# When Trees Are Not Enough: Learning Mixed-Topology Feature Graphs with Adaptive Graph Sparse Autoencoders

arXiv: https://arxiv.org/abs/2609.36294 · HTML: https://mlsanity.com/paper/2609.36294/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in SAE features and their applications, one of 13 directions in Mechanistic interpretability.
- Builds on 36 of its 46 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- Mechanistic interpretability / [SAE features and their applications](https://mlsanity.com/mechinterp/sae-features-and-their-applications/)

## Builds on (most cited first, 30 of 36)

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- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Sparse Feature Circuits: Discovering and Editing Interpretable Causal Graphs in Language Models](https://mlsanity.com/paper/2403.19647/) (2024)
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- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [k-Sparse Autoencoders](https://mlsanity.com/paper/1312.5663/) (2013)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
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- [Do I Know This Entity? Knowledge Awareness and Hallucinations in Language Models](https://mlsanity.com/paper/2411.14257/) (2024)
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- [The MiniPile Challenge for Data-Efficient Language Models](https://mlsanity.com/paper/2304.08442/) (2023)
- [Identifying Functionally Important Features with End-to-End Sparse Dictionary Learning](https://mlsanity.com/paper/2405.12241/) (2024)
- [Interpreting CLIP with Hierarchical Sparse Autoencoders](https://mlsanity.com/paper/2502.20578/) (2025)
- [The Geometry of Concepts: Sparse Autoencoder Feature Structure](https://mlsanity.com/paper/2410.19750/) (2024)
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- [From Flat to Hierarchical: Extracting Sparse Representations with Matching Pursuit](https://mlsanity.com/paper/2506.03093/) (2025)
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- [SynthSAEBench: Evaluating Sparse Autoencoders on Scalable Realistic Synthetic Data](https://mlsanity.com/paper/2602.14687/) (2026)

## Abstract (arXiv)

Sparse autoencoders (SAEs) expose interpretable features in large language model activations, yet existing structured SAEs impose single-parent trees or forests, while post-hoc graphs permit multiple parents but neither guide feature learning nor ensure reliable relation recovery. We introduce the Adaptive Graph Sparse Autoencoder (AG-SAE), a structure-guided training paradigm that treats each feature's complete parent set as an atomic structural hypothesis and lets evidence select zero, one, or multiple parents. By competing complete parent sets against null, subset, and alternative explanations, AG-SAE identifies jointly necessary multi-parent relations while rejecting redundant or spurious alternatives and verifying that each child contributes beyond its parents. The induced topology over SAE features then defines a differentiable structural loss that guides SAE training, while topology-guided refinement mitigates feature absorption and uses persistent reconstruction gaps exposed by the learned structure to initialize new features. The entire graph is then induced again from the revised dictionary by reassessing every feature's complete parent set, closing the dictionary-graph self-consistency cycle. Experiments demonstrate exact mixed-topology recovery in a controlled toy model, greater relational reliability and semantic validity than structured and post-hoc baselines on real LLM activations, and stronger feature-level causal interventions than conventional SAE features. AG-SAE thereby turns recovered mixed-topology feature structure into an unsupervised training signal that improves the dictionary, enables reliable feature organization beyond the topological limitations of trees, and exhibits stronger causal control beyond reconstruction.