S-RAS: Spectral Riemannian Alignment Score

• The paper introduces S-RAS as a novel measure that summarizes local Fisher metrics to quantify small-stimulus discriminability in neural representations.
• It employs an affine-invariant log-spectral distance on SPD matrices to compare the gain and shape of local sensitivity ellipsoids under noise.
• The method demonstrates practical advantages across applications such as network layer correspondence, training regime differences, and neural sensitivity in biological data.

Spectral Riemannian Alignment Score (S-RAS) is a representational similarity measure introduced in “Beyond Activation Alignment: The Geometry of Neural Sensitivity” to compare systems at the level of local decodable information rather than only activation-space agreement. The framework is motivated by the observation that activation-alignment measures such as Representational Similarity Analysis (RSA), Canonical Correlation Analysis (CCA), and Centered Kernel Alignment (CKA) assess agreement between optimal linear readouts over broad families of global tasks, yet representations may align in activation space while differing in their sensitivity to small perturbations. S-RAS addresses this by summarizing each representation through an expected projected pullback/Fisher metric over a specified stimulus-coordinate subspace, then comparing those summaries with an affine-invariant log-spectral distance on the manifold of symmetric positive definite (SPD) matrices (Yavari et al., 4 May 2026).

1. Motivation and scope

The central distinction drawn by the framework is between global readout agreement and local stimulus sensitivity. RSA, CCA, and CKA compare activation point clouds or global readout similarity: RSA uses representational dissimilarity matrices between all pairs of stimuli; CCA finds shared linear subspaces maximizing correlation; and CKA measures normalized Hilbert–Schmidt inner products of Gram matrices (Yavari et al., 4 May 2026). These methods are informative about population-level alignment, but they do not determine how a system uses local stimulus evidence.

The motivating limitation is stated sharply: two representations may support identical global decoder performance yet differ in which stimulus directions they amplify or suppress. They may therefore align activation clouds but disagree on local sensitivity to small perturbations. S-RAS is designed as a complementary framework that focuses on which small stimulus changes are most discriminable under noise. This yields a dataset-level, subspace-restricted comparison of local sensitivity ellipsoids.

A common misconception, directly addressed by the framing of the method, is that strong activation alignment is sufficient to establish representational equivalence. The paper instead treats agreement in local Fisher geometry as a distinct question. This suggests that representational comparison can be decomposed into at least two regimes: one governed by global readout structure and another governed by local sensitivity structure.

2. Formal definition

The framework begins with a differentiable layer representation

$$f:\mathbb{R}^d \to \mathbb{R}^m,\qquad x\sim\mathcal{D}.$$

Its pullback (Jacobian) metric at $x$ is

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$

When stimuli are parameterized by coordinates $s\in\Omega\subset\mathbb{R}^p$ with $\psi:s\mapsto x$, the induced representation is

$$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$

Under additive Gaussian noise $\Sigma$ on the representation, the pointwise local Fisher metric is

$$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$

To restrict perturbations to a $k$-dimensional stimulus subspace $\mathrm{span}(P)$, with $x$0 orthonormal and $x$1, the projected expected Fisher operator is

$x$2

In the isotropic noise case $x$3, up to the scalar $x$4 this reduces to the pullback form

$x$5

To compare two PSD summaries $x$6 and $x$7, each is lifted to strictly SPD by

$x$8

The affine-invariant log-spectral distance on $x$9 is

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$0

Thus, for two representations $M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$1 and $M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$2,

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$3

A normalized similarity score is then obtained by

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$4

In this construction, the operator $M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$5 or $M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$6 is the primary object, while the exponentiated score is a derived similarity. The formalism is therefore simultaneously metric-geometric and task-oriented (Yavari et al., 4 May 2026).

3. Derivation from local discrimination tasks

The derivation starts from a two-gaussian local discrimination task,

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$7

Under small $M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$8 and linearization, the squared discriminability is

$M_f(x)=J_f(x)^\top J_f(x)\in\mathbb{S}_+^d.$9

Averaging over a second-moment family with $s\in\Omega\subset\mathbb{R}^p$0 gives the trace form

$s\in\Omega\subset\mathbb{R}^p$1

Restricting perturbations to $s\in\Omega\subset\mathbb{R}^p$2, with $s\in\Omega\subset\mathbb{R}^p$3, and averaging over stimuli $s\in\Omega\subset\mathbb{R}^p$4 yields the projected operator $s\in\Omega\subset\mathbb{R}^p$5.

This derivation is important because it identifies the operator as a summary over a family of local tasks, rather than as a heuristic descriptor. The paper states that the resulting operator provides a minimal, complete dataset-level summary of expected discriminability for the induced second-moment family of local discrimination tasks.

The choice of log-spectral geometry is then justified by the geometry of SPD matrices. The manifold of SPD matrices carries a canonical affine-invariant Riemannian metric (AIRM) whose geodesic distance is exactly

$s\in\Omega\subset\mathbb{R}^p$6

This distance is invariant to congruence transforms $s\in\Omega\subset\mathbb{R}^p$7, matching how metric operators transform under changes of basis. The method therefore compares both gain and shape of discriminability ellipsoids rather than only aggregate scale.

4. Theoretical properties

Three theoretical properties are emphasized in the summary: affine-invariance, a uniform multiplicative certificate, and minimal completeness (Yavari et al., 4 May 2026).

Affine-invariance follows directly from the choice of metric. By construction, $s\in\Omega\subset\mathbb{R}^p$8 is invariant under congruence transforms $s\in\Omega\subset\mathbb{R}^p$9 and $\psi:s\mapsto x$0. This is the invariance class naturally associated with Fisher and pullback operators.

