Woodbury Influence: Effective Evaluation of Unseen Sample Familiarity Before Training

Influence Functions (IFs) provide a powerful framework for quantifying the impact of individual training samples on the trained model using gradient and Hessian information. Most IFs primarily focus on the effect of removing samples rather than adding novel ones, but this unseen-sample regime arises in many real-world settings that must determine how familiar a new sample is to the current optimal model. Extending IFs to this regime presents two fundamental challenges: (1) the prohibitive computational cost of estimating influence based on Hessian, and (2) the absence of validated labels in IF frameworks for unseen data. To overcome these challenges, we propose Woodbury Influence (WIF) as a measure of an unseen sample's familiarity to the entire model parameters before training on it, leveraging pseudo-labeling and the Woodbury matrix identity. Empirical analyses reveal the underlying mechanisms driving the effective separation of unfamiliar samples from familiar ones, accurate approximation of the true optimal model, and competitive computational cost. Furthermore, we propose practical methods for applying WIFs to representative tasks in this regime, including Out-of-Distribution (OOD) detection, Open-Set Recognition (OSR), and Continual Learning (CL) : offline class-order selection. WIF achieves superior performance across all three benchmarks, establishing a robust, influence-based framework for these settings.