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.

翻译：学习随时间演化对象（如多元时间序列和动态网络）需要开发新颖的知识表示机制与神经网络架构，以捕捉数据中隐含的时间依赖信息。这类信息通常无法直接观测，但对学习任务性能至关重要。然而，面向时变数据的知识编码机制若缺少时间维度，将导致频繁模型更新、学习性能低下以及次优决策。本文提出一种新颖的时间感知知识表示机制，重点关注沿多几何维度的隐式时间相关拓扑信息。具体而言，我们提出一种名为时间多维持续性（Temporal MultiPersistence, TMP）的新方法，通过利用现有单参数拓扑摘要生成数据的多维拓扑指纹。TMP的核心思想是融合拓扑表示学习中两个最新方向：多维持续性（同时描述数据形状沿多个关键参数的演化）与之字形持续性（提取随时间变化的最显著数据形状信息）。我们推导了TMP向量化的理论保证，并在交通流量基准、以太坊区块链及心电图数据集上的预测应用中验证其效用，展示了竞争性性能，尤其在数据记录有限的情况下。此外，TMP方法将现有最优多维持续性摘要的计算效率提升了59.5倍。