Sheaf Theory as a Mathematical Foundation for Distributed Applications Involving Heterogeneous Data Sets

Mansourbeigi, Seyed M.H.

doi:10.1109/waina.2018.00059

Cited by 3 publications

(3 citation statements)

References 7 publications

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“…Sensor networks benefit from other applications of homology groups’ computation such as tracking [ 13 ] or network management [ 31 ]. Moreover, discrete vector fields are also useful within the sheaf approach applied to networks [ 32 ] as this theory appears to be well-suited for the integration of heterogeneous and non-localized data [ 33 , 34 , 35 , 36 ]. Our objective is to extend our solution in order to compute relevant information extracted from a sheaf in a distributed and adaptive way based on the work [ 37 ].…”

Section: Discussionmentioning

confidence: 99%

Adaptive Discrete Vector Field in Sensor Networks

Zhang

Goupil

2018

Sensors

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Homology groups are a prime tool for measuring the connectivity of a network, and their computation in a distributed and adaptive way is mandatory for their use in sensor networks. In this paper, we propose a solution based on the construction of an adaptive discrete vector field from where, thanks to the discrete Morse theory, the generators of the homology groups are extracted. The efficiency and the adaptability of our approach are tested against two applications: the detection and the localization of the holes in the coverage, and the selection of active sensors ensuring complete coverage.

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Section: Discussionmentioning

confidence: 99%

Adaptive Discrete Vector Field in Sensor Networks

Zhang

Goupil

2018

Sensors

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“…Many sheaf encodings of standard models (such as dynamical systems, differential equations, and Bayesian networks) have been catalogued [ 45 ]. Furthermore, these techniques can be easily applied in several different settings, for instance in air traffic control [ 46 , 47 ] and in formal semantic techniques [ 48 ]. Sheaf-based techniques for fundamental tasks in signal processing have also been developed [ 1 ].…”

Section: History and Contextmentioning

confidence: 99%

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

Joslyn

Charles

DePerno

et al. 2020

Sensors

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Integration of multiple, heterogeneous sensors is a challenging problem across a range of applications. Prominent among these are multi-target tracking, where one must combine observations from different sensor types in a meaningful and efficient way to track multiple targets. Because different sensors have differing error models, we seek a theoretically justified quantification of the agreement among ensembles of sensors, both overall for a sensor collection, and also at a fine-grained level specifying pairwise and multi-way interactions among sensors. We demonstrate that the theory of mathematical sheaves provides a unified answer to this need, supporting both quantitative and qualitative data. Furthermore, the theory provides algorithms to globalize data across the network of deployed sensors, and to diagnose issues when the data do not globalize cleanly. We demonstrate and illustrate the utility of sheaf-based tracking models based on experimental data of a wild population of black bears in Asheville, North Carolina. A measurement model involving four sensors deployed among the bears and the team of scientists charged with tracking their location is deployed. This provides a sheaf-based integration model which is small enough to fully interpret, but of sufficient complexity to demonstrate the sheaf’s ability to recover a holistic picture of the locations and behaviors of both individual bears and the bear-human tracking system. A statistical approach was developed in parallel for comparison, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies. When the observations are normalized into a common coordinate system, the structure of the dynamic linear observation model recapitulates the structure of the sheaf model, demonstrating the canonicity of the sheaf-based approach. However, when the observations are not so normalized, the sheaf model still remains valid.

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“…Many sheaf encodings of standard models (such as dynamical systems, differential equations, and Bayesian networks) have been catalogued [42]. Furthermore, these techniques can be easily applied in a number of different settings, for instance in air traffic control [43,44] and in formal semantic techniques [45]. Sheaf-based techniques for fundamental tasks in signal processing have also been developed [1].…”

Section: Sheaf Geometry For Fusionmentioning

confidence: 99%

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

Joslyn¹,

Charles²,

DePerno³

et al. 2019

Preprint

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Integration of multiple, heterogeneous sensors is a challenging problem across a range of applications. Prominent among these are multi-target tracking, where one must combine observations from different sensor types in a meaningful and efficient way to track multiple targets. Because different sensors have differing error models, we seek a theoretically-justified quantification of the agreement amongst ensembles of sensors, both overall for a sensor collection, and also at a finegrained level specifying pairwise and multi-way interactions amongst sensors. We demonstrate that the theory of mathematical sheaves provides a unified answer to this need, supporting both quantitative and qualitative data. Furthermore, the theory provides algorithms to globalize data across the network of deployed sensors, and to diagnose issues when the data do not globalize cleanly. We demonstrate and illustrate the utility of sheaf-based tracking models based on experimental data of a wild population of black bears in Asheville, North Carolina. A measurement model involving four sensors deployed amongst the bears and the team of scientists charged with tracking their location is deployed. This provides a sheaf-based integration model which is small enough to fully interpret, but of sufficient complexity to demonstrate the sheaf's ability to recover a holistic picture of the locations and behaviors of both individual bears and the bear-human tracking system. A statistical approach was developed in parallel for comparison, a dynamic linear model which was estimated using a Kalman filter. This approach also recovered bear and human locations and sensor accuracies. When the observations are normalized into a common coordinate system, the structure of the dynamic linear observation model recapitulates the structure of the sheaf model, demonstrating the canonicity of the sheaf-based approach. But when the observations are not so normalized, the sheaf model still remains valid.

show abstract

Sheaf Theory as a Mathematical Foundation for Distributed Applications Involving Heterogeneous Data Sets

Cited by 3 publications

References 7 publications

Adaptive Discrete Vector Field in Sensor Networks

Adaptive Discrete Vector Field in Sensor Networks

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

A Sheaf Theoretical Approach to Uncertainty Quantification of Heterogeneous Geolocation Information

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