University of Amsterdam Breaks SHAP's Two Biggest Limits With Three Axioms
A new axiomatic framework extends Möbius inversion and Shapley values to vector-valued functions on directed acyclic multigraphs, unlocking attribution for embeddings and non-lattice structures.
PRO- Forré and Jansma generalize Möbius inversion and Shapley values to directed acyclic multigraphs and vector-valued functions.
- Classical axioms fail to uniquely determine Shapley values outside lattices, so two new axioms are introduced.
- Weak elements and flat hierarchy axioms, plus linearity, force a unique closed-form attribution formula.
- Attributions become path-proportional splits: Sh_r(v) = Σ (π(r,y)/π(y)) · synergy(y) across all nodes.
- Framework recovers classical Shapley on power sets and handles previously unreachable non-lattice cooperative games.
- Opens vector-based attribution for transformer embeddings, syntax trees, and higher-order Ising models in physics.
Shapley values are the mathematical machinery behind SHAP, one of the most popular tools for explaining model predictions. The classical formulation has two hard limits: the value function has to output a real number, and the underlying structure has to be a lattice (usually the power set of features). A new paper from Patrick Forré and Abel Jansma at the University of Amsterdam breaks both restrictions at once, and derives a surprisingly clean formula from just three axioms.
The generalization moves in two orthogonal directions: from real-valued functions to functions valued in any abelian group (in particular, vector-valued functions), and from partial orders and lattices to directed acyclic multigraphs (DAMGs) and weighted versions thereof. In plain terms, you can now compute Shapley-style attributions when your model outputs vectors (like transformer embeddings), and when the coalition structure isn't a tidy power set.
Why the classical axioms broke
The standard Shapley recipe rests on four axioms (linearity, efficiency, null player, and symmetry) that together pick out a single attribution formula on a Boolean lattice. Move to a more general graph and this uniqueness collapses. Symmetry, as classically stated, doesn't say enough when the graph has irregular structure, and you can construct multiple attribution rules that all satisfy the four axioms but disagree on how to split synergy among players.
Two new axioms and a projection trick
This story is for Pro members
You've reached the end of the free preview. Upgrade to AlphaSignal Pro to read the full article - and everything else behind the paywall.
Comments
