标题： 森林稀疏性在多通道压缩感知中的应用 显示英文标题

标题： Forest Sparsity for Multi-channel Compressive Sensing

摘要： 本文研究了一种新的压缩感知模型，用于多通道稀疏数据，其中每个通道可以表示为一个分层树，并且不同通道之间高度相关。因此，完整数据可以遵循森林结构，我们称此性质为\emph{森林稀疏性}。该模型利用了通道内和通道间的相关性，并丰富了现有的基于模型的压缩感知理论家族。提出的理论表明，对于具有森林稀疏性的多通道数据，仅需要$\mathcal{O}(Tk+\log(N/k))$次测量即可，其中$T$是通道数量，$N$和$k$分别是每个通道的长度和稀疏度。 这个结果远优于树稀疏性的$\mathcal{O}(Tk+T\log(N/k))$、联合稀疏性的$\mathcal{O}(Tk+k\log(N/k))$，并且远远优于标准稀疏性的$\mathcal{O}(Tk+Tk\log(N/k))$。 此外，我们将森林稀疏性理论扩展到多测量向量问题中，其中测量矩阵是一个块对角矩阵。 结果表明，当数据在每个通道中具有相同能量时，所需的测量界限可以与密集随机测量矩阵的情况相同。 提出了一种新算法，并将其应用于四个示例应用中以验证所提出的模型的优势。 大量的实验展示了所提出的理论和算法的有效性和高效性。 显示英文摘要

摘要： In this paper, we investigate a new compressive sensing model for multi-channel sparse data where each channel can be represented as a hierarchical tree and different channels are highly correlated. Therefore, the full data could follow the forest structure and we call this property as \emph{forest sparsity}. It exploits both intra- and inter- channel correlations and enriches the family of existing model-based compressive sensing theories. The proposed theory indicates that only $\mathcal{O}(Tk+\log(N/k))$ measurements are required for multi-channel data with forest sparsity, where $T$ is the number of channels, $N$ and $k$ are the length and sparsity number of each channel respectively. This result is much better than $\mathcal{O}(Tk+T\log(N/k))$ of tree sparsity, $\mathcal{O}(Tk+k\log(N/k))$ of joint sparsity, and far better than $\mathcal{O}(Tk+Tk\log(N/k))$ of standard sparsity. In addition, we extend the forest sparsity theory to the multiple measurement vectors problem, where the measurement matrix is a block-diagonal matrix. The result shows that the required measurement bound can be the same as that for dense random measurement matrix, when the data shares equal energy in each channel. A new algorithm is developed and applied on four example applications to validate the benefit of the proposed model. Extensive experiments demonstrate the effectiveness and efficiency of the proposed theory and algorithm.