An Enhancement of Effective Models of Distributed Compressed Sensing

In addition, real world data are uncertain in nature and they may be represented as uncertain data. Compressed sensing is a thriving research field covering a class of problems where a large sparse signal is reconstructed from a few random measurements. In the presence of several sensor nodes measuring correlated sparse signals, improvements in terms of recovery quality or the requirement for a fewer number of local measurements can be expected if the nodes cooperate. In this paper, we provide an overview of the current literature regarding distributed compressed sensing; in particular, we discuss aspects of network topologies, signal models and recovery algorithms. I.Introduction With the increased availability of cheap sensors, applications with sensor networks are of major interest to the community. In such networks, data signals (possibly correlated) are gathered by sensors and processed either distributively (at the local sensor nodes) or at some fusion center (centralized). Natural signals are often compressible; a signal may be of a large dimension, but the information content is small. Instead of collecting such large signals, fewer measurements can be used to reconstruct the signal. To this end, compressed sensing (CS) [1,2] has been an active research area over the past few years. CS relies on few random measurements being sufficient to reconstruct a large signal where the information content is sparse. Comparing CS to traditional data compression systems, the complexity is moved away from the information source, and instead, we find a challenge in the signal reconstruction part of the system. Two main classes of reconstruction methods are dominant; the convex relaxation-based solvers in which a relaxed problem is solved exactly, and the greedy pursuit (GP)based solvers in which the exact IJCSIET--International Journal of Computer Science information and Engg. , Technologies ISSN 2277-4408 || 01062016-013 IJCSIET-ISSUE6-VOLUME2-SERIES2 Page 2 problem is solved sub-optimally. In a CS system, if signals measured at different sensors are correlated, the necessary number of measurements can be further reduced by exploiting correlations in the reconstruction process. Therefore, we have recently seen a growing interest in different attempts to improve CS performance by designing new reconstruction algorithms for multiple signals. For such problems, which we refer to as distributed compressed sensing (DCS) problems, there are few specific analytical results available. For the results that are available, the gap between what is practically achieved and the prediction of the analytical result is often large; which is true already at the single sensor. We endeavor to provide a moderate level of overview on practical DCS algorithms. The paper is arranged by first a description of DCS problems along with fundamental tools for relevant mathematical justifications, basic types of networks considered in the literature where DCS can be realized, various signal models for data correlation , several convex relaxationbased approaches to DCS problems and, finally, GP algorithms for DCS problems. II.Related Work A network is an infrastructure where data is shared through connections between different sensing and/or computational nodes. For analysis of distributed algorithms, the choice of network model plays an important role. Some models closely resemble practical random networks, while others are fixed and structured to provide controlled performance comparison. If every node in a network can transmit information to every other node through one or multiple hops, the network is said to be connected. If there is a direct link between every node in the network, the network is said to be fully connected. If no connection can be found between two nodes (either directly or through multiple hops), the network is disconnected, and then, we can regard the nodes as forming two or more independent networks. Formally, we say that sensor node l has a set of neighbors to which it can transmit information (outgoing set of neighbors), defined as Lout l , and a set of incoming neighbors (from which it receives information), defined as Lin l . These sets are not necessarily the same, implying that all connections are not IJCSIET--International Journal of Computer Science information and Engg. , Technologies ISSN 2277-4408 || 01062016-013 IJCSIET-ISSUE6-VOLUME2-SERIES2 Page 3 two-way. We also define a complete network with all nodes as L. To model networks, there are two main categories of network models: fixed network models and random network models. The fixed network models are completely determined, while the random networks have some parameters determining a random layout. We will discuss both types of network models and start with the fixed ones. Fixed Network Models In a fixed network, the connections among the sensor-nodes are manually formed. For example, we can imagine sensors equidistantly aligned horizontally and vertically to form a lattice. This may closely model the connections formed by independent sensors placed uniformly over some area. Observe that the network connections can be formed this way, even when the sensors are spatially located according to some other topology. In the task of analyzing the performance of DCS algorithms, this fixed network model may not be the best suited one, since it is hard to understand how to modify the network connectivity. Figure 1. Lattice and circular network topologies. The forward circular network model in [9] is an example of a fixed network that is designed specifically to make modifications of connectivity easily amenable. In this network model, nodes are artificially placed in a circle, and connections among the nodes are realized by one-way communication links. The network having the least possible connections, such that it is fully connected, is thus formed by letting each node connect to one other node according to Figure 1b. By forming this circular topology of nodes and forwardly adding new connections in a systematic way, we can study how the overall performance of an algorithm improves as the network connectivity increases. In Figure 1c, we show a network where each node is connected to two other nodes (referred to as a degree two network). By adding more connections according to the forward circular model, IJCSIET--International Journal of Computer Science information and Engg. , Technologies ISSN 2277-4408 || 01062016-013 IJCSIET-ISSUE6-VOLUME2-SERIES2 Page 4 we can, in a controlled manner, systematically improve the connectivity until the network is fully connected (where each node is connected with every other node). Another common circular network topology approach is to allow for bidirectional communication links, and we refer to such a model as a circular network model (compared to the previous forward circular network model). In Figure 1d, we see a typical example of a standard circular network model based on bidirectional connections. Random Network Models When the setup grows large and the focus is on practical applications, the category of random network models may be of more relevance. Of several popular random models, we first present the random graph network model developed by Erd ̋os and R ́enyi in [10]. Erd ̋os and R ́enyi take an analytical approach, deriving probabilities for full network connectivity and connectivity distributions for the nodes. From an engineering perspective, we can utilize their model to generate a network based on a parameter, p, which specifies the probability that any two nodes are connected. An example of the random graph network model with probability parameter p = 0:2 is shown in Figure 2a. Note that the circular placement of the nodes has nothing to do with the topology of the connections. A (semi)random network model utilizing two tuning parameters, l and p, was presented by Watts and Strogatz in [11]. The first tuning parameter, l, specifies that each node in the network should be connected with l neighbors structurally (i.e., as in the forward circular model). Then, with probability p, each such connection is rewired to another node, where the other node is chosen uniformly at random. This model has the strength that if p = 0, it represents a completely structured model, and if p = 1, it represents a (uniformly) random model. Thus, this network model may be suitable for understanding how a system behaves for the intermediate networks. We show an example of the Watts– Strogatz network model in Figure 2b for ten sensor nodes, where l = 2 and p = 0:2. In the random geometric graph model, presented by Penrose in [12], nodes are placed randomly (according to some distribution) within a unit-length square. Then, every node connects with IJCSIET--International Journal of Computer Science information and Engg. , Technologies ISSN 2277-4408 || 01062016-013 IJCSIET-ISSUE6-VOLUME2-SERIES2 Page 5 every other node located within distance d of it. We show an example of such a network in Figure 2c. Here, the distribution chosen is uniform, and we have chosen d = 0:2. Any of these three random models may lead to disconnected networks. Figure 2. Random network topologies. The network models we have mentioned are general in the sense that they are not intentionally created to model a specific application. We notice that in the literature, there are many examples of such specific network models. One such example is presented in [13] by Barab ́asi and Albert, which models hyper-link connections on the Internet. A Comment on Fundamental Limits in Networks In a sensor network, assume nodes 1 and 2 want to communicate some information and we are interested in the maximum rate at which communication can take place. If finding the precise rate between the nodes is not possible, we can still upper-bound it using the maxflow min-cut theorem [14,15]. By cutting the network into two parts, so that the first node resides in one part and the second node in