Comparing Linear Probes with Mahalanobis Cosine Similarity
Authors: Zhuofan Josh Ying, Peter Hase, Nikolaus Kriegeskorte
Organizations: Departments of 1Psychology, 2Neuroscience, 3Electrical Engineering · 4Zuckerman Mind Brain Behavior Institute; Columbia University, New York, NY · 5Stanford University, Stanford, CA · 6Schmidt Sciences, New York, NY
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.
It is becoming increasingly necessary to have monitors check for harmful behaviors during language model interactions, but text-only monitoring has not been sufficient. This is because models sometimes exhibit strategic deception and sandbagging, changing their behavior during evaluation. This motivates the use of white-box monitors like linear probes, which can read the model internals directly. Currently, such probes can fail under distribution shift, limiting their usefulness in real settings. We study whether there exists a low-dimensional subspace of the model internals that captures harmful behaviors more robustly, while leaving out spuriously correlative features. Inspired by the Assistant Axis and Persona Selection Model, we construct persona axes for deception and sycophancy using contrastive persona prompts. The first principal components, obtained by unsupervised PCA of the persona-specific vectors, cleanly separate harmful and harmless personas. Across 10 evaluation datasets, we show that persona-derived directions transfer non-trivially and probes trained on persona-PC projections generalize better than probes trained on raw activations. We also find that a unified axis consisting of multiple harmful and harmless behaviors improves generalization across behaviors and datasets. Overall, persona vectors provide a useful inductive bias for building more transferable behavior probes.
Linear probes trained on LLM activations are increasingly proposed as deception-detection metrics, yet report AUROC exceeding 0.96 on clean benchmarks while collapsing under distributional shift. This paper systematically pressure-tests probe-based metrics across the Gemma 3 model family (1B-27B parameters), diagnosing why they fail rather than merely documenting that they fail. We test four hypotheses about deception encoding: (1) single linear direction, (2) multi-dimensional subspace, (3) convex conic hull, (4) entropy proxy. Our design includes cross-domain transfer matrices, multi-dimensional probe analysis with permutation null baselines, entropy-residualization tests, and distractor evaluations across 8 stylistic shifts. We find that: (a) probes achieve near-perfect AUROC (>=0.998) on clean data but collapse under stylistic shifts; style-augmented probes recover near-perfect detection (mean AUROC 0.979-0.983) on unseen styles; (b) the single-direction hypothesis is rejected (k=1 captures only 0.61-0.80 AUROC), with cross-domain transfer failure confirmed as geometric rather than layer-mismatch-driven; (c) the entropy-proxy hypothesis is rejected (max |rho|=0.454, max Delta-AUROC after residualization=0.004); and (d) deception does not form a significant linear subspace (per-domain k*=0), yet multi-dimensional probes (k>=5) recover the signal through distributed sub-threshold features. Probe fragility reflects distributional narrowness rather than an architectural limitation: style-augmented probes recover near-perfect detection at both 4B and 27B, establishing that the inverse scaling pattern is a training-distribution artifact rather than a genuine scale-dependent phenomenon.
Neural representations are not unique objects. Even when two systems realize the same downstream computation, their hidden coordinates may differ by reparameterization. A probe family intended to reveal structure already present in a representation should therefore be stable under the relevant representation symmetries rather than be tied to a particular basis. We study this group action in the tractable exact setting of the final readout layer, where equivalent realizations induce affine changes of hidden coordinates. The resulting symmetry principle singles out a unique hierarchy of shallow coordinate-stable probes, with linear probes as its degree-1 member. We also show that a natural object for cross-model probe transfer is a shared probe-visible quotient--the representation modulo directions invisible to the probe family--rather than the full hidden state. Experiments on synthetic and real-world tasks support both predictions, showing where degree-2 probes help beyond linear ones and how quotient-based transfer enables coverage-aware monitor portability across model families. These results point toward a broader geometric representation theory of neural probing, with coverage-aware monitor transfer as a concrete operational consequence.