352 research outputs found
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 arithmetic operations, where 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 -interleavings of persistence
modules. -interleavings induce a pseudometric on (isomorphism
classes of) persistence modules, the interleaving distance. The definitions of
-interleavings and 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,
is equal to the bottleneck distance . 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 -interleaving relation on multidimensional
persistence modules. This expresses transparently the sense in which two
-interleaved modules are algebraically similar. Third, using this
characterization, we show that when we define our persistence modules over a
prime field, satisfies a universality property. This universality result
is the central result of the paper. It says that satisfies a stability
property generalizing one which is known to satisfy, and that in
addition, if is any other pseudometric on multidimensional persistence
modules satisfying the same stability property, then . We also show
that a variant of this universality result holds for , over arbitrary
fields. Finally, we show that 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 TPC filled with CF 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 particle detection are also discussed. Finally, the 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
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