Building machine learning systems that are both effective and trustworthy relies on understanding how a model sees and uses its training data. This project approaches that question geometrically, using the magnitude of a metric space, an isometric invariant capturing multi-scale structure in data, as the foundation for a family of tools that make these properties explainable and measurable. We first introduced the magnitude distance, a novel dissimilarity measure between finite datasets with a tunable scale parameter, proving its theoretical properties and demonstrating its use as a training objective for push-forward generative models, where it outperforms standard baselines in training stability. We also introduced the magnitude potential ratio to study neural network training dynamics, showing that it correlates with established memorization scores, detects structural changes in decision boundaries, and provides a geometric indicator of delayed generalization, called grokking, all without requiring model retraining.

Building on these results, we propose three potential extensions to this work which we intend to pursue; First, we propose to address magnitude's memory intensive nature through aggregation of one-dimensional projections of the data. From this we will obtain provably bounded sensitivity, which facilitates its use in applications such as differentially private release. Second, we consider the applicability of Spread, a computationally lighter relative of magnitude, as a diagnostic for mode perturbations in generative and representation-learning settings. Third, we use magnitude potential to characterize which training points are most exposed to privacy attacks such as membership-inference attacks and how easily their influence can be removed via machine unlearning. Our goal is to combine the results from these extensions into a geometric framework that connects representation structure, memorization, and privacy risk, offering lightweight, retraining-free alternatives to the existing expensive procedures that standard privacy auditing and unlearning verification currently rely on.