park-2025-causal-inner-product-whitening-formula
IN premise — summaries/2026/08/24/park-2024-categorical-hierarchical-concepts-s2-using-this-result-we-show-that-semantic-hierarchy-between-co-chunk-1.md
Created 2026-08-25T02:58:24+00:00
The Causal Inner Product is constructed by whitening the unembedding space: g(y) = Cov(γ)^(−1/2) · (γ(y) − E[γ]), after which the Euclidean inner product equals the causal inner product.
Summary
The causal structure hidden in the geometry of the representation space can be made visible by a single coordinate transformation: subtract the mean and rescale the axes so all directions have equal spread. Once that is done, the system no longer needs a special weighted metric — a plain dot product between two points directly gives their causal relationship, turning what would be a complicated calculation into ordinary Euclidean geometry.