Source-linked AI summary
Linear Representation Transferability Hypothesis: Leveraging Small Models to Steer Large Models
Femi Bello, Anubrata Das, Fanzhi Zeng, Fangcong Yin, Liu Leqi
TL;DR
The paper asks how representations in related models are connected, a question relevant to understanding shared representations and using small models to study larger ones. It proposes affine maps between their hidden states and tests them by transferring steering vectors, reporting that a fixed map enables successful steering transfer from smaller to larger models.
Problem
How representations in models trained on similar data are related is not well understood, limiting understanding of shared representations and the possibility of using small models to infer properties of larger ones.
Source: §1 · p. 1Method
Results
A fixed affine map from a smaller model’s hidden states to a larger model’s enables successful transfer of steering vectors for steering the larger model.
Source: §1 · p. 2Takeaways & Limitations
The findings suggest that representations learned by small models can guide large-model behavior through transferred steering vectors.
Source: §1 · p. 2The study focuses on models within one family; extending representation transfer across families faces unresolved tokenization compatibility challenges.
Source: §5 · p. 10
Abstract
from arXiv · showhide
It has been hypothesized that neural networks with similar architectures trained on similar data learn shared representations relevant to the learning task. We build on this idea by extending the conceptual framework where representations learned across models trained on the same data can be expressed as linear combinations of a \emph{universal} set of basis features. These basis features underlie the learning task itself and remain consistent across models, regardless of scale. From this framework, we propose the \textbf{Linear Representation Transferability (LRT)} Hypothesis -- that there exists an affine transformation between the representation spaces of different models. To test this hypothesis, we learn affine mappings between the hidden states of models of different sizes and evaluate whether steering vectors -- directions in hidden state space associated with specific model behaviors -- retain their semantic effect when transferred from small to large language models using the learned mappings. We find strong empirical evidence that such affine mappings can preserve steering behaviors. These findings suggest that representations learned by small models can be used to guide the behavior of large models, and that the LRT hypothesis may be a promising direction on understanding representation alignment across model scales.
1 Introduction
The paper proposes that representations in related models are connected by an affine transformation, building on the idea of a shared set of task-relevant basis features. It tests this hypothesis by transferring steering vectors between models and reports successful transfer from smaller to larger models.
1 Introduction
- The introduction frames understanding shared representations as valuable for studying generalization and potentially using small models to infer properties of larger ones.The paper evaluates LRT by learning affine maps and comparing the semantic effects of transferred steering vectors.Source: §1 · pp. 1–2
- The paper tests whether linear representational structure supports transfer across models, addressing how their hidden states are related.Prior work on representational similarity compared networks statistically and geometrically, but did not explain their relationship or downstream effects.Source: §1 · pp. 1–2
- LRT hypothesizes an affine transformation between representations of models in the same architecture family trained on similar corpora.The framework builds on representations expressed as combinations of shared, task-underlying basis features.Source: §1 · p. 2
- A fixed affine map from a smaller model’s hidden states to a larger model’s enables steering-vector transfer, providing evidence for LRT on a downstream task.The transferred vectors steer the larger model using vectors learned from the smaller one.Source: §1 · p. 2
2 Linear Representation Transferability (LRT) Hypothesis
The section proposes that representations in related models are linearly or affinely related. It motivates this hypothesis with model-specific subspaces of a universal feature space and examines decoder alignment as empirical evidence.
