Feature Attribution Methods

• Feature Attribution Methods are algorithmic procedures that assign quantifiable influence scores to individual input features on a model's output.
• They incorporate methodologies such as gradient-based, game-theoretic, and density estimation techniques to ensure robust and interpretable attributions.
• These methods enhance model transparency by measuring contribution, ensuring fidelity, and addressing challenges like feature interactions and computational tractability.

A feature attribution method is any algorithmic procedure that assigns to the individual input features (such as pixels, words, or variables) of a model instance a quantification of their influence on the model’s output. The magnitude of each attribution score reflects, per a relevant formalism (gradient, perturbation, game-theoretic index, distributional contrast, etc.), the extent to which the corresponding feature contributes to the predicted outcome. Recent advances have addressed foundational questions regarding mathematical definitions, evaluation frameworks, geometric and distributional underpinnings, and the robustness, specificity, and tractability of attributions in complex machine learning models.

1. Mathematical Foundations and Formal Definitions

Contemporary feature attribution begins with rigorous formalizations rooted in probability, information theory, and cooperative game theory. For an instance $x \in \mathbb{R}^d$ and a model $f: \mathbb{R}^d \to \mathbb{R}^K$, an attribution method $A(x, f) \in \mathbb{R}^d$ returns a vector whose $i$th entry quantifies the influence of $x_i$ on $f(x)$. Definitional desiderata include:

• Completeness: The sum of attributions over a set of "true" predictive features $F_C$ should approach unity if all model signal passes through $F_C$: $$\text{Attr}\%(F_C) = \sum_{i\in F_C} |A_i(x, f)| / \sum_{i=1}^d |A_i(x, f)| \approx 1$$.
• Irrelevance: Features independent of the output (set $F_N$) receive vanishing attribution as the model accuracy rises: $\text{Attr}\%(F_N)\to 0$.
• Structural Axioms: Properties such as monotonicity (adding features cannot reduce functional dependence), and complementary dependence (attributions for a fixed feature subset are stable over all inputs that share those features) (Afchar et al., 2021).

Cooperative game theory yields foundational indices such as the Shapley value, defined for feature $a$ by: $$\phi(a) = \sum_{S \subseteq N \setminus \{a\}} \frac{|S|! (n - |S| - 1)!}{n!} \left( f(S \cup \{a\}) - f(S) \right)$$ where $N$ is the feature set and $f(S)$ denotes the model's output under the coalition $S$, frequently instantiated by replacing features not in $S$ with a baseline or via expected values (Barceló et al., 4 Jan 2025, Jiang et al., 2023).

Recent formalizations demand that explanations be rooted in the data-generating distribution $\mathbb{P}$, prohibiting synthetic or unsupported perturbations and ensuring that attributions are empirically meaningful (Li et al., 12 Nov 2025).

2. Methodological Approaches

Feature attribution methods are typically categorized by the underlying mechanism:

• Gradient-based: Compute $\partial f / \partial x_i$ or its smoothed/integrated variants, requiring differentiable models. Integrated Gradients (IG) computes line-integrals from a baseline $x'$ to the input: $$\mathrm{IG}_i(x) = \int_0^1 \frac{\partial f(x' + t(x-x'))}{\partial x_i} (x_i - x'_i)\,dt$$ (Zaher et al., 16 May 2024, Zhuo et al., 16 Jun 2024).
• Path-based generalizations: Manifold Integrated Gradients (MIG) replace linear interpolation with on-manifold geodesics, accumulated via Riemannian geometry on data manifolds, mitigating off-manifold noise and adversarial manipulation (Zaher et al., 16 May 2024). IG² further adapts the integration path and counterfactual baseline by iterative gradient-guided steps towards a reference, aligning both path and endpoint with model-specific decision boundaries (Zhuo et al., 16 Jun 2024).
• Distributional and density-based: Methods such as DFAX define attribution by contrasting empirical one-dimensional kernel density estimates for each feature in the predicted class and others, thus quantifying how characteristic a feature value is for the decision relative to the data distribution (Li et al., 12 Nov 2025).
• Game-theoretic: In addition to the classical Shapley value, modern frameworks explore the full space of Weighted Möbius Scores, which provide linear combinations of Harsanyi dividends (pure interactions of feature sets) and capture both single-feature and higher-order attributions (Jiang et al., 2023).
• Learning submodular and ensemble-based attributions: Algorithms such as SEA-NN learn monotone submodular set functions from collections of attribution maps, encoding diminishing returns and increasing specificity by computing marginal gains under the learned scoring function (Manupriya et al., 2021).
• Generative and counterfactual models: Generative adversarial approaches (e.g., VA-GAN) learn to produce maps that morph a target-class instance into the closest baseline-class sample, directly optimizing for realism and completeness of the attributed region (Baumgartner et al., 2017). Instance-level counterfactuals for anomaly attribution in climate models or outlier detection perturb features to reference values and quantify reconstruction loss changes as attribution scores (Ale et al., 11 Feb 2025, Shen et al., 2023).
• Contextual and argumentation-based: Emerging frameworks incorporate context-aware user modeling, treating each feature as a supporting or attacking argument in a tripolar argumentation graph, integrating explicit user, context, and feature interactions for transparent, interpretable attributions (Zhong et al., 2023).

