lin-0070

4.9 Natural LINCS: intrinsic tangent repair

A tangent diagnosis identifies admissible directions in which an obstruction changes, but it does not yet say which corrective direction is smallest. A coordinate-dependent Euclidean norm on parameters answers that question only after choosing a presentation. Natural LINCS instead equips the model space with an intrinsic geometry and uses that geometry to select among typed repairs.

Let (M) be a smooth model object and let a typed obstruction be represented locally by

\[ \mathcal O:M\longrightarrow E, \]

where \(E\to M\) is an obstruction bundle. At \(\theta \in M\), its linearized response is

\[ D\mathcal O_\theta :T_\theta M\longrightarrow E_\theta . \]

Suppose that \(g\) is a declared Riemannian metric on model directions. A first-order natural repair is a solution of

\begin{equation} \xi _\theta ^\star \in \operatorname *{arg\, min}_{\xi \in T_\theta M} \frac12 g_\theta (\xi ,\xi ) \quad \text{subject to}\quad D\mathcal O_\theta [\xi ]=-\mathcal O(\theta ). \end{equation}

4.1

Thus the obstruction remains typed and potentially vector- or bundle-valued; the metric merely chooses the least intrinsic change that cancels it to first order. When exact cancellation is unavailable, a metric \(h\) on the obstruction fibers gives the regularized problem

\begin{equation} \operatorname *{arg\, min}_{\xi \in T_\theta M} \frac12\left\| D\mathcal O_\theta [\xi ]+\mathcal O(\theta )\right\| _{h_\theta }^2 +\frac{\lambda }{2}\| \xi \| _{g_\theta }^2. \end{equation}

4.2

Definition 4.8 Natural LINCS structure

A Natural LINCS realization consists of a model object \(M\), a typed obstruction object or bundle \(E\), declared tangent and obstruction metrics \(g\) and \(h\), a policy for quotienting null directions, and a retraction or integration map that returns an admitted tangent proposal to the model space. Its local repair rule is Equation 4.1, or an explicitly declared relaxation such as Equation 4.2.

For a scalar observation \(L:M\to \mathbb R\), the metric induces the musical maps

\[ g^\flat _\theta :T_\theta M\to T_\theta ^*M, \qquad g^\sharp _\theta =(g^\flat _\theta )^{-1}, \]

and the familiar natural-gradient vector is

\[ \operatorname {grad}_g L(\theta ) =g^\sharp _\theta (dL_\theta ). \]

Natural gradient is therefore a scalar special case of the more general typed repair problem; it is not the definition of non-compositionality.

Design principle

The sketch declares what is broken; tangent diagnosis says how it changes; geometry chooses a presentation-invariant repair direction; admission decides whether to accept the resulting proposal.

Neither tangent-category axioms nor Cartesian closedness automatically supply a metric, a cotangent dual, an expectation operator, or an invertible Fisher matrix. These are additional semantic declarations. In an internal smooth setting, a metric may be represented by a bundle morphism \(g^\flat :TM\to T^*M\), but its inverse may exist only after localization or quotienting. Singular directions are especially informative: they often mark parameter symmetries or observationally indistinguishable models. Natural LINCS therefore quotients such null directions before inversion, in the sense developed in Chapter 9, or uses an explicitly audited pseudoinverse or damping rule.