature\-importance measure, even when compared to model\-based benchmarks\.

Onsynthetic data, SHAP andΨΔ\\Psi\_\{\\Delta\}both performed well across the configurations\. SHAP obtained the best correlation with the ground truth importance in several settings \(ρ=0\.819\\rho=0\.819forK=3K=3andρ=0\.811\\rho=0\.811forK=5K=5\), which demonstrated its effectiveness\. Nevertheless,ΨΔ\\Psi\_\{\\Delta\}always fell in the same performance category as SHAP, permutation importance, and Parr impact, with correlations between0\.7480\.748and0\.8180\.818\.ΨΔ\\Psi\_\{\\Delta\}also followed the ANOVA F\-score, which also captures univariate signal strength, but is more sensitive to linear mean\-shift assumptions and lacks robustness tuning\.

Regarding the methods based purely on data, mutual information performed well in certain settings but still remained behindΨΔ\\Psi\_\{\\Delta\}and SHAP, especially for largerKKor complex correlation structures\. Despite being algorithm\-agnostic, Drop\-column degraded substantially in some configurations from what we have seen \(e\.g\.,ρ=0\.514\\rho=0\.514forK=5K=5\), likely due to the need to refit models on reduced feature sets and the resulting instability in high\-dimensional, correlated designs\.

The synthetic experiments above have shown that the proposedΨΔ\\Psi\_\{\\Delta\}method estimates the ground truth with similar success to advanced model\-based approaches at a lower computational expense\.

To examine the relation toΨΔ\\Psi\_\{\\Delta\}in the context ofreal\-world datawhere the ground truth is not available, we calculated the Spearman rank correlation coefficient between the ranking of SHAP values andΨΔ\\Psi\_\{\\Delta\}features \(Table[2](https://arxiv.org/html/2610.02299#S4.T2)\)\. These correlations are positive and in many cases high, for instanceρ=0\.964\\rho=0\.964for the Diabetes 130\-US dataset \(as seen in Figure[3](https://arxiv.org/html/2610.02299#S4.F3)\) andρ=0\.904\\rho=0\.904for the Covertype dataset \(as seen in Figure[3](https://arxiv.org/html/2610.02299#S4.F3)\)\. Even on other challenging, high\-dimensional or noisy datasets, the agreement remained moderate to strong \(ρ\\rhoin the range0\.6450\.645–0\.8110\.811\)\. This pattern is encouraging, since in the regimes in which SHAP is trusted to reproduce meaningful model behaviour,ΨΔ\\Psi\_\{\\Delta\}is in agreement while being computed without ever fitting a predictive model\.

![Refer to caption](https://arxiv.org/html/2610.02299v1/diabetes.png)Figure 2:Correlation ofΨ\\Psi\-Resilience and SHAP scores for the Diabetes dataset, scaled to\[0,1\]\[0,1\]\.
![Refer to caption](https://arxiv.org/html/2610.02299v1/covertype.png)Figure 3:Correlation ofΨ\\Psi\-Resilience and SHAP scores for the Covertype dataset, scaled to\[0,1\]\[0,1\]\.

In summary, the above experiments demonstrated thatΨ\\Psi\-Resilience is able to compete in ranking accuracy on synthetic benchmarks with known ground truth, as well as maintaining consistent agreement with SHAP on real data, while offering distinctive advantages: model independence, robustness control and auditability\. All of these properties illustrate thatΨΔ\\Psi\_\{\\Delta\}would be a useful addition to the toolbox of global feature\-importance methods, especially in scenarios where data\-grounded explanations are needed\.

