Remark. Product symmetry is larger than the isometric gauge [ftip-00B5]
Remark. Product symmetry is larger than the isometric gauge [ftip-00B5]
The full \(\mathrm {GL}(k)\) action in Definition [ftip-00AX] preserves \(W\), but ordinary Euclidean gradients are covariant under the orthogonal specialization used in Lemma [ftip-00B1]. A theorem for \((UQ,VQ)\) must therefore not be silently promoted to every product-preserving reparameterization.
The source calls the orthogonal subgroup the gauge in its optimizer classification while also noting the larger function-preserving group. These are compatible statements only when their different geometric scopes remain explicit.