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Projecting Assumptions: The Duality Between Sparse Autoencoders and Concept Geometry
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Frames sparse autoencoders (ReLU, JumpReLU, TopK) as projection-based bilevel optimization, shows each implicitly assumes a data geometry, and proposes SpaDE, a simplex-projection prototype-distance SAE that handles nonlinear separability and heterogeneous intrinsic dimension.
Problem
Sparse autoencoders are assumed to uncover all concepts a model uses, but their architectures may implicitly bias them toward certain concept geometries, so different SAEs can find different concepts.
Approach
SAE encoders are orthogonal projections onto architecture-specific constraint sets, which induce receptive fields and implicit assumptions about concept geometry (linear separability, uniform intrinsic dimension). Designing an SAE whose projection set (probability simplex over Euclidean distances to prototypes) matches these properties recovers concepts other SAEs miss.
Key results
- On the 2D separability experiment, ReLU and JumpReLU reach F1 of 1 for the linearly separable concept but are bounded by about 0.5 for the nonlinearly separable one, while SpaDE reaches F1 1.0 for both.
- In the heterogeneity experiment, TopK's normalized MSE falls below 20% for a concept only when k exceeds that concept's intrinsic dimension, whereas ReLU, JumpReLU and SpaDE adapt sparsity per concept.
- On DINOv2 Imagenette activations, SpaDE has the most monosemantic latents across all classes, with latent co-occurrence reduced relative to other SAEs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SpaDE instead of ReLU or JumpReLU | SAE concept discovery · Synthetic Gaussian clusters, nonlinear concept | F1: ~0.5 -> 1.0 | Helped |
| Use ReLU or JumpReLU instead of SpaDE | SAE concept discovery · Synthetic Gaussian clusters, linearly separable concept | F1: 1 -> 1 | No clear effect |
| Use SpaDE rather than ReLU or JumpReLU | SAE concept discovery · English PCFG, parts of speech | F1 | Helped |
| Use TopK with fixed sparsity K | SAE reconstruction · Synthetic Gaussian clusters, heterogeneous concept dimensions | normalized MSE: below 20% | Hurt |
| Set TopK sparsity k to 8 for the 6-dimensional concept | TopK SAE · Synthetic Gaussian clusters, 6-dimensional concept | normalized MSE: below the dashed line only after k=8 | Helped |
| Use SpaDE rather than ReLU, JumpReLU, or TopK | SAE concept specialization · DINOv2-base activations on Imagenette | F1 | Helped |
Surprising findings
- Standard SAEs such as ReLU and JumpReLU show cross-concept latent co-occurrence even when their reconstruction is good, suggesting lack of concept specialization.
- Hyperparameters tuned for SpaDE on synthetic Gaussian clusters transferred to the formal-language task, while other SAEs' hyperparameters did not transfer.
Limitations
- SpaDE implicitly assumes concepts are separated by Euclidean distance, which may still lead to latent co-occurrence otherwise.
- Overly aggressive sparsity in SpaDE can yield overly specialized latents that capture special cases.
- Analysis focuses on mutually exclusive concepts; co-occurring concepts may behave differently.
Experimental setup
Datasets: synthetic Gaussian clusters PCFG formal language strings Imagenette
Benchmarks: Imagenette
Metrics: F1 score of top monosemantic latents normalized MSE L0 sparsity latent co-occurrence cosine similarity
Baselines: ReLU SAE JumpReLU SAE TopK SAE BatchTopK SAE
Builds on: sparse dictionary learning TopK SAE JumpReLU SAE K-Deep Simplex sparsemax
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