Topological adaptive learning over cell complexes

This paper introduces a novel approach to adaptive learning from streaming flow signals defined over cell complexes, utilizing a topology-based least mean squares (LMS) strategy. By harnessing the principles of Hodge theory, we develop a topological LMS algorithm that efficiently gathers and integrates flow data across various edges and their neighboring cells at mul- tiple levels. Through comprehensive theoretical examination, we elucidate the algorithm’s stochastic behavior, outlining conditions that ensure stability in terms of mean and mean-square error. Furthermore, we derive explicit formulas for assessing the mean- square performance, highlighting how it is influenced by the underlying topological structure, sampling techniques, and data characteristics. Our empirical evaluations, using both synthetic and real-world network traffic datasets, validate our theoretical finding and demonstrate the superiority of our topological approach over traditional graph-based adaptive learning methods that overlook higher-order topological elements.