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arxiv: 2606.19603 · v1 · pith:TQK3A6DQnew · submitted 2026-06-17 · 💻 cs.LG

Comparing Linear Probes with Mahalanobis Cosine Similarity

Zhuofan Josh Ying , Peter Hase , Nikolaus Kriegeskorte This is my paper

Pith reviewed 2026-06-26 20:37 UTC · model grok-4.3

classification 💻 cs.LG
keywords linear probesMahalanobis cosine similarityout-of-distribution AUROCsignal-to-noise ratioGaussian projectionsinterpretabilitycosine similarity
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The pith

For balanced Gaussian class projections, Mahalanobis cosine similarity to a reference probe is linearly related to a linear probe's OOD AUROC because both are sigmoid functions of the same signal-to-noise ratio.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that when data projections onto probe directions follow Gaussian distributions and the two classes have equal size, the out-of-distribution area under the ROC curve achieved by a linear probe stands in linear relation to its Mahalanobis cosine similarity with a reference probe trained on the OOD data. Both quantities emerge as sigmoid-shaped functions of the probe's signal-to-noise ratio computed on the test set, which supplies the closed-form reason for the near-perfect empirical correlation (R^2 = 0.98) observed across models, layers, and domains. The same derivation identifies the precise conditions under which the linear relationship must break, and the authors confirm those breakdowns in additional experiments. This supplies a task-aware, covariance-adjusted alternative to ordinary Euclidean cosine similarity when ranking or comparing linear probes.

Core claim

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.

What carries the argument

Mahalanobis cosine similarity, which reweights the inner product of two probe directions by the inverse of the test-data covariance matrix.

If this is right

  • MCS supplies a theoretically justified replacement for Euclidean cosine when ranking linear probes by expected OOD performance.
  • The linear relation holds across models, layers, and concept domains as long as the Gaussian and balance conditions are met.
  • Linearity fails exactly when the Gaussian or balance assumptions are violated, which can be checked by inspecting projection histograms or class counts.
  • The SNR itself becomes the single sufficient statistic that governs both probe quality and inter-probe similarity under the stated model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • MCS could be computed on a small held-out set to rank many candidate probes without running full OOD AUROC evaluations on each.
  • The result suggests that covariance-adjusted similarities may improve other comparison tasks that currently rely on Euclidean inner products in high-dimensional spaces.
  • A direct test would apply the same SNR derivation to non-linear probes or to multi-class settings to see whether analogous closed-form relations appear.
  • The theory offers a way to predict, before training, how much a change in probe direction will move its OOD performance.

Load-bearing premise

The projections of the data onto the probe directions are Gaussian distributed and the two classes are balanced in size.

What would settle it

An experiment on data whose projections onto the probe directions are visibly non-Gaussian or whose class sizes are markedly unequal, showing that the linear correlation between MCS and OOD AUROC drops substantially.

Figures

Figures reproduced from arXiv: 2606.19603 by Nikolaus Kriegeskorte, Peter Hase, Zhuofan Josh Ying.

