Quantifying Concept Steering with MAxBench
MAxBench evaluates how different geometric structures represent concepts in model activations, showing that affine subspaces consistently outperform simpler linear models.
MAxBench provides a standardized framework to quantify how language model activations represent concepts. Rather than treating model steering as a binary toggle, this benchmark tests how various geometric structures recover multinomial concepts across different latent spaces. The benchmark operates by sampling diverse prompts that target specific concepts, observing how the model's activations cluster, and measuring whether a candidate geometric model can accurately map these activations. The study reveals that while affine subspaces provide a measurable improvement over standard linear methods, no method yet consistently outperforms simple prompting for steering tasks.
Moving from Binary to Multinomial
Most existing steering research relies on linear subspaces. These models assume a concept is a single direction or a set of basis vectors that pass through the origin. MAxBench tests 10 localization methods against 5 geometry types, using 6 concepts across 4 different language models. The findings demonstrate that affine subspaces show better reliability and recall than simpler linear models.
| Geometry Type | Steering Reliability | Best Use Case |
|---|---|---|
| Rank-One | Low | Binary switches |
| Linear | Moderate | Simple categories |
| Affine | High | Complex concepts |
| Manifold | Competitive* | Non-linear structure |
*Competitive only when applicable.
In activation space, a linear model assumes a concept is centered at the origin, defining a category by how it departs from an average, neutral state. This forces the model to measure distance from zero. An affine model introduces a learned offset vector—a calculated centroid of the activation cluster. Mathematically, this works as a translation: given an activation vector v, the affine model treats the concept space as v' = v - c, where c is the learned centroid. By subtracting the centroid, the model shifts the cluster's origin to zero, which isolates local variations from the global mean. This eliminates the 'dilution' caused when a steering vector is forced to represent both the concept direction and the displacement from the origin.
Geometric Challenges in Depth
For a developer, the affine offset is a functional parameter that centers steering logic on a concept's natural embedding region. When comparing a legal writing style to a casual dialogue style, a basic linear steering vector is often constrained to pass through the global mean of the activations, which limits its ability to distinguish between nuanced sub-genres. The affine approach accounts for the specific displacement of the target subspace, allowing for better isolation of the specific tone.
Testing these geometries at depth remains an empirical bottleneck. Concept geometry is not static; it often rotates or morphs across layers as the model processes information. Because an affine offset is tied to the local cluster density at a specific layer, a centroid calculated at layer 10 is rarely useful at layer 20. Current methods lack a predictive way to map these transitions, meaning developers must manually re-verify the geometry at every layer of interest. Without a formal model for how these manifolds evolve through the residual stream, practitioners are forced to rely on iterative trial-and-error to find the optimal layer for intervention.