539 research outputs found
Subsampling Methods for Persistent Homology
Persistent homology is a multiscale method for analyzing the shape of sets
and functions from point cloud data arising from an unknown distribution
supported on those sets. When the size of the sample is large, direct
computation of the persistent homology is prohibitive due to the combinatorial
nature of the existing algorithms. We propose to compute the persistent
homology of several subsamples of the data and then combine the resulting
estimates. We study the risk of two estimators and we prove that the
subsampling approach carries stable topological information while achieving a
great reduction in computational complexity
Persistent Homology of Delay Embeddings
The objective of this study is to detect and quantify the periodic behavior
of the signals using topological methods. We propose to use delay-coordinate
embeddings as a tool to measure the periodicity of signals. Moreover, we use
persistent homology for analyzing the structure of point clouds of
delay-coordinate embeddings. A method for finding the appropriate value of
delay is proposed based on the autocorrelation function of the signals. We
apply this topological approach to wheeze signals by introducing a model based
on their harmonic characteristics. Wheeze detection is performed using the
first Betti numbers of a few number of landmarks chosen from embeddings of the
signals.Comment: 16 pages, 8 figure
Towards Emotion Recognition: A Persistent Entropy Application
Emotion recognition and classification is a very active area of research. In this paper, we present
a first approach to emotion classification using persistent entropy and support vector machines. A
topology-based model is applied to obtain a single real number from each raw signal. These data are
used as input of a support vector machine to classify signals into 8 different emotions (calm, happy,
sad, angry, fearful, disgust and surprised)
Towards Emotion Recognition: A Persistent Entropy Application
Emotion recognition and classification is a very active area of research. In
this paper, we present a first approach to emotion classification using
persistent entropy and support vector machines. A topology-based model is
applied to obtain a single real number from each raw signal. These data are
used as input of a support vector machine to classify signals into 8 different
emotions (calm, happy, sad, angry, fearful, disgust and surprised)
Computing Topological Persistence for Simplicial Maps
Algorithms for persistent homology and zigzag persistent homology are
well-studied for persistence modules where homomorphisms are induced by
inclusion maps. In this paper, we propose a practical algorithm for computing
persistence under coefficients for a sequence of general
simplicial maps and show how these maps arise naturally in some applications of
topological data analysis.
First, we observe that it is not hard to simulate simplicial maps by
inclusion maps but not necessarily in a monotone direction. This, combined with
the known algorithms for zigzag persistence, provides an algorithm for
computing the persistence induced by simplicial maps.
Our main result is that the above simple minded approach can be improved for
a sequence of simplicial maps given in a monotone direction. A simplicial map
can be decomposed into a set of elementary inclusions and vertex collapses--two
atomic operations that can be supported efficiently with the notion of simplex
annotations for computing persistent homology. A consistent annotation through
these atomic operations implies the maintenance of a consistent cohomology
basis, hence a homology basis by duality. While the idea of maintaining a
cohomology basis through an inclusion is not new, maintaining them through a
vertex collapse is new, which constitutes an important atomic operation for
simulating simplicial maps. Annotations support the vertex collapse in addition
to the usual inclusion quite naturally.
Finally, we exhibit an application of this new tool in which we approximate
the persistence diagram of a filtration of Rips complexes where vertex
collapses are used to tame the blow-up in size.Comment: This is the revised and full version of the paper that is going to
appear in the Proceedings of 30th Annual Symposium on Computational Geometr
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
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