Pairwise matrices for sparse autoencoders: single-feature inspection mislabels causal axes

Technical Summary: Pairwise Matrices for Sparse Autoencoders

Problem Statement
The standard protocol for interpreting Sparse Autoencoders (SAEs) in Large Language Models (LLMs) relies on two steps: assigning a label to a feature based on its top-activating contexts and validating that label via single-feature steering (activating or suppressing the feature at a single magnitude). This paper argues that this "one-corner" inspection protocol is systematically incomplete. It posits that a feature's causal role cannot be fully understood by inspecting a single activation regime or a single feature in isolation. Specifically, the standard protocol misses non-monotonic behavioral shifts along the coefficient axis and the collective, structural roles of near-orthogonal features that are masked when steered individually.

Methodology: The Pairwise Matrix Protocol
The authors propose a "pairwise matrix protocol" that expands the standard inspection into a two-dimensional space:

1. Coefficient Axis: Sweeping a single feature across a range of steering coefficients ($c$) to detect mode-switches or non-monotonic dose-responses that a single-point inspection would miss.
2. Joint-Condition Axis: Steering multiple features simultaneously at a fixed coefficient. This tests whether features are individually substitutable or jointly necessary for specific behaviors, a role often hidden because unsuppressed neighbors compensate for the suppressed feature.

The protocol is applied to three post-trained models: Qwen3-1.7B-Instruct, Gemma-2-2B-it, and Llama-3.1-8B-Instruct. The experimental pipeline involves four phases:

• Phase 1 (Behavioral Pilot): Generating samples from introspective prompts (e.g., "What fascinates you?") and procedural control prompts (e.g., recipes, engine explanations).
• Phase 2 (Cluster Identification): Identifying "register collapse" phenomena where models peak on a narrow vocabulary (e.g., philosophy-of-mind terms) for introspective prompts but not controls.
• Phase 3 (Feature Ranking): Ranking SAE features based on activation differences between introspective and control pools, using bootstrap and permutation controls to ensure stability.
• Phase 4 (Decoder Steering): Applying the matrix protocol. Crucially, the authors employ matched-geometry controls: they compare single-feature steering, joint-feature steering, and random-direction perturbations that produce identical geometric distortions in the residual stream (matching norm ratios and cosine similarities) to isolate whether output degradation is due to magnitude or direction pattern.

Key Findings

1. Coefficient Axis: Top-Context Labels Miss Causal Axes
On Qwen3-1.7B, Feature #26221 is labeled "AI self-disclaimer" based on its top-activating contexts. However, a coefficient sweep reveals an inverted U-shape in disclaimer rates.

• At baseline ($c=0$), the model produces standard disclaimers.
• At high amplification ($c=+500$), the model does not merely increase disclaimers; it substitutes the disclaimer phrase with a fluent "contemplative-philosopher voice" (e.g., "I am deep in contemplation...").
• At extreme amplification ($c=+1000$), the model loops on these markers.
• Implication: The standard label captures only one surface form (the training-data regime) and fails to identify the feature as a directional axis capable of producing distinct, coherent modes at different magnitudes. Two control features (#22082 and #2932) confirmed that monotonic behavior or pure breakdown are distinct from this non-monotonic, coherent mode-switch.

2. Joint-Condition Axis: Individual Substitutability vs. Joint Necessity
Three near-orthogonal features in Qwen (#29108, #26221, #4405) individually steer a "philosophy-of-mind" register.

• Single-Feature Suppression: Suppressing any one of these features at $c=-500$ leaves procedural controls (recipes, engine explanations) intact; the model substitutes content from the remaining active features.
• Joint Suppression: Suppressing all three simultaneously at $c=-500$ causes a "register collapse" on all prompts, including controls. The model outputs syntactic skeletons filled with placeholder tokens (e.g., "BASIC TOMOATO SOUP RECIOPLEY... Vc. 100+").
• Implication: These features are individually substitutable but jointly necessary for grounded composition. The standard protocol, which inspects features one by one, misses this collective structural role.

3. Geometric Distortion Does Not Predict Coherence
To rule out the hypothesis that joint suppression simply pushes the model "off-manifold" due to magnitude, the authors compared single, joint, and random-direction perturbations that matched the residual stream's geometric distortion (norm ratio $\approx 1.55$, cosine $\approx 0.64$).

• Single Feature: Produced coherent strategy-filler answers.
• Random Direction: Produced coherent answers with diverse content substitutions.
• Joint Suppression: Produced placeholder text with high regex-degeneration rates.
• Implication: Coherence loss is direction-pattern-dependent, not magnitude-dependent. The specific combination of near-orthogonal features triggers a structural failure that random perturbations of the same magnitude do not.

Cross-Model Replication
These findings were replicated on Gemma-2-2B-it and partially on Llama-3.1-8B-Instruct:

• Gemma: Replicated the joint-condition finding. The damage signature differed (injecting "human/I/AI" tokens into recipe slots rather than Qwen's placeholder codes), but the structural mechanism (joint necessity for grounded composition) remained. The matched-geometry control on Gemma showed a $\approx 10\times$ gap in degeneration rates between joint steering and random direction.
• Llama: A top causally responsible feature (#38565) was identified for an "encyclopedic-science" register, confirming the existence of sharp register collapse and causal features across models, though full matrix-level tests were not run due to SAE availability constraints.

Significance and Claims
The paper claims that the standard SAE interpretability protocol is insufficient because it inspects only one corner of a feature's behavioral matrix. The authors argue that:

1. Labels are Local, Not Global: A label derived from top-activating contexts captures a specific activation regime but may miss the feature's full causal axis (e.g., the shift from disclaimer to philosophical voice).
2. Features are Context-Dependent: Near-orthogonal features can mask each other's roles; single-feature steering fails to detect structural dependencies required for grounded composition.
3. Geometry is Not Sufficient: Output coherence is determined by the specific direction of perturbation in the feature space, not just the magnitude of the geometric distortion.

The authors conclude that a robust interpretability protocol must include coefficient sweeps with coherence checks and joint-condition probes of near-orthogonal neighbors to accurately map the causal roles of SAE features. They do not claim these findings apply to every feature, but demonstrate that the standard protocol systematically misses these specific classes of behaviors in the tested models.