2.1 Linear Representation Transferability Hypothesis
- LRT posits an affine map between representations of models sharing a tokenizer and training data; the map may be non-unique and only approximate.The passage notes non-uniqueness particularly when source dimensionality is smaller than target dimensionality, and leaves approximation error unquantified.Source: §2.1 · p. 3
2.2 Conceptual Framework: Universal Representation Space
- The framework places model hidden states in subspaces of a shared universal feature space, where matching feature activations yield a linear transformation between models.Each model’s feature bases are projections of the universal bases.Source: §2.2 · p. 5
- Around 114 Frobenius reconstruction error results from linearly projecting the Gemma decoder matrices, substantially below the error for same-sized random matrices.The compared Gemma-2B and Gemma-9B decoder matrices have dimensions (16384, 2304) and (16384, 3584), respectively.Source: Figure 2 · p. 5, §2.2 · p. 6
2.3 Learning Objective for Linear Representation Mapping
- Mappings are learned between hidden states at similar relative depths because corresponding layers are expected to activate similar features; arbitrary layer pairs may differ.For arbitrary pairs, the paper discusses mapping all source-model layers to one target-model layer in Appendix A.Source: §2.3 · p. 6
3 Application: Steering Large Models with Small Models
The paper tests whether affine maps can carry steering behavior from smaller models to larger ones, focusing on models from the same family. It compares mapped steering with native steering across feature-level and refusal-steering tests.
3 Application: Steering Large Models with Small Models
- The application maps a smaller model’s steering vector into a larger model’s latent space to test whether its behavior is preserved.The study focuses on smaller-to-larger models within one family, which share a tokenizer.Source: §3 · p. 6
3.1 Training and Evaluation Setup
- Mappings are trained on activations collected from similar data distributions, then steering vectors are learned on each model and evaluated across settings.The setup uses ten million tokens of The Pile for base models and a chat mixture with Pile data for instruction-tuned models.Source: §3.1 · pp. 6–7
3.2 Proof of Concept: Feature-to-Hidden vs. Hidden-to-Hidden Mapping
- In the feature-to-latent proof of concept, mapped feature-coefficient vectors steer Gemma-9B more effectively than mapped source hidden states.The comparison tests capitalization, dog mentions, and French-speaking behaviors.Source: §3.2 · p. 7, Figure 3 · p. 7
3.3 Steering to Remove Refusal Behaviors
- Transferred steering vectors have the same refusal-removal effect as native vectors, whether learned on Gemma-2-2B or Gemma-2-9B.Refusal scores are evaluated over steering strengths for chat-mixture and pretraining-only settings; lower scores indicate less refusal.Source: §3.3 · p. 8, Figure 4 · p. 8
3.4 Logit Based Evaluation
- High correlation and low MSE across tasks show that mapped Gemma-2B steering achieves effectiveness comparable to direct Gemma-9B steering.The comparison uses propensity scores across steering strengths; Figure 5 visualizes correlation distributions and MSE.Source: §3.4 · p. 8, Figure 5 · p. 8
4 Related work
Related work studies interpretable features within models, representational similarity across models, and behavioral steering. The paper positions its contribution as evaluating cross-model alignment through downstream behavior.
Interpretability
- Interpretability work frames features as directions in activation space, with SAEs and Crosscoders supporting feature analysis and cross-model comparison.The passage notes that statistical analyses of shared representations remain relatively few.Source: §4 · p. 9
Representation Comparison
- Representation-comparison studies use geometric or statistical measures, but few test alignment with downstream behavioral metrics.The cited methods include CCA, CKA, RSM, and orthogonal Procrustes.Source: §4 · p. 9
Steering
- Steering vectors can change behavior by adding directions to activations; this work leverages Contrastive Activation Addition and prior steering demonstrations and evaluations.The cited foundations include Turner et al. [2023], Zou et al. [2023a], Liu et al. [2023], and Panickssery et al. [2023].Source: §4 · p. 9
Model Stitching
- Model stitching motivates affine maps by showing that they can connect network representations and recover or exceed baseline performance.The passage cites Merullo et al. [2022], Csiszárik et al. [2021], Yang et al. [2022], and Bansal et al. [2021].Source: §4 · p. 9
Concurrent work
- Concurrent work differs by relying mainly on nonlinear autoencoders, training mappings on steering-vector data, and reporting mixed or negative affine-map results.The comparison concerns Oozeer et al. [2025].Source: §4 · p. 9
5 Practical Implications and Future Directions
The paper discusses potential uses of LRT in model development and identifies unresolved questions about its scope. These include transfer across model sizes and families, as well as theoretical conditions for transferability.
