Dualities in persistent (co)homology

We consider sequences of absolute and relative homology and cohomology groups that arise naturally for a filtered cell complex. We establish algebraic relationships between their persistence modules, and show that they contain equivalent information. We explain how one can use the existing algorithm for persistent homology to process any of the four modules, and relate it to a recently introduced persistent cohomology algorithm. We present experimental evidence for the practical efficiency of the latter algorithm.Comment: 16 pages, 3 figures, submitted to the Inverse Problems special issue on Topological Data Analysi

Persistent Homology Over Directed Acyclic Graphs

We define persistent homology groups over any set of spaces which have inclusions defined so that the corresponding directed graph between the spaces is acyclic, as well as along any subgraph of this directed graph. This method simultaneously generalizes standard persistent homology, zigzag persistence and multidimensional persistence to arbitrary directed acyclic graphs, and it also allows the study of more general families of topological spaces or point-cloud data. We give an algorithm to compute the persistent homology groups simultaneously for all subgraphs which contain a single source and a single sink in $O(n^4)$ arithmetic operations, where $n$ is the number of vertices in the graph. We then demonstrate as an application of these tools a method to overlay two distinct filtrations of the same underlying space, which allows us to detect the most significant barcodes using considerably fewer points than standard persistence.Comment: Revised versio

Heartbeat Classification in Wearables Using Multi-layer Perceptron and Time-Frequency Joint Distribution of ECG

Heartbeat classification using electrocardiogram (ECG) data is a vital assistive technology for wearable health solutions. We propose heartbeat feature classification based on a novel sparse representation using time-frequency joint distribution of ECG. Fundamental to this is a multi-layer perceptron, which incorporates these signatures to detect cardiac arrhythmia. This approach is validated with ECG data from MIT-BIH arrhythmia database. Results show that our approach has an average 95.7% accuracy, an improvement of 22% over state-of-the-art approaches. Additionally, ECG sparse distributed representations generates only 3.7% false negatives, reduction of 89% with respect to existing ECG signal classification techniques.Comment: 6 pages, 7 figures, published in IEEE/ACM International Conference on Connected Health: Applications, Systems and Engineering Technologies (CHASE

Categorification of persistent homology

We redevelop persistent homology (topological persistence) from a categorical point of view. The main objects of study are diagrams, indexed by the poset of real numbers, in some target category. The set of such diagrams has an interleaving distance, which we show generalizes the previously-studied bottleneck distance. To illustrate the utility of this approach, we greatly generalize previous stability results for persistence, extended persistence, and kernel, image and cokernel persistence. We give a natural construction of a category of interleavings of these diagrams, and show that if the target category is abelian, so is this category of interleavings.Comment: 27 pages, v3: minor changes, to appear in Discrete & Computational Geometr

The Theory of the Interleaving Distance on Multidimensional Persistence Modules

In 2009, Chazal et al. introduced $\epsilon$-interleavings of persistence modules. $\epsilon$-interleavings induce a pseudometric $d_I$ on (isomorphism classes of) persistence modules, the interleaving distance. The definitions of $\epsilon$-interleavings and $d_I$ generalize readily to multidimensional persistence modules. In this paper, we develop the theory of multidimensional interleavings, with a view towards applications to topological data analysis. We present four main results. First, we show that on 1-D persistence modules, $d_I$ is equal to the bottleneck distance $d_B$. This result, which first appeared in an earlier preprint of this paper, has since appeared in several other places, and is now known as the isometry theorem. Second, we present a characterization of the $\epsilon$-interleaving relation on multidimensional persistence modules. This expresses transparently the sense in which two $\epsilon$-interleaved modules are algebraically similar. Third, using this characterization, we show that when we define our persistence modules over a prime field, $d_I$ satisfies a universality property. This universality result is the central result of the paper. It says that $d_I$ satisfies a stability property generalizing one which $d_B$ is known to satisfy, and that in addition, if $d$ is any other pseudometric on multidimensional persistence modules satisfying the same stability property, then $d\leq d_I$. We also show that a variant of this universality result holds for $d_B$, over arbitrary fields. Finally, we show that $d_I$ restricts to a metric on isomorphism classes of finitely presented multidimensional persistence modules.Comment: Major revision; exposition improved throughout. To appear in Foundations of Computational Mathematics. 36 page