Uniform multiplicative certificate is obtained from the log-spectral distance. If $\psi:s\mapsto x$1, then

$\psi:s\mapsto x$2

Consequently, for any probe covariance $\psi:s\mapsto x$3,

$\psi:s\mapsto x$4

The distance therefore controls the worst-case log-multiplicative disagreement across all lifted second-moment tasks.

Minimal complete summary is established through a trace-separation lemma. Knowing $\psi:s\mapsto x$5 determines all expected discriminabilities $\psi:s\mapsto x$6; conversely, if two representations agree on all $\psi:s\mapsto x$7, then their $\psi:s\mapsto x$8 operators coincide. Hence $\psi:s\mapsto x$9 or $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$0 is the unique minimal dataset-level summary of the chosen local tasks.

These properties distinguish S-RAS from similarity indices that aggregate pairwise or subspace relations without an explicit task certificate. A plausible implication is that the method is intended not only as a scalar similarity score but also as a representation of what local evidence is decodable under the chosen noise model and coordinate restriction.

5. Computational procedure and numerics

The computational recipe is explicit and finite-dimensional.

1. Choose a set of $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$1 stimulus samples $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$2 and a $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$3-dimensional orthonormal subspace $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$4.
2. For each $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$5, compute via Jacobian-vector products the $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$6 matrix $$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$7.
3. Form the empirical average

$$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$8

1. Lift to SPD:

$$h_f(s)=f(\psi(s)),\qquad J_{h_f}(s)=J_f(\psi(s))\cdot D\psi(s).$$9

1. Repeat for a second representation $\Sigma$0 to obtain $\Sigma$1.
2. Compute the eigenvalues $\Sigma$2 of

$\Sigma$3

1. Evaluate the distance

$\Sigma$4

and optionally normalize or exponentiate.

The numerical profile is also specified. The cost is $\Sigma$5 JVPs per sample, or one JVP with $\Sigma$6-vector batch. Building $\Sigma$7 is $\Sigma$8 to form $\Sigma$9 and $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$0 to accumulate. Eigen-decomposition of a $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$1 SPD matrix costs $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$2. For regularization, the summary states that one may add $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$3 before inversion or lift to ensure strict SPD.

This procedure makes the method subspace-centric: the computational burden depends on the chosen coordinate dimension $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$4, not directly on the ambient stimulus dimension $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$5. This suggests that coordinate-restricted comparisons are an intended operating regime rather than a secondary option.

6. Relationship to RSA, CCA, and CKA

The paper places S-RAS alongside established activation-alignment measures rather than presenting it as a replacement (Yavari et al., 4 May 2026). The comparison is organized around what each family of methods measures.

Method	Comparison target	Limitation identified
RSA	Representational dissimilarity matrices between all pairs of stimuli	Does not determine how local stimulus evidence is used
CCA	Shared linear subspaces maximizing correlation	May miss differences in amplified or suppressed stimulus directions
CKA	Normalized Hilbert–Schmidt inner products of Gram matrices	Can align activation clouds while local sensitivity differs
S-RAS	Local Fisher geometry over a specified stimulus-coordinate subspace	Designed to compare local sensitivity ellipsoids

The key conceptual distinction is that RSA, CCA, and CKA compare activation point clouds or global readout similarity, whereas S-RAS focuses on local Fisher geometry. It asks which small stimulus changes are most discriminable under noise, then compares the resulting operators after averaging over stimuli and restricting to a chosen coordinate family.

The framework is therefore complementary to methods based on global decoder agreement. It is intended for settings in which local stimulus perturbations, coordinate systems, and noise models are central to the scientific question. In biological datasets, this includes explicitly parameterized stimulus families; in artificial networks, it includes controlled perturbation subspaces and class-conditional contrasts.

7. Empirical applications

The summary lists four empirical applications of S-RAS, spanning artificial neural networks and mouse visual cortex (Yavari et al., 4 May 2026).

Layer correspondence in Tiny10 CNNs: S-RAS recovers the architecturally corresponding layers across independently trained models. Although linear CKA has higher raw top-1, S-RAS shows stronger off-diagonal suppression and coherent layer-distance decay as task subspace dimension $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$6 grows.

Class-conditional diagnostic probes: For two model families $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$7 versus $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$8, class-conditional $$I_{f,\psi}(s)=J_{h_f}(s)^\top \Sigma^{-1}J_{h_f}(s)\in\mathbb{S}_+^p.$$9 contrasts $k$0 are formed. Their leading eigenvectors yield finite smooth perturbations. These “contrast-derived probes” separate held-out images and held-out models better than random or pooled-sensitivity probes, demonstrating transfer of local-sensitivity differences.

Standard versus robust training dissociation: Comparing ERM versus PGD/TRADES/MART ResNet-18 banks under fixed architecture, S-RAS contrasts reveal regime-specific shifts in local evidence use. Contrast probes separate held-out images between training objectives, even within robust families.

Mouse visual cortex static gratings: Stimuli parameterized by $k$1 define a natural $k$2 coordinate system. For each experiment $k$3, the procedure estimates condition-mean response $k$4 and noise covariance $k$5, computes local Fisher and pullback metrics, and averages over $k$6 or coordinate-restricted families. In this setting, S-RAS retrieves matching cortical areas across animals and reveals depth- and stimulus-family-dependent structure, outperforming activation-based, decoder, and mapping baselines in area retrieval.

Taken together, these applications situate S-RAS as a geometry-aware measure of representational similarity that quantifies how two systems agree on making small stimulus perturbations discriminable under noise. The empirical results suggest that local sensitivity can separate models, training regimes, and cortical populations even when activation-based comparisons are less discriminative.