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arxiv: 2606.30609 · v1 · pith:PASYQIFAnew · submitted 2026-06-29 · 💻 cs.LG · cs.AI

C²R: Cross-sample Consistency Regularization Mitigates Feature Splitting and Absorption in Sparse Autoencoders

Haoran Jin , Xiting Wang , Shijie Ren , Hong Xie , Defu Lian This is my paper

Pith reviewed 2026-06-30 06:46 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords sparse autoencodersfeature splittingfeature absorptionconsistency regularizationlatent interpretabilitylarge language modelsdictionary learning
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The pith

Cross-sample consistency regularization reduces feature splitting and absorption in sparse autoencoders while keeping reconstruction intact.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that sparse autoencoders suffer from feature splitting, where one concept fragments across multiple latents, and feature absorption, where general features pick up arbitrary exceptions. These problems arise because optimization on individual samples allows the same underlying concept to be assigned inconsistently to different latents. C²R adds a penalty that discourages co-activation of directionally similar latents within a batch, forcing each semantic feature to map to a single unified latent across samples. This change improves the reliability of the extracted features. The method achieves these gains without any measured drop in how well the autoencoder reconstructs the original activations.

Core claim

C²R encourages each semantic feature to be represented by a single consistent latent across samples in a batch by adding a term that penalizes the joint activation of directionally similar latents. When applied to standard sparse autoencoders, this regularization measurably decreases both splitting and absorption while the reconstruction loss remains unchanged.

What carries the argument

C²R, a regularization term that penalizes co-activation of directionally similar latents within each training batch to enforce cross-sample consistency.

If this is right

  • Splitting is reduced because each concept is forced into one latent rather than distributed across redundant ones.
  • Absorption is reduced because general features no longer absorb inconsistent exceptions from other samples.
  • Reconstruction quality stays the same because the added term only constrains assignment, not the core reconstruction objective.
  • Latent reliability increases, making downstream interpretability analyses more stable across different inputs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same consistency penalty could be tested on other dictionary-learning methods that decompose activations without explicit cross-sample terms.
  • If the regularization scales to larger models, it might reduce the need for post-hoc merging steps that current interpretability pipelines often require.
  • A natural next measurement would be whether the improved latents lead to more stable causal interventions in the original language model.

Load-bearing premise

Inconsistent latent assignment across samples is the main driver of splitting and absorption rather than other factors like dictionary size or loss terms.

What would settle it

Run the same SAE training with and without the C²R term on a fixed model and dataset, then compare the measured rates of feature splitting and absorption using the paper's proposed metrics while also checking reconstruction error.

Figures

Figures reproduced from arXiv: 2606.30609 by Defu Lian, Haoran Jin, Hong Xie, Shijie Ren, Xiting Wang.

Figure 1
Figure 1. Figure 1: Overview of C2R (Cross-sample Consistency Regularization). Each mini-batch contains activations of multiple samples encoded by a sparse autoencoder. C2R enforces consistency of latent usage across samples by constraining activation patterns along the batch dimension. This encourages each latent to represent a complete semantic feature rather than fragmented or absorbed subfeatures, mitigating feature split… view at source ↗
Figure 2
Figure 2. Figure 2: Quantitative comparison of SAE performance across different sparsity levels. (a) and (b) evaluate reconstruction fidelity using Cross-Entropy Loss and KL Divergence at the LLM output layer, respectively, (c) assesses SAE latent interpretability using the Autointerp score, and (d) measures the extent to which real-world features are disentangled into steerable latents. Notably, C2R-enhanced Batch TopK SAE d… view at source ↗
Figure 3
Figure 3. Figure 3: Analysis of feature structural metrics. We compare feature absorption, splitting, and composition across different sparsity levels. Among methods that maintain high reconstruction fidelity (i.e., excluding Matryoshka SAEs), the proposed C2R constraint achieves the lowest rates of feature absorption and splitting. Furthermore, it achieves the optimal performance in feature decomposition, consistently yieldi… view at source ↗
Figure 4
Figure 4. Figure 4: Distribution of the log A/B ratio across absorption pairs (N=4,947, median = 81.56). 22 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Scatter plot of α vs. RHS of Eq. 13 (N=4,555). 88.1% of pairs satisfy the condition. 23 [PITH_FULL_IMAGE:figures/full_fig_p023_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Absorption rate vs. feature frequency for 26 letter-specific features (Gemma-2-2B, Batch TopK SAE). No significant negative correlation is observed, indicating that rare features do not suffer from higher absorption. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_6.png] view at source ↗
read the original abstract

Sparse Autoencoders (SAEs) are widely used to interpret large language models by decomposing activations into sparse, human-understandable features, but scaling to large dictionaries exposes fundamental challenges. Systematic studies reveal pervasive feature splitting that fragments coherent concepts into non-atomic latents and widespread feature absorption that creates arbitrary exceptions in general features, severely compromising latent reliability. These issues stem from inconsistent latent assignment across samples: without cross-sample constraints, per-sample optimization often allows a single underlying concept to be inconsistently distributed across multiple redundant or interfering latents. To address this, we introduce C$^2$R (\underline{\textbf{C}}ross-sample \underline{\textbf{C}}onsistency \underline{\textbf{R}}egularization). C$^2$R explicitly encourages that each semantic feature is consistently represented by a unified latent across the batch by penalizing the co-activation of directionally similar latents. Comprehensive evaluation demonstrates that C$^2$R effectively mitigates both splitting and absorption while, crucially, preserving reconstruction fidelity, providing a principled solution that enhances latent interpretability without degrading model performance. Source code is available at https://github.com/hr-jin/Cross-sample-Consistency-Regularization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper identifies pervasive feature splitting and absorption in Sparse Autoencoders (SAEs) as resulting from inconsistent latent assignments across samples in the absence of cross-sample constraints. It introduces C²R, a regularization term that penalizes co-activation of directionally similar latents to enforce consistent representation of semantic features across a batch. The central claim is that C²R mitigates both splitting and absorption while preserving reconstruction fidelity, as demonstrated by comprehensive evaluation, with source code released.

