1 Introduction↩︎

Sparse autoencoders have become a standard tool in mechanistic interpretability, especially for language models, by decomposing neural network activations into a sparse linear combination of interpretable dictionary atoms [2], [3]. A well-known practical challenge is instability: two SAEs trained on identical data with different random seeds learn substantially different dictionaries, making it unclear which features are meaningful rather than accidental. Recent work has treated this as a serious problem, with [4] and [5] both independently identifying and measuring it.

Archetypal SAEs [1] are part of a broader and valuable effort to place SAEs within the older dictionary-learning tradition, alongside methods such as k-means, PCA, NMF, and archetypal analysis [6], [7]. This perspective is especially useful for mechanistic interpretability, where SAEs are sometimes treated as the de facto standard for unsupervised feature discovery rather than as one point in a larger design space. Archetypal SAEs address instability by constraining dictionary atoms to lie within the convex hull of a set of data points (the “archetypes”), initialized via k-means clustering. The authors demonstrate improved endpoint stability and attribute it to the archetypal constraint. We argue this attribution rests on an experimental design that does not isolate the archetypal constraint from its deterministic initialization.

First, the conceptual issue is a conflation of stability with stabilization. Stability, which also called consistency [4], is achieved by having two independently trained instances agree at the end of training. On the other hand, stabilization – a stronger claim than stability – requires two different initialization converge over the course of training. The former can be achieved trivially by fixing the starting point; the latter cannot. Archetypal SAEs report the former while never testing the latter, because both seeds start from the same k-means centroids. Inter-run distance is zero before a single gradient step is taken, and “stability” at the end is simply that initialization being preserved. Figure 1 makes this distinction visible by plotting the full inter-run trajectory rather than only the endpoint.

Second, a more subtle issue concerns the geometry induced by preprocessing and cosine distance. Cosine distance, the standard stability metric, can be deflated when dictionary atoms cluster far from the origin, while element-wise standardization defines the archetypal hull in a coordinate-dependent transformed space. This preprocessing ambiguity complicates interpretation of endpoint stability metrics. Our main conclusion does not depend on this metric critique: even under the authors’ preferred geometry, a stability comparison that shares the entire decoder initialization across archetypal runs cannot establish stabilization.

The stakes are not merely methodological. Findings of dictionary learning with SAEs ¹ rest on the assumption that the learned decomposition accurately reflects the underlying mechanisms of the model, rather than artifacts of the training run. If two SAEs trained on the same data with different random seeds learn substantially different dictionaries, it is unclear which atoms are genuine features and which are noise. End-of-training stability has therefore been proposed as a proxy for feature validity: an atom that reappears across seeds is more likely to correspond to real computational structure, not to a local optimum specific to one run. We argue that this proxy is only meaningful, if stability is evidence of convergence. A model that is stable by construction (because its initialization is fixed rather than because its training dynamics converge) offers no such evidence. The distinction matters for every downstream claim that uses stability to validate interpretability findings.

We make three contributions:

• Stabilization diagnostic: plotting inter-run dictionary distance as a function of training step. A flat trajectory at zero is not stability—it is the absence of a test. When the test is administered, classical TopK stabilizes faster than the archetypal variant.

• Initialization ablation: two controlled conditions (archetypal with the convex-hull projection disabled; classical TopK initialized from the same k-means centroids) that isolate the contribution of the archetypal mechanism from that of shared initialization.

• Metric analysis: exposing the uncentered-cosine confound in language-model settings and discussing how preprocessing choices affect the geometry in which the archetypal hull and cosine stability metric are defined.

2 Related Work↩︎

2.0.0.1 Concept extraction as dictionary learning.

Concept-based interpretability has long studied unsupervised feature discovery through classical dictionary-learning methods, including k-means, PCA, NMF, and archetypal analysis. [6] give a particularly useful unifying framework: concept extraction is cast as dictionary learning, while concept importance estimation is cast as attribution in the learned concept space. We adopt this broad view. SAEs should not be treated as the only possible unit of analysis for neural representations, including in NLP; rather, they are one scalable neural dictionary-learning method among many. Our contribution is not to reject this program, but to examine whether one proposed intervention in that program—the archetypal constraint—actually supports the stability claim made for it.

