Anatomy of Linearized Group Influence: From a Bregman Geometric Perspective

Standard first-order group influence aggregates member-wise responses into a single estimate, obscuring how individual contributions reinforce or cancel one another. We show that standard group influence is the Bregman centroid of scaled member-wise parameter responses, in the geometry induced by the linearized PBRF objective. Bregman information measures the distortion of representing these responses by their centroid. This characterization supports a decomposition into multiple components whose contributions sum to the original group estimate. A two-component Dogfish case study exposes opposing contributions hidden by the aggregate. Experiments on small training groups from Dogfish, binary MNIST, and binary CIFAR-10 also show a positive rank association between target-projected Bregman information and squared approximation error relative to the nonlinear PBRF. Together, these results provide a geometric understanding of what additive group influence retains and a basis for analyzing the within-group structure it omits.