# HiPACE: Hierarchical Phase-Boundary Analysis and Controlled Evaluation of Feature Absorption in Sparse Autoencoders

arXiv: https://arxiv.org/abs/2609.29551 · HTML: https://mlsanity.com/paper/2609.29551/

## 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 13 of its 20 references that are in the map.
- First posted on arXiv 26 Aug 2026.

## Placed in

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

## Builds on (most cited first, 13 of 13)

- [Pythia: A Suite for Analyzing Large Language Models Across Training and Scaling](https://mlsanity.com/paper/2304.01373/) (2023)
- [Sparse Autoencoders Find Highly Interpretable Features in Language Models](https://mlsanity.com/paper/2309.08600/) (2023)
- [Toy Models of Superposition](https://mlsanity.com/paper/2209.10652/) (2022)
- [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)
- [A is for Absorption: Studying Feature Splitting and Absorption in Sparse Autoencoders](https://mlsanity.com/paper/2409.14507/) (2024)
- [Learning Multi-Level Features with Matryoshka Sparse Autoencoders](https://mlsanity.com/paper/2503.17547/) (2025)
- [SAEBench: A Comprehensive Benchmark for Sparse Autoencoders in Language Model Interpretability](https://mlsanity.com/paper/2503.09532/) (2025)
- [Sparse Autoencoders Do Not Find Canonical Units of Analysis](https://mlsanity.com/paper/2502.04878/) (2025)
- [Archetypal SAE: Adaptive and Stable Dictionary Learning for Concept Extraction in Large Vision Models](https://mlsanity.com/paper/2502.12892/) (2025)
- [Universal Sparse Autoencoders: Interpretable Cross-Model Concept Alignment](https://mlsanity.com/paper/2502.03714/) (2025)
- [Feature Hedging: Correlated Features Break Narrow Sparse Autoencoders](https://mlsanity.com/paper/2505.11756/) (2025)

## Abstract (arXiv)

Sparse autoencoders (SAEs) decompose LLM activations into sparse dictionary atoms, so that each distinct concept gets its own feature. One recurring behavior complicates this premise: feature absorption, in which a parent concept and its children--fruit and apple, banana, pear, say--collapse into a shared family direction. Prior work documents absorption empirically; missing is a closed-form prediction of when the shared direction is the cost-optimal representation of an active semantic family. This paper closes that gap. For a hierarchical Bernoulli generator with k active children and residual scale α, the L0-penalized reconstruction objective admits a closed-form phase boundary λc(k,α)=α2 k/(k-1): above it, pure parent absorption is strictly cheaper than pure child coding. Building on this boundary, we introduce HiPACE, an evaluation protocol that tests the boundary's structural consequence in real SAE dictionaries--measuring parent--child decoder structure over WordNet families, freezing the discovery-selected statistic before testing on unseen families, and contrasting genuine families against randomized sibling nulls. The boundary proves sharp in its native regime, predicting the synthetic transition within ±15% on all 30 tested cells. In Pythia-160m SAEs, the parent--child decoder gap recovers the predicted ordering with partial correlations up to -0.93 that sustain on the locked holdout and exclude sibling nulls (p=0.002). Controlled activation composition connects the theory's active-child count to the recovered family directions, and residual-stream interventions show that signed family directions increase parent-category logits, reversing under sign flip and vanishing under random controls--establishing causal sufficiency at the family-subspace level.