2.0.0.2 SAE stability.

The instability of sparse autoencoders was independently identified and measured by [5] and [4]. The basic measurement idea is older than these papers: the first public SAE interpretability report used mean maximum cosine similarity (MMCS) to compare learned dictionaries to ground-truth features and to compare dictionaries of different sizes [8]. Hungarian matching and PW-MCC are stricter or differently normalized variants of the same general recipe: compare dictionary atoms by pairwise similarity while accounting for permutation invariance. Paulo and Belrose use a cosine Hungarian loss to document that different training runs produce substantially different dictionaries; Song et al.frame the same phenomenon as a feature consistency problem and argue it undermines the validity of circuit-level interpretability claims. Both works vary initialization across seeds, the convention we adopt here. Our contribution is orthogonal: we are not measuring how unstable classical SAEs are, but asking whether the claimed fix (the archetypal constraint) actually stabilizes anything. In concurrent work, [9] address instability by reparameterizing the encoder to enforce a unit inner product between each encoder row and its corresponding decoder column, achieving Pareto improvements on reconstruction, dead features, and cross-seed stability without additional hyperparameters. That work demonstrates that constraining the encoder–decoder geometry is a viable route to stabilization; ours shows that the archetypal constraint—which operates on the decoder alone—does not constitute one. [4] also report feature-consistency curves over training; we build on this temporal perspective to separate endpoint agreement from stabilization, i.e.convergence from genuinely different initial conditions. This distinction is the central issue in the archetypal SAE setting, where shared decoder initialization can make endpoint agreement look like stability without testing convergence at all.

2.0.0.3 Archetypal SAEs.

The archetypal constraint draws on archetypal analysis [7], a classical technique that represents data as convex combinations of extreme points (archetypes) lying on the boundary of the data cloud. [1] is the work we directly examine. Our critique is specifically about the stability claim; the archetypal construction has genuine virtues independent of that claim. Constraining atoms to the convex hull of a fixed reference set provides a principled geometric interpretation (atoms are literally combinations of representative data points) and enforces bounded atom norms as a regularizer. These properties may be valuable for visualization, for controlling atom scale, or for aligning learned features with semantically meaningful prototypes. We leave open whether the archetypal constraint provides benefits beyond stability, and make no claim about the broader value of the method.

2.0.0.4 Decoder centering and initialization.

The practice of initializing the decoder bias at the geometric median of the training distribution, as used by [3] and [10], is conceptually related to our centering correction. Both interventions ground the dictionary in the data distribution before training begins. The key distinction is the reason. Prior work motivates centering as a convergence aid: starting near the data mean reduces the distance atoms must travel early in training. Our argument is different: centering is required for the cosine stability metric to be a valid measurement instrument. Without it, any method whose atoms remain near the data mean will appear stable by cosine distance, regardless of the underlying learning dynamics. The same preprocessing step serves two purposes, and the measurement-validity purpose is the one that has not previously been stated.

3 Methods and Metrics↩︎

Let \(X \in \mathbb{R}^{n \times d}\) denote a matrix of activations. A sparse autoencoder learns an encoder \(f_\theta : \mathbb{R}^d \to \mathbb{R}^m\) and a decoder dictionary \(D \in \mathbb{R}^{m \times d}\) such that \[\hat{x} = zD, \qquad z = f_\theta(x),\] with \(z\) constrained or regularized to be sparse. A TopK SAE enforces sparsity by retaining only the \(k\) largest activations in \(z\) and setting the rest to zero. In a classical TopK SAE, the decoder dictionary \(D\) is a free learned parameter, usually initialized randomly.

Archetypal SAEs replace this free dictionary with a convex-hull parameterization. Let \(\mathbf{C} \in \mathbb{R}^{r \times d}\) be a fixed set of candidate archetypes, typically k-means centroids of the activation dataset. The archetypal dictionary is \[D = W\mathbf{C}, \qquad W_i \in \Delta^{r-1},\] where each row \(W_i\) lies on the probability simplex. Thus each dictionary atom is a convex combination of the candidate points. The scalable implementation of [1] initializes \(W\) as the identity (or the corresponding rectangular identity when \(m \leq r\)), so when the same \(\mathbf{C}\) is shared across seeds the decoder dictionary is identical at step 0.

The relaxed archetypal SAE (RA-SAE) adds a bounded residual term, \[D = W\mathbf{C} + \Lambda, \qquad W_i \in \Delta^{r-1}, \qquad \|\Lambda_i\|_2 \leq \delta.\] The hyperparameter \(\delta\) controls how far each atom may move away from the convex hull of \(\mathbf{C}\). When \(\delta = 0\), the atoms remain exactly inside the hull. Larger \(\delta\) permits more reconstruction flexibility while keeping the dictionary close to the archetypal reference set.