## 5Conclusion

We have introducedΨ\\Psi\-Resilience, a new framework for model\-free global feature importance based on one\-dimensional topological signals extracted from class\-conditional densities\. The resulting importance scores are purely data\-driven, do not require any predictive model, explicitly parameterised by robustness, and come with a natural visual audit trail via 1D signals and persistence\. On synthetic benchmarks of known ground\-truth importance,ΨΔ\\Psi\_\{\\Delta\}achieves competitive Spearman rank correlations with the true feature importance, usually falling in the same performance band as SHAP, permutation importance, and partial\-dependence impact, and outperforming or matching baselines based only on data, like mutual information or ANOVA F\-scores\. On real datasets, for which no ground truth is available,ΨΔ\\Psi\_\{\\Delta\}shows strong consistency with SHAP\-based rankings across a wide collection of classification tasks, suggesting that the captured robust separation patterns are closely aligned with how strong tree\-based models take advantage of\.

Ψ\\Psi\-Resilience can compete with, but also complements existing methods for intrinsic and algorithm\-agnostic variable importance and model\-based global attribution methods\. While loss\-based intrinsic importance quantifies how predictive performance changes when features are removed or masked,ΨΔ\\Psi\_\{\\Delta\}quantifies how much robust one\-dimensional class separation is present along each feature axis, at a specified resolutionΔ\\Delta\.

We mention several limitations of our work\. First,ΨΔ\\Psi\_\{\\Delta\}is fundamentally univariate: it operates on one feature at a time and captures the marginal class\-separation structure along that axis\. While this is shared by many screening and ranking approaches \(MI, F\-scores, univariate PD\-based impact\), it is indicative that purely interaction\-driven importance will not be fully captured\. In principle, one could extend the resilience homology to the multivariate disagreement fields of disagreement, but this would introduce major computational and statistical challenges\.

Second, we considered a kernel density estimation with a fixed bandwidth rule\. Although the method is well\-understood and practical in one dimension, the resulting disagreement signals and persistence diagrams are sensitive to bandwidth choice, sample size, and class imbalance\. We have adopted a set of standard heuristics \(Scott’s rule, fixed grids\) for robustness and simplicity, but a more systematic treatment of bandwidth selection and uncertainty quantification forΨΔ\\Psi\_\{\\Delta\}remains to be developed\.

Finally, we have focused on classification tasks\. Adaptation to regression and structured outputs are not trivial\.

In summary, we believe thatΨ\\Psi\-Resilience is a good example of how tools from topological data analysis can be adapted to obtain simple, robust and auditable feature\-importance measures that speak directly to the structure of the data, and that such model\-free perspectives play an increasingly important role in building trustworthy and interpretable machine learning systems\.

