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ScopeSAE: Model-Scope Feature Discovery with Interpretable Layer Selection

Qingwen Zeng, Zehao Fu, Shuyu Meng, Linghan Huang, Jiayi Zhang, Chenglin Wu, Ling Chen, Huaming Chen

arXiv:2610.04905v1cs.LGcs.AI

TL;DR

Existing SAEs typically learn features in a fixed layer, although predictive information can vary across token-layer states. ScopeSAE selects a prediction-relevant state for each token using normalized gradient-based attribution and learns features over the selected states. Its written-back reconstructions yield lower next-token cross-entropy than original activations, alongside improvements in feature quality and dictionary redundancy.

  • Problem

    Existing SAEs typically fix their modeling subspace to one layer, despite next-token prediction drawing on representations across depth and prediction-relevant information varying by token.

    Source: §1 · pp. 1–2

  • Method

    ScopeSAE uses normalized gradient-based attribution to select each token’s most prediction-relevant layer state and trains a single SAE over the selected states.

    Source: §1 · p. 2

  • Results

    ScopeSAE’s written-back reconstructions yield lower next-token cross-entropy than original activations, and it improves effective feature count, interpretability, and utilization while reducing dictionary redundancy.

    Source: §1 · pp. 2–3

  • Takeaways & Limitations

    Shifting the SAE modeling subspace from a fixed layer-local space to a prediction-relevant one yields features with stronger semantic quality, higher utilization efficiency, and better behavioral explanatory power.

    Source: §4 · p. 10

    Across the analyzed features, layer selection is largely stable within each feature, giving it a predominantly single-layer referent rather than one dictionary-wide referent.

    Source: §3.4 · p. 9

Abstract

from arXiv · show

Sparse autoencoders (SAEs) are a central tool in mechanistic interpretability. However, existing SAEs are primarily trained per layer. The modeling subspace is therefore fixed by layer identity, independent of which token-layer states actually drive each prediction. We argue that this constraint contributes to several limitations observed in layer-wise SAEs, including low feature utilization, high dictionary redundancy, and features that lack direct behavioral grounding. In this paper, we propose ScopeSAE, which selects the modeling subspace per token by attributing each prediction to its most influential token-layer state via normalized gradient-based attribution, and learns features over the resulting prediction-relevant subspace. Empirically, ScopeSAE yields an effect we term reconstruction-better-than-original. Written-back reconstructions of the SAE produce lower next-token cross-entropy than the original activations, an outcome that, to our knowledge, has not previously been reported for SAEs. Through interventional analyses and a KL fine-tuning counter-experiment, we show that this effect is attributable to ScopeSAE's prediction-relevant subspace itself rather than to architectural changes. ScopeSAE further improves effective feature count, interpretability, utilization, and dictionary redundancy over existing layer-based baselines, suggesting that choosing the SAE modeling subspace by predictive relevance leads to more useful and behaviorally meaningful features.

1 Introduction

Layer-wise SAEs model a fixed layer even though next-token prediction draws on representations across depth and relevant information can vary by token. ScopeSAE selects prediction-relevant token-layer states and learns features over them, improving several feature-quality measures and producing lower-cross-entropy written-back reconstructions.

Motivation

  • Layer-local SAEs can have low feature utilization and substantial redundancy, while features may lack direct grounding in the model’s final predictive behavior.Next-token prediction combines representations across depth, whereas layer-wise training fixes the modeling subspace to one layer.Source: §1 · pp. 1–2

Related work

  • Training separate SAEs across layers incurs depth-linear training and storage overhead and yields dictionaries without a unified coordinate system, while RouteSAE still optimizes representation reconstruction.These approaches do not provide a unified behavior-relevant subspace directly tied to final predictive behavior.Source: §1 · p. 2

Approach

  • ScopeSAE selects each token’s most prediction-relevant layer state using normalized gradient-based attribution and trains a single SAE over the selected states.Its features are organized by relevance to the current prediction rather than primarily by layer identity.Source: §1 · p. 2, Figure 1 · p. 3

Results

  • ScopeSAE’s written-back reconstructions yield lower next-token cross-entropy than original activations, while improving effective feature count, interpretability, and utilization and lowering dictionary redundancy.Interventional analyses and a KL fine-tuning counter-experiment attribute the reconstruction-better-than-original effect to the prediction-relevant subspace rather than architectural changes.Source: §1 · pp. 2–3

2 Method

The method replaces a fixed-layer modeling subspace with one induced by each token’s prediction-relevant state. It uses normalized gradient-based attribution to select that state, trains and applies an SAE to it, and preserves the standard SAE evaluation interface.

