It is known that, for an $m$ dimensional space and an $n\times m$ dimensional random matrix $U$ whose entries are iid Gaussian, then $\|I-(1/n)U^TU\|$ is bounded by $\sqrt{m/n}$ when $n>m$.
If a set $\mathcal{A}\subset \mathbb{R}^m$, when intersecting with unit sphere $\mathcal{S}\subset\mathbb{R}^m$, has Gaussian width $w(\mathcal{A}\cap \mathcal{S}) = w$, then can we say $\max_{x\in\mathcal{A}} \{\|(I-(1/n)U^TU)x\|/\|x\|\}\le w/\sqrt{n}$ with high probability?
