Linear-Size Approximations to the Vietoris-Rips Filtration

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration is often too large to construct in full. We show how to construct an O(n)-size filtered simplicial complex on an $n$-point metric space such that its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time depend only on the doubling dimension of the metric space and the desired tightness of the approximation. For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees

Time-Aware Knowledge Representations of Dynamic Objects with Multidimensional Persistence

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

Interleaving by parts for persistence in a poset

Metrics in computational topology are often either (i) themselves in the form of the interleaving distance $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})$ between certain order-preserving maps $\mathbf{F},\mathbf{G}:(\mathcal{P},\leq)\rightarrow (\mathcal{Q},\leq)$ between posets or (ii) admit $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})$ as a tractable lower bound, where the domain poset $(\mathcal{P},\leq)$ is equipped with a flow. In this paper, assuming that $\mathcal{Q}$ admits a join-dense subset $B$, we propose certain join representations $\mathbf{F}=\bigvee_{b\in B} \mathbf{F}_b$ and $\mathbf{G}=\bigvee_{b\in B} \mathbf{G}_b$ which satisfy $d_{\mathrm{I}}(\mathbf{F},\mathbf{G})=\bigvee_{b\in B} d_{\mathrm{I}}(\mathbf{F}_b,\mathbf{G}_b)$ where each $d_{\mathrm{I}}(\mathbf{F}_b,\mathbf{G}_b)$ is relatively easy to compute. We leverage this result in order to (i) elucidate the structure and computational complexity of the interleaving distance for poset-indexed clusterings (i.e. poset-indexed subpartition-valued functors), (ii) to clarify the relationship between the erosion distance by Patel and the graded rank function by Betthauser, Bubenik, and Edwards, and (iii) to reformulate and generalize the tripod distance by the second author.Comment: Major revision; exposition has improved throughout and previous results have been significantly extended. 30 pages, 10 figure

Basis-independent partial matchings induced by morphisms between persistence modules

In this paper, we study how basis-independent partial matchings induced by morphisms between persistence modules (also called ladder modules) can be defined. Besides, we extend the notion of basis-independent partial matchings to the situation of a pair of morphisms with same target persistence module. The relation with the state-of-the-art methods is also given. Apart form the basis-independent property, another important property that makes our partial matchings different to the state-of-the-art ones is their linearity with respect to ladder modules