SHAP Scores for Column Weights

Updated 16 January 2026

SHAP Scores for Column Weights is a framework that applies Shapley value theory to quantify both local and global feature importance in datasets.
The methodology involves computing local SHAP values, aggregating them into normalized global weights, and integrating these into clustering, ranking, and auditing tasks.
Empirical results demonstrate improved clustering metrics and robust feature selection, underscoring the framework’s practical application in diverse modeling scenarios.

SHAP (SHapley Additive exPlanations) scores for column weights constitute a quantitative framework derived from cooperative game theory, specifically the Shapley value, to assess and assign global or local importance to each feature (column) in a dataset. These scores are widely used in machine learning explainability and, recently, as principled global feature-weighting mechanisms for downstream analytic and modeling tasks such as clustering, ranking, and supervised learning. The approach rigorously extends from attribution for prediction to general parameter- and column-weight importance, grounded in theoretical properties and approximation algorithms.

1. Theoretical Foundations of SHAP Scores

SHAP scores assign to each feature (or, in generalized settings, each parameter or column-weight) a contribution value using the Shapley value:

$\phi_j(f, x) = \sum_{S \subseteq F \setminus \{j\}} \frac{|S|! (d - |S| - 1)!}{d!} \left[ E\left[f(x_{S \cup \{j\}})\right] - E\left[f(x_S)\right] \right]$

Here, $f$ is the model, $x$ the instance, $j$ the feature index, and $x_S$ denotes $x$ with features outside $S$ marginalized or replaced by background values. The expectation is with respect to some joint or marginal background distribution, a choice that is nontrivial and governs what explanatory question the SHAP value answers (Herren et al., 2022).

This formulation is equivalent to the Shapley value in cooperative game theory, guaranteeing the axioms of efficiency, symmetry, dummy, and additivity. For interpreting column weights, SHAP values quantify either the average causal contribution of each column (for per-instance scores) or, when aggregated globally, the overall influence of each column in the behavior or output of the function or model (Letoffe et al., 2024).

In ranking models and parameterized families, the Shapley value extends to column-weight parameters by treating each column-weight as a "player" and the effect of varying its value as the "payoff" in the Shapley setting (Standke et al., 9 Jan 2026).

2. Methodologies for Computing SHAP-Based Column Weights

2.1 Classical Algorithmic Pipeline

The canonical computation of global column weights from SHAP scores follows a sequence:

Model Construction: Train the primary model $f$ or, in clustering/unsupervised contexts, generate pseudo-labels via an initial clustering step and fit a surrogate classifier to these labels (e.g., RandomForest) (Galis et al., 12 Mar 2025).
Local SHAP Value Computation: For each instance, compute the SHAP value $\phi_{i, j, c}$ (contribution of feature $j$ to class $c$ for sample $i$ ). Tree-based models allow for efficient computation via TreeExplainer or similar polynomial-time algorithms (Herren et al., 2022, Galis et al., 12 Mar 2025).
Aggregation: Aggregate the local scores into global feature weights by averaging the absolute SHAP values over all samples (and, in multiclass settings, over all classes):

$w_j = \frac{1}{N K} \sum_{i=1}^N \sum_{c=1}^K |\phi_{i, j, c}|$

Normalize to ensure $\sum_j \hat{w}_j = 1$ :

$\hat{w}_j = w_j / \sum_\ell w_\ell$

Integration: Rescale data columns by $\hat{w}_j$ before downstream modeling. In clustering applications, these can directly serve as column (feature) weights in Euclidean, Mahalanobis, or other distance computations (Galis et al., 12 Mar 2025).

A similar approach aggregates per-instance SHAP values in supervised contexts for regression or classification to produce a global feature-importance vector (Letoffe et al., 2024, Letoffe et al., 2024).

