Attribution Flow

Attribution Flow is governed by a strict conservation law: the total relevance entering a neuron must equal the total relevance exiting it. This is not a heuristic—it's a mathematical constraint derived from the network's topology.

• Kirchhoff's Analogy: Just as current is conserved at a junction in a circuit, relevance is conserved at each neuron.
• Completeness Guarantee: Conservation at every layer ensures the sum of input attributions equals the model's output difference from a baseline.
• Unified Framework: Methods like Layer-wise Relevance Propagation (LRP), Deep Taylor Decomposition, and Excitation Backpropagation are all specific instances of flow with different propagation rules.

Attribution Flow fundamentally differs from gradient-based methods by redistributing the output value itself, not the local sensitivity.

• Value Redistribution: Flow methods decompose the output score ( f(x) ) into contributions from each input feature, satisfying the Completeness Axiom by construction.
• Gradient Limitation: Raw gradients measure infinitesimal change, not contribution. A saturated feature with zero gradient may be critical to the prediction—flow methods correctly assign it high relevance.
• Path Independence: Unlike Integrated Gradients, which requires choosing an integration path, flow methods propagate through the network's actual computational graph, avoiding the baseline selection problem.

The specific rule for redistributing relevance at each layer defines the flavor of Attribution Flow. These rules encode assumptions about how information mixes.

• LRP-0: Redistributes relevance proportionally to positive weighted activations. Simple but can produce noisy attributions.
• LRP-ε: Adds a small stabilization term ε to the denominator, absorbing weak or contradictory relevance and improving signal-to-noise ratio.
• LRP-αβ: Separates positive and negative contributions with weights α and β. Setting α=2, β=1 emphasizes excitatory pathways and suppresses inhibitory noise.
• Deep Taylor Decomposition: Derives propagation rules by linearizing the neuron's activation function at a root point, providing a theoretical justification for LRP rules.

Deep Taylor Decomposition provides the rigorous mathematical backbone for many flow-based attribution methods by applying Taylor expansion at a functionally relevant root point.

• Root Point Selection: For a ReLU neuron, the root point is chosen where the neuron's output is zero and minimally distant from the actual activation, ensuring the decomposition is local and meaningful.
• First-Order Redistribution: The function value is redistributed to inputs using the partial derivatives at the root point, naturally yielding the LRP-α1β0 rule for ReLU networks.
• Handling Non-Linearities: Each activation function (ReLU, tanh, softmax) requires a tailored root point strategy, making the framework extensible to arbitrary architectures.

Flow-based methods excel in domains requiring high-resolution, trustworthy explanations where gradient noise is unacceptable.

• Medical Imaging: LRP heatmaps on MRI and CT scans provide radiologists with pixel-level evidence that respects the image's spatial structure, unlike noisier gradient saliency maps.
• Natural Language Processing: Relevance scores assigned to individual tokens reveal which words drove a sentiment classification or entailment decision, enabling debugging of transformer models.
• Regulatory Compliance: The built-in conservation property provides a complete audit trail—every unit of the output decision is accounted for by a specific input feature, satisfying GDPR right to explanation requirements.

Attribution Flow serves as a meta-framework that reveals the structural relationships between seemingly disparate explanation techniques.

• LRP as Flow: LRP is explicitly a flow method with hand-designed propagation rules per layer type.
• Excitation Backpropagation: Equivalent to a flow where only excitatory (positive) relevance is propagated, corresponding to an LRP variant with α=1, β=0.
• Gradient × Input Connection: Under specific conditions (linear activations, zero biases), Gradient × Input approximates a flow decomposition, explaining why it sometimes satisfies completeness.
• SHAP Relationship: Deep SHAP can be interpreted as a flow method that uses expected gradients over a background distribution, bridging game-theoretic and conservation-based approaches.

A method that explains predictions by performing a Taylor expansion of the model's output at a root point. It redistributes the function value to input features via propagation rules that are derived analytically for each layer type.

• Provides a theoretical bridge between gradient-based methods and LRP
• Identifies root points where the function value is zero
• Enables custom propagation rules for complex architectures

An axiomatic method that computes feature importance by accumulating gradients along a straight-line path from a baseline to the actual input. This path integral directly embodies the flow concept, where attributions are the accumulated gradient flow.

• Satisfies Sensitivity-n, Implementation Invariance, and Completeness
• Requires careful selection of a baseline input (often a black image or zero embedding)
• The path integral is approximated via Riemann summation in practice

The principle that the sum of feature attributions must equal the difference between the model's output and a baseline reference output. This conservation law is the mathematical bedrock of the Attribution Flow framework.

• Guarantees that attributions fully account for the model's decision
• Violated by simple gradient methods like Gradient × Input
• Enables attributions to be interpreted as additive credit assignments

A class of attribution techniques that define feature importance by integrating gradients along a specified path in the input space. Attribution Flow generalizes this by viewing the path as a streamline in a relevance field.

• Integrated Gradients uses a straight-line path
• Guided Integrated Gradients adapts the path to avoid regions of high non-linearity
• The choice of path encodes prior assumptions about feature interactions

An extension of Integrated Gradients that measures the importance of a specific hidden neuron by computing the flow of the gradient through that neuron along the integration path. This directly quantifies the internal flow of relevance.

• Decomposes the total attribution into per-neuron contributions
• Identifies critical bottleneck neurons in the network
• Enables debugging of internal representations and dead filters