Layer loss by environment substitution

Read the equation

The layer objective is defined by substituting that layer’s environment Eℓ into the global loss. This is a restriction of a shared objective, intended to reuse the same logic locally. It does not imply that all layers have equal inputs or equal solutions.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read L_\ell^\*(S',D'; U,E_\ell) as the quantity being defined or updated by this relation.

2. Substitute the source-defined variables or recorded values and follow the displayed arithmetic and indexing exactly.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.

Original context

(Πℓ is some projection/filter to “layer ℓ data”.)

Define:

L_\ell^\*(S',D'; U,E_\ell)
L^\*(S',D'; U,E)\big|\_{E \mapsto E_\ell}

Explicitly:

L_\ell^\*(S',D') =
\|D_\ell'(U,E_\ell)-B_\ell(S',E_\ell)\|
+ W_\ell(U,E_\ell)\cdot \text{Cost}_\ell(S',S)
+ R_\ell(S',E_\ell)

L_\ell^\*(S',D'; U,E_\ell)
=
L^\*(S',D'; U,E)\big|\_{E \mapsto E_\ell}

Related builds