The field of compressed sensing has become a major tool in high-dimensional analysis, with the realization that vectors can be recovered from relatively very few linear measurements as long as the vectors lie in a low-dimensional structure, typically the vectors that are zero in most coordinates with respect to a basis. However, there are many applications where we instead want to recover vectors that are sparse with respect to a dictionary rather than a basis. That is, we assume the vectors are linear combinations of at most s columns of a $d\times n$ matrix D, where s is very small relative to n and the columns of D form a (typically overcomplete) spanning set. In this direction, we show that as a matrix D stays bounded away from zero in norm on a set S and a provided map Φ comprised of i.i.d. subgaussian rows has number of measurements at least proportional to the square of $w(\mathbf{D}S)$, the Gaussian width of the related set $\mathbf{D}S$, then with high probability the composition $\mathbf{\Phi }\mathbf{D}$ also stays bounded away from zero. As a specific application, we obtain that the null space property of order s is preserved under such subgaussian maps with high probability. Consequently, we obtain stable recovery guarantees for dictionary-sparse signals via the ${\ell }_1$-synthesis method with only $O(s\log (n/s))$ random measurements and a minimal condition on D, which complements the compressed sensing literature.

压缩感测领域已经成为高维分析的主要工具，因为认识到只要向量处于低维结构中，就可以从相对很少的线性测量中恢复向量，通常在向量为零时关于基础的大多数协调。但是，在许多应用程序中，我们希望恢复相对于字典而不是基础而言稀疏的向量。也就是说，我们假设向量是a的最多s列的线性组合$d\times ñ$矩阵D，其中s相对于n很小，D的列形成一个（通常是超完备的）扩展集。在这个方向上，我们显示出矩阵D在集合S的范数上始终远离零，并且所提供的由iid次高斯行组成的映射Φ的测量数量至少与平方成正比。$w（\mathbf{d}\mathrm{小号}）$，相关集的高斯宽度 $\mathbf{d}\mathrm{小号}$，那么很有可能 $\mathbf{\Phi }\mathbf{d}$也保持远离零的界限。作为一个特定的应用，我们获得了在这种亚高斯映射下以高概率保留阶数为s的零空间属性。因此，我们可以通过以下方式为字典稀疏信号获得稳定的恢复保证：${\ell }_{\mathrm{1个}}$-合成方法 $Ø（s\mathrm{日志}（ñ/s））$随机测量和D的最小条件，它补充了压缩的传感文献。