Induced Metric from Loss Embedding

The pullback metric $G = \gamma I + \xi\, \nabla L\, \nabla L^\top$ on parameter space, obtained by embedding $\theta \mapsto (\theta, L(\theta))$ into $\mathbb{R}^{N+1}$ with block-diagonal ambient metric $\mathrm{diag}(\gamma I_N, \xi)$. The Sherman-Morrison inverse gives a Riemannian gradient flow $\dot\theta = -\nabla L / (\gamma + \xi \|\nabla L\|^2)$, which is a smoothed variant of gradient clipping. This is the base construction underlying the learnable, off-diagonal, Newton-target, and output-embedding (Gauss-Newton / KFAC) variants tracked in the rest of the project.