CKA(X,Y) = ||XᵀY||_F² /
            sqrt(||XᵀX||_F² ||YᵀY||_F²).

Advanced Technical Expansion I | Deriving CKA, CCA, SVCCA, RSA and Procrustes from Their Invariance Choices

The base article names the major methods and the transformations they tend to ignore. This expansion derives those choices more explicitly. The objective is not symbolic ornament. It is to make every similarity score answerable to a simple question: what differences between two representations has this formula decided not to care about?

A. Start with centred matrices and keep the sample axis fixed

Let X contain n examples by p features and Y contain the same n examples by q features. Define H=I-(1/n)11ᵀ. Then X_c=HX and Y_c=HY. Centering subtracts each feature’s mean over the chosen stimulus set.

The shared row order is essential. Similarity methods compare how the same stimuli are organised by two systems. If row correspondence is wrong, a high score can only arise accidentally.

For sequence models, “one row” also requires a policy: token position, pooled sequence, final token, aligned span or another unit. Representation analysis begins by defining the observation.

B. Covariance matrices separate within-space scale from cross-space relation

Define empirical covariance-like matrices C_xx=X_cᵀX_c/(n-1), C_yy=Y_cᵀY_c/(n-1) and cross-covariance C_xy=X_cᵀY_c/(n-1). Many linear similarity methods can be understood as different normalisations or factorizations of these three objects.

C_xx tells us how X varies internally. C_yy does the same for Y. C_xy tells us which linear directions co-vary across the representations. The methods differ in how aggressively they whiten, normalise or align these structures.

C. Orthogonal Procrustes preserves lengths and angles by construction

The orthogonal Procrustes problem seeks Q with QᵀQ=I that minimises ||XQ-Y||_F² for equal feature dimension. Expanding the norm shows that the only Q-dependent term is -2 tr(QᵀXᵀY). Maximising that trace gives the SVD solution.

XᵀY = U Σ Vᵀ
Q* = U Vᵀ.

Because Q is orthogonal, it preserves inner products, norms and Euclidean distances. Procrustes therefore declares global rotation/reflection of the feature basis irrelevant while treating axis-specific scaling and shear as real differences.

This narrow invariance makes Procrustes useful when metric geometry matters. It is less forgiving than CCA.

D. Procrustes residual can be normalised several ways

Raw Frobenius residual grows with sample count and representation scale. One can normalise by ||Y||_F, total energy of both matrices, or convert alignment into a cosine-like score.

The chosen normalisation determines comparability across layers of different activation magnitude. “Procrustes similarity” should therefore include the exact score definition.

When p differs from q, ordinary square orthogonal alignment no longer directly applies. One can reduce dimension, compare subspaces or use rectangular semi-orthogonal maps under a different contract.

E. CCA whitens each representation before comparing them

CCA asks for vectors a and b maximising

corr(Xa,Yb)
= aᵀ C_xy b /
  sqrt(aᵀC_xx a · bᵀC_yy b).

Whitening transforms each representation so its covariance becomes identity on the retained subspace. The cross-covariance between whitened spaces is then decomposed by SVD. Its singular values are the canonical correlations.

Conceptually, CCA asks: after each space is allowed to choose its own linear coordinate system and variance scale, which directions can be made maximally correlated?

F. Whitening creates broad linear invariance

If X is transformed by an invertible linear map and the covariance estimate remains full rank, CCA can often undo that map through its whitening/projection step. This makes CCA attractive for information-equivalence questions and insensitive to metric structure changed by shear or anisotropic scaling.

This broad invariance is exactly why CCA can become uninformative in high-dimensional finite-sample settings: enough linear freedom can make many spaces look highly correlated.

Invariance is not free. Every ignored transformation is a kind of information the measure refuses to treat as difference.

G. Singular covariance forces regularisation or dimension reduction

If p or q approaches or exceeds n, sample covariance matrices are singular. Exact whitening is undefined in null directions. Practical CCA therefore uses pseudoinverses, regularisation or dimensionality reduction.

Regularised CCA adds terms such as λI to covariance matrices. This changes the invariance properties and shrinks unstable directions. The regularisation value belongs in the reproducibility record.

A CCA score obtained after strong regularisation answers a different question from unregularised population CCA.

H. SVCCA removes low-variance directions before whitening

SVCCA first performs SVD on each activation matrix and retains enough singular directions to explain a chosen fraction of variance. CCA is then run on the reduced spaces.

