riesz-map-as-key-value-semantic-bridge
OUT derived (depth 5)
Created 2026-08-25T03:45:56+00:00 · Reviewed 2026-08-25T04:28:09+00:00
The Riesz isomorphism under the causal inner product is the mathematical "compiler" that translates between ROME's key representation (MLP input activation) and value representation (MLP output projection), unifying the parametric knowledge storage mechanism as a single dual-geometric object rather than two independent weight matrices.
Justifications
SL — The Riesz map establishes that embedding and unembedding are isomorphic under the causal inner product; the parametric duality identifies key (input) and value (output) as functionally distinct channels. Together they show the MLP's W_V (key projection) and W_U (value projection) are Riesz-dual objects, meaning the "key-value memory" of ROME is a single geometric entity with two representations. The duality claim alone doesn't explain WHY the two sides are related; the Riesz map alone doesn't ground it in the operational MLP mechanism.
Antecedents (all must be IN):
- OUT riesz-map-as-canonical-semantic-isomorphism — The Riesz map under the causal (whitened) inner product is the unique canonical isomorphism that makes embedding and unembedding semantically identical, thereby elevating "meaning" from a model-specific activation pattern to a well-defined algebraic object in a model-independent vector space
- OUT parametric-contextual-knowledge-duality — LLMs maintain two functionally independent knowledge channels: parametric (editable via ROME's rank-one weight update) and contextual (supplied via retrieved passages that invert accuracy trends), which operate on separate mechanisms and can be independently manipulated.
Dependents
These beliefs depend on this one:
- OUT riesz-map-as-unified-operational-framework — The Riesz map under the causal inner product is the unique canonical object that simultaneously defines the evaluation coordinate system (cosine/Spearman as inner-product measurement), the editing mechanism (ROME's key-value bridge via pre-/post-activation isomorphism), and the semantic equivalence of embedding and unembedding—making all three operations different projections of a single canonical structure.