= Quadratic form net bound
{title2=$\|H\|_{\mathrm{op}}\leq(1-2\eta)^{-1}\max_{u\in U}|u^\top Hu|$}
For a <symmetric matrix> $H$ and an $\eta$-<metric net> $U$ of the <unit sphere>, with $0<\eta<1/2$, approximate a maximizing unit vector $x$ by $u\in U$. Expanding $x^\top Hx-u^\top Hu=(x-u)^\top Hx+u^\top H(x-u)$ bounds its absolute value by $2\eta\|H\|_{\mathrm{op}}$. Rearrangement proves the bound. The <volumetric bound for Euclidean metric nets> limits the number of needed directions, allowing a <union bound> to control a random matrix.
Back to article page
