Distributed information fusion in tangle networks

Abstract

The potential of a decentralized/distributed system to behave intelligently as a whole hinges on the capacity of its constituents to exchange and process information. In sensor networks and multiagent platforms this may be realized by means of distributed information fusion techniques. The very nature of these approaches demands specifying, or at least approximating, the statistical interdependencies between individual entities in the network. This, however, becomes impractical with the increase in network size. In the past years a number of scalable techniques have been devised, which relax this constraint while yet maintaining a number of desired statistical properties like consistency and convergence to consensus. Here, we present a methodological approach for designing and analyzing information fusion in potentially large-scale networks. Tangle networks, the objects of study in our formalism, are flexible diagrammatic models that capture key properties of scalable information fusion. A tangle network comes equipped with a natural notion of equivalence: two networks are equivalent if they can be transformed one into the other by successive application of local deformations. Any such deformation preserves the information content and consistency of estimators in the network. We derive novel particle-filtering-based algorithms for distributed information fusion over tangle networks and analyze their performance in various settings. The algebraic properties of tangle networks are shown to bear resemblance to algebraic properties of graphs. In particular, we show that the agreement between estimators in the network is governed by the spectral gap of the network’s associated matrix, the analog of a graph Laplacian. The utility of the framework is demonstrated through comparison with state-of-the-art distributed information fusion techniques.

Introduction

Intelligent systems make online decisions by fusing information from diverse sources. For example a robotic system performing autonomously in a dynamic environment acquires data from onboard sources such as laser ranging devices, acoustic and proximity sensors, and cameras. The raw data streams flowing from these devices are handled by a filtering algorithm which is capable of integrating the data into a single coherent and useful piece of information, perhaps estimates of the robot’s position and posture.

When multiple intelligent platforms are present in networked/multiagent systems, it is a greater challenge to fuse and exchange data between them for enhancing and coordinating their actions. The decisions made by each entity must take into account information received from other decision-making entities. Nowadays, the definition of a network system encompasses engineered systems e.g. multi-robotic platforms (Beni and Wang, 1993, Mataric, 1995, Reif and Wang, 1999) and sensor networks (Chong & Kumar, 2003), social and natural systems e.g. crowds and flocks, as well as biological networks e.g. neural and cellular networks.

Network systems are expected to be resilient to varying environmental parameters and uncertainties as well as to potential internal faults which may occur. This resilience will usually be manifest in a design with the capacity to alter the underlying network topology in response to external variations and possible malfunctions (Bahramgiri et al., 2006, Parker, 1998). The network topology is the communication infrastructure by which the entities “talk” with one another and hence governs the emergence of intelligent behavior of the network as a whole (Bornholdt & Thimo, 2000). Many insights about adaptation, resilience, and self-organization come from nature herself, for such design principles are abundant in biological networks (Achard et al., 2006, Kitzbichler et al., 2009, Tero et al., 2010).

Adaptive network systems require equally flexible information fusion paradigms (Brooks et al., 2003, Chong and Kumar, 2003, Julier and Uhlmann, 2009, Olfati-Saber and Shamma, 2005, Xiong and Svensson, 2002, Yang et al., 2008). There may be many factors that come into play in designing such schemes depending upon the global objective of the system and on its computational resources. Nevertheless, we note two prominent attributes underlying any such scheme.

• Efficiency. Energy requirements impose a limit on the computational power of entities in the system. This implies that most of the entities will be able to carry out only limited information processing. Fusion of data from other entities will be required to be computationally efficient meaning that not all of the data nor all of its statistics will be available to every entity.

• Resilience. Environmental variations and faults are compensated for and overcome by altering the network topology in a way that guarantees that the goal of the overall system will still be met.

In large-scale sensor networks the above concepts are particularly important (Chong & Kumar, 2003). Such a network may consist of thousand of sensors each having a modest information processing capability. Typically, each sensor will run its own information fusion algorithm such as a Kalman filter (Julier and Uhlmann, 2009, Olfati-Saber and Shamma, 2005) or a particle filter (Li & Nehorai, 2018). This approach is known as decentralized or distributed information fusion. Decentralized architectures normally involve some nodes to which information flows from a subgroup of neighboring nodes (Chong & Kumar, 2003). Such nodes fuse the data that they receive with their own measurements to yield improved estimates which may then be distributed to other nodes in the network. The global objective of this system is to detect an anomaly. Its ability to meet this objective depends on the network topology. The more energy is diverted towards that part of the network which is near the location of an anomaly, the more sensitive the network becomes, which allows better detection. Sensors that encounter suspicious behavior may signal central nodes to start change the network topology such that they will be able to occupy a greater bandwidth compared with other far away and temporarily less effective nodes.

