Layer-wise Relevance Propagation

The total relevance score assigned to the input vector must exactly equal the model's output prediction score. This is the completeness axiom in action. No relevance is created or destroyed during backpropagation; it is merely redistributed.

• Sum Rule: (\sum_i R_i^{(l)} = \sum_j R_j^{(l+1)} = f(x))
• This ensures the explanation is a true decomposition of the prediction, not a heuristic approximation.
• Violating conservation leads to 'leakage' where the explanation fails to account for the full output magnitude.

LRP rules are designed to ensure that neurons contributing positively to the target class receive positive relevance. This avoids the gradient shattering problem seen in raw gradients.

• z⁺-Rule: Only positive contributions are propagated, ignoring inhibitory connections.
• This property is critical for generating clean, non-noisy heatmaps where only supporting evidence is highlighted.
• It prevents negative relevance from canceling out positive evidence in different regions of the input.

LRP can be mathematically derived as a series of Deep Taylor Decompositions applied at each layer. This provides a theoretical justification for the propagation rules.

• Each neuron's output is decomposed using a first-order Taylor expansion around a root point in the input space.
• The choice of root point (e.g., nearest root, rescaled root) directly determines the specific LRP rule (e.g., (\alpha\beta), (w^2)).
• This connects heuristic propagation rules to a rigorous functional decomposition framework.

Relevance is conserved locally between adjacent layers, not just globally between input and output. This layer-wise conservation is a stricter constraint than simple input-output conservation.

• (\sum_i R_i^{(l)} = \sum_j R_j^{(l+1)}) for all layers (l).
• This prevents intermediate layers from acting as 'relevance sinks' or 'relevance sources'.
• It ensures that the attribution is stable and interpretable at every stage of the network's hierarchy.

LRP is not a single algorithm but a composable framework of propagation rules. Different layers can use different rules to optimize for stability and interpretability.

• Deep Taylor LRP: Uses (\alpha\beta)-rules for hidden layers and (z^+)-rules for the final layer.
• Composite LRP: Applies (LRP-\epsilon) for flat layers and (LRP-\alpha\beta) for convolutional layers.
• This composability allows practitioners to tailor the explanation to the specific architecture, avoiding numerical instability in deep networks.

Unlike gradient-based methods, LRP does not treat the model's bias terms as contributing to relevance. The bias is considered a baseline shift that does not explain the input's specific contribution.

• Biases are explicitly excluded from the relevance propagation formula.
• This prevents a constant offset from being distributed across the input, which would dilute the explanation of the actual input features.
• The result is a sparser, more focused attribution that highlights the true signal in the data.

The theoretical framework defining the mathematical properties a feature attribution method must satisfy to be considered reliable. LRP is explicitly designed to satisfy the conservation property (completeness), ensuring the sum of relevance scores across all input features equals the model's total output score. Other key axioms include sensitivity (a feature receiving zero attribution must not affect the prediction) and implementation invariance (functionally equivalent networks must yield identical attributions).

The mathematical framework underpinning the most common LRP rules. It propagates relevance by performing a first-order Taylor expansion of the neuron's function at a carefully chosen root point in the input space. This provides a principled way to derive propagation rules for different layer types:

• LRP-α1β0: Emphasizes positive contributions
• LRP-ε: Absorbs weak or contradictory relevance with a small stabilizer
• LRP-γ: Favors positive over negative contributions

An axiomatic attribution method that computes feature importance by accumulating gradients along a straight-line path from a baseline input (typically a black image or zero vector) to the actual input. Unlike LRP, which uses a backward pass with modified gradient rules, Integrated Gradients relies on standard gradients and satisfies the completeness axiom through path integration. It requires careful baseline selection, whereas LRP's reference point is implicitly defined by the propagation rules.

Quantitative evaluation criteria that measure how accurately an attribution map reflects true feature importance. The core principle is perturbation-based evaluation:

• Deletion Metric: Remove pixels from most to least important and measure prediction decay
• Insertion Metric: Add pixels from most to least important into a blurred baseline and measure probability increase
• ROAR (RemOve And Retrain): Iteratively retrain the model after removing top-attributed features to assess if the method identifies genuinely used features

A backpropagation-based attribution method that explains predictions by comparing neuron activations to a reference activation. It distributes contribution scores according to differences from that reference state using a summation-to-delta property analogous to LRP's conservation rule. DeepLIFT uses multiplier-based rules (Rescale and RevealCancel) to handle nonlinearities, making it closely related to LRP in spirit but differing in its reference-dependent formulation.

A localization technique that produces coarse heatmaps by using the gradients of a target concept flowing into the final convolutional layer. Unlike LRP, which propagates relevance through all layers to produce fine-grained pixel-level attributions, Grad-CAM is restricted to convolutional architectures and generates class-discriminative but spatially coarse explanations. It is often combined with Guided Backpropagation to achieve higher resolution through pointwise multiplication.