This is an interactive piece. Read the full version with live charts — drag the steering rank and watch the law emerge — here:

→ The Steerability Spectrum of Visual Attributes

Here is a small puzzle that turns out to say something general about vision models.

Take a frozen vision encoder — DINOv2, CLIP, SigLIP — and a set of rendered objects where you vary exactly one thing: an object's size, or its orientation. Train a linear probe to read that attribute back. For both attributes the probe is nearly perfect — the information is plainly there, linearly, in almost every layer.

Now try to change the model's mind with the classic steering vector — the difference of class means, added to the activations. For size, it nudges the readout. For orientation, it barely moves. The same direction that reads orientation perfectly fails to steer it. Decodability and steerability come apart.

The interactive explainer works through why, and introduces a number — k*, the steerability dimension — that measures how many directions an attribute is spread across, predicts how hard it is to steer, and prescribes the fix.

The headline findings:

  • Decodable ≠ steerable. A feature you can read off perfectly can still resist a single-vector push if it is geometrically wide.
  • k* measures that width — and prescribes the intervention. Generalize the rank-1 concept vector to a rank-K subspace edit. Steering success peaks exactly when the rank K reaches k*: for orientation on DINOv2, the classic concept vector closes only 11% of the gap, but a rank-k* edit closes 96% — with object identity perfectly intact. Push past k* and identity breaks. You need k* dimensions, and no more.
  • The law is objective-universal — it holds across self-distillation (DINOv2), contrastive (CLIP), and sigmoid (SigLIP) encoders.
  • But it is bounded by the readout. Inside PaliGemma, the same edit steers the representation (following the k* law) yet a frozen language model downstream ignores it — the caption doesn't move. Steerability is gated by two things: the encoding geometry (k*) and whether the readout listens to the subspace you edited.

The full piece has two interactive figures driven by the real measurements — switch attributes and encoders, drag the rank, and see the dissociation break and re-form for yourself.

→ Read the interactive version