论文标题
得出稀疏词典的RIP传感矩阵
Deriving RIP sensing matrices for sparsifying dictionaries
论文作者
论文摘要
压缩感应涉及映射$ SD \ in \ Mathbb {r}^{m \ times n} $的反转，其中$ m <n $，$ s $是一个感应矩阵，而$ d $是一种宽松的词典。受限制的等轴测特性是反转的强大条件，可确保通过凸优化从其低维嵌入到欧几里得空间中的高维稀疏向量的恢复。但是，确定$ SD $是否具有给定率词典的限制等轴测属性是一个NP硬性问题，从而阻碍了压缩感应的应用。本文提供了一种解决此问题的新方法。我们证明，有可能为任何具有限制性等轴测特性的稀疏字典而得出一个传感矩阵。在具有K-SVD，Parseval K-SVD和小波的传感矩阵的数值实验中，我们的恢复性能与使用Gaussian和Bernoulli随机传感矩阵获得的基准相当。
Compressive sensing involves the inversion of a mapping $SD \in \mathbb{R}^{m \times n}$, where $m < n$, $S$ is a sensing matrix, and $D$ is a sparisfying dictionary. The restricted isometry property is a powerful sufficient condition for the inversion that guarantees the recovery of high-dimensional sparse vectors from their low-dimensional embedding into a Euclidean space via convex optimization. However, determining whether $SD$ has the restricted isometry property for a given sparisfying dictionary is an NP-hard problem, hampering the application of compressive sensing. This paper provides a novel approach to resolving this problem. We demonstrate that it is possible to derive a sensing matrix for any sparsifying dictionary with a high probability of retaining the restricted isometry property. In numerical experiments with sensing matrices for K-SVD, Parseval K-SVD, and wavelets, our recovery performance was comparable to that of benchmarks obtained using Gaussian and Bernoulli random sensing matrices for sparse vectors.
