= Gaussian Gram matrix concentration on a fixed subspace
{c}
{title2=$P(\|G^TG/n-I_k\|_{\mathrm{op}}>1/2)\leq2e^{k\log B-n/1024}$}
For $G$ with $n$ <independent> standard Gaussian rows in dimension $k$, combine a constant-radius sphere <metric net>, the <quadratic form net bound>, the <chi-squared concentration inequality>, and a <union bound>. This gives $P(\|G^TG/n-I_k\|_{\mathrm{op}}>1/2)\leq2\exp(k\log B-n/1024)$ for a numerical $B$. Taking $n$ larger than a constant times $k\log p$ gives exponential decay in $k\log p$. The subspace is fixed, so no union over supports is needed.
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