3.0.0.1 Stability metrics.

Given two trained dictionaries \(D^{(1)}, D^{(2)} \in \mathbb{R}^{m \times d}\), the standard endpoint stability metric is the optimal cosine matching between their atoms. Let \[c(i,j) = 1 - \frac{\langle D^{(1)}_i, D^{(2)}_j\rangle}{\|D^{(1)}_i\|_2\|D^{(2)}_j\|_2}\] be cosine distance. The normalized cosine Hungarian distance is \[H(D^{(1)}, D^{(2)}) = \frac{1}{m} \min_{\pi \in S_m} \sum_{i=1}^{m} c(i,\pi(i)), \label{eq:hungarian}\tag{1}\] where \(S_m\) is the set of permutations of the \(m\) atoms. Lower values indicate more similar dictionaries. This is the metric we report for endpoint stability in Table 3.

For training trajectories, computing the Hungarian assignment at every logging step is unnecessarily expensive. We therefore use the mean minimum cosine distance \[M(D^{(1)}, D^{(2)}) = \frac{1}{m}\sum_{i=1}^{m}\min_j c(i,j).\] This drops the bijection constraint and asks how close each atom in one dictionary is to its nearest neighbor in the other. [5] argue that Hungarian matching is more principled, but empirically find that maximum-cosine nearest-neighbor scores closely track the Hungarian scores because most atoms are matched to their nearest neighbors. We use \(M\) only as a linear-time trajectory diagnostic; endpoint numbers use the Hungarian metric.

4 Clarifying Confusion↩︎

The concept of SAE stability is less well-defined in the literature than it might appear, and the evidence for archetypal stability draws on two distinct measurement pathologies. The first is an inconsistency in what randomness is varied across seeds; the second is a geometric pathology of the cosine metric on uncentered activations. Together they inflate reported stability scores for any method whose dictionary atoms remain near the data mean.

4.1 Inconsistent sources of randomness↩︎

Randomness can enter SAE training in two distinct ways: parameter initialization, which determines the starting point of optimization, and batch order, which affects the trajectory of stochastic gradient descent. Conflating these leads to incomparable experimental results.

Table 1 organizes the assumptions made by recent papers on SAE stability. [5] and [4] both vary initialization across seeds while fixing batch order, whereas [1] vary batch order but fix initialization: k-means always produces the same centroids given the same data and seed. This inconsistency means that archetypal SAEs enjoy a structural advantage in every stability comparison against classical SAEs. They are never exposed to the primary source of instability.

Recent literature contains a direct conflict about whether TopK SAEs are more or less stable than vanilla SAEs with L1 regularization: [5] find TopK SAEs more seed-dependent than ReLU/L1 SAEs, whereas [4] find TopK SAEs more consistent than Standard SAEs in their selected settings. This disagreement was also raised in discussion of [5] ², where the authors noted that several recent works report stable L1-trained SAEs and suggested that differences in dead-latent rates may explain part of the discrepancy. We do not adjudicate that architecture-level question here; our point is that stability comparisons are sensitive to the experimental definition of “independent runs,” including which sources of randomness are varied and which features are counted.

4.2 The uncentered-cosine confound↩︎

Cosine distance is a meaningful stability metric only when the objects being compared are centered near the origin. In LLM residual streams, activations can have a non-negligible mean component: the distribution sits away from zero. Under k-means clustering of uncentered activations, every resulting centroid inherits this mean offset. All centroids, and therefore all initial dictionary atoms of an archetypal SAE, lie inside a narrow cone around \(\mu\).

The cosine distance between two vectors in such a cone is dominated by \(\mu\), not by the directions that distinguish them. This is the galaxy-far-away effect: all stars in a distant galaxy have nearly identical angles as seen from Earth, even though they may be far apart in absolute terms. [11] independently characterize this large-scale anisotropy of SAE feature clouds and use the same galaxy metaphor to describe it.

Figure 2 illustrates the geometry. Panel (a) shows k-means centroids clustered near the mean vector \(\mu\); the angular spread seen from the origin is tiny, so any two atoms score as highly similar regardless of what they encode. Panel (b) shows the same atoms after centering: the mean offset is removed, atoms scatter around the origin, and cosine distance now reflects genuine directional differences.

