Graph-Based Information Processing:Scaling Laws and Applications

We live in a world characterized by massive information transfer and real-time communication. The demand for efficient yet low-complexity algorithms is widespread across different fields, including machine learning, signal processing and communications. Most of the problems that we encounter across these disciplines involves a large number of modules interacting with each other. It is therefore natural to represent these interactions and the flow of information between the modules in terms of a graph. This leads to the study of graph-based information processing framework. This framework can be used to gain insight into the development of algorithms for a diverse set of applications. We investigate the behaviour of large-scale networks (ranging from wireless sensor networks to social networks) as a function of underlying parameters. In particular, we study the scaling laws and applications of graph-based information processing in sensor networks/arrays, sparsity pattern recovery and interactive content search. In the first part of this thesis, we explore location estimation from incomplete information, a problem that arises often in wireless sensor networks and ultrasound tomography devices. In such applications, the data gathered by the sensors is only useful if we can pinpoint their positions with reasonable accuracy. This problem is particularly challenging when we need to infer the positions based on basic information/interaction such as proximity or incomplete (and often noisy) pairwise distances. As the sensors deployed in a sensor network are often of low quality and unreliable, we need to devise a mechanism to single out those that do not work properly. In the second part, we frame the network tomography problem as a well-studied inverse problem in statistics, called group testing. Group testing involves detecting a small set of defective items in a large population by grouping a subset of items into different pools. The result of each pool is a binary output depending on whether the pool contains a defective item or not. Motivated by the network tomography application, we consider the general framework of group testing with graph constraints. As opposed to conventional group testing where any subset of items can be grouped, here a test is admissible if it induces a connected subgraph. Given this constraint, we are interested in bounding the number of pools required to identify the defective items. Once the positions of sensors are known and the defective sensors are identified, we investigate another important feature of networks, namely, navigability or how fast nodes can deliver a message from one end to another by means of local operations. In the final part, we consider navigating through a database of objects utilizing comparisons. Contrary to traditional databases, users do not submit queries that are subsequently matched to objects. Instead, at each step, the database presents two objects to the user, who then selects among the pair the object closest to the target that she has in mind. This process continues until, based on the user’s answers, the database can identify the target she has in mind. The search through comparisons amounts to determining which pairs should be presented to the user in order to find the target object as quickly as possible. Interestingly, this problem has a natural connection with the navigability property studied in the second part, which enables us to develop efficient algorithms

Adaptive Content Search Through Comparisons

We study the problem of navigating through a database of similar objects using comparisons. This problem is known to be strongly related to the small-world network design problem. However, contrary to prior work, which focuses on cases where objects in the database are equally popular, we consider here the case where the demand for objects may be heterogeneous. We show that, under heterogeneous demand, the small-world network design problem is NP-hard. Given the above negative result, we propose a novel mechanism for smallworld design and provide an upper bound on its performance under heterogeneous demand. The above mechanism has a natural equivalent in the context of content search through comparisons, and we establish both an upper bound and a lower bound for the performance of this mechanism. These bounds are intuitively appealing, as they depend on the entropy of the demand as well as its doubling constant, a quantity capturing the topology of the set of target objects. Finally, based on these results, we propose an adaptive learning algorithm for content search that meets the performance guarantees achieved by the above mechanisms