arxiv: 2605.03222 · v2 · submitted 2026-05-04 · 💻 cs.LG · stat.ML

Recognition: 3 theorem links

· Lean Theorem

Beyond Activation Alignment: The Geometry of Neural Sensitivity

Amirhossein Yavari , Farnaz Zamani Esfahlani

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:25 UTC · model grok-4.3

classification 💻 cs.LG stat.ML

keywords neural representationsFisher informationlocal sensitivityrepresentational geometryactivation alignmentRiemannian distanceSPD matricesdiscriminability

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The pith

Neural representations aligned on global tasks can still differ in how they discriminate small local stimulus changes under noise.

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

Standard measures like RSA, CCA, and CKA assess agreement between optimal linear readouts on broad global tasks, yet leave open whether representations use local stimulus evidence in the same way. The paper introduces a complementary summary based on the expected projected Fisher metric over a chosen stimulus-coordinate subspace, which encodes a representation's sensitivity to small perturbations under an additive noise model. This operator serves as a minimal complete dataset-level description of expected discriminability across a second-moment family of local tasks. Comparisons are performed via log-spectral distance on the manifold of symmetric positive definite matrices, producing the Spectral Riemannian Alignment Score together with a uniform multiplicative certificate on lifted task values.

Core claim

By summarizing each representation with the expected projected pullback of the Fisher metric over a stimulus subspace, the framework yields a regularized signature operator that minimally and completely captures expected discriminability for the induced family of local discrimination tasks; these signatures are compared on the SPD manifold using log-spectral distance to obtain the S-RAS score and its associated uniform certificate.

What carries the argument

The expected projected Fisher metric over the stimulus-coordinate subspace, which acts as the regularized signature operator for the family of local tasks and is compared via log-spectral distance on symmetric positive definite matrices to produce the Spectral Riemannian Alignment Score.

If this is right

The method recovers corresponding layers between independently trained artificial networks.
It supports class-conditional probes that transfer across networks.
It produces controlled dissociations between standard and robustly trained networks.
It detects stimulus-coordinate family effects in mouse visual cortex recordings from the Allen Brain Observatory.

Where Pith is reading between the lines

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

Because the score supplies a uniform multiplicative bound over the lifted local-task family, it could be used to certify robustness margins when small input changes are known to matter.
The same geometric construction might be applied to sequence models by defining stimulus subspaces over token embeddings or attention patterns.
If the subspace choice is varied systematically, the resulting family of S-RAS matrices could serve as a fingerprint for how a network allocates sensitivity across different stimulus dimensions.

Load-bearing premise

The chosen stimulus-coordinate subspace together with the additive noise model for local perturbations correctly reflects the discrimination tasks of interest, and the resulting projected Fisher metric isolates sensitivity differences that global alignment measures do not already capture.

What would settle it

Two representations with high global alignment scores but substantially different S-RAS values would be shown to have identical performance on a battery of concrete local perturbation-discrimination tasks under the same noise model.

Figures

Figures reproduced from arXiv: 2605.03222 by Amirhossein Yavari, Farnaz Zamani Esfahlani.

**Figure 1.** Figure 1: Comparing representations via dataset-level sensitivity summaries. (A) Input space contains structured data on a data manifold, contrasted with off-manifold noise. (B) Representation maps f and g transform the same data into feature spaces where local neighborhoods have different geometries. (C) Pulling these local geometries back to input/stimulus coordinates yields Jacobianinduced sensitivity ellipses, … view at source ↗

**Figure 2.** Figure 2: Held-out class-conditional diagnostic probes. (A) Example probes at K = 32, two probes per side. (B) Mean held-out image separation across task-family dimensions and controls. (C) Split-class diagnostic visualization against pooled sensitivity; formal tests aggregate to the held-out model-split level. sensitivity, not pooled sensitivity shared by both groups. We compare them with random-contrast probes, po… view at source ↗

**Figure 3.** Figure 3: Controlled regime shifts under robust training. (A) Example probes from the standardversus-PGD contrast. (B) Held-out image separation at K = 32, comparing contrast-derived probes to random-contrast, pooled-sensitivity, and label-permutation controls. Error bars in (B) are 95% percentile bootstrap intervals over held-out model-split means. (C) Separation as a function of task-family dimension K. Full Fish… view at source ↗

**Figure 4.** Figure 4: Biological sensitivity geometry in mouse visual cortex. (A) Cross-area identity under donor-distinct matched-count comparison. (B) VISp-depth identity is harder. Full Fisher remains competitive with CKA and above naive geometry. (C) Matching depends on the grating-coordinate family. Dashed curves show trace-normalized shape-only summaries for families with dimension at least two. 4.4 Biological static-grat… view at source ↗

