Feature Geometry & Linearity

How literally should we take 'features are directions'? The linear representation hypothesis claims most features in trained networks correspond to approximately linear directions — meaning the network reads them with linear projections and writes them with additive updates.

The evidence is broad: embedding arithmetic works; linear probes decode thousands of properties; steering vectors control behavior along single axes; sparse autoencoders — linear encoders — recover monosemantic features. If the representation were highly nonlinear, none of these cheap linear tools would work as well as they demonstrably do.