In recent decades, with the increasement of the system aquisition and data services, the explosion of data poses challenges to storage, transmission and processing, as well as device design. Meanwhile, in the background of big data, the network technology is not only characterized by the data transmission, but developed by exploration the inherent structures of the data. The burgeoning theory of sparse recovery reveals the outstanding sensing ability on the large scales of high-dimensional data. Based on this consideration, how to deepen the understanding of the unique theory and combine it with the networks in order to improve the performance, is becoming one of the hot research issues in the field of information and signal processing in networks.This dissertation starts from the fundamental theory of the sparse recovery, including compressive sensing and low rank matrix completion, and then focuses on the research on the application of the high-demansional signal sensing in networks. The detailed contents elaborate from two aspects:one is the study on the basic recovery conditions of the sparse high-dimensional signals under various definitions; the other is the novel high-dimensional signal processing technologies involving network architectures and the dimension reducing technologies towards the nature of the data itself and various performance requirements in networks. The contributions of this doctoral dissertation mainly can be summarized as follows.1. Suppose Î¦âˆˆRMÃ—N is a known sensing matrix and M: RmÃ—nâ†’RK is a sensing operator. For any nonzero vector h and nonzero matrix H in the null space of Î¦ and M, the definitions of S-largest mutilated vector of h and generalized r-largest mutilated vector of H are proposed. Then the concepts of concentration in lp space (0<pâ‰¤1) associated with S-largest mutilated vector and generalized r-largest mutilated vector are defined and some of its important properties are developed. As an application of these important properties, it is proved that, there always exists a positive interger So (r0), for any S<S0(r<r0), the concentration of S-largest mutilated vector (the generalized r-largest mutilated vector) with lp-quasinorm is less than1/2. Moreover, any S-sparse signal (S<S0) and r-rank matrix (r<r0) can be exactly recovered via (lp) minimization in the noiseless case. This is a complement and generalization of a result proposed by D. L. Donoho. Also, an upper bound condition of Î´s(Î¦) for exact recovery of a class of sparse signals via (l1) minimization is also derived from this result, which is an improvement of a conclusion proposed by T. Cai etc.. Further, it is can be proved that every S sparse signal could be recovered by this upper bound condition of Î´r(M). Besides, the aforementioned results could be generalized to the scenario of low rank matrix completion, which reveals the potential relationship between compressive sensing and low rank matrix completion.2. Network monitoring is an important module in the operation and management of networks in which the performance characteristics should be monitored. In order to efficiently sensing, transmit and infer the internal characteristics vector ensemble generated by monitoring, a measurement model (operator) based on the routing topology and dimensional reduction operator is proposed by a combination of the topic of Network Tomography. In addition, the problem of sensing the network characteristics vector ensemble boils down to the sparse recovery from combined fusion frame measurements. Then, for some specified routing matrices and dimensional reduction operators, the reconstruction sufficient conditions in noiseless and noise scenarios are proposed. The derivation utilizes the property of concentration of S-largest mutilated vector ensembles in the null space of the linear measurement operator. Note that these reconstruction conditions are satisfied for a class of routing matrices and dimensional reduction operators, the characteristics of the routing matrices and dimensional reduction operators can be also perceived.3. With new high-definition video technologies and data-intensive applications, massive high-dimensional correlative data gets rise to the information transmission and processing obstacle in access and core networks. In this context, it seems possible to modeling the data structure as a low-rank matrix and apply the low-rank matrix completion to reduce data dimensionality in the procedure of information capture and propagation in photonic domain. A spatio-temporal high-dimensional correlative data sensing approach is porposed via a variant of low-rank matrix completion. Also the feasibility of the sensing approach is proved with the constraint that the data obeys the existence form in optical networks. Furthermore, an all-optical measurement structure is designed, to realize a high-dimensional vector ensemble encoding scheme in finite field and multicasting transmission for passive optical networks or wireless optical networks.4. Mobile online social network (mOSN) is a burgeoning research area. However, most existing works referring to mOSNs deal with static network structures and simply encode whether relationships among entities exist or not. In contrast, relationships in signed mOSNs can be positive or negative and may be changed with time and locations. Applying certain global characteristics of social balance, a data model formed as a tensor is used to represent the relationship data (known or unknown) in dynamic signed mOSNs and formulates this sign inference problem as a tensor recovery problem, which can be translated into a low-rank matrix cluster estimation problem. Specifically, motivated by the Singular Value Thresholding (SVT) algorithm, a compact dictionary is selected from the observed dataset (tensor). Based on this compact dictionary, the relationships in the dynamic signed mOSNs are estimated via solving the formulated problem. Furthermore, the estimation accuracy is improved by employing a dictionary self-updating mechanism. The results of numerical experiments indicate the feasibility and validity of the proposed model and inferring mechanism. |