**Figure 5.** Figure 5: Layer-matching diagnostics. (A) Mean similarity as a function of absolute layer distance for CKA and S-RAS under random-basis and PCA-basis families at K = 768. (B) Top-margin statistic, defined as the correct-match similarity minus the strongest incorrect-match similarity. (C) Layer-identification accuracy as a function of task-family dimension K, with linear CKA shown as a dashed reference. 0.75 0.80 0.8… view at source ↗

**Figure 6.** Figure 6: Task-family dependence of layer-identification performance and off-diagonal separability. A, diagonal-versus-off-diagonal separability, measured by AUC, for linear CKA, S-RAS under the random-basis family, and S-RAS under the PCA-basis family. B, layer-identification accuracy for the PCA-basis family as a function of cumulative variance explained by the retained principal directions. C, layer-identificati… view at source ↗

**Figure 7.** Figure 7: Random-basis task-family sweep for S-RAS layer identification. Each panel shows the pair-averaged S-RAS similarity matrix for a random-basis subspace-restricted local perturbation task family of dimension K, together with the corresponding layer-identification accuracy. As the task-family dimension increases from K = 16 to K = 1024, accuracy rises from 71.7% to 84.7%, with the highest value reached at K = … view at source ↗

**Figure 8.** Figure 8: PCA-basis task-family sweep for S-RAS layer identification. Each panel shows the pair-averaged S-RAS similarity matrix for a PCA-basis subspace-restricted local perturbation task family of dimension K, together with the corresponding layer-identification accuracy. Here the admissible perturbations are restricted to the top-K principal directions of the dataset, but weighted isotropically within that span. … view at source ↗

**Figure 9.** Figure 9: Additional ablations for held-out class-conditional diagnostic probes. A, held-out model classification AUC as a function of task-family dimension K. B, low-amplitude version of the held-out image-separation analysis. C, high-amplitude version of the held-out image-separation analysis. D, probe-count ablation at the main setting (K = 32). E, distribution of split-class rows at the main setting. F, split-cl… view at source ↗

**Figure 10.** Figure 10: All pairwise regime comparisons at the main setting. Held-out image separation at K = 32 for all six pairwise comparisons among standard, PGD, TRADES, and MART-trained ResNet-18 models, comparing contrast-derived probes to the same three controls used in the main text. The three standard-versus-robust comparisons are strongest, TRADES versus MART is smaller but positive, and PGD versus MART shows little o… view at source ↗

**Figure 11.** Figure 11: Within-robust qualitative example. Example probes for PGD versus TRADES at the main setting. Even though the quantitative effect is smaller than in the standard-versus-robust comparisons, the two probe sets still differ in visible character. Fn a t 1 Fn a t 2 Fn a t 3 Fn a t 4 Fn a t 5 Fn a t 6 Fn a t 7 Fn a t 8 view at source ↗

**Figure 12.** Figure 12: Examples from the learned family basis. Representative family directions used to instantiate finite probes. These basis elements are smooth local deformations rather than noisy pixel patterns, which helps interpret the learned probe sets as structured perturbations within a coherent family. TRADES remains clearly positive (0.28), TRADES versus MART is smaller but still reliable (0.22), and PGD versus MART… view at source ↗

**Figure 13.** Figure 13: Supporting analyses: direct model-space separation and layer ablation. (A), direct model-space score separation at the pre-logit layer for S-RAS, Log-Euclidean distance, and activationbased baselines. (B), layer ablation for contrast-derived probes across block 6, block 8, and pre-logits. Direct model-space separation is already strong for several metrics, whereas the held-out probe effect is not confine… view at source ↗

