ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk
ℓ2/ℓ2-Foreach Sparse Recovery with Low Risk
复制标题
ℓ2/ℓ2-Foreach 低风险稀疏恢复
DOI：
10.1007/978-3-642-39206-1_39
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发表时间：
2013
影响因子：
3.4
通讯作者：
M. Strauss
中科院分区：
文献类型：
--
作者：
A. Gilbert;H. Ngo;E. Porat;A. Rudra;M. Strauss
在本文中，我们考虑了问题p的“ foreach”稀疏恢复问题。概率至少1-p $$ \ | \ MATHBF {x} -a（\ phi \ phi \ mathbf {x}）\ | _2 \ leqslant c \ | \ | \ Mathbf {x} - \ Mathbf {x} $$ xk是x的最佳k-sparse近似。
我们的两个主要结果是：（1）我们证明了m的下限，数字测量值是ω（klog（n/k）+log（1/p）），价格为$ 2^{ - \ theta（n）} \ Leqslant P <1 $。 （1）我们的结果是Gilbert等人的延伸。 [6]信息理论有限的对手的结果\ | _2，$$，其中xk是x的最佳k-sparse近似。
我们的两个主要结果是：（1）我们证明了m的下限，数字测量值是ω（klog（n/k）+log（1/p）），价格为$ 2^{ - \ theta（n）} \ Leqslant P <1 $。 （1）我们的结果是Gilbert等人的延伸。 [6]信息理论有限的对手的结果\ | _2，$$，其中xk是x的最佳k-sparse近似。
我们的两个主要结果是：（1）我们证明了m的下限，数字测量值是ω（klog（n/k）+log（1/p）），价格为$ 2^{ - \ theta（n）} \ Leqslant P <1 $。 （1）我们的结果是Gilbert等人的延伸。 [6]信息理论有限的对手的结果\ | _2，$$，其中xk是x的最佳k-sparse近似。
我们的两个主要结果是：（1）我们证明了m的下限，数字测量值是ω（klog（n/k）+log（1/p）），价格为$ 2^{ - \ theta（n）} \ Leqslant P <1 $。 （1）我们的结果是Gilbert等人的延伸。 [6]信息理论上有限的对手的结果。
In this paper, we consider the "foreach" sparse recovery problem with failure probability p. The goal of the problem is to design a distribution over m ×N matrices Φ and a decoding algorithm A such that for every x∈ℝN, we have with probability at least 1−p$$\|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x.
Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x.
Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x.
Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries. $$|\mathbf{x}-A(\Phi\mathbf{x})\|_2\leqslant C\|\mathbf{x}-\mathbf{x}_k\|_2,$$ where xk is the best k-sparse approximation of x.
Our two main results are: (1) We prove a lower bound on m, the number measurements, of Ω(klog(n/k)+log(1/p)) for $2^{-\Theta(N)}\leqslant p <1$. Cohen, Dahmen, and DeVore [4] prove that this bound is tight. (2) We prove nearly matching upper bounds that also admit sub-linear time decoding. Previous such results were obtained only when p=Ω(1). One corollary of our result is an an extension of Gilbert et al. [6] results for information-theoretically bounded adversaries.