## References

- \[1\]P\. Baldi, P\. Sadowski, and D\. Whiteson\(2014\)Searching for exotic particles in high\-energy physics with deep learning\.Nature Communications5,pp\. 4308\.External Links:[Document](https://dx.doi.org/10.1038/ncomms5308)Cited by:[item 7](https://arxiv.org/html/2610.02299#S4.I1.i7.p1.1)\.
- \[2\]J\. A\. Blackard and D\. J\. Dean\(1999\)Comparative accuracies of artificial neural networks and discriminant analysis in predicting forest cover types from cartographic variables\.Computers and Electronics in Agriculture24\(3\),pp\. 131–151\.External Links:[Document](https://dx.doi.org/10.1016/S0168-1699%2899%2900046-0)Cited by:[item 3](https://arxiv.org/html/2610.02299#S4.I1.i3.p1.1)\.
- \[3\]L\. Breiman\(2001\)Random forests\.Machine learning45\(1\),pp\. 5–32\.Cited by:[5th item](https://arxiv.org/html/2610.02299#S4.I3.i5.p1.1)\.
- \[4\]T\. Chen and C\. Guestrin\(2016\)XGBoost: a scalable tree boosting system\.InProceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining,pp\. 785–794\.Cited by:[4th item](https://arxiv.org/html/2610.02299#S4.I3.i4.p1.1)\.
- \[5\]I\. Covert, S\. M\. Lundberg, and S\. Lee\(2020\)Understanding global feature contributions with additive importance measures\.Advances in neural information processing systems33,pp\. 17212–17223\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p5.1),[§2](https://arxiv.org/html/2610.02299#S2.p2.1)\.
- \[6\]P\. Dlotko\(2025\)Cubical complex\.InGUDHI User and Reference Manual,External Links:[Link](https://gudhi.inria.fr/doc/3.11.0/group__cubical__complex.html)Cited by:[§4](https://arxiv.org/html/2610.02299#S4.p1.1)\.
- \[7\]H\. Edelsbrunner and J\. Harer\(2010\)Computational topology: an introduction\.American Mathematical Soc\.\.Cited by:[§3\.2](https://arxiv.org/html/2610.02299#S3.SS2.p1.1)\.
- \[8\]Google\(2026\)Google Colaboratory\.Note:[colab\.research\.google\.com](https://colab.research.google.com/)\[Accessed: 07/05/2026\]Cited by:[§4](https://arxiv.org/html/2610.02299#S4.p1.1)\.
- \[9\]I\. Guyon, K\. Bennett, G\. Cawley, H\. J\. Escalante, S\. Escalera, T\. K\. Ho, N\. Macià, B\. Ray, M\. Saeed, A\. Statnikov,et al\.\(2015\)Design of the 2015 chalearn automl challenge\.In2015 International joint conference on neural networks \(IJCNN\),pp\. 1–8\.Cited by:[item 5](https://arxiv.org/html/2610.02299#S4.I1.i5.p1.1)\.
- \[10\]M\. Hopkins, E\. Reeber, G\. Forman, and J\. Suermondt\(1999\)Spambase \[dataset\]\.Note:UCI Machine Learning RepositoryExternal Links:[Document](https://dx.doi.org/10.24432/C53G6X),[Link](https://doi.org/10.24432/C53G6X)Cited by:[item 4](https://arxiv.org/html/2610.02299#S4.I1.i4.p1.1)\.
- \[11\]C\. Huang and V\. R\. Joseph\(2025\)Factor importance ranking and selection using total indices\.Technometrics,pp\. 1–17\.Cited by:[§2](https://arxiv.org/html/2610.02299#S2.p5.1)\.
- \[12\]S\. M\. Lundberg and S\. Lee\(2017\)A unified approach to interpreting model predictions\.InAdvances in Neural Information Processing Systems,pp\. 4765–4774\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p1.1),[§1](https://arxiv.org/html/2610.02299#S1.p5.1),[§2](https://arxiv.org/html/2610.02299#S2.p1.1),[4th item](https://arxiv.org/html/2610.02299#S4.I3.i4.p1.1),[§4](https://arxiv.org/html/2610.02299#S4.p1.1)\.
- \[13\]R\. M\. Mohammad, F\. Thabtah, and L\. McCluskey\(2012\)An assessment of features related to phishing websites using an automated technique\.InInternational Conference for Internet Technology and Secured Transactions \(ICITST\),pp\. 492–497\.Note:IEEE; article number 6470857External Links:ISBN 978\-1\-4673\-5321\-2Cited by:[item 6](https://arxiv.org/html/2610.02299#S4.I1.i6.p1.1)\.