read the original abstract

Activation-alignment measures such as Representational Similarity Analysis (RSA), Canonical Correlation Analysis (CCA), and Centered Kernel Alignment (CKA) are widely used to compare biological and artificial neural representations. Recent theoretical work interprets many of these methods as assessing agreement between optimal linear readouts over broad families of global tasks. However, agreement at the level of global readouts does not determine how a system uses local stimulus evidence. Specifically, representations may align in activation space yet differ in their sensitivity to small perturbations. To address this challenge, we introduce a complementary framework based on local decodable information, which focuses on a representation's ability, under noise, to discriminate small perturbations within a specified stimulus-coordinate subspace. Building on Fisher information and local representation geometry, we summarize each representation using the expected projected pullback/Fisher metric over that subspace. This formulation induces a second-moment family of local discrimination tasks, for which the resulting operator provides a minimal, complete dataset-level summary of expected discriminability. We compare these regularized signatures using a log-spectral distance on the manifold of symmetric positive definite (SPD) matrices, yielding the Spectral Riemannian Alignment Score (S-RAS) and a uniform multiplicative certificate over the corresponding family of lifted task values. Empirically, this framework enables the recovery of corresponding layers across independently trained artificial neural networks, supports transferable class-conditional probes, reveals controlled dissociations between standard and robust training, and uncovers stimulus-coordinate family effects across mouse visual cortex using the Allen Brain Observatory static gratings dataset.

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.

Desk Editor's Note private letter to a colleague

The paper gives a geometric local sensitivity score using projected Fisher metrics and log-spectral distances on SPD matrices that could complement global alignment measures, but its claims rest on untested choices for subspaces and noise.

read the letter

This paper introduces a measure of local neural sensitivity based on projected Fisher information and compares representations using a Riemannian distance on the resulting matrices. It aims to capture how networks differ in handling small stimulus perturbations, something global alignment scores like CKA do not address directly. The new part is the Spectral Riemannian Alignment Score, or S-RAS, which summarizes local decodable information via expected pullback metrics over a stimulus subspace. The authors derive that this gives a minimal complete summary for a family of discrimination tasks and use the log-spectral distance to get a uniform certificate. Empirically, it recovers corresponding layers in separately trained networks, distinguishes standard from robust training, and shows effects in mouse visual cortex data from the Allen Observatory. These results suggest the measure picks up meaningful differences. The approach is grounded in existing Fisher geometry ideas but applies them freshly to the alignment setting. The theoretical framing around second-moment families and the certificate is a solid step. The soft spots center on the modeling assumptions. The framework requires that the chosen subspace and additive noise model reflect the relevant local tasks. If they do not, the signatures and their distances may not correspond to actual sensitivity differences. The paper does not appear to test robustness to other subspaces or noise models, so the advantage over global methods remains tied to these choices. Checking the full derivations for how the completeness is established would also help. This work is for researchers focused on comparing representations in deep networks and biological systems who want tools beyond broad task agreement. A reader already using geometric methods will see the value in the experiments. It deserves a serious referee because the idea is distinct and the applications are concrete, though revisions on the assumptions and validation would strengthen it. I would recommend sending it for peer review.

Referee Report

1 major / 1 minor

Summary. The paper proposes a complementary framework to global activation-alignment measures (RSA, CCA, CKA) that focuses on local sensitivity to small perturbations within a stimulus-coordinate subspace. It summarizes each representation via the expected projected pullback/Fisher metric under additive noise, asserts that this induces a second-moment family of local discrimination tasks for which the operator is a minimal complete dataset-level summary, and defines the Spectral Riemannian Alignment Score (S-RAS) as the log-spectral distance on the SPD manifold, which supplies a uniform multiplicative certificate over the lifted task values. Empirical results are reported on layer recovery across ANNs, dissociations between standard and robust training, transferable probes, and stimulus-coordinate effects in mouse V1 from the Allen Brain Observatory gratings dataset.

Significance. If the modeling assumptions hold, the work supplies a principled geometric tool for quantifying local discriminability differences that global readout agreement measures miss, with the certificate property and dataset-level minimality as notable formal strengths. The empirical demonstrations on both artificial networks and biological data suggest practical utility for comparing representations across training regimes and species.

major comments (1)

[Abstract / theoretical framework] Abstract and theoretical construction: the assertion that the projected Fisher metric 'provides a minimal, complete dataset-level summary of expected discriminability' and that the log-spectral distance yields a 'uniform multiplicative certificate over the corresponding family of lifted task values' depends on the chosen stimulus-coordinate subspace and additive Gaussian noise model accurately spanning the relevant local discrimination tasks. These are modeling choices whose mismatch would make the S-RAS distance uninformative about actual sensitivity differences; no first-principles derivation or robustness check to alternative subspaces/noise models is provided to support the minimality/completeness claim.

minor comments (1)

[Method] The regularization strength for the signatures is listed as a free parameter; its effect on the claimed uniformity of the certificate and on empirical stability should be quantified or bounded.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive summary of the work and for identifying this important point about the scope of the theoretical claims. We respond to the major comment below.

read point-by-point responses

Referee: [Abstract / theoretical framework] Abstract and theoretical construction: the assertion that the projected Fisher metric 'provides a minimal, complete dataset-level summary of expected discriminability' and that the log-spectral distance yields a 'uniform multiplicative certificate over the corresponding family of lifted task values' depends on the chosen stimulus-coordinate subspace and additive Gaussian noise model accurately spanning the relevant local discrimination tasks. These are modeling choices whose mismatch would make the S-RAS distance uninformative about actual sensitivity differences; no first-principles derivation or robustness check to alternative subspaces/noise models is provided to support the minimality/completeness claim.