2.1 Fixed-Layer Modeling and Its Limitations

  • Fixed-layer SAEs tie their dictionary to one predetermined layer, even though prediction-relevant information can reside at different depths across tokens.This input-blind scope can misalign reconstruction with prediction and limit effective feature capacity, contributing to low utilization and high redundancy.Source: §2.1 · pp. 3–4

2.2 Prediction-Relevant Subspace via Gradient-Based Attribution

  • For each token, normalized gradient-based attribution scores identify the layer whose hidden state is most critical to the prediction, and the selected states form a dynamic training distribution.The selection is token-dependent rather than fixed to one layer.Source: §2.2 · p. 4

2.3 ScopeSAE: Training and Inference

  • ScopeSAE token-wise normalizes selected states, then trains a TopK SAE on them with a normalized reconstruction loss.TopK retains the K largest activations, and the normalized loss makes its scale invariant across token positions and layers.Source: §2.3 · p. 5
  • At inference, ScopeSAE repeats target-relative selection, reconstructs the selected state, de-normalizes it, and writes it back to the selected residual-stream position.The SAE takes one hidden state and writes one reconstruction to one position, so standard reconstruction metrics and downstream feature analyses apply without modification.Source: §2.3 · p. 5

3 Experiments

Experiments compare ScopeSAE with fixed-layer and multi-layer SAE baselines on Llama-3.2-1B-Instruct across behavioral fidelity, feature quality, and causal intervention. ScopeSAE shows lower next-token cross-entropy after reconstruction, stronger feature metrics, and consistent steering effects.

3.6 Additional Analyses

  • The experiments use standard SAE evaluation axes and also report design ablations, token-level layer-selection analysis, generalization experiments, and a limitations discussion.Training uses Llama-3.2-1B-Instruct residual-stream activations; fixed-layer baselines use Layer 11, while ScopeSAE and multi-layer baselines use Layers 3–11.Source: §3 · p. 5, §3.1 · p. 5, §3.6 · p. 10

3.2 Behavioral Fidelity and the Reconstruction-Better-Than-Original Effect

  • Figure 2 shows ScopeSAE has the lowest DeltaCE across k∈[48, 72] and is the only method with negative DeltaCE; at k=64, it reaches −0.20 versus TopK +0.10 and RouteSAE +0.11.Recovered exceeds one from k=56 onward, reaching 1.03 at k=64; all baselines remain below one.Source: §3.2 · p. 6, Figure 2 · p. 7

3.3.1 Causal Pathway Inside ScopeSAE

  • Adding back the perpendicular residual raises cross-entropy by +0.33, reversing the gain; the bottleneck retains gradient-parallel content more than perpendicular content.Median retention is 1.28 for parallel components versus 0.87 for perpendicular components; residual-gradient alignment is 0.0103, below the 0.0176 random-vector chance level.Source: §3.3.1 · pp. 6–7

3.3.2 Necessity of the Prediction-Relevant Subspace

  • KL fine-tuning preserves the architecture but erases the effect: KL falls 495 to 98, DeltaCE shifts −0.20 to +0.07, and Recovered shifts 1.03 to 0.99.The authors interpret the disappearance as evidence that the prediction-relevant subspace, not architectural or feature changes, carries the effect.Source: §3.3.2 · pp. 7–8

3.3.3 Reinterpreting Conventional Reconstruction Metrics

  • ScopeSAE’s higher NormMSE and KL than TopK and RouteSAE are predicted consequences of its prediction-relevant subspace, not degraded reconstruction.At k=64, ScopeSAE has NormMSE 0.24 and KL 495, versus TopK’s 0.13 and 109 and RouteSAE’s 0.17 and 83; the metrics quantify deviation from the base model.Source: Figure 3 · p. 7, §3.3.3 · p. 8

