arXiv subjects
Publications and source records attributed to Jinming Wen.
We propose an algorithm with expected complexity of $\bigO(n\log n)$ arithmetic operations to solve a special shortest vector problem arising in computer-and-forward design, where $n$ is the dimension of the channel vector. This algorithm is more efficient than the best known algorithms with proved complexity.
The Korkine-Zolotareff (KZ) reduction has been used in communications and cryptography. In this paper, we modify a very recent KZ reduction algorithm proposed by Zhang et al., resulting in a new algorithm, which can be much faster and more numerically reliable, especially when the basis matrix is ill conditioned.
The common method to estimate an unknown integer parameter vector in a linear model is to solve an integer least squares (ILS) problem. A typical approach to solving an ILS problem is sphere decoding. To make a sphere decoder faster, the well-known LLL reduction is often used as preprocessing. The Babai point produced by the Babai nearest plan algorithm is a suboptimal solution of the ILS problem. First we prove that the success probability of the Babai point as a lower bound on the success probability of the ILS estimator is sharper than the lower bound given by Hassibi and Boyd [1]. Then we show rigorously that applying the LLL reduction algorithm will increase the success probability of the Babai point. Finally we show rigorously that applying the LLL reduction algorithm will also reduce the computational complexity of sphere decoders, which is measured approximately by the number of nodes in the search tree in the literature
In this paper, we show that, under the assumption that $\|\e\|_2\leq ε$, every $k-$sparse signal $\x\in \mathbb{R}^n$ can be stably ($ε\neq0$) or exactly recovered ($ε=0$) from $\y=\A\x+\e$ via $l_p-$mnimization with $p\in(0, \bar{p}]$, where \beqnn \bar{p}= \begin{cases} \frac{50}{31}(1-δ_{2k}), &δ_{2k}\in[\frac{\sqrt{2}}{2}, 0.7183)\cr 0.4541, &δ_{2k}\in[0.7183,0.7729)\cr 2(1-δ_{2k}), &δ_{2k}\in[0.7729,1) \end{cases}, \eeqnn even if the restricted isometry constant of $\A$ satisfies $δ_{2k}\in[\frac{\sqrt{2}}{2}, 1)$. Furthermore, under the assumption that $n\leq 4k$, we show that the range of $p$ can be further improved to $p\in(0,\frac{3+2\sqrt{2}}{2}(1-δ_{2k})]$. This not only extends some discussions of only the noiseless recovery (Lai et al. and Wu et al.) to the noise recovery, but also greatly improves the best existing results where $p\in(0,\min\{1, 1.0873(1-δ_{2k}) \})$ (Wu et al.).
In this letter, we first construct a counter example to show that for any given positive integer $K\geq 2$ and for any $\frac{1}{\sqrt{K+1}}\leq t<1$, there always exist a $K-$sparse $\x$ and a matrix $\A$ with the restricted isometry constant $δ_{K+1}=t$ such that the OMP algorithm fails in $K$ iterations. Secondly, we show that even when $δ_{K+1}=\frac{1}{\sqrt{K}+1}$, the OMP algorithm can also perfectly recover every $K-$sparse vector $\x$ from $\y=\A\x$ in $K$ iteration. This improves the best existing results which were independently given by Mo et al. and Wang et al.