Completeness Axiom

A formal framework for evaluating feature importance methods against a set of mathematical axioms. Completeness is the primary axiom, ensuring the sum of attributions equals the output difference from a baseline. Other key axioms include Sensitivity-n (zero attribution for irrelevant features) and Implementation Invariance (identical attributions for functionally equivalent networks). Methods like Integrated Gradients and DeepLIFT are explicitly designed to satisfy these axioms, providing theoretically grounded explanations.

A path method that satisfies the Completeness Axiom by design. It computes feature importance by accumulating the gradients of the model's output along a straight-line path from a baseline input (e.g., a black image) to the actual input. The integral of the gradients exactly accounts for the total change in output, guaranteeing completeness. This method is a cornerstone of the axiomatic approach and is widely used for its theoretical rigor.

An attribution method that explains predictions by comparing the activation of each neuron to a reference activation and assigning contribution scores based on the difference. DeepLIFT satisfies the Completeness Axiom through its Summation-to-Delta property, which ensures the sum of all neuron contributions equals the difference in the model's output. It uses a linear approximation via multipliers, avoiding the gradient saturation problem that plagues simple gradient-based methods.

An attribution method that satisfies the Completeness Axiom by decomposing the model's output into the contributions of both the input features and the intermediate network biases. Standard input-gradient methods ignore bias terms, leading to an incomplete decomposition. FullGrad aggregates the input-gradient with the gradients of all bias parameters across every layer, providing a mathematically complete and faithful attribution of the network's entire output.

A decomposition technique that redistributes a model's prediction score backwards through the network using a conservation property. LRP defines specific propagation rules for each layer type that ensure the total relevance is preserved from the output layer down to the input features. This conservation directly satisfies the Completeness Axiom, as the sum of the final input relevances exactly equals the model's output score for the target class.

A metric that quantifies how faithfully an attribution method satisfies the Completeness Axiom in practice. It measures the mean squared error between the actual change in the model's output (when a significant perturbation is applied) and the dot product of the attribution with that perturbation. A low infidelity score indicates that the attributions accurately predict the model's behavior under perturbations, validating the completeness property empirically.