2.2 Variations and Enhancements

Axiomatic SHAP: Modifications to the classical characteristic function (basis for Shapley values) have been proposed to ensure compliance with feature-relevancy properties and class-invariance, e.g., similarity- and explanation-based functions. This yields SHAP scores that nullify for truly irrelevant features and respect label symmetry (Letoffe et al., 2024).
Extended Distribution Averaging: For sound global importance, SHAP values must be aggregated over the product of the marginals (“extended support”); otherwise, features with true effect may be missed when the data distribution is lower-dimensional. Practically, this is implemented by independently permuting columns and recomputing SHAP values, ensuring the resulting averages have a rigorous population-level justification (Bhattacharjee et al., 29 Mar 2025).
Attention-Weighted SHAP: In deep models with attention mechanisms (e.g., CNN-TFT for time series), SHAP values are modulated (“multiplied”) by normalized attention scores to yield composite column weights, integrating model explainability with intrinsic allocation of model capacity (Stefenon et al., 8 Oct 2025).
Interaction SHAP (SHAP-IQ): For capturing feature interactions, sampling-based estimators (e.g., SHAP-IQ) generalize the computation to any-order Shapley indices, encompassing the standard feature-attribution as a special case (Fumagalli et al., 2023).

3. Applications and Empirical Results

3.1 Clustering and Unsupervised Feature Selection

In unsupervised clustering, SHAP-based feature weighting improves clustering quality by up to 22.7% in ARI compared to unweighted data or other weighting schemes across benchmark datasets (Iris, Wine, Breast Cancer, Digits, Vehicle), as demonstrated in (Galis et al., 12 Mar 2025). The pipeline requires only an initial cluster assignment, surrogate classifier fitting, and TreeExplainer SHAP computation; the resulting weights can be directly injected into any clustering algorithm.

Key results from (Galis et al., 12 Mar 2025) include:

Dataset	Algorithm	ARI_unweighted	ARI_SHAP	Improvement
Iris	HDBSCAN	0.564	0.568	+0.7%
Wine	GMM	0.897	0.947	+5.6%
Breast Cancer	Ward	0.586	0.719	+22.7%
Digits	Ward	0.664	0.700	+5.4%
Vehicle	k-means	0.075	0.096	+28.0%

3.2 Supervised Feature Selection and Trustworthy Global Rankings

Global SHAP-based column weights are widely utilized in supervised pipelines for model auditing, variable selection, dashboarding, and interpretation. However, recent theoretical developments have shown that for provably sound feature elimination, aggregation must be performed over the extended distribution, ensuring detection of any nontrivial functional dependence (Bhattacharjee et al., 29 Mar 2025). This principle underpins robust feature selection in high-stakes domains such as medicine, biology, and scientific modeling.

3.3 Ranking Systems and Parameter Analysis

SHAP values generalize to the analysis of column-weight parameters in ranking functions (SUM, MAX, MIN, LEX). This setting involves computing the Shapley value of a parameter in a score aggregation or ranking pipeline, quantifying the expected effect of perturbing a weight under a specified distribution. This enables audits of informativeness and fairness in information retrieval or multi-criteria ranking systems (Standke et al., 9 Jan 2026).

The computational complexity of calculating SHAP scores for column weights in ranking depends heavily on function class and “effect function”; some combinations admit polynomial-time algorithms, while others are #P-hard.

4. Computational Complexity and Approximation

The exact computation of SHAP scores or their derived column weights is, in general, exponentially expensive in the number of features due to the summation over $2^d$ subsets. For tree-based models, TreeExplainer achieves polynomial-time computation leveraging model structure (Galis et al., 12 Mar 2025). For general models, several scalable approximation methods are standard:

KernelSHAP: Weighted least-squares on a sampled collection of coalitions, guaranteeing unbiasedness in the limit and calibration to the Shapley value system of equations (Herren et al., 2022, Fumagalli et al., 2023).
SHAP-IQ: Unified unbiased Monte Carlo estimators for any-order interaction index, with explicit variance control and convergence guarantees (Fumagalli et al., 2023).
Fully Polynomial-Time Randomized Approximation Scheme (FPRAS): For SHAP in ranking-based parameter analysis, an additive FPRAS is always available via Monte Carlo sampling, even when exact computation is #P-hard (Standke et al., 9 Jan 2026, Letoffe et al., 2024).
Complexity results: For basic ranking/parameter SHAP (MAX, LEX ranking with global/local effect, or MAX-desc with top- $k$ ), exact computation is polynomial; for SUM ranking with binary-encoded numbers and most top- $k$ /MD/Hamming effects, the computation is #P-hard (Standke et al., 9 Jan 2026).

