342 research outputs found
A statistical approach to persistent homology
Assume that a finite set of points is randomly sampled from a subspace of a
metric space. Recent advances in computational topology have provided several
approaches to recovering the geometric and topological properties of the
underlying space. In this paper we take a statistical approach to this problem.
We assume that the data is randomly sampled from an unknown probability
distribution. We define two filtered complexes with which we can calculate the
persistent homology of a probability distribution. Using statistical estimators
for samples from certain families of distributions, we show that we can recover
the persistent homology of the underlying distribution.Comment: 30 pages, 2 figures, minor changes, to appear in Homology, Homotopy
and Application
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
Graph products of spheres, associative graded algebras and Hilbert series
Given a finite, simple, vertex-weighted graph, we construct a graded
associative (non-commutative) algebra, whose generators correspond to vertices
and whose ideal of relations has generators that are graded commutators
corresponding to edges. We show that the Hilbert series of this algebra is the
inverse of the clique polynomial of the graph. Using this result it easy to
recognize if the ideal is inert, from which strong results on the algebra
follow. Non-commutative Grobner bases play an important role in our proof.
There is an interesting application to toric topology. This algebra arises
naturally from a partial product of spheres, which is a special case of a
generalized moment-angle complex. We apply our result to the loop-space
homology of this space.Comment: 19 pages, v3: elaborated on connections to related work, added more
citations, to appear in Mathematische Zeitschrif
Random geometric complexes
We study the expected topological properties of Cech and Vietoris-Rips
complexes built on i.i.d. random points in R^d. We find higher dimensional
analogues of known results for connectivity and component counts for random
geometric graphs. However, higher homology H_k is not monotone when k > 0. In
particular for every k > 0 we exhibit two thresholds, one where homology passes
from vanishing to nonvanishing, and another where it passes back to vanishing.
We give asymptotic formulas for the expectation of the Betti numbers in the
sparser regimes, and bounds in the denser regimes. The main technical
contribution of the article is in the application of discrete Morse theory in
geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete &
Computational Geometr
Testis and Antler Dysgenesis in Sitka Black-Tailed Deer on Kodiak Island, Alaska: Sequela of Environmental Endocrine Disruption?
It had been observed that many male Sitka black-tailed deer (Odocoileus hemionus sitkensis) on Kodiak Island, Alaska, had abnormal antlers, were cryptorchid, and presented no evidence of hypospadias. We sought to better understand the problem and investigated 171 male deer for phenotypic aberrations and 12 for detailed testicular histopathology. For the low-lying Aliulik Peninsula (AP), 61 of 94 deer were bilateral cryptorchids (BCOs); 70% of these had abnormal antlers. Elsewhere on the Kodiak Archipelago, only 5 of 65 deer were BCOs. All 11 abdominal testes examined had no spermatogenesis but contained abnormalities including carcinoma in situ–like cells, possible precursors of seminoma; Sertoli cell, Leydig cell, and stromal cell tumors; carcinoma and adenoma of rete testis; and microlithiasis or calcifications. Cysts also were evident within the excurrent ducts. Two of 10 scrotal testes contained similar abnormalities, although spermatogenesis was ongoing. We cannot rule out that these abnormalities are linked sequelae of a mutation(s) in a founder animal, followed by transmission over many years and causing high prevalence only on the AP. However, based on lesions observed, we hypothesize that it is more likely that this testis–antler dysgenesis resulted from continuing exposure of pregnant females to an estrogenic environmental agent(s), thereby transforming testicular cells, affecting development of primordial antler pedicles, and blocking transabdominal descent of fetal testes. A browse (e.g., kelp) favored by deer in this locale might carry the putative estrogenic agent(s)
A convenient category of locally preordered spaces
As a practical foundation for a homotopy theory of abstract spacetime, we
extend a category of certain compact partially ordered spaces to a convenient
category of locally preordered spaces. In particular, we show that our new
category is Cartesian closed and that the forgetful functor to the category of
compactly generated spaces creates all limits and colimits.Comment: 26 pages, 0 figures, partially presented at GETCO 2005; changes:
claim of Prop. 5.11 weakened to finite case and proof changed due to problems
with proof of Lemma 3.26, now removed; Eg. 2.7, statement before Lem. 2.11,
typos, and other minor problems corrected throughout; extensive rewording;
proof of Lem. 3.31, now 3.30, adde
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
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
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 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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