3. Algorithmic Procedures and Computational Tractability

Algorithmic construction varies by approach but generally involves:

• Gradient path discretization: IG/MIG/IG² approaches compute grids of points along a chosen path and accumulate gradients, requiring as many backpropagations as grid steps. For MIG, manifold geodesics are numerically estimated by minimizing discrete path energy in latent space, and decoded to data space using deep generative models (Zaher et al., 16 May 2024).
• Density estimation: DFAX preprocesses class-conditional and global kernel mean maps for each feature, allowing rapid attributions using inner products with feature embeddings—scaling linearly with dimensionality and embedding size (Li et al., 12 Nov 2025).
• Sampling and combinatorics: Game-theoretic schemes require exponentially many model calls for exact computation but often admit efficient Monte Carlo approximations or tractable closed forms for special cases (e.g. Bernoulli or Banzhaf indices), reducing to a constant or polynomial number of expectation evaluations under certain product distributions (Barceló et al., 4 Jan 2025).
• Learning submodular functions: Submodular ensemble attributions train a neural network scoring function to fit input maps and respect cardinality constraints, then use greedy rank marginal gains for final attribution (Manupriya et al., 2021).
• Counterfactual perturbations: For each feature, reconstruct the model’s output after replacing the feature with a reference (median, baseline), and attribute the resulting loss change to that feature (Ale et al., 11 Feb 2025, Shen et al., 2023).

Tractability crucially depends on the model’s structure and the class of indices considered. For model classes closed under product mixtures, the complexity of computing all simple power indices is polynomially equivalent to the feasibility of expected value computation (Barceló et al., 4 Jan 2025).

4. Evaluation Metrics, Robustness, and Human Factors

Evaluation of feature attribution methods leverages both automatic and human-centered metrics:

• Fidelity metrics: Deletion/insertion curves, faithfulness (output drop when high-attribution features are removed), and infidelity measure how well the attributions align with the model’s actual predictive dependencies (Li et al., 12 Nov 2025, Zaher et al., 16 May 2024, Zhuo et al., 16 Jun 2024, Liu et al., 2 Apr 2025).
• Robustness metrics: Sensitivity, adversarial attributional attacks, and Output Similarity-based Robustness (OSR) assess how stable attributions are with respect to small changes or similar inputs as judged by the model’s own output distribution; high OSR corresponds to stable, reliable explanations (Kiourti et al., 7 Dec 2025).
• Specificity and redundancy reduction: Submodular function learning reduces overlap and improves the sharpness of heatmaps; marginal gains under the learned function attribute less to redundant features (Manupriya et al., 2021).
• Human subject studies: Direct user testing on real tasks (e.g., classification with and without attributions, fine-grained or adversarial settings) often reveals that attribution maps do not reliably improve, and can even degrade, human decision-making—sometimes less effective than nearest-neighbor prototypes (Nguyen et al., 2021). Correlations between automatic proxy metrics and human utility are weak, highlighting the need for redesigned evaluation strategies.
• Ground-truth benchmarking: Synthetic data with injected ground-truth signals, label reassignment, and manipulation allow measurement of completeness and exclusion, precision/recall, and compliance with Shapley axioms, surfacing systematic failure modes in saliency, attention, and rationale-based methods (Zhou et al., 2021, Afchar et al., 2021).

