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 $1\leq K\leq m\leq N$ be positive integers, let $\Omega\subset\mathbb Z_N$ be any fixed set with $|\Omega|=m$, 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 $$m\geq C\delta^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/\eta)\}.$$ 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 $\log(em)$.