Integrated Hessians

Integrated Hessians is an attribution method that captures feature interactions by integrating second-order derivatives (the Hessian) along a path from a baseline to the input — extending Integrated Gradients to detect pairwise feature interactions that first-order methods miss.

How Integrated Hessians Works

• Interaction Attribution: $IH_{ij} = (x_i - x_i')(x_j - x_j') \int_0^1 \frac{\partial^2 F}{\partial x_i \partial x_j} dalpha$ along the interpolation path.
• Pairwise: Captures how pairs of features jointly influence the prediction (cross-terms).
• Completeness: Integrated Hessians + Integrated Gradients together fully decompose the prediction.
• Approximation: Computed using finite differences or automatic differentiation of the Hessian.

Why It Matters

• Interaction Detection: Reveals which feature pairs interact — critical for semiconductor processes where variables interact strongly.
• Beyond Additivity: First-order methods (IG, SHAP) assume additive contributions — Integrated Hessians captures non-additive effects.
• Process Insight: In pharmaceutical/semiconductor processes, interaction effects often dominate main effects.

Integrated Hessians is the second-order attribution — capturing how pairs of features jointly influence predictions beyond their individual contributions.