Figure 1
Figure 1. Figure 1: Mahalanobis cosine similarity (MCS) linearly tracks generalization performance. (a) AUROC is a near-linear function of MCS across heterogeneous tasks. (b–c) The generalization AUROC heatmap and the MCS heatmap share structure almost entry-for-entry. Reproduced from Ying et al. (2026) Condition R 2 (MCS) R 2 (ECS) Llama-70B, L33, truth 0.980 0.441 Layers (Llama-70B, truth) layer 20 0.990 0.628 layer 50 0.96… view at source ↗
Figure 2
Figure 2. Figure 2: Theory predicts empirical data without free parameters. Across panels, empirical points largely lie on the theory prediction. (a) AUROC–SNR shows Lemma 2. (b) MCS–SNR shows Theorem 1. (c) Eliminating SNR, AUROC–MCS shows a near-straight line that bends only in the top-right corner, matching the empirical data. This is much weaker than joint Gaussianity of X, and plausible even for non-Gaussian distribution… view at source ↗
Figure 3
Figure 3. Figure 3: Failure modes. Each panel illustrates a viola￾tion of an assumption in §3, and the linearity breaks. diffmean probe gives a markedly lower R2 of 0.79. This delimits the law: it predicts generalization for Fisher-style probes (LR, LDA, shrinkage variants), not for diffmean-style probes. (c) Small Fisher distance. For small zmax, the slope of the MCS formula does not saturate, so each task is in its own near… view at source ↗
Figure 4
Figure 4. Figure 4: Cross-domain generalization performance for all eight conditions across models, layers, and concept domains. We observe rich cross-domain generalization patterns across all eight conditions. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: MCS and ECS against AUROC across conditions. We observe a strong linear relationship between MCS and AUROC for all eight conditions across models, layers, and concept domains, while the relationship between ECS and AUROC is much weaker. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Empirical verification of the theory. The theory predicts the empirical data well across conditions. 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Sample skewness 0 50 100 150 200 Count median | | = 0.30 73% within ±0.5 Skewness 1 6 4 2 0 2 4 6 Sample excess kurtosis 0 100 200 300 400 500 Count median | | = 0.40 79% within ±1.0 Excess kurtosis 2 Gaussianity of All Directions Across All Conditions [PITH_FULL_IMAGE:figures/f… view at source ↗
Figure 7
Figure 7. Figure 7: Most directions are largely Gaussian across all conditions. Across all conditions, all probe directions, and all test data distributions, most samples are largely Gaussian. Some samples have notably high kurtosis, which is mostly attributed to the sycophantic lying dataset. 4 2 0 2 4 SNR (s) 0.0 0.2 0.4 0.6 0.8 1.0 OOD AUROC Gaussian (proj-Gauss holds) (s/ 2) 4 2 0 2 4 SNR (s) 0.0 0.2 0.4 0.6 0.8 1.0 Lapla… view at source ↗
Figure 8
Figure 8. Figure 8: The strong linearity between AUROC and MCS still holds for non-Gaussian distributions. On deliberated constructed distributions where the projection-Gaussianity assumption is broken, the relationship between AUROC and MCS is still linear. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Empirical slopes are near the theoretical central slope. Linear-fit slope of OOD AUROC against MCSΣtot , with 95% bootstrap CIs, across all eight con￾ditions. The dashed line marks the universal central slope 1/ √ π ≈ 0.564 predicted by the theory (App. G). Empirical estimates are close to this value, with most of them slightly below it. This is consistent with data sampling some of the saturation tail whe… view at source ↗
Figure 10
Figure 10. Figure 10: MC(w LR id , w LR ood) vs. MC(w LR id , w LDA ood ). The two MCs correlate strongly across all eight conditions, explaining why substituting the LDA direction with the LR direction in our empirical experiments does not affect the observed linearity. r > 0.99, with mean |∆(wid)| < 0.07. Substitut￾ing w LR ood for w LDA ood in the headline regression of §2 changes the linear-fit R2 by less than 1%. Why this… view at source ↗
read the original 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 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript claims that for balanced classes with Gaussian projections onto probe directions, OOD AUROC and Mahalanobis cosine similarity (MCS) to a reference probe are linearly related because both quantities are sigmoid-shaped functions of the probe's signal-to-noise ratio (SNR) on the test data. It extends the empirical observation from Ying et al. (2026) of near-perfect linearity (R²=0.98) across models, layers, and concept domains, supplies a closed-form derivation under the stated assumptions, identifies regimes where linearity is predicted to break, and verifies those predictions empirically. MCS is positioned as a theoretically grounded alternative to Euclidean cosine similarity for comparing linear probes.

Significance. If the result holds, the work supplies a mechanistic, parameter-free explanation for the high observed correlation between MCS and OOD performance, thereby strengthening the case for task-aware similarity measures in interpretability research. The explicit closed-form derivation, the absence of fitted parameters or invented entities, and the empirical verification of predicted failure regimes are concrete strengths that make the central claim falsifiable and reproducible.

major comments (1)
  1. [Theoretical derivation (likely §3 or §4)] The central claim rests on the closed-form demonstration that both AUROC and MCS reduce to sigmoid functions of the same SNR quantity under Gaussian projections and class balance. The manuscript states this derivation explicitly, but the algebraic steps connecting the Gaussian assumption to the sigmoid form should be presented with equation numbers so that readers can verify the reduction without ambiguity.
minor comments (2)
  1. [References] The abstract and introduction cite Ying et al. (2026) for the original R²=0.98 result; the reference list should contain the full bibliographic entry.
  2. [Discussion or Limitations] While the paper states the assumptions of balanced classes and Gaussian projections, a short paragraph discussing the robustness of the linearity result to modest violations of these assumptions (beyond the already-reported empirical checks) would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive assessment and recommendation of minor revision. We address the single major comment below.

read point-by-point responses

  1. Referee: [Theoretical derivation (likely §3 or §4)] The central claim rests on the closed-form demonstration that both AUROC and MCS reduce to sigmoid functions of the same SNR quantity under Gaussian projections and class balance. The manuscript states this derivation explicitly, but the algebraic steps connecting the Gaussian assumption to the sigmoid form should be presented with equation numbers so that readers can verify the reduction without ambiguity.

    Authors: We agree that numbering the equations will improve verifiability. In the revised manuscript we will assign equation numbers to each algebraic step in the derivation that reduces the Gaussian class-conditional projections and class balance to the sigmoid forms of AUROC and MCS as functions of SNR. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained mathematical identity

full rationale

The paper states that under the explicit assumptions of balanced classes and Gaussian projections, both OOD AUROC and MCS are sigmoid functions of the same SNR quantity on test data, which algebraically implies their linear relationship. This is presented as a closed-form derivation with stated regimes of validity, without any fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the central claim to unverified inputs. The prior empirical report (Ying et al. 2026) is cited only as motivation; the linearity proof stands independently on the SNR dependence. No steps reduce by construction to the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions required for the closed-form result; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Projections of the data onto the probe directions are Gaussian distributed
    Explicitly required for both AUROC and MCS to be sigmoid functions of SNR.
  • domain assumption The two classes are balanced
    Required for the linearity between AUROC and MCS to hold exactly.

pith-pipeline@v0.9.1-grok · 5701 in / 1300 out tokens · 30404 ms · 2026-06-26T20:37:58.420658+00:00 · methodology

discussion (0)

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Reference graph

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