Non-Linear Alignment Map (ϕ_nonlin)

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

• Non-linear alignment map ϕ_nonlin achieves near-optimal IIA across all layers on hierarchical equality task, eliminating layer-dependent degradation seen with linear mapsfinding0.866
  Key empirical result: non-linear maps overcome linear maps' failure in deeper layers
• Alignment Map (ϕ)concept0.834
  The bijective function mapping DNN inner neurons to latent variables in causal abstraction; its complexity is the central variable studied
• Identity Alignment Map (ϕ_id)method0.796
  Simplest alignment map ϕ(h)=h, equivalent to assuming privileged bases hypothesis
• Linear Map (a ⊸ b)framework0.764
  Semantic domain for linear transformations; denotation as actual linear function; Category instance generated from homomorphism principle.
• Linear alignment map ϕ_lin shows substantial IIA decrease in third layer for both equality relations and left equality relation algorithms in hierarchical equality taskfinding0.762
  Replicates Geiger et al. 2024b pattern of layer-dependent IIA degradation with linear maps
• Non-linear ϕ_nonlin achieves near-perfect IIA on distributive law task for both And-Or and And-Or-And algorithms, eliminating linear/identity map differencesfinding0.758
  Corroborating result on additional task confirming main paper findings
• Variational Family V for Alignment Mapsconcept0.751
  Generalised notion restricting alignment maps to a family V; linearity is special case
• Alignmentconcept0.743
  The goal of making model behavior match human values and intentions, often addressed during post-training.