Statistical cross-dependencies between nodes are an important piece of information without which a fusion algorithm may yield statistically inconsistent estimates. However, it may become practically infeasible to account for the cross-dependencies as the network grows in size. In such cases alternative, cross-correlation-free, methods may be employed. Many state-of-the-art techniques in this class rely on Chernoff fusion. An application of the Chernoff fusion for Gaussian mixtures is introduced in Julier (2006), for non-Gaussian continuous distributions in Farrell and Ganesh (2009) and in Li and Nehorai, 2018, Ong et al., 2008 and Ong et al. (2006) for a sample-based numerical distribution, namely particle filters. An extension of the technique to random finite sets is described in Üney, Houssineau, Delande, Julier, and Clark (2019).

In previous work we have developed a diagrammatic formalism for describing information flow in information networks which we call tangle machines (Moskovich and Carmi, 2014, Moskovich and Carmi, 2015). Tangle machine graphically capture certain properties of information fusion and of computation (Carmi & Moskovich, 2015).

The tangle machine formalism is underpinned by the mathematical discipline of low-dimensional topology, from which it inherits two key features: (1) A tangle machine description is flexible. There are many different tangle machines which achieve the same global objective, each of which has its own local features, and there is a simple mechanism to switch between any two such descriptions. (2) A tangle machine is characterized by quantities called invariants, each of which tells us something intrinsic about how information flows inside the network. Invariants may provide insight into the qualitative workings of information fusion in the network (Carmi & Moskovich, 2014).

The above features were proposed as basic principles for designing large-scale fault-tolerant adaptive information fusion networks (Moskovich & Carmi, 2014), referred to as tangle networks in the present work. Tangle networks abstract the key properties of the class of correlation-free information fusion techniques that appeared in the literature (Farrell and Ganesh, 2009, Hurley, 2002, Julier, 2006, Julier and Uhlmann, 2009), all of which are applicable in large-scale settings where the nodes’ statistical interdependencies (correlations) are mostly unavailable, hence the namesake. As demonstrated by Moskovich and Carmi (2014), the filtering consistency and other information-theoretic properties of correlation-free fusion reflect the basic symmetries of tangle networks. Indeed, a tangle network may be thought of as a paradigm for orchestrating the information processing of many different nodes in a statistically consistent manner.

We consolidate and extend our previous works (Tslil et al., 2018, Tslil and Carmi, 2018) where a novel approach is introduced for fusing arbitrary empirical probability densities whose statistical interdependencies are unspecified. Such densities may, for example, constitute the output of Monte Carlo sampling techniques, e.g., particle filters. The proposed technique is an application of the generalized Chernoff fusion, which was shown to satisfy the information processing axioms of tangle networks already in Moskovich and Carmi (2014). In the present paper the same technique is amended for large scale tangle networks where data synchronization issues may arise. An analysis is provided of various performance criteria such as consistency, information redundancy, and convergence to consensus. The algebraic properties of tangle networks are shown to bear resemblance to algebraic properties of graphs. In particular, we show that the agreement between estimators in the network is governed by the spectral gap of the network’s associated matrix, the analog of a graph Laplacian. The viability of the tangle networks formalism is demonstrated through comparison with state-of-the-art information fusion techniques in such applications as distributed object tracking and cooperative robots localization.

This paper is organized as follows. The next section describes an algebraic structure abstracting key properties of correlation-free information fusion. Its diagrammatical interpretation as tangle networks is the subject of Section 3. Both these sections are a concise summary of the main ideas presented in Moskovich and Carmi (2014), Tslil et al. (2018) and Tslil and Carmi (2018). The following sections make the main contribution of this work. Section 4 deals with some of the issues arising in the design of large scale tangle networks, in particular, the concept of synchronization and the decomposition of large tangle networks from smaller ones. Consistency and consensus as well as other information-theoretic properties of tangle networks are studied in Section 5. Section 6 illustrates the utility of the formalism for distributed information fusion through comparison with state-of-the-art techniques. Conclusions are offered in the last section.