"Compressed" Compressed Sensing

The field of compressed sensing has shown that a sparse but otherwise arbitrary vector can be recovered exactly from a small number of randomly constructed linear projections (or samples). The question addressed in this paper is whether an even smaller number of samples is sufficient when there exists prior knowledge about the distribution of the unknown vector, or when only partial recovery is needed. An information-theoretic lower bound with connections to free probability theory and an upper bound corresponding to a computationally simple thresholding estimator are derived. It is shown that in certain cases (e.g. discrete valued vectors or large distortions) the number of samples can be decreased. Interestingly though, it is also shown that in many cases no reduction is possible

Compressive sensor networks : fundamental limits and algorithms

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 85-92).Compressed sensing is a non-adaptive compression method that takes advantage of natural sparsity at the input and is fast gaining relevance to both researchers and engineers for its universality and applicability. First developed by Candis et al., the subject has seen a surge of high-quality results both in its theory and applications. This thesis extends compressed sensing ideas to sensor networks and other bandwidth-constrained communication systems. In particular, we explore the limits of performance of compressive sensor networks in relation to fundamental operations such as quantization and parameter estimation. Since compressed sensing is originally formulated as a real-valued problem, quantization of the measurements is a very natural extension. Although several researchers have proposed modified reconstruction methods that mitigate quantization noise for a fixed quantizer, the optimal design of such quantizers is still unknown. We propose to find the optimal quantizer in terms of minimizing quantization error by using recent results in functional scalar quantization. The best quantizer in this case is not the optimal design for the measurements themselves but rather is reweighted by a factor we call the sensitivity. Numerical results demonstrate a constant-factor improvement in the fixed-rate case. Parameter estimation is an important goal of many sensing systems since users often care about some function of the data rather than the data itself.(cont.) Thus, it is of interest to see how efficiently nodes using compressed sensing can estimate a parameter, and if the measurements scalings can be less restrictive than the bounds in the literature. We explore this problem for time difference and angle of arrival, two common methods for source geolocation. We first derive Cramer-Rao lower bounds for both parameters and show that a practical block-OMP estimator can be relatively efficient for signal reconstruction. However, there is a large gap between theory and practice for time difference or angle of arrival estimation, which demonstrates the CRB to be an optimistic lower bound for nonlinear estimation. We also find scaling laws 'for time difference estimation in the discrete case. This is strongly related to partial support recovery, and we derive some new sufficient conditions that show a very simple reconstruction algorithm can achieve substantially better scaling than full support recovery suggests is possible.by John Zheng Sun.S.M

Compressive sensing for signal ensembles

Compressive sensing (CS) is a new approach to simultaneous sensing and compression that enables a potentially large reduction in the sampling and computation costs for acquisition of signals having a sparse or compressible representation in some basis. The CS literature has focused almost exclusively on problems involving single signals in one or two dimensions. However, many important applications involve distributed networks or arrays of sensors. In other applications, the signal is inherently multidimensional and sensed progressively along a subset of its dimensions; examples include hyperspectral imaging and video acquisition. Initial work proposed joint sparsity models for signal ensembles that exploit both intra- and inter-signal correlation structures. Joint sparsity models enable a reduction in the total number of compressive measurements required by CS through the use of specially tailored recovery algorithms. This thesis reviews several different models for sparsity and compressibility of signal ensembles and multidimensional signals and proposes practical CS measurement schemes for these settings. For joint sparsity models, we evaluate the minimum number of measurements required under a recovery algorithm with combinatorial complexity. We also propose a framework for CS that uses a union-of-subspaces signal model. This framework leverages the structure present in certain sparse signals and can exploit both intra- and inter-signal correlations in signal ensembles. We formulate signal recovery algorithms that employ these new models to enable a reduction in the number of measurements required. Additionally, we propose the use of Kronecker product matrices as sparsity or compressibility bases for signal ensembles and multidimensional signals to jointly model all types of correlation present in the signal when each type of correlation can be expressed using sparsity. We compare the performance of standard global measurement ensembles, which act on all of the signal samples; partitioned measurements, which act on a partition of the signal with a given measurement depending only on a piece of the signal; and Kronecker product measurements, which can be implemented in distributed measurement settings. The Kronecker product formulation in the sparsity and measurement settings enables the derivation of analytical bounds for transform coding compression of signal ensembles and multidimensional signals. We also provide new theoretical results for performance of CS recovery when Kronecker product matrices are used, which in turn motivates new design criteria for distributed CS measurement schemes