Concentration of Measure for Block Diagonal Matrices With Applications to Compressive Signal Processing

Abstract

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.

Partitioned compressive sensing with neighbor-weighted decoding

“…In the compressive sensing field, partitioned encoding has been studied for block encoding of natural images [19], with basis-specific enhancements used to improve reconstruction quality. Authors have also recently proven sufficiency of blockdiagonal matrices for signal recovery [20], and extended these results to analyze signals heterogeneous across partitions [21].…”

Reconstruction of super-resolution STORM images using compressed sensing based on low-resolution raw images and interpolation

“…When a measurement matrix has a good RIP, its columns and rows have good incoherence, its row and column norms are approximately equal, and the elements of columns and rows are random to a certain degree. The incoherence of columns plays a leading role in the RIP of the measurement matrix [25][26][27][28][29]. Existing research results indicate that a measurement matrix with good column incoherence can improve the reconstruction results of various reconstruction algorithms.…”

Study on adaptive compressed sensing & reconstruction of quantized speech signals

“…It has been shown in some literatures that a dense In this section, a sensing matrix is constructed according to the characteristics of voiced speech signals. In [25][26][27][28][29], a kind of structured random matrix called block diagonal matrix is applied to achieve CS in wireless communication and image processing. In [25,26], a lot of identical blocks are used to construct a block diagonal matrix as a sensing matrix for image processing with no proof of its property to meet RIP.…”

“…From a view of information theory, [27] proposes the block diagonal matrix for natural images also with no proof of its property to meet RIP. In addition, [28,29] present RIP for block diagonal matrices.…”

“…However, in this work, a specific block diagonal matrix with just two different blocks called two-block diagonal (TBD) matrix is constructed for voiced speech signals and a simple proof of its RIP is given although some proofs of RIP for block diagonal matrices have been given in [28,29].…”