Integrated Hessians

Unlike first-order methods that assume independent features, Integrated Hessians explicitly models feature interactions by integrating the Hessian matrix along the path from baseline to input.

• Captures pairwise interactions between features that first-order methods miss
• Essential for models where features combine non-additively (e.g., XOR patterns)
• Decomposes the model output into main effects and interaction effects
• Provides a complete functional decomposition satisfying the completeness axiom

The method computes attributions by integrating both the gradient (first derivative) and the Hessian (second derivative) along a straight-line path.

• Path integral: ∫ (∇f + H·(x - x')) dα from α=0 to α=1
• The Hessian matrix H captures local curvature and feature coupling
• Interaction between features i and j is quantified by the off-diagonal Hessian entries ∂²f/∂xᵢ∂xⱼ
• Satisfies the completeness axiom: sum of all attributions equals f(x) - f(baseline)

Integrated Hessians provides a full decomposition of the model output into individual feature contributions and all pairwise interaction terms.

• Main effects: Attributions for individual features acting alone
• Interaction effects: Attributions for pairs of features acting jointly
• The sum of all main and interaction effects exactly reconstructs the output difference
• Enables auditing of synergistic and antagonistic feature relationships
• Critical for domains like genomics where gene-gene interactions drive outcomes

While Integrated Gradients provides only first-order attributions, Integrated Hessians reveals the full interaction structure.

• Integrated Gradients: Captures main effects only; assumes feature independence
• Integrated Hessians: Captures main effects + all pairwise interactions
• Trade-off: Higher computational cost due to Hessian computation (O(n²) vs O(n))
• Use Integrated Gradients for quick feature ranking; use Integrated Hessians when interactions matter
• Both methods share the same axiomatic properties (completeness, sensitivity, implementation invariance)

Computing the full Hessian for high-dimensional inputs is computationally expensive, requiring optimization strategies.

• Hessian-vector products avoid materializing the full n×n matrix
• Use stochastic sampling along the integration path to reduce computation
• Leverage automatic differentiation frameworks (PyTorch, JAX) for efficient second-order gradients
• Sparsity assumptions: Often only top-k interactions are practically significant
• Approximation techniques like Hessian diagonal can trade accuracy for speed

Integrated Hessians is particularly valuable in domains where feature interactions drive predictions.

• Healthcare: Identifying drug-drug interactions and gene-environment effects
• Finance: Modeling non-linear interactions between market factors in risk models
• NLP: Understanding how word pairs jointly influence sentiment or entailment
• Computer vision: Explaining how image regions interact to form object representations
• Scientific ML: Validating that models learn known physical interaction laws

The square matrix of second-order partial derivatives of a scalar-valued function. In neural networks, it captures the local curvature of the loss landscape with respect to the input features.

• Encodes pairwise feature interactions
• Diagonal elements represent the rate of change of gradients
• Off-diagonal elements quantify how one feature's gradient changes with another feature

The phenomenon where the effect of one input variable on a model's prediction depends on the value of another variable. Integrated Hessians explicitly quantifies these non-additive effects.

• Captured by off-diagonal Hessian terms
• Essential for explaining models with multiplicative or logical operations
• Missed entirely by first-order attribution methods like vanilla Integrated Gradients

The mathematical framework that defines desirable properties for feature attribution methods. Integrated Hessians extends these axioms to the domain of interaction attribution.

• Completeness: Total attributions sum to the output difference
• Implementation Invariance: Identical functions yield identical attributions
• Interaction Completeness: The sum of all pairwise interactions plus main effects equals the total output

A class of attribution techniques that define feature importance by integrating derivatives along a continuous path in the input space. Integrated Hessians generalizes this to second-order path integration.

• Straight-line paths are the default for simplicity
• Alternative paths can encode domain-specific priors
• The choice of baseline anchors the explanation context

A phenomenon in deep networks where the raw gradient with respect to the input resembles white noise, providing no coherent saliency map. Integrated Hessians partially mitigates this by examining second-order structure.

• Caused by highly non-convex loss surfaces
• Averaging over paths reduces noise
• Hessian-based analysis can reveal structure invisible to gradients alone