Time-Aware Knowledge Representations of Dynamic Objects with
  Multidimensional Persistence
Baris Coskunuzer Ignacio Segovia-Dominguez Yuzhou Chen Yulia R. Gel
Learning time-evolving objects such as multivariate time series and dynamic
networks requires the development of novel knowledge representation mechanisms
and neural network architectures, which allow for capturing implicit
time-dependent information contained in the data. Such information is typically
not directly observed but plays a key role in the learning task performance. In
turn, lack of time dimension in knowledge encoding mechanisms for
time-dependent data leads to frequent model updates, poor learning performance,
and, as a result, subpar decision-making. Here we propose a new approach to a
time-aware knowledge representation mechanism that notably focuses on implicit
time-dependent topological information along multiple geometric dimensions. In
particular, we propose a new approach, named \textit{Temporal MultiPersistence}
(TMP), which produces multidimensional topological fingerprints of the data by
using the existing single parameter topological summaries. The main idea behind
TMP is to merge the two newest directions in topological representation
learning, that is, multi-persistence which simultaneously describes data shape
evolution along multiple key parameters, and zigzag persistence to enable us to
extract the most salient data shape information over time. We derive
theoretical guarantees of TMP vectorizations and show its utility, in
application to forecasting on benchmark traffic flow, Ethereum blockchain, and
electrocardiogram datasets, demonstrating the competitive performance,
especially, in scenarios of limited data records. In addition, our TMP method
improves the computational efficiency of the state-of-the-art multipersistence
summaries up to 59.5 times.