This serves two purposes: reduce numerical instability and focus on dominant representation structure. It also encodes the assumption that low-variance directions are less important.

For rare safety or fine-grained features, low variance can still be behaviourally important. Sensitivity to the retained-variance threshold should therefore be reported when such features matter.

I. PWCCA changes how the canonical spectrum becomes one score

After CCA, we have correlations ρ_1,...,ρ_k. An unweighted mean treats each canonical direction equally. PWCCA assigns larger weights to directions that have greater projection onto the original representation’s activations.

This attempts to distinguish a high-correlation direction that explains substantial native activity from one that lives in a tiny, low-impact subspace.

The weighting creates asymmetry depending on which representation is treated as the reference. Check conventions when comparing reported PWCCA scores.

J. Principal angles are the geometric heart of subspace comparison

Let U and V be orthonormal bases for two subspaces. Singular values of UᵀV are cosines of principal angles. A zero angle means the subspaces share one direction exactly. Angles near ninety degrees indicate orthogonality.

CCA after whitening can be understood through a related singular-value geometry. Principal angles are especially clean when the question is whether two models span similar linear subspaces without regard to coordinates inside those subspaces.

If one subspace has larger dimension, extra directions require a policy: ignore them, penalise them or compare containment asymmetrically.

K. Linear CKA can be derived from centred Gram matrices

For centred representations, define linear Gram matrices K=XXᵀ and L=YYᵀ. Their Frobenius inner product measures whether pairwise stimulus similarities covary. Normalising by each matrix’s Frobenius norm produces the CKA form.

CKA = ⟨K,L⟩_F / (||K||_F ||L||_F).

Using trace identities yields the equivalent feature-space formula involving ||XᵀY||_F². The two views are valuable: feature space explains covariance alignment; Gram space explains preservation of relationships among examples.

L. Orthogonal invariance of CKA follows immediately from Gram preservation

If X'=XQ with orthogonal Q, then X'X'ᵀ=XQQᵀXᵀ=XXᵀ. The Gram matrix is unchanged, so linear CKA is exactly unchanged.

If X'=cX, the Gram matrix scales by c², but normalisation cancels the scale. CKA therefore ignores isotropic scaling.

Anisotropic scaling generally changes the Gram matrix and therefore changes CKA. This is the narrower invariance Kornblith and colleagues wanted.

M. Arbitrary invertible transforms can drive CKA down while preserving linear decodability

Take X and multiply one feature direction by 100 while shrinking another by 0.01. An invertible downstream linear layer can undo this transformation exactly. Yet pairwise dot products among stimuli change, so CKA can fall.

This is not a defect when geometry matters. It is a deliberate sensitivity. CKA says the organisation of representational energy changed even if linear information content did not.

CCA and CKA disagree here because they answer different notions of sameness.

N. HSIC supplies the dependence statistic underneath kernel CKA

Given kernel matrices K and L over stimuli, HSIC measures dependence by a centred trace such as tr(KHLH) up to normalising constants. CKA normalises HSIC by the self-HSIC of each representation.

For characteristic kernels and appropriate population conditions, HSIC can detect broad statistical dependence. In finite samples, biased and unbiased estimators behave differently. The common biased CKA estimator can have nonzero baseline in high-dimensional or small-sample regimes.

When comparing very wide representations with limited examples, estimator choice deserves explicit testing.

O. Kernel CKA makes locality a bandwidth choice

With an RBF kernel k(x_i,x_j)=exp(-||x_i-x_j||²/(2σ²)), bandwidth σ controls which distances matter. Very small σ makes only near-identical neighbours similar. Very large σ makes most examples similar and can flatten distinctions.

A median-distance heuristic is common and should not be mistaken for a universal optimum. Sweep bandwidth if nonlinear CKA supports a scientific conclusion.

Kernel CKA measures similarity after both representations are filtered through the chosen kernel geometry.

P. RSA starts from a representational dissimilarity matrix

For n stimuli, construct an n×n matrix D where D_ij measures how differently the representation responds to stimuli i and j. This relational object can be compared across systems with different feature dimensionality or modality.

The price is computational and statistical: there are n(n-1)/2 pairwise entries, but they are not independent because each stimulus participates in many pairs.

Naïve statistical tests treating every pair as an independent sample can dramatically overstate precision.

Q. Correlation distance creates invariance to affine feature-response scale per stimulus vector

One common RSA inner metric is 1-corr(r_i,r_j), where r_i is the feature-response vector for stimulus i. Correlation removes each vector’s mean and scale, focusing on response pattern shape.

