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Title: Topological invariants for high-dimensional time series Authors:  Tullia Padellini - Sapienza University of Rome (Italy) [presenting]
Pierpaolo Brutti - University of Rome - Sapienza (Italy)
Abstract: When dealing with increasingly complex data, the need for identifying them through a few, interpretable features grows considerably. Topology has proven to be a useful tool in this quest for ``insights on the data'', since it characterises objects through their connectivity structure, i.e. connected components, loops and voids. This topological approach to data analysis (TDA) can be exploited in the case of high dimensional time series, where, in addition to investigating the relation between observations at each time, we are also interested in summarizing its time evolution. We introduce a new topological summary that takes into account both dimension of interest. Our method allows for an intuitive visualization of complex dependency structures, and, as it is based on persistent homology, it also allows for a quantitative measure of their relevance in explaining the data, i.e. persistence. We investigate the theoretical properties (such as convergence and stability) of our proposal and finally we show it in action.