This is a measurement artifact, not a property of the learned dictionary. Any method that constrains its atoms to lie near the data mean (whether through the archetypal convex hull, through fixed k-means initialization, or simply through an objective that does not push atoms away from \(\mu\)) will appear stable by cosine distance on uncentered activations. Centering the activations before running k-means, and subtracting the activation mean from the decoder bias during training, eliminates this pathology. We apply this correction to both experimental settings (§5, §6).

A natural question is whether [1] apply such a correction. The paper states that activations are element-wise standardized, which would eliminate the mean offset and thereby fix the cosine metric. However, neither the public Colab notebook nor the Overcomplete library applies any centering or standardization before k-means or during training, so the reported stability numbers are computed without this correction. Moreover, element-wise standardization—dividing each coordinate by its estimated standard deviation—is not geometrically motivated in LLM settings: it privileges the standard basis of the weight-storage coordinate system, which is arbitrary for residual stream activations. Centering alone is sufficient to make the cosine metric valid; per-coordinate rescaling by standard deviation adds an unjustified coordinate-dependent assumption. See Appendix 8 for the detailed code audit supporting these claims.

4.3 Empirical verification↩︎

Table 2 reports the key statistics for both experimental settings, using the centroid sets used in our stability experiments.

The key observation is the before/after change in centroid cosine similarity. Before centering, mean pairwise cosine similarity among centroids is 0.362 (LLM) and 0.298 (vision), which is not negligible. After centering, both collapse to approximately zero, consistent with centroids that are isotropically distributed around the origin. This positive bias directly inflates reported stability: cosine distance is 1 minus cosine similarity, so any two centroids appear systematically closer by 0.30–0.36 units when activations are not centered. A stability method that keeps atoms near the uncentered data mean will benefit from this inflation regardless of whether it has learned anything useful.

Notably, DinoV2 vision centroids show the same positive cosine bias before centering (0.298), confirming that the same confound applies in the vision setting. We therefore apply the centering correction to both experimental settings in §5 and §6.

5 Stabilization, not Stability↩︎

End-of-training stability and stabilization are not the same thing. A model is stable if two independently trained instances agree at the end. A model stabilizes if those instances, started from genuinely different points, converge over the course of training. The former can be achieved trivially by fixing the starting point; the latter cannot. We argue stabilization is the right desideratum—it is evidence that the data distribution is constraining the learned representation, rather than an artifact of the experimental protocol.

Any method that reduces variance in the starting point of optimization will look stable by an end-of-training measure, regardless of whether the underlying learning dynamics converge. This is the identification problem for archetypal SAE stability claims. The decoder dictionary is \(D = W\mathbf{C}\), where \(\mathbf{C}\) is the fixed centroid matrix (computed once, shared across all seeds) and \(W\) is initialized to the identity. At step 0, before any gradient is applied, the two seeds produce identical decoder dictionaries: \(D = \mathbf{C}\). Inter-run decoder distance is not merely small at initialization—it is exactly zero by construction. End-of-training stability then reflects this identity being approximately preserved, not any convergence that happened during training.

The diagnostic we propose is to track inter-run distance throughout training, not just at the end. This temporal view is related to the training-dynamics analysis of [4], but the conceptual target here is different: stabilization, not stability. If two independently trained SAEs are genuinely converging to the same solution, their dictionary distance should decrease over the course of training, starting high (different initializations) and falling as the shared structure of the data constrains the solution. A method that merely preserves a shared initialization shows endpoint agreement without stabilization. In the limiting case relevant here, the runs begin identical: the two runs were never meaningfully separated to begin with.

Figure 1 shows this trajectory for the conditions we study, measured as mean minimum cosine distance between paired dictionaries at regular training intervals (defined in §3). Classical TopK with random initialization starts with high inter-run distance and decreases monotonically over training: genuine stabilization. The two seeds begin from unrelated random points in a high-dimensional parameter space and converge as training progresses. This is what stabilization looks like: the data distribution has a basin of attraction, and both runs find it independently.

Archetypal TopK with fixed k-means initialization is flat at zero from step 0. As established above, both seeds produce identical decoder dictionaries at initialization (\(D = \mathbf{C}\)), and this identity is largely preserved throughout training. The relaxation term \(\delta\) bounds how far atoms can drift from the centroid set, so inter-run distance at convergence is also bounded by \(2\delta\), small by design. This is not convergence. The two runs were never apart.