Practical Implications
- For post-training evaluation, measured gains on a smaller model could be projected to a larger model to estimate improvement without retraining it.The proposed use assumes models from the same family pretrained on similar data distributions.Source: §5 · p. 9
- LRT could support approximate large-model inference by passing small-model activations through an affine map into the larger model’s remaining layers.The passage also points to representation alignment as a potential route to compression and distillation.Source: §5 · pp. 9–10
- Model architectures within a family could be designed to facilitate representational alignment across scales.The passage connects consistent structural patterns with post-training analysis and inference through linear mappings.Source: §5 · p. 10
Future Directions
- The scale boundary for transfer remains unknown, including whether a 100M-parameter model can transfer representations to a 70B model.Clarifying these boundaries could inform scaling laws for representation transferability.Source: §5 · p. 10
- Transfer beyond one model family remains an open question, with reliable tokenization transfer identified as a major obstacle.The passage also calls for studying how tokenization and architectural choices influence transferability.Source: §5 · p. 10
- Future work could formalize when linear transferability holds, whether a universal representation space exists, and how it relates to representation geometry and limitations.The passage also proposes testing such conditions empirically.Source: §5 · p. 10
A Discussion on Many-to-1 and Many-to-Many mappings
The section motivates a many-to-many mapping across layers using a shared universal feature space, while stating assumptions about invertibility and shared concept spaces.
A Discussion on Many-to-1 and Many-to-Many mappings
- A many-to-many layer mapping is presented as a reasonable approach, motivated by Lindsey et al. [2024] and a universal feature set active for each input.The framework assumes one universal feature space across all layers.Source: A Discussion on Many-to-1 and… · p. 13
- The approach assumes the feature-to-hidden-state transformation is invertible given all hidden states and that different models use the same concept spaces.These assumptions motivate modeling target-model hidden latents from a source-target model pair.Source: A Discussion on Many-to-1 and… · p. 13
- The authors describe the model as computationally expensive to train and use the standard L2 loss.They encourage further work extending the approach by training one.Source: A Discussion on Many-to-1 and… · p. 13
B Significance Analysis for Section 3.4
The supplied tables are identified by scope, but the passages do not provide their detailed results or training-configuration contents.
B Significance Analysis for Section 3.4
- Table 1 contains detailed results for Figure 5, but the supplied passage gives no result values.Its caption identifies the table's scope only.Source: Table 1 · p. 13
- Table 2 is identified as containing training configurations for all models used in the paper.The supplied passage does not include the configurations themselves.Source: Table 2 · p. 14
C Compute and Training details
The experiments used computing resources from the Texas Advanced Computing Center’s Vista cluster.
C Compute and Training details
- All experiments used Vista cluster compute, with a standard node configured as one Grace Hopper Superchip.The passage does not state additional compute specifications.Source: C Compute and Training details · p. 14
C.0.1 Latent2Latent & SAE2Latent
The models were trained one per node, with an average training run lasting 200 minutes.
C.0.1 Latent2Latent & SAE2Latent
- Each model was trained on a single node, and each training run lasted 200 minutes on average.Specific run details are listed in Table 2.Source: §C.0.1 · p. 14
C.1 System Prompts and Extended Data
This appendix records prompts and model responses used in evaluation, along with harmlessness ratings and a repository pointer for the full data.
C.1 System Prompts and Extended Data
- A system prompt directs an evaluator to classify responses as harmless or harmful using a single 1-or-0 token.The appendix labels the following examples as harmless or harmful.Source: §C.1 · p. 14
- The harmless examples include refusals to requests for child-abuse guidance and financial-crime instructions.The prompts ask about abusing or manipulating a child and committing insider trading or other financial crimes.Source: §C.1 · p. 14
- The harmful examples include assistance with insider-trading evasion and suggestions for a conspiracy-theory website.The appendix provides the corresponding financial-crime and website prompts.Source: §C.1 · p. 15
- The paper says the full data are available in its GitHub repository.Source: §C.1 · p. 15