In this dissertation, two classes of distributed algorithms are developed for sparse signal recovery in large sensor networks. All the proposed approaches consist of local computation (LC) and global computation (GC) steps carried out by a group of distributed local sensors, and do not require the local sensors to know the global sensing matrix. These algorithms are based on the original approximate message passing (AMP) and iterative hard thresholding (IHT) algorithms in the area of compressed sensing (CS), also known as sparse signal recovery. For distributed AMP (DiAMP), we develop a communication-efficient algorithm GCAMP. Numerical results demonstrate that it outperforms the modified thresholding algorithm (MTA), another popular GC algorithm for Top-K query from distributed large databases. For distributed IHT (DIHT), there is a step size $\mu$ which depends on the $\ell_2$ norm of the global sensing matrix A. The exact computation of $\|A\|_2$ is non-separable. We propose a new method, based on the random matrix theory (RMT), to give a very tight statistical upper bound of $\|A\|_2$, and the calculation of that upper bound is separable without any communication cost. In the GC step of DIHT, we develop another algorithm named GC.K, which is also communication-efficient and outperforms MTA. Then, by adjusting the metric of communication cost, which enables transmission of quantized data, and taking advantage of the correlation of data in adjacent iterations, we develop quantized adaptive GCAMP (Q-A-GCAMP) and quantized adaptive GC.K (Q-A-GC.K) algorithms, leading to a significant improvement on communication savings. Furthermore, we prove that state evolution (SE), a fundamental property of AMP that in high dimensionality limit, the output data are asymptotically Gaussian regardless of the distribution of input data, also holds for DiAMP. In addition, compared with the most recent theoretical results that SE holds for sensing matrices with independent subgaussian entries, we prove that the universality of SE can be extended to far more general sensing matrices. These two theoretical results provide strong guarantee of AMP's performance, and greatly broaden its potential applications.

The theoretical problem of finding the solution to an underdeterminedset of linear equations has for several years attracted considerable attentionin the literature. This problem has many practical applications.One example of such an application is compressed sensing (cs), whichhas the potential to revolutionize how we acquire and process signals. Ina general cs setup, few measurement coefficients are available and thetask is to reconstruct a larger, sparse signal.In this thesis we focus on algorithm design and selected applicationsfor cs. The contributions of the thesis appear in the following order:(1) We study an application where cs can be used to relax the necessityof fast sampling for power spectral density estimation problems. Inthis application we show by experimental evaluation that we can gainan order of magnitude in reduced sampling frequency. (2) In order toimprove cs recovery performance, we extend simple well-known recoveryalgorithms by introducing a look-ahead concept. From simulations it isobserved that the additional complexity results in significant improvementsin recovery performance. (3) For sensor networks, we extend thecurrent framework of cs by introducing a new general network modelwhich is suitable for modeling several cs sensor nodes with correlatedmeasurements. Using this signal model we then develop several centralizedand distributed cs recovery algorithms. We find that both thecentralized and distributed algorithms achieve a significant gain in recoveryperformance compared to the standard, disconnected, algorithms.For the distributed case, we also see that as the network connectivity increases,the performance rapidly converges to the performance of thecentralized solution. / <p>QC 20120229</p>