Infidelity Measure

Infidelity is formally defined as the expected squared difference between two quantities when a significant perturbation I is applied to the input:

• Term 1: The actual change in the model's output: f(x) - f(x - I)
• Term 2: The dot product of the attribution map Φ with the perturbation: Φ(x)ᵀ · I

A lower infidelity score indicates that the attribution method more accurately predicts how the model's output responds to meaningful input changes. The metric is computed over a distribution of perturbations, not just a single instance.

Unlike random noise, infidelity perturbations must be semantically significant to expose attribution weaknesses:

• Gaussian noise is insufficient because it explores trivial, off-manifold directions the model never encounters
• Meaningful perturbations include blurring, occlusion by superpixels, or interpolation toward a baseline image
• The perturbation distribution defines what 'faithfulness' means in practice—a method faithful under one perturbation set may fail under another

This design choice directly tests whether the explanation captures the model's behavior along data-manifold directions rather than arbitrary vector spaces.

Infidelity generalizes the Completeness Axiom from axiomatic attribution frameworks:

• Completeness requires that attributions sum to the output difference f(x) - f(baseline) for a single baseline
• Infidelity measures this property across an entire distribution of baselines and perturbations simultaneously
• An attribution method with zero infidelity perfectly satisfies completeness for all perturbations in the test set

This makes infidelity a more rigorous, empirical complement to the theoretical completeness guarantee provided by methods like Integrated Gradients.

For a given perturbation distribution, there exists an optimal attribution map that minimizes infidelity:

• The optimal attribution equals the expected gradient of the model integrated over the perturbation distribution
• This connects infidelity directly to Expected Gradients, which averages Integrated Gradients over a background dataset
• Methods like SmoothGrad and VarGrad can be understood as approximations to this optimal attribution under specific perturbation choices

This theoretical link provides a principled way to design new attribution methods by explicitly specifying the perturbation distribution they optimize for.

Infidelity is closely related to the sensitivity of an explanation to small input changes:

• A fragile attribution map that changes dramatically under tiny perturbations will exhibit high infidelity
• The metric penalizes explanations that rely on shattered gradients—noisy, high-frequency patterns that don't reflect true model behavior
• By averaging over perturbations, infidelity naturally smooths out gradient noise, favoring methods like SmoothGrad and Integrated Gradients over raw Gradient × Input

This property makes infidelity a practical tool for comparing the robustness of different attribution techniques.

Infidelity serves as a standardized benchmark for comparing explanation techniques:

• Integrated Gradients typically achieves low infidelity due to its axiomatic completeness property
• Gradient × Input often scores poorly because it captures only first-order effects and suffers from gradient saturation
• Guided Backpropagation may exhibit high infidelity as it violates the completeness axiom by filtering negative contributions

When reporting infidelity scores, always specify the perturbation distribution, number of samples, and baseline choice to ensure reproducibility across studies.

An axiomatic attribution principle requiring that if a model's output is mathematically independent of a feature, that feature must receive an attribution of zero.

• Directly counters a source of infidelity: if a perturbation to a zero-importance feature changes the explanation, the method is unfaithful.
• A method violating Sensitivity-n will exhibit high infidelity when that irrelevant feature is perturbed.
• Core component of the axiomatic framework used to evaluate attribution quality.

A robustness metric that measures the maximum change in an explanation under small, adversarial input perturbations.

• Quantifies the stability of a saliency map, which is closely related to infidelity: an unstable explanation is highly unfaithful under minor noise.
• A high Local Lipschitz Estimate indicates that the dot product between the attribution and a perturbation vector will vary wildly, directly increasing the Infidelity Measure.
• Used to certify that explanations do not shatter under imperceptible input changes.

A phenomenon where the gradient of a deep network with respect to its input resembles white noise, providing no visually coherent saliency map.

• A classic symptom of high infidelity: the raw gradient is a poor explanation because it fails to capture the true decision boundary.
• Occurs due to the highly non-linear loss surface, making the local gradient a noisy and unfaithful approximation of the model's global behavior.
• Techniques like SmoothGrad and Integrated Gradients are designed specifically to address this failure mode.

A regularization term added to a model's training objective that encodes a desired property of the feature attributions, such as smoothness or sparsity.

• Trains models to be inherently more interpretable and faithful, rather than just explaining a black box post-hoc.
• By penalizing high infidelity during training, the model learns to have a simpler, more linear decision boundary where the gradient is a faithful explanation.
• Examples include penalizing the variance of gradients or encouraging attributions to be concentrated on a few features.

An attribution method that computes feature importance by averaging the gradients of the model's output over a distribution of background samples.

• Unifies Integrated Gradients and SHAP to reduce the variance and noise that contribute to infidelity.
• By integrating over many reference points, it avoids the pathological failure where a single baseline path yields a misleading, high-infidelity explanation.
• Provides a more robust and faithful attribution by accounting for the model's behavior across the entire data manifold.

A technique that finds the smallest smooth mask over an input image that maximally preserves a model's prediction.

• Directly addresses the perturbation component of the Infidelity Measure by searching for the most impactful region to modify.
• Instead of measuring error from random perturbations, it identifies the minimal sufficient explanation, providing a complementary view of faithfulness.
• The resulting mask is a high-fidelity saliency map that highlights only the pixels absolutely necessary for the prediction.