Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing
Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing
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DOI：
10.1109/tsp.2011.2166546
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发表时间：
2011-12
影响因子：
5.4
通讯作者：
J. Park;H. L. Yap;C. Rozell;M. Wakin
中科院分区：
文献类型：
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作者：
J. Park;H. L. Yap;C. Rozell;M. Wakin
随机压缩算子的理论分析通常依赖于所讨论算子的测度不等式的集中。通常，这样的不等式量化了一个随机矩阵在相乘后保持信号范数的可能性。测量结果的集中很好地建立了非结构化压缩矩阵，由独立和同分布（i.i.d）随机条目填充。然而，许多现实世界的采集系统受到体系结构的限制，使得这样的矩阵不切实际。本文导出了两类块对角压缩矩阵的测度界集中，一类是主对角上的块是随机而独立的，另一类是随机但相等的。对于这两种类型的矩阵，我们表明范数保留的可能性取决于被测量信号的某些属性，但对于最佳情况信号，两种类型的块对角矩阵都可以提供与其非结构化、i.i.d.对应物相当的集中性能。我们通过说明性模拟以及对几个信号类别的分析和实证研究来支持我们的理论结果，这些信号类别非常适合使用块对角矩阵进行测量。我们还讨论了这些结果在确保各种信号族的稳定嵌入以及在直接在压缩域中解决各种信号处理任务（如检测和分类）建立性能保证方面的应用。
Theoretical analysis of randomized, compressive operators often depends on a concentration of measure inequality for the operator in question. Typically, such inequalities quantify the likelihood that a random matrix will preserve the norm of a signal after multiplication. Concentration of measure results are well established for unstructured compressive matrices, populated with independent and identically distributed (i.i.d.) random entries. Many real-world acquisition systems, however, are subject to architectural constraints that make such matrices impractical. In this paper we derive concentration of measure bounds for two types of block diagonal compressive matrices, one in which the blocks along the main diagonal are random and independent, and one in which the blocks are random but equal. For both types of matrices, we show that the likelihood of norm preservation depends on certain properties of the signal being measured, but that for the best case signals, both types of block diagonal matrices can offer concentration performance on par with their unstructured, i.i.d. counterparts. We support our theoretical results with illustrative simulations as well as analytical and empirical investigations of several signal classes that are highly amenable to measurement using block diagonal matrices. We also discuss applications of these results in ensuring stable embeddings for various signal families and in establishing performance guarantees for solving various signal processing tasks (such as detection and classification) directly in the compressed domain.