- \[14\]S\. Oka, N\. Inoue, and Y\. Takefuji\(2025\)Beyond predictive accuracy: statistical validation of feature importance in biomedical machine learning\.Computer Methods and Programs in Biomedicine,pp\. 109085\.Cited by:[§2](https://arxiv.org/html/2610.02299#S2.p2.1),[§2](https://arxiv.org/html/2610.02299#S2.p7.1),[2nd item](https://arxiv.org/html/2610.02299#S4.I3.i2.p1.1)\.
- \[15\]T\. Parr, J\. Hamrick, and J\. D\. Wilson\(2024\)Nonparametric feature impact and importance\.Information Sciences653,pp\. 119563\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p2.1),[§1](https://arxiv.org/html/2610.02299#S1.p5.1),[§2](https://arxiv.org/html/2610.02299#S2.p1.1),[§2](https://arxiv.org/html/2610.02299#S2.p7.1),[7th item](https://arxiv.org/html/2610.02299#S4.I3.i7.p1.1)\.
- \[16\]F\. Pedregosa, G\. Varoquaux, A\. Gramfort, V\. Michel, B\. Thirion, O\. Grisel, M\. Blondel, P\. Prettenhofer, R\. Weiss, V\. Dubourg, J\. Vanderplas, A\. Passos, D\. Cournapeau, M\. Brucher, M\. Perrot, and E\. Duchesnay\(2011\)Scikit\-learn: machine learning in Python\.Journal of Machine Learning Research12,pp\. 2825–2830\.Cited by:[§4](https://arxiv.org/html/2610.02299#S4.p1.1)\.
- \[17\]M\. Saarela and S\. Jauhiainen\(2021\)Comparison of feature importance measures as explanations for classification models\.SN Applied Sciences3\(2\),pp\. 272\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p2.1),[§2](https://arxiv.org/html/2610.02299#S2.p1.1),[§2](https://arxiv.org/html/2610.02299#S2.p2.1)\.
- \[18\]C\. Spearman\(1904\)The proof and measurement of association between two things\.The American journal of psychology15\(1\),pp\. 72—101\.Cited by:[§4\.3](https://arxiv.org/html/2610.02299#S4.SS3.p1.1)\.
- \[19\]B\. Strack, J\. P\. DeShazo, C\. Gennings, J\. L\. Olmo, S\. Ventura, K\. J\. Cios, and J\. N\. Clore\(2014\)Impact of hba1c measurement on hospital readmission rates: analysis of 70,000 clinical database patient records\.BioMed Research International2014,pp\. 781670\.External Links:[Document](https://dx.doi.org/10.1155/2014/781670)Cited by:[item 2](https://arxiv.org/html/2610.02299#S4.I1.i2.p1.1)\.
- \[20\]Y\. Takefuji\(2025\)Beyond shap: reliable feature selection methods for clinical prediction models\.Archives of Gerontology and Geriatrics,pp\. 105873\.Cited by:[§2](https://arxiv.org/html/2610.02299#S2.p2.1),[§2](https://arxiv.org/html/2610.02299#S2.p7.1),[2nd item](https://arxiv.org/html/2610.02299#S4.I3.i2.p1.1)\.
- \[21\]B\. D\. Williamson, P\. B\. Gilbert, M\. Carone, and N\. Simon\(2021\)Nonparametric variable importance assessment using machine learning techniques\.Biometrics77\(1\),pp\. 9–22\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p2.1),[§2](https://arxiv.org/html/2610.02299#S2.p4.1)\.
- \[22\]B\. D\. Williamson, P\. B\. Gilbert, N\. R\. Simon, and M\. Carone\(2023\)A general framework for inference on algorithm\-agnostic variable importance\.Journal of the American Statistical Association118\(543\),pp\. 1645–1658\.Cited by:[§1](https://arxiv.org/html/2610.02299#S1.p2.1),[§1](https://arxiv.org/html/2610.02299#S1.p3.1),[§1](https://arxiv.org/html/2610.02299#S1.p5.1),[§2](https://arxiv.org/html/2610.02299#S2.p1.1),[§2](https://arxiv.org/html/2610.02299#S2.p4.1),[6th item](https://arxiv.org/html/2610.02299#S4.I3.i6.p1.1)\.
- \[23\]W\. Wolberg, O\. Mangasarian, N\. Street, and W\. Street\(1993\)Breast Cancer Wisconsin \(Diagnostic\)\.Note:UCI Machine Learning RepositoryDOI: https://doi\.org/10\.24432/C5DW2BCited by:[item 1](https://arxiv.org/html/2610.02299#S4.I1.i1.p1.1)\.