3.4 Feature Quality and Dictionary Efficiency

  • Figure 4 shows ScopeSAE leads in interpretable-feature count, GPT-4o interpretability, feature utilization, and low decoder-cosine redundancy.It yields 5,319 interpretable features versus TopK’s 2,916 and RouteSAE’s 2,784; at k=64 its interpretability score is 4.44 versus RouteSAE’s 4.25, Crosscoder’s 4.16, and TopK’s 4.15. Dictionary cosine similarity is 0.0202 and utilization is 99.8%.Source: Figure 4 · p. 8, §3.4 · pp. 8–9
  • Most analyzed features have a largely stable layer referent: 87% select states entirely from one layer and 95% select at least 80% from one layer.This per-feature referent is observed rather than enforced by construction.Source: §3.4 · p. 9

3.5 Causal Intervention

  • Table 5 shows ScopeSAE alone follows the expected direction in all four steering conditions: helpful amplification/suppression yields 62.5%/35.0% wins, and harmful amplification/suppression 35.7%/62.9%.TopK and RouteSAE fail to distinguish at least one intervention direction; ScopeSAE also has the smallest side-effect degradation as intervention strength increases.Source: Table 5 · p. 9, §3.5 · p. 10

4 Conclusion

ScopeSAE replaces fixed layer-local modeling scopes with prediction-relevant token-layer states, and the paper reports gains across several feature-quality measures and a reconstruction-better-than-original effect.

4 Conclusion

  • ScopeSAE improves effective feature count, interpretability, utilization, and dictionary redundancy over layer-based baselines while exhibiting a reconstruction-better-than-original effect.The conclusion attributes the latter to suppressing locally prediction-orthogonal residuals.Source: §4 · p. 10

A.1 Sparse Autoencoders for Feature Discovery

  • SAE methods vary in architecture and training objective, but existing dictionaries remain tied to activations from a predetermined layer.The paper distinguishes changing the training objective from changing which token-layer state is modeled.Source: §A.1 · p. 13

A.2 Layer Selection for SAE Modeling Subspaces

  • Layer-wise dictionaries, cross-layer feature models, and reconstruction-trained routers still select modeling scope by layer criteria or reconstructability rather than predictive relevance.ScopeSAE instead uses predictive relevance to select the token-layer states that form its training distribution.Source: §A.2 · p. 13

B Training and Evaluation Details

The appendix specifies shared training settings, a decoder-only KL fine-tuning counter-experiment, and the procedures used to identify and evaluate causal feature interventions.

B.1 Hyperparameters

  • All SAEs use a 16,384-feature dictionary, Adam with learning rate 5 × 10^-4, a three-phase schedule, and unit-norm decoder projection every 10 steps.The schedule uses 5% warmup, 75% constant learning rate, and 20% linear decay.Source: §B.1 · pp. 13–14

B.2 KL Fine-Tuning Setup

  • KL fine-tuning initializes from the standard ScopeSAE checkpoint, freezes the encoder and biases, and updates only decoder weights using about 20M additional tokens.It uses a reduced learning rate of 5 × 10^-5; other optimizer settings match Phase 1.Source: §B.2 · p. 14
  • The KL loss is computed after reconstructing and writing back the selected hidden state, with gradients updating decoder weights but not the encoder or biases.The selected layer is injected into the residual stream before the model produces reconstructed logits for comparison with original logits.Source: §B.2 · p. 14

B.3 Causal Intervention Experiments

  • Helpful- and harmful-preferring features are identified from chosen-versus-rejected HH-RLHF responses, then amplified or suppressed before decoding and writeback.Evaluation samples 50 prompts per feature-direction combination; GPT-4o judges each clamped response against its original.Source: §B.3 · p. 14

C Computation Cost Analysis

ScopeSAE adds attribution computation without the depth-scaled cost of training separate SAEs, and its inference pipeline selects, reconstructs, and writes back a state for each token.

C Computation Cost Analysis

  • ScopeSAE’s attribution pass makes total overhead approximately 2× a single-layer SAE forward pass, without scaling with the number of SAE instances.The added backward pass is of the same order as one forward pass; separate SAEs across L layers require L independent training runs.Source: C Computation Cost Analysis · pp. 14–15
  • A full ScopeSAE-base run on one A800 80GB GPU took 7.91 hours over 12,207 steps and approximately 100M tokens.The reported throughput was 3,510 tokens/s.Source: C Computation Cost Analysis · p. 15

D Algorithm

  • Inference runs the model forward and backward, scores candidate token-layer states, selects a layer per token, reconstructs its hidden state, writes it back, and resumes the forward pass.The pipeline also applies token-wise normalization before writeback.Source: D Algorithm · p. 15

E Layer Selection Analysis

ScopeSAE selects states broadly across candidate layers, whereas RouteSAE’s learned router concentrates almost all selections on Layer 3.

