Improved RIP Bounds for Gaussian Partial Circulant Matrices

Abstract

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 $\hat{I}\text{c◯}\subset {\mathbb{Z}}_N$ be any fixed set with $|\hat{I}\text{c◯}|=m$, and let $g\sim \mathcal{N}(0,I_N)$. For every $\hat{I}´,\hat{I}\cdot \in (0,1)$, the normalized partial circulant matrix generated by $g$ has the RIP of order $K$ with constant at most $\hat{I}´$, with probability at least $1-\hat{I}\cdot$ over the draw of $g$, provided \[ m\geq Cδ^{-2}K \max\{\log^2(eK)\log(2N)\log(em),\log(2/η)\}. \] 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 $\log (em)$.

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