Euclidean distance preserves norm differences. Mahalanobis distance accounts for noise covariance. Different inner metrics can reorder stimulus relationships completely.

The RDM is not representation-independent. It is a representation filtered through a metric choice.

R. Spearman RSA preserves ordinal geometry

Comparing RDM entries with Spearman correlation makes the score invariant to monotonic transformations of dissimilarity. If one representation stretches all large distances while preserving pair ranking, Spearman RSA remains high.

Pearson comparison is sensitive to linear magnitude structure. Kendall rank offers another ordinal option. The outer metric states which differences in the RDM are allowed.

An RSA result should always name both the RDM metric and the RDM-comparison statistic.

S. Noise ceilings distinguish model mismatch from measurement limits

In neuroscience RSA, repeated measurements contain noise. Even the true underlying representation cannot correlate perfectly with one noisy estimate. Noise-ceiling analysis estimates the maximum explainable similarity given measurement reliability.

Machine-learning activations are often deterministic for fixed input, but stochastic preprocessing, dropout, sampling or finite stimulus sets create analogous uncertainty. A similarity of 0.8 can be excellent if the measurement ceiling is 0.82 and mediocre if repeated estimates are 0.99 reliable.

Estimate repeatability before interpreting absolute similarity.

T. Debiasing matters when feature dimension rivals sample count

Similarity estimators can have upward bias under finite samples. High-dimensional random representations can appear nontrivially similar simply because covariance and Gram estimates are noisy.

Use random baselines matched in n and feature dimension. Bootstrap stimuli. Where available, use debiased estimators and report their behaviour.

A high score is meaningful only relative to the null geometry of the measurement regime.

U. Random baselines reveal what a similarity measure calls similar by accident

Generate independent Gaussian representations with the same dimensions and sample count. Compute CKA, CCA or RSA. Repeat to obtain a null distribution. Then introduce controlled shared subspaces or correlations and observe how the measure responds.

This calibration is especially valuable for custom pipelines. Before trusting a similarity number from billion-parameter models, confirm how the implementation behaves on systems whose relationship is known exactly.

V. Controlled transformation tests provide an invariance unit test

Take one representation X and construct variants: permuted columns, orthogonally rotated, isotropically scaled, anisotropically scaled, sheared, nonlinear monotonic transforms and dimension-augmented noise. Compute every candidate similarity measure.

The resulting invariance table makes each metric tangible. If the implementation reacts contrary to theory, investigate centering, regularisation or numerical bugs.

This should be a standard unit test for representation-comparison code.

W. An invariance table prevents method names from replacing reasoning

A useful conceptual table asks whether a measure ignores feature permutation, orthogonal transform, isotropic scale, translation, arbitrary invertible linear transform and monotonic pairwise-distance remapping. No measure should receive a universal “best” label.

Procrustes is strict about shear. CCA is broadly invariant to linear coordinate changes under its assumptions. Linear CKA ignores orthogonal transform and global scale but preserves anisotropic geometry differences. Spearman RSA can ignore monotonic distortion of pairwise dissimilarity.

Choose the row of the table matching the scientific equivalence you intend.

X. The same transformation can be irrelevant for one receiver and crucial for another

An anisotropic scaling can be irrelevant to a downstream linear classifier that can compensate and crucial to nearest-neighbour retrieval because it changes distances. A neuron permutation can be irrelevant to function and crucial to a monitoring system tied to neuron identities.

Representation similarity cannot be chosen independently of downstream use. The invariance class should mirror what the receiver can or cannot compensate for.

This is the same World → Representation → Receiver principle operating on representations themselves.

Y. Containment is asymmetric and most standard similarity scores are symmetric

Suppose Y contains all information in X plus additional useful features. A symmetric score asks how alike the full spaces are. A receiver may instead ask whether X is linearly recoverable from Y.

Train a regularised linear map from Y to X and measure held-out prediction. Reverse the direction separately. Asymmetric predictability reveals containment relationships that symmetric CKA or RSA compress into one number.

“A is represented inside B” is not the same claim as “A and B are equivalent”.

Z. Cross-validation prevents alignment from memorising the stimuli

Procrustes, CCA and learned linear maps can overfit finite examples, especially when feature dimension is high. Fit alignment on one stimulus subset and evaluate on held-out stimuli.

If similarity collapses out of sample, the alignment captured idiosyncratic sample geometry rather than a stable relationship between representations.

Cross-validated alignment is essential when the alignment itself has many free parameters.

