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Pairwise gossip in CAT(k) metric spaces

  • Bellachehab, Anass
Publication Date
Nov 10, 2017
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This thesis deals with the problem of consensus on networks. Networks under study consists of identical agents that can communicate with each other, have memory and computational capacity. The network has no central node. Each agent stores a value that, initially, is not known by other agents. The goal is to achieve consensus, i.e. all agents having the same value, in a fully distributed way. Hence, only neighboring agents can have direct communication. This problem has a long and fruitful history. If all values belong to some vector space, several protocols are known to solve this problem. A well-known solution is the pairwise gossip protocol that achieves consensus asymptotically. It is an iterative protocol that consists in choosing two adjacent nodes at each iteration and average them. The specificity of this Ph.D. thesis lies in the fact that the data stored by the agents does not necessarily belong to a vector space, but some metric space. For instance, each agent stores a direction (the metric space is the projective space) or position on a sphere (the metric space is a sphere) or even a position on a metric graph (the metric space is the underlying graph). Then the mentioned pairwise gossip protocols makes no sense since averaging implies additions and multiplications that are not available in metric spaces: what is the average of two directions, for instance? However, in metric spaces midpoints sometimes make sense and when they do, they can advantageously replace averages. In this work, we realized that, if one wants midpoints to converge, curvature matters. We focused on the case where the data space belongs to some special class of metric spaces called CAT(k) spaces. And we were able to show that, provided initial data is "close enough" is some precise meaning, midpoints-based gossip algorithm – that we refer to as Random Pairwise Midpoints - does converge to consensus asymptotically. Our generalization allows to treat new cases of data spaces such as positive definite matrices, the rotations group and metamorphic systems

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