報告時間：2018年11月30日上午9:00-10:00

報告地點： 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 $m\approx4K\ln(n/\delta)$.

報告人介紹：溫金明，2015年6月畢業(yè)于加拿大麥吉爾大學數(shù)學與統(tǒng)計學院，獲哲學博士學位。從2015年3月到2018年9月，溫教授先后在法國科學院里昂并行計算實驗室、加拿大阿爾伯塔大學、多倫多大學從事博士后研究工作。從2018年9月至今，他是暨南大學網(wǎng)絡空間安全學院的教授。他的研究方向主要是整數(shù)信號和稀疏信號恢復的算法設計與理論分析。他以第一作者在IEEE Communications Magazine（2篇）、Applied and Computational Harmonic Analysis (中科院數(shù)學一區(qū)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等頂級期刊和會議發(fā)表25篇（含三篇ESI高被引論文）, 以通訊作者和合作者身份發(fā)表期刊和會議發(fā)表14篇。 目前他擔任IEEE Access（中科院二區(qū)）期刊的編輯。