Comparing Linear Probes with Mahalanobis Cosine Similarity
Provides a theoretically grounded metric for comparing linear probes in interpretability research, replacing ad-hoc Euclidean cosine similarity.
The authors prove that Mahalanobis cosine similarity (MCS) between a probe and a reference probe linearly predicts OOD AUROC (R^2 = 0.98) under Gaussian assumptions, and extend this finding across models, layers, and domains. They also identify conditions where linearity fails.
Linear probes are widely used in interpretability research and often compared by cosine similarity. The Mahalanobis cosine similarity (MCS) between two directions, which reweights the inner product by test data covariance, is a natural task-aware refinement. Ying et al. (2026) report that a probe's MCS to a reference probe trained on the out-of-distribution (OOD) data near-perfectly linearly predicts the probe's OOD AUROC (R^2 = 0.98). Here, we extend this empirical finding across models, layers, and concept domains, and prove this general phenomenon in closed form: For balanced classes whose projections are Gaussian, OOD AUROC and MCS to the reference probe are linear because both are sigmoid-shaped functions of the probe's signal-to-noise ratio (SNR) on the test data. The theory also predicts when this linearity fails, which we verify empirically. MCS offers a theoretically grounded and empirically effective alternative to Euclidean cosine similarity for comparing linear probes.