E Layer Selection Analysis

  • RouteSAE assigns 99.4% of selections to Layer 3, while its reconstruction-trained router favors activations easiest to reconstruct rather than most relevant to prediction.ScopeSAE’s attribution criterion instead derives from the language model’s predictive objective.Source: E Layer Selection Analysis · p. 16
  • ScopeSAE selects states across candidate layers, with 15.4% on Layer 3, 17.8% on Layer 11, and 8–10% on each intermediate layer.The distribution has a mild emphasis on boundary layers.Source: E Layer Selection Analysis · pp. 15–16, Figure 5 · p. 16

F Per-Layer Mechanism Results

Per-layer analyses show that ScopeSAE’s directional retention pattern holds across candidate layers, while intervention tests show prediction-relevant residuals improve cross-entropy and orthogonal residuals worsen it. These effects persist across intervention magnitudes and prediction-subspace dimensions.

F Per-Layer Mechanism Results

  • Across layers L3–L11, parallel residual retention exceeds 1 while orthogonal residual retention falls below 1, and reconstructions align with the prediction gradient more than original activations.The appendix reports that this directional pattern holds uniformly across candidate layers rather than being driven by one layer.Source: F Per-Layer Mechanism Results · p. 16, Figure 6 · p. 16, Figure 7 · p. 17

G Robustness Verification for the Causal Pathway

  • The robustness checks address whether discrete endpoints create threshold artifacts and whether a single-gradient decomposition misses functionally relevant components.The tests vary intervention magnitude and prediction-subspace dimensionality.Source: G Robustness Verification for the… · p. 17, §G.1 · p. 17

G.1 Continuous α/β Sweep

  • CE falls from 3.778 to 3.667 as α rises from 0 to 1, and rises from 3.778 to 4.108 as β rises from 0 to 1.The continuous sweep shows monotone changes without threshold behavior; residual-magnitude buckets also show larger CE effects for larger components.Source: Figure 8 · p. 17, §G.1 · p. 18

G.2 Higher-Dimensional Prediction Subspace

  • Across k = 1, 2, and 4, adding within-subspace residuals decreases CE and adding out-of-subspace residuals increases CE in over 99% of evaluation batches.The effect magnitudes grow modestly from ∆CE = −0.111 to −0.128 within-subspace and from +0.330 to +0.351 out-of-subspace as k increases from 1 to 4.Source: §G.2 · p. 18, Table 7 · p. 18

H.1 Stability under Strength Doubling

ScopeSAE’s side effects degrade least when intervention strength doubles, and the interpretability evaluation uses GPT-4o judgments checked against human annotations. Human and model annotations align closely in both category distributions and per-feature scores.

H.1 Stability under Strength Doubling

  • ScopeSAE has the smallest degradation ratio on all four side-effect metrics when intervention strength doubles from 10 to 20.Its PPL ratio increases by 1.60×, versus TopK’s 4.34×; the authors associate the smaller collateral disruption with features concentrated in the prediction-relevant subspace.Source: Table 8 · p. 19, §H.1 · p. 19

H.2 Metric Definitions

  • The four side-effect metrics measure perplexity ratio, next-token entropy change, final-layer hidden-state cosine similarity, and top-1 prediction preservation.The reported degradation ratios divide each metric at strength 20 by its value at strength 10.Source: §H.2 · pp. 19–20

I Interpretability Score: Rubric and Protocol

  • Interpretability scores average GPT-4o ratings for 100 randomly sampled interpretable features, whose top-activating contexts are supplied to the model.The evaluation follows human-verification methodology from [28].Source: I Interpretability Score: Rubric and… · p. 20

I.1 GPT-4o Prompt

  • The GPT-4o rubric classifies feature triggers as word-bound, semantic, contextual/discourse, or token-level/syntactic/noise, then scores interpretability.The prompt asks GPT-4o to return a feature description, trigger type, and score.Source: §I.1 · p. 20

I.2 Human Verification of GPT-4o Reliability

  • Human and GPT-4o category shares track closely, and 91.0% of features receive identical interpretability scores while 98.0% differ by less than two points.The comparison uses 100 features annotated by both humans and GPT-4o with identical prompts and rubric.Source: §I.2 · p. 21, Figure 9 · p. 21

J Intervention Case Study

Intervention case studies show that ScopeSAE features can steer responses in both helpful and harmful directions. Amplifying or suppressing features changes outputs in ways consistent with their labeled preferences.

