1,218 research outputs found
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
Comparing persistence diagrams through complex vectors
The natural pseudo-distance of spaces endowed with filtering functions is
precious for shape classification and retrieval; its optimal estimate coming
from persistence diagrams is the bottleneck distance, which unfortunately
suffers from combinatorial explosion. A possible algebraic representation of
persistence diagrams is offered by complex polynomials; since far polynomials
represent far persistence diagrams, a fast comparison of the coefficient
vectors can reduce the size of the database to be classified by the bottleneck
distance. This article explores experimentally three transformations from
diagrams to polynomials and three distances between the complex vectors of
coefficients.Comment: 11 pages, 4 figures, 2 table
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
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 Intersection Homology for the Analysis of Discrete Data
Topological data analysis is becoming increasingly relevant to support the
analysis of unstructured data sets. A common assumption in data analysis is
that the data set is a sample---not necessarily a uniform one---of some
high-dimensional manifold. In such cases, persistent homology can be
successfully employed to extract features, remove noise, and compare data sets.
The underlying problems in some application domains, however, turn out to
represent multiple manifolds with different dimensions. Algebraic topology
typically analyzes such problems using intersection homology, an extension of
homology that is capable of handling configurations with singularities. In this
paper, we describe how the persistent variant of intersection homology can be
used to assist data analysis in visualization. We point out potential pitfalls
in approximating data sets with singularities and give strategies for resolving
them.Comment: Topology-based Methods in Visualization 201
Topological characteristics of oil and gas reservoirs and their applications
We demonstrate applications of topological characteristics of oil and gas
reservoirs considered as three-dimensional bodies to geological modeling.Comment: 12 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
Gait-based gender classification using persistent homology
In this paper, a topological approach for gait-based gender recognition is presented. First, a stack of human silhouettes, extracted by background subtraction and thresholding, were glued through their gravity centers, forming a 3D digital image I. Second, different filters (i.e. particular orders of the simplices) are applied on ∂ K(I) (a simplicial complex obtained from I) which capture relations among the parts of the human body when walking. Finally, a topological signature is extracted from the persistence diagram according to each filter. The measure cosine is used to give a similarity value between topological signatures. The novelty of the paper is a notion of robustness of the provided method (which is also valid for gait recognition). Three experiments are performed using all human-camera view angles provided in CASIA-B database. The first one evaluates the named topological signature obtaining 98.3% (lateral view) of correct classification rates, for gender identification. The second one shows results for different human-camera distances according to training and test (i.e. training with a human-camera distance and test with a different one). The third one shows that upper body is more discriminative than lower body
Optimal topological simplification of discrete functions on surfaces
We solve the problem of minimizing the number of critical points among all
functions on a surface within a prescribed distance {\delta} from a given input
function. The result is achieved by establishing a connection between discrete
Morse theory and persistent homology. Our method completely removes homological
noise with persistence less than 2{\delta}, constructively proving the
tightness of a lower bound on the number of critical points given by the
stability theorem of persistent homology in dimension two for any input
function. We also show that an optimal solution can be computed in linear time
after persistence pairs have been computed.Comment: 27 pages, 8 figure
Hierarchies and Ranks for Persistence Pairs
We develop a novel hierarchy for zero-dimensional persistence pairs, i.e.,
connected components, which is capable of capturing more fine-grained spatial
relations between persistence pairs. Our work is motivated by a lack of spatial
relationships between features in persistence diagrams, leading to a limited
expressive power. We build upon a recently-introduced hierarchy of pairs in
persistence diagrams that augments the pairing stored in persistence diagrams
with information about which components merge. Our proposed hierarchy captures
differences in branching structure. Moreover, we show how to use our hierarchy
to measure the spatial stability of a pairing and we define a rank function for
persistence pairs and demonstrate different applications.Comment: Topology-based Methods in Visualization 201
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