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        <title>Comparing Linear Probes with Mahalanobis Cosine Similarity</title>
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    <abstract>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 a nearly perfect linear relationship between a probe’s MCS to a reference probe trained on out-of-distribution (OOD) data and its OOD AUROC (R⌃2=0.98). Here, we extend this empirical finding across models, layers, and concept domains, and prove it in closed form: For balanced classes whose projections are Gaussian, OOD AUROC and MCS to the reference probe are near-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.</abstract>
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%0 Conference Proceedings
%T Comparing Linear Probes with Mahalanobis Cosine Similarity
%A Ying, Zhuofan
%A Hase, Peter
%A Kriegeskorte, Nikolaus
%Y Lee, Hung-Yi
%Y Dev, Sunipa
%Y Li, Jessy
%Y Elliott, Desmond
%S Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing
%D 2026
%8 October
%I Association for Computational Linguistics
%C Budapest, Hungary
%@ 979-8-89176-459-0
%F ying-etal-2026-comparing
%X 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 a nearly perfect linear relationship between a probe’s MCS to a reference probe trained on out-of-distribution (OOD) data and its OOD AUROC (R⌃2=0.98). Here, we extend this empirical finding across models, layers, and concept domains, and prove it in closed form: For balanced classes whose projections are Gaussian, OOD AUROC and MCS to the reference probe are near-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.
%U https://aclanthology.org/2026.emnlp-main.946/
%P 21793-21810

Markdown (Informal)

[Comparing Linear Probes with Mahalanobis Cosine Similarity](https://aclanthology.org/2026.emnlp-main.946/) (Ying et al., EMNLP 2026)

• Comparing Linear Probes with Mahalanobis Cosine Similarity (Ying et al., EMNLP 2026)

ACL

• Zhuofan Ying, Peter Hase, and Nikolaus Kriegeskorte. 2026. Comparing Linear Probes with Mahalanobis Cosine Similarity. In Proceedings of the 2026 Conference on Empirical Methods in Natural Language Processing, pages 21793–21810, Budapest, Hungary. Association for Computational Linguistics.