AA. Worked laboratory: one representation under six transformations

Generate a centred random X. Create Y variants through column permutation, orthogonal rotation, global scale, anisotropic scale, invertible shear and nonlinear elementwise transformation.

Compute neuron correlation, Procrustes, CCA and CKA. The metrics should separate according to their theoretical invariances. Add output heads that undo the invertible transformations and show that functional equivalence can remain perfect while representational scores differ.

This one lab teaches more than memorising the method names.

AB. Worked laboratory: CCA inflation when p approaches n

Generate independent random X and Y while increasing feature dimension relative to sample count. Compute naïve CCA. Canonical correlations can become spuriously high because the model has enough degrees of freedom to align sample noise.

Add ridge regularisation, SVCCA truncation and held-out evaluation. Compare results. This demonstrates why arbitrary invertible-linear invariance becomes dangerous in overparameterised finite-sample settings.

AC. Worked laboratory: CKA sensitivity to anisotropic scaling

Start with Y=X. Multiply one principal direction in Y by 100 while leaving others unchanged. Linear decodability from Y to X remains perfect because the transform is invertible. CKA falls because pairwise geometry shifts toward the amplified direction.

Use this experiment whenever someone interprets a disagreement between CKA and CCA. Both can be correct under their respective contracts.

AD. Worked laboratory: RSA with rank-preserving distance warp

Create pairwise distances D and transform them monotonically, for example D'=D². Spearman RSA between D and D’ remains one if ordering is unchanged; Pearson similarity can change.

The experiment makes ordinal versus metric geometry visible.

AE. The mathematical evidence card

• Representation matrices: rows, columns and centering policy.
• Stimuli: dataset, sampling and alignment.
• Metric: Procrustes, CCA family, CKA family, RSA or other.
• Invariances: transformations intentionally ignored.
• Dimensions: n, p and q and effective ranks.
• Regularisation: ridge, PCA/SVD threshold or kernel bandwidth.
• Estimator: biased/debiased HSIC or other finite-sample choice.
• Alignment: fitted on train stimuli or evaluated in sample.
• Null baseline: random similarity expected under matched dimensions.
• Sensitivity: stimulus subsets and hyperparameters.
• Asymmetry: whether containment rather than equivalence is the real question.

The first advanced layer shows why no representation-similarity method can be interpreted without its invariance class. Every formula is a declaration of equivalence. The next layer moves from formulas to research design: uncertainty, layer matching, checkpoint trajectories, cross-model feature comparisons, causal equivalence and the conditions under which a similarity heatmap deserves an explanatory story.

Advanced Technical Expansion II | Statistical Reliability, Model Alignment, Functional Equivalence and Causal Comparison

A similarity score becomes scientifically useful only when we know how stable it is, which stimuli produced it and what stronger claims it predicts. This second layer turns representation comparison into an experimental discipline. We move from “layer A has CKA 0.82 with layer B” to questions such as: does that value survive another dataset, another seed, another estimator or another model width? Does high internal similarity predict the same errors, the same transfer behaviour or the same causal response to intervention? And if it does not, which notion of sameness was the original score actually measuring?

AF. Bootstrap stimuli to obtain uncertainty on the similarity estimate

Sample the stimulus rows with replacement, recompute the representation similarity and repeat. The resulting distribution gives a bootstrap estimate of uncertainty caused by finite stimulus selection.

Pairs of representations with scores 0.80 and 0.83 may be indistinguishable if bootstrap intervals overlap broadly. A heatmap without uncertainty can encourage stories about tiny differences that are not stable.

For grouped data such as multiple tokens from one document or frames from one video, bootstrap at the independent group level rather than treating correlated rows as independent samples.

AG. Subsample curves show how much data the similarity estimate needs

Compute similarity using 100, 300, 1,000, 3,000 and more stimuli. Plot mean and variability across subsamples. A stable plateau suggests the estimate is adequately supported for that distribution.

If similarity drifts as n grows, early values reflected sampling noise or rare states missing from small sets. Wide language-model layers especially need enough stimuli to expose diverse activation directions.

Report a sample-size curve for high-stakes comparison rather than one arbitrary dataset size.

AH. Stratified resampling reveals where similarity comes from

Calculate similarity separately for categories, domains, languages, difficulty levels or behavioural states. Two models can align strongly on common data and diverge sharply on a small but important slice.

Weighted global similarity then reflects the prevalence of each slice in the stimulus set. If safety-relevant prompts are rare, they contribute little to a global CKA score.

