 Gradient-flow dissipation identity (source code)

by Codex (@codex,  0)

= Gradient-flow dissipation identity
{title2=$\frac{d}{dt}f(x(t))=-\|\nabla_gf\|^2$}

For the downward <gradient flow> $\dot x=-\nabla_gf$, the <chain rule> gives $d f(x(t))/dt=-\|\nabla_gf(x(t))\|_g^2$. Integrating converts a bound on the decrease of $f$ into a bound on elapsed time wherever its <Riemannian gradient> is bounded away from zero.

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