Technical Summary: C2R (Cross-sample Consistency Regularization)

1. Problem Statement

Sparse Autoencoders (SAEs) are a primary tool for interpreting Large Language Models (LLMs) by decomposing dense activations into sparse, human-understandable features. However, as SAEs scale to large dictionaries, they exhibit two pervasive pathologies that compromise the reliability of learned latents:

• Feature Splitting: Coherent, high-level concepts are fragmented into multiple non-atomic, redundant latents (e.g., a single "Mathematics" concept splitting into separate "Algebra" and "Geometry" latents). This obscures the true functional concepts used by the model and wastes dictionary capacity.
• Feature Absorption: Specific latents capture activations that should belong to general features, creating arbitrary exceptions (e.g., a "starts with S" latent failing to activate on "short" or "small" because token-specific latents absorb the signal). This distorts the intended semantic pattern.

The authors argue these issues stem from inconsistent latent assignment across samples. Standard sparsity objectives (such as $\ell_1$ regularization or TopK constraints) operate on a per-sample basis. Without cross-sample constraints, the optimizer can minimize the sparsity penalty by distributing a single semantic concept across multiple disjoint latents or allowing specific latents to "steal" activations from general ones, as long as the reconstruction error remains low for that specific sample.

2. Methodology: C2R

To address these failures, the paper introduces C2R (Cross-sample Consistency Regularization). This method enforces that each semantic feature is consistently represented by a unified latent across the entire batch.

Theoretical Foundation

C2R is grounded in the Minkowski inequality and the strict convexity of the $\ell_2$ norm. The core insight is that for non-aligned vectors $u$ and $v$, the sum of their norms strictly exceeds the norm of their sum:
$$\|u\|_2 + \|v\|_2 > \|u + v\|_2$$
In the context of SAEs, if a semantic feature is split across multiple latents, the sum of the $\ell_2$ norms of their activations across a batch will be higher than if the feature were consolidated into a single latent. C2R leverages this to penalize the splitting of semantic information.

The C2R Objective

The method extends the pairwise analysis to a full dictionary by identifying redundant latent pairs and penalizing their co-activation.

1. Redundancy Detection: For each latent $i$, the algorithm identifies its nearest neighbor $j^*(i)$ in the decoder weight space based on cosine similarity. This targets pairs that likely represent split fragments or absorbed variations.
2. Gated Penalty: The regularization loss is weighted by the squared rectified cosine similarity ($\rho^2$) between the decoder weights of the pair. This acts as a gate:
   • If latents are distinct (orthogonal), $\rho \approx 0$, and no penalty is applied.
   • If latents are redundant (high alignment), $\rho$ is large, and the penalty is applied.
3. Loss Formulation:
   $$L_{C2R}(X) = \frac{1}{k} \sum_{i=1}^{k} \rho_{i, j^*(i)}^2 \cdot \left( \|Z_{:,i}\|_2 + \|Z_{:,j^*(i)}\|_2 \right)$$
   Where $Z_{:,i}$ represents the batch of activations for latent $i$. The total training objective adds this term to the standard SAE loss:
   $$L(X) = L_{SAE}(X) + \lambda_{C2R} L_{C2R}(X)$$

Gradient Dynamics

The gradient of C2R provides two simultaneous forces:

• Consistency Pressure: Scales with similarity ($\rho^2$) to minimize the sum of norms, pushing split activations into a single latent.
• Implicit Orthogonality: Scales with the activation magnitude ($S$) to minimize cosine similarity between decoder weights. Unlike uniform orthogonality constraints, this scales the penalty by feature frequency, preventing the premature destruction of rare or developing features.

3. Key Contributions

The paper outlines three primary contributions:

1. Theoretical Diagnosis: A unified formal analysis identifying the lack of cross-sample consistency in per-sample sparsity objectives as the root cause of feature splitting and absorption.
2. Principled Objective: The proposal of C2R, a novel regularization that utilizes decoder geometry and batch-level statistics to enforce consistent latent selection, distinguishing between true polysemanticity and harmful redundancy.
3. Empirical Validation: Demonstration that C2R significantly mitigates splitting and absorption while preserving reconstruction fidelity, outperforming state-of-the-art baselines like Batch TopK, Matryoshka SAEs, and OrtSAE.

4. Experimental Results

Experiments were conducted on Gemma-2-2B (and validated on Qwen3-8B and Llama-3-8B) using the SAEBench framework.

• Mitigation of Pathologies: C2R achieved the lowest rates of feature absorption and splitting among methods that maintained high reconstruction fidelity. While Matryoshka SAEs showed lower splitting/absorption scores, they suffered significant degradation in reconstruction quality.
• Reconstruction Fidelity: Unlike Matryoshka SAEs, C2R-enhanced models maintained high "Loss Recovered" and "KL Divergence" scores, indicating that the LLM's original behavior was preserved.
• Interpretability & Disentanglement: C2R did not negatively impact the "AutoInterp" score (human-understandability of features) or "Disentanglement" scores (causal control of attributes).
• Feature Composition: C2R yielded the lowest maximum cosine similarity between decoder weights, indicating a more atomic and distinct feature set.
• Downstream Tasks: In causal intervention tasks (Spurious Correlation Removal and Targeted Probe Perturbation), C2R outperformed TopK, Batch TopK, and OrtSAE, achieving performance comparable to Matryoshka SAEs.
• Robustness: The method remained effective across different random seeds, model scales (8B), layers, and training data compositions (OpenWebText vs. The Pile).

5. Significance and Claims

The paper claims that C2R provides a principled solution to the trade-off between feature atomicity and model fidelity. By enforcing cross-sample consistency, it addresses the fundamental limitation of per-sample optimization in SAEs.

The authors emphasize that their approach:

• Enhances Latent Interpretability: By consolidating fragmented concepts and removing arbitrary exceptions, the resulting dictionary better reflects the hierarchical nature of language model features.
• Preserves Performance: Crucially, these improvements are achieved without degrading the SAE's ability to reconstruct the original model's activations, a common failure point in previous structural SAE attempts.
• Offers Theoretical Guarantees: Under specific conditions (empirically satisfied in 88.1% of absorption pairs), the method provides a theoretical guarantee against splitting and absorption derived from the Minkowski inequality.

The work positions C2R as a general-purpose regularizer compatible with various SAE architectures (TopK, Batch TopK, OrtSAE, AbsTopK), offering a robust path forward for analyzing large language model internals.