Authors: We agree that the minimality, completeness, and certificate properties are established relative to the family of local discrimination tasks induced by second-moment statistics under the additive Gaussian noise model within the chosen stimulus subspace. The derivation proceeds by showing that expected discriminability for tasks in this family reduces to quadratic forms governed by the projected Fisher metric (via the local expansion of the log-likelihood ratio under Gaussian perturbations), making the metric a sufficient statistic for that family; this is a direct consequence of information-geometric properties of the Fisher metric as the local Hessian of KL divergence. The log-spectral distance on SPD matrices then supplies the uniform multiplicative bound on distortion of these quadratic forms. The manuscript states these relations in the theoretical framework, but we acknowledge that the text does not include explicit robustness checks against alternative noise models or subspaces. In the revision we will (i) add a clarifying paragraph in the introduction and discussion that explicitly delimits the claims to the stated modeling assumptions and (ii) include a supplementary robustness analysis on synthetic data using alternative noise distributions (e.g., Laplace) to verify that the main empirical conclusions remain qualitatively stable. This constitutes a partial revision. revision: partial

Circularity Check

0 steps flagged

No significant circularity detected in the derivation

full rationale

The paper constructs the projected Fisher metric and S-RAS directly from the definitions of local decodable information, Fisher information, pullback metrics, and the log-spectral distance on SPD matrices. The claim that the resulting operator is a minimal complete summary of expected discriminability follows from the induced second-moment family by the stated geometric construction, without any reduction to fitted parameters on target data, self-citation chains, or renaming of prior results as new derivations. No equations or steps equate a prediction to its own inputs by construction. The framework remains independent of the alignment scores it computes.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The framework rests on standard properties of Fisher information under noise and Riemannian geometry of SPD matrices; the subspace selection and regularization are domain choices without independent external grounding.

free parameters (1)

regularization strength for signatures
Abstract refers to regularized signatures whose strength must be chosen to produce the operator summary.

axioms (2)

domain assumption Local perturbations are modeled with additive Gaussian noise whose covariance allows Fisher information to be well-defined.
Invoked when defining the expected projected pullback/Fisher metric over the stimulus subspace.
standard math The manifold of symmetric positive definite matrices with the log-spectral distance forms a valid metric space for comparing the summary operators.
Used to define the S-RAS distance and the uniform multiplicative certificate.

invented entities (1)

Spectral Riemannian Alignment Score (S-RAS) no independent evidence
purpose: To provide a distance between representations based on their local sensitivity operators.
Newly constructed score whose independent evidence is limited to the paper's own empirical demonstrations.

pith-pipeline@v0.9.0 · 5573 in / 1496 out tokens · 35345 ms · 2026-05-08T18:25:33.800897+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Foundation/AlphaCoordinateFixation.lean (cosh/log-coordinate cost in RS uses ratio symmetry x↔x⁻¹, not SPD/AIRM) CostAlphaLog / Jcost unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We compare lifted signatures with the affine-invariant Riemannian metric on S^k_++, d_AIRM(A,B) := ||log(A^{-1/2} B A^{-1/2})||_F
Foundation/ArithmeticFromLogic.lean (RS cost-uniqueness is washburn_uniqueness_aczel for J(x)=½(x+x⁻¹)−1; paper uses Fisher quadratic form, no ratio symmetry) washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the projected expected Fisher operator F_{f,ψ,P,Σ} := E_{s∼ν}[P^⊤ J_{h_f}(s)^⊤ Σ^{-1} J_{h_f}(s) P] ... is exactly the minimal complete dataset-level summary of expected local discriminability.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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Random-contrast probes:random directions sampled in the same family and ranked by the same quadratic contrast criterion
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Label-permutation null:probes derived after randomly permuting discovery-set group labels, rebuilding the contrast from those permuted labels, and deriving probes from that null contrast. Main result.At the main setting ( K= 32 , two probes per side), contrast-derived probes achieve mean held-out image separation 1.10, compared with 0.43 for random-contra...