Sparse Feature Geodesics: A Topology of Attention

Interpretability work on large language models has settled on sparse autoencoders (SAEs) as a way to decompose activations into human-comprehensible features. The common move is to take a trained SAE, find the features that activate on a given input, and read those activations like a map. The map is useful, but it is static: a single snapshot per layer, as if the model thought in still photographs.

We suspected the model thought in motion. This paper treats SAE features not as points but as trajectories — how a feature’s activation moves across the attention heads that process a given token. We call the resulting paths feature geodesics, and we report two findings that justify the framing.

Approach

We trained SAEs on the residual streams of Mantus, our balanced model, using a standard sparse autoencoder objective with a wide dictionary and a sparsity penalty. Rather than extract activations at each layer independently, we tracked the activation of each feature across the full set of layers that contribute to a single token’s representation, connecting them in order.

For each feature, we then computed a small set of invariants: path length (how far the feature’s activation travels), direction change (how much the trajectory bends), and endpoint stability (whether the trajectory converges to a consistent value across similar inputs). We validated the setup on a set of 2,000 prompts spanning reasoning, coding, and conversation, and compared trajectories against layer-wise single-point analyses on the same data.

Results

Two results stand out.

First, feature geodesics are substantially more discriminative than single-point activations. On a held-out set, trajectory shape alone classified attention patterns across our prompt categories with 82% accuracy, against 61% for the best single-layer activation threshold. The gain came almost entirely from direction change: features that bend sharply correlate with attention heads that shift context.

Second, trajectories are stable. Across 50 paraphrases of the same prompt, endpoint stability was high (mean relative variance 0.12), but path shape was even more stable — suggesting that the geometry is a property of the computation, not of surface phrasing. Features whose trajectories were unstable were overwhelmingly features we had previously flagged as poorly understood, which is a useful diagnostic signal on its own.

Discussion

A feature that occupies one point at one layer and another point two layers later is two different facts about the model. Tracking the path between them changes what interpretability can say: instead of asking “what does this feature mean?”, we can ask “what role does this feature play as the computation unfolds?” That second question is closer to what a model actually does.

We are cautious about the limits. SAE reconstruction is approximate, trajectories are sensitive to dictionary choice, and we have only studied one model family. The finding is a suggestion, not a conclusion: attention has a topology, and it may be worth studying directly rather than through static snapshots.

References

1. Bricmont, L., & Kwan, A. (2025). Sparse autoencoders as lossy compressors of model behavior. Journal of Interpretable Systems, 12(3), 44–61.
2. Delacroix, M. (2024). Feature superposition and the geometry of residual streams. ICLR 2024.
3. Fernandez, R., et al. (2025). Trajectory-based analysis of attention heads. NeurIPS 2025.
4. Holtz, J. (2023). The topology of language models. Workshop on Mechanistic Interpretability, NeurIPS.
5. Okada, Y., & Singh, P. (2025). Path stability as a probe of learned structure. Transactions on Machine Learning Research, 2025(8).
6. Weber, J., et al. (2025). Feature dictionaries beyond the layer. Curos Internal Report 2025-04.

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