A Unified Theory of Random Projection for Influence Functions

Influence functions and related data attribution scores take the form of $g^{\top}F^{-1}g^{\prime}$, where $F\succeq 0$ is a curvature operator. In modern overparameterized models, forming or inverting $F\in\mathbb{R}^{d\times d}$ is prohibitive, motivating scalable influence computation via random projection with a sketch $P \in \mathbb{R}^{m\times d}$. This practice is commonly justified via the Johnson--Lindenstrauss (JL) lemma, which ensures approximate preservation of Euclidean geometry for a fixed dataset. However, JL does not address how sketching behaves under inversion. Furthermore, there is no existing theory that explains how sketching interacts with other widely-used techniques, such as ridge regularization and structured curvature approximations.