一般p-范数的限制等距性质
Restricted Isometry Property for General p-Norms
AI总结：
本文研究了一般p-范数下限制等距性质的最优行数界，揭示了p=2时的特殊性，并讨论了其对稳定稀疏恢复问题的影响。
摘要：
限制等距性质（RIP）是使稀疏恢复可行的矩阵基本属性。非正式地说，一个m×n矩阵满足ℓ_p范数下顺序k的RIP，如果对于每个至多有k个非零坐标的向量x，‖Ax‖_p≈‖x‖_p。对于每个1≤p<∞，我们获得了RIP成立所需最小行数m的几乎紧界。此前仅研究了p=1、1+1/logk和p=2的情况。有趣的是，我们的结果表明，当p=2时，最优行数m为~Θ(k^p)，对于所有p∈[1,∞)\{2}，与p=2时的~Θ(k)不同。我们还获得了RIP矩阵列稀疏度的几乎紧界，并讨论了我们的结果对稳定稀疏恢复问题的影响。
英文摘要：
The Restricted Isometry Property (RIP) is a fundamental property of a matrix which enables sparse recovery. Informally, an $m \times n$ matrix satisfies RIP of order $k$ for the $\ell_p$ norm, if $\|Ax\|_p \approx \|x\|_p$ for every vector $x$ with at most $k$ non-zero coordinates. For every $1 \leq p < \infty$ we obtain almost tight bounds on the minimum number of rows $m$ necessary for the RIP property to hold. Prior to this work, only the cases $p = 1$, $1 + 1 / \log k$, and $2$ were studied. Interestingly, our results show that the case $p = 2$ is a "singularity" point: the optimal number of rows $m$ is $\widetildeΘ(k^{p})$ for all $p\in [1,\infty)\setminus \{2\}$, as opposed to $\widetildeΘ(k)$ for $k=2$. We also obtain almost tight bounds for the column sparsity of RIP matrices and discuss implications of our results for the Stable Sparse Recovery problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
