lin-0066

4.5 The running equivariance example

Return to the equivariance declaration from Chapter 3. Let \(X\) and \(Y\) be finite-dimensional normed vector spaces, and let the model \(F_\theta :X\to Y\), input action \(a_g:X\to X\), and output action \(b_g:Y\to Y\) be smooth. The base representative is

\[ E_{g,\theta }(x) = F_\theta (a_gx)-b_gF_\theta (x). \]

For an input perturbation \(v\), the first tangent representative is

\[ DE_{g,\theta }(x)[v] = DF_\theta (a_gx)[Da_g(x)v] - Db_g(F_\theta (x))\, DF_\theta (x)[v]. \]

If \(a_g\) and \(b_g\) are linear, this reduces to

\[ DF_\theta (a_gx)[a_gv]-b_gDF_\theta (x)[v]. \]

The base audit asks whether the two finite routes agree. The tangent audit asks whether their local sensitivities intertwine the declared actions. A model can have a small base error at sampled points while its tangent error exposes a rapid departure nearby.

Example 4.5 Parameter and input probes

If parameters also move by \(\dot\theta \), the full differential contains

\[ \partial _\theta F_\theta (a_gx)[\dot\theta ] - b_g\partial _\theta F_\theta (x)[\dot\theta ]. \]

Input probes \(v\) and parameter probes \(\dot\theta \) answer different questions and should occupy different admission blocks.