Archetypal TopK with random initialization provides the decisive test of whether the convex-hull constraint itself exerts a convergence force, independent of the shared starting point. Here the decoder initialization is randomized: the convex combination weights \(W\) and relaxation offsets are drawn randomly per seed, while the centroid set \(\mathbf{C}\) remains fixed and centered. Both seeds therefore start from genuinely different decoder dictionaries (inter-run distance \(\approx 0.88\) at step 0), and the archetypal constraint is the only structural difference from the classical condition.

The trajectory does decrease (the constraint does exert some convergence force) but at roughly half the rate of unconstrained classical TopK. After 50 million tokens, the archetypal random-init condition reaches a distance of \(\approx 0.35\), compared to \(\approx 0.17\) for classical TopK. Rather than accelerating stabilization, the convex-hull constraint slows it. This reversal is decisive: atoms that are free to move wherever the data pulls them converge to shared solutions faster than atoms constrained to the interior of the centroid hull. In this setting, the archetypal mechanism does not enhance stabilization; it impedes it. The endpoint stability reported by [1] therefore cannot be interpreted as evidence that independently initialized archetypal dictionaries converge to a shared solution.

6 Ablating Archetypes↩︎

We now isolate the mechanism behind the stability claim with two ablations, using the minimal working example provided by the authors of archetypal SAEs as a public Google Colab notebook.³ This setup uses DinoV2 patch embeddings extracted from a single image class (rabbits). We additionally replicate all experiments on residual stream activations from layer 6 (the middle layer) of Pythia-160m [12], using WikiText-103 as the text source. In both settings, stability is measured as the normalized cosine Hungarian loss between the dictionaries of two independently trained SAEs (Eq. 1 ); a lower score indicates more stable (similar) dictionaries across seeds. Unlike the original notebook, we fix torch.manual_seed before each training run to ensure that any difference between conditions is attributable to initialization rather than encoder weight randomness. Our code is available at https://github.com/MichalBrzozowski91/ablating-archetypes. The repository provides two notebooks: one replicating the original Colab’s preprocessing exactly (uncentered, for direct comparison), and one with the centering correction applied (matching Table 3). See the repository README for details.

Both settings center activations before running k-means and subtract the activation mean from the decoder bias before training. As documented in §4.3, this correction is necessary for the cosine stability metric to reflect genuine directional differences rather than proximity to the uncentered data mean.

6.1 Ablation 1: Disabling the Convex-Hull Projection↩︎

The defining feature of archetypal SAEs is the projection that keeps dictionary atoms within the convex hull of the data points. If this constraint is responsible for stability, disabling it should degrade stability toward the randomly initialized classical SAE baseline. We test this by training an archetypal SAE with the convex-hull projection disabled, retaining the k-means initialization but removing the convex-hull enforcement during training. Without the projection, both the convex-hull constraint and the \(\delta\) bound on the residual term \(\Lambda\) are disabled: \(W\) is frozen, the norm constraint \(\|\Lambda_i\|_2 \leq \delta\) is lifted, and \(\Lambda\) becomes a free, unconstrained learnable dictionary matrix initialized directly to \(\mathbf{C}\). It is this combination — removing the projection and releasing the residual bound — that makes the ablated model structurally identical to a classical TopK SAE whose decoder has been initialized from the k-means centroids, which is exactly Condition 4 (§6.2). The numerical identity of Conditions 3 and 4 in Table 3 (identical Hungarian distance and R\(^2\) under consistent seeding) follows by construction, not coincidence: they are running the same forward and backward pass, implemented two ways.

As shown in Table 3, the ablated model is more stable than the full archetypal SAE in both experimental settings (vision: 0.068 vs.; Pythia-160m: 0.112 vs.).

The R\(^2\) gap shows that the convex-hull projection hurts reconstruction. This is the default expectation under any constraint: restricting the feasible set can only reduce reconstruction capacity unless the constraint happens to be a well-suited inductive bias for the data. Whether the archetypal parameterization constitutes such a bias is an open question, and the burden of proof lies with its proponents. Our results are consistent with the reduced reconstruction performance acknowledged by [1] themselves. Crucially, the projection is not necessary for the observed stability either, so it incurs a reconstruction cost with no stabilization benefit in our setting.

6.2 Ablation 2: Fixing the Initialization of a Classical SAE↩︎

If stability comes from initialization rather than the archetypal constraint, then a classical TopK SAE initialized from the same k-means centroids should achieve the same stability as an archetypal SAE. We construct this condition by initializing the dictionary of a classical TopK SAE directly from the first \(K\) k-means centroids, with no archetypal constraint applied at any point during training.

