Axiomatic Attribution

The sum of all feature attributions must exactly equal the difference between the model's output for the input and a baseline. This ensures no importance is created or destroyed during the explanation process.

• Mathematical Form: Σᵢ ϕᵢ(f, x) = f(x) - f(x')
• Practical Impact: Guarantees a full accounting of the prediction. If the model output is 0.8 and the baseline is 0.0, the attributions must sum to 0.8.
• Violation Example: A method that only highlights the top 3 pixels but ignores background context fails this axiom.

If an input differs from a baseline in exactly one feature, and that difference causes the model's output to change, that feature must receive a non-zero attribution.

• Also Known As: The 'Dummy' property.
• Contrast with Implementation Invariance: Sensitivity cares about the mathematical function, not the network weights.
• Failure Mode: A constant function that ignores the input would give zero attribution to the differing feature, violating this axiom.

Two functionally equivalent models must always produce identical attributions, regardless of their internal architecture or parameterization.

• Definition: If f₁(x) = f₂(x) for all inputs x, then ϕ(f₁, x) = ϕ(f₂, x).
• Critical Distinction: This axiom is violated by methods that rely on network weights or gradients, as two different networks can compute the exact same function.
• Example: A 2-layer ReLU net and a 3-layer ReLU net computing the same linear function must yield the same explanation.

If a model is a linear combination of two sub-models, the attribution for the combined model must be the same linear combination of the individual attributions.

• Mathematical Form: ϕ(af₁ + bf₂, x) = aϕ(f₁, x) + bϕ(f₂, x).
• Ensemble Justification: This axiom is crucial for explaining ensemble methods. The explanation of a random forest should be the average of the explanations of its trees.
• Path Methods: Integrated Gradients uniquely satisfies this among path-based methods.

Two features that play exactly the same role in the model's mathematical function must receive identical attribution scores.

• Formal Definition: If swapping two features does not change the function output, their attributions must be equal.
• Fairness Guarantee: Prevents the explanation method from arbitrarily favoring one input dimension over another when the model treats them identically.
• Game Theory Origin: Derived directly from the symmetry axiom of Shapley values.

A stronger form of sensitivity requiring that if a feature's contribution is strictly larger in one setting than another, the attribution must reflect this ordering.

• Monotonicity: The attribution method must preserve the ordinal ranking of feature importance across different inputs or baselines.
• Practical Relevance: Ensures that if a pixel becomes more critical to the prediction, its attribution score must increase, not decrease.
• Axiomatic Uniqueness: Integrated Gradients is the unique path method satisfying this property.

Also known as Summation-to-Delta, this property requires that the sum of all feature attributions equals the difference between the model's output for the input and a baseline. It ensures that the prediction is fully accounted for.

• Violation Example: Methods that normalize attributions post-hoc often break completeness.
• Satisfying Methods: Integrated Gradients, DeepLIFT, and SHAP are designed to inherently satisfy this axiom.

If an input feature differs from a baseline and changes the prediction, it must receive a non-zero attribution. Conversely, if the feature is mathematically identical to the baseline, it receives zero attribution.

• Purpose: Prevents irrelevant features from being highlighted.
• Contrast: Standard Gradient*Saliency violates this when gradients saturate, missing features that are critical but have a near-zero local gradient.

Two functionally equivalent models—networks that always produce the same output for every possible input—must yield identical attributions, regardless of their internal architecture.

• Key Insight: This axiom rejects methods that rely on inspecting internal weights or activations, such as Guided Backpropagation.
• Practical Test: A deep network and its functionally equivalent wider variant must produce the same explanation map.

Linearity requires that the attribution for a linearly composed model is the weighted sum of attributions from its components. Symmetry (or Dummy) dictates that two features playing the exact same role in the model receive identical attributions.

• Game Theory Link: These directly mirror the Shapley axioms of additivity and equal treatment of equals.
• Failure Mode: Random baselines in perturbation methods often violate symmetry by assigning different scores to identical features due to noise.