Metric/Property	Attribution Goal	Papers
Completeness, Correctness	Target feature focus	(Zhou et al., 2021, Afchar et al., 2021)
Faithfulness, Infidelity, Deletion	Output drop/robustness	(Li et al., 12 Nov 2025, Zaher et al., 16 May 2024)
Robustness, OSR	Stability to input	(Kiourti et al., 7 Dec 2025, Zaher et al., 16 May 2024)
Specificity, Non-redundancy	Focus, pruning	(Manupriya et al., 2021)
Fidelity to human users	Human-AI task utility	(Nguyen et al., 2021)

5. Extensions, Open Challenges, and Future Directions

Major open directions and limitations recognized in the literature:

• Distributional and geometric improvements: MIG shows that enforcing on-manifold geodesic paths denoises attributions and greatly reduces adversarial vulnerability, but quality is sensitive to generative model fit and may require domain-specific latent spaces for non-Euclidean or hierarchical data (Zaher et al., 16 May 2024).
• Feature interactions and higher-order effects: Weighted Möbius Score, Shapley-Taylor indices, and submodular models generalize attribution to interaction and redundancy structures between features, with associated sampling and computational trade-offs (Jiang et al., 2023, Manupriya et al., 2021).
• Unsupervised and anomaly settings: Counterfactual-based methods identify anomaly sources without labels, and inverse-multiscale occlusion yields interpretable attributions for OOD outliers (Ale et al., 11 Feb 2025, Shen et al., 2023).
• Context and user-specificity: Argumentation-based methods enable transparent, context-aware explanations, but typically model only additive effects and rely on explicit item attributes and context factors (Zhong et al., 2023).
• Tractability and approximations: The complexity of exact attribution is exponential in the number of features in the general case; restricted power indices and approximate sampling mitigate this but may omit high-order interactions or fine structure (Barceló et al., 4 Jan 2025).
• Practical guidance: Practitioners are urged to exhaustively benchmark attributions on synthetic data with ground truth, explicitly validate structural axioms, and optimally select hyperparameters (e.g., reference points, integration steps). In sensitive domains, human-in-the-loop evaluation and caution regarding possible misleading or spurious explanations are essential (Zhou et al., 2021, Nguyen et al., 2021).

6. Representative Implementations and Practical Considerations

• Computational demands: Path-based methods (IG/IG²/MIG) require tens to hundreds of integration steps per attribution; geodesic optimization in MIG and gradient descent in IG²/GradPath add significant per-instance overhead. Efficient manifold learning (VAE, flow) and vectorization (GPU batch processing) are critical (Zaher et al., 16 May 2024, Zhuo et al., 16 Jun 2024).
• Baselines and references: The selection and construction of baselines (zero-input, counterfactuals, class prototypes) strongly impact the meaningfulness of attributions; manifold-respecting or counterfactual-driven baselines are preferred for stability and interpretability (Zaher et al., 16 May 2024, Zhuo et al., 16 Jun 2024, Baumgartner et al., 2017).
• Ensembling and aggregation: Submodular ensembling, density-based smoothing, and contextual aggregation (LAFA) can substantially reduce noise, improve faithfulness, and recover rare but relevant features, particularly in text and NLP settings (Zhang et al., 2022, Manupriya et al., 2021).
• Hyperparameter tuning: For methods relying on kernels, step sizes, or number of integration steps, cross-validation on held-out data is generally necessary for robustness (Li et al., 12 Nov 2025, Manupriya et al., 2021, Zaher et al., 16 May 2024).

7. Summary

The field of feature attribution has advanced from heuristic saliency maps to theoretically grounded, distributionally robust, and even context-sensitive frameworks leveraging geometry, density estimation, game theory, and generative modeling. Contemporary methods not only provide finer faithfulness and specificity but also exhibit improved resistance to adversarial perturbations, more principled human-alignment testing, and greater computational tractability in relevant settings. However, challenges persist: accurate modeling of interactions, generalization to non-tabular modalities, real-world user effectiveness, and the reconciliation of fidelity, stability, and interpretability remain active research frontiers. These ongoing developments are documented across the technical literature (Zaher et al., 16 May 2024, Li et al., 12 Nov 2025, Zhuo et al., 16 Jun 2024, Zhou et al., 2021, Nguyen et al., 2021, Manupriya et al., 2021, Jiang et al., 2023, Barceló et al., 4 Jan 2025, Kiourti et al., 7 Dec 2025, Liu et al., 2 Apr 2025, Zhong et al., 2023, Shen et al., 2023, Ale et al., 11 Feb 2025, Zhang et al., 2022, Baumgartner et al., 2017).