The results support the initialization account. In both settings, the classical SAE with fixed initialization matches the ablated archetypal SAE exactly (vision: 0.068; Pythia-160m: 0.112) and substantially outperforms both the randomly initialized classical SAE (vision: 0.433; Pythia-160m: 0.200) and the full archetypal SAE with its convex-hull projection intact (vision: 0.243; Pythia-160m: 0.170).

This result has a natural interpretation: k-means initialization places dictionary atoms near dense regions of the activation space, reducing the effective distance each atom must travel during optimization and thereby making training less sensitive to the stochastic batch order. This is conceptually related to the decoder bias centering technique proposed in [3] and used in [10], where subtracting the geometric median of activations from the decoder bias serves a similar role of grounding the dictionary in the data distribution before training begins.

Limitations↩︎

Our experiments are conducted with a set of hyperparameters drawn from the reference implementation of [1] (number of concepts, sparsity level, training epochs). We do not perform a hyperparameter sweep, and it is possible that the archetypal constraint provides some benefit at other settings or in other regimes. The ablation experiments (§6) follow the reference implementation and train for 20 epochs; the stability trajectory experiment (§5) trains for 50 million tokens. In both cases, longer training may shift the relative ordering of conditions. The vision experiment uses a single image class (rabbits) from the authors’ reference Colab, and the language experiment a single layer of a single model; broader coverage would strengthen the empirical claim.

7 Hyperparameters: Stability Trajectory Experiment↩︎

Table 4 lists the full hyperparameter configuration for the LLM stability trajectory experiment (§5). All values are script defaults; no sweep was performed.

8 Code Audit Notes↩︎

We examined the public codebases of both papers whose methodology we discuss. This appendix records the specific findings that bear on the claims in the main text.

8.0.0.1 Song et al.([4]) — batch order is fixed across seeds.

The repository is available at https://github.com/xiangchensong/sae-feature-consistency. In examples/run_single.py (lines 63–65), the global random state is fixed once before the training loop begins:

random.seed(demo_config.random_seeds[0])
t.manual_seed(demo_config.random_seeds[0])
np.random.seed(demo_config.random_seeds[0])

Individual SAE trainers subsequently receive different seeds (defaulting to 42, 43, and 44) via the random_seeds loop, which controls weight initialization. The ActivationBuffer draws training batches via t.randperm() (line 74 of examples/dictionary_learning/buffer.py), which reads from the global PyTorch random state. Because that state is fixed to random_seeds[0] before the loop, all runs see the same sequence of activation batches and differ only in their weight initialization. This is consistent with our Table 1 classification: Song et al.vary initialization across seeds and hold batch order fixed.

8.0.0.2 Fel et al.([1]) — paper and code disagree on preprocessing; standardization is not geometrically motivated.

The reference implementation is the public Colab notebook linked in the paper (https://colab.research.google.com/drive/1TmAtUhIdFGSMlDhKr2ndXGR8GU4R4aTq) and the accompanying Overcomplete package (https://github.com/KempnerInstitute/Overcomplete).

Paper vs.code discrepancy. The paper states (p. 7): “The data matrix \(A\) was element-wise standardized.” Element-wise standardization—subtracting the per-coordinate mean and dividing by the per-coordinate standard deviation—does eliminate the galaxy-far-away effect, because subtracting the mean centers the distribution around the origin. However, neither the Colab notebook nor the Overcomplete library applies any centering or standardization before running k-means or during SAE training. The stability numbers reported in the reference notebook are therefore computed without the preprocessing described in the paper.

Standardization is not geometrically motivated. Even setting aside the paper–code discrepancy, element-wise standardization carries an additional assumption that centering alone does not: dividing each coordinate by its estimated standard deviation privileges the standard basis of the ambient space. For LLM residual stream activations, no such privileged coordinate system exists—the basis in which weights happen to be stored is arbitrary and changes under linear reparameterization. Centering (subtracting the mean) is geometrically well-founded and sufficient to make the cosine metric a valid stability instrument; the per-coordinate scaling by standard deviation is an additional, unjustified assumption in this setting. Our correction applies centering only, without per-coordinate rescaling.

We verify the practical effect empirically in Table 2: before centering, mean pairwise cosine similarity among k-means centroids is 0.362 (LLM) and 0.298 (vision); after centering it collapses to approximately zero in both settings. Any stability metric based on cosine distance is therefore deflated for any method—including archetypal SAEs—whose dictionary atoms remain near \(\mu\). We apply the centering correction to both experimental settings (§5, §6).

References↩︎