The persistence landscape and some of its properties

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine learning. Furthermore, the mapping from persistence diagrams to persistence landscapes is stable and invertible. We introduce a weighted version of the persistence landscape and define a one-parameter family of Poisson-weighted persistence landscape kernels that may be useful for learning. We also demonstrate some additional properties of the persistence landscape. First, the persistence landscape may be viewed as a tropical rational function. Second, in many cases it is possible to exactly reconstruct all of the component persistence diagrams from an average persistence landscape. It follows that the persistence landscape kernel is characteristic for certain generic empirical measures. Finally, the persistence landscape distance may be arbitrarily small compared to the interleaving distance.Comment: 18 pages, to appear in the Proceedings of the 2018 Abel Symposiu

Sub MeV Particles Detection and Identification in the MUNU detector ((1)ISN, IN2P3/CNRS-UJF, Grenoble, France, (2)Institut de Physique, Neuch\^atel, Switzerland, (3) INFN, Padova Italy, (4) Physik-Institut, Z\"{u}rich, Switzerland)

We report on the performance of a 1 m$^{3}$ TPC filled with CF$_{4}$ at 3 bar, immersed in liquid scintillator and viewed by photomultipliers. Particle detection, event identification and localization achieved by measuring both the current signal and the scintillation light are presented. Particular features of $\alpha$ particle detection are also discussed. Finally, the ${54}$Mn photopeak, reconstructed from the Compton scattering and recoil angle is shown.Comment: Latex, 19 pages, 20 figure

Analyzing Collective Motion with Machine Learning and Topology

We use topological data analysis and machine learning to study a seminal model of collective motion in biology [D'Orsogna et al., Phys. Rev. Lett. 96 (2006)]. This model describes agents interacting nonlinearly via attractive-repulsive social forces and gives rise to collective behaviors such as flocking and milling. To classify the emergent collective motion in a large library of numerical simulations and to recover model parameters from the simulation data, we apply machine learning techniques to two different types of input. First, we input time series of order parameters traditionally used in studies of collective motion. Second, we input measures based in topology that summarize the time-varying persistent homology of simulation data over multiple scales. This topological approach does not require prior knowledge of the expected patterns. For both unsupervised and supervised machine learning methods, the topological approach outperforms the one that is based on traditional order parameters.Comment: Published in Chaos 29, 123125 (2019), DOI: 10.1063/1.512549

Background discrimination capabilities of a heat and ionization germanium cryogenic detector

The discrimination capabilities of a 70 g heat and ionization Ge bolometer are studied. This first prototype has been used by the EDELWEISS Dark Matter experiment, installed in the Laboratoire Souterrain de Modane, for direct detection of WIMPs. Gamma and neutron calibrations demonstrate that this type of detector is able to reject more than 99.6% of the background while retaining 95% of the signal, provided that the background events distribution is not biased towards the surface of the Ge crystal. However, the 1.17 kg.day of data taken in a relatively important radioactive environment show an extra population slightly overlapping the signal. This background is likely due to interactions of low energy photons or electrons near the surface of the crystal, and is somewhat reduced by applying a higher charge-collecting inverse bias voltage (-6 V instead of -2 V) to the Ge diode. Despite this contamination, more than 98% of the background can be rejected while retaining 50% of the signal. This yields a conservative upper limit of 0.7 event.day^{-1}.kg^{-1}.keV^{-1}_{recoil} at 90% confidence level in the 15-45 keV recoil energy interval; the present sensitivity appears to be limited by the fast ambient neutrons. Upgrades in progress on the installation are summarized.Comment: Submitted to Astroparticle Physics, 14 page

Topological Machine Learning with Persistence Indicator Functions

Techniques from computational topology, in particular persistent homology, are becoming increasingly relevant for data analysis. Their stable metrics permit the use of many distance-based data analysis methods, such as multidimensional scaling, while providing a firm theoretical ground. Many modern machine learning algorithms, however, are based on kernels. This paper presents persistence indicator functions (PIFs), which summarize persistence diagrams, i.e., feature descriptors in topological data analysis. PIFs can be calculated and compared in linear time and have many beneficial properties, such as the availability of a kernel-based similarity measure. We demonstrate their usage in common data analysis scenarios, such as confidence set estimation and classification of complex structured data.Comment: Topology-based Methods in Visualization 201