Path Methods

The core mechanism involves defining a continuous, differentiable path ( \gamma(\alpha) ) from a baseline input ( x' ) to the target input ( x ). The most common path is a straight line: ( \gamma(\alpha) = x' + \alpha (x - x') ). The attribution is the path integral of the model's gradient along this trajectory. The choice of path is not arbitrary; it fundamentally determines the attribution result. A non-straight path can encode specific priors about how features should be compared, but the straight-line path is the default because it preserves the linearity axiom and provides a symmetric transition between the baseline and the input.

Path methods are explicitly designed to satisfy the Completeness Axiom, meaning the sum of all feature attributions exactly equals the difference between the model's output for the input and the baseline: ( \sum_i A_i = F(x) - F(x') ). This is a non-negotiable property for financial auditing and regulatory compliance. They also guarantee Implementation Invariance, ensuring that two functionally identical networks yield identical attributions regardless of internal architecture. This is a critical failure point for simpler methods like Guided Backpropagation, which break this axiom.

The baseline represents the 'absence' of information and is the most critical hyperparameter. A black image is a common baseline for vision tasks, while a zero-embedding vector is used for text. The attribution explains the change from this baseline to the input. A poorly chosen baseline leads to counterfactual explanations that are meaningless. For instance, using a white image as a baseline for object recognition would attribute importance to dark pixels rather than object-specific features. The baseline must encode a neutral, information-free state for the domain.

The canonical path method that integrates gradients along a straight-line path. IG is defined as: ( (x_i - x'i) \int{\alpha=0}^{1} \frac{\partial F(x' + \alpha (x - x'))}{\partial x_i} d\alpha ). In practice, the integral is approximated by a Riemann sum over ( m ) discrete steps. The number of steps is a critical accuracy-efficiency trade-off; too few steps (e.g., < 50) can cause the approximation to violate the completeness axiom due to gradient saturation in the saturated regions of the network's non-linearities.

A probabilistic extension that removes the dependency on a single, arbitrary baseline. Instead of integrating from one baseline, it computes the expected value of Integrated Gradients over a distribution of baselines ( D ). This is defined as: ( \int_{x'} IG(x, x') \cdot p_D(x') dx' ). By averaging over many background samples from the training data, Expected Gradients unifies the path method framework with the SHAP value framework, providing attributions that are both axiomatically sound and grounded in the data manifold.

A fundamental limitation that path methods are designed to solve. In deep networks, the gradient ( \partial F / \partial x_i ) can approach zero for features that strongly activate the correct class. A raw gradient at the input point would incorrectly assign zero importance to these critical features. By integrating along a path from a baseline where the gradient is non-zero, path methods accumulate the signal before it saturates. This is why Gradient × Input fails for deep classifiers while Integrated Gradients succeeds.

Path methods satisfy the completeness axiom, meaning the sum of all feature attributions exactly equals the prediction difference from a baseline. This mathematical property is critical for ECOA and EU AI Act compliance, where auditors require a full, auditable decomposition of every automated decision. Financial institutions use Integrated Gradients to generate adverse action reasons for credit denials that are both exhaustive and legally defensible.

Radiologists require pixel-level justification for AI-assisted diagnoses. Path methods applied to 3D CT scans and histopathology slides provide high-fidelity attribution maps that highlight the exact tissue regions influencing a tumor classification.

• Integrated Gradients avoids the gradient saturation problem common in simple saliency maps
• Attributions are computed relative to a black-image or blurred baseline, isolating pathological features
• Enables human-in-the-loop verification before clinical intervention

When a transformer model misclassifies a document, path methods decompose the prediction across token embeddings. By integrating gradients from a zero-embedding baseline to the final token representation, engineers can identify which words or subwords drove the error.

• Detects spurious correlations where models latch onto formatting tokens instead of semantic content
• Used to validate that sentiment models rely on adjectives, not author demographics
• Provides token-level conductance scores for attention head analysis

In financial crime investigations, analysts must understand exactly which transaction attributes triggered a fraud alert. Path methods provide instance-level explanations that decompose the anomaly score across features like transaction amount, location, time, and merchant category.

• Baseline is typically a 'normal' transaction profile
• Attributions reveal whether the model flagged velocity patterns or amount deviations
• Supports SAR (Suspicious Activity Report) documentation with quantitative evidence

For ISO 26262 functional safety certification, perception models must provide interpretable evidence for object detection decisions. Path methods applied to LiDAR point clouds and camera inputs generate 3D saliency maps showing which sensor returns influenced a pedestrian detection.

• Baseline is an empty scene or uniform noise field
• Enables failure mode analysis when models miss occluded objects
• Integrates with safety case documentation for regulatory submission

Medicinal chemists use path methods to understand which molecular substructures drive a predicted binding affinity. By integrating gradients from a null pharmacophore baseline to the candidate molecule, Integrated Gradients highlights critical functional groups.

• Guides structure-activity relationship (SAR) analysis
• Identifies which atoms contribute positively vs. negatively to predicted potency
• Accelerates lead optimization by focusing synthetic efforts on high-attribution moieties