Concentration of Measure for Block Diagonal Matrices with Applications to Compressive Sensing

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. When this likelihood is very high for any signal, the random matrices have a variety of known uses in dimensionality reduction and Compressive Sensing. 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. Finally, we discuss applications of these results in establishing performance guarantees for solving signal processing tasks in the compressed domain (e.g., signal detection), and in establishing the Restricted Isometry Property for the Toeplitz matrices that arise in compressive channel sensing.