Improved RIP Bounds for Gaussian Partial Circulant Matrices

We prove an improved restricted isometry bound for Gaussian partial circulant matrices with arbitrary prescribed sampling sets. There is a universal constant $C>0$ such that the following holds. Let $1\le K\le m\le N$ be positive integers, let $\mathrm{\Omega }\subset {\mathbb{Z}}_N$ be any fixed set with $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=m$, and let $g\sim \mathcal{N}(0,I_N)$. For every $\delta ,\eta \in (0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\delta$, with probability at least $1-\eta$ over the draw of $g$, provided $$m\ge C{\delta }^{-2}K\max \{{\log }^2(eK)\log (2N)\log (em),\log (2\mathrm{/}\eta )\}\mathrm{.}$$ The proof refines the Maurey entropy step in the chaos-process argument by combining a noncommutative Khintchine inequality with a Schatten moment estimate controlled by $m$, replacing one factor $\log (2N)$ in the Krahmer–Mendelson–Rauhut bound by $\log (em)$.