Layer-wise Relevance Propagation (LRP)

LRP is fundamentally governed by a conservation of relevance axiom. The total relevance score assigned to the output neuron (the prediction) must be exactly redistributed backwards through the network without loss or amplification. At each layer, the sum of relevance scores of the neurons equals the relevance of the layer above. This ensures that the final heatmap on the input variables is a true decomposition of the model's decision, not an approximation. This property mathematically guarantees that Σ R_output = Σ R_input.

LRP rules are theoretically grounded in Deep Taylor Decomposition (DTD). This framework views the redistribution process as a series of local Taylor expansions applied at each neuron. By identifying a root point in the input space where the neuron's activation is zero, DTD provides a principled way to derive propagation rules that satisfy the conservation axiom. This connection to Taylor series gives LRP a stronger theoretical footing than purely heuristic gradient-based methods, explaining why a neuron fires in terms of its inputs.

A single rule is insufficient for deep networks. LRP employs a composite strategy, applying different propagation rules to different layer types:

• LRP-ε: Stabilizes division by adding a small constant ε to the denominator, absorbing weak or contradictory relevance.
• LRP-γ: Favors positive contributions by adding a parameter γ to positive weights, increasing the robustness and visual quality of explanations.
• LRP-αβ: Separates positive and negative weights explicitly, allowing fine-grained control over the treatment of inhibitory connections.
• LRP-0 (Flat): A basic rule used in upper layers where no specific adjustments are needed.

Unlike methods that produce coarse localization maps, LRP computes a relevance score for every single input dimension. For an image, this means a pixel-level heatmap where each pixel's color intensity directly corresponds to its contribution to the final classification score. This granularity allows for the identification of precise evidence and counter-evidence. For example, in a 'cat' classification, LRP can highlight the specific fur texture pixels that supported the decision and the background pixels that acted against it, providing a complete audit trail.

LRP explanations exhibit continuity, meaning similar inputs produce similar explanations. This is a critical property for trust and debugging. If two nearly identical images of a stop sign receive vastly different relevance heatmaps, the explanation method is unreliable. LRP's deterministic, rule-based redistribution ensures that the explanation varies smoothly with the input, making it a stable tool for comparing model behavior across a dataset and identifying anomalous predictions that warrant further investigation.

The relevance scores produced by LRP form a canonical decomposition of the prediction function f(x). This means the explanation is not just a local sensitivity measure but a structured breakdown of the function's value at a specific point. By separating the positive relevance (evidence for a class) from negative relevance (evidence against it), LRP provides a signed, additive decomposition. This allows users to understand not just where the model is looking, but whether that region is providing supporting or contradictory evidence for the final decision.