Mahalanobis Cosine Similarity Improves Linear Probe Comparison

Linear probes are a common tool in AI interpretability research, often compared using standard cosine similarity to understand how well they capture specific concepts. However, a more refined metric, Mahalanobis cosine similarity (MCS), reweights the inner product by the test data covariance, making it more task-aware. Previous work suggested that a probe's MCS to a reference probe trained on out-of-distribution (OOD) data could nearly perfectly predict the probe's OOD AUROC (Area Under the Receiver Operating Characteristic curve). This new research expands on that empirical observation, demonstrating its generality across various models, layers, and concept domains. The study provides a closed-form proof for this phenomenon, explaining that for balanced classes with Gaussian projections, both OOD AUROC and MCS are sigmoid-shaped functions of the probe's signal-to-noise ratio (SNR) on the test data, thus exhibiting a linear relationship. The theory also predicts conditions under which this linearity might fail, which was empirically verified. MCS therefore offers a theoretically grounded and empirically effective alternative to Euclidean cosine similarity for comparing linear probes, enhancing the reliability of interpretability studies.