# Decompose Sparsely Where You Should, Absorb Densely Where You Should No

arXiv: https://arxiv.org/abs/2606.14040 · HTML: https://mlsanity.com/paper/2606.14040/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Sparse autoencoder training and evaluation, one of 13 directions in Mechanistic interpretability.
- Builds on 30 of its 33 references that are in the map.
- First posted on arXiv 12 Jun 2026.

## Placed in

- Mechanistic interpretability / [Sparse autoencoder training and evaluation](https://mlsanity.com/mechinterp/sparse-autoencoder-training-and-evaluation/)

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

- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Gemma Scope: Open Sparse Autoencoders Everywhere All At Once on Gemma 2](https://mlsanity.com/paper/2408.05147/) (2024)
- [Jumping Ahead: Improving Reconstruction Fidelity with JumpReLU Sparse Autoencoders](https://mlsanity.com/paper/2407.14435/) (2024)
- [Massive Activations in Large Language Models](https://mlsanity.com/paper/2402.17762/) (2024)
- [Representation Degeneration Problem in Training Natural Language Generation Models](https://mlsanity.com/paper/1907.12009/) (2019)
- [AxBench: Steering LLMs? Even Simple Baselines Outperform Sparse Autoencoders](https://mlsanity.com/paper/2501.17148/) (2025)
- [All-but-the-Top: Simple and Effective Postprocessing for Word Representations](https://mlsanity.com/paper/1702.01417/) (2017)
- [Not All Language Model Features Are One-Dimensionally Linear](https://mlsanity.com/paper/2405.14860/) (2024)
- [Transcoders Find Interpretable LLM Feature Circuits](https://mlsanity.com/paper/2406.11944/) (2024)
- [Scaling Monosemanticity: Extracting Interpretable Features from Claude 3 Sonnet](https://mlsanity.com/paper/2605.29358/) (2026)
- [Improving Dictionary Learning with Gated Sparse Autoencoders](https://mlsanity.com/paper/2404.16014/) (2024)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
- [Quantizable Transformers: Removing Outliers by Helping Attention Heads Do Nothing](https://mlsanity.com/paper/2306.12929/) (2023)
- [Llama Scope: Extracting Millions of Features from Llama-3.1-8B with Sparse Autoencoders](https://mlsanity.com/paper/2410.20526/) (2024)
- [Learning Multi-Level Features with Matryoshka Sparse Autoencoders](https://mlsanity.com/paper/2503.17547/) (2025)
- [BatchTopK Sparse Autoencoders](https://mlsanity.com/paper/2412.06410/) (2024)
- [SAEBench: A Comprehensive Benchmark for Sparse Autoencoders in Language Model Interpretability](https://mlsanity.com/paper/2503.09532/) (2025)
- [Are Sparse Autoencoders Useful? A Case Study in Sparse Probing](https://mlsanity.com/paper/2502.16681/) (2025)
- [BERT Busters: Outlier Dimensions that Disrupt Transformers](https://mlsanity.com/paper/2105.06990/) (2021)
- [Sparse Autoencoders Trained on the Same Data Learn Different Features](https://mlsanity.com/paper/2501.16615/) (2025)
- [Sparse Autoencoders Do Not Find Canonical Units of Analysis](https://mlsanity.com/paper/2502.04878/) (2025)
- [Identifying Functionally Important Features with End-to-End Sparse Dictionary Learning](https://mlsanity.com/paper/2405.12241/) (2024)
- [Spectral Filters, Dark Signals, and Attention Sinks](https://mlsanity.com/paper/2402.09221/) (2024)
- [Decomposing The Dark Matter of Sparse Autoencoders](https://mlsanity.com/paper/2410.14670/) (2024)
- [Efficient Dictionary Learning with Switch Sparse Autoencoders](https://mlsanity.com/paper/2410.08201/) (2024)
- [Automated Interpretability Metrics Do Not Distinguish Trained and Random Transformers](https://mlsanity.com/paper/2501.17727/) (2025)
- [Transcoders Beat Sparse Autoencoders for Interpretability](https://mlsanity.com/paper/2501.18823/) (2025)
- [Dense SAE Latents Are Features, Not Bugs](https://mlsanity.com/paper/2506.15679/) (2025)

## Abstract (arXiv)

Sparse autoencoders (SAEs) are typically trained to reconstruct the entire residual stream through a sparse dictionary, implicitly assuming that all activation content is amenable to sparse, monosemantic decomposition. We question this assumption and hypothesize that activations contain a low-rank, dense component that is computationally important to the model yet inherently unsuitable for sparse representation, which serves as a major source of the persistent dense latents widely observed in trained SAEs. To test this, we add a small rank-r linear bottleneck in parallel with standard SAEs (BatchTopK and Matryoshka), allowing dense structure to be absorbed before sparse reconstruction. On Gemma-2-2B layer 12, a rank-24 bottleneck reduces dense latent count by up to 84% while improving sparse probing and targeted probe perturbation on both architectures at matched sparsity. The absorbed component is (i) structurally identifiable as the top principal components and outlier dimensions; (ii) causally necessary, with removing it raising next-token cross-entropy by 7.5×, far exceeding the 2.8× from removing the geometrically near-identical top-24 PCA directions; and (iii) redundantly encoded by sparse dictionaries, with ablating 787 maximally aligned sparse features raising cross-entropy by only 2.9× and ablating 2,048 topic-aligned features leaving MMLU topic classification virtually unchanged, whereas removing the scaffold drops it from 98.7% to chance. Together, our findings identify a compact, semantically informative and causally important component of residual stream activations (which we term a computational scaffold) that standard sparse dictionaries represent inefficiently, suggesting that the scope of sparsity-based interpretability methods warrants careful re-examination.