5. Axiomatic and Robustness Properties

SHAP scores, when applied to column weighting, partially inherit classical properties: efficiency (sum equals model output minus baseline), symmetry (features of identical effect have equal score), dummy (irrelevant features receive zero), and additivity.

Recent developments strengthen robustness and theoretical compliance:

Feature relevancy compliance: Axiomatic modifications ensure true irrelevance (as in logical or abductive explanations) yields exactly zero weight (Letoffe et al., 2024, Letoffe et al., 2024).
Permutation invariance: Column weights derived via similarity-based characteristic functions or coverage-based indices become independent of label values or arbitrary output relabelling (Letoffe et al., 2024).
Class independence and duality: Certain new indices guarantee that abduction-based and contrastive-based scores yield consistent feature rankings (Letoffe et al., 2024).
Soundness theorems for feature removal: When averaged over the extended distribution, the mean absolute SHAP value for a column is zero if and only if the model output is identical (almost everywhere) when that column is removed, yielding a rigorous criterion for feature elimination (Bhattacharjee et al., 29 Mar 2025).

6. Practical Considerations and Recommendations

For clustering and unsupervised tasks, a single SHAP-based column weighting step offers statistically significant separation improvements, requiring only an initial pseudo-labeling, surrogate training, and SHAP aggregation (Galis et al., 12 Mar 2025).
In supervised and post-hoc model auditing, practitioners are advised to perform column-wise permutation before SHAP-aggregation for robust feature selection (Bhattacharjee et al., 29 Mar 2025).
Tree-based surrogates and sampling-based SHAP algorithms (e.g., KernelSHAP, SHAP-IQ) offer the most computationally practical solutions for high-dimensional data or large models (Fumagalli et al., 2023).
When strict feature relevancy compliance, class-label independence, or dual explanation needs arise, preference should be given to formal-axiomatic SHAP variations (Formal-SHAP, coverage FIS) (Letoffe et al., 2024, Letoffe et al., 2024).
In attention-based neural architectures, combining SHAP with normalized attention produces interpretable, composite column weights intertwining model-attribution and model-focus mechanisms (Stefenon et al., 8 Oct 2025).

7. Open Issues and Future Directions

Scalability under combinatorial explosion: Efficient scalable computation of higher-order interaction SHAP scores and SHAP values for large models, possibly via structural model compression or hybrid surrogates, remains an open area (Fumagalli et al., 2023).
Feature interaction and redundancy: Distinguishing between true functional interaction, redundancy, and collinearity in SHAP-based weighting is not fully resolved.
Axiomatic generalization: Recent research proposes new formally justified feature-importance indices in the SHAP family, including Banzhaf-style, Deegan–Packel, and coverage based, whose downstream impact and comparative behavior require further empirical and theoretical investigation (Letoffe et al., 2024).
Integration with causal and counterfactual analysis: Extending column weight interpretations from model-centric to data-generating process-centric (and dealing robustly with non-i.i.d. structure) is an active frontier.
Ranking-specific parameter SHAP: For complex ranking and preference aggregation functions, deeper connections between coalitional effect, fairness, and interpretability via SHAP column weights are expected to be elaborated in future literature (Standke et al., 9 Jan 2026).

The use of SHAP scores for column weights continues to expand across machine learning, AI explainability, and decision analytics, underpinned by rigorous theory, scalable approximation, and empirically validated gains in practical downstream tasks.