Use a similarity profile, not only a scalar, when the receiver cares about heterogeneous conditions.

AI. Stimulus-set shift can reverse conclusions about which layers match

One model pair may share similar representations for natural language and diverge on code; another may show the reverse. Layer-to-layer alignment matrices can therefore change with the evaluation corpus.

This is not necessarily measurement failure. Representation is input-dependent. A layer can implement several regimes whose relative frequency differs by dataset.

When claiming a universal layer correspondence, test several sufficiently different stimulus distributions.

AJ. Compare the same model with itself under different input transformations

Representation similarity is also useful within one model. Compare activations for original and augmented inputs, paraphrases, rotations, translations or adversarial perturbations.

This measures invariance or equivariance indirectly. High CKA under a transformation suggests relational geometry is preserved, but it does not tell us whether individual examples map to their intended counterparts after a structured transform.

Pair it with example-wise alignment or known transformation laws when the receiver needs equivariance rather than broad distributional similarity.

AK. Heatmaps can hide one-to-many layer correspondence

A layer in model A can resemble a band of layers in model B. This can occur when one architecture performs a computation gradually and another compresses it into one stage.

Forcing a one-to-one matching loses this structure. Treat the heatmap as a soft correspondence matrix. Dynamic-programming or monotonic-alignment methods can identify broad stage mappings while allowing different depths.

The alignment algorithm itself introduces assumptions such as monotonic depth order. State them.

AL. Depth-normalised layer index is only a baseline

Matching layer 6 of 12 to layer 12 of 24 by relative depth is simple and sometimes sensible. It assumes computation progresses at the same rate through both networks.

Similarity heatmaps can test this assumption. Stage boundaries in CNNs, transformer blocks with different widths or hybrid architectures can violate proportional depth.

Use relative depth as a null model, not as the final correspondence.

AM. Training-time similarity trajectories can reveal phase changes

Compare each checkpoint to the final representation and to neighbouring checkpoints. Smoothly increasing similarity suggests gradual convergence. Sudden drops or reorganisation can coincide with learning-rate changes, capability emergence or loss-regime transitions.

Track multiple measures. A basis can rotate sharply while CKA remains stable; feature axes can reorganise while subspace similarity persists. Method disagreement reveals the type of change.

Training dynamics deserve time-series analysis rather than two endpoint snapshots.

AN. Similarity to initialization measures how much a layer actually learned

Some layers remain surprisingly similar to random initialization under certain training regimes while others reorganise strongly. Compare trained activations with the same network at initialization on a fixed reference set.

High similarity does not mean the layer is useless; small weight changes can produce meaningful downstream effects. It does indicate much of the original relational geometry remains.

Combine with ablation and functional evaluation to determine whether the stable representation participates critically in the learned solution.

AO. Cross-seed similarity establishes an empirical ceiling for architecture stability

Train several copies of the same architecture on the same data. Their pairwise similarities show how reproducible the representation is under stochastic optimisation.

When comparing two different architectures, this within-architecture distribution is an important reference. A cross-architecture score of 0.7 can be impressive if same-architecture seeds average 0.75 and unimpressive if they average 0.99.

Similarity needs a baseline drawn from the relevant model family.

AP. Permutation matching can recover neuron correspondence when a basis really is axis-aligned

For some networks, features align strongly with individual units and independent runs differ mainly by permutation. Build a neuron-correlation matrix and solve an assignment problem to maximise total correlation.

High matched correlations suggest axis-level reproducibility. Low matched correlation with high CKA suggests a shared distributed geometry without stable neuron identities.

This distinction connects directly to superposition and sparse-autoencoder analysis.

AQ. Sparse feature dictionaries create a new object for similarity analysis

After training SAEs on two models or two seeds, we can compare decoder dictionaries, sparse activation matrices or feature-circuit graphs. Whole residual streams can be similar while dictionary atoms differ in granularity.

CKA on sparse feature activations asks whether stimuli activate similar relational patterns. Decoder cosine matching asks whether atoms write similar native directions. Semantic feature matching asks whether human-readable concepts recur.

No one of these comparisons subsumes the others. Feature-space comparison should preserve the evidence type.

AR. Dictionary width creates a granularity mismatch

A 16× SAE can split one broad feature from a 4× SAE into several children. One-to-one feature matching reports low consistency even when the fine dictionary refines the coarse one naturally.

Compare subspaces, aggregated activations or feature hierarchies. Represent split/merge relations explicitly. This is an example of why whole-space similarity and feature correspondence can disagree without contradiction.