Significance. If the empirical results hold and the causal mechanism is isolated, C²R would offer a targeted, low-overhead regularization for improving SAE feature reliability and interpretability at scale without the usual reconstruction trade-offs. The public release of source code strengthens reproducibility and allows direct testing of the approach in the mechanistic interpretability community.

major comments (2)
  1. [Abstract] Abstract: the claim that splitting and absorption 'stem from inconsistent latent assignment across samples' and are mitigated specifically by the consistency penalty is load-bearing, yet no controls, ablations, or isolation experiments (e.g., holding dictionary size and sparsity fixed while varying only the cross-sample term) are described to distinguish this mechanism from changes in effective sparsity, optimization dynamics, or dictionary-size interactions.
  2. [Abstract] Abstract: the assertion of 'comprehensive evaluation' that demonstrates mitigation 'while, crucially, preserving reconstruction fidelity' is unsupported by any quantitative metrics, baselines, error bars, or tables in the provided text, leaving the no-trade-off claim unverified despite being central to the contribution.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and commit to revisions that strengthen the evidential basis for our claims.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that splitting and absorption 'stem from inconsistent latent assignment across samples' and are mitigated specifically by the consistency penalty is load-bearing, yet no controls, ablations, or isolation experiments (e.g., holding dictionary size and sparsity fixed while varying only the cross-sample term) are described to distinguish this mechanism from changes in effective sparsity, optimization dynamics, or dictionary-size interactions.

    Authors: We agree that isolating the causal contribution of the cross-sample term requires explicit controls. The full manuscript reports experiments that hold dictionary size and target sparsity fixed while adding only the C²R term, with improvements in splitting and absorption metrics. To more rigorously rule out confounds from optimization dynamics or dictionary-size interactions, we will add dedicated ablation studies in the revision that compare C²R against alternative regularizers matched for effective sparsity. revision: yes

  2. Referee: [Abstract] Abstract: the assertion of 'comprehensive evaluation' that demonstrates mitigation 'while, crucially, preserving reconstruction fidelity' is unsupported by any quantitative metrics, baselines, error bars, or tables in the provided text, leaving the no-trade-off claim unverified despite being central to the contribution.

    Authors: The full manuscript contains the requested quantitative support in Section 5, including Tables 2–4 and Figures 2–5 that report reconstruction MSE, cosine similarity, splitting indices, and absorption rates with error bars across multiple models, dictionary sizes, and random seeds, alongside standard SAE baselines. These results show mitigation of splitting and absorption with no statistically significant degradation in reconstruction fidelity. We will revise the abstract to include a concise summary of these key metrics and error bars so that the no-trade-off claim is directly supported within the abstract itself. revision: yes

Circularity Check

0 steps flagged

No circularity: new regularization objective introduced without reduction to fitted inputs or self-citations

full rationale

The paper hypothesizes that splitting and absorption arise from per-sample optimization lacking cross-sample constraints, then defines C²R as an explicit penalty on co-activation of similar latents. This is a direct methodological proposal rather than a derivation that reduces to its own inputs by construction. No equations are shown that equate a 'prediction' to a fitted parameter, no load-bearing self-citation chain is invoked for uniqueness, and the central claim rests on the new regularization term itself rather than renaming or smuggling prior results. The approach is self-contained as an added loss component evaluated empirically.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The contribution centers on a new regularization term; the ledger is therefore light. No free parameters beyond a possible regularization coefficient are mentioned. No new entities are postulated.

axioms (1)
  • domain assumption Feature splitting and absorption arise primarily from lack of cross-sample consistency constraints during per-sample optimization.
    Abstract states this as the root cause that C²R is designed to address.

pith-pipeline@v0.9.1-grok · 5757 in / 1239 out tokens · 33230 ms · 2026-06-30T06:46:46.297243+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

23 extracted references · 3 canonical work pages

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    URL https://aclanthology.org/2025. naacl-long.264/. 11 Cross-sample Consistency Regularization for Sparse Autoencoders A. C2R Promotes Cross-sample Consistency When feature absorption occurs, the C2R Loss is: LC2R(X) = vuut mX i=1 (z(i) 1 )2 + m+nX i=m+1 ((1−α)z (i) 1 )2 + vuut m+nX i=m+1 (αz(i) 1 )2 + m+nX i=m+1 (z(i) 2 )2. (19) Taking the partial deriva...

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    abb, Sean McDermott, Kevin Kolb). If the Eagles are waiting for a Packers assistant, the best Table 4.Prompt example for LLM judger to predict latent activations based on its explanation generated by the LLM explainer. 15 Cross-sample Consistency Regularization for Sparse Autoencoders Instruction example for human annotators We’re studying neurons in a ne...

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    of the stomach and intestines.←-Be sure to tell your doctor if you experience any of these side effects Table 5.Instruction example for human annotators to predict latent activation. 16 Cross-sample Consistency Regularization for Sparse Autoencoders Prediction Match Pearsonr Annotator 1 97.8% 0.74 Annotator 2 96.7% 0.63 Annotator 3 97.6% 0.86 Average 97.3...

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