Linear alignment map ϕ_lin shows substantial IIA decrease in third layer for both equality relations and left equality relation algorithms in hierarchical equality task

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

• Over 80% IIA achieved using complex non-linear alignment maps on randomly initialised MLPs in hierarchical equality taskfinding0.844
  Demonstrates that high IIA can be obtained even when model cannot solve the task
• Linear alignment map ϕ_lin IIA tracks DNN accuracy during Pythia-410m training progression on IOI taskfinding0.813
  Suggests linear maps may be better correlated with genuine task implementation than non-linear maps
• Identity of first argument algorithm IIA consistently hovers around 50% for all alignment map types on hierarchical equality taskfinding0.807
  Exception to the general trend; attributed to insufficient RevNet capacity rather than algorithm not being implemented
• Best localist alignment achieves IIA of 0.73 on hierarchical equality Both Equality Relations in Layer 1finding0.804
  Shows localist alignment fails to capture the distributed structure found by DAS.
• Linear Alignment Map (ϕ_lin)method0.803
  Alignment map ϕ(h)=W_orth*h using orthogonal matrix; assumes linear representation hypothesis
• The effect of alignment map ϕ complexity on IIA in causal abstraction is an analogue of the probing complexity–accuracy trade-offclaim0.787
  Authors connect their finding to the prior probing literature debate
• 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 differencesfinding0.773
  Corroborating result on additional task confirming main paper findings
• Brute-force search achieves best IIA of 0.60 on hierarchical equality Both Equality Relations in Layer 1finding0.766
  DAS substantially outperforms brute-force search (1.00 vs 0.60 IIA) on the hierarchical equality task.