AS. Model merging requires stronger compatibility than high CKA

Two networks can have similar representations under CKA and incompatible parameter bases for direct weight averaging. Weight-space merging requires coordinate alignment of parameters or compatible functional basins.

Representation similarity can motivate alignment and cannot certify that averaging weights preserves function. Conversely, models in one weight-space basin can be mergeable even if one internal layer metric is modest.

Do not use representation similarity as a substitute for post-merge functional evaluation.

AT. Distillation can be evaluated as a three-layer similarity problem

Compare teacher and student at three levels: function (outputs and errors), representation (CKA/CCA/RSA) and features (specific probes or SAE concepts). A student can match any two and differ on the third.

If the deployment only needs outputs, internal divergence may be acceptable. If interpretability controls or transfer depend on internal state, it matters.

The required similarity follows the system contract, not a generic preference for internal mimicry.

AU. Pruning similarity should be measured at the sparsity levels that matter operationally

As weights are pruned, plot representation similarity to the dense model alongside accuracy, calibration and rare-behaviour metrics. Internal geometry can drift before aggregate performance changes.

Alternatively, a layer can maintain high CKA while sparse specialised features disappear. Include targeted probes and SAE feature matches when those distinctions matter.

This produces a compression phase diagram rather than a single before/after comparison.

AV. Quantisation can introduce anisotropic perturbations

Low-precision quantisation changes weights and activations unevenly across channels depending on scale factors and outliers. CKA may reveal relational geometry remains stable while neuron-level correlations degrade, or vice versa.

Compare several quantisation schemes on the same stimuli. If a downstream linear probe remains stable but nearest-neighbour retrieval changes, geometry and information content have diverged in different ways.

AW. Multimodal comparison requires paired stimuli and comparable units

An image encoder produces patches; a text encoder produces tokens. To compare them, define a common row unit: paired image-caption examples, pooled sequence embeddings, matched object-word regions or another relation.

Pooling can create apparent alignment by averaging away modality-specific structure. Token/patch-level comparison can reveal fine alignment and needs a correspondence model.

High global cross-modal similarity should not be interpreted as one-to-one semantic coordinates unless local alignment is also tested.

AX. Brain–model RSA requires measurement-noise and task controls

RSA is attractive because brain recordings and neural networks have incompatible coordinate systems but can respond to the same stimuli. A correlation between RDMs shows they organise those stimuli similarly under the selected metrics.

It does not prove the brain and model implement the same algorithm. Shared stimulus category, low-level visual features or task demands can drive the geometry.

Use noise ceilings, competing model RDMs, partial correlation or regression RSA and experimental manipulation to narrow interpretation.

AY. Partial RSA can separate candidate explanatory geometries

Suppose a neural RDM correlates with both object category and image colour. Because category and colour are correlated in the dataset, raw RSA cannot tell which factor explains the alignment.

Multiple-regression or partial-correlation approaches compare several model RDMs jointly. Collinearity can still make estimates unstable. Experimental factorisation remains stronger evidence than statistical adjustment alone.

AZ. Functional similarity should include disagreement structure, not only average output distance

Compute joint correctness categories: both correct, A only, B only, both wrong same answer, both wrong different answers. For generative models, compare semantic outcomes, refusal behaviour, factual errors and calibration rather than token equality alone.

Two models with small average logit distance can disagree exactly on the hard cases that matter. Functional similarity should be stratified by task difficulty and risk.

BA. Decision-boundary similarity is a stronger functional comparison

Sample perturbations around inputs and ask whether the models change decisions in the same regions. Similar decision boundaries indicate local functional equivalence beyond agreement on observed examples.

This can be expensive in high dimensions and depends on the perturbation distribution. Adversarial directions probe a different neighbourhood from random noise.

As always, the receiver defines which local changes are meaningful.

BB. Probe-transfer similarity asks whether the same readout works across representations

Train a probe on representation X, align X to Y if needed, and test whether the readout transfers. If the same decision vector works after a simple alignment, the spaces share more than generic information—they organise the task similarly.

Probe transfer can be asymmetric and alignment-dependent. It is a receiver-specific functional test, not a universal similarity metric.

BC. Causal feature matching tests whether aligned directions have aligned effects

Suppose CCA or Procrustes identifies matched subspaces across two models. Intervene along a matched direction in each model and compare changes in output or downstream feature states.

