In this paper, we discuss the compressed data separation problem. In order to reconstruct the distinct subcomponents, which are sparse in morphologically different dictionaries ${\mathit{D}}_1\in {\mathbb{R}}^{n\times d_1}$ and ${\mathit{D}}_2\in {\mathbb{R}}^{n\times d_2}$, we present a general class of convex optimization decoder. It can deal with signal separation under the corruption of different kinds of noises, including Gaussian noise ($p=2$), Laplacian noise ($p=1$), and uniformly bounded noise ($p=\infty$).

Although the restricted isometry property adapted to frames is a commonly used tool, the measurement number is suboptimal when $p>2$. Furthermore, the ${\ell }_p$ robust nullspace property adapted to $\mathit{\Psi }$, which is constructed by ${\mathbf{D}}_1$ and ${\mathbf{D}}_2$, may fail to work on data separation problem. Here we introduce the modified ${\ell }_p$ robust nullspace property adapted to $\mathit{\Psi }$ (abbreviated as the modified (${\ell }_p$, $\mathit{\Psi }$)-RNSP). First of all, we show the robust recovery of signals based on the modified (${\ell }_p$, $\mathit{\Psi }$)-RNSP and the mutual coherence between ${\mathbf{D}}_1$ and ${\mathbf{D}}_2$. Besides, we find that Gaussian measurements meet the modified (${\ell }_p$, $\mathit{\Psi }$)-RNSP for any $1\le p\le \infty$, provided with the optimal number of measurements $\mathcal{O}(slog(d∕s))$, where $s$ is the sparsity level and $d=d_1+d_2$.

Furthermore, we introduce another properly constrained ${\ell }_1$-analysis optimization model, called the Split Dantzig Selector. It can recover signals which are approximately sparse in terms of different frame representations, when the measurement matrix satisfies the modified (${\ell }_p$, $\mathit{\Psi }$)-RNSP. As a special case, when considering Gaussian white noise, the recovery error by the Split Dantzig Selector is $\mathcal{O}\left(\frac{\sqrt{slogd}}{\sqrt{m}}\right)$. It outperforms the ${\ell }_2$-constrained model, whose recovery error is $\mathcal{O}\left(\sqrt{logm}\right)$, if the sparsity level is small.

在本文中，我们讨论了压缩数据分离问题。为了重建不同的子成分，这些子成分在形态上不同的词典中稀疏${\mathit{d}}_{\mathrm{1个}}\in {\mathbb{[R}}^{ñ\times d_{\mathrm{1个}}}$ 和 ${\mathit{d}}_2\in {\mathbb{[R}}^{ñ\times d_2}$，我们提出了凸优化解码器的一般类。它可以处理各种噪声（包括高斯噪声）的破坏下的信号分离（$p=2$），拉普拉斯噪声（$p=\mathrm{1个}$）和均匀界噪声（$p=\infty$）。

尽管适合于框架的受限等轴测特性是一种常用工具，但是当 $p>2$。此外，${\ell }_p$ 健壮的nullspace属性适用于 $\mathit{\Psi }$，由 ${\mathbf{d}}_{\mathrm{1个}}$ 和 ${\mathbf{d}}_2$，可能无法解决数据分离问题。这里我们介绍修改后​​的${\ell }_p$ 健壮的nullspace属性适用于 $\mathit{\Psi }$ （缩写为修改后的（${\ell }_p$， $\mathit{\Psi }$）-RNSP）。首先，我们展示了基于修改后的（${\ell }_p$， $\mathit{\Psi }$）-RNSP与之间的相干性 ${\mathbf{d}}_{\mathrm{1个}}$ 和 ${\mathbf{d}}_2$。此外，我们发现高斯测量值满足修改后的（${\ell }_p$， $\mathit{\Psi }$）-RNSP $\mathrm{1个}\le p\le \infty$，并提供最佳的测量数量 $\mathcal{Ø}（s日志（d∕s））$，在哪里 $s$ 是稀疏度， $d=d_{\mathrm{1个}}+d_2$。

此外，我们引入了另一个适当约束的 ${\ell }_{\mathrm{1个}}$分析优化模型，称为拆分Dantzig选择器。当测量矩阵满足修改后的值时，它可以恢复在不同帧表示形式上近似稀疏的信号（${\ell }_p$， $\mathit{\Psi }$）-RNSP。作为一种特殊情况，当考虑高斯白噪声时，Split Dantzig选择器的恢复误差为$\mathcal{Ø}\left(\frac{\sqrt{s日志d}}{\sqrt{米}}\right)$。它的表现优于${\ell }_2$约束的模型，其恢复误差为 $\mathcal{Ø}（\sqrt{日志米}）$，如果稀疏度较小。