报告时间：2018年11月30日上午9:00-10:00
报告地点： X7503
主持人：周正春教授
Title: Exact Sparse Signal Recovery via Orthogonal Matching Pursuit with Prior Information
Abstract: Exact recovery of $K$-sparse signals $\x\in \mathbb{R}^{n}$ from linear measurements $\y=\A\x$, where $\A\in \mathbb{R}^{m\times n}$ is a sensing matrix, arises from many applications. The orthogonal matching pursuit (OMP) algorithm is a widely used algorithm for reconstructing the $\x$ based on $\y$ and $\A$ due to its excellent recovery performance and high efficiency. A fundamental question in the performance analysis of OMP is the characterizations of the probability that it can exactly recover $\x$ for random matrix $\A$ and the minimal $m$ to guarantee a satisfactory recovery performance. Although in many practical applications, in addition to the sparsity, $\x$ usually also has some additional properties (for example, the nonzero entries of $\x$ independently and identically follow the Gaussian distribution, and $\x$ has exponential decaying property), as far as we know, none of existing analysis uses these properties to answer the above question. In this talk, we first show that the prior distribution information of $\x$ can be used to provide an upper bound on $\|\x\|_1^2/\|\x\|_2^2$. Then, we explore this upper bound to develop a better lower bound on the probability of exact recovery with OMP in $K$ iterations. Furthermore, we develop a lower bound on $m$ to guarantee that the exact recovery probability of $K$ iterations of OMP is not lower than a given probability. We further show that, if $K$ is sufficiently small compared with $n$, when $K$ approaches infinity, $m\approx 2K\ln(n)$, $m\approx K$ and $m\approx 1.6K\ln(n)$ are enough to ensure that OMP has a satisfactory recovery performance for recovering any $K$-sparse $\x$, $K$-sparse $\x$ with exponential decaying property and $K$-sparse $\x$ whose nonzero entries independently and identically follow the Gaussian distribution, respectively. This significantly improves Tropp {\em{et. al.}}'s result which requires $m\approx4K\ln(n/\delta)$.
报告人介绍：温金明，2015年6月毕业于加拿大麦吉尔大学数学与统计学院，获哲学博士学位。从2015年3月到2018年9月，温教授先后在法国科学院里昂并行计算实验室、加拿大阿尔伯塔大学、多伦多大学从事博士后研究工作。从2018年9月至今，他是暨南大学网络空间安全学院的教授。他的研究方向主要是整数信号和稀疏信号恢复的算法设计与理论分析。他以第一作者在IEEE Communications Magazine（2篇）、Applied and Computational Harmonic Analysis (中科院数学一区期刊，2篇)、IEEE Transactions on Information Theory（2篇）、 IEEE Transactions on Signal Processing（2篇）、IEEE Transactions on Wireless Communications（2篇）、 IEEE Transactions on Communications等顶级期刊和会议发表25篇（含三篇ESI高被引论文）, 以通讯作者和合作者身份发表期刊和会议发表14篇。 目前他担任IEEE Access（中科院二区）期刊的编辑。