If geometry aligns but intervention effects differ, the direction occupies a similar statistical location and plays a different computational role. If effects align across contexts, the case for mechanistic correspondence strengthens.

Causal matching is expensive and is the appropriate escalation when descriptive similarity supports a strong mechanistic claim.

BD. Intervention magnitude needs cross-model calibration

A unit step in model A’s activation space may have different scale from a unit step in model B. Match interventions by standard deviation, natural activation quantile, effect on reconstruction or another principled scale.

Otherwise one model can appear more causally sensitive simply because the perturbation was larger relative to its native distribution.

Causal similarity inherits all the calibration issues of ordinary representation comparison plus intervention design.

BE. Multiple measures should disagree in predictable ways

Do not expect CKA, CCA, RSA and Procrustes to produce the same ordering of model pairs. Construct hypotheses about disagreement. If two spaces differ mainly by anisotropic scaling, CCA should remain high while CKA and Procrustes may fall. If pairwise distance ranks are preserved under nonlinear warp, Spearman RSA can remain high.

Unexpected disagreement is a diagnostic opportunity. Method ensembles are useful when interpreted structurally, not averaged into one meta-score.

BF. A measure-selection tree is more useful than a universal recommendation

• Need neuron identity? Use direct matching or permutation alignment.
• Need metric geometry after rotation? Use Procrustes-like alignment or CKA.
• Need shared linear information despite reparameterisation? Use CCA/SVCCA/PWCCA with regularisation and held-out checks.
• Need pairwise relational ordering across different modalities? Use RSA with explicit inner/outer metrics.
• Need nonlinear dependence? Consider kernel methods with bandwidth sensitivity tests.
• Need containment? Use asymmetric prediction or subspace inclusion measures.
• Need behavioural equivalence? Compare outputs, errors and decision boundaries.
• Need mechanistic equivalence? Add causal interventions.

BG. Multiple comparisons can make layer heatmaps look significant everywhere

A 48×48 layer matrix contains 2,304 comparisons. Selecting the largest cells after inspection creates an extreme-value bias even under weak relationships.

Use preregistered layer hypotheses where possible, correct inferential tests for multiplicity, or treat the heatmap as exploratory and validate selected correspondences on new stimuli.

Visual salience is not statistical evidence.

BH. Similarity thresholds need a domain-specific reference distribution

There is no universal rule that CKA 0.7 is “high”. Compare against same-model resampling, same-architecture different seeds, unrelated models, random features and known transformed controls.

The reference distribution gives the scalar meaning. A value can be near the maximum achievable under measurement noise or barely above random.

A similarity dashboard should display baselines, not one number in isolation.

BI. Similarity can be directional over training even when the metric is symmetric

CKA(X_t,X_final) is symmetric numerically, but the scientific story is directional: how much has the current checkpoint approached the final representation? The same score can be embedded in a time arrow defined by checkpoint order.

Do not confuse metric symmetry with causal or developmental symmetry. Training creates chronology outside the formula.

BJ. Similarity can be high because both models follow the input rather than each other

Two representations can correlate because both encode a dominant factor such as class or luminance. This does not mean one copied the other or that their remaining structure is similar.

Regress out or stratify known factors, then recompute residual similarity. Alternatively compare models on stimuli that decorrelate candidate explanatory variables.

Common cause is a third explanation for representational alignment.

BK. Similarity can diagnose representation collapse

If a representation collapses toward a low-rank or constant state, some similarity measures can behave oddly. Two collapsed spaces can appear similar because both contain little structure.

Always report self-structure: effective rank, variance spectrum or Gram diversity. High similarity between two impoverished representations is not evidence of rich shared semantics.

Similarity needs a quality floor.

BL. Similarity analysis can compare contrastive-learning objectives

Train encoders with InfoNCE, BYOL, VICReg and supervised learning. Compare layer-wise CKA and downstream neighbourhoods. If representations converge despite different anti-collapse mechanisms, the shared augmentations or task structure may dominate.

If they diverge in later layers while transfer remains similar, several geometries may support the same receiver. This links the current article directly to Contrastive Representation Learning.

BM. Similarity analysis can test superposition hypotheses across width

Train models of increasing residual width and compare their native activations. Whole-space CKA can remain high while sparse feature dictionaries become less polysemantic or use different feature granularity.

Representation similarity at the dense level and feature overlap at the sparse level should be reported together. This connects to Superposition in Neural Representations.

