Persistent topology for natural data analysis - A survey

Natural data offer a hard challenge to data analysis. One set of tools is being developed by several teams to face this difficult task: Persistent topology. After a brief introduction to this theory, some applications to the analysis and classification of cells, lesions, music pieces, gait, oil and gas reservoirs, cyclones, galaxies, bones, brain connections, languages, handwritten and gestured letters are shown

Topological Comparison of Some Dimension Reduction Methods Using Persistent Homology on EEG Data

In this paper, we explore how to use topological tools to compare dimension reduction methods. We first make a brief overview of some of the methods often used in dimension reduction such as isometric feature mapping, Laplacian Eigenmaps, fast independent component analysis, kernel ridge regression, and t-distributed stochastic neighbor embedding. We then give a brief overview of some of the topological notions used in topological data analysis, such as barcodes, persistent homology, and Wasserstein distance. Theoretically, when these methods are applied on a data set, they can be interpreted differently. From EEG data embedded into a manifold of high dimension, we discuss these methods and we compare them across persistent homologies of dimensions 0, 1, and 2, that is, across connected components, tunnels and holes, shells around voids, or cavities. We find that from three dimension clouds of points, it is not clear how distinct from each other the methods are, but Wasserstein and Bottleneck distances, topological tests of hypothesis, and various methods show that the methods qualitatively and significantly differ across homologies. We can infer from this analysis that topological persistent homologies do change dramatically at seizure, a finding already obtained in previous analyses. This suggests that looking at changes in homology landscapes could be a predictor of seizure

Mind the Gap: A Study in Global Development through Persistent Homology

The Gapminder project set out to use statistics to dispel simplistic notions about global development. In the same spirit, we use persistent homology, a technique from computational algebraic topology, to explore the relationship between country development and geography. For each country, four indicators, gross domestic product per capita; average life expectancy; infant mortality; and gross national income per capita, were used to quantify the development. Two analyses were performed. The first considers clusters of the countries based on these indicators, and the second uncovers cycles in the data when combined with geographic border structure. Our analysis is a multi-scale approach that reveals similarities and connections among countries at a variety of levels. We discover localized development patterns that are invisible in standard statistical methods

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

A Stable Multi-Scale Kernel for Topological Machine Learning

Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes