lin-0107

8.3 Deep learning sketches

Definition 8.2 Deep learning sketch

A deep learning sketch is a symmetric monoidal learning sketch \(\mathbb S^\otimes \) together with a strong symmetric monoidal candidate model

\[ D:\mathsf{FMon}(\Sigma )\longrightarrow \mathbf{Para}, \]

where \(\Sigma \) is the typed generating signature and the assigned arrows are parametrized smooth modules.

The free symmetric monoidal category \(\mathsf{FMon}(\Sigma )\) records both serial composition and parallel placement. Declared equations generate a quotient

\[ q:\mathsf{FMon}(\Sigma ) \longrightarrow \mathsf{FMon}(\Sigma )/{\sim }. \]

The candidate is compositional when it descends through this quotient and realizes any designated cones or cocones.

Definition 8.3 Base DLINCS obstruction

For parallel paths \(p,q:x\to y\), let \(\mathcal O_{\mathrm{eq}}(D(p),D(q))\) be the universal obstruction for the registered declaration that their images agree. Define

\[ \mathcal O_0(D;p,q) = \mathcal O_{\mathrm{eq}}(D(p),D(q)). \]

For the whole sketch,

\[ \mathcal O_0(D) = \mathcal O_{\mathbb S^\otimes }(D). \]

The symbol \(\simeq \) is interpreted by the realization: equality of maps, an isomorphism, or an equivalence after quotienting a registered gauge. A supported statistical test may observe one of these declarations, but does not define the relation. A coordinate residual is likewise an observation of the obstruction, not its definition.

Diagram illustrating 8.3 Deep learning sketches.

None

A deep sketch retains forward route obligations and reverse learner semantics.