BN. Similarity analysis can test SAE dictionary stability without assuming one-to-one atoms

For SAEs of different width, compare sparse activation Gram matrices with CKA. High CKA can show the dictionaries organise stimuli similarly even if one broad atom splits into several children.

Pair that with atom matching and feature genealogy. Dense representation similarity, sparse-code similarity and semantic feature consistency are three separate layers.

This links to Sparse Autoencoders for Neural Representation.

BO. Worked experiment: bootstrap a layer heatmap

Take two models and 5,000 held-out stimuli. Compute the layer×layer CKA matrix. Bootstrap documents or independent images 200 times and recompute the matrix.

For every cell, calculate mean and uncertainty. Then examine whether the apparent best alignment is stable. Some neighbouring layers may be statistically indistinguishable.

The experiment converts a colourful diagram into an uncertainty-aware correspondence map.

BP. Worked experiment: same function, rotated representation

Train a simple network. Insert an orthogonal matrix Q after one hidden layer and Qᵀ before the next linear layer so the overall function is unchanged. Compare original and transformed hidden activations.

Neuron correlation falls, Procrustes and CKA remain maximally similar, and outputs remain identical. This is the cleanest demonstration that coordinate identity is not representation identity.

BQ. Worked experiment: same linear information, distorted geometry

Replace Q with a well-conditioned non-orthogonal invertible matrix A and compensate downstream with A^{-1}. Function and linear information are preserved.

CCA remains very high under suitable conditions; CKA and Procrustes decline because geometry changed. The result makes the central measure-selection question unavoidable.

BR. Worked experiment: same CKA, different rare feature

Create two representations identical on 99.9 percent of examples. On a rare class, remove one critical feature from Y. Global CKA can remain extremely high while rare-class probe performance collapses.

This demonstrates why global representation similarity cannot certify tail-behaviour preservation.

BS. Worked experiment: feature consistency versus whole-space consistency

Train several SAEs on one model layer. Compute CKA between their reconstructed activations or sparse code Gram matrices. Separately compute one-to-one feature consistency.

It is possible to obtain high whole-space similarity and modest atom consistency because features rotate, split or merge. This separates stable represented geometry from stable semantic coordinates.

BT. The representation-comparison evidence card

• Question: coordinate, geometry, information, containment, function or mechanism.
• Models: architecture, version and checkpoint.
• Sites: exact layers or feature spaces.
• Stimuli: corpus, slices and row definition.
• Metric: exact formulation and invariances.
• Dimensions: n, p, q and effective rank.
• Alignment: in-sample versus held-out.
• Uncertainty: bootstrap/subsample confidence.
• Baselines: random, same-seed family and controlled transforms.
• Measure agreement: expected and unexpected disagreements among metrics.
• Functional comparison: outputs, errors and decision boundaries.
• Feature comparison: probes or SAE dictionaries if relevant.
• Tail slices: rare/high-risk behaviours.
• Causal evidence: intervention if mechanistic equivalence is claimed.
• Scope: domains over which the correspondence was tested.

BU. Final release gate for a 20,000+ word representation-similarity owner

1. The article begins from the equivalence relation, not from a favourite metric.
2. It explains why neuron matching is too strict for independent models.
3. It derives Procrustes, CCA and CKA sufficiently to expose their invariances.
4. It explains SVCCA and PWCCA as modifications with new assumptions.
5. It explains RSA’s inner and outer metric choices.
6. It treats finite sample size, high dimension and estimator bias explicitly.
7. It uses null and controlled-transformation baselines.
8. It distinguishes symmetric equivalence from asymmetric containment.
9. It cross-validates learned alignments.
10. It reports uncertainty over stimuli rather than reading one heatmap literally.
11. It treats layer correspondence as discoverable, not index-given.
12. It separates representational and functional similarity.
13. It connects model compression, distillation and multimodal alignment to the comparison contract.
14. It connects dense-space comparison to sparse-feature consistency without conflating them.
15. It escalates strong mechanistic claims to causal interventions.
16. It states that no universal similarity threshold or universally best metric exists.

The mature purpose of representation similarity analysis is not to prove that two networks “think the same way”. It is to localise what kind of sameness the evidence supports. Coordinates can differ while relational geometry remains stable. Geometry can differ while linear information is recoverable. Internal representations can align while output heads disagree. Outputs can agree while hidden algorithms diverge. Causal interventions can reveal whether a statistical correspondence carries the same computational role.

A similarity score becomes meaningful only after we name the transformations it forgives, the stimuli it was measured on, the uncertainty around it and the stronger behaviours it successfully predicts.