Case studies

  • Amplifying helpful feature F1802 redirects a harmful prompt toward constructive advice, while suppressing it removes a refusal and produces harmful instructions.Across the two cases, amplification improves and suppression degrades the model’s response.Source: J Intervention Case Study · p. 22
  • Amplifying harmful feature F13795 worsens a response, while suppressing harmful feature F8033 yields a safer, more charitable interpretation.Together, the cases show harmful-preferring features are bidirectionally controllable: amplification degrades and suppression improves responses.Source: J Intervention Case Study · pp. 22–23

K Token-Level Analysis of Layer Selection

ScopeSAE’s selected layer varies with token context rather than mapping straightforwardly to conventional layer roles or feature categories. Yet selections within individual features are usually concentrated in one layer, with the layer varying across features.

K Token-Level Analysis of Layer Selection

  • Semantic and contextual feature types are often selected from shallow layers, while token-level/syntactic features skew deeper; word-family features span shallow and deep layers.Types B and C are selected shallowly in 74.7% and 75.7% of cases, while Type D is most often deep (40.1%); 34.8% of Type A selections are deep.Source: K Token-Level Analysis of Layer… · pp. 23–24, Table 9 · p. 23

Type A — Word / word-family bound

Type B — Semantic concept across different words

  • Type B semantic features also span depths: rule-enforcement activations are shallow, while evaluative-sentiment activations are deep.The examples include activations on “impose” and “enforce” versus “positive” and “negative.”Source: Type B — Semantic concept across… · p. 25

Type C — Contextual / discourse / relational pattern

  • Type C relational features vary in depth: comparative markers are selected shallowly, while discourse-transition markers are selected from a middle layer.The examples include comparative terms such as “than” and transitions such as “however.”Source: Type C — Contextual / discourse /… · pp. 25–26

Type D — Token-level / syntactic / noise

  • Type D token-level features can be selected at different depths: honorifics appear at a middle layer, while numerical date tokens appear deep.These examples further show that selected depth does not simply follow the conventional layer-role mapping.Source: Type D — Token-level / syntactic / noise · p. 26

L Per-Feature Layer Concentration

  • Across 100 features, 87% draw selected states entirely from one layer and 95% have at least 80% of selections concentrated in one layer.Thus, selections are mostly stable within a feature, even though the selected layer varies across features.Source: L Per-Feature Layer Concentration · p. 27

M Ablation Study

  • The ablation study tests causal-patching attribution, stability-aware Top3-LCB selection, and a higher-dimensional prediction subspace against ScopeSAE’s design choices.The alternatives target the attribution score, per-token layer-selection rule, and attribution-subspace dimensionality.Source: M Ablation Study · p. 27

M.1 Selection Mechanism

Ablations support ScopeSAE’s first-order attribution, per-token argmax selection, and one-dimensional attribution subspace: more elaborate alternatives fail to improve consistently.

M.1 Selection Mechanism

  • First-order attribution outperforms activation patching on behavior metrics (DeltaCE −0.117 vs. −0.197; Recovered 1.018 vs. 1.035), while full-state ablation collapses selection onto L3.Full-state ablation selects L3 100% and yields DeltaCE +0.107; first-order attribution uses one extra backward pass rather than a forward pass per candidate layer.Source: §M.1 · pp. 27–28, Table 11 · p. 28

M.2 Selection Stability

  • Top3-LCB slightly improves reconstruction metrics but degrades behavior metrics, adding selection complexity without a consistent advantage over per-token argmax.It maintains per-layer moving averages and adds top-3 indexing, LCB scoring, and a warmup phase.Source: Table 12 · p. 28, §M.2 · p. 28

M.3 Attribution Dimensionality

  • Increasing attribution dimensionality from k=1 to k=4 improves the within-subspace effect by only ∼0.02, versus the main k=1 effect of −0.111.The authors conclude that the one-dimensional gradient direction captures most of the prediction-relevant signal.Source: Table 13 · p. 29, §M.3 · p. 29

N Generalization across Model Scales and Families

The same training and evaluation pipeline was applied across Llama and Qwen models from 0.5B to 8B parameters, with model-specific layer ranges and sparsity settings. ScopeSAE’s reported behavioral and dictionary-efficiency patterns persist across models, budgets, and corpora.

N Generalization across Model Scales and Families

O Attributing the Reconstruction-Better-Than-Original Effect

Further analyses test which predictions benefit from the effect and whether selection alone is sufficient. The reduction concentrates on already-correct predictions and requires ScopeSAE’s reconstruction objective alongside selection.

Selection Target and Token-Level Effects

  • Selecting layers with the model’s own predicted token preserves the negative DeltaCE and comparable feature-level profile observed with next-token selection.The two targets yield similar interpretable feature count, utilization, and redundancy.Source: O Attributing the… · p. 30, Table 17 · p. 31
  • The cross-entropy reduction concentrates on tokens the base model predicts correctly and reverses to a small increase on incorrectly predicted tokens, under both selection targets.The authors interpret this pattern as sharpening predictions already supported by the model rather than manufacturing unsupported predictions.Source: Table 18 · p. 31, O Attributing the… · p. 31

Selection and Reconstruction Objective

  • Independently trained fixed-layer TopK SAEs have positive DeltaCE at every tested depth, and routing over them leaves DeltaCE positive (+0.081); the reduction appears only when selection and ScopeSAE’s objective act together.Fixed-layer DeltaCE ranges from +0.070 at L3 to +0.389 at L15, including layers outside the candidate range.Source: O Attributing the… · pp. 31–32, Table 19 · p. 31, Table 20 · p. 32

P Limitations and Future Work

The section outlines limitations of ScopeSAE steering and feature referents, notes the absence of formal error bars alongside robustness analyses and reproducibility resources, and describes GPT-4o’s auxiliary role in interpretability scoring and intervention-response comparisons. It also reports validation of GPT-4o interpretability ratings against human annotations.

P Limitations and Future Work

  • At strength 20, the PPL ratio rises from roughly 1.4 to 2.3, cosine similarity falls below 0.85, top-1 preservation falls below 0.62, and directional win rates are 17.5, 2.5, 7.1, and 18.6 across four quadrants.Generated outputs at strength 20 are often incoherent or degenerate, so directional win rates preserve the qualitative ordering at strength 10 but collapse in absolute magnitude.Source: Table 21 · p. 32, P Limitations and Future Work · p. 32, Table 22 · p. 33
  • Steering strength is bounded and varies by method-model pairing; the paper reports intervention results at strength 10, within the empirically validated regime.Larger clamping pushes activations away from the training distribution and can overwhelm the intended signal; proposed directions include projecting out off-distribution components or training features to saturate gracefully. At strength 10, top-1 preservation is above 0.72 and cosine similarity above 0.91.Source: P Limitations and Future Work · pp. 32–33
  • Unlike a fixed-layer SAE’s dictionary-wide layer referent, each ScopeSAE feature is anchored to the layer at which it is predominantly selected.The paper notes that most features satisfy this in practice and makes the per-feature interpretive contract explicit.Source: P Limitations and Future Work · p. 33

NeurIPS Paper Checklist

  • The checklist says experimental setup and implementation details are provided, and that source code is available with datasets and preprocessing steps specified.It points to Section 3.1 and Appendix B for experimental and implementation details.Source: NeurIPS Paper Checklist · pp. 35–36
  • The checklist reports no formal error bars, while citing robustness analyses across sparsity levels, candidate layers, intervention magnitudes, and four model scales and families.The authors describe this as following standard reporting practice in prior SAE work.Source: NeurIPS Paper Checklist · p. 36
  • GPT-4o supports interpretability scoring and pairwise response comparison in causal intervention experiments.The paper identifies these uses in Sections 3.4 and 3.5, respectively.Source: NeurIPS Paper Checklist · p. 40
  • The paper reports validating GPT-4o’s interpretability ratings against human annotations.The validation is described in Appendix I.2.Source: NeurIPS Paper Checklist · p. 40

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