Linear Alignment Map (ϕ_lin)

Entities in the same semantic neighborhood but without a typed relation to this one — candidates for new edges or unrecognized duplicates.

• Alignment Map (ϕ)concept0.857
  The bijective function mapping DNN inner neurons to latent variables in causal abstraction; its complexity is the central variable studied
• Linear Map (a ⊸ b)framework0.834
  Semantic domain for linear transformations; denotation as actual linear function; Category instance generated from homomorphism principle.
• Non-linear alignment map ϕ_nonlin achieves near-optimal IIA across all layers on hierarchical equality task, eliminating layer-dependent degradation seen with linear mapsfinding0.808
  Key empirical result: non-linear maps overcome linear maps' failure in deeper layers
• Linear alignment map ϕ_lin shows substantial IIA decrease in third layer for both equality relations and left equality relation algorithms in hierarchical equality taskfinding0.803
  Replicates Geiger et al. 2024b pattern of layer-dependent IIA degradation with linear maps
• Identity Alignment Map (ϕ_id)method0.800
  Simplest alignment map ϕ(h)=h, equivalent to assuming privileged bases hypothesis
• linear directionconcept0.781
  A straight vector in activation space, traditionally used for concept manipulation; claimed to be insufficient when true concept geometry is curved.
• Linear alignment map ϕ_lin IIA tracks DNN accuracy during Pythia-410m training progression on IOI taskfinding0.773
  Suggests linear maps may be better correlated with genuine task implementation than non-linear maps
• Alignmentconcept0.772
  